## 3 Orthogonal Test Scheme and Numerical Model  
### Orthogonal Test Scheme  
The purpose of this paper was to study the effects of fracture inclination, dip angle, trace length, and bulk density (relating to spacing) on the UCS and elastic modulus of fractured rock masses. Due to a large number of influencing factors and levels, it is quite troublesome to calculate the scheme. The orthogonal test method can not only effectively reduce the workload of calculation, but also obtain satisfactory sample results [28; 29]. As a result, an orthogonal test scheme with four influencing factors and three levels was selected, and \(L_{9}\left(3^{4}\right)\) was chosen as the orthogonal table for the design of this test scheme, as shown in Table 3. There are nine group test protocols in the table. The column number indicates the test factor, while the value under that column number indicates the level used for that factor. The level values of each influencing factor based on the range of the main influence factors also are given in Table 3. The level values of the influencing factors of inclination are 47.24\({}^{\circ}\), 94.49\({}^{\circ}\), and 141.73\({}^{\circ}\), respectively. To dip the angle, trace length, and bulk density, the level values were 28.60\({}^{\circ}\), 57.20\({}^{\circ}\), 85.80\({}^{\circ}\), 0.987 m, 1.973 m, 2.960 m, 0.05838 m\({}^{-3}\), 0.07784 m\({}^{-3}\), and 0.09730 m\({}^{-3}\) in the order of levels 1, 2, and 3. Therefore, the test protocols for each main control factor were established based on the orthogonal test scheme.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Material Type**} & \multicolumn{1}{c}{**1\# Fracture Group**} & \multicolumn{1}{c}{**2\# Fracture Group**} & \multicolumn{1}{c}{**3\# Fracture Group**} \\ \hline \multirow{3}{*}{Inclination/\({}^{\circ}\)} & Distribution type & Normal & Lognormal & Normal \\  & Mean value & 235.15 & 351.34 & 94.73 \\  & Standard deviation & 6.48 & 0.15 & 13.68 \\  & Distribution type & Normal & Lognormal & Normal \\ Dip angle/\({}^{\circ}\) & Mean value & 25.98 & 28.16 & 85.80 \\  & Standard deviation & 8.41 & 0.83 & 9.92 \\  & Distribution type & Lognormal & Normal & Lognormal \\ Trace length/m & Mean value & 3.42 & 3.12 & 2.96 \\  & Standard deviation & 2.05 & 1.48 & 1.39 \\  & Distribution type & Negative exponential & Lognormal & Negative exponential \\ Spacing/m & Mean value & 1.35 & 1.70 & 0.69 \\  & Standard deviation & 1.21 & 1.55 & 0.7 \\ Bulk density/m\({}^{-3}\) & Mean value & 0.01734 & 0.02020 & 0.05838 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Probability distribution and geometrical parameters of the fracture groups [24; 27].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Test Number**} & \multicolumn{4}{c}{**Influencing Factors**} \\ \cline{2-5}  & **Inclination/\({}^{\circ}\)** & **Dip Angle/\({}^{\circ}\)** & **Trace Length/m** & **Bulk Density/m\({}^{-3}\)** \\ \hline T-1 & Level 1 (47.24) & Level 1 (28.60) & Level 1 (0.987) & Level 1 (0.05838) \\ T-2 & Level 1 & Level 2 (57.20) & Level 2 (1.973) & Level 2 (0.07784) \\ T-3 & Level 1 & Level 3 (85.80) & Level 3 (2.960) & Level 3 (0.09730) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Orthogonal arrays \(L_{9}\left(3^{4}\right)\) of the test scheme.