Abstract: To describe two-phase flows in porous media, a two-scale mathematical model is developed using the homogenization method applied to the coupled system of Navier–Stokes and Cahn–Hilliard equations. This system is based on the assumption that the fluid phases are separated by a diffusion layer. We prove that the macro-equations represent a generalized Darcy law with cross-coupling permeabilities. It implies that the seepage velocity of each phase depends on pressure gradients of both the phases. Micro-equations serve for determination both of the permeability tensors and the capillary diffusion energy. It is established that a formal sharpinterface limit justifies the empirical concept of relative phase permeabilities. To illustrate the capabilities of the new two-scale model, the problem of counter-current capillary imbibition is solved. We show that the imbibition rate is lower compared to that predicted by traditional equations based on the empirical concept of relative phase permeability.

Cite: Shelukhin V. , Trusov K. , Youcef A.
Cross-coupling permeabilities in two-phase flows through porous media: Spontaneous counter-current capillary imbibition
International Journal of Engineering Science. 2026. V.221. 104471 :1-15. DOI: 10.1016/j.ijengsci.2026.104471

Dates:

Submitted:	Aug 22, 2025
Accepted:	Jan 8, 2026
Published online:	Jan 10, 2026

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