Characteristics of blasting vibration induced by short-delay cylindrical charges based on Heelan model

Abstract

As the electronic detonators can control the detonation error within 1 ms, short-delay blasting was nowadays proposed to replace multi-hole simultaneous blasting to reduce the vibration hazards. In this study, the Heelan model was used to study blast-induced vibration of short-delay cylindrical charges. The influence of delay time on the vector peak particle velocity (VPPV) of superimposed vibration waveforms was analyzed and the calculating formula of the shortest delay time between adjacent holes based on vibration reduction was established. It is found that as the delay time increases, the overlapping area of VPPV becomes further away from the explosion source. The main influencing factors for the shortest delay time between holes include the distance between the monitoring point and the explosion source, the difference between P wave and S wave velocities, and the duration of S wave in the first and second explosion holes. As the distance from the explosion source increases, the shortest inter hole delay time based on vibration reduction also increases. The research results have guiding significance for the control of blasting vibration in underground mines.

Keywords

Blast-induced vibration short-delay blasting delay time cylindrical charge

Introduction

The use of blasting for hard rock fragmentation is widely adopted in mining and quarrying engineering. In rock blasting, only a portion of the explosive energy is used to break the rock mass medium within the target range, while another portion of the energy is transmitted to the surrounding rock mass in the form of seismic waves, causing damage and destruction to the surrounding rock mass and structure (Huo et al., 2023; Wang et al., 2022). In underground mining, raise excavation blasting and cut blasting in stopes often associate with the situation of only a single free surface. At this point, multi-hole simultaneous blasting is frequently used to achieve the best rock crushing effect, but it also leads to the problem of high blasting vibration (Huang et al., 2019; Shi et al., 2016a). As the electronic detonators can control the detonation error within 1 ms, short-delay blasting was nowadays proposed to replace multi-hole simultaneous blasting to reduce the vibration hazards (Qiu et al., 2018).

With the improvement of the delay accuracy of detonators, the use of the stress wave superposition effect to improve the rock breaking effect has received widespread attention. Yamamoto (1999) stated that simultaneous blasting (0 ms delay) can generate explosive stress wave superposition at the midpoint of two adjacent holes, and the optimal rock fragmentation occurred in the delayed tensile stress superposition area. Rossmanith and Kouzniak (2004) pointed out that compared to conventional long delay blasting, the additional stress wave superposition generated by short-delay blasting was crucial for rock fragmentation. Vanbrabant and Espinosa (2006) believed that the delay time between adjacent holes could promote rock blasting fragmentation when it could achieve negative phase tail superposition of particle vibration velocity. However, some studies found that the rock fragmentation of delayed blasting without stress wave superposition was better than that of extremely short-delay blasting. Katsabanis et al. (2014) conducted blasting experiments covering the entire delay range using a small step grouting model, and the results showed that the rock fragmentation with a delay range of 4–10 ms/m resistance line was the best, while the extremely short-delay blasting was the worst. Johansson and Ouchterlony (2013) compared the rock fragmentation with delays in the time range or not of shock wave interaction through the blasting test of a small-scale bench model with four holes and showed no statistical difference. Yi et al. (2017) studied the effects of short-delays on the effect of blasting fragmentation by means of numerical simulation, indicating that the cases with delays of 6 ms and 3 ms result in better fragmentation than that with delays of 0 ms and 1 ms. Zhang et al. (2022) studied the influence of detonator delay scatter on rock fragmentation by bunch-holes blasting.

Compared with the studies on the mechanism of rock fragmentation, there are few studies focused on the mechanism of vibration mitigation related to short-delay blasting. Gao et al. (2011) found that the short-delay blasting initiated by electronic detonators could effectively achieve peak control of blasting vibration through outdoor bench blasting experiments. Blair (2010a) found that compared to long delay blasting, short-delay blasting was not conducive to spectrum control due to a significant increase in low-frequency vibration components. Yue et al. (2022) numerically investigated the effect of delay time on rock damage and vibration attenuation in multi-hole blasting. Zhang et al. (2023) studied the influence of different delay time on blast-induced PPV by carrying out multi-delay time of double-hole and multi-hole blasting experiments. Shi et al. (2016b) analyzed the effect of the vibration mitigation of short-delay blasting with the HHT energy spectrum.

