Research on the theory and method of reduced-hole blasting for large cross-section tunnel based on explosive energy dissipation

Abstract

The traditional tunnel drilling and blasting method places cut holes at the lower center of the excavation face, resulting in an excessive number of blasting holes. With the continuous increase in cross-section area, this design concept can no longer meet the requirements of safe and efficient tunnel boring for large cross-section tunnels. This paper puts forward the theory and method of reduced-hole blasting for large cross-section tunnels, as an alternative to the traditional drilling and blasting method of the “more holes, less charge” design concept. Based on the explosion energy dissipation law and rock’s critical crushing energy dissipation characteristics, the calculation method of the extrapolation distance of the wedge-cut holes is given. The optimum extrapolation distance of the wedge-cut holes was verified using numerical simulation and field tests. The results show that the number of drilling holes can be reduced by about 15.8% using the theory and method proposed in this paper, and at the same time, the damage of retained rock can be effectively controlled. The results of this study can provide a reference for the design of blast network parameters for similar large cross-section tunnels.

Article Highlights

• This paper put forward the theory and method of reduced hole blasting (RHB) for large cross section tunnels, the number of blasting holes was reduced by an average of 15.8%.

• This paper gave the wedge-cut hole extrapolation distance calculation method based on the explosion energy dissipation law and rock’s critical crushing energy dissipation characteristics.

• This paper proposed a computational expression for energy dissipation under coupled in-situ stress and blast loading, determined the critical crushing energy dissipation density of gray sandstone under different combined dynamic and static loading conditions.

1 Introduction

The drilling and blasting method has the advantages of being economic and efficient, and is currently the most commonly used construction method for highway and railroad tunnels, mine shafts, and underground space hard rock excavation. With China’s socio-economic development, as well as the continuous promotion of the Belt and Road initiative and the construction of new infrastructure, large cross-section tunnels have appeared frequently. However, the design of blasting parameters still follows the traditional design concept of small cross-section tunnels, resulting in too many blasting holes, which restricts the quality and duration of tunnel construction. Therefore, there is an urgent need to put forward an innovative theory and method of hole-reduction blasting for large cross-section tunnels, to maximize the outward push of the wedge-cut holes, and to reduce the number of holes while ensuring the quality of blasting, so as to realize the safe and efficient blasting of large cross-section tunnels.

The parameter design of wedge-cut holes is the soul of whether large cross-section tunnels can be successfully blasted to break rocks. Many scholars have conducted research on the hollowing parameters of large cross-section tunnel blasting. Yang et al. (2024) showed that in large cross-section hard-rock tunnel blasting, using the frames on both sides of the drilling and blasting cart as a reference for the main wedge-cut holes can reduce the number of drilled holes, but the theoretical basis of this design method was not reflected in their study. Ji et al. (2020) proposed an inverted T-shaped hollowing and blasting method for very large cross-section tunnels, which improved the blasting effect. Mei et al. (2021) carried out studies on hollowing methods, blast hole placement, charge volume, and charge structure to achieve safe and efficient blasting excavation. Zhang et al. (2022) proposed an innovative hollowing and blasting technique based on cavity hollowing and debris throwing. Li et al. (2023) proposed a hollowing out and blasting technique with decentralized charging and staged blasting. Qin and Zhang (2020) proposed a “three-step + inverted” tunnel blasting excavation method, which effectively controlled the blasting vibration. Cheng et al. (2023) proposed a medium-deep hole-hollowing and blasting method based on directional pre-cracking with slit charging to improve the tunneling progress. Gao et al. (2022) improved the hollowing and blasting initiation method and proposed a hybrid initiation method based on the reverse transmission of neighboring holes. Zhang et al. (2020a; b) analyzed the hollow-hole effect of hollowing and blasting and gave a method for determining the parameters of hollow-hole straight-eye hollowing and blasting. Zhang et al. (2020a; b) proposed a second-order two-stage hollowing and blasting technique, which effectively improved the efficiency of blasting excavation. Wu et al. (2023) proposed a zonal blasting excavation scheme for small-spaced large-section tunnels to effectively control blasting vibration and perimeter rock damage. The above research has effectively improved the blasting quality and construction efficiency through the optimization of segmental hollowing blasting technology and hollowing parameters, which is of great significance for the safe and efficient blasting of large cross-section tunnels. However, its parameter design methods are not free from the traditional design concept of small cross-section roadway blasting, the number of blasting holes laid is still high. With the increasing cross-section area of the tunnel excavation, it is possible to extrapolate the placement position of the wedge-cut holes, and utilize the bidirectional critical surface formed after blasting the wedge-cut holes to break the rock effectively. The transformation of design concepts is crucial.

Currently, the wedge-cut holes spacing, angle, and other parameters are determined by the radius of the crushed zone, the radius of the fractured zone, and the geometric relationship, which leads to smaller spacing of the wedge-cut holes and limited hollowing effect. The accumulation, propagation, and dissipation of energy are the driving factors for the destabilization of the medium, and it is more scientific to determine the optimal spacing of the wedge-cut holes based on the explosive energy dissipation. Li et al. (2018a; b) analyzed the energy evolution law of deep buried tunnels under explosive loading. Leng et al. (2016) discussed the process of explosive energy transfer during side detonation and end detonation. Sanchidrián et al. (2007) analyzed the energy distribution pattern within the rock mass after explosive blasting based on single-hole blasting tests. Xia et al. (2020) analyzed the effect of charge structure on blast energy transfer. Leng et al. (2019) analyzed the explosion energy transfer law under the condition of double-point detonation in the hole. The above research has conducted a detailed exploration of the transmission and distribution of explosion energy, which is of great significance for exploring the mechanism of rock damage and failure. However, there are few research results on the design of hole network parameters for large cross-section tunnel blasting from the perspective of explosive energy dissipation.

This paper is based on the significant demand for safe and efficient tunnel boring in large cross-section tunnels, and offers the method of wedge-cut hole extrapolation, featuring additional center holes, and adjusts the detonation sequence of large cross-section tunnels according to reduced-hole blasting (RHB) theory and method, completely changing the traditional tunnel blasting "more holes, less charge" design concept. Based on the explosion energy dissipation law and the rock’s critical crushing energy dissipation characteristics, the calculation method of the extrapolation distance of the wedge-cut hole is given. A three-dimensional numerical simulation model was established to compare and analyze the hollowing effect and the damage law of the retained rock body after being subjected to different wedge-cut holes with different extrapolation distances, and the optimal extrapolation distance of the wedge-cut holes was determined. Based on the Gonghe Village Tunnel of the Ludian-Qiaojia Expressway, five on-site blasting tests were carried out to verify the scientific validity of the theory and method proposed in this paper. The results of this study can provide a reference for the design of blast network parameters for similar large cross-section tunnels.

2 Theory and method

2.1 Engineering background

The Gonghe Village Tunnel of the Ludian-Qiaojia Expressway is located in Chongxi Township, Qiaojia County, Yunnan Province. The total length of the right tunnel is 10,698 m, with a maximum depth of about 1320 m, which is a two-way four-lane detached extra-long tunnel. The buried depth of the section used in this study is about 400 m, the surrounding rock level is class III, the surrounding rock lithology is dominated by gray sandstone, and the excavated section area is 99.28 m², which belongs to the large cross-section tunnel. The section used in this study was excavated by blasting using the upper and lower step method, as shown in Fig. 1.

