A fast solution method for the velocity of Krauklis waves in a fluid-containing fracture system

Abstract

The present invention provides a method for quickly solving the Krauklis wave velocity in a fluid-containing fracture system, which mainly includes the following steps: (1) Using the frequency reverse recurrence method, recursively solve the Krauklis wave velocity in the non-viscous fluid-containing fracture system from the high-frequency cut-off frequency to the low-frequency cut-off frequency; (2) Using the non-viscous-viscous recurrence method, recursively solve the Krauklis wave velocity vector in the viscous fluid-containing fracture system from the Krauklis wave velocity in the non-viscous fluid-containing fracture system. The present invention can quickly and stably solve the propagation velocity of the Krauklis wave in the full frequency band range in the non-viscous and viscous fluid-containing fracture systems. For fractures of any scale including faults, cracks, and microcracks, regardless of whether fluid viscosity is considered, the Krauklis wave velocity can be efficiently and robustly obtained in the entire frequency band where the Krauklis wave exists.

Description

Technical Field

[0001] The present invention relates to the technical field of quantitative interpretation of fracture geometric parameters, and particularly to a method for rapidly solving the Krauklis wave velocity in a fluid-containing fracture system. Background Art

[0002] The quantitative interpretation of the geometric parameters of a fluid-containing fracture system (including faults, fractures, and microfractures) is of great practical significance for oil and gas reservoir exploration and development and hydraulic fracturing monitoring. The Krauklis wave is a guided wave propagating in a fluid-containing fracture system with extremely strong dispersion. Its propagation characteristics are closely related to the geometric parameters of the fracture system. Moreover, there is a resonance effect in the Krauklis wave, which can cause the frequency-dependent characteristics of seismic body waves. Using this characteristic, the geometric parameters of the fluid-containing fracture system, including length, height, and aperture, can be quantitatively interpreted. Therefore, the Krauklis wave is an important medium for carrying out the quantitative interpretation of fracture system geometric parameters. In particular, the quantitative interpretation technology of fracture system geometric parameters based on the Krauklis wave resonance effect is considered a key technical means for the quantitative interpretation of fracture system geometric parameters.

[0003] The success or failure of this technology depends on whether the Krauklis wave propagation velocity can be obtained quickly and stably. The Krauklis wave propagation velocity is a function of the observation frequency, fracture system aperture, surrounding rock, and fluid physical property parameters. The dispersion equation involved is a transcendental equation with extremely complex form. In addition, both the natural fluid existing in the underground rock space and the fracturing fluid used in engineering have viscosity, so the fluid viscosity needs to be taken into account. When the fluid viscosity is introduced, the solutions of the Krauklis wave dispersion equation are distributed in the complex plane, and the convergence rate of the conventional numerical method is extremely low and the stability is extremely poor. When quantitatively interpreting the geometric parameters of the fracture system based on the Krauklis wave resonance effect, it is necessary to repeatedly solve this complex-domain dispersion equation for different parameters. The extremely low convergence efficiency and extremely poor stability will bring serious numerical challenges, thus restricting the development and application of this technology. Summary of the Invention

[0004] Aiming at the technical problems existing in the above background art, the present invention proposes a method for rapidly solving the Krauklis wave velocity in a fluid-containing fracture system, which can rapidly and stably solve the Krauklis wave propagation velocity in the entire frequency band in a fluid-containing fracture system with non-viscous and viscous fluids. For fractures of any scale, including faults, fractures, and microfractures, regardless of whether the fluid viscosity is considered, the Krauklis wave velocity can be obtained efficiently and robustly in the entire frequency band where the Krauklis wave exists.

[0005] To solve the above technical problems, a method for quickly solving the Krauklis wave velocity in a fluid-containing fracture system provided by the present invention mainly includes the following steps:

[0006] (1) Using the frequency reverse recurrence method, recursively solve the Krauklis wave velocity in the non-viscous fluid-containing fracture system from the high-frequency cut-off frequency to the low-frequency cut-off frequency;

[0007] (2) Using the non-viscous-viscous recurrence method, recursively solve the Krauklis wave velocity vector in the viscous fluid-containing fracture system from the Krauklis wave velocity in the non-viscous fluid-containing fracture system

[0008] For the method for quickly solving the Krauklis wave velocity in the fluid-containing fracture system, the specific process of the step (1) is as follows:

