## Abstract

The pressure surge at the moment of blasting seriously threatens the safety of pipeline structures and other buildings. This article established two numerical simulation models of the hydraulic transition process and water–air two-phase flow based on the one-dimensional transient flow theory and considering the actual flow characteristics of water and air. The hydraulic characteristics and blasting parameters of the blasting system were studied under two different conditions of the tunnel with and without the pressure flow. The observed parameters of an actual blasting engineering prototype verified the accuracy of the mathematical model. The maximum pressure of pressure blasting was 2.55 times as high as hydrostatic pressure. The maximum pressure of non-pressure blasting was 1.77 times as high as hydrostatic pressure. Pressure blasting impact was 2.17 times greater than nonpressure blasting. Discrepancies between actual and simulated data for gushing and overflow heights ranged from 9 to 14%. The model can provide a reference for blasting engineering on the simulation calculation of hydraulic characteristics such as impact pressure, overflow time, and overflow of the underwater rock plug blasting system.

## HIGHLIGHTS

The simulation model of the hydraulic transient process in underwater rock plug blasting engineering was established by method of characteristics and water–air two-phase flow equation, respectively.

The observed parameters of an actual blasting prototype verified the accuracy of the mathematical model.

The blasting should be carried out in the nonfull flow in the tunnel when the impact pressure could meet the pressure standard of the system.

## INTRODUCTION

Rock plug blasting is a controlled blasting under the water. The tunnel inlet is built on the reservoir or slope protection. After the completion of the tunnel construction, the rock plug blasting enables an unobstructed connection between the reservoir and the tunnel. Rock plug blasting technology has been widely used for saving investment, shortening the construction period, and its relatively mature construction technology (Ding 1998; Sharafat *et al.* 2019). Researchers have done much research on blasting monitoring technology, blasting method, blasting device, blasting technology, and explosive dosage, thus accumulating rich experience in underwater rock block blasting technology (Zhao *et al.* 2016; Choudhary & Agrawal 2022; Li *et al.* 2022). Many underground caverns were also excavated by blasting (Van Kien *et al.* 2022). A new blasting method was proposed to prevent premature emission of explosive gas, reduce unit consumption of explosives, and improve the utilization rate (Yang *et al.* 2018). The Fluent–EDEM coupling algorithm simulated the block motion (Wu *et al.* 2023). To comprehensively consider the deformation and failure mechanism caused by the strain rate effect and energy flow, a Johnson-Holmquist-Rock (JHR) constitutive model was proposed to simulate the failure process of rock under a blasting load (Xie *et al.* 2019). Hu *et al.* (2016) revealed that the numerical simulation and penetration test of rock plug blasting confirmed each other. Also, the difference in the actual blasting depth increased significantly with the increase in the drilling depth.

For the hydraulic characteristics of underwater rock plug blasting, the primary performance of the rock plug blasting in blasting dynamics was shown as a peak in an instant on the inlet system produced by shock wave blasting. The pressure of the building, such as the water conveyance tunnel and the gate shaft in the system, would rise rapidly after a few seconds when blasting occurred (Wu *et al.* 2023; Xie *et al.* 2019). Too much pressure could cause dangerous accidents in each building of the water supply system (Qasim *et al.* 2022). The water hammer pressure in the water supply system was mainly caused by the sudden opening and closing of the valve in the water supply or the impact of pressurized water on the retained air mass during the flushing process of the tunnel (Aminoroayaie Yamini *et al.* 2021; Wang *et al.* 2022; Zhang *et al.* 2022). For a closed pipe containing an air pocket in the end or air released in an exhaust hole, the air pocket impacted by pressurized water with a valve opening would produce several times direct water hammer pressure (Zhou *et al.* 2011). The transient theory and models have been applied with substantial progress and achievement for traditional elastic pipes, such as one-dimensional (1D) and two-dimensional (2D) water hammer models and unsteady friction or turbulence models (Duan *et al.* 2018, 2020). Computational fluid dynamics techniques were now routinely employed for unsteady hydraulic engineering problems (Kyriakopoulos *et al.* 2022). It has been widely used for studying two-phase pressurized flows such as the free surface-pressurized flow (Cong *et al.* 2017; He *et al.* 2022), transient cavitating flow (Zhou *et al.* 2018), transient flows with entrapped air pockets (Huang & Zhu 2021; Zhai *et al.* 2021), and geyser formation in vertical shafts (Chang & Wei 2022; Liu *et al.* 2022).

