Published online 16 September 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/hyp.7449
Radial groundwater flow to a finite diameter well in a leaky
confined aquifer with a finite-thickness skin
Shaw-Yang Yang
1and Hund-Der Yeh
2*
1Department of Civil Engineering, Vanung University, Chungli, Taiwan 2Institute of Environmental Engineering, National Chiao Tung University, Hsinchu, Taiwan
Abstract:
A mathematical model that describes the drawdown due to constant pumpage from a finite radius well in a two-zone leaky confined aquifer system is presented. The aquifer system is overlain by an aquitard and underlain by an impermeable formation. A skin zone of constant thickness exists around the wellbore. A general solution to a two-zone leaky confined aquifer system in Laplace domain is developed and inverted numerically to the time-domain solution using the modified Crump (1976) algorithm. The results show that the drawdown distribution is significantly influenced by the properties and thickness of the skin zone and aquitard. The sensitivity analyses of parameters of the aquifer and aquitard are performed to illustrate their effects on drawdowns in a two-zone leaky confined aquifer system. For the negative-skin case, the drawdown is very sensitive to the relative change in the formation transmissivity. For the positive-skin case, the drawdown is also sensitive to the relative changes in the skin thickness, and both the skin and formation transmissivities over the entire pumping period and the well radius and formation storage coefficient at early pumping time. Copyright 2009 John Wiley & Sons, Ltd.
KEY WORDS groundwater; skin effect; Laplace transforms; leaky confined aquifer; sensitivity analysis
Received 27 November 2007; Accepted 17 July 2009
INTRODUCTION
Hantush and Jacob (1955) presented the non-steady drawdown solution in a leaky confined aquifer due to constant well discharge. They assumed that the rate of leakage from an aquitard is proportional to the drawdown at any point and the storage of the aquitard is negligible. In addition, the early-time and late-time approximate solutions were also developed by Hantush and Jacob (1955). Hantush (1960) dealt with a flow system in which the effect of storage in the semi-pervious layers was taken into consideration. The solutions to the boundary-value problems were obtained using the Hankel and Laplace transforms. The investigations of Hantush and Jacob (1955) and Hantush (1960) did not consider the effect of well radius and hence might not accurately describe the early-time drawdown response. Cheng and Morohunfola (1993) presented an analytical drawdown solution for the problem of radially convergent flow towards a well pumping at a constant rate in a multi-layered leaky aquifer system. They modelled the aquifer system based on the methodology of Neuman and Witherspoon (1969) and Herrera (1970), and utilized a numerical inversion algorithm to evaluate the drawdowns in associated layers. Sekhar et al. (1994) presented a procedure for the determination of flow parameters in an anisotropic aquifer in which the direction of principal axes is unknown. They used a modified parameter perturbation technique to
* Correspondence to: Hund-Der Yeh, Institute of Environmental Engineering, National Chiao Tung University, Hsinchu, Taiwan. E-mail: [email protected]
determine the sensitivity coefficients. Shan et al. (1995) studied the problem of saturated water flow to a sin-gle pumping test well in an aquifer-fault-aquifer sys-tem. They developed analytical solutions and presented methods in determining the fault transmissivity from pumping test data. Zlotnik (2004) introduced a new con-cept of maximum stream depletion rate (MSDR), defined as a maximum fraction of pumping rate contributed by the stream depletion. The MSDR was determined from the aquifer hydro-stratigraphic conditions, geom-etry of recharge and discharge zones, and locations of pumping wells. Yeh and Huang (2005) employed the extended Kalman filter to determine aquifer parameters in leaky aquifer systems with and without considering stor-age effect in the aquitard. Copty et al. (2006) assessed the effect of leakage on equivalent transmissivity for a steady-state radial flow in heterogeneous leaky aquifers. Zhan and Bian (2006) provided the analytical and semi-analytical solutions for use in calculating leakage rate and volume in a leaky confined aquifer bounded by a rela-tively thin aquitard. Yeh et al. (2007) developed a novel approach based on global optimization methods such as simulated annealing or a genetic algorithm to determine the best-fit aquifer parameters for leaky aquifer systems. Hunt and Scott (2007) obtained an approximate solution for the aquifer–aquitard–aquifer problem from numerical inversions of exact analytical solutions for Laplace trans-forms and reduced it to a well-known aquifer–aquitard problem. Li (2007a) presented a new analytical solution to investigate the aquifer horizontal movement driven by hydraulic forces. His solution described the aquifer
radial transient movement caused by well discharge and recharge in a leaky confined aquifer. In addition, Li (2007b) also developed new analytical solutions in the velocity and cumulative displacement fields describing transient radial movement of an unconfined leaky aquifer. He indicated that the large leakage is important in slow-ing radial movement and reducslow-ing aquifer deformation. Li and Neuman (2007) interpreted the pumping test in the Oxnard basin by coupling the Neuman solution (Neuman, 1968) with a numerical inversion algorithm in a five-layer system. Trinchero et al. (2008) developed a double inflection point (DIP) method for the interpretation of pumping tests in the leaky aquifers. Their DIP method does not involve any curve fitting, requiring the estima-tion of the posiestima-tion of three points on the time-drawdown curve instead.
