Prospects of Improving the Vibroacoustic Method for Locating Buried Non-Metallic Pipelines

Abstract

Acoustic methods are a promising direction when determining the position of buried non-metallic pipelines. Under difficult soil conditions, one of the most effective methods is the vibroacoustic method, which has a maximum range of application when acoustic waves propagate through the transported medium. However, due to limited energy input into the pipeline, signal detection at significant distances from the source becomes challenging. This article considers the mechanism of acoustic oscillations attenuation in pipes and suggests possible directions for optimization of the investigated technology. The evaluation of mathematical modeling methods for the investigated process is conducted, and the key signal attenuation relationships are presented. The analysis allowed us to establish that the vibroacoustic method has the potential of increasing the efficiency by approximately 10–20%.

1. Introduction

With the widespread adoption of non-metallic pipelines as part of urban infrastructure, there has been a growing demand for reliable means of locating and detecting these pipelines. While public communications are generally well documented and mapped, private communications often lack comprehensive records, resulting in incomplete or absent mapping. This issue is prevalent worldwide, with countries such as the USA, the UK, China, and Russia being among those most affected due to the extensive length of their buried pipeline networks. At present, approximately 70% of non-metallic gas pipelines in the Russian Federation do not have the integrated means for detection or location. This situation represents a significant challenge for ensuring the safe operation and maintenance of these assets. The absence of effective location methods leads to risks of emergency situations, particularly in the context of intensive construction of new infrastructure facilities. Additionally, the problem of illegal taps into gas pipelines using non-metallic materials is also widespread. One potential solution to these challenges is to increase investments in R&D focused on developing new technologies for locating underground utilities [1]. Among the most effective methods for locating such assets, ground-penetrating radar and acoustic techniques are commonly highlighted. Acoustic methods are of particular interest to operational organizations due to their versatility and ability to provide high accuracy combined with straightforward implementation. However, despite its advantages, the vibroacoustic method has a significant drawback, namely the rapid attenuation of the signal as the distance from the oscillation source increases [2,3,4]. The further development and application of this method largely depends on addressing its key issues and defining the scope of its most effective use.

2. Acoustic Methods for Locating Buried Non-Metallic Pipelines

The operating principle of acoustic methods for locating buried pipelines involves controlled excitation of the pipeline or its segment, generating acoustic waves that propagate into the surrounding medium and are detected by specialized equipment [5]. These methods are typically categorized based on whether direct access to the pipeline is required. Among the methods requiring direct access to the pipeline is the vibroacoustic location method, commonly referred to as the pipeline excitation method, along with its various modifications [6]. In the vibroacoustic method, acoustic waves are generated in the pipeline body or in the transported medium using a device connected to the pipeline or placed inside it. The characteristics of the generated waves can be widely adjusted, allowing for signal optimization based on the specific measurement conditions [7]. The choice of the transmitted signal largely depends on the user’s objectives and the characteristics of the pipeline under investigation. The transmitted signal is recorded on the soil surface using geophones or accelerometers. The measuring device is capable of operating both as an individual unit, necessitating multiple measurement points for pipeline route determination, and as part of a synchronized array, which greatly minimizes the required number of measurements for accurate pipeline location [8]. Pipeline location can be performed either by determining the maximum amplitude of the recorded acoustic signal or by determining the wave travel time and utilizing phase information. In cases where there are minor differences in signal amplitudes, the use of phase information from the recorded wave enables a more accurate determination of the pipeline route location [9]. Recent advancements in vibroacoustic methods have primarily focused on three-dimensional location of buried pipelines and improved signal processing techniques. Li et al. proposed a new signal denoising method based on wavelet transforms, combined with cross-correlation analysis, to estimate time delays between sensors [10]. This method enables an accurate estimation of pipeline depth and location; however, its effectiveness is influenced by the placement of both sensors and the signal exciter.

The limitations of this method are primarily associated with the natural attenuation of acoustic oscillations and external sources of vibrations [11]. To ensure optimal sound propagation, the oscillation source generates signals in the frequency range of 10 to 500 Hz. According to researchers, one of the most commonly used materials for buried non-metallic pipelines—low-density polyethylene (LDPE)—is a poor medium for sound propagation. Therefore, to enhance signal propagation distance, it is more effective to excite oscillations in the transported medium rather than in the pipe material itself [12,13].

Methods for locating buried non-metallic pipelines without the need for direct access to the pipe are divided into seismic methods and point vibration measurements. Seismic methods are based on the same theoretical principles that are applied in seismic exploration of oil and gas fields [14,15]. An acoustic signal, excited by a special device on the surface, propagates into the ground, where it is scattered and reflected by objects with mechanical properties differing from those of the soil. The resulting composite signal is recorded by an array of geophones on the surface, followed by mathematical processing to determine the spatial location of underground inhomogeneities. Shear and compression waves differ primarily in their propagation velocity and corresponding wavelength at a given frequency, which in turn influences the achievable spatial resolution [6]. A shorter wavelength corresponds to higher detail, and vice versa. Among seismic methods, the time-domain superposition method and the frequency-domain cross-correlation method are distinguished. In the time-domain superposition method, information about the arrival times of reflected waves is used to construct an image of the subsurface, with the signal source being a short-duration impulse. The frequency-domain cross-correlation method employs a linear frequency modulation signal, while the essence of image formation remains identical to that of the time-domain superposition method [6]. One of the recent advancements in seismic methods involves the use of time delay estimation for wave reflection analysis [16]. This estimation is achieved by analyzing waveform differences across frequency bands. This approach enables researchers to determine the pipeline position by intersecting the possible location regions obtained from different geophones. Additionally, a recent technique improves the frequency-domain cross-correlation method by applying superimposed imaging based on cross-correlation coefficients between signals from all geophones and multiple sources [17]. Compared to the conventional “single-point transmit, multi-point receive” approach, this method demonstrates a notable improvement in location accuracy.

