# Experimental Study of Pulsed Discharge Underwater Shock-Related Properties in Pressurized Liquid Water

Authors: D. C. Bian; D. Yan; J. C. Zhao; S. Q. Niu
Journal: Advances in Materials Science and Engineering
 (2018)

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## Abstract

Engineering background of hydraulic fracturing is applied to improve the permeability of unconventional gas wells, such as coal seams and shale gas wells, by a pulsed discharge mechanism. We studied the general relations between water shock wave properties (the maximum pressure, wave velocity, and energy conversion efficiency), the discharge voltage, and hydrostatic pressure during high-voltage pulsed discharge experiments in pressurized liquid water. The following observations were made: (1) when the discharge voltage increased from 7 kV to 13 kV, the maximum pressure increased from 12.6 MPa (hydrostatic pressurePH = 12 MPa) to 40 MPa (PH = 6 MPa), wave velocity increased from 1418 m/s (PH = 12 MPa) to 1454 m/s (PH = 6 MPa), and energy conversion efficiency increased from 9% to 11%, and (2) when hydrostatic pressure increased from 0 MPa to 12 MPa, the maximum pressure and wave velocity augmented and then diminished slowly (the critical hydrostatic pressure occurs in the 3 to 6 MPa range), whereas the change of energy conversion efficiency was not obvious. Their properties are explained by the variation of electrical parameters during the pulsed discharge.


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## Body

## 1. Introduction

The most direct and effective method to improve oil and gas production is reservoir fracturing, which can substantially improve reservoir permeability [1, 2]. Because of significant variations in the characteristics of these unconventional gas reservoirs (e.g., coal gas and shale gas) [3, 4], the traditional fracturing techniques are not efficient to improve reservoir permeability. Problems such as low efficiency, poor degree of control, damage caused by the fracturing fluid to the reservoir and underground environment, and large water consumption are very common [5, 6]. Therefore, the natural gas mining industry urgently needs a better fracturing technology for these unconventional types of extraction.The phenomenon of high-voltage pulsed discharge in water (HVPD) has been applied to several domains, such as mechanical processing, lithotripsy, and waste water treatment [7–9]. In recent years, there has been some investigation into the application of HVPD to remove oil and gas well plugs and for reservoir fracturing improvements [6, 10]. The water shock waves (WSWs) produced by HVPD are the main power source behind the fracturing of gas reservoirs, the coal and rocks of the gas reservoirs will exhibit different fracturing characteristics under different WSWs. [11]. Thus, any research of this technology must consider the properties of WSW. Considering the depth of oil and gas wells may range from hundreds to thousands of meters, it is important to consider the effect of hydrostatic pressure (PH) on plasma breakdown, WSW excitation, and transfer processes.There are many researchers investigating the characteristics of WSWs produced by HVPD [12–19], but few of them considered the effects of PH on WSWs. Furthermore, other researchers carried out some experiments to study the electrical characteristics of pulsed discharge in pressurized water [20–22]. The results showed that PH increased the breakdown voltage of water dielectric and hindered the processes of HVPD and the formation of plasma channels. Schaefer’s results showed that the larger the PH, the smaller the bubble maximum radius and the higher the peak frequency [23]. Jeffrey conducted a theoretical study revealing that the bubble period and the minimum rarefaction pressure were dependent on PH, while the first peak pressures of WSW were unrelated to PH when PH < 0.9 MPa [24]. Based on Jeffrey’s results, Lu presented a new model including bound-bound transitions in the calculation of the thermal radiation power [25]. His calculations agreed with the experimental results of Jeffrey. However, researchers determined the influence of law of PH on WSWs through theoretical study of the HVPD in pressurized liquid water. Moreover, comparison with experimental results was restricted to PH < 0.9 MPa. Therefore, it is unclear if the first maximum pressure of the WSW is unrelated to PH values > 0.9 MPa [23]. Investigating the relation between the pulsed discharge voltage UD, hydrostatic pressure PH, and water shock wave properties (e.g., the maximum pressure PM, wave velocity DW, and energy conversion efficiency η) is important to better understand the phenomenon of HVPD and characterize the impact load of WSWs. This can lead to improvements in the construction technologies for reservoir fracturing.This study establishes the functional relations betweenPM, DW, and η and UD and PH during HVPD experiments. The characteristics of WSW were then controlled by adjusting UD and PH. Finally, we generated controlled and repeatable optimized shock loads that conformed to the requirements of fracturing rock mass.


