Nonhysteretic Capillary Pressure in Two‐Fluid Porous Medium Systems: Definition, Evaluation, Validation, and Dynamics

Abstract

A closure relation for capillary pressure plays an important role in the formulation of both traditional and evolving models of two‐fluid‐phase flow in porous medium systems. We review the traditional approaches to define capillary pressure, to describe it mathematically, to determine parameters for this relation, and to constrain the domain of applicability of this relation. In contrast to the traditional approach, we provide a rigorous, multiscale definition of capillary pressure, define the state domain of interest in practice, summarize computational and experimental approaches to investigate the system state, and apply the methods for two‐fluid states in a model ink bottle system, the classical Finney pack of spheres, and a synthetic sphere pack system. The results of these applications show that a state equation exists that describes capillary pressure without hysteresis. This state equation parameterizes a function that describes the nonwetting phase volume fraction in terms of the capillary pressure, the interfacial area, and the specific Euler characteristic of the nonwetting phase. Furthermore, this state equation applies over the complete range of conditions encountered in practice, and it applies under both equilibrium and dynamic conditions. This state equation involving capillary pressure forms an important foundation for the development of the next generation of macroscale two‐fluid‐phase flow models in porous medium systems.

Fronts in two‐phase porous media flow problems: The effects of hysteresis and dynamic capillarity

“…24. Using thermodynamically constrained averaging theory (TCAT), 25,26 one can eliminate hysteresis in capillary pressure altogether by introducing interfacial area and Euler characteristics as additional unknowns. A mathematical study of such models is undoubtedly interesting.…”

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

“…This equation is different from the traditional definition, even at equilibrium [59], and it also applies under dynamic conditions; the quantities in this equation are firmly connected to pore-scale physics.…”

“…These hysteretic models introduce additional empiricism to describe the volume fraction occupied by the disconnected, or entrapped, non-wetting fluid phase [57,58]. A parameter in these relations is the irreducible wetting-phase saturation, which is the minimum wetting-phase saturation that can be obtained by manipulating boundary pressures in a laboratory cell [59]. Lastly, these relations assume that an equilibrium state exists between the fluid pressures and the wetting-phase saturation, and laboratory data is typically collected to approximate this equilibrium state.…”

“…It has been shown in recent work that a state equation exists for p c that is hysteresis-free, does not rely upon an empirical model for entrapment of the non-wetting phase or specification of an irreducible saturation, and applies under both equilibrium and dynamic conditions [59,60]. This state equation has been derived from integral geometry and has been validated against a range of ideal and real porous media [59][60][61]. The state equation can be written in a linear form as…”

“…where ns denotes the interface between the non-wetting fluid and the solid, a i are constants, F is a nonlinear model specified by a spline fit resulting from a generalized additive model (GAM), and χ n is a measure of non-wetting phase fluid connectivity called the specific Euler characteristic. Details on the derivation of these equations, and non-dimensional forms, are available in the literature [59]. The state equations shown in Equations (9) and (10) appear to introduce three additional unknowns to the model given by Equations (1)- (7): ns , J ns s , and χ n .…”