CN111474586A - Frequency domain multi-scale crack weakness inversion method - Google Patents

Frequency domain multi-scale crack weakness inversion method Download PDF

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CN111474586A
CN111474586A CN202010373072.6A CN202010373072A CN111474586A CN 111474586 A CN111474586 A CN 111474586A CN 202010373072 A CN202010373072 A CN 202010373072A CN 111474586 A CN111474586 A CN 111474586A
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潘新朋
李林
陈怀震
张广智
柳建新
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China University of Petroleum East China
Central South University
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    • G01—MEASURING; TESTING
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    • G01V1/00—Seismology; Seismic or acoustic prospecting or detecting
    • G01V1/40—Seismology; Seismic or acoustic prospecting or detecting specially adapted for well-logging
    • G01V1/44—Seismology; Seismic or acoustic prospecting or detecting specially adapted for well-logging using generators and receivers in the same well
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    • G01—MEASURING; TESTING
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Abstract

The invention provides a frequency domain multi-scale crack weakness inversion method. The method comprises the following steps: establishing a forward model of a frequency domain; establishing an inversion target function by maximizing a posterior probability distribution function of the model parameters; selecting frequency components with high signal-to-noise ratio, and dividing the frequency components into different frequency groups according to the sequence of frequencies from low to high; and (4) taking the inversion result of the low-frequency group as an initial model for inversion of the next high-frequency group, and obtaining a crack parameter inversion result through successive inversion. The model parameters established by the method can realize automatic decoupling in the frequency domain, so that each frequency component can be independently used for inverting the model parameters, and a stable and reliable crack weakness inversion result can be obtained only by utilizing partial effective frequency component information. By adopting a multi-scale inversion strategy and sequentially inverting from low frequency to high frequency by using the optimal effective frequency components, the nonlinearity of an inversion problem is favorably relieved, and the accuracy of the stability of the fracture weakness parameters is improved.

Description

Frequency domain multi-scale crack weakness inversion method
Technical Field
The invention relates to the technical field of seismic exploration, in particular to a frequency domain multi-scale crack weakness inversion method.
Background
Natural fractured reservoirs contain abundant oil and gas resources and are extremely valuable economic targets in seismic exploration, and underground fractures can provide high-permeability channels for fluid flow and help to accelerate the flow of fluid in the stratum. Two non-negative dimensionless fracture parameters (i.e., normal weakness and tangential weakness) are directly related to fracture-induced anisotropy and can be used to characterize fluid filling conditions in the fracture. Therefore, fracture parameter inversion is of great significance for seismic fracture characterization and fluid identification. AVOAz (seismic amplitude changes with offset and azimuth) inversion is an effective method for fracture parameter prediction using azimuth seismic reflection data. The conventional AVOAz inversion is performed in a time domain, full frequency component information of seismic data is utilized, and under the condition that the signal-to-noise ratio of the seismic data is low, the stability and the reliability of the conventional time domain AVOAz inversion are possibly low due to the fact that the conventional time domain AVOAz inversion utilizes the full frequency component information of the seismic data and the use of low signal-to-noise ratio seismic data information.
In view of the above, there is a need for a frequency domain multi-scale fracture weakness inversion method to solve the problems in the prior art.
Disclosure of Invention
The invention aims to provide a frequency domain multi-scale fracture weakness inversion method to improve the stability and prediction accuracy of fracture weakness inversion.
In order to achieve the purpose, the invention provides a frequency domain multi-scale fracture weakness inversion method, which comprises the following steps:
step one, establishing a forward model of a frequency domain;
step two, establishing an inversion target function by maximizing a posterior probability distribution function of the model parameters;
selecting frequency components with high signal-to-noise ratio, and dividing the frequency components into different frequency groups according to the sequence of frequencies from low to high;
and step four, taking the inversion result of the low frequency group as an initial model for inversion of the next high frequency group, and obtaining the inversion result of the fracture parameters through successive inversion.