After the rock plug blasting, the air and water are the two-phase unsteady flow. The transition process of water and gas is highly complex, which causes the difficulty in setting up a mature theory. Most blasting projects obtained the blasting effect and impact pressure through onsite blasting prototype observation and the hydraulic model test. However, a few studies did numerical simulation calculations on the transition process after underwater rock plug blasting. Numerical simulation has the advantages of fast computation and saving test costs and time. The pressure generated at the rock plug blasting may cause harm to the pipeline system and related buildings. It is necessary to calculate the hydraulic dynamic characteristics of the water flow in the pipeline to ensure the safe and stable operation of the whole water conveyance system and downstream buildings after rock plug blasting. Chen *et al*. introduced the principle and application of air cushion underwater rock plug blasting technology, put forward the numerical simulation model of the hydraulic transition process of air cushion underwater rock plug blasting, and discussed the sensitivity analysis of blasting parameters on hydraulic characteristics (Chen *et al.* 2022). Zhang *et al.* (2020) established a 1D mathematical model of the underwater rock plug blasting system using the method of characteristics (MOC) based on the transient flow theory. The changes in water hammer pressure and flow were studied, generated by the system after blasting when the end of the tunnel was blocked and full of flow and pressure. The results showed that the negative pressure along the system is severe after blasting. It was necessary to increase the area of the ventilation shaft for water hammer protection. A reasonable diameter of the ventilation shaft (*D* = 10.0 m) was proposed to ensure the safe operation of the system after blasting. This article only studied the blasting condition of the tunnel with full pressure flow. The pressure safety should be met at all locations in the system, which caused the diameter of the ventilation shaft to be expanded to 10 m, which was too high in construction cost. Whether the safety structure of the foundation meets the requirements needs further discussion.

Therefore, based on previous studies and saving engineering costs, a method of blasting under nonfull flow conditions without pressure was put forward. Based on 1D transient flow theory and considering the mechanical characteristics of water and air, a mathematical model of a water–air two-phase flow in the rock plug blasting system was proposed in this article, and the changes of water–air impact pressure, overflow flow, overflow time, and other parameters produced by blasting in a tunnel with no pressure and nonfull flow were simulated. Specifically, this study focuses on:

- (1)
Two different simulation models of transition processes in the tunnel with full and nonfull pressure flow were established according to the 1D MOC and the water–air two-phase flow equation.

- (2)
The tunnel with and without pressure blasting was simulated. The hydraulic characteristics were studied, such as impact pressure fluctuation, flow rate, water level, velocity, and overflow time at different locations of the blasting system. The changes in hydraulic characteristics of key buildings were analyzed in detail.

- (3)
Comparing the difference between the simulation results of two different models, a better blasting scheme was proposed.

- (4)
The numerical simulation results of the water–air two-phase flow model were compared with the actual engineering prototype observation results to illustrate the reliability of the mathematical model.

## MATHEMATICAL MODEL

In rock plug blasting, two mathematical models were proposed to meet different calculation conditions according to the different flow movement states in the tunnel downstream of the blasting point. The first mathematical model was based on MOC, which was used to simulate the impact pressure, water level change, overflow flow, overflow time, and other parameters of the burst water flow in the tunnel. The second mathematical model was based on the actual flow characteristics of water and air, which was used to simulate the impact pressure, overflow flow, air pressure, air mass, and other parameters after blasting in the tunnel with no pressure and nonfull flow. The two mathematical models are described in the following sections.

