## 2 Materials and Methods
### Heat Transfer in Coaxial WBHEs  
In coaxial WBHEs, the steel downward is in contact with the hole in the well. The energy balance of the fluid in the injection pipe can be expressed with the following equation [14; 35]:  
\[\frac{\partial\Big{(}(\rho c)_{f}A_{0}T_{fo}\Big{)}}{\partial\tau}+\frac{ \partial\Big{(}(\rho c)_{f}A_{0}v_{f}T_{fo}\Big{)}}{\partial z}=-\frac{dQ}{dz }+\frac{dQ_{i0}}{dz} \tag{1}\]  
where \(A_{0}\) and \(v_{f}\) are the outer pipe area and fluid velocity, respectively, \(T_{fo}\) is the fluid temperature in the outer pipe and \(dQ/dz\) is the heat extraction from the formation at unit well depth (W/m) (Table 2).  
Although insulation is used to prevent heat loss from the inner-pipe fluid, heat is partly transferred between the two pipes; thus, \(dQ_{io}/dz\) represents the heat flux from the inner pipe to the outer pipe. Therefore, the energy equation for the inner pipe can be given as follows.  
\[\frac{\partial\Big{(}(\rho c)_{f}A_{i}T_{fi}\Big{)}}{\partial\tau}+\frac{ \partial\Big{(}(\rho c)_{f}A_{i}v_{f}T_{fi}\Big{)}}{\partial z}=-\frac{dQ_{i0 }}{dz} \tag{2}\]  
By assuming steady heat transfer and constant heat flux in wellbore components (e.g., insulation, casing and cement), heat extraction from the formation \(dQ/dz\) can be assumed to be equal to the heat flux through the outside surface of the wellbore (interface of the wellbore and rock formation) to the injected fluid [14; 35]:  
\[\frac{dQ}{dz}=2\pi r_{w}k_{w}\Big{(}T_{f0}-T_{w}\Big{)}=\Big{(}T_{f0}-T_{w} \Big{)}/R_{w} \tag{3}\]  
where \(T_{w}\) is the temperature at the interface of the wellbore and the formation, \(k_{w}\) is the heat transfer coefficient between the outer pipe fluid and wellbore exterior and \(R_{w}\) is the resistance between the outer pipe and surrounding rocks.  
At the well bottom, the heated fluid is forced to enter and flow through the internal pipe of the coaxial WBHE. Proceeding upwards to the wellhead, heat transfer occurs only through the wall of the internal pipe. Thus, \(dQ_{i0}/dz\) is determined by considering  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Coaxial Wellbore Heat Exchanger-Geometric Parameters** & **Symbol** & **Unit of Measure** \\ \hline Outer pipe area & \(A_{0}\) & [m\({}^{2}\)] \\ Inner pipe area & \(A_{i}\) & [m\({}^{2}\)] \\ Radius of outside wellbore & \(r_{w}\) & [mm] \\ External radius of the external casing & \(r_{c}\) & [mm] \\ Internal radius of the external casing & \(r_{i}\) & [mm] \\ Radius of the internal casing & \(r_{0}\) & [mm] \\ Thicknesses of the pipe exchanger & \(d\) & [mm] \\ Depth & \(z\) & [m] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Coaxial WBHE–geometric parameters.  
the temperature difference between the outer-pipe and inner-pipe fluids, as well as the estimated thermal resistance of the insulation:  
\[\frac{dQ_{i0}}{dz}=2\pi\tau_{0}k_{i0}\Big{(}T_{fi}-T_{f0}\Big{)}=\Big{(}T_{fi}-T_ {f0}\Big{)}/R_{i0} \tag{4}\]  
where \(T_{fi}\) is the fluid temperature in the inner pipe, \(k_{i0}\) is the heat transfer coefficient between the outer pipe and inner pipe and \(R_{i0}\) is the thermal resistance between the outer pipe and inner pipe.