Multiphase Flows - Nonlinear Monotone FV Scheme and Dynamic Grids

Abstract

We consider development of the nonlinear monotone FV scheme and its application to two- and three-phase flow models. The scheme is applicable for full anisotropic discontinuous permeability tensors and arbitrary conformal polyhedral cells. The nonlinear scheme is compared with conventional linear approaches: two-point and O-scheme multipoint flux approximations. The new nonlinear scheme has a number of important advantages over the traditional linear discretizations. Compared to the linear TPFA, the nonlinear scheme demonstrates low sensitivity to grid distortions and provides appropriate approximation in case of full anisotropic permeability tensor. For non-orthogonal grids or full anisotropic permeability tensors the conventional linear TPFA 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 TPFA, yet it is rather competitive. Compared to MPFA, the new scheme provides sparser algebraic systems and thus is less computational expensive. Moreover, it is monotone which means that the discrete solution preserves the non-negativity of the differential solution. We consider using of the dynamic octree-based grids for better recovery of pressure gradient and saturation fronts propagation.