A Fourier finite element model for unsaturated flow in porous media

Abstract

A new numerical methodology for solving the partial differential equation governing the unsaturated flow of water in porous media is developed and applied to approximate the hydraulic head and to solve for the flux functions that obey the one-dimensional unsaturated flow equation. The algorithms are based on a Fourier finite element type of formulation, which combines the efficiency of the Fourier series expansion as a refinement tool and the finite element method, where the spatial domain is divided into discrete elements and nodal points. A truncated Fourier series expansion was used as the approximating function over orthogonal discrete elements. The Fourier cosine series was selected for the expansion series and the same denomination is used for the numerical integration techniques using polynomials of a high order as approximating functions. The solution to the partial differential equation is obtained by marching through time in discrete steps, beginning with a known solution of Fourier series coefficients at time zero. A Fourier finite element computer program based on the new methodology was written and applied to modeling a one-dimensional flow of water into porous media. The nonlinear solution behavior was well simulated by the high order approximating functions. The accuracy of the numerical solution used in this model was evaluated by analyzing a test problem for which the solution is available, and a good agreement was obtained.