## 2. Experimental Instruments and Design

### 2.1. Principles of Experiments

Our studies were based on the fundamental principle of the electrohydraulic effect. A conventional 220 V single-phase alternating current will charge the pulsed energy storage capacitor bank after filtering, boosting, and rectifying by corresponding circuits. The capacitance of the capacitor bank was fixed, and the stored energy was controlled byUD. During the HVPD experiments, a trigger signal was generated by the control system to close the discharge switch. The energy stored in the capacitor bank was loaded instantaneously into the water body located between the discharge electrodes. A high-energy density plasma channel was generated and expanded rapidly from the inside to the outside. The excited pulse was transmitted through water in a given direction forming a shock wave. The relations between UD, PH, and water shock wave properties were investigated by monitoring and analyzing the voltage, current, and WSW waveforms.


### 2.2. Structure of the Experimental Instruments

The experimental equipment was composed of a HVPD system, a pressure-bearing pipeline system, and a measuring system (Figure1).Figure 1
Schematic diagram of the experimental instruments.The capacitance of the capacitor bank in the pulsed discharge cabinet was set atC = 60 μF, and its rated operational voltage was 15 kV. The discharge voltage of two MFM30-15 pulse capacitors was continuously adjusted within the prescribed range. A coaxial copper configuration was set in the electrode to make the WSW propagate along the pipeline horizontal direction and avoid electrode high-voltage ablation. This set up ensures the stability of the HVPD and meets the strength of the electrode. The electrode was 300 mm long, and the high- and low-voltage electrode spacing (l) was 5 mm. The discharge chamber presents a tubular structure (Figure 2). The axle sleeve and electrode structure were sealed and fixed at one end of the discharge chamber by a threaded sleeve and clamping cap. The flange plate was located at the other end of the discharge chamber, which was connected to the pressure-bearing pipeline of the WSW. Figure 3 illustrates the pressure-bearing pipeline system. It contains two long pipes (1 m in length) and one short pipe (0.5 m length) connected by flange plates. The initial PH in the pipeline system was supplied by a BD-4DSB-16 pressure testing pump producing a maximum pressure of 16 MPa. Two pressure sensors, numbered 1 and 2, were installed on the pipeline wall. The CY400 high-frequency dynamic piezoresistive pressure sensor was selected for data acquisition. The sensor provided a range between 0 and 50 MPa with a 150% overload capacity. The maximum sampling frequency was 1 MHz. The bandwidth and response time of the pressure sensor are 1/(400 kHz) and 0.3 μs, respectively. The signals collected by the sensors were amplified and processed using a TST6250 instantaneous signal recorder. The measured precision was 0.01 MPa for dynamic pressure and 0.1 m/s for velocity.Figure 2
Schematic diagram of the discharge chamber structure.Figure 3
Photograph of the pressure-bearing pipeline system.The external trigger interface of the instantaneous signal recorder was connected to a discharge control box. The current signal was monitored by a self-integrating Rogowski coil with a sensitivity of 38 kA/mV. The voltage signal was detected using a Tektronix P6015A high-voltage probe. The current and voltage signals were loaded into an Agilent DSO6014A digital oscilloscope.


### 2.3. Anti-Interference Measures of the Experimental Instruments

A significant energy conversion resulted from the breakdown of the water gap by HVPD. This was accompanied by strong electromagnetic radiations and a voltage/current break in the loop of the instruments [26]. We thus applied the following anti-interference measures:(1)
Power supply: a single-phase alternating current (220 V, 50 Hz) was supplied to the HVPD system. Internal direct current rechargeable batteries were used for the measuring system, that is, the signal recorder and computer, which was sensitive to interference signals. Therefore, a conduction coupling interference formed by the power supply was avoided.(2)
Grounding measures: the pressure-bearing pipeline was connected firmly to the pulsed capacitor grounding point, and the measuring system is insulated at this grounding point. The charge–discharge control box, digital oscilloscope, and high-voltage probe must be grounded separately. Floating was used in the case of the signal recorder [27].(3)
Electromagnetic shielding measures: we used piezoresistive pressure sensors with metal shielding providing a clear and strong resistance to interference. We also selected an instantaneous signal recorder with a complete metal shell. The instantaneous signal recorder and the Rogowski coil were placed in an aluminum shielding box.(4)
Wires: we kept the wires in the experiment as short as possible. The two wires located between the capacitor and the electrode are coaxial cables. Radiation coupling and electrical loop inductance were reduced as much as possible.


