Convergence of a TPFA finite volume scheme for nonisothermal immiscible compressible two-phase flow in porous media

This paper presents and analyzes a finite volume scheme for a fully degenerate parabolic system modeling the compressible two-phase flow, such as water-gas, problem in porous media considering the effects of temperature. The problem is written in terms of the phase formulation, i.e. the saturation of one phase, the pressure of the second phase and the temperature are primary unknowns. We use a first-order time implicit Euler scheme. The spatial discretization uses a fully coupled fully implicit cell-centered two-point flux approximation scheme where upwinding of the flux across an interface between two control volumes is performed for each fluid phase. Under physically relevant assumptions on the data, we prove the maximum principle for both saturation and temperature. We obtain a priori estimates and we show that the discrete problem is well posed. Also, we exploit various a priori estimates as well as compactness arguments to establish the convergence of the numerical scheme toward the weak solution. The open-source platform DuMu(X) was employed for the implementation of the developed algorithm based on the above scheme. Numerical result are presented to demonstrate the efficiency of this scheme. The test addresses the evolution in 2D of CO2 injection into a layered aquifer. The algorithm's convergence with first order is validated by our numerical results, which support the theoretical results. To our best knowledge, this is the first convergence result of a finite volume scheme in the case of nonisothermal immiscible compressible two-phase flow in porous media.