Analysis of hydrologic performance tests in unconfined aquifers
Item Type Thesis-Reproduction (electronic); text
Authors Chaudhri, Ata-ur-Rehman,1932-
Publisher The University of Arizona.
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Link to Item http://hdl.handle.net/10150/191469 ANALYSIS OF HYDROLOGIC PERFORMANCE TESTS
IN UNCONFINED AQUIFERS
by
Ata - ur - Rehman Chaudhri
A Thesis Submitted to the Facultyof the
DEPARTMENT OF HYDROLOGY
In Partial Fulfillment of theRequirements For the Degree of
MASTER OF SCIENCE
In the Graduate College THE UNIVERSITY OF ARIZONA
1966 STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of requirements for an advanced degreeat The University of Arizona and is depositedin the University Library to be made available to borrowers under rules of the Library. Brief quotations from this thesis areallowable without special permission, provided thataccurate acknowledgment of source is made. Requests for permission for extended quotation from orreproduction of this manuscript in whole or in part maybe granted by the head of the majordepartment or the Dean of the Graduate College when inhis judgment the proposed use of the materialis in the interests ofscholarship. In all other instances, however,permission must be obtained from the author.
SIGNED: /i: 4 /
APPROVAL BY THESIS DIRECTOR
This thesis has been approved onthe date shown below:
2t1-Kk/*' at (I J. G. FERRIS Professor of Hydrology ACKNOWLEDGMENTS
The author is greatly indebted to Professor
John G. Ferris, thesis director, for his valuable suggestions and guidance in accomplishing this task.
Special thanks are due to R. W. Stallman of the
Water Resources Division, U. S. Geological Survey,
Denver, Colorado, for his letter with valuable
suggestions in attacking the solution of unconfined
aquifer problems. The author is grateful to Dr. John W.
Harshbarger, Chairman of the Hydrology Committee, and to
Dr. Eugene S. Simpson for their guidance and
encouragement. The author is indebted to Water and Soil
Investigation Division of Water and Power Development
Authority, and the United States Agency forInternational
Development for providing the opportunityand facilities
for training the author in hydrology.
311 TABLE OF CONTENTS
Page
LIST OF TABLES Vi
LIST OF ILLUSTRATIONS Vii
ABSTRACT
INTRODUCTION 1
Location 1
General Features 2
Geology of Indus Plains
History of Investigation 6
Initial Pumping Test Program and Preliminary Analysis 7
Previous Studies 8
Nature of the Problem and Purpose of Present Study 8
FLOW REGIME OF A PUMPING WELL 11
General 11
Theis' Model 13
Effects of Anisotropic Conditions l'-I
Partial Penetration and Its Effects 15
Non-instantaneous Release Effects 21
Artesian Flow Equations and Their Application to Unconfined System 2i
iv V
Page
AN APPROACH TO PUMPING TEST ANALYSIS 26
General 26 Analysis of BR T/W 9 29
Partial Penetration Analysisof Early Data 31 Probable Magnitude of Anisotropyand Its Effect on Early DataAnalysis 36
Effect of Water TableConditions on Early Data Analysis 39
Delayed Yield Analysis 142 Distance Drawdown Analysis 43
Evaluation of the Magnitude ofPerturbations 46
Results of Selected Tests 51
Behaviour of Shallow ObservationWells 53
AUXILIARY TECHNIQUES 58 Time Drawdown Semi-log Plots 58 Estimation of Probable Errorin Specific Yield 63
Specific Yield CalculationOflBasis of Volume of Cone ofDepression 65 70 CONCLUSIONS 73 APPENDIX A DETAILED DATA OF TEST WELL
83 APPENDIX B GLOSSARY OF SYMBOLS
87 REFERENCES LIST OF TABLES
Table Page
Vertical Head Difference Between Deep and Shallow Wells at 21000 Minutes 30
Results of Early Pumping Test Data After Hantush (1961) 35 Partial Penetration Correction for Observation Wells 145
52 -i. Results of Selected Tests Initial Storage Coefficient asDetermined from Deep and Shallow Wells 57
Drawdown of Deep and Shallow Wells After First 10 Minutes 57
Slope Per Log Cycle ofTime-Drawdown Graphs at Later Times 62
vi LIST OF ILLUSTRATIONS
Figure Page
Location Map of West Pakistan 3
Sub Surface Formations of Rechna Doab 5
Potential Distribution Around a Non-penetrating Well 16 Ii. Potential Distribution Around a Well Penetrating Upper Half of 125 Feet Thick Sand 16
Transition from Initial Artesian Response to Water Table Response Due to Delayed Yield 23
Type Curves for Selected Observation Wells Taking into Account Partial Penetration Effects for Analysis of Early Pumping Test Data 33
Plots of Early Pumping Test Data of Selected Observation Wells 34
Type Curves Taking into Account the Partial Penetration and AnisotropiC Effects for a Deep Well 50 Feet from BR-T/W 9 38
Type Curves Indicating theDifference if the Effect of Vertical Componentof Flow due to Lowering WaterTable is Neglected 41
Partial Penetration Correction Incorporated in Distance-Drawdown Plot (BR-T/W 9) 147
vii viii
Figure Page
Observed and Estimated Drawdown for the Deep Observation Well 50 Feet from the Pumped Well 48 Magnitude of Departures from Theis'Initial Artesian Model with Time for Variou Conditions for the Deep Well 50Feet from BR-T/W 9 50 Time Drawdown Plot of Deep andShallow Observation Wells at 100 Feetfrom BR-T/W 8 60 Time Drawdown Plot of 300,500 and 1000 Feet Observation Wells(BR-T/W 8) for Estimation of "ds/dt."
15, Plot of CumulativePumping and the Volume of the Cone of Depressionfor Estimation of Specific Yield 69 ABSTRACT
Evaluation of aquifer characteristics was one of the facets of hydrologicalinvestigations started in Rechna and l95 in the Indus plains of WestPakistan. Chaj Doabs (area between Ravi andChenab rivers; Fig. 1) were the first to beinvestigated. Analysis of the pumping tests in these areas wasmade after methods of
Theis and Jacob. The values of storagecoefficients obtained indicated artesianconditions but this was in conflict with the availablegeological evidence. TransmissibilitieS were alsoquestionably high, and there was a disparity betweenresults calculated from time-th'awdown versusdistance-dX'aWdoWfl analyses. The purpose of thepresent study was to under-
stand the flow regimearound a pumping wellin the area inapplicability of of study; to identifythe causes of
Theis' analysis method,and to evaluatealternative
approaches for analysisof pumping tests inthis area. Analysis of fourspecially designed long components duration pumping testsindicates that vertical variable of flow due topartial penetration and a the two storage coefficientdue to slow drainage are
ix x
major factors which control the aquifer response and make Theis' method inapplicable. In the initial pumping period the effect of partial penetration on drawdownis very significant whereas the effectof delayed yield is inappreciable. In general, the effect of partial penetration is more pronounced throughout thepumping period as compared to delayed yield. However, with increasing distance from the pumping well, thepartial penetration effects decrease rapidly andthereby become
less important.
It was found that analysis of thefirst 20 - '#0
minutes of pumping data, with allowancefor partial penetration, gave an accurate valueof "P" and artesian
storage. In most cases the effectivedepth of aquifer could also be estimated. Observation wells at about
l.5b or farther from thepumped well, with adjustment for delayed yield, gave goodvalues of specific yield
and transmissibility. Close agreement was observed between the values of"T" from both methods. The use of
data of nearer wells for thepartial penetration analysis,
and the farther well data atlater times for the
delayed yield analysisprovided the best approach for
aquifer characteristicdetermination. xi
In the present analysis the effects of anisotropy and vertical component of flow due to lowering water table are so small relative to the partialpenetration
influence that they do not affect the early dataanalysis.
However, a method has been suggested to accountfor these
factors if in any test their magnitude appears
significant. The average value found forpermeability was
7.5 x 1O ft/sec, the artesian storage coefficient was in the range of 0.0001 - 0.005, andspecific yield was
0.2 - 0.25. The effective depth of aquifertaking part ranged from 720 feet to over 1500feet. INTRODUCTION
BanDoab (doab means the land between two rivers) and the Bhawalpur area, where these tests were
conducted, form a small part of the IndusPlains, West
Pakistan. In order to identify the hydrologicenviron- ment, a brief description of WestPakistan seems
appropriate.
