Simulation of Spark Source Wavelet Under Multibubble Motion

WEI Jia , YANG Huiliang , , FENG Jing , and LI Yang

1) Qingdao Institute of Marine Geology, CGS, Qingdao 266071, China

2) Laboratory for Marine Resources, Qingdao National Laboratory for Marine Science and Technology, Qingdao 266071, China

Abstract Marine spark sources are widely used in high-resolution marine seismic surveys. The characteristic of a wavelet is a critical part in seismic exploration; thus, the formation and numerical simulation of spark source wavelets should be explored. In studies on spark source excitation, the acoustic field generated by the interaction between bubbles constitutes the near-field wavelet of a source. Therefore, this interaction should be revealed by studying complex multibubble motion laws. In this study, actual discharge conditions were combined to derive the multibubble equation of motion. Energy conservation, ideal gas equation, and environmental factors in the discharge of spark source wavelets were studied, and the simulation method of an ocean spark source wavelet was established. The accuracy of the simulation calculation method was verified through a comparison of indoor-measured signals using three electrodes and the spark source wavelet obtained in the field. Results revealed that the accuracy of the model is related to the number of electrodes. The fewer the number of electrodes used, the lower will be the model’s accuracy. This finding is attributed to the statistical hypothesis factor introduced to eliminate the coupling term of the interaction of the multibubble motion equation.This study presents a method for analyzing the wavelet characteristics of an indoor-simulated spark source wavelet.

Key words spark source; wavelet simulation; multibubble motion; statistical hypothesis factor

1 Introduction

A marine spark source is widely used in the exploration of shallow seabed geological structures and energy because of its high primary frequency, wide frequency band,and fast charge and discharge process (Peiet al., 2007). In the 1960s and the 1970s, spark sources gradually replaced small-capacity airguns and became a well-known small-energy ocean source (Duchesneet al., 2007). In the 1980s, a spark source was used for the first time in China to acquire the marine seismic profiles of a length of nearly 90000 km. Since then, a series of studies on the composition and mechanism of spark sources have been conducted(Huang and Chen, 1981), and the influence of a virtual reflection on a spark source wavelet has also been explored(Zhu, 1985). Studies have shown that the signal-to-noise ratio of seismic data acquired with a spark source is lower than that obtained with an airgun (Zhanget al., 2009), but it plays a vital role in multichannel high-precision exploration technology (Xing, 2012; Ramanaet al., 2014). Spark sources are also used to obtain images of natural gas hydrates (Luo and Cai, 2017) and characterize ocean undercurrent boundaries (Müller-Michaelis and Uenzelmann-Neben, 2015). Existing research on the application of spark sources has mainly focused on plasma discharge theory and high-resolution seismic data acquisition and analysis but only a few studies have explored wavelet formation and numerical algorithms for ocean spark sources.

Spark sources are theoretically based on a plasma discharge model. Eubanket al. (1993) studied the material exchange process in plasma models. Cooket al. (1997)considered the problem of material flow caused by bubbles in steam, compression, and ablation and examined the effects of ionization and blackbody radiation on the discharge model. Chinese scholars conducted a series of studies on the discharge of plasma water. Luet al. (2000)found that the radiation characteristics of plasma in a pulsed discharge in water can be regarded as blackbody radiation when channel pressure and temperature simultaneously meet the necessary criteria. Luet al. (2002a) used high-speed cameras to study the plasma channel characteristics of the discharge in water and the bubble rupture process. They found that the discharge of a plasma channel is not uniform along the axial direction. Luet al.(2002b) introduced the boundary constraint transformation problem of thermal radiation and the electric field interaction between ionized particles to the model and established a high-voltage plasma water discharge model. Yanet al. (2014) analyzed the underlying phenomena of highpressure discharge in water and the mechanical simulation of the effects of water shock and bubble pulse. Huanget al.(2014) constructed an indoor single-electrode discharge pulse model using the ideal gas equation and single-bubble motion theory. They also studied parameters such as pulse bubble radius, bubble wall velocity, and bubble temperature. As one of the critical parameters in seismic exploration, source wavelet directly affects the quality of seismic data (Lu, 1993). The volume and numbers of bubbles produced by spark excitation are smaller than those formed by airguns, and more bubbles are formed at the top of electrodes in the former than in the latter. The interaction between a large number of small bubbles is also more complex in spark excitation than in airguns. Therefore, the multibubble motion law is an important part in the numerical simulation of spark source wavelets.

