## 7 Theoretical Verification of Minimum Overburden Thickness  
The minimum thickness \(L\) of a karst tunnel under the blasting load consists of two parts: the blasting damage area thickness \(L_{\rm c}\) and the hydraulic fracture area thickness \(L_{\rm w}\). During blasting construction, the stress waves generally disturb the front of the palm face to a depth of no more than 1.5 m, as shown in Figure 14. In this paper, Because \(L_{\rm c}\) represents the initial crack before the action of water pressure, it is considered as the crack length generated by blasting when \(L_{\rm c}\) is taken, and this data has been analyzed and obtained in Section 5.1; therefore, \(L_{\rm c}\) = 1.18 m.  
Figure 13: Stress peak analysis of measuring point #1.  
The minimum outburst prevention thickness \(L\) of a karst tunnel under the action of the blasting load consists of two parts, namely, the thickness of the blasting damage area \(L_{\rm c}\) and the thickness of the hydraulic fracturing area \(L_{\rm w}\). During blasting construction, the disturbance depth of the stress wave to the front of the tunnel face generally does not exceed 1.5 m. The calculation diagram is shown in Figure 14. In this paper, \(L_{\rm c}\) is set as 1.18 m according to the actual blasting damage thickness in the calculation.  
When \(P_{\rm w}<P_{\rm jj}\), the thickness \(L_{\rm w}\) of the hydraulically fractured zone is calculated as:  
\[\left\{\ln\lambda-\ln\!\left[\lambda-\frac{2 fP_{\rm w}\sqrt{\pi a}-\sqrt{3}K_{ \rm HC}-2c\sqrt{\pi a}+2\tau_{1}\big{|}K_{\rm II}^{(2)}\big{|}\sqrt{\pi a}}{ \gamma H\sqrt{\pi a}(f-f\cos 2\beta+\sin 2\beta)}\right]\right.-\left.\frac{f+f\cos 2 \beta-\sin 2\beta}{f-f\cos 2\beta+\sin 2\beta}\right\} \tag{7}\]  
In the formula, \(R\) is the radius of the tunnel equivalence circle, \(R=(B+h)/4\), \(B\) is the span of the tunnel, \(h\) is the height of the upper step of the tunnel excavation, \(\lambda\) is the lateral pressure coefficient, \(f\) is the friction coefficient, a is one-half of the crack length, \(K_{\rm HC}\) is the type II compression-shear extension stress intensity factor, \(c\) is the cohesion of the crack surface, \(\tau_{1}\) for the maximum shear stress on the wavefront surface, \(K_{\rm II}^{(2)}\) for the dynamic stress intensity factor, \(\gamma\) is the surrounding rock weight, \(H\) is the tunnel depth, and \(\beta\) is the vertical direction to the center of the fissure to establish the angle of the coordinate system X-axis.  
The buried depth of Dejiang tunnel at the YK9 + 915 section is 312 m, the tunnel section span \(B\) is 10.8 m, and the height \(h\) is 5.6 m. There is an aquifer in this area. The head height of the tunnel roof is 34 m, the water pressure \(P_{\rm w}\) is 0.32 MPa, and the limestone type II pressure. The shear expansion stress intensity factor is selected according to the literature, which is taken as 7.94 \(\times 10^{6}N/\rm m^{3/2}\), the lateral pressure coefficient \(\lambda\) is taken as 1.5, the crack length a is taken as the initial crack thickness of 1.18 m, the friction coefficient of the crack surface is taken as 0.7, and the cohesion coefficient is taken as 0.7. The force c is taken as 0.3 MPa, the dynamic stress intensity factor \(K_{\rm II}^{(2)}\) is taken as 0.4, and \(\beta\) is taken as 12\({}^{\circ}\).  
According to the parameters of the Dejiang tunnel project, the thickness \(L_{\rm w}\) of the hydraulic fracturing zone under the blasting load can be obtained to be 1.60 m; then, the minimum outburst prevention thickness of the karst tunnel under the blasting load calculated by the formula is \(L=2.78\) m, and the numerical model L determined by the mathematical analysis of the data is 3.08 m, and the error between the two is 9.7%. This proves that the minimum outburst prevention thickness of a karst tunnel under blasting action obtained by numerical model analysis is reliable.  
Figure 14: Schematic diagram of minimum overburden thickness.