### Mathematical model of the blasting system with pressure flow in the tunnel

The blasting point model of the rock plug was like the valve model. The blasting moment was like the sudden opening of the valve. From the beginning of the blasting to the end, the corresponding valve was closed at the beginning. This state was instantaneous. The reservoir, blasting point, tunnel, ventilation shaft, and terminal maintenance gate model were established according to the composition of the rock plug blasting project's water conveyance system. The specific mathematical models were as follows:

#### Water hammer equation of pressurized pipeline

^{3}/s); are coefficients, which can be obtained from the pipeline parameters and head flow before and after the node at the previous moment.

#### Upstream reservoir model

*C*is the constant.

#### Blasting point model

*τ*= 0) to fully opening (

*τ*= 1). According to the flow equation of the valve node, combined with the characteristic line Equations (1) and (2), the control equation of the rock plug blasting point can be established as follows:where is the blasting node flow (m

^{3}/s); is the head difference when the valve orifice is fully open (m); is the overflow when the valve is fully open (m

^{3}/s); and is the flow coefficient of the dimensionless valve.

#### Ventilation shaft model

### Mathematical model of the blasting system in the tunnel with water and air two-phase flow

When there is no pressure flow in the tunnel behind the rock plug, when the rock plug blasting occurs, because there is a large amount of air in the tunnel, it is necessary to establish an air valve model to simulate the intake and exhaust characteristics. The corresponding water–air two-phase mathematical model is based on the actual characteristics of the water and air in the tunnel and branch hole.

#### Air valve model

Air valve boundary conditions are divided into the following four cases:

*p*is the pressure in the pipe.

*i*at time of ; is the flow out of section

*i*at time of

*t*; is the flow into the cross section

*i*at time of ; is the mass of the air in the cavity at time of ; is the mass of the air flow into and out of the cavity at time of ; and is the mass of the air flow into and out of the cavity at time of

*t*.

#### Water–air two-phase flow calculation equations

Based on the actual flow characteristics of actual water and air, according to the flow continuity equation, the relationship between the flow and the water level, the head balance equation, the air valve exhaust equation, and the ideal gas state equation, the calculation equations of water–air two-phase flow are established. After simplification, the differential equations can be obtained as follows:

^{2}); is the actual flow area of the tunnel (m

^{2});

*V*is the pipe flow rate (m/s); is the flow in the tunnel (m

^{3}/s); is the local atmospheric pressure (1.013 × 10

^{5}Pa);

*P*is the air pressure in the tunnel (Pa);

*g*is the gravitational acceleration (9.81 m/s

^{2}); and is the actual flow area at the valve (m

^{2}).

## CASE STUDY

Condition . | Water level of the reservoir . | Water level of the tunnel . | Flow regime of the tunnel . | Calculation values . |
---|---|---|---|---|

1 | 316.0 m | 267.85 m | Pressure flow | Water pressure, flow velocity, flowrate, surge change, and total overflow in the ventilation shaft, pressure tunnel, and 1# branch hole. |

2 | 316.0 m | 266.00 m | Nonpressure flow | Exhaust process, air pressure, water pressure, air mass, surge height, flow rate changes in the ventilation shaft, 1# branch hole. |

Condition . | Water level of the reservoir . | Water level of the tunnel . | Flow regime of the tunnel . | Calculation values . |
---|---|---|---|---|

1 | 316.0 m | 267.85 m | Pressure flow | Water pressure, flow velocity, flowrate, surge change, and total overflow in the ventilation shaft, pressure tunnel, and 1# branch hole. |

2 | 316.0 m | 266.00 m | Nonpressure flow | Exhaust process, air pressure, water pressure, air mass, surge height, flow rate changes in the ventilation shaft, 1# branch hole. |

