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The scaling relation between the drainage area and stream length (Hack's law), along with exceedance probabilities of drainage area, discharge, and upstream flow network length, is well known for channelized fluvial regions. We report here on a laboratory experiment on an eroding unconsolidated sediment for which no channeling occurred. Laser scanning was used to capture the morphological evolution of the sediment. High-intensity, spatially nonuniform rainfall ensured that the morphology changed substantially over the 16-hr experiment. Based on the surface scans and precipitation distribution, overland flow was estimated with the D8 algorithm, which outputs a flow network that was analyzed statistically. The above-mentioned scaling and exceedance probability relationships for this overland flow network are the same as those found for large-scale catchments and for laboratory experiments with observable channels. In addition, the scaling laws were temporally invariant, even though the network dynamically changed over the course of experiment.

Plain Language Summary In spite of different climates, vegetation, and land properties, the geometry of river networks is characterized by near-identical scaling laws. Can we expect similar statistical metrics for surface flow over unchanneled morphologies at small scales? To answer this question, the morphology of uncohesive sediment was measured at high resolution in a laboratory flume and under nonuniform rainfall. Based on this morphology, the overland flow network was determined. The results showed that even at small scale (2m by 1m) and in absence of rills, the flow network has the same statistical characteristics as large-scale river networks. In other words, the shallow overland flow could be represented by a network that dynamically changed while preserving catchment scaling laws.