Acoustic characteristics of rocks

Sound wave is a form of matter movement, which is generated by mechanical vibration of matter, and the vibration is transmitted from near to far through the interaction between particles. The frequency of sound that the human ear can hear is between 20 Hz and 20 kHz. Waves with a frequency greater than 20 kHz are called ultrasonic waves. Generally, the acoustic frequency used in acoustic logging is 15 ~ 20kHz.

For acoustic waves emitted by acoustic logging, underground rocks can be regarded as elastic media, which can produce shear elastic deformation and compression elastic deformation under the action of acoustic vibration. Therefore, rocks can propagate both shear waves and longitudinal waves, and the propagation speed is closely related to the elasticity of rocks.

2. 1. 1 rock elasticity

An object that is deformed by an external force and can be restored to its original state after the external force is removed is called an elastomer, and the deformation of an elastomer is called elastic deformation. An object that cannot be restored to its original state after the external force is cancelled is called a plastic body.

Whether an object is elastic or plastic is not only related to the nature of the object itself, but also related to the magnitude of external force, the length of action time, the mode of action and other factors. Generally speaking, the external force is small, the action time is short, and the object is elastic.

In acoustic logging, the acoustic energy emitted by the sound source is small, and it acts on the rock for a short time, so for sonic logging, the rock can be regarded as an elastic body. Therefore, we can use the propagation law of elastic waves in media to study the propagation characteristics of sound waves in rocks.

In a uniform and infinite rock, the sound wave velocity mainly depends on the elasticity and density of the rock. As an elastic medium, the elasticity of rock can be described by the following parameters.

(1) Young's modulus

The deformation per unit length of elastic body is called strain, the elastic force per unit cross-sectional area is called stress, and Young's modulus is the ratio of stress to strain, expressed by E, and the unit is N/m2.

(2) Poisson's ratio

Under the action of external force, the elastic body extends longitudinally and contracts transversely. The ratio of transverse relative shrinkage to longitudinal relative elongation is called Poisson's ratio, expressed by σ, and its dimension is 1. Poisson's ratio is only a coefficient representing the geometric deformation of an object. For all substances, σ is between 0 and 0.5.

(3) Shear modulus

Shear modulus refers to the ratio of shear stress to shear strain of elastic body under shear stress, expressed in μ, and the unit is N/m2.

(4) Elastic modulus of volume deformation

Under the action of external force, the ratio of volume strain to stress is called the elastic modulus of volume deformation, which is expressed in k and the unit is N/m2. The reciprocal of the elastic modulus of volume deformation is called the volume compression coefficient, which is expressed by β.

2. 1.2 Propagation characteristics of sound waves in rocks

Rock can be regarded as elastic body, so the propagation characteristics of sound wave in rock can be studied by the propagation law of elastic wave in medium.

The propagation of elastic wave in medium is essentially the sequential propagation of particle vibration. The wave whose propagation direction is consistent with the particle vibration direction is called longitudinal wave. In the process of longitudinal wave propagation, the medium undergoes volume deformation of compression and expansion, so longitudinal wave is also called compression wave. Waves whose propagation direction and particle vibration direction are perpendicular to each other are called shear waves. During the propagation of shear wave, the medium undergoes shear deformation, so shear wave is also called shear wave. Usually, these two waves propagate simultaneously in medium, but shear waves cannot propagate in liquid and gas.

The propagation speed of sound wave in elastic medium mainly depends on the elastic modulus and density of the medium. In homogeneous isotropic media, the relationships between P-wave velocity vp, S-wave velocity vs and Young's modulus of elasticity E, Poisson's ratio σ and density ρ are as follows:

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Comparing formula (2. 1. 1) and formula (2. 1.2) shows that the P-wave velocity is always greater than the S-wave velocity. When the Poisson's ratio σ of rock is 0.25, the velocity of P-wave is 1.73 times that of S-wave, and P-wave is always received earlier than S-wave.

