Figure
1.
Comprehensive test platform of HVPD rock fragmentation.
| Citation: |
Shijie HUANG, Yi LIU, Yong ZHAO, Youlai XU, Fuchang LIN, Hua LI, Qin ZHANG, Liuxia LI. Stress wave analysis of high-voltage pulse discharge rock fragmentation based on plasma channel impedance model[J]. Plasma Science and Technology, 2023, 25(6): 065502. DOI: 10.1088/2058-6272/acb136
|
High-voltage pulse discharge (HVPD) rock fragmentation controls a plasma channel forming inside the rock by adjusting the electrical parameters, electrode type, etc. In this work, an HVPD rock fragmentation test platform was built and the test waveforms were measured. Considering the effects of temperature, channel expansion and electromagnetic radiation, the impedance model of the plasma channel in the rock was established. The parameters and initial values of the model were determined by an iterative computational process. The model calculation results can reasonably characterize the development of the plasma channel in the rock and estimate the shock wave characteristics. Based on the plasma channel impedance model, the temporal and spatial distribution characteristics of the radial stress and tangential stress in the rock were calculated, and the rock fragmentation effect of the HVPD was analyzed.
Scenarios such as oil and gas, geothermal resource development in deep or complex strata, and pile foundation construction in infrastructure require fragmentation of hard rock. Mechanical rotary drilling suffers from slow speed, low efficiency and wear of various components. It is urgent to explore new and efficient rock fragmentation technologies [1, 2]. At present, new rock fragmentation technologies such as high-voltage pulse discharge (HVPD), jets, lasers, and microwave rock fragmentation have received extensive attention, all these methods being safe, efficient and pollution free [3–6]. According to the different forms of rock fragmentation, HVPD rock fragmentation can be divided into hydroelectric rock fragmentation and electric pulse rock fragmentation. Compared with hydroelectric rock fragmentation, electric pulse rock fragmentation has high efficiency and low energy consumption. Therefore, this work is mainly aimed at electric pulse rock fragmentation. There are two necessary conditions for HVPD rock fragmentation: first, the electrodes and the rock are both in a liquid environment and the electrodes are placed on the rock surface. By adjusting the parameters of the high voltage pulse, electrode type, etc, the plasma channel is formed inside the rock. Second, with the rapid injection of external electrical energy, the temperature and pressure inside the plasma channel rise sharply, and the channel expands rapidly, radiating shock waves outward, and generating strong mechanical stress in a very short time, which stimulates rock cracks and fractures [7, 8]. Since the plasma channel is formed inside the rock during the HVPD rock fragmentation process, it is difficult to measure the shock waves and stress waves. By establishing an impedance model of the plasma channel in the rock to predict the shock wave and stress wave characteristics, the connection between the circuit electrical parameters and the mechanical parameters can be established, which has important research value in power supply design and rock fragmentation effect optimization.
The plasma channel is the 'tool' for HVPD rock fragmentation. In order to analyze the wave propagation process in rock fragmentation, it is necessary to analyze the characteristics of the plasma channel. The energy deposited in the plasma channel is mainly converted into mechanical energy, internal energy, and a part of the energy that is lost in the form of radiation. Since the channel is formed inside the rock, it is difficult to directly obtain the shock wave and channel expansion characteristics in the test. The shock wave intensity and the transient characteristics of the discharge channel can be predicted by establishing an accurate time-varying impedance model of the plasma channel [9]. In past studies, the time-varying plasma channel resistance in solids was mostly estimated by empirical formulas. In the derivation process, only the influence of the internal energy of the channel was considered, and the rest of the energy was regarded as a fixed value, thus introducing errors [10, 11]. During the discharge process, due to the rapid injection of energy, the plasma channel expands rapidly under the Joule heating effect, the temperature rises rapidly, and the channel radius and conductivity change with time. By establishing a more accurate plasma channel time-varying impedance model, HVPD shock waves and channel transient characteristics can be calculated. The shock waves generated by the expansion of the plasma channel load on the surrounding rock stimulate cracks and cause fragmentation.
In the stage of rapid energy deposition, due to the small radius of the plasma channel in the rock, the energy density inside the plasma channel is extremely large, and the rock fragmentation process is similar to the explosion process [12]. During the wave propagation process, due to the reduction of energy density on the wave front per unit area and the energy loss, the shock waves gradually decay into stress waves with smaller amplitude and steepness [13]. Based on the elastic–plastic model of rock fragmentation, Burkin et al regarded the deformation of the material around the plasma channel as the material motion, and established a physical and mathematical model of the HVPD rock fragmentation by combining the circuit electrical equation and the energy balance equation. It was found that the rock was mainly cracked and broken by the shear stress when high energy was continuously injected, while under low energy, the stress wave and the reflected wave worked together to form a tensile fracture and form a fragmentation cavity [14, 15]. Cho et al compared the observation and simulation results of fractures of rock sample fragmentation by HVPD, and found that the rock fragmentation process can be simulated by a quasi-static multi-fracture model [16]. Kuznetsova et al carried out an electric explosion test of copper wire in a polyethylene concrete medium, and found that the shock wave on the borehole wall and the stress wave in the medium were determined by the energy deposition rate of the plasma channel. Higher energy deposition rates produced compression stress with larger amplitudes, while lower energy deposition rates produced radial and tangential tensile stresses with higher amplitudes [17]. According to the von Mises criterion, when rock is subjected to radial or tangential stress greater than its own compressive or tensile strength, cracks and fractures occur [13].
