Analysis of acoustic transmission characteristics and transmission model prediction of oil and gas production strings

Summary

Acoustic wave transmission is a promising technology in the field of logging data transmission while drilling due to its strong environmental adaptability and high transmission rate, which can meet the needs of data transmission while drilling. Firstly, a finite element simulation model of the acoustic transmission of the string system is established based on COMSOL, and the influence of the number of cascades of tubing and the size of the coupling of tubing connection on the acoustic transmission characteristics of the string is studied through parametric numerical simulation, taking the sound pressure level attenuation and sound pressure amplitude ratio as the evaluation indexes. The LightGBM algorithm was used to predict the acoustic transmission characteristic curve of the string. On the basis of the prediction model, taking the maximum transmission distance and the minimum transmission attenuation as the objective functions, the NSGA-II optimization algorithm is used to obtain the optimal parameter combination of string system structure and transmission frequency. The research results provide a reference for the selection of acoustic transmission channels for downhole strings and the optimization of string system structure.

Keywords: acoustic transmission; COMSOL； LightGBM method; NSGA-II algorithm; Multi-objective optimization

1 Introduction

Oil and gas resources are non-renewable energy sources, and after the onshore resources have been fully developed, human beings have begun to pay attention to the potential of the deep ocean. There are a large number of deep wells and low-permeability oil and gas resources around the world, which are thousands of meters deep and extremely risky and dangerous to exploit. Therefore, it is increasingly critical to accurately obtain information about the conditions and formations of the downhole operation. Due to the unique advantages of acoustic signal transmission, many scholars have carried out research on it. The acoustic wave transmission technology while drilling has the ability to adapt to a variety of formations, is not limited by external conditions such as drilling fluid and formation damping, has a high transmission rate, and can realize the real-time transmission of downhole data, which is of great significance for the development of intelligent drilling and logging technology. Acoustic wireless telemetry technology overcomes the limitations of traditional cables that need to pass through packers, making it possible to install pressure and temperature monitoring instruments at critical locations in single-layer, multi-layer, and different depths, with significant advantages and potential.

Many scholars have made important contributions to the field of downhole acoustic wave transmission systems and devices. In the early 90s of the 20th century, Kostek et al. established a borehole acoustic field model to study the acoustic signal excited by the piezoelectric transducer and its propagation. Then, in 1993, Drumheller patented an improved electromechanical transducer. In 1994, J. V. Leggett and B. Leggett R. Green developed the SDAS surface data acquisition system and successfully tested the data transmission in the production well. In 1995, Tochikawa T et al. developed a new MWD system based on magnetostrictive technology and successfully conducted data transmission tests downhole. In 1996, Japan National Oil Corporation further developed an acoustic wave transmission system based on magnetostrictive transducers and conducted successful field tests. At the end of the 20th century, Halliburton began to study acoustic wave transmission methods, and in 2004 published the results of the transmission depth test of its acoustic wave telecommunication system. In 2001, Sandia National Laboratory and a number of units developed the acoustic telemetry MWD system AT-MWD, which was extensively tested in the field and basically reached the level of commercial application. In 2002, Halliburton Energy & Oil successfully applied the sonic telemetry system ATS in Netherlands, demonstrating the practical utility of sonic data transmission technology in oilfield production. During the period from 2005 to 2009, various technical studies were carried out, including the research of Luo et al. on the stress wave data transmission technology, L. Gao's calculation of the acoustic wave transmission theory of drill string system, and He Lai's discussion on the single-carrier frequency domain equilibrium in the deep well channel, which further promoted the development of the application of acoustic wave transmission technology in the oilfield. In 2017, Zheng Lichen of PetroChina Research Institute of Petroleum Exploration and Development conducted a field test of signal remote transmission of vibration wave downhole communication technology in Daqing Oilfield, with a test well depth of 900 meters, and it is predicted that downhole communication can be achieved within 3,000 meters. In the same year, in the field test of stratified water injection in Jilin Oilfield, the test well was 426 meters deep, and the operation of water distributor opening adjustment was successfully realized. In 2020, Xu Fei deeply studied the principle of downhole acoustic wave wireless transmission, analyzed the transmission characteristics of sound waves in the drill pipe, and established a relay bidirectional transmission model based on amplification and forwarding. A simulation model is designed and experiments are carried out to verify the ability of the relay transmission system to extend the transmission distance. In the same year, Li Mingjun established an acoustic wave transmission model of drill pipe through multi-physics coupling analysis, and analyzed the transmission characteristics of acoustic signals in drill pipe. At the same time, Gao Jun et al. studied the problem of acoustic wave transmission of measurement data while drilling in the process of conventional rotary drilling, and proposed corresponding solutions. In 2021, Cao Shufan designed an experimental system for acoustic signal transmission, studied the propagation characteristics of acoustic signals in the tubing string, and verified the characteristics of the comb filter and the selection of acoustic frequency through experiments to ensure the efficiency and quality of signal transmission. In 2022, Zheng Zhong and Geng Yanfeng developed an acoustic wireless communication system based on orthogonal frequency division multiplexing (OFDM), completed the hardware and software design of the system, and verified that the transmission efficiency of the system can reach 250 kbit/s through experiments. These studies have not only promoted the development of acoustic wave transmission technology, but also achieved local applications in oil exploration and development.

