# Evolution of a Geothermal System with Fluid Re-Injection
## 6 Conclusions  
The main conclusion of our investigation is that stable stationary states of the liquid/vapor interface are possible either with liquid or vapor recharge combined with vapor extraction. The main parameters \(\omega_{2}\) (natural liquid recharge), \(\beta_{v}<0\) (vapor extraction) and \(B_{v}>0\) (vapor meteoric recharge) need to be combined conveniently to maintain a stable interface. Increasing the vapor extraction moves the interface towards the bottom of the reservoir, which at the limit is shown to be increasingly vapor-dominated. Interestingly, stable interfaces can generally be obtained only if \(\omega_{2}\) is less than a critical value; that is, only if the natural liquid recharge is sufficiently large. The results obtained considering uniform porosity and permeability provide a good qualitative description of the possible steady configurations of a vapor-dominated basin such as Larderello. An important characteristic of the model is that stationary stable states are compatible with the presence of natural and artificial recharging mechanisms. Such a conclusion agrees with the results of other geological studies [18], despite the drastic simplifications introduced in our model: 1D geometry, an absence of chemicals dissolved in water and uniform geological parameters (which, in a real geothermal field, are far from being homogeneous and may change significantly due to local \begin{table}
\begin{tabular}{l c c} \hline \hline
**Year** & \(\mathbf{Q_{\text{extr}}}\) (kg/s) & \(\mathbf{Q_{\text{field}}}\)/ (\(\mathbf{C_{2}V_{\text{field}}}\)) \\ \hline
1960 & 412 & \(-\)92 \\
1970 & 287 & \(-\)64 \\
1975 & 241 & \(-\)54 \\
1985 & 255 & \(-\)57 \\
1995 & 266 & \(-\)59 \\
2000 & 275 & \(-\)62 \\
2010 & 294 & \(-\)66 \\
2015 & 315 & \(-\)69 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Production history of the Larderello field from 1960 to 2015. Artificial re-injection started in 1975. \begin{table}
\begin{tabular}{l l l c} \hline \hline
**Symbol** & **Description** & **Units** & **First Used in Formula** \\ \hline \(L\) & Reservoir thickness & m & (2) \\ \(v\) & Vapor velocity & m s\({}^{-1}\) & (3) \\ \(v_{\ell}\) & Liquid velocity & m s\({}^{-1}\) & (8) \\ \(d_{v}\) & Vapor density rate & kg m\({}^{-3}\)s\({}^{-1}\) & (4) \\ \(d_{\ell}\) & Liquid water source density rate & s\({}^{-1}\) & (8) \\ \(\delta\) & Dirac delta & - & (4) \\ \(D_{\ell}\) & Total liquid recharge rate & m s\({}^{-1}\) & (12) \\ \(P^{*}\) & Clapeyron pressure & Pa & (15) \\ \(\beta_{v}\) & Pressure source term & - & (30) \\ \(Q_{\text{field}}\) & Reservoir flow rate & kg s\({}^{-1}\) & (56) \\ \(q_{\text{mod}}\) & Total discharge per unit volume & kg m\({}^{-3}\)s\({}^{-1}\) & (57) \\ \(\sigma\) & Liquid/vapor interface & - & (22) \\ \(N\) & Number of injection heads & - & (4) \\ \(P_{\text{inj}}\) & Re-injection pressure & Pa & (4) \\ \(d_{\ell,\text{inj}}^{(i)}\) & \(i\)-th re-injection rate order of magnitude & m s\({}^{-1}\) & (13) \\ \(\beta_{\ell}^{(i)}\) & \(i\)-th liquid injection to meteoric rate ratio & - & (28) \\ \(\beta_{v}^{(i)}\) & \(i\)-th vapor injection to meteoric rate ratio & - & (31) \\ \(V_{\text{field}}\) & Reservoir volume & m\({}^{3}\) & (60) \\ \(T_{o}\) & Injection well depth & - & (47) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Nomenclature and other symbol definitions. temperature and pressure). The differential Equation (19), 4, has been proven to be quite satisfactory for modeling the vapor drawdown from a reservoir and for recharging from both aquifers and injection wells. We also think that our analysis is likely to produce physically meaningful results also for equilibrium configurations of water-dominated fields. In conclusion, we believe that, despite the many simplifications made in the model, this study can be an useful step towards improving our understanding of the dynamics of geothermal fields. [PERSON]. ([PERSON]), [PERSON] ([PERSON]), [PERSON] and F.R.