A skin is usually developed near the wellbore due to an extensive well development or the intrusion of drilling mud into the adjacent formation during well construction. A positive wellbore skin (also called positive skin or low-conductivity skin) has a lower permeability than that of the original formation. In contrast, a disturbed formation with a higher permeability near the bore well is referred to as a negative bore well skin (also called negative skin or high-conductivity skin). Novakowski (1989) mentioned that the thickness of the skin zone might range from a few millimetres to several metres and thus should be con-sidered in the pumping-test data analysis. The effect of wellbore skin on the results of pumping tests had been investigated by Barker and Herbert (1982) without con-sidering the well radius effect and by Novakowski (1989), Novakowski (1990), and Yeh et al. (2003) accounting for the well radius effect. However, these papers did not consider the leaky condition in a two-zone confined aquifer system. Moench (1985) developed the concep-tual models combining the Hantush theory of the storage in the aquitard (Hantush, 1960) with the Papadopulos and Cooper theory of a large-diameter well (Papadopu-los and Cooper, 1967). The solution for the dimensionless drawdown in the comprising aquifer and aquitard due to pumpage was provided in Laplace domain and inverted numerically. However, his model treated the skin effect as a factor of head loss and therefore, the skin thickness was neglected.
The drawdown solution, which accounts for the effects of the skin zone, finite radius well, and storage in the aquitard, has not been developed before for the case of a leaky confined aquifer. The purpose of this paper is to present a mathematical model that describes the draw-down due to constant pumpage from a finite radiuswell in a two-zone leaky confined aquifer accounting for the effects of storage of the aquitard and the finite thickness skin. The drawdown solutions in the skin and formation zones are developed in Laplace domain and evaluated to the time-domain solutions by a numerical inversion algorithm. Hypothetical leaky aquifer systems are used to illustrate the effects of the skin zone and leakage on draw-down distribution in a two-zone leaky confined aquifer system. In addition, the sensitivity analysis is performed
to investigate the aquifer drawdown in response to the relative changes of parameters of the aquifer and aquitard.
THEORY
A schematic cross-section of an idealized leaky confined aquifer system is depicted in Figure 1. The aquifer of constant thickness is overlain by an aquitard and underlain by an impermeable formation. A skin zone of finite thickness is assumed to exist around the wellbore. The pumping well penetrates the entire thickness of the aquifer and the pumping rate is maintained constant. In this study, the assumptions made for the conceptual model are:
1. The upper aquifer is highly permeable with an infinite amount of water supply such that it maintains a constant head at any time.
2. The formation zone is homogeneous, isotropic, of a constant thickness, and infinite in radial extent. 3. The skin zone is also homogeneous, isotropic, and of
a constant thickness around the wellbore.
4. The flow direction is vertical in the aquitard and horizontal in the confined aquifer.
Mathematical model
Based on the above assumptions, the governing equation describing the drawdown distribution, sr, t, for the skin and formation zones are, respectively,
∂2s1r, t ∂r2 C 1 r ∂s1r, t ∂r C q0 T1 D S1 T1 ∂s1r, t ∂t , rwr r1 1 and ∂2s2r, t ∂r2 C 1 r ∂s2r, t ∂r C q0 T2 D S2 T2 ∂s2r, t ∂t , r1r < 1 2
Figure 1. The schematic cross-sectional diagram of an idealized two-zone leaky confined aquifer system
where subscripts 1 and 2 respectively denote the skin and formation zones; q0 is the leakage of the aquitard; T is
the transmissivity; S is the storage coefficient; r is the radial distance from the centerline of pumping well; rw
is the radius of pumping well; r1 is the radial distance
from the centerline of the well to the outer skin envelope; and t is the time from the start of the pumping test.
The drawdown of the aquifer is initially zero within the skin and formation zones. Thus, the initial conditions for Equations (1) and (2) are written as
s1r,0 D s2r,0 D 0 3
As r approaches infinity, the drawdown in the forma-tion zone tends to be zero. Therefore, the outer boundary condition at an infinite distance is specified as
s21, t D0 4
Applying the Darcy law, the boundary condition for a constant flow rate across the screen is assumed to be uniform and expressed as
QwD 2T1r
∂s1r, t
∂r , r D rw 5 where Qw is a constant pumping rate.
