Modeling Groundwater Flow under Transient Nonlinear Free Surface

Abstract

This article presents a procedure to simulate groundwater flow subject to a nonlinear moving boundary resulting from periodic recharge and significant vertical hydraulic gradients. Under these conditions the Dupuit assumptions are not valid, and the governing equation is an elliptic partial differential equation that reduces to Laplace’s equation in homogeneous isotropic aquifers. This equation is subject to a transient nonlinear free-surface boundary condition. To overcome the mathematical difficulties of this boundary-value problem with a nonlinear moving boundary condition, the method of decomposition in combination with successive approximations is proposed. Decomposition series provide a simple systematic approach to the approximate solution of the nonlinear boundary-value problem without linearization or discretization. Components of the decomposition solution are shown to converge fast to an exact solution and are compared with corresponding linearized and finite-difference solutions with good agreement. The model is demonstrated with an application to a perched aquifer in the Jackson Purchase region of Kentucky, which experiences annual fluctuations in the water table of over 5 m and strong vertical gradients. Using bulk values of hydraulic conductivity, an initial condition fitted to heads known at a few piezometers, and a generalized approximate model of periodic mean monthly recharge, approximate expressions for the time-dependent water table, and hydraulic heads are derived and compared with limited measured events. The simplified model reproduced the monthly heads and their seasonal variability. As expected, periodic recharge from rainfall functionally affects the hydraulic head. As a corollary, periodic recharge is imbedded in hydraulic gradients and pore velocities.