### 2.4. Experimental Process

U
D values of 7, 9, 11, and 13 kV were used, respectively. PH was increased in 1 MPa steps in the 0 to 4 MPa range and in 2 MPa steps in the 4 to 12 MPa range. The distance between No. 1 sensor and the electrode, and between Nos. 1 and 2 sensors, is 1 m. Several tests were repeated for each UD and PH to eliminate random system errors [20, 28]. Water in the pipeline was replaced before each test to ensure constant temperature (T = 20°C) and conductivity (ρ = 340 μS/cm) [29, 30]. The voltage, current, and WSW pressure waveforms were monitored by the measuring system. Note that the shock wave peak pressure measured by the sensor included PH. In the following discussion, PM does not include PH.


## 2.1. Principles of Experiments

Our studies were based on the fundamental principle of the electrohydraulic effect. A conventional 220 V single-phase alternating current will charge the pulsed energy storage capacitor bank after filtering, boosting, and rectifying by corresponding circuits. The capacitance of the capacitor bank was fixed, and the stored energy was controlled byUD. During the HVPD experiments, a trigger signal was generated by the control system to close the discharge switch. The energy stored in the capacitor bank was loaded instantaneously into the water body located between the discharge electrodes. A high-energy density plasma channel was generated and expanded rapidly from the inside to the outside. The excited pulse was transmitted through water in a given direction forming a shock wave. The relations between UD, PH, and water shock wave properties were investigated by monitoring and analyzing the voltage, current, and WSW waveforms.


## 2.2. Structure of the Experimental Instruments

The experimental equipment was composed of a HVPD system, a pressure-bearing pipeline system, and a measuring system (Figure1).Figure 1
Schematic diagram of the experimental instruments.The capacitance of the capacitor bank in the pulsed discharge cabinet was set atC = 60 μF, and its rated operational voltage was 15 kV. The discharge voltage of two MFM30-15 pulse capacitors was continuously adjusted within the prescribed range. A coaxial copper configuration was set in the electrode to make the WSW propagate along the pipeline horizontal direction and avoid electrode high-voltage ablation. This set up ensures the stability of the HVPD and meets the strength of the electrode. The electrode was 300 mm long, and the high- and low-voltage electrode spacing (l) was 5 mm. The discharge chamber presents a tubular structure (Figure 2). The axle sleeve and electrode structure were sealed and fixed at one end of the discharge chamber by a threaded sleeve and clamping cap. The flange plate was located at the other end of the discharge chamber, which was connected to the pressure-bearing pipeline of the WSW. Figure 3 illustrates the pressure-bearing pipeline system. It contains two long pipes (1 m in length) and one short pipe (0.5 m length) connected by flange plates. The initial PH in the pipeline system was supplied by a BD-4DSB-16 pressure testing pump producing a maximum pressure of 16 MPa. Two pressure sensors, numbered 1 and 2, were installed on the pipeline wall. The CY400 high-frequency dynamic piezoresistive pressure sensor was selected for data acquisition. The sensor provided a range between 0 and 50 MPa with a 150% overload capacity. The maximum sampling frequency was 1 MHz. The bandwidth and response time of the pressure sensor are 1/(400 kHz) and 0.3 μs, respectively. The signals collected by the sensors were amplified and processed using a TST6250 instantaneous signal recorder. The measured precision was 0.01 MPa for dynamic pressure and 0.1 m/s for velocity.Figure 2
Schematic diagram of the discharge chamber structure.Figure 3
Photograph of the pressure-bearing pipeline system.The external trigger interface of the instantaneous signal recorder was connected to a discharge control box. The current signal was monitored by a self-integrating Rogowski coil with a sensitivity of 38 kA/mV. The voltage signal was detected using a Tektronix P6015A high-voltage probe. The current and voltage signals were loaded into an Agilent DSO6014A digital oscilloscope.


## 2.3. Anti-Interference Measures of the Experimental Instruments

A significant energy conversion resulted from the breakdown of the water gap by HVPD. This was accompanied by strong electromagnetic radiations and a voltage/current break in the loop of the instruments [26]. We thus applied the following anti-interference measures:(1)
Power supply: a single-phase alternating current (220 V, 50 Hz) was supplied to the HVPD system. Internal direct current rechargeable batteries were used for the measuring system, that is, the signal recorder and computer, which was sensitive to interference signals. Therefore, a conduction coupling interference formed by the power supply was avoided.(2)
Grounding measures: the pressure-bearing pipeline was connected firmly to the pulsed capacitor grounding point, and the measuring system is insulated at this grounding point. The charge–discharge control box, digital oscilloscope, and high-voltage probe must be grounded separately. Floating was used in the case of the signal recorder [27].(3)
Electromagnetic shielding measures: we used piezoresistive pressure sensors with metal shielding providing a clear and strong resistance to interference. We also selected an instantaneous signal recorder with a complete metal shell. The instantaneous signal recorder and the Rogowski coil were placed in an aluminum shielding box.(4)
Wires: we kept the wires in the experiment as short as possible. The two wires located between the capacitor and the electrode are coaxial cables. Radiation coupling and electrical loop inductance were reduced as much as possible.


