Fully decoupled energy-stable numerical schemes for two-phase coupled porous media and free flow with different densities and viscosities

In this article, we consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation.  This model consists of three parts: a Cahn-Hilliard-Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn-Hilliard-Navier-Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit.  Based on the separate Cahn-Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the two-phase systems of the two regions and the seven interface conditions between them, and the corresponding energy law is proved for the model. A fully decoupled numerical scheme, including the novel decoupling of the Cahn-Hilliard equations through the four phase interface conditions, is developed to solve this coupled nonlinear phase field model. An energy-law preservation is analyzed for the temporal semi-discretization scheme.  Furthermore, a fully discretized Galerkin finite element method is proposed. Six numerical examples are provided to demonstrate the accuracy, discrete energy law, and applicability of the proposed fully decoupled scheme.