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Parameter estimation in an aquifer-water table aquitard system
Journal of Hydrology, 136 (1992) 177-192
177
Elsevier Science Publishers B.V., Amsterdam
N
Parameter estimation in an aquifer-water table aquitard system M. Sekhar, M.S. Mohan Kumar and K. Sridharan Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India (Received 16 November 1990; accepted 13 November 1991)
ABSTRACT Sekhar, M., Mohan Kumar, M.S. and Sridharan, K., 1992. Parameter estimation in an aquifer-water table aquitard system. J. Hydrol., 136: 177--192. The weighted least squares approach is used for the estimation of parameters in an aquifer-water table aquitard system. Six parameters are to be evaluated, namely: the equivalent transmissivity, degree of anisotropy and storage coefficient of the aquifer, ana the leakage coefficient, specific storage and specific yield of the aquitard. The coupled aquifer-aqtfitard equations are solved by an iterative numerical procedure and the optimisation problem is solved by the sensitivity analysis technique. This method is applied to one hypothetical problem and two field pumping tests of 7 days duration.
INTRODUCTION
A numerical model for the analysis of an aquifer-water table aquitard system was presented by Sridharan et al. (1990). This paper is an extension of the above study wherein the numerical model is coupled wit~ an optimisation algorithm for estimation of parameters. Such a coupling becomes important as the number of aquifer and aquitard parameters to be estimated is six, making the graphical matching procedure highly subjective and extremely difficult. Considerable work has beer., done concerning the application of optimisation methods for characterising subsurthce hydrologic systems. Parameter estimation techniques for saturated flow problems were recently reviewed by Yeh (1986) and Carrera and Neuman (1986). Computational techniques based on the least squares method have been used to evaluate aquifer parameters by Mania and Sucche (1978), McElwee (1980), Chander Correspondence to: K. Sridharan, Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India.
0022-1694/92/$05.00
© 1992 - - Elsevier Science Publishers B.V. All rights reserved
178
M. SEKHAR ET AL.
et aL ( 1981 ), Cobb et al. (1982) and Sridharan et al. (1985, 1987). These studies are all based on the use of existing analytical solutions for the drawdown and the number of parameters estimated does not exceed four. There is no analytical solution available for the general case of aquifer-compressible water table aquitard system in which the number of parameters is six: the aquifer transmissivities in the x and y directions, the aquifer storage coefficient, the leakage coefficient, and the specific storage and specific yield of the aquitard. Aquifer-aquitard or two aquifer systems have been studied by Hantush (1964), Neuman and Witherspoon (1969), Boulton and Streltsova (1975), Streltsova (1976), Walton (1979) and Javendel and Witherspoon (1980, 1983). Cooley and Case (1973), first gave an approximate solution for the aquiferwater table aquitard problem by showing that Boulton's (1963) convolution integral can be used to represent the velocity of flow at the base of an incompressible water table aquitard overlying a pumped aquifer. In the solution of Bou!ton and Streltsova (1975) and Streltsova (1976), the water table aquitard is treated as incompressible and, further, the flux boundary condition on the water table is applied on the initial horizontal surface. Recently, Sridharan et al. (1990) have accounted for the compressibility of the aquitard and also satisfied the flux and varying head boundary conditions at the water table, resorting to numerical analysis. The objective of the present work is to extend the simulation-optimisation methodology of parameter estimation to the case of an aquifer-water table aquitard system accounting for compressibility and gravity drainage in the aquitard, treating the water table as an unknown boundary. A numerical model is used to solve the direct problem for the coupled aquifer-aquitard equations by an iterative procedure in which water table elevations and flow transfer terms are updated at each iteration. The inverse problem for estimation of parameters is solved based on the least squares approach using the sensitivity analysis technique. DIRECT PROBLEM
Figure 1 is a schematic diagram of the aquifer-water table aquitard system. The aquifer is anisotropic with transmissivity Tx and T,. along the principal Cartesian directions x and y and with a storage coefficient S,. The aquitard has a vertical hydraulic conductivity K', specific storage S~ and specific yield Sy. A well fully penetrating the aquifer is pumped at a constant discharge Q. Flow is lateral in the aquifer and vertical in the aquitard. Both elastic storage release and gravity drainage in the aquitard are considered. In particular, the
PARAMETER ESTIMATION:AQUIFER-WATERTABLEAQUiTARD SYSTEM
179
.._~.Pumping well (Q) Ground level Initial water table
,//••Nater t\
i ho
/
/
"Aquifer head under pumping
,,
I
Aqi,,.r°
table under pumping
/
//~
qY
Aquifer
:! --
lTx. Tv,S,]
//////t'///z////ll/i/,'111z."/)~'111"/'1~'IP/ll/1,'IA'11,'/ll/f11
Im >ermeable
Fig. 1. Schematic diagram of the aquifer-aquitard model.
water table is treated as an unknown boundary at the current simulation time. The governing equations of flow are as follows: Aquifer:
02h 02h oh T~-ff'~x2 + Ty t~y2 = S, Ot
K,
k--Oz, -o- + Q3(x)6(y)
(1)
Aquitard:
K'
B2h" ~g 2
ah" S;
"-
67~
(2)
In eqns. (1) and (2), h is the piezometric head in the aquifer, h' is the piezometric head in the aquitard, z is the vertical coordinate and 6( ) is the Dirac delta function. Equation (1) may be simplified to an equivalent isotropic problem by making the coordinate transformation: 4
X = x\-~]
,
r
= y
(3)
The equations are rewritten in terms of the drawdowns defined by: s = h0 -
h;
s' -
h0 -
h'
(4)
where h0 is the prepumping head assumed to be the same in aquifer and aquitard (Fig. l), s is the drawdown in the aquifer and s' is the drawdown in
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M. SEKHAR ET AL.
