In the deeply buried small clear distance tunnel projects, the distance between the excavated tunnel and the existing tunnel will be less than the buried depth of the tunnel. The length of the resistance line is the distance from the center of the blasting charge to the outer surface of the side wall lining of the existing tunnel. (In practical deeply buried small clearance tunneling projects, where the distance between the excavated tunnel and the existing tunnel is less than the tunnel depth, the length of the resistance line usually corresponds to the distance from the package to the lining of the existing tunnel sidewall.) To prevent damage to the existing tunnel lining caused by blasting vibrations during tunnel excavation, it is essential that the influence of the resistance line is considered in the actual predictions. The physical quantities associated with tunnel blasting and excavation are presented below. And through theoretical analysis of the magnitudes, a formula is derived for calculating the vibration velocity of tunnel blasting excavation, considering the effects of free face and resistance lines (Yang et al., 1994; Song et al., 2013).

In tunnel excavation blasting vibration velocity analysis, there are two main variables: one is the dependent variable, that is, the variable brought about by the different blasting design, mainly including the peak vibration velocity, single section blasting charge, and blast source distance, and the second is the independent variable, which includes the characteristics of the rock itself, rock density, and wave velocity. The main variables involved in the calculation of the model for studying the decay of vibration velocity in deeply buried small clearance tunnel blasting excavation are summarized in Table 1.

The above influencing factors were selected as the main physical quantities affecting the peak velocity of blast vibration to be studied. It can be seen that the seven physical quantities affecting the propagation law of blast vibration velocity are all related to time T , length L , and mass M . Therefore, the expression of the basic gauge system is the following equation: [ X ] = L α M β T y . Here, [ X ] represents the dimension of the variable X ; L , M , and T represent the dimension of length, mass, and time, respectively; and α , β , and γ are real-type indices.

The scale matrix θ , based on the scale of each physical quantity listed in Table 1, is given by the following equation: [ V   Q   D   ρ c c w d b A ] ↓ θ = [ 1 0 1 0 1 0 − 1 0 0 − 3 1 0 1 0 0 0 − 1 0 2 0 0 ] → [ L M T ] .

In order to determine the rank of the dimensional matrix θ , the maximum charge of one section D , distance to the explosive’s center D , and rock wave velocity c were selected as independent dimensions, and the following determinant was formed from them: Δ = | 0 1 1 1 0 0 0 0 − 1 | = 1 .

Calculate the determinant △≠0, so the rank of the dimensional matrix r ( θ ) = 3 , which indicates that the dimensional matrix is a non-linear correlation matrix. The number of dimensionless matrices is k = n − r ( θ ) = 7 − 3 = 4 , and the expression of each dimensionless matrix is as follows: { π 1 = [ M ] α 1 [ L ] β 1 [ L T − 1 ] y 1 [ L T − 1 ] , π 2 = [ M ] α 2 [ L ] β 2 [ L T − 1 ] y 2 [ M L − 3 ] , π 3 = [ M ] α 3 [ L ] β 3 [ L T − 1 ] y 3 [ L 2 ] ,                                                                                   π 4 = w d b .

It is known that, in the above equation, π 1 ∼ π 4 are all dimensionless. And it follows from the volley principle of dimension that the power of any parameter in the fundamental system of magnitudes should be 0 (Li, 1992). Therefore, the parameters M , L , and T of π 1 in the above equation have the following relationship: { α 1 = 0 , β 1 + γ 1 + 1 = 0 − γ 1 − 1 = 0. ,

The above equation corresponds to the sum of powers of M , L , and T , respectively, to 0. Solving the above equation gives α 1 = 0 , β 1 = 0 , and γ 1 = − 1 . Similarly, the values of α 2   , β 2 , γ 2 , α 3 , β 3    , and γ 3 are − 1,3,0   and 0 , − 2,0   for π 2 and π 3 , respectively. By bringing the calculated indices back to the expressions π 1 ∼ π 4 , the dimensionless expressions can also be expressed as follows: { π 1 = V c , π 2 = ρ c Q D − 3 , π 3 = A D 2 , π 4 = w d b .

According to the theory of dimensional analysis (π theorem), the physical relations can be transformed into dimensionless form and the number of independent variables in the function is reduced. So, the relationship between the vibration velocity and the physical quantities in the tunnel when blasting excavation is carried out can be expressed as follows: V c = Φ ( ρ c Q D − 3 , A D 2 , w d b ) .

In practical engineering, the density of the rock mass ρ c and the propagation velocity of the rock mass for vibration waves c can be viewed as constants (Cai, 2013). Therefore, the following expression is derived for the function between the vibration velocity of tunnel blasting excavation and other physical quantities: In V= α 0 + α 1 In 1 Q D -3 + α 2 In A D 2 + α 2 In w d b .

