Method and System For Calculating Extension Length of Main Fracture In Hydraulic Fracturing, and Medium

Description

CROSS REFERENCE TO RELATED APPLICATION

This patent application claims the benefit and priority of Chinese Patent Application No. 202411933226.7, filed with the China National Intellectual Property Administration on Dec. 26, 2024, the disclosure of which is incorporated by reference herein in its entirety as part of the present application.

TECHNICAL FIELD

The present disclosure belongs to the technical field of hydraulic fracturing, and in particular, to a method and system for calculating an extension length of a main fracture in hydraulic fracturing, and a medium.

BACKGROUND

Hydraulic fracturing, as a crucial measure for enhancing oil and gas production, has been widely adopted worldwide. This technology involves injecting high-pressure fluid into formations to create a complex fracture network, thereby improving the permeability and productivity of oil and gas reservoirs. During hydraulic fracturing, the main fracture serves as one of the most critical fluid pathways, and its extension length is of significant importance for reservoir development. On one hand, the evaluation of the extension length of the main fracture directly impacts fracture stability and productivity, playing a vital role in optimizing fracturing design and increasing oil and gas output. On the other hand, it indirectly determines the quantity of proppant required, helping to avoid unnecessary proppant waste and the costs associated with re-fracturing. This maximizes the protection of groundwater resources and the ecological environment, enabling sustainable development in oil and gas extraction.

Currently, methods for evaluating the extension length of main fractures primarily include microseismic monitoring, electromagnetic detection, and other techniques. Microseismic monitoring tracks fracture propagation by recording microseismic events generated during fracturing from the surface or within wells. However, this method is limited by environmental noise, formation conditions, and the arrangement of monitoring equipment, and has high monitoring costs. Electromagnetic detection involves charging underground fluids via wellbores to enable fluids in fractures to generate electromagnetic fields for fracture detection. However, the detection resolution is low, making it difficult to capture full extension information of main fractures accurately in real time. Meanwhile, the extension behavior of main fractures during fracturing is influenced by multiple factors, including in-situ stress distribution, rock mechanical properties, and fracturing fluid characteristics. The complexity and uncertainty of these factors pose significant challenges to the accurate evaluation of the extension length of main fractures.

Furthermore, with the advancement of unconventional oil and gas resource development and continuous innovation in fracturing techniques, traditional evaluation methods struggle to meet the demand for efficiency and convenience in new fracturing processes. Particularly in shale gas development, due to strong formation heterogeneity and complex stress environments, the applicability of existing evaluation methods is severely limited. Therefore, there is an urgent need to develop a new method for evaluating the extension length of main fractures to improve accuracy and practicality.

SUMMARY

In view of this, the present disclosure provides a method and system for calculating an extension length of a main fracture in hydraulic fracturing, and a medium, which can solve the problem of lacking accurate evaluation methods for the extension length of main fractures, reduce the influence of factors such as in-situ stress distribution, rock mechanical properties, fracturing fluid properties and monitoring equipment in traditional evaluation methods, improve detection accuracy, and reduce monitoring costs.

The present disclosure is implemented as follows:

The present disclosure provides a method for calculating an extension length of a main fracture in hydraulic fracturing, including the following steps:

• • S10: acquiring a water hammer wave signal at a wellhead after pump shutdown during hydraulic fracturing using a high-frequency pressure gauge;
  • S20: converting the water hammer wave signal into a complex cepstrum and performing cepstrum analysis on the water hammer wave signal;
  • S30: calculating a minimum period between positive pulse extremum points in a curve of the complex cepstrum and taking half of the minimum period as a time for a water hammer wave to travel from the wellhead to a bridge plug;
  • S40: calculating an average velocity of the water hammer wave in a well based on the time for the water hammer wave to travel from the wellhead to the bridge plug and a distance from the wellhead to the bridge plug;
  • S50: performing linear trend removal on a local signal of the water hammer wave during an ascending phase or a descending phase of the water hammer wave along a wellbore;
  • S60: performing a Fourier transform on the local signal with linear trend removed, to obtain a frequency spectrum of the local signal of the water hammer wave;
  • S70: extracting a fundamental frequency corresponding to maximum value points in the frequency spectrum of the local signal of the water hammer wave as a response frequency in a main fracture;
  • S80: calculating a propagation time of the water hammer wave in the main fracture based on the response frequency in the main fracture; and
  • S90: calculating an extension length of the main fracture based on the average velocity of the water hammer wave in the well and the propagation time of the water hammer wave in the main fracture.