The advantages of seismic methods include a strong theoretical foundation and the ability to determine the location in three-dimensional space. Among their disadvantages, the complexity of data interpretation and the lack of differentiation of detected inhomogeneities are often highlighted.

For determining the location of shallowly buried pipelines, some researchers propose the use of Rayleigh waves (surface waves), which, theoretically, allow for the location of an object buried at a depth of no more than one wavelength. This limitation arises from the exponential nature of Rayleigh wave attenuation with increasing distance from the surface. Despite the lack of extensive research dedicated to the application of surface waves for detecting buried non-metallic pipelines, this method holds significant potential, as it is already employed for identifying underground inhomogeneities and cavities [5].

Point vibration measurements involve low-frequency harmonic excitation of the soil surface, causing the soil to behave similarly to a “mass-spring-damper” model. According to this model, the soil, at any point within the investigated area, exhibits its own resonance characteristics, which depend on the elastic properties of the soil and the excitation radius [5]. The presence of any inhomogeneities in the underground medium affects the soil’s resonance characteristics, which enables their detection. The advantage of this method lies in the simplicity of its use and data processing; however, it has numerous limitations that significantly narrow its scope of application. Point vibration measurements are not yet mature, and their application is most feasible as an auxiliary tool for locating large-diameter pipelines at shallow depths (no more than half a meter) [18]. A schematic illustration of the operating principles of the listed methods is presented in Figure 1.

The classification of acoustic methods for locating buried non-metallic pipelines, developed based on a literature review, is shown in Figure 2. Examples and a brief description of commercial devices used in these methods are provided in

Table 1. Examples of commercial devices utilizing the investigated technologies (Compiled by the authors).

Technology	Name of Commercial Device	Key Characteristics of the Device	Device Limitations
Excitation of oscillations in the transported gas	Gas Tracker 2, MADE S.A., La Farlède, France	The location accuracy is up to 15 cm from the pipeline route. Operating frequencies range from 400 to 500 Hz. The generator produces a sound pressure level of 110 dB. The oscillation sensor features various gain- and noise-filtering settings. The supplied software and hardware enable the device to be used “out-of-the-box”.	Requires connection to the pipeline system. Operation is possible without interrupting gas supply, but with reduced efficiency. Optimal performance is achieved within a pressure range of 0.021 to 4 bar. The effective location range is up to 200 m from the connection point. The device is not suitable for pipelines enclosed in casings and is susceptible to noise interference from road traffic and railway lines. The measuring device displays only the sought-after pipeline. Turns and tees significantly reduce the operational range of the device. The accuracy and location range of the pipeline are reduced in loose and heterogeneous soils.
Excitation of oscillations in the pipe body	The UM-112M impact mechanism as part of the Uspekh TPT-522N system, TECHNO-AC, Kolomna, Russian Federation	The location accuracy is within 20 cm of the pipeline route. The impact mechanism excites acoustic oscillations of the pipeline body at an adjustable frequency of 0.5, 1, or 2 strikes per second. The impact force is regulated by a voltage of 12/24 V. The acoustic sensor measures the received signal in the range of 0.09 to 2.2 kHz. The supplied software and hardware enable the device to be used “out-of-the-box”.	The maximum depth of the pipeline for location is up to 3 m. The method requires a connection to the surface of the pipe and is not applicable to pipes with a diameter of less than 50 mm. The effective location range extends up to 100 m from the connection point. The accuracy and location range of the pipeline are reduced in loose and heterogeneous soils. The measuring device displays only the sought pipeline.
Seismic method	Ultra-Trac APL, SENSIT Technologies, Valparaiso, IN, USA	The location accuracy is no less than 45 cm of the pipeline route. The device operates in two modes with frequencies of 500 and 900 Hz, depending on the estimated location depth of the target object and soil characteristics (such as the velocity of wave propagation within it). The supplied software and hardware enable the device to be used “out-of-the-box”.	The depth of the pipeline for location ranges from 30 cm to 250 cm, depending on the diameter. The device’s location efficiency decreases under conditions of heterogeneous and loose soils. Differentiation of the detected utilities is performed manually. The location range along the pipeline route is unlimited.
Point vibration measurements	Electromagnetic shaker system F4/Z820WA, Wilcoxon Sensing Technologies, Frederick, MD, USA	The location accuracy is within 20 cm of the pipeline route [18]. The device allows for determining the resonant frequency and amplitude of oscillations of soil at a specific point. The device requires proprietary software for data processing and interpretation but is supplied without it.	The maximum depth of the pipeline for location is up to 30 cm. The device requires a large number of point measurements and loses efficiency under heterogeneous soil conditions. Differentiation of the detected utilities is performed manually. The location range along the pipeline route is unlimited.

