Numerical analysis of an interior penalty discontinuous Galerkin scheme for two phase flow in heterogeneous porous media with discontinuous dynamic capillary pressure effects

Abstract

We present an interior penalty discontinuous Galerkin scheme for a two-phase flow model in heterogeneous porous media. The model includes dynamic effects and discontinuities in the capillary pressure. We define the interface conditions arising across material interfaces in heterogeneous media and show how to account for capillary barriers. We numerically approximate the mass-conservation laws without reformulation, i.e. without introducing a global pressure. We prove the existence of a solution to the emerging fully discrete systems, show the convergence of the numerical scheme, and obtain error-estimates for sufficiently smooth data. We also present a linearization scheme for the non-linear algebraic system resulting from the fully discrete discontinuous Galerkin approximation of the model. The linearization scheme does not require any regularization step. Additionally, in contrast with Newton or Picard methods, the linearization scheme does not involve computation of derivatives. Finally, to validate our theoretical findings and to show the scope of the applicability of the scheme, we present 1D and 2D numerical examples in realistic settings for homogeneous as well as heterogeneous porous media. We rigorously prove that the scheme is robust and linearly convergent.