The continuities of the drawdown and the flux between the skin and formation zones, respectively, require that
s1r1, t D s2r1, t 6 and T1 ∂s1r1, t ∂r DT2 ∂s2r1, t ∂r 7 Aquitard flow
Considering the effect of the aquitard storage, the governing equation describing drawdown distribution within the aquitard is
b0K0∂ 2s0 z, t ∂z2 DS 0∂s 0 z, t ∂t 8
where s0z, t is the drawdown of the aquitard; z is
the vertical distance from the lower impermeable layer; b0 is the thickness of the aquitard; and K0 and S0 are
the hydraulic conductivity and storage coefficient of the aquitard, respectively.
The drawdown in the aquitard is initially assumed zero and expressed as
s0z,0 D 0 9 The boundary condition at the interface between the aquitard and the lower aquifer is
s0z, t D s1r, t D s2r, t, z D b 10
In addition, the boundary condition on the top of the aquitard is
s0z, t D0, z D b C b0 11
Applying Laplace transforms to Equations (8), (10) and (11) leads to d2s0z, p dz2 D˛ 02 s0z, p, ˛02D pS 0 b0K0 12 s0z, p D s1r, p D s2r, p D sr, p, for z D b 13 and s0z, p D0, for z D b C b0 14 The general solution to Equation (12) is
s0z, p D C1sinh˛0z C C2cosh˛0z 15
where C1 and C2 are undetermined constants.
Substituting Equation(15) into Equations (13) and (14), one obtains C1 D cosh˛0b C b0 sinh˛0b0 sir, p 16 and C2 D sinh˛0b C b0 sinh˛0b0 sir, p 17 where sir, p is the drawdown within the skin for
i D1 and within the formation for i D 2. The solution for drawdown within the aquitard can be obtained by substituting Equations (16) and (17) into Equation (15) after some manipulations as
s0z, p D sinh˛
0
b C b0z
sinh˛0b0 sir, p 18 Based on the mass conservation, the leakage of the aquitard is q0DK0 ds 0 z, p dz zDb 19 Substituting Equation (18) into Equation (19), the leakage to the confined aquifer is
q0D K0˛0coth˛0b0sir, p 20
Drawdown of leaky aquifer
The solutions to Equations (1) and (2) with respect to boundary conditions Equations (4)–(7) can be found using Laplace transform method. The detailed derivation for the Laplace-domain solutions is given in Appendix A and the drawdown solutions within the skin and formation zones are respectively
s1r, p D Qw 2rwT1 1I0˛1r C 2K0˛1r ˛1p[1I1˛1rw C 2K1˛1rw] 21 and s2r, p D Qw 2rwr1 K0˛2r ˛1p[1I1˛1rw C 2K1˛1rw] 22
with 1 DT2˛2K0˛1r1K1˛2r1 T1˛1K1˛1r1K0˛2r1 23 and 2DT2˛2I0˛1r1K1˛2r1 C T1˛1I1˛1r1K0˛2r1 24 where p is the Laplace variable; sr, p is the transformed drawdown; ˛2
1DS1/T1p C K0˛0/T1coth˛0b0; ˛22D
S2/T2p C K0˛0/T2coth˛0b0; I0(.) and I1(.) are the
modified Bessel functions of the first kind of order zero and one, respectively; and K0(.) and K1(.) are the
modified Bessel functions of the second kind of order zero and one, respectively.
Dimensionless solutions
Define dimensionless variables as following: tDD T2t S2rw2 , rDD r rw , r1DD r1 rw , b0DD b 0 rw , TDD T1 T2 T0DD K 0 b0 T2 , SDD S1 S2 , S0DD S 0 S2 , s1DD 2T2 Qw s1, s2DD 2T2 Qw s2 25
Using the above-defined dimensionless variables, Equations (21) and (22) can be expressed in dimension-less forms as s1DrD, p D 1 TD˛1D 1DI0˛1DrD C 2DK0˛1DrD p[1DI1˛1D C 2DK1˛1D] 26 and s2DrD, p D 1 ˛1Dr1D K0˛2DrD p[1DI1˛1D C 2DK1˛1D] 27 with 1DD˛2DK0˛1Dr1DK1˛2Dr1D TD˛1DK1˛1Dr1DK0˛2Dr1D 28 and 2DD˛2DI0˛1Dr1DK1˛2Dr1D CTD˛1DI1˛1Dr1DK0˛2Dr1D 29 where ˛02 DDS0D/T0Dp, ˛1D2 DSD/TDp C T0D˛0D
/TDb0Dcoth˛0Db0D, and ˛22D Dp C T0D˛0D/b0Dcoth
˛0 Db0D.
