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 di¤usion 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. The research that led to the present paper was partially supported by a grant of the group GNFM of INdAM.