## 3 Finite Element Simulations
### Electrical Resistivity Simulation Results  
The concrete slab model used for the ER simulation is the same one utilized for the HCP simulation and is represented as a block measuring 1 m \(\times\) 1 m \(\times\) 0.2 m. The center of the ER model contains the probe contact area, represented by four electrodes spaced 5 cm apart. The ER technology simulation was carried out by mimicking the operation of the Resipod Proceq probe (according to the Resipod Proceq manual, 2017). This probe injects a current of 200 \(\upmu\)A from the two outer electrodes, while the two inner electrodes measure the potential difference in the resulting electrical field, as illustrated in Figure 5. Specifically, probe No. 1 has a current intensity of +200 \(\upmu\)A, and probe No. 4 has a current intensity of \(-\)200 \(\upmu\)A. Table 1 displays the electrical conductivity values for concrete, water, and air used in the model. Concrete's conductivity of 0.002 \(S\)/m corresponds to a resistivity of 500 \(\Omega\).m.  
The electrical potential distribution measured in (mV) at the C-C cross-section (Figure 1) for the sound concrete is shown in Figure 6a. The models containing air-filled delamination (AFD) and water-filled delamination (WFD) are shown in Figure 6b,c, respectively. The simulation results indicate that the measurement of ER technology is highly affected by the level of saturation, the presence and depth of delamination, and the moisture condition of the delamination. As a result, several models were developed to assess the effects of these factors. The ER modeling results highlight a significant impact of these parameters. Generally, the ER values are inversely proportional to the degree of saturation. However, the ER measurements are directly proportional to the depth of delamination for the WFD case and inversely proportional to the AFD case.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Materials**} & \multicolumn{2}{c}{**Properties**} \\ \cline{2-3}  & **Electrical Conductivity** & **Relative Permittivity** \\  & **(S/m)** & **Relative Permittivity** \\ \hline Concrete & 0.002 & 4.5 \\ \hline Water * & 0.5 & \(88.1\) \\ \hline Air * & \(3\times 10^{-15}\) & \(1\) \\ \hline \hline \end{tabular}  
* used to fill the delamination.  
\end{table}
Table 1: Material properties.  
Figure 5: C-C cross-section.  
The change in electrical resistivity with the change in the degree of saturation is illustrated in Figure 7a for the sound concrete model, where the electrical resistivity decreases as the degree of saturation increases. The impact of air-filled delamination (AFD) on electrical resistivity (ER) measurements is demonstrated in Figure 7b by presenting the ER values for different delamination depths and degrees of saturation. The trend in electrical resistivity lines is similar to those in Figure 7a, with a drastic decrease in resistivity in the low degree of saturation range from 0.2 to 0.3. Furthermore, the resistivity decreases as the depth of the delamination increases. This can be explained by the fact that AFD serves as an insulator as the electrical conductivity of air is close to zero. In contrast, the electrical resistivity (ER) measurements for water-filled delamination (WFD) are directly proportional to the delamination depth, as demonstrated in Figure 7c. This can be attributed to the higher electrical conductivity of water compared to concrete. Again, Figure 7c shows a sharp decrease in ER values for all delamination depths in the low degree of saturation range from 0.2 to 0.3.  
Figure 6: Concrete potential distributions from ER simulations: (**a**) sound concrete; (**b**) AFD at 70 mm; (**c**) WFD at 70 mm.