Study on the Propagation Law and Waveform Characteristics of a Blasting Shock Wave in a Highway Tunnel with the Bench Method

Abstract

In the bench method of tunnel excavation, the blasting impact from upper bench blasting poses significant risks to personnel and equipment. This study employed dynamic analysis software, ANSYS/LS-DYNA, and field testing to examine the propagation characteristics and attenuation behavior of tunnel shock waves. The findings revealed that, near the central axis of the tunnel, shock wave overpressure was lower compared to areas near the tunnel wall due to reflections from the wall. As the shock wave traveled a distance six times the tunnel diameter, it transitioned from a spherical wave to a plane wave. The attenuation coefficient for the plane wave ranged from 1.03 to 1.17. A fitting formula for shock wave overpressure attenuation, based on field test results, was proposed, and it showed good agreement with the numerical simulation results. This provided valuable theoretical insights for predicting shock wave overpressure during bench method tunnel excavation.

1. Introduction and Background

Blasting is a cost-effective and efficient method widely utilized in tunnel construction, hydraulic engineering, mining, and various other civil engineering projects [1,2,3,4]. Approximately 70% to 80% of the energy released during a blast is dissipated in the form of vibration and air blasts, with only a small amount of the energy utilized for rock breakage [5]. Simultaneously, while crushing rock, the blast strongly compresses the air near the tunnel face, leading to an instantaneous rise in temperature and air pressure and forming a blasting shock wave. When this shock wave propagates in a high-pressure and high-speed state along the tunnel, it can cause serious harm to both field equipment and personnel. Hence, studying the laws of shock wave propagation from tunnel blasting is crucial.

Numerous research projects have focused on analyzing air blast waves (or shock waves) induced by the explosive charges. Via experimental methods, Brode [6] and Henrych [7] determined that the primary factors affecting the peak overpressure of an air shock wave are the quantity of explosive material used and the distance from the point of detonation. The overpressure formula of a free field explosion was proposed by Brode and Henrych. Baker et al. [8] provided an investigation table of shock wave parameters, including the overpressure peak, propagation distance, and propagation time. In addition, numerical simulation methods to study fluid motion have great advantages [9,10]. For example, Chapman et al. [11] used numerical methods to simulate the transmission of a shock wave through an air medium. Zukas et al. [12] and Luccioni et al. [13] investigated the effect of the grid size on blasting results in computational fluid dynamics (CFD) and concluded that a grid size of 100 mm is appropriate for accurately simulating shock wave propagation.

The blasting shock wave is constrained in narrow spaces, such as tunnels, resulting in slower attenuation and farther blasting in open spaces [14,15,16]. Smith et al. [17] designed several equally scaled tunnel models with different section shapes, and tested the overpressure values inside the tunnels through a series of blasting experiments. Rodriguez et al. [18] proposed a semi-empirical approach to predict the air pressure at the tunnel exit during propagation of the blast shock wave. Benselama et al. [19] classified the propagation of shock waves in tunnels into two distinct modes: a three-dimensional overpressure attenuation mode near the explosion point and a one-dimensional overpressure attenuation mode at greater distances from the explosion. Uystepruyst et al. [20] observed that when an explosion occurs in a rectangular tunnel, the shock wave propagation follows a two-dimensional pattern. By means of numerical simulation and model tests, Pennetier et al. [21] proved that shock waves in subway stations can propagate multiple times through reflection and refraction in connecting channels. Wu et al. [22] investigated wave propagation characteristics in a confined environment, providing a comprehensive description of wave dynamics during the blasting process and valuable insights into wave propagation. Figuli et al. [23] examined the attenuation behavior of shock waves across different explosion modes. Fang et al. [24] conducted field tests to study the propagation behavior of shock waves in a highway tunnel and discovered that the attenuation patterns varied across different regions. Many researchers have explored the propagation characteristics of shock waves resulting from explosions in air, military tunnels, and coal mine roadways, and have identified the attenuation laws for these shock waves. However, there is a lack of research specifically on the propagation characteristics of shock waves following blasting in highway tunnels. Due to the larger excavation section of highway tunnels compared to military and coal mine tunnels, a greater quantity of explosives is required for blasting operations, leading to a more pronounced dynamic impact from shock waves [25,26,27]. In particular, despite the widespread use of the bench method for tunnel excavation in complex geological conditions, research on blasting shock waves associated with this technique is limited.

