# Correlation Analysis of Macro-Micro Parameters of Sandstone Based on PFC3D
## 3 Results and Discussion
### Analysis and Discussion of Results  
The correlation and regression analysis results indicate significant differences in the influence of different mesoscopic parameters on the macroscopic mechanical properties. Among them, the effective bond modulus \(\overline{E}_{c}\) and elastic modulus \(E\) exhibit the strongest positive correlation (r = 0.876), which suggests that the bond stiffness between particles is a key factor in determining the overall deformation stiffness of the material. The negative correlation between the filler ratio \(R_{f}\) and \(E\) reflects that an increase in the proportion of weak contacts weakens the overall continuity and stiffness of the structure, thereby reducing its deformation resistance.  
Regarding [PERSON]'s ratio \(\
u\), the positive correlation between the filler ratio \(R_{f}\) and the bond normal stiffness ratio \(k\) indicates that a higher proportion of weak contacts and  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Macroscopic** & \(R^{2}\) & **Variance Inflation** & **Maximum Condition** & **Jarque–Bera Test** & **Durbin–Watson** \\
**Parameters** & & **Factor** & **Number** & **Test** \\ \hline \(E\) & 0.9 & 108.5 & 924 & 0.658 & 0.983 \\ \(v\) & 0.891 & 62.44 & 970 & 0.958 & 2.272 \\ \(\sigma_{c}\) & 0.819 & 19.0 & 2660 & 0.736 & 1.747 \\ \(\varphi\) & 0.793 & 21.08 & 1180 & 0.397 & 1.825 \\ \(c\) & 0.903 & 31.16 & 4180 & 0.824 & 2.168 \\ \(\sigma_{cd}/\sigma_{c}\) & 0.843 & 22.59 & 1080 & 0.892 & 1.506 \\ \(\sigma_{c}/\sigma_{t}\) & 0.855 & 32.31 & 1010 & 0.417 & 0.971 \\ \hline \end{tabular}
\end{table}
Table 7: Statistical test results of each macro parameter.  
stronger normal resistance help suppress particle sliding, leading to more significant lateral deformation. In contrast, a larger \(\overline{E}_{c}\) limits particle micro-motion, resulting in a decrease in Poisson's ratio.  
The uniaxial compressive strength \(\sigma_{c}\) is influenced by multiple factors, with the negative correlation between the moment distribution coefficient \(\overline{\beta}\) and \(R_{f}\) being particularly prominent. This suggests that unreasonable moment transfer and a high proportion of weak contacts may reduce the overall load-bearing capacity between particles. The positive correlation between the tensile strength \(\sigma_{t}\) and strength ratio \(R_{\sigma}\) further confirms that high strength and a uniform structure contribute to improving compressive performance.  
The strong positive correlation (r = 0.569) between the compressive-tensile strength ratio \(\sigma_{c}/\sigma_{t}\) and the friction angle \(\varphi\) reveals the decisive role of frictional resistance in determining the failure mode of the structure. An increase in frictional force suppresses crack propagation, enhancing the overall compressive strength of the structure. The negative correlation between the moment distribution coefficient \(\overline{\beta}\) and \(R_{\sigma}\) also suggests that mechanical heterogeneity within the structure can reduce the compressive-tensile strength ratio.  
This study established logical relationships between mesoscopic parameters and macroscopic responses through orthogonal design and quantitative analysis methods, providing theoretical support for parameter inversion in rock particle flow modeling. The statistical test results of the regression models indicate a high fitting accuracy, with R\({}^{2}\) values exceeding 0.79, meeting the needs for engineering prediction and analysis. Compared with traditional empirical calibration methods, the process proposed in this paper is systematic and repeatable, providing a more reliable parameter basis for numerical rock simulations.