Fig. 1

The Gonghe village tunnel

The originally proposed Gonghe Village Tunnel was constructed by using the traditional drilling and blasting method, which led to the problems of a large number of blasting holes, high drilling intensity, excessive crushing of stones, and serious over excavation, as shown in Fig. 2. The number of boreholes for conventional tunnel drilling and blasting is estimated on the basis of the tunnel cross-section area S and the rock solidity coefficient f of $N = 3.3 \cdot \sqrt[3]{{fS^{2} }}$. The rock solidity coefficient f of the study section of the Gonghe Village Tunnel is 10, the tunnel section area S is 99.28 m², and the number of blasting holes to be drilled in the whole section is 152, which is a huge drilling workload. The current tunnel rock drilling cart is expensive, has a high failure rate, and is limited by the tunnel space, which leads to the automated drilling technology not being mature and still mainly relying on manual drilling. Taking a 4 m depth of the blast hole as an example, the drilling efficiency of the workers is generally about 3 holes/h, and a drilling shift is calculated with 13 people, which takes about 3.9 h of drilling time per cycle, making drilling intense and long, and seriously affecting the construction progress. At the same time, the excessive number of blasting holes laid resulted in high usage of explosives, excessive damage to retained rock, and severe over excavation. The number of blasting holes, drilling time, explosives used, and average over-digging value per cycle of blasting in the upper bench of the study section of the Gonghe Village Tunnel were counted, as shown in Table 1. Therefore, the traditional drilling and blasting method could not meet the requirements of safe and efficient construction of large cross-section tunnels, and there is an urgent need to put forward innovative theories and methods that can reduce the number of drilled holes, reduce the unit consumption of explosives, and ensure the quality of blasting.

Fig. 2

Technical difficulties caused by the traditional drilling and blasting method

Table 1 Relevant data collection after blasting in the original program

2.2 Theory and method of RHB

A typical compound wedge hollowing blast is used as an example to discuss the design method of traditional tunnel blasting parameters, as shown in Fig. 3. In Fig. 3, the orifice spacing, bottom spacing, row spacing, hollowing angle, and vertical depth of the first-order wedge-cut holes are S₁, c, a, β₂, and h, respectively. The orifice spacing, bottom spacing, row spacing, hollowing angle, and vertical depth of the second-order wedge-cut holes are S₂, d, b, β₁, and H, respectively. To determine the above blasting parameters, the following three steps are usually performed: (1) determine the bottom of hole spacing and row spacing based on the radius of the crushed zone and fractured zone, and further determine the orifice spacing based on the geometric relationship. (2) Determine the hollowing angle based on the force relationship of the slot cavity. (3) Determine the rest of the blasting parameters based on the geometry of the slot cavity volume by associating (1)(2).

Fig. 3

Duplicate wedge hollowing blast analysis model

The destruction of rock in the slot cavity is based on the principle that the fractured zones of adjacent wedge-cut holes are interconnected, therefore, the row spacing of the wedge-cut holes a should be satisfied:

$$R_{T} \le a \le 2R_{T}$$

(1)

where R_C and R_T are the radius of the crushed zone and fractured zone, respectively.

When determining the spacing between the bottoms of the holes in the wedge-cut holes, the principle of overlapping fractured zones is used to ensure that the rock at the bottom of the slot cavity is sufficiently fractured. Therefore, the bottom spacing c and d of the wedge-cut holes should be satisfied:

$$c = 2^{(1 + \alpha )/\alpha } R_{C}$$

(2)

$$R_{T} \le d \le 2R_{T}$$

(3)

Combined with the geometric relationship in Fig. 3, it can be seen that the orifice spacing S₁ and S₂ of the wedge-cut holes should be satisfied:

$$S_{1} = c + 2h/\tan \beta_{2}$$

(4)

$$S_{2} = d + 2H/\tan \beta_{1}$$

(5)

The force state of the slot cavity lumen is further analyzed. When the wedge-cut hole charging section explosives explosion, the rock around the blasting hole in the strong compression of the explosive shock wave to form a crushed zone. Since the row spacing a of the wedge-cut holes is small, a through surface is formed in the charging section. Subsequently, the rock forms a fractured zone outside of the crushed zone under the combined effect of the blast stress wave and the blast-generated gas. Due to the presence of the hollowing angle, the explosion of the explosive generates a force perpendicular to the free surface outward, forcing the slot cavity to move in the direction normal to the free surface. As a result, the rock in the slot cavity will be damaged by shear with the surrounding rock, forming shear surfaces B₁C₁I₁H₁, A₁D₁J₁G₁, A₁B₁H₁G₁, and D₁C₁I₁J₁. The bottom of the slot cavity is subjected to large tensile stresses, forming damage surface H₁I₁J₁G₁.

The total resistance to blasting in the direction of the line of least resistance for the wedge-cut holes is:

$$\begin{aligned} Q = & Q_{{B_{1} C_{1} I_{1} H_{1} }} + Q_{{A_{1} D_{1} J_{1} G_{1} }} + (Q_{{A_{1} B_{1} H_{1} G_{1} }} + Q_{{D_{1} C_{1} I_{1} J_{1} }} )\sin \beta_{1} + Q_{{H_{1} I_{1} J_{1} G_{1} }} \\ = & (c + \sigma_{V} \tan \varepsilon )\frac{{(d + S_{2} )H}}{{2\sin \beta_{1} }} + 2(c + \sigma_{H} \tan \varepsilon ) \cdot bl_{c} \sin \beta_{1} + db\sigma_{td} \\ \end{aligned}$$

(6)

where c and ε are rock cohesion and angle of internal friction, respectively. l_c is the length of the charge.

The combined force acting on the explosive gas in the direction of the line of least resistance is:

$$F = 2Pl_{c} r_{0} \cos \beta_{1}$$

(7)

where P is the average burst pressure in the blast hole, P = ρ₀D²/8.

When the following conditions are met, the wedge-cut holes blasting is able to throw out the slot cavity smoothly and form a critical surface:

$$F \ge Q$$

(8)

The value of the hollowing angle β₁ can be determined by associating Eqs. (6)–(8).

Ideally, the following relationship exists between the rock volumes V₁ and V₂ for first-order hollowing and second-order hollowing blasts:

$$V_{1} /V_{2} \le K_{s} /(K_{s} - 1)$$

(9)

From the geometrical relations in Fig. 3, the values of the remaining parameters can be further determined:

$$\left\{ {\begin{array}{*{20}l} {V_{1} = (aS_{1} + ac + \sqrt {a^{2} S_{1} c} )\frac{h}{3}} \hfill \\ {V_{2} = (bS_{2} + bd + \sqrt {b^{2} S_{2} d} )\frac{H}{3}} \hfill \\ {V_{1} /V_{2} \le K_{s} /(K_{s} - 1)} \hfill \\ {S_{1} = c + 2h/\tan \beta_{2} } \hfill \\ {S_{2} = d + 2H/\tan \beta_{1} } \hfill \\ \end{array} } \right.$$

(10)

After determining the parameters of the wedge-cut holes according to the above blasting parameter calculation method, the peripheral holes are laid out according to the requirements of contour molding, and finally the auxiliary holes are evenly arranged according to the size of the remaining section area and its location. The above blasting parameters determination method is more applicable to small cross-section tunnels, but with the increase of the tunnel excavation cross-section area, the above blasting parameters will lead to too many blasting holes.