SIMPLIFIED SOLUTIONS Solution without considering skin effect
The aquifer formation is a single-layer system if the skin zone is absent. Under this condition, the aquifer properties T1 DT2DT and S1 DS2DS and the
vari-ables 1D0, 2DT/r1, and ˛1D˛2D˛. Then both
Equations (21) and (22) reduce to sr, p D Qw 2rwT K0˛r ˛pK1˛rw , ˛ DS/Tp C K0˛0/Tcoth˛0b0 30
If the well radius is negligible (i.e. rw!0), the
modified Bessel function K1˛rw approaches 1/˛rw
and Equation (30) reduces to sr, p D Qw 2T K0˛r p 31 which is the drawdown solution in Laplace domain given by Hantush and Jacob (1955) for a leaky confined aquifer under the assumption of an infinitesimal radius well. Solution without considering leakage effect
If the aquitard is impervious (i.e. K0D0), then one can write ˛2
1DS1/T1p and ˛22 DS2/T2p in
Equations (21) and (22), which, consequently, lead these two solutions to those presented in Yeh et al. (2003) for the two-zone nonleaky aquifer case. After neglecting the well radius (rw!0), the solutions of Yeh et al. (2003)
reduce respectively to s1r, p D Qw 2T1 1I0˛1r C 2K0˛1r p2 32 and s2r, p D Qw 2T1 T1K0˛2r r1p2 33 Note that Equations (32) and (33) were given in different forms by Barker and Herbert (1982).
As the aquitard is impervious and the skin zone is absent, both Equations (21) and (22) reduce to
sr, p D Qw 2rwT K0˛r ˛pK1˛rw , ˛ DS/Tp 34 which is the drawdown equation in Laplace domain for a single-layer confined aquifer.
NUMERICAL INVERSION OF THE SOLUTIONS The Laplace-domain solutions to Equations (21) and (22) for drawdowns consist of the products of the Bessel func-tions. These Bessel functions can be approximated by the formulas given in Watson (1958) and Abramowitz and Stegun (1964). The application of the Shanks method (Shanks, 1955; Wynn, 1956) will be computational effi-cient to numerically evaluate the Bessel functions. Simi-lar approximations of the Bessel functions can be found
in the works of Yeh and Yang (2006) and Yang and Yeh (2007).
The analytical inversion of the Laplace-domain solu-tions Equasolu-tions (21) and (22) may not be possible. There-fore, the method of numerical Laplace inversion such as the Crump (1976) algorithm is employed. The rou-tine INLAP of IMSL (2003), developed on the basis of the Crump (1976) algorithm, can be used to evaluate the time-domain solutions to Equations (21) and (22) with accuracy to the fourth decimal place in comparison to the analytical inversion. This routine had also been success-fully applied to the groundwater problems mentioned in the study by Yang and Yeh (2005) or Yang et al. (2006).
RESULTS AND DISCUSSION Effect of leakage
Several examples with hypothetical data are used to illustrate the effects of the skin and leakage in a two-zone leaky confined aquifer system. The parameters of the formation zone are T2 D103 m2 s1, S2 D103,
and the thickness of the confined aquifer b D 30 m. The storage coefficient of the aquitard is S0D103. The well
radius rw is 0Ð05 m and the pumpage is maintained
constant at Q D 103 m3 s1.
Figure 2 illustrates the effects of the conductivity and thickness of the aquitard on drawdowns in a pump-ing well for the positive-skin case. Figure 2a displays the time-drawdown curves for T1D104 m2 s1, r1 D
0Ð50 m, b0D5 m, and K0D0 (no leakage), 108, 107
or 106 m s1 when the time ranges from 101 to 106 s. Figure 2b displays the time-drawdown curves for T1 D
104 m2 s1, r
1D0Ð50 m, K0D107 m s1, and b0D
5, 10 or 20 m for the same time. The time-drawdown curves shown in Figure 2a are the same at early pump-ing time, and the drawdown increases with the decrease of K0 at a later pumping time (say, t ½ 2 ð 103 s). The
drawdown in a leaky aquifer is less than that in a non-leaky aquifer after t ½ 2 ð 103 s. Furthermore, the
aquifer without having the skin and leakage produces the least drawdown among these five drawdown curves for the positive-skin case. The drawdown tends to be stabilized when the pumping time is larger than 105 s (1Ð157 day). Figure 2b also shows that the drawdowns are the same at early pumping time and increase with b0
at the later pumping time. These results indicate that a smaller K0and/or a larger b0results in a larger drawdown for the positive-skin case. In addition, the influence of K0
and b0 is slightly more profound on drawdown at later
time.