## 2.4. Experimental Process

U
D values of 7, 9, 11, and 13 kV were used, respectively. PH was increased in 1 MPa steps in the 0 to 4 MPa range and in 2 MPa steps in the 4 to 12 MPa range. The distance between No. 1 sensor and the electrode, and between Nos. 1 and 2 sensors, is 1 m. Several tests were repeated for each UD and PH to eliminate random system errors [20, 28]. Water in the pipeline was replaced before each test to ensure constant temperature (T = 20°C) and conductivity (ρ = 340 μS/cm) [29, 30]. The voltage, current, and WSW pressure waveforms were monitored by the measuring system. Note that the shock wave peak pressure measured by the sensor included PH. In the following discussion, PM does not include PH.


## 3. Analysis of Experimental Data

The voltage, current, and WSW pressure waveforms from a typical HVPD process are illustrated in Figure4. The transfer process of the WSW in the pipeline is monitored by two pressure sensors on the pipeline wall. Typical pressure waveforms are illustrated in Figure 5.Figure 4
Diagram of the voltage, current, and pressure waveforms versus time for a typical HVPD process (PH: 1 MPa; UD: 7000 V).Figure 5
Typical WSW waveforms obtained by two pressure sensors (PH: 1 MPa; UD: 7000 V).WSWs are generated after the discharge gap breakdown of the electrode and propagated from the end of the electrode to the bottom of the pipeline. The pressure was measured by the No. 1 sensor (Time-consuming T1). After transmission on 1 m, the signal was captured by No. 2 sensor (Time-consuming T2). The WSW is propagated and reaches the bottom of the pipeline after transmission on 1 m. Then, it is reflected by the end flange and successively propagated to No. 2 (Time-consuming T3) and No. 1 sensor (Time-consuming T4).


### 3.1. Relationship betweenUD and WSW Properties

The voltage and current waveforms were analyzed to determine the values of breakdown voltageUB, discharge breakdown time delay TB, loop resistance RL, peak current IP (the first half-wave), and peak power PP [31].The plots revealTB gradually decreasing with increasing UD values when PH remained constant (Figure 6). Theoretically, increasing the UD values will enhance the field strength difference between the two ends of the gap, accelerating the water ionization and gasification velocity [32]. Then, an ionization avalanche is started, increasing the growth speed of the main plasma channel and decreasing the TB values during the discharge process. The energy input into the plasma channel after the breakdown was enhanced by increasing the UD values. As a result, the plasma temperature grew, inducing a higher ionization rate and conductivity. The channel pressure also augmented due to the higher plasma temperature, accelerating the expansion of the channel, enhancing the conductive cross section, and finally leading to a decrease in channel resistance [33]. The plasma is the energy-transforming component of the HVPD. Augmenting the UD values reduced the resistance of the plasma, which increased the IP and PP values in the plasma channel.Figure 6
Variation diagram of (a)TB, (b) RL, (c) IP, and (d) PP versus PH for a range of UD and PH values.
(a)
(b)
(c)
(d)P
M and DW increased with increasing UD when the PH values were fixed. This is explained by the empirical equation of the water shock wave peak pressure (PM) [34]:(1)PM=kQBα,where k is a constant related to the experimental environment, QB is the breakdown energy, and α forms the attenuation coefficient of the water shock wave associated with the electrode structure.Therefore, thePM value is enhanced by increasing the UD value when all other parameters are constant. For a weak water shock wave (PM < 0.1 GPa), the transfer process is isentropic, producing the following relation between wave velocity (DW) and PM [35] such as(2)DW=c01+n+12nPMB1/2≅c01+n+14nPMB,where c0 is the sound velocity of water before perturbation, n = 7.15, and B = 299 MPa.Figure7 indicates similar increasing trends of DW and PM values with the augmentation in UD.Figure 7
Variations betweenUD, PM, and velocity DW under four different PH settings: (a) 0 MPa, (b) 4 MPa, (c) 8 MPa, and (d) 12 MPa.
(a)
(b)
(c)
(d)The experience formula for computing the energy of WSW is as follows [36, 37]:(3)Es=Sρv∫0+∞P2dt,where ES is the energy of WSW in J, S is the wavefront area in m2, ρ is the density of water in kg/m3, v is the velocity of WSW in m/s, and P is the pressure of WSW in Pa.The ratio between the WSW energy (ES) to the electrical energy stored in the capacitors (E) is defined as the energy conversion efficiency (η):(4)η=ESE×100%=2SρvCUD2∫0+∞P2dt×100%.From Figure8, we observed a η value is roughly around 10% increasing slightly with the UD value when the PH values were fixed. This is principally related to the increase in UD that reduces TB and then lowers the leakage loss of energy and improves the energy conversion efficiency.Figure 8
Variation diagram ofη versus PH for a range of UD.


