# Study on the Stage Characteristics of Noise Dispersion in the Deformation Evolution of Red Sandstone

Authors: Yimin Song; Deshun Yuan; Junhui Zhang; Hailiang Xu; Dong An; He Ren
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The uniaxial compression experiment of red sandstone is observed simultaneously by using acoustic emission and digital speckle correlation methods. The deformation evolution of red sandstone is divided into microfracture random expansion stage, deformation localization stage, subinstability stage, and instability failure stage. Green’s function and dispersion curve of each stage are obtained from the noise data picked up by acoustic emission equipment, and the dispersion characteristics of each evolution stage are analyzed. The results show the following: (1) In the stage of random propagation of microcracks, the noise in the low frequency range passes through at a higher phase velocity, the phase velocity changes periodically, the correlation coefficient is high at the initial time, and the variation trend of frequency dispersion curve is relatively consistent. (2) In the deformation localization stage, the frequency range without zero phase velocity moves to the high frequency range, and the phase velocity changes periodically. (3) In the subinstability stage, dense phase velocity zeros appear on the dispersion curve image, the dispersion curve tends to be disordered, various indicators change obviously, and the correlation coefficient decreases rapidly. (4) In the unstable failure stage, the fracture evolution is completed, and the variation trend of each index of the frequency dispersion curve is consistent. (5) The size and difference of the sensitive kernel function of the two layers are related to the evolution region of the fracture. The sensitivity kernel value of the medium layer where the fracture evolves is high, and the longer the evolution time, the greater the difference.


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## Body

## 1. Introduction

Rock is the main geological material of geotechnical engineering, such as mining, transportation, water conservancy, and hydropower. Under the action of external load, the internal damage of rock, such as crack initiation, propagation, and intersection, is continuously accumulated, which makes the rock material deform and gets destroyed, and even induces the instability of rock engineering. The rock deformation stage directly reflects different damage states in the rock. It has important theoretical and practical significance for the accurate identification of the rock deformation stage and the prediction and early warning of rock engineering disasters [1–5].Liu et al. [1] studied and examined the influence of LN2 cold leaching with different coal grades on their pore structure and fractal characteristics. Zhou et al. [2] established a mechanical model to analyze the stability of roadway floor heave by comparing it with deep foundation pit base heave. The research results provide a scheme and guidance for roadway support optimization. Wang et al. [3] analyzed the movement and deformation of the roof of the ultra-high working face, obtained the working resistance of the ultrahigh mining face by introducing an equivalent direct roof, and proposed measures to improve the stability of the support, providing guidance for the safe mining of coal in the ultrahigh mining face. Li et al. [4] studied the stress distribution law of each deep mine in Linyi mining area, and the research results have certain guiding significance for mine disaster prevention and safe production. Liu et al. [5] studied the failure mechanism and control technology of gob side entry retaining in close coal seams, providing guidance for the design of gob side entry retaining under similar mining geological conditions. Sarfarazi et al. [6] studied the influence of joint separation on the shear behavior of plane nonpersistent joints under high normal load by PFC2D. In the uniaxial compression test, Haeri et al. [7] applied a compression load to the CTT at a compression rate of 0.02 MPa per second on a 30 ton hydraulic jack with load measuring elements. The results showed that the indirect tensile strength was significantly lower than the Brazilian test strength. Haeri et al. [8] also have studied the influence of normal load on the failure mechanism of trapezoidal joints by using PFC2D. The results show that the failure mode is mainly affected by normal load, while the shear strength is related to the failure mechanism. Haeri et al. [9] proposed a new method to measure the fracture toughness of rock samples by using the central crack horseshoe disk (CCHD) method for radial compression tests, and observed that the crack initiation and propagation patterns between the experiment and numerical simulation are very consistent. Fu et al. [10] studied the damage location and convergence curve based on differential evolution algorithm for precracked concrete specimens with central cracks of different dip angles, and considered the noise effect.Xuhailiang et al. [11] studied the deformation and failure evolution characteristics of composite rock specimens with weak interlayer, and theoretically analyzed the interlayer effect. In order to explore the time-dependent deformation characteristics of deep engineering rock mass under disturbed action, Liumin et al. [12] carried out dynamic disturbed triaxial loading test on red sandstone by using disturbed servo triaxial loading system. In order to explore the deformation and failure characteristics of engineering rock mass under cyclic disturbance, Jin et al. [13] carried out uniaxial cyclic disturbance tests of marble with different stress levels and amplitudes. Bo et al. [14] studied the compression deformation and failure behavior of two rock fissures with different lithology under dry and saturated conditions through uniaxial compression test and elastoplastic contact numerical calculation, and collected and analyzed the acoustic emission signals during the test. Lvxiangfeng and Songyimin [15] used the uniaxial compression test method to carry out the research on the correlation between the stability and deformation localization evolution in the whole process of deformation and failure of coal samples, and selected a coal as a sample to analyze the tensile displacement and rate, sliding displacement and rate evolution of the deformation localization zone, the evolution relationship of the shear stress displacement of the deformation localization zone, and the energy evolution of surrounding rock during the deformation process. The noise data is used to calculate the dispersion curve and identify the indicators. The noise contains a lot of available information and is widely used to study the formation model and deep velocity structure.In the process of signal processing, the background noise recorded by the probe often brings a lot of trouble to waveform analysis. Scholars use various signal processing technologies to reduce the impact of noise and improve the signal-to-noise ratio of records [6–8]. For example, Qi and Zhang [16] summarized in detail how to use seismic interferometry and array processing technology to extract various body wave signals for studying different detection scales (local, regional, and global). Based on wavelet theory, Fan et al. [17] proposed using wavelet scattering decomposition transform to extract the characteristics of microseismic events and noise signals, calculate the characteristic coefficients of the two types of signals, and form the corresponding characteristic matrix. Yanggui et al. [18] used the wireless microseismic sensor and acoustic emission monitoring technology developed for rock mass collapse monitoring to simulate the gradual change of natural stress during the preparation process of rock mass collapse by means of graded creep loading, and studied the acoustic signal characteristics during the laboratory simulation test of dolomite pull apart collapse. However, the recent interdisciplinary research shows that there is a lot of useful information in the noise. Through the cross-correlation operation of the noise, the Green’s function between the receiving points can be extracted, so as to image the noise and obtain the understanding of the structure.Scholars have carried out a lot of research work in noise research. For example, in terms of noise processing, noise signal extraction, and dispersion curve analysis, Hedaxi et al. [19] designed a typical symmetrical three-layer rock coal rock model, calculated multiple sets of Rayleigh trough wave dispersion curves by using the generalized reflection transmission coefficient algorithm, and extracted the dispersion curves for verification. Jia et al. [20] improved the calculation efficiency of the original method by tracking the surface wave group on the shot set record, and verified the correctness and effectiveness of the improved MSc method. Dai et al. [21] Used the frequency Bessel transform method to process the background noise data collected from Chao Hu beach in Anhui Province, and used the quasi Newton method for inversion. Luo et al. [22] provided a theoretical basis for concrete surface crack detection in practical projects by discussing the correlation between the “catastrophe point” and the cut-off frequency in the phase spectrum and frequency dispersion curve of SASW method due to the existence of cracks. Wang et al. [23] tested the practicability of the inversion method by selecting BP neural network to inverse the dispersion curves obtained by various typical geological models in the case of no noise and noise. Zhang et al. [24] used the active passive source seismic surface wave exploration method combined with the dispersion curve extracted from the active passive source surface wave data to broaden the frequency band and improve the resolution of low-frequency signals. Xu et al. [25] proposed an improved dragonfly algorithm based on adaptive weight, which enables the weight parameters of individual aggregation, collision avoidance, and formation to be self-adjusted according to fitness.The noise dispersion characteristics in the process of rock deformation evolution are analyzed, and few studies have been conducted by scholars at home and abroad. Zhao et al. [26] and others explored the influence of water content on the characteristic stress and acoustic emission response characteristics of red sandstone through dispersion curve and other indicators. Gao et al. [27] and others explored the acoustic emission dispersion characteristics of tight oil reservoirs in Erdos Basin, but did not explore the stage characteristics of each stage of deformation evolution.In this paper, taking red sandstone as the research object, the digital speckle correlation method is used to analyze the speckle pattern in the loading process, determine the stage time of deformation evolution, and find out the noise signal information collected by acoustic emission equipment at the correlation time. The cross-correlation analysis of the processed noise is carried out, and the dispersion characteristics of red sandstone in different deformation evolution stages are calculated. The dispersion characteristics are analyzed, and then the sensitive kernel function is analyzed. The fracture evolution trend is determined by the sensitive kernel function, which corresponds to the deformation evolution cloud diagram of the speckle surface, and provides a new method for studying the evolution process of rock stress and deformation.


