# Numerical Simulation Analysis Method for Rockburst Control in Deeply Buried Caverns
## 4 Numerical Analysis Method for Relieving Rockburst through Borehole Stress Release
### Mathematical Model of the Implicit Hole Element  
Stress relief holes with a diameter of 45 mm, a depth of 1-3 m, and a spacing of tens of centimeters can be drilled perpendicular to the tunnel face to release part of the energy, to alleviate the stress concentration. Due to the need to compare the borehole schemes during the design process, multiple models need to be established according to conventional methods. Owing to the small size of the boreholes, it is difficult to align them with the rock mass elements. Therefore, the hole elements can be independently modeled and embedded in the rock mass elements and the energy release effect of the borehole can be equivalently reflected through interpolation. The mathematical model of the borehole element is similar to the anchor column element. But, considering the effect of the borehole, the rock mass within a distance of three times the diameter of the hole is treated as an anisotropic material to calculate its stiffness. Its radial elastic modulus is calculated according to the rock mass elastic modulus E and the axial elastic modulus is taken as (0.5-0.7) E based on the influence of the borehole diameter. Then, the stiffness loss of the borehole element is calculated according to Equation (6). The anchor rod element increases the stiffness of the surrounding rock while the borehole element weakens the stiffness, which is different from the anchor rod element. The existence of boreholes makes the anisotropy of rock mass along the axial direction more obvious so the rock mass element with boreholes must be calculated by forming a stiffness matrix according to the anisotropy. When arranging more than two stress relief boreholes within the same rock mass element, the anisotropy caused by the two boreholes needs to be vector superimposed. As shown in Figure 2, the vectors in the axial direction of the two boreholes A and B can be superimposed first to determine the normal vector \(\widetilde{AB}\) in the layered direction of the rock mass element and then the stiffness matrix of the two boreholes A and B can be converted to the main plane \(\widetilde{AB}\) according to the following formula:  
\[[K_{Ri}]=[R_{i}]^{T}[K_{zi}][R_{i}] \tag{12}\]  
where \(K_{zi}\) represents the element stiffness generated by the \(i\)-th borehole element along its axis direction, \(K_{Ri}\) represents the stiffness of the i-th borehole element superimposed to the stiffness in the rock mass element, \([R_{i}]\) is the three-dimensional coordinate transformation matrix between the axis direction of the \(i\)-th borehole element and the direction of the synthetic principal vector \(\widetilde{AB}\), and \(i\) represents the number of the borehole element in the rock mass element. The borehole element is independently implicit in the rock mass element, similar to the stiffness of the anchor rod element. According to the interpolation principle of the finite element, the stiffness of the borehole element shall be superimposed into the rock mass element according to Equation (7) and then the influence of the borehole element shall be deducted from the stiffness matrix of the rock mass element. So, the combined stiffness of the rock mass element with boreholes \([K_{RZ}]\) is  
\[[K_{RZ}]=[K_{E}]-\sum_{1}^{i}[N]^{T}[K_{Ri}][N] \tag{13}\]  
where \([K_{E}]\) is the anisotropic rock mass element stiffness calculated according to the principal axis vector direction synthesized by the number of boreholes and \([N]\) is the conversion matrix between the borehole and rock mass element calculated in reference [32]. The use of implicit borehole elements not only simplifies the analysis and calculation of different borehole arrangements but also accurately represents the reduction in stiffness of rock elements due to drilling holes and effectively demonstrates their role in mitigating rockburst. Figure 2: Determination of the normal direction of the rock mass element layer with boreholes.