An energy stable and positivity-preserving computational method for compressible and immiscible two-phase flow in porous media

Multiple coupled physical processes of compressible and immiscible two-phase flow in porous media give rise to substantial challenges in the development of effective computational methods preserving relevant physical laws and properties. In this paper, we propose an energy stable, positivity-preserving and mass conservative numerical method for compressible and immiscible two-phase flow in porous media with rock compressibility. In order to design this method, we first propose an alternative formulation of the model by taking fluid densities as the primary variables rather than pressures as well as chemical potential gradients instead of pressure gradients as the primary driving forces. We introduce two-phase free energy functions accounting for fluid compressibility, the interfacial energy for capillary effect and rock energy for rock compressibility, which allow to prove that the alternative model follows an energy dissipation law. Applying the convex-splitting approach to treat the energy functions, we design the discrete chemical potentials, which are the keys to preserve the positivity of densities and saturations. We take some subtle treatments for coupled relationships between multiple variables and physical processes; in particular, we introduce proper implicit and explicit mixed treatments to construct the approximations of two phase pressures and the effective pore pressure. Both semi-discrete and fully discrete forms of the scheme are proved to preserve the original energy dissipation law. Moreover, we prove that the fully discrete scheme can guarantee the boundedness of saturations and the positivity of porosity and two phase densities without extra operations and restrictions on time steps and mesh sizes. Additionally, the scheme has the ability to conserve the mass of each phase even in the presence of the changes of fluid densities and porosity. Numerical results are also provided to demonstrate that the performance of the proposed scheme is in agreement with the theoretical analysis.