A method for calculating the dispersive attenuation of longitudinal waves incident from any direction in a horizontal layered medium

Abstract

The present invention discloses a method for calculating the dispersion attenuation of a longitudinal wave incident in any direction in a horizontal layered medium, comprising the following steps: establishing a periodic layered medium model; setting a stiffness matrix of the periodic layered medium model; calculating various parameters of the stiffness matrix; calculating the longitudinal wave velocity at any incident angle according to the stiffness matrix; and calculating the longitudinal wave velocity dispersion and attenuation at any incident angle according to the longitudinal wave velocity. The present invention calculates the change of the longitudinal wave velocity in the model by calculating the stiffness matrix of the periodic layered medium model, thereby analyzing the velocity dispersion and energy attenuation characteristics of the longitudinal wave when incident at any angle.

Description

Technical Field

[0001] The invention relates to the field of seismic exploration, and in particular to a method for calculating the dispersion attenuation of incident longitudinal waves in any direction in a horizontal layered medium. Background Art

[0002] In the article "The influence of mesoscopic flow on the P-wave attenuation and dispersion in a porous media permeated by aligned fractures" published by Deng Jixin et al. in 2012, they mainly studied the influence of fluid flow on the dispersion and attenuation of P-waves in porous media permeated by aligned fractures. They analyzed the influence of fluid flow on the dispersion and attenuation of P-waves when the P-waves are perpendicular to the fractures. In fact, when P-waves propagate in layered media, their propagation direction can be in any direction; it is of great significance to study the dispersion and attenuation of P-waves propagating in any direction, but there is no record of this in the existing literature. Summary of the invention

[0003] In order to solve the above technical problems, the present invention provides a method for calculating the dispersion attenuation of incident longitudinal waves in any direction in a horizontal layered medium with a simple algorithm and high accuracy.

[0004] The technical solution of the present invention to solve the above technical problem is: a method for calculating the dispersion attenuation of an incident longitudinal wave in an arbitrary direction in a horizontal layered medium, comprising the following steps:

[0005] Step 1: Establish a periodic layered medium model;

[0006] Step 2: Set the stiffness matrix of the periodic layered medium model;

[0007] Step 3: Calculate the parameters of the stiffness matrix;

[0008] Step 4: Calculate the longitudinal wave velocity at any incident angle based on the stiffness matrix;

[0009] Step 5: Calculate the P-wave velocity dispersion and attenuation at any incident angle based on the P-wave velocity.

[0010] The above method for calculating the dispersion attenuation of incident longitudinal waves in any direction in a horizontal layered medium, in the step 1, the periodic layered medium model is composed of two horizontal porous medium layers A and B that are staggered and cyclically extended infinitely in the horizontal direction, and an equivalent volume element is selected at the junction of the A and B layers. The equivalent volume element is located exactly in the middle of the A and B layers, the upper surface of the equivalent volume element is located in the middle of the A layer, and the lower surface is located in the middle of the B layer. A uniform horizontal solid displacement is applied to the left and right sides of the equivalent volume element, and the other boundaries of the equivalent volume element, namely the top and bottom boundaries, are restricted; other influencing factors are not considered, and fluid is not allowed to flow into and out of this model, and shear stress is not applied. Under this strain condition, the upper and lower layers of the equivalent volume element satisfy the Biot equation.

[0011] In the above method for calculating the dispersion attenuation of incident longitudinal waves in any direction in a horizontal layered medium, in step 2, for the established periodic layered medium model, in the long-wave limit, a layered medium composed of multiple layers of transversely isotropic materials is equivalently anisotropic, and its equivalent stiffness is:

[0012]

[0013] Where <·> indicates that the attributes in brackets are weighted averaged according to volume ratio; is the elastic constant of the porous medium, Represents the stress component in the x-direction of a certain stress combination that causes the model to undergo unit strain in the x-direction; Represents the stress component in the y direction of a certain stress combination that causes unit strain in the model in the x direction; Represents the stress component in the z direction of a certain stress combination that causes unit strain in the model in the x direction; M represents the stress component in the z direction of a certain stress combination that causes the model to have unit strain in the z direction; 44 、M 66 are the elastic parameters of periodic layered media in different directions, For the corresponding M 44 , M 66 Shear modulus in the direction; M 44 =M 66 =G, G is the shear modulus of the solid skeleton.

