Fractal nature of viscous fingering in two‐dimensional pore level models

Abstract

Abstract Use of saturation‐dependent relative mobilities leads to linear flow; however, experiment and theory show that, in the limit of very large viscosity ratio, the flow is not linear but fractal. Generally, fractional flows and relative mobilities depend on both saturation and time. Use of a standard pore‐level model of 2‐D flow in the limit of infinite capillary number shows that this flow is fractal for large viscosity ratios (M = 10,000) and the sauration and fractional flows agree with the results of our general arguments. For realistic viscosities (M = 3 → 300), our modeling of the unstable flow shows that, although the flows are initially fractal, they become linear on a time scale, τ increasing as τ = τ 0 M 0.17 . Once linear, the saturation front advances as x ≈ v 0 M 0.068 t; the factor M 0.068 acts as a 2–D Koval factor.