Hybrid Upwind Discretization for the Implicit Simulation of Three-Phase Coupled Flow and Transport with Gravity

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 counter-current flow due to buoyancy, the coupling between flow (pressure) and transport (saturations) is often the cause of nonlinear problems when single-point Phase-Potential Upwinding (PPU) is used. To overcome such convergence problems in practice, the time step is reduced and Newton's method is restarted from the solution at the previous converged time step. Here, we generalize the work of Lee, Efendiev and Tchelepi [Advances in Water Resources, 2015] to propose an Implicit Hybrid Upwinding (IHU) scheme for coupled flow and transport. In the pure transport problem, we show that the numerical flux obtained with IHU is differentiable, monotone and consistent for two and three-phase flow. For coupled flow and transport, we prove saturation physical bounds as well as the existence of a solution to our scheme. Challenging two- and three-phase heterogeneous multi-dimensional numerical tests confirm that the new scheme is non-oscillatory and convergent, and illustrate the superior convergence rate of our IHU-based Newton solver for large time steps.