Capillary Network Model: Capillary Power and Effective Permeability

A simple model of two-phase flow in porous media is presented. A connection is made to statistical mechanics by applying capillary power as a constraint. Stochastic sampling is then used to test the validity of this approach. Good agreement is found between stochastic sampling and time stepping for flow-rates above a transition value.

💡 Research Summary

The paper introduces a minimalist yet powerful representation of two‑phase flow in porous media, called the Capillary Network Model (CNM). In this framework each pore throat (or “capillary”) is abstracted as a binary element that can be either saturated with the wetting phase or unsaturated. The key physical quantity governing the state of each element is the capillary power – the work done by surface tension forces as fluid moves across the meniscus. By treating capillary power as a global constraint, the authors map the flow problem onto a statistical‑mechanical ensemble.

Mathematically, the state of the entire network is described by a set of binary variables σi (i = 1…N). The total capillary power Π(σ) is the sum of the individual contributions Πi = γ cosθ · 2πri σi, where γ is the interfacial tension, θ the contact angle, and ri the radius of throat i. The probability of finding the system in a particular configuration σ is postulated to follow a Boltzmann‑like distribution
 P(σ) = Z⁻¹ exp