A method for calculating the reflection coefficient of spherical longitudinal waves in viscoelastic media

Abstract

The present invention belongs to the technical field of geophysical exploration, and particularly relates to a method for calculating the spherical wave reflection coefficient of viscoelastic media. A method for calculating the spherical P-wave reflection coefficient of viscoelastic media includes the following steps: calculating the frequency-dependent P-wave and S-wave velocities of viscoelastic media according to the quality factor model; substituting the frequency- and quality factor-dependent P-wave and S-wave velocities into the spherical wave reflection coefficient equation of elastic media to obtain the complex spherical wave reflection coefficient of viscoelastic media; calculating the P-wave velocity reflection coefficient dependent on the minimum P-wave quality factor; calculating the S-wave velocity reflection coefficient dependent on the minimum P-wave quality factor; and deriving an approximate equation for the spherical P-wave reflection coefficient of viscoelastic media. While maintaining a high accuracy of the spherical wave reflection coefficient of viscoelastic media, the present invention characterizes the reflection characteristics of the spherical wave of viscoelastic media, reduces the number of inversion parameters of viscoelastic media, and helps to improve the accuracy and stability of pre-stack seismic viscoelastic media parameter inversion.

Description

Technical Field

[0001] The present invention belongs to the technical field of geophysical exploration, and particularly relates to a method for calculating the spherical wave reflection coefficient of a viscoelastic medium. Background Art

[0002] When seismic waves propagate in a subsurface viscoelastic medium, especially in hydrocarbon-bearing reservoirs, attenuation always exists. The quality factor or attenuation factor can be used to quantify seismic wave attenuation and has become an important hydrocarbon indicator. Based on the plane-wave decomposition algorithm of spherical waves, Haase (2004) gave a method for calculating the spherical wave reflection coefficient in a layered homogeneous medium and discussed the amplitude and phase characteristics of the reflection coefficients of spherical PP and PS waves corresponding to the first and third types of AVO. Ursenbach (2007) introduced a special form of wavelet, enabling the integral over frequency in the calculation of the spherical wave reflection coefficient to be analytically obtained, greatly improving the calculation efficiency. Ayzenberg et al. (2007, 2009) gave the form of the spherical wave reflection coefficient expressed in terms of the effective reflection coefficients (ERCs).

[0003] The relationship between the plane-wave reflection coefficient in a viscoelastic medium and the longitudinal and transverse wave quality factors has been widely used in quality factor inversion in plane-wave seismic inversion. However, for the viscoelastic wave field excited by a point source, the plane-wave reflection coefficient is inaccurate, and its meaning involves some fundamental difficulties. The spherical wave reflection coefficient in an elastic medium and the plane-wave reflection coefficient in a viscoelastic medium have been widely studied. Currently, there are many pre-stack seismic plane-wave parameters to be inverted in a viscoelastic medium, reducing the inversion accuracy and stability. Summary of the Invention

[0004] The purpose of the present invention is to solve the problems existing in the prior art and provide a method for calculating the spherical longitudinal wave reflection coefficient of a viscoelastic medium.

[0005] To achieve the above purpose, the technical solution adopted by the present invention is: a method for calculating the spherical longitudinal wave reflection coefficient of a viscoelastic medium, which includes the following steps:

[0006] Step (1), calculating the frequency-dependent longitudinal and transverse wave velocities of the viscoelastic medium according to the quality factor model;

[0007] Step (2), substituting the frequency- and quality factor-dependent longitudinal and transverse wave velocities into the spherical wave reflection coefficient equation of an elastic medium to obtain the complex spherical wave reflection coefficient of the viscoelastic medium;

[0008] Step (3), calculating the longitudinal wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor;

[0009] Step (4), calculating the transverse wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor;

[0010] Step (5), using the longitudinal wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor and the transverse wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor, to derive an approximate equation for the spherical longitudinal wave reflection coefficient in viscoelastic media.

[0011] Further, in the said step (3), the calculation method of the longitudinal wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor includes:

[0012] Expand the longitudinal wave quality factor with an approximately constant value into a function of frequency, set the critical frequency as the reference frequency, and establish the relationship between the frequency-dependent quality factor and the maximum and minimum values of the longitudinal wave modulus.

[0013] The longitudinal wave quality factor reaches its minimum at the critical frequency, which is called the minimum longitudinal wave quality factor.

[0014] Substitute the minimum longitudinal wave quality factor into the frequency-dependent longitudinal wave velocity in viscoelastic media to obtain the longitudinal wave velocity dependent on the minimum longitudinal wave quality factor.