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3. Challenges and Features of the Vibroacoustic Method for Locating Buried Non-Metallic Pipelines

The vibroacoustic method is based on the transmission of oscillation energy, which attenuates as it propagates along the pipeline. According to the law of conservation of energy, the amount of transmitted energy is limited and is gradually expended as it overcomes various types of resistance during propagation. Thus, the maximum signal transmission range and its amplitude at the ground surface are strictly constrained, depending on the properties of the investigated system and the characteristics of the transmitted signal. Due to the complexity and high cost of altering the physical properties of the investigated system, the primary approach to improving this technology involves optimizing the transmitted signal and the methods for its acquisition and processing. Various researchers have identified three types of waves that most effectively transmit energy and propagate in buried pipelines [19,20,21,22]:

• Axisymmetric wave with predominant propagation in the transported medium and some radial movement associated with the compliance of the pipe and soil (n = 0; s = 1);
• Axisymmetric wave with predominant propagation in the pipe body and some accompanying radial wall movement, influenced by Poisson’s ratio and the bulk modulus of the transported medium (n = 0; s = 2);
• Axisymmetric torsional wave, practically not accompanied by radial movement of the pipe wall (n = 0; s = 0).

In accordance with the rules for designating oscillation modes, the mode n = 0 in cylindrical coordinates indicates that the system oscillates symmetrically relative to the axis, and its oscillations in the transverse cross-section depend solely on the radial position and are independent of the azimuthal angle [21,23]. In other words, n is the number that denotes how many full oscillation periods are observed when analyzing the transverse cross-section in the azimuthal direction. The number s was introduced to differentiate between various wave types. In earlier works, the waves n = 0; s = 1 and n = 0; s = 2 were referred to as n = 0; m = 0 and n = 0; m = 1, respectively [24,25]. According to the rules for designating modes in shells, m represents the order of the axial mode, and n represents the order of the circumferential mode [23,26,27]. The oscillations n = 0; s = 0 are torsional and clearly fall outside the nomenclature of cylindrical shell oscillations (n, m). If we adopt the notation used by Silk and Bainton, the oscillations n = 0; s = 0 would be denoted as T(0, m) (m = 1, 2, 3, 4…), where m is the number of oscillation nodes in the radial direction of the transverse cross-section [28,29]. A visualization of the described oscillation modes is presented in Figure 3.

The axisymmetric torsional wave is not applicable to the problem at hand due to its limited detection range [30]. The axisymmetric wave s = 1 propagates with less attenuation compared to the s = 2 wave. However, the axisymmetric s = 1 wave induces smaller oscillations in the surrounding soil, resulting in a lower amplitude of the measurable signal on the surface [14]. According to the mathematical model developed by Ying Liu et al., the axial propagation distance of the n = 0; s = 1 wave increases under conditions of weak coupling between the transported medium and the pipeline–soil system (e.g., high acoustic impedance mismatch) and at low excitation frequencies, as less oscillation energy is transmitted into the surrounding soil [19].

Waves n = 0; s = 2 (with predominant propagation in the pipe body) travel at a significantly higher speed compared to waves n = 0; s = 1 (with predominant propagation in the transported medium). According to Snell’s law, for a modal wave with a phase velocity exceeding the bulk wave velocity in the soil during propagation in the pipe, energy is transferred to the surrounding soil as an incident bulk wave at a characteristic angle [19]. The propagation speed of the longitudinal wave n = 0; s = 2 in LDPE is over 2000 m/s, whereas the propagation speed of the wave n = 0; s = 1 in methane is 446 m/s and its propagation speed of oscillations in soils rarely exceeds 2000 m/s [31]. Due to this, waves n = 0; s = 1 are capable of having a phase velocity lower than the bulk wave velocity of the surrounding medium, while for waves n = 0; s = 2, such situation is an exception, occurring only in particularly dense dry soils.

If the phase velocity of the wave is lower than the bulk wave velocity of the surrounding medium, the characteristic energy transfer angle to the surrounding medium becomes imaginary, and the energy is confined within the pipe. As a result, soil displacement rapidly attenuates in the radial direction [32]. For this reason, the wave mode n = 0; s = 2 exhibits significantly reduced dependence on soil properties and can generate ground oscillation amplitudes that are orders of magnitude higher than those of the n = 0; s = 1 mode [19]. Additionally, soil displacement amplitudes for PVC pipes are found to be greater than those for cast iron pipes. This is attributed to the fact that PVC pipes are easier to excite due to their lower mass and, consequently, lower inertia.

Analysis of the properties of n = 0; s = 1 and n = 0; s = 2 waves leads to the conclusion that the location of buried non-metallic pipelines represents a complex optimization problem, where the selection of optimal signal parameters directly depends on the characteristics of the pipeline system under study and the objectives set for the researcher. Within the scope of this publication, we will address the problem of determining the pipeline route at the maximum possible distance from the signal source using n = 0; s = 1 waves.

