Reply to comment by O. D. L. Strack on “Steady two‐dimensional groundwater flow through many elliptical inhomogeneities”

Abstract

[1] We thank Strack [2005] for demonstrating that his expression for the potential for an elliptical inhomogeneity [Strack, 1989] may be reduced to our expression [Suribhatla et al., 2004] when the poles are removed and the limit of his summation is taken to infinity. Strack arrives at his result through an approach based on conformal mapping and complex power series expansions, which indeed has the advantage that it may be applied to model single-aquifer, two-dimensional flow through inhomogeneities of other smooth shapes. In contrast, our expression is derived through separation of variables in elliptical coordinates. Our approach may be applied to solve other types of flow problems; these include multiaquifer flow [Bakker, 2004] and unsaturated flow [Bakker and Nieber, 2004] through elliptical inhomogeneities, as well as three-dimensional saturated flow through ellipsoidal inhomogeneities [Janković and Barnes, 1999], and unsaturated flow through a spherical inhomogeneity [Warrick and Knight, 2004]. [2] The expression for the potential of an elliptical inhomogeneity contains a number of coefficients that need to be computed through application of the boundary condition along the boundary of the inhomogeneity. Strack [1989] computed these coefficients through application of the boundary condition of continuity of head at collocation points along the boundary of the inhomogeneity and solved the system of equations for all coefficients simultaneously. We derived exact expressions for these coefficients, in integral form, through the application of Fourier expansions [Suribhatla et al., 2004, equations (25) and (26)]. This method has four main advantages. First, the convergence to the exact solution is guaranteed. Second, the coefficients represent the optimal solution, in a least squares sense, at a given truncation level of the series solution. Third, the method allows for the use of many terms in the series solution. Such high-order solutions are needed for an accurate solution when the spacing between inhomogeneities is small. And fourth, it facilitates the simulation of flow through large numbers of elliptical inhomogeneities. We applied an iterative solution algorithm on a massively parallel supercomputer cluster, and accurately simulated flow through a field of 10,000 inhomogeneities, which required the computation of 1,010,000 coefficients (see example 2 of Suribhatla et al. [2004]).