Figure 3 displays the distance-drawdown curves in a leaky confined aquifer with r D 0Ð05 m, b0D5 m, T
1 D
104 m2 s1, and r
1 D0Ð50 m for K0D0 (no leakage),
108, 107 or 106 m s1. The figure shows that the
drawdown decreases with time and radial distance. In addition, the drawdown decreases rapidly within the skin zone and slowly within the formation zone. The drawdowns in the skin and formation zones near the
Figure 2. The time-drawdown curves with r D 0Ð05 m, T1D104m2 s1, and r1D0Ð50 m for (a) b0D5 m and various K0D0 (no leakage), 108, 107, or 106m s1and (b) K0D107m s1and various b0D5,
10, or 20 m
interface behave differently. The drawdown is larger within the skin zone and smaller within the formation zone when compared with that of the aquifer without skin and leakage. The figure also indicates that a larger K0 results in a smaller drawdown. Figure 4 also displays the distance-drawdown curves in a leaky confined aquifer for K0D0 (no leakage) or 107 m s1, r D 0Ð05 m, T1D104 m2 s1, and r1 D0Ð50 m when b0D5, 10, or
20 m. For the positive-skin case, the drawdown decreases rapidly within the skin zone and slowly within the formation zone. In addition, a larger b0 leads to a larger
Figure 3. The distance-drawdown curves for r D 0Ð05 m, T1D104 m2 s1, r1D0Ð50 m, b0D5 m and various K0D0 (no leakage), 108, 107,
or 106m s1
Figure 4. The distance-drawdown curves for r D 0Ð05 m, T1D104 m2 s1, r1D0Ð50 m, K0D107 m s1and various b0D5, 10, or 20 m Effect of wellbore skin
Figure 5 shows the time-drawdown curves for rwD
0Ð05 m in the aquifer with the skin zone of T1D
104, 103 (no skin) or 102 m2 s1 and r
1 D0Ð25
or 0Ð50 m and the aquitard of K0D0 (no leakage)
or 107 m s1 and b0D5 m when the time ranges
from 101 to 106 s. Note that the case of K0D0
corresponds to the solution of a non-leaky confined aquifer system and the case of K0D0 and T
1 DT2D
103 m2 s1 corresponds to the solution for a
single-layer non-leaky confined aquifer system. In this figure, the time-drawdown curves for the aquifer with a positive skin when T1D104 m2 s1, without skin when T1D
103 m2 s1, and with a negative skin when T 1D
102 m2 s1. The figure shows that the aquifer with a
Figure 5. The drawdown-time curves within a pumping well (rwD 0Ð05 m) for K0D0 (no leakage) and 107m s1, b0D5 m, r1D0Ð25 and 0Ð50 m, and T1D104in the positive-skin case, 103in the no skin
case or 102m2s1in the negative-skin case
positive skin produces the largest drawdown, the no skin is the second, and the negative skin yields the smallest drawdown. In addition, the difference in drawdown between the aquifer with a positive skin and a single-layer aquifer is larger than that between the aquifer with a negative skin and a single-layer aquifer. It also shows that a larger r1 has a larger drawdown for the
positive-skin case and a smaller drawdown for the negative-positive-skin case. This indicates that the effect of a positive skin on drawdown is larger than that of a negative skin. The results demonstrate that the drawdown is significantly influenced by a positive skin than by a negative skin. In addition, a thicker skin zone has a more profound effect on drawdown. The time-drawdown curves also show that the drawdown of a non-leaky curve is proportional to natural logarithm of time.
Sensitivity analysis
Sensitivity analysis is a technique to assess the effects of uncertainty in input parameters on the model result. This method is helpful in assessing how a model responds to the change in certain parameters. The sensitivity of a dependent variable in response to the change in a parameter is defined as (Liou and Yeh, 1997)
Xi,j D
∂Vi
∂Pj
35 where Xi,j is the sensitivity coefficient of the jth
param-eter (Pj) at the ith time and Vi is the dependent variable
of the model, e.g. the drawdown distribution. Huang and Yeh (2007) provided a normalized sensitivity to assess the effect of relative changes in parameters on depen-dent variable. The normalized sensitivity of a dependepen-dent
variable to the relative change in a given parameter is defined as X0i,j DPj ∂Vi ∂Pj 36 where X0i,j is the normalized sensitivity of the jth parameter (Pj) at the ith time. The partial derivative in
Equation (36) may be laborious to evaluate. A finite-difference formula may be used to approximate the differentiation (Yeh, 1987). That is
∂Vi
∂Pj
D ViPjCPj ViPj Pj
37 where Pj is a small increment chosen as 102ðPj.