4. Possible Approaches to Optimizing the Vibroacoustic Method to Increase the Operational Range for Locating Buried Non-Metallic Pipelines

To solve the optimization problem, it is first necessary to identify all parameters that have a significant impact on the resulting outcome and assess the possibility of controlling these parameters. The effective range of the vibroacoustic method for locating buried non-metallic pipelines is influenced by the following: the characteristics and power of the emitted signal, properties of the transported medium, properties of the pipeline material, pipeline dimensions and shape, soil properties and homogeneity, configuration and characteristics of the measuring devices, and data processing methods. The signal, as well as data acquisition and processing processes, are under the full control of the researcher. The properties of the transported medium and soil can be partially modified within the framework of the problem, but the pipeline itself and the surrounding soil must remain unchanged. Hence, soil is the most uncontrollable factor, strongly affecting the propagation and damping characteristics of vibrational waves.

The mechanical properties of soils are strongly influenced by their granulometric composition [33,34,35]. When acoustic waves pass through the soil, they cause particles to move within a certain range, leading to a change in their initial positions. A high amplitude compensates for the influence of large inter-particle gaps. Conversely, smaller gaps can reduce the dependence of acoustic wave propagation on signal amplitude [36]. It is important to note that although soil compaction (reduction in inter-particle gaps) mitigates the effect of low-amplitude oscillations transmitted by n = 0; s = 1 waves, it also increases the propagation velocity of oscillations, which impairs the energy transfer of n = 0; s = 1 waves into the surrounding medium. This effect is also relevant in the context of changes in the attenuation coefficient, which, according to various studies, increases with compaction and higher water content for all soils except extremely sandy ones [37,38]. Additionally, an increase in soil water saturation allows for a significant reduction in the propagation velocity of shear waves, which positively impacts the energy transfer of oscillations.

Thus, it can be concluded that manipulating soil properties by altering their degree of compaction and water saturation can increase the operational range of the vibroacoustic method for locating buried non-metallic pipelines. Recent rainfall and soil compaction along the pipeline route are expected to reduce the effective range of the studied method; therefore, it is most advisable to perform pipeline route location under dry weather conditions.

It should be noted that any modification of soil properties is a rather resource-intensive process and may be inapplicable under the conditions where most pipelines are located.

In some cases, it may be feasible and acceptable to modify the properties of the transported medium. In the presence of impurities, the attenuation coefficient of gas at low frequencies tends to increase significantly due to acoustic relaxation processes [39]. Under specific conditions, pipeline location may be more effective when performed with high-purity gas, which exhibits better acoustic properties than industrial natural gas [40]. Theoretically, gases with higher molecular weight, offering lower sound velocity and increased molecular kinetic energy, may also extend the acoustic wave propagation range. Nevertheless, in most cases, replacing the transported medium is unacceptable as it disrupts the technological process of gas supply to consumers and requires additional costly measures. Therefore, it is necessary to consider options for modifying the properties of the transported medium, such as its pressure and temperature.

According to theory, the speed of sound in a gas increases with temperature, but the acoustic impedance does not show such a straightforward dependence [41]. Depending on various conditions, the same change in temperature can have completely opposite effects on the attenuation of acoustic oscillations in a gas. The characteristic specific acoustic impedance can be expressed as the product of density and the speed of sound [42]. As temperature increases, the speed of sound increases linearly, while density decreases according to a law that is nearly quadratic. Thus, as temperature rises, the characteristic specific acoustic impedance of the gas decreases. However, the experimental results indicate an increase in viscothermal losses in channels as temperature rises [43]. Therefore, the effect of temperature on the efficiency of sound wave transmission through a gaseous medium under different conditions requires separate investigation and represents one of the potential reserves for optimizing the studied technology. The influence of humidity and pressure on the attenuation of acoustic oscillations in methane has been scarcely studied, making it another area of particular interest as a potential optimization reserve.

Among all the components of the system, the creation of a signal, as well as data acquisition and processing, are under the greatest control of the researcher. It is crucial to introduce a signal into the pipeline that can propagate as far as possible while maintaining the ability to be reliably detected on the surface. Increasing the amplitude of the signal is only feasible up to a certain threshold; therefore, it is necessary to adapt the frequency of the emitted oscillations to the parameters of the pipeline. One method for determining the optimal source frequency is the identification of the resonant frequencies of the pipeline–soil system. These frequencies can be determined by injecting white noise into the pipeline and detecting it on the surface. The frequency with the highest amplitude will correspond to the most efficient energy transfer from the pipeline to the surface. This is particularly important given that the characteristic specific acoustic impedance of methane is approximately 300 Pa·s/m, whereas the characteristic specific acoustic impedance for longitudinal and transverse waves in soils exceeds that of methane by thousands of times [42]. Acoustic impedance characterizes the resistance that a medium exerts on an oscillating surface. As previously noted, the acoustic coupling between the transported gas and the pipeline–soil system is very weak, resulting in the retention of acoustic energy within the pipeline and minimal transmission to the surface, potentially leading to undetectable signal levels [19]. A potential solution to this issue is the use of resonant frequencies of the pipeline–soil system.

The main objective of signal acquisition is to ensure the maximum sensitivity of the measuring device. The more sensitive the measuring device (e.g., a geophone or an accelerometer), the farther the signal can travel along the pipeline route before becoming undetectable. In this context, the use of cascades of measuring devices is permissible, as they can enhance the accuracy of pipeline detection by providing a larger amount of data [44]. During signal processing, the primary optimization reserve for the studied technology lies in the algorithms for filtering the useful signal and estimating the three-dimensional location of the pipeline using triangulation based on the phase (time) information of the measured oscillations.

