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Multiphase flows in porous medium systems are typically modeled at the macroscale by applying the principles of continuum mechanics to develop models that describe the behavior of averaged quantities, such as fluid pressure and saturation. These models require closure relations to produce solvable forms. One of these required closure relations is an expression relating the capillary pressure to fluid saturation and, in some cases, other topological invariants such as interfacial area and the Euler characteristic (or average Gaussian curvature). The forms that are used in traditional models, which typically consider only the relationship between capillary pressure and saturation, are hysteretic. An unresolved question is whether the inclusion of additional morphological and topological measures can lead to a nonhysteretic closure relation. Relying on the lattice Boltzmann (LB) method, we develop an approach to investigate equilibrium states for a two-fluid-phase porous medium system, which includes disconnected nonwetting phase features. A set of simulations are performed within a random close pack of 1964 spheres to produce a total of 42 908 distinct equilibrium configurations. This information is evaluated using generalized additive models to quantitatively assess the degree to which functional relationships can explain the behavior of the equilibrium data. The variance of various model estimates is computed, and we conclude that, except for the limiting behavior close to a single fluid regime, capillary pressure can be expressed as a deterministic and nonhysteretic function of fluid saturation, interfacial area between the fluid phases, and the Euler characteristic. To our knowledge, this work is unique in the methods employed, the size of the data set, the resolution in space and time, the true equilibrium nature of the data, the parametrizations investigated, and the broad set of functions examined. The conclusion of essentially nonhysteretic behavior provides support for an evolving class of two-fluid-phase flow in porous medium systems models.
McClure et al. (Fri,) studied this question.