The study of downhole acoustic wave transmission characteristics is essential for the development of downhole acoustic wave transmission systems and devices. In the 70s of the 20th century, Barnes and Kirkwood analyzed the torsional and longitudinal harmonic zero-mode transmission in an ideal drill string, revealing the structural characteristics of comb filters with alternating passband and stopband in acoustic wave transmission. They found that the longitudinal wave pass band was about twice as wide as the torsion wave pass band and proposed a calculation method. In the 80s and 90s of the 20th century, Drumheller's research team conducted an in-depth study of the acoustic wave transmission characteristics in the drill string. In 1985, they analyzed the fine spectral structure in each passband of an ideal drill string structure and discussed the influence of the drill string structure size on the acoustic propagation characteristics. In 1989, they used the central finite difference method to analyze the propagation characteristics of sound waves in a periodic drill string structure and verified the correctness of the model. In 1991, they designed an analog control circuit for an acoustic transducer array, and proposed a method for calculating the acoustic spectrum based on characteristic curves, which was used to analyze the response characteristics of the drill string under harmonic action. In the early 90s of the 20th century, H. Y. Lee studied the effects of drill strings, drilling fluids, and formations on acoustic wave propagation, and found that these factors were usually negligible. In 1999, Carcione and Poletto developed the basic theory and algorithm for simulating the propagation of different vibration modes in the drill string, realizing the simulation of the stop band in the propagation of sound waves and the generation of stress waves in piezoelectric transducers. At the beginning of the 21st century, Wang Chunyan used the transmission matrix method to study the acoustic wave transmission characteristics in the drill string, and showed that the numerical calculation results of this method are more accurate than the finite difference method. In 2008, Xiaohua Che et al. also used the transfer matrix method to numerically simulate the transmission characteristics of longitudinal waves in the drill string, which confirmed the reliability of the theoretical estimate. Before 2010, Zhao Guoshan established an ideal drill string model based on the acoustic permeable layer element model, analyzed the acoustic wave transmission characteristics of drill pipe and joint combinations of different sizes by using the transfer matrix method, and established an experimental system to verify the near-drill bit transmission ability of acoustic signals. In 2019, Gaoli conducted a test on the acoustic transmission characteristics of drill pipes, and found that the acoustic signal strength in the 2000~4000Hz frequency band was significantly higher than that in other frequency bands. In the same year, Lu Haiqi verified the feasibility and stability of the system by analyzing the channel model of deep wells and establishing a deep well acoustic wave OFDM transmission system, which supported the development of intelligent logging technology. In 2020, Liu Guiwen analyzed the characteristics of the comb filter for acoustic wave transmission of drill pipe, and confirmed that the change of drill pipe length will affect the position and bandwidth of the on-stop band, and its theory is basically consistent with the experimental data. In 2021, Gao Tianlin applied the improved singular value decomposition and cascaded random resonance algorithm to filter the acoustic signal, and the experimental results showed that the proposed method could effectively improve the signal-to-noise ratio and optimize the acoustic signal processing. In 2024, Lei Peng carried out the acoustic repeater test, explored the transmission law of sound waves along the pipe string, and found that the sound signal intensity decreases exponentially with the increase of tubing length, and the medium in the well significantly affects the carrier frequency, which provides important data support for the development of acoustic remote transmission technology and real-time efficient testing technology. By understanding the propagation characteristics of sound waves in drill strings and strings, scientists were able to optimize signal transmission efficiency and reduce energy loss.

