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Immiscible two-phase flow in porous media: a statistical mechanics approach

Alex Hansen, Santanu Sinha

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.

Immiscible two-phase flow in porous media: a statistical mechanics approach

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.
Paper Structure (10 sections, 55 equations, 4 figures)

This paper contains 10 sections, 55 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Phase diagrams
  3. Pore scale phases
  4. Darcy scale phases
  5. Non-thermal statistical mechanics
  6. Non-thermal thermodynamics at the Darcy scale
  7. Agiture
  8. Thermodynamic velocities and the co-moving velocity
  9. Is there an equivalent to the co-moving velocity in ordinary thermodynamics?
  10. Discussion and conclusion

Figures (4)

  • 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 phase. The transition line coincides with the onset of power law behavior, see Equation (\ref{['eq6']}). The figure is adapted from sinha2026glassy.
  • 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 $Q_p$. The wetting fluid covers an area $A_w$ of the REA and the non-wetting fluid an area $A_n$. The total pore area of the REA is $A_p$. Adapted from hansen2025thermodynamics.
  • Figure 3: Measured relative permeability curves, $k_{rw}(S_w)$ and $k_{rn}(S_w)$ plotted against $S_w$, and the corresponding co-moving velocity $v_M/v_0$ plotted against $(1/v_0)dv_p/dS_w= \mu/v_0$, where $v_0$ is a velocity scale. From berg2026from.
  • Figure 4: Co-molar volume ($\nu_m$) as a function of partial molar volume difference ($w = d\nu/dx_1$) for the mixtures. Mixture 1 is stable, whereas mixtures 2 and 3 are unstable in that they have a tendency to form homogeneous clusters. From olsen2025new.