## 2 Materials and Methods
### Heat Transfer in Coaxial WBHEs
#### 2.3.1 Coaxial WBE: Coefficient of Heat Exchange between Outer-Pipe Fluid and the Wellbore Exterior  
In an analysis of the energy balance equation for the fluid in the outer pipe (injection pipe) of a coaxial WBE, a careful estimate of the parameter \(k_{w}\) is fundamental for proper evaluation of the heat exchange between the outer-pipe fluid and the drilled geological formations.  
For a coaxial WBE, the heat exchange coefficient for the injection pipe can be expressed as the sum of heat transfer components, expressed in terms of thermal resistance values (\(R_{w}\)) [22]:  
\[R_{w}=\ R_{s}+R_{a}+\ R_{c} \tag{5}\]where \(R_{s}\) is a function of time that represents the thermal resistance due to conductive heat transfer in the rock, \(R_{a}\) is the thermal resistance due to convective heat transfer into the pipe and \(R_{c}\) is the thermal resistance due to conductive heat transfer through the casings of the well.  
In the evaluation of total thermal resistance, the conductive term prevails; consequently, the thermal exchange is directly proportional to the convective transfer coefficient.  
Conductive thermal resistance (\(R_{s}\)) can be expressed as follows:  
\[R_{s}=\ \frac{1}{2\lambda_{s}}In\frac{2\sqrt{\alpha_{s}t}}{r_{w}} \tag{6}\]  
where \(\lambda_{s}\) (W/mK) is the thermal conductivity of the rock and \(\alpha_{s}\) (m/s) is the thermal diffusivity of the rock. In Equation (6), the relationship in the numerator of the second term represents the time-dependent radius of the thermal influence of the well (\(r_{s}\)).  
Convective thermal resistance (\(R_{a}\)) can be determined by the following equation:  
\[R_{a}=\ \frac{1}{2r_{c}h_{f}} \tag{7}\]  
where \(r_{c}\) is the external radius of the external casing and \(h_{f}\) is the convective heat transfer coefficient, which was calculated by using the Nusselt number (Nu) and a form of the Dittus-Boelter equation that assumes turbulent flow inside the tubes (Reynolds number \(\geq\) 104) [23]:  
\[h_{f}=\ \frac{Nu\lambda_{f}}{2r_{c}} \tag{8}\]  
\[Nu=0.023 Re^{0.8}Pr^{0.4} \tag{9}\]  
with \(Pr=\frac{\rho_{c}f^{\mu}}{\lambda_{f}}\) and \(Re=\frac{\rho_{0}r_{2}r_{c}}{\mu}\).  
Finally, the thermal resistance to heat conduction through the casings of the well can be determined as follows:  
\[R_{c}=\sum_{i=1}^{n}R_{\lambda i}=\ \frac{1}{2}\sum_{i=1}^{n}\frac{1}{\lambda_{i }}In\frac{r_{c,i+1}}{r_{c,i}} \tag{10}\]  
where \(\lambda_{i}\) is the thermal conductivity of the rock in correspondence with the different casings of the well. Generally, due to the high thermal conductivity of the steel piping, the total thermal resistance of the casing is negligible compared to the rock thermal resistance [13].  
As a result, the heat exchange coefficient \(k_{w}\) can be correctly determined as follows:  
\[\frac{1}{k_{w}}=\ \frac{2r_{c}}{2\lambda_{s}}In\frac{4\sqrt{\alpha_{s}t}}{2r_{ w}}+\frac{1}{h_{f}} \tag{11}\]  
where \(rc=rw\) as the thickness of the external tube is negligible.  
2.3.2 Coaxial WBHE: Coefficient of the Heat Exchange between the Outer-Pipe Fluid and the Inner Pipe  
Unlike in the injection pipe, the total heat flux in the upward pipe (extraction pipe) is determined by a conductive component of the composite pipe and by two convective components, one on the internal wall and one on the external wall of the WBHE.  
Consequently, the total heat exchange coefficient \(k_{i0}\) for the extraction pipe can be calculated as follows:  
\[\frac{1}{k_{i0}}=\ \frac{r_{0}}{r_{0+d}}\frac{1}{h_{i}}+r_{0}\sum_{i=1}^{n} \frac{1}{\lambda_{i}}In\left(\frac{r_{i+1}}{r_{i}}\right)+\frac{1}{h_{0}} \tag{12}\]where \(r_{0}\) is the radius of the inner pipe, \(d\) is the thicknesses of the pipe exchanger, \(h_{0}\) and \(h_{i}\) are the coefficients of convective heat transfer to the inner and outer wall, respectively, and \(\lambda_{i}\) is the thermal conductivity of the pipe material.