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Emergence of Darcy's law and apparent violation of Onsager symmetry in multiphase transport

Omid Tavakkoli1, Dick Bedeaux2, Signe Kjelstrup2, Steffen Berg3,4, Marcel Moura5, Ying Da Wang6, Peyman Mostaghimi1, and Ryan T. Armstrong1,*

  • *Contact author: ryan.armstrong@unsw.edu.au
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Phys. Rev. E 114, 015103 – Published 13 July, 2026

DOI: https://doi.org/10.1103/79nz-zp2f

Abstract

We introduce a nonequilibrium thermodynamic (NET) framework for immiscible two-phase flow in porous media, in which the total flux of both phases is driven by the gradient of an effective pressure. We show that the classical two-phase Darcy formulation emerges as a projected limit of the full linear flux-force relations under steady-state saturation constraints, with cross-coupling terms arising naturally. Large-scale lattice Boltzmann simulations reveal that the full Onsager transport matrix remains symmetric at the fundamental level, while its reduced Darcy representation can appear asymmetric because the standard force-flux pairs are not independent. Projecting onto the physically admissible subspace restores reciprocity and recovers the total phase mobility from the main- and cross-coupling coefficients. These results resolve the long-standing debate over apparent violations of Onsager symmetry and establish Darcy's law as an emergent thermodynamic limit for multiphase porous-media flow.

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Outline

Information

Phys. Rev. E 114, 015103– Published 13 July, 2026
  • Received 8 December 2025
  • Accepted 22 June 2026
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