An upwind finite volume method on non-orthogonal quadrilateral meshes for the convection diffusion equation in porous media

Abstract

The aim of this work is to solve the two-dimensional convection diffusion equation on non-rectangular grids formed only by quadrilaterals honoring the internal structures of a reservoir (preferential flow channels, faults, areas of high permeability contrast, changes in sediment type, etc.), taking into account different physical configurations of the porous medium. To take advantage of the good representation of the domain through these meshes, the finite volume method was used, which is conservative and facilitates the treatment of the boundary conditions. In this method, the convection diffusion equation is integrated on each quadrilateral (control volume) of the mesh, thus obtaining the integral form of the equation. The velocity value in the face of each quadrilateral is determined according to the direction of the flow (upwind scheme). After approximating the integrals involved and taking into account the boundary conditions, a discrete equation in each control volume showed up. Finally, a large sparse linear system is obtained, generally non-symmetric and ill-conditioned, which can be solved by iterative methods such as GMRES with incomplete LU preconditioning. Different scenarios were considered varying boundary conditions (Dirichlet and Neumann type), source term, and diffusion constant fluid velocity. The results are consistent with the physical interpretation of each configuration.