Convergence of Discontinuous Galerkin Methods for Incompressible Two-Phase Flow in Heterogeneous Media

Abstract

A class of discontinuous Galerkin methods with interior penalties is presented for incompressible two-phase flow in heterogeneous porous media with capillary pressures. The semidiscrete approximate schemes for fully coupled system of two-phase flow are formulated. In highly heterogeneous permeable media, the saturation is discontinuous due to different capillary pressures, and therefore, the proposed methods incorporate the capillary pressures in the pressure equation instead of saturation equation. By introducing a coupling approach for stability and error estimates instead of the conventional separate analysis for pressure and saturation, the stability of the schemes in space and time and a priori hp error estimates are presented in the $L^2(H^1)$ for pressure and in the $L^\infty(L^2)$ and $L^2(H^1)$ for saturation. Two time discretization schemes are introduced for effectively computing the discrete solutions.