Further, in the case of M incident angles, n interfaces, and M effective frequency components, the forward model of the frequency domain is represented as:
d＝Gx (8)
wherein the content of the first and second substances,
Figure BDA0002478876100000021
Figure BDA0002478876100000022
Figure BDA0002478876100000023
Figure BDA0002478876100000024
Figure BDA0002478876100000025
theta is the angle of incidence,
Figure BDA0002478876100000026
for observing the azimuth angle phi and the symmetric azimuth angle phi of the cracksymAzimuth angle in between;
Figure BDA0002478876100000027
and W (ω) is the spectrum of the seismic reflection amplitude and seismic wavelet respectively,
Figure BDA0002478876100000028
the azimuth longitudinal wave reflection coefficient at the time point tau is shown, and omega represents angular frequency; deltaNAnd ΔTRespectively representing the normal weakness difference and the tangential weakness difference of two sides of the interface.
Further, equation (8) is rewritten by the real and imaginary parts of each parameter as:
d'＝G'x (9)
wherein the content of the first and second substances,
Figure BDA0002478876100000029
wherein, the symbols real [. cndot ] and imag [. cndot ] are real part operator and imaginary part operator respectively.
Further, the initial inversion objective function is:
Figure BDA00024788761000000210
in the formula, symbol T represents the transpose of the matrix, σeIs the variance of the noise in the frequency domain, σxIs the variance of the model parameters.
Further, the low frequency constraints of the model parameters are introduced into equation (14) to compensate for the low frequency components and enhance the stability and lateral continuity of the inversion, resulting in:
Figure BDA0002478876100000031
in the formula, λ1And λ2Constraint coefficients of a normal weak term and a tangential weak term are respectively;N0andT0low frequency models representing normal and tangential weaknesses, respectively, L represents an integration matrix.
Further, an iterative reweighted least square algorithm is adopted to solve equation (15), and the direction weakness and the tangential weakness are obtained by performing the following transformation:
N＝LΔN； (16)
T＝LΔT。 (17)
the technical scheme of the invention has the following beneficial effects:
(1) the model parameters established by the method can realize automatic decoupling in the frequency domain, so that each frequency component can be independently used for inverting the model parameters, and therefore, the method can obtain stable and reliable crack weakness inversion results only by utilizing partial effective frequency component information (frequency components with high signal-to-noise ratio).
(2) The method adopts a multi-scale inversion strategy, and sequentially carries out stage-by-stage inversion from low frequency to high frequency by using the optimal effective frequency components, so that the nonlinearity of the inversion problem is favorably alleviated, and the accuracy of the stability of the fracture weakness parameter is improved.
In addition to the objects, features and advantages described above, other objects, features and advantages of the present invention are also provided. The present invention will be described in further detail below with reference to the drawings.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this application, illustrate embodiments of the invention and, together with the description, serve to explain the invention and not to limit the invention. In the drawings:
FIG. 1(a) is a composite azimuth gather with a signal-to-noise ratio of 5;
FIG. 1(b) is a composite azimuth gather with a signal-to-noise ratio of 2;
FIG. 2(a) is the results of the crack parameter inversion for the case of inversion using 5-15Hz frequency components at a signal-to-noise ratio of 5;
FIG. 2(b) is the results of the crack parameter inversion for the case of inversion using 15-35Hz frequency components at a signal-to-noise ratio of 5;
FIG. 2(c) is the results of the fracture parameter inversion for the case of inversion using 35-55Hz frequency components at a signal-to-noise ratio of 5;
FIG. 3(a) is the results of the crack parameter inversion for the case of inversion using 5-15Hz frequency components at a signal-to-noise ratio of 2;
FIG. 3(b) is the results of the crack parameter inversion for the case of inversion using 15-35Hz frequency components at a signal-to-noise ratio of 2;
FIG. 3(c) is the results of the crack parameter inversion for the case of inversion using 35-55Hz frequency components at a signal-to-noise ratio of 2;
FIG. 4(a) is a comparison graph of final inversion results obtained by the method of the present invention and a conventional time domain method, respectively, with a signal-to-noise ratio of 5;
FIG. 4(b) is a comparison graph of final inversion results obtained by the method of the present invention and the conventional time domain method, respectively, when the signal-to-noise ratio is 2;
FIG. 5(a) shows an azimuth angle φ1A 40 ° stacked seismic section;
FIG. 5(b) is an azimuth angle φ2130 ° stacked seismic sections;
FIG. 6(a) is the normal weakness results of inversion using 10-20Hz frequency components;
FIG. 6(b) is the tangential weakness results of inversion using 10-20Hz frequency components;
FIG. 7(a) is the normal weakness results of inversion using 10-20Hz and 25-45Hz frequency components;
FIG. 7(b) is the tangential weakness results of inversion using 10-20Hz and 25-45Hz frequency components;
FIG. 8(a) is a normal weakness result of a conventional time domain inversion;
FIG. 8(b) is the tangential weakness result of a conventional time domain inversion;
FIG. 9 is a comparison of inversion results obtained at a well location using the method of the present invention and a conventional time domain inversion method, respectively.
