Scaling structure of viscous fingering

Abstract

When a fluid in a porous medium attempts to displace one which is more viscous, an initially flat boundary between them is unstable and fingers of the inviscid liquid penetrate the other. We model the medium numerically using a lattice of capillary tubes of random radii. Previous studies by one of the authors (P.R.K.) have already shown that the displacement is compact, but that the boundary between the two fluids is fractal. In this paper we study the distribution of velocities normal to the interface. We find that the distribution is consistent with the hypothesis that, for a system of size L, $L^{f\left(\alpha \right)}$ sites have velocities scaling as $L^{\mathrm{-}\alpha }$. The scaling function f(α) is measured and its variation with the viscosity ratio and randomness of the medium is found.