On the basis of the above technical solution, the method for calculating an extension length of a main fracture in hydraulic fracturing according to the present disclosure may be further improved as follows:

S20 specifically includes:

• • performing a Fourier transform on the water hammer wave signal to obtain a signal frequency spectrum;
  • performing a modulo operation on the signal frequency spectrum to obtain a signal amplitude spectrum;
  • performing a natural logarithm operation on the signal amplitude spectrum to obtain a logarithmic spectrum;
  • performing an inverse Fourier transform on the logarithmic spectrum to obtain the complex cepstrum of the water hammer wave signal;
  • extracting an amplitude of the complex cepstrum using a square-root operation of a sum of squares of a real part and an imaginary part; and
  • extracting maximum value points of the amplitude of the complex cepstrum using a peak detection method.

Further, S30 specifically includes:

• • extracting a frequency corresponding to a maximum value of the signal amplitude spectrum of the water hammer wave signal and taking a reciprocal of the frequency as a motion period of the water hammer wave during the ascending phase or the descending phase along the wellbore;
  • identifying a positive pulse region in the curve of the complex cepstrum based on the motion period;
  • performing a local extremum search within the positive pulse region and marking positions of extremum points;
  • calculating time intervals between adjacent extremum points to obtain multiple period values;
  • averaging the multiple period values to obtain an average period value; and
  • taking half of the average period value as the time for the water hammer wave to travel from the wellhead to the bridge plug.

Further, S40 specifically includes:

• • obtaining an actual distance from the wellhead to the bridge plug from drilling data as a propagation distance of the water hammer wave; and
  • dividing the propagation distance of the water hammer wave by the time taken to travel from the wellhead to the bridge plug to calculate the average velocity of the water hammer wave in the well.

Further, S50 specifically includes:

• • identifying a rising segment and a falling segment of the water hammer wave signal to extract the local signal;
  • fitting a linear trend of the local signal using a least squares method;
  • calculating linear trend coefficients, comprising slope and intercept, and generating a trend signal having a same length as the local signal; and
  • subtracting the linear trend from the local signal to obtain the local signal with linear trend removed.

Further, S60 specifically includes:

• • performing framing and windowing on the local signal with linear trend removed;
  • performing a fast Fourier transform on a windowed signal;
  • taking a modulus of a fast Fourier transform result to obtain an amplitude of each frequency component; and
  • taking frequencies and amplitudes within a Nyquist frequency range as the frequency spectrum of the local signal.

Further, S70 specifically includes:

• • performing peak detection on the frequency spectrum of the local signal of the water hammer wave;
  • searching for frequency points satisfying peak conditions in the frequency spectrum;
  • sorting the frequency points in ascending order and extracting only a smallest frequency point exhibiting a multiple relationship as the fundamental frequency; and
  • selecting the fundamental frequency as the response frequency in the main fracture.

Further, S80 specifically includes:

• • taking a reciprocal of the response frequency in the main fracture and dividing the reciprocal by 2 to obtain the propagation time of the water hammer wave in the main fracture.

