Environmental Isotopes in the Hydrological Cycle Principles and Applications

Vol. 6 Water R esources Programme Water Atoms for Peace

International Atomic Energy Agency and United Nations Educational, Scientific and CAtomsultural for Peace Organization (reprinted with minor corrections) Ar H He

C Kr

C ENVIRONMENTAL ISOTOPES in the H HYDROLOGICAL CYCLE O

C N S

He Ar

VOLUME VI

modelling

Co-ordinating editor Y. Yurtsever Isotope Hydrology Section International Atomic Energy Agency, Vienna

Contributing Authors

A.Zuber, P.Małoszewski; M.E.Campana, G.A.Harrington, L.Tezcan; L.F.Konikow

Preface to volume VI

This last volume in the series of textbooks on en- completed Co-ordinated Research Project by the vironmental isotopes in the hydrological cycle IAEA entitled “Use of isotopes for analysis of provides an overview of the basic principles of flow and transport dynamics in groundwater sys- existing conceptual formulations of modelling ap- tems” will also soon be published by the IAEA. proaches. While some of the concepts provided in This is the reason why the IAEA was involved in Chapter 2 and Chapter 3 are of general validity the co-ordination required for preparation of this for quantitative interpretation of isotope data; the volume; the material presented is a condensed modelling methodologies commonly employed overview prepared by some of the scientists that for incorporating isotope data into evaluations were involved in the above cited IAEA activities. specifically related to groundwater systems are This volume VI providing such an overview was given in this volume together with some illustra- included into the series to make this series self- tive examples. sufficient in its coverage of the field of Isotope Development of conceptual models for quantita- Hydrology. A special chapter on the methodolo- tive interpretations of isotope data in hydrogeolo- gies and concepts related to geochemical mod- gy and the assessment of their limitations and field elling in groundwater systems would have been verification has been given priority in the research most desirable to include. The reader is referred and development efforts of the IAEA during to IAEA-TECDOC-910 and other relevant publi- the last decade. Several Co-ordinated Research cations for guidance in this specific field. Projects on this specific topic were implemented Valuable contributions in the preparation of this and results published by the IAEA. Based on volume were accomplished by A.Zuber (Poland), these efforts and contributions made by a num- P.Maloszewski (Germany), M.E.Campana (USA), ber of scientists involved in this specific field, the G.A.Harrington (Australia), L.Tezcan (Turkey), IAEA has published two Technical Documents and L.F.Konikow (USA), as acknowledged with entitled “Mathematical models and their appli- each chapter cations to isotope studies in groundwater studies, IAEA TECDOC-777, 1994” and “Manual on Vienne, mars 2000 Mathematical models in isotope hydrogeology – IAEA TECDOC-910, 1996”. Results of a recently Yuecel Yurtsever

491

CONTENTS OF volume VI

1. modelling INTRODUCTION...... 495

2. lumPED PARAMETER MODELS...... 497 2.1. Introduction...... 497 2.2. Basic principles of the lumped-parameter approach for constant flow systems...... 499 2.3. Models...... 500 2.3.1. The piston flow model...... 500 2.3.2. the exponential model...... 500 2.3.3. The combined exponential-piston flow model...... 501 2.3.4. the dispersion model...... 502 2.4. Cases of constant tracer input...... 502 2.5. Cases of variable tracer input...... 503 2.5.1. the tritium method...... 503 2.5.2. The 3H-3He method...... 505 2.5.3. The krypton-85 method...... 506 2.5.4. the carbon-14 method...... 506 2.5.5. The oxygen-18 and deuterium method...... 506 2.5.6. other potential methods...... 507 2.6. examples of 3H age determinations...... 508 2.7. determination of hydrogeologic parameters from tracer ages...... 508 2.8. The lumped-parameter approach versus other approaches...... 511 2.9. Concluding remarks...... 512

3. COMPARTMENTAL MODEL APPROACHES TO GROUNDWATER FLOW SIMULATION.517 3.1. Introduction...... 517 3.2. A simple compartmental model: theory and application to a regional groundwater flow system...... 517 3.2.1. Theory...... 517 3.2.1.1. tracer mass balance...... 518 3.2.1.2. Transient flow...... 518 3.2.1.3. Age calculations...... 519 3.2.2. Application to the Nevada test site flow system...... 520 3.2.2.1. Introduction...... 520 3.2.2.2. Hydrogeology...... 520 3.2.2.3. model development and calibration...... 521 3.2.2.4. results and discussion...... 522 3.2.2.5. Concluding remarks...... 524 3.3. Constraining regional groundwater flow models with environmental isotopes and a compartmental mixing-cell approach...... 524 3.3.1. Introduction...... 524 3.3.2. governing equations...... 525 3.3.3. model design, input data and calibration procedure...... 526 3.3.4. Application to the Otway basin, south Australia...... 526 3.4. mixing-cell model for the simulation of environmental isotope transport...... 529 3.4.1. Introduction...... 529 3.4.2. Mixing-cell model of flow and transport dynamics in karst aquifer systems...... 531 3.4.2.1. Physical framework of the model...... 532 3.4.2.2. Hydrologic model...... 533 3.4.2.3. transport model...... 534 3.4.3. Conclusions...... 535 3.5. summary and conclusions...... 535

4. USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT...... 541 4.1. Introduction...... 541 4.2. Models...... 542 4.3. Flow and transport processes...... 543 4.4. governing equations...... 543 4.4.1. Groundwater flow equation...... 543 4.4.2. seepage velocity...... 544 4.4.3. solute-transport equation...... 544 4.5. numerical methods to solve equations...... 547 4.5.1. Basics of finite-difference methods...... 550 4.5.2. Basics of finite-element methods...... 552 4.5.3. matrix solution techniques...... 553 4.5.4. Boundary and initial conditions...... 553 4.6. model design, development and application...... 554 4.6.1. grid design...... 555 4.6.2. model calibration...... 556 4.6.3. model error...... 557 4.6.4. mass balance...... 559 4.6.5. sensitivity tests...... 559 4.6.6. Calibration criteria...... 560 4.6.7. Predictions and postaudits...... 560 4.6.8. Model validation...... 561 4.7. Case history: local-scale flow and transport in a shallow unconfined aquifer...... 561 4.8. Available groundwater models...... 564 1. modelling INTRODUCTION

Y. Yurtsever (1) Qualitative information, pertaining to system boundaries, hydraulic discontinuities International Atomic Energy Agency, and stratifications, hydraulic interconnec- Division of Physical and Chemical Sciences, tions, origin of water, presence and process Iotope Hydrology Section of replenishment, sources of pollution (in- During the last four decades the use of isotopes, cluding water salinization), and cause-effect either naturally occurring (environmental isotopes) relationships of different processes involved or intentionally injected (artificial isotopes) have during flow and circulation of water in hy- proved their value in studies related to water re- drological systems; sources assessment, development and management. (2) Quantitative information concerning wa- The applications in a large variety of hydrological ter fluxes (i.e. rate of direct replenishment problems are based on a general concept of trac- of groundwater, fluxes into the system from ing. The term Isotope Hydrology is now realised as boundaries), mixing proportions of compo- a scientific discipline comprising proven method- nent flows originating from different sourc- ologies applied to a wide spectrum of hydrological es, travel times involved in hydrological sys- problems as an integral part of the investigations in tems and characteristic dynamic parameters water resources and related environmental studies. related to mass-transport processes. The potential role and contributions of isotope Quantitative evaluations to be made from envi- methods in the water resources sector can be ronmental isotopes require conceptual mathemat- grouped into the following general categories: ical models, so as to describe both the tracer distribution within the system or isotope input/out- (1) Determination of physical parameters re- put relationships under given flow and transport lated to flow, the dynamics of transport and conditions. Since temporal/spatial variations of the structure of the hydrological system, the environmental isotopes can not be controlled (2) Process tracing — delineation of processes by the investigator; quantitative information to be involved in circulation of water and mass derived from observed isotope concentrations in transport of dissolved constituents, any given hydrological system will have to rely on specific input pattern of different isotopic (3) Identification of the origin (genesis) of wa- species. ter, The most commonly used modelling formulations (4) Component tracing — determination of in this regard can be classified into three broad cat- pathways and mixing ratios of component egories as follows: flows, (1) lumped-parameter models, based on linear (5) Determining the time scale involved in hy- systems approach (transfer-function type of drological events. modelling) for a tracer case Due to the natural labelling of water in the hy- (2) compartmental simulation models, which drological cycle, the environmental isotopes have can be considered as a quasi-physical distrib- the distinct advantage of facilitating the study of uted-parameter modelling approach water movement and hydrological processes on much larger temporal/spatial scales than inten- (3) models involving mathematical formulations tionally injected, artificial tracers which are often of advective transport with dispersion and used for site-specific, local scale hydro-engineer- their analytical and/or approximate numeri- ing problems. cal solutions. The type of information that can be obtained from In most cases, tracer input and the observed output isotope techniques in hydrological systems is: (isotope data collected on spatial and/or temporal

495 modelling variations) are available, and quantitative inter- Basic data concerning the natural isotope content pretation, in mathematical terms, is an inverse of precipitation, as an input to the hydrological problem. While the formulations to be discussed systems, has been collected by the IAEA from in later sections are of general validity for any a Global Network of Isotopes in Precipitation tracer case, their use, particularly with the most (GNIP) since 1960. The measurements made on commonly employed environmental isotopes are 18O, 2H and 3H content of monthly composite pre- emphasized in this Volume VI. cipitation samples regularly being collected from It is evident that the first requirement in the ap- this global network, provide the basic data needed plication of natural isotopes for this purpose is in this regard. Furthermore, data from national net- determination of the tracer input into the system works being operated in some countries supplement under investigation, which is the isotopic compo- this global network; the overall data are published sition of the inflow(s). For most commonly used environmental isotopes, natural processes gov- regularly by the IAEA and are also available at erning their occurrence result in time- and space- http://www.iaea.org/programs/ri/gnip/gnipmain. dependent variations. The time-variant input(s) htm”. for a given system and the known decay rate (in This volume provides an overview of the basic the case of radioactive isotopes) enable quantita- concepts and formulations of modelling approach- tive estimates, based on the observed isotope varia- es cited above with some illustrative examples of tions within the system or on observed concentrations at the outflow of the system, through the use their applications. The reader is referred to the ref- of the mathematical modelling approaches cited erences given in each chapter for in-depth cover- above. The case of a constant input concentration, age of the above cited modelling approaches, as which may need to be adopted for some systems, is applied for quantitative evaluation of isotope data a sub-set of this general approach. in hydrology.

496 2. LUMPED PARAMETER MODELS

A. ZUBER tem by observing the behaviour of a specific substance, the tracer, which has been added (injected) Institute of Nuclear Physics, Cracow, Poland to the system. Environmental tracers are added by natural processes whereas their production is P. MAŁOSZEWSKI either natural or results from the global activity GSF – Institute for Hydrology, Neuherberg, of man. German An ideal tracer is a substance behaving in the system exactly as the traced material, at least as far as the sought parameters are concerned, 2.1. Introduction and which has one property that distinguishes it A comprehensive description of the lumped-pa- from the traced material. For an ideal tracer, there rameter models applicable to the interpretation of should be neither sources nor sinks in the system environmental tracers in groundwater systems is other than those related to the sought parameters. given. It will be shown that the lumped-parame- In practice a substance which has other sources ter models are particularly useful for interpreting or sinks can also be regarded as suitable tracer, if the tracer data which were obtained at separate they can be properly accounted for, or if their in- sampling sites, when it is neither possible, nor fluence is negligible within the required accuracy. justified, to use distributed-parameter models, A conservative tracer is an ideal tracer without a s the latter require more detailed knowledge of sinks (there is no decay, sorption or precipitation). the investigated system, which is often unavail- able. A more detailed description of the approach A conceptual model is a qualitative description of and a number of examples can be found in Ref. a system and its representation (e.g. description of [1] and in other references given further. A user- geometry, parameters, initial and boundary condi- friendly programme (FLOWPC) for the inter- tions) relevant to the intended use of the model. pretation of environmental tracer data by several A mathematical model is a mathematical repre- most commonly used models is available from sentation of a conceptual model for a physical, the IAEA. chemical, and/or biological system by expres- For a better understanding of the tracer method sions designed to aid in understanding and/or pre- and the interpretation of the tracer data, several dicting the behaviour of the system under speci- definitions are recalled. Some of these definitions fied conditions. are more or less generally accepted and frequently In a lumped-parameter model (black-box model) used (e.g., [2–5]); whereas remaining are unfortu- spatial variations of parameters are ignored and nately used only occasionally. As a consequence the system is described by adjustable (fitted) pa- of infrequent use of adequate definitions, a lot of rameters. misunderstandings occur in literature, especially when radioisotope ages versus water ages are Verification of a mathematical model, or its com- considered, or when mathematical models equiv- puter code, is obtained when it is shown that alent to the behaviour of a well-mixed reservoir the model behaves as intended, i.e., that it is are used for groundwater systems in which good a proper mathematical representation of the con- mixing never occurs. As explained further, some ceptual model and that the equations are correctly misunderstandings also result from a common encoded and solved. identification of tracer ages with water ages in Model calibration is a process in which the mathe- fractured rocks whereas in fact these two physical matical model assumptions (e.g., type of the mod- quantities differ considerably. el) and parameters are varied to fit the model to The tracer method is a technique for obtaining observations. Usually, calibration is carried out information about a system or some part of a sys- by a trial-and-error procedure, and it can be quan-

497 modelling

titatively described by the goodness of fit. Model the mobile water volume (Vm) to the volumetric calibration is a process in which the inverse flow rate (Q) through the system: problem (ill-posed problem) is solved, i.e., from known input-output relations the values of param- tw = Vm/Q (2.1) eters are determined by fitting the model results For vertical flow in the recharge area, especially to experimental data. Sought (fitted, matched) in the unsaturated zone, Q in Eq. 2.1 can be ex- parameters are found in the process of calibra- pressed by recharge rate (I): tion. The direct problem is solved if for known or assumed parameters the output results are cal- tw = Vm/I (2.1a) culated (model prediction). In the FLOWPC programme an option is included (when no observa- If a system can be approximated by unidimen- tions exist) which serves for direct calculations. sional flow pattern, this definition yieldst w = x/vw, Testing of hypotheses is performed by compari- where x is the length for which tw is determined, son of model predictions with experimental data. and vw is the mean water velocity, defined below. Darcy’s velocity (vf) is defined as the ratio of Q/S, Validation is a process of obtaining assurance S being the cross-section area perpendicular to that a model is a correct representation of the pro- flow lines. The effective porosity is understood cess or system for which it is intended. Ideally, as that in which the water movement takes place validation is obtained if the predictions derived [4]. Consequently, the mean water velocity (vw) is from a calibrated model agree with new observa- defined as the ratio of Darcy’s velocity to the ef- tions, preferably for other conditions than those Q/S, S étant l’aire de la section traversée perpendiculaire aux lignes de flux. La porosité fective porosity, vw = vf/ne (other equivalent terms: used for calibration (e.g., larger distances and efficacepore velocity est définie, interstitial comme velocity la porosité, travel dans velocity laquelle, l’eau se déplace (Lohman et al. 1972). En longer times). Contrary to calibration, the valida- conséquence,transit velocity la). vitesseOther definitionsmoyenne de of l’eau the effective (vw) est définie comme le rapport de la vitesse de tion process is qualitative and based on the mod- Darcyporosity sur are la also porosité common. efficace, For instance, vw = vf /itn eis (ou com par- des termes équivalents: vitesse de pore, eller’s judgement. In the case of the tracer method vitessemon to interstitielle,define the effective vitesse porosityde transport, as that vitesse which de transit). D’autres définitions de la porosité the validation is often performed by comparison is effective to a given physical process, e.g., dif- efficace sont aussi utilisées. Par exemple, il est coutumier de définir la porosité efficace of the values of found parameters with the val- fusion. Of course, in such cases, the effective po- ues obtainable independently from other meth- commerosity differs une porosité from that qui which est efficace is directly pour related un pr ocessusto physique donné, par ex., la diffusion. ods. In such a case it is perhaps more adequate Evidemment,Darcy’s law. dans de tels cas, la porosité efficace diffère de celle directement relative à la loi to state that the model is confirmed, or partially de Darcy. confirmed. In spite of contradictions expressed by The mean tracer age (tt; other terms: mean transit some authors (e.g., [6]), the difference between L’âgetime of moyen tracer du, mean traceur travel (tt; timeet les of autres tracer termes:) can be temps de transit moyen du traceur, temps de validation and confirmation is rather verbal, and transportdefined as:moyen du traceur) peut être défini comme: mainly depends on the definitions used and their ∞ understanding (e.g., some authors by the working ∫t'(CI )(t') dt' definition of validation understand the process of 0 tt = ∞ (2.2) (2.2) calibration). ∫CI (t') dt' Partial validation can be defined as validation 0 performed with respect to some properties of Oùwhere CI estC I lais concentration the tracer concentration du traceur observée observed sur le site de mesure (l’exutoire d’un système) a model. For instance, in the modelling of artifi- commeat the measuring résultat d’une site injection(the outlet instantanée of a system) à l’entrée. as cial tracer tests or pollutant transport, the disper- the result of an instantaneous injection at the en- sion equation usually yields proper solute veloci- L’âgetrance. moyen du traceur est égal à l’âge moyen de l’eau, seulement s’il n’y a pas de zones ties (i.e., can be validated in that respect), but sel- stagnantes dans le système, et si le traceur est injecté et mesuré dans le flux. L’injection et la The mean tracer age is equal to the mean water dom adequately describes the dispersion process mesureage only du if fluxthere signifient are no stagnant qu’à la foiszones à l’entréein the sys et- à la sortie, les teneurs en traceur le long des in predictions at much larger distances. lignestem, and d’écoulement the tracer issont injected proportionnelles and measured à leur in débit. Cette condition est automatiquement

The turnover time (tw; other terms: age of water satisfaiteflux. Flux dans injection les systèmes and measurement naturels pour mean les thattraceurs entrant dans le système par de l’eau leaving a system, mean exit age, mean residence d’infiltrationat both the entrance et mesurés and dans outlet les the flux amounts sortants. of Quoi qu’il en soit, si l’échantillonnage est time of water, mean transit time, hydraulic age, réalisétracer inà uneparticular certaine flow profondeur lines are dans proportional un forage, tocette condition peut, peut-être, être satisfaite kinematic age) is usually defined as the ratio of their volumetric flow rates. That condition is au- dans la gamme de débits échantillonnés, mais sûrement pas pour le système entier. Très probablement, dans quelques cas, le carbone radioactif ne satisfait pas les conditions d’une 498 injection dans le flux, parce qu’il pénètre dans les systèmes aquifères principalement suite à la

production de CO2 par les racines des plantes. Donc, son injection naturelle n’est pas nécessairement proportionnelle aux débits. Le problème d’une injection et d’une mesure convenables est plus important pour le traçage artificiel, cependant, il faut être conscient que même un traceur environnemental idéal peut dans certains cas donner un âge qui diffère de l’âge de l’eau. Le problème des zones stagnantes, qui est d’une importance particulière pour les roches fissurées, sera discuté plus loin. Les systèmes stagnants ne concernent pas ce travail, mais pour la cohérence des définitions de l’âge, ils doivent être mentionnés. L’âge de l’eau d’un système immobile est d’ordinaire considéré comme la durée pendant laquelle le système a été séparé de l’atmosphère. Dans de tels cas, l’âge d’un radio-isotope volatile, qui n’a pas d’autres sources et pertes que la

91 désintégration radioactive, peut être assimilé à l’âge de l’eau. L’âge radio isotopique (ta) est défini par la décroissance radioactive:

C(ta)/C(0) = exp(−λta)

Où C(ta) et C(0) sont respectivement les concentrations actuelles et initiales, et λ la constante de désintégration radioactive. Il y a malheureusement peu de traceurs radio-isotopes disponibles pour dater à la fois les systèmes aquifères anciens mobile et immobile. Ainsi, pour de tels systèmes, on emploie plutôt l’accumulation de certains produits de désintégration (par ex., 4He et 41Ar). De la même LUMPEfaçon,D PARA la MErelationTER MO entreDELS δ2H et δ18O dans les eaux météoriques peut fournir des informations sur l’âge des systèmes mobiles et immobiles en terme de périodes géologiques dont les tomatically satisfied in natural systemsclimats for - trac sont2.2. connus, Ba avecsic pdesrin conditionsciples climatique of s qui existaient au moment de la recharge. ers entering the system with infiltrating waterEvidemment, and les âgesthe lumpe des systèmesd- pimmobilesarame toue rdes systèmes ayant été un temps immobiles, measured in outflows. However, if nesampling doivent is pas êtrea interprétéspproach directement for cons en tatermesnt deflo paramètresw hydrodynamiques. performed at a certain depth of a borehole, that systems condition may perhaps be satisfied for the sam- 2.2 PRINCIPESIn the lumped-parameter DE BASE approach DE the L’APPROCHE groundwa- EMPIRIQUE pled flow line, but surely not for the whole sys- ter system is treated as a whole and the flow pat- tem. Radiocarbon most probably does not satisfy POURtern is assumed DES SYSTÈMESto be constant. UsuallyÀ FLUX the flowCONSTANT in some cases the flux injection because it enters rate through the system is also assumed to be con- groundwater systems mainly due to the Dansproduc l’approche- stant empirique,because variations le système in theaquifère flow estrate traité through dans son intégralité et le mode de tion of CO2 by plant roots. Therefore, itsflux natu est- supposéthe systemconstant. and La valeur changes du influx its est volume aussi couramment were supposée constante parce ral injection is not necessarily proportionalqu’il ato été démontréshown to que be sa negligible variation à when travers distinctly le système shorter et des changements dans son volume the volumetric flow rates. The problem of a prop- étaient négligeablesthan the quand mean ilsage sont [7]. nette Detailedment plus description courts que of l’âge moyen (Zuber et al. 1986). er injection and measurement is more acute in ar- Une descriptionthe lumped-parameter détaillée de l’approche approach empi canrique be foundpeut êtrein trouvée dans de nombreuses tificial tracing, however, one should be aware that a number of papers [1, 8–10]. For the most com- publications (Amin et Campana 1996, Małoszewski et Zuber 1982, 1996, Zuber 1986). Pour even an ideal environmental tracer may in some monly applied models, the schematic presentation cases yield an age which differs from theles water modèles of les underground plus communément water systems employés, is given une in Fig.présentation 2.1, schématique des systèmes age. The problem of stagnant zones, which is of aquifères estand donnée the relationdans la Fig.between 2.1, etthe la variablerelation entreinput les (C concentrationsin) variables à l’entrée particular importance for fissured rocks, will be (Cin) et à la sortieand output (C) est: (C ) concentrations is: discussed further. t Immobile systems are beyond the scope of this = − −λ − C(t) = ∫Cin (t') g(t − t')exp[−λ(t − t')]dt' (2.4) (2.4) work, but for the consistency of age definitions −∞ they should mentioned. The water age of an im- Ou une forme équivalente: mobile system is usually understood as the time An equivalent form is: span for which the system has been separated ∞ form the atmosphere. In such cases, the radioiso - C(t) = ∫ Ci n (t − t') g(t')exp(−λ t')dt' (2.5) (2.5) tope age of an airborne radioisotope, which has 0 no other sources and sinks than the radioactiveOù t’ est lewhere temps t’ d’entrée, is time of t –entry,t’ est t –let’ tempsis the transit de transit, time, etand la fonction g(t–t’) est appelée la decay, can be identified with the age of water. fonction de the réponse g(t-t’), qui function décrit is la called distribution the response à la sortie func- d’une substance conservative The radioisotope age (ta) is defined by the radio- tion, which describes the output distribution of (traceur) injectée de manière instantannée à l’entrée, et l’intégration à partir de ou vers l’infini active decay: a conservative substance (tracer) injected instan- signifie quetaneously toute la atcourbe the inlet, d’entrée and the (C integrationin) doit être from prise or en compte pour obtenir une C(ta)/C(0) = exp(−λta) concentration(2.3) de sortie correcte (Cout dans la Fig. 2.1). to infinity means that the whole input curve (Cin) has to be included to get a correct output concen- where C(ta) and C(0) are the actual and initial ra- tration (C in Fig. 2.1). dioisotope concentrations, respectively, and 92 λ is out the radioactive decay constant. Other common terms for the g(t) function are: Unfortunately, few radioisotope tracers are avail- transit time distribution, residence time distribu- able for dating both mobile and immobile old tion (RTD) of tracer, tracer age distribution, and groundwater systems. Therefore, for such sys- weighting function. As discussed further the RTD tems, the accumulation of some decay products is of tracer is not necessarily equivalent to the RTD rather used (e.g., 4He and 41Ar). Similarly, the deof the investigated fluid. pendence of δ2H and δ18O in meteoric waters Sometimes it is convenient to express Eq. 2.4 on the climatic conditions which existed when or 2.5, as a sum of two convolution integrals, the recharge took place may supply information or two input functions. The most common case on the age of both mobile and immobile systems is that one component is either free of tracer, or in terms of geological periods of known climates. the tracer concentration can be regarded as being Obviously, the ages of immobile systems, or sys- constant. As shown further in some cases such tems which were immobile for some time, should approach is justified by independent informa- not be interpreted directly in terms of hydraulic tion, which defines the fraction of tracer free (or parameters. constant) component. In other cases, the fraction

499 En ingénierie chimique, la fonction de réponse est souvent identifiée à la fonction E(t) qui décrit la distribution du temps de sortie (ou celle du temps de résidence, RTD) du fluide étudié. Par définition, la valeur moyenne de la fonction E est égale au volume du système divisé par la valeur du débit, et est égale à l’âge moyen de sortie du fluide (i.e., au temps de résidence moyen du fluide). Dans le cas de systèmes aquifères, la fonction de réponse décrivant la distribution de l’entrée du traceur, peut être assimilée à la distribution du temps de sortie du flux d’eau, uniquement dans les conditions favorables qui excluent la présence de zonesmo stagnantesdelling dans le système exploré. Lorsque des zones stagnantes sont présentes, même un traceur idéal peut être retardé par rapport à l’écoulement de l’eau, en raison d’un échange par diffusiongated entre system. les zones When mobiles stagnant et immobile zones ares. Ce present, problème sera développé plus loin de manière pluseven détaillée. an ideal tracer may be delayed in respect to the water flow due to diffusion exchange between mobile and immobile zones. That problem will be discussed further in more detail. 2.3 MODELES 2.3. Models 2.3.1 MODELE2.3.1. The piston « PISTON flowFLOW model »

Fig. 26.1. Schematic presentations of groundwater sys- In the piston flow model (PFM) approximation the flow lines are assumed to have the same tran- tems in the lumped-parameter approach. Dans l’approximation du Modèle « Piston Flow » (PFM) on suppose que les lignes de flux ont sit time, and the hydrodynamic dispersion and le même tempsdiffusion de transitare negligible. et que laT herefore,dispersion the hydrodynamique tracer et la diffusion sont of tracer free component is used as an additional négligeables.moves Ainsi, from le traceur the recharge se déplace area depuis as if it la was zone in dea can. recharge comme s’il était dans une fitting parameter. In the FLOWPC programme an boîte. La fonctionThe response de réponse function est donnéeis given par by thela fonction well-known delta de Dirac bien connue, g(t’) = option is included for an older fraction of water Dirac delta function, g(t’) = δ(t’ – t ), which in- (b) which either contains a constant tracerδ(t ’ con– tt),- qui introduite dans l’Eq. 2.4, donne: t serted into Eq. 2.4 yields: centration, or is free of tracer.

The response function represents the normalised C(t) = Cin (t − tt )exp(−λ tt ) (2.6) (2.6) output concentration, i.e., the concentration di- Eq. 2.6 means that for the PFM the output con- vided by the injected mass, which results L’Eq.2.6from an signifie que pour le PFM, la concentration de sortie à un instant donné est égale à la centration at a given time is equal to the input instantaneous injection of a conservative tracer concentrationconcentration d’entrée au attemps the time tt antérieur, t earlier, et and n’est changed modifiée que par la désintégration at the inlet. It is impossible to determine the re- t radioactive auonly cours by dethe la radioactive durée tt. Le decaytemps duringde transit the du time traceur (tt) est le seul paramètre du sponse functions of groundwater systems experi- modèle, et spanla forme t . The de transit la fonctiontime of the de tracer concentration (t ) is the onlyà l’entrée est semblable pour la mentally. Therefore, functions known from other t t parameter of the model, and the shape of the in- fields of science are used. The response function,concentration de sortie. Il sera montré plus loin que le PFM est applicable seulement dans les put concentration function is followed by the out- which is either chosen by the modeller, orsystèmes found avec un apport de traceur constant. Les trois modèles présentés dans les parties put concentration. It will be shown further that by calibration, defines the type of thesuivantes model sont les plus couramment utilisés. the PFM is applicable only to systems with con- whereas the parameters of the model are found by stant tracer input. The most commonly used are calibration. Calibration means finding a good fit 2.3.2 LEthe MODELEthree models EXPONENTIEL considered in the following sec- of concentrations calculated by Eq. 2.4 or 2.5, to tions. the experimental data, for a known or estimatedDans l’approximation du modèle exponentiel (EM), les lignes de flux sont supposées avoir la input function (time record of Cin). Usually,distribution when exponentielle des temps de transit i.e., la ligne la plus courte a un temps de transit referring to a model of good fitting, the type of 2.3.2. The exponential model théorique égal à zéro, et la ligne la plus longue un temps de transit égal à l’infini. Par the model and the values of its parameters are re- In the exponential model (EM) approximation, ported. the flow lines are assumed to have the exponen- In chemical engineering, the response function 94 is tial distribution of transit times, i.e., the shortest often identified with the E(t) function which de- line has the theoretical transit time equal to zero,

scribes the exit time distribution (or the residence and the longest line has the transit time equal to time distribution, RTD) of the investigated fluid. infinity. It is assumed that there is no exchange of By definition, the mean value of the E-functionhypothèse, iltracer n’y abetween pas d’échanges the flow delines, traceur and thenentr ethe les follow lignes- de flux et on obtient ainsi la is equal the volume of the system dividedfonction by de réponseing response suivante: function is obtained: the volumetric flow rate, and is equal to the mean = −1 − exit age of the fluid (i.e., to the mean residence g(t') tt exp( t'/ tt ) (2.7) (2.7) time of the fluid). In the case of groundwater systems, the response function, which describesCette relation T hisest relationshipmathématiquement is mathematically équivalente equivalent à la fonction to de réponse d’un réservoir bien the exit distribution of the tracer, can bemélangé, identi- connuethe response en ingéniérie function chimique. of a well-mixed Quelques opérateursreservoir, rejettent le EM car en principe fied with the exit time distribution of waterle bon flow mélange known n’existe in chemical pas dans engineering. les aquifères, Some alors investiga que d’autres- défendent l’utilisation du only under favourable conditions, whichEM exist comme torsun bon reject indicateur the EM des because conditions in principle de mélange no gooddans un aquifère. when there are no stagnant zones in the investi- mixing may occur in aquifers whereas others Ces deux opinions sont erronées car, comme indiqué, le modèle est basé sur une supposition de non-échange de traceur entre des lignes de flux individualisées (Eriksson 1958, 500 Małoszewski et Zuber 1982, 1996, Zuber 1986). Si le traceur s’échange entre les lignes de flux avec une distribution exponentielle du temps de transport, sa distribution tendra à être décrite par le modèle de dispersion discuté plus loin. Les effets attendus sont similaires aux effets connus pour des distributions de traceur dans un flux laminaire dans un capillaire (Małoszewski et Zuber 1996, Fig. A.1). La compréhension de tous les effets, pouvant conduire à des différences entre la fonction de réponse du traceur et la distribution des lignes de flux, est vraiment utile pour une interprétation convenable des données du traceur. Pour l’approximation du modèle exponentiel, le mélange se produit seulement au site d’échantillonnage (source, puits d’exploitation, ruisseau ou rivière). En général, les systèmes aquifères ne sont jamais bien mélangés, et ils peuvent contenir des eaux mélangées seulement si deux, ou plus, des flux d’eau se rencontrent, ou dans les zones de transition où la dispersion hydrodynamique et la diffusion jouent un rôle important. De manière similaire au PFM, le temps de transit moyen (âge) du traceur est le seul paramètre du EM qui définit sans ambiguité la distribution globale du temps de transit (Fig. 2.2). Donc, lorsque l’on donne l’âge du traceur, le modèle utilisé ou la fonction de réponse devrait aussi être donnés. La fonction de réponse du EM montre que le modèle est inapplicable aux systèmes dans lesquels des lignes de flux infinitésimalement courtes ne peuvent exister. En d’autres termes, le EM n’est pas applicable quand les échantillons sont pris bien en dessous de la surface du sol, par ex., à partir de forages crépinés à de grandes profondeurs, de mines ou de sources artésiennes. L’expérience montre que très souvent, en raison d’un enregistrement trop court de la donnée du traceur, le modèle expérimental produit un bon ajustement bien que son usage ne soit pas justifié. Dans de tels cas, on doit retenir que le résultat obtenu est une approximation grossière et que la situation réelle peut être décrite de manière plus adéquate par l’un des modèles discutés dans les prochaines parties. Evidemment dans de tels cas, aucune solution unique n’est disponible. Le EM et les autres modèles avec une large distribution des âges décrivent des situations dans lesquelles seules les plus courtes lignes de flux fournissent au site d’échantillonnage un traceur radioactif (par ex., tritium ou 3H), ou un traceur non radioactif avec la fonction

95 LUMPED PARAMETER MODELS claim the applicability of the EM to be indicative at large depths, mines, and artesian outflows. of good mixing conditions in a groundwater sys- Experience shows that very often, due to a too tem. Both opinions are wrong because, as men- short record of the tracer data, the exponential tioned, the model is based on an assumption of model yields a good fit though its use is not justi- no exchange (mixing) of tracer between particular fied. In such cases, it should be remembered that flow lines [1, 9–11]. If tracer exchanges between the obtained result is a rough approximation, and the flow lines with an exponential distribution of the real situation can be described more adequate- travel times, its distribution will tend to be de- ly by one of the models discussed in the next sec- scribed by the dispersion model discussed further. tions. Evidently in such cases no unique solution Expected effects are similar to the effects shown is available. for tracer distributions in a laminar flow in a cap- The EM and other models with a broad distri- illary ([1], Fig. A.1). Understanding of all effects bution of ages describe situations in which only which may lead to differences between the tracer the shortest flow lines supply to the sampling site response function and the distribution of flow a decaying tracer (e.g., tritium or 3H), or a non- lines is very useful for a proper interpretation of decaying tracer with the input function starting tracer data. from zero (e.g., freons). Therefore, in the case of For the exponential model approximation, mixing a large value of the mean tracer age, no informa- occurs only at the sampling site (spring, abstraction is in fact available on the part of the system tion well, stream or river). In general, groundwa- with flow lines without tracer. In consequence, ter systems are never well mixed, and they may the knowledge on the whole system is derived contain mixed waters only if two, or more, water from the information available for its frac- flows meet, or in transition zones where the hy- tion with low ages (short transit times). In other drodynamic dispersion and diffusion play an im- words, the remaining part of the system, which portant role. does not supply tracer to the sampling site, may Similarly to the PFM, the mean transit time (age) have a quite different distribution of flow lines of tracer is the only parameter of the EM, which than that assumed in the model. unambiguously defines the whole transit time distribution (Fig. 2.2). Therefore, when report- 2.3.3. The combined exponential- ing the tracer age, the model used, or the response piston flow model function should also be given. The response func- In the exponential-piston flow model (EPM) ap- tion of the EM shows the model to be inappli- proximation, the aquifer is assumed to consist of cable to systems in which infinitesimally short two parts in line, one with the exponential distri- flow lines do not exist. In other words, the EM is bution of transit times, and another with the dis- not applicable when samples are taken well be- tribution approximated by the piston flow. The re- low ground surface, e.g., from boreholes screened sponse function of the EPM is:

g(t’) = (η/tt) exp(−ηt’/tt + η − 1) for t’ ≥ tt(1 − η-1) (2.8)

g(t’) = 0 for t’ < tt(1 − η -1) where η is the ratio of the total volume to the volume with the exponential distribution of transit times, i.e., η = 1 means the exponential flow model. The response function is independent of the sequence in which EM and EPM are combined. The EPM has two fitting (sought) parameters, i.e.,

tt and η. Examples of the response functions are Fig. 26.2. Examples of response functions of the expo- shown in Fig. 2.3. For low values of η that model nential model (EM). is close to the EM whereas for large values of η

501 modelling

Fig. 26.4. Examples of response functions of the dis- Fig. 26.3. Examples of response functions of the expo- persion model (DM) for typical values of the disper- nential-piston flow model (EPM). sion parameter.

2.4. Cases of constant tracer it is somewhat similar to the dispersion model input with a low value of the apparent dispersion parameter. That model is somewhat more realistic The lumped-parameter approach is applicable for than the exponential model because it allows for any tracer with variable input. It is also applicable the existence of a delay of the shortest flow lines. for radioisotopes with a constant input concentration. However, in the latter case, a unique interpretation is possible only for models with a single 2.3.4. The dispersion model sought parameter, because two unknown values In the dispersion model (DM), the following uni- cannot be found from a single equation. The most dimensional solution to the dispersion equation typical solutions to Eq. 2.4 for a constant input for a semi-infinite medium is used as the response (Co) are: function [12]:

C = Co exp(−λtt) for PFM (2.10) −1/ 2 −1 2 g(t') = (4πPDt'/ tt ) t' exp[−(1− t'/tt ) /(4PDt'/tt )] (2.9) (2.9) C = Co/(1 + λtt) for EM (2.11)

Où PD est le paramètre de dispersion apparent (réciproque du nombre de Peclet), lequel–1 est 1/2 where PD is the apparent dispersion parameter C = Co exp{(2PD) ×[1 – (1 + 4λPDtt) ]} sans rapport(reciprocal avec la of dispersivité the Peclet number), habituelle which des is systèmes unre- aquifères,for DM et dépendent (2.12) principalementlated de to la the distribution common dispersivity des temps de of transport. groundwater Plus la valeur du paramètre de The investigators who apply Eq. 2.10 for dating dispersion est élevée, plus la distribution des temps de transport est large et asymétrique. Des systems, and mainly depends on the distribution and understand its limitations often use the term exemples deof fonctions travel times. de réponse The higher sont pr theésentés value sur of la the Fig. disper 2.4, pour- desapparent valeurs tracer de PD deage 0,05 for the PFM tracer age (e.g. et 0,5, qui représententsion parameter, les situations the wider les and plus the courantes. more asymmet Quoi qu’il- en[14]). soit, dansEqs. des 10–12 études demonstrate de that the radioiso- cas publiées,rical l’interprétation the distribution des enregistrements of the travel times. de 3H E indiquentxamples des topevaleurs age du found paramètre from deEq. 2.3 is a correct represen- of the response functions are shown in Fig. 2.4, dispersion jusqu’à 2,5 (Małoszewski et Zuber 1982, Zuber 1986, Zubertation et al. of 2000), the mean alors tracer que age (tt) only for the PFM, for the PD values of 0.05 and 0.5, which bracket which, as mentioned, is equal to the mean water des valeurs theplus mostfaibles common que 0,05 situations.sont moins pr However,obables. Quelques in some auteurs, au lieu de l’Eq. 2.9, age (tw) under favourable conditions. In spite 3 appliquent lapublished solution casede l’équation studies, thede dispinterpretationersion pour of un H milieu re- infini,of a numberqui est inadéquate, of works, in which differences be- surtout danscords les casyielded de valeurs the values élevées of the du dispersion paramètre parame de dispersion- tween (Kreft the et ages Zuber resulting 1978, from particular models Małoszewskiter et as Zuber high 1982, as 2.5 Zuber [9, 10, 1986) 13]whereas. lower values and the radioisotope age were shown, it is a quite than 0.05 are rather unexpected. Some authors, common mistake to identify the radioisotope age, instead of Eq. 2.9, apply the solution to the dis- given by Eq. 2.3 with the mean tracer age. It is persion equation for an infinite medium, which is especially common in the case of 14C measure- inadequate, especially in cases of high values of ments of samples taken from systems with either the dispersion parameter [9, 10, 12]. an unknown flow pattern, or with a flow pat-

502

Fig. 2.4. Exemples de fonctions de réponse du modèle de dispersion (DM) pour des valeurs typiques du paramètre de dispersion.

2.4 CAS DE L’APPORT CONSTANT DE TRACEUR

L’approche empirique peut s’appliquer pour des traceurs avec un signal d’entrée variable. Elle s’applique aussi pour des radio-isotopes à concentration d’entrée constante. Cependant, dans ce dernier cas, une interprétation unique n’est possible que pour des modèles avec un seul

98 2.5 CAS DE L’APPORT VARIABLE DU TRACEUR

2.5.1 LA METHODE TRITIUM 3 Les concentrations en Tritium ( H; T1/2 = 12,43 ans) dans les eaux atmosphériques étaient constantes et très faibles (5–10 TU) avant les essais sur les bombes à hydrogène, qui ont LUMPEdébutéD PARA enME 1952.TER MOLesD ELSconcentrations les plus élevées, d’environ 6000 TU durant les mois d’été dans l’hémisphère nord, ont été observées en 1962–63. Depuis lors, les concentrations tern described by evidently another modelatmosphériques than with décroissentvery low dispersivity, exponentiellement the 3H methodatteignant yields 10–20 TU dans les années 90, avec the PFM. For a graphical presentation ofdes C/C teneurso ages maximales of waters caractéristiques recharged after pendant 1952 becauseles mois for du printemps et de l’été et des values yielded by different models see Fig.teneurs 2.2 in minimalesolder waters au cours the des present mois d’automneconcentrations et d’hiver. are close Ref. [9], Fig. 4 in Ref. [1], or Fig. 27 in Ref. [10]. to zero. However, for systems approximated by 3 As it is impossible to get a unique solutionLes if twoconcentrations the exponential en H élevées model, dans even les theprécipitations ages of the des or soixante- dernières années offrent or more sought parameters are used, the ageune can opportunité- der of unique 1000 pouryears dater can be les determined. systèmes aquifères For typi récents- dans une gamme d’âges not be found from C/Co. Therefore, if no otherrelativement in- cal large. dispersive Dans systems, le cas du the flux ages piston, of 100–200 ou des years systèmes avec une dispersivité très formation is available, at least bracket age values faible, la méthodeare often du observed. 3H donne T lesherefore, âges des the eauxenvironmental rechargées après 1952 car pour des eaux yielded by the PFM and EM should be given. 3 plus vieilles,H is les still concentrations the most useful sont tracer proches for dating de zéro.young Cependant, pour des systèmes waters, especially in the northern hemisphere. In general, when the flow pattern isapproximés unknown, par le modèle exponentiel, même les âges de l’ordre de 1000 ans peuvent être the interpretation should be performed for differ- Unfortunately, in the tropics the atmospheric 3H déterminés. Pour des systèmes typiquement dispersifs, les âges de 100–200 ans sont souvent ent models, and the ages obtained can be regarded peak was3 much lower, and in the southern hemi- as brackets of the real values. That problemobservés. is se- Donc,sphere le itsH wasenvironnemental even more dampedest encore and le delayedtraceur le plus utile pour dater les eaux 3 rious only for large relative ages. As mentionedrécentes, surtout[15], dans which l’hémisphère makes the nord.dating Malh moreeureusement, difficult orsous les tropiques, le pic de H above, it can easily be shown that if the traceratmosphérique age even a impossible.été bien plus faible, et dans l’hémisphère sud, il a même été plus amorti et is lower than the half-life of the radioisotope (t < retardéa (Gat 1980), ce qui rend la datation3 plus difficile voire même impossible. T = 0.693/l), all the models yield close values of Seasonal variations of the H concentration in 1/2 3 ages independently of the assumed flowLes pattern variations precipitation saisonnières as well de laas concentrationvariations in the en precipiH dans- les précipitations ainsi que la [1, 9], 10]. variabilité destation précipitations and infiltration et de l’infiltration rates cause rendent difficulties difficiles in l’estimation de la fonction the estimation of the input function, i.e., C (t). d’entrée, i.e., Cin(t). Pour chaque année calendaire, la valeurin du signal d’entrée peut être In the case of constant tracer input, the age can For each calendar year the value of the input can be found from a single measurement. Theexprimée only par: be expressed as: way to validate, or confirm, a model is to compare its results with other independent data, if avail- 12 12 12 12 able. However, the environmental tracers are par- Cin = ∑∑Ci Ii / Ii =∑∑Ciαi Pi / α i Pi (2.13) (2.13) ticularly useful in investigations of little known ii==1 1 ii==1 1 3 systems, where no other data are available for where Ci, Pi and Ii are the3 H concentration in où Ci, Pi et Ii sont les concentrations en H des précipitations, le hauteur de précipitation, et le comparisons. Therefore, the general validity of precipitation, precipitation rate, and infiltration taux d’infiltration pour le i-ème mois, respectivement. Le coefficient d’infiltration (α = I /P ) particular models is judged on the basis of vast rate for the i-th month, respectively. The infiltra- i i i literature of the subject. représente la fraction de précipitation qui entre dans le système aquifère au cours du i-ème tion coefficient (αi = Ii/Pi) represents the fraction mois. Les of valeurs precipitation de Cin, which calculées enters pour the chaque groundwater année antérieure à la dernière date 3 2.5. Cases of variable tracerd’échantillonnage, system in représentent the i-th month. la fonction The record d’entrée. of C Pourin val - l’interprétation de données H input anciennes, lesues, données calculated de Cforin doiteach inclure year prior les valeursto the constanteslatest de Cin observées avant le début du picsampling de 1954, date,causé represents par les essais the nucléaireinput function.s dans l’atmosphère,For dans les autres cas les 2.5.1. The tritium method 3 calculs de lathe fonction interpretation d’entrée of peuvent old H débuterdata, the à partirrecord de of 1954. Cin 3 Tritium ( H; T1/2 = 12.43 years) concentrations in should include constant Cin values observed prior atmospheric waters were constant and very low to the beginning of the rise in 1954 caused by (5–10 TU) before the hydrogen bomb tests, which hydrogen bomb test in the atmosphere, in other started in 1952. The highest concentrations, 100 cases the calculations of the input function can be up to about 6000 TU during summer months in started since 1954. the northern hemisphere, were observed in 1962– 1963. Since then, the atmospheric concentrations Some authors tried to estimate the infiltration co- exponentially decrease reaching 10–20 TU in efficients for particular months [16, 17]. In gen- late 90-ties, with characteristic maximum con- eral, these coefficients usually remain unknown, tents in spring and summer months and minimum and approximations have to be applied. If it is as- contents in autumn and winter months. High 3H sumed that the infiltration coefficient in the sum- concentrations in the precipitation of early six- mer months (αs) of each year is the same frac- ties offer a unique opportunity for dating young tion of the infiltration coefficient in the winter groundwater systems in a relatively wide range month (αw), i.e., α = αs/αw, Eq. 2.13 simplifies into of ages. In the case of piston flow, or systems Eq. 2.14 [18].