## 2. Evolution Experiment of the Red Sandstone Deformation Stage

### 2.1. Experimental System and Method

The size of red sandstone specimen is 100 mm × 50 mm × 50 mm, the size of speckle surface and the layout of measuring points are shown in Figure1, and the specific location coordinates are shown in Table 1. The experimental system includes loading system and acquisition system. The loading system uses rljw-2000 hydraulic servo testing machine to load the test piece in the displacement control mode of 0.1 mm/min.Figure 1
Schematic diagram of speckle size and measuring point layout.Table 1
Position coordinates of measuring points (mm).
StationXYZ103585201550303515415085535050The experimental acquisition system includes digital image acquisition system and acoustic emission acquisition system. The digital image acquisition system collects the speckle surface in the whole loading process through the Basler A641f CCD camera, with an image resolution of 1600 pixels × 1200 pixels. The soft island series AE system is used to collect the AE signals of the specimen in the whole loading process. A total of five sensors are arranged on the face and side of the speckle of the specimen to collect acoustic emission information. Vaseline is used as a coupling agent to enhance the coupling effect between the specimen and the sensor. The sampling frequency is 3 MHz. The layout of the experimental system is shown in Figure2.Figure 2
Experimental system.


### 2.2. Experimental Process

The two ends of the specimen shall be treated to ensure that the force on the specimen is uniform without sliding friction. The surface shall be uniformly covered with black paint by painting, and then the white paint shall be uniformly sprayed on the surface in the form of dots. The speckle pattern should be high contrast, random point distribution, consistent point size, and black-and-white density distribution. Before the experiment, the digital image acquisition system, acoustic emission acquisition system and loading system are calibrated to ensure the strict consistency of the time of the whole experimental system. At the beginning of the experiment, the digital image acquisition system, acoustic emission acquisition system, and loading system are triggered at the same time. We carry out uniaxial compression loading on the specimen, the camera collects the speckle image on the surface of the specimen, and the acoustic emission system collects the acoustic emission signal of the specimen until the specimen is damaged. We close the data loading system and the acquisition system at the same time. After the experiment, the digital image acquisition system, the acoustic emission acquisition system, and the loading system are shut down at the same time, and the test data are analyzed.


### 2.3. Experimental Results

The stress evolution curve of displacement loading on the specimen is shown in the Figure3. During the loading process, the peak stress of the specimen is 166.53 MPa. The stress rises slowly in the 0–100 s time interval, and the stress growth rate is uniform in the 0–700 s time interval. This interval is the elastic stage, and the calculated elastic modulus is 5.462 × 103 MPa. After the peak stage, the rock instability stress decreases rapidly. In order to find the time interval of each deformation evolution stage conveniently, we take 1 s as a time interval, analyze and collect data through Vic2D speckle analysis software, and analyze the deformation evolution stage.Figure 3
Stress evolution curve.Through analysis and according to the characteristics of the loading curve, seven typical moments are selected for identification, of which the identification points 1 and 2 are located in the random fracture development stage of the loading curve, the identification points 3 and 4 are located in the deformation localization zone stage of the loading curve, the identification point 5 is located in the subinstability stage, and the identification points 6 and 7 are in the instability stage.


## 2.1. Experimental System and Method

The size of red sandstone specimen is 100 mm × 50 mm × 50 mm, the size of speckle surface and the layout of measuring points are shown in Figure1, and the specific location coordinates are shown in Table 1. The experimental system includes loading system and acquisition system. The loading system uses rljw-2000 hydraulic servo testing machine to load the test piece in the displacement control mode of 0.1 mm/min.Figure 1
Schematic diagram of speckle size and measuring point layout.Table 1
Position coordinates of measuring points (mm).
StationXYZ103585201550303515415085535050The experimental acquisition system includes digital image acquisition system and acoustic emission acquisition system. The digital image acquisition system collects the speckle surface in the whole loading process through the Basler A641f CCD camera, with an image resolution of 1600 pixels × 1200 pixels. The soft island series AE system is used to collect the AE signals of the specimen in the whole loading process. A total of five sensors are arranged on the face and side of the speckle of the specimen to collect acoustic emission information. Vaseline is used as a coupling agent to enhance the coupling effect between the specimen and the sensor. The sampling frequency is 3 MHz. The layout of the experimental system is shown in Figure2.Figure 2
Experimental system.


