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Type curves for recovery of a discharging well with storage
Journal o f Hydrology, 33(1977) 341--348 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
341
TYPE CURVES FOR RECOVERY OF A DISCHARGING WELL WITH STORAGE
P.R. FENSKE Desert Research Institute, Water Resources Center, University o f Nevada System, Reno, Nev. 89507 (U.S.A.)
(Received June 6, 1976; revised and accepted July 23, 1976)
ABSTRACT Fenske, P.R., 1977. Type curves for recovery of a discharging well with storage. J. Hydrol., 33: 341--348. The analysis of water level recovery data is usually carried out using the straight line or late time solution. Recently Case et al. presented a method based upon the reversion of the series representing the exponential integral whereby recovery data could be analyzed if only two points were known on the recovery curve. This analysis was based upon the Theis equation and consequently subject to all of the assumptions to the Theis equation. Important of these assumptions is the lack of storage in the discharging well. Papadopulos and Cooper solved the problem and presented appropriate type curves for well recovery after instantaneous charge of water when the well contained storage. Fenske presented an extension of the Theis equation which considers both discharging and observation well storage. The simpler mathematics makes the derivation of type recovery curves for radial flow systems with discharging well storage readily feasible. It is found that as pumping time becomes very short relative to recovery time, the type recovery curves approach as a limit the curves of Cooper, et al. for recovery after instantaneous drawdown. Recovery data from a test at Dawsonville, Georgia, is used as an example.
INTRODUCTION T h e analysis o f w a t e r level r e c o v e r y d a t a is usually carried o u t using the straight-line or late-time s o l u t i o n due t o the d i f f i c u l t y in a n a l y z i n g early time r e c o v e r y data. R e c e n t l y Case et al. ( 1 9 7 4 ) p r e s e n t e d a m e t h o d derived f r o m the Theis e q u a t i o n and based u p o n reversion o f the series r e p r e s e n t i n g the e x p o n e n t i a l integral w h e r e b y r e c o v e r y d a t a c o u l d be a n a l y z e d if o n l y t w o p o i n t s were k n o w n o n the r e c o v e r y curve. This analysis is s u b j e c t to the a s s u m p t i o n s used in d e v e l o p i n g the Theis e q u a t i o n . I m p o r t a n t in these a s s u m p t i o n s is the lack o f c o n s i d e r a t i o n o f storage in the discharging well. P a p a d o p u l o s a n d C o o p e r ( 1 9 6 7 ) solved the p r o b l e m f o r d r a w d o w n in a well with storage a n d P a p a d o p u l o s ( 1 9 6 7 ) p r e s e n t e d an analysis f o r d r a w d o w n at a n y l o c a t i o n in the c o n e o f depression o f a discharging well with storage.
342 Fenske (1974) was able to reproduce all well hydraulics analyses for isotropic h omo g en eo u s confined aquifers of infinite areal e x t e n t by a simple extension of the Theis equation. The simpler mathematics makes derivation of type recovery curves for discharging well with storage feasible. In the following paper a brief review of Fenske's extension of the Theis equation is presented and t ype recovery curves for a discharging well are developed. Data from the Dawsonville, Georgia test, presented by Cooper et al. (1967}, is used for demonstration o f the method. REVIEW OF THE THEIS EQUATION EXTENSION The extension of the Theis equation to the more general application is based upon the simple relationship that the volume of the region of the aquifer and all of the wells or other storages that are emptied by the discharging well at any instant in time divided by the average discharge is equal to the time required to develop that volume and its associated drawdown. The volume o f the cone of depression and associated storages is given by the following equation:
V = 2~S
/
rsdr + nrcwSw2 + (1-S)nr~mS m
(1)
rw
where r = radius to any location on cone of depression, rcm = casing radius of observation well, rcw = casing radius of discharging well, r w = effective radius of discharging well, S = storage coefficient, s = drawdown at any location in cone of depression, s m = drawdown at the observation well, s w = drawdown at the discharging well, V = volume of cone of depression plus volumes of discharging and observation wells. To integrate the first term on the right of the instantaneous volume expression, an equivalent to the drawdown, s, in terms of the radius to any location within the radial flow field is required. In a non-leaky isotropic h o m o g e n e o u s confined aquifer of uniform thickness and infinite areal extent, equipotential surfaces are vertical and concentric a b o u t the discharging well. The cone of depression described by the Theis equation is also the cone of depression associated with a well of finite diameter, and the desired expression for drawdown is obtained from the Theis equation. After integrating eq. 1, dividing the volume of the aquifer and associated storages by the average discharge and some additional manipulation (Fenske, 1974), the following equation set that describes dimensionless drawdown vs. dimensionless time at any location in the radial flow field is obtained:
343
4Tt r2ca
_
4~T
Qa (xweXw)_ Q
1
+ 1 El(Xmi]
Qa
(3)
--SQ =~-El(X )
Qa --=lO
Qw Q
(2)
1[ a
e
-x W
+ E, (x w )
(xweXw)-1 - (1 - a)El(Xw) 1
El(Xw) [e_Xw( 1 +--l ) + l e - x m xw
a
+ ~1 El(Xm)
]
/3m
(4)
[{xweXw)-l- (1-1)El(Xw)+ l-l--El(xm)l2 a {Jm J where El(X ) = exponential integral of x, Qa = instantaneous discharge from the aquifer, r = radial distance, T = transmissivity, t = time, r m = distance between discharging well and observation well, x m = (rm/rw) Xw, argument of exponential integral referred to the observation well, x w argument of exponential integral at discharging well, and:
a = (rw/rcw)2S,
{Jm = (rw/rcm)2S/(1-S)
The equations are linked together by the argument of the exponential" integral. By applying appropriate limiting conditions, the type curves of Theis (1935), Hantush (1964), Papadopulos (1967), and Papadopulos and Cooper (1967), are readily obtained. In addition, a type curve can be derived for constant drawdown analysis. Calculation of type curves requires use of.the exponential integral which is readily obtained from tables or calculated. No numerical integration is required.
344 CALCULATION
OF TYPE RECOVERY
CURVES
The recovery of a discharging well after the pump has been turned off is duplicated mathematically by superimposition of an equal recharging well upon the discharging well. The drawdown caused by the discharge continues but is offset to an ever increasing degree by the increase in water level due to the superimposed recharging well, The fractional residual drawdown is obtained from the following equation:
SN = (SD- SR)/So :
4nT
4~T
) 4~T
SD . . . . .Q SR / --Q- So
(5)
where S D = drawdown of discharging well, S N = normalized residual drawdown, S O = drawdown at end o f discharging episode, and S R = drawdown of superimposed recharging well. The drawdown of the discharging well, SD, and corresponding dimensionless time (Tt/r2c) are readily calculated using eqs. 2--4. Any number of values with any desired spacing can be calculated by incrementing the argument of the exponential integral. Values for specific dimensionless times, as required to superimpose the recharging well during the recovery period, can only be calculated by iteration, since eqs. 2 and 4 have not been solved explicitly for the exponential integral. The most efficient approach appears to be to increment the argument of the exponential integral for the discharging well and calculate the exponential integral by iteration for the specific dimensionless times for the superimposed recharging well. Convergence can be made rapid for the iteration procedure, and an entire sequence of calculations can be carried out in a few minutes of computer time. Fig. 1 is typical and represents the results of a calculation of type recovery curves for a large range of dimensionless discharging times and a = 10 -4. No storage other than that of the discharging well was considered. LIMITING CONDITIONS Examination of Fig.1 shows that as the dimensionless time of the pumping episode becomes shorter, the curves representing recovery converge and further shortening of the pumping time provides curves that essentially overlay one another. The limiting curves for the usual range of a values are presented in Fig.2. These curves represent recovery of a well in which an amount of water has been removed rapidly or essentially instantaneously.: However, a finite rather than instantaneous period o f drawdown is implied from the nature of the analysis presented. Type curves were developed by Cooper et al. (1967) for the recovery of a finite-diameter well after an instantaneous charge of water, and the mathematical analysis uses an instantaneous change in head within the well as one of the boundary conditions. The curves of Cooper et al. are essentially the same as the limiting curves presented in Fig.2.
345
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O"
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-~.so
-~,oo
-'1.so
-'l.oo
-h.so
LOG DIMENSIONLEB$ TIME
o'.oo
o'.so
{.oo
t'.so
2'.oo
Fig.2. Limiting curves for a = 1 0 - ' , 10 -4, 1 0 - ' , 10 -2 a n d 1 0 - ' .