The sensitivity analyses of parameters of a two-zone leaky confined aquifer system are performed using the hypothetic data. The aquifer has b D 30 m, T2 D
103 m2 s1, and S2D103for the formation zone while
the aquitard has b0D5 m, K0D107 m s1, and S0D
103. The well radius r
w is 0Ð05 m and the pumping
rate is maintained constant at Q D 103 m3 s1. For a
skin zone with the storage S1 D103, two cases are
considered; that is, T1 D102 m2 s1 for the
negative-skin case and T1 D104 m2 s1 for the positive-skin
case. Figure 6a and b plots the time-drawdown curves and the normalized sensitivities of the parameters K0,
S0, b0, T
1, S1, r1, T2, S2, and rw. Figure 6a shows
that a relative change in b0 produces a minor positive
effect on drawdown and the other parameters produce negative effects on drawdown for the negative-skin case with T1D102 m2 s1. The normalized sensitivity of
drawdown with respect to b0starts with a slight increase
after 103 s and reaches a constant value of 0Ð08 m after 105 s. In contrast, the normalized sensitivity of
drawdown with respect to K0 produces a similar pattern
but has a negative effect. A relative change in r1 has a
negative effect on drawdown and the sensitivity curve shows a slight increase with time at early time and stabilized at a value of 0Ð14 m after t ½ 3 s. The normalized sensitivity of drawdown with respect to T2
decreases with time and approaches a constant value of 0Ð90 m when t ½ 105 s. Furthermore, the normalized
sensitivities of drawdown with respect to K0, b0, r 1, and
T2 maintain constant values of 0Ð08, 0Ð08, 0Ð14, and
0Ð90 m, respectively, at a later time (say, t ½ 105 s) because the leaky confined aquifer system reaches a steady-state condition. The figure also shows that the drawdown in response to the change of T2 produces
the largest normalized sensitivity in the magnitude, the parameter r1 is the second, and the other parameters
such as S0, T
1, S1, and S2, rw are relatively less in the
analyses. Those results indicate that the drawdowns are very sensitive to the change in T2, slightly less sensitive
to the change in r1, and the parameters b0 and K0 give
minor influences on drawdown at a later time in the negative-skin case. Figure 6b plots the time-drawdown curves and the normalized sensitivities for the positive-skin case with T1D104 m2 s1. The figure shows that
the normalized sensitivities of drawdown with respect to
Figure 6. Plots of the time-drawdown curve and the normalized sen-sitivities of the parameters K0, S0, b0, T1, S1, r1, T2, S2, and rw versus time for (a) T1D102 m2s1 in the negative-skin case and
(b) T1D104 m2s1in the positive-skin case
b0, r1, and rware positive and those with respect to other
parameters are negative. The normalized sensitivity to the relative change in r1 starts to increase after about
0Ð5 s and is stabilized at a value of 1Ð4 m after about 10 s. The normalized sensitivity of drawdown in response to the change in T1 decreases with time and keeps a
constant value of 3Ð6 m after about 5 s. The normalized sensitivity to the relative change in T2 becomes larger
after 1 s and approaches a constant value (0Ð9 m) when t ½105 s. The normalized sensitivity of drawdown with
respect to rw is relatively small (say 0Ð36) at t D 0Ð1 s
and then decreases with time and approaches zero when t ½600 s. This figure also shows that the normalized sensitivity to the relative change in S2produces a negative
effect, gradually decreases in magnitude with time from 1 s, and approaches zero at about 5 s. It indicates that the effect of the formation storage on drawdown occurs only at early time. Furthermore, the relative changes in b0, K0, S0, and S1 yield very small effects on drawdown.
The influences of relative changes in parameters K0 and
b0 on drawdown become profound at a later time. In
short, the results of sensitivity analyses indicate that the effects of relative changes in T1, T2, and r1 on
drawdown are very significant and the parameters rw
and S2 are fairly sensitivity only at an early time
in the positive-skin case. In addition, the effect of relative change in r1 on drawdown in the
positive-skin case is much larger than those in the negative-positive-skin case.
CONCLUSION
A mathematical model describing the drawdown distri-bution for a radial flow to a fully penetrating well of finite radius in a two-zone leaky confined aquifer is devel-oped, accounting for the effect of the finite-thickness skin zone. The general solution to a two-zone leaky confined aquifer system in Laplace domain is developed and the time-domain solution is obtained by the modified Crump (1976) algorithm. The conclusions can be drawn as fol-lows:
1. An aquitard with larger hydraulic conductivity and smaller thickness leads to a smaller drawdown in the positive-skin case. Both hydraulic conductivity and thickness of the aquitard influence the drawdown after a large pumping time in a two-zone leaky confined aquifer system.
2. The effect of relative change in formation transmissiv-ity T2 to the drawdown is significant for the
negative-skin case. On the other hand, the effects of relative changes in the transmissivities of the skin and forma-tion zones (i.e. T1 and T2) and the radial distance from
the centerline of the well to the outer skin envelope (r1) on the drawdown are significant for the
positive-skin case. In addition, the drawdown in response to the relative change in the well radius (rw) and the
forma-tion storage coefficient (S2) is sensitive only at early
time.
