REVERSE TIME MIGRATION IMAGING METHOD FOR CASED-HOLE STRUCTURE BASED ON ULTRASONIC PITCH-CATCH MEASUREMENT

Abstract

A reverse time migration imaging method for cased-hole based on ultrasonic pitch-catch measurement, including: calculating a theoretical dispersion curve; expanding original Lamb data of two receivers into array waveform data based on phase-shift interpolation; establishing a two-dimensional migration velocity model including density, P-wave velocity and S-wave velocity of a target area; generating and storing a forward propagating ultrasonic wavefield for each time step; reversing a time axis; generating and storing a reversely propagating ultrasonic Lamb wavefield for the two receivers after phase-shift interpolation; calculating envelopes of the forward propagating ultrasonic Lamb wavefield and the reversely propagating ultrasonic Lamb wavefield; applying a zero-lag cross-correlation imaging condition to obtain reverse time migration imaging results; and applying Laplace filtering to suppress low-frequency imaging noises in the imaging results.

Claims

1. A reverse time migration imaging method for a cased-hole structure based on ultrasonic pitch-catch measurement, comprising: (S1) inputting original ultrasonic Lamb waveform data and related parameter files; calculating a theoretical dispersion curve of A0-mode waveforms; and expanding original Lamb data of two receivers into array waveform data based on phase-shift interpolation; (S2) according to background information, establishing a two-dimensional migration velocity model including initial density, P-wave velocity and S-wave velocity of a target area; (S3) based on two-dimensional high-order staggered grid finite difference and non-split perfectly matched layer, generating and storing a forward propagating ultrasonic Lamb wavefield for each time step; (S4) reversing a time axis; and generating and storing a reversely propagating ultrasonic Lamb wavefield; (S5) based on Hilbert transform, calculating an envelope of the forward propagating ultrasonic Lamb wavefield and an envelope of the reversely propagating ultrasonic Lamb wavefield; (S6) applying a zero-lag cross-correlation imaging condition to the forward propagating ultrasonic Lamb wavefield and the reversely propagating ultrasonic Lamb wavefield to obtain reverse time migration imaging results for ultrasonic pitch-catch measurement; and (S7) applying Laplace filtering to suppress low-frequency imaging noises in the reverse time migration imaging results.

2. The reverse time migration imaging method of claim 1, wherein in step (S1), the theoretical dispersion curve of the A0-mode waveforms is calculated according to borehole fluid properties, casing thickness, casing elastic parameters and central frequency recorded in the related parameter files; based on A0's phase velocity, a waveform propagating forward or backward from one of the two receivers to a certain distance is calculated by using a phase shift method through the following equation: $\begin{array}{l}g\left(t\right)={\displaystyle \int F\left(w\right)}H\left(w\right)e^{-jwt}dw;\\ H\left(w\right)=e^{-jk\left(w\right)x_0}\end{array}$ wherein g(t) is a waveform travelling at a certain distance x.sub.0; F(w) is a frequency spectrum of an original A0-mode waveform at near or far receiver; w is angular frequency; -j is imaginary number; e is natural logarithm; H(w) is a propagation matrix, and k represents wavenumber; and due to dispersion, k is a function of phase velocity v of the A0-mode waveforms.

3. The reverse time migration imaging method of claim 1, wherein in step (S2), the two-dimensional migration velocity model is established through steps of: setting grid spacing and model size; and according to density and velocity of borehole fluid, steel casing, and cementin the target area, establishing the two-dimensional migration velocity model.

4. The reverse time migration imaging method of claim 1, wherein in step (S3), the generation and storage of the forward propagating ultrasonic Lamb wavefield are performed through steps of: selecting a Ricker wavelet as an ultrasonic source; generating the forward propagating ultrasonic Lamb wavefield by using a high-order staggered grid finite difference algorithm; absorbing and attenuating a reflection at an artificial boundary based on the non-split perfectly matched layer; and storing the forward propagating ultrasonic Lamb wavefield for each time step.

