Blast-induced ground vibration propagation and prediction in a sandstone slope with soft mudstone interlayer

Abstract

Accurate prediction of blast-induced vibration propagation in rock slopes is critical for ensuring safe and efficient excavation, particularly in geologically complex settings where soft mudstone interlayers fundamentally modify vibration attenuation mechanisms. This study presents an integrated methodology combining field-scale blast experiments, advanced dynamic finite-element simulations, and wavelet packet energy analysis to systematically evaluate how varying dip angles and thicknesses of soft mudstone interlayers govern vibration transmission patterns and spectral energy redistribution in rock slopes. The in situ blasting test focuses on the Pinglu Canal Qingnian Hub project, where multi-point vibration monitoring was implemented across rock slope profiles. A dimensional analysis-driven predictive framework was formulated to quantify interlayer geometric effects on vibration velocity propagation. The results demonstrate three critical insights: First, vertical vibration components consistently exhibit dominant characteristics, manifesting the highest peak velocities and predominant frequencies across all monitored locations. Second, increasing soft mudstone interlayer inclination angles and thicknesses significantly amplify vibration attenuation rates along slope surfaces while inducing notable energy redistribution – enhanced dissipation of high-frequency vibrational components coupled with increased proportional contributions from low-frequency bands. This frequency-dependent energy transformation mechanism elevates resonance risks as spectral characteristics progressively align with slope structures’ intrinsic vibrational modes. Third, the developed prediction model effectively generalizes the relationship between changes in the thickness and dip of the soft mudstone interlayer and the blast-induced vibration velocity, and compares it with the classical Sadowski’s formula with much improved accuracy. By establishing quantitative relationships between soft mudstone interlayer configurations and vibration transmission behaviors, this research advances fundamental understanding of soft mudstone interlayer while providing actionable guidelines for optimizing blast designs in interlayered rock masses.

Keywords

blast-induced vibration propagation rock slope soft mudstone interlayer wavelet packet energy analysis in situ blasting test prediction model

Introduction

Rock slopes with soft mudstone interlayers are common in large-scale engineering projects. With the accelerated development of infrastructure construction, particularly the advancement of large-scale water conservancy and transportation projects, blasting technology is being increasingly applied in slope excavation processes (Deng and Chen 2021). Blasting vibration can adversely affect the stability of rock slopes containing soft mudstone interlayers. Accurate prediction of blast-induced vibration propagation is a prerequisite for effective vibration control in slopes. The presence of soft mudstone interlayers introduces new characteristics to the attenuation of blast-induced vibration. Therefore, analyzing and predicting the propagation patterns of blast-induced vibration in rock slopes with soft mudstone interlayers has significant practical implications for engineering applications (Zhai et al., 2025; Zhang et al., 2022).

In recent years, most researchers have focused on studying the dynamic response of rock slopes under blast-induced vibration through field experiments and numerical simulation software. Blast loading often induces geological hazards such as slope instability and soil sliding, making it crucial to investigate the propagation and attenuation patterns of blast-induced vibration velocity in rock slopes.

Existing research often combines field monitoring and theoretical analysis to investigate blast-induced vibrations on slopes. However, conventional prediction formulas do not adequately account for topographic factors such as elevation differences, leading to limited accuracy in predicting vibrations in slope areas. With advances in numerical simulation technology, software such as ANSYS/LS-DYNA has been increasingly applied to study blast-induced vibrations on slopes. Nevertheless, further validation of their consistency with field data and clarification of vibration variation patterns under different working conditions are still required. Man et al. (2021) conducted field experiments and determined the threshold for maximum blast-induced vibration velocity at a distance of 10 m based on vibration attenuation patterns and rock damage characteristics. Haghnejad et al. (2019) employed a three-dimensional discrete element method program to develop various models to explore the influence of discontinuities and rock properties on blast-induced vibration velocity. Yan et al. (2022) found through numerical simulations that slopes can be regarded as convex parts of strata, subjected to fewer constraints and lower loads compared to flat strata, resulting in lower stability and a pronounced “whiplash effect.” This effect is particularly significant at the crest of steep slopes, where vibration amplification is more evident. As the slope angle decreases, these effects weaken, and the vibration attenuation rate increases. Zhao et al. (2025) established slope models with different cross-sections by introducing the Weibull random distribution function to investigate the dynamic wave propagation characteristics and dynamic response of heterogeneous layered slopes under blast-induced vibrations. Cui et al. (2022, 2023) conducted experiments to explore the influence of soft interlayers at different locations and thicknesses on the blasting characteristics of natural rocks. They found that the closer the interlayer is to the blast hole, the greater the damage to the surrounding rock. Yang et al. (2022) pointed out that blast-induced damage and vibrations in rock slopes are influenced by various factors, including explosive properties, blast hole layout, detonation sequence, rock characteristics, and slope geometry. They noted that the blast-induced vibration velocity limits used in existing standards for rock damage control are mostly semi-empirical, highlighting the need for further theoretical and simulation studies in the future.

