ahc - Adaptive Homotopy Continuation for Two-Phase Flow in PorePy.

This repository extends the functionality of PorePy with the following:

• an incompressible & immiscible two-phase flow model in the fractional flow formulation
• a posteriori error estimators for the two-phase flow model
• an adaptive homotopy continuation algorithm based on the error estimators

The code does not work with the current PorePy version, but requires this fork, which adds some functionalities for homotopy continuation to work properly.

If you use this code, please cite:

P. von Schultzendorff, J. W. Both, J. M. Nordbotten, T. H. Sandve, and M. Vohralík, "Adaptive homotopy continuation solver for incompressible two-phase flow in porous media"

Installation and reproducing the examples

Local setup (requires Python 3.12.11):

git clone https://github.com/pschultzendorff/porepy_hc
pip install -e ./porepy_hc[development,testing]
git clone https://github.com/pschultzendorff/ahc
pip install -e ./ahc
sh ./ahc/run_all.sh

Docker:

git clone https://github.com/pschultzendorff/ahc
cd ahc
docker compose up

Content

Two-phase flow model

Implemented in the fractional flow formulation.

A posteriori error estimates

The implementation closely follows [1] and [2]. The pressure reconstruction reuses code of Jhabriel Varela [3].

• Implements functions to calculate global and complementary pressure, equilibrate fluxes, and post-process and reconstruct pressures.
• Guaranteed error estimates.
• Decomposition of the error into discretization, homotopy continuation, and linearization error.

Adaptive Homotopy Continuation

The adaptive homotopy continuation (AHC) algorithm gradually deforms a simple, easy-to-solve problem (linear rel. perm., zero capillary pressure) into the target nonlinear problem using adaptive stepping criteria:

• Newton corrector loop: The linearization error estimator in ahc.models.error_estimate provides a stopping criterion that allows early termination of the Newton corrector loop when the linearization error becomes sufficiently small relative to the HC error estimator.
• Homotopy continuation loop: After each Newton solve, the HC error estimator evaluates whether the homotopy step is small enough to proceed to the next step, or should be retried with a reduced step size. This adaptive step control, implemented in ahc.models.homotopy_continuation, prevents both over-refinement (excessive homotopy steps) and solver failure (oversized steps).

The combined effect is a globally adaptive algorithm that balances computational efficiency with robustness by terminating iterations early when convergence criteria are met.

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References

[1] C. Cancès, I. Pop, and M. Vohralík, "An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow," Math. Comp., vol. 83, no. 285, pp. 153–188, Jan. 2014, doi: 10.1090/S0025-5718-2013-02723-8.

[2] M. Vohralík and M. F. Wheeler, "A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows," Comput Geosci, vol. 17, no. 5, pp. 789–812, Oct. 2013, doi: 10.1007/s10596-013-9356-0.

[3] J. Varela, C. E. Schaerer, and E. Keilegavlen, "A linear potential reconstruction technique based on Raviart-Thomas basis functions for cell-centered finite volume approximations to the Darcy problem," Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, vol. 11, no. 1, Jan. 2025, doi: 10.5540/03.2025.011.01.0332.

[4] M. A. Christie and M. J. Blunt, "Tenth SPE Comparative Solution Project: A Comparison of Upscaling Techniques," SPE Reservoir Evaluation & Engineering, vol. 4, no. 4, pp. 308–317, 2001, doi: 10.2118/72469-PA.

[5] J. M. Nordbotten, M. A. Ferno, B. Flemisch, A. R. Kovscek, and K.-A. Lie, "The 11th Society of Petroleum Engineers Comparative Solution Project: Problem Definition," SPE Journal, vol. 29, no. 5, pp. 2507–2524, 2024, doi: 10.2118/218015-PA.

[6] X. Wang and H. A. Tchelepi, “Trust-region based solver for nonlinear transport in heterogeneous porous media,” Journal of Computational Physics, vol. 253, pp. 114–137, 2013, doi: 10.1016/j.jcp.2013.06.041.

AI disclosure

Generative AI (GitHub Copilot in VS Code, ChatGPT, and Microsoft Copilot) was used to create scripted figures with Matplotlib.

TODO

Fix some of the tests