The size of the freshwater zone below an elongated island with infiltration

An exact solution is derived for two-dimensional, steady interface flow below an elongated island with infiltration. The exact solution is obtained with the hodograph method in combination with conformal mapping; the complex specific discharge function and the reference function are used as auxiliary functions. A new procedure is presented for the solution of the parameter problem in the conformal map of the upper half plane onto the domain in the specific discharge plane. The exact solution is used to compute the depth of the interface for both isotropic and anisotropic aquifers. The depth of the interface at the center of the island is independent of the vertical hydraulic conductivity; the length of the outflow face at the beach is independent of the horizontal hydraulic conductivity. The exact solution can only be evaluated numerically for values of the infiltration rate that are at least an order of magnitude larger than may be expected realistically. For practical problems it is proposed to approximate the shape of the interface with a semiellipse; explicit equations are given for the size of the ellipse in terms of the aquifer parameters. The approximate solution is compared to the exact solution for a number of cases with large infiltration rates (so that the exact solution could be evaluated). The approximate and exact solutions produce results that are identical for at least six significant digits for all cases investigated, which suggests that it may be reasonable to use the approximate formulas outside the range of comparison. Solutions for isotropic and anisotropic aquifers obtained with the approximate formulas are compared to results obtained with a Dupuit solution for a set of realistic aquifer parameters; the interface is approximated well with the Dupuit solution.