In summary, although there have been abundant studies on the acoustic transmission characteristics of the downhole acoustic transmission system and the downhole string system, most of the current studies focus on the acoustic transmission characteristics of the drill pipe with periodic drill string distribution, and have not been considered from the whole string system, and the overall system including casing, base pipe, tubing and the fluid medium inside and outside the pipe has been studied rarely. The existing studies consider the single-tube system, and do not consider the influence of the coupling and transmission distance of the string connection. In this context, this paper analyzes the acoustic transmission characteristics of oil and gas production strings and predicts them by model.

Numerical analysis of acoustic transmission characteristics of 2 columns

2.1 The overall scheme of acoustic transmission of the string

The overall system scheme for studying acoustic signal transmission in this paper is shown in Figure 1. The body of the well is a horizontal well, with a vertical depth of 280 meters and a depth of 900 meters. There are three layers of pipelines under the seabed: casing, base pipes and tubing. The pipeline structure is all steel pipes. The casing is the outermost pipe, nested in the borehole, which plays the role of supporting the borehole, protecting the formation, and isolating different pressure layers. In this study, the production of casing specifications is 9-5/8 inch casing (outer diameter 244.5mm, inner diameter 220.5mm). The base pipe is located between the casing and the tubing, the main function is to connect the casing and the tubing, and provide a channel for the production of oil and gas, the base pipe can help support the weight of the tubing, prevent the tubing from deforming or breaking due to its own gravity. In addition, the base pipe can play a role in isolating different formations and prevent formation water or other fluids from entering the tubing. In this study, the base tube size is 5-1/2 inch (outer diameter 139.7mm, inner diameter 121mm). The tubing is the innermost pipe, which is the key channel for transporting oil and gas from the ground to the surface. In this study, the outer diameter of the coupling is 107.95mm, the gas production pipeline is a 3-1/2 inch tubing (outer diameter 88.9mm, inner diameter 76mm,), the length of a single connection is 9.62m, the length of the short connection is 3m, and the short connection of the tool connection is 1.5m. The 3-1/2 inch tubing extends to 10m above the sand control screen packer, and the 2-3/8 inch tubing (outer diameter 60.3mm, inner diameter 50.3mm) is connected to the variable buckle.

Fig.1 Schematic diagram of acoustic signal transmission in underwater drilling

The importance of a periodic string system lies in its ability to precisely control the wave propagation characteristics by adjusting the periodic structure of the string and the properties of the medium. Such systems exhibit significant frequency selectivity, allowing only waves in a specific frequency range to be transmitted, while waves of other frequencies exhibit reflection or attenuation. The finite element simulation model of the acoustic transmission of periodic strings is established, and the transmission effect of acoustic signals can be studied in depth by changing the arrangement spacing of repeaters, and the transmission efficiency can be evaluated by measuring the acoustic wave transmission loss. In addition, the influence of couplings on acoustic wave transmission losses can be evaluated and losses under different cascaded strings can be modeled to better understand and optimize the characteristics and effects of acoustic wave transmission in string systems.

2.2 Numerical simulation of acoustic transmission of pipe strings

2.2.1 Establishment of finite element model of acoustic transmission

Because the overall length of the pipe is much larger than the diameter, you can model the pipe as a straight pipe and ignore its curvature. Tubing can be thought of as a periodic string acoustic transmission system consisting of multiple tubing connected by couplings. The linear array method can be used to control the number of tubing in cascade by setting the array parameter n. To simplify the simulation, each of the 10 cascades (each 9.62 meters long) is considered as a group. The system is divided into 5 groups, with 10 tubing between each group, which facilitates the processing and analysis of the acoustic characteristics of the system. From the influence of cascading logarithm on the acoustic transmission of the string, it can be seen that the sound wave transmission effect is the best when n=30. When performing a string acoustic-structure interaction simulation, all structures need to be considered, including tubing couplings, casings, and base pipes. The simplified study model is shown in Figure 2.

