Three-Layer Model of Gas–Liquid–Solid Multiphase Transient Flow After Rock Plug Blast

Abstract

Underwater rock plug blasting involves a highly complex, transient gas–liquid–solid multiphase flow that is difficult to simulate accurately with conventional single-phase models. To address this gap, a novel three-phase three-layer mathematical model is presented in this study. This model represents the stratified flow behavior by decomposing the conduit into an upper gas layer, a middle gas–liquid–solid mixture layer, and a lower consolidated bed layer. Governing equations for mass, momentum, and energy conservation are derived and solved using the finite volume method. The model is validated against physical model tests, showing a maximum gate shaft surge deviation of only 0.27%, a Pearson correlation coefficient of 0.965, and a relative RMSE of 4.2%. A sensitivity analysis is performed to quantify the influence of key operational water levels, including the reservoir, gate shaft, and slag pit, on critical transient loads. The results demonstrate that a decrease in the reservoir water level from 106 m to 86 m concurrently reduces both surge height and impact pressure. A smaller reservoir–shaft water level difference (5–15 m) increases the initial cushion pressure and amplifies the surge. In contrast, a larger level difference (20–30 m) suppresses surge but increases impact pressure. Furthermore, an excessively high water level in the slag pit (exceeding 47.8 m) weakens the cushioning effect, thereby lowering the impact pressure. The proposed multiphase model provides an improved approach for predicting hydraulic transients during underwater rock plug blasting.

1. Introduction

Hydropower is a clean, renewable energy source that plays a key role in reducing carbon emissions and supporting sustainable development [1,2,3]. It generates electricity by converting the gravitational and kinetic energy of flowing water [4,5,6,7,8]. In a hydropower system, the water conveyance network delivers water steadily from the intake to the turbine-generator units [9]. Conventional construction methods for underwater intakes often involve extended timelines and high costs. A more efficient alternative is underwater rock plug blasting. In this approach, a tunnel is excavated from downstream toward the reservoir while a rock mass, referred to as the rock plug, is intentionally retained at the inlet [10,11]. Once the tunnel is complete, the plug is blasted open, forming the final intake connection. Yang et al.’s [12] recent work on hydraulic characteristics of rock plug blasting directly motivates the need for improved predictive models, as their study identified critical gaps. Therefore, accurate prediction of the blasting-induced pressure is essential for the structural design of both the intake and outlet.

Underwater rock plug blasting is a critical phase in constructing water conveyance and power generation systems. Zhang et al.’s [13] pressure pulsation analysis of an underwater rock plug blasting system provides a valuable background for this paper. The diagram of the underwater rock plug is shown in Figure 1, where two primary blasting schemes are commonly used. In the first scheme (see Figure 1a), a plug is typically installed within the tunnel prior to blasting. Ding et al. [14] and Sharafat et al. [15] discussed the use of rock plugs as barriers to protect downstream units by preventing reservoir water intrusion. Alternatively, if a gate shaft exists downstream (see Figure 1b), a separate plug may be unnecessary. Lowering the gate before detonation provides the required barrier. Importantly, Cooper [16] specifies that a slag pit is usually constructed downstream of the rock plug, a design consideration that directly impacts the efficiency of subsequent operations. The function of slag pit is to collect blast debris and prevent it from entering the tunnel. The upper section of this pit contains an air cushion chamber. Filling the tunnel with water before blasting forms this cushion, which reduces shock wave pressure. Zhang et al.’s [17] research shows that slag pit water injection treatment can inhibit the high-speed movement of rock slag.

The hydraulic transient induced by rock plug blasting consists of two distinct phases: the blast dynamics phase and the hydraulic transient phase. Jiang et al.’s [18] experimental study on the hydraulic characteristics of a rock plug inlet/outlet states its rationality. The blast dynamics phase is characterized by a detonation shock wave. This wave is accompanied by high-temperature, high-pressure gases and rapidly moving rock slag [19]. During the blast dynamics phase, the primary manifestation is the explosion shock wave generated by the explosion. The peak of the blast shock wave represents a transient process. The shock wave peak is extremely brief. It imposes an instantaneous pressure load on the conduit and plug but has little effect on bulk water motion. Detonation gases expand toward both the reservoir and the slag pit. Gas entering the reservoir escapes through the water surface as visible pockets [20,21]. Wan and Guo et al. [22,23] studied the gas after the barrier blasting. Their research shows that gases reaching the intake cushion chamber cool and depressurize, and push the water column between the pit and the gate shaft into motion. When the air cushion pressure exceeds the reservoir pressure, gas escapes rapidly through the rock plug opening. This process occurs in several discrete bursts [24]. Once the slag pit pressure drops below the reservoir pressure, water rushes into the breach. The resulting head difference then drives the conduit flow, initiating the hydraulic transient [25,26]. This transient becomes evident within seconds to tens of seconds after the blast. The gate shaft water level rises sharply to a peak and then recedes, producing periodic surge oscillations [27,28]. Therefore, it is very important to simulate these processes for the sustainable and safe implementation of rock plug blasting in hydropower projects [29].

1.1. State-of-the-Art Review

Rock plug blasting has a long international history spanning over a century and has been widely used in natural lake development across several European countries [30]. In China, its application began in the 1970s, and more than thirty underwater projects have been successfully completed to date. Early implementations include the Fengman Hydropower Project in 1979. A notable recent example is the 2015 blasting for the Taohekou desilting tunnel and intake at the Liujiaxia Hydropower Station expansion, which marked China’s first large-diameter, high-head rock plug blast conducted through a thick silt–sand layer for combined sediment discharge and power generation [31].