503

Fig. 2.5. Concentration en 3H (échelle logarithmique) des précipitations (α = 1,0) et fonctions d’entrée pour α = 0,7 et 0,0 calculées pour la station de Cracovie (Pologne). Les fonctions d’entrée pour la station de Świeradów (Montagnes des Sudètes, Pologne), et une des fonctions d’entrée corrigée pour la décroissance radioactive en 1998 sont indiquées pour comparaison.

Des auteurs ont essayé d’estimer les coefficients d’infiltration pour des mois particuliers (Andersen et Sevel 1974, Przewłocki 1975). En général, ces coefficients demeurent inconnus, et des approximations doivent être appliquées. Si on suppose que le coefficient d’infiltration dans les mois d’été (αs) de chaque année est la même fraction du coefficient d’infiltration dans les mois d’hiver (αw), i.e., α = αs/αw, l’Eq. 2.13 se simplifie à l’intérieur de Eq. 2.14 (Grabczak et al. 1984). modelling

9 3 9 3 Cin = [(α ∑Ci Pi )s + ( ∑Ci Pi )w ]/[(α ∑∑Pi )s + ( Pi )w ] (2.14) i=4 i =10 ii ==4 10 (2.14)

In the northern hemisphere the summer months are from April to September (from the fourth to the ninth month), and the winter months are from October to March (from the tenths to the third month of the next calendar year). Monthly precipitation amounts should be taken from the nearest meteorological station, and the 3H data should be taken from the nearest station of the IAEA net- Fig. 26.5. 3H concentration (logarithmic scale) in pre- work. As complete records are usually unavail- cipitation (α = 1.0) and input functions for α = 0.7 and able, the record of a given station has to be com- 0.0 calculated for Cracow101 (Poland) station. Shown for pleted by extrapolating correlation with another comparison is the input function for Świeradów station station for which a complete record exists, either (Sudetes Mts., Poland), and one of the input functions in original or correlated form [19]. Experience corrected for the radioactive decay to 1998. shows that in a rough approximation, the input functions from distant stations, with climatic conditions similar to those of the investigated area, can be used, especially for ages larger than about 20 years. 3H concentrations in precipitation and examples of the input functions are given in Figs. 2.5 and 2.6. The logarithmic scale of Fig. 2.5 gives a better idea about the concentrations which have been observed since 1954, and the long tail 3 of the H pulse whereas the linear scale of Fig. 2.6 serves for a better understanding the pulse character of the 3H input. That pulse character and low values of the tail gave reasons to opinionspréliminaires that le coefficientα était supposé égal à zéro ou 0,05. Cependant, la composition the 3H method would be of little use in nearisotopique fu- Fig. de l’eau26.6. The souterraine same as in superficielle Fig. 2.5, but the est 3 H d’ordinaire concen- égale, ou proche, de la ture. However, it seems that the 3H methodcomposition will tration isotopique is in linear moyenne scale. annuelle pondérée des précipitations, même dans les zones remain the best method for dating youngoù waters l’évapotranspiration potentielle domine les précipitations au cours des mois d’été. Cela for two decades at least. It is also evident signifiethat for qu’en été, l’évapotranspiration remobilise partiellement l’eau stockée dans la zone large values of α no drastic changes in the input represents the winter and summer precipitation. non saturée à la fois pendant les mois d’été et d’hiver. En conséquence, l’eau résiduelle, qui function are observed. When local precipitation and isotope data ex- atteint la nappeists, or phréatique,if they are available représente from les a nearbyprécipitations station, d’été et d’hiver. Quand les It is well known that under moderate climatic con- précipitationsthe locales value etof les the données α coefficient isotopiques can ex istent, be estimated ou si elles sont disponibles à partir ditions the recharge takes place mainly in winter from Eq. 2.15. months. Therefore, in some early publicationsd’une station proche, la valeur du coefficient α peut être estimée à partir de l’Eq. 2.15. the α coefficient was assumed to be equal to zero 3 3 9 9 or 0.05. However, the isotopic composition of α = [(∑∑Piδi )w −δ ( Pi )w ]/[δ (∑ Pi )s − ( ∑Piδi )s ] (2.15) shallow groundwaters is usually equal, or close, ii−=10 10 i=4 i=4 (2.15) to the yearly mean weighted isotopic compositionDans cette équation, δis, δiw sont les compositions en isotope stable des précipitations des mois of precipitation, even in areas of prevailing poten- d’été et des Imoisn that d’hiver, equation respectivement; δis, δiw are etthe δ est stable la composition isotope isotopique moyenne de l’eau tial evapotranspiration over precipitation in sum- compositions18 of the2 precipitation in the summer months. It means that in the summer souterrainemonths mer locale months (δ O and ou δ winterH) (Grabczak months, etrespectively; al. 1984). L’Eq. 2.15 est utile si des the evapotranspiration partly removesenregistrements water and δ suffisamment is the mean longsisotopic (une composition petite année) of delocal la composition isotopique et des stored in the unsaturated zone both in the summerhauteurs degroundwater précipitations (δ 18 sontO or disponibles δ2H) [18].. Eq. De 2.15 toute is façon,use- pour des climats tropicaux and winter months. In consequence, the tempérésremain- et fulhumides, if sufficiently le coefficient long α est (a few couramment years) records compris of dans la gamme de 0,4–0,8, et ing water, which reaches the groundwater table, the isotopic composition and precipitation rates l’expérience montre qu’à l’intérieur de cette gamme la précision de la modélisation dépend seulement faiblement de la valeur présumée de α, si les âges sont plus grands que 10–20 ans. 504 En général, si la fonction d’entrée n’est pas trouvée indépendamment, le coefficient α est arbitrairement choisi par le modélisateur, ou tacitement utilisé comme un paramètre (recherché) d’ajustement inconnu. Comme cela a été mentionné, plus le nombre de paramètres recherchés grand est, plus faible est la fiabilité de la modélisation. Donc, le nombre de paramètres recherchés devrait être gardé aussi faible que possible. Dans tous les cas, la méthode utilisée pour le calcul de la fonction d’entrée devrait également être indiquée. Supposer α=0 sur la base des observations hydrologiques conventionnelles, qui indiquent le manque de recharge nette dans certaines zones durant les mois d’été, est une erreur courante car, comme indiqué plus haut, cela ne signifie pas l’absence de 3H estival dans la recharge.

2.5.2 LA METHODE 3H-3HE Les concentrations en 3H dans l’atmosphère sont maintenant beaucoup plus faibles que durant le pic des essais nucléaires et elles décroissent encore, ce qui rend la méthode 3H moins intéressante pour un futur proche que durant les quatre dernières décades. En conséquence, d’autres méthodes de traçage sont considérées comme des outils potentiels, qui peuvent remplacer la méthode 3H ou prolonger son applicabilité (par ex., Plummer et al. 1993). Comme 3H décroît en donnant 3He, les mesures de 3He tritiogénique accumulé dans les systèmes aquifères prolongent potentiellement la gamme de datation résultant du pic de 3H 103 (Małoszewski et Zuber 1983). Dans la méthode 3H–3He, on considère généralement le rapport 3He sur 3H, qui donne pour le modèle PFM une formule bien connue dans laquelle l’âge du traceur est indépendant de l’apport (Torgersen et al. 1979):

−1 3 3 tt = λT ln[1+ HeT / H] (2.16)

3 3 où λT est la constante de décroissance radioactive pour H (1/λT = T1/2/ln2 = 17,9 a), H est la 3 3 3 3 3 teneur en H, et HeT est la teneur en He tritiogénique exprimée en unités H (pour He exprimé en ml STP de gaz par gramme d’eau, le facteur est de 4,01 × 1014 pour obtenir la 3 LUMPEteneurD PARA enME HeTE enR MO TU).DELS Malheureusement, l’Eq. 2.16 n’est pas applicable à d’autres modèles de flux. Si l’Eq. 2.5 est 3 are available. However, for moderate andutilisée humid pouring le equationcalcul de shouldla fonction be used de sortieor the théorique daughter du3He H, l’équation suivante doit être tropical climates, the α coefficient is utiliséecommonly avec theoretical la sortie théorique output [21]: de 3He fille (Małoszewski et Zuber 1983): within the range of 0.4–0.8, and experience shows ∞ that within this range the accuracy of modelling CHe = ∫CT in (t − t') g(t')[1− exp(−λT t')]dt (2.17) (2.17) only slightly depends on the assumed α value, if 0 the ages are greater than 10–20 years. In general, where C is the 3H input3 function, and C is if the input function is not found independentOù CTin- est la fonctionTin d’entrée du H, et CHe est la concentrationHe d’hélium exprimée dans les ly, the α coefficient is either arbitrarilymêmes chosen unités the que helium dans l’Eq.concentration 2.16. expressed in the same by the modeller, or tacitly used as a hidden fit- units as in Eq. 2.16. Plusieurs études récentes on montré l’applicabilité de l’Eq. 2.16 pour le transport vertical à ting (sought) parameter. As mentioned, the larg- Several recent studies showed the applicability er the number of sought parameters, thetravers lower les ofzones Eq. 2.16non saturéefor vertical ou saturée,transport où through les échantillons the un- sont prélevés à différentes 3 3 the reliability of modelling. Therefore, theprofondeurs num- saturated d’un profil or choisi, saturated et où zone, la disper wheresivité samples est négligeable. are Ainsi, la méthode H– He ber of sought parameters should be kept asdans low asl’approximation taken at different PFM estdepths avantageuse of a chosen par profile, rapport andà la méthode 3H car plusieurs 3 3 possible. In any case, the method used for échantillonsthe cal- the seulement dispersivity pris isà différentesnegligible. profondeThen, theurs H-prochesHe de la surface fournissent la culation of the input function should also be re- method in the PFM3 approximation is advanta- même information que le pic H et permet de déterminer le taux de recharge (Eq. 2.1a) ported. It is a common mistake to assume α = 0 geous to the 3H method because only several on the basis of conventional hydrologicalcomme obser- cela samplesa été montré taken par at Cookdifferent et Solomon depths close (1997). to the Ceci sur est- particulièrement important car 3 vations, which indicate the lack of net rechargedans la plupart face des supply cas, lethe pic same de H,information qui correspond as the au 3H pic peak atmosphérique de 1963, a disparu, in some areas during summer months, becauseou est itpréservé and dansallow les to profilsdetermine verticaux recharge uni ratequement (Eq. sous2.1a) desas conditions exceptionnellement does not mean the lack of the summer 3Hfavorables, in re- etshown à de bygrandes Ref. [23].profondeurs. That is especiallyLa méthode important peut aussi être employée pour les flux 3 charging water, as mentioned above. horizontaux asdans in mostla zone cases saturée, the H si peak, les lignewhichs decorresponds flux individuelles sont étudiées à l’aide to the atmospheric peak in 1963, has disappeared, d’échantillonneurs multi-niveaux. au moyen. Dans ce dernier cas, la méthode 3H–3He s’est or is preserved in vertical profiles only under ex- 2.5.2. The 3H-3He method montrée particulièrementceptionally favourable utile pour conditions, calibrer le sand modèles at large de flux et de transport dans les 3H concentrations in the atmosphere areaquifères now superficielsdepths. The. Comme method déjàcan alsomentionné, be used inl’autre horizontal avantage de la méthode est qu’elle much lower than during the bomb test peakpourra and être utiliséeflow in plus the longtempssaturated zone, que la if méthode the particular 3H. flow they still decrease, which cause the 3H method lines are observed with the aid of multi-lever to be less useful in near future than in Lathe nécessitélast samplers. de séparer In l’thehélium latter tritiogéniquecase the 3H-3 Hede methodl’hélium has issu d’autres sources (dissolution four decades. In consequence, other traceratmosphérique, meth- been production shown to radiogéniquebe particularly et useful air en to excès calibrate) constitue une limitation spécifique ods are considered as potential tools,de which la méthode flow 3He, and comme transport cela models est discuté in shallow en détail aquifers. par TorgersenAs et al. (1979), Weise et 3 may either replace the H method or prolongMoser its (1987), mentioned, et Schlosser another et al. advantage (1989). Pourof the l’approximation method is its du PFM, les incertitudes sur 3 3 applicability (e.g., [20]). As H decays to He, 3 3 1 l’âge causéespotentially par ces sources, longer applicabilityet par la diffusion in near rapide future de inHe comparée à celle de H HO, the measurements of the tritiugenic 3He accumu- comparison with the 3H method. lated in groundwater systems potentially ontprolong été réexaminées succinctement par Solomon et al. (1998). Specific limitations of the 3He method result the dating range resulting from the 3H peak 104 [21]. 3 3 from the need to separate the tritiogenic helium (MałoszewskiIn etthe Zuber 3H–3 He1983). method Dans the la méthode3He to 3H H–ratioHe, is onusu considère- généralement le rapport from helium originating from other sources (at- 3He sur 3H, allyqui considered,donne pour whichle modèle for thePFM PFM une yields form ulea well- bien connue dans laquelle l’âge du mospheric solubility, excess air and radiogenic known formula in which the tracer age is indepen- traceur est indépendant de l’apport (Torgersen et al. 1979): production) as discussed in detail in Refs [22, 24, dent of the input [22]: 25]. For the PFM approximation, age uncertain- −1 3 3 ties caused by these sources, and by fast diffusion tt = λT ln[1+ HeT / H] (2.16) (2.16) of 3He in comparison with the diffusion of 3H1HO, where l is the radioactive decay constant for 3H were shortly reviewed in Ref. [14]. λ T 3 λ 3 où T est la (1/constantel = T de/ln2 décroissance = 17.9 a), 3 Hradioactive is the 3H content, pour H and (1/ T = T1/2/ln2 = 17,9 a), H est la 3 T 3 1/2 3 Other difficulties3 are common3 to all gaseous trac- teneur en H,3He et isHe theT esttritiugenic la teneur 3He en contentHe tritiogénique expressed expriméein en unités H (pour He T ers and they are mainly related to possible es- exprimé en 3mlH unitsSTP (forde gaz 3He par expressed gramme in d’eau, ml ST leP offacteur gas per est de 4,01 × 1014 pour obtenir la capes or gains by enhanced diffusion when wa- 3 gram of water, the factor is 4.01 × 1014 to obtain teneur en He en TU). ter is in contact with air in the unsaturated zone the 3He content in TU). Malheureusement, l’Eq. 2.16 n’est pas applicable à d’autres modèlesor in de karstic flux. Sichannels. l’Eq. 2.5 For est instance, Grabczak et Unfortunately, Eq. 2.16 is not applicable to other al. [26] determined the models and 3H ages for utilisée pour le calcul de la fonction de sortie théorique du 3H, l’équation suivante doit être flow models. If Eq. 2.53 is used for the calculation withdrawal wells exploiting an unconfined aqui- utilisée avecof la the sortie theoretical théorique 3H de output He fille function, (Małoszewski the follow et Zuber- fer 1983): with thick loess and sandy covers, and for sev-

∞ CHe = ∫CT in (t − t') g(t')[1− exp(−λT t')]dt (2.17) 0 505

3 Où CTin est la fonction d’entrée du H, et CHe est la concentration d’hélium exprimée dans les mêmes unités que dans l’Eq. 2.16. Plusieurs études récentes on montré l’applicabilité de l’Eq. 2.16 pour le transport vertical à travers les zones non saturée ou saturée, où les échantillons sont prélevés à différentes profondeurs d’un profil choisi, et où la dispersivité est négligeable. Ainsi, la méthode 3H–3He dans l’approximation PFM est avantageuse par rapport à la méthode 3H car plusieurs échantillons seulement pris à différentes profondeurs proches de la surface fournissent la même information que le pic 3H et permet de déterminer le taux de recharge (Eq. 2.1a) comme cela a été montré par Cook et Solomon (1997). Ceci est particulièrement important car dans la plupart des cas, le pic de 3H, qui correspond au pic atmosphérique de 1963, a disparu, ou est préservé dans les profils verticaux uniquement sous des conditions exceptionnellement favorables, et à de grandes profondeurs. La méthode peut aussi être employée pour les flux horizontaux dans la zone saturée, si les lignes de flux individuelles sont étudiées à l’aide d’échantillonneurs multi-niveaux. au moyen. Dans ce dernier cas, la méthode 3H–3He s’est montrée particulièrement utile pour calibrer les modèles de flux et de transport dans les aquifères superficiels. Comme déjà mentionné, l’autre avantage de la méthode est qu’elle pourra être utilisée plus longtemps que la méthode 3H. La nécessité de séparer l’hélium tritiogénique de l’hélium issu d’autres sources (dissolution atmosphérique, production radiogénique et air en excès) constitue une limitation spécifique de la méthode 3He, comme cela est discuté en détail par Torgersen et al. (1979), Weise et Moser (1987), et Schlosser et al. (1989). Pour l’approximation du PFM, les incertitudes sur l’âge causées par ces sources, et par la diffusion rapide de 3He comparée à celle de 3H1HO, ont été réexaminées succinctement par Solomon et al. (1998). 104 modelling eral karstic springs. In all the cases the concentra- 85 3 tions of Kr, He and freon-11 (CCl3F) were in disagreement with the values expected on the basis of the 3H models. These disagreements were explained as diffusion losses or gains caused by sharp differences in concentration between water and air either in the unsaturated zone of the recharge areas or in channels partly filled with water near the outflows from a karstic aquifer. In the case of 3H, the age is counted from the moment of recharge at the surface whereas for gas tracers it starts rather at the water table [14, 27] Fig. 26.7. Specific activity of85 Kr in the air of the north- which makes additional difficulty in other appli- ern hemisphere [23, 29, 30], directly applicable as cations than the observations of vertical profiles the input function. for recharge studies. the calculations of the output concentrations show 2.5.3. The krypton-85 method a need of prolonged records of sampling [21]. For short tracer ages, say, up to about 5 to 10 years, The presence of radioactive 85Kr (T = 10.76 1/2 the differences between particular models are years) in the atmosphere results from emissions slight, similarly to the constant tracer inputs. For from nuclear power stations and plutonium pro- larger ages, the differences are not negligible. duction for military purposes. In spite of large spatial and temporal variations, the input function based on yearly averages is quite smooth as 2.5.4. The carbon-14 method shown in Fig. 2.7 for the northern hemisphere. Usually the 14C content is not measured in young For the southern hemisphere, the specific activity 3 3 85 waters in which H is present unless mixing of is about 0.2 Bq/m lower [28]. The Kr concen- components having distinctly different ages is tration is expressed in units of the specific activ- investigated. However, in principle, due to a dis- ity, and, therefore, it is independent of the krypton tinct bomb peak of 14C concentration, the lumped- solubility in water, and of the possible excess of parameter approach for variable input can be ap- air in water, which is related to a common effect plied. A high cost of 14C analyses and a low accu- of incorporation of air bubbles in the recharge racy related to the problem of the so-called initial area. The 85Kr method was initially hoped to re- carbon content make that approach impractical. place the 3H method in near future. However, However, it is suggested that when the 14C data serious limitations result from large samples re- are available, the lumped-parameter approach can quired due to low solubility of Kr and low con- be used to check if they are consistent with the re- centrations of 85Kr), and possible excess or deficit sults obtained from the 3H modelling. of 85Kr caused by exchange with the atmosphere, especially in karstic channels and thick unsaturated zones, similarly to the discussed earlier 3He 2.5.5. The oxygen-18 and deuterium tracer. In spite of these limitations the krypton-85 method method is probably the most promising replace- Seasonal variations of δ18O and δ2H in precipita- 3 ment of the H method in future. Other potential tion are under favourable conditions observed in gaseous tracers are discussed further. outlets of small catchments with the mean ages up Depth profiles for vertical flow or multi-level to about 4 years (a common definition of a small samplers make the method useful for studies of catchment is that with the surface area up to recharge rates [23]. However, for the typical ap- 100 km2 [31]. Due to a strong damping of the sea- plications of the lumped parameter models (inter- sonal input variations in outflows, a frequent sam- pretation of data obtained in abstraction well and pling over several years is usually required both springs), the solutions of the direct problem, i.e., at the input and the outlet. The input data should

506 LUMPED PARAMETER MODELS signal d’entrée doivent être récupérée à partir d’un pluviomètre local et les données de sortie be taken from a local precipitation collector and a frequent sampling, which makes it costly. d’un site dethe drainage outlet choisi,data form i.e., aune chosen source drainage ou un écoulement site, i.e., drainantTherefore, le bassin in the de caserétention of small retention basins signalétudié (Bergmand’entréea spring doivent et al. or 1986, streamêtre récupéréeMa drainingłoszewski à the partir etinvestigated al. d’un 1992) pluviomètre. reten- localits applicability et les données is limited de sortie to research purposes. In signal d’entréetion doivent basin [32, être 33]. récupérée à partir d’un pluviomètre localthe case et les of données bank filtration, de sortie the method is undoubt- d’unsignal site d’entrée de drainage doivent choisi, être récupérée i.e., une source à partir ou d’un un écoulementpluviomètre drainant local et leles bassin données de rétentionde sortie d’un site de drainage choisi, i.e., une source ou un écoulement drainantedly cheaper le bassin than de a rétention number of drilled wells need- étudiéd’un site (Bergman de drainage et al. choisi, 1986, Mai.e.,ł oszewskiune sourcen et al.ou 1992)un écoulement. drainant le bassin de rétention étudié (Bergmanδ (t) et= δal.+ 1986,[α P (Maδ −łoszewskiδ )]/( α etP al./ n1992)) . ed to obtain data for(2.18) construction of a numerical étudié (Bergmanin et al. 1986,i i Mai łoszewski∑ eti al.i 1992). (2.18) i=1 flow and transport model. n n whereδ (t) =`δδ is+ [theα P mean(δ −δ input)]/( whichα P / mustn) be equal 18 (2.18)2 où ⎯δ est l’apportδin (t) =moyenδ +[α quii Pi ( δdoiti −δ être)]/( égal∑n α auxi Pi / sortiesn) moyennes des valeurs δ O ou(2.18) δ H et n toin the mean outputi i i of δ18O ∑ior=1 δi2Hi values and n is 2.5.6. Other potential methods δin (t) = δ +[αi Pi (δi −δ )]/(∑i=1 αi Pi / n) (2.18) est le nombrethe denumber mois of(ou months de semaines, (or i= weeks,1 ou de or périodes two-weeks de deux semaines, car dans cette Among other18 environmental2 tracers with variable oùméthode ⎯δ est unel’apportperiods, unité moyen debecause temps qui doitin plus that être courte methodégal estaux pr asorties éférable)shorter moyennes time pour lesquels des valeurs les observations δ18O ou δ2H sontet n où ⎯δ est l’apport moyen qui doit être égal aux sorties moyennes desinput valeurs the most δ18O promising ou δ2H et nfor age determinations oùdisponibles. ⎯δ est l’apportunit Quand is moyen preferable) l’information qui doit for être sur which égalα n’est aux the sorties pasobservations disponible, moyennes ildes peut valeurs être δ remplacéO ou δ parH et α n, est le nombre de mois (ou de semaines,i ou de périodes de deuxof young semaines, waters car are dans freons cette (chlorofluorocarbons), commeméthodeest le nombre pour uneare unité la deavailable. méthode mois de temps (ou When3 H. de plus α semaines,the est courte information alors est ou calculé pr deéférable) périodeson α à i is partir pour not de dedeux lesquels l’Eq. semaines, 2.15, les observations ou car supposé, dans cette sont et méthode une unité de temps plus courte est préférable) pour lesquelsparticularly les observations freon-12 (CCl sont2F 2), and sulphur hexa- méthode uneavailable, unité de it temps can be plus replaced courte estby prα,éférable) similarly pour to lesquels les observations sont corresponddisponibles. au Quand coefficient3 l’information pour les surprécip αi itations n’est pas des disponible, mois d’été, ilfluoride alors peut êtreque (SF α remplacé6) =which 1 est has parutilisé beenα, shown to be a good disponibles.the Quand H method. l’information Then α sur is αeitheri n’est calculated pas disponible, from il peut être remplacé par α, disponibles. Quand l’information3 sur α n’est pas disponible, ilatmospheric peut être remplacé tracer. T parheir α input, functions mono- commepour les mois pourEq. d’hiver. la 2.15, méthode or assumed, 3H. α est and alors i appears calculé as a à coeffi partir- de l’Eq. 2.15, ou supposé, et comme pour la méthode 3H. α est alors calculé à partir de tonically l’Eq. 2.15, increase ou supposé, due to etthe global contamina- commecorrespond pour cientau lacoefficient for méthode the precipitation pourH. αles estprécip of alors theitations calculésummer des àmonthsmois partir d’été, de l’Eq.alors 2.15,que α ou = 1 supposé, est utilisé et correspondLa méthode audes coefficient isotopes stables pour lesest précipaussi utileitations pour des déterminer mois d’été, la tion fractionalors of que the d’eau αatmosphere = de 1 rivière,est utilisé by ou industry (Fig. 2.8). In correspond whereasau coefficient α = 1 ispour put lesfor précipthe winteritations months. des mois d’été, alors que α = 1 est utilisé pourde lac, les pompée mois d’hiver. dans les puits proches des rivières (lacs), et le tempsthe desouthern-hemisphere transport de cette eau their concentrations pour les mois d’hiver. Lade méthode la rivièreT des he (lac) isotopesstable au puits,isotope stables si method laest composition aussi is utilealso pour usefulisotopique déterminer for de de- l’eaula arefraction de somewhat la d’eau rivière de lower. (lac)rivière, Freonsvarie ou enter groundwater La méthodetermining des isotopes the stables fraction est aussi of river, utile orpour lake, déterminer water la systemsfraction d’eausimilarly de rivière, to other ou gases with infiltrating desuffisammentLa lac,méthode pompée des de dansisotopes manière les puits stables saisonnière. proches est aussi des La utile rivières com pourposition (lacs), déterminer isotopique et le temps la fraction dans de transport le d’eau puits de dede rivière, pompagecette eau ou de lac, pompéeflowing dans toles pumpingpuits proches wells desnear rivières rivers (lacs),(lakes), et and le tempswater de intransport which theyde cette are eaudissolved in low concen- decorrespond lac, la rivière pompée au (lac) mélange dans au les puits, depuits la sirivièreproches la composition et des de rivièresl’eau isotopiquesouterraine (lacs), et dele(Stichler temps l’eau de et transport laal. rivière1986, de Hötzl (lac) cette et varie eau al. de la rivièrethe (lac) travel au time puits, of sithat la water composition from the isotopique river (lake) de l’eautrations. de la As rivière mentioned, (lac) varie exchange with the air in 1989,de la Ma rivièrełoszewski (lac) auet al. puits, 1990) si: la composition isotopique de l’eau de la rivière (lac) varie suffisammentto dethe manièrewell, if the saisonnière. isotopic composition La composition of the isotopique riv- the dans unsaturated le puits de zone pompage makes the input function suffisammenter de (lake) manière water saisonnière. sufficiently La com variesposition seasonally. isotopique dans le puits de pompage 3 correspond au mélange de la rivière et de l’eau souterraine (Stichlerless accuratelyet al. 1986, defined Hötzl et than al. for the H. Under ex- 1989, correspond Małoszewski Theauδ (mélanget )isotopic= etpδ al. (det )composition1990)+ la(1 rivière−: p)δ (ett in) de the l’eau pumping souterraine well is(Stichler tremely et al. favourable 1986, Hötzl(2.19) conditions et al. (low filtration rate 1989, Małoszewskiw et al.r 1990): g 1989, Małoszewskithe mixture et al. of 1990) the river: and groundwater [34–36]: and high diffusion coefficient in the unsaturated où p est la fraction de l’eau de la rivière et les indices w, r et g représententzone), the responserespectivement function l’ should probably start δ w (t) = pδ r (t) + (1− p)δg (t) (2.19) (2.19) δ w (t) = pδ r (t) + (1− p)δg (t) at the water table. (2.19)The use of freons is also lim- eau souterraineδ w ( tpompée,) = pδ r ( tl’eau) + (1 −dep la)δ grivière(t) et l’eau souterraine locale. La valeur de p (2.19)peut être where p is the fraction of the river water and ited due to sorption effects, which are still little trouvéeoù p est enla fractionréarrangeant de l’eau l’Eq. de 2.19 la rivière et en etutilisant les indices les compositionsw, r et g représentent isotopiques respectivement moyennes de l’ où p est la fractionsubscripts de l’eauw, r andde la g rivièrestay for et theles indicespumped, w, river r et g représententknown. Another respectivement difficulty l’ results from the depen- chaqueoù p est composant: la fraction de l’eau de la rivière et les indices w, r et g représentent respectivement l’ eau souterraineand pompée,local groundwater, l’eau de la respectively.rivière et l’eau The souterraine value of locale.dence La of valeur the input de p function peut être on their solubility, i.e., eau souterraine pompée, l’eau de la rivière et l’eau souterraine locale. La valeur de p peut être trouvée en réarrangeantp can be found l’Eq. by 2.19 rearranging et en utilisant Eq. 2.19 les andcompositions us- isotopiques moyennes de chaquetrouvée composant:en réarrangeanting the mean l’Eq. isotopic 2.19 compositionset en utilisant of les particular compositions isotopiques moyennes de chaque composant:p = (δ w −δ g ) /(δ r −δ g ) (2.20) chaque composant:components:

La compositionp = ( isotopiqueδ −δ ) /(δ de− δ l’eau) souterraine locale (δg) est constante ou très légèrement(2.20) p = (δ w −δ g ) /(δ r −δ g ) (2.20) w g r g (2.20) variable comparéep = (δ w à− laδ g ) composition/(δ r −δ g ) isotopique de l’eau de rivière (δr). En conséquence,(2.20) le tempsLa composition de transportThe isotopiqueisotopic de la composition rivière de l’eau au souterraine captage of local est groundwaterlocale obtenu (δ eng) est ajustant constante l’Eq. ou 2.21 très alors légèrement que la La composition isotopique de l’eau souterraine locale (δg) est constante ou très légèrement La composition(δg) is isotopique either constant de l’eau or souterraine only slightly locale varies (δ ) inest constante ou très légèrement fractionvariable de comparée l’eau de àla la rivière composition est obtenue isotopique à partir de de l’eau l’Eq. g de 2.20 rivière (Stichler (δr). et En al. conséquence, 1986, Hötzl et le variable comparéecomparison à la with composition the isotopic isotopique composition de l’eau of river de rivière (δr). En conséquence, le al.tempsvariable 1989, de comparéeMa transportłoszewski à de la et la composition al. rivière 1990) . au isotopiquecaptage est de obtenu l’eau ende rivièreajustant ( δ l’Eq.r). En 2.21 conséquence, alors que lale temps de transportwater (δ r de). In la consequence, rivière au captage the travel est obtenu time from en ajustant l’Eq. 2.21 alors que la fractiontemps de de transport l’eau de dela rivière la rivière est obtenue au captage à partir est obtenude l’Eq. en 2.20 ajustant (Stichler l’Eq. et 2.21al. 1986, alors Hötzl que et la fraction de l’eauriver de to la the∞ rivière withdrawal est obtenue well à is partir found_ de by l’Eq. fitting 2.20 (Stichler et al. 1986, Hötzl et fraction de l’eauδ (t) de= pla δrivière(t − t 'est) g (obtenuet') dt' + (à1 partir− p)δ deg l’Eq. 2.20 (Stichler et al. 1986,(2.21) Hötzl et al. 1989, MaEq.łoszewskiw 2.21 whereas∫ etr al. 1990) the fraction. of the river water is al. 1989, Małoszewski0 et al. 1990). obtained from∞ Eq. 2.20 [34–36]. ∞ _ _ Fig. 26.8. Atmospheric concentrations of freons and δ w (t) = p ∞∫δ r (t − t') g(t') dt' + (1− p)δ_ g (2.21) δ w (t) = p ∫δ r (t − t') g(t') dt' + (1− p)δ g –12 (2.21) δ (t) = p 0 δ (t − t') g(t') dt' + (1− p)δ g SF6 in pptv (10 parts(2.21) per volume). F-11 and F-12 w ∫0 r (2.21) 0 after [23, 30, 37]; F-113 after107 [23, 30], and SF6 after [30], 38, 39]. The input functions are obtainable by The stable isotope method used for small reten- applying appropriate gas solubility for the recharge tion basins or bank filtration usually requires temperature and pressure. 107 107 107

507 modelling on the pressure and temperature at the recharge area, which is especially serious when the altitude of the recharge area remains unknown. However, the most serious difficulties are related to possible local contamination of shallow groundwaters by industry, and legal and illegal disposal sites (e.g., disposal of refrigerators into sinkholes in karstic areas). Therefore, chlorofluorocarbons are more commonly used to observe the contaminant transport in groundwater systems, and to calibrate numerical transport models, than to determine the age of water. Due to stripping effects, all gas- Fig. 26.9. 3H data and fitted models, a spring in eous tracers are not applicable in investigations of Szczawina, Sudetes Mts., Poland. waters rich in CO2 and CH4. Increased 3H concentrations in groundwaters is a temporary phenomenon due to a short half-life of that radioisotope and a short duration of the atmospheric peak. Theoretically, the atmospheric peak of bomb produced 36Cl, with half-life of about 3.01 × 105 years, should be an ideal tracer for relatively young groundwaters. However, the spatial differences in peak concentrations make this tracer difficult to apply in a similar way to 3H. An exception was for early recharge rate studies where the position of the peak in vertical profiles was measured and interpreted by the PFM approximation (see Ref. [40] for a review). Fig. 26.10. 3H data and fitted models, a spring in Łomnica Nowa, Sudetes Mts., Poland. 2.6. Examples of 3H age determinations

Examples of 3H age determinations for relatively the basis of the geology of the area and isotopic long records of 3H data can be found in references altitude effect as discussed in next section. given earlier whereas in Figs. 2.9 and 2.10 two other examples are given after Ref. [41] who gave a number of examples with short records of data. 2.7. Determination of In the first case a large number of models can be hydrogeologic parameters fitted whereas in the second case an infinite num- from tracer ages ber of models can be fitted, considering the ac- Principles of the interpretation of 3H data, espe- curacy range of the experimental data. However, cially in combinations with other environmen- the models shown are not inconsistent because if tal tracer data, can be found in a number of text a given model yields a lower age of the 3H com- books, manuals, and reports. However, the hy- ponent, a larger fraction (b) of the 3H-free water is drologic meaning of the tracer age in double po- obtained. The total mean ages are given by mod- rosity rocks (fractured rocks), or triple porosity els with b = 0. There is no doubt that for the spring rocks (karstic rocks) differs from that in granular in Szczawina the total mean age is of the order of rocks where it is related directly to the flow rate. 150 years and for the spring in Łomnica Nowa The difference in the meaning of the tracer age this age is of the order of 1 ka. In both cases between single porosity and double porosity rocks the final selection of the model was performed on is schematically shown in Fig. 2.11. For fractured

508 LUMPED PARAMETER MODELS

where nf and np are the fracture and matrix porosities, respectively. The approximate form of Eq. 2.23a is the result of the fracture porosity being usually negligibly low in comparison with

the matrix porosity (nf << np). A similar equation applies for triple porosity rocks (karstic-fractured- porous), where the karstic porosity is usually low in comparison with the fracture porosity [44]. The approximate form of Eq. 2.23a and the fol- Fig. 26.11. Schematic presentation of the tracer trans- lowing similar approximations are of great practi- port in fractured rocks at large scales when the tracer cal importance because the fracture porosity (plus is able to penetrate fully into the stagnant water in karstic porosity in triple porosity rocks) usually the matrix. remains unknown whereas np is easily measurable on rock samples (taken from unweathered rock at the outcrops, or from drill cores). Matrix poros- rocks, due to diffusion exchange between the mo- ity based on literature data can be used when no bile water in fractures and stagnant or quasi stag- samples are available. If the dimensions of the in- nant water in the micropores of matrix, the tracer vestigated system are known from the geologi- transport at large scales can be regarded as if it cal map and cross-sections, the rock volume is were flowing through the total open porosity [42, also known, and it can serve for the verification 43]. Unfortunately, that problem is tacitly omit- of the age by comparison with the volume found ted in a number of research papers and text books. from Eq. 2.23a. Therefore, basic formulas relating the mean tracer ages obtained from lumped-parameter models When the mean distance (x) from recharge area to with hydrologic parameters are recalled below. the sampling site is known, the following relation However, it should be remembered that these applies for single porosity rocks: simple relations for the fractured rocks are of approximate character, and they are valid only at tt = tw = x/vw =x/vt (2.25) large scales and for dense fracture networks. which means that the tracer age (travel time) and The volume of water (Vw) in the part of a given velocity are equal to those of water. system discharged by a spring is given as: For fractured rocks (double porosity), instead of Eq. 2.25 the following relations are applicable Vw = Q × tt (2.22) (Fig. 2.11): where Q is the outflow rate. That volume of water in granular systems is practically equal to the vol- tt = x/vt = tw(nf + np)/nf = (x/vw)(nf + np)/nf (2.25a) ume of mobile water because fraction of water in the micropores of grains is negligibly low). For In consequence, the tracer travel time is 1 + np/nf fractured rocks, that volume is equal to the total times longer than the travel time of water (i.e., volume (mobile water in fractures and stagnant tracer velocity is 1 + np/nf times slower than wa- or quasi stagnant water in matrix (Fig. 2.11). ter velocity). The fracture porosity is difficult to Consequently, for a single porosity rock, the rock estimate, and, therefore, if the tracer velocity is volume (Vr) occupied by Vw is given as: known, the water velocity remains unknown, and vice versa, if the water velocity is known,

Vr = Vw/ne (2.23) the tracer velocity remains unknown. where ne is the effective porosity, which is close For single porosity rocks, Darcy’s velocity (vf) is to the open porosity and total porosity (n). For related to water and tracer velocity by effective fractured rocks, the following equation applies: porosity:

Vr = Vw/(nf + np) ≅ Vw/np (2.23a) vf = nevw = nevt (2.26)

509 modelling

For fractured rocks, Darcy’s velocity is related in The 3H ages shown in Figs. 2.9 and 2.10 are re- a good approximation to water velocity by frac- lated to two springs discharging at the foot of ture porosity, because that porosity is usually the same morphological unit of a gneiss forma- close to the effective porosity: tion [41]. The matrix porosity was assumed to be 0.007, i.e., similar to the values measured on rock vf = nevw ≅ nfvw = nfvt(nf + np)/nf = vt(nf + np) samples taken from another gneiss formation of

≅ vtnp = (x/tt)np (2.26a) the same geological age in the Sudetes. The mean 3H ages estimated from the simplest models The approximate form of Eq. 2.26a means that are about 160 and 1000 years for Szczawina from tracer velocity (or age) it is easy to calcu- and Łomnica, respectively. However, accord- late Darcy’s velocity without any knowledge on ing to the stable isotope data, the recharge takes the fracture system. The hydraulic conductivity place mainly on a plateau at the top of the unit. (K) is defined by Darcy’s law, i.e.,v f = (DH/Dx)K, Therefore, the models should be selected in ac- where (DH/Dx) is the hydraulic gradient. In con- cordance with that information. It seems that sequence, Darcy’s law yields the following rela- most adequate are the exponential models (EM) tions: for the local component recharged directly above the springs on the slope of the unit, with a domi- 3 K = nex/[(∆H/∆x)tt] (2.27 nance of the H-free component recharged at the plateau. The following equation holds for for a single porosity rock, and a two-component mixing: K ≅ (n + n )x/[(∆H/∆x)t ] ≅ n x/[(∆H/∆x)t ] p f t p t t = (1 − β) × t + β × t (2.28 (2.27a) t, moyen t, jeune t, vieux where b is the fraction of the 3H-free compo- for a fractured rock. nent, and subscripts young and old correspond 3 3 In both Eq. 2.27 and 2.27a, the hydraulic gradient to the H and H-free components, respectively. 3 represents the mean value along the flow distance. From Eq. 2.28 the age of the H-free (old) compo- 3 The simplified form of Eq. 2.27a is of great prac- nent can be estimated, if the mean and H (young) tical importance because it allows estimation of component ages have been correctly determined. the regional hydraulic conductivity from the trac- The parameters of the two flow systems are sum- er age without any knowledge on the fracture marised in Table 2.1. For these calculations, network [45]. A number of examples of different the flow distances were estimated from the mor- applications of the lumped parameter approach phology map. For the 3H components, they were can be found in references given above as well as taken as the half slope distance, and for the 3H-free in other works. Two selected examples related to component they were taken as the distance from fractured rocks are given below. the adequate part of the plateau (for details see

Table 2.1 Parameters of the Szczawina and Łomnica systems [41]. Age Q V V K Site w r (a) (m3/hour) 106 m3 108 m3 (10–8 m/s) Szczawina 158 0.72 1.0 1.4 0.9

Szczawnina Young fraction 65 0.32 0.2 0.3 1.0 Old fraction 230 0.40 0.8 1.1 0.8 Łomnica Young fraction 70 a) 0.5 Old fraction 1000 0.1 a) unmeasurable due to a partial discharge of the spring in a stream.

