## 2 Model Test Design
### Derivation of Similarity Criteria  
The heat conduction differential equation of soil freezing is as follows:  
\[\frac{\partial\theta_{n}}{\partial\tau}=a_{n}\bigg{(}\frac{\partial^{2}\theta _{n}}{\partial\tau^{2}}+\frac{1}{r}\frac{\partial\theta_{n}}{\partial r}\bigg{)} \ (0<r_{0}<r<\infty,\ \tau>0,\ n=1\,\ 2) \tag{1}\]  
where \(r\) is radial coordinate, \(\mathbf{e}\) is time; \(\theta\) is soil temperature, which is the temperature of unfrozen area when \(n=1\) and that of frozen area when \(n=2\), \(r_{0}\) is the outer radius of freezing pipes, and \(a\) is the thermal diffusivity of soil. The initial and boundary conditions of soil freezing temperature field [30] are  
\[\left\{ \begin{array}{l}\theta(r,0)=\theta_{0}\\ \theta(\infty,\tau)=\theta_{0}\\ \theta(\rho,\tau)=\theta_{f}\\ \theta(r_{0},\tau)=\theta_{y}\end{array} \right. \tag{2}\]  
where \(\theta_{0}\) is the initial temperature of soil, \(\rho\) is the external surface coordinates of frozen wall, \(\theta_{f}\) is the freezing temperature of soil, \(\theta_{y}\) is the wall temperature of freezing pipes. The heat balance equation at the freezing front (\(r=\rho\)) is  
\[\lambda_{2}\frac{\partial\theta_{2}}{\partial r}\big{|}_{r=\rho}-\lambda_{1} \frac{\partial\theta_{1}}{\partial r}\big{|}_{r=\rho}=L\frac{d\rho}{d\tau} \tag{3}\]  
where \(\lambda\) is the thermal conductivity of soil, \(L\) is the latent heat of soil. According to Equations (1)-(3), the similarity criterion equation of soil freezing temperature field can be obtained by equation analysis method. \[F(F_{0},K_{0},R,\theta)=0 \tag{4}\]  
where \(F_{0}=\frac{\partial\tau}{r^{2}}\) is the Fourier criterion of temperature field; \(K_{0}=\frac{L}{\partial\theta}\) is Kosovic criterion and \(c\) is the specific heat of soil; \(R\) is geometric criterion, and \(\theta\) is temperature criterion. In the model test, the soil was taken from the site, \(C_{\rho}=1\), \(C_{a}=1\), \(C_{c}=1\), \(C_{L}=1\). The water content of soil is the same, so the latent heat released during freezing is equal, which is substituted into Equation (4) to obtain:  
\[C_{\tau}={C_{r}}^{2} \tag{5}\]  
\[C_{\theta}=1 \tag{6}\]  
Equation (5) shows that the time similarity ratio in the model test is the square of the geometric similarity ratio; according to Equation (6), the temperature of each point in the model test is the same as that of each corresponding point in the prototype. Water migration occurs in the soil freezing process, and its essence is the humidity field problem in the freezing process. The mathematical model is:  
\[\frac{\partial h}{\partial\tau}=b\left(\frac{\partial^{2}h}{\partial r^{2}}+ \frac{1}{r}\frac{\partial h}{\partial r}\right) \tag{7}\]  
The boundary condition is:  
\[\left\{ \begin{array}{l}h(r,0)=h_{0}\\ h(\infty,\tau)=h_{0}\\ h(\rho,\tau)=0\end{array} \right. \tag{8}\]  
where \(h\) is humidity, and \(b\) is moisture conductance. According to Equations (7) and (8), the similarity criterion equation of soil moisture field can be obtained by the equation analysis method as follows:  
\[F(F_{h},R,H)=0 \tag{9}\]  
where \(F_{h}=\frac{b\tau}{r^{2}}\) is the Fourier criterion of humidity field, \(H\) is humidity criterion. It shows that the water transfer process and soil freezing process are similar in the mathematical model, and they all obey the Fourier criterion. Therefore, under the condition of geometric similarity, as long as the temperature field is similar, the humidity field is also similar to the temperature field. In this paper, the similarity ratio of geometry (\(C_{l}\)) in the model test was determined to be 30; this ratio can be used to calculate the similarity ratio of other parameters (Table 1) according to the aforementioned similarity criteria.