[0015] Substitute the longitudinal wave velocity dependent on the minimum longitudinal wave quality factor into the longitudinal wave velocity reflection coefficient, omit the high-order terms, and obtain the longitudinal wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor.

[0016] In the said step (4), the calculation method of the transverse wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor includes:

[0017] In the lower frequency range, establish the relationship between the longitudinal wave quality factor and the transverse wave quality factor through Poisson's ratio.

[0018] Substitute the relationship between the longitudinal wave quality factor and the transverse wave quality factor, and the relationship between the longitudinal wave quality factor and the minimum longitudinal wave quality factor into the frequency-dependent transverse wave velocity in viscoelastic media to obtain the transverse wave velocity dependent on the minimum longitudinal wave quality factor.

[0019] Substitute the transverse wave velocity dependent on the minimum longitudinal wave quality factor into the transverse wave velocity reflection coefficient, omit the high-order terms, and obtain the transverse wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor.

[0020] Further preferably, in the said step (4), Poisson's ratio is obtained through the ratio of the longitudinal and transverse wave velocities in the seismic frequency band.

[0021] Further preferably, the lower frequency range is 3 - 100 Hz in the seismic frequency band.

[0022] Compared with the prior art, the present invention has the following beneficial effects: The approximate equation of the spherical P-wave reflection coefficient of the viscoelastic medium obtained by the method of the present invention is a function of the reference P-wave velocity reflection coefficient, the reference S-wave velocity reflection coefficient, the density reflection coefficient, the minimum P-wave quality factor perturbation term, the P-wave incident angle, the frequency, the P-wave to S-wave velocity ratio, and the reflection interface depth. While maintaining a high accuracy of the spherical wave reflection coefficient of the viscoelastic medium, it depicts the characteristics of the spherical wave reflection of the viscoelastic medium, reduces the number of inversion parameters of the viscoelastic medium, and helps to improve the accuracy and stability of the pre-stack seismic viscoelastic medium parameter inversion. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 It is a schematic flow chart of the method for calculating the spherical P-wave reflection coefficient of the viscoelastic medium of the present invention;

[0024] Figure 2 It is a schematic diagram showing the relationship between the amplitude and phase of the spherical wave reflection coefficient of the viscoelastic medium calculated in Example 1 of the present invention with the quality factor; wherein, (a) is a schematic diagram showing the relationship between the amplitude of the spherical wave reflection coefficient and different quality factors (Q1, Q2, Q3), and (b) is a schematic diagram showing the relationship between the phase of the spherical wave reflection coefficient and different quality factors (Q1, Q2, Q3);

[0025] Figure 3 It is a schematic diagram showing the relationship between the amplitude and phase of the spherical wave reflection coefficient of the viscoelastic medium calculated in Example 2 of the present invention with the quality factor; wherein, (a) is a schematic diagram showing the relationship between the amplitude of the spherical wave reflection coefficient and different quality factors (Q1, Q2, Q3), and (b) is a schematic diagram showing the relationship between the phase of the spherical wave reflection coefficient and different quality factors (Q1, Q2, Q3). DETAILED DESCRIPTION OF THE EMBODIMENTS

[0026] For the convenience of understanding the present invention, the present invention will be described in more detail below with reference to the accompanying drawings and specific embodiments. The preferred embodiments of the present invention are shown in the accompanying drawings. However, the present invention can be implemented in many different forms and is not limited to the embodiments described in this specification. On the contrary, the purpose of providing these embodiments is to make the understanding of the disclosed content of the present invention more thorough and comprehensive.

[0027] The method for calculating the spherical P-wave reflection coefficient of the viscoelastic medium provided in this embodiment has a flow as Figure 1 shown, and specifically includes the following steps:

[0028] Step (1), calculating the frequency-dependent P-wave and S-wave velocities of the viscoelastic medium

[0029] Using the approximate constant Q (quality factor) model, the frequency-dependent P-wave and S-wave velocities of the viscoelastic medium can be calculated.

[0030] The frequency-dependent P-wave velocity of the viscoelastic medium is:

[0031]

[0032] The shear wave velocity dependent on the viscoelastic medium is as follows:

[0033]

[0034] where, v p (f) is the compressional wave velocity dependent on frequency, v s (f) is the shear wave velocity dependent on frequency, f is the seismic wave frequency, f r is the reference frequency, v pr is the compressional wave velocity at the reference frequency, v sr v pr is the shear wave velocity at the reference frequency, Q p is the compressional wave quality factor, Q s is the shear wave quality factor.