To obtain the best delay time between holes is always the focus of blasting technicians. Aiming at better rock fragmentation by blasting, Henrych and Abrahamson (1979) stated that the reasonable delay time between holes should include three parts: the time needed for the explosive stress wave to propagate to the free surface and return to the explosive, the time required for the crack to expand to the free surface, and the time required for the crack to continue expanding and form a new free surface with a certain crack width. Chiappetta (2010) argued that the optimal delay times should be in the range where the shock waves interact with each other between the holes. Based on the energy distribution theory of explosive explosion in rock mass, Zhong et al. (2015) derived the formula for calculating the reasonable delay time between holes. In terms of reducing blasting vibration, many scholars stated that the reasonable delay time should just make two vibration waves with half period difference are superimposed on each other, so that the main phases of the blasting vibration waves are staggered (Gou et al., 2021; Huang et al., 2019; Chen et al., 2015). Through a lot of experiments, Sanchidrian et al. (1992) put forward a formula for calculating reasonable delay time, which was well verified in practice.

It is fairly costly to study the influence of delay internals on the vibration characteristics of delay blasting by field tests (Yang et al., 2018). In addition, although many scholars used numerical simulation methods to study the vibration of explosive charges blasting (Huo et al., 2022; Ainalis et al., 2017), there are still discrepancies between the explosion load sources added in numerical simulation and engineering practice (Lu et al., 2011). Some scholars have developed theoretical models to study the vibration effects of charge blasting in order to obtain more accurate results. Holmberg and Persson (1979) developed a near-field blasting vibration peak model, but this model only used proportional charge to calculate PPV and did not consider any vibration waveform information, so it cannot truly reflect blasting vibration information (Blair and Minchinton, 2006). Meredith et al. (1993) provided the most accurate full-field solution for calculating the vibration of explosive charges. However, this model had too many calculation parameters and low computational efficiency (Blair, 2007, 2010b). Heelan (1953) provided a mathematical solution model for the blasting vibration, which considered the blasting vibration excited by a cylindrical charge as the superposition of countless small length explosive units. The Heelan model has higher computational efficiency than the global model, and can still provide more accurate approximate solutions for simulating the waveform excited by explosive charges (Blair, 2007) Therefore, the Heelan model has been widely used in studying the vibration effects of explosive charges (Blair, 2015; Chen et al., 2015).

The cylindrical charge, which is defined as a charge column with length-diameter ratio more than 6, has important applications in opencast mining as well as underground exploiting (Yang et al., 2016). This study focused on the study of blast-induced vibration of short-delay cylindrical charges based on Helen model. Firstly, based on the propagation laws of P and S waves in cylindrical charge blasting, qualitative analysis was conducted on the vibration superposition mechanism of two hole charge delayed blasting; Secondly, a Heelan model of cylindrical charge was established and a reasonable length of explosive unit was determined; Then, the Heelan model was used to analyze the influence of delay time on the vector peak particle velocity (VPPV) of superimposed vibration waveforms, and a calculating formula of the shortest delay time between adjacent holes based on vibration reduction was established.

2. Vibration superposition mechanism of double-hole charge short-delay blasting

The vibration wave induced by charge blasting includes a P wave and an S wave. Blair depicted the vibration wave induced by charge blasting when the detonation point (DP) is at the bottom (Blair, 2014), as shown in Figure 1. The charge in the figure is 5 m long, with a detonation wave velocity of 5200 m/s and a P wave velocity of 3800 m/s. As shown in the figure, the bottom charge at the initiation point radiates loading phases of P wave (PL) and S wave (SL), while the top charge radiates unloading phases of P wave (PU) and S wave (SU).

Figure 2 shows the sketch of a vibration wave induced by the simultaneous detonation of a double-hole charge with a spacing of 1 m. As shown in the figure, the superposition of the vibration waves induced by the simultaneous detonation of the double-hole charge is extremely complicated. The superposition of the vibration waves at different positions mainly includes the following types: PP wave superposition occurring near the axis of two holes and in the upper region of the two hole cylindrical charge; SS wave superposition occurring along the axis of two holes and in the upper region of the two hole cylindrical charge; PS wave superposition occurred in the right area of hole #2 and the left area of hole #1. It can be inferred from the figure that the PS waves are only superposed within a certain range from the charge column because the P wave velocity is greater than the S wave velocity. In contrast, the same PP and SS waves can superpose in a large range.

