IAEA-TECDOC-777

Mathematical models and their applications to isotope studies in hydrology

Proceedings of a final Research Co-ordination Meeting held Vienna,in June1-4 1993

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MATHEMATICAL MODEL THEID SAN R APPLICATIONO ST ISOTOPE STUDIE GROUNDWATESN I R HYDROLOGY IAEA, VIENNA, 1994 IAEA-TECDOC-777 ISSN 1011-4289 Printed by the IAEA in Austria December 1994 FOREWORD

During the last few decades, the use of tracer techniques in dealing with a variety of hydrological and hydrogeological problems have proved their value in improving the assessment, developmen d managemenan t f wateo t r resources n thiI . s regarde th , methodologies based on observations of temporal and spatial variations of naturally occurring isotopes, often referre 'environmentas a o dt l isotope techniques' widele ar , y employen a s da integra e routinl th par f o te investigations relate o variout d s hydrological systemsd an , particularl regionan yi l ground water . A substantial amoun f o isotopt r ecollectefa dat o d s publisheaan d d from hydrogeological applications of natural isotopes, however, is often used for qualitative inferences to be made of the system under study, and unproved understanding of processes dynamicd an f wateso r circulation e nee r improveTh .dfo d methodologie r quantitativfo s e evaluations to be made from isotope data as regards the relevant physical parameters of the system has been recognized and a co-ordinated research programme (CRP) was initiated by the IAEA for this purpose in 1990. The main objective of the CRP during 1990-1993 was to intensify co-ordinated international efforts into appraisal of existing models and development of further mathematical formulations as a basis for quantitative interpretation of isotope data in groundwater hydrology. This publication is the result of the CRP and compiles papers summarizing the results and findings of the work undertaken by the participating institutes. Both theoretical aspects of mathematical modelling approache theid san r application isotopo st e dat actuan ao l field result coverede sar . . YurtseverY . Mr , Divisio Physicaf no Chemicad an l l Sciences IAEe th s A wa office, r in charge of designing and implementing the CRP. It is expected that the information provided will be useful guidance material to scientists involved in research/development of isotope applications in hydrology, as well as to engineers and professionals involved in field applications of environmental isotopes in groundwater systems. EDITORIAL NOTE

In preparing this document for press, staff of the IAEA have made up the pages from the original manuscripts submittedas authors.the viewsby The expressed necessarilynot do reflect those of the governments of the nominating Member States or of the nominating organizations. Throughout the text names of Member States are retained as they were when the text was compiled. The use of particular designations of countries or territories does not imply any judgement by publisher,the legalthe IAEA,to the status as of such countries territories,or of their authoritiesand institutions or of the delimitation of their boundaries. The mention of names of specific companies productsor (whether indicatednot or registered)as does not imply any intention to infringe proprietary rights, nor should it be construed as an endorsement or recommendation on the part of the IAEA. The authors responsibleare havingfor obtained necessarythe permission IAEAthe to for reproduce, translate materialuse or from sources already protected copyrights.by CONTENTS

SUMMARY OF THE CO-ORDINATED RESEARCH PROGRAMME ...... 7

calibration O validatiod nan f mathematicano l modelinterpretatioe th r sfo f no environmental tracer data in aquifers ...... 11 ZuberA. approacw Atranspore ne th o ht t problem ...... 3 4 . N. Limic, T. Legovic Ellam: An effective tool for modelling sharp fronts in quantitative isotopic hydrology ...... 7 6 . . Herrera/ Assessmen f groundwateo t r fluxe transmissivitied an s environmentay sb l tracers: Summar f theoryyo , applicatio sensitivitd nan y analysis ...... 3 9 . EM. Adar Fluid flosolutd wan e transpor fracturen i t d media ...... 3 12 . Moreno,L. NeretnieksI. Synthesis of geochemical, isotopic and groundwater modelling analysis to explain regional flow in a coastal of southern Oahu, Hawaii ...... 147 C.I. Voss, W.W. Wood Calibration and verification of a regional groundwater flow model by comparing simulate measured dan d environmental isotope concentrations: An applicatio Alnare th o nt p aquifer syste southwestermn i n Sweden ...... 9 17 . G. Barmen Applicatio f hydrogeochemicano l modellin validatior gfo f nhydrologio e flow modellin Tucsoe th n gi n basin aquifer, Arizona, United State f Americso a ....9 20 . R.M. Kalin, A. Long Cyclic mode r seasonafo l l recharg dischargd ean grounf eo sprind dan g water Valdae inth y experimental basin (humid zone) using environmental isotope data ...... 5 25 . V.l. Ferronskij, V.V. Romanov, L.S. Vlasova, G.P. Kolesov, S.A. Zavileiskij, V.T. Dubinchuk

List of Participants ...... 283 SUMMARY OF THE CO-ORDINATED RESEARCH PROGRAMME

INTRODUCTIO BACKGROUND NAN D

Environmental isotope methods in water sciences have already reached a stage of routine applicatio a wid o t ne spectru f hydrologicamo l problems encountere n watei d r resources assessment, developmen managementd an t fielde Th . , often referre 'isotops a o dt e hydrology' presentls i , recognizeya d scientific disciplin methodologied ean developer fa o ss d employee ar integran a s da l part f investigationo f wateso r resources. presentt A environmentae ,th l isotope data collected from field applications, particularly for regional ground water systems, is often employed for improved understanding of the various processes involve occurrencee th n di , sourc flod ewan dynamic groundwatere th f so . Neemora r d'informatioe fo th efficien f o e us t n content isotope th f o ' e dat estimatinr afo g e relevanth t physical parameter e hydrologicath f o s l system through formulatiod an n developmen f appropriato t e mathematical model r isotopfo s e transpor n hydrologicai t l systems is recognized. This has been the main motivation for the IAEA to initiate and implemen a Co-ordinatet d Research Programme (CRP n Mathematicao ) l Modelr fo s Quantitative Evaluatio f Isotopo n e Dat n Hydrologyi a e programm Th e .mai th nd ha e objective of co-ordinating joint efforts of interested national institutions in development of modelling formulations for isotope transport processes in hydrological systems and verifying their applicability. The scope and modality of carrying out the CRP was initially designed at a consultants meeting organize IAEAe th y db fro 8 Decemberm4- ,topice 198th n 9,o during whice hth present state-of-the-art in available model formulations was reviewed and further research needs were delineated. Subsequently topie IAEe th th ,c n A o durin initiateP CR g e 1990dth . A total of ten institutions took part in the CRP. The first co-ordination meeting of the programm hels Viennn edwa i a from 18-22 November, 199 whico 1t l chiehal f scientific investigators contracts/agreemente (CSIth f o ) s attended revieo t , progrese wth s madd ean plan further required work. The second (final) co-ordination meeting was also held in Vienna (1-4 June 1993), during whic overale hth l achievements were reviewee th detain di d an l programme was concluded. The final manuscripts prepared by each participating institute on resulte th f theiso r work were also presente discussed dan t thida s final meeting finae Th .l manuscript compilee sar d into this technical document.

SCOP ACHIEVEMENTD EAN S

Considering that model formulations for different hydrological subsystems (i.e. surface waters, unsaturated zone moisture transport, saturated groundwater flotranspord wan r fo t different media, etc.) would need different approaches due to different nature of transport processes s limite wa e subjec th o saturate,P t d CR t e mattedth flof o rgroundwaten wi r systems, wher naturae eth l isotope mose ar s t frequently employed generae Th . l modelling formulations include aredP withi:CR e scope th nth f eo

Linear lumped parameter models, Stochastic model r fracturesfo d media transport, Distributed parameter numerical models of flow and transport with geochemical coupling, Compartmental modelling approach, Numerical solution algorithm transporr sfo t modelling. The scope of the programme included all above theoretically possible types of formulations, and work undertaken by the participating institutions was related to one or several aspects of these formulations. The work undertaken was both related to theoretical considerations of development of conceptual models and relevant mathematical formulations as o theit wel s ra l applicatio o actuant l field dat o assest a s their applicabilit d datan ya requirements. Results of the work undertaken and detailed discussions led to the following conclusions overvien a e capabilities a th f wo contributiond an s f differeno s t mathematical modelling approaches in isotope data evaluations for groundwater systems. Lumped parameter models, which are based on tracer input-output concentration relationships, have been developed usesufficientle b practicn dn i ca interpretatior d efo yan n of environmental tracer data. While the simple one parameter models of this type can be employed for systems with inadequate basic hydrogeological data, more complex multi- parameter model adoptee basib e n th availablf sca o n do e information approace Th . alss hi o useful for systems with multi-component flows when data on more than one environmental trace available rar e (i.e. tritiu stabld man e isotop componeno e tw dat r afo t flow system)e Th . output information of this type of model is always related to the mean transit time of the tracer through the system. Thus, additional data (tracer interactions in the system for non- conservative tracers, porosities of mobile and immobile fractions in double porosity systems) are required to relate transit time obtained from the tracer to that of water. While improvement approace th n so h coul envisagee db d through further research methodologe th , y of lumped-parameter modelling is sufficiently developed now to be widely used in practice. Preparation of user friendly computer codes are most desirable for this purpose. Distributed parameter models have the advantage of offering a capability for many different and complex processes to be included into isotope transport process description, and thei e shoulus r e considereb d d wheneve e majorth f feasiblero limitatione On e . th f o s approach is the high spatial and temporal density of the required data. Model development and data collection should be interactive processes linked to each other. Incorporation of the double porosity medium concept into distributed parameter modelling, particularly for isotope data interpretations would be required, and development of efficient algorithms to handle such formulation desirables i s . Ther presentle ear gooya d numbe f computeo r r codes that coul e readildb y transferre r widefo d rpracticen i scal e eus . Strong requirement botn o s h computer capabilitie datd an sa collection impose distributey db d parameter modele b o t s develope isotopr dfo e data evaluation significana e b y sma t limitatio wids it r en fo scal e use. Considerin e overalth g l trace d isotopan r e method n conjunctioi s n with data requirement r modellinsfo g purposes date th ,a would probably neve sufficiene b r trulo t t y characteriz e systeeth m (costs involved, complex spatial distributions observe naturan di l systems principle,n i etc. d )an , validatio modet possibla f nno o stric s s it i l n eti sense, since date alessentialle th l aar y use create/improvo dt modesysteme e eth th f o l . Best strategr yfo data collection should include reconnaissance review of all available data and preliminary assessment (discriminating identifo )t y data that includes real informatio systee th d n mno an processes under investigation (i.e. multi-variate statistical analyses). Data collection should follow suc hpreliminara objectivye th stag d ean modef eo l making shoul developmene db t of a consistent explanation (or explanations) that will work for all types of data collected, through model simulation trials. As regard couplee sth d geochemical transport modelling through distributed parameter modelling, equilibrium model alreade sar useye lonr b availabl dfo n g ca tim d ean reactions. Sufficient data and understanding of basic systems exist. These do not include, however, thennodynamic data, such as ion exchange, high concentration trace metals, and isotopes. For systems with chemical transition zone, kinetic controlled reaction models are required. However, lack of data on reaction kinetics in natural settings limits the usefulness of these

8 model t presentsa . Today reliance th , e woul equilibriun o d e neeb o dt m models that enable mass balance computations. In this regard, evaluation of data for commonly used natural isotopes of C-14 should be based on understanding of the geochemical processes which control the movement of this isotope in a given hydrological system. Mathematical formulation of the adjustment of C-14 activities should be based on geochemical equilibrium process description. Multi-compartmenta r multi-cell(o l ) modelling formulation facilitates incorporatiof no available data (hydrogeological, hydrochemical, natural isotopes) into quantitative assessment of hydrological systems. The particularle yar y usefu caser fo l s wher amoune eth f existino t g datdistributea af o woul e t justifus dno e parameteth y r numerical modelling approach. Thus methodologe th , mors yi e appropriat regionar efo l scale studie whico st h detailed flow and transport models canno applie e laco b tt f sufficienke o demployedu e b n tca datat i d r o , for preliminary quantitative assessment prio suco rt h detailed modelling presene th t A . t stage of development, this approach shoul consideree db meana s da f quantitativo s e evaluation methodology rather than an approach to provide a validated transport model for prediction purposes. Advantage multi-cele th f so l approach are:

it can easily be adopted to non-steady state cases, essential parameters related to mass transport (i.e. dispersivities) are implicitly included approache inth , e methodologth alsn employee yca ob r solutio e inversdfo th f no e problem, i.eo t . estimate the distribution of aquifer parameters from tracer data.

This methodology, at the present stage of development, can readily be transferred to objectivee th usee r th rfo s cited above. Incorporatio non-conservativf no e processes inte oth approac d developmenan h f improveo t d solution algorithms a wouldesirabl e b d e improvemen thin o t s approach. theoreticaw Ne l approache regards sa formulatioe sth f conceptuano l model f maso s s transpor classicate t basewhicth no n e do har l dispersion equation, involve incorporatiof no tracer data in stochastic transport models, use of stochastic differential equations with averaging procedure, Monte Carlo simulatio channelind nan g concep describo t t e solute transport in fractured media and role of multi-variate statistical analyses. The new theoretical development simulato t s e mass transpor groundwaten i t r system provy ma s e effectivo t e overcome e difficultiesomth f eo limitationd an s s inheren e classicath n i t l formulatiof no transport process description, i.e. random character of the observed transit time distributions of wate naturan i r l systems, descriptio f variouno s retardation processes and/or retention phenomen theid an a r adequate incorporatio explicin a n i n t form into formulationsd an , descriptio f compleno x and/or non-stationary boundary conditions. Based on the results achieved during this CRP, the following are the most important issues delineated to be highest priority areas for further development and research needed in this field:

Isotopes are necessary mainly to understand the system and enable quantitative estimate f relevanso t physical parameter f dynamicso f transportso methodologe Th . y should be made more readily available, through field projects, and applied demonstration studies. A particularly important aspect will be to apply different models n actuao l 'test cases', where sufficient date availablear a . This will also enable intercomparison of different approaches. Consideration of non-steady state cases is an important aspect to be further developed formulationse inth thin I . s regard palaeoclimatic evidence point vaso st t changee th n si hydrological cycle on temporal scales ranging from diurnal to millennium. Isotopic studieonle th ye toosar l presently available, whic verifn hvaliditca e steadye yth th f yo - state assumption temporan si l scale. Incorporation of geochemical-chemical processes during transport, particularly for kinetic controlled reactions, needs further experimental data and research. Continued work on the use of isotopes as a tool not only for verification but also for calibratio f continuuno mixing-celd man l (compartmental) model neededs si . Improved understandin f transporgo t processe unsaturatee th n i s d zone modeld an , s coupling unsaturated and saturated flow will be a desirable development.

expectes i t I d tha resulte wore th tth kf so presente thin di s publication usefua wil e b l l contribution to the scientific community in reflecting the present stage of development in isotope transport modellin s welga s indicatina l g importane somth f o e t issuese b stil o t l considered for further research and development.

10 ON CALIBRATION AND VALIDATION OF MATHEMATICAL MODELS FOR THE INTERPRETATION OF ENVIRONMENTAL TRACER DAT AQUIFERN AI S

A. ZUBER Institute of Nuclear Physics, Cracow, Poland

Abstract

Calibratio d validationan f mathematicano l e interpretatiomodelth r sfo n environmentaf o l tracer dat aquifern i a e reviewedsar . Definition f basiso c terms are recalled and the importance of calibration and validation processes for obtaining satisfactory results is discussed. Due to insufficient data, it is usually difficul obtaio t t uniquna e calibration. Eve properlna y calibrated model doe t necessarilsno y suppl requiree th y d informatioe th e ndu influence of hidden parameters. The ages of ideal tracers are understood as those corresponding to the ages of water molecules, whereas the water ages are understood as those defined by the water flux in Darcy's law. In fissured rocks, flo d tracewan r velocitie r exi(o st ages) diffe retardatioy b r n factor caused by matrix diffusion (R ) which is equal to the ratio of total to fissure porosity, and in a good approximation to the ratio of matrix to fissure porosity. Thus, tracer age e sinterpreteb must no t termn i d f wateo s r ages. In consequence, though the environmental tritium as well as the stable isotopes of oxygen and hydrogen can be regarded as ideal groundwater tracers, their ages in fractured rocks may exceed by orders of magnitudes the water ages. Useful interpretation and validation possibilities are offered by a modified versiogivea t a n Darcy') f whic n o i t tracee ( w th he sla rag relatee hydraulie b distancth n o ca t d ) c(x e conductivit d gradienan ) (k yt (i t regiona)a l matrie scaleth f i x, porositknows i ) n n ( froy m laboratory p determinations (k s n x/t i). In the case of C, additional retardation is p t caused by interactions between dissolved carbonate species and solid carbonates in the matrix. Due to high values of the interaction coefficients in carbonate rock systems, the movement of 14C is much delayed by factors difficult to estimate, which makes quantitative age determinations by the C method unreliable. In porous rocks the tracer age obtained from calibrated models may also differ from the water age due to the use of an inadequate model and/or to other reasons, e.g. due to exchange of tracer(s) between the aquifer and aquiclude. Therefore, model validation should be attempted whenever possible.

11 1. INTRODUCTION e presenTh t pape bases i r severan o d l works performed withie th n Coordinated Research Programm Mathematicae th IAEe (CRP n th o A f )o l Modelling of Environmental Tracer Data in and devoted to the interpretatio f eitheno r environmental trace r artificiao , r 2] dat, [I al tracer experiments and pollutant movement at different scales [3, 4], or both types of data [5, 6]. Parameters of fissured rocks, which can be determined from artificial tracer experiments and/or pollutant movement, are also of importance for the interpretation of environmental tracer data. Therefore, properly designed artificial tracer experiment expectee b e usefun b sca o t dl in som ee interpretatio th case r sfo environmentaf no l tracer data. Both group f paperso e alssar o relate discussioy b d differencee th f no s between tracer ager velocities(o s d wate)an r agevelocities)r (o s , f whico e har particular importance for fissured rocks. Following the original works [1, 2, 5, 6], the present paper is addressed to the modellers of environmental tracer dat groundwaten i a r e systemth f thoso o t s wel o mak sa e s wh ea lus e tracet sufficientlno e r ar data t bu ,y familiar wit e limitationth h e th f o s mathematical models commonly applied. The aim of the paper is to discuss the major difficultie d pitfall an scalibratioe th f o s d validationan n processes clarifo t s wel a s a l y problems whic e oftehar n omittemodellere th y b d s and, which, consequently, escape the attention of the users of tracer data. The common sins of the modellers are the application of inadequate models, satisfaction with obtaine t withoufi d t proving thacalibratioe th t s i n unique e lac f distinctioth ,o k n between fitted (sought d know)an n parameters, the applicatio f modelno s wito largto h e number f soughso t parameters (which yields ambiguous solutions), and no attempts at performing the validation process. r determininFo parametere th g floe th w f systemso r predictio fo r o , f no its response from environmental tracer data quantitativa , e approacs hi needed and a mathematical model must be employed. The parameters of the model e relatee input-outpuar th o t d t tracer relation f onemakiny o sb r e ,o us g followine moreth f o , g principles decreas) (a : f traceeo r concentratioe ndu radioactive tth o e decay along flow paths transfe) (b , f variablro e tracer input by the system, and (c) mass balance of flow and tracer(s) components. Principle (a) is usually employed when individual tracer determinations are availabl e traceth d r an einpu constants i t . Principl employes i ) r (b e fo d tracers with variable concentrations at the entry to a system, and a time record of tracer concentrations at the outlet is available. Principle (c), ofte combination i n d (b)n an particularls wit,i ) (a h y applicable when availabilit a larg f o ye numbe f tracero r data distribute spacn i d e permito t s

12 mode e structurth l systeme th f eo , e.g networa .y b f y cellko b r so analytical or numerical solutions to the solute transport equation(s) in whic parametere hth e allowesar varo t dspacen i y . Models base principlen do s e usuallar ) (b y simpled an ) ,(a especially when employe solvino t d e th g inverse problem, i.e. the values of parameters are sought by calibration (fitting). Models based on principle (c) are usually characterized by a large numbe f parameterso r numbee th f f independenI o r. t model equation equas i s l to the number of parameters, or if parameters can be determined by independent methods, their larg e advantage th e modelnumbeo th t s f i .eo r However, if the number of sought (fitted) parameters is larger than the number of independent equations a unique solution is not available, even if a s obtainedwa goot dfi betteA . obtainet fi r d n owinincreasea o t g d numbef o r fitted parameter r improvin trivias fo i s y d suc calibratioe wa an lhth a g n should not be attempted, unless additional parameters are unavoidable and a unique calibratio stils i n l available. Some of the terms and/or their definitions discussed within this paper t generallno e ar y know acceptedr no , and, therefore, thee recalleyar e th n i d following section. 7] s , afte3 , [I r

2. DEFINITIONS RELATED TO CALIBRATION AND VALIDATION PROCESSES Conceptual modelqualitativa s i e descriptios systea it f d no man representation (e.g. geometry, parameters, initia d boundaran l y conditions) relevant to the intended use of the model. Mathematical model is a mathematical representation of a conceptual model for a physical, chemical, and/or biological system by expressions understandinn i designe d ai o t d g and/or predictin behavioue th ge th f o r system under specified conditions. Verification of a mathematical model, or its computer code, is obtained when it is shown that the model behaves as intended, i.e. that it is a proper mathematical representation of the conceptual model and that the equations are correctly encoded and solved. A model should be verified prior to calibration. Model calibration procesa s i whicn i smathematicae th h l model assumptions and parameters are varied to fit the model to observations. Usually, calibratio trial-and-erroa carries y i nb t ou d r proceduree Th . calibration process can be quantitatively described by the goodness of fit. Model calibration is a process in which the inverse problem is solved, i.e. from known input-output relation valuee th s f parametero s e determinedsar . The direct problem is solved if for known or assumed parameters the output results are calculated (model prediction).

13 Validation procesa s i obtaininf so g assurance tha modea tcorreca s i l t representatio e procesth f no r syste so r s intendedwhici mfo t i h . Ideally, validatio obtaines i npredictione th f i d s derived fro mcalibratea d model agree wit w observationshne , preferabl other yfo r conditions than those used r calibrationfo . Contrar calibrationo yt validatioe th , n procesa s si qualitativ modeller' e e baseth on en o d s judgmente trace casth e th f ero n I . method the validation is often performed by comparison of the values of parameters obtained fromodele th m s with those obtainable independently (e.g. flow velocity obtained fro modema l fitte traceo t d r dat shows i a agreo nt e with that calculated frohydraulie th m c gradien d conductivitan t y known from conventional observations [I, 3, 4, 5]). Partial validation definee b n s validatioca a d n performed with respect to some properties of a model [1, 3, 4, 5]. For instance, models represented by solutions to the transport equation yield proper solute velocities (i.e. can be validated in that respect - a partial validation), but usually do not yield proper dispersivitie r predictionsfo t largesa r scales. Another useful term usee whicb dn eithehca r instea validatiof o d n i r no additio o thant t ter identif'labilitys i m [8]. A lac f identiflabilityko means that it is not possible to corroborate or refute all the constituent hypothese modela modea f so f I .l canno e identifietb d there wil e littlb l e confidence in the model's parameter values [8].

Definition validatiof so validatioe th d nan n procesf so arist lo ea controversies discussed recentl excellenn a n i y t pape ] whic[9 rs i h recommended to all those interested in mathematical modelling and/or in the results of modelling, though in the present paper somewhat different approach is accepted. Konikow and Bredehoeft [9] stated: "We believe the terms validation d verificationan have littlplaco n ground-waten r i eo e r science; these terms lead to to a false impression of model capability. More meaningful descriptor procese th f so s include model testing, model evaluation, model calibration, sensitivity testing, benchmarking, history matching, and parameter estimation." In the opinion of the present author, e criticisth e validatioth f o m n proces mainls si y relate o impropet d r definition f thaso t term (which lea falso t d e impressions thae modeth t l represents reality) and to a wrong practice in which the calibration process (or its part) is applied as validation (i.e. improper operational definitions of validation are commonly applied in hydrology). The cited authors claim that the models cannot be validated, but only invalidated. However, it is difficul agreo t e wit opinione h th som f eo d argument san s expresse than i d t paper. First of all, if the hydrogeologists do not see the difference between calibratio d validationnan efforn a , needes i tpropea n di r educatiot no t nbu in the reduction of the vocabulary. Secondly, if the models cannot be validated but only invalidated, as claimed by the cited authors, it means that either all the models are invalid (can one use invalid models?), or some models canno showe e invalidtb b o nt d suc an ,h models shoul appliede b d . This semantia seeme b o st c problem. Finally strongese th , t argument against validation is based on its unprecise definitions related to the modeller's judgment. Therefore, Konikow and Bredehoeft [9] are right in stating that model testing, model evaluation, sensitivity testing, d parameteran

14 estimation shoul e include b de proces th n i df validationso . However seemt i , s that model calibration should be regarded as a separate process which must precede model validation. History matching tera r calibratios i fo m n when time record f dat so e available aar . Validatio obtainee b n nca eithef i d e th r parameters of a model found by calibration agree with the values of parameters found independently (within the accuracy satisfying the modeller's needs), or the prediction yielded by a model is checked against new data. However, themodee th n l result d experimentasan l dat e comparet aar no t bu d matched. In other words, it is stressed once more that the calibration process should not be identified with validation, which seems to be a common mistake in hydrology. In order to avoid possible misunderstandings which can be caused by the above given definition tha modea correca " t s i l t representatioe th f no process or system for which it is intended", the following definition can be suggested: Validation is a process of obtaining assurance that a model satisfie e modeller'th s procese s th need r r systeso sfo r s whici mfo t i h intended, within an assumed or requested accuracy. If such freedom of choice is not allowed, all the efforts directed at the development of models which "canno validatede b t " shoul regardee b d e models uselesa dth d ss an develope d e appliedshoulb t no d .

Tracer methods are usually applied in reconnaissance stages of aquifer investigations when neither result t largesa r time and/or space scaler sfo comparison with predictions r resultno , s obtaine othey b d r methode sar available sucn I . h cases validatioe th , n process canno performede b t e th n I . lac othef o k r choices assurance th , e tha modea t l satisfactorily describes the investigated system must be based on experience gained in earlier studies of other systems. However, when new data are available, either recalibration of the model or its validation should be performed.

3. DEFINITIONS RELATED TO THE TRACER METHOD AND THE RELATIONS BETWEEN TRACER AND HYDRAULIC PARAMETERS 3.1. General e tracerTh method techniqua s i r obtaininefo g information aboua t syste somr o m e systea par f o tobserviny mb behavioue th g specifia f ro c substance tracere th , , thas beeha t nsystee addeth o mt d case [10]th f eo n .I environmental tracers the e addeyar d (injected systee th naturay o )b mt l processes, whereas their production is either natural or results from the global activity of man. An ideal tracer is a substance that behaves in the system exactly as the traced material as far as the sought parameters are concerned, and which has one property that distinguishes it from the traced material [11]. This definition mean n ideasa thar l fo ttrace r there shoul neithee b d r sourcer sno systee sinkth n i sm other than those adheren soughe th o t parametersn I . practice even substances which have other source sinkr s o e regardesar s a d tracers thesf i , e source properle d sinkb san n sca y accounted forf i ,r o their influenc negligibls i e e withi measuremene th n t accuracy [11] e mai.Th n

15 f somo edifficulte environmentaus e th n i y l tracers results froe th m existenc f additionaeo l source d sinkssan . Hydrochemical models whice har supposed to account for some of these sources and sinks are usually not well complexite th defineo t e naturaf o ydu d l e systemlac f sufficienth o k d san t data. The definitions of the turnover time (water age), conservative tracer age, and radioisotope age are commonly accepted, though it is not necessarily always well understoo y thed wh ywhe d e notarenan ar ,,r o compatible . The turnover time of water in a steady state system (or mean age of water leavin ga system r meano , transit time water,f o r meano residence time of mobile water) is defined as:

t = V /Q ( 1 ) w m

e volumetrith whers i eQ c flow rate throug volume th systeme s f hth i o e V , m mobile water r unidimensionaFo . l flow systems, approximatee b whic n hca y b d piston flo r dispersiowo n model (discussed further) x/v= e t ,,th whers i ex meae th ns i transisysteme lengt v th d f ho tan , velocit f watero y e samTh .e relation holds if the age of water along a chosen flow line is considered. This latter definition is useful for direct comparisons of water ages deduced from tracer ages with flow parameters available from conventional methods (e.g. with flow velocity calculated as Darcy's velocity divided by porosity). r variablFo e flow systems functioa s i meae ) th , f timen t no r ( wate Fo .e rag w large groundwater systems which contain water recharged under different climatic conditions, Eq. (1) is usually understood as that representing eithe e presenth r t situation reconstructioe th r o , e timth eo t nbefor n ea intensive exploitation has been started. The mean transit time of a conservative tracer (t ), or the age of tracer leaving the system, or the mean residence time of tracer, is defined by:

(2)

concentratioe th s i ) wher(t etracef c no r resulting froinstantaneous it m s injection at the entrance to the system at t = 0. The tracer age defined by y mode an f flow o r l fo , ) watee (1 equas th definee i . rag o ) Eq t l (2 y b d . Eq if the tracer is injected and measured in flux of water, i.e. if it is injecte d measuredan d proportionall volumetrie th o t y c flow ratef so individual flow path o stagnan n therf i s, 14]e d 13 [11ear an ,t, ,12 zone s

16 accessible to tracer. In groundwater systems with active flow but containing old waters recharged during the Pleistocene, periods of no recharge probably existed sucn I . h case expectes tracee i s th e largee b rag o t dr tha e watenth r ages define. (1)Eq . y b dThen meae th ,n a traces i ) r t velocitx/ = v ( y apparent quantity, lower tha meae nth n water velocity. Equation (2) is of basic importance in artificial tracing, whereas in environmenta lusefue bettea b tracinr n fo lca r t understandini g f somgo e problems. For a radioisotope tracer, Eq. 2 yields the same value of age as for a conservative tracer, if the correction factor for radioactive decay is introduced. case Isteadf nth eo y flow throug hgroundwatea r system outpue th , t concentration, c(t), can be related to the input concentration (c ) of any i n tracer by the well known convolution integral:

CO c(t) = f c (t-f) g(t') exp(-At') df (3) J in 0

where systee g(t'th s )mi response function (g(te (t)Q/Mc th )= s i M , injected mass or activity of the tracer) which describes the residence time ) distributio(f outlef e traceno th radioactive t tra th [11s i ,A 14]d ean , decay constant modee th e typ l.Th f eo (e.g pistoe th . n flow modelr o , dispersion model defines g(t'e i ) th )y b d functio n chosemodellere th y nb . The main paramete g(ty an ' functio) f mearo e th n s transini t tim tracef o e r (age) which is denoted here as t for both a conservative tracer and a radioisotope tracer corrected for the radioactive decay [6]. e radioisotope usualls Th i ) t y( definee radioactive ag th y b d e decay 3. equatio systea r nfo m wit constanha t input concentration:

) (4 c/ exp(-A= c ) t o a

where c and c are the measured and initial radioisotope tracer o concentrations, respectively. Equation (4) is meaningful only in two cases: (a) if a portion of water s beeha n separated sinc pistoe recharge th eth f i n flo) e (b timew d modean , l (PFM) can be accepted as a good approximation of flow in a given aquifer. self-evidents i Cas) (a e ,easiln whereaca t showe i yb cas n ) i sn (b e fro. mEq diffusiof i , t f = traceno t ) = rtha (3 frot m mobil stagnano t e t water w t a zone negligibles i s . In unconfined homogeneous aquifers the transit times between the recharge area and discharge site along different flow lines have the

17 exponential distribution, i.e e exponentiath . l model applies (EM)e Th . radioisotope tracer age (t') which is supposed to represent the mean transit 3. time value is then given by [11, 14, 15, 16]:

c/c = 1/(1 + At*) (5) o a

It may be shown again that t' = t = t , if diffusion is negligible. a t w Evidently, the same c/c ratio values yield different ages (t or t' ) o a a depending on the type of flow. It is also evident that the radioisotope age definee b n differenn ca i d t ways dependin floe th w commoa n modelgo s i t nI . practice to regard Eq. (4) as the definition of the radioisotope age, which differs from the water age when the piston flow model (PFM) does not apply. applicables i M PF e th Eve, f diffusioi n n shoul negligible b d s showa e n further. Therefore, it is strongly suggested that a radioisotope age, or more generally a tracer age, should be defined by both the tracer and the model employed. For instance, "the piston flow C age is....", or "the exponential model tritiu e is..."mag . Whe reportes ntracea i e rag d withou referenca t e to the model it is understood that Eq. (4) was employed, but then the age value may have little meaning. Similarly, if the initial tracer concentration (c ) is not directly measurable but estimated from a special model, and if o that model is not reported, the age may also be of little value. Unfortunately commoa s i t ni ,practic reporo t e ageC t s without reporting 14 the concentrations measured, the hydrochemical model applied for the initial concentration, and the model of flow pattern in the system. For large groundwater systems which had periods of no recharge in the past, the tracer age is always larger than the water age defined by Eq. (1), unless the mean value of t would be calculated for the time span considered. w Unfortunately, the data needed for the calculation of the mean water age are not available. Therefore r suc,fo h systems y yiel tracee ma th ,e d rag lowe r flow velocities than the real values observed for the flow field which existed befor n intensivea e exploitation. Exploitatio n aquifea f no y rma change the flow velocities and directions, which makes comparison with tracer velocities even more difficult.

3.2. Fissured rocks A number of recent studies have shown that stagnant water in the micropores of matrix material of fissured rocks is an important hold-up reservoir for solutes transported by mobile water in fissures and fractures. e delaTh f solutyo e movemen causen i t diffusioy db n exchange betweee nth fissure d matrixsan .

18 The conservative tracer age is much greater than the water age, as shown e followinbth y g equation [17, 18]:

) (6 n )/ n + n ( = n - n )/ n + n ( = V )/ V + V ( = t / t = R p t w p f f p f f p p f f

where R is the retardation factor resulting from matrix diffusion, n and n P P f e matrith d fissure an x ar e porositiese th , e respectivelyar V d an V ; p f stagnant water volume micropore th mobile n matrie th i eth d ef san xo wate r volum fissuresn i e , respectively tracee th e .) ag rAccordin (6 . Eq o t g corresponds to the total water volume accessible to tracer, whereas the water defines e mobili th e y ag b ed water volume ,fissuren i assume e b o st d onlyf I . a non-ideal tracer exchanges with the solid matter, a large surface available r exchang fo matrie th n i xe contribute increases it o st d dela d greatean y e rag as discussed further for the 14C method. Theoreticall ) doe scalee depent (6 th sno . n .o dyEq However meae th ,n transit time of tracer (t ) is not measurable at small scales due to a high asymmetry of tracer response curves (the g(t) functions), which is caused by matrix diffusion vers i yt I .difficul defino t e scal) th e(6 t whica e . Eq h becomes practically applicable [6]. Theoretical tracer curves calculated for different assumed parameters [6, 18] and experience gained in some case studies [5] suggest that for densely fissured rocks, say, several fissures 2 e timth e, scalm r f severaeo pe l month probabls i s y sufficient r sparselFo . y fissured rocks a ,larg edifficule b scal y ma e obtaio t t floe nth weve r nfo along the whole system. Equation (6) holds for conservative substances. For decaying tracers its applicabilit s limitei y d because thee usuallyar y unabl penetrato t e e th e matrix deep enough to occupy homogeneously the whole matrix. The reader is derivatione th referrer fo ] f formulaso [17so 19 t d , ,18 , give Appendin i n x I, meant whicI . s h t tha sho> fissuren i t wt > tha dt rocke sth taw radioisotope tracer age (t ) is always greater than the water age (t ) but a w usually smaller than the conservative tracer age (t ). For long-lived radioisotope denseln i s y fissured rockrockn i sC s case (e.g th f eo n .i 14 2 with several fissures per m ), t = t in a good approximation [2, 19]. 3, t Theoretically r short-livefo , d radioisotopes ,pre-bome e.gth r a .fo ber smalles i ) t' tritiur r r (5)(o o m) t , (4 interprete . Eq f o d d ai wit e th h a d e difficulb n relatios ca it determino t t d o thaan t n nt e t[19] . w applies i ) mentioneds determino A (3 t d a . r Eq e trace fo f th ei , e rag variable inpuradioisotopa f o t e (e.g. tritium) correctioe th , n factor rfo the radioactive decay cause conservativa f work) i s (3 s thasa . Eq t e tracer were used yielddan meae th sn conservative valuth f o e e tracer age. ,t

19 Fro. (6followt mEq ]i s that Darcy's velocitrelates e i th ) o v t d( y conservative tracer velocity by [5, 18, 20]:

) (7 t x/ n = n +n)( v n = = v n = v t p t p t f p f f

which means that contrary to the flux of water (mass transport), which takes plac fissuren i e s containin mobile th g e water stagnane th , t e wateth n i r maie matrith n s reservoii x tracer rfo r transpore Th t. ) (usualln » yn p f following formula results directly from Eq. (7) inserted into Darcy's law [5, 20]:

) (8 x/(n = ) ti i / v n = i / v ) n + n ( = i / v = k f p f t pt pt

e hydraulith whers i ek c e hydrauliconductivitth s i i c d gradientyan . Matrix porosities are usually distinctly larger than 1 % whereas fissure porosities are very often much smaller. Therefore, in practice, the approximate form of . (8)Eq , i.e. withouapplicables i , n t . Then hydraulie th , c conductivitn yca be found from the solute velocity (or tracer age) and matrix porosity, withou y informatioan t fracture th n no e system. Equatio ) differ(8 n s froe th m

common for n whic f Darcy' mo y t b (ag y hw b f etracerreplace la so d an , , dn P ft mean transit time of tracer) which replaced t (age of water, mean transit tim f water)o e . It is evident that Eqs (7) and (8) are particularly useful for validation of models by comparison of Darcy's velocity or hydraulic conductivity obtained from mathematical modellin f traceo g r data with those obtaine y conventionab d l methods. e traceTh e modelrag e commonlsar y calibrated without taking into ) becausaccoun(6 e influenc. eth Eq t matrif o e x diffusio foune b t n dou nca only in the process of validation. In general, a number of examples can be given to illustrate the importance of matrix diffusion for solute transport in fissured rocks, e.g .21-23], [2-718 importancs , ,It .17 e ag r fo e determination validatioe th modele r th fo f d sno san employe e th r fo d interpretation of environmental tracer data has been exemplified in several recent studie , 24][5 s. Similarly seemt i , s that "tracer ages thay exceema t d the computed hydrologie ages by orders of magnitude" reported in [25] and [26] also result from the matrix diffusion, though other reasons were originally y casegiven referrean e n th ,I . d works supply additional arguments thatracee th t r age d hydraulisan c age diffey sma ordery rb magnitudesf so , and therefore their definitions must be well understood, and the models employed for their determinations properly calibrated and validated.

20 Unfortunately, the influence of matrix diffusion on the tracer ages, known since early worknumbea f so authorf ro d an s ] (e.g22 ., 21 [17 , 18 , approximated for fissured rocks by Eqs (6) and (7), is commonly neglected by modellers. Such practic y lea seriouo ma et d s errore interpretatioth n i s f no tracer dat termn i a f hydrogeologiso c parameters.

14 3.3. The C method in fissured carbonate rocks In spit f mano e y limitations importance datinC a th , e s a gth f eo 14 routine method is beyond any doubts for the majority of aquifer lithologies. e maiTh n limitatio methoe th relates f exchange i do n th o t d e betweee nth dissolved carbonate species and solid carbonates of the rock material. Low contents of carbonate material in many rocks allows to apply the 14C method wit sufficiena h t degre confidencef eo . Unfortunately fissuren i , d carbonate rocks the surface area available for exchange is so large in small micropores matrie th f xo cannoC thadelae th tneglectede f yb to r denselFo . y fissured 14 rocks with neglected exchange in the fissures, the total retardation factor (R), e watei.eth e e ratio rt th C-PF. th agee f o : Mag is , 14

t /t = R s [n + (R + R )n ]/n = [n + R n ]/n (9) f apakp w a f fapf

where R and R are the retardation factors caused by instantaneous and ap ak kinetic exchange reactions, respectively (Appendix II). The R -factor may ap have valueR -facto e th s d rabou an abou moderat3 a troca 0 [2] 2 tf r ko Fo . e ak porosity, say, 0.05, and the fissure porosity of about 0.01, the retardation due to matrix diffusion only is equal to 6 [R = (0.01+0.05)70.01]. However, p for a carbonate rock, if R = R + R = 20, the total retardation factor is a ap ak about 100 chalkr Fo .d marls san , which have average porosities about 0.40, the retardation factoconservativa r rfo e trace equas i r abouo t l 0 (fo4 t r fissure porosity as above), and for 14C it is equal to about 800. For lower fissure porosities than that assumed above retardatioe th , n factore b n sca larger. The 14C method has another limitation which is usually tacitly neglected by the modellers. When water flows with differert concentrations of dissolved carbon species meet, the resulting concentration depends not only on 14C concentration volumetrin o d san c flow concentratione rate th alst n sbu o f so total inorganic carbon (TIC), or total organic carbon (TOC). In the lumped- parameter approach this effect cannot be taken into account. However, in the MCM approach and the distributed-parameter models it should not be neglected.

21 3.4. Granular aquifers e intergranulaTh r porosit granulaf yo r aquifer usualls i s y much larger than the microporosity of grains, and, therefore the R -factor is negligible. p However, it may happen that the grains are of high porosity and the R -factor p should be taken into account (then, the intergranular porosity, n, replaces the fissur e. However porosit6) . granular Eq fo ,n i y r aquiferse th , retardation facto eass i rdetermino yt e because bot e intergranulahth r porosity and matrix porosity of grains are measurable. In confined aquifer f largso e exten d sloan d w flow diffusioe th , n exchange between the aquifer and aquiclude may also influence the concentratio f traceno r alon floe th gw lines [27 , 29]28 , , and, thene th , piston flow mode yiely ltracee ma represente th d) e whic (4 rag . h Eq y db differs from the water age as discussed further.

MODELE TH THEI D . SAN 4 R CALIBRATIO VALIDATIOD NAN N An effort was made to use terms more or less generally applied. The classification of the models given below is a modified version of that chosen in [1] to fit the aims of the paper.

4.1. Lumped-parameter models [11, 14] In the lumped parameter approach the groundwater system is treated as a whole and the variations of parameters within the system are not allowed (e.g. [11, 14])constana r Fo . t tracer input, only single parameter models can yield unique solutions, i.e. Eqs (4) and (5), for the piston flow model exponentiae (PFMth d )an l flow model (EM), respectively) . (4 Whe s nEq and (5) are applied to interpret single determinations, unique values of tracer age e directlsar y obtainable, withou y calibrationan t , providine th g initial concentratio knowns i n . Calibratio performes i n d whe numbena f ro individual determination givea n i sn syste e interpretemar obtaio t d n isochrones and/or average velocities along chosen flow lines. In general e applicabilitth , e lumped-parameteth f yo re modelth r fo s interpretation of tracer data was proved in chemical engineering. In hydrogeology, the validity of Eq. (4) was obtained in a number of case studie confinedf so , granular non-carbonatd ,an e aquifers. Validatioe b n nca regarde s obtaineda ageC sr casee agreefo dth whicn i se th hd either with 14 conventional flow data (e.g. [31]) witr o , h 10,000 years' isochrone supposed to be represented by the position of the middle of the ô1 80 and oD shifts cause e lasclimatiy db th t f glaciao d c en change le th perio t sa d (e.ge se . [32]). Note that for the 14C PFM age a constant input is assumed and the

22 decay of the tracer along the flow lines, whereas for the stable isotopes their step-lik transfes it ed changan r o througag e a abouaquifere k th h0 1 t . For fissured rocks, it is easy to check that in none of the case studies the C ages (or velocities) agreed well with the conventional ages (or 14 velocities). In all these cases an effective porosity was put in Darcy's law to obtain "a good comparison". That effective porosity was never defined nor measured e effectivvaluee Th .th f so e porosity usually assumede th wer f eo order of 10 %, whicch clearly shows that they were close to the matrix porosity. Therefore, in fact, the approximate form of Eq. 8 was applied, though its meaning was never explained. In the case of the 14C method in carbonate rocks, the effective porosity most probably includes in a hidden forretardatioe th m n factor resulting from exchange reaction, s 9) (se . eEq and, consequently, can even be larger than the matrix porosity. In the case of the 14C ages an additional difficulty results from the nonconservative behaviour of that tracer and complex hydrochemistry of carbon species e probleTh . usualls i m y simplifie determinatioe th o t d e th f o n initial activit , which) C ( y, sometimes s includei ,e calibratio th n i d n o process, as discussed in the next section for the *3fiCl method. For a variable tracer input, numerical solutions to Eq. (3) are sought, i.e. the type of the g(t') function and the values of its parameters have to be found by fitting (calibration) of the calculated output concentrations to the experimental values. In addition to the PFM and EM, two other models model e commonlsar y applied, i.e dispersioe .th n mode le combine th (DM d )an d exponentia d pistoan l n flow mode lonle (EPM)th y s meae i Th .) n t trace( e rag parameter of the single parameter models and the main parameter of other models. The dispersion parameter which describes the distribution of transit e secontimeth s propeA i ds . parameteDM r e solutioth f ro o unidimensionat n l transport equatio dispersioe nth givet g(t'e bu th s, n)DM functioe th r nfo parameter, i.e distributioe .th f flono w times, mainly results from different flow path length d velocitiesan s betwee recharge th n sampline eth are d aan g site, and has nothing to do with the dispersivity known from artificial tracer tests or pollutant transport (exceptions are discussed in Sect. 4.8). The ratio of the total volume of the groundwater system to the volume of its part wit e exponentiath h l distributio f flono wsecone lineth s di s parameter EPMe th .f o In general, the longer the time record of tracer data, the more reliable valuee th modef e so ar l parameters foun calibrationy b d reliabla r Fo . e interpretation of tritium data with the aid of lumped-parameter models, several years' record of output concentrations is required. Unfortunately, in practice, hydrogeologists request from modellers the determination of water

23 age either from a single tritium analysis or from a short record of data, t possiblwhicno s uniqua i h n y becausi e ewa numbeea f modelro equalln ca s y well be fitted (calibrated). Even when several years' record of data is available uniqua , e calibration canno obtainee b tnumbee th f f fittei ro d d (sought) parameters exceeds two numbee .Th parameterf ro sometimes i s s increase modellee th y db r whe nmora e sophisticated mode assumes i l d (because of the lack of a good fit for two-parameter models [30], or if the input function is not well known and its determination is included into the calibration process [30]) caso .Tw e studies discusse [30]n revisea i d d ,an d interpretation of one of them given in [1], demonstrate the difficulties encountered in searching for a unique calibration even when several years' recor f tritiudo m dat availables i a generaln e lumped-parameteI .th r fo , r approach, the lower the number of fitted parameters, the more reliable the model [33]. Lumped-parameter models develope variabla r fo d e flow [34] were showo nt yield better fits [35] thamodele nth s develope steada r fo dy flod wan applied for variable flow conditions. However, when the periods of variations are much shorter than the mean tracer age, the gain in the accuracy of calibration is not sufficient to justify the effort required. Reference exampleo s t validatioe M werth PF ee f so giveth f no n abovr efo granular aquifers. Other examples of the validations of the PFM, EPM and EM foune b [5]n n i d ca , ) wher (8 regionae th e. Eq f lo hydraulid ai wit e th h c conductivities of fissured rocks calculated from the tracer ages were shown to be equal, or close, to those determined by other methods. One of the example briefls i s ] give[5 y n discussei n d below. Consider the carbonate system of the Czatkowice karstic springs near Cracow, Poland [5], wit ha lon g recor f tritiudo Nowe m th datae r springFo . , the EPM tritium age was 300 years, which for the known matrix porosity of 0.030±0.005 (mea Carboniferoue n th valu r efo s dolomite Jurassid san c limestones weighe contributioy db f thesn o formationo etw e lengtth o f st ho the flow system) meae th ,n flow distanc 8,00012,00f o meae th n d an , 0m hydraulic gradien 0.00610.00f o t 1 yielde= (4.211.5)xlO~ k ) d(8 fro. Eq m 6 m/s ,gooa n whicdi agreemens i h t wit111. .(4 h DxlO s estimatem/ ~ d from meae th n Darcy'r flofo ww ratsla e throug middle hth aquiferf o e youna r gFo . componen Chuderske th n i t i springs 111tritiuM E wa e 4e yearsth ,ag m , which knowe th nr matrifo x porosit 0.02110.002f yo meae ,th n flow distancf eo 15001300 m, and the mean hydraulic gradient of 0.07 yielded the hydraulic conductivity of (1.310.6)xlO —R m/s which is in a good agreement with k = 1.7x1 s founm/ 0d— Rfrom numerical modellin f inflo go nearba o t w y quarry.

24 The carbonate system of Czatkowice is an exception because not only the matrix porosity is known but also the fissure porosity of the Carboniferous dolomites, whic equas i h abouo t l t 0.0015 [5]. Therefore -factoR e th ,s i r p (n + n )/n = (0.021 + 0.0015)70.0015 = 15, which means that the age of f f p water for the groundwater system of the Chuderski is t = t /R = W t p 11/15 = 0.7 a. Assuming the fissure porosity of the Jurassic limestones to be equa thao t lf dolomites o t Nowe systee -factoR th th e e f r th ,sprinmo rfo g p is (0.030 + 0.0015)70.0015 = 20, and the water age is 300/20 = 15 a. Of course, these two water ages applied in the common version of Darcy's law yield similar values of the hydraulic conductivity as those obtained from Eq. (8). However applies tracee commoe i th th e f n i rag ,i dn versio f Darcy'no s law, the hydraulic conductivity completely disagrees with other estimates. The case Czatkowic e studth f yo e springs proves that serious errore sar introduce tracef i d r age fissuren i s d rock e identifiesar d with water ages without taking into account the retardation factor caused by matrix diffusion alss i ot I .obviou s tha a largt e retardation observee factob n rca d relativela eve r nfo matriw lo y x porosity. Calibration curves] show[1 n ni demonstrate tha uniqua t e calibratio sometimes i n s impossibl obtaio t e n even for long records of tritium data. The selection of the proper mathematical model can be obtained only by identification of the conceptual model. In that particular case the identification was obtained on the basis of geological data which allowe rejeco t d earliee th t r hypothesi presence th a n so f eo tritium free componen botn i t h springs assumo t d ean , thaoldee th t r water component in the Chuderski spring contains the same water as the Nowe spring . 51 , [1

4.2. Lumped-parameter models combined with hydrochemical models Single parameter models, especiall PFMe yth e ofte ,ar n combined with hydrochemical models which take into accoun undergroune th t d productiod nan other losses than that radioactive causeth y db e decay mentioneds A . e th r fo , 1 4 C method hydrochemicaa , initiae l th mode contenC r lfo l sometimes i t s 14 included in the process of calibration, i.e. the type of the model and its parameter e chosesar obtaio nt bettena r fit. The PFM combined with simple hydrochemical models is also applied for oo datinl thC e g metho Greae d th case [36 tf 38]th , eo , 37 n .I n correctes BasiM (Eqwa PF ) . ne 4 secular th [35fo d ] ,sitn 36 i r u production op of Cl, and the calibration procedure consisted of the selection of the most adequate model of Cl~ and Cl" hydrochemistry to obtain a good fit of the Cl isochrones to the isochrones calculated from the conventional hydrologie data. Different hydrochemical models yielded "the best fits"

25 in particular recharge areas, supplying informatio origie th f n salinitno o y d theian r relative importanc whole th er basinfo e . However than i , t case eth conventional ages were applied to calibrate the models, therefore the comparison with them cannot be regarded as validation. The agreement between tracer and conventional ages up to 1 million years means that the system was recharged without major breaks under different climatic conditions of the Quaternary, whic difficuls i h explaio t n considering constan d littlan t e scattere observeD <5 d dan d alon0 value floe ô th gwf so line s 1Q [39]. _ _ _ D O s mentionedA ) combine(4 . Eq ,d l witC simplha d an el C mode f o l hydrochemistry, whic s assumen basea wa h n filtration do io d n enrichments wa , also e Milapplieth k r Rivefo d r Aquifer [38]. That model yielde gooa dt fi d e Cl/Cth f o l contour experimentae th o st l data, whereas other simple models yielded unacceptable resultsfiltration io e Th . n mode s alswa lo shown to explain changes in <5D values along the flow lines. The ages obtained yielded flow velocities in agreement with hydraulic determinations. All these facts coule model ag regardee b dl validatioe C . th However e s a dth f no , OG other mathematical models base quitn do e different conceptual models yielded equally good fits, which makes the problem of validation unsolved, as discussed further.

4.3. Continuum approache unidimensionan si l flow Some groundwater systemapproximatee b n sca y unidimensionab d l flow with exchange of mass along a chosen flow line. Tracer exchange without any change in water flow may result from diffusion between the aquifer and aquiclude ] wherea29 , [2728 s, changes tracen boti flo n d i h rwan concentration result either from seepage throug aquitare hth confinen i d d aquifers [40]y b ,r o recharge and in unconfined systems [41], Changes in tracer content also result, as mentioned earlier, from retardation and/or underground production, which are most properly considered as continuous processes in differential transport equations (e.g. [42]). The models discussed within this section could be equally well regarded as lumped parameter models. However, it seems that a separate category is convenient for models based on the assumption of a unidimensional flow with a constant, or linearly decreasin r increasingo g velocity, i.e r modelfo . s with lumped flow parameters d reactionan , s occurring alon floe th gw lines. e MilTh k River Aquifer (MRA), whic s investigatehwa severay b d l scientific teams y servgooa ,ma s dea exampl difficultief eo s encounteren i d e validatioth f mathematicao n l identification modeli d an s f conceptuao n l models. The MRA was initially regarded as a simple hydrodynamic and hydrochemical system becaus dipt i e s gently frorecharge th m e area d soman ,e

26 its chemical component e eithesar r constan changr o t e their concentrationn i s a monotonie way [28]. The following discussion does not include some earlier conceptual models of the MRA hydrochemistry based on mixing of two different type megascopi o t f wateso e rdu c dispersio d limitenan d recharge area [43]. The ion filtration model for the distribution of op Cl/Cl_ and ô1 O 0 in the MRA s alreadwa y discusse Sectn i d . 4.2 conceptuae Th . l mode aquicludf o l e diffusion served as a basis for two similar mathematical models, which were calibrated in different ways. Hendry and Schwartz [28] considered diffusion of OO Cl and 1 Q0 whereas Nolte et al. [29] also included the underground production of oo Cl in the aquifer and diffusion of inactive Cl _ from the aquiclude botn I . h case transiene sth t stat s consideredewa , active i.eth . e flow through the aquifer was assumed to start in the past (the step input function) active th e f fitte eo aquifee Th . fron e th ag dn i tr correspondo st that past event. The age obtained in [28] for the active inflow of meteoric water was 1-2 Ma whereas the age for inflow of water with a different isotopic composition than those originally containe aquifee th n s i drwa 0.25-0.5 Ma. This discrepancy in ages is difficult to explain. The model of Nolte et al. [29] with the flow parameters found originally was also calibrated to ô1 80 and oD data by Maloszewski and Zuber [l] by fitting only the diffusion coefficient of water molecules in the aquiclude (the diffusion coefficien *1 ? — t wa*2 s 1x10 —1 Om/sP for H 0 and 6.3x10 m/s for 2 Cl~) n [28 I dispersivite . ]th s accounteywa d for, thoug influencs it h r efo the assumed value of 100 m was negligible, whereas in [29] the dispersivity s alreadwa y omittedeparture th n i d e equation e meaTh .n flow velocities found in [28 calibratioy ]b n werr flofo e w a 0.1patm/ 0 (eastern1 h d 0.0)an a 7m/ r flofo w pat h2 (western) . Thelowee yar r thavaluee nth s known from hydraulic estimate s0.1d an (0.5 a m/a3m/ , respectively), which shoule b d expected considerin possible th g e period rechargo n f so e durine th g Quaternary shoult I . notee b d d thacalibratioe th t n obtaine n [29i dd ]an discrepance th fres i f eo ] dat[1 D Ô n yi ad mentionean extende 0 ô o dt d 18 abovf infloo activ f e eo watef wo ag because e erag th flo e witd eth wan ha different isotopir fo c a compositioM 0 same 1. th e d r pate an (0. fo nar 1 h a 7M path 2). Uranium dating is particularly difficult even for aquifers approximated by the PFM because a number of geochemical parameters should be known from independent determinations as shown by the general model of Fröhlich et al. [42] spitn a largI .f eo e numbe parametersf ro calibratioe th , east no ys i n to obtain becaus a larg f eo e scatte experimentaf ro l data e [42]Th . comparison give [42n i n ]uraniuf o m concentrations along flow lines wite hth 14C ages indicate that 234 U and 238 U concentrations depend on the age only in

27 the oxidizing and transition zones of the investigated aquifer (up to about 2 reducine th ka) n I . g zone their concentration t see e depenno o t mth o s d n o d age up to about 14 ka. Much more promising results were obtained for the MRA [44], where for the age range of about 0.01-1.5 Ma the uranium method was shown to yield good agreement with hydraulic data. For the flow path 1, it 3 m/a0. , s 0.2-0.d respectively wa an floe s a th wwa 6m/ r t pati fo 1 hd an , 0.1-0.4 m/a and 0.15 m/a, respectively (The uranium ratio method, which rather yields the uranium migration velocity than the flow velocity [42], yielded 0.1-0.2 m/a [44]). However, that "good agreement" was obtained for the rate constants determined for another aquifer [45]. Without the knowledge of these rate constants the flow velocity cannot be obtained even for a calibrated model. Therefore evidens i t e i uraniu, th te cas th f tha eo n mi t metho calibratioe th d n cannot yiel floe th dw velocity (ag f watereo ) even ni granular aquifers without an independent determination of the rate constants. In general r larg,fo e system MRAe tracee th s tha s,a th f o t re b age n sca large possiblo t r e tha du watee enth e periodrag o rechargn f so e durine th g glaciations, i.e. the tracer ages yield mean apparent tracer velocities whic e lowehar r tha meae nth n water velocity s explainea , Sect.3.1n i d . Two models considered by Nolte et al. [29] for opCl and several other models for other radioisotopes [46] yield similar tracer ages which slightly conceptuae depenth n o dhydrochemistryA MR le modeth f o l . This means thae th t tracer ages obtaine e reliable conceptuae ar d th t bu , le modeth f o l hydrochemistry remains unidentified. It should be noted that Fabryka-Martin . [47eal t] proposed another conceptual model, mainlye baseth n o d concentrations and concentration ratios of halogens in the MRA and in the Bow Island Sandstone (BIS). They conclusiocama o t e n thahydrochemistre th t f yo the MRA can be explained by diffusion exchange with shale beds within the aquifer, which contain halogens derived from the diagenesis of organic matter sedimentse ith n e argumentTh . f thesso e authors agains exchange th t f eo water and its constituents between the MRA and the BIS are very convincing. Unfortunately, no data were given from the Colorado shales about 600 m thick, whic d froPakowke an th mhS BI separat i froA e formatioMR th m e eth n abou0 12 t m thick, which overlie MRAe modele sth . Th Hendrf so d Schwartyan z [28d an ] . [29Noltal ] t e wer e assumptio e baseth n do diffusionaf no l exchange with aquicludes in which the initial concentrations of Cl— , 1R 0 and D were assumed to be similar to those in the BIS, which does not imply that exchange with water in the BIS takes place. There is no doubt that the conceptual model of diffusion exchange with the aquicludes [28] and the mathematical model developed in [29] and its calibration presented both in [29] and [1] give the best quantitative description available for several species in the MRA. On

28 the other hand, all the models applied so far can be regarded as validated in respect to tracer ages. However, as mentioned above, they are not sensitive enough to other parameters to identify the conceptual model of the MRA hydrochemistry factn I .modele th , s develope [28n i dd [29]an ] also describe the conceptual model of hydrochemistry governed by diffusion from shale beds within the MRA [47], especially if the beds are sufficiently thick to support diffusional exchange for the time span involved. In such a case, the main proble r obtaininfo m uniqua g e calibratio moss i n t probably relateo t d determining the initial concentrations in the shale beds. Unfortunately, no dat e availablaar e froColorade th m o shale d Pakowksan i formatio s welna s a l from shale beds supporwithio available t th A f rejecr nMR o o t y e an tmodels . The required data would be the diffusion coefficients and tracer profiles in the discussed formations, including observed or extrapolated initial concentrations. Most probabl discussee th l yal d phenomena A goverMR e nth hydrochemistry, i.e. ion filtration through the Pakowki formation, diffusion exchang aquiclude(s)e th d ean betweeA d diffusioMR ,an e th n n exchange within the shale beds and more permeable parts of the MRA. Only if one of these processes dominates over the others, there is a possibility for its identificatio y modellinnb tracee th g r data.

4.4. Distributed-parameter models with lumping for time records of data e principleTh multi-cele th f o s l model se interpretatio (MCMth r fo ) f no environmental tracer data in hydrology were described in [48]. By modelling the cell size d flosan w route possibls i t i s accouno t e r somfo t e variations f parametero systeme th n i s. Therefore, this categor f modelyo calles i s d here the distributed models with lumping. Its popularity results from an easy conceptual and mathematical formulation. On the other hand, an easy formulation of a model with a large number of cells may lead to too large numbe f fittinro g parameters, which turnn n unnoticei ,a y lea ma o ,t d d ambiguous calibration. The use of this approach for solving the inverse problem from time recor givef a dato d t a n site e seemlittlb o st e justified because one- and two-lumped-parameter models give equally good fits [14]. The multi-cell models are also applicable to tracer data in variable flow. They seem to have a distinct advantage over the lumped-parameter models because the are able to model both the concentration and flow variations [49, 50].

4.5. Distributed-parameter models with lumping for space records of data The multi-cell models (MCM) are particularly suitable for modelling the tracer and inflow (recharge) distributions in space. It is the only method

29 available at the operational level, for the interpretation of scarce data in large systems which due to the complexity of flow networks cannot be approximate continuouy b r o M dPF s eithee floth wy rb model s discussed earlier, or by distributed parameter models discussed further. The design of a given cell network is determined by a prior knowledge of the flow system, or by calibration procedure, or by both [48]. However, several early case studie e probablsar t fre yno doubtf eo s relatee lac th f uniqu ko o t d e calibration [1]. e powe Th d somran e limitation e learn b MCM-e n th tsca f o sfro recena m t study of a regional carbonate-alluvial system in Nevada [51]. Conservative tracer, deuterium s usewa ,d under assumption f invarianso t tracer concentration and recharge rates. Thirty-four deuterium values were used for recharge componentrepreseno t 0 4 meae d th tsan n value cellsn i s shoult I . d be positively stressed that contrary to many earlier studies, the number of cells was smaller than the number of deuterium determinations in groundwater. Due to to the lack of constrains three scenarios (models) were considered with somewhat different cell arrangement three-dimensionan i s l cell networks (17 to 20 cells arranged in 2 tiers), recharge distributions and flow routes. e arrangementTh cele th l f snetworkso , cell volume d recharge/dischargsan e rates were based on earlier hydrogeological investigations. The flow model was calibrated with deuterium to obtain the flow rates between the cells. Onc e modeleth s were calibrate s possiblwa t i d calculato et e mead th en an median ages as well as age distributions. However, it should be noted that due to the conservative character of the tracer and the assumed steady state, the ages obtained depen known o d assumer no d cell volumes, i.et . no the o yd represent independent estimates radioisotopA . e tracer conservativa r o , e tracer in a transient state would yield independent age determinations. carbonate th r Fo systeme e th par effective f o tth , s e wa porosit % 3 f yo assumed. As stated in [1], if this value is close to the mobile water porosity (fractures, solution channels, conducting fault zones), the calculated ages correspon watee th ro dt age defines presense a th n i dt paper. However, if this value is close to the matrix porosity, the calculated ages correspond to the tracer ages. In the latter case, the mobile water volumes in the carbonate cells are much smaller than those originally estimated. In spita larg f o ee numbe tracef ro r data three th , e scenarios considered demonstrate that unique calibration was not possible. It was also clearly stated that it was virtually impossible to verify (validate in the terminology of the present paper) the numbers obtained, especially for the carbonat, ka e0 ordesystee e 1 systeth th f ro f f m o o mea e me th [51]nag r Fo . and maximum ages in.some cells approaching 100 ka, the assumptions on

30 invariant recharge rate concentrationd san e versar y weak. Thereforee th , parameters obtaine y calibratiob d n were viewe s firsa d t approximations, which can serve as starting points for more sophisticated models or planning purposes [51].

4.6. Multi-tracer multi-cell models (MTMCM) A novel approach to the multi-cell modelling of environmental tracer dat s presenteawa seriea n i d 56], paperf principle so 55 Th . , s54 [52, e ,53 of this approach is as follows [53]: "The aquife divides i r d into cells within whic e isotopehth d dissolvean s d constituent e assumesar undergo t d o complete mixing r eacFo .h mixing cell, mass balance equations expressin conservatioe gth watef no r isotoped san dissolved chemicals are written. These equations are solved simultaneously r unknowfo n rate f rechargso e intvarioue th o s cell quadratiy b s c programming e degreTh . whico t e h individual dissolved constituente b y sma considered conservative is tested a priori by means of equilibrium model such s WATEQFa . Constituents t paswhicno so hd thi s tese eithear t r disregarder do suitably assigned a small weight in the quadratic program." The idea is to obtain the flow model for systems without hydrologie information where the hydrochemical data are available from a sufficient number of wells. The number of equations must exceed the number of unknown aquifee flowsth r rFo . parameter estimation from additional incorporatiof no periodic variation watee th rf so table welle e th als, ar s o use measuro t d e the hydraulic heads [55]. The method is applied in a constrained way, which means that the number of cells cannot exceed the number of sampling sites. M approachSimilarlMC e th e dispersio th ,o t ye aquife th n i s ni r automatically modelled by the number of cells and their arrangement. The general validity of multi-tracer multi-cell modelling (MTMCM) was proved on synthetic data [53]. Similarly to other tracer methods and models, the MTMCM approach is particularly applicable to systems where conventional data hydraulic data are unavailable or inadequate. Therefore, it would be unrealistic to expect a possibilit validatios it r fo y eacn i n h case seemt I . s thae validitth t a f o y given MTMCM approach, whether teste r noto d , will mainly e depenth n o d presence of a sufficient number of conservative constituents, on the accuracy of assumptions related to the conservative behaviour of nonconservative constituents, and on the validity of assumptions on invariant flow and concentration inputs. In addition, even such conservative tracers as f1 i 0, D, Cl and Br" may change their concentration due to either ion filtration through ion membranes or diffusion exchange with aquicludes. The MTMCM approach in its present form does not account for such effects, which in some case y lea sfalso ma t d e result spitn uniqua i s f o e e calibration.

31 4.7. Distributed-parameter models based on flow equations Numerical solution equatione th o t s f floso w conservatio d tritiunan m balance were obtained for a phreatic aquifer, which was approximated by unidimensional flow with inflow d outflowsan s cause rechargy b d d ean irrigation [41] e mode s welTh . wa ll calibrated, similarl e earlieth s a yr developed unidimensional MCM approach [57], and both models yielded the same recharge rates agreemenn i , t with conventional estimates. Numerical model of flow in the carbonate fissured formations of the Paris Basin was partially validated by the comparison of 14C concentrations calculated from the equation of advective flow (a modified version of the PFM) wit observee th h d values [58]. However floe th ,w rats calculateewa r fo d adjusted effective porosity of 15 %, which was not defined. According to Eq. (7) this effective porosity should be understood as n + n . As stated in f P [1], depending on the modeller's needs and required accuracy, that porosity can be assumed either to be known and the model regarded as validated, or unknown and its value obtained from calibration. However, in the latter case one cannot claim that the model was validated. Quite to the contrary, if laboratory measurements on core samples yielded, say, n = 1 % = p n (effective) modee th , l woul e invalidatedb d states A . Sectn i d . 3.3, total the radiocarbon method in fissured carbonate formations yields tracer ages larger than the conservative tracer ages due to the delay caused by exchange reactions. Consequently, in the discussed case the total porosity is most probably much less tha adjustee nd th tha an t, d% valu 5 valu1 ef o einvolve s the R -factor (see Eq. 9). Undoubtedly, this interesting study clearly a demonstrates some pitfall calibratioe th f so d validationan n processes adherent to the tracer method in general, and to the 14C method in particular.

4.8. Distributed-parameter models based on transport equation As mentioned distributed-parametee th , r approach requiref dato t a lo sa which can be obtained only in favourable situations. For instance, for a small granular aquifer situate floa t wa d divide betwee watershedso ntw a , dense lin f boringe installatioo eth d san bundle-typf no e piezometers, each consisting of nine individual piezometers with short slotted and screened tips, permitte obtaio t d n detailed observation hydraulie th f so co t head d san perform hydraulic test d tritiusan m samplin desiret a g d depths [58].A numerical flow mode constructes wa l observee th r dfo d cross-sectioa s na prerequisite for the modelling of tritium transport. A known analytical unidimensional solution to the dispersion equation was used and calibrated by selecting the flow velocities and dispersivities along chosen flow lines to obtain fits with the spatial distribution of tritium data along flow lines at

32 different sites e dispersioTh . nregardee b mode n ca ls validate da d becaust i e yielded flow velocities close to those known from the flow model. The best fit was obtained for the dispersivity of 0.02 m, which for flow velocities of tha correspondem/ orde1 f ro approximation si coefficiene th o nt f o t molecular diffusion in granular medium (D/T = D = a v, where D is the p p L coefficien f moleculao t r diffusio e tortuositth s fren i i n 5 e 1. ywater = T , p factor) e agreemenTh .observee th f to d dispersivity with that expected from the coefficient of diffusion can also be regarded as validation. Note that the parameters of the dispersion model were lumped, however the use of the model for individual flow lines defined by the numerical flow model allows to consider the whole approach as the distributed type. Such low dispersivity is obtained only for individual flow lines. Other types of sampling would lead to much higher dispersivities. e samTh e aquife s alsrwa o modelle applyiny b d g two-dimensional solution e dispersiotth o n equation. Calibratio d validationan n were obtainee th n i d way as in the case of the analytical model. Point samplin r [59K ]f go alonocg chosen flow line s alswa s o applien i d the Bordon aquifer. The piston flow model was used to calculate the ages for these lines. Two-dimensional model dispersion model was also applied. Similar piston flow mode s user environmentawa ldfo l Fréons observed along chosen flow lines in another aquifer [60]. thesl Ial n e case e tracesth rt applie methono s orden i dwa fino rt d some hydrologie parameters, but rather to validate the transport models which turn i e use nr b prediction fo dca f pollutanno t movement.

5. CONCLUSIONS Environmental tracer data in groundwater systems can be used in mathematical models eithe r findinfo r g some flow and/or rock parameterf so the system or for calibration of flow and/or pollutant transport models. Mathematical modellin e improveb n ca gexecutioy b d propef no r calibration process, and, whenever possible validatioy b , partiar no l validatioe th f no model employed. The relations between flow and tracer parameters should be well understood and taken into account either directly in models or in the interpretatio resulte th f nmodellingf o so fissuren I . d rock e influencsth e of matrix diffusion leads to tracer ages which may exceed by orders of magnitude watee sth r ages. This effect, know morr nfo e tha decadena s i , unfortunately seldom taken into accoun practicn e interpretatioi t th r efo f no environmental tracer data, though its importance is commonly accepted for pollutant transport modelling.

33 For the interpretation of tracer data in fissured rocks, the matrix porosit s bee mose yha nth t showe b importan o nt t parameter because tracer data can be related to Darcy's velocity or hydraulic conductivity without any knowledge on the fissure network parameters. Therefore, matrix porosity data f typicao l rocks shoul e interpretatio usefue th b d r fo l environmentaf no l tracer data. Unfortunately, there is little understanding among the hydrogeologists on the importance of the matrix porosity as the most important transport paramete fissuren i r d rocks.