5. The reverse time migration imaging method of claim 1, wherein in step (S4), the reversely propagating ultrasonic Lamb wavefield is expressed through the following equation: $\rho \stackrel{.Math.}{\text{u}}\text{-}\left(\lambda +2\mu \right)\nabla \nabla \cdot \text{u+}\mu \nabla \times \nabla \times \text{u=d}\left(x,z=0,T,t\right);$ wherein d is actual ultrasonic Lamb waveforms measured by the receiver; ρ indicates density; u indicates the reversely propagating ultrasonic Lamb wavefield; ü is a second derivative of the reversely propagating ultrasonic Lamb wavefield with respect to time; ∇ represents a spatial derivative operation; ∇. represents an operation for solving divergence degree; V × represents an operation for solving curl; T is a total receiving time; t represents time step; and λ and .Math. are elastic parameters.

6. The reverse time migration imaging method of claim 1, wherein in step (S5), the envelope of the forward propagating ultrasonic Lamb wavefield and the envelope of the reversely propagating ultrasonic Lamb wavefield are calculated by the following equation: $\widehat{u}\left(t\right)=\frac{1}{\pi }{\displaystyle {\int }_{-\infty }^{+\infty }\frac{u\left(\tau \right)}{t=\tau }d\tau };$ wherein û indicates Hilbert transform of a wavefield u; τ is time delay; and an envelope ũ of a propagating wavefield is calculated based on a modulus of the Hilbert transform, expressed as: $\tilde{u}\left(t\right)=\left|\widehat{u}\left(t\right)\right|;$ wherein | | represents a modulo operation.

7. The reverse time migration imaging method of claim 1, wherein in step (S6), the reverse time migration imaging results are expressed as follows: $I\left(z,x\right)={\displaystyle {\int }_0^{T}S\left(z,x,t\right)R\left(z,x,t\right)dt};$ wherein S(z, x, t) is the envelope of the forward propagating ultrasonic Lamb wavefield at spatial position (z,x) in a t.sup.th time step; R(z,x,t) is the envelope of the reversely propagating ultrasonic Lamb wavefield at the spatial position (z,x) in the t.sup.th time step; and T indicates the number of sampling points.

8. The reverse time migration imaging method of claim 1, wherein in step (S7), the Laplace filtering is expressed as: $\tilde{I}\left(z,x\right)={\nabla }^{2}I\left(z,x\right)=\frac{{\partial }^{2}I\left(z,x\right)}{\partial z^2}+\frac{{\partial }^{2}I\left(z,x\right)}{\partial x^2}{}_{;}$ wherein I is an original reverse time migration imaging result; ĩ is an imaging result after applying the Laplace filtering; z indicates a depth coordinate, and x indicates a distance coordinate.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0037] FIG. 1 is a flowchart of a cased-hole reverse time migration imaging method based on ultrasonic pitch-catch measurement according to an embodiment of this application;

[0038] FIG. 2a is a schematic diagram of a measured waveform at a near receiver for ultrasonic pitch-catch measurement;

[0039] FIG. 2b is a schematic diagram of a measured waveform at a far receiver for ultrasonic pitch-catch measurement;

[0040] FIG. 3 shows a theoretical dispersion curve of A0-mode waveform;

[0041] FIG. 4 shows the array A0-mode waveform obtained based on phase-shift interpolation;

[0042] FIG. 5 illustrates the forward velocity model used in synthetic case study;

[0043] FIG. 6 depicts the measured ultrasonic Lamb waveform at the far receiver at a certain depth point and its envelope calculated using Hilbert transform;

[0044] FIG. 7 shows reverse time migration imaging results using the zero-lag cross-correlation imaging condition for the simulated ultrasonic pitch-catch measurement data;

[0045] FIG. 8 schematically illustrates the enhanced imaging results obtained by applying Laplace filtering algorithm;

[0046] FIG. 9 illustrates the established migration velocity model for the cased-hole reverse time migration in a calibration well;

[0047] FIG. 10 shows the measured time-depth waveform at the near receiver for the ultrasonic pitch-catch measurement in the calibration well; and

[0048] FIG. 11 illustrates the final reverse time migration imaging result for the measured ultrasonic pitch-catch measurement data in the calibration well.