[0014] In the above method for calculating the dispersion attenuation of incident longitudinal waves in any direction in a horizontal layered medium, in step 3, the parameters of the stiffness matrix are have:

[0015]

[0016] Among them, λ1 and λ2 are intermediate quantities:

[0017] For the stiffness matrix parameters and have:

[0018]

[0019]

[0020] Where l1 and l2 are the distances from the upper and lower interfaces of the equivalent element to the junction, and the height of the equivalent element is L = l1 + l2. are the uniaxial strain moduli under drainage conditions in the upper and lower media, are the uniaxial strain moduli under non-drainage conditions in the upper and lower media, η1 and η2 are the dynamic viscosities of the pore fluids in layers A and B, and the intermediate are the M corresponding to the porous media in layers A and B respectively. 11 , are the M corresponding to the porous media in layers A and B respectively. 12 , a1 and a2 represent the Biot coefficients of the porous media in layers A and B, respectively, k1 and k2 represent the rock permeabilities of the porous media in layers A and B, respectively, i is an imaginary unit, ω is the frequency, γ1 and γ2 are the Skempton coefficients of the porous media in layers A and B under uniaxial strain conditions, respectively.

[0021] The above method for calculating the dispersion attenuation of incident longitudinal waves in any direction in a horizontal layered medium, in step 4, the velocity of the longitudinal wave in the equivalent medium is for:

[0022]

[0023] The intermediate amount θ is the angle between the incident longitudinal wave and the z-axis, and ρ is the average density of the equivalent volume element.

[0024] The above method for calculating the dispersion attenuation of incident longitudinal waves in any direction in a horizontal layered medium, in step 5, the longitudinal wave phase velocity V p and the inverse quality factor of the longitudinal wave attenuation Calculated by:

[0025]

[0026] Where Re represents the real part of the corresponding quantity, Im represents the imaginary part of the corresponding quantity, and ρ represents the average density of the equivalent volume element.

[0027] The beneficial effects of the present invention are as follows: the present invention first establishes a periodic layered medium model; then sets the stiffness matrix of the periodic layered medium model; then calculates various parameters of the stiffness matrix, and calculates the longitudinal wave velocity at any incident angle according to the stiffness matrix; finally, calculates the longitudinal wave velocity dispersion and attenuation at any incident angle according to the longitudinal wave velocity. The present invention calculates the change of the longitudinal wave velocity in the model by calculating the stiffness matrix of the periodic layered medium model, thereby analyzing the velocity dispersion and energy attenuation characteristics of the longitudinal wave when incident at any angle. BRIEF DESCRIPTION OF THE DRAWINGS

[0028] Figure 1 It is a flow chart of the present invention.

[0029] Figure 2 Schematic diagram of velocity dispersion in porous media when longitudinal waves are incident in any direction.

[0030] Figure 3 Schematic diagram of the attenuation characteristics of longitudinal waves in porous media when they are incident in any direction. DETAILED DESCRIPTION

[0031] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0032] like Figure 1 As shown, a method for calculating the dispersion attenuation of an incident longitudinal wave in a horizontal layered medium in any direction comprises the following steps:

[0033] Step 1: Establish a periodic layered medium model.

[0034] The periodic layered medium model consists of two horizontal porous medium layers A and B that are interlaced and infinitely extended in the lateral direction. An equivalent volume element is selected at the junction of the A and B layers. The equivalent volume element is located exactly in the middle of the A and B layers. The upper surface of the equivalent volume element is located in the middle of the A layer, and the lower surface is located in the middle of the B layer. Uniform horizontal solid displacement is applied to the left and right sides of the equivalent volume element, while the other boundaries of the equivalent volume element, namely the top and bottom boundaries, are restricted. Other influencing factors are not considered, and fluid is not allowed to flow into and out of this model, and shear stress is not applied. Under this strain condition, the upper and lower layers of the equivalent volume element satisfy the Biot equation.

