Wellbore flow-rate solution for a constant-head test in two-zone
finite confined aquifers
Chi-Shan Tsai and Hund-Der Yeh*
Institute of Environmental Engineering, National Chiao Tung University, Hsinchu, Taiwan
Abstract:
The solution describing the wellboreflow rate in a constant-head test integrated with an optimization approach is commonly used to analyze observed wellboreflow-rate data for estimating the hydrogeological parameters of low-permeability aquifers. To our knowledge, the wellboreflow-rate solution for the constant-head test in a two-zone finite-extent confined aquifer has never been reported so far in the literature. This article isfirst to develop a mathematical model for describing the head distribution in the two-zone aquifer. The Laplace domain solutions for the head distributions and wellboreflow rate in a two-zone finite confined aquifer are derived using the Laplace transform, and their corresponding time domain solutions are then obtained using the Bromwich integral method and residue theorem. These new solutions are expressed in terms of an infinite series with Bessel functions and not straightforward to calculate numerically. A large-time solution for the wellboreflow rate is therefore developed by employing the relationship of small Laplace variable versus large time variable and L’Hospital’s rule. The result shows that the large-time solution is identical to the steady-state solution obtained after applying the Tauberian theorem into the Laplace domain solution. This large-time solution can reduce to the Thiem equation in the case of no skin. Finally, the newly developed solution is used to investigate the effects of outer boundary distance and conductivity ratio on the wellboreflow rate. Copyright © 2011 John Wiley & Sons, Ltd.
KEY WORDS analytical solution; aquifer test; Laplace transform; Bromwich integral; residue theorem; composite aquifer; Thiem equation
Received 16 April 2010; Accepted 30 August 2011
INTRODUCTION
The observed data of wellboreflow rate from a constant-head test is commonly analyzed to determine the aquifer hydrogeological parameters of low-permeability aquifers. The analysis of wellboreflow rate data usually relies upon an approach which includes the analytical solution of the wellbore flow rate coupled with an optimization scheme (see, e.g. Yeh et al., 2007a,b; Yeh and Chen, 2007). A number of studies have been presented the wellbore flow-rate models for describing field constant-head tests (e.g. Mishra and Guyonnet, 1992; Markle et al., 1995; Chen and Chang, 2002). Interestingly, the transient solution for the wellbore flow rate expressed in slightly different formats also appears in a variety of disciplines such as heat transfer (Carslaw and Jaeger, 1959) and electrochemistry (e.g. Aoki et al., 1985; Szabo et al., 1987; Fang et al., 2009; Britz et al. 2010; Bieniasz, 2011). Based on the heat conduction solution of Smith (1937), Jacob and Lohman (1952) presented an analytical expression for the wellbore flow rate to a constant-head test in an infinite aquifer. Moreover, the transmissivity and storativity were determined by plotting the ratio offlow rate to the constant drawdown at the test well against time to squared well radius. Later, Carslaw and Jaeger provided a formula for the heat flux
across the inner boundary in a radial heat conduction problem (1959, p. 336, Equation (8)) which can also be used to describe the groundwaterflow problems of the constant-head test. Van Everdingen and Hurst (1949) developed the transient pressure head and wellboreflow rate solutions for the constant-flux and constant-head tests in finite and infinite confined aquifers without considering the skin zone. For the well surrounded by a skin, Uraiet and Raghavan (1980) examined the transient flow rate at a well and pressure behavior infinite and infinite aquifers when the test well maintains at a constant drawdown. They demonstrated the concepts of the infinitesimally thin skin and effective wellbore radius are applicable to describe the skin region around a well producing at constant pressure. In addition, they also plotted the simulation results obtained from afinite difference model to demonstrate the effects of skin thickness and permeability ratio between the skin and formation zones on the wellboreflow rate. Yang and Yeh (2002) provided an analytical solution of the wellboreflow rate for the constant-head test performed in an infinite-extent aquifer with considering the effects of the finite well radius and skin zone. They further developed the transient analytical solution of the hydraulic head distributions in the patch and outer regions for the constant-head test in a patchy aquifer of infinite extent (Yang and Yeh, 2006). To our knowledge, the transient analytical solution for the head distribution or the wellboreflow rate in a two-zone finite-extent aquifer has never been reported so far in the literature.
*Correspondence to: Hund-Der Yeh, Institute of Environmental Engin-eering, National Chiao Tung University, Hsinchu, Taiwan.
