The fractal nature of viscous fingering in porous media

Abstract

Viscous fingering in porous media is an instability which occurs when a less viscous fluid displaces a more viscous one. An interface between the fluids is unstable against small perturbations and gives rise to a fingered configuration. In the oil industry viscous fingering can be a serious problem when displacing viscous oil by a more mobile fluid because it leads to poor recovery of the hydrocarbon. Recent work suggests an analogy between viscous fingering at an infinite viscosity ratio and diffusion-limited aggregation (DLA) and hence that the fingers may be fractal with a fractal dimension of around 1.7 (in two dimensions). This leaves unanswered the question of the nature of the fingered patterns at a finite viscosity ratio. To answer this a network model of the porous medium has been used. The rock is modelled as a lattice of capillary tubes of random radius through which miscible displacement occurs. At a high viscosity ratio and in the presence of a large amount of disorder the model reproduces DLA fingering patterns. The results of this model provide evidence that at a finite viscosity ratio the displaced area is compact with a surface fractal dimension between 1 and a DLA result of 1.7 with increasing viscosity ratio.