[0035] Step 2: Set the stiffness matrix of the periodic layered medium model.

[0036] For the established periodic layered medium model, in the long-wave limit, a layered medium composed of multiple layers of transversely isotropic materials is equivalently anisotropic, and its equivalent stiffness is:

[0037]

[0038] Where <·> indicates that the attributes in brackets are weighted averaged according to volume ratio; is the elastic constant of the porous medium, Represents the stress component in the x-direction of a certain stress combination that causes the model to undergo unit strain in the x-direction; Represents the stress component in the y direction of a certain stress combination that causes unit strain in the model in the x direction; Represents the stress component in the z direction of a certain stress combination that causes unit strain in the model in the x direction; M represents the stress component in the z direction of a certain stress combination that causes the model to have unit strain in the z direction; 44 、M 66 are the elastic parameters of periodic layered media in different directions, For the corresponding M 44 , M 66 Shear modulus in the direction; M 44 =M 66 =G, G is the shear modulus of the solid skeleton.

[0039] Step 3: Calculate the parameters of the stiffness matrix.

[0040] For the parameters of the stiffness matrix have:

[0041]

[0042] Among them, λ1 and λ2 are intermediate quantities:

[0043] For the stiffness matrix parameters and According to the article "The influence of mesoscopic flow on the P-wave attenuation and dispersion in aporous media permeated by aligned fractures" published by Deng Jixin et al. in 2012, we know that:

[0044]

[0045]

[0046] Where l1 and l2 are the distances from the upper and lower interfaces of the equivalent element to the junction, and the height of the equivalent element is L = l1 + l2. are the uniaxial strain moduli under drainage conditions in the upper and lower media, are the uniaxial strain moduli under non-drainage conditions in the upper and lower media, η1 and η2 are the dynamic viscosities of the pore fluids in layers A and B, and the intermediate are the M corresponding to the porous media in layers A and B respectively. 11 , are the M corresponding to the porous media in layers A and B respectively. 12 , a1 and a2 represent the Biot coefficients of the porous media in layers A and B, respectively, k1 and k2 represent the rock permeabilities of the porous media in layers A and B, respectively, i is an imaginary unit, ω is the frequency, γ1 and γ2 are the Skempton coefficients of the porous media in layers A and B under uniaxial strain conditions, respectively.

[0047] Step 4: Calculate the longitudinal wave velocity at any incident angle based on the stiffness matrix.

[0048] The velocity of longitudinal waves in the equivalent medium for:

[0049]

[0050] The intermediate amount θ is the angle between the incident longitudinal wave and the z-axis, and ρ is the average density of the equivalent volume element.

[0051] Step 5: Calculate the P-wave velocity dispersion and attenuation at any incident angle based on the P-wave velocity.

[0052] P-wave phase velocity V p and the inverse quality factor of the longitudinal wave attenuation Calculated by:

[0053]

[0054] Where Re represents the real part of the corresponding quantity, Im represents the imaginary part of the corresponding quantity, and ρ represents the average density of the equivalent volume element.

[0055] This is used to analyze the velocity dispersion of longitudinal waves ( Figure 2 ) and attenuation ( Figure 3 ).from Figure 2 It can be seen that the dispersion of the longitudinal wave velocity changes with the incident angle of the longitudinal wave. When the incident angle is 0 degrees, the dispersion of the longitudinal wave velocity is most obvious, and when the incident angle increases to 45 degrees, the dispersion of the longitudinal wave velocity decreases significantly. When the incident angle further increases to 90 degrees, the dispersion of the longitudinal wave velocity can be ignored. It can be seen that the dispersion of the longitudinal wave velocity changes significantly with the incident angle, that is, the longitudinal wave velocity has obvious frequency-dependent anisotropy characteristics. Correspondingly, Figure 3The variation of attenuation with incident angle is given. Similarly, at different incident angles, the attenuation is different. When the incident angle is 0 degrees, the attenuation is the largest, and when the incident angle increases, the attenuation gradually decreases. When the incident angle is 90 degrees, the attenuation is negligible. Therefore, similar to the longitudinal wave velocity, the attenuation also has obvious frequency-dependent anisotropy characteristics.