## 2.2. Experimental Process

The two ends of the specimen shall be treated to ensure that the force on the specimen is uniform without sliding friction. The surface shall be uniformly covered with black paint by painting, and then the white paint shall be uniformly sprayed on the surface in the form of dots. The speckle pattern should be high contrast, random point distribution, consistent point size, and black-and-white density distribution. Before the experiment, the digital image acquisition system, acoustic emission acquisition system and loading system are calibrated to ensure the strict consistency of the time of the whole experimental system. At the beginning of the experiment, the digital image acquisition system, acoustic emission acquisition system, and loading system are triggered at the same time. We carry out uniaxial compression loading on the specimen, the camera collects the speckle image on the surface of the specimen, and the acoustic emission system collects the acoustic emission signal of the specimen until the specimen is damaged. We close the data loading system and the acquisition system at the same time. After the experiment, the digital image acquisition system, the acoustic emission acquisition system, and the loading system are shut down at the same time, and the test data are analyzed.


## 2.3. Experimental Results

The stress evolution curve of displacement loading on the specimen is shown in the Figure3. During the loading process, the peak stress of the specimen is 166.53 MPa. The stress rises slowly in the 0–100 s time interval, and the stress growth rate is uniform in the 0–700 s time interval. This interval is the elastic stage, and the calculated elastic modulus is 5.462 × 103 MPa. After the peak stage, the rock instability stress decreases rapidly. In order to find the time interval of each deformation evolution stage conveniently, we take 1 s as a time interval, analyze and collect data through Vic2D speckle analysis software, and analyze the deformation evolution stage.Figure 3
Stress evolution curve.Through analysis and according to the characteristics of the loading curve, seven typical moments are selected for identification, of which the identification points 1 and 2 are located in the random fracture development stage of the loading curve, the identification points 3 and 4 are located in the deformation localization zone stage of the loading curve, the identification point 5 is located in the subinstability stage, and the identification points 6 and 7 are in the instability stage.


## 3. Analysis of Deformation Field Evolution Characteristics of Red Sandstone

### 3.1. Rock Stage Division Method

According to the characteristics of time-space evolution of rock deformation field and loading curve, the deformation and failure of coal and rock can be divided into four stages: random evolution stage of microfracture, deformation localization stage, subinstability stage, and instability stage.In the first and second stages before the peak, the deformation field develops from microcrack random evolution to deformation localization. Among them, the selection of the starting point of rock deformation localization becomes the key to stage division. The concept of deformation localization is that the difference of principal strain presents obvious local concentration compared with the surrounding area, and the localization area must include the fault trace. The significance of the study is that in terms of the nonuniform evolution information of the deformation field of red sandstone in the whole experimental process, the study of localization can further analyze the corresponding relationship between the evolution of deformation localization and the noise dispersion index, as well as the influence of the post peak localization evolution of rock on the mechanical properties of rock.At present, the starting of rock deformation localization is judged by qualitative analysis method. It is judged that the starting of deformation localization is due to the local concentration of the maximum shear strain compared with other regions. This localized region must contain the trace of the fault. At the beginning of deformation localization, the corresponding acoustic emission rate increased significantly. The other is the evolution process of rock deformation localization, which presents a nonuniform and dynamic evolution process. At present, there are few related experimental studies, and most of them are qualitative descriptions.To solve the above problems, firstly, the quantitative analysis of each stage of the deformation localization process is carried out, and the quantitative characterization parameters of each stage of localization evolution are given. On this basis, the characterization indicators of the initiation of deformation localization are studied, and the prepeak deformation stages of rocks are divided.Through the evolution of rock deformation, it is found that the initial deformation localization occurs in the prestress peak stage. The characteristics of deformation localization are mainly reflected in two aspects. The first is the spatial characteristics of deformation localization, that is, deformation is concentrated on one or more bands, showing the spatial characteristics of deformation concentration; The second is the numerical characteristic of deformation localization, that is, the magnitude in the deformation localization band is much larger than that outside the band.In the third and fourth stages after the peak value, it is divided according to the characteristics of the loading curve. From the loading curve shown in Figure4, subinstability stage and instability stage can be divided after the stress unloading time point (mark point 5).Figure 4
Green’s function.


