# Compression-driven viscous fingering in a radial Hele-Shaw cell

## TL;DR

This study investigates how gas compressibility influences viscous fingering in a radial Hele-Shaw cell, revealing that higher compressibility delays finger formation and reduces pattern severity, with experiments and simulations providing comprehensive insights.

## Contribution

It introduces the compressibility number as a key parameter affecting viscous fingering, expanding understanding beyond traditional incompressible models.

## Key findings

- Increasing compressibility number delays finger onset.
- Higher compressibility reduces fingering pattern severity.
- Experiments and simulations show consistent effects across parameters.

## Abstract

The displacement of a viscous liquid by a gas within a Hele-Shaw cell is a classical problem. The gas-liquid interface is hydrodynamically unstable, forming striking finger-like patterns that have attracted research interest for decades. Generally, both the gas and liquid phases are taken to be incompressible, with the capillary number being the key parameter that determines the severity of the instability. Here, we consider a radially outward displacement driven by the steady compression of a gas reservoir. The associated gas-injection rate is then unsteady due to the compressibility of the gas. We identify a second nondimensional parameter, the compressibility number, that plays a strong role in the development of the fingering pattern. We use an axisymmetric model to study the impact of compressibility number on the unsteady evolution of injection rate and gas pressure. We use linear stability analysis to show that increasing the compressibility number delays the onset of finger development relative to the corresponding incompressible case. Finally, we present and compare a series of experiments and fully nonlinear simulations over a broad range of capillary and compressibility numbers. These results show that increasing the compressibility number systematically decreases the severity of the fingering pattern at high capillary number. Our results provide an unprecedented comparison of experiments with simulations for viscous fingering, a comprehensive understanding of the role of compressibility in unstable gas-liquid displacement flows, and insight into a new mechanism for controlling the development of fingering patterns.