APPENDI. I X RELATIONS BETWEEN RADIOISOTOPE TRACER AGES AND CONSERVATIVE TRACER AGES Equatio usualls i ) (4 ny regarde hydrogeologisty db generaa s sa l definition of the tracer age, and often wrongly identified with the water age. As discussed in the main text, that equation is meaningful only for aquifers in which the flow pattern can be approximated by the piston flow model. In such cases, the radioisotope age is equal to the mean transit time of water (t ) only for nonsorbable tracers, and if there are no stagnant water zones into which the tracer is able to diffuse. In fractured rocks, t a t >becaus f stagnaneo t microporee wateth rocn e i rth k f matrixso e th f I . fractured network is approximated by a model of parallel fractures of equal apertur d spacingean solutioe advectioe th , th f no n equation combined with matrix diffusion equatio ) lead (4 d wit o [16t s. nan eq h , 7 18]1 : tanh(p)/(n + 1 = ) nt p / t (I.1) aw p f

with 1 /? p = U/D ) (L/2 - b) (1.2) p

where, D is the coefficient of molecular diffusion in the matrix, L is the p distance between the fractures (axis to axis) and b is the half fracture aperture ^p 0.25r Fo . . (I.I,Eq ) ) witsimplifie(6 t h replace . Eq o t sy b d t 2 [17 , : 2 wherea ,p 18] r s:fo a

/(pn + nl ) = t t/ (1.3) aw p f

In the case of piston flow and an instantaneous injection the solution is [18]:

n (L-2b) tanh(p) _2 t /t = -5- + O.Scosh (p) (1.4) td w 2b 2p

34 meae th n s transiwheri t e t ^p tim decayind f o ean b 2 g » tracer L r Fo . td 0.25 . (1.4Eq , ) simplifie . (6)Eq o ,t wherea s *p 3 r fo s

/(2pn + ) nl = t / t (1.5) td w p f

Equations (I.I o (1.5t ) ) diffe ) becausa case r (6 f th fro eo . n i eEq m decaying tracer its particles decay during the diffusion process in the matrix and are unable to penetrate the whole depth [17, 18, 19, 27]. Therefore generaln i whic,, t h> t ,mean ^ t s tha radioisotopa t s i e eag t W a larger than the the water age, but it can be lower than the conservative tracer age. A simple estimation of the condition p ^ 0.25 shows that for the radiocarbon method any spacing not greater than about 0.8 m leads to t /t = a w alsn accepteThereforee ca . ob ) n s a (6 m s . da Eq ,D 0 ^ 1 f i , R —11 2 —1 p P approximatio radiocarboe th r nfo n metho denseln i d y fissured rocks [2]f .O course, for rocks with higher values of the diffusion coefficient than that assumed above fissure th , e spacine largerb n gca . Equation (5) was also sometimes applied for the tritium of the pre-bomb n easilca erat showe I yb . n thar tritiufo t conditioe th m ^np 0.2 mors 5i e difficul satisfo t t radiocarbone yth tha r nfo . However, sinc beginnine th e g e thermonucleath f o r bomb test e tritiuusualls i sth e ag my determined with mai e . (3)th nEq s ,parametef a o g(te d thth eai 'functio ) f ro n founy db fitting correctioe th . radioactive o t th Then e r n,du fo s i e) decay(6 . Eq , applicable, i.e same th .e result e obtainesar r decayinfo d d nondecayingan g tracers. It means that similarly to the 14C method, the R -factor must be p know orden ni calculato t r watee e th (mea rag n transit time) froe mth tritium retardatioe ageth .s A n facto seldos i r m known watee th e ,usuall rag y remains unknown. However, other interpretation possibilities are offered by approach discussed in the main text.

APPENDIX II. RETARDATION FACTORS CAUSED BY EXCHANGE REACTIONS Assuming that the exchange reactions are governed in part by an instantaneous equilibrium and in part by a kinetic process, the retardation factors caused by exchange reactions in the matrix can be expressed as follow , 61][2 s : n / k )p n - 1 ( + 1 = R (II) .1 ap p 3 p

where p is the density of the rock matrix and k is the distribution constant •J

35 for an instantaneous equilibrium reaction, expressed in units of water volume per unit weight of the solid material [k = (C in solid material in g/g)/(C O in water in g/ml) at instantaneous equilibrium]. The retardation factor caused by kinetic reactions is [2, 61]:

k / k = R (11.2) ak 12 forware th backward e dan ar k dd wherratan eek constants (dimension- s i 2 T ), respectively. The distribution coefficient (k ) obtained at the final d equilibrium is:

n k -g-.——?——-+ k = L k - (II.3) d 3 (1 - n )p k P 2 which meanR uses facto e i n (IIsth i d ) insteak tha, .1 r f k i t f o d d 3 ap represent previoue R s th facto. (9)+ e choice Eq th Th .n sR i r f eo ap ak approach is a matter of convenience depending on which parameters are easier to measure. For instance, in a laboratory experiment the equilibrium was not reached even after 800 hours [62]. It seems that the reaction constants and retardation factors can be measured either in laboratory on rock samples, or in field experiments with artificial tracers [61]. However, it remains an unsolved problem if such experiments can be useful for obtaining conservative tracer ages from 14 C ages. Considering large values of the adsorption retardation factors and the expecte accuracw lo d f theiyo r determinations satisfactora , y solution does not ease b seeobtaino yt o t m .

REFERENCES [1] MALOSZEWSKI, P. and ZUBER, A., Principles and practice of calibration d validatioan mathematicaf no l models e interpretatioapplieth r fo d f no environmental tracer data in aquifers, Adv. Water Resour. 16 (1993) 173- 190. ] MALOSZEWSKI[2 d ZUBER an Influenc, . A. ,P matrif o e x diffusio d exchangnan e reaction radiocarbon so n age fissuren i s d carbonate aquifers, Water Resour. Res. 27 8 (1991) 1937-1945. [3] MALOSZEWSKI, P. and ZUBER, A., On the calibration and validation of mathematical e interpretatiomodelth r sfo f traceno r experimentn i s groundwater, Adv. Water Resour. 15 (1992) 47-62. [4] MALOSZEWSKI, P. and ZUBER, A., Tracer experiments in fissured rocks: Matrix diffusio validite th d modelsf nan yo , Water Resour. Res8 9 2 . (1993) 2723-2735.

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37 [21] GRISAK, G.E., PICKENS, J.F. d CHERRYan , , J.A .Solut, é transport through fractured media, 2. Column study of fractured till, Water Resour. Res. 20 (1980) 731-739. [22] FOSTER, S.S.D. chale ,Th k groundwater tritium anomal possibla - y e explanation, J. Hydfol. 25 (1975) 159-165. [23] BIBBY, R., Mass transport of solutes in dual-porosity media, Water Resour. Res (19817 .1 ) 1075-1081. [24] CIEZKOWSKI GRÖNING, W. , LESNIAK, . M , , P.M., WEISE S.M. d ZUBER,an , A. , f thermao e Origiag l d waternan Cieplicn i s e Spa, Sudeten, Poland, inferred from isotope, chemical and noble gas data, J. Hydrol. 140 (1992) 89-117. [25] MAZOR, d BOSCHE.an , Nobl , ,A. e gase formation i s n fluids from deep sedimentary basins reviewa : , Applied Geochemistr y2 (1987 ) 621-627. [26] MAZOR, E., and KROITORU, L., Phreatic-confined discontinuities and restricted flo confinen i w d groundwater systems, Isotope Technique Waten i s r Resources Development 1987, IAEA, Vienna (1987) 427-437. [27] SUDICKY, E.M d FRIND.an , E.M., Carbo datin4 1 n groundwatef go n i r confined aquifers: Implication aquitarf o s d diffusion, Water Resour. Res7 .1 (19814 ) 1060-1064. [28] HENDRY, M.J. and SCHWARTZ, F. W. , An alternative view on the origin of chemica d isotopian l c pattern groundwaten i s r froMile th mk River aquifer, Canada, Water Resour. Res (19884 .2 ) 1747-1763. [29] NOLTE, E., KRAUTHAN, P., KORSCHINEK, G. , MALOSZEWSKI, P., FRITZ, P. and WOLF, M. , Measurements and interpretations of •"3G Cl in grounwater, Milk River aquifer, Alberta, Canada, Applied Geochemistry 6 (1991). 435-445. [30] GRABCZAK MALOSZEWSKI, ,J. RÖZANSKI, ,P. d ZUBER an Estimatio , . K ,A. f no the tritium input functio stablf o d n ai ewit e isotopeshth , Caten 1 (1984a1 ) 105-114. [31] PEARSON, F.J.Jr d WHITEan . , D.E., Carbo age 4 1 d nflo san w rate watef so r in Carizo Sand, Atascosa County, Texas, Water Resour. Res. 3 (1967) 251-261. [32] BLAVOUX, B. and OLIVE, Ph. , Radiocarbon dating of groundwater of the aquifer confine Lowee th rn i dTriassi c sandstone Lorraine th f so e region, France, J. Hydrol. 54 (1981) 167-183. [33] HIMMELBLAU, D.M. and BISCHOFF, K.B. , Process Analysis and Simulation: Deterministic Systems, Wiley Yorw Ne ,k (1968). [34] ZUBER, A., On the interpretation of tracer data in variable flow systems, J. Hydrol. 86 (1986) 45-57. [35] ZUBER MALOSZEWSKI, ,A. HERRMANN, ,P. d STICHLER an . A ,Trace, W. , r relations in variable flow, 5th International Symposium on Underground Water Tracing, IGME (Inst Geolf o .Minerad an . l Explor.), Athens (1986) 355-360.

38 [36] BENTLEY, H.W., PHILLIPS, F.M d DAVISan . , S.N., Chlorine-3e th n i 6 terrestrial environment, Hanboo Environmentaf ko l Isotope Geochemistry, Vol. 2, Part B (Eds P. FRITZ and J.Ch. FONTES) Eisevier, Amsterdam (1986) 427-480. [37] BENTLEY, H. W. , PHILLIPS, F.M., DAVIS, S. N. HABERMEHL, M.A., AIREY, P.L., ELMORE, CALFM. . GOVE, G , D. , , H.E d TORGERSENan . , Chlorin,T. 6 datine3 f go ver groundwaterd yol e GreaTh . t1 , Artesian Basin, Australia, Water Resour. Res. 22 (1986) 1991-2001. [38] PHILLIPS, F.M., BENTLLE DAVIS, . W . H Y, S.N., ELMOREd SWANICKan . D , , Chlorin, G. .B 6 datin3 e f ver o ggroundwaterd yol . Mil2 , k River aquifer, Alberta, Canada, Water Resour. Res 2 (19862 . ) 2003-2016. [39] AIREY, P.L., CALF, G.E., CAMPBELL, B.L., HARTLEY, P.E., ROMAN, D. and HABERMEHL, M.A., Aspect e isotopth f so e hydrolog Greae th tf yo Artesia n Basin, Australia, Isotope Hydrology 1978, Vol. I, IAEA, Vienna (1979) 205-219. [40] GEY, M. A., and BACKHAUS, G., Hydrodynamic aspects of carbon-14 dating, Isotope Hydrology 1978, Vol. II, IAEA, Vienna (1979) 631-643. [41] COLVILLE, J.S., Estimatio aquifef no r recharg flon eo w from natural tritium content of groundwater, J. Hydrol. 67 (1984) 195-222. [42] FRÖHLICH, K., GELLERMAN, R. and HEBERT, D., Uranium isotopes in a sandstone aquifer, Isotope Hydrology 1983, IAEA, Vienna (1984) 447-466. [43] HENDRY, M.J., SCHWARTZ, F.W. and ROBERTSON, C. , Hydrogeology and hydrochemistr Mile th k f Riveyo r aquifer system, Alberta, Canada reviewa : , Applied Geochem. 6 4 (1991) 369-380. [44] IVANOVICH, M., FRÖHLICHh, K. and HENDRY, M. J., Uranium-series radionuclide fluidn i s d solidssan , Milk River aquifer, Alberta, Canada, Applied Geochem. B 4 (1991) 405-418. [45] FRÖHLICH, K., Uranium isotope studies combined with groundwater dating as a natural analogue, Natural Analogues in Performance Assessments for the Disposal of Long-lived Radioactive Wastes, IAEA, Tech. Rept Ser. No 304, Vienna (1989) 46-50. [46] IVANOVICH , FRÖHLICHM. , , HENDRYK. , , M.J .ANDREWS, , J.N., DAVIS, S.N. , DRIMMIE, R.J., FABRYKA-MARTIN, J. , FLORKOWSKI, T., FRITZ, P., LEHMANN, B.E., LOOSLI, H.H. and NOLTE, E., Evaluation of isotopic methods for the dating of very old groundwaters, a case study of the Milk River Aquifer, Isotope Techniques in Water Resources Development 1991, IAEA, Vienna (1992) 229-244. [47] FABRYKA-MARTIN, J., WHITTEMORE, D.O., DAVIS, S.N., KUBIK, P.W. and SHARMA, P., Geochemistry of halogens in the Milk River aquifer, Alberta, Canada, Applied Geochem. 6 4 (1991) 447-464.

39 [48] SIMPSON, E.S d DUCKSTEINan . Finit, L. , e state mixing-cell models, Karst Hydrology and Water Resources, Vol. 2, Water Resources Publications, Fort Collins, Colorado (1976) 489-508. [49] YURTSEVER, Y. and PAYNE, B.R. , A digital simulation approach for a tracer cas hydrologican i e l systems: mult i-compartmental mathematical model, Second Intern. Conf. on Finite Elements in Water Resources, London (1978) 4.165. [50] YURTSEVER d PAYNEan . ,Y B.R., Mathematical models basen do compartmental simulation approach for quantitative interpretation of tracer dat hydrologican ai l systems h Intern5t , . Symp Undergrounn o . d Water Tracing, IGME (Inst Geolf o .d Minera an . l Explor.), Athens (1986) 341-353. [51] KIRK, S.T d CAMPANA.an . M.E deuterium-calibrateA ., d groundwater flow model of a regional carbonate-alluvial system, J. Hydro1. 119 (1990) 357-388. [52] ADAR, E.M. and NEUMAN, S.P., The use of environmental tracers (isotopes d hydrochemistryan quantificatior )fo naturaf no l recharg d floean w component arin i s d basins h Intern,5t . Symp Undergrounn o . d Water Tracing, IGME (Inst. Of Geol. and Mineral Explor.), Athens (1986) 235-253. [53] ADAR, E.M., NEUMAN, S. P. and WOOLHISER, D.A. , Estimation of spatial recharge distribution using environmental isotope d hydrochemicasan l data. I , Mathematical model and application to synthetic data, J. Hydrol. 97 (1988) 251-276. [54] ADAR, E.M. and NEUMAN, S.P., Estimation of spatial recharge distribution using environmental isotopes and hydrochemical data, II. Application to Avaralpa Valle Southern i y n Arizona, USA Hydrol. J , 7 (19889 . ) 279-302. [55] ADAR, E.M d SOREKan .Mul, ,S. t i-compartmental modellin r aquifefo g r parameter estimation using natural tracer non-steadn i s y flow, Adv. Water Resour. 12 (1989) 84-89. [56] ADAR, E.M. , ROSENTHAL, E., ISSAR, A. S. and BATELAAN, 0., Quantitative assessment of the flow pattern in the southern Arava Valley (Israel) by environmental tracer mixina d san g cell model Hydrol. J , 6 (199213 . ) 333-352. [57] ALLISON, G.H. and HUGHES, M. W., The use of environmental to estimate recharge to a South-Australian aquifer, J. Hydrol. 26 (1975) 245-254. [58] ROBERTSON d CHERRYan . D ,. ,W J.A f recharg o .Tritium, n a d s ea an dispersion in groundwater system in Central Ontario, Water Resour. Res. 25 (1990) 1097-1109. [59] SMETHIE, W. M. Jr., SOLOMON, O.K., SCHIFF, S.L. and MATHIEU, G.G., Tracing groundwater flow in the Bordon aquifer using krypton-85, J. Hydrol. 130 (1992) 279-297.

40 [60] BUSENBERG, E. and PLUMMER, L. N. , Use of chlorof luorocarbons (CCI F and ki CCI F ) as hydrologie tracers and age-dating tools: the alluvium and teracce 2 2 system of central Oklahoma, Water Resour. Res. 28 9 (1992) 2257-2283. [61] MALOSZEWSKI, P. and ZUBER, A., Mathematical modeling of tracer behavior in short-term experiment fissuren i s d rocks, Water Resour. Res7 (1990 6 2 . ) 1517-1528. [62] MOZETO, A.A., d REARDONFRITZan . P , , E.J., Experimental observationn so carbon isotope exchang carbonate-waten i e r systems, Geochim. Cosmochim. Acta 48 (1984) 495-504.

Next page(s) left blank 41 A NEW APPROACH TO THE TRANSPORT PROBLEM

N. LIMIC . LEGOVIT , C Rudjer Boskovic Institute, Zagreb, Croatia

Abstract

e firsInth t parassumptione tth stochastif o s c transport, from whic conventionae hth l dispersion equatio deriveds i n e ,ar reconsidered. From more general assumptions on velocity fluctuation n equatioa s n that generalize e conventionath s l dispersion equatio derived s equationsno i tw e .Th , conventional and generalized dispersion equations, are compared and properties discussed. Condition e examinear s d undef ro whice us h conventional dispersion equation is allowed. In the second part, the velocity fluctuations are assumed o havt e bounded realizationw clas ne f model o sa f d o san s transport in the mean is derived. This new class essentially differs froe conventionath m l dispersion equation e basiTh . c featur modelf e o finit a s si e velocit spreadinf yo substancf go e from source. Finall transpore yth substancf to groundwatern ei s is studied using the new class of models. A simpler subclass of models is derived and applied for the description of tracer in karstic groundwaters of the Istra peninsula. Lis symbolf to s

bfc, , bjj) ,k b (X,t,(X ) amplitude f dispersioso n velocity D dispersion constant v, v(t), v(t,x) random velocity fields w, w(t) , w(t,x) mean fields w = E(v) B operato meae th n f ro ove r ensemble f, f(t), f(t,x) velocity fluctuationw - v = sf c, c(t), c(t,x) random concentration field C, C(t), C(t,x) mean concentration field C = E(c) C(X), C(X,t), C(X,t,x) concentration field wite hth probability p(X) dX p, p(X) probability density distribution q, q(t,x) distribution of input *k, *k(x) derministic velocity fields 9k/ 9k(t) random processes S(X) spectral function

1. INTRODUCTION In the present article the term conventional dispersion equation is reserved for the linear dispersion equation with advection e conventionaTh . l dispersion equatio s i usualln y derived froe masth ms balance equatio a y applyinb nt i o t g relationship connecting the flux and concentration gradient by the Pick's law. This approach is basically deterministic because e dispersioth describes ni averagy db e quantities e averag:th e flux of substance and average concentration. On the other side, Pick's law is basically phenomenological. In a fundamental approac o transportt h muse ,w t start with transport modeln i s

43 random velocity fields knoe .W w tha conventionae tth l dispersion equation describes an average motion in the case that the random velocity field is defined by the Brownian motion. Hence, the random counterpart of the dispersion equation is

-fc*(t) = v(t,r(t)), (1.1)

describing the random motion r(t) of a particle in the random velocity field v. So far no other stochastic model is known that e resultdispersioth n i s n equation aftee statisticath r l averagin s carriei g d out. Therefore e musw , te th thin f o k Brownian motion in a wider sense than in the case of molecular diffusion. Since, here e dispersioth , n equatio s user i n fo d describing transpor muca n hto larger scale muse ,w t anticipate that the Brownian motion is the underlying random motion of water dispersioe th s a masses r nfa equatios .A basie th cs ni mode r lfo describing transport in the mean, the Brownian motion is the basic model for the corresponding random motion of water. In this wider sens tere eth m Brownian motio uses ni d here. There are several drawbacks of the conventional dispersion equation that are often noted and discussed. After an analysis of assumptions abou randoe tth m natur transportf eo , from which the conventional dispersion equation is derived, the following drawbacko tw particularle b n sca y marked: ) Spreadin(a f traceo g r around location f sourceo s s i.e. immediately after the release of substance, has been observed to follow the linear law, while the conventional dispersion equation predicts the quadratic law. (b) Substance spreads with a finite velocity from a source through the space. This apparent fact contradicts the infinite velocity that follows from the conventional dispersion equation. It is well known that the conventional dispersion equation does not contain information about the spectral function of velocity fluctuations. This fact can be related closely to the drawback (a). The encountered shortcomings of the dispersion equation have forced researcher othey tr ro s t methods. Monte Carlo methodo (t s be denote M.C.y db mose )ar t often exploited amon approachew gne s o thit s problem. Unfortunately, froe theoreticath m l poinf o t view, a stethi s e poverali s th bac n i kl stud f transporo y t problem explais u t .Le n this statement. A M.C. e appliemethob n f necessarca di d y statistice ar s defined. Schematically, this step is illustrated by the first box in Figur . Afte1 e r defining necessary statistice th f o s stochastic transport one can use a M.C. method to simulate paths of particles and calculate the average concentration in a portion of the medium. The other way is to derive an equation for the average concentration field. The equation must be derived from a stochastic model of transport and defined statistics of velocity fluctuations. This stes theoreticali p . Aften a r equation is derived, the average concentration field is obtained by solving the equation as illustrated by the box "MODEL SOLUTION".

44 BASICS: definition of statistics of velocity fluctuations

SIMULATION: MEAN FIELD MODEL: simulatio transporf no t derivatio modee th f lno method. C b. yM s for the averaged concentration field

MODEL SOLUTION: descriptio transporf no t by solvin modee gth l

Figure schematiA . 1 c illustratio approacheo tw f e o nth o st transport proble randoa n i m m velocity field.

Obtaining an efficient and accurate mean field model, i.e. e steth p "MEAN FIELD MODEL" mose th ts i ,difficul t i par d tan surprisint isno presene th g o that tp tu onl exace yon t modes lha been found and used systematically. Unfortunately, this model is e conventionath l dispersion equation that exhibits undesirable features which are mentioned in (a) and (b). accepo T rejecr to conventionae tth l dispersion equatios na a transport mode e matteth l f experience seemo rb o t s e rather than some basic theoretical principle. Therefore, it is of a considerable interes o lootheoreticar t fo k l criteri mako t a e a better judgement about a possible applicability of the conventional dispersion equation. This article is devoted to 1) reconsideration of principles thae unavoidablar t r cognitioou n i e f transporo n randoa n i t m velocity field, 2) understanding consequences of any simplification tha s mad i tn ordei eo derivt r e tractable transpor tattempn a models) 3 o deriv t d ,an e transport models without simplification orden si relato rt e drawback d (b)an ) (a s usualle th o t y supposed simplification basie th cf s o principles . The Brownian motion is defined by the random process with independent increments. Each incremen normaa s tha l distribution with zero mead variancan n e proportionao t o time t le Du . independent increments we have the drawback a) (Lirnio [1,2]). Because increment e independenar s e covariancth t e functiof o n velocity fluctuations is proportional to the Dirac S-function. By removin e independencgth incrementf yo drawbace n sth ca ) (a k be eliminated. Unfortunately, some other nice featuree th f o s conventional dispersion equation are lost. Therefore, in the firs r tstud ou pare considew yf o t e transporth r randoa n i t m velocity field for which fluctuations are normally distributed but the covariance function is non-trivial, i.e. we assume that the spectral function of velocity fluctuations is general. By keeping unchanged the assumption about normal distribution of velocity fluctuations we are allowed to call the derived equation

45 the generalized dispersion equation. In Section 3. this generalized equatio s studiedi ne th r emphasi Ou n .o s i s comparison of properties of solutions of conventional and generalized dispersion equations rather than their derivations so that details of derivation are omitted. drawbace causeface s Th i th t) y db tha(b k t incremente sar normally distributed (see [2]). It is naturally, therefore, to omit the assumption about normally distributed velocity fluctuations in order to have a finite velocity of transport. In the new approach the content of steps "BASICS", "MEAN FIELD MODEL d "MODEan " L SOLUTION f Figuro " . diffe1 e r froe th m classical derivatio dispersioe th f no n equation thin .I s approach randoe th m proces non-trivias sha l covariance functione th d san fluctuations are bounded almost surely. Due to these features the drawbac avoideds i ) (b k . Thi clasw s ne model f so studies si n i d remainine th g par thif to s work. w clasne f model o se Th s tha s derivei t d here resembles formall wave telegrapyth r e o h equation seemt .I s that Goldstein e firs th o utiliz t ts [3 wa e ]telegrap th e h equatior fo n description of transport of substance in a turbulent medium. The model was generalized to the case of more dimensions by Bourett [4]. Resulte presenth f o st wore morar ke general. Thee ar y derived from a unique supposition on velocity fluctuations by a method which is developed in [2]. The method is quite general so tha t reproducei t e conventionath s l dispersion equatioa s a n special case. 2. CONVENTIONAL DISPERSION EQUATION AND BROWNIAN MOTION To simplify notation d focu an r sdiscussio ou s o basit n c principles of transport in a random velocity field it is admissibl consideo et transpore rth space on n eti dimension. remins u t dLe firs tBrowniae whath s ti n motiospace on en ni dimension and what is the meaning of velocity implied by the Brownian motion. The Brownian motion is the random process x(t) on [0,<») , x(t) e R, that is defined by (i) x(0) = 0,

(ii) For 0 < t, < t2 < ••• < tn, the random variables x(tk) - x(tk_!) are normally distributed and independent, (iii) For any pair s, t, such that t > s > 0, the random variable x(t) - x(s) has the zero mean and variance Var[x(t) - x(s)] = 2D(t-s) ,

a positiv whers i D ee number tha s callei t e dispersioth d n constant. The covariance function of this process is B(t,s) = min{s,t}. The finite-dimensional characteristic functions can be easily derived. Let t0 = 0, n e N, and t0 < tj < t2 < ••• < tn. Then

e BrowniaTth o n motion x(t) there correspond a velocits y fluctuation f(t dx(t)/dt= ) . Fropropertiee mth s (iid (iiian ) )

46 we have C(t,s )E[f(t)f(s)= ]2D5(t-s= ) wher eDirae Sth (t) cs i delta function. Hence, the spectral function of f(t) is S(X) = 2D for all X e (-«>,«>) . Realizations of the Brownian motion x(t) are continuous in R, and start form the origin. Let us call them paths. However, time derivative thesf o s e paths, representing velocity fluctuation, are defined only as generalized functions. To get a feeling about smoothness of paths that are generate Browniae th y db n motion conside followine rth g example x(t, 1 generatee = )b D e Brownia t th Le y b d. 2 innR motiod nan les considetu r another process y(t) whicr covariance fo , hth e and spectral function K(t[7r(pe / sar 2p ) S(Xd 2= +t an 2 )2exp()= ] - p|X|). An illustration of paths for both processes is given in Figure 2. For the second process the choice of parameter is p = spectrae 2.Th l function valu e tendth tendp eo s t s S(0a s 2 )= zeroo smallea t o .T r valu thif eo s parameter there corresponds a less smooth path in Figure 2. velocite Ith f Equation i v y n (1.1, f defines )+ i w y b d meae th whern s velocite velocityi th ew s i f E[v]= yf o ,w d ,an the Brownian motion, the correspondine nth g transpor meae th nn ti is describe conventionae th y db l dispersion equation

f I ^* \"5t c(t\ k tx)= (2.1)

This equation must be supplied with an initial condition at t = orden i havo 0 rt uniquea e solution. x-independend e meaan I th f- n t fiels i w d t solutionf o s (2.1) can be easily represented by means of the fundamental solution Y that is also known as the Green function. For a constan fundamentae th tw l solutiofore th m s nha

t Je - (2.2)

Figure 2. A path (left) generated by the Brownian motion with D =1, and a path (right) generated by the Gaussian process having the spectral function S(X) = 2exp(-2|X|).

47 e fundamentaTh l solutioe dispersioth f o n n equatioe b n ca n interpreted as the mean concentration field resulting from the source at x = 0, differing from zero only for t = 0, i.e. q(t,x) = £(t)5(x) .A genera l solutio representee f (2.1b o n n ca ) y b d e fundamentameanth f o s l solution e initiat cth 0Le .(x e b )l condition at t = t0. Then the corresponding solution of (2.1) has the following form

«e Y(t-tc(t,xy d f a)= ,x-y) c0(y)+ „ f <2'3> j dy I ds Y(t-s,x-y) q(s,y) .

In the case of two space dimensions, instead of (2.2) we have

whercomponente componentar e u ewx 2 e d ar 2 w u th an f x so f so drift velocit e expressioTh . w y n (2.3 e two th s vali i )-r fo d dimensional cas wells ea , afte (2.4f o r Y substitute s )i e th d dan integration wit hs changei respec y do t t inte doublth o e integration with respect to yl and y2. Drawbacks a) and b) can be directly seen from (2.4). Let us conside e procesth r = {w,0}n R i 2sw ,d assum an , e thae th t "puffa process i thao t s a "q t, f s0 I c 0start= (x . t =0 )t sa t = 0 and x = 0, i.e. q(t,x) = q05(t) 0, the resulting concentration field is c(t,x) = q0Y(t,x). This concentration field has isolines in the x,,x2- plane, that differ slightly from curvee th s definey b d

4Dt

parametee th whers i f isolinesea ro . These curve circlee ar s s having centres at {wt,0} e R2, moving downstream with the velocity w, and having redia 2(Dta)1/2. This behaviour can be easily observed further from source. Around source the radii are proportiona rathet o lt r tha square ndrawbacth e Th e . root kf to (b) can be demonstrated even by simpler arguments. Let the input distribution q(t,x) be zero for t < 0 and localized to a bounded . Thee 0 solutio th n> t nl al domai(2.3 r f fo (2.1no ) s i ) positive on the whole R for any t > 0. The only possible explanation of this fact is by concluding that transport in the mean carries away substance with velocities tha arbitrarile tar y large. Heuristically tha y dispersioe sa t th o t e us ne ,equatiow n allows an infinite velocity of spreading.

3. GENERALIZED DISPERSION EQUATION velocite Th y fluctuatio (t,xnf agais )i Gaussiana n field, i.e. for fixed t e t0/00) random variables f(t,x) are normally distributed with zero means. In contrast to the previous section,

48 randoe th m field t relatef(t,xe Browniano th s o )i t d n motion. s generaIe sensi t th en i l tha s covarianceit t s C(s,t,x,y= ) e E[Th f (t,x. y )(s,ye d smootf ar an ) ] x h , functions , t f o s velocity fluctuatio defines ni y db

f(t,x) £= gk(t) q^Or), (3.1) k

e velocitwherar k 0 e y mode d g k(te an randos )ar m processen o s [0,oo) for which statistics must be defined. The representation (3.1) can be understood as a decomposition of velocity fluctuation int lineaa o r combinatio locaf no l fluctuation. k s0 Each local fluctuatio randoa s nha m amplitude gk(t) dependinn go time. Statisticdefinee ar k : g by dr sfo

f pU)dX (laU,^,^, ...,tn> A wher reae ar l eb valued) s functions, . seta ^„(X. d s . ,an i , , t , A

are k statistical moments of a Gaussian process. Again, for the sake of simplicity, the integral over A is omitted in the course discussionr oou f finae th lt . A steintegrae pth e b l n oveca rA include a genera t orden ge i d l o t rrepresentatio f resultso n . Furthermore, the stationär ity of processes gk in the strict sense s assumei o thas e dstatistica th t l moment depenn ju s d onln o y differences tj - tj. Then the functions bj, become constant, i.e. real numbers. In this way, statistical moments, to be used in our derivations, have the form: nJ.1

The real numbers bk are still unspecified. In order to match the conventional dispersion equatio s closela n s possibla y e th e number definee ar k sb imposiny db conditioga covariance th n no e function of velocity fluctuations. The second order statistical moment of velocity fluctuations or covariance function is defined by

Cov(s.t.x.y) K(t-s)= JEv^OO jbj!bi(y) , (3.3)

where K(t-s) = /z2(t,s) . The covariance function (3.3) does not depend on space variables if the numbers bk are chosen in such a thay wa t Sbk0k(x constans )i Therefor. R n to choose ew mako t k eeb the function Ebk0k(x) equal to 1 on R. Then the standard deviation of velocity fluctuations is t- and x- independent, a = (Mat0))1'2- In the velocity field v, either deterministic or random, the transport proble knowa s nmha form tha equivalens ti mase th s o tt balancconcentratioe th r efo consideref no d substance: -Jj-C(t,x) + -j-(v(t,x)C(t,x)) = q(t,x}, dt dx (3-4)

49 By inserting the representation (3.1) into (3.4) we get Jlc(t,x) + -j-(w(t,x)C(t,x}} -»- -j-(f(t,x}C(t,x)) = at ox ox (3.5) Q(t,x) .

Solutions of (3.4) can be obtained by using a family of stochastic evolution operators U(t,s) . The operators U(t,s) are uniquely defined by the following system , J U(t,U(s,t)-i= ) t U(t,s),= U(u,T)£7(T,s) =U(t,s), ^ a (3-6) -jLu(t,s) = --j^ [(w(t,x) + f(t,x)) U(t,s)1 ,

Their random nature follows from the randomness of f(t,x). After the evolution operators U(t,s) are calculated, any solutions of (3.4) is represented by

= U(t,t0) C0U )+ U(t,s) q(s,x) . ds (3.7)

In particular, the mean value c(t,x) = E[C(t,x)] can be obtained as t

C(t,x) =Z[I7

e representatioTh n (3.8 )s veri y simila o (2.3)t r . Much more, thes o expressiontw e e velocit th e ssam f th musi ee yb t fluctuation f(t,x) is defined by a Brownian motion. Hence, in this particular case we would have

X[U(t,s}]q(t,x) Y(t-a.x-y)dy f = q(s,y),

i.e. an integral operator with the kernel equal to the fundamental solution Y of the conventional dispersion equation. velocite Forth y fluctuations f(t,x thif )o s section E[U(t,s)] is again an integral operator. Its kernel is proportional to the fundamental solution Z of the following differential equation:

-jLz(t,x) = h(t) D J?--Z(t x) , t > 0, (3.9) 2 dx dt l where

eff t K(T= D ), dvJC(t)f h(t) — = , (3.10) J ? Z oy o

50 We hav o defint e e another function: t t e(t) = 2D fdt2 = x Jdfc2 K(tz. ) (3.11) 0 00 With notations, just introduced, the solution of (3.9) is

Z(t,x) exp - (3.12) 2e(t) J'

The scaling factor (2ir)'1'2 is introduced in order to have

•» J Z(t.x) dx = 1, Z(Q.x) =6(x).

The function X(t,x) = exp(-it) (3.13)

is evidently a solution of

~ + W-- - hit) + k \ X(t,x) « 0, t>0, (3.14) J

and can be called the fundamental solution of this equation. The resemblance of the function (3.13) with the fundamental solution of dispersion equation (2.2 )s strikingi e functioTh . n (3.13) would be equal to the fundamental solution of dispersion equation functioe th (2.2 f i ) n e(t fore )n (3.12i th m d e(t)ha 2Dt= ) . Three particula re covarianc caseth f o s e functioe ar K n interesting to discuss:

(a) K(s)ds, i > t~v.~ p o 0 > (JbD K(s)f = ) s d o t (c) j* K(s)ds -* ».

In all three cases e(t) downstream -» behaves as t5 24 fo 0 4 r 5 smal3 0 3 l 5 t2 . 0 2 As 5 1 0 1 5 time tend o infinityt s e th , l l l I I l l l I function e(t) in (a) tends to a positiv t Figureei number) . 3 (b Example n i , f isolineo s s t correspondini ) (c thren e i th d eo gt an case behave, Dt s s a s increases to the infinity. The in (3.15) case (b) corresponds to the conventional dispersion law.

51 The number D can be interpreted as an asymptotic dispersion constant solutioe .Th n (3.12) exhibit smalr lineae fo sth lw la r e standarth time d an sd quadrati r largefo w rla ctimes . This property stresse non-trivialite sth obtainee th f yo d model (3.9) e clasith n f exactlo s y derivabl ee transpor modelth r fo sf o t substanc randoa n ei m velocity field. Thus, (3.9extension a s )i n e conventionaoth f l dispersio r whic fo e covarianc th hw la n e functio velocitf no y fluctuation smoota s si h function. An illustratio gives i propertief Figurn ) no i n (c - e ) (a s

3. Let K(t) be the covariance functions, defined by K(t) = (27r)" jexp(itX) S(X) dX, where 1 (a)

(Jb) s(A.) = A __jL_, e2(t) = -l(t - 1 + exp(-fr))

= 6U), e3(t)=

= {w,0} w straightforware d th , an 0 Wit= k h d extensioe th f no function space (3.13th eo ) t dimensio s i 2 = n

-Ytf-TcA V <-1 -"~\ i -"-~X 2 \' = 2ne(t)

The curves X(t,Xj,x2) = const are drawn in Figure 3. By usin fundamentae gth l solution (3.13 (3.8n )i )solution, s o termstw f oo ,f m transporwrittee su b e meaa n th s ca nna n i t e includinon e initiath g le othe th stat rd an eincludin g input distribution assums u t .eLe tha procese tth - starte d = s ha t t da oo, so that an initial state is absent. Then we have t • c(t,x)exp{-k(t-s)s d f = Z(t-s,x-y-w(t-s)y d j } ) q(s,y) .163 ( ). —#» —w

Thisolutioa s followine si th f no g integro-differential equation fl ffi \ & C _ *_ a tt^_£ - n_£L_ - lr\_4 r-ff v\° = srl +tn - kv\- }• c(t,x) = q(t,x) - D

[1 - h(t-s) ] Z(t-s,x-w(t-s) -y) exp(-Jc(t-sr)) g(s,y) . e lefTh t han firse d th sid righte d th ean ter tn mo han d sidf eo this equation form a conventional dispersion equation. The second e righth ter tn o m han d additiona n a sid s i e l term e b whic y hma interprete a correctio s e a conventionadth o t n l dispersion equation.

4. COMPARISON OF EQUATIONS An advantag conventionae th f eo l dispersion equatioe th o nt generalize obviously, dis simplicite ,th formee th f yro one. This cannot be the crucial point for ruling out the generalized dispersion equation. Their properties must be compared before we e makother th a decisioe r A .comparisoo e on n e wetheus n o t r

52 reduce n analysi a e additiona th o t sf o s l ter f (3.17)o m A . general discussion is not necessary in order to expose basic features of the term. Although the conventional dispersion equation (2.1) and the generalized dispersion equation (3.17) are very similar in structure they differ significantl importann i y t features that are listed in Table I.

TABLE I.

prop- conventional dispersion generalized dispersion erty equation (c.d.e.) equation 1 time-locality time non-locality 2 quadratif o w la c f lineao w la r spreading around source spreading around source 3 D = S(0) asymptotically tends to c.d.e. with D =S(0), after switching off the input term 4 time non-reversibility time non-reversibility discuss u t Le s these feature oney b .e son Time non-reversibility. We start our discussion with the fourth feature because only this feature is shared by both equations. Time non-reversibility means generally that the past cannot be reconstructed from the present (or future). Let us point out that this property is not a mere consequence of taking statistical averag meanr e (o stochasti f )o c transport modelse .Th clas modelf so transporf so meandevelopee e b th o e n ,t ti th n i d following section, are time reversible. Hence, the time non- reversibility is a consequence a random nature of fluctuations. Fluctuation normalle sar y distribute thao s d t ther positivs ei e probability that their amplitudes are larger than any fixed number. Speaking vaguely, a portion of substance is removed beyon boundary an d d cannoan y recoverede tb . Time locality A .syste m fulfils time localits it f i y knowledge at t, is sufficient for its description at any later timt> othen t .I 2 et r words histore ,th systef yo m until, t = lt does not need to be known for its reconstruction at t > tt. Only s knowledg it s necessar= ti j t t d a sufficiene an y r sucfo th reconstruction. Systems e thadefinear t y b differentiad l equations are obviously time local. The generalized dispersion a differentia equatiot no s i n l n integroequationa s i t -I . differential equation that is not equivalent to a system of a finite number of differential equations. Hence, time non-locality s inhereniit s t property. This fac known s i tyearca r nfo d san be traced back to the fundamental work of Taylor [5], There, the turbulent dispersion (or diffusion), is described by means of covariance functio f locationo n f fluio s d elements (particles) Lagrangeae inth n pictur motionf eo maie .Th n poin thas ti e tth complete histor f motioo y f fluio n d elements mus e knowb t n i n order to calculate the covariance function.

53 For the present purpose we demonstrate the non-locality of e generalizeth d dispersion equatio usiny nb particulaa g r form e covariancofth e functio velocitf no y fluctuations choice .Th e enable o makt es u s analytical calculation n closi s e ford an m obtain solution seriesa s sa . Therefore assume ,w e that K(t= ) pD exp(j-p|t|) so that h(t,s) = (l-exp(-p | t-s | ) and h(t,s) - 1 = - exp(-pIt-s|) .A genera l covariance functio representee b n ca n d by usin e Laplacth g e transform, K(t = )j P(p)exp(-p|X|)dpo s , that the considered covariance function is a basis for a general discussion. The use of the formal operator calculus has some advantag presenn ei t derivation defins u t e.Le + A + dx dt A = JLdx*, Lk = 4- v4- - -° * L^= qc t represenanle d t formall expressioe yth n c » exp{-*(t-sr)s c(t,x)d f = Y(t-s,x-y-w(t~s))y d }J q(s.y), » —* —*»

i.e. a solution of (2.1) on R. In terms of the defined formal operators the equation (3.17) can be rewritten as

Lk cAc^oD = kr - ,

k remindc wher d e inden Ls an tha^i th eu e s extinctio k xth t n constan thesn ti e expression unique Th . ek s solutiosi thif no s equatio defines e serieni th y sb d a n (j>Aiw1) «r. (4.D n«0 m"0 thero s , ex d exist an e functio s t Th it s l defines i al k nc r fo d Fourier transform

(4<2) 1 + + J.(p+wp)p i Em £ l- c + fk l+ T7——— — T-T

The non-locality follows from this expression. The representation (4.2) proves that it is impossible to define a differential e functioequatioth r fo nn ck(t,x) e existencTh .a f o e differential equation would imply the existence of a polynomial p(Po/P) i-n Poi with coefficients tha generae tar l function, p f so such that

Obviously that (4.2) doe allot sno w such polynomial. Law of spreading around a source. This law has been discusse previoue th n di s sectioconventionae th r nfo l dispersion equation discussioe .Th completee b describiny n b nca w dno e gth behaviou f solutiono r e generalizeth f o s d dispersion equation

54 around a source. The law of spreading is implied by the form of exponent of the fundamental solution. In the present case, Equation (3.13) mus usede b t . Assumptions about parametere sar e previouth f o sd sectionen e e same th th thoss th a er t Fo .a e proble, isoline 2 e R fundamenta th n i mf o s l solutio e verar ny curvee closth o est

x2 = 2e(fc) = «4jDJ'dfc1 f dtz h(tz)

where a is a parameter. The right hand side behaves as t2 for small t, implying that the defined curves are circles, moving downstream with velocity havin d w,an g radii proportionat o lt r smalfo . lt Dispersion constant spectrald an function. From comparison of two equations in terms of the third property in Table I., we can drae mosth wt serious critical remark regardine th g applicability of conventional dispersion equation. We dare say that this comparison is fundamental in determining verges beyond which an application of the conventional dispersion equation is meaningless. The present discussion tries to answer the following basic question: Let velocity fluctuations be normally distributed and have the stationary covariance function K(t) with spectral function conventionalthe S(\)Is . dispersion equation adequatean tool for describing transport in the mean? With random nature of velocity fluctuation so assumed, there is no doubt that the average concentration field is given by Expression e (3.16)otheth rn e O conventiona .th hand f i , l dispersion equation were valid, then the average concentration field woul givee db y (2.3)nb , i.es a . t •> ccda(t,x) = I ds J dy Y(t-s,x-y) q(s,y) .

4— 0 » —*

Hence, the difference of c of (3.16) and ccde must be estimated in order to answer the formulated basic question. Two situations are discussed separately, one admitting the applicabilit e conventionath f o y l dispersion equatioe th d an n other conflicting it. firse Th t situatio characterizes ni switchiny db inpuf gof t at t = 0. A simple estimate can be derived if we assume that the covariance function has the same form as in discussion of the previous property spatialls i ,q y localize origine th t da , i.e. q(t,x )q= 0 (t)0(-t)

V Y(t-s,x)ds f =

tha, t positiv s ti y Thenan largr d e,fo an e enough, ccde(x )c(x> ) anhav e dw e followin eth g estimate

55 t'X) - c(ttx) = f ds [ Y(t-s,x} - «*-•.*>

For t positive and large enough we have X < Y so that

_ 8pDte

From the definition of dispersion constant D in (3.10) there follows the identity 2D = |k(t) dt = S(0) and, thus, the property s establishef i Tablo 3 . I e r thifo ds particular casf o e covarinece function. However, the same is true for a general form of covariance functio correspondine nth K(tr Fo ). g proof details e asymptotioth f c behaviou f X(t,xo r ) /Y(t,x) mus e deriveb t d first. In the second situation, contrary to the first one, the inpu differens i tassume W . 0 e > ttha e t fro tth l mal zerr fo o inpu constans ti t with respecthao s tt stationaro tt y conditions of the transport can be achieved at t = °°. Again, an example of the covariance function is discussed because general case can w essentiane giv o n e l features e assumptioTh . n aboue th t covariance function, drift velocity w, and input q are the same as in the previous case. Only the input term is not switched off Wit. 0 accepteo hs = a tt d parameter generalizef so d dispersion equation, its solutions tend to time independent functions as t -»• oo. Let us denote the corresponding limit by ck(x) . Formally, ck(x) is represented by (4.1) where 1^ = -DA+k. Hence, the Fourier transfor cf mo k (xs i )

Becaus q(pf eo ) 2nq= have w of e

** « > 2*0) [ dp —— ^— + 2n<7 f dp y°J ^ +a *°0 J ^ (k+Dp2)

The first term on the right hand side is the unique solution of stationary, conventional dispersion equation I^c^,. = q, so that

56 A striking conclusion is implied by this inequality. The relative error between stationary solution e conventionath f o s d an l generalized dispersion equation close b 50%o n et sca . o T larger p valuether f o es correspond a bettes r approximation by the conventional dispersion equation. As p tends to infinity the spectral function tends to a constant, i.e. to e spectrath l functio f velocito n y fluctuations inducee th y b d Brownian motion. There is another interesting result of the present analysis. same th e r randoFo m mode velocitf lo y fluctuations,e th i.e r .fo same p in terms of our example, reliability of results of the conventional dispersion equation depends on the extinction constant k. Let |U = p/k. If /z > 100, the relative error (4.3) is less than 10%. On the other hand, for /n « 1, the relative error can be up to 30%. e followinTh g exampl illustratn ca transporf y eo ba a e n ti better this dependenc n extinctiono e A .spectra l functiof o n velocity fluctuations in a bay cannot be fitted with the function S(X) = 2Dp2/ (X2+p2) because of many local minima and maxima in a real spectrum. Such a fit can be understood, as a zero order approximation. With this assumption, we can say that the paramete e rangvalua th s n ei e ha 1 h"0.0 p r0. 1 . A colifor 5- m extinctioe th s ha a whil1 nbacterise h" constan e 3 th 0. n = ai tk hydrocarbons (dissolve l componentsoi d ) hav= 0.00 k e 5 h't 1I . follows from our analysis that the conventional dispersion e descriptioth e useequatior b fo d f n transporo nca n f o t hydrocarbon e sea f thith o n ,se i s us equatiowhil e th er fo n transpor coliforf to m bacteri dubiouss i a worthwhils i t I . o et poin t tha ou ta reliablt e descriptio e transporth f o n t around source r short-livesfo d pollutant crucias si applicationsn i l . Results obtained so far regarding comparison of the conventional and generalized dispersion equation were derived by using one class of covariance functions, K(t) = pDexp(-p|t|) . Now we can derive indirectly a conclusion which is valid for general covariance function consideree .Th d clas covariancf so e functions is defined by two parameters, D and p. The first parameter, D, dispersioe isth n constant, whil secone eth d parametera s i . ,p measure of extinction of the correlation between fluctuations at a time s and time s+t. This is a simple consequence of the fact that the correlation coefficient is exp(-p|t|). Hence T = In2/p is the half-life of correlation in the same way as E = In2/k is the half -life of extinction of the dissolved substance. The ratio p. = p/k can be expressed in terms of half-lives:

H= - (4-4)

Thus f thii , s rati s largi o e thee approximatioth n y b n conventional dispersion equatio s goodi n . Such conclusios i n wholly plausible. In terms of the transport it follows that to a larger ratio there correspond sa large r amoun f substanco t e survivinw realizatione a n i g f velocito n y fieldt whicno s i h correlated with the previous realization. The conventional dispersion equation describe e transporth s t precisel n suci y h realizations of velocity fields, i.e. in velocity fields with uncorrelated fluctuations generalizatioa w .No thif no s simple result is straightforward. It suffices to determine the half-life e covariancoth f e functio f velocito n y fluctuations. This

57 quantit obtainee b n yca variouy db s methods proposes i e .On d here by definin e e spectrazergth th o f o orde lt fi rfunction . After the half-life of correlation is determined the dimensionless paramete n rindicato a s use i p.s a d f reliabilito r f resulto y s obtained by the conventional dispersion equation.

5. A SIMPLIFIED APPROACH Let us consider the transport of substance through a duct spreading alon x-axise gth assume .W e tha crose tth s sectiof no duce th t doe changt sno tha o parameterel s tal fieldd san s depend only on time, t, and one space variable, x. Here, we consider a simplified stochastic transport. A particle is moved through the duct with a random velocity v(t) depending only on time. This is the first step in simplification becaus egenerallv y depend timn sspaco d ean e variablesn ca e .W defin meae eth n valu velocitf eo y w(t E(v(t))= fluctuatiod )an n f (t) = v(t) - w(t) . The next simplification consists in defining the fluctuation by f(t) = v w(t) where v is a random variable. Thus e assumw , e thae randoth t m natur f f(to e )s completeli y defined by the time-independent random variable v. This simplification may appear drastic since we are aware that general velocity fluctuations depen tim n spaco d mor a e an n ei complex e randoth wayt m Le .variabl hav v ee probabilit eth y densit, p y whic positivs i he interva th n o e l (-1,1) probabilite .Th y that e fluctuatioth values sub-intervaha a nf n i s l (Xi,X equas 2i ) l

cBlpU) .

With this simple definitio f velocito n y fluctuation f(t)e th , velocite forth m s v(tha y = )(l+i>)w(t s realizationIt ). s have alway directioe sth amplitudee w(tf th no d )an proportionae sar l to w(t). Now we can describe the transport in the following way. A particle can have the velocity (l+X)w(t) with the probability A clou p(X . particlef o ddX ) s tha s initialli t y definee th y db distribution C0(x) at t=0, changes in t so that its realization at a time moment t>0 has the form:

C(A,t,x )= [C0U-(l-X)z(t) )C+ 0 U-(l+X)z(t)>], b t (5.1) z(t) = f w(a) ds,

with the probability p(X) dX. Hence, the expected concentration s i fielt t a d i c(t.x) -JT(C(t,x) | fdXp(A- )- ) Ctt,t,x). (5.2) 2 -i e initiath f I l distributio proportionas i 0 C n e Dirath - S co t l function, this expression has a simple form. Let C0(x) = A

58 c(t,x) = (5.3) 2 z(t)

casconservative a I nth f eo e tracer meae ,th n transient tim, 0 et positioa t a n defines xi , y db fc(t,x)dt t fc(t,x}dt

Assuming the symmetry of the function p we have the following expressions i fc(t,x) 2£= dt t J w2

fc(t,x) t dt = ^ f J w J

and consequently

K =

Since (1-X e (-1,1 > )(1-X)X . 1 e havr )w > fo 2 K e Let us illustrate the obtained results by two simple examples. Relevant quantitie e firs th second tan r fo sd example are denoted by subscript 1 and 2, respectively. Let:

(5.4) -|) ec|X|-d),

parametea wher s i ed r definin intervae gth l [-d,d] outside which e densitieth se zero par e ;graph Th .e illustratef par ;o s n i d Figure 4. The corresponding functions Cj(t,x) can be expressed by mean rationaf so l functions have .W e

(3 „ r, ('-!». *£ 4d 2 t fc 2 3 (5.5) 0 otherwise

, (5.6) otherwise

59 -0.9 0.9 -0.9 0.9

Figure 4. Probability densities pt and p2. e graphth thesf 9 o s = ew x/ = wher x/w= r w=0.r r ed Fo . an 1 function e illustratesar Figurn i d . 5 e The expression (5.1) represents a travelling wave along the x-axis. The wave is initially defined by the distribution c0 and after initial momen splitt ti s int componentso otw componene .On t travels wit velocite hth y (1+X othee ) th w(t) rd an travel, s with velocity (l-X)w(t). Therefore, it is not surprising that the function (5.1) is a solution of the following telegraph (wave) equation: The construction of the concentration field (5.1) was easily carried out because the velocity v(t) was x-independent. For a

--)2 CU,t,x)- CU,t,x)- (5.7) (-A 4. (t)-A =0. dx dt v velocity fluctuation depending on x the concentration field c(t,x) cannot be constructed by the present simple technique. The derivatio e fielth df o nmus t follo e pathth w : "BASICS", "MEAN

concentration concentration

t ime 11 me

Figure 5. Graphs of concentration fields (4.5), (4.6) measured at the point x = 0.9 units downstream from the source.

60 FIELD MODEL" d "MODE,an L SOLUTION Figurf "o Thi. e1 s means that a famil modelf yo s suc s (5.7ha ) mus derivee tb d firs d thetan n solutions C(X,t,x obtainee b n numericaa )ca y b d l method. This program can be carried out for a general case of transport problem in a random velocity field as described in [1,2]. Due to a larger numbe detailf o r mathematicaf so l natur e prefeew o rt avoi exposition a d thif no s rather theoretical problem.

6. A CASE STUDY Results of an experiment [6] in which tracer is transported through groundwaters of the Istra peninsula (Figure 6.) are analyzed. The terrain is karstic. Tritium was used as a tracer. Tritium was injected on 16.03.1976. into a cave at (5i2e (Figure 7). At the end of experiment 99.87 % of the injected tracer was recovere Gradole th t da e spring. First occurrenc tracee th f ero appeared 5.72 days after release. The peak of the tracer concentration was recorded 7.42 days after the injection (Figure 8) . Tracer was also monitored at two other springs in the vicinity (100 m and 250 m) of Gradole but no tritium was found. This indicates thae floth tw resemble undergrounn sa d "river". Geographical distance betweee Th n. £i2km d Gradol 4 an e1 s i e corresponding differenc followt I elevation . ei m s0 26 fro s ni m this data that the velocity of water corresponding to the peak

Figure Norther. 6 n Mediterranean Sea. Istra peninsul s locatei a d in the rectangle. Figure 7. Istra peninsula. Tracer was injected into cave at CIZE and recoveree sprinth t ga d GRADOLE.

is equal or larger than 1.96 cm/s. On 16.03.1991. the flow rate of water at Gradole was 7.58 m3/s while at the end of 5 m3experimen3. /s e flos .s nearlTh wa wa w t i yt constantly decreasing durin experimene gth t (Figur. e8)

61 appls u t y Le result previoue th f so s sectio dato nt a which are illustrated in Figure 8. We try to find out the probability density p of (5.4) so that the solution (5.6) reproduces results of the measurement. The assumptions of the model (5.1)-(5.3) are practically satisfied: (a) the measured flow rate during the experiment was changing slowly with nearly constant gradient, and (b) the transport is not branching i.e the total amount of tracer, injecte beginnine th pathe t da th carrie s ,f i g o d unti poine lth t of measurement. The model is used to calculate the probability density function, p, (Figure 4) . From this result a general theoretical model for p (5.4) is proposed. A model for p is necessary for our further study. Sinc e wateth e r outflo t Gradola w e sprin s decreasinwa g g during the measurements with a constant gradient, initially the mean velocity w(t) is approximated by w(a + a T,-; < t . < 0 (6.1)

wher = 1.7 0 e4w Neithe. h cm/s = 1.52 0 37 ,a r = a, =T -0.d an 5 e precisth e distance , betweex , e injectioth n n poind an t measuremen e knownar 0 e tvaluw . th poinf Becauso er no te th e solution in (5.6) depends on r = x/w0 it is sufficient to determine one of the two parameters, x or w0, while the other parametedeterminee b o t s rha d frod man datam k assume 4 .W 1 = ex obtained w0 = 1.5 cm/s. The criterion for the determination of w0 is the symmetry of probability density p, i.e. the constraint:

V -L Ap = | J cfA, p(A) - y cfA min -1 0

obtainee W d this velocity wit0.01= p hA , i.e. wit relative hth e grape Th probabilitf o h. erro1% f ro y densit s illustratei yp d in Figure 9.

nCI I 600

400

200

4 1 day 3 1 s2 1 1 )9 01 8 1 B 5 .0 .5 Tritiu) Figur(a . e8 m concentratio) (b d nan Figure 9. Probability density flow rat watef eo t ra function from data. Gradole spring.

62 From the obtained results the following model of the probability densit proposes i yp d

Ixl P(x= ) " 6 2 0 otherwise wit a hsingl e paramete , representina r e maximuth g m ratif o o diffusion to mean velocities. w sucHo a hsimpl e mode f fluctuationo l s abl si o givt e e plausible results? The model (5.3) is not trivial, although its derivation is based on a simplified model of fluctuations. The e deriveb mode n ca ld fro a generam l transport a mode r fo l stochastic velocity field and with all parameters depending on spac timd ean e [1,2] n illustratio.A thif o n s general approach is give n FigurA restrictioi n . 10 ee genera th f no l transport model to the one-dimensional transport, where the mean velocity is independen externad an x f to l source omittede sar defines ,i d by (5.3 assumptione Th ). modee practicalle th lar f s o y satisfied: (a) data on flow rate suggest that velocity is a slow changing function during the experiment, and (b) branchings of flow are absent. Therefore reasonabls i t ,i assumo et e tha floe tth w rate is also a slowly changing with position x.

simplified fluctuations a class of general (unrealistic covariances) fluctuations (realistic covariances)

corresponding famil generaf o y l transport problems

an approximatio parameterf no s modele ofth s

a famil telegrapf o y h equations

Figure derivatioe Step. th 10 n e samsi th ef no typ f eo model from different assumptions on statistics

7. DISCUSSION Main properties of solutions c(t,x) of (5.3) or (5.7) are briefly describe recapitulats u Section di t Le moso . tw n5 t e eth important properties.

63 e (AfunctioTh ) n c(t,x), definin e meath gn concentration field, is equal identically to zero if the quantity jx - tw(t,x) | is sufficiently large, i.e e velocit.th f spreadino y f "tho g e cloud" c(t,x) is finite. This feature is illustrated in Figure 5. e functioTh ) (B n c(t,x) describe e averagth s e motiof o n "fluid elements". Each "fluid element" travels along the x- directio velocityn n ow wit s hit , w(t,x Xw(t))+ . Actually, "fluid elements" travel in pairs, one with the velocity w + Xw and the averagine otheTh . rXw witg- procedurw h carries i e witt ou dh respec weighe possiblth e o th tove t p l ral e value velocitf so y Xw. Therefore, the velocity Xw(t) is called the dispersion velocity witprobabilite hth y dXp(X).

Conclusion In principle, deterministic transport models describe transport in the mean. To each model of velocity fluctuation there correspond a modes f o transpore lmean th n .i t Unfortunately, this choic strictls ei y limited usea f .rI accepts e hypothesith s that velocity fluctuation e normallar s y distributed thee conventionath n d generalizean l d dispersion equations are at the user's disposal for a description of transport in the mean. If the hypothesis on normally distributed fluctuations must be rejected then only one limited class of models of transport in the mean can be used. The other possibilit withdrao t s i y w from modelling transpor meae th nn ti and turn bac o M.Ct k . method d considean s e stochastith r c transport problem. Normally distributed velocity fluctuation. Normall) 1 y distributed velocity fluctuation none b n - ca s correlated and correlated. They define a Brownian motion in the former case and a general Gaussian process in the latter case. The conventional dispersion equation describes transport in the mear whic fo e nfluctuation th h e definee Browniaar sth y b d n motion r fluctuation.Fo s define generaa y db l Gaussian process the transport problem is described by the generalized dispersion equatio Sectiof no . n4 e sourceth e e f switcheconventiona2I th ar )s f of d l dispersion equation describe transpore sth t reliably. conventionae 3)Th l dispersion equation cannot describe eth transport reliabl vicinite th n i ysourcesf yo . e conventionaTh ) 4 l dispersion equation induces highly underestimate t e inputcas ne dth f smal o en i s l value n i f p,o s (4.4), /i ~ 1. In such cases the conventional dispersion equations cannot describ transpore eth t reliably. 5) If ß is large, ß > 100, the net balance by the conventional dispersion equation is in a good agreement with the net balance of exact model for the assumed velocity fluctuations. In this case the conventional dispersion equation is an acceptable model. 6) The generalized dispersion equation describes the transport correctly for normally distributed velocity fluctuations. Unfortunately, the modelling is complex because the equation is not a differential equation and solutions must be constructe y iterationsb d , solvin a conventionag l dispersion equatio eact na h iteration step.

64 Uniformly bounded velocity fluctuation. For a class of uniformly bounded velocity fluctuations and the case of one space dimension, the model of transport in the mea s beenha n derived defines i partiaa t I .y b d l differential equatio hyperbolie th f no c type modee .beeTh s le ha n th user dfo descriptio tracef no r experiment groundwatersn i s e floth wf I . rate changes slowly o thas e velocit, th te b f floo n y ca w approximated by a function which does not depend on position, predicted concentrations are in excellent agreement with data. For flows which change significantly during the experiment, predictions cannot describe all changes in concentration field. In this case models with spatially varying velocity muse b t used.

REFERENCES Limic] [1 , Model,N. f velocito s y fluctuation d exactlan s y derivable diffusion laws, Appl. Math Model 4 (19901 . ) 549-552. [2] Limic, N., A new approach to the problem of transport in aquatic ecosystems, FIZIK B (1992I A ) 7-31. [3] Goldstein, I., On diffusion by discontinues movements and e telegrapth on h equation, Quart Mech. J .d Appl an . . Math4 . (1951) 129-156. Bourett] hypothesi[4 n A , ,R. s concerning turbulent diffusion, Canad. J. Phys. 38 (I960) 665-676. [5] Taylor, The statistical theory of isottropic turbulence, J. Aeronaut. Sei., 4 (1937) 311-315 [6] V. Kubelka, Application of isotopes for determining quality of karstic groundwater Istran si , ReporRudjee Th f to r Boskovic Institute, 1979

Next page(s) lef5 t6 blank ELLAM: AN EFFECTIVE TOOL FOR MODELLING SHARP FRONTN SI QUANTITATIVE ISOTOPIC HYDROLOGY

. HERRERI A National Universit f Mexicoyo , Mexico City, Mexico

Abstract

mainThe topic studyof Quantitativeof Isotopic Hydrology transportisthe isotopesof waterby flowing a porous in medium, theirand interactions. mathematicalThe modelsof such processes advection-diffus the based are in ion equation or systems of such equations. Until recently, effective mass conservative algorithms capable modelingof advection-dominated transport were lacking. However, many of the difficulties encountered previously have been overcome by ELLAM methods, recently developed by the author and coworkers. Here, the different implementations of ELLAM methods that exist at present, are presented and evaluated, withpurposethe makingof them more readible available to the scientific community working in Quantitative Isotopic Hydrology.

1. INTRODUCTION The numerical solution of the advection-diffusion equation, is a proble f greao m t importanc e stud th f transpor o yn i e f soluteo t s a liqui y b d phase particulaA . r cas thif eo s general probleme th s i , study of the transport of a tracer by water flowing in a porous medium. The central problem of Quantitative Isotopic Hydrology, is precisely this problem. e numericaTh l treatmen e advection-diffusioth f o t n equation, when advectio s dominanti n s bee ha ,a challenginn g problea r fo m long time, specially if sharp fronts are present. A feature that is required from algorithm e n ablorde b i so mode t o et r l effectively advection dominated transport, is that its performance be independen e Couranth f o t number a larg o t ,e extent. Another feature which is essential, specially in Quantitative Isotopic Hydrology, is that the algorithms be mass-conservative, even when significant boundary behavior is present. A general clas f methodo s s thas beeha t n quite successfud an l is being applied extensively, is the Eulerian-Lagrangian Localized Adjoint Method (ELLAM)[1-20] e importanOn . t featur f o ELLAe M methods s tha i ,e onl tth y the e characteristiar y c methods thur fa s developed, e thamasar ts conservative. This property enhances

67 further the potential applications of ELLAM methods to mathematical model Quantitativf o s e Isotopic Hydrology. This pape s devotei r o explait d d discusan n e ELLAth s M methodology with the intention of making it more readily available to the scientific community working in Quantitative Isotopic Hydrology additionn I . briea , f critical compariso differene th f no t ELLAM implementations that have been developed mades i , . e methodTh s availabl o t treae eadvective-diffusiv th t e transport equation, are usually classified into: Eulerian, Lagrangia d Eulerian-Lagrangianan n A metho. s callei d d Eulerian, whee spatiath n l gris kepi d t fixes i callen i time dt I d. Lagrangian or characteristic method, when the time derivatives are discretized followin e e fluimotioth th g df o ns i particle t i d an s called Eulerian-Lagrangian, when the fluid particles are tracked, but the spatial grid is kept fixed through time. When applied to advection dominated transport, the salient features of approximations which derive from an Eulerian approach, e b summarize y ma e s tima followsTh d e) (i truncatio: n error dominate e solutionsth s , (ii e solutionTh ) e characterizear s y b d significant numerical diffusio d soman n e phase errors, (iiie Th ) Courant number (Cu = V-j—)A t is generally restricted to be less than one, and sometimes much less than one. Among such procedures, one y distinguisma h Optimal Spatial Methods (OSM) whicn i , n accurata h e solution of the spatial problem is developed. Other Eulerian methods seek to cancel the errors introduced by the time discretization with the errors produced by the spatial discretization (see r ,exmplfo e [21-24]). Som f suco e h methods actually improv o somt e e extent e inconvenienth , t featuref o s Eulerian methods in general, mentioned above. However, they still suffer from severe Courant number limitation. ] [ s In Lagrangian methods in general, the problem is solved step by step in time. The process of obtaining the solution at the new time level froe solutio e th previoum th t a n s one n turni , s carriei , d out in two teps: one in which fluid particles are tracked and a second one in which a purely spatial elliptic problem is solved. This latter ste frequentls i p y calle e "diffusivth d e step", because the elliptic characte problee th f s induceo re presenci m th y b df o e diffusion (usually Fickian). Methods whic e purelar h y Lagrangia e particln th carr t ou ye tracking forward in time. This introduces distorsions of the spatial

68 grid, which complicat e implementatioth e e diffusivth f o n e sted an p lead to inacuracies of the solution. In Eulerian-Lagrangian approaches the grid is kept fixed at all times, avoiding in this manner the grid distorsions. To this end, the particles are tracked backward n timei s . Thus, such procedures profit froe structurth m e of characteristic curves when carrying out the time-discretization, bun additioi t n they profi f havino t g kept fixe e spatiath d l grid, when carrying out the diffusive step. Eulerian-Lagrangian methods have the significant advantage that Courant number restrictions of Eulerian methods are overcome to a large extent, since the advection term is eliminated from the elliptic problems to be solved at each time step. On the other hand, the Localized Adjoint Method (LAM) is a methodolog r discretizinfo y g partial differential equations which s introducewa e authoth y rb d [25-30]. This procedur s basei e n o d Herrera's Algebraic Theory of Boundary Value Problems [31-35] (also [25]). Applications have successively been made to ordinary differential equations r whicfo , h highly accurate algorithms were developed [25-27], multidimensional steady state problems [28] and optimal spatial method advection-diffusior fo s n equations [29-30]. Recently, Localized Adjoint Method (LAM) has been applied in space-time n a Eulerian-Lagrangia n i , n manne o t problemr f o s advective-diffusive transport, using specialized test functions [1-3,7-9]. These functions locally satisfy the homogeneous adjoint equation within each element e generaTh . l methodolog o obtaines y s i d the Eulerian-Lagrangian Localized Adjoint Method (ELLAM). Like characteristic methods in general [36-43], ELLAM methods have advantagth e e that Courant number restriction f purelo s y Eulerian methods are removed to a large extent, but in addition they present other important advantages. Until ELLAM was developed, characteristic methodd threha d e sha kind f limitationsso : inability to rigorously treat boundary fluxes when characteristics intersect inflow or outflow boundaries, inability to ensure mass conservation e introductioth d an f numericao n l dispersio r somfo n e methodse du , ordew tlo o r interpolatio r integrationo n [44], othee th rn O hand generae th , l framewor s beeELLAMf o kha ns a , presente ] (se[2 en i alsd o [6]) quits i , e wide contrasn I . otheo t r characteristic methods, ELLAM allows systematic treatmentf o s boundary e conditionresultinth d an sg algorithm e masar ss

69 conservative [1] n .I addition t providei , a unificatios f o n characteristic methods (CM's). The general methodology of ELLAM [2], can be implemented in many differen kindo tw f implementation to sw mannersno o t p U . s have been developed. They derive from the application of two different

classe f teso s t functions n [1]I . , bilinear functions whice ar h define s "chapeaua d " function t levesa l tim d econstan an t*n t along characteristic curves, were applied and in this manner the first mass conservative Eulerian-Lagrangia e ngenera th schem r lfo e transport equations s developedwa , . This metho s referrei ds a o t d BELLAM [7]. An alternative manner of implementing ELLAM, is to use test functions whic e piece-wisar h e constant e advectear d an ,d wite th h transport velocite problemth f n o y[7,8]I . , unde e titlth rf o e "ELLAM Cells" (CELLAM), a very effective implementation of ELLAM using this kind of test functions, has been developed. CELLAM has the advantages of ELLAM methods described above, but in addition it ensures local mass conservatio yieldd nan s algorithms whic more har e convenien r existinfo t g solute-transport codes whic e basear hn o d finite differences. Thus far, the numerical performance of CELLAM s beeha n slightly better than tha f BELLAo t M [7-9]. Alsoe th , simplicity of the implementation of the method, is appealing. In passing e mentiow , n tha n implementatioa t n (FVELLAM) using similar test functions was intended in [45], but the authors reported numerical difficulties which severely e limith t applicability of their results.

2. BILINEAR ELLAM (BELLAM) This approac s firswa h t presente a sequenc papero n i dtw f so e [1,2]. Consider the one-dimensional transient advec t ion-diffusion equation in conservative form:

*U = - -(D -VU ) + RU = f(x't}> in ^ (2'1} = [0,1 ]ß xe x T1 n+ . n i r te• nn = [t , t ] (x,t)€ß = n x n X t subject to initial conditions u(x, tn) = un(x), (2.2) and suitable boundary conditions, at x=0 and J. The following development accommodate y combinatiosan boundarf no y conditionse Th .

70 manne e initia whicn th i r e regio d th hlan conditionQ n e chosear s n in Eqs. (2.1) and (2.2), is suitable for applying a step by step solution procedure. To make the exposition more readable, only the case of constant coefficients will be explained here, although variable coefficients have already been treated (seexamplr efo e [11]) r simplicityFo . e w , proceed in an ad-hoc manner. More systematic expositions placing the procedures discussed in this article in the general frame-work of the Localized Adjoint Method (LAM), are given in [2] and [5]. For the case when the coefficients of Eq. (2.1) are constant, the source term vanishe se partitio th (R=0 d an )s uniform i n e th , test functions used are: t - t x x- ,n+l + Ax V Ax ' ^"---"j x -x t. n+l -t, w*(x,t)= i +1 f i U^AYv AU^Yk ' ' ?£t 0, all other (x,t) shows i Fign s i n a Suc. . 1 e har d &weightinwher an Q e g functions 1 2 e continuouar d satisfan s0 y (i.= £w e * i[w]=0) t havbu ,e discontinuous first derivatives (i.e.; [dw/dx]*0) e jump.Th e sar rEui = 1_ • r—i = —• r—i = — (24) Lax-Ii- x 'A i Lox-l ' " Ax LaxJAx i i i+ > *

DISCRETIZATION IN THE INTERIOR OF Q Whee regioth n n ß doet intersecno s e laterath t l boundaries, integration oveyield, Q r s

u(x,t" "Jw'Cx.t - T-Z.1 u(c x )d "r (t),t)d- t

„tn+1 tn+1 2f u((r (t),t)dt+ f u(cr (t),t)c tn l tn 1+1 * =^J +1 u(x,tn)wi(x,tn)d x+ f f^dxdt, (2.5) x i-l

wher e unknownth e s have been collecte e left-hanth n i d d membef o r the equation while the data is included in the right one. In Eq. (2.5), it is assumed that x=o- (t) is the characteristic curve passing throug t tima t ehx (Fig. 1) . i n+l

71 i+i

Figure 1.- Test functions used in BELLAM.

Notice that the unknown function u(x,t) has not yet been approximate y specifian y db c functional form e timTh .e integraly sma be approximated using a Backward-Euler (fully implicit) scheme. Then the spatial integrals that appear in Eq. (2.5), may be approximated in many different ways, usin nodae e discretth gth lt a value eu f o s , exclusivelytim + n t e d levelan o n thae s unknownt ,s e th t th n i s equation ultimately correspond to nodal values at time tn . Different approximations of these integrals lead to different CM algorithms reporte e literaturth n i d e [1]r exampleFo . , piecewise linear spatial interpolation of u at time levels tn and tn , coupled with a one-point (at t=t ) fully implicit approximation to n the temporal integral e , modifieth lead o t sd methof o d characteristics of Douglas and Russell [40].

72 BOUNDARY CONDITIONS Whe a region intersectQ n e infloth s w boundary, several cases 1 can occur. As an example, we discuss the case illustrated in Fig. 2. Then, integratin . (2.1Eq g ) overegioe th r obtaineds ni Q.t i , :

+l

Figur - Cas2. e e whetesdomaie e th th nt f functionno s intesecte th s inflow boundary (BELLAM).

