# Petrophysics Parameter Inversion and Its Application Based on the Transient Electromagnetic Method
## 3 Numerical Simulation
### Inversion of Rock Physical Properties  
The above analysis confirms the reliability of the numerical simulation algorithm. To achieve the inversion of the formation of petrophysical parameters, the resistivity of each geoelectric model was obtained using the Gauss-Newton inversion method. As shown in Figures 8 and 9, the blue solid line represents the inverted resistivity profile, and the black solid line corresponds to the true geoelectric model. Overall, the inversion results accurately reflect the actual resistivity structure. The depth and resistivity values of each layer show good agreement with the true model, demonstrating the validity and effectiveness of the proposed inversion approach.  
Figure 10 presents the porosity profiles obtained from the inversion. The black solid line represents the true porosity model, and the blue solid line represents the inverted porosity. The results indicate a clear response in the target layer, and the inverted porosity  
Figure 8: Resistivity inversion results corresponding to different porosity values: (**a**) 15%, (**b**) 20%, (**c**) 25%, (**d**) 30%, (**e**) 35%, and (**f**) 40%.  
Figure 9: Resistivity inversion results corresponding to different water saturation values: (**a**) 20%, (**b**) 30%, (**c**) 40%, (**d**) 50%, (**e**) 60%, and (**f**) 70%.  
values consistently increase with increasing model porosity. Figure 11 shows the inverted water-saturation profiles. The black solid line represents the true model, and the blue solid line denotes the inverted water saturation. A comparative analysis reveals that the inversion results closely follow the overall trend of the true model, indicating good inversion performance.  
Figure 12 presents the inverted water saturation profile. The black solid line represents the true water saturation model, and the blue solid line shows the inversion result. The inversion results indicate a clear response in the target layer, with the inverted saturation values increasing consistently as the model saturation increases. Figure 13 shows the inverted porosity profiles. The black solid line corresponds to the true porosity model, and the blue solid line represents the inversion results. While the overall trend of the inverted porosity closely matched the true model, noticeable deviations were observed at the target layer. This discrepancy is likely due to the coupling effect between water saturation and  
Figure 11: Water saturation inversion results for \(S_{w}=20\%\). **(a)** porosity 15%, **(b)** porosity 20%, **(c)** porosity 25%, **(d)** porosity 30%, **(e)** porosity 35%, and **(f)** porosity 40%.  
Figure 10: Porosity inversion results for different porosity values: **(a)** 15%, **(b)** 20%, **(c)** 25%, **(d)** 30%, **(e)** 35%, and **(f)** 40%.  
porosity during inversion; when the error in water saturation inversion is large, it can propagate and amplify the error in porosity inversion.  
The results indicate that the inverted resistivity values closely match the actual geoelectric models. Based on the inverted resistivity and using [PERSON]'s equation, the subsequent inversion of water saturation and porosity yields results that follow the overall trend of the true model, with both showing distinct responses at the target layer. However, due to the parameter coupling between water saturation and porosity, as well as the limitations of the inversion algorithm, errors in one parameter may propagate and influence the inversion accuracy of the other. Despite these challenges, the inversion results effectively capture the morphological variations and underlying patterns of formation petrophysical properties.  
Figure 12: Water saturation inversion results for different saturation values: (**a**) 20%, (**b**) 30%, (**c**) 40%, (**d**) 50%, (**e**) 60%, and (**f**) 70%.  
Figure 13: Inversion result for porosity \(\varphi=20\%\). (**a**) water saturation 20%, (**b**) water saturation 30%, (**c**) water saturation 40%, (**d**) water saturation 50%, (**e**) water saturation 60%, and (**f**) water saturation 70%.