Control volume function approximation methods and their applications to modeling porous media flow II: the black oil model

This is the second paper of a series where we introduce control volume function approximation (CVFA) methods and present their applications to modeling porous media flow. In the first paper, we introduced these methods to solve linear partial differential equations (PDE) in two-dimensional space and applied them to modeling two-phase flow. In this paper we extend them to solving coupled nonlinear PDEs in three-dimensional space and present their applications to reservoir simulation using the black oil model. The numerical results show that the CVFA methods are efficient and accurate for solving ‘crossing bubble point’ and three-phase coning problems. For a large model problem with over 20,000 nodes, the computational cost of the CVFA methods is virtually the same as that of the control volume finite element methods, slightly more than that of the 5-point finite difference (FD) method (the 7-point in three dimensions), and less than that of the 9-point FD method (the 11-point in three dimensions).