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

¹ Department of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an Shaanxi 710129, P.R. China
² Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA
³ Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
⁴ Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

^* Corresponding author: hex@mst.edu

Received: 28 May 2022
Accepted: 2 February 2023

Abstract

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.