## 3 Numerical Simulation
### Model Validation  
Prior to numerical simulations, the developed coupled hydro-thermal model is validated by a comparison with the experimental test of [PERSON] et al. [42] A schematic illustration of [PERSON] et al. [42]'s laboratory model is shown in Figure 5a. Three vertical freeze pipes with an inner diameter of 0.21 m were installed in a watertight and insulated box (L \(\times\) W \(\times\) H = 1.3 \(\times\) 1.0 \(\times\) 1.2 m\({}^{3}\)) filled with sand soil. Two opposite faces in the \(x\)-direction were set as inlet and outlet surfaces to simulate the seepage flow. The initial temperature of sand soil and seepage is 18.4 \({}^{\circ}\)C. The temporal temperature applied in the outer boundaries of freeze pipes is approximated as \(T_{\mathrm{pipe}}(t)\ =\ T_{\infty}+15\Big{(}t-\frac{15}{T_{\infty}-T_{ \mathrm{ini}}}\Big{)}^{-1}\), depicted in Figure 6a. Figure 4: (**a**) A typical realization of soil thermal conductivity random field (\(\mu_{\mathrm{\lambda s}}\) = 2.9 W/(m\(\cdot\)K) and COV = 0.25); (**b**) mesh size and temperature field at 10 days when the seepage velocity is 0.5 m/d, and the model’s soil thermal conductivity is shown in (**a**). Figure 5: (**a**) Schematic illustration of [PERSON] et al. [42]’s laboratory model; (**b**) mesh size of validation model and temperature field at 40 h under 1.5 m/d seepage flow. In order to monitor the temperature development, 15 thermistors are distributed along the measuring lines (ML1 and ML2) at the middle plane (\(z\) = 0.6 m), which is the selected plane for validation study in this work. The specific locations of thermistors and material parameters involved can refer to [42]. The seepage velocity of 1.5 m/d is selected, and the numerical analysis is conducted using the developed coupled hydro-thermal model. Figure 5b shows the mesh size of the validation model and temperature field at 40 h. The red dots denote where the thermistors are placed. In Figure 6, the simulation results of the validation model are compared with the experimental data (sourced from [42]) at five typical freezing durations (i.e., 0 h, 1 h, 5 h, 20 h, 40 h). It can be observed that the simulation results are basically in good agreement with the experimental data, except at the boundary of the laboratory model (see Figure 6c, when \(y\) = 1.0 m). This is because it is difficult to achieve completely heat insulated boundary conditions in the lab, leading to overestimated results of the freezing effect in numerical calculations. Table 4 summarizes the computational errors of numerical analysis to experimental tests at different freezing durations. The total error measure \(E_{\text{tot}}\) considers the errors of all temperature monitoring points along the ML1 and ML2 at each moment. It is defined as:  
\[E_{\text{tot}}\ =\ \sqrt{\sum{(T_{\text{s}}-T_{\text{e}})^{2}}/\sum{(T_{\text{e }})^{2}}} \tag{15}\]  
where \(T_{\text{s}}\) and \(T_{\text{e}}\) are the temperature values of simulated and experimental results, respectively. The comparison results show that the calculation error along the ML1 is within 19.85%, which serves to validate the developed numerical model. The validation study is mainly used to verify the effectiveness of developed coupled hydro-thermal model in simulating the AGF process. To investigate the effects of seepage flow and spatial variability of soil thermal conductivity, a ser \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Typical Duration** & **0 h** & **1 h** & **5 h** & **20 h** & **40 h** \\ \hline \(E_{\text{tot}}\) along ML1 (\%) & 0.87 & 6.59 & 14.9 & 15.58 & 19.85 \\ \(E_{\text{tot}}\) along ML2 (\%) & 0.59 & 2.75 & 21.21 & 29.18 & 37.32 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Errors of numerical analysis to experimental test under 1.5 m/d seepage flow. Figure 6: (a) Thermal boundary conditions of freeze pipes, and comparison between simulation and experimental results along (b) ML1, and (c) ML2 under 1.5 m/d seepage flow. are established.