Trust-region based solver for nonlinear transport in heterogeneous porous media

We analyze the discrete nonlinear transport equation obtained using finite-volume discretization with phase-based upstream weighting. Then, we prove convergence of the trust-region Newton method irrespective of the timestep size 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 scheme is unconditionally convergent for 1D transport. That is, for a given choice of timestep size, the unique discrete solution is found independently of the initial guess. For problems dominated by buoyancy and capillarity, the trust-region 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 reservoir models with strong heterogeneity, we compare it with state-of-the-art nonlinear solvers for two-phase flow and transport using the SPE 10 model. Compared with existing nonlinear solvers, our trust-region solver results in superior convergence performance and achieves reduction in the total Newton iterations by more than an order of magnitude together with a corresponding reduction in the overall computational cost.