Although water infiltration through unconsolidated rocks and fractured rock formations control flow and transport to groundwater, spatial distributions of flow paths are poorly understood. Infiltration experiments conducted on packs of rocks showed that a well-constrained distribution of fluxes develops despite differences in rock type (angular diabase and sandstone, and subangular serpentinite), rock size (30 to 200 mm), and packing (up to 42 rock layers). Fluxes stabilize into a geometric (exponential) distribution that keeps about half of the system depleted of flow, retains a small fraction of high flow regions, and has a characteristic scale determined by the rock size. Modification of a statistical mechanical model shows that gravity-directed, random flow paths evolve to the observed flux distribution, and that it represents the most probable distribution. Key similarities between infiltration in rock deposits and fractured rock formations indicate that the geometric flow distribution may also apply in the latter systems.