73 r*!*! , , n+1 n+ , 1 .. , i D t )w (x, t \f u(eru(l * J 1 l * J * * ^ t t n+1 i-l i t * -ti-1 i gu + u(o- + Vu(0,t)}.d- = t +l(t)' i » J J * dx '

* * \ ( 'ti-i fti l r i D u(0,t)dt - u(0,t)dU + f w dxdt (2.6) Ax J * J * J o *•• t t J " i i+1 The integrals along characteristics appearing in Equ. (2.6), can again be evaluated by means of a fully implicit approximation. However, the fact that each of these three integrals has a different length, introduces problem r achievinfo s g consistencn i y the orde f accurac o re approximations th f o y r somfo , e classef o s boundary conditions t a least, . Suitable combinatione th f o s integrals just mentioned, wite lasth ht integrae left-hanth f o l d side of Eq. (2.6), may overcome the problem. However, whether this is feasible or not, depends on the type of boundary conditions to be satisfied o exhibiT . t this problem s necessari t i , o develot y a p more careful derivatio whicn e error i ne orde th th hf so r introduced at each step s explicitli , y stated. Thus e readeth , s referrei r o t d Sectio , whern5 mora e e careful derivatio a simila f no r equatios i n presente CELLAMr fo d . The last term in the left-hand side of Equ. (2.6) must be handled with special care, to obtain an algorithm with satisfactory properties. If we simply apply the Backward-Euler scheme to the unknown boundary flux along the time direction, the discretization wil e unsatisfactorb l r largfo y e Courant numbers (Cu=VAt/Ax), since many characteristic lines will be crossed. Thus, instead, one can evaluate the contribution to the integral of the term containing u(0,t), since thi Dirichles i s t data e d rightransposth an , to t t i e equatione th sid f eo n [1]I remainine . ,th e integrag th par f s o t lwa approximate y whichwa a s indicate a ,n i d n [8]i d equivalens i , : to t

t*- Xi 1 + 1 '" « 5 J w'!|

This approximation however, as pointed out in [7], is not necessarily consistent with the order of approximation that is required in the formulation: OfAxAt ). This latter order of

74 approximatio achievede b n nca , using relations simila. (2.7)Eq o t ,r only if the expressions under the integrals, are suitably combined wite integralth h s along characteristics presen. (2.6)Eq d n i an ,t thi possibles i s s beeha ns a ,alread y been mentioned, onlr somfo ye kinds of boundary conditions [7]. For outflow boundary conditions of Dirichlet type, the outflow boundary contributions vanish for all the test functions. This is due to the fact that all the weighting functions vanish in the characteristic Z , which passes through (x ,tn ), and beyond it. E E Also, the system of equations that is obtained in the manner explained above, is closed, because u is datum. If additional information is desired at the outflow boundary, it can be obtained applying procedures which amount essentiall post-processino t y g [1].

3. SOME REMARK DISCRETN SO E METHODS o basitw Ther ce ar taske s that every numerical methor fo d partial differential equations has to accomplish [2,6,7]: i).- Gathering information about the sought solution; and ii).- Interpolatin , moror ge generally, processing such information. These two processes are distinct, although in many numerical methods they are not differentiated clearly. In procedures which are derived froe methoth m f weighteo d d residuals e informatioth , n aboue th t exact solution that is gathered, is determined mainly by the weighting functions used. Since this information does not determine uniquely the sought solution, some procedure for extending it is required, in order to fill the gaps of information and exhibit at the end, a unique approximate solution. Different methods of solution follow different strategies for accomplishing this latter task of extending the information that is available n generalI . , interpolatio d extrapolatioan n n procedures e appliedar r exampleFo . n finiti , e element methods some basis functions are chosen and the approximate solution is assumed to be a superpositio sucf no h functions thin I . s case informatioe th , n about e exacth t solution whic gatheres i hweightine th y b d g functionss i , interpolate mannea n i d r whic determines i h e familth f basiy o yb d s functions chosen. Clearly, it is not convenient to carry out the process of extendin e informatioth g n blindly, ignorin e gactua th wha s li t information that is available. However, this is what is usually

75 e contrarydoneth advantageous n i O . t e insighi , th f o tmako st e eus gained when the available information has been identified, since the selectio e besth t f o nprocedur r extendin fo s estrongli , it g y dependent on the information that is at hand. Due to these facts, in recent works [6,7] the author has advocate n approaca d r developinhfo g numerical methods whicn i ,e hth processes i) and ii) are clearly separated. Firstly, the information abou e soughth t t solutio nt hand a s thaidentifiei s ,i t d an d secondly, using that insight, a procedure for extending such informatio defineds i n . Herrera's Algebraic Theory of Boundary Value Problems [25,31-35], which permits localizing the adjoint, has shown to be quite suitabl r identifyinfo e e informatioth g n containen i d approximate f thio solutions se us theor e s cleaTh .ha y r advantages over other options, such as the standard theory of distributions, because of two reasons at least: the use of the algebraic theory permit e localizatioth s adjointe simultaneoue th th f d o n f an ,o e sus discontinuous tria d tesan lt function feasibles i s . Then, depending on the information that is identified, interpolation procedures suitabl r handlinefo efficientlyt i g e selectear , d appliedan d . This is what should be properly called Localized Adjoint Method. The introduction of basis functions is not required and, even more, s inconvenieni thei e us r d n som[8]i an t e,] thi[7 cases sn I . approac s appliehwa d quite successfull derivo t y e CELLAM.

4 ELLAM CELLS (CELLAM) This method was presented originally in [7] and [8] (see also [9]). The notations adopted conform with those which are usual for cell approaches. A partition {x ,x , x ,...,x_ , x >, is introduced, which induces a partition of £2 into subregions {n1,^2, . . . ,QE}, if for each i=2,...,E-l, n1 is defined as the subregio e , limitecharacteristith fi y f b o dn c curved san Z (see Fig. 3), s thawhili t Q n ewhicpar f o te h th lie o t s e subregioth s I d Œf fan io n whicf E h lie o right s f o t]T n . The subregions of the partition are called "cells" and they t. if d are said to be "uniform" when

r f h = ° i=2,...,E-li-i/> 2 ; X3/2-xi=h/2,XE-xE_i/2=h/2 (4.1)

A syste f constano m t weighting function applieds i s . These ar e the characteristic functions of the subregions that constitute this

76 .n+i

X0=0

Figure 3.- Test functions used in CELLAM.

partition. Actually, not all of them are required. The system of weighting functions applie derivo t d e CELLAM [7]: is , f (x,t)e1i , Q (4.2) Vf (X,t)= a f (x,t)«sri , 0 i

DISCRETIZATIO E INTERIOTH N Q I N F RO e casIth ne when Q doe t intersecsno e laterath t l boundaries of the region Q=[0,J]X[t , t ], integration of Eq. (2.1) over Qa, n n+1 yields: r , 1 a+ r u n+idx, + r tn+l tn+l L J I dt a-i tn a+1/2 tn a-1/2 x ~ a+i i n I x d u # = (4.3) cc-i Equation a (4.3modifie d an ) d t versiodesignei f o o nt d incorporate boundary terms when the lateral boundaries of £2 are intersected by Q , is the starting point of the numerical treatment. Observe that * x x a+i a+i [ u n+la x, - | 35 u ndx,= 0(hk) (4.4a) X a-i Xa-i and

tn+l -tn+l dt = O(hk) (4.4b) tn a+1/2 J tn a-i/2

77 where h = max (h -h ) and k = t -t . Thus, in the i+l/2 i-l/2 n+l n developments it is required that integrals such as those appearing in Equs. (4.4), be evaluated to a precision of 0(hk ), at least. It wil assumee b l d that h«k ,o thas t 0(hk2)=0(h2k)=0(k3)=0(h3). Equation (4.3) supplies Information about the sought solution —in —th—e —————interva—l [xa-i/ 2, xa-t-i/ 2 ] —a t ——tim—e tn+ i and ——abou— t —it—s x-derivative on the characteristics Z and E . In the CELLAM ——————— — ———————————— — — a-i/ — a+i/2— 2 approach e informatio[7]th e goath f ,o l n processino t s i g e solutio e value th "celth th f o n elt i a n concentrat, it f o l al e o thiT . s . (4.3)centerend t timEq t a e integraln t= , ei ,th x " s (X n+l e firstlar , y t approximate o t fro t m a full n i dy implicit manner n n+l (i.e., by a one-step Backward-Euler approximation at t ). Thus n+l

(DÔX J+n t _V. I a+1/2 a-1/2

(4 5) - -- •a-i/2J "

For a uniform spacing and constant coefficients, a central difference approximation yields: n+l n+l „ n+l u - U l U n+ „ -v „n+n+ll u 2 - „n+U + l U A (|ü ) - (|H ) \k = a+1 «-1———«k+0(h3k) (4.6) 9x h a+i/x o a-i/22 j The extension of this formula to the case of a non-uniform partition a similae don b n i en ca ,r manner. However e ordeth ,f o r precision is reduced by one and the overall error in (4.6), becomes 0(h2k). In characteristic methods e ,numerica th mos f o t l diffusios i n due to the interpolations in space, which are required because in general, characteristic t t crostim no e eth o s d sleve t nodesa l . n Thus, all the approximations in space have to be carried out with special care. A special feature of the approximations used in the derivatio f CELLAo n M [7] s tha i ,o assumption t s madi n e aboue th t solutione shapth f eo . e firsTh t integra n (4.3)i l approximates ,i y b d

x Ot+l „ „2. n+l', n+l3 , 1, U n+la li , f , . _,. U dx = U h + h (4 7) 1x vox ' aa ' a-i and only the second order derivative requires a numerical approximation, since the information is being concentrated in the "cell a centers"tri-diagona t ge o T . l structur e matrixth r t i ,fo e

78 is necessary to use three-point approximations only. In the case of a "uniform partition", a central difference approximation yields 1 1 „n+ n+ _ l n+ X u 22 N + u + u , , l a i a+ i f_ a- n+ii a+ l , , _, . , . u dx = ————— 24———— — h (4.8) 5Ca-1 If the partition is non-uniform, the approximation to the second order derivative by a three-point scheme is only first order, and the error in the evaluation of the integral in (4.8), is only order four. There is greater freedom for the choice of the approximations to be used in the evaluation of the integrals at time t , since they n do not affect the structure of matrix of the final system of algebraic equations. In [7], the integral appearing in the right-han . d(4.3 Eq sid s approximatef o wa )e d usin n approaca g h similar to the one used for deriving Equ. (4.7); i.e., integrating the Taylor series expansio n u aroun e f mid-poino e nth d th f o t . However] intervax , _ , x [ sincl e suca "cel ht lno poin s i t center", un is not known there and an interpolation must be used to evaluate it. Using three-point formulas, u" and its second order derivativ e evaluateb n ca e o ordert d s thre d onean e , respectively. This yields an approximation which is fourth order in h.

5. BOUNDARY CONDITIONS The numerical approximations presented thus far, apply only when the subregion S3 c£2 does not intersect the lateral boundaries e regio Qu9th 9 f Qno , Q .e caseth Whe ,t n no boundarthi s i s y o 0 t conditions must be included. This Section is devoted to presenting the CELLAM procedures for dealing with them. In Eulerian-Lagrangian approaches e analysth , t doet havno s e controe discretizatioth f o l n infloa t a nw boundary, s sinci t i e completely determined by the spatial discretization. However, the situatio thin i n s respec a littl s i t e betten outfloa t ra w boundary. Thus, when dealing with boundary conditions, speciall n infloa t a yw boundary, numerica a llarg o diffusiot e e extente facdu th s t i no t , that characteristic t cros no e boundarie th so d se space-tim th f o s e region Q, at times levels belonging to the partition of the time interval. Thuse interio e spatiath ,th n i jusf o rs l a t regione th , approximatino n spaci s ee performeb hav o t e d with special caro t e minimize numerical diffusion, when dealing with boundary conditions the time integral boundariee th n so se treate b hav o t e d with special care.

79 There is an additional reason which enhances this effect at an inflow boundary e informatioTh . n thas suppliei tn inflo a t a dw boundary has a larger effect on the solution than that coming from n outfloa w boundary, speciall e cas th f advection-dominateo en i y d transport, becaus e formeth e s transmittei r e interioe th th o t df o r spatial region by advection and diffusion, while the latter is only transmitte diffusiony b d . In [7], it was pointed out that in some cases, it is more difficult to achieve the desired degree of accuracy in the integrals with respect to time at the boundary, than in the integrals with e differenth t a respec , tx timo t t e levels r DirichleFo . t boundary conditions e differenth , t terms a occucombinatio n i r n whics i h suitabl r obtaininfo e desiree th g d degre f accurayo e . However, when the total flu prescribes i x wher o d n considering boundary conditions of Neuman type, this is not the case [7].

A .Dirichle- t Conditions For 2 tesE- thit se functionsus case e w , ; namely, those associated with subregions Ü , . . . ,Q ~ . In particular, no test function is applied on the first subregion (Q ) or on the last one (£2E). See Fig. 4. Inflow Boundary Dirichlet boundary condition e incorporatear s numericae th n i d l equation o mannerstw n i s : directly, throug e boundarth h y termd an s indirectly, imposin e conditioth g e numericanth n thai t l approximations e variable, th som f o e s take prescribeth e d boundary values. Assume Q intersects the inflow boundary, as illustrated in Fig.4. Then x - - a+l/2 „tn+l t „ _ -v 1 U O U O I I 1 n+ I I xudcr = u dx - \ (D-^—) - (D—) kit - V Jx V 1 5x2W/ axSa-i/2/ a-i/2 a-i/2 2

a+i/2 a+i/2 e firs o Th integral tw e tright-han th f o s d membe e likar re those appearing in Equs. (4.8) and (4.5), and can be handled similarly. In addition, the last one in this equation is easy to deal with, since e prescribeu(0,tth s i ) d boundary value e thirTh . d integral, however, require speciasa l treatment.

80 Firstly, observe that (D) - (D) _ is 0(h). Thus, for —x 0 X O OÎt fi C a+i/2 s ha e on , ] x , x [ € x y an y a-i/2 a+i/2 '

a+i/2 (5.2) a+i/2 * * where, for brevity, we have written t instead of t (x). The approximation implied by Equ. (5.2), has the property that the * values of the functions involved, at time t , are approximated by their values at time t , on the same characteristic. In this n+l manner, crossing of characteristics is avoided. Such property is important in order to preserve the advantages of characteristic methods. Equ. usee (5.2) b obtaio t dn ,ca n

*• f tt-l/2/. Su, rr,Su* l4- 4-r 1^4* - (D = (t t ' V a+i/ t 2J J, " V^" ôx°x=oZ a-i/} 2- a+i/2 Ot+1/2 a+i/2

Xa+i/2 n+i ,* I. Cx)a5T(x)dx + 0(hk2) (5'3) Xa-i/2 As an illustration of the numerical implementation for this equation, we explain the case of constant coefficients. In this case t*(x) = t - -£- (5.4) n+l V so that

dt 1 dx* V (5.5) and Equ. (5.3), becomes

dt1 - JV'1 a+i/2 ^ ~ a+i/2

D, n+l n+l x nr-i-i 2i /r- ^^ TT(V a+i/U 2 -a-i/u 2 ) + 0(hk ) (5.6)

In Equ. (5.6) derivative th , e Sus x t xa (

81 Outflow Boundary Observe thae lasth tt test functio e applieb e w o Th " .t n s i d E-l suppor f thio t s test functio s SÏi n , which doe t E—intersecno sl e th t lateral boundary x=l. Thus boundar,e th non f eo y terms involvine th g outflow boundary numericae occuth n i r l equation prescribee th d san d boundary values are incorporated in the numerical equations in an indirect manner exclusively; i.e., introducing them instean u f o d in approximations suc s (4.6ha d (4.8))an . B.- Flux Conditions For this case, we use E test functions. Thus, the test functions associated with region d Qs an ,f i lwhic h werE e omitted when dealing with Dirichlet boundary conditions, are applied when dealing with this kin boundarf o d yt zer a condition o valuee u th f d so an , + n and at 1 are treated as unknowns. Inflow Boundary Fig.4, illustrates a case in which Q intersects the inflow boundary. For flux boundary conditions, it is more convenient to write Equ. form e (5.1th :n i )

cc+i/2 tn+i

a-1/2 a-1/2 a+i/2 a-i/2

?-| 1/d- t2 F(t)dt (5.7) a+i/2 a+i/2 a+i/2 Here, F= s prescribedi , .

XE-1/2XE

Figur - Case . 4 e s whee tesdomaie th th nt f functiono n s intersects the lateral boundaries (CELLAM).

82 Since F(t a datum )s e correspondini th , g integral offero n s special difficulty in being approximated to any desired order of accuracy. The other integrals can be treated in a manner similar to case wha Dirichles th f done o twa n i e t conditions e approximatioTh . n *

1 - -- *"*- A * dt = (t -t )(D^ ) (5. , „„ ^ a-i/2 --- ~ °" It a+i/2 a+i/2 is similar to one used in Equ. (5.3). However, the term (D^ —) which appear Equn i s . (5.3) missins ,i g here, becauss wa t i e incorporated in the flux F. Due to this fact, approximation (5.8) is t consistenwhic, no ) onls k i h 0( y t wite orde th hf approximatio o r n that has been used in all other terms. However, suc a hshortcomin t beeno n s manifesteha g e th n i d numerical applications, which use Equ. (5.8). In particular, all the numerical experiments reported in this article, use this approximation developmene Th . algorithn a f o t m fully consistent with the e beginninordeth r f accuracou o rt a f o gt yse thas wa t discussion, would requir a proceduree , considerably more elaborate, which would not be justified at this point. In addition, it must be mentioned that when the support of a weighting function e cornere domaiintersectth th f , f o Q no s y an s the treatment presented requires slight modifications, whose details e leavw ee infloth out t wA . boundary, this o teshappentw t r fo s functions. One is w and there is one more, whose support intersects the corner (0, t ), as shown in Fig. 4. n Outflow Boundary The only weighting function whose support intersects the outflow boundary is w . Applying it, we get

r r* r^ Ptn+i - x uud nd - x £ud

Using three points approximations, one can evaluate the first two integrals of the right-hand side of this equation to 0(h ). The third one offers no difficulty, since F(t) are data. Finally, an approximating procedure analogou o Equt s . e applie(5.8)b n o ca ,t d the last integral. However, the comments that were made immediately after Equ.(5.8), apply here too.

83 NUMERICA6 L COMPARISONS The efficiencies of the BELLAM and CELLAM procedures have been compared n I [7]. , numerical examples that involve significant boundary behavior were solved, using thes methodso tw e e resultTh . s obtained there, are described next. The following change in the notation is noticed: In this Section, instead of taking the spatial region to be the interval [0,1], as we did in the theoretical discussions, we set Q=[a,b]. X Conside advancinn ra g Gaussian hill thay crostma n inflosa r wo an outflow boundary. Its general expression is 2 , , , t

and the initial and boundary conditions are chosen in such a way that the exact solution of the problem is (6.1). Thus, the initial conditions are u (x) = exp(-îrx ) (6.2) whil boundare th e y conditione sar

u(a,t (a,tu )= ) (6.3a) a and

u(b,t) = u (b,t) (6.3b) a whenever Dirichlet condition e consideredsar additionn I . , thee ar y

and f An} f fll,,^ b) ,t (6.4b)

whenever total flux boundary condition e consideredar s e th n I . numerical examples, only Dirichlet and total flux boundary conditions will be treated. Several combinations of initial and boundary conditions are prescribe Equr fo d . sucn (2.1)i manneha t f the o ,bu my r an thar fo t the exact solutio gives i n y Equnb . (6.1). Also, domains considered were: I=[2V ,9], 0=[-3,2 d N=[-3,9]an V] r them e pulsFo .th , e 3 3 crosses an inflow boundary, an outflow boundary and neither, respectively.

A,, Comparison based on the Euclidean Norm s explaineA n previoui d s sections e CELLAth n Mi , methoe th d information abou soughe th t t solutio concentrates i n d exclusiveln i y the cell-centers and no base functions are used. This implies that o assumptionn made sar e aboue solution e shapth th tf eo . n i Thi s i s

84 contrast with other ELLAM procedures n e whic i ,e th shap th hf o e solutio assumes i n d [I]. One consequence of this way of handling the information, is that some of the standard procedures for measuring the errors of approximate solutions, are not appropriate. In [1], for example, in which bilinear basis functions were used (the space of such functions, whic piece-wise har e linea d continuouran wil, e Q b l n si X denote e approximatL erro th e e if),b df th o r e solution that BELLAM yields (and such approximate solution necessarily belongs to ^), was compared with the L error of the projection of the exact solution n ifo . This rati necessarils i o y greate r equao r o onet l , becausf eo the minimal propert projectione th f yo . When all the information is concentrated at the cell-centers, the best we can do is to obtain the exact values at those points. This, however, does not define a function of the space !f, and a direct comparison using the L norm, is not possible. Of course, one could try to use linear interpolation of the approximate values at the cell-centers, to associate an element of y to the approximate solution. However e proceedon f i ,n tha i s t manner, evee optimath n l solution (i.e., that whose cell-centervaluee th t sa e exac th e tsar values) would give an L error that in general, will be greater than projectione th tha f o t , again becausminimae th f o el properte th f o y projection. To illustrate this fact and its importance in the different cases tested, Table 1 compares the L error of the linear interpolation, when the values at the cell-centers are the exact ones, projectioe wit e erroth th he exacf o rth tf no solution n $.o , It can be seen that in all cases, the L error of the linear

Table 1, Comparison of L3 errors between the projection of the exact solution on the space of ptecewise linear functions and the function of this space, whose values are exact at the nodes ("interpolation").

ERROR DOMAIN AX PROJECTION INTERPOLATION

N 0.267 0.715E-02 0.1S9E-01 N 0.053 0.265E-03 0.646E-03 I 0.267 0.715E-02 0.159E-01 I 0.053 0.26SE-03 0.646E-03 O 0.267 0.497S-02 0.102E-01 O 0.053 0.177E-03 0.429E-03

85 interpolation of the exact values, is at least twice that of the projectio n ^o .n Thus f thii , s measure erro th s used i rf e o eon , e abl b woulo discriminat t et no d e between different methode th n o s basi performancef so . Before leaving this point e woulw , d lik o remart e k that when the information abou e exacth t t solution consiste exacth tf o s e celvalueth l t a scenter s exclusively e extensioth , f thio n s information to the entire interval can be done in manners which are more efficient than using linear interpolation. For example, one a higcoul e h us dorde r interpolation procedure r solvo , a eloca l problem (thikina f post-processing)s o di s mentioo t , nw jusfe a t of the possibilities for processing such information. Therefore nora , m that directly compare valuecele e sth th l t sa center s use wa o scompart d e differene errorth th e f o s t methods. nore Th m "average choseth s nwa e Euclidean norm" , E N 1/2 llu-ul„ l.. = A

Here, u are the values of the exact solution at the cell centers, while u are those of the approximate one. Tabl summarizee2 numericae th s l results n [1]i finae s ,A .th l time t =0.5, Ax is taken to be 4/15=0.267 (Pe=26-) and Ax=-^ so. 0533 5 7 3 f (Pe=5-). For Ax=—, At=0.25 was used (Cu=9-). For Ax=4/75, the 8 o 1 o values 0.25, 0.05 and 0.01 of At, were used, which correspond to Cu= 46-7, 3 9- and 11-, respectively. The integrals involving initial or 88 7 boundary conditions were evaluated using Gauss-Kronrod rulea o t s high degree of precision. In Table 2, the Euclidean errors associated wite approximatth h e solutions that were derived using CELLA e comparear M d with those obtained with BELLAM l [1]al .n I case errore sth s listed correspon finae th lo t dtim =0.5t e .

TADLE 2. Coiiipruiscm of errors between ELLAM*Cclls aod BUincal-ELLAM Bouud.Cond Eudidcan Error Mnxinmn Value Hun !X)M IN OUT fX AT Cell« Piii'lkUl Ccllx Uilin F.wcl 1 N D F 0,267 0.25 0.50IE-02 0.6S2IÎ-02 O.R04 O.KI3 0.7X4 2 N D F 11.053 0.25 0.-M2E.OZ 0.440E-02 O.W)3 O.H03 0.7K4 3 N D F 0.053 O.OS O.I03E-02 , IÜSB-00 2 0.7H« 0.78« fl,7«4 4 N D ¥ 0.053 0,01 «usM-oa 0.265E-03 0.7HS L.7KS «,7K4 5 D F 0.267 0,25 0.315E-02 0,Gy5ß-02 0.7'.) 1 0.80« 0.7R4 (, 1 D F 0.053 0.25 O.Z7ÛE.02 0.2S9E-02 1' II.WJ 0.01 (>.2,12!-:-03 (l.2KVe-»,1 0,7JM H.7ÄS 0,7X4 y K F ».267 0.23 0.3U25-Q2 O.CS41ÎXI2 0.7DI 0.807 0.7H4 H) P F 0.053 (1.25 0.27SE-02 0.287B-02 0.7« 0.793 07S4 11 K I» 0.05Ï 0.05 0.7IRE-03 O.Ä2 113-03 0.7KÖ 0.78Ü 0,784 12 F F 0.1153 0.01

86 In general terms, one may conclude that in these examples, CELLAM performed slightly better than BELLAM. Runs 1 to 4 do not involve significant boundary contributions. Whe e ncase th thi ,s i s BELLAM and the Modified Method of Characteristics (MMOC), become identical [1]. From the results shown in Table 2, it follows that in the examples treated, CELLA slightls i M y more precise than MMOCn O . the other hand, when the boundary contributions are important, MMOC is considerably less accurate than CELLA d BELLAMMan .

B. Comparison Based on the Maximum Value In [1] maximue numericae th ,th f o m l solutio s comparenwa d with the maximuexace th tf mo solution . Thus r completenessfo , e samth ,e compariso s mad i ne result eth here alsd ar san e o illustraten i d Table 2. Inspecting this table, it is seen that also when the performanc s judgei e d accordin o thit g s critérium e resultth , s obtained with least CELLAa goos e ta BELLAMs Mar a d . Observe that whee domaith n s 0=[-3,2-]i n e maximue th ,th f o m O Gaussian distribution (6.1) has already crossed the outflow boundary e spatiaoth f l domain o thas , t whe e boundarth n y conditionf o e sar Dirichlet type maximue th , m e approximatvalueth e exacf so th td an e solution e equalar s . Hence, this compariso t informativno s i n n i e those cases.

7 DISCUSSION AND CONCLUSIONS e centraTh l subjec f o Quantitativt e Isotopic Hydrology, consists in studying the transport of tracers by water flowing in a porous medium and the interactions that take place between the solid e solutesth matri e d correspondinTh an .x g mathematical models, derive from the advection-diffusion equation. The numerical solution of this equation, is a problem of great importance in many other scientific and technical fields, as well and it is being the subject of intensive research. e numericaTh l treatmen e advection-diffusioth f o t n equation when advectio s dominanti n s bee ha ,a challenginn g problea r fo m long time, speciall a shar f i yp fron presents i t A featur. e thas i t required from algorithms in order to be able to model effectively advection dominated transport s s i thaperformancit , t e b e independent of the Courant number, to a large extent. Another feature which is essential, specially in Quantitative Isotopic Hydrology s thae i algorithm, th t e mass-conservativeb s , even when significant boundary behavior is present.

87 It has been recognized that in order for the performance of an algorithm to be independent of the Courant number, it is necessary to incorporate in its formulation the structure of characteristic lines. Methods which are based in the characteristic structure of the differential equations, are known as "characteristic" or "Lagrangian" methods. Until recently, characteristic methodd ha d sha three important limitations: inabilit ensuro t y e mass conservation, inabilit o treat y t boundary fluxes effectivel e introductioth d an y n of numerical dispersion, due to low order interpolation or integration. With the developmenmt by the author and coworkers, of Eulerian-Lagrangian Localized Adjoint Methods (ELLAM), which are e generabaseth M methodologn i dLA l y introduce e authorth e y b dth , limitations of characteristic methods mentioned above, have been overcome to a large extent. e ELLATh M approac e implementeb n hca severan i d l mannerso t p U . now two such implementations have been developed: BELLAM and CELLAM. Evidenc s beeeha n presented, which indicates that CELLAt leasa s ti M as accurate as BELLAM and in some cases, more accurate. In addition, CELLAM is easier to implement and more general, since it is applicable to the general advection-diffusion equation with non-constant coefficients. Here BELLAM and CELLAM have been explained and discussed, making them more readibly availabl e scientifith o t e c community which works in Quantitative Isotopic Hydrology. REFERENCES 1 n Celia, Rüssel , Herrerad "A . Ewing. A an F . , M . , . T R ,,I. , Eulerian-Lagrangian Localized Ad.joint Method for the Advection-Diffusion Equation". Advance n Watei s r Resources, 13(4), pp.187-206, 1990.

2 Herrera, I., Ewing., R.E., Celia, M.A., and Russell, T.F., an-Lagrani r e l u E n Localizea gi d Ad.joint Method e TheoreticaTh : l Framework.- Numerical Methods for Partial Differential Equations 9(4) 431-457p p , , 1993.

3 Herrera , Loca. I , l ized Adjoint Methods w DiscretizatioNe A : n Methodology .e book th Chapte :f o "Computationa 6 r l Methodn i s Geosciences" . FitzgibboE . . WheeleW F , . M & rn Eds., SIAMp p , 66-77, 1992 (Invited paper).

4 Herrera, I. , Local i zed Ad.loint Methods in Water Resources Problems. In Computational Methods in Surface Hydrology, G. Gambolati . RinaldA , d C.Aan o . Brebbia, Eds., Springer-Verlag, 433-440, 1990 (invited paper).

88 5 Herrera, I.,"Localized Adjoint Method: An Overview". Advances in Computer Methods for Partial Differential Equations VII, R. Vichnevetsky, et al. (Editors), International Association for Mathematics and Computers in Simulation (IMACS), pp 342-348, 1992 (invited paper).

6 Herrera , "InnovativI. , e Discretization Methodologies Basen o d LAM", FEMIF93, Barcelona, 1993. (To appear).

. Herrera7 I Herrera d n Eulan "A an-Lagran ,i e. r G , n Methoa gi f o d Cells ,e Localize Baseth n o d d Adjoint Method", Num. Methodr fo s Partial Differential Equations, 199 press)n 4(I .

8 Herrera, G. , "Tratamiento Numeri co de Transporte Dominado por Adveccion. Thesis, Institut Geofisicae d e , UNAM, Mexico, 1992.

9 Herrera, G.S. Herrer,. . GalindoI A d aan , "ELLAM Procedurer fo s Advection Dominated Transport", Advance Computen si r Methodr sfo Partial Differential Equations VII, R. Vichnevetsky, et al. (Editors), International Association r Mathematicsfo d an Computers in Simulation (IMACS), pp 333-341, 1992 (invited paper).

Herrera. I 0 d 1 Herreraan ,. S "Applicatio. G , Advectioo t M LA f o n Dominated Transport", FEMIF93, Barcelona, 1993.(To appear).

11 Russell, T.F and R.V. Trujillo. , "Eulerian-Lagrangian Localized Ad.joint Methods with Variable Coefficient n i Multipls e Dimensions", Computational Method Surfacn i s e Hydrology, Eds. G . Gambolati et al., Computational Mechanics Publications, Springer Verlag 357-363. pp , . 1990.

12 .,"OperatoE Ewing. R , r Splittin d Eulerian-Lagrangiaan g n Localized Adjoint Methods for Multiphase Flow". J. Whiteman Ed., MAFELAP 1990, Academic Press Diegon Sa ,215-232. pp , , 1991.

. E Ewing.."Localize. R , 3 1 Herrer. I a, d Adjoint Methods: Application o Multiphast s e Flow Problems." Proceedings Fifth Wyoming Enhance l RecoverOi d y Symposium, Mayo 10-11, 1989, Enhanced Oil Recovery Institute, University of Wyoming, pp.155-173, 1990.

14 Ewing, R.E. and Celia. M.A., "Multiphase Flow Simulation in Groundwater Hydrolog d Petroleuan y m Engineering". Computational Methods in Subsurface Hydrology, Eds, G. Gambolati et al. , Computational Mechanics Publications, Springer Verlag, pp. 195-202. 1990.

15 Zisman , "SimulatioS. , f contaminano n t transpor groundwaten i t r systems using Eulerian-Lagrangian localized adjoint methods.S M " Thesis, Dept. Civil Eng., MIT, 1989.

16 d ZismaCeliaan "Eulerian-LagrangiaA . S nM. , n Localized Adjoint Method for Reactive Transport in Groundwater" Computational Methods in Subsurface Hydrology, Eds, G. Gambolati et al. , Computational Mechanics Publications, Springer Verlag. pp , 383-390. 1990.

89 7 Neuman1 , S.P.,"Adjoint Petrov-Galerkin Method with Optimum Weight d Interpolatioan n Functions Define n Multi-dimensionao d l Nested Grids". Computational Method n Surfaci s e Hydrology. G s Ed , Gambolat t al.e i , Computational Mechanics Publications, Springer Verlag 347-356. pp , , 1990.

18 Espedal, N.S. d Ewingan , , R.E., "Characteristic Petrov-Galerkin subdomain method r two-phasfo s e immiscible flow", Comp. Meth. Appl. Mech. Engng ., (64 p 113-135)p , 1987.

9 1 Dahle, H.K. , Espedal, N.S. d Ewingan , , R.E., "Characteristic Petrov-Galerkin subdomain methods for convection diffusion problems". IMA Numerical Simulation in Oil Recovery, M. F. , Wheeler (ed), Springer-Verlag, Berlin, (11) 77-88p p , , 1988.

20 Wang H. , R.E. Ewing, and T. F. Russell, "Eulerian-Lagrangian localized adjoint method r convection-diffusiosfo n equationd san their convergence analysis" . NumerJ A IM ., Anal., 1992o (t . appear).

1 2 Cantekind Westerinan . E . "Non-DiffusivJ . M . ,J k 2 degreN+ e e Petrov-Galerkin methods for two-dimensional transient transport computations, Int. J. Num. Meth. Engrg., 30 (1990), 397-418. 2 2 Westerink, J.J.. D Shea d ,,an "Consistent higher degree Petrov-Galerkin e Solutioe MethodTransienth th r f o fo ns t Convection-Diffusion Equation, Int . NumJ . . Meth. Engng.8 2 , (1989), 1077-1102.

3 2 Bouloutas, n analysiE.T"A d a ClasCelia, . an . f A f o s. o s M , Petrov Galerkin and Optimal Test Functions Methods", Numerical Method r Transporfo s d Hydrologian t e Processes, Vol. , M.A2 . Celi t al.e a , Eds. e Serie,th f Volso Developmen6 3 . t in Water Science, Elsevier, Amsterdam, 1988, pp 15-20, 1988. 4 2 Bouloutas, n E.Timprove"A d ., an Celia. A d. M ,cubi c Petrov-Galerkin Metho r Advection-Dominatefo d d flown i s rectangularly decomposable domains", to appear in Comp. Meth. Appl. Mech. & Engng., 1993.

5 2 Herrera . ChargoL . Alduncin, G I. ,y y ., "Unifie d Approaco t h Numerical Methods. Part 3. Finite Differences and Ordinary Differential Equations, Journa Numericaf o l l Method r Partiasfo l Differential Equations, 1,241-258 (1985)

26 Herrera, I., T he algebr a ic theory approach for ordinary differential equations: Highly accurate finite differences, Journa f Numericao l l Method Partiar sfo l Differential Equations, 3(3) 199-218p p , , 1987.

7 2 Celia, M.A. HerreraI d .an , Solutio f generao n l differential equations using the Algebraic Theory approach. Journal of Numerical Method r Partiafo s l Differential Equations, 3(1),p p 117-129, 1987

28 Celia, M . A. , Herrera, I., and Bouloutas, E. T. , "Ad.lolnt Petrov-Galerkin Methods for Multi-Dimensional Flow Problems", In Finite Element Analysis in Fluids, T.J. Chung and Karr R., Eds., UAH Press, Huntsville Alabama, pp. 953-958, 1989.

90 9 2 Celia.,M,A. Herrera,I., Bouloutas I.E Kindredd .an , J.S.,"w Ne A Numerical e AdvectivApproacTh r fo he Diffusive Transport Equations". Numer. Method r Partiafo s l Differential Equations, 5(3), 203-226, 1989.

0 3 Celia, M.A., Kindred, J.S. d Herreraan , , "ContaminanI. , t Transpor d Biodégradationan t A Numerica . 1 : l Mode r Reactivfo l e Transport in Porous Media", Water Resources Research, 25(6) PP 1141-1148, 1989.

1 3 Herrera , "Boundar. I , y methods n algebraiA : c theory", Pitman Advanced Publishing Program, Boston, London, Melbourne, 1984

32 Herrera, I. , "Unified Approach to Numerical Methods. Part 1. Green's Formula Operatorr sfo Discontinuoun i s s Fields", Journal of Numerical Methods for Partial Differential Equations, 1(1), pp 12-37, 1985.

33 Herrera, I., "Unified Approach to Numerical Methods, Part 2. Finite Elements. Boundary Methodss coupling"it d an , , Numerical Method r fo Partias l Differential Equations p 159-186p , 3 , , 1985.

4 3 Herrera "Som, I. , e unifying concept applien i s d mathematics"n E . "The Mergin f Disciplineso g w DirectionNe : n Purei s , Applied, and Computational Mathematics". Edited by R.E. Ewing, K.I. Gross and C.F. Martin. Springer-Verlag, New York, pp 79-88, 1986 (invited paper).

35 Herrera, I. , "On Operator Extensions: The Algebraic Theory Approach". Proceedings of VII Taller IIMAS-UNAM, Oaxaca, January, 1992 (invited paper).

6 3 Neuman, S.P. Eulerian-Lagrangian ,"A n Numericale Schemth r fo e Dispersion-Convection Equation using Conjugate Space-time Grids" Computationa. Jr . l Phys 270-294, 41 . , 1981.

37 Neuman, S. P ., "Adaptive Eulerian-Lagranglan Finite-element Method for Advection-dispersion. International Journal Numerical Meth Engng. 20, 321-337, 1984.

38 Garder, A. 0 . , Peaceman, D.W. and Pozzi, A. L. , "Numerical Calculations of multidimensional miscible displacement by the metho Characteristics"f o d . Soc. Pet. Eng. Journal 4(1), 26-36, 1964

39 Konikow, L.F., and Bredehoeft, J.D., " Computer Model of Two-dimensional Solute Transpor Dispersiod tan Groundwater"n i n , Journa f Teco lf Water-Resources ho U.Se th .f o Geol v .In . Surv, Book 7 Chapter C2, 90p 1978. d 0 Russell4 an Douglas , Jr , J ,T.F ., Numerica l Methodr fo s Convection-dominated Diffusion Problems Base n Combinino d e th g Metho f o Characteristicd s with Finite Elemen r o Finitt e Difference Procedures. SIAM, J Numerical Analysis 19, 871-885, 1982.

91 41 Russell, T. F. Wheeler, M.F., and Chiang, C.Y., "Large-scale Simulation of Micible Displacement by Mixed Characteristic Finite Element Methods. in Mathematical and Computational Methods in Seismic Exploration and Reservoir Modeling". W. E. Fitzgibbon, Piladelphia, SIAM 85-107, 1986.

42 Espedal, M.S., and Ewing, R.,E., "Characteristic Petrov-Galerkin Subdomain Method r Two-phassfo e Immersclble Flow". Computational Methods. Appl. Mech. Engng. 64, 113-136, 1987.

43 Finder, G.F. , and Cooper, H.H. "A Numerical technique for calculatin e transienth g t positioe saltwateth f o n r front, Water Resources Research, 6 (3), pp 875-882 1970. 4 4 Healy, R.W. Russelld ,an , T.F., "Efficient Implementatioe th f o n Modified Metho f Characteristico d n Finite-Differenci s e Models of Solute Transport", Proceedings of Conference in Solving Ground Water Problems with Models, . J Lehredite y ,b d Indianapolis, NWWA, pp 483-491, 1989.

5 4 Healy, R.W. d ,Russellan , T.F. A Finite-Volum" , e Eulerian- Lagrangian Localized Ad.joint Method for Solution of the Advection-Dispersion Equation", Water Resour. Res. o appeart , , 1993.

92 ASSESSMEN GROUNDWATEF TO R FLUXES AND TRANSMISSIVITIE ENVIRONMENTAY SB L TRACERS: SUMMARY OF THEORY, APPLICATION AND SENSITIVITY ANALYSIS

E.M. ADAR Ben-Gurion University of the Negev, Sede Boker, Israel

Abstract

This paper demonstrate implementatioe sth novea f no l mathematical model for assessing groundwater fluxe d distributioan s f transmissivitieo n n i s groundwater basin with several source f rechargeo se s basei th t n I o d. assumption that each source is associated with unique characteristics of isotopic ratio d dissolvean s d constituents modee Th . l applie basino t s s with complex hydrogeological structures for which scarce physical hydrologie information is available modee Th .s base i l spatian do l distributio f environmentano l tracers and relies heavily on stable isotopes of oxygen and hydrogen. It is assumed that spatial variation f dissolveso d constituent isotopid an s c aquifee ratioth n n si ca r be attributed to mixing and dilution of several sources of groundwater recharge. Tracer assumee sar conservative b o dt e alon floe gth w pathfloe wTh . domais ni discretized into mixing cells based on the distribution of the dissolved constituents. Environmental tracer e thear s n f wateuseo d writt o dt an r se ea mass balance equations in a compartmental flow system, such that the unknowns are transmissivities and groundwater fluxes. Quadratic programming is used for a unique assessment of the unknowns. The model has been tested for multi-celo tw l flow systems with synthetic dat r whicfo a hprecisa e analytical solutio availables ni modee lates Th . wa lr implemente steada r dfo y flow system to estimate recharge, internal fluxes and transmissivities in an arid alluvial basin e southeroth f n Arava Valley, Israel. Results indicate thae calculateth t d transmissivities for the alluvial aquifer of the southern Arava are close to what obtaines wa d with interference pumping tests. Sensitivity analysis indicates that (1). the model is extremely sensitive to the accuracy of the chemical and isotope constituents assigned to the external flow components and to each aquifer compartment, and (2). the accuracy of the assessed transmissivities is highly related boto t h precise tracers' concentration assignee th o t d san geometr e th f yo cells.

1. INTRODUCTION

To properly manage groundwater resources, thera nee r accurats i edfo e information about inflows (recharge), outflows (discharge) and the physical characteristic bees aquifersf ha so nt i commot Ye . exploio nt t aquifers based only on partial information concerning the hydrologie characteristics of many semi- arid and arid basins. A major source of uncertainty stems from the hydrologist's inability to reliably estimate the spatial and temporal distribution of recharge rates, groundwater fluxes, transmissivities and storativities. It is highly related obscure th o t e knowledg subsurface th f eo e geohydrological system. Quantitative assessmen f rechargo t d groundwateean r flow components i s often complicated in basins with scarce hydrological information and having a

93 puzzling geological structure. Therefore, an analytical solution is usually out of e questionth . However lace f precisth ,ko e geohydrological information along external and internal boundaries exclude even the implementation of numerical solutions. Knowledg e spatiath f o el distributio f groundwateo n r fluxes, recharge components and the physical parameters of an aquifer such as transmissivities, e essentia e developmenar th r fo l f groundwateo t r resources. Fluxes through external boundaries (recharge and discharge) and along stream lines can be assessed by numerical solution of the flow equation, provided that the piezometri e aquifer'th d an cs parameters distributio e knowar n r welfo n l determined subsurface flow system. In general, transmissivity can be calculated from interferenc d recoveran e y pumping tests. Various methods have been developed for the interpretation of pumping tests, and most rely on the Theis [1] metho r applyinfo d e flogth w equatio o simplt n e infinite confined aquifers. However, methods can vary according to the geometry and type of aquifer and also accordin geometre th pumpine o gt th f y o monitorin d gan g wells maie Th .n method listee sar Kruseman di Riddee D d rn an assumption e [2]Th . s behine dth analytical Theis solution, and the interpretation of the well function from pumping test data, restric utilizatioe th t f sucno h method vero t s y simpld ean local hydrological systems. Another, more practical, approach for the evaluation of the spatial distribution of transmissivities relies on the so-called inverse method of numerical solution for the flow equation (i.e., Carrera and Neuman, d [4]) [3thin an I ] .s method, precise boundary condition requiree ar e s th r dfo assessmen f transmissivitieso t accuratd an , e initial condition needee e sar th r dfo evaluatio f storativitieso n . Wittmeye d Neumaan r ] use[5 n d quadratic estimatio f hydraulio nt criterise a cr aconductivitfo y dat meeo at hydraulie th t c head distribution n adaptivA . e method r applie fo n elliptia E o t dPD c simultaneous identificatio f aquifeno r parameter s suggesteswa Hoffmay db . et n Dirichler fo ] al[6 . t boundary conditions. Carrer Gloriosd aan demonstrate] o[7 da geostatistical formulation of the inverse problem of groundwater flow. Whi, Leodou Marsild xan ] use inverse y[8 dth e metho r parametedfo r estimation ni deep aquifers in Paris basin via a calibration process of a numerical model of a transient flow equation above th l e Al mentione. d methods, however rareln ca , y be implemented in remote and undeveloped basins due to lack of basic hydrological information. In basins with a limited number of wells and hence limited knowledge of the hydrogeological structure oftes i t i ,n difficul preciselo t t y defin floe eth w system e boundarth r o y conditions. Furthermore e lac f th observatio,ko n welld an s pumping test data eliminate the possibility of proper calibration processes. To improve the ability of modelling groundwater flow systems, environmental tracers have long been use furtheo dt r illuminat subsurface th n eo e flow pattern. In the past, hydrologists have used chemical and isotope data for recharge studies primaril a qualitativ n i y e sense. Environmental isotopes playea d dominant role in such studies, as exemplified by the works of Verhagen, et al. [9], GaDansgaard an t d [10], Mazor . [11]al . , et , Levin . [12] al Rosenthad . ,an et , . al . et l othen I [13].r studies, tritiu s use mwa obtaio dt n quantitative estimatef o s recharge (Eriksson [14], Bredenkamp t al.,[15]e , , Vogel t al.e , , [16] Allisod ,an d nan Hughes [17]). Different conceptual modelin f hydrologicago l systems reliea n so lumped parameter modelin a continuu n i g m approach usin e so-calleth g d convolution integral. This concept can be exemplified by a series of papers by Maloszewski and Zuber (i.e. [18] and [19]), and Zuber et al. [20]. A comprehensive

94 review of the implementation of the above mentioned approach can be found in r [21]Ni , Zube rGoblen i [22 d [23]d an ]an ,t [24]. These method onle b yn ca s implemented for a pronounced stream line for the evaluation of an average flow velocity, storage coefficien d transmissivityan t n attempA . o incorporatt t e natural tracers in hydrological modeling in a combined hydrologie and hydrochemical mathematical purpose modeth r fo lf sourc eo e identificatiod an n quantification was presented by Gorelick, et al. [25] and Wagner and Gorelick [26]. They deal with the question of identifying the location and magnitude of pollution sources that might have contributed to the contamination of an aquifer r thisFo . , they utiliz a etwo-dimensional , numerical mode f soluto l e transport in the aquifer, coupled with various optimization techniques. Attempt extraco t s t quantitative information concernin e groundwategth r flow system from hydrochemical data often rel statistican yo l analysesn a s A . example, Lawrenc d Upchurcan e h [27] used factor analysi o identift s e th y recharge source for certain groups of dissolved chemical species in an aquifer. In all such works, the authors either evaluate the magnitude of a given recharge source, or evaluated the potential of recharge without providing quantitative estimates of recharge rates. Rosenthal et al, [13] and Adar et al. [28] used cluster analyse classifo t s y groundwater latee unitb o rt s incorporate numericaa n di l mode quantitativa r fo l e assessmen rechargef o t . Environmental tracers have long been use elaborato dt groundwaten eo r flow systems. Recently, however, environmental tracers have been appliea o t d quantitative hydrological modelin regionaa f go l aquifer r this so-callee Fo th ,. d Mixing Cell Approach (Simpso d Ducksteian n n [29]) s beeha , n adaptedr Fo . basins with a complex geological structure and with scarce hydrologie information, this paper introduce methoa s r incorporatindfo g environmental tracer d mixinan s g cell approace quantitativth r fo h e assessmenf o t transmissivitie fluxesd san . This paper briefly describe theore sth present d yan s results obtained from synthetic datd fro n an alluviaama l basin witn a h extremely complicated hydrogeological system souther,e sucth s ha n Arava Rift Valley, eighty kilometers north of the Gulf of Eilat (Figure 4).

2. THEORY

Adar and Neuman [30] and Adar et al. [31] have demonstrated the utilization of environmental tracer r quantitativfo s e evaluatio f variouo n s sourcef o s recharge. Adar and Sorek [32] extended this approach and presented a theory which enable o assest d e physicath s l parameter e aquifeth f o sr froe th m piezometric head distribution and the spatial distribution of natural dissolved tracers. The model relies on two types of conceptual approaches applied in groundwater hydrology: 1. Evaluatio e motioth f f no waten o d dissolvean r d constituent a multi n i s - compartmental mixing cell model s suggestea , d initiall y Simpsob y d an n Duckstein [29[ and Yurtsever and Payne [33]. 2. Quantitative assessmen groundwatef o t r flow component optimiziny sb t se ga of mass balance equation waterf so , ionenvironmentad san l isotope multia n si - cell aquifer system. This approach assume a steads y flow syste d fullman y conservative dissolved tracers (Adar et al., [31]). Campan d Simpsoan a n [34d Campanan ] d Mahian a n [35] usee th d distribution of stable isotopes in multi-compartmental modeling to calculate

95 internal fluxes, rate f rechargo s d groundwatean e r storage, respectivelyn Va . Ommen [36], foun gooda d agreement betwee nmixing-cella model wit oneha - dimensional convection/diffusion equatio n transpori n n f reactivo tno d an e reactive constituents in groundwater flow system. Yurtsever and Payne [37] and Yurtsever et al. [38] formulated a mathematical models for the interpretation of isotope distribution in a regional compartmental aquifer to quantitatively evaluate the groundwater flow system. A similar model was successfully implemente e Bangkoth r dfo k Basi Thailann ni d (Yurtseve Buapend an r g [39]). In the following model, the steady flux for every flow component can be estimated providing thagenerae th t l flow spatiae patterth d ln an distributio f no environmental tracer e numbee knownth ar s d f maso an r, s balance equations ove entire th r e flow syste mmucs i h greater tha numbee nth f unknowo r n fluxes. A solution is obtained utilizing an optimizing quadratic programming scheme, as suggested by Woolhiser et al. [40]. For a periodic transient flow system, Adar Sored an k [32 d [41]an ] showed addition thai f tracee i t th o nt r distributione th , spatial hydraulic head distribution is also available, the model can be used to asses cell'e sth s storage coefficien transmissivitd an t y across every permeable cell boundary. This paper describes a method for assessing the transmissivities in a steady flow system where the Darcian expression is directly embedded into the mass balance expressions. Results fro e southermth n Arava Valley, Israee ar l compared with transmissivity values obtained from evaluation of pumping tests. Comprehensive sensitivity analyses is followed, taking into account possible error isotopin si chemicad can l data assessmene wels th a , s a geometrle th f o t f yo cell'e th s configuration. The model is designed in order to account for natural tracers, such as dissolved chemicals and minerals (ions and trace elements), electrical conductivity, and environmental stable isotopes. Knowledge of the accurate flow pattern and the piezometric water head potential in every cell is essential for obtainin e transmissivitiegth s acros e permeablth s e boundaries. These water characteristics and hydrogeological information are relatively easily attained, even in undeveloped basins with limited hydrogeological data. This model conceptualizes an aquifer system that can be discretized into a finite number of cells, such that eac assignee hb celn ca l d wit huniqua e representative value th f eo piezometric hea concentrationd dan f tracerso s . This implies complete dilution and, hence completa , e mixin l wateal f go r sources entering each cell. Wite hth mixing cell approach, we assume that at each segment (compartment) of the aquifer gradients of ions and isotopes concentrations and water head potentials are small and hence negligible. Therefore, the concentrations of solutes within cells and the hydraulic head are assumed to be constant for a specific time step. For each potentiaceld an l l source (contributor) shoule on , able d b determin o et e the same representative values of all accounted tracers and hydraulic heads. Similarly, in every compartment, the same information must be known for sinks sourcesd an thef i , y exist. Therefore assumes i t i , d that enough observation wells and/or spring available sar hear efo d measurement wated san r sampling. Mass balance expression r wate fo d ssolute an r e writtear s r eacfo n h compartment, taking into accoune potentiath l al t l source f wateo s r y thama t contribute to the subsurface reservoir enclosed by each cell's boundaries. Therefore, all the flow components entering and leaving the specific compartment must be known, at least qualitatively. Most sink or point source terms can be assessed quantitatively. Fluxes are described by a Darcian type of

96 expression that includes terms related to transmissivities via the geometry of the compartments.

2.1 Quantitative assessment of transmissivities

With regard to the above ideas, we will now write a set of balance equations flue soluted foth ran x s withi ngivea n tim eacr s i fo e h t I period cel . an n lt dA assumed that the flow system is steady at least within the time period At. For a fluid with constant density, the mass balance for the n-th compartment is expresse followine th y db g equation:

(1) 1=1 mn]]=i n dt

wher J d denotan e I numbee eth sourcef r compartmento r /o d san s from which n

the flon w enters and leaves the n-th compartment, respectively. Qin and Qnj denote the fluxes from the i-th source (or compartment) into the n-th , and from the n-th compartment intj-te oth h compartment,n respectivelyPm d an n So . denote the fluid sources and sinks (pumping), respectively, in the n-th compartment. Sn* represent storage th s e capacity withi n denoteh d n celan sn l the hydraulic head associated with that compartment. heae Ath s d (h r eacnfo ) h compartmen tidentify variesma e yw , pointn i s (t2>ti)2 Ï d t whican a , i hydraulitimet e hy th sa , csame headth e e s(e.g.ar th t a , specifia f beginnino d c en season) d gan . This means that over that time interval T=t2-ti (regardles sequence th f so f changeeo headn si s during this time interval),

the total magnitude of the derivatives dhn/dt in each compartment has not changed. Hence, since Sn**Sn*(t), we obtain: 1 2 S: JU =O ; hn(t])shj I/ IT (2)

Thus, by integrating equation (2) over a time period T=t2-ti and dividing by T we obtain accordingly:

_ ° J _ _ !" _ ) (3 Son-Pmn+^Qm-^Qnj =0 1=1 j=i

where Qin and Q„j denote the average values of Qin and Qnj, respectively, i.e.:

Ldt (4)

Likewise, the time averaged values of the source and sink fluxes in cell n are

So and Pm respectively. Note that equation (3) expresses a quasi-steady state n

situation n resulting fro time mth e averaging process. Following Darcy's flow expressio a stead r nfo y flow, every fJux tern mi equation (3) leaving or entering cell n can be written in terms of the cell's

97 geometry, transmissivit d hydraulian y c head difference acros e activth s e boundary.

where Tnj is the transmissivity across the boundary between cells n and j, respectively. wnj is the width of the boundary normal to the flow path, and Inj is lengte streae th th f hmo line betwee geometrie nth e c Th center . j d celln si an sn term Fnj includes hydraulic head difference between cell n and j, and all the known parameters relate geometre th celle o dt th .f yo From this point, let In designate only the number of inflows into cell n from external sources designatM d an , numbee eth inflowf ro s into cel fron l m nearby

s assumecellsi t I . d ntha addition i t pumpingo nt l outfloal , w components from cel enteln nearbe rth y cel j (j=l,2,.....Jl n). Therefore froe mth ) equation(5 d an ) s(3 water balance expressio rewrittee b n nca followss na :

) (6 So |Q+ n in +£FmnT £ran= n j -n (^IQPm + nj

where Fmn is a factor which includes the geometric parameters between cells m and n and the head difference across the m-n boundary as expressed in equation (5). En is an error term for a possible deviation from the water balance due to the misunderstandin appropriate th f go e flow pattern and/or disregardinr o e gon more of the flow components. Water balance (equation (6)) might not hold because of an error in identifying and measuring of fluxes or rates of pumping. Assuming only small variations in the concentrations of the dissolved constituents durin writtime n gth ca e masea e perioon s, balancdAt e expression average foth r e concentratio everf no y species (tracer celn i . n l)k For quasi steady-state variations of concentrations, when the mixing cell concep applieds i t , usin above gth e mentioned assumption equatiod san n (6)e ,w writ masea s balance expressio dissolvea r nfo d constituen celn i . ln k t

j / \ n M I CsnkSon + ECilnkQin +XCtmnkFmnTmn- CnkIQnj + Pmn = enk (7)

where Csnk is the average concentration of k associated with source Son;

is the average concentration of solute k entering cell n together with the flux

h

coming from cell I. Ctmrtk is the concentration of k entering n through the mt

h boundary Cnfd an ,c denotes average concentratiot k constituene th f no t within cel . When l n constructing mass balance equations from field measurementsk 6n , deviatioe th s i n fro solute mth e balanc mase Th sceln ei . balanc n l f soluteeo s (equation 7) may be affected by analytical errors in measuring concentrations, especially when quantifying cell concentrations. For every cell n, there are K+l watee equationsth r r fo balanc more K ever:r on d efo e an yspeciek s (k=l,2,.....,K). Thi demonstrates si equation di . n8

98 In +So

(8)

nfe

Upon combining equations (6) and (7) to a matrix form for each compartment obtaine n,w :

= E CnXn n (9)

where Cn is a matrix with known concentrations associated with every flux enterin r leavinvectoe unknowa go th s i f o n rg X celn . n lfluxe s througe hth boundaries of cell n. D.n is a vector containing elements that are measured and known quantitativel celn yi , suc n l s knowa h n fluxe f sino s k (pumpingd an ) sources. En is the error vector in cell n. A comprehensive description of the above mentioned vector matriced san gives si equation i . n10

ßln !+,,+ _ 1 * So Pm Mn n , ,F+ t , ,4-+ +C C0. Q, C F Cs So 1 1 1 \ ln 11 "l ~ *MnlMn In n n\ r n\ n Ci +, ,+Ci +Ct F +,. +Ct F -Cn 2 n M 2 Mn 1 2 1 2 In 12

+ T Mn n (10) Q . n\ Ci +, ,+Ct +Ct F + +a F -Cn Cs So U Ink \k l Mnk Mn k nk r nk n

n Jn

E D X C All flux components in the aquifer can now be estimated by a minimization square oth f e error sum . SimilasJ procedura o rt e suggeste Ada. (31)y dal b t y r,e b virtu assembliny b f equatio eo d an ) square n(6 gth e error terms cellN ove l sal r we obtain:

T J = J Bn) w(cnD x + n (11)

wher denoteT ) e( s transpose matrix representW . diagonasa l matrix comprised of weighting values for estimated errors (independent of each other) expected for e termeacth f sho buildin e masgth e dissolves th balanc e fluid th an dr dfo e constituents weightine Th . g matrix , alsW ,o reflect degree th s f confidenceo o et

99 which the tracers are assumed conservative and the degree of accuracy of the chemical and isotope analyses. For the solution of equation (11) for Xn (a vector with In+Mn+Jn unknowns), the objective function J is decomposed into linear and non-linear parts. For example, J can now be minimized to evaluate Xn by quadratic programming wit algorithn ha m develope Woly db f [42].

3. TESTING THE MODEL AND THE COMPUTER CODE WITH SYNTHETIC DATA

The conceptua le compute modeth d an l r code were testef o d t witse a h synthetic data which was generated for a hypothetical aquifer divided into mixing cells. These tests were conducted in order to evaluate the ability of the computer cod solvo et e simultaneousl mathematicae yth l mode r unknowfo l n fluxed an s transmi v i sifirss e tiesTh t . s conducteteswa t d using four cellsconfiguratioe Th . n cele o th flassignee set-uth d pan d fluxes, sinks, sources (volum r uniepe t time), and transmissivities (area per unit time) for the internal boundaries are given in Figure. 1. Cell I has a sink (pumping) term and receives four inflows through external boundaries without a source term. Fluxes leaving Cell I contribute water

LEGEND OUTFLOW 100.0= IT S§*S7 assigned transmissivity value li„24!5> calculated transmissivity 20.0 assigned flux value 20.173 calculated flux value ^______3100______^ distance

Figure 1 Schematic compartmental aquifer used for testing the mathematical algorithm. Pm is the rate of discharge (in volume per unit time). The upper numbers are the assigned values and the lower numbers designat resule eth t obtaine quadratiy db c programming.

100 (!0 32 /t)d , an respectively] 6 [3 I II d Cello t an .I sI Cel I alsI l o receives water from five external sources (Nos. 5 to 9 in Figure 1) and has both sink and source terms . Subsurface fluxes leave Cell II and flow into Cells III [5(l3/t)} and IV [45(l3/t)]. Cell s threha eI II external sources wit sinha k term wated an , r leaves this d celan l flows only into Cell IV [40(l3/t)j. Cell IV has two external sources with a sink totae th ter ld subsurfacman e outflo (!0 3 /t)Figur10 n I uppee w.s i th e1 r numbers designate the assigned fluxes or transmissivities used to calculate the concentratio tracere th everr f no sfo lowe e yth celld r numberan , resulte th e ssar obtained from the model. The result indicates a precise evaluation of both fluxes transmissivitiesd an . To tesabilite modee th tth f evaluato yo t l e transmissivitie fluxed san s across active boundaries in a more realistic flow pattern, a complicated synthetic aquifer structure with a multi-cell flow system was used. In this test the aquifer was divided into twelve cells in a 3-D flow pattern, as illustrated in Figure. 2. Fifteen unknown internal fluxe d transmissivitiean s s were assigne e internath r fo dl active (permeable) boundaries. Fourteen different type watef so r were assignes da source externar sfo l recharge sourceo Tw . s contribut same eth e typ f wateeo o t r more than one cell. Sources #13 and #16 are the same type of water which recharge XII d Cellan , respectivelysX als e sam.e ar Sourceo th 5 e #1 d s an #10 4 ,#1 and contribute the same type of water to cells IX, XI and XII, respectively. In total seventeen unknown external sources totae Th l. numbe f unknowno r7 (1 7 4 s si external inflows interna5 1 , ltransmissivities)5 1 fluxe d an s . e cellsSomth f :eo werI XI , VII ed alsVI an , oVIIII X assigne , ,X d sink (pumping rates) termse Th . number e ellipsoidth n i s s designate rate f inflowo s s (recharge) from external sources uppee Th . r number assignee th e ar s d values, whil lowee eth r onee sar computee th d results. Als Figur n onumberi e th trapezoide , e2 th n si s specife yth rates of internal fluxes between cells, and in Figure 3 these numbers specify the values of transmissivities. Here also the upper numbers designate assigned values, whil lowee eth r ones sho calculatee wth d values. Though internal fluxes and transmissivities are illustrated in different figures, it is important to emphasize that they were both obtained simultaneously. These values, while arbitrarily chosen (so as to maintain an isotopic and chemical balance for each specie neverthelese sar accordancn si e with actual data fro Aravaipe mth a Valley in southern Arizona (Adar et al. [28]). With the flow rates in Figures 1 and 2, and with an appropriate concentration in the cell that follows the mixing cell approach e erroth , r term ) shoul e (7 n equationi th s e zerod b dd an an ) , (6 s minimizatio equation i . J f no n (11) should yiel dzera o optimue valuth r efo m solution e noticeableTh . , small variations between assigne d calculatean d d values for fluxes and transmissivities are attributed to computer round off errors, thess a e small deviations vary slightly among different computers.

. RESULT4 S FRO SOUTHERE MTH N AR RIFA V T VALLEY.

The model has been used in an arid alluvial basin in the southern Arava Valley southere Th . n Arava Valle narroa s yi w down faulted rift valley abou0 2 t km wide and extending about 80 km north of the Gulf of Eilat. It is an extremely arid basin with an average annual precipitation of about 50 mm. Due to its geological and geophysical nature, the rift forms a low base level into which surface wels a ,s subsurface a l , flows drain fro surroundine mth g mountainsn I . the valley, water was found in (a) sandstone of Paleozoic age, (b) sandstone of

101 4.0 OUTFLOW

Cell No.XII \ Internal flow Inflow No.3 m between cells External inflow Pumpin2 1 g rate (sink) 6.0 -assigned flux Outflow -&52V 6.043-calculated flux

Figur . Estimatine2 g fluxes versus assigned value obtaines sa d from a schematic multi-cell compartimentai aquifer.

102 4.0

LEGEND £479 ? Transmissivities: 5.4-assigned, 5.479-calculated 574 ) Externa8L l inflows: 5.5-assigne d5.481-calculate, d

12.0 @CellNo.XII Pumpin PiezometriM g rate c hea) d(m and outflow ^-^

Figure 3. Estimated transmissivities versus assigned values as calculated by the model for internal boundaries of the aquifer system.

103 Lower , (c) limestone of Cretaceous age, and (d) alluvial fill of Quaternary age. Figure 4 illustrates the geography, various geological outcrops and location of well fields along the valley. Due to extremely complex hydrogeologic structures cause tiltey db d faults, followe upwary b d d and/or downward movement of blocks, it was almost impossible to obtain detailed and reliable informatio physicae th n no l propertie f eacso h structural componenf o t the aquifer system. The hydrogeological complexity is exemplified with two geological cross sections given in Figure 5. An attempt to determine the spatial distributio f transmissivitieo n s from pumpin t reveagno testd l di sreliabl e results. Interference tests which were performed in one pumping well related to several observation holes provided a wide range of transmissivity values. A review of the draw down and the recovery curves versus time suggests the existence of unknown impermeable and/or permeable boundaries that interfere wit classie hth c Theis testssame Th e. phenomeno observes nwa severan di l well fields alon e valleyth g . Hences impossiblwa t i , o assest e e appropriatth s e transmissivity values. Therefor quantitativa e e assessmen floe th w f o tsyste y mb numerical modelling is not possible at this stage. Groundwate varyinf o r g chemica isotopid an l c qualitie exploitee sar welly db s drillevallee th alond n ydan i s margins git . Wells drilled into different blocks and layers and springs showed that due to differences in lithology and mineralogy, each source of recharge provides the alluvial aquifer with water of a specific chemical composition. Also, the isotopic ratios of oxygen-18 to oxygen-16, and deuterium to hydrogen are determined by the geographic location, including prevailing temperatures and the altitude of the area in which recharge occurs. The spatial isotopic and ionic distribution within the alluvial aquifer along the southern Arava Valley seem affectee b o st d mainlrelative th y yb e proportiof no recharge contribution from each source. Hence, dilutio mixind nan assumee gar d e majoth e rb mechanismo t s which contro e hydrochemicath l d isotopian l c compositio alluviae th f no l groundwater reservoir. The temporal distribution of dissolved ions have revealed almost constant concentrations ove lase th rt twelve years. Furthermore piezometrie th , c head distributio pumpine th d nan g regime also constane seeb mo t thar fo tt period. Hence, changes in heads and concentrations of most species within cell n during time periovere b yturnet o t dA smal t d thudou an ls negligibl r modelinefo g purposes. This implie almosn sa t steady state hydrological flow regime t leasa , t for the past decade. For a further hydrogeological description and for the detailed flow pattern, as suggested by the distribution of environmental tracers, the reader is referred to Rosenthal et al. [13]. Six ions, TDI, deuterium (D) and oxygen-18 (18O) parameters were finally isolate multi-variably db e cluster analyse o characterizt s majo2 1 e r potential sources of recharge and divide the alluvial aquifer into 6 homogeneous compartments. Several close configurations of cells and potential inflows were modeled. For only four close configurations, the Wolf algorithm provided a solutionremaindere th r Fo . , unbounde solutiono n r do s were obtainedA . comprehensive descriptio e quantitativth f no e assessmen f subsurfaco t e fluxes withi southere nth n Arava basi gives ni . Ada[28]n ni al t .re Figure 6 presents the results for 12 potential inflows obtained with 7 dissolved isotopes2 iond an s . Subsurface recharge component internad an s l fluxee ar s give 10n ni 6m3/year. Transmissivitie e givear s mn ni 2 /dayrange Th f .eo calculated fluxes as appears in Figure 6, represent the minimum and maximum

104 Mediterranean Sea

ERUSALEM • Israel

YOTVATT, A NEGEV Jubell s i n a i

ARAVA A\V VALLEY

----- INTERNATIONAL BORDER LINE 678» MAIN FAULT LINE ELA> T10 GEOLOGICAL CONTACT LINE -——c; WADD BE I RoaeoX | 8..'ELAT 16 ?f5> SABHAH (SALT MARSH) 0-«221 *> 9 t 2 %OJ0 / p CPrecambria- n rocks NS - Nubian Sandstone JG - Judea Group (Carbonate) QA - Quaternary alluvium t - Observation hole nea productiora n well s - Shallow; d- Deep southere th f o p n FigurAravma A ae4 basin showin locatioe gth e th f no main well fields and a schematic appearance of major geological units (After Rosenthal et al. [13]).

105 TIMNA 2

Gravel Upper Sandstone Ore layers TagHI formation Lover Sandstone

N

KETORA Yotvata YAALON 3 Yotvata 29/t YOTVATA SAMAR 2 2 1 Yotvat5 a 160 3 10 160 1 l/t 100 100

60 60

20 20

-20 -20

-60 -60 -100 -100

-140 -140

-180 -180

-220 -220

-260 -260 km

Figure 5 Complex hydrological structure along (A) and across (B) the southern Arava Rift Valley (After Adar et al. [28]).

106 H CELL 1: YAALON H 1 .46-1 .66 AREA 2.56-3.0 JYAALON 3,4- P =2.43 JORDAN

FLOODS 3. 1-3.4 CELL 2: GROFIT AREA BIP. 0.7-1 .3 6.9P= 3 BAI DA YOTVATA 9,9t, 1 2"

YOTVATA SABHA NEGEV c___S o.o [GROFIT 3,4.5! ^ CEL : 3 5AMR/TIMNL A 0.97-6 1. AREA i 0.9P= 8 DOME BRINE 0-0.9 Timna FB81.FB78 PALED >., FB248F8 2 ^8 1 , Ï 0.0 .38-2. CELL 3: TIMNA J N AREA i0.1 2-07 .1 P= 1 .64 TIMNA BRINE Timna it, 5,6,7 WATER T" il 0.5-0.73 BEER-ORA AREA CELL 5: EVRONA Beer-Or ,2,2a1 A AREA 0 0. P= 0.1 2-0.36

7 1 . 0

CEL : 6 ELAL T AREA P = 0.08 0. 1 4-0.3

0.37

Gulf of Elat A -(CLUSTER) T = 2@4-34i Transmlssivltles

I-CELL SOURCE 0.97-1.6 ———fr- FLOW DIRECTION D RANGAN F CALCULATEO E D FLUXES

Figure 6. Schematic flow pattern for the southern Arava Valley (fluxes are given in 106 m3/day transmissivities in m2/day).

results obtained from a slightly modified set-up of the assigned concentrations to the end members. It also stems from the fact that the assigned pumping values were adjusted followin change gth configuration ei celle th .f no Onl y small non- significant changes coul implementede db morA . e notable modification caused an unbounded solution, in other words, no unique solution could be obtained. This issue wil furthee b l r addressed whe sensitivite nth y analyse describee sar n di followine th g sections.

107 . SENSITIVIT5 Y ANALYSI S: EFFEC ERRORF TINPUO E TH TSN I DATA

In real field situation inpue th s t knowt no dat e nar a with accurace th d yan assumptions behind the model are not always satisfied. The model, therefore, should not be expected to provide precise estimates of flow rates. To examine the effec f erroneouo t s inpu qualite t datth n f asuco yo h estimates optimizatioe ,th n scheme with synthetic input data was repeated after corrupting with various levels of uncorrelated Gaussian noise. The data was corrupted according to a certain realistic hydrochemical and isotopic constraints which is further discussed in this chapter. Adar and Neuman [30] examined the effect of erroneous chemica isotopid an l c input data witnois0 h25 y dat assessmene a th set n o s f o t flow rates. Results show tha strictee th t constraintse rth closee ,th average th r o et true th e flow ratesface tTh .tha modee th t l yields good results when noisy data satisfy constraints similar to those that real data usually satisfy suggests that the model should be applied to real hydrologie conditions in the field. In the following the sensitivity analysis is extended in order to account for uncertainties which are associated with vague geometry and the configuration of the cell. The rational particulae th r efo r method use generato dt r noiseou e stems from assumptioe th n thamajoe th t r caus laborator) f erroreo (1 e sar y error isotopin si c and hydrochemical analysis, and (2) inaccurate assessment of the compartmental configuratio cell'e th sd geometrynan firse Th t . generatsteo t s pi e normal errors of zero mea d uninan t variance N(o,i y meane followinb ) th f so g formula (Box and Muller, 1958, as cited in Bard, [43]:

5 NIog2 (0>1(- 10= ) \]1}°- cos(2nl]2) (12)

e independenar 2 U d wheran t i randoU e m variables drawn from uniform distribution. For perturbation of hydrochemical and isotopic values, each "true" Ck value entering into the model is transformed into a noisy concentration Ck according to:

) as c;=ck[i+&N(6>1)]

weightina wher s i k eß g parameter controllin magnitude gth e corrupteth f eo d concentratio relativ* nCk "truee thath o e t f "o t concentratio . ThinCk s causee sth erro increaso t r e linearly with concentratio mannea n i r simila thao rt t assumed Woolhisery b . [40]al t .e , This seem realistie b o st laboratorr cfo y data. Whee nth noise was made independent of concentration, the Wolf algorithm at times failed to converge. It appears that errors in dissolved components have greater effect than errors associated with the measuring of sink or sources when the number of flow rate equation modee th n s lesi si l s tha numbee nth f chemicao r l balance equations. e sensitivitTh y s performeteswa t whicn di h differen k valueß t beed ha sn assigned to the various isotopic and chemical species k according to:

=?*- (14)

108 Here ^k is the mean of a large number of concentrations determined for laborator associatee yth standards i speciesk k e O dth d standarf so an , d deviation. n thiI s manne k becomeß r e coefficienth s f variatioo te erroth n i f ro n determinin laboratore gth y standarspeciesk valuee ß th e e eacr r th Th . sdfo fo f h o K tracers used in this test are listed in Table 1.