DETAILED DESCRIPTION OF EMBODIMENTS

[0049] This application will be described in detail below with reference to the accompanying drawings and embodiments to make the technical solutions, objectives and beneficial effects of this application clearer.

Embodiment 1

[0050] Referring to an embodiment shown in FIG. 1, a reverse time migration imaging method for a cased-hole based on ultrasonic pitch-catch measurement is provided, which is performed as follows.

(s1) Original Data Input and Phase-Shift Interpolation of A0-Mode Wave

[0051] Original ultrasonic Lamb waveform data and parameters including borehole fluid velocity, casing thickness, casing longitudinal wave velocity, casing transverse wave velocity and central frequency are input. A theoretical dispersion curve of the A0-mode waveforms is calculated. Based on A0's phase velocity, a waveform propagating forward or backward from one of the two receivers to a certain distance is calculated by using a phase shift method through the following equation:

$g\left(t\right){\displaystyle \int F\left(w\right)H\left(w\right)e^{-jwt}dw};$

$H\left(w\right)=e^{-jk\left(w\right)x_0};$

where g(t) is a waveform travelling at a certain distance x.sub.0; F(w) is a frequency spectrum of an original A0-mode waveform at a near or far receiver; w is angular frequency; -j is imaginary number; e is natural logarithm; H(w) is a propagation matrix, and k represents wavenumber; and due to dispersion, k is a function of phase velocity v of the A0-mode waveforms.

[0052] FIG. 2a is a schematic diagram of a measured waveform at a near receiver for the ultrasonic pitch-catch measurement. FIG. 2b is a schematic diagram of a measured waveform at a far receiver for the ultrasonic pitch-catch measurement. FIG. 3 shows a theoretical dispersion curve of A0-mode waveforms. FIG. 4 shows the array waveform obtained based on phase-shift interpolation.

(S2) Establishment of Two-Dimensional Migration Velocity Model

[0053] Grid spacing and model size are set. According to density and velocity of borehole fluid, steel casing and cement in the target area, the two-dimensional migration velocity model is established. FIG. 5 illustrates the forward velocity model used in synthetic case study.

(S3) Generation and Storage of Forward Propagating Ultrasonic Lamb Wavefield

[0054] A Ricker wavelet is selected as an ultrasonic source. The forward propagating ultrasonic Lamb wavefield is generated by using a high-order staggered grid finite difference algorithm. The reflection at an artificial boundary is absorbed and attenuated based on the non-split perfectly matched layer, and the forward propagating ultrasonic Lamb wavefield for each time step is stored.

(S4) Generation and Storage of Reversely Propagating Ultrasonic Lamb Wavefield

[0055] A time axis is reversed, and a reversely propagating ultrasonic Lamb wavefield is generated and stored, which is expressed through the following equation:

$\rho \stackrel{.Math.}{\text{u}}-\left(\lambda +2\mu \right)\nabla \nabla \cdot \text{u+}\mu \nabla \times \nabla \times \text{u=d}\left(\text{x,z=0,}T,t\right);$

where d is actual ultrasonic Lamb waveforms measured by the receivers; ρ indicates density; u indicates the reversely propagating ultrasonic Lamb wavefield; ü is a second derivative of the reversely propagating ultrasonic Lamb wavefield with respect to time; ∇ represents the spatial derivative operation; ∇. represents an operation for solving divergence degree; ∇ × represents an operation for solving curl; T is a total receiving time; t represents time step; and λ and .Math. are elastic parameters.