Location
The State of Pakistan emerged on theworld map
as an independentsovereign state on August lii-,l9L.7, as
a result of thepartition of the subcontinent ofIndia.
It has two parts, namely, WestPakistan and East Pakistan, separated by 1,000 milesof Indian territory.
(Fig. 1) West Pakistan occupies thewesterly portion of
the Indo-Pakistansubcontinent between longitudes 61 to 75 degrees east, andlatitudes 2 to 37 degrees north.
On the west it is boundedby Iran, on the north and
northwest by Afghanistan,Oflthe northeast by Azad-
Kashmir and the disputedterritory of Jammu andKashmir,
1 2 and on the south by the Arabian Sea. Its area is
310,'403 square miles and according to the 1961 census, its population is '42,968,000.
General Features
West Pakistan on its north touchesthe Himalayan
foothills and the Hindukush mountains. The rugged mountainous region gives way to theplains and fertile
land of the Indus valley and thenceto barren deserts in
the south and west. The Indus plains constitutethe central portion of the Indus valleyand have an area of
about 50 million acres. West Pakistan has three welldefined seasons:
winter, summer, and monsoon orrainy season. In some
parts the winter is extremelycold but generally dry.
The summer begins inmid-April, and during the nextthree months the temperature in theplains may reach 120° F.
Between July and Septemberthe monsoon provides an
average rainfallof about 10 inches in theplains to
about 60 inches in thehills. Figuxe 1
Location map of Pakistan. 80 64° 120 68° K II 72° 76 80° LEGE ND 4° ITS SR / S S I. PROVINCIALCAPITALPROVOWLIAL Cliv bOUNDARYCAPITALS 360 36° UNDEFI 50 0 50 100 JAMMU SrinOgOr F/RE CA LOCATION MAP PeshY RowaIpindl L)z RIVER 32° Bonnu /iasokot. Jhe/um 'LAHOREE/ TERRITORY 32° P/shin Dero ,,,Montgomery /cy Lyolipur \'' ,Quetta Ghozi 88° LAST PAKISTAN 92° 28° KALAT DINAJPUR BRA HMAPU1W4 Sokkur 24° Hyde rabod JUNAGADH & MANAVADAR 4 Karachi MANAVADAI...\/JUNAG4D ARABIAN SEA 64° 680 72° ARAB/AN SEA 76° 86° 92° Geology of Indus Plains
Between the Tibetan plateau, with its mighty mountain chains along its southern edge, and the Indian peninsula block, is a broad expanse of flat valleys known as the Indo-Gangetic plains, curving northerly from east to west for approximately 2,000 miles. It conceals a great rift, or fracture, manythousand feet deep in the earth's crust. This depression, over the past few million years, has becomefilled with alluvium that was washed down from the mountainsand swept into the rift by the Indus, Ganges, andtheir tributaries.
(A. V. Karpove, 19614) The depth of the alluvium filleddepression is not known, but is expected to be afew thousand feet at places. Test drilling has not extendedbeyond 1,500
feet. Often lenticular clay lensesinterspersed in the aquifer are found but there is noevidence of artesian
conditions in the area under study. Figure 2 gives a subsurface section of Rechna Doab,and it is expected
that similar conditions occurin other doabs of the
Indus plain. Deep test holes indicate ageneral de- creasing trend in the permeabilitybeyond a few hundred
feet. Figure 2
Fence diagram showing sub surface formations of Rechna Doab in West Pakistan.
Clay lenses are indicated by darkshading. 41
5
(p
Figure 2. 6
History of Investigation
Agriculture is the mainstay of West Pakistan's economy, and is practiced in the Indusplain. Very low rainfall led to the gradual development of canalirri- gation system since 1900. At present the canal system
is about 38,000 miles in length and diverts 714M. A. F.
of river water annually to irrigatethis area. This additional source of recharge imposed by man onthe hydro-
logic system of the area, withoutproviding an equivalent
discharge facility, has disturbedthe hydrologic balance.
A rapid rise of the watertable has brought the country
face to face with the twin menaceof waterlogging and
salinity. About 100,000 acres arebeing lost to culti- vation annually in addition to thedecrease in production
of the remaining land. Open drains were tried for some
time but without much success.
In 19514, with the assistanceof the U. S. Inter-
national Co-operationAdministration (now named U. S. A.
I. D.), an era ofhydrologic studies started. The main
purpose was to studythe hydrologic systemof various
areas, diagnosethe causes ofwaterlogging, and search
for the necessary remedy. A comprehensive program was (now started by "Ground WaterDevelopment Organization"
named Water and SoilInvestigation Division) tocollect 7
the basic data needed by planners and to research some of the problems encountered in this program. Pumping tests to evaluate the aquifer characteristics is one of the many facets of its investigations.
Initial Pumping Test Program and Preliminary Analysis
In the initial phases of investigation more than
100 tests were performed, principally in Rechna andChaj
Doabs (area between Ravi and Chenab rivers, Fig.1) of
Indus plains. The pumping test set up in general con-
sisted of a 300-foot pumped well, screened between120-
300 feet, and four or five observation wells about200
feet deep, installed at different distances fromthe
pumped well. Arif and Rehman (1960) published the resultsin
their preliminary analysis of these tests. The analysis
was carried out aftermethods of Theis (1935) and Cooper and Jacob (196). It was considered that the field conditions of these testsclosely matched the
assumed conditions for which thesemethods are applicable.
It was observed that the valuesof storage
coefficients were very smallindicating artesian con-
ditions for which there was nogeological evidence. Transmissibilities were alsoquestionably high, and 8 their values as calculated from time-drawdown analysis and distance-drawdown analysis did not agree. There was no apparent cause to explain thisdiscrepancy.
Previous Studies
Arif (1964) analyzed some of these testswith a
view to locate the cause of these discrepanciesand find
a method to improveinterpretation. He assumed that delayed yield, partial penetration,and anisotropy may
be the factors affecting results. In his analysis of tests, after Boulton (1963), heobserved that results of observation wells beyond 400 feet didnot agree with the
results of nearer wells. He also tried a solutionof
one of the tests on anelectric analog assuming both the aquifer thickness and the ratioof flPz/PrtT. It was
observed that usefulness of themethod lies in the
accurate knowledge ofaquifer thickness and Pzh/Prtt
ratio. These are generally unknownto start with and their incorrect estimation mayaffect the conclusions.
Nature of the Problemand Purpose of PresentStudy
The effective thicknessof the Indus plain
aquifers may be from 750feet to more than 1,500feet. 9
Interspersed clay lenses are often found in these aquifers (Fig. 1). It is generally difficult and also uneconomical to determine the aquifer thicknessby test
drilling at each pumping site, and even whendetermined,
the effective depth of aquiferthroughout the flow
system may be different due tothe presence of inter-
spersed clay lenses. Without specific information onthe effective
aquifer depth, the magnitudeof "z'r"' and the drainage delay of the sediments,the modeling of all
aquifer factors is virtuallyimpossible. These may be
the reasons that Arif(l96) observed that theanalog model results did not agreewith analytical results. At present no analyticalformulation is avail-
able which accounts forall of these factors. In view
of these conditions thepurpose of thepresent study around a pumping is: (1) to understandthe flow regime separate the per- well in the area;(2) to identify and turbations which renderconventional methods
inapplicable, and toestimate their magnitudes and approach for (3) to select the bestpossible analytical
estimating aquifercharacteristics. In order to makethis study possible,special
aquifer tests weredesigned. The pumped well, as 10
previously stated, was about 300 feet deep and screened between 120 feet and 300 feet, Sets of deep and shallow observation wells were used to observe the drawdown. The deep observation wells were approximately 150 feet deep with 5 - 10 feet of strainer, and the shallow wells only tapped the water table. The distance of observation wells ranged from 20 feet to 2,000 feet from the pumped well. In comparison to earlier tests which were only of six days duration, these tests were run for about 20 days.
A complete layout plan of one such test site BR-T/W.9, along with constructional detail of the pumped well, the observation wells and their lithology are given in
Appendix A. FLOW REGIME OF A PUMPING WELL
General
Flow around a pumping wellamong other things is
highly dependent upon anisotropy,partial penetration of
the pumped well, confinementor unconfinement of the
aquifer, and variation of the storage coefficientwith time.