EXAMPLE
Cooper et al. (1967) demonstrated the use of type recovery curves for instantaneous charge of water with data from a test at DawsonviUe, Georgia. Drawdown in the well was created by rapidly removing a long weighted float. Considering this drawdown to be essentially instantaneous and using the dimensions o f the well and the weight of the displaced water to calculate the maximum drawdown, they normalized the recovery data, matched it to their curves for instantaneous drawdown, and reached the conclusion that a was 10 -s and T, the transmissivity, was 5.3 cm2/s. To use the type curves presented in this paper, the recovery data for ~ e Dawsonville, Georgia test was plotted on log paper. In this case, the actual
347
head measurements were used rather than measurements normalized with the maximum drawdown, since the normalization procedure is an unnecessary step, when log plots are used, that possibly introduces additional errors attributable to the estimation of the maximum drawdown attained. The data was found to most closely match the type curve for a = 10 -4 and a dimensionless discharging time of 10-'. For purposes of comparison, the dimensionless discharging time curves o f 10-' for a = 10 -4 and a = 10 -3 are presented in Fig.3. The Dawsonville, Georgia data is plotted on the curve for a = 10 -4 as Z
°-
CD C~
i
CL
bd
I
10°~
O:2 IxA
t~ Ct~C:D
Oc~ /
r
.@@
i
0.50
@.00
LO0
0.50
DINENS[ONLESS
1.00
1 .50
Z'.O0
TIbiE
F i g . 3 . D i s c h a r g i n g t i m e c u r v e s w i t h d i s c h a r g i n g t i m e o f 1 0 - ' f o r a = 1 0 -4 a n d a -- 1 0 -3.
the best match. For this match the transmissivity, T, was f o u n d to be 5.8 cm2/s. Using the type curves presented in this paper, the transmissivity was about 10% higher and the a about an order o f magnitude lower than values arrived at considering instantaneous drawdown. Further, drawdown was n o t instantaneous but required about 1 s. Examination o f the drawdown recovery record, presented by Cooper et al. {1967), substantiates a drawdown time o f this length rather than instantaneous drawdown. The maximum drawdown was estimated from the curves to be 0 . 5 2 0 rather than 0 . 5 6 0 m determined by the displacement of the float calculation. The probable explanation for the discrepancy in maximum drawdown estimates is that during the one second of drawdown, some water entered the well from the aquifer. SUMMARY AND CONCLUSION
Type recovery curves have been developed for analyzing recovery data
348
from wells where the discharging times can vary from essentially instantaneous to any length of time desired. To use the curves to estimate transmissivity and storage it is only necessary to have the recovery data. Discharge, maximum drawdown, and discharge duration are not required to obtain a solution. The curves are not as sensitive to storage as to transmissivity. It is possible to be in error by an order of magnitude in estimation of storage, but the estimate of transmissivity should be within 10%. As the duration of the drawdown period decreases, smaller volumes o f the aquifer are tested and the aquifer parameters estimated through recovery analysis contain increasingly larger errors associated with well completion, aquifer inhomogeneities and anisotropicity, and partial penetration and become increasingly less useful for general application. ACKNOWLEDGEMENTS
The work upon which this report is based was supported by the U.S. Energy Research and Development Administration under Contract No. AT {29-2) 1253.
REFERENCES Case, C.M., Pidcoe, W.W. and Fenske, P.R., 1974. Theis equation analysis of residual drawdown data. Water Resour. Res., 10(6): 1253--1256. Cooper, Jr., H., Bredehoeft, J.D. and Papadopulos, I.S., 1967. Response of a finite-diameter well to an instantaneous charge of water. Water Resour. Res., 3(1 ): 263--269. Fenske, P.R., 1974. Radial flow with discharging well and observation welt storage. EOS, (Trans. Am. Geophys. Union), 55(12): 1118 (Abstr.H46). Hantush, M.S., 1964. Hydraulics of wells. In: V.T. Chow (Editor), Advances in Hydroscience, Vol. 1, Academic Press, New York, N.Y., pp. 318, 319 and 340. Papadopulos, I.S., 1967. Drawdown distribution around a large diameter well. Proc. AWRA on Ground Water Hydrology, pp. 157--168. Papadopulos, I.S. and Cooper, Jr., H.H., 1967: Drawdown in a well of large diameter. Water Resour. Res., 3(1): 241--244. Theis, C.V., 1935. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage. Trans~ Am. Geophys. Union, 16: 519--524.
341
TYPE CURVES FOR RECOVERY OF A DISCHARGING WELL WITH STORAGE
P.R. FENSKE Desert Research Institute, Water Resources Center, University o f Nevada System, Reno, Nev. 89507 (U.S.A.)
