Abstract
Multiphase flow in porous medium systems is typically modeled using continuum mechanical
representations at the macroscale in terms of averaged quantities. These models require closure
relations to produce solvable forms. One of these required closure relations is an expression relating
fluid pressures, fluid saturations, and, in some cases, the interfacial area between the fluid phases,
and the Euler characteristic. An unresolved question is whether the inclusion of these additional
morphological and topological measures can lead to a non-hysteretic closure relation compared to the
hysteretic forms that are used in traditional models, which typically do not include interfacial areas,
or the Euler characteristic. We develop a lattice-Boltzmann (LB) simulation approach to investigate
the equilibrium states of a two-fluid-phase porous medium system, which include disconnected now-
wetting phase features. The proposed approach is applied to a synthetic medium consisting of 1,964
spheres arranged in a random, non-overlapping, close-packed manner, yielding a total of 42,908
different equilibrium points. This information is evaluated using a generalized additive modeling
approach to determine if a unique function from this family exists, which can explain the 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
non-hysteretic function of fluid saturation, interfacial area between the fluid phases, and the Euler
characteristic. 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 parameterizations investigated, and
the broad set of functions examined. The conclusion of essentially non-hysteretic behavior provides
support for an evolving class of two-fluid-phase flow in porous medium systems models.