Tabl . e1 coefficien variatiof o t n (ßk) value ionsr sfo , TDI, D 18d Oan constituents averagn date a s Th ai . e from five laboratorie Israeln si , USA, Austri Germanyd aan .

TDI HCOs SO4 Cl Ca Mg N a 18O D

0.0122 0.0315 0.0622 0.0163 0.0411 0.0520 0.0855 0.0231 0.0647

When applied even to a simple flow system, results with the aforementioned algorithm for data corruption, seldom yield an optimal solution. Most of the difficulty stems from having neglected to maintain the ionic balance between the chemical fixee specieth dd empiricasan l ratios betwee 18electricad f i O an r ,o nD l conductivity does not vary linearly with the total dissolved ion (TDI). We had to do the same with the artificial concentrations obtained when generating our noise. To do so, the ionic balance was maintained within 5%, the ratio between electrical conductivity and TDI ranged from 35 to 55, and the ratio between D and 8.5o t froO 5 18 , whicm6. meae closs hi th o net world meteoric line. Resultf so this typ f teseo t wit hsimpla e synthetic cell configuratio illustratee nar Adan di r . [30al . ]. et Thes(Figure6) d ean figure5 s s sho meae wth n computed valuf eo fluxes vary wit e numbeth h f realizationso r d convergan , e towar e "trueth d " value. The same error generator (equatio s alsi ) o n12 use r introducindfo g noiso et the geometry of the flow system, for this, equation 13 is modified accordingly:

(15)

weightina whers i m eX, g parameter controllin magnitude gth corruptee th f eo d length or width Lm* of the mth active cells' boundary relative to that of the "true" length Lm. The flow configuration of the synthetic, 3-D compartmental aquifer presente s n Figurusei d o examinwa t d 1 e e sensitivitth e f boto y h calculated fluxe transmissivitied an s expecteo st d errorinpue th n tsi data. Errors e cell'th suc s a hconcentration e lengt th d n widtactiv a d an h f an so he boundaries were introduce fore th m n d i describe r fo 5 1 equationy r db o 4 1 , 13 s fluxes and transmissivities, respectively. For this purpose, the used ß values are liste equa s maximuassigo t % XTabld wa n d0.1 i o 10 t 0lan n, a e 1 m error. Figure 7 shows the mean computed fluxes between cells obtained by a Monte- Carlo simulation applie typeo tw perturbef so o d t d datasolie Th d. lines illustrate e variationth f fluxeo s s versus increasing numbe f realizationo r s derivey b d perturbation of chemical and isotopic data. The dashed line was derived when

109 the error generator was also applied to perturb the assigned length of active boundaries and distances between cells. Results show that under severe, but nevertheless realistic chemical and isotopic constraints, the average inflow estimate almose sar t identica assignee th o t l d (true) flux values. Some deviations are due to the non negativity requirement of the optimizer algorithm. Figure 8 show variatione th s meae th nn i s computed transmissivities obtainee th y db above mentioned tests. Solid lines represent results obtain from perturbed tracers, whil e dasheth e e resultd th line e sar s from perturbed tracerd an s geometry of cells. The true values for both fluxes and transmissivities are posted Fign i . .Figur1 illustrate9 distributioe eth meaf no n value inflowf so s (recharge) for random errors employed to the ionic and isotopic concentrations (solid lines) as well as to the geometry and the magnitude of the compartments (dashed lines). It is important to note that although the mean values of fluxes, transmissivities and inflows are presented in a separate figures, results were obtained fro ma singl e Monte-Carlo typ f simulatioeo n witrealizations0 h50 . Results indicate that fluxes and transmissivities converge into stable values close to the "true" (assigned) solution after 50 to 100 realizations. Also, the later the unknown variable is solved by the optimization scheme and the higher is the magnitude of the variable, the effect of the errors is more significant. Therefore, e meath n calculated values converge after more realizations with higher deviation fro e "truemth " value e lasTh t .thre e figures also demonstrate that increasing the computation effort by introducing further errors associated with cell's boundaries, has only a minor effect on the final solution. In general, the mean values converge even faster towar dstabla e solution e facTh t .tha e th t model yields good results when noisy data satisfy constraints similar to those that real data usually satisfy, suggests thae modeth t l shoul e applieb d reao t d l hydrologie conditions in the field.

5.1 Sensitivity test for the Southern Atava Valley.

Sensitivity analysi s performewa s n resulto d s obtained fro e southermth n Arava Rift samVallee th en y o flo w configuration presente Figurn di . eResult6 s with 1992/93 rates of pumping as presented in Table 2 (see the following section) wer esensitivite useth n di y test whic performes h wa stages o tw Rando ) n di (1 : m errors were applied to tracers associated with external sources of inflows and to concentration f cellso s Rando) (2 ; m errors were employe dgeometre alsth o ot y and the magnitude of the cells' boundaries. The effect of random errors on inflows (component f subsurfacso e recharge), fluxes withi e alluvianth l aquifer evaluatee th an n do d transmissivities were studied l simulational n I . valuek sß s equa coefficieno t l f variatioo t n evaluated from laboratory standards i X d an s equal of 0.15 to account for up to 15% error in the length of the flowing boundarie cell'd san s magnitude. Figur 0 show 1 e abov e effecth th sf o et mentioned error n rechargo s e components ( 1, 3, 4 and 12). Figures 11 and 12 show the same error effects on internal fluxes betwee associatee th n no celld dan s transmissivities respectively. Those figures show that simila resulto t r s obtained with synthetic datae th , average computed fluxes (inflows) converge realizations0 15 d o aftet 0 4 rt too I . k

110 0 20 0 18 0 16 0 14 0 12 0 10 0 8 0 6 0 4 0 2 0 Realizations I I I INFLOW 3

0 20 40 60 80 100 120 140 160 180 200 Realizations

0 20 40 60 80 100 120 140 160 180 200 Realizations 088

080 0 20 40 60 80 100 120 140 160 180 200 Realizations ——•—— Random errors including boundaries -•-••o—- Random errors applie tracero dt s only

Figure 7. Results of Monte Carlo simulation for inflows calculated for the southern Arava Valley:

111 1.2

nM O) " 1.1 o O

X l

0 20 0 18 0 16 0 14 0 12 0 10 0 8 0 6 0 4 0 2 0 Realizations 0.

0.92 ce eu 0. 0.88 0.86 x 0.84 3 Flow IV-V (g) l—H 0.82-fi Flow IV-V 0.80 0 20 0 18 0 16 0 14 0 12 0 10 0 8 0 6 0 4 0 2 0 Realizations 0.81

0 20 0 18 0 16 0 14 0 12 0 10 0 8 0 6 0 4 0 2 0 Realizations

Figur . Resulte8 f Montso e Carlo simulatio r fluxenfo s calculatee th r fo d southern Arava Valley:

112 TRANSMISSIVITY I - II

H 800 0 20 0 18 0 16 0 14 0 12 0 10 0 8 0 6 0 4 0 2 0 Realizations

TRANSMISSIVITY IV-V

170 0 20 0 18 0 16 0 14 0 12 0 10 0 8 0 6 0 4 0 2 0 Realizations 290 TRANSMISSIVITY V-VI 280

270 .5s en 260

250

240 0 20 40 60 80 100 120 140 160 180 200 Realizations -•—— Random, errors applied to tracers and cells' geometry -*—— Random errors applied onl tracero yt s

Figure 9. Results of Monte Carlo simulation for transmissivities calculated e southerfoth r n Arava Valley:

113 Tabl e2 Results obtaine unknown2 1 r dfo s with weighr t fo facto l W= r oxygen-1 deuteriud 8an W=0.d man ionsr 5fo . A-Inflow steadr sfo y rates of pumping until 1990; B- Estimated inflows for expected ______pumping rates for 1992/1993______The unknown inflows are: A B Nam f eo Pumpin g Rat inflof eo w Pumping Rat inflof eo w ______inflow______fin 1Q6- m3/y)______fin IQ6- m&y)______Cell 1 2.433 2.72 BIRBAIDA = 3.0131 3.877 YAALON 3/4 = 0.0000 0.000 Cel l2 6.93 7.75 YOTVTA 9t = 1.7687 2.986 YOTVTA 12 = 2.7656 1.736 YOTVTA 9 = 0.4490 2.525 Cell 3 0.979 1.10 YOTVT = t A9 1.6349 1.143 Cel l4 1.64 0.90 YOTVT = t A9 1.7297 1.256 BEERORA1 = 0.5245 0.387 TIMNA 6 = 0.1183 0.033 Cel 5 l 0.00 1.00 BEERORA = 1 0.1788 0.298 YAALO = N3 0.0765 0.503 Cel 6 l 0.08 1.09 YAALO = N3 0.2220 0.832

Total estimated inflows 12.481110 % (100.0%6nr )

3 Qout + Total pumpage: 12.4307 10 6 m/y; Absolute difference: 0.0504 (0.4054%)______

more realizations until the average values converged into the "true" value for the first unknowns to be solved. However, when random errors were applied to concentrations and to boundaries, the perturbed realizations converged faster than those applied onl cell'o yt s concentrations additionae Th . l errors employed e flowinth o t g boundarie e cellth sf o ssimpl y further perturbe e resultsth d . However, the solutions converged into the same values as obtained when errors applied ha dconcentrationse onlth o yt samcalculatee e founs th Th r e.wa dfo d internal fluxes and transmissivities (Figures 11 and 12 respectively). For cells with relatively short flowing boundaries (i.e. V-VI in Figure 12) the effect of errors on the cell's geometry is more pronounced. In such alluvial aquifer, the e flowinwidtth f ho g boundaries were assigne de surfacbaseth n o se contact between the alluvium and the side base rocks as taken from maps and air photos. This is a rough estimation of the true magnitude of the flowing boundary. Results indicate that when transmissivitie simultaneousle ar s y calculated with internal fluxes and recharge components, the solution is quite stable. It is importan noticeo t t , however, thadeviatioe th t f transmissivitieno s betweee nth assessed values and the assigned transmissivities are heavily dependent upon e rateth s assigne e knowth o t dn outflows e absolutTh . e value f theso s e deviations have no real meaning aside from the fact that for more reliable flow configurations, one may expect to obtain lower deviations.

114 45.0

co

0 50 0 40 0 30 0 20 0 10 0 0 50 0 40 0 30 0 20 0 10 0

1.55

eO 1.35 0 50 0 40 0 30 0 20 0 10 0 0 50 0 40 0 30 0 20 0 10 0

f.J" | » #7 (4.0) Tr. "»»A ^W- ^_ _ - —— w 2 •yi- xw-v •w» ,,,» O Tr. &^Geo. "o > *T» J.

9 Tr. C 4.0- ^«Vvs flv/oc \?*.v)/ ^ f\\ Tr.

•**• •vv J.V.VO #1 ()(15.(J) —— #1 1 (!().()) 15.04- s | ^ 10.06- S » ' Tr.H-Geo. ~ *5 10.04- 15.00- 'wv"^^^S, "0 Tr. fGeo. i-r****^ •-. . . - |_i j -',--.- ' b 10.02- fy/v ""•^ox^ji^ 14.98- ff Tr. r^_n*»ir%^. ULrt—fj-^ ••"-.r"' • * « o ' 'll /v i _-• 0 10.00- Tr. C 14.96- '.«'"'•I' c c 14.94- M OOR- (> 10 10 2CK» 3C« 0 4«50 10 0 5040 0 30 0 20 0 10 0

w.j- ^u.^- ft #13 (0.0) 1 (20.04 #1 ) *' k 0.2- Tr. 7n i - i - "--,-% f^— "-^ ^ Tr.+Geo. i_*n. L- - .. „. \v^-^^ 1-— _ ---..-•- .v^v r^vr^>&L: >•'•—• ~^.,-.— 0.1- 20.0- Tr._ eo Tr.+Geo

nn- IOO- 0 100 200 300 400 500 0 100 200 300 400 500 Realizations Realizations Figure 10. Results of Monte Carlo simulation for inflows calculated for the southern Arava Valley:

115 3 19.5 u_ Perturbed tracers & geometry Perturbed tracer geometrs& y 19 Perturbed tracers Perturbed tracers 18.5 1 51 101 151 201 251 301 351 401 451 501 1 51 101 151 201 251 301 351 401 451 501 Realization Realizations

5.6 F(ll-lll) 44.9 cr" "3T 5 2 .i - I 44-8 S 5 J44.7 i« ' 34.6 |44.6 "-4.4 Perturbed tracer geometrs& y u_ Perturbed tracer geometrs& y 44.5 4.2 Perturbed tracers Perturbed tracers 4 1 51 101 151 201 251 301 351 401 451 5C1 51 101 151 201 251 301 351 401 451 501 Realizations Realization

40.4 T—————— ) IV - F! (II 40.' 3 Perturbed tracer geometrs& y

Perturbed tracers

1 50 1 45 1 40 1 35 1 30 1 25 1 20 1 15 1 10 1 5 1 Realizations

Figur 1 e1 Result f Montso e Carlo simulatio r fluxenfo s calculatee th r fo d southern Arava Valley:

116 T(l II)

V 7 V M ^"^ '-~~—~~^*s ~ ~- — "^~ • ~ Perturbed tracers & geometry

"068 Perturbed tracers

X Z3 i*- G 6 Perturbed tracer geometrs& y

Perturbed tracers _ 1 51 101 151 201 251 1 3050 1 1 45 35 1 1 4040 1 1 4535 1 1 5030 1 1 25 1 20 1 15 1 10 1 5 1 Realization Realizations

264 T(ll IV)

:26 3 Perturbed tracer geometrs& y

Perturbed tracers 0 262

Perturbed tracers & geometry 261

Perturbed tracers 26 1 51 101 151 201 251 301 351 401 451 501 1 51 101 151 201 251 301 351 401 451 501 Realizations RcTlization

154 T( II! IV)

V1535 E Perturbed tracers & geometry 0153 Perturbed tracers

15 25

15 2 1 51 101 151 201 251 301 351 401 451 501 Realizations

Figur 2 e1 Result f Monteso Carlo simulatio r transmissivitienfo s calculated e southerfoth r n Arava Valley:

117 . SUMMAR6 DISCUSSIOND YAN . In this model onlthe , y data tha requireis t d aside froqualitativmthe e knowledg floe th w f eo syste r solvinmfo time-averagee gth d spatia e fluxeth e lar s distributio f environmentao n l tracers. However e calculatioth r fo , f o n transmissivities shoule on , d obtai distributioe nth hydraulif no c heads wels a , s a l e dimensionth e cellsth f ;o s mainl e siz f th yactivo e e boundariese Th . combinatio f heano d distributio d celnan l configuration suggests whice th f ho boundarie a flowin s i s g boundary (active boundary) s boundara ; y whics i h situated paralle streaa o t l m line norma piezometrie th o t l c surface, shoule db considere floo n ws d a boundary . The concentrations and the hydraulic heads which represent each compartment are assigned to the geometric center of the cell. In other words, an imaginary wel poste s i geometrile th t da c cente clustea f o r observatiof ro n holes presenting a similar type of water. Therefore, in a real case, the actual location of boundaries between cells is not precisely known. In fact, the most one can say is that the boundary should be somewhere between the extreme (marginal) wells of nearby clusters forming the cells. The location of the representative imaginary wells are assigned arbitrarily to e geometrith c well'e centeth f so r cluster entire e cente(noth th f o tet o r cell's area). Therefore exace th , t locatio cellse th f 'n o boundarie e effeco th n n s o t sha distanc n betweeli e e centere cellsth nth f . o s However e positioe th , th f no boundaries, which are always normal to the flow trajectory, affects the width Win of permeable boundaries (Equatioe boundariee widtth Th f ho e . th 5) n n i s southern Arava Valley were assigned accordin e contacth o t g t betweee th n alluviu surroundine th d man g hills determines a , d from satellite r imageai d san photos. Sensitivity analyses show thae modeth t s extremeli l y sensitive th o t e assigned concentration values associated with potential inflows and internal fluxes less i t sI .sensitiv exace th o tet cell's geometry mainl r activyfo e (flowing) boundaries with high rate f fluxesso lons watee A g.th masd an r s balance ear maintaine uniquda e solutio obtainede b n nca . This implies thaoptimizee th t d solution is highly related to the assigned known fluxes, mainly sinks and/or sources. Those assigned fluxes hav direcea twatee effecth n ro tbalanc e within each cell. Therefore orden i , maintaio t r e watenth r balance increasn a , n i e pumping rate must be adjusted by an appropriate increase in inflows and internal fluxes. This issue is further demonstrated in Table 2 where two sets of calculated fluxes with different set f pumpinso g value comparede sar (A)n I inflowe . ,th s were assesse averagn a r dfo e steady rate pumpinf so g prevailed until 1990 (B)n I . , f tracere samo th t ese s were used with higher rate f pumpino s g assumer dfo 1992/93. Due to an increase demand for water, pumping was increased in Cell I, e pumpinTh . VI g locaa d rats decrease, IIIo an II ewa t l ,V e du Celn di V I l increas watef eo r salinity. Tabl showe2 s that followin increase gth pumpinn ei g rates a respective increase of inflows and internal fluxes were predicted for all cell but Cell No. III. It seems that the increasing pumping in Cell HI, attracts more water from Cell II than from external inflow. Transmissivity values that were obtaine variouy db s type pumpinf so g testn si wells alon vallee g th liste e transmissivitye ar Tabln di Th . e3 y values assessey db the above-mentioned model are in fairly good agreement with the values found from the pumping tests. As listed in Table 3, the measured transmissivity value were obtained by different methods such as draw down (DD) test, recovery (R)

118 Tabl . e3 Calculate d versus measure transmissivitie) d(* thren si e sections along the southern Ar a va Valley (m-^/day).

Section Measured Calculated Yaalon-Grofit area. 997-1016 Ketor (DD4 )a 1849 (R) 1209 Yotvata 2 (R) 2665 (ITS2 ) 1400 Yotvatall (DD) 980 11 (DD) 1000 ) 11(R 1060 Yotvatal ) 2(R 2010

Samar-Timna area 104-350 Samar 2 (?) 400 Timna 4 (R) 370 Timna FC (IT?) 807 Timn a4 (ITS) 4600 (IT1) 2800

Eilat area 284-345 % 4 ) (R Eila 0 1 t 5 10 Eilat m.s) (R . ______Eilat 16 (R) 273______*(R) - Recovery test ; (DD) - Draw down test ; (IT) - Interference test and interference (IT) tests. Data were obtained from the Water Commissionery - Hydrological Service, Jerusalem. Also in Table 3, for those wells where two or more methods were differenceemployed% 50 o t p su , were found importans i t I . t to notice, however, that this model allows the evaluation of transmissivities for flowing segments of the aquifer with steady fluxes. Though the transmissivity is a physical propert e aquifeth f yo r independen e hydraulith f o t c head thin i , s model transmissivitie e assesseb t no dn unlesca s s ther e heaar e d gradients between cells. Therefor cellr fo , s with heavy rate f pumping so centrae th n i ls a , section of the southern Arava basin (Fig. 6), transmissivities can not be evaluated sinc e inter-celth e l fluxe e zeroar s . Consequently s necessari t i , o assest y s simultaneously both transmissivities and fluxes. Similar to results obtained with the inverse numerical solution, the calculated transmissivities are average values betwee differenco t nodese e nth Du .magnitudn i e compartmentaa n ei l scheme, these value e assorar s f conductanceo t s across permeable boundary between cells othen I . r words t describei , abilite boundare sth th f yo transmio yt t a certain volume of water per unit length of the boundary per unit time.

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120 [19] MALOSZEWSKI v ZUBERP , , PrincipleA. , practicd san f calibratioeo f no mathematical model interpretatiofor s environmentanof l tracer datin a aquifers. Advances Wat. Resour. (1993) (in press). [20] ZUBER , MALOSZEWSKIA. , , STICHLERP. , , HERRMANNW. , , TraceA. , r relation n i variabls e flow h t 5 Internationa, l Symposiun o m Groundwater Tracing, Athens (1986) 355-360. [21] NIR, A., Role of tracer methods in hydrology as a source of physical information. Basic concepts and definitions, Mathematical Models for Interpretation of Isotope Data in Groundwater Hydrology, IAEA-TECDOC- 381,IAEA,Vienna, (1986) 7-44. [22] ZUBER , MathematicaA. , l modele interpretatioth r sfo f environmentano l isotopes in groundwater systems, Handbook of Environmental Isotope Geochemistry . FontsCh . J . Frit, P ,d Eds. zan , Elsevier, Amsterdam (1986- 1 ) 59. [23] ZUBER , RevieA. , f existinwo g mathematical model r interpretatiosfo f no tracer data in hydrology, Mathematical Models for Interpretation of Tracer Dat Groundwaten ai r Hydrology, IAEA-TECDOC-381, IAEA, Vienna, (1986) 69-116. [24] GOBLET , P. Genera, l revie f methodologwo d approachean y n i s mathematical model r interpretatiofo s f traceo n r dat n hydrologicai a l systems, IAEA-TECDOC-381, IAEA, Vienna, (1986) 45-68. [25] GORELICK, S.M., EVANS REMSON& . B , , I. ,Identifyin g sourcef o s groundwater pollution: an optimization approach, Wat. Resour. Res. 19 (3) (1983) 779-790. [26] WAGNER, B.J. AND GORELICK, S., Optimal groundwater quality management under parameter uncertainty, Wat. Resour. Res3 .2 (1986) 1162-1174. [27] LAWRENCE, F.WUPCHURCHD AN . , S.B., Identificatio f rechargo n e areas using geochemical factors analysis., Ground Water 20(6) (1982) 260-287. [28] ADAR, E.M., ROSENTHAL , ISSARE. , , A.S., BATELAAN, O., Quantitative assessment of flow pattern in the southern Arava valley (Israel) by environmental tracer mixina d san g cell model . HydrolI . (19926 13 . ) 333- 354. [29] SIMPSON, DUCKSTEINE.SD .AN . Finit , L. , e state mixing-cell models: In . V. Yevjevich (Editor), Karst Hydrology and Water Resources. Vol. 2. Water Resource Publications Collins. Ft , , Co., (1976) 489-508. [30] ADAR, E.M., NEUMAN, S.P., WOOLHISER, D.A., Estimatio f spatiano l recharge distribution using environmental isotope d hydrochemicaan s l data. I. Mathematical model and application to synthetic data. J. Hydrol. 97 (1988) 251-277. [31] ADAR, E.M., NEUMAN, S.P., Estimation of spatial recharge distribution isotope hydrochemicad san l data . ApplicatioII . Aravaipo nt a Vallen yi Southern Arizona, USA. J. Hydrol , (198897 . ) 297-302. [32] ADAR SOREK& . E ,, S. Multi-compartmenta, l modellin r aquifefo g r parameter estimation using natural tracers in non-steady flow. Advances Wat. Resour. 12(2) (1989) 84-88.

121 [33] YURTSEVER, Y. AND PAYNE, B.R., A digital simulation approach for a tracer case in hydrological systems: Multi compartmental model, Finite Elements in Water Resources, Brebbia, Gray and Finder Eds., London, (1978). [34] CAMP ANA, M.F. & SIMPSON, E.S., Groundwater residence time and discharge rates using discrete-state compartment moded Can l data. J. Hydrol. 72 (1984) 171-185. 14 [35] CAMP ANA, M.E., MÄHEN, D.A., Model derived estimates of groundwater mean ages, recharge rates, effective porosities and storage in a limestone aquifer Hydrolf o . J , (19856 .7 ) 247-264. OMMENN VA [36] , H.C. mixing-cele Th , l concept applie transporo dt n no f o t reactiv reactivd ean e component soigroundwaterd n si an l Hydrolf o . J , 8 7 . (1985) 201-213. [37] YURTSEVER , PAYNEY. , , B.R., Mathematical models basen o d compartmental simulation approach for quantitative identification of tracer data in hydrological systems, 5th International Symposium on Groundwater Tracing, Athens, (1986) 341-353. [38] YURTSEVER, Y., PAYNE, B.R., GOMEZ, M., Use of linear compartmental simulation approac r quantitativhfo e identificatio f isotopno e data under time variant flow conditions, Mathematical Models for Interpretation of Isotope Dat Groundwaten ai r Hydrology, IAEA-TECDOC-381,IAEA,Vienna, (1986) 203-231. [39] YURTSEVER . BUAPENGY , , CompartmentaS. , l modelling approacr hfo simulatio f spatiao n l isotopic variation n studdini s g groundwater dynamics: A case study of a multi-aquifer system in Bangkok Basin, Thailand, Isotope Technique Waten i s r Resources Development, IAEA, Vienna (1991) 291-308. [40] WOOLHISER, D.A GARDNER.& , H.R., Estimatio f multiplno e inflowo t s a stream reach using water chemistry data. Trans. ASAE, 25(3) (1982) 616- 622 [41] ADAR, E. & SOREK, S., Numerical method for aquifer parameter estimation utilizing environmental tracers in a transient flow system, MODELCARE 90, International Conference on Calibration and Reliability in Groundwater Modelling Haguee Th , , Holland Kovar. K , . IAH. Ed , No S 195 (1990) 135-148. [42] WOLF, P. Methods of Non-linear Programming, Chap. 6. In: Interscience J. Wiley Yorkw ,Ne , (1967) 97-131. [43] BARD, Y., 1974. Nonlinear parameter estimation (Appendix D). Academic Press, (1974) 332 p.

122 FLUID FLOW AND SOLUTE TRANSPORT FRACTUREN I D MEDIA

. MORENOL . NERETNIEKI , S Royal Institut f Technologyeo , Stockholm, Sweden

Abstract A model is proposed to describe flow and transport in fractured rocks. It is based on concepe th networa f to channelsf ko . This approac backes hi observationy db driftn si s and tunnels that flow in fractured rocks takes place in sparse narrow effluent locations with widths typicall channea yd lesan slm thafrequencc 0 w nchanne1 e fe on a f r yo lpe square meters to one channel per more than a hundred square meters. Observations in boreholes also indicate that there are large distances, tens to hundreds of meters, betwee mose nth t conductive section boreholesn si .

For visualization purposes the model is displayed on a rectangular grid. The individual channels are given stochastically selected conductances. Flowrate calculations have been performed in grids of sizes 20*20*20 channels in most cases but larger grids have also been used r largFo . e standard deviation conductancesn si , greatee th rn i tha 6 n1. log normal distribution (base 10), channeling becomes pronounced with most of the water flowing in a few paths. The effluent patterns and flowrate distributions obtained in the simulations have been compared to three different field measurements of flowrate distributions in drifts and tunnels. Standard deviations of channels conductances were between 1.6 and 2.4 or more in some cases. Channel lengths were found to vary between 1.2 m and 10.2 m in the different sites.

A particle tracking technique was used to simulate solute transport in the network. Nonsorbing as well as sorbing tracer transport can be simulated and by a special technique also tracers that diffuse int roce oth k matrisimulatede b n xca .

Tracer measurement sitee on , Stripan si , were use comparo dt e dispersivities. These were large, Peclet numbers less tha botn5 simulationn hi fiele th dd resultssan . From the Stripa tracer data it was also found that the tracers were taken up into the rock matrix by molecular diffusion.

INTRODUCTIO BACKGROUND NAN D

In this paper we develop a model for solute transport in sparsely fractured rock. We also show some propertie modee th f so l that affect solute migration. Furthermoree w , attempt to estimate values for the important parameters from large-scale field experiments and observations. Solute transport is also calculated for solutes that diffuse int roce oth k matrix.

Flo solutd wan e transpor fracturen i t d roc bees kha n founpoorle b o dt y describey db the advection-dispersion concept and equations. Field observations show that there are strong channeling effect thad an ts when attempt made sar evaluato et dispersioe eth n coefficient, it appears to increase with distance (Neretnieks et al., 1987). Field observations in drifts and tunnels also show that flow channels are sparse and that the

123 flow-rate distribution is very large among channels. For short distances, it has been proposed tha floe solutd tth w an e transport might bette describee rb takins da g placn ei a bundle of independent channels (Neretnieks et al., 1987). For longer distances, channels hav a greatee r chanc f meetino enetwora d an g k concept seems more appropriate. For very long distances, it is conceivable that the mixing between channels is large enough for the transport to behave as described by the advection-dispersion model. This does not seem to have been observed in fractured rocks so far.

Fracture network models have been propose tested d an fiel n do d data (Robinson, 1984; Long et al., 1985; Dverstorp and Andersson, 1989) for flow calculations and in some cases als tracer ofo r transport (Dverstorp, 1991; Caca t al.se , 1990b). These models need information on fracture trace lengths, orientations, frequencies, and transmissivities, plus some assumptions, thadifficule ar t proveo ende t t th , n theI . y have to be calibrated to field observations.

In this pape attempe rw simplea t r approach assumd an , e that flosolutd wan e transport can be described as taking place in a network of channels. This simplification allows us to develo psimpla e model that doe t neesno d very detailed information.e b Dat n aca obtained from borehole transmissivity measurement observationd an s fracturn o s e widths. Calibrations can be made with observations in drifts and tunnels. One important aspect of the model is that it is simple enough to accommodate the transport of solutes that diffuse into the rock matrix. This is very important when assessing retardatio thesf no e solutes. Below summarize w , e some observations that have been the basis for the formulation of this model.

Abelin et al. (1985) analyzed flow and solute movement in three natural fractures intersecting tunnels in the granite in the Stripa mine. They found that the fractures were closed or very tight, with few open parts between the areas where two fracture surfaces contactn i e ar . Observation drift n stunneli d an s s strengthen this impression (Neretnieks et al., 1987). In two investigations (Neretnieks, 1987; Abelin et al., 1991a.b) it was found that fracture intersections often mak higp eu h flow-rate conduits.

In the 3-D tracer test experiment performed in the Stripa mine by Abelin et al. (1991a,b), it was found that water flows into the drift with a very uneven spatial distribution. The results also show that the tracers are unevenly distributed in the drift. Very different concentrations were found in sheets near each other. Low values alternate with high values in nearby locations. In some cases, much of the tracer was found in locations far away from the injection hole.

Detailed observation f floo s w distribution otheo tw rn i slon g drift tunnelsd an s , Kymmen (Palmqvist and Stanfors, 1987) and SFR (Neretnieks et al., 1987), show that watee th r outflow occur narron si w channels, mos thef o t m les widesm e thac 0 Th .n1 channel density is about 1 per 20 to 100 m^. Experiments that specifically aim to study channeling in individual natural fractures on the scale of 2-2.5 m in the Stripa mine (Abeli t al.ne , 1990) also show that wate s conductei r onln di smalya l a par f o t fracture. Channels have typical width f les so . Hydrauli s cm tha 0 n1 c conductivities measure e fracturth n i d e planes show that ther e verar e y strong variationn i s conductivit fracturesplane e th th f n eyo i .

Network models have been used to describe the flow rate distribution in the Stripa experiment (Geie al.t re , 1990; Dverstorp, 1991). They describ stochastie eth c naturf eo

124 the areal flow distribution but do not account for the channeling nature of the flow. None of the models have addressed the question of the interaction of solutes with the rock surfaces. Nor have any of the models accounted for the observations (Abelin et al., 1990; Neretnieks, 1987) that fracture intersections play an important role in conductin watee gth r solutes.

In a recent study, it was assumed that the flow in fractures that intersect each other can be described as the flow through channels in the planes of the fractures (Cacas et al., 1990a). These channels connect the centres of the fractures through a point at the intersection line between the fractures. The fracture network may then be simplified to a networ connectef ko d channels.

One drawback of these models is that they need a large amount and very detailed data on fracture orientations, fracture size distributions and fracture conductivity distributions. Some assumptions must also be made in interpreting the field observation o translatt s e them inte datth o ae models useth n i d . e Manth f o y assumption difficule sar validateo t t modele Th . s wit l theihal r built-in assumptions must finally be validated by comparison with field data. The model described here needs much less data but also needs to be validated by field observations.

CONCEPTUAL MODEL

In this approach a three dimensional channel network model is developed. It can accoun stochastie th r fo t c naturfloe th w f eo distribution channeline th , g nature th f eo flow and the interaction of reactive solutes with the flow-wetted rock surfaces. The solute channele th n s i als n srocca oe diffusth kf matrixo t e ou intd .oan avoio T d difficultiemane th f yo s that fracture networks exhibit choose w , differenea t starting approach. It is assumed that the flow paths make up a channel network in the rock. Every channel member can connect to any number of other channel members, but we choose an upper limit of six members intersecting at a point, for reasons that are described below.

The use of six channel members is partly based on the observation that both fracture intersection and channels in the fracture planes play an important role in conducting flow. For two fractures that intersect, there will be one channel in every fracture plane continuy thama t e ove intersectioe rth n line wit othee hth r fracture thin I . s wayo t p ,u four member fracture th n si e plan intersecy nodee ema r thoson t Fo .a t e intersections wher fracture eth e intersection itsel conductings i f moro tw , e member addede b y ,sma formin six-membega r intersection.

Although the model concept is based on the above considerations, this does not mean that we consider a channel to lie only in one fracture or at a fracture intersection. A channel in our concept may have been formed by individual channel members in series. Thus a channel may consist of 1, 2, 3 or even more channel members. The actual numbe t verno y s ri important propertiee th s channele a , th f so s wil obtainee b l y db calibration to field measurements.

For visualization purposes, the network is depicted as a rectangular grid. The hydraulic propertie e member th e generate b f o sn ca s includiny db effecte gth f channeo s l

125 member differenf so t lengths, different hydraulic conductivitie othed san r propertief so interest eass i t varo yI .channet e yth l member densit differenn yi t regions (e.g., along fracture zones) and to include effects of anisotropy in a simplified manner. In this paper, we develop and show some important basic properties of the model. They include the ability to describe the uneven spatial distribution of flow, the residence time distributions of tracers, and the retardation due to matrix diffusion and sorption in the rock matrix. channele t thinth no f ko o d necessarils a se W y being clearly identifiable physical features fractura n I . e with varying apertures watee th , r will trac t differeneou t paths, depending on the gradient that exists at a given moment. When the conductivity variation largee sar ,path w therfe sa e wher r wilo e l eonlwateon e mosth e yb f ro t flows. Thi bees sha n foun simulationn di s (Moren t al.oe , fiel1988e th d weln s i )a s a l (Abelin et al. 1990). The flow paths may also be actual physical channels along fracture intersection r wherso e dissolution processes have formed channels. Thehavy yma e properties that vary along the flow path, but an average conductivity, volume and flow- wetted surface may be assigned to every channel member in the model. All properties of the channel members used in our model are thought to have a stochastic nature. The average transmissivity along a member and its length define the conductance. This is the only entity needed to calculate the flow, if the pressure difference between the two ends of the member is known. If the residence time is to be calculated for non-interacting solutes then the volume of the channel members is needed as well r soluteFo . srocke thasor"flow-wettedth e n n th ,ca tb o " surface are alss ai o needed (Neretnieks 1980) roce th kf I .matri solutee porous xi th d ssan have acceso st the interior porosity, the matrix diffusion properties must be known as well. We assume that it is possible to average these properties along a channel member. We also assume, in the present paper, that there is ideal mixing at channel intersections. Other mixing modes will be explored in later work. residence Th e time distribution (RTD oftes i ) n describe termn d i mea a f so n residence time and dispersion. These are obtained from the first and second moments of the RTD. Higher moments, which describe exceptionally long tail earlr so y arrivals usualle ar , y not considered. There are several mechanisms that cause dispersion of tracers: dispersion in the individual members of the network, dispersion caused by increase in the spread of residence differen o timet e sdu t velocitie channelse th n s i spreadin d an , g causey db matrix diffusion effects. In preliminary calculations, we found that the dispersion in the individual member negligibls si e otheo comparetw re causesth e b o d wilt d t an ,no l used furthe thin i r s study. Mixing at intersections will also influence the RTD of the network. Several possible reasonable assumption mixine made b th f n eo g sca processe t nonsbu e have been actually tested in real networks. We will only use the full mixing assumption in this paper. Matrix diffusion effects and sorption in the interior of the matrix are assumed to be active. Fluid flod solutwan e transpor e simulatedar t r traceFo . r transport, additional information on channel volumes and flow-wetted surfaces is needed. No independent dat thin availablee ao sar thereford an , e various assumption use e compared sar dan d with the field results.

126 SOLUTE TRANSPOR CHANNEA N TI L

The equation for solute transport in one channel, neglecting longitudinal dispersion is:

3c 3c _ 2D ac Ra+u-~e p

The first term is the accumulation of solute and includes the accumulation in the water channee solute inth th d e an sorbelchannee th n do lretardatio a walls s i a R . n factor defined as

Ra=l+2-Ka (2)

The second term in Equation (1) accounts for the advection in the channel. The third term accounts for the exchange rate of solute with the rock matrix by molecular diffusion.

diffusioe solutroce Th e th th kf n eno i matri describes xi y db

) (3 ^ = De

wher firse eth t term account accumulatioe th r sfo f solutpore no th en e i watee th n ri matrix as well as on the micropore surfaces. The sorption capacity of the solid and the pore water is

For the situation where there is no solute in the system initially, and the input concentratio solute th f nconstanteo s i solutioe th , (Carslas ni Jaegerd wan , 1959)

c -

travee th wher s i l soluta N timeI r efo e that doe t diffussno e introce oth k matrix. Equation (4) shows that the outlet concentration from one channel is a function of the channel volume, matrix properties, flow-wetted surfac flod wean rate.

diffusioe th f I n int roce oth k matri pronounceds xi travee th , solute l timth r muces ei fo h longer than solut a tha r tfo e that doe interact sno t wit matrixoutlee e hth th , t So .solut e concentration is almost independent of the channel volume. The concentration is then mainly a function of the matrix properties and the ratio between the flow-wetted surface and the flow rate.

CALCULATION PROCEDURES

Generatio f networno k

The use of a rectangular network does not mean that the channel members are of equal lengt r thaho t they form suc hnetworka onls i t f visualizini simplya :o y ewa e gth

127 network. Each membe networe th f ro assignes ki hydraulida c conductance tracer Fo . r transport voluma , surfacd ean alse ear o needed orientatioe Th . locatiod e nan th f no members only become important when the comparee yar d with real geometry.

The flow calculations only need the information on the conductances of the channel members and the boundary conditions. The conductance the ratio between the flow in a channel membe pressure th d ran e difference betwee endss nit . When solute transpors i t included, the volume of the member has to be known. If sorption onto the fracture surfac diffusior eo n int matrie oth x wil includee lb modele th surfacn de i ,th ee areth f ao flow-wetted surface must also be included. Then some properties of the rock are neede s wellda , suc rocs ha k matrix porosity, diffusivity sorptiod an , n capacitr yfo sorbing species. individuae Th l channel givee sar n stochastically selected conductances presene th n I . t simulations, the conductances of the channel members are assumed to be lognormally distributed, with mean \LQ and standard deviation GC- ft is also assumed that the conductance t correlateno e ar s spacen di volume Th . e wil estimatee b l usiny db g various assumptions, owin lace dataf th k o o gt .

Figur showe1 schematia s cmes viethe hwof use thidin s report outlinAn . the eof approach use calculato dt fluie eth d flosolutd wan e transport throug networe hth f ko member presentes si d below methoe Th . describes di morn di e detai Morenn i l . al t oe (1988).

Figure 1 At a fracture intersection up to six channel members may intersect.

Fluid flow calculations

For laminar conditions, the flow through a channel member is proportional to the pressure gradient writtee b floe y ws Th .n a ma betwee" "j d pointo an n tw " s"i

Qij = Cij(Pj - PI) (5)

128 wher conductance e th Cj s i j e connectin "j. d pressure nodee an "g th Th " "i s e fiels di calculate writiny db mase gth s balanc eact ea h intersection point

) (6 I Qr0 i j fo= l ral j solutioe Th f thino s syste f equationmo s yield pressure sth t eacea h node floe wTh . between adjacent node thes si n calculate usiny db g Equation (5).

Solute transport calculations

solute Th e transpor simulates i t usiny db particle-followinga g technique (Robinson, 1984; Moren t al.oe , 1988). Many particle introducede sar timea t a ,e inte on , oth known flor moro we efiel on locations t da . Particles arrivin intersection a o gt e nar distributed in the outlet channel members with a probability proportional to their flow rates. Thi equivalens si assumino t g total mixin intersectionse th t ga . Each individual particl followes ei d throug networke hth residence Th . e timgivea n ei n channer fo l non-sorbing tracer determines si floe th w y dthrougb channee hth ls it membe y b d ran volume. The residence time of an individual particle along the whole path is determined as the sum of residence times in every channel member that the particle has traversed. The residence time distribution is then obtained from the residence times of a multitude of individual particle runs.

From the RTD, the mean residence time and variance can be calculated. They may be used to determine the Peclet number, which is a dimensionless measure of the dispersivity (Levenspiel, 1972)

When dispersion in the channel and/or diffusion into the rock matrix are considered, different particle same th en si channel member will have different residence times. Here, residenc particlese th f o e D particle e ,timedescribee th b RT r y e sfo th s ma y db expresse probabilita s da y density function outlee thoughe th b s y a t, f pdfma o t t I . concentration for a pulse injection. If this curve is integrated over all the possible residence times, the cumulative distribution of the residence times is obtained.

When diffusion from the moving water into and out of the rock matrix takes place, a particl residy matrie th e ma somr n ei x fo e tim addition ei residencs it o nt ee timth n ei water in the channel member. For a flat channel from which the diffusion is perpendicula channee th o rt l surface simpla , e analytical solutio availabls ni e th r efo RTD. The cumulative curve, F, for the residence times is obtained as

0.5 F =

for times greater tha water-plug-floe nth w residence tim . Otherwisw et value eth s ei zero. Equation (8) considers only advection in the channel and diffusion into the rock matrix. Longitudinal dispersion is neglected.

129 rectangulaa r Fo r channel member water-plug-floe ,th w residence tim obtaines ei d from the ratio between the channel volume and the flow rate through it. It may be calculated by LWÔ/Q. Introducing this expression into Equatio yield) n(8 s

5 (t-tw)°- Q

particlr Fo e following folloe w ,techniqu e wth e use Yamashity db Kimurd aan a (1990). The travel time for each particle in a channel member is determined by choosing a uniform random numbe intervae th n ri l [0,1] travee .Th particlee l timth r thes ei fo , t ,n calculated by solving for t in Equation (10).

SOME PROPERTIES OF THE MODEL

simulatee Th d domai cubia s cnwa grid wit channe0 h2 l member eacn si h directione .Th pressure gradien imposes i t d perpendicularl sidee on . o yNo-flot w conditions were imposed on the four sides parallel to the pressure gradient. The outflow is through the 400 channels in the low-pressure side. Grids with a larger number of channel members were also simulated.

Flow rate distributions

Figure 2 shows the number of channels, of the 400 at the outflow face of the cubic grid, which carry a given flow rate (relative) for two ac- Note that the flow rate decreases to the right in the graph. The flow rates have been grouped in "bins" with the upper bound twic lowe e largs ea th s era bound. This form geometria s c progression with a factor of 2 between the bins. This mode of representation has been chosen because in practice the larger channels will always be seen, whereas the measurement limit will determine the smallest flow rates that can be observed. In field observations, onl e binflo8 yth 6- w s e wit rateth e larges hth n i s t flo e wuseb r ratedfo y ma s calibration purposes. This means flow rates in 2-3 orders of magnitude. In histograms showing the fraction of flow in a given interval, only the largest flow rates contribute significantly to the total flow. Figure 3 shows the cumulative fraction of flow for flow rates less than a given value, for two GO The total flow is normalized to the total flow larges8 e th tn i intervals contributioe Th . intervae th f no l wit smallese hth t flow rats ei very smal botr lfo h <7c. This typ grapf eo h will late usee rb calibrato dt modee eth l with field data.

As shown in Figure 2, the distribution of the flow rates from the grid, on a logarithmic scale, shows an asymmetric tail extending out to smaller flow rates. This tail is not observe fiele th dt d a observation s becaus vere eth y smal measuredt l flowno e sar e Th . flow rates cove order3 r2- magnitudf so mostt ea .

130 300 n Std Dev = 0.2

« •S 200-

«M O

,0 E 100-

•* es oo O o o O o o es rt «n es es' o o CJ o in O tn es es -H '• \d C3 o o o o o es u-i CX5 o o o >o es "-1 es es es W-) en o Relative flow intervals

300-

es 200- u

01 100-

Relative flow intervals

Figur e2 Histogram numbee th r f channelfo so r s with flow givea rat n ei n standare th n i ) d(b deviatio8 0. d intervalan f no ) (a valuer 2 fo ,0. f so

the conductance ac. The flow at the left is large, and decreases toward righte sth .

131 1 .0- o3: «£• M.. 0 . 8 - o

0 . 6 - o ni 0.8= v 0De —d —St «0.4- 1.6= v 0De •""•••d St • 2.4= v 0 De d St - - -

0.2- ü

O.O-I I I 23456 Log2(Relaîive flow rate)

Figure 3 Cumulative curve for fraction of flow with flow rate less than a given valu differenr efo t standard deviatio conductancen i s

Solute residence time distributions for non-interacting solutes

It will later be shown that the RTD for non-interacting solutes is not and cannot be f sorbino D gRT relatesolutese th o dt . Nevertheless, non-interacting solutee ar s commonly used for tracer tests. Information on spacial distribution of flow and of mixin spreadind gan g processe networe th n howevei sn kca obtainede b r r thiFo .s propertieD reasonRT e non-interactinr th , sfo g solute discussee sar d here somd an ,f eo the limitations in our understanding are pointed out.

Although a few flow paths carries most of the flow all channel members contribute to some extent. This implies tha distributioe tth residence th f no e timbecomy ema e very broad. The fastest paths will be those with large flows, but the tail of the RTD will be

determine low-floy db w channel members expectee b y largr ma Fo . det i value , c a f so that the difference in residence times between the fast and the slow paths is considerable influences i D floe RT onlt th w e dy no channe e y rateb .Th th n si l members but als theiy ob r volume.

reliablThero n e ear e relationships known presentt a , , betwee conductance nth e th d ean volume. Several approaches were explored maie Th .n difficult determininn i y g which possible ofth e approache represeny sma volume lacth te fielf th k o s ei d data have W . e tested some approaches between conductance and volume. From these simulations, we expect to find which approach(es) could be used, by comparisons with field observations of the distribution of tracer residence times.

132 Four approaches were chosen:

- Volum proportionals ei hydraulie th o t c conductivity.

Volum constant.s ei

- Volum proportionas ei cubico lt root channef o l conductance.

Volume is determined from an independent random distribution.

To test these approaches, solute transport was simulated. Two types of injection were used. First, a large number of particles was injected on the high-pressure side of the bloc collected kan opposite th n do e side numbee Th . particlef ro s that flow into each inlet channel is proportional to the flow rate in these channels. This type of injection was chosen because result independene sar injectioe th f o t n location (Moren t al.oe , 1990). secone th r dFo injection type, many particle injectee sar e meshe poina th th t d a n i tan , particles reaching the outlet side are collected at their respective locations. The relative concentratio determines ni d fro numbee mth particlef ro unir spe t time that arriva t ea collection point, divided by the flow rate in the same location. This type of injection uses comparo wa dt date eth a with thos traceD e 3- fro e r mtesth t experiment donn ei Stripa (Abeli t al.ne , 1991a,b).

As will be shown later, values of oc obtained from comparisons with field observations are between 1.6 and 2.4.

Peclet number mead san n residence time calculatee sar d fro particle mth e residence time distribution. Peclet numbers when the particles are injected in one of the block sides are show Tabl n differenr i fo e1 t volume approaches. Ther insufficiens ei t informatioo nt accep f thes rejecr o o t y e an tapproaches . However considee w f i , r tha vera t y large dispersion (Peclet numbers much less than 1.0 uncommos )i fieln ni d observations,

Table 1 Peclet number as a function of the standard deviation CTc for different assumptions used for channel volume. Average values from 20 simulations.

Assumption for Standard deviation GC channel volume 0.80 1.60 2.40

Proportional to Cond 2.63 0.19 0.13 Constant volume 3.90 0.56 0.09 Prop, to Cond1/3 10.45 4.41 1.13 Prop Condo t , 2/3 8.41 0.79 0.20 Random wit ac/= h3a 3.19 0.28 0.80 Randomc 0 wit = ha 0.41 0.04 0.06

133 then some approaches seem less likely, e.g. when the volume is chosen from a new distribution or when the volume of the channel is proportional to the channel conductances. The other approaches give residence time distributions found in field experiments. The largest Peclet number (or the smallest dispersion) is obtained when the channel volume is taken to be proportional to the cubic root of the conductance. The same result obtainee sar d when point injectio useds ni .

Solute transport for matrix interacting solutes

If the solutes also have access to the matrix porosity by diffusion, the residence times of the solute particles are larger than the residence time without diffusion into the matrix. For low flow rates, Q, in relation to the flow-wetted surface, LW, in a channel member vere solute b th ,yn muceca h retarded compare watee th o drt flow residence flotracee d th w an n timeI r experiment. w t , Stripn si a (Abeli . 1991a,b)al t ne , where water flow residence differen e timeth r sfo t tracers varied from month yearso st , they found that matrix diffusion may have contributed noticeably to the retardation of the tracers have W . e therefore incorporated this mechanis laten i e moder r us ou r mn i fo l studie tracef so r transport. residence Th e tim eacr efo h particle that passe channee sth l membe stochasticalls ri y determined by Equation (10). The capability of the particle following method to describe the diffusion in the rock matrix was tested by generating a network with equal channel members (zero standard deviation). In these channels, we used the stochastic process to simulate the diffusion in the rock matrix. The results agree very well when compared with the analytical solution if a few thousand particles are used. The procedure was found to be fast and accurate in network calculations as well.

CALIBRATION AND COMPARISON WITH FIELD OBSERVATIONS

The calibration of the model, to a given site, involves the determination of the conductance distribution used to generate the channel and the channel length. If a lognormal distributio s usedni , the e mea standard nth an n d deviation woule db determined. The mean works as a scale factor for flowrate, so if the flux is known the evaluatede b mea y nma . Procedure determino st standare eth d deviatio showe nar n below.

determino T e these parameters, observation watef so r inflo tunnelwo t e b drifr y so ma t used. Borehole data may be also used to estimated channel lengths (Moreno and Neretnieks, 1993. The water inflow to tunnels will be at isolated points on the tunnel walls. This will look lik numbeea f "channelsro roce th " k y e beinfaceb th t f gcu o s tunnel outfloe Th . w networe facth f eo k model will also hav numbeea f "channelsro " with flowing water. floe wTh rate frot simulatee sou m th d cubic grid cove verra y wide interval larga f i , e value of <7c is used. On the other hand, the field results often show variations of only a few orders of magnitude in flow rates. This aspect is schematically presented in Figurhistograe Th . e4 m show numbee sth f channelo r s with flow rategivea n si n interval, if even the channels with a very small flow rate could be monitored. From field observations obtainede , b onl biny 8 y- s ma dat6 .r Theafo showe yar n with gray figuree bar th white n si Th . e bars represen unknowe th t n data whole Th . e histogram could also represent the results of a simulation.

134 CO •*—o» Q. CO

13

Flow rate

Figur e4 Histogram showin numbee gth channelf ro s wit hfloa w ratgivea n ei n interval. The gray bars represent hypothetical data from field observations. The white bars represent channels with a flow rate below the detection limit.

The calibration procedure is as follows. The inflow measurements to the drift are compared wit simulatee hth d results. Wit available hth e data (the gray bars standare )th d deviation GC mav be estimated. Then, when the value of GC is known, the number of channels in the observable flow rate interval for the field data is compared with that of the model for the same interval. The total number of channels in the field (observable and non-observable channels or gray and white bars in the histogram) is obtained by comparing the field data with the results from simulations.

Total number of Numbe channelf ro s observen di channels in rock rock abov choseea n flowrate Total number of Number of channels in the simulations (11) channels in simulations spannin same gth e flowrate range

The same procedure can be used for the packer test data in boreholes if these are analyse numbee th r f intersectionsdfo o r , provided channe l borehole widthth d san e diameter are known. This will be described later.

The channel network model gives no inherent information on channel member widths or lengths. Nor is the distribution function for the conductances known a priori. If the latter is assumed to be lognormal, then the mean and standard deviation can be found by fitting them to measured data. It has been found, by several investigators, that the standard deviatio transmissivitief no s obtaine boreholen di s often o betwee s ni 3 d an n1 a logarithmic (bas scal) e10 e (Cacas, 1990a; Geie t al.re , 1990). Dverstorp (1991) used <7c values ranging from 0.4-2.4 in simulations of the Stripa experiments with the best results for ac = 1.7. Cacas et al., (1990a,b) found a value of ac = 3.2 for the Fannay Augère experiments in their fracture network model calibration.

135 Using data from drift tunneld an s s siteR ,SF there floth e t ear w A rate measurements availabl individuar efo l spots oven ra 2 (Neretnieksm 0 00 4 are1 f ,a o 1987). Tabl summarizee2 data e floe sth wTh . rate ranges have been chosen as a decreasing geometrical progression, such that each range "bin hals previoui "e th f s onereasoe Th . thas wile ni on tl undoubtedly observe eth largest flow rate spots t wilbu ,l hav decreasinea g accurac detectinn yi g spots with lower flow rates alss i t oI . generall possiblt yno knoo et totae wth l inflolowe th -wo t flow rate spots, so that a cumulative flow rate curve cannot be constructed to start at the low-flow end. However, a cumulative curve can be started from the high-flow end and

Table 2 Flow rate distribution at the different spots at SFR.

Flow rate Numbef ro Cumulative Fraction of range, 1/min spots numbe spotf ro s Flow rate

>=1.6 2 2 0.13 0.8-1.6 4 6 0.15 0.4-0.8 12 18 0.21 0.2-0.4 41 59 0.30 0.1-0.2 38 97 0.13 <0.1 67 164 0.08

comparee b n modee ca th o dt l results f graphI . s wit cumulative hth e fractio f flono w experimentae rate th use e d sar dan l results plotte thosen di estimatn a , standarde th f eo deviation of conductances can be made. The cumulative flow rate increases little when more than 5-6 "bins" are used. The method can thus be expected to be robust, in the sense that the most important paths have been accounted for. For the SFR data, GC is found to be about 1.6 with this method. The cumulative flow rate curves obtained from the simulations and the cumulative curve from the field observations are shown in choicFigure t totallTh no . s e5 i y clear, becaus differencee eth s betwee curve nth e obtaine smalle ar r 4 observationFo .R 2. d d witSF curvee an e hth th 6 Gr d 1. C sfo s an = this reason alternativn a , e procedur discusses ei d below.

Tablcumulativn e I th , e3 e number f channelso given e ratie threr ar th sfo f C of o eI G . the cumulative number of channels in the model to the number of channels in the SFR observations is calculated, the model and observed results agree best for GC = 1.6. The agreement is based on obtaining the same ratio along the various flow rate intervals. This is shown in Figure 6, where this ratio is plotted as a histogram. The figur emora showf o e e evesar than6 bare Gr 1. heighth t Csfo = othee t th tha r rnfo GC, meaning thaobservee th t moded dan l distributions agree bes r thiThiO fo ts G s confirms the value of GC obtained with the cumulative flow rates. The values used in Tabl correspone3 averagdo t e valuerealizations0 2 r sfo .

136 bees Oncha nC eestablishedG numbee th , channelf ro foune b "aligningn y db sca e "th two columns, "Numbe channelf ro modele th "Observen d si "an d numbe channels,f ro " starting with the highest flow rate range, see Table 3. The ratio of the number of channels in the model to the observed number of channels was about 0.99, which means that the model has 1 % fewer channels than the real drifts and caverns at SFR.

1 .0- 3 o

H_ 0.8- o

•;= 0.6 H o 03 1.6= v 0De d St - - 0) 0.4------Std Dev = 2.40 •*• _J8 - SFR data E 0.2- 3 Ü Q.O-\ I I \ \ 1 23450 6 -Log2(Relative flow rate)

Figure 5 Cumulative fraction of flow. Solid line shows values for SFR data, dashed lines show simulated results.

Table 3 Matc modef ho observed an l d flow rate gria t r SFRdsa fo wit, 0 h2 channels in each direction.

Cumulative numbe channelf ro s from moder lfo Observed channels 0.8 1.6 2.4 R aSF t

Flow rate category 1 11 2 1 2 1/2 53 8 6 6 1/4 119 27 17 18 1/8 194 57 34 59 1/16 258 90 56 97 1/32 351 163 103 164 0 400 400 400

137 SFR data

3 4 -Iog2 (Flow rate)

Figure6 Ratio of model to observed number of channels in SFR for different

standard deviations, cc.

modee presene th Th n i l t simulation channel0 40 outfloe d th t sha s R a wSF facee Th . thu 400/0.9s sha inflo4 40 9w= channels thin I . s channele figurth l eal includede sar . Thes channel4 e40 . Thifoune surfaca m sar s n 0 meando 00 e 4 are1 s f thaao t every 2 average n average o th 2 d m channe 5 an e, 3 lengtchannee s th ha l f ho l membere th s si square roo . Whethif o m t simulatione s9 nfigureth 5. = Z , s were done usin griga d with 40 channels in each direction, the ratio of the number of channels in the model to the observed numbe channel f totare o th ls 4.1 s numbeA .s wa channelf ro modee th n si l 600/4.1 1 inflo0 s 39 ha wr o , R channelsSF s 1600e i th , . This give densita s 1 f yo channel per 36 m2 and a length of 6.0 m. The results not are only influenced by the grid size.

same Th e metho appliee b observatione n th dca o d t Stripan si assuminy ,b g that every sheet has only one channel. The best agreement is obtained for

Channel widths have been measured in drifts and tunnels. Recently Palmqvist and Lindström (1991) analysed earlier observation Kymmee th n si n tunne found an l d that channele th f o 99.s% 7 hav widtea h smalle . Channer m tha 1 n0. l werwidthR SF e t sa also a few tens of centimetres, at most, with a few exceptions. A large number of them was point spots, found at fracture intersections and as small holes (Neretnieks 1987). In the 3D drift at Stripa, they also found that the water collection sheets collected more

138 water in areas where there were more fracture intersections (Abelin et al., 1991a,b). A recent experimental investigation that specifically measured channelin Stript ga a (Abelin t al.e , 1990) also found that channels were typicall centimeterw fe ten yw a f sfe o a o st "typicala centimeters a m c " 0 widt2 schannele r widehus fo e .W thi n si s paper.

Tabl e4 Summar resulte th SFRr f yo s fo , Strip Kymmed aan n

°c A Z(drift) m2 m

SFR 1.6 35 5.9 Stripa 2.4 2.7 1.6 Kymmen rock mass 2.4 105 10.3 Kymmen zones 2.4 21 4.5

Using borehole dat estimato at e channel lengths

In practice measuremene th , t limi packen i t r tests determines whe packena r sectios ni assume "non-conducting.e b o dt " Typically transmissivite th , mose th f ty o conductin g section roce th k n order6 i smas 4- f s magnitudi so s e more transmissive thae nth measurement limit. The measurements are usually available in the form of a number of packer intervals with measured transmissivities for the borehole tests. These measurements can be used to estimate the total number of conductive fractures (channels) intersecte borehole(s)e th y db . Thi s beesha n don datn eo a from Stripa (Geier et al., 1990). The information then needs to be translated into the number of channels of a given width W and length Z per rock volume. A method for doing this is developed below.

The rock contains a large number of channels randomly orientated, with an average averagn a lengtd an e h Z widttherf I severa. e ehar W sucf o l h channel voluma n si f eo rock, a borehole that is drilled in the rock may intersect some of them. In this analysis, we assume thaaperture th channele t th f eo vers i s y small compare othee th o rdt dimension channele th f so tha d width e channele san tth th f s o smal e sar relation li o nt the average distance between channels.

Conside boreholra e with diamete channea w rho D^ld musan locatee tb orden di o rt be in contact with the hole. Figure 7 shows a vertical borehole and a channel with different angles to the horizontal. The average distance, Wav, is obtained by integrating over all angles a

Jl/2 — W = c do 6 0 n si W (11} 7C/2J K V ' Jo

In the same way the average distance at which a channel will intersect the hole in the other direction, Zav, can be obtained.

139 r Z sin (12) •=-Tu/2H1 K Ja

Thu average sth e are whicn ai channeha l contacmuso t e holtli e tth s ei

(13)

average Ith f e distance between channels intersecte borehole th y dbeeb s eha n showo nt

, the roce H nth e k b volume containin channee gon H-As i l systea avn .I m wit cubiha c grid of channels, every cube with sides Z is delimited by 12 channels. Every channel is shared by four other cubes, so there are three channels per volume of size 7?. Thus

(14)

Substituting Equations (11 (12d )an ) into (14) gives

(15) knowe ar W n d froan m h independenDb , IfH t measurements, theobtaines i nZ d from equation (12). Whe » D^hn firsZ e ,th t ter equatiomn i neglectene s b (12 i n Z )ca d dan obtained from

(16) K 71

Average distance from borehole in which channels intersect hole W

Borehole

Figur e7 Vie f boreholwo e from above, indicatin t whaga t distance froe mth borehole a channel must lie, on average, in order to be in contact with a borehole.

140 Geier et al., (1990) found that in three horizontal north-south orientated boreholes, drilled nearl drifyD Stripa parallen 3- i t e standare th ,th o lt d deviatio transmissivitief no s logarithmia wa5 1. s c conductiv(base scal) th d e10 ean e fracture frequenc 0.5s 7ywa m~l. This fracture frequency corresponds to the total number of conductive fractures. In two horizontal east-west orientated boreholes, starting at the 3-D drift, the same values were found for the fracture density and the standard deviation. The mean transmissivity, however, was 10 times higher in the latter holes.

With a width W = 0.1 m and a borehole diameter of 0.076 m, Equation (13) above gives an average channel grid length Z of 1.2 m. This agrees reasonably well with the results obtained from the observations of flow rate distribution in the 3D drift, where Z . m wa6 1. s

Thus measurements from drift tunneld an s wels sa fros a l m borehole usee b o dn t sca estimate the channel lengths. To use borehole information, channel widths are needed. However vert resulte no th ,y e sensitivsar channee th o et l samwidthe th f e o thef si e yar magnitud borehole th s ea e diameter seee . b Thi nn frosca m Equatio takes i n (13)W n f I . to be equal to D^ (although it is much less in reality), Z would have been overestimated by 51 %. This is not a very large error compared to many other entities used for the simulation of flow in the network. However, W has a large effect on the interacting tracers, because it directly influences the flow-wetted surface. It must, for this reason determinee b , d accurately have .W e found vermeasurementsw yfe , beyond those mentioned, that coul usee db determino dt flow-wettee eth d surfac channer eo l width more precisely.

nexe Inth t subsection attempn a , sommade s i t us eo e t trace r least data o at chece th kf i value W ~ 0.1 m is at all reasonable.

SAMPLE CALCULATIONS FOR SOLUTE TRANSPORT

To illustrate the potential impact of matrix diffusion on the transport, some sample calculation presentede sar date chosee Th a.ar coveo nt widra e interval, from species that do not diffuse into the rock matrix to species that have a small sorption capacity. Simulations were done on a mesh of 20x20x20 nodes. The member length was ; thiassumem s 5 mean e b o dt travesa channeA . l m distanc 0 m l widt2 10 0. f eo f ho an channeda chosene ar Darce lm aperturTh .m y1 velocit0. f eo y throug mese hth s hi assumed to be 0.0001 m3/m2a. Therefore is the water mean residence time 2.4 years.

For the species that diffuse into the rock matrix, three values were chosen for De K^ pp. This product describes the diffusion and sorption properties of the rock. For species that are not sorbed within the matrix the term Kj pp is equal to the porosity. In the first simulation, small value e assume porosite ar seffective th th r d dfo y an e diffusion coefficient for the rock matrix (0.003 and 10"13 m2/s respectively). These values are similar to those found in crystalline rock. In the second simulation large values are used 10~d (0.0an 132 mlase 2th /s)t simulationn I . , transpor slightla f o t y sorbing species si calculated. Here, a value of 3.0 is assumed for the term IQ pp.

141 simulatione Th s were done wit standardha deviation f 1.6o , .c Figura , showe8 e sth breakthrough curves for these realizations. The transport of species that interact with the matrix is thus determined mainly by the diffusion and sorption properties of the rock, the flow distribution, and the flow-wetted surface area of the channels. The impac channee th f to l volum negligibls ei e sinc residence eth e tim interactinf eo g solutes in a channel is usually much longer than the water residence time, see Equation (4).

In an earlier paper, simulations were done for sorbing species using different volume- conductance relations (Moreno and Neretnieks, 1993). The breakthrough curves had the same shape irrespective of the volume-conductance relations used. This suggests tracerf o D s RT tha thae tth t have accesroce th ko st matri x wil determinee b l e th y db conductance distributio flow-wettee th d flos nan it wy b dt surfac no rocke d th f an ,e o porosity. For this reason the calculations were carried out with a constant aperture.

1.2-1

0.0

10 10 10 1 0 Time, years

Figure8 Breakthrough curve r solutesfo s that interact wit matrixe hth e Th . water residence years 4 tim figure2. e s ei Th . s attache curvee th o dt s 5 correspon valuee th (Df o dt so e IQ pp)°

So, to simulate the transport of species that diffuse into and out of the rock matrix, the following dat primarile aar y neede modele th n di :

conductance distribution ([4,, ac) channel lengths (L) channe) (W l widths

The conductance distribution, together with the average hydraulic gradient, gives the flow rate distribution in the channel network. The channel lengths and widths determine flow-wettee th d surfac channee th f eo l members. Thi sufficiens si simulatioe th r fo t f no

142 the migration over the distance of interest. Note that no information is needed on flow porosity or "dispersion". The data needed can be obtained from field observations and experiments.

DISCUSSIO CONCLUSIOND NAN S

modee Th l propose intentionalls di y made simpl thaeass o i es t incorporatti o y t e tracer transport and to accommodate interacting tracers. The model must be calibrated to field measurements. Borehole transmissivity data obtained by "narrow" packer intervals should principlen i , , suffic obtaio et effective nth e channel lengths, provided channel widths can be determined independently. The comparisons and calibrations with field observation rathee sar r straightforwar havd dan e shown tha modee tth exhibin lca e tth propertie channelinf so g observe fielde th n .di Tracer residence time distributiond san patterns also agree. bees ha nt I shown thaflow-wettee th t d surface f primareo s ai e importance th r efo migratio f soluteno s that diffuse int roce oth k matrix .enougt Therno e ehar detailen di situ observations that can be used to assess the flow-wetted surface area. Special techniques and tests must be developed and applied to obtain more accurate information.

NOTATION

A Area m^ a Wetted surface nvVm^ C Conductance m^s/kg c Concentration mol/m^

De Effective diffusion coefficient m^/s Hole diameter m Sorption coefficient m^/kg m L Length P Pressure kg/(m-s2) Pe Peclet number R Retardation factor Q Water flow rate m^/s m W Channel width m r Radial distance s w t Water residence time Z Channel length m a Angle 0 Aperture m e Porosity

|Lio Mean logarith mconductancf o e pp Densit bulf yo k rock kg/m^ a Standard deviatio lognormae th n i l distributiof no conductances O Standard deviation in the residence time distribution s

143 REFERENCES

Abelin . NeretnieksI , H. , . TimbrantS , . MorenoL d an , , Final Repor Migratioe th f o t n in a Single Fracture - Experimental Results and Evaluation, Stripa Project Technical Report 85-03, OECD/NEA, SKB 1985.

Abelin, H., L. Birgersson, H. Widen, T. Âgren, L. Moreno, and I. Neretnieks, Channeling experiment to study flow and transport in natural fractures, Stripa Project Technical Report 90-13, OECD/NEA, SKB 1990.

Abelin . BirgerssonL , ,H. . GidlundJ , . NeretnieksI d an , large-scalA , e flotraced wan r experiment in granite, 1. Experimental design and flow distribution, Water Resour. Res., 27, 3107-3117, 1991a.

Abelin . BirgerssonL , H. , . MorenoL , . WidenH , . Âgren. NeretnieksT I , d an , A , large-scale flow and tracer experiment in granite, 1. Results and interpretation, Water Resour. Res., 3119-313527 , , 1991b.

Cacas, M.C.e Marsilyd . G ,. TillieB , . BarbreauA , . DurandE , . FeugaB , d an , P. Peaudecerf, Modelling fracture flow with a stochastic discrete fracture network: calibration and validation - 1 The flow model, Water Resour. Res., 26, 479-489, 1990a.

Cacas, M.C., E. Ledoux, G. de Marsily, A. Barbreau, P. Calmels, B. Gaillard, and . MargrittaR , Modelling fracture flow wit hstochastia c discrete fracture network: calibration and validation - 2 The transport model, Water Resour. Res., 26, 491-500, 1990b.

Carslaw, H. S. and J. C. Jaeger, Conduction of Heat in Solids. Oxford University Press, 1959.

Dverstorp, B., Analyzing flow and transport in fractured rock using the discrete fracture network concept. Ph.D. Thesis, Royal Institute of Technology, Dept. of Hydraulic Engineering TRITA-VBI-151 Stockholm 1991.

Dverstorp, B. and J. Andersson, Application of the discrete fracture network concept with field data: Possibilities of model calibration and validation, Water Resour. Res. 25(3), 540-550, 1989. Geier, J., W. Dershowitz, and G. Sharp, Prediction of inflow into the D-holes in the Stripa mine. Stripa Project Technical Report 90-06. OECD/NEA, SKB April 1990.

Levenspiel, O., Chemical Reaction Engineering, 2nd ed., p 275. John Wiley and Sons, New York, 1972.

Long, J.C.S., H.K. Endo, K. Karasaki, L. Pyrak, P. MacLean, and P.A. Witherspoon, Hydrological Behavio Fracturf o r e Networks. Hydrogeolog Rockf yo s Permeabilityw oLo f ConferencH IA , 7-12n Ja e , 1985, Tucson Arizona. Volume XVII 44-468p , , 1985.

144 Moreno , Y.WL. , . Tsang, C.F. Tsang, F.V .. Neretnieks HaleI d an , , Flo traced wan r transpor a singl n i t e fracture stochastiA . c s relatiomodeit d somo an lnt e field observations, Water Resour. Res. , 2033-304824 , , 1988.

Moreno , C.FL. , . Tsang, Y.W. Tsang . NeretnieksI d an , , Some anomalous featuref so flow and solute transport arising from fracture aperture variability, Water Resour. Res., 26, 2377-2391, 1990.

Moreno, L. and I. Neretnieks, Fluid flow and solute transport in a network of channels, Journa Contaminanf o l t hydrology, accepte publicationr dfo , 1993

Neretnieks, L, Diffusion in the Rock Matrix: An Important Factor in Radionuclide Retardation? J. Geophys. Res. 85, 4379-4397, 1980.

Neretnieks , ChannelinL , g effect flon transpord i s wan fracturen i t d rock sSom- e recent observation modelsd san , Paper presente GEOVAt da L symposium, Stockholm, Proceedings, 315-335, 1987.

Neretnieks, L, H. Abelin, and L. Birgersson, Some recent observations of channeling in fractured rocks. Its potential impact on radionuclide migration, In DOE/AECL conference Sept 15-17, 1987, San Francisco, Proceedings p 387-410, 1987.

Palmqvist, K. and R. Stanfors, The Kymmen power station TBM tunnel. Hydrogeological mapping and Analysis, SKB Technical Report TR 87-26, 1987.

Palmqvist. LindströmM d an . K , , Channel widths TechnicaB SK , l Repor 91-14R T t , 1991.

Robinson, P.C., Connectivity, flow and transport in network models of fractured media. ThesisD . . Catherine'Ph , St , s College, Oxford University 1072P T y f Ma ,Re , 1984.

Yamashita, R. and H. Kimura, Particle-tracking technique for nuclide decay chain transport in fractured porous media, Journal of Nuclear Science and Technology, 27, 1041-1049, 1990.

KBS/SKB - Technical Reports can be obtained from: IMS CLEARING HOUSE, International Atomic Energy Agency, P.O. Box 100, A-1400 VIENNA, AUSTRIA.

