Development of Discontinuous Galerkin Methods and a Parallel Simulator for Reservoir Simulation

Abstract

Abstract The classical discontinuous Galerkin (DG) methods are designed for elliptic (parabolic) problems and hyperbolic problems. For reservoir simulations, the pressure equation from the black oil model is elliptic (parabolic), while the equations for saturations are hyperbolic. Due to this special property, it is difficult to directly apply the discontinuous Galerkin methods to the black oil model. In this paper, we extend the discontinuous Galerkin methods to reservoir simulations. In our schemes, the local discontinuous Galerkin (LDG) method is used to discretize the black oil model. The upwind concept is combined with the numerical flux term of the LDG method to simulate the direction of propagation of the multiphase flow in reservoirs to avoid the unphysical solutions. We also extend the Peaceman model to the discontinuous Galerkin methods on unstructured grids. Based on the extended discontinuous Galerkin methods, we employ the iterative implicit pressure-explicit saturation (iterative-IMPES) and fully implicit (FIM) methods to solve the coupled nonlinear black oil model. A parallel simulator is implemented using the parallel adaptive finite element package, Parallel Hierarchical Grid (PHG), and validated by testing the first and ninth SPE Comparative Solution Projects. The parallel scalability of our simulator is also tested by a large scale case.