the aquitard. Using eqns. (3) and (4), the governing equations and boundary conditions for the aquifer and aquitard are written as follows: Aquifer: +
\Or 2
10 ) =
7 -&
S'ot
\OzL=o
(5)
t = O; s = 0 OS
t>0;
r~OO0r
= 0
Os) r -~ O; lira r -~ =
(6) Q 2nTr
Aquitard: K , Ozs"
a--7~
=
e , Os"
o, Ot
(7)
t = O; s' = 0 t>0;
z =
z = r/,
0,
s' =
s
0s' Or/ (i) K ' d z = S~o-i (ii) s' =
h0-
t/
(8)
In eqns. (5)-(8), q is the water table elevation (Fig. 1), T~ is the equivalent transmissivity of the aquifer given by
rr -
x/(SlxTy)
(9)
and r is the radial coordinate in the equivalent isotropic domain,
r 2 = X 2 + y2 = Di/2x 2 + D-J/2y2 where the degree of anisotropy, D is v~"'~,~O =
T~/T~
(10)
by (l l)
The problem is governed by six parameters which are represented by the vector/~ whose components are T,, S,, K'/ho, S~, S~ and D. The degree of anisotropy, D is required for finding the radial distance at which the observation well is located from the pumping well, given x and y coordinates of the observation well. Since the parameter, D, is not present in eqns. (5)-(8), the analysis model for the direct problem does not require it. But this parameter
PARAMETERESTIMATION:AQUIFER-WATERTABLEAQUITARDSYSTEM
181
is required in the inverse problem wherein the computed drawdowns are matched with the observed drawdowns at the observation wells. Accordingly, for the same T,, there will be different drawdowns for different values of D, at the observation well. The direct problem for the coupled aquifer-aquitard equations is solved numerically by the finite difference method, using an iterative procedure to determine the unknowv~ water table elevation and flow transfer term from aquitard to aquifer. The details of the computational procedure are presented in Sridharan et al. (1990). INVERSE PROBLEM
In the inverse problem, the governing equations subject to initial and boundary conditions defined by eqns. (5)-(8) are solved for the parameter set fl, by comparing the computed drawdown, s(xi, yi, z~, ty) with observed drawdown Se(Xi, yi, zi, tj). Here (x~, y~, z~) define the discrete points in space and tj the discrete times at which the drawdown is measured in the field. The computed drawdowns from the solution of the direct problem are compared with observed drawdowns using a suitable error criterion, in this case the sum of weighted squares of difference between observed and computed drawdowns. g
--
M N(i) Z X ~ij [Se(Xi, Yi, zi, tj) -- s(xi, Yi, zi, tj)] 2 i=l j=l
(12)
Here W,j are suitable weights, M and N(i) are the number of space and time observations available. The spatial location of the observation well is represented by (xi, y~) while z,. refers to the location in depth where drawdown measurements are made; z~ = 0 refers to measurements in the aquifer an~d 0 < z~ ~< r/refers to measurements in the aquitard.
Choice of weights Kool et al. (1987) discuss the choice of weights in the weighted least squares method with a view to ensuring that the most accurate measurements receive the most weight. The ordinary least squares (OLS) method is based on the use of W~j = 1 in eqn. (12). While such an approach is simple, it tends to provide a bias in favour of fitting the long time drawdowns better, at the cost of possible poor fit for small time drawdowns. A similar anomaly may also occur between aquifer and aquitard drawdowns. In the graphical matching procedure, an attempt is made to give due importance in matching the drawdown data over the different time cycles. In the present study, this is achieved by suitable choice of g'~j values.
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M. SEKHAR ET AL.
Alternative choices of weights are studied by varying the exponent n in the following equation. ~ij --
Se(Xi,yi, Z,, tj)
(13)
The choice n = 0 gives the OLS approach which, for a field situation, tends to provide a better fit at long times. The choice n = 2 gives a weighting wherein all observations are given equal relative importance. This resembles the form of a suitable logarithmic transformation of the drawdowns in the OLS problem. However, a choice 0 < n < 2 may be desirable considering that short time observations may have larger relative errGrs in view of the smaller values of drawdowns. This approach is adopted considering that the deviation between comwRed and observed drawdowns occur due to two causes: measurement errors and the difference between the conceptual model and field situation. In an ideal situation wherein the correct constitotive model with error-free data is used, the choice of weights is immaterial. It mast be noted that for different choice of n in eqn. (13), the dimension of E differs.
Solution methodology Corresponding to the initial trial values or previous iteration values of parameter set fl,, the drawdown s, at any instant can be computed using the direct problem. If Afl)s (t' = 1-6) are the respective changes in the pararaeter values in each iteration, the improved drawdown can be written using the first order Taylor series as follows. 6
s = s, + Y' UtAflt g=l
(14)
where Ut are the sensitivity coefficients given by (OslOfle) and subscript t' refers to the parameter T,, S,, K'/ho, S~, Sy and D. The sensitivity coefficients in eqn. (14) are evaluated with the current values of the parameters. To obtain the minimum value of the objective function, eqn. (14) is substituted into eqn. (12) and the partial derivatives of the objective function with respect to Aflt are equated to zero. The resulting equation is: M N(i) 6 ! M N(i) 1 E E WqUeUm Afire = Z Z WijUg [Se(Xi, Yi, zi, tj) -- s$(xi, Yi, zi, tj)] m=l " ;=1 j=l i=1 j = l
E
for,e =: 1,2,...6
(15)
In eqn. (15), Ut and Umare evaluated at (xi, Yi, zi, tj). The sensitivity coefficients
PARAMETER ESTIMATION: AQUIFER-WATER TABLE AQUITARD SYSTEM
183
are evaluated using forward difference approximation given by OsU
~
s(x,, yi, z,, tj;
+ 6 m) -- S(X , Yi, Zi, tj;
(16)
The sensitivity coefficients are evaluated to the required degree of accuracy with the finite difference increments set equal to 5fl,, = 0.005-0.01 tim- Since the parameter, D does not occur explicitly in the direct problem governed by eqns. (5)-(8), it is enough if we solve the direct problem (5 + 1) = 6 times, in each iteration. The sensitivity coefficients for the parameter, D are obtained usir, g the solution of the unperturbed parameters, as a variation in D only changes the locations ri for specified values of (xi, y~) of the observation wells. Equation (15) provides six linear equations for At3~,s and they are solved by the Gaussian elimination procedure. The changes are used to update the values of parameters in the successive iteration using the following equation flkm+' = ffm + Aft,.; m =
1,2,...6
(17)
where the superscript k refers to the iteration number. The procedure is continued until the value of the corrections AflmS simultaneously satisfy a specific convergence criterion when the iteration is terminated. As the objective function does not exhibit a convex nature uniformly over the entire parametric space, it is found necessary to impose some constraints on the corrections to ensure covergence to the global minimt~m, for the initial assumptions in any region of the parametric space. The constraints proposed by Cobb et al. (1982) for a simple case are found to be successful for the present problem also. Thus, constraints on the relative claanges of the parameter given by
a/ m
- 0 . 2 < tim ~< 0.5
(18)
are imposed in the computations, it is found that with these constraints imposed on corrections to the parameters, convergence to true values can be obtained even when the initial assumptions differ from the true values of the parameters by orders of magnitude, as verified from a correct constitutive model with error-free data. However, for a fieia" problem, in order to reduce the computational time, it is desirable to provide reasonable initial assumptions based on the graphical method or a knowledge of the local hydrogeology.