E-mail: [email protected]
Published online 10 January 2012 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/hyp.8322
The objective of this article is first to develop a mathematical model to describe the groundwater flow in the skin and formation zones for the constant-head test in two-zone finite confined aquifers. The head solution of the model is obtained by applying the Laplace transform technique, Bromwich integral method, and residue theorem. The wellbore flow rate solution is obtained by first applying Darcy’s law to the Laplace-domain head solution in the skin zone and then transforming the result to the time domain using the Bromwich integral method and residue theorem. In addition, a large-time solution for the wellboreflow rate is also developed by employing the relationship of small Laplace-domain variable p with a large time-domain variable t and L’Hospital’s rule. The result of large-time wellbore flow happens to be the steady-state solution which can reduce to the Thiem equation if neglecting the skin effect. Finally, the dimensionless forms of the solutions for hydraulic head and well bore flow rate are also presented for practical uses or engineering applications. The wellbore flow-rate solution is also used to study the effects of boundary distance and conductivity ratio on the estimatedflow rate for confined aquifers of finite extent.
The solutions for the head distribution and wellboreflow rate developed herein are mainly for a two-zone aquifer which can also be called as patchy aquifer (Baker and Herbert, 1988), nonuniform aquifer (Butler, 1988), com-posite formation (Novakowski, 1989), or aquifer with a skin zone (Yang and Yeh, 2002). It is well recognized that the groundwaterflow is analogous to the heat flow. Therefore, the solutions developed in this paper can naturally be regarded as an extension of the work in Carslaw and Jaeger (1959, p. 333, Equation (10) for the case that k1 ¼ k’1= 0) for
heatflow in a composite hollow cylinder. MATHEMATICAL DEVELOPMENT Mathematical model for a two-zone confined aquifer
The assumptions for the mathematical model describing the head distribution for the constant-head test in a two-zone
confined aquifer are: (1) the test well is of a finite radius and fully penetrates the aquifer thickness; (2) the well has a finite-thickness skin zone with different hydrogeological properties from the formation zone; (3) the aquifer is homogeneous in each zone and bounded by afinite outer boundary in the formation zone; and (4) the water level in the test well is maintained constant.
Figure 1 shows the schematic diagram of the two-zone aquifer. The governing equations describing the hydraulic head h(r, t) for the skin zone and formation zone are (Yang and Yeh, 2002), respectively,
@2h 1 @r2 þ 1 r @h1 @r ¼ S1 T1 @h1 @t rw< r < r1 (1) and @2h 2 @r2 þ 1 r @h2 @r ¼ S2 T2 @h2 @t r1< r < R (2)
where subscripts 1 and 2 denote the skin and formation zones, respectively, the variable r is the radial distance from the central line of the test well, rwis the well radius, r1is the
radial distance from the central line to the outer boundary of the skin zone, R is the distance from the central line to the outer boundary of the formation zone, t is the time, S is the storage coefficient, and T is the transmissivity.
The initial hydraulic head of the aquifer is considered to be zero. The initial head conditions for Equations (1) and (2) are therefore
h1ðr; 0Þ ¼ h2ðr; 0Þ ¼ 0 (3)
The hydraulic head is assumed to equal zero at the outer boundary. On the other hand, a constant well water level (or hydraulic head) hwis maintained at the wellbore.
Thus, the hydraulic heads at the outer and inner boundaries are given, respectively, as
h2ðR; tÞ ¼ 0 (4)
and
h1ðrw; tÞ ¼ hw (5)
The continuity conditions for the hydraulic head and flow rate at the interface between the skin and formation zones require, respectively,
h1ðr1; tÞ ¼ h2ðr1; tÞ (6) and T1@h 1ðr1; tÞ @r ¼ T2@h 2ðr1; tÞ @r (7)
The Laplace-domain solutions for head distribution and wellboreflow rate
The Laplace-domain solution for head distributions in the skin and formation zones obtained by applying the Laplace Transform to Equations (1) and (2) with Equations (3–7) are
h1¼ 1 p hwðΦ1I0ðq1rÞ Φ2K0ðq1rÞÞ Φ1I0ðq1rwÞ Φ2K0ðq1rwÞ (8) and h2¼ hw p Φ1I0ðq1r1Þ Φ2K0ðq1r1Þ Φ1I0ðq1rwÞ Φ2K0ðq1rwÞ ½ I0ðq2RÞK0ðq2rÞ K0ðq2RÞI0ðq2rÞ I0ðq2RÞK0ðq2r1Þ I0ðq2r1ÞK0ðq2RÞ (9)
where p is the Laplace variable, q1¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pS1=T1 p , q2¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pS2=T2 p
, I0and K0are the modified Bessel functions of
thefirst and second kinds of order zero, respectively, and I1
and K1 are the modified Bessel functions of the first and
second kinds of orderfirst, respectively. The variables Φ1
andΦ2in Equations (8) and (9) are defined as
Φ1 ¼ f ffiffiffiffiffiffiffiffiffi S2T2 S1T1 r K0ðq1r1ÞK0ðq2r1ÞK1ðq1r1ÞK0ðq2r1Þ (10) and Φ2¼ f ffiffiffiffiffiffiffiffiffi S2T2 S1T1 r I0ðq1r1ÞK0ðq2r1ÞþI1ðq1r1ÞK0ðq2r1Þ (11) with f ¼I0ðq2RÞK1ðq2r1Þ þ I1ðq2r1ÞK0ðq2RÞ I0ðq2RÞK0ðq2r1Þ I0ðq2r1ÞK0ðq2RÞ (12)
Applying Darcy’s law to Equation (8) and setting r = rw,
the Laplace-domain solution for the wellboreflow rate can then be obtained as Q rð Þ ¼ 2prw wT1 q1hw p Φ1I1ðq1rwÞ þ Φ2K1ðq1rwÞ Φ2K0ðq1rwÞ Φ1I0ðq1rwÞ (13) If the outer boundary distance approaches infinity, i.e. R! 1, Equation (12) becomes f =K1(q2r1)/K0(q2r1).
