The fluid dynamics of the well-documented eruptive episodes at Pu'u "O'o, Kilauea are used to investigate quantitatively the size and shape of the shallow conduit system beneath the vent. Cooling calculations are employed to study the long-term survival of conduits and the consequences of multiple dike injection events. We find that the subvent conduit must have a planar geometry at depths greater than a few tens to at most a few hundreds of meters, with a width of less than a few meters, a length of the order of 100 m, and a height of the order of 1 km. This structure is clearly the residue of the preeruption dikes and we propose that it be referred to as a "planar magma storage zone'.