# Numerical Simulation Analysis Method for Rockburst Control in Deeply Buried Caverns
## 4 Numerical Analysis Method for Relieving Rockburst through Borehole Stress Release
### Energy Release Calculation of the borehole Element  
The use of boreholes to control rockburst has two main functions: one is releasing part of the energy of the rock mass through drilling and the other is adjusting the stress of the surrounding rock mass by weakening the stiffness to slow down the stress concentration. For part of the energy released by the borehole element, the energy release can be calculated using the following equation based on the stress intensity \(\sigma_{0}\) of the rock mass element that was drilled and the ratio of the borehole element volume to the rock mass element volume, as follows:  
\[\Delta W_{k}=\frac{V_{k}}{V_{R}}\times\frac{0.5\sigma_{0}^{2}}{E} \tag{14}\]  
where \(V_{k}\) represents the volume of the borehole element, \(V_{R}\) represents the volume of the rock mass element, and \(E\) is the elastic modulus of the rock mass element. The element stress intensity can be calculated according to the principal stress of the rock mass element where the borehole is located:  
\[\sigma_{0}=\sqrt{\left[\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma_{ 2}-\sigma_{3}\right)^{2}+\left(\sigma_{3}-\sigma_{1}\right)^{2}\right]/2} \tag{15}\]  
Drilling increases the energy dissipated in rock mass cracking and expansion during unloading. Therefore, when using Equation (3) to determine the rockburst, the released energy caused by drilling holes should be added to \(W\)s. Then, the \(W_{sk}\) consumed by the rock mass element for cracking and expansion after drilling holes is  
\[W_{sk}=W_{s}+\sum_{i=1}^{n}\Delta W_{ki} \tag{16}\]  
where \(n\) represents n boreholes in the rock mass element. We can first calculate the load release generated by boreholes according to the size of the borehole element and then calculate the relief effect on the rock mass stress concentration according to the load release. Considering the drilling as a plane deformation problem, according to the circular pipeline stress and displacement elastic theory, under the action of stress \(\sigma_{0}\), the radial displacement of the borehole radius \(R\) is [34]  
\[u_{r}=\frac{r_{0}(1+\mu)}{E}\sigma_{0} \tag{17}\]  
where \(\mu\) is the Poisson's ratio of the rock mass. Considering the radial symmetry of the borehole element, the stiffness of the two-dimensional borehole element can be calculated according to Equation (6) which can be converted into three-dimensional stiffness according to two mutually perpendicular radial directions, as follows:  
\[[K_{Z3}]=[S]^{T}[K_{Z2}][S]= \begin{bmatrix}k_{rr}&0&k_{rz}\\ 0&k_{rr}&k_{rz}\\ k_{zr}&k_{zr}&k_{zz}\end{bmatrix}  \tag{18}\]  
where \([S]\) is the conversion matrix between two-dimensional and three-dimensional stiffness determined based on the directional cosines of the two-dimensional and the converted three-dimensional coordinates. Assuming that the axial displacement of the borehole element is not taken into account and \(u_{z}=0\), the release load of the borehole element can be obtained as  
\[\{f\}=[K_{Z3}][u]= \begin{bmatrix}k_{rr}&0&k_{rz}\\ 0&k_{rr}&k_{rz}\\ k_{zr}&k_{zr}&k_{zz}\end{bmatrix}\begin{bmatrix}u_{r}\\ u_{r}\\ u_{z}\end{bmatrix}  \tag{19}\]According to the location of the borehole element in the rock mass element, using the finite element interpolation principle, the load \(\{f\}\) released from the borehole element can be converted into the node load \(\{F\}\) of the rock mass element using the following equation:  
\[\{F\}=[N]^{T}\{f\} \tag{20}\]  
where \([N]\) is the node interpolation shape function of the borehole element. The node loads of rock mass elements generated by the borehole element are all reversely applied to the structure and are substituted into the overall equilibrium Equation (10). Then, a balance calculation is performed and we can calculate the relief \(\sigma\)sm of the stress concentration and the energy release \(\Delta\)_We_ caused by stress and strain changes.