Extensive research has focused on construction schemes, blasting parameter control, and vibration in rock plug blasting engineering. Zhang et al.’s [32] experimental study on pressure characteristics of direct water hammer, Ding et al.’s [33] multi-objective optimization method for transient flow oscillation, and Liu et al.’s [34] sensitivity analysis of hydraulic transient simulations all contribute to the understanding of transient phenomena. Duan et al.’s [35] incorporation of nonlinear turbulent friction into transient pipe flow modeling represents an advancement in single-phase transient analysis. However, their single-phase or two-phase approaches do not capture the full complexity of gas–liquid–solid interactions that occur during rock plug blasting. This is particularly evident in the blasting processes at inlets and outlets of pumped-storage power stations. The underlying mechanisms of the transient flow are not yet fully understood, and the process governing peak pressure generation still requires further investigation. The high complexity of the blasting hydrodynamics leads to significant simplifications in conventional calculation models. Consequently, their results can exhibit considerable deviations [36]. Therefore, more investigation into the hydraulic transient characteristics of rock plug blasting is essential. This process can be studied through physical model tests or numerical analysis. Numerical methods offer distinct advantages, including no site restrictions, shorter modeling and computation cycles, and greater flexibility for modifications [37].

However, these models predominantly assume a single-phase (water) flow, neglecting the intrinsically multiphase (water–gas–solid) nature of the blasting process [25]. Moreover, due to the lack of multiphase flow technology, numerical models that ignore the multiphase flow effect, based instead on single-phase flow, are still far from successful application in simulation accuracy. The comparison reveals that, except for the maximum pressure peak, the simulated and measured results differ significantly in fluctuation frequency [36]. Furthermore, the simulated timing of the peak pressure and its occurrence in the sequence of post-blast peaks do not match the measured data.

The adoption of underwater rock plug blasting is increasing in engineering projects. Consequently, numerical models based on single-phase flow assumptions no longer meet practical requirements, especially for such complex multiphase processes. Simulating underwater rock plug blasting requires a multiphase flow model capable of describing interactions among gas, liquid, and solid phases [37,38]. Currently, however, no software exists that can accurately simulate this entire transient process. Therefore, research on the transient characteristics of rock plug blasting must integrate advanced multiphase flow simulation technology [12].

1.2. Aim of the Work

The work is aimed to develop an accurate hydraulic mathematical model to describe the hydraulic transient analysis of the liquid–gas–solid three-phase flow process after underwater rock plug blasting. The main contributions are follows:

(1)

A three-phase, three-layer model is proposed based on the stratified-flow assumption, enabling the coupled motion and interaction of the water, gas, and solid phases to be simulated simultaneously. The governing equations are derived, and the corresponding discretization scheme and solution procedure are presented.

(2)

The influence of reservoir water level, gate shaft water level, and initial water level of slag pit on blasting transient hydraulic load is systematically analyzed, which provides a theoretical reference for engineering safety design.

The structure of the remainder of the paper is as follows. Section 2 describes the mathematical model of multiphase flow model and its governing equations. Section 3 presents theoretical analysis and sensitivity analysis of key parameters. Section 4 outlines conclusions.

2. Mathematical Model and Numerical Solution Scheme

2.1. Governing Equations of the Three-Phase Distributed Flow Model

In the process of rock plug blasting, there are significant differences in the mass density and specific gravity of the three material forms of liquid, gas, and solid. Therefore, in most cases, they tend to distribute in a layered pattern within the flow channel. This pattern consists of an upper pure gas layer, a middle liquid–gas–solid mixture layer, and a lower consolidated solid layer. The three-phase three-layer distribution model is shown in Figure 2. This stratified distribution pattern of gas, mixture, and bed layers forms the basis of what is termed the three-phase, three-layer model in this study. In the model, the boundary between these three layers is constantly changing in the dynamic process. Thus, the term immobile, describing the immobile solid layer (bed layer), is relative. When water flow velocity increases, its surface undergoes erosion and the solid layer thins. Conversely, deposition occurs and the solid layer thickens when flow velocity decreases. The surface of the immobile solid layer (bed layer) undergoes erosion when the water flow velocity increases, leading to a thinning of the solid layer. Conversely, the layer thickens due to deposition when the flow velocity decreases.

It is worth noting that the proposed three-layer stratified model differs fundamentally from interface-capturing methods such as level set or phase field [39,40,41], which solve the full Eulerian multiphase equations and can handle arbitrarily deforming interfaces. In the present model, the layered topology is assumed a priori, which enables efficient one-dimensional simulation of large-scale hydraulic transients but precludes the resolution of complex interface breakup or mixing. This methodological choice is justified by the physical conditions following rock plug blasting, where the three phases tend to stratify rapidly due to large density contrasts. For scenarios involving strong interfacial deformation, such as the immediate detonation stage or highly turbulent mixing zones, interface-capturing methods would be more appropriate. The two approaches are therefore complementary, and the present model is best suited for system-level transient analysis where computational efficiency and long-duration simulation are required.

To derive the governing equations based on the finite volume method, a fluid control volume is defined within the stratified flow channel. Based on the assumption above, the momentum conservation equations for the gas, liquid, and solid phases within their respective layers are derived as follows. Specially, the three-phase three-layer distribution is shown in Figure 2.

$${\displaystyle \frac{\partial m_g{u}_g}{\partial t}}=-{\displaystyle \frac{\partial m_g{u}_g{u}_g}{\partial x}}-{\alpha }_g{\displaystyle \frac{\partial p}{\partial x}}-F_{g-w}\left(u_g-u_w\right)-F_g{u}_g-{\alpha }_g{\rho }_g\mathit{gsin}\theta$$