[0009] (1.1) Solve the Scholte wave velocity on the fluid-solid interface

[0010] Set the opening parameter h in the dispersion equation Eq1(h,f) of the Krauklis wave in the non-viscous fluid-containing fracture system to 0; at this time, the equation degenerates into the Scholte wave dispersion equation Eq1(0), and the Scholte wave dispersion equation Eq1(0) is independent of frequency, denoted as Eq1(0); the solution of the Scholte wave dispersion equation Eq1(0) is the Scholte wave velocity on the fluid-solid interface; use the iterative algorithm to directly solve the Scholte wave dispersion equation Eq1(0), and select the fluid acoustic velocity cp as the initial value of the iterative method to obtain the Scholte wave velocity V scholte , V scholte =solve[Eq1(0)| cp ;

[0011] (1.2) Estimate the high-frequency cut-off frequency f max

[0012] Set the opening parameter h in the dispersion equation Eq1(h,f) of the Krauklis wave in the non-viscous fluid-containing fracture system to the actual opening of the required fracture system. At this time, h is a known quantity, and estimate the high-frequency cut-off frequency; and the process of estimating the high-frequency cut-off frequency is as follows:

[0013] (1.2.1) Select Taking V scholte as the initial value, use the iterative method to solve the Krauklis wave dispersion equation Eq1(h,f) to obtain the solution

[0014] (1.2.2) Calculate the solution result in the above step (1.2.1) and the initial value Vscholte Numerical error If diff ≤ 1, then If diff > 1, then Increase at a rate of 20%, and repeat the above steps (1.2.1) until an appropriate one is selected Make diff ≤ 1, then

[0015] (1.3) Frequency backward recursion

[0016] Select the required low-frequency cut-off frequency f min And the corresponding frequency interval df, then define the frequency vector f(n):

[0017] f(n) = f min +(n - 1)df, n = 1, 2, …, N; where

[0018] For the opening h in the non-viscous fluid fracture system, the Krauklis wave velocity vector corresponding to the frequency vector f(n) in the non-viscous fluid fracture system Can be solved by recursively calculating from high frequency to low frequency, and the corresponding frequency backward recursion formula is:

[0019]

[0020]

[0021] The method for quickly solving the Krauklis wave velocity in the fluid-containing fracture system, where the dispersion equation Eq1(h, f) of the Krauklis wave in the non-viscous fluid fracture system is in the following form:

[0022]

[0023]

[0024] Where h is the fracture system opening Is the Krauklis wave velocity in the non-viscous fluid fracture system Is the Krauklis wave number in the non-viscous fluid fracture system, and there is Where ω = 2πf, f is the frequency; cp is the fluid acoustic wave velocity, ρ f Is the fluid density, VP s Is the longitudinal wave velocity of the surrounding rock, VS s Is the longitudinal wave velocity of the surrounding rock, ρ S Is the density of the surrounding rock

[0025] The method for quickly solving the Krauklis wave velocity in the fluid-containing fracture system, wherein the dispersion equation of the Krauklis wave in the fracture system with an aperture of h and containing viscous fluid is Eq2(h,f), then the Krauklis wave velocity vector in the fracture system with viscous fluid corresponding to the frequency vector f(n) can be solved by the method of recursion from the non-viscous model to the viscous model, and the non-viscous-viscous recursion formula is:

[0026]

[0027] The method for quickly solving the Krauklis wave velocity in the fluid-containing fracture system, wherein the form of the dispersion equation Eq2(h,f) of the Krauklis wave in the fracture system with viscous fluid is as follows:

[0028] (1 - c) 2 m1m2n1n2 - (b - c) 2 m1n1 - c(1 - a)(1 - b)(m1n2 + m2n1) - (1 - ac) 2 m2n2 + (ca - b) 2 =0;

[0029]

[0030]

[0031]

[0032]

[0033]

[0034]

[0035] Wherein, is the Krauklis wave velocity in the fracture system with viscous fluid; is the wave number of the Krauklis wave in the fracture system with viscous fluid, and there is where ω = 2πf, f is the frequency; ξ is the compression viscosity coefficient, and η is the shear viscosity coefficient.

[0036] Adopting the above technical solution, the present invention has the following beneficial effects:

[0037] The method for quickly solving the Krauklis wave velocity in the fluid-containing fracture system of the present invention is reasonably conceived and can quickly and stably solve the propagation velocity of the Krauklis wave in the fluid-containing fracture systems with non-viscous and viscous fluids in the full frequency band. For fractures of any scale including faults, cracks, and microfractures, regardless of whether the fluid viscosity is considered, the Krauklis wave velocity can be efficiently and robustly obtained within the frequency band where the Krauklis wave exists.