184
M. SEKHAR ET AL.
APPLICATION OF THE M E T H O D
Hypothetical data The parameter estimation procedure outlined in the previous section is used for a hypothetical problem as well as two field pumping tests. Figure 2 presents schematic diagrams of the three well fields showing the p,~.mping well, PW, and the two observation wells, OW~, and OW2. For the hypothetical problem, both aquifer and water table drawdowns are used for the parameter ~A~':--~:^LIIII~LIUII, The well is pumped at a constant discharge of 15/'s -~ The principal axes are taken along east-west (x axis) and north-south (y axis) directions. In order to generate hypothetical data consisting of 'measurements' of aquifer and water table drawdowns, the direct problem (eqns. (5) and (8)) is solved for a set of assumed parameters. Simulated aquifer and water table drawdowns are sampled at the two observation wells (Fig. 2(a)) at discrete times for a pumping duration of 7 days. These error free data are used as input to the parameter estimation algorithm and are shown in Figs. 3 and 4 as 'observed' data. Table I presents the initial guess values and the final estimated values of the parameters for four cases. In the first three cases, the initial values of
OW2
+
N
t
PW ~q W ' ~I,2~-3d OWl
(a) Hypothetical
(c) Haradagere PW- Pumping well OW - Observation well
:)W
°OWl (b) Haragonadona Fig. 2. Location of wells for test problems.
PARAMETER ESTIMATION: AQUIFER-WATER TABLE AQUITARD SYSTEM
185
101
aquifer
E
lo~
3= 3=
O
'
m L_ t't
'
161 o
lo 0
,. . . . . .
I wte t I,,,
0;r0:
°
101
10z Time (rain.)
10~
104
Fig. 3. Aquifer and water table drawdowns at OW~ for the hypothetical problem.
parameters are over/under estimated by one order of magnitude, while in the fourth case the initial values are underestimated by two orders of magnitude, except for the degree of anisotropy. The final estimated parameter values are identical to the true values in all the four cases. Figures 3 and 4 present a comparison between the 'observed' and computed values of aquifer and water table drawdowns. The inverse problem for the hypothetical data is tested with different values of n in the weighting function in eqn. (13). It is found that the value of n does not influence the parameter estimates or the matching between the computed and measured drawdowns. This shows that for error-free data with the correct constitutive model it is possible to estimate the parameters accurately using an OLS procedure. Figure 5 presents the convergence rate ol the iteration algorithm for cases 1-3. It is observed from Fig. 5 and Table 1 that the convergence is more rapid for the case of all parameters underestimated. Ramaswamy (1983)studied the convexity of the objective function of the OLS problem for simple confined and semi-confined systems, for both isotropic and anisotropic cases, and concluded that the objective function is convex in the parametric space wherein all the parameters are underestin~ated. While the complexity of the present problem makes a study of convexity practically unfeasible, it is interesting to note that the convergence is found to
186
M. SEKHAR ET AL.
lo 0
161
er table
to al r'~
/ o -'Observed"drawdown o/ ~ Computed drawdown
10 0
101
10 z
10 a
1--------0 t,
Time (mir~.) Fig. 4. Aquifer and water table drawdowns at OW 2 for the hypothetical problem.
be most rapid for underestimated parameter values. It is found that for underestimation of all parameters, convergence occurred even when the constraints on corrections imposed by eqn. (18) are not applied. The convergence in such cases is found to be more rapid, as seen by the number of iterations presented in Table 1 for cases 2 and 4. However, convergence was not obtained for cases l and 3 (over and mixed estimates) if the limits as per eqn. (18) were not applied. FieM data The method is also applied to two long-duration pumping tests reported by Sarma et al. (1986), in the Haragonadona and Haradagere well fields in the Vedavati River Basin in the State of Karnataka in India. The test well fields are in schist and granite regions, respectively. Based on tracer test and the nature of drawdown observations in the pumping and recovery phases, Sarma et al. (1986) concluded that the well fields can be represented by an aquiferwater table aquitard system. Figures 2(b) and 2(c) present relative locations of pumping wells and the observation wells for both cases. The wells are pumped at a constant discharge of 15 f s-t and 3 d s -t , respectively, for 7 days duration. A knowledge of hydrogeology of the region clearly indicates that
PARAMETER ESTIMATION: AQUIFER-WATER TABLE AQUITARD SYSTEM
~ X
[..
T
X
X
187
T
X
X
X
0
O 0
0
O 0
o
dc~
d
c~o
T ~
T
T
m
~
~m
X
X
o
d~
X
X
o
dd
~mm
imm
~mm
X
X
X
~mmm
X
~
~mmm
X
X
d
do
X
X
X
X
X
d
c~d
I
~
"0
E
X
N
X
X
w~
4m~
"7 ~
E
ob
X
X
~7
'~
X
X
o
X
X o~ m
0
6
0
? T m
N
~
? ~
m
m m
•=
X
X
d
dd
X
X
X
d
do
X
~
°~
0
°.,i
E L.
0
.1
<
[..
E c~ Q.
~,~
~
~
Z
188
M. SEKHAR ET AL.
II ¢1 If D i O
D
O
A
o
°
o
II
2 b
161
u
o
I t,
E :E r,, o
u
o-0ver estimation (case 1) a-Under estimation (case2) A-Mixed (case 3) n-Under estimation without constraints (case 1) B
I= O O
1~6
I
0
/.
8 Iterations
6 A
a
1.2
16
Fig. 5. Convergence rate for the hypothetical problem.
