## 5 Discussion
### Laboratory Experiment: Prediction of Contamination Evolution  
Predicted apparent electrical resistivity values were inverted using the open-source software ResIPy for retrieving correspondent predicted resistivity sections [62]. The area under investigation was discretized using a node-centered, 2D, triangular mesh and dividing the area into two regions: the outer one, characterized by fixed very high resistivity values to account for tank finite boundaries; and the inner region for the sand body, characterized by a starting resistivity of 100 \(\Omega\)m.  
Figure 9a shows the starting resistivity model before the salt solution injection. The measured resistivity values range between 1 and 50 \(\Omega\)m. Higher values are probably associated to noisy electrodes. In Figure 9b, CH-ERT imaging following the solution injection are presented as resistivity change (%) relative to the pre-tracer condition (0% for no change between datasets, negative values for electrical resistivity decrease). Moreover, an example of a predicted resistivity model is shown retrieved from the previous \"historical data\" used for training the network. Figure 8: (**a**) Some images of resistivity variation (d, %) after tracer injection. Black dots are electrodes, and P1, P2, and B5 are piezometers; (**b**) electrical resistivity variation (d, %) curves along the piezometers P1, B5, and P2 from the CH-ERT dataset. Values at days 29 and 31 were estimated from CH-ERT predicted datasets. Electrical resistivity method is an effective tool to determine the Total Dissolved Solids (TDS) concentration if electrical resistivity of the formation and porosity are known. The possibility to correlate TDS with electrical resistivity is showed by [PERSON] [70], who proposed an empirical relationship between formation electrical resistivity (\(\rho_{b}\), \(\Omega\)m) and pore water electrical resistivity (\(\rho_{w}\), \(\Omega\)m) according to the following equation:  
\[\rho_{b}=a\varphi^{-m}S^{-n}\rho_{w} \tag{10}\]  
where \(a\) is the pore geometry coefficient (usually \(a=1\)), \(\varphi\) is the porosity, \(m\) is the cementation factor (dimensionless, \(1.14\leq m\leq 2.9\)[71]), \(S\) is the saturation degree, and \(n\) is the saturation exponent [72]. Figure 9: (**a**) True resistivity model obtained using ResIPy software of pre contamination condition; (**b**) images of resistivity variation (d, %) after salt solution injection. Black dots are electrodes, C and D are piezometers. In fully saturated conditions (\(S=1\)), this equation can be rearranging introducing the formation factor \(F\) defined as:  
\[F=\frac{a}{\varphi^{m}}=\frac{\rho_{b}}{\rho_{w}}. \tag{11}\]  
Assuming for the simulated sandy aquifer a constant value for \(F\) (homogenous sand), it is possible to define the saturation degree as:  
\[S^{-n}=\frac{\rho_{b}}{\rho_{w}F}. \tag{12}\]  
The electrical conductivity of groundwater (normalized to 25 \({}^{\circ}\)C) is directly related to the _TDS_ based on the assumption that _TDS_ in the water consist mainly of ionic constituents that conduct electricity [73]. In detail, the correlation of _TDS_ and solution electrical conductivity (\(\sigma\) in S/m = 1/\(\rho\)) at the specified temperature T = 25 \({}^{\circ}\)C can be estimated by the following equation:  
\[TDS=k_{e}\sigma \tag{13}\]  
where \(k_{e}\) will increase along with the increase of ions in water _TDS_ is expressed in g/L and \(\sigma\) in mS/cm. Therefore, considering \(k_{e}\) = 0.7 (saline water with \(\sigma\) from 1 till 45 mS/cm [74]), _TDS_ (g/L) in the saturated zone can be obtained with the relation:  
\[TDS=7\frac{FS^{-n}}{\rho_{b}}. \tag{14}\]  
For the studied case, \(\rho_{b}\) (\(\Omega\)m) was obtained by CH-ERT monitoring, porosity was 0.45, \(m=1.14\), and \(n=\)1.5 [64; 65], while saturation degree in the vadose zone was estimated by numerical models [66]. Figure 10 shows the 3D representation of _TDS_ > 10 g/L allowing us to delineate the salt plume evolution under 30 cm depth at 10, 18, and 24 days after the NaCl solution injection.