Based on this analysis, a list of the most promising parameters for optimizing the studied technology was created, as presented in

Table 2. Optimization reserves for the studied technology (Compiled by the authors).

Potential Optimization Reserve	Description
Variation in gas temperature, humidity, and pressure	Adjusting these parameters changes the physical characteristics of the sound propagation medium without significantly deviating from the normal operational limits of the gas supply process, making it potentially acceptable for most gas distribution companies. However, the impact of these parameters on the efficiency of buried gas pipeline location is not yet sufficiently understood and requires further investigation.
Increase in the amplitude of the transmitted signal and optimization of its characteristics	The frequency of the signal introduced into the pipe directly affects the ratio between the energy transmitted by the gas to the surface and the energy propagated further through the gas inside the pipe. Higher frequencies result in larger soil oscillation amplitudes, which reduce the sound propagation distance along the pipeline route [19]. Conversely, lower frequencies exhibit the opposite effect. Therefore, during pipeline location, when the signal is lost, it may be advisable to perform repeated measurements at both higher and lower frequencies, as this could allow the signal to be re-detected due to changes in the balance between surface signal strength and its propagation distance. Special attention should also be paid to the resonant frequency of the pipeline–soil system, which enables a more efficient transmission of oscillations across the phase boundary with poor acoustic coupling.
Optimization of measuring devices, signal filtering, and data processing	The use of a greater number of highly sensitive devices and more advanced data processing algorithms will enable signal filtering and processing at lower signal-to-noise ratios.

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5. Mechanisms of Acoustic Energy Attenuation in Pipes and Its Transmission into the Soil

Acoustic wave propagation from the gas medium to the surface involves processes that differ in both spatial scale and physical principles. To fully account for the interaction of these physical processes with each other, it is necessary to understand their nature and the characteristics of energy transfer.

The primary mechanisms of sound attenuation in a gas pipe in the absence of flow are viscous and thermal losses [45]. During wave propagation in a pipe, the particle velocity at the stationary boundary of the medium will be zero, while particles located further from the boundary will have velocities that increase with distance. The resulting velocity difference creates shear stress, which is counteracted by viscous forces dissipating part of the acoustic wave energy [46]. As the distance from the boundary increases, the velocity difference decreases to negligible values. The region of significant velocity differences is commonly referred to as the boundary layer.

Thermal losses in the boundary zone occur in a similar manner. Acoustic oscillations, through pressure fluctuations, induce temperature oscillations, which initiate heat transfer processes in directions normal to the medium boundary, i.e., heat exchange with the wall. Since the boundary has a high heat capacity, the amplitude of temperature oscillations is zero at the boundary and increases with the distance away from it [47]. Heat exchange is most pronounced at the boundary interface, while its influence diminishes rapidly with distance, thereby characterizing the spatial limits of the boundary zone.

For polyatomic gases, a significant contribution to the attenuation of acoustic energy is also made by bulk viscosity (relaxation processes). The origin of bulk viscosity in continuous media is related to the fact that pressure in a non-equilibrium state (i.e., mechanical pressure) does not coincide with pressure in equilibrium (i.e., thermodynamic pressure) [48]. The magnitude of bulk viscosity depends on temperature and pressure: it typically decreases with increasing temperature and increases with increasing pressure. The bulk viscosity of monatomic gases is in the order of 10⁻¹⁰ Pa·s at atmospheric pressure, which is approximately five orders of magnitude lower than the dynamic viscosity of polyatomic gases [49]. For this reason, bulk viscosity can be neglected in attenuation calculations for monatomic gases, but for polyatomic gases, its value is generally comparable to dynamic viscosity and must be taken into account [50]. In the case of methane, the bulk viscosity is comparable to or even exceeds the dynamic [51].

Bulk viscosity is one of the physical qualities characterizing relaxation processes. The essence of acoustic relaxation lies in the conversion of translational motion energy in an acoustic wave into energy associated with the vibrational and rotational degrees of freedom of molecules during molecular collisions. Upon excitation, molecules tend toward local equilibrium (the equipartition of energy effect), and processes restoring equilibrium, known as relaxation processes, arise in the medium. These processes are accompanied by the irreversible transition of mechanical energy into heat. Attenuation caused by relaxation processes is the main factor influencing acoustic attenuation at low frequencies [39]. The time required for the energy state of molecules to reach equilibrium is referred to as the relaxation time. The rotational relaxation time is typically very short, and equilibrium can be achieved within a few molecular collisions.

