# Experimental and Theoretical Study of the Influence of Saline Soils on Frozen Wall Formation
## 2 Experimental Study on the Effect of Salinity on Soil Thermophysical Properties
### Results and Processing of Experimental Measurements
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Soil**} \\ \cline{2-9}  & \multicolumn{3}{c}{**Clay**} & \multicolumn{3}{c}{**Chalk**} & \multicolumn{3}{c}{**Sand**} \\ \cline{2-9}
**Temperature (\({}^{\circ}\)C)** & \multicolumn{6}{c}{NaCl (g/100 g)} \\ \cline{2-9}  & **0.00** & **3.46** & **6.90** & **0.00** & **3.46** & **6.90** & **0.00** & **5.20** \\ \hline \multicolumn{9}{c}{Thermal Conductivity (W/(m\({}^{\circ}\)C))} \\ \hline
20 & - & - & - & - & - & - & 1.84 & 1.87 \\ \hline
10 & 1.10 & 1.104 & 1.12 & 1.50 & 1.49 & 1.52 & - & - \\ \hline
0 & - & - & - & 1.51 & 1.51 & 1.53 & - & - \\ \hline \(-1\) & 1.44 & 1.104 & 1.12 & 1.86 & 1.51 & 1.53 & 2.18 & 1.87 \\ \hline \(-2\) & 1.54 & 1.10 & 1.12 & - & - & - & - & - \\ \hline \(-3\) & 1.61 & 1.13 & 1.12 & 2.08 & 1.59 & 1.56 & 2.38 & 1.87 \\ \hline \(-4\) & 1.66 & 1.40 & 1.12 & - & - & - & - & - \\ \hline \(-5\) & 1.69 & 1.47 & 1.12 & 2.15 & 1.83 & 1.77 & - & - \\ \hline \(-6\) & 1.72 & 1.53 & 1.13 & - & - & - & 2.51 & 2.17 \\ \hline \(-7\) & 1.73 & 1.57 & 1.38 & - & - & - & - & - \\ \hline \(-8\) & 1.75 & 1.59 & 1.43 & - & - & - & - & - \\ \hline \(-10\) & 1.77 & 1.65 & 1.51 & 2.19 & 2.00 & 1.92 & 2.57 & 2.30 \\ \hline \(-15\) & 1.81 & 1.73 & 1.62 & 2.20 & 2.07 & 1.99 & 2.59 & 2.37 \\ \hline \(-20\) & 1.85 & 1.78 & 1.69 & 2.21 & 2.10 & 2.01 & 2.60 & 2.42 \\ \hline \(-25\) & 1.87 & 1.81 & 1.73 & - & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Experimental measurements of the dependence of the thermal conductivity on temperature and salinity. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Soil**} \\ \cline{2-9}
**Temperature (\({}^{\circ}\)C)** & \multicolumn{6}{c}{**Clay**} & \multicolumn{6}{c}{**Chalk**} & \multicolumn{3}{c}{**Sand**} \\ \cline{2-9}  & \multicolumn{6}{c}{NaCl (g/100 g)} \\ \cline{2-9}  & **0.00** & **3.46** & **6.90** & **0.00** & **3.46** & **6.90** & **0.00** & **5.20** \\ \hline \multicolumn{9}{c}{Unfrozen Water Content (kg/kg)} \\ \hline \(-0.5\) & 0.981 & - & - & 0.488 & - & - & 0.168 & - \\ \hline \(-1.0\) & 0.835 & - & - & 0.216 & - & - & 0.080 & - \\ \hline \(-2.0\) & 0.650 & - & - & 0.132 & - & - & 0.053 & - \\ \hline \(-3.0\) & 0.523 & 0.881 & - & 0.108 & 0.760 & - & - & - \\ \hline \(-4.0\) & 0.446 & 0.758 & - & - & - & - & 0.779 \\ \hline \(-5.0\) & 0.38 where \(\gamma\) is the unfrozen water content (kg/kg), \(T_{liq}\) is the freezing temperature of porewater (\({}^{\circ}\)C), \(T\) is the temperature (\({}^{\circ}\)C), \(\omega\) is the characteristic cooling temperature associated with the most common pore radius (\({}^{\circ}\)C), and \(m\) is the index indicating the distribution of the pore radius relative to the average radius. The unknown parameters of Function (7), \(\omega\) and \(m\), were determined via experimental measurements using the least squares method. The obtained values for various types of soils and salinity are presented in Table 5. Later, during the process of numerical simulation, the dependence of the value \(\omega\) on salinity, according to Table 5, was approximated by a linear function, and the value of \(m\) was given as a constant due to its small change in the considered range of salinities. Its value is equal to the mean of m values for three soil samples with different salinities. The functional dependence of the thermal conductivity was set by a known dependence on the unfrozen water content [30]:  
\[\lambda=\lambda_{s}^{1-n}\lambda_{w}^{m\gamma}\lambda_{i}^{n(1-\gamma)} \tag{8}\]  
where \(\lambda\) is the thermal conductivity of the soil (W/(m\(\cdot\)\({}^{\circ}\)C)), \(n\) is the soil porosity, the \"\(s\)\" index corresponds to the dry skeleton, the \"\(w\)\" index corresponds to the water, and the \"\(i\)\" index corresponds to ice.