Energy Stable and Mass Conservative Numerical Method for Gas Flow in Porous Media with Rock Compressibility

Abstract

Taking rock compressibility into consideration brings up a quite challenging issue to modeling and numerical simulation of gas flow in porous media. To develop efficient numerical methods for such a problem, on the basis of the existing model, we, for the first time, propose an alternative formulation by introducing a free energy accounting for rock compressibility and by using chemical potential gradient instead of pressure gradient as the primary driving force. This formulation has a distinct feature that it is proved to satisfy an energy dissipation law. We further propose an efficient energy stable time discrete scheme using the improved energy factorization approach to tackle the Helmholtz free energy density determined by the Peng--Robinson equation of state. The discrete pressure is taken carefully using the discrete chemical potential and free energy such that the discrete energy dissipation law can be preserved. The fully discrete scheme is constructed based on the cell-centered finite difference method with the upwind strategy, which features full mass conservation and positivity preservation of molar density under proper conditions. Numerical results are also provided to show the performance of the proposed scheme and validate theoretical analysis.