# Evolution of a Geothermal System with Fluid Re-Injection
## 1 Introduction  
In volcanic areas, geothermal energy represents an important resource. For instance, the Tuscan basins of Larderello and Monte Amiata provide almost 30% of the energetic need of the whole region (see, for example, [1, 2]). Several studies have been devoted to geothermal reservoirs with the aim of providing mathematical models--some of them very sophisticated--for extraction wells [3, 4, 5, 6, 7, 8, 9, 10]. Here we refer to a specific situation--a vapor-dominated system [11, 12]--in which the reservoir is sealed at the top by a nearly impermeable rock layer, under which water is in the vapor state up to a depth at which the pressure and temperature are such that it is present as liquid. Wells are drilled through the impermeable layer to reach the high-pressure vapor zone. The vapor then leaves the reservoir, and extraction is sustained by the vaporization of deep water (sometimes called plutonic or juvenile water). This is the case for the Larderello geothermal system (one of the largest in the world), which is characterized by a reservoir pressure that is much lower than hydrostatic pressure. This is a consequence of its natural evolution from an initial liquid-dominated to the current vapor-dominated system [3, 10]. The genesis of vapor-dominated reservoirs has intrigued researchers for many years. In 1973, [PERSON] and [PERSON] [6] reviewed all the principal conceptual models proposed until then for the reservoir and possible mechanisms for superheating the steam. These models aimed to give a qualitative description of the physical processes leading to the development of a vapor-dominated reservoir. However, all efforts notwithstanding, many open questions remain. A reservoir is always formed by a complex of fractured permeable rocks confined between impermeable layers and saturated by water, in a liquid and/or vapor phase. As far as the Larderello geothermal field is concerned, the heat source is a magmatic intrusion located at a relatively small depth (5-10 km) producing temperatures of \(500\div 700\)\({}^{\circ}\)C in the rocks surrounding the reservoir. Both the lower and upper boundaries of the reservoir depend on the local lithology. The so-called K horizon (a seismic reflecting surface), strictly related to a temperature of about 450 \({}^{\circ}\)C, is consideredto represent the bottom of the geothermal reservoir [10; 13; 14]. The top of the productive reservoir is set in correspondence to the first fractured level in the central areas of the field, where a great amount of well data are available. In the peripheral areas, where well data are lacking, the top is set in correspondence to the 250 degC isotherm [13; 14; 15; 16]. Usually, the shallow carbonate-anhydrite reservoir shows a temperature of \(220\div 250\) degC and pressure of about \(15\div 20\) bar at 1000 m depth, while the deep metamorphic reservoir has a temperature of \(300\div 350\) degC and pressure around 70 bar at 3000 m depth [13]. The sustainable management of geothermal resources basically involves setting up and maintaining a specific long-term production scheme [17; 18]. It also involves the basic ingredients of successful geothermal resource management, i.e., re-injection, monitoring and modeling [14; 19; 20]. However, the term \"sustainable development\" has many other important implications and has received ever-increasing attention over the last decade or two [21; 22; 23; 24; 25; 26; 27; 28; 29]. Vapor-dominated geothermal fields are still little understood, and their production histories are so short that premonitory indications of exhaustion are not yet generally recognized. In the 1970s, [PERSON] [30] concluded that the production of steam from the Larderello fields at that time was approaching a steady state and could have been maintained for many years. However, [PERSON] and [PERSON] [6] came to quite different conclusions, saying that the Larderello field was approaching exhaustion. This appears to have been sustained 30 years later, as the time series of production shows, in the papers by [PERSON] et al.