Thermodynamic approach to effective stress in partially saturated porous media

The form of the equilibrium effective stress acting on the solid phase of a porous medium containing two immiscible fluid phases is derived. The derivation makes use of the postulation of the thermodynamics of the system at the macroscale, a scale on the order of tens of pore diameters. The postulation at this scale facilitates the identification of the fraction of the solid surface in contact with each fluid phase as being the appropriate coefficient weighting each of the fluid phase pressures analogous to the Bishop parameter. In addition, the curvature of the surface of the solid phases is shown to impact the pressure exerted on the solid phase by the fluid. For the special case of low saturations when the wetting phase may be considered to be present only as a film on the solid phase, the macroscale disjoining pressure is found to modify the equilibrium form of the effective stress. In addition to the equilibrium effective stress, which is related to the forces acting on the interface between the solid phase and the fluids, the appropriate relation between the fluid pressures at the fluid-fluid interface is obtained. This analysis leads to the expression for the capillary pressure as function of the phase pressures and the disjoining pressure.