Continuous-time finite element analysis of multiphase flow in groundwater hydrology

Abstract

Summary. A nonlinear differential system for describing an air-water system in ground water hydrology is given. The system is written in a fractional fl o w formulation, i.e., in terms o f a saturation and a gl oba l pressure. A continuous-time version o f the finite element method is developed and analyzed for the approximation o f the saturation and pressure. The saturation equation i s treated by a Galerkin finite element method, w h il e the pressure equation is treated by a mixed fi n i te 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 o f optimal order in the L 2 -norm and almost optimal order in the L °°- nor m can be obtained i n the nondegenerate case. In the degenerate case w e consider a regularization o f the saturation equation by perturbing the diffusion coefficient. The norm o f error estimates depends on the severity o f the degeneracy in diffusivity, with almost optimal order convergence for non-severe degeneracy. Existence and uniqueness o f the approximate solution is also proven.