there is an anisotropy, with the principal axes identified along northeastsouthwest (x axis) an,4 northwest-southeast (y axis) directions, with maximum transmissivity being in the y direction. Since water table drawdown is not available in both the well fields, only aquifer drawdown is used for the purposes of least squares minimisation. The parameters are estimated using the inverse problem algorithm described in the previous section, with weights assigned in the least squares minimisation, using n = 1.0 in eqn. (13). The input (guess) parameters and estimated parameters for the Haragonadona and Haradagere tests are given in Tables 2 and 3, respectively. For the Haragonadona test, parameter estimates based
PARAMETER ESTIMATION: AQUIFER-WATER TABI E AQUITARD SYSTEM
189
TABLE 2 Results of parameter estimates for Haragonadona well field data
Parameter
Initial values
Estimates from inverse procedure
Estimates from graphical procedure
Transmissivity, T , ( m 2 d a y - l ) Storage coefficient, S, Leakage coefficient, K'/ho (day -I )
0.40 x 10 ! 0.50 x l0 -5 0.20 x l0 -3
0.167 x 102 0.17 x l0 -4 0.41 x l0 -2
0.206 x 102 0.50 x l0 -4 0.21 x 10-3-0.46 x l0 -3
Specific storage, S, Specific yield Sy Degree of Anisotropy, D
0.5 x l0 -6 0.3 × 10 -3 0.1 x 10 m
0.80 x l0 -5
0.167 x l0 -4
0.58 × 10 -2
0.50 X 10 -2
0.37 x 10 m
0.795 x 10 ~
r.m.s, error (m) r.m.s./Smax
22.03 1.366
0.248 0.015
0.788 0.049
No. of iterations
20
on the graphical method were obtained by Sridharan et al. (1990). These results are also presented in Table 2 along with the r.m.s, values for both graphical and computed parameter estimates. The r.m.s, value for the computed parameter estimates is clearly better than for graphical parameter estimates. The absolute value of r.m.s, for the Haragonadona test is larger than for the Haradagere test (Tables 2 and 3) because of the larger drawdowns in the former case. If the r.m.s, value is normalised with respect to maximum drawdown, results for the two tests are comparable. The matching between TABLE 3 Results of parameter estimates for Haradagere well field data
Parameter
Initial values
Estimated values
Transmissivity, T, ( m 2 day- I) Storage coefficient, S, Leakage coefficient, K'/ho (day -l )
0.i x 102 0.2 x 10 -3 0.5 x 10 -2
0.207 x 102 0.238 x 10 2 0.39 × 10 -2
Specific storage, S, Specific yield, Sy Degree of anisotropy, O
0.8 x 10 -4 0.6 x 10 -3 0.4 x 101
0.81 x 10 ---3 0.51 x 10 -2 0.176 x 101
r.m.s, error (m)
1.13 1.41
0.0108 0.0135
r.m.s./Smax
No. of iterations
26
190
M. SEKHAR ET AL
10x2] OWl
f
/
¢0 0 0
0
0
o° °°
o -Observed
drawdown
J Computed drawdown
10
10z Time
104
10 (rain.)
Fig. 6. Aquifer drawdowns for Haragonadona well field.
10o
OW2
=.-,..
E
,.=,,,
=~ 161 0
L.
Q
I
I
I
I
I
II ~I
102
]
I
a
103 Time
(min.)
Fig. 7. Aquifer drawdowns for Haradagere well field.
i
I
l
a
I
I
104
PARAMETER ESTIMATION: AQUIFER-WATER TABLE AQUITARD SYSTEM
191
computed and observed drawdowns is presented in Figs. 6 and 7 for the two tests. CONCLUSION
The weighted least squares approach is used for estimation of parameters in an aquifer-water table aquitard system. There are six parameters to be evaluated for both the aquifer and aquitard. The direct problem governed by the coupled partial differential equations for the aquifer and the aquitard, is solved by an iterative numerical procedure to determine the unknown water table elevation and flow transfer term. The inverse problem is solved by a sensitivity analysis technique. The convergence characteristics of the algorithm are studied using a correct constitutive model with error-free data. Application of the method to two field pumping tests of 7 days duration is presented. REFERENCES Boulton, N.S., 1963. Analysis of data from non-equilibrium pumping tests allowing for delayed yield from storage. Proc. Ins* Civ. Eng., 26: 469-482. Boulton, N.S. and Streltsova, F.D., 1975. New equations for determining the formation constants of an aquifer from pvmping test data. Water Resour. Res., 1I: 148-153. Carrera, J. and Neuman, S.P., 1986. Estimation of aquifer parameters under transient and steady state conditions, I, Maximum likelihood method incorporating prior information. Water Resour. Res., 22: 199-210. Chander, S., Kapoor, P.N. and Goyal, S.K., 1981. Analysis of pumping test data using Marquardt algo,;thm. Ground Water, 19: 275-278. Cobb, P.M., McElv~ee, C.D. and Butt, M.A., 1982. Analysis of leaky aquifer pumping test data: AI~ automated n::::lerical solution using sensitivity analysis. Ground Water, 20: 325-333. Cooley, R.M. and "Case, C.M., 1973. Effect of water table aquitard on drawdown in an underlying pumped aquifer. Water Resour. Res., 9: 434-447. Hantush, M.S., 1964. Hydraulics of wells. In: V.T. Chow (Editor), Advances in Hydroscience. Academic Press, New York, Vol. 1, pp. 282-430. Javandel, I. and Witherspoon, P.A., 1980. A semianalytical solution for partial penetration in two-layered aquifers. Water Resour. Res., 16:1099-1106. Javendel, I. and Witherspoon, P.A., 1983. Analytical solution of partially penetrating well in two-layer aquifer. Water Resour. Res., 19: 567-578. Kool, J.B., Parker, J.C. and van Genuchten, M.TH., 1987. Parameter estimation for unsaturated flow and transport models - - a review. J. Hydroi., 9!" 225-293. Mania, J. and Sucche, M., 1978. Automatic analysis of pumping test data - - Application of the Boulton and Hantush hypothesis. J. Hydrol., 37: 185-194. McEiwee, C.D., 1980. Theis parameter evaluation from pumping tests by sensitivity analysis. Ground Water, 18: 56-60. Neuman, S.P. and Witherspoon, P.A., 1969. Theory of flow in a confineo two aquifer systems. Water Resour. Res., 5: 803-816.
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Ramaswamy, R., 1983. Optimisation technique for evaluation of aq~fifer parameters, Ph.D. Thesis, Indian Institute of Science, Bangalore, India, 352 pp. Sarma, K.V.N., Sridharan, K., Achuta Rao, and Sarma, C.S.S., 1986. Computer model for Vedavati ground water basin, Part 1. Well field model. Sadhana, Proc. Indian Sci., 9(1): 31-42. Sridharan, K., Ramaswamy, R. and Lakshmana Rao, N.S., 1985. Identification of parameters in unconfined aquifers. J. Hydrol., 79: 73-81. Sridharan, K., Ramaswamy, R. and Lakshamana Rao, N.S., 1987. Identification of parameters in semiconfined aquifers. J. Hydrol., 93: 163-173. Sridharan, K., Sekhar, M. and Mohan Kumar, M.S., 1990. Analysis of an aquifer-water table aquitard sytem. J. Hydrol., 114: 175-189. Streltsova, T.D., 1976. Analysis of aquifer-aquitard flow. Water Resour. Res., 12: 415-522. Walton, W.C., 1979. Review of leaky artesian aquifers test evaluation methods. Ground Water, 17: 270-283. Yeh, W.W.-G., 1986. Review of parameter identification procedures in ground water hydrology: The inverse problem. Water Resour. Res., 22: 95-108.