(Received June 6, 1976; revised and accepted July 23, 1976)
ABSTRACT Fenske, P.R., 1977. Type curves for recovery of a discharging well with storage. J. Hydrol., 33: 341--348. The analysis of water level recovery data is usually carried out using the straight line or late time solution. Recently Case et al. presented a method based upon the reversion of the series representing the exponential integral whereby recovery data could be analyzed if only two points were known on the recovery curve. This analysis was based upon the Theis equation and consequently subject to all of the assumptions to the Theis equation. Important of these assumptions is the lack of storage in the discharging well. Papadopulos and Cooper solved the problem and presented appropriate type curves for well recovery after instantaneous charge of water when the well contained storage. Fenske presented an extension of the Theis equation which considers both discharging and observation well storage. The simpler mathematics makes the derivation of type recovery curves for radial flow systems with discharging well storage readily feasible. It is found that as pumping time becomes very short relative to recovery time, the type recovery curves approach as a limit the curves of Cooper, et al. for recovery after instantaneous drawdown. Recovery data from a test at Dawsonville, Georgia, is used as an example.
INTRODUCTION T h e analysis o f w a t e r level r e c o v e r y d a t a is usually carried o u t using the straight-line or late-time s o l u t i o n due t o the d i f f i c u l t y in a n a l y z i n g early time r e c o v e r y data. R e c e n t l y Case et al. ( 1 9 7 4 ) p r e s e n t e d a m e t h o d derived f r o m the Theis e q u a t i o n and based u p o n reversion o f the series r e p r e s e n t i n g the e x p o n e n t i a l integral w h e r e b y r e c o v e r y d a t a c o u l d be a n a l y z e d if o n l y t w o p o i n t s were k n o w n o n the r e c o v e r y curve. This analysis is s u b j e c t to the a s s u m p t i o n s used in d e v e l o p i n g the Theis e q u a t i o n . I m p o r t a n t in these a s s u m p t i o n s is the lack o f c o n s i d e r a t i o n o f storage in the discharging well. P a p a d o p u l o s a n d C o o p e r ( 1 9 6 7 ) solved the p r o b l e m f o r d r a w d o w n in a well with storage a n d P a p a d o p u l o s ( 1 9 6 7 ) p r e s e n t e d an analysis f o r d r a w d o w n at a n y l o c a t i o n in the c o n e o f depression o f a discharging well with storage.
342 Fenske (1974) was able to reproduce all well hydraulics analyses for isotropic h omo g en eo u s confined aquifers of infinite areal e x t e n t by a simple extension of the Theis equation. The simpler mathematics makes derivation of type recovery curves for discharging well with storage feasible. In the following paper a brief review of Fenske's extension of the Theis equation is presented and t ype recovery curves for a discharging well are developed. Data from the Dawsonville, Georgia test, presented by Cooper et al. (1967}, is used for demonstration o f the method. REVIEW OF THE THEIS EQUATION EXTENSION The extension of the Theis equation to the more general application is based upon the simple relationship that the volume of the region of the aquifer and all of the wells or other storages that are emptied by the discharging well at any instant in time divided by the average discharge is equal to the time required to develop that volume and its associated drawdown. The volume o f the cone of depression and associated storages is given by the following equation:
V = 2~S
/
rsdr + nrcwSw2 + (1-S)nr~mS m
(1)
rw
where r = radius to any location on cone of depression, rcm = casing radius of observation well, rcw = casing radius of discharging well, r w = effective radius of discharging well, S = storage coefficient, s = drawdown at any location in cone of depression, s m = drawdown at the observation well, s w = drawdown at the discharging well, V = volume of cone of depression plus volumes of discharging and observation wells. To integrate the first term on the right of the instantaneous volume expression, an equivalent to the drawdown, s, in terms of the radius to any location within the radial flow field is required. In a non-leaky isotropic h o m o g e n e o u s confined aquifer of uniform thickness and infinite areal extent, equipotential surfaces are vertical and concentric a b o u t the discharging well. The cone of depression described by the Theis equation is also the cone of depression associated with a well of finite diameter, and the desired expression for drawdown is obtained from the Theis equation. After integrating eq. 1, dividing the volume of the aquifer and associated storages by the average discharge and some additional manipulation (Fenske, 1974), the following equation set that describes dimensionless drawdown vs. dimensionless time at any location in the radial flow field is obtained:
