  Models that describe two-fluid flow in porous media suffer from a
widely-recognized problem that the constitutive relationships used to predict
capillary pressure as a function of the fluid saturation are non-unique, thus
requiring a hysteretic description. As an alternative to the traditional
perspec- tive, we consider a geometrical description of the capillary pressure,
which relates the average mean curvature, the fluid saturation, the interfacial
area between fluids, and the Euler characteristic. The state equation is
formulated using notions from algebraic topology and cast in terms of measures
of the macroscale state. Synchrotron-based X-ray micro-computed tomography
({\mu}CT) and high- resolution pore-scale simulation is applied to examine the
uniqueness of the proposed relationship for six different porous media. We show
that the geometric state function is able to characterize the microscopic fluid
configurations that result from a wide range of simulated flow conditions in an
averaged sense. The geometric state function can serve as a closure
relationship within macroscale models to effectively remove hysteretic behavior
attributed to the arrangement of fluids within a porous medium. This provides a
critical missing component needed to enable a new generation of higher fidelity
models to describe two-fluid flow in porous media.