### 3.2. Characteristic Analysis of Each Deformation Evolution Stage

The nephogram as shown in Figure5 is obtained by calculating the speckle surface of red sandstone specimen, and the stage characteristics of rock deformation evolution stages divided by the above method are analyzed. Due to the naturalness and heterogeneity of red sandstone rock material, there are a large number of microstructures (microcracks, weak layers, micropores, etc.) in it. Under the action of external load, microcracks will be generated in the rock specimen. The development of these microcracks is random, as shown in Figure 6(a). There are random, partial, and slight deformation areas corresponding to the random development stage of the micro grain. The deformation evolution cloud of identification point 1 shown in Figure 6(a). At this time, the corresponding load is 16.83 MPa, a small amount of deformation appears in the cloud diagram, and the maximum deformation is about 2.5 × 10−3 mm. When the load reaches the corresponding moment of identification point 2, the corresponding load is 64.34 MPa. At this moment, there are many deformations in the cloud map, and the maximum deformation is about 3.5 × 10−3 mm.Figure 5
Dispersion curve.Figure 6
Cloud diagram of displacement of identification points. (a) Point 1. (b) Point 2. (c) Point 4. (d) Point 5. (e) Point 6. (f) Point 7.
(a)(b)(c)(d)(e)(f)When the rock specimen is loaded to a certain stage, the randomly developed microcracks will gradually develop on one or more bands in space, and the deformation value on the bands is much greater than that in other nearby areas. It can be seen from the cloud diagram of the deformation field of mark point 4 shown in Figure6(c), a localized zone running up and down through the specimen has been preliminarily formed. At this time, the load is 166.53 MPa, and the maximum deformation is about 5 × 10−3 mm. In the evolution process, the growth rate of the upper shear deformation variable is faster than that of the bottom, and the upper and lower bottom gradually extend to the middle part to form a connection. At the moment of reaching the mark point 5, an obvious localization zone has evolved. At this moment, the load is 164.74 MPa, and the maximum deformation is about 6.5 × 10−3 mm.After the peak point of the rock specimen, the rock specimen will propagate along the deformation localization zone to form a macro crack, and the rock becomes a structure with interface. This deformation stage is defined as the macro crack propagation stage. The load of identification point 6 at this stage is 157.22 MPa. It can be seen that theX coordinate 40 mm y coordinate 50 mm in this displacement cloud chart is the center of the maximum deformation dark area, and the maximum deformation is 8 × 10−3 mm, as shown in Figure 6(e). Under the action of external load, the rock slides along the interface, resulting in the sliding instability of the rock specimen. Because the rock specimen at this stage is already a structural body with interface, and its failure essence is the structural instability of the rock specimen, this deformation stage is defined as the interface sliding stage. The displacement nephogram of stage deformation evolution is shown in Figure 6(f). The maximum displacement has evolved from a small part of the area into a belt, reflecting that an obvious crack appears on the speckle surface, and the internal structure interface slides. At this time, it is engraved at the identification point 7, with a load of 146.51 MPa.According to the stage characteristics of rock deformation evolution, six time points are distinguished on the loading curve according to the deformation evolution stage. Among them, mark points 1 and 2 are the random expansion stage of microcracks, mark points 4 and 5 are the deformation localization stage, mark point 6 is the subinstability stage, and mark point 7 is the instability failure stage.


## 3.1. Rock Stage Division Method

According to the characteristics of time-space evolution of rock deformation field and loading curve, the deformation and failure of coal and rock can be divided into four stages: random evolution stage of microfracture, deformation localization stage, subinstability stage, and instability stage.In the first and second stages before the peak, the deformation field develops from microcrack random evolution to deformation localization. Among them, the selection of the starting point of rock deformation localization becomes the key to stage division. The concept of deformation localization is that the difference of principal strain presents obvious local concentration compared with the surrounding area, and the localization area must include the fault trace. The significance of the study is that in terms of the nonuniform evolution information of the deformation field of red sandstone in the whole experimental process, the study of localization can further analyze the corresponding relationship between the evolution of deformation localization and the noise dispersion index, as well as the influence of the post peak localization evolution of rock on the mechanical properties of rock.At present, the starting of rock deformation localization is judged by qualitative analysis method. It is judged that the starting of deformation localization is due to the local concentration of the maximum shear strain compared with other regions. This localized region must contain the trace of the fault. At the beginning of deformation localization, the corresponding acoustic emission rate increased significantly. The other is the evolution process of rock deformation localization, which presents a nonuniform and dynamic evolution process. At present, there are few related experimental studies, and most of them are qualitative descriptions.To solve the above problems, firstly, the quantitative analysis of each stage of the deformation localization process is carried out, and the quantitative characterization parameters of each stage of localization evolution are given. On this basis, the characterization indicators of the initiation of deformation localization are studied, and the prepeak deformation stages of rocks are divided.Through the evolution of rock deformation, it is found that the initial deformation localization occurs in the prestress peak stage. The characteristics of deformation localization are mainly reflected in two aspects. The first is the spatial characteristics of deformation localization, that is, deformation is concentrated on one or more bands, showing the spatial characteristics of deformation concentration; The second is the numerical characteristic of deformation localization, that is, the magnitude in the deformation localization band is much larger than that outside the band.In the third and fourth stages after the peak value, it is divided according to the characteristics of the loading curve. From the loading curve shown in Figure4, subinstability stage and instability stage can be divided after the stress unloading time point (mark point 5).Figure 4
Green’s function.


## 3.2. Characteristic Analysis of Each Deformation Evolution Stage

The nephogram as shown in Figure5 is obtained by calculating the speckle surface of red sandstone specimen, and the stage characteristics of rock deformation evolution stages divided by the above method are analyzed. Due to the naturalness and heterogeneity of red sandstone rock material, there are a large number of microstructures (microcracks, weak layers, micropores, etc.) in it. Under the action of external load, microcracks will be generated in the rock specimen. The development of these microcracks is random, as shown in Figure 6(a). There are random, partial, and slight deformation areas corresponding to the random development stage of the micro grain. The deformation evolution cloud of identification point 1 shown in Figure 6(a). At this time, the corresponding load is 16.83 MPa, a small amount of deformation appears in the cloud diagram, and the maximum deformation is about 2.5 × 10−3 mm. When the load reaches the corresponding moment of identification point 2, the corresponding load is 64.34 MPa. At this moment, there are many deformations in the cloud map, and the maximum deformation is about 3.5 × 10−3 mm.Figure 5
Dispersion curve.Figure 6
Cloud diagram of displacement of identification points. (a) Point 1. (b) Point 2. (c) Point 4. (d) Point 5. (e) Point 6. (f) Point 7.
(a)(b)(c)(d)(e)(f)When the rock specimen is loaded to a certain stage, the randomly developed microcracks will gradually develop on one or more bands in space, and the deformation value on the bands is much greater than that in other nearby areas. It can be seen from the cloud diagram of the deformation field of mark point 4 shown in Figure6(c), a localized zone running up and down through the specimen has been preliminarily formed. At this time, the load is 166.53 MPa, and the maximum deformation is about 5 × 10−3 mm. In the evolution process, the growth rate of the upper shear deformation variable is faster than that of the bottom, and the upper and lower bottom gradually extend to the middle part to form a connection. At the moment of reaching the mark point 5, an obvious localization zone has evolved. At this moment, the load is 164.74 MPa, and the maximum deformation is about 6.5 × 10−3 mm.After the peak point of the rock specimen, the rock specimen will propagate along the deformation localization zone to form a macro crack, and the rock becomes a structure with interface. This deformation stage is defined as the macro crack propagation stage. The load of identification point 6 at this stage is 157.22 MPa. It can be seen that theX coordinate 40 mm y coordinate 50 mm in this displacement cloud chart is the center of the maximum deformation dark area, and the maximum deformation is 8 × 10−3 mm, as shown in Figure 6(e). Under the action of external load, the rock slides along the interface, resulting in the sliding instability of the rock specimen. Because the rock specimen at this stage is already a structural body with interface, and its failure essence is the structural instability of the rock specimen, this deformation stage is defined as the interface sliding stage. The displacement nephogram of stage deformation evolution is shown in Figure 6(f). The maximum displacement has evolved from a small part of the area into a belt, reflecting that an obvious crack appears on the speckle surface, and the internal structure interface slides. At this time, it is engraved at the identification point 7, with a load of 146.51 MPa.According to the stage characteristics of rock deformation evolution, six time points are distinguished on the loading curve according to the deformation evolution stage. Among them, mark points 1 and 2 are the random expansion stage of microcracks, mark points 4 and 5 are the deformation localization stage, mark point 6 is the subinstability stage, and mark point 7 is the instability failure stage.