(S5) Calculation of Envelops of the Forward Propagating Ultrasonic Lamb Wavefield and the Reversely Propagating Ultrasonic Lamb Wavefield

[0056] Based on Hilbert transform, envelopes of the forward propagating ultrasonic Lamb wavefield and the reversely propagating ultrasonic Lamb wavefield are calculated by the following equation:

$\widehat{u}\left(t\right)=\frac{1}{\pi }{\displaystyle {\int }_{-\infty }^{+\infty }\frac{u\left(\tau \right)}{t-\tau }dt};$

where û indicates Hilbert transform of a wavefield u; τ is time delay; and an envelope ũ of a propagating wavefield is calculated based on a modulus of the Hilbert transform, expressed as:

$\tilde{u}\left(t\right)=\left|\widehat{u}\left(t\right)\right|;$

where | | represents a modulo operation.

[0057] FIG. 6 depicts the measured ultrasonic Lamb waveform at the far receiver at a certain depth point and its envelope calculated using Hilbert transform.

(S6) Application of Reverse Time Migration Imaging Condition

[0058] A zero-lag cross-correlation imaging condition is applied to the forward propagating ultrasonic Lamb wavefield and the reversely propagating ultrasonic Lamb wavefield to obtain reverse time migration imaging results for the ultrasonic pitch-catch measurement, expressed as follows:

$I\left(z,x\right)={\displaystyle {\int }_0^{T}S\left(z,x,t\right)}R\left(z,x,t\right)dt;$

where S(z, x, t) is the envelope of the forward propagating ultrasonic Lamb wavefield at spatial position (z, x) in a t.sup.th time step; R(z, x, t) is the envelope of the reversely propagating ultrasonic Lamb wavefield at the spatial position (z, x) in the t.sup.th time step; and T indicates the number of sampling points.

(S7) Laplace Filtering

[0059] The Laplace filtering is applied to suppress low-frequency imaging noises in the reverse time migration imaging results, where the Laplace filtering is expressed as:

$\tilde{I}\left(z,x\right)={\nabla }^{2}I\left(z,x\right)=\frac{{\partial }^{2}I\left(z,x\right)}{\partial z^2}+\frac{{\partial }^{2}I\left(z,x\right)}{\partial x^2};$

where I is an original reverse time migration imaging result; Ĩ is an imaging result after applying the Laplace filtering; z indicates a depth coordinate, and x indicates a distance coordinate.

[0060] The Laplace filtering is configured to highlight the high-frequency boundary by means of a second derivative of imaging with respect to space to eliminate low-frequency image noises.

[0061] In order to verify that the reverse time migration imaging method provided herein has better imaging effect of the cased-hole structure acoustic interface, the simulated ultrasonic pitch-catch measurement data and the measured ultrasonic Lamb wave calibration well data are respectively subjected to trial calculation, as shown in Embodiment 2 and Embodiment 3.

Embodiment 2

[0062] In this embodiment, the ultrasonic pitch-catch measurement data are simulated through the following steps. [0063] (1) The ultrasonic pitch-catch measurement simulated wavefield record (SimulatedLamb.dat) is read in. The borehole fluid velocity, borehole fluid density, casing density, casing P-wave velocity, casing S-wave velocity, cement density, cement P-wave velocity, cement S-wave velocity, formation density, formation P-wave velocity and formation S-wave velocity are input. [0064] (2) A theoretical dispersion curve of the A0-mode waveforms is calculated. The original Lamb data of two receivers is expanded into array waveform data based on phase-shift interpolation. [0065] (3) The grid spacing and the model size are set to establish an initial velocity model of the target area. [0066] (4) Based on the two-dimensional high-order staggered grid finite difference and the non-split perfectly matched layer, a forward propagating ultrasonic Lamb wavefield and a reversely propagating ultrasonic Lamb wavefield for each time step are generated and stored. [0067] (5) Based on the Hilbert transform, envelopes of the forward propagating ultrasonic Lamb wavefield and the reversely propagating ultrasonic Lamb wavefield are calculated. [0068] (6) A zero-lag cross-correlation imaging condition is applied to obtain reverse time migration imaging results, and then the Laplace filtering is applied to suppress low-frequency imaging noises in the reverse time migration imaging results.

