Dynamics of growing interfaces from the simulation of unstable flow in random media

Abstract

Viscous fingering in random porous media is encountered in many applications of two-phase flow, where the interface is unstable because the ratio of the viscosity of the displaced fluid to that of the injected fluid is large. In these applications, including enhanced oil recovery, characterization of the width of the interface is an important concern. In the limit of stable flow, the interfacial width had been found to grow as w≊${\mathit{t}}^{\mathrm{\beta }}$, where β≊0.66, approximately independent of capillary number. To study the same behavior for the unstable case, we have simulated flow in two-dimensional random porous media using a standard model with different viscosity ratios and zero capillary pressure. When the injected fluid has zero viscosity, viscosity ratio M=∞, the interfacial width has the expected self-similar diffusion-limited-aggregation-like behavior. For smaller viscosity ratios, the flow is self-affine with β=0.66±0.04, which is the same value that had been observed in studies of stable flow. Furthermore, the crossover from self-similar fractal flow to self-affine fractal flow is observed to scale with the same ‘‘characteristic’’ time, τ=${\mathit{M}}^{0.17}$, that had been found to scale the average interface position. This ‘‘fractal’’ scaling of the crossover leads to definite predictions about the viscosity-ratio dependence of the amplitudes associated with interfacial position and interfacial width.