Fully Discrete Finite Element Analysis of Multiphase Flow in Groundwater Hydrology

Abstract

This paper deals with the development and analysis of a fully discrete finite element method for a nonlinear differential system for describing an air-water system in groundwater hydrology. The nonlinear system is written in a fractional flow formulation, i.e., in terms of a saturation and a global pressure. The saturation equation is approximated by a finite element method, while the pressure equation is treated by a mixed finite element method. The analysis is carried out first for the case where the capillary diffusion coefficient is assumed to be uniformly positive, and is then extended to a degenerate case where the diffusion coefficient can be zero. It is shown that error estimates of optimal order in the $L^2$-norm and almost optimal order in the $L^\infty$-norm can be obtained in the nondegenerate case. In the degenerate case we consider a regularization of the saturation equation by perturbing the diffusion coefficient. The norm of error estimates depends on the severity of the degeneracy in diffusivity, with almost optimal order convergence for nonsevere degeneracy. Implementation of the fractional flow formulation with various nonhomogeneous boundary conditions is also discussed. Results of numerical experiments using the present approach for modeling groundwater flow in porous media are reported.