Trust-Region Based Nonlinear Solver for Multiphase Flow in Heterogeneous Porous Media

We describe a nonlinear solver for immiscible two-phase transport in heterogeneous porous media, where viscous, buoyancy, and capillary
forces are all present and significant. The flux (phase velocity) is a nonlinear function of saturation and typically has an inflection
point. With strong buoyancy effects, the flux can be non-monotonic with additional inflection points. The non-convexity and non-monotonicity of the flux function are major sources of difficulty
for nonlinear solvers of transport in natural porous media. We present a modified-Newton algorithm for the numerical simulation of two-phase flow that employs trust-regions of the flux function to guide the Newton iterations. The flux function is divided into saturation trust
regions. The delineation of these regions is dictated by the inflection, unit-flux, and end points. The updates are performed such that two successive iterations cannot cross any trust-region boundary. If a crossing is detected, we chop the saturation value back the
appropriate trust-region boundary.

We analyze the nonlinear transport equation with finite-volume discretization and phase-based upstream weighting. Then, we prove
convergence of the trust-region based Newton method for arbitrary timestep sizes for single-cell problems. Numerical results across the
full range of the parameter space of viscous, gravity and capillary forces indicate that our trust-region based scheme is unconditionally
convergent for 1D transport. That is, for a given choice of timestep size, the unique saturation solution is found independently of the initial guess. For problems dominated by buoyancy and capillarity, the trust-region based Newton solver overcomes the often severe limits on
timestep size associated with existing methods. To validate the effectiveness of the new nonlinear solver for large-scale,heterogeneous reservoir models, we compare it with the current
state-of-the-art nonlinear solvers for two-phase flow and transport using the SPE 10 model problem (highly heterogeneous 3D model with more than 1 million cells). Compared with the existing nonlinear solvers, our trust-region based nonlinear solver results in superior
convergence performance and achieves significant reduction in the total Newton iterations by more than an order of magnitude along with a corresponding reduction in the overall computational cost.