(1)

$$\begin{array}{c}{\displaystyle \frac{\partial m_w{u}_w}{\partial t}}=-{\displaystyle \frac{\partial m_w{u}_w{u}_w}{\partial x}}-{\alpha }_w{\displaystyle \frac{\partial p}{\partial x}}-{\alpha }_w\left({\rho }_w-{\rho }_g\right)gsin(\theta ){\displaystyle \frac{\partial h}{\partial x}}+F_{g-w}\left(u_g-u_w\right)-F_{w-s}(u_w-u_s)\\ -F_{w-w}{u}_w+{\alpha }_w{\rho }_{w}gsin\left(\theta \right)\end{array}$$

(2)

$${\displaystyle \frac{\partial m_s{u}_s}{\partial t}}=-{\displaystyle \frac{\partial m_s{u}_s{u}_s}{\partial x}}-{\alpha }_s{\displaystyle \frac{\partial p}{\partial x}}+F_{w-s}(u_w-u_s)+{\alpha }_s{\rho }_{s}gsin(\theta )$$

(3)

$$u_b=0$$

(4)

where m_g, m_w, and m_s represent the mass per unit volume of gas, water, and blast slug within the control body, respectively; u_g, u_w, u_s, and u_b denote the axial velocities of the gas, water, blast slug, and immobile bed layer; a_g, a_w, and a_s are the area fractions occupied by the gas, liquid, and solid phases over the total cross-sectional flow area. ρ_g, ρ_w, and ρ_s are the densities of the gas, liquid, and solid phases, respectively. p is the pressure. F_g, F_g-w, F_w-w, and F_w-s represent the interfacial friction coefficients between the gas layer and wall, gas layer and liquid layer, liquid layer and liquid layer, and liquid layer and solid layer, respectively. g is the gravitational acceleration, and θ is the angle between the flow direction of the fluid control volume and the horizontal direction.

Equation (1) is the motion equation of gas in the gas layer, Equations (2) and (3) are the motion equations of water and slug in the liquid layer, respectively. Equation (4) is the motion equation of the immobile layer.

To account for the relative motion between the dispersed phase (e.g., slag particles) and the carrying fluid, a drift flux model [42] was adopted. The drift velocity was defined as:

$$u_{F,l}=c_{F,l}{u}_l+v_{F,l}$$

(5)

where F represents the phase index; l denotes the layer index; u_l represents the axial velocity of l-layer fluid; C_F,l is the distribution coefficient; and V_F,l refers to the drift velocity.

There are six equations in the mass equations (continuity equations), which were given as follows:

$${\displaystyle \frac{\partial m_{G,g}}{\partial t}}=-{\displaystyle \frac{\partial m_{G,g}{u}_{G,g}}{\partial x}}+S_{G,g}-E_{G,w}+D_{G,w}$$

(6)

$${\displaystyle \frac{\partial m_{G,w}}{\partial t}}=-{\displaystyle \frac{\partial m_{G,w}{u}_{G,w}}{\partial x}}+S_{G,w}+E_{G,w}-D_{G,w}$$

(7)

$${\displaystyle \frac{\partial m_{W,g}}{\partial t}}=-{\displaystyle \frac{\partial m_{W,g}{u}_{W,g}}{\partial x}}+S_{W,g}+E_{W,g}-D_{W,g}$$

(8)

$${\displaystyle \frac{\partial m_{W,w}}{\partial t}}=-{\displaystyle \frac{\partial m_{W,w}{u}_{W,w}}{\partial x}}+S_{W,w}-E_{W,g}+D_{W,g}$$

(9)

$${\displaystyle \frac{\partial m_{S,w}}{\partial t}}=-{\displaystyle \frac{\partial m_{S,w}{u}_{S,w}}{\partial t}}+S_{S,w}+E_{S,w}-D_{S,w}$$

(10)

$${\displaystyle \frac{\partial m_{S,b}}{\partial t}}=-{\displaystyle \frac{\partial m_{S,b}{u}_{S,b}}{\partial x}}+S_{S,b}-E_{S,w}+D_{S,w}$$

(11)

where S represents the source term, while E and D denote the interlayer mass transfer terms for entrainment and deposition, respectively. Subscripts G, W, and S represent gas, water, and solid phases. Subscripts g, w, and b represent gas, water, and solid layers. ${\displaystyle \frac{\partial m_{F,l}}{\partial t}}$ denotes the rate of change of the F phase mass per unit volume in the l layer with time. ${\displaystyle \frac{\partial m_{F,l}{u}_{F,l}}{\partial x}}$ represents the convection term, which represents the net outflow rate due to the flow of F phase (whose velocity is u_F,l) in the l layer. $S_{F,l}$ represents the F mass rate of the F phase directly added to the l layer. F represents the phase index; l denotes the layer index. $E_{G,w}$, $E_{W,g}$, and $E_{S,w}$ represent the rate at which the gas phase in the gas layer enters the water layer, the water phase in the water layer enters the gas layer, and the solid phase in the solid layer enters the water layer, respectively. $D_{G,w}$, $D_{W,g}$, and $D_{S,w}$ represent the rate at which the gas phase in the water layer returns to the gas layer, the water phase in the gas layer returns to the water layer, and the solid phase in the water layer precipitates to the solid layer, respectively.

Equations (6)–(11) represent the change of gas phase mass in the gas layer, gas phase mass in the water layer, water phase mass in the gas layer, water phase mass in the water layer, solid phase mass in the water layer, and solid phase mass in the solid layer, respectively.

The energy equations were given as follows:

$${\displaystyle \frac{\partial m_G{e}_G}{\partial t}}=-{\displaystyle \frac{\partial m_G{u}_G{H}_G}{\partial x}}+S_G{H}_G-q_{g,w}-q_g$$

(12)

$${\displaystyle \frac{\partial m_W{e}_W}{\partial t}}=-{\displaystyle \frac{\partial m_W{u}_W{H}_W}{\partial x}}+S_W{H}_W+q_{g,w}-q_w$$

(13)

$${\displaystyle \frac{\partial m_S{e}_S}{\partial t}}=-{\displaystyle \frac{\partial m_S{u}_S{H}_S}{\partial x}}+S_S{H}_S$$

(14)

$$T_W=T_S$$

(15)

where e denotes the energy per unit mass; H represents the enthalpy; q_g,w is the heat exchange between the gas and liquid phases; q_g and q_w stand for the heat transfer from the gas and liquid phases to the surroundings, respectively; T_W and T_S indicate the temperatures of the liquid and solid phases.

