New Solutions to the Constant-Head Test Performed at a Partially Penetrating Well

Journal of Hydrology 369 (2009) 90–97

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Journal of Hydrology

journal homepage: www.elsevier.com/locate/jhydrol

New solutions to the constant-head test performed at a partially penetrating well

Y.C. Chang, H.D. Yeh *

Institute of Environmental Engineering, National Chiao-Tung University, No. 1001, University Road, Hsinchu, Taiwan article info summary

Article history: The mathematical model describing the aquifer response to a constant-head test performed at a fully Received 18 February 2008 penetrating well can be easily solved by the conventional integral transform technique. In addition, Received in revised form 16 October 2008 the Dirichlet-type condition should be chosen as the boundary condition along the rim of wellbore for Accepted 8 February 2009 such a test well. However, the boundary condition for a test well with partial penetration must be considered as a mixed-type condition. Generally, the Dirichlet condition is prescribed along the well screen This manuscript was handled by P. Baveye, and the Neumann type no-flow condition is specified over the unscreened part of the test well. The model Editor-in-Chief, with the assistance of for such a mixed boundary problem in a confined aquifer system of infinite radial extent and finite ver- Chunmiao Zheng, Associate Editor tical extent is solved by the dual series equations and perturbation method. This approach provides analytical results for the drawdown in the partially penetrating well and the well discharge along the screen. Keywords: The semi-analytical solutions are particularly useful for the practical applications from the computational Semi-analytical solution point of view. Groundwater Ó 2009 Elsevier B.V. All rights reserved. Dual series equations Perturbation method Confined aquifer Mixed boundary value problem

Introduction Neuman boundaries (Robin boundary). The well skin is usually of a finite thickness and thus should be considered as a different for- The constant-head test is generally performed in a confined mation zone (see, e.g., Yang and Yeh, 2002; Yeh et al., 2003; Yeh aquifer to yield field scale characteristic parameters including and Yang, 2006a) instead of neglecting its thickness and using a hydraulic conductivity, the specific storativity and possibly the factor to represent its effect. leakage factor. These parameters can be used to quantify ground- Many physical problems can be described by partial differential water water resources (Cassiani and Kabala, 1998). The analysis equations with various types of initial and boundary conditions. At of data obtained from the test is used to determine the aquifer present time, the analytical solutions to the mixed-type boundary hydrogeological parameters. The test well may be fully or partially value problems in well hydraulics are very limited. The techniques penetrated and the wellbore may have well skin. The fully pene- used to solve the mixed-type boundary value problems analytically trating well can be considered as a Dirichlet (also called first type) include the dual integral/series equation (Sneddon, 1966), Weiner- boundary condition in the constant-head test, and the resulting Hopf technique (Noble, 1958), and Green’s function (Hung and model can be solved by the conventional integral transform tech- Chang, 1984). However, most of solutions to the mixed-type niques (Hantush, 1964). If the effect of well skin is negligible, the boundary value problems are obtained numerically (Yedder and Dirichlet boundary condition is suitable to describe the drawdown Bilgen, 1994) or by approximate methods such as asymptotic anal- (or well water level) along the well screen and the Neuman (also ysis (Bassani et al., 1987) or perturbation techniques (Wilkinson called second type) boundary condition of zero flux is specified and Hammond, 1990). along the casing for a partially penetrating well. Thus, the bound- For the mathematical model subject to the mixed-type boundary condition specified for the partially penetrating well is a ary condition in a confined aquifer of semi-infinite thickness, Wil- mixed-type condition. The term ‘‘mixed-type” boundary is used kinson and Hammond (1990) used the perturbation method to give to distinguish this boundary condition from the ‘‘uniform” condi- an approximate solution for drawdown changes at the well. Cassi- tions of Dirichlet and Neuman or a combination of Dirichlet and ani and Kabala (1998) used the dual integral equation method to derive the Laplace-domain solutions for the constant rate pumping

* Corresponding author. Fax: +886 3 5725958. test and slug test performed at partially penetrating wells that ac- E-mail address: [email protected] (H.D. Yeh). count not only for wellbore storage, inﬁnitesimal skin, and aquifer

