## 3 Results and Discussion
### Wbhe Configurations  
The temperature profiles associated with the described Coaxial and U-tube WBHE system configurations were obtained by making use of the specific ground properties of the selected case study (Trecate4 hydrocarbon well) reported in Table 1. Values relating to the thermal properties of the different rock formations have been attributed by following those proposed in [31,32] (Table 1). Moreover, the thermal conductivity value of the insulating material was set to 0.025 W/mK. Working fluid inlet flow rate (water) and temperature values were considered as 3.0 kg/s and 50 \({}^{\circ}\)C, respectively. Selected inlet temperature represents a typical value for direct applications such as production cycles in manufacturing and agricultural districts [1]. Subsequently, an analysis was conducted on the term vector fluid temperature at the outlet as the inlet flow rate varies. For the coaxial WBHE configuration, the sizings of the inner and outer tubes, as well as the final casing size, were fixed according to the values proposed in Table 4a. The characteristic configuration of the U-Tube WBHE is shown in Table 4b. For the Coaxial WBHE configuration, over the first 1200 m downwards, the selected thermo-vector fluid decreases in temperature. As the downward profile line crosses the underground temperature line, the real heating process begins, and the ground contribution became positive. Due to the presence of the insulating material, the heat exchange coefficient between tubes turns out to be low, and the increase in working fluid temperature can be associated with the ground (Figure 7a). Figure 6: Soil resistance over time with different ground density (\(\rho\)–kg/m\({}^{3}\)) (a); specific heat (pc\({}_{s}\) rock-J/kg/K) (b); conductivity (\(\lambda_{s}\)–W/mK) (c); thermal diffusivity (\(\alpha\)) (d). Figure 7: Temperature profile associated with the coaxial WBHE (**a**) and U-tube WBHE (**b**) configurations considering site-specific stratigraphy. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**a** & **Tube Sizing** & **ID** & **OD** \\ \hline  & & mm & mm \\ \hline  & 3 inches & 77.9 & 88.9 \\ \hline  & 5 inches & 121.4 & 139.7 \\ \hline  & Casing 7 inches & 150.4 & 177.8 \\ \hline
**b** & **Well Radius** & **Tube Radius** & **W Distance** \\ \hline  & mm & mm & mm \\ \hline  & 75 & 21.5 & 80 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: (**a**). Coaxial WBHE tube sizing. ID: internal diameter; OD: external diameter [24]; (**b**). U-tube WBHE parameters [25]. In coaxial WBHE, the maximum recorded temperature is equal to 98.6 \({}^{\circ}\)C. The inlet flow rate strongly influences the temperature of the wellhead thermal fluid and, consequently, the heat power amount. As observed in Figure 8a, by considering inlet flow rate values between 0.5 and 1 kg/s, the output fluid temperature increases up to about 155 \({}^{\circ}\)C. Consequently, for the specific case study and associated plant configuration, it is possible to identify an inlet flow rate value that potentially allows an optimization of the wellhead temperature. Considering the U-tube WBHE configuration and its associated temperature profile, it was possible to identify how the ground's contribution was responsible for a sizable temperature variation, both in the downward and upward tube. Over the first borehole section (approximately 1200 m), the fluid was cooled by the ground. Subsequently, as the temperature profile line crossed the underground temperature line, the trend was inverted, and the heat carrier fluid temperature began to increase, reaching a maximum temperature value of 84 \({}^{\circ}\)C (Figure 7b). As for the considered coaxial-WBHE configuration, the wellhead thermal fluid temperature in U-tube WBHE changes as the input flow rate parameter varies; these were estimated (Figure 8b). For a flow rate value of 1.5 kg/s, the fluid reaches the surface at a maximum temperature of 89.4 \({}^{\circ}\)C. For higher flow rate values, the wellhead temperatures recorded are progressively lower.