Abstract: In response to recent experiments by Richard Packard's group at Berkeley, we construct a model of superflow through an array of apertures. We find that as the disorder becomes weak there is a transition from a regime where phase slips in the various apertures are largely independent to a regime where interactions lead to system-wide avalanches of phase slips. We explore the flow dynamics in both regimes, drawing analogies to the Burridge--Knopoff Earthquake model and the random field Ising model, and make connections to the experiments. Due to lack of viscosity, the flow of a superfluid through an aperture (at a constant applied pressure difference) exhibits dynamics distinct from those of a normal fluid. While a normal fluid may exhibit a steady flow in which the dissipative forces balance the applied pressure difference, this situation is inherently impossible in a superfluid. Instead, a superfluid is always accelerated by the pressure difference. However when the superflow velocity exceeds the critical velocity the superflow collapses and the velocity drops by a quantum in a phase-slip process. Our model for the superflow through an array of apertures incorporates two basic ingredients: (1) disorder associated with each aperture having its own random critical velocity, and (2) effective interaperture coupling, mediated through the bulk superfluid.