Hybrid Upwinding for Two-phase Flow in Heterogeneous Porous Media with Buoyancy and Capillarity

Numerical simulation of subsurface flow requires an efficient solution strategy for the partial differential equations governing coupled multiphase flow and transport in porous media. A common characteristic of geological porous media is their highly heterogeneous structure. Heterogeneity is a challenge for numerical simulation, as variations by orders of magnitude in the permeability and porosity fields create a wide range of time scales in the transport problem. Therefore, in the solution strategy, the temporal discretization of choice is often the unconditionally stable fully implicit method. However, the nonlinear systems arising from this discretization – often solved with Newton’s method with global damping –, are highly nonlinear and difficult to solve. Thus, the overall computational cost is strongly dependent on the nonlinear convergence rate, and enhancing this nonlinear convergence property is key to speed up flow simulation. We improve the robustness and nonlinear convergence with an efficient fully implicit, finite-volume scheme for immiscible two-phase flow in the presence of viscous, buoyancy, and capillary forces. Each term in the numerical flux is treated separately based on physical considerations to obtain a differentiable and robust discretization. Following the Implicit Hybrid Upwinding strategy (Lee et al., Advances in Water Resources, 2015), the viscous term is upwinded based on the sign of the total velocity, whereas the directionality of the gravity part is determined by the density differences. In this paper, the emphasis is on the discretization of the capillary flux, which is composed of a rock- and geometry-dependent transmissibility, a nonlinear diffusion coefficient, and a saturation gradient. We propose a discretization that yields a consistent, bounded, and differentiable numerical flux. Importantly, the numerical flux in the presence of capillary forces is monotone. We show that the monotonicity of the numerical flux is critical for the robustness of the scheme when applied to heterogeneous porous media. Our numerical examples, which range from buoyancy-driven flow with capillary barriers to viscous-dominated flow, demonstrate that the Implicit Hybrid Upwinding scheme leads to significant reductions in the number of nonlinear iterations compared with the standard phase-based upwinding schemes.