Trust-region Based Nonlinear Solver for Counter-current Two-phase Flow in Heterogeneous Porous Media

We describe a new nonlinear solver for immiscible two-phase flow where viscous, buoyancy, and capillary forces are significant. The flux function is a nonlinear function of saturation and typically has inflection points and a unit-flux point. The non-convexity of flux function is a major source of convergence difficulty for nonlinear solvers. We describe a modified Newton solver that employs trust-regions of the flux function to guide Newton iterations and solution updating. The flux function is divided into saturation trust regions. The delineation of these regions is dictated by the inflection and unit-flux points. Newton update is performed such that two successive iterations cannot cross any trust-region boundary. If a crossing is detected, we "chop back" the saturation value at the appropriate trust-region boundary. This development is a significant generalization of the inflection-point approach of Jenny et al. (JCP, 2009) for viscous dominated flows. Mathematically we prove the global convergence of the trust-region based nonlinear solver. Numerically we test it for multiphase flow and transport in large-scale heterogeneous problem. Using our new nonlinear solver, we achieved significant reduction in the total Newton iterations by more than an order of magnitude together with a corresponding reduction in the overall computational cost.