[0035] Step (2), deriving the complex spherical wave reflection coefficient of the viscoelastic medium

[0036] Substituting the compressional and shear wave velocities dependent on frequency and quality factor into the spherical wave reflection coefficient equation of the elastic medium, the complex spherical wave reflection coefficient of the viscoelastic medium can be obtained:

[0037]

[0038] where,

[0039]

[0040]

[0041] v p1 (f) is the compressional wave velocity dependent on frequency in the upper layer, z = h is the height between the source and the geophone from the reflection interface, i is the imaginary unit, f is the seismic wave frequency, r is the offset, θ p1 is the compressional wave incident angle, J 0 is the Bessel function of the first kind of order zero, J 1 is the Bessel function of the first kind of order one, x is the integration variable; is the plane wave reflection coefficient, which is a function of frequency, depth, and the compressional and shear wave velocities and quality factors at the reference frequencies on both sides of the reflection interface, and is expressed as:

[0042]

[0043] where,

[0044]

[0045]

[0046]

[0047]

[0048] and

[0049]

[0050] θ p2 is the longitudinal wave conversion angle, θ s1 is the longitudinal wave conversion angle, θ s2 is the shear wave reflection angle; v p1 is the frequency-dependent longitudinal wave velocity in the upper layer of the reflection interface, v p2 is the frequency-dependent longitudinal wave velocity in the lower layer of the reflection interface; v s1 is the frequency-dependent shear wave velocity in the upper layer of the reflection interface, v s2 is the frequency-dependent shear wave velocity in the lower layer of the reflection interface; ρ 1 is the density of the upper layer of the reflection interface, ρ 2 is the density of the lower layer of the reflection interface; △v p = v p2 - v p1 , △v s = v s2 - v s1 , △ρ 1 = ρ 2 - ρ 1 , wherein, the shear wave reflection angle and the longitudinal and shear wave transmission angles can be obtained through Snell's law.

[0051] Step 3, calculate the longitudinal wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor

[0052] Expand the approximately constant longitudinal wave quality factor into a function of frequency, set the critical frequency as the reference frequency, and a relationship between the frequency-dependent quality factor and the maximum and minimum values of the longitudinal wave modulus can be established:

[0053]

[0054] wherein, M ∞ and M 0 are the high-frequency and low-frequency longitudinal wave moduli respectively.

[0055] At this time, the longitudinal wave quality factor reaches a minimum at the critical frequency (reference frequency):

[0056]

[0057] wherein, Q pm = Qp (f r ) is called the minimum P-wave quality factor, which is obtained by the P-wave quality factor at the reference frequency f r . Substituting Equation (6) into Equation (5), the relationship between the P-wave quality factor and the minimum P-wave quality factor can be obtained, expressed as:

[0058]

[0059] Substituting Equation (7) into the frequency-dependent P-wave velocity in the viscoelastic medium (i.e., Equation (1)), the P-wave velocity dependent on the minimum P-wave quality factor can be obtained:

[0060]

[0061] Further substituting the P-wave velocity dependent on the minimum P-wave quality factor into the P-wave velocity reflection coefficient, the P-wave velocity reflection coefficient dependent on the minimum P-wave quality factor can be finally obtained:

[0062]

[0063] Omitting the high-order terms and small terms, the relationship between the P-wave velocity reflection coefficient and the P-wave velocity reflection coefficient at the reference frequency and the perturbation term of the minimum P-wave quality factor can be obtained:

[0064]

[0065] Step (4), calculating the S-wave velocity reflection coefficient dependent on the minimum P-wave quality factor

[0066] Within the lower seismic frequency band range (3 - 100 Hz), the relationship between the P-wave quality factor and the S-wave quality factor is established through the Poisson's ratio:

[0067]

[0068]

[0069]

[0070] Among them, the Poisson's ratio can be obtained through the P-wave to S-wave velocity ratio within the seismic frequency band. Substituting the relationship between the P-wave quality factor and the S-wave quality factor (Equation 11), and the relationship between the P-wave quality factor and the minimum P-wave quality factor (Equation (7)) into the frequency-dependent S-wave velocity in the viscoelastic medium (i.e., Equation (2)), the S-wave velocity dependent on the minimum P-wave quality factor can be obtained:

[0071]

[0072] Step 5, Derive the approximate equation of the spherical P-wave reflection coefficient in viscoelastic media

[0073] Substitute the calculated P-wave velocity reflection coefficient dependent on the minimum P-wave quality factor and the S-wave velocity reflection coefficient dependent on the minimum P-wave quality factor into the spherical wave reflection coefficient in viscoelastic media (Equation (4)), and the approximate equation of the spherical P-wave reflection coefficient in viscoelastic media can be derived, which is approximately expressed as:

[0074]

[0075] This approximate reflection coefficient equation is basically consistent with the accurate spherical wave reflection coefficient equation in viscoelastic media, and is a function of the reference P-wave velocity reflection coefficient, the reference S-wave velocity reflection coefficient, the density reflection coefficient, the perturbation term of the minimum P-wave quality factor, the P-wave incident angle, the frequency, the P / S wave velocity ratio, and the depth of the reflection interface. F(~) represents the non-linear relationship between the approximate spherical wave reflection coefficient in viscoelastic media and the parameters to be inverted.

