Jin Xiong , Kiyun Liu , Xingjun Liu , Lixi Ling , Chongyng Zhng

a State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, China

b Geophysical Technology Research Center of BGP INC., China National Petroleum Corporation, Chengdu, China

ABSTRACT The acoustic response characteristics of shales were investigated by the acoustic transmission experiment, which is the basis of solving geological and engineering problems using the seismic or logging information during the process of the exploration and development of shale gas reservoirs.Based on the theory of acoustic wave and the background of acoustic transmission experiment, the initial condition,vibration source condition, boundary condition and stability condition were constructed, and the numerical simulation of acoustic transmission experiment of shales were completed through Matlab programming.The results show that under the same bedding angle, the acoustic time and attenuation coefficient of shales shown positive correlation with the bedding density; whereas under the same bedding density, the variation laws of the acoustic time and the attenuation coefficient of shales were more complex with the change of the bedding angle,that is,the acoustic time and attenuation coefficient of shales increased first,then decreased and then increased again with the increase of the bedding angle.

Keywords:Shale Bedding characteristics Acoustic wave Numerical simulation

1.Introduction

With the adjustment of China’s energy structure and the increasingly prominent environmental problems, natural gas, as a clean energy source, has received more and more attention, especially the unconventional oil and gas resources,among which shale gas has become a hot spot in the global exploration and exploitation of unconventional oil and gas resources.At the same time,the U.S.Energy Information Administration (EIA) released the results report of shale gas resource evaluation in 42 countries including the US[1],which pointed out that the technical recoverable amount of shale gas in the world is 220.73×1012m3,indicating there are the great potential of shale gas resources in the world.At present, the petrophysical characteristics of the shales were the hot spots of research, and the research work mainly focused on mineralogical characteristics of shales [2-5], pore structure characteristics of shales [6-8], physical and chemical properties of shales [9-12],mechanical properties of shales[13-15]and so on.However,there were a few studies on the acoustic wave propagation characteristics of shales [16-18].The research methods of acoustic wave propagation of rock mainly include acoustic transmission experiment and numerical simulation method.Domnesteanu et al.[16], Chen et al.[17], and Xu et al.[18] analyzed the P-wave and S-wave velocity, attenuation and frequency response of shales by means of ultrasonic transmission experiment and numerical simulation method.Baechle et al.[19] and Weger et al.[20] using different resolution electron microscopy including X - ray resolution of 3.1 μs,EPMA of 1 μs,and SEM of 0.1 μs,and the casting thin sections digital image to analysis and extract the pore size,spherical degree,porosity,the complexity of the pore space network,which could be used to describe the pore space quantitatively,and on this basis,the numerical simulation method of acoustic wave propagation of rocks was established.Chen et al.[21,22], Wang et al.[23], and Liang et al.[24] carried out research on acoustic attenuation coefficient of fracture model based on the two-dimensional acoustic numerical simulation method.These research results have gained certain understandings of acoustic wave propagation characteristics of rocks, which also shows that two-dimensional acoustic numerical simulation technology is also an effective method to investigate the acoustic wave propagation characteristics of rocks.

Fig.1.Some shale samples under different bedding angle (the solid line is axial direction and the dashed line is bedding plane direction).

At present, some researchers had mainly studied the acoustic wave propagation characteristics of rocks by the acoustic transmission method,and some understandings had also gained.As the complexity of the bedding structure of shales, it is not enough to just investigate the acoustic characteristics of shales only by the physical simulation models.Therefore, it is necessary to use numerical simulation technology to investigate the propagation characteristics and response characteristics of acoustic wave of shales.So, this paper intends to take the shales as the research object, the numerical simulations of the acoustic transmission experiment based on the wave theory were carried out, and the influences of the bedding density and bedding angle on the acoustic wave propagation characteristics of shales were also discussed.

2.Numerical simulation method

2.1.Laboratory test

The shale samples are collected from the lower Silurian Longmaxi Formation in the Changning area of Sichuan Basin, and the cores are shown in Fig.1.From Fig.1,we can note that the bedding can be observed clearly.According to the statistics of the bedding density of the Longmaxi Formation shales in this area,the bedding density of the shales ranges from 0.155 piece/mm to 0.889 piece/mm (simplified pc/mm), with an average of 0.449 pc/mm.The acoustic wave transmission method is used to measure the acoustic wave of the Longmaxi Formation shales at different bedding angles,and the results are shown in Fig.2.From Fig.2,we can be note that the changing rules of the acoustic time of shales are complexity with the increase of the bedding angle.The acoustic time of shales first increases with the increase of the bedding angle, and at the angle of 30°, the acoustic time of shales reaches the maximum;after that the acoustic time of shales go down with the increase of the bedding angle,and at the angle of 70°,the acoustic time goes to the minimum,then the acoustic time increases with the increase of the bedding angle.This may be related to the inconsistent bedding density of each test sample.It shows that the acoustic anisotropy of shales is relatively significant.The bedding characteristics,such as the bedding density and the bedding angle, have an important influence on the acoustic wave propagation characteristics of shales,whereas how the influences of the bedding density and the bedding angle on the acoustic wave propagation characteristics of shales were not clear enough.Therefore, two-dimensional numerical simulation method of acoustic wave propagation of rock is adopted to further investigate the influences of the bedding characteristics on the acoustic characteristics of shales.