In this study, the numerical simulation method is utilized to perform a dynamic analysis of the tunnel’s blasting scheme, simulating the propagation process of a shock wave generated by upper bench blasting. The study investigates the fundamental characteristics of shock wave propagation on the tunnel’s upper bench, focusing on changes in the shock wave flow field, cross-sectional overpressure distribution, and longitudinal overpressure attenuation. Based on field tests, the empirical formula of a blasting shock wave in a single tunnel is modified to make it more suitable for similar engineering environments.

2. Engineering Background

Dual-lane highways predominate in China, and there is widespread use of the bench blasting method for excavating tunnels in Grade IV and V rock masses. The Shengli tunnel serves as a crucial component of the Jinshajiang expressway in Southwest China, connecting Ningnan City to Panzhihua City. The tunnel has a semicircular arch cross-section of 12 m × 8 m (width × height), with a tunnel length of 1.7 km and a maximum buried depth of 178 m. The surrounding rock has low strength and fractures are developed, and 78.4% of the total length of the tunnel has been classified as Grade IV.

The tunnel was excavated with the bench blasting excavation scheme. The upper bench had an excavation height of 6.5 m and a length ranging from 130 to 150 m. The lower bench had an excavation height of 2.5 m and a length of 20 m. The excavation area of the upper bench was 2.2 times greater than that of the lower bench, and the charge quantity for the upper bench was significantly larger. Consequently, this paper primarily focused on the propagation process and overpressure attenuation law of shock waves generated by blasting in the upper bench.

Figure 1 displays the blasting schemes utilized for bench excavation in the tunnel. In order to ensure the blasting effect, an emulsion explosive and a segmented electronic millisecond detonator were adopted in the blasting. The diameter of the contour holes was 42 mm, the length of the cartridge was 200 mm, and the single-quantity explosive was 200 g. The diameter of the remaining blasting holes was 42 mm, the length of the cartridge was 300 mm, and the single-quantity explosive was 300 g. The blasting parameters are shown in

Table 1. Blasting parameters.

Hole Classification	Segment/#	Hole Number	Hole Depth/m	Single-Hole Cartridge Number	Single-Hole Charge/kg	Delay Time/(ms)	Segment Charge/kg
Cut hole	1	12	4.05	8	2.4	0	28.8
Stope hole	3	12	3.6	7	2.1	50	25.2
Stope hole	5	10	3.5	6/7	1.8/2.1	110	19.8
Stope hole	7	4	3.5	5	1.5	200	6
Stope hole	9	19	3.5	4	1.2	310	22.8
Stope hole	11	25	3.5	3/4	0.9/1.2	460	26.1
Bottom hole	13	14	3.7	5	1.5	650	21
Contour hole	15	45	3.5	3/4	0.6/0.8	880	31.4
Total		141					181.1

.

3. Establishment of Numerical Calculation Model

Numerical simulations are highly effective for analyzing changes in airflow patterns and shock wave propagation. The explicit dynamic analysis finite element software, ANSYS/LS-DYNA, provides high precision and reliability in simulating the explosion process. Establishing a three-dimensional model enhances the ability to observe variations in the waveform and propagation characteristics of the shock wave. The ALE algorithm can effectively solve the deformation problem of the large mesh and is widely used in the study of fluid–solid coupling problems. The explosion involves two materials, namely, explosives and air, and the shock wave causes the mesh elements to undergo huge deformation. Therefore, using the multi-material ALE algorithm for simulation is suitable for explosives and air [28,29].

