A single mesh approximation for modeling multiphase flow in heterogeneous porous media

Abstract

The control volume finite element (CVFE) approach is based on the discretization of primary flow and transport unknowns on two different meshes. The element mesh, used to represent the physical properties of the medium, differs from the control volume mesh necessary to ensure a mass conservative solution. The inherent two mesh feature of the CVFE approximation introduces inconsistency in the transport solution due to the averaging of physical and computed quantities between neighboring elements in the mesh. In this work, we present a consistent approach for modeling multiphase flow and transport in heterogeneous porous media. The approach is applied in the CVFE framework by enabling the discretization of primary unknowns on a single mesh. We combine the discretization of an element-wise, discontinuous pressure approximation with a first-order, discontinuous velocity approximation to resolve the elliptic or parabolic flow problem. The effectiveness of the formulation is achieved by exploiting the same finite element mesh as the flow problem, when updating the saturation solution. This direct mapping between the flow and transport mesh is simple yet effective for establishing a consistent solution in addition to circumventing the non-physical mass leakage exhibited in the classical CVFE method. We describe the interface approximations and the discontinuous terms needed for consistent solutions. The method is well suited to model flow in complex geometrical subsurface domains and is shown to be numerically stable while providing locally and globally mass conservative solutions. We apply the approach to several domains with complex geometrical features to emphasize the superiority of the approximation over conventional methods. The analysis shows over two orders of magnitude reduction in solution error compared to the classical CVFE. The new formulation effectively captures accurate transport solutions, even in the presence of varying material properties, without the need for mesh refinement.