## 3 Engineering Case Study
### Construction of Heterogeneous Mechanical Parameter Block Model
By integrating the key geological parameters of the rock mass (\(GSI\), \(m_{i}\), \(D\), \(H\)) with the laboratory-derived mechanical properties (\(\rho\), \(\sigma_{ci}\)), the equivalent mechanical parameters (\(\sigma_{t}\), \(E_{m}\), \(\phi^{\prime}\), \(c^{\prime}\)) are computed using the generalized Hoek-Brown criterion (Section 2.2). This integration enables the development of a comprehensive mechanical parameter block model that effectively characterizes the spatial variability (Figure 12c,d). The specific values of \(J(Cond_{89}\), \(m_{i}\), \(\rho\), and \(\sigma_{ci}\) are detailed in Table 2. Considering the implementation of controlled blasting techniques at the site, \(D\) is conservatively assigned a value of 0.8 [26]. The vertical distance between each block and the ground surface is automatically computed through Python 3.10 programming to determine \(H\). As shown in Figure 13, the statistical regression of the lithology-specific mechanical parameters--obtained by first interpolating the RQD data and then converting the results via the generalized Hoek-Brown criterion--demonstrates that the \(E_{m}-GSI\) relationship obeys an exponential law,  
\[\ln E_{m}=\alpha_{0}+\alpha_{1}GSI, \tag{21}\]  
with an excellent goodness of fit on a logarithmic scale (\(R_{\log}^{2}\approx 1\)). The slope remains nearly invariant across the lithologies (\(\alpha_{1}\approx 0.0576\)), meaning that every 10-point increase in the \(GSI\) multiplies \(E_{m}\) by \(\exp(10\alpha_{1})\approx 1.778\), while the lithologic differences are  
Figure 11: Construction and cross-validation of RQD block model: (**a**) spherical semivariogram model fitting for RQD; (**b**) RQD block model; (**c**) leave-one-out cross-validation (LOOCV)—predicted vs. observed; (**d**) quantile–quantile plot of LOOCV standardized residuals; (**e**) experimental semivariogram of LOOCV residuals. chiefly reflected in the intercept \(\alpha_{0}\). The depth dependence of the effective cohesion \(c\prime\) is approximately linear,  
\[c^{\prime}=\beta_{0}+\beta_{1}H, \tag{22}\]  
with \(\beta_{1}\) ranging from about 0.122 to 1.406 kPa m\({}^{-1}\) among the lithologies, indicating lithology-dependent sensitivity to confinement and overburden. The increasing scatter with depth suggests heteroscedasticity. The regression coefficients and diagnostics are summarized in Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Lithologies** & _J_**_Cond\({}_{89}\)_ & \(m_{i}\) & \(\rho\) (kN/m\({}^{3}\))** & \(\sigma_{ci}\) (MPa)** & **Poisson’s Ratio** \\ \hline Slate & 20 & 8 & 28.6 & 50.44 & 0.26 \\ Weathered slate & 11 & 7 & 25.5 & 21.19 & 0.27 \\ Dolomite & 25 & 12 & 30.1 & 86.92 & 0.21 \\ Mica schist & 20 & 7 & 29.9 & 54.40 & 0.19 \\ Limestone & 16 & 6 & 26.5 & 28.73 & 0.28 \\ Fault & 7 & 4 & 23.3 & 7.25 & 0.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geological and mechanical parameters of different lithologies. Figure 12: Block model: (**a**) lithology; (**b**) GSI; (**c**) Young’s modulus, \(E_{m}\) (GPa); (**d**) cohesion, \(c\prime\)(kPa). Figure 13: Parametric regression analysis: (**a**) \(E_{m}-GSI\) by lithology (log-linear fits); (**b**) \(c^{\prime}-H\) by lithology (linear fits). The lithology-based color scheme in (**b**) corresponds to the same legend as shown in (**a**).