Figure 3 shows the sketch of the vibration wave induced by the short-delay detonation of a double-hole charge. Charge #1 on the left is detonated before charge #2 on the right. Under short-delay condition, the types of superposition of the vibration waves of the explosives at different positions are as follows: PP wave superposition between PL2 and PU1; SS wave superposition between SL2 and SU1; PS wave superposition between PL1 and SL1 and between PU2 and SU2; PS wave superposition between PL2 and SL1 and between PU2 and SU1. The superposition of different waveforms only exists between the S wave (SL1 and SU1) induced by charge #1 and the P wave (PL2 and PUS) induced by charge #2. It is conceivable that the superposition between the two is, in essence, the process in which the P wave induced by charge #2 with the faster propagation velocity catches up to the S wave induced by charge #1 with a slower propagation velocity. For an extremely short-delay (0.263 ms in this case) that the S wave induced by charge #1 fails to reach charge #2 when propagating to the right when charge #2 is detonated, the PS wave superposition occurred only between two charges and to the left of charge 1#., as shown in Figure 3(a). While for short-delays that can be realized in the current engineering blasting (millisecond class), the PS wave superposition did not occur between two charges, but on either side of the two charge, as shown in Figure 3(b). It should be noted that the extremely short-delay shown in Figure 3(a) has a strict limit on the delay time. It is impossible to achieve this under current detonator delay accuracy and is thus only possible theoretically.

3. The Heelan model of vibration induced by cylindrical charge

3.1 Heelan model

Figure 4 shows the sketch of Heelan model for single-hole cylindrical charge with length L and diameter d. The vibration wave at monitoring point M (r, z) is the superposition of vibration waves induced by N explosive elements with length l. Thus the relationship is as follows:

l = L / N

(1)

The explosive pressure $P (t)$ exerted on the hole wall of the explosive element at time $t$ can be expressed as:

P (t) = {H (t) P}_{0} {t^{n} e}^{- γ t}

(2)

where

P_{0}

is the initial explosive pressure,

γ

is the explosive attenuation coefficient,

n

is a constant, and

H (t)

is Heaviside function which is calculated as follows:

H (t) = {\begin{cases} 0 t \leq 0 \\ 1 t > 0 \end{cases}

(3)

The propagation times of P wave and S wave of the $i$ th explosive element to the monitoring point M are as follows:

t_{P} = γ (t - \frac{z_{i}}{V_{D}} - \frac{R_{i}}{α})

(4)

t_{S} = γ (t - \frac{z_{i}}{V_{D}} - \frac{R_{i}}{β})

(5)

where

V_{D}

is the velocity of detonation.

α

and

β

are the P and S wave velocities, respectively.

According to the Heelan model (Heelan, 1953), the radial and circular velocity time histories $V_{R i}$ and $V_{\emptyset i}$ induced by the $i$ th explosive element (0, z_i) at monitoring point M are as follows:

V_{R i} = \frac{C}{R_{i}} H (t_{P}) g (t_{P}) (1 - 2 \frac{β^{2}}{α^{2}} {\cos \emptyset}^{2})

(6)

V_{\emptyset i} = \frac{C}{R_{i}} H (t_{S}) g (t_{S}) (\frac{β}{α} \sin 2 \emptyset)

(7)

where

C

is a constant for a particular explosive model, which is calculated as follows:

C = \frac{γ^{2} {d l (e / n)}^{n} P_{0}}{2 μ α}

(8)

where

μ

is the Poisson's ratio of rock.

g (t)

is the wave velocity function, which is defined by:

g (t) = e^{- t} [t^{n} - 2 n t^{n - 1} + n (n - 1) t^{n - 2}]

(9)

According to the geometry shown in Figure 1, the horizontal and the vertical velocity time histories $V_{r i} (t)$ and $V_{z i} (t)$ induced by the $i$ th explosive element (0, z_i) at monitoring point M are as follows:

V_{r i} (t) = V_{R i} \sin \emptyset + V_{\emptyset i} \cos \emptyset

(10)