Based on the demand for safe and efficient construction of large cross-section tunnel blasting, this paper proposes the theory and method of RHB for the blasting construction of the tunnel using a step method with more than three levels of surrounding rock, as shown in Figs. 4 and 5. Central to this method are the extrapolation of wedge-cut holes, the addition of center holes, and the adjustment of the detonation sequence. The methodology is as follows:

• (1) The wedge-cut hole is pushed outward. The conventional drilling and blasting method places the wedge-cut holes in the lower middle of the tunnel excavation face, as shown in Fig. 4a. The spacing of the wedge-cut holes is small, and a large number of auxiliary holes need to be laid to ensure the uniform distribution of the explosive energy, and the number of blasting holes is large. The RHB method pushes the placement of the wedge-cut holes outward to the maximum extent possible and to the minimum distance from the tunnel design contour line. This distance should ensure that blasting of the wedge-cut holes will not cause damage to the retained rock outside the tunnel design contour line, as shown in Fig. 4b. At this time, the wedge-cut holes blasting forms a two-way critical surface (Lou et al. 2022): one provides a critical surface for auxiliary hole blasting; the other provides a critical surface for center holes blasting. From Fig. 4, it can be seen that the wedge-cut holes are pushed out, which reduces the deployment of auxiliary holes in large quantities and reduces the drilling workload of the up-stage blasting excavation. To ensure the effectiveness of surface blasting, the two methods are consistent in terms of the placement of peripheral holes.

• (2) Addition of center holes. In this paper, the blasting holes laid in the center of the tunnel excavation face are defined as the center holes, which need to be loaded with explosives, and are able to break the larger volume of rock in the center of the excavation face by blasting through 2–3 center holes. As a result of the extrapolation of the wedge-cut holes, a large area of rock will be left in the center of the tunnel excavation face waiting to be blasted. After blasting the main wedge-cut holes, the rock in the center of the tunnel excavation face is "isolated" and forms three free surfaces simultaneously, so 2–3 center holes can be used to break this part of the rock. At this point, even if the charge of the center holes is increased, there will be no damage to the retained rock outside the design contour line of the tunnel, as shown in Fig. 4b. In addition, the deployment of center holes can effectively solve the phenomenon of the "bulging belly" of the tunnel excavation face and improve the digging footage.

• (3) Adjustment of the detonation sequence. The traditional tunnel drilling and blasting method in terms of the detonation sequence is as follows: cut hole detonation, auxiliary hole detonation, peripheral hole detonation, bottom hole detonation. This paper optimizes the order of detonation adjustment. The adjusted detonation sequence is as follows: the main wedge-cut holes are detonated first, forming a large volume critical surface in the center of the working face, and providing a free surface for the secondary and tertiary wedge-cut holes, center holes, and collapse holes to be blasted. Then, the secondary wedge-cut holes and center holes are detonated at the same time, the secondary wedge-cut holes blasting for the tertiary level of wedge-cut holes and the collapse holes blasting to provide airspace. After a center hole is detonated, it is possible to fully break up the rock that has not been fully exploded at the bottom of the main wedge-cut holes, providing a larger critical surface. The tertiary wedge-cut holes and collapse holes are then detonated simultaneously to provide a critical surface for auxiliary hole blasting. Finally, the auxiliary holes, peripheral holes, and the bottom holes are detonated in turn, as shown in Fig. 5.

Fig. 4

Wedge-cut hole extrapolation theory and method

Fig. 5

Adjustment of the detonation sequence

Compared with the traditional tunnel blasting construction process, the technical advantages of the theory and method proposed in this paper are mainly as follows: (1) the number of blasting holes and the drilling intensity have been reduced, and the construction efficiency has been improved; (2) reduction in the number of blasting holes, improvement in the phenomenon of wrong drilling and leakage, and improvement in the quality of blasting; (3) to curb the phenomenon of bulging belly, this method improves the digging footage to ensure the smoothness of the working face; (4) the order of detonation has been adjusted to change the sequence of energy release from the explosives, and the energy is utilized more efficiently; (5) reduced damage to retained rock outside the tunnel design contour.

2.3 Calculation of extrapolation distance for wedge-cut hole

Energy is an essential feature of the deformation response of rock and is the driving factor for the occurrence of destabilizing damage in rock (Xie et al. 2004), explosive blast rock is the result of the joint action of shock wave energy and explosive gas expansion energy (Zhao et al. 2019). After the explosion of the explosives, the energy consumed mainly consists of shock waves that expand the explosive cavity of the energy consumption W₁, stress waves that produce radial fissures of the energy consumption W₂, stress waves caused by elastic deformation of the energy consumption W₃, bursting gas that expands the cavity of the energy consumption W₄, and bursting gas that provides the energy consumption of the thrown rock debris W₅. Several studies (Zhou and Zhong 2022; Leng 2020; Raina and Trivedi 2019) show that the energy utilization rate of explosive blast crushing rock is only 20–30% of the total release of chemical energy W₀, and the vast majority of the remaining energy is consumed in other forms. Thus, this paper took 30% W₀ as the total energy of explosive blast crushing rock. The total energy consumed after the explosion of the explosive is W_d, and when the residual energy W_r (0.3W₀-W_d) is greater than the critical crushing energy dissipation density W_a0 when the rock is damaged by impact loading, the rock will continue to be damaged.

Therefore, only W_d and W_a0 need to be determined to calculate the minimum distance of the wedge-cut holes from the tunnel design contour line. In Sect. 3.1, the explosive energy dissipation law under the coupled action of in-situ stress and explosive loading is discussed, and the computational expression for the energy dissipation of each part under the action of the coupled stress field is proposed. As discussed in Sect. 3.2, rock impact compression tests under different dynamic static combination loading conditions were designed to determine the critical crushing energy dissipation density W_a0 of gray sandstone.

3 Parameter calculation

3.1 Explosive energy dissipation law analysis

3.1.1 Calculation of the extent of the crushed and fractured zone under coupled stress field

Before the tunnel is excavated, the surrounding rock is already in a three-dimensional stress state due to the self-gravitational stresses of the overlying rock. The in-situ stress has an inhibitory effect on the explosive load (Yan et al. 2015; Bastante et al. 2012; Hamdi et al. 2011; Mandal and Singh 2009), and the in-situ stress field of the original rock and the dynamic stress field formed after the detonation of explosives are superimposed onto each other to form a secondary coupled stress field. The expression for the stress distribution based on the coupled effect of blast load and in-situ stress is given by:

$$\left\{ {\begin{array}{*{20}l} {\sigma_{r} = \frac{{\sigma_{V} + \sigma_{H} }}{2}\left( {1 - \frac{{r_{0}^{2} }}{{r^{2} }}} \right) + \frac{{\sigma_{V} - \sigma_{H} }}{2}\left( {1 + 3\frac{{r_{0}^{4} }}{{r^{4} }} - 4\frac{{r_{0}^{2} }}{{r^{2} }}} \right)\cos 2\theta + P_{d} \left( {\frac{{r_{0} }}{r}} \right)^{\alpha } } \\ {\sigma_{\theta } = \frac{{\sigma_{V} + \sigma_{H} }}{2}\left( {1 + \frac{{r_{0}^{2} }}{{r^{2} }}} \right) - \frac{{\sigma_{V} - \sigma_{H} }}{2}\left( {1 + 3\frac{{r_{0}^{4} }}{{r^{4} }}} \right)\cos 2\theta - \lambda P_{d} \left( {\frac{{r_{0} }}{r}} \right)^{\alpha } } \\ {\tau_{r\theta } = \tau_{\theta r } = - \frac{{\sigma_{V} - \sigma_{H} }}{2}\left( {1 - 3\frac{{r_{0}^{4} }}{{r^{4} }} + 2\frac{{r_{0}^{2} }}{{r^{2} }}} \right)\sin 2\theta } \\ \end{array} } \right.$$

(11)

where r, r₀ are the distance from the calculation point to the center of the packet and the radius of the blast hole, respectively. θ is the angle between the line between any point in the rock and the center of the blast hole in the horizontal direction. P_d is the homogeneous force acting on the borehole wall, MPa. α is the pressure attenuation coefficient, for the shock wave region α ≈ 3 or α = 2 + μ/(1—μ) and for the stress wave region β = 2 − μ/(1 − μ).