The rock fragmentation mechanism of HVPD involves electricity and solid mechanics. When comprehensive analysis of the discharge process and rock fragmentation process is carried out, the electrical equation and energy conversion equation of the discharge circuit need to be solved, and then the shock wave generated by the discharge can be obtained. The connection between electricity and force can be established by applying a shock wave as a load to the rock. When solving solid mechanics problems, it is necessary to establish mechanical equilibrium equations, geometric equations, and constitutive equations linking stress and deformation. When using the analytical method to solve the wave process of rock fragmentation under impact load, the elastic failure criterion can be used to simplify the solving process. The elastic failure criterion considers that the rock is homogeneous and the failure of the rock is caused by the stress in it exceeding the stress limit, and rock is elastic and brittle before fragmentation [13].
In order to deeply study the rock fragmentation mechanism of HVPD and analyze the stress wave effect during fragmentation process, a comprehensive test of HVPD rock fragmentation was carried out, and the electrical parameters such as plasma channel voltage and current were measured. Based on the time-varying plasma channel impedance model in the rock, the shock wave intensity of the discharge under specific test conditions was predicted. By considering the rock as a homogeneous elastic and isotropic medium, and the shock wave calculated by the model as the load, the variation law of the stress wave in the rock medium with time and space was obtained and analyzed. Based on the von Mises criterion, the radius of the crushing zone and fracture zone of rock under single HVPD is estimated, which can provide theoretical guidance for the practical application of HVPD rock fragmentation.
The comprehensive test platform of the HVPD rock fragmentation system is shown in figure 1, and is mainly composed of the Marx generator unit, the pulse discharge unit and the measurement unit. The Marx generator has five stages, the capacitance of each stage CS is 0.2 μF, and the rated charging voltage is 100 kV. The charging isolation resistance Ri is 5 kΩ, the wave front resistance Rf is 0.6 Ω per stage, and the wave terminal resistance Rt is 170 Ω per stage. The Marx generator has five spark gaps with spherical electrodes; the first spark gap is broken down by the trigger signal Trig1, and the rest of the spark gaps are operated under a self-breakdown regime. The discharge electrodes are in the form of plate–plate electrodes, the electrode materials are tungsten–copper alloys (tungsten content is 90%). The diameter and thickness of the electrodes are 30 mm and 15 mm, respectively, and the electrode distance is adjustable. The electrodes are placed on the surface of the rock to be broken. The electrodes and rock are immersed in a tank filled with lightly filtered tap water with a conductivity of about 34.7 mS m−1. The size of the water tank is length × width × height = 500 mm × 500 mm ×500 mm, the material is stainless steel, and there are round Plexiglas observation windows with a diameter of 190 mm on both sides. For a typical liquid–solid composite medium, the path of the discharge channel depends on the waveform of the applied high-voltage pulse (mainly the rise time), the electrode type and the liquid–solid insulation properties, etc. It is generally believed that when the rise time of the voltage applied to the electrode is less than 500 ns, and the electric field strength is greater than the rock breakdown field strength, the plasma channel is formed inside the rock [18, 19].
A high-voltage divider (model: North Star VD-200) and a current monitor (model: Pearson 1330) were used to measure the voltage U(t) across the electrode gap and the current I(t) of the discharge channel, and the waveforms were stored in an oscilloscope (model: Tektronix MDO3054). A high-speed camera (model: Phantom V1612) equipped with an 18–200 mm focal length lens was used to capture the images of the channel morphological development process of the HVPD. The sampling speed of the camera was set to 3.93 μs/frame, and the exposure time was 0.76 μs. The high-speed camera was interconnected with the oscilloscope through the photoelectric isolation device, and was triggered synchronously by the signal Trig2 given by the oscilloscope, with a trigger delay of about 200 ns. Considering that the plasma luminescence was strong, it was not possible to distinguish the plasma channel shape, so a 400 times neutral gray mirror was used for light reduction. Considering that the HVPD might cause the ground potential to rise, the measurement equipment was independently powered by an uninterruptible power supply.
The charging voltage of each stage capacitor was set to 70 kV, and the electrode gap distance was 3 cm. Three HPVD rock fragmentation tests were carried out. The voltage and current waveforms of rock breakdown obtained in the third test are shown in figure 2. The peak value of the voltage across the discharge gap was about 253.12 kV, the rise time from 10% to 90% was about 262 ns, and the discharge had no obvious pre-breakdown delay. After the breakdown, the voltage across the gap dropped and the current rose rapidly. The current oscillated in a second-order underdamped damping. The peak value and oscillation period of the channel current were about 11.05 kA and 5.63 μs, respectively.
Figure 3(a) shows the placement of the plate electrodes. Corresponding to the discharge conditions in figure 2, the plasma channel image of the HVPD rock fragmentation captured by a high-speed camera is shown in figure 4. When rock breakdown occurred, the streamer started at the edge of the high-voltage electrode, developed in the water for a certain distance, and connected to the low-voltage electrode through the interior of the rock. A strong electric field distortion point was formed at the junction of electrode, rock formation and water, which was favorable for streamer initiation. After the plasma channel formed, the strong luminescence of the plasma near the electrodes could be photographed. With the discharge process, the energy injected into the plasma channel decreased, the channel luminescence gradually darkened, and the plasma channel gradually decayed until it disappeared.