0022-1694/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2009.02.016 Y.C. Chang, H.D. Yeh / Journal of Hydrology 369 (2009) 90–97 91 anisotropy, but also for the mixed-type boundary condition. Cassi- tions, and the modified Bessel functions of second kind, where the ani et al. (1999) further used the same mathematical method to de- integrals are in terms of trigonometric functions multiplying the rive the Laplace-domain solutions for constant-head pumping test associated Legendre functions. The series in the solution developed and double packer test that treated as the mixed-type boundary in Laplace domain are difficult to accurately evaluate due to the value problems. Slim and Kirkham (1974) used the Gram–Schmidt oscillatory nature and slow convergence of the multiplied func- orthonormalization method to find a steady state drawdown solutions. Therefore, Shanks’ transform method (Shanks, 1955) is used tion in a confined aquifer of finite horizontal extent. Furthermore, to accelerate the evaluation of the Laplace-domain solution and the similar mixed-type value problems also arise in the field of heat numerical inversion scheme, Stehfest algorithm (Stehfest, 1970), is conduction. Among others, Hung (1985) used the Weiner-Hopf used to obtain the time domain solution. technique to find the solution in a semi-infinite slab and Hung and Chang (1984) combined the Green’s function with conformal mapping to develop the solution in an elliptic disk. Mathematical model For the real world problem, the thickness of aquifer is generally finite. As mentioned above, Cassiani andKabala (1998) and Cassiani Fig. 1 shows a partially penetrating well in a confined aquifer of et al. (1999) developed the solutions to the mixed boundary prob- finite extent with a thickness of b. The drawdown at the distance r lem by assuming infinite aquifer thickness. These solutions are from the well and the distance z from the bottom of the aquifer at appropriate for the case where the pressure change caused by time t is denoted as sðr; z; tÞ. The well screen extends from the top the pumping test has not reached the bottom of the aquifer or of the aquifer ðz ¼ bÞ to z ¼ d with a length of l. The hydraulic the screen length is significantly smaller than the aquifer thick- parameters of the aquifer are horizontal hydraulic conductivity ness. Chang and Chen (2002) considered an aquifer with a finite Kr, vertical hydraulic conductivity Kz, and specific storage Ss. The thickness and a skin factor accounting for the well skin effect. They governing equation for the drawdown can be written as (Yang treated the boundary along the well screen as a Cauchy (third type) et al., 2006) ! boundary condition and handled the wellbore flux entering @2s 1 @s @2s @s through the well screen as unknown. In addition, they changed K þ þ K ¼ S ð1Þ r @r2 r @r z @z2 s @t the mixed boundary into homogeneous Neumann boundary and then discretized the screen length into M segments. Thus, their A Dirichlet boundary condition for a fixed drawdown specified solution may be inaccurate for the case where the size of segments along the well screen is: is coarse.

The purpose of this study is to develop a new solution to the sðrw; z; tÞ¼sw d 6 z 6 b ð2aÞ constant-head test performed at a partially penetrating well in an aquifer with a finite thickness. The mathematical model with A Neumann boundary condition of zero flux is specified as: the mixed boundary condition at the well is directly solved via @s ¼ 006 z 6 d ð2bÞ the methods of dual series equations and perturbation method. @r This solution contains single and double infinite series involving r¼rw the summations of multiplication of integrals, trigonometric func- Moreover, the initial condition and other boundary conditions are:

Fig. 1. The cross-section configuration of the aquifer system involving the mixed boundary value problem. 92 Y.C. Chang, H.D. Yeh / Journal of Hydrology 369 (2009) 90–97 () pffiffiffi Z n Z dp u sðr; z; 0Þ¼0 ð3Þ 2 pffiffiffi b d sð1; z; tÞ¼0 ð4Þ Cðn; k; pÞ¼ pHð0; pÞCð0; k; pÞ f2ðvÞdv f4ðuÞ du p 0 0 du ffiffiffi () p Z nd and ffiffiffi b p 2 p nd pHð0; pÞCð0; k; pÞ f2ðuÞduf4 p @s p 0 b ¼ 0; z ¼ 0; z ¼ b ð5Þ () @z ffiffiffi Z n p dp 2 1 b nd Eq. (1) may be expressed in dimensionless terms as: þ f3ðu; kÞduf4 p p k 0 b 2 2 ffiffiffi () @ s 1 @s 2 @ s @s p Z nd Z b p u 2 þ þ a 2 ¼ ð6Þ 2 1 d @q q @q @n @s f3ðv; kÞdv f4ðuÞdu ð17Þ p k 0 0 du subject to the boundary and initial conditions written in dimension- sinða=2Þ less terms as f1ðaÞ¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð18Þ cosðaÞcosðpn =bÞ sðq; n; s ¼ 0Þ¼0 ð7Þ d f ðaÞ¼af ðaÞð19Þ sðq ¼1; n; sÞ¼0 ð8Þ 2 1 f3ða; bÞ¼sinðabÞf1ðaÞð20Þ