3. This solution predicts the drawdown distribution in a leaky confined aquifer system with the skin zone around the wellbore. It is useful in preliminary design for a constant-flux pumping system in a two-zone leaky confined aquifer.
ACKNOWLEDGEMENTS
Research leading to this paper has been partially sup-ported by the grants from Taiwan National Science Coun-cil under the contract number NSC 96— 2221— E— 238— 019 and NSC 96— 2221— E— 009— 087-MY3. The authors would also like to thank two anonymous reviewers for their valuable and constructive comments.
APPENDIX A. DERIVATION OF THE LAPLACE-DOMAIN SOLUTIONS TO
EQUATIONS (21) AND (22)
Applying Laplace transforms to Equations (1) and (2) with Equation (20) yields the following subsidiary equations, respectively, d2s1r, p dr2 C 1 r ds1r, p dr D˛ 2 1s1r, p, rwr r1 A1 and d2s2r, p dr2 C 1 r ds2r, p dr D˛ 2 2s2r, p, r1 r < 1 A2 where p is the Laplace transform variable correspond-ing to the time variable t; sr, p is the transformed drawdown; ˛21DS1/T1p C K0˛0/T1coth˛0b0; ˛22D
S2/T2p C K0˛0/T2coth˛0b0; and ˛02 DpS0/b0K0.
The boundary conditions of Equations (4) and (5) in Laplace domain are
s21, p D0 A3 and Qw p D 2T1rw ds1rw, p dr A4
Moreover, the continuity conditions of drawdown and flux between the skin and formation zones after applying Laplace transform yield
s1r1, p D s2r1, p A5 and T1 ds1r1, p dr DT2 ds2r1, p dr A6
The general solutions to Equations (A1) and (A2) are s1r, p D D1I0˛1r C D2K0˛1r A7
and
s2r, p D D3I0˛2r C D4K0˛2r A8
where D1, D2, D3, and D4 are undetermined constants.
Substituting Equations(A7) and (A8) into Equations(A3)–(A6), the undetermined constants can then be determined as D1D Qw 2rwT1˛1p 1 1I1˛1rw C 2K1˛1rw A9 D2D Qw 2rwT1˛1p 2 1I1˛1rw C 2K1˛1rw A10 D3D0 A11
and D4 D Qw 2rwT1˛1p 1I0˛1r1 C 2K0˛1r1 1I1˛1rw C 2K1˛1rwK0˛2r1 A12 with 1DT2˛2K0˛1r1K1˛2r1 T1˛1K1˛1r1K0˛2r1 A13 and 2 DT2˛2I0˛1r1K1˛2r1 C T1˛1I1˛1r1K0˛2r1 A14 Consequently, the solutions of drawdowns within the skin and formation zones can then be respectively obtained by substituting the constants in Equations (A9) and (A10) into Equation (A7) and the constants in Equations (A11) and (A12) into Equation (A8). The final results are given as Equations (21) and (22) in the text.
REFERENCES
Abramowitz M, Stegun IA. 1964. Handbook of Mathematical Functions
with Formulas, Graphs and Mathematical Tables, National Bureau of Standards. Dover: Washington, DC.
Barker JA, Herbert R. 1982. Pumping tests in patchy aquifers. Ground
Water 20(2): 150– 155.
Cheng AH-D, Morohunfola OK. 1993. Multilayered leaky aquifer systems: 1. Pumping well solutions. Water Resources Research 29(8): 2787– 2800.
Copty NK, Sarioglu MS, Findikakis AN. 2006. Equivalent transmissivity of heterogeneous leaky aquifers for steady state radial flow. Water
Resources Research 42: W04416. DOI:10.1029/2005WR004673.
Crump KS. 1976. Numerical inversion of Laplace transforms using a Fourier series approximation. Journal of the Association for Computing
Machinery 23(1): 89– 96.
Hantush MS. 1960. Modification of the theory of leaky aquifers. Journal
of Geophysical Research 65(11): 3713– 3725.
Hantush MS, Jacob CE. 1955. Non-steady radial flow in an infinite leaky aquifer. Transactions AGU 36(1): 95– 100.
Herrera I. 1970. Theory of multiple leaky aquifers. Water Resources
Research 6: 185– 193.
Huang YC, Yeh HD. 2007. The use of sensitivity analysis in on-line aquifer parameter estimation. Journal Hydrology 335(3– 4): 406– 418. Hunt B, Scott D. 2007. Flow to a well in a two-aquifer system. Journal
of Hydrologic Engineering 12(2): 146– 155.