510 LUMPED PARAMETER MODELS

Ref. [41]). The mean hydraulic gradients were as- 2.8. The lumped-parameter sumed to follow the morphology. approach versus other approaches Considering approximate character of the age, distance and hydraulic gradient estimations, The multi-cell approach has been introduced to the accuracy of the parameters given in Table the tracer method in hydrology by Simpson and 2.1 is probably not better than about 50%. In any Duckstein [46], and Przewłocki and Yurtsever case a comparison with other crystalline rock [47]. When uni-dimensional arrangement of cells systems suggests their correctness. They are also is applied the method can be regarded as a less ver- internally consistent because the slope above satile version of the lumped-parameter approach. the Łomnica spring is distinctly shorter than that For a single cell, it is equivalent to the EM, and above the Szczawina spring, which results in for a very large number of cells, it approaches 3 a lower fraction of the H component in the for- the PFM. However, when more complicated ar- mer. It is difficult to say if the difference between rangements are applied (e.g., different volumes 3 the hydraulic conductivity of both H systems is of cells, two- and three-dimensional cell arrange- significant. However, a distinctly lower value of ments) the number of sought (fitted) parameters 3 the hydraulic conductivity of the H-free system increases and unique solutions are not available. in Łomnica than in Szczawina most probably re- Therefore, the multi-cell models can be regarded sults from a larger part of the plateau covered by as a distributed parameter approach with lumping. less permeable Cretaceous sediments. When interrelated tracer data distributed in time In the urbanised area of Lublin city, eastern and space are available, the multi-cell modelling Poland, groundwater is exploited from Cretaceous is definitely advantageous over the lumped pa- marls which are fractured to the depth of about rameter approach. Unfortunately, quite frequently 100–200 m. In spite of dense urbanisation, the wa- publications appear in which a single 3H determi- ter is of a good quality. A number of wells and nation, or a mean value of several samples taken springs were sampled twice for 3H determinations in a short period of time, is interpreted either with at the beginning of 1995 and end of 1997. All the aid of the EM or the multi-cell approach. Such the concentrations were below 10 TU with slow publications should be regarded as examples of declines. Models fitted to the data were similar to incorrect interpretation. those shown in Fig. 2.10, and the greatest mean 3H ages were in the range of 250–500 years. The re- As mentioned, the lumped parameter models are gional hydraulic conductivity in the range of 4–15 particularly useful when no sufficient data exist to m/d was obtained from Eq. 2.27a for the matrix justify the use of multi-cell models, multi-tracer porosity of 0.40 and estimated distances of flow multi-cell models [48], or numerical solutions to yielded. Hydrodynamic modelling and pumping the transport equation. They are also very useful tests yielded the hydraulic conductivity of 2.5–10 in early investigations of little known systems. m/d in most of the watershed, and 50–300 m/d at For a separate sampling site (e.g., a spring, or the tectonic zones of the valley axes. Therefore, in a withdrawal well), only the use of the lumped spite of a low accuracy of age determinations re- parameter models is sufficiently justified. Some sulting from a low number of 3H determinations, investigators express opinions that in the era of the hydraulic conductivity obtained is in a gen- numerical models, the use of a lumped-parameter eral agreement with that derived from the con- approach is out of date. However, it is like trying ventional methods. Large values of regional to kill a fly with a cannon, which is neither effec- 3H ages, which result from the matrix diffusion tive nor economic. Experience shows that a num- (Eq. 2.25a), explain the good quality of water in ber of representative hydrologic parameters can that densely urbanised area. However, when some be obtained from the lumped-parameter approach non-decaying pollutants appear in the groundwa- to the interpretation of environmental tracer data ter, their removal will also take a very long time. in a cheap and effective way.

511 modelling

2.9. Concluding remarks found. The modelling procedure should always The lumped parameter approach is particularly start with the simplest models. More sophisti- useful for the interpretation of 3H data in ground- cated models with additional parameters should water systems with separate sampling sites as, be introduced only if it is not possible to obtain for instance, in investigations of the dynam- a good fit with a simple model, or if other infor- ics of small catchments [49]. 18O has also been mation excludes a simpler model. However, it shown to be applicable in investigations of small should be remembered that if a single parameter retention basins and bank filtration from rivers model yields a good fit, an infinite number of two and lakes. As mentioned, the 3H–3He method is parameter models also yield equally good fits. advantageous over the 3H method for recharge Therefore, in such situations other available in- rate measurements. The use of 85Kr is still trou- formation should be used for the final selection of blesome and costly, and its advantages have not the most adequate model. As the inverse solutions been proved so far. Measurements of freons have belong to the category of ill-posed problems, and become routinely used in some countries (espe- the record of the experimental data is usually very cially in the USA), though, most probably, due to short, exact and unique solutions are in general a lower accuracy inherent to their character, they not available. However, even non-unique and/or cannot so far compete with the 3H method. non-exact solutions are better than a lack of any quantitative, or semi-quantitative, information. As mentioned a user-friendly computer-programme for the interpretation of environmental An additional difficulty results from hetero- tracer data by the lumped-parameter approach geneity of groundwater systems. As shown in is available free from the IAEA (FLOWPC). In Refs [51, 52], in highly heterogeneous systems, that programme the PFM, EM, EPM, LM (linear the mean tracer age may considerably differ from model), LPM (combined linear-piston flow mod- the mean water age. In some cases the tracer age el), and DM are included, and it contains options practically represents the upper, more active part for the applications of stable isotopes and other of the system, whereas in strongly stratified sys- tracers (excluding 3He), as well as for using any tems, dispersive losses of tracer to deeper layers α value, and any b value with a chosen constant may result in an apparent value of the b coeffi- tracer concentration. In addition, ASCII files of cient. Similarly other parameters do not neces- the response, input and output functions are yield- sarily represent properly the system investigated. ed. Curves shown in Figs. 2.2 to 2.6, 2.9 and 2.10 However, in spite of all the limitations, experi- were calculated with the aid of the FLOWPC. ence shows the lumped-parameter approach to When solving the inverse problem it should be re- the interpretation of environmental tracer data is membered that in general the lower the number of practical importance and usually yields repre- of fitted (sought) parameters, the more reliable sentative results. Experience also shows that even the results of modelling [50]. A better fit obtained such heterogeneous systems as karstic rocks can with a larger number of parameters does not nec- effectively be interpreted by that approach (e.g., essarily mean that a more adequate model was [33, 53].

512 LUMPED PARAMETER MODELS

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[25] Schlosser, P., Stute, M., Sonntag, C., Munnich, K.O., Tritiogenic 3He in shallow groundwater, Earth Planet. Sci. Lett. 89 (1988) 353–362. [26] Grabczak J., Zuber, A., Małoszewski, P., Różański, K., Weiss, W., Śliwka, I., New mathematical models for the interpretation of environmental tracers in groundwaters and the combined use of tritium, C-14, Kr-85, He-3 and freon-11 methods, Beitr. Geol. Schweiz. – Hydrologie 28 (1982) 395–405. [27] Solomon, D.K., Schiff, S.L., Poreda, R.J., Clarke, W.B., A validation of the 3H/3He method for determining groundwater recharge, Water Resour. Res. 29 9 (1993) 2851–2962. [28] Sartorius, H., 20 Jahre Kr-85-Messungen in Freiburg, BfS Bundensamt für Strahlenschutz, Freiburg (1993) 122–123. [29] Sartorius, H., Der Krypton-85 Untergrundpegel in der nördlichen Hemisphäre, BfS Bundensamt für Strahlenschutz, Freiburg (1998). [30] Current Greenhouse Gas Concentrations, http://cdiac.esd.ornl.gov/pns/current_ghg. html, January 1999. [31] Buttle, J.M., “Fundamentals of small catchment hydrology”, Isotope Tracers in Catchment Hydrology (C. Kendall, C., McDonnel, J.J., Eds), Elsevier, Amsterdam (1998) 1–49. [32] Bergman, H., Sackl, B., Małoszewski, P., Stichler, W., “Hydrological investigations in a small catchment area using isotope data series”, 5th International Symposium on Underground Water Tracing. Institute of Geology and Mineral Exploration, Athens (1986) 255–271. [33] Małoszewski, P., Harum, T., Zojer, H., Modelling of environmental tracer data: Transport Phenomena in Different Aquifers, Steirische Beiträge zur Hydrologie, Band 43 (1992) 116–136. [34] Stichler, W., Małoszewski, P., Moser, H., Modelling of river water infiltration using oxygen-18 data, J. Hydrol. 83 (1986) 355–365. [35] Hötzl, H., Reichert, B., Małoszewski, P., Moser, H., Stichler, W., “Contaminant transport in bank filtration – determining hydraulic parameters by means of artificial and natural labelling”, Contaminant Transport in Groundwater (Kobus, H.E., Kinzelbach, W., Eds), A.A. Balkema, Rotterdam (1989) 65–71. [36] Małoszewski, P., Moser, H., Stichler, W., Bertleff, B., Hedin, K., “Modelling of groundwater pollution by river bank filtration using oxygen-18 data”, Groundwater Monitoring and Management, IAHS Publ. No. 173 (1990) 153–161. [37] Oak Ridge National Laboratory, Trends’93: A Compedium of Data on Global Change, Carbon Dioxide Information Analysis Center, Oak Ridge, Etats-Unis (1993).

[38] Mais, M, Levin, I., Global increase of SF6 observed in the atmosphere, Geophys. Res. Let. 21 7 (1994) 569–572. [39] Mais, M., Steele, L.P., Francey, R.F., Fraser, P.J., Langefelds, R.L., Trivet, N.B.A., Levin, I., Sulfur hexafluoride — a powerful new atmospheric tracer, Atmos. Environ. 30 (1996) 1621–1629. [40] Bentley, H.W., Philips, F.M., Davis, S.N., “Chlorine-36 in the terrestrial environment”, Handbook of Environmental Isotope Geochemistry, Vol. 2, Part B, Elsevier, Amsterdam (1986) 427–479. [41] Zuber, A., Ciężkowski, W., “A combined interpretation of environmental isotopes for analyses of flow and transport parameters by making use of the lumped-parameter approach”, Use of Isotopes for Analyses of Flow and Transport Dynamics in Groundwater Systems, Results of a co-ordinated research project 1996–1999, IAEA, Vienna (2002) CD-ROM. [42] Neretnieks I., Age dating of groundwater in fissured rock: Influence of water volume in micropores. Water Resour. Res. 17 (1980) 421–422. [43] Małoszewski, P., Zuber, A., On the theory of tracer experiments in fissured rocks with a porous matrix, J. Hydrol. 79 (19850 333–358. [44] Zuber, A., Motyka, J., Hydraulic parameters and solute velocities in triple-porosity karstic-fissured- porous carbonate aquifers: case studies in southern Poland, Environ. Geol. 34 2/3 (1998) 243–250. [45] Zuber, A., Motyka, J., Matrix porosity as the most important parameter of fissured rocks for solute transport at large scales, J. Hydrol. 158 (1994) 19–46. [46] Simpson E.S., Duckstein, L., “Finite state mixing-cell models”, Karst Hydrology and Water Resources, Vol. 2, Water Resources Publications, Fort Collins, Colorado (1976) 489–508. [47] Przewłocki, K., Yurtsever, Y., “Some conceptual mathematical models and digital simulation approach in the use of tracers in hydrological systems”, Isotope Techniques in Groundwater Hydrology 1974, Vol.2. IAEA, Vienna (1974) 425–450.

514 LUMPED PARAMETER MODELS

[48] Adar, E.M., “Quantitative evaluation of flow systems, groundwater recharge and transmissivities using environmental tracers”, Manual on Mathematical Models in Isotope Hydrology, IAEA-TECDOC-910, IAEA, Vienna (1996) 113–154. [49] Kendall, C., McDonell, J.J., Isotope Tracers in Catchment Hydrology. Elsevier, Amsterdam (1998). [50] Himmelblau, D.M., Bischoff, K.B., Process Analysis and Simulation: Deterministic Systems, Wiley, New York, N.Y. (1968). [51] Varni, M., Carrera, J., Simulation of groundwater age distributions, Water Resour. Res. 34 12 (1998) 3271–3281. [52] Małoszewski, P., Seiler, K.P., “Modeling of flow dynamics in layered ground water system – comparative evaluation of black box and numerical approaches”, Isotope Techniques in Water Resources Development and Management, Proc. IAEA Symp. Vienna 1999, CDRom IAEA-CSP-2C (1999). [53] Rank, D., Völkl, G., Małoszewski, P. Stichler, W., “Flow dynamicsin an Alpine karst massifstudied by means of environmental isotopes”, Isotope Techniques in Water Resources Development 1991, IAEA, Vienna (1992) 327–343.

515

3. COMPARTMENTAL MODEL APPROACHES TO GROUNDWATER FLOW SIMULATION

MICHAEL E. CAMPANA Compartmental models have been used to solve the inverse problem (estimating aquifer properties Department of Earth and Planetary Sciences and and recharge boundary conditions) [1–5]. Other Water Resources Program, applications have sought to determine groundwa- University of New Mexico, Albuquerque ter ages and residence times [6–10], or analyze New Mexico, USA tracer data and delineate groundwater dynamics [11–13]. Other investigators have used them as GLENN A. HARRINGTON transport models [14–15]. A recent pioneering ap- CSIRO Land and Water, proach uses a compartmental model to constrain Glen Osmond, Australia a finite-difference regional groundwater flow model [16]. LEVENT TEZCAN The three compartmental models described herein Department of Hydrogeological Engineering, represent different approaches and levels of so- Hacettepe University, Ankara, Turkey phistication. The first, a relatively simple model by Campana, is calibrated with the spatial distribution of the environmental isotope deuterium. 3.1. Introduction Calibration with deuterium yields estimates of groundwater flow rates and residence times with- Compartmental or mixing-cell models have been in a regional aquifer system. The second approach applied to groundwater flow systems by a num- by Harrington uses a compartmental model, cali- ber of investigators. Note that the expressions brated with 14C, to constrain a finite-difference ‘compartment’, ‘cell’ and ‘mixing cell’ are syn- regional groundwater flow model of the Otway onymous and used interchangeably in this paper. Basin in South Australia. To our knowledge The compartmental model represents the ground- this represents a first. The final application, by water system as a network of interconnected cells Levent Tezcan, describes a distributed mixing- or compartments through which water and one or cell model that can simulate groundwater flow more dissolved constituents (tracers) are trans- and transport; he applies it to a karst aquifer on ported. Within a given cell, perfect or complete the Mediterranean coast of Turkey. Tezcan’s mod- mixing of the tracer occurs, although some model also functions like a watershed model in that els relax this constraint. Flow rates of water and it simulates the surface hydrology in addition to tracer between cells can be calculated by: subsurface flow. (1) use of a flow model that solves the partial differential equations of groundwater flow, 3.2. A simple compartmental model: theory and (2) calibration with observed tracer data, application to a regional (3) a flow algorithm based on linear or non-lin- groundwater flow system ear reservoir theory, or 3.2.1. Theory (4) some combination of the preceding. We use a numerical compartmental or mixing-cell Each cell in the model depicts a region of the hy- model [6, 17] to simulate flow in a subsurface flow drogeological system; regions are differentiated system. The code has been applied to a variety of based upon their hydrogeological uniformity, subsurface flow systems [6–10, 18]. The com- the availability of data, the degree of resolution partmental model represents the groundwater sys- desired, and constraints imposed by numerical so- tem as a network of interconnected cells or com- lutions. partments through which water and a dissolved

517 modelling constituent (tracer) are transported. Each cell in case holds for discharge from the system (SBDC the model depicts a region of the hydrogeologi- and SBDV). cal system; regions are differentiated based upon The mass balance equation, Eq. 3.1, is applied their hydrogeological uniformity, the availability successively to each cell during a given itera- of data, and the degree of resolution desired. Cells tion; discharge (BDV and BDC) from an ‘up- can be of any desired size and can be arranged stream’ cell becomes recharge (BRV and BRC) in a one-, two-or three-dimensional configuration. to a ‘downstream’ cell. The BDC(N) term on The model can be used as a ‘stand-alone’ model the right-hand side of Eq. 3.1 is the only unknown or coupled to a flow model. and can be determined from one of two mixing Our compartmental model permits the user to rules, the simple mixing cell (SMC), which simu- specify the flow paths between cells and the dis- lates perfect mixing, or the modified mixing cell charge from the system. Discharge can also be (MMC), which simulates some regime between calculated using linear reservoir theory. To do so perfect mixing and piston flow. For the SMC: requires an initial estimate of the flow system, such that an initial set of specifications can be BDC(N) = [S(N – 1) + BRV(N) × BRC(N)]/[VOL established. During the calibration process, these + BRV(N)] (3.2 parameters are adjusted by the modeller to obtain For the MMC: agreement between the simulated and observed tracer concentrations. We use environmental iso- BDC(N) = S(N – 1)/VOL (3.3) topes as tracers. where: VOL = volume of water in the cell, equal to The following sections describe the equations the cell’s total volume times its volumetric mois- governing the flow of water and tracer in a net- ture content (for unsaturated flow) or its effective work of cells or compartments. porosity (for saturated flow). Note that the MMC simulates pure piston flow as BRV → VOL and 3.2.1.1. Tracer mass balance perfect mixing as BRV → zero. Although pure pis- The basic equation, applied to each cell, or com- ton flow within a cell is possible, pure piston flow partment, is [17]: for the entire array of cells is not implied because some degree of mixing occurs between cells [8]. S(N) = S(N – 1) + [BRV(N) × BRC(N)] The same mixing rule must be used for each cell during a given model run. We used the MMC rule – [BDV(N) × BDC(N)] (3.1) in the model described herein. where: S(N) = cell state at iteration N, the mass of tracer within the cell; BRV(N) = boundary re- 3.2.1.2. Transient flow charge volume, the input volume of water at iteration N; BRC(N) = boundary recharge concen- The above equations cannot account for chang- tration, the input tracer concentration; BDV(N) = es in storage within the groundwater system. boundary discharge volume, the output volume Previous compartment models have treated tran- of water leaving the compartment or cell; and sience [13]. Following previous workers [6–13] BDC(N) = boundary discharge concentration, we simulate transient flow by assuming that the output tracer concentration. the outflow from a groundwater reservoir is proportional to the storage in the reservoir [19–20]: Tracer concentrations and water volumes cross- ing model boundaries and entering/leaving a cell S = KQ (3.4) on the boundary of the model are given the pre- fix ‘system’ or ‘S’. Thus, recharge entering a cell where: S = storage above a threshold, below from outside the model boundaries has a charac- which the outflow is zero; K = storage delay time teristic tracer concentration SBRC (system bound- of the compartment; and Q = volume rate of out- ary recharge concentration) and volume SBRV flow from the element. Eq. 3.4 describes a con- (system boundary recharge volume). The similar ceptual element known as a linear reservoir.

518 L’Eq. 3.5 ne prend pas en compte la présence d’un seuil dans le compartiment, mais peut être adapté pour un tel cas en écrivant l’Eq. 3.5:

L’Eq. 3.5 neVOL prend(N) pas– PHI en =compte K × BDV la (présenceN) d’un seuil dans le compartiment, mais(3.6) peut être adaptéoù PHI pour = volume un tel cas seuil en duécrivant compartiment, l’Eq. 3.5: en dessous duquel sa décharge est considérée comme égaleVOL à zéro.(N) – SiPHI VOL = (KN )× estBDV inférieur(N) ou égal à PHI, alors BDV(N) est pris égale(3.6) à zéro. où PHI = volume seuil du compartiment, en dessous duquel sa décharge est considérée commeSi K est égalemaintenu à zéro. constant Si VOL pour( Ntout) est N , inférieuralors le système ou égal décrit à PHI par, lesalors équations BDV(N ci-) est dessus pris estégale à zéro.un système linéaire, constant dans le temps; si K est une fonction du temps ou du nombre d’itération, i.e., K = K(N), alors le système est un système linéaire, constant dans le temps Si K est maintenu constant pour tout N, alors le système décrit par les équations ci- dessus est (Mandeville et O'Donnell 1973). un système linéaire, constant dans le temps; si K est une fonction du temps ou du nombre d’itération,Si l’Eq. 3.5 i.e.,est réécriteK = K( Npour), alors l’itération le système N+1 etest substituée un système dans linéaire, l’Eq. 3.7, constant une équation dans le de temps (Mandevilleconservation etdu O'Donnell volume pour 1973). un compartiment ou une cellule donnés: Si l’Eq. 3.5VOL est (réécriteN + 1) = VOLpour( Nl’itération) + BRV(N N ++1 1) –et BDV substituée(N + 1) dans l’Eq. 3.7, une(3.7) équation de conservationle résultat est du volume pour un compartiment ou une cellule donnés: VOL((NN ++ 1)1) = = VOL VOL(N(N) +) +BRV BRV(N( N+ 1)+ 1)– [ VOL– BDV(N (+N 1)/ + K1)] (3.8) (3.7) lequi résultat se simplifie est en: VOL((NN ++ 1)1) = = [ KVOL/K +( N1]) [+VOL BRV(N(N) + + BRV 1) –( N[VOL + 1)]( N + 1)/K] (3.9) (3.8) quiA l’itérationse simplifie N en:+ 1, toutes les quantités sur le côté droit de l’Eq. 3.9 sont connues, donc VOL(N + 1)VOL peut( êtreN + calculé.1) = [K /BDVK + (1]N +[VOL 1) peut(N) alors+ BRV être(N calculé + 1)] à partir de l’Eq. 3.5. (3.9) A l’itération N + 1, toutes les quantités sur le côté droit de l’Eq. 3.9 sont connues, donc 3.2.1.3 CALCULS DES AGES VOL(N + 1) peut être calculé. BDV(N + 1) peut alors être calculé à partir de l’Eq. 3.5. Pour le modèle à compartiments quand la recharge ne varie pas en fonction du temps, le 3.2.1.3calcul de l’âgeCALCULS moyen ouDES du A tempsGES de résidence moyen de l’eau dans un compartiment ou une cellule est relativement simple (Campana 1975; 1987). COMPARTMENTAL MODEL APPPourROACH le ESmodèle TO GR OUNà compartimentsDWATER FLOW quand SIMUL laAT IONrecharge ne varie pas en fonction du temps, le calculPour la de SMC: l’âge moyen ou du temps de résidence moyen de l’eau dans un compartiment ou une cellule est relativement simple (Campana 1975; 1987). In the context of the compartment model, Eq. 3.4 ⎡VOL + BRV ⎤ k AGE = DELT + ∑ FBRV AGEFBRV (3.10) for a single compartment is: Pour la SMC: ⎢ ⎥ []i i ⎣ BRV ⎦ i = 1 VOL(N) = K × BDV(N) où(3.5) AGE = âge moyen⎡VOL de l’eau + BRV dans la cellule;⎤ kDELT = temps(3.10) réel entre les itérations; FBRV = AGE = DELT + ∑ FBRV AGEFBRV (3.10)i ⎣⎢ BRV ⎦⎥ []i i Eq. 3.5 does not account for the presencefraction of de toutewhere l’eau AGE entrante = mean dans age la of cellule thei = water1 (BRV in) depuis the cell; la cellule i; AGEFBRVi = âge DELT = real time between iterations; FBRV = a threshold in the compartment, but can bemoyen adapt -de FBRVi; et k = nombre de cellules amont, qui apportenti l’eau directement à la où AGE = âge moyen de l’eau dans la cellule; DELT = temps réel entre les itérations; FBRVi = ed for such a case by rewriting Eq. 3.5: cellule. fraction of all incoming water to the cell (BRV) fraction de toute l’eau entrante dans la cellule (BRV) depuis la cellule i; AGEFBRVi = âge which is from cell i; AGEFBRVi = mean age of VOL(N) – PHI = K × BDV(N) moyenPour(3.6) la deMMC: FBRVFBRV i;i; etand k k= = nombrenumber deof upgradientcellules amont, cells whichqui apportent l’eau directement à la cellule. contribute water directly to the cell. where PHI = threshold volume of the compart- ⎡VOL ⎤ k AGE = ⎢ DELT ⎥ + ∑ []FBRV AGEFBRVi (3.11) ment, below which the discharge from thePour com la -MMC:For the⎣ MMC:BRV ⎦ i=1 i partment is defined as zero. IfVOL(N) is less than 92 ⎡VOL ⎤ k or equal to PHI, then BDV(N) is defined as zero. AGE = ⎢ DELT ⎥ + ∑ []FBRV AGEFBRVi (3.11) ⎣ BRV ⎦ i=1 i If K is held constant for all N, then the system de- 92 (3.11) scribed by either of the above equations is a linear, time-invariant system; if K is a function of time or The age distribution and cumulative age distribu- iteration number, i.e., K = K(N), then the system tion of each cell can be simulated by an impulse La distribution des âges et la distribution cumulée des âges de chaque cellule peuvent être is a linear, time-variant system [20]. -response method [7]. The mean age of the wa- simulées parter une in eachimpulsion cell can — méthodebe calculated de réponse by an instanta(Campana- 1987). L’âge moyen de l’eau If Eq. 3.5 is rewritten for iteration N+1 and sub- dans chaqueneous cellule injection peut être of tracer calculé through par une the SinjectionBRV inputs instantanée de traceur à travers les stituted into Eq. 3.7, a volume conservation equa- entrées de SBRVto each pour cell, chaquei.e., the cellule, recharge i.e., water l’eau of de age recharge zero. d’âge zéro. L’âge moyen est tion for a given compartment or cell: alors obtenuT par:he mean age is then found by: VOL(N + 1) = VOL(N) + BRV(N + 1) N – BDV(N + 1) (3.7) ∑ iC(N) i = 1 i A = DELT (3.12) N (3.12) the result is ∑ C(N)i i = 1 VOL(N + 1) = VOL(N) + BRV(N + 1) – où: C = concentrationwhere: C = du tracer traceur concentration dans la cellule; in cell; et A–and = âgeA = moyen de l’eau dans la cellule, – [VOL(N + 1)/K] (3.8) équivalent àmean AGE age (équations of the water 10 ou in 11)the cell,sauf equalpour l’erreurto AGE de troncature associée à C. La (Equations 10 or 11) except for truncation error which simplifies to distribution de l’âge A peut être obtenue à partir de C(N–) car la concentration du traceur dans associated with C. The age distribution of A can chaque cellulebe obtainedaux itérations from (C(N)N) après because l’injection the concentration est la mesure de la quantité fractionnaire de VOL(N + 1) = [K/K + 1] [VOL(N) l’âge de l’eauof the(N ×tracer DELT in) dans each cette cell cellule. at iterations La distribution (N) after cumulative des âges peut être + BRV(N + 1)] (3.9) facilement obtenuethe injection à partir is ade measure la distribution of the fractional de l’âge (Campana amount 1987). of water age (N × DELT) in that cell. The cumu- At iteration N+1, all quantities on the right-handIl y a des moments où le modèle à compartiments peut être utilisé sous des conditions de lative age distribution can be easily determined side of Eq. 3.9 are known, so VOL(N+1)régime can be permanent (pour une cellule donné VOL = constant) mais la recharge au niveau du calculated. Once this has been accomplished, from the age distribution [7]. modèle peut varier, par exemple pour simuler des changements dans le régime hydrologique BDV(N+1) can be calculated from Eq. 3.5. There are times when the compartment model induits par des variations climatiques. Sous de telles conditions, les équations données plus may be operated under steady state conditions haut ne peuvent pas être utilisées pour calculer les âges moyens. Des relations plus 3.2.1.3. Age calculations (for a given cell VOL = constant) but the recharge compliquéesto doivent the model être may employées; vary, perhaps dans unto simulatebut de concision chang- celles-ci ne seront pas données When the recharge to the compartmentalici model mais peuventes in êtrethe trouvéeshydrologic dans regime Campana induced (sous bypresse). climate does not vary with time, calculation of the mean change. Under such conditions, the equations giv- age or mean residence time of the water inUne a com copie- du code et le manuel d’utilisation sont mis à disposition par l’auteur en above cannot be used to calculate mean ages. partment or cell is relatively straightforward([email protected]). More complicated relationships must be used; for [6, 7]. the sake of brevity these will not be given here but For the SMC: 3.2.2 canAPPLICATION be found in [22]. AU SYSTEME D’ECOULEMENT DU SITE TEST DU NEVADA

Le modèle à compartiments réalise principalement un519 bilan massique sur le système d’écoulement pour déterminer les flux, les âges des eaux souterraines et les temps de résidence; il faut pour cela un traceur. Un traceur idéal est un traceur qui se déplace à la vitesse de l’eau, facile à échantillonner et à détecter, chimiquement inerte une fois dans la zone saturée, et qui affiche une variabilité spatiale. Les isotopes stables 2H (deutérium) et 18O, partie intégrante de la molécule d’eau, se rapprochent le plus des traceurs idéaux. Nous présentons un modèle à compartiments des systèmes d’écoulements souterrains du site test du Nevada, sud-ouest des Etats-Unis, calibré avec la distribution spatiale du deutérium.

93 modelling

A copy of the code and user’s manual are avail- gitude and covers 19 000 km2 (Fig. 3.1). The area able from the author ([email protected]). is in the southern Great Basin section of the Basin and Range physiographic province, with a topography of north-trending, block-faulted mountain 3.2.2. Application to the Nevada ranges separated by alluvial basins. Elevations in test site flow system the study area range from about 3500 meters (m) The compartmental model essentially performs above mean sea level in the Spring Mountains to a mass balance on the flow system to determine below sea level in Death Valley. flow rates, groundwater ages and residence times; Precipitation, temperature, and plant communi- it needs a tracer to do this. An ideal tracer is one ties in the area are generally a function of eleva- that moves with the velocity of the water, is easy tion. The average annual precipitation increases to sample for and detect, does not react chemically as a function of elevation, from less than 8 cm in once in the saturated zone, and displays spatial vari- Death Valley and the Amargosa Desert to great- ability. The stable isotopes 2H (deuterium) and 18O, er than 70 cm in the upper reaches of the Spring both of which occur as part of the water molecule, Mountains. Annual pan evaporation rates range come about as close to ideal tracers as there are. from 58 to over 71 cm. Therefore, the climate can We present a compartmental model of the Nevada be arid on the valley floors while sub-humid at Test Site groundwater flow system, which under- higher elevations. Most of the precipitation occurs lies a portion of the south-western USA, calibrated during winter as a result of Pacific Ocean fronts, with the spatial distribution of deuterium. but some occurs during summer as high intensity thunderstorms. Winters are short and mild, while 3.2.2.1. Introduction summers are long and hot except at the higher al- Four decades of nuclear testing have served as an titudes. Because of the primarily arid conditions, impetus for numerous studies of the groundwater no major perennial streams exist in the study area flow system beneath the Nevada Test Site (NTS) except local drainage from major springs. and vicinity, south-western USA. A more recent Groundwater is recharged by precipitation in impetus is the possible location of a high-level the higher elevations in the north, east, and south- nuclear waste disposal site at Yucca Mountain, east and by stream-channel infiltration during adjacent to the western boundary of the NTS. the infrequent flow events. It also enters the sys- Possible radionuclide migration to the accessible tem as subsurface inflow from the north and east. environment is a concern; therefore, knowledge Generally, groundwater flows to the south toward of the nature and extent of the NTS groundwater areas of discharge in Oasis Valley, Ash Meadows, flow system, hereafter referred to as the NTSFS, Death Valley and Franklin Lake playa (southern is of paramount importance. We use the envi- Amargosa Desert). ronmental stable isotope ratio 2H/1H (deuterium/ The geology of the region is complex and has hydrogenium) to calibrate a simple compartmen- been well-described [25]. Eleven hydrogeologic tal model of the ground-water system beneath units in the region have been identified, ranging the NTS and vicinity. 2H has the advantage of in age from being stable and essentially conservative once in the saturated zone. Our model is based upon Precambrian to Quaternary; five of these are previous ones [23, 24], but encompasses a larger aquitards, six are aquifers. Of these units, two of area, provides better information on groundwater the most important are the lower clastic aquitard, residence times, and, more importantly, attempts comprised of Precambrian to Cambrian quartzites a transient simulation by treating each compart- and shales, and the lower carbonate aquifer, com- ment as a linear reservoir. prised of Cambrian through Devonian limestones and dolomites. The great thickness of the former, an aggregate of about 3000 m, and areal extent 3.2.2.2. Hydrogeology make it a major element in controlling regional The study area lies between 36 and 38 degrees groundwater flow; the latter (aggregate thickness north latitude and 115 and 117 degrees west lon- of about 4500 m) underlies much of the study area

520 COMPARTMENTAL MODEL APPROACHES TO GROUNDWATER FLOW SIMULATION

in the basin fill overlying a deeper regional system in carbonate and other rocks. In some cases, perched flow systems occur in the basin fill. Downward flow occurs from the basin-fill aquifers and tuff aquitards to the carbonate rocks and interbasin groundwater flow occurs [25]. The NTS flow system is not an isolated regional system but one of a number of such systems in the carbonate-rock province of Nevada, Utah and adjacent states [26–28].

3.2.2.3. Model development and calibration Two sets of δ2H values (versus the VSMOW standard; see for definitions Volume I) were used: ‘local’ values from high-altitude springs and shallow wells with signatures of –90 to –102‰, representing recharge water; and ‘regional’ values from large, low-altitude springs and deep wells with signatures of –98 to –117‰, representing Fig. 27.1. The study area and model network, with the flow system groundwater. The latter were used the Nevada Test Site indicated by heavy line. in model calibration and, along with the hydrogeology (e.g., hydrostratigraphy, structure), used to subdivide the flow system into 30 compartments and is the major conduit for regional groundwater or cells (Fig. 3.1). The difference between the re- flow in the area. The upper clastic aquitard and gional and local δD values can be explained by upper carbonate aquifer are similar to the lower relatively depleted subsurface inflow from higher clastic aquitard and lower carbonate aquifer, re- latitudes and possibly past climatic regimes en- spectively, but are less important because of their tering the study area and becoming gradually en- lower thicknesses and more limited areal ex- riched along flow paths by water recharged within 2 tents. Other hydrogeologic units worth noting are the study area. The overall trend of regional δ H the basin-fill aquifer, important in the Amargosa values is gradual enrichment from north to south Desert, and the various volcanic-rock aquifers with the north-west area being the most deplet- and aquitards (Tertiary tuffs and associated lithol- ed and the south-west being the most enriched. 2 ogies), which are up to 4000 m thick in the west- Some recharge δ H values were estimated from 2 ern portion of the study area. the δ H values in precipitation. Complete data can be found in [24]. The area’s structural geology has played a major role in shaping the hydrogeology of the region. Initial estimates of the SBRV (both as recharge Mesozoic and Tertiary folding and thrust fault- and subsurface inflow) were based on previous ing significantly deformed the Precambrian and data [29–31]. The initial recharge estimates were Palaeozoic rocks; Tertiary normal block faulting used as starting points and references during cali- produced classic basin and range topography. bration. These same forces were responsible for fractur- Flow routing inputs to the model are expressed as ing the aforementioned carbonate rocks, produc- the percentage of total discharge from each cell ing a very transmissive regional aquifer. The in- to each of its receiving cells or out of the model termontane basins filled with sediments derived boundaries. Initial flow routing values were based from the surrounding mountain ranges, in many on the aforementioned publications on the hydro- cases producing a two-tiered flow system charac- geology of the area. Total discharge from a cell teristic of the region: a shallow system developed was calculated by Eq. 3.5.

521 modelling

The parameter VOL equals the volume of active water in a cell. Cell areas were measured with a planimeter from a 1:250 000 scale map. An effective porosity of 2%was used for the carbonate aquifer, which is the mean value of 25 samples presented in [25]. The 5% effective porosity measured for the welded tuff aquifer was chosen for the volcanic strata based on the observation that most of the flow is transmitted through the welded tuffs; the higher porosities of the non-welded tuffs were not used. The upper clastic aquitard (cells 13 and 19) was assigned an effective porosity of 4%, the mean values of 22 samples presented in. The basin-fill aquifer was characterised as being generally poorly sorted and was assigned an effective porosity of 15%. The model utilised a 100-year iteration interval and the MMC (modified mixing cell) option. To simulate climate change we increased recharge to the model by 50% and decreased its δ2H by 5‰ during the period 23 000 to 10 000 years before present [32]. Model calibration was accomplished by adjusting SBRV and intercellular flow routing values until the difference between the observed and simulated δ2H values was within ±1‰, the analytical error for 2H. Fig. 27.2. Recharge rates in mm/year.

3.2.2.4. Results and discussion The areal distribution of average annual recharge 20 are thought to divide the Alkali Flat-Furnace is shown in Fig. 3.2. In general, higher recharge Creek and Ash Meadows subbasins. The highest rates are present in the northern region of the modflow rates correspond to a major potentiometric el with lower rates in the southern; however, qua- trough in the carbonate aquifer immediately up- si-isolated areas which do not follow this trend gradient (cell 21) from the Ash Meadows area have the highest recharge rates within the model [33], the terminus of the Ash Meadows subbasin area. The high recharge areas in Fig. 3.2 corre- (cell 27), and the constriction and termination of spond to areas of relatively enriched δ2H values: the Alkali Flat-Furnace Creek subbasin (cells 26 eastern Pahute Mesa (cell 8); Stockade Wash (cell and 30). 12); Fortymile Canyon/Wash (cell 18); the Spring Mountains (cell 28); and the Sheep Range (cell Other regional models of the area [34, 35] sug- 23). gested the possibility of, but did not simulate, The total flow rate through the system averages subsurface inflow from northern and north-west- 58.9 × 106 m3/a. Broad divisions of average flow ern areas. Therefore, all flow through these pre- rates are shown in Fig. 3.3. The lowest flow rates vious models is comprised of locally-recharged correspond to cells that are dominated by the pres- water and not a combination of locally-recharged ence of an aquitard (cells 13 and 19), cells im- and underflow water as in the present model. Our mediately downgradient from an aquitard (cells model indicates that a substantial amount (40%, 9, 14, 20, and 25), and a cell which is a moder- or 23.6 × 106 m3/a) of the average total system ate recharge area (cell 3). Cells 3, 13, 19, and throughflow is derived from subsurface inflow.

522 COMPARTMENTAL MODEL APPROACHES TO GROUNDWATER FLOW SIMULATION

Fig. 27.4. Mean groundwater residence times in years Fig. 27.3. Volumetric flow rates in 106 m3 per year.