Many equations of flow to a pumping wellare
available in the literature, with each takingcare of one factor or the other, but none is available which
accounts for all the factors complicating the aquifer
response. The selection of any given flow-system model,
to represent the field conditions, is therefore a matter
of judgment left to the analyst. When field conditions
are complicated, and one system cannot fully represent
the response, then the best procedure is to select a
system which is nearest to the field conditions, and
estimate thereby the aquifer characteristics. It is also possible that a given factor may be more pronounced at
one time of pumping or within a certain distance from the pumping well, whereas others may have a different
influence both in space and time. During the same
11 12
pumping test, therefore, the response of the aquifer may change in time and space. In view of the variety of possibilities, the knowledge of characteristics of various flow conditions around a pumping well and their mutual relation can be of great help in weighing the field observations to find a flow-system model which can represent them in the best possible manner.
Theis'(1935)equation is an idealized model conceived for a confined aquifer but is applicable to an unconfined aquifer under certain limiting conditions. A comparison between the responses of confined and un- confined aquifers has been made possible by Boulton's (195k)treatment of unconfined aquifers. Due to the paucity of models for unconfined flow, artesian flow equations are used in this analysis, but corrections wherever necessary are applied using (195k) integral. In order to study the causes of departures and their magnitude from the Theis model, abrief summary of Theis' flow modeland other flow systems which come into play due toviolation of one or the other of Theis' assumptions aregiven in the following pages. This brief treatment will also helpto understand
the analysis procedureadopted later in this report. 13
Theis' Model
Theis (1935) using a heat analogy and Jacob
(l96) from the hydraulics point of view derived the nonsteady state equation, which gives drawdown after time "t at a point "r" feet from a wellpumping con- tinuously at constant discharge as,
e Q (2.1) s du flT U
r25 Tt
where "T" is the transmissibilityof aquifer, "Sn is
storage coefficient, andHQHis the discharge of the
pumping well. Various assumptions involvedin this derivation are summarized (afterJ. G. Ferris) as
follows:
a. Pumping Well Infinitesimal diameter
Fully penetrates theaquifer
Uniform flux distribution
. Constant discharge 114
b. The Aquifer
Infinite areal extent
Homogenous and isotropic
Uniform thickness
c. Flow System Instantaneous release from storage
Radial flow Darcy's law is applicable
No vertical flow component
Most of these assumptions arefulfilled in the field but several modify the aquifer responsein some instance to such an extent thatTheis' equation is not applicable.
Effect of Anisotropic Conditions
Theis' equation of drawdown around apumping well is a particular solutionof the general flow equation,
+ 2h 2h . h (2.2) Pr ) + Pz ( r 6r àz2 subject to the assumptionslisted in the previous section. If all other assumptions arevalid except "° h' that of isotropy, even then theterm remains àz2 15
zero for strictly radial flow conditions, and anisotropywould not affect Theis' flow model. How- ever, if either partial penetration or water table conditions prevail, then vertical components of flow exist, and " will not be zero, and anisotropy will àz2 accentuate the response. Anisotropy magnifies the influence of vertical flow components and the nature of its effect will be discussed under partial penetration effects.
Partial Penetration and Its Effects
If the pumped well does not penetrate the full saturated thickness of the aquifer or if only a portion of a fully penetrating well is screened, the well is said to be partially penetrating.
Figure 3 and Figure (Muskat, pp. 270-271) depict the potential distribution for two cases:
A non-penetrating pump-well; A pump-well penetrating only 50 percent
(upper half).
From the figures it follows that under the circumstances vertical flow components cannot be avoided and their magnitude increases with decreasing distance from the pumped well. The head registered in the vicinity of 16
/) o.. 1rç 02 01 0 OS 07 Sf3
0
0 w 0
0
Figure 3. Potential distribution around a non-penetrating well. (After Muskat,p.270)
02 0 05 06 07 08 09 _J0 0 0
02 w 0
01
Figure Potential distribution around a well penetrating upper half of 125 feet thick sand (after Muskat, p. 270). 17
the pumped well is not only a function of screen location of the pumped well but also of the screen portion and depth of the observation well. The greater the magnitude of vertical flow, the larger is the departure from the radial flow assumption and from the applicability of Theis' ideal model.
Hantush (1961) observed that the average draw- down registered by an observation well, however close it may be to a partially penetrating pumped well, isgiven by Theis' equation, provided the observation well is screened throughout the aquifer. The same is true in the case of an observation well located at adistance greater than l.5b (P/P)2regardless of the space position of its screen. In cases other than mentioned, the drawdownis not given by the Theis equation. During the early period of pumping and before the curve inflection appears, time drawdown curves have the same shapeand general appearance as for the caseof complete penetration. The
Theis formula is not applicable evenin the very early period of pumping, when one might assumethat the aquifer ends at the bottom of the pumpedwell unless the geometry of flow is such thatthe drawdown equation 18
for partially penetrated case for relatively short time reduces to the form
s M (u, L/r) (2.3) IT PL in which case the validity is assured onlyin the range LS5 where is less than 20
The general equation of drawdownregistered by
a piezometer at depth"z" in the case of a partially penetrated well, screened between depthtiLt? and "d"
(L greater than d) is given byHantush (1961, p. 85)
either of the two equations:(assuming P
z/b)] (2.) s Q/4Pb)[W(u)+ f(u,r/b, L/b, d/b,
where (nTTd/b) f 2b/JT(L-d) > (1/n) sin (nTTL/b - sin cos (nTTz/b;W(u, nTTr/b)
or L -Z [Mug L+ Z + M(u, )+ S Q/8p.(L-d) ) r r
d Z i(u, b/r, L/r, z/r) -M(u,
(2.5) M(u,d -Z)-f(u, b/r, dir, z/r) r ] 19
where 2nb + x + z f'(u, b/r, L/r, z/r) M(u, - r
2nb -X - Z 2nb - x - z M(u, + M(u, ) r r
M(u,2nb - x+ Z r (2b - L - For relatively short time, "t" less than 20 P the equation reduces to an approximate form as
Q S (2.6) 8TTP CL - d) where
L+z L -Z d +Z M(u, ) + M(u, -M(u, r r r d - z - M(u ) r and for large times, "t" greater thanbS/2P the equation
can be written as
Q (2.7) s [Wu + f3(r/b, L/b, a/b,z/b)] 4 iT Pb
where 0 (fur) niTz 4b I/n K0 C OS (L - d) b b
nffd nTTL - sin b b 20
The effect of partial penetration resembles the effect of leakage from storage in thick semi-pervious confining layer (Hantush 1960). Also if the curve inflection is apparent, but the period of observation is not long enough to establish the ultimate straight line variation on a semi-log, the effect may resemble that of a recharge boundary. The effect of partial penetration on drawdownis therefore to cause a rapid increase in drawdown tobegin with, but with time the effect of partialpenetration reaches its maximum value and the time-drawdown curve assumes a slope comparable tofull penetration conditions, if there is rio other influencing factor. If anisotropy is coupled withpartial penetration, its effect will be to increase thedrawdown still further
and much more time will be neededbefore partial penetration effect becomes constant; thus,much longer
distance would be required beforepartial penetration
effects could be considerednegligible (1-lantush 196k,
p. 353; R. W.Stailman 1965). 21
Non-instantaneous Release Effects
Slow draining of saturated soil samples is the general phenomena noted by L. K. Wenzel (19142, W. S. P.
887) and mentioned by R. W. Staliman (19614, professional paper 14l1_E) with reference to sand columnsof F. H.
King. Slow draining phenomena is of particular significance in unconfined flow study.
L. K. Wenzel (19142) observed thatits effect on
the determination of "T" from the steady stateformula
is not very significant, but in non-steadystate cases
which also involve determination of "5"it does present
difficulty. As mentioned by Boulton (1963),the initial
response of an unconfinedaquifer resembles the artesian
case until thedelayed yield effect becomessignificant.
The phenomena of slowdraining, therefore, can also be
looked upon as that ofincreasing storage coefficient
with time; and if the initialsystem be taken as our
reference, then increasing storagecoefficient is the
departure of the assumedconstant "S" condition from
Theis' ideal model. The effect of slowdraining can be looked upon
in two ways: if the initialartesian response is taken 22
as the reference, one can say that the effect of slow
draining on departure from the Theismodel is negligible
in the early period of pumping. If reference is taken
as the Theis water table response, then the effectcan be interpreted as maximum in the early period of pumping (see curve 1 and 2, Fig. 5).
Therefore, the effect of slow draining on the
drawdown in the former case can be consideredas negligible in the early period of pumping, and in my
analysis I selected this initial system as the reference point.
