Multiphase flow in porous media is an extreme case in colloid and interface science. The large surface area amplifies solid-fluid interactions and the complex pore space causes a wide range of flow regimes with rich spatio-temporal dynamics posing a major challenge for deriving transport equations. Historically, macroscopic two-phase flow is described through phenomenological extensions of Darcy's law, which - besides many other shortcomings and inconsistencies - covers strictly only the connected pathway flow regime at very low flow rates while regimes with moving interfaces and associated topological changes are entirely implicit. Developing a description for all flow regimes by upscaling from pore to Darcy scale represents a long-standing challenge. Over the past decades, the field advanced by introducing thermodynamic approaches, geometric state variables for capillarity and capturing non-equilibrium effects. Experimental insights, enabled by advances in pore-scale imaging and modeling, has motivated several novel recent approaches which inherently include fluctuations and intermittency, and thereby avoid previous limiting assumptions. They cover the physics of the three dominant flow regimes: (I) the capillary-dominated regime, consisting of connected pathway flow with capillary fluctuations is covered by the space-time averaging approach and by the extended nonequilibrium thermodynamic theory (NET), resulting in linear laws; (II) the nonlinear flow regime, where capillary states become increasingly accessible by viscous mobilization leading to ganglion dynamics and intermittency, is described by the statistical thermodynamics approach; (III) the viscous limit consisting of drop-traffic, is described by the NET approach, which utilizes the fluctuation-dissipation theorem and Onsager reciprocal relationships leading again to a linear law, or the statistical thermodynamics approach. Most applications reside in regime I which is the most complex and least intuitive because it is a "frozen state". A better starting point is regime III which is from the perspective of dynamics, and then approaching successively regime II and I. We conclude with open questions and invite to contribute steering the theoretical advances towards application. The most immediate is using the co-moving velocity, which utilizes inherent symmetries in the 2-phase Darcy equations, to constrain the functional form of relative permeability and thereby simplify measurement protocols. The choice of state variables and the statistical thermodynamics approach that establishes relationships between them can be used to replace empirical hysteresis models. Grounding transport laws in thermodynamic concepts opens new possibilities for describing coupled transport phenomena in many relevant applications.
Berg et al. (Mon,) studied this question.