Journal of Applied Mathematics
Volume 2003 (2003), Issue 8, Pages 377-396

Variational analysis for simulating free-surface flows in a porous medium

Shabbir Ahmed1 and Charles Collins2

1Big Cypress Basin, South Florida Water Management District, 6089 Janes Lane, Naples 34109, FL, USA
2Department of Mathematics, University of Tennessee, 121 Ayres Hall, 1403 Circle Drive, Knoxville 37996, TN, USA

Received 29 January 2003; Revised 25 March 2003

Copyright © 2003 Shabbir Ahmed and Charles Collins. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A variational formulation has been developed to solve a parabolic partial differential equation describing free-surface flows in a porous medium. The variational finite element method is used to obtain a discrete form of equations for a two-dimensional domain. The matrix characteristics and the stability criteria have been investigated to develop a stable numerical algorithm for solving the governing equation. A computer programme has been written to solve a symmetric positive definite system obtained from the variational finite element analysis. The system of equations is solved using the conjugate gradient method. The solution generates time-varying hydraulic heads in the subsurface. The interfacing free surface between the unsaturated and saturated zones in the variably saturated domain is located, based on the computed hydraulic heads. Example problems are investigated. The finite element solutions are compared with the exact solutions for the example problems. The numerical characteristics of the finite element solution method are also investigated using the example problems.