On the coupling of two-phase free flow and porous flow

The coupling of free flow with porous flow is of special interest in a wide range of environmental phenomena and industrial applications. In this work, we extend the classical single-phase two-domain model to a laminar two-phase coupling flow system. The free fluid region can be considered as separated two-phase flow for simplicity, which is modeled by using the Navier-Stokes and Cahn-Hilliard equations. And the mathematical model of two-phase flow in porous media is based on Darcy's theory. The main challenge is how to introduce specific interface conditions to couple these two models. To this end, the normal continuity conditions of flux and forces are developed, and an extended Beavers-Joseph-Saffman condition for two-phase flow system is also proposed as a Cauchy boundary condition based on consistent phenomenological explanations. These lead to a simple and solvable coupling model, and an efficient finite element numerical scheme is developed. The numerical results show that the developed model is capable to capture the macroscopic flow characteristics of laminar two-phase coupling flow system by comparing to the experimental results. Our model can be used to model the related two-phase flow process in karstic aquifers and fractured reservoirs, and wind-driven evaporation from soil.