Boulton (1955, p. 475) recommended the following
equation for calculating the correction to be applied to
the drawdown calculated by Theis' equation with reference to the initial system:
Q - E. (-t) + -5 = [n E.(t1 )j (2.8) 4/TT
Limitation of this formula is that it is applicable for small distances from the pumped well where
J(r 1 and 'igure 5
Transition from initial artesian response to water table response due to delayedyield. 0 I I I 00 TIME (TS) I I III 1000 I I I 10000 -WTER TABLE RESP ONSE ARTESIAN RESPONSE CBSERVED RESPONSE
DATA OF HYPOTHETSyIO.13,T:O.75, O=3CFS bs000 ICAL CURVE, r=5o, S0.0055 2l.
Artesian Flow Equations and
Their Application to Unconfined System
In the case of a confined aquifer the discharge of the pumping well is sustained by compaction of aquifer and expansion of water due to the lowering of pressure.
If the well fully penetrates the aquifer and there is no dewatering, flow is radial and no vertical componentof flow is involved. In an unconfined aquifer the pumping well discharge is supplied by actualdewatering of the sediments and the lowering of water tabledevelops vertical components of head, even if thepumping well is fully penetrating. Boulton (195L1.) observed that theTheis expo- nential integral as applied to the watertable conditions is theoretically incorrect becauseit tales no account of the vertical component of pore waterflow approaching the well. This criticism applies particularlyto the early stages of a pumping testin a deep well, when themotion
of water in the upper partof aquifer is more vertical
than horizontal. He observed that after atime factor t Pat/Sb, has become greater than 5,only then does
exponential integral becomeapplicable. Before that time 25
the drawdown at a point is given by
sz Q (2.9) 2TTT v(e,'t) where
J0 (A,) V(t°,t) = - exp( -tAtanhA)dA]
This treatment provides a means for testing the applica-
bility of various flow equations derived strictly for
artesian conditions. If "f" is greater than 5, no correction is needed in those equations. But for smaller times the correction can be calculated as follows:
s Q (2.10) 4TTT [W(u)- 2v(e,'t)J
nowP andt:Pt u= b 'Tt
For each value of "u,"' being known, "V" can be calculated and also the value ofV(', jt). AN APPROACH TO THE PUMPING TEST ANALYSIS
General
In order to decide the best possible approach to pumping test analysis, it is of vital importance to identify and assess the magnitude of system noise. Most of Theis' assumed conditions are fulfilled in the field within practical limits. The principal departures are: the partial penetration of pumped well andobservation wells, the anisotropy of the aquifer, the slowdrainage of sediments, and the confined or unconfined natureof the aquifer. A brief theory of the aquifer response and the effect on Theis' flow model has beendiscussed in the previous section. The perturbations relative to model as the frame of reference may be due to one ormore of the following causes: Anisotropy of the aquifer,
Vertical flow components due topartial
penetration,
Slow drainage of sediments, Vertical flow component due to watertable
conditions.
26 27
At present no analytical formulation is available which takes into account all of the above-mentioned factors. The problem, therefore, is one of selecting the flow regime which best represents the field con- ditions. The method of analysis to be followed is to select a condition where or when one factor isdominant and other influences are either absent or negligible.
From a preliminary evaluation of aquifercharacteristics, an effort is made to reproduce atirne-drawdown curve for one of the observation wells. A method of successive trial is then followed to reduce thedeparture between the theoretically predicted andactually observed time- drawdown curve. The magnitude of perturbationdue to each factor is calculated toevaluate the analysis
approach or the probable error tobe expected in results.
The flow regime changeswith space and time in
the same pumping test. For the long pumpingduration of these pumping tests, it waspossible to select data so
as to satisfydifferent boundaryconditions necessary
for minimizing certaineffects and thus isolate them. A summary of theboundary conditions as given
by R. W. Staliman (inhis letter dated July 21,1965)
is as follows: 28
equation or others of artesian radial flow can be applied to the draw downs observed in vicinity of partially or fully penetrating pumped well (except where the restrictions noted) if:
1) "t" is greater than "; (Boul-ton, 1954). At
smaller times the vertical flow components due to lowering water table are significant.
"r" is greater than l.5b r'z'2 (Jacob, 1945; Hantush, 1961, and 1964, p. 355). At smaller radii partial penetration produces vertical flow components.
HTt/r2Stt is to be greater than 0.5 "Tt/b2S" to be greater than 5; "rIb. " to be greater than 0.8 for draw down at water table near a fully penetrating well. (Boulton, 1964, p. 608).
"TtIr2S" to be greater than 1.0; "rIb z/Pr2t0 be greater than 3; approximately for draw downs anywhere in the vicinity of a partially pene- trating well. (Boulton, 1964, p. 603).
11L\s/ it" is to be less than 0.002P; (R. W. Stallman's letter). If water table declines at a faster rate, specific yieldwill likely to be in error by more than 20%.
" t" greater than 1.8 + 2.6 rIB; where this condition is met, delay yield effects are negligible. Using some of the above mentioned conditions and other flow characteristics discussedin the second chapter along with the process of successivetrial, an analysis of
BR-T/W 9 is given in detail in thefollowing treatment.
Results of other tests have beensummarized in Table 4. 29
Analysis of B R - T/W 9
In the previous section under non-instantaneous release effects, it was observed that if Theis' artesian response is considered as the reference, the effect of slow draining is very small in the early part of the pumping period. Keeping this in view, early pumping data may have the disturbances due to anisotropy, partial penetration, and vertical flow component due to lowering water table.
Table 1 gives the drawdowns registered by deep
and shallow observation wells after 21000 minutesof pumping. Visual observation of the data indicate the presence of partial penetrationeffects. As a first approximation, assume that the other two effects are
comparatively small, and analyze the earlier datafor
partial penetration only. 30
TABLE 1
VERTICAL HEAD DIFFERENCE BETWEEN DEEP ANDSHALLOW WELLS
@ "t" 21000 (minute)
r BR-T/W 8 BR-T/W 9 (ft) 33 Sd SS Sd - SS Sd 3d -
1.05 20 10.1'4 7.48 5.79 5.64 '4.59
14.46 2.49 50 7.48 4.09 3.39 6.95 0.66 100 5.36 3.72 1.64 '4.69 '4.03 0.22 200 3.65 3.06 0.59 3.03 2.81 1.89 0.56 300 2.56 2.31 0.25 2.45
500 1.73 1.62 0.11 - - - 1.11 -0.12 1000 0.87 0.75 0.12 0.99 0.44 -0.02 2000 0.18 0.21 -0.03 0.42 31
Partial Penetration Analysis of Early Data:
For a relatively short period of pumping the drawdown around a partially penetrating well is given by equation (2.6). According to Hantush (1961) if the semi- log time drawdown curve of the observed data exhibits an inflection, and if the number anddistribution of observed points are such that the details of the curve prior to attainment of ultimate straight line are discernible, then the type curve method (as givenbelow) can be used to determine theformation coefficients and often its thickness.
