## 3 Evaluation of the Model Under Controlled Synthetic Conditions
### Large Vertical Rod Driven into a Tri-Layered Soil  
The second proposed synthetic example consists of a long bare copper rod, with a length of L = 12 m and a radius of r = 5 mm, vertically driven into a soil modeled as three homogeneous horizontal layers. The layers have resistivities and thicknesses (from the surface) of \(\rho_{1}\) = 50 \(\Omega\cdot\)m with a thickness of \(h_{1}\) = 2.1 m, \(\rho_{2}\) = 20 \(\Omega\cdot\)m with a thickness of \(h_{2}\) = 0.9 m, and \(\rho_{3}\) = 100 \(\Omega\cdot\)m. The fault current injection point, with a magnitude of 10 kA, is located at the upper end of the rod, which corresponds to the point (0, 0, 0). As for the previous application example, each electrode rod is discretized into several segments per meter to enable the calculation of electrical parameters and the temperature distribution. The algorithm for computing the temperature profile is applied to the soil surrounding each segment. The simulation follows the procedure outlined in the flowchart of Figure 4 and is executed over 20 time steps, corresponding to a total fault duration of 0.5 s. The initial soil temperature is set to \(T_{s}(0)\) = 20 \({}^{\circ}\)C. In each case, the three models described in Expression (19) for the temperature dependence of soil resistivity are implemented. Figure 10 shows the time evolution of the electrode potential, which corresponds to the potential at the fault current injection point. The left panel of Figure 10 displays the electrode potential versus fault duration for the three temperature-dependent soil resistivity models, using a discretization of seven segments per meter along the electrode rods. In contrast, the right panel presents the corresponding results obtained with a finer discretization of 11 segments per meter. A saturation of the potential overtime is once again observed, resulting from the segment equivalent radius reaching its predefined limit, set to one-fourth of the segment length. In both panels of Figure 10, it can be observed that the  
Finally, Figure 12 displays the nodal potential along the rod in the left panel, while the right panel shows the temperature of each rod segment for the partition with seven segments per meter. Both panels correspond to Model 2, which accounts for the temperature-dependent variation of soil resistivity, using a segmentation of seven segments per meter. Figure 11: Time-dependent behavior of the maximum electrode temperature for each model. Figure 10: Grounding potential versus time of the considered electrode for two different segmentations. The right panel of Figure 12 shows higher temperatures in the rod segments located within the low-resistivity soil layer. This is because these segments inject a higher current into the ground, thereby generating more heat due to the Joule effect and increasing the temperature. Although this temperature rise is partially mitigated by the reduction in soil resistivity, it is not sufficient to prevent these segments from reaching higher temperatures than the rest. The left panel of Figure 13 displays the soil resistivity profile as a function of radial distance from electrode segments located in different soil layers. The blue line corresponds to a segment within the 20 \(\Omega\cdot\)m layer, the red line corresponds to a segment in the 50 \(\Omega\cdot\)m layer, and the orange line corresponds to a segment situated in the highest-resistivity layer of 100 \(\Omega\cdot\)m, at the final time instant. The right panel shows the temperature as a function of the radial distance from the previously mentioned segments. Both panels display a zoomed-in view of the region closest to the electrode, covering the radial distance from r = 5 mm (the electrode surface) to r = 10 mm. The data shown corresponds to Model 2, which accounts for the temperature-dependent soil resistivity.