# Correlation Analysis of Macro-Micro Parameters of Sandstone Based on PFC3D
## 3 Results and Discussion
### Macro-Mesoscopic Parameter Regression Modeling  
Based on the results of the aforementioned Pearson correlation coefficient analysis, to further investigate the quantitative relationship between the microscopic parameters and macroscopic parameters, the microscopic parameters with an \(|\)r\(|\)\(\geq\) 0.3 were identified as the primary factors influencing the macroscopic parameters and are presented in Table 6.  
We established a multiple linear regression model using the main influencing factors from Table 6 as the independent variables and the macroscopic parameters as the dependent variables. The regression analysis was conducted using SPSS Statistics 26.0 software, with significant variables selected through the stepwise regression method. The linear regression model was fitted based on the ordinary least squares method (OLS), resulting in the regression equations for each macroscopic parameters as follows:  
\[E=0.087\overline{E}_{c}-9.5351R_{f}+21.9868 \tag{4}\]  
\[\
u=-0.0003\overline{E}_{c}+0.0183k+0.1011R_{f}+0.1252 \tag{5}\] \begin{table}
\begin{tabular}{c c} \hline
**Macroscopic Parameters** & **Main Influencing Factors** \\ \hline Elastic modulus & \(\overline{E}_{c}\), \(R_{f}\) \\ Poisson’s ratio & \(R_{f}\), \(k\), \(\overline{E}_{c}\) \\ Compressive strength & \(\overline{\beta}\), \(R_{f}\), \(\overline{\sigma}_{t}\), \(\overline{c}/\overline{\sigma}_{t}\), \(R_{r}\) \\ Internal friction angle & \(\overline{\beta}\), \(R_{f}\), \(\mu\overline{E}_{c}\) \\ Cohesion & \(\overline{\sigma}_{t}\), \(\overline{E}_{c}\), \(\overline{c}/\overline{\sigma}_{t}\), \(R_{f}\), \(R_{r}\), \(\overline{\phi}\) \\ \(\sigma_{cd}/\sigma_{c}\) & \(\overline{\sigma}_{t}\), \(R_{f}\), \(\overline{c}/\overline{\sigma}_{t}\), \(\overline{\beta}\), \(\overline{\phi}\) \\ Compression–tension ratio & \(\overline{\phi}\), \(\overline{\beta}\), \(R_{r}\), \(R_{f}\) \\ \hline \end{tabular}
\end{table} 
Table 6: Main influencing factors corresponding to each macroscopic parameter. \[\sigma_{c}=0.3330\overline{\sigma}_{t}+32.0619\overline{c}/\overline{\sigma}_{t}-86.7 826\overline{\beta}-78.4979R_{f}+278.3672R_{\sigma}+15.275 \tag{6}\]  
\[\phi=0.0239\overline{E}_{c}+6.9954\mu-8.7006\overline{\beta}-7.1117R_{f}+39.9752 \tag{7}\]  
\[c=-0.0214\overline{E}_{c}+0.0274\overline{\sigma}_{t}+2.1228\overline{c}/ \overline{\sigma}_{t}-0.0875\overline{\beta}-4.7258R_{f}+20.0655R_{\sigma}+5.9 245 \tag{8}\]  
\[\sigma_{cd}/\sigma_{c}=-0.0013\overline{\sigma}_{t}-0.1209\overline{c}/ \overline{\sigma}_{t}+0.0048\overline{\phi}+0.2414\overline{\beta}+0.3008R_{f }+0.3587 \tag{9}\]  
\[\sigma_{c}/\sigma_{t}=0.150\overline{\phi}-5.3925\overline{\beta}+5.6370R_{f }-19.0706R_{\sigma}+5.4365 \tag{10}\]  
Table 7 presents the statistical test results of the regression models for each macroscopic parameters, showing the following:  
(1) The \(R^{2}\) values of all models are greater than 0.79, with \(E\) (\(R^{2}\) = 0.9) and \(c\) (\(R^{2}\) = 0.903) demonstrating the best fit and relatively strong fitting effects; \(\varphi\) (\(R^{2}\) = 0.793), though slightly lower, still has good explanatory capability. (2) The multicollinearity test indicates that the maximum condition number reaches 4180 (cohesion \(c\) model), but the variance inflation factors (VIFs) for all models are below five, suggesting that although there is some degree of correlation among the independent variables, it remains within an acceptable range. (3) The normality of the residuals is confirmed by the Jarque-Bera test (\(p\) > 0.05), with the models for Poisson's ratio \(v\) (\(p\) = 0.958) and the critical deviatoric stress ratio \(\sigma_{cd}/\sigma_{c}\) (\(p\) = 0.892) demonstrating optimal residual normality. (4) The autocorrelation tests show that except for models \(E\) (DW = 0.983) and \(\sigma_{c}/\sigma_{t}\) (DW = 0.971), which exhibit slight autocorrelation, the DW values for the other models range between 1.5 and 2.3, satisfying the independence assumption.