## 3 Numerical Simulation
### Random Field of Soil Thermal Conductivity  
[PERSON] et al. [29] analyzed the stochastic thermal regime of frozen soil by taking the uncertain thermal parameters of frozen soil as random variables, stochastic processes and random fields, respectively. The results suggested that the situation of random fields was more rational. Therefore, the spatial variation of soil thermal conductivity is taken into consideration by the concept of random field in this work.  
In various methods of simulating random fields, the spectral representation method (SRM) is one of the most often used methods to generate second-order stationary random fields of soil property discretization [40,41]. This research generates lognormal random fields reflecting the spatial variability of soil thermal conductivity by SRM, and groups of Monte-Carlo simulations are carried out to analyze the AGF process in such spatially variable media. The SRM usually discretizes the random field into a series of trigonometric functions with random phase angles; for this, two independent \(N\)-order random number matrices \(\mathbf{R_{1}}\) and \(\mathbf{R_{2}}\) satisfying uniform distribution are first generated. The frequencies are set to:  
\[\omega_{i}\ =\ i\times\Delta\omega-\Delta\omega/2;\ \omega_{j}\ =\ j\times\Delta\omega-\Delta\omega/2;\ \text{for}\ i,j\ =\ 1,2,\ldots,N \tag{11}\]  
where \(\omega_{i}\) and \(\omega_{j}\) are frequency coordinate values; \(\Delta\omega\) is the discrete interval on the frequency coordinate axis. The power spectral density functions, \(G_{i}\) and \(G_{j}\), are real non-negative functions of \(\omega_{i}\) and \(\omega_{j}\), defined as below:  
\[G_{i}\ =\ G_{j}\ =\ (l_{x}\times l_{y})\cdot\exp\left\{-\frac{l_{x}^{2}\omega_{i }^{2}}{4\pi}-\frac{l_{y}^{2}\omega_{j}^{2}}{4\pi}\right\}/\pi^{2},\ \text{for}\ i,j\ =\ 1,2,\ldots,N \tag{12}\]  
where \(l_{x}\) and \(l_{y}\) are correlation distances in the \(x\)- and \(y\)-directions, respectively. The standard deviations, \(\sigma_{i}\) and \(\sigma_{j}\), are expressed by:  
\[\sigma_{i}\ =\ \sqrt{G_{i}\times\Delta\omega^{2}};\ \sigma_{j}\ =\ \sqrt{G_{j}\times\Delta\omega^{2}}\text{for}\ i,j\ =\ 1,2,\ldots,N \tag{13}\]  
Therefore, the random field \(f\) (\(x,y\)) is obtained as:  
\[f(x,y)\ =\ \sum\limits_{i\ =\ 1}^{N}\sum\limits_{j\ =\ 1}^{N}[\sigma_{i}\cos(\omega_{i}\times x+ \omega_{j}\times y+2\pi\mathbf{R}_{1})+\sigma_{j}\cos(\omega_{i}\times x- \omega_{j}\times y+2\pi\mathbf{R}_{2})] \tag{14}\]  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Description** & **Symbol** & **Value** & **Unit** \\ \hline Initial temperature & \(T_{\text{ini}}\) & 18 & \({}^{\circ}\)C \\ Latent heat & \(L_{\text{f}}\) & 334 & kJ-kg\({}^{-1}\) \\ Gravity acceleration & g & 9.8 & m-s\({}^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: \(Cont.\)  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Time (Day)** & **0** & **1** & **5** & **10** & **15** & **20** & **40** & **80** \\ \hline Temperature (\({}^{\circ}\)C) & 18 & 0 & \(-10\) & \(-20\) & \(-25\) & \(-30\) & \(-30\) & \(-30\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Cooling plan of the AGF program.  
Referring to [PERSON] et al. [38]'s work, the soil thermal conductivity ranges from 1.87 to 3.40 W/(m\(\cdot\)K). For this reason, the mean value of \(\lambda_{\mathrm{s}}\) is selected as 2.9 W/(m\(\cdot\)K) and the coefficient of variation (COV) as 0.25 in this study; the correlation distances along the horizontal and vertical directions are set to 20 m and 5 m, respectively. Figure 4a illustrates a typical realization of soil thermal conductivity random field. Figure 4b shows the mesh size of established finite element model and temperature field at 10 days when the seepage velocity is set as 0.5 m/d.