### 3.2. Relation betweenPH and WSW Properties

We cannot ignorePH when applying HVPD during unconventional gas well fracturing and in deep sea environments. Kao [38] and Korobeinikov [39] developed a bubble initiation theory to explain the plasma breakdown process, in which the environmental pressure significantly affected the breakdown field strength of a liquid. The influence of PH was determined by HVPD experiments in pressurized liquid water. During the pulsed discharge process, PH shows “inhibiting” and “enhancing” effects. The inhibiting effect is caused by the compression and hindering of the plasma channel expansion shock, since PH lowers the initial PM and DW. The enhancement is generated by a reduction of the attenuation coefficient during the transfer of the WSW through the water medium. When the PH values increase, both of these effects will change PM and DW starting by an augmentation followed by a decrease (Figure 9). The critical hydrostatic pressure (PCH) occurs in the 3 to 6 MPa range.Figure 9
Binary diagrams showing the correlations betweenPH, PM, and DW under four different values of UD: (a) 7 kV, (b) 9 kV, (c) 11 kV, and (d) 13 kV.
(a)
(b)
(c)
(d)The inhibiting mechanism works as follows: PH and the internal pressure within the channel and the pressure on the inner wall of the channel caused by the electric field were all applied during the channel breakdown process. The combination of these pressures causes a change in the channel radius, and PH hinders the channel expansion [40, 41].The experimental data (Figure9) and theoretical models indicated that the compressive action of the external pressure on the plasma due to PH delayed the water breakdown and caused an increase in TB. Therefore, when the UD value remains constant, the increase in TB with increasing PH values produces a lower energy input into the plasma channel. More energy is wasted as heat and infrared radiation before the plasma channel formed [42], and its electrical resistivity also increased. However, the plasma channel wall was extruded by a PH increase. The channel expansion was lowered, leading to a conductive cross section decrease relative to a situation without PH; this also results in a growth in electrical resistivity and IP, whereas the channel PP declines accordingly (Figure 6). These observations indicate that TB and PH increase simultaneously when UD is constant. The maximum outward expansion pressure of the plasma also decreases.The enhancing mechanism works as follows: The propagation of weak WSW (p ≤ 100 MPa) is isentropic. The propagation and characteristics are similar to that of sound waves in water. The research of Saul and Wagner [43] showed that the shock wave maximum pressure and velocity generated by a similar source increased with increasing water hydrostatic pressure at room temperature.P
M and DW augmented with increasing PH when 0 ≤ PH ≤ PCH. Inhibiting and enhancing effects are present during this process, but the former is limited due to a low water pressure. In this case, the enhancing effect related to PH played a primary role.P
M and DW declined slowly with increasing PH when the inhibiting effect of PH played a dominant role and significantly weakened PM and DW. Although an increased pressure also enhanced the water shock wave transfer, PM and DW values began to slowly diminish when PCH ≤ PH because of the initial lowering of PM and DW.Figure8 also reveals that PH will reduce η when pressure is applied. However, due to the increase in instability of the discharge process with increasing PH, the change of η has no regularity.