Given that the sum of the volume fractions occupied by the gas, liquid, and solid phases within the control body equals unity, the following relationship holds:

$$\sum {\displaystyle \frac{m_F}{{\rho }_F}}\left(p,T\right)=1$$

(16)

By differentiating the above formula, the following equations are obtained:

$$\sum \left[{\displaystyle \frac{1}{{\rho }_F}}{\displaystyle \frac{\partial m_F}{\partial t}}-{\displaystyle \frac{m_F}{{\rho }_F^2}}\left({\displaystyle \frac{\partial {\rho }_F}{\partial p}}{\displaystyle \frac{\partial p}{\partial t}}+{\displaystyle \frac{\partial {\rho }_F}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}}\right)\right]=0$$

(17)

The density of liquid and solid is assumed to be constant, so only the density of gas in the above equation needs to be obtained by the equation of state.

2.2. Boundary Conditions for Transient Events

The hydraulic transient model incorporates the rock plug, reservoir, and gate shaft, with their boundary conditions defined as follows:

(1)

Rock plug

Before blasting, it is treated as an immobile solid layer. The blast is modeled as a brief source of high-pressure, insoluble gas, with a specified fraction rushing into the tunnel. Post-blast, the section is considered fully penetrated and contains only gas and liquid layers.

(2)

Reservoir

Water level variation is neglected relative to tunnel pressure, so the level is held constant:

$$H=H_{res}$$

(18)

where H_res represents the water level of reservoir.

(3)

Gate shaft

It was treated as a surge tank with inflow only (gas ingress from bottom neglected). Its dynamics are governed by:

$$Q_s=A{u}_w$$

(19)

$$\rho g{p}_s=\mathrm{Z}+R_k{Q}_s\left|Q_s\right|$$

(20)

$${\displaystyle \frac{dz}{dt}}={\displaystyle \frac{Q_s}{A_s}}$$

(21)

where Q_s represents the flow of water flowing into the gate shaft; A represents the cross-sectional area of the tunnel upstream of the gate shaft; u_w denotes the water velocity at the terminal section of the upstream tunnel; p_s represents the bottom pressure of the gate shaft; Z is the water level of gate shaft; R_k is the resistance coefficient of the gate shaft, and A_S indicates the cross-sectional area of the gate shaft itself.

2.3. Numerical Solution Scheme

Under the assumption of ignoring the diffusion effect, the one-dimensional multiphase flow motion model [42] can be expressed as follows:

$${\displaystyle \frac{\partial X}{\partial t}}=-{\displaystyle \frac{\partial F}{\partial x}}+S$$

(22)

where X denotes a conserved quantity; t represents time; F represents its flux; and S stands for the source term.

A series of differential elements (also known as volume elements) are obtained by discretizing the flow channel into a differential grid. This solution method, referred to as the finite volume method, is essentially the same as the finite difference method used in single-phase flow numerical solutions. The difference form of the partial differential equation of the above equation is shown as follows:

$${\displaystyle \frac{X_j^{n+1}-X_j^n}{\Delta t}}=-{\displaystyle \frac{F_{j+1/2}^{n+1}-F_{j-1/2}^{n+1}}{\Delta x}}+S$$

(23)

where the x of the right hand represents the coordinate; the superscript n denotes the n-th time level; the subscript j represents the j-th control body, which is also known as the grid cell.

Time discretization is handled using a semi-implicit scheme. The convective terms are evaluated explicitly using values from the previous time level, while the pressure-related terms are treated implicitly to enhance numerical stability. This approach avoids the severe time step restrictions associated with fully explicit schemes for multiphase flow problems.

The time step size Δt is determined based on the Courant–Friedrichs–Lewy (CFL) condition, which ensures that the numerical domain of dependence encloses the physical domain of dependence. Specifically, the maximum Courant number is defined as:

$$C=\max ({\displaystyle \frac{\left|u_g\right|\Delta t}{\Delta x}},{\displaystyle \frac{\left|u_w\right|\Delta t}{\Delta x}},{\displaystyle \frac{\left|u_s\right|\Delta t}{\Delta x}})$$

(24)

C needs to be maintained below 0.5 throughout all simulations [43]. A constant time step of Δt = 0.001 s is adopted. The CFL number calculated using the maximum flow velocity u_max = 10 m/s is 0.01, which fully satisfies the CFL condition. No additional time step restrictions related to interphase mass or momentum transfer were found necessary under the simulated conditions.

Furthermore, the present model uses a semi-implicit finite volume method with a first-order upwind scheme for convective terms. Stability is also governed by the Courant–Friedrichs–Lewy (CFL) condition.

With Δx = 1.0 m. Δt = 0.001 s, and maximum flow velocity u_max ≈ 10 m/s, the Courant number is C = 0.01, which satisfies the stability requirement.

The spatial discretization follows the finite volume method on a staggered grid. For each conserved variable X, the numerical flux at the cell interface j + 1/2 is computed using a first-order upwind scheme to ensure stability:

$$F_{j+1/2}=\left\{\begin{array}{l}{X}_j{u}_{j+1/2},u_{j+1/2}\ge 0\\ X_{j+1}{u}_{j+1/2},u_{j+1/2}<0\end{array}\right.$$

(25)

where u_j+1/2 is the interface velocity (taken as the upwind value). This upwind treatment introduces numerical diffusion but effectively suppresses spurious oscillations, which is acceptable for engineering-scale predictions where the primary interest lies in the overall transient behavior.