From the image, it could be judged that the plasma channel had passed through the inside of the rock, and the plasma discharge image inside the rock could not be recorded. Since the length lch of the plasma channel could not be directly obtained by high-speed images, this work used gypsum powder to reconstruct the broken area to obtain the shape of the broken area. As shown in figure 3(b), after three HVPD, an irregular crushing pit was formed in the middle and bottom of the electrodes, and the crushed particles were of different sizes. Using gypsum powder to reconstruct the crushing pit in figure 3(b), the shape of the obtained crushing pit is shown in figure 3(c). The maximum length of the crushing pit along the direction of the two pairs of electrodes was 54.81 mm, and the crushing depth was 14.67 mm. The path of the plasma channel is shown in figure 3(d); assuming that the plasma channel passed through the center of the crushing pit, the length of the plasma channel in this test was lch ≈ 46 mm.
The discharge equivalent circuit of the HVPD rock fragmentation system is shown in figure 5, where C is the equivalent discharge capacitor, C = 40 nF and Uc is the discharge voltage. R0 is the inherent resistance of the outer loop, and L0 is the inherent inductance of the outer loop. Through the short-circuit experiment, R0 = 4.95 Ω and L0 = 19.03 μH. S is the switch, which controls the energy of the discharge capacitor to be applied to the electrodes. Rpl(t) and Lpl(t) are the time-varying resistance and time-varying inductance of the plasma channel in the rock, respectively. The Kirchhoff equation of the equivalent discharge circuit is:
| (R0+Rpl(t))i(t)+d[Lpl(t)i(t)]dt+L0di(t)dt+∫τ0i(τ)dτC=Uc | (1) |
where i(t) is the current across the electrode gap. The value of the resistance component of the plasma channel in the solid is about a few ohms [11, 20], while the inductive component of the spark plasma impedance accounts for less than 20% and is usually ignored [21]. Therefore, compared with the external loop inductance L0 and the resistance component Rpl(t) of the plasma channel, the proportion of the channel inductance Lpl(t) is small and can be ignored [22], so equation (1) can be simplified as
| (R0+Rpl(t))i(t)+L0di(t)dt+∫τ0i(τ)dτC=Uc | (2) |
Rpl(t) has time-varying characteristics in the process of rock breakdown, from the initial hundreds of ohms to the minimum value at the peak moment of channel current. The experimental value of Rpl(t) can be obtained by processing the measured voltage and current waveforms. In order to better describe the dynamic behavior of the plasma channel and analyze its transient characteristics, it is necessary to establish a more accurate time-varying impedance model of the plasma channel. Zinoviev et al assumed that in the energy conversion process, the energy in the plasma channel did not change with time except for the internal energy, and gave an empirical formula for the time-varying resistance of the plasma channel [11]:
| Rpl(t)=Alch√∫t0i2dt | (3) |
where A is a constant, which is related to the material of the solid medium. Solving equation (2) and loop electrical equation (3) simultaneously can obtain the current and resistance of the discharge channel. However, equation (3) only considered the first oscillation period of the discharge current in the derivation process, and was an empirical formula, ignoring the effects of parameters such as temperature, electrical conductivity, and electromagnetic radiation [11]. Using equation (3) to solve the development process and transient characteristics of the electric pulse discharge plasma channel brings certain errors. Considering the basic resistance formula and approximating the plasma channel as a cylinder, the time-varying resistance Rpl(t) can be expressed as
| Rpl(t)=lchA(t)σ(t) | (4) |
where lch is the length of the plasma channel, A(t) = π·(r(t))2 is the cross-sectional area of the plasma channel, r(t) is the radius of the plasma channel, and σ(t) is the average conductivity of the channel plasma cross-section. The channel conductivity changes with temperature, and in the presence of a magnetic field, the conductivity of a fully ionized plasma is proportional to (T(t))3/2. T(t) is the temperature of the plasma channel. An exponential factor is added and the time-varying conductivity expression is [23, 24]
| σ(t)=ξ(T(t))3/2e−5000/T(t) | (5) |
where ξ is a constant, and ξ can be adjusted to make it more in line with the properties of plasma channels in rocks. Analogous to the liquid discharge theory, the energy deposited into the plasma channel can be divided into three parts: the mechanical work done by the expansion of the plasma channel to the surrounding solid medium, the internal energy of the plasma channel, and the electromagnetic radiation generated as the temperature of the channel rises. The energy balance equation of HVPD in the solid can be expressed as [25]
| Win(t)+Wm(t)+Wrad(t)=Epl(t) | (6) |
where Win(t) is the internal energy of the plasma channel, Wm(t) is the mechanical energy of the plasma channel expansion, Wrad(t) is the electromagnetic radiation, and Epl(t) is the electric energy deposited by the plasma channel. The deposition energy of the plasma channel is mainly determined by the impedance characteristics of the channel and the outer loop, and Epl(t), Win(t) and Wm(t) are expressed as [25, 26]
| Epl(t)=∫(i(t))2Rpl(t)dt | (7) |
| Win (t)=WV(t)V(t) | (8) |
| Wm(t)=∫Pch(t)dV(t) | (9) |
where V(t) is the volume of the curved cylindrical plasma channel, V(t) = π·(r(t))2 lch; Pch(t) is the shock wave generated by the expansion of the plasma channel; and WV(t) is the internal energy per unit volume in the channel, and is proportional to the plasma channel pressure Pch(t) [27],
| WV(t)=Pch(t)γ−1 | (10) |
where γ is the adiabatic coefficient, and the electromagnetic radiation Wrad(t) is [24]
| Wrad(t)=∫(1−f)σS(T(t))4S(t)dt | (11) |
