1 INTRODUCTION
Vibration of weak geological structures and existing buildings induced by rock mass blasting excavation poses significant risks to the safety and stability of underground infrastructure to a large extent (Xie et al., 2020). With the continuous development of infrastructure construction, the construction of deep-buried tunnels is becoming increasingly prevalent, with more dense distributions. As a result, existing tunnels are often impacted by blasting excavation. Especially owing to faults present within the rock mass, stress waves generated from blasting excavation will separate and attenuate, rendering propagation processes arduous to foresee. Therefore, elucidating blast stress wave propagation in a deep rock mass containing faults is a key scientific issue to ensure the safety and stability of underground projects.
The vibration of existing tunnels induced by blasting excavation is commonly observed and predicted using peak particle velocity (PPV) (Hendron, 1977; Langefors & Kihlstrom, 1978). Based on this, Li et al. (2013) proposed a theoretical method incorporating consideration of PPV and stress distribution, analyzing tunnel wall vibration under combined blast and ground reflected motions. Li et al. (2016) studied the tunnel vibration from unloading, and the results showed that the PPV induced by the unloading action could be as high as the PPV induced by a blast wave. Based on field testing and LS-DYNA numerical simulations, Wang et al. (2022) discussed the propagation characteristics and attenuation prediction equation of blast-induced vibration on closely spaced rock tunnels. Considering the existence of in situ stress in a rock mass, Li et al. (2020) applied theoretical and discrete element numerical methods to study the dynamic response of existing tunnels subjected to unloading waves in a high-stress rock mass and found that a high-stress environment is more likely to cause dynamic failure of the surrounding rock. Li et al. (2023) established a blasting vibration prediction equation based on the geological strength index of surrounding rock onsite and made an approximate prediction of the blast-induced vibration of the existing adjacent tunnel. Furthermore, on the basis of practical engineering scenarios, many scholars have studied the vibration problems for existing tunnels caused by blasting construction of new tunnels through theoretical analysis, field tests, and numerical simulation methods (Dang et al., 2018; Duan et al., 2019; Liu et al., 2020; Xie & Tang, 2018).
However, most of the above studies do not fully consider the influence of faults within a rock mass on blast waves. The stress waves will be transmitted and reflected at faults, generating transmitted and reflected waves on both sides. Currently, the dominant theoretical methods for analyzing stress wave propagation at faults are grounded upon the displacement discontinuity theory (Schoenberg, 1980), including the characteristic line method proposed by Zhao and Cai (2001) and the time-domain recursive method (TDRM) proposed by Li et al. (2010). The TDRM has been extensively utilized by scholars to study various fault models, such as a linear model (Chai et al., 2016), a nonlinear model (Chai et al., 2021), and a slippery model (Li et al., 2011). Furthermore, on the basis of modified TDRM (Li, Li, et al., 2012), Chai et al. (2023) proposed a three-element model accounting for filled layer thickness and deduced plane P- and S-wave propagation equations at structure interfaces. The results indicated that the filled layer thickness profoundly impacts stress wave transmission. However, these above studies only consider the attenuation caused by a fault or structural plane, while the factors causing wave attenuation in practical engineering scenarios are multifaceted.
Deep rock masses contain not only faults but also a vast number of microscopic fractures, rendering both rock materials and faults viscoelastic and capable of attenuating stress wave propagation (Walsh, 1966). Prior studies have confirmed that almost all natural rocks show viscoelastic behavior (Johnson & Rasolofosaon, 1996; Lucet & Zinszner, 1992). In view of this, many scholars have researched the influence of rock mass viscoelasticity on stress wave propagation. For example, Li et al. (2018) applied the Kelvin viscoelastic model to analyze the ground vibration caused by a seismic wave passing through one fault. On this basis, Chai et al. (2017) studied the influence of wave frequency, quality factors, and other parameters on transmission and reflection coefficients. On the basis of the modified TDRM, Wang et al. (2017, 2021) used an SLS model to study P- and SH-wave propagation at viscoelastic filled joints. Collectively, the above studies show that the rock mass or fault viscoelasticity cannot be discounted from stress wave propagation. In particular, the viscoelasticity of a natural rock mass will change with the change of the external environment. Especially in deep rock masses, the closure of small cracks in rock masses under the action of in situ stress will change, thus affecting the attenuation of blast waves. There have been studies on the relationship between rock depth and rock viscoelasticity. Generally, with the increase of depth or in situ stress, the viscoelasticity of rock masses gradually decreases (Dvorkin, 2005; Winkler & Nur, 1982), which is more conducive to the propagation of stress waves. Therefore, a deep-buried tunnel is more prone to damage under the action of blast stress waves.
Most of the previous studies have analyzed the tunnel vibration generated by plane waves. However, waves arising from practical blasting excavation scenarios typically propagate initially as cylindrical fronts within nearfields. Cylindrical wave propagation through rock undergoes not only viscoelastic attenuation but also geometric damping. Li, Ma, et al. (2012) analyzed the vibration of existing tunnels caused by underground blast waves using a simplified model. On this basis, Chai, Tian, et al. (2020) and Chai et al. (2021) used numerical and theoretical methods to analyze the propagation law of cylindrical waves in a rock mass containing structural plane and subsequent surface motions. However, theoretical research on cross-fault existing tunnel vibrations caused by cylindrical waves remains scarce.
Therefore, in this study, the TDRM is used to analyze the vibration of a cross-fault adjacent existing tunnel induced by blasting excavation of a new deep-buried tunnel. The rock mass and fault are assumed to satisfy the Kelvin model, the three-element fault model and the blast cylindrical wave propagation model are established by introducing quality factors, and the propagation equation of a blast wave at the fault is derived to obtain the PPV of the adjacent existing tunnel wall. Finally, the influences of fault dip angle, fault thickness, blast wave frequency, and tunnel burial depth on the PPV are analyzed. This paper provides a valuable addition to theoretical research of blast wave propagation in deep engineering rock masses with faults, and provides a theoretical basis for better analysis of the dynamic response of existing tunnels under blasting, prediction of the existing tunnel stability, and protection of tunnel construction safety.
2 THEORETICAL DERIVATION
2.1 Presentation of the problem
As shown in Figure 1, a fault exists within the natural rock mass, with a deep-buried tunnel situated on one side. Blast waves are generated when the charge tunnel is excavated on the opposing fault side. The blast stress waves diffuse in the form of cylinders and propagate in the rock mass, some of which will pass through the fault and continue to propagate, and finally impact on the existing tunnel and induce tunnel wall vibration.
Diagram for the influence of cross-fault tunnel excavation on an existing tunnel.
Since the natural rock mass contains abundant microscopic fractures, minerals, and impurities, the rock mass in Figure 1 can be regarded as a viscoelastic body, and the amplitude attenuation effect will occur when blast waves propagate. Since the natural faults usually contain various filled materials, the fault in Figure 1 is regarded as a filled structural plane with a certain thickness. The filled materials also show viscoelasticity under long-term weathering and extrusion. The blasting wave will be transmitted and reflected at the contact interfaces between the fault and the rock mass, and the stress waves will be reflected many times between the two contact interfaces, which makes the propagation path of blasting waves very complicated. In addition, the rock mass is exposed to a certain amount of in situ stress, which is generated by gravity stresses and tectonic stresses. The deep-buried tunnels are usually located in high in situ stress rock masses. The magnitude of in situ stress will affect the mechanical properties of faults and microscopic fractures, and then affect blasting wave propagation. To sum up, the vibration on the adjacent existing tunnel is related to various factors, and the propagation path of blast waves reaching the adjacent existing tunnel is very complicated. The specific analysis process and influencing factors are shown in Figure 2.
Specific analysis process and influencing factors.
2.2 Analysis of the cylindrical P-wave propagation path
The vibration problem of the adjacent existing tunnel wall caused by blasting excavation is simplified into a plane problem, and a rectangular coordinate system is established as shown in Figure 3, where the explosion source and the center of the adjacent existing tunnel are assumed to be on the same horizontal level. If there is a certain level difference between two tunnels, the model in Figure 3 is still applicable, and the calculation can be performed again only by changing the direction of the rectangular coordinate system. The coordinates of O1 and O2 are (−L,0) and (L,0), respectively, the radius of the adjacent existing tunnel is R, the angle between the fault and the X-axis is γ, and the maximum value range of the cylindrical P-wave angle η is (−180°, 180°).
Diagram of cylindrical P-wave propagation.
In Figure 3, it is assumed that the fault length is long enough and the value range of fault dip angle γ is (γmin, 180°−γmin), where
. According to the magnitude of γ, the value range of cylindrical P-wave angle η can be divided into the following three cases, where
.
As shown in Figure 4a, in this case, the critical stress waves reaching the upper and lower parts of the existing tunnel are the P- and the S-wave, respectively.
Diagram of cylindrical P-wave propagation. (a)
, (b)
, and (c)
.
As shown in Figure 4b, in this case, the critical stress waves reaching the upper and lower parts of the existing tunnel are both S-waves.
As shown in Figure 4c, in this case, the critical stress waves reaching the upper and lower parts of the existing tunnel are the S-wave and the P-wave, respectively.
2.3 Attenuation of a cylindrical wave
The attenuation of a cylindrical wave in a rock mass includes geometric attenuation and physical attenuation. The geometric attenuation is due to the increase of the spatial distribution of stress wave energy, which causes the cylindrical wave emitted from the explosion source to decay at the ratio of (1/
r)
1/2. According to the law of energy conservation on wave fronts, the total energy of cylindrical waves from the same wave source passing through different wave fronts in unit time is the same, that is,
(1)
where
I
j and
I
1 are the corresponding energy intensities when the radii of the wave front are
r
j and
r
1, respectively. According to the kinetic energy theorem:
(2)
where
m
j and
V
j represent the mass and the relative volume of rock, respectively, and
v
j is the particle vibration velocity of the cylindrical wave at the radius of wave front
r
j. According to Equations (
1) and (
2),
(3)
Assuming that the particle vibration velocity at the initial wave front of the cylindrical P-wave is
v
0, the incident wave at the fault can be expressed as
(4)
where
r
0 is the radius of the initial wave front;
l is the distance that the cylindrical P-wave propagates from the initial wave front to the fault.
The physical attenuation of the cylindrical wave is caused by viscoelasticity, that is, by the friction effect of microscopic fractures and mineral impurities in the rock mass. According to a previous study (Chai et al.,
2023), the rock mass can be equivalent to the Kelvin model, and the stress waves attenuate in the form of a negative exponent in the rock mass. Since the time delay effect of stress waves caused by rock mass viscoelasticity has little influence on tunnel vibration, this study ignores the time delay of cylindrical waves. Then, the particle vibration velocity of incident waves at the fault can be expressed as
(5)
where
a
m is the attenuation coefficient of stress wave propagation in the viscoelastic body. If the quality factors
Q
p and
Q
s of stress waves are introduced, then the parameter
a
m can be expressed as
(6)
where m can denote p or s, corresponding to P- and S-waves, respectively;
ρ is the density of the viscoelastic body;
E
m is the elastic modulus of material; and
ω is the wave angular frequency.
In summary, when considering both geometric attenuation and physical attenuation of cylindrical waves in a rock mass simultaneously, the particle vibration velocity of the incident waves at the fault can be expressed as
(7)
Due to the separation of the cylindrical wave caused by the fault, the wave front of transmitted waves is not unique. In addition, blasting waves in engineering generally only propagate in the near field in the cylindrical form, and as the propagation distance increases, cylindrical waves gradually transform into plane waves. Therefore, this paper assumes that the transmitted waves at the fault continue to propagate in the form of plane waves to the existing tunnel, which means that the attenuation of transmitted waves is only physical attenuation.
2.4 Cylindrical wave propagation at the fault
2.4.1 Theoretical model of stress wave propagation in a filled fault
Considering the attenuation effect of the fault on stress waves and the multiple reflections of stress waves inside the filled layer, based on the research of Chai et al. (2023), the three-element model shown in Figure 5 is adopted to calculate the transmission and reflection of stress waves at the fault in this paper. The three-element model includes two sides of rock material and the intermediate filled material, where the contact interfaces of the rock mass and the filled layer are simplified into thick-less interfaces. It is assumed that the contact interfaces satisfy the discontinuous displacement method, that is, the stresses on both sides are continuous but the displacements are discontinuous. Both the rock mass and the filled layer are viscoelastic and there are initial in situ stresses around them.
Three-element model of a fault.
In Figure 5, Ip represents the incident P-wave at the fault, Rp and Rs represent the reflected P- and S-waves at the fault, respectively, and Tp and Ts represent the transmitted P- and S-waves at the fault, respectively. σV and σH represent the vertical and horizontal in situ stress respectively.
As shown in Figure 6, the contact interfaces between the fault and the rock mass adopt the B–B model with nonlinear hyperbolic deformation characteristics in the normal direction and a linear elastic model in the tangential direction. In Figure 6, “−” and “+” indicate the left and right sides of the interface, respectively.
Scheme of the nonlinear contact interface.
In Figure
6, the contact interfaces satisfy the displacement discontinuous boundary condition, that is, the total normal stress and tangential stress on both sides of the contact interfaces can be written as
(8)
In the normal B–B model,
d
n is the normal closure of the contact interface;
σ
n is the normal effective stress;
d
max is the maximum allowable closure of the contact interface; and
k
n0 is the normal stiffness of the contact interface under initial in situ stress. The relationship between the normal effective stress
σ
n and maximum allowable closure
d
max of the B–B model is as follows:
(9)
where
and
are the normal displacements on the left and right sides of the contact interfaces, respectively.
The tangential stress and displacement of the contact interfaces satisfy the linear elastic model, and the tangential displacement of the contact interfaces can be expressed as
(10)
where
k
s is the tangential stiffness,
and
are the tangential displacements at fault contact interfaces, where – and + represent the left and right sides, respectively.
2.4.2 The propagation equation of stress waves at the fault contact interface
Transmission and reflection will occur when the blast waves propagate to the contact interface 1 on the left side of the fault. According to the modified TDRM (Li, Li, et al., 2012), there are four directions of stress waves at contact interface 1, as shown in Figure 7, where α and β are the incident angles of P- and S-waves, respectively.
Scheme of the stress wave in contact interface 1.
When the four directions of blast waves reach the contact interface, there will be eight tiny units that include the stress wave, the stress wave front, and the contact interface, as shown in Figure 8.
Scheme of stresses and waves on the two sides of the contact interface.
In Figure 8, υ is the Poisson's ratio,
and
represent the normal stresses of the right- and left-running P-waves on their wave fronts, respectively, and
and
represent the tangential stresses of the right- and left-running P-waves on their wave fronts, respectively, where w represents “−” and “+” symbols, representing the left and right sides of the contact interface, respectively.
According to Figure
8, the normal and tangential stress components of left- and right-running waves on the contact interface are
(11)
(12)
(13)
(14)
When the stress waves act on the fault in a deep rock mass, the initial in situ stresses will affect the total stress on the contact interface, and then affect the deformation behavior of the fault. Therefore, considering the distribution of in situ stress, the total normal stress and tangential stress on the left and right sides of the contact interface are
(15)
(16)
(17)
(18)
According to the TDRM (Li et al.,
2010), the momentum conservation is satisfied on the wave front, that is,
(19)
z
p and
z
s are the P-wave impedance and the S-wave impedance, respectively, that is,
,
;
ρ is the density of the propagation medium.
By substituting Equation (
19) into Equations (
15-18), the total normal stress and tangential stress on both sides of the contact interface can be written as:
(20)
(21)
(22)
(23)
where
and
are particle vibration velocities of the right- and left-running P-waves, respectively;
and
are particle vibration velocities of the right- and left-running S-waves, respectively.
According to Figure
8, the total normal and tangential particle vibration velocities on the left and right sides of the contact interface are
(24)
(25)
(26)
(27)
Since the contact interfaces satisfy the discontinuity boundary condition, Equations (
9) and (
10) are differentiated to time
t, respectively, as follows:
(28)
(29)
where
is the time interval;
i is the time parameter.
As shown in Figure
5, the properties of materials on the left and right sides of the contact interface are differentiated, which means that the additional stress of the contact interface caused by stress waves is also different. For the fault in Figure
5, the left side of contact interface 1 is the rock mass and the right side is the filled material, while the left side of contact interface 2 is the filled material and the right side is the rock mass. For contact interface 1, based on the derivation by Chai et al. (
2023) and considering the additional stress caused by stress waves and the in situ stress of the rock mass, the total stress on the left and right sides of the contact interface 1 should be rewritten as:
(30)
(31)
(32)
(33)
where
and
are P-wave impedance and S-wave impedance in the rock material, respectively, that is,
and
,
is the density of rock, and
and
are the propagation speeds of P- and S-waves in rock, respectively.
and
are the wave impedances of P- and S-waves in filled material, respectively, that is,
and
,
is the density of filled materials, and
and
are propagation speeds of P- and S-waves in the filled layer. By substituting Equations (
30)–(
33) into Equation (
8), we obtain
(34)
where
(35)
(36)
(37)
(38)
By substituting Equations (
24)–(
27) into Equations (
28) and (
29), where the normal stress is
σ
+ and the tangential stress is
τ
+, we obtain
(39)
where
C,
D,
E, and
F are represented as follows:
(40)
(41)
(42)
(43)
k
ni is represented as follows:
(44)
Equations (34) and (39) are the propagation equations of stress waves at contact interface 1, and the propagation equations of stress waves at contact interface 2 can be obtained in the same way.
2.4.3 Multiple reflection of stress waves inside a fault
According to previous studies (Chai et al., 2023), stress waves will be reflected several times inside the filled layer of the fault, that is, the stress waves at contact interface 1 (contact interface 2) will pass through the filled layer and propagate to contact interface 2 (contact interface 1), and the stress waves reflected by contact interface 2 (contact interface 1) will pass through the filled layer and return to contact interface 1 (contact interface 2), as shown in Figure 9.
Propagation of stress waves inside the filled layer.
The filled materials are generally subjected to long-term weathering and extrusion, have more internal cracks than rocks, and the viscoelasticity is more obvious. In order to facilitate the operation, it is assumed that the wave quality factor in the filled layer is 1/3 of the rock mass, that is,
,
.
and
are wave quality factors of P- and S-waves in the filled layer, respectively. The attenuation coefficients
and
of P- and S-waves in the filled layer can be obtained from Equation (
6). The equations of stress wave propagation between two contact interfaces can be expressed as
(45)
(46)
(47)
(48)
where
n
p and
n
s are the integers of time steps for stress waves to propagate through the filled layer, that is,
(49)
(50)
where
and
are the propagation times of P- and S-waves between the two contact interfaces, respectively, which can be obtained from the following equations:
(51)
The propagation of stress waves at contact interface 1 can be calculated using Equations (34) and (39). By interchanging the materials on both sides of the contact interface, the stress wave propagation equation at contact interface 2 can also be calculated from the above analysis. The transmitted waves at contact interface 2 are the transmitted waves of the fault.
2.5 Motion equation of the adjacent tunnel wall
After the cylindrical P-wave passes through the fault, the transmitted P- and S-waves continue to propagate in the rock mass, and then impact on the adjacent existing tunnel wall. It is assumed that the particle vibration velocities of P- and S-waves at the adjacent existing tunnel wall are
v
p(
t) and
v
s(
t), respectively. The angle between the propagation direction of the stress wave and the radius and circumferential direction of the tunnel is
α′ and
β′, respectively. The radial and circumferential components of the vibration velocities of P- and S-waves on the tunnel wall can be calculated using the following formula, respectively:
(52)
(53)
In the rock mass, the speed of the P-wave is about 1.7 times that of the S-wave (Achenbach,
1973), and the stress wave generated by blasting excavation is close to the half-sine wave and the period is particularly small, so it is difficult for P- and S-waves transmitted through the fault to generate superposition in the half-period when propagating to the adjacent tunnel. In addition, due to the complexity of propagation paths, it is difficult to analyze the superposition of vibration caused by all waves propagating through different paths to the same particle of the existing tunnel. Considering these aspects, PPV caused by the P-wave and PPV caused by the S-wave are analyzed separately in this paper, and the results can accurately indicate the actual situation. The particle vibration velocities induced by P- and S-waves are defined as
and
, respectively; then, the radial and circumferential components of PPV induced by P- and S-waves on the tunnel wall can be expressed as
(54)
(55)
In addition, the blast wave will be reflected between the two tunnels and act on the adjacent tunnel wall several times. However, in addition to the stress waves that first arrive at the adjacent existing tunnel, other multiple reflected waves must pass through at least two faults to cause tunnel vibration. According to Li et al. (2018), the amplitude of stress waves passing through fault two or more times is relatively small compared with the amplitude of the incident wave, that is, less than 1%, so the tunnel vibration caused by repeated reflected stress waves is very small. Meanwhile, under the conditions of this paper, according to China's “Highway Tunnel Engineering Code” (JTGD20-2006), the minimum medium distance between two tunnels should not be less than 1/4 of the maximum perimeter of the tunnel. When the tunnel distance and the stress wave propagation path are both minimum values, it can be calculated from the above derivation that the stress wave amplitude reflected back to the charge tunnel is only 0.15, that is, the stress wave attenuates at least 80%-90%, so the actual attenuation will be larger and can be ignored. Therefore, this paper only calculates the stress waves that first arrive at the adjacent existing tunnel. Similarly, stress waves reflected repeatedly in the tunnel and the fault have little influence on PPV in adjacent existing tunnels, so the superposition effect of multiple reflected waves is not considered in this paper.
3 SPECIAL CASES
In special cases, the half-cycle sinusoidal cylindrical P-wave with an amplitude of 1 and a frequency of 100 Hz is selected as the initial blast wave. The initial wave front is set as
r
0 = 1 m,
L is set as 15 m, the radius of the adjacent existing tunnel is set as
R = 6 m, and the burial depth of the tunnels is set as 1500 m. The in situ stresses can be obtained by
and
. In order to more accurately characterize the horizontal stress, according to the variation rule of horizontal stress with
H (Brown & Hoek,
1978),
k can be calculated by
k = 2 −
H/1000. According to Chai, Wang, et al. (
2020), the density of rock and filled materials was set as 2430 and 1912 kg/m
3, respectively. The propagation speeds of P- and S-waves in the rock mass are 3368 and 1704 m/s, respectively. The propagation speeds of P- and S-waves in filled materials are 2381 and 1205 m/s, respectively. It is assumed that the initial normal stiffness of the contact interface is
k
n0 = 3.5 GPa/m, the tangential stiffness is
k
s = 3.5 GPa/m, the fault thickness is 4 mm, and the maximum allowable closure of the contact interfaces is
d
max = 1 mm. It is assumed that
Q
p is 20 (Chai et al.,
2023), and
Q
s can be calculated by Equation (
56) (Clouser & Langston,
1991).
(56)
It is generally believed that cylindrical waves in the far field, that is, the propagation distance is greater than five times the blast source distance, will decay into plane waves while propagating (Henrych, 1979). Hence, it is assumed that no geometric attenuation occurs after the column wave passes through the fault in this paper, and only the physical attenuation is considered. When the fault dip angle is 45°, 90°, and 135°, respectively, the PPVs of P- and S-waves at the adjacent existing tunnel wall are shown in Figures 10 and 11.
Peak particle velocity at the adjacent existing tunnel wall caused by P-waves. (a)
γ = 45°, (b)
γ = 90°, and (c)
γ = 135°.
Peak particle velocity at the adjacent existing tunnel wall caused by S-waves. (a)
γ = 45°, (b)
γ = 90°, and (c)
γ = 135°.
As can be seen from Figures 10 and 11, in the cases where the fault dip angles are 45° and 135°, the PPVs on the tunnel wall are equal, but the angular coordinates of the monitoring points are exactly opposite. However, when the fault dip angle is 90°, the PPVs on the tunnel wall show considerable differences from the other two cases. This indicates that the fault dip angle has a great influence on the vibration of the existing tunnel, including the influence on the PPVs and the monitoring points at the tunnel wall.
By comparing Figures 10 and 11, it can be seen that when the initial blast wave is a cylindrical P-wave, the vibration of the existing tunnel wall is mainly caused by transmitted P-waves. The S-waves experience significant amplitude attenuation after propagating through the fault, causing relatively small vibration of the existing tunnel wall. This is in complete agreement with previous research (Chai et al., 2023; Li et al., 2010; Li, Li, et al., 2012), that is, the transmitted wave of the incident P-wave at the structural plane is mainly the transmitted P-wave, and the transmitted S-wave is very weak. Therefore, in practical engineering, when the blast wave is the P-wave, more attention should be paid to the compressive stress at the adjacent existing tunnel wall. When the fault dip angle is 45°, the PPVs near the upper side of the tunnel wall are larger, whereas when the fault dip angle is 135°, the PPVs near the lower side of the tunnel wall are larger. In addition, the vibration effect of the S-wave on the adjacent tunnel gradually increases from the middle (θ = 0°) to the two sides (θ = ±90°).
4 PARAMETRIC ANALYSIS
The parametric analyses include the effect of the fault dip angle, fault thickness, blast wave frequency, and tunnel burial depth on the vibration of the existing tunnel. As described in Section 3, the vibration of the adjacent existing tunnel wall caused by the blast cylindrical P-wave is mainly generated by the transmitted P-waves, and the vibration caused by the transmitted S-waves is very small. Therefore, the parameter analyses in this section mainly consider the PPVs of the existing tunnel wall caused by P-waves. Unless otherwise specified, the parameter values are the same as those in Section 3.
4.1 Effect of the fault dip angle
To explore the effect of the fault dip angle on the vibration of the existing tunnel wall, the monitoring point angle θ is set as ±60° and ±30°, and the value range of fault dip angle γ is set as (γmin, 180°−γmin). The changes of PPV(P) with the fault dip angle under different θ are shown in Figure 12.
Variation of PPV(P) of the existing tunnel with the fault dip angle. (a)
θ = ±60° and (b)
θ = ±30°.
It can be seen from Figure 12 that with the increase of the fault dip angle, the radial and circumferential components of PPV(P) decrease gradually when θ = 60° and θ = 30°, but increase gradually at the same rate when θ = −60° and θ = −30°. This is because the increase of the fault dip angle will lead to greater geometric attenuation of the initial cylindrical P-wave reaching the upper side of the existing tunnel wall, that is, the propagation distance of the initial P-waves before reaching the fault becomes longer, while the stress waves reaching the lower side of the existing tunnel wall show just the opposite trend. Since the changing trends are completely opposite, in other parameter analysis processes, only the PPV(P) when θ > 0 is analyzed, and the PPV(P) when θ < 0 can be obtained accordingly.
In addition, by comparing Figures 12a,b, it can be found that when θ = ±60°, the circumferential component of PPV(P) is significantly larger than the radial component, and the shear effect on the existing tunnel wall is greater. However, when θ = ±30°, the compress effect on the existing tunnel wall is greater. The previous study (Li, Ma, et al., 2012) also shows that when the existing tunnel is subjected to blasting load, the normal component of PPV in the middle of the tunnel wall is larger and the tangential component of PPV on the upper and lower sides is larger. Therefore, in practical engineering, key protection should be provided to the extrusion failure on the middle of the tunnel wall and the shear failure on the upper and lower sides of the tunnel wall.
4.2 Effect of fault thickness
In order to explore the effect of fault thickness on the vibration of the existing tunnel wall, the fault dip angle γ is set as 45°, 90°, and 135°, the monitoring point angle θ is set as 60° and 30°, and the fault thickness is set as 4–50 mm. The variation curve of PPV(P) with fault thickness is shown in Figure 13.
Variation of PPV(P) of the existing tunnel with fault thickness. (a)
θ = 60° and (b)
θ = 30°.
As can be seen from Figure 13, PPV(P) at the two monitoring points gradually decrease with the increase of fault thickness. This is because, on the one hand, the increase of fault thickness will reduce the overall stiffness of the fault and limit the transmission of stress waves; on the other hand, the propagation distance of stress waves in the filled layer increases, which intensifies the attenuation degree of stress waves caused by the viscoelasticity of the filled layer.
In addition, the reduction degree of the existing tunnel PPV(P) varies under different fault dip angles. The reduction degree of PPV(P) is the largest when the fault dip angle is 135° and the smallest when the fault dip angle is 90°. This is because when γ is 135°, the initial P-wave has the largest incident angle α at the fault, resulting in the longest propagation distance of the stress wave in the filled layer, and the stress wave is subject to the largest physical attenuation. When γ is 90°, α is the smallest and the stress wave has the shortest propagation distance in the filled layer, so the attenuation degree is the smallest. In the practical project, according to the trend of the fault, the appropriate location of the explosion source can be selected on the basis of the above conclusions.
4.3 Effect of blast wave frequency
In this section, the fault dip angle γ is set as 45°, 90°, and 135°, the monitoring point angle θ is set as 60° and 30°, and the blast wave frequency is set as 50–200 Hz. The variation of the PPV(P) with the blast wave frequency is shown in Figure 14.
Variation of PPV(P) of the existing tunnel with blast wave frequency. (a)
θ = 60° and (b)
θ = 30°.
As can be seen from Figure 14, the PPV(P) on the existing tunnel wall decrease with the increase of blast wave frequency; this is because the fault passes the stress wave with low frequency and blocks the stress wave with high frequency, resulting in smaller vibration of the existing tunnel caused by a blast wave with high frequency. If there are faults in the practical engineering rock mass, low-frequency blast waves can be selected appropriately in order to ensure efficient excavation.
4.4 Effect of tunnel burial depth
A change of tunnel burial depth will cause a change of in situ stress around the tunnel, and a change of in situ stress will change rock mass viscoelasticity, thus affecting the blast wave propagation in the rock mass. Therefore, in this section, tunnel burial depth H is used as a variable to study the effect of in situ stress on PPV of the existing tunnel, and H ranges from 100 to ~1500 m. The quality factors of waves in the rock mass also change with the change of in situ stress and are approximately positively correlated with in situ stress (Dvorkin, 2005). According to previous studies (Brown & Hoek, 1978), the change of Qp with in situ stress is simplified as Qp = 10 + 3/70H. The fault dip angle γ is set as 45°, and the monitoring point angle θ is set as 60°, 30°, and 0°, respectively. The initial normal stiffness kn0 is set as 3.5 and 1 GPa/m. The variation of PPV(P) with tunnel burial depth is shown in Figure 15.
Variation of PPV(P) of the existing tunnel with tunnel burial depth. (a)
k
n0 = 3.5 GPa/m and (b)
k
n0 = 1.0 GPa/m.
It can be seen from Figure 15 that with the increase of burial depth, the PPV(P) of the existing tunnel gradually increases and then becomes stable. This is because the tunnel burial depth changes the rock mass viscoelasticity by changing in situ stresses. On the one hand, when the in situ stresses increase, the compression of the in situ stresses causes the fault to close gradually, increasing the overall stiffness of the fault and facilitating the propagation of stress waves. On the other hand, the higher in situ stress leads to closure of microscopic fractures in the rock mass, reducing the viscoelasticity of the rock mass, and therefore reducing the attenuation of stress waves.
In addition, compared with Figure 15a and Figure 15b, it can be seen that when the tunnel burial depth is less than 500 m, the PPV is significantly larger when the initial normal stiffness is 3.5 GPa/m. When the tunnel burial depth is greater than 500 m, the PPV under the two initial normal stiffness is almost the same. This is because when the burial depth is small, the blast wave propagation is greatly affected by the initial normal stiffness of the fault, and the greater the stiffness, the more favorable the wave transmission. However, when the burial depth is larger, the higher ground stress will induce closure of the fault, and the overall stiffness of the fault will increase, weakening the influence of the initial normal stiffness.
5 CONCLUSIONS
This study theoretically investigates the blast cylindrical wave propagation in a deep rock mass with a fault, and revealed the dynamic response of a deep-buried tunnel under blasting excavation, which provides insight for safety and stability analyses of deep rock masses under dynamic load disturbance. Based on the time-domain recursive method of stress wave propagation at the fault, the cylindrical P-wave propagation equation across a fault to the adjacent existing tunnel in a deep viscoelastic rock mass was derived by introducing the quality factor, and then parameters affecting the PPVs on adjacent existing tunnel wall were analyzed. The main conclusions are as follows:
1.
The vibration of a cross-fault existing tunnel is mainly generated by transmitted P-waves. With the increase of the fault dip angle, the PPVs on the upper side of the adjacent existing tunnel gradually decrease, while the PPVs on the lower side gradually increase. The closer the vibration is to the upper and lower sides, the stronger the shear effect on the tunnel wall, while the closer the vibration is to the middle, the stronger the pressure effect on the tunnel wall.
2.
With the increase of fault thickness, on the one hand, the overall stiffness of the fault decreases, which limits the transmission of stress waves. On the other hand, the increase of the wave propagation distance in the filled layer increases the attenuation degree of stress waves caused by filled layer viscoelasticity. The combined action leads to the decrease of the PPVs at the adjacent existing tunnel wall.
3.
The fault acts as a filter of “passing low frequency and blocking high frequency” on blast cylindrical P-wave propagation, so the higher the blast wave frequency, the smaller the PPVs on the adjacent existing tunnel wall.
4.
The in situ stresses of rock masses increase with the increase of tunnel burial depth. The deeper the burial tunnel depth, the more obvious the closure of the fault and microscopic fractures, which results in more blast wave propagation, and larger PPVs of the adjacent existing tunnel wall.
It should be noted that there are many assumptions in the process of the theoretical analysis; for example, the value of in situ stress is simplified and the relationship between in situ stress and rock mass viscoelasticity is simplified to linear, which will be different from a practical rock mass. The contact interface between the fault and the rock mass is simplified, without considering the influence of some important parameters such as roughness coefficient JRC and matching coefficient JMC. The initial blast wave is considered as a half-cycle sinusoidal cylindrical P-wave, but the blast wave in practical engineering will be more complex and variable. It is hoped that researchers can consider more of the above simplification problems to make the analysis more accurate.
ACKNOWLEDGMENTS
This study was supported by the National Natural Science Foundation of China (No. 42172302 & No. 41902277), the Natural Science Basic Research Plan in Shaanxi Province (No. 2023-YBGY-085), and Fundamental Research Funds for the Central Universities, CHD (No. 300102282201).
CONFLICT OF INTEREST STATEMENT
The authors declare no conflict of interest.
Biographies
Shaobo Chai, PhD, is mainly engaged in research on rock joint mechanical characteristics and propagation of explosion vibration waves in discontinuous rock masses. He has presided over more than 10 scientific research projects, including the National Natural Science Foundation of China. He has published 40 papers as the first or corresponding author.
Kai Liu, PhD, specializes in Triaxial Hopkinson Bar (THB) techniques, dynamic behaviors of rocks under high confining pressure and strain rates, digital rock technology, and mechanical failure of solid material (e.g., rock, concrete, and composite) under temperature and pressure conditions. He has authored over 30 papers in leading journals within the field, while also serving as a reviewer for more than ten journals and research projects.
