A three dimensional finite element method and multigrid solver for a Darcy-Stokes system and applications to vuggy porous media

Abstract

A vuggy porous medium is one with many small cavities called vugs, which are interconnected in complex ways forming channels that can support high flow rates. Flow in such a medium can be modeled by combining Darcy flow in the rock matrix with Stokes flow in the vugs. We develop a finite element for the numerical solution of this problem in three dimensions, which converges at the optimal rate. We design a multigrid method to solve a saddle point linear system that comes from this discretization. The intertwining of the Darcy and Stokes subdomains in a natural vuggy medium makes the resulting matrix highly oscillating, or ill-conditioned. The velocity field we are trying to compute is also very irregular and its tangential component might be discontinuous at the Darcy-Stokes interface. This imposes a difficulty in defining intergrid transfer v operators. Our definition is based on mass conservation and the analysis of the orders of magnitude of the solution. A new smoother is developed that works well for this ill-conditioned problem. We prove that coarse grid equations at all levels are well posed saddle point systems. Our algorithm has a measured convergence factor independent of the size of the system. We then use our solver to study transport and flow properties of vuggy media by simulations. We analyze the results of our transport simulations and compare them to experimental results. We study the influence of vug geometry on the macroscopic flow properties of a three dimensional vuggy porous medium.