Next page(s) left blank 5 14 SYNTHESIS OF GEOCHEMICAL, ISOTOPIC AND GROUNDWATER MODELLING ANALYSIS TO EXPLAIN REGIONAL FLOW IN A COASTAL AQUIFER OF SOUTHERN OAHU, HAWAII

C.I. VOSS, W.W. WOOD United States Geological Survey, Reston, Virginia, United States of America

Abstract

Regional geohydrologic characterizatio floe w th coastae f nfielo th dn i l stratified basalt aquifer of the Pearl Harbor area in southern Oahu, Hawaii, is provided by a synthesis of geochemical and isotopic information, collected in vertical profiles, and numerical simulation of the variable-density ground- water flowuppermose Th . thickm t 5 wate, 12 consist o rt layerm f wateso 5 7 , r recharged from local rainfall and irrigation over the past few decades. Below this is the core of the freshwater lens, 100 m to 150 m thick, containing waters with an apparent carbon-14 age of 1800 years. The freshwater lens floats on a third saltwater layer that likely extends to the bottom of the aquifer. The apparent carbon-14 age of the saltwater is between 6000 years and 9000 years, and the difference in ages between an inland well an coastada l well suggests that before aquifer development bega 1880se th nn i , saltwater flowed inland at a velocity of about 2 m/yr. Assuming a constant lateral velocity in the saltwater body, the saltwater fro m observatioe rechargk m th 4 2 o t e m thuareny k estimate e wells1 aresn ab ma 1 a ae n i b , o dt locate offshorm southerk e 4 th d 1 f e o betweed nan Oahm k n1 u coast. Post-bomb carbon-1 tritiud 4an m data indicat eresidenca e tim watef eo r withi freshwatee nth rmor o lenn f es o ten thaw f yearsfe so n a . If total recharge estimates assumed here are correct, the 1800-year apparent age of this water can only be explained by long residence time in recharge area compartments the central plateau of Oahu and the dike zone in the Koolau Mountains before entering the lens, although the storage volume is not sufficient accoun o t entire th r e fo tdelay additionn I . , greae somth f teo apparene tb freshwatey ma e ag r accounte componena y b r organid dfo ol f o t c carbon introduce ground-watee th dn i r recharge area. The simple geochemical reaction thin si s aquifer system allow clear interpretatio f wateno r chemistr isotopid yan c chemicadatae Th . stabld an l e isotope compositio l sampleal f no wells si - explained by simple mixing of the three types of water. The major reaction in the saltwater is sodium- calcium exchange, which increases calcium concentration over that of seawater, but which has negligible effect on carbon-14 dating. There is no evidence of calcite dissolution affecting the freshwater. Extrapolated aquifer saltwater carbon-13 conten somewhas i t t lighter than oceanic values, probably indicativ f oxidatioeo f organino c carbon. Assuming this organic carbo devois nwa f carbon-14do , measured carbon-14 value r saltwatesfo r mus adjustee b t d upwar t mosa y factoda b t f 1.24o r . Thus, carbon-14 values wit withour ho t adjustment give similar absolute age traved san l time watef so r flow between measurement points. Unfortunately, a number of samples collected show a high degree of scatter suggestin possibilite gth f contaminatioyo modery nb n carbon-14 during handlin r analysisgo , reducin quantite gth verticaf yo date aag l available from profiling. The characterization of both the freshwater and saltwater flow field obtained from geochemical analysi consistens i t with regional hydrologie behavio Peare th f lo r Harbor area representee th y db cross-sectional variable-density flosolutd wan e transport mode Vosf o Souzld san a (1993). This model was based only on analysis of hydraulic data and vertical salinity profiles through the freshwater lens and saltwater transition zone. Thus simply-structuree th , d numerical model, which include estimatn da f eo regional effective porosity for flow (0.04) that could not be verified on the basis of data existing at that time, can explain all currently-available geochemical and hydraulic field information. This may be the first analysis in which it was possible to corroborate ground-water model-calculated density-driven saltwater velocities in a coastal aquifer with flow velocity determined from chemical and isotopic field data.

1. INTRODUCTION

Hydrologie, geologic, geochemica othed an l r informatio synthesizee b n nca do t obtain a satisfactory explanation of the occurrence and large-scale behavior of

147 subsurface fluid complen i s x hydro-geologic environments mose Th t. satisfactory explanatio simplese th n s thai e ton simultaneously describe typel sal availablf so e data. Solute geochemistry constrains the possible interpretations of the hydrogeology of a syste addd man s direct measuremen f floo t w velocit identificatiod yan source th f no e of water often unavailabl hydrologien i e investigations. Most regional ground-water flow system three-dimensionale sar , exhibitint gno only areal variability t verticabu , l variabilit challengina s i s well t a yI . g taso t k satisfactorily explai verticae nth l variation sucn si h systems; most wells provide data ove limitea r d depth cosintervale f th drillin o t d an ,welga l make definitioe sth f no vertical variability even more sparse tha f areano l variability. Vertical variatione sar important because ground-water system highle sar y stratified; recharge from highera - elevation area generall founys i greatet da r depth than locally-recharged water. Thus, vertical variations contain information on recharge areas and flux of water from these areas. In coastal aquifer systems, vertical variations are generally better characterized because of the threat of seawater intrusion from below. In Hawaii, coastal aquifers have been characterized in a vertical sense primarily in terms of salinity-depth profiles in deep observation wells. This information, together with hydraulic information collected since aquifer development bega 1880n ni s bee,ha n use numerican di l simulations by Souza and Voss (1987) and Voss and Souza (1993), to explain the vertical distribution of regional groundwater flow in the Pearl Harbor area of the southern Oahu aquifer, near Honolulu. For these studies, vertical profiles of isotopic othed an r geochemical field data yielding informatio fluin no d velocitie sourcd san e area unavailables spossiblt wa no s thud directlewa o t ,san t i y test these aspecte th f so model. However, the areal distribution of stable and radioactive environmental isotopes as well as major ion chemistry in southern Oahu aquifers was well-characterized in a comprehensive series of studies by Hufen and others (1972), Hufen (1974), Hufen and others (1974a), Hufe otherd nan s (1974b) Hufed otherd an , nan s (1980) resulte Th . s demonstrated significant difference sourcn si e area aged sfreshwatef an s o r pumped from various aquifer southern i s n Oahu. Hydrologie explanatio findinge th f no s concerned mainl areae yth l difference isotopesn si moss a , t datcollectes awa y db pumping water from existing wells. Three generic layers of water in the aquifers were postulated to explain some of the areally-measured variations: water recharged from irrigation, (apparen tenf o years)f se o tag , freshwater lens core (age ranging from tens of years to hundreds of years), and intruded seawater (age as great as ten-thousand years). In the immediate vicinity of the Waipahu monitor well where one profile was collected for the present work, Hufen and others (1980) found carbon-14 concentrations from 87 pmC to more than 100 pmC in samples collected from water pumped from active supply wells. It will be seen in this paper that the vertical profile collecte r thidfo s Waipahe studth t ya u well showe relativelda y smooth carbon-14 variation frouppermose mth n i mor C C epm pm ttha 0 water9 0 s n10 a valueo ,w t lo s sa in the core of the freshwater lens, to 30 pmC in the saltwater. This illustrates the difficulty of hydrologie interpretation of isotope data collected as single-depth samples at or near active wells. On the other hand, it will be seen that our analysis confirms muc f Hufen'ho s general vie f verticawo l structure. Further, becaus r analysieou s i based on continuous vertical geochemical and isotopic profiles, it refines the proposed structur allowd ean s explanatio hydrogeologie th f no c origiverticae th f no l structureo t be propose quantitativa n do e basis.

148 In the present study, vertical profiles of geochemical and isotopic information extending through the freshwater lens into saltwater, were collected in two observation wells located approximately alon floga areae w th linn l e (i planePear e th n li ) Harbor area. Analysi f thesso e datcorroboratiod aan n witexistine hth g vertical hydraulic systemodee th f o ml (Souz Vossd aan , 1987; VosSouzad an s , 1993) allowsa quantitative explanation of water occurrence and flow at the regional scale that satisfies all data typeproposede b o st .

2. HYDROGEOLOGY

The major population cente volcanie th f ro c Hawaiian island chain locatee th dn i north Pacific Ocea cite f Honolulislanye welth o s n a th s a lf Oahn do o us i u (FIGURE 1). Oah buils lavay ub twa s extruded fro shielo mtw d volcanoe theid san r associated rift zones (FIGURE 2), the Waianae on the west (approximately 3 million years old) and the younger Koolau on the east (approximately 2 million years old). See MacDonald and others (1983) for a complete discussion of Oahu geology. The rift zones are characterized by a concentration of sub-vertical dikes through which basalts were extrude surfacee islanth e bult e dth a f kdTh o . mas composes i f dike-fredo e basalts forme sub-aeriay db l lava flows which mak gently-dippina p eu g stratified aquifer fabric extending from land surface to considerable depth. These layers rest upon a base of submarine extruded pillow basalts, which, because of subsidence of the island, are depta t beloa f m ovehk o levea w w2 se r lno (Andrew d Bainbridgean s , 1972). Although some interfingering of Waianae and Koolau basalts is thought to have occurred, the bulk of Koolau basalts rest upon the weathered flank of the older Waianae volcano. While the eastern side of the Koolau volcano and both sides of the Waianae volcan severele oar y eroded Koolae ,th u basalt centraflowe th n si l saddlf eo the island have not been eroded extensively. Much of the coastal portions of the island are covered by a sedimentary wedge (referred to as 'caprock') composed of layer f marinso terrestriad ean l sediment clayd swelan s scoraa s a l l organireed an f c debris largese Th . t caproc soutm k f Oah0 ho 4 k o uextendt (FIGURm k 0 s3 . E3)

160° 159° 158° 157° 156° 155'

22° Kauaf Niihau ^Apahu Molokai

LanaiJ-1 ^ iMaui Kahoolawe 20'

FigureJ, - Map showing location and relative positions of Hawaiian islands (after

Hunothersd an t , 1988).

149 The layered basalts are highly transmissive to ground water with hydraulic conductivity m/dtypicall0 ordee 50 th .f o n rThe yo als e yar o highly vesiculad an r porou 50%o t t connectep )bu s(u d porosity through which significant water flon wca occu less i r s than 10% largese .Th t lav tenae b flowf kilometer so n sca s longo t p u , 1 km wide and up to 10 m thick. Current examples of such flows may be found on the volcanically-active island of Hawaii (e.g. see Lockwood and Lipman, 1987, and Holcomb, 1987) well-connectee Th . d highly conductive zones occu rubbln ri y material between overlapping lava flows, while the body of the lava flows themselves are relatively impermeable stace 100o t Th f tabula0 k.o 0 10 time e rb s unit y lessma s conductive vertically than horizontally and the caprock is 1000 to 10000 times less conductive than the layered basalts (Souza and Voss, 1987). A number of geologic barriers control the regional flow of ground water on Oahu (FIGURE 2). The rift zones contain swarms of intersecting vertical dikes cutting across layered basalts which hav permeabilitw elo whicd yan h impound wate numeroun i r s compartments (Takasak Minkd an i , 1985; Hunothersd an t , 1988). Sediment-filled valleys scoured intlayeree oth d basalt barrierss a als t o ac flo so t w throug layeree hth d basalts does a ,contace sth t zone between Waiana Koolad ean u basalts, whereie nth weathered surface Waianath f o e e basalts likely have lower conductivity than unweathered layered basalts. The caprock acts to impede ground-water discharge nea coaste rth , raising hydraulic headlayeree th sn i d basalt aquifers. Except beneath the caprock, aquifers are unconfined. Two other barriers to flow are inferred from precipitous drop groundwatesn i r levels (discusse followinge th dn i ) belo centrae wth l saddl f Oaheo u (Dal Takasakid ean t bee, no 1976) n s dateo determineT ha .t i , d whether these barrier dikese sar , weathered ridgeWaianae th f so e volcano somr o , e other low-permeability structures.

OAHU 4 MILES J 5 KILOMETERS

Figure 2 - Map showing Hydrogeologie barriers on Oahu (after Hunt and others, 1988). Map also shows location of measurement points (A: Waipio monitor well, B: Waipahu monitor well, C,D,E,F: Soil-gas samples, G: Waihee high-level water and soil-gas samples.)

150 Oahu S*\

B B'

co 600 - QC LU 300 - K 1- IU 0- 2 -300 - ^•^^CAPRTCtfX^ OCEAN ' z -600- -900 - aASAtT AQUIFER -1200 - -1500 -

.EVATION , -1800 LU ! l l i i i C) 5 10 15 20 25 30 3

DISTANCE KILOMETERN I . S

Figure 3 - Generalized geologic cross-section of southern Oahu showing offshore exten f caproco t k (modified from Gregory, 1980).

Ground-water recharg concentrates ei hige th h n di rainfal l areas alone gth eroded volcanic ridges (FIGUR arean I . sE 4) wher e rainfal mors i l e tha m/yr3 n1. , roughl rainfalye recharghalth y f o f ma l aquifere eth s (Dal Takasakid ean , 1976)e Th . high-recharge areas are thus coincident with the rift zones and central saddle area of Oahu resultine Th . g ground-water levels display sharp elevation change t geologisa c barriers Waiana(FIGURe th n I . eE 5) rift zon highese eth t measured water0 leve49 s i l m, in the Koolau rift zone the highest level is 300 m, the central saddle area has a water level of 85 m, while between the coast and inland geologic barriers, water levels rarely exceed 10m. Within compartments formed by barriers, water levels are nearly eacd flaan th compartmen 'isopiestireferren s i ta s a do t c area'. Thus, various water bodie barrierd san s have been delineate Oahn do u largelbasie th watef sn o y o r level (FIGUR. E6) The geohydrologic relations of compartmentalized and basal water bodies are show cross-section i FIGURn coastai e Th . El7 water bodies with low-elevation water table callee sar d 'basal-water bodies' basal-wateA . r bod freshwatea s yi r lens that freely floats on intruded seawater (FIGURE 8), with an intervening zone of transition containing a mixture of freshwater and saltwater. The Honolulu water bodies are separated from each other and from the Pearl Harbor body, the largest continuous water bod Oahun yo valley-fily b , l barriers resultin partian gi l compartmentalization (FIGURE 2, FIGURE 5, FIGURE 6). The compartmentalized waters are the Koolau and Waianae dike-impounded water bodies in the rift zones, and the Schofield high-level water body in the central saddle. Because of the high water elevation in the rift zones and Schofield water body, saltwater would be expected only at depths of several kilometers (Dale and Takasaki, 1976); and thus, these compartments must contain freshwater to the aquifer bottom. This study focusses on relations in the Pearl Harbor basal-water body.

151 isa'is isyoo

OAHU

*>e » i i - o i ) t • i i

EXPLANATION

Fàtnfal Z n excesi l l t o s n'etres Der year; Ai ^a ot potential deep infiltration

L mes are in metres , interva s variabli l e

158*15 E ISLANTH RAINFALF OAHUR O D FO P , 1931-6MA L 0 Figure 4 - Rainfall map for Oahu (after Dale and Takasaki, 1976). N EXPLANATION

OAHU f — Wate g _ r level contour meterm , s aDove mea a levese n l 0 4 MILES (contour interval variable)

0 5 KILOMETERS 85 Point observalion of water level, in melers above mean sea level Representative water level where waler o tablto s i e Hat lo be contoured, meterm s above mean a levese l Generalized directio l groundno - water (tow

Figur Wate- 5 e r level principan i s l Oahu aquifer d inferrean s d freshwater flow directions (after Hunt and others, 1988).

GEOHYDROLOGIC MAP OF THE ISLAND OF OAHU Figur Majo- e6 r water bodie Oahn so u defined from water levels (after Dald ean Takasaki, 1976).

153 EXPLANATION Lin f sectioo e n shown i n figure 6 » 0 l 3 3 4 Ml L l s

> 0 I î ] 4 J K l l OM t T • I $

FEET METRES 3000-, A r—1000

AIAHOIE TBANSMISSION TUNNEL

COUNTY//X/;WAIANAD CITAN Y E 400- TUNNtl __^_ e'-K 1I' M'P6 ÉO N' Ö ü

800 VERTICAL EXAGGERATIO N13.X 2 Line of section shown in figure 6 r-400

-200

1200 VERTICAL EXAGGERATIO 13.NX 2

Figore 7 - Schematic cross-sections of ground-water bodies on Oahu (after Dale and Takasaki, 1976 Figure r locationse ;fo 6 e f sections)o s .

FRESHWATER BETWEEN DIKES

Figur Typica- e8 l cross-sectio basaf no dike-impounded an l d water bodies showing occurrence and movement of ground water (modified from Souza and Voss, 1987).

154 Natural recharge to the Pearl Harbor area aquifer occurs mainly in the upper- elevation areas as a result of direct infiltration, discharge from the Schofield high-level

water body dischargd an , e fro Koolae mth u dike-impounded water body (FIGUR. E5)

3 Total recharg Peare th o elt Harbor are5 mabouas 10 i / dx 9 (24t 0 Mgal/d) (Mink, 1980). Natural discharge occurs alon ling a springf e o inlan e th t sda boundare th f yo caprock near Pearl Harbor, and likely as diffuse leakage through the caprock to Pearl Harbor. Saltwater discharge likely occurs diffusely through the caprock (FIGURE 8). Pumping fro aquifee mth r begaearle th yn i 1880'purposee th r sfo f sugaso r cane irrigation. Rates of withdrawal increased continuously from the early 1900's to 1980 with resultant reductions in head and upward and landward movement of the transition zone between freshwate saltwaterd ran . Most well publir s fo irrigatio d can n supple yar located in a band near the coast (FIGURE 9). Some wells were abandoned as a result of increasing salinity. Pre-development heads in the Pearl Harbor aquifer measured about 10 m near the spring discharge area, and are estimated to have been over 12 nea m upstreae th r m boundary pre-developmene Th . t freshwate rPeare lenth n lsi

HAWAII OAHU

STUDY AREA'

21'45

^ OAHU v^

10 KILOMETERS PACIFIC

10 MILES OCEAN I 158'IS 158- 157'45

Figure 9 - Map showing Pearl Harbor area aquifer with Hydrogeologie boundaries, hydraulic heads from 1958 (after Mink, 1980), location of monitor wells (A: Waipio Waipahu): ,B locatiod ,an coastaf no l ban pumpinf o d g (modified from Souza and Voss, 1987).

155 thicm k0 Harbo (Mink50 o t r ,m are thu1980s 0 aswa 40 ) assuming condition statif so c equilibrium with seawater 1990n I . , freshwater headaquifee th sn i r ranged from about 5 m to 7 m above sea level, and measured salinity profiles showed the freshwater lens thicm k0 30 (Vos Souzao d t san m onle 0 ,b y20 1993)o t .

3. SAMPLE COLLECTION

Vertical profile f solutso isotopid ean c chemistr aquifee th n yi r located within Koolau basalts were obtaineo observatiotw r fo d naparm k well 5 t 4. slocate d approximately along a regional flow line (Well A and Well B, FIGURE 2, FIGURE 9). belom levela 0 wse 37 . s i ) W " 41 ' 59 7 15 wele , Th N l " botto16 ' t mWaipi26 a « 1 o(2 The well bottom at Waipahu (21 23' 40" N, 158 01' 20" W) is 296 m below sea level. Samples were collected from open uncased holes. The well water is representative of the aquifer water at each sample depth because there is little comminglin f watego r from ove r underlyino r g samplzoney an t sa e welle pointh .n i t This condition is clearly satisfied in the large scale view as shown by temperature in FIGURE 10 (Voss, unpublished data extracted from continuous log) and chloride in FIGURE 11 (this study) which change in a regular manner. Vertical flow measurements in the wells (Voss, unpublished data) show that water enters and leaves the boreholes through conductive zones approximately every 20 m. This may be seen in FIGURE 10, where temperature changes abruptly at a number of points, located where outflow occurs fro wele mth l bore. Point f infloso locatee war d along smooth sectione th f so profile, between the sharp shifts. Thus, there is a local disturbance by vertical borehole flow on the order of about ±20 m. On the scale of the entire profile, however, the resulting data adequately describe verticae sth l distributio f fluino d composition. Samples were collected sequentially from the top down in the uncased wells by a 4.5-liter down-hole motorized sampler attached to the cable of a borehole logging system. The sampler had a valve at the top that could be opened and closed from the surface. Samples were collecte 1992n e i 199 n di everd Th m ever .0m an 5 y2 0 y5 chamber of the sampler was filled with nitrogen gas prior to each descent in the hole.

0 50 100

150 I 200 j 250 e 300 350

400 0 19.20.2 5 1 521.2 2 522.2 3 523.2 5 T«smp«ratureC ° n i ,

Figur - Dept 0 1 e h profil f fluio e d temperature, Waipio monitor well, 1988.

156 u 0 "" A 50 -O A • I O A --••* ., Titltmln nhn* 100 - 3D A - C A 150 - O /v\ - 0 A Sm _ 1 200 OO A *A ^0 Vater 250 - ^X^

Figur Dept- 1 1 eh profile f chloridso e concentratio grounn i n d water, Waipid an o Waipahu monitor wells, based on two rounds of sample collection, 1990 and 1992, also showing seawater chloride concentration.

Motion in the hole was less than 7 m/min, and less than 2 m/min in the vicinity of the sampling depth. After lowering the closed nitrogen-filled sampler to the desired depth, it was opened for ten minutes and occasionally shaken or closed and re-opened to release trapped bubbles fro samplee mth r chamber water-fillee Th . d chambes wa r then closed and brought to the surface. The top sampler valve was then opened and the water was drained downwards by a pump through a manual valve and fitting at the sampler bottom to a nitrogen-filled 3.9 liter glass container fitted with a two hole rubber stoppe r carbofo r n isotope analysis e inle Th rubbee t .th sid f eo r stoppes wa r diamete connectelengtm c m c 1 0 f h2 o r a polyvinydo t l chloride tubing that connected to the motorized sampler (pump) and was closed by a pinch clamp. The discharge inlee th tf tubo place s bottod e esamplwa e th en t dth a mf o e bottl alloeo t w gentle filling thud an s minimizing potential degassing outlee Th . trubbee holth f eo r stoppes wa r fitted with a20cmx'\ cm glass tube filled with granular CO2-sorbing material (Askerite) and connected to 2 cm of polyvinyl chloride tubing and a pinch clamp at the end. This arrangement was designed to prevent any atmospheric CO2 from entering the sample bottle durin filline gth g process connecteinles e twa Th . samplere th do t inle e d ,th an t outlet pinch clamps were remove sample th d edan bottles were completely fillede Th . stopper assembly was removed and the bottle quickly capped with a conical-shaped "poly-seal" cap. Additionally, the following samples were collected: a one-liter sample in a plastic bottle for tritium, two 250 ml (milliliter) filtered samples (.45 micron) in plastic bottle standarr sfo d inorganic chemical analysis (one sampl acidified)s 0 6 e wa a d an , ml sampl glasa en i s bottl deuteriur efo oxygen-18d man cape Th s. were taped, then bottles were wrapped with bubble wrap, storecheste ic shippe d n di s an d overnight express to our home office in Reston, Virginia. Specific conductance, temperature, pH and alkalinity determinations were madtime collectiof th e t o e a remainine th n no 0 g10 mf sampleo l 1992n I . , sample r organisfo c carbon analyses were removed froe mth glass bottles.

157 Carbonat carbon-1r efo 4 analyse precipitates swa d unde nitrogera n atmosphere as SrCO precipitatine Th 3. g solutio f Sr(OH)no s preparewa 2 y dissolvindb 2 g2.

kilograms of SrCI in 5 liters of NHOH. After overnight settling, any precipitate was 4

removed by filtratio2 n in a nitrogen glove box using 0.45 micrometer filter. One hundred ml of the clear stock solution was added to 3 liters of the sample. After overnight settling, the solution was filtered through a 0.45 micrometer filter designed for high pH solutions, washed with one liter of carbonate-free deionized water, dried, placed into plastic Petri dishe shipped an s d overnight expresd e laboratoran th C o st 13 r fo y carbon-14 determination. Methods of inorganic analyses are those given in Skougstad and others (1989).

4. GEOCHEMISTRY

Based upo solute nth isotopid ean c analyse collectee th f so d samples (TABLE 1), the vertical profiles of the aquifer can be divided into three zones: an uppermost zon irrigation-returf eo n water middl,a e zon pristinf eo e freshwater lowea d ran , mixing zone grading from freshwater to seawater. The relative position and thickness of each inferree b zon n t Waipiedca a o (FIGUR whicn i ) groune Eh10 th d water temperature profile illustrates the presence of warmer irrigation return water (0 to 100 m below sea saltwatee th f o p r to mixin freshwatee e th Th f g o . zonp m) level to r0 e n (1020 o ) 0o t occurs at approximately 200 m and extends to the bottom of the well. This analysis is consistent with the chloride concentration in Waipio (FIGURE 11). The higher concentrations of chloride, nitrate, dissolved solids, alkalinity, isotopically-heavier deuterium and oxygen-18, and some tritium activity in the water Waipit a abovm o0 (TABLe10 consistene ar ) E1 t wit presence hth f retureo n irrigation water. Local ground water is pumped for irrigation and reapplied to the ground surface. When freshwate withdraws i r r irrigationnfo , evaporation increasee th s concentration of solutes and dissolves fertilizers and any soil conditioners applied to watee fieldth e warmeds d rth i san . Thus, irrigation-return distincwatea s ha r t isotopic signature from interaction wit atmosphere hth biospherd ean d an O e18 makin, 2H e gth 513C isotopically heavier and increasing tritium and carbon-14 activity. At Waipahu, the irrigation return wate completels ha r y maske pristine dth e water zonreturd ean n irrigation water merges directly wit saltwatee hth r mixing zone. solutee transitioe Th th sn i n zone between freshwate seawated ran r result largely from a simple conservative mixing of seawater and water from the freshwater lens. This is clearly indicated in FIGURE 12 which illustrates the strong correlation between chloride and 618O (VSMOW). The scatter in the upper portion of this diagram is the result of return-irrigation water which has experienced some evaporation. In additio simplo nt e mixing exchangn io , e occursaltwatee th n i s r moving beneat e islanhth d (Mink, 1961; Vishe Minkd an r , 1964), altering solute ratios somewhat from that expected according to simple mixing. FIGURE 13, based on data from Waipahu, illustrates that lesse sodiuma o t rd degree,an , potassium seawaten i , r has exchanged with calciu magnesiud man m fro caproce mth k resultin increasn a gn i e in calcium and magnesium and a decrease of sodium and potassium in solution. The dat FIGURn ai plottee ar 3 Ed1 relativzere th o o et lin e generate conservativy db e mixing of the freshwater and seawater end members. To carry out age-interpretation of carbon-14 data, it is necessary that potential sources of dissolved carbon be considered. In the freshwater, possible dissolved carbon source Peare th o lst Harbor area aquifer are: plant respiration (soil gas), dissolved organic carbon from decaying vegetation, and calcite dissolution. Solutes

158 TABLE 1

Sample ID Depth ; Date Ca Mg i Na Kr S , LI Alk *» CI SO4 Br ! NOS H4SIO4 pH mBSL m/d/y ' C«CO3 i PIO-410 0 ' 5/27/99 22 i 9. 9 98. ' 1.9 '0.07! 0.01! 58 34 ; 8 8 1 1 0 1 6.2 ' 105 '7.36 PIO-49S 25 5/27/96 29 i i 8.9 29 9 1. i 10.07' 0.01 : 59 32 7.6 0.16 1 3.5 5 7.310 7 PIO-577 1 50 4/17/90s i • 7. 87 6. : 1.3 0.02 59 20 4.0 71 0 : 4.3 1~ 90 7.92 PIO-577 ; 50 5/27/92 7.2 I 6.8 19 1.5 ,0.04 0.01 5 4 : ! 20 5. 0 i 0.12 ! 2.4 95 ,7.56 PIO-655 97 , 5/27/92 7.0 i 6.6 16 9 1. i '0.04 0.01 45 1——20 i 5.0 ! 0.12 3. 0 i 95 7.51 PIO-741 100 4/17/90 7.0 • 6.2 i 162 1. i 0 50 : 1 9 1 4. ! ! 0.09 3.0 88 !?.S3 PIO-741 ' 100 5/28/92 7 1 5.2 9 4. 1.5 0.03 0.01 32 21 3.3 1 0.08 <0.01 70 J7.85 PIO-825 312 i 5/28/92 5.3 I 4.8 17 1.5 I0.04I 0.03 3 1! ; 21 1 3.3 0.09 <0.01 70 7.97 PIO-005 i 150 4/17/90 6.0 i 5.1 i 16 : 1.20 l i 43 9 1 2.9 0.07 0.27 74 17.85 PIO-005 150 5/28/92 5.1 4.7 17 JO. 03 0.01 33 20 3.2 i 0.10 <0.01 70 .7.91 PIO-985 717 | 5/28/95 23 | 7. 3 9 7. i 4 2. ! 0.70I 0.02 32 58 1 8.0 I 0.27 <0.01 5 6 17.8| 9 PIO-106» 200 4/17/90 7.0 1 6.1 26 1.9 0 37 43 ' 5.4 0.15 <0.01 66 ;8. 15 PIO-1069 200 5/28/96 23 8. 2 4 7. ! 2.4 I0.70I 0.02 33 60 0 8. 1 0.27 <0.01 i 60 !7.80 PIO-1151 225 5/28/92 42 I 49 335 | 15 10.50 0.14 34 650 ' 88 3.3 <0.01 5 5 7.9i 9 PIO-1233 250 4/17/97 09 ! '1120 64 : 18 iO.70 ! 42 ; 1310 ' 176 4.7 <0.01 , 61 (7.88 PIO-1233 I 250 5/28/92 95 , 115 ! 670 , 23 1.1 0.31 40 1420 200 6.9 l<0.01 0 '7.86 | 6 PIO-1315 I 275 5/28/92 465 580 2530 110J 5.0 1.7 53 5930 790 28 <0.01 60 '7.59 PIO-1397 | 300 4/17/90 056 ; 900 ' 5530 140 6.0 1 8 ' 11100 1290 38 !<0.01| 70 I7.45 PIO-1397 300 5/29/92 600 , 885 57001195 9.0 2.2 ; 74 11840 1630 54 <0.01 5 6 17.6l 1 PIO-1479 325 5/29/92 810 •'1065 74101220 9.0 2.5 80 14380 2000 62 (<0.01 65 17.52 PIO-1360 135 I 5/29/92 ï~675 111601 8100 240 10 2.4 ! 87 15800 2010 64 i<0.01 65 '7.33 PIO-1627 370 4/17/90 720 11200! 8100 21 1 7.2 97 16000 1820 54 <0.01| 45 ,7.50 PIO-1827 ! 370 5/29/92 670 11190! 8710!260 10 2.5 94 16850 2150 70 <0.01 65 7.39 i ! ' i AHU-28 ' 0 5/30/92 1 9 17 75 3.9|0.15| 0.090 i 52 158 i 22 0.82 <0.1 14 8.22 AHU-110 25 . 5/30/92 24 21 67 3.3 0.16 0.090 49 160 22 0.80 0.22 85 7.62 AHU-192 50 4/19/90 26 21 65 3.1 0.1 ! 52 157 21 0.80 1.3 78 7.54 AHU-182 50 i 5/30/92 24 21 68 3.510.16 0.094 48 165 2 2 | 5 8 0.67.4 9! 5 83. | ————————————— ,

AHU-274 75 t 5/30/92 25 22 68 3.3 0.16 0.097 47 160 ' 21 0.70 1.28 85 7.50 AHU-356 0 '10 4/19/90 26 21 64 3.1 0.1 47 153 ! 20 0.80 5.1 77 '7 56 AHU-358 100 5/30/92 248 6 1 ' 2 3.3 0.16 0.097 48 160 21 0.68 . 2.47 85 I7.53 AHU-438 125 5/30/92 27 3 7 3 '2 3.5 0.20 0.109 45 185 24 0.83' 2.14 85 '7.51 AHU-520 150 4/19/90 64 54 | 180 8.8 0.4 28 476 58 • <0 .01 67 7.54 AHU-520 150 ' 5/30/92 74 67 ' 195 8.9 0.50I 0.241 29 560 75 2.4 • <0 .01 70 '7.56 AHU-602 175 5/31/92 315 345 815 30 3.5 1.1 34 2550 350 1 1 <0 .01 75 7 36 AHU-648 200 ' 4/19/90 560 650 ! 2100 60 4.9 47 5490 650 <0 015 i3 7 67 AHU-648 200 ' 5/31/92 58 ; 2410 0 072 7' 9 0 6. 2.3 50 5920 980 3; 2 6/1/92 21°30'00 157°5N " 8 12" W 29Qm -7.1 1103.0 t O.e 3.9 0.09J , ! PIO-S E -21.4 , 6/1/92 !21°30'40" N 157°57'30" W 500m -10.15 0. 92.± 8 1.1 I -0.12 PtO-4F | -13.9 i 6/1/92 J21°30'45" Nil57°58'08" W 1000m -5 5 83.5 * 0.6 1 4 i-0.15 HEE-5O ' -26.7 [ 6/2/92 Î21°26'42" NH157°52'08" W ! 1900ml -7.6 88.3 ± 0.5 ! 0.9 '-0.04

159 TABL (continuedE1 )

Sample ID : Depth DbMolve - Temp C .TO d' O D DensitC 13 yl da do tO 218 H l de , Tritium 14C mBSL •olW» •c «t 20*0 TU proC PIQ-410 0 186 75 12 , 21.3 0 9981' -10.5 -2 90 ' •0.39 ± 0 1V PIO-4M 25 181 2.0 1 21.0 0 9981 ' -14.0 -3.15 1 1 1 0. .0.3 ± 6 PIO-570 5 7 144 l 20.7 ! -13.5 -3.15 -18.'0.093 ± 0 17' 112.7* 2.7 PIO-577 50 137 5.0 2.2 20.7 ! 0.9982 i -10.5 , -3.05 0.08 ± 0.14l PIO-659 75 137 3.2 1.8 20.7 0.9981 -9.0 ' -3.00 ' 0.3± 0.137 1 PIO-740 10 1 ' 130 l 19.9 0.9979 -15.0 -3.20 3 -19.2. 96.l ± 417 0 3 0.2* 1 PIO-740 10 1 107 3.0 14 ' 19.9 0.9981 | -8.5 ' -3.0-16.0 4 0.0± 0.134 ! 73.4 ±.58 PIO-823 125 108 8 3. 2. 8i 20.0 0.9981 -10.5 -2.75 -2t8 . 0.13 ± 0.131 80 ± .61 PIO-905 1 50 113 : 20.0 ' 0.9980 -10.0 , 2 -3.31 -2 0 0.27 ± 0.17i 96.4 ± 2.3 j Pl 0-905 ! 150 106 2.8 ! 2.5 20.0 0.9981 -9.5 -2.80 -21.1 l 82.3 .6 7 * PIO-987 175 172 19. j 4 3. 20.1 0.9981 -9.0 -2.95 -22.0 9 .5 * 9 7 i PIO-1069 200 144 20.1 0.9988 -10.0 -3.10 ' -20.8 0.12 * 0.111 84.9 ± 2.3 PIO-106» 200 173 2.6 ; 5.7 20.1 0.9982 -9.5 -2.90 -22.0 79.i 1.22 9 PIO-1151 225 1231 3.1 5.4 • 20.9 0.9986 -7.0 -2.60 • -19.7 ' 75.6 ± .59 pra-t23s 250 2414 ; 21.2 -9.0 -2.95 , -19.3 0.1± 0.131 i 76.8 1. 6 ± PIO-1233 250 2585 2.4 • 21.2 0.9996 -6.5 -2.60 -20.9 . 70 ± .84 PtO-1315 275 10501 2.2 ' 21.8 1.0057 -5.0 -2.00 ! -17.2 6 .5 • 53.± 4 PIO-1397 i 300 19648 , 22.3 1.0129 -3.5 -1.10 ; -9. 3 558.5 .1 1. 0.1 3* ± 1 PIO-1387 300 20992 ' 1.0 i i 22.3 1.0132 0.5 -0.65 -12.5 68.1 * .50

PIO-147Ö 325 25839 1.0 l ; 22.7 1.0165 0.0 -0.60 ; -11.4 9 .5 40.* 7 PIO-1561 350 28146 1.2 23.1 1.0188 3.0 ' -0.1. -9.0 6 8 .3 i 41.* 2 PIO-1627 370 28193 i 23.2 1.0189 1.5 -0.3• -7.5 3 0.10.16± 4 l0 26.1. 0* PIO-1627 370 30001 , 0.6 23.2 1.0201 1.5 -0.05 ; -11.9 64.4 ± .60

AHU-28 0 334 1.0 1.2 24.8 0.9981 •6.0 -2.35 0.7± 0.12 1 AHU-110 25i 0 37 2.2 : 0.54 22.5 0.9981 -10.5 -2.9. 0 0.32 ± 0.101

l AHU-182 50 365 l 0.9985 -11.0 -3.25 j -16.7 0.19 ± 0.17rl09.4 ± 3.8 AHU-192 50 379 2. 3i 0.8 5 23.1 0.9980 -10.5 -2.65 > AHU-274 75 372 2.2 : 0.54 22.5 0.9981 -11.0 -3.0' 0 0.2± 0.102 : AHU-35« 100 360 0.9985 -13.0 -3.30 : -16.7 0.22 ± 0.16 99.2 ± 4.1 AHU-356 100 372 2.3 ' 0.47 i 23.0 0.9981 -10.0 -2.90 ! 0.40.1l± 2 ! AHU-438 125 408 2.5 0.46 23 5 0.9983 -10 5 -2.80 0.30 ± 0.12' AHU-520 15: 0 1 89 0.9980 -9.5 -3.15 : -22.2 0.0± 0.175 1 : 2. 94.± 4 AHU-520 150 1035 2.3 0.44 i 23.0 0.9986 -9.5 -2.85 -19.4 88.3 ± 1.35 AHU-602 175 447! 8 2.1 ' 1 23.2 1.0015 -7.5 -2.45 -22.0 75.1 ± .68 AHU-S48 200 9577 1.0052 -7.0 -2.30 -17.2 !0.04 ± 0.13 60.5 ± 15 AHU-648 200 10794 1.7 ; 23.0 1.0061 -2.5 -1.75 -17.3 8 .4 58± AHU-766 225 17237 ! 1.4 23.5 1.0105 -1.5 -1.15 -14.0 5 »•62.0.8 * 5 AHU-848 250 19479 -3.5 -1.30 -11.1 '0.08 * 0.15 38.9 * 1.2 AHU-848 250 21088 1-0 ', . 24.5 1 0139 -0.5 -0.80 -12.3 i 40.0 * .38 AHU-930 275 25531 ' 1.0 ' . 24.0 1.0170 1 2.4 5-1 0 -03 i 75.0 ± .63 AHU-1000 296 27734 , 1.0196 -2.0 -0.45 • -12. ± 0.16 33 •'00 34.0 1. 94 AHU-1000 j 296 29027 l 1.0 ' 24.0 1.0195 2.5 0.05 -12.8 ' 59.6 ± .49

! | l ——— 1

' l l ; ' : :

160 20,000 -3.3 -2.- 5 52 -1.- 5 1 50. -0.- 0 5 Oxygen 18, in per mil

Figure 12 - Mixing diagram showing oxygen-18 and chloride for Waipahu monitor well; based on two rounds of sample collection, 1990 and 1992.

300 -80 -60 -40 -20 0 20 40 80 Differences between observed and calculated, In meq/L

Figur - Catio 3 1 e n imbalance depth profil r samplefo e s collecte t Waipaha d u

monitor wel n 1992i l .

freshwatee inth r zon generatee ear carboniy db c acid weatherin silicate th f go e minerals in the volcanic rock. The result is a dilute sodium-calcium-magnesium- bicarbonate type water and high silica with dissolved solids of approximately 110 mg/L (milligram literr spe ) (PIO-74 PIO-9871o t concentratiow lo , e TABLTh . E1) calciuf no m (~5 mg/L ,freshwate e TABLth n i ) E1 r zon consistenes i t with weatherin piagioclasf go e pyroxend t calcitean no d describes ea ,an d below

161 (-20 %o) is in approximate equilibrium with the soil gas observed (-26 %o) in the probable recharge zone, discussed below, suggesting the absence of calcite dissolution. However, ther indicatioes i dissolvef no d organic carbon d (TABLol d whic) ad E 1 y hma carbo freshwatere th no t . Unfortunately 6e 13C,th value doe t allosno w discrimination of this source from plant-respiration-derived carbon. Delineation of recharge area is an important consideration in the hydrology of aquifer systems. Becaus hig d flae han tth f wateeo r beelevels ha nt si proposed that Schofiele th d dike free arelarga s ai e reservoir from which water enter Peare sth l Harbor aquifer (Dal Takasakid ean , 1976). However t cleano f mosi rs i t i , t wates i r recharged withi Schofiele nth d dikareo t r eao compartment adjacene th n si t Koolau Range which has a higher topographic position and higher rainfall. We attempted to gain some insight intlocatioe oth f rechargno e are sampliny ab r

162 ü

1 50 Waipahu i 100 1992 • . g * I 150-

1 200m £ „ E g "- 250 'S o Qnn -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 Bicarbonate differ««» mac^n i , L

Figur - Bicarbonat 4 e1 e imbalance depth profil r sampleefo s collecte t Waipahda u

monitor well in 1992.

u Waipio 50 • s ^Dotomtt« • 199L 0 \ " \ 100 - Catofta «A Ç ' ./ \ 150 - * +MGypui m7

200 - > */ ^«/ x--^^" ___ ^— 250 - *f* ^- •" " / / -^ 300 A. I *' C ! / ,' 350 i > i

Ann . . . . i i . i . . i . . . 2 - 1 - 0 -3 -4 LOG IAP/KT

Figur Saturatio- 5 1 e n index depth profile r calcitfo s e dolomit gypsumd ean r ,fo samples collecte t Waipia d o monitor wel 1990n i l . Saturation indices calculated using PHREEQE (Parkhurst and others, 1980).

saltwatere Inth , probable dissolved carbon source Peare th o slt Harbor area aquifer are: bicarbonate from calcite equilibriu mseawatern i organid an , c carbon i sea-bottom sediments and within the caprock though which recharging seawater flows. saltwatere Inth smal,a l amoun f calcitto e precipitation result exchangsn froio e mth e mentioned above. The addition of calcium to the solution from exchange caused precipitation of calcite, thereby removing calcium and an equivalent amount of bicarbonate from solution sees a ,actua e FIGURn ni Th l. amounE14 f calcito t e precipitate smallds i milliequivalent w onls fe ,a ya liter bicarbonatf srpe o lose ear t from

163 solution. The expectation of calcite precipitation is supported by thermodynamic calculation using the code PHRQPITZ (Plummer and others, 1988). FIGURE 15 shows that the log of the saturation indices for calcite is near zero, indicating equilibrium. The 613C of saltwater reaching both the Waipio and Waipahu wells is isotopically lighter (-6 %o) than seawater hypothesizes i t I . d that this results from dissolutio f organino c material in the caprock (with a

5. CARBON-14 CHEMISTRY AND DATING

Ideall woule w y d lik obtaio et n different residence time thin si s hydrologie system tim) 1 : e required from saltwate movo rt e fro oceae mth wellse eacno t th f h,o 2) time required for freshwater to move from the recharge area to each of the wells. These times would also provide: 3) time for freshwater and saltwater flow between the o welltw s whic approximatele har same th en o yflo w path. initiaBasen a n ldo hydrologie model of the system, our belief was that carbon-14 dating of the dissolved bicarbonate would be a suitable method by which to attempt dating for items 1 and 2 above. It was felt that difference in freshwater residence time between wells might be so low as to preclude obtaining a freshwater time for item 3 from carbon-14. Additionally clearly-definea lace f th ,o k d pristine-wate rWaipah e zonth t ea u well preclude chemicay san isotopir lo c dating techniqu freshwater efo r travel time between the wells. The SrC03 precipitate, described earlier, was analyzed for carbon isotopes. The samples collected in 1990 were analyzed using a specially designed miniature cell and long counting times (Mebus Geyh, NLB, Hannover, Germany, personal communication) thosd an e collecte 199n di 2 were analyzed using accelerator mass spectroscopy (AMS) methods at the University of Arizona. Accuracy of analyses are about ±2 pmC for samples dominated by freshwater and about ± 1 pmC for samples dominated by seawate r botfo r h methods. 613C values were obtained fro same mth e samples. Unfortunatel carbo199e S yth 2AM n isotope data (both S 13pmCd Can ) showa large degree of scatter relative to the traditional counting method and depth. Our AMS control sample, an Ordovician age marble, was dissolved in acid under nitrogen

atmospher precipitated ean same th t eda time wit same hth e reagent othee th s rsa samples, gave the expected values of greater then 45,000-year age and

164 apparen determinatioe ag t necessarns i calcitr yfo e precipitatioo n e therb d n nan eca dissolutio caproce dead th f no ad d ko carbot systeme th no t . However, assuming that the organic carbon added during seawater recharge is dead and seawater has a

of 0 %o, the pmC values for carbon-14 in the saltwater must be increased by a factor of 1.24 based on extrapolation of observed

6. AGE RELATIONS The objective here is to date the three water types and to determine travel times of these waters betwee pointe nth sprofileo whertw e seth were collecteds i s A . commo hydrologin i e systems, each sampl mixtura s ei f wateeo r type r 'ens(o d members'), in this case, up to three types. To date each water type occurring in a mixture, it would be necessary to theoretically unmix the sample, a calculation that would require knowledge of a number of unknown quantities. To avoid this dilemma, an assumptio made b n en thaca t every sample collecte simpla ds i e mixtur f distinceo t water types, or end members, and that samples collected along each observation well diffe relative r th onl y b ye amount f eacso h wate r mixturee Peare typth th n ei n l I . Harbor area, a different set of end members may be defined for each of the two profiles. This may be motivated as follows. Of the three water types, only irrigation-return water contains finite (but small) amounts of tritium. Simultaneously, this water has high carbon-14 levels (up to 110 pmC). Thus watee ,th r contains post-bomb carbo tritiumd nmusan d lese an , b t s than 50 years old. The low tritium content in this young water may be due to the use of already-old ground water as irrigation water (Hufen and others, 1972) which is imprinte modery db n carbon during percolation throug soie hth l zone. The second water type, the core of the freshwater lens, contains no tritium. This implies thawatee th t r mus greatee b t r tha year0 n5 s old. Howeve hydrologia r e calculation based on accepted values of recharge and pumping records implies that water must travel throug entire hth e len ten w onl n syearsf i fe s o ya . Rough calculation with recharge of 106 m3/d, aquifer width of 26 km, lens thickness of 300 m, and porosit f 10%yo , give freshwatesa r velocit m/d3 1. f , yo where : velocit rechargy= e / (porosity x width x thickness). The resulting travel time through a 10 km long aquifer is only abou year0 som2 td san e tritium activity woul expectee db thidn i s water. This hydrologically-determined residence freshwate e timth en i r lens conforms wit briee hth f residenc eirrigation-reture timth f eo n water determined geochemically similaA . r velocity of irrigation-return water and lens water would indeed be expected because the hydraulic gradient affecting samebote th hs i . This apparent contradiction between isotope dating (ag f moreo e tha years0 n 5 hydrolog d an ) y (residence time less than 20 years) is resolved when the significant storage of ground water in high-level dike compartments is recognized. Ground-water flow through a large compartment could take hundreds or thousands of years, resulting in a large component of old water recharging the freshwater lens. Moreover, acceptance of a brief residence time of freshwater implies that the carbon-14 content of freshwater is practically constant throughou lense th t . Inspection of carbon-14 values for deep samples predominantly composed of pmC)seawate0 3 o t ,0 indicate(2 r s residence timeabouo t half-liveso p su tw t 4 10 r o , years. Thus, the saltwater ages as it moves inland, and for the purposes of a simple mixing mode assumes i l movdo t e inlan unifora dn i m manner, with depth-independent age.

165 For the simple mixing model applied to carbon-14 content, there is only one end membe r irrigatiofo r r nfreshwate fo wate d an rr botfo r h profiles r saltwaterFo . a , different end member must be allowed for each profile, as a significant travel time betwee pointe nth s would resul differena n ti t carbon-14 valu eact ea h location. Based on the mixing fraction defined by chloride content of each sample, the carbon-14 content as pmC of a binary mixture is defined as follows:

where: A CI 4 =

Cl-Clf, û)= cis-ciF

AC=CS-CF

pmCd an , pmCmixturee pmCd th Fe an ar s, freshwate saltwated an r r end-member percent modern carbon freshwatee th ,e respectively ar saltwate d s C ran d an r F end,C - member inorganic carbon concentrations. <•> is the saltwater end-member fraction in freshwatee th e ar d saltwate s an thr Cl e d mixturer an end-membe F CI , r chloride concentrations 0 mg/ 2 ,d 19,35 an l 0 mg/l e chlorid,th respectively s i I eC d an , concentration of the mixture. Because pmC is a relative concentration and the total carbo mixture th f no e depend seawaten so r fraction. mixin,a vs C g plopm a lin f n to e o o> is non-linear (FIGURE 16a). This mixing relation is used to estimate end-member carbon-14 concentrations for each profile collected o thisd ,o T alkalinit. y profile n eaci s h welfirse ar tl extrapolated to obtain freshwater and saltwater end members. Because of the pH of these waters, the alkalinity is assumed to be equivalent to the total inorganic carbon concentration. Thus, total inorganic carbon value Waipahr sfo Ce Fu=3.ar 1 mg/d an l Cs=12.0 mg/l, and for Waipio are CF=4.0 mg/l and Cs=12.4 mg/l. Then the mixing curve for each well is moved upwards on the carbon-14 vs. saltwater fraction plot (FIGURE 16ab) by adjusting end-member pmC values in the above mixing relation, until the curve rests agains largese tth t possible numbe lowermose th f ro t data points. This result end-memben si r carbon-14 value Waipahr sfo f pmCuo F=10 pmCd 0an s=30, Waipir fo d f pmCoo an F =8pmCd 0an s=23. Use of only the lowermost points may be justified as follows. The scatter relative simple th o t e contaminatiomixino t e du g e curveb y samplef nsma o s with atmospheric carbo othee nth dioxidef ro isotopil al s chemicad ,a can l species collecte same th den i water samples closely follow simple mixing lines betweet membersd no nen d di d an , vary significantly between the 1990 and 1992 sampling rounds (TABLE 1) except for the small upwards shift in 1992 described earlier. Rather, the variability may be due contaminatioo t f sampleno s during handlin modery gb n carbon which would raise

166 14 . SaltwateCvs r Fraction Waipio and Waipahu Wells - April 1990 and May 1992

0.2 0.4 0.6 0.8 Saltwater Fraction

Figure 16a - Carbon-14 (pmC) relation to saltwater fraction for Waipahu monitor well profile (X is 1990, inverted-Y is 1992) and for Waipio monitor well (Square is 1990, and Triangle is 1992).

values above 'true values' which would fall close to the simple mixing curve in FIGURE 16ab. Indeed thin i , s system, there doe reasonablt appeaa se no b o rt e mechanism for lowering carbon-14 measurements below actual aquifer values; thus, interpretation ma mose yb t soundly lowesbasee th n dto value relation s i mixin e th no t g curve. Considerin additioe gth f deano d carbo oxidatioy nb f organino c carbon indicatey db end-member carbon-13 values for saltwater, the maximum possible adjustment to carbon-1 saltwatee th f 4o memberd en r increase th s i pmC en factoi a y b s r 1.24s a , discussed earlier. This adjustment result valuen si Waipahr sfo f pmCuo r fo s =3d 7an Waipi f pmCoo s=28. Using a half-life of carbon-14 of 5730 years, the Waipio freshwater carbon-14 has undergone t recharg pmC2 decaa ± C ,0 8 y givinpm eo frot 0 apparenmn g10 a t age ranging from 1600 to 2100 years. The Waipahu freshwater end member (FIGURE highea s 16ha r) carbon-14 conten f abou o tpmC 0 10 t, which would suggest that water gettine sar g 'younger' alon floe gwth line from Waipi Waipahuoo t . Howevers ,a discussed earlier, the freshest water at Waipahu is a mixture of irrigation-return water pristind an e freshwater; thus, this hig irrigation-returo t h e valudu es i n componenf o t the mixture and is not a true freshwater end member. In contrast, the saltwater end-member carbon-14 content at Waipahu is relatively well-constraine daty db withiaC binary-mixin(FIGURe pm nth 1 ± 0 E3 16be gb )o t model. Seawater recharge to the aquifer system occurs offshore likely through the

167 14 . SaltwateCvs r Fraction Waipi Waipahd oan u Well s- Apri l 199Mad 0yan 1992 120

100

80

y 3 60 u

40

20

0 0.00001 0.0001 0.001 0.01 0.1 Saltwater Fraction

FigurCarbon-1- b e16 4 (pmC) relatio o Toglt n f saltwateo O r fractio r Waipahfo n u monitor well profile (X is 1990, inverted-Y is 1992) and for Waipio monitor well (Square is 1990, and Triangle is 1992).

caprock, which is at a depth of about 500 m south of Oahu. An initial seawater carbon-14 content is required in order to estimate the age of saltwater under Oahu. In orde characterizo t r rechargine eth g seawater verticaa , l profil f seawateeo s wa r collected for this project at the Kahe-1 station roughly 25 km west of Pearl Harbor by GOFS (Global Ocean Flux Study) in October 1992. Carbon-14 and other isotope analyses are presented in TABLE 1 (carbon analyses by AMS at University of Arizona, stable isotopes by USGS, Reston, Virginia). A deeper oceanic GEOSECS profile (Östlund and others, 1979) collected about 1000 km southwest of Oahu (FIGURE 17) may give additional guidance regardin initian ga l carbon-14 conten f seawateo t r that entere aquifee dth r thousand f yearso GEOSECe s th ago n I . S profile minimue th , m ocean carbon-14 value is about 80 pmC and we may guess that in the pre-bomb profile, this value may have existed up to a depth as shallow as 500 m (which is the approximate submarine depth of the caprock, see FIGURE 3). The greatest possible value at 500 m depth at the time of recharge would be the present value, about 90 pmC. Thus the range of possible initial seawater activities at recharge is roughly from baseC botn pm do GEOSECe 0 hth 9 8o 0t SGOFe profilth d S ean profile e Th . resulting apparent saltwate Waipaht a e rag betweeus i n 780 940d 0an 0 years. Using «S13C value adjustind san dear gfo d organic carbon input resultine ,th g saltwatee ag r at Waipahu is between 6200 and 7600 years.

168 Figure 17 - Profile of carbon-14 in open ocean south of Oahu, Hawaii (Station 235). After Ostlund and others (1979).

The end-member saltwater carbon-14 content at Waipio (23 ±1 pmC) is less tha Waipaht n a pmC 1 ± 0 )u (3 consisten t wit inlann ha dsaltwatere floth f wo . Because the chang carbon-1n ei 4radioactivo t contene du s i t e decay only associatee th , d

saltwater travel time between the wells is between 1600 and 2800 years. This travel tim unaffectes ei saltwatee th y db r C$ 13 adjustment which only changes both end- member carbon-14 values by the same fraction. The associated estimate of saltwater velocity to travel 4.5 km from the Waipahu Waipie m/ywel8 th 2. o t lo .d welNotan betwee s i ly e tham/ 6 tn 1. thi s velocity actually describes condition e aquifeth n i sr before development bega 0 year10 n s ago. Considering that the age of the saltwater is much greater, the difference in radiocarbon age betwee profileo tw establishe s e nsth wa d well before aquifer development began. Present-day saltwate mucs a r fiv s e velocitha b e y timema y s greater than before development (Vos Souzad san ongoine ,th 1993o t e g )shrinkagdu lense e th Th f .e o simple assumption thae predevelopmenth t t velocit s constanwa y t betweee th n Waipah recharge uth weld an l e are seawater afo r allows estimatio distanca f no e th eo t recharge area. Using the above saltwater age range at Waipahu, the distance is between 14 km and 24 km (or with the adjusted ages, between 11 km and 19 km). From Oahu's southern coast seawatee th , r rechargem arek 9 thereforas i o t m k e1 offshorm k 4 1 eo t usin offshorem k g4 5r (o C-adjuste d ages). 13The 1800-year radiocarbon age of freshwater in the Pearl Harbor area lens is in apparent contradiction to the brief residence time spent in the lens of only a few tens of years possiblA . e explanatio freshwatea r nfo greatee ag r r tha residence nth e time Peare inth l Harbor are tha ahigh-leves i e tth l ground-water compartments contain large volumes of fresh ground water which is impounded for long times before the water

169 enter freshwatee sth r lens. This water become significansa t amoun totae th f l o lent s recharge, quickly flows to the coast and discharges. This idea is not without precedent. Dal Takasakd ean i (1976) evaluate possible dth e effect f increaseso d pumpage Schofiele inth d high-level water bod reducinn yi g recharg Peare th eo lt Harbor area aquifer. Takasaki and Mink (1985) evaluated the water resources of the Oahu dike zones and discussed significant discharge to adjoining basal aquifers. Hufen and others (1980) foun arean a dn i l sampling progra mf shalloo w waters that some Pearl Harbor area water westere th sn i n portio aquifee th f no r wer few-hundreea d years old. They suggested that some older recharge water may derive from the Schofield high- level water body. They also found waters in some Honolulu aquifers separated by valley fills with apparent age nearlf so y 1000 year suggested san d that some recharge may have been impounded in dike compartments. Lacking vertical profiles through the freshwater lens, however, Hufe otherd nan s (1980) concludo wert d ele e that most ground watePeare th n lri Harbor are experienced aha d short residencs e timewa d san young water contrasn i , t wit oldee hth r freshwater lens foun thidn i s study. The total compartment volume availabl impouno et d ground water befort ei enters the Pearl Harbor area freshwater lens does not, according to a simple calculation, suffic creato et delaea lons ya 180s ga 0 years largese Th . t possible volume of the Koolau rift zone that may contribute recharge to the Pearl Harbor area begins alongside the Schofield high-level water body and continues southeast along the upstream boundar areatotae e th f Th l.y o lengt thif ho s maximal zon aboues i 0 2 t km and width is about 10 km. A maximum thickness of subaerial basalts intruded by dikes in this area is about 2 km. Allowing that the maximum possible volumetric porosit vesiculaf yo r basalts, abou participatn %0 ,ca t5 wateen i r storage, give totasa l absolute maximum stored water volum f aboueo x 10 . 3 2 Witt 1 m 1earliere hth - mentioned natural recharge rate to the Pearl Harbor area aquifer of about 9 x 105 m3/d (240 Mgal/d), the mean residence time in the rift-zone compartment would thus be, at most, 600 years. This is equivalent to a considerable ground-water recharge rate of about 1.7 m/y over the rift-zone area. Considering an additional compartment volume Schofiele th n i d high-leve deepaream m k k 2 4 ,, y provideb l watem k r8 s bodn a n yi onl additionan ya maximuyeare 0 th 10 l o st m compartment residence time shoull dal rift-zone water flow through the Schofield area before entering the Pearl Harbor area lens. The two to three times discrepancy between apparent freshwater radiocarbon age (1600 years to 2100 years) and maximum possible residence time in the compartments (700 years) may be presently explained in two ways. Either the natural recharg Peare th eo lt Harbor water bod less yi s than factoo a assume tw o t f ro p u y db threecarbon-1e o t th r o , 4 activit freshwatee th n yi r lenbees sha n decreasen a y db undetermined sourc carbod ol f eo n during recharge possiblA . e carbon sourcs ei additio organid ol f no c matte rechargn ri e areas discussed earlier have W e. not thit ,a s time, further considered this discrepancy, elucidation of which requires additional field reconnaissance in recharge areas to determine initial carbon-14 activity and carbon sources.

7. COMPARISON WITH MODEL

numericae Th l modesystee th discussef s o lm wa Vosy dSouzb d san a (1993) and by Souza and Voss (1987) to which the reader is referred for details on the model development. Onl brieya f review sufficien demonstrato t e model results applicableo t the present geochemical and age relationship analysis is given here. The 20 km long

170 two-dimensional cross-sectional model (based on the finite-element method) represents the entire Pearl Harbor area aquifer fro dike-zone mth e boundar Schofield yan d high- level water body boundary to the inland edge of the caprock (10 km) and an additional distance into the ocean south of Oahu (10 km). The width of the section is 26 km, and depth is 1.8 km. The numerical model simulates variable-density ground-water flow and solute transport (of total dissolved solids) in the section. Note that the rift zone Schofield an d wate t includer bodno e yar thid n i s model. The inland and bottom model boundaries are impermeable (FIGURE 18), and the vertical sea boundary is held at hydrostatic seawater pressure allowing either inflow of seawate r outfloo r f aquifewo r water uppee Th . r boundar watea s yi r table witha specified amoun rechargf to e inlancaprocke th hel specifies i f a d o t d a ,an d pressure of zero above the caprock to reflect the presence of the Pearl Harbor water and the ocean with concomitant inflow or outflow. Along the inland portion of the water table, time-independent natural recharge (from rai compartmenr no t spillage specifies )i s da increasing linearly toward inlane sth d boundary naturae Th . l recharg assignes ei da seawater concentration (dissolved solids zerof )o . Between thi caproce s th are d aan k edge, spatially-constant recharge from irrigation is specified to occur on a time-varying schedule from 188 1990o t determines 0a d from pumping records departura n I . e from the previously-published modeling results, the irrigation recharge is arbitrarily assigned a non-zero concentration in order to track the movement of this water in the section.

RECHARGE i P""—- l , FROM IRRIGATION ZONF EO t i t t i t « « t t f t t t t t PUMPING

-300 f

LU C/3 o tr

1o -1200h 3 LU -1500h

-1800 -6-4-2 02 4 6 10 DISTANCE, IN KILOMETERS FROM CAPROCK BOUNDARY

Figur - Pear 8 1 el Harbor aquifer cross-sectional model showing finite-element grid (schematic), boundary conditions, pumping region, basal caprocd tan k regions (after Voss and Souza, 1993).

171 Nea caproce rth k belo edgm deptwatea e 0 wt 4 eth a f ho r basaltable th en i t aquifer, coastae th l ban wellf do specifies i pumdo t p water fro aquifee m th time-varyin a t a r g rate from 188 199determineo s t 0a 0 d from pumping records. The model structure is very simple. The layered-basalt aquifer is anisotropic with horizonta time0 20 sd lowelan conductivit d r verticam/ 7 45 f ly o conductivity e Th . caproc s isotropii k c with conductivit 4 time10 y s less tha e basalnth t (horizontal). Specific yield and porosity (and effective porosity) are 0.04, and there is considerable compressive storag layeree th en i d basalts (matrix compressibilit f 2.5x10"yo 9 Pa"r o 1, equivalently, specific storage of 2.37x10"5 m~1). In the model, longitudinal dispersivities for horizontal flow and vertical flow are respectively, 250 m and 50 m, while transverse dispersivity is only 0.25 m. The following model results derive from new simulations wit earlier-reportee hth d numerical model (Vos Souzad san , 1993) without change with the exception noted above to trace the irrigation-return water. The modeled distribution of the three water types and the flow field for predevelopment conditions (1880) and for 1990 is shown in FIGURE 19abcd. The shrinkage of the freshwater lens in this period is apparent, and even given constant recharge and pumping after 1990, the shrinkage would continue (Souza and Voss, 1989). In the 1990 simulation, the upstream monitor well (Waipio) intercepts all three water types, while at the downstream monitor well (Waipahu) the irrigation-return water and the saltwater overlap and mix across the freshwater lens (FIGURE 19c). This conforms with the spatial distribution found in the geochemical analysis. To further compar e field-measureth e e distributio ag e dmodel th d ,an n isochrones (lines of equal residence time) were calculated in the model. Using the 1880 steady-state flow field, an extra simulation was run at steady state for an imaginary species having zero concentratio inflowt na witt ,zero-ordebu h a r production of this specie fluie th d n si (se e Voss, 1984definitior fo , f thino s source) wit valuha f eo one per year. This numerical trick produces isochrones directly from the transport model, accurately indicating residenc emodee timth n ei l sectio r regionnfo s witw hlo

75 59 8 9 50 09

-10.0 -a.O -6.0 -4.0 -2.0 0.0 2.0 4.0 8.0 8.0 10.0

FigurModele- a 19 e d cross-sectio f Pearo n l Harbor area aquifer showing seawater distributio n 1880i n . Contour percenn i e ar st seawater. Upper hal f modeo f l section is shown.

172 m / s 4.0000E-08 .0QOOE-07 4.000QE-06 4.0000E-05 , , t • ' t * * ' * ' t V 11 • • • •. KLt ' ' » f .

-10.0 -8.Q -6.0 -1.0 -2.0 0.0 2.0 4.0 8.0

Figure 19b - Modeled cross-section of Pearl Harbor area aquifer showing velocity distribution in 1880. Upper half of model section is shown.

9098 7595

Figure 19c - Modeled cross-section of Pearl Harbor area aquifer showing seawater distribution contours)(lowef o t rse d irrigatio,an n recharge water distribution f contours(uppeo t se r n 199i ) unitn 0i f perceno s f irrigatioo t n waterd an , location of monitor wells. Contours are in percent seawater. Upper half of model sectio s showni n .

173 s / m 4.0000E-08 4.0000E-07 4.0000E-06 4.0DOOE-05

-10.D -9.0 -6.0 -1.0 -2.0

Figure 19d - Modeled cross-section of Pearl Harbor area aquifer showing velocity distributio n 1990i n . Upper hal f modeo f l sectio s i shownn .

solute dispersion. The result (FIGURE 20) in both the freshwater lens and the deep saltwater gives model residence times. Isochrones within the zone of mixing between freshwater and seawater are ignored. Total residence time of water in the freshwater lens is less than 20 years, in conformance with the tritium findings for irrigation-return freshwatee waterth d an , r would exhibi constana t t carbon-1 throughoute 4ag e Th . saltwater takes approximately 6000 years to flow 20 km from the sea boundary to the inland boundar modele th n yi , equivalen velocita o t t f abouyo m/y3 3. t . This si somewhat greater than but similar to the carbon-14-based saltwater transit velocities from the Waipahu well to the Waipio well of between 1.6 m/y and 2.8 m/y. The inland flow of saltwater in a steady-state system is driven by the amount of salt entrained in and subsequently discharged with the freshwater flow towards the coast. This, in turn, depends on two factors, the flux of freshwater and the dispersion process. To slow down the modeled saltwater inflow to better conform with the isotope-based velocity would require eithe decreasa r modee th en i l recharge th eo t aquifer by up to one half, or a decrease in the model dispersion coefficients used to reproduc seawatee eth r transition zone. Another factor t considereno , modee th dn i l calculations thas effectivi ,e th t e porosity experience slow-movine th y db g saltwater may be greater than the value of 0.04, assigned to the entire aquifer, a value which beed ha n determined mainl freshwatee daty th y b n ao r zone slow-movinn I . g saltwater the entire volumetric porosity, up to about 0.40, may participate in solute transport by both solute diffusio advectiod nan n through block f basalso t alongsid welle eth - connected conductive rubbly beds for which the regional effective porosity is

174 apparently onl effectivcasn e ya th f 0.04eo n I e. porosit f 0.40same yo th , e saltwater flux as determined in the simulations can be obtained with a ten-times lower saltwater velocity. Thus there is ample flexibility in this process to account for the discrepancy in modeled and isotope-determined saltwater velocities even with only a small portion less-conductive oth f e basalt porosity participatin saltwategn i r zone solute transport.

-10.0 -8.0

Figure 20 - Modeled cross-section of Pearl Harbor area aquifer showing contours

f equao l travel time (note: NOT absolute ages) from entry into model arer fo a conditions in 1880. Upper half of model section is shown. Travel times are in years.

8. CONCLUSIONS

Samplin verticagn i l wellprofilo tw sen i alon regionaga l ground-water flow line and geochemica isotopid an l c analysi resultes sha proposadn i hydrologia f lo e model of the Pearl Harbor area aquifer in southern Oahu, Hawaii. The hydrologie model includes ages and flow velocities of the three major water types found. These have been compared with the predictions of a numerical variable-density ground-water solute transport model developed independently of the present geochemical analysis from hydraulic data and from vertical profiles of salinity through the freshwater lens and into seawater. Within margin f uncertainto s e chemicath n i ye ag l datd an a interpretations floe th w, behavior represente numericae th y db l mode goon i s i dl agreement with the geochemically-derived model. This agreement was not necessarily expecte outse studyexperience r e th th ou t df s a to ,a thas etwa most numerical models need to be revised significantly as new data becomes available. Perhaps most interesting is the similarity in the velocity of saltwater moving inland as determined by carbon-14 dating (abou m/y2 t thad )tan occurrin numericae th gn i l model (abou3 t m/y). While this agreement may at worst be only a coincidence, to our knowledge, it is the first time that a numerical variable-density ground-water model prediction of saltwater circulation in a coastal aquifer has been independently corroborated by geochemical field data.

175 The uppermost water consisting of recharge from irrigation is found to be only a few tens of years old and moves through the aquifer at a velocity on the order of 1 km/y. The freshwater core of the freshwater lens has a great apparent age of about 1800 years. From hydrologie considerations, once this water enter coastae sth l system from the compartments, it must move through the aquifer as quickly as the irrigation recharge water greae Th . t apparen satisfactorilte freshwateb y ma e yrag explaines da lono beint ge gstoragdu ground-wateen i r compartment recharge th sn i e areae th f so island. Simple volumetric calculations show, however, that this storage can account for considerably less than one-half of the apparent age. The excess apparent age may b r presene partlou e ytdu inabilit quantifyo t removd yan possibl y ean e contributiof no old dissolved organic carbon added during recharge, which would ten makdo t e eth water appear olde deepese rTh tha . is tnt i wate intrudes ri d seawater wit apparenn ha t ag f aboueo t 6000 year 900o st 0 years nea inlane th r d edg f Peareo l Harbor. Older saltwate founs i r d further inland frod difference man ,th agesn ei approximate th , e velocities mentioned above were calculated. It would have been difficult to propose a quantitative model of the hydrologie syste basiareae n a th f smn lo geochemicao l distributio vertically-mixef no shallor do w samples collected from springs and active pumping wells. Mixed samples would have obscured much of the primary information on water type, recharge area and age, while shallow samples would only characteriz uppermose eth t laye waterf ro . This pointso t importance th f vertically-stratifieeo d geochemical sampling throug entire hth e aquifer section. In our view, it is of utmost importance to understanding of regional aquifer system o collect s t hydrogeologic, geochemica wels s a hydraulila l c information. Further, we believe that these data must be analyzed simultaneously, with the objective being the discovery of a descriptive model that gives a consistent description of all data typesimpls a t quantitativsn i e bu manneea possibles ra presene th n I . t investigation, the corroboratio f hydrogeologicno , geochemica isotopid an l c data with predictionf so a numerical ground-water flow model has lent credence to the viability of a simply- structured quantitative mode describo t l Peare eth l Harbor area aquifer system.

ACKNOWLEDGEMENTS

The analysi f carbon-1so carbon-1d 3an 199e th 0r 4 datfo provides awa Profy d. b Dr . Mebus Geyh, and the Niedersächsisches Landesamt für Bodenforschung, Hannover, Germany contributioa s ,a Coordinatee th no t d Research Progra Internationae th mf o l Atomic Energy Agenc "Mathematican yo l Model Quantitativr sfo e Evaluatio Isotopf no e Data in Hydrology", of which this study was a part. We are most indebted to Dr. Geyh for this gracious support. We thank Dr. Gordon Tribble of the U.S. Geological Survey,

Hawaii District office, for providing a core sample of Oahu caprock, and for leading a

18 windward Oahu reconnaissance trip. Tyler Coplen, USGS provided O an d 2 H analyses, and Robert Michel, USGS provided tritium analyses for which we are most grateful alse oW . than Hawaie kth i District offic logisticar efo equipmend an l t support, botd State an hth f Hawaieo Board an i Watef do r Supply, CitCountd yan f Honolulyo u for providing access to monitor wells. Finally, we thank the Global Ocean Flux Study (GOFS) through David Karl and Dale Hebel at the University of Hawaii for generous suppor arranginn i t g special samplin collectinr fo d gseawatean e gth r isotope profile reported in this project.

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177 Parkhurst, D.L, D.C. Thorstenson, and N. Plummer, PHREEQE - A computer program for geochemical calculations, U.S. Geological Survey Water-Resources Investigations Report 80-96 pp.5 19 , (revised August 1990), 1980.

Plumrner, L.N. L ParkhurstD. , , G.W. Flemin S.Ad gan . Dunkle, PHRQPIT computeA Z- r program incorporating Pitzer's equation calculatior sfo geochemicaf no l reaction brinessn i , U.S. Geological Survey Water-Resources Investigations Report 88-4153 pp.0 , 31 , 1988.

Skougstad, M.W., M.J. Fishman . FriedmanLC , , D.E. Erdman S.Duncand nan , eds., Methodr sfo determinatio f inorganino c substance waten s i fluvia d an r l sediments, Technique f Water-Resourceso s Investigations of the U.S. Geological Survey, Book 5, Chapter A1, 626 pp., 1989.

Souza, W.R. and C.I. Voss, Analysis of an anisotropic coastal aquifer system using variable-density fiow and solute transport simulation, Journal of Hydrology, 92, 17-41, 1987.

Souza, W.RC.Id an . Voss, Assessmen f potablo t e groundwate freshwatea n i r r lens using variable- density flow and transport simulation, in proceedings of NWWA conference on Solving groundwater problems with models, Feb7-9, Indianapolis, Indiana, National Water Well Association, 1023-1043,1989.

Takasaki, K.J. and J.F. Mink, Evaluation of major dike-impounded ground-water reservoirs, Island of Oahu, U.S. Geological Survey Water-Supply Paper 2217, 77 pp., 1985.

Visher, F.N. and J.F. Mink, Ground-water resources in southern Oahu, Hawaii, U.S. Geological Water- Supply Paper 1778pp.9 3 , , 1964.

Voss, C.I., SUTRA: A finite element simulation model for saturated-unsaturated, fluid density dependent ground-water flow with energy transpor r chemicallo t y reactive single species solute transport. U.S. Geological Survey Water-Resources Investigations Report 84-4369 pp.9 , 40 , 1984.

Voss W.R,d C.Ian .. Souza, Dynamic regionaa f so l freshwater-saltwater transition anisotropizonn a en i c coastal aquifer system, Journa f Hydrologyo l pressn i , , 1993.