V_{z i} (t) = {- V}_{R i} \cos \emptyset + V_{\emptyset i} \sin \emptyset

(11)

Therefore, the horizontal and the vertical velocity time histories $V_{r} (t)$ and $V_{z} (t)$ induced by the cylindrical charge at monitoring point M are as follows:

V_{r} (t) = \sum_{1}^{N} V_{r i} (t)

(12)

V_{z} (t) = \sum_{1}^{N} V_{z i} (t)

(13)

The vector velocity time history ( $V_{v e c t o r}$ ) is defined as:

V_{v e c t o r} = \sqrt{{V_{r}}^{2} (t) + {V_{z}}^{2} (t)}

(14)

And the vector peak particle velocity (VPPV) is defined as:

V P P V = \max \sqrt{{V_{r}}^{2} (t) + {V_{z}}^{2} (t)}

(15)

3.2 Influence of charge element length on blasting vibration

It is unquestionable that a small charge element length leads to more precise wave results, but it also requires longer computation time. Blair noted that the Heelan model provided similar results with the full-field solution only when l/d was less than 2 (Blair, 2007). In order to gain the influence of the charge element length on vibration waves, six charge element lengths varying from 0.01 m to 1 m are used to calculate the vibration wave by a charge column with d of 89 mm and a length of 5 m. For a charge element length of 0.01 m, the element number is 500, and for a charge element length of 1 m there are only five elements. Figure 5 shows the radial waves at (10, 10) for different element lengths. It can also be seen that the element lengths of 0.05 m and 0.1 m provide very close results to the 0.01 m case. Compared with the case of l/d = 0.11, the errors of peak wave velocity for l/d of 0.56 and 1.12 are 0.25% and 0.34%, while the errors of peak wave velocity for other cases are all larger than 1%. Therefore, an element length of 0.1 m (l/d = 1.12) is used in this study.

4. Influence of the delay time on the peak particle velocity (PPV) of the superposed signal

4.1 Superposed vibration waveforms induced by the short-delay blasting of a double-hole charge

Figure 6 shows the vibration waves in the r direction at the monitoring point M (10,7) induced by the detonation of the adjacent double-hole charges. From this figure, the vibration waves induced by the double-hole charges at point M are similar. Charge #1 is further away from the monitoring point M than charge #2 is, the time its wave arrives at point M is slightly late, and its amplitude is also slightly low. The superposition of the vibration waves between the adjacent double-hole charges can be summarized as one of two types: the first type is a superposition of the same waves, namely, PP wave superposition and SS wave superposition, which are generally found in simultaneous detonation or extremely short-delay blasting (Figure 7(a)); the second type is a superposition of different waves, namely, the superposition of an S wave induced by the charge blasted first and a P wave induced by the charge blasted later, which is generally found in long delay blasting (Figure 7(b)).

To analyze the superposition patterns of the vibration waves induced by short-delay blasting of the adjacent double holes, the monitoring points are placed every 5 m on a virtual free surface with a resistance line of 2 m to record the vibration waveform. Figure 8 shows the vibration waveforms monitored at different distances from the double holes with different delays. It can be determined from Figure 8 that only the same vibration waveforms induced by blasting a double-hole charge with a 1 ms delay can superpose, but same waveforms do not superpose for a 5 ms delay. Instead, the different waveforms superpose between the P wave induced later and the S wave induced first. As the delay time increases from 5 ms to 10 ms, the distance between the position where the P wave and the S wave begin to superimpose as well as the cylinder charge increase from 15 m to 45 m. It can be seen that as the delay time increases, different types of waves between the P waves induced later and the S waves induced first superpose at a farther distance.