[0038] The present invention can simultaneously calculate the Krauklis wave velocity in the fluid-containing fracture systems with non-viscous and viscous fluids; and has less dependence on the iterative solution method, and has good adaptability to conventional iterative solution algorithms such as the fixed-point method, Newton's method, and Newton downhill method; the present invention is also applicable to parallel computing, and during the process of parallel programming implementation, no complex splitting is required and the operation is simple.

[0039] The present invention can quickly and stably solve the propagation velocity of the Krauklis wave in the fluid-containing fracture systems with non-viscous and viscous fluids, greatly improving the operation speed and stability of the quantitative interpretation method of fracture geometric parameters based on the Krauklis wave, improving the accuracy of quantitative interpretation of fracture geometric parameters, and enhancing the practical application effect of the quantitative interpretation method of fracture geometric parameters based on the Krauklis wave. This will help improve the exploration accuracy of fractured reservoirs and provide support for increasing oil and gas reserves and production; at the same time, it improves the accuracy of hydraulic fracturing monitoring and provides strong guarantees for fracturing effect evaluation, fracturing plan optimization, etc. Description of the Drawings

[0040] In order to more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following will briefly introduce the drawings required for use in the description of the specific embodiments or the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings.

[0041] Figure 1 It is the flowchart of the method for quickly solving the Krauklis wave velocity in the fluid-containing fracture system of the present invention;

[0042] Figure 2 It is the diagram of the Krauklis wave velocity and Scholte wave velocity in the fracture containing viscous water with an aperture of 3 mm involved in the method for quickly solving the Krauklis wave velocity in the fluid-containing fracture system of the present invention.

[0043] Figure 3 It is the diagram of the Krauklis wave velocity in the fractures containing viscous water with different apertures involved in the method for quickly solving the Krauklis wave velocity in the fluid-containing fracture system of the present invention.

[0044] Figure 4 This is the Krauklis wave velocity diagram in a crack with a width of 3 mm containing viscous water, oil, and gas in the fluid fracture system of the present invention for the rapid solution method of Krauklis wave velocity. Specific embodiments

[0045] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0046] The present invention will be further explained below in conjunction with specific embodiments.

[0047] As Figure 1 shown, the rapid solution method of Krauklis wave velocity in the fluid fracture system provided in this embodiment specifically includes the following steps:

[0048] (1) Solving the Krauklis wave velocity in the fracture system containing non-viscous fluid by the frequency reverse recurrence method

[0049] (1.1) Solving the Scholte wave velocity at the fluid-solid interface

[0050] Set the aperture parameter h in the dispersion equation Eq1(h,f) of Krauklis wave in the fracture system containing non-viscous fluid to 0; at this time, the equation degenerates into the Scholte wave dispersion equation. The Scholte wave dispersion equation is independent of frequency, and the Scholte wave dispersion equation is denoted as Eq1(0); its solution is the Scholte wave velocity at the fluid-solid interface. Use the iterative algorithm to directly solve the Scholte wave dispersion equation Eq1(0), and select the fluid acoustic wave velocity cp as the initial value of the iterative method to obtain the Scholte wave velocity V scholte where V scholte = solve[Eq1(0)| cp .

[0051] Among them, fractures will occur in underground rocks during geological tectonic movements. These fractures have scale differences and can be divided into faults, fractures, and microfractures from large to small in scale. The three are interconnected and jointly constitute the fracture system of underground rocks; and the fracture system is the storage space and seepage channel of fluids in rocks, the main exploration target of fractured hydrocarbon reservoirs, and also the key monitoring object during the development of unconventional energy such as shale gas; when considering the viscosity of underground fluids, it is called a fracture system containing viscous fluid; when not considering the viscosity of underground fluids, it is called a fracture system containing non-viscous fluid.

[0052] (1.2) Estimate the high-frequency cut-off frequency f max

[0053] Set the aperture parameter h in the Krauklis wave dispersion equation Eq1(h,f) of the non-viscous fluid fracture system to the actual aperture of the required fracture system (at this time h is a known quantity), and estimate the high-frequency cut-off frequency; the process of estimating the high-frequency cut-off frequency is as follows:

[0054] (1.2.1) Select Take V scholte as the initial value, and solve the equation Eq1(h,f) by the iterative method to obtain the solution

[0055] (1.2.2) Calculate the numerical error between the solution obtained in the above step (1.2.1) and the initial value V scholte If diff ≤ 1, then If diff > 1, then Increase at a rate of 20%, and repeat step (1.2.1) until an appropriate is selected such that diff ≤ 1, then

[0056] Among them, the Krauklis wave disappears after exceeding the high-frequency cut-off frequency f max