## 4. Extraction of the Dispersion Curve of Acoustic Emission Background Noise

### 4.1. Noise Data Preprocessing

Noise data preprocessing is to transform the original noise waveform data extracted from the sensor into noise waveform data that can be used to calculate subsequent cross-correlation and Green’s function through format conversion, de-averaging and de-trending, band-pass filtering, time-domain normalized spectrum whitening, and other processing work.We export the waveform time domain image in the software and export the text format. Filter the data of the derived waveform, and select the noise waveform data that meets the threshold value. In the image shown in Figure7, the waveform with small amplitude and high frequency before the signal waveform is a noise waveform.Figure 7
Noise waveform.De-averaging means that each dimension subtracts the mean value of the corresponding dimension, so that each dimension of the input data is centered to 0. The reason for de-averaging is that it is not easy to fit without de-averaging. Detrended processing can eliminate the influence of the offset generated by the sensor when acquiring data on the later calculation. Deleting a trend from the data can focus the analysis on the fluctuations of the data trend itself. The processed data image is shown in Figure8.Figure 8
De-mean and de-trend.Filtering the frequency of a specific band in the waveform data is an important measure to suppress and prevent interference. Here, the noise signal is transformed into a waveform signal that meets the requirements, as shown in Figure9.Figure 9
Bandpass filtering.


### 4.2. Noise Data Cross Correlation

Before data cross-correlation processing, the maximum and minimum amplitudes of time domain maps are attributed to the set [−1, 1], and the scales of each feature are controlled within the same range, so that the optimal solution can be easily found. Data whitening is to reduce the dimension of data by discarding the dimensions with less information and retaining the main feature information. A few representative and unrelated features are used to replace the original large number of features with certain correlation, which is easy to analyze.Data whitening is to reduce the correlation between features. Features have the same variance and the covariance is 1. Normalization is to unify data to the same level of quantity. In the process of noise data whitening, the PCA operation is carried out first, and then the normalization operation is carried out. The processed image is shown in Figure10.Figure 10
Domain normalization and spectral whitening.In signal analysis, cross-correlation represents the degree of correlation between the values of the time series between two time series at any two different times, that is, the cross-correlation function describes the degree of correlation between the values of random signalsx (t) and y (t) at any two different times t1 and t2.The discrete calculation formula of cross-correlation is as follows:(1)Rxyn∆τ=1N−n∑1N−nxryn+r,where N is the number of sampling points and n is the delay sequence.We perform cross-correlation calculation on the noise waveform data collected by the two stations, and the calculation results are shown in Figure11.Figure 11
Cross-correlation function.The Green’s function is expressed as the relationship between the “field” and the “source” that produces the field. By using the superposition principle, the field of any source can be obtained. The field generated by the point source is called Green’s function. The Green’s function of the station and the surrounding stations is calculated through the cross-correlation function. The results are shown in Figure4.


### 4.3. Extraction and Analysis of the Station Noise Dispersion Curve

The dispersion curve represents the relationship between the period of the dispersion wave and the wave velocity. The dispersion curve of noise in red sandstone is extracted to prepare for the subsequent study of its deformation stage. Here, we use theF−k change method to extract the dispersion curve.t−x record φt,x, the number of channels is M, the number of sampling points is N, the records in f−k domain are Ff,k, and the collected discrete records are as follows:(2)fm,n=fx0+m∆x,t+n∆t,where m=0,…,M−1 and n=0,…,N−1;(3)Ff,k=∑m=0M−1∑n=0N−1fm,nWnf/M+mk/NW=e−2πj.Here,f=0,1,2,…,M−1,k=0,1,2,…,N−1, f represents frequency, k represents wavenumber, and Ff,k is the frequency wavenumber spectrum of ut,x. Using the relation v=f/k, the f−k domain is transformed into the f−v domain, and the calculation results are shown in Figure 5.


## 4.1. Noise Data Preprocessing

Noise data preprocessing is to transform the original noise waveform data extracted from the sensor into noise waveform data that can be used to calculate subsequent cross-correlation and Green’s function through format conversion, de-averaging and de-trending, band-pass filtering, time-domain normalized spectrum whitening, and other processing work.We export the waveform time domain image in the software and export the text format. Filter the data of the derived waveform, and select the noise waveform data that meets the threshold value. In the image shown in Figure7, the waveform with small amplitude and high frequency before the signal waveform is a noise waveform.Figure 7
Noise waveform.De-averaging means that each dimension subtracts the mean value of the corresponding dimension, so that each dimension of the input data is centered to 0. The reason for de-averaging is that it is not easy to fit without de-averaging. Detrended processing can eliminate the influence of the offset generated by the sensor when acquiring data on the later calculation. Deleting a trend from the data can focus the analysis on the fluctuations of the data trend itself. The processed data image is shown in Figure8.Figure 8
De-mean and de-trend.Filtering the frequency of a specific band in the waveform data is an important measure to suppress and prevent interference. Here, the noise signal is transformed into a waveform signal that meets the requirements, as shown in Figure9.Figure 9
Bandpass filtering.