For the gas and water phases in the mixture layer, where sharp gradients may occur, a van Leer flux limiter is optionally applied to reduce numerical diffusion while maintaining monotonicity. No explicit artificial viscosity is added, as the upwind scheme provides sufficient stability for the flow conditions considered.

Taking the gas phase as an example, the gas mass at a specific point and time was substituted for the corresponding term in the general equation. This yielded the discrete form of the gas continuity equation. Specifically, the discrete form is expressed as follows:

$$X_j^{n+1}=m_{G,g,j}^{n+1}$$

(26)

The iterative flux term was expressed as follows:

$$F_{j+1/2}^{n+1}=\left\{\left({\displaystyle \frac{1}{2}}\right)\left({\displaystyle \frac{u_{G,g,j}^{n^{\prime }}}{\left|u_{G,g,j}^{n^{\prime }}\right|}}+1\right)m_{G,g,j}^{n+1}+\left({\displaystyle \frac{1}{2}}\right)\left(1-{\displaystyle \frac{u_{G,g,j}^{n^{\prime }}}{\left|u_{G,g,j}^{n^{\prime }}\right|}}\right)m_{G,g,j+1}^{n+1}\right\}{u}_{G,g,j}^{n^{\prime }}$$

(27)

where $u_{G,g,j}^{n^{\prime }}$ denotes the iteratively updated gas velocity. Therefore, this constitutes a semi-implicit solution procedure. The solution flow chart is shown in Figure 3a.

The overall research framework, from model formulation to validation, is summarized in Figure 3b.

3. Numerical Simulation Analysis

Based on the parameters of a Chinese power station, this section establishes a mathematical transient model for rock plug blasting. A sensitivity analysis is conducted regarding the influence of three key operational parameters.

3.1. Description of the Rock Plug and Inlet System

The rock plug has an outer diameter of 14.6 m and an inner diameter of 10.0 m, with a thickness of 12.5 m. The inclination angle of its central axis relative to the horizontal plane is 43°. A connecting section is installed between the rock plug and the slag pit. The slag pit adopts an air cushioned sloping layout where the lower portion serves as the sloping slag pit, and the upper portion functions as an air cushion chamber during blasting and converts to the water-passing cross-section after blasting. The slag pit section has a length of 78.9 m, with a pit volume of 4600 m³. The maximum volume of the air cushion chamber is 5300 m³. A 110 m long horizontal tunnel connects the pit to the gate shaft. During the rock plug blasting operation, the reservoir water level is 96 m, approximately.

The power plant adopted an open air cushioned underwater rock plug blasting scheme. A temporary plug was installed approximately 25 m downstream of the gate shaft. Prior to the initiation of blasting, the slag pit was filled with water and pressurized with air. The final stabilized levels were 45.8 m in the slag pit and 68.8 m in the gate shaft.

The mechanical characteristics of the three phases are shown in

Table 1. The mechanical characteristics of the three phases.

Phase	Temperature	Density
Gas	19 °C	1.255 kg/m3
Water	19 °C	1000 kg/m3
Solid	\	2800 kg/m3

Notes: “\” indicates that the solid temperature is not an independent variable in the model; it follows the water temperature as per Equation (15).

.

3.2. Numerical Simulation Results

During the rock plug blasting process, the power plant conducted real-time monitoring of water and air pressure using pressure points. Point 1 was installed at the crown of the slag pit to monitor its top pressure, while Point 2 was positioned in the horizontal tunnel section to record the pressure there. The specific locations of these transducers and a schematic diagram of the established computational model are presented in Figure 4.

The following computational assumptions were adopted:

(1)

At standard atmospheric pressure, 0.8 m³ of gas is produced per kilogram of explosive.

(2)

Considering that a small portion of the blasting gases may escape into the reservoir, a conservative assumption is adopted from a perspective of enhancing safety margins. It is assumed that 100% of the gases generated by the explosive enter the gas cavity and are subsequently driven into the slug plug and tunnel under the action of high external pressure.

It should be noted that this model is mainly aimed at the gas expansion-driven liquid–gas–solid three-phase flow and the subsequent hydraulic oscillation stage after the explosion, rather than simulating the detonation of the explosive and the transient process of rock fragmentation.

The calculation applied boundary conditions corresponding to an actual hydropower station rock plug blast. The reservoir water level was 96 m, the slag pit level 45.8 m, and the gate shaft level 68.8 m. The blasting was assumed to begin at 3.0 s and conclude at 3.7 s. Gas and water flow rates at the intake after blasting are shown in Figure 5a. A large gas volume was released during the 3.0–3.7 s blast period. After blasting ended, the reservoir connected to the tunnel, allowing some gas to escape upward. Once the air-cushion pressure dropped below the reservoir head, water rushed into the tunnel. Rising water in the pit then compressed the remaining gas, gradually raising its pressure. This pressure increase slowed the inward water flow. When cushion pressure exceeded the reservoir head, flow reversed and water discharged outward, accompanied by further gas release. The cycle repeated: as cushion pressure again fell below reservoir head, water re-entered. After several oscillations, all cushion gas was expelled by about 20 s. As a result, the tunnel was filled with water completely.

The pressure variation at Point 1 following the rock plug blast was obtained and is depicted in Figure 5b. During the first 20 s, the pressure at Point 1 corresponded to air cushion pressure. After the air cushion dissipated at 20 s, the pressure transitioned to hydrostatic pressure. The pressure rise in the slag pit was mainly caused by gas generation during blasting and subsequent water compression of the cushion. The peak pressure at Point 1 reached 88.38 m at 18 s. Following cushion expulsion, periodic pressure oscillations occurred due to water reciprocating between the reservoir and gate shaft.

