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

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.