## RESULTS

### Results of pressure flow in the tunnel

The maximum and minimum distribution of water hammer pressure generated at each node of the water conveyance system under the blasting condition of a tunnel's full flow state is shown in Figure 5. The more considerable pressure of the whole pipeline was distributed between the No. 4 node (ventilation shaft) and No. 30 node (1# branch hole). The pressure value was more significant than 130 m. Most nodes' negative pressure extreme value distribution area was −3.82 to −10 m. The minimum pressure of the whole pipeline was also distributed between the No. 4 node (ventilation shaft) and No. 30 node (1# branch hole). The negative pressure value was −10.0 m (under the vaporization pressure), which indicated that the water body in this area had vaporized. The pressure behind 1# branch hole was almost unchanged, stable at 3.65 m. Affected by the water level fluctuation and water hammer wave reflection of the two surge wells of the ventilation shaft and 1# branch hole, the node between the ventilation shaft and 1# branch hole produced a considerable water hammer pressure. Due to the large area of 1# branch hole, the pressure wave propagating from the upstream to the downstream was effectively cut off, and the pressure wave was prevented from propagating downstream, equivalent to effectively shortening the distance of the pressure pipeline.

^{3}/s, and a gradual decrease in the water level difference after that. The flow of the tunnel decreased gradually, and the flow returned after 297.1 s, with a minimum value of −5.38 m

^{3}/s. The pressure value of the water hammer changed periodically. Although the area of 1# branch hole was significant, which weakened part of the water hammer pressure, the elimination effect of the water hammer wave was not strong due to the considerable pressure generated by blasting, and considerable water hammer pressure was still generated at No. 19 node.

From Figure 7(a), the maximum water hammer pressure in the ventilation shaft was the extreme value of the first wave pressure rise, 131.6 m, which occurred within 1 s after blasting. Because the ventilation shaft was close to the rock plug blasting mouth, the pressure in the blasting ventilation shaft rose to the maximum value at the beginning. Then, the subsequent water hammer pressure shows a periodic change. There was a trend of gradual decrease from the maximum value of the second wave water hammer pressure of 56.74–53.17 m. Figure 7(b) shows that the water level of the ventilation shaft rose rapidly to 323 m after blasting, exceeding the overflow water level of 320 m. Similar to the water hammer pressure fluctuation, the ventilation shaft's water level variation presented periodicity. After a period, the water level gradually decreased to 316 m, tended to be stable, and remained unchanged. The relationship between the inflow and the outflow of the ventilation shaft and the time is shown in Figure 7(c). The relative flow rate was the inlet flow value of the ventilation shaft at a specific time minus the flow value of the outlet of the ventilation shaft. Due to the ventilation shaft's small area and the blasting's front end, the inflow of the ventilation shaft quickly reached the maximum of 42.12 m^{3}/s, which occurred at 0.67 s after the blasting. The outlet flow reached the maximum of 17.03 m^{3}/s, which happened in 6.66 s after the blasting. The relationship between the overflow of the ventilation shaft and time is shown in Figure 7(d). The ventilation shaft overflows with the maximum overflow of 27.98 m^{3}, which occurred at 1.8 s after the blasting. The first overflow lasted for 1.3 s. With the subsequent fluctuation of the water hammer pressure, water gushed out from one shaft after another at each time point in the later period. The total number of gushes was three. The overflow of the first overflow was the largest. No overflow occurred after 16 s. From the aforementioned analysis, because the ventilation shaft was close to the location of the blasting point, the blasting point was similar to the valve opening process. An enormous water hammer pressure was generated at the ventilation shaft. After the blasting began, the ventilation shaft began to overflow, and the overflow in the shaft was large. However, due to the small area of the ventilation shaft, the water hammer wave was not well eliminated, and the pipeline was still in a more dangerous state.

^{3}/s, which occurred 56 s after blasting. The maximum outlet flow was 25.69 m

^{3}/s, which happened in 90.81 s after blasting. Figure 8(d) shows that 1# branch hole exceeds the overflow water level by 320 m, and an overflow occurred. The overflow occurred at 77.64 s, and the overflow time lasted 227.45 s. After 305.09 s, there was no overflow. The maximum overflow during the overflow process was 248.8 m

^{3}. After blasting, the more considerable water hammer pressure propagated from upstream to downstream, and the reflection period was short when it propagated to the end gate well and then returned to 1# branch hole.