s 1; n; 1; n 6 n 6 b 9a f4ðaÞ¼½Pnðcos aÞþPn1ðcos aÞ ð21Þ ðq ¼ sÞ¼ d ð Þ n n @s 6 6 f ðaÞ¼ln 1 cos d þ a ln 1 cos d a ð22Þ ¼ 0; 0 n nd ð9bÞ 5 b b @q q¼1 Hðn; pÞ¼K1ðknÞ=K0ðknÞð23Þ @s Iðn; pÞ¼n k Hðn; pÞð24Þ ¼ 0; n ¼ 0; n ¼ b ð10Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin @n npa 2 where s s=s is the dimensionless drawdown, tK = S r2 is kn ¼ þ p ð25Þ ¼ w s ¼ r ð s wÞ b 2 the dimensionless time, a ¼ Kz=Kr is the anisotropy ratio of the

aquifer, b ¼ b=rw is the dimensionless aquifer thickness, q ¼ r=rw where K0 and K1 are the modiﬁed Bessel functions of the second

and n ¼ z=rw are dimensionless spatial coordinates, nd ¼ d=rw is kind with order zero and one, respectively, and the Pnðcos aÞ is the the dimensionless depth at the bottom of the well screen. Note that associated Legendre function (Abramowitz and Stegun, 1970,p. Eqs. (6)–(10), construct a mixed-type boundary value problem. 335). The detailed development for the solution of Eq. (6) with Eqs. The flux entering the well screen and the total well discharge (7)–(10) using dual series equation and perturbation method is gi- obtained using Eq. (11) are, respectively, given as: ven in Appendix. The solution for the drawdown in Laplace domain @s ðq; n; pÞ can be written as: qð1; n; pÞ¼ ÀÁffiffiffi @q p X1 ÀÁ 1 K0 pq K ðk qÞ pffiffiffi ÀÁffiffiffi 0 n ffiffiffi X1 s ðq; n; pÞ¼ Bð0; pÞ p þ Bðn; pÞ 1q¼1 p K1ÀÁp K1ðknÞ 2 K p K0ðknÞ ¼ Bð0; pÞ p pffiffiffi þ Bðn; pÞkn 0 n¼1 2 K k K0 p n¼1 0ð nÞ n cos n p ð11Þ n b cos n p ð26Þ b with and X1 Z Bð0; pÞ¼ð0ÞBð0; pÞþ Iðk; pÞBðk; pÞCð0; k; pÞð12Þ b Q p q 1; n; p dn=k k¼1 ð Þ¼ ð Þ X1 nd ÀÁpffiffiffi Bðn; pÞ¼ð0ÞBðn; pÞþ Iðk; pÞBðk; pÞCðn; k; pÞð13Þ ffiffiffi X1 1 p K1ÀÁp b K1ðknÞ ¼ Bð0; pÞ p pffiffiffi Bðn; pÞkn k¼1 2 n k K k ffiffi n K0 p p 0ð nÞ p R dp n¼1 1 2 b nd f1ðuÞf5ðuÞdu þ 21 p 2p 0 b nd ð0ÞBð0; pÞ¼ ð14Þ sin n p ð27Þ pffiffi ffiffiffi R nd b 2 p b p 1 þ pHð0; pÞ f2ðuÞdu ffiffiffi p ()0 k = p Z nd Z where ¼ l rw is the dimensionless length of screen. ffiffiffi b p u ð0Þ 2 p d Bðn; pÞ¼ pHð0; pÞBð0; pÞ f2ðvÞdv f4ðuÞ du 2p 0 0 du Numerical evaluations ffiffiffi () p Z nd ffiffiffi b p 2 p nd pHð0; pÞBð0; pÞ f2ðuÞduf4 p Eq. (11) contains single and double infinite series which consist 2p 0 b ()of the summations of multiplication of integrals, trigonometric ffiffiffi Z n p dp 2 2 1 b n functions, and the modified Bessel functions of second kind. The f u f u duf d þ 2 5ð Þ 1ð Þ 4 p integrals are in terms of trigonometric functions multiplying p p 0 b () pffiffiffi Z n Z associated Legendre functions. This solution involves numerous dp u 2 2 1 b d complicated mathematical functions. Therefore, numerical ap- f1ðvÞf5ðvÞdv f4ðuÞdu p2 p du proaches including the Gaussian quadrature (Gerald and Wheatley, 0 0 2 1 1 n 1989), Shanks’ transform and Stehfest method are proposed to sin n d p ð15Þ p p n b evaluate the solution. The Gaussian quadrature with six terms (Yang and Yeh, 2007) is first utilized to evaluate the integrals in pffiffi R nd 1 2 2 b p f3ðu; kÞdu Eq. (11). Since the oscillation and slow convergence of the multipli- Cð0; k; pÞ¼ k p 0 ð16Þ pffiffi ffiffiffi R nd cation terms, the summations are difficult to evaluate accurately 2 p b p 1 þ p pHð0; pÞ 0 f2ðuÞdu and efficiently. Therefore, the Shanks’ transform method (Shanks, Y.C. Chang, H.D. Yeh / Journal of Hydrology 369 (2009) 90–97 93

