[论文解读] Immiscible two-phase flow in porous media: a statistical mechanics approach

Alex Hansen, Santanu Sinha|arXiv (Cornell University)|Mar 10, 2026

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一句话总结

该论文审阅了一种 Jaynesian、非热统计力学框架，用以将不可混两相流在多孔介质中从孔隙尺度上升尺度至 Darcy 尺度，引入 agiture、流动导数和共动速度等关键涌现变量。

ABSTRACT

The central problem in the physics of immiscible two-phase flow in porous media is to find a proper description of the flow at scales large enough so that the medium may be regarded as a continuum: the scale-up problem. So far, the only workable approach to the multiphase flow scale-up problem has been a set of phenomenological equations that have obvious weaknesses. Attempts at going beyond this relative permeability theory have so far not led to practical applications due to exploding complexity. Edwin T. Jaynes proposed in the fifties a generalization of statistical mechanics to non-thermal systems based on the information theoretical entropy of Shannon. This approach is used to construct a description of immiscible two-phase flow in porous media at the continuum scales, which is directly related to the physics at the pore scale, and at a level of complexity that is manageable. The approach leads to a thermodynamics-like formalism at the continuum scale with all the relations between variables that "normal" thermodynamics has to offer. New emergent variables appear. Among these, the co-moving velocity stands out as a key variable with implications for ordinary thermodynamics. We present here a short review of this approach.

研究动机与目标

• Motivate the Darcy-scale upscaling problem for immiscible two-phase flow in porous media.
• Review a statistical mechanics framework that connects pore-scale physics to Darcy-scale continuum descriptions.
• Introduce and interpret emergent intensive variables (agiture, flow derivative, flow pressure) and co-moving velocity.
• Highlight phase behavior (glassy vs non-glassy) and its thermodynamics-like structure at the Darcy scale.

提出的方法

• Use Jaynes maximum entropy approach to derive a probability distribution over pore-scale configurations given constraints on average flow and areas.
• Define a Representative Elementary Area (REA) and decompose flow into wetting and non-wetting components with associated areas.
• Derive a partition function and perform a Legendre transform to obtain a thermodynamics-like framework with agiture (theta), flow derivative (mu), and flow pressure (pi).
• Introduce thermodynamic velocities and the co-moving velocity to relate macro- and pore-scale flows and to connect v_p, v_w, and v_n.
• Discuss phase behavior (glassy vs non-glassy) and how steady-state flow can be described by a non-thermal thermodynamics.

Figure 1: Phase diagram for immiscible two-phase flow in porous media for a viscosity ratio $M=1$ . The abscissa shows the non-wetting saturation $S_{n}$ and the ordinate $\log{\rm Ca}$ . The color shows the value of the Edwards-Anderson order parameter, revealing a glassy phase and a non-glassy pha

实验结果

研究问题

• RQ1 How can a Jaynes-inspired information-theoretic approach describe immiscible two-phase flow at the Darcy scale?
• RQ2 What emergent intensive variables arise in this non-thermal thermodynamics, and how do they relate to pore-scale quantities?
• RQ3 How does the co-moving velocity influence the relationship between average and phase-velocity fields in steady-state flow?
• RQ4 What is the nature of the phase diagram (glassy vs non-glassy) for immiscible flow, and how does it relate to the onset of power-law versus linear velocity–pressure relations?

主要发现

• A thermodynamics-like formalism emerges at the Darcy scale from information-theoretic upscaling, with emergent variables theta (agiture), mu (flow derivative), and pi (flow pressure).
• A co-moving velocity v_m relates thermodynamic and seepage velocities, and the velocities obey a two-way mapping with v_p and v_m determining v_w and v_n.
• There exists a glassy flow phase (phase Ib) and a non-glassy phase (phase II/III) with a critical line coinciding with the onset of power-law v_p vs. |∇p| behavior.
• The framework distinguishes between thermodynamic velocities and laboratory-measured seepage velocities, requiring v_m to account for interface dynamics and saturation changes.
• The co-moving velocity and co-molar volume concepts illustrate analogous structures between porous media and molecular mixtures, suggesting broader applicability of co-moving-type variables.

Figure 2: We define a Representative Elementary Area (REA) within a cut orthogonal to the average flow direction through the cylider-shaped porous medium sample. There is a wetting fluid flow rate $Q_{w}$ and a non-wetting fluid flow rate $Q_{n}$ passing through the REA. The total flow rate is $Q_{p