Assuming a constant curvature along the tunnel axis, a long, straight tunnel model was established due to the tunnel’s axisymmetric shape. To simplify the calculations, a 1/2 tunnel model was created along the tunnel’s centerline. Fixed boundary conditions were applied at the tunnel face and at the contact boundary between the air and the tunnel wall. The tunnel’s end was assigned a transmission boundary condition to permit the passage of shock wave energy. Therefore, only the explosive and air were considered in the model, where the explosive was simulated by the 3DSOLID164 solid element and air was simulated by Eulerian meshes. The model utilized hexahedral solid elements for meshing, with a maximum grid size of 0.45 m and a minimum grid element size of 0.1 m. The entire calculation model operates using the cm-g-μs unit system. As shown in Figure 2, the geometric model and mesh of the tunnel were mirrored. The tunnel model had a width of 12 m, a height of 6.5 m, and a length of 150 m. The cross-sectional area was 62.54 m², and the equivalent diameter was 7.87 m.

The maximum overpressure peak of the shock wave was influenced by the TNT equivalent of the explosive used in the cut hole [30]. Therefore, the parameters of the emulsion explosive need to be reduced by the TNT equivalent coefficient and the shock wave energy conversion coefficient. When the peak overpressure of the shock wave generated by the explosive and TNT was the same, the ratio of the explosive charge mass to TNT mass was referred to as the equivalent coefficient, which had a value of 0.582. The shock wave energy conversion coefficient was 0.4 [18,21]. The 28.8 kg of emulsion explosive used in the upper bench cutting section of the tunnel was equivalent in energy to the blasting of 6.5 kg of TNT explosive in the open air of the tunnel.

The TNT explosive was set at 3 m above the bottom of the upper bench as the detonation point. Blasting holes were symmetrically arranged according to the tunnel centerline, so a semi-tunnel model was established, as shown in Figure 3. The explosive size was 20 cm × 10 cm × 10 cm (length × width × height), which formed a rectangular charge shape. The size of the TNT explosive was very small for the tunnel, which can be regarded as an ideal point explosion source, and the detonation was completed instantly to reach the detonation pressure.

The material parameters for the explosive were defined using the *MAT_HIGGINS_EXPLOSIVE_BURN keyword in LS-DYNA. As detailed in

Table 2. Explosive material parameters.

Parameters	ρ/(kg/m3)	D/(m/s)	PCJ/(MPa)
Value	1.63 × 103	6.93 × 103	2.55 × 104

, ρ represents the density of the explosive, D denotes the detonation velocity, and PCJ indicates the detonation pressure of the explosive.

The equation of state for the explosive is given by *EOS_JWL, as shown in Equation (1). The state equation parameters of the explosive are listed in

Table 3. State equation parameters of the explosive.

Parameters	${\mathit{\rho }}_0$/(kg/m3)	C0	C1	C2	C3	C4	C5	C6	E/(J/m3)
Value	1.29	0	0	0	0	0.4	0.4	0	2.5 × 105

. Here, A, B, R₁, and R₂ are the parameters to be determined, E represents the internal energy per unit volume of the explosive, and V denotes the relative volume at the initial moment of the explosion:

$$P=A\left(1-{\displaystyle \frac{\omega }{R_{1}V}}\right)e^{-R_{1}V}+B\left(1-{\displaystyle \frac{\omega }{R_{2}V}}\right)e^{-R_{2}V}+{\displaystyle \frac{\omega E}{V}}$$

(1)

The material parameters for air were defined using the LS-DYNA keyword *MAT_NULL, and the state equation was specified by *EOS_LINEAR_POLYNOMIAL. The state equation followed a linear polynomial form, given as:

$$P=C_0+C_1\mu +C_2{\mu }^2+C_3{\mu }^3+\left(C_4+C_5\mu +C_6{\mu }^2\right)E$$

(2)

where $\mu ={\displaystyle \frac{\rho }{{\rho }_0}}-1$, $\rho$ is the current density of air, ${\rho }_0$ is the initial density of air, C₀, C₁, C₂, C₃, C₄, C₅, and C₆ are constants, and E is the internal energy per unit volume of air. The selection of parameters is shown in

Table 4. State equation parameters of air.

