Growth activity during fingering in a porous Hele Shaw cell

TL;DR

This study investigates the dynamics of fingering during unstable drainage in a 2D porous medium, revealing scale-invariant invasion activity and power-law relationships with capillary number, supported by experimental and analytical insights.

Contribution

It provides new experimental evidence and analytical models linking invasion probability, pressure fields, and growth dynamics in viscous fingering within porous media.

Findings

Invasion occurs within a screening length from the finger tip.

Invasion probability density depends only on distance to the tip.

Power-law relationships are observed between finger growth metrics and capillary number.

Abstract

We present in this paper an experimental study of the invasion activity during unstable drainage in a 2D random porous medium, when the (wetting) displaced fluid has a high viscosity with respect to that of the (non-wetting) displacing fluid, and for a range of almost two decades in capillary numbers corresponding to the transition between capillary and viscous fingering. We show that the invasion process takes place in an active zone within a characteristic screening length from the tip of the most advanced finger. The invasion probability density is found to only depend on the distance to the latter tip, and to be independent of the value for the capillary number Ca. The mass density along the flow direction is related analytically to the invasion probability density, and the scaling with respect to the capillary number is consistent with a power law. Other quantities characteristic of the displacement process, such as the speed of the most advanced finger tip or the characteristic finger width, are also consistent with power laws of the capillary number. The link between the growth probability and the pressure field is studied analytically and an expression for the pressure in the defending fluid along the cluster is derived. The measured pressure are then compared with the corresponding simulated pressure field using this expression for the boundary condition on the cluster.