In addition to studying the variation patterns of blast-induced vibration velocity, analyzing the characteristics of vibration velocity signals is also a crucial aspect of investigating vibration propagation modes. In frequency analysis of vibration signals, Song et al. (2020) proposed a time-frequency joint analysis method to study the propagation characteristics of blast-induced vibration waves in slopes and their impact on slope dynamic response. Chen et al. (2025), based on practical engineering backgrounds, found that when the size of unstable rock masses is close to or exceeds half the wavelength of blast-induced vibrations, different parts of the rock mass experience different phases of the vibration wave (peak, trough), resulting in inconsistent force magnitudes and directions. This leads to a “load counteraction” effect, where the total load actually acting on the unstable rock mass is smaller than the value calculated by traditional methods, resulting in a higher stability coefficient. Li et al. (2022) developed geological models of different slopes to study their natural frequencies, damping characteristics, and dynamic acceleration responses under blast-induced vibrations. Regarding energy analysis in different frequency bands, Zhao et al. (2025) used wavelet packet energy analysis and found that the peak particle velocity (PPV) of slopes attenuates with increasing distance from the blast source, with the vertical component > horizontal component. They also confirmed that the elevation-modified Sadovsky formula can accurately predict this attenuation. Vibration energy is primarily distributed in the 11–100 Hz range, with high frequencies (51–100 Hz) attenuating most significantly, while low frequencies (11–25 Hz) exhibit the most pronounced amplification at the step crest. Sun et al. (2022) applied wavelet packet theory to analyze the energy distribution and attenuation patterns across frequency bands. They found that the blast-induced vibration energy in counter-tilt layered rock slopes is mainly distributed in the 0–80 Hz range (accounting for over 85%) and the 115–160 Hz range (accounting for 5%–15%), with low-frequency bands being the core influencing frequencies. Additionally, analyzing the influence of soft interlayers on blast-induced vibration energy is crucial. Zhang et al. (2020) found based on experimental results that when bottom blasting and multi-hole blasting are applied to slopes with soft interlayers, the dynamic responses of different layers tend to synchronize, and the degree of inconsistency gradually decreases. This finding indicates that when the blast-induced energy is small (below the critical blast energy), the dynamic response of each layer in the slope is primarily influenced by its internal structure. Nan et al. (2021) systematically studied the influence mechanism of soft interlayers on bench blasting effects through numerical simulations and field tests. They found that the thickness and wave impedance of soft interlayers are the main factors affecting bench blasting outcomes. Soft interlayers influence blasting effects by altering the direction of the minimum resistance line and causing energy leakage.

Current research predominantly focuses on unidimensional investigations of blast-induced vibration velocity propagation in rock slopes and frequency-band energy analysis of vibration signals, while neglecting the coupled effects of soft mudstone interlayers. To advance the analysis and prediction of blast-induced vibration propagation in rock slopes with soft mudstone interlayers, this study transcends conventional single-parameter paradigms through an integrated tripartite research methodology combining in situ field testing, numerical simulation, and dimensional analysis. Centered on the Pinglu Canal Qingnian Hub project, we conducted full-scale field experiments to measure three-component blast-induced vibration velocities and frequency spectra. Complementing experimental data, finite element modeling established slope models incorporating soft mudstone interlayers with varying dip angles (15°-30°) and thicknesses (2.0-4.0 m). Wavelet packet decomposition was employed to quantify energy propagation characteristics across frequency sub-bands, revealing distinct energy attenuation patterns modulated by interlayer geometry. Ultimately, a generalized prediction model for blast-induced vibration velocity was developed.

The field blasting tests

Project overview

The Qingnian Hub is located at Longwan Reservoir in Qinzhou, serving as the lowermost cascade of the three planned tiers of the Pinglu Canal. It is the first domestic 5000-ton-class dual-line water-saving ship lock at a river-sea interface. Its geographical location is illustrated in Figure 1. The stratigraphic distribution within the hub area is summarized in Table 1. Among these strata, the Fourth Section of the Liantan Group of the Lower Silurian is widely distributed, featuring complex rock mass compositions including mudstone, argillaceous sandstone, and sandstone. The slope is primarily composed of argillaceous soft rock and sandstone, with sandstone accounting for an average of 83% and mudstone making up about 17%. Most slopes contain soft mudstone interlayers, and their key mechanical properties are provided in Table 2.

Figure 1.

Geographical location of the Qingnian Hub.

Table 1.

Strata distribution in the hub area.