where σS is the Stefan–Boltzmann constant, σS = 5.67 × 10−8 W m−2 K−4, and S(t) is the surface area of the plasma channel, S(t) = 2π·r(t)·lch. Since the plasma channel is formed inside the rock, part of the electromagnetic radiation does not actually escape from the plasma channel. As denoted by factor (1 − f), there is a thin layer of fluid at the boundary of the plasma channel. f represents the fraction of electromagnetic radiation absorbed by the thin layer around the plasma channel. The thin layer of fluid is then evaporated and reincorporated into the plasma channel during channel development. Therefore, the electromagnetic radiation is returned into the plasma channel along with the recently absorbed thin layer of fluid [24]. Plasma in rocks is described by the ideal gas law:
| Pch(t)=n(t)kT(t) | (12) |
where k is the Boltzmann constant k = 1.380649 × 1023 J K−1, and n(t) is the number of particles in the plasma channel. Since the air gap may play a key role in rock breakdown [28, 29], it is feasible to apply equation (12) in the rock impedance model. Substituting equations (7)–(11) into equation (6), the energy balance equation of the HVPD in the rock is obtained by differentiation as
| (I(t))2σ(t)2π2(r(t))3=(Pch(t)+WV(t))dr(t)dt+r(t)2dWV(t)dt+(1−f)σS(T(t))4 | (13) |
According to the conservation of mass and momentum [25],
| dn(t)dt=2r(t)(fσs(T(t))4εsublimation −n(t)dr(t)dt) | (14) |
| dr(t)dt=(√7α1/143√ρ0)((Pch(t)+β)3/7−β3/7) | (15) |
where εsublimation is the energy required to sublime the surrounding solid medium into plasma, ρ0 is the density of the solid medium in the undisturbed region, and α and β are coefficients: α = 300.1 MPa, β = 300 MPa [20, 25]. Solving equations (2), (4), (5), (10), (12)–(15) simultaneously, the time-varying impedance model of the plasma channel can be established, and the development process and the transient characteristics of the plasma channel can be solved. The model considers the effects of temperature, channel expansion and electromagnetic radiation, assumes that the channel radius r(t) expands uniformly in the radial direction, and uses the time-varying conductivity σ(t) that is affected by the channel temperature.
| {(R0+Rpl(t))i(t)+L0di(t)dt+∫τ0i(τ)dτC=UcWin(t)+Wm(t)+Wrad(t)=Epl(t)dn(t)dt=2r(t)[fσs(T(t))4εsublimation −n(t)dr(t)dt]dr(t)dt=(√7α1/143√ρ0)((Pch(t)+β)3/7−β3/7) | (16) |
Based on the analysis above, the established time-varying impedance model of the plasma channel of the HVPD rock fragmentation is shown in equation (16). By combining the electrical equation and the energy balance equation of the equivalent circuit, the development process and transient characteristics of the plasma channel can be solved.
After the discharge electrodes break down, the plasma channel is formed inside the rock and bridges the two electrodes, as shown in figure 6(a). The energy stored in the capacitor is rapidly injected into the plasma channel through the electrodes. The energy density of the plasma channel in the rock is about 25–30 kJ cm−3, and the channel temperature and pressure reach 104 K and 109 Pa respectively in a very short time [12, 16, 30]. The plasma channel expands rapidly, generating shock and stress waves inside the rock, crushing the surrounding rock medium and inducing cracks.
It is assumed that after the plasma channel is formed, the cylindrical cavity in the rock fills with water immediately. As shown in figure 6(b), when the shock wave reaches the rock boundary, transmission and reflection of the shock wave occur due to the difference in medium density. The relationship between the transmitted shock wave PT(t) and the incident shock wave Pch(t) can be expressed as [31]
| PT(t)=2ρcpρcp+ρ0cwPch(t) | (17) |
where ρ and cp represent the density of rock and the velocity of the longitudinal wave, ρ0 represents the density of water, and cw = 1500 m s−1 represents the velocity of the shock wave in water.
The shock wave generated by the expansion of the plasma channel loads on the rock medium around the channel, and the shock wave and stress wave change with time and space during the propagation inside the rock. Under the loading of radial stress and tangential stress, the stress state of the rock changes [13, 32]. Taking a cross section of the plasma channel perpendicular to the channel axis, since the expansion of the plasma channel in the rock is more difficult, the volume of the plasma channel is small relative to the rock and can be approximated as a curved cylinder [11]. Based on the elastic failure criterion, the fragmentation process of rock under the loading of channel expansion is simplified as: there is a cylindrical cavity in the homogeneous elastic and isotropic rock, and the surface of a cylindrical cavity is subjected to the shock wave Pch(t) uniformly distributed in the radial direction. Under the loading of shock waves and stress waves, the rock is broken and cracked. The formation of the plasma channel and the process of channel expansion and crushing of surrounding rocks to form the cylindrical cavity are not considered. Figure 6(b) shows the different state regions of the rock after HVPD. The red region represents the plasma channel, and r0 is the channel radius. The reaming of the cylindrical cavity under the action of the shock wave is very small [33]; therefore, r0 is regarded as a constant during the calculation process and decided by the plasma impedance model. It should be noted that the value of r0 does not affect the calculation accuracy; with the change of r0, the stress wave at n·r0 from the center of the cavity is the same (n is a constant and n ≥ 1). Ⅰ, Ⅱ and Ⅲ stand for crushing zone, fracture zone, and vibration zone, respectively. σR and σθ are the radial stress and tangential stress at any position in the rock, respectively. One of the stress states is shown in figure 6(b); the radial stress and the tangential stress are both the tensile stress, σR < 0 and σθ < 0.
