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

Based on the phenomenological extension of Darcy's law, two-fluid flow is dependent on a relative permeability function of saturation only that is process/path dependent with an underlying dependency on pore structure. For applications, fuel cells to underground $CO_2$ storage, it is imperative to determine the effective phase permeability relationships where the traditional approach is based on the inverse modelling of time-consuming experiments. The underlying reason is that the fundamental upscaling step from pore to Darcy scale, which links the pore structure of the porous medium to the continuum hydraulic conductivities, is not solved. Herein, we develop an Artificial Neural Network (ANN) that relies on fundamental geometrical relationships to determine the mechanical energy dissipation during creeping immiscible two-fluid flow. The developed ANN is based on a prescribed set of state variables based on physical insights that predicts the effective permeability of 4,500 unseen pore-scale geometrical states with $R^2 = 0.98$.