Stratum	Composition and distribution
Artificial fill of Holocene (Q4ml)	Plain Fill①1: Composed of silty clay, containing a small amount of plant roots in the surface layer. Exposed by boreholes XY07 and ZT17, with a thickness of 0.5–1.1 m
Alluvial-diluvial deposits of Holocene (Q4al+pl)	Silt②1-1: Mainly composed of clay particles, with a thickness of 1–2 m, distributed at the bottom of the downstream approach channel pond
	Gravelly Sand②3: Primarily composed of quartz, with medium-coarse sand content of 50–60%, gravel content of 30–40%, and fine silt content of less than 10%. Distributed in the Qin river bed, exposed by borehole QNZ03, with a thickness of 6.5 m
	Silty Clay③2-1: Brownish-yellow, hard plastic, with a thickness of 3.5–8.0 m, distributed in the upper part of the first terrace on both banks of the Qin river
	Silty Clay③2-3: Brownish-yellow, soft plastic, exposed by borehole QNZ02, with a thickness of 4.0 m
Eluvial-diluvial Deposits (Qel+dl)	Silty Clay④1: Distributed on hill slopes on both banks, with a thickness of 0.7–5.0 m
Upper member of upper Cretaceous (K22b)	Silty mudstone interbedded with argillaceous siltstone, with silty mudstone accounting for about 92% and argillaceous siltstone for about 8%
Lower Jurassic (J2)	Interbedded medium-thick argillaceous siltstone and mudstone with sandstone interlayers
Fourth section of liantan group, lower Silurian (S1lnd)	Medium-thick sandstone interbedded with thin-layer mudstone, with sandstone accounting for about 83% and mudstone for about 17%

Table 2.

Mechanical properties of rocks.

Rock type	Longitudinal wave velocity V_P/(m/s)	Transverse wave velocity V_S/(m/s)	Dynamic Poisson’s ratio μ_d	Dynamic modulus of elasticity E_d/(GPa)	Density ρ/(g/cm³)
Sandstone	2585	1378	0.30	13.35	2.62
Mudstone	2058	1009	0.34	7.10	2.72

Rock slopes with soft mudstone interlayers in this area are generally gently inclined, with interlayer dip angles mostly ranging between 2° and 28°. The overall thickness of mudstone layers typically varies from 1.2 m to 5.0 m.

Field blasting tests

In order to further explore the propagation patterns of blast-induced vibration in rock slopes containing soft mudstone interlayers, full-scale physical model tests were conducted on rock slopes with geological conditions consistent with those at the Phase I construction area of the Qingnian Hub. The selected experimental slope has a height of approximately 60 m and a slope gradient of about 40°∼50°. The slope is divided into three experimental areas, each with different soft mudstone interlayer thicknesses and dip angles, based on varying stratigraphic conditions. The experimental slope and its surface are shown in Figure 2.

Figure 2.

Experimental slope.

After the experimental slope was selected, boreholes were drilled at a distance of 6 m from the toe of the slope for explosive loading, with additional boreholes spaced at 4-m intervals for the same purpose. Each borehole had a diameter of 100 mm. Emulsion explosives were used, with a cartridge diameter of 90 mm, loaded from the slope toe outward. The charge weights per borehole were 48 kg, 42 kg, and 36 kg, corresponding to a linear charge density of 7.84 kg/m. The arrangement of boreholes is illustrated in Figure 2. After loading the explosives, the boreholes were stemmed with a mixture of clay and drill cuttings to lengths of 3.88 m, 4.65 m, and 5.41 m, respectively. Detonation was initiated from Borehole #1, followed by subsequent holes with a delay time of 1 second between successive detonations.

The TC-4850 blasting seismograph was used as the vibration monitoring equipment, with monitoring points arranged at different elevations along the three slope surfaces. The arrangement of the monitoring points is shown in Figure 2. This monitoring device can measure blast-induced vibration velocity in three directions. For ease of data recording, the horizontal direction along the borehole was selected as the X direction, the direction perpendicular to X in the horizontal plane as the Y direction, and the vertical direction perpendicular to the XY plane as the Z direction.

The peak values of the blast-induced vibration velocity at each monitoring point obtained from the field tests are shown in Table 3.

Table 3.

Peak blast-induced vibration velocity at each monitoring point in the field test.

Slope number	Monitoring point	Explosive center distance/m	Elevation difference/m	Peak blast-induced vibration velocity/(cm’s^-1)
Slope number	Monitoring point	Explosive center distance/m	Elevation difference/m	X-direction	Y-direction	Z-Direction
Ⅲ	V7	13.40	0.11	12.24	16.40	35.20
		9.90		13.40	16.84	34.12
		6.10		18.36	22.63	38.54
	V8	23.30	9.40	7.04	6.14	10.07
		19.60		4.09	4.06	3.93
		16.30		7.54	9.45	10.2
	V9	73.30	52.63	0.67	0.64	3.02
		70.70		0.16	0.20	0.80
		68.10		0.42	0.38	2.74

From Table 3, it can be observed that the peak blast-induced vibration velocity gradually decreases with the increase in blasting distance and elevation. Comparing the blast-induced vibration velocities in the three directions, the peak vibration velocity in the Z direction is generally the largest. This indicates that the blast-induced vibration velocity in the Z direction has the greatest impact on slopes with soft mudstone interlayers.