Mean residence times are shown in Fig. 3.4. Residence time distributions (RTDs) provide The youngest values are found in the cells with more information on the cells’ waters than simply high recharge versus subsurface flow from up- mean or median values [7]. Cumulative RTDs for gradient cells (cell 3, 18, and 28). Cells 8 and 12 six regions are shown in Figs.3.5 and 3.6. Fig. 3.5 have relatively young waters due to their high re- shows the RTDs [F(N)] for cells 16 (Oasis Valley/ charge rates, while cell 24 receives its relatively Beatty Wash), 17 (Crater Flat) and 18 (Fortymile young water from cells 28 and 23. The oldest Canyon/Wash); Fig. 3.6 shows F(N) for cells 28 mean residence times are found in the upper clas- (north-western Spring Mountains), 29 (Furnace tic aquitard cells (cells 13 and 19), downgradient Creek Ranch region of Death Valley) and 30 from aquitards (cells 9, 14, and 15) and in areas (Franklin Lake playa and vicinity). where most of the flow originates directly or in- Fortymile Canyon/Wash has the highest areally- directly as underflow (cells 5, 15, 21, 22, 27, 1, distributed recharge rate (29.4 mm/year) and and 6). A decrease in mean residence times along a volumetric flow rate second only to the Spring flow paths occurs in many areas and is caused by Mountains. Most of the groundwater beneath this relatively large amounts of recharge in the direc- region is very young -- 60% of the water is fewer tion of flow. The means represent all of the water than a few thousand years old. Contrast this fact in a given cell and may include a mixture of very with the other two regions on Fig. 3.6 – Oasis young water recharged locally and very old water Valley and Crater Flat – where 60% of the waters received from upgradient cells. are at least 15 000 years old. Both of these areas

523 modelling

an area of approximately 19 000 km2 in southern Nevada-California, USA. This model consists of a network of 30 compartments delineated through the integrated interpretation of general hydrogeologic characteristics of the area and deuterium data from approximately 300 sites. The model shows the significant contribution of subsurface inflow −40% of the average total system throughflow- to the NTS regional groundwater system. This flow enters from the north and east. The eastern subsurface inflow is undoubt- Fig. 27.5. Cumulative groundwater residence time dis- edly from the White River regional groundwater tribution F(N) for cells 16 (Oasis Valley/Beatty Wash), flow system. Previous workers [36] estimated 17 (Crater Flat), and 18 (Fortymile Canyon/Wash). that about 7.4 × 106 m3/a flowed from Pahranagat Valley (part of the White River flow system and just east of cell 5) to the NTS system; an earlier compartmental model [10] showed as much as 5.4 × 106 m3/a discharged as underflow from Pahranagat Valley. We estimate the underflow to be between 11.1 × 106 m3/a and 16.8 × 106 m3/a. High recharge areas within the flow system boundaries are the Fortymile Canyon/Wash-Stockade Wash area, the Spring Mountains, the Sheep Range, and Pahute Mesa. Recharge accounts for Fig. 27.6. Cumulative groundwater residence time 60% of the average system throughflow. distribution F(N) for cells 28 (north western Spring Mountains), 29 (Furnace Creek Ranch area), and 30 The model provides detailed information on (Franklin Lake playa). groundwater residence times. The position of a region in the flow path does not necessarily cor- are minor recharge areas; indeed, Oasis Valley is relate with mean residence time as recharge can more important as a discharge area. mask the effects of old subsurface inflow to a region. Fig. 3.6 contains the cumulative RTDs of the area’s major recharge area (Spring Mountains – cell 28) and two major discharge areas (Furnace 3.3. Constraining regional Creek Ranch – cell 29; and Franklin Lake playa groundwater flow models – cell 30). These latter two cells are major dis- with environmental isotopes charge regions and have much older ground and a compartmental mixing- waters. Note that even though cell 30 is farther cell approach downgradient than cell 17 (Crater Flat) its F(N) is 3.3.1. Introduction shifted slightly to the left relative to Crater Flat’s, indicating slightly younger waters. This appar- Numerical groundwater flow models such as ent discrepancy is easily explained by noting that MODFLOW [37] are often used to interpret hy- Franklin Lake playa receives relatively young draulic head data and physical properties (e. g., water (via other cells) from the Spring Mountains porosity, conductivity) of regional aquifer sys- and Fortymile Canyon/Wash. tems. Once calibrated, these models can provide important quantitative information about groundwater recharge, lateral flow and leakage between 3.2.2.5. Concluding remarks aquifers. However, proper calibration of regional We used a deuterium-calibrated mixing-cell mod- groundwater models usually requires a greater de- el to simulate regional groundwater flow beneath gree of spatial parameterisation than is available

524 les communications entre les aquifères sont déterminées en modifiant les flux entre les cellules du modèle, jusqu’à ce que les concentrations en traceurs simulées soient égales à celles observées sur le terrain. Dans la partie suivante, nous présentons une nouvelle approche pour interpréter quantitativement les données de traceurs environnementaux et contraindre les modèles de flux d’eaux souterraines régionaux. Alors que les précédents modèles CMC nécessitaient que les flux entre les cellules du modèle soient modifiés manuellement, l’approche adoptée ici est d’utiliser le modèle de flux d’eaux souterraines MODFLOW de la U.S.Geological Survey (McDonald et Harbauch 1988) pour obtenir des flux intercellulaires et des hauteurs COMPARTMENTAL MODEL APPROACHpiézométriques.ES TO GROUNDWAT De EplusR F LOamplesW SIMUL détailsATION du développement et de l’application du modèle sont présentés dans Harrington et al. (1999). from field data. Hence, many of the input param- aquifer undergoes complete mixing with inputs eters for these models have to be estimated, thus3.3.2 of waterEQUATIONS and tracer DE mass BASE (e.g., via recharge or reducing confidence in the final calibrated model. lateral inflow) over a designated time step. This L’approche CMC est basée sur l’hypothèse que chaque compartiment ou cellule de mélange The benefits of incorporating environmental trac- assumption can be justified providing the size of d’un modèle est le siège d’un mélange complet entre les apports d’eau et la masse du traceur er techniques into hydrogeological investigations individual mixing cells and time steps are chosen are well known. For example, the stable isotope(par ex.,sensibly. via la I frecharge one also ou assumes des apports that volumetric latéraux) au-delàin- d’un pas de temps choisi. Cette composition (2H/1H and 18O/16O) of water molhypothèse- flows peut over être a time justifiée step si are la negligibletaille des comparedcellules individuelles et les pas de temps sont ecules is often used to identify palaeo-rechargejudicieusement to the volume choisis. of Sithe on mixingsuppose cell, égalemen and changest que les apports volumétriques au cours d’un in fluxes of tracer into each cell are linear across water in aquifers by comparing groundwaterpas de temps sont négligeables comparés au volume de la cellule de mélange, et que les compositions with those of present-day rainfall time steps, then the following equation may be modifications des flux de traceur à l’intérieur de chaque cellule sont linéaires au cours des pas [38, 40]. Radioactive isotopes such as 14C and 36Cl used to determine the concentration of tracer in are commonly employed to infer mean groundde- temps,a cell alors after al’équation certain time, suivante t [16]: peut être utilisée pour déterminer la concentration du water residence times in regional aquifers [41,traceur dans une cellule après un certains temps, t (Harrington et al. 1999): 0 1 2 2 43]. Nevertheless, environmental tracer data are c = c + c t + c t (3.13 = 0 + 1 + 2 2 generally only used in a qualitative or semi-quan - c c c t c t (3.13) where: titative manner. Hence, there is a need to developavec: and apply techniques for interpreting tracer data c is the concentration of tracer in the ‘mixed’ –3 simultaneously with hydrogeologic data to pro- c cell,la concentration [ML–3]; du traceur dans la cellule "homogénéisée", [ML ] vide more quantitative information about ground- c0 is the initial concentration of the cell at t=0, c0 la concentration initiale de la cellule à t = 0, [ML–3] water processes such as lateral flow and leakage. [ML–3]; n The compartmental or mixing-cell (CMC) ap- 0 0 0 0 0 ∑Qi (ci − c ) − λV c proach is one of the most straightforward ways c1 = i=1 , [ML–3T–1] ; in which environmental tracer, hydraulic and hy- V 0 drogeologic data can be analysed simultaneously. n isle thenombre number d’apports of inputs à la to cellule the cell; Used by numerous authors over the past three 0 c i are the input concentrations to the cell at decades [8, 11–13, 16, 17, 44–47], the CMC ap- 0 –3 c les concentrations–3 des apports à la cellule à t = 0, [ML ] proach uses linear mass-balance equations to sim- i t=0, [ML ]; Q 0 are the input fluxes to the cell at t=0, [L3T–1]; ulate the transport of conservative or radioactive i0 3 –1 Qi les débits des apports à la cellule à t =–1 0, [L T ] tracers through an aquifer system. Quantitative λ is the decay constant of the tracer, [T ]; 0 3 estimates of physical processes such as lateral λV la is constante the volume de ofdésintégration the cell at t=0, du [traceur,L ] [T–1] flow and leakage between aquifers are determined c2=0 3 by altering fluxes between model cells until simu- Vn le volumen de la cellule à t = 0, [L ] Q0 (c1 − c1) + Q1(c0 − c0 ) − λ(V 1c0 +V 0c1) −V 1c1 lated tracer concentrations match those observed ∑ i i ∑ i i c2 = i=1 i=1 [ML–3T–2] in the field. 0 104 V [ML–3T–2]; In the following text, we present a new approach1 –3 –1 ci les modifications des concentrations de l’apport par unité de temps, [ML T ] for quantitatively interpreting environmental 1 c i are the changes in input concentrations per tracer data and constraining regional groundwa- 1 –3 –1 3 –2 Qi les modificationstime, [ML desT flux]; d’entrée par unité de temps, [L T ] ter flow models. Whereas previous CMC models 1 Q i are the changes in input fluxes per time, required the fluxes between model cells to be al- 1 3 –1 V la modification3 –2 du volume de la cellule par unité de temps, [L T ] tered manually, the approach adopted herein is [L T ]; 1 to use the U.S.Geological Survey’s groundwater V is the change in cell volume per time, 3 –1 flow model MODFLOW [37]to obtain inter-cel- [L T ]; lular fluxes and aquifer heads. Further detailsL’Eq. of3.13 Eq.permet 3.13 allowsde déterminer the concentration la concentration of each de inter chaque- cellule de mélange the model development and application areinterconnectée pre- connected dans un mixing domaine cell du in modèle, a model sous domain des conditions to be d’écoulement transitoire sented in Ref. [16]. et/ou sous determineddes concentrations under d’entrée transient variable flow s conditionsdu traceur. Ceci and/ est particulièrement utile pour modéliseror varying les systèmes tracer aquifèresinput concentrations. régionaux dans Tlesquelshis is le gradient hydraulique a particularly useful for modelling regional aquifer changé au cours d’une longue période de temps (par ex., 103–105 ans). 3.3.2. Governing equations systems in which the hydraulic head gradient has The CMC approach is based on the assumption changed over a long period of time (e.g., 103–105 that each compartment or mixing cell in a 3.3.3model years).GRANDES LIGNES DU MODELE, DONNEES D’ENTREE ET PROCEDURE DE CALIBRATION

Le modèle CMC décrit par l’Eq. 3.13 a été directement relié525 à MODFLOW, un modèle d’écoulement en différences finies développé par l’U.S.Geological Survey (McDonald et Harbauch 1988). Le domaine du modèle et la configuration des cellules des éléments de MODFLOW et du CMC pour une application particulière doivent être la même pour faciliter leur liaison. Au départ, les données hydrogéologiques, telles que les valeurs mesurées ou estimées de la porosité et de la perméabilité de l’aquifère, sont précisées dans MODFLOW, avec les taux de recharge, les conditions aux limites et le choix du pas de temps. MODFLOW est alors exécuté pour fournir à la fois les niveaux statiques et les flux intercellulaires (horizontaux et verticaux) pour chaque cellule du modèle de l’aquifère. La concentration initiale d’un traceur (chimique ou isotopique) dans chaque cellule, et sa concentration dans l’eau de recharge qui entre dans chaque cellule (à chaque pas de temps), sont spécifiées dans un fichier d’entrée pour le modèle CMC. Le modèle CMC est alors exécuté afin d’obtenir une distribution des concentrations du traceur à travers le système aquifère en utilisant les flux et les niveaux obtenus par MODFLOW. Le modèle combiné est calibré en utilisant une procédure itérative qui modifie les paramètres hydrogéologiques d’entrée estimés pour MODFLOW jusqu’à obtenir des niveaux et des concentrations simulés équivalents aux distributions observées. On obtient finalement un modèle d’écoulement régional calibré des eaux souterraines à partir duquel des estimations quantitatives de processus tels que les écoulements latéraux et les transferts verticaux peuvent être obtenues avec une fiabilité meilleure que si on n’avait pas utilisé les données des traceurs de l’environnement.

105 modelling

Fig. 27.7. Location of transect A–A’, Otway Basin, South Australia.

3.3.3. Model design, input data and tain a distribution of tracer concentrations through calibration procedure the aquifer system using the fluxes and head data obtained from MODFLOW. The CMC model described by Eq. 3.13 has been directly linked to MODFLOW, a finite-differ- The combined model is calibrated using an it- ence groundwater flow model developed by erative procedure, whereby the estimated hy- the U.S.Geological Survey [37]. The model do- drogeologic input parameters for MODFLOW main and cell configuration of the MODFLOW are altered until the simulated aquifer heads and and CMC components of a particular applica- the simulated tracer concentrations match the ob- tion must be the same to facilitate the linkage served distributions. The end result is a calibrated between the two models. Initially, hydrogeologic regional groundwater flow model from which data such as field and estimated values of aqui- quantitative estimates of processes such as lateral fer porosity and hydraulic conductivity are speci- flow and vertical leakage of groundwater can be fied in MODFLOW, along with recharge rates, obtained with greater confidence than if the envi- boundary conditions and time step information. ronmental tracer data had been excluded. MODFLOW is then executed to provide both hydraulic heads and inter-cellular fluxes (horizon- 3.3.4. Application to the Otway tal and vertical) of water for each model cell in basin, south Australia the aquifer. The combined hydraulic/environmental tracer The initial concentration of a tracer (chemical or approach outlined above has successfully been isotopic) in each cell, and the tracer concentration applied to the Otway Basin of South Australia in recharge water entering each cell (at each time (Fig. 3.7) to quantify mixing between two re- step), are specified in an input file for the CMC gional Tertiary aquifers: the Gambier unconfined model. The CMC model is then executed to ob- limestone aquifer and the Dilwyn confined sand

526 COMPARTMENTAL MODEL APPROACHES TO GROUNDWATER FLOW SIMULATION aquifer [16]. Rates of leakage from the Gambier Radiocarbon (14C) was chosen as the tracer be- aquifer into the underlying Dilwyn aquifer are cause it has a half life (~ 5730 years) that enables required to determine sustainable rates of extrac- hydrologic processes to be traced over time scales tion for the relatively fresh (< 1000 mg/L) Dilwyn commensurate with the simulation period for groundwater resource. the model (27 000 years). Over the simulation period, the elevation and horizontal position of Both the Gambier unconfined and Dilwyn con- the western boundary condition was varied to ac- fined aquifers are connected to the sea in the south- count for eustatic sea level changes. The 14C con- west. Hence, the flow regime of the groundwater centration of recharge water was also varied over in each aquifer has varied over the last 30 000 the simulation period. years (and beyond) due to eustatic sea level variations [42]. Potential leakage between the two MODFLOW simulations were preformed initial- aquifers occurs where relative head gradients fa- ly until modelled aquifer heads matched the ob- vour upward or downward movement. To the east served distribution (Fig. 3.9a). Intercellular fluxes of the zero head difference (ZHD, Fig. 3.7), and heads from the calibrated MODFLOW model the water table in the unconfined aquifer is higher were then used as input data for the CMC model than the potentiometric head in the confined aqui- to simulate the observed distribution of radiocarbon concentrations in the Dilwyn confined fer. Hence there is potential for downward leak- aquifer. From the plot shown in Fig. 3.9b, it was age (confined aquifer recharge) to occur in this obvious that the calibrated MODFLOW model area. Conversely, to the west of the ZHD there was not accounting for enough leakage of rela- is potential for upward leakage (confined aquifer discharge). Because eustatic sea level variations have altered the hydraulic head distributions within both aquifer systems in the past, the position of the ZHD, and hence size and position of the potential recharge and discharge zones, has also varied. The model domain selected for application of the hydraulic/tracer approach was a two-dimensional vertical slice along a transect (A-A’, Fig. 3.7) that runs perpendicular to potentiometric contours for both the unconfined and confined aquifers. The slice was divided into 30 columns, each of length and width 8660 m (Fig. 3.8).

A ZHD coast A'

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 200 )

D 100 A H ?

m 0 ( ?

on SEA i t -100 ? ? v a ? ? e ?

E l ? -200 ? ? -300 0 50 Gambier Aquifer Dilwyn Fig. 27.9. Original and final input parameters used Aquitard (Unconfined) Aquifer kilometres (Confined) to model the observed aquifer head distributions in the Otway Basin. Hydraulic conductivity values relate Fig. 27.8. Hydrogeological cross-section along tran- to the Gambier unconfined aquifer; Leakance (Kv/ sect A–A’, Otway Basin. The transect was divided thickness) values relate to the regional confining aqui- into 30 cells of length and width 8660 m for both tard; and transmissivity values relate to the Dilwyn the MODFLOW and CMC model applications. confined aquifer.

527 modelling tively high-14C water from the unconfined aquifer area where the model was unable to ideally match into the confined aquifer. Hence the MODFLOW the trend in observed radiocarbon data is in those model had to be re-calibrated using the iterative cells beyond the present day ZHD (116 kilometres procedure outlined above. This was achieved from A) towards the coast. One explanation for by increasing the vertical hydraulic conductiv- this is that our model has used values of hydrau- ity of the regional confining aquitard that sepa- lic conductivity for the Dilwyn confined aquifer rates the two aquifers, and altering the horizontal system that are higher than those in reality. This hydraulic conductivity of both aquifers. Fluxes would result in modelled flow rates being higher and heads from the final calibrated MODFLOW and hence modelled radiocarbon concentrations model (Fig. 3.9c) provided a much better match decreasing less rapidly. A previous investigation between the observed 14C distribution and that ob- [48] has suggested that the confined system may tained from the CMC model (Fig. 3.9d). be receiving upward leakage of relatively ‘older’ water from an underlying Cretaceous aquifer near Although the peak 14C concentration was not the coast. This would result in the measured 14C modelled exactly, we considered it more impor- activities being lower than the modelled output, tant to model the shape of the observed 14C distri- as observed. bution rather than the absolute values. The reason for this is that observed concentrations are from Using the combined hydraulic/environmen- the uppermost portion of the confined aquifer and tal tracer approach in the Otway Basin has re- hence do not represent the ‘average’ concentra- sulted in quite different estimates of hydraulic tion of the entire aquifer thickness, as calculated conductivity being used in the initial and final in the CMC model. From the comparison the only MODFLOW calibrations, particularly for verti-

140 35 Observed (Uncon ned) Modelled 120 Modelled (Uncon ned) 30 Observed Observed (Con ned) 100 Modelled (Con ned) 25

80 20

60 15 Head (m AHD) 40 Carbon-14 (pmc) 10

20 5

0 0 0 100 200 300 a) 0 50 100 150 200 Distance from A (km) b) Distance from A (km)

140 35 Observed Observed (Uncon ned) Modelled 120 Modelled (Uncon ned) 30 Observed (Con ned) 100 Modelled (Con ned) 25

80 20

60 15 Head (m AHD) 40 Carbon-14 (pmc) 10

20 5

0 0 0 100 200 300 0 50 100 150 200 c) Distance from A (km) d) Distance from A (km)

Fig. 27.10. (a, b) Modelled and observed aquifer heads and radiocarbon distribution in the Dilwyn confined aquifer obtained using the initial MODFLOW model. (c, d) Modelled and observed aquifer heads and radiocarbon distribution in the Dilwyn confined aquifer obtained using the final MODFLOW model.

528

Dans des conditions de mélange total pour chaque cellule, les équations récursives pour chaque traceur et pour un intervalle de temps Δt peuvent être représentées par:

Masse de traceur (t) = Masse de traceur (t – Δt) + Flux de masse entrant(t) – Flux de masse sortant (t)

L’équation ci-dessus représente la masse du traceur à chaque pas de temps en fonction des flux de masse entrant et sortant. Cette méthode a été largement utilisée pour simuler le transport d’isotopes et de substances chimiques réactives (Campana et Mahin 1985; Yurtsever et Payne 1978; Van Ommen 1985; Simpson et Duckstein 1976).

COMPARTMENTAL MODEL APPROACHES TO GROUNDWATER FLOW SIMULATION

cal conductivity between the aquifers (Fig. 3.10). The rates of groundwater flow and inter-aquifer mixing obtained from the final model are therefore more realistic than those calculated from the initial model. For example, leakage from 4 3 the Gambier unconfined aquifer into the Dilwyn 2 1 confined aquifer was determined to be less than 1 ∆z mm/a along transect A-A’ using the initialFig. model, 3.11. Représentation du système hydrogéologique par des cellules volumétriques interconnectées. whereas a range in leakage rates of between 2 ∆ ∆y x and 9 mm/a was obtained from the final model. The latter estimates of leakage rates compareL’équation well partielle différentielle représentant le transport par advection (en négligeant la Fig. 27.11. Representation of the hydrogeologic sys- with previous estimates of around 1 mm/adispersion, for latem diffusion, by interconnected l’adsorption, volumetric les réactions cells. et les dégradations) peut être exprimée par: sites near the ZHD where the head difference, and hence potential for leakage, between the aquifers would be lower [49]. ∂C ⎛ ∂()qxCx ∂()qyCy ∂()qzCz ⎞ n = −⎜ + + ⎟ (3.14) (3.14) One of the greatest difficulties encountered when ∂t ⎝ ∂x ∂y ∂z ⎠ the CMC model was applied in transient mode for 27 000 years was the lack of informationoù about C est la Tconcentrationhe partial dudifferential soluté (M/L equation3), t le temps describing (T), n la porosité efficace, et q le débit tracer input functions. 14C was considered to be the advective transport (neglecting dispersion, spécifique/dans la direction des x,y,z croissant(L/T). L’apport et la sortie peuvent aussi être the most robust tracer for regional-scale field ap- diffusion, adsorption, reaction and decay) can be intégrés dans l’équation. La solution de cette équation partielle différentielle à trois plications because variations in the 14C concentra- expressed as: dimensions (3-D) peut être obtenue par de nombreux modèles numériques basés sur les tion of the atmosphere were probably more global in which C is solute concentration (M/L3), t is méthodes Eulérienne ou Lagrangienne. Avec l’approche Eulérienne l’équation de transport est than most other environmental tracers. time (T), n is effective porosity, and q is the spe- résolue aveccific un maillage discharge fixé in comme the direction pour lesof increasingméthodes en x,y,z différences finies (DF) et éléments 3.4. Mixing-cell model finis (EF). (LaL/T taille). The du source maillage and etsink les canpas alsode tempsbe included doivent être suffisamment petits afin for the simulation of d’éviter la dispersionin the above numérique equation. et Tleshe oscillasolutiontions. of thisAvec three- la méthode Lagrangienne l’équation environmental isotope dimensional (3-D) partial differential equation 111 transport can be achieved by numerous numerical models based on either Eulerian or Lagrangian methods. 3.4.1. Introduction In the Eulerian approach the transport equation is solved with a fixed grid such as in finite-dif- The mixing-cell method is the simplest solution ference (FD) or finite-element (FE) methods. In to the advective transport equation. The method order to avoid numerical dispersion and oscilla- is based on the discretisation of the flow domain tions, restrictively small grid size and time steps into a finite number of cells (compartments) are required. In the Lagrangian method the equa- (Fig. 3.11) in which perfect mixing for the tracer tion is solved in either a deforming grid or de- takes place over discrete time intervals. Under forming co-ordinate in a fixed grid through par- complete mixing conditions in each cell, the re- est résolue ticlesoit tracking.avec un Thismaillage method variable is free soit of avec numerical des coordonnées variables dans un cursive state equations for each tracer for a Dt dispersion. time interval can be represented by: maillage fixe avec un repérage des particules. Cette méthode n’est pas affectée par la dispersion numérique.In a continuous flow domain, the mixing-cell al- Tracer Mass(t) = Tracer Mass (t–Dt) + Incoming gorithm is either an explicit or implicit backward Dans un domaine d’écoulement continu, l’algorithme de la cellule de mélange est une Mass Flux(t) – Outgoing Mass Flux(t) finite-difference approximation of the advec- approximation,tive enterm différences in the solute finies rétrogradestransport equation, implicite [50]. ou explicite, du terme représentant The equation above represent the mass of thel’advection trac- Tdanshe 1-l’équationD advective de transport solute transport de masse equation (Bajracharya can et Bary 1994). L’équation 1-D er at each time step as a function of the incomingde transport be de expressedmasse par as:advection peut être exprimée par: and outgoing mass flux. The method has been used widely for the simulation of isotopic and ∂C qx ∂C = − (3.15) (3.15) chemically-reactive transport [9, 11, 14, 17]. ∂t n ∂x où C = C(x, t) est la concentration et (qx/n) la vitesse moyenne des eaux souterraines. Le terme de cette équation représentant l’advection peut 529être approché par la formule en différence finie suivante:

t+Δt t qxΔt t t Ci = Ci + ()Ci−1 − Ci (3.16) nΔx où (qxΔt/nΔx) est le nombre de Courant qui devrait être inférieur à 1 pour rendre compte de la stabilité de la solution en différence finie. Cette formule fournit une solution identique à la solution analytique de l’équation de transport de masse par advection (Van Ommen 1995). L’Eq. 3.14, pour chaque cellule dite (i,j,k), peut être approchée par les valeurs des concentrations au niveau de la limite des cellules adjacentes par:

C t+Δt − C t q C − q C n i, j,k i, j,k = − i, j+1/ 2,k i, j+1/ 2,k i, j−1/ 2,k i, j−1/ 2,k Δt Δx j q C − q C − i+1/ 2, j,k i+1/ 2, j,k i−1/ 2, j,k i−1/ 2, j,k (3.17) Δyi q C − q C − i, j,k+1/ 2 i, j,k+1/ 2 i, j,k−1/ 2 i, j,k−1/ 2 Δz k

où Δxj, Δyi, Δzk sont les dimensions des cellules, et j+1/2, i+1/2, et k+1/2 indiquent les interfaces des cellules perpendiculaires aux directions x, y, z (Fig. 3.12). La concentration à l’interface de la cellule entre deux nœuds voisins suivant une direction particulière est considérée égale à la concentration au niveau du nœud amont suivant la même direction (Bear 1979; Zheng et Bennett 1995). Cette approche est appelée le système amont pondéré et fournit des solutions non affectées par l’oscillation:

⎧Ci, j−1,k if qi, j-1/2,k > 0 C = ⎨ (3.18) i, j−1/ 2,k C if q < 0 ⎩ i, j,k i, j-1/2,k

112 est résolue soit avec un maillage variable soit avec des coordonnées variables dans un maillage fixe avec un repérage des particules. Cette méthode n’est pas affectée par la estdispersion résolue numérique. soit avec un maillage variable soit avec des coordonnées variables dans un Dansmaillage un fixedomaine avec d’écoulementun repérage descontinu, particules. l’algorithme Cette méthodede la cellule n’est depas mélange affectée est par une la dispersion numérique. est résolue soitapproximation, avec un maillage en différences variable finiessoit avecrétrogrades des coordonnées, implicite ou variables explicite, dans du terme un représentant maillage fixe Dansl’advectionavec unun domainerepérage dans l’équation d’écoulementdes particules. de transport continu, Cette de masseméthodel’algorithme (Bajracharya n’est de pas la etaffectéecellule Bary 1994). depar mélangela L’équation est 1-Dune dispersion numérique.estapproximation,de transportrésolue desoit masseen avec différences parun advectionmaillage finies peut variablerétrogrades être expriméesoit, implicite avec par: des ou coordonnées explicite, du variablesterme représentant dans un modelling maillagel’advection fixe dans∂C avec l’équation qun∂ Crepérage de transport des particules. de masse (BajracharyaCette méthode et Bary n’est 1994). pas affectéeL’équation par 1-D la Dans un domaine d’écoulement= − continu,x l’algorithme de la cellule de mélange est une (3.15) dispersionde transport numérique. de masse par advection peut être exprimée par: approximation, en différences∂t finiesn rétrogrades∂x , implicite ou explicite, du terme représentant where C = C(x,t) is the concentration and (qx/n) l’advection dansDans l’équation un domaine∂ deC transport qd’écoulement∂C de masse continu, (Bajracharya l’algorithme et Bary 1994).de la L’équationcellule de 1-Dmélange est une où C = C(x,is t) theest= − averagela x concentration groundwater et ( qxvelocity./n) la vitesse The advec moyenne- des eaux souterraines.(3.15)i,j,k Le-1 de transport determe approximation, masse de par cette advectiontive∂t enterméquation différencesn ofpeut∂ thisx êtrereprésentant equation finiesexprimée rétrogrades can l’advecpar: be approximatedtion, implicite peut êtreouby explicite, approché du par terme la représentantformule en l’advection dans l’équation de transport de masse (Bajracharya et Bary 1994). L’équation 1-D ∂Cdifférenceoù C q= C∂(C xfinie,the t) estfollowingsuivante: la concentration finite-difference et (qx/ nscheme:) la vitesse moyenne des eaux souterraines. Le i-1,j,k = − x (3.15) ∂ttermede transport nde∂ xcette de masse équation par advectionreprésentant peut l’advecêtre expriméetion peut par: être approché par la formule en t+Δt t qxΔt t t Ci = Ci + ()Ci−1 − Ci (3.16) (3.16) différence finie∂C suivante:q ∂Cn Δx où C = C(x, t) est la concentration= − x et (qx /n) la vitesse moyenne des eaux souterraines. Le (3.15) ∂t n ∂x terme de cetteoù équation(qxΔt/nΔ wherexreprésentant)t +estΔt le (qx nombret Dt/nq xl’advecΔD tx)de Courantt istion the t peutCourant qui devraitêtre numberapproché être inférieur and par laà 1formule pouri, jrendre-1 ,ken comptei,j ,dek la Ci = Ci + ()Ci−1 − Ci (3.16) différence finie où suivante: C = C(x ,should t) est bela lessconcentrationn Δthanx 1 to affirm et (qx /then) finite-differencela vitesse moyenne des eaux souterraines. Le stabilité de la solution en différence finie. Cette formule fournit une solution identique à la i,j+1,k termesolution de analytique cettesolution équation de is l’équationstable. représentant This de scheme transport l’advec provides detion masse thepeut par same êtreadvection approché (Van parOmmen la formule 1995). en toù+Δt (qxΔt t/nqΔxxΔ)t est tle nombret de Courant qui devrait être inférieur à 1 pour rendre compte de la Ci différence= Ci + finiesolution() Csuivante:i−1 −asC ithe analytic solution of the advective (3.16) L’Eq.stabilité 3.14, den Δsolutelapourx solution transportchaque en différence celluleequation dite [14].finie. (i,j,k Ce),tte peutformule être fournit approchée une solutionpar les identique valeurs àdes la concentrationssolution analytiquet+ auΔt niveau det l’équationq dexΔ tla limitet de transport dest cellules de masse adjacentes par advection par: (Van Ommen 1995). où (qxΔt/nΔx) est le nombreEq.Ci de 3.14 Courant= C ati + any qui cell, ()devraitCi −say1 − C( i,j,kêtrei ) inférieur can be approximat à 1 pour rendre- comptei+1,j, kde la (3.16) nΔx stabilité de la L’Eq. solution 3.14, en eddifférencepour byt+Δ tthechaque concentration t finie. cellule Cette formule ditevalues (i,j,k atfournit ),the peut neighbour une être solution approchée- identique par à lesla valeursi,j, k+des1 Ci, j,k − Ci, j,k qi, j+1/ 2,kCi, j+1/ 2,k − qi, j−1/ 2,kCi, j−1/ 2,k solution analytiqueconcentrationsoù (qx deΔ tl’équation/nΔingxn) est aucell leniveau de facesnombre transport deas:= la−de limite deCourant masse des quiparcellules devraitadvection adjacentes être (Van inférieur par:Ommen à 1 pour1995). rendre compte de la Δt Δx j stabilité de la solutiont+Δt t en différence finie. Cette formule fournit une solution identique à la L’Eq. 3.14, pour chaque celluleC − Cdite (i,j,kqq+ ), + peutCC+ +être− −qapprochéeq− − CC− −par les valeurs des solution analytiquen i, j,k de l’équationi, j,k =−− i 1i,/ j2de,1j/,k2 transport,k i i1,/j2,1j/,k2,k dei masse1i,/j2,1j/,k2,k ipar1i,/j2, 1advectionj/,k2,k (Van Ommen 1995).(3.17) concentrations au niveau de la limiteΔt des cellules adjacentesΔΔy par:x Fig. 27.12. Cell (i,j,k) and indices for the adjacent i j cells. L’Eq.t+Δt 3.14,t pour chaque celluleq diteC (i,j,k− ),q peutC être approchée par les valeurs des C − C q C qi,+−j1,/k2q+,1j/,k2Ci,+j1,/kC2+,1j/,k2 − qi,−j1,/k2−,1j/,k2Ci,−j1,/k2−,1j/,k2 n concentrationsi, j,k i, j,k = − aui ,niveauj+1/ 2,k i,dej+−1 /la2,k limitei, j− 1des/ 2,k cellulesi, j−1/ 2,k adjacentes par: (3.17) Δzyk Δt Δx j i t+Δt t − form if they represent the end of the time step Cqi, j,k − CCi, j,k q−iq, jiq,kj+1/ 2,CkCi, Cji,kj+1/ 2,k −qiq, ji,kj−1/ 2,CkCi, ji,kj−1/ 2,k où Δxj, Δyi,n −Δzki+ 1sont/ 2, j,k lesi+1/ 2=,−dimensionsj,−k i−1/ 2, j,k desi−1/ 2,cellules,j,k et j+1/2, i+1/2,(t + D (3.17)ett). kThe+1/2 explicit indiquent form les of the equation can be Δt ΔΔzkx interfaces des cellules perpendiculairesΔyi aux direcj tions x, y, z (Fig.solved 3.12). directly La concentration for C(t + Dt )ài,j,k . The implicit form q C − q C l’interface de qlai, j,kcellule+1/ 2Ci, j ,kentre+1/ 2 −i+ 1q/deux2i,,jj,,kk−1 /i2+nœudsC1/ 2i,,jj,,kk−1/ 2voisii−1/ 2,nsj,k suivanti−1/ 2, j,k unerequires direction the particulièresimultaneous est solutions of the equa- où Δxj, Δyi, −Δzk sont les −dimensions des cellules, et (3.17)j+1/2, i+1/2, et k+1/2 indiquent(3.17) les Δz Δyi tion for all nodes by using a matrix solver. interfacesconsidérée des égale cellules à la concentration perpendiculairesk au niv auxeau direc du nœudtions amontx, y, z suivant(Fig. 3.12). la même La concentrationdirection (Bear à l’interface1979; Zheng de et laBennett cellule 1995) entreq.i ,Cette j,kdeux+1/ 2C approche i,nœudsj,k+1/ 2 − qvoisiesti, j, kappelée−1ns/ 2C suivanti, j,k −le1/ 2système uneT hedirection amont solution pondéré particulière of E etq. fournit3.17 est requires the incoming où Δx , Δy , Δz sont les wheredimensions Dx , Ddesy−, cellules,Dz are etcell j+1/2, dimensions, i+1/2, et andk+1/2 indiquent les j i k j i k Δz and outgoing specific discharge rates between considéréedes solutions égale non à affectées la concentration par l’oscillation: au niveau kdu nœud amont suivant la même direction (Bear interfaces des cellules perpendiculairesj + 1/2, i + 1/2, aux and direc k + 1/2tions denote x, y, zthe (Fig. cell 3.12). inter La- concentrationthe cell i,j,k andà the neighbouring cells. In the ear- 1979; Zhengfaces et Bennett normal 1995) to the. Cette x, y, approche z directions est appelée(Fig. 3.12). le système amont pondéré et fournit l’interface de oùla Δcellulexj, Δy i,entre Δzk sontdeux ⎧ lesCnœudsi, j−dimensions1,k if voisi qi, j-1/2,ns k des>suivant0 cellules, une etdirection j+1/2, iparticulière+1/2,ly applications et k+1/2 est indiquent of the mixing-cell les method to iso- C = ⎨ (3.18) des solutionsThe noni, j− 1/ cell2affectées,k interfaceC par if l’oscillation: concentrationq < 0 between two topic transport, the flow was assumed as steady, considérée égaleinterfaces à la concentration des cellules au⎩ perpendiculaires nivi, j,keau du i,nœudj-1/2,k auxamont direc suivanttions x,la mêmey, z (Fig. direction 3.12). (Bear La concentration à neighbouring nodes in a particular direction is set so that incoming and outgoing flow rates were 1979; Zheng etl’interface Bennett 1995) deequal la. Cette tocellule the ⎧approcheC i concentration, jentre−1,k if estdeux qi, jappelée-1/2, knœuds at> 0the le systèmevoisiupstreamns suivantamont node pondéré une direction et fournit particulière est C = ⎨ equal and there was(3.18) no change in the volume of des solutions nonconsidérée affectées égalealong pari, j−1 l’oscillation:/à 2the, kla concentrationsameC direction if q au [51, niv 0 gimes, numerical groundwater flow models such = i, videsj−1,k oscillationi, j-1/2,k free solutions: (3.18) Ci,desj−1121/ 2 ,solutionsk ⎨ non affectées par l’oscillation: as MODFLOW [37] can calculate the cell-by-cell ⎩Ci, j,k if qi, j-1/2,k < 0 flow rates. ⎧Ci, j−1,k if qi, j-1/2,k > 0 Ci, j−1/ 2,k = ⎨ (3.18) Groundwater flow(3.18) models generally require 112 C if q < 0 ⎩ i, j,k i, j-1/2,k the continuous representation of the flow domain

Eq. 3.17 represents the 3-D recursive state equa- in terms of hydraulic parameters (K,T,S). In some cases, it is difficult to obtain all the parameters 112 tion for each cell in the flow domain consider-

ing the tracer mass in the cell and incoming and required by the numerical groundwater models. outgoing mass fluxes. Since the mixing cell al- Additionally, in a karst aquifer system, such mod- 112 gorithm is the either an explicit or implicit back- els cannot be used because of the discontinuities ward FD approximation of the advective term in in the flow domain. In such cases the flow terms the solute transport equation, Eq. 3.17 can be con- can be calculated by the flow routing technique based on linear reservoir theory (Tezcan, in press). sidered as an explicit form if the concentration terms at the right side of the equation represent The flow routing technique uses the reservoir wa- the beginning of the time step (t) or as an implicit ter balance equation. Each model cell represents

530 COMPARTMENTAL MODEL APPROACHES TO GROUNDWATER FLOW SIMULATION

pas être utilisés à cause de la discontinuité du système d’écoulement. Dans de tels cas les linear or non-linear reservoirs in which the rela- (R ) recharge rates (Fig. 3.13). The parameter K termes représentant l’écoulement peuvent être calculést-D tpar la technique de l’écoulement tionship between the storage (S) and outflow (Q) is the storage constant that has the dimension of orienté, basée sur la théorie du réservoir linéaire (Tezcan, sous presse). pas être utilisés à cause de la discontinuitéis given du by: système d’écoulement. Dans de tels cas lestime. Under the no recharge condition, the equa- Cette technique utilise l’équation de bilan hydrologiquetion becomes du réservoir. the well-known Chaque cellule Maillet du equation of termes représentant l’écoulement peuvent êtren calculés par la technique de l’écoulement S = KQ (3.19) the hydrograph recession curves. orienté, basée sur la théorie du réservoirmodèle linéaire représente (Tezcan, unsous réservoir presse). linéaire ou non dans lequel la relation entre l’emmagasinement (whereS) et le K débit and n sortant represent (Q) the est constants donnée par: for the phys- Cette technique utilise l’équation de icalbilan process. hydrologique The water du balanceréservoir. or Chaquethe mass cellule con- du3.4.2. Mixing-cell model of flow n (3.19) modèle représente un réservoir linéaire servationou non da withinnsS lequel= KQ a timela relation interval entre Dt l’emmagasinementfor each cell and transport dynamics in (S) et le débit sortant (Q) est donnée par:oùcan K be et expressedn représentent as: les constantes pour le processus physique.karst Le bilan aqu hydrologiqueifer systems ou la n conservationTotal Inflow de– Totalmasse Outflow au cours = Changed’un intervalle in Storage de Wetemps develop Δt peuvent a distributed être exprimés conceptual pour model to S = KQ (3.19) chaqueor: cellule par: analyse groundwater flow and transport dynam- où K et n représentent les constantes pour le processus physique. Le bilan hydrologique ou laics in large-scale karst aquifer systems by using Apport Total – Sortie Totale = Changement de stock conservation de masse au cours d’un intervalle ded Stemps Δt peuvent être exprimés pourthe mixing-cell approach and the power of terrain R(t) − Q(t) = (3.20) chaque cellule par: dt d S analysis. The model is developed for aquifer sys- ou: R(t) − Q(t) = (3.20) d t tems in which knowledge about the hydraulic and Apport Total – Sortie TotaleThis = Changement relation can debe stockrearranged for an input (R) to the transport characteristics is limited. The aqui- Cettethe linear relation reservoir peut (êtren = 1)réajustée by continuity: pour un entrant (R) dans le réservoir linéaire (n = 1) en d S fer system is discretised into a finite number of ou: R(t) − Q(t) = continuité: (3.20) d t dQ the cells in three dimensions, and the transport K + = R dQ (3.21) process is simulated by the mixing-cell approach Cette relation peut être réajustée pour un dentrantt K (+R) dans= Rle réservoir linéaire (n = 1) en (3.21) d t whereas the surface and subsurface flow process- continuité: L’écoulementThe flow between entre lesthe cellulescells is estthen alors expressed exprimé by par l’équationes are simulated suivante: by flow routing. Special charac- dQ the following equation: K + = R (3.21) teristics such as the distribution of the recharge, d t flow and storage properties of the karst terrain are −ΔT ⎡ K ⎛ −ΔT ⎞⎤ ⎡ K ⎛ −ΔT ⎞ −ΔT ⎤ K ⎜ K ⎟ ⎜ K considered⎟ K in the dynamics of the flow. L’écoulement entre les cellules est alorsQ exprimét = e Q part−Δ tl’équation+ ⎢1− ⎜suivante:1− e ⎟ ⎥ Rt + ⎢ ⎜1− e ⎟ − e ⎥ Rt−Δt (3.22) ⎢ ΔT ⎝ ⎠⎥ ⎢ΔT ⎝ ⎠ ⎥ ⎣ ⎦ ⎣ Groundwater⎦ flow models are generally based on −ΔT ⎡ K ⎛ −ΔT ⎞⎤ ⎡ K ⎛ −ΔT ⎞ −ΔT ⎤ the prediction of the consequences of a proposed K ⎜ K ⎟ L’équation⎜ 3.22K ⎟ est Kla forme discrète de l’équation de l’écoulement orienté représentant Qt = e Qt−Δt + ⎢1− ⎜1− e ⎟⎥ Rt + ⎢ ⎜1− e ⎟ − e ⎥ Rt−Δt (3.22) action to the flow system. In complex cases, mod- ⎢ ΔT ⎝ ⎠⎥ ⎢ΔT ⎝ ⎠ ⎥ ⎣ ⎦ l’écoulement⎣ sortant (Q⎦ t) du réservoir au(3.22) temps t elsen canfonction be used de asl’écoulement an explanatory sortant tool to pro- précédentEq. 3.22 is (Q thet-Dt ),discrete les taux form de recharge of the flowactuel routing (Rt), et précédentvide additional (Rt-Dt) (Fig. information 3.13). Le and paramètre interpretation on L’équation 3.22 est la forme discrète de l’équation de l’écoulement orienté représentant Kequation est la representingconstante d’emmagasinement the outflow (Qt) of qui the ares la- dimensionthe flow d’un domain, temps. which En is l’absence especially de important in l’écoulement sortant (Qt) du réservoirrecharge,ervoir au attemps timel’équation t asen a fonctionfunctiondevient l’expressionofde thel’écoulement previous (bie outn sortantconnue)- karst de terrain Maillet where relative the knowledgeaux courbes about de the flow précédent (Q ), les taux de recharge actuel (R ), et précédent (R ) (Fig. 3.13). Le paramètre t-Dt tarissement.flow (Qtt-Dt), the present t-Dt (Rt), and the previous and transport system is limited. K est la constante d’emmagasinement qui a la dimension d’un temps. En l’absence de recharge, l’équation devient l’expression (bien connue) de Maillet relative aux courbes de 3.4.2 MODÈLE D’ÉCOULEMENT A CELLULES DE MELANGE ET tarissement. DYNAMIQUER DE TRANSPORT DANS LES SYSTÈMES AQUIFÈRES KARSTIQUES ΣRm 3.4.2 MODÈLE D’ÉCOULEMENT A CELLULESR DEQ MELANGER ET DYNAMIQUE DE TRANSPORTOn développe DANS un LES modèle SYSTÈMES conceptuel AQUIFÈRES distribué pour analyser les écoulements souterrains et la KARSTIQUES dynamique du transportQ dans les systèmes aquifères karstiques à grande échelle, en utilisant l’approche à cellules de mélange et l’apport des analyses de terrain. Le modèle est développé On développe un modèle conceptuel distribué pour analyser les écoulements souterrains et la pour des systèmes aquifères pour lesquels la connaissance des caractéristiquesΣQm d’écoulement dynamique du transport dans les systèmeset de aquifètransportres karstiquesest limitée. à Le grande système échelle, aquifère en utilisantest discrétisé en un nombre fini de cellules en l’approche à cellules de mélange et l’apport des analyses de terrain. Le modèle est développé pour des systèmes aquifères pour lesquels 114 la connaissance des caractéristiques d’écoulement et de transport est limitée. Le système aquifère est discrétisé en un nombre fini de cellules en

114 Fig. 27.13. The convolution of the recharge to the discharge through interconnected reservoirs.

531 modelling

The complex organisation of the flow domain and resenting the flow and transport in the cell are the heterogeneous recharge distribution causes defined as 3-D functions of the geographic (topo- difficulties in understanding groundwater flow graphic) co-ordinate system. An equal grid spac- pattern in the karst aquifers. Groundwater flow ing is used in x-y surface, whereas the thickness models developed for the granular aquifers, based of the layers can vary and be assigned by the user on Darcy’s law, are not applicable for karst aqui- for each layer at the beginning of the simulation. fers, where the groundwater generally moves Then the number of the layers is calculated for through the conduits, and the velocity is higher each grid according to the topographic elevation than that through the granular systems. The dis- of the grid. An additional layer may be located at continuity in the flow domain limits the expres- the bottom of the system to consider dead storage, sion of the flow system by the differential equa- submarine discharge, or deep percolation. tions based on the continuity and the representative elementary volume approach. Each cell is attributed by a cell type code (‘GeoCode’) representing the hydrogeologic The model developed in this study is a first attempt properties of the cell based on the lithology to identify the karst groundwater flow system by (Fig. 3.14). The cell type code is taken as nega- a terrain-based distributed parameter hydrologic tive for impervious (non-active) units, zero for model coupled with the mixing-cell approach for constant head cells such as sea, and positive for environmental isotope transport. This distributed pervious (active) units. Infiltration or groundwa- flow and transport model considers the spatial ter circulation does not take place over the nega- variations of the parameters in three dimensions tive coded cells; instead, overland flow occurs ac- and is applied to the highly karstified Beydaglari cording to the aspect and the slope of the terrain Aquifer system located at the Mediterranean coast in these cells. The ‘GeoCode’ is a 3-D array and of Turkey (Tezcan, in press). it is read for all the layers as input to the model. The deeper extinction of the lithology that is out- 3.4.2.1. Physical framework of the model cropping can be estimated by using the depth of the cell/layer and the thickness, dip, and strike. The hydrogeologic system is simulated as though it is composed of interconnected reservoir sys- Structural features (faults, folds, and linea- tems. The flow system is discretised into volumet- ments) are defined by their directions. The direc- ric cells (DV = Dx × Dy × Dz) in a 3-D co-ordinate tions of these features are limited to eight direc- system and all the variables and parameters rep- tions: North, Northeast, East, Southeast, South,

-8 -8 -8 -8 -8 -8 -8 -8 -8 -8 -7 -7 -7 -7 -7 -7 -7 -8 -8 -8 -8 -8 -8 -8 -8 -8 -7 -7 -7 -7 -7 -7 -7 -7 -8 -8 -8 -8 -8 -8 -8 -7 -7 -7 -7 -7 -7 -7 -7 -7 1 -8 -8 -8 -8 -8 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 1 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 1 1 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 1 1 1 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 1 1 1 1 1 -7 -7 -7 -7 -7 -7 -7 -7 -7 -7 1 1 1 1 1 1 1 1 1 1 -7 -7 -7 -7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -7 -7 -7 1 1 1 1 1 1 1 1 1 1 1 1 1 -7 -7 -7 -7 1 1 1 1 1 1 1 1 1 1 1 1 -7 -7 -7 -7 -7 1 1 1 1 1 1 1 1 1 1 1 1 -7 -7 -7 -7 -7 1 1 1 1 1 1 1 1 1 1 1 1 -7 -7 -7 -7 -7 1 1 1 1 1 1 1 1 1 1 1 1 -7 -7 -7 -7 -7 1 1 1 1 1 1 1 1 1 1 1 1 -7 -7 -7 -7 -7

Fig. 27.14. The discretisation of the active (+) and non-active (-) cells by ‘Geocode’.

