# Correlation Analysis of Macro-Micro Parameters of Sandstone Based on PFC3D
## 2 Methodology
### Orthogonal Experimental Design  
Orthogonal experimental design is a method that optimizes experimental schemes to effectively quantify the impact of multiple factors on the target response. In this study, the improved parallel bond model (PBM) involved 12 microscopic parameters. Conducting a full-factor test would have resulted in an excessive number of trials, so an orthogonal experimental design was employed to reduce the number of experiments while ensuring the validity of the results. For the 12 microscopic parameters, three levels were selected per parameter for the experimental design. The specific factors and levels are presented in Table 3. Based on these factor levels, an \(L_{27}\) (3\({}^{12}\)) orthogonal array design was utilized to generate 27 sets of experimental schemes, detailed in Table 4.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline
**Number** & \(\overline{E}_{c}\) **(GPa)** & \(E_{d}\) **/ \(\overline{E}_{c}\)** & \(k\) & \(\overline{\sigma}_{t}\) **(MPa)** & \(\overline{\sigma}\overline{\sigma}_{t}\) & \(\mu\) & \(\overline{\Phi}\) & \(\overline{\beta}\) & \(R_{f}\) & \(R_{e}\) & \(R_{E}\) & \(R_{k}\) \\ \hline
1 & 75 & 0.2 & 2 & 50 & 1.5 & 0.5 & 30 & 0.3 & 0.3 & 0.05 & 0.1 & 1 \\ \hline
2 & 75 & 0.2 & 2 & 50 & 2 & 0.7 & 40 & 0.5 & 0.5 & 0.1 & 0.2 & 2 \\ \hline
3 & 75 & 0.2 & 2 & 50 & 2.5 & 0.9 & 50 & 0.7 & 0.7 & 0.15 & 0.3 & 3 \\ \hline
4 & 75 & 0.3 & 3 & 100 & 1.5 & 0.5 & 30 & 0.5 & 0.5 & 0.15 & 0.3 & 3 \\ \hline
5 & 75 & 0.3 & 3 & 100 & 2 & 0.7 & 40 & 0.7 & 0.7 & 0.05 & 0.1 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Orthogonal experimental design.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Microscopic Parameters** & **Macroscopic Parameters** \\ \hline Effective bond modulus \(\overline{E}_{c}\) & Elastic modulus \(E\) \\ \hline Particle effective modulus/effective bond modulus \(E_{c}/\overline{E}_{c}\) & Poisson’s ratio \(\
u\) \\ \hline Bond stiffness ratio \(k\) & Uniaxial compressive strength \(\sigma_{c}\) \\ \hline Tensile strength \(\overline{\sigma}_{t}\) & Uniaxial compressive strength/uniaxial tensile strength \(\sigma_{c}/\sigma_{t}\) \\ \hline Cohesion/tensile/strength \(\overline{c}\) / \(\overline{\sigma}_{t}\) & Crack damage stress \(\sigma_{cd}\) \\ \hline Friction coefficient \(\mu\) & Internal friction angle \(\varphi\) \\ \hline Friction angle \(\overline{\phi}\) & Cohesion \(c\) \\ \hline Moment distribution coefficient \(\overline{\beta}\) & \\ \hline Packing ratio \(R_{f}\) & \\ \hline Strength ratio \(R_{e^{\prime}}\) & \\ \hline Modulus ratio \(R_{E}\) & \\ \hline Stiffness ratio \(R_{k}\) & \\ \hline \end{tabular}
\end{table}
Table 2: Macro- and microscopic parameters of the improved linear parallel bond model.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**Number** & \(\overline{E}_{c}\) **(GPa)** & \(E_{d}\) / \(\overline{E}_{c}\) & \(k\) & \(\overline{\sigma}_{t}\) **(MPa)** & \(\overline{\sigma}\overline{\sigma}_{t}\) & \(\mu\) & \(\overline{\Phi}\) & \(\overline{\beta}\) & \(R_{f}\) & \(R_{e}\) & \(R_{E}\) & \(R_{k}\) \\ \hline
1 & 75 & 0.2 & 2 & 50 & 1.5 & 0.5 & 30 & 0.3 & 0.3 & 0.05 & 0.1 & 1 \\ \hline
2 & 125 & 0.3 & 3 & 100 & 2.