To further investigate the dynamic response characteristics of slopes with soft mudstone interlayers under blast-induced vibration, a set of typical blast-induced vibration velocities was selected for Fourier transformation, yielding the frequency spectra of different measurement points after detonation of different boreholes. The spectra are shown in Figure 3.

Figure 3.

Frequency spectra of blast-induced vibration velocities at different measurement points.

From Figure 3, it can be further observed that the amplitude corresponding to different frequencies in the Z direction is generally the largest at each measurement point, and the dominant frequencies are mostly below 50 Hz. The occurrence of dominant frequencies greater than 50 Hz and amplitudes larger than those in the Z direction for the X and Y directions may be due to wave reflection and superposition during the propagation of blast-induced vibration waves, leading to such discrepancies. Additionally, the amplitude of the blast signals decreases significantly as the vibration wave propagates.

During the blast-induced process, explosive energy primarily propagates through compressional waves (P-waves) and shear waves (S-waves). Compressional waves exhibit the fastest propagation velocity and concentrated energy, with their vibration direction aligning parallel to the wave propagation direction. When the blast source is located at the slope toe, the vertical release of explosive energy is intensified due to the combined effects of gravitational forces and in-situ rock stress. This superposition leads to a significant amplification of the vertical vibration component (Z-direction) (Chong, 2018; Gong and Wang et al., 2005). Overall, this further illustrates that the blast-induced vibration velocity in the Z direction has the greatest impact on slopes containing soft mudstone interlayers. Therefore, the subsequent research in this study focuses exclusively on the analysis and prediction of blast-induced vibration velocity in the Z direction.

Finite element numerical model and reliability verification

Finite element numerical modeling

Based on a typical slope selected for this study and incorporating topographic data from field tests, a simplified slope model containing a soft mudstone interlayer was established using ANSYS/LS-DYNA numerical simulation software, as shown in Figure 4. Due to technical challenges in achieving full-scale replication of the field-tested slope geometry in numerical modeling, the numerical simulations adopted an average slope height of 60 m, derived from field investigation reports, the slope gradient is 45°, the stratigraphic structure is simplified as the interlayer structure of moderately weathered sandstone and moderately weathered mudstone, with the dip angle of 28°, and the layer thickness of 3.0 m. The setup of the holes is consistent with the field test. The model is a quasi two-dimensional slope model with the same coordinate direction as that of the field test, the Z-axis is along the height direction of the model, and the unit of calculation adopts the cm-g-μs system. The model is set as non-reflective boundary except the free top surface. Explosive material model, in the numerical simulation using *MAT_HIGH_EXPLOSIVE_BURN material model (Jiang et al., 2020; Peng et al., 2025). The relationship between pressure and specific volume in the simulated explosive detonation process is the JWL equation of state as in equation (1) (Jiang et al., 2023; Xie et al., 2017):

p = A (1 - \frac{ω}{R_{1} V}) e^{- R_{1} V} + B (1 - \frac{ω}{R_{2} V}) e^{- R_{2} V} + \frac{ω E_{0}}{V}

(1)

Where A、B、R₁、R₂、ω denote material constants, p denotes pressure, V denotes relative volume, and E₀ denotes initial specific internal energy. The relevant parameters of the explosives are shown in Table 4.

Figure 4.

Simplified model for numerical simulation.

Table 4.

Explosive parameters.

Density/g⋅cm	Explosive velocity/cm⋅μs^-1	A/GPa	B/GPa	R₁	R₂	$ω$	Initial specific internal energy E₀/GPa
1.1	0.54	214	18.2	4.2	0.9	0.15	4.19

In order to facilitate the calculation, the field rocky slopes are assumed to be continuous and isotropic elastic-plastic materials. In this paper, the *MAT_PLASTIC_KINEMATIC material model is selected for sandstone and mudstone layers, and this model is applicable to isotropic plastic randomly reinforced materials with strain rate effect, which can simulate the dynamic stress-strain behavior of the rock body(Jiang and Zhou, 2012; Ren et al., 2025; Yang et al., 2024).The material parameters are shown in Table 5.

Table 5.

Material parameters.