Fig.2. Acoustic transmission model of tubing (20-2000(Hz))

The column system studied in this paper is a typical axisymmetric model, and the spatial dimension of the geometry model selected for COMSOL modeling is 2D axisymmetry. For parametric study, it was established as a periodic element structure with two halves of tubing connected to the coupling. When setting up the physics, pressure acoustics is used in liquids, solid mechanics is used in structural steels, and acoustic-structure interaction is used to study the entire content. The piping system as well as the couplings are defined as structural steel, and the medium inside the casing and inside the base pipe is set to water. In this study, the frequency range is 20Hz-2000Hz, and the boundary meshing is carried out by setting the unit size of the specified coupling area to 0.005m and the specified tubing at the boundary of the axis of symmetry to 0.01m. The rest of the area is divided by a map, so that the mesh draws a regular quadrilateral grid. A total of 172816 (n=30) were divided. For the left boundary of the gas-water compound in the tubing, the axisymmetric boundary condition is set to ensure that the 2D axisymmetric model takes effect. The upper and lower boundaries of the pipeline fluid are set as plane wave radiation, which means that the sound wave can propagate and absorb on the upper and lower sides of the pipeline liquid and will not be reflected. The upper and lower boundaries of the tubing, base pipe, and casing are set as free boundary conditions, which means that their displacement is not constrained, and the sound wave propagation will not be disturbed by the edges. The outside of the casing is set as a fixed constraint, indicating that it is buried in the formation and is constrained by the formation. The specific boundary conditions are shown in Figure 3.

Figure 3 a) Encryption mesh at the coupling b) Mapping grid for the rest of the area

2.2.2 Establishment of evaluation indicators for acoustic transmission characteristics

1) Sound pressure amplitude ratio

In order to measure the transmission loss at different frequencies, the ratio of sound pressure amplitude at the outlet and incident ends is defined to evaluate the transmission loss.

2) Tubing displacement ratio

In order to reflect the eigenfrequency of the domain, the displacement ratio of the outlet end and the incident end of the tubing is defined, and the eigenfrequencies of the tubing are viewed from this.

3) Sound pressure attenuation

Defines the sound pressure attenuation of the tubing, which is defined as the ratio of incident and outgoing sound energy in dB. The formula for sound energy attenuation or transmission loss is:

where and represents the incident power of the inlet and the output power of the outlet, respectively, and these quantities can be calculated by integrating at the corresponding boundaries of the inlet and outlet. Suppose the propagation expression for a plane wave is:

2.3 Parameters for the acoustic transmission analysis of the string

Increasing the number of cascades of tubing increases the length of the sound wave propagation path, but it also increases transmission loss and delay. Couplings are structures that are abruptly altered to produce acoustic attenuation. The acoustic transmission frequency of the string system has a comb-like structure, which affects which frequencies of waves can propagate and which frequencies of waves are reflected or absorbed. Therefore, optimizing these factors is critical to the design of the string system. In this paper, we will explore the impact on acoustic transmission performance from factors such as the number of tubing cascades, the size of the coupling, and the frequency response. Table 1 shows the structural parameters and their variation ranges.

Table 1 Definitions and value ranges of parameter variables

2.4 Analysis of the simulation results of acoustic transmission of pipe strings

2.4.1 Acoustic pressure, sound pressure level contour and line diagram in the string at different frequencies (n=30)

In the column, the calculation of sound pressure at different frequencies can be described by the wave equation and the transmission matrix method, and the sound pressure level at different frequencies can be converted by the calculation formula of sound pressure. Here's the basic formula.

For the propagation of a one-dimensional plane wave in an undestructive medium, the sound pressure can be described by the following wave equation:

Where: is the sound pressure at the position, at the time. is the speed at which the sound waves propagate in a medium.

By converting the sound pressure from the time domain to the frequency domain by means of the Fourier transform, we can get the sound pressure representation in the frequency domain:

Where: is the amplitude of the incident sound pressure. is the angular frequency, , is the frequency. is the wavenumber,

。

Considering the propagation of sound waves within the string, the transmission of sound pressure can be expressed as:

For the sound pressure calculation at different frequencies, it can be expressed as:

If the losses within the string are taken into account, the transmission formula for sound pressure can be modified as:

Where: is the absorption coefficient, which indicates the attenuation of sound waves in the medium. is the phase coefficient, which is usually equal to the number of waves.

Sound pressure level is defined as:

Where: is the sound pressure level, expressed in decibels (dB). is the sound pressure at the measurement point. is the reference sound pressure, typically 20 μPa (20 microPa) in air.

By substituting the above sound pressure calculation results into the definition formula of sound pressure level, the sound pressure level in the string at different frequencies can be obtained.

