## 4 Numerical Simulation Test
### Different Porosity Model Test
The compressive strength values obtained from numerical simulation tests on rock pore structure reconstruction models with varying porosities were compared against corresponding theoretical strength values derived from model curves, as illustrated in Figure 16. Table 5 presents the correlation coefficients quantifying the agreement between porosity-compressive strength simulation data and different theoretical model curves. Figure 16 demonstrates remarkable consistency between experimental simulation results and theoretical predictions. Quantitative analysis reveals that the power-law model exhibits the strongest correlation (R\({}^{2}\) = 0.989) in describing the porosity-compressive strength relationship of reconstructed rock pore structures, followed closely by the exponential model (R\({}^{2}\) = 0.980). Furthermore, the logarithmic and linear models also demonstrate statistically significant correlations with R\({}^{2}\) values of 0.933 and 0.916, respectively. These robust correlations across multiple mathematical models collectively validate the reliability of the proposed pore structure reconstruction methodology. Figure 16: Numerical and theoretical UCS vs. porosity. Based on the above two models, a combined mathematical relationship between compressive strength and porosity of Sichuan sandstone is found as follows, and the comparison between this mathematical relationship and the numerical simulation results is shown in Figure 17:  
\[\sigma=\sigma_{0}\gamma^{p}-3 pe^{-2.5p} \tag{5}\]  
where \(\gamma\) is the control factor, whose value range is 0.98322 \(\pm\) 9.8639 \(\times\) 10\({}^{-5}\). The correlation coefficient between the porosity-compressive strength model curve of the rock pore structure reconstruction model proposed in this study and the porosity-compressive strength simulation test data is 0.995, which is 0.006 higher than that of the power-law model curve and the porosity-compressive strength simulation test data. This indicates that the porosity-compressive strength model can accurately describe the correlation between the porosity and compressive strength of the rock pore structure reconstruction model. The elastic modulus of porous materials, such as porous sandstone, is also closely related to porosity. Various theoretical models have been proposed by scholars to characterize the relationship between elastic modulus and porosity. The Maxwell-type model [24] is expressed as  
\[E=E_{0}\times\frac{1-p}{1+p} \tag{6}\]  
where \(E\) is the [PERSON]'s modulus of porous material and \(E_{0}\) is the [PERSON]'s modulus of non-porous material. The self-consistent model [25] is expressed as  
Figure 15:0  
Figure 15:1 The proposed model for compressive strength versus porosity. \begin{table}
\begin{tabular}{c c} \hline
**Theoretical Model** & **Correlation Coefficient** \\ \hline Power model & 0.98852 \\ Exponential model & 0.97995 \\ Logarithm model & 0.93288 \\ Linear model & 0.91594 \\ \hline \end{tabular}
\end{table} 
Table 5: The correlation coefficient of the porosity–UCS theoretical model. The differential scheme model [26] is expressed as  
\[E=E_{0}\times(1-p)^{2} \tag{8}\]  
The exponential model [27] is expressed as  
\[E=E_{0}\times e^{\frac{-2p}{1-p}} \tag{9}\]  
The porosity-elastic modulus simulation test data and theoretical model curves are presented in Figure 18, with the corresponding correlation coefficients provided in Table 6. Generally, the elastic modulus decreases as porosity increases. The elastic moduli obtained from the simulation tests on rock pore structure reconstruction models with different porosities are in good agreement with those at the corresponding positions on the theoretical model curves. Furthermore, the correlation between the porosity-elastic modulus simulation test data and the theoretical model curves indicates that the simulation data closely follow the exponential model, with a correlation coefficient of 0.933. For the other three model curves, the correlation coefficients also reach 0.932, 0.924, and 0.897, respectively.