178 CALIBRATIO VERIFICATIOD NAN A F NO REGIONAL GROUNDWATER FLOW MODEL BY COMPARING SIMULATE MEASURED DAN D ENVIRONMENTAL ISOTOPE CONCENTRATIONS applicationAn Alnarpthe to aquifer systemin southwestern Sweden

. BARMEG N Lund Institut f Technologyeo , Lund University, Lund, Sweden

Abstract Environmental isotope studies and computerized groundwater flow modelling and radioisotope transport modelling have been applie large th eo d t syste f reservoirmo sedimentare th n si y deposits of southwestern Scania, Sweden. The stable isotopes 2H, 18O and 13C and the radioactive 3H and 14C have been measured and the results obtained can improve the estimations of the periods of recharge and the average circulation times of the groundwater reservoirs studied. A groundwater flow model based on finite difference techniques and a continuum approach has been modified by data from traditional hydrogeological studies. A calibration against piezometric records has been made for 1970, assuming steady-state conditions, and also for the transient evolution from 184 19880o t computee Th . r code NEWS A alss Mha o been useo dt simulate steady-stat transiend ean t isotope transpor aree th a n studiedi t , taking into account advective transport with radioactive decay interactine .Th g groundwater reservoirs studied have been represented by a three-dimensional grid in the numerical model. A major meri thif to s combinatio isotopf no e hydrogeolog regionad yan l flotranspord wan t modelling is that the isotope transport simulations help to demonstrate where zones particularly vulnerabl pollutioo et situatede nar . However, ther severae ear l difficulties concernine gth detailed, quantitative interpretations of the results, and these difficulties must be the focus of future work. Even if the aquifer system studied here can be classified as unusually homogeneou well-definedd san numbee th , isotopf ro e measurement smals i l comparee th o dt extension of the system. Furthermore, the calculated concentrations are similar over large areas. Therefore, it is at present not possible to calibrate the isotope transport model by measured isotope concentration orden si verifo rt rejecr yo result e underlyine th tth f so g groundwater flow model.

. INTRODUCTIO1 N Environmental isotope technique computerized san d mathematical flotranspord wan t modelling have become relatively widespread and frequently used. Yet, these tools have been developed sid sidy eb e with very little interaction. Recent years, however, have witnesse tendencda y towards greater collaboration and interaction among the different techniques and disciplines dealing with groundwater majoe .On r reaso expectatio n thir a ns fo s i n tha knowledge tth e about a syste f maquifero s will often increas becomd ean e more reliabl resulte th f ei s from many independent method f investigatioso interpretatiod nan usee n ar combination n di . Thif o s si particular interest when little informatio availables ni . Data fromethoe mon then e dca nb expecte compensatdo t somo et e lacextene datf th k o r atfo from another method. followst I , therefore, tha woult ti speciaf o environmentae e db us l intereso t y tr o tt l isotope techniques together with computerized distributed parameter, conceptual flotranspord wan t modellin regionaf go l aquifer systems author'e th o .T s present knowledge, Robertson (1974s )i

179 the only one who has published a scientific paper on the application of such an approach. In addition, Markussen, Möller, Villumsen, Mortensen & Selchau presented a report founded on a similar idea in 1991.

. PURPOSE 2 DELIMITATIOND SAN S

The main purpose of this work is to investigate whether a combination of the results from deterministic flow and advective transport modelling with observations of environmental isotope contents can improve the understanding of a chosen regional groundwater system. The investigation shoul carriee db t wit overale dou hth l objectiv improvo t e basie r eth fo s maintainin gquantitativela qualitativeld yan y safe long-tergroundwatee th f o e mus r resources studied maie Th .n purpos brokee b interactin n w intp enfe ca u o a subordinatd gan e aims. It should be investigated whether comparisons between observed temporal and spatial isotopic variations and corresponding calculated values can be used as an independent way of verifying the results of the numerical, groundwater flow model, which underlies the isotope transport model used. Another aspec thif to s combinatio resultf no thas si t isotope data might improve estimations of the properties relevant to the flow and transport processes which are e groundwateactivth n i e r system modelled. Therefore e potentiath , l contributiof o n environmental isotop e identificatio th dat o t ae physica th f o n l parametere th f o s hydrogeological syste studiede b mo t , should als investigatede ob . A second subordinat obtaio t s i bettena m eai r knowledge concernin hydrogeologicae gth l system choseapplicatioe th r methodologne fo th f no y outlined selectee .Th d syste msituates i d sedimentare inth y rock glaciofluviad san l sediment southernmosf so t Sweden (see d figuran ) e1 includes the Alnarp valley aquifer. The major reasons for choosing this system are that it has been the subject of comparatively thorough investigations and observations over more than a century, that all parts of it are situated at moderate distances from Lund and that the volumes of groundwater withdraw thir nfo s regio substantiae nar l (se examplr efo e Brinck, Leande& r Winqvist, 1969, and Hydén, Leander & Voss, 1980). Financial and temporal restrictions have set limits also for this work. In general, priority has been give equipmeno nt techniqued an t s which were easily availabl cheapd e an whict bu , h satisfied basic requirements of quality and performance. The following is an account for the most decisive delimitations. The use of only one hydrogeological system obviously implies an important restriction concerning the generality of the conclusions. In principle, it is possible to draw conclusions related onl thio yt s particular syste vero t r ymo similar ones. Thoug greaha t amounf o t information exists about the groundwater system of southwestern Scania compared to other aquifers, only a small and unequally distributed number of useful data in relation to the extensio systee th f nmo have bee calibratioe hano th n t r dfo n work. Thicoursf o s si generaea l and realistic situation t limiti t possibilite bu ,sth f drawinyo g methodological conclusions concernin combinatioe gth f environmentano l isotope investigations with regional flod wan transport modelling. deterministie Th c flotranspord wan t model applied assume environmenn sa t equivalena o t porous medium. Moreover maie th , n modelling wor bees kha n don meany eb f onlse o yon computer code, NEWSAM (Ledoux, 1986b). This finite difference computer code has its limits in simulating groundwater flow and isotope transport phenomena as quasi-three-dimensional and only taking into account advective transport with radioactive decay. Thus numericae th , l representatio regionae th f no l hydrogeological system used doe t admisno accuratn a t e simulatio isotope th f no e concentratio evern i y point

. PREVIOU3 S WORKS In this work a deterministic continuum approach with distributed parameters is followed to combine environmental isotope data with a numerical flow and transport model for a regional

180 system of aquifers. This approach comprises a very conceptual representation of isotope transport through hydrogeological systems physical .Al l and/or chemical phenomena whicn hca includee describee occub ar n d ca r dan beins da g continuou deterministicd an s . Tha, is t advection, molecular diffusion, hydrodynamic dispersion, adsorption, ion-exchange and other forms of retention as well as chemical reactions and radioactive decay can be considered. In many real, field situation spatiae sth l variation propertiee th f studiee so th f so d groundwater system make the mathematical representation too complex to be solved analytically. Instead a numerical metho r solvindfo transpore gth t appliede equatiob o t s . nha Ther e exist several numerical methods whic suitable har solvinr efo governine gth g equatio solutr nfo e transport, i.e. the advection-dispersion equation. For instance, finite differences, finite elements, the metho characteristicsf do , includin methode gth particlf so celln ei s (PIC randod an ) m walkd ,an also combinations of these, have been most commonly used. A large number of computer codes for applicatio thesf no e methods have been developed durin pasdecadese o gth tw t . However, only a few results of the application of these models for comparison with environmental isotope data have been published. Robertson (1974) seems to have been the first to document an isotope-verification of a two- dimensional, deterministic groundwater flotranspord wan t model base n numericado l techniques e solveH . floe th sw equatio meanfinite y nb th f eo s difference methode Th . calculated groundwater velocitie subsequentle sar y use simulato dt e advective transport, two- dimensional dispersion, radioactiv eexchangn decaio methoe d th y an y echaracteristicsf b d o . result e modee Th th f so l were compared with measured concentration tritiumf so , strontium-90 and chlorid Snake th n eei River Plain aquife proved satisfactorye an rb o dt isotopee Th . d san solutes studied originate from a radioactive waste disposal and are thus not environmental. Herweijer Luijn va Appel, n& o (1985) have applie dsimilaa r approac calibrato ht oneea - dimensional vertical flow and transport model with measured concentrations of environmental tritium. They use randoe dth m walk metho solutioe transpore th r th dfo f no t equation instead (see, e.g., Kinzelbach . 227-24pp , Custodion 5i , Gurgu Lob& i o Ferreira, 1987)e Th . calibrated mode subsequentls lwa simulatioye useth r dfo contaminatiof no n originating from heavily manured areas. Markussen, Möller, Villumsen, Mortense Selchan& u (1991) have applie computee dth r code SHE, European Hydrological System simulato t , three-dimensionae eth l flod wan transpor environmentaf o t l tritiu systea mn i maquiferf o s below Copenhagen, Denmark. This code solves the flow and transport equations by means of finite difference techniques and the version applied has been developed by the Danish Hydraulic Institute. Markussen et al. have compared the simulated and the measured tritium concentrations in the main aquifer of the studied area and they have adjusted the porosity values of the model to obtain a better fit. The resulting average circulation groundwatetimee th f so vicinite th n r i somf yo e wells producing drinking-water have subsequently been used to estimate the propagation of various pollutants threatening the groundwater quality in the Copenhagen area.

. HYDROGEOLOGICA4 L CONDITIONS

The investigated area is situated in southwestern Scania (see figure 1) at the edge of a geological border zone between archaean rocks in the north and thick layers of sedimentary rocks in the south Fennoscandiae .Th n Border calleds Zonei t i s ,a , form tectonisa c buffer zone betweee nth essentially rigid Fennoscandian Shield and a large area of subsidence, which comprises the Danish-Polish Trough bordee .Th r zone crosses Scani norta n ai h west-southeast directiod nan includes several crystalline f themhorstso Romeleâe e th , On . s hörst, constitutee th s northeastern are e limi interestf th a o f to . The geological evolution of southwestern Scania has been characterized by considerable tectonic movements. Most of the faults and flexures follow the northwest-southeast trend of the border zone. Fault othen si r direction also sd obed-roce occudivideth e b d n ran k ca d into blocks, define tectoniy b d c zone f weaknessso generaln I . blocke th , s southwese th f o t Fennoscandian Border Zone have subside consequenca s da alternatinf eo g downward tectonic

181 GERMANY

Figure 1: The hatched domain in southwestern Scania, the southernmost province of Sweden, is the area being investigated.

movement, marine transgressio sedimentationd nan . Today, these sediments have been petrified and for msequenca f sedimentareo y rockthreo t p esu kilometres thick. This sequencs ei dominate deposity db s fro Uppee mth r Cretaceous, probablcomparativela o t e ydu y rapid subsidenc sedimentatiod ean n during this period (Bjelm, Haïtien, Röshoff, Bennet, Bruch, Persson & Wadstein, 1977). The upper bed-rock consists mainly of Danian limestone. morphologe Th uppee th f yro bed-rock, show figurn almosi s i , e2 t completely maskey db Quaternary deposits. These deposits are only a few metres thick at the Romeleâs hörst and alon coastse gth . Toward Alnare sth p valle Quaternare yth y deposits become e thicketh d ran observed maximum is 183 m at Lemmestrotorp, situated in the southeastern part of this depression (Gustafsson, 1978). deee Th p sediments withi Alnare nth p valley consist mainl fluviaf yo l deposits, called Alnarp sediments. At the bottom of the depression they consist predominantly of gravel and sand. The sediments become progressively more fine-grained upwards and the main part of the Alnarp sediment mads finf i o ep e u san siltd dyan fine sand assumes i t .I havdo t e been depositea y db big river flowing from the south-east in the Alnarp depression (Hoist, 1911). This happened about 2500 30000o t 0 years ago, durin interstadialn ga warme,a r substage latese th f t, o glacia l period (Weichselian). At that time the sea-level is assumed to have been about 60 m lower than today, which would explain the position of the sediments. Later, silt and clay have been layere th deposite finf f so o ep santo n do (Ringberg, 1980).

182 Figure 2: Map showing the relief of the upper surface of the bed-rock in southwestern Scania (after Gustafsson Teeling,& 1973). valuesAll metresin above sea- level.

Alnare Th p sediments bed-roce wels th a , s a l botn ko h Alnar e sideth f so p depressionn i e ,ar general overlai Laty nb e Weichselian tills (se compositioee figurTh . differene3) e th f no t beds totae th l d f tilthickneso an l f tilso l vary markedly, both verticall horizontallyd yan . This variation is an effect of the complex glacial history in this area. Ice masses probably arrived from the north as well as from the east and the south only during the last part of the Weichselian, approximately 2500 1300o 0t 0 years ago. Furthermore, ther sometimee ear s fluvia glaciofluviar o l l sediments betwee differene nth t bed tillf so . These sediments have been deposited during ice-free periods when temporary déglaciation occurred. maie Th n groundwater reservoir withi hydrogeologicae nth l syste southwestermf o n Scanis ai formed by the uppermost part of the limestone bed-rock, which is relatively fissured. The comparatively coarse-grained Alnarp sediment hydraulicalle sar y connected wit limestone hth e reservoi the d therefore ran yar e also include thidn i s confined aquifer principae Th . l inflowo st

183 SW NE

m.a.s.1. 50_

-70—1 — -70 J Bed-rocU k '•A4 Coarse silt Fffl Block of chalk Clay ClaO G yclayed tilan l y till Alluvial sediments E3 Gravel and sand P-™ 0 5km E3 Fine sanI d I I I I I

Figur : 3 e Simplified geological cross-section (SW-NE) northwesterne th n i parte oth f Alnarp valley close Staffanstorpto (after Ringberg, 1980). outflowd an s fro maie mth n aquife aree th f investigatio ao f o r showe nar n schematicalln yi figure 4. The aquifer is chiefly supplied by: 1) Infiltration on and along the Romeleâs hörst. It is probably the relatively high altitude, the thin cover of comparatively permeable till, the fractured nature of the rocks and the steeply tilted sedimentary layers which together make this area one of the major zones of recharge. 2) Vertical leakage from small and shallow aquifers situated in the uppermost deposits of the hummocky moraine region in the centre of the area of interest. Here, the altitude is relatively high, the hummocky morphology reduces the surface run-off and the uppermost Quaternary deposits consist mainl f comparativelyo y permeable sand r clayeyo y sandy tills l thesAl . e features facilitäte the recharge of the main aquifer in this region. 3) Vertical leakage from small and shallow aquifers in the lowlands outside the two zones of recharge mentioned above. However thesn ,i e area rate infiltratiof sth e o n toward maie sth n aquife vers i r y sensitiv changeo et groundwaten si r exploitatio therefors ha t i d en an varie d greatly with time. Nevertheless, most of the recharge of the main aquifer seems to occur here today. principae Th l aquifer chiefly discharge: sby 1) Groundwater extraction in wells situated in the Alnarp sediments and in the upper part of the limestone bed-rock. Apar tfluctuations frow fe ma exploitatioe th , increases nha d since eth beginnin thif go s century. Today, this exploitation cause biggese sth t outflow groundwatef so r from the main aquifer. 2) Upward leakage from the main aquifer towards shallow, phreatic aquifers, which gives ris marsho et y tract springr so locan si l depressions. This phenomeno relativels nwa y common in large parts of southwestern Scania at the beginning of this century. Today, this type of outflow occurs onl smaln yi l flazonee th t f landso s north, wes hummocke soutd th an tf ho y central area. 3) Groundwater discharge into the Baltic Sea and the Sound, as the gradient of the hydraulic maie heath nn d i aquife r cause flosa w directed toward semipervioue coasth e d sth tan s layers overlying the aquifer are usually relatively thin in off-shore coastal areas. As mentioned above size inflow e somf ,outflowth e d o th an f eo o st s fro maie mth n aquifer have varie witt lo dha time majoe .Th r reaso increase thir th nfo s i groundwatef eo r extraction last century, which has resulted in a corresponding decrease of the hydraulic heads.

184 However, the head of the main reservoir has chiefly been lowered in the vicinity of the wells exploited, that is, in particular around the northwestern part of the Alnarp valley, but also in the wester southerd nan n lowlands. Abou hundree on t d years ago, ther mainls ewa y upward leakage fro maie mth n aquife thesn i r e areas. Today ,recharg e mosth aquifee f e to th th f eo n ri Alnarp sediments and the uppermost Danian limestone seems to occur here. The lowered hydraulic heads have probably also reduce groundwatee dth r discharge int seae oth . In the central parts of the area of investigation the long-term piezometric records concerning maie th n aquifer display only relatively small fluctuations. Consequently recharge th , e conditions have probably not changed very much here.

Romele Hummocky hörst moraine region

Extraction 25

In/Outflows in Mm3/year

Figur : 4 e Schematic representation principale oth f outflowsinflowsd an o t frome th main aquifer southwesternin Scania, approximating situationthe aboutin 1970.

5. ENVIRONMENTAL ISOTOPE CONCENTRATIONS 5.1 Oxygen-18 and deuterium 37 samples of groundwater from 32 wells have been analysed with respect to the stable isotopes of oxygen-18 and deuterium (Barmen, 1989). 25 of the sampled wells yield groundwater from the main aquifer. Burgman, Galle Westma& s n (1987 Called an ) Westmas& n (1989) have studiee dth isotopic composition of precipitation in Sweden. They have recorded mean annual values of about -9.7%c for 518O and -69%o for 52H at Arup, some ten kilometres northeast of the investigation area. The major part of this area has a more maritime climate than at the sampling station. Therefore, the isotopic composition of the precipitation in the area of interest is probably slightly heavier than at Arup. Burgman, Eriksso Westman& n (1983) claim thae isotopith t c compositioe th f no groundwater in southern Sweden should essentially reflect that of the mean annual precipitation. The major recharge in the area of interest is assumed to take place during the autumn, when the

185 isotopic compositio f precipitatioo n s usualli n y e annuaclosth o t e l th mea d an n evapotranspiration loss is relatively small. The isotope contents of the analysed groundwaters agree very well with this argumentation. Almost all 8i8Q values are close to -9%o (see figure 5) and the Ô^H values are about -65%o. The spatial variations and the differences between the reservoirs are very small. Therefore, the stable isotope content t seevere b no m yo t o suitablsd regionar efo l groundwater flow modellinn gi southwestern Scania expectee b .o t Thi s thers d sa comparativelwa e ear y small variatione th n si are interesf ao regards ta s climat altituded ean .

OXYGEN-18

Figur : 5 e Distribution oxygen-18e oth f contents groundwatersn i sampled Barmeny b (1989). Figures boxesin represent shallow aquifers, while figures italicsin represent groundwater from Alnarpthe sediments. veryvalueslow Two are underlined and discussed in section 5.4. The other values are for water from the rest of the main aquifer. All values are given as &S()SMOW in per mil.

Tritiu2 5. m

Ther ver e measurementew ar yfe naturae th f so l tritium content precipitation si n from before 1952. Since that year tritiue th , m from nuclear weapon test overshadowes sha naturae dth l productio somy nb e order magnitudef so . However assumee b n ca t di , fro generae mth l data presented in Fritz & Fontes (1980) that the background level in southwestern Sweden should be

186 Perer. abouTU Johanssos0 & t2 n (1980 Seveld )an , Kelstru Binzep& r (1981) have carriet dou hydrogeological investigations in or in the vicinity of the area of interest. They claim that the tritium concentratio precipitation ni probabls nwa y relatively constant before 1952, witn ha averag . SaxeneTU clos0 a1 opinioe (1990o eth t f o ns )i that these pre-bomb values might have accordin, abous TU a 4 w beet2- estimate lo go t s na s base analysen do winesd ol f so . In this study an average tritium concentration of 18 TU in precipitation before 1952 has been caseadoptedw alternativfe n sa a alss n I . ha o beeU eT nvalu0 1 used f eo . Accordine th o gt references mentioned, both of these values might be somewhat too high. However, whether 4, 10 or 20 TU is adopted as an average for the period before 1952 has a comparatively small influenc interpretatioe th n eo today'f no s tritium concentration maie th nn si aquifer. Also as regards the tritium contents in precipitation after 1952 in the area being studied, unfortunately only relativel measurementw yfe s exist onle .Th y ones availabl thir efo s work have been undertaken on precipitation collected at Skurup in the southeastern part of the investigation area. Nevertheless, it is usually possible to make highly valid extrapolations from other sampling sites with similar geographical conditions. By means of such extrapolations the annual average tritium concentrations in precipitation in southwestern Scania have been estimated. The estimations are based on correlations between records from the sampling stations Skurup and Huddinge in Sweden and Taastrup and Odum in Denmark. Values for years without measurements in those places have been extrapolated from the long tritium records in Ottawa and Vienna. Groundwater aree th a n sbeini g studied have been sample analysed dan d with respec tritiuo t m since the early sixties, as reported by Nilsson in 1966. The general results of these analyses and the detailed results of some 100 other tritium analyses have been available for this work. Most of the latter data have been collected in the hydrogeological mapping of southern Sweden (Gustafsson, 1972,1978,1981 and 1986) or as a part of the groundwater monitoring program Swedise th f o h Geological Survey (SGU) additionn I . furthea , well2 2 r southwestern si n Scania have been sample tritiur dfo m content (Barmen, 1989). Finally, samples from three wells at Grevie, taking groundwater from the Alnarp sediments for the Malmö water-works, were analysed in 1991. As before, the conditions within the main aquifer were the main focus and the vast majority of the sampled wells yield groundwater from this aquifer. A comparatively large number of samples for tritium analyses were collected in about 1971. The results of the analyses regarding groundwater from the main aquifer are shown in figure 6. Tritium data also from shallow aquifer available sar archivee th Swedis e n ei th f so h Geological Survey. 24 samples were collected in 9 wells in the southeastern part of the investigation area ove yeare rth s 1973-1975 sample e mosn .th I f tritiuto e sth m concentratioe n Th exceed . TU 0 s5 maximum tritium content is 209 TU and the average of these samples is about 75 TU. Furthermore, the Danish Geological Survey (DGU et al, 1975) have also collected samples for tritium analysis around the Danish part of the Alnarp valley. They have found no measurable tritiu groundwatemn i r fro Alnare mth p sediment limestone th d sothean e eth belorn O . wit hand tritiue ,th m concentratio watee th Lakf n ri o e Esru bees m. ha n determineTU 4 10 e b do t The data on the distribution of tritium in groundwater may be summarized^ and interpreted as follows. The outlines of this interpretation are more or less in accordance with the opinions of, among others, Nilsson (1966), Gustafsson (1972,1978,1981 and 1986), Perers & Johansson (1980) and Brinck & Leander (1981). The mentioned average groundwater circulation times are roughly estimate one-dimensionaa y db l piston flow model, usually leadin under-estimatesgo t . The concentrations are comparatively high in samples from lakes and shallow aquifers. High tritium values have also been found in groundwater from the main aquifer along the coasts and along the northeastern boundary of the area in question. In these zones the Quaternary deposits are comparatively thin and in the southwestern coastal district the extraction of groundwater from the main aquifer is extensive. Some samples from the central zone southwest of the Alnarp valley also showed somewhat enhanced tritium contents, in particular in 1971, but also in 1982. At most of these sampling points the Quaternary deposits were locally comparatively thin and also permeable. In the cases mentioned the tritium concentrations were at least 5-10 TU and often more than 20-3. 0TU

187 TRITIUM CONTENTS

OBSERVED

.4 contents in TU

U T 2 « f •

Figure 6: Tritium contents measured in ground-water from the main reservoir in southwestern Scania in 1970 and 1972 (from Gustafsson, 1972 and 1978). Four values from 1974 threeand from 1975 alsoare included (data from Gustafsson, 1978 1981, and also and unpublished data from Swedishthe Geological Survey). The interval 3-24 TU, marked for a sampling site in the southwestern part of the area, indicates that the tritium contents have varied between these values in seven samples taken between 1970 and 1975.

The conclusions to be drawn from observations of enhanced tritium contents in the main developee aquifeb y rma d furthe placet a r s wher locae eth l hydrogeological condition wele sar l known. In certain zones, it is plausible that the high tritium values indicate that groundwater withdrawal mora s o havt e d rapiele d turn-ove stimulatioa o t e rdu rechargf no e from shallow aquifers and surface waters.

5.3 Carbon-14 Ther mane ear y processes whic affecn carbohe ca th t n budge groundwatera f to . Consequently, usualls i straight-forwart a i t yno triviad dan l tas interpreo kt resulte th t carbon-1f so 4 analyses of groundwater samples. Several attempt overcomo st e these problems have been made during

188 pase th t thirty numbeyeara d s an possibl f ro e correction procedures have been presentey db different authors. Methods for carbon-14 corrections are usually applicable on either "closed-" or "open system" conditions and in some cases also on mixed conditions, which is perhaps the most realistic descriptio manf no y real systems open a n .I system with respec CC>2(g)o t , carbonate dissolution occurs while the solution remains in contact with an abundant gaseous environment of constant pCU2, such as the atmosphere. In contrast to this, the dissolution proceeds in the absenc CO2(gf eo closea n )i d system. Consequently equilibriue ,th m pCC>groundwatee th f 2o r will decrease, when carbonic acid is consumed in this latter case. maie Th n aquife soutf o r h westernmost Swede generalln nca characterizee yb closea s da d system consistin closn i r eo , contacgof t with limestone. Accordin Fonteo gt s (1985)e th , correction methods presented in Pearson & Hanshaw (1970) and in Fontes & Gamier (1979) shoulmose th te db suitabl e one thin si s typ situationf emethodo o tw e s.Th give similar results for most samples collected in the area of investigation. 14C activities estimated by the method of Fontes & Gamier with feasible input parameters are presented in figure 13. A few values from wells penetrating deep into the limestone and with very high carbon-13 contents have been calculated by the correction process outlined by Wigley, Plummer & Pearson (1978). More informatio inpue th n t no value r thessfo e figures wels a , s soma l e alternative resultse ar , presented in Barmen (1992). sampleSome th f eo d wells penetrate deep int limestone oth maie th nf e o reservoi shod ran w high content f 13o sC . These feature y indicatma s e that incongruent dissolutiod an n reprecipitatio calcitf no e occu vern ri y slowly circulating groundwater sucn I . hsituatioa e nth correction procedure presente Wigleyy db , Plumme Pearsor& n (1978) woul adequate db d ean a slightly simplified version of it has been applied to the samples here. To estimate the effects of incongruent dissolutio reprecipitatiod nan n better, more detailed geochetnical informatioe th n no groundwater system being studie neededs di influence th f .I sucf eo h processe substantials si , many of the corrected 14C activities presented in figure 13 are likely to be too low. effece fractionatioTh f o t carboisotopie e th th soie d n no th lan c n compositioi 2 CO e th f no dissolve groundwatee th n di probabls rha reachet yno d equilibriu watee th mn ri rechargine gth main aquifer, accordin generao gt l description Pearsoy b s Hanshan& w (1970)a s A . consequence, the activities obtained by the method of Fontes & Gamier (1979) and presented in figure 13 may be slightly too low. Finally, anothe wordw rfe f cautionso increasee Th . d exploitatio maie th nf n o aquifen i r southwestern Sweden ove pase th r t centur generan i s yha l change directioe dth f flono w through the aquitardeous layers. Therefore, the groundwater extracted may first have been moving slowly upwards for a very long time and then increased withdrawal has caused a principally downward migration toward exploitation sa n well. This means thameasuree th t d 14C activities in the sampled groundwaters do not reflect today's hydrogeological situation. The activities are lower and the estimated "ages" are older than they would have been had they represente actuae dth l average circulation maitimee th nf s o aquifer . This proble mmoss i t severe for the interpretation of radioisotopes with relatively long half-lives, a point which has been

well describe Geyy db Backhauh& s (1978 Sudickd )an Friny& d (1981). These authors also trea relatee th t d difficulty that water with 14veractivitieC w ylo n si stagnant groundwater bodie diffusy sma e int orelativela y young groundwater. Such relatively wated ol r might residsmallese th n ei t pore fissurer sclao e th y f limestonestillso d san . Thereby, groundwatee th r velocities will lowee seeb mo t r than what coul assumee db d froe mth hydrodynamic conditions and the kinematic porosity. However, these problems shoul t affec de comparisonno th t s betwee correctede nth , measured C activitie simulatee th d san d ones discusse followinge th n di simulatione Th . f so transiene th t evolutio 14 floe transpord th w an f no t phenomena will star calculatiny tb gsteadya - state situation in 1840, prior to any exploitation of the main reservoir. Furthermore, the simulations shoul properte dus y sets with porosities higher than normally estimate beins da g kinematic ones, to compensate for diffusion effects from groundwater residing in the smallest pore fissured clae san th y f tilllimestoneso d san aree th a f beinso g studied.

189 5.4 General results from the isotopic investigations Very shallow aquifer likele sar havo yt e short, mean turn-over times, years perhapw fe a , f so contaid an n recently recharged water average .Th e circulation time aquifern si imer-morainin si c sediment estimatee sar somewhae b do t hundret w longer ordefe e a th f df ro , o years . groundwatere Th maie th nn si aquife uppee th limestonn e ri parth f to e outsid Alnare eth p Alnare th vallen i p d sedimentyan s usually see mhavo t emuca h lower rat renewalf eo . Mean turn-over times range from more than 40 years, estimated from tritium measurements, to 500 - 2500 years, accordin correctego t d carbon-14 activity. Nevertheless coastan ,i l zones, alone gth northeastern boundary and also in a few other localities in the area being investigated, where the Quaternary deposit relativele sar y thipermeabled nan , recently recharged groundwaters have been detecte majoe th n dri aquifer. These high content f radioisotopeso indicaty sma e that groundwater withdrawal mora s o havt e d rapiele d turn-ove stimulatioa o t e rdu rechargf no e from shallow aquifers and surface waters and that the risk of pollution of the main aquifer has increased. Conclusions base datn do a from locations wher hydrogeologicae eth l conditione sar better known could obviousl furthee yb r elaborated. Groundwater sample deedn i p wells, penetratin least ga t several ten metref Daniaso e th f so n limestone, often see represenmo t t reservoirs with very long circulation times chemicae .Th l contents are high and the concentration of radioisotopes is very low. Estimates based on 14C activity indicate average turn-over times of 4000 -12000 years. Moreover, two samples from this typ reservoif eo r belo Alnare wth p valley hav (seO e18 e ver d figurcontentw an y lo eH 2 f so . Thi5) s feature indicates that these groundwaters least a , t partly, have been recharged duringa period with generally colder climate numbee Th . f observationo r e th smalo d to s an lsi interpretation tools are not accurate enough to make these observations convincing. Nevertheless speculatn ca e ,on e whether groundwater influence rechargy db e durin finae gth l stage of the latest glaciation might have been retained in hydrogeological "pockets" within the main aquifer. contraste Th 18d s Qbetweean concentrationH 2 e nth groundwatee th n si r from different aquifee partth f so r smalsystesuitablo e followine b to th o e t lmr ar e fo g regional transport modelling. There is not even an indication of an admixture of sea-water, rich in these stable isotopes groundwaten ,i r from wells situated coasts close th o et . As regards tritium, there are significant contrasts between the different parts of the investigated groundwater system. However, the contrasts within large areas of the main aquifer are small. Due to the short half-life of tritium and to the usually, long mean turn-over time of this aquifer, most samples from it show very low contents of tritium. This may reduce the possibilitie drawinf so g reliable conclusions fro mcomparisoa n wit result e transpore hth th f so t modelling. Furthermore woult ,i vere db y valuabl colleco et t sample tritiur sfo m analysis from several different levels withi aquitardeoue nth s layers. Such results could probably elucidate eth recharge processes in the main aquifer. Finally bees ha n t i ,emphasize triviaa t dno l thas tasi t interprei to k t contentC t s froma system including limestones and 14 cla y tills comparatively rich in lime. In spite of the many uncertainties related to such an interpretation, contrasts in the concentrations of the same type as for tritium may be detected. Using both of these radioisotopes in comparisons with the subsequent isotope transport modellin probablgs i benefio t besy e yth wa t fro resulte mth f so isotope investigations.

6. GROUNDWATER FLOW MODELLING

The computer code NEWSAM is founded on a classical phenomenological, macroscopic description of groundwater flow through porous media. It is assumed that the flow within an aquifer takes place onl quasi-horizontaa yn i l plane thad t,an water passes aquitardse onlth yn i perpendicular direction. With this approximatio avoin ca de treatinnon reaga l three-dimensional flow problem, whic muca s hi h more laborious task. NEWSA usee b obtaio dy t Mma n na

190 approximate solution by means of finite difference techniques. The code utilises nested squares of variable size for the subdivision of the area of interest into discrete cells. The mean value of the hydraulic head is calculated for each cell or mesh and for each time step considered. For a given distributio aquifef no r propertie defined san d initia boundard lan y conditions, there exists only one solution to the flow equation above. The reader is referred to Ledoux (1986a and b) and Marsily, Ledoux, Levassor, Poitrina Salel& m (1978 mora r )efo detailed descriptioe th f no principles behind the finite difference computer code NEWSAM.

Subdivisio1 6. n into layer discretd an s e meshes majoe Th r aquife studiee th f ro d hydrogeological syste mconfines i situated uppede an th n dri limestone parth f o t eAlnare bed-rocth n i p d sedimentskan . This groundwater aquifen i s i r general separated from surface water and local, shallow aquifers by thick layers of till, which functio aquitardss na shalloe .Th w reservoir representee smodelar e th f o ,laye 1 y whil d. b rno e the botto modele th m f o layer , 2 correspond . ,no maie th no s t aquifer elemente .Th layef so . rno assignee ar 2 d leakage coefficients, which gover verticae nth l exchang watef eo r betweee nth two strata. In this manner the variable thickness and vertical hydraulic conductivity of the almost impervious clayey clatilld syan till takee sar n into account. The two superposed layers of the model have been subdivided into meshes of up to four different sizes. In general, the small mesh sizes have been used in zones where the available informatio relativels ni y dense and/or particulawherf o s i t ei r interes obtaio t n comparatively detaile reliabld dan e results fro modee mth l calculations meshesg othee bi th e rn .Th o ,hand , have chiefly been positione zonen di s wher existine eth g hydrogeological dat relativele aar y scarce and/or where it is less important to obtain detailed results from the model calculations. As regards the bottom stratum, the grid-net has been designed with the situations of wells wit groundwateg hbi r withdrawal welld san s with long piezometric record basise a s th sa n .I first case, big piezometric gradients in the vicinity of the pumped wells motivate small elements. secone Inth d case meshee ,th s smalhave b mako eo t lt e possible relatively reliable comparisons betwee measuree nth d piezometric level thosd san e derived fro modele mth . Areas withoue th t above types of wells are subdivided into big meshes (see figure 7).

6.2 Assumptions and boundary conditions Average values of transmissivity, the leakage coefficient and storage coefficient concerning each mesh in the model are required for the numerical calculations. In addition, an estimate of the real, mean piezometric leve eacf lo eachr mesfo hd timhan e step considere necessars di r yfo comparison wit correspondinge hth , calculated valuemako t d es an possibl calibratioe eth f no floe th w model. However numbee th , f measurementro f hydrogeologicaso l propertied san variables of state is very small compared to the number of elements in the model. Therefore, it necessars i makyo t e considerable assumption approximationd san able b estimat o et o t s a o ess the mentioned input dat eacr afo h mesh genera.A l proble mtha s sizcomplexiti e d tth e an e th f yo hydrogeological system demand denssa e subdivisio modee th f no l into small elements, while the existing density of observations is not sufficient to permit such a high mesh frequency. estimatione Th inpuf so t dat eacr afo h mesh have mainly been don han y somewhaea b n di t subjective manner. As the measurements of hydrogeological properties are relatively few, the estimates have been based on qualitative hydrogeological information, mainly derived from Gustafsson (1972, 1978,1981 and 1986). Another major source of information are the almost 3000 drilling record wellf so southwestern si n Scania, whic storeWele e har th ldn i Archive f so the Swedish Geological Survey. They usually comprise a fairly good description of the geological stratification along the bore-holes. Piezometric observations and results from well capacity tests are also often included. Assisted by this additional information all meshes of the model have been assigned tentative and approximate, but hydrogeologically reasonable values of transmissivity, leakage coefficient, storage coefficien initiad an t l piezometrye Th . assessments of the boundary conditions have been made in a corresponding way.

191 N

10 20km

Figure 7: Subdivision of the bottom layer into 1238 meshes of four sizes. This stratum of the numerical groundwater flow model represents the main aquifer in the area of interest.

Another simplificatio modee th n nthas i li seasona o tn l variation instancer fo , sof , pumping rates, infiltratio piezometrid nan c level takee sar n into consideration. Only annual averages have been judge s relevana d r thifo ts investigation, which mainly considers long-term hydrogeological processe regionaa n so l scale additionn I . , detailed information about seasonal variation availabls si e limiteonla r yfo d numbe placef ro variabled san s over relatively short periods. As wil discussee lb d below recharge th , e processe highle sar y simplifie thin di s modele .Th piezometri cuppermose leveth f o l t laye regardes ri constans da t with time. Therefore, apart fro leakage mth e coefficients onls i differenc e t i y,th e between this constane t valuth d ean calculated piezometric mai e leveth nf lo aquifer , which determin amoune eth rechargf to e th o et bottom layer of the model. Interactions between surface wate groundwated ran almoss ri t neglecte floe th w dn i model . Nevertheless, the piezometric levels are prescribed where there are lakes and rivers. It should als mentionee ob d tha bottoe tth m mode e layeth f ro l doe havt hydrauliy sno ean c contact with reservoirs situated below it. In reality, a certain groundwater exchange of this type may take place, in particular along fissured fault zones. The reservoirs situated at great depths have been neglected, as their global influence on the system of interest is most likely relatively small. Moreover, ther hardls ei informatioy yan n availabl thein eo r hydraulic propertied san hydrogeological status. One of the major boundary conditions of the groundwater flow model is that the water table of the upper layer is prescribed in each mesh. The piezometric level of this layer varies spatially but it is regarded as stationary. The average ground surface level of every element has been

192 approximate mean topographicay e db th f so lscal e mapth en so 1:50000 meae .Th n piezometric level has been determined somewhat subjectively to be located 1 - 6 m below the ground surface level of the mesh. No horizontal flow occurs in the local phreatic aquifers due to the completely prescribed piezometry of the upper layer. In fact, the main purpose of layer no. 1 of the model is to simulate the recharge of the principal aquifer. If the hydraulic head on the bottom stratum is situated below the water table in certaia n place there downwara wil e b l d vertica modelle floth n wi , tha , rechargeis t e Th . prescriptio piezometrie th f no c uppelevele th f sro stratum means tha mesta willayef 1 h o . l rno be supplied with water from "outside" until flow and mass balance is obtained within the whole syste meshesmf o hydraulie th f .I cbottoe heath n mdo stratu msituates i d abov prescribee eth d water table in a certain position, groundwater will be transported upwards and leave the model from the upper layer in the same manner. The magnitude of the recharge or discharge will be proportional to the vertical piezometric gradient and also to the leakage coefficient representing the aquitardeous layers between the two strata of the model. While the piezometric levels of all the meshes of the upper layer are prescribed, there are no meshes with completely prescribed piezometric botto e heath dn mi layer northeastere .Th n limit bottoe oth f m laye defines ri boundara s da y wher horizontao en l groundwater pasfloy wsma the border. However, the area on and along the Romeleâs hörst is a preferred zone of infiltration and recharge. The hydrogeological situation along this border, with a relatively big vertical inflomaie th nwo t aquifer reproduces i , modee th meany y db l b hig a f sho elevatiof no prescribee th d piezometric additionleveln I . layef 1 s o . meshe e boundare rno th , th t sa e yar assigne leakagg dbi e coefficient tako t s e into accoun face th tt thaaquitardeoue th t s layere sar comparatively thin and permeable there.

Result3 6. s from groundwater flow modelling Boundary conditions with prescribed piezometric level drainagd san e meshe accordancn si e with the previous description have been applied in the following. Furthermore, the storage coefficients of the main aquifer were assumed to be uniform and equal to 5 • 10- in the transient simulations. After various tests initian a , l time-ste hour8 bees 16 sf ha p no founo dt 4 giv satisfactorea y convergenc non-stationare th f eo y numerical calculations time-stee .Th s pha been increased progressivel calculatione squarth e th n i y yo eb rooeacr tw sf fo o th period- normall yyeara . After some adjustment estimatee th f so d tentative transmissivitie leakagd san e coefficients, the resulting distribution of values was assumed to represent the real groundwater system relatively well, as a fairly good correspondence between model-derived and observed steady-state and transient piezometric values was obtained in most parts of the area of the investigation difference Th see e . b 9 figure n n i d eca an betwees s8 a , n calculate observed dan d values is usually less than two metres in these figures. However, as mentioned earlier, several different distributions of transmissivities and leakage coefficients could probably have led to an acceptable piezometric fit, due to the complexity of the modelled system and the relatively small numbe f observationso r . Nevertheless alternatine th , g calibration procedure considering steady-stat s welea s transiena l t conditions, significantly limit numbee th s f adjusteo r d combinations possible. alsIs ti o interestin compargo t piezometrie eth representinp cma g steady-state conditionn si 1970 derived fro modele mth , wit correspondine hth obtainee gon d afte transiena r t simulation of the piezometric evolution 1840-1970. The two maps are very similar, which strengthens the conclusion that the groundwater flow model is fairly well adjusted. Another way to try to verify the results of the flow model is to compare the different component watee th f sro balanc groundwatee th f eo r system wit correspondine hth g estimates. However, these estimates are relatively rough and it is therefore not possible to make a very accurate verification. Nevertheless figuree th , s presente tabln di eindicatI e thaadjustee tth d groundwater flow model gives reasonable result alsd osan tha resulte tth s from steady-statd ean transient simulation similare sar modellee .Th d outflow perhape ar a s se sint e slightloth o yto

193 small, whil modellee eth d inflow northeastere th t sa n boundary seeslightle b m o bigt o yto . This type of flow result can also be used to illustrate how the overall inflow-outflow pattern and magnitude have changed over the past century.

Figure 8: Piezometric map of the main aquifer, corresponding to the situation in 1970. The dashed lines with figures in italics show the results obtained by steady-state calculations with adjustedthe groundwaterflow model. Theyare compared piezometricthe to based observationsmap on (continuous lines). piezometricAll values givenare metresin above sea-level. calculatedThe piezometric curves are interrupted at the shore-line to make the presentation clearer.

194 m.a.ï.l. Löddckoping* '0. »-— — ~° 8: ^~> - 0 ^v^ u »- •N^ je 6: S s .2 *: ^, 3 9- s, o 2~- \i \ \ T3 o: >N i \ / F I : tyf -

1B90 1910 1930 19501970 1990 Year 2 m.a.».l. Awrp 18: •^— —. ^ T,'*' ^ 0 12; D '' - ^\ ^s "-in: "^v. V- Ü lA "5 fl- Xvu*k Ü «: ia|\ ^^ X >A[ 2: fur /^ /- *VP \^. o-i ïSP nj JLr 1890 1910 1930 1950 1970 1990 Year

m.o.».l. Ekaholmwi 48

o : !«y 4•2 •038 36 3**5Ç= -5 1890 1910 1950 1970 1990 1890 1910 1930 1950 1870 Year Year

m.a.s.l. Klagitofp Sv»dolo 30- -§28

_O 24- 3 -, 0 22~

16- 189O 1910 1930 19SO1970 199O 1890 1910 1930 1950 197O 1990 Year Year

Figur : 9 e Piezometric variations somen i selected observation wells (thin lines) compared with values obtained transientby model calculations (thick lines). Measurements markedare smallas filled squares.

195 TABLEI Calculate estimated dan d magnitude differene th f so t componenet watee th f rso balance concernin maie gth n aquifer figurel Al . givee sar Mmn i /year. 3

Components of the Steady-state Transient Estimated total water balance model results model results values maie ofth n aquifer for 1970 for 1970

Limhamn limestone quarry 1.2 1.2 1.2 - 1.5 Klagshamn limestone quarry 0.3 0.3 0.3 Outflow to the sea 5.3 5.4 5-8 Extraction from wells 26.8 26.8 26.8 Total outflows 33.6 33.7 33. 336.- 6 Northeastern boundary 8.0 7.9 6-8 Net recharge elsewhere 25.6 25.8 21-33 Total inflows 33.6 33.7 27-41

7. GROUNDWATER TRANSPORT MODELLING

Accordin objectivee th go t thif s o conclusion e s worth d kan interesr sectiof so ou 4 n5. t wile b l focussed on the transport of the radioactive isotopes tritium and carbon-14 through a regional groundwater system reasonabls i t I . assumo et e that commonly occurring concentrationf so tritium and carbon-14 neither change the density of the groundwater, nor modify the groundwater flow. Consequently, the flow equation and the transport equation may be solved one after the other and not with full interaction between the two phenomena. In general, tritiu subjecme changesassumes y i b an o t o t t t dno , exchanges reactionr o , s during migration, other than radioactive decay. On the other hand, carbon-14 is known to be influence severay db l physical, geochemical, radiologica biologicad an l l mechanismsn A . attemp correco tt measuree tth concentrationC d thesr sfo e effect bees sha n mad mentiones ea d in section 5.3. 14 Ala-Eddin & Magnusson (1991) have shown that in a hydrogeological environment like the one of southwestern Scania, variations in the dispersivity have a relatively big influence on solute transport on the local scale. However, the author's experience from preliminary tests with the computer code METIS (Goblet, 1985) is that the sensitivity to variations in the dispersivity is relatively small on a regional scale. Solute transport is more sensitive to variation sucf so h propertie hydraulis sa c conductivit porosityd yan . Dispersivitie likele sar o yt differ very much among the different geological materials in the area of interest Furthermore, dispersion coefficient vere ar sy difficul measuro t t estimatr eo regionaa n eo l scal tenf eo f so kilometres. Thi particularls si y true whe time nth e spa interesf no t covers more thayear0 n15 s groundwatee th d an r flow patter changes nha d markedly during this period consequencea s .A , omitting explicitly dispersion from the transport equation will probably give a satisfactory representation of the transport phenomena if our interest is focussed on the major characteristics and behaviour of the regional hydrogeological system over the past century. simplificatione Th s mentioned will followinresule th n ti g transport equation:

where concentration, as mass of solute per unit volume of solution V Darcy velocity vector n porosity t time K radioactive decay constant

196 According to this description, the transport of the dissolved substances will be in the same directio wit d same nhan th e velocit transpore th groundwate e s ya th f to r moleculesa , thavi , tis advective transport. The isotopes will decrease by radioactive decay as they are transported. The computer code used for the groundwater flow calculations may also be used for simplified transport simulations principlen .I code ,th e calculates solute transpor accordancn ti e wit equatioe hth n above, approximate finity db e differences. Ledoux (1986b) argues that NEWS AM give satisfactorsa y representatio transporf no t phenomen purpose th studf ao i t s ei y regional hydrogeological characteristics. In general, it does not provide an accurate simulation of the solute concentrations at every point of an aquifer system. Nevertheless, calculations with this computer codhely gaipo e t ma understandin n na regionae th f go l transport mechanismsn .I general, finite difference approximation f solutso e transport equation causy s ma relativele a y large numerical dispersion. Therefore, it should be emphasized once again that the results presented in the following are to be regarded only as a first approximation of the regional solute transport phenomena.

Subdivisio1 7 . n into layer discretd an s e meshes The subdivision used earlier has been extended to a somewhat more fully three-dimensional representatio orden i modeo rt transpore lth isotopef to s throug groundwatee hth r system being studied. The shallow reservoirs are represented by layer no. 1 of the model, in the same way as previoue inth s sections, whil almose eth t impervious clayey tillclamodelled w syan no till e sar d as 14 strata, no. 2-15, each 5 metres thick. They are of various sizes to take into account the variable total thicknesse aquitarde th f so southwestern si n Scania bottoe .Th m stratum, laye. rno 16 of the model, represents the main aquifer. It is 30 metres thick, which corresponds to the fissured, upper part of the limestone bed-rock and the sandy sediments at the bottom of the Alnarp valley. The positions of some of the 16 layers are shown schematically in figure 10. It must be emphasized that the model representation of the variable total thicknesses of the aquitard layers is very much simplified and schematized. Only strata classified as clay, clay till and clayey till from drill-hole record takee sar n into account determinatioe th r ,fo abovee th f no - mentioned number of layers. The transport through other geological materials situated within and below these strat assumes ai occuo dt r instantaneously compare transpore th o dt t times throug aquitare hth d layers. This assumptio strengthenes ni resulte th y morf db so e detailed, finite-element calculations execute Ala-Eddiy db Magnusso& n n (1991 a simila r fo ) r environment. However dilutioe ,th n isotopeffecte th n o se concentration verticaf so l transport run the risk of being under-estimated, as the volume of "old" groundwater remaining in the more coarse-grained layers is neglected by the model.

xWatertable

Semipermeabte

J ______Aquifer

Figur : 10 e Schematic positions layers6 1 e o th numericalfe oth f model of groundwater flowtransportand model.

197 Leakage coefficient assignee ar s meshe l al o dt layerf so . 2-16sno , regarding leakage from or to the layer situated above. Their values have been chosen to give the same leakage through thaquitar4 1 e d layer betwees aquifeo a s tw e rnth layers used earlier. Furthermoree th , coefficients have been assumed to be equal in meshes overlying each other. Thus, vertical variations in the leakage properties have not been modelled. However, zones with relatively thin and/or permeable aquitardeous layers in the real geological environment have been represented layerbw yfe s wit leakagg hbi e coefficient numericae th n si l model. In addition to the prescribed piezometric levels of the uppermost layer, also the isotopic concentration prescribee sar d when simulating transport tritiume Th . concentratione th n si meshes of the uppermost layer have been fixed at 18 TU before 1952. As regards the period after 1952, the tritium concentrations of the uppermost layer have been set equal to the average concentration precipitation si estimates na section di n carbon-1e 5.2Th . 4 concentratione th n si meshes of the uppermost layer have been fixed at 100 pmc in most of the simulations. simulatione Th isotopf so e transpor aquifee th n ti r syste minteresf o t have been done under mor r leseo s identical condition wers sa e applie groundwatee th n di r flow simulationsn I . general e "adjustedth , " propert t mentioneyse section di s bee ha n n6 used situatioA . n correspondin long-tergo t m steady-state condition 184n s i bee s 0ha n estimated steady-state .Th e calculations have been initialized with zero-concentration isotopef so l mesheal n si s excepr tfo those of the uppermost layer. From the resulting situation the changes until 1988 have been simulate transieny db t calculations. After 195 prescribee 2th d isotopic concentratione th n si uppermost layer have been changed annually.

Result2 7. s from transport modelling When simulating isotope concentration accordancn si e wit equatioe hth n given above, onle yth porosity and the radioactive decay constant will be additional input properties compared to the groundwater flow simulations. The latter is in general determined satisfactorily. On the other hand, the porosity values of interest as regards isotope transport are usually much more difficult to determine. Most porosity measurements concern the total porosity. However, it is the kinematic porosity whic oftes hi nmose judgeth te b adequat do t solutr efo e transport. Unfortunatelys i t i , more complicate determino dt e this porosity r instanceFo . plausibls i t i , e thakinematie th t c porosity varies with the hydraulic gradient. In principle, there is a big difference between the total porosity and what is usually assessed as the kinematic porosity of clay till. The groundwater flow crosses this material relatively slowly. Thus, there will probably be time for equilibration between the truly mobile water and those water molecules, which are instancer fo , ,surface e fixeth o dt f geologica so l materiae ar r o l stagnant in dead-end pores. Consequently, a much bigger fraction of the porosity than the strictly kinematic one will probably be "effective" in the migration of substances. This latter porosity whic e valuon moss e hbelieves i ei th te b reasonabl simulationn o di t e us r efo f so isotope transport. On the whole, this discussion shows that it is very difficult to assign porosity values to the meshe transpore th f so t model. Therefore sensitivite th , modee th f variationo yo t l e th n si porosity has been investigated. Due to the lack of detailed information and also for simplicity, the porosities have been assumed to be uniform within both the aquitards and the main aquifer. Values within a comparatively wide interval have been tested in the simulations of isotope transpor aree interesf th a o n ti t Moreover, the calculated tritium contents are close to 0 TU and the calculated carbon-14 contents are between 80 and 100 pmc at almost all sampling points. Therefore, plots of the calculated isotope values agains observee tth d ones give very little information (see Barmen, 1990). theso t e e difficultiesDu result e transpore th ,th f so t model wil presentee lb d mainl maps ya s of the calculated tritium and carbon-14 distributions in the main aquifer for the selected years.

198 The corresponding, observed values will also be shown for general comparison. The maps will concern the years around 1971 and 1988, when the isotope measurements were comparatively frequent (se e12)d figuran .1 1 e generae Th l pattern vere sar y simila tritiue th n rmi distributions presente figuren di la-c1 s . Both the calculate measuree th d dan d value enhancee sar certain di n zones alon coaste g alonth d san g the northeastern boundary, wher aquitare eth d layer thine tritiue sar Th . m value alse sar o relatively high centraher e therd th ean n ei l areas. However lattee th rn ,i cas positione eth e th f so high, measured and of the high, calculated concentrations, do not correspond very well. The majorit botf ymeasureo e hth calculated dan d value muce sar theso ht lowee e Du . r thaTU n2 features, the figures do not make it possible to determine which of the sets of effective porosities used, best correspon realityo dt . Nevertheless, results from simulations with other set propertief so s mak likelt ei y thae t effectivth n i cla% e eth 15 y porositie n d tili lan % 30 f so main aquifer are the smallest ones which would result in relatively reasonable simulated tritium

TRITIUM CONTENTS CALCULATED § >10 TU S 3-10 TU U T 3 < D OBSERVED .4 contents in TU

+ «2 TU

Figure lia: Calculated observedand tritium contents mainthe in aquifer correspondingto the situation aboutin 1971. effectiveThe porosities calculationsthe in are mainthe in aquifer. claythe 15% tillin and 30%

199 TRITIUM CONTENTS

CALCULATED

B >1»TU

S 3-10 TU

CD <3 TU

OBSERVED 4 . contentU T n I s

<2 TU

Figure lib: Calculated observedand tritium contents mainthe in aquifer, corresponding situationtothe aboutin 1971. effectiveThe porosities calculationsthe in are 30% in the clay till and 30% in the main aquifer. concentrations. However, ther alse ear o some clear differences betwee resulte nth s presentedn i figures lla-c effective th . f Wheo y e nporositiean increaseds si zonee th , s with enhanced, calculated tritium content maie th nn s i aquife r become smalle lesd sran numerous. The general patterns in figure 12 are very similar to those in figures 1 la-c. However, the zones with enhanced, calculated tritium content maie th nn s i aquifer have become large mord ran e numerous in 1988. The smaller the porosities are, the stronger this tendency is. Similar conclusion drawe b n nsca fro resulte mth simulationf so transpore th r sfo carbonf to - 14 with varying effective porosities. Both calculated and measured isotope concentrations are enhanced in certain zones along the coasts and along the northeastern boundary. Increased effective porositie numericae th n si l model imply zone thae areae th th tf so with high isotope concentrations in the main aquifer decrease. alsIs ti o difficul determino t t e whicporosite th f ho y set ssimulatione useth n di s resule th n ti best agreement with observations maie .Th n reaso thir nfo s proble moncs i e agai smale nth l number of measured isotope concentrations. In particular, there are hardly any measured 14C values in the large areas southwest of the Alnarp valley with enhanced, calculated activities.

200 TRITIUM CONTENTS

CALCULATED

M >10 TÜ

3-1@ U 0T

D <3 TU

OBSERVED .4 contents In TU

<2TU

Figure lie: Calculated observedand tritium contents mainthe in aquifer, corresponding situationtothe aboutin effective 1971.The porosities calculationsthe in are 50% in the clay till and 30% in the main aquifer.

These peculiarly high concentrations may at least partly be due to an overestimation of the leakage coefficients and/or transmissivitie numericae th n si l model. On the other hand, the measured contents of 14C in the central part of the Alnarp valley are much lowecalculatee th f ro thay dnan values. Her t shoulei repeatee db d thameasuree th t d activities have been corrected for various dilution effects. These figures may thus be unreliable, especiall vere ones w discrepancyreasoe th e y lo th Th .r nfo alsy thae leakagoe yb ma th t e coefficients and/or transmissivitie overestimatee sar thidn i s area. There are very small differences between the simulation results concerning 1 C activities in 1988n i . s origid it floa transpor an d s Figur184wn nan i 0ha 3 1 e t situation befory an e 4 exploitation of the main reservoir had started in southwestern Scania. In other words, the low isotope concentrations in large areas probably do not reflect the present circulation times in this groundwater system.

201 TRITIUM CONTENTS CALCULATED M >10 TU S 3-10 TU D <3 TU OBSERVED .4 contents In TU

+

Figur : 12 e Calculated observedd an tritium contents maine th n i aquifer, corresponding situationtothe aboutin 1988. effectiveThe porosities calculationsthe in are mainthe in aquifer. claythe 15% tillin and 30%

8. GENERAL CONCLUSIONS AND RECOMMENDATIONS A major meri thif o t s combinatio f isotopno e hydrogeolog regionad yan l flotranspord wan t modelling is that the isotope transport simulations help to demonstrate where zones particularly vulnerabl pollutioo et situatede nar . These location chiefle sar resule yth hydrogeologicaf o t l characteristics traditionally examined thet revealee ,bu y ar meantranspory e db th f so t model. Subsequent, more detailed investigations can then be focussed primarily on these vulnerable zones. groundwatee Th r flotranspord wan t model applie updatee b n dca d progressivel results ya s of further investigations become available deterministie .Th timy usece an e modeb t do a t n lca judge the quantitative and qualitative effects of, for instance, increased groundwater extractions or an inflow of a pollutant, on the basis of the present state of knowledge about the groundwater syste minterestf o . Ther alse ear o several difficulties concernin detailede gth , quantitative interpretatione th f so results, and these difficulties must be the focus of future work. Even if the aquifer system

202 CARBON-14 CONTENTS

CALCULATED

>99 pmc 96-99 pmc 90-9c 6pm

80 - 90 pmc pm0 c<8 J L

OBSERVED

Figure 13: Calculated and observed carbon-14 contents in the main aquifer, corresponding to the situation in about 1988. The effective porosities used in mainthe in aquifer. clay the 30% calculations tillthe in and 50% are studie dclassifiee b her n eca unusualls da y homogeneou well-definedd san numbee th , f ro isotope measurements is small compared to the extension of the system. Furthermore, the calculated concentrations are similar over large areas. The concentrations observed in sampling wells often represent mixture groundwaterf so s from different depth witd san h different average circulation times, whil calculatee eth d concentrations correspon average th elemen n do a t n ei f to the mode certaia t la n level. Occasionally measuree ,th d isotope content represeny sma smalleta r portio groundwatee th f no r reservoir tha calculatee nth d mean entirvalun a r efo element. Peak concentrations will be smoothed out during the vertical transport due to the relatively rough subdivision into layers. Therefore, it is at present not possible to verify or reject the results of the flow mode calibratioa y lb isotope th f no e transport model. Furthermore isotopie th , c investigations include thin di s wor improvn kca generae eth l estimation propertiee th f so s relevan flo e transpord th w an o t tpossiblt processesno s i t i e t ,bu to make any reliable, detailed quantitative determinations. The contrasts between the 2H and i«O concentration differenn si t aquife e partth f so r systesmalsuitablo e b to e o lm t transpor r ar efo t modelling regards A . , ther significane s3H e ar t contrasts betwee differene nth t e partth f so

203 investigated groundwater system, but within large areas of the main reservoir the concentrations usualle ar y ver ylonga smallo t ,e average,du , circulation time measuree .Th valueC d14 s have to undergo a substantial correction before they can be compared with the simulation results. In spite of the many uncertainties related to such an interpretation, contrasts of the same type as for tritium can be detected. recommendes i t I d thacombinatioe th t isotopf no e hydrogeology with regional flod wan transport modelling shoul developee db d furthe improvo t r fundamentae eth l knowledge th d ean investigation methods essentia maintaininr fo l gquantitativela qualitativeld yan y safe, long-term groundwatef o e us r resources. Other modelling concept techniqued san alsd soan other arear sfo application shoul triede db . In general, the applied combination of various investigation methods will result in a wider understandin floe transpord th w an f go t conditions withi nstudiea d groundwater system. Zones which seem to be particularly vulnerable to pollution or where a small amount of additional information could improve the hydrogeological survey substantially, can easily be identified. Detailed investigations with measurements and more accurate modelling should be conducted at these locations e informatioTh . n gained fro e detailemth d studies shoul e usedb r dfo adjustment regionae th f so l flotranspord wan t model recalibratee Th . d regional model will serve to detect other zones in which additional, detailed investigations should be carried out, etc. This iterating procedure will gradually lead to improved knowledge about the groundwater system studied. treatmene Th interpretatiod an t available th f no e data, concerning, among other things, geological logs from drill-holes, groundwater hydraulic hydrochemistryd an s , have been undertaken mainly without automatization spatiae temporaTh d . an l l distributions have been determined by hand and by means of simple statistical methods. It is recommended that data are treated by more advanced methods, such as computerized kriging and factorial analysis, to obtai starting-poinna calibratioe th r fo t flof transpornd o wan t models more swiftly. However, the results of such operations have to be checked against qualitative information from stratigraphie descriptions, drill-hole logs, local hydrogeological experience, etc. Most samples from the main groundwater reservoir of southwestern Scania show very low content tritiumf so , reducin possibilite gth drawinf yo g reliable conclusions fro mcomparisoa n with the results of the transport modelling. It would be very valuable to collect samples for tritium analysis from several different levels betwee shalloe nth w aquifermajoe th d r aquifesan r betteo t r elucidat verticae eth l transport processe thin si s groundwater system. It is also recommended that future research should be devoted to estimations of porosities suitable for isotope transport simulations, in general as well as in southwestern Scania in particular relationshie Th . p betwee isotope nth e compositio f precipitationo wated nan r rechargin maie gth n aquifer mus studiee b t d further. Additional attempt estimato st C 14 e eth contents of the water recharging the main reservoir after all types of dilution mechanisms and processes should als carriee ob d out.

ACKNOWLEDGEMENTS This study has partly been financed by grants from the Swedish Environmental Protection Agency, the Swedish National Board for Technical Development and the Swedish Natural Science Research Council, which hereby are gratefully acknowledged. The author is also very grateful to professor G. de Marsily, Paris School of Mines, and professor J. Ch. Fontes, Paris University XI, for their generosity and benevolent guidance during his stays in their laboratories for mathematical modelling and isotope analysis respectively. REFERENCES

Ala-Eddin, M. and Magnusson, E. (1991): FE-modellering av amnestransport i grundvatten. Maste Sciencf ro e Thesis, Departmen Engineerinf o t g Geology, Lund Institut Technologyf eo , LUTVDG/TVTG-5029, Lund, Sweden, 55pp.

204 Barmen . (1989)G , : Transpor löstv a t a airme isotopeh noc i rgrundvattensysteme t kring Alnarpsdalen. Mätnin modelleringh goc . Report LUTVDG-TVTG-7013, Departmenf o t Engineering Geology, Lund Institut Technologyf eo , Lund, Sweden, 23pp.

Barmen, G. (1990): Combining environmental isotope investigations with flow and transport modellin regionaf go l groundwater systems frorepor 6 NordiP e 2 m. th .NH tno c Hydrological Conference 199Kalmarn i 0 Sigurdsson,n editeGu y db , Nordic Hydrological Association, ISBN 91-87996-02-2 Norrköping, Sweden . 104-113pp , .

Barmen . (1992)combinatioe G , th n O : f isotopno e hydrogeology with regional flod wan transport modelling. An application to the groundwater system of southwestern Sweden. Doctoral Thesis, Department of Engineering Geology, Lund Institute of Technology, LUTVDG/TVTG-1006, ISBN 91-628-0587-8 Lund, Sweden, 201pp.

Bjelm , HaïtienL , , RöshoffJ , , BennetK , , BruchJ , , PerssonH , ,Wadsteind P-Gan . . P , (1977): Geotermisk energiutvinnin gi Skâne . Slutrapport Etap . Förutsättningarn1 p r fo a utvinning i Skâne. Department of Engineering Geology and Department of Soil Mechanics and Foundation Engineering, Lund Institute of Technology, LUTVDG/TVGL-5013, Lund, Sweden, 60pp.

Brinck Leanderd an . S , . (1981),B : Alnarpsströmmen. Sammanställnin utredningav ga r 1970- 1980. The Alnarp committee, VBB 10520, Malmö, Sweden, 45pp. Brinck , Leander S ,Winqvisd an . B . ,(1969) G t : Alnarpsströmmen. Utredning rörande vattentillgâng och dess lämpliga utnyttjande. The Alnarp committee, VBB, Malmö, Sweden, 56pp. Burgman, J.O, Galles, B. and Westman, F. (1987): Conclusions from a ten year study of oxygen-18 in precipitation and runoff in Sweden. In the proceedings of an international symposiu Isotopmn o e technique waten si r resources development, International Atomic Energy Agency, Vienna, Austria, pp. 579-590.

Burgman, J.O, Eriksson, E. and Westman, F. (1983): A study of oxygen-18 variations in river waters and monthly precipitation in Sweden and determination of mean residence times. Internal report, Division of Hydrology, University of Uppsala, Uppsala, Sweden, 19pp.

Galles, B. and Westman, F. (1989): Oxygen-18 and Deuterium in precipitation in Sweden. Repor , Divisio47 t . serie no f Hydrology no , sA , Universit f Uppsalayo , Uppsala, Sweden, 20pp. Custodio, E, Gurgui, A. and Lobo Ferreira, J.P. (editors, 1987): Groundwater Flow and Quality Modelling. Proceeding NATe th f sOo Advanced Research Worksho Advancen po n si Analytica Numericad lan l Groundwater Flo Qualitd wan y Modelling, Lisbon, June 2-6, 1987. NATO ASI ser. C: Mathematical and Physical Sciences, vol. 224. D. Reidel Publishing Company, Dordrecht, Netherlands, 843pp.

DGU and Kemp & Lauritzen (1975): Nordvand. Grundvandsundersogelse ved Esrum so. The Danish Geological Surve Kemd yan Lauritzep& n A/S, Frederiksborg Amtskommune, Denmark, 88pp.

Fontes, J.Ch. (1985): Some considerations on groundwater dating using environmental isotopes. Memoir 18te th h f so congres Internationae th f so l Associatio Hydrogeologistsf no , Hydrogeolog service th yn i manf eo , Cambridge, Great Britain . 118-154,pp .

205 Fontes, J.Ch. and Garnier, J-M. (1979): Détermination of the initial activity of the total dissolved carbon. A review of existing models and a new approach. Water Resources Research, vol. 15, no. 2, pp. 399-413. Fritz, A.P. and Fontes, J.Ch. (editors, 1980): Handbook of Environmental Isotope Geochemistry, Volume 1, The Terrestrial Environment, A. Elsevier, Amsterdam, Netherlands, 545pp. Geyh, M.A Backhausd .an . (1978)G , : Hydrodynamic aspect f carbon-1so 4 groundwater dating proceedinge th n I . internationan a f so l symposiu Isotopn mo e Hydrology, vol, .II International Atomic Energy Agency, Vienna, Austria . 631-643,pp . Goblet . (1985)P , : Programme METIS. Notice d'utilisation. Centre d'Informatique Géologique, Ecole Nationale Supérieur Mines ede Parise sd , LHM/RD/85/41, Fontainebleau, France, 68pp. Gustafsson, O. (1972): Beskrivning till hydrogeologiska kartbladet Trelleborg NV och Malmö SV (Description of the Hydrogeological Map Trelleborg NV and Malmö SV). Swedish Geological Survey, SOU, ser. Ag, no. 4, Stockholm, Sweden, 37pp. Gustafsson, O. (1978): Beskrivning till hydrogeologiska kartbladet Trelleborg NO/Malmö SO (Descriptio Hydrogeologicae th o nt Trelleborp Ma l g NO/Malmö SO). Swedish Geological Survey, SGU, ser. Ag, no. 6, Stockholm, Sweden, 72pp.

Gustafsson, O. (1981): Beskrivning till hydrogeologiska kartbladet Malmö NV (Description to the Hydrogeological Map Malmö NV). Swedish Geological Survey, SGU, ser. Ag, no. 13, Uppsala, Sweden, 51pp. Gustafsson . (1986)O , : Beskrivning till hydrogeologiska kartbladet HelsingborV gS (Description to the Hydrogeological Map Helsingborg SV). Swedish Geological Survey, SGU, ser. Ag, no. 14, Uppsala, Sweden, 58pp. Gustafsson, O. and Teeling, M. (1973): Sydvästra Skanes geologi: Jordlager - Berggrund - Grundvatten. Swedish Geological Survey, SGU, Lund, Sweden. Herweijer, J.C Luijnn ,va , G.A Appelod .an , C.A.J. (1985): Calibratio masa f no s transport model using environmental tritium. Journa Hydrologyf lo . 1-17, volpp , ..78 Hoist, N.O. (1911): Alnarps-floden svensn E . k "Cromer-flod". Swedish Geological Survey, SGU, . 237serno , , .StockholmC , Sweden, 64pp.

Hydén, H., Leander, B. and Voss, C.I. (1980): The Alnarp Groundwater System. A Mathematical Model Study SpeciaB .VB l Report 02:80.1, Stockholm, 25pp. Ledoux . (1986a)E , : Modèles mathématique hydrogéologien e s . Centre d'Informatique Géologique, Ecole Nationale Supérieur Mines ede Parise sd , LHM/RD/86/12, Fontainebleau, France, 120pp. Ledoux, E. (1986b): Programme NEWSAM. Simulation des aquifères multicouches en mailles de taille variable. Not présentatione ed . Centre d'Informatique Géologique, Ecole Nationale Supérieure des Mines de Paris, Fontainebleau, France, 33pp. Markussen, L.M, Möller, H-M.F, Villumsen , MortensenB , , J.K Selchaud .an . (1991)T , : Frederiksberg Kommune. Sikring av drikkevandsressourcen. Ramböll & Hannemann Bulletin, , Copenhagenno27 . , Denmark, 85pp.

206 Marsily, G. de, Ledoux, E, Levassor, A, Poitrinal, D. and Salem, A. (1978): Modelling of large multilayered aquifer systems: Theory and applications. Journal of Hydrology, vol. 36, pp. 1-34. Nilsson . (1966)groune ,K th n d:O water condition sedimentare th n si y rock Scaniaf so e th n .I proceeding internationan a f o s l symposium hel n Stockholmi d , Wenner-Gren Center International Symposium Series, vol. 11. Ground Water Problems, edited by E. Eriksson, Y. Gustafsson and K. Nilsson. Pergamon Press, Great Britain, 1968, pp. 43-56.

Hanshawd an Pearson r J . , FJ ,B.B . (1970): Source f dissolveo s d carbonate specien i s ground water and their effects on carbon-14 dating. In the proceedings of an international symposiu Isotopmn o e Hydrology, International Atomic Energy Agency, Vienna, Austria. ,pp 271-286. Perers, J. and Johansson, I. (1980): Trelleborgs och Vellinge kommuner. Grundvatten. Utredning over tillgâng, behov, utnyttjande och skydd. Sammanfattning. VIAK 5812.1518, Malmö, 29pp. Ringberg, B. (1980): Beskrivning till jordartskartan Malmö SO (Description to the Quaternary map Malmö SO). Swedish Geological Survey, SGU, ser. Ae, no. 38, Uppsala, Sweden, 179pp. Robertson, J.B. (1974): Applicatio f digitano l modellin predictioe th o gt f radioisotopno e migratio groundwatern ni proceedinge th n I . internationan a f so l symposiu Isotopn mo e Technique Groundwaten si r Hydrology, vol Internationa, II . l Atomic Energy Agency, Vienna, Austria, pp. 451-478.

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Wigley, T.M.L, Plummer, L.N. and Pearson, FJ. Jr (1978): Mass transfer and carbon isotope evolution in natural water systems. Geochimica et Cosmochimica Acta, vol. 42, pp. 1117- 1139.

Next page(s) left blank 207 APPLICATION OF HYDROGEOCHEMICAL MODELLING FOR VALIDATION OF HYDROLOGIC FLOW MODELLIN TUCSOE TH GN I N BASIN AQUIFER, ARIZONA, UNITED STATES OF AMERICA

R.M. KALIN University of Georgia, Athens, Georgia A. LONG University of Arizona, Tucson, Arizona United States of America

Abstract

This work examine hydrogeochemicae dth l evolutio f Tucsono n basin groundwater, including isotope hydrology, geochemistr determinationse ag d yan . Result f minéralogiso e investigatio n basino n fill constrain water-rock geochemical reactions. Examination of 45 years of water quality data shows that groundwater mining has affected water quality. Stable isotopes of carbon, oxygen, hydrogen, sulfur, and chlorine and radiocarbon, tritium [1] and radon determinations [2] refine the interpretation of hydrogeochemical evolution of Tucson basin groundwater as modelled with NETPATH [3]. Two distinct sampling periods, the first in 1965 [4] and the second between 198 1989d 4an , resulte determinatioe th n di f groundwateno r age r watesfo r mine decadeo dtw s apart. Isotope hydrolog geochemicad yan l modelling suggest tha t watee mucth f rho presently mined froe mth Tucson componenbasia s nha t recharged durin las e years0 gth 5 t . Increased sulfate concentrations suggest that heavy pumping in the northeastern basin induced increased leakage from lower units. Results of geochemical modelling indicate an average of 5 percent mountain-front recharge to the Ft. Lowell fm. along the northern aquifer margin increasn A . dissolven ei d solids alon basie gth n margin implies that this componen rechargo t t e has increased in the past decade. The radiocarbon age of the basin groundwater was compared with the temporal movement of water as modelled with MODFLOW and PATH3D [5]. In general, the hydrologie simulation agrees with both the distribution of tritium and the exponentially modelled water age, as determined with bomb-derived radiocarbon, for areas of the Tucson basin that contain water less than 50 years in age. Hydrologie modelling failed to predict the antiquity of recently sampled water in the central basin but is similar to age determinations on waters collected in 1965.

