In this paper, we consider the numerical approximation for a phase field model of the coupled two-phase free flow and two-phase porous media flow. This model consists of Cahn-Hilliard-Navier-Stokes equations in the free flow region and Cahn-Hilliard-Darcy equations in the porous media region that are coupled by seven interface conditions. The coupled system is decoupled based on the interface conditions and the solution values on the interface from the previous time step. A fully discretized scheme with finite elements for the spatial discretization is developed to solve the decoupled system. In order to deal with the difficulties arising from the interface conditions, the decoupled scheme needs to be constructed appropriately for the interface terms, and a modified discrete energy is introduced with an interface component. Furthermore, the scheme is linearized and energy stable. Hence, at each time step one need only solve a linear elliptic system for each of the two decoupled equations. Stability of the model and the proposed method is rigorously proved. Numerical experiments are presented to illustrate the features of the proposed numerical method and verify the theoretical conclusions.
B111However, most of the existing works on the Stokes-Darcy model are devoted to single-phase flows, and hence they cannot be used for the applications of multiphase flow. Recently, researchers [19,28,48,51,53] started to develop the models to couple the porous media flow with the two-phase free flow, for which the phase field method is adopted. In [19], Chen, Sun, and Wang utilized the Cahn-Hilliard-Navier-Stokes (CHNS) equation to govern the two-phase fluid system, and the two-phase Darcy's law for the two-phase porous medium flow. The coupling between these two equations is through the interface conditions including the generalized Navier boundary condition [80,81,82]. In [28], a modified Cahn-Hilliard equation is coupled with an unsteady Darcy-Stokes model, which uses the same equation with different coefficients for both Darcy and Stokes flows without interface transmission conditions. In [48,51,53], a Cahn-Hilliard-Stokes-Darcy (CHSD) system and a Cahn-Hilliard-Navier-Stokes-Darcy (CHNSD) system are developed for two-phase incompressible flows in the karstic geometry. It is remarkable that the CHNS equation is utilized to govern the two-phase free fluid, which is similar to [19]. However, in the porous media region, the Cahn-Hilliard-Darcy (CHD) equation is used instead of the twophase Darcy's law. Therefore, the interface conditions in [48,51,53] couple these two equations together that are different from those of [19]. Furthermore, two coupled, unconditionally stable numerical algorithms, which combine two Cahn-Hilliard equations in the Stokes and Darcy domains into one Cahn-Hilliard equation on the whole domain, are proposed and analyzed for the CHSD system in [48]. A "partially" decoupled and nonlinear scheme, which still directly solves one Cahn-Hilliard equation on the whole domain but decouples the Stokes and Darcy...