Charts for free surfaces in steady-state seepage flow through homogeneous isotropic rectangular dams

Locating the free surface in steady-state seepage flow through a homogeneous isotropic rectangular dam is a well-posed free-boundary problem. It has often been used as a benchmark problem to validate sophisticated numerical schemes developed for solving more complex seepage problems encountered in groundwater hydrology. This problem has been solved analytically using the hodograph method and the conformal mapping technique. However, very few explicit results are available on free-surface profiles, primarily owing to mathematical complexities and computational implementation difficulties. To ameliorate this situation and enhance the practical value of the analytical solution to the benchmark problem, this paper presents charts for determining the free-surface profiles in rectangular dams. The charts are created based on the celebrated Polubarinova-Kochina analytical solution, which consists of a set of parametric equations involving elliptic integrals with two unknown parameters, namely a and beta, that characterize the relative positions of boundary singular points in the hodograph plane. An efficient random search algorithm is developed and used to find the values of these two parameters and, hence, locate the free-surface profiles for a wide range of dam widths and headwater and tailwater levels. These results are well organized and plotted in charts for ease of use. The accuracies of the proposed algorithm and chart for the two parameters are verified by comparing the computed results of the total outflow rates with those calculated using an exact closed-form formula. The algorithm and charts proposed for the free-surface profiles are validated with the results available in the literature. Moreover, the angle between the line tangent to the free surface and the horizontal direction is shown to change abruptly at its entry and exit points. This finding offers an explanation of the paradox between the theoretically expected and graphically observed values of the entry or exit angle of the free surface.