Robust and Efficient Reduced Newton (Seidel) Methods Based on Residual Ordering

Abstract The Sequential Fully Implicit formulation for reservoir simulation allows for specialized solution methods for the flow and transport subproblems. The transport subproblem is highly nonlinear, with flux functions that are non-monotonic and non-convex. It is also characterized by local support for error propagation across the domain, and uni-directionality along streamlines in the absence of gravity and capillarity. By reordering the cells along the directed dependency graph corresponding to inter-cell flux direction, which can be achieved by topological sorting, the system is guaranteed to converge in a single sweep of non-linear single-cell updates (Gauss-Seidel). This has been extended to handle gravity-driven countercurrent flow by grouping cyclically dependent cells into clusters to be solved simultaneously, such that the entire system can still be solved in a single pass of nonlinear block Gauss-Seidel iterations. Capillarity leads to global but geometrically decaying support, fusing the entire system into a single cluster. The Gauss-Seidel method can still be effective for this problem but would typically require multiple sweeps. We present a residual-error-based Seidel scheme that improves convergence speed for general Advection-Diffusion problems. We extend the Seidel scheme to residual-based block method, and show its robustness to increased diffusion, which otherwise leads to severe oscillations that slow than Seidel convergence. In our 1D numerical experiments with a relatively fine grid and high CFL number, the computational speedup of the proposed residual-based block methods over spatially ordered blocks with optimal spatial block sizes range from 5 in a low diffusion case to 30 in a high diffusion case. The proposed reduced Newton methods generally outperform Newton's method by orders of magnitude. We further validate the proposed scheme by simulating water-flooding problems on a highly heterogeneous reservoir with speed-ups ranging from a factor of 2 to 5, depending on the extent of gravitational influence.