Understanding the motion law of multibubbles is the basis for establishing the wavelet numerical model of spark sources. Multibubble motion is more complicated and essential than single-bubble motion (Rayleigh, 1917). Introduced in 1971, double-bubble motion provided a theoretical basis for subsequently studying three-bubble motions (Shima, 1971; Morioka, 1974). The multibubble equation of motion in incompressible fluids (Garipov, 1973; Lianget al.,1998), the interaction between bubbles in the external force field (Fujikawa and Takahira, 1986), the acoustic radiation law during bubble collapse (Puet al., 2005), and the multibubble coupling of nonspherical properties (Jianget al.,2015) have been studied in accordance with double-bubble theory. The multibubble interaction under different initial conditions has also been explored in detail. However,the number of bubbles in spark source discharge is greater than that of previous studies, and environmental constraints are more complex in the former than in the latter.

In this study, multi-electrode synchronous excitation with equal energy was set as a condition to simplify the coupling term in a multibubble motion equation based on a previous research. Then, the kinematic equation of the multibubble interaction under this condition was derived.A theoretical model of the spark source wavelet was established under multibubble motion conditions by combining the plasma discharge model, the wave propagation attenuation law, the acoustic wave tuning effect, and different inter-interface effects. The spark source wavelet was numerically simulated, and the accuracy of the model was verified through a comparison with actual data. This model verification provided a theoretical basis for the practical application of the spark source.

2 Model Assumptions and Initial Conditions

2.1 Model Assumptions

Corona pulse discharge in water is a process by which plasma is generated at the tip of an electrode. According to Ohm’s law, when the conductivity of water is sufficiently high, the voltage applied to the tip of an electrode can produce a strong current density. The tip of the electrode then heats the surrounding water because of energy transfer to form water vapor, which is continuously injected into bubbles; water vapor subsequently ionizes to produce plasma (Ushakovet al., 2007; Bruggeman and Leys,2009).

The model assumptions of the actual process are described below (Zel’dovich and Raizer, 1967; Loffe and Naugol’nykh, 1968; Fujikawa and Akanatsu, 1980; Plesset and Prosperetti, 2003):

1) The gravity of the bubble is disregarded, and the bubble is considered a spherical ball with good symmetry.

2) Water is incompressible such that density and ambient pressure are considered constant.

3) The temperature and pressure in the bubble are continuous.

4) No mass exchange occurs between the bubble and the surrounding environment.

5) The gas in the pulse bubble satisfies the ideal state equation.

6) The surface tension and viscous resistance of the bubble are ignored.

In the acoustic pulse signal simulation of single-electrode water discharge, only the gas state equation, bubble vibration theory equation, and energy conservation principle are considered on the basis of the above assumptions.Then, the acoustic pulse pressure field can be calculated.

2.2 Initial Conditions

Plasma is usually generated in a few microseconds, so the initial temperature and pressure of the plasma cannot be measured experimentally. However, the generally considered initial temperature and pressure are water vapor temperature atT0= 373 K and ambient static pressure atP0=105Pa. According to the energy conversion of the liquid electric effect, the following can be determined (Huanget al., 2014):

whereV0is the initial bubble volume,Edisis the total energy of the discharge system, and the conversion constantηis the experimental calibration parameter, which ranges from 0.05% to 0.3%. The initial number of water moleculesn0of the bubble, the initial radius of the bubblerb,the initial temperature of the bubbleT0, and the initial massm0of water vapor can be estimated using the ideal gas equationPV=mRmTand the other assumptions described in Section 2.1 (Table 1). The energy input calculation range is 5–30 J, andηis 0.06%.

