A free boundary model for the evolution of a geothermal system

The evolution of a geothermal system is studied. A mathematical model is proposed and the corresponding free boundary problem is formulated in a one-dimensional geometry. A situation corresponding to the geothermal field in Larderello, Tuscany (Italy) is considered, showing that the problem has two characteristic time scales, related to the motion of interface and diffusion of vapor. Since the former is much slower, a quasi-steady approximation can be introduced and solved, obtaining a qualitative description of the evolution of the Larderello basin from a liquid-dominated to a vapor-dominated situation. This is in agreement with the geological results.