INTRODUCTION Groundwater chemistry is determined by the initial chemistry of water entering a groundwater system and subsequent chemical reactions with minerals comprising the aquifer materials. It is possible to model chemical reactions (water-rock) that contro chemicae th l l compositio f groundwateno followine th f i r e gar known ) basiA : c knowledg hydrologe th f aquifee eo th f yo r system ) initiaB , l chemistr f rechargyo e waters, ) knowledgC f mineraleo s most prevalen ) chemicaaquifee D th d n i t an r l analyse f majoso minod an r r chemical constituent groundwatern i s . Tucson City, Pima County and Arizona State planning is based on estimates of population growth, economic growth, national trends and other factors such as availability of an ample water supply. The metropolitan Tucson area will continue to depend on ground water as a natural resource. The Central Arizona Project (CAP) can supply up to an estimated 18 million m of Colorado river water to Tucson annually, but the

long-term master-plan for Tucson projects continuous growth3 , and consequently, the use of CAP water will only b stop-gaea p solutio minine th o nf Tucsot g o n basin groundwater. quantite Th f wateyo r recharge Tucsoe th o dt n basin from infiltratio f locano l precipitatio bees nha n estimated at 6.3 million m per year [6]. The volume of water recharging the basin as leakage along the

mountain front has been estimate3 d by Davidson [6] at 13,000 to 25,000 m3 per kilometer of perimeter per year. Marra [7] applied the finite difference model MODFLOW to movement of groundwater in the Tucson basin. Mathematical models are based on estimates of aquifer parameters and may not fully represent the actual hydrology. This study of the hydrogeochemical evolution and isotope hydrology of Tucson basin groundwater is used to validate our understanding of the movement of water in the subsurface independent of hydrologie modelling.

209 SIGNIFICANCE OF HYDROGEOCHEMICAL MODELLING The Tucson basin is well studied with both hydrological and geochemical methods [6 - 57], yet a full understanding of the temporal and spatial flow of water in the Tucson basin is not fully realized. This is due, in part, to a high degree of large-scale inhomogeneity of the aquifer, chemistry of the water and boundary condition wels a s s severa l e anthropogenic impachydraulie th n o t c system. Understanding the chemical evolution of groundwater requires identification of chemical reactions and subsequent mass-transfer. Use of mass-balance calculations in conjunction with water-mineral equilibria calculations allow identification of controlling geochemical reactions along a flow path. Knowledge of reactant and product phases constrains geochemical reactions. Proposed mass-transfer reactions are examined to determine which reaction path best predicts the mass of dissolving and precipitating phases occurring. Isotope balance usee sar validato dt e mass-balance calculations. Isotope particularle sar y well suiter dfo this problem becaus almose th f eo t unique solutions require reproduco dt measuree eth d values. These isotopic constraint appliee b mass-transfee th n so ca dt r model calculateo t f Wigle so Plumme] d . [58[3 al an ]. t ye al t e r

thC/eC anCd of the original water, place limits on the possible admixture and reactions along the flow path

12 14

and I3 to accurately estimate of the age of the water.

PHYSIOGRAPHIC SETTING The Tucson basin, southeastern Arizona, USA (Figure 1) is part of the Basin and Range geological provinc southwestere th f eo n United State broaa ss i [59]dt I northwes. t trending alluvial valley basie Th s .ni bounde mountainy db s that rang elevation ei 280no t fro0 0 m90 meter s (mea levela nse , m.s.L) elevatioe Th . f no basie th n floor ranges fro mnorthwese th 610 n i m 1070o t t m southeaste m.s.lth n i .entire Th . e Tucson basin covers approximately 3600 km2. e TucsoTh n basin drainnorthweste th o st . Major rivers consis Sante th f ao t Cruz river, which flows generally northward alon westere gth n basin e edg th Pantan e f eo th ; o river, which drain eastere sth n basin mountain-front along the Rincon mountains and joins Tanque Verde creek and Rillito river which drains the Catalina mountains alon northere gth n basin edg el majo (figurAl . r 1) drainage e feature currentle ar s y intermittent but may have been perennial during the Holocene [60,61], Recharg basie th no et occurs mainl leakags ya e through riverbeds (figur . Additiona2) e l recharge th o et groundwater system occurs as recharge along the mountain-front. These components of recharge flow toward the basin center. Water leaves the aquifer system through evapotranspiration, flow out of the basin to the northeast and by mining of groundwater. Mining of groundwater is defined here as an excess of water removed over water recharged resulting in a net loss of water in the aquifer. Modern precipitation consist majo2 f so r rainfall patterns. Winter rainfall arrive frontas a s l storms that pass eastward over Arizona from October to May. Summer precipitation occurs as intense local convective "monsoon" storms that for moiss ma r moveai t s into Arizona from Mexico. Periodically "cut-off" lows, intense low pressure storms that form in the low latitudes of the pacific ocean, move across Arizona and bring significant rainfall occurrence Th . f theseo l Nino/SoutherE e relatee e eventb th y o dt sma n Oscillation (ENSO) [62]. f raisnod o nan Rainfalw m r yea m (wate pe 0 r m higs ovel65 a basie rm averags hd a th r0 nan 30 s ei equivalent) on the surrounding mountains [63,64]. The potential evaporation, ca. 2000 mm per year [19,45], currently far exceeds precipitation in the basin. Metropolitan Tucson lies withi north-centrae nth basine thire th f on l do . Tucson's populatios ni continuously growing (Table 1) and is expected to exceed 1.5 million by the year 2025 [46-49]. Tucson presently depend groundwaten so watea s a r r supply. Groundwater minin resultes gha waten di r table declines exceedin meter0 g5 somn si e portion Tucsoe th f so n aquifee basinth s dewatereds i A r . , potentiae therth s ei r fo l basin subsidence [50]. The arrival of CAP water can offset groundwater mining but will not provide all of the water resources needed. The Tucson Water Resources Plan [17] calls for a decrease in groundwater withdrawals from 123

million m in 1991 to approximately 18 million m in 1995. With population growth, groundwater mining will 3

increase to3 56 million m per year by the year 2030. Fifty percent of Tucson's water needs are expected to be

met by CAP water. Non-potabl3 e effluent is presently used for irrigation of parks and golf courses to offset water demands. Effluen presentls i t y recharge groundwatee th o dt r system and, with recharge waterP d CA s expecte i , d to be recovered, treated and used as drinking water before 2050. These plans have been made without detailed evaluation of the fate of recharged water or the temporal movement of water in the Tucson basin aquifer system. Results of this study should be used to consider the fate and reclaimation of recharged water.

GEOLOG HYDROGEOLOGD YAN Y basie Th n consist f Oligocèneso Pleistoceno t alluviue eag m [65-69] deposite deee th pn di graben s produced from high-angle normal faulting during Basin-and-Range crustal extension (figure 2). The most

210 centrally-located grabens within the Tucson basin extend to over 3000 meters depth below land surface. The location of these graben faults as mapped by Anderson [68] are in figure 3. e lowesTh t uni f Tucsoo t n basin sediment Pantane th s si o formation [66]. This formatios ni consolidate containd dan clastw sfe f Catalin o s a gneiss suggesting Cataline thauplife th th t f Rincod o t aan n core complexes is post-Oligocene [65]. It is interbedded with volcanic flows which have potassium-argon dates ranging from 31.4 to 24.9 million years (m.y.) [67]. The Pantano fm. outcrops on the periphery of the basin as up-thrown blockBasin-and-Range th f so e faulting.

Santa Catalina, Tanque Verde and Rincon Mountains

Figure 1. Geographical location of study area

211 Mountain front recharge

STREA1 I M1 GRAVEALLUVIU[~ L M M CLAY, SILT & EVAPORATES E3 PRE-BASIN & RANGE SEDIMENTARY ROCKS I I FINE SAND ffefl PRE-BASIN & RANGE SEDIMENTARY ROCKS. INTERBEDDED AN D VOLCANIC ROCK I SANSI D l~~] COARSE SAND & GRAVEL P^l BEDROCK HI VOLCANIC

Figure 2. Idealized hydrologie system of Tucson basin and cross-section of aquifer system

Table 1. Projected population for the Tucson Metropolitan Area after Arizona Departmen f Economio t c Security, December 1981

Year Population Projection [46] Projected Water [46] Growtw Lo h- Hig h Growth (millione Us ) 3 m s 2000 829,000 - 851,600 183. - 1883 3

2010 1,019,700- 1,101,000 225.5 - 243.5 2020 1,176,000- 1,316,600 2600-301.1 2030 1,318,60 0- 1,624,10 0 291 6 - 359.1 2040 1,433,60 0- 1,891,70 0 317.0-418.3

2050 1,525,600 - 2,146,060 337. 4474.- 6

212 Tucson Basin Groundwater Study The Universit Arizonf yo a Catalin Rincod aan n Mountains N

1040 T

305

Figur . eEstimate3 d locatio f grabeno n Tucsoe faultth n i s n basi east-to-wesd nan t cross section

The central units of sediments, the Tinaja beds [18,68], are Miocene to Pliocene in age and range from 100 meters thickness along the basin perimeter to 600 meters thickness in the basin center. The sediments comprising these units include clasts from the Tucson Mountain volcanic and the Catalina and Rincon metamorphic core complexes interbedded with clay-rich anhydrite/gypsum beds. Andersen [68] subdividee dth Tinaj . intafm o three unconformable units uppee th , r Tinaja, middle Tinaj lowed an a r Tinaja beds. Some production wells penetrat uppee penetratw eth fe rt Tinajmiddle bu e. th lowed fm a ean r units. Most Tucsoe wellth n i s n basin draw water fro uppee mth r geologi . Lowelc Ft unit e th ,l fm., which unconformally overlies the upper Tinaja beds [68]. The Ft. Lowell fm. was deposited during the early to mid- Pleistocene. Sediments of the Ft. Lowell fm. range between 60 and 125 meters thick and consist of gravel to clayey silt. Overlayin veneea . Lowel s Ft i . e f Pleistoceno rgth fm l Holocend ean e alluvial sediments whice har generally abov watee eth r table. These deposit typicalle sar y very coarse sangraved therefordan d an l n eca allow significant recharge alon intermittene gth t rivers. Rogers [25] reported the hydraulic conductivities of the Middle and Upper Tinaja beds at between 0.3 d whilm/ an0 e1 d transmissivit y ranges betwee 150e averagd 0 an Th 6 mn1 2e/d . saturated . thicknesFt e th f so reportee Th Lowel s 30mi d.. hydraulifm l c conductivit f thiyo s formation [25,7d rangem/ 0 ]s6 betweed an 3 n0.

213 with traiismissivities between 270 and 15,500 m/d. Regions of high transmissivity and "shoestring" aquifers act

as recharge conduits of mountain-front recharge 2 water [9] and are hypothesized here as paleo-alluvial channels and fans fro streame mth riverd san s that drai surroundine nth g mountains. Figur showe4 percene sth t clar yfo the Tinaja and Ft. Lowell fms. respectively (after Anderson [68]). It is envisioned that during the closed-basin formatio Tinaje th f nao beds seriea , f alluviaso l fans stretched acros northere sth n basin (Agua Caliente, Tanque Verde, Bear Canyon, Sabino Canyon and Ventana Canyon) depositing sediments of high transmissivity. This trend can also be seen in the Ft. Lowell fm. which contains very little clay at the pediment-basin interface,

Tucson Basin Groundwater Study The University of Arizona Catalina and Rincon Mountains N A

Tucson Basin Groundwater Study The University of Arizona Catalin Rincod aan n Mountains TV A

>80 60-80 40-60 20-40 <20

Figure 4. Percent clay in Tinaja (lower confined) and Ft. Lowell (upper unconfined) aquifers

214 suggestin ghigh-energa y depositional environment hydrogeologe Th . hydrogeochemistrd yan y suggest that water presently recharging and flowing in these zones of higher transmissivity travel at greater velocity than in regions of lower transmissivity. This boundary must be taken into consideration when choosing the flow path for geochemical reactions. Mixin f oldego r with younger waters shoul consideree db d along these boundaries. Figures 5 shows the head distribution in the Tucson basin for the years 1940 and 1988 respectively. The effect of pumping can be seen in the west-central portion of the basin where hydrologie head has lowered significantly in the last 50 years of population growth and expansion of water mining. This extensive

Tucson Basin Groundwater Study Universite Th Arizonf yo a Catalina and Rincon Mountains N A

Tucson Basin Groundwater Study The University of Arizona Catalina and Rincon Mountains N A

Figure 5. Potentiometric surface of unconfined aquifer during 1940 and 1988

215 perturbatio naturae th f no l groundwater system increase complexite sth f groundwateyo r resource analysi boty sb h hydro-geological and hydrogeochemical methods. However, this strong transient state also provides the opportunity to examine changes in the hydrogeology and hydrogeochemistry of the system as it accentuates changes in recharge sources, mixing and the groundwater flow characteristics near transmissivity boundaries.

ISOTOPIC COMPOSITIO BASIF NO N PRECIPITATION The stable isotopic composition of water can be used for identification of different sources of water [70- 87]. Samples of most precipitation events in Tucson have been collected by A. Long and R. Kaiin during the time period 1981 until present. The majority of these samples were analyzed for stable oxygen and hydrogen isotopes [88-94] through the combined efforts of many individuals, most notably Christopher Eastoe. The number of samples represented in this data set are far greater than those in the work of Simpson et al. [33], but the same basic interpretation of the seasonality of recharge applies. Figure 6 plots the summer (May through October) precipitation events and winter (November through April) respectively. The average 8D values of these samples are -58%c SMOW for winter and -38%o SMOW for summer. These values are close to the values of -61%o and -42%c reported by Simpson et al.. [33] in 1970. Oxygen and hydrogen isotopic information defines local meteoric water lines for winter and summer precipitation. Figure 7 plots the d-parameter [95] of these samples against the rainfall amount. A value of d = 10 indicates a sample population that lies on the globally averaged meteoric water line. A d-parameter greater than indicatey ma 0 1 , among other effects sourca , f precipitatioeo n that originate solia s da d precipitate (snowfall). Value lesd f sso tha indicat0 1 n e local/regional evaporative processes acting upo precipitatione nth sourcee Th . s of moisture for the Tucson basin are winter cyclonic storms that are expected to produce precipitation with d- parameters equal to or greater than 10, and for the summer, local convective storms that are expected to produce precipitation with d-parameters average lesTh s tha . e n10 d-paramete r summefo r r rainfalr 4.9s fo i l d 8an winter rainfall is 11.14. These results confirm the hypothesized range of values. The coefficient of fit (r2) for both graphs suggest no relationship of d-parameter with precipitation amount.

ISOTOPIC COMPOSITIO MOUNTAIF NO N PRECIPITATION Summe winted an r r precipitation samples fro Sante mth a Catalina mountain plottee sar figurn di . 8 e e valueTh s measure thin di s study compare favorably summee witTh h . thos] r 3 f Simpsomountaieo [3 . al t ne n precipitation is similar to that at the basin floor with the exception that most of the mountain events lie near or abov globae eth l meteoric water average lineTh . e d-parameter - (12.5.3+/ 6f summe) o r rainfale th n o l mountain s nea si globa e th r apparene lTh averag . 10 t distributiof eo d-parametee th f no r nea globae th r l average suggests that moisture for summer convective storms does not undergo exchange with local evaporative moisture. discrepance Th y betwee d-parametee nth f higho r elevation precipitatiod-parametew lo e th d f rainfalnan o r n o l the basin floor may reflect "below cloud" evaporative effects that occur as raindrops evaporate during their travel to the basin floor. This distribution may also reflect the sparse nature of rainfall events collected on the mountains e winteTh . r precipitatio s isotopicallni y lighter than summer precipitatio averagn a s ha - ed d nan parameter (24. - 4.40+/ ) abov globae eth l meteoric average, consistent wit source hth f moistureeo . The seasonality of the sources of precipitation, and the effect of the El Nino - Southern Oscillation regionan o Inde I xSO l precipitation patterns [62] prompte investigation da f seasonano l differences reflecten di stable th e isotopic compositio f rainfallno . Figur showe9 monthle spline-fisa th 5e f Do th yo t t averaged precipitation, and the monthly average values for ÔD, 818O and d-parameter. An annual cycle in the data shows effece th f differino t g source f moistureso . Figur show0 e1 frequence sth y distribution oxygee th f d so nan hydrogen stable isotopic composition f rainfalso l events distributioe Th . f valueno s sugges differen4 t t precipitation patterns for the rainfall events. Pattern 1 is cyclonic storms that originate in the North Pacific and are swept across Arizona by the jet stream. Months that represent this pattern are October, November, February, March, April, and possibly May. The isotopic composition of these events is characterized by light isotopic averagn valuea d san e d-parameter nea. 10 r Precipitation pattern 2 is dominated by "cut-off" low pressure systems that originate in the low latitudes Pacifie o th fmov d an c e across Arizona fro southweste mth . These storms generally occur durin monthe gth f so Decembe Januaryd an r . Figur showe9 s distinct difference precipitatior fo s n during these months average Th . e 8D and 818O are heavier and the d-parameter for this pattern averages 7.5. Pattern 3 is from local convective systems that begin to appear in June and are fully represented during monthe th f Julyso , Augus o'd isotopicalle 8Septemberd Oan ar an tD S e yTh . averagheaviee th d - an red parameter, 4.98, is lower than other precipitation. The source of this moisture is the lower latitudes of the Pacific and/o Gule th rf California o f .

216 Ô180 SMOW -16 -12 -8

Rainfall Stable Isotope througy Ma s- h October

Best-fit least squares equation to points

D/ H645"180/16= 2 36 O- R Square9 8 0 d= Average 18O/16O =-543 Averag3 6 e8 D/-3 H= Average D-Excess = 4 98 — -40

07 O U)

— -80

Global Meteoric Water Line

l— -120

518OSMOW -16 -12 -8

Rainfall Stable Isotope sNovembe- r through April

Best-fit least squares equatio pointr nfo s

D/53*7 H= 1 BO/160 1 7 + 0 R Square2 8 0 d= Average 18O/160 = -862 Average D/H = -57 76 Average D-Excess = 1114 -40

aO, o

— -80

Global Meteoric Water Line

"— -120

Figur . eStabl6 e isotopic compositio f summeno winted an r r precipitation 198 1- 199 2

217 Pattern 4 is represented as a series of precipitation events during the summer months that are isotopically lighter than most summer convective events. These events also have d-parameters near and above isotopie Th . c 10 signatur f thieo s patter similas ni precipitatioe th o t r n event source patterf so Th f eo . n1 moisture for these events is cyclonic storms that move across Arizona during the summer months.

40 — | * 30 — — • 20 — • • 4 • v % •* * • *• r • CD ( • • ^Î^ * * 4v 0 * * •+-• 10 —

2 0 — •;•' ;%#i'ii/ 2V '' ça Q. -10 — '.:•--'/:• • . .-- '" Q • • • • -20 — * -30 — • Summer precipitation l l l l I I — 0 -4 I II I 0 1. 10.0 100.0 rainfalm m evenr pe l t

50 — !

40 — •

30 — eu ' • . •s 20 — • • • • _ . • _ — • • 1 10 — O. * * * • • *• Q * t * * 0 — • •

-10 — , Winter precipitation

— 0 -2 II l l l l l l l 1.0 10.0 100.0 mm rainfall per event

Figure 7. d-parameter of summer and winter precipitation against rainfall (mm)

218 <51ÖOSMOW -16 -12 -8 -4

Mountain Precipitation

Best-fit summer precipitation D/H = 7 20* 18O/160 + 4 19 Summer^ R Squared = 0 92 Average 18O/16O = -720 6 6 7 Averag-4 = H eD/ Average D-Exces6 12 s=

Best-fit winter precipitation — -40 D/H = 5 81 M8O/16O-210 R Square3 6 0 d= Average 18O/16O = -1193 o, Average D/H = -71 44 a Average D-Exces0 4 2 s= w Wmter~ Global Meteoric Water Line

-80

-120

40

" I Snowfall events 20 —

mean= 126+1- 53

a 0 —

-20 —

-40 n—i—rr I I I I I I 0 o10 1 0 1 1000 mm rainfal evenr pe l t

Figure 8. Stable isotopic composition of mountain precipitation and d-parameter against rainfall (mm)

219 -20 — — -20

-40 — — -40

-60 — — -60

-80 — -80

-100 — — -100

-120 -120 81 82 83 84 85 86 87 88 89 90 91 92 93 Year

ou _, « § ^« » - 20 -5 2 ° — X !^— -4^^ ! 10 _jp-f^jr^c T ^^SJL^ ?^ « -i t p- 0 -2 ^ o ft r = CD -10 -| t- w 0 | > -20 -£ - 3 d-paramete r 0 S 3 -30 (S

6 7 10 11 12 Month

Figur . e9 Seasona d-parameted l O , 8variationan 8D e th f Tucsoo rn si n basin precipitation 18

220 8 —,

0) ü c CO 3 ü Running average ü O 4-

/ CD eZ5r CD

0

-100 -80 -60 -40 -20 0 20 SMOW

10

üCD l ü Running average oü 5 — fr 3CD er o

0

-20 -15 -10 -5 0 518O SMOW

Figure 10. Frequency distribution of oD and 618O of precipitation 1981 - 1992

ISOTOPIC COMPOSITION OF RIVER WATER River water samples were aperiodically collected from major Tucsorivere th n i s n basin resulte Th . f so these samples are plotted in figure 11. The findings of this study are similar to those of Simpson et al. [33] for the Rilhto River system that drain Sante sth a Catalina, Tanque Verd Rincod ean n mountains alon eastere gth n and northern boundar basine th f yo . Winter precipitatio mountaine snod th nan n w o s provid ewatee mosth f ro t sampled fro Rillite mth otributariess Riveit d an rresule Th .isotopicalls i t y light waters wit averagn ha - ed parameter slightly above the global meteoric water line. The Santa Cruz River samples are significantly more enriched in heavy isotopes than those of the Rillito, and have an average d-parameter below the global meteoric line. This suggests that summer rainfall and precipitation from lower elevations comprise the greater proportion of water sampled fro Sante mth a Cruz River. Infiltration through river bottom primare th s si y mechanisf mo

221 <518OSMOW

-16 -12 -8

Bes least tfî t squares line pointso t s

Santa Cruz River D/ H6.2= 6 M8O/16O - 9.2 8 R Square 0.9d= 2 Average 180/160 =-8.60 Average D/H = -63.10 Average D-parameter = 7.25

Rillfto River -40 D/H = 6.35*180/160-2.82 R Squared = 0.91 Global Meteoric Water Line Average 18O/16 O-9.2= 0 X w Averag -60.7= H 2eD/ Average D-paramete 12.9r= • 1 w w CO Q

- -80

• Rillito River • Santa Cruz River W Winter S Summer L- -120

-16 -12 -8

Best-fit least squares line through data

D/H = 6.26 M8O/16O - 6.93 R Squared = 0.75 Average 180/160 = -9.11 Averag -63.9= H e7D/

— -40 Global Meteoric Water Line

O CO Q

-80

>— -120

Figur . Stable11 e isotopic compositio Tucsof no n basin river wate groundwated an r r samples

222 recharge to the Tucson basin aquifer. From these data, the stable isotopic composition of groundwater recharged along the Rillito River and its tributaries is expected to average O5 = -9.2 and OD= -64, and for groundwater

that recharged alon Sante gth a Cruz River -56,= D 5. 18Ô -8. = 0d 6an 18

STABLE ISOTOPE OXYGEF SO HYDROGED NAN GROUNDWATEN NI R Simpson et al. [33] reported 8D values of groundwater samples in the Tucson basin and concluded that most recharge to the system was from winter precipitation. An additional 142 samples of Tucson basin groundwater were analyze r thidfo s study. Figur als1 e1 of groundwate o showD Ô ploa s Ov f 8o t r along

with the global meteoric water line. Figure 12 show18 s the distributions of Oo and SD in the basin groundwater.

isotopie Th c compositio f groundwateno generan i s i r l agreement wit meae hth n isotopic signature f Riveso r I8

Tucson Basin Groundwater Study The Universit Arizonf yo a Catalin Rincod aan n Mountains N A

Stable Oxygen Isotope 10 Water j Contour Inverval = 04 permil SMOW Kilometers

Tucson Basin Groundwater Study Universite Th Arizonf yo a Catalina and Rincon Mountains N A *

Stable HydrogeS Waten i r Contour Inverva permi3 = l l SMOW Kilometers

Figur . Spatiale12 distributio 5d 18 an OTucso n i D Ô f no basin groundwater

223 waters recharging the basin. Isotopic values are heaviest along the lower Santa Cruz River, O5 = -8.0 and

OD= -57, where summer precipitation and precipitation from lower elevations dominate recharge18 . Some distinct zones of groundwater are isotopically lighter than River waters. These zones either represent a relatively large proportion of mountain-front recharge (high elevation mountain precipitation) or represent paleowaters. Only samples used for determining the geochemical evolution of groundwater are plotted. The least-squares line calculated through these point s similasi locae th lo t rmeteori c wate l rprecipitatio al lin f eo n events analyzed. Figure 13 plots of the frequency and cumulative frequency of these values. The distributions imply that for most wells sampled, recharg derives ei d from winter precipitation. Ther , howevereis smala , l grouf po values indicativ f summeeo r precipitatio r isotopicallno y heavy "cut-off precipitatiow lo " n possibly relateo dt ENSO activity [62], These isotopically heavy values generally parallel the Santa Cruz River along the western edge of the Tucson basin. The Santa Cruz River has a large drainage system that extends into Mexico. It is plausible that summer rainfall does not greatly contribute to recharge of the basin groundwater system, but rather large-scale ENSO related storm events during October, November, and December produce significant recharge during hig hSante floth n wai Cruz drainage system. 10 -n 8 I 3 r- ,- Ü Ü O — -i o c CD —I — —

0 il i i l i l l i n i i ri i IT Tri i n i ri i ri i r i i

-8»0 4 - -75 5 -4 70 0 -5 -65 5 -5 -6 D (5DSMOW 25 -q _ 8 = § 2°- o ou 1-i5* ~i *o - o* 10 -£ eu I — er c Q) O — ul = 0 — I 1 M 1 1 n I 1 1n M 1 i I 1 1M 1 1\ i \ i i i I M i n-nn

-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 <518OSMOW

Figur . Frequence13 y distribution Tucson basin groundwate S5d an D 8 r 34

224 RECHARGE ESTIMATE TRITIUA SVI M Figur summarize4 e1 resulte sth f Lindquiso s depict d an areae ] sth [1 t l exten f detectablo t e tritiue th mn i near-surface groundwate Tucsoe th f o r n basin. basie Areath f n so suspecte f havindo g modern water containing detectable tritium, but for which we have no measurement, are represented with question marks in figure 14. Tritium is a naturally-occurring radioactive isotope of hydrogen with a half-life of 12.42 years. The concentration of naturally-produced tritium in precipitation prior to 1954 was estimated to be 5 to 24 tritium units (TU) [118]. (1 TU is defined as 1 3H in 1018 H atoms) During the 1950's and early 1960's, thermonuclear bomb testing injected tritium intatmospheree oth . This "bomb pulse" increased tritiu Arizonmn i a precipitation

Tucson Basin Groundwater Study The Universit Arizonf yo a Catalina and Rincon Mountains N A

? - suspected tritium ? but not measured

Tucson Basin Groundwater Study The University of Arizona Catalin Rincod aan n Mountains N A

Radon Concentration in Water Contour Inverval = 150 pCi/l

Figure 14. Spatial distribution of tritium and 222Rn in Tucson basin groundwater

225 levelo t average greas sa Th 4,40s a t. e 0TU tritiu m concentratio Arizonn ni a precipitatio perioe th r dnfo 1962- 1965 was 1140 TU [119]. During the period since the cessation of atmospheric thermonuclear testing, the tritium concentration in precipitation has decreased due to radioactive decay and mixing in the hydrosphere. The tritium concentratio southwestern ni n North American precipitatio perioe th r dnfo 198 3- 198 8 averaged 10.- 6+/ 3.8 TU [119]. Post-1954 recharge to groundwater can be distinguished from older waters by the presence of measurable tritium. Recharge rates were calculated from location of the post-1954 recharge front (samples containing measurable quantities of tritium) [3] using the following equation: V*W*T|= R * (b*Cm/Ci) average wherth s i eR annual recharge rate average (mth 3s /yr)i eV , linear flow velocit hypotheticae th n i y l flow tube leadin wele th lwidte o gt (m/yr)th f flos hi o wW , tube (m) effective th , s r\i e porosit aquifee th f yo r thicknese th aquife e s i th (0.30) ) f o sr(m sampleb , d alon floe gth w tube (total saturated thickness samplea s da result f perforationso productioe th n i s measuree nth well)s i m dC , tritiume concentratioth s i Q wele d th an lt na 40-year decay corrected average input concentration of tritium (24 TU) (table 2). Burkham [12] estimated recharge rates (table 2) along the rivers in the Tucson basin using flow-duration curves that related infiltration rates to recharge during the period 1936 to 1963. Anderson [8] developed an electrical-analog model of the Tucson basin hydrologie system. Anderson's recharge estimate 194e th 0r sfo steady-stat e modelled perion i e dar tabl . e2 Hadj-Kaddou r [54] calculated recharge rates alon Rillite gth o Creek through deconvolutio welle th f -no response function for 1960 data (table 2).

Table 2. Comparison of Recharge Rates from rivers in the Tucson basin Arizona

River Recharge Average Recharge Rate ( x 106 m3 yfl km'1 ) Study Rillito Tanque Verde Pantano Wash River River Lindquist [1] 0.92 0.69 0.84 Burkham [5] 0.63 0.33 0.18 Anderson [1] 0.15 0.14 0.11 Hadj-Kaddour [61] 0.92- 1.9 - -

RADO TUCSON NI N BASIN GROUNDWATER resulte Th 222f so R n analyse f basio ] n[2 s groundwate plottee processee ar r Th figurn di . whicy 14 esb h 222 Rn enters the groundwater from aquifer minerals are weathering, diffusion, indirect radioactive recoil and direct radioactive recoil. The effect of these processes are referred to as the emanation power of a radium- bearing mineral, defined by Tanner [120]. The relationship between the hydrogeology and radon concentrations in groundwater has been documented [120-133]. Semprini [128] and Kalin [2] focused on the relationship between radon concentration in groundwater and the hydrogeology of an alluvial aquifer. They concluded that radon concentration groundwaten si derivee ar r d directly from aquifer mineral thad san t radon concentrations increase with decreasing particle size of aquifer material. Fine grained sediments have greater surface area and shorter pathways of radon to the groundwater. Therefore, fine grained sediments have an enhanced potential that direct radioactive recoil will resul radon i t n exitin minerae gth enterind an l groundwatere gth spatiae Th . l variability of radon in Tucson basin groundwater is most likely controlled by differences in lithology. Lithologie change Tucsoe th n si n basin represent change depositionae th n si l environment, from high energy alon basie gth n energ edgew centrae lo th o n syt i l basin. Graben faults have up-thrown lower stratigraphie units alone gth westerr easterfa d nan basin margins suspectee ar d an ,f producin do g very fine-grained fault brecci gouged aan . The interior well field of mid-Tucson represents background concentrations of 222Rn, ranging from 125 to 483 pCi/1. The anomalous 222Rn concentrations, approaching 2,000 pCi/1, along the western basin are changehypothesizeo t e du litholog n si e b o Kaliy t d b ] y n[2 associate d wit Sante hth a Cruz graben fault. These results suggest that regional lithologie compositions control radon values. The chemical composition of groundwater will be in-part controlled by the lithology of the aquifer system. Therefore, the chemical compositio f Tucsono n basin groundwate expectee b n ca r chango dt westere th n ei n basin.

226 STABLE ISOTOPES OF SULFUR AND OXYGEN IN DISSOLVED SULFATE The distribution of the measured S5 values is presented in figure 15. The results of these analyses

suggest multiple source f sulfato s basin ei n groundwater 6e 34STh 34 . values alon Rillite gth o River show variable mixing between SO4 sources. As the ground waters evolve down-gradient, geochemical reactions and admixture of shallow water with water from deeper stratigraphie units progressively increas 6e 34eth S isotopic value. An attemp mads wa tcollec o et t end-member sulfate phases from drill cutting archives t durinbu , g drilling of production wells in the basin, the cuttings are washed and contaminated with drilling mud, yielding samples of questionable value. The 034S values of the sulfate sources were estimated empirically using results

on dissolved sulfate. Figure 15 shows the results plotted as SS vs 1/[SOJ. There are three hypothesized 34

Tucson Basin Groundwater Study Universite Th Arizonf yo a Catalina and Rincon Mountains TV A

Stable sulfur isotopic composition Contour interva permi2 T = l lCD

0.00 0.05 0.10 0.15 0.20 1 / [S04] 12.0 —

10.0 -j I 8.0 —

oo 6.0 -^

40 -H-

4 , 10. 0 0 8. 12. 0 6. 0 14.4.0 0 16.0 T CD S S

34 34 34 Figur . Distributioe15 f 5no Tucson Si n basin groundwate relationshid an r f 0po S with [SO d 54]an S

227 mechanisms that produc distributioe eth f data) reactionno 1 , f riveso r wate r mountain-frono r t water (56 34 S= estimateT CD r bothdfo ) with sulfate phase basin si n mineral estimated)3 s1 (5S= ) mixin2 ; f recharggo e

water (low sulfat CDT e6 mountain-frond S5= an ) t recharge wate34 r (high sulfat) 3 CDT)e6 d S8= an ; 34

values between the previou34 s trends due to mixing of the two water types, either intrawell or along flow paths. The hypothesized values of S5 will be used when modelling the geochemical evolution using NETPATH.

Figure 15 also plots S34 5 vs O8 in dissolved sulfate for 6 groundwater samples. These waters were 18

selected based on the results of34 the S5 analyses. These samples represent O8 in sulfate values of the sulfate 18

mineral phase basine th n si , recent recharge basine wateth o t r, 34 waters havy thama te anthropogenic sulfata d an e water sample which is a mixture of 634S end-members. The 018O of sulfate is directly correlated to 834S and therefore, further study of the 518O in dissolved sulfate species may not enhance interpretation of the geochemical evolutio f basino n groundwater. O0 value f sulfatso hel y trackinn pma ei g potential

anthropogenic sources of sulfate for specifi18 c regions within the basin. Further study of this is warranted.

STABLE CHLORINE ISOTOPIC DATA Six samples were collected for analysis of stable 637C1. Christopher Eastoe performed the analysis of these samples at the University of Arizona, Laboratory of Isotope Geochemistry. The results of these samples hav uncertaintn ea f 0.09%yo havd oan e been publishe . Lon[134y db al t ]ge without interpretation. average Th e chloride concentratio groundwatee th f no basie th lowmg/1s 0 e sourcen i i r" <3 Th ,Cl . f o s initiae ar l meteoric chlorid rechargn ei e water, fluid inclusion mineralsn si , stratigraphically controlled evaporite deposits and recharge of secondary treated effluent and CAP water. There is no compelling evidence that suggests what source produces the small increase in chloride during the hydrogeochemical evolution in the Tucson hality basinan . Lowelef I Ft .existe e th lons l n dfm.i ha g t i ,sinc e been dissolved selectioe Th . f no samples, representing different sources of chloride, was based on the preliminary geochemical model of the basin ground waters. Three samples were selected that represented post-bomb recharge waters. These waters all have either meteoric chlorid r anthropogenieo c source f dissolveso l positival e dar salted witsan a h Se respec e th o t t Water Standard define permi0 s a d l 8 C1. Three waters were selecte thad dan t represent water contacn si t with 37 lower stratigraphie formations. All of these water samples have negative CS1 values. Long et al. [134]

suggested thaisotopicalle th t y light value r brinesfo s could potentiall usee yb identifo dt y leakage into37 potable groundwater supplies inferres i t I . d that 837C f recentl1o y recharged meteori r anthropogenicallco y derived chloride may be isotopically different from C81 of chloride in the mineral phases or deep, high TDS water.

Figur show6 e1 plosa C . f 1/[CIS1o t vs C Sd 1 an plotte] d against S0. Give uncertaint37 e nth - +/ f yo

37 34

0.09%o Cr thes81fo e measurements37 hypothesizes i t i , d thadate th t a represents onlsourceo ytw f chlorinso e

isotopes, 37 one isotopically heavier than sea water (meteoric or anthropogenically derived) and one isotopically depleted with respect to sea water (lithological source). The results of figure 16 generally support this hypothesis lightee Th . r value r 837sfo C 1 coincide with heavier value f 8o s34 S. 837C1 isotopic e valueb y ma s potentially useful when interpreting the geochemical evolution of groundwater in alluvial basins and the effects of anthropogenic contamination, but due to the high cost of analysis and small difference in values between chloride sources, other isotopic data should first be examined.

STABLE AND RADIOACTIVE CARBON ANALYSIS RESULTS Over 100 analyses of 14C from various groundwater and unsaturated zone studies in the Tucson basin are now available. The samples were collected and analyzed according to published methods [135-140]. The distributions of 813C and 14C in the basin are presented in figure 17. The results of the stable carbon isotopic analyses of the DIG show the influence of active recharge/carbonate dissolution zones along the Rillito River system. As water moves down-gradient, the stable carbon isotopic value of groundwater approaches an average value between -10 and -11 permil PDB. This is due to reactions between recharge water and the carbonate and dissolved CO phases in the recharge zone. Concentrations of dissolved organic carbon (DOC) in

Tucson basin groundwater rang2 e between 0 and 7 mg/1. The influence of oxidation of DOC oCn and C8 13

values of DIG is estimated when modelling with NETPATH, but DOC has little effect due to th14 e small mass- balance of carbon from this source in down-gradient wells. Further study of DOC in Tucson basin groundwater neededs i e distributioTh . f radiocarbo no , in-pardilutio o is t e G du tDI n n nwiti h depleted carbon through reactions with carbonate phases and, in-par radioactivo t t e decaf yo C . 14 Constraint of end-member carbon phases is crucial for reconciliation of both radiocarbon and stable carbon isotopes during geochemical modellin f groundwatergo . Tabl summarizee3 s radiocarbo stabld nan e carbon isotopic results on solid and gas phase carbon. The data are results of unpublished carbonate analyses from samples collecte rechargn di e zones throughout southern Arizona average Th . e 8e valu13n th i C f eo 4 recharge zone carbonate %0 01. . - -4.s Ther +/ 5larga swa s ei e rang f valueeo the'n si C conten carbonatef o t s from recharge zones. This study assumes a conservative estimate of 10 percent Modern Carbon (pMC), 1950 atmosphere = 100.0 pMC, for carbonates in regions in the Tucson basin where rapid groundwater velocities and

228 geochemical evolution under open condition assignes carbonate wa th sC o existdt pM valu A e5 . f phaseo n ei regions of the Tucson basin where groundwater ages are suspected to be less than 100 years and for where the movement of water parallels the recharge zone. Measured values of carbonate phases from well cuttings in the central basin have an average of -2.5%o 513C and 1 pMC.

MINERALOGY OF BASIN SEDIMENTS In an attempt to constrain controlling hydrogeochemical reactions in the Tucson basin groundwater the author reviewed the literature [4, 18, 21, 65-69, 142-157] and performed experimental analysis on basin fill sediments using X-ray diffraction, X-ray fluorescence, microscopic and pétrographie analysis. Quartz, plagioclase, calcite, orthoclase, muscovite, biotite, chlorite, and clay are the dominant minerals in the bulk fraction. Garnet, epidote, pyroxene amphiboled san s were detecte somn di e sample x-rad san y florescence data suggeste presence dth f apatiteeo , ilmenit dolomitd ean somn ei e samples. Minerals identifie thin di n sectioe nar

u. ou — —— -r 0.20 — ~~ o ~~ e g 0.10— ce ^ o __ co 0.00 — CD CO I i £j -0.10 — o

^r~v — o -0.20 — —L -

-0.30 — I I I 0.00 0.10 0.20 0.30 / [Ci 1 ]

0.2 , — 0

0.10

co 0-0] 0— CD CO .^ -0.10 —

-0.20

4.0 5.0 7.0 8.0 34 ' T CD S ô

Figure 16. 537C1 relationship with [Cl] and 034S

229 plagioclase, biotite, quartz calcited an , . Microscopic investigatio f sand-sizno e grains show presence sth f eo quartz, plagioclase, ferrihydroxides, orthoclase, chlorite, ilmenite?, biotite, and garnet. X-ray diffraction mineral identificatio lese th s n thano n 2-micrometer fraction f basio s n fill use standare dth d procedur f Dixoeo d nan Weed [156]. The samples were mounted on quartz slides and scanned at a rate of 2 degrees 2 theta. The samples were then treated with ethylene glyco reanalyzedd an l . Peaks that shifted after treatment with ethylene glycol indicate montmorillonite. Other peaks correspon micaso dt , calcite, illit kaolinitd ean e [157]s A . suspected, the major clay component was smectite.

Tucson Basin Groundwater Study Universite Th Arizonf yo a Catalina and Rincon Mountains N A

Stable Carbon Isotopes of DIG | Contour Inverva permi1 B = l PD l Kilometers

Tucson Basin Groundwater Study The Universit Arizonf yo a Catalin Rincod aan n Mountains N A #

Radiocarbon of DIC ContouC r InvervapM 0 1 = l

Figur . Spatia Tucsoe17 n i C l ndistributiod basian nC groun0 f no d wateC DI r M B

230 Table 3. Values of carbon phases used to constrain reaction path modelling.

Carbon Phase 513C Measured MeasureC 14CPM d Recharge Zone Carbonates11411 Average Valu -4.5%oe= Average Value = 10 pMC Mixing Zone Carbonates Estimated Value = -3.5%o EstimateC dpM Valu5 = e Deep basin Carbonates Average Valu -2.5%e= o AveragC epM Valu1 = e

11401 Soil Gas CO2 Deep Tucson Average Valu -18%e= o Range 115.9- 61.6 pMC basin Surface - Depth

[1401 SoiCOs Ga l2 Recharge Average Value = -19.1%o Estimated Value = Modern Zones Soil Gas COj11411 Recharge Average Valu -20.3%e= o Estimated Value = Modern Zones Southern Arizona River Water 1965 N/A C 142.pM 6

The author performed X-ray fluorescence analyses on 14 samples of both 120 mesh and less than 62 micrometer size fractions e stoichiometriTh . c compositio f eacno h minera constraines i l X-rae th y ydb florescence results, and is used in NETPATH when modelling water-mineral reactions.

CONTROLLING GEOCHEMICAL REACTIONS Table 4 shows stoichiometrically determined plausible reactions that control the geochemical evolution basie oth f n groundwater majoe baseth n rdo mineral basin-fil n i sthermodynamie th d an l c stabilit f eacyo h mineral. Selection of mineral phases and stoichiometry must agree with the chemistry of the groundwater, because the water chemistry is controlled, in part, by the chemistry of water-rock interactions. In the recharge 20 zones, high pCO . 10"2ca , [24] sourca dissolutioe s i th , f r Heo *fo incongruene f calcitno th d ean t weathering of silicate minerals. Silica concentrations in the recharge zones are oversaturated with most silica mineral phases. The geochemical evolution of groundwater chemistry is controlled both thermodynamically and

Table 4. Plausible chemical reactions occurring during the geochemical evolution of groundwater in the Tucson basin.

(1) Congruent dissolution of Tucson basin high-magnesian calcite: 2 2 Ca, 3Mg0,(C03)2 + 2H20 « 1.3Ca* + 0.7 Mg* + 2HCO3' + 2OH"

) Formatio(2 dissociatiod nan f carbonino c acid: + CO2 + H2O « H2CO3« H + HCO, * H* + CO/

) Incongruen(3 t dissolutio f Oligoclasno formatiod ean f Tucsoo n n basin smectite clay: 2 + Ca05Na90Al2Si3Os + 12H2O -» [(Ca.167>Mg«,K,,Fe05)]Al2Si367O10(OH)2H2O + (Ca* )+ (Na )

) Incongruen(4 t dissolutio f Biotiteno : +2 K(MgFe)3(AlSi30,0)(OH)2 + 3H2O + 8H* -> K* + 3-*Mg + xFeOOH + A1(OH)3 + 3H4SiO4

(5) Incongruent dissolution of Orthoclase and formation of Tucson basin Smectite clay.

KAlSi 3K(Ca.> O - 7H+ s* H 2 O167+ ,Mg45,K,,Fe05)Al2Si367O10(OH)2H2O -f JVH* K 2 O+

(6) Hydrolysis of silica: SiO2 + 2H2O « H4SiO4

(7) Congruent dissolution of Gypsum/Anhydrite: = CaSO4° 2H2O * Ca" + SO4

(8) Formation of Goethite orFerrihydroxides:

FeO(OH)> -- 0 2 H + 2 2 O + e F

(9) Ion Exchange: t2 (Na,K)2 - clay + (Ca,Mg) -> (Ca.Mg) - clay + 2 (Na,K)*

231 stoichiometrically. Reactant and product phases modelled in the groundwater system are constrained by the solubility of each phase. The speciation of dissolved constituents and the fugacity of gas phases are calculated to determin minerae eth l saturation index.

less i I s Imeasure e S ftha th thermodynamicalls i n0 P dIA y undersaturated with respec givee th no t phase. Conversely thermodynamicalls greates i i I P S IA f ri , e thath n0 y oversaturated phasa IAPe f o Th e,I . S K r ^o doe t predicsno reactione th t s that will occur rathet bu , potentiaa r givea r fo ln reactive phase. Kinetic hindrance r competino s g reaction inhibiy sma phasa t e from taking parchemicaa n i t l reaction. Data require accurate th r dfo e determinatio I includS f no e accurate chemical analyse f majoso d an r minor dissolved constituents, field pH, temperature, alkalinity, and when electro-chemical reactions are expected, accurate Eh.

CHEMISTRY OF TUCSON BASIN GROUNDWATER e Cit f TucsonTh yo , Water Department provide year5 d4 f chemistrso y dat e welle r manth th afo f n si yo Tucson basin. The Flowing Wells water company, which operates a small nest of wells in the Tucson basin, provided this study with chemical analyse f wellso s sampled ove lase th rt decade. These data, with data from ] wer 3 3 e , compile27 , 26 r geochemica , dfo 24 , the23 literatur, 21 l , modelling20 e, [4,1017 , .16 , Rose [158] and Cheng [159] present descriptive discussions of the general chemical trends of Tucson basin groundwater. Complete discussion of the chemistry of all data is beyond the scope of this work. In general, the basin groundwate rathes i r r dilute, suggesting that most water presently mined fro basie mundergons th nha e water-rock reaction unconsolidatee th n i s . Lowel dminoFt a ; fm lr amoun f wateo t r mine reactes dha d wit Tinaje hth a bed. The Tinaja fm. contains evaporites in lower stratigraphie units, but the Ft. Lowell fm. contains no evaporites. chemicae Th l compositio f Tucsono n basin groundwate controlles i r weatherine th y db g reactions listed in tabl . eThes4 e reactions refin r understandineou source th f f dissolvego eo d constituents [160]. Hydrogen , requireion+ incongruene H ,th r dfo t dissolutio f silicatno e mineral congruend san t dissolution of carbonates, derives fro dissociatioe mth f carbonino c acid equilibriun thai s i t m wit partiae hth l pressurf eo

CO in the unsaturated zone. Oxidation of pyrite/marcasite and the hydrolysis of Fe to Fe(OH) also produces 2 3 3

H groundwaten *i majoa t r systemno r s contributoi t sbu Tucsoe th r rfo n basin. + Sodium and calcium ions in Tucson basin groundwater, Na* & Ca*2, originate from the incongruent dissolutio f plagioclasno foro et m clay minerals. Cation exchang f divaleneo evaporitd an t + cationNa er sfo deposits are additional potential sources of Na+. Where carbonate minerals exist, Ca+2 is controlled by the solubility of calcite or high magnesium calcite. Gypsum/anhydrite is also a source of Ca.

Mg and K , magnesium and potassium ions, result from the incongruent weathering+2 of biotite.

2 +

Additional source+ f Mg*so include ferromagnesian minerals (pyroxene amphibolesd an s r dolomite/higo ) h

magnesium calcite. Som2 e K+ originates fro incongruene mth t weatherin f feldspargo . Dissolved silica species derive fro incongruene mth t dissolutio f silicatno e minerals. Quartz, becausf eo s neait r inertness, contributes insignificant dissolved silica even wit ubiquitoue hth s natur f quarteo basin zi n fill material. Solubility constraints may be controlled by clay formation or perhaps chalcedony. Naturally occurring chloride, Cl" controlles i , concentratioe th y db f chloridno meteorin ei c precipitation, river water, releas f fluieo d inclusions durin weatherine gth f mineralgo frod san m discharg f watereo contacn i s t with evaporite minerals. Sulfate, SO/2, ions derive from meteoric precipitation, dissolution of gypsum/anhydrite or other sulfate minerals, and from the oxidation of sulfide minerals. Bicarbonate, HCO3", forms from the dissolution of calcite and other carbonate minerals as well as the dissolution of CO2 gas and the dissociation of carbonic acid, H2CO3. The concentration of aluminum in groundwater is kept low by the solubility constraints of gibbsite and other aluminum hydroxide rangspeciesH p f Tucsoe eo Th . n basin groundwatee betwees 7.9i d rTh .an 7 n6. dissolved aluminum wil betweee b l n 10"10"d an 7 6M. Measured aluminum concentrations were betwee 10"x n6 5 and 2 x 10"4 M. This range lies within the gibbsite stability field. Therefore, aluminum is considered conservativ solie th dn e i l chemicaphas al r efo l weathering reactions that involve aluminum. Likewise concentratioe th , f irono n (FeTucson i ) n basin groundwate controlles i r stabilite th y db f yo

goethit ferrihydroxidesd ean . Dissolved oxyge+3 ubiquitous ni basin si n groundwate f Tucsoo h r E [158] ne Th . basin groundwater is between 0.3 and 0.9 volts and the pH range is between 6.7 and 7.9. This places the water

in the stability field of the mineral goethite. This concentration of iron, between 10" and 10" M, is in the range 8

of values presente . Therefore5b d tablen di an dissolutioe a s5 th , f irono n from biotit ferromagnesiad ean n 7 mineral alsn consideree sca ob d conservativ iros ea n wil solie l stath dn y i phase .

232 SELECTED WATER CHEMISTRY Tabl givea e5 s typical chemical concentration r differensfo t waterTucsoe th n si n basin. Wel5 C8 l represents recharge water. Two chemical analyses of water from a well drilled by the USGS in 1966 were provide Fredericy db k Robertso USGSe th f no . Thes analyseo etw s represent water fro unconsolidatee mth . dFt Lowell fm. and the consolidated Tinaja fm. There is a difference in water chemistry between the two formations. Austin Long collected water from the Keim well and the water was analyzed at the Laboratory of Isotope Geochemistr Universite th d yan y Analytical Center.

Table 5a. Chemistry of selected waters in the Tucson basin

Well . LowelFt l Tinaj. fm a Tucson Well Mountain Santa 160- fmm40 . 200-235m C85 Front Keim Cruz Screen Depth Screen Depth Recharge Well River 60m 220m Water [194] Calcium (mg/1) 37 488 30 13.6 25 Magnesium (mg/1) 5.2 3.0 2.7 2.6 2 Sodium (mg/1) 40 482 32 155 20

Potassium (mg/1) 4.2 N/A 1.4 7.51 0.95

Alkalinity (mg/1) 174 30 110 146 98 Sulfate (mg/1) 41 2010 37 317 9

Chloride (mg/1) 10 105 10 34.3 7.9 Fluoride (mg/1) 0.4 5.4 0.3 - 0.4 Iron (mg/1) 0.02 0.06 - 0

Silica (mg/1) 30 17 40 29.8 0.5

pH 8.6 7.9 7.4 8.1 7.0

Tabl . Chemistre5b f selecteyo d waterTucsoe th n si n basin

Well Flowing Flowing Flowing Flowing Effluent Wells Wells Wells Wells Recharge #70 11/02/89 #70 06/16/86 #70 08/25/83 #70 Ponds 08/06/80 Calcium (mg/1) 90 108 88 75 53 Magnesium (mg/1) 7.4 10.0 6.6 5.0 5.0 Sodium (mg/1) 57.9 61 56 46 120 Potassium (mg/1) - - - - 12.0 Alkalinity (mg/1) 117 112 68 90 150 Sulfate (mg/1) 336 154 155 81 105 Chloride(mgA) 139 124 118 77 111 Fluoride (mg/1) 0.13 - ND 0.12 0.6

Iron (mg/1) 0.07 ND 0.15 0.16 ND Silica (mg/1) - - - - -

pH 7.61 7.7 7.9 7.9 7.1

233 Tabl . Chemistre5c f yselecteo d waterTucsoe th n si n basin

Well Rillito River Tanque Verde Santa Cruz Pantano CAP River River River Water Calcium (mg/1) 32 18 47 74 53 Magnesium (mg/1) 5.0 3.0 7 15 27 Sodium (mg/1) 11 20 28 46 88 Potassium (mg/1) 0.5 0.5 - - 6 Alkalinity (mg/1) 119 59 137 218 100 Sulfate (mg/1) 9 11 65 143 252 Chloride (mg/1) 4 10 16 14 70 Fluoride (mg/1) 0.3 0.3 0.5 0.6 0.4 Iron (mg/1) - - - - - Silica (mg/1) - - - - - pH 6.9 6.8 7.1 7.1 8.4

Tabl presentb e5 s change chemistre th n si f Flowinyo g Wells Irrigation District welove0 #7 ldecad a r e of groundwater minin welgas chemicaas l l analyse effluensof t presently infiltrating alon Santgthe a Cruz River (Pima County Wastewater Management). The increase in dissolved sulfate and chloride over the decade 1980 - 1989 may represent a decreasing proportion of dilute water from the Ft. Lowell fm and an increase in higher wateS TD r fro Tinaje m th . (tabl afm e 5a)depte Th .f thi ho , suggest s m. well 0 23 ,s thachemicae th t l compositio f wateno r from this welproduca s i l f intrawelo t l mixing between wate . d Lowelr Ft froan e . mth fm l Tinaje th . fm a Table 5c presents the average chemical composition of water infiltrating the groundwater system through riverbeds in the Tucson basin and the average chemical composition of Central Arizona Project (CAP) water delivered to Tucson. The major source of recharge to the basin aquifer system is the Rillito River and its tributaries. The composition of river water defines the chemistry of infiltrating water prior to water-rock reaction and modelling with NETPATH. The present chemical composition of water in the Pantano River suggests a component of anthropogenic sources of dissolved constituents. The CAP water has higher TDS than natural recharge water but is similar in chemistry to reclaimed effluent. Mining of groundwater in the Tucson basin has affected the chemistry. Figure 18 shows the spatial

distribution of SO in groundwater measured in 1983 and 1992 along with contours of increased sulfate 4

concentrations betwee= n 198 1992d 3an . Three distinct area f increasso sulfatn ei evidente ar e . Three hypothetical sources of increased sulfate concentrations are; 1) In the eastern basin, draw-down of the water table has increased the contribution of mountain-front recharge and water from the upper Tinaja fm. to water mined in this region. Intensive water mining has reduced the head in the Ft. Lowell fm. and may have induced increased leakage fro mountain-fronte mth structurae Th . l geology also suggests thaTinajshallos e i th t. fm aw along the up-thrown side of high-angle normal faults in this area. The reduced hydrostatic head above this unit increasy ma e leakag f wateeo r upward fro lowee mth r unitsresule Th . t woulincreasen a e db d percentagf eo high sulfate water entering the Ft. Lowell fm. flow system. 2) In the western basin, the mining of groundwater continue removo st e shallow water deepee . Lowels Th Ft fro . e rm th watefm l r that reside up-thrown si n unitf o s Tinaje th woulm af d increasin n thea e nb g (over time) percentag f wateeo r mined. Recharg f reclaimeeo d sewage effluent at the City of Tucson Pilot Recharge Project ponds, at the Rodger road site, and in the Santa Cruz Riverbed also increased during this time period combinatioe Th . f thesno e effects probably accounte th r sfo observed increase. 3) In the southwestern extent of the study area, water residing in the Tinaja fm. may be an increasing (with time) contributio mineo nt d wate resula s a rf dewaterin o t . LowelFt . e Thisgth fm l , along with anthropogenic influence due to prescribed groundwater restoration, may account for the observed increased sulfate concentration.

234 Tucson Basin Groundwater Study The Universit Arizonf yo a Catalin Rincod aan n Mountains N A

Sulfate Concentration 1983 Contour Inverval = 20 mgfl

Tucson Basin Groundwater Study The Universit Arizonf yo a Catalina and Rincon Mountains N A

Sulfate Concentration 1992 Contour Inverval = 20 mg/l

Figur . Affecte18 groundwatef o s r minin [SOn go TucsoJn i n basin groundwater 198 3- 198 9

GEOCHEMICAL MODELLIN REACTIOF GO N TUCSOE PATHTH N SI N BASIN AQUIFER The conceptual model of hydrogeochemical reactions in the Tucson basin is similar to Robertson [142] where rather dilute recharge waters react with primary silicate mineral incongruentld san y form secondary mineral phases. This study differs from previous geochemical studie f Tucsoso n basin groundwate, 14 , 10 , [4 r 16, 17, 18, 23, 26, 27, 34, 38, 158, 159] in that, for previous studies, either little minéralogie data was available to constrai defind nan e water-rock reaction phasecompositioe th r so f minerano l phase speculatives swa n I . recharge zones, water-rock chemical reaction controllee sar presence carbonatd th y an db 2 f soieo CO le species (open-system) oncd an ,e wate moves ha r d down-gradient, theradditionao n s ei l sourc f COeo COd 2,an s 2i

235 consume procese th n di f silicat so e weathering (closed-system). Though Tucsoe wateth n i r n basin ultimately enter phasa s f geochemicaeo l evolution close COo dt 2, watee mucth f rho presently t evolve mineno a s o dha t aluminasilicate mineral phase boundary. Through examination of basin sediments and the chemical nature of water along selected flow paths, this study determined stoichiometrically correct reactions occurring in the subsurface that produce the observed changes in water chemistry from point to point along the flow path.

MASS TRANSFER MODELLING WITH NETPATH Geochemical reaction path controllind an s g geochemical reactions were modelle groundwaters5 5 r dfo , analyzed during 1984-1989 as part of this study. In addition, 13 groundwaters analyzed in 1965 were reevaluated. Many studies have been dedicated to the geochemistry of groundwater and the correction of radiocarbon age groundwaten so r [167-193] numbeA . f modelo r s have been develope , 16616 , , 170d[3 , 172, 174, 175, 176, 184, 191] that can be used when correcting groundwater radiocarbon ages for geochemical mass- balance, isotope mass-balance and mass-transfer. Hydrogeochemical modelling in this study considers two chemical systems: open-system, in which the aquifer chemistr controlles yi availabilite th y db f atmospheriyo c gases particularn i , , carbon dioxide down- gradient from the recharge point: closed-system, in which the aquifer chemistry is controlled only by water-rock interaction t influenceno s i d atmospheriy b dsan c gases. The new computer code, NETPATH [3], determines mass transfer between two waters when stoichiometric constraint placee sar reactivn do e phase initiad finad an s an ll water chemistry. NETPATH consist codes3 firste f so Th ,. DB.EXE databasa managemens e i , th r efo f chemicao t physicad an l l information about each sample. The second, WATEQ, is an aqueous speciation model modified after Plummer et al. [171]. thermodynamie Th c databas thir efo s versio f WATEno Q doe t contaisno complets na databasea WATEQ4s ea F [166]). This progra uses mthin wa di s stud determino yt equilibriue eth m stat f aqueoueo s solute waten si r with respect to mineral phases. The third program is the NETPATH code which is an extension of BALANCE [177]. This code determines potential mixing ratio mass-transfed san r reactions between initia finad an ll water chemical and isotopic compositions. NETPATH is an interactive program that can be used to interpret the net geochemical mass-balance reactions along a hydrologie flow path. The program uses defined chemical and isotopic data for waters fro mhydrochemicaa l syste constraimo t solutione nth . NETPATH examine l possiblsal e geochemical mass-balance reaction models between selecte f chemicao t d se waterisotopid a an lr fo s c f constraintso t se a d an , plausible phases in the system. The results from NETPATH are useful in interpreting geochemical reactions, mixing proportions, evaporatio dilutiod nan f waterno minerad an s l mass-transfe chemicae th n i risotopid an l c evaporation of groundwater. Rayleigh distillation calculations are applied to each mass-balance model that satisfies the constraints to predict carbon, sulfur and strontium isotopic compositions at the final water, including radiocarbo calculationse nag . The chemical and isotopic data to be modelled were entered into DB, the equilibrium state with respect minerao t l phase determines swa d automatically with WATEQ NETPATd an , uses Hmodeo wa dt l reasonable hydrologie flow paths (constrained by the 1910, 1940 and 1988 potentiometric surfaces) in the basin. This entailed a series of permutations that resulted in the analysis of hundreds of potential reaction path models. Results of these models, coupled with the results of preliminary modelling efforts with WATEQ4F constrained a series of flow paths and resulted in convergence of reaction pathways.

HYDROGEOCHEMICAL EVOLUTIO TUCSOE TH F NO N BASIN The flow of water across the entire basin is controlled by the potentiometric surface. The general flow path follows from recharg ease th t n ebasii northwese exith o n t n i t t basin. Riverbed infiltratio mountaind nan - front recharge alon uppee gth r reache Tanque th f so e Verde River flow betwee confluence nth f thieo s rived ran the Pantano River, under the Pantano River, to the center of the basin and then to the northwest exit of the basin. Hydrogeochemical modelling of this flow path included both open-system reactions, where carbon chemistry is controlling the geochemical evolution of water and closed-system reactions, where recrystillization reactions with carbonate phases and incongruent dissolution of silicates become the controlling reactions. Figures 19 to 21 plot the chemical and isotopic trends that support the conceptual model of this flow path four-stageA . d system (opene d- close d- opene dclosed- evidens i ) t alon floe gth w supportes i patd han d bpCOe ytotad Th figuran 2. l carbo19 e groundwaten ni t eaca r h point alon reactioe gth n path shows changa s ea

availabilite resulth f o t f soiyo l opee COTh .n reactiostagee th f so n pat zonee har f activso e recharge where 2 water with high pCO2 infiltrates throug soie hth l groundwatee zonth o et r system. pCO2 value highese ar s n i t water fro decreasd man wel5 eC7 l down-gradien dissolves a t d carbon dioxid uses sourca ei s da f protoneo s during the incongruent dissolution of primary silicate and carbonate phases. These reactions consume available dissolutioe th d an 2 f calcitno CO e continues until saturation with respec calcito t achieves ei t Welda l D-34 (1.0 kgradienp mu t from Pantano wash). Betwee D30d nan wel,4 recharg D3 lgroundwatee th o et r system occurs from infiltration along the Pantano River. The influence of open-system conditions near the Pantano wash is

236 0 6 8 10 12 14 16 18 20 22 Km along flow path

130 C85 C75 120 H D37 110 - 100 -

90 - D30 80 -

70 - o 60 -

50 -

40 -

30 - A35 20 -

10 -

-16 -15 -14 -13 -12 -11 -10 -9 -8 (PDB)

Figure 19. Changing carbon chemistry and isotopes during geochemical evolution of Tucson basin groundwater

237 Dolomite

8>

•1

-1

8> -2

~3 iiiiiii i Tiiiiiiiiiir

0 2 4 6 8 10 12 14 16 18 20 22 10.0 —, Km along flow path

ü w

0 2 8 1 6 1 4 1 2 1 0 1 8 6 4 2 0 Km along flow path

Figure 20. Changing sulfur chemistry and isotopes, and the staturation index of calcite, gypsum and dolomite durin geochemicae gth l evolutio f Tucsono n basin groundwater

238 ~1—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I—I— 2 2 0 2 8 1 6 1 4 1 2 1 0 1 8 6 4 2 0 Km along flow path

o

——|——|——|——|——|——|——|-r 0 2 4 6 8 10 12 14 16 18 20 22 20 — Km along flow path

15

—l— 2 2 0 2 8 1 6 1 4 1 2 1 0 1 8 Km along flow path

Figur . Changine21 ration mass-transfegd io san r durin geochemicae gth l evolutio f Tucsono n basin ground water

239 evident from the increase of pCO2 in well D-30 (150 meters down-gradient from the Pantano wash). Down- gradient from the wash, the saturation index of calcite again falls below 0 and calcite dissolves as consumption of CO2 takes place. The influence of the high-magnesian calcite discussed previously is evident in figure 20. Slow reaction kinetics produce hydrogeochemical control calciun so magnesiud man m concentrations actinn go the decade to century time scale. This geochemical pathway has a striking influence on the 14C and S13C as seen in figure 19. Plotting 5I3C values agains valueC 14 t s relate three reactions tha tisotopese contror botth o f e ho on l . First openn i , - 14 s phaswatee systemsga th e f er o th control dissolutioe (heren C th i , 2 e , sth bomCO f no b carbone th d an ) dissolution of carbonate mineral phases controls the 813C of DIG. This results in a nearly constant 14C of DIG and an increasingly heavy 513C trend. Second, in the closed-system, the dissolution of silicate minerals controls the carbon chemistry. The pH of water evolving in a closed-system increases, thereby oversaturating the water with respect to calcite and precipitating carbonate phases. Reactions in the closed-system have little effect on the o' s groundwate3f DIGCA o . r moves down-gradient, radioactive occurdecaC 14 f yo s with timdecreased ean e sth watere th f o . C Third14 effece th , f "recrystallizatioo t n reactions closed-systea n "i concurrena ms i t enrichment of the 513C due to exchange with the carbonate phase and a decrease in the14C content due to dilution with carbon with much lowe activityC r14 . Three wells, C-75, C-85 and D-37, define the bomb pulse, open-system dissolution of calcite and 14 13 consumptio f COno 2. Wells D-3D-3o t 74 show evidenc decayC trenf A eo . d toward heavier 5 C values suggests that true closed-system condition t exist t ratheno bu o , sd r wells C-75, D-3d C-8an 7 5 represena t transition phase between open and closed. At well D-30, open-system conditions reappear, the pCO2 increases 14 13 and water infiltratin Pantane th t ga o River add2 witshCO moder lighted an n e levelC rth 8o f t Cso + 13 groundwater system increasee Th . d pCO2 provides additionar Hfo l dissolutio f calciteno 5e Th .C becomes heavie resulta s a r . Water flowing down-gradient undergoe transitioe th s n from ope closo nt e system reactionn si the vicinity of well D-l, and when it reaches well B43, the system is essentially closed to pCO2. e relationshiTh f 8po 34 S with distance alon reactioe gth n path suggests thainitiae th t l 634 f wateSo r averages 6.0 %o CDT (figure 20). Intrawell mixing of water from deeper stratigraphie units or dissolution reactions with sulfate mineral phases (S34S = 13%o CDT) along the flow path increases the 634S. The dissolved sulfate component has been diluted with isotopically lighter sulfate from active recharge (from the Pantano River at well D34 to well Dl). The final 534S of the reaction path is a result of additional dissolution of sulfate mineral phases or again from intrawell mixing of water from lower stratigraphie units. Chlorid s conservativeei , increasing slightly along this flow pathfluctuatione Th . concentration" Cl n i s s are assumed to be due to variation in the mixing ratio of mountain-front recharge water, variable inputs of Cl" from precipitatio riversd nan r froo , m possible leakage from lower stratigraphie unitse tren Th f increasin.do g C1/SO4 alon floe gth w path befor Pantane eth o River indicate increasinn a s g mountain-front componen morn i t e recently recharged groundwater (figure 21). Cl/Ca tracks the dissolution of carbonate phases during open-system conditions and is nearly constant in locations where closed-system reactions prevail. The large increase in Cl/Ca t welindicate5 a A3 l s that this wate eithes ha r inpua r t from lower stratigraphi es undergon unitha r o s e significant Ca/N exchangen aio . Ca/Mg (figure 21) is low in active recharge zones due to the influence of the initial Ca/Mg of river and mountain-front waters. Dissolution of calcium carbonate phases in an open-system results in an increased Ca/Mg. This may be the mechanism affecting the Ca/Mg in well D34 near the recharge zone of the Pantano River. During closed-system reactions, the precipitation of calcite again decreases the Ca/Mg. The high ratio at well A35 again suggests a change in lithology or an increased input of water from lower stratigraphie units. The Na/K is problematic (figure 21). Na is controlled by two reactions along the flow path; incongruent dissolution of oligoclase and Ca/Na and Mg/Na ion exchange on clays. The sources of K are incongruent dissolutio f orthoclaseno , river water, mountain-front recharge wate intraweld an r l mixin f watego r from deeper stratigraphie units. Illit smectitd ean e clay sinke sar f potassiumso interpretatioe Th . f Na/no K ratio dependans si reactione th n o t s controlling [K]. Recently recharged wateNa/a s K ha r indicativa f eo substantial mountain-front recharge component. Na/K values decline to well D35. The rise in Na/K between suggest0 D3 d influn san a wellagais f rive 4 i . o x Wel 5 D3 sNa nr watern A3 lanomaloui w lo , s further substantiating potential intrawell mixing between two stratigraphie units. Figure 21 also presents the mass of dissolving and precipitating mineral phases along the flow path in Case Study Four. The geochemical reactions involving groundwater between the confluence of the Tanque Verde and Pantano Rivers (well C75 to well D34) progress from an open-bicarbonate buffered system to a closed silicate-weathering controlled system. This is evident as change in mass-transfer of the Ca-Mg carbonate phase from dissolutio precipitationo nt . Influ f rechargo x Pantane th t ea o River changes mass-transfer reactiono t s open-system condition . Mass-transfeDl d an t well sa 0 sD3 r reaction agaie sar n controlle silicate-weatheringy db , closed-system conditions down-gradient mass-transfee Th . f silicato r e phases increases when closed-system, silicate-weathering geochemical reactions dominate.

240 DETERMINATIO DISTRIBUTIOD NAN TUCSOF NO N BASIN GROUNDWATER AGES

35 of the 55 groundwater samples analyzed foCr and C6 resulted in values that contain water that is 13

modern (<50 years) after carbon mass-balance modelling I4 with NETPATH. A full description of the equations used to determine corrections to the'C of DIG using the mass-balance approach are presented by Plummer et al

[3]. 8d 13Analysian C C involve14 f o s s considerable tim effortd ean 4 ; therefore, this study will l al attemp e us o t t o'd 3placCan o t result recharge C e14 th f so f eaceo h sampl timen ei . Modellin f groundwatego r ages using bomb-derivedH has been used extensively in hydrology. The use of the exponential model has proven effective

interpretatioe foth r f ageno 3 s from tritium applicatioe Th . exponentiae th f no l mode bomb-deriveo t l d radiocarbon has been problematic due to the complex chemical reactions that affect the'C of DIG. The use of

isotope r validatiosfo f mass-transfeno r modellin determinatiod gan f geochemicano l evolutio f groundwateno r 4 increases ha d confidenc correctionn ei valueso C st presene W . simplifiea t d exponential model basen do

groundwate correcteC 14 r d with mass-transfe14 geochemicad an r l modelling. This approach essentially simplifies influe th intf C o xo groundwater durin e 1950'g th earl d san y subsequene 1960'th d an s t declin thin ei s function

afte14 r the atmospheric levels described by MacKay et al. [196].

defino T inpue C th e t functio r rechargfo n e water minimua , f thremo Ce value e needesar o dt 14

represent 14 groundwate recharge th n i r C e zone over timethree Th e. years chose defino nt groundwatee th e r

input function are 195614 , 1965 and 1989. The 1956 value of 14C in recharged groundwater was defined as 100 +/- pMC4 . This represent atmosphere th averagn sa f o C e14 e between 195 maximu1957e d 4an Th n .i C m14 recharge waters occurred during 1964 - 1965, the height of above-ground thermonuclear testing. Bennett [197] measure dsampla f riveeo rdurin C wate14 earle r gth fo r y par f 1965 o resulte Th f thi. o t s analysi s 142.swa - 6+/ 4.0 pMC. NETPATH was used to confirm the use of this value for the predicted maximumC of recharge water

in 1965.C and C5 used for this prediction includeC: = 180 pMC and C8 = -20.3%«, PDB for 14 soil CO

2

I3 14 13

8d I3 an C = -4.5% C carbonatr [141pM o fo 0 ,1 195]= 0 ed C I4 13 minera14 ;an C C pM l phase0 18 = s C (tablI4 ; e3) = -19 %o for river water and 14C = 16.5 pMC and 813C = -10.7%o for mountain-front recharge to the Ft. Lowell fm (table 3) and well C75-1984 chemistry. ThCe of mountain-front recharge is considered valid because the

source, Keim Well less i , s fro tham Ague mth n1k a Caliente Wash. Thes 14 e value definee sar d wit date hf th a o Bennett [197], Robertson [141], and results from this study. The 14C of groundwater recharged in 1965 as predicted with NETPATH mass-transfer reaction modellin s 148.gi 7 pMC. The radiocarbon activity of DIG in groundwater recharged during 1989 was measured at well C75 as 115.9 pMC. This well is less than 500 meters from the Tanque Verde creek and represents open-system groundwater that receives infiltration fro Tanque mth e Verde creek. determino T bomb-derivee eth d exces t eaca s hC well carboe th , n isotope value r NETPATsfo H

modelling of mass-transfer reactionI4 s were assigned pre-bomb values oCf =100 pMC and C5 = -19%o for

13 14 13 14 I4 13 river water, C = 100 pMC 8 C = -20.3%o for soil CO2 and C = 100 pMC and 8 C = -25%0 for dissolved organic carbon (DOC). The 14C and 8I3C of carbonate mineral phases were previously defined. The 14C concentratio atmosphere th n nyeai e th rr 195conventioney fo b assignes 0i C pM 0 dvalua , 10 thu f eo s 195s 0i radiocarbor yeafo 0 r n ages. NETPAT uses Hmodeo wa dt carboe th l n mass-transfer during hydrogeochemical evolution of each groundwater sample based on the constraints described in Case Studies One to Four. The mass-transfe finae f carboo rth ld carbonan n isotopic compositio f eacno h water sampl validates ewa d usine gth measured 813C of DIG. NETPATH uses mass-transfer, Rayleigh distillation and exchange reactions to predict

welle f waterth o l se C Al thath . t contain measurable tritium contain bomb-derive difference CdTh . e 14

14 between the predicted pre-bomb 14C value as determined by NETPATH and the measured 14C content of the wate assumes i r equao dt excese th l s bomb-derive abovC pMCd 0 14 line o 10 e Tw s. throug three hth e points defin bomb-derivee eth inpuC d14 t functio Tucsoo t n n basin groundwater (figure 22). Paralle- d an la line+l t sa 1er represen uncertainte th t inpue th f tyo function predictee Th . d bomb-derive r eacdfo hC well resulto tw n i s

potential recharge years intercepe Th . t year selecte r eacdfo h samplin14 g poin bases unifora wa tn do m relationship of increasing age down-gradient. Tritium values in figure 22 confirm the exponential radiocarbon function. Tritium measurement waten o s r sample r radiocarbodfo n follo same wth e trend represend an , t bomb- pulse tritiu basimn i n groundwater. Re-evaluation of samples analyzed in 1965 by Bennett [197] and interpreted by Wallick [4] were modelled with NETPATH usin same gth e assumption minerad san e 198lth phaser 9 fo dat s sa set. f o Thi t sse data is interesting because it was collected before severe anthropogenic effects on the hydrological system had occurred suspectes i t I . d that Tucsoe wateth n i r n basin aquifer syste t onlmno y increases witdowe hag n gradient, but also increases in age with increasing depth in the basin. This vertical stratification of ages increases the complexity of interpretation of the isotope hydrology of the Tucson basin system. 13 of the samples collecte Bennety db t [197] were use further dfo r study. Other samples wer t evaluate laca e no f Co kSt o e ddu

data, reproducible results, questionable laboratory note lacr so f complet ko e 13 chemical analysi watee th r f rfo so the year 1965. The predictedC of 8 of the 13 water samples suggests a component of water with bomb-

derived radiocarbon. These wate I4 r sample consideree sar d recently recharged water (<50 years).