532 COMPARTMENTAL MODEL APPROACHES TO GROUNDWATER FLOW SIMULATION

Southwest, West, and Northwest. The cells are overland flow, surface storage, and groundwater attributed by the code (‘StrCode’) for structural flow. The time step in the model is selected as one features. If the cell is not crossed by any struc- day, but selection of more frequent time steps is tural feature, the ‘StrCode’ of the cell is assigned also possible. The general structure of the flow to zero. The ‘StrCode’ is attributed with numbers model is indicated in Fig. 3.15. from 1 to 8 (N, NE, E, SE, S, SW, W, NW) rep- The precipitation observed in the meteorological resenting the direction of the groundwater flow observation stations is considered the recharge controlled by the structure. source of the model area. The stations are defined The location of the sinkholes and springs are by their topographic co-ordinates in the model. expressed by similar codes (‘SnkCode’ and The model checks the records of the precipita- ‘SprCode’). The cells including a sinkhole/spring tion stations at all time steps during the simula- are assigned by the value 1, whereas all the re- tion to determine whether point rainfall events maining are expressed as zero. are recorded for these particular time steps. For The terrain slope represents the slope at any grid every time step, if precipitation happens in one or node on the surface and is reported in degrees more stations in the model area, the values are in- from –90 (vertical downhill) to 90 (vertical up- terpolated for each cell. Potential evapotranspira- hill). tion is also given to the model for the observation sites and then extrapolated over the model area. For a particular point on the surface, the terrain The net recharge is calculated as precipitation slope is based on the direction of steepest descent surplus over evapotranspiration. In case of greater or ascent at that point so that across the surface, evapotranspiration than the precipitation amount, the gradient direction can change. the deficiency is supplied from the surface storage The aspect of the terrain is the direction of (if any). the steepest slope at each grid node. It represents Infiltration takes place over the pervious geo- the direction that water would flow over the sur- logic units as defined by positive ‘Geocode’ val- face or the angle that is exactly perpendicular to ues. The sources of infiltration are the precipita- the contour lines on the surface. Terrain aspect tion surplus, the amount of water flowing from values are calculated as azimuth, where 0 degrees upstream cells as overland flow, and the surface points due North, and 90 degrees points due East. In calculation of the overland flow the flow direc- storage of the previous time step in the cell. tion is expressed as one of eight major directions: N (337.5º – 22.5º), NE (22.5º – 67.5º), E (67.5º – 112.5º), SE (112.5º – 157.5º), S (157.5º – 202.5º), SW (202.5º – 247.5º), W (247.5º – 292.5º), and NW (292.5º – 337.5º).

3.4.2.2. Hydrologic model The model is designed to simulate the surface water and groundwater circulation on the karst terrain. The terrain is described by topographic, geologic, and morphologic aspects as outlined above. The hydrologic system is expressed by the recharge, storage, and discharge events. The model considers precipitation and evaporation, infiltration, overland flow, surface storage, percolation, groundwater storage, and flow processes. The water balance is calculated for each cell at every time step by considering the recharge from precipita- Fig. 27.15. The structure of the model of karst ground- tion, infiltration, percolation, evapotranspiration, water flow and transport processes.

533 modelling

Infiltration is simulated in the model as either volumes are lower than the volume of the cell a concentrated or diffuse process. If the cell of interest, the flow partition to the cell is esti- consists of sinkholes, all the recharge water will mated by the percentage of the total gradient with infiltrate to the top layer in that particular time the neighbouring cells. The flow does not occur interval as point/concentrated recharge, and to the neighbouring cell whose volume is higher the surface storage will be zero. In absence of the sinkholes, Hortonian infiltration will take than the cell of interest. If the volume of the any place for the diffuse recharge. neighbouring cell is at its maximum value, flow will not take place to that cell. Overland flow may occur in both negative and positive geocoded cells where surface storage The porosity of the cell is an important factor to is available. The negative ‘Geocode’ means that define the karst circulation path. If a cave, con- the geologic unit does not allow the water to in- duit, or similar secondary porosity feature ex- filtrate, so all the water will flow over the surface. ists in the cell, the porosity may be as high as In positive geocoded cells, overland flow occurs 100%. Any known value for the secondary po- if there is water remaining after the infiltration event in that particular time step. The water at rosity value may increase the representativeness the surface may be the excess of the infiltration, of the model. but it may be the amount coming from upstream The model allows downward movement of cells. The direction of the overland flow is due to groundwater from the lowest cell to outside the aspect value of the cell. The water may flow to the one of the eight neighbouring cells in the di- of the model region, but does not allow an up- rection of the aspect of the cell. The overland ward recovery. The release of the water from flow for per unit width of the cell is calculated by the model region to the outside may be inter- the routing equation given above. preted as the deeper circulation, or dead storage The volume of the water in a particular time step for the modelled region. To simulate the interre- remaining after all the hydrologic events is called lations with the neighbouring aquifers, the outer the surface storage, and is calculated by the bal- cells accept recharge as specified flux through ance equation at each time step: the horizontal model boundaries.

Sstor = Sstor(t – ∆t) + P(t) + OFlowUp(t) – Et(t) The flow model outputs the discharge hydro- – I(t) – OFlowDown(t) graphs of the cells representing the springs of in- Overland flow coming from upstream cells terest. The calibration parameters are the storage (OFlowUp(t)) , precipitation (P(t)) and the surface constant (K) and the infiltration constant. storage of the previous time step (Sstor(t–Dt)) are the gains of the balance equation, whereas evapo- 3.4.2.3. Transport model transpiration (Et(t)), infiltration (I(t)), and overland flow to downstream cells (OFlowDown(t)) Karst groundwater flow is simulated by a flow are the losses at the surface of the terrain. routing technique whereas the transport pro- The cell-by-cell flow process is simulated by cess is simulated by the mixing-cell approach. Eq. 3.22. The flow between the cells as a function The model is designed to simulate the spatial and of time is controlled by the storage constant (K), temporal distribution of the up to five conserva- which represents the turnover time for the res- tive (non-reactive) tracers (isotopes, chlorofluo- ervoir. Each cell can be recharged from or dis- rocarbons, electrical conductivity, chloride, etc.) charge to the neighbouring six cells. The parti- in the groundwater. The model simulates only tioning of the flow between the neighbouring cells is estimated by the volumetric gradient between advective transport and does not consider dis- the cells. The model calculates the current vol- persion and diffusion processes. The transport umes of neighbouring cells and compares them process is simulated by considering Eq. 3.17 in to the current cell’s volume. If the neighbouring explicit form:

534 Le modèle prévoit un mouvement des eaux souterraines descendant, au niveau de la cellule la plus basse vers l’extérieur de la zone du modèle, mais ne prévoit pas le mouvement inverse. La libération d’eau de la zone du modèle vers l’extérieur peut être interprétée comme une circulation profonde, ou stockage inactif dans la région modélisée. Pour simuler les relations avec les aquifères voisins, les cellules externes prennent en compte une recharge sous forme Le modèle prévoit un mouvement des eaux souterraines descendant, au niveau de la cellule la d’un flux imposé au niveau des limites horizontales du modèle. plus basse vers l’extérieur de la zone du modèle, mais ne prévoit pas le mouvement inverse. LaLe libérationmodèle d’écoulement d’eau de la génèrezone dules modèle hydrographes vers l’extérieur de débit despeut cellules être interprétée représentant comme les une circulationsources intéressantes. profonde, Lesou stockageparamètres inactif de calage dans sont la régionla constante modélisée. d’emmagasinement Pour simuler (K les) et relationsla avecconstante les aquifères d’infiltration. voisins, les cellules externes prennent en compte une recharge sous forme d’un flux imposé au niveau des limites horizontales du modèle. 3.4.2.3 MODÈLE DE TRANSPORT Le modèle d’écoulement génère les hydrographes de débit des cellules représentant les sourcesL’écoulement intéressantes. souterrain Les dans paramètres le karst estde calagesimulé sontpar la la techniqueconstante de d’emmagasinement l’écoulement orienté, (K ) et la constantetandis que d’infiltration. le processus de transport est simulé par l’approche à cellules de mélange. Le modèle est conçu pour simuler les distributions spatiales et temporelles pour un nombre de 3.4.2.3traceurs allantM ODÈLEjusqu’à DEcinq TRANSPORT (isotopes, chloro fluorocarbones, conductivité électrique, chlorure, etc.) conservatifs (non-réactifs) dans les eaux souterraines. Le modèle ne simule que le L’écoulement souterrain dans le karst est simulé par la technique de l’écoulement orienté, transport par advection et ne prend pas en compte les processus de dispersion et de diffusion. tandis que le processus de transport est simulé par l’approche à cellules de mélange. Le Le processus de transport estCOMP simuléART àMEN partirTA deL MOl’EqD EL3.17 APP sousROACH sa formeES TO explicite: GROUNDWAT ER FLOW SIMULATION modèle est conçu pour simuler les distributions spatiales et temporelles pour un nombre de traceurs allant jusqu’à cinq (isotopes, chlorofluorocarbones, conductivité électrique, chlorure, ⎡ qt+Δt C t − qt+Δt C t ⎤ Discrete-state compartmental models such as etc.) conservatifs (non-réactifs)⎢− i , jdans+1/ 2,k iles, j+1/ 2,eauxk i , jsouterraines.−1/ 2,k i, j−1/ 2,k ⎥ Leproposed modèle ne by simule [9, 11, que 14, 17]arele the initial works Δx transport par advection et ne⎢ prend pas en compj te les processus⎥ deof dispersion distributed et deparameter diffusion. modelling of environ- ⎢ t+Δt t t+Δt t ⎥ Le processus de transport Δestt simuléq + àC partir+ de− ql’Eq− 3.17C − sous sa formemental explicite: isotope data. These studies have provided C t+Δt = C t + ⎢− i 1/ 2, j,k i 1/ 2, j,k i 1/ 2, j,k i 1/ 2, j,k ⎥ (3.23) i, j,k i, j,k n ⎢ Δy ⎥ turnover time and dynamic volume of the com- ⎢ i ⎥ t+Δtt+Δt t t t+Δt t+Δt t t partments representing part of the groundwater ⎢ ⎡ qi,qj,ik, +j1+/12/C2,ki,Cj,ki+, 1j+/ 21/−2,kqi−, j,qk −i1, /j2−C1/ 2i,,jk,Ck −1i,/ 2j−⎥1/ 2,k ⎤ ⎢−⎢− ⎥ ⎥system by calibrating the isotopic contents and ΔzkΔx ⎣ ⎢ j ⎦ ⎥additionally, the flow rate at the single outlet of ⎢ t+Δt t t+Δt t ⎥the system. The model proposed in this study dif- Δt qi+1/ 2, j,kCi+1/ 2, j,k − qi−1/ 2, j,(3.23)kCi−1/ 2, j,k C t+Δt = C t + ⎢− ⎥fers from these earlier(3.23) works in that it produces Cette forme explicitei, j,k ide, j,k l’équationn ⎢ peut être résolueΔy directement pour⎥ C(t + Δt)i,j,k. La valeur The explicit form⎢ of the equation cani be solved ⎥output of the isotopic contents and flow rates de la concentration est calculée pourt+Δt chaquet cellulet +àΔ t chaquet pas de temps et le modèle directly for C(t+D⎢ t)i,j,kqi,.j ,kT+1he/ 2C iconcentration, j,k +1/ 2 − qi, j,k −1 / 2valueCi, j,k − 1/ 2 ⎥for all the cells in the system and allows the es- − produit des graphesis calculated de concentrations for every⎢ cellen fonction at each du time temps step pour and des cellules⎥timation déterminées. of the calibration variables at multiple ⎣ Δzk ⎦ La formule theexplicite model de outputs l’équation the de concentration-time transport par advection relation (Eq. 3.23)outlet est affectée points par(springs, de la wells, etc.). The structure for specified cells. of the model is similar to the finite-difference Cettedispersion forme numérique explicite dedue l’équation à l’erreur peutde troncat être résolueure de l’équationdirectement partielle pour Cdifférentielle(t + Δt) . Lade valeur flow modelsi,j,k such as MODFLOW [37]in terms transport (Eq. T 3.14).he explicit L’utilisation scheme d’un of maillagethe advective fin et de transport pas de temps plus courts permet de de la concentration est calculée pour chaque cellule à chaque ofpas the de gridtemps design et le andmodèle the discrete structure of equation (Eq. 3.23) is subject to numerical disper- produitminimiser des cette graphes dispersion de concentrations numérique. Les en pas fonction de temps du dutemps transport pour devraientdesthe cellules flow être: and déterminées. transport equations. As in the deter- sion introduced by the truncation error of the par- ministic groundwater flow and transport models, La formule explicite de1 l’équation de transport par advection (Eq. 3.23) est affectée par de la tialΔt ≤ differential transport equation (Eq. 3.14). In the model imposes(3.24) large requirements for data to q qy q dispersion numériqueorder tox +minimise due+ àx l’erreurnumerical de dispersion troncature a define l’équation grid define partielle all thedifférentielle parameters deat all nodes of the grid, Δx Δy Δz transport (Eq.and 3.14). smaller L’utilisation time steps d’un are required.maillage Tfinhe ettransport de pas de tempsand it plusis inherently courts permet impossible de to obtain a unique minimiser cettetime dispersion steps should numérique. be: Les pas de temps du transportsolution devraient and être:the parameters and the variables obtained by the model are not the real and unique 1 120 Δt ≤ (3.24) values representing(3.24) the physical system. They are q q q x + y + x approximate values based on limited knowledge Δx Δy Δz about the system. Due to the size and the complexity of the aquifer system, knowledge of The grid spacing and the model time interval are the spatial and temporal distribution of the input determined at the beginning of the simulation by data (recharge, concentration) may not be satis- 120 the user and the specific discharge components factory. Therefore, the model calibration does not calculated by these specified intervals. The trans- necessarily produce an exact match of the ob- port time steps are then estimated by the model served data. Instead, a good representation of according to Eq. 3.24 at each model time step. the observed discharge and concentration distri- Depending upon the grid size and the groundwa- butions should be achieved at all the observation ter velocity, the number of the transport time steps points. The calibration may also be constrained by may increase enormously. using several transport variables (isotopes, CFCs, hydrochemical variables, etc.) and by considering several outlets of the system. 3.4.3. Conclusions The model program and sample data set can be The mixing-cell approach together with the flow requested by contacting the author at tezcan@ routing technique is used for the simulating hacettepe.edu.tr. the groundwater flow and transport dynamics in large-scale karst aquifer systems. The model is developed as compatible with the data processing 3.5. Summary and conclusions structure of the GIS approach. The cell configu- The three models described above illustrate ration, geologic and hydrogeologic system and the utility of and level of sophistication in com- the recharge and discharge events are all defined partmental or mixing-cell models. In this section in space co-ordinates. This will help in further as- we will summarise the results and speculate on sociations of the model to a GIS system. the future of the compartmental model approach.

535 modelling

Campana’s compartmental model is the simplest proach for transport, linear reservoir theory for to use, but it is not as well-constrained as the oth- flow, surficial processes (infiltration, runoff, etc.), er models. It can be used as a ‘stand-alone’ model and terrain analysis, all in a GIS-type framework. or coupled to a flow model. How successful it is Tezcan’s model is particularly well-suited to karst relies heavily upon the skill of the modeller and and fractured-rock aquifers, where Darcy’s Law his/her familiarity with the area being modelled. may not apply and traditional REV-based ap- The model is appropriate to apply in an area with proaches are questionable. A full-scale GIS ap- few data, perhaps to guide future data-collection proach is the next step for this approach. and model-building efforts. Certainly, the model is improved and afforded a stronger physical ba- Future research efforts should involve the follow- sis by using a linear-reservoir routing algorithm to ing: calculate intercellular discharges; use of a Darcy- (1) continued investigation of the linear and type equation could also be used. Adar and col- leagues have used such an approach as well as non-linear reservoir approach for the treat- linear and non-linear programming to further con- ment of transient effects; strain compartmental models. Campana’s model (2) more rigorous constraint of the Campana has the advantage in that it can calculate age and model parameter estimates; residence time distributions, although the calculation of these must be extended to transient flow. (3) development of transient analogues to the steady-state residence-time distributions; Harrington’s approach shows one of the potential strengths of the compartmental model - environ- (4) further application of the Harrington ap- mental isotope approach: its utility in constrain- proach, including its extension to three di- ing a physically-based groundwater flow model. mensions and application of multi-isotopic However, when conducting transient simulations tracers; covering thousands of years, the modeller may have difficulty estimating the 14C inputs over that (5) investigation of the compartmental model’s length of time. This adds additional uncertainty utility in palaeoclimatic and palaeohydrolog- to the results. The use of other environmental ic investigations; isotopes will present similar problems. The next (6) use of compartmental models as contaminant steps for Harrington’s approach are a three-di- transport models; mensional simulation and the use of other environmental isotopes (perhaps multiple ones). (7) coupling of geochemical models to compartmental models; Tezcan’s model is a highly-integrated, powerful model in that it combines the mixing-cell ap- (8) availability of user-friendly software.

536 COMPARTMENTAL MODEL APPROACHES TO GROUNDWATER FLOW SIMULATION

References

[1] Adar, E.M., Neuman, S.P., “The use of environmental tracers (isotopes and hydrochemistry) for quantification of natural recharge and flow components in arid basins”, Proc. 5th International Symposium on Underground Tracing, Athens, Greece (1986) 235–253. [2] Adar, E.M., Neuman, S.P., Estimation of spatial recharge distribution using environmental isotopes and hydrochemical data; II: Application to Aravaipa Valley in Southern Arizona, Etats-Unis, J. Hydrol., 97 (1988) 297–302. [3] Adar, E.M., Neuman, S.P., Woolhiser, D.A., Estimation of spatial recharge distribution using environmental isotopes and hydrochemical data; I: Mathematical model and application to synthetic data, J. Hydrol. 97 (1988) 251–277. [4] Adar, E., Sorek, S., Multi-compartmental modeling for aquifer parameter estimation using natural tracers in non-steady flow, Adv.Water Resour. 12 (1989) 84–89. [5] Adar, E., Sorek, S., “Numerical method for aquifer parameter estimation utilizing environmental tracers in a transient flow system: MODELCARE 90”, Proc. Intern. Conf. on Calibration and Reliability in Groundwater Modeling (Kover, K., Ed.), The Hague, Holland, IAHS Publ. No. 195 (1990) 135–148. [6] Campana, M.E., Finite-state models of transport phenomena in hydrologic systems, Ph.D. dissertation, University of Arizona, Tucson (1975) 252 pp. [7] Campana, M.E., Generation of ground-water age distributions, Ground Water 25 1 (1987) 51–58. [8] Campana, M.E., Simpson, E.S., Groundwater residence times and recharge rates using a discrete state compartment model and C-14 data, J. Hydrol. 72 (1984) 171–185. [9] Campana, M.E., Mahin, D.A., Model-derived estimates of groundwater mean ages, recharge rates, effective porosities and storage in a limestone aquifer, J. Hydrol. 76 (1985) 247–264. [10] Kirk, S.T., Campana, M.E., A deuterium-calibrated groundwater flow model of a regional carbonate- alluvial system, J. Hydrol. 119 (1990) 357–388. [11] Yurtsever, Y., Payne, B.R., “A digital simulation approach for a tracer case in hydrological system (multi-compartmental mathematical model)”, Proc. Intern. Conf. on Finite Elements in Water Resources, London (1978). [12] Yurtsever, Y., Payne, B.R., “Time-variant linear compartmenal model approach to study flow dynamics of a karstic groundwater system by the aid of environmental tritium (a case study of south- eastern karst area in Turkey)”, Proc. Symp. on Karst Water Resources, Ankara-Antalya, July 1985, IAHS Pub. No. 161 (1985) 545–561. [13] Yurtsever, Y., Payne, B.R., “Mathematical models based on compartmental simulation approach for quantitative interpretation of tracer data in hydrological systems”, Proc. 5th Intern. Symp. on Underground Water Tracing, Inst. Geol. and Min. Explor., Athens, Greece (1986) 341–353. [14] Van Ommen, H.C., The “mixing-cell” concept applied to transport of non-reactive and reactive components in soils and groundwater, J. Hydrol. 78 (1985) 201–213. [15] Rao, B., Hathaway, D., A three-dimensional mixing-cell solute transport model and its application, Ground Water, 27 4 (1989) 509–516. [16] Harrington, G.A., Walker, G.R., Love, A.J., Narayan, K.A., A compartmental mixing-cell approach for quantitative assessment of groundwater dynamics in the Otway Basin, South Australia, J. Hydrol. 214 (1999) 49–63. [17] Simpson, E.S., Duckstein, L., “Finite-state mixing-cell models”, Karst Hydrology and Water Resources, Vol. 2 (Yevjevich, V., Ed.), Water Resources Publications, Ft. Collins, CO (1976) 489–512. [18] Campana, M.E., Byer, R.M., A conceptual evaluation of regional groundwater flow, southern Nevada-California, Etats-Unis, Environ. Engin. Geosci. II 4 (1996) 465–478. [19] Dooge, J.C.I., The routing of groundwater recharge through typical elements of linear storage, Publ. 52, General Assembly of Helsinki, Intern. Assn. of Sci. Hydrology 2 (1960) 286–300. [20] Dooge, J.C.I., Linear theory of hydrologic systems., Technical Bulletin 1468, U.S. Dept. of Agriculture (1973) 327 pp. [21] Mandeville, A.N., O’Donnell, T., Introduction of time variance to linear conceptual catchment models, Water Resour. Res. 9 2 (1973) 298–310.

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[22] Campana, M.E., “Compartment model simulation of ground-water flow systems”, Use of Isotopes for Analyses of Flow and Transport Dynamics in Groundwater Systems, Results of a co-ordinated research project 1996–1999, IAEA, Vienna (2002) CD-ROM. [23] Feeney, T.A., Campana, M.E., Jacobson, R.L., A deuterium-calibrated groundwater flow model of western Nevada Test Site and vicinity, Water Resources Center, Desert Research Institute, Reno, Nevada, Publ. No. 45057 (1987) 46 pp. [24] Sadler, W.R., A deuterium-calibrated discrete-state compartment model of regional groundwater flow, Nevada Test site and vicinity, M.S. thesis, University of Nevada, Reno (1990) 249 pp. [25] Winograd, I.J., Thordarson, W., Hydrogeologic and hydrochemical framework, south-central Great Basin, Nevada-California, with special reference to the Nevada Test Site, U.S.Geol. Surv. Profess. Paper 712-C (1975) 92 pp. [26] Mifflin, M.D., Hess, J.W., Regional carbonate flow systems in Nevada, J. Hydrol. 43 (1979) 217–237. [27] Burbey, T.J., Prudic, D.E., 1991. Conceptual evaluation of regional ground-water flow in the carbonate- rock province of the Great Basin, Nevada, Utah, and adjacent states. U.S.Geol. Surv. Profess. Paper 1409-D: pp 84. [28] Plume, R.W., Hydrogeologic framework of the Great Basin region of Nevada, Utah, and adjacent states. U.S.Geol. Surv. Profess. Paper 1409-B (1996) 64 pp. [29] Rush, F.E., Regional groundwater systems in the Nevada Test Site area, Nye, Lincoln, and Clark Counties, Nevada, Nevada Dep. of Conservation and Natural Resources, Ground Water Resource Reconnaissance Series Rep. 54 (1970) 25 pp. [30] Walker, G.E., Eakin, T.E., Geology and groundwater of Amargosa Desert, Nevada-California, Nevada Dep. of Conservation and Natural Resources, Ground Water Resource Reconnaissance Series Rep. 14 (1963) 45 pp. [31] Malmberg, G.T., Eakin, T.E., Ground-water appraisal of Sarcobatus Flat and Oasis Valley, Nye and Esmeralda Counties, Nevada, Nevada Dep. of Conservation and Natural Resources, Ground Water Resource Reconnaissance Series Report 10 (1962) 39 pp. [32] White, A.F., Chuma, N.J., Carbon and isotopic mass balance of Oasis Valley-Fortymile Canyon groundwater basin, southern Nevada, Water Resour. Rese. 23 4 (1987) 571–582. [33] Winograd, I.J., Pearson, F.J. Major carbon-14 anomaly in a regional carbonate aquifer: possible evidence for megascale channeling, south central Great Basin, Water Resour. Res. 12 6 (1976) 1125–1143. [34] Rice, W.A., Preliminary two-dimensional regional hydrologic model of the Nevada Test Site and vicinity, Sandia National Laboratories, Albuquerque, New Mexico, Report SAND 83-7466 (1984) 44 pp. [35] Waddell, R.K., Two-dimensional, steady-state model of groundwater flow, Nevada Test Site and vicinity, Nevada-California, U.S.Geol. Surv.-Resources Investigations Report 81-4085 (1982) 71. [36] Winograd, I.J., Friedman, I. Deuterium as a tracer of regional groundwater flow, southern Great Basin, Nevada-California, Geol. Soc. of America Bull., 83 12 (1972) 3691–3708. [37] McDonald, M.G., Harbauch, A.W., A modular three-dimensional finite-difference ground-water flow model, Techniques of Water Resources Investigations of the U.S.Geol. Surv., Book 6, Ch. A1 (1988) 586 pp. [38] Edmunds, W.M., Wright, E.P., Groundwater recharge and palaeoclimate in the Sirte and Kufra Basins, Libya, J. Hydrol. 40 (1979) 215–241. [39] Clark, I.D., Fritz, P., Quinn, O.P., Rippon, P.W., Nash, H., Sayyid Barghash Bin Ghalib Al Said, “Modern and fossil groundwater in an arid environment: a look at the hydrogeology of Southern Oman”, Isotope Techniques in Water Resources Development, IAEA, Vienna (1987) 167–187. [40] Fontes, J.Ch., Andrews, J.N., Edmunds, W.M., Guerre, A., Travi, Y., Palaeorecharge by the Niger River (Mali) deduced from groundwater geochemistry, Water Resour. Res. 27 2 (1991) 199–214. [41] Mazor, E., Verhagen, B.T., Sellschop, J.P.F., Robins, N.S., Hutton, L.G., “Kalahari groundwaters: their hydrogen, carbon and oxygen isotopes”, Isotope Techniques in Groundwater Hydrology, IAEA, Vienna (1974) 203–225. [42] Love, A.J., Herczeg, A.L., Leaney, F.W., Stadter, M.F., Dighton, J.C., Armstrong, D., Groundwater residence time and palaeohydrology in the Otway Basin, South Australia: 2H, 18O and 14C data, J. Hydrol. 153 (1994) 157–187. [43] Bentley, H.W., Phillip, F.M., Davis, S.N., Habermehl, M.A., Airey, P.L., Calf, G.E., Elmore, D., Gove, H.E., Torgersen, T., Chlorine-36 dating of very old groundwater, I: The Great Artesian Basin, Australia, Water Resour. Res., 22 13 (1986) 1991–2001.

538 COMPARTMENTAL MODEL APPROACHES TO GROUNDWATER FLOW SIMULATION

[44] Allison, G.B., Hughes, M.W., The use of environmental tritium to estimate recharge to a South Australian aquifer, J. Hydrol. 26 (1975) 245–254. [45] Przewlocki, K., Yurtsever, Y., “Some conceptual mathematical models and digital simulation approach in the use of tracers in hydrological systems”, Isotope Techniques in Groundwater Hydrology, IAEA, Vienna (1974) 425–450. [46] Yurtsever, Y., Buapeng, S., “Compartmental modelling approach for simulation of spatial isotopic variations in the study of groundwater dynamics, A case study of a multi-aquifer system in the Bangkok Basin, Thailand”, Isotopic Techniques in Water Resources Development, IAEA, Vienna (1991) 291–308. [47] Yurtsever, Y., Payne, B.R., Gomez, M., “Use of linear compartmental simulation approach for quantitative identification of isotope data under time variant flow conditions”, Mathematical Models for Interpretation of Isotope Data in Groundwater Hydrology, IAEA, Vienna (1986) 203–222. [48] Love, A.J., Groundwater Flow Systems: Past and Present, Gambier Embayment, Otway Basin, South- East Australia, MSc thesis, School of Earth Sciences, Flinders University of South Australia (1992). [49] Love, A.J., Herczeg, A.L., Walker, G.R., “Transport of water and solutes across a regional aquitard inferred from porewater deuterium and chloride profiles: Otway Basin, Australia”, Isotopes in Water Resources Management. IAEA, Vienna (1996) 73–86. [50] Bajracharya, K., Bary, D.A., Note on common mixing cell models, J. Hydrol. 153 (1994) 189–214. [51] Bear, J., Hydraulics of Groundwater, McGraw Hill (1979) 567 pp. [52] Zheng, C., Bennett, G.D., Applied Contaminant Transport Modeling: Theory and Practice, Van Nostrand Reinhold (1995) 440. [53] Tezcan, L., “Distributed modeling of flow and transport dynamics in large scale karst aquifer systems by environmental isotopes”, Use of Isotopes for Analyses of Flow and Transport Dynamics in Groundwater Systems, Results of a co-ordinated research project 1996–1999, IAEA, Vienna (2002) CD-ROM.

539

4. USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT

L.F. KONIKOW1 Because isotopes move through groundwater systems under the same driving forces and by US Geological Survey, Reston, Virginia, USA the same processes as do dissolved chemicals, it is natural that the groundwater flow and solute- transport models applied to groundwater con- 4.1. Introduction tamination problems be linked to and integrated In the past, the main driving force for hydrogeo- with isotopic measurements and interpretations. logic studies has been the need to assess the wa- Many previous applications of isotopic analyses ter-supply potential of aquifers. During the past to groundwater systems, however, have assumed 20 years, however, the emphasis has shifted from overly simplified conceptual models for groundwater-supply problems to water-quality problems. water flow and transport of dissolved chemicals- This has driven a need to predict the movement -either plug flow (with piston-like displacement of contaminants through the subsurface environ- and no mixing) or a well-mixed reservoir (which unrealistically overestimates the mixing effects of ment. One consequence of the change in emphasis dispersion and diffusion). If the interpretations of has been a shift in perceived priorities for scientif- isotopic analyses are coupled with more realistic ic research and data collection. Formerly, the fo- conceptual models of flow and transport, then it cus was on developing methods to assess and is anticipated that the synergistic analysis will measure the water-yielding properties of high- lead to a more accurate understanding of the hy- permeability aquifers. The focus is now largely drogeologic system being studied. Dinçer and on transport and dispersion processes, retardation Davis [4] provide a review of the application of and degradation of chemical contaminants, the ef- environmental isotope tracers to modelling in hy- fects of heterogeneity on flow paths and travel drology, and Johnson and DePaolo [5] provide an times, and the ability of low-permeability materi- example of applying such a coupled approach in als to contain contaminated groundwater. their analysis of a proposed high-level radioactive waste repository site. The past 20 years or so have also seen some major technological breakthroughs in groundwater The purpose of this chapter is to review the state of hydrology. One technological growth area has the art in deterministic modelling of groundwater been in the development and use of determinis- flow and transport processes for those who might tic, distributed-parameter, computer simulation want to merge the interpretation of isotopic analy- models for analysing flow and solute transport ses with quantitative groundwater model analysis. This chapter is aimed at practitioners and is in- in groundwater systems. These developments tended to help define the types of models that are have somewhat paralleled the development and available and how they may be applied to com- widespread availability of faster, larger memory, plex field problems. It will discuss the philosophy more capable, yet less expensive computer sys- and theoretical basis of deterministic modelling, tems. Another major technological growth area the advantages and limitations of models, the use has been in the application of isotopic analyses and misuse of models, how to select a model, and to groundwater hydrology, wherein isotopic mea- how to calibrate a model. However, as this chap- surements are being used to help interpret and deter is only a review, it cannot offer comprehensive fine groundwater flow paths, ages, recharge areas, and in-depth coverage of this very complex topic; leakage, and interactions with surface water [3]. but it does guide the reader to references that pro-

1 Chapter largely based on Konikow [1] and Konikow and Reilly [2]; Y.Yurtsever and T.E.Reilly are gratefully acknowledged for contributions and support..

541 modelling vide more details. Other recent comprehensive re- ternal properties, boundaries, and stresses of views of the theory and practice of deterministic the system are approximated. Deterministic, dis- modelling of groundwater processes are provided tributed-parameter, numerical models can relax by Refs [6, 7]. the rigid idealised conditions of analytical models or lumped-parameter models, and they can there- 4.2. Models fore be more realistic and flexible for simulating field conditions (if applied properly). The word model has so many definitions and is so overused that it is sometimes difficult to discern The number and types of equations to be solved the meaning of the word [8]. A model is perhaps are determined by the concepts of the dominant most simply defined as a representation of a real governing processes. The coefficients of the equa- system or process. A conceptual model is a hy- tions are the parameters that are measures of pothesis for how a system or process operates. the properties, boundaries, and stresses of the sys- This hypothesis can be expressed quantitatively tem; the dependent variables of the equations as a mathematical model. Mathematical models are the measures of the state of the system and are abstractions that represent processes as equa- are mathematically determined by the solution tions, physical properties as constants or coeffi- of the equations. When a numerical algorithm cients in the equations, and measures of state or is implemented in a computer code to solve one potential in the system as variables. or more partial differential equations, the result- Most groundwater models in use today are de- ing computer code can be considered a generic terministic mathematical models. Deterministic model. When the grid dimensions, boundary con- models are based on conservation of mass, mo- ditions, and other parameters (such as hydraulic mentum, and energy and describe cause and effect conductivity and storativity), are specified in an relations. The underlying assumption is that given application of a generic model to represent a par- a high degree of understanding of the processes ticular geographical area, the resulting computer by which stresses on a system produce subsequent program is a site-specific model. The ability of responses in that system, the system’s response to generic models to solve the governing equations any set of stresses can be predetermined, even if accurately is typically demonstrated by example the magnitude of the new stresses falls outside of applications to simplified problems. This does the range of historically observed stresses. not guarantee a similar level of accuracy when Deterministic groundwater models generally re- the model is applied to a complex field problem. quire the solution of partial differential equations. If the user of a model is unaware of or ignores Exact solutions can often be obtained analytically, the details of the numerical method, including but analytical models require that the parameters the derivative approximations, the scale of dis- and boundaries be highly idealised. Some deter- cretisation, and the matrix solution techniques, ministic models treat the properties of porous me- significant errors can be introduced and remain dia as lumped parameters (essentially, as a black undetected. For example, if the groundwater flow box), but this precludes the representation of het- equation is solved iteratively, but the convergence erogeneous hydraulic properties in the model. criterion is relatively too coarse, then the numeri- Heterogeneity, or variability in aquifer properties, is characteristic of all geologic systems and is cal solution may converge, but to a poor solution. now recognised as playing a key role in influenc- The inaccuracy of the solution may or may not be ing groundwater flow and solute transport. Thus, reflected in the mass-balance error. Unrecognized it is often preferable to apply distributed-param- errors in numerical groundwater models are be- eter models, which allow the representation of coming more possible as ‘user-friendly’ graphical more realistic distributions of system properties. interfaces make it easier for models to be used Numerical methods yield approximate solutions (and to be misused). These interfaces effective- to the governing equation (or equations) through ly place more ‘distance’ between the modeller the discretisation of space and time. Within and the numerical method that lies at the core of the discretised problem domain, the variable in- the model.

542 USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT

4.3. Flow and transport est. Stochastic approaches have resulted in many processes significant advances in characterising subsurface The process of groundwater flow is generally as- heterogeneity and dealing with uncertainty [12]. sumed to be governed by the relations expressed in Darcy’s law and the conservation of mass. 4.4. Governing equations Darcy’s law does have limits on its range of applicability, however, and these limits must be evalu- The development of mathematical equations that ated in any application. describe the groundwater flow and transport processes may be developed from the fundamental The purpose of a model that simulates solute principle of conservation of mass of fluid or of transport in groundwater is to compute the con- solute. Given a representative volume of porous centration of a dissolved chemical species in an medium, a general equation for conservation of aquifer at any specified time and place. The the- mass for the volume may be expressed as: masse de fluide ou de soluté. Etant donné un volume représentatif de milieux poreux, oretical basis for the equation describing solute transport has been well documented in thel’équation litera- générale(rate of massde conservation inflow) – (rate de masseof mass pour outflow) ce volume + peut être exprimée sous la ture [9–11] provide a conceptual frameworkforme: for (rate of mass production/consumption) = (rate of masse de fluide ou de soluté. Etant donné un volume représentatif de milieux poreux, analysing and modelling physical solute-transport mass accumulation) (4.1) processes in groundwater [11]. Changes in(massel’équation chemi -entrante) générale – (masse de conservation sortante) + de(masse masse produite/consommée) pour ce volume peut = (masse être exprimée accumulée) sous la This statement of conservation of mass (or con- cal concentration occur within a dynamic groundforme: - (4.1) water system primarily due to four distinct pro- tinuity equation) may be combined with a math- cesses: (massePour obtenir entrante)ematical une – équation (masse expression sortante) différentielle of +the (masse relevantdécrivan produite/consommée) t processl’écoulement to ou = le (masse transport, accumulée) ce bilan de conservation(4.1)obtain de masse a differential (ou équation equation de conti describingnuité) peut flow être or combiné avec une expression (1) advective transport, in which dissolved mathématiquetransport du processus [9, 10, correspondant13]. (Bear 1997; Domenico et Schwartz 1998; Freeze et chemicals are moving with the Pourflowing obtenir une équation différentielle décrivant l’écoulement ou le transport, ce bilan de Cherry 1979). groundwater; conservation de masse (ou équation de continuité) peut être combiné avec une expression 4.4.1. Groundwater flow equation (2) hydrodynamic dispersion, in which molecumathématique- du processus correspondant (Bear 1997; Domenico et Schwartz 1998; Freeze et 4.4.1 ÉQUATIONThe rate of flow DE L’HYDRODYNAMof water through a porousIQUE media SOUTERRAINE lar and ionic diffusion and small-scaleCherry varia -1979). is related to the properties of the water, the prop- tions in the flow velocity through theComme porous l’énonce la loi de Darcy, la quantité d’eau qui s’écoule à travers un milieu poreux media cause the paths of dissolved molecules erties of the porous media, and the gradient of dépend des propriétés de l’eau et de celles du milieu poreux, ainsi que du gradient de charge and ions to diverge or spread from the4.4.1 aver - ÉQUATIONthe hydraulic head,DE L’HYDRODYNAM as represented by Darcy’sIQUE law,SOUTERRAINE hydraulique,which ce qui can peut be être written formulé as: de la manière suivante: age direction of groundwater flow; Comme l’énonce la loi de Darcy, la quantité d’eau qui s’écoule à travers un milieu poreux (3) fluid sources, where water of one composidépend- des propriétés∂ deh l’eau et de celles du milieu poreux, ainsi que du gradient de charge qi = −Kij (4.2) (4.2) tion is introduced into and mixed withhydraulique, water ce qui peut∂x être formulé de la manière suivante: of a different composition; j −1 où q est le débit unitaire,∂h LT ; Kij est la conductivité−1 hydraulique (perméabilité) du milieu (4) reactions, in which some amount of a pari - where qi is the specific discharge, LT ; Kij is qi = −Kij −1 (4.2) ticular dissolved chemical species mayporeux be (un tenseurthe hydraulic de∂ secondx j conductivity ordre), LT of ;the et hporous est la chargemedium hydraulique, L. −1 added to or removed from the groundwater (a second-order tensor),−1 LT ; and h is the hydrau- oùOn qpeutest obtenir le débit une unitaire, forme généraleLT ; Kij de est l’équati la conductivitéon qui décrit hydraulique l’écoulement (perméabilité) en régime transitoiredu milieu as a result of chemical, biological, and physii - lic head, L. poreuxd’un fluide (un tenseurcompressible de second dans ordre), un aquifère LT−1; etnon h esthom laogène, charge anisotrope, hydraulique, en L combinant. la loi de cal reactions in the water or between the wa- A general form of the equation describing Darcy et l’équation de continuité. Une équation générale de l’hydrodynamique souterraine ter and the solid aquifer materials orOn other peut obtenirthe transient une forme flow générale of a compressible de l’équation fluid qui décritin a non- l’écoulement en régime transitoire separate liquid phases. peut s’écrire en notation tenseur Cartésien de la manière suivante: d’un fluide homogeneouscompressible dans anisotropic un aquifère aquifer non may hom beogène, derived anisotrope, en combinant la loi de The subsurface environment constitutes Darcya com -et l’équationby combining de continuité. Darcy’s Une law équa withtion the généralecontinuity de l’hydrodynamique souterraine plex, three-dimensional, heterogeneous hydrogeo- equation.∂ ⎛ ∂ h A ⎞general∂h groundwater flow equation peut s’écrire en ⎜notationK ⎟ tenseur= S Cartésien+W * de la manière suivante: logic setting. This variability strongly influences may⎜ beij written⎟ ins Cartesian tensor notation as: ∂xi ⎝ ∂xi ⎠ ∂t groundwater flow and transport, and such a real- (4.3) ity can be described accurately only through care- ∂ ⎛ ∂h ⎞ ∂h ⎜ ⎟ * −1 où S est l’emmagasinement⎜ Kij ⎟ = S spécifique,+W L ; t le temps,(4.3) T; W* est le flux volumétrique par ful hydrogeologic practice in the field. RegardlessS ∂x ∂x s ∂t i ⎝ i ⎠ of how much data are collected, however,unité uncer de- volume (positif pour les écoulements sortant et négatif pour les écoulements (4.3)entrant), −1 −1 tainty always remains about the propertiesT ; andet x sontwhere les coordonnéesS is the specific Cartésiennes, storage,−1 LTL. La, t conventionis time, T; d’additivité relative à l’analyse où S esti l’emmagasinementS spécifique, L ; t le temps, T; W* est le flux volumétrique par S W* is the volumetric flux per unit volume (posi- boundaries of the groundwater system ofpar inter tenseur- Cartésien est sous-entendue dans les équations 4.2 et 4.3. L’équation 4.3 peut unité de volume (positif pour les écoulements sortant et négatif pour les écoulements entrant), généralement−1 être appliquée si: les conditions isothermes dominent, le milieux poreux se T ; et xi sont les coordonnées Cartésiennes, L. La convention d’additivité relative à l’analyse déforme uniquement verticalement, les volumes des grains543 constitutifs reste constant pendant par tenseur Cartésien est sous-entendue dans les équations 4.2 et 4.3. L’équation 4.3 peut la déformation, la loi de Darcy s’applique (et si le gradient hydraulique constitue la seule généralement être appliquée si: les conditions isothermes dominent, le milieux poreux se force motrice), et les propriétés du fluide (densité et viscosité) sont homogènes et constantes. déforme uniquement verticalement, les volumes des grains constitutifs reste constant pendant Les propriétés aquifères peuvent varier dans l’espace, et les contraintes sur le fluide (W*) la déformation, la loi de Darcy s’applique (et si le gradient hydraulique constitue la seule peuvent varier dans l’espace et le temps. force motrice), et les propriétés du fluide (densité et viscosité) sont homogènes et constantes. Les propriétés aquifères peuvent varier dans l’espace, et les contraintes sur le fluide (W*) peuvent varier dans l’espace et le temps. 93

