# Evolution of a Geothermal System with Fluid Re-Injection
## 5 Sustainability of Equilibria and Re-Injection  
The main environmental and industrial issue concerning a geothermal field is the long-term sustainability of its power production capability. Since the middle of the last century, this question has been deeply discussed in the specialized literature [40; 45; 47; 49; 50], and the primary role of re-injection has been clearly emphasized. It is interesting to express the equilibrium conditions investigated in this study in dimensional variables and to compare them with some historical field data referring to the Larderello reservoir. Due to the severe approximations considered, such a comparison does not have any specific intention but to meet, at least, the order of magnitude of the most significant and available data. An estimate of the volume \(V_{\rm field}\) of the Larderello reservoir [10; 11; 14] is about \(8\times 10^{11}\) m\({}^{3}\). Assuming the porosity of Table 1, the volume \(V_{\rm fluid}\) of fluid in the reservoir is on the order of \(1.6\times 10^{10}\) m\({}^{3}\). The meteoric recharge from the shallow aquifer surrounding the reservoir is not constant in time; it depends on a variety of factors and can only be roughly estimated [40]. The rate of steam extraction \(Q_{\rm extr}\) has changed considerably during the last decades; for example, it was about 300 kg/s in the 1950s and declined to about 150 kg/s in the 1980s [11]. Precise historical series are available from ENEL, the company holding the rights of exploitation of the Larderello geothermal fields [2]. In the 1990s, the natural recharge \(Q_{\rm nat}\) from the shallow aquifers in the area was estimated to be about one quarter of the extraction rate \(Q_{\rm extr}\)[40]. Due to re-injection, it is likely that the ratio of  
Figure 6: (a) Steady statein the case of a single injection well in the liquid domain. (b) The critical value of \(\sigma\) is \(\approx\)1.053 and \(\omega_{3,cr}\approx 3.3\times 10^{-4}\). All steady states with \(\sigma\in(\sigma_{\rm cr},1+\Gamma)\) and \(\omega_{3}<\omega_{3,cr}\) are stable. In dimensional variables, the interface is located at \(-1723\) m.  
Figure 7: Check of the compatibility condition in Equation (18) for case 3.  
extraction/natural recharge has not changed very much [10; 13]. To solve these urnnoe,s, we assume \(Q_{\text{nat}}=0.25\ Q_{\text{extr}}\) at any time. Then, we define  
\[Q_{\text{field}}(t)=-Q_{\text{extr}}(t)+Q_{\text{nat}}(t)=-0.75\ Q_{\text{extr}}(t) \tag{56}\]  
Consider, now, \"case 2\" as discussed in Section 4.2. According to the model, the total discharge per unit volume (that is, in \(\text{kg}/(\text{m}^{3}\text{s})\)) is given by  
\[q_{\text{model}}=\rho_{\ell}d_{\ell,\text{met}}+d_{v,\text{met}}+d_{v,\text{ inj}}, \tag{57}\]  
where \(d_{\ell,\text{met}}\), \(d_{v,\text{met}}\) and \(d_{v,\text{inj}}\) are determined by the dimensionless parameters \(\omega_{2}\), \(\hat{\beta}_{v}\), and \(B_{v}\) through (45), 2, (46), 2, and (53), 2, i.e.,  
\[d_{\ell,\text{met}}=C_{1}\omega_{2}^{-1},\quad d_{v,\text{met}}=C_{2}B_{v}, \quad d_{v,\text{inj}}=C_{2}\hat{\beta}_{v}, \tag{58}\]  
where, by using data from Table 1,  
\[C_{1}=\frac{\kappa_{0}P_{\text{s}}^{2}\gamma^{2}}{\phi_{v}\sigma_{\rho_{\ell} }\mu_{0}T_{\text{s}}^{3}}\approx 2.6\times 10^{-13}\ \text{s}^{-1},\qquad C_{2}=\rho_{\ell}C_{1}\approx 2.1\times 10^{-10}\ \text{kg}\,\text{m}^{-3}\text{s}^{-1}. \tag{59}\]  
For fixed values of \(\hat{\beta}_{v}<0\) and \(B_{v}>0\), any stable steady state is represented by a point \((\sigma,\omega_{2})\), with \(0<\omega_{2}<\omega_{\text{crit}}\) and \(\sigma>\sigma_{\text{crit}}\), on the curve \(\omega_{2}=\tilde{\mathfrak{F}}(\sigma;\hat{\beta}_{v},B_{v})\). Therefore the real location of the interface depends upon the effective value of \(\omega_{2}\).