## 4.2. Noise Data Cross Correlation

Before data cross-correlation processing, the maximum and minimum amplitudes of time domain maps are attributed to the set [−1, 1], and the scales of each feature are controlled within the same range, so that the optimal solution can be easily found. Data whitening is to reduce the dimension of data by discarding the dimensions with less information and retaining the main feature information. A few representative and unrelated features are used to replace the original large number of features with certain correlation, which is easy to analyze.Data whitening is to reduce the correlation between features. Features have the same variance and the covariance is 1. Normalization is to unify data to the same level of quantity. In the process of noise data whitening, the PCA operation is carried out first, and then the normalization operation is carried out. The processed image is shown in Figure10.Figure 10
Domain normalization and spectral whitening.In signal analysis, cross-correlation represents the degree of correlation between the values of the time series between two time series at any two different times, that is, the cross-correlation function describes the degree of correlation between the values of random signalsx (t) and y (t) at any two different times t1 and t2.The discrete calculation formula of cross-correlation is as follows:(1)Rxyn∆τ=1N−n∑1N−nxryn+r,where N is the number of sampling points and n is the delay sequence.We perform cross-correlation calculation on the noise waveform data collected by the two stations, and the calculation results are shown in Figure11.Figure 11
Cross-correlation function.The Green’s function is expressed as the relationship between the “field” and the “source” that produces the field. By using the superposition principle, the field of any source can be obtained. The field generated by the point source is called Green’s function. The Green’s function of the station and the surrounding stations is calculated through the cross-correlation function. The results are shown in Figure4.


## 4.3. Extraction and Analysis of the Station Noise Dispersion Curve

The dispersion curve represents the relationship between the period of the dispersion wave and the wave velocity. The dispersion curve of noise in red sandstone is extracted to prepare for the subsequent study of its deformation stage. Here, we use theF−k change method to extract the dispersion curve.t−x record φt,x, the number of channels is M, the number of sampling points is N, the records in f−k domain are Ff,k, and the collected discrete records are as follows:(2)fm,n=fx0+m∆x,t+n∆t,where m=0,…,M−1 and n=0,…,N−1;(3)Ff,k=∑m=0M−1∑n=0N−1fm,nWnf/M+mk/NW=e−2πj.Here,f=0,1,2,…,M−1,k=0,1,2,…,N−1, f represents frequency, k represents wavenumber, and Ff,k is the frequency wavenumber spectrum of ut,x. Using the relation v=f/k, the f−k domain is transformed into the f−v domain, and the calculation results are shown in Figure 5.


## 5. Research on the Identification Index of Deformation Stage

Select the frequency dispersion curve of four stages. Figure12(a) shows the frequency dispersion curve image of the fracture random expansion stage, and the corresponding time is the identification point 2. In this stage, the phase velocity of the noise signal of each station has a similar change trend within a certain frequency range, and the first frequency fluctuation range is about 500 Hz. Figure 12(b) shows the dispersion curve image of the deformation localization stage. For the time point 4, the variation trend of the phase velocity of the noise signal at each station in this stage is similar in a certain frequency range, and the first frequency variation fluctuation range is about 1000 Hz. Figure 12(c) shows the dispersion curve image of the subinstability stage, and the corresponding time is the identification point 6. The phase velocity change of the noise signal at each station at this stage gradually tends to be disordered, which is reflected in the change of the phase velocity value at the highest point of the first fluctuation change. The phase velocity value at the highest point of station 3 and station 4 is about 4000 m/s, while the phase velocity value at the highest point of station 1, station 2, and station 5 is about 10000 m/s, with a large difference. Figure 12(d) shows the dispersion curve image of instability and failure stage, and the corresponding time is the identification point 7. The phase velocity change of noise signals at each station is disordered, which is specifically reflected in the disordered change of frequency interval corresponding to the fluctuation of phase velocity change.Figure 12
Dispersion image. (a) Dispersion image of fracture random propagation stage. (b). Deformation localization phase dispersion image. (c) Sub instable phase dispersion image. (d) Frequency dispersion image in the instability stage.
(a)(b)(c)(d)Based on the above analysis, we can know the abnormal dispersion change in the late stage of deformation localization and the early stage of subinstability. We further take the identification point between the identification point 4 and the identification point 5 in the late stage of deformation localization and the early stage of subinstability to further explore the dispersion change between these two time points. Divide the interval between the identification point 4 and the identification point 5 equally, and take the three time points as time point 1, time point 2, and time point 3, respectively. According to Figure13(a), at this time, the point is in the late stage of deformation localization, and the peak value of the fluctuating phase velocity of the first change is large, of which station 1 is close to 10000 m/s, and the dispersion curves of the five stations have the same change trend and smooth curve. In Figure 13(b), the peak value of phase velocity generally decreases, and the change trend is gradually inconsistent, which is reflected in the unsmooth curve and different zero frequency values of phase velocity. Figure 13(c) the curve tends to be disordered. In conclusion, the variation trend of dispersion curve from the late stage of deformation localization to the stage of subinstability is from consistent to disordered.Figure 13
Time point dispersion image. (a) Time point 1. (b) Time point 2. (c) Time point 3.
(a)(b)(c)