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Elsevier Science Publishers B.V., Amsterdam
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Parameter estimation in an aquifer-water table aquitard system M. Sekhar, M.S. Mohan Kumar and K. Sridharan Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India (Received 16 November 1990; accepted 13 November 1991)
ABSTRACT Sekhar, M., Mohan Kumar, M.S. and Sridharan, K., 1992. Parameter estimation in an aquifer-water table aquitard system. J. Hydrol., 136: 177--192. The weighted least squares approach is used for the estimation of parameters in an aquifer-water table aquitard system. Six parameters are to be evaluated, namely: the equivalent transmissivity, degree of anisotropy and storage coefficient of the aquifer, ana the leakage coefficient, specific storage and specific yield of the aquitard. The coupled aquifer-aqtfitard equations are solved by an iterative numerical procedure and the optimisation problem is solved by the sensitivity analysis technique. This method is applied to one hypothetical problem and two field pumping tests of 7 days duration.
INTRODUCTION
A numerical model for the analysis of an aquifer-water table aquitard system was presented by Sridharan et al. (1990). This paper is an extension of the above study wherein the numerical model is coupled wit~ an optimisation algorithm for estimation of parameters. Such a coupling becomes important as the number of aquifer and aquitard parameters to be estimated is six, making the graphical matching procedure highly subjective and extremely difficult. Considerable work has beer., done concerning the application of optimisation methods for characterising subsurthce hydrologic systems. Parameter estimation techniques for saturated flow problems were recently reviewed by Yeh (1986) and Carrera and Neuman (1986). Computational techniques based on the least squares method have been used to evaluate aquifer parameters by Mania and Sucche (1978), McElwee (1980), Chander Correspondence to: K. Sridharan, Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India.
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et aL ( 1981 ), Cobb et al. (1982) and Sridharan et al. (1985, 1987). These studies are all based on the use of existing analytical solutions for the drawdown and the number of parameters estimated does not exceed four. There is no analytical solution available for the general case of aquifer-compressible water table aquitard system in which the number of parameters is six: the aquifer transmissivities in the x and y directions, the aquifer storage coefficient, the leakage coefficient, and the specific storage and specific yield of the aquitard. Aquifer-aquitard or two aquifer systems have been studied by Hantush (1964), Neuman and Witherspoon (1969), Boulton and Streltsova (1975), Streltsova (1976), Walton (1979) and Javendel and Witherspoon (1980, 1983). Cooley and Case (1973), first gave an approximate solution for the aquiferwater table aquitard problem by showing that Boulton's (1963) convolution integral can be used to represent the velocity of flow at the base of an incompressible water table aquitard overlying a pumped aquifer. In the solution of Bou!ton and Streltsova (1975) and Streltsova (1976), the water table aquitard is treated as incompressible and, further, the flux boundary condition on the water table is applied on the initial horizontal surface. Recently, Sridharan et al. (1990) have accounted for the compressibility of the aquitard and also satisfied the flux and varying head boundary conditions at the water table, resorting to numerical analysis. The objective of the present work is to extend the simulation-optimisation methodology of parameter estimation to the case of an aquifer-water table aquitard system accounting for compressibility and gravity drainage in the aquitard, treating the water table as an unknown boundary. A numerical model is used to solve the direct problem for the coupled aquifer-aquitard equations by an iterative procedure in which water table elevations and flow transfer terms are updated at each iteration. The inverse problem for estimation of parameters is solved based on the least squares approach using the sensitivity analysis technique. DIRECT PROBLEM
Figure 1 is a schematic diagram of the aquifer-water table aquitard system. The aquifer is anisotropic with transmissivity Tx and T,. along the principal Cartesian directions x and y and with a storage coefficient S,. The aquitard has a vertical hydraulic conductivity K', specific storage S~ and specific yield Sy. A well fully penetrating the aquifer is pumped at a constant discharge Q. Flow is lateral in the aquifer and vertical in the aquitard. Both elastic storage release and gravity drainage in the aquitard are considered. In particular, the
PARAMETER ESTIMATION:AQUIFER-WATERTABLEAQUiTARD SYSTEM
179
.._~.Pumping well (Q) Ground level Initial water table
,//••Nater t\
i ho
/
/
"Aquifer head under pumping
,,
I
Aqi,,.r°
table under pumping
/
//~
qY
Aquifer
:! --
lTx. Tv,S,]
//////t'///z////ll/i/,'111z."/)~'111"/'1~'IP/ll/1,'IA'11,'/ll/f11
Im >ermeable
Fig. 1. Schematic diagram of the aquifer-aquitard model.
water table is treated as an unknown boundary at the current simulation time. The governing equations of flow are as follows: Aquifer:
02h 02h oh T~-ff'~x2 + Ty t~y2 = S, Ot
K,
k--Oz, -o- + Q3(x)6(y)
(1)
Aquitard:
K'
B2h" ~g 2
ah" S;
"-
67~
(2)
In eqns. (1) and (2), h is the piezometric head in the aquifer, h' is the piezometric head in the aquitard, z is the vertical coordinate and 6( ) is the Dirac delta function. Equation (1) may be simplified to an equivalent isotropic problem by making the coordinate transformation: 4
X = x\-~]
,
r
= y
(3)
The equations are rewritten in terms of the drawdowns defined by: s = h0 -
h;
s' -
h0 -
h'
(4)
where h0 is the prepumping head assumed to be the same in aquifer and aquitard (Fig. l), s is the drawdown in the aquifer and s' is the drawdown in
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M. SEKHAR ET AL.