343
4Tt r2ca
_
4~T
Qa (xweXw)_ Q
1
+ 1 El(Xmi]
Qa
(3)
--SQ =~-El(X )
Qa --=lO
Qw Q
(2)
1[ a
e
-x W
+ E, (x w )
(xweXw)-1 - (1 - a)El(Xw) 1
El(Xw) [e_Xw( 1 +--l ) + l e - x m xw
a
+ ~1 El(Xm)
]
/3m
(4)
[{xweXw)-l- (1-1)El(Xw)+ l-l--El(xm)l2 a {Jm J where El(X ) = exponential integral of x, Qa = instantaneous discharge from the aquifer, r = radial distance, T = transmissivity, t = time, r m = distance between discharging well and observation well, x m = (rm/rw) Xw, argument of exponential integral referred to the observation well, x w argument of exponential integral at discharging well, and:
a = (rw/rcw)2S,
{Jm = (rw/rcm)2S/(1-S)
The equations are linked together by the argument of the exponential" integral. By applying appropriate limiting conditions, the type curves of Theis (1935), Hantush (1964), Papadopulos (1967), and Papadopulos and Cooper (1967), are readily obtained. In addition, a type curve can be derived for constant drawdown analysis. Calculation of type curves requires use of.the exponential integral which is readily obtained from tables or calculated. No numerical integration is required.
344 CALCULATION
OF TYPE RECOVERY
CURVES
The recovery of a discharging well after the pump has been turned off is duplicated mathematically by superimposition of an equal recharging well upon the discharging well. The drawdown caused by the discharge continues but is offset to an ever increasing degree by the increase in water level due to the superimposed recharging well, The fractional residual drawdown is obtained from the following equation:
SN = (SD- SR)/So :
4nT
4~T
) 4~T
SD . . . . .Q SR / --Q- So
(5)
where S D = drawdown of discharging well, S N = normalized residual drawdown, S O = drawdown at end o f discharging episode, and S R = drawdown of superimposed recharging well. The drawdown of the discharging well, SD, and corresponding dimensionless time (Tt/r2c) are readily calculated using eqs. 2--4. Any number of values with any desired spacing can be calculated by incrementing the argument of the exponential integral. Values for specific dimensionless times, as required to superimpose the recharging well during the recovery period, can only be calculated by iteration, since eqs. 2 and 4 have not been solved explicitly for the exponential integral. The most efficient approach appears to be to increment the argument of the exponential integral for the discharging well and calculate the exponential integral by iteration for the specific dimensionless times for the superimposed recharging well. Convergence can be made rapid for the iteration procedure, and an entire sequence of calculations can be carried out in a few minutes of computer time. Fig. 1 is typical and represents the results of a calculation of type recovery curves for a large range of dimensionless discharging times and a = 10 -4. No storage other than that of the discharging well was considered. LIMITING CONDITIONS Examination of Fig.1 shows that as the dimensionless time of the pumping episode becomes shorter, the curves representing recovery converge and further shortening of the pumping time provides curves that essentially overlay one another. The limiting curves for the usual range of a values are presented in Fig.2. These curves represent recovery of a well in which an amount of water has been removed rapidly or essentially instantaneously.: However, a finite rather than instantaneous period o f drawdown is implied from the nature of the analysis presented. Type curves were developed by Cooper et al. (1967) for the recovery of a finite-diameter well after an instantaneous charge of water, and the mathematical analysis uses an instantaneous change in head within the well as one of the boundary conditions. The curves of Cooper et al. are essentially the same as the limiting curves presented in Fig.2.
345
2
mCD I1 O2 Z off_
? 0 U
-OdQImJ
oX
8
-C,J (j~ LLJ LO~
Z L~J ~Z
"a 0
£ CD
-o
[ 00"0
-c3
8
-~
> 0
o OS" 01--
NHOOMU~O
00" ]'_
3Un01938
O~ • ~I
]AI/WO38
00' ~ T
008
M
346
O"
5
-..I
>~.
I.U o"
g.
°3.00
-~.so
-~,oo
-'1.so
-'l.oo
-h.so
LOG DIMENSIONLEB$ TIME
o'.oo
o'.so
{.oo
t'.so
2'.oo
Fig.2. Limiting curves for a = 1 0 - ' , 10 -4, 1 0 - ' , 10 -2 a n d 1 0 - ' .
