The First- and Second-Order Energy Stable, Mass Conservative and Bounds-Preserving Schemes for Two-Phase Incompressible Flow with Rock Compressibility

In this paper, we construct a unified framework of first- and second-order schemes in time for thermodynamically consistent modeling of two-phase incompressible and immiscible flow in porous media with rock compressibility. We rigorously prove that the proposed schemes can preserve the energy dissipation law, conserve the mass of each phase as well as pore volumes and are bounds-preserving for both phases without any restrictions on the time step size. Moreover, the proposed schemes can achieve local mass conservation for both phases and preserve the pore volumes when the saturation of the whole region is within the bounds, and we only need to solve one linear system and several linear algebraic equations, whereas for points where saturation is outside the bounds, we just need to solve an additional nonlinear algebraic equation and hence the proposed schemes can achieve global mass conservation. The developed schemes exhibit high efficiency owing to a substantial reduction in the scale of nonlinear computations and their straightforward implementation. The key point is that we propose a new Lagrange multiplier method that is based on the judgment of saturation bound. This addresses the limitation of the classical Lagrange multiplier method, which is only capable of ensuring global mass conservation. Finally, a variety of illustrative numerical examples including several benchmark problems are provided to validate the accuracy and efficiency of the proposed schemes.