93 modelling

tive for outflow and negative for inflow), T−1, and in space or time. This may occur where water xi are the Cartesian co-ordinates, L. The summa- temperature or dissolved-solids concentration tion convention of Cartesian tensor analysis is im- changes significantly. When the water properties plied in Eqs. 4.2 and 4.3. Eq. 4.3 can generally be are heterogeneous and (or) transient, the relations applied if isothermal conditions prevail, the po- among water levels, hydraulic heads, fluid pres- rous medium only deforms vertically, the volume sures, and flow velocities are neither simple nor of individual grains remains constant during de- straightforward. In such cases, the flow equation formation, Darcy’s law applies (and gradients of is written and solved in terms of fluid pressures, hydraulic head are the only driving force), and fluid densities, and the intrinsic permeability of fluid properties (density and viscosity) are homo- the porous media [14]. geneous and constant. Aquifer properties can vary spatially, and fluid stresses (W*) can vary in space and time. 4.4.2. Seepage velocity If the aquifer is relatively thin compared to its lat- The migration and mixing of chemicals dissolved eral extent, it may be appropriate to assume that in groundwater will obviously be affected by groundwater flow is areally two-dimensional. the velocity of the flowing groundwater. The spe- Si l’aquifèreThis est relativementallows the three-dimensional peu épais par rappor flowt à son equation extension cific latérale, discharge il est préférable calculated de from Eq. 4.2 is some- considérer queto be l’écoulement reduced to the souterrain case of two-dimensionalse fait sur un plan, areal à deux timesdimensions. called theCeci Darcy permet velocity. de However, this nomenclature can be misleading because q does not réduire l’équationflow, forde l’hydrodynamiquewhich several additional souterra simplificationsine tri-dimensionnelle à un écoulement plan i are possible. The advantages of reducing the di- actually represent the speed of water movement. bi-dimensionnel, plusieurs autres simplifications sont alors possibles.Rather, La réductionq represents du nombre a volumetric flux per unit mensionality of the equation include less stringent i de dimensionsdata présenterequirements, l’avantage smaller d’un computer besoin memory moins strictre- decross-sectional données, de area.capacité Thus, de to calculate the actual mémoires informatiquesquirements, and plus shorter réduite, computer et d’un execution temps d’exécutiontimes seepage relativement velocity court of groundwater, pour one must ac- obtenir les solutionsto achieve numériques. numerical solutions. l’eau souterraine,count onfor doitthe actualprendre cross-sectional en compte la surfacearea through réelle de la section à travers laquelle l’eau s’écoule,which de laflow façon is suivante:occurring, as follows: Une expressionAn expression similaire similarà l’équation to Eq. 4.3 4.3 peutmay beêtre derived obtenue pour l’écoulement en deux for the two-dimensional areal flow of a homoge- dimensions d’un fluide homogène dans un aquifère captif, et écrite sousq ila forme:Kij ∂h neous fluid in a confined aquifer and written as: Vi = = − (4.5) (4.5) ε ε ∂x j ∂ ⎛ ∂h ⎞ ∂h ⎜ ⎟ où Vi est lawhere vitesse V d’écoulement is the seepage (aussi velocity comm (alsounément commonly appelée vitesse moyenne linéaire ou ⎜Tij ⎟ = S +W (4.4) i ∂x ∂x ∂t −1 i ⎝ i ⎠ vitesse moyennecalled interstitielle), average linear LT (4.4) velocity; et ε est or la average porosité intersti efficace- du milieu poreux. tial velocity), LT−1, and ε is the effective porosity where Tij is the transmissivity, L2T−1, and Tij = Kij 2 −1 of the porous medium. où Tij est lab; transmissivité, b is the saturated L T thickness; et Tij = ofK ijthe b; aquifer,b est4.4.3 l’épaisseur L ; S ÉQUATION mouillée de l’aquifère, DE TRANSPORT L; S DE MASSE est le coefficientis the d’emmagasinementstorage coefficient (sans(dimensionless); dimension); andet W W = Wb est le flux volumique par On peut obtenir, à partir du principe de conservation de masse, une équation décrivant le unité de surface,= W*b LT is−1 the. volume flux per unit area,LT −1. 4.4.3. Solute-transport equation transport et la dispersion d’un composé chimique dissous dans les écoulements souterrains Quand l’équationWhen 4.4Eq. est 4.4 appliquée is applied à toun an système unconfined aquifère(Eq. (water- libre 4.1), (nappe), toutAn comme equation on suppose on describingl’a fait que pour le fluxthe l’équation transport de and l’hydrodynamique disper- souterraine (Bear 1979; table) aquifer system, it must be assumed that sion of a dissolved chemical in flowing ground- est horizontal et que les équipotentielles sont verticales,Domenico que et Schwartzle gradient 1998; hydraulique Konikow et Grove 1977; Bear 1972; Bredehoeft et Pinder 1973; flow is horizontal and equipotential lines are ver- water may be derived from the principle of con- horizontal est égal à la pente de la nappe, et que le coefficientReddell d’emmagasinementet Sunada 1970). Le est principe égal à lade conservation de masse exige que la masse nette de tical, that the horizontal hydraulic gradient equals servation of mass (Eq. 4.1), just as a general flow capacité d’égouttement (S ) (Anderson et Woessner 1992). Notons que dans un aquifère libre, the slope of they water table, and that the solutéstorage entrant equation ou sortant was d’un so derived volume [9, d’aquifè 10, 14–17].re donné, The pendant prin- un intervalle de temps donné, l’épaisseur coefficientmouillée varie is equalsuivant to lesthe specificvariations yield de la soit(S yhauteur) égale[6]. àde ciplela laperte nappe of ouconservation au(ou gain charge). de ofmasse massPar stock requiresée dans that ce the volume net pendant l’intervalle de temps conséquent,Note la transmissivité that in an unconfined peut aussi system,changer the dans saturatedconsidéré. l’espace Cettemasset le relationoftemps solute (i.e.peut entering T alorsij = Korêtreij bleaving; exprimée a specified mathématiquement vol- en considérant tous les thickness changes as the water-table elevation (or b(x,y,t) = h − hb, et hb est l’altitude du mur de l’aquifère.flux entrantsume et sortantsof aquifer d’un during volume a given élémen timetaire interval représentatif must (VER), comme le décrit Bear head) changes. Thus, the transmissivity also can equal the accumulation or loss of mass stored in Dans certains cas, les propriétés du fluide telles que(1972, la p.19).densité et la viscosité peuvent change over space and time (i.e. Tij = Kijb; b(x,y,t) that volume during the interval. This relationship - considérablement= h hvarierb, and dans hb is l’espace the elevation ou le tempof thes. bottomCelaUne peut formeof avoir maygénéralisée lieu then dans be dedes expressed l’équation zones où mathematically lade transport de bymasse con -est présentée par Grove (1976), températurethe ou aquifer).la concentration de matière dissoute varientdans laquellefortement.sidering des Quand termes all lesfluxes représentent propriétés into and de les out réactionsof a representative chimiques et la concentration de soluté l’eau sont hétérogènesIn some field et (ou) situations, variables, les fluid relations properties aussientre bienles such niveaux danselementary le fluideaquifères, volume interstitiel les (REV),charges que sur as ladescribed surface solide:inr Ref. hydrauliques,as les density pressions and viscosityde fluides, may et varyles vitesses significantly d’écoulement [15] (19).ne sont ni simples ni ∂(εC) ∂ ⎛ ∂C ⎞ ∂ franches. Dans ces cas là, l’équation de l’hydrodynamique souterraine est= établie⎜εD et résolue⎟ − en ()εCV − C′W * + CHEM (4.6) ∂t ∂x ⎜ ij ∂x ⎟ ∂x i terme de pressions544 de fluides, de densité de fluide et de perméabilité intrinsèquei ⎝ duj ⎠milieui poreux (Konikow et Grove 1977). où CHEM est égal à:

∂C 4.4.2 VITESSE D’ÉCOULEMENT − ρ pour l’adsorption contrôlée par un équilibre linéaire ou les réactions b ∂t La migration et le mélange de composés chimiques dissous dans les eaux souterraines d’échange d’ions, dépendent de la vitesse d’écoulement de l’eau. Le débit spécifique calculé à partir de l’Eq. 4.2 s est parfois appelé vitesse de Darcy. Cependant, cette nomenclature peut être trompeuse car en ∑ Rk pour les réactions chimiques contrôlées, et (ou) fait qi ne représente pas la vitesse de déplacement de l’eau, qi estk= 1un flux volumétrique par unité de surface de la section traversée. Ainsi, pour calculer la vitesse réelle d’écoulement de − λ()εC + ρb C pour la décroissance,

94 2 −1 et où Dij est le coefficient de dispersion hydrodynamique (un tenseur d’ordre 2), L T , C' la concentration de solutés dans l’eau entrante ou sortante, C C– la concentration de l’espèce

chimique absorbée sur le solide (masse de solutés/masse de solide), ρb la densité globale du −3 −3 −1 sédiment, ML , Rk le taux de production de soluté dans la réaction k, ML T , et λ la −1 constante de décroissance (égale à ln2/T1/2), T (Grove 1976). 95

l’eau souterraine, on doit prendre en compte la surface réelle de la section à travers laquelle l’eaul’eau souterraine,s’écoule, de laon façon doit suivante:prendre en compte la surface réelle de la section à travers laquelle l’eau s’écoule,souterraine, de laon façon doit prendresuivante: en compte la surface réelle de la section à travers laquelle q Kij ∂h V = i = − (4.5) l’eau s’écoule,i deε la façonε ∂suivante: l’eau souterraine,q ion doitKij prendre∂xhj en compte la surface réelle de la section à travers laquelle Vi = = − (4.5) l’eau s’écoule, deε la façonε suivante:∂x j où Vi est la vitesseqi d’écoulementKij ∂h (aussi communément appelée vitesse moyenne linéaire ou Vi = = − −1 (4.5) oùvitesse Vi est moyenne la vitesse interstitielle),ε d’écoulementε ∂x j LT (aussi; et ε est comm la porositéunément efficace appelée du milieuvitesse poreux. moyenne linéaire ou q K ∂h V = i = − ij −1 vitesse où V est moyenne la vitessei interstitielle), d’écoulement LT (aussi; et ε estcomm la porositéunément efficace appelée du vitesse milieu moyenne poreux. linéaire (4.5) ou i ε ε ∂x j vitesse4.4.3 moyenneÉQUATION interstitielle), DE TRANSPORT LT−1; et ε est DEla porosité MASSE efficace du milieu poreux. où Vi est la vitesse d’écoulement (aussi communément appelée vitesse moyenne linéaire ou 4.4.3On peut obtenir,ÉQUATION à partir DEdu principeTRANSPORT de conservation DE MASSE de masse, une équation décrivant le vitesse moyenne interstitielle), LT−1; et ε est la porosité efficace du milieu poreux. On4.4.3transport peut obtenir,et ÉQUATIONla dispersion à partir DEd’undu TRANSPORTprincipe composé de chimi conservation DEque MASSEdissous de dans masse, les écoulementsune équation souterrains décrivant le transport(Eq. 4.1), ettout la commedispersion on l’a d’un fait pourcomposé l’équation chimi deque l’hydrodynamique dissous dans les souterraine écoulements (Bear souterrains 1979; 4.4.3On peut obtenir,ÉQUATION à partir DEdu TRANSPORTprincipe de conservation DE MASSE de masse, une équation décrivant le Domenico et Schwartz 1998; Konikow et Grove 1977; Bear 1972; Bredehoeft et Pinder 1973; transport(Eq. 4.1), ettout la commedispersion on l’a d’un fait composépour l’équation chimique de l’hydrodynamiquedissousLe premier dans termeles écoulements souterraine à droite dans (Bearsouterrains l’équation 1979; 4.6 représente la variation de concentration causée OnReddell peut etobtenir, Sunada à 1970) partir. Ledu principeprincipe de de conservation conservation de demasse masse, exige une que équation la masse décrivant nette de le (Eq.Domenico 4.1), tout et Schwartz comme on 1998; l’a fait Konikow pour l’ équationet Grove de1977; l’hydrodynamiquepar Bear la dispersion1972; Bredehoeft souterraine hydrodynamique. et Pinder (Bear 1979;1973; Cette expression est analogue à celle de la loi de Fick qui transportsoluté entrant et la ou dispersion sortant d’un d’un volume composé d’aquifè chimire donné,que dissous pendant dans un intervalleles écoulements de temps souterrains donné, DomenicoReddell et etSunada Schwartz 1970) 1998;. Le Konikowprincipe deet Groveconservation 1977;décrit Bearde masse les1972; flux exige Bredehoeft de diffusion.que la etmasse PinderCe modèlenette 1973; de de Fick suppose que le gradient de concentration est la soluté(Eq.soit égaleLe4.1), entrant premier toutà la oupertecomme termesortant ou onauà d’un droitel’again fait volumede dans pourmasse l’équation l’d’aquifè équationstockéere dans4.6 dedonné, l’hydrodynamiquere ceprésenteforce volume pendant motrice lapendant unvariation intervalleet quesouterraine l’intervalle lede fluxdeconcentration temps dispersif(Bearde temps donné, 1979; va causée des concen trations les plus fortes vers les plus faibles. Reddell et Sunada 1970).USE Le OprincipeF NUMER deICA conservationL MODELS TO SIMULde masseATE GRexigeOUN queDWAT laE masseR FLOW nette AND TRAde NSPORT soitDomenicoconsidéré. égalepar la à etCette dispersionla Schwartz perte relation ou hydrodynamique. au1998; peut gain Konikowalors de masse être et exprimée Cettestock Grove éeexpression dans1977;mathématiquement Cependant,ce Bear volume est 1972;analogue pendant cette Bredehoeft en àconsidéranthypothèse cellel’intervalle deet Pinderla n’est tousloi de de tempsles1973;pas Fick toujours qui vérifiée dans les observations et fait encore soluté entrant ou sortant d’un volume d’aquifère donné,Le pendant premier un terme intervalle à droite de tempsdans l’équation donné, 4.6 représente la variation de concentration causée considéré.Reddellflux décritentrants et CettelesSunada et flux sortants relation de1970) diffusion. d’un .peut Le volume principealors Ce modèleêtre élémen de exprimée conservation detaire Fick représentatif mathématiquement supposel’objet de masse que de(VER), recherches leexige gradient comme en que considérant etdela le d’études. masse concentrationdécrit nettetousBear Le lesdedeuxième est la terme représente le transport par advection et soit égale à la perte ou au gain de masse stockée dans parce volume la dispersion pendant hydrodynamique. l’intervalle de temps Cette expression est analogue à celle de la loi de Fick qui soluté(1972, entrant p.19).A ougeneralised sortant d’un form volume of the d’aquifè solute-transportre donné, pendantequa- untion. intervalle The fourth de temps term donné, lumps all of the chemical, considéré.flux forceentrants motriceCette et sortants relation et que d’un peutle flux volumealors dispersif être élémen exprimée va destaire concen représentatifmathématiquementdécrittrations les les(VER), déplacements plus en commefortes considérant vers lede décrit lessolutés tousplus Bear lesfaibles. en considérant une vitesse moyenne d’écoulement soit égale à tionla perte is presented ou au gain in de Ref. masse [18], stock in whichée dans terms décritce volume are les flux pendantgeochemical, de diffusion. l’intervalle and Ce biologicaldemodèle temps de reactionsFick suppose that quecause le gradient de concentration est la (1972,fluxUne Cependant,entrantsforme p.19). généralisée et sortantscette hypothèsede d’un l’équation volume n’est de élémen pastransport toujourstaire de représentatif massevésouterrain.rifiée est dans présentée(VER), Le les troisième observationscomme par Grove termele décrit (1976) et représente fait Bear, encore les effets d’un mélange au niveau d’une zone de incorporated to represent chemical reactionsLeforce premier and motrice transferterme et que à droitele of flux mass dans dispersif between l’équation va des the 4.6 concenliquid représente trations and solidla les variation plus fortes de concentrationvers les plus faibles.causée considéré.l’objet Cette de recherches relation peut et d’études. alors être Le exprimée deuxième mathématiquement termerecharge représente ou d’injection, leen transport considérant avec par une tousadvection source les de et fluide ayant une concentration différente de celle Une(1972,dans formelaquelle p.19). solutegénéralisée des termesconcentration de représentent l’équation both indeles the transportréactions pore fluid dechimiques masse and onest et présentéelaphases concentration or par conversion Grove de soluté (1976) of dissolved, chemical spe- flux entrants et sortants d’un volume élémentaire représentatifparCependant, la dispersion (VER), cette hydrodynamique.hypothèsecomme le décritn’est pasBearCette toujours expression vérifiée est analoguedans les àobservations celle de la loiet defait Fick encore qui dansaussi décrit laquellebien danstheles des solidledéplacements fluidetermes surface, interstitiel représentent as: de quesolutés sur les la réactions ensurface considérant solide: chimiquesde l’eau une et souterraine.ciesvitesse la concentrationfrom moyenne one Le form quatrièmeded’écoulement to soluté another. terme T he chemicalenglobe attoutes- les réactions chimiques, (1972,Une forme p.19). généralisée de l’équation de transport de décritl’objetmasse les deest flux recherchesprésentée de diffusion. paret d’études. Grove Ce modèle (1976) Le deuxième de, Fick suppose terme représente que le gradient le transport de concentration par advection est laet aussisouterrain. bien dans leLe fluide troisième interstitiel terme que représente sur la surface les effets solide:géochimiques d’un mélangetenuation et biologiques au ofniveau inorganic d’unequi engendrent chemicalszone de uncan tr ansfertoccur byde masse entre le liquide et les dans laquelle des termes représentent les réactions chimiquesforcedécrit motriceles etdéplacements la et concentrationque le flux de dispersif desolutés soluté va en des considérant concentrations une les vitesse plus fortes moyenne vers les d’écoulement plus faibles. recharge ou d’injection,⎛ avec⎞ une source de fluide ayant une sorption/desorption,concentration différente precipitation/dissolution, de celle or Une forme généralisée∂(εC) ∂ ⎜de l’équation∂C ⎟ ∂ de transport de* phasesmasse estsolides présentée ou les par passages Grove (4.6) entre(1976) différentes, formes chimiques dissoutes. La diminution aussi bien dans le fluide= interstitielεDij − que ()surεCV lai surface− C′W +solide:Cependant,souterrain.CHEM Lecette troisième hypothèse terme n’est représente pas toujours les effets vérifiée d’un dans mélange les observations au niveau d’uneet fait zone encore de de l’eau∂ t souterraine.∂x ⎜⎛ ∂Lex ⎟⎞quatrième∂x terme desenglobe concentrations toutesoxidation/reduction; les desréactions composés organicchimiques, chimique chemical s caninorganiques adsorb peut s’expliquer par des dans laquelle∂ (εdesC) termes∂i ⎝⎜ représentent∂Cj ⎠⎟ ∂i les réactions chimiques*recharge ou et d’injection,la concentration avec unede (4.6) solutésource de fluide ayant une concentration différente de celle = εDij − ()εCVi − C′W l’objet+ CHEM de recherchesor degrade et byd’études. microbiological Le deuxième processes. terme représente There le transport par advection et géochimiques et biologiques⎜ ⎟ qui engendrent unphénomènes transfert de massed’adsorption/désorption, entre le liquide et lespréci pitation/dissolution, ou des phénomènes aussi bien dans ∂ tle fluide∂xi ⎝⎛interstitiel∂x j ⎠⎞ que∂x isur la surface solide:(4.6) où CHEM est∂ (égalεC) à: ∂ ⎜ ∂C ⎟ ∂ *dedécrit l’eau les souterraine.déplacementshas been considerable Le de quatrièmesolutés (4.6) progress en terme considérantin modelling englobe une these toutes vitesse les moyenne réactions d’écoulement chimiques, phases solides= ou lesε Dpassagesij − entre()ε CVdifférentesi − C′W formes+ CHEM chimiques dissoutes. La diminution ∂t ∂x ⎜ ∂x ⎟ ∂x d’oxydoréduction;reaction les processes; composés however, organiques a comprehensive eux, peuvent re être- adsorbés ou décomposés au où CHEM estwhere égal CHEMà: i ⎛⎝ equalsj :⎞⎠ i souterrain.géochimiques Le ettroisième biologiques terme qui représente engendrent les effetsun transfert d’un mélange de masse au entreniveau le d’uneliquide zone et lesde des concentrations∂(εC∂)C ∂ des ∂Ccomposés∂ chimiques* coursinorganiques de processus peut microbiologiques.s’expliquer par Ildes y a eu un progrès considérable en matière de = ⎜εD ⎟ − ()εCV − C′W phases+ CHEM solides view ou of les the passages reaction (4.6) entreprocesses différentes and their formes represen chimiques- dissoutes. La diminution où CHEM est− ρégalb à: pour⎜ l’adsorptionij ⎟ contrôléei par unrecharge équilibre ou d’injection,linéaire ou lesavec réactions une source de fluide ayant une concentration différente de celle phénomènes∂t ∂ t d’adsorption/désorption,∂xi ∂x j ∂xi précipitation/dissolution,modélisationtation de ces in outransportprocessus; des modelsphénomènes cependant, is beyond un btheilan scope complet of de ces processus et de leur ∂C ⎝ ⎠ des concentrations des composés chimiques inorganiques peut s’expliquer par des − ρb pour l’adsorption contrôlée parde un l’eauéquilibre souterraine.this linéaire chapter. ou Le les quatrième réactions terme englobe toutes les réactions chimiques, oùd’échange CHEMd’oxydoréduction; estd’ions, égal∂ tà: les composés organiques eux,représentation peuvent être dans adsorbés les modèles ou décomposés de transport au va au delà de l’objectif de ce chapitre. ∂C phénomènesgéochimiques etd’adsorption/désorption, biologiques qui engendrent préci unpitation/dissolution, transfert de masse ouentre des le liquidephénomènes et les cours de−s ρprocessusb pour microbiologiques.l’adsorption contrôlée Il y para eu un un équilibre progrès linéaireconsidérable ou les enréactions matière de d’échange d’ions,∂ t Si les réactionsIf reactions sont limitées are limited à des échanges to equilibrium-controlled ou adsorptions contrôlées par un équilibre et des for Rlinear pour equilibrium les réactions controlled chimiques sorptioncontrôlées,d’oxydoréduction;phases or etion- (ou) solides ou lesles passagescomposés entre organiques différentes eux, formes peuvent chimiques être adsorbés dissoutes. ou décomposés La diminution au modélisation∑s k∂ Cde ces processus; cependant, un bilan completsorption de ces or processus exchange et and de first-order leur irreversible d’échange d’ions,exchange−k=1ρb pourreactions, l’adsorption contrôlée parcoursdesréactions un équilibreconcentrations de irréversiblesprocessus linéaire microbiologiques.des deou premier lescomposés réactions ordre chimique (décroiIl y assance), seu inorganiques un alorsprogrès on peutconsidérablepeut écrire s’expliquer l’équation en matière par de base desde représentation∑ Rk∂ tpour dans les les réactions modèles chimiquesde transport contrôlées, va au delà et de (ou) l’objectif rate (decay) de ce reactions,chapitre. then the general governing ks=1 phénomènesmodélisation(Eq. 4.6) de equationla de façond’adsorption/désorption, ces suivante:(Eq.processus; 4.6) can cependant, be written préci unas:pitation/dissolution, bilan complet de cesou processusdes phénomènes et de leur d’échange d’ions,− λR()εC + ρb C pour la décroissance, Si les réactions∑ k pour sont les limitées réactions à des chimiques échanges contrôlées, ou adsorptionsreprésentation et (ou) contrôlées dans les par modèles un équilibre de transport et des va au delà de l’objectif de ce chapitre. ks=1 d’oxydoréduction; les composés⎛ organiques⎞ eux, peuvent être adsorbés ou décomposés au − λ()εC + ρb C pour la décroissance, 2 −1 * et oùréactions D est le irréversibles coefficient de de dispersion premier ordre hydr odynamique(décroissance), (un alorstenseur on∂C d’ordre peutρb écrire∂ 2),C L l’équationT∂ ⎜, C' la∂ deC base⎟ ∂ C′W ρb ij ∑ Rk pour les réactions chimiques contrôlées,coursSi les et réactionsde (ou) processus +sont limitéesmicrobiologiques.= à ⎜desD échanges ⎟Il− y oua() CVadsorptionseu un+ progrès contrôlées− λ Cconsidérable− λparC un équilibreen matière (4.7) et desde fork=1 s chemical rate-controlledpour la décroissance, reactions, and (or) ∂t ε ∂t ∂x2 −1ij ∂x ∂x i ε ε etconcentration où(Eq. D 4.6)est− le λde ()coefficientε Clasolutés +façonρb C dans suivante: de dispersionl’eau entrante hydr ouodynamique sortante, C (un C– tenseurla concentration d’ordre 2), de Ll’espèceT⎜ , C' la ⎟ ij réactionsmodélisation irréversibles de ces processus;de premieri ⎝ cependant,ordrej (décroi⎠ uni ssance), bilan completalors on peutde ces écrire processus l’équation et de baseleur concentrationchimique absorbée de solutés sur le danssolide l’eau (masse entrante de solutés/masse ou sortante,représentation de C solide), C– la ρ concentrationdansb la densité les modèles globale de2 l’espèce−de1 du transport va au delà de l’objectif de ce chapitre. et où Dij est− leλ ()coefficientεC + ρb C pourdefor dispersion decay, la décroissance,⎛ hydr⎞odynamique L’évolution(Eq. (un4.6) tenseurde* ladans façon d’ordre le temps suivante: 2), de L la T concentration, C' la adsorbée dans l’Eq. 4.7 peut être représentée en −3 ∂C ρb ∂ C ∂ ⎜ ∂C ⎟ ∂ C′W ρb −3 −1 chimiquesédiment, absorbéeML , R ksur +le letaux solide de= production(masseD de desolutés/masse −solutéCV dans + dela solide),réaction− λC ρ bk− , laML densitéλCT ,(4.7) globaleet λ la du concentration de solutés dans l’eau⎜ entranteij ⎟ou sortante,()fonctionSii les C réactions C– de lala concentration sont limitées dude à2 dessolutél’espèce−1 échanges (4.7) en utilisant ou adsorptions la chaîne de contrôlées calcul, de par la façonun équilibre suivante: et des et où D estand le− 3 coefficientwhere∂t εD ∂deist thedispersion∂ xcoefficient⎜ ∂hydrx −⎟1 odynamiqueof ∂hydrodynamx (un-ε tenseur d’ordreε 2),−3 L−1T⎛ , C' la ⎞ * sédiment,constanteij deML décroissance, Rk le tauxij (égale de àproduction ln2/i ⎝ T1/2), Tj de⎠ (Grove solutéi 1976). dans la réaction∂C ρ k, ∂MLC T∂ ⎜, et λ∂ Cla ⎟ ∂ C′W ρ chimique absorbée sur le solide (masse de solutés/masseréactions2 −1 de solide), irréversibles +ρb lab densité de =premier globaleD ordre du (décroi− ssance),CV + alors −onλ Cpeut− écrireb λC l’équation de base concentrationic de dispersion solutés dans (a second-order l’eau entrante tensor), −ou1 sortante, L T , CC’ C– lad CconcentrationdC ∂C de ⎜l’espèceij ⎟ ()i (4.7) constante de décroissance−3 (égale à ln2/T1/2), T (Grove 1976). ∂t ε ∂t −3∂x−1⎜ 95∂x ⎟ ∂x ε ε sédiment,L’évolution MLis the, dans Rconcentrationk le le tauxtemps de de productionof la theconcentration solute de insoluté adthesorbée (Eq. sourcedans 4.6) dansla deréaction l’Eq.Tlahe façon =temporal 4.7 k ,suivante:peut ML être changeTi représentée⎝, et inλ sorbedlaj ⎠ en concentrationi in (4.8) chimique absorbée sur le solide– (masse de solutés/masse de solide),dt ρb ladC densité∂t globale du constantefonction deor décroissance de sink la concentrationfluid, C(égale is the àdu ln2/ concentrationsolutéT ), en T −utilisant1 (Grove of the la 1976). chaînespe- deE calcul,q. 4.7 cande la be façon represented suivante:95 in terms of the solute −3 1/2 L’évolution dans le temps −de3 −la1⎛ concentration⎞ adsorbée dans l’Eq. 4.7 peut être représentée en sédiment, MLcies , adsorbedRk le taux on de the production solid (mass de of soluté solute/massPour dans des la réactionsréactionconcentration d’échangesk, ML usingT ,et theet d’adsorption λ chain la rule àof l’équilibre calculus, * asd C dC , dC–/dC ainsi que C– C, ∂C ρb ∂C ∂ ⎜ ∂C95 ⎟ ∂ C′W ρb constante de décroissancedC dC (égale∂C à ln2/T ), T −1 (Grovefonction 1976). de follows:la concentration+ = du ⎜solutéD en⎟ utilisant− ()CV la chaîne+ de− calcul,λC − deλ laC façon suivante: (4.7) of solid), =ρb is the bulk density1/2 of the sediment,dépend uniquement ∂t ε de∂ tC. Par∂x conséquent,⎜ ij (4.8)∂x ⎟ la∂ xrelationi d’équilibreε pourε C– et dC–/dC peut être −3 dt dC ∂t i ⎝ j ⎠ i ML , Rk is the rate of production of the solutesubstituée in dansdC l’équationdC ∂C de base pour95 formuler une équation partielle différentielle ne reaction k, ML−3T−1, and l is the decay constant = (4.8) (4.8) Pour des réactions d’échanges et d’adsorption à L’évolutiondépendantl’équilibre qued dansdCt dCde le d,C tempsdC. CL’unique–/d∂t Cde ainsi la concentrationéquation que C– Cde, transport adsorbée obtenuedans l’Eq. est 4.7 résolue peut êtreen fonctionreprésentée de enla (equal to ln2/T ), T−1 [18]. dépend uniquement de1/2 C. Par conséquent, la relationconcentrationfonction d’équilibre de la concentrationde pour soluté. C– etLa d duCconcentration–/d solutéC peut en utilisantêtre adso rbée la chaîne peut dealors calcul, être decalculée la façon en suivante: utilisant la Pour des réactionsFor equilibrium d’échanges sorption et d’adsorption and exchange à l’équilibre reactions d C dC , dC–/dC ainsi que C– C, substituéeThe dansfirst l’équationterm on the de right base side pour of Eq.formul 4.6relation errepre une- d’équilibre.équation– partielle La réaction différentielle d’adsorption– ne linéaire considère que la concentration de soluté dépend uniquementddC C /dC,d asCde ∂C wellC. Par as conséquent, C is a function la relation of Cd’équilibre alone. pour C– et dC–/dC peut être sents the change in concentration due to hydrody- = – – dépendant que de C. L’unique équation de transportadsorbée obtenue surT herefore,le est milieu résolue theporeux equilibriumen fonction est directement relationde la forproportionnelle C and dC / à la concentration du (4.8) soluté namic dispersion. This expression is analogoussubstituée to dansdt l’équationdC ∂t de base pour formuler une équation partielle différentielle ne concentration de soluté. La concentration adsorbéedans peutle fluide alorsdC contenu canêtre be calculée danssubstituted les en pores, utilisantinto suivant the governingla la relation equation Fick’s Law describing diffusive flux. ThisPourdépendant Fickian des réactions que de Cd’échanges. L’unique et équation d’adsorption de transport à l’équilibre obtenue dC estdC résolue, dC–/d Cen ainsi fonction que Cde– Cla, relation d’équilibre. La réaction d’adsorption linéaire considèreto developque la concentration a partial differential de soluté equation in terms model assumes that the driving force is theconcentration con- de soluté. La concentration adsorbée peut alors être calculée en utilisant la dépend uniquementofC =C Konly.d C de T Che. Par resulting conséquent, single latransport relation equationd’équilibre pour C– et dC–/dC (4.9)peut être adsorbéecentration sur le milieu gradient poreux and est that directement the dispersive proportionnelle flux à la concentration du soluté relationsubstituée d’équilibre. dansis solved l’équation La for réaction solute de based’adsorption concentration. pour formul linéai Serorbedre uneconsidère conéquation- que partiellela concentration différentielle de soluté ne dans leoccurs fluide contenuin a direction dans les from pores, higher suivant concentrations la relation adsorbée surcentration le milieu can poreux then estbe calculateddirectement using proportionnelle the equi- à la concentration du soluté towards lower concentrations. However, dépendantthis96 as- que de C. L’unique équation de transport obtenue est résolue en fonction de la librium relation. The linear-sorption reaction sumptionC = Kis Cnot always consistent with fieldconcentrationdans leob -fluide contenude soluté. dans La les concentration pores, (4.9) suivant adso la relationrbée peut alors être calculée en utilisant la d considers that the concentration of solute sorbed servations and is the subject of much ongoingrelation d’équilibre. La réaction d’adsorption linéaire considère que la concentration de soluté toC =theK porousC medium is directly proportional to research and field study. The second term repre- d (4.9) 96 adsorbée surthe le concentrationmilieu poreux of est the directement solute in the proportionnelle pore fluid, à la concentration du soluté sents advective transport and describes the move- dans le fluideaccording contenu todans the les relation pores, suivant la relation ment of solutes at the average seepage velocity 96 of the flowing groundwater. The third term rep- C = K C (4.9) (4.9) resents the effects of mixing with a source fluid d 3 –1 that has a different concentration than the ground- where Kd is the distribution coefficient, L M . water at the location of the recharge or 96 injec- This reaction is assumed to be instantaneous and

545

3 −1 où Kd est le coefficient de partition, L M . Cette réaction est supposée instantanée et irréversible. La courbe de la concentration adsorbée en fonction de la concentration dissoute correspond à un isotherme. Si cette relation est linéaire, la pente (dérivée) de l’isotherme, dC–

/dC, correspond au coefficient d’équilibre de partition, Kd. Ainsi, dans le cas d’un isotherme linéaire,

dC dC ∂C ∂C = = K (4.10) dt dC ∂t d ∂t Après avoir substitué cette relation dans l’Equation 4.7, on peut réécrire cette dernière sous la forme:

∂C ρ K ∂C ∂ ⎛ ∂C ⎞ ∂ C'W * ρ K b d = ⎜ ⎟ + − − b d λ + ⎜ Dij ⎟ − ()CVi λC C (4.11) ∂t ε ∂t ∂xi ⎝ ∂x j ⎠ ∂xi ε ε

En factorisant le terme (1+ ρb K d ε ) et en définissant un facteur de retard, Rf (sans dimension), comme:

ρ K R =1+ b d (4.12) f ε et en substituant cette relation dans l’Equation 4.11, on obtient:

∂C ∂ ⎛ ∂C ⎞ ∂ C'W * = ⎜ ⎟ − + − λ Rf ⎜ Dij ⎟ ()CVi Rf C (4.13) ∂t ∂xi ⎝ ∂x j ⎠ ∂xi ε

Comme Rf est constant sous ces hypothèses, la solution de cette équation de base est identique à la solution de l’équation de base sans phénomènes d’adsorption, sauf que la

vitesse, le flux dispersif, et les sources d’impulsions sont réduits par un facteur Rf. Le processus de transport apparaît ainsi comme retardé à cause de l’équilibre instantané d’adsorption sur le milieu poreux. Dans la formulation conventionnelle de l’équation de transport de masse (Eq. 4.6), on définit le coefficient de dispersion hydrodynamique comme la somme de la dispersion mécanique et de la diffusion moléculaire (Bear 1997). La dispersion mécanique est fonction, à la fois des propriétés intrinsèques du milieu poreux (comme des conductivités hydrauliques et des porosités hétérogènes), mais aussi de l’écoulement du fluide. La diffusion moléculaire dans un

milieumo poreuxdellin gsera différente de celle avec de l’eau libre du fait du rôle de la porosité et de la

tortuosité. Ces relations se présentent généralement sous la forme:

