Closure conditions for two-fluid flow in porous media

Modeling of multiphase flow in porous media requires that the physics of the phases present be well described. Additionally, the behavior of interfaces between those phases and of the common lines where the interfaces come together must be accounted for. One factor complicating this description is the fact that geometric variables such as the volume fractions, interfacial areas per volume, and common line length per volume enter the conservation equations formulated at the macroscale or core scale. These geometric densities, although important physical quantities, are responsible for a deficit in the number of dynamic equations needed to model the system. Thus, to obtain closure of the multiphase flow equations, one must supplement the conservation equations with additional evolutionary equations that account for the interactions among these geometric variables. Here, the second law of thermodynamics, the constraint that the energy of the system must be at a minimum at equilibrium, is used to motivate and generate linearized evolutionary equations for these geometric variables and interactions. The constitutive forms, along with the analysis of the mass, momentum, and energy conservation equations, provide a necessary complete set of equations for multiphase flow modeling in the subsurface.