Influence of pore-scale disorder on viscous fingering during drainage

TL;DR

This study investigates how pore-scale disorder influences viscous fingering during drainage in porous media, revealing scale-dependent fractal dimensions and an effective growth law related to pressure gradients.

Contribution

It introduces a detailed analysis of pore-scale disorder effects on viscous fingering, identifying a crossover length and linking growth dynamics to the Dielectric Breakdown Model.

Findings

Lateral width of the central zone is 40% of system width, independent of capillary number.

Fractal dimension transitions from 1.83 to 1.53 at a scale proportional to Ca^{-1}.

Effective growth law v ∝ (∇P)^2 when averaging over disorder.

Abstract

We study viscous fingering during drainage experiments in linear Hele-Shaw cells filled with a random porous medium. The central zone of the cell is found to be statistically more occupied than the average, and to have a lateral width of 40% of the system width, irrespectively of the capillary number $Ca$. A crossover length $w_f \propto Ca^{-1}$ separates lower scales where the invader's fractal dimension $D\simeq1.83$ is identical to capillary fingering, and larger scales where the dimension is found to be $D\simeq1.53$. The lateral width and the large scale dimension are lower than the results for Diffusion Limited Aggregation, but can be explained in terms of Dielectric Breakdown Model. Indeed, we show that when averaging over the quenched disorder in capillary thresholds, an effective law $v\propto (\nabla P)^2$ relates the average interface growth rate and the local pressure gradient.