Analysis of Hybrid Upwinding for Fully-Implicit Simulation of Three-Phase Flow with Gravity

Abstract

The systems of algebraic equations arising from implicit (backward-Euler) finite-volume discretization of the conservation laws governing multiphase flow in porous media are quite challenging for nonlinear solvers. In the presence of countercurrent flow due to buoyancy, the numerical flux obtained with single-point Phase-Potential Upwinding (PPU) is not differentiable, which causes convergence difficulties for nonlinear solvers. Recently, [Lee, Efendiev, and Tchelepi, Adv. Water Resour., 82 (2015), pp. 27--38] proposed a hybrid upwinding strategy for two-phase flow that gives rise to a differentiable numerical flux across the entire viscous-gravity parameter space. Here, we first present an Implicit Hybrid Upwinding (IHU) scheme for hyperbolic conservation laws, extending the work of Lee, Efendiev, and Tchelepi to an arbitrary number of fluid phases. We show that the numerical flux obtained with the IHU is consistent, and a monotone function of its own saturation. It is also a differentiable function of the saturations in one spatial dimension. In addition, we generalize the IHU numerical scheme to solve the elliptic-hyperbolic partial differential equations governing coupled flow and transport in multiple dimensions. For this problem, we derive pressure and saturation estimates, and prove the existence of a solution to the scheme. Finally, we apply the IHU scheme to the Buckley--Leverett problem with buoyancy. Our numerical experiments confirm that IHU is nonoscillatory, convergent, and improves upon Newton's method convergence rate for two- and three-phase flow.