### Results of nonpressure flow in the tunnel

^{3}. The pressure and mass of the air in the tunnel changed with time, as shown in Figure 11. From Figure 11(a), after blasting, due to the high water level of the upper reservoir, the water pressure entering the tunnel was considerable, and the air in the tunnel was squeezed into the 1 # branch hole. At 119 s after the blasting, the air pressure in the tunnel reached a maximum of 75.11 m, and then the air pressure decreased slightly with time. When the time reached 1,500 s, the air pressure decreased suddenly, from 62.03 m to the local atmospheric pressure directly. After 1,550 s, the air in the tunnel was discharged entirely through the ventilation shaft, and the tunnel was filled with water. The relationship between the air change in the tunnel and time is shown in Figure 11(b). The air mass in the first 1,500 s after the rock plug blasting was 40,220 kg of the initial air mass, which did not change. After 1,550 s, the air mass in the tunnel decreased rapidly to 0 kg, indicating that the air in the pressure tunnel had been discharged through the ventilation shaft and the tunnel had been filled with water.

The numerical simulation results of pressure blasting and nonpressure blasting in this project were different. When the pressure blasting was carried out, the whole pipeline would produce enormous water hammer pressure. The maximum positive pressure was 131.6 m. Also, the negative pressure was below the vaporization pressure. The extreme value was distributed between the ventilation shaft and the 1 # branch hole node. If the blasting was carried out according to this scheme, the water conveyance system and related buildings may be destroyed. However, when the nonpressure blasting was carried out, the positive pressure was significantly reduced. The maximum positive pressure of the pipeline was 60.43 m. Also, there was no negative pressure along the line. The pressure value of the pipeline met the pressure standard. Therefore, it was recommended to select nonpressure blasting.

## DISCUSSION

This study established mathematical models for the full and nonfull flow modes in the tunnel. Compared with previous studies, this article compared the pressure surge, overflow flow, overflow time, and other parameters generated by the two models under different flow modes in the tunnel. The flow state of the tunnel after the burst point was the critical parameter affecting the impact pressure of rock plug blasting. When the tunnel was filled with water, the impact pressure generated at the blasting moment was much larger than in the state of no pressure. The maximum pressure reached 2.55 times the static pressure of the tunnel, while the maximum pressure in the state of nonpressure blasting was only 1.17 times the static pressure of the tunnel. The maximum impact pressure of the tunnel under pressure blasting was 2.17 times that of nonpressure blasting. Negative pressure occurred along the first half of the blasting system when the tunnel was under pressure. No negative pressure occurred along the system when the tunnel was under no pressure.

In the mathematical model of the blasting system with the pressure flow in the tunnel, it was considered that the valve was opened instantaneously, which meant that the valve opening time was completed instantaneously. Therefore, a significant negative pressure was generated in the first half of the blasting system, and the values were all under the vaporization pressure. Although the ventilation shaft in the system could eliminate part of the water hammer wave, the water hammer wave cannot be eliminated due to the small area of the ventilation shaft. In the author's previous research results, it is also mentioned that to effectively reduce the positive pressure of the water hammer and make the negative pressure meet the system requirements, the diameter of the ventilation shaft should be expanded to 10 m. This approach increased the engineering cost and the need to demonstrate the bearing capacity of rock mass and foundation. Therefore, a mathematical model of the rock plug blasting system was proposed in this article when the tunnel presented the nonpressure flow, which was a new model applicable to the simulated pressure surge after the tunnel shows the nonpressure flow and the changes of the overflow time and the overflow height of the building after the explosion. Compared with the calculation results of the pressure flow in the tunnel of the rock plug blasting system proposed earlier, the tunnel's nonpressure blasting advantages were illustrated. The actual prototype observation data demonstrated the reliability of the mathematical model.

In the mathematical model of the blasting system in the tunnel with water and air two-phase flow, the nonpressure flow contains many different water levels. So long as the wet circumference is not greater than the wet circumference of the full flow, it can be called the nonpressure flow. This article did not propose the change of pressure surge after the explosion of the nonpressure flow under different water levels, which made the research in this article have certain limitations. Moreover, only the actual water and air movement were studied in these two mathematical models. However, in actual engineering, there is a three-phase flow of water, gas, and rock slag (solid) after rock plug blasting. The model proposed in this article ignores the hydraulic transition process law of the three-phase flow, so it is necessary to study further the three-phase flow characteristics of liquid, solid, and gas phases. This research contributes that the model can be extended to the simulation of the hydraulic characteristics of the underwater rock plug blasting system, such as impact pressure, overflow time, and overflow rate. It can predict the parameters under various working conditions before blasting in advance and has a specific reference role for blasting engineering.