241 100

1960 1965 1970 1975 1980 1985 1990 Yea f Rechargro e

40

•*-to' 'E Z) 20

0 1960 1965 1970 1975 1980 1985 1990

Figure 22. Exponetially modelled radiocarbon and tritium in Tucson basin groundwater

RADIOCARBO ESTIMATEE NAG WATEF SO R Figur present3 e2 s isochron r watesfo r collecte 198n di 9 that contained bomb-deriveCd, isochronf so

the radiocarbon age of groundwater collected in 1989 and the ages of water 14 collected in 1965 respectively. Comparison of these figures suggests that groundwater withdrawals have increased the age of water mined parallel to recharge zones (rivers) in the basin. Vertical stratification of groundwater ages is consistent with this observation. The interpretation of these results may also be an artifact of sampling. Present groundwater mining extends ovemeter0 10 r s intaquifee oth r resultin lesn gi s than ideal sampledeterminatione ag r sfo e antiquitTh . y of some groundwater agereflecsmay t intrawell mixing between pleistocen holoceneand e groundwaterThe . distribution of young water, ages less than 50 years, determined from the 1989 dataset reflect the rapid turnover of this water component with time. Hydrogeochemical modelling of samples yielded radiocarbon ages ranging from modern to 6300 years. Any inconsistencies between young (<50 years olded an ) r water results fro effece m th f krigin o t g data spacet da decadaa l scal datd ean a space t millenniada l scale. From evidens i figur t i 3 e2 t that much basie wateth s n i i r less than 250 years old. The antiquity of some water presently mined from the basin cannot be disputed given resulte hydrogeochemicae th th f so l model. Potential intrawell mixin f youngego olded an r r waters suggests that water wit oldese hth t radiocarbo woule nag d hav componenea f mixino t g greater tha determinee nth d value.

242 Tucson Basin Groundwater Study The University of Arizona Catalin Rincod aan n Mountains N A •*-

Contour Inverval =10 years

Tucson Basin Groundwater Study The University of Arizona Catalina and Rincon Mountains A

Contour Inverval = 500 years Kilometers

Figur . Isochrone23 f Tucsoo s n basin groundwater, 1989 radiocarbon ages

Reviewing figure 4 shows that the particle size distribution and transmissivity in the Tucson basin aquifer decreases from east to west. This change in transmissivity will slow water movement in the central Tucson basin. Shoe-string aquifers of high transmissivity run parallel to the Rillito River and act as high- velocity conduits for groundwater flow, bypassing the central basin. Recent recharge has not replaced water mined from the Ft Lowell fm. in the central basin. Thus, groundwater mined in the central basin will continue to increase in radiocarbon age. This hypothesis is supported by re-evaluation of samples analyzed by Bennett [197]

and reinterpretation of Wallick [4] for groundwater foCr and C8 samples in 1965. Figure 24 presents I3

isochrons of the radiocarbon age of water mined in 196514 . Comparison of the 1989 data and 1965 data shows

243 that the radiocarbon age of groundwater mined in the central Tucson basin has increased 5000 years during the last two decades. The influence of vertical stratification of ages is further supported by measured radiocarbon age196n i 1989 d t wel1 sa 5an A3 l, wher radiocarboe eth water fo e r nag mine d during 196 s 1005wa 0 yeard san in 198 s 1509wa 0 years. These values represen trena t f increasindo g radiocarbo f mineo e dnag water over time. The result is that Tucson depends on water that was recharged many thousands of years in the past. Paralle Rillitoe th o t l , Pantan Tanqud oan e Verde Rivers, groundwater velocitie sufficientle sar y higo ht offset the mining of the young water component and replace it with recently recharged water. However, in the central and southern basin, lower groundwater velocities and decrease in the potentiometric surface has depleted young, shallow water from the aquifer.

DETERMINATION AND DISTRIBUTION OF TUCSON BASIN GROUNDWATER AGES BASED ON MODFLOW AND PATHSD1™ HYDROLOGIC MODELLING prograe Th m PATH3 uses conjunctiowa n di ] D[5 n wit MODFLOe hth W result f Marrrelato o st ] ea[3 the temporal flow of water as determined witCh to the mathematically defined temporal flow of water. Marra

[7] chose MODFLOW, the Modular Three-Dimensiona 14 l Finite-Difference Ground-Water Flow Model, to develop an extensive two-layer computer model of groundwater flow in the north-central Tucson basin aquifer system. The MODFLOW program, as released by the USGS, consists of many integrated packages that are used to represent recharge, water removal, spatial head and transmissivity variations in three-dimensions, evapotranspiration, river boundard san y conditions modee Th . l reporte Marry db ] usea[7 d MODFLOWs a , publishe USGSe th y dsimulatb o t , movemene eth Tucsof e wateo tth n i r n basin. This study reconstructed the 1940 steady-state modelling efforts of Marra [7] from appendices and model output. Additional packages have been writte r MODFLOnfo Wboty b h publi privatd can e organizations. e versioTh f MODFLOno W use defino dt heae eth d distributio r 194nfo 0 steady-stat thin ei s studs ywa purchased from S.S. Papadopolou Associatesd san . This versio MODFLOf no W overcomes e somth f eo weaknesses of the original version, including the ability to re-wet dewatered cells during transient analysis. Marra [7] describes the refined MODFLOW model of the north-central Tucson basin as a "fresh look" at layer geometry, aquifer parameter previoud san s hydrologie studie reduco st e recyclin f interpretationgo d san circular reasoning. This modefinea s rha l grid (figur ) thae24 n many previous modelling efforts finee Th r. grid describes aquifer heterogeneities bette mord an r e specifically control locatione sth f naturaso l hydrologie boundaries. The mountain-front recharge boundary was constructed to be hydrologically connected to the flow s placesystemwa t dsufficiena bu , t distance Rillit e nortth f ho Rive dampeo t r effece nth flun o tx variations. The scarcity of both hydrologie and geochemical data along the basin-pediment interface severely limits the abilit modeo yt hydrologe th l hydrogeochemistrd yan f Catalinyo Rincod aan n mountain pediments layero Tw .s were incorporated intMODFLOe oth W mode f Marro l a uppe e [7]Th . r layer represent unconsolidatee sth d sediments of the Ft. Lowell fm. and overlying sediments. Layer two generally represents the consolidated sediment uppee th s undei f rso d Tinajran variabl. afm e semi-confined conditions. Leakage from stratigraphie units belo uppee wth r Tinajconsideres i . fm a d negligible [7]. PATH3 uses Dtraco wa dt movemen e kth f particleo t thren si e dimension timd san e e baseth n do velocity vector field calculated from the potentiometric surface results of MODFLOW. PATH3D includes the hydrologie constraint sMODFLOe useth n di W mode required an l s supplemental data arrays files needer dfo particle tracking. PATH3D consist majoo tw f r so segments firsvelocite a s i Th t. y interpolator use convero dt t hydraulic heads calculate t nodada l points int ovelocita y fiel r whicdfo velocite hth givea n yi n time stes pi determined as a function of position (x,y,z). The second is numerical solving routines used when tracking the movement of fluid particles in groundwater flow systems. The effective porosity and saturated thickness of each layer/node is needed by PATH3D to calculate the mass-balanc f water eo effective Th . assignes e. Lowel porositFt wa . e th fm ldvalu a f yo f 0.2eo 5 throughoue th t basin. The effective porosity of the consolidated upper Tinaja fm. was defined as 0.15 throughout the basin. e saturateTh d thicknes 567e f eacth so 0f ho node r botsfo h layelayed an I wer I rI r e calculated fro resulte mth s of Marra [7]. Given these data, PATH3D was used to calculate the mass-flux passing each node with time as constrained by the transmissivity, heterogeneity, potentiometric surface and stresses on the aquifer system. The PATH3D mode reconstructee bases th wa ln do d MODFLOW mode f Marro l a [7], wit revisione hth s requirer dfo the expanded version of MODFLOW and for PATH3D. The grid for the MODFLOW model of Marra [6] was digitized at the same scale as all other work in this study. PATH3D was used to track the temporal movement of water based on both the measured head distribution for 1940 (from Marra [7] appendix e) and the modelled head distribution (from MODFLOW output) for the 1940 steady-state analysis. The particles tracked with PATH3D in the Tucson basin flow system were introduced at the Santa Cruz, Rillito, Tanque Verde, Agua Calient Pantand ean o Rivers sub-seA . f particleo t alss oswa introduced alone gth southern boundary of the model to track the movement of water entering the grid from recharge sources not represented within the MODFLOW grid. The particles are envisioned here as specific water molecules that have

244 Tucson Basin Groundwater Study The Universit Arizonf yo a Catalin Rincod aan n Mountains N A

cson Basin Groundwater Study Universite Th Arizonf yo a talin Rincod aan n Mountains N A

Figure 24. Isochrons of Tucson basin ground water, 1965 radiocarbon ages and hydrological model grid infiltrated throug riverbee hth groundwatee th o dt r systegivey movd an nt man a tim , e0 down-gradiene= t affected onl hydrologiy yb e stresses definemathematicae th y db l model. Particle tracking represents forward modelling of groundwater flow. The results of hydrogeochemical modelling and the corrected exponential model of radiocarbon concentrations is an inverse model. To reconcile the two models, time = 0, is arbitrarily assigned the year 1989. Isochrons representing both the radiocarbon age and the movement of the recharge front over tim definee ear years da s before 1989. To represent the movement of water modelled with PATH3D, the location of each particle in three- dimensional spac projectes ewa plane Y locatiode X- ontTh .e oeacf th no h particl saves r eeacwa dfo h yeae rth particle moved along flow paths constrained from MODFLOW and PATHS D. The isochrons resulting from

245 recharge alon Rillitoe gth , Tanque Verd Agud ean a Caliente River isochrond san r watesfo r rechargee th n di Pantano and Santa Cruz Rivers are presented in figure 25. Different contour intervals denote spatial differences groundwaten i r flow-rate Tucsoe th n i s n basin aquifer. Each isochron define spaceY fronte X- f th so n ,i , recharge Tucsoe wateth n i r n basin system high-transmissivite Th . y regions that extend alon northere gth n aquifer margin presen conduia t r groundwatefo t r flow. Hydrologie modelling results show tha percen3 6 t f o t water recharged (from mass-balanc f particleo e tracking) alon northere gth n aquifer margi residenca s nha e time aquifee inth f leso r s tha years0 n5 . Onl percen0 y3 f wateo t r recharge Sante th n adi Cru Pantand zan o Rivers has a residence time less than 50 years (from mass-balance of particle tracking). Water recharged along the

Tucson Basin Groundwater Study Universite Th Arizonf yo a Catalin Rincod aan n Mountains N A

Contour Inverva year0 1 l= s

Tucson Basin Groundwater Study Universite Th Arizonf yo a Catalin Rincod aan n Mountains N A *

Contour Inverval = 50 years

Figure 25. Isochrons of Tucson basin groundwater age modelled hydrologically

246 Pantano River near the confluence of this river and the Rillito River, flows in the high-transmissivity zones that parallel the northern basin margin. Water recharged in the Santa Cruz River flows parallel to the river and does not extend more than a few kilometers into the basin. Compariso show5 2 d an remarkabl f figure4 no 2 , 23 s y similar groundwater ages from both hydrological modellin geochemicad gan l modelling frone Th .t definin spatial-distributioe gth f groundwateno r recharged over the last 50 years, as derived by geochemical modelling, lies within a couple of kilometers of the front derived from hydrological modelling. The variations between the geochemical model and the hydrologie model may only reflec uncertainte th t y when contourin resultse gth . convergence methodo Th tw e r moderth sfo f eo n (<50 years) recharge water suggests thasteadye th t - state hydrologie model reproduce basie sth c flow pattern basie th f nso flow system. However, MODFLOd Wan PATH3D modelling efforts failed to predict any waters greater than 300 years in age. The age of groundwater mined during 1989 approaches 6000 year determines sa d with radiocarbon dating morA . e accurate undisturbed representation of the age of groundwater in the Tucson basin aquifer is shown the 1965 radiocarbon data. This data represents the spatial distribution groundwater radiocarbon ages on samples analyzed in 1965 and showed maximue th f groundwateo e mag r mined during 196 s 1005wa 0 years. Compariso 196e th f 5no geochemical dataset with hydrological modellin temporae th f go l movemen f wateo t r result onln si factor-of-threya e difference. This comparison again suggests thasteady-state th t e hydrologie model reproduce pre-stressee dth d Tucson basin flow system. Potentia groundwatey l reasonwh o t s sa rcentrae ageth n si t completel l basino d ndi y converge include; MODFLOe 1Th ) W mode incompletn a s i l e representatio basie th nf n o flo w system. Extendin modee gth o t l include a larger area by adding a longer reach of the Santa Cruz River will enlarge the convergence zone between flow from the Santa Cruz recharge and flow from the Rillito River recharge. At the convergence of flow paths there is a stagnation of the water movement. The waters that show antiquity in the central basin may reflect these stagnation points. 2) The mining of groundwater prior to 1965 removed groundwater originally less than 500 years old, and continual mining of groundwater removes water that increases in age over time. 3) The hydrogeochemical evolution model for central basin groundwater has flaws. The increasing radiocarbon age of water mined fro same mth e well over time suggests vertical stratificatio f groundwateno r age. Geochemical modelling did not account for intrawell mixing of water having disparate radiocarbon ages. It is likely that a resulte3 combinatio discrepanciee d th an n di 2 1, f no s between resulthydrologie th f so e flow-modee th d an l hydrogeochemical model resulte Th .f bot o s h modelling efforts suggest tha higa t h degre f confidenceo e b n eca applied to the 1940 steady-state simulation of Marra [7], and that bomb-derived radiocarbon can be effectively modelled if combined with a thorough geochemical investigation of the hydrological system.

SUMMARY AND CONCLUSIONS This study included examinatio f isotopno e hydrology, hydrogeochemical change Tucson si n basin groundwater, minéralogie investigatio f basin o relationshi e th n d fillan , p between radiocarbo f groundwateo e nag r and the temporal movement of water determined by hydrologie modelling. The collection of many new chemical and isotopic data combined wit alreadn ha y extensive geochemica isotopid an l c dataset make Tucsoe sth n basin an intensively sampled aquifer system. Tucson will continu dependane b o et groundwaten o t watea s a r r supply resulte Th .f thi so s study suggest that some water now used as a water resource in the Tucson basin has a component recharged in the last years0 5 e diluteTh . , pristine water presently mined finit a fro s . LowelmeFt ha . limite mucth fm l f .ho Water from deeper stratigraphie units has much lower water quality. Investigation of the change in water quality with tim s showeha n tha s watea t r continue minee b o st d fro Tucso e increaseds mth ha S nmose basinTD Th .t e th , plausible hypothesis for a rise in TDS is the gradual dewatering of the Ft. Lowell fm. and subsequent increased percentag f wateeo r mined fro Tinaje mth a fm.. The study of isotope hydrology in the Tucson basin confirmed the results of Simpson et al. [33]. The new dataset f boto s h stable isotope precipitation si n (1981-1992 stable th d e an )isotopi c compositio f Tucsono n basin groundwater have expanded previous interpretations of recharge to the basin. Isotope hydrology confirms that recharg basie th no et alon Ague gth a Caliente, Tanque Verde, Rillit Pantand oan o river dominates si y db winter precipitatio run-ofd nan f fro surroundine mth g mountain ranges westere Th . n basin groundwatea s ha r heavier isotopic composition than along the eastern and northern basin margins. These results reflect evidence of recharge from either low-elevation, large storm events (potentially related to ENSO activity) or from summer precipitation. Evaluation of the mineralogy and stoichiometry of minerals that comprise the basin-fill has allowed for detailed mass-transfer modellin hydrogeochemicae th f go l reactions occurrin Tucsoe th n gi n basin aquifer. This t conclusivno stud s investigatios yit wa n ei chemicae th f no l composition, structur spatiad ean l (3-D) distribution of the clay mineralogy within the Tucson basin aquifer. Mineralogie investigation was sufficient to constrain the major water-rock reactions occurring during the hydrogeochemical evolution of Tucson basin groundwater. As

247 expected, the dominant clay component was smectite. Recalculation and réévaluation of previously published activity diagrams substantiate this finding relationshie Th . catioe th f pno exchange capacity wit structure hth f eo the Tucson basin clay mineralogy may increase the understanding of the relationship of dissolved constituents with particle size distributio basine clae th n Th yni . mineralogy will also pla importann ya e t rolth n ei movemen fatd f organican eo t subsurfacee th n i s . Furthe rwarrantede studb y thin yo ma s . Isotope e versar y useful tool r validatiosfo conceptuae th f no l understandin f groundwatego r movement, hydrogeochemical reaction-pat constraine hag modellinn a groundwates n a o t d gan r movement. Stable isotopes of carbon and sulfur were used to constrain potential mass-transfer reactions modelled with the USGS program NETPATH. The nearly unique solutions needed for convergence of mass-balance and isotope-balance constrained mixing and mass-transfer reactions modelled in the Tucson basin. The chemical and isotopic modelling indicated that groundwater mining may have increased leakage from lower stratigraphie units, or from mountain-front recharge into the Ft. Lowell fm. along the northern-eastern basin margin. This result is most apparent as an increase in sulfate with time. This trend may also result from an increase in intrawell mixing. The large screened interval within production wells results in less than ideal sampling for isotopic and chemical interpretation. This, combined with the inability of NETPATH to geochemically model a multi-component mixing system affecy interpretatioe ma , th t resulte th f f thino so s study. NETPAT uses Hcorreco wa dt t radiocarbon determination Tucson so n basin groundwatere DIGTh . stable isotopic composition of carbon (DIG) and sulfur (SO) together with the chemical composition of

groundwater, were used to constrain mixing and water-rock4 reactions. The stoichiometric composition of minerals that comprise the basin-fill were well defined. Of the 55 wells sampled foCr, 35 contained some

component of bomb-derived radiocarbon. Correction of mixing and isotope-dilution 14 reactions with NETPATH allowe approximate dth e yea f rechargo r determinee b o et d base simpla n do e exponential model spatiae Th . l distribution of groundwater with a "young" recharge component (<50 years) follows zones of high transmissivity within the Tucson basin aquifer system. The radiocarbon age of water sampled during 1989 from the central-basin approached 6500 years after mass-balanc isotopd ean e validation with NETPATH. Thirteen groundwater samples were collecte analyzed dan d for 14C by Bennett [197] in 1965. These samples were reanalyzed using the results and interpretations of this study e radiocarboTh . f groundwateo e nag r mined durin centrae g th 196n i 5l basin only approached 1500 years. Analysis of well A-31 in both 1965 and 1989 resulted in an increase of the age of mined water by 500 years. Therefore, Tucson is becoming increasingly dependant on water that recharged many thousands of years in the past thad an ,t muc f thiho s water canno easile b t y replace rechargey db . The results of the Tucson basin hydrogeochemical modelling effort were compared to results of hydrologie modellin same th f ego system . This allowe evaluatioe dth temporae th f no l movemen f wateo t y b r two independent methods MODFLOe Th . W finite-difference hydrologie mode extendes f Marro wa l ] r a[7 d fo wite us h PATH3D particla , e tracking software package. Water-recharg inpus ewa t alon Rillitoe gth , Pantano, Agua Caliente, Tanque Verde and Santa Cruz Rivers and tracked along the steady-state (1940 estimate) velocity vector field as determined from the results of the MODFLOW model. The movement of these recharge "fronts" with time compared favorably wit spatiae hth l distributio recharge th f no e "front" determined with bomb-derived 14C. The age of water in the central basin determined with hydrologie modelling (1940 steady-state) and the radiocarbo f wateo e rnag mine 196n di 5 have onl facto ya difference 3 f o r convergence Th . f radiocarboeo e nag and hydrologie age increases confidence in both the hydrogeochemical modelling effort and the steady-state analysi f Marro s a [7] attempo .N s mad wa treconcil o et transiene eth t modelling effor f Marro t ] wite a[7 hth antiquit f groundwateyo e 198r th age9 r sfo isotopi c dataset. This effort would require evaluatio multia f no - component mixing/mass-balance geochemical model. No hydrogeochemical model is presently available for this. In addition resulte th , f Marrso ] suggesa[7 t that transient modellin Tucsoe th f go n basin syste t welno lms i defined usin MODFLOe gth W model. Refinemen MODFLOe th f o t W mode requires i l d befor attempy ean s i t made to reconcile both transient models. Further work to reconcile the transient data is warranted. Studies of bomb-derived C1, fréon nobld an s emovemenfatd e gasesth ean d an , f dissolveo t d organic matteC r d ( an C 36 14 13 isotope balances Tucson i ) n basin groundwater should provide additional constraints needeo dt hydrogeochemically model a multi-phase mixing system. hydrologie Th hydrogeochemicad ean l modelling show that recharge alon Tucsorivere e gth th n i s n basin may have a relatively short mean residence-time (<50 years). Any artificial recharge to the aquifer system should consider the modelled movement and temporal fate of water in the aquifer. Artificial recharge may increas velocite eth f wateyo certain i r n region basine th f o s, ultimately resultinrechargee th lose f th o s n gi d water frorr basie th i n syste recovered.me b beforn ca t ei Finally, climatic change will affect the present sources of recharge to the Tucson basin aquifer. A comprehensive evaluatio paleohydrologe th f no Tucsoe th Avrf d yo n an a Valley aquifers shoul consideree db o dt assist in long-term planning of water resources. The results of this study should be further evaluated to provide information on century-to-millennia time scale recharge to the Tucson basin aquifer.

248 ACKNOWLEDGMENTS

authore Th s would lik acknowledgo et dedicatioe eth stafe graduatd th an ff no e research assistante th t sa Laboratory of Isotope Geochemistry, University of Arizona, in particular the efforts of Christopher Eastoe. This work was funded by the State of Arizona, Geohydrology Program. We acknowledge helpful dicussions to this work made during cooperation with the IAEA Coordinated Research Program on Mathematical Models for the Evaluation of Isotope Data in Hydrology. Finally, we would like to acknowledge the contributions made by Jean-Charles Fontes understandine th o t , radiocarboe th f go n datin f groundwatego r systems.

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254 CYCLIC MODEL FOR SEASONAL RECHARGE AND DISCHARG GROUNF EO SPRIND DAN G WATERN I VALDAE TH Y EXPERIMENTAL BASIN (HUMID ZONE) USING ENVIRONMENTAL ISOTOPE DATA

V.I. FERRONSKIJ, V.V. ROMANOV, L.S. VLASOVA Water Problems Institute, Russian Academ f Sciencesyo , Moscow O.P. KOLESOV, S.A. ZAVILEISKIJ State Institute of Hydrology, St. Petersburg V.T. DUBINCHUK Research Institute of Hydrogeology and Engineering Geology, Moscow Russian Federation

Abstract The seasonal recharge and discharge of ground .and spring water withi smalna l experimental basi humin ni d zone near Lake Valday (56N, 33E studies )wa d using environmental tritiume .Th contributio transid nan t tim differenf o e t source rechargf so e and discharg grounf eo d sprindan g water were identifiede .Th sources followss appearea e b o :t d precipitatiod nan direct surface runoff with zero transit time, sandy layer with a transit time of 0.25yr, the morain sandy loam with locae th ld swaman , pyr wit0 2 hf wateo tim e watef ro eag r . exchangyr 5 f eo The process of seasonal recharge and discharge of ground and spring sourcy watewholb a d s ra eean component s cyclisha c regime n thi.I s connectio cyclie nth c mode r descriptiolfo n procese th f proposes o si d discusseddan .

1. Introduction The problem of groundwater formation through infiltration recharg permanenf o s ei t interes hydrologisto t d soisan l researchers because of its practical importance. The other aspect of the task is the regime of groundwater discharge to- wards surface spring riversd san e consideratio.Th botf no h the infiltration rechargsurface th d eean spring dischargf o e groundwater withi catchmenna t areelaboratiod aan corresf no -

255 ponding physical and mathematical modela setß up the complete formulatio probleme th f universano o thert n . Bu s i e l solution of this problem because of the variety of natural hydrogeolo- gical and climatic conditions. Therefore, an experimental app- roach is searched for generalized but typical hydrogeological and geographical conditions. The present work was carried out on Valday experimental ..stati- State th eof no Institut Hydrologyf o e statioe .Th situas ni - ted near Lake Valday, at a distance of 350 km from Moscow, in the directio St.Peterburgo nt humida n ,i ' zone typicar lfo the North-West Russia, The scop goald ean experimenta f so analiticad lan ls a wor e kar follows : estimato t - e seasonal infiltration recharg dischard ean - ge of groundwater within the studied catchment basin; - to determine the parameters of soil moisture transport through the unsaturated zone; physicae builo th .t p du - l picturrunofe th f fo e formati- on during the flood and non-flood seasons; - to comprehend the mechanism of groundwater discharge into local springs; - to elaborate physical and mathematical models of gro- undwater regime for the studied basin taking into acco- unt its infiltration recharge and discharge into springs.

. 2 Hydrological condition basie th nf o s Valdae Th y hydrological statio representativs ni Northe th f -e-o West regioEuropeae th f no n par Russiaf o tlocates i t .I n di the central part of the Valday Upland (56N, 33E). The studied area cover ssmala l meadow catchment basin withi fiele nth f o d the statiocalles i Hollo e d th dnan w Usadievsky e sprin.Th n go the bottom appear left-bane b locae o st th lf o kPonik a River flowing intLake oth e Valday catchmen e (Pig Th Hole . th . -1) f to low Usadievsky is elliptic in shape and has an area of 0.44 kin2 , stretching Soute frolonm Nort e k beinth d mth gho an 8 ht g 0. and 0.4 width5m k aree characterizes .Th ai morainy db e hilly landscape. The exess of the hill tops over the Hollow thalweg does not exceed 4 m. The only part exceeding 18-20 m is its wes- tern region. The mean slope of the catchment is equel to 4?k>. The slope of the Hollow bed is 26#o. Positive forms of the landscape are formed by solid moraine loam coverd by a layer of sandy lo-

256 am from 0.3 to 0.5 m thick. An exception to this general pictu- re occurs on the most northern and southern parts of the catch- ment which are composed by sand varying in grain size and in- cluding pebble and gravel of 3-4 m thick.

Pig Schem1 . experimentaf o e l basi,n Usadievskd yan sampling point location figuree .Th s indicate borehole numbers.

The podzolic soils make up 75% of the basin area. The peat lay- er on the swanp is of 1-2 m thick. More than 80% of the catch- ment is occupied by a field and a meadow. A patch of forest is locateNorth-Ease th basie n i dth d occupie nf an o t moro sn e than 2% of the area. The swamp constitutes 11% of the area and is locateNorth-Ease th Ease n th i dt d partstan .

257 The annual runoff of the Hollow Usadievsky is equal to 120 000 vr whil totae eth l precipitation reaches 600-700 mm/yre .Th mean monthly flow rate varies from 0.1 to 25 I/sec. The total duration of the runoff absence due to winter freezing and sum- mer dry seasons is close to 250 days. The floods occur in late March or early April every year.

3. Methodology The collection and accumulation of experimental isotope and conventional data on infiltration recharge and discharge con- ditions of the basin was the main goal of the first stage of the work environmentae .Th l tritiu e tracea s use mth wa s da f ro study process tracee .Th r carries both geneti d timcan e infor- mation and is the best tool for this particular purpose. Perio- dic but systematic sampling of water was carried out at fixed points. Pig. show1 locatioe ssketca th f o hf observationa no l boreholes, stream d othesan r point samplingf o s . The geological structure of the studied basin is shown in Pig.2.

m , 0 10

Pig .Geologica2 l sectio'experimentae th f o n l basin: borehol ) numbers (1 it ) d sand(2 en ,e y loam, (3) clayish sand, (4) clay, (5) sand.

The data on tritium content in precipitation and its variation in time are needed for hydrogeological interpretation of the tracer behaviour and for the elaboration of physical and ma- thematical models. The collection of those data in the region was started at the beginning of 1985» mainly on the monthly ba- long-tersise Th . m mean annual value tritiuf so precipitan mi -

258 tion are determined by the data of the near located meteorolo- gical station Europeaf so n Russia where such observations have taken place, namely in Arkhangelsk, Tver, Vologda, Minsk and Moscow. Those particular data are presented in Table 1. The re- ported result previoue th f so s years were recalculated relative to 1986 taking into account the radioactive decay of tritium.

Table 1 Mean annual concentration of tritium in precipitation over Central European Russia and recalculated figures relative to 1986

Year Tritium Tritium Year Tritium Tritium measured relativo et measured relativo et in T.U. 1986 in T.U. in T.U. 1986 in T.U

1960 207 48 1974 127 65 1961 246 61 1975 127 68 1962 1340 349 1976 101 58 1963 3963 1090 1977 99 60 1964 2325 678 1978 99 63 1965 934 288 1979 77 52 1966 752 245 1980 52 37 1967 404 139 1981 75 43 1968 274 100 1982 49 39 1969 296 114 1983 52 44 1970 198 81 1984 41 40 1971 137 58 1985 36 35 1972 155 71 1986 38 38 1973 90 43 1987 33 1990 27 1988 35 1991 31 1989 28

During 1986-198 sampline 7th s carrieboree gwa th frot -l ou dmal hole d springtime san spring-summef th o e t sa autumn-winted ran r recharge seasons e dat.Th a were user identificatiofo d e th f no genetid an e ag c typ waterf o e s which tak recharge e th par n i t e springe oth f d groundwatesan differenf ro t soil layerse .Th data were also interpreted with respect to water dynamics in the various seasons of the year.

259 Starting from 1988 monitorine ,th continues gwa d using limited numbe representativa f ro e borehole within basii e d nth nan the mouth ranges of the spring and its tributaries. The samp- les were collected during spring-autumn floods, after individu- rainfalll a durind s an wintee gth r thaws.

. 4 recharged-dischargef o Genesie ag d san d water Pig demonstrate3 . resulte th s summef so r 1986 observationse .On thae camaie nse th t n surface th par d subsurfacf o tean e samp- les have tritium content close to its concentration in the mean annual precipitation varying from 25 to 40 T.U. Besides, there are three places where tritium values are different from the Eorth-Ease th n i aboves i basite e th par .On nf o t- withi up e nth per swamps and is characterized by higher concentrations which vary from 52 to 54 T.U. This fact indicates that meteoric water of the previous years has stored here. Taking into account the data reported in Table 1, it is possible to calculate the time of water delay which is equal to 6-7 years and the measured values of tritiu groundwaten mi r represent precipitatio 1979f no .

f

Pig. 3 Mean values of tri tium measuren i d strea) 1986(1 : m sampling place, (2) areas with higher concentra- tions relative to precipitation.

260 The second place with higher tritium concentration (from 41 to 53 T.U. s discover)wa d over sand boreholeye th loa, n 12 mi , s11 Her . average 18 eth d an e tim delaf o e groundwaten i y r movement is equel to 3 or 4 years and related to moraine sandy loam. The third area is characterized by lower concentration of tri- fixes T.U.8 wa boreholn o i dd t tiu )an 7 wher n ( m e30 e artesian carboniferoue wateth f ro s sediment e dischargedsar . This water involvet no s i d intgroune oth surfacd dan e water dischargeo t d the springs. A histogram in Pig. 4 shows the frequences of tri- tium concentration watee th rl sampleal n si s take 198n ni d 6an 1987 and confirms the above mentioned statement.

10

Pig. 4 Histogram of tritium concentration distribution in groun surfacd an d e Hollowatere th f wso precipitation Usadievski d an ) (1 y n ovee th r experimenta 198n l i 1987d are) 6an (2 a.

The mai groune n th parsurfacd f dan o t e waters (78$ rechare )ar - ged from precipitation of sampling year. And water with tritium concentration higher tha T.U0 n5 . (16# relate s )origii e th o ndt

261 of previous seasons. The water with tritium content less than T.U0 1 . forms distribution maximum connectet whicno s hi d with the main mass of water. Let us consider time variation of tritium concentration in two group waterf o s "olde "younge th :th "d "onesan . mose th Boreholt s representativi n 53 e firse th tf o egroup t .I is located on the east slope of the spring bed, and its depth is 4."4'-ffl from the surface. The borehole reveals groundwater in the covering sandy loasandd man y lens below (see Pig. 2) . Pig show5 . tim e sth e variatio tritiuf no m concentration ni water froborehole mth whicaccordancn n i 53 es hi e wite hth water level change witd san h tritium values variatioe th n ni monthly precipitation during all the periods of observation. The average tritium contenbote th h n typei t same watef th o se s i r and equals to 34 T.U.

50 T.U. +—°>(ca) 30 S 2 ,——C-~O' 30 ; 4 e

40. (b) 23 4

20 !.*W IC^ V J1 -———-*—,"

io:l 'YJj/u 'v 1 »i ixi 'jv' H ' iv ' %v 'v i xrx/h 'vcv/i I ' f ' 1986 1987 Pig Variatio5 . tritiuf no m concentratio precipitan ni - tion (a), borehole 53n (b), and borehole 11 (c) relative to vater table variation. (1) tritium concentration, (2) water table relative to the surface, (3) the peak of concentration.

Buy correlatiotan n between tritium concentratio groundwan ni - ter Cg and water table Hw . as well as between Cg and precipita- tion C calculated by the least square method is practically, absent (r=0. 0.0d 2an 1 respectively) followt .I s from this

262 that despitface th t f thao e t groundwate recharges ri prey db - cipitation of the same year, there is a delay in time between the fallout and appearance in the borehole. One can see from the plot that the delay does not exceed one season, i.e. three months morr .Fo e correct estimatio infiltratiof no n delae yw compare time successive occurencf ^th th o e l al f eeo monodirec- tiona Fign ld i extremean ) bot.e 5 (b th h d f o curvean s ) s(a calculated difference in time. For the parallel extreme seri- fro9 sprine - th m e summee 1 s198f go th 198f 6o ro t e 7th value time difference is equal to 60, 37, 7, 10, 18, 16, 27, 10 and 62 days, accordingly. The maximum values of time delay equal to 1 and 2 months are related to the period of May and June when winter precipitation appears in groundwater with max- imum delay.For the rest of the year the average time delay is close to 12 days. Taking into account the depth of the perfora- ted casing of the borehole, it was found that the average velo- cit verticaf o y l infiltratio sande th yn ni loa d sandman y sec- tio s 1 closcm/dayi 2 n o t e . Groundwater of the second group with higher tritium concentra- tio studies ni detain i d l together with precipitatio boren ni - borehole holTh p (se1 u 1 e . m locates 3) eei 0 Fig3 d an n .i d2 to the slope from borehole 53n and intersects only sandy loaia. Its depth from the surface is 2.76 m, and the lower end of the perforation is situated at the depth of 2.64 m. The tritium concentratio wated nan r table variatio e demonstratenar n i d Fig. 5c. The mean value of tritium concentration in the water durin perioe gth f observatioo d equas 3 T.Uni 5 o .t excelt I - meae edth sn valu tritiuf eo precipitation i m same th e r fo n perio T.U9 y notb de . W interestinn ea g fact thatritiue th t m content change has reverse run with water table in the bore- hole e ris .Th watef eo r tabl accompanes i e loweriny db f go tritium conten d vican t e versalowee th rt . A wate re tablth t a e from surface c th m5 deptmeae 13 eth f n o h valu f tritiueo m con- tent in water is equal to 60.4 T.U., and at a higher level, at the depth of 27 cm, the tritium concentration drops to 46.8 T.U, This fact allows to assume that there are two sources of water recharg differenf eo feedine tag boreholee gth f o e .On them supplie lowee th s r sandy loam layer with retardatiof no movement and the other is connected with the upper layer where the modern water is moving along the moraine sandy loam. We as- sume that the upper source has tritium concentration similar to

263 watee th thaboreholn ri f to e 53n, i.e. equa 33.o lt 6 T.Ü., whic similas meahi e th no rvalut e durin time observaf gth o e - tion, and this water is mainly responsible for the water table variation in borehole 11. Then it is possible to write a system of balance equations whose solution allows to find the propor- tio mixinf no g water component tritius it d msan concentration lowee th rn i laye sandf ro y loam. We writ systeme eth :

Cup.w.t+ . X "1 C Clow.w.tf 1 C .- 1 = y + X or Cup.w.t. - C1X + C2(1 - x) Clow.w.t. - V* + C2(1 - x> meae th n e valuear . f tritiuso d CU an m con. wher- eC centration in water of boreholes 11 at the upper and lower water table, respectivelye tritiuar p C m d concentrationan L ;C e th n si both, water sources "younge th , n i.ei ". water simila thao rt f o t borehole 53n, and in the sandy loam of the lower layer; x and y are portions of both waters; it is the coefficient of variation of water proportion of the first source which is proportional to the variation of water level in borehole 11 and equal to . (270.5135)7(27: - 6cm 7 ) 27 6- The numerical form of the system is as follows:

) x C+ 2- 46.(33.= 1x 86

60.4 = 33.6.0.57 x + C2(1 - 0.57 x) The solution of the system gives the following figures:

x = 0.71; y » 0.29; C2 «= 78.5 T.U. ( x'» 0.4 = 0.6= 0.5 ) ;0.5x - 0 x 7 7 1 y '= The calculated figures show that durin flooe gth d period (higher water table y "young)b borehol3 2/ " filles o i watert 1 e1 p du . In the dry season 60$ of the borehole water is represented by the "old" one from the lower layer of sandy loam. In order to thif o estimat se "oldag e "eth watercomparn ca calcue e ,w th e - lated value of 78.5 T.U. of tritium concentration with the in- put function of tritium in precipitation corrected by radioac- tive decay relativ 198o t e 6 (see thin I Tabl se . casus 1) e e ew

264 piston flow approximation. The precipitation with tritium con- tent of 80 T.U. corresponds to 1970. So, water age from the lo- am sediments of 2.76 m depth is close to 16 year, and the ver- tical velocity of infiltration in sandy loam is 276 : 16 = 17.2 cm/yr. Additional soil samples near borehol wer1 1 e e take meany nb f so auger in November 1986 and in June 1987» The boring depth was 4.23 m and sampling was fulfilled at interval of 0.5 - 1.0 m. The pore water was evaporated at the temperature of 300°C and tritium conten measures twa d (Pig . Variatio.6) tritiuf no m concentratio pore th e explainee n wateb ni n ca r d onl pistoy yb n emplacemen e "oldth f "o t meteoric takewatere on sf .I int - oac count the diffusion washing of tritium and its radioactive de- cay in time, in this case the maximum value of 120 T.U. in pore connectee b n ca wate m d depta 3 wit t rf a o hh penetratiof no meteoric dept e wateth ho t rsinc e 1963 when thermonuclear tests took place in the atmosphere. This give stima e interva year3 2 f slo relativ 198o et 6 yeaf ro observation. Prom this the velocity of water infiltration is froe se m n «3 botcm/yr3 =2 1 ca : he 0 .calculationOn 30 s thae th t ordewatee th figure f rsamo e r d th movemenean s sande si th yn i t 5 order2. y b sloa s lowei m r tha sandn ni . Q 1P JT.U . . . . . o _, g ,_» ,

1986 100

200

300.

Pig. 6 Tritium profile of pore water in sandy 400 loam near borehole 11 measure n Novi d . cm 1986 and June 1987.

265 The calculated values of filtration velocity seem to be limited for infiltration properties of the waterbearing sediments. For example, the velocity of filtration in carbonate and fractured rocks estimated by analogous isotope methods is 63 cm/yr. This correlates with the same interval and by 4 times exceeds the ve- locity in the sandy loam. froe above se th m n One ca edat a that groundwatef tritiuo e mag r in two different deposits varies from 0 to 3 months in sand and from 0 to 23 years in the moraine sandy loam. Using experimental results, it is possible to estimate seasonal age variatio watef no r withi yearna . Tabl containe2 s initial data and calculation results of water proportions of different age (a,x,y) for the groundwater of sandy lens sampled from bore- frod an mhol n Sprine53 g Usadievsky calculatioe .Th carries nwa d out relativ seasonae th o t e l balanc watef o e r whic s acceptehwa d xmitys a . Fig .demonstrate7 same th se dat absolutn i a e valuef so of water amount of the particular age . The relative data of Tab- le 2 represent the formation conditions of water in a season while Pig show.7 s their changes durin yeare th g . The age characteristics of water properties were selected from the calculations on the basis of the previous analysis of isotope data. In fact, water with t = 0 within the season transit time and seasonal delay t = 3 months relative to the time of precipi- tatio takes basia i n s a nc fallout watee prin .Th i r 0 -wit = ht specifiee nciplb surface n th ca e s da e runoff. It follows from Tabl tha2 emaximue th t m precipitation proporti- groundwatee th n i o n0 wit = r ht appear sprinn i s summerd gan , and the minimum one - in autumn and winter. Accumulation of gro- undwater in the cold half of the year by 70 or 80% takes place at the expense of previous year precipitation. Groundwater recharg wara n mi e season goes frosurface mth e runoff. It was found that groundwater balance in the spring of 1986 and usee on s f onli 198fractionso t tw y7 fi doe t sno . Evidentlye ,th third component frodeee th mp sandy loam takes par n thii t s pro- cess. The old water is displaced into the sandy lens by the pre- ssur floof eo d water. Tritium conten sucn i t h wate s equari o t l 78.5 averagee T.U. casth e three-componenf th n o e,o n .I t task, for the calculation of three ages it is possible to combine two balance equation sprinf o s g season d generasan l water balance equation assuming small hydrological difference between the same

266 seasons of two years. The result of the calculation shows that the fraction of "old" groundwater with the age of from 16 to 23 years in the sandy lens in a spring season does not exceed 1056. By analogy, the balance of the main surface spring Usadievsky at the closing range was calculated (Table 2 and Fig 7). Cont- grounwatere th rar o yt isotope ,th e balanc r sprinefo g does not fit given that two components are used for spring as well r autumnfo s attempe a e tritiu.Th us o t t m concentratiof no

Table 2 Seasonal variation of ages of ground and spring basiwatere th 1986-198n ni f so ?

Season Discharge, Tririum content Portion of water Mean mea ncomponente valuth n ei s a x y age y C x C a C C s in1/ t»0 t=0.4 t=20 in yr Groundwater, borehole 53n 1986 Spring 0.93 43 40 37.578.5 0.57 0.33 0.1 2.13 Summer 1.01 41 38 43 - 0.4 0.6 - 0.24 Autumn 0.49 31 36 30 — 0.2 0.8 - 0.32 Winter 1.03 29 24 31 - 0.3 0.7 - 0.28 Mean weited ralue 36 - - 0.37 0.61 0.025 0.77 1987 Spring 1.20 9 32 2 6 2 78.5 0.57 0.33 0.1 2.13 Summer 0.75 35 37.2 53 - 0.55 0.45 - 0.18 Spring Usadievsky 1986 Spring 11.32 8 43 1 0 4 43 0.28 0.23 0.49 2.54 Summer 0.6 3 4 0 8 3 - 0.6 0.4 - 0.16 Autumn 7.45 37 0 3 6 39 0.49 0.05 0.46 2.32 Winter 1.92 29 24 31 - 0.29 0.71 - 0.28 Mean weited value 38.5 - - 0.36 0.22 0.42 2.19 1987 Spring 10.45 35 9 2 26 43 0.28 0.23 0.49 2.54 Summer 4.26 26 37.2 3 5 22 0.19 0.12 0.59 3.0 Autumn - 26.5 15 53 39 0.49 0.05 0.46 2.32

267 (a) I/B 9

5 4

A » . / \\ 3

2 v / xV' -/ v\ /

I I I V I I II I I I I II III IV I II 1986 1987 1986 1987

Pig. 7 Seasonal variation of different age water for groundwate sprind e an th ) gf (a ro wate ) (b r basin: (1) water table and discharge, (2) sur- face runoff (t=0), (3) groundwater with t=3 months, (4) "old" water with t=20 yr (a) and t=5 yr (b).

eandy loam (78.5 T.U.) with a longer delay time (20 yr) as the third component does not provide the balance and the equation unsolvede b systeo t s i m. Prophysicae th m l poin viewf to t ,i is understandable due to the fact that the proportion of the "old "soie wateth l n doei rt excee sno durin% 2 d yeare th g , and because of dilution of this water its tritium concentra- tion appears to be within analytical errors. In this particu- possibls r i cas la t ei assumo t e e tha addition i t precipio nt - tation (portion "a") and groundwater (portion "x"), the third watee componenth re b uppe e fro e swamn th th m ca tr f po par f o t the basin. Isotope characteristi swame th representes i pf co d Sprine byth borehol d drainagge an th Daln n d 26 eyan e trench. The mean value of tritium concentration here is close to 43

268 T.U9 sprine 3 th automn d .i T.Ur g an .fo n (t= fro5r y m piston flow model) resulte calculatioe .Th th f so e presentenar n i d Table 2. and Fig. 7. e formatioTh runofe th f fno with basie n th s characterizeni d by the opposite tendencies relative to those of the groundwa- ter. Increas n contr-tbutioei runofe th f o fn gro(t=0e th -n )i undwater leads to decrease in the spring, i.e. at continu- ous saturation of porous media of the unsaturated zone by pre- cipitatio melter o n d snoproportioe th w f thino s "young" water runofe th lowes n f i i vic d ran e versa r instance.Po e th n ,i similar spring seaso 198rainf e no th d 1986 yan d 7summean f ro minimue 198th 7 m proportio surface th f no e sprie runofth -f fo soile maximus e characteristicth it sar r d fo man g n . Maximum surface oth f e runoff occur autumn si n when 80 soif %o l pours e fillear d wit hprevioue wateth f ro s precipitation (t=0., 4yr see Table 2). And winter is the only season when there is clo- se agreement of water component ages for the spring and the soil. During the rainy seasons of 1987 the percentage of swamp water in the spring runoff is high (42% of the mean annual runoff). But this wate completels ri y sande absenth yn i tlens e .Th runoff from the swamp is discontinued in dry seasons of the year (the summer and the winter of 1986). Because of high pro- portion of swamp water with t=5 yr the groundwater component with t=0.4 yr in the spring runoff is lower than in the unea- turate meas dit n zon d annuaan e l valu 22%.s ei meat Bu n annual surface runoff of the spring and soil water is the same (31%). firse basiyeare o th th tw tf observatiof o so - n O fo s wa t ni d thaun t wate fouf ro r age balancins si recharge th n gi d ean discharge within the basin: surface water of precipitation origin (t=0), groundwate sandf ro y deposits with residence 0.2o t swam, p 5yr timu pf o ewate exchangf o r s wityr e5 h time, and water of morain sandy loam with mean age of 20 yrs. The contribution of different age waters to the recharge of Spring Usadievsk varies yi d seasonally. Mean componentvaluee th f so s in 1986-1987 were 36, 22, 41»8 and 0.2 accordingly. The mechanism of groundwater discharge into the spring is well identifien simply b d e piston mode displasemenf lo watef o t r accumulate sedimente th n i d s under pressur enteree th f eo d pre- cipitation mode x waterbo le give.Th ssprinf o mea e gnag water . yr 2 equa2. o lt

269 5. Seasonal cyclin rechargef o g d dischargean d d basin water The results of tritium measurements of groundwater in boreholes 11, 18, 26, 26n and 53n obtained in 1989-1991 are presented in . 8 Fig.

T.U» 50 .

40.

30

20-

10.

m c 0 20 0 10 50 Water table

Relationshi8 Pig. p between tritium concentration of groundwater and its water table (mean value r 1989-1991sfo )

The depth of boreholes Taries from 3 to 5 m. Perforation of the boreholes 18 and 26n was accomplished on all the length. The borehol reache1 1 e s only sandy loam n come, 53 sand o t s y layer an 6 locateswam2 e d th pn si are opend aan s peat water. One can see from Fig. 8 that the sandy loam profile (26n, 18) tritium concentration of water increases with decrease of gro- undwater table. This relation demonstrates the fact that after flood water has gone the recharged into the borehole water previoue th f o appear se b year o st s origin with higher tritium concentration s founwa t d.I earliedate boreholth f ao y rb 1 1 e that mean water age of this is about 20 yrs. For sandy sedi- ments in borehole 53n tritium concentration practically does not watee depenth rn o dtable . Thi becauss si higf o e h value movement velocit watef o y sandsn ri .

270 A reverse pictur s observe ei swame th pn i dborehol wher, 26 e e tritium concentration of water is sharply droped with water ta e decreasbl d visean e versa consequenca s A » e eth peat water on the depth has very low tritium concentration. On the contrary uppe sectioe r th par f o tn peat wate s higherha r tritium concentration of the previous years precipitation. Pig show9 . s this phenomenon more clearly.

0 10 30 50 T.U, 1 .

3 -

5-

Pig Distributio9 . f no 7- tritium concentra- tio swamn i n p bore- m hole (mean valur efo 1990)

The plot was built up using 1990 observational data of one mo- n upstreaa n i e boreholr p u mnea e t swamth ese r f locateo p d Tver region. The profile of tritium concentration is divided in- to two parts. The upper one up to 2 m has high concentration whicr mea e fo e th hobtainenag mode x bo abou s li y db yrs5 t . Th edepte loweth hf ro show par m - st8 wa thao fros t it t2 m ter age is not less then 100 yrs due to diffusion process of transfer e sam.Th e characte watef ro r dynamics seeme b o st responsibl r tritiufo r mfo e distributio borelole th n ni 6 2 e relativ watee th ro t etable relationshi, .So p between tritium conten wated an t r level shows that after spring floodd san rainfalls process of replacement of previous year water in sandy loam and peat soils takes place. Sandy soil does not re- taiwaterd nol . Taking into account distribution of the observational boreholes within the basin it is assumed that the mean tritium concentra-

271 samplee same th th e l accepte e n montb i stioal n f no hca s a d mean isotope characteristi r groundwatecfo basine th f n .ro I this case isotope variatio groundwatef no r from mont monto t h h is applie s indicatoa d dynamicss it f o r . Pig .show0 1 s this variatio d dynamicsnan .

T.U.

40

05 •o 20 Fig Relationshi0 1 . p bet- ween tritium concentration ni groundwater and precipitation (mon- thly mean values for 1989-199D. Figures indicate ths i e t mont d han 0 3 0 2 10 40 T.U. residence time. Precipitation

Theoretical lines reflecting general direction tritiuf so m con- tent changegroundwatee th n si n connectiori n wit varis hit - atio precipitation ni alse nar o drawn line .Th defineeI s tri- tium distribution in groundwater of zero age. This is possib- le when tritium conten groundwaten i t equas ri thao lt n i t precipitation. Theoretical lin I correspondeI grounde th o st - yrs5 t= .f Isotopiwateo e rag c compositio thif no s water reve- weas al k reaction with respec seasonao t l variatio trif no - tium in precipitation and the line of its tritium concentrati- on passes in parallel with the abscissa axis. Let us consider seasonal distribution od tritium concentration in groundwater shown in Fig, 10. During winter time tritium content does not depend on its concentration in precipitation, thi becauss si frozef o e n soi d snolan w cover t concentra.Bu - tio tritiuf no m here equaT.U.1 3 o lt , wha lowes ti r then ni groundwater during 198 4 T.U.8(3 averagen )i - ex .e b Thin sca plained by infiltration of small portion of water of 1989 win- ter precipitation with low tritium concentration (10 T.U.) into

272 groundwater. Perhap happens ha t si d during wintere du thaw d san to variatio freezinf no g soie deptn connectioth li f o h n with winter temperature variation meae .Th n valu thif eo s tha- win fluece winter water in soil is estimated by tritium balance wher, portios tha e i a) 34(+ th ea w- a 1 9 f wate no = 1 r3 ( of 1989) d equa,an 12%o t l . Prom March to April tritium content in groundwater sharply dro- pped from 34 T.U. to 24 T.U. due to the spring flood. The amo- unt of thaw water in the soil in April was 32$ (24 = 9a + 31(1- a). In June tritium concentration in groundwater was 29.5 T.U. while in precipitation its mean monthly content was 31 T.U. Prom isotope balance it follows that in the beginning of Summer Sprinf o onl % y1 g flood water presente groundwatere th n i d e .Th remainin wit x managoldet e mi g hno th o 93t rd e $wate di d ran discharged into the spring. During Summer variation of tritium concentratio smaln i s l nwa rang e (28-31 T.U.comparison i ) n with broad change mean i s n monthly precipitatio thesf o n e mon- ths (17-31 T.U.). The mean monthly tritium content of groundwa- ter in summer was 30 T.U. and in precipitation was about 24 T.U. It means that the groundwater content of tritium is weakly de- penden summen o t r precipitation. Using tritium - balancta d ean king into account tritium conten groundwatef o t winten ri s ra 31 T.U., one can find that summer precipitation here forms por- tio 14f no waterf same $ o th e t .timA e summer precipitation ni that regio35-40p annuae u th t e f nse $th lo o onet .e du Thi s si surface runoff and évaporation-transpiration process. In December when negativ temperature is set up, tritium content in the groundwater was close to the summer value (29 T.U.) but with small shift relative to the preautumn flood of 1989 (31 T.U.). The valu meaf eo n annual tritium conten grounwaten i t e th f ro basin is close to 29 T.U. and in the precipitation is 24 T.U, Prom this using isotope balance equation the portion of preci- pitation in the groundwater is equal to 29$. It means that one third of old groundwater was replaced by this year precipitati- on water. Mean residence time of groundwater here is close to 3 years. It is worth to draw attention that this is mean residence time. It varies from clos zero r t sane somo fo t d e tent yearf o h s r sandfo yy obtai loamma e n.On froequatioe th m isotopif no c

273 balance that only 16 groundwaterf %o representee sar watey db r from the sandy loam and loamy sediments. Let us consider now seasonal variation of isotopic composition of Spring Usadievsky in its mouth range. Pig. 11 shows tritium content changes relative to the seasonal water discharge in 1989-1991. demonstratea Fi11 g regime th s Sprinf eo g discharge lona timn r i g fo e tha Februarn i w beginnine th y n 1989I f .go the thadischarge th w e s increase Th e1/swa 7 .o t s d 1/ fro5 m1. tritium content reached the value of 38 T.U. exceeding its le- groundwatee th ven li 1 T.ÏÏ. (3 rmucd )an snohe th morw n pacei k (9 T.Ü.). That time tritium conten watef o t tributarn ri y Bli- zhny flowing out of swamp was 43 T.U., and in the borehole 26 (on the swamp) was 49 T.U. So that the possible source of high tritium concentration was swamp water with that of about 46 T.U, With the increased flood up to its maximum of 9 February the portion of melting snow water in accordance with proportion portioe th t f no Bu . increase) a M% - o t 1 p 9( du + a 38 = 3 3 swamp water is still high because the tritium content of spring water was 33 T.U. which is higher then that of groundwater (31 T.U.). In the end of February when the thaw was finished the discharge of spring dropped to 33 1/s» the tritium concentrati- on in its water was decreased lower of its value in the gro- undwater and reached the figure of 23 T.U. The calculations show that this was a stage of maximum portion of melted snow water in the spring (42$). In general during the thaw the dec- reas tritiuf eo m conten sprinn i t g meltin o watet e rdu g water discharge is observed. But in this particular case the main contribution belonge swame th po t dwate r discharge. showb same Fig11 sth .e distribution durin sprine gth g flood in ï.Iarch-April 1991. In the end of March when melting started the value of tritium content in the spring water was 2? T.U. This is close to that of mean monthly content of groundwater (26 T.U.). The mean tritium concentration in snow water of win- ter precipitatio T,U6 1 . s Wit nwa increase hth dischargf eo e f sprino g wate tritius rit m content decrease dilusioo t e ddu n by melting water, but after 23 March started slowly to increase. After 4 April the discharge dropped from 107 1/s to 50 1/s but tritium concentration starte increaso t d reached ean d maximum value of 45 T.U, on 6 April. And to the end of flood both pa- rameters returne initiae th o t dl values makin gcyclea .

274 i 1 T. U, (a) T.U. (b) Swamp 50 . 50. Swamp ^6.04 40 . ^.02 40- / ^v^^

~*~—--»^

«^ ^ Groundwater^-- —~ 20 «^8.02 20 _ 1^.04^*-2T.03 Snow+rein 04.1991

10 . Snow 10

0 10 0 5 0 s l/ 0 5 0 4 0 2 c) l/s

T.U. (c) 50 Precipitation 08.1991 40

30 6.08.1989 J^recipitation 08.1989 12 20«.2"L —.-,-,19.08 20 . Fig. 11 Relationship between TS.~08.1991 tritium concentrati- on and dischage of spring water during 10 - February thaw of 1989 (a), March- 10 30 50 70 08.1^89 Apri f 199lo 1 flood (b), and separate 0 T———F 08.1993 1 rainfall 198n si 9 l/s and 1991 (c).

Between the results of numerous analyses the maximum tritium concentrations were observed in the water in the water of Spr- ins Dalny and Blizhny from 4 to 9 April with values of 4-4-60 T.U. So that the maximum concentration of tritium during the period has swamp water with value of around 52 T.U. In this co- nnectio increase nth tritiuf eo m explanevaluee b - y in sma y b d crease of discharged swamp water, If it is true then the swamp

275 contribution to the droped discharge of the spring water from determinee b n ca 26( + proportio ) figury a a b d 1 - 52 f = o e 5 n4 7396. Fig. 11c demonstrates the spring dynamics during separate rain- fall differenf so t intensity intensive Th . e rainfall during 5-7 August 1989 increased the spring discharge up to 50 1/s. The precipitation wate d tritiurha m concentratio T,U9 2 f .no The mean value of that of groundwater in August was 39 T.U., and the spring water at mouth range showed 30 T.U. It follows from that the surface runoff of the rainfall in this situation reached 70%, Late tritiun ro m contensprine th n gi t increased to 38 T.U. and water discharge in the spring decreased to 35 assumes i 1/s t I , d thatritiue th t m content increase corres- ponds to swamp water of Spring Dalny with concentration of 42 T.U, Its amount is accounted by figure of 66%. Some different pictur observes ewa d durin poo a longt rbu g rainfal 17-2n li 0 August precipitatioe ,Th n tritiu concend mha - tration of 43 T.U,, and the spring water with discharge of 0.5 1/s had 24 T.U,, and groundwater content showed 24 T.U« The increas accompanies sprinf wa o e s g1/ y discharg5 b d2. o t p eu weak increas tritiuf o e m conten 5 T.U2 o y isotop.t B e balance only 14% of the spring discharge was determined by surface run- off. In the first case the spring discharge was increased by 16, 6 secontimese 35, th t portiotimesy n e db .i Bu th d f ,an no surfacn i firs% e 14 th e t o n runoft i cas d % s equaean 66 f wa o lt the second one. It follows from those that during weak but long rainfall the water saturates the soil. During heavy rainfall the surfacy wateoverfloe ma th d n ro swame ean run wth f psof which discharges with som espringe delath o t y. n ordeI havo t r e more representative pictur f isotopo e e dyna- mics in the spring water the mean tritium data of the basin from 198 199o 6 t e presente 1ar Pign o dTritiu. .12 m concentra- tions of the spring water here in January-February are exceeded tha precipitationn ti sprine ,Th g recharge occured mainlt ya the expense of groundwater with higher tritium content. During the spring flood which starts in the region in the end of Marc beginninr o h Aprif go l tritium concentratio sprine th f gno - water is decreased due to their dilution by melting water wit tritiuw hlo m content procese .Th mixinf so snowf go , swamp and groundwater of zero, five and three year age is observed on the plot.

276 The main experimental resultfollowss a e sar : residenc- e tim groundwatef o e sandn ri y loa determines i m y b d d everan ys yeayr 3 r fresh precipitation water displacee on s

T.U. \ v *^vv v 30_ *7^Lx - --^

20.

10.

V I I II I I I I VIIX VI X I II V V I XXI I 1986-1991 Pig. 12 Variation of tritium concentrations in preci- pitation (1). in spring water (2), and in groundwater (3). Mean values for 1986-1991 are normalised to 1986. waterd thirol f ;o d - residence time of groundwater in swamp is equal to 5 yrs and it discharges during the spring and autumn floods; - during winter thaw groundwater is recharged by melting water and its portion is accounted by 12-17$; - the .'recharge of springs is performed by surface runoff, gro- undwate d swamran p water; - during 6 yrs of observation the water balance of spring dis- charge is accounted by 50% of precipitation, 30% of ground- swamf o % waterp20 water d ,meae an .Th n residence timf o e spring water is close to 2 yrs. . 6 Cyclic mode seasonar lfo l recharg dischargd ean e of ground and spring water using the approach of maltycompartmental hydrological systems Seasonal variatio watef no r cycl sinchronous i e s with variation of isotopic composition of precipitation. In this connection a bescyclie descriptior th tfo e cb modeo t d simulatio s nan li n ef natural processes like cyclic recharg dischargd ean f eo groun sprind dan g water.

277 3 represent1 g Fi a block-schems e syste th mode f mo l under consideration. In order to preserve a symmetry of the analytical presentatio e modeth f lo n and ,n turni , avoi n ovea d r complication of equatio e deriveb o suppost e n w d e tha l blockal t s Ccompartments, q M ro PRECIPITATION

n qdi Cdi

SURFACE RUNOFF

"rz Crz

SWAMP

q C ar ar d3

GW, 1st HORIZON

q C r* r* V »* »* d GWHORIZO2n , N

v v v v out out DRAIN

Fig 13 A block-scheme of the multi compartment model of a hydrological dranaige system with sinusoidal inputs:

q . Ct3*q .simot. + q .

± i

"out +q di -«-Mq. M

•••^q .

réservai r sD of the system are intercbnnected each with others and back-feeded itself. Then a basic model equations can be written as follows:

k=l

CE) k=l

278 where k , i indexe particulaf so r compartments, n total number of the compartments, C , , C isotope tracer concentration in the i-th and k— th com- partment respectively, i.e. variables to be sought, V. water volum storage!eC f i-t)o h compartment, q wate, q r flow rates froi-te mth h into k-th compartment and vice versa respectively, q . recharge flow rate to the i— th compartment from the external systems C input, a precipitation share used to recharge the i-th compartment} , q discharge flow rate of the i-th compartment to external systems Coutput, discharge into a main drain of the watershed, i.e into a creak, river, lake, etc.}, X isotope tracer decay constant. o thiT s equation some initial conditions concernin e wateth g r storage d tracean s r concentration n everi s y compartments muse b t added, for instance: eV,Ct=O:>, V C3D , oi i C4D C oi sC.CtsODi , Generall e modeth ) f lC2 yo equation d speakingt se an e D th C s, appeae non-stationarb o t r l y variableal one s a s s incorporatee ar d depended on time. Opening the derivative in the right side of C2D gives

C5> VidCi/dt+C.dVi/dt= Zqk.Cki - £qikCik+qriCri-qdiCdl-XaViCi l i= k=l

Further substituting dV /dt from C13 into C5D yields Chere and farther the summation indexes are omitted as they have been fixed!)

C6> V

where

C7!> V±- o Thus, the general set of the basic model differential equations has been get. It is seen that an analytical solution to such a set is not so directle easb o yt y solved especiall takes i t ni intf yi o accounta real dependenc f inpuo e t function d interrelatioan . q s n functions e currenth d «I*n o an q, t.. time. Generally, there might be used any relevant known calculation techniques. Due to a huge complexity the general solution to this problem in an explicit form is practically impossible to be given in this short communication. There is a reason to give here some approaches how this solution could be get and demonstrate some illustratio morf no e simplified cases.

The first simplification can be made assuming that water and an isotope tracer in each compartment is fast and completely mixed less during a time period which is to be much less than a mean residence time of water and the component respectively. This is so-called a perfect mixing model CPMM3. In that case, a current output concentration in any i-th compartm is to be equal'ed to that inside of this compartment. It means that

279 C83 C C9D C..Ct>=C.Ct) ik i C10D C..CO-C. CO di i

Using this relationship n C6i s D gives u s

C113

or C12D ri

where the similar members are underlined. Further rearranging yields

C135 V,dC

V^C./d + (Zqt q+ kir + i

Nevertheless, this variangenerae th f to l equatio beet ye n s nha stayed non-stationar d indefinitan e on y e relatine wateth ro t g storage CtsV D depende n currendo t time solutios It . principalls ni y possibl f therei s dat i en curreno a t temporal variation f wateo s r :flow rates related Cseq ee

Makin e seconth g d simplificatio e assum w nthaw no te temporal changes of all water storages V. are negligible compared with their mean averaged values, s i i.esuppose t i . d that definitels a , or y less approximately

C15D V±Ct) «const and consequently

C163 dV./dt»O. cleas Ii t r thar confinefo t d ground water aquifer, this supposition is rather reliable surfaco t s a , e reservoir d unconfinean s d shallow horizon s likeli t si a strony g suppositio e controlleb o e t n th y b d experimental data. n sucd I onl an hsucn yi a cas he generath e l model equatioe b n ca n convolute stationare th o t d e wityon h respecstoragee th o t t s related:

C17D dC±/dt + Introducing the following constant kinetic coefficients ciw C19> eq C173 can be rewritten in a new more symmetrical form

C213 dCi/dt + (zKki + Xr± * Xa]c± - ZXkiCk - XriCri Opening sums containe n C21di 3 gives furthermore introducing the following denotation for ii-th kinetic coe eni fc fti s

280 it can b© derived a very symmetric and convenient equation i r i r n i n i l i C2-i z 2 i Oil dC / i , /dt-X . C -X . C -. . . -A. . C. -. . . -X . C «X . C , Moreover, applying the Laplace transformation to this set of differential equations we can derive a set of algebraic equations which can be solved immediately. Indeed, in the Laplace transformation C24D is written as i r i r n i n 1 i i 2 i z i C25 o ii i 5 sC .i Cs5-C . -X .C Cs3-X .C CsD-. . . -A. . C . Cs3-. . . -X .C Cs5"X .C , Cs3

where s is a parameter of the Laplace transformation. t Onrearranginye e l similagal r member n C25 i obtaie s w D n

C255 -Xili2 n matrii r o x form

r t c^DJ l A,ki|, * i i ^**l «k ^ - ^-* |i ~~ li '*»• oi* ~vf v ri» ^» ri• Thus have ,w e obtaine matrida x presentatio C17q e tern i 3f f no mo the Laplace transformation by Vi=const. . a solutioI nt ordege o thit o nt rs matrio xt equatios ha e on n multiply both its side by the inverse matrix of the kinetic coefficient matrix as follows

C26X I D X ki ki ci

As a product of the direct and inverse matrix is a unit matrix the final solution in the Laplace form is given by -i C273 C oi,+ X ri. Cr i,Cs3

It is seen that to reach the final solution to the problem we have o assumt — n explicia e t dependenc e inputh tf o econcentration s C .CO on time and find out their Laplace presentations as well, - to define or nominate all kinetic coefficients X , and X, . rememberi i k i r that ii—th values are determined by the eq C233, - to construct a matrix equation in accordance with C273 which represent a solution to be sought in the laplace form, then — to make an inverse Laplace transition to the explicit original functions being sought. s i knowIt n e thatritiuth t m concentratio s a wels na l precipitation flow rate recharge q can be represented as a periodical functions versus current time t. In the simplest case, it migh assumee tb d that there exis annuan ta l periodicit forn yi m C28 q .Ct5«) q .sinw. q + t•

" C2O r C 3 where q . and C . represent amplitudes of the water flow rate and isotope tracer concentratio reflect. C n changesd san . mea ,q n annual values of the water flow rate and tritium concentration in e i-tth h compartment external input respectively, furthermore, $ denote sphasa e deviatio nconcentratioe anglth f eo n input function relatively to the flow rate function which is considered as synchronizing one. Thus totae ,th l input mass flow rat tritiuf eo s mi

281 C3O3

C .sinCwt+S.^sincoai .sinCo>t+$C , q * • t» . .C D+.sinwC . q, q + t

. |ClX25IcosCot+* C . q . -wt^-cosCwt+ i nCos . tC $+«.: +wt+ >q + D1

q C ri* ai ai jClX23(cos$1-cosC2wt+$1Dj

C sinwt ri *

It is completely evident that

Thus, the last member in the right side matrix in eq C27D can explicitly be written in form

C C32X D e LaplacTh e transformatio e takeb n n ca n fro e standarth m d Laplace transformation tables and used to solve the matrix eq C27D. Even at the present level of consideration, it is evident that output concentrations of an isotope tracer exiting a hydrogeological system wite atmospherith h c precipitation ca n consis f severao t l periodical components having different phase angle deviations eac othef ho differend ran t amplitud variationf eo s caused by — non—synchron d variationize f rechargo s e water flow e rateson n o , hanisotopd dan e tracer concentration variations anothen ,o r hand, - difference f residenco s e time f wate o sd trace an r n particulai r r compartments of the system under interest, the latter fact has already been demonstrated in recent publications [1,2,3]. e superpositioTh l thesal ef o nsinusoida l componente b n ca s e approac analyze th e base futur th f th o e n n ho ei d developed an d relative long— term observational dat f boto a h isotop d commoan e n hydrology character. By the way, in the publications mentioned obove, some total solutions to a stationary one compartment model with a sinusoidal tracer input was given and analyzed.

1. Kusakabe M. et al., J. Geoph. Res. .vol. 75, No. 3O. , C197OD, S941-59SO. 2. Isotope Technique Grounn si d Water Hydrology, 1974. IAEA, C1974D, Vol. 1, 399-428. Dubinchuk V.T. et al. 3. Radioisotope Techniques in Engineering Geology and Hydrogeology, C. 19773, Atomizdat, Moscow, pp. 3O4, Ci n Russian}

282 LIST OF PARTICIPANTS

Adar, E.M. Jacob Blaustein Institut r Deserefo t Research, Ben-Gurion University of the Negev, Water Resources Center, Sede-Boker Campus, 84993 Israel Barmen. G , Lund Institut f Technologyeo , Department of Engineering Geology, P.O. Box 118, S-22Lund0 0 1 , Sweden Ferronskij, V. Water Problems Institute, Isotope Hydrology Laboratory, Sadovaja-Chernogriazskaja 13/3, 103064 Moscow, Russian Federation Herrer a, I. Institute of Geophysics, UN AM, Ap. Postal 22-582, 14000P C , Mexico City, Mexico Kaiin, R. Laboratory of Isotope Chemistry, University of Arizona, Building 77, Gould-Simpson Building, Tucson, AZ 85721, United States of America Limic, N. Department of Physics, Energy and Applications, Ruder Boskovic Institute, P.O. Box 1016, 41001 Zagreb, Croatia Maloszewski, P. Institute of Hydrology, Ingolstaedter Landstrass, el D-85758 Neuherberg, Germany Moreno, L. Chemical Engineering Deparment, Royal Institute of Technology, Teknikringe, n26 S-100 44 Stockholm, Sweden Voss, C.I. US Geological Survey, Water Resources Center, 431 National Center, Reston, VA 22092, United States of America Yurtsever. Y , Division of Physical and Chemical Sciences, (Scientif Secretary) International Atomic Energy Agency, 100x P.O,Bo . A-1400 Vienna, Austria Zuber. A , Institute of Nuclear Physics, Radzikowskiego 152, 31-342 Cracow 23, Poland

283