When absorption losses are not considered:

When considering absorption losses, we only care about the attenuation part:

Total sound pressure graph

Looking at the sound pressure contours and line diagrams of the incident end and the outlet end at different frequencies in Figure 4, it can be seen that although the acoustic wave generator is installed on the base tube, the existence of sound pressure can be detected in the three liquid media at the bottom, which means that the vibration signal generated by the acoustic wave generator can propagate the sound signal in the fluid-filled string system. It can be seen from the sound pressure contour diagram that at the incident end, the position where the acoustic repeater is installed, that is, the water between the tubing and the base pipe, produces a larger sound pressure, followed by the sound pressure inside the tubing, and the water in the casing has the smallest sound pressure. At the incident end of the sound source, transmission losses occur at different frequencies. And as the frequency increases, the wavelength of the sound pressure fluctuation in the three media becomes smaller. At the same time, in three separated media, under the action of the same sound source, the wavelength of the three is different, and at the same sound source frequency, the wavelength of the water located in the base tube is the shortest of the three. At the outlet end, the sound pressure shows a certain attenuation at different frequencies, and the attenuation becomes more obvious with the increase of frequency. At the same time, it can be seen that at the outlet end, due to the inconsistency of the wavelengths in the three media, the sound pressure of the three at the outlet end shows a phase difference, which is more obvious at high frequency.

It can be seen from the line diagram that from the incident end to the outlet end, the sound pressure decreases with the increase of the transmission distance of the sound wave, and with the increase of the transmission frequency, the frequency and amplitude of the sound wave attenuation increase significantly, and the attenuation is more obvious, and the sound wave enters a relatively stable transmission state earlier. When the transmission frequency is 2000Hz, the attenuation of the sound wave after 40 meters at the incident end is very small, indicating that the sound wave has good transmission performance at this frequency.

b1) Incident (500HZ) b2) Outlet (500HZ) b3) Incident - Exit (500HZ)

c1) Incident (2000HZ) c2) Outlet (2000HZ) c3) Incident - Outlet (2000HZ)

Fig.4 Acoustic pressure contours and lines at different frequencies at the incident and outlet ends

Total sound pressure level diagram

Sound pressure level (SPL) is a physical quantity used to express the intensity or intensity of sound intensity or sound pressure, usually expressed in decibels (dB). Sound pressure level is a commonly used way to quantify the intensity of sound in acoustics and is used to describe the loudness of a sound or the intensity level of the auditory perception.

The conclusion in Figure 5 is similar to that in Figure 4 above, and will not be repeated here.

d1) Incident (100HZ) d2) Outlet (100HZ) d3) Incident - Outlet (100HZ)

e1) Incident (500HZ) e2) Outlet (500HZ) e3) Incident - Outlet (500HZ)

f1) Incident 2000HZ f2) Outlet 2000HZ f3) Incident - Outlet 2000HZ

Fig.5 Contour and line plots of sound pressure levels at the incident and outlet terminals at different frequencies

2.4.2 Acoustic transmission characteristics evaluation index diagram

As can be seen from Figure 6, the ratio of sound pressure amplitude between the outlet end and the incident end of the pipeline system cascaded by 30 tubing shows different degrees of loss, and with the increase of frequency, the greater the loss, when the more than 2000Hz, most of the attenuation has reached 90%.

As can be seen from Figure 7, the displacement ratio of the tubing is around the eigenfrequency, and even the displacement exceeds the source side due to resonance, which means that in the domain, in the frequency range of the crest, its sound propagation will obtain a more efficient transmission effect. At the same time, it can also be seen that with the increase of the transmission frequency, the vibration amplitude at the outlet end will still be lower, and even if it is located near the eigenfrequency, the wave peak value is much lower than the amplitude of the low frequency. This means that in the acoustic transmission of tubing, it is more reasonable to use a transmission frequency lower than 1500Hz. From several peaks, it can also be seen that the suitable frequencies for transmission are 230Hz, 460Hz, and 690Hz, respectively.

As can be seen from Figure 8, the sound pressure attenuation is less attenuated at low frequencies and increases as the frequency increases, reaching a maximum sound pressure attenuation of 19 dB at 1640 Hz. This indicates that this frequency is not suitable for acoustic transmission.

