Numerical treatment of two-phase flow in capillary heterogeneous porous media by finite-volume approximations

Abstract

This paper examines two-phase flow in porous media with heterogeneous capillary pressure functions. This problem has received very little attention in the literature, and constitutes a challenge for numerical discretization, since saturation discontinuities arise at the interface between the different homogeneous regions in the domain. As a motivation we first consider a one-dimensional model problem, for which a semi-analytical solution is known, and examine some different finite-volume approximations. A standard scheme based on harmonic averaging of the absolute permeability, and which possesses the important property of being pressure continuous at the discrete level, is found to converge and gives the best numerical results. In order to investigate two-dimensional flow phenomena by a robust and accurate numerical scheme, a recent multi point flux approximation scheme, which is also pressure continuous at the discrete level, is then extended to account for two-phase flow, and is used to discretize the two-phase flow pressure equation in a fractional flow formulation well suited for capillary heterogenity. The corresponding saturation equation is discretized by a second-order central upwind scheme. Some numerical examples are presented in order to illustrate the significance of capillary pressure heterogeneity in two-dimensional two-phase flow, using both structured quadrilateral and unstructured triangular grids.