## 2 Materials and Methods
### WBHE Model Assumptions  
Python is a high-level programming language with an object-oriented approach described in [39]. As the Python software allows algorithms to be solved easily, it was used to perform the analysis of the WBHEs by implementing the described models (coaxial and U-tube-WBHE models) in different codes that have the structure illustrated in Figure 5.  
According to both selected models, the propagation of heat in the reservoir occurs through conduction; the propagation of heat inside the wellbore tubes takes place through conduction and convection phenomena. The reservoir model was built by assuming a single well located at the center of a circular reservoir.  
Figure 4: Geometrical model characteristics for calculating the equivalent diameter (modified from [22]. Copyright Elsevier, 2014).  
Figure 5: Python codes simplified research flowchart.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**U-Tube Wellbore Heat exchanger-Geometric Parameters** & **Symbol** & **Unit of Measure** \\ \hline Borehole diameter & \(D_{b}\) & (mm) \\ External U-pipe diameter & \(D_{pi}\) & (mm) \\ External U-pipes radius & \(r_{i}\) & (mm) \\ Equivalent pipes surface & \(S_{qq}\) & (m\({}^{2}\)) \\ Equivalent pipes diameter & \(D_{eq}\) & (mm) \\ Shank spacing (center-to-center) & \(W\) & (mm) \\ Depth & \(z\) & (m) \\ \hline \hline \end{tabular}
\end{table}
Table 3: U-tube WBHE geometric parameters.  
The temperature profile in the radial direction was assumed to be constant. Therefore, there was no temperature gradient in the annulus or the inner tube. Due to the turbulent flow, enhanced mixing phenomena occurred, which decreased the radial gradient. The temperature changed only in the annulus and in the vertical direction of the inner tube. Consequently, the temperature profile was unidirectional (vertical temperature profile).  
The properties of the heat carrier fluid were assumed to be constant. As the fluid used in this study was water (100 \({}^{\circ}\)C, 2 bar), no variations occurred due to pressure or/and temperature gradients.  
The analytical models were built under steady-state conditions; there were no temperature variations over time, with each point in the tubes (annulus and inner tube) maintaining the same temperature for the lifecycle of the system. In addition, the models considered the resistance associated with tube thickness to be negligible. The tube material had very high conductivity (15 W/mK), so its resistance could be considered small compared to the other resistances in the system. For the estimation of resistance associated with the rock (see Equation (6)), the time value used was 3 years. The analysed system, in the period preceding the three years (1-3 years), turns out to ensure larger heat exchange phenomena with the possibility of causing overestimation in energy performances.  
Considering the U-tube configuration, heat exchange was assumed to take place in an area that was half the area of the casing pipe, while the interaction between the downward and upward tubes was neglected.  
Both proposed analytical models follow the path of the working fluid with an approach that could be called step-by-step. In detail, they considered intervals of length \(dz\) in which the inlet and outlet temperatures were calculated by solving the energy balance equation for each considered volume \(dv\). For the estimation of the energy exchange in the radial direction, the mean value of the temperature in the volume \(dv\) was used and calculated by using the arithmetic mean. All energy exchanged in the radial direction in the volume \(dv\) was absorbed by water (selected working fluid).