4.2 Impact of the delay time on the vector peak particle velocity

Figure 9 shows the VPPV contour with high resolution induced by 5 m cylindrical charge with bottom initiating. Figure 10 shows the VPPV contours induced by the simultaneous detonation and short-delay detonation of double-hole charges. For a short-delay detonation, the hole charge on the left is detonated first. It can be seen from Figure 10(a) that the VPPV contours induced by the simultaneous detonation of double-hole charges are bilaterally symmetrical. The shape of the contour is similar to that of the VPPV contour induced by a single-hole cylindrical charge. The difference is that the VPPV amplitude of the simultaneous detonation of two holes is greater than that of a single hole, which is because a superposition of the vibration waves of two adjacent holes exists at any position for the simultaneous detonation of two holes. By contrast, the VPPV contour under the short-delay detonation in Figure 10(b)-(f) shows an abrupt change in different positions. This is due to the superposition of S waves (which is induced first and is on the left side of the region with an abrupt change) and P waves (which is induced later and is on the right side), thus resulting in an increase in the VPPV in this region. It can be determined through a comparison between Figure 10(a) and (b)-(f) that the VPPV amplitude under the short-delay detonation drops significantly compared to a simultaneous detonation. The shapes of the VPPV contours outside the region with an abrupt change under different delay times are similar (Figure 10(b)-(f)). This indicates that there is no great change in the VPPV amplitude outside the superposition region under the different delay times. In addition, the superposition region where the VPPV amplitude has an abrupt change becomes much farther away from the charges as the delay time increases.

To further analyze the superposition features of the vibration waves via short-delay blasting at different distances, Figure 11 shows the trend of the VPPV of two holes over r under different delay times when the resistance line is 2 m. It can be seen from the figure that VPPV increases at different positions under the different delay times due to the superposition effect of the vibration waves. The VPPV increases when r is within 19–23 m and the delay time is 4 ms. The increases in the VPPV are 26–34 m, 40–45 m, and 52–58 m at delay times of 6 ms, 8 ms, and 10 ms, respectively. However, outside the range of the increased VPPV, the VPPV amplitude is the same under the different delay times. It can be seen that as the delay time increases, the superposition region of the VPPV is increasingly farther away from the explosive source. This indicates that the superposition position between the S waves, induced first, and the P waves, induced later, is increasingly farther as the delay time increases.

To quantify the reduction in amplitude between a short-delay detonation and a simultaneous detonation, most scholars currently employ the drop rate to measure the effect of the vibration reduction in short-delay blasting, namely, the ratio of the difference in the VPPV amplitude between a simultaneous detonation and short-delay blasting to the VPPV under a simultaneous detonation. The definition of the drop rate $δ$ of the VPPV is:

δ = \frac{{V P P V}_{0} - {V P P V}_{t}}{{V P P V}_{0}} \times 100 %

(16)

where

{V P P V}_{0}

represents the VPPV of double-hole charges under a simultaneous detonation, and

{V P P V}_{t}

is the VPPV of the short-delay detonation with a delay time of t.

Figure 12 shows the drop rate contour of the VPPV under short-delay blasting and simultaneous blasting. A drop rate greater than 0 indicates that the short-delay blasting effectively reduces the VPPV, while a drop rate less than 0 indicates that the VPPV under short-delay blasting is greater than that of a simultaneous detonation. $δ < 0$ implies that the short-delay blasting fails to reach an effective vibration reduction due to the superposition of the vibration waves of two adjacent holes in this region. As shown in Figure 12, the short-delay blasting has a desirable vibration reduction effect at the upper and lower parts of charges. In particular, the drop rate at the upper part of the charges is more than 50%. However, on the left and right sides of the charges, the vibration reduction effect of the short-delay blasting is limited, and the drop rate in some regions is even smaller than 0. In particular, the drop rate is as low as −40% at the 45° region on the low right part when the delay times are 2 ms and 3 ms. It can be determined that when the delay times are 1 ms, 2 ms, and 3 ms, the regions with the worst vibration reduction effect of the VPPV under the short-delay blasting are at the four 45° directions of the charges. Since the S waves induced first in the short-delay blasting superpose with the P waves induced later, the crest of the superposition waves increase; thus, the VPPV under the short-delay blasting is greater than that of simultaneous detonation. However, in the four regions along 45°, the amplitude of the S waves induced first is very close to that of the P waves induced later. Therefore, the superposition of these waves enables the maximum amplitude to increase in the VPPV, and the vibration reduction effect is the worst.