Fig.6 Acoustic pressure amplitude ratio curve Fig.7 Tubing displacement ratio curve

Fig.8. Acoustic pressure attenuation curve of tubing

2.4.3 Effect of the cascade number of tubing on the acoustic transmission performance

Studying the influence of the cascade number of tubing on the acoustic wave transmission performance is helpful to realize the rationality of the arrangement interval of the repeater. The cascade of 10, 20, 30, 40 and 50 tubing was parametrically simulated, and the sound pressure amplitude ratios at different frequencies were obtained to evaluate their attenuation performance. As can be seen from Figure 9, with the increase of the number of string cascades, that is, with the increase of the sound wave transmission distance, the amplitude attenuation caused by the structural damping loss becomes more and more serious, and for a specific string channel, the amplitude attenuation degree of different passbands is different, the higher the frequency of the passband, the greater the amplitude attenuation, therefore, it is not recommended to use high-frequency acoustic signals for data transmission.

Fig.9. Ratio of sound pressure amplitude at different frequencies

As can be seen from Figure 9, when the frequency exceeds 500Hz, the sound pressure amplitude ratio is lower than 0.4, so it can be seen that for tubing transmission, the operating frequency below 500Hz is the key interval, and the curve is amplified as follows:

Fig.10. Ratio of sound pressure amplitude at different frequencies (0-500Hz)

As can be seen from Figure 10, the change in the number of cascades of tubing will affect the eigenfrequency of sound wave transmission to a certain extent. When the number of tubing cascades is low, the number of peaks is less, and when the number of tubing cascades is more, more peaks are displayed, and the eigenfrequency obtained by the solution is higher. From the figure, the number of tubing cascades can also be determined by finding its amplitude ratio according to the transmission frequency of the acoustic signal emitted by the acoustic repeater.

2.4.4 The influence of the size of the coupling on the acoustic wave transmission performance

In the tubing acoustic transmission, the influence of the tubing coupling on the acoustic propagation is mainly reflected in the reflection characteristics of the sound wave at the coupling. According to the basic principle of sound wave reflection, reflection and transmission occur when a sound wave encounters the interface of two mediums with different impedances. For this study, the influence of the threaded connection structure of the coupling and the tubing on the acoustic transmission was not considered, and the wall thickness and length of the coupling were not considered. The wall thickness of the coupling is lkg and the length of the coupling is HKG, and the parametric scanning is carried out by setting it as a parametric variable.

Firstly, the length of the coupling was fixed, the thickness of the coupling was changed, and the ratio curve of the sound wave amplitude was observed under different thicknesses. It is not difficult to see from Figure 11 that the change in the thickness of the coupling has little effect on the attenuation curve of the acoustic amplitude. This is because the coupling cannot exceed the inner diameter of the base tube, so the change in cross-sectional area is small compared to the wavelength. Moreover, the propagation of sound waves in tubing is dominated by longitudinal waves, and the change of cross-section is small, and the change of sound wave transmission characteristics is limited.

hkg=0.2 b) hkg=0.4

c) hkg=0.6

Fig.11. Acoustic pressure amplitude ratio curve under different coupling thicknesses

Then, fix the thickness of the coupling, change the length of the coupling, and observe the ratio curve of the acoustic wave amplitude under different coupling lengths. It is not difficult to see from Figure 12 that the change of the length of the coupling will significantly change the transmission characteristics of the sound wave, which is mainly reflected in the increase of the length of the coupling, which will shift the eigenfrequency to the right, that is, the eigenfrequency will increase. At the same time, increasing the length of the coupling, at certain peaks, also lowers its peak, resulting in greater sound pressure attenuation, which is evident at low frequencies, especially at the first peak.

a) lkg=0.4 b) lkg=0.6

c) lkg=0.8 d) lkg=1.0

e) lkg=1.2

Figure 12 Acoustic pressure amplitude ratio curves for different coupling lengths

3. Prediction and parameter optimization of the acoustic transmission model of the pipe string

3.1 Prediction of acoustic transmission characteristic curves of strings based on LightGBM

3.1.1 Principles of LightGBM algorithm

The prediction of the acoustic transmission characteristic curve of the string was carried out by the LightGBM (Light Gradient Boosting Machine) method, and the algorithm flow diagram is shown in Figure 13. To solve this problem, the scikit-learn and lightgbm libraries in Python are used to implement the LightGBM algorithm. The main principle steps of the algorithm include:

1) 梯度提升决策树（Gradient Boosting Decision Tree, GBDT）

LightGBM is based on the Gradient Boosting Decision Tree (GBDT) algorithm, which makes predictions by integrating multiple decision trees. The basic idea of GBDT is to train decision trees iteratively, with the goal of each training being to reduce the gradient direction of the loss function.

