This is a treatment of the phenomenon of energy storage by melting in an enclosure where the heating is provided by an invading line of higher temperature. We show that the melting process consists of two phases: "invasion" thermal diffusion along the invading line, which is followed by "consolidation" as heat diffuses perpendicularly to the invading line. We report the duration of both phases and the evolution of the melt layer around the invading line during the two-dimensional and three-dimensional invasion. The amount of melted material increases in time according to a curve shaped as an S. Diffusion sweeps the territory faster as the complexity of the tree design increases meaning the S-curve of the history of the process becomes steeper.