# Experimental and Theoretical Study of the Influence of Saline Soils on Frozen Wall Formation
## 3 Numerical Simulation of Heat and Mass Transfer on the FW Formation
### Mathematical Model
The relative permeability depends on the unfrozen water content and is determined through the following ratio [28; 29]:  
\[k_{r}=\sqrt{\gamma}\Big{[}1-\left(1-\gamma^{\frac{1}{m}}\right)^{m}\Big{]}^{2}, \tag{22}\]  
The pressure field was calculated based on the assumption of the incompressibility of porewater:  
\[\
abla\cdot\mathbf{v}=-\frac{K}{\mu_{w}}\frac{1}{r}\frac{\partial}{\partial r }\left(rk_{r}\frac{\partial p}{\partial r}\right)=0, \tag{23}\]  
It is worth noting that the calculation of porewater flow was implemented only for the sand layer. For clay and chalk, the flow of porewater was not considered, because these layers are weakly permeable. Thus, the absolute permeability of sand, \(K\), was assumed to be 0.04\(\cdot\)10\({}^{-12}\) m\({}^{2}\). An important feature of the physical process of freezing saline soil is that only pure water crystallizes, while dissolved salt accumulates in unfrozen water. This process can be mathematically expressed by the following expression:  
\[s=\frac{s_{0}}{\gamma}, \tag{24}\]  
where \(s_{0}\) is the initial salinity of porewater in an untouched soil mass (g/100 g). The dissolved salt accumulates in the unfrozen water until the eutectic point is reached [36; 37]. After that, the excess salt precipitates. The solution itself (water + salt) in this case is saturated. At the same time, it is also possible to release the additional heat of the phase transition, which, according to estimates [37; 38], is 216-235 J per 1 g of the eutectic solution involved in the chemical reaction. In the case of NaCl, the saturated solution has two ways to reach the eutectic point: (1) when salinity reaches 23.2% [37]; or (2) when water with a salt solution freezes to a temperature of \(-\)21.2 \({}^{\circ}\)C [37]. However, this is true only in the case where water with a salt solution is in an unstressed state. According to [39], it is difficult to reach the eutectic point in porewater, which is in a confined state. For this reason, the process of salt precipitating upon reaching the eutectic point was not considered in the mathematical model. To calculate the change in salinity over time, Equation (24) was differentiated by time, and the corresponding term was added to the resulting equation to account for the convective transport of porewater:  
\[\frac{\partial s}{\partial t}+\frac{1}{r}\frac{\partial}{\partial r}(\mathbf{v} rs)=-\frac{s_{0}}{\gamma^{2}}\frac{\partial\gamma}{\partial t}, \tag{25}\]  
The system of Equations (20)-(25) was supplemented by the following initial and boundary conditions:  
\[T|_{t=0}=T_{0},\ s\left.s\right|_{t=0}=s_{0}, \tag{26}\]  
\[-\lambda\left.\frac{\partial T}{\partial\mathbf{n}}\right|_{r=R_{f}}=q, \tag{27}\]  
Figure 8:0  
Figure 8:1  
where \(\mathbf{n}\) is the normal vector to the boundary (m); \(R_{1}\) and \(R_{2}\) are the internal and external radii of the calculated area (see Figure 9), respectively; and \(p_{1}\) and \(p_{2}\) are the pressures at the boundaries of the computational domain (Pa). For a numerical discretization of Equations (20)-(29) in space, the finite volume method was used [40]; for discretization in time, an explicit Euler [41] with a dynamic time step was used. The computational mesh was uneven, with a decrease near the freeze pipe (see Figure 10). Table 7 shows the main numerical parameters that were further used in the simulation. Figure 8:2 Schematic representation of boundary conditions.