## 3.1. Relationship betweenUD and WSW Properties

The voltage and current waveforms were analyzed to determine the values of breakdown voltageUB, discharge breakdown time delay TB, loop resistance RL, peak current IP (the first half-wave), and peak power PP [31].The plots revealTB gradually decreasing with increasing UD values when PH remained constant (Figure 6). Theoretically, increasing the UD values will enhance the field strength difference between the two ends of the gap, accelerating the water ionization and gasification velocity [32]. Then, an ionization avalanche is started, increasing the growth speed of the main plasma channel and decreasing the TB values during the discharge process. The energy input into the plasma channel after the breakdown was enhanced by increasing the UD values. As a result, the plasma temperature grew, inducing a higher ionization rate and conductivity. The channel pressure also augmented due to the higher plasma temperature, accelerating the expansion of the channel, enhancing the conductive cross section, and finally leading to a decrease in channel resistance [33]. The plasma is the energy-transforming component of the HVPD. Augmenting the UD values reduced the resistance of the plasma, which increased the IP and PP values in the plasma channel.Figure 6
Variation diagram of (a)TB, (b) RL, (c) IP, and (d) PP versus PH for a range of UD and PH values.
(a)
(b)
(c)
(d)P
M and DW increased with increasing UD when the PH values were fixed. This is explained by the empirical equation of the water shock wave peak pressure (PM) [34]:(1)PM=kQBα,where k is a constant related to the experimental environment, QB is the breakdown energy, and α forms the attenuation coefficient of the water shock wave associated with the electrode structure.Therefore, thePM value is enhanced by increasing the UD value when all other parameters are constant. For a weak water shock wave (PM < 0.1 GPa), the transfer process is isentropic, producing the following relation between wave velocity (DW) and PM [35] such as(2)DW=c01+n+12nPMB1/2≅c01+n+14nPMB,where c0 is the sound velocity of water before perturbation, n = 7.15, and B = 299 MPa.Figure7 indicates similar increasing trends of DW and PM values with the augmentation in UD.Figure 7
Variations betweenUD, PM, and velocity DW under four different PH settings: (a) 0 MPa, (b) 4 MPa, (c) 8 MPa, and (d) 12 MPa.
(a)
(b)
(c)
(d)The experience formula for computing the energy of WSW is as follows [36, 37]:(3)Es=Sρv∫0+∞P2dt,where ES is the energy of WSW in J, S is the wavefront area in m2, ρ is the density of water in kg/m3, v is the velocity of WSW in m/s, and P is the pressure of WSW in Pa.The ratio between the WSW energy (ES) to the electrical energy stored in the capacitors (E) is defined as the energy conversion efficiency (η):(4)η=ESE×100%=2SρvCUD2∫0+∞P2dt×100%.From Figure8, we observed a η value is roughly around 10% increasing slightly with the UD value when the PH values were fixed. This is principally related to the increase in UD that reduces TB and then lowers the leakage loss of energy and improves the energy conversion efficiency.Figure 8
Variation diagram ofη versus PH for a range of UD.


## 3.2. Relation betweenPH and WSW Properties

We cannot ignorePH when applying HVPD during unconventional gas well fracturing and in deep sea environments. Kao [38] and Korobeinikov [39] developed a bubble initiation theory to explain the plasma breakdown process, in which the environmental pressure significantly affected the breakdown field strength of a liquid. The influence of PH was determined by HVPD experiments in pressurized liquid water. During the pulsed discharge process, PH shows “inhibiting” and “enhancing” effects. The inhibiting effect is caused by the compression and hindering of the plasma channel expansion shock, since PH lowers the initial PM and DW. The enhancement is generated by a reduction of the attenuation coefficient during the transfer of the WSW through the water medium. When the PH values increase, both of these effects will change PM and DW starting by an augmentation followed by a decrease (Figure 9). The critical hydrostatic pressure (PCH) occurs in the 3 to 6 MPa range.Figure 9
Binary diagrams showing the correlations betweenPH, PM, and DW under four different values of UD: (a) 7 kV, (b) 9 kV, (c) 11 kV, and (d) 13 kV.
(a)
(b)
(c)
(d)The inhibiting mechanism works as follows: PH and the internal pressure within the channel and the pressure on the inner wall of the channel caused by the electric field were all applied during the channel breakdown process. The combination of these pressures causes a change in the channel radius, and PH hinders the channel expansion [40, 41].The experimental data (Figure9) and theoretical models indicated that the compressive action of the external pressure on the plasma due to PH delayed the water breakdown and caused an increase in TB. Therefore, when the UD value remains constant, the increase in TB with increasing PH values produces a lower energy input into the plasma channel. More energy is wasted as heat and infrared radiation before the plasma channel formed [42], and its electrical resistivity also increased. However, the plasma channel wall was extruded by a PH increase. The channel expansion was lowered, leading to a conductive cross section decrease relative to a situation without PH; this also results in a growth in electrical resistivity and IP, whereas the channel PP declines accordingly (Figure 6). These observations indicate that TB and PH increase simultaneously when UD is constant. The maximum outward expansion pressure of the plasma also decreases.The enhancing mechanism works as follows: The propagation of weak WSW (p ≤ 100 MPa) is isentropic. The propagation and characteristics are similar to that of sound waves in water. The research of Saul and Wagner [43] showed that the shock wave maximum pressure and velocity generated by a similar source increased with increasing water hydrostatic pressure at room temperature.P
M and DW augmented with increasing PH when 0 ≤ PH ≤ PCH. Inhibiting and enhancing effects are present during this process, but the former is limited due to a low water pressure. In this case, the enhancing effect related to PH played a primary role.P
M and DW declined slowly with increasing PH when the inhibiting effect of PH played a dominant role and significantly weakened PM and DW. Although an increased pressure also enhanced the water shock wave transfer, PM and DW values began to slowly diminish when PCH ≤ PH because of the initial lowering of PM and DW.Figure8 also reveals that PH will reduce η when pressure is applied. However, due to the increase in instability of the discharge process with increasing PH, the change of η has no regularity.