The expression for calculating the radius of the crushed zone R_C when considering the effect of in-situ stress is:

$$\begin{aligned} & 3\cos 2\theta \cdot \frac{{\sigma_{V} - \sigma_{H} }}{2} \cdot \left( {\frac{{r_{0} }}{{R_{C} }}} \right)^{4} + P_{d} \cdot \left( {\frac{{r_{0} }}{{R_{C} }}} \right)^{3} - \left[ {\frac{{\sigma_{V} + \sigma_{H} }}{2} + 2(\sigma_{V} - \sigma_{H} ) \cdot \cos 2\theta } \right] \cdot \left( {\frac{{r_{0} }}{{R_{C} }}} \right)^{2} \\ & + \frac{{\sigma_{V} + \sigma_{H} }}{2} + \frac{{\sigma_{V} - \sigma_{H} }}{2} \cdot \cos 2\theta - \frac{\sqrt 2 }{A}\sigma_{cd} = 0 \\ \end{aligned}$$

(12)

where σ_cd is the dynamic compressive strength of the rock mass, MPa. Let $\frac{{r_{0} }}{{R_{C} }} = x$, then Eq. (12) becomes:

$$\chi x^{4} + \vartheta x^{3} + \varsigma x^{2} + \psi x + \zeta = 0$$

(13)

where

$$\left\{ {\begin{array}{*{20}l} {\chi = 3\cos 2\theta \cdot \frac{{\sigma_{V} - \sigma_{H} }}{2}} \hfill \\ {\vartheta = P_{d} } \hfill \\ {\varsigma = - \left[ {\frac{{\sigma_{V} + \sigma_{H} }}{2} + 2(\sigma_{V} - \sigma_{H} ) \cdot \cos 2\theta } \right]} \hfill \\ {\psi = 0} \hfill \\ {\zeta = \frac{{\sigma_{V} + \sigma_{H} }}{2} + \frac{{\sigma_{V} - \sigma_{H} }}{2} \cdot \cos 2\theta - \frac{\sqrt 2 }{A}\sigma_{cd} } \hfill \\ \end{array} } \right.$$

(14)

Equation (13) can be solved using the MATLAB program, and excluding the limited understanding, the calculation expression of the crushed zone radius R_C is obtained:

$$x = \frac{{r_{0} }}{{R_{C} }} = \frac{ - \vartheta }{{4\chi }} \pm \frac{1}{2}\sqrt {\frac{{\vartheta^{2} }}{{4\chi^{2} }} - \frac{2\varsigma }{{3\chi }} + \Delta } \pm \frac{1}{2}\sqrt {\frac{{\vartheta^{2} }}{{2\chi^{2} }} - \frac{4\varsigma }{{3\chi }} - \Delta \pm \frac{{ - \frac{{\vartheta^{3} }}{{\chi^{3} }} + \frac{4\vartheta \varsigma }{{\chi^{2} }} - \frac{8\psi }{\chi }}}{{4\sqrt {\frac{{\vartheta^{2} }}{{4\chi^{2} }} - \frac{2\varsigma }{{3\chi }} + \Delta } }}}$$

(15)

where

$$\left\{ {\begin{array}{*{20}l} {\Delta = \frac{{\sqrt[3]{2}\Delta 1}}{{3\chi \sqrt[3]{{\Delta 2 + \sqrt { - 4\Delta_{1}^{3} + \Delta_{2}^{2} } }}}} + \frac{{\sqrt[3]{{\Delta 2 + \sqrt { - 4\Delta_{1}^{3} + \Delta_{2}^{2} } }}}}{{3\sqrt[3]{2}\chi }}} \hfill \\ {\Delta_{1} = \varsigma^{2} - 3\vartheta \psi + 12\chi \zeta } \hfill \\ {\Delta_{2} = 2\varsigma^{3} - 9\vartheta \varsigma \psi + 27\chi \psi^{2} + 27\vartheta^{2} \zeta - 72\chi \varsigma \zeta } \hfill \\ \end{array} } \right.$$

(16)

The expression for calculating the radius R_T of the fracture zone when considering the effect of in-situ stress is (Ge 2020):

$$R_{T} = R_{C} \cdot \left( {\frac{{\sigma_{cd} }}{{m\sigma_{td} }}} \right)^{{\frac{1}{\beta }}}$$

(17)

where σ_td is the dynamic tensile strength of the rock mass, MPa. m is the coefficient of rock tensile strength enhancement caused by in-situ stress.

Further discussion on the enhancement factor m. The force state of the microelementary point under the action of coupled stress field is analyzed with reference to the research results of Ge (2020), as shown in Figs. 6 and 7. According to Dai and Qian (2007), the initial static load on the rock mass actually increases the dynamic compressive or tensile strength of the rock mass indirectly. It is assumed that when the microelement point is compressed by a concentrated load (σ_Va) in the vertical direction, the dynamic tensile strength of the rock in that direction is enhanced by 100%, i.e., the dynamic tensile strength in that direction becomes 2σ_td.

Fig. 6

Force analysis of rock microelementary points under in-situ stress action (λ is the lateral pressure coefficient)

Fig. 7

Force analysis of rock microelementary points under explosive load

From the force analysis of the microelement points in Figs. 6 and 7, it can be seen that the tensile stress on each microelement point is (σa/λ·sinθ₁− σa·cosθ₁). Cracks are most likely to expand when the dynamic tensile strength of the rock is not enhanced, i.e., the crack length is longest when θ₁ = arctanλ. Taking the burial depth of 600 m as an example for analysis, when the lateral pressure coefficients λ are 0.2, 0.4, 0.6, 0.8, and 1.0, the longest cracks are in the directions of 79°, 68°, 59°, 51°, and 45° of the blasting hole, respectively. The values of the enhancement coefficient m in each direction of the borehole are shown in Table 2, the variation rule of the enhancement coefficient m with θ under different lateral pressure coefficients λ is shown in Fig. 8, and the variation rule of the enhancement coefficient m with the lateral pressure coefficient λ in the same direction of the blasting hole is shown in Fig. 9.

