Kinetic Roughening in Two-Phase Fluid Flow through a Random Hele-Shaw Cell

A nonlocal interface equation is derived for two-phase fluid flow, with arbitrary wettability and viscosity contrast, c=(μ₁-μ₂)/(μ₁+μ₂), in a model porous medium defined as a Hele-Shaw cell with random gap b₀+δb. Fluctuations of both capillary and viscous pressure are explicitly related to the microscopic quenched disorder, yielding conserved, nonconserved, and power-law correlated noise terms. Two length scales are identified that control the possible scaling regimes and which scale with capillary number Ca as ₁~b₀(cCa)^-1/2 and ₂~b₀Ca^-1. Exponents for forced fluid invasion are obtained from numerical simulation and compared with recent experiments.