Energy stable modeling of two-phase flow in porous media with fluid–fluid friction force using a Maxwell–Stefan–Darcy approach

Abstract

Two-phase incompressible flow in porous media plays an important role in various fields including subsurface flow and oil reservoir engineering. Due to the interaction between two phases flowing through the pores, the fluid–fluid friction force may have a significant effect on each phase velocity. In this paper, we propose an energy stable (thermodynamically consistent) Maxwell–Stefan–Darcy model for two-phase flow in porous media, which accounts for the fluid–fluid friction. Different from the classical models of two-phase flow in porous media, the proposed model uses the free energy to characterize the capillarity effect. This allows us to employ the Maxwell–Stefan model to describe the relationships between the driving forces and the friction forces. The driving forces include the pressure gradient and chemical potential gradients, while both fluid–solid and fluid–fluid friction forces are taken into consideration. Thermodynamical consistency is the other interesting merit of the proposed model; that is, it satisfies an energy dissipation law and also obeys the famous Onsager's reciprocal principle. A linear semi-implicit numerical method is also developed to simulate the model. Numerical simulation results are provided to show that the fluid–fluid friction force can improve the oil recovery substantially during the oil displacement process.