Claims

1. A method for calculating the dispersive attenuation of an incident longitudinal wave in a horizontal layered medium, characterized in that: The following steps are involved: Step 1: Establish a periodic layered medium model; In the step 1, the periodic layered medium model is composed of two horizontal porous medium layers A and B that are interlaced and cyclically extended infinitely in the lateral direction. An equivalent volume element is selected at the junction of the A and B layers. The equivalent volume element is located exactly in the middle of the A and B layers. The upper surface of the equivalent volume element is located in the middle of the A layer, and the lower surface is located in the middle of the B layer. A uniform horizontal solid displacement is applied to the left and right sides of the equivalent volume element, and the other boundaries of the equivalent volume element, that is, the top and bottom boundaries, are restricted. Other influencing factors are not considered, and fluid is not allowed to flow into and out of this model, and shear stress is not applied. Under this strain condition, the upper and lower layers of the equivalent volume element satisfy the Biot equation; Step 2: Set the stiffness matrix of the periodic layered medium model; Step 3: Calculate the parameters of the stiffness matrix; Step 4: Calculate the longitudinal wave velocity at any incident angle based on the stiffness matrix; Step 5: Calculate the P-wave velocity dispersion and attenuation at any incident angle based on the P-wave velocity.

2. The method for calculating the dispersion attenuation of incident longitudinal waves in any direction in a horizontal layered medium according to claim 1, characterized in that: In the step 2, for the established periodic layered medium model, in the long-wave limit, a layered medium composed of multiple layers of transversely isotropic materials is equivalently anisotropic, and its equivalent stiffness is: Where <·> indicates that the attributes in brackets are weighted averaged according to volume ratio; is the elastic constant of the porous medium, Represents the stress component in the x-direction of a certain stress combination that causes the model to undergo unit strain in the x-direction; Represents the stress component in the y direction of a certain stress combination that causes unit strain in the model in the x direction; Represents the stress component in the z direction of a certain stress combination that causes unit strain in the model in the x direction; M represents the stress component in the z direction of a certain stress combination that causes the model to have unit strain in the z direction; 44 、M 66 are the elastic parameters of periodic layered media in different directions, For the corresponding M 44 、M 66 Shear modulus in the direction; M 44 =M 66 =G, G is the shear modulus of the solid skeleton.

3. The method for calculating the dispersion attenuation of incident longitudinal waves in any direction in a horizontal layered medium according to claim 2, characterized in that: In step 3, the parameters of the stiffness matrix are have: Among them, λ1 and λ2 are intermediate quantities: For the stiffness matrix parameters and have: Where l1 and l2 are the distances from the upper and lower interfaces of the equivalent element to the junction, and the height of the equivalent element is L = l1 + l2. are the uniaxial strain moduli under drainage conditions in the upper and lower media, are the uniaxial strain moduli under non-drainage conditions in the upper and lower media, η1 and η2 are the dynamic viscosities of the pore fluids in layers A and B, and the intermediate are the M corresponding to the porous media in layers A and B respectively. 11 , are the M corresponding to the porous media in layers A and B respectively. 12 , a1 and a2 represent the Biot coefficients of the porous media in layers A and B, respectively, k1 and k2 represent the rock permeabilities of the porous media in layers A and B, respectively, i is an imaginary unit, ω is the frequency, γ1 and γ2 are the Skempton coefficients of the porous media in layers A and B under uniaxial strain conditions, respectively.

4. The method for calculating the dispersion attenuation of incident longitudinal waves in any direction in a horizontal layered medium according to claim 3, characterized in that: In step 4, the velocity of the longitudinal wave in the equivalent medium is for: The intermediate amount θ is the angle between the incident longitudinal wave and the z-axis, and ρ is the average density of the equivalent volume element.

5. The method for calculating the dispersion attenuation of incident longitudinal waves in any direction in a horizontal layered medium according to claim 4, characterized in that: In step 5, the longitudinal wave phase velocity V p and the inverse quality factor of the longitudinal wave attenuation Calculated by: Where Re represents the real part of the corresponding quantity, Im represents the imaginary part of the corresponding quantity, and ρ represents the average density of the equivalent volume element.

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