Detailed Description
Embodiments of the invention will be described in detail below with reference to the drawings, but the invention can be implemented in many different ways, which are defined and covered by the claims.
Example 1:
referring to fig. 1(a) to 9, a frequency domain multi-scale fracture weakness inversion method specifically includes the following steps:
the HTI medium longitudinal wave reflection coefficient expression is as follows:
Figure BDA0002478876100000041
wherein, theta is an incident angle,
Figure BDA0002478876100000042
for observing the azimuth angle phi and the symmetric azimuth angle phi of the cracksymIn azimuth of (c). Riso(θ) is a conventional AVO (amplitude versus offset) polynomial expression that can be expressed as:
Figure BDA0002478876100000043
wherein α and ρ are the longitudinal and transverse wave velocity and density of the isotropic background, respectively, the superscript represents the average property of the upper and lower media, the symbol Δ represents the property difference of the upper and lower media,
Figure BDA0002478876100000051
Figure BDA0002478876100000052
represents the anisotropic perturbation term, which can be expressed as:
Figure BDA0002478876100000053
wherein the content of the first and second substances,
Figure BDA0002478876100000054
Figure BDA0002478876100000055
delta in equation (3)NAnd ΔTRespectively representing the normal weakness difference and the tangential weakness difference of two sides of the interface.
From equations (1) and (3), the reflection coefficient difference expression for two orthogonal azimuth lines can be derived as follows:
Figure BDA0002478876100000056
wherein the content of the first and second substances,
Figure BDA0002478876100000057
Figure BDA0002478876100000058
the convolution model of the frequency domain can be expressed as:
Figure BDA0002478876100000059
wherein the content of the first and second substances,
Figure BDA00024788761000000510
and W (ω) is the spectrum of the seismic reflection amplitude and seismic wavelet respectively,
Figure BDA00024788761000000511
denotes the azimuthal longitudinal wave reflection coefficient at time τ and ω denotes the angular frequency.
The frequency domain expression of the quadrature azimuth line seismic response difference is as follows:
Figure BDA00024788761000000512
substituting equation (4) into equation (6) yields:
Figure BDA0002478876100000061
in the case of M incident angles, n interfaces, and M effective frequency components, the forward model of the frequency domain can be expressed as:
d＝Gx (8)
wherein the content of the first and second substances,
Figure BDA0002478876100000062
Figure BDA0002478876100000063
Figure BDA0002478876100000064
Figure BDA0002478876100000065
Figure BDA0002478876100000066
equation (8) can be further rewritten by the real and imaginary parts of each parameter as:
d'＝G'x (9)
wherein the content of the first and second substances,
Figure BDA0002478876100000067
wherein, the symbols real [. cndot ] and imag [. cndot ] are real part operator and imaginary part operator respectively. Since the kernel function incorporates the fourier transform operator E, it makes it possible to invert the fracture parameters in the frequency domain with only a limited number of effective frequency components.
Bayesian theory is widely applied to seismic inversion, and an inversion target function is established by maximizing a posterior probability distribution function of model parameters. According to the forward modeling of the frequency domain in equation (9), the posterior probability distribution function p (x ' | d ') can be represented by a likelihood function p (d ' | x ') and a prior probability function p (x '):
Figure BDA0002478876100000071
the likelihood function is assumed to follow a gaussian distribution:
Figure BDA0002478876100000072
in the formulaThe symbol T represents the transpose of the matrix, σeIs the variance of the noise in the frequency domain.
