# Evolution of a Geothermal System with Fluid Re-Injection
## 3 Steady-State Solutions
### Case 1. _Recherage Due to Liquid Groundwater Recharge and a Single Vapor Extraction/Injection Well_  
We now prove that, also in this case, the equation \(F=0\) can be solved explicitly as in Equation (38) provided that the injection rate and the well depth fulfill suitable conditions. In this case, \(D_{\ell}\) is still given by Equation (43) while \(d_{\mathrm{v}}=d_{\mathrm{v,inj}}\delta\left(x-x_{\mathrm{o}}\right)\). Thus, \(\beta_{\mathrm{v}}\) given by (30) becomes a constant  
\[\beta_{\mathrm{v}}\left(P,T\right)=\hat{\beta}_{\mathrm{v}},\ \ \ \ \ \mathrm{with}\ \ \ \ \hat{\beta}_{\mathrm{v}}=\frac{\phi_{\mathrm{o}}}{\kappa_{\mathrm{o}}}\frac{r\ \mu_{\mathrm{o}}T_{\mathrm{s}}^{3}}{\gamma^{2}P_{ \mathrm{s}}^{2}}\ d_{\mathrm{v,inj}}. \tag{46}\]  
Integrating Equation (29), 1 once, we obtain  
\[\frac{P}{T}\left(\frac{\partial P}{\partial T}-\eta\frac{P}{T}\right)=-\hat{ \beta}_{\mathrm{v}}\theta\left(T-T_{\mathrm{o}}\right)+C_{1}\, \tag{47}\]  
where \(C_{1}\) is an integration constant, \(\theta\) is the Heaviside function and \(T_{\mathrm{o}}\) is the injection well depth. Setting \(z=P^{2}/2\), Equation (47) becomes  
\[z^{\prime}-2\eta\frac{z}{T}=C_{1}T-\hat{\beta}_{\mathrm{v}}T\theta\left(T-T_{ \mathrm{o}}\right),\]  
whose general solution is  
\[z\left(T\right)=C_{2}T^{2\eta}-\frac{C_{1}T^{2}}{2\left(\eta-1\right)}-\frac{ \hat{\beta}_{\mathrm{v}}}{2\left(\eta-1\right)}\left[T^{2\eta}\int_{1}^{T} \tau^{2-2\eta}\delta\left(\tau-T_{\mathrm{o}}\right)d\tau-T^{2}\theta\left(T-T _{\mathrm{o}}\right)\right],\]where \(\theta(T-T_{o})\) is the Heaviside function. By imposing boundary conditions (29), 2, and (29), 3, we get  
\[P^{2}\left(T\right) = \underbrace{T^{2\eta}-A\left(\sigma\right)\left(T^{2\eta}-T^{2} \right)}_{\text{for }\hat{\beta}_{v}=0}+\underbrace{\frac{\hat{\beta}_{v}}{\eta-1}\left[ \left(\frac{\sigma^{2\eta}T_{o}^{2-2\eta}-\sigma^{2}}{\sigma^{2\eta}-\sigma^{ 2}}\right)\left(T^{2\eta}-T^{2}\right)\right.}_{\text{additional term for }\hat{\beta}_{v}\
eq 0}\] \[- \underbrace{T^{2\eta}\int_{1}^{T}\tau^{2-2\eta}\delta\left(\tau- T_{o}\right)d\tau+T^{2}\theta\left(T-T_{o}\right)}_{\text{additional term for }\hat{\beta}_{v}\
eq 0},\]  
with \(A\left(\sigma\right)\) given by Equation (40), which reduces to Equation (39) when \(\hat{\beta}_{v}=0\). In particular,  
\[\frac{\partial P}{\partial T}\bigg{|}_{\sigma}=\frac{1}{\sqrt{f \left(\sigma\right)}}\left[\eta\sigma^{2\eta-1}+A\left(\sigma\right)\left( \sigma-\eta\sigma^{2\eta-1}\right)\right] \tag{49}\] \[+ \frac{\hat{\beta}_{v}}{\left(\eta-1\right)\sqrt{f\left(\sigma \right)}}\left[\frac{\sigma^{2\eta}T_{o}^{2-2\eta}-\sigma^{2}}{\sigma^{2\eta}- \sigma^{2}}\left(\eta\sigma^{2\eta-1}-\sigma\right)-\left(\eta\sigma^{2\eta -1}T_{o}^{2-2\eta}-\sigma\right)\right],\]  
so that \(\frac{\partial P}{\partial T}\bigg{|}_{\sigma}\) tends to (41) when \(\hat{\beta}_{v}\to 0\) and when \(T_{o}\to 1\) (as physically expected). To prove that the equation \(F=0\) can be solved explicitly, notice first that, because of (49), \(\Omega\) has a different form, namely  
\[\Omega=\Omega_{1}(\sigma)+\hat{\beta}_{v}\mathcal{G}\left(\sigma,T_{o}\right), \tag{50}\]  
where \(\Omega_{1}\) is given by Equation (42) and  
\[\mathcal{G}\left(\sigma,T_{o}\right)=\frac{\left[\sigma^{2\eta}T_{o}^{2-2\eta }-\sigma^{2}\right.\left(\eta\sigma^{2\eta-1}-\sigma\right)-\left(\eta\sigma^ {2\eta-1}T_{o}^{2-2\eta}-\sigma\right)\right]}{(\eta-1)f(\sigma)}. \tag{51}\]  
To use the explicit form in Equation (36), we need \(\Omega(\sigma,T_{o})\
eq 0\). It is convenient now to fix an upper limit to the variability of \(T_{o}\)--say \(\mathfrak{T}_{o}\).