Parameters	${\mathit{\rho }}_0$/(kg/m3)	C0	C1	C2	C3	C4	C5	C6	E/(J/m3)
Value	1.29	0	0	0	0	0.4	0.4	0	2.5 × 105

.

4. Calculation Results Analysis

4.1. Measuring Point Arrangement

In the calculation model, 15 measuring points were positioned along the longitudinal direction of the upper bench tunnel model to analyze the attenuation of overpressure from the blasting shock wave along the tunnel’s axis. Additionally, three measuring points were placed radially at four cross-sections located 30 m, 40 m, 50 m, and 60 m from the explosion center to observe the formation of the plane wave. In total, 27 measuring points were arranged, as illustrated in Figure 4.

4.2. Shock Wave Flow Field Changes

The shock wave pressure cloud diagram following the explosion of the upper bench explosive is depicted in Figure 5. As illustrated in Figure 5a, the shock wave initially propagated as a spherical wave after the explosion. As the shock wave encountered the tunnel wall, it was constrained by the tunnel’s enclosed structure, and multiple reflections occurred with the wall, leading to a reflected overpressure significantly greater than the incident overpressure, as shown in Figure 5b. Figure 5c demonstrates that the collision of the shock wave with the tunnel wall created a reflected wave and compressive wave superimposition, resulting in the Mach effect. Subsequently, a stable plane wave formed at the front of the shock wave and continued to propagate forward, while regular oblique and Mach reflections persisted at the tail of the shock wave.

4.3. Cross-Section Overpressure Analysis

The equivalent diameter of the upper bench in the tunnel was 7.87 m. The blasting shock wave developed into a plane shock wave at a distance of approximately 4 to 8 times the equivalent diameter from the explosion center, as illustrated in Figure 5. To pinpoint the exact location where the plane shock wave formed, four sections spaced 10 m apart were selected, starting from 30 m away from the explosion center. The following formula was applied:

$$\left\{\begin{array}{l}{{\displaystyle P}}_{\max }/{{\displaystyle P}}_{\min }\le {{\displaystyle \vartheta }}_P\\ {{\displaystyle I}}_{\max }/{{\displaystyle I}}_{\min }\le {{\displaystyle \vartheta }}_I\end{array}\right.$$

(3)

where P_max and P_min are the maximum and minimum overpressure values of a section, respectively. I_max and I_min are the maximum and minimum impulses of a section. When ${\vartheta }_P=1.5$, the spherical wave was restored to a plane wave, and when ${\vartheta }_I=1.05$, a plane wave was completely formed.

The overpressure and impulse values at various measuring points on the cross-section, located at different distances from the explosion center, are presented in

Table 5. Overpressure ratio of the radial measuring point in the cross-section.

Distance/(m)	Point a/(kPa)	Point b/(kPa)	Point c/(kPa)	Point d/(kPa)	${\mathit{P}}_{\mathbf{max}}/{\mathit{P}}_{\mathbf{min}}$
30	49.26	49.24	55.97	76.96	1.563
40	44.32	44.79	51.02	63.31	1.428
50	37.73	37.79	40.05	52.48	1.391
60	35.71	35.68	36.53	49.70	1.393

and

Table 6. Impulse ratio of the radial measuring point in the cross-section.

