An energy stable, conservative and bounds-preserving numerical method for thermodynamically consistent modeling of incompressible two-phase flow in porous media with rock compressibility

In this paper, we consider modeling and numerical simulation of incompressible and immiscible two-phase flow in porous media with rock compressibility. Using the second law of thermodynamics, we rigorously derive a thermodynamically consistent mathematical model, which characterizes the two-phase capillarity and rock compressibility through free energies, thereby following an energy dissipation law. We also derive a general and thermodynamically consistent formulation for the effective pore fluid pressure acting on rocks, which is a fundamental problem for two-phase flow with rock compressibility. To solve the model effectively, we propose an energy stable numerical method, which can preserve multiple physical properties, including the energy dissipation law, full conservation law for both fluids and pore volumes, and positivity of porosity and saturations. Benefiting from the newly-developed energy factorization approach and careful treatments for the effective pressure and porosity, the proposed scheme can inherit the energy dissipation law at the discrete level. The fully discrete scheme is constructed using a locally conservative cell-centered finite difference method. The implicit strategy is applied to treat the upwind phase mobilities and porosity in the phase mass conservation equations and the porosity equation so as to conserve the mass of each phase as well as pore volumes. The positivity of porosity and saturations is proved without any restrictions on time step and mesh sizes. An efficient sequential iterative method is also developed to solve the nonlinear system resulting from the scheme. Finally, numerical results are given to verify the features of the proposed method.