From Interface Dynamics to Darcy Scale Description of Multiphase Flow in Porous Media

TL;DR

This article surveys four cutting-edge frameworks for upscaling immiscible two-phase flow in porous media from pore to Darcy scale, driven by advances in pore-scale imaging and modeling. It foregrounds the geometric (capillary) state, topology (Minkowski functionals and Euler characteristic), nonequilibrium thermodynamics, and a statistical thermodynamics perspective with space-time averaging, integrating concepts like interfacial area, co-moving velocity, agiture, and REA/REV construction. The work demonstrates that capillary fluctuations, topology, and nonequilibrium effects underpin flow regimes and can yield linear Darcy-type transport when properly averaged, offering principled constraints on relative permeability and a route to predict hysteresis and traveling-wave phenomena from first principles. By linking pore-scale dynamics to macroscopic transport through rigorous averaging and emergent thermodynamic variables, the paper lays the groundwork for a unified, predictive framework that can leverage imaging data, coarse-grained theory, and AI to improve upscaling and design in applications ranging from groundwater to CCS and energy storage.

Abstract

An outstanding characteristic of porous media, desired in many applications, is the large surface area, which facilitates solid-fluid interactions, making porous media an extreme case in colloid and interface science. In two-fluid systems, wetting and the balance of capillary and viscous forces control fluid displacement processes, leading to a wide range of complex flow regimes with rich spatio-temporal dynamics. Macroscopic two-phase flow is historically described through the phenomenological extensions of Darcy's law. Besides many other shortcomings and inconsistencies, it covers only connected pathway flow in the capillary-dominated flow regime in a rigorous manner while other flow regimes with moving interfaces and associated topological changes are entirely implicit. Given the lack of adequate descriptions, upscaling multiphase flow from pore to Darcy scale represents a long-standing challenge paving into the fields of thermodynamics, statistical mechanics and integral geometry. In this review, we compare novel concepts which have been largely motivated by experimental insights, enabled by significant advances in pore-scale imaging and modeling over the last decade.

Figures (59)

• Figure 1: Overview of selected examples in science and technology where multiphase flow in porous media is relevant maasViscousFingeringCCS2024maasViscousFingeringCCS2024simonInfluenceGasDiffusion2017zhangSimilaritiesDifferencesGas2024Hochberg2019yueMultiphaseFlowProcessing2018miriNewInsightsPhysics2015.
• Figure 2: Multiphase flow in porous media from molecular to Darcy scale. While the length scales associated with the structure of the porous medium itself also generate a multi-scale problem, there are two upscaling steps where physical concepts change: (1) the upscaling of the fluids from molecular to continuum hydrodynamic scale and (2) the upscaling from the pore scale where pores are discrete to a continuum mechanics (Darcy scale) description of the porous medium. This review focuses on (2), i.e. the upscaling of the multiphase flow from the pore to the Darcy scale. Four novel approaches are introduced, which are indicated in the light-blue boxes, including the respective sections (S5-S7) Ruecker2021. The key novelty are the consideration of the geometric (capillary) state as starting point (S2) and the explicit consideration of fluctuations at the capillary energy scale (S3) armstrongSubsecondPorescaleDisplacement2014.
• Figure 3: Overview of approaches for developing descriptions for Darcy-scale multiphase flow in porous media. While the approaches on the left have the postulated 2-phase Darcy equations as the starting point, approaches towards the right often involve upscaling from the pore to the Darcy scale. The focus of this review is on the paths indicated in yellow.
• Figure 4: Assuming a connected pathway flow regime, upscaling from pore to Darcy scale for multiphase flow can be conceptually achieved by assuming a capillary tubes model with the size distribution linked to the pore size distribution tullerHydraulicConductivityVariably2001. Assuming a log-normal pore size distribution, the exponent of the log-normal pore size distribution is linked to the exponent $\lambda$ in the Brooks-Corey model brooksPropertiesPorousMedia1966 for the relative permeability and capillary pressure-saturation functions (for drainage). Adapted from tullerHydraulicConductivityVariably2001
• Figure 5: Illustration of the key results of the DeProF theory by Valavanides valavanidesReviewSteadyStateTwoPhase2018a: Energy efficiency map relating steady-state relative permeability $k_r$ as a function of flow rate ratio of non-wetting and wetting fluid phases $r=q_n/q_w$ and capillary number $Ca$ to energy efficiency expressed by $f^*_{EU}$. For explanations of the figure in detail we refer to the original work by Valavanides valavanidesReviewSteadyStateTwoPhase2018a.