Equations (10) and (11) can then reduce to the correspond-ing equations given in Yang and Yeh (2002, Equations (10) and (11)). Equations (8) and (9) for the head distribution therefore reduce to the solutions represented in Yang and Yeh (2002, Equations (8) and (9)) for the infinite aquifer. In addition, Equation (13) for the wellboreflow rate reduces to the one given in Yang and Yeh (2002, Equation (12)) for the infinite aquifer.
Time-domain solution
The time-domain solution for the head distributions in the skin and formation zones can be obtained by applying the Bromwich integral method and residue theorem to Equations (8) and (9). The detailed derivation is shown in Appendix A, and the results are
h1ðr; tÞ ¼ hw lnðr1=rÞ þ T1=T2ln Rð =r1Þ lnðr1=rwÞ þ T1=T2ln Rð =r1Þ þ pX1 n¼1 J0ðanrwÞY0ðanrÞ Y0ðanrwÞJ0ðanrÞ ½ exp -a2 ntT1=S1 xn2 ðzAnÞ2þ B2nþ z Bðð nCnþ AnDnÞ þ AnBn=anÞ=r1 h i 1 8 < : 9 = ; (14) h2ð Þ ¼ hr; t w ( T1=T2ln Rð =rÞ lnðr1=rwÞ þ T1=T2ln Rð =r1Þ þpX1 n¼1 J0ðanrwÞY0ðanr1Þ Y0ðanrwÞJ0ðanr1Þ ½ Y½ 0ðxanRÞJ0ðxanrÞ Y0ðxanrÞJ0ðxanRÞexp -a2ntT1=S1 xn2 ðzAnÞ2þ B2nþ z Bðð nCnþ AnDnÞ þ AnBn=anÞ=r1 h i 1 n o Bð nÞ ) (15)
These two solutions are significantly different from the solutions presented in Yang and Yeh (2002, Equations (12) and (13)) which are in terms of integrals with lower and upper limits from zone to infinity for the two-zone aquifer of infinite extent. In addition, Equations (14) and (15) are developed by the Bromwich integral along a contour of infinite poles and residue theorem, while the solutions of Yang and Yeh (2002) were obtained based on the Bromwich integral with a single branch point in the integrand. The transient solution for the wellbore flow rate can also be developed in a similar manner to the one presented in Appendix A when transforming Equation (13) into the time domain. The result is
with Bn ¼ J0ðanrwÞ zAnJ0ðanr1Þ BnJ1ðanr1Þ (17) An¼ J1ðxanr1ÞY0ðxanRÞ J0ðxanRÞY1ðxanr1Þ (18) Bn¼ J0ðxanRÞY0ðxanr1Þ J0ðxanr1ÞY0ðxanRÞ (19) Cn ¼ xR½J1ðxanr1ÞY1ðxanRÞ J1ðxanRÞY1ðxanr1Þ xr1Bn An an (20) Dn¼ xR J½ 1ðxanRÞY0ðxanr1Þ J0ðxanr1ÞY1ðxanRÞ xr1An (21)
where J0and Y0are the Bessel functions of thefirst and
second kinds of order zero, respectively, J1and Y1are the
Bessel functions of the first and second kinds of order first, respectively, x= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT1S2=T2S1
p
, z= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS2T2=S1T1
p
, and an
are the roots of
J1ðxar1ÞY0ðxaRÞ J0ðxaRÞY1ðxar1Þ
½
z Y½ 0ðarwÞJ0ðar1Þ Y0ðar1ÞJ0ðarwÞ
þ J½ 0ðxar1ÞY0ðxaRÞ J0ðxaRÞY0ðxar1Þ
Y½ 1ðar1ÞJ0ðarwÞ J1ðar1ÞY0ðarwÞ ¼ 0
(22)
By neglecting the presence of skin zone, Equation (16) reduces to
where an are the roots of J0(arw)Y0(aR) Y0(arw)J0
(aR) = 0. Equation (23) is exactly the same as the formula represented in Wang and Yeh (2008, Equation (5)). The large-time wellboreflow-rate solution in a finite confined aquifer
The approximations of I0(x) ~ 1/Γ(1), I1(x) ~ x/2Γ(2), K0
(x) ~ ln(x), and K1(x) ~ 1/x can be made when the
arguments of Bessel functions are small (Abramowitz and Stegun, 1979, p. 375). The Laplace domain solution for the large-time wellboreflow rate can therefore be obtained from Equation (13) after employing the relationship of small
p versus large t (Yeh and Wang, 2007) and L’Hospital’s rule as Q rð w; pÞ ¼ 2pT1hw p 1 ln rw r1 þT1 T2ln R r1 (24) where the negative sign in Equation (24) represents withdrawal in the test well.
