Efficient design of continuation methods for hyperbolic transport problems in porous media

Abstract

Full-physics modeling of multiphase flow in porous media, e.g., for carbon storage and groundwater management, requires the nonlinear coupling of various physical processes. Industry standard nonlinear solvers, typically of Newton-type, are not unconditionally convergent and computationally expensive. Homotopy continuation solvers have recently been studied as a robust and versatile alternative. They tackle challenging nonlinear problems by first solving a simple auxiliary problem and then tracing a solution curve towards the more complex target problem. Robustness and efficiency of the method depends on the iterative numerical curve tracing algorithm as well as on careful design of the auxiliary problem. We assess the traceability of the solution curve for different choices of the auxiliary problem. For the Buckley-Leverett equation, modeling two-phase flow in one dimension, we exemplarily compare the previously introduced vanishing-diffusion and linear constitutive laws homotopy continuation, and a new approach based on the entropy solution of the problem. This provides insight toward systematically and robustly designing homotopy continuation methods for solving complex multiphase flow in porous media.

Figures (5)

• Figure 1: Curvature and relative convergence metrics for Case 1 ($M=1$) and Case 2a ($M=10$).
• Figure 2: Curvature and relative convergence metrics for Case 2b ($M=10, S_i=1.0$) and Case 2c ($M=10, S_i=0.8$).
• Figure 3: Target, linear relative permeabilities, and convex hull flow functions for the different cases.
• Figure 4: Evolution of the solution along the HC curve beginning from the different auxiliary problems for case 2a.
• Figure 5: Solution curve for Case 2b using the convex/concave hull HC.