# 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 problem of artificial freezing of the soil volume in the form of a circular cylinder was considered. Freezing was technically accomplished using 39 freeze pipes (see Figure 8a). Geometrically, the pipes were considered as circular cylinders with equal radii, at an equal distance relative to each other, and with zero deviation from the vertical direction. The temperature distribution of the brine moving in the freeze pipes was assumed to be homogeneous. In view of these assumptions, a transition was made from a discrete set of heat sources in the form of freeze pipes to a continuous heat source distributed along the freezing contour (see Figure 8b). This transition was made by specifying a distributed source of heat flux density in the form of the ratio of the total specific heat flux from all pipes to the radius of the freezing contour:  
\[q=\alpha\frac{N_{f}r_{f}}{R_{f}}\Big{(}T_{f}-T\Big{)}, \tag{19}\]  
where \(q\) is the heat flux density (W/m\({}^{2}\)), \(\alpha\) is the heat transfer coefficient between the freeze pipes and the soil (W/(m\({}^{2.\circ}\)C)), \(N_{f}\) is the number of freeze pipes, \(r_{f}\) is the radius of the freeze pipes (m), \(R_{f}\) is the radius of the freezing contour (m), \(T_{f}\) is the temperature on the wall of the freeze pipes (\({}^{\circ}\)C), and \(T\) is the soil temperature (\({}^{\circ}\)C). It was assumed that the considered soil layers were sufficiently extended in the vertical direction, and therefore, the influence of the vertical heat flows on the temperature field of the median horizontal section of the layers would be negligible over a long period of time. The considered soils were assumed to be homogeneous and isotropic, and heat transfer in them occurs due to thermal diffusion and the convective transport of the porewater. The heat transfer processes between the soil and the freeze pipes were also considered. Figure 8: Part of the computational domain with a discrete set of heat sources: with freeze pipes (**a**) and with a distributed heat source (**b**). In addition, the release of latent heat during the phase transition of the porewater was considered. This assumption was made about the local thermodynamic equilibrium of phases at each point of the given cylindrical domain. The effect of supercooling the porewater was not considered. In view of all the accepted model assumptions, we can say that the problem has angular symmetry, and thermodynamic processes proceed mainly in a radial direction. For these reasons, the problem can be considered using cylindrical coordinates, with its reduction to the one-dimensional case along the radial coordinate. In this case, the transient temperature field in the soil layers is described by the law of energy balance in the following form [32]:  
\[\frac{\partial H}{\partial T}\frac{\partial T}{\partial t}+\frac{1}{r}\frac{ \partial}{\partial r}(\mathbf{v}rH)=\frac{1}{r}\frac{\partial}{\partial r} \left(r\lambda\frac{\partial T}{\partial r}\right), \tag{20}\]  
where \(H\) is the soil specific enthalpy (J/m\({}^{3}\)), \(t\) is the time (s), \(r\) is the radial coordinate (m), and \(\mathbf{v}\) is the vector of groundwater velocity (m/s). It was assumed that the flow of porewater is plane-radial. This type of flow is typical, for example, in situations when water bulges from the front of the phase transition due to the expansion of freezing water [33]. The Darcy equation was used to simulate porewater flow [34; 35]:  
\[\mathbf{v}=-\frac{K_{r}}{\mu_{w}}\
abla p=-\frac{K_{r}}{\mu_{w}}\frac{ \partial p}{\partial r}\hat{\mathbf{r}}, \tag{21}\]  
where \(K\) is the soil absolute permeability (m\({}^{2}\)), \(k_{r}\) is the soil relative permeability (m\({}^{2}\)/m\({}^{2}\)), \(\mu_{w}=0.001236\) Pa-s is the porewater dynamic viscosity, \(p\) is the soil density (Pa), and \(\hat{\mathbf{r}}\) is the basis vector of the radial coordinate.