Summary: We consider the numerical solution of the partial differential equations governing multiphase flow in porous media. For highly nonlinear problems, the temporal discretization of choice is often the unconditionally stable fully implicit method. However, the nonlinear systems, often solved with Newton’s method, are difficult to solve. Thus, the computational cost is strongly dependent on the nonlinear convergence rate, and enhancing this convergence property is key to speed up subsurface flow simulation.
We focus on the case of spatially discontinuous capillary pressure between rock regions. To efficiently and accurately simulate the flow dynamics in heterogeneous porous media, the flux computation combines Implicit Hybrid Upwinding with transmission conditions between different rock regions. This leads to a scheme that correctly represents the trapping mechanisms while improving the nonlinear convergence.
We extend our previous results [“Hybrid upwinding for two-phase flow in heterogeneous porous media with buoyancy and capillarity”, in: Proceedings of the15th European conference on the mathematics of oil recovery, ECMOR XV. Red Hook, NY: Curran Associates, Inc. (2016; doi:10.3997/2214-4609.201601851)] by generalizing the scheme to fully implicit coupled flow and transport to address realistic problems in multiple dimensions. The generalized scheme is supported by an analysis of its mathematical properties. Our multidimensional numerical examples, which range from buoyancy-driven flow with capillary barriers to viscous-dominated flow, demonstrate that the Implicit Hybrid Upwinding scheme improves the accuracy compared to the standard phase-based upwinding scheme, while leading to significant reductions in the number of nonlinear iterations in multiple dimensions.