3.1 5 Q 8 iT P(L-d)
The equation is of Theis' typein composition, wherein "E11 in addition includescharacteristics of the pumped well and observation wells:and is a function of
"u." A type curve therefore can beplotted between "E" and"i/U" iflthe same manner as can beplotted between "W(u)" and "i/u." Values of can be calculated by equation (2.6). 32
In the present test, the data used to evaluate
"E" is as follows:
180 ft; 'td" 60 ft; "Zd" 110 ft;
S = 0.0 ft
Semi-log plots of the observation well data
indicated that wells up to 500 feet had enoughpoints
to define the curve inflection, thereforeonly the
nearer wells were analyzed. Figures 6 and 7 give the type curves and early data for theobservation wells
used in analysis. Values of "P" and the artesian storage as obtained by thisanalysis are given in
Table 2. The average value oftransmissibility is 0.52 (ft2/sec) and of artesianstorage, 0.0057. The
aquifer thickness taking part wascalculated as 720
feet. It was observed that thevalue of storage coefficient as obtained fromshallow wells was much
greater than that calculatedfrom the deep wells. The method used here isapplicable only when "(2b - L -Z)2 S "t" is less than The time 20 T calculated is about 11 minutes. The data used for
analysis was within thistime limit. I/u - l0 I I I II tOO r=5o till 1000 I I I till r: 300 -z zw 0., 0.01 Figure 6. penetrationType curves effectsfor selected for analysis observation of early wells pumping taking testinto data.account partial Ti ME (MIS) I 10 J00 I I I III I I I I 1 1 I I I 1000I I I I I - S Iii S S S S Ii ...... a p P p . a 4 0 a 0a 0 0 0- o 10 - 0 0 0 0 .00 00000 00 .0- .0.0W000°0J -R- 0 0' - zo 0 0 0 0.i - 0 0 . r:5o - S 0 0 rr:200 oo 0.01 Figure 7. Plots of early pumping test data of selected observation wells. 35
TABLE 2
RESULTS OF EARLY PUMPING TEST DATA ANALYSIS AFTER HANTUSH (1961)
r (ft) P (ft/see) b (ft)
Deep Wells 50 6.6 x lO 1.36 x iü 5'0 100 8.6 x l0 8.f io-6 695 200 7.2 x lO 8.x l0 595
300 6.6 x l0 7.'x
Avg. 7.2 x 8 x i06
Shallow Wells
20 7.8 x10' 2.68 x 10' 590
50 7.2 x 2.82 x 10 655
100 6.4 x 2.00 x 10 790 200 6.6 x l0 2.52 x 10 750
Avg. 7.0 x 10 2.5 x 10 720 36
Probable Magnitude of Anisotropy and Its Effect on Early Data Analysis:
In the partial penetration analysis it was assumed
that the aquifer is isotropic. Next an estimate of the probable magnitude of "r'Z" is made and itseffect on
the analysis is examined. Four special tests, whose analysisis presented
here, included sets of deep and shallowobservation wells
up to 2000 feet. The shallow wells just penetratedthe water table and deep wells wereabout 150 feet deep. General observation of the headdifference registered by
the deep versus shallow wellsindicate that beyond 1000
feet the difference is negligiblysmall. Table 1 gives
the drawdown observed at differentdistances from the
pumping well both in the deepand shallow wells of test
BR-T/W, 8 and 9. Analysis of these tests forpartial penetration gave the effectivedepth of aquifer as 1000
feet and 750 feetrespectively. If from the data it is
now assumed thatin both cases the verticalhead differ-
ence vanishes at1500 feet, the value of "r'Z"can be estimated by the followingequation:
2.25 Pr(b/rv) (3.2) 37
The ratio "Pr/PZ" for BR-T/W 8 appears to be one, and for BR-T/W 9, two. These values indicate that anisotropy in the area under study is of quitelimited degree. In order to study the effect of anisotropy on the early data analysis ofBR-T/W 9, assume that in this test is '. Equation (3.1) still holds, except that the function giving the valueof "E" is slightly modified. The value of "E" is given bythe following
equation: (Hantush, l96, p. 353)
M(u, a'.L + Z) + M(u, a',L - Z) - r r
M(u, a' d + Z) M(u, a'. d - Z) (3.3) r r
where 2 a' =
As before, type curve canbe prepared to take be analyzed. account of anisotropyand the early data can of BR-T/W 9 Figure 8 gives type curvesfor a 50-foot well It was found that the for "Pr/Pzt''t and 1 respectively. cent, if the effect error in results wasless than 15 per I/u - I I IIJj-1 I I 1 uhF 10 1 1 111111 100 I 11111' 1 1 1 fIJI z w
Figure 8. Typeanisotropic curves taking effects into for account a deep thewell partial 50 feet penetration from BR-T/W and 9. I I 39
of vertical permeability was neglected. The value oftTptt would be slightly smaller and that of storage coefficient would be slightly larger than actual values.
Effect of Water Table Conditions on Early Data Analysis:
The effect of lowering water table upon the early data of observation wells near the pumping well is quite significant, and can be neglected only when "t" is greater than 5. In order to study its effect on analysis,it was assumed that p, and the effect of delay yield was negligible in early period. Boulton's (l95) solution for theunconfined aquifer provides a means to modifythe artesian formula of partial penetration to suit thewater table condition and thus estimate the effect ofneglecting this correction. Drawdown around a pumpingwell in a water table aquifer is given by equation(2.9), and in a confined aquifer by equation (2.1).
Q 5wt 2TTT
S = Q "W( u) t L1TT 40
The correction "s" to be applied to the draw- down given by partial penetration equation (3.1) will be:
Q s. - s (W(u) - 2V) (2. 10) 4TTT
Drawdown around a partially penetrating well in L Z2 a water table aquifer for "t" less than Sb - - 2b 5P
is then given by the following equation:
b "E" 5 = 5 Q - W(u) + 2V) (3.4) pp- Sc 41TT 2L - d)
Partial penetration analysis of the early pumping
data indicates the aquifer thickness to be 720 feet,
ttb/2(L- d)" then reducesto "3" and equation (3.4)
simplifies to
Q (3E - W(u) + 2V) (3.5) 5= 4TTT
Now and "W(u)" are all related to"u" and a modified type curve can be prepared foranalysis of early
pumping data observed in a partiallypenetrating water
table aquifer. Figure 9 gives such type curves forobservation
wells at 50 feet and 200 feetfrom BR-T/W 9. On the same
figure are plotted in dotted linethe type curves U - I I I I 1111 too I I I I IJIL I I I I 1111 I I I II Ill I - r5o ------r = 200
Figure 9. Typecomponent curves ofindicating flow due theto loweringdifference water if thetable effect is neglected. of vertical I-J '42 neglecting the water table conditions. From the type curves it can be seen that the departure is not very significant, and the magnitude of the partial penetration
effects appear so dominant that the water table condition
does not affect the results from the initial dataanalysis.
However, in cases where the partial penetration effects
are so significant, the method suggestedcould be used
successfully. In the present analysis, however,it was
neglected. A table giving values of for selected values of "u" is given by R. W. Stailman(1961, U.S.G.S,
prof. paper, '42'4-C) but they are notsufficient for
detailed preparation of such curves.
Delayed Yield Analysis:
With reference to Table 1 it canbe seen that the
effect of partial penetrationis negligible, at 1000 feet
and beyond. Therefore, the analysisof data for wells at 1000 feet and 2000 feet cangive an estimate of "T" and
the specific yield ifanalyzed according to Boulton(1963)
by using "Type B" curves. In the present test, thewell
at 2000 feet was probablyregistering some additional
influence and could notbe analyzed. Analysis of the
1000-foot well gave thefollowing values: Ll.3
QL4 S 0.19 rIB 0.8 o 1.5xio6
The value of Boultonts (l95'4) t factor for "C"
21000 minutes is 5.7 which precludes the effect of
vertical components of flow due to lowering of the water table.
Distance Drawdown Analysis:
As pumping begins, the hydraulic gradient, which
is essentially an equilibrium gradient or enough to transmit the required amount of water to the pumping
well, is established in the immediate vicinity. As
pumping continues, the equilibrium gradient continues to
be established at greater distances from the wellCL. K.
Wenzel, W.S.G.S., Wsp. 887, 00. 98-99). Thus the
portion of the cone nearer to the well reaches steady
flow condition at a much earliertime, and as such, Wenzel's argument for applying thedistance-drawdown
analysis to reduce the delayed yieldeffect is appli-
cable. In order to apply this approach,the partial
penetration correction is needed. At a relatively large
pump time, the drawdownaround a partially penetrating well is given by equation (7)ofHantush (1961,P. 173). The partial penetration correction is therefore given by:
us U - Q (3.) PC 1TT
This drawdown is to be deducted from the observed draw- down to get the drawdown free of partial penetration effects. At the very large times used in this analysis, the effect of the vertical component of flow due to lowering water table is also absent. The only effect remaining is delayed yield which as noted above according to Wenzel will not affect the results to any great extent. Partial penetration corrections have been calcu- la-ted for deep wells using values of "M(u,,& )function given by Hantush (professional paper 102, 1961). The corrections calculated are given in Table 3. The values used are:
Itbu= 720 ft. "P" 7.2 xl0' 180 ft.
" d" = 60 ft. '-1.5
TABLE 3
PARTIAL PENETRATION CORRECTION FOR OBSERVATION WELLS
Observed d/d t21000 Corrected r Spc mt d/d
20 15.4 6.22
50 7.6 3.10 6.95 3.85
100 4.06 1.68 '4.69 3.01
200 1.714 0.70 3.03 2.33
300 0.93 0.38 2.45 2.07
500 0.292 0.12 1.70 1.58
1000 - 0.008 0.99 0.99
2000 - - 0.2'4 0.24 46
The corrected drawdowns have been plotted on semi- log graph (Fig. 10) and have been found to lie on a straight line. The value of transmissibility and storage coefficient calculated are as given below:
0.49; "S 0 19 y
These values confirm the values of "T" and already calculated.