Distance/(m)	$\mathbf{Point} \mathbf{a}/\left(\mathbf{k}\mathbf{g}\cdot \mathbf{m}/\mathbf{s}\right)$	$\mathbf{Point} \mathbf{b}/\left(\mathbf{k}\mathbf{g}\cdot \mathbf{m}/\mathbf{s}\right)$	$\mathbf{Point} \mathbf{c}/\left(\mathbf{k}\mathbf{g}\cdot \mathbf{m}/\mathbf{s}\right)$	$\mathbf{Point} \mathbf{d}/\left(\mathbf{k}\mathbf{g}\cdot \mathbf{m}/\mathbf{s}\right)$	${\mathit{I}}_{\mathbf{max}}/{\mathit{I}}_{\mathbf{min}}$
30	4080.70	4075.15	4072.64	5510.93	1.353
40	4054.98	4036.54	4041.83	4643.83	1.150
50	4008.58	4005.20	4006.37	4137.37	1.033
60	3955.87	3954.16	3953.47	4008.819	1.014

. These tables allowed for the determination of overpressure and impulse ratios at various distances from the explosive, as shown in Figure 6. Further analysis revealed that the ratio of maximum to minimum overpressure of the shock wave was less than 1.5 at a distance of 40 m from the tunnel face. Similarly, the ratio of maximum to minimum impulse was less than 1.05 at distances ranging from 40 m to 50 m from the tunnel face. Beyond 48.5 m from the explosion center, the ratio of maximum to minimum impulse was also less than 1.05, as determined by linear interpolation. Thus, the plane wave formed by the blasting shock wave on the upper bench of the tunnel occurred at a distance of 48.5 m, approximately 6 times the equivalent diameter of the tunnel.

4.4. Analysis of Shock Wave Overpressure Attenuation

As shown in Figure 7, the shock wave exhibited multiple peaks during its propagation, followed by pronounced serrated attenuation and noticeable oscillations before eventually returning to atmospheric pressure. The peak overpressure was higher near the explosion center, resulting in more rapid attenuation. As the shock wave propagated further, the rate of attenuation decreased, the time required for reduction increased, and the duration of the overpressure also increased.

Following the explosion, the overpressure peak at 5 m rapidly reached 90.3 kPa, before quickly attenuating. Due to shock wave reflections, a smaller overpressure peak of 9.3 kPa was observed at 55 ms. At 10 m, two closely spaced overpressure peaks of 75.9 kPa and 76.5 kPa were recorded. The first peak represented the initial shock wave front pressure immediately after the explosion, while the second peak resulted from the superposition of air compression waves generated by multiple reflections of the shock wave from the tunnel wall. At 20 m, the peak overpressure was slightly higher than the maximum overpressure observed at 10 m. This increase was attributed to the higher pressure and propagation speed of the reflected wave compared to the incident shock wave. Both regular oblique and Mach reflections occurred, intensifying the shock wave at this distance.

The shock wave overpressure generated at the tunnel face decreased rapidly near the explosion center. As the shock wave propagated farther, the rate of attenuation of the overpressure peak changed. To quantify this attenuation rate, the attenuation coefficient of the shock wave overpressure is defined by Equation (4):

$$\delta ={\displaystyle \frac{\Delta P}{\Delta P_1}}$$

(4)

where ΔP is the shock wave overpressure peak at the previous position, and ΔP₁ is the shock wave overpressure peak at the latter position.

As shown in Figure 8, the shock wave initially underwent rapid attenuation. Within 0 m to 20 m from the explosion center, the reflected shock wave compressed the air again and quickly caught up with the wave front of the initial shock wave. This superposition, which was due to multiple reflections of the shock wave off the tunnel walls and floor, resulted in an increase in the shock wave overpressure peak. After this phase, at 30 m from the detonation center, the shock wave overpressure decreased dramatically, and the attenuation coefficient rose sharply to 1.56. Beyond 30 m, the attenuation of the shock wave overpressure peak stabilized, with the attenuation coefficient ranging from 1.17 to 1.03.

5. Blasting Shock Wave Field Test and Overpressure Prediction

5.1. Test Instruments

The setup for the field test of shock waves in the upper bench of the Shengli highway tunnel is shown in Figure 9. A shock wave overpressure sensor (PCB 113B28 SN, produced by PCB Piezotronics Inc., New York, NY, USA) and a shock wave acquisition instrument (Blast-PRO, produced by Chengdu Taice Technology Co., LTD, Chengdu, China) were used to test the blasting shock wave.