The large-time solution for the wellbore flow rate is then obtained after taking the inverse Laplace transform of Equation (24) as Q rð w; tÞ ¼ 2pT1hw 1 ln rw r1 þT1 T2ln R r1 (25) which is independent of time and indeed a steady-state solution. In fact, this solution can also be obtained if applying the Tauberian theorem (Yeh and Wang, 2007) to Equation (13). This result indicates that the wellbore flow-rate solution for a two-zone confined aquifer of finite extent can reach steady state when the time is large. In addition, Equation (25) can be simplified to the Thiem equation if neglecting the skin zone, i.e. setting r1equals rw.
Dimensionless solutions
The dimensionless variables defined for simplifying the developed solutions are k = T2/T1, g = S2/S1, t¼ T2t=S2r2w,
r = r/rw, r1= r1/rw, rR= R/rw, hD= h/hw and QD= Q(rw)/
(2pT2hw). The variable k represents the conductivity ratio,
g represents the ratio of storage coefficients of the skin and formation zones, r represents dimensionless distance, r1
Q rð w; tÞ ¼ 2phwT1 1 lnðr1=rwÞ þ T1=T2ln Rð =r1Þ þ 2X1 n¼1 exp -a2 ntT1=S1 xn2 ðzAnÞ2þ B2nþ z Bðð nCnþ AnDnÞ þ AnBn=anÞ=r1 h i 1 8 < : 9 = ; (16) Q rðw; tÞ ¼ 2phwT 1 lnðR=rwÞ þ 2X1 n¼1 exp -Ta2nt=S J2 0ðanrwÞ J02ðanRÞ =J2 0ðanRÞ ( ) (23)
represents dimensionless distance of the outer boundary of the skin zone, hD represents the dimensionless head
distribution in the Laplace domain, hD represents the
dimensionless head distribution in time domain, QD represents the dimensionlessflow rate in Laplace domain, and QD represents the dimensionless flow rate in time
domain.
The time domain solution for the dimensionless head distributions in the skin and formation zones can then be written as
In addition, the time-domain solution of the dimen-sionless wellbore flow rate is
where bn= rwan are the roots of
½J1ðxbr1ÞY0ðxbrRÞ J0ðxbrRÞY1ðxbr1Þ z Y½ 0ðbrwÞJ0ðbr1Þ Y0ðbr1ÞJ0ðbrwÞ þ J½ 0ðxbr1ÞY0ðxbrRÞ J0ðxbrRÞY0ðxbr1Þ Y½ 1ðbr1ÞJ0ðbrwÞ J1ðbr1ÞY0ðbrwÞ ¼ 0 (29) with BDn¼ J0ð Þbn zanJ0ðbnr1Þ bnJ1ðbnr1Þ (30) an ¼ J1ðxbnr1ÞY0ðxbnrRÞ J0ðxbnrRÞY1ðxbnr1Þ (31) bn ¼ J0ðxbnrRÞY0ðxbnr1Þ J0ðxbnr1ÞY0ðxbnrRÞ (32) cn¼ krR½J1ðxbnr1ÞY1ðxbnrRÞ J1ðxbnrRÞY1ðxbnr1Þ xr1bn an bn (33) dn ¼ xrR½J1ðxbnrRÞY0ðxbnr1Þ J0ðxbnr1ÞY1ðxbnrRÞ xr1an (34)
The numerical evaluations of Equation (28) can be achieved by finding the roots of Equation (29) first by
Newton’s method and then adding the summation term for n up to 100. The accuracy of the results can be made at least to thefifth decimal.
