Laboratory investigation of a passive acoustic method for measurement of underwater gas seep ebullition

Passive acoustic methods have been used in industrial applications to monitor gas production during chemical reactions and to measure sizes and spatial distributions of bubbles throughout processes such as fermentation.^1–3 Similar methods show promise in environmental monitoring applications, such as the measurement of gas flow from underwater seeps, where the low power requirements and accurate nature of passive acoustic bubble size measurement⁴ may be used to conduct long-term monitoring of gas release. Passive acoustic bubble sizing has been reported;^1–5 however, the authors are not aware of the use of acoustics to log or measure gas volume flow rates of continuous ebullition. In situ recordings have confirmed the presence of acoustically classifiable, naturally oscillating bubbles exiting methane seeps,⁵ but the associated gas flow rate was not measured. Flow rates from these seeps are of interest for their potential contribution to atmospheric methane levels, and knowledge of time-varying changes in flow rates may help identify mechanisms of seep activity.

Experiments described in this paper relate the acoustically detected natural frequencies of individual bubbles emitted into a tank to their volumes using the formula developed by Minnaert.⁶ Bubbles of radius r₀ ≈ 1 mm were investigated as they flowed from a model seep at rates of 0–10 bubble/s, corresponding to flow rates up to 200 ml/h. This range of flow rates is coincident with that reported for some natural seeps.⁷ Gas trap and optically derived volumes were also acquired contemporaneously, and the measurement uncertainty for each technique was analyzed. The reported acoustic technique shows promise as a potentially viable tool for long-term monitoring of gas seeps from which single streams of bubbles are emitted. Additional work necessary to fully validate this technique is also discussed.

A tabletop apparatus was constructed to generate bubbles and record their radiated acoustic signals. A 35 cm × 35 cm × 13 cm tank with 6.25 mm thick walls was filled with distilled water. Medical-grade breathing air was directed through copper tubing into a 10 cm long, 26 gauge, vertically oriented, stainless steel needle placed at the bottom of the tank. A small hydrophone (Brüel and Kjær 8103) was placed 26 mm from the needle tip, oriented approximately 30° off-vertical, pointed at the needle tip, as shown in Fig. 1. The hydrophone cable was ensheathed in a water-filled stainless steel tube to decouple it from the acoustic system. The hydrophone signal was conditioned with a charge amplifier then bandpass filtered (between 1.3 and 4.5 kHz) before capture by the data acquisition system. An example of a recorded acoustic signal and analysis is shown in Fig. 2.

To verify the acoustic measurements, a gas trap constructed of an inverted water-filled graduated cylinder was placed over the area of surfacing bubbles with its open end approximately 1 cm below the surface of the water. Gas flow was measured as the change of water level in the cylinder over the duration of acoustic recording. A third measurement of gas flow was obtained optically. A camera with a shutter speed of 1/1000 s was attached to a microscope placed along the broad side of the tank to record uncompressed video at a rate of 30 frame/s. An example still image recorded by the video capture system is shown in Fig. 3. A MATLAB script was developed to import video frames singly, fit an ellipse to the outline of each bubble, and calculate the volume of each bubble assuming spheroidal geometry. The needle tip, visible near the bottom of each frame, was used as a size reference. Optical, gas trap, and acoustic measurements of gas ebullition rates were obtained simultaneously for direct comparison.

The focus of this investigation was to exploit the gas bubbles’ natural frequencies of oscillation to measure underwater gas flow rates. In a manner similar to a plucked guitar string, bubbles in this experiment were excited into oscillation [seen as a damped sinusoid in Fig. 2(b)] as they broke free from the needle tip of the model seep. Acoustic signals generated by each bubble were analyzed in the frequency domain to determine the natural frequency f₀. Zero-padding of the fast Fourier transform (FFT) was implemented to achieve a resolution bandwidth of 6 Hz for recordings where ebullition rates were approximately 1 bubble/s. For tests in which the gas flow rate neared 10 bubble/s, the shorter time between bubbles resulted a resolution bandwidth of 12 Hz. Bubble volumes were calculated by Eq. (2) and summed to obtain a total gas flow measurement for each recording. Audio recordings of two gas flow rates captured by the hydrophone are provided in Mms. 1 and 2.

The tank used in the current experiment is the same tank described in Ref. 8, and the bubbles investigated in both experiments were of similar shape and size with radii between 0.8 and 1.5 mm. It was reported in Ref. 8 that effects of the tank walls on bubble resonance frequency were negligible and hence are considered negligible in this work as well. No corrections for the effects of the tank are necessary, and none were applied. The water level and temperature remained constant throughout the measurements.

The natural frequency of a bubble undergoing low-amplitude volume oscillations was first calculated by Minnaert in 1933.⁶ As shown by Leighton,⁹ the Minnaert frequency can be obtained through the analogue of a simple harmonic mechanical oscillator. The angular natural frequency ω₀ of a mass m on a spring of stiffness k is given by

In an analogous acoustic system, a gas bubble of equilibrium radius r₀ and polytropic exponent ν is described by stiffness k_a ≈ 12πνr₀P, where P is the average hydrostatic pressure on the bubble. Terms accounting for surface tension at the gas-liquid interface are absent due to the size of the bubbles investigated in the present experiment. For bubbles in clean water, surface tension may be neglected for r₀ > 0.1 mm. Note that bubbles from natural seeps are typically larger than 1 mm in radius⁷ but may have surfactants that alter the surface tension at the bubble wall, and hence surface tension effects may need to be included for certain natural seeps. The air bubbles used in this experiment were determined to have a polytropic exponent of ν = 1.18, calculated using Eq. (27) of Ref. 10. This value is also in agreement with values for air bubbles of similar size described in Ref. 11.

