## 1. Introduction

[2] The constant drawdown pumping test is particularly useful in low-transmissivity aquifers [*Jones et al.*, 1992; *Jones*, 1993]. Also, pumping under fixed drawdown is a common means of controlling off-site migration of contaminated groundwater [*Hiller and Levy*, 1994] or recovering the free product of light nonaqueous phase liquids [*Murdoch and Franco*, 1994; *Abdul*, 1992]. A pumping well may fully or partially penetrate the aquifer and may be subject to well skin effects. In well hydraulics a flowing fully penetrating well can be appropriately simulated as a Dirichlet (the first type) boundary of a prescribed potential, and the associated models can be solved with the conventional integral transform technique [e.g., *Jacob and Lohman*, 1952; *van Everdingen and Hurst*, 1949; *Hantush*, 1964; *Clegg*, 1967; *Hurst et al.*, 1969; *Ehlig-Economides and Ramey*, 1981; *Olarewaju and Lee*, 1989; *Mishra and Guyonnet*, 1992]. On the other hand, a flowing partially penetrating well must be considered as a mixed boundary that involves two different types of boundary conditions. For example, the flowing partially penetrating well with infinitesimal skin shown in Figure 1 is a typical mixed boundary such as

where the Cauchy (the third type) condition is prescribed along the well screen extending from ξ_{d} to ξ_{l}, while the Neumann (the second type) condition of no flux is stipulated over the remaining unscreened part (all variables are defined in the Notation section unless otherwise noted.) If there is no skin (i.e., *S*_{k} = 0), equation (1a) becomes a Dirichlet condition, but equations (1a) and (1b) still remain as a mixed boundary. Not directly solvable by the integral transform technique or the method of separation of variables, mathematical models involving a mixed boundary are generally dealt with by special solution techniques [*Sneddon*, 1966; *Noble*, 1958; *Fabrikant*, 1991].

[3] For the well hydraulics model that involves equations (1a) and (1b) in a confined aquifer of infinite thickness, *Cassiani et al.* [1999] used the dual integral equation method to develop the Laplace domain solution. Neglecting skin effect in equations (1a) and (1b), *Selim and Kirkham* [1974] used the Gram-Schmidt orthonormalization method to find the steady state solution where the aquifer has a finite horizontal extent. For similar mixed boundary problems in heat conduction, *Huang* [1985] used the Weiner-Hopf technique to find the solution in a semi-infinite slab, and *Huang and Chang* [1984] combined the Green's function approach with conformal mapping to determine the solution in an elliptic disk.

[4] There are other groundwater problems associated with a mixed boundary. *Dagan* [1978] employed the Green's function method to find the solutions for the slug test in a partially penetrating well. For constant rate pumping in a partially penetrating well, *Gringarten and Ramey* [1975] used the source function approach, *Wilkinson and Hammond* [1990] used the perturbation method, and *Cassiani and Kabala* [1998] used the dual-integral equation method to determine the solutions for different conditions. However, a partially penetrating well under a constant rate pumping can also be formulated as a homogeneous Neumann boundary [*Hantush*, 1961; *Javandel*, 1982; *Dougherty and Babu*, 1984; *Moench*, 1985; *Streltsova*, 1988]. *Ruud and Kabala* [1997] noted that the use of the homogeneous Neumann boundary as the substitute for the mixed boundary should be exercised with extreme caution. Nevertheless, whether a mixed boundary or a homogeneous Neumann boundary should be used is subject to debate, for *Hantush* [1964] pointed out “neither a uniform flux nor a uniform drawdown is really conceived along the face of the well, because of several involved field and operational conditions.” Yet it is very clear that equations (1a) and (1b) are a mixed boundary condition.

[5] Aquifer thickness is finite under field conditions. As stated above, *Cassiani et al.* [1999] have determined the Laplace domain solution for the problem involving equations (1a) and (1b) by assuming infinite aquifer thickness. In the solution, infinite thickness cannot be reduced to finite thickness. Thus this solution may be appropriate for early times during which pressure change caused by the constant drawdown pumping has not reached the bottom of the aquifer or for the special condition where the screen length is significantly shorter than aquifer thickness. Such limitations can be removed by assuming finite aquifer thickness. Considering that the dual integral equation method as employed by *Cassiani et al.* [1999] is not suitable for finite aquifer thickness and the integral transform technique is rather prevalent, the purpose of this paper is to develop an analytical approach using appropriate integral transforms to solve the problem involving equations (1a) and (1b) with finite aquifer thickness.