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Based on the phenomenological extension of Darcy's law, two-fluid flow isdependent on a relative permeability function of saturation only that isprocess/path dependent with an underlying dependency on pore structure. Forapplications, fuel cells to underground $CO_2$ storage, it is imperative todetermine the effective phase permeability relationships where the traditionalapproach is based on the inverse modelling of time-consuming experiments. Theunderlying reason is that the fundamental upscaling step from pore to Darcyscale, which links the pore structure of the porous medium to the continuumhydraulic conductivities, is not solved. Herein, we develop an ArtificialNeural Network (ANN) that relies on fundamental geometrical relationships todetermine the mechanical energy dissipation during creeping immiscibletwo-fluid flow. The developed ANN is based on a prescribed set of statevariables based on physical insights that predicts the effective permeabilityof 4,500 unseen pore-scale geometrical states with $R^2 = 0.98$.

Effective permeability of an immiscible fluid in porous media determined from its geometric state