Toward a New Generation of Two-Fluid Flow Models Based on the Thermodynamically-Constrained Averaging Theory

Traditional models of two-fluid flow through porous media at the macroscale have existed for nearly a century. These phenomenological models are not firmly connected to the microscale; thermodynamic constraints are not enforced; empirical closure relations are well known to be hysteretic; fluid pressures are typically assumed to be in a local equilibrium state with fluid saturations; and important quantities such as interfacial and curvilinear geometric extents, tensions, and curvatures, known to be important from microscale studies, do not explicitly appear in traditional macroscale models. Despite these shortcomings, the traditional model for two-fluid flow in porous media has been extensively studied to develop efficient numerical approximation methods, experimental and surrogate measure parameterization approaches, and convenient pre- and post-processing environments; and they have been applied in a large number of applications from a variety of fields. The thermodynamically constrained averaging theory (TCAT) was developed to overcome the limitations associated with traditional approaches, and we consider here issues associated with the closure of this new generation of models. It has been shown that a hysteretic-free state equation exists based upon integral geometry that relates changes in volume fractions, capillary pressure, interfacial areas, and the Euler characteristic. We show an analysis of how this state equation can be parameterized with a relatively small amount of data. We also formulate a state equation for resistance coefficients that we show to be hysteretic free, unlike traditional relative permeability models. Lastly, we comment on the open issues remaining for this new generation of models.