Convergence of a finite volume scheme for immiscible compressible two-phase flow in porous media by the concept of the global pressure

Abstract

This paper deals with development and analysis of a finite volume (FV) method for the coupled system describing immiscible compressible two-phase flow, such as water-gas, in porous media, capillary and gravity effects being taken into account. We investigate a fully coupled fully implicit cell-centered “phase-by-phase” FV scheme for the discretization of such system. The main goal is to incorporate some of the most recent improvements in the scheme and the convergence of the numerical approximation to the weak solution of such models. The spatial discretization uses a TPFA scheme and a new strategy for handling the upwinding. Based on a priori estimates and compactness arguments, we prove the convergence of the numerical approximation to the weak solution. The particular feature in this convergence analysis of the classical engineering scheme based on the “phase-by-phase” upwinding on an orthogonal mesh relies on the global pressure–saturation fractional flow formulation as was defined relatively recently for immiscible compressible flow in porous media. We have developed and implemented this scheme in a new module in the context of the open source platform DuMuX. Two numerical experiments are presented to demonstrate the efficiency of this scheme. The first test addresses the evolution in 2D of gas migration through engineered and geological barriers for a deep repository for radioactive waste. The second test case is chosen to test the ability of the method to approximate solutions for 3D problems modeling scenarios of CO2 injection in a fully water-saturated domain.