Nonlinear analysis of multiphase transport in porous media in the presence of viscous, buoyancy, and capillary forces

Based on our understanding of the nonlinearity, a nonlinear solver is developed, referred to as the Numerical Trust Region (NTR) solver. The solver is able to guide the Newton iterations safely and efficiently through the different saturation 'trust-regions' delineated by the kinks and inflections. Specifically, overshoots and oscillations that often lead to convergence failure are avoided. Numerical examples demonstrate that our NTR solver has superior convergence performance compared with existing methods. In particular, convergence is achieved for a wide range of timestep sizes and Courant-Friedrichs-Lewy (CFL) numbers spanning several orders of magnitude. In addition, a discretization scheme is proposed for handling heterogeneities in capillary-pressure-saturation relationship. The scheme has less degree of nonlinearity compared with the standard Single-point Phase-based Upstream weighting scheme, leading to an improved nonlinear convergence performance especially when used together with our NTR solver. Our proposed numerical solution strategy that is based on the numerical flux and handles capillarity extends the previous work by Jenny et al. (2009) [6] and Wang and Tchelepi (2013) [7] significantly.