the aquitard. Using eqns. (3) and (4), the governing equations and boundary conditions for the aquifer and aquitard are written as follows: Aquifer: +
\Or 2
10 ) =
7 -&
S'ot
\OzL=o
(5)
t = O; s = 0 OS
t>0;
r~OO0r
= 0
Os) r -~ O; lira r -~ =
(6) Q 2nTr
Aquitard: K , Ozs"
a--7~
=
e , Os"
o, Ot
(7)
t = O; s' = 0 t>0;
z =
z = r/,
0,
s' =
s
0s' Or/ (i) K ' d z = S~o-i (ii) s' =
h0-
t/
(8)
In eqns. (5)-(8), q is the water table elevation (Fig. 1), T~ is the equivalent transmissivity of the aquifer given by
rr -
x/(SlxTy)
(9)
and r is the radial coordinate in the equivalent isotropic domain,
r 2 = X 2 + y2 = Di/2x 2 + D-J/2y2 where the degree of anisotropy, D is v~"'~,~O =
T~/T~
(10)
by (l l)
The problem is governed by six parameters which are represented by the vector/~ whose components are T,, S,, K'/ho, S~, S~ and D. The degree of anisotropy, D is required for finding the radial distance at which the observation well is located from the pumping well, given x and y coordinates of the observation well. Since the parameter, D, is not present in eqns. (5)-(8), the analysis model for the direct problem does not require it. But this parameter
PARAMETERESTIMATION:AQUIFER-WATERTABLEAQUITARDSYSTEM
181
is required in the inverse problem wherein the computed drawdowns are matched with the observed drawdowns at the observation wells. Accordingly, for the same T,, there will be different drawdowns for different values of D, at the observation well. The direct problem for the coupled aquifer-aquitard equations is solved numerically by the finite difference method, using an iterative procedure to determine the unknowv~ water table elevation and flow transfer term from aquitard to aquifer. The details of the computational procedure are presented in Sridharan et al. (1990). INVERSE PROBLEM
In the inverse problem, the governing equations subject to initial and boundary conditions defined by eqns. (5)-(8) are solved for the parameter set fl, by comparing the computed drawdown, s(xi, yi, z~, ty) with observed drawdown Se(Xi, yi, zi, tj). Here (x~, y~, z~) define the discrete points in space and tj the discrete times at which the drawdown is measured in the field. The computed drawdowns from the solution of the direct problem are compared with observed drawdowns using a suitable error criterion, in this case the sum of weighted squares of difference between observed and computed drawdowns. g
--
M N(i) Z X ~ij [Se(Xi, Yi, zi, tj) -- s(xi, Yi, zi, tj)] 2 i=l j=l
(12)
Here W,j are suitable weights, M and N(i) are the number of space and time observations available. The spatial location of the observation well is represented by (xi, y~) while z,. refers to the location in depth where drawdown measurements are made; z~ = 0 refers to measurements in the aquifer an~d 0 < z~ ~< r/refers to measurements in the aquitard.
Choice of weights Kool et al. (1987) discuss the choice of weights in the weighted least squares method with a view to ensuring that the most accurate measurements receive the most weight. The ordinary least squares (OLS) method is based on the use of W~j = 1 in eqn. (12). While such an approach is simple, it tends to provide a bias in favour of fitting the long time drawdowns better, at the cost of possible poor fit for small time drawdowns. A similar anomaly may also occur between aquifer and aquitard drawdowns. In the graphical matching procedure, an attempt is made to give due importance in matching the drawdown data over the different time cycles. In the present study, this is achieved by suitable choice of g'~j values.
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M. SEKHAR ET AL.
Alternative choices of weights are studied by varying the exponent n in the following equation. ~ij --
Se(Xi,yi, Z,, tj)
(13)
The choice n = 0 gives the OLS approach which, for a field situation, tends to provide a better fit at long times. The choice n = 2 gives a weighting wherein all observations are given equal relative importance. This resembles the form of a suitable logarithmic transformation of the drawdowns in the OLS problem. However, a choice 0 < n < 2 may be desirable considering that short time observations may have larger relative errGrs in view of the smaller values of drawdowns. This approach is adopted considering that the deviation between comwRed and observed drawdowns occur due to two causes: measurement errors and the difference between the conceptual model and field situation. In an ideal situation wherein the correct constitotive model with error-free data is used, the choice of weights is immaterial. It mast be noted that for different choice of n in eqn. (13), the dimension of E differs.
Solution methodology Corresponding to the initial trial values or previous iteration values of parameter set fl,, the drawdown s, at any instant can be computed using the direct problem. If Afl)s (t' = 1-6) are the respective changes in the pararaeter values in each iteration, the improved drawdown can be written using the first order Taylor series as follows. 6
s = s, + Y' UtAflt g=l
(14)
where Ut are the sensitivity coefficients given by (OslOfle) and subscript t' refers to the parameter T,, S,, K'/ho, S~, Sy and D. The sensitivity coefficients in eqn. (14) are evaluated with the current values of the parameters. To obtain the minimum value of the objective function, eqn. (14) is substituted into eqn. (12) and the partial derivatives of the objective function with respect to Aflt are equated to zero. The resulting equation is: M N(i) 6 ! M N(i) 1 E E WqUeUm Afire = Z Z WijUg [Se(Xi, Yi, zi, tj) -- s$(xi, Yi, zi, tj)] m=l " ;=1 j=l i=1 j = l
E
for,e =: 1,2,...6
(15)
In eqn. (15), Ut and Umare evaluated at (xi, Yi, zi, tj). The sensitivity coefficients
PARAMETER ESTIMATION: AQUIFER-WATER TABLE AQUITARD SYSTEM
183
are evaluated using forward difference approximation given by OsU
~
s(x,, yi, z,, tj;
+ 6 m) -- S(X , Yi, Zi, tj;
(16)
The sensitivity coefficients are evaluated to the required degree of accuracy with the finite difference increments set equal to 5fl,, = 0.005-0.01 tim- Since the parameter, D does not occur explicitly in the direct problem governed by eqns. (5)-(8), it is enough if we solve the direct problem (5 + 1) = 6 times, in each iteration. The sensitivity coefficients for the parameter, D are obtained usir, g the solution of the unperturbed parameters, as a variation in D only changes the locations ri for specified values of (xi, y~) of the observation wells. Equation (15) provides six linear equations for At3~,s and they are solved by the Gaussian elimination procedure. The changes are used to update the values of parameters in the successive iteration using the following equation flkm+' = ffm + Aft,.; m =
1,2,...6
(17)
where the superscript k refers to the iteration number. The procedure is continued until the value of the corrections AflmS simultaneously satisfy a specific convergence criterion when the iteration is terminated. As the objective function does not exhibit a convex nature uniformly over the entire parametric space, it is found necessary to impose some constraints on the corrections to ensure covergence to the global minimt~m, for the initial assumptions in any region of the parametric space. The constraints proposed by Cobb et al. (1982) for a simple case are found to be successful for the present problem also. Thus, constraints on the relative claanges of the parameter given by
a/ m
- 0 . 2 < tim ~< 0.5
(18)
are imposed in the computations, it is found that with these constraints imposed on corrections to the parameters, convergence to true values can be obtained even when the initial assumptions differ from the true values of the parameters by orders of magnitude, as verified from a correct constitutive model with error-free data. However, for a fieia" problem, in order to reduce the computational time, it is desirable to provide reasonable initial assumptions based on the graphical method or a knowledge of the local hydrogeology.
