## 2 Materials and Methods
### U-Tube WBE: Thermal Resistances Model  
In U-tube WBE, the hole between the tubes and the well is filled with grout (bentonite) in order to prevent direct leakage between the ground and tubes, avoiding a connection between the ground and Earth's subsurface.  
Several approaches have been proposed for reproducing the thermal behaviour of different U-tube WBE configurations [37]. In this work, to compute the temperature profile in the U-tube configuration using a one-dimensional heat exchange, the model with a set of equivalent thermal resistances described in [25] was applied. Six thermal resistances were considered at each depth, including the thermal properties of the ground, the grout and the pipes (Figure 3).  
Figure 3: Thermal resistances definition steps: (**a**) borehole resistance, (**b**) parallel borehole resistances, (**c**) convective and conductive resistances and (**d**) final resistances configuration (reproduced from [22]. Copyright Elsevier, 2014).  
The thermal resistances between the grout (bentonite) and pipe depend on the overall borehole thermal resistance \(R_{blue}\). This parameter is the average thermal resistance between the fluid in the pipe and the borehole wall, and it is usually determined after experimental tests. The grout area is divided into two zones according to the pipe numbers; thus, \(R_{blue}\) was also considered as divided into two parallel resistances that connected each pipe with the corresponding grout zone. This parameter can be further divided into a convective (\(R_{h}\)) and a conductive (\(R_{c}\)) term (Equation (13)):  
\[2R_{blue}=R_{h}+R_{c} \tag{13}\]  
where \(R_{c}\) represents conductive thermal resistance, considering the total conductive resistance between the pipes and the borehole wall. As the grout node can be located at a certain distance \(D_{x}\), \(R_{c}\) is divided into two different resistances (Equations (14) and (15)):  
\[R_{c}=R_{b}+R_{x} \tag{14}\]  
\[R_{b}=R_{b1}=R_{b2} \tag{15}\]  
where \(R_{b}\) represents the conductive resistance between the pipe and the considered node.  
\(R_{h}\) is calculated by following Equation (16), where \(r_{i}\) is the internal pipe radius, \(h_{f}\) is the convective heat coefficient and \(Nu\) is the dimensionless Nusselt number that is calculated using Equation (9).  
\[R_{h}=\frac{1}{2\pi r_{i}h_{f}dz}=\frac{1}{\pi\lambda_{f}Nudz} \tag{16}\]  
The \(R_{c}\) value is indeed estimated using a calculation method that requires the estimation of the pipe equivalent surface \(S_{eq}\) and its diameter \(D_{eq}\) (Equation (17)) (Table 3).  
\[D_{eq}=2\sqrt{\frac{S_{eq}}{\pi}} \tag{17}\]  
The pipe equivalent surface (\(S_{eq}\)) can be calculated by following the approach proposed in [38], considering the sum of the areas \(S_{gg}\) and \(S_{p}\) as indicated in Figure 4. Consequently, \(D_{eq}\) is obtained by following Equation (18).  
\[D_{eq}=D_{pe}\sqrt{\frac{4W}{\pi D_{pe}}}+1 \tag{18}\]  
Using the equivalent diameter, the conductive thermal resistances \(R_{b}\) and \(R_{x}\) are calculated by considering a semi-cylindrical conductive heat transfer:  
\[R_{b}=R_{b1}=R_{b2}=\frac{ln\left(D_{x}/D_{eq}\right)}{k_{b}\pi dz} \tag{19}\]  
\[R_{x}=\frac{ln\left(D_{b}/D_{x}\right)}{k_{b}\pi dz} \tag{20}\]  
where \(k_{b}\) is the borehole grout thermal conductivity.  
Usually, when the pipes are near the borehole wall, \(D_{x}\) can be located at the same distance from the borehole wall; thus \(D_{x}=D_{b}\), and, as a consequence, the \(R_{x}\) part of the conductive resistance can be neglected.  
Finally, the correlation as reported in Equation (6) is used for the estimation of ground thermal resistances. The total resistance value related to the analysed single tube includes convective (\(R_{a}\)), grout conduction (\(R_{b}+R_{x}\)) and ground resistance (\(R_{s}\)) terms.