## 3 Results and Discussion  
### Influence of Degree of Saturation and Porosity on Electrical Resistivity--Laboratory Investigations  
Research in the column apparatus determined the impact of the degree of saturation and porosity on the electrical resistance of sandy soils. When assessing the impact of the degree of saturation vs. electrical resistivity, the soil samples had the relative density, \(\mathrm{D_{r}}\), in the range of 0.48 to 0.52, to ensure that the porosity of the material is relatively constant. The estimated relationship between the soil electrical resistivity and the saturation degree is:  
\[\mathrm{S_{r}}=\left(\frac{\rho_{\mathrm{bSAT}}}{\rho_{\mathrm{b}}}\right)^{ \frac{1}{\beta}}, \tag{4}\]  
here \(\mathrm{S_{r}}\) is the saturation degree [-], \(\rho_{\mathrm{bSAT}}\) is the electrical resistivity of soil in fully saturated conditions [\(\mathrm{\Omega\cdot m}\)], \(\rho_{\mathrm{b}}\) is the electrical resistivity of material/soil [\(\mathrm{\Omega\cdot m}\)], and \(\beta\) is the empirical constant. The empirical constants used in this equation and depending on the grain size of sand are shown in Table 1. It should be noted that the saturation exponent \(\beta\) is often adopted. The saturation exponent \(\beta\) most often occurs in the literature in the value range of 1.00 \(\div\) 2.50 [42; 43]. At the same time, it should be mentioned that hysteresis occurs during sample saturation and desaturation processes and is influenced by the sample preparation method [44]. The obtained values of parameter \(\beta\) in the tested sand samples are within the value range found in the literature. Figure 6: A modified oedometer for measuring electrical resistivity of clayey soils; scheme of apparatus and view of oedometer ring with electrodes and apparatus during the test. Figure 7a shows the relation between electrical resistivity and saturation degree of fine sand moistened with different KCl solutions. This graph shows how important the electrical conductivity of a fluid filling the pore space is, and what error can be made in in situ tests when trying to estimate the soil saturation degree when the electrical resistivity of this soil in fully saturated conditions is not known. In addition, this error can be greater at higher soil moisture contents [2]. To assess the effect of porosity on soil electrical resistivity, a number of tests were carried out on fully saturated sand samples compacted with a relative density (D\({}_{\text{r}}\)) between 0.14 and 0.87. The empirical constants for use in Equation (5) are presented in Table 2, while Figure 7b shows an example of the relationship between electrical resistivity and porosity for fine sand. \[\text{n}=\left(\frac{1}{\text{F}}\right)^{\frac{1}{m}}=\left(\frac{\rho_{ \text{f}}}{\rho_{\text{bSAT}}}\right)^{\frac{1}{m}}, \tag{5}\]  
where n is the porosity [-], F is the formation factor [-], \(\rho_{\text{f}}\) is the electrical resistivity of fluid in pore spaces [\(\Omega\cdot\text{m}\)], \(\rho_{\text{bSAT}}\) is the electrical resistivity of soil in fully saturated conditions [\(\Omega\cdot\text{m}\)], and m is the empirical constant. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Soil** & **Effective Size d\({}_{50}\) [mm]** & **[-]** & **R\({}^{2}\) [-]** \\ \hline Fine sand (FSa) & 0.22 & 1.63 & 0.93 \\ Medium sand (MSa) & 0.40 & 1.61 & 0.93 \\ Coarse sand (CSa) & 0.80 & 1.53 & 0.95 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Empirical constants in the relationship between the degree of saturation and electrical resistivity according to Equation (4) [45]. Figure 7: Analytical relation between electrical resistivity of fine sand (FSa) moistened with different KCl solution vs. saturation degree (**a**) and electrical resistivity vs. porosity (**b**). For practical purposes, direct correlation between porosity and electrical resistivity is in common usage. It is assumed that the tortuosity value is equal to 1.