1955), a non-linear iterative algorithm based on the sequence of Eqs. (28) and (29) are identical to the solutions of drawdown partial sums, is used to compute the summations in Eq. (11). This and flow rate in Laplace domain given in Chen and Stone (1993); method has been successfully devoted to efficiently computing the Yang and Yeh (2005). The solutions of the aquifer drawdown and solutions arisen in the groundwater area (see, e.g., Peng et al., well flux can be determined by inverting Eqs. (11) and (26) by 2002; Yeh and Yang, 2006b). In addition, the Stehfest algorithm the Stehfest (1970) method with eight weighting factors. (Stehfest, 1970) is further employed to inverse the Laplace-domain The validity of the proposed solutions can be assessed by exam- solution into time domain solution. The proposed numerical ap- ining the sensitivity of the boundary conditions in (9) in the calcu- proaches can accurately evaluate the drawdown solution to the lation. Fig. 2 shows the drawdown for b ¼ 100 and nd ¼ 50 with mixed-type boundary value problem for a flowing partially pene- different q values at s ¼ 1; 100; 104 and 106. As indicated in the trating well and the results are demonstrated in the following figure, the drawdown is constant along the well screen and de- section. creases with increasing radial distance at s ¼ 1. In addition, the drawdown increases with dimensionless time along the unscreened part of the well. Fig. 3 shows the plots of the flux along 4 6 Results and discussion the well screen for b ¼ 100 and nd ¼ 50 at s ¼ 1; 100; 10 and 10 . The flux is non-uniformly distributed and larger at the screen edge, When the well fully penetrates the entire thickness of the for- due to the vertical flow induced by the presence of well partial mation, i.e., nd is zero, the drawdown and the well discharge can penetration. be obtained using Eqs. (11) and (17), respectively, as In order to explore the effect of partial penetration on the well ÀÁpffiffiffi discharge, Fig. 4 illustrates four different penetration ratios 1 K0 pq sðq; n; pÞ¼ ÀÁpffiffiffi ð28Þ x ¼ k=b with k ¼ 50. The well discharges for those four cases are p K0 p the same at the small time; however, it decreases with increasing and penetration ratio at large time. If the penetration ratio is smaller ÀÁpffiffiffi than 0.01, the well discharge of this study agrees with that of con- K1 p stant-head pumping test in Cassiani et al. (1999) in an aquifer of QðpÞ¼pffiffiffi ÀÁpffiffiffi ð29Þ pK0 p semi-infinite thickness. In other words, if the aquifer thickness is

Fig. 2. The drawdown distribution at dimensionless time s ¼ 1; 100; 104 and s ¼ 106 for different q. 94 Y.C. Chang, H.D. Yeh / Journal of Hydrology 369 (2009) 90–97

forms is first used to reduce the original partial differential equation with mixed-type boundary and initial conditions for a partially penetrating well in an aquifer of finite thickness to the dual series equations. The dual series equations are then solved via the perturbation method. The present solutions for a fully penetrating well in an aquifer of finite thickness are identical to the solutions of the drawdown and well discharge given in Chen and Stone (1993). It is found that the solution of Cassiani et al. (1999) for well response to a constant-head pumping test in a semi-infinite aquifer approximates the solution for the case where the aquifer thickness of a finite aquifer is 100 times greater than the length of well screen. In addition, the flux is non-uniformly distributed along the screen and with a local peak at the edge, due to the vertical flow induced by well partial penetration. The new semi-analytical solutions provide accurate description of the response of the aquifer system to a constant-head pumping test performed at a partially penetrating well in a confined aquifer Fig. 3. The distribution of flux along the well screen at different dimensionless of infinite radial extent and finite vertical extent. In addition, those time. solutions are particularly attractive for practical applications since they can be used to evaluate the sensitivities of the input parameters in a mathematical model, to identify the hydraulic parameters if coupling with an optimization approach in the analysis of aquifer data, and to validate a numerical solution.