Table 1 Initial conditions of each parameter of a bubble under different charging energies

2.3 Energy Conservation

Corona pulse discharge in water follows the law of energy conservation (Fig.1). The total discharge energyEdisis the total input of a system, and it includes the circuit loading energyELand the circuit energy consumptionEcwhich are expressed in the following equations:

and whereCis the capacitance of the charging system,Uis the voltage of the charging system,p(t) is the potential of the circuit, andi(t) is the current of the circuit.

Bubbles are formed when electrical energy is released.At this point, the internal conductivity of plasma is much larger than the conductivity of the surrounding water. As such, the Joule heat dissipation and the plasma resistance effect are negligible. Therefore, the energy of plasma is di-

Fig.1 Energy conservation during electrode discharge.

vided into two parts,i.e., the change in the internal en thalpy of the bubble and the energy of the radiant heat. Differential equations are expressed as follows (Huanget al.,2014):

where dU/dtis the internal energy change,PdV/dtis the potential energy change, and ΦTis the heat radiation power;therefore,p(t)i(t) can be measured experimentally (Huanget al., 2014).

3 Multibubble Motion Equation

3.1 Multibubble Motion Equation

The case of multiple bubbles with an equal size is considered (Fig.2) on the basis of double-bubble motion equation (Fujikawa and Takahira, 1986).

Fig.2 Schematic of multibubble motion coordinate (modified after Fujikawa and Takahira, 1986).

The expansion and contraction of a bubble wall in a liquid and the surrounding liquid satisfy the wave equation; according to the premise mentioned earlier, the bubble wall velocity potential Φ is expressed as follows (Fujikawa and Takahira, 1986):

whereliis the bubble and reference point distance,nis the number of bubbles,tis the time, andφi(li,t) is the wall velocity potential of the bubblei. Therefore, the multibubble motion satisfies the wave equation (Fujikawa and Takahira, 1986):

wherecis the acoustic wave velocity. Then, the boundary conditions are described as follows (Fujikawa and Takahira, 1986):

where ˜iis the first derivative of the bubbleiradius,piis the ambient pressure of the bubblei,picis the internal pressure of the bubblei,p0is the hydrostatic pressure,ρ0is the fluid density, andδis the fluid tension coefficient.

In discussing bubblei, the remaining adjacent bubble contribution values should be estimated as

Similarly, boundary conditions can be estimated (Fujikawa and Takahira, 1986):

where

and

is the difference between the inside and outside of the bubble wall, including vapor pressurepv, hydrostatic pressurep0, vapor pressure variable in the adiabatic processfluid tension pressurea nd fluid viscous resistance pressure

3.2 Simplified Condition

The simplified conditions in this study are multi-electrode linear alignment and equal-energy synchronous excitation. If all the bubbles have the same radius and its difference, the radius of theibubble and its first derivative should be equal,

and

then Eq. (7) can be simplified as

whereεijis summed as whereγ= 0.577216 is the Euler constant in a harmonic series, andε = R/Lis a distance-proportional parameter.Eq. (8) is a single-bubble motion equation in the equalenergy synchronous excitation of multibubbles in a linear arrangement.

4 Other Influencing Factors

4.1 Seismic Ghost Reflection

The sea level is a stable water–air interface. When the near-field waveletu(t) reaches this interface, a negative reflection waveu(t− ∆t) is generated (Fig.3). Therefore,the near-field wavelet and the reflected wave are superimposed with a time delay ∆tto form a far-field waveletw(t). Hence, the far-field wavelet can be expressed as

where the reflection of the interface is −1 ≤α＜ 0.

The filter factor of a ghost reflection is obtained after the Fourier transform of Eq. (9):

Then, its amplitude spectrum is

The ghost reflection has a frequency suppression effect on the wavelet signal, as described in Eq. (10). This phenomenon is called a notch.

Fig.3 Schematic of the relative positional relationship between excitation and reception in the observation system(modified after Zhu, 1985).

