A new, simple, and easily reproducible experiment was designed to simulate the production, accumulation, and transport of melt within rock. The transport was found to be of the self-organized critical type. The emergence of self-organized criticality is explained by the availability of hydrofracture propagation as a rapid or ballistic transport mechanism. This mechanism also serves as a mechanism for stepwise accumulation. These findings are confirmed by a numerical model, which shows the emergence of self-organized critical behavior when Darcian transport cannot accommodate transport and the dormant transport mechanism of hydrofracture propagation is activated. Ballistic and self-organized critical transport may play a significant role in the transport and accumulation of geological fluids, such as melt and hydrocarbons. This conclusion has a profound impact on the modeling of many transport processes in geology (e.g., accumulation of melt, oil, and gas).