A two-step finite volume method for the simulation of multiphase fluid flow in heterogeneous and anisotropic reservoirs

Abstract

Efficient and accurate simulation of multiphase flow and transport in porous media plays a key role in optimizing recovery of hydrocarbon resources. Two-Point Flux Approximation (TPFA) is widely used in the industry for discretizing the pressure equation because of its robustness and efficiency despite the fact that it introduces O(1) flux errors on non-K-orthogonal grids. Multi-Point Flux Approximation (MPFA) provides a consistent discretization of the flow equations at the cost of larger flux stencils which leads to a denser, asymmetric system of linear equations requiring more computer memory and work for linear solvers. This drawback of MPFA becomes more pronounced for generally unstructured triangular grids and is even worse in 3D applications. In this work, we develop a two-step finite volume method (TSFVM) that is consistent for the anisotropic pressure equation on triangular and tetrahedral grids with improved efficiency as compared to MPFA methods. During each time step, the pressure equation is solved implicitly and saturation is then updated explicitly (IMPES). To discretize the pressure equation with full permeability tensors, Galerkin finite element method (FEM) is employed in the first step to compute pressure at grid vertices. Pressures at grid vertices are then utilized in the second step to derive a two-point flux stencil with a constant for each grid face. Pressure at cell centers and flux across grid faces can then be solved using a TPFA approach. Finally, the flux field at the new time level is used to update saturations explicitly. Although during each time step our TSFVM needs to solve two systems of linear equations, results of the numerical experiments clearly demonstrate that it is still less expensive than the classical MPFA-O method while providing high-quality numerical solutions matching that of MPFA-O. The gained efficiency is especially significant for 3D tetrahedral grid. As the scale of the problem increases, MPFA becomes less appealing because of its large flux stencils that result in much denser matrices while our TSFVM still remains computationally tractable thanks to the flexibility of FEM in the first step and the two-point flux stencil in the second step. On the other hand, the first step of the TSFVM is reminiscent of the Control Volume Finite Element Method (CVFEM) but with important differences. Unlike CVFEM, our TSFVM does not need to construct a dual control volume mesh around grid vertices on the basis of the primary mesh and the saturation unknowns are not associated with node-centered control volumes but are piece-wise constant on grid cells in which reservoir properties are distributed. This ensures that a saturation value does not straddle across material discontinuities which CVFEM suffer from.