The process of shock wave loading on rock medium is simplified as an axisymmetric, linear elastic plane strain solution. In polar coordinates, the governing equation of stress wave propagation in a rock medium is [32]
| {∂2u∂R2+1R∂u∂R−uR2=1c2p∂2u∂t2, (a) σR=(λ+2μ)(∂u/∂R)+λu/R, (b) σθ=λ(∂u/∂R)+(λ+2μ)u/R, (c) | (18) |
where u is the radial displacement of the medium particle and its variable form is u(R, t), and R is the distance between the medium particle and the center of the plasma channel. σR and σθ are the radial stress and tangential stress in the rock medium, respectively; their variable forms are σR(R, t) and σθ(R, t). λ and μ are Lame constants, which are related to the properties of the rock medium. cp is the P-wave velocity of the rock medium, cp=√(λ+2μ)/ρ, where ρ is the density of the rock. The initial and boundary conditions are:
| {u(R,0)=∂u(R,0)∂t=0,(R⩾ | (19) |
Equation (18) is difficult to solve directly; let \bar{u}=\int_0^{\infty} u \mathrm{e}^{-s t} \mathrm{d} t, and apply a Laplace transform to equation 18(a):
| R^2 \frac{\partial^2 \bar{u}}{\partial R^2}+R \frac{\partial \bar{u}}{\partial R}-\left[1+\left(\frac{s \cdot R}{c_{\mathrm{p}}}\right)^2\right] \bar{u}=0 | (20) |
Let x = s·R/cp, and substitute into equation (20):
| x^2 \frac{\partial^2 \bar{u}}{\partial x^2}+x \frac{\partial \bar{u}}{\partial x}-\left[1+x^2\right] \bar{u}=0 | (21) |
Equation (21) conforms to the modified first-order Bessel equation, and its general solution is:
| \bar{u}=A I_1(x)+B K_1(x) | (22) |
where, I1(x) and K1(x) are first-order Bessel functions of the first kind and second kind, respectively. Considering the boundary conditions, A = 0 is obtained, otherwise the solution does not converge. Therefore, equation (22) is deduced as
| \bar{u}=B K_1(x) | (23) |
Applying the Laplace transform to radial stress σR(R, t) and combining with equation (23), we get
| \bar{\sigma}_R=(\lambda+2 \mu) B K_1^{\prime}(x) \frac{s}{c_{\mathrm{p}}}+\lambda B K_1(x) / R | (24) |
Similarly, the Laplace transform of the tangential stress σθ(R, t) is
| \bar{\sigma}_\theta=\lambda B K_1^{\prime}(x) \frac{s}{c_{\mathrm{p}}}+(\lambda+2 \mu) B K_1(x) / R | (25) |
Considering the boundary conditions at the interface between the surface of the plasma channel and the rock medium, there is σR(r0, t) = PT(t), and the Laplace transform is:
| \bar{\sigma}_R\left(r_0, t\right)=P_{\mathrm{T}}(s) | (26) |
By combining equations (23)–(26), the Laplace expression of radial stress σR(R, t) and tangential stress σθ(R, t) in HVPD rock fragmentation can be obtained. Using the Stehfest algorithm to calculate the inverse Laplace transform [34], the time domain solutions of the radial stress and the tangential stress can be obtained.
In order to analyze the wave process of HVPD rock fragmentation, the time-varying impedance model of the plasma channel in the rock is first established, as in equation (16). Since it is difficult to directly measure the plasma channel pressure and temperature, this work uses the current and resistance measured by the test to compare with the model calculation results to ensure the rationality of the model solution. The initial values of parameters and variables are determined iteratively, so that the calculated I(t) and Rpl(t) are more consistent with the measurement results. Figure 7 shows the calculation flow chart.
Using the discharge conditions and measurement results in figure 2 to verify the accuracy of the model, the end criteria for calculating current and resistance through model iteration is:
| \left\{\begin{array}{l} \left|I_{\mathrm{em}}-I_{\mathrm{m}}\right| \leqslant 3 \% I_{\mathrm{em}} \\ \left|T_{\mathrm{e}}-T_{\mathrm{m}}\right| \leqslant 3 \% T_{\mathrm{em}} \\ \left|R_{\mathrm{es}}-R_{\mathrm{ms}}\right| \leqslant 3 \% R_{\mathrm{es}} \end{array}\right. | (27) |
The comparison between the current calculated by the plasma channel impedance model and the experimental measurement results is shown in figure 8. The current amplitude calculated by the model is 11.24 kA, the oscillation period is 5.53 μs, and the errors between the current amplitude and period and the measurement results are 1.72% and 1.78%, respectively, which are in good agreement.