Table 7. Test instrument parameters.

Test instrument	Parameter	Technical Index
Shock wave overpressure sensor	Sensitivity	14.5 mV/kPa
	Resolution	0.07 kPa
	Measuring range	344.7 kPa output in $\pm 5 \mathrm{V}$
		689.4 kPa output in $\pm 10 \mathrm{V}$
Shock wave test instrument	Channel number	2
	Sampling rates	500 k~4 MHz
	A/D precision	24-bit
	Bandwidth	>700 Hz

shows the parameters of the test instruments.

5.2. Analysis Results

We conducted 21 tests on the upper bench and obtained 15 sets of valid data. Before the blast, a tripod was setup at the measurement point, and a sensor was fixed above the tripod, parallel to the ground. The measurement system was checked to ensure it was in normal working condition. When the test system was started, the shock wave data collection began, with measurement data recorded during the tunnel face blasting.

The test results are shown in

Table 8. Blasting shock wave test results.

Number	Q (kg)	R (m)	S (m2)	V (m3)	d (m)	ΔP (kPa)
1	174	74	62.8	4647.2	7.87
2	180	74.5	62.8	4678.6	7.87	39.17
3	174	76	62.8	4772.8	7.87	36.49
4	180	80	62.8	5024	7.87	31.56
5	168	83	62.8	5024	7.87	26.57
6	168	87.6	62.8	5501.3	7.87	24.72
7	174	90	62.8	5652	7.87	23.80
8	168	96	62.8	6028.8	7.87	23.16
9	186	99	62.8	6217.2	7.87	21.77
10	168	100	62.8	6280	7.87	20.60
11	186	104.5	62.8	6374.2	7.87	19.28
12	180	110	62.8	6908	7.87	18.03
13	174	124	62.8	7787.2	7.87	17.46
14	180	130	62.8	8164	7.87	16.33
15	180	142.5	62.8	8949	7.87	14.82

: Q is the total explosive charge of the tunnel face blasting, R is the distance from the shock wave measuring point to the tunnel face, S is the tunnel section area, V is the volume of the shock wave propagation area, d is the equivalent diameter of the tunnel section, and $\Delta P$ is the peak value of the blasting shock wave overpressure at the measuring point.

Since the tunnel was a single-ended excavation tunnel, the empirical formula for the peak value of shock wave overpressure in roadways with one-end openings, proposed by Pokrovsky, could be applied for fitting. Then, field-measured data were combined with the Pokrovsky formula. Finally, a prediction formula for blasting shock wave overpressure suitable for the tunnel was proposed.

(1)

Attenuation formula fitting

The formula proposed by Pokrovsky is provided in Equation (5). The overpressure data at the measuring point were considered valid only when the propagation distance was equal to or greater than 6 times the equivalent diameter of the tunnel. Since the distance from the measuring point to the tunnel face exceeded 47.22 m (where 6d = 6 × 7.87 = 47.22 m), the overpressure data for these measuring points were deemed valid.

$$\Delta P=\left[{\displaystyle \frac{8Q}{SR}}+14.6\sqrt[3]{{\left({\displaystyle \frac{Q}{SR}}\right)}^2}+1.81\sqrt[3]{{\displaystyle \frac{Q}{SR}}}\right]10Q,\left(R\ge 6d\right)$$

(5)

