# Energy Generation and Attenuation of Blast-Induced Seismic Waves under In Situ Stress Conditions
## 3 Numerical Study of Blasting Seismic Energy Generation and Attenuation under In Situ Stress
### Numerical Model and Simulation Method
#### 3.2.4 Numerical Model and Boundary Conditions  
The explicit dynamic analysis software AUTODYN is employed to simulate the multiple-hole blasting of the circular tunnel excavation under varied in situ stress conditions. In real tunnel blasting, cylindrical blastholes are commonly used, and the explosives placed at different depths are detonated at different times because of the limited detonation velocity. Therefore, real tunnel blasting is a complicated three-dimensional (3D) problem. In order to study the explosion seismic wave energy attenuation with distance, the numerical model needs to extend to a region with a radius of at least 50 m. Furthermore, to ensure simulation accuracy, the element size should not exceed 1/12\(-\)1/6 of the shortest wavelength [8]. Under these requirements, the number of elements will exceed one billion if a fully 3D model is used. The computation at this scale is too time-consuming, especially in implementing stress initialization. Therefore, in this numerical study, the explosives at different depths are assumed to be fired simultaneously, and the blashhole pressure is considered to be consistent along the blashhole axis. Under these assumptions, the multiple-hole blasting can be treated as a plane strain issue.  
A plane model with a width of 100 m and a height of 100 m is developed for the multiple-hole blasting of the circular tunnel excavation, as shown in Figure 5. The twenty stopping holes in the fourth round are located 3.2 m away from the tunnel center. The free surface created by the blasting of the blastholes in the previous rounds has a radius of 2.2 m. In most cases, the blast-induced fractured zone extends to a range within 100 of the blasthole radius. Therefore, the explosive and the rock within 5.2 m from the tunnel center are discretized into SPH particles. In order to improve computational efficiency, the rock in the other region is discretized into FEM elements. The size of the FEM elements near the blashholes is set to 0.005 m to satisfy the relative relationship between the element size and the shortest wavelength. As the seismic waves spread outward, the wavelength gradually increases. Accordingly, the FEM mesh size gradually increases from 0.005 m near the blashholes to 0.5 m at the outer boundaries.  
During the modeling, the horizontal stress \(\sigma_{x}\) and the vertical stress \(\sigma_{y}\) are first applied to the external boundaries of the numerical model to perform stress initialization. When the static stress reaches a stable state, the JWL EOS is then activated to simulate the explosion loading. To investigate the effects of the in situ stress level and anisotropy on the blasting seismic energy, hydrostatic (\(\sigma_{x}=\sigma_{y}\)) and non-hydrostatic (\(\sigma_{x}\
eq\sigma_{y}\)) in situ stress conditions are considered in the numerical study. Under hydrostatic stress conditions, the stress levels varying from 0 to 60 MPa are considered. Under non-hydrostatic stress conditions, the vertical stress \(\sigma_{y}\) keeps constant at 10 MPa, and the lateral coefficient \(\sigma_{x}/\sigma_{y}\) is set to 2 and 3, respectively.  
Figure 4: Numerical modeling of rock fracture caused by single-hole blasting: (a) without initial stress; (b) with a uniaxial stress of 5 MPa.