We present a series of similarity solutions to describe the temperature field as liquid spreads from a line source into a porous rock saturated with liquid of higher temperature. We identify slow and fast flow regimes. In the slow flow regime, the liquid is heated to the far-field temperature by conduction of heat from the far field. In the fast flow regime, there is negligible conduction of heat from the far field. Instead, the liquid is heated to the far-field temperature by cooling a region of the host rock near the source, and an internal boundary layer develops within the newly injected liquid. We successfully test our quantitative theoretical predictions with a series of laboratory experiments in which water was injected into a consolidated bed of sand filled with liquid of different temperature. We extend our model to describe the vaporization of liquid as it spreads slowly from a central source into a superheated porous rock. A further family of similarity solutions shows that the rate of vaporization depends upon the injection rate as well as upon the initial superheat of the reservoir. For high injection rates, the liquid is typically heated to the interface temperature long before reaching the interface. The rate of vaporization then becomes independent of the initial liquid temperature, and depends mainly on the reservoir superheat. For lower injection rates, heat is conducted from ahead of the boiling front into the liquid. As a result, for progressively smaller injection rates, an increasing fraction of the liquid vaporizes, until virtually all the liquid boils, and only a very small liquid zone develops in the rock. Again, we successfully test our theoretical predictions with a laboratory experiment in which liquid water was injected into a superheated layer of permeable sandstone.