The model test usually uses a TNT explosive to detonate directly in the air or tunnel. The tunnel is blasted by an emulsion explosive, and Q in Equation (5) is the mass of the TNT explosive. Therefore, it is necessary to convert the explosive mass used in tunnel blasting into the equivalent TNT explosive and increase the TNT equivalent conversion coefficient, $\gamma$. In addition, most of the energy of the blasting explosives in tunnel excavation is used to crush rock mass, and the remaining part is converted into shock wave energy, so a shock wave energy conversion coefficient, η, is added. Combined with the above analysis, the parameters in Equation (5) were modified and expressed by parameters a₀, b₀ and c₀ to establish a new empirical formula for shock wave overpressure:

$$\Delta P=\left[{\displaystyle \frac{a_{0}Q\gamma \eta }{SR}}+b_0\sqrt[3]{{\left({\displaystyle \frac{Q\gamma \eta }{SR}}\right)}^2}+c_0\sqrt[3]{{\displaystyle \frac{Q\gamma \eta }{SR}}}\right]Q$$

(6)

In Equation (6), SR is the volume of the shock wave propagation area, which can be replaced by V, which can be written as Equation (7):

$${\displaystyle \frac{\Delta P}{Q}}=a_0\gamma \eta {\displaystyle \frac{Q}{V}}+b_0{\left(\gamma \eta \right)}^{{\displaystyle \frac{2}{3}}}{\left({\displaystyle \frac{Q}{V}}\right)}^{{\displaystyle \frac{2}{3}}}+c_0{\left(\gamma \eta \right)}^{{\displaystyle \frac{1}{3}}}{\left({\displaystyle \frac{Q}{V}}\right)}^{{\displaystyle \frac{1}{3}}}$$

(7)

Let $y={\displaystyle \frac{\Delta P}{Q}}$, $x={\left({\displaystyle \frac{Q}{V}}\right)}^{{\displaystyle \frac{1}{3}}}$, and Equation (7) can be written as:

$$y=a_0\gamma \eta x^3+b_0{\left(\gamma \eta \right)}^{{\displaystyle \frac{2}{3}}}{x}^2+c_0{\left(\gamma \eta \right)}^{{\displaystyle \frac{1}{3}}}x$$

(8)

Since $a_0$, $b_0$, $c_0$, $\gamma$, and $\eta$ are constants, we can use new parameters to replace the parameters in Equation (9), as follows:

$$y=a_1{x}^3+b_1{x}^2+c_{1}x$$

(9)

Through the least-squares method, the field shock wave test data were fitted to obtain $a_1=99.2$, $b_1=-55.37$, and $c_1=8.05$, and finally, to fit the tunnel shock wave empirical formula:

$${\displaystyle \frac{\Delta P}{Q}}=99.2{\displaystyle \frac{Q}{V}}-55.37{{\displaystyle \left(\frac{Q}{V}\right)}}^{{\displaystyle \frac{2}{3}}}+8.05{\left({\displaystyle \frac{Q}{V}}\right)}^{{\displaystyle \frac{1}{3}}}$$

(10)

Comparing the fitted shock wave overpressure prediction formula with the field-measured data, taking ${\left({\displaystyle \frac{Q}{V}}\right)}^{{\displaystyle \frac{1}{3}}}$ as the X-axis and ${\displaystyle \frac{\Delta P}{Q}}$ as the Y-axis, the relationship curve was drawn, as shown in Figure 10. The fitting degree, R² = 0.99, shows that the curve fitting was good. Equation (10) can be used as the empirical formula for predicting the attenuation of shock wave overpressure in the tunnel.

(2)

Applicability analysis

By substituting various total charge quantities into Equation (10), the attenuation curves of shock wave overpressure for different charges could be derived. As shown in Figure 11, the closer the blasting distance was to the tunnel working face, the higher the shock wave overpressure. With the increasing propagation distance, the shock wave overpressure gradually decreased. The attenuation law of shock wave overpressure predicted by the empirical formula aligned well with the results of the numerical simulations. The empirical formula indicated a faster attenuation rate of overpressure in the 70 m to 100 m range, with a slower attenuation after 100 m, while the numerical simulation showed a more gradual attenuation curve. The discrepancies between the empirical formula and numerical simulation results may have arisen because the empirical formula was based on field test data, which accounted for a more complex tunnel environment (e.g., trolleys, vehicles, and other obstacles), whereas the numerical simulation assumed a simpler tunnel environment.