5. Optimal delay time between holes for vibration reduction

Figure 13 shows the variation of the vibration wave crest (rPPV) induced by double-hole charges in the r direction over the delay time. It can be seen that in the range of 0–0.8 ms, the superposition of the same type of waveform enables rPPV to increase more than that in single-hole blasting. However, the superposition between the S waves induced first and the P waves induced later causes another increase in rPPV in the range of 2.0–2.5 ms. However, in actual blasting, the duration of the blasting vibration wave is long, and there is no obvious boundary between the P and S waves in single-hole blasting. Therefore, in a delay time range that covers the superposition of the same type of wave to the superposition of different types of waves (PS waves), the PPV might increase. To avoid the superposition of vibration waves of two adjacent holes, the delay time should be greater than the time for the crest superposition between the S waves induced first and the P waves induced later.

In addition, the shortest delay time when the vibration waves induced by a double-hole are completely staggered at point M is 4.45 ms. However, when the delay time reaches 2.58 ms the crest of the superposed vibration waves of the double-hole is consistent with that of the single-hole. This indicates that the delay time between the holes to achieve a vibration reduction is not necessarily long enough to allow for the vibration waves induced by the double-hole to be completely staggered. In fact, in engineering blasting, the blasting vibration waves tend to last long due to the influence of a surrounding rock mass medium, fracture, crack, structure, etc. Thus, it is not feasible to have the vibration waves of double-hole blasting be completely staggered. Therefore, considering vibration reduction, the shortest delay time between holes should be between the two times, namely, it should be greater than the crest superposition time between the S waves induced first and the P waves induced later, but smaller than the time to have the vibration waves of the adjacent holes be completely staggered.

Considering the vibration reduction, the shortest delay time $Δ t$ between holes is made up of three parts: the propagation time difference $t_{1}$ of the vibration waves under the simultaneous detonation of two adjacent holes, the superposition time $t_{2}$ for the crest of the P waves induced later to catch up to the crest of S waves induced first, and the time difference $t_{3}$ when the crest of the superposed vibration waves of the double-hole due to the crest superposition of the PS waves is consistent with that of single-hole blasting:

Δ t = t_{1} + t_{2} + t_{3}

(17)

where the propagation time difference

t_{1}

of the vibration waves under the simultaneous detonation of two adjacent holes can be represented as the propagation time difference of the P waves caused by the distance difference between the two holes and the monitoring point:

t_{1} = \frac{R 1 - R 2}{α}

(18)

where

R 1

is the distance of the first blasted hole to the monitoring point,

R 2

is the distance of the latter blasted hole to the monitoring point, and

α

is the P wave velocity in the rock. If the hole far away from the monitoring point is detonated first,

t_{1}

is greater than 0. However, if the hole closer to the monitoring point is detonated first,

t_{1}

is smaller than 0. It can be seen that the propagation time difference of the vibration waves under the simultaneous detonation of two adjacent holes is mainly influenced by the distance between the adjacent double-hole charges and the monitoring point.

The superposition time $t_{2}$ for the crest of the P waves induced later to catch up to the crest of S waves induced first can be simplified as the time difference between the P waves and S waves via single-hole blasting. The expression is:

t_{2} = \frac{R 2}{α - β}

(19)

Due to the complexity of the blasting vibration waveform, it is not easy to quantify the time difference $t_{3}$ when the crest of the superposed vibration waves of the double-hole due to the crest superposition of the PS waves is consistent with that of single-hole blasting, but it is obvious that the time is related to the duration of the S waves induced first and duration of the P waves induced later. Therefore, $t_{3}$ can be defined as:

t_{3} = a * T_{S} + b * T_{P}

(20)

where

a

and

b

are quantization coefficients of the durations of the S and P waves, respectively,

T_{S}

is the duration of S waves induced first, and

T_{P}

is the duration of the P waves induced later.

It is hard to control the vibration near the blasting region. In engineering blasting, the region covered by the blasting vibration control is usually the middle and far explosion region. For the middle and far explosion region, the propagation time difference of the P waves induced by the distance between adjacent holes is reduced as the distance from the explosive source increases. This has an increasingly smaller impact on the shortest delay time between holes. Therefore, when ignoring $t_{1}$ , the shortest delay time $Δ t$ can be simplified as:

Δ t = \frac{R 2}{α - β} + a * T_{S} + b * T_{P}

(21)

It can be seen that the influence factors of the shortest delay time between holes when considering the vibration reduction mainly include the distance between the monitoring point and the explosive source, the difference between the velocity of the P and S waves, and the durations of the S waves induced first and the P waves induced later. The farther the monitoring point is from the explosive source, the longer the shortest delay time is between holes when considering a vibration reduction. The larger the difference between the velocity of the P and S waves, the shorter the shortest delay time between holes when considering a vibration reduction. Finally, the longer the duration of the S waves induced first and the P waves induced later, the longer the shortest delay time between holes when considering a vibration reduction.