2.2.Numerical model setup

2.2.1.Wave theory

Suppose the displacement of any point (x, y) in the twodimensional space at time t is U, then the two-dimensional acoustic equation is:

Where:V2(x,y) is the velocity of the P-wave.

And the difference expression of two-dimensional wave equation can be obtained by discretization:

Where: Δx,Δyare the horizontal and vertical spacing of the difference grid,respectively;M,Nis the number of grid in thexandydirection, respectively.When 0 ≤x≤a0 ≤y≤b, the Δx=a/MandΔx=b/N.

Fig.2.Relationship between the acoustic time and the bedding angle.

Presuming that:

Then the discretization form of the apparent difference approximation for the two-dimensional acoustic wave equation is:

2.2.2.Initial conditions

Presuming a vibration source function and the initial condition is:

The discretization description of the above expression is:

When t=0,the displacement velocity is zero,that is,when the t = 0, the discretization description is:

And combination Eq.(7) with Eq.(4), the initial equation for difference calculation can be gain:

2.2.3.Boundary conditions

If the upper and lower boundaries (y= 0,y=b) adopt the absorption boundary[25],then the expression of the upper and lower boundaries is:

The left and right boundary is the normal reflection boundary,the displacement of the particle on the boundary(x=0,x=a)is 0,then the boundary conditions can be expressed as:

2.2.4.Stability and convergence conditions

When the error of the results of difference equation does not increase with the time,the results of difference equation is stable.In two dimensions,using the Von Neumann method based on Fourier Expansion, for a space step (Δx,Δz) when t = 0, the difference equation and the solution of differential equation are defined as:

Wherenis imaginary unit,m=(0,1,2,···),qrepresents the error of the results of difference and differential equation in a time step.According to the study of Mitchell and Griffiths[26],Eq.(13)satisfy the disposition equation,and associate Eq.(4)can be used to obtain the results ofq, that is:

Fig.3.The numerical calculation models under different bedding angles.

Fig.4.Relationship between the acoustic time and the bedding density.

If|q|≤1,the error of solution of difference equation would not increase with the time step, indicating that the results of the difference equation is stable.And in order to satisfy the condition that|q|≤1, the following equation would be true.

2.2.5.Attenuation treatment

The attenuation phenomenon exists when the acoustic wave propagates in the medium, and the attenuation coefficient is a physical quantity used to measure the inelastic attenuation rate of acoustic wave vibration or wave energy in the medium.

The amplitude of vibration changes with the increase of distance.According to the exponential decay law,the equation can be as:

WhereAis the amplitude of acoustic wave after a certain distance;A0is the maximum amplitude of acoustic wave on the surface of the emitting probe; a is the attenuation coefficient of acoustic wave;L is the length of the core.

At present, the methods characterizing the acoustic attenuation coefficient of rocks in laboratory experiments include the long and short rock sample comparison method, the standard sample comparison method and the signal comparison method.Because the rock matrix part in the numerical model is isotropic linear elastomer,and the hole part is air filled, the attenuation in this paper is the comprehensive effect of diffusion attenuation and scattering attenuation,andtheabsorptionattenuationisnotconsidered.Inthispaper,the first wave amplitude of incident wave and transmission wave is read to get the attenuation coefficient through the signal comparison method.The calculation formula of acoustic attenuation coefficient of rock sample calculated by signal comparison method is as follows:

2.3.Numerical calculation model

Based on the wave theory and the finite difference method,the simulation experiment of acoustic transmission of shales is programmed.The complex shale structure models can be constructed through changing the angel of bedding and the density of plies(shown in Fig.3).The long of the models is 50 mm,and the wide of the models is 25 mm, the bedding plane density ranges from 0.2 piece/mm to 1 piece/mm (simplified pc/mm), and the bedding plane angle ranges from 0°to 90°,including a total of 50 numerical models.In the process of ultrasonic transmission simulation,isotropic linear elastomer is filled in the matrix part and bedding part of the two-dimensional numerical model, with the time accuracy of 10ns and the spatial accuracy of 0.1 mm.The excitation signal of the 250 KHz in the laboratory was used as the numerical simulation vibration source.

Fig.5.The snapshot of wave field under the bedding angle of 90° with different bedding density.