2) Histogram-based decision tree algorithm

An important innovation of LightGBM is the use of histogram-based decision tree algorithms. Traditional decision tree algorithms need to iterate through all data points to find the best split point at each node split, which is inefficient on large-scale datasets. LightGBM builds a histogram to approximate the optimal splitting point, which greatly reduces the amount of computation.

Specifically, LightGBM's histogram algorithm consists of the following steps:

Step 1 Discretize the eigenvalues into a specified number of histogram bins, each containing multiple eigenvalues.

Step 2 According to the histogram bin, the gradient histogram and data distribution of the eigenvalues are counted.

Step 3 Quickly find the best split point based on the statistics of the histogram.

3) Leaf-wise-based decision tree growth strategy

Different from the traditional depth-first strategy, LightGBM adopts the Leaf-wise decision tree growth strategy. The leaf-wise strategy selects the leaf node with the largest splitting gain for each growth, which allows each tree to grow deeper, thus improving the accuracy of the model. However, to prevent overfitting, LightGBM introduces a parameter that controls the number of leaf nodes.

4) Feature parallel training and optimization

LightGBM supports feature parallel training, and only part of the features can be used for histogram construction and splitting calculation in each iteration during the training process, which can further improve the training speed.

Figure 13 Flow chart of the LightGBM algorithm

Fig.14 LightGBM histogram algorithm

Let's say we have a training dataset where the feature vector of the first sample is the corresponding label. The goal of GBDT is to progressively reduce the gradient of the loss function by integrating multiple decision trees. For regression problems, the loss function usually chooses the squared loss function, i.e.:

where is the value predicted by the model. For classification problems, commonly used loss functions include logistic loss.

GBDT iteratively learns a series of decision trees, each of which fits residuals based on the predictions of all previous trees. The goal of the first round is to optimize the following loss functions:

Wherein, is the front wheel, which is the prediction result of the model of the front wheel.

On top of the histogram algorithm, LightGBM is further optimized and uses a leaf-wise algorithm with depth limitation, and its algorithm formula is:

where is the optimal segmentation point found in the first iteration, is the feature used for the best segmentation in the first iteration, is the value that is segmented for the feature in the first iteration, this symbol represents the value of finding the minimum loss function, and is the loss function, which is used to measure the difference between the predicted value and the actual value, where represents the model prediction in the first iteration, and represents the operation of segmenting the data with features and values at the point.

where is the operation of splitting using the best split point, feature, and split value found in the first iteration.

3.1.2 Analysis of predictive model results

The coefficients of the following model show that the model has high accuracy in predicting (sound pressure amplitude ratio and sound pressure level attenuation), and the R2 scores are 0.8854125133152095 and 0.8481855194466482, respectively, indicating that the model can explain the corresponding data variance of 88.54% and 84.8%, and the overall performance is excellent. LightGBM's optimal parameter setting ensures that the model achieves a good balance between complexity and training speed, so that the model can effectively capture the features in the data while avoiding overfitting.

Table 3 Model 1 Table 4 Model 2

(Sound Pressure Amplitude Ratio) (Sound Pressure Level Attenuation) The model coefficient for identification

As can be seen from the parameter identification results of the following model, most of the points in the graph are distributed near the reference line (red dotted line), indicating that there is a strong linear relationship between the predicted value and the actual value. The data points are close to the diagonal, indicating that the model has good prediction accuracy. The scatter plot shows that the relationship between the actual and predicted values is relatively close, but some data points deviate from the diagonal, indicating that the prediction accuracy of the model is slightly skewed in some cases.

As can be seen from the residual histogram of the model below, most of the residuals are concentrated around 0, indicating that the error between the predicted value and the actual value of the model is small. In most cases, the model can accurately predict the sound pressure amplitude ratio, but there are still some errors in extreme cases.

Overall, the model performs well in predicting the sound pressure level, sound pressure amplitude ratio, and sound pressure level attenuation of the upper base tube. Both the residual distribution and the scatter plot show the high accuracy and robustness of the model.