Table 2 Table of values of enhancement factor m for different lateral pressure coefficients λ

Fig. 8

Law of change of m with θ for different λ

Fig. 9

Law of change of m with λ in the same direction of the blasting hole

3.1.2 Calculation of explosion energy under coupled stress field

After the explosive explodes, the rock around the blast hole in the shock wave produced by the violent compression decays, the hole wall continues to expand outward, and the blast cavity expands. The cavity expansion process ends when the shock wave propagates to the edge of the crushed zone. In the crushed zone, the energy dissipation of the shock wave is equal to the work W₁ performed by the shock wave in the process of cavity expansion:

$$W_{1} = \int_{{r_{0} }}^{{R_{1} }} {2\pi r\sigma_{r} } dr$$

(18)

where the explosion cavity radius $R_{1} = \left[ {R_{C}^{2} - (R_{C}^{2} - r_{0}^{2} )\frac{{\rho_{m} }}{{\rho_{r} }}} \right]^{\frac{1}{2}}$. r₁ is the blast cavity radius corresponding to r. ρ_m is the original rock density, kg/m³. ρ_r is the rock density behind the shock wave front at the blast hole wall, $\rho_{r} = \frac{{(a + bV_{0} )\rho_{m} }}{{a + (b - 1)V_{0} }}$, kg/m³. a and b are the rock test constants, and V₀ is the initial velocity of the rock particle at the blast hole wall. Let $\frac{{r_{0} }}{{R_{1} }} = x_{1}$, the calculation expression of shock wave expansion cavity energy dissipation W₁ can be obtained:

$$W_{1} = \int_{{r_{0} }}^{{R_{1} }} {2\pi r\sigma_{r} } dr = - 2\pi r_{0}^{2} \left[ {\frac{\chi }{2}\left( {x_{1}^{2} - 1} \right) + \vartheta (x_{1} - 1) + \varsigma \ln x_{1} + \frac{{A\zeta + \sqrt 2 \sigma_{cd} }}{2A}(1 - x_{1}^{ - 2} )} \right]$$

(19)

The shock wave expanding cavity consumes a large amount of energy and then decays into a stress wave, which has a stretching effect on the rock around the blast hole and forms a large number of radial fractures. The energy dissipation W₂ for the generation of radial cleavage by the stress wave can be expressed as (Zhang 2007):

$$W_{2} = n\int_{{r_{0} }}^{{R_{T} }} {\delta_{w} dr} = n\int_{{r_{0} }}^{{R_{T} }} {\frac{{2\sqrt 2 K_{1}^{2} (1 - \mu^{2} )}}{{\pi E_{m} }}dr}$$

(20)

where n is the number of radial fractures, generally taken as 10. K₁ is the stress intensity factor, $K_{1}^{2} = \pi r\sigma_{\theta }^{2}$. E_m is the dynamic modulus of elasticity, GPa. Let $\frac{{r_{0} }}{{R_{T} }} = x_{2}$, the calculation expression of W₂ of the radial crack generated by stress wave can be obtained:

$$\begin{gathered} W_{2} = \frac{{20\sqrt 2 (1 - \mu^{2} )}}{{E_{m} }}\int_{{r_{0} }}^{{R_{T} }} {r\sigma_{\theta }^{2} } dr \hfill \\ \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} \\ \end{array} = \frac{{20\sqrt 2 (\mu^{2} - 1)r_{0}^{2} }}{{E_{m} }}[\frac{{\chi^{2} }}{6}x_{2}^{6} + \frac{2\lambda \chi \vartheta }{{2 + \beta }}x_{2}^{2 + \beta } - \frac{\chi \varphi }{4}x_{2}^{4} + \frac{\lambda \vartheta \varphi }{3}x_{2}^{3} + \left( {\chi \omega + \frac{{\varphi^{2} }}{8}} \right)x_{2}^{2} \hfill \\ \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} \\ \end{array} - \frac{{\left( {\lambda \vartheta } \right)^{2} }}{{2\left( {1 - \beta } \right)}}x_{2}^{{2\left( {\beta - 1} \right)}} - \frac{\lambda \vartheta \varphi }{\beta }_{2}^{\beta } - \frac{2\lambda \vartheta \omega }{{2 - \beta }}x_{2}^{\beta - 2} - \varphi \omega \ln x_{2} - \frac{{\omega^{2} }}{2}x_{2}^{ - 2} - B] \hfill \\ \end{gathered}$$

(21)

where

$$\left\{ {\begin{array}{*{20}l} {\varphi = \sigma_{V} + \sigma_{H} } \hfill \\ {\omega = \frac{{\sigma_{V} - \sigma_{H} }}{2}\cos 2\theta - \frac{{\sigma_{V} + \sigma_{H} }}{2}} \hfill \\ {B = \frac{{\chi^{2} }}{6} + \frac{2\lambda \chi \vartheta }{{2 + \beta }} - \frac{\chi \varphi }{4} + \chi \omega + \frac{{\varphi^{2} }}{8} - \frac{{\left( {\lambda \vartheta } \right)^{2} }}{{2\left( {1 - \beta } \right)}} - \frac{\lambda \vartheta \varphi }{\beta } - \frac{2\lambda \vartheta \omega }{{2 - \beta }} - \frac{{\omega^{2} }}{2}} \hfill \\ \end{array} } \right.$$

(22)

Outside the fractured zone, with the attenuation of the stress wave, the stress wave can only cause elastic deformation of the rock mass. At this time, the elastic deformation energy of the rock in unit volume is:

$$\delta_{E} = \frac{1}{2}(\sigma_{r} \varepsilon_{r} + \sigma_{\theta } \varepsilon_{\theta } ) = \frac{1}{{2E_{m} }}(\sigma_{r}^{2} + \sigma_{\theta }^{2} )$$

(23)

Let $\frac{{r_{0} }}{{R_{E} }} = x_{3}$, the calculation expression of elastic deformation energy dissipation W₃ caused by stress wave can be obtained as:

$$\begin{aligned} W_{3} = & \int_{{r_{0} }}^{{R_{E} }} {2\pi r\delta_{E} } dr \\ \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} \\ \end{array} = & - \frac{{\pi r_{0}^{2} }}{{E_{m} }}[\frac{{\chi^{2} }}{3}x_{3}^{6} + \frac{(1 + \lambda )2\chi \vartheta }{{2 + \beta }}x_{3}^{2 + \beta } + \frac{\chi (2\varsigma - \varphi )}{4}x_{3}^{4} + \left( {\chi \kappa + \chi \omega + \frac{{4\varsigma^{2} + \varphi^{2} }}{8}} \right)x_{3}^{3} \\ \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} \\ \end{array} & - \frac{{\left( {1 + \lambda^{2} } \right)\vartheta^{2} }}{{2\left( {1 - \beta } \right)}}x_{3}^{{2\left( {\beta - 1} \right)}} + \frac{{\left( {2\varsigma - \lambda \varphi } \right)\vartheta }}{\beta }x_{3}^{\beta } - \frac{2\vartheta (\kappa + \lambda \omega )}{{2 - \beta }}x_{3}^{\beta - 2} \\ \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} \\ \end{array} & + (2\varsigma \kappa - \varphi \omega )\ln x_{3} - \frac{{\kappa^{2} + \omega^{2} }}{2}x_{3}^{ - 2} - B - C] \\ \end{aligned}$$

(24)

where

$$\left\{ {\begin{array}{*{20}l} {\kappa = \zeta - \frac{\sqrt 2 }{A}\sigma_{cd} } \hfill \\ {C = \frac{{\chi^{2} }}{6} + \frac{2\chi \vartheta }{{2 + \beta }} + \frac{{\varsigma^{2} - \kappa^{2} + \chi \varsigma + 2\chi \kappa }}{2} - \frac{{\vartheta^{2} }}{{2\left( {1 - \beta } \right)}} + \frac{2\vartheta \varsigma }{\beta } - \frac{2\vartheta \kappa }{{2 - \beta }}} \hfill \\ \end{array} } \right.$$

(25)