If the relaxation time is significantly longer or significantly shorter than the pressure variation time in a sound wave, the influence of energy transformation from translational to vibrational and rotational molecular motion on sound propagation is negligible. This behavior arises from the medium remaining effectively stationary with respect to the sound wave, which can occur either due to a slow relaxation process or as a result of the establishment of an “equilibrium” between internal states in the case of rapid relaxation [52]. However, if the relaxation time is comparable to the transit time of the acoustic wave, the energy of translational motion is converted into vibrational or rotational energy and then back into translational energy, simultaneously with the arrival of the low-pressure region. In this case, the wave amplitude attenuates because the acoustic energy is transformed either into random molecular motion (heat) or into pressure that is out of phase. The relaxation frequencies and the extent of amplitude attenuation depend on the type of molecules, the oscillation modes, and the presence of other molecules. If the gas contains several different types of molecules, each will have its own relaxation times, including separate relaxation times for collisions with molecules of another type. This leads to the appearance of multiple frequencies at which increased sound attenuation is observed, as well as an overall increase in the attenuation coefficient [39]. The relaxation frequency of methane is above 100 kHz [53]. A schematic representation of acoustic energy losses in a pipe is shown in Figure 4.

When an acoustic wave propagating in a gas encounters the wall of a pipeline, refraction and reflection effects occur. Additionally, a phenomenon known as mode conversion is observed. This arises when a wave encounters an interface between materials with different impedances, and the angle of incidence is not normal to the boundary. If a longitudinal wave from a continuous medium (liquid or gas) strikes a solid, it refracts and reflects depending on the angle of incidence. However, if part of the energy induces particle motion in the transverse direction, a secondary shear (transverse) wave is generated, which can also refract and reflect as it propagates. In such cases, Snell’s law of refraction can be formulated as the following equation [54]:

$${\displaystyle \frac{\sin {\theta }_1}{v_{L1}}}={\displaystyle \frac{\sin {\theta }_2}{v_{L2}}}={\displaystyle \frac{\sin {\theta }_3}{v_{S1}}}={\displaystyle \frac{\sin {\theta }_4}{v_{S2}}},$$

(1)

where θ_i is the angle of deviation of the i-th wave from the normal to the surface, v_Li is the velocity of longitudinal wave propagation in medium i, v_Si is the velocity of shear wave propagation in medium i.

Thus, the longitudinal oscillations of the gas in the transported medium are capable of inducing transverse oscillations in the pipeline and soil. A visualization of this process is presented in Figure 5. The acoustic properties of the investigated media are provided in

Table 3. Acoustic properties of the media (Compiled by the authors).

Medium	Density, kg/m3	Velocity of Elastic Wave Propagation, m/s	Characteristic Specific Acoustic Impedance (ρ ∙v), Pa·s/m	Source
Methane	0.657 (at 25 °C and 1 atm)	446 (at 25 °C and 1 atm)	293 (at 25 °C and 1 atm)	[55,56]
HDPE	941–965	Longitudinal 2401–2493 Transverse 982–1024	Longitudinal 2.26·106–2.41·106 Transverse 0.92·106–0.99·106	[57,58]
LDPE	910–930	Longitudinal 2073–2142 Transverse 657–716	Longitudinal 1.89·106–1.99·106 Transverse 0.6·106–0.66·106	[57,58]
Dry sand	1500–1750	Longitudinal 350–600 Transverse 200–400	Longitudinal 0.53·106–1.05·106 Transverse 0.30·106–0.70·106	[59,60]
Saturated sand	1900–2200	Longitudinal 1750–2000 Transverse 175–250	Longitudinal 3.33·106–4.40·106 Transverse 0.33·106–0.55·106	[59,60]
Loose backfill soils	1400–1700	Longitudinal 100–300 Transverse 70–150	Longitudinal 0.14·106–0.51·106 Transverse 0.10·106–0.26·106	[61]
Loams	1600–2100	Longitudinal 300–1400 Transverse 140–700	Longitudinal 0.48·106–2.94·106 Transverse 0.21·106–1.47·106	[61]
Clay soils, moist, plastic	1700–2200	Longitudinal 500–2800 Transverse 130–1200	Longitudinal 0.85·106–6.16·106 Transverse 0.22·106–2.64·106	[61]
Clay soils, dense, semi-hard, and hard	1900–2600	Longitudinal 2000–3500 Transverse 1100–2000	Longitudinal 3.80·106–9.10·106 Transverse 2.09·106–5.20·106	[61]
Concrete	2250–2400	Longitudinal 3000–4600 Transverse 1750–2600	Longitudinal 6.75·106–11.04·106 Transverse 3.93·106–6.24·106	[62,63]

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6. Models Describing the Investigated Physical Process

To describe the propagation of acoustic oscillations in a pipeline, the approach proposed by G. Kirchhoff is often employed, which simultaneously accounts for the viscosity and thermal conductivity of the sound propagation medium. Equations (2)–(5)—used to model acoustic wave propagation with consideration of the aforementioned losses—include the linearized Navier–Stokes equation, continuity equation, ideal gas equation of state, and energy equation [64].

$${\rho }_0\partial \overline{v}/\partial t=-\overline{\nabla }\overline{p}+\left({\displaystyle \frac{4}{3}}\mu +\eta \right)\overline{\nabla }\left(\overline{\nabla }·\overline{v}\right)-\mu \overline{\nabla }·\left(\overline{\nabla }·\overline{v}\right)$$

(2)

$${\rho }_0\left(\overline{\nabla }·\overline{v}\right)+\partial \overline{\rho }/\partial t=0,$$

(3)

$$p=\rho R_0\overline{T}$$

(4)

$${\rho }_0{C}_p\partial \overline{T}/\partial t=\lambda \overline{∆}\overline{T}+\partial \overline{\rho }/\partial t$$

(5)

where ρ₀ is the background (stationary) density, v is the velocity, t is the time, p is the pressure, μ is the dynamic viscosity, η is the bulk viscosity, ρ is the density, R₀ is the gas constant, T is the temperature, C_p is the heat capacity at constant pressure, and λ is the thermal conductivity. An overlined variable denotes a vector field.