### 5.1. Correlation Coefficient

In order to describe the distribution of fractures in red sandstone during deformation evolution through dispersion curve, the correlation statistical index of dispersion data is introduced according to the different dispersion degree of dispersion curve data of each station and identification point and the concept of correlation coefficient as follows:(4)Xt=corrX1,X2,…Xn,=∑1n∑1icorrXn,Xi/in,i=1,2,3…,n.The function corr (x, y) uses the Pearson product moment correlation coefficient to measure the relationship (linear correlation) between the two groups of data. The value range is [−1, +1]. The two groups of data used are the data matrix of the dispersion curve.(5)corrX,Y=covX,YσXσY,=∑i−1nXi−X¯Yi−Y¯∑i−1nXi−X¯2∑i−1nYi−Y¯2.It can be seen from Figure14 that the correlation coefficient of random fracture expansion stage is close to 1 in all periods of time point distribution, indicating that the correlation of each station is high in the early stage of random fracture development stage and deformation localization stage. In the middle and late period of deformation localization, the correlation coefficient in the subinstability stage decreases rapidly and reaches a relatively low point, indicating that the development of visible fractures in the subinstability stage affects the similarity of frequency dispersion characteristics between stations.Figure 14
Correlation coefficient.In the random fracture expansion stage of deformation evolution, due to the short loading time and the small loading stage, the red sandstone has small deformation inside and on the surface, resulting in random small fractures. The interference ability to the dispersion curve between stations is at a low level, so the correlation value is large. However, with the increase of displacement loading, the microfractures increase, and the correlation coefficient curve decreases to a certain extent in the middle and later stages of this stage. With the increase of time and load, the deformation evolution of red sandstone has entered the deformation localization stage. At this stage, the correlation coefficient of frequency dispersion curve decreases significantly, and it decreases rapidly in the later stage of localization stage. At this stage, observable cracks are formed on the surface of red sandstone, indicating that the generation of cracks has a great interference on the frequency dispersion of stations. In the subinstability stage, at the post peak stage of the loading curve, the load drops sharply, the damage to the red sandstone increases, the fractures continue to develop, the internal fractures increase, and the dispersion degree of the dispersion curve of each station increases, indicating that the dispersion characteristics of each station are different at this stage.On the back of the speckle surface, i.e., the surface of stations 1, 2, and 3, the correlation coefficient value is always at a high level, which is less affected by the fracture development in the deformation evolution of red sandstone. On the surface of stations 4 and 5, i.e., the left side of the speckle surface, the correlation coefficient value gradually decreases in the sub instability and instability stages, but at a relatively high level. For the stations crossing the plane, we give the correlation coefficient values between 1 and 4 stations and between 3 and 4 stations. Since 1 and 4 stations are not in the same plane with 3 and 4 stations, and their straight path passes through the red sandstone, it can be seen that the correlation coefficient values of these two curves reach a relatively low level in the subinstability stage and instability stage, indicating that the correlation coefficient between stations crossing the plane is low, it also shows that the correlation coefficient value of the station group in which the straight line passes through the red sandstone is low due to the internal fracture evolution in the deformation localization stage, subinstability stage, and instability stage. The outer surface of the red sandstone where 1, 2, and 3 stations are located is parallel to the speckle surface and perpendicular to the fracture. Compared with the correlation coefficients of all stations, stations 1, 2, and 3 are at a higher level, and the dispersion data error measured by the three stations is small. The outer surface of the red sandstone where stations 4 and 5 are located is parallel to the fracture surface, and the correlation coefficient is at a medium level. The linear path of 4 stations and the fracture surface have a certain angle value, and the linear path of 3 and 4 stations and the fracture surface have a certain angle value. The angle values of both are within the interval [0,π/2]. The correlation coefficients of these two groups of stations are at a low level.


### 5.2. Sensitive Kernel Function

In order to study the sensitivity of noise dispersion characteristics of background stations and calculate the sensitive kernel function of media at different depths, it is known from the above that the surface correlation coefficient of red sandstone where stations 1, 2, and 3 are located is high, so the test specimen is divided into two layers of media vertically up and down according to the horizontal plane where stations 1, 2, and 3 are located.Sensitive kernel function of phase velocity inversion of noise dispersion curve:(6)f=1m∑kaknk∑iciksVS−ciko2.Here,i and k are sampling frequency points and dispersion curve order, ciks is the phase velocity of the k th order dispersion curve at the i th frequency point, ciko is the phase velocity of the observed dispersion curve, m is the order of the participating dispersion curve, nk is the sampling points of the inverse k-th dispersion curve, ak is the dispersion curve weight, where the dispersion curve weight is 1. Using the dispersion curves of the four stages in Figure 12, the sensitive kernel function curve is drawn.It can be seen from the sensitive kernel function image in Figure15 that the variation trend of the frequency dispersion sensitive kernel coefficient collected at the measuring point at the same deformation evolution stage is relatively consistent in the two layers of media. In the four deformation evolution stages, the sensitive nuclear function value decreases gradually from low frequency to high frequency. It can be clearly observed in the instability stage that the value of sensitive kernel function of layer 1 medium is significantly higher than that of other stages, and there are periodic fluctuations.Figure 15
Sensitivity function of rock deformation evolution stage. (a, b) Sensitivity function of random crack growth stage. (c, d) Sensitivity function of deformation localization stage. (e) Sensitivity function of metastable stage. (f) Sensitivity function of unstable stage.
(a)(b)(c)(d)(e)(f)The program algorithm is used to find out the maximum difference of the sensitive kernel function corresponding to the same frequency of the two-layer medium, and the corresponding frequency is output. In the stage of random crack expansion, the sensitive kernel function curves of the two layers of red sandstone media are closely fitted, and the maximum difference is at a low level in the whole deformation evolution process. The red sandstone surface where stations 1, 2, and 3 are located is parallel to the speckle surface and perpendicular to the cracks evolved in the deformation localization stage of the speckle nephogram. In Figures15(a)–15(d), the peak value of the sensitive kernel function of the two layers of media is greater than the peak value of the sensitive kernel function of the one layer of media, and the difference gradually increases, It shows that there are great differences in frequency dispersion characteristics between 2 and 3 stations, and the fissures are mainly developed between 2 and 3 stations, corresponding to the evolution of the marked points 1, 2, 4, and 5 of the speckle cloud map. In the stage of subinstability and instability, the numerical peak value of layer 1 sensitive kernel function is significantly greater than that of layer 2 sensitive kernel function, and the corresponding speckle nephogram evolution is that the cracks mainly evolve between 1 and 2 stations.


