a numerical method for a model of two phase flow in a coupled free flow and porous media system
DESCRIPTION
A Numerical Method for a Model of Two Phase Flow in a Coupled Free Flow and Porous Media SystemTRANSCRIPT
1 BACKGROUND !
Jie Chen1, Shuyu Sun1,2, Xiaoping Wang3!1School of Mathematics and Statistics, Xi'an Jiaotong University, China!
2Division of Mathematics and Computer Science and Engineering, King Abdullah University of Science and Technology, Kingdom of Saudi Arabia!3Department of Mathematics, The Hong Kong University of Science and Technology, Hong Kong!
A numerical method for a model of two phase flow in a coupled free flow and porous media system!
3 METHODOLOGY!
We study two-phase fluid flow in
a coupled free flow and porous media regions.!
The problem arises in several industrial applications, such as well-reservoir coupling in petroleum engineering, fluid-organ interactions
and industrial filtering where a fluid passes through a filter to remove unwanted particles.
The model we use consists of a coupled Cahn-Hilliard and Navier-Stokes equations for the
free flow region and a two-phase Darcy law for the two-phase flows in the porous medium
region.!
2 Why / PURPOSE!
4 RESULTS!
T. Qian, X. P. Wang, and P. Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rew. E 68(2003) 016306. M. Gao, and X. P.Wang, A gradient stable scheme for a phase field model for the moving contact line problem, J. Comput. Phys. 231 (2012) pp. 1372–1386. G. S. Beavers and D. D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30(1967), pp. 197–207.
In Ωf , we assume that the two flow is governed by a coupled system of Cahn-
Hilliard Navier-Stokes equations.!
The velocity along the solid boundary satifies the generalized Navier boundary condition (GNBC).
where!
For the porous media domain Ωp, we assume that the flow is governed by mass balance equation and Darcyʼs law.
To couple the two flow systems, several transmission conditions are imposed on the interface boundary .
1. Solve the Cahn-Hilliard equations with the semi-implicit method and finite element discretization to obtain ϕn+1 and μn+1 with a convex splitting scheme.!
2. With the obtained ϕn+1 and μn+1, the Navier-Stokes equations are solved with a P2-P0 mixed finite element method. !
3. The Darcyʼs law is solved by IMPES method.!
The Robin-Robin iterative method is used to decouple the system
Given initial values ηf and ηp which may be taken to be zeroes. ηf and ηp are updated in the following manner:
A droplet passing through interface
Static contact angle is 60 degree at interface and the top and bottom boundaries!
Static contact angle is 90 degree at interface and 120 degree at the top and bottom boundaries!
Static contact angle is 60 degree at interface and the top and bottom boundaries!
A layer of non-wetting phase passing through an interface
Conclusion!The two-phase flow crossing the interface between free flow region and porous media region is considered. Based on a semi-implicit finite element method for the coupled Cahn-Hilliard and Navier-Stokes system and IMPES method for the two-phase Darcy law, we propose a Robin-Robin domain decomposition method for the coupled system with the generalized Navier interface boundary condition at the interface between the free flow and the porous media regions. The effectiveness of this method is illustrated with several numerical examples.
Why!1. Coupling problem of two-phase flow is a challenging
problem in fluid dynamics.!2. The problem has industrial applications.!
Purpose!1. Build a model for the coupling problem of two-phase
flow!2. Find a effective numerical method to solve this
problem!