(An) immersed finite element method for computing a two-phase flow in porous media

Abstract

We develop three numerical schemes for two-phase flows in heterogeneous porous media based on immersed finite element method (IFEM). First one is based on the implicit pressure-explicit saturation procedure. To solve the pressure equation, we use edge average degree of freedom based IFEM and mixed finite volume (MFVM) frameworks, where the Darcy velocity is computed locally from the pressure variables. To enhance the accuracy the saturation variables are solved by control volume methods with upwinding. Moreover, multigrid solver are applied where the performance of multigrid is optimal in scalability for the examples we tested. Second scheme is based on the Euler backward scheme. Both the pressure and saturation variables are solved in IFEM space where the degree of freedoms are defined on nodes. We show that the proposed scheme has exact solution and proved optimal error estimates in energy like norms. The last scheme is made for the special case where the capillary functions are discontinuous along the interface. When the capillary pressures are discontinuous, both the saturation and pressures become discontinuous. To deal with the discontinuity, we applied discontinuous bubble IFEM. All of the methods are implemented on a structured grid which is independent of the underlying heterogeneous porous media. There are many advantages of using structured grid, since the arising discrete system have the simple data structure. For all the schemes, we provide various numerical results. We observe optimal convergence rates for pressure and velocity variables for analytic problems. We also tested our schemes to well-known examples such as "five-spot flooding" and DNAPL infiltration. We observe plausible numerical solutions without oscillation phenomena are obtained. Thus, we are able to provide accurate and stable numerical solutions for various heterogeneous media problems.