## 2 System Description
### Buried Pipe Model  
As shown in Figure 3, the buried pipe is a coaxial casing structure in which water, as the working medium, flows into the annulus, absorbs heat from the surrounding rocks and soil, and then flows out of the inner pipe. Besides the constant temperature difference between the water and formation, some basic parameters applied to the operation mode are listed in Table 1. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Drilling depth & 2000 m \\ Drilling diameter & 0.28 m \\ Outer diameter of outer tube & 177.8 mm \\ Inner diameter of outer tube & 159.42 mm \\ Outer diameter of inner tube & 110 mm \\ Inner diameter of inner tube & 90 mm \\ Thermal conductivity of outer tube & 41 W/(m\(\cdot\)K) \\ Thermal conductivity of inner tube & 0.4 W/(m\(\cdot\)K) \\ Thermal conductivity of backfill material & 1.8 W/(m\(\cdot\)K) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic parameters of buried pipe. Figure 2: The building model in TRNSYS. The governing equations applied in the buried pipes are expressed as follows: Energy equation of the ring cavity:  
\[\rho q_{\mathrm{\gamma}}c\frac{\partial T_{\mathrm{f1}}}{\partial z}=\frac{T_{ \mathrm{s}}-T_{\mathrm{f1}}}{R_{1}}+\frac{T_{\mathrm{f2}}-T_{\mathrm{f1}}}{R_{2}} \tag{1}\]  
Energy equation of the internal cavity:  
\[\rho q_{\mathrm{\gamma}}c\frac{\partial T_{\mathrm{f2}}}{\partial z}=\frac{T_{ \mathrm{f1}}-T_{\mathrm{f2}}}{R_{2}} \tag{2}\]  
where \(\rho\) is the density of the circulating mass, kg\(\cdot\)m\({}^{-3}\). \(q_{\mathrm{\gamma}}\) is the flow rate of the circulating mass, m\({}^{3}\cdot\)s\({}^{-1}\). \(c\) is the specific heat capacity of the circulating mass, J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\). \(R_{1}\) is the thermal resistance of the underground rock to the ring cavity fluid, m\(\cdot\)K\(\cdot\)W\({}^{-1}\). \(R_{2}\) is the thermal resistance of the ring cavity fluid to the inner cavity fluid, m\(\cdot\)K\(\cdot\)W\({}^{-1}\). The thermal resistances \(R_{1}\), \(R_{2}\) are calculated by Equations (3) and (4):  
\[R_{1} =\frac{1}{h_{1}\pi d_{\mathrm{i1}}}+\frac{1}{2\pi\lambda_{\mathrm{ p1}}}\ln\left(\frac{d_{\mathrm{1}o}}{d_{\mathrm{i1}}}\right)+\frac{1}{2\pi \lambda_{\mathrm{b}}}\ln\left(\frac{d_{\mathrm{b}}}{d_{\mathrm{1o}}}\right) \tag{3}\] \[R_{2} =\frac{1}{h_{2}\pi d_{\mathrm{2i}}}+\frac{1}{2\pi\lambda_{ \mathrm{p2}}}\ln\left(\frac{d_{\mathrm{2o}}}{d_{\mathrm{2i}}}\right)+\frac{1}{ h_{1}\pi d_{\mathrm{2o}}} \tag{4}\]  
where \(d_{\mathrm{b}}\) is the drilling diameter, m. \(d_{\mathrm{i1}}\) and \(d_{\mathrm{1o}}\) are the inner diameter of the outer tube and the outer diameter of the outer tube, respectively, m. \(d_{\mathrm{2i}}\) and \(d_{\mathrm{2o}}\) are the inner diameter of the inner tube and the outer diameter of the inner tube, respectively, m. \(\lambda_{\mathrm{p1}}\), \(\lambda_{\mathrm{p2}}\), and \(\lambda_{\mathrm{b}}\) are the thermal conductivity of the outer tube, the inner tube, and the backfill material, respectively, W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\). \(h_{1}\) and \(h_{2}\) are the convective heat transfer coefficient of the ring cavity fluid and inner cavity fluid, respectively, W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\). The convection heat transfer coefficient is calculated from the Nussle number:  
\[h=Nu_{\mathrm{f}}\frac{\lambda_{\mathrm{f}}}{d} \tag{5}\]  
Figure 3: Schematic diagram of the casing buried pipe heat exchanger (red arrows—fluid with high temperature; blue arrows—fluid with low temperature). where \(h\) is the convective heat transfer coefficient of the fluid, W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\). \(\lambda_{\ell}\) is the thermal conductivity of the fluid, W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\). \(d\) is the characteristic length, m. \(Nu_{\ell}\) is calculated by the Sider-Tate equation [33]:  
\[Nu_{\ell}=\left\{