After the end of the shock wave effect, the explosion gas continues to act in the form of quasi-static pressure on the cavity wall, so that the explosion cavity continues to expand. The energy dissipation W₄ of the expression gas can be expressed as (Zhang 2007):

$$W_{4} = \left\{ {\begin{array}{*{20}l} {\frac{{\pi P_{0} r_{0}^{2} }}{2}\begin{array}{*{20}c} {\left( {\frac{{r_{0} }}{{R_{1} }}} \right)^{4} } \\ \end{array} \left[ {1 - \left( {\frac{{R_{1} }}{{R_{2} }}} \right)^{4} } \right]\begin{array}{*{20}c} {} \\ \end{array} (\sigma_{cd} \ge P_{k} )} \hfill \\ {\frac{{4\pi P_{k} r_{0}^{2} }}{3}\left( {\frac{{P_{0} }}{{P_{k} }}} \right)^{\frac{4}{9}} \begin{array}{*{20}c} {\left( {\frac{{r_{0} }}{{R_{1} }}} \right)^{\frac{2}{3}} } \\ \end{array} \left[ {1 - \left( {\frac{{R_{1} }}{{R_{2} }}} \right)^{\frac{2}{3}} } \right]\begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} \\ \end{array} (\sigma_{cd} < P_{k} )} \hfill \\ \end{array} } \right.$$

(26)

where R₂ is the final radius of the blast chamber, $R_{2} = \left\{ {\begin{array}{*{20}l} {r_{0} \left( {\frac{{P_{0} }}{{\sigma_{cd} }}} \right)^{\frac{1}{6}} \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} \\ \end{array} (\sigma_{cd} \ge P_{k} )} \hfill \\ {r_{0} \left( {\frac{{P_{0} }}{{P_{k} }}} \right)^{\frac{1}{6}} \left( {\frac{{P_{k} }}{{\sigma_{cd} }}} \right)^{\frac{3}{8}} \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {} \\ \end{array} (\sigma_{cd} < P_{k} )} \hfill \\ \end{array} } \right.$. P₀ is the explosive gas pressure at the beginning of the expansion, P₀ = ρ₀D²/4, where ρ₀ is the density of the explosive, kg/m³, and D is the bursting speed of the explosive, m/s. P_k is the critical pressure, MPa.

In addition to blast cavity expansion, the expression gas also throws the broken rock fragments due to the action of the blast stress wave. The energy dissipation W₅ of the expression gas throwing rock fragments can be expressed as (Zhang 2007):

$$W_{5} = \int_{0}^{{r_{a} }} \frac{1}{2} V_{1}^{2} \left[ {1 - \left( {\frac{t}{{r_{a} }}} \right)^{k} } \right]^{2} \frac{2}{3}\rho_{m} Wtdt = \frac{{\rho_{m} n_{1}^{2} \pi W^{3} V_{1}^{2} }}{3}\left( {\frac{1}{2} + \frac{1}{2 + 2k} - \frac{1}{2 + k}} \right)$$

(27)

where n₁ is the throwing action index of the blasting funnel, $n_{1} = \frac{{r_{a} }}{W}$, r_a is the blasting funnel radius, and W is the minimum resistance line. For a standard blasting funnel, n₁ takes 1. $V_{1} = \frac{{\sigma_{cd} }}{{0.38\rho_{m} C_{p} }}$, C_p is the longitudinal wave velocity. k is a constant related to explosives.

The total chemical energy released per unit length of explosive when it explodes is:

$$W_{0} = \pi r_{0}^{2} \rho_{0} Q$$

(28)

where Q is the explosion heat of the explosive, MJ/kg.

3.2 Critical crushing energy dissipation density of rock

The lithology of the surrounding rock in the study section of the Gonghe Village Tunnel is gray sandstone, and static and kinetic tests were carried out after on-site core drilling and sampling to determine the critical crushing energy dissipation values of the gray sandstone under different stress states. According to the specimen sizes recommended by the International Commission on Rock Dynamics (Zhou et al. 2012), the uniaxial compression test and triaxial compression test rock samples were machined as 50 mm × 100 mm cylinders, the Brazilian split test rock samples were machined as 50 mm × 50 mm cylinders, and the impact dynamics test rock samples were machined as 50 mm × 25 mm cylinders. The unevenness and non-perpendicularity of the rock samples were less than 0.02 mm, and the deviation of the end face normal was less than 0.25°. The rock samples are shown in Fig. 10.

Fig. 10

Rock samples

The static tests were carried out using a TAJW-2000 microcomputer-controlled electro-hydraulic servo triaxial rock test system as shown in Fig. 11a, and the static parameters of the gray sandstone are shown in Table 3. The impact dynamics tests were carried out on a detached Hopkinson lever test system, ALT100, as shown in Fig. 11b.

Fig. 11

The test systems used in this study

Table 3 The static parameters used in the experiments

Based on the results of the static tests of the gray sandstone, the axial pressure (σ_V) was set to 0, 10 MPa, 20 MPa, 30 MPa, and 40 MPa, which are 0%, 9.1%, 18.3%, 27.4%, and 36.6% of the static compressive strength, respectively, and the confining pressure (σ_H) was set to 0, 4 MPa, 8 MPa, and 12 MPa.

The main purpose of this test is to determine the critical crushing energy density of gray sandstone under different dynamic and static combination loading conditions. Li et al. (2010) identified the state when the rock is broken into exactly 2–4 pieces as the critical damage state. At the beginning of the test, we set a certain air pressure to impact the rock sample, and if the rock sample is intact or only produces a small amount of cracks, we need to increase the impact air pressure. If a certain impact air pressure, the rock sample is broken into 2–4 pieces, this is the critical damage state. Continue to increase the impact air pressure, if the rock sample crushed, the previous impact air pressure is the critical impact air pressure. So we take the impact air pressure when the rock is exactly broken into 2–4 pieces as the critical impact air pressure, and we take the crushing energy density at this time as the critical crushing energy dissipation density.

During the test, to ensure the reliability of the test results, it is necessary to ensure that the stress on both ends of the rock samples reaches a state of dynamic equilibrium. The stresses, σ₁ and σ₂, which are applied to the two ends of the rock sample can be calculated from the incident strain signal, $\varepsilon_{I} \left( t \right)$, the reflected strain signal, $\varepsilon_{R} \left( t \right)$, and the transmitted strain signal, $\varepsilon_{T} \left( t \right)$, according to the following equation:

$$\begin{gathered} \sigma_{1} = \frac{{A_{0} E_{0} }}{A}[\varepsilon_{I} (t) + \varepsilon_{R} (t)] \hfill \\ \sigma_{2} = \frac{{A_{0} E_{0} }}{A}\varepsilon_{T} (t) \hfill \\ \end{gathered}$$

(29)

Figure 12 shows the dynamic stress equilibrium curve of gray sandstone (with an axial pressure of 10 MPa, a confining pressure of 4 MPa, and an impact air pressure of 0.4 MPa). The combined stress curves of the incident and reflected waves are fundamentally consistent with the stress curves of the transmitted waves, indicating that the stresses at both ends of the rock samples are virtually in equilibrium during the test, which verifies the validity of the test process and the test results.

Fig. 12

The dynamic stress balance curve

The critical crushing energy dissipation density of gray sandstone under the different dynamic and static combinations of loading conditions is shown in Table 4. The curves of the critical crushing energy dissipation density of gray sandstone according to confining pressure and axial pressure are shown in Figs. 13 and 14, respectively.

