## 3 Numerical Simulations
### Mathematical Model
#### 3.2.2 Governing Equations  
(1) Water flow and heat transfer in the far-field heat extraction zone \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & **Value** & **Items** & **Value** \\ \hline Density of rock(kg/m\({}^{3}\)) & 2700 & Inner radius of annulus(m) & 2 \\ Thermal capacity of rock(J/(kg\({}^{.9}\)C)) & 1000 & Outer radius of annulus(m) & 3 \\ Thermal conductivity of rock(W/(m\({}^{.9}\)C)) & 3.5 & Outer radius of tunnel (m) & 3 \\ Porosity w.r.t. SRV & 0.23 & Radius of shattered rock zone & 30 \\ Permeability w.r.t. SRV (m\({}^{2}\)) & \(10^{-10}\) & Outer radius of models(m) & 200 \\ Length of models(m) & 2000 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Input parameters of the models. For the far-field heat extraction zone, the water flow in the porous medium follows [PERSON]'s law, and the mass conservation equation can be expressed as [34]:  
\[\frac{\partial\Big{(}\rho_{wf}\varphi\Big{)}}{\partial t}+\
abla\cdot\Big{(}\rho _{wf}u_{f}\Big{)}=0 \tag{4}\]  
where \(\rho_{wf}\) is the density of water and \(\varphi\) is the porosity of the thermal reservoir. The subscripts w and f denote the working fluid as water and the far-field heat extraction zone, respectively. \(t\) is the time. \(u_{f}\) is Darcy velocity, which can be expressed as:  
\[u_{f}=-\frac{k}{\mu_{f}}\
abla p \tag{5}\]  
where \(k\) is the permeability, \(p\) is the pressure, \(\mu_{f}\) is the viscosity of water. The heat transfer process in porous media is described by the following energy conservation equation [35],  
\[(\rho c)_{eff}\frac{\partial T_{f2}}{\partial t}+\rho_{wf}c_{p,wf}u_{f}\cdot \
abla T_{f2}-\
abla\cdot\Big{(}\lambda_{eff}\
abla T_{f2}\Big{)}=0 \tag{6}\]  
where \(T_{f2}\) is the fluid temperature, \(c_{p,wf}\) is the heat capacity of water, \((\rho c)_{eff}\) and \(\lambda_{eff}\) is the equivalent heat capacity and thermal conductivity, respectively,  
\[(\rho c)_{eff}=\varphi\rho_{wf}c_{p,wf}+(1-\varphi)\rho_{r}c_{r} \tag{7}\]  
\[\lambda_{eff}=\varphi\lambda_{wf}+(1-\varphi)\lambda_{r} \tag{8}\]  
where \(\rho_{r}\) is the density of rock, and the subscript \(r\) stands for rock. \(c_{r}\) is the heat capacity of the rock, \(\lambda_{wf}\) is the thermal conductivity of water, and \(\lambda_{r}\) is the thermal conductivity of the rock. The heat transfer process in the surrounding rock is mainly heat conduction, which can be expressed as,  
\[\rho_{r}c_{r}\frac{\partial T_{f1}}{\partial t}=\
abla\cdot\Big{(}\lambda_{r }\
abla T_{f1}\Big{)} \tag{9}\]  
where T\({}_{f1}\) is the temperature of the surrounding rock. Due to the coupling heat transfer between the surrounding rock and the porous medium, the conditions of equal interface temperature and conservation of heat flow should be satisfied,  
\[T_{f1}=T_{f2} \tag{10}\]  
\[-\lambda_{r}\
abla T_{f1}=-\lambda_{eff}\
abla T_{f2} \tag{11}\]  
(2) Water flow and heat transfer in the near-field heat extraction zone  
For the near-field heat extraction zone, the water flow and heat transfer process in the tunnel are described by a 1D non-isothermal pipeline model. The momentum and mass equations can be expressed as [36],  
\[\frac{\partial(A\rho_{wn})}{\partial t}+\
abla\cdot(A\rho_{wn}u_{n})=0 \tag{12}\]  
\[\frac{\partial\Big{(}\rho_{wf}\varphi\Big{)}}{\partial t}+\
abla\cdot\Big{(}\rho _{wf}u_{f}\Big{)}=0 \tag{4}\]0  
where \(\rho_{wn}\) is the water density, and the subscript \(n\) denotes the near-field heat extraction zone.