Kinetic roughening in two-phase fluid flow through a random Hele-Shaw cell

TL;DR

This paper derives a nonlocal interface equation for two-phase fluid flow in a random Hele-Shaw cell, identifying key length scales and comparing numerical simulation exponents with experimental data.

Contribution

It introduces a novel nonlocal interface equation accounting for microscopic disorder, wettability, and viscosity contrast in two-phase flow through a porous medium.

Findings

Two characteristic length scales scale with capillary number.

Numerical exponents are consistent with recent experimental results.

Explicit relations between fluctuations and microscopic disorder are established.

Abstract

A nonlocal interface equation is derived for two-phase fluid flow, with arbitrary wettability and viscosity contrast c=(mu_1-mu_2)/(mu_1+mu_2), in a model porous medium defined as a Hele-Shaw cell with random gap b_0+delta b. Fluctuations of both capillary and viscous pressure are explicitly related to the microscopic quenched disorder, yielding conserved, non-conserved and power-law correlated noise terms. Two length scales are identified that control the possible scaling regimes and which scale with capillary number as ell_1 ~ b_0(c Ca)^{-1/2} and ell_2 ~ b_0 Ca^{-1}. Exponents for forced fluid invasion are obtained from numerical simulation and compared with recent experiments.