Table 4 The critical crushing energy dissipation density of gray sandstone under different dynamic and static combination loading conditions

Fig. 13

The variation curve of the critical crushing energy dissipation density according to confining pressure

Fig. 14

The variation curve of the critical crushing energy dissipation density according to axial pressure

The maximum burial depth of the Gonghe Village Tunnel is about 1320 m. The rock density of the study section is 2.56 g cm⁻³ and the static Poisson's ratio is 0.37. Assuming that the lithology of the study section is homogeneous, the burial depths corresponding to axial pressures of 0 MPa, 10 MPa, 20 MPa, 30 MPa, and 40 MPa are 0 m, 391 m, 781 m, 1172 m, and 1563 m, respectively, when considering only the effect of gravitational stress. The dynamic Poisson's ratio of the lithology of the study section is 0.296, and the lateral pressure coefficient is 0.42, from which the corresponding confining pressures are 0 MPa, 4 MPa, 8 MPa, 12 MPa, and 16 MPa, respectively. Due to the equipment, the confining pressure of 16 MPa could not be loaded. Setting up the study conditions according to Table 4 can provide a basis for the design of blasting parameters for large cross-section tunnels under different stress states.

From Fig. 13, it can be seen that the critical crushing energy dissipation density of gray sandstone increases with an increase in the confining pressure. The logarithmic function was used to fit the relationship between the changes; the fitting model and the fitting results are listed in Table 5. It is evident that the confining pressure exerts a reinforcing effect on the gray sandstone samples; the greater the confining pressure, the more significant the reinforcing effect. The destruction of rock samples is a process of transition from a stable state to an unstable state under energy-driven action. Therefore, to destroy the gray sandstone samples, they must store more energy to reach their energy storage limit. Macroscopically speaking, this is manifested as the critical crushing energy dissipation density of the gray sandstone, which increases along with the increase in the confining pressure. As can be seen from Table 5, the model’s fits at the axial pressures of 0 MPa and 20 MPa are relatively poor, which may be due to the discrete nature of the gray sandstone samples.

Table 5 The fitting model and fitting results of the critical crushing energy dissipation density versus confining pressure for gray sandstone samples

From Fig. 14, it can be seen that the critical crushing energy dissipation density of the gray sandstone samples increases and then decreases with the increase in axial pressure. A quadratic function is used to fit the relationship between its changes, and the fitting model and fitting results are listed in Table 6. From the previous analytical results, it is evident that axial pressure exerts a reinforcing effect on the gray sandstone samples when the axial pressure is small, while when the axial pressure continues to increase, the effect of the axial pressure on the gray sandstone samples is converted from a reinforcing to a deteriorating effect. Therefore, when the axial pressure is small, more energy needs to be absorbed to force the gray sandstone to destabilize. When the axial pressure continues to increase, the axial pressure has already caused internal damage within the gray sandstone; therefore, less energy needs to be absorbed in order to force the gray sandstone to destabilize.

Table 6 The fitting model and fitting results of the critical crushing energy dissipation density versus axial pressure for the gray sandstone samples

From Table 6, it can be calculated that the critical crushing energy dissipation density of gray sandstone reaches a maximum value of 7.01 J cm⁻³ when the axial pressure is 25.00 MPa, along with a peripheral pressure of 4 MPa. The critical crushing energy dissipation density of gray sandstone reaches a maximum value of 7.38 J cm⁻³ when the axial pressure is 28.33 MPa, along with a peripheral pressure of 8 MPa. The critical crushing energy dissipation density of gray sandstone reaches a maximum value of 8.40 J cm⁻³ when the axial pressure is 20.00 MPa, along with a peripheral pressure of 12 MPa.

3.3 Determination of extrapolation distance for wedge-cut hole

The main wedge-cut holes in the study section of the Gonghe Village Tunnel were loaded with a single charge of 3.0 kg in each hole and a charge length of 3.0 m. From Eq. (28), W₀ is calculated as 32.85 MJ, and the total chemical energy density is 32.85 J cm⁻³, in which the heat of detonation Q of No. 2 rock emulsion explosives is taken as 4.5 MJ/kg. In this section, 30% W₀ is used as the total chemical energy released by the explosives when the main wedge-cut hole explodes, which is 9.86 J cm⁻³.

The burial depth of the study section is 400 m, σ_V = 10.24 MPa, and σ_H = 4.3 MPa. The wedge-cut hole has a radius of 25 mm, a depth of 3.5 m, and a charge length of 3.0 m. The site uses No. 2 rock emulsion explosives, explosive density of 1.24 g cm⁻³, and bursting speed of 4200 m s⁻¹. From the results of the study in Sect. 3.1, it can be seen that the radius of the crushing zone R_C is 58.28 mm, the radius of the fissure zone R_T is 261.73 mm, the radius of the bursting cavity R₁ is 43.68 mm, and the final radius of the bursting cavity R₂ is 48.05 mm. W₁, W₂, W₄, and W₅ were 1.98 MJ m⁻³, 0.59 MJ m⁻³, 0.14 MJ m⁻³, and 2.08 MJ m⁻³, respectively. At this point, the residual energy is 5.07 MJ m⁻³. From the test results in Sect. 3.2, the critical energy dissipation density of gray sandstone under this condition is 4.10 MJ m⁻³. The residual energy is greater than the critical energy dissipation density and the rock mass will continue to be destroyed. Therefore, further consumption of explosive energy is required. Backcalculating the joint Eqs. (19), (21), (24), (26), and (27), the R_E should be a minimum of 2.8 m when the residual energy is 4.10 MJ m⁻³. It can be seen that for the upper bench blasting in the study section of the Gonghe Village Tunnel, the minimum distance of the main wedge-cut holes from the tunnel design contour line is 2.8 m. Compared with the conventional tunnel blasting method, the main wedge-cut holes of the RHB method were laid 1.4 m outward, as shown in Fig. 15.

Fig. 15

Schematic diagram of the extrapolation distance of the main wedge-cut holes of the RHB method

4 Parameter validation

4.1 Model building

ANSYS/LS-DYNA numerical simulation software was used to establish a three-dimensional numerical analysis model to compare and analyze the effect of hollowing out and the damage law of the retained rock body after blasting, so as to verify the reasonableness of the theoretical calculation results in Sect. 3.3. For the main wedge-cut holes 2.4 m, 2.6 m, 2.8 m, 3.0 m, and 3.2 m from the tunnel design contour line, the overall dimensions of the model are 22 m × 18 m × 3.7 m (X × Y × Z), and the radius of the blasting holes and the radius of the pill rolls are 25 mm and 16 mm, respectively. The following simplifications are made to reduce the computational effort of the model: (1) only half of the overall model size is created when modeling, and after the model is calculated, the keyword *CONSTRAINED_GLOBAL is used to analyze the model after symmetry; (2) only primary wedge-cut holes, secondary wedge-cut holes, and center holes are modeled. The mesh size directly affects the accuracy of the simulation results. In this paper, we first refer to the research results of the literature to determine the preliminary grid size (Wang et al. 2016), and then comprehensively consider the calculation effect and calculation time, and finally select the rock grid size of 5 mm within the design contour line of the tunnel. The total number of modeled units is approximately 1.93 million. A simplified model is shown in Fig. 16a, the grid division diagram is shown in Fig. 16b.

