# Numerical Simulation of Water Migration during Soil Freezing and Its Resulting Characterization
## 2 Theory of Water Migration
After simplifying Equations (6) and (7), the differential equation governing fluid transfer in frozen soil is expressed as follows:  
\[\frac{\partial\theta_{u}}{\partial t}+\frac{\rho_{l}}{\rho_{w}}\frac{ \partial\theta_{i}}{\partial t}=-\frac{\partial}{\partial x}\left[k(\theta_{ u})-D(\theta_{u})\frac{\partial\theta_{u}}{\partial x}\right] \tag{8}\]  
\(\mathrm{D}(\theta_{u})\) is calculated by the following equation:  
\[D(\theta_{u})=\frac{k(\theta_{u})}{c(\theta_{u})}.I \tag{9}\]  
Here, k(\(\theta_{u}\)) is the soil permeability coefficient of unfrozen water (m/s); \(\theta_{i}\) is the pore ice volume content; \(\theta_{u}\) is the unfrozen water content in the frozen soil (1/m); \(\mathrm{D}(\theta_{u})\) is the specific water capacity, determined by the hysteresis model; and I is the impedance factor [13]. Transient strains, phase changes in moisture, and soil strains due to migration need to be considered in the calculation of frozen soil strains. Mechanical properties of frozen soil such as modulus of elasticity \(E\), Poisson's ratio \(v\), and shear strength \(C\) are temperature dependent. Relevant parameters \(a_{i}\) and \(b_{i}\) of clay can be obtained from the literature, and \(m\) is taken as 0.6 [23; 24]. Soil deformation is initiated when the ice content in the soil reaches the threshold necessary for frost formation. Even in the absence of external loading, frost heaving is analyzed based on the theory of linear elasticity. The coefficient of deformation \(\eta\), which is a function of the volumetric ice content, is determined using the following approach [13]:  
\[\eta(x,y)=\left\{ \begin{aligned} 0.089\omega_{\theta_{i}}(x,y)-0.03,\omega_{\theta_{i}}(x,y)>0.03\\ 0,\omega_{\theta_{i}}(x,y)<0.03\end{aligned} \right. (clay) \tag{10}\]  
where \(\omega_{\theta_{i}}(x,y)\) is the mass fraction of ice content,  
\[\omega_{\theta_{i}}(x,y)=0.9\theta_{i}(x,y)/\rho(clay) \tag{11}\]  
The driving force of water migration is the difference in pore water pressure. In this context, the formula for pore water pressure deviates slightly from the traditional Clapeyron equation. The derivation of the pore water pressure formula is as follows:  
In frozen soil, when there is no external loading, the Gibbs free energy of the ice and water phases becomes equal as the system reaches a new equilibrium due to a temperature change. \[G_{w}(T+\Delta T,p_{w}+\Delta p_{w})=G_{i}(T+\Delta T,p_{i}+\Delta p_{i}) \tag{12}\]  
Simplifying this equation yields the basic form of the Clausius-Clapeyron [25]:  
\[\frac{P_{w}}{\rho_{w}}-\frac{P_{i}}{\rho_{i}}=L_{f}ln\frac{T+273.15}{T_{0}+273.15} \tag{13}\]  
where \(\rho_{w}\) and \(\rho_{i}\) are the densities of water and ice; L\({}_{f}\) is the latent heat of fusion; T\({}_{0}\) is the initial freezing temperature of the soil in degrees Celsius; and T is the soil temperature in degrees Celsius, where P\({}_{w}\) and P\({}_{i}\) are the pressure of pore water and ice. Equation (13) can also be rewritten as  
\[P_{w}=\frac{\rho_{w}}{\rho_{i}}P_{i}+L_{f}ln\frac{T+273.15}{T_{0}+273.15} \tag{14}\]Based on the continuum medium mechanics, the total stress of saturated frozen soil is expressed as [26]  
\[\sigma_{ij}=\sigma_{ij}^{s}+n(1-s_{i})p_{w}+ns_{i}p_{i} \tag{15}\]  
[PERSON] and [PERSON] characterize the difference between the external load and the effective stress as the neutral pressure. According to their definition, when the neutral pressure is less than the external load, an ice lens does not form [27]. Additionally, at the frozen fringe, the pore water pressure is insufficient to produce an ice lens [28]. Pore pressure is the weighted value of water pressure and ice pressure:  
\[P_{por=}(1-s_{i})p_{w}+s_{i}p_{i} \tag{16}\]  
Joining Equations (14) and (16), the ice pressure can be eliminated. Equation (13) will be processed as  
\[P_{w}=