3 −1 où Kd est lereversible. coefficient The de curve partition, relating L3 Msorbed−1 . Cette concentra réaction- est supposée instantanée et où Kd est le coefficient de partition, L M . Cette réaction est supposéeVm Vinstantanéen et Dij = αijmn + Dm , i, j, m, n = 1, 2, 3 i , j(4.14), m, n = 1,2,3 (4.14) irréversible.tion La courbe to dissolved de la concentration concentration ad issorbée known en asfonction an de la concentrationV dissoute irréversible. La courbe de la concentration adsorbée en fonction de la concentration dissoute isotherm. If that relation is linear,3 −1 the slope (de- 3 −1 correspondoù Kd est àle un coefficient isotherme. de Si cettepartition, relation L–M est .lin Cetteéaire, réactionla pente (dérivée)est supposée de l’isotherme, instantanée d Cet– où Kd est le coefficient de partition, L M . Cette réaction est supposée instantanée et rivative) of the isotherm, dC 3 /dC−1, is knownoù α ijmn as est la dispersivité dans le milieu poreux (un tenseur d’ordre 4), L; Vm et Vn sont les where αijmn is the dispersivity of the porous me- irréversible.où/dC ,K correspondd est le La coefficient courbeau coefficient de dela concentrationpartition, d’équilibre L deMad sorbéepartition,. Cette en réaction Kfonctiond. Ainsi, estde dans lasupposée concentrationle cas d’uninstantanée isotherme dissoute et irréversible. La courbe de la concentration adsorbée en fonctionthe deequilibrium la concentration distribution dissoute coefficient, Kcomposantesd. Thus, de la vitesse d’écoulement du fluide respectivement dans les directions m et n, linéaire, dium (a fourth-order tensor), L; Vm and Vn are linéaire,correspondirréversible. àin La un the courbeisotherme. case ofde a la Silinear concentration cette isotherm, relation adestsorbée linéaire, en lafonction− 1pente (dérivée) de la concentration de l’isotherme, dissoute dC– 2 −1 correspond à un isotherme. Si cette relation est linéaire, la pente (dérivée) de l’isotherme, dC– LT ; Dm estthe le componentscoefficient deof diffuthe flowsion moléculairevelocity of theeffective, fluid L T ; et |V| est la norme du /dC, correspond au coefficient d’équilibre de partition,correspond/dC, correspond K . Ainsi, à un isotherme.dansau coefficient le cas Si d’un cette d’équilibre isotherme relation de est partition, linéaire, laKd pente. Ainsi, (dérivée)in dans the lem de casand l’isotherme, d’un n directions, isotherme dC– respectively, LT−1; D d dC dC ∂C ∂C m 97 linéaire, /dC, correspond =au coefficient= Kd d’équilibre de partition, Kd. Ainsi,is dans the effectivele cas d’un coefficient isotherme (4.10) of molecular diffusion, linéaire, dt dC ∂t d ∂t (4.10) (4.10) linéaire, dt dC ∂t ∂t L2T−1, and |V| is the magnitude of the velocity vec- Après avoir substitué cette relation dans l’Equation 4.7, on peut réécrire −cette1 dernière sous la dC dC ∂C ∂C Après avoir AfterdsubstituéC substitutingdC cette∂C relation this∂C dans relation l’Equation into Eq. 4.7, 4.7, on wepeut réécriretor, LT cette, defined dernière as sous la = = Kd (4.10) = = Kd forme: ddCt ddCC ∂∂Ct ∂∂Ct (4.10) dt dC ∂t ∂t forme: can then rewrite Eq. 4.7 as: −1 2 2 2 = = Kd V = V +V +V (4.10)[9, 10, 19]. dt dC ∂t vecteur∂t vitesse, LT , définie comme x y z (Bear 1979; Domunico et Schwartz Après avoir substitué cette relation dans l’Equation 4.7,* on peut réécrire cette dernière sous la Après avoir substitué cette relation dans l’Equation 4.7, on peut réécrire cette ⎛dernière⎞ sous la * ∂C ρb Kd ∂C ∂ ⎛ ∂C ⎞ ∂ C'W ρb Kd ∂C ρb Kd ∂C ∂ ⎜1998;∂C Scheidegger⎟ ∂ 1961)C'W. La dispersivitéρb Kd dans un milieu poreux isotrope peut être caractérisée forme:Après avoir+ substitué cette= relation⎜ Dij dans⎟ − l’Equation()CVi + 4.7, on− λ peutC − réécrireλC cette dernière (4.11) sous la forme: + = ⎜ Dij ⎟ − ()CVi + − λC − Theλ dispersivityC of (4.11) an isotropic porous medi- ∂t ε ∂t ∂xi ⎜⎝ ∂x j ⎟⎠ ∂xi ε ε forme: i ⎝par deuxj ⎠ constantes.i Il s’agit de laum dispersivité can be defined longitudinale, by two αL , constants.et de la dispersivité These are ⎛ ⎞ * ⎛ ⎞ * ∂C ρb Kd ∂C ∂ ∂C ∂ α C'W ρb Kd ∂C ρ K ∂C ∂ ∂C ∂ EnC'W factorisantρ leK terme (⎜1transversale,+ ρb K d⎟ ε ) et T.en Celles-cidéfinissant sont un liéesfacteurthe longitudinalaux de coefficientsretard, dispersivity Rf (sansde dispersion of the medium, transversale αL, et b d ⎜ ⎟ En factorisant+ ble d terme= (1D+ijρb K d −ε ) et()CV eni +définissant* − λC un− facteurλC de retard, (4.11)Rf (sans + = Dij − ()CVi + − λC − λC ⎛⎜ (4.11)⎞⎟ (4.11) ⎜ ⎟ ∂∂Ct ρbεKd ∂∂Ct ∂∂xi ⎝longitudinale∂xCj ⎠ ∂∂x ipar D =C α'εW|V| et D =ρ αbεandK|Vd |. the La transverseplupart des dispersivity applications of de the modèles medium, de αtransportT. ∂t ε ∂t ∂xi ⎝ ∂x j ⎠ ∂xi dimension), ε + comme:ε = ⎜ D ⎟ − ()CV L+ L − λCT− T λC (4.11) dimension), comme: ⎜ ij ⎟ i These are related to the longitudinal and trans- ∂t ε ∂t ∂xi ⎝à des∂x problèmesj ⎠ ∂xi hydrogéologiquesε quiε ont été établies jusqu’à maintenant se sont basées sur En factorisant le terme (1+ ρb K d ε ) et en définissant un facteur de retard, Rf (sans En factorisant le terme (1+ ρb K d ε ) et en définissant un facteur de retard, Rf (sans verse dispersion coefficients byD L = αL|V| and DT ρb Kd Factoringρ boutKd thecette term formulation (1 + ρb K d /conventionnelle.ε) and defining En factorisantRf =le1 +terme (1+ ρ K ε ) et en définissant un facteur de retard, (4.12)Rf (sans dimension), Rcomme:f =1+ b d = αT|V|. Most applications (4.12) of transport models to dimension), comme: a retardationε factor, Rf (dimensionless), as: dimension), comme: Bien que la théorie conventionnellegroundwater prétende queproblems α est that généralement have been document une propriété- et en substituant cette relation dans l’Equation 4.11, on obtient: L ρ K et en substituant cetteρ brelationKd dans l’Equation 4.11, on obtient: ed to date have been based on this conventional b d Rf =1+ intrinsèque de l’aquifère, il (4.12)est démontré dans la pratique (4.12)qu’elle dépend et est proportionnelle Rf =1+ (4.12) ρbεKd formulation. ε * Rf ∂=C1+ ∂ ⎛ à∂ l’échelleC ⎞ ∂ de mesure.C'W La* plupart des valeurs de αL indiquées (4.12) se situent dans l’intervalle de ∂C ∂ε ⎜ ∂C ⎟ ∂ C'W Although conventional theory holds that αL is et en substituantRf cette= relation⎜ Dij dans⎟ − l’Equation()CVi + 4.11, on− obtient:Rf λC (4.13) et en substituant cette relation dans l’Equation 4.11, on obtient:andR f substituting= ⎜ Dij this ⎟relation− ()CV intoi +Eq. 4.11,− Rresultsf λC (4.13) ∂t ∂xi ⎜ 0,01∂x j ⎟à 1,0∂x ifois l’échelleε de mesure, biengenerally que le rapportan intrinsic αL à l’échelleproperty de of mesure the aquifer, ait tendance it à et en substituantin: ∂ tcette∂ relationxi ⎝ ∂ xdansj ⎠ l’Equation∂xi 4.11,ε on obtient: décroître avec des échelles* plus grandesis found(Anderson in practice 1984; Gelharto be dependent et al. 1992). on Laand dispersion pro- à * ∂C ∂ ⎛ ∂C ⎞ ∂ C'W ∂C ∂ ⎛ ∂C ⎞ ∂ C'WComme Rf est constant⎜ sous ces⎟ hypothèses, la solution de cette équation de base est ⎜ ⎟ Comme Rf Restf constant= D sousij ces− hypothèses,()CVi + la * solution− Rf λC de portionalcette équation to the de scale (4.13)base of estthe measurement. Most Rf = Dij − ()CVi + − Rf λC ⎛⎜ l’échelle⎞⎟ (4.13) du terrain (communément appelée macrodispersion) est due à des variations spatiales ⎜ ⎟ identique à la ∂∂solutionCt ∂∂xi ⎝de l’équation∂xCj ⎠ ∂∂xi de baseC sans'εW phénomènes d’adsorption, sauf que la ∂t ∂xi ⎝ ∂x j ⎠ ∂xi εidentique à Rla solution= ⎜deD l’équation⎟ − ()deCV base+ sans −phénomènesR λC reported d’adsorption, values saufof (4.13)α quefall lain a range from 0.01 to f ∂t ∂x ⎜ ijà∂ xgrandes⎟ ∂x échellesi desε propriétésf hydrauliques. En conséquence,L l’utilisation de valeurs de vitesse, le flux dispersif,i ⎝ et lesj ⎠ sourcesi d’impulsions sont réduits1.0 timespar unthe facteurscale of R thef. Le measurement, although Comme R est constant sous ces hypothèses, lavitesse,Comme solution leR f flux deest cette dispersif,constant équation souset les deces sourcesbase hypothèses, est d’impulsions la solution sont deréduits cette paréquation un facteur de base Rf. estLe f dispersivité relativement grandes et thede ratiopropriétés of α tohydrauliques scale of measurement uniformes (tendsKij et to ε) est identiqueCommeprocessus R àf de laest transportsolution constant deapparaît sous l’équation ces ainsi hypothèses, de comme base sans retardéla solutionphénomènes(4.13) à cause de cette d’adsorption,de l’équilibreéquationL desauf instantané base que estla identique à la solution de l’équation de base sans phénomènes d’adsorption,inapproprié sauf que pour la décrire les phénomènesdecrease de transport at larger dans scales les systèmes[20, 21]. géologiquesField-scale dis (Smith- et vitesse,identiqued’adsorption le à flux lasur solution ledispersif, milieu de poreux etl’équation les. sources de based’impulsions sans phénomènes sont réduits d’adsorption, par un facteur sauf queRf. Lela vitesse, le flux dispersif, et les sources d’impulsions sont Becauseréduits parRf isun constant Schwartzfacteur underR 1980)f. Le these . Si unassumptions, modèle, appliqué persion à un système (commonly dans calledlequel macrodispersion)la conductivité hydraulique re- vitesse,processus le defluxthe transport solutiondispersif, toapparaît etthis les governing sourcesainsi comme equationd’impulsions retardé is identi sontà cause- réduits de parl’équilibre un facteur instantané Rf. Le processus de transport apparaît ainsi comme retardéDans la àformulation cause de conventionnellel’équilibreest variable, instantané de l’équatiutilise ondes de valeurs transport moyennes desults masse fromet (Eq.par large-scale conséquent 4.6), on définit spatial ne représente variations pas in hydrauexplicitement- d’adsorptionprocessus decal sur transportto le the milieu solution apparaît poreux for. theainsi governing comme equation retardé withà cause de l’équilibre instantané d’adsorption sur le milieu poreux. le coefficient de dispersion hydrodynamiquecette variabilité, lacomme calibration la somme du modèle delic la properties.dispersion va probablement mécaniqueConsequently, donner et desthe usevaleurs of relatively de coefficients d’adsorptionno sur sorption le milieu effects, poreux except. that the velocity, dis- large values of dispersivity together with uniform deDans la ladiffusion formulation moléculaire conventionnelle de(Bear dispersivité 1997). de l’équati La supérieures dispersionon de transport mécaniqueà celles de qui masseest seraient fonction, (Eq. localement4.6), à la on fois définit desmesurées sur le terrain. De la Dans la formulation conventionnelle de l’équation de transportpersive de masse flux, (Eq. and 4.6), source on définit strength are reduced by hydraulic properties (Kij and ε) is inappropriate leDanspropriétés coefficient la formulation intrinsèques de dispersion conventionnelle du hydrodynamiquemilieu poreux de l’équati (comme commeon de la destransport somme conductivités de lamasse dispersion hydrauliques (Eq. 4.6), mécanique on définitet des et le coefficient de dispersion hydrodynamique commepropriétés la somme intrinsèquesa factorde la dispersion Rf. Tduhe mêmetransportmilieu mécanique façon,poreux process etreprésenter (comme thus appears desun domaineconductivités to ford’écoulem describing hydrauliquesent transitoiretransport et des parin ungeological domaine systemsd’écoulement porosités hétérogènes),be ‘retarded’ mais because aussi deof l’écoulemthe instantaneousent du fluide. equi La- diffusion moléculaire dans un de la diffusion moléculaire (Bear 1997). La dispersiondeleporosités coefficientla diffusionmécanique hétérogènes), de moléculaire dispersion est fonction, mais hydrodynamique moyen(Bearaussi à la de1997). foispermanent, l’écoulem des La dispersioncomme entcomme du lafluide. sommemécaniqueon leLa faitdediffusion[22]. la esten dispersion Ifonction,généraf moléculairea modell, mécaniqueignoreà appliedla foisdans fondamentalement des to unet a system having les varivariations- milieu poreuxlibrium sera différentesorption onto de celle the porousavec de medium. l’eau libre du fait du rôle de la porosité et de la propriétés intrinsèques du milieu poreux (commedemilieupropriétés lades diffusion poreux conductivités intrinsèques sera moléculaire différente hydrauliques du possibles(Bearmilieu de celle 1997). poreuxet avecde des Lavitesse de (comme dispersion l’eau et libre doitdes mécanique du conductivitésêtre fait compenséduable rôleest hydraulic fonction, dehydrauliques lapar porosité l’utilisation conductivityà la foisetet de des lade uses valeurs mean plusvalues grandes and de tortuosité.porositéspropriétés hétérogènes), CesintrinsèquesIn relationsthe conventional maissedu présentent dispersivitémilieuaussi formulationde poreux l’écoulemgénéralement (essentiellement (comme ofent thedu sous desfluide.solute- la conductivitésforme:la La dispersivité diffusion thereby hydrauliques moléculairedoes transversale) not explicitly danset des un(Goode represent et theKonikow variabil -1990). porosités hétérogènes), mais aussi de l’écoulement du fluide. Latransport diffusion equation moléculaire (Eq. 4.6),dans theun coefficient of hy- ity, the model calibration will likely yield values milieuporosités poreux hétérogènes), sera différente mais Globalement,aussi de celle de l’écoulem avec plus de l’eau entle modèle du libre fluide. du pourra faitLa diffusiondu représenter rôle de moléculaire la deporosité façon dans etprécise de un la la distribution des vitesses milieu poreux sera différente de celle avec de l’eau libre du faitdrodynamic du rôle Vdem V dispersionlan porosité iset de defined la as the sum of VmVn for the dispersivity coefficients that are larger Dij = αijmn + Dm , i, j, m, n = 1,2,3 tortuosité.milieu poreux CesDij serarelations= α ijmndifférente se présentent+réelles Ddem ,celle dans généralement avec l’espace de l’eau et sous libre i,lej, m latemps,du, nforme: fait= 1,2 ,dumoins3 rôle ilde y la aura porosité d’incertitude (4.14) et de la sur la représentation des tortuosité. Ces relations se présentent généralement sous la forme:mechanical Vdispersion and molecular diffusion than would be measured locally in the field area. tortuosité. Ces[9]. relations The mechanical se présententprocessus dispersion généralement de dispersion. is a function sous la both forme: Similarly, representing a transient flow field by où α est la dispersivitéVmVn dans le milieu poreux (un tenseur d’ordre 4), L; V et V sont les VmVn où αijmn est la dispersivité dans le milieu poreux (uni ,tenseurj, m, n = 1d’ordre,2,3 4), L; Vm et Vn sont les ijmn = ofDij the= α ijmnintrinsic+ propertiesDm , of the porous media a mean steady-statem n (4.14) flow field, as is commonly Dij = αijmn + Dm , i, j, m, n 1,2,3 VVV Une forme (4.14) particulière de l’équation de transport de masse peut être utilisée pour la V composantes(such de la as vitesse heterogeneitiesm n d’écoulement in hydraulicdu fluide conductivrespectivement- dans les directions m et n, composantes D deij = laα ijmnvitesse d’écoulement+ Dm , du fluide respectivement i, j, m, n = 1,2,3done, dans inherentlyles directions ignores (4.14) m et some n, of the variability in −1 V simulation directe/de l’âge des eaux (Goode2 −1 1996; 1999). Ceci s’effectue en ajoutant un terme oùLT −α1; D mest est ityla le dispersivitéand coefficient porosity) dans de and diffu le of milieu sionthe fluidmoléculaire poreux flow. (un effective,tenseurMolecular d’ordre L2velocityT−1 ;4), et L |Vand; | Vest mustet la V norme besont compensated lesdu for by using α LT ;ijmn Dm est le coefficient de diffusion moléculaire effective, L T ; et |V| estm la normen du −3 −1 où ijmn est la dispersivité dans le milieu poreux (un tenseur diffusiond’ordre 4), in La; porousVmde et croissance V median sont leswill d’ordre differ nul, from qui that représente increasedrait la valuesproduction of dispersivity interne de (primarilysolutés ML transT -. Pour composantesoù αijmn est la de dispersivité la vitesse d’écoulementdans le milieu du poreux fluide (un respectivement tenseur d’ordre dans 4), les L; directionsVm et Vn sontm et les n, composantes de la vitesse d’écoulement du fluide respectivement dans les directions m et n, 97 −1 in free water becauseformuler of une the effectséquation of de porosity transport 2versed’âge,−1 dispersivity) on remplace [23]. les Overall, concentrations the more par accu les- âges −1 composantesLT ; Dm est dele2 coefficientla− 1vitesse d’écoulement de diffusion dumoléculaire fluide respectivement effective, L Tdans; et les |V directions| est la norme m et dun, LT ; Dm est le coefficient de diffusion moléculaire effective,and L tortuosity.T ; et |V| Testhese la relationsnorme du are commonly ex- rately a model can represent or simulate the true −1 correspondants, ce qui correspondant 2à −un1 âge pour un volume moyen d’eau dans l’aquifère; LT ; Dm estpressed le coefficient as: de diffusion moléculaire effective, LvelocityT ; et |distributionV| est la norme in space du97 and time, the less of le taux de croissance97 d’ordre nul a une valeur unitaire; on considère les réactions de 97 dégradation et d’adsorption comme inexistantes; et en général, l’âge de l’eau entrante 546 (analogue à C') est notée égale à zéro. Ce type d’analyse permet une comparaison directe entre les résultats de la modélisation hydrogéologique et l’information apportée par les traceurs de l’environnement en rendant compte des effets de dispersion et autres processus de transport.

98 USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT a problem will be the uncertainty concerning rep- ferential equations can be approximated numeri- resentation of dispersion processes. cally. In so doing, the continuous variables are replaced with discrete variables that are defined at A special form of the solute-transport equation grid blocks (or nodes). Thus, the continuous dif- can be used for direct simulation of groundwater age [24, 25]. This is accomplished by adding ferential equation, which defines hydraulic head a zero-order growth term, which would represent or solute concentration everywhere in the system, internal production of the solute, ML–3T–1. In de- is replaced by a finite number of algebraic equa- veloping an age transport equation, concentrations that defines the hydraulic head or concentrations are replaced with corresponding ages, which tion at specific points. This system of algebraic represent a volume-averaged groundwater age in equations generally is solved using matrix tech- the aquifer; the zero-order growth rate has a unit niques. This approach constitutes a numerical value; decay and sorption reactions are assumed model. to be not present; and, in general, the age of in- The equations describing groundwater flow and coming water (analogous to C’) is specified as solute transport are second-order differential zero. This type of analysis allows a direct com- equations, which can be classified on the basis of parison of groundwater modelling results with their mathematical properties. There are basically environmental tracer information while account- three types of second-order differential equations, ing for effects of dispersion and other transport which are parabolic, elliptic, and hyperbolic [32]. processes. Such equations can be classified and distinguished based on the nature and magnitude of the coeffi- 4.5. Numerical methods to solve cients of the equation. This is important because equations the numerical methods for the solution of each type have should be considered and developed The partial differential equations describing separately for optimal accuracy and efficiency in groundwater flow and transport can be solved the solution algorithm. mathematically using either analytical solutions or numerical solutions. The advantages of an ana- Two major classes of numerical methods have lytical solution, when it is possible to apply one, come to be well accepted for solving the ground- are that it usually provides an exact solution to water flow equation. These are the finite-differ- the governing equation and is often relatively ence methods and the finite-element methods. simple and efficient to obtain. Many analytical Each of these two major classes of numerical solutions have been developed for the flow equa- methods includes a variety of subclasses and tion; however, most applications are limited to implementation alternatives. Comprehensive well hydraulics problems involving radial sym- treatments of the application of these numerical metry [26–28]. The familiar Theis type curve methods to groundwater problems are presented represents the solution of one such analytical by [33, 34]. Both of these numerical approaches model. Analytical solutions are also available to require that the area of interest be subdivided by solve the solute-transport equation [9, 29–31]. In a grid into a number of smaller subareas (cells or general, obtaining the exact analytical solution elements) that are associated with node points (ei- to the partial differential equation requires that ther at the centres of peripheries of the subareas). the properties and boundaries of the flow system Finite-difference methods approximate the first be highly and perhaps unrealistically idealised. derivatives in the partial differential equations as For most field problems, the mathematical ben- difference quotients (the differences between val- efits of obtaining an exact analytical solution are ues of variables at adjacent nodes, both in space probably outweighed by the errors introduced by and time, with respect to the interval between the simplifying assumptions of the complex field those adjacent nodes). There are several advanced environment that are required to apply the ana- text books that focus primarily on finite-difference lytical model. methods [32, 33, 35]. Finite-element methods use Alternatively, for problems where the simplified assumed functions of the dependent variables analytical models are inadequate, the partial dif- and parameters to evaluate equivalent integral

547 modelling formulations of the partial differential equations. to specify the spatial co-ordinates of each node, Huyakorn and Pinder [36] present a comprehen- is valuable to effectively utilise the advantageous sive analysis and review of the application of features of a finite-element model. Fig. 4.1 illus- finite-element methods to groundwater problems. trates a hypothetical aquifer system, which has In both numerical approaches, the discretisation impermeable boundaries and a well field of inter- of the space and time dimensions allows the con- est (Fig. 4.1a), which has been discretised using tinuous boundary-value problem for the solution finite-difference (Fig. 4.1b) and finite-element of the partial differential equation to be reduced (Fig. 4.1c) grids. Figs.4.1b and 4.1c illustrate con- to the simultaneous solution of a set of algebraic ceptually how their respective grids can be adjust- equations. These equations can then be solved used to use a finer mesh spacing in selected areas of ing either iterative or direct matrix methods. interest. The rectangular finite-difference grid ap- Each approach has advantages and disadvantages, proximates the aquifer boundaries in a step-wise but there are very few groundwater problems for manner, resulting in some nodes or cells outside which either is clearly superior. In general, the fi- of the aquifer, whereas sides of the triangular ele- nite-difference methods are simpler conceptually ments of the finite-element grid can closely fol- and mathematically, and are easier to program for low the outer boundary using a minimal number a computer. They are typically keyed to a rela- of overall nodes. tively simple, rectangular grid, which also eases The solute-transport equation is in general more data entry tasks. Finite-element methods gener- difficult to solve numerically than the ground- ally require the use of more sophisticated math- ematics but, for some problems, may be more ac- water flow equation, largely because the math- curate numerically than standard finite-difference ematical properties of the transport equation vary methods. A major advantage of the finite-element depending upon which terms in the equation are methods is the flexibility of the finite-element dominant in a particular situation. When solute grid, which allows a close spatial approximation transport is dominated by advective transport, as of irregular boundaries of the aquifer and (or) of is common in many field problems, then Eq. 4.6 parameter zones within the aquifer when they are approximates a hyperbolic type of equation (simi- considered. However, the construction and speci- lar to equations describing the propagation of fication of an input data set is much more diffi- a wave or of a shock front). But if a system is cult for an irregular finite-element grid than for dominated by dispersive fluxes, such as might oc- a regular rectangular finite-difference grid. Thus, cur where fluid velocities are relatively low and the use of a model preprocessor, which includes aquifer dispersivities are relatively high, then a mesh generator and a scheme to efficiently Eq. 4.6 becomes more parabolic in nature (similar number the nodes and elements of the mesh and to the transient groundwater flow equation).

a b c

WELL FIELD ACTIVE CELL ELEMENT AQUIFER BOUNDARY INACTIVE CELL

Fig. 28.1. Hypothetical application of finite-difference and finite-element grids to an irregularly bounded aquifer [1].

548 USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT

The numerical methods that work best for para- mensional, transient, finite-difference models that bolic partial differential equations are not best simultaneously solve the fluid pressure, energy- for solving hyperbolic equations, and vice versa. transport, and solute-transport equations for non- Thus, no one numerical method or simulation homogeneous miscible fluids include Refs [41, model will be ideal for the entire spectrum of 42]. A two-dimensional finite-element transport groundwater transport problems likely to be en- model is documented by Voss (1984). Because countered in the field. Further compounding this none of the standard numerical methods are ideal difficulty is the fact that in the field, the seepage for a wide range of transport problems, there is velocity of groundwater is highly variable, even currently still much research on developing better if aquifer properties are relatively homogeneous mixed or adaptive methods that aim to minimise (because of the effects of complex boundary con- numerical errors and combine the best features ditions). Thus, in low permeability zones or near of alternative standard numerical approaches stagnation points, the velocity may be close to [44–47]. zero and the transport processes will be domi- The conventional solute-transport equation is nated by dispersion processes; in high permeabil- a Fickian model. However, most mechanical dis- ity zones or near stress points (such as pumping persion actually arises from variations in velocity wells), the velocity may be several meters per about the mean, so at least partly is an advective- day and the transport processes will be advection based process. Transport in stratified porous me- dominated. In other words, for the same system, dia may be non-Fickian in nature [48, 49]. Thus, the governing equation may be more hyperbolic no matter how accurately we can solve the gov- in one area (or at one time) and more parabolic erning solute-transport equation, that equation it- in nature in another area (or at another time). self is not necessarily a definitive and sufficient Therefore, no matter which numerical method description of the processes controlling solute is chosen as the basis for a simulation model, it transport at the scale of most field problems. In will not be ideal or optimal over the entire domain general, the more accurately a model can rep- of the problem, and significant numerical errors resent or simulate the true velocity distribution, may be introduced somewhere in the solution. the less of a problem will be the uncertainty con- The transport modelling effort must recognise cerning representation of dispersion processes. this inherent difficulty and strive to minimise and control the numerical errors. There are additional complications when the solutes of interest are reactive. The reaction terms Although finite-difference and finite-element included in Eq. 4.6 are mathematically simple models are commonly applied to transport prob- ones. They do not necessarily represent the true lems, other types of numerical methods have complexities of many reactions. Also, particularly also been applied to transport problems, includ- difficult numerical problems arise when reaction ing method of characteristics, particle tracking, terms are highly non-linear, or if the concentra- random walk, Eulerian-Lagrangian methods, and tion of the solute of interest is strongly dependent adaptive grid methods. All of these have the abil- on the concentration of other chemical constitu- ity to track sharp fronts accurately with a mini- ents. In reality, isotherms may not be linear and mum of numerical dispersion. Documented mod- may not be equilibrium controlled. Ref. [50] dis- els based on variants of these approaches include cusses and classifies the chemical nature of reac- Refs [37–40]. tions and their relation to the mathematical prob- Finite-difference and finite-element methods lem formulation. Ref. [51] compare kinetic and also can be applied to solve the transport equa- local equilibrium formulations for solute trans- tion, particularly when dispersive transport is port affected by surface reactions. For field prob- large compared to advective transport. However, lems in which reactions are significantly affect- problems of numerical dispersion and oscillations ing solute concentrations, simulation accuracy is may induce significant errors for some problems. less limited by mathematical constraints than by The numerical errors can generally be reduced by data constraints. That is, the types and rates of using a finer discretisation (either time steps or reactions for the specific solutes and minerals in spatial grid). Examples of documented three-di- the particular groundwater system of interest are

549 ⎛ ∂h ⎞ h − h ⎜ ⎟ ≈ 0 1 (4.15) modellin⎝ ∂gx ⎠d Δx Notons que les puits d’observation sont séparés par des distances égales. De façon similaire, ⎛ ∂h ⎞ h − h 2 2 rarely known⎜ and⎟ ≈ require0 1 an extensive amountune bonne of approximationat point 0 (the delocation la dérivée of the d’ordre centre deux, well) ð canh/ð xbe, au point 0 (localisation du puits ⎛ ∂h ⎞ h0 − h1 (4.15) ⎝ ∂x ⎠d Δx ⎜ ⎟ ≈ (4.15) data to assess accurately. Ref. [52] reviewcentral) hydro- peutgiven⎛ être∂h ⎞ donnéeas: h0 − par:h ⎜⎝ ∂x ⎟⎠d ≈ Δx 1 (4.15) Notonsgeochemical que les transport puits d’observation models and sontdiscuss séparés various par des⎝ distances∂x ⎠d égales.Δx De façon similaire, Notons que2 les puits2 ⎛d’observation∂h ⎞ ⎛ ∂h ⎞ sonth séparés− h par− des distances égales. De façon similaire, unemathematical bonne approximation approaches de to la modellingdérivée d’ordre transport deux, ð h/ðx , au ⎜point⎟ 0− (localisation⎜ ⎟ 2 du 0puitsh0 h1 Notons que les2 puits d’observation sont séparés− par des2 distances2 égales. De façon similaire, une bonne approximation⎛ ∂ h ⎞ ⎝ ∂x ⎠ dee la⎝ ∂ dérivéex ⎠d d’ordreΔx deux,Δx ð h/ðh1x+, hau2 − point2h0 0 (localisation du puits of multiple reacting species. ⎜ ⎟ ≈ = 2= 2 (4.16) central) peut être donnée par: une bonne approximation⎜ 2 ⎟ de la dérivée d’ordre deux, ð h/ðx , au point2 0 (localisation du puits central) peut⎝ être∂x ⎠donnée par:Δx Δx ()Δx central) peut être donnée par: ⎛ ∂h ⎞ ⎛ ∂h ⎞ − 4.5.1. Basics of finite− -differenceh2 hSi0 onh 0 considère− h1 aussi les puits 3 et 4 représentés sur la figure 4.2b, localisés sur une ligne 2 ⎜ ⎟ ⎜ ⎟ − ⎛ ∂h ⎞ ⎛ ∂h ⎞ h − h h − h ⎛ ∂ h ⎞ ⎝ ∂x ⎠ ⎝ ∂x ⎠ h1 + h2 −⎜2h ⎟ − ⎜ ⎟ 2 0 0 1 2 2 met⎜ hods⎟ ≈ e d = ΔxparallèleΔx à l’axe= 2 des y,⎛ ∂ onh0⎞ peut⎛ ∂deh ⎞ façonh similaire (4.16)− h − h faire− h une approximation de ∂ h/∂y au point ⎜ 2 ⎟ ⎛ ∂ h ⎞ ⎝2 ∂x ⎠e ⎝ ∂x ⎠d 2Δx 0 0Δx 1 (4.16)h1 + h2 − 2h0 ∂x Δx Δx ⎜ 2 ⎟Δ≈x ⎜ ⎟ − ⎜ ⎟ = − = (4.16) ⎝ ⎠ ⎜⎛ ∂ h2 ()⎟⎞ ⎝ ∂x ⎠ ⎝ ∂x ⎠ h + h −22h The partial differential equations describing0 (le même point⎝⎜ ∂x ⎠⎟0 que sure Δlax figured 4.2a)Δ suivantx Δx (BennettΔx 11976):()Δ2x 0 ⎜ 2 ⎟ ≈ = = 2 (4.16) Sithe on flow considère and transportaussi les puitsprocesses 3 et in4 représengroundwatertés sur la⎝ ∂figurex ⎠ 4.2b, localisésΔx sur une ligneΔx ()Δx If we2 also consider wells 3 and 4 shown in Si on considère⎛ ∂ aussi⎞ h les+ h puits− 2h 3 et2 4 représen2 tés sur la figure 4.2b, localisés sur une ligne parallèleinclude àterms l’axe desrepresenting y, on peut derivativesde façon similaire of con faire- une⎜ happroximation⎟ 3 4 de0 ∂ h/∂y au point Si on considèreFig.⎜ 2 4.2b,aussi⎟ ≈ locatedles puits2 on 3 a et line 4 représenparallel tésto thesur y-axis,la figure 4.2b, localisés2 sur 2une (4.17) ligne tinuous variables. Finite-difference methodsparallèle are à l’axe∂y des y, onΔ peuty de façon similaire faire une approximation de ∂ h/∂y au point 0 (le même point 0 que sur la figure 4.2a) suivant (Bennettwe⎝ 1976):can⎠ similarly () approximate ð2h/ðy2 at point 0 2 2 based on the approximation of these derivatives0parallèle (le même à l’axe point des 0 que y, onsur peut la figure de façon 4.2a) similaire suivant (Bennettfaire une 1976): approximation de ∂ h/∂y au point (the same point 0 as in Fig. 4.2a) as [53]: Δ = Δ = 0Si (le l’espacement même point des0 que puits sur surla figure la figure 4.2a) 4.2b suivant est uniforme(Bennett 1976):(soit, x y a ), alors on peut (or slopes ⎛ of∂ 2h curves)⎞ h + byh discrete− 2h linear changes ⎜ ⎟ ≈ 3 4 0 formuler l’approximation2 suivante: (4.17) over small⎜ discrete2 ⎟ intervals2 of space or time. ⎛ ∂ h ⎞ h3 + h4 − 2h0 ⎝ ∂y ⎠ ()Δy ⎜ 2 ⎟ ≈ (4.17 (4.17) ⎜⎛ ∂ h2 ⎟⎞ h3 + h4 −22h0 If the intervals are sufficiently small, then all of ⎝⎜ ∂2 y ⎠⎟ ≈2 ()Δy (4.17) ⎜∂ h 2 ⎟∂ h h + 2h + h + h − 4h Sithe l’espacement linear increments des puits will sur represent la figure a4.2b good est ap uniforme- ⎝ ∂y (soit,+⎠ Δ≈x()Δ=1yΔy2= a ),3 alors4 on0 peut (4.18) 2 2 2 Δx = Δy = a proximation of the true curvilinear surface.Si l’espacementIf∂ x the des ∂spacingy puits sur of thela figure awells 4.2b in Fig. est 4.2buniforme is uni (soit,- ), alors on peut formuler l’approximation suivante: Si l’espacement des puits sur la figure 4.2b est uniforme (soit, Δx = Δy = a ), alors on peut formuler l’approximationform (that is, ∆suivante:x = ∆y = a), then we can develop Considering the observation wells in a confinedOn peut également obtenir ces approximations en utilisant les expansions en séries de Taylor. ∂ 2h ∂ 2h h + h + h + h − 4hformuler l’approximationthe following approximation:suivante: aquifer, as illustrated+ ≈in Fig.1 24.2a,3 Ref.4 [53]Une0 shows approximation 2 2des dérivées par les différences(4.18) finies introduit une certaine erreur, mais 2 2 2 ∂ h ∂ h h1 + h2 + h3 + h4 − 4h0 ∂x ∂y a 2 2 celle-ci va généralement2 + 2 ≈ diminuer+ + 2pour+ −des valeurs de a (ou de Δx et Δy) décroissantes. (4.18) On that a reasonable approximation for the derivative ∂xh ∂yh h1 h2 ha3 h4 4h0 2 + 2 ≈ 2 ( 4 . 1 8 (4.18) Onof peuthead, également ðh/ðx , atobtenir a point ces (d) approximations midway betweenappelle en utilisant cette∂ xerreur, les ∂expansionsy une erreur en desériesa troncatur de Taylor.e car remplacer une dérivée par une différence wells 1 and 0 is: On peut également obtenir ces approximations en utilisant les expansions en séries de Taylor. Une approximation des dérivées par les différencesOnfinie peut revient égalementfinies à introduitutiliser obtenir une une cessérie certaine approximations tronquée erreur, de mais Taenylor, utilisant de sorteles expansions que la solution en séries exacte de Taylor. d’une Une approximationThese approximationsdes dérivées par canles différencesalso be obtainedfinies introduit une certaine erreur, mais celle-ci va généralement diminuer pour deséquation valeurs de différencea (ou de Δ soitx et différenteΔy) décroissantes. de la solution On de l’équation différentielle correspondante ⎛ ∂h ⎞ h0 − h Une approximationthrough desthe dérivéesuse of parTaylor les différencesseries expansions. finies introduit une certaine erreur, mais ⎜ ⎟ ≈ 1 celle-ci(4.15) va généralement diminuer (4.15) pour des valeurs de a (ou de Δx et Δy) décroissantes. On appelle cette erreur, une erreur de troncaturcelle-ci(Peacemane car remplacerva généralementA1977). certain uneAussi, dérivéeerror diminueril n’estis par involved peut-être unepour différence des inpas valeurs approximating possible de ad’atteindre (ou de Δx uneet Δ solutiony) décroissantes. “exacte” Onde ⎝ ∂x ⎠d Δx appelle cette erreur, une erreur de troncature car remplacer une dérivée par une différence finie revient à utiliser une série tronquée del’équation Taylor, dedethe différence sortederivatives que laà cause bysolution finite-differences, des exacte limites d’une de précision but this en- er ce qui concerne le stockage de Notons que les puits d’observation sont séparés par desfinieappelle distances revient cette égales.àerreur, utiliser Deune unefaçon erreur série similaire, detronquée troncatur dee Tacarylor, remplacer de sorte une que dérivée la solution par une exacte différence d’une équationNote that de différence the observation soit différente wells arede la spaced sonombreslution an de dansl’équationror willles generallyordinateurs. différentielle decrease Lors correspondante asde ala (or résolution Δx and Δ yd’un) is gros fichier d’équations, de une bonne approximation de la dérivée d’ordre deux, ðéquationfinie2h/ð xrevient2, au de point différence à utiliser 0 (localisation soitune différentesérie du tronquée puits de la sodelution Taylor, de l’équationde sorte que différentielle la solution correspondante exacte d’une (Peacemanequal distance 1977). apart. Aussi, Similarly, il n’est peut-êtrea reasonable pasnombreuses possibleap- given d’atteindreopérations smaller unearithmétiques and solution smaller “exacte” sontvalues. eff deectuées,This error et isdes erreurs successives peuvent central) peut être donnée par: 2(Peacemanéquation2 de 1977).différence Aussi, soit il différente n’est peut-être de la so paslution possible de l’équation d’atteindre différentielle une solution correspondante “exacte” de l’équationproximation de différencefor the second à cause derivative, des limites ðparfois h/ðxde précision ,s’accumuler. called en cea ‘truncationqui concerne error’ le stockage because de the replace- l’équation(Peaceman de1977). différence Aussi, à il cause n’est despeut-être limites pas de possibleprécision d’atteindre en ce qui uneconcerne solution le stockage“exacte” de nombres dans⎛ ∂h ⎞les ⎛ordinateurs.∂h ⎞ Lors de la résolution d’un gros fichier d’équations, de h2 − h0 h0 − h1l’équation de différence à cause des limites de précision en ce qui concerne le stockage de 2 ⎜ ⎟ − ⎜ ⎟ − nombres dans les ordinateurs. Lors de la résolution d’un gros fichier d’équations, de nombreuses⎛ ∂ ⎞ ∂opérations∂ arithmétiques sont effhectuées,+ h − 2eth des erreurs successives peuvent ⎜ h ⎟ ⎝ x ⎠e ⎝ x ⎠d Δx Δx nombres1 2 dans0 les ordinateurs. Lors de la résolution d’un gros fichier d’équations, de ⎜ 2 ⎟ ≈ = nombreuses= 2 opérations arithmétiques (4.16) sont effectuées, et des erreurs successives peuvent parfois∂x s’accumuler.Δx Δx ()Δx ⎝ ⎠ parfoisnombreuses s’accumuler. opérations arithmétiques sont effectuées, et des erreurs successives peuvent Si on considère aussi les puits 3 et 4 représentés surparfois la figure s’accumuler. 4.2b, localisés sur une ligne parallèle à l’axe des y, on peut de façon similaire faire une approximation de ∂2h/∂y2 au point 0 (le même point 0 que sur la figure 4.2a) suivant (Bennett 1976):

⎛ ∂ 2 ⎞ h + h − 2h ⎜ h ⎟ 3 4 0 ⎜ 2 ⎟ ≈ 2 (4.17) ⎝ ∂y ⎠ ()Δy 104 Si l’espacement des puits sur la figure 4.2b est uniforme (soit, Δx = Δy = a ), alors on peut formuler l’approximation suivante:

104∂ 2h ∂ 2h h + h + h + h − 4h + ≈ 1 2 3 4 0 104 (4.18) ∂x2 ∂y 2 a2 104 Fig. 28.2. Schematic cross section through confined aquifer to illustrate numerical approximation to derivatives of On peut égalementhead, ðh/ðx obtenir (a) and ces ðh/ðy approximations (b) (modified en from utilisant Ref. [53]). les expansions en séries de Taylor. Une approximation des dérivées par les différences finies introduit une certaine erreur, mais celle-ci va généralement diminuer pour des valeurs de a (ou de Δx et Δy) décroissantes. On 550 appelle cette erreur, une erreur de troncature car remplacer une dérivée par une différence finie revient à utiliser une série tronquée de Taylor, de sorte que la solution exacte d’une équation de différence soit différente de la solution de l’équation différentielle correspondante (Peaceman 1977). Aussi, il n’est peut-être pas possible d’atteindre une solution “exacte” de l’équation de différence à cause des limites de précision en ce qui concerne le stockage de nombres dans les ordinateurs. Lors de la résolution d’un gros fichier d’équations, de nombreuses opérations arithmétiques sont effectuées, et des erreurs successives peuvent parfois s’accumuler.

104

Fig. 4.2. Coupe transversale schématique à travers un aquifère captif pour illustrer les approximations Fig. 4.2. Coupe transversale schématique à travers un aquifère captif pour illustrer les approximations numériques des dérivées de la charge hydraulique, ∂h/∂x (a) et ∂h/∂y (b) (modifié à partir numériques des dérivées de la charge hydraulique, ∂h/∂x (a) et ∂h/∂y (b) (modifié à partir de Bennett, 1976). de Bennett, 1976).

Nous devons aussi considérer la discrétisation du temps, qui peut être assimilé à une autre USE OF NUMERICAL MODELS TOdimension, SIMULATE GRet deOUN ceDWAT fait EreprésentéR FLOW AN Dpar TRA unNSPO autrRTe indice. Considérons un segment représentatif d’un hydrogramme (voir Fig. 4.3), pour lequel la charge est représentée en fonction du temps dans le cas d’un système d’écoulement en régime transitoire, n étant l’indice utilisé pour dans le cas anyd’un point système is the d’écoulementderivative of headen régi withme respect transitoire, to n étant l’indice utilisé pour Approximate slope at t n indiquer le tempstime, andauquel it can on beobserve approximated une valeur as de∂h /∂charget ≈ ∆h donnée./∆t. La pente de l’hydrogramme h n+1 h est en tout point la dérivée de la charge en fonction du temps, et peut être approchée suivant ∆h est en tout pointIn terms la dérivée of the deheads la charge calculated en fonction at specific du temps, time et peut être approchée suivant ∂h/∂t ≈ Δh/Δt. En fonction des charges calculées pour des accroissements de temps h n ∂h/∂t ≈ Δhincrements/Δt. En fonction (or time des nodes), charges the slopecalculées of the pour hy - des accroissements de temps spécifiques drograph(ou temps at nodal), time n on can peut be approcherapproximated la pente by: de l’hydrogramme au temps n par: ∆ h spécifiques (ou temps nodal), on peut approcher la pente de l’hydrogramme au temps n par:

h n-1 ⎛ ∂h ⎞ hn+1 − hn ⎜⎛ ∂h ⎟⎞ ≈ hn+1 − h ⎜ ⎟ ≈ n (4.19) (4.19) ∆t ∆t ⎝ ∂t ⎠nΔt Δt ⎝ ∂t ⎠nΔt Δt t t t n-1 n n+1 ou or time ⎛ ∂h ⎞ hn − hn−1 ⎜⎛ ∂h ⎟⎞ ≈ hn − h ⎜ ⎟ ≈ n−1 (4.20) (4.20) Fig. 28.3. Part of a hydrograph showing that the de- ⎝ ∂t ⎠nΔt Δt ⎝ ∂t ⎠nΔt Δt rivative (or slope, ∂h/∂t) at time node tn may be approximated by Δh/Δt [1]. We are calculating the derivative at t = n∆t in

Eq. 4.19 by taking a ‘forward difference’ from ment of a derivative by a difference quotient is time n to time n + 1, and by taking a ‘back- equivalent to using a truncated Taylor series, so ward difference’ in Eq. 4.20. In terms of solving that the exact solution of a difference equation the groundwater flow equation for a node (i,j) of differs from the solution of the corresponding dif- a finite-difference grid, we have to consider heads 105 105 ferential equation [32]. Also, it may not be possi- at five nodes and at two time levels, as illustrat- ble to achieve an ‘exact’ solution of the difference ed in Fig. 4.4. In Fig. 4.4a, we have expressed equation because of limits of precision in storing the spatial derivatives of head at time level n, numbers in a digital computer. In solving a large where all values are known, and the time deriva- set of difference equations, many arithmetic op- tive as a forward difference to the unknown head erations are performed, and round-off errors may at time step n + 1. Then for every node of the grid sometimes accumulate. we will have a separate difference equation, each We must also consider the discretisation of time, of which contains only one unknown variable. which may be viewed as another dimension, and Thus, these equations can be solved explicitly. hence represented by another index. If we consid- Explicit finite-difference equations are thus sim- er a representative segment of a hydrograph (see ple and straightforward to solve, but they may Fig. 4.3), in which head is plotted against time for have stability criteria associated with them. That a transient flow system, n is the index or subscript is, if time increments are too large, small numeri- used to denote the time at which a given head cal errors or perturbations may propagate into value is observed. The slope of the hydrograph at larger errors at later stages of the computations.

ti m e ti m e h i,j+1,n t h n+1 i ,j,n+1 t n hi -1,j,n h i,j,n h i+1,j,n

h i,j+1,n hi ,j-1,n t n t hi -1,j,n hi ,j,n h i+1,j,n n-1 hi ,j,n-1

hi ,j-1,n EX PL A N A TI ON Kno wn head

a Unkno wn head b

Fig. 28.4. Grid stencil showing discretisation of time at node (i,j) in two-dimensional finite-difference grid: (a) explicit (forward-difference) formulation and (b) implicit (backward-difference) formulation [1].

