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

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New Solutions to the Constant-Head Test Performed at a Partially Penetrating Well Journal of Hydrology 369 (2009) 90–97 Contents lists available at ScienceDirect 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 con- sidered 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 ana- lytical 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 bound- ary 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 solu- tions. 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 À 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 à à 2 à à 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 À 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 sÃðq; n; s ¼ 0Þ¼0 ð7Þ d f ðaÞ¼af ðaÞð19Þ sÃðq ¼1; n; sÞ¼0 ð8Þ 2 1 f3ða; bÞ¼sinðabÞf1ðaÞð20Þ sà 1; n; 1; n 6 n 6 b 9a f4ðaÞ¼½Pnðcos aÞþPnÀ1ðcos aÞ ð21Þ ðq ¼ sÞ¼ d ð Þ Ã n n @s 6 6 f ðaÞ¼ln 1 À cos d þ a À ln 1 À cos d À a ð22Þ ¼ 0; 0 n nd ð9bÞ 5 b b @q q¼1 Hðn; pÞ¼K1ðknÞ=K0ðknÞð23Þ @sà Iðn; pÞ¼n À k Hðn; pÞð24Þ ¼ 0; n ¼ 0; n ¼ b ð10Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin @n npa 2 where sà 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.
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