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

TL;DR

This paper investigates the design of homotopy continuation methods to improve robustness and efficiency in solving nonlinear multiphase flow problems in porous media, with a focus on the Buckley-Leverett equation.

Contribution

It compares different auxiliary problems for homotopy continuation, including a new entropy-based approach, to enhance solution traceability and robustness.

Findings

Entropy-based homotopy improves robustness.

Linear and vanishing-diffusion homotopies have varying traceability.

Systematic design principles for auxiliary problems are proposed.

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.