Fractal nature of viscous fingering in three-dimensional pore-level models

Abstract

Using three-dimensional pore-level models of random porous media, we have studied viscous fingering in ‘‘two-phase,’’ miscible flow. When the viscosity ratio M (=${\mathrm{\mu }}_{\mathit{D}}$/${\mathrm{\mu }}_{\mathit{I}}$, the ratio of the viscosity of the displaced fluid to that of the injected fluid), is infinite, the flow is known to be modeled by diffusion-limited aggregation (DLA). We have observed and characterized the crossover from fractal flow at short times or large viscosity ratios to compact (i.e., Euclidean) flow at later times. In our pore-level model of three-dimensional flow in the limit of infinite capillary number (zero surface tension), the low viscosity fluid is injected at constant pressure on one face of the model porous medium. This modeling shows that this flow is fractal for large viscosity ratios (M=10 000), consistent with DLA. For realistic viscosities (M=30–1000), 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=${\mathrm{\tau }}_0$ ${\mathit{M}}^{0.16}$. This characteristic crossover time predicts that the flow become compact for patterns larger than a characteristic length, which increases with viscosity ratio as l=${\mathit{l}}_0$ ${\mathit{M}}_{\mathit{f}}^{0.16/(\mathit{D}}$-2), where ${\mathit{D}}_{\mathit{f}}$ is the fractal dimension. Once compact, the saturation front advances as x≊${\mathit{v}}_0$ ${\mathit{M}}^{0.16}$t; the factor ${\mathit{M}}^{0.16}$ acts as a three-dimensional Koval factor. © 1996 The American Physical Society.