The pressure variation at Point 2 following the rock plug blast is shown in Figure 6a. Its pressure exhibited a similar pattern to that at Point 1, with a maximum value of 99.84 m occurring at 15.7 s.

The variation process line of the gate shaft water level is presented in Figure 6b. Following the rock plug blast, reservoir water surged into the tunnel. The air cushion pressure exceeded the pressure at the bottom of the gate shaft, while the water level in the shaft rose continuously. After 14.0 s, the pressure at the shaft bottom exceeded the cushion pressure. Consequently, the water level in the gate shaft began to fall. Once the air cushion dissipated at 20 s, water from the reservoir began to flow into/out of the tunnel periodically. This resulted in corresponding periodic rises/falls in the gate shaft water level. The maximum surge level in the gate shaft was 102.78 m, occurring at 29.8 s.

3.3. Sensitivity Analysis of Key Blast-Related Design Parameters

The key factors influencing rock plug blasting are the reservoir water level, water level difference of reservoir well, and the initial water level in the slag pit. The optimal combination of these three parameters is critically important. Based on the actual conditions of this project, hydraulic transient calculations of the surge of the gate shaft and pressure of Point 1 were conducted. Noteworthily, the pressure of measuring Point 1 represents the pressure of the slag pit. These calculations considered various combinations of the three water levels mentioned above.

3.3.1. Water Level of Reservoir

With a constant head difference between reservoir and gate shaft, a higher reservoir level raises the gate shaft water level accordingly. This also increases the initial air cushion pressure in the slag pit. The elevated initial cushion pressure during and after blasting generates higher shock pressure. This enhanced pressure promotes more effective fracturing of the rock plug. Figure 7a,b illustrates the influence of reservoir level on the maximum gate shaft surge and the peak slug pit pressure. These results assume an identical head difference and a fixed slug pit water level. The maximum surge in the gate shaft rises by about 12.4% with reservoir elevation increases from 86 m to 106 m. Conversely, the peak gas pressure in the slag pit first decreases and then increases. The maximum increase is 12.5%. The lowest peak occurred at 91–96 m, while the 86 m case produces a higher peak than 91 m and 96 m. The underlying mechanisms of these water level effects can be explained by analyzing the surge and pressure time-history curves.

Figure 8 illustrates the influence of reservoir water level on the hydrodynamic responses during rock plug blasting. The results reveal distinct behaviors in the gate shaft surge and slag pit pressure, which are critical for optimizing blasting conditions.

As depicted in Figure 8a, the surge amplitude in the gate shaft increases with rising reservoir level. This phenomenon can be attributed to the higher initial air cushion pressure and greater total gas mass at elevated water levels, which intensify gas expansion and subsequent water flow oscillations. Similar gas–water interaction effects have been reported in studies of surge tanks and air chambers (Guo et al., 2020 [23]). However, previous investigations primarily focused on steady-state or slowly varying flows, whereas the present work captures the transient dynamics under extreme blasting-induced gas release. The finding that the maximum surge level escalates with reservoir level underscores the necessity of conducting blasting at lower water levels to mitigate overtopping risks—a practical insight not systematically addressed in the prior literature.

Overall, the pressure inside the slag pit still increased with the rising reservoir water level, as shown in Figure 8b. When the reservoir water level is 86 m, the pressure at Point 1 displays multiple distinct peaks (at 8 s, 13 s, and 16 s), with the maximum occurring at the latest peak. At 106 m, the pressure peaks earlier (~12 s) and decays slowly, with nearly equal peak values.

Higher reservoir water levels (106 m) correspond to a greater total gas mass. After the plug is penetrated, the combined gas pressure immediately exceeds the inlet hydrostatic pressure, causing a single rapid gas ejection. Each gas release event also tends to be more complete. This releases most of the energy early, producing an early pressure peak (~12 s) followed by gradual decay. The gate shaft surge rises quickly due to the sudden water displacement. This behavior aligns with the theory of pressurized gas discharge in confined spaces (Wang et al., 2021 [20]).

Conversely, at lower water levels (e.g., 86 m), the initial gas release mass is insufficient to overcome the inlet hydrostatic pressure; as the pressure differential builds, subsequent releases involve progressively larger gas masses, leading to successively higher pressure peaks (at 8 s, 13 s, and 16 s), followed by rapid decay. This is the phenomenon of intermittent release. The surge in the gate shaft also builds gradually due to the repeated gas–water exchanges.

At intermediate levels (e.g., 91 m and 96 m), the peak pressure is less than 86 m and 106 m because the gas release mode is converted between the two modes of intermittent release and complete release. This conversion results in a smaller peak pressure.

According to the analysis above, lower levels do not necessarily reduce peak pressure in the slag pit. The 86 m case produced a higher peak than 91–96 m due to staged releases, indicating that very low water levels may still generate significant impact pressures. This challenges the intuitive assumption that lower water levels always imply lower risks. In addition, an intermediate reservoir level (91–96 m) offers a balanced performance: moderate surge height, reduced peak pressure, and controlled gas release dynamics. This range minimizes both surge-related risks and excessive impact pressures.

3.3.2. The Water Level Difference of Reservoir Well

Water level difference of reservoir well refers to the difference between the water levels of the reservoir and the gate shaft. This parameter directly affects the interaction strength between gas and water after rock plug blasting. From the perspective of the pressure balance mechanism, the smaller the water level difference, the smaller the back pressure of the bottom of the gate well on the air cushion, and the easier it is for gas to release quickly after the rock plug is penetrated. However, as the water level difference increases, the gas needs to overcome the higher water column pressure to discharge, thus forming intermittent release.

