## 3 Engineering Case Study
### Construction of Heterogeneous Mechanical Parameter Block Model  
In this study, a block model was developed to characterize the spatial variability in slope mechanical parameters, establishing a robust parametric foundation for precise  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multirow{2}{*}{Intact slate block} & Young’s modulus (GPa) & \multirow{2}{*}{Poisson’s ratio} & Cohesion (Mpa) & Friction angle (\({}^{\circ}\)) & Tensile strength (Mpa) \\ \cline{3-6} \cline{5-6} \cline{2-6} \cline  
slope stability assessment. The research employed a systematic modeling methodology comprising two sequential stages: first, a RQD block model was constructed with block dimensions corresponding to the REV. Subsequently, this model was transformed into a spatially variable mechanical parameter block model using the generalized Hoek-Brown criterion. Through the analysis of 4402 digital images from core samples obtained from 40 boreholes in the southern slope of the main mining area, the RQD data were systematically integrated into the borehole information database (Figure 10). Due to exploration cost constraints, boreholes in the mine are typically spaced at regular intervals. The RQD values for the unobserved areas were spatially interpolated using Ordinary Kriging. As illustrated in Figure 11a, an experimental semivariogram of borehole RQD data was computed with a 10 m lag distance. A spherical semivariogram model was fitted to the discrete experimental values, yielding the following optimal parameters: a nugget of 60, sill of 420, and range of 230 m. The resulting RQD block model (Figure 11b), constrained by the mine surface's triangulated irregular network, has overall dimensions of 1610 m \(\times\) 798 m \(\times\) 490 m and comprises 169,576 regular block units. A leave-one-out cross-validation (LOOCV) was performed to evaluate the predictive performance of Ordinary Kriging. To balance the computational cost with the dense down-hole sampling and short-range vertical correlation, three representative depths--approximately 50 m, 100 m, and 150 m--were selected for each of the 40 boreholes, providing 120 validation points. Each point was withheld in turn, the remaining data were used to predict the RQD at that location, and the prediction, local Kriging standard deviation \(\sigma_{K}\), and residual \(e\) (observed--predicted) were recorded. Figure 11c compares the LOOCV predictions with the observations along the 1:1 line; the points cluster tightly around the diagonal, indicating good agreement with negligible bias, and the resulting root mean square error (RMSE) is 14.05. Figure 11d displays the quantile-quantile (Q-Q) plot of the standardized residuals \(r=e/\sigma_{K}\), which adheres closely to the theoretical standard normal line except for minor tail deviations; the root mean squared standardized error (RMSSE) is 0.916, close to the nominal value of 1, supporting the assumption of residual normality. Figure 11e shows the experimental semivariogram of the LOOCV residuals (using the same lag configuration as in Figure 11a), which remains essentially flat with distance, indicating that the prediction errors are uncorrelated and that the adopted spherical variogram adequately captures the principal spatial structure of RQD. Figure 10: Borehole RQD information database. Text labels indicate borehole names. Collectively, the cross-validation results demonstrate minimal interpolation bias, well-characterized prediction variance, and spatially unstructured residuals, providing no evidence that a trend-based method such as Universal Kriging is required. At the investigated scale, Ordinary Kriging is therefore appropriate. Based on the lithological zoning patterns (Figure 3b), distinct lithological attributes are systematically assigned to the block model (Figure 12a). Utilizing the structural plane condition coefficient (\(J(Cond_{89})\)) obtained from the field investigations, the GSI block model of the rock mass is quantitatively estimated through Equation (9), as illustrated in Figure 12b.