IMSL. 2003. IMSL Fortran Library User’s Guide Math/Library Volume
2 of 2 , version 5Ð0. Visual Numerics: Houston.
Li J. 2007a. Transient radial movement of a confined leaky aquifer due to variable well flow rates. Journal of Hydrology 333: 542– 553. Li J. 2007b. Analysis of radial movement of an unconfined leaky aquifer
due to well pumping and Injection. Hydrogeology Journal 15(4): 1063– 1076.
Li Y, Neuman SP. 2007. Flow to a well in a five-layer system with application to the Oxnard basin. Ground Water 45(6): 672– 682.
Liou TS, Yeh HD. 1997. Conditional expectation for evaluation of risk groundwater flow and transport: one-dimensional analysis. Journal of
Hydrology 199: 378– 402.
Moench AF. 1985. Transient flow to a large-diameter well in an aquifer with storative semiconfining layers. Water Resources Research 21(8): 1121– 1131.
Neuman SP. 1968. Transient flow of ground water to wells in
multiple-aquifer systems. Ph. D. dissertation, Department of Civil Engineering,
University of California: Berkeley, California.
Neuman SP, Witherspoon PA. 1969. Theory of flow in a confined two-aquifer system. Water Resources Research 5: 803– 816.
Novakowski KS. 1989. A composite analytical model for analysis of pumping tests affected by wellbore storage and finite thickness skin.
Water Resources Research 25(9): 1937– 1946.
Novakowski KS. 1990. Analysis of aquifer tests conducts in fractured rock: a review of the physical background and the design of a computer program for generating type curves. Ground Water 28(1): 99– 107. Papadopulos IS, Cooper HH Jr. 1967. Drawdown in a well of large
diameter. Water Resources Research 3(1): 241– 244.
Sekhar M, Mohan Kumar MS, Sridharan K. 1994. Parameter estimation in an anisotropic leaky aquifer system. Journal Hydrology 163: 373– 391.
Shan C, Javandel I, Witherspoon PA. 1995. Characterization of leaky faults: study of water flow in aquifer-fault-aquifer systems. Water
Resources Research 31(12): 2897– 2904.
Shanks D. 1955. Non-linear transformations of divergent and slowly convergent sequences. Journal of Mathematical Physics 34: 1– 42. Trinchero P, Sanchez-Vila X, Copty N, Findikakis A. 2008. A new
method for the interpretation of pumping tests in leaky aquifers.
Ground Water 46(1): 133– 143.
Watson GN. 1958. A Treatise on the Theory of Bessel Functions (2nd edn). Cambridge University Press: Cambridge.
Wynn P. 1956. On a device for computing the em(Sn) transformation.
Mathematical Tables and other AIDS to Computation 10: 91– 96.
Yang SY, Yeh HD. 2005. Laplace-domain solutions for radial two-zone flow equations under the conditions of constant-head and partially penetrating well. Journal of Hydraulic Engineering, ASCE 131(3): 213– 214. DOI:10.1061/(ASCE)0733-9429(2005)131:3(209): 209– 216.
Yang SY, Yeh HD. 2007. On the solutions of modeling slug test performed in a two-zone confined aquifer. Hydrogeology Journal 15(2): 301– 302. DOI: 10.1007/s10040-006-0100-x: 297.
Yang SY, Yeh HD, Chiu PY. 2006. A closed-form solution for constant-flux pumping test under the effect of well partial penetration. Water
Resources Research 42(5): W05502. DOI:10.1029/2004WR003889.
Yeh HD. 1987. Theis’ solution by nonlinear least-squares and finite-difference Newton’s method. Ground Water 25: 710– 715.
Yeh HD, Huang YC. 2005. Parameter estimation for leaky aquifers using the extended Kalman filter and considering model and data measurement uncertainties. Journal of Hydrology 302(1– 4): 28– 45. Yeh HD, Lin YC, Huang YC. 2007. Parameter identification for leaky
aquifers using global optimization methods. Hydrological Processes 21: 862– 872. DOI:10.1002/hyp.6274.
Yeh HD, Yang SY. 2006. A novel analytical solution for a slug test conducted in a well with a finite-thickness skin. Advances in Water
Resources 29(10): 1479– 1489. DOI:10.1016/j.advwatres.2005.11.002.
Yeh HD, Yang SY, Peng HY. 2003. A new closed-form solution for radial two-layer drawdown equation under constant-flux pumping in a finite-radius well. Advances in Water Resources 26(5): 747– 757. Zhan H, Bian A. 2006. A method of calculating pumping induced
leakage. Journal of Hydrology 328: 659– 667.
Zlotnik VA. 2004. A concept of maximum stream depletion rate for leaky aquifers in alluvial valleys. Water Resources Research 40: W06507. DOI:10.1029/2003WR002932.