Evaluation of the Magnitude of Perturbations
Partial penetration analysis of the early data, delayed yield analysis after Boulton (1963), and the distance-drawdown analysis indicate the following values for ttT,fl "Syfl and other constants.
0.48; "S" 0.00575; "S " 0.19 y = i06 ttII b = 720 ft. ; ci ' 1.5 11-- 34
Using the above values, the departuresfrom response were calculatedwith equations 2.1, 2.6, 2.8, 2.10 and 3.6. In figure 11, the observedand estimated drawdown for the 50-foot well fromBR-T/W 9 were plotted to check Figure 10
Partial penetration correction incorporated in distance-drawdown plot (BR.-T/W 9). I Ofl 2700 0000 TIME (MIS) /o 0 CORRECTEDOBSERVED 0 T:AS: 2.05 z.3.752x2.05 =0.49 o0z / 7' If / /0 0 (p700)2 - G 0 I II11 I I III1l I It 11111 1 I I 11111 I -- Figure 11
Observed and estimated drawdownfor the deep observation well 50 feet from thepumped well. I 11111 I0 I I 11111 I TIM E(MTS)I I IIII 11111 10000 o OBSERVED I I IIIII 0 ESTIMATED 0 S0 C 0 0 z S S0 0 S 0 - 0do
OD I I 111111 I I I 11111 I I I 11111 I 1 11111 1 I I 11111 Ltg
the values of the estimated aquifer characteristics. It was observed that the estimated and actual field measure- ments agreed closely for times larger than 1200 - 1500 minutes. At smaller times the departure is significant and can be attributed to the approximate estimation of
the partial penetration effect in the transition zone.
In Figure 12, departure from Theis' model were plotted for each cause. It can be seen from this graph
that the partial penetration effect becomes significantly
great, as the pumping starts, and after about 100 minutes,
it tends to become almost constant. The delayed yield
effect is quite small in the early periods and the storage coefficient can be assumed constant for all practical
purposes during the early pumping period. However, at later times, its effect is quite significant and cannot
be ignored. The vertical component of flow is quite important in the early periods, but decreases rapidly with time. After about three minutes of pumping, the effect of this relative to partial penetrationis hardly
15 percent. This analysis clearly proves that partial pene-
tration is the most important factor incontrolling the aquifer response in the early period, and atlater times
delayed yield becomes quite effective andimportant. 'N TIME MTS) i a a a
2 : 110 FT rd: =b: 60 720 50 Figure 12. forNagnitude various of conditions departures for from the Theis' deep wellinitial 50 feetartesian from modelBR-T/W with 9. time 17= = 34 .5X106 51
Results of Selected Tests
Three other long range and specially designed aquifer tests were analyzed as per the proceduredeveloped and their final results are given in Table 4.
It was observed that this analysis procedure gives good results and the estimated values fromdifferent portions of data match closely. However, this analysis procedure can be further improved ifit is possible to
evaluate the ratio betweenvertical and horizontal permeability. If ratio of z'r is known,then type curves
can be prepared takinginto account both the effectof partial penetration and ofanisotropy by using Hantush
(equation 82, 1964, p. 353). By coupling thisequation
with vertical flow componenteffect as suggested by the the author in previous section,it is possible to analyze
early data with great confidence. 52
TABLE L.
RESULTS OF SELECTED TESTS
Partial Penetration Analysis of Distant Test Analysis of Wells for Delayed Site Early Data Yield Effects
S b T S Tr y
BR-TW 9 0.52 5.75 x l0 720 ft. O.L1l4 0.19
BR -T 1W 8 0.75 L x i0- 1000 ft. 0.66 0.20
BR-T/W 10 0.95 1 x l0 1600- 0.95 0.18 1700ft.
BR-T/W 12 0.7 5 x 10 1200 ft. 0.78 5.5 x i0 53
Behaviour of Shallow Observation Wells
In the course of this analysis it was observed that the value of initial storage coefficient as
calculated from early pumping data of the shallow
observation wells was much higher than that calculated
from the deep wells, and often was within the water
table range (see Table 5). A quantitative explanation of this phenomenon was not developed, but qualitatively
it may be explained as follows: Pumped wells in general are not screened through
the first 50 to 60 feet below the watertable because
the aquifer may be capped by finematerial. This is
true also in case of test wellsanalyzed here. In the early pumping period of a water table casethe free
surface boundary equation isaccording to Boulton
(l95-, p. 568)
)2 2 Sy r (_ôs -PZ[( - Initially the water table iseverywhere hori-
zontal. Before water levels begin todecline along the surface of the saturated zone, ahead gradient between
the water table and wellface is produced in response to
the reduced water levelin the pumped well, Along the 54
water table, this gradient is directed downward initially and perpendicular to the water table. In the early period the water table may be considered virtually horizontal over most of the area, ós/èr is almost zero, and the term bs/z is much larger than the term (s/z)2. Also in the unconfined system, the total effect on the relation between drawdown and time caused by the storage changes due to the change in the water table position is generally of a much greater magnitude than that resulting from decompression of the water below the water table. Therefore, equation 3.7 can be approximated as (R. W. Staliman, 1963, p. 215):
S s (3.8) y t ôz
Now suppose that it is possible to varythe magnitude of "PZ" keeping?S/Z'constant. is the aquifer characteristic andis also constant; "ôs/' therefore will vary directly as "Pa". The smaller the value of "'' the smallerwill be the lowering rate of the water table. According to Wyllie and Gardner(1958) the hydraulic conductivity can beestimated as where
is the decimal fractionof the pore space filled with 55
liquid removable by gravity--or, the decimal fraction of specific yield held in the pore space. As an approxi- mation it may be noted that the water held at some point in the pore space cannot join the moving water table if it is located where the hydraulic conductivity is less than the velocity of water table. Thus, above a water table moving at a constant velocity, the water content will not be lowered to a value less than
'/4 0 / as/At (3.9) Pz
Let "Sa" be the apparent specific yield established and "Sy" be the final specific yield, then
Sa/Sy (1 - (3.10)
1/4 (3.11) Sa/Sy 1 -(ts/L\t )
It follows that the smallerthe value of the second term, the smaller isthe difference between the initial and final storage values. From equations 3.7 to3.11, the following con- clusions can be made:
1. Shallow wells in unconfinedaquifers are expected to respond to watertable conditions from the 56
beginning unlike the deep wells whose initial response is essentially artesian. On account of the process of slow drainage, the value of storage coefficient from initial data may be much smaller than the final specific yielddepending
upon the rate of water tablelowering during the initial period as compared to the finallowering rate. Vertical permeability of the layer between
the water table and the top of thepumping well screen
does affect the lowering rateof water table and may
cause variation between theratio of initial to final specific yield from one site to another as canbe
noted from Tables 5 and 6. 57
TABLE 5
INITIAL STORAGE COEFFICIENTS AS
DETERMINED FROM DEEP AND SHALLOW WELLS
Initial Storage Coefficients Test No. Deep Wells Shallow Wells Sy
BR-T/W 8 0.004 0.02 0.20
BR-T/W 9 0.00575 0.13 0.19
TABLE 6
DRAWDOWN OF SHALLOW WELLS AFTERFIRST 10 MINUTES
Distance From Pumped Well(ft.) Pumping Rate (cfs) Test No. 20 50 100 200
BR-T/W 8 0.48 0.36 0.22 0.10 3.0 2.75 BR-T/W 9 0.39 0.17 0.11 0.12 AUXILIARY TECHNIQUES
In addition to the methods already described, the following techniques may be used to estimate the values of "T" and "Sr." Their application, however, depends upon the duration of pumping and aquifer response and may not be used indiscriminately.
Time Drawdown Semi-log Plots
After a large time "t" greater than "bS/2P" the effect of partial penetration reaches its maximum value (Eantush 1961). Also, if "t" is greater than 1.8 + 2.6 rIB, the final specific yield has almost been established (R. W. Stailman, letter). When both these conditions are met and also nutt is less than 0.02, then the observed drawdowns should plot onsemi-log graph as a straight line. Only from the shape of the plot it then becomes possible to see whether these conditions arefulfilled or not. In the course of the present analysissemi-log plots of BR-TIW 8 (deep wells) behaved in the same
58 59 manner and after about 5000 to 6000 minutes the data plotted on a straight line for the remaining pumping period (Fig. 13).