The time-varying resistance of the plasma channel can be obtained through the voltage and current measured by the test. Since it is difficult to directly measure the voltage of the plasma channel, the measured voltage U(t) obtained by the high-voltage resistance-capacitive divider in the test actually includes the resistive and inductive voltage components of the plasma channel and the resistive and inductive voltage components of part of the cables. The channel inductance Lpl(t) is small and negligible relative to the inductance Lc of the cables, and the resistance of the cables is small and negligible relative to the resistance of the plasma channel Rpl(t). The current I(t) measured by the current transformer can be regarded as the current flowing through the plasma channel; therefore, the expression of the measured voltage U(t) is
| U(t)=R_{\mathrm{pl}}(t) I(t)+L_{\mathrm{c}} \frac{\mathrm{d} I(t)}{\mathrm{d} t} | (28) |
where Lc is the inherent inductance of part of the cables, which is a constant value. The derivation of the actual measured voltage U(t) is described in detail in appendix A1. After the measurement Lc is about 2.79 μH, the time-varying resistance Rpl(t) of the plasma channel can be expressed as
| R_{\mathrm{pl}}(t)=\frac{U(t)-L_{\mathrm{c}} \frac{\mathrm{d} I(t)}{\mathrm{d} t}}{I(t)} | (29) |
When calculating the time-varying resistance Rpl(t) from the voltage and current measured by the experiment, it is necessary to strip the inductive component of the measured voltage. From the measured data in figure 2, the test and calculated plasma channel resistance in the rock is shown in figure 9. The plasma channel resistance at the initial stage of discharge is described in appendix A2. After rock breakdown, the plasma channel resistance Rpl(t) decreases rapidly from the initial value, showing a time-varying characteristic. The stable value of the channel resistance obtained under the test condition is about 1.68 Ω. Due to the influence of the zero-crossing point of current, the channel resistance periodically increases at the zero-crossing point, and then decreases to a stable value. With the development of the discharge, the energy injected into the plasma channel decreases, and the stable value of the channel resistance also increases gradually. The resistance Rpl(t) obtained by experimental calculation has oscillation and some distortion points. This is because the inductance component of the measured voltage is regarded as a constant value during the calculation process. In fact, the inductance has a time-varying characteristic and cannot be completely stripped, and there is some interference in the device, which causes distortion and oscillation of the calculated resistance. The variation trend of the resistance calculated by the plasma impedance model is close to the experimental results. After the plasma channel is formed, the resistance drops rapidly, and the stable value is about 1.66 Ω, with an error of 1.19%.
Table 1 shows several important parameters and initial values of variables in the model calculation, which remain unchanged in subsequent calculations. By determining the appropriate parameters and initial values of variables, the time-varying plasma impedance model conforms to the actual situation of the discharge in the rock, and can be in good agreement with the test results.
| Parameters | Values | Units |
| r(0) | 0.22 × 10−3 | m |
| n(0) | 5 × 1024 | particles m−3 |
| T(0) | 19 000 | K |
| γ | 1.12 | dimensionless |
| εsublimation | 9.74 × 10−19 | J/particle |
| ξ | 0.003 85 | K−1.5 Ω−1 m−1 |
| f | 0.2 | dimensionless |
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With the model parameters unchanged, the radius, temperature, shock wave and particle density of the plasma channel are calculated as shown in figure 10. At 25 μs, the radius reaches 1.37 mm in the rock. During the development of the plasma channel, the maximum particle density is 8.95 × 1026 m−3, the maximum temperature is 70.89 × 103 K, the peak value of the shock wave is Pm = 808.6 MPa, and the peak time tr = 0.86 μs. The model calculation results are of the same order of magnitude as in some studies [10, 11].
The shock wave Pch(t) generated by the channel expansion can be obtained through the impedance model. Since the calculation results of the model are numerical solutions, equation (30) can be used to fit the time domain waveform of the transmitted shock wave [35]:
| P_{\mathrm{T}}(t)=4 P_{\mathrm{m}}\left(\mathrm{e}^{-\alpha t / \sqrt{2}}-\mathrm{e}^{-\alpha \sqrt{2} t}\right) | (30) |
where α is the empirical coefficient, and the relationship between the value of α and the peak time tr of the shock wave is [35]
| \alpha=-\frac{\sqrt{2} \ln \left(\frac{1}{2}\right)}{t_{\mathrm{r}}} | (31) |
The incident shock wave PT(t), the transmitted shock wave PT(t) and its fitting results are shown in figure 11. The peak value of the transmitted shock wave is about 1438.1 MPa. The peak value and pre-peak time of the fitted shock wave are in good agreement with the model calculation results.
As shown in figure 5, the plasma channel radius r0 is regarded as a fixed value in the stress wave calculation, and the value is the channel radius when the shock wave drops to 10% of the peak value. Under this test condition, r0 = 1.1 mm. Granite is used as the test rock. After testing, the physical and mechanical parameters of the rock are shown in table 2.
| Parameters | Values |
| ρ (kg/m3) | 2639 |
| Tensile strength (MPa) | 5.18 |
| Compressive strength (MPa) | 67.7 |
| cp (m/s) | 4563 |
| λ (GPa) | 2.19 |
| μ (GPa) | 1.65 |
| Poisson ratio v | 0.285 |
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The stress wave versus time at different distances is shown in figure 12. In the figure, the compressive stress is positive and the tensile stress is negative. The radial stress and the tangential stress attenuate with increasing distance, and the stress at a certain distance increases to the peak value and then decays gradually. At a certain distance, the radial stress is mainly compressive and may convert to tensile; the tangential stress changes from compressive stress to tensile stress, and the tensile stress is dominant.