ADVANTAGES AND APPLICATIONS OF THE SOLUTIONS
Advantages over the existing solutions
To our knowledge, there are only two articles, i.e. Yang and Yeh (2002) and the present one, in the groundwater literature to provide the transient analytical solutions (in time domain) of the wellbore flow rate for the constant-head test in two-zone aquifer systems. The present article has following three advantages over Yang and Yeh (2002). First, the present solutions can reduce to those given in Yang and Yeh (2002) when the outer boundary goes infinity. In other words, the solutions presented in Yang and Yeh (2002) can be considered as a special case of the present solutions. Second, the solution in Yang and h1D¼ ( lnðr1=rÞ þ 1=k ln rð R=r1Þ lnðr1Þ þ 1=k ln rð R=r1Þ þ pX1 n¼1 J0ð ÞYbn 0ðbnrÞ Y0ð ÞJbn 0ðbnrÞ ½ exp -gb2 nt=k xDn2 ðzanÞ 2þ b2 nþ z bðð ncnþ andnÞ þ anbn=bnÞ=r1 h i 1 ) (26) h2D¼ ( 1=k ln rð R=rÞ lnðr1Þ þ 1=k ln rð R=r1Þ þpX1 n¼1 J0ð ÞYbn 0ðbnrÞ Y0ð ÞJbn 0ðbnrÞ ½ Y½ 0ðxbnrRÞJ0ðxbnrÞ Y0ðxbnrÞJ0ðxbnrRÞexp -gb 2 nt=k xDn2 ðzanÞ 2þ b2 nþ z bðð ncnþ andnÞ þ anbn=bnÞ=r1 h i 1 n o bð nÞ ) (27) QD¼ 1 k 1 lnðr1Þ þ 1=k ln rð R=r1Þ þ 2X1 n¼1 exp -gb 2nt=k xDn2 ðzanÞ2þ b2nþ z bðð ncnþ andnÞ þ anbn=bnÞ=r1 h i 1 8 < : 9 = ; (28)
Yeh (2002) for the head distributions in the skin and formation zones is only in Laplace domain while that in the present article is in time domain. Finally and most importantly, the wellboreflow-rate solution in Yang and Yeh (2002, Equation (19)) is in terms of an integral from zero to infinity with the variable u in the denominator of the integrand, posing the problem of singularity at the origin for the integration as indicated in Yang and Yeh (2002, p. 178, Figure 2). Due to the presence of singular point, the results of numerical evaluation for the integral are accurate only to the second decimal as shown in Yang and Yeh (2002, p. 179, Tables 1 and 2). On the other hand, the present wellboreflow solution is composed of infinite series and can be easily evaluated with accuracy to at leastfive digits after the decimal.
Potential applications
The properties of an aquifer with the presence of skin zone may be characterized byfive parameters, i.e. the outer radius of the skin zone and the transmissivity and storage coefficient for each of the skin and aquifer zones. If those parameters are known, the presented solution can be used to predict the wellboreflow rate and head distributions in both the skin and formation zones and explore the physical insight of the constant-head test in a two-zone aquifer system. On the other hand, if the parameters are not available, the determination of thosefive parameters from analyzing measured data is a subject of inverse problem. It may not be possible or too complicate to develop type curves for parameter estimation because the unknowns are too many. Feasible ways of solving such an inverse problem of involving five unknown parameters are to adopt the presented solution and couple it with the algorithm of extended Kalman filter (e.g. Leng and Yeh 2003; Yeh and Huang 2005) or with an optimization
approach such as the nonlinear least-squares (e.g. Yeh, 1987) or simulated annealing (e.g. Lin and Yeh, 2005; Yeh et al., 2007a,b).
The present solution can also be used as a tool to design a field constant-head or to verify newly developed numerical codes for simulating theflow in two-zone aquifer systems. Generally speaking, the sensitivity analysis (Liou and Yeh, 1997) can be performed to investigate the effect of changing input parameters (i.e. thefive parameters) on the output (i.e. head distribution or wellbore flow rate). It works as an indicator in assessing the influences of parameter uncer-tainty on the predicted head or wellboreflow rate. If the head (or wellbore flow rate) is very sensitive to a specific parameter, a small change in that parameter will then markedly affect the predicted head (or wellboreflow rate). In contrast, the change in a less sensitive parameter will have little impact on the predicted result. This indicates that a less sensitive parameter is much more difficult to be estimated. With the present solution, the sensitivity analysis can be easily performed to the targeted parameters (e.g. Huang and Yeh, 2007), and the results will provide useful information about the degrees of sensitivity among targeted parameters.