Transmissibility being a function of slope only,
can be evaluated by using Jacob's time-drawdown formula.
The storage coefficient is affected because of the
translation of the graph due to the partial penetration
effect. The correction for this has been suggested by
Hantush (1961); and the storage coefficient is calcu-
lated by the following equation:
2.25 T t exp (f ) 5 Sy (L..1) r2
where f5 is evaluated from equation 2.7.
Special feature noted in the present analysis
is the variation of the final slopeof time-drawdown graph with distance, and also the greaterslope of a
shallow observation well as compared toits counterpart
deep well (Table 7). The decrease of final slope with distance may be attributed to one or moreof the follow-
ing causes: (a) increase of transmissibilitywith distance due to increasingdepth of effective aquifer thickness; (b) variation of storagecoefficient with
distance. Figure 13
Time drawdown plot of deep and shallow observation wells at 100 feet from BR-T/W 8. I I - I -. I I II TIME (MTS) I III I I I I SI BR.T.W. 8 I 1 ( ** * 0z * 4. * .SHALLOW WELL 0 - DEEP WELL c) 61
In view of the above variation it has been observed that well at 1000 feet on analysis gave "T" as 0.70, and "Sy' as 0.19. These values agree with the
already calculated values. This indicates that the above method if used for farther wells can give good values. 62
TABLE 7
SLOPE PER LOG-CYCLE OF TIME-DRAWDOWN GRAPHS AT LATER TIMES
Deep Wells Shallow Wells I, Slope Slope
20 1.6 ft/log cycle 1.75 50 l.4 1.8 100 1.35 1.70 200 1.32 1.6 300 1.0 1. 35 500 1.0 1.3 1000 0.8 0.8 63
Estimation of Probable Error in Specific Yield
Sediments take a very long time to drain com- pletely and thus for the limited durationof most field tests the final value of specific yieldis not likely to be established. R. W. Staliman (his letter1965) suggested a method to estimate the probable errorin the value of specific yield from testsof limited duration. A summary of his methodfollows. Assume we could lower the watertable at any desired constant velocity, andconsider first an infinitesimal rate of lowering. The liquid content in the pore space at somedistance above the water table is but asmall fraction of the pore volume andconsequently the hydraulic conductivity is small comparedwith the vertical permeability below the watertable. If the downward velocity of the watertable is small compared with theconductivity above the water table, water in and abovethe capillary fringe can easily followthe lowering watertable and thereby contribute tostorage removal instan- taneously, as per thedefinition of specific yield. However if we were tolower the water table at a velocity muchgreater than the vertical permeability, mostof the water would remain stranded above becausethere is available a maximumdownward hydraulicgradient of 1, and the maximum downwardvelocity attainable in a gravity field is thereforeP. With the latter there would be no freedrainage possible as long as wemaintained a rate oflowering greater and the observablespecific yield would than P theory with be zero. Coupling unsaturated flow observed drawdown ratesand value of Paffords of error in a means forestimating the magnitude delayed yield. According S possibly arising from to Wyllie andGardener (1958) thehydraulic conductivity can beestimated as where0 is 6'4 the decimal fraction of the pore space filled with liquid removable by gravity--or, the decimal fraction of S held in the pore space. As an app.roximation"we cansay that water held at some point in the pore space cannot join the moving water table if it is located where the hydraulic conductivity is less than the velocity of the water table. Thus above a water table moving at constant velocity, the water content will not be lowered to a value less than l/'+ 0 As /P = L\t ) approximately. Also it can be seen that the apparent specific yield will be
S (1-0) =Sa y y
Following table gives ratio of apparent tofinal specific yield.
S IS ya y (s/ LtYP
o
0.2 0.41
0.1-I 0.13
0.6 .026
0.8 .0016
0.9 .0001 65
The knowledge of vertical permeability is of
great value in such estimates and as pointed out before
can be fairly estimated as suggested by Staliman (1963)
and Weeks (1964).
Figure 14 gives the plotting of drawdown versus
time for three observation wells selected from BR-T/W 8.
The average lowering rate of these wells at later times is from 1.94 x lO to 2.8 x 10 ft/minute. Tests, as indicated above, show that the value
of Sy estimated as 0.2 may be in error as much as
33 percent. The value of vertical permeability used has
been assumed equal to radial permeability which is
5.5 x i0 ft/sec.
Specific Yield Calculation on Basis of Volume of Cone of Depression
A semi-log plot of a set of deep and shallow
observation wells is given in Figure 13. From inspection
it appears that both are lowering almost at the same rate.
This response of the aquifer was used by Bennet,
Rehman, and others (1964) to estimate theprobable
specific yield. A brief review and summary is given below. Figure 1'
Time drawdown plot of 300, 500, and 1000 feet observation wells (BR-T/W 8) for estimation of "ds/dt." 14000 16000 I TIME (tIlTS) I e000 2000 0
VERTICAL SCALE I': I. FT 67
Deep wells initially show an artesian response until the slow draining becomes significant. From the behaviour of deep and shallow wells it is clear that the artesian response of deep wells begins to change slowly towards water table response in accord with what has been termed as delayed yield by Boulton (1954, 1963). It is believed that the cause of change of the response from artesian to water table in effect relates to vertical percolation through the strata lying between the water table and the top of pumped well screen and its effect is registered by shallow wells. When the rate of lower- ing of deep and shallow wells becomes almost the same, it can be concluded that the verticalpercolation from upper strata is supplying the pumped water and thecontribution
from artesian response is negligible. If the cone of depression is plotted on semi-log graph paper for various times, then volume of the cone
of depression is given by:
7T (K1 -19 r2 r2 (K2- 1<3) . I I 4.6 L 1
(K - K ) 2 r2 (4.2) n-i r + 1 and "r" gives the various radii at which the slopes change (Bennet and others, l96i). If the cumulative dewatered volume thus calculated for different times is plotted against the cumulative pumping, the limiting slope of the graph gives an estimated value of the specific yield achieved during the pumping period. Such an estimate has been made for BR-T/W 8, as shown in Figure 15, and gives an estimate of tiSyt?as 0.21. This method has a decided advantage over that suggested by Ramsahoye and Lang (1961) in the sense that at latter times when delayed yield effect is almost vanishing, the slope gives the final yield established during the last period. The effect of partial pene- tration is also eliminated automatically because at later times its effect is constant and by taking the slope its effect vanishes from the results.However, the main drawback of the method lies in the fact that the determination of zero drawdown radius is subjective and may lead to appreciable error. Despite this drawback, if wells at great enough distances are used, it can serve as an additional test for thevalue of "Sy" already estimated. 