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

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 (preclosure) and leveling (postclosure) 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 prefactor, which is 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 prefactor 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 prefactor calculated, both simulations and experiments are shown to obey the predicted self-similar scalings. For both the pre- and postclosure regimes, we provide detailed asymptotic (analytical) characterization of the universal current profiles that arise as self-similarity of the second kind.