# The Generalized Skin Depth for Polarized Porous Media Based on the Cole-Cole Model
## 4 Skin Depth of Porous Media with Metallic Particles  
The skin depth of polarized media depends on not only the frequency and conductivity, but also the chargeability and other parameters. Therefore, when calculating the skin depth of porous media, we need to understand the chargeability and other characteristics. Here, the polarization parameters of metal particles can be fitted by the Cole-Cole model. The polarization parameters of graphite, pyrite, silver, copper, and magnetite were obtained from the experiments conducted in Reference [17]. The range of polarization parameters of frozen rock was given in Reference [18]. The specific parameters selected in this paper are shown in Table 1.  
\(\delta_{\text{{Graphite}}}\), \(\delta_{\text{{Pyrite}}}\), \(\delta_{\text{{Silver}}}\), \(\delta_{\text{{Copper}}}\), \(\delta_{\text{{Magnetite}}}\), and \(\delta_{\text{{Frozen rocks}}}\) were used to represent the skin depth of graphite, pyrite, silver, copper, magnetite and frozen rock, respectively. Here, we took the actual orebody model as an example to calculate the skin depth using the proposed method and the classical skin depth calculation method, as shown in Figure 12a-f. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Metallic** & **Chargeability** & **Conductivity at Infinite** & **Frequency** & **Characteristic Time** \\
**Particles** & \(\eta\) & **Frequency \(\sigma_{\infty}\) (S/m)** & **Dependence c** & **Constant \(\tau\)(s)** \\ \hline Graphite & 0.308 & 0.0479 & 0.6 & 2 \\ Pyrite & 0.275 & 0.0472 & 0.78 & 0.0155 \\ Silver & 0.037 & 0.0325 & 0.62 & 0.004 \\ Copper & 0.221 & 0.0289 & 0.41 & 0.0042 \\ Magnetite & 0.13 & 0.0307 & 0.38 & 0.01 \\ Frozen rocks & 0.46 & 0.01 & 0.8 & 0.00005 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Polarization parameters of common metal particles. Figure 11: The effect of the characteristic time constant and the chargeability on the skin depth ((**a**) 2D curve, (**b**) 3D view). Compared with the results of the classical skin depth calculation method, when polarization parameters, such as the chargeability, characteristic time constant, and frequency dependence, were taken into account, the predicted skin depth of the polarized media was deeper. Among the samples characterized here, the frozen rock had the largest skin depth and the graphite had the smallest skin depth. The skin depth of a polarized medium is not determined by a specific parameter. That is, the skin depth of an EM wave in a porous polarized medium depends not only on conductivity, but also on polarization parameters, such as chargeability, characteristic time constant, and frequency dependence. The influence of these parameters cannot be ignored. Therefore, for the depth calculation of the porous polarized medium model, accurate results can be obtained by using the generalized skin depth formula. The relative error between the generalized epidermal depth and classical epidermal depth is shown in Figure 13.  
\[\delta_{\text{relative error}}=\left|\frac{\delta_{\text{pole-cole}}-\delta_ {\text{classical}}}{\delta_{\text{classical}}}\right|\times 100\% \tag{16}\]  
Figure 12: Six models were used to calculate the skin depth (among them, the black curve is the result of the classical skin depth formula and the curve in colour is the result of the generalized skin depth formula). In the range of 10\({}^{-1}\)-10\({}^{4}\) Hz, the maximum relative errors of silver, graphite, copper, pyrite, magnetite, and frozen rock were 1.84%, 5.145%, 11.88%, 16.89%, 5.63%, and 26.51%, respectively. With decreasing frequency, the relative error increased gradually, as did the effect of the polarization parameters on the skin depth. Therefore, in low-frequency EM detection, to obtain accurate depth interpretation imaging, when implementing the porous polarized metal particle model, the generalized skin depth must be used. As shown in Figure 13, at the same frequency, the relative error of frozen rock was the largest and that of silver was the smallest.