Pre-charging the gate shaft with water before blasting provides a cushioning effect. This reduces slag velocity, shock force, and post-blast flow speed in the tunnel. It also limits the maximum surge height in the shaft. Adequate filling helps confine the blast slag scatter, leading to a more optimized muck pile profile. A higher pre-charge level creates greater bottom pressure in the shaft. This pressure helps expel both the original cushion gas and blast gases through the rock plug opening, shifting shock wave generation to the reservoir surface and protecting downstream structures. However, the pre-charge level should not be excessive. An overly high level risks exceeding the surge height limit and increases water and air filling costs. Under fixed reservoir and slag pit levels, the influence of the water level difference of the reservoir well on hydraulic transients is shown in Figure 9a,b. As the water level difference increases from 10 m to 30 m, the highest surge of the gate shaft, the pressure of Point 1 decreases by 7.7% and increases by 63.2%, respectively. The following explanation combines the shaft surge level and the pit pressure variation process.

The influence of the water level difference of the reservoir well on the maximum surge level in the gate shaft is relatively complex. This complexity stems from the differing variation processes of the surge level under various water level differentials, as illustrated in Figure 10a.

When the head difference between the reservoir and gate shaft is small (5–15 m), the initial air cushion pressure is high because of limited water column compression. This results in a larger total gas mass inside the cushion. After the rock plug is penetrated, cushion gas mixes with blast-generated gas. The resulting high-pressure mixture exceeds the inlet hydrostatic pressure, causing a single rapid gas ejection toward the reservoir. The expelled gas carries tunnel water with it. This leads to an initial drop in the surge of the gate shaft. When the high-pressure gas is vented, the water in the reservoir flows into the tunnel due to the decrease in pressure. The surge of gate shaft rises. In this scenario, a larger water level difference between the reservoir and the gate shaft corresponds to a lower initial air cushion pressure and a smaller initial total gas mass. Consequently, the induced water flow fluctuations are reduced, leading to a decrease in the flow rate entering and leaving the tunnel and a reduction in the surge amplitude within the gate shaft. Accordingly, the maximum surge water level in the gate shaft also declines.

When the water level difference of the reservoir well is large (20–30 m), the initial pressure of the air cushion is low, and the mixed gas is mixed after the rock plug is blasted. The pressure still does not reach the hydrostatic pressure of the rock plug inlet, so the water in the reservoir will flow into the tunnel and compress the air cushion. This leads to the surge of gate shaft rising in the initial stage. At this stage, the larger the water level difference of the reservoir well, the lower the initial pressure of the air cushion. After the blasting of the rock plug, the greater the pressure difference, and the more water flows into the tunnel, leading to an increase in the amplitude of the surge in the gate shaft and the rise of the highest surge.

Similarly, the pressure variation in the slag pit also changes with the head difference, as illustrated in Figure 10b. For a small head difference (5–15 m), the mixed gases discharge rapidly in a single event upon rock plug penetration. This causes a sharp pressure drop, followed by quick stabilization once water inflow begins, without repeated cushion compression. A large difference (20–30 m) leads to intermittent gas release. It produces intense, slowly decaying pressure oscillations that persist even after the air cushion dissipated. In both cases, the maximum pressure in the slag pit increases, with a larger head difference. This occurs because the lower initial cushion pressure and gas mass resulted in successively larger gas releases during the multi-stage discharge process, leading to higher pressure peaks.

At the same time, due to the reverse movement of water flow after each gas release, the formation of surge in gate shaft also presents multi-peak characteristics. Therefore, the water level difference not only affects the initial surge amplitude, but also determines the oscillation mode and energy dissipation path of the whole transient process.

3.3.3. Water Level of Slag Pit

The water level of the slag pit determines the initial volume and pressure of the air cushion. From the perspective of the air cushion buffering mechanism, the lower water level means a larger air cushion volume, which can more effectively absorb the explosion impact energy and slow down the impact of water flow on the downstream structure. However, too much air cushion volume also means that the total mass of gas increases, and the gas release process is more complicated, which may lead to longer oscillation. Under fixed reservoir level and fixed head difference conditions, the influence of the slag pit water level on hydraulic transient characteristics in the range of 43.8 m to 49.8 m was analyzed and investigated. The results are shown in Figure 11a,b. It is worth noting that when the water level of the slag pit rises gradually, the surge height of the gate well and the pressure of Point 1 both show a trend reversal at 47.8 m. This indicates that when the water level of the slag pit rises, 47.8 m is the critical point of the buffering effect from strong to weak.

When the slag pit water level was below 47.8 m (41.8–47.8 m), the maximum surge of the gate shaft decreased by 6.1% as the pit water level rose. This is because a higher water level meant a smaller air cushion volume and lower initial pressure, which reduced the post-blasting reciprocating flow at the inlet and thus lowered the gate shaft surge. Concurrently, the maximum pressure of the slag pit increased by 17.2%. This is because the smaller gas mass in the cushion was more easily compressed by the incoming reservoir flow.

However, when the water level exceeded 47.8 m, the initial air cushion pressure became too low to provide effective cushioning. Even combined with the blast gases, the total pressure remained below the hydrostatic pressure at the inlet. Consequently, a large volume of reservoir water surged into the tunnel immediately after rock plug penetration. This caused the static water pressure to drop by 30.8%. The gases were then expelled from the tunnel in one rapid discharge. Due to the absence of water–gas oscillation exchange before the dissipation of the air cushion, the surge of the gate shaft increased significantly, by 10.9%. With minimal compression of the air cushion, the resulting pressure increase was limited. The peak pressure only reached the inlet hydrostatic level, indicating a loss of cushioning effect. The distinct variation processes under this condition are illustrated in Figure 12a,b.