[0069] FIG. 7 shows reverse time migration imaging results using the zero-lag cross-correlation imaging condition for the simulated ultrasonic pitch-catch measurement data. The casing interface can be clearly observed, but the imaging of the casing-cement interface, and the cement-formation interface are not clear enough.

[0070] FIG. 8 schematically illustrates the enhanced imaging results obtained by applying the Laplace filtering algorithm. The casing position, and the imaging of the casing-cement interface and the cement-formation interface can be clearly observed in FIG. 8, and the imaging results are highly consistent with the forward model, verifying the effectiveness of the method provided herein.

Embodiment 3

[0071] In this embodiment, ultrasonic Lamb wave calibration well data are measured as follows. [0072] (1) The ultrasonic measurement data for a calibration well is read in. The velocity and density of a borehole fluid, casing density, casing P-wave velocity, casing S-wave velocity, cement density, cement P-wave velocity, cement S-wave velocity, formation density, formation P-wave velocity and formation S-wave velocity are input. [0073] (2) A theoretical dispersion curve of the A0-mode waveforms is calculated, and the original Lamb wave data of the two receivers is expanded into array waveform data. [0074] (3) The grid spacing and the model size are set to establish the initial velocity model of the target area. [0075] (4) Based on the two-dimensional high-order staggered grid finite difference and the non-split perfectly matched layer, a forward propagating ultrasonic Lamb wavefield is generated and stored for each time step, and a reversely propagating ultrasonic Lamb wavefield for the two receivers after phase-shift interpolation is generated and stored. [0076] (5) Based on the Hilbert transform, envelopes of the forward propagating ultrasonic Lamb wavefield and the reversely propagating ultrasonic Lamb wavefield are calculated. [0077] (6) A zero-lag cross-correlation imaging condition is applied to obtain reverse time migration imaging results, and then the Laplace filtering is applied to suppress low-frequency imaging noises in the reverse time migration imaging results.

[0078] FIG. 9 is a schematic diagram of the established velocity model for the cased-hole reverse time migration in a calibration well. FIG. 10 shows the measured time-depth waveform at the near receiver for the ultrasonic pitch-catch measurement in the calibration well. FIG. 11 is a schematic diagram of the final reverse time migration imaging result for the measured ultrasonic pitch-catch measurement data of the calibration well. Referring to FIGS. 9-11, the casing position and the imaging of the casing-cement interface and the cement-formation interface can be clearly seen, and the imaging results are highly consistent with the forward model, verifying the effectiveness of the method provided herein.

[0079] In conclusion, with respect to the reverse time migration imaging method provided in this embodiment, the original waveform data of the two receivers are expanded into array waveforms by phase-shaft interpolation, and the envelopes of the forward propagating ultrasonic Lamb wavefield and the reversely propagating ultrasonic Lamb wavefield are calculated based on the Hilbert transform, so as to suppress the image artifacts caused by the A0's dispersion. Moreover, the method provided herein allows the A0-mode reflection waves and diffraction waves to converge to their real positions, which is not affected by the interface inclined angle, casing eccentricity and tool eccentricity, enabling the accurate imaging of the casing, the casing-cement interface, and the cement-formation interface, and the reliable cementing quality evaluation.

[0080] Described above are basic principles, technical features and beneficial effects of this application. It should be understood that the above embodiments are merely illustrative of this application, and are not intended to limit this application. It should be noted that various changes and modifications made by those skilled in the art without departing from the spirit and scope of this application shall fall within the scope of this application defined by the appended claims.