Cauchy probability distribution helps to improve inversion resolution and produce sparse solutions. Assuming that the prior probability function obeys the Cauchy distribution, it is expressed as:
Figure BDA0002478876100000073
in the formula, σxIs the variance of the model parameters.
Substituting equations (11) and (12) into equation (10) yields:
Figure BDA0002478876100000074
by maximizing the posterior probability density function in equation (13), the initial inversion objective function is obtained:
Figure BDA0002478876100000075
low frequency components in seismic data are typically missing or contaminated with noise. To compensate for the low frequency components and enhance the stability and lateral continuity of the inversion, the low frequency constraints of the model parameters are introduced into equation (14) to yield:
Figure BDA0002478876100000076
in the formula, λ1And λ2The constraint coefficients are normal and tangential weakness terms, respectively.N0AndT0the low-frequency model respectively represents normal weakness and tangential weakness, L represents an integral matrix, an iterative reweighted least square algorithm is adopted to solve equation (15), and the directional weakness and the tangential weakness are obtained by performing the following transformation:
N＝LΔN(16)
T＝LΔT(17)
the method adopts a frequency domain multi-scale strategy to realize the inversion of the fracture parameters. Firstly, selecting frequency components with high signal-to-noise ratio, and dividing the frequency components into different frequency groups according to the sequence of frequencies from low to high; and secondly, sequentially inverting from the low-frequency group to the high-frequency group, namely, taking the inversion result of the low-frequency group as an initial model for inverting the next high-frequency group, and obtaining a stable and accurate crack parameter inversion result through sequential inversion.
The feasibility and the superiority of the method in the crack weakness inversion are verified by using single well data. And (3) making a synthetic azimuth gather by utilizing 35Hz Rake wavelets and well data through a convolution model, and simulating and observing the seismic data by adding Gaussian noise with signal-to-noise ratios of 5 and 2 into the synthetic data respectively. The effective frequency component range used for inversion is 5-55Hz, and the inversion is further divided into three frequency groups, namely a low frequency group (5-15Hz), a medium frequency group (15-35Hz) and a high frequency group (35-55 Hz). The initial model is obtained by smoothing the original real fracture parameters. And starting inversion from the low-frequency set, and taking the inversion result of the low-frequency set as an initial model of the next high-frequency set inversion. Fig. 1(a) and 1(b) are synthetic azimuth gathers for signal-to-noise ratios of 5 and 2, respectively. Fig. 2(a) to 2(c) show the inversion results of fracture parameters when inversion is performed using different frequency groups at a signal-to-noise ratio of 5. Fig. 3(a) to 3(c) show the inversion results of the fracture parameters when the inversion is performed using different frequency groups when the signal-to-noise ratio is 2. The dashed line represents the inversion results, the solid line the true values and the dotted line the initial model. It can be seen that as the frequency set used for inversion increases, the resolution and accuracy of the inversion result steadily improve, and the final inversion result (using 5-55Hz data) is more consistent with the true value of the fracture parameter. The method of the present invention is further compared to conventional time domain methods. Fig. 4(a) and 4(b) are comparison graphs of final inversion results obtained by the method of the present invention and the conventional time domain method, respectively, under different signal-to-noise ratios. The dotted line represents the inversion result obtained by the method of the invention, the dotted line represents the inversion result obtained by the conventional time domain method, and the solid line represents the true value of the fracture parameter. The method can better realize the crack parameter estimation, and the inversion result obtained by the method has higher resolution and precision, thereby further verifying the feasibility and the superiority of the method.