3.Results and discussions

3.1.The effects of the bedding density

The relationships between the acoustic time and the bedding density under different bedding angles are shown in Fig.4.From Fig.4, it can be seen that under the same bedding angle, the acoustic time of shales increases linearly with the increase of the bedding density.When the angle is 20°,the acoustic time increases the sharpest,and compared with horizontal bedding direction,the wave velocity in the vertical direction decline quite small.It can be seen in Fig.5(the snapshot of wave field under the bedding angle of 90°with different bedding density)that in the direction of acoustic wave propagation,as the bedding density increases,the number of the acoustic waves passing through the interface between the bedding and the skeleton gradually increases,resulting in a gradual decrease of the wave velocity, that is, the acoustic time gradually increases.This indicates that the bedding density has a great influence on the acoustic time,and the higher the bedding density is,the higher the acoustic time would be.

Under different bedding angles, the variation laws of the attenuation coefficient with the bedding density can be seen in Fig.6.From Fig.6, we can note that the attenuation coefficient increases linearly with the increase of the bedding density under the condition of the same bedding angle.This is because that with the increase of the bedding number under the same length of core skeleton, the heterogeneity increases, and the modulus between the core and the skeleton are different,causing that a certain strain phase difference could be generated between the skeleton and the bedding.As a result, internal friction is generated at the interface between the skeleton and the bedding,which turns the mechanical energy into the thermal energy,indicating that the lot of the energy could be greatly consumed.At the same time, as the bedding density increases, the propagation path of the acoustic wave increases, resulting in the increase of the attenuation coefficient of the acoustic wave.

3.2.The effects of the bedding angle

Fig.6.Relationship between the attenuation coefficient and the bedding density.

Fig.7 shows the numerical results of the acoustic time with bedding angle under different bedding densities.From Fig.7,it can be observed that under the same bedding density, the variation laws of the acoustic time with the bedding angle is complex.Under the same bedding density, at a low bedding angle (less than 20°),the effects of bedding angle on acoustic time are obvious, namely there is a trend of increase in acoustic time with the increase of the bedding angle,and when the bedding density is larger,the acoustic time increase faster.This may be due to with the increase of the bedding angle,the angle between the ultrasonic wave propagation direction and the bedding plane becomes larger, the passage through the matrix part becomes smaller during the process of the acoustic wave propagation, the number of bedding planes passing through increases, and the path of the acoustic wave propagation increases,resulting in the decrease of the wave velocity or increase of the acoustic time.With the increase of the bedding angle, the wave velocity increases and the acoustic time decreases.When the bedding angle is 80°,the acoustic time reaches the minimum,then the acoustic time increase again.This may be because when the bedding angle increases to a certain extent, the existence of the bedding may not necessarily lead to the decrease of the wave velocity.Different wave impedance interfaces (layer surface structures) may have an impact on reflection and refraction during the process of the propagation of the acoustic wave, resulting in the increase of the wave velocity with the increase of the bedding angle.The variation laws of acoustic time of shales with the bedding angle is similar to the experimental result,indicating that the influences of the bedding angle on the propagation of the acoustic wave is quite complicated.

Under the conditions of different bedding densities, the variation laws of the attenuation coefficient with the bedding angle can be seen in Fig.8.Under the same bedding density,the attenuation coefficient of shales increases first and then decreases slightly with the increase of the bedding angle.This is because that with the increase of the bedding angle,the angle between the acoustic wave propagation direction and the bedding plane becomes larger, the part passing through the matrix is reduced during the process of the acoustic wave propagation, the number of bedding planes passing through increases, the total area of wave impedance interface contact during the process of the acoustic wave propagation increases,and the attenuation effect caused by the bedding angle increases.When the bedding angle increases to 70°, the acoustic time shows a downward trend.This may be because the bedding angle increases to a certain extent, the effect of the bedding angle on the attenuation coefficient of shales gets weak.

4.Conclusions

(1) Based on the wave theory,the numerical simulation method is established by setting the corresponding initial conditions,vibration source conditions, absorbing boundary conditions and stability conditions, and using Matlab to solve the calculation of high-order finite-difference to simulate the ultrasonic transmission experimental environment.

Fig.7.Relationship between the acoustic time and the bedding angle.

Fig.8.Relationship between the attenuation coefficient and the bedding angle.

(2) Under the same bedding angle, the acoustic time and attenuation coefficient of shales show positive correlation with the bedding density.

(3) Under the same bedding density, the variation laws of the acoustic time and the attenuation coefficient are rather complex, that is, the acoustic time increases first, then decreases and then increases again with the increase of the bedding angle,and the attenuation coefficient increases first and then decreases slightly with the increase of the bedding angle.

Declaration of competing interests

The authors declare that they have no conflict of interests.

Acknowledgements

This research is supported by the National Science and Technology Major Project(Grant No.2019A-3307),the National Natural Science Foundation of China (Grant No.41872167).