184
M. SEKHAR ET AL.
APPLICATION OF THE M E T H O D
Hypothetical data The parameter estimation procedure outlined in the previous section is used for a hypothetical problem as well as two field pumping tests. Figure 2 presents schematic diagrams of the three well fields showing the p,~.mping well, PW, and the two observation wells, OW~, and OW2. For the hypothetical problem, both aquifer and water table drawdowns are used for the parameter ~A~':--~:^LIIII~LIUII, The well is pumped at a constant discharge of 15/'s -~ The principal axes are taken along east-west (x axis) and north-south (y axis) directions. In order to generate hypothetical data consisting of 'measurements' of aquifer and water table drawdowns, the direct problem (eqns. (5) and (8)) is solved for a set of assumed parameters. Simulated aquifer and water table drawdowns are sampled at the two observation wells (Fig. 2(a)) at discrete times for a pumping duration of 7 days. These error free data are used as input to the parameter estimation algorithm and are shown in Figs. 3 and 4 as 'observed' data. Table I presents the initial guess values and the final estimated values of the parameters for four cases. In the first three cases, the initial values of
OW2
+
N
t
PW ~q W ' ~I,2~-3d OWl
(a) Hypothetical
(c) Haradagere PW- Pumping well OW - Observation well
:)W
°OWl (b) Haragonadona Fig. 2. Location of wells for test problems.
PARAMETER ESTIMATION: AQUIFER-WATER TABLE AQUITARD SYSTEM
185
101
aquifer
E
lo~
3= 3=
O
'
m L_ t't
'
161 o
lo 0
,. . . . . .
I wte t I,,,
0;r0:
°
101
10z Time (rain.)
10~
104
Fig. 3. Aquifer and water table drawdowns at OW~ for the hypothetical problem.
parameters are over/under estimated by one order of magnitude, while in the fourth case the initial values are underestimated by two orders of magnitude, except for the degree of anisotropy. The final estimated parameter values are identical to the true values in all the four cases. Figures 3 and 4 present a comparison between the 'observed' and computed values of aquifer and water table drawdowns. The inverse problem for the hypothetical data is tested with different values of n in the weighting function in eqn. (13). It is found that the value of n does not influence the parameter estimates or the matching between the computed and measured drawdowns. This shows that for error-free data with the correct constitutive model it is possible to estimate the parameters accurately using an OLS procedure. Figure 5 presents the convergence rate ol the iteration algorithm for cases 1-3. It is observed from Fig. 5 and Table 1 that the convergence is more rapid for the case of all parameters underestimated. Ramaswamy (1983)studied the convexity of the objective function of the OLS problem for simple confined and semi-confined systems, for both isotropic and anisotropic cases, and concluded that the objective function is convex in the parametric space wherein all the parameters are underestin~ated. While the complexity of the present problem makes a study of convexity practically unfeasible, it is interesting to note that the convergence is found to
186
M. SEKHAR ET AL.
lo 0
161
er table
to al r'~
/ o -'Observed"drawdown o/ ~ Computed drawdown
10 0
101
10 z
10 a
1--------0 t,
Time (mir~.) Fig. 4. Aquifer and water table drawdowns at OW 2 for the hypothetical problem.
be most rapid for underestimated parameter values. It is found that for underestimation of all parameters, convergence occurred even when the constraints on corrections imposed by eqn. (18) are not applied. The convergence in such cases is found to be more rapid, as seen by the number of iterations presented in Table 1 for cases 2 and 4. However, convergence was not obtained for cases l and 3 (over and mixed estimates) if the limits as per eqn. (18) were not applied. FieM data The method is also applied to two long-duration pumping tests reported by Sarma et al. (1986), in the Haragonadona and Haradagere well fields in the Vedavati River Basin in the State of Karnataka in India. The test well fields are in schist and granite regions, respectively. Based on tracer test and the nature of drawdown observations in the pumping and recovery phases, Sarma et al. (1986) concluded that the well fields can be represented by an aquiferwater table aquitard system. Figures 2(b) and 2(c) present relative locations of pumping wells and the observation wells for both cases. The wells are pumped at a constant discharge of 15 f s-t and 3 d s -t , respectively, for 7 days duration. A knowledge of hydrogeology of the region clearly indicates that
PARAMETER ESTIMATION: AQUIFER-WATER TABLE AQUITARD SYSTEM
~ X
[..
T
X
X
187
T
X
X
X
0
O 0
0
O 0
o
dc~
d
c~o
T ~
T
T
m
~
~m
X
X
o
d~
X
X
o
dd
~mm
imm
~mm
X
X
X
~mmm
X
~
~mmm
X
X
d
do
X
X
X
X
X
d
c~d
I
~
"0
E
X
N
X
X
w~
4m~
"7 ~
E
ob
X
X
~7
'~
X
X
o
X
X o~ m
0
6
0
? T m
N
~
? ~
m
m m
•=
X
X
d
dd
X
X
X
d
do
X
~
°~
0
°.,i
E L.
0
.1
<
[..
E c~ Q.
~,~
~
~
Z
188
M. SEKHAR ET AL.
II ¢1 If D i O
D
O
A
o
°
o
II
2 b
161
u
o
I t,
E :E r,, o
u
o-0ver estimation (case 1) a-Under estimation (case2) A-Mixed (case 3) n-Under estimation without constraints (case 1) B
I= O O
1~6
I
0
/.
8 Iterations
6 A
a
1.2
16
Fig. 5. Convergence rate for the hypothetical problem.
