## 3 Methodology
### Variation in Ground Temperature and Heat Transfer Process  
The variation in ground temperature is caused by seasonal fluctuations of air temperature on the surface that can reach depths of around 20 m [20; 21]. Equation (2) is used to describe the transient heat transfer in a semi-infinite solid, where \(T\) is temperature (\({}^{\circ}\)C), \(t\) is time (s), \(\alpha\) is the thermal diffusivity (m\({}^{2}\)/s), and \(Z\) is depth (m):  
\[\frac{\partial T}{\partial t}=\alpha\frac{\partial^{2}T}{\partial Z^{2}}, \tag{2}\]  
The shallow thermal perturbation behaves like a sinusoidal wave with time, and, for solving Equation (2), a sinusoidal temperature mathematical model is used [21; 22]:  
\[T(Z,t)=T_{m}+A_{z}\sin\biggl{[}\frac{2\pi}{P}(t-t_{0})-\gamma Z\biggr{]}+ \frac{\partial T}{\partial Z}, \tag{3}\]  
where \(T\)(Z,\(t\)) is the ground temperature at time \(t\) (h) and depth \(Z\) (m); \(T_{m}\) is average surface temperature (\({}^{\circ}\)C); \(A_{z}\) is the amplitude of the ground annual surface temperature wave (obtained from Equation (4)); \(t_{0}\) is the time lag needed (day of the year) for the ground surface temperature to reach the \(T_{m}\) value; and \(\partial T/\partial Z\) is the geothermal gradient of 0.43 \({}^{\circ}\)C/m, which was calculated between the average of \(T_{m}\) = 25 \({}^{\circ}\)C (\(Z\) = 0) and \(SFT\) = 46.6 \({}^{\circ}\)C to \(Z\) = 50 m depth. Equations (4) and (5) was used to estimate the \(A_{z}\) value:  
\[A_{z}=\ A_{0}exp^{-\gamma z}, \tag{4}\]  
\[\gamma=\sqrt{\frac{\pi}{\alpha P}}, \tag{5}\]  
where \(\gamma\) is the inverse of the damping depth, and \(P\) is the period of the oscillation (365 days) in hours.  
For the numerical heat transfer analysis between well W1 and the U-tube heat exchanger, the convective heat transfer process was evaluated, so for an estimation of the maximum water temperature to the heat exchanger outlet, the following empirical relationship for pipe and tube flow was used [21; 23]:  
\[\dot{Q}=h\pi dL\biggl{(}T_{W}-\frac{T_{1}+T_{2}}{2}\biggr{)}=\dot{m}C_{P}(T_{ 2}-T_{1}), \tag{6}\]  
where \(h\) is the convective heat transfer coefficient (W/m\({}^{2}\)\({}^{\circ}\)C), \(d\) is the tube diameter (m), \(L\) is the total length of the well (m), \(T_{W}\) is the wall temperature along the depth of the tube, \(\dot{m}\) is the mass flow (kg/s), \(Cp\) is the fluid heat capacity (J/kg \({}^{\circ}\)C), and \(T_{1}\) and \(T_{2}\) are the inlet and outlet system temperatures. In this study, \(T_{2}\) corresponds to \(T_{max}\) or the heating temperature of water at depth. The turbulent flow in the inner part of U-tube was calculated using the empirical relation by [PERSON] and [PERSON] [23; 24]:  
\[Nu_{d}=0.023\ R_{d}^{0.8}\ Pr^{n}, \tag{7}\]  
where \(Re_{d}\) and \(Pr\) correspond to Reynolds and Prandtl numbers, respectively, and the exponent \(n\) is equal to 0.4 for heating and 0.3 for cooling [24]. The thermal properties of water for temperatures between 20 and 100 \({}^{\circ}\)C were calculated from the following polynomial regression equations [21]:\[k=0.5576\ +\ 0.0021T-0.0296\times 10^{-4}T^{2}\ +\ 6.1869\times 10^{-8}T^{3}, \tag{8}\]  
\[\mu=0.0018-5.805210\times 10^{-5}T+1.145710\times 10^{-6}T^{2}-1.220410\times 10^{- 8}T^{3}+5.183910\times 10^{-11}T^{4}, \tag{9}\]  
\[\rho=999.8556+0.0366T-0.0064T^{2}+0.00002T^{3}, \tag{10}\]  
\[C_{P}\ =\ 4225.9949-5.2960T\ +\ 0.2410T^{2}-0.0055T^{3}\ +\ 6.2127\times 10^{-5}T^{4}- 2.6538\times 10^{-7}T^{5}. \tag{11}\]