## 5 Analysis and Discussion
### Deviation Angles at Different Spatial Scales  
In the assessment of crack space characteristics, area and volume were used to evaluate the size of the crack space. In addition, the directional characteristic is one of the significant parameters that must be evaluated. Based on the total microcrack distribution, the directions of the theoretical distribution pattern space are fixed, and do not vary with the spatial scales Therefore, they were set as the basis axes to analyze the deviation angles with the MVEE results.  
It can be inferred from Section 3.3 that the theoretical distribution pattern space can be represented by a three-dimensional matrix, which is the covariance matrix \(\mathcal{C}_{S^{*}}\). The direction of each axis of this crack space is represented by the eigenvector corresponding to the eigenvalue. In the same way, the ellipsoid \(\varepsilon\) from the MVEE method represented by a positive definite matrix \(E_{m}\) can be obtained, as well as the direction of its three axes. Thus, for the same spatial scale, the deviation angle of the MVEE ellipsoid can be obtained by calculating the angle between its principal axis and the principal axis of theoretical distribution pattern space. The deviation angles in each spatial scale were calculated and displayed in Figure 9.  
It can be observed in Figure 9 that the \(X\)-axis deviation angles of three specimens were generally less than 10 degrees, with an average of 6.24 degrees, 1.01 degrees, and 6.17 degrees, respectively. They reached the relatively smallest values before the 85% scale,  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Specimen** & \multicolumn{2}{c}{**S2**} & \multicolumn{3}{c}{**S3**} & \multicolumn{3}{c}{**S5**} \\ \hline
**Method** & **TDPS** & **3 DCH** & **MVEE** & **TDPS** & **3 DCH** & **MVEE** & **TDPS** & **3 DCH** & **MVEE** \\ \hline Area & 0.970 & 0.986 & 0.971 & 0.988 & 0.982 & 0.988 & 0.969 & 0.984 & 0.968 \\ Volume & 0.983 & 0.982 & 0.976 & 0.988 & 0.983 & 0.985 & 0.981 & 0.984 & 0.9670 \\ \hline \hline \end{tabular} Note: TDPS means the theoretical distribution pattern space, 3 DCH means the 3D convex hull, and MVEE means the minimum volume enclosing ellipsoid.  
\end{table}
Table 1: The goodness-of-fit \(R^{2}\) of area and volume.  
Figure 9: Results of deviation angle evolution. (**a**) Deviation angle evolution of Specim S2. (**b**) Deviation angle evolution of Specim S3. (**c**) Deviation angle evolution of Specim S5.  
where the scale of the relative minimum value was reached at 90% in Figure 9c. After an increase, it began to show a decreasing trend again. Similar evolutionary trends were found on the other two axes. However, there was a tendency for the deviation angle to increase near 65%.  
Considering the fact that microcracks are not evenly distributed within the specimen according to a theoretical distribution pattern, and a more dense distribution would be observed in the severely damaged zone [9; 13; 19], as the spatial scale decreases, there would be a situation where the crack space contains more densely distributed areas, with a lower share of other spatial regions [20; 22]. Therefore, a reduction in spatial scale results in a loss of information about the crack space. In addition, the area and volume of the crack space are significantly underestimated, if the crack scale drops considerably. Therefore, a reliable crack space cannot be approached by continually reducing the spatial scale to approximate the minimum deviation angle. Summarizing the evolution of the evaluated values for the area, volume, and deviation angle, the appropriate spatial scale range for the crack space is proposed from 65% to 85%. This result is applicable only for these tests. The evaluation of different rock types requires further application to more rocks.