Acknowledgements

This study was partly supported by the Taiwan National Science Council under the Grant NSC 96-2221-E-009-087-MY3. The authors would like to thank the associate editor and three anony- mous reviewers for their valuable and constructive comments.

Appendix

The Laplace and finite cosine Fourier transforms are first used to solve the mixed-type boundary value problem. The definition of Laplace transform is (Sneddon, 1972): Z 1 ps s ðq; n; pÞ¼Lp½s ðq; n; sÞ; s ! p¼ s ðq; n; sÞe ds ðA1Þ 0

where sðq; n; pÞ is the dimensionless drawdown in Laplace domain. Taking the Laplace transform of Eqs. (6) and (8)–(10) and using the Fig. 4. The influence of the penetration ratio on the flux. initial condition in Eq. (7), the problem reads: greater than 100 times of the screen length, the aquifer can be con- @2s 1 @s @2s sidered as semi-infinite. As the penetration ratio equals unity, the þ þ a2 ps ¼ 0 ðA2Þ @q2 q @q @n2 well discharge of this study is identical to that of Chen and Stone (1993) for a fully penetrating well. In addition, well discharges of sðq ¼1; n; pÞ¼0 ðA3Þ this study agree with those of Chang and Chen (2003) for x ¼ 0:01 and x ¼ 0:001 when k ¼ 50. As indicated in Fig. 4, there 1 sðq ¼ 1; n; pÞ¼ ; n 6 n 6 b ðA4aÞ are no obvious differences between the well discharges for different p d penetration ratios until s ¼ 104. For the cases of x ¼ 0:1 and x ¼ 0:01, the flow caused by the partial penetrating well has not @s 6 6 4 ¼ 0; 0 n nd ðA4bÞ reached the bottom of the aquifer before s ¼ 10 . The aquifer thick- @q q¼1 ness has an influence on groundwater flow after s is greater than 104. The well discharge for x ¼ 0:01 becomes steady as s increases @s 6 ¼ 0; n ¼ 0; n ¼ b ðA5Þ to 10 and the well discharge for x ¼ 0:5 continues to decrease. @n

Conclusions In order to eliminate the n coordinate, the finite cosine Fourier transform is used as follows (Sneddon, 1972): This paper developed a new semi-analytical solution for the Z b npn aquifer system in response to the constant-head test at a partially ^sðq; n; pÞ¼F ½sðq; n; pÞ; n ! n¼ sðq; n; pÞ cos dn c b penetrating well in a confined aquifer of infinite radial extent and 0 finite vertical extent. The Laplace and finite cosine Fourier trans- ðA6Þ Y.C. Chang, H.D. Yeh / Journal of Hydrology 369 (2009) 90–97 95

