Fully Discrete Finite Element Analysis of Multiphase Flow in Groundwater Hydrology

This paper deals with development and analysis of a fully discrete nite element method for a nonlinear di erential system for describing an air water system in groundwater hydrol ogy The nonlinear system is written in a fractional ow formulation i e in terms of a saturation and a global pressure The saturation equation is approximated by a nite element method while the pressure equation is treated by a mixed nite element method The analysis is carried out rst for the case where the capillary di usion coe cient is assumed to be uniformly positive and is then extended to a degenerate case where the di usion coe cient can be zero It is shown that error es timates of optimal order in the L norm and almost optimal order in the L norm can be obtained in the nondegenerate case In the degenerate case we consider a regularization of the saturation equation by perturbing the di usion coe cient The norm of error estimates depends on the severity of the degeneracy in di usivity with almost optimal order convergence for non severe degeneracy Implementation of the fractional ow formulation with various nonhomogeneous boundary condi tions is also discussed Results of numerical experiments using the present approach for modeling groundwater ow in porous media are reported