EXAMPLE
Cooper et al. (1967) demonstrated the use of type recovery curves for instantaneous charge of water with data from a test at DawsonviUe, Georgia. Drawdown in the well was created by rapidly removing a long weighted float. Considering this drawdown to be essentially instantaneous and using the dimensions o f the well and the weight of the displaced water to calculate the maximum drawdown, they normalized the recovery data, matched it to their curves for instantaneous drawdown, and reached the conclusion that a was 10 -s and T, the transmissivity, was 5.3 cm2/s. To use the type curves presented in this paper, the recovery data for ~ e Dawsonville, Georgia test was plotted on log paper. In this case, the actual
347
head measurements were used rather than measurements normalized with the maximum drawdown, since the normalization procedure is an unnecessary step, when log plots are used, that possibly introduces additional errors attributable to the estimation of the maximum drawdown attained. The data was found to most closely match the type curve for a = 10 -4 and a dimensionless discharging time of 10-'. For purposes of comparison, the dimensionless discharging time curves o f 10-' for a = 10 -4 and a = 10 -3 are presented in Fig.3. The Dawsonville, Georgia data is plotted on the curve for a = 10 -4 as Z
°-
CD C~
i
CL
bd
I
10°~
O:2 IxA
t~ Ct~C:D
Oc~ /
r
.@@
i
0.50
@.00
LO0
0.50
DINENS[ONLESS
1.00
1 .50
Z'.O0
TIbiE
F i g . 3 . D i s c h a r g i n g t i m e c u r v e s w i t h d i s c h a r g i n g t i m e o f 1 0 - ' f o r a = 1 0 -4 a n d a -- 1 0 -3.
the best match. For this match the transmissivity, T, was f o u n d to be 5.8 cm2/s. Using the type curves presented in this paper, the transmissivity was about 10% higher and the a about an order o f magnitude lower than values arrived at considering instantaneous drawdown. Further, drawdown was n o t instantaneous but required about 1 s. Examination o f the drawdown recovery record, presented by Cooper et al. {1967), substantiates a drawdown time o f this length rather than instantaneous drawdown. The maximum drawdown was estimated from the curves to be 0 . 5 2 0 rather than 0 . 5 6 0 m determined by the displacement of the float calculation. The probable explanation for the discrepancy in maximum drawdown estimates is that during the one second of drawdown, some water entered the well from the aquifer. SUMMARY AND CONCLUSION
Type recovery curves have been developed for analyzing recovery data
348
from wells where the discharging times can vary from essentially instantaneous to any length of time desired. To use the curves to estimate transmissivity and storage it is only necessary to have the recovery data. Discharge, maximum drawdown, and discharge duration are not required to obtain a solution. The curves are not as sensitive to storage as to transmissivity. It is possible to be in error by an order of magnitude in estimation of storage, but the estimate of transmissivity should be within 10%. As the duration of the drawdown period decreases, smaller volumes o f the aquifer are tested and the aquifer parameters estimated through recovery analysis contain increasingly larger errors associated with well completion, aquifer inhomogeneities and anisotropicity, and partial penetration and become increasingly less useful for general application. ACKNOWLEDGEMENTS
The work upon which this report is based was supported by the U.S. Energy Research and Development Administration under Contract No. AT {29-2) 1253.
REFERENCES Case, C.M., Pidcoe, W.W. and Fenske, P.R., 1974. Theis equation analysis of residual drawdown data. Water Resour. Res., 10(6): 1253--1256. Cooper, Jr., H., Bredehoeft, J.D. and Papadopulos, I.S., 1967. Response of a finite-diameter well to an instantaneous charge of water. Water Resour. Res., 3(1 ): 263--269. Fenske, P.R., 1974. Radial flow with discharging well and observation welt storage. EOS, (Trans. Am. Geophys. Union), 55(12): 1118 (Abstr.H46). Hantush, M.S., 1964. Hydraulics of wells. In: V.T. Chow (Editor), Advances in Hydroscience, Vol. 1, Academic Press, New York, N.Y., pp. 318, 319 and 340. Papadopulos, I.S., 1967. Drawdown distribution around a large diameter well. Proc. AWRA on Ground Water Hydrology, pp. 157--168. Papadopulos, I.S. and Cooper, Jr., H.H., 1967: Drawdown in a well of large diameter. Water Resour. Res., 3(1): 241--244. Theis, C.V., 1935. The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground-water storage. Trans~ Am. Geophys. Union, 16: 519--524.