A quasi-incompressible Cahn–Hilliard-Darcy model for two-phase flows in porous media

Two-phase flows in porous media is known as the Muskat problem. The Muskat problem can be ill-posed. In this talk we introduce a quasi-incompressible Cahn-Hilliard-Darcy model as a relaxation of the Muskat problem.  We establish global existence of weak solution to the model.  We then present a second-order accurate bound-preserving and unconditionally stable numerical method for solving the equations.