Fig. 16

Schematic of the simplified model and the grid division diagram

The in-situ stress σ_V = 10.24 MPa was applied to the upper and lower faces of the model, and the in-situ stress σ_H = 4.3 MPa was applied to the left and right faces (burial depth 400 m). The method of applying the in-situ stress is as follows: Define the load curve CURVE with the keyword *DEFINE, load from 0 to σ_V (or σ_H), and use the keyword *INTERFACE to output the DYINA file with the ground stresses, replacing the original k file, to achieve the effect of applying the in-situ stress indicated in the model. The rock material is defined as a solid, the explosive and air material is defined as a fluid, the fluid is meshed in a co-nodal way, and the solid and the fluid are connected by fluid solid coupling. All faces except the free surface are set as non-reflective boundary conditions. The RHT model was used for the rock constitutive model, and the parameters of the RHT constitutive model are shown in Table 7 (Li et al. 2018a; b). The explosives, air, and gun clay material models were modeled using *MAT_HIGH_EXPLOSIVE_BURN, *MAT_NULL, and *MAT_SOIL_AND_FOAM material models, respectively, and the material models were taken with reference to the results of several studies in the literature (Liu et al. 2020; Wang 2020; Li et al. 2018a, b). The blast parameters of the simplified model are shown in Table 8.

Table 7 RHT model parameters

Table 8 Simplified model blast parameters

4.2 Analysis of results

After the wedge-cut hole is extrapolated, whether the slot cavity can be successfully hollowed out is key to the success of blasting. The hollowing effect with different wedge shaped hollowing hole extrapolation distances is shown in Fig. 17. Due to the clamping function of the rock, the rock in the slot cavity needs to overcome shear resistance and tensile resistance when being hollowed out. As can be seen from Fig. 17, the hollowing effect gradually deteriorates as the wedge-shaped hollowing hole is extrapolated by an increasing distance. When the distance between the main wedge-cut hole and the design contour line of the tunnel is greater than or equal to 2.8 m, the explosive load can overcome the shear resistance and tensile resistance, and the rock in the slot cavity can be thrown out smoothly. At the same time, due to the role of center holes, the bottom of the hole will not show signs of the bulging belly phenomenon. When the distance between the main wedge-cut holes and the tunnel design contour line is less than 2.8 m, the spacing between the bottom of the wedge-cut holes is large, and the wedge-cut holes are able to form pre-cracking cracks after blasting, but are unable to overcome the tensile resistance of the rock at the bottom of the holes. Even if the center holes blasting can break part of the rock, it cannot also successfully hollow out the slot cavity, resulting in a serious bulging belly phenomenon, or even lead to blasting failure.

Fig. 17

Effect of hollowing out

Damage to the retained rock outside the tunnel design contour after the wedge-cut holes have been extrapolated is another area of concern. The damage to the retained rock at different wedge-cut hole extrapolation distances is shown in Fig. 18. As can be seen from Fig. 18, when the distance of the main wedge-cut hole from the tunnel design contour line is greater than or equal to 2.8 m, blasting basically will not cause damage to the retained rock outside the tunnel design contour line. However, when the distance of the main wedge-cut hole from the tunnel design contour line is less than 2.8 m, blasting can cause extensive damage to the retained rock in the arch waist area. Peripheral hole blasting and post-blast drainage will result in more severe over excavation and affect the quality of blasting. Therefore, the optimum distance from the main wedge-cut hole to the tunnel design contour line in the study section of the Gonghe Village Tunnel is 2.8 m. The numerical simulation results are in agreement with the theoretical calculations.

Fig. 18

Damage to retained rock

5 Field tests

Adopting the theory and method of RHB for large cross-section tunnel blasting proposed in this paper, five on-site blasting tests were carried out in the research section of the Gonghe Village Tunnel, and the plan view of the hole network design, the actual blast hole layout plan, the hole network design section plan, and the schematic diagram of the over-undercutting measurement point layout are shown in Fig. 19a–d. In order to ensure the hollowing effect, the collapse holes were drilled at a downward inclination of 5°–10°, and the blasting parameters are shown in Table 9. After the completion of blasting, the quality of blasting was comprehensively evaluated in terms of the excavation effect of the working face, the number of blasting holes, the drilling time, the utilization rate of the blasting holes, the unit consumption of explosives, and the over and under excavation.

Fig. 19

Schematic diagram of aperture network design and monitoring point placement

Table 9 Blasting parameters

Using the theory and method proposed in this paper for on-site blasting tests, the slot cavity can be successfully hollowed out, and the work surface is relatively flat, with basically no bulging belly phenomenon. After blasting was completed, the value of over undercut was measured at each measurement point, as shown in Fig. 20. Data on the number of blasting holes, drilling time, utilization rate of blasting holes, unit consumption of explosives, and over and under excavation were collected, as shown in Table 10.

Fig. 20

Measurement of values for over and under excavation

Table 10 Relevant data collection table after the adjustment of blasting program

As can be seen from Table 10, by adopting the theory and method of RHB for large cross-section tunnel blasting proposed in this paper, the number of blasting holes per cycle of step blasting in the study section is reduced by an average of 21 holes, the drilling time is saved by an average of 0.5 h, the explosive unit consumption is reduced by an average of 0.16 kg cm⁻³, and the value of over excavation is reduced by an average of 6.6 cm. The main reasons for the better contouring results of the RHB method compared to the conventional blasting method are as follows: (1) the wedge-cut hole pushed out to form a two-way critical surface after blasting, which can effectively block the stress wave during the center holes blasting and reduce the damage of the retained rock mass. (2) The number of auxiliary holes deployed was reduced, effectively minimizing the cumulative damage to the retained rock mass. (3) When the RHB method is applied in the field, the peripheral holes are loaded with axial spacing, and the explosive energy is distributed more evenly.

The cost of explosives on site is RMB 8.4 per kg, and the cost of the digital electronic detonator is RMB 16 each, whereas the research section of the upper step blasting can reduce the digital electronic detonator cost by RMB 320 per cycle while reducing explosives cost by RMB 302.4 per cycle. The total length of the right width of the Gonghe Village Tunnel is 10,698 m; by adopting the theory and method proposed in this paper, it is expected to save 1783 h of drilling time and about RMB 2.22 million on the cost of explosives. After the optimization of the program, the unit consumption of explosives is still high, which is due to the higher compressive strength (109.33 MPa) and better integrity of the rock in the study section, which requires more energy to be consumed to form crushed and fractured zones.

6 Conclusions

• (1) This paper put forward an RHB theory and method for large cross-section tunnels. The calculation method of explosion energy dissipation under the action of coupled stress field is given, the critical energy dissipation density of rock under different stress states is determined, and the calculation method of the extrapolation distance of wedge-cut holes is constructed.

• (2) The critical crushing energy dissipation density of gray sandstone increases with the increase in peripheral pressure, and the change rule conforms to the logarithmic function relationship. The critical crushing energy dissipation density of the gray sandstone samples increases and then decreases with the increase in axial pressure, and the change rule conforms to the quadratic function relationship. The critical crushing energy dissipation density of the gray sandstone samples under the conditions of confining pressure of 4 MPa axial pressure of 25.00 MPa, confining pressure of 8 MPa axial pressure of 28.33 MPa, and confining pressure of 12 MPa axial pressure of 20.00 MPa reached maximum values of 7.01 J cm⁻³, 7.38 J cm⁻³, and 8.40 J cm⁻³, respectively.

• (3) The optimum distance of the main wedge-cut holes in the study section of the Gonghe Village Tunnel from the tunnel design contour is 2.8 m. The number of drill holes per cycle of step blasting in the section used in this study was reduced by about 15.8%, the drilling time was reduced by an average of 0.5 h, the unit consumption of explosives was reduced by an average of 0.16 kg cm⁻³, the value of over excavation was reduced by an average of 6.6 cm, and the cost of explosives was reduced by an average of RMB 622.4.