Therefore, under the combined loading of radial stress and tangential stress, the rock medium around the plasma channel is mainly damaged by compressive shear stress and tensile shear stress, resulting in fragmentation and crack propagation. A schematic of radial stress and tangential stress at R = 10r0 is shown in figure 13. By comparing the relationship between radial stress and tangential stress with time at the same distance, it can be found that the stress state of the rock medium around the channel is in the state of compression shear or tension shear stress, which can be roughly divided into three stages: S1: the compressive-shear stress state under the combinatorial loading of radial compressive stress and tangential compressive stress; S2: the tensile-shear stress state under the combinatorial loading of radial compressive stress and tangential tensile stress; and S3: the tensile-shear stress state under the combinatorial loading of radial tensile stress and tangential tensile stress [32, 36].
The stress wave versus distance at different times is shown in figure 14. The radial stress and the tangential stress attenuate with increasing distance. The radial stress is mainly compressive and may convert to tensile as the distance increases; the tangential stress is mainly tensile and may convert to compressive as the distance increases. The crushing zone in rock blasting is mainly generated by compressive-shear stress, while the fracture zone is mainly generated by tensile-shear stress [37]. According to the wave process analysis of HVPD rock fragmentation, the stress state of the rock changes with time and distance, and the rock is mainly subjected to tensile-shear and compressive-shear stress, so the HVPD has a better rock breaking effect.
Rocks are crushed and cracked under the action of shock waves and stress waves, and the fragmentation effect depends on the characteristics of the rock and the actual stress state [13]. Granite is brittle rock, and its tensile strength is much less than its compressive strength. It is considered that the radial stress is compressive stress and the tangential stress is tensile stress. The peak values of radial stress σRm(R) and tangential stress σθm(R) at any point in rock can be expressed as, respectively [38]
| \sigma_{R \mathrm{m}}(R)=P_{\mathrm{Tm}} \cdot\left(R / r_0\right)^{-\lambda}, R \geqslant r_0 | (32) |
| \sigma_{\theta \mathrm{m}}(R)=-b \cdot P_{\mathrm{Tm}}\left(R / r_0\right)^{-\eta}, R \geqslant r_0 | (33) |
where PTm = 1.438 GPa is the peak value of the transmission shock wave, λ and η are the attenuation coefficients, and b is the lateral stress coefficient. Figure 15 shows the peak value of the calculated stress wave at any point and its fitting results through equations (32) and (33).
According to the fitting results, λ = 1.077, b = 1.23, η = 1.645. The failure criterion of materials under external load depends on the properties of materials and the actual stress state [39]. The stress strength at any point in rock σim can be expressed as
| \sigma_{\mathrm{im}}=\frac{1}{\sqrt{2}}\left[\left(\sigma_{R \mathrm{m}}-\sigma_{\theta \mathrm{m}}\right)^2+\left(\sigma_{\theta \mathrm{m}}-\sigma_{z \mathrm{m}}\right)^2+\left(\sigma_{z \mathrm{m}}-\sigma_{R \mathrm{m}}\right)^2\right]^{\frac{1}{2}} | (34) |
where σRm, σθm, and σzm are the radial, tangential and axial stress at any position in the rock. According to the analysis in B, the strength of the stress wave in the rock varies with time and space, so the effect of the peak value of the stress wave on the rock is considered here. For plane strain problems, σzm can be expressed as
| \sigma_{z \mathrm{m}}=v_{\mathrm{d}}\left(\sigma_{R \mathrm{m}}+\sigma_{\theta \mathrm{m}}\right) | (35) |
where vd is the dynamic Poisson ratio of the rock medium. During the blasting process, it can be considered that vd = 0.8 v [13]. According to the von Mises criterion, the failure condition of rock is
| \sigma_{\mathrm{im}} \geqslant \sigma_0 | (36) |
| \sigma_0=\left\{\begin{array}{l} \sigma_{\mathrm{c}}, \text { (crush zone) } \\ \sigma_{\mathrm{f}}, \text { (fracture zone) } \end{array}\right. | (37) |
where σ0 is the failure strength of rock under uniaxial stress, and σc and σf are the uniaxial dynamic compressive strength and uniaxial dynamic tensile strength of rock, respectively. The dynamic strength of rock increases with an increase in loading strain rate, and different rocks have different sensitivities to the strain rate. Under the condition of dynamic loading, the dynamic strength of rock may be 5–10 times the static strength [31]. According to table 2, under the action of HVPD, it is assumed that the dynamic compressive strength range of rock is 338.5–677 MPa, and the dynamic compressive strength range of rock is 25.9–51.8 MPa. The relationship between the stress strength and the dynamic strength of rock can be obtained by solving equations (32)–(36), as shown in figure 16.
According to figure 15, under the test conditions, the radius of the crushing area is about 3.1–5.4 mm, and the radius of the fracture area is about 25.6–46.7 mm. HVPD can achieve a good fragmentation effect on rocks through multiple actions. In fact, there are many factors affecting the fragmentation effect of HVPD, including power supply parameters, plasma channel deposition energy, electrode distance, electrode form, and solid–liquid properties, which will be further analyzed in the future.
