Adaptive homotopy continuation solver for incompressible two-phase flow in porous media
Abstract
Convergence of iterative linearization methods for multiphase flow in porous media simulations is often hindered by strong nonlinearities in constitutive relations. Mitigation strategies, such as trust-region methods and tailored flux discretizations, exist but their implementation for complex multiphysics coupling, for example in the strongly-coupled viscous, capillary, and buoyancy regime, is not straightforward. We employ the homotopy continuation method as a versatile and robust nonlinear solver for immiscible, incompressible two-phase flow simulations. The method first solves a simplified problem - with linear constitutive laws - and then iteratively traces a solution curve toward the fully nonlinear target system. The predictor-corrector method for curve tracing couples an outer continuation loop with an inner nonlinear solver loop. Efficiency depends on suitable stopping criteria for both loops. Our main contribution is the integration of homotopy continuation with a posteriori error estimates. We derive a computable error estimator that distinguishes error contributions from discretization, the outer continuation loop, and the inner nonlinear solver loop. Based on this error decomposition, we formulate adaptive stopping criteria that terminate either loop once its respective error contribution becomes negligible. This novel adaptive homotopy continuation method converges robustly for large time steps while reducing the cumulative number of nonlinear iterations. The method is validated through numerical simulations on the SPE10 and SPE11 benchmark geometries.
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BibTeX
@misc{schultzendorff2026adaptive,
title = {{Adaptive homotopy continuation solver for incompressible two-phase flow in porous media}},
author = {Peter von Schultzendorff and Jakub Wiktor Both and Jan Martin Nordbotten and Tor Harald Sandve and Martin Vohralík},
howpublished = {HAL},
year = {2026},
month = oct,
url = {https://inria.hal.science/hal-05598136v2},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Adaptive homotopy continuation solver for incompressible two-phase flow in porous media
AU - Peter von Schultzendorff
AU - Jakub Wiktor Both
AU - Jan Martin Nordbotten
AU - Tor Harald Sandve
AU - Martin Vohralík
PY - 2026
DA - 2026/10/01
PB - HAL
UR - https://inria.hal.science/hal-05598136v2
AB - Convergence of iterative linearization methods for multiphase flow in porous media simulations is often hindered by strong nonlinearities in constitutive relations. Mitigation strategies, such as trust-region methods and tailored flux discretizations, exist but their implementation for complex multiphysics coupling, for example in the strongly-coupled viscous, capillary, and buoyancy regime, is not straightforward. We employ the homotopy continuation method as a versatile and robust nonlinear solver for immiscible, incompressible two-phase flow simulations. The method first solves a simplified problem - with linear constitutive laws - and then iteratively traces a solution curve toward the fully nonlinear target system. The predictor-corrector method for curve tracing couples an outer continuation loop with an inner nonlinear solver loop. Efficiency depends on suitable stopping criteria for both loops. Our main contribution is the integration of homotopy continuation with a posteriori error estimates. We derive a computable error estimator that distinguishes error contributions from discretization, the outer continuation loop, and the inner nonlinear solver loop. Based on this error decomposition, we formulate adaptive stopping criteria that terminate either loop once its respective error contribution becomes negligible. This novel adaptive homotopy continuation method converges robustly for large time steps while reducing the cumulative number of nonlinear iterations. The method is validated through numerical simulations on the SPE10 and SPE11 benchmark geometries.
ER -