X1 where ^sðq; n; pÞis the dimensionless drawdown after finite cosine 1 pffiffiffi ð0ÞBð0; pÞ pHð0; pÞþ nð0ÞBðn; pÞ cosðnxÞ¼0; 0 6 x Fourier transform. Substituting Eq. (A6) into Eqs. (A2), (A3) and 2 n¼1 (A5) results in the Bessel differential equation as 6 nd @2^s 1 @^s p ðA16bÞ þ k2^s ¼ 0 ðA7Þ b @q2 q @q n The coefficients of ð0ÞBð0; pÞ and ð0ÞBðn; pÞ can be determined by with the boundary condition following the process given in Sneddon (1966). Assume that when 0 6 x 6 n p=b ^sðq ¼1; n; pÞ¼0 ðA8Þ d X1 1 ð0Þ ð0Þ where kn is defined in Eq. (25). Bð0; pÞþ Bðn; pÞ cosðnxÞ 2 The general solution of Eq. (A7) with the boundary condition Eq. n¼1 Z n (A8) is (Carslaw and Jaeger, 1959, p. 193) dp x b h ðyÞdy ¼ cos pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 ðA17Þ ^ s ðq; n; pÞ¼Aðn; pÞK0ðknqÞðA9Þ 2 x cos x cos y where Aðn; pÞ can be found from using the mixed-type boundary The coefficient ð0ÞBð0; pÞ and ð0ÞBðn; pÞ in Eq. (A17) are, respectively, condition Eq. (A4). The inverse of the finite cosine Fourier transform given by the equations (Sneddon, 1966, p. 161, Eqs. (5.4.56) and is (Sneddon, 1972, p. 425) (5.4.57)) "# Z n Z X1 dp p 1 2 npn 2 b 1 s ðq; n; pÞ¼ ^s ðq; 0; pÞþ ^s ðq; n; pÞ cos ðA10Þ ð0Þ pffiffiffi b b b Bð0; pÞ¼ p h1ðyÞdy þ dy ðA18Þ n¼1 p nd p 2 0 b p Thus, the solution in n domain obtained by inserting Eq. (A9) into ( Z nd Eq. (A10) is b p ð0Þ 2 p Bðn; pÞ¼ pffiffiffi h1ðyÞ½Pnðcos yÞ X1 1 ffiffiffi 2 p 2 2 0 p ) s ðq; n; pÞ¼ Að0; pÞK0ðÞpq þ Aðn; pÞK0ðknqÞ Z b b p n¼1 1 þPn1ðcos yÞdy þ cosðnyÞdy ðA19Þ npn ndp p cos ðA11Þ b b where Pnðcos uÞ is the associated Legendre function. with its derivative with respect to q given by Substituting Eq. (A19) for the coefficients in the integrated @s 1 pffiffiffi pffiffiffi 2 equivalent of Eq. (A16b) obtains ðq; n; pÞ¼ Að0; pÞ pK1ðÞpq @q b b 1 pffiffiffi X1 ð0ÞBð0; pÞ pHð0; pÞx þ ð0ÞBðn; pÞ sinðnxÞ¼0 ðA20Þ X1 npn 2 Aðn; pÞk K ðk qÞ cos ðA12Þ n¼1 n 1 n b n¼1 We can find that h1ðyÞ satisfies the following equation: (Sneddon, Substituting Eq. (A11) into Eq. (A4a) and Eq. (A12) into Eq. (A4b) re- 1966, p. 161, Eq. (5.4.58)) Z n sults in a system of the dual series equations (DSE) dp X1 b 1ffiffiffi 1 pffiffiffi 2 X1 npn 1 h1ðyÞ p ½Pnðcos yÞþPn1ðcos yÞ sin nxdy Að0; pÞK ðÞþp Aðn; pÞK ðk Þ cos ¼ ; 0 2 n¼1 b 0 b 0 n b p Z n¼1 1 pffiffiffi 2 X1 p 1 ¼ pHð0; pÞBð0; pÞx cosðnyÞdy sin nx ðA21Þ n 6 n 6 b A13a n d ð Þ 2 p d p n¼1 b p 1 pffiffiffi pffiffiffi 2 X1 npn The summation term on the left-hand side of Eq. (A21) can be ex- Að0; pÞ pK ðÞþp Aðn; pÞk K ðk Þ cos b 1 b n 1 n b pressed as (Sneddon, 1966, p. 59, Eq. (2.6.31)) n¼1 ÀÁ 6 6 1 X1 cos x Hðx yÞ ¼ 0; 0 n nd ðA13bÞ ffiffiffi 2 p ½Pnðcos yÞþPn1ðcos yÞ sin nx ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA22Þ 2 cos y cos x We define that n¼1 where HðXÞ is the Heaviside unit step function which has different Bðn; pÞ¼2Aðn; pÞK0ðknÞ=b ðA14Þ value for different range of X as The DSE of (A13) can be arranged as (Sneddon, 1966, p. 161): 8 <> 0 X < 0 X1 1 1 nd Bð0; pÞþ Bðn; pÞ cosðnxÞ¼ ; p < x 6 p ðA15aÞ HðXÞ¼> 1=2 X ¼ 0 ðA23Þ 2 p b : n¼1 1 X > 0