69 2 0. IS 16 12 VOLUME OF CONE OF DEPRESSION (GFTXIO) I I I 4 8 12 16 2o Figure 15. Plot of cumulative pumping and the volume of the cone of depression for estimation of specific yield. CONCLUSIONS In the course of analysis it has been observed that the effect of vertical components of flow is the single most dominant factor which controls the drawdown around the pumping well and seriously affects the determination of transmissibility and artesian storage from the earlier pumping data. Boulton's delayed yield curves are the most appropriate for determination of specific yield but can be used only where there is no other dominating factor influencing the observed drawdown. As such, their application to wells in the vicinity of partially penetrating wells will always lead toinconclusive results. Hantushts (1961) partial penetration analysis can be used quitesuccessfully in determination of permeability from early pumping data. The general value of thepermeability in the area under analysis appearsto be 7.5 x lO and the artesian storage is in the rangeof 0.0001 to 0.005; the specific yield is 0.2 to 0.25. The depth of aquifer taking part is 750 feet to 1500 feet. 70 71 5. The analog solution may prove to be successful but at the present moment it has not been possible to model the delayed yield (Stailman, 1965). The other major unknowns are the depth of aquifer and ratio of vertical permeability to radial permability. In view of this analysis, the following points, if noted during performance and analysis of tests in unconfined aquifer, may improve the estimation: (1) Design of aquifer test should provide at least three or four observation wells in the range of 800 feet to 1500 feet. Wells between 50 to 200 feet can serve as a check on results and may permit evaluation of effective depth of aquifer taking part. Wells nearer than 50 feet may not be very useful. Lithologic samples collected from each test should be carefully analyzed for permeabilityand specific yield determination and compared with this analysis to support it. In the past no such effort has been made and I hope this can improveunderstanding of the aquifer behaviour. The axiom of analysis is toremain out- side the area of trouble wheneverpossible and the various limits have already been given on page28. 72 (iv) According to Stallman, if all plots are made on the same graph paper usingt/r2, then many difficulties can be avoided. AP1ENDIX A DETAILED DATA OF TEST WELL 74 Constructional Detail of Pumped Well BR - T/W 9 N. S.L. 97 ft housing pipe, Diameter of 14 inches diameter drilling hole is 22 inches '. 120 ft strainer, 12 inches diameter 1' -. I - 5 ft bail plug -I F, :': :':: fV ',, I f F 315 ft WATER AN SOft VvAFOA) C \'.E. 5 EA L1 r\ LI U ' WELL LOG (L[Ti-OLOGCAL) LITh S1tTON D;VSON C! iUU\V )\I Sheet No. I Form WELL NO SR\./frfr 7,2' ' TYPE - CD-ODNATELOCTON I t TABLE Tk iU DLLIN CO1ENCED LEPTn TO WTEP DLLC CO ETE2 \ GEO:C: LE 'EL EE'A ON p . 3 1' DRiL TYPE c'iPSe7 Pe7 a4 DEPTH FEET GRA- OG TH FEETrss FCRMATCNC / '--4S s' C ::- ESCR PT:C'N F E 1 A F. KS - Iy A __ d , i x7/-ce. 1/y- 7v - -3 I Tj-eif V! R yVt( o 5(ô - / -(/J f! yL'(I .c j__c,_ 3 / /- I- - [;7qy - -fl_.- C_L. 21 H- - cLy / y If/A ., ars- cO - C 2 // -u,-.-' -tI.td.--- Z -iu /7A/ - (1 - 0 '-I a- i-Ic'I,f L15,C - - ,-iI (& c -30 4ItfcL..-i' L. 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Million Acre-feet M(u,) (eY erf(1 dy JYci P Isotropic permeability Radial permeability Pg Permeability along 0 direction Vertical permeability = Pumping rate of well (cfs) Sy Specific yield Sya Apparent specific yield coefficient S Instantaneous storage S' Delayed storage coefficient S Specific storage (S/b) TS Transmissibility of aquifer(ft2/sec) W(u) Well function of"U"; [ e du. J(J U Z Depth of piezoineter(deep) below water table (ft Z Depth of shallowpiezometer below water table (ft) Reciprocal of delayindex (see Boulton, 1963) 86 9 L Tt/r2S 0' LI. Tt/r2S' s+s, SI Pt Sb RE FE RENCES Arif, A. H., and Rehman, A. R., 1960,Analysis of aquifer tests in Rechana and Chaj Doags:West Pakistan Water and Power DevelopmentAuthority, Water and Soil Investigation Division,Technical Paper No. 2. Arif, A. H. ,1964, Analysis of selectedtests of aquifer characteristics, West Pakistan: M.S. thesis, University of Arizona, Tucson. Bennet, G. D., Rehman, A. R.,Shiekh, I. A., and All,S., 1964, Analysis of aquifertests in the Punjab region of West Pakistan: WestPakistan Water and Power Development Authority,Water arid Soil Investigation Division. in Bental, R., 1963, Shortcutsand special problems aquifer tests: U. S. G. S.Water Supply Paper No. 1545-C. table under Boulton, N. S., 1954, Thedrawdown of water non-steady conditions near apumped well in an Proc. unconfined formation:Inst. Civil Engineers London, v.3, pt. 3, p.564-579. pumped well 1955, Unsteadyradial flow to a allowing for delayedyield from storage:Pub. Association Internatioflaled' No. 37, de i' Tome II, Hydrologie, Assembleegenerale de Rome, p.472-477. non_equilibrium 1963, Analysisof data from allowing for delayedyield from pumping tests London, storage: Inst.Civil Engineers Proc., v. 26, p.469-482. of "Analysis of datafrom non- 1964,DiscusSiOfl delayed yield equilibrium pumpingtests allowing Proc., from storage" Inst. Civil Engineers London, v. 28, p.603-610. 87 88 Cooper, H. H.,and Jacob, C. E., 1946, Ageneralized graphical method of evaluatingformation constants and summarizing wellfield history: A. G. U. Transactions, v.27, p. 526-534. Ferris, J. G., Knowles, D. B., Brown,R. H., and Stallinan, R. W., 1962, Theory ofaquifer tests, U. S. G. S. Water supply paper1536-E. Hantush, M. S., 1954, Planepotential flow of ground water with linear leakage: A.G. U., Transactions, v. 35, p. 917-936. 1955, Analysis of data frompunping tests in leaky aquifers; A. G. u.,Transactions, v. 37, no. 6, p. 702-714. 1957, Non-steady flow to awell partially penetrating an infinite leakyaquifer: Proc. Iraqi Science Society,V.I, p. 10-19. 1960, Modification of theoryof leaky aquifer: J. G. R., v. 65, no. II, p.3713-3725. 1961, Drawdown around apartially penetrating well: Proc. A. S. C. E.,Hy, 4, p. 83-97. 1961, Aquifer tests onpartially penetrating wells: Proc. A. S. C.E., Hy, 5, p.171-195. 1961, Tables of thefunction M(u,B), Inst. of Professional paper102: New Mexico Mining and Technology,Soccorro. 1962, On thevalidity of theDupuit- Forchheimer well dischargeformula: J. G. R., V.67, no. 6, p.2417-2420. in 1964, Hydraulicsof wells, in Advances HydrosCienCe, ed. VenTe Chow:Academic Press, Inc., New York, p.281-442. and Jacob, C.E., 1955,Non-steady Hantush, N. S., aquifer: A. G. radial flow in aninfinite leaky 36, no. I, p. 95 -100. U., Transactions,V. 1955, Threedimensional flow to a well______ma twolayered aquifer: A.G. U., 286-292. TransaCtiofl6, v.36, no. 2, p. 89 Hydrologicsh Colloquium,1964, Steady flow of ground water towards wells:Committee for Hydrological Research, T. N. 0., P. 0. Box297, The Hague, Netherlands. Jacob, C. E., 1940, Flow of waterin an elastic artesian aquifer: A. G. U., Transactions, v.21, p. 574- 586. 1946-b, Radial flow in a leakyartesian aquifer: A. G. U., Transactions,V.27, p. 198-205. Karpov, A. V., 1964, Indusvalley--WestPakistafl!s Life Line, Journal: HydraulicDivision, A. S. C. E. (Jan. 1964), p. 216. Kirkham, Don, 1959, Exact theoryof flow into apartially penetrating well: J. G. R.,V.64, no. 9, p. l317 1327. Muskat, N. S., 1937, The flowof homogeflioUsfluids through porous media:McGraw Hill, p.270-271. S. M., 1961, Asimple method Ramsahoye, L. E., and Lang, pumping tests: for determiningspecific yield from U. S. G. S., Watersupply paper no.1536-H. for pumping Staliman, R. W., 1961a,Boulton's integral Geological SurveyResearch, test analysis, in 424-C, 1961: U. S. G. S.,Professional Paper p.24-29. 1961b, The significanceof vertical flow of pumping wells in un- components in vicinity Professional confined aquifer:in U. S. G. S., Paper 424-B, p.41-43. dimensional flow 1963, Electricanalog of three its applicationto unconfined to wells and Supply paper 1536-H, aquifers: U. S. G.S., Water p.205-242. conditions on 1965, Effectsof water table changes nearpumping wells: A. G. U., water level 2, p. 295- Water resourcesresearch, v. I, no. 312. 90 1965, A letter to theauthor on analysis of tests in unconfinedaquifers (dated July 21, 1965). lowering of Theis, C. V., 1935, Therelation between the piezometric surface and rateand duration of discharge of a well usingground water storage: A. G. U., Transactions,V.16, p. 519-524. Wenzel, L. R., 1942, Methodsfor determining permeability of water bearingmaterials: U. S. G. S., Water Supply paper887. Weeks, E. P., 1964, Fieldmethods fordetermining aquifer anistrOpY vertical permeability and 193- in U. S. G. S., ProfessionalPaper SOl-D, p. 198. G. H. F., 1958,The Wyllie, M. R. J., and Gardner, equation: Its applica- generalized Kozeny_Carman Porous tion to problems ofmultiphase flow in seC.V.146, media; pt. 2: WorldOil, Production, p.210-228.