# 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 Hydrostatic In Situ Stress
#### 3.3.2 Seismic Energy Attenuation with Distance  
By extracting the velocity histories at different distances, the seismic energy at various locations can be obtained. It is acknowledged that during seismic wave propagation, the relative reduction in seismic energy is proportional to the travel distance [55]. The seismic energy at a distance \(r\) from the elastic boundary is denoted as \(E_{\mathrm{s}}\). The seismic energy reduction over a travel distance \(\mathrm{d}r\) is represented as \(-\mathrm{d}E_{\mathrm{s}}\). Then,  
\[-\frac{\mathrm{d}E_{\mathrm{s}}}{E_{\mathrm{s}}}=\alpha\mathrm{d}r \tag{13}\]  
where \(\alpha\) is the attenuation index of seismic energy. A greater \(\alpha\) value means that the seismic energy decays faster with distance. Figure 6: Distributions of the fractured zones caused by blasting under varied hydrostatic in situ stress levels: (**a**) \(\sigma_{x}=\sigma_{y}=0\) MPa; (**b**) \(\sigma_{x}=\sigma_{y}=10\) MPa; (**c**) \(\sigma_{x}=\sigma_{y}=30\) MPa; (**d**) \(\sigma_{x}=\sigma_{y}=60\) MPa. Figure 7: Blasting seismic efficiency variation with in situ stress levels. The seismic energy variation with distance is obtained by the integration of Equation (13). According to the boundary condition \(E_{\rm s0}=\eta E_{\rm e}=\eta Q_{\rm e}W_{\rm e}\) at \(r=0\), the seismic energy \(E_{\rm s}\) is given by:  
\[E_{\rm s}=\eta Q_{\rm e}W_{\rm e}e^{-\alpha r} \tag{14}\]  
Through taking logarithms on both sides, Equation (14) becomes a linear function as follows:  
\[\ln E_{\rm s}=-\alpha r+\ln(\eta Q_{\rm e}W_{\rm e}) \tag{15}\]  
Figure 8 presents the seismic energy versus distance on a logarithmic plot. The attenuation index \(\alpha\) of seismic energy can be acquired through linear regression of the scattered data pairs. In the linear regression, the slope of the best fit straight line represents the attenuation index \(\alpha\). In this figure, the symbol \(R^{2}\) denotes the squared correlation coefficient between the scattered data pairs and the best fit straight line. The squared correlation coefficients are all greater than 0.95. This indicates that the seismic energy attenuation with distance does follow the exponential function described in Equation (14). Figure 8 only shows the regression analysis for the in situ stress conditions \(\sigma_{x}=\sigma_{y}=10\), 30, and 60 MPa. The attenuation indexes obtained by the regression analysis for the other in situ stress conditions are summarized in Table 2. As the hydrostatic in situ stress level increases from 0 to 30 MPa, the attenuation index of seismic energy decreases from 0.0940 to 0.0443. The blasting seismic energy attenuation with distance becomes slower with increasing in situ stress levels. However, when the stress continues to increase from 30 to 60 MPa, the attenuation index turns to increase instead from 0.0443 to 0.0648. The attenuation of seismic energy accelerates after the in situ stress reaches 30 MPa. Anyway, the attenuation is still slower than that of the case without in situ stress. Overall, the attenuation index of seismic energy first decreases and then increases with the increase in the hydrostatic in situ stress. Notably, since the seismic energy attenuation with distance is described by an exponential function, a small value is assigned to the attenuation index. Nevertheless, from the relative change of the attenuation index, the variation of the hydrostatic in situ stress level causes a significant change in the seismic energy attenuation rate. From the analysis in Section 2.2, in situ stress mainly affects the viscoelastic attenuation of seismic waves. The viscoelastic attenuation is specified by the seismic \(Q\). The inverse of the seismic \(Q\) is defined as [43]:  
\[Q^{-1}=\frac{1}{2\pi}\frac{\Delta E}{E_{\rm max}} \tag{16}\]  
where \(\Delta E\) is the amount of energy dissipated per cycle of a harmonic excitation in a certain volume, and \(E_{\rm max}\) is the peak elastic energy in the same volume.