Based on this approach, researchers distinguish several types of sound energy loss processes in pipes [65,66]:

• Narrow pipelines (viscothermal losses are significant across the entire cross-section);
• Wide pipelines (viscothermal losses are significant only in the layer adjacent to the pipe wall).

The differences in the physics behind the underlying processes necessitate a corresponding approach to modeling, which is reflected in the variations in mathematical models for different types of pipelines. It is worth noting that G. Kirchhoff’s approach assumes several simplifications, particularly the assumption of medium homogeneity, which imposes a limitation on the frequency of the investigated sound wave. The frequency must be such that the dimensions of the medium under study and the wavelength are significantly larger than the mean free path of the molecules [64]. The boundary conditions of the model include the velocity at the pipe wall (no-slip condition), heat exchange with the surrounding environment (isothermal or adiabatic walls), and pressure at the ends of the pipe. The original approach can be modified by introducing additional assumptions or accounting for supplementary factors, leading to the development of models that are most effective under specific conditions. Particular attention should be paid to the model known as the Low Reduced Frequency Model. This model is based on the assumption that the wavelength is significantly larger than both the pipe wall thickness and its cross-sectional dimensions. Due to this assumption, pressure can be considered a function of distance, as it takes a constant value across the cross-section. Dimensionless parameters can be applied to assess the compliance of the studied system with the optimal conditions for calculations using the Low Reduced Frequency Model or any other model [64]. The use of this model has repeatedly demonstrated its effectiveness, and therefore, it can be considered the most promising method for calculating acoustic energy losses within a buried pipeline during its vibroacoustic location.

To model the transmission of oscillations into the surrounding soil, elements of shell theory are applied. The assumption of continuity of the radial velocity of the medium filling the pipe and its wall is used [67]. Additionally, equations of motion and strength of materials are employed to evaluate the movement of the pipe wall [21]. Significant progress in describing soil vibrations excited by oscillations within a pipeline was achieved by the research team of J.M. Muggleton et al. [22]. By transitioning from shell oscillations to the vibrations of the surrounding soil, using the same assumption of continuity of the radial velocity of the medium filling the pipe and its wall, this team was able to estimate energy losses, confirming their hypothesis experimentally [68,69]. Among the limitations of this approach, the neglect of soil porosity and heterogeneity stands out, which may be a potential cause of incomplete convergence between calculated results and experimental data [70]. It should also be noted that the high computational complexity of this approach significantly limits a researcher’s ability to quickly evaluate various systems. Therefore, modeling sound propagation processes in pipes using the finite element method in various software packages becomes a more preferable option for studying the impact of various system changes on the efficiency of the vibroacoustic method for locating buried non-metallic pipelines.

The model proposed by Muggleton et al. relies on Equation (6) for the determination of the wavenumber of a low-frequency axisymmetric wave in a pipe [70].

$$k_1^2=k_f^2\left(1+{\displaystyle \frac{\beta }{1-{\mathsf{\Omega }}^2+\alpha }}\right)$$

(6)

Here, $k_f^2$ denotes the fluid wavenumber, Ω is the non-dimensional frequency, and α and β represent the parameters characterizing the influence of soil and fluid loading on the pipe wall. This model is relatively complex and involves an extensive mathematical formulation; however, it provides the most accurate results when describing acoustic attenuation in buried pipes, particularly when soil and fluid properties are taken into account. The approach demonstrates high accuracy for frequencies exceeding 200 Hz, as confirmed by the comparison of the calculated and experimental data.

While the modeling of signal attenuation presents a significant challenge, the use and simulation of time delays is a more commonly addressed task. Xerri et al. proposed a model for estimating pipeline depth based on time delay measurements in two propagation media, referred to as M2 [44]. This model employs Equations (7) and (8) to calculate the pipeline depth using time delay data.

$$t_{1i}\left({\mathsf{\Theta }}_{M2}\right)={\displaystyle \frac{\left|S{P}_i\right|-\left|S{R}_1\right|}{v_0}}+{\displaystyle \frac{\left|P_i{R}_i\right|}{v_1}}$$

(7)

$${\displaystyle \frac{S_Z-P_{iZ}}{\left|S{P}_i\right|}}={\displaystyle \frac{P_{iZ}}{\left|P_i{R}_i\right|}}{\displaystyle \frac{v_0}{v_1}}$$

(8)

In these equations, t_1i denotes the relative time delay between sensors R₁ and R_i, S P_i represents the wave travel time from the signal source S to the interface point P_i, v_i is the wave propagation velocity in medium i, S_z indicates the depth of the signal source, and P_iZ corresponds to the depth of the interface point. These equations enable the estimation of pipeline depth, provided the sensor positions and the location of the medium transition are known. The main challenge lies in isolating the target signal through noise filtering and the application of cross-correlation or any other suitable signal processing technique.

