  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 $R^2 = 0.98$.