RESULTS AND DISCUSSION
Figure 2 shows the curves of the dimensionless wellbore flow rate versus dimensionless time t for various conductivity ratios k and outer boundary distance r with dimensionless radial distance r1= 5. Note that k< 1
denotes for the negative, skin case while k > 1 for the positive skin case. The figure indicates that the dimen-sionless wellboreflow rate for an aquifer with a positive skin is always smaller than that with a negative one. For small values of k, the wellbore flow-rate solution for an aquifer offinite extent is equal to that of infinite extent
Figure 2. The curves of the dimensionless wellboreflow rate versus dimensionless pumping time for various conductivity ratio k and outer boundary distance rRwith r1 = 3. The solid line represents the solution for the infinite-extent aquifer, while the dash line with the symbols represents the solution
only at early test time (say, roughly t< 40 when k = 0.1 and t< 100 when k = 1) indicating that wellbore flow rate will reach steady state quickly for aquifers with negative skins. In the period of moderate time (100< t < 1000), these twoflow-rate solutions deviate from one another, indicating that thefinite-extent solution can no longer be used to approximate to the infinite-extent solution. In other words, the outer boundary distance has an effect on the wellbore flow rate when the pumping time is not short. Finally, thefinite-extent solution tends to reach an asymptotic limit, the steady-state solution, revealing the significance of boundary effect on the wellbore flow rate while the infinite-extent solution decreases endlessly with dimensionless time. The effect of boundary distance on the wellbore flow rate is very small for a high value of k (say, 10). Figure 2 also indicates that the dimensionless time when the outer boundary starts to affect the wellbore flow rate increases with the dimensionless outer boundary distance rR.
CONCLUSIONS
A mathematical model for describing the head distributions in a two-zone confined aquifer bounded by a finite outer boundary for a constant-head test has been presented. The Laplace-domain solution for the head distributions in the skin and formation zones arefirst developed using the Laplace transform, and the Laplace-domain solution for the wellbore flow rate is then developed based on the head solution of the skin zone and Darcy’s law. Both the Laplace-domain solutions for the head distribution and wellboreflow rate can reduce to the solutions for the two-zone aquifer with an infinite boundary when the outer boundary distance approaches infinity. The transient solutions of the head distribution and wellboreflow rate are then developed from their Laplace-domain solutions by the Bromwich integral method and residue theorem. Finally, the relationship of small Laplace variable versus large time variable and L’Hospital’s rule are used to develop the large-time wellbore flow rate which turns out to be the steady-state solution. This result indicates that the wellboreflow rate can reach steady state quickly for a finite-domain aquifer. In addition, this steady-state result reduces to the Thiem equation if the skin effect is negligible.
The wellboreflow-rate solution for the constant-head test is used to investigate the effects of conductivity ratio and outer boundary distance on theflow rate across the wellbore in a two-zone confined aquifer bounded by a finite outer boundary. The result indicates that the dimensionlessflow rate for an aquifer with a positive skin is always smaller than that with a negative skin. For small conductivity ratios, the flow-rate solution for an aquifer of finite extent equals that of infinite extent only at early test time, indicating that wellbore flow rate can reach steady state quickly for aquifers with the negative skin. The wellboreflow-rate solution for a finite aquifer tends to an asymptotic limit at large time while that solution for an infinite aquifer decreases endlessly with dimensionless time. The effect of outer boundary distance
on the wellbore flow rate is small for an aquifer of high conductivity ratio and the dimensionless time when the outer boundary begins to influence the wellbore flow rate increases with the dimensionless boundary distance.
ACKNOWLEDGEMENTS
Research leading to this work has been partially supported by the grants from Taiwan National Science Council under the contract numbers NSC 99-2221-E-009-062-MY3, NSC 100-2221-E-009-106, and NSC100-3113-E-007-011. The authors would like to thank two anonymous reviewers for their valuable and constructive comments that help improve the clarity of our presentation.
REFERENCES
Abramowitz M, Stegun IA. 1979. Handbook of Mathematical Functions. Dover Publications: New York.
Aoki K, Honda K, Tokuda K, Matsuda H. 1985. Voltammetry at microcylinder electrodes: Linear sweep voltammetry. Journal of Electroanalytical Chemistry 182(2): 267–279.
Baker JA, Herbert R. 1988. Pumping test in patchy aquifer. Ground Water 20(2): 150–155.
Bieniasz LK. 2011. A highly accurate, inexpensive procedure for computing theoretical chronoamperometric current at cylindrical wire electrodes. Electrochimica Acta 56: 6982–6988.
Britz D, Osterby O, Strutwolf J. 2010. Reference values of the chronoamperometric response at cylindrical and capped cylindrical electrodes. Electrochimica Acta 55: 5629–5635.
Butler JJ Jr. 1988. Pumping tests in nonuniform aquifers-The radially symmetric case. Journal of Hydrology 101: 15–30.
Carslaw HS, Jaeger JC. 1959. Conduction of Heat in Solids, 2nd edn. Oxford University Press: London.
Chen CS, Chang CC. 2002. Use of cumulative volume of constant-head injection test to estimate aquifer parameters with skin effects, Field experiment and data analysis. Water Resources Research 38(5): 1056.