4.2 Propagation Distance

The propagation distance of an acoustic signal depends on the depth of a hydrophone (Fig.3). From the coordinates of the excitation point and the hydrophone, the acoustic pulse propagation distance is

According to the Rayleigh equation under incompressible conditions (Rayleigh, 1917), the pressure field from the bubblercan be expressed as follows (Rayleigh, 1917):

wherePris the pressure from the bubbler. Therefore, the pressure sound energy at that point can be described as

The propagation distance directly affects the pressure field of the acoustic signal, which influences the magnitude of the acoustic energy. The peak of the acoustic pulse that arrives first and the peak of the bubble pulse decrease as the propagation distance increases. With the increasing distance, the ratio of the peak of the first pulse to the peak of the bubble pulse is 0.279. However, when the propagation distance increases to 0.3 m, the peak drop is 96.7%(Fig.4). Therefore, the hydrophone receives the acoustic signals propagating at different distances, and the time of the first arrival of the received signal pulses varies and directly affects the bubble superposition.

4.3 Superposition Principle

In the analysis of bubblei, the superimposed pressure field of the remaining bubbles is regarded as the pressure acting on bubble 1 and having a polarity opposite to the first external pressure field. With this simplified method,the external environmental pressure field of bubbleiremains hydrostatic. Bubble 1 still exists in an independent oscillation manner. The maximum radius of the excitation bubble in the electrode water is usually less than one wavelength, so the bubble model is still considered spherically symmetric. Therefore, the pressure signal of the electrode combination can be calculated by superimposing the pressure wavefield of all individual bubbles after abstraction and the corresponding virtual reflection on the sea surface (Ziolkowski, 1970).

Fig.4 Variation in acoustic signal pulse amplitude with distance.

Given the center coordinate of theith bubble (xi,yi,zi)the distance between theithandkthbubbles is expressed as follows (Ziolkowski, 1970):

The ‘Abstract’ method can be used to infer that the pressure difference in thekthbubble at the bubble wall is described by calculating the corresponding pressure fieldpi(rik) and the ghost reflection pressure fieldpi(gik) (Ziolkowski, 1970):

wherePkis the independent vibration sound pulse pressure of thekthbubble, andp0is the ambient hydrostatic pressure; hence, the inner pressure of thekth bubble is(Ziolkowski, 1970):

Then, the superposition principle described in Eq. (21)can be applied to calculate the acoustic pulse wavelets received by the hydrophone at the (x,y,z) position in water:

The distance between theithbubble and the receiver is given in Eq. (22), and the distance between the ghost reflection of theithbubble and the receiver is shown in Eq.(23):

and

5 Establishment of the Simulation Flow

Plasma spark sources in water typically have dozens or even hundreds of electrodes arranged to form an effective excitation source. A plasma spark source can be observed as a combination of different linear arrangements regardless of the kind of arrangement. Therefore, the linear arrangement observed in this study forms the basis for simulating the spark source wavelet. The methodological flow of the synchronous excitation of multibubbles with equivalent energy in a linearly arranged spark wavelet simulation is described as follows:

1) The initial radius of the simulation calculation, the initial bubble wall velocity, the initial temperature, and the ambient pressure are selected on the basis of different excitation energies (Table 1).

2) The spacing of electrodes, the number of electrodes,the positional relationship of the electrode arrangement,the depth of electrodes, the positional parameters of the hydrophone, and the distance between the electrodes and the hydrophone are determined.

3) The multibubble equation of motion,i.e., Eq. (12), the ideal gas state equation, and the multi-electrode discharge energy conservation equation,i.e., Eq. (5), the positional relationship of the electrodes, and the pressure pulse function of the bubble wall corresponding to each electrode are solved in sequence.

4) The positive acoustic pulse function received by the hydrophone is calculated according to the incompressible condition of the Rayleigh equation shown in Eq. (16) and the distance of the electrodes and the hydrophone.

5) Similar to step 4, the virtual reflection,i.e., the negative acoustic pulse function received by the hydrophone,is calculated.

6) The far-field wavelet is obtained in accordance with the superposition principle. Combining the actual sampling interval and the filtering range, we ensure that the simulated far-field wavelets are consistent with the parameters of the actual signal.

