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