## 3 Numerical Simulation  
### Model Description  
[PERSON] et al. [3, 37] proposed that compared with the axial heat flux between ground strata, the horizontal heat flux between freeze pipes and ground structure is more dominant. Therefore, the AGF model in this study simplified the shield tunnel of the Nanjing Metro Line 10 from a general three-dimensional model to a two-dimensional geometry [38]. As shown in Figure 3a, this tunnel, with a radius of 5.82 m, rests in a soil layer with an initial temperature of 18 \({}^{\circ}\)C. A stable seepage flows from the left to the right inside saturated soil. Fifty-two freeze pipes with a diameter of 0.108 m and a final cooling temperature of \(-30\)\({}^{\circ}\)C are distributed on the circumference of the pipe circle, the radius of which is 6.15 m. In order to monitor the freezing process and the development of the frozen body, 17 monitoring points are set up to analyze the heat transfer process, illustrated in Figure 3b. Their specific coordinates are listed in Table 1. It can be noted that the monitoring points (except origin) form a ring with a thickness of 1.5 m, which is the minimal thickness of frozen wall to meet the demand of frozen soil strength [39]. The material parameters involved in this study are listed in Table 2.  
Figure 2: Relation curves during freezing. (**a**)Unfrozen water content against temperature; (**b**) relative permeability against unfrozen water content.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Position No.**} & \multicolumn{2}{c}{**Coordinate (m)**} \\ \cline{2-3}  & \(\mathbf{x}\) **(Horizontal)** & \(\mathbf{y}\) **(Vertical)** \\ \hline \(A\) & \(-6.9\) & \(0\) \\ \(B\) & \(-5.4\) & \(0\) \\ \(C\) (Origin) & \(0\) & \(0\) \\ \(D\) & \(5.4\) & \(0\) \\ \(E\) & \(6.9\) & \(0\) \\ \(O_{1}\) & \(-4.88\) & \(4.88\) \\ \(I_{1}\) & \(-3.82\) & \(3.82\) \\ \(O_{2}\) & \(0\) & \(6.9\) \\ \(I_{2}\) & \(0\) & \(5.4\) \\ \(O_{3}\) & \(4.88\) & \(4.88\) \\ \(I_{3}\) & \(3.82\) & \(3.82\) \\ \(O_{4}\) & \(4.88\) & \(-4.88\) \\ \(I_{4}\) & \(3.82\) & \(-3.82\) \\ \(O_{5}\) & \(0\) & \(-6.9\) \\ \(I_{5}\) & \(0\) & \(-5.4\) \\ \(O_{6}\) & \(-4.88\) & \(-4.88\) \\ \(I_{6}\) & \(-3.82\) & \(-3.82\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Coordinates of temperature monitoring points.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline \multicolumn{1}{c}{**Description**} & **Symbol** & **Value** & **Unit** \\ \hline Density of soil & \(\rho_{\text{s}}\) & \(1900\) & \(\text{kg}\cdot\text{m}^{-3}\) \\ Density of water & \(\rho_{\text{w}}\) & \(1000\) & \(\text{kg}\cdot\text{m}^{-3}\) \\ Density of ice & \(\rho_{\text{i}}\) & \(910\) & \(\text{kg}\cdot\text{m}^{-3}\) \\ Thermal conductivity of soil solid & \(\lambda_{\text{s}}\) & \(2.9\) & \(\text{W}\cdot\text{m}^{-1}\,\text{K}^{-1}\) \\ Thermal conductivity of water & \(\lambda_{\text{w}}\) & \(0.6\) & \(\text{W}\cdot\text{m}^{-1}\,\text{K}^{-1}\) \\ Thermal conductivity of ice & \(\lambda_{\text{i}}\) & \(2.4\) & \(\text{W}\cdot\text{m}^{-1}\,\text{K}^{-1}\) \\ Heat capacity of soil solid & \(\text{C}_{\text{s}}\) & \(1300\) & \(\text{J}\cdot\text{kg}^{-1}\,\text{K}^{-1}\) \\ Heat capacity of water & \(\text{C}_{\text{w}}\) & \(4200\) & \(\text{J}\cdot\text{kg}^{-1}\,\text{K}^{-1}\) \\ Heat capacity of ice & \(C_{\text{i}}\) & \(2100\) & \(\text{J}\cdot\text{kg}^{-1}\,\text{K}^{-1}\) \\ Material constant & \(m\) & \(0.7\) & / \\ Porosity & \(n\) & \(0.4\) & / \\ Material constant & \(w\) & \(0.5\) & \({}^{\circ}\text{C}\) \\ Material constant & \(z\) & \(0.7\) & / \\ Intrinsic permeability & \(\kappa\) & \(1\times 10^{-11}\) & \(\text{m}^{2}\) \\ Dynamic viscosity (at \(T_{\text{ini}}\)) & \(\mu_{\text{w}}\) & \(0.001\) & Pa\(\cdot\)s \\ Freezing point & \(T_{0}\) & \(0\) & \({}^{\circ}\text{C}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Material parameters involved in the hydrothermal coupling model.  
Figure 3: Schematic illustrations of (**a**) simulated model of AGF; (**b**) monitoring points.