It should be pointed out that the quantization coefficients $a$ and $b$ highly depend on the field conditions during the vibration wave propagation. And the calculation formula of the optimal delay time needs further verification by field tests.

6. Conclusions

Based on the Heelan model, this paper focused on the blast-induced vibration of short-delay cylindrical charges. The influence of delay time on the vector peak particle velocity (VPPV) of superimposed vibration waveforms was analyzed and the calculating formula of the shortest delay time between adjacent holes based on vibration reduction was established. The main conclusions are as follows:

1. As the P wave propagation velocity is faster than S wave, the superposition of blasting waves in adjacent holes at short-delay intervals is mainly caused by the P wave of the later initiated hole catching up with the S wave of the firstly initiated hole. The P/S wave superposition only occurs within a certain distance range from the cylindrical charge.

2. As the delay time increases, the overlapping area of VPPV becomes further away from the explosion source. The vibration reduction effect on the left and right sides of the cylindrical charge under short-delay detonation is worse than that in the upper and lower areas. Because the amplitudes of the S wave induced by the firstly initiated hole and the P wave induced by the later initiated hole in this area are at the same level. These two are more likely to overlap, leading to an increase in VPPV.

3. The main influencing factors for the shortest delay time between holes include the distance between the monitoring point and the explosion source, the difference in P wave and S wave velocities, and the duration of S wave in the first and second explosion holes. As the distance from the explosion source increases, the shortest inter hole delay time based on vibration reduction also increases.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (52374152), the National Key R&D Program of China (2022YFC2904602), and the Guangxi Key R&D Plan (2022AB31023).

ORCID iDs

Xianyang Qiu

Xiuzhi Shi

References

Ainalis

Kaufmann

Tshibangu

, et al. (2017) Modelling the source of blasting for the numerical simulation of blast-induced ground vibrations: a review. Rock Mechanics and Rock Engineering 50: 171–193.

Blair

(2007) A comparison of Heelan and exact solutions for seismic radiation from a short cylindrical charge. Geophysics 72(2): 33–41.

Blair

(2010a) Limitations of electronic delays for the control of blast vibration and fragmentation. Proceedings of the 9th International Symposium on Rock Fragmentation by Blasting. Granada: 171–184.

Blair

(2010b) Seismic radiation from an explosive column. Geophysics 75(1): 55–65.

Blair

(2014) Blast vibration dependence on charge length, velocity of detonation and layered media. International Journal of Rock Mechanics and Mining Sciences 65(1): 29–39.

Blair

(2015) The free surface influence on blast vibration. International Journal of Rock Mechanics and Mining Science 77: 182–191.

Blair

Minchinton

(2006) Near field blast vibration models. Fragblast 8: 152–159.

Chen

Zhang

(2015) Influence of millisecond time, charge length and detonation velocity on blasting vibration. Journal of Central South University 22(12): 4787–4796.

Chiappetta

(2010) Combining electronic detonators with stem charges and air decks. https://www.iqpc.com/redForms.aspx?id=414354&sform_id=473344.

10.

Gao

Yang

Xiao

, et al. (2011) Deep-hole controlled blasting technology based on precise time delay. Journal of China Coal Society 36(S2): 386–390.

11.

Gou

Shi

Qiu

, et al. (2021) Assessment of induced vibrations derived from the wave superposition in time-delay blasts. International Journal of Rock Mechanics and Mining Sciences 144(2): 104814.

12.

Heelan

(1953) Radiation from a cylindrical source of finite length. Geophysics 18(3): 685–696.

13.

Henrych

Abrahamson

(1979) The Dynamics of Explosion and its Use. Elsevier Scientific Publishing Company.

14.

Holmberg

Persson

(1979) Design of tunnel perimeter blasthole patterns to prevent rock damage. Proceedings of IMM Tunnelling’79 Conference. London.

15.