## CONCLUSIONS

Rock plug blasting technology has been applied increasingly in developing and utilizing water and hydropower resources and flood control and disaster reduction projects. The pressure surge of blasting seriously threatens the safety of pipeline structures and other buildings. Predicting the pressure surge and other parameters before rock plug blasting is necessary. This article proposed two mathematical models to simulate the impact pressure of the underwater rock plug blasting system for the pressure flow and the nonpressure flow in the tunnel. The variations of hydraulic characteristics such as pipeline pressure, water flow velocity, flow rate, surge, and overflow after rock plug blasting in the upstream reservoir when the downstream tunnel of a practical project was under pressure and no pressure were simulated. Although the two mathematical models cannot entirely reflect the actual blasting, they can fully reflect the pressure surge and overflow situation of the blasting system after rock block blasting. The relevant conclusions can be applied to predicting transient pressure changes in similar underwater rock plug blasting projects and provide a reference for selecting actual blasting methods in similar blasting projects.

The reliability of the numerical simulation results was illustrated by comparing the measured results after engineering blasting with the numerical simulation results. The main conclusions are as follows:

- (a)
A mathematical model of the hydraulic transition process based on MOC was proposed for the pressure flow in the tunnel. In the mathematical model, the pressure blasting of the tunnel was approximately an instantaneous valve opening. The valve opening time was instantaneous, so a considerable water hammer pressure was generated along the blasting system, with the maximum pressure being 2.55 times the hydrostatic pressure. A large negative pressure was generated in the first half of the blasting system. The values were all under the vaporization pressure. Although there was a ventilation shaft in the system to eliminate part of the water hammer wave, due to the small area of the ventilation shaft not eliminating the water hammer wave, it is not recommended to blast the tunnel in the condition of full flow and pressure for the actual project.

- (b)
When there was no pressure and nonfull flow in the tunnel, a mathematical model of the water–air transition process based on the actual motion process of the water–air two-phase flow was proposed. When the tunnel behind the blasting point was in a state of no pressure, the impact pressure after blasting was discharged to the maximum 27 min after blasting, and the maximum pressure was only 1.17 times the static water pressure of the tunnel. The maximum impact pressure of the tunnel under pressure blasting was 2.17 times that of the nonpressure blasting, and there was no negative pressure along the system. Under this condition, the difference between prototype observation and numerical simulation was 9–14%. As a result, selecting the tunnel as a nonpressure state for actual blasting was recommended.

The mathematical model proposed in this article provides a valuable and reliable guide for predicting the impact pressure of the underwater rock plug system after blasting to ensure the safety of the tunnel, ventilation shaft, branch tunnel, gate shaft, and other buildings in the blasting system. In the mathematical model proposed in this article, only the actual movement of water and air is considered. In engineering, there is a three-phase flow of water, gas, and rock slag (solid) after rock plug blasting. For the model of rock plug blasting with water and gas two-phase flow, only the nonpressure state in the tunnel is considered, and the sudden change of pressure of the nonpressure flow after blasting at different water levels is not considered, which leads to certain limitations in the research in this article. It is necessary to study further the three-phase flow characteristics of the liquid phase, solid phase, and gas phase, as well as the actual flow characteristics and pressure changes of the tunnel after blasting at different water levels.

## ACKNOWLEDGEMENTS

The authors gratefully thank the support of the Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant No. 2021D01B57), Major science and technology projects in Xinjiang Uygur Autonomous Region (Grant No. 2022A02003-4), and Xinjiang Key Laboratory of Hydraulic Engineering Security and Water Disasters Prevention Open Project in 2022 (Grant No. ZDSYS-JS-2022-01).

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

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