Material	Mass density	Young’s modulus	Poisson’s ratio	Yield stress	Tangent modulus	Hardening parameter
Mudstone	2.72	0.30	0.34	0.0028	0.04	0.5
Sandstone	2.62	0.13	0.30	0.003	0.04	0.5
Clogging	2.60	0.21	0.20	0.001	0.04	0.5

Validation of numerical simulation results

This study focuses on the blast-induced vibration velocities at Slope III. A comparative analysis between the numerically simulated frequency spectra and field-measured spectra is presented in Figure 5. As shown in Figure 5, the simulated spectra closely match the experimental spectra, with peak amplitude frequencies below 50 Hz for both cases, falling within the low-frequency range. Notably, the numerical results exhibit smoother amplitude decay and greater stability beyond 100 Hz due to the absence of noise interference inherent in field tests. These observations validate the high reliability of the numerical model’s construction.

Figure 5.

Comparison of numerical results and field test blast-induced frequency spectra.

Blast-induced ground vibration propagation of slopes with soft mudstone interlayers

Blast-induced vibration velocity propagation under the influence of soft mudstone interlayers

In order to further analyze the blast-induced vibration velocity propagation law under different soft mudstone interlayer thicknesses, combined with the actual project, the use of ANSYS/LS-DYNA finite element numerical simulation software in the original model to change the other four groups of different soft mudstone interlayer thicknesses, respectively, for the 2.0 m, 2.5 m, 3.5 m, 4.0 m, as shown in Figure 6. Because the change of the thickness of the interlayer has little effect on the blast-induced vibration velocity at the nearest measurement point from the source, only the two measurement points above and below the soft mudstone interlayer in the numerical simulation are selected to calculate and analyze the blast-induced vibration velocity, and the peak blast-induced vibration velocity in the Z-direction is shown in Figure 8.

Figure 6.

Model diagram of different soft mudstone interlayer thicknesses.

Similarly, to examine the effect of the soft mudstone interlayer dip angle on the propagation of blast-induced vibration velocity, four additional numerical models with dip angles of 30°, 25°, 20°, and 15° were established based on a fixed interlayer thickness of 3 m, in accordance with actual engineering conditions. These models are illustrated in Figure 7. The peak blast-induced vibration velocities in the Z-direction at each measurement point under different dip angles, as obtained from numerical simulations, are presented in Figure 9.

Figure 7.

Model diagram of dip angle of different soft mudstone interlayer.

As shown in the calculations in Figures 8 and 9, as the distance from the blast center increases, the blast-induced vibration velocity corresponding to different thicknesses of the soft mudstone interlayer decreases and eventually stabilizes. Before the vibration wave reaches the lower side of the interlayer, the peak blast-induced vibration velocity is relatively high even as the thickness of the interlayer increases. Beyond the upper side of the interlayer, however, the peak vibration velocity becomes significantly smaller with increasing interlayer thickness.

Figure 8.

Numerical simulation results of different soft mudstone interlayer thicknesses.

Figure 9.

Numerical simulation results of different soft mudstone interlayer dip angels.

This phenomenon can be attributed to the fact that the soft mudstone interlayer in the slope model consists of mudstone, which exhibits a strong vibration attenuation effect that becomes more pronounced with greater thickness. The mudstone layer, being a softer medium, possesses a higher capacity for absorbing and attenu vibrational energy. Therefore, increasing the thickness of the mudstone layer—while keeping other conditions constant—reduces the peak vibration velocity at the slope surface, which is consistent with the numerical simulation results.

Similarly, the blast-induced vibration velocity under different interlayer dip angles follows the attenuation trend wherein vibration velocity decreases with increasing distance from the blast center. Moreover, as the dip angle of the soft mudstone interlayer increases, the overall vibration velocity at measurement points located above the interlayer is further reduced. This may be due to the fact that steeper dip angles bring the soft mudstone interlayer closer to the upper part of the slope, thereby enhancing the attenuation effect on blast-induced vibrations in the upper layers and resulting in lower recorded velocities.

Relevant studies have pointed out that the material density is the most important factor affecting the attenuation rate of blast-induced vibration wave propagation in soft mudstone interlayers (Bhagade et al., 2021; Gou et al., 2019; Wang et al., 2021; Yu et al., 2020). The attenuation rate is directly proportional to the material density, i.e., the greater the material density, the faster the attenuation of the stress wave in the propagation process. This is because denser materials have a stronger damping effect on the stress wave, resulting in an increase in energy loss, which makes the amplitude of the blast-induced vibration wave decrease rapidly in the propagation process. In this actual project, the density of mudstone is greater than that of sandstone, thus further explaining the reason for the attenuation of the peak blast-induced vibration velocity.

Blast-induced vibration energy attenuation under the influence of soft mudstone interlayers

As shown in Figure 3, the blast-induced vibration waves exhibit complex frequency content in both low- and high-frequency ranges. To analyze the vibration velocity components across these frequencies, wavelet packet decomposition was applied to the numerical simulation results. A sampling rate of 2048 Hz, consistent with the field test, was selected, resulting in a Nyquist frequency of 1024 Hz according to the sampling theorem. The numerical simulation outputs were decomposed into six layers using the db8 wavelet basis, divided into seven frequency bands: 0–16 Hz, 16–32 Hz, 32–64 Hz, 64–128 Hz, 128–256 Hz, and 256–1024 Hz. In wavelet packet decomposition, the energy at each node can be calculated using the squared norm of its coefficients, as given by the following equation:

E_{i} = \sum_{j = 1}^{N} c_{i j}^{2}

(2)

where E_i denotes the energy of the ith node, c_ij denotes the jth coefficient of the ith node, and N denotes the number of coefficients of the node.

The energy values of different soft mudstone interlayer dip angels and different soft mudstone interlayer thicknesses and different frequency bands in their corresponding measurement points can be obtained by calculation, as shown in Figures 10 and 11. Since the energy is obtained by calculating the squared paradigm (i.e., the square of the Euclidean paradigm) of the wavelet packet coefficients. This calculation is a quantification of the signal energy, but it does not have a specific physical unit, therefore, the “energy” unit is dimensionless, they represent the relative magnitude of the signal amplitude, used to analyze the spectral content of the blast-induced vibration velocity signal.

Figure 10.

Energy values for different soft mudstone interlayer dip angels.

Figure 11.

Energy values for different soft mudstone interlayer thicknesses.

The calculation results clearly indicate that regardless of whether the variable property of the soft mudstone interlayer is its dip angle or thickness, the attenuation of blast-induced vibration energy is highly significant, with the total energy attenuation exceeding 80%. Variations in the thickness of the soft mudstone interlayer lead to considerable changes in energy values, preliminarily suggesting that thickness has a greater influence on energy attenuation of blast-induced vibrations. Furthermore, the attenuation of energy with increasing thickness and dip angle of the soft mudstone interlayer follows a trend consistent with the corresponding attenuation law of peak vibration velocity. This demonstrates that although blast-induced vibration energy is related to vibration velocity, frequency, and duration, the peak vibration velocity remains the most representative indicator of the energy magnitude.

In order to better analyze the impact of soft mudstone interlayers on the energy of different frequency bands, the energy share of different frequency bands is plotted, as shown in Figures 12 and 13.

Figure 12.

Energy share of each frequency band for different soft mudstone interlayer dip angels.

Figure 13.

Energy share of each frequency band for different soft mudstone interlayer thicknesses.

Overall, the energy is mainly concentrated in the low-frequency band, whose main vibration frequency band is at 16 Hz ∼ 32 Hz, and the energy in the high-frequency band accounts for less than 1%. Combined with the calculation results in Figures 10 and 12, the soft mudstone interlayer absorbs more energy in the 4th and above frequency bands, especially in the 5th frequency band, so the energy share of the 2nd and 3rd frequency bands is larger in V9 measurement points, and with the increase of the dip angle of the soft mudstone interlayer, the energy share in the 2nd frequency band is larger, and the energy share is more than 40% when the dip angle is 30°. When the dip angle is 30°, the energy share of band two reaches more than 40%.

Combined with the calculation results in Figures 11 and 13, the increase of the thickness of the soft mudstone interlayer absorbs the energy of the high-frequency band more significantly, and the energy share of the 2nd frequency band increases sequentially, and the energy share of the main vibration band reaches more than 45% when the thickness of the layer is 4.0 m. Therefore, combined with the above analysis, the energy share of the main vibration band is more than 40% when the dip angle is 30°. Therefore, combined with the above analysis, the thickness of the soft mudstone interlayer has a greater impact on the blast-induced vibration energy. Overall, with the increase of the dip angle and thickness of the soft mudstone interlayer, the energy of the low-frequency band is more significant, which is closer to the self-oscillating frequency of the slope, and is prone to resonance damage, so it has certain significance in guiding the actual engineering construction (Sun and Li 2023).

The rock mass is a non-homogeneous, anisotropic discontinuous structure randomly intersected by faults, joints, fracture zones and other different types of soft surfaces. When the explosion seismic wave propagates in the jointed rock body, the attenuation of the explosion vibration will be affected by the characteristics of the soft and soft surfaces, as well as by the rock wave impedance on both sides of each discontinuity, the angle of the discontinuity and the propagation direction. In the existing related research (Abdelwahed and Abdel-Fattah 2015; Babanouri et al., 2020; Farrokhi et al., 2016; Kumar et al., 2018), scholars have found that when the blast-induced vibration wave passes through the soft mudstone interlayer surface of the rock body, the energy of the vibration wave will be dissipated, and when the vibration wave passes through the soft mudstone interlayer surface to produce transmitted and reflected waves, the energy of the vibration wave will be transmitted. While the attenuation of blasting vibratory waves in discontinuous media is affected by wave reflection, frictional sliding and viscous damping in saturated media, the rock body contains small faults, joints, and softer sections, which divide the rock body into blocks. Depending on the size and separation distance, these discontinuities may not affect the propagation of low-frequency long waves if the propagation wavelength is much longer than the separation distance of the discontinuities. This further explains that changes in the thickness and dip of the soft mudstone interlayer have little effect on the energy in the low-frequency band.

Prediction of blast-induced vibration velocity propagation law for slopes with soft mudstone interlayers

Predictive modeling of blast-induced vibration velocity propagation

Based on the analysis of the corresponding characteristics of the slope dynamics under the action of blast-induced vibration in Section 3, the blast-induced vibration wave propagation process affects factors including the distance between the measurement point and the source of the blast, i.e., the center of the blast distance R, the difference in elevation H, the maximum section of the charge Q, as well as the nature of the rock, etc., the specific physical quantities are shown in Table 6, in which M, L and T are the magnitude of the mass, length and time, respectively (Han et al., 2024; Jiang et al., 2022).

Table 6.

Physical quantities of blast-induced vibration velocity influencing factors.

Influencing factor	Dimension	Influencing factor	Dimension
Explosive Quality Q	M	Elevation difference H	L
Explosive center distance R	L	Peak blast-induced vibration velocity v	LT-1
Rock density ρ	ML-3	Blast-induced vibration wave propagation velocity c	LT-1

From Buckingham’s theorem (π theorem), the influences in Table 6 are analyzed quantitatively, where the peak blast-induced vibration velocity v can be expressed as:

v = φ (Q, H, R, ρ, c, θ)

(3)

According to π-theorem, independent measures Q, R and c are taken to create dimensionless variables denoted by π:

π = \frac{v}{c}, π_{1} = \frac{ρ}{{Q R}^{- 1}}, π_{2} = \frac{θ}{1}, π_{3} = \frac{H}{R}

(4)

This can be obtained by substituting equation (4) into (3):

\frac{v}{c} = φ (\frac{ρ}{{Q R}^{- 3}}, θ, \frac{H}{R})

(5)

Since the product and multiplier of different dimensionless numbers are still dimensionless, the above dimensionless numbers π₁, π₂ and π₃ are taken to further construct the dimensionless numbers:

π_{4} = {(π_{1})}^{- \frac{1}{3}} π_{2} π_{3} = \frac{\sqrt[3]{Q}}{R \cdot \sqrt[3]{ρ}} \cdot \frac{H}{R} \cdot θ

(6)

In contrast, the rock-related properties ρ and c are determined in different working conditions, so they can be considered as constants, leading to the following equation:

v = k {(\frac{\sqrt[3]{Q}}{R})}^{α} {(\frac{H}{R})}^{β_{1}} {(\frac{θ}{π})}^{β_{2}}

(7)

Prediction of blast-induced vibration velocity propagation

Various researches and tests have found that the traditional prediction model is not accurate in slopes containing soft interbedded slopes with complex topographic conditions, so many scholars have proposed a prediction model that can respond to the correction of elevation amplification effect to improve the accuracy of prediction (Jiang et al., 2017; Tang and Li, 2011), as shown in the following equation:

v = k {(\frac{\sqrt[3]{Q}}{R})}^{α} {(\frac{H}{R})}^{β}

(8)

In the predictive model proposed in this study, k represents the site coefficient related to the physical properties of the rock mass, α denotes the attenuation coefficient for blast-induced vibration velocity, and β is the influence coefficient accounting for elevation differences.

The fitting results of the traditional Sadowski predictive model are summarized in Table 7.

Table 7.

Fitting results of the traditional Sadowski predictive model.

Predictive model	Fitting results	r ²
$v = k {(\frac{\sqrt[3]{Q}}{R})}^{α}$	$v = 73.67 {(\frac{\sqrt[3]{Q}}{R})}^{1.055}$	0.772
$v = k {(\frac{\sqrt[3]{Q}}{R})}^{α} {(\frac{H}{R})}^{β}$	$v = 80.37 {(\frac{\sqrt[3]{Q}}{R})}^{1.038} {(\frac{H}{R})}^{0.136}$	0.853

Equation (7) can be regarded as the traditional Sadowski’s formula to take into account the elevation amplification effect and the influence of the dip angle of the soft mudstone interlayer on the peak velocity of blast-induced vibration at the measurement point. Combined with the use of numerical simulation software in Section 3 to obtain different soft mudstone interlayer dip angle of the peak velocity of blast-induced vibration at different measurement points, Equation (7) is fitted, the fitting results are shown in Table 8, where r² is the fitting coefficient of determination.

Table 8.

Fitting results considering the effect of the dip angel of the interlayer.

Consideration	Predictive model	k	α	β ₁	β₂	r ²
θ	$v = k {(\frac{\sqrt[3]{Q}}{R})}^{α} {(\frac{H}{R})}^{β_{1}} {(\frac{θ}{π})}^{β_{2}}$	16.4	2.21	0.64	0.57	0.944

In order to further investigate the influence of soft mudstone interlayer thickness on the propagation law of blast-induced vibration velocity, on the basis of equation (7), the introduction of interlayer thickness influence coefficient φ, the meaning of which is the ratio of the peak vibration velocity of the measurement point at the same location under the different interlayer thicknesses under the study to the peak vibration velocity of the measurement point at a certain location of the slope surface under the same thickness of the interlayer, with the following expression:

φ \frac{v}{v_{0}}

(9)

Therefore, equation (8) can be rewritten as the following equation, taking into account the effect of the thickness of the soft mudstone interlayer:

v = φ k {(\frac{\sqrt[3]{Q}}{R})}^{α} {(\frac{H}{R})}^{β}

(10)

Based on the results of the numerical simulation, the coefficients of the influence of the thickness φ of five groups of different soft mudstone interlayer thickness models at the two measurement points are linearly fitted, and the fitting effect is shown in Figure 14.

Figure 14.

Linear fit of the coefficient of influence of the thickness of the soft mudstone interlayer.

With the above fitting results, taking the soft mudstone interlayer thickness as D, equation (10) can be rewritten as follows:

v = (- 0.14284 D + 1.46086) k {(\frac{\sqrt[3]{Q}}{R})}^{α} {(\frac{H}{R})}^{β}

(11)

Combining equation (11) and the results calculated by the numerical simulation software for fitting analysis, the fitting results are obtained as shown in Table 9:

Table 9.

Fitting results considering the effect of soft mudstone interlayer thickness.

Consideration	Predictive model	k	α	β	r ²
D	$v = (- 0.14284 D + 1.46086) k {(\frac{\sqrt[3]{Q}}{R})}^{α} {(\frac{H}{R})}^{β}$	14.99	2.63	0.95	0.952

The fitting results presented in Tables 8 and 9 demonstrate that the predictive formulas incorporating soft mudstone interlayer dip angles and thicknesses achieve coefficients of determination around 0.95, indicating high overall prediction accuracy. This validates the reliability of equations (7) and (11) for predicting blast vibration attenuation patterns in slopes with varying interlayer geometries. A comparative analysis of results from Tables 7 –9 further reveals that the enhanced predictive model, which accounts for interlayer dip angles and thicknesses, exhibits superior accuracy compared to conventional approaches.

Conclusions

With the background of Pinglu Canal Qingnian Hub Phase I Project, the dynamic response of slopes containing soft mudstone interlayer under the action of blast-induced vibration is analyzed by full-size test model and using ANSYS/LS-DYNA numerical simulation software to establish models with different parameters, and the attenuation models under the action of considering the thickness of the interlayer and considering the effect of the dip angel of the interlayer are fitted and analyzed, and the following conclusions are drawn.

(1) Z-direction blast-induced vibration velocity peak value is the largest, X, Y, Z-direction blast-induced vibration signal main frequency are roughly less than 50 Hz, and Z-direction amplitude is the largest, the main vibration direction.

(2) Before the lower layer of the soft mudstone interlayer, with the increase of the thickness of the soft mudstone interlayer, the peak value of its blast-induced vibration velocity is overall large, but after the upper layer of the soft mudstone interlayer, with the increase of the thickness of the soft mudstone interlayer, the peak value of its blast-induced vibration velocity is overall small.

(3) As the dip angle of the soft mudstone interlayer increases, the soft mudstone interlayer is closer to the upper portion of the slope, which has a greater attenuation effect on the vibration velocity of the upper layer, thus leading to a reduction in the vibration velocity.

(4) With the increase of the soft mudstone interlayer dip angel and thickness, the low-frequency band energy accounted for a more significant proportion of the slope with the more close to the self-oscillation frequency, easy to resonate damage, and the thickness of the soft mudstone interlayer has a greater impact on the vibration energy of the blasting.

(5) The modified Sadowski’s formula, which takes into account the dip angel, thickness, and elevation amplification effects of the soft interbedded layer, is fitted with higher accuracy, which reaches about 95% in all cases.

Footnotes

ORCID iDs

Nan Jiang

Chuanbo Zhou

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is supported by the Natural Science Foundation of Hubei Province (Grant No. 2024AFA092), the National Natural Science Foundation of China (Grant No. 52578584 and No. 52478525), and the Hubei Province's Special Fund for Centrally Guided Local Scientific and Technological Development (Grant No.2024CSA094).

Declaration of conflicting interests

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

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