Huang

Qiu

Shi

, et al. (2019) Experimental and numerical investigation of blast-induced vibration for short-delay cut blasting in underground mining. Shock and Vibration 2019: 1–13.

16.

Huo

Shi

Qiu

, et al. (2022) A study on raise blasting and blast-induced vibrations in highly stressed rock masses. Tunnelling and Underground Space Technology 123: 104407.

17.

Huo

Qiu

Shi

, et al. (2023) Attenuation characteristics of blasting stress under decoupled cylindrical charge. Rock Mechanics and Rock Engineering 56(6): 4185–4209.

18.

Johansson

Ouchterlony

(2013) Shock wave interactions in rock blasting: the use of short delays to improve fragmentation in model-scale. Rock Mechanics and Rock Engineering 46(1): 1–18.

19.

Katsabanis

Omidi

Rielo

(2014) Examination of timing requirements for optimization of fragmentation using small scale grout samples. Blasting and Fragmentation 8(1): 25–35.

20.

Yang

Chen

, et al. (2011) An equivalent method for blasting vibration simulation. Simulation Modelling Practice and Theory 19(9): 2050–2062.

21.

Meredith

Toksoz

Cheng

(1993) Secondary shear waves from source boreholes. Geophysical Prospecting 41(3): 287–312.

22.

Qiu

Shi

Gou

, et al. (2018) Short-delay blasting with single free surface: results of experimental tests. Tunnelling and Underground Space Technology 74(APR): 119–130.

23.

Rossmanith

Kouzniak

(2004) Supersonic detonation in rock mass: Part 2: particle displacements and velocity fields for single and multiple non-delayed and delayed detonating blast-holes. Fragblast 6(2): 95–117.

24.

Sanchidrian

Pesquero

Garbayo

(1992) Damage in rock under explosive loading: implementation in DYNA2D of a TCK model. International Journal of Surface Min and Reclamation 6(3): 109–114.

25.

Shi

Qiu

Zhou

, et al. (2016a) Technology and case study of ultra-large section and high shaft excavation by short-millisecond spherical-like blasting. Chinese Journal of Rock Mechanics and Engineering 35(8): 1659–1667.

26.

Shi

Qiu

Zhou

, et al. (2016b) Application of Hilbert-Huang transform based delay time identification in optimization of short millisecond blasting. Transactions of Nonferrous Metals Society of China 26(7): 1965–1974.

27.

Vanbrabant

Espinosa

(2006) Impact of short delays sequence on fragmentation by means of electronic detonators: theoretical concepts and field validation. Proceedings of the 8th international symposium on rock fragmentation by blasting. Santiag 2006: 326–331.

28.

Wang

Zheng

, et al. (2022) Full waveform prediction of blasting vibration using deep learning. Sustainability 14(13): 8200.

29.

Yamamoto

(1999) Experimental and theoretical study on smooth blasting with electronic delay detonators. Fragblast 3: 3–24.

30.

Yang

Jiang

, et al. (2016) A study on the vibration frequency of blasting excavation in highly stressed rock masses. Rock Mechanics and Rock Engineering 49(7): 2825–2843.

31.

Yang

Jiang

Zhang

, et al. (2018) Dynamic stress adjustment and rock damage during blasting excavation in a deep-buried circular tunnel. Tunnelling and Underground Space Technology 71: 591–604.

32.

Sjöberg

Johansson

(2017) A numerical study of the impact of short delays on rock fragmentation. International Journal of Rock Mechanics and Mining Sciences 100: 250–254.

33.

Yue

, et al. (2022) The effect of blast-hole arrangement, delay time, and decoupling charge on rock damage and vibration attenuation in multihole blasting. Shock and Vibration 2022: 2110160.

34.

Zhang

Qiu

Shi

, et al. (2022) Influence of detonator delay scatter on rock fragmentation by bunch-holes blasting. Geomechanics and Geophysics for Geo-energy and Geo-resources 8(136): 11–21.

35.

Zhang

Yan

(2023) Experimental investigations of the effect of millisecond-delay time on the blast vibration reduction with electronic detonators. Journal of Vibration and Control 29(17–18): 4204–4215.

36.

Zhong

Cao

, et al. (2015) Experimental study of reducing vibration intensity based on controlled blasting with precise time delay. Journal of China Coal Society 40(s1): 107–112.