Equations (2. 1. 1) and (2. 1.2) show that the velocities of longitudinal and shear waves of rocks depend on the Young's modulus and density of the medium, and the velocities of longitudinal and shear waves will increase with the increase of Young's modulus.

For sedimentary rocks, acoustic velocity is related to the following geological factors in addition to the above basic factors.

(1) lithology

Because the elastic modulus of different minerals is different, and the elastic modulus of the medium is the main factor affecting the sound speed of the medium, the sound speed of rocks composed of different minerals is also different. The P-wave velocities of some common media and sedimentary rocks are shown in Table 2. 1. 1 (the time difference in the table refers to the reciprocal of the velocity).

Table 2. 1. 1 P-wave velocity and time difference of common media and sedimentary rocks

(2) Porosity

Rock pores are usually filled with fluid media such as oil, gas and water, and the elastic modulus and density of these pore fluids are different from those of rock skeleton. Obviously, the porosity of rock and the elastic modulus and density of pore fluid have obvious effects on the sound velocity of rock. It can be seen from Table 2. 1. 1 that the pore fluid is a low-speed medium relative to the rock skeleton, so the greater the porosity, the smaller the sound velocity of rocks with the same lithology and constant pore fluid.

(3) the geological age of rock strata

Many practical observation data show that rocks with the same depth and similar composition have different sound velocities at different geological ages. The velocity of sound in the old stratum is higher than that in the new stratum.

(4) Buried depth of rock stratum

Many practical observation results show that under the same lithology and geological age, the sound speed increases with the deepening of rock burial depth. This change is due to the increase of Young's modulus of elasticity of rock due to the increase of overlying strata pressure. When the buried depth of shallow stratum increases, the sound velocity changes dramatically; In deep strata, with the increase of buried depth, the sound velocity changes little.

From the above analysis, it can be seen that we can study the rock stratum according to the sound velocity of the rock and determine the lithology and porosity of the rock stratum.

2. 1.3 Propagation characteristics of sound waves on the medium interface

Sound waves will be reflected, transmitted and refracted when they pass through the interface of two media with different propagation speeds. As shown in Figure 2. 1. 1, two media with sound velocity of v 1 and v2 are in close contact, and there is a sound source in the upper medium. When the sound wave emitted by the sound source reaches the interface at the incident angle α, it will produce reflected waves, transmitted waves, refracted waves, sliding waves and total reflection waves with different incident angles α.

Fig. 2. 1. 1 sound wave propagation on medium interface

When the incident angle α of incident wave is less than a certain critical angle I, reflected wave and transmitted wave appear. After the incident wave reaches the interface, part of it is reflected by the interface to form a reflected wave. According to the law of reflection, the reflection angle is equal to the incident angle. Another part of the incident wave propagates through the interface in the lower medium, forming a transmitted wave. According to the transmission law, the transmission angle β is not only related to the incident angle, but also related to the wave velocity ratio of the two media, and they satisfy the following relationship:

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Because v 1 and v2 are constant values for a certain medium, the transmission angle β also increases with the increase of incident angle α, such as V2 >: under the condition of v 1, β >; α. When the incident angle increases to a certain angle i, the transmission angle reaches 90. At this time, the transmitted wave will slide along the interface at the speed of v2 in the lower medium, which is called sliding wave in acoustic logging, and the angle I at this time is called critical angle.

The propagation of sound wave is essentially the propagation of particle vibration. When the sliding wave slides forward along the interface, any point it passes can be regarded as a new point vibration source from that moment on. Due to the close contact between the upper and lower media, the vibration of any point where the sliding wave passes will inevitably cause the vibration of particles in the upper media, and a new wave will be formed in the upper media, which is called refracted wave. The refraction angle of the refracted wave is equal to the critical angle I, and the refracted wave is a cluster of parallel straight lines.

When the incident angle α is greater than the critical angle I, all the incident waves are reflected back to the upper medium, forming a total reflection wave.