It is important to note that, despite the availability of various complex and detailed models for describing sound attenuation in buried non-metallic pipeline locaction, many of these models still do not provide consistently accurate results. This discrepancy can lead to significant deviations—often exceeding 50%—between simulated and measured acoustic signal attenuation [71]. These differences may arise from numerous factors influencing the propagation of acoustic waves both underground and within the pipeline.

Currently, validation of various models is being conducted at the Empress Catherine II Saint Petersburg Mining University. To support this objective, a dedicated test facility has been constructed, incorporating polyethylene pipelines with different configurations, including simulated illegal connections made from a variety of materials. This facility serves as a platform for evaluating the performance of systems designed to locate underground non-metallic pipelines. The process of locating buried non-metallic pipeline using the seismic method is illustrated in Figure 6.

7. Results of Preliminary Modeling

To evaluate the quantitative characteristics of optimization reserves for wave attenuation in a straight pipe, the COMSOL Multiphysics 6.2 software package was utilized. The software supports the calculation of sound propagation in channels with constant pressure within a cross-section perpendicular to the axis of sound propagation. A similar simplification is applied in the Low Reduced Frequency Model described above. Using the Pressure Acoustics module and the Narrow Region Acoustics node, a series of calculations were performed for a pipe with a diameter of 45 mm and a length of 20 m. Additional input parameters used in the model are summarized in

Table 4. Input data for the model (Compiled by the authors).

Parameter	Value
Material	CH4 (methane) [gas]
Bulk viscosity	14·10−6 Pa·s at 1 atm, 25 °C calculated with [51]; 17.1·10−6 Pa·s at 1 atm, 75 °C calculated with [51]; 14.2·10−6 Pa·s at 10 atm, 25 °C estimated with [72].
Ratio of specific heats	1.31 under all simulation conditions
Speed of sound	446 at 1 atm, 25 °C [55]; 485 at 1 atm, 75 °C calculated with [73]; 447 at 10 atm, 25 °C estimated with [74].
Input boundary condition	Pressure = 6.3 Pa
Output boundary condition	Circular port

. As a result, the data presented in

Table 5. Preliminary results of modeling sound attenuation in a pipe with a diameter of 45 mm and a length of 20 m (Compiled by the authors).

Varied Parameter	Relative Change in Attenuation
Increase in sound frequency to 1 kHz	+49.91%
Decrease in sound frequency to 100 Hz	−41.69%
Increase in pressure to 10 atm	+15.12%
Increase in temperature to 75 °C	+20.24%

were obtained. The baseline calculation conditions were a pressure of 1 atm, a temperature of 25 °C, and a sound frequency of 500 Hz. The results of the baseline calculation and the model mesh data are shown in Figure 7.

The pressure function, with respect to distance in the considered case of a plane wave, can be described as shown in Equation (9) [75].

$$p\left(x,t\right)=p_0{e}^{j(2\pi ft-kx)}$$

(9)

where p₀ is the initial amplitude of the acoustic wave, f is the frequency of the acoustic wave, x is the propagation distance, k = 2πf/c₀, and c₀ is the speed of acoustic wave propagation. This simplification reduces the problem of describing the acoustic front to a one-dimensional form, significantly reducing its computation time.

Based on the results of the conducted study, it can be confidently stated that variations in the physical parameters of the transported gas within fairly wide limits have only a minor effect on sound attenuation in pipes. Therefore, this direction of technology optimization can be considered unpromising. To achieve a higher location range, it is necessary to optimize the emitted signal and the methodology for its acquisition and processing, taking into account the specific design features of individual pipelines. This will be further explored by the authors in subsequent works.

8. Conclusions

Based on the analysis of research and practical experience, the efficiency of the vibroacoustic method for locating buried non-metallic pipelines has been substantiated. In support of this, a list and comparison of commercial devices utilizing various acoustic location methods for determining the underground position of non-metallic pipelines is provided. This study highlights the main advantages and disadvantages of each method listed.

For the vibroacoustic method of locating buried non-metallic pipelines, the main mechanisms of acoustic signal transmission to the surface are described, and the processes of signal attenuation within the pipe and its transfer into the surrounding environment are analyzed. This paper also outlines the main methods of mathematical modeling of the studied process and the dependencies of signal attenuation in the soils under various conditions. As a result of analyzing the aforementioned information, three most promising directions for optimizing the investigated technology have been identified:

• Increase in the amplitude of the transmitted signal and optimization of its characteristics;
• Optimization of measuring devices, signal filtering, and data processing;
• Variation in gas temperature, humidity, and pressure.

It is quite challenging to increase the amplitude of the transmitted signal while maintaining the mobility of the equipment. More stationary configurations, such as vehicle-mounted systems or combined excitation setups, have the potential to significantly increase the acoustic energy transmitted into the pipeline. At the same time, the use of signal modulation and optimization techniques can facilitate signal detection at the surface.

In terms of signal processing, the application of modern techniques such as wavelet transforms or machine learning has the potential to enhance researchers’ capabilities in identifying various dependencies and improving the signal-to-noise ratio after processing. The use of modern high-precision quartz vibration transducers and accelerometers can improve the obtained signal quality, which will positively impact the overall efficiency of the technology. Finally, working with specific gas characteristics can further enhance the signal propagation distance before it becomes undetectable.