To further verify the feasibility of the method of the invention, it was applied to the actual data of a certain gas-bearing fractured reservoir in southwest. The seismic data is classified into common azimuth gathers, and each azimuth gather is then converted from an offset domain to an angle of incidence domain. FIGS. 5(a) and 5(b) show the azimuth angles φ, respectively140 ° and phi2130 ° stacked seismic section. The method comprises the steps of firstly obtaining azimuth seismic amplitude differences of two azimuth seismic sections, and predicting fracture parameters by using the azimuth seismic amplitude differences. The effective frequency components for inversion include two frequency groups (10-20Hz and 25-45 Hz). FIGS. 6(a) and 6(b) show the results of the crack parameter inversion using frequency components of 10-20 Hz. FIGS. 7(a) and 7(b) show the results of the crack parameter inversion using frequency components of 10-20Hz and 25-45 Hz. It can be seen that the resolution of the inversion results increases as the set of frequencies used for the inversion increases (as shown by the black circles). Fig. 8(a) and 8(b) show the results of crack parameter inversion of a conventional time domain inversion method. From fig. 7 and 8, it can be seen that the method of the present invention displays more high resolution information (as shown by the black circles) than the conventional time domain inversion method. FIG. 9 shows a comparison of inversion results obtained at a well location using the method of the present invention and a conventional time domain inversion method, respectively. The solid line represents the true value of the fracture parameter, the dotted line represents the inversion result of the conventional time domain inversion method, and the dotted line represents the inversion result of the method. It can be seen that the inversion results of the conventional time domain inversion method and the inversion method of the invention are similar to the true values of the fracture parameters, but the inversion result of the method of the invention has higher resolution and precision, and the feasibility and the superiority of the method of the invention in the aspect of fracture parameter prediction are further verified.
The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention, and various modifications and changes may be made by those skilled in the art. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (6)

1. A frequency domain multi-scale fracture weakness inversion method is characterized by comprising the following steps:
step one, establishing a forward model of a frequency domain;
step two, establishing an inversion target function by maximizing a posterior probability distribution function of the model parameters;
selecting frequency components with high signal-to-noise ratio, and dividing the frequency components into different frequency groups according to the sequence of frequencies from low to high;
and step four, taking the inversion result of the low frequency group as an initial model for inversion of the next high frequency group, and obtaining the inversion result of the fracture parameters through successive inversion.
2. The frequency domain multi-scale fracture weakness inversion method according to claim 1, wherein in the case of M incidence angles, n interfaces and M effective frequency components, the forward model of the frequency domain is represented as:
d＝Gx (8)
wherein the content of the first and second substances,
Figure FDA0002478876090000011
Figure FDA0002478876090000012
Figure FDA0002478876090000013
Figure FDA0002478876090000014
Figure FDA0002478876090000015
theta is the angle of incidence,
Figure FDA0002478876090000016
for observing the azimuth angle phi and the symmetric azimuth angle phi of the cracksymAzimuth angle in between;
Figure FDA0002478876090000017
and W (ω) is the spectrum of the seismic reflection amplitude and seismic wavelet respectively,
Figure FDA0002478876090000018
the azimuth longitudinal wave reflection coefficient at the time point tau is shown, and omega represents angular frequency; deltaNAnd ΔTRespectively representing the normal weakness difference and the tangential weakness difference of two sides of the interface.
3. The frequency domain multi-scale fracture weakness inversion method according to claim 2, wherein equation (8) is rewritten by real parts and imaginary parts of each parameter as:
d'＝G'x (9)
wherein the content of the first and second substances,
Figure FDA0002478876090000021
wherein, the symbols real [. cndot ] and imag [. cndot ] are real part operator and imaginary part operator respectively.
4. The frequency domain multi-scale fracture weakness inversion method according to claim 3, wherein the initial inversion objective function is:
Figure FDA0002478876090000022
in the formula, symbol T represents the transpose of the matrix, σeIs the variance of the noise in the frequency domain, σxIs the variance of the model parameters.
5. The frequency domain multi-scale fracture weakness inversion method according to claim 4, characterized in that the low frequency constraint of the model parameters is introduced into equation (14) to compensate the low frequency component and enhance the inversion stability and lateral continuity, resulting in:
Figure FDA0002478876090000023
in the formula, λ1And λ2Constraint coefficients of a normal weak term and a tangential weak term are respectively;N0andT0low frequency models representing normal and tangential weaknesses, respectively, L represents an integration matrix.
6. The frequency domain multi-scale fracture weakness inversion method according to claim 5, characterized in that an iterative reweighted least squares algorithm is adopted to solve equation (15), and the weak orientation and the tangential weakness are obtained by performing the following transformation:
N＝LΔN； (16)
T＝LΔT(17) 。
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