# Effects of Vibration on Adjacent Pipelines under Blasting Excavation
## 3 Blast Vibration Attenuation Law  
A robust regression analysis method was used to reflect the propagation law of the tunnel blast vibration. First, the classical [PERSON]'s _PPV_ empirical formula is introduced as follows. The _PPV_ (\(v\)) value is expressed in the following form using mathematical analysis:  
\[v=\Phi(Q,\mu,c,\rho,\gamma,d,a,f,t) \tag{1}\]  
The right side of Equation (1) shows the main variables for attenuating blast vibration propagation on the ground (Table 3). \(Q\), \(r\), and \(c\) are independent, and the following equation can be drawn from [PERSON]'s pi theorem in dimensional analysis [22]:  
\[\left\{ \begin{array}{c}\pi_{0}=\frac{v}{Q^{4}\eta_{0}c^{3}\eta},\pi_{1}=\frac {\mu}{Q^{4}\eta_{1}c^{3}\tau^{4}},\pi_{2}=\frac{\rho}{Q^{4}\eta_{2}c^{3}\tau^{ 2}}\\ \pi_{3}=\frac{d}{Q^{4}\eta_{3}c^{3}\eta},\pi_{4}=\frac{d}{Q^{4}\eta_{4}c^{3}\tau ^{4}}\\ \pi_{5}=\frac{f}{Q^{4}\eta_{5}c^{3}\eta_{5}c^{3}},\pi_{6}=\frac{t}{Q^{4}\eta_{ 6}c^{3}\eta}\end{array} \right. \tag{2}\]  
Given the dependent variable dimensionless form of \(\pi_{n}\), where \(n=1,2,\ldots,6\), and where the indices, \(\alpha_{n}\), \(\beta_{n}\), and \(\gamma_{n}\), are also dimensionless. In addition, given the formula for \(\pi_{n}\):  
\[\left\{ \begin{array}{c}\pi_{0}=\frac{v}{c},\pi_{1}=\frac{\mu}{r},\pi_{2}= \frac{\rho}{Q^{r-3}}\\ \pi_{3}=\frac{d}{r},\pi_{4}=\frac{a}{r^{-1}c^{2}}\\ \pi_{5}=\frac{f}{r^{-1}c^{2}},\pi_{6}=\frac{t}{r^{-1}}\end{array} \right. \tag{3}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Points**} & \multirow{2}{*}{**Horizontal Distance**} & \multirow{2}{*}{**Distance from the Source (m)**} & \multicolumn{4}{c}{_PPV_ (cm/s)} \\ \cline{3-6}  & & & **X** & **Y** & **Z** & **Resultant** \\ \hline T1 & \(-\)39 & 45 & 0.62 & 0.73 & 0.54 & 1.10 \\ T2 & \(-\)28 & 36 & 0.69 & 0.79 & 0.57 & 1.20 \\ T3 & \(-\)25 & 33 & 0.73 & 0.84 & 0.62 & 1.27 \\ T4 & \(-\)17 & 28 & 0.77 & 0.86 & 0.68 & 1.33 \\ T5 & 16 & 27 & 0.79 & 0.90 & 0.65 & 1.36 \\ T6 & 20 & 30 & 0.71 & 0.83 & 0.59 & 1.24 \\ T7 & 28 & 36 & 0.66 & 0.77 & 0.52 & 1.15 \\ T8 & 37 & 43 & 0.59 & 0.71 & 0.46 & 1.03 \\ T9 & 51 & 56 & 0.42 & 0.54 & 0.40 & 0.79 \\ T10 & 58 & 62 & 0.39 & 0.50 & 0.36 & 0.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data monitored from the pipeline vibration levels survey. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **Symbol** & **Dimension** \\ \hline Displacement & \(\mu\) & L \\ Peak particle velocity & v & LT\({}^{-1}\) \\ Particle vibration acceleration & a & LT\({}^{-2}\) \\ Main frequency & f & T\({}^{-1}\) \\ Charges & Q & M \\ Distance from blast source & r & L \\ Depth of blast source & d & L \\ Density of surrounding rock & \(\rho\) & ML\({}^{-3}\) \\ Phase velocity & c & LT\({}^{-1}\) \\ Time & t & T \\ \hline \hline \end{tabular} Note: L is the dimension of length, M is the dimension of mass, and T is the dimension of time.  
\end{table} 
Table 3: Variables associated with surface _PPV_ affected by the blasting excavation of the water conveyance tunnel. In substituting Equation (3) into Equation (1), we can arrive at the following Equation:  
\[\frac{v}{c}=\Phi\left(\frac{\mu}{r},\frac{\rho}{Qr^{-3}},\frac{d}{r},\frac{a}{r^{ -1}c^{2}},\frac{f}{r^{-1}c},\frac{t}{rc^{-1}}\right) \tag{4}\]  
Next, combining the above given, \(\pi_{2}\), \(\pi_{3}\), and \(\pi_{4}\), a new dimensionless parameter \(\pi_{7}\) can be derived:  
\[\pi_{7}=\left(\pi_{2}^{\frac{1}{2}}\right)^{\beta 1}\pi_{3}^{\beta 2}=\left( \frac{\sqrt[3]{\rho r}}{\sqrt[3]{Q}}\right)^{\beta 1}\left(\frac{d}{r}\right)^{ \beta 2} \tag{5}\]  
where \(\beta_{1}\) and \(\beta_{2}\) are the correlation coefficients of \(\pi_{1}\) and \(\pi_{2}\).