Non-isothermal Transport of Multi-phase Fluids in Porous Media. The Entropy Production

We derive the entropy production for transport of multi-phase fluids in a non-deformable, porous medium exposed to differences in pressure, temperature, and chemical potentials. Thermodynamic extensive variables on the macro-scale are obtained by integrating over a representative elementary volume (REV). Contributions come from porous media specific properties, phase volumes, surface areas, and contact lines. Curvature effects are neglected. Using Euler homogeneity of the first order, we obtain the Gibbs equation for the REV. From this we de fine the intensive variables, the temperature, pressure, and chemical potentials, and, using the balance equations, we derive the entropy production for the REV. The entropy production de fines sets of independent conjugate thermodynamic fluxes and forces in the standard way. The transport of two-phase flow of immiscible components is used to give a first illustration of the equations.