## 2 Similarity Model Test
### Test Modeling Based on Similarity Criteria  
#### 2.1.1 Similarity Criteria  
Based on the similarity theory, the local frozen pipe in the typical soil layer is selected for freezing simulation test, and the following similarity criteria are listed by combining the relevant mathematical models of temperature field, moisture field, and stress field during the freezing process. (1) Temperature field criterion  
According to the basic equation of transient heat conduction, the heat conduction equation suitable for artificial freezing heat transfer is listed:  
\[\frac{\partial t_{n}}{\partial\tau}=a_{n}\left(\frac{\partial^{2}t_{n}}{ \partial r^{2}}+\frac{1}{r}\frac{\partial t_{n}}{\partial r}\right) \tag{1}\]  
When \(\tau>0,0<r<\infty\), the heat balance equation at the frozen front is as follows:  
\[\lambda_{2}\frac{\partial t_{2}}{\partial r}\bigg{|}r=\zeta-\lambda_{1}\frac{ \partial t_{1}}{\partial r}\bigg{|}r=\zeta=Q\frac{d\zeta}{d\tau} \tag{2}\]  
The initial conditions and boundary conditions are  
\[t=0,\ t(r)=t_{0} \tag{3}\]  
\[t>0,\ t(r=0)=t_{y},\ t(\infty)=t_{0},\ t(r=\zeta)=t_{d} \tag{4}\]  
where \(r\) is the radial distance from each point in the soil plane to the center of the freezing pipe; \(t_{n}\) is the temperature at the position \(r\), where \(n=1\) represents unfrozen soil and \(n=2\) represents frozen soil; \(\tau\) is the freezing time; \(a_{n}\) is the thermal conductivity coefficient; \(\lambda_{1}\), \(\lambda_{2}\) are the thermal conductivity of unfrozen soil and frozen soil, respectively; \(\zeta\) is the boundary position coordinate of the frozen wall, which is the expansion radius of the frozen wall; \(Q\) is the latent heat release during the freezing process of unit soil; \(t_{0}\), \(t_{d}\), \(t_{y}\) are the initial soil temperature, freezing temperature, and cold source temperature (prime temperature), respectively. The similarity criterion of the temperature field is obtained by analogical derivation of Equations (1) and (2):  
\[\varphi(K_{0},F_{0},\theta,R)=0\]where \(K_{0}\) is Kosovich criterion; \(K_{0}=Q/Ct\), \(Q\) is the latent heat of soil mass; \(C\) is the soil specific heat; \(t\) is the temperature; \(F_{0}\) is Fourier criterion; \(F_{0}=a\tau/r^{2}\); \(a\) is the soil thermal conductivity; \(r\) is the radial distance from the freezing front to the center of the freezing pipe; \(\theta\) is the temperature criterion; \(R\) is the geometric criterion. Since the geotechnical material used in the test is the same as the prototype,\(C_{c}\) and \(C_{a}\) are both 1, and \(C_{\tau}=C_{I}^{2}\) is deduced. As long as the test meets the requirements that the geometry of the frozen wall, the temperature of the cold source, the initial temperature of the soil, and the freezing time of the model are similar to those of the prototype, the similar simulation of the temperature field can be realized. (2) Water field criterion  
Based on the water migration law and seepage equation of soil during freezing process, the suitable water field equation is listed:  
\[\frac{\partial h}{\partial\tau}=b\left(\frac{\partial^{2}h}{\partial r^{2}}+ \frac{1}{r}\frac{\partial h}{\partial r}\right) \tag{6}\]  
The initial conditions and boundary conditions are  
\[t=0,\ h=h_{0} \tag{7}\]  
\[\tau>0,\ H(h=\infty)=h_{0},\ h(h=\zeta)=0 \tag{8}\]  
where \(h\) is the soil moisture content, and \(b\) is the soil moisture conductivity coefficient. After similarity derivation, the similarity criterion of water field is as follows:  
\[\varphi(K_{0},W,R)=0 \tag{9}\]  
where \(W\) is the humidity criterion. Because the moisture migration process and freezing process are similar in the mathematical model, both contain Fourier criterion, so under the condition of geometric similarity, as long as the temperature field is similar, the moisture field can be \"self-simulated\" to achieve similarity. (3) Stress field criterion  
In the freezing process, the stress of unfrozen soil and frozen soil is quite complex, and it is very difficult to realize accurate simulation.