Fang YM, Sun JJ, Chen, GN. 2009. A simple approach to solution of the diffusion equation at the microcylinder electrode-an inspiration from the film projector. ChemPhysChem 10: 2393–2396.
Huang YC, Yeh HD. 2007. The use of sensitivity analysis in on-line aquifer parameter estimation. Journal of Hydrology 335: 406–418. Jacob CE, Lohman SW. 1952. Nonsteady flow to a well of constant
drawdown in an extensive aquifer. Transactions-American Geophysical Union 33(4): 559–569.
Kreyszig E. 1999. Advanced engineering mathematics. John Wiley & Sons Inc: New York.
Leng CH, Yeh HD. 2003. Aquifer parameter identification using the extended Kalmanfilter. Water Resources Research 39(3): 1062. Lin YC, Yeh HD. 2005. Trihalomethane species forecast using
optimization method: genetic algorithm and simulated annealing. Journal of Computing in Civil Engineering 19(3): 248–257. Liou TS, Yeh HD. 1997. Conditional expectation for evaluation of risk
groundwater flow and solute transport: one-dimensional analysis. Journal of Hydrology 199: 378–402.
Markle JM, Rowe RK, Novakowski KS. 1995. A model for the constant-head pumping test conducted in vertically fractured media. International Journal for Numerical and Analytical Methods in Geomechanics 19: 457–473.
Mishra S, Guyonnet D. 1992. Analysis of observation-well response during constant-head testing. Ground Water 30(4): 523–528. Novakowski KS. 1989. A Composite Analytical Model for Analysis of
Pumping Tests Affected by Well Bore Storage and Finite Thickness Skin. Water Resources Research 25(9): 1937–1946.
Smith LP. 1937. Heatflow in an infinite solid bounded internally by a cylinder. Journal of Applied Physics 8(6): 441–448.
Szabo A, Cope DK, Tallman DE, Kovach PM, Wightman RM. 1987. Chronoamperometric current at hemicylinder and band
microelectrodes-theory and experiment. Journal of Electroanalytical Chemistry 217: 417–423.
Uraiet AA, Raghavan R. 1980. Unsteadyflow to a well producing at a constant pressure. Journal of Petroleum Technology 32(10): 1803–1812. Van Everdingen AF, Hurst W. 1949. The application of the Laplace transformation toflow problems in reservoirs. Transactions. American Institute of Mining, Metallurgical and Petroleum Engineers 186: 305–324.
Wang CT, Yeh HD. 2008. Obtaining the steady-state drawdown solutions of constant-head and constant-flux tests. Hydrological Processes 22 (17): 3456–3461.
Yang SY, Yeh HD. 2002. Solution forflow rates across the wellbore in a two-zone confined aquifer. Journal of Hydraulic Engineering 128(2): 175–183.
Yang SY, Yeh HD. 2006. A novel analytical solution for constant-head test in a patchy aquifer. International Journal for Numerical and Analytical Methods in Geomechanics 30(12): 1213–1230.
Yeh HD. 1987. Theis’ solution by nonlinear least-squares and finite-difference Newton’s method. Ground Water 25: 710–715.
Yeh HD, Chen YJ. 2007. Determination of skin and aquifer parameters for a slug test with wellbore-skin effect. Journal of Hydrology 342: 283–294. Yeh HD, Huang YC. 2005. Parameter estimation for leaky aquifers using the extended Kalmanfilter and considering model and data measure-ment uncertainties. Journal of Hydrology 302(1–4): 28–45.
Yeh HD, Wang CT. 2007. Large-time solutions for groundwater flow problems using the relationship of small p versus large t. Water Resources Research 43(6): W06502 DOI: 10.1029/2006WR005472 Yeh HD, Chang TH, Lin YC. 2007a. Groundwater contaminant source
identification by a hybrid heuristic approach. Water Resources Research 43: W09420. DOI: 10.1029/2005WR004731
Yeh HD, Lin YC, Huang YC. 2007b. Parameter identification for leaky aquifers using global optimization methods. Hydrological Processes 21: 862–872.
APPENDIX: DERIVATION OF EQUATION (14)
The head distributions in time domain, h1, can be obtained
by Bromwich integral method (Carslaw and Jaeger, 1959) as h1ð Þ ¼ Lt 1½h1ð Þp ¼ 1 2pi Z reþi1 rei1 epth1ð Þdpp (A1)
where i is an imaginary unit and re is a very large real
constant that all of the real parts of the poles are smaller than it. The graph of the Bromwich integral contains a close contour with a straight line parallel to the imaginary
axis and a semicircle. According to Jordan’s Lemma, the value of the integration for the semicircle tends to zero when its radius approaches infinity. Based on the residue theorem, the head distribution in the skin zone (Equation (A1)) can be expressed as
h1ð Þ ¼t
X1 n¼1
Re s e½ pth1ð Þ; gp n (A2)
where gnare the poles in the complex plane.