[0057] (1.3) Frequency reverse recursion

[0058] First, select the required low-frequency cut-off frequency f min and the corresponding frequency interval df according to the actual needs; the selection of the low-frequency cut-off frequency f min is arbitrary and can be completely determined according to the needs; the frequency interval df can first select 1000 frequency points and carry out numerical tests to determine;

[0059] Then define the frequency vector, which has the following form: f(n) = f min +(n - 1)df, n = 1, 2,..., N; in the above formula

[0060] For the non-viscous fluid fracture system with aperture h, the Krauklis wave velocity vector corresponding to the frequency vector f(n) in the non-viscous fluid fracture system can be solved by the method of reverse recursion from high frequency to low frequency, and the corresponding frequency reverse recursion formula is:

[0061]

[0062] ​​

[0063] (2) Non - viscous - viscous recurrence method for solving the Krauklis wave velocity vector in a fractured system containing viscous fluid

[0064] The dispersion equation of Krauklis wave in a fractured system with an aperture of h and containing viscous fluid is Eq2(h, f). Then, the Krauklis wave velocity vector in the fractured system containing viscous fluid corresponding to the frequency vector f(n) can be solved by the method of recurrence from the non - viscous model to the viscous model. The non - viscous - viscous recurrence formula is as follows:

[0065]

[0066] The form of the above - mentioned dispersion equation Eq1(h, f) of Krauklis wave in a fractured system containing non - viscous fluid is as follows:

[0067]

[0068]

[0069] where h is the aperture of the fractured system, is the Krauklis wave velocity in a fractured system containing non - viscous fluid, is the Krauklis wave number in a fractured system containing non - viscous fluid, and there is where ω = 2πf, f is the frequency; cp is the acoustic wave velocity of the fluid, ρ f is the fluid density, VP s is the longitudinal wave velocity of the surrounding rock, VS s is the longitudinal wave velocity of the surrounding rock, ρ S is the density of the surrounding rock.

[0070] The form of the above - mentioned dispersion equation Eq2(h, f) of Krauklis wave in a fractured system containing viscous fluid is as follows:

[0071] (1 - c) 2 m1m2n1n2-(b - c) 2 m1n1 - c(1 - a)(1 - b)(m1n2 + m2n1)-(1 - ac) 2 m2n2+(ca - b) 2 = 0;

[0072]

[0073]

[0074]

[0075]

[0076]

[0077]

[0078] Among them, is the Krauklis wave velocity in the viscous fluid-containing fracture system; is the Krauklis wave number in the viscous fluid-containing fracture system, and there is where ω = 2πf, f is the frequency; ξ is the compression viscosity coefficient, and η is the shear viscosity coefficient.

[0079] The appendix of the present invention Figure 1 includes key technical links such as the solution of the Scholte wave velocity on the fluid-solid interface, the estimation of the high-frequency cut-off frequency, the frequency reverse recursion, and the non-viscous-viscous recursion. The arrows indicate the technical process and the recursion direction.

[0080] The appendix of the present invention Figure 2 The abscissa is the logarithmic frequency, and the ordinate is the velocity magnitude; the figure shows the curves of the Krauklis phase velocity (solid line, the velocity calculated by the present invention) and the group velocity (dashed line, a physical quantity further calculated based on the velocity calculated by the present invention) varying with the frequency in the crack containing viscous water with an opening of 3 mm in the frequency band of 1 Hz - 1 MHz.

[0081] The appendix of the present invention Figure 3 The abscissa is the logarithmic frequency, and the ordinate is the velocity magnitude; the figure shows the curves of the Krauklis phase velocity (solid line, the velocity calculated by the present invention) and the group velocity (dashed line, a physical quantity further calculated based on the velocity calculated by the present invention) varying with the frequency in the fracture systems containing viscous water with openings of 0.001 m, 0.01 m, 0.1 m, and 1 m respectively in the frequency band of 1 Hz - 1 MHz.

[0082] The appendix of the present invention Figure 4 The abscissa is the logarithmic frequency, and the ordinate is the velocity magnitude; the figure shows the curves of the Krauklis phase velocity (solid line, the velocity calculated by the present invention) and the group velocity (dashed line, a physical quantity further calculated based on the velocity calculated by the present invention) varying with the frequency in the cracks containing viscous water, viscous oil, and viscous gas with an opening of 3 mm respectively in the frequency band of 1 Hz - 1 MHz.

[0083] The concept of the present invention is reasonable. For fractures of any scale including faults, cracks, and microfractures, regardless of whether fluid viscosity is considered, the Krauklis wave velocity can be efficiently and robustly obtained within the entire frequency band where the Krauklis wave exists.