4. Model Validation

In this section, the reliability of the proposed numerical model is assessed from two perspectives. First, a grid convergence study based on the CFL condition is conducted to verify the numerical consistency of the discretization scheme. Second, the simulation results are compared with physical model test data to validate the physical accuracy of the three-phase three-layer model.

4.1. Grid Convergence Based on CFL Condition

The proposed numerical scheme uses a finite volume method with semi-implicit time integration. The convergence and stability are governed by the Courant–Friedrichs–Lewy (CFL) condition, which requires that the Courant number $C=\left|u\right|\Delta t/\Delta x$ does not exceed a stability limit. For the first-order upwind scheme, a conservative limit of C_max = 0.5 is adopted.

With a constant time step Δt = 0.001 s and a uniform grid spacing Δx = 1.0 m, the resulting Courant number, based on the characteristic wave speed of the flow, is 0.01, approximately. Hence the CFL condition is strictly satisfied (below 0.5), ensuring numerical stability and convergence. The adequacy of this numerical setup is further supported by the good agreement between the simulated results and experimental data (Section 4.2).

4.2. Comparison with Experimental Data

In this section, the correctness of the numerical calculation model based on the prototype parameters of the power station is verified by comparing the measured values with the calculated values. The explosive parameters used in the calculation are listed in

Table 2. Explosive parameters.

Parameters	Unit	Value
Total explosives	kg	1122.51
Blasting volume	m3	563.18
Unit explosive consumption	kg/m3	1.99
Maximum single blow blasting charge	kg	199.66

. The blasting was assumed to commence at 10.0 s, with a total initiation duration of 0.7 s. The post-initiation complete combustion ratio of the explosive was assumed to be 0.94. Furthermore, it was assumed that the gas products generated per kilogram of explosive occupied a volume of 0.8 m³ under standard conditions. After the explosion, 100% of these gas products were assumed to be propelled downward into the tunnel.

Based on the stated assumptions, the rock plug blasting transient was simulated for a reservoir water level of 130 m, gate shaft water level of 120 m, and slag pit water level of 80 m.

Figure 13 presents the water level variations in the gate shaft and the trash rack shaft, and the measured and calculated values of the gate shaft are compared. The proximity of gate shaft and trash rack shaft ensured rapid pressure wave transmission and nearly synchronous level changes. The numerical results demonstrate good agreement with the model test data. The results demonstrate that the numerical calculation effectively predicted both the maximum surge height and the surge fluctuation period for the gate shaft.

Furthermore, the root mean square error (RMSE) between the calculated and test value of water levels of the gate shaft is analyzed. First, the calculated data were screened based on the time points of the test values to ensure alignment of the time nodes between the two datasets. The results show that the Pearson correlation coefficient is 0.965, indicating a strong positive correlation between the two datasets. The model accurately captures the water level’s variation trend.

However, discrepancies existed between the calculated and the test value regarding the blasting and gas escape processes. The RMSE is 5.29, which corresponds to 4.2% of the average water level. These discrepancies are primarily attributed to the simplification of the three-dimensional flow. The 1D framework has inherent limitations in capturing near-field three-dimensional flow effects.

To further validate the accuracy of the numerical model, the maximum surge water levels under a water level difference of 10 m between reservoir and gate shaft was calculated for a reservoir water level of 140 m and a slag pit water level of 80 m, as presented in

Table 3. The maximum surge water level of gate shaft.

Working Condition	Calculated Value (m)	Model Test Value (m)	Relative Deviation (%)
140–130–80	146.4	146.0	0.27

Notes: Relative deviation in the table is presented in two forms. The first value is calculated as M_E − M_S, and the second value is determined using (M_E − M_S)/M_E. Here, M_E represents the calculated value, M_S represents the model test value.

. The results indicate that the numerical simulation reliably predicted the recovery of the gate shaft water level following the rock plug blasting.

In summary, the numerical simulation results based on the three-phase three-layer model are in good agreement with the model test data in peak data and periodicity. The numerical calculations can effectively represent the movement patterns of different phases during the rock plug blasting process. When physical blasting tests are not feasible, numerical simulation can be employed for preliminary analysis and optimization of key parameters.

5. Conclusions

This study developed a novel three-phase three-layer mathematical model for hydraulic transients during rock plug blasting, based on one-dimensional multiphase flow theory. By incorporating interfacial mass, momentum, and energy exchange mechanisms, the model provides a more realistic description of the coupled processes involving explosive gas expansion, water flow, and rock slag transport. The key factors influencing the rock plug blasting process are analyzed. The main conclusions are as follows:

(1)

A one-dimensional multiphase flow model for rock plug blasting is developed based on three-phase, three-layer theory. The model accounts for interphase interactions as well as mass and energy fluctuations induced by blasting, addressing the deficiencies of conventional 1D models that oversimplify phase interactions and ignore stratified flow characteristics, and the flow domain is discretized into differential elements. A semi-implicit numerical scheme is formulated to solve the resulting governing equations.

(2)

The parametric analysis indicates that lower reservoir water levels reduce both surge height and impact pressure. A small reservoir–gate shaft water level difference increases initial cushioning pressure and amplifies surge, whereas a larger difference promotes staged air–water release, increasing impact pressure while suppressing surge height. An excessively high slag pit water level weakens the cushioning effect and lowers impact pressure, demonstrating that the proposed model enables quantitative optimization of operational parameters for hydraulic load control and offers clear replicability in similar hydropower projects.

However, the 1D framework cannot capture near-field three-dimensional effects or local phase interactions. Especially at the initial moment of explosion and in the strong mixing region, the gas–liquid–solid interface may be highly deformed or even broken, and the strict stratification assumption is a simplification. Future work should integrate 3D multiphase CFD-DEM simulations and systematic physical experiments to refine interfacial relations, thereby extending the model’s predictive capability for full-scale blasting scenarios.