There are infinite poles in h1ð Þ and obviously one polep
at p = 0. To determine other poles, the denominator and numerator in the brackets of Equation (8) are written, respectively, as
Δ ¼ Φ1I0ðq1rwÞ Φ2K0ðq1rwÞ (A3)
Ψ ¼ hw½Φ1I0ðq1rÞ Φ2K0ðq1rÞ (A4)
LetΔ = 0, the roots anin p¼ pn¼ T1a2n
=S1can be
determined from Equation (A3) with q1= ian and
q2¼ xa2n. Substituting pn¼ T1a2n
=S1 into Equation
(A3) yields Equation (22). The residue of the pole at p = 0 can then be obtained from the following formula (Kreyszig, 1999)
Res e½ pth1ð Þ; 0p ¼ lim p!0h1ð Þep
ptðp 0Þ (A5)
Substituting Equation (8) into Equation (A5) and applying L’Hospital’s rule, the result is
Res e½ pth1ð Þ; 0p ¼ hw
lnðr1=rÞ þ T1=T2ln Rð =r1Þ
lnðr1=rwÞ þ T1=T2ln Rð =r1Þ
(A6)
The other residues at the poles pn¼ T1a2n=S1 can be
written as
Res e½ pth1ð Þ; pp n ¼ limp!p
n
h1ð Þep ptðp pnÞ (A7)
Applying L’Hospital’s rule to Equation (A7), the denominator of Equation (8) becomes
where the variables Φ1 and Φ2 are defined in Equations
(10) and (11), respectively, and Φ’1 and Φ’2 are the differentiations of Φ1andΦ2, respectively.
A variable Bnintroduced based on Equation (A3) and
Δ = 0 to simplify Equation (A8) is defined as pdΔ dpp¼T1a2 n=S1 ¼ 1 2q dΔ dq q1¼ian;q2¼ikan ¼1 2q1 Φ ’ 1I0ðq1rwÞ Φ2’K0ðq1rwÞ þ rw½Φ1I1ðq1rwÞ þ Φ2K1ðq1rwÞ " (A8) Bn¼ I0ðq1rwÞK0ðxq1r1Þ Φ2½ðI0ðxq1RÞK0ðxq1r1Þ I0ðxq1r1ÞK0ðxq1RÞ (A9)
¼
K0ðq1rwÞK0ðxq1r1Þ Φ1½ðI0ðxq1RÞK1ðxq1r1ÞþI1ðxq1r1ÞK0ðxq1RÞIn addition, following two recurrence formulas (Carslaw and Jaeger, 1959, p. 490) are used to eliminate the imaginary unit in Equation (8):
Kv ze 1 2pi ¼ 1 2pie 1 2vpiJ vð Þ iYz vð Þz ½ (A10) and Iv ze 1 2pi ¼ e1 2vpiJ vð Þz (A11)
Substituting Equations (A10) and (A11) into Equation (A9) results in J0ðanrwÞ zAnJ0ðanr1Þ BnJ1ðanr1Þ ¼ Y0ðanrwÞ zAnY0ðanr1Þ þ BnY1ðanr1Þ ¼ Bn (A12)
With Equations (A9) and (A12), Equation (A8) is given as
where the constants shown on the right-hand side of Equation (A13) have been defined in Equations (17–21). Similarly, the numerator of Equation (8) can also be obtained as
Ψ ¼ 1
2Bnfphw½Y0ðanrwÞJ0ðanrÞ þ J0ðanrwÞY0ðanrÞg (A14)
The residues at the poles pn¼ T1a2n=S1 can be
obtained from Equations (A13) and (A14) as
Therefore, Equation (A2) can be expressed as
h tð Þ ¼ Res e½ pth1ð Þ; 0p þ Res e½ pth1ð Þ; pp n (A16)
Finally, the head distribution in the skin zone can be obtained from Equations (A6) and (A15) and given as Equation (14). pdΔ dp p¼T1a2n=S1 ¼ 1 2q dΔ dq q1¼ian;q2¼ikan ¼ 1 2Bn Bn2 ðzAnÞ 2þ B2 nþ z Bðð nCnþ AnDnÞ þ AnBn=anÞ=r1 h i 1 n o (A13) Res e½ pth1ð Þ; pp n ¼ hwp X1 n¼1 J0ðanrwÞY0ðanrÞ Y0ðanrwÞJ0ðanrÞ ½ exp -a2 ntT1=S1 xn2 ðzAnÞ2þ B2nþ z Bðð nCnþ AnDnÞ þ AnBn=anÞ=r1 h i 1g (A15)