1 pffiffiffi X1 Substituting Eq. (A22) into Eq. (A21) one gets Bð0; pÞ pHð0; pÞþ nBðn; pÞ cosðnxÞ Z n dp 2 b h ðyÞHðx yÞ n¼1 p1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy X1 cos y cos x nd 0 ¼ Iðk; pÞBðk; pÞ cosðkxÞ; 0 6 x 6 p ðA15bÞ ffiffiffi b x 1 p 0 k¼1 ¼ sec pHð0; pÞð ÞBð0; pÞx 2 2 ) Our goal is to determine the coefficients Bð0; pÞ and Bðn, pÞ appear- Z 2 X1 p 1 ing in Eq. (A15). The pair of DSE (A15) when Iðk; pÞ¼0 can be ex- cosðnyÞdy sin nx ðA24Þ p p pressed as n¼1 ndp=b 1 X1 1 n Using the property of Heaviside unit step function in Eq. (A23) an ð0ÞBð0; pÞþ ð0ÞBðn; pÞ cosðnxÞ¼ ; d p < x 6 p ðA16aÞ 2 p b equivalent integral equation of Eq. (A24) can be obtained n¼1 96 Y.C. Chang, H.D. Yeh / Journal of Hydrology 369 (2009) 90–97 Z x h ðyÞ Eq. (A29) can be expressed as a general pair of DSE by defining new pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 dy cos y cos x coefficients Cð0; k; pÞ and Cðn; k; pÞ as 0 ffiffiffi X1 x 1 p ð0Þ 1 n ¼ sec pHð0; pÞ Bð0; pÞx Cð0; k; pÞþ Cðn; k; pÞ cosðnxÞ¼0; d p < x 6 p ðA30aÞ 2 2 2 b Z ) n¼1 2 X1 p 1 n cos ny dy sin nx 0 6 x < d A25 ð Þ p ð Þ ffiffiffi X1 p n p=b p b 1 p n¼1 d Cð0; k; pÞ pHð0; pÞþ nCðn; k; pÞ cosðnxÞ 2 Then, the function h1ðyÞ can be obtained based on Sneddon (1966, n¼1 p. 41, Eq. (2.3.5)) as n Z ¼ cosðkxÞ; 0 6 x 6 d p ðA30bÞ 1 pffiffiffi d y x sinðx=2Þ b h ðyÞ¼ pHð0; pÞð0ÞBð0; pÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx 1 p dy cos x cos y 0Z where X1 y 4 1 n p d sinðx=2Þ sinðnxÞ þ sin n d pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ðA26Þ B0ð0; pÞ p2 pn b dy cos x cos y Cð0; k; pÞ¼ ðA31Þ n¼1 0 Iðk; pÞBðk; pÞ By integrating Eq. (A26) and substituting it into Eqs. (A18) and and (A19), the coefficients Bð0; pÞ and Bðn; pÞ can be expressed as Eqs. (14) and (15), respectively. B0ðn; pÞ Cðn; k; pÞ¼ ðA32Þ Based on the perturbation method (Boridy, 1990), one needs to Iðk; pÞBðk; pÞ find the coefficients of the following DSE when Iðk; pÞ–0 Thus the coefficients Bð0; pÞ and Bðn; pÞ when I – 0 in Eq. (A15) can 1 X1 n n B0ð0; pÞþ B0ðn; pÞ cosðnxÞ¼0; d p < x 6 p ðA27aÞ be, respectively, written as (Boridy, 1990, Eq. (13)) 2 b n¼1 X1 1 pffiffiffi X1 Bð0; pÞ¼ð0ÞBð0; pÞþ Iðk; pÞBðk; pÞCð0; k; pÞðA33Þ B0ð0; pÞ pHð0; pÞþ nB0ðn; pÞ cosðnxÞ 2 k¼1 n¼1 X1 n and ¼ Iðk; pÞBðk; pÞ cosðkxÞ; 0 6 x 6 d p ðA27bÞ b X1 k¼1 Bðn; pÞ¼ð0ÞBðn; pÞþ Iðk; pÞBðk; pÞCðn; k; pÞðA34Þ The DSE above can be further separated into infinite pairs of DSE as k¼1 (Boridy, 1990, Eqs. (11) and (12)) 8 Consequently, the coefficients of Cð0; k; pÞ and Cðn; k; pÞ are, respec- > P1 > 1 0 0 