6. Discussion

In accordance with ‘Blasting Safety Regulations’, the overpressure peak of a shock wave is categorized into five grades based on its impact on the human body, as illustrated in

Table 9. Damage level of shock wave overpressure peaks to the human body.

Level	Overpressure Peak Value (kPa)	Extent of Damage to the Body
I	<2	None
II	20~30	Minor bruises
III	30~50	Hearing and organ damage; fractures
IV	50~100	Internal organs have suffered severe damage; possibly death
V	>100	Most individuals die

.

As shown in Table 9, the overpressure peak value should be less than 2 kPa to ensure personnel safety. Considering the law of pressure wave overpressure attenuation and the attenuation coefficient, we obtained the minimum safe distance for personnel from the blast source in the tunnel as 236.8 m.

Blasting Safety Regulations provide Equation (11) for determining the air shock wave overpressure during tunnel blasting:

$$\Delta p=\left(3270{\displaystyle \frac{q{m}_y}{RS}}+780\sqrt{{\displaystyle \frac{q{m}_y}{RS}}}\right){\mathrm{e}}^{{\displaystyle \frac{\beta R}{d}}}$$

(11)

where $\Delta P$ is the shock wave overpressure value (kPa), R denotes the distance from the measuring point to the explosion point (m), S is the tunnel cross-sectional area (m²), q is the mass of the explosive charge (kg), $m_y$ is the shock wave conversion factor, taken as 0.007, $\beta$ is the roughness factor of the tunnel wall, taken as 0.014, and d is the equivalent diameter of the tunnel section.

Figure 12 illustrates the attenuation law of shock waves, based on field test data, numerical simulation results, the overpressure empirical formula proposed in this study, and the empirical formula specified in the Blasting Safety Regulations. Our analysis showed that the empirical formula proposed in this study closely aligned with both the field test results and the numerical simulation outcomes. In contrast, the overpressure values obtained from the empirical formula in the ‘Blasting Safety Regulations’ were significantly smaller.

7. Conclusions

In this paper, the explicit dynamic analysis finite element software, ANSYS/LS-DYNA, was utilized to simulate the propagation process of a blasting shock wave within a tunnel. The study analyzed the changes in the shock wave flow field and the attenuation behavior of overpressure across the tunnel’s cross-section. Based on these simulations and field test data, a predictive formula for blasting shock wave overpressure suitable for tunnels was proposed. The main conclusions were as follow:

(1)

Initially, the shock wave propagated spherically, but it was subsequently affected by the enclosed structure of the tunnel. This interaction caused multiple reflections of the shock wave with the tunnel walls, resulting in a significant increase in overpressure values at the vault, side wall, and arch foot positions. At a distance of 6 times the equivalent diameter from the explosion center (48.5 m from the tunnel face), the spherical shock wave transitioned into a plane wave and continued to propagate forward.

(2)

As the shock wave reached a specific position, its overpressure instantaneously peaked before undergoing multiple-peak oscillation, zigzag attenuation, and gradual restoration to initial atmospheric pressure. Within 0~20 m from the tunnel face, repeated reflection between the shock wave and tunnel wall caused an increase in the overpressure value at 20 m, while exhibiting significant fluctuations in the attenuation coefficient. Beyond 50 m, propagation occurred as a stable plane wave, with an overpressure attenuation coefficient ranging from 1.17 to 1.03. The minimum safe distance between personnel and the explosion source in the tunnel was 236.8 m.

(3)

Through on-site testing of shock wave overpressure values, Pokrovsky’s empirical formula for shock wave overpressure has been refined based on parameters derived using the least-squares method. This paper proposed an empirical attenuation formula for shock wave overpressure with a good fitting degree (R² = 0.993). When excavating tunnels using drilling and blasting methods, this formula can be utilized to predict plane shock wave overpressures in the upper bench of the tunnel.