Further, S90 specifically includes:

• • multiplying the average velocity of the water hammer wave in the well by the propagation time of the water hammer wave in the main fracture to obtain the extension length of the main fracture.
  • an expression of complex cepstrum conversion of the water hammer wave signal is as follows:

x ˆ ( t ) = F - 1 [ ln ⁢ ❘ "\[LeftBracketingBar]" F [ x ⁡ ( t ) ] ❘ "\[RightBracketingBar]" ] ;

• • where {circumflex over (x)}(t) is the complex cepstrum of the water hammer wave signal (unit: s); F⁻¹[·] is an inverse Fourier transform operator; F[·] is a Fourier transform operator; x(t) is a time-domain signal of the water hammer wave (unit: MPa); t denotes time (unit: s); In is a natural logarithm function.

a calculation expression of the average velocity of the water hammer wave in the well is as follows:

v ¯ = d T ;

• • where v is the average velocity of the water hammer wave in the well (unit: m/s); d is a propagation distance of the water hammer wave from the wellhead to the bridge plug along the wellbore (unit: m); T is the time for the water hammer wave to propagate from the wellhead to the bridge plug along the wellbore (unit: s).

A calculation expression for linear trend removal for the local signal of the water hammer wave is as follows:

y k ( t ) = x k ( t ) - ( at + b ) ; where : a = n ⁢ ∑ i = 1 n t i ⁢ x k ( t i ) - ∑ i = 1 n t i ⁢ ∑ i = 1 n x k ( t i ) n ⁢ ∑ i = 1 n t i 2 - ( ∑ i = 1 n t i ) 2 ; b = ∑ i = 1 n x k ( t i ) - a ⁢ ∑ i = 1 n t i n

• • where γ_k(t) is data after linear trend removal of a k-th local signal of the water hammer wave (unit: MPa); x_k (t) is the k-th local signal of the water hammer wave (unit: MPa); t is a time series of the local signal (unit: s); n is a length of the time series.

A calculation expression for the frequency spectrum of the local signal of the water hammer wave is as follows:

y k ( ω ) = ❘ "\[LeftBracketingBar]" F [ y k ( t ) ] ❘ "\[RightBracketingBar]" ;

• • where γ_k(o) is the frequency spectrum of the k-th local signal of the water hammer wave; ω is an angular frequency (unit: rad/s).

A calculation expression for the propagation time of the water hammer wave in the main fracture is as follows:

t ′ = 1 2 ⁢ ω ′ ;

• • where t′ is the propagation time of the water hammer wave in the main fracture (unit: s); ω′ is the response frequency in the main fracture (unit: Hz).

A calculation expression for the extension length of the main fracture is as follows:

l = v ¯ ⁢ t ′ ;

• • where l is the extension length of the main fracture (unit: m).

The principles and significance of these equations are as follows:

1. The complex cepstrum conversion uses the inverse Fourier transform of the logarithmic spectrum, which can effectively enhance the fracture response in the signal and reduce noise interference.

2. The average velocity calculation considers the influence of wave velocity changes of the water hammer wave in different cycles, improving the calculation accuracy.

3. The linear trend removal better highlights the characteristic frequency of the main fracture signal, making it easier to extract the fundamental frequency of the signal.

4. The calculation of the frequency spectrum of the local signal considers the fact that the fracture response frequency is easily submerged in the overall water hammer wave signal, and by extracting local signals, the spectral analysis is made more efficient and accurate.

5. The calculation of the propagation time considers the reciprocal relationship between the signal frequency and the signal period, improving the convenience of time estimation.

6. The calculation of the main fracture extension length separately considers the average propagation velocity of the water hammer wave in the wellbore and the time rules of propagation in the main fracture, making the results more reliable.

Compared with the prior art, the method of the present disclosure:

• • 1. extracts local signals of the water hammer wave for spectral analysis, highlighting the characteristics of fracture response;
  • 2. uses complex cepstrum analysis to improve the accuracy of signal processing;
  • 3. considers the dynamic changes of the water hammer wave; and
  • 4. establishes a complete mathematical model system that can perform quantitative calculations.

The method for calculating an extension length of a main fracture in hydraulic fracturing proposed by the present disclosure analyzes the water hammer wave signal, combines cepstrum analysis and spectral analysis techniques, and establishes a complete evaluation system for the extension length of the main fracture. The method has the following technical effect:

First, the method of the present disclosure uses a high-frequency pressure gauge to obtain the water hammer wave signal. Compared with traditional monitoring methods, it has the advantages of simple equipment and low cost. Through complex cepstrum analysis of the water hammer wave signal, the periodic components in the signal can be effectