Introduction

Drill and blast method (D and B) is a cost-effective and flexible way in the excavation of underground structures including mining, quarrying and tunneling. But the approach has an unavoidable drawback because it leads to a blast-induced damage zone (BIDZ) in the surrounding rock due to explosive shock. Besides, there are frequently different types of discontinuities including foliation, faults and joints in natural rock mass (Azarafza et al. 2017). When the shock wave encounter these discontinuities, it will be reflected and generate tensile stress wave, thus cause more extended cracks in the surrounding rock (Trisugiwo et al. 2016). A series of problems will also be provoked:

potentially unstable rocks (Wang et al. 2020), increasing permeability (Zhang et al. 2021), secondary treatment costs, and severe rock mass damage can even cause the tunnel destabilization. Thus, it is of great importance to investigate the dynamic response for tunnels containing numerous discontinuities.

Many research efforts have been made to conduct studies on tunnel damage issues. For example, Holmberg (1979) developed the smooth blasting technique which can reduce the rock damage and form a flatter profile. Many field investigations and model predictions are also used to study the stability of jointed rock mass (Azarafza et al. 2018; Dey and Murthy 2012; Foderà et al. 2020; Han et al. 2016; Koopialipoor et al. 2019). These methods are helpful to improve the effectiveness and safety of tunnel excavation. But due to the difficulty of setting joints in laboratory experiment and the limitation of monitoring capability in field test, the research on jointed tunnel damage under internal blasting influence is not extensive.

Numerical methods have been widely applied for blasting damage analysis for its cost performance and convenience. The commonly implemented programs for jointed rock tunnel include Finite Element Method (FEM), Discrete Element Method (DEM) and hybrid Finite Discrete Element Method (FDEM), as shown in Table 1. The FEM programs including AUTODYN, LSDYNA and ABAQUS are able to carry out the study of blasting, impact and collision with high calculational efficiency due to its explicit solving functionality. But since these programs are continuous numerical forms, they are not suitable for simulating the large deformation or failures during blasting as well as difficult to model discontinuities. In comparison, the DEM or FDEM is more applicable for modeling of cracking and casting process because it allows a large-scale sliding of the elements. The branches of DEM mainly include Particle Flow Code (PFC), Discontinuous Deformation Analysis (DDA), Universal Distinct Element Code (UDEC) and 3DEC (three-dimensional version of UDEC). For instance, Li et al. (2014) studied the dynamic reaction caused by the release of in situ stress during tunnel excavation through PFC. Zhao et al. (2011) investigated the effects of time delay and rock properties on tunnel blast-induced crack propagation by DDA method. Varma et al. (2019) evaluated the influence of geometrical properties of joints on tunnel stability under seismic loading based on UDEC. Deng et al. (2014) focused on the damage zones and peak particle velocity (PPV) around tunnel surface and studied the stability of tunnel subject to ground blasting shock wave under varied geometric and mechanical properties of joints through UDEC. Monsalve et al. (2019) used terrestrial laser scanning technology to statistically characterize the tunnel discontinuities and defined a discrete fracture network in 3DEC. Moreover, some hybrid FDEM codes have also been developed in the simulation of jointed tunnel such as FLAC-PFC (Saiang 2010) and Y-HFDEM IDE (Zhao et al. 2011).

Table 1 Numerical methods commonly used for jointed rock tunnel
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Nevertheless, in the numerical methods above, although the coupled FDEM programs were used to analyze the influence of damage zones on tunnel, the operation of joints is very complex in these codes (Xiao and Quan 2002), and they usually need to consume more computing time (Cai et al. 2013). Besides, there also are some limitations in DEM codes. For PFC, the determination of the rock mechanics parameters and joints attitude are complicated. The treatment of natural discontinuities in DDA are set as rigid (Ning et al. 2012). UDEC is a two-dimensional simulation process, but D and B in tunnel usually has a certain advance length. Comparatively, 3DEC is a three-dimensional numerical code and has an acceptable computational efficiency, while also applied to analyze the propagation behavior of shock waves in jointed rock mass (Malmgren et al. 2006), but it has not been sufficiently explored to investigate tunnel blasting mechanism and formation of damage zones.

The research aims to develop a tunnel excavation model based on 3DEC to study the surrounding rock damage subjected to blasting shock in jointed rock mass. The blast load is considered as a stress time history applied on the tunnel contour boundary. The BIDZ is employed as the evaluation indicator. The effects of some parameters on the tunnel damage are investigated. For finding the most significant factor affecting tunnel damage, a sensitivity analysis of these parameters is also performed. In the discussion, the results of this study are compared with an existing empirical formula.

Methodology

Research route

The main procedure for the research content is shown in Fig. 1. First, the equivalent blast shock wave and loading boundary are introduced. Then the modeling process, material properties and damage evaluating indicator are described and so the tunnel excavation model with D and B is formed. Subsequently, parametric studies were conducted, which mainly involved the effects of joints, stress distribution and advance length because these parameters have a relatively important influence on the stability of tunnel structure. In the latter section, a sensitivity analysis of these parameters was performed, and the results of this model are also compared with an empirical formula and some suggestions are provided for jointed tunnels excavated with blasting.

Fig. 1
figure 1

Flow chart for the model generation and modeling process

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The blast load on equivalent elastic boundary

As shown in Fig. 2, the rock between the contour holes and buffer holes is the burden layer. After detonation of the charges, the stress waves generated by the contour blastholes are reflected at the free surface and transform to tensile stress waves. Then the burden rock is shattered and a series of crater boundaries are formed. The tunnel profile is established by these crater boundaries, and the connecting line of the contour blastholes is referred as the equivalent elastic boundary.

Fig. 2
figure 2

Diagram of the equivalent elastic boundary

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Different from specialized dynamic analysis codes such as LSDYNA and AUTODYN, the parameters of explosive material cannot be added directly in 3DEC, thus the equivalent stress time-history of blasting load is widely adopted. The blast time-history is ideally obtained by monitoring the pressure around blastholes in the field experiment. But due to the extremely fast blasting process, it is difficult to monitor and extract the blasting load time-history from the explosion site. Hence, Lu et al. (2011) developed a semi-empirical formula for the blasting load versus time based on the Jones-Wilkinese-Lee (JWL) state equation. As shown in Fig. 3, the green dashed line is the equivalent blasting load. To use this time-history in 3DEC, it was simplified to the triangular load function \(P_{{\text{D}}} {(}t)\) as shown in the solid red line, and its govern equations are given in Eqs. (1) and (2) (Yan et al. 2016).

$$P_{{\text{D}}} {\text{(t}}) = P_{0} f(t)$$
(1)
$$f(t) = \left\{ {\begin{array}{*{20}c} {\frac{t}{{t_{{\text{r}}} }}(0 \le t \le t_{{\text{r}}} )} \\ {\frac{{t_{{\text{z}}} - t}}{{t_{{\text{z}}} - t_{{\text{r}}} }}(t_{{\text{r}}} \le t \le t_{{\text{z}}} )} \\ \end{array} } \right.$$
(2)
Fig. 3
figure 3

Blasting load applied on the equivalent elastic boundary

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According to the Chapman-Jouguet model in a condensed explosive, for the uncoupled charge structure, the peak of initial explosion pressure \(P_{0}\) on the blasthole wall is given by Eq. (3)

$$P_{0} = \frac{{\rho_{0} D^{2} }}{2(\gamma + 1)}\left( \frac{a}{b} \right)^{2\gamma }$$
(3)

where \(a\) is the charge diameter, \(b\) is the blasthole diameter, \(D\) is the detonation velocity, \(\rho_{0}\) is the explosive density, \(\gamma\) is the ratio of the specific heats for the detonation gasses(\(\gamma = 3\)).

The blasting load applied to the equivalent elastic boundary meets the Saint–Venant’s principle (Lu et al. 2012), so it does not affect the dynamic response of far-filed rock mass. The mimetic blasting load \(P_{{\text{e}}} (x,t)\) on the equivalent elastic boundary is shown in Eq. (4)

$$P_{{\text{e}}} (x,t) = \frac{{2r_{0} }}{{r_{1} }}P_{{\text{D}}} (t)$$
(4)

where \(r_{0}\) is the radius of the blasthole, \(r_{1}\) is the distance between two contour blastholes.

The 2# emulsion explosive was adopted in this study and the detailed parameters are shown in Table 2. The initial explosion pressure \(P_{0}\) can be calculated by substituting the data in Table 2 into Eq. 3. According to the consequence of Lu et al., the rising time of triangular load \(t_{{\text{r}}}\) and the total action time \(t_{{\text{z}}}\) should be adopted as 0.7 ms and 8 ms respectively for reasonably considering the movement of detonation gas (Lu et al. 2012).

Table 2 Explosive parameter
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Numerical model and calculation process

Bench blasting technique is a typical blasting method used in tunnel construction (Yu 2018). Some studies have shown that the over-breaking and rock damage of upper step is more severe under blasting load due to gravity and stress redistribution (Zhang et al. 2020). Accordingly, the numerical model in this study is designed as upper bench of semi-circular tunnel. The advance length variation is achieved by the change of model thickness. Figure 4 is the model diagram of the tunnel with an assumed width of 7 m, a height of 3.5 m. Considering the influence of boundary effect on dynamic analysis, the width and height of the entire model are both 75 m because it can minimize the interference of boundary on blasting process when the ratio of model size to tunnel diameter is more than 10. The setup of the joints is described in details in the section of parameter studies. The modeling procedure is mainly divided into three stages, as shown in Fig. 4:

  1. (a)

    Initial stress was generated throughout the model, where fixed boundary condition was applied at the bottom, and zero velocity in X-direction was applied at lateral boundaries, while the top boundary was treated as stress boundary.

  2. (b)

    The blastholes inside the tunnel contour usually cause minor damage to the surrounding rock. Thus, after the completion of initial balance calculation, the rock mass inside the contour blastholes were removed to simulate the tunnel excavation.

  3. (c)

    The blasting load time-history was applied to the tunnel profile surface (also as equivalent elastic boundary), and at this stage all the boundaries of model were replaced with non-reflection boundaries.

Fig. 4
figure 4

3DEC tunnel model in jointed rock masses subject to blasting load: a initial equilibrium b removing rock block in inner blastholes c application of blasting load

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Rock and joint properties

The response of the rock masses under dynamic loading is different from that under static loading, so the mechanical parameters of rock should be set as the dynamic parameters. Besides, after the excavation of underground opening, a certain extent of damage will be incurred within the surrounding rock (Read 2004). Research from the underground laboratory also shown that for tunnels constructed by the drill and blast method, the rock in the vicinity of the tunnel profile is subject to both blast shock waves and in-situ stress redistribution, usually yielding more severe damage (Martino and Chandler 2004). In contrast, the rock relatively far from the contour is only affected by the ground stress redistribution. In the numerical model, the blasting process of the blastholes inside the tunnel contour is simplified by removing the interior rock mass. Therefore, prior to the application of blast shock wave, excavation damage zones (EDZ) have been formed in the surrounding rock around the equivalent elastic boundary. To consider the effect of internal rock excavation, we classify the rock around tunnel excavation perimeter into two thin weak layers as inner EDZ and outer EDZ, respectively.

The depth of inner and outer weak layers is supposed to be respectively 0–1.0 m and 1.0–2.0 m adjacent to the tunnel contour, as shown in Fig. 5. The detailed rock mass parameters are shown in Table 3, corresponding to granite rock mass, and are obtained through laboratory and numerical tests (Lu et al. 2011).

Fig. 5
figure 5

Diagram of inner and outer EDZ

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Table 3 Rock properties in different zones around the tunnel surface (Lu et al., 2011)
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For the joint stiffness value, many studies have shown that it changes significantly with the normal stress (Cui, 2019; Jing et al. 1994). But because of the complexity of the jointing and uncertainty of the normal stress distribution, comprehensive stiffness data is still not available for joints in various rock types (Kulatilake et al. 2016). Thus, it is more common to adopt empirical formula for the estimation of the joint stiffness. From the back-calculation based on deformability information of intact rock and jointed rock mass, the following formula proposed by (HART 1993) is used for approximating the normal stiffness of joints \(k_{{\text{n}}}\)

$$k_{{\text{n}}} = \frac{{E \times E_{{\text{r}}} }}{{s(E_{{\text{r}}} - E)}}$$
(5)

where \(E\) is the elastic modulus of intact rock material; \(E_{{\text{r}}}\) is the elastic modulus of rock mass; \(s\) is the joint spacing. In the study, \(E\) is taken as the dynamic elastic modulus of virgin rock in Table 3. \(E_{{\text{r}}}\) is assumed to be one-tenth of the elastic modulus of intact rock (Barton 1972). Thus, if the joint spacing is 0.5 m, \(k_{{\text{n}}}\) is calculated to be approximately as 10GPa/m by Eq. (5). The joint shear stiffness is assumed to be 0.5 of corresponding normal stiffness (Burdekin et al. 1979). In addition, for granite rock mass, the joint friction angle and cohesion are assumed to be 36° and 0.5 MPa, respectively (Horváth et al., 2012; Özvan et al., 2014).

Tunnel damage evaluating indicator

The blast-induced damage zones (BIDZ) were adopted as the evaluation indicators of tunnel damage in the study. BIDZ usually refers to the area where the strength or stiffness of rock mass is significantly attenuated under the blasting shock (Shen et al. 1997), and it may be characterized by cracks, sliding or even falling. The depth of the BIDZ has been widely used to evaluate the extent of damage to the surrounding rock (Sjoberg 1979; Sjoberg et al. 1997).

In jointed rock masses, the joint displacement has an important influence on the tunnel stability. According to the movement of joints, Shen et al. (1997) divided the rock damage zones near tunnel contour into failure zone, open zone and shear zone. The failure zone refers to the area where the rock blocks has been detached and falling into the tunnel while open zone is the rock zones where joints are displaced, and the shear zone where a certain shear displacement occurs. Among the three zones, the failure zone has a direct impact on tunnel stability, while the open zone and the shear zone are related to the groundwater seepage in the rock masses (Barton et al. 1985; Makurat et al. 1991). Since most open zones are close to the tunnel contour, even if the rock blocks in the zone do not fall off instantly after blasting, it will leave potential threats to the tunnel stability, such as the seepage causing pressure on the primary lining or further expansion of cracks activated under seismic load. Accordingly, the failure zone and the open zone are considered as the BIDZ, in which the open zones where joints displacement exceeds 0.01 m are regarded as BIDZ. As shown in Fig. 6, where the red zone is the failure zone and green zone is the open zone.

Fig. 6
figure 6

Diagram of blast-induced damage zone in this study

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Parameter studies

In the section, some parameters are investigated, including joint inclination angle, joint spacing, joint strike angle, joint normal stiffness, tunnel depth and advance length, which appear to be significant factors affecting tunnel stability (Chakraborty et al. 1994a; Deng et al. 2014; Jia and Tang 2008; Shen and Barton 1997). The followings will provide a more detailed description of the value range for these factors.

As shown in Fig. 8, three intersected persist- tent joints are added to the model. In the study, the joint inclination angle is defined as the angle between the two sets of joints in the XY plane, and the joint strike angle is referred to the angle between joint strike and tunnel axis. To adequately consider the influence of the joint inclination, its range is assumed from 15° to 90° while fixing the level of one set of joints in XY plane.

For the joint strike, in general, when the direction of tunnel excavation is the same as the joint strike, it is usually referred to as driving with strike as shown in Fig. 7a, and the inverse is as driving against strike, as shown in Fig. 7b. The RMR rock mass classification system proposed by the South African Council of Scientific and Industrial Research (CSIR) indicates that tunneling with the joint strike is more beneficial than driving against joint strike (Bieniawski 1979). As shown in Table 4, effects of joint strike can be considered as favorable or very favorable when driving with strike. Therefore, to consider the adverse effect of joint strike angle on excavation damage, assuming that the tunnel excavation is driving against strike. The strike angles are set from 15° to 75°. Noted that in the case of strike angle 0o and 90°, there will be an overlap of two sets of joints in YZ plane, so the strike angle 0o and 90° are not analyzed in the part.

Fig. 7
figure 7

The relationship between the direction of tunnel excavation and joint strike

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Table 4 Effect of Discontinuity Orientation and Strike in tunneling (Bieniawski, 1984)
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For rock mass with natural joints, the joint spacing differs widely. The suggestion from the China railway sector that rock mass with single set of joints is classified according to the difference in joint spacing between 0 m and 1.0 m, as shown in Table 5. But if the joint spacing is set 0.1 m in the 3DEC model, it will consume enormous computational resources and even lead to breakdown. Hence, the joint spacing ranges are from 0.25 m to 1.0 m with an interval of 0.25 m (Fig. 8). Besides, the normal stiffness also varies for joints with different normal stress, roughness or fillings (Bandis et al. 1981; Muralha et al. 2014). In the model, we assumed that the joint normal stiffness is 2.5GPa ~ 40GPa to consider different strength joints.

Table 5 Classification of rock mass based on joint spacing
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Fig. 8
figure 8

Diagram of joint inclination angle and strike angle

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At present, more than 50% of tunnels in China were buried in challenging geotectonic belts with depths of more than 100 m, and most were excavated using conventional drill and blast method (Zhou et al. 2021). Besides, the trial calculations revealed that there was very little blast-induced damage when the burial depth exceeded 600 m due to in situ stress, as shown in Fig. 13. Therefore, we assumed the tunnel burial depth range as 100 m-600 m with an interval of 100 m.

In the numerical model, the thickness of 3DEC model was changed to simulate different advance lengths. The commonly used advance length of tunnels constructed by D and B is approximately 2 m-5 m (Costamagna et al. 2018; Foderà et al. 2020; Zare and Bruland 2007). Additionally, it is notable that a blast round length-to-diameter ratio exceeding 2.0 was accomplished in the Canada Underground Research Laboratory (Kuzyk et al. 1995). However, due to the complexity of geologic conditions, the quality of boreholes is very hard to be guaranteed in the field application. Thus, we only added extra model thickness of 6 m and 7 m to denote larger round lengths.

In summary, the value range of the selected parameters is shown in Table 6. To analyze the effect of single factor, it is also necessary to choose a typical value for these parameters, i.e., when one variable is studied, a constant value for other parameters. For the joint inclination, after trial calculation, it was found that when the angle exceeded 60°, the tunnel damage increased significantly, so the inclination angle 60o was selected as the typical value. For joint strike angle, the typical value is assumed to be 0 o, i.e., the joint strike is parallel to the tunnel axis. The constant value of joint spacing is assumed to be 0.5 m to consider the calculation efficiency. The corresponding joint normal stiffness is 10GPa from Eq. (5) as mentioned before. The typical values of tunnel buried depth and advance length are assumed to be 100 and 3 m respectively (Iverson et al. 2008; Zhao et al. 2013).

Table 6 The range of values for different parameters
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Besides, since wave attenuation in rock masses is mainly caused by joints in this model, the damping of original rock was neglected (King et al. 1986; Pyrak‐Nolte et al. 1990).

Effect of joints on the damage of tunnel

Joint inclination angle

Figure 9 shows the BIDZ distribution after blasting loading at different joint inclination angles. It can be seen that when the joint angle is 15°, there is no visible damage zone around the contour. With the increase of the joint angle, the BIDZ gradually enlarges. When the joint angle is 30° to 60°, the failure zone is concentrated in the vault and shoulder positions. Furthermore, it can be clearly revealed that BIDZ range reaches the maximum at the joint angles of 75°and 90°, and destabilization disruption occurs in the vault rock masses.

Fig. 9
figure 9

Tunnel damage in terms of different joint inclination angles

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Joint strike angle

The tunnel damage at different joint strike angles is shown in Fig. 10. It can be observed that the depth of tunnel damage zone is increased gradually as the strike angle approaches more vertical in relation to the tunnel axis. When the joints are close to horizontal (i.e., dip angle 15 o), the failure zone is mainly in the tunnel vault. As the joint dip angle increases to 30 o ~ 75 o, the failure zone is transferred to the right side the tunnel wall.

Fig. 10
figure 10

Tunnel damage in terms of different joint strike angles

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Joint spacing

Figure 11 presents damage condition with changing the joint spacing. It can be observed that the maximum depths of BIDZ at joint spacing of 0.25 m and 1 m are very similar without obvious failure zones. The damage depth at joint spacing of 0.5 m is at the middle value and there is a noticeable failure zones on the left shoulder. The tunnel damage is most severe when the joint spacing is 0.75 m. It is worth noting that the damage extent is not increased or decreased with increasing joint spacing.

Fig. 11
figure 11

Tunnel damage in terms of different joint spacings

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Joint normal stiffness

Figure 12 show the tunnel damage for different joint stiffness. It can be found that the range of the failure zones gradually decreases when the joint stiffness varies from 2.5 GPa to 10 GPa. The tunnel is relatively stable when the joint stiffness is 20GPa ~ 40GPa because there are almost no obvious failure zones in the surrounding rock.

Fig. 12
figure 12

Tunnel damage in terms of different joint normal stiffness

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Effect of initial stress distribution and advance length

Tunnel depth

Figure 13 displays the distribution of tunnel damage under blasting loading in the varied tunnel depths. It can be seen that the damage area gradually decreases with the rising of tunnel depth, but the magnitude of the change is not obvious. When the buried depth increases to 600 m, only a very few rock blocks fall from the left shoulder and there are almost no failure zones in the surrounding rock.

Fig. 13
figure 13

Tunnel damage in terms of different buried depths

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Advance length

Figure 14 shows the surrounding rock damage at different advance lengths. It can be clearly seen that when the advance length is 2 m, there is no visible failure zones around the tunnel surface. With the increase in the advance length, the tunnel damage range extends considerably. When the advance length reaches 7 m, the vault rock mass has the tendency to fall down.

Fig. 14
figure 14

Tunnel damage in terms of different advance lengths

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Sensitivity analysis

Figure 15 shows the arithmetic mean of maximum damage depth and its standard deviation for different calculated cases above. The standard deviation reflects the dispersion of the individual data in every dataset, and it is used to show the impact extent of a change in one of the above conditions on the tunnel damage, where the smaller the standard deviation means that the impact of the factor on tunnel damage is relatively insignificant. It can be seen that the arithmetic mean and standard deviation of maximum damage depth in different advance length are greater than other factors, which reveals that the tunnel damage by increasing the advance length is more severe compared to the geometrical and mechanical properties of joints and tunnel depth.

Fig. 15
figure 15

Arithmetic mean of maximum damage depth and standard deviation in different cases

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Figure 16 shows the peak particle velocity (PPV) at the tunnel left shoulder in different advance lengths. A remarkable increase of PPV can be found when the advance length is changed from 2 to 3 m, which is also consistent with Fig. 14 where the failure zone begins to develop when the advance length is above 2 m. Besides, damage depth and standard deviation as increasing tunnel buried depths are the smallest among these factors, which means that the buried depth could have more minor effect on tunnel damage. Besides, in the joint geometric factors including inclination, strike and spacing, the damage depth under different strike angles is higher, but its standard deviation is relatively small. It can also be found from Fig. 10 that the tunnels with varied joint strike angles all contain visible damage zones.

Fig. 16
figure 16

Peak particle velocities at tunnel left roof in terms of different advanced length

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Discussion

For the estimation of tunnel damage zones excavated by drill and blast method, researchers have proposed some empirical formulas from different aspects, many of which are based on a single factor such as the charge density, maximum charge per delay or peak particle velocity (Dowding 1984; Kwon et al. 2006; Pusch and Stanfors 1992; Zhou 2011).

Comparatively, the damage extent estimation formula developed by Verma et al. (2018) integrated the explosive charge, borehole depth, cross-sectional area and rock mass quality. Thus, the formula is used to contrast with the results of this study:

$$D_{{\text{d}}} = 0.96\left[ {\frac{{q_{{\text{p}}}^{0.15} \sqrt {W + q} }}{{Q^{0.33} }}\left( {\frac{{C_{{\text{f}}} }}{{A_{{\text{f}}} }}} \right)^{0.15} } \right] - 1.28$$
(6)
$$C_{{\text{f}}} = \frac{d}{a}$$
(7)
$$A_{{\text{f}}} = \frac{l}{d}$$
(8)

where \(D_{{\text{d}}}\) is the damage depth, \(q_{{\text{p}}}\) is the perimeter charge factor, \(W\) is the maximum charge per day, \(q\) is the specific charge, \(Q\) is the rock mass quality index(Barton’ Q-system), \(d\) is the hole depth, \(l\) is the advance length, \(a\) is the tunnel cross-sectional area.

In comparison with the Q system, the RMR system considers the effects of joint inclination, strike and spacing on the tunnel structure. Therefore, we use the Eq. (9) to convert the rock mass scores from RMR system to Q system (Bieniawski 1989).

$$Q = 10^{{\frac{{\text{RMR - 44}}}{{{10}}}}}$$
(9)

The tunnel damage depth for different joint inclination angles, strike angles, spacings and advance lengths were compared with the Eq. (6). The detailed parameters used for the calculation are shown in Table 7, which refer to the commonly adopted drill and blast schemes in the tunnel excavation (Wang 2015). The tunnel cross-sectional area \(a\) in the model is 19.24m2. The value of the confinement factor \(C_{{\text{f}}}\) varies with the advance length \(l\). The advancement factor \(A_{{\text{f}}}\) is assumed to be 1 (i.e., the hole depth \(d\) equal to the advance length \(l\)). The scoring values of RMR system for different joint inclination angles, spacings and strike angles are shown in Table 8. These values are acquired according to the rock and joint characteristics of the model, and referenced to the RMR classification system (Bieniawski 1989), and then converted to Q system by Eq. (9).

Table 7 Calculation parameters of the empirical formula
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Table 8 Rock mass scores for different joint configurations
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From Fig. 17a, it can be seen that the damage depth at different joint inclination angles is consistent with the empirical method of Verma et al. Moreover, when the inclination angle is less than 45°, the damage depth is in the range of 1 m, and if the inclination is greater than 60°, the damage depth may be deeper than 2 m. It is also comparable to the RMR system (Bieniawski 1993), where inclination angles of 20° ~ 45° and 45° ~ 90° are considered as fair and unfavorable, respectively. It should also be observed that when the joints parallel to the tunnel axis, the failure zones are mainly found in the upper part of the tunnel as shown in Fig. 9, indicating that the rock reinforcement in proximity to the tunnel top should be given enough attention in the condition.

Fig. 17
figure 17

Comparison of the results of this study with an empirical formula: a Joint inclination angle; b Joint strike angle; c Joint spacing; d Advance length

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However, the damage depth between the model and estimated value shows significantly opposite tendency as the strike angle increasing, as shown in Fig. 17b. This is because the RMR system regards that a larger joint strike is relatively beneficial to tunnel stability. But under the internal blasting effect, when the joints are more perpendicular to the tunnel axis, the blast wave is less affected by the joints and so is prone to cause more damage in the surrounding rock (Chakraborty et al. 1994a, 1994b). Thus, as the joint strike angle increases, the number of blastholes or explosive density should be decreased correspondingly. It is noted that this study concentrates more on the tunnel damage under the internal blasting. If blasting charges are well controlled or if excavation is conducted by mechanical methods (e.g., Tunnel Boring Machine), driving against joint dip is still detrimental to the tunnel structure (Delisio et al. 2013; Einstein et al. 2006).

In the computational model, there is a significant dispersion for tunnel damage depth at joint spacing from 0.25 m to 1.0 m, as shown in Fig. 17c.In general, a smaller spacing of joints implies more numbers of joints and thus is detrimental to the tunnel stability (Barton 2013; Wang et al. 2013). But on another side, more joints will absorb more blast wave energies and reduce the blast-induced surrounding rock damage (Lanari and Fakhimi 2015). Therefore, for tunnels excavated by drill and blast, the increase in the joint spacing does not induce in increasing or decreasing tendency of tunnel damage zones. This phenomenon was also found for the jointed tunnel response subject to external blast shock (Deng et al. 2014).

The increased joint stiffness makes the tunnel more stable, as shown in Fig. 12. This is mainly because the greater stiffness limits the opening of the joints (Bandis et al. 1983). Furthermore, the increasing buried depth (i.e., in-situ stress) also restricts the crack growth (Tao et al. 2020; Xie et al. 2016), and the model obtained similar results, as shown in Fig. 13. However, it should be noted that in the excavation of deeper tunnel, the BIDZ may be further extended and the surrounding rock may be damaged persistently due to stress redistribution, underground pressurized water or creep of rock masses(Hedayat and Weems 2019; Wang et al. 2016). Therefore, the range of damage zones may need to further explored for some larger buried tunnels.

The tunnel damage is significantly more severe with the increase in the advance length, and this is also consistent with the empirical estimated results, as shown in Fig. 14 and Fig. 17d. It indicates that if the drill and blast method is used to excavate jointed tunnels, the advance length should be cautiously controlled.

Conclusions

This study develops a numerical approach to evaluate the damage extent for a jointed tunnel excavation subjected to internal blasting shock using DEM-based code 3DEC. Some parameters that may affect tunnel stability were investigated and a sensitivity analysis for these factors was performed. Lastly, several comparisons were conducted with existing empirical evaluation formulas for tunnel damage. The main conclusion are presented as follows:

  1. (1)

    The numerical model considers both excavation and internal blast effects by converting the blasting load into a stress-time history on the tunnel contour. The evaluation of tunnel damage for various factors is performed through failure and open zones.

  2. (2)

    Under the internal blasting load, the tunnel damage is more critical as the joint inclination and strike angle increases. The vault rocks of the tunnel are liable to fall when joints are parallel to tunnel axis, and as the strike angle increases, the failure zones are mainly in the sidewall, so the rock mass at these positions should be reinforced carefully in above conditions.

  3. (3)

    The magnitude of tunnel damage could not be evaluated simply by increasing or decreasing the joint spacing in the dynamic loading condition. An increase in buried depth (i.e., higher in-situ stress) or joint stiffness is beneficial in reducing tunnel blast damage. But the tunnel depth in this study is limited within 600 m, and the damage zones at deeper levels should be further investigated.

  4. (4)

    The result of the sensitivity analysis shows that the increase in the advance length will have a more significant adverse effect on tunnel damage than the joint parameters and buried depth. Thus, for jointed tunnels, increasing advance length per round should be considered cautiously.

  5. (5)

    In the research, tunnel damage was only evaluated for a single factor variation. For tunnel excavation in the more complex geological situation, some unfavorable factors probably occur in combination, so it is necessary to evaluate the combined effect using real case in the future.