6 Model Testing and Discussion

The observation data of the acoustic pulse generated by the three electrodes in the laboratory (Fig.5) and the hydrophone are placed atD= 130 mm from the middle of the three electrodes. The three-electrode excitation energy is 5 J and the intervalLis 5 mm. The sampling interval ∆Tis 1 µs, and the hydrophone filtering range is 0–8 kHz. Generally, the results indicate that the actual acoustic pulse wavelet and the indoor-simulated acoustic pulse wavelet(Fig.6) fit well in the initial and bubble pulses. The waveform of the actual acoustic pulse shows more oscillations because of the tightness and limitation of the experimental device space. The actual acoustic pulse waveform collected in the laboratory is affected by the environmental noise and multiple reflections of the container wall. Therefore, we consider only the pulse that arrives first and the bubble pulse portions.

Fig.5 Schematic of the experimental device (modified after Huang et al., 2014).

Fig.6 Comparison between analog and actual waveform signals from the three electrodes.

The pulse that arrives first and the bubble pulse portions are subjected to normalized power spectrum analysis (Fig.7). The results indicate that the normalized power spectrum of the initial sound pulse is similar in the range of 0–500 Hz. The frequency energy of the actual waveform in the range of 500–3000 Hz is smaller than the energy of the simulated waveform. The results of the power spectrum of the bubble pulse portion indicate that the analog sound pulse is confined around 400 Hz, whereas the actual waveform is around 500 Hz. Moreover, the actual acoustic pulse energy is slightly lower than the analog sound pulse over the entire range,i.e., 0–3000 Hz. The waveform in the experimental device is subjected to multiple reflections, the effects of ambient noise, and the depth of the electrodes from the water surface. As such, the inaccurate electrode depth measurement causes an offset by a notch point. The most significant difference is that the first break peak on the right side of the bubble pulse has a poor fit with the actual waveform. The waveform of the bubble pulse is affected not only by the multiple reflections of the first break arrival but also by the functional relationship between the statistical hypothesis factorεijand the number of more sensitive electrodes. Therefore,the accuracy of the simulation method is not satisfactory when few electrodes are present.

Fig.7 Spectral comparison between analog and actual signals.

The fishbone electrode of the SIG spark source has 100 electrodes on both sides with a spacing of about 2 cm, a total simulated energy of 5 kJ, and a spark source wavelet with a depth of 3 m. The spark source wavelet is compared with the actual wavelet. The reliability and accuracy of the method are verified by comparing the simulated and actual waveforms (Fig.8). The source depth of the actual observation data is not accurately measured, so the pulse width between the theoretical model and the actual wavelet is different. Similarly, the corresponding position of the first notch slightly varies. However, this difference does not affect the accuracy of the source wavelet model. The source depth parameter can be modified by calibrating the actual data. Additionally, the waveform may have been affected by environmental factors because of the lighter weight and smooth rotation during the discharge of the spark source.

Fig.8 Comparison between the simulation and actual signals of the wavelet and spectrum of the spark.

7 Conclusions

The multibubble equation of the motion model was derived using a combination of double-bubble oscillation theory in the free field and the actual excitation process.Environmental factors, ghost reflection, propagation distance, and acoustic wave superposition principle were analyzed. Then, the flow of the simulation calculation of the spark source wavelet was established by combining energy conservation and ideal gas equations.

The results reveal that the accuracy of the proposed model is related to the actual number of electrodes compared with that of the actual wavelet model. Generally,the fewer the electrodes, the lower the accuracy of the model. This finding is attributed to the statistical hypothesis factor introduced to eliminate the coupling term of the interaction of the multibubble motion equation. Therefore,the proposed multi-electrode spark source wavelet model exhibits high accuracy and credibility. It also provides a method for analyzing the characteristics of indoor-simulated spark source wavelets.

Acknowledgements

The indoor test data and technical support were provided by Drs. Y. F. Huang (Shenzhen Institutes of Advanced Technology) and L. C. Zhang (Zhejiang University). This study was supported by the Geological Survey of China(No. DD20191003), and the National Key Research and Development Plan (No. 2016YFC0303901).