Shape of spreading and leveling gravity currents in a Hele-Shaw cell with flow-wise width variation

TL;DR

This study develops self-similar solutions to describe the spreading and leveling of gravity currents in a Hele-Shaw cell with variable width, validated through simulations and experiments, revealing universal profiles and regime transitions.

Contribution

The paper introduces second-kind self-similar solutions for gravity currents in variable-width Hele-Shaw cells, linking theory, simulations, and experiments for both spreading and leveling regimes.

Findings

Self-similar solutions accurately describe current shapes.

Simulations and experiments agree with theoretical scalings.

Universal profiles emerge in both regimes.

Abstract

We study the spreading and leveling of a gravity current in a Hele-Shaw cell with flow-wise width variations as an analog for flow {in fractures and horizontally heterogeneous aquifers}. Using phase-plane analysis, we obtain second-kind self-similar solutions to describe the evolution of the gravity current's shape during both the spreading (pre-closure) and leveling (post-closure) regimes. The self-similar theory is compared to numerical simulations of the partial differential equation governing the evolution of the current's shape (under the lubrication approximation) and to table-top experiments. Specifically, simulations of the governing partial differential equation from lubrication theory allow us to compute a pre-factor, which is \textit{a priori} arbitrary in the second-kind self-similar transformation, by estimating the time required for the current to enter the self-similar regime. With this pre-factor calculated, we show that theory, simulations and experiments agree well near the propagating front. In the leveling regime, the current's memory resets, and another self-similar behavior emerges after an adjustment time, which we estimate from simulations. Once again, with the pre-factor calculated, both simulations and experiments are shown to obey the predicted self-similar scalings. For both the pre- and post-closure regimes, we provide detailed asymptotic (analytical) characterization of the universal current profiles that arise as self-similarity of the second kind.