## 4. Data Fitting and Contrast

Data from the literature [20, 34, 40] and experimentations reveal that TB is a function of PH, or TB = h(PH). Thus, the breakdown energy QB is given by following equation:(5)QB=12CUD2exp−2hPHRC,where k and α in the empirical formula (1) are parameters related to the experimental environment and electrode geometry [44, 45]. These two parameters represent the energy conversion characteristics of the WSW in different experimental conditions [42]. We then assume k and α are related to PH. The functions can be written as k = f(PH) and α = g(PH).Substituting (5) into (1), we produce an expression on PM:(6)PM=fPH12CUD2exp−2hPHRCgPH.Finally, the empirical formula (7) representing PM in pressurized liquid water can be obtained by fitting of the experimental data. DW can also be obtained by substituting (7) into (2); thus,(7)PM=0.7PH+5.59⋅CUD22000exp−2RC43.14P−H25033.140.41PH+6.140.02PH+0.31,where PM is the water shock wave peak pressure in MPa, PH forms the hydrostatic pressure in MPa, UD is the discharge voltage in kV, C defines the capacitance of the capacitor bank in μF, and R gives the equivalent resistance of the water gap in kΩ. In our experiments, C = 60 μF and R = 8.5 kΩ.The functionsf(PH) and g(PH) represent the energy conversion of the WSW in different experimental situations in (7). f(PH) and g(PH) are all linear increasing functions. This indicates that the intensity of the WSW increases with PH, confirming the conclusions of Saul and Wagner [43] and Lu [46]. The function h(PH) represents the compression and delaying of the plasma channel expansion shock by PH. The expression reveals that the h(PH) function increases regularly; the greater the h(PH), the more compressed the plasma channel. This is also consistent with the conclusions of Jia et al. [20], Zhang et al. [21], and Liang et al. [22].The experimental data and the fitting curves (Figure9) indicate PCH is not constant. PCH increases when UD grows.Figure10 is a comparative diagram showing the experimental data, the fitted curve, and the data of Touya et al. [34] and Sun [42], when PH = 0 MPa and the distance d between the discharge electrode and pressure sensor is 1 m. The experimental data and fitted curve are very different from the data of Touya and Sun Bing, but their trends are similar. The differences are explained principally by the settings of Touya’s experiment, in which the distance of the electrode gap l is 10 mm, the curvature radius r is 15 mm, α = 0.35, and k = 9000/d (d being the distance between the discharge electrode and pressure sensor). The equation of the maximum pressure PT obtained by Touya is(8)PT=9000dQB0.35,where PT is in bar, QB is in kJ, and d is in mm.Figure 10
Comparative diagram showing the experimentalUD versus the water shock peak pressure with the fitted curve and the data of Touya and Sun Bing, using PH of 0 MPa and d of 1 m.The pulsed discharge experiment of Touya was conducted in an open environment. One of the water boundaries was not restricted. This generates an energy loss during plasma expansion and impact on the free water surface, and the larger the distanced, the more prevalent is the drop in maximum pressure attributed to the loss. In this paper, the pulsed discharge experiments were carried out in an airtight pipeline which was fully filled with water, and d = 1 m (Touya’s d values were  345–555 mm). The different discharge environments produce large deviations in Touya’s calculations relative to our experimental values. The Touya’s empirical equation is no longer applicable for water with all restricted boundaries (have no free surface), such as oil and gas wells filled with water.The pulsed discharge experiments of Sun Bing were carried out with rod-rod electrodes in an airtight pipeline (5 m in length and 300 mm in diameter) filled with water (310 μS/cm). The empirical formula of peak pressurePS by Sun Bing is(9)PS=2590QD⋅e−0.4836×10−3d,where PS is in bar, QD is in kJ, and d is in mm.The maximum discharge energy of Sun Bing’s experiment was 22.5 J (C = 50 nf and UD = 30 kV). However, the minimum discharge energy of our experiments is > 1 kJ (QD ≈ 1.5–5 kJ). Furthermore, the peak pressure PS shows a linear correlation with the energy value QD when the distance d is constant.Figure11 shows that the deviation in the calculation results of the two formulas is small when QD < 50 J. However, if QD > 50 J, the deviation is more pronounced. The main reason was the difference of discharge parameters between this two experiments (our experiment: C = 60 μF and UD = 7–13 kV. Sun Bing’s experiment: C = 50 nF and UD = 22–30 kV). Sun Bing’s fitting formula was based on those experimental data which were obtained from high voltage (UD > 20 kV) and small energy (Q < 22.5 J) discharge experiments. This led the efficiency of transforming the electrical energy into mechanical energy to be far higher than that of our experiment. For HVPD with larger energy (e.g., 1.5 kJ), the theoretical discharge voltage of Sun Bing was 200 kV (only 7 kV in our experiment). The calculations used Sun Bing’s fitting formula (formula (9)), where the intensity of WSW was about 240 MPa and the energy conversion efficiency was about 180% (suppose the WSW width was 2 μs). Therefore, the empirical formula for peak pressure calculation by Sun Bing does not apply for a pulsed discharge process with larger QD.Figure 11
Comparison diagram showing the discharge energy (QD) versus water shock peak pressure using Sun Bing’s formula and ours.


## 5. Discussion

The relationship between the WSW properties (PM, DW, and η) and UD and PH was studied by experimentation. Results show that an increase of UD can enhance the WSW properties (PM, DW, and η), but the processes of increasing PH on the WSW properties (PM, DW, and η) are more complicated. The results obtained in this paper after fitting the experimental data are compared with that of other studies, and we explained discrepancies with the other published data. Through this study, we understood the basic method of controlling PM and DW. As a result, we can specifically modify the UD and PH values and then adjust the WSW properties (PM and DW) during the process of deep well fracturing.In the process of HVPD, Jones found that the conductivity of water almost had no effect on the discharge process [28]. This is mainly because the discharge voltage of Jones’ experiment was relatively higher (UD > 50 kV), which caused the discharge breakdown process to be in nanoseconds. Thus, the influence of the conductivity on breakdown time delay was negligible basically. But when the discharge voltage was lower (UD < 5 kV), the higher conductivity of water would lead to the shorter breakdown time delay [7, 30]. For the intensity of WSWs, Zhu et al. found that when UD = 4 kV, with the conductivity increasing from 0 mS/cm to 120 mS/cm, PM declined rapidly from 3.5 MPa to 0.62 MPa [29]. While Zhuang Jiasheng found that when UD = 7 kV (HVPD was carried out in a pipe which had a parabolic reflector), with the conductivity increasing from 0.0177 mS/cm to 109.7 mS/cm, PM increased from 20 MPa to 100 MPa [47]. Thus, under low discharge voltage, the influence of conductivity was huge and uncertain on intensity of WSWs. This means that a better fracturing effect of oil and gas wells can be achieved when using a fracturing fluid with suitable conductivity. So, the influence rules of conductivity on the intensity of WSWs need to be further studied.Large capacity pulse capacitors were adopted in our experiment to store the electric energy. The capacitance value cannot be modified. If it was adjustable, we would need multiple parallel capacitors of small capacitance. However, this would bring the discharge process of each capacitor out of sync, release the pulse energy separately, and reduce the intensity of the first pulse shock wave [27].The rise time of WSW obtained in this paper is significantly larger relative to other studies, and thePH influence on the η value is unclear. This needs further research and analysis in the future.


## 6. Conclusions

WhenPH is constant, TB and RL decreased, and IP and PP all increased. Furthermore, the WSW properties (PM, DW, and η) augmented with increasing UD. A larger UD always strengthened the effect of HVPD and the WSW properties.Moreover, whenUD is constant, TB and RL increased and IP and PP all decreased with increasing PH. The PH augmentation demonstrated its “inhibiting” effect on the HVPD process. However, increasing PH will partially reduce the attenuation coefficient of WSW and enhance the transfer of WSW. This demonstrates the “enhancing” effect on PM and DW. Thus, when UD is constant, PM and DW first grow and then diminish with increasing PH. This is caused by the inhibiting and the enhancing effects. Moreover, the η value of HVPD in pressurized liquid water is slightly smaller to that in stress-free water, but the change of η has no regularity.The experimental data fitting enabled a partial application of the empirical formula which would provide a basis for theoretical calculation to further improve the permeability using pulsed discharge hydraulic fracturing.



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*Source: 8025708-2018-01-23.xml*