# The Generalized Skin Depth for Polarized Porous Media Based on the Cole-Cole Model
## 3 Generalized Skin Depth Characteristic Analysis Based on the Cole-Cole Model
### Characteristic Analysis of the Generalized Skin Depth at Low Frequency  
Four polarization parameters affected the generalized skin depth, and all of them were concentrated in the low-frequency band (approximately f \(<\) 10 Hz). According to the existing transmitter frequency in the laboratory, we chose a frequency of 6.25 Hz. The characteristics of the generalized skin depth at low frequencies were analysed, with the conductivity range \(10^{-2}<\sigma_{\infty}<10^{0}\), c = 0.5, and \(\tau=0.1\) s. The chargeability values were \(\eta=0.2\), \(\eta=0.4\), \(\eta=0.6\), and \(\eta=0.8\). The effect of the conductivity at infinite frequency (\(\sigma_{\infty}\)) and the chargeability (\(\eta\)) on the skin depth is shown in Figure 8 for a  
Figure 6: Generalized skin depth under different frequency dependence values.  
Figure 7: Generalized skin depth curves with different characteristic time constants.  
chargeability range of \(0.1<\eta<0.9\) and frequency dependence values of c =0.2, c =0.4, c =0.6, and c =0.8. The effect of the chargeability (\(\eta\)) and frequency dependence (c) on the skin depth is shown in Figure 9 for a characteristic time constant range of \(10^{-2}\) s \(<\tau<10^{0}\) s and \(\eta=0.6\). Figure 10 shows the effect of the characteristic time constant (\(\tau\)) and frequency dependence (c) on the skin depth for the same characteristic time constant range, c =0.5 and \(\tau=0.1\) s. Figure 11 shows the effect of the characteristic time constant (\(\tau\)) and chargeability (\(\eta\)) on the skin depth.  
Figures 8-11 show that in the low-frequency band, the conductivity at infinite frequency decreased gradually and the effects of the chargeability on the generalized skin depth became increasingly obvious. When the chargeability and frequency dependence changed simultaneously, the chargeability was proportional to the generalized skin depth, and the frequency dependence was inversely proportional to the generalized skin depth. When the characteristic time constant and the frequency dependence changed simultaneously, the characteristic time constant was inversely proportional to the generalized skin depth.  
Figure 8: The effect of the conductivity at infinite frequency and the chargeability on the skin depth ((a) 2D curve, (b) 3D view).  
Figure 10: The effect of the characteristic time constant and the frequency dependence on the skin depth ((a) 2D curve, (b) 3D view).  
Figure 9: The effect of the chargeability and the frequency dependence on the skin depth ((a) 2D curve, (b) 3D view)  
skin depth, and the decreasing rate was determined by the frequency dependence. At the same conductivity at infinite frequency, the chargeability was proportional to the generalized skin depth. When the chargeability was small, the frequency dependence had little effect on the generalized skin depth. With increasing chargeability, the frequency dependence had a more obvious effect on the generalized skin depth. When the chargeability and characteristic time constant changed simultaneously, the chargeability was proportional to the generalized skin depth, and the characteristic time constant was inversely proportional to the generalized skin depth. The characteristic time constant was relatively small, and the effect of chargeability on the generalized skin depth was prominent.