551 modelling

In Fig. 4.4b we have expressed the time deriva- constant and uniform within each cell but may be tive as a backward difference from the heads different between cells. Other types of means for at time level n, which are thereby the unknown interblock transmissivity may be more appropri- heads, whereas the heads at the previous time ate for other assumptions about the transmissivity

level, n – 1, are known. The spatial derivatives of distribution, such as smoothly varying transmis- head are written at time level n, where all values sivity [54]. are unknown, so for every node of the grid we Sur Surla figure la figure 4.4bwill 4.4b la dérivéedérivée have la dérivée one dudu temps difference tempsdu temps aa étéété a equationexprimée expriméeété exprimée comme that comme comme contains une une différence unedifférence différence4.5.2. amont amontBasics amont à àpartir partir àof partir des desfinite des -element five unknowns, which cannot be solved directly. methods chargescharges au tempsau temps n, quiqui n, sontquisont sont parpar conséquentparconséquent conséquent lesles chargeschargesles charges inconnues, inconnues, inconnues, alors alors alorsque que les queles charges chargesles charges au au au However, for the entire grid, which contains N The finite-element method (FEM) is a numeri- tempstemps précédent, précédent,nodes, n–1, n sont–1, wesont sont would connues.connues. connues. have LesLes a Lesdérivéessystemdérivées dérivées of dede Nla la de equationscharge charge la charge dans dans dansl’espace l’espace l’espace sont sont écritessont écrites écrites au au au cal analysis technique for obtaining approximate tempstemps n, où n, toutesoùcontaining toutes lesles valeursvaleursles avaleurs total sontsont of sont inconnue inconnueN unknowns. inconnues,s, doncdoncs, Suchdonc pour pour a pour chaque systemchaque chaque nœud nœud nœud du du maillage maillagedu maillage on on aura auraon aura solutions to a wide variety of problems in phys- une uneéquation équation deof différencede simultaneous différence àà cinqcinq à equations, cinq inconnues,inconnues, inconnues, together quiqui nequine with pourra pourrane specipourra paspas- être pasêtre résolue êtrerésolue résolue directement. directement. directement. ics and engineering. The method was originally fied boundary conditions, can be solved implic- Cependant,Cependant, si on si considèreon considère lele maillage maillagele maillage entier,entier, entier, quiqui contientquicontient contient N N nœuds, nœuds, N nœuds,applied on on a aonun unto asystème structuralsystèmeun système à à N mechanics N à N but is now used itly. Although implicit solutions are more com- équationséquations avec avec un total un total dede NN de inconnues.inconnues. N inconnues. UnUn tel telUn système systèmetel système d’équations d’équations d’équations simultanées, simultanées,in simultanées, all fields associé associé associéof à àdescontinuum des à des mechanics. Ref. [55] plicated, they also have the advantage of gener- describes four different approaches to formulate conditionsconditions aux auxallylimites limitesbeing spécifiques,spécifiques, unconditionally spécifiques, peutpeut peut être êtrestable. êtrerérésolusolu Mré ostsolu implicitement.implicitement. available implicitement. Bien Bien Bienque que les queles solutions lessolutions solutions the finite-element method for a problem, which implicitesimplicites soient groundwatersoient plusplus plus compliquées,compliquées, flow compliquées, models solveelleselles elles anont ontimplicit ontquandquand finite-quand mêmemême même l’avantagel’avantage l’avantage d’êtred’être d’être are: the direct approach, the variational approach, inconditionnellementinconditionnellementdifference stables.stables. stables.approximation LaLa plupartLaplupart plupart todesdes the desmodèlesmodèles flow modèles equation. d’écoulementd’écoulement d’écoulement souterrain souterrain souterrain disponibles disponibles disponibles the weighted residual approach, and the energy utilisentutilisent une uneapproximationWe approximation may next impliciteimplicite consider implicite auxaux a two-dimensional différences auxdifférences différences finies finies finiesground pour pour pourla -la résolution résolution labalance résolution de de approach.l’équation l’équation de l’équation de Inde groundwaterde problems, l’écoulementl’écoulement souterrain.water souterrain. flow equation for a heterogeneous, aniso- the approach frequently used is either the weight- tropic aquifer, in which the co-ordinate system is ed residual or variational approach. ConsidéronsConsidérons maintenant maintenant uneune uneéquationéquation équation dede l’éc l’écde oulementoulementl’écoulement souterrain souterrain souterrain à àdeux deux à deuxdimensions dimensions dimensions pour pour unpour un un aligned with the major axes of the transmissivity The finite-element method (FEM) uses a concept aquifèreaquifère hétérogène hétérogènetensor. etet anisotrope,Thisanisotrope, et anisotrope, may be dansdans approximated dans laquelaque laquellelle le lelle système systèmeby le thesystème followde de coordonnées coordonnéesde- coordonnées se se superpose superpose se superpose avec avec avec of ‘piecewise approximation’. The domain of les lesaxes axes majeurs majeursing dudu finite-difference tenseurdutenseur tenseur dede transmissivdetransmissiv transmissivequationité.ité. forCeciité.Ceci Cecirepresentative peutpeut peut êtreêtre êtreapprochéapproché approché pourpour pourun un nœud unnœud nœud the problem, that is the extent of the aquifer to représentatifreprésentatif (i,j)node (i,j)parpar l’équationpar(l’équationi,j) l’équationas: auxaux différencesauxdifférences différences finies finies finies suivante: suivante: suivante: be simulated, is divided into a set of elements or pieces. In theory, the elements can be of differ- ⎛⎛hh ⎛ h −−hh − h⎞⎞ ⎞ ⎛⎛hh ⎛ h−−hh − ⎞h⎞ ⎞ T ⎜⎜ ii−−1⎜1,,jj,,nni−1, j,nii,,jj,n,n⎟⎟i, j,n ⎟ +T ⎜⎜ i+i1+,⎜1j,,jn,ni+1, j,in,ij,,nj,n⎟⎟i, j,n ⎟ Txx[][]ii−−11/xx/22,[],ijj−1⎜/ 2, j ⎜ 22 2⎟ ++TT⎟xxxx[]i[]+i+11/ xx2/ 2, []j,ij+⎜1/ 2, j ⎜ 2 2 2⎟ ⎟ ent shapes and sizes. Most FEM computer pro- ⎜ Δx Δx ⎟ ⎜ Δx Δx ⎟ ⎝⎝ ⎝()()Δx ()⎠⎠ ⎠ ⎝⎝ ⎝()()Δx ()⎠⎠ ⎠ grams use one shape element, most commonly ⎛⎛hh ⎛ h −−hh − h⎞⎞ ⎞ ⎛⎛hh ⎛ h−−hh −⎞h⎞ ⎞either triangular or quadrilateral elements. In + T +T ⎜⎜ ii,,j⎜j−−11,n,ni, j−1,ni,i,j,jn,n⎟⎟i,+j+,nT⎟ +T ⎜⎜ i,ij,+⎜j1+,1n,ni, j+1i,n, ij,,nj,n⎟⎟i , j ,n ⎟ (4.21) (4.21) (4.21) Tyyyy[][]ii,,jj−−yy11//[]2i2, ⎜j⎜−1/ 2 ⎜ 22 2⎟⎟ Tyy⎟yy[]i[],ij,+j1+yy/12/[]i2,⎜j⎜+1/ 2 ⎜ 2 2 2⎟⎟ ⎟the groundwater model MODFE [56, 57] trian- ⎝⎝ ⎝()()ΔΔyy ()Δy ⎠⎠ ⎠ ⎝⎝ ⎝()Δ()Δyy ()Δy ⎠⎠ ⎠ gular elements are used, whereas in the ground- ⎛⎛hhi, j⎛,nh−−i, jhh,ni, j−,n−h1i,⎞j⎞,n−1 ⎞qqi, j qi, jK K i, ⎜j,n i, j,n−1 ⎟ i, j Kz z z water model SUTRA [43] quadrilateral elements = S ⎜⎜= S ⎟⎟++ + −− −()H()Hs[si[,()ij,H]j]−s−[hi,hij,]ij,,−nj,nhi, j,n ⎜⎜ ⎜ Δt ⎟⎟ ⎟ ΔxΔy m ⎝⎝ ⎝ ΔΔtt ⎠⎠ ΔΔ⎠xxΔΔyy mm are used. Point values of the dependent variable (for example, head, pressure, or concentration) où q le taux volumétrique (flux) d’apport ou de perte(4.21) au nœud 3i,j,3 − 1L−13T−1. Cette formulation où qi,j lei,j taux volumétriquevolumétrique (flux)(flux) d’apportd’apport ouou de de perte perte au au nœud nœud i,j, i,j, L LareTT calculated. .Cette Cette formulation formulation at nodes, which are the corners or supposesuppose de cede faitce faitqueque quetoutetoute toute perturbation,perturbation, perturbation, commecomme comme cellecelle celle queque représentequereprésente représente q q , ,s’appliqueq s’applique, s’applique à à à where qi,j is the volumetric rate of withdrawal or verticesi,j i,jof thei,j elements, and a simple equation is l’ensemblel’ensemble de lala derecharge surfacesurface la surface dedeat lala thede cellulecellule lai,j cellule node, plutôtplutôt plutôtL 3qu’en Tqu’en–1. qu’enT un hisun point pointformulationun point donné donné donné (ou (ou nœud (ouusednœud nœud i,j).to i,j). describe Ceci i,j).Ceci impliqueCeci implique the implique value of the dependent vari- que quesi un si puitsun puits inherentlyde pompagepompagede pompage assumes figurefigure figure surthatsur le surleany nœudnœud lestresses, nœud i,j,i,j, la la i,j, suchcharge charge la ascharge repsera sera- calculéesera calculéeable calculée withinen en considérant considéranten theconsidérant element. la la la This simple equation sectionsection passante passanteresented dudu débitdébitdu bydébit de deqi,j ,sortie sortiedeare sortieapplied commecomme comme over égaleégale the égale àentireà lala à surface surface lasurface surface de de la ladeis cellule cellulecalledla cellule plutôt aplutôt basis plutôt que quefunction de quede de and each node that is area of cell i,j rather than at a point (or at node part of an element has an associated basis func- considérerconsidérer sa valeur sa valeur réelle.réelle. réelle. DansDans Dans l’équationl’équation l’équation 4.214.21 4.21les les termes termesles termes transmissivité transmissivité transmissivité correspondent correspondent correspondent aux aux aux i,j). This implies that if a pumping well is repre- tion. The simplest basis functions that are usually moyennesmoyennes harmoniques harmoniquessented at des desnode destransmissivitéstransmissivités i,j transmissivités, then the head desdes dewilldesdeuxux debe cellules cellulesux calculated cellules adjacentes. adjacentes. adjacentes.used On Onare peut Onpeutlinear considérerpeut considérer functions. considérer The solution to the dif- la moyennela moyenne harmoniqueas harmonique if it were commecomme being comme withdrawnappropriéeappropriée appropriée from etet significative significative eta wellsignificative that had sisi on onsi faitferentialfaiton l’hypothèse faitl’hypothèse l’hypothèseequation que que for la que la flow la (Eq. 4.3) or transport transmissivitétransmissivité estesta borehole constanteconstanteest constante surface etet uniformeuniforme et areauniforme equal auau sein seinau to seinΔ dedex Δ chaque chaqueyde rather chaque cellule cellulethan cellule mais mais(Eq. maisqu’en qu’en4.6) qu’en revancheis revanche approximated revanche elle elle elle by a set of elements in peutpeut être êtredifférente différenteits actual entreentre entre value. lesles cellules.lescellules. In cellules.Eq. 4.21D’auD’au D’autrestresthe trestypestransmissivitytypes types dede moyennesdemoyennes moyenneswhich sontsont the sont peut-êtrepeut-être dependent peut-être plus plus variable plus only varies linearly terms represent the harmonic means of the trans- within the element, but the entire set of elements appropriéesappropriées pour pour exprimer exprimer lala transmissivitétransmissivité la transmissivité dans dans dansle le cas cas le d’autres casd’autres d’autres hypothèses hypothèses hypothèses de de distribution distribution de distribution des des des missivity of the two adjacent cells. The harmonic approximates the complex distribution of head or transmissivités, comme leur variation progressive (Goode et Appel 1992). transmissivités, meancommecomme can leurleur be variationvariation shown progressive toprogressive be appropriate (Goode (Goode andet et Appel Appel con- 1992). 1992).concentration. Refs [34, 36, 55, 57, 58] provide sistent with the assumption that transmissivity is more comprehensive explanations of the method.

552 107107 107 USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT

4.5.3. Matrix solution techniques tion factors are used. Unfortunately, the definition of best values for these factors commonly is prob- As indicated, the finite-difference and finite-ele- lem dependent. In addition, iterative approaches ment approximations lead to an algebraic equation for each node point. The set of algebraic require that an error tolerance be specified to stop equations may be solved numerically by one of the iterative process. An optimal value for the tol- two basic methods: direct or iterative. In direct erance, which is used to evaluate when the itera- methods, a sequence of operations is performed tive calculations have converged on a solution, only once to solve the matrix equation, providing may also be problem dependent. If the tolerance a solution that is exact, except for machine round- is set too large, then the iterations may stop be- off error. Iterative methods arrive at a solution fore adequate numerical accuracy is achieved. If by a process of successive approximation. They the tolerance is set too small, then the iterative involve making an initial guess at the solution, process may consume excessive computational then improving this guess by some iterative pro- resources in striving for numerical precision that cess until an error criterion is satisfied. Therefore, may be orders of magnitude smaller than the pre- in these techniques, convergence and the rate of cision of the field data, or the iterative process convergence are of concern. may even fail to converge. Direct methods can be further subdivided into: More recently, a semi-iterative method, or class of methods, known as conjugate-gradient meth- (1) solution by determinants, ods, has gained popularity. One advantage of (2) solution by successive elimination of the un- the conjugate-gradient method is that it does not knowns, and require the use or specification of iteration parameters, thereby eliminating this partly subjective (3) solution by matrix inversion. procedure. Direct methods have two main disadvantages. The first problem is one of computer resource 4.5.4. Boundary and initial requirements, including large storage (memory) conditions requirements and long computation times for large problems. The matrix is sparse (contains To obtain a unique solution of a partial differen- many zero values) and to minimise computational tial equation corresponding to a given physical effort, several techniques have been proposed. process, additional information about the physical However, for finite-difference and finite-element state of the process is required. This information methods, storage requirements may still prove to is supplied by boundary and initial conditions. be unavoidably large for three-dimensional prob- For steady-state problems, only boundary condi- lems. The second problem with direct methods is tions are required, whereas for transient problems, round-off error. Because many arithmetic opera- boundary and initial conditions must be specified. tions are performed, round-off errors can accumu- Mathematically, the boundary conditions include late for certain types of matrices. the geometry of the boundary and the values of Iterative schemes avoid the need for storing large the dependent variable or its derivative normal to matrices, which make them attractive for solv- the boundary. In physical terms, for groundwater ing problems with many unknowns. Numerous model applications, the boundary conditions are schemes have been developed; a few of the more generally of three types: commonly used ones include successive over- (1) specified value (head or concentration), relaxation methods, iterative alternating-direction implicit procedure, and the strongly implicit pro- (2) specified flux (corresponding toa specified cedure. gradient of head or concentration), or Because iterative methods start with an initial esti- (3) value-dependent flux (or mixed boundary mate for the solution, the efficiency of the method condition, in which the flux across a bound- depends somewhat on this initial guess. To speed ary is related to both the normal derivative up the iterative process, relaxation and accelera- and the value) [59].

553 modelling

The third type of boundary condition might be Once a decision to develop a model has been used, for example, to represent leakage or ex- made, a code (or generic model) must be select- change between a stream and an adjacent aqui- ed (or modified or constructed) that is appropri- fer, in which the leakage may change over time ate for the given problem. Next, the generic code as the head in the aquifer changes, even though must be adapted to the specific site or region be- the head in the stream might remain fixed. A no- ing simulated. Development of a numerical deter- flow boundary is a special case of the second type ministic, distributed-parameter, simulation model of boundary condition. The types of boundaries involves selecting or designing spatial grids and appropriate to a particular field problem require time increments that will yield an accurate solu- careful consideration. tion for the given system and problem. The analyst must then specify the properties of the system The initial conditions are simply the values of (and their distributions), stresses on the system the dependent variable specified everywhere in- (such as recharge and pumping rates), boundary side the boundary at the start of the simulation. conditions, and initial conditions (for transient Normally, the initial conditions are specified to be problems). All of the parameter specifications and a steady-state solution. If, however, initial condi- boundary conditions are really part of the overall tions are specified so that transient flow is occur- conceptual model of the system, and the initial ring in the system at the start of the simulation, numerical model reflects the analyst’s conceptual it should be recognised that heads will change model of the system. during the simulation, not only in response to It must always be remembered that a model is the new pumping stress, but also due to the initial an approximation of a very complex reality, and conditions [60]. a model is used to simplify that reality in a manner that captures or represents the essential features 4.6. Model design, development and processes relative to the problem at hand. In and application the development of a deterministic groundwater model for a specific area and purpose, we must se- The first step in model design and application is lect an appropriate level of complexity (or, rather, to define the nature of the problem and evaluate simplicity). Although finer resolution in a model the purpose of the model. Although this may seem will generally yield greater accuracy, there also obvious, it is an important first step that is some- exists the practical constraint that even when ap- times overlooked in a hasty effort to take action. propriate data are available, a finely discretised This step is closely linked with the formulation of three-dimensional numerical model may be too a conceptual model, which again is required prior large to run on available computers, especially to development of a simulation model. In formu- if transport processes are included. The selection lating a conceptual model, the analyst must evalu- of the appropriate model and appropriate level ate which processes are significant in the system of complexity remains subjective and dependent being investigated for the particular problem at on the judgement and experience of the analysts, hand. Some processes may be important to con- the objectives of the study, and level of prior in- sider at one spatial or temporal scale of study, but formation on the system of interest. The trade-off negligible or irrelevant at another scale. The an- between model accuracy and model cost will al- alyst must similarly decide on the appropriate ways be a difficult one to resolve, but will always dimensionality for the numerical model. Good have to be made. In any case, water managers and judgement is required to evaluate and balance other users of model results must be made aware the trade-offs between accuracy and cost, with re- that these trade-offs and judgements have been spect to both the model and to data requirements. made and may affect the reliability of the model. The key to efficiency and accuracy in modelling In general, it is more difficult to calibrate a solute- a system probably is more affected by the for- transport model of an aquifer than it is to calibrate mulation of a proper and appropriate conceptual a groundwater flow model. Fewer parameters model than by the choice of a particular numeri- need to be defined to compute the head distribu- cal method or code. tion with a flow model than are required to com-

554 USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT pute concentration changes using a solute-trans- typically be accomplished using head-dependent port model. Because the groundwater seepage leakage (third type) boundary conditions. These velocity is determined from the head distribution, boundaries should also be placed as far as pos- and because both advective transport and hydro- sible away from the area of interest and areas of dynamic dispersion are functions of the seepage stresses on the system, so as to minimise any im- velocity, a model of groundwater flow is often pact of conceptual errors associated with these ar- calibrated before a solute-transport model is de- tificial boundary conditions. veloped. In fact, in a field environment perhaps In designing the grid, the length to width ratio (or the single most important key to understanding aspect ratio) of cells or elements should be kept as a solute-transport problem is the development close to one as possible. Long linear cells or ele- of an accurate definition (or model) of the flow ments can lead to numerical instabilities or errors, system. This is particularly relevant to transport and should be avoided, particularly if the aspect in fractured rocks, where simulation is common- ratio is greater than about five [6]. ly based on porous-media concepts. Although the potential (or head) field can often be simu- In specifying boundary conditions for a particular lated, the required velocity field may be greatly problem and grid design, care must be taken to in error. not overconstrain the solution. That is, if dependent values are fixed at too many boundary nodes, at either internal or external nodes of a grid, 4.6.1. Grid design the model may have too little freedom to calcu- The dimensionality of the model should be select- late a meaningful solution. At the extreme, by ed during the formulation of the conceptual mod- manipulating boundary conditions, one can force el. If a one- or two-dimensional model is selected, any desired solution at any given node. While this then it is important that the grid be aligned with may assure a perfect match to observed data used the flow system so that there is no unaccounted for calibration, it is of course not an indicator of flux into or out of the line or plane of the grid. model accuracy or reliability, and in fact is mean- For example, if a two-dimensional areal model is ingless. applied, then there should be no significant verti- To optimise computational resources in a model, cal components of flow and any vertical leakage it is generally advisable to use an irregular (or or flux must be accounted for by boundary condi- variably-spaced) mesh in which the grid is fin- tions; if a two-dimensional profile model is ap- est in areas of point stresses, where gradients plied, then the line of the cross section should be are steepest, where data are most dense, where aligned with an areal streamline, and there should the problem is most critical, and (or) where great- not be any lateral flow into or out of the plane of est numerical accuracy is desired. It is generally the cross section. advisable to increase the mesh spacing by a fac- To minimise a variety of sources of numerical tor no greater than about two between adjacent errors, the model grid should be designed using cells or elements. Similarly, time steps can often the finest mesh spacing and time steps that are be increased geometrically during a simulation. possible, given limitations on computer memory At the initial times or after a change in the stress and computational time. To the extent possible, regime, very small time steps should be imposed, the grid should be aligned with the fabric of as that is when changes over time are the greatest. the rock and with the average direction of ground- With increased elapsed time, the rate of change in water flow. The boundaries of the grid also should head typically decreases, so time steps can often be aligned, to the extent possible, with natural hy- be safely increased by a factor of two or more. drologic and geologic boundaries of the system of Because transmissivity is a property of the porous interest. Where it is impractical to extend the grid media, the cross-product terms of the transmissiv- to a natural boundary, then an appropriate bound- ity tensor can typically be dropped out of the gov- ary condition should be imposed at the edge of erning flow equation that is solved ina model the grid to represent the net effects of the con- by aligning the model grid with the major axes tinuation of the system beyond the grid. This can of the transmissivity tensor. However, this can-

555 modelling not typically be done for the dispersion tensor in Even when the match to historical data is good, the transport equation because it is related to, and the model may still fail to predict future responses depends on, the flow direction, which changes accurately, especially under a new or extended set orientation over space and time. There is, in gen- of stresses than experienced during the calibra- eral, no way to design a fixed grid that will always tion period. be aligned with a changing flow field. The calibration of a deterministic groundwater model is often accomplished through a trial and 4.6.2. Model calibration error adjustment of the model’s input data (aqui- Deterministic groundwater simulation models fer properties, sources and sinks, and boundary impose large requirements for data to define all and initial conditions) to modify the model’s out- of the parameters at all of the nodes of a grid. put. Because a large number of interrelated fac- To determine uniquely the parameter distributors affect the output, this may become a highly tion for a field problem would require so much subjective and inefficient procedure. Advances in expensive field testing that it is seldom feasible parameter identification procedures help to elimi- either economically or technically. Therefore, nate some of the subjectivity inherent in model the model typically represents an attempt, in ef- calibration [61–65]. The newer approaches tend fect, to solve a large set of simultaneous equations to treat model calibration as a statistical proce- having more unknowns than equations. It is in- dure. Thus, multiple regression approaches al- herently impossible to obtain a unique solution to low the simultaneous construction, application, such a problem. and calibration of a model using uncertain data, so that the uncertainties are reflected as estimat- Uncertainty in parameters logically leads to a lack ed uncertainties in the model output and hence of confidence in the interpretations and predic- in predictions or assessments to be made with tions that are based on a model analysis, unless the model [66]. it can be demonstrated that the model is a reason- ably accurate representation of the real system. Even with regression modelling, however, the hy- To demonstrate that a deterministic groundwater drologic experience and judgement of the mod- simulation model is realistic, it is usual to com- eller continues to be a major factor in calibrat- pare field observations of aquifer responses (such ing a model both accurately and efficiently, even as changes in water levels for flow problems or if automated procedures are used. In any case, changes in concentration for transport problems) the modeller should be very familiar with the spe- to corresponding values calculated by the model. cific field area being studied to know that both The objective of this calibration procedure is to the data base and the numerical model adequately minimise differences between the observed data represent prevailing field conditions. The mod- and calculated values. Usually, the model is con- eller must also recognise that the uncertainty in sidered calibrated when it reproduces historical the specification of sources, sinks, and boundary data within some acceptable level of accuracy. and initial conditions should be evaluated during What level is acceptable is, of course, determined the calibration procedure in the same manner as subjectively. Although a poor match provides the uncertainty in aquifer properties. Failure to evidence of errors in the model, a good match in recognise the uncertainty inherent both in the in- itself does not prove the validity or adequacy of put data and in the calibration data may lead to the model [8]. ‘fine-tuning’ of the model through artificially pre- cise parameter adjustments strictly to improve Because of the large number of variables in the match between observed and calculated vari- the set of simultaneous equations represented in ables. This may only serve to falsely increase a model, calibration will not yield a unique set the confidence in the model without producing an of parameters. Where the match is poor, it sug- equivalent (or any) increase in its predictive ac- gests (i) an error in the conceptual model, (ii) an curacy. error in the numerical solution, or (iii) a poor set of parameter values. It may not be possible to dis- Fig. 4.5 illustrates in a general manner the use tinguish among the several sources of error [8]. and role of deterministic models in the analysis

556 USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT

match (minimises deviations) between observed Hypothesis testing data and model calculations. Least squares devia- tion is usually chosen as a criteria. The minimisa- Management decisions tion procedure uses sensitivity coefficients that are based on the change in calculated value divided Predictions Postaudits by the change in the parameter. For groundwater flow, for example, this may take the specific form of ∂h/∂t, that is the change in head with changing Calibration model & sensitivity tests transmissivity. The sensitivity coefficients themselves may be useful in the consideration of ad- Models of groundwater flow ditional data collection. and transport A recently documented computer program, UCODE, performs inverse modelling, posed as Site specific Conceptual models of governing a parameter-estimation problem, using non-linear Data Processes regression [67]. UCODE is extremely general and Fig. 28.5. The use and role of models in the analysis powerful because it can be used with almost any of groundwater problems [1]. application model or set of models. An estimated parameter can be a quantity that appears in the input files of the application model(s), or that can be of groundwater problems. The value of the mod- used in conjunction with user-defined functions to elling approach is its capability to integrate site- calculate a quantity that appears in the input files. specific data with equations describing the rel- UCODE calculates sensitivities as well as statis- evant processes as a quantitative basis for pre- tical measures that evaluate estimated parameter dicting changes or responses in a groundwater values and quantify the likely uncertainty of mod- system. There must be allowances for feedback el simulated values. Ref [68] documents meth- from the stage of interpreting model output both ods and guidelines for modern model calibration to the data collection and analysis phase and to using inverse modelling such that the resulting the conceptualisation and mathematical definition model is as accurate and useful as possible. Hill of the relevant governing processes. One objec- notes that obtaining useful results with inverse tive of model calibration should be to improve modelling depends on (i) defining a tractable in- the conceptual model of the system. Because verse problem using simplifications appropriate the model numerically integrates the effects of to the system under investigation and (ii) wise use the many factors that affect groundwater flow or of statistics generated using calculated sensitivi- solute transport, the calculated results should be ties and the match between observed and simu- internally consistent with all input data, and it can lated values. be determined if any element of the conceptual model should be revised. In fact, prior concepts or interpretations of aquifer parameters or variables, 4.6.3. Model error such as represented by potentiometric maps or Discrepancies between observed and calculated the specification of boundary conditions, may be responses of a system are the manifestation of revised during the calibration procedure as a re- errors in the mathematical model. In applying sult of feedback from the model’s output. groundwater models to field problems, there are Automated parameter-estimation techniques im- three sources of error [8]. One source is concep- prove the efficiency of model calibration and tual errors--that is, theoretical misconceptions have two general components--one part that cal- about the basic processes that are incorporated culates the best fit (sometimes called automatic in the model. Conceptual errors include both ne- history matching) and a second part that evaluates glecting relevant processes as well as representing the statistical properties of the fit. The objective inappropriate processes. Examples of such errors of automatic history matching is to obtain the es- include the application of a model based upon timates of system parameters that yield the closest Darcy’s Law to media or environments where

557 modelling

Darcy’s Law is inappropriate, or the use of a two- 1.2 dimensional model where significant flow or D 1.0 transport occurs in the third dimension. A second source of error involves numerical errors arising 0 .8 B A in the equation-solving algorithm. These include 0 .6

truncation errors, round-off errors, and numerical CONCENTRATION

V 0 .4 dispersion. A third source of error arises from un- I T A certainties and inadequacies in the input data that L C E 0 .2 reflect our inability to describe comprehensively R and uniquely the aquifer properties, stresses, and 0.0 0 2 4 6 8 10 12 boundaries. In most model applications concep- RELATIVE DISTANCE tualisation problems and uncertainty concerning the data are the most common sources of error. Fig. 28.6. Representative breakthrough curves for a simple flow and transport problem to illustrate types Numerical methods in general yield approximate of numerical errors that may occur in numerical solu- solutions to the governing equations. There are tion to transport equation: (A) plug flow having no dis- a number of possible sources of numerical error in persion, (B) ‘exact’ solution for transport with disper- the solution. If the modeller is aware of the source sion, (C) numerical solution for case B that exhibits ef- and nature of these errors, they can control them fects of numerical dispersion, and (D) numerical solution for case B that exhibits oscillatory behaviour [1]. and interpret the results in light of them. In solving advection dominated transport problems, in which a relatively sharp front (or steep concen- of a sharp front for a case having no dispersion tration gradient) is moving through a system, it (plug flow). Curve B represents an exact analyti- is numerically difficult to preserve the sharpness cal solution for a nonzero dispersivity. Curve C of the front. Obviously, if the width of the front illustrates the breakthrough curve calculated us- is narrower than the node spacing, then it is in- ing a numerical method that introduces numerical herently impossible to calculate the correct val- dispersion. ues of concentration in the vicinity of the sharp Numerical dispersion can be controlled by reduc- front. Even in situations where a front is less ing the grid spacing (Δx and Δy). However, reduc- sharp, however, the numerical solution technique tion to a tolerable level may require an excessive can calculate a greater dispersive flux than would number of grid points for a particular region to occur by physical dispersion alone or would be be simulated and render the computational costs indicated by an exact solution of the governing unacceptably high [32]. It may also be controlled equation. That part of the calculated dispersion in finite-element methods by using higher- or introduced solely by the numerical solution algo- der basis functions or by adjusting the formula- rithm is called numerical dispersion, as illustrated tion of the difference equations (using different in Fig. 4.6. Because the hydrologic interpretation combinations of forward, backward, or centred of isotopic data is sensitive to mixing phenomena in time and/or space, or using different weighting in an aquifer, numerical mixing (or dispersion) functions). Unfortunately, many approaches that can have the same effect on the interpretation of eliminate or minimise numerical dispersion in- model-calculated isotopic values. Therefore, care troduce oscillatory behaviour, causing overshoot must be taken to assess and minimise such nu- behind a moving front and undershoot ahead of merical errors that would artificially add ‘numeri- the front (see curve D in Fig. 4.6), and vice versa. cal’ mixing to the calculated mixing attributable Undershoot can result in the calculation of nega- to physical and chemical processes. tive concentrations, which are obviously unreal- Fig. 4.6 illustrates calculated breakthrough curves istic. However, overshoot can introduce errors of for a hypothetical problem of uniform flow and equal magnitude that may go unnoticed because transport to the right, at some time and distance the value is positive in sign (although greater after a tracer having a relative concentration of than the source concentration, so still unrealistic). 1.0 was injected at some point upstream. Curve Oscillations generally do not introduce any mass- A represents the breakthrough curve and position balance errors, and often dampen out over time.

558 USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT

In some cases, however, oscillatory behaviour can 4.6.4. Mass balance become unbounded, yielding an unstable solution One measure of model accuracy is how well or failure to converge numerically. the model conserves mass. This can be measured In solving the advective-dispersive transport by comparing the net fluxes calculated or speci- equation, some numerical errors (mainly oscilla- fied in the model (e.g. inflow and sources minus tions) can be related to two dimensionless param- outflow and sinks) with changes in storage (accumulation or depletion). Mass-balance calculations eter groups (or numbers). One is the Peclet num- should always be performed and checked during ber, P , which may be defined as P = Δl/α, where e e the calibration procedure to help assess the nu- Δl is a characteristic nodal spacing (although it merical accuracy of the solution. As part of these should be noted that there are several alterna- calculations, the hydraulic and chemical fluxes tive, though essentially equivalent, ways to define contributed by each distinct hydrologic compo- Pe). Anderson and Woessner [6] recommend that nent of the flow and transport model should be ite- the grid be designed so that Δl < 4α (or Pe < 4), mised separately to form hydrologic and chemical although Ségol [69] recommends a criteria of budgets for the system being modelled. The bud-

Pe ≤ 2. Similarly, time discretisation can be related gets are valuable assessment tools because they to the Courant number, Co, which may be defined provide a measure of the relative importance of each component to the total budget. as Co = VΔt/Δl [6]. Anderson and Woessner [6] also recommend that time steps be specified so Errors in the mass balance for flow models should that Δt < Δl/V (or Co < 1.0), which is equivalent generally be less than 0.1%. However, because to requiring that no solute be displaced by advec- the solute-transport equation is more difficult tion more than one grid cell or element during one to solve numerically, the mass-balance error for time increment. The deviations of curves C and D a solute may be greater than for the fluid, but this from the exact solution can be significant in some will depend also on the nature of the numerical locations, although such errors tend to be mini- method implemented. Finite-difference and finite- mal at the centre of a front (relative concentration element methods are inherently mass conservative, while some implementations of the method of 0.5). of characteristics and particle tracking approaches In solving the transport equation, classical numer- may not be (or their mass-balance calculations ical methods exhibit the proportionately largest themselves are only approximations). It must also numerical errors where the relative (or dimension- be remembered that while a large mass-balance error provides evidence of a poor numerical solu- less) concentrations (C/Cmax) are lowest. Ref. [70] shows that the error-to-signal (or noise-to-signal) tion, a perfect mass balance in itself does not and ratio can become quite large (>0.1) where the rel- cannot prove that a true or accurate solution has been achieved or that the overall model is valid. ative concentrations are less than 0.01. In isotope That is, a perfect mass balance can be achieved analyses of groundwater systems, the samples if the model includes compensating errors. For from areas of interest frequently reflect concen- example, the solutions C and D in Fig. 4.6 that trations less than 0.01 of the source concentration, exhibit significant numerical dispersion or oscil- so caution is warranted. latory behaviour arise from solutions that show In transport models there may also be a grid-ori- a near-perfect mass balance, but they are still entation effect, in which the solute distribution, wrong. calculated for the same properties and boundary conditions, will vary somewhat depending 4.6.5. Sensitivity tests on the angle of the flow relative to the grid. This Assuming various values for given parameters phenomena is largely related to the cross-product also helps to achieve another goal of the calibra- terms in the governing equation, and generally is tion procedure, namely to determine the sensitiv- not a serious source of error, but the model user ity of the model to factors that affect groundwater should be aware of it. flow and transport and to errors and uncertainty

559 modelling in data. Evaluating the relative importance of cesses and at some time accept the model as being each factor helps determine which data must be adequately calibrated (or perhaps reject the mod- defined most accurately and which data are al- el as being inadequate and seek alternative ap- ready adequate or require only minimal further proaches). This decision is often based on a mix definition. If additional data can be collected in of subjective and objective criteria. The achieve- the field, such a sensitivity analysis helps you de- ment of a best fit between values of observed and cide which types of data are most critical and how computed variables is a regression procedure and to get the best information return on the costs of can be evaluated as such. That is, the residual er- additional data collection. If additional data can- rors should have a mean that approaches zero and not be collected, then the sensitivity tests can help the deviations should be minimised. Ref. [71] to assess the reliability of the model by demon- discusses several statistical measures that can be strating the effect of a given range of uncertainty used to assess the reliability and ‘goodness of fit’ or error in the input data on the output of the mod- of groundwater flow models. The accuracy tests el. The relative sensitivities of the parameters that should be applied to as many dependent variables affect flow and transport will vary from prob- as possible. The types of observed data that are lem to problem. Furthermore, sensitivities may most valuable for model calibration include head change over time as the stress regime imposed on and concentration changes over space and time, a system evolves. Thus, one generalisation is that and the quantity and quality of groundwater dis- a sensitivity analysis should be performed during charges from the aquifer. the early stages of a model study. While it is necessary to evaluate the accuracy of The sensitivity of the solution to the grid design the model quantitatively, it is equally important to (or spacing), time-step criteria, nature and place- assure that the dependent variables that serve as ment of boundary conditions, and other numerical a basis for the accuracy tests are reliable indica- parameters should also be evaluated, even if an in- tors of the computational power and accuracy of verse or regression modelling approach has been the model. For example, if a particular dependent used. This is frequently overlooked, but failure to variable was relatively insensitive to the govern- do so may cause critical design flaws to remain ing parameters, then the existence of a high cor- undetected. For example, parameter-estimation relation between its observed and computed val- models cannot calculate the sensitivity to grid ues would not necessarily be a reflection of a high spacing or certain boundary conditions that are level of accuracy in the overall model. fixed in the model by the user. A general approach Similarly, caution must be exercised when the ‘ob- that works is after a preliminary calibration has served data’ contain an element of subjective in- be achieved with a model, it should be rerun for terpretation. For example, matching an observed the same stresses and properties using a finer grid, potentiometric surface or concentration distribu- smaller time steps, and perhaps alternative bound- tion is sometimes used as a basis for calibrating ary conditions. If such a test yields significantly groundwater models. However, a contoured sur- different results, then the model should be recali- face is itself interpretive and can be a weak basis brated using design criteria that yield a more ac- for model calibration because it includes a vari- curate numerical solution. If such a test yields no ability or error introduced by the contouring pro- significant differences, then the coarser design is cess, in addition to measurement errors present in probably adequate for that particular problem. the observed data at the specific points.

4.6.6. Calibration criteria 4.6.7. Predictions and postaudits Model calibration may be viewed as an evolution- As model calibration and parameter estimation ary process in which successive adjustments and are keyed to a set of historical data, the confidence modifications to the model are based on the re- in and reliability of the calibration process is pro- sults of previous simulations. The modeller must portional to the quality and comprehensiveness decide when sufficient adjustments have been of the historical record. The time over which pre- made to the representation of parameters and pro- dictions are made with a calibrated model should

560 USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT also be related to, and limited by, the length of of a subsequently revised model. Revised predic- the historical record. A reasonable guideline is to tions can then be made with greater reliability. predict only for a time comparable to the period that was matched. 4.6.8. Model validation The accuracy of a model’s predictions is the best It is natural for people who apply groundwater measure of its reliability. However, predictive models, as well as those who make decisions accuracy can be evaluated only after the fact. based on model results, to want assurance that Ref. [6] summarise several published studies in the model is valid. Groundwater models are em- which the predictive accuracy of a deterministic bodiments of various scientific theories and hy- groundwater model was evaluated several years potheses. Karl Popper [73] argues that ‘as sci- after the prediction had been made. The results entists we can never validate a hypothesis, only suggest that extrapolations into the future were rarely very accurate. Predictive errors often were invalidate it’. The same philosophy has been ap- related to having used a time period for history plied specifically to groundwater models [8, 74]. matching that was too short to capture an impor- The criteria for labelling a model as validated are tant element of the model or of the system, or to inherently subjective. In practice, validation is at- having an incomplete conceptual model. For ex- tempted through the same process that is typically ample, processes and boundary conditions that and more correctly identified as calibration--that are negligible or insignificant under the past and is, by comparing calculations with field or labora- present stress regime may become nontrivial or tory measurements. However, the non-uniqueness even dominant under a different set of imposed of model solutions means that a good comparison stresses. Thus, a conceptual model founded on can be achieved with an inadequate or erroneous observed behaviour of a groundwater system model. Also, because the definition of ‘good’ is may prove to be inadequate in the future, when subjective, under the common operational defini- existing stresses are increased or new stresses are tions of validation, one competent and reasonable added. A major source of predictive error is some- scientist may declare a model as validated while times attributable primarily to the uncertainty of another may use the same data to demonstrate future stresses, which is often controlled by de- that the model is invalid. To the general public, mographic, political, economic, and (or) social proclaiming that a groundwater model is vali- factors. But if the range or probability of future dated carries with it an aura of correctness that stresses can be estimated, then the range or prob- many modellers would not claim [72]. Because ability of future responses can be predicted. An labelling a model as having been ‘validated’ has encouraging trend is that many analysts are now very little objective or scientific meaning, such attempting to place confidence bounds on predic- ‘certification’ does little beyond instilling a false tions arising out of the uncertainty in parameter sense of confidence in such models. Konikow and estimates. However, these confidence limits still Bredehoeft [8] recommend that the term valida- would not bound errors arising from the selection tion not be applied to groundwater models. of a wrong conceptual model or from problems in the numerical solution algorithms [72]. 4.7. Case history: local-scale If a model is to be used for prediction relating flow and transport in to a problem or system that is of continuing in- a shallow unconfined terest or significance to society, then field moni- aquifer toring should continue and the model should be periodically postaudited, or recalibrated, to in- Reilly et al. [75] combined the application of corporate new information, such as changes in environmental tracers and deterministic numeri- imposed stresses or revisions in the assumed cal modelling to analyse and estimate recharge conceptual model. A postaudit offers a means to rates, flow rates, flow paths, and mixing proper- evaluate the nature and magnitude of predictive ties of a shallow groundwater system near Locust errors, which may itself lead to a large increase in Grove, in eastern Maryland, U.S.A. The study the understanding of the system and in the value was undertaken as part of the U.S. Geological

561 modelling

Survey’s National Water Quality Assessment analyses. The first-level calibration of a ground- Program to provide flow paths and travel time es- water flow model (second task) provided the ini- timates to be used in understanding and interpret- tial system conceptualisation. The third task was ing water-quality trends in monitoring wells and a second-level calibration and analysis involving stream base flows. The study area encompassed simulation of advective transport, which pro- about 2.6 × 107 m2 of mostly agricultural land on vided quantitative estimates of flow paths and the Delmarva Peninsula. The surficial aquifer in- time of travel to compare with those obtained cludes unconsolidated permeable sands and grav- from the CFC analyses. The fourth task involved el that range in thickness from less than 6 m to the application of a solute-transport model to sim- more than 20 m. This surficial aquifer is underlain ulate tritium concentrations in the groundwater by relatively impermeable silt and clay deposits, flow system as influenced by the processes of ad- which form a confining unit. vection, dispersion, radioactive decay, and time- varying input (source concentration) functions. In this study, chlorofluorocarbons (CFCs) and tritium were analysed from a number of water The sampling wells were located approximately samples collected from observation wells to es- along an areal flow line, and a two-dimensional timate the age of groundwater at each sampling cross-sectional model was developed for the sim- location and depth. Because errors and uncer- ulation of processes occurring along this flow tainty are associated with estimates of age based line. The MODFLOW model [76] was used to on environmental tracers, just as errors and un- simulate groundwater flow and advective trans- certainty are associated with deterministic mod- port. The finite-difference grid consisted of 24 els of groundwater flow and transport, the authors layers and 48 columns of nodes, with each cell applied a feedback or iterative process based on having dimensions of 1.14 by 50.80 m, as shown comparisons of independent estimates of travel in Fig. 4.8, which also shows the wells that lie in time. Their approach is summarised and out- the cross section. The simulation was designed to lined in Fig. 4.7. Each task shown was designed represent average steady-state flow conditions. to improve either the estimates of parameters or After the flow model was calibrated, pathline the conceptualisation of the system. and travel time analysis was undertaken and The preliminary calculations (first task) were used comparisons to CFC age estimates were made. to set bounds on the plausibility of the results Fig. 4.9 shows the pathlines calculated using of the more complex simulations and chemical MODPATH [77] after the second-level calibra-

Task: Preliminary Calculations Method: Calculation of ranges of travel times to shallow wells using known ranges Plausible simulated advective flow system. of recharge and porosity. Purpose:T o check consistency of CFC ages. Task: Simulation of observed tritium concentrations. MOC (Konikow & Bredehoeft, 1978) Task: First-level ground-water flow model calibration. Method: Method: MODFLOW (McDonald & Harbaugh, 1988) Purpose:T o simulate the transport of tritium with Purpose:T o calibrate a ground-water flow model to radioactive decay, and test the sensitivity of known heads and flows. the system to dispersion. Also to corroborate the plausible advective flow system.

Task: Second-level calibration of flow model and pathline analysis. Method: MODFLOW & MODPATH (Pollock, 1988, 1989, & 1990) Purpose:T o recalibrate the ground-water flow model with the Evaluation of conceptualization of additional information of travel times based on CFC transport and flow system. age data. And to determine flow paths and time of travel in the ground-water system.

Fig. 28.7. Flow diagram of the steps taken to quantify the flow paths in the Locust Grove, Maryland, groundwater flow system (modified from Ref. [75])

562 USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT

Fig. 28.8. Model grid used to simulate Locust Grove cross section, showing well locations (modified from Ref. [75]).

Fig. 28.9. Pathlines (calculated using MODPATH after second-level calibration) in Locust Grove cross section to observation wells showing time of travel (in years) from the water table (modified from Ref. [75]). tion with MODFLOW. The comparison with CFC the stream (i.e., excluding wells 159, 160, and estimates were generally good. However, Reilly 161) was 3.4 years. et al. [75] note that close to the stream, many flow 3H concentrations of recharge waters have varied lines converge, and the convergence of pathlines considerably over the last 40 years. Thus, the time representing the entire range of travel times pres- of travel would not always be readily apparent ent in the aquifer causes waters of different ages from the 3H concentration in a water sample. to be relatively near each other. Thus, at the scale Also, mixing of waters recharged during periods and grid spacing of the model, in the area near of these relatively sharp changes of input con- the stream the convergent flow lines cannot be centrations can make the interpretation of time of travel from 3H concentrations even more un- readily differentiated in the model and the loca- certain. Thus, the investigators simulated solute tions of individual well screens cannot be ac- transport of 3H within the system using a model curately represented directly under the stream. that accounts for mixing (dispersion), radioactive After the second-level calibration, the root decay, and transient input functions, which also mean squared error between the simulated ages allowed a further evaluation of consistency with and the CFC ages for the 10 wells farthest from the results of the previous flow and advective

563 modelling

tium values that were observed. The MOC model was advantageous for this problem because it minimises numerical dispersion and it can solve

the governing equations for αL of 0.0, which transport models based on finite-difference or finite- element methods generally cannot do. The results of the solute-transport simulation are consistent with the advective flow system determined by the second-level calibration and thus strengthen the case for the conceptual model. The coupling of the 3H analyses and the transport model indicates where discrepancies between the measured and simulated concentrations occur, where additional data collection would be most useful, and where refinement of the conceptual model may be warranted. This case study illustrates that environmental tracers and numerical simulation methods in combination are effective tools that complement each other and provide a means to estimate the flow rate and path of water moving through a groundwater system. Reilly et al. [75] found that the environmental tracers and numerical simulation methods also provide a ‘feedback’ that allows a more objective estimate of the uncertainties in the estimated rates and paths of movement. Together the two methods enabled a coherent explanation of the flow paths and rates of movement while identifying weaknesses in the understanding of the system that require additional data collection and refinement of conceptual models of Fig. 28.10. Simulated 3H distribution at the end of the groundwater system. 1990: (A) with dispersivity αL = 0.0 m and αT = 0.0 m, and (B) with dispersivity αL = 0.15 m and αT = 0.015 m. Contour interval 25 TU. Measured concentrations 4.8. Available groundwater from samples obtained from wells in November 1990 models are given for their location in bold italics (modified from Ref. [75]). A large number of generic deterministic groundwater models, based on a variety of numerical methods and a variety of conceptual models, are transport model. They applied the MOC solute- available. The selection of a numerical method or transport model of Konikow and Bredehoeft [37] generic model for a particular field problem de- and Goode and Konikow [78] for this purpose. pends on several factors, including accuracy, effi- The results of the simulations of the 3H distri- ciency/cost, and usability. The first two factors are bution assuming (i) no dispersion and (ii) αL of related primarily to the nature of the field prob-

0.15 m and αT of 0.015 m are shown in Fig. 4.10. lem, availability of data, and scope or intensity of The limiting case simulation of no dispersion the investigation. The usability of a method may yielded acceptable results and was used as the best depend partly on the mathematical background of estimate of the 3H distribution in November 1990 the modeller, as it is preferable for the model user [75]. This case reproduces the sharp concentra- to understand the nature of the numerical meth- tion gradients required to reproduce the low tri- ods implemented in a code. It may be necessary

564 USE OF NUMERICAL MODELS TO SIMULATE GROUNDWATER FLOW AND TRANSPORT to modify and adapt the program to the specific list 19 separate software distributors and provide problem of interest, and this can sometimes re- brief descriptions of several codes. The availabil- quire modifications to the source code. In select- ity of models on the internet is growing. Some ing a model that is appropriate for a particular ap- World Wide Web sites allow computer codes to plication, it is most important to choose one that be downloaded at no cost whereas other sites pro- incorporates the proper conceptual model; one vide catalogue information, demonstrations, and must avoid force fitting an inappropriate model to pricing information. The International Ground a field situation solely because of the model’s con- Water Modelling Center, Golden, CO (http:// venience, availability, or familiarity to the user. www.mines.edu/igwmc/) maintains a clearing- Usability is also enhanced by the availability of house and distribution centre for groundwater pre-processing and post-processing programs or simulation models. Many U.S. Geological Survey features, and by the availability of comprehensive public domain codes are available over the inter- yet understandable documentation. net (http://www.thehydrogeologist.com/). A large number of public and private organisa- The Hydrogeologist’s Home Page (http://www. tions distribute public domain and (or) proprietary thehydrogeologist.com/) is an example of a gen- software for groundwater modelling. Anderson et eral groundwater-oriented Web site that provides al. [79], in their review of groundwater models, links to a large number of software resources.

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