there is an anisotropy, with the principal axes identified along northeastsouthwest (x axis) an,4 northwest-southeast (y axis) directions, with maximum transmissivity being in the y direction. Since water table drawdown is not available in both the well fields, only aquifer drawdown is used for the purposes of least squares minimisation. The parameters are estimated using the inverse problem algorithm described in the previous section, with weights assigned in the least squares minimisation, using n = 1.0 in eqn. (13). The input (guess) parameters and estimated parameters for the Haragonadona and Haradagere tests are given in Tables 2 and 3, respectively. For the Haragonadona test, parameter estimates based
PARAMETER ESTIMATION: AQUIFER-WATER TABI E AQUITARD SYSTEM
189
TABLE 2 Results of parameter estimates for Haragonadona well field data
Parameter
Initial values
Estimates from inverse procedure
Estimates from graphical procedure
Transmissivity, T , ( m 2 d a y - l ) Storage coefficient, S, Leakage coefficient, K'/ho (day -I )
0.40 x 10 ! 0.50 x l0 -5 0.20 x l0 -3
0.167 x 102 0.17 x l0 -4 0.41 x l0 -2
0.206 x 102 0.50 x l0 -4 0.21 x 10-3-0.46 x l0 -3
Specific storage, S, Specific yield Sy Degree of Anisotropy, D
0.5 x l0 -6 0.3 × 10 -3 0.1 x 10 m
0.80 x l0 -5
0.167 x l0 -4
0.58 × 10 -2
0.50 X 10 -2
0.37 x 10 m
0.795 x 10 ~
r.m.s, error (m) r.m.s./Smax
22.03 1.366
0.248 0.015
0.788 0.049
No. of iterations
20
on the graphical method were obtained by Sridharan et al. (1990). These results are also presented in Table 2 along with the r.m.s, values for both graphical and computed parameter estimates. The r.m.s, value for the computed parameter estimates is clearly better than for graphical parameter estimates. The absolute value of r.m.s, for the Haragonadona test is larger than for the Haradagere test (Tables 2 and 3) because of the larger drawdowns in the former case. If the r.m.s, value is normalised with respect to maximum drawdown, results for the two tests are comparable. The matching between TABLE 3 Results of parameter estimates for Haradagere well field data
Parameter
Initial values
Estimated values
Transmissivity, T, ( m 2 day- I) Storage coefficient, S, Leakage coefficient, K'/ho (day -l )
0.i x 102 0.2 x 10 -3 0.5 x 10 -2
0.207 x 102 0.238 x 10 2 0.39 × 10 -2
Specific storage, S, Specific yield, Sy Degree of anisotropy, O
0.8 x 10 -4 0.6 x 10 -3 0.4 x 101
0.81 x 10 ---3 0.51 x 10 -2 0.176 x 101
r.m.s, error (m)
1.13 1.41
0.0108 0.0135
r.m.s./Smax
No. of iterations
26
190
M. SEKHAR ET AL
10x2] OWl
f
/
¢0 0 0
0
0
o° °°
o -Observed
drawdown
J Computed drawdown
10
10z Time
104
10 (rain.)
Fig. 6. Aquifer drawdowns for Haragonadona well field.
10o
OW2
=.-,..
E
,.=,,,
=~ 161 0
L.
Q
I
I
I
I
I
II ~I
102
]
I
a
103 Time
(min.)
Fig. 7. Aquifer drawdowns for Haradagere well field.
i
I
l
a
I
I
104
PARAMETER ESTIMATION: AQUIFER-WATER TABLE AQUITARD SYSTEM
191
computed and observed drawdowns is presented in Figs. 6 and 7 for the two tests. CONCLUSION
The weighted least squares approach is used for estimation of parameters in an aquifer-water table aquitard system. There are six parameters to be evaluated for both the aquifer and aquitard. The direct problem governed by the coupled partial differential equations for the aquifer and the aquitard, is solved by an iterative numerical procedure to determine the unknown water table elevation and flow transfer term. The inverse problem is solved by a sensitivity analysis technique. The convergence characteristics of the algorithm are studied using a correct constitutive model with error-free data. Application of the method to two field pumping tests of 7 days duration is presented. REFERENCES Boulton, N.S., 1963. Analysis of data from non-equilibrium pumping tests allowing for delayed yield from storage. Proc. Ins* Civ. Eng., 26: 469-482. Boulton, N.S. and Streltsova, F.D., 1975. New equations for determining the formation constants of an aquifer from pvmping test data. Water Resour. Res., 1I: 148-153. Carrera, J. and Neuman, S.P., 1986. Estimation of aquifer parameters under transient and steady state conditions, I, Maximum likelihood method incorporating prior information. Water Resour. Res., 22: 199-210. Chander, S., Kapoor, P.N. and Goyal, S.K., 1981. Analysis of pumping test data using Marquardt algo,;thm. Ground Water, 19: 275-278. Cobb, P.M., McElv~ee, C.D. and Butt, M.A., 1982. Analysis of leaky aquifer pumping test data: AI~ automated n::::lerical solution using sensitivity analysis. Ground Water, 20: 325-333. Cooley, R.M. and "Case, C.M., 1973. Effect of water table aquitard on drawdown in an underlying pumped aquifer. Water Resour. Res., 9: 434-447. Hantush, M.S., 1964. Hydraulics of wells. In: V.T. Chow (Editor), Advances in Hydroscience. Academic Press, New York, Vol. 1, pp. 282-430. Javandel, I. and Witherspoon, P.A., 1980. A semianalytical solution for partial penetration in two-layered aquifers. Water Resour. Res., 16:1099-1106. Javendel, I. and Witherspoon, P.A., 1983. Analytical solution of partially penetrating well in two-layer aquifer. Water Resour. Res., 19: 567-578. Kool, J.B., Parker, J.C. and van Genuchten, M.TH., 1987. Parameter estimation for unsaturated flow and transport models - - a review. J. Hydroi., 9!" 225-293. Mania, J. and Sucche, M., 1978. Automatic analysis of pumping test data - - Application of the Boulton and Hantush hypothesis. J. Hydrol., 37: 185-194. McEiwee, C.D., 1980. Theis parameter evaluation from pumping tests by sensitivity analysis. Ground Water, 18: 56-60. Neuman, S.P. and Witherspoon, P.A., 1969. Theory of flow in a confineo two aquifer systems. Water Resour. Res., 5: 803-816.
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Ramaswamy, R., 1983. Optimisation technique for evaluation of aq~fifer parameters, Ph.D. Thesis, Indian Institute of Science, Bangalore, India, 352 pp. Sarma, K.V.N., Sridharan, K., Achuta Rao, and Sarma, C.S.S., 1986. Computer model for Vedavati ground water basin, Part 1. Well field model. Sadhana, Proc. Indian Sci., 9(1): 31-42. Sridharan, K., Ramaswamy, R. and Lakshmana Rao, N.S., 1985. Identification of parameters in unconfined aquifers. J. Hydrol., 79: 73-81. Sridharan, K., Ramaswamy, R. and Lakshamana Rao, N.S., 1987. Identification of parameters in semiconfined aquifers. J. Hydrol., 93: 163-173. Sridharan, K., Sekhar, M. and Mohan Kumar, M.S., 1990. Analysis of an aquifer-water table aquitard sytem. J. Hydrol., 114: 175-189. Streltsova, T.D., 1976. Analysis of aquifer-aquitard flow. Water Resour. Res., 12: 415-522. Walton, W.C., 1979. Review of leaky artesian aquifers test evaluation methods. Ground Water, 17: 270-283. Yeh, W.W.-G., 1986. Review of parameter identification procedures in ground water hydrology: The inverse problem. Water Resour. Res., 22: 95-108.