A Monotone Non-linear Finite Volume Method for Advection-diffusion Equations and Multiphase Flows

We present a new nonlinear monotone finite volume method for diffusion and convection-diffusion equations and its application to two-phase black oil models. We consider full anisotropic discontinuous diffusion/permeability tensors and discontinuous velocity fields on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which reduces to the conventional 7-point stencil for cubic meshes and diagonal tensors. The approximation of the advective flux is based on the second-order upwind method with the specially designed minimal nonlinear correction. We show that the quality of the discrete flux in a reservoir simulator has great effect on the front behavior and the water-breakthrough time. We compare the new nonlinear two-point flux discretization with the conventional linear two-point scheme. The new nonlinear scheme has a number of important advantages over the traditional linear discretization. First, it demonstrates low sensitivity to grid distortions. Second, it provides appropriate approximation in the case of full anisotropic permeability tensor. For non-orthogonal grids or full anisotropic permeability tensors the conventional linear scheme provides no approximation, while the nonlinear flux is still first-order accurate. The computational work for the new method is higher than the one for the conventional dicretization, yet it is rather competitive.