# Evolution of a Geothermal System with Fluid Re-Injection
## 2 The Mathematical Model
As \(\rho\ll\rho_{\ell}\), we can approximate this relation with  
Table 3.1  
Thus, recalling Equations (6) and (9), we obtain  
Table 3.2  
where  
Table 3.3  
which (with a dimension of m s\({}^{-1}\)) is the total recharge rate in the liquid domain. In particular, we assume that we can identify two distinct effects:  
1. [label=()]
2. \(\mathrm{Recharge}\) due to the meteoric waters, denoted by \(d_{\ell,\mathrm{met}}\), which, as \(d_{\
u,\mathrm{met}}\), may depend on the local pressure;
3. \(\mathrm{Recharge}\) due to clusters of re-injection wells which, for simplicity, we assume to be concentrated in \(M\) locations. Thus, setting  
Table 3.4  
where, as in (4), the injection rates depend on the difference between the re-injection pressure \(P_{\mathrm{inj}}\) and the local pressure. Indeed, \(d_{\ell,\mathrm{inj}}^{(i)}\), \(i=1,2,\,...,\,M\), gives the injection rate order of magnitude,while \(J_{\ell}^{(i)}\left(P\left(x_{i}\right)-P_{\text{inj}}^{(i)}\right)\), which is dimensionless, gives the dependence on the pressure difference. We thus have  
Table 3.5 Table 3.6  
where  
Table 3.7  
A second free boundary condition is provided by [PERSON]'s law, which gives the relationship between \(P\) and \(T\) at the liquid/vapor interface. In the range of temperature relevant to our problem, this law takes the following empirical form:  
Table 3.8  
where \(T\) is in Kelvin, \(P\) in Pascal and \(a\) and \(b\) have values of \(a=9.617\times 10^{2}\,\text{Pa}\), \(b=17.35\). Consequently, the second condition on the free boundary is  
Table 3.9  
and (11) rewrites as  
\[ \begin{array}{rcl}\dot{s}&=& D_{\ell}\left[1+\frac{P^{*}}{rT \rho_{\ell}}\right]_{x=s(t)}\\ \\ &+&\left[\frac{P^{*}}{rT\rho_{\ell}}\frac{\kappa}{\phi\mu}\left( \frac{\partial P}{\partial x}+\frac{P^{*}}{rT}\mathcal{S}\right)\right]_{x=s( t)}.\end{array}  \tag{17}\]  
We remark that (16) requires the following condition be fulfilled:  
\[P\left(x,t\right)<P^{*}\left(T\left(x\right)\right),\text{in the vapor domain; i.e., }s(t)<x\leq L_{s}, \tag{18}\]  
in order to obtain a physically consistent solution. To complete the information, we need to give the pressure at the reservoir top, \(P\left(L_{s},t\right)=P_{s}\) (see Table 1) and the initial conditions \(s\left(0\right)=s_{o}\), \(L_{i}<s_{o}<L_{s}\), \(P\left(x,0\right)=P_{o}\left(x\right)\), \(s_{o}<x<L_{s}\), fulfilling Equations (18) and (16); i.e., \(P_{o}\left(x\right)<P^{*}\left(T\left(x\right)\right)\) and \(P_{o}\left(s_{o}\right)=P^{*}\left(T\left(s_{o}\right)\right)\). Thus, the mathematical problem to be solved is the following: find a time \(\tau>0\), a function \(s\left(t\right)\in C\left[0,\tau\right]\cap C^{1}\left(0,\tau\right)\) such that \(s\left(t\right)\in\left(L_{i},L_{s}\right)\) and a function \(P\left(x,t\right)\in C^{2,1}\left(D_{\tau}\right)\), continuous in \(\overline{D}_{\tau}=\{(x,t)\mid s(t)\leq x\leq L_{s},\ 0\leq t\leq\tau\}\), where \(\partial_{x}P\) is continuous up to \(x=s(t)\), \(t\in(0,\tau)\), so that, for \(T(x)\) given by Equation (1), the following system is fulfilled:  
\[\left\{ \begin{array}{l}\frac{\partial P}{\partial t}-\frac{T}{ \phi}\frac{\partial}{\partial x}\left[\frac{xP}{T\,\mu}\left(\frac{\partial P }{\partial x}+\frac{g}{r}\frac{P}{T}\right)\right]=rTd_{v}(P),\qquad(x,t)\in D _{\tau},\\ P(L_{s},t)=P_{s},\qquad t\in[0,\tau]\,,\\ P(s(t),t)=P^{*}(T\,(s(t))),\qquad t\in[0,\tau]\,,\\ s=D_{\ell}\,\left[1+\frac{P^{*}}{rT\rho_{\ell}}\right]_{x=s(t)}\\ \quad+\left[\frac{P^{*}}{rT\rho_{\ell}}\frac{\kappa}{\phi\mu} \left(\frac{\partial P}{\partial x}+\frac{P^{*}}{rT}g\right)\right]_{x=s(t)},\quad t\in(0,\tau)\,,\\ P(x,0)=P_{o}(x),\qquad x\in[s_{o},L_{s}]\,,\\ s(0)=s_{o},\qquad s_{o}\in(L_{i},L_{s})\,,\end{array} \right. \tag{19}\]  
According to the results in [43,44], the problem has one unique solution provided that \(P_{o}\in C\left[s_{o},L_{s}\right]\).