# Evolution of a Geothermal System with Fluid Re-Injection
## 2 The Mathematical Model  
We consider a geothermal fluid composed of water in a gaseous and liquid state. We denote by \(x\) the vertical axis pointing upward. The bottom and the top of the geothermal reservoir correspond to \(x=L_{i}\left(L_{i}<0\right)\) and to \(x=L_{s}\left(L_{s}<0\right)\), respectively. The ground surface corresponds to \(x=0\), so that the domain of interest is \(\left[L_{i},L_{s}\right]\). We assume that the width of the capillary fringe is negligible with respect to the vertical dimensions of the reservoir, and thus that there is a clear separation between the liquid and the vapor phase through a sharp interface, \(x=s(t)\), that may vary in time. So, the lower part of the domain \(L_{i}\leq x\leq s\left(t\right)\), is saturated with liquid, while in the upper part of the domain, \(s\left(t\right)\leq x\leq L_{s}\), there is only vapor. At \(x=L_{s}\) the vapor pressure is known, and we denote it as \(P_{s}\). The latter is a measured datum and essentially depends on the exploitation of the basin. In Table 1, we list the values of the constants involved in the model. These parameters are compatible with those available for some areas of the geothermal field of Larderello, Tuscany [10,14]. According to the experimental data [37, 38, 39], the temperature within the reservoir is linear:  
\[T(x)=T_{s}+\gamma\left(L_{s}-x\right),\ \ \ \ L_{i}\leq x\leq L_{s} \tag{1}\] \begin{table}
\begin{tabular}{c l l l} \hline \hline
**Symbol** & **Description** & **Value** & **Units** \\ \hline \(L_{s}\) & Reservoir top depth & \(-1000\) & m \\ \(L_{i}\) & Reservoir bottom depth & \(-3500\) & m \\ \(T_{s}\) & Temperature at the top of the reservoir & \(550\) & K \\ \(\gamma\) & Thermal gradient & \(0.04\) & K/m \\ \(P_{s}\) & Vapor pressure at the top of the reservoir & \(9\times 10^{5}\) & Pa \\ \(r\) & Universal gas constant over H\({}_{2}\)O molar mass & \(4.6\times 10^{2}\) & J/\(\left(\mathrm{kg\,K}\right)\) \\ \(\rho_{l}\) & Liquid water density (at local conditions) & \(\approx\)\(8\times 10^{2}\) & kg/m\({}^{3}\) \\ \(\phi\) & Medium porosity & \(2\times 10^{-2}\) & - \\ \(g\) & Standard gravity & \(9.8\) & m/s\({}^{2}\) \\ \(\mu\) & Vapor viscosity & \(2\times 10^{-5}\) & Pa s \\ \(\kappa\) & Medium permeability & \(\approx\)\(5\times 10^{-15}\) & m\({}^{2}\) \\ \(K_{T}\) & Thermal conductivity & \(\approx\)\(2\) & W/\(\left(\mathrm{m\,K}\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of parameters referring to the reservoir of Larderello [10,14]. All dimensionless constants used in the present work are evaluated by using these values. A full list of additional simbols is reported in Table 3.  
where \(\gamma>0\), constant in time, is the known thermal gradient (see Table 1). We also set  
\[L=L_{s}-L_{i}(=2500\,\mathrm{m}). \tag{2}\]  
**Remark 1**. : _The given value of \(\gamma\) in Table 1 implies that the temperature at depth \(L_{i}=-3500\,\mathrm{m}\) is \(650\,\mathrm{K}\). Assuming a surface temperature of \(300\,\mathrm{K}\), we have a geothermal gradient in the caprock of \(\gamma_{cap}=0.25\,\mathrm{K}/\mathrm{m}\). This may be surprising, since this value is about three times the value commonly indicated in textbooks as the typical gradient in a geothermal field (generally \(0.08\,\mathrm{K}/\mathrm{m}\)) and actually requires that the average thermal conductivity of the caprock be rather low--say \(0.32\,\mathrm{W}/\left(\mathrm{mK}\right)\). A simple explanation is that the mean gradient of the field is indeed based upon the estimated depth of the \(K\) horizon (in the present case, around \(-5000\,\mathrm{m}\)) and the isotherm which identifies the \(K\) horizon itself (about \(750\,\mathrm{K}\)), leading, correctly, to \(\gamma_{mean}=0.08\,\mathrm{K}/\mathrm{m}\). On the other hand, the question is much more complex. In a very recently published article [39], the great variability of the thermal conductivity \(K_{T}\) of rocks at Larderello and their strong dependence on temperature and, to a much lesser extent, on pressure were clearly emphasized.