nd 6 tively, given by (Sneddon, 1966, p. 161, Eqs. (5.4.56) and (5.4.57)) > 2 B ð0; pÞþ B ðn; pÞ cosðnxÞ¼0 b p < x p <> n¼1 ffiffi n ffiffiffi P1 p R dp 0 p 0 1 2 2 b 1 f3ðu; kÞdu > 2 B ð0; pÞ pHð0; pÞþ nB ðn; pÞ cosðnxÞ¼Ið1; pÞBð1; pÞ cosðxÞ k p 0 > Cð0; k; pÞ¼ ffiffi n ðA35Þ > n¼1 p pffiffiffi R dp : n 2 b 0 6 x 6 d p 1 þ p pHð0; pÞ 0 f2ðuÞdu 8 b > P1 > 1 B0ð0; pÞþ B0ðn; pÞ cosðnxÞ¼0 nd p < x 6 p and > 2 b () < n¼1 pffiffiffi Z n Z dp u pffiffiffi P1 2 pffiffiffi b d 1 0 0 Cðn; k; pÞ¼ pHð0; pÞCð0; k; pÞ f ð Þd f ðuÞ du > 2 B ð0; pÞ pHð0; pÞþ nB ðn; pÞ cosðnxÞ¼Ið2; pÞBð2; pÞ cosð2xÞ 2 v v 4 > n¼1 p 0 0 du :> () nd ffiffiffi Z n 0 6 x 6 p p dp b 2 pffiffiffi b n pHð0; pÞCð0; k; pÞ f ðuÞduf d p . 2 4 8. p 0 b P1 () ffiffiffi Z n > 1 0 0 nd p d > B ð0; pÞþ B ðn; pÞ cosðnxÞ¼0 p < x 6 p b p > 2 b 2 1 nd < n¼1 þ f3ðu; kÞduf4 p pffiffiffi P1 p k 0 b > 1 B0ð0; pÞ pHð0; pÞþ nB0ðn; pÞ cosðnxÞ¼Iðk; pÞBðk; pÞ cosðkxÞ ffiffiffi () > 2 p Z nd Z > n¼1 b p u :> 2 1 d 0 6 x 6 nd p f3ðv; kÞdv f4ðuÞdu ðA36Þ b p k 0 0 du ðA28Þ where the function f3 is defined in (20). Rearranging Eq. (A28) obtains 8 Series in Eqs. (A33) and (A34) can be considered as a perturba- 0 P1 0 > 1 B ð0;pÞ B ðn;pÞ nd 6 tion series in Iðn; pÞ. In the zeroth-order approximation, the second < 2 Ið1;pÞBð1;pÞ þ Ið1;pÞBð1;pÞ cosðnxÞ¼0 b p < x p n¼1 terms on the right-hand side of Eqs. (A33) and (A34) are ignored so ffiffiffi P1 > B0ð0;pÞ p B0ðn;pÞ n ð0Þ : 1 pHð0;pÞþ n cosðnxÞ¼cosðxÞ 0 6 x 6 d p that coefficients Bð0; pÞ and Bðn; pÞ are simply given by Bð0; pÞ and 2 Ið1;pÞBð1;pÞ Ið1;pÞBð1;pÞ b ð0Þ 8 n¼1 Bðn; pÞ. In the first-order approximation Bð0; pÞ and Bðn; pÞ of the P1 0 0 0 second member of Eqs. (A33) and (A34) are replaced by ð ÞB 0; p > 1 B ð0;pÞ B ðn;pÞ nd 6 ð Þ < 2 Ið2;pÞBð2;pÞ þ Ið2;pÞBð2;pÞ cosðnxÞ¼0 b p < x p ð0Þ n¼1 and Bðn; pÞ, respectively. Thus, in this approximation, coefficients ð1Þ ð1Þ > 0 pffiffiffi P1 0 Bð0; pÞ and Bðn; pÞ are denoted by Bð0; pÞ and Bðn; pÞ, respec- :> 1 B ð0;pÞ B ðn;pÞ 6 6 nd 2 Ið2;pÞBð2;pÞ pHð0;pÞþ n Ið2;pÞBð2;pÞ cosðnxÞ¼cosð2xÞ 0 x b p n¼1 tively, as (Boridy, 1990, Eq. (16)) . X1 8. ð1Þ ð0Þ ð0Þ P1 Bð0; pÞ¼ Bð0; pÞþ Iðk; pÞ Bðk; pÞCð0; k; pÞðA37Þ > B0 0;p B0 n;p n > 1 ð Þ ð Þ d 6 k¼1 < 2 Iðk;pÞBðk;pÞ þ Iðk;pÞBðk;pÞ cosðnxÞ¼0 b p < x p n¼1 ffiffiffi P1 and > B0ð0;pÞ p B0ðn;pÞ n : 1 pH 0;p n cos nx cos kx 0 6 x 6 d 2 Iðk;pÞBðk;pÞ ð Þþ Iðk;pÞBðk;pÞ ð Þ¼ ð Þ b p X1 n¼1 ð1ÞBðn; pÞ¼ð0ÞBðn; pÞþ Iðk; pÞð0ÞBðk; pÞCðn; k; pÞðA38Þ A29 ð Þ k¼1 Y.C. Chang, H.D. Yeh / Journal of Hydrology 369 (2009) 90–97 97

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