Figure 15-1 Parameter identification results of model 1 Figure 15-2 Histogram of the residual of model 1

Figure 16-1 Parameter identification results of model 2 Figure 16-2 Histogram of residuals of model 2

3.2 Multi-objective optimization of acoustic transmission parameters based on NSGA-II

3.2.1 NSGA-II algorithm principle

For this study, it was desirable to have the maximum transmission distance and the least attenuation of the sound waves. It is the number of tubing cascades n that determines the transmission distance, that is, the larger n is hoped, the better. At the same time, it is hoped that the sound pressure amplitude ratio is large, the top sound pressure level is large, and the sound pressure level attenuation is small, which means that the acoustic transmission performance is excellent. The decision variables can be set for the thickness of the coupling, the length of the coupling, and the transmission frequency. The objective function is: 

Enter the number of objective functions, the number of decision variables, and the range space of decision variables. Define your own objective function by evaluate_objective function, and the input parameters of a multi-objective optimization function include pop population size and gen algebra. The objective function contains the dimension M of the objective space, the dimension V of the decision variable space, and the range min_range and max_range of the variables in the decision variable space. The implementation process of the algorithm is shown in Figure 17: firstly, the initial population of N scale is randomly generated, and the first-generation offspring population is obtained through the selection, crossover and mutation of genetic algorithm after non-dominant sorting. Starting from the second generation, the parent and child populations were merged, and the non-dominant ranking was rapid, and the crowding degree of individuals in each non-dominant layer was calculated at the same time, and a new parent population was formed according to the non-dominant ordering. Finally, the basic operation of genetic algorithm was used to generate new progeny populations. Until the conditions for the end of the program are met.

In the NSGA-II algorithm, non-dominant ranking, crowding and elite strategies are the core content congestion, and the formula for calculating congestion is as follows:

where: is the number of solutions in a dominant layer; the number of objective functions; is the congested distance.

Figure 17 Algorithm flow chart

3.2.2 Analysis of optimization results

As can be seen in Figure 18, the points represent the Pareto optimal solution with different trade-offs. The abscissa represents the Sound pressure level at the top of the basement tube (maximized) for one of the objective functions, and the ordinate represents the Attenuation Sound Pressure (minimized) for the second objective function. The red intersection indicates the optimal solution chosen, which has the best balance when considering the two objectives. As shown in the table, the optimal set of Pareto decision variables was selected as the optimal combination of decision variables: [1.30166434, 0.00683223631, 12.5164776, 111.19275].

Figure 18 Optimization results

Table 5 Pareto optimal decision variables

4. Conclusions and recommendations

By studying the acoustic propagation characteristics and prediction models of the bottom hole, the following conclusions can be drawn:

(1) In the acoustic transmission part of the string, with the increase of the sound wave frequency, the attenuation of the acoustic transmission becomes more obvious. At frequencies above 600 Hz, the sound pressure amplitude attenuation is obvious. It is recommended to work below 600Hz for the transmission frequency of the sound source.

(2) The distance between the two relay generators depends on the gain of the repeater, and the larger the gain, the longer the distance that can be transmitted. If 2x gain is used, the acoustic attenuation amplitude ratio is required to be greater than 0.5, and according to the simulation results, it is recommended that the number of drill pipe cascades should not exceed 20. If the gain is 3 times, it is recommended that the number of drill pipe cascades should not exceed 30.

(3) In the acoustic transmission part of the string, the influence of the thickness of the coupling on the acoustic transmission is small, and the change of the length of the coupling will significantly change the transmission characteristics of the sound wave, which is mainly reflected in the increase of the length of the coupling, which will make the eigenfrequency move to the right, that is, the eigenfrequency will increase. At the same time, increasing the length of the coupling, at certain peaks, will also reduce its peak, i.e., resulting in greater sound pressure attenuation. It is recommended to reduce the length of the coupling as much as possible while ensuring the strength of the connection.

(4) Based on the LightGBM algorithm, the acoustic transmission model of the column is predicted, and the predicted value is very close to the actual value, and the error curve fluctuates in a small range near zero, indicating that the prediction error of the model is small, indicating that the prediction results have high reliability. On the basis of the prediction model, taking the maximum transmission distance and the minimum transmission attenuation as the objective functions, the NSGA-II optimization algorithm is used to obtain the optimal parameter combination of string system structure and transmission frequency. The results of this study provide a reference for the selection of acoustic transmission channels for downhole strings and the optimization of string system structure.

Glossary

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