[0084] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those of ordinary skill in the art should understand that they can still modify the technical solutions described in the foregoing embodiments, or perform equivalent replacements for some or all of the technical features; and these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for quickly solving the velocity of Krauklis waves in a fluid-containing fracture system, characterized in that, It mainly includes the following steps: (1) Using the frequency reverse recurrence method, from the high-frequency cut-off frequency to the low-frequency cut-off frequency, recursively solve the Krauklis wave velocity in the non-viscous fluid fracture system; (2) Using the non-viscous-viscous recurrence method, the Krauklis wave velocity vector in the fractured system containing viscous fluid is recursively solved from the Krauklis wave velocity in the fractured system containing non-viscous fluid. The specific process of step (1) is as follows: (1.1) Solve the Scholte wave velocity on the fluid-solid interface Set the aperture parameter h in the dispersion equation Eq1(h,f) of Krauklis waves in a fracture system containing non-viscous fluid to 0; at this time, the equation degenerates into the Scholte wave dispersion equation Eq1(0), and the Scholte wave dispersion equation Eq1(0) is independent of frequency, denoted as Eq1(0); the solution of the Scholte wave dispersion equation Eq1(0) is the Scholte wave velocity on the fluid-solid interface; use an iterative algorithm to directly solve the Scholte wave dispersion equation Eq1(0), and select the fluid acoustic wave velocity cp as the initial value of the iterative method to obtain the Scholte wave velocity V on the fluid-solid interface scholte , V scholte =solve[Eq1(0)| cp ; (1.2) Estimate the high-frequency cut-off frequency f max Set the aperture parameter h in the Krauklis wave dispersion equation Eq1(h,f) in the non-viscous fluid fracture system to the actual aperture of the required fracture system. At this time, h is a known quantity, and estimate the high-frequency cut-off frequency. The high-frequency cut-off frequency estimation process is as follows: (1.2.1) Select With V scholte as the initial value, solve the Krauklis wave dispersion equation Eq1(h,f) by the iterative method to obtain the solution (1.2.2) Calculate the solution result in the above step (1.2.1) and the initial value V scholte for the numerical error If diff ≤ 1, then If diff > 1, then Increase at a rate of 20%, repeat the above step (1.2.1) until an appropriate is selected such that diff ≤ 1, then (1.3) Frequency reverse recurrence Select the required low-frequency cut-off frequency f min and the corresponding frequency interval df, and then define the frequency vector f(n): f(n)=f min +(n - 1)df, n = 1, 2, …, N; where, In a non-viscous fluid fracture system with an opening of h, the Krauklis wave velocity vector corresponding to the frequency vector f(n) in the non-viscous fluid fracture system can be solved by recursively calculating from high frequency to low frequency. The corresponding reverse frequency recurrence formula is:

2. The method for quickly solving the Krauklis wave velocity in the fluid fracture system according to claim 1, wherein The form of the dispersion equation Eq1(h,f) of the Krauklis wave in the non-viscous fluid fracture system is as follows: where h is the fracture system aperture, is the Krauklis wave velocity in the fracture system containing non-viscous fluid, is the Krauklis wave number in the fracture system containing non-viscous fluid, and there is where ω = 2πf, f is the frequency; cp is the acoustic wave velocity of the fluid, ρf is the fluid density, VP s is the longitudinal wave velocity of the surrounding rock, VS s is the shear wave velocity of the surrounding rock, ρ S is the density of the surrounding rock.

3. The method for rapidly solving the Krauklis wave velocity in the fluid fracture system according to claim 1, characterized in that , the dispersion equation of Krauklis waves in the fracture system with viscous fluid and an opening of h in step (2) is Eq2(h,f). Then, the Krauklis wave velocity vector in the fracture system with viscous fluid corresponding to the frequency vector f(n) can be solved by recursively deriving from the non-viscous model to the viscous model. The non-viscous to viscous recurrence formula is:

4. The method for quickly solving the Krauklis wave velocity in the fluid fracture system according to claim 3, characterized in that The form of the dispersion equation Eq2(h,f) of the Krauklis wave in the viscous fluid fracture system is as follows: (1-c) 2 m1m2n1n2-(b-c) 2 m1n1-c(1-a)(1-b)(m1n2+m2n1) -(1 - ac) 2 m2n2+(ca - b) 2 = 0; μ f = -iωη, μ S = ρ S VS s 2 ; wherein, is the Krauklis wave velocity in the viscous fluid-containing fracture system; is the Krauklis wave number in the viscous fluid-containing fracture system, and where ω = 2πf, f is the frequency; ξ is the compressional viscosity coefficient, and η is the shear viscosity coefficient.

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