Sigma mapping for drainage problems with a time-dependent water table

Abstract

Regional groundwater flow forecasting is frequently accomplished using model equations based on Darcy's law, the continuity equation, and kinematic and dynamic boundary conditions for a time-dependent water table, to predict hydraulic heads and fluxes. For homogeneous and isotropic aquifers, the model reduces to Laplace's equation for the hydraulic head and a transient non-linear free surface boundary condition. These equations are solved in the literature and in computer packages by a variety of numerical methods, including finite difference, finite element, and boundary element methods. The solution of this model is not simple and subject to instability problems. Computations in a vertical plane using finite-difference methods are not easy due to the irregular mesh needed near a highly curved water table, frequently involving smoothing of the numerical results to ensure stability. A technique to avoid the complications related to mesh generation near a curved boundary and its tracking entails using mappings to produce transformed domains. However, this is not so frequent to compute unsteady seepages. In particular, the so-called sigma mapping is widely used to model irrotational water waves; it applies to any flow involving a time-dependent free surface boundary. This mapping transforms the domain into a rectangle, so that the free surface becomes static in the transformed plane. There is close similarity between the mathematical model describing irrotational water waves and that of unsteady seepage flow, e.g., both are governed by Laplace's equation involving relevant boundary conditions at the moving boundary. Both moving free surfaces are different in character, e.g., the seepage is dissipative whereas the water wave is dispersive. However, it is possible to transfer mathematical approximations between both fields of research. This transfer is exploited in this work and the sigma mapping is newly and successfully applied to seepage flows involving both a steady and unsteady water table either receiving, or not, a recharge from rainfall or an artificial source.