## 5.1. Correlation Coefficient

In order to describe the distribution of fractures in red sandstone during deformation evolution through dispersion curve, the correlation statistical index of dispersion data is introduced according to the different dispersion degree of dispersion curve data of each station and identification point and the concept of correlation coefficient as follows:(4)Xt=corrX1,X2,…Xn,=∑1n∑1icorrXn,Xi/in,i=1,2,3…,n.The function corr (x, y) uses the Pearson product moment correlation coefficient to measure the relationship (linear correlation) between the two groups of data. The value range is [−1, +1]. The two groups of data used are the data matrix of the dispersion curve.(5)corrX,Y=covX,YσXσY,=∑i−1nXi−X¯Yi−Y¯∑i−1nXi−X¯2∑i−1nYi−Y¯2.It can be seen from Figure14 that the correlation coefficient of random fracture expansion stage is close to 1 in all periods of time point distribution, indicating that the correlation of each station is high in the early stage of random fracture development stage and deformation localization stage. In the middle and late period of deformation localization, the correlation coefficient in the subinstability stage decreases rapidly and reaches a relatively low point, indicating that the development of visible fractures in the subinstability stage affects the similarity of frequency dispersion characteristics between stations.Figure 14
Correlation coefficient.In the random fracture expansion stage of deformation evolution, due to the short loading time and the small loading stage, the red sandstone has small deformation inside and on the surface, resulting in random small fractures. The interference ability to the dispersion curve between stations is at a low level, so the correlation value is large. However, with the increase of displacement loading, the microfractures increase, and the correlation coefficient curve decreases to a certain extent in the middle and later stages of this stage. With the increase of time and load, the deformation evolution of red sandstone has entered the deformation localization stage. At this stage, the correlation coefficient of frequency dispersion curve decreases significantly, and it decreases rapidly in the later stage of localization stage. At this stage, observable cracks are formed on the surface of red sandstone, indicating that the generation of cracks has a great interference on the frequency dispersion of stations. In the subinstability stage, at the post peak stage of the loading curve, the load drops sharply, the damage to the red sandstone increases, the fractures continue to develop, the internal fractures increase, and the dispersion degree of the dispersion curve of each station increases, indicating that the dispersion characteristics of each station are different at this stage.On the back of the speckle surface, i.e., the surface of stations 1, 2, and 3, the correlation coefficient value is always at a high level, which is less affected by the fracture development in the deformation evolution of red sandstone. On the surface of stations 4 and 5, i.e., the left side of the speckle surface, the correlation coefficient value gradually decreases in the sub instability and instability stages, but at a relatively high level. For the stations crossing the plane, we give the correlation coefficient values between 1 and 4 stations and between 3 and 4 stations. Since 1 and 4 stations are not in the same plane with 3 and 4 stations, and their straight path passes through the red sandstone, it can be seen that the correlation coefficient values of these two curves reach a relatively low level in the subinstability stage and instability stage, indicating that the correlation coefficient between stations crossing the plane is low, it also shows that the correlation coefficient value of the station group in which the straight line passes through the red sandstone is low due to the internal fracture evolution in the deformation localization stage, subinstability stage, and instability stage. The outer surface of the red sandstone where 1, 2, and 3 stations are located is parallel to the speckle surface and perpendicular to the fracture. Compared with the correlation coefficients of all stations, stations 1, 2, and 3 are at a higher level, and the dispersion data error measured by the three stations is small. The outer surface of the red sandstone where stations 4 and 5 are located is parallel to the fracture surface, and the correlation coefficient is at a medium level. The linear path of 4 stations and the fracture surface have a certain angle value, and the linear path of 3 and 4 stations and the fracture surface have a certain angle value. The angle values of both are within the interval [0,π/2]. The correlation coefficients of these two groups of stations are at a low level.


## 5.2. Sensitive Kernel Function

In order to study the sensitivity of noise dispersion characteristics of background stations and calculate the sensitive kernel function of media at different depths, it is known from the above that the surface correlation coefficient of red sandstone where stations 1, 2, and 3 are located is high, so the test specimen is divided into two layers of media vertically up and down according to the horizontal plane where stations 1, 2, and 3 are located.Sensitive kernel function of phase velocity inversion of noise dispersion curve:(6)f=1m∑kaknk∑iciksVS−ciko2.Here,i and k are sampling frequency points and dispersion curve order, ciks is the phase velocity of the k th order dispersion curve at the i th frequency point, ciko is the phase velocity of the observed dispersion curve, m is the order of the participating dispersion curve, nk is the sampling points of the inverse k-th dispersion curve, ak is the dispersion curve weight, where the dispersion curve weight is 1. Using the dispersion curves of the four stages in Figure 12, the sensitive kernel function curve is drawn.It can be seen from the sensitive kernel function image in Figure15 that the variation trend of the frequency dispersion sensitive kernel coefficient collected at the measuring point at the same deformation evolution stage is relatively consistent in the two layers of media. In the four deformation evolution stages, the sensitive nuclear function value decreases gradually from low frequency to high frequency. It can be clearly observed in the instability stage that the value of sensitive kernel function of layer 1 medium is significantly higher than that of other stages, and there are periodic fluctuations.Figure 15
Sensitivity function of rock deformation evolution stage. (a, b) Sensitivity function of random crack growth stage. (c, d) Sensitivity function of deformation localization stage. (e) Sensitivity function of metastable stage. (f) Sensitivity function of unstable stage.
(a)(b)(c)(d)(e)(f)The program algorithm is used to find out the maximum difference of the sensitive kernel function corresponding to the same frequency of the two-layer medium, and the corresponding frequency is output. In the stage of random crack expansion, the sensitive kernel function curves of the two layers of red sandstone media are closely fitted, and the maximum difference is at a low level in the whole deformation evolution process. The red sandstone surface where stations 1, 2, and 3 are located is parallel to the speckle surface and perpendicular to the cracks evolved in the deformation localization stage of the speckle nephogram. In Figures15(a)–15(d), the peak value of the sensitive kernel function of the two layers of media is greater than the peak value of the sensitive kernel function of the one layer of media, and the difference gradually increases, It shows that there are great differences in frequency dispersion characteristics between 2 and 3 stations, and the fissures are mainly developed between 2 and 3 stations, corresponding to the evolution of the marked points 1, 2, 4, and 5 of the speckle cloud map. In the stage of subinstability and instability, the numerical peak value of layer 1 sensitive kernel function is significantly greater than that of layer 2 sensitive kernel function, and the corresponding speckle nephogram evolution is that the cracks mainly evolve between 1 and 2 stations.


## 6. Conclusion

From the deformation evolution nephogram and frequency dispersion curve of red sandstone, the following can be concluded:(1)
In the stage of random propagation of microcracks, the rock will make the waveform pass through the low frequency range at a higher phase velocity.(2)
In the deformation localization stage, the noise dispersion curve of red sandstone appears periodic fluctuation, and the fluctuation periodic frequency interval is larger than that in the random fracture development stage.(3)
From the deformation evolution stage and dispersion curve of red sandstone, the change trend is from highly consistent (smooth curve) to gradually disordered, and then to completely disordered, which proves that the dispersion property of noise in the loading stage develops from orderly to disordered.(4)
In the instability stage, the cracks of red sandstone have evolved, and the noise dispersion curve data collected at the measuring points are more disordered under the influence of cracks.(5)
The size and difference of the sensitive kernel functions of two layers of stations are related to the evolution region of fractures. The sensitive kernel value of the medium layer where the fracture develops is high. The longer the evolution time, the greater the difference, which corresponds to the fracture development of the speckle surface displacement evolution cloud map.



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*Source: 2149273-2022-12-26.xml*