Comparison of Two-Phase Darcy’s Law with a Thermodynamically Consistent Approach

Abstract

The extended Darcy's law is a commonly used equation for the description of immiscible two-phase flow in porous media. It dates back to the 1940s and is essentially an empirical relationship. According to the extended Darcy's law, pressure gradient and gravity are the only driving forces for the flow of each fluid. Within the last two decades, more advanced and physically based descriptions for multiphase flow in porous media have been developed. In this work, the extended Darcy's law is compared to a thermodynamically consistent approach which explicitly takes the important role of phase interfaces into account, both as entities and as parameters. In this theoretically derived approach, forces related to capillarity and interfaces appear as driving/resisting forces, in addition to the traditional terms. It turns out that the extended Darcy's law and the thermodynamically based approach are compatible if either (i) relative permeabilities are a function of saturation only, but capillary pressure is a function of saturation and specific interfacial area or (ii) relative permeabilities are a function of saturation and saturation gradients. Theoretical considerations suggest that the former alternative is only valid in case of reversible displacement while in the general case (irreversible displacement), the latter alternative is relevant. Keywords

Real-time 3D imaging of Haines jumps in porous media flow

“…By following an argument based on reversibility and dissipation of drainage work in the framework of a thermodynamically based approach (40), it has been shown that relative permeability in the traditional two-phase Darcy formulation (3) has contributions originating from hydraulic conductivity and from dissipative porescale events. This leads to a saturation gradient and, implicitly, a rate dependency for the relative permeability (14), which has been observed many times (e.g., ref. 41).…”

“…When the associated intrinsic relaxation time scale is comparable to the time scale for the general advancement of fluid front propagation or an externally imposed main flow rate (11), the fluid configuration is not in capillary equilibrium (12), leading to the well-known rate dependency of the flow parameters (11,12), which is not captured by the commonly used two-phase extended Darcy description (13,14).…”

“…These rapid pore-scale events are essential to the upscaling of multiphase flow because they make up a significant fraction of the energy dissipation within the system (15,16) and contribute to relative permeability (14) and to macroscopic nonequilibrium effects (12). An estimate of the Helmholtz free-energy balance (15), ΔF, of the Haines jumps, using…”

“…Niessner et al (14) have shown that pore-scale dissipative displacement events are important contributors to macroscopic transport coefficients, leading to rate dependency. By following an argument based on reversibility and dissipation of drainage work in the framework of a thermodynamically based approach (40), it has been shown that relative permeability in the traditional two-phase Darcy formulation (3) has contributions originating from hydraulic conductivity and from dissipative porescale events.…”

On the consistency of scale among experiments, theory, and simulation

“…Despite the fact that interface reconfiguration has an important role in determining the properties and behavior of a multi-fluid porous medium system, attention to this feature is extremely limited (Gray and Miller, 2013;Gray et al, 2015). In some cases, models of two-fluid-phase flow in porous media have been proposed that do not account for either system kinematics or for interfacial stress (e.g., Niessner et al, 2011). Both are necessary components of physically realistic, highfidelity models.…”