HVPD rock fragmentation technology mainly uses rapid expansion of the plasma channel in the rock. In this work, a time-varying impedance model suitable for plasma channels in rocks is established, which can predict the development characteristics and shock wave intensity of plasma channels. Through HVPD rock fragmentation tests, the accuracy and applicability of the model are verified, and some parameters of the model are determined. Based on the impedance model, the rock to be broken is simplified as a homogeneous and linear elastic material, and the variation law of the stress wave in the rock is analyzed. The calculation results show that the stress wave changes with time and space, and the radial stress is mainly compressive stress, the tangential stress is mainly tensile stress, and the stress state may change. The HVPD rock fragmentation mainly causes compressive-shear and tensile-shear damage to the rock, and has high efficiency. Based on the von Mises criterion and under the test conditions in this paper, the radius of the crushing zone is about 3.1–5.4 mm, and the radius of the fracture zone is about 25.6–46.7 mm for single action. The rock fragmentation effect of HVPD is affected by many factors, which will be studied in future work.
Figure A1 shows a photo of the test platform. When measuring the voltage of the plasma channel, the voltage divider was attached with relatively long wires to the electrodes. The cathode was connected to the ground of the circuit through a copper busbar. Due to the large size of platform, the length of the copper busbar was about 3.6 m. The diagram of the measuring part of the divider is shown in figure A2.
According to figure A2, the measuring circuit of the divider includes: w1, w2, w3 (plasma channel) and w4. Since the impedance of the voltage divider is extremely large, it can be considered as open circuit. The current on w1 and w4 is close to 0, which contributes little to the measured voltage. While w2 and w3 belong to the main circuit, their contribution to the measured voltage should be considered.
The resistance and inductance of w3 are Rc and Lc, respectively. Rc and Lc are part of the inherent resistance R0 and inductance L0 of the loop. The resistance and inductance of plasma channel w2 are Rpl(t) and Lpl(t), respectively. As the long wires form a closed circuit, the main discharge circuit has mutual inductance on the measurement circuit. The mutual inductance with the main discharge circuit is M. The current monitor measures the current I(t) flowing through the plasma channel. The voltage U(t) measured by the voltage divider is:
| U(t)=\left[R_{\mathrm{pl}}(t)+R_{\mathrm{c}}\right] I(t)+\left(L_{\mathrm{c}}+L_{\mathrm{pl}}(t)+M\right) \frac{\mathrm{d} I(t)}{\mathrm{d} t} | (A-1) |
Rpl(t) is usually several ohms, and copper busbars are used in the circuit, so Rc can be ignored relative to Rpl(t). The inductance of the spark channel is very small [21], so the channel inductance Lpl(t) is small and negligible relative to the inductance Lc. Due to the high discharge voltage and current, in order to avoid the influence of electromagnetic interference on the measurement, the circuit formed by the divider wire, the discharge circuit wire and the Marx generator were vertically arranged, and the mutual inductance between them was close to zero. Therefore, the measured voltage of the divider is
| U(t)=R_{\mathrm{pl}}(t) I(t)+L_{\mathrm{c}} \frac{\mathrm{d} I(t)}{\mathrm{d} t} | (A-2) |
Even if M was not zero, the inductive voltage it brought was also included in the voltage U(t) measured by the divider. The inductive component of the measured voltage due to Lc could be considered as a constant, which was mainly caused by the long wires of the circuit. The value of Lc can be calculated by using the zero-crossing point of the current I(t). Under this test condition, Lc = 2.79 μH. The voltage of the plasma channel can be obtained by stripping the inductive component.
Due to the electromagnetic noise recorded by the oscilloscope, it is difficult to obtain an accurate resistance during the initial stage of discharge. Figure A3 shows the voltage U(t), current I(t) and resistance Rpl(t) at the initial stage.
According to figure A3, the electromagnetic noise was mainly caused by the rapid increase in voltage. The plasma channel was not broken down when the voltage was rising. The pre-breakdown time of the pulse discharge in the rock was not obvious, so the current rose directly and there was also large electromagnetic interference in the initial stage. Therefore, it is practically impossible to accurately determine the plasma resistance during the initial stage. The initial value of resistance is relatively large, about several hundred ohms, and also depends on the channel length, rock properties, etc. After the channel breakdown, the voltage and the resistance dropped rapidly. This paper showed the plasma resistance after channel breakdown. In order to clearly show the resistance of the plasma channel after breakdown, the coordinate scale was appropriately reduced, so the results in figure 9 were obtained.
The authors gratefully acknowledge the support of National Natural Science Foundation of China (No. 52177144).
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Figures(19) / Tables(2)
Supported by: Beijing Renhe Information Technology Co., Ltd. 
| Parameters | Values | Units |
| r(0) | 0.22 × 10−3 | m |
| n(0) | 5 × 1024 | particles m−3 |
| T(0) | 19 000 | K |
| γ | 1.12 | dimensionless |
| εsublimation | 9.74 × 10−19 | J/particle |
| ξ | 0.003 85 | K−1.5 Ω−1 m−1 |
| f | 0.2 | dimensionless |
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| Parameters | Values |
| ρ (kg/m3) | 2639 |
| Tensile strength (MPa) | 5.18 |
| Compressive strength (MPa) | 67.7 |
| cp (m/s) | 4563 |
| λ (GPa) | 2.19 |
| μ (GPa) | 1.65 |
| Poisson ratio v | 0.285 |
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CSV
