# Numerical Simulation of Water Migration during Soil Freezing and Its Resulting Characterization
## 2 Theory of Water Migration  
As shown in Figure 2, soil freezing is a complex process of heat and fluid transport, including water migration and the ice-water phase transition. Throughout this process, the principle of heat balance and mass conservation are upheld. Solving the controlling partial differential equations is crucial for accurately modeling the coupled thermal-hydrodynamic process in frozen soil. To simulate the moisture movement in soil during freezing, we employed a mathematical model comprising two fundamental control equations. Considering the one-dimensional hydrothermal coupling problem, according to [PERSON]'s law [20], the latent heat of phase change generated by the phase change of ice and water is regarded as the heat source, and according to the principle of thermodynamic equilibrium,  
\[Q_{1}-Q_{2}=Q_{3} \tag{1}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Type of Soil**} & \multicolumn{5}{c}{**Temperature Settings (\({}^{\circ}\)C)**} & \multirow{2}{*}{**Weight Water Content(\%)**} & \multirow{2}{*}{**High (cm)**} \\ \cline{2-2} \cline{5-7}  & **Warm End** & & & **Cold End** & & **Chamber Temperature** \\ \hline Kaolin clay & +3 & \(-5\) & \(-10\) & +3 & 50 & 56 & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Boundary and initial condition for the indoor freezing of artificial kaolin clay. Figure 1: The apparatus of the one-dimensional freezing test. Here, \(Q_{1}\)--the increase in thermal energy at the freezing front (\(J\)); \(Q_{2}\)--the difference between the heat transferred into and out of the freezing front (\(J\)); and \(Q_{3}\)--the amount of heat generated by the internal heat source, i.e., the latent heat of the phase change (\(J\)). Simplifying Equations (1) and (2), the differential equation of heat transfer in frozen soil can be written as  
\[C_{v}\frac{\partial T}{\partial t}-L_{f}\rho_{i}\frac{\partial\theta_{i}}{ \partial t}=\lambda(\theta)\frac{\partial}{\partial x}\bigg{(}\frac{\partial T }{\partial x}\bigg{)} \tag{3}\]  
Here, T is the transient temperature of the soil; t is time (h); \(\theta\) is the volumetric unfrozen water content; \(\theta_{i}\) is the pore ice volume content; \(\rho\) is the density of the soil; \(C_{v}\) is the volumetric heat capacity (\(W/(\text{m}\cdot^{\circ}C)\)); \(\lambda_{s}\) is the thermal conductivity (\(J/(\text{kg}\cdot^{\circ}C)\)); and \(L_{f}\) is latent heat of fusion [21]. \[C_{v}=\rho_{s}c_{s}\theta_{s}+\rho_{w}c_{w}\theta_{w}+\rho_{i}c_{i}\theta_{i} \tag{4}\]  
Here, \(s\), \(w\), \(i\) represent soil particles, water and ice, respectively, and \(c_{(s,w,i)}\) is mass specific heat capacity (MSHC) \(J/(\text{kg}\cdot^{\circ}C)\)[1]. \[\lambda(\theta)=\lambda_{s}^{\theta_{S}}\cdot\lambda_{w}^{\theta_{w}}\cdot \lambda_{i}^{\theta_{i}} \tag{5}\]  
According to the law of conservation of mass, per unit time,  
\[I_{1}-I_{2}=I_{3} \tag{6}\]  
\[\left\{ \begin{array}{l}I_{2}=\rho_{i}\frac{\partial\theta_{i}}{\partial x} dx\\ I_{1}=\rho_{w}\bigg{(}k\frac{\partial\varphi}{\partial x}\bigg{)}dt\\ I_{3}=\rho_{w}\frac{\partial\theta_{w}}{\partial x}dx\end{array} \right. \tag{7}\]  
Here, the meaning of each symbol is as follows: \(I_{1}\)--difference between incoming and outgoing mass at the freezing front; \(I_{2}\)--change in unfrozen water content at the freezing front; \(I_{3}\)--change in ice content at the freezing front; \(\psi\)--soil water potential (m); and \(k\)--hydraulic conductivity (m/s). Figure 2: Diagram of water migration at freezing fringe. In unfrozen unsaturated soils, the pattern of water migration bears similarities to that in frozen soils. However, the obstructive effect of pore ice on the flow of unfrozen water must be considered when applying [PERSON]'s equation during ice-water phase transitions. In this context, soil water potential is utilized instead of pressure head [21; 22]. These considerations are integrated into the differential equations for unsaturated water migration in frozen soils to more accurately replicate the process.