[0076] In Example 1, the method provided by the present invention is used to calculate the spherical wave reflection coefficient in viscoelastic media of Formation Model 1, and the parameters are shown in Table 1:

[0077] Table 1 Formation Model 1

[0078]

[0079] As Figure 2 shown, (a) is a schematic diagram of the variation relationship of the spherical wave reflection coefficient amplitude with different quality factors (Q1, Q2, Q3), and (b) is a schematic diagram of the variation relationship of the spherical wave reflection coefficient phase with different quality factors (Q1, Q2, Q3); it can be seen that the spherical wave reflection coefficients in viscoelastic media with different quality factors in Example 1 vary greatly.

[0080] In Example 2, the method provided by the present invention is used to calculate the spherical wave reflection coefficient in viscoelastic media of Formation Model 2, and the parameters are shown in Table 2:

[0081] Table 2 Formation Model 2

[0082]

[0083]

[0084] As Figure 3As shown, (a) is a schematic diagram of the variation of the amplitude of the spherical wave reflection coefficient with different quality factors (Q1, Q2, Q3), and (b) is a schematic diagram of the variation of the phase of the spherical wave reflection coefficient with different quality factors (Q1, Q2, Q3); it can be seen that there are also certain differences in the spherical wave reflection coefficients of viscoelastic media with different quality factors in Example 2. This feature indicates that different seismic reflection information can be used to estimate the quality factor, thereby guiding the prediction and evaluation of oil and gas reservoirs.

Claims

1. A method for calculating the reflection coefficient of spherical longitudinal waves in viscoelastic media, characterized in that, it includes the following steps: Step (1), calculate the longitudinal and transverse wave velocities dependent on frequency in viscoelastic media according to the quality factor model; Step (2), substitute the longitudinal and transverse wave velocities dependent on frequency and quality factor into the spherical wave reflection coefficient equation of elastic media to obtain the complex spherical wave reflection coefficient of viscoelastic media; Step (3), calculate the reflection coefficient of longitudinal wave velocity dependent on the minimum longitudinal wave quality factor: expand the longitudinal wave quality factor with an approximate constant value into a function of frequency, set the critical frequency as the reference frequency, and establish the relationship between the quality factor dependent on frequency and the maximum and minimum values of the longitudinal wave modulus; The longitudinal wave quality factor reaches a minimum at the critical frequency, which is called the minimum longitudinal wave quality factor; Substitute the minimum longitudinal wave quality factor into the longitudinal wave velocity dependent on frequency in viscoelastic media to obtain the longitudinal wave velocity dependent on the minimum longitudinal wave quality factor; Substitute the longitudinal wave velocity dependent on the minimum longitudinal wave quality factor into the longitudinal wave velocity reflection coefficient, omit the high-order terms, and obtain the longitudinal wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor; Step (4), calculate the reflection coefficient of transverse wave velocity dependent on the minimum longitudinal wave quality factor: within the range of 3 - 100 Hz, establish the relationship between the longitudinal wave quality factor and the transverse wave quality factor through the Poisson ratio; Substitute the relationship between the longitudinal wave quality factor and the transverse wave quality factor, and the relationship between the longitudinal wave quality factor and the minimum longitudinal wave quality factor into the transverse wave velocity dependent on frequency in viscoelastic media to obtain the transverse wave velocity dependent on the minimum longitudinal wave quality factor; Substitute the transverse wave velocity dependent on the minimum longitudinal wave quality factor into the transverse wave velocity reflection coefficient, omit the high-order terms, and obtain the transverse wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor; Step (5), use the longitudinal wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor and the transverse wave velocity reflection coefficient dependent on the minimum longitudinal wave quality factor to derive the approximate equation of the spherical longitudinal wave reflection coefficient in viscoelastic media.

2. The method for calculating the reflection coefficient of spherical longitudinal waves in viscoelastic media according to claim 1, characterized in that: The Poisson ratio is obtained from the ratio of longitudinal and transverse wave velocities within the seismic frequency band.

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