Simplifying the equation above, we get V = β 0 ( 1 Q D − 3 ) β 1 ( A D 3 ) β 2 ( w d b ) β 3 .

Observe that the first term of the above equation is found to be consistent with the proportional charge quantity of Sadowski formula ρ , so collating the above equation leads to the following equation: V = γ 0 ( Q D 3 ) γ 1 ( A D 2 ) γ 2 ( w d b ) γ 3 .

The corresponding relationships between the coefficients in the above equations are as follows: { β 0 = γ 0 = e α 0 α 1 = β 1 = − 1 / 3 γ 1 α 2 = β 2 = γ 2 α 3 = β 3 = γ 3

Analyzing the final expression, the first term on the right-hand side of the equation, γ 0 ( Q 3 D ) γ 1 , is identical to the traditional Sadowski formula and shows the relationship between the blast vibration velocity and the proportional charge; the second term, ( A D 2 ) γ 2 , integrates the effect of the area of the free face of the blasting excavation on the blast vibration intensity; the last term, ( w d b ) γ 3 , considers the effect of the length of the minimal resistance line on the blast intensity and is more applicable to the prediction of blasting vibration velocity for deeply buried small clearance tunnel projects.

A sample of actual blasting data from Badaling Great Wall Station was used for analysis. The station has many levels, many cavities, and complex cavity types, making it the most complex concealed cavity group station in China. There are multiple spatial relationships in the construction of sections and areas affected by close blasting construction.

Mountain tunnels with a burial depth H exceeding 50 m (conservatively estimated at 100 m) can be classified as deeply buried tunnels (Rao and Wang, 2013). In contrast, a small clear distance tunnel is a special form of tunnel arrangement where the thickness of the intermediate rock column between the tunnels is less than the recommended value in Table 2 (Specification for Design of Highway Route JTG D70). The Badaling tunnel project is a mountainous tunnel with a burial depth of over 100 m and a maximum tunnel spacing of 6 m<2.0 × 7 m (excavated section of Grade III surrounding rock), so it is a deeply buried small clearance tunnel project.

The three separate cavern sections are part of a three-line parallel tunnel group, with the right line 2 months behind the mainline and the mainline 3 months behind the left line. Their actual excavation dimensions and chamber spacing are shown in Figure 1. As the net distance between the rock columns of the three parallel tunnels is only 2.27–6 m, blasting in the mainline will cause vibration effects or even damage to the left and right line tunnel structures on both sides where the initial support or secondary lining has been completed. Blasting vibration control is required to ensure that the support effect of the existing support is not affected.

The section of the Great Wall Station at Badaling is 398 m in length, and the surrounding rock level is mainly Grade III. In the actual blasting construction, the section of Grade III surrounding rock is constructed by the step method, with a step length of 5∼8 m, in the order of up- and down-blasting construction. The holes were drilled by hand-held wind drill with a diameter of 42 mm, and the blasting explosive was Φ32 mm rock emulsion explosive. The layered explosive loading structure and uncoupled explosive loading structure are adopted.

Physical measures of blast vibration intensity include the mass vibration velocity, displacement, acceleration, and energy ratio. Following the Safety Regulations for Blasting, this paper uses mass point vibration velocity as the basis for measurement. TC-4850 blasting vibration meter is shown in Figure 2. By importing the vibration data into the software and analyzing the analysis, the vibration velocity of the mass at each monitoring point can be directly obtained as X (horizontal radial), Y (horizontal tangential), and Z (vertical ground direction), and the combined velocity can be analyzed by vector operation of the three velocity values.

The vibration test was used to monitor the changes in blast vibration velocity and acceleration at the foot and waist of the left-hand side of the excavated chamber and at various points in the same plane as the palm face during the blasting construction of the cavern. The actual measurement points are shown in Figure 3 and Figure 4, with a total of seven measurement points, of which measurement point 2 is facing the blast charging section of the tunnel palm face; measurement point 1 is placed in the excavated section and measurement point 2 is 5 m away; measurement points 3, 4, and 5 are placed in the unexplored area and the distance from measurement point 2 is 5, 13, and 23 m; and measurement points 6 and 7 are in the same plane as measurement point 2 but 0.8 m lower than measurement point 2. The blast vibration velocities at different locations were analyzed by monitoring the vibration velocities of one line and the same plane of measurement points.

The measured vibration sensors were recorded in all three directions at the same time, and the measured vibration data are listed in Table 3.