Assuming the system under investigation is in the long-wavelength regime such that r₀ ≪ λ, where λ is the acoustic wavelength, the appropriate mass term for use in Eq. (1) is given by the effective inertia of the liquid oscillating in the system, $ma≈4ρ1πr03$⁠, where ρ₁ is the density of the liquid surrounding the bubble. For a spherical bubble of stiffness k_a with radiation mass m_a, Eq. (1) relates the bubble’s natural frequency f₀ to its volume V as

Previous work employing Minnaert’s formula has most often relied upon an assumption that the bubbles of interest behave as spheres of ideal gas. However, where the compressibility factor Z of a gas deviates from unity by more than 1%, the gas must be treated as a real gas by modifying Eq. (2) to include Z. The real gas formulation of Minnaert’s model is obtained by letting the stiffness of the bubble be k_a,real ≈ 12πZνr₀P. The expression for bubble volume as a function of natural frequency is then

The compressibility factor of a gas at a given temperature and pressure may be obtained through an appropriate equation of state for the gas of interest. In Ref. 12, the van der Waals equation of state is shown to be adequate for describing the acoustic behavior of real gas bubbles in water. Reference 13 details a procedure for calculating the compressibility factor of a gas using the van der Waals equation of state. For methane gas at 10 °C, it is seen that Z falls below 0.99 at a marine depth of 33 m; thus, seeps located below this depth should be modeled by the form of Minnaert’s expression presented in Eq. (3).

Gas volumes measured using the inverted graduated cylinder, video analysis, and passive acoustic techniques are presented in Table I. All volume measurements are in agreement within the uncertainty associated with each measurement technique. Passive acoustic methods offer the lowest measurement uncertainty of the three techniques presented here. The optical system provided insight into the shapes and behaviors of the bubbles; however, the method of direct sizing using images carried with it several potential sources of error. Optical measurements were plagued by uncertainties including the potential for errors resulting from brightness thresholding as it affected the determination of bubble outlines. Pixelation errors may also be present, both in the measurement of the bubble radii and the measurement of the needle tip that was used as a size reference. Further, these errors were cubed upon conversion from radii to volumes. Gas volumes obtained using the graduated cylinder were intended for direct comparison to acoustic measurements; however, it was observed that these values suffered from ±0.4 ml uncertainty due to lingering unbroken bubbles that often obfuscated the reading of the meniscus.

Uncertainty in the acoustic measurements arises from the 6 to 12 Hz frequency resolutions of the FFTs. A second source of measurement uncertainty stems from the approximation implicit in Eq. (2) that the bubbles are spherical in shape and pulsate omnidirectionally. Image analysis shows elliptical bubble profiles with an average major-to-minor-axis ratio of 1.15. According to Strasberg, the spherical assumption here should result in no more than 0.6% volume measurement error.¹⁴ In total, acoustic measurement uncertainty is ±0.8% for the recordings of lower gas flow rates and ±1.1% for the higher gas flow rate recordings. A detailed uncertainty analysis is given in Ref. 15.

Passive acoustic measurement techniques to determine the volume flow rate of underwater gas ebullition may benefit industrial and environmental monitoring applications. The measurement technique presented in this paper is well-suited for inclusion in deployable remote sensing systems as it is capable of precise measurements of bubble volumes for single-bubble-stream seeps, and power requirements for passive acoustic recording instrumentation are low. Results of these laboratory experiments in which 0–10 bubble/s were emitted from a single-orifice model seep indicate a level of precision that is unmatched by other common measurement techniques. Though the range of volumetric flow rates investigated here is limited, in situ measurements of seep flow are challenging using any measurement technique. For example, gas-trap methods do not allow for long-term monitoring and are further limited by the rapid dissolution of methane bubbles as they rise through the water column.

A number of issues relating to the use of the acoustic measurement technique in nature still warrant investigation. Aspects of the acoustic signal radiated by a new bubble are dependent on specifics of the orifice. For example, the amplitude of the radiated acoustic signal depends on the orientation of the orifice, and bubbles generated by near-horizontal channels may emit very little sound and hence might not get counted. Although not observed for the flow rates reported here, bubble fragmentation and coalescence can occur at higher flow rates, which might also lead to miscounting. The range of flow rates for which this technique is useful has not been completely explored. Given the duration of the typical individual bubble signature, it is possible that flow rates two to three times higher than that reported here could be measured. It is also possible that the technique could be used for seeps emitting multiple streams of bubbles of sufficiently low individual flow rates. The experiments described in this paper were performed in distilled water; however, Eq. (3) is generalized to account for seawater through use of the appropriate density term. Surfactants found in natural waters can alter the natural frequency of a bubble the radius of which is less than 0.1 mm; however, seep bubble radii reported in the literature⁷ typically exceed ten times this size. Hence, effects of surface tension have been neglected in this work but could be accounted for if warranted. Assuming these issues can be resolved, the reported technique can potentially be used to make long-term acoustic remote sensing measurements of gas flow rates for a class of underwater seeps. In their current state, the methods reported here are suitable for employment in long-term monitoring of relative changes in seep flow rates.

Finally, despite a longstanding convention of assuming ideal gas behavior when employing the Minnaert formula, it is suggested that the real-gas formulation presented in this paper offers a more appropriate model of the acoustic behavior of bubbles the compressibility factor of which deviates by more than 1% from unity. Specifically, for applications of this technique to measure underwater methane gas flux, the real gas formulation is recommended for seeps located at depths exceeding 30–40 m.