# 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
### Seismic Energy Generation and Attenuation under Non-Hydrostatic In Situ Stress
#### 3.4.2 Seismic Energy Density Attenuation with Distance  
A series of observation points along the principal stress directions are chosen to investigate the seismic energy density attenuation with distance along different directions. It is assumed that the seismic energy density also follows the attenuation law of an exponential function as the seismic energy. Figure 10 gives the seismic energy density versus distance on a logarithmic plot for the observation points in different directions. The best fit straight line for solving the attenuation index of seismic energy density is also plotted in this figure. At \(\sigma_{x}=2\sigma_{y}=20\) MPa, the attenuation indexes of the seismic energy densities in the maximum and minimum principal stress directions are 0.2058 and 0.1811, respectively. At \(\sigma_{x}=3\sigma_{y}=30\) MPa, the attenuation indexes are 0.2523 and 0.1846, respectively. Under non-hydrostatic in situ stress conditions, the seismic energy density decays at different rates in each direction. The seismic energy density attenuation with distance exhibits anisotropy. The relative magnitude of the attenuation rates in different directions depends on the state of rock defects under in situ stress. If the initial defects are closed under in situ stress, the attenuation index becomes smaller compared to the case in the absence of in situ stress. If the initial defects are expanded, or new cracks are induced due to higher in situ stress, which means stress-induced rock mass damage, the attenuation index will increase. In this numerical example, the rock mass nearby the free surface in the maximum principal stress direction is damaged due to stress redistribution and concentration. Therefore, the seismic \(Q\) of the rock mass in the maximum principal stress direction decreases and the seismic energy density in this direction decays faster with increasing distance.  
Figure 10: Regression analysis of seismic energy density attenuation with distance under non-hydrostatic in situ stress conditions: (**a**) \(\sigma_{x}=2\sigma_{y}=20\) MPa; (**b**) \(\sigma_{x}=3\sigma_{y}=30\) MPa.  
Figure 9: Distribution of the fractured zone caused by blasting under a non-hydrostatic in situ stress condition \(\sigma_{x}=3\sigma_{y}=30\) MPa.  
When the lateral coefficient \(\sigma_{x}/\sigma_{y}\) is equal to 2, the attenuation index of the seismic energy density differs by 13.6% between the maximum and minimum principal stress directions. For \(\sigma_{x}/\sigma_{y}=3\), the difference reaches 36.7%. With the increase in the lateral coefficient, the anisotropy of the seismic energy density attenuation becomes more pronounced. In the numerical modeling, the minimum principal stress \(\sigma_{y}\) keeps constant at 10 MPa, and the maximum principal stress \(\sigma_{x}\) increases from 20 to 30 MPa. It is seen that as \(\sigma_{x}\) changes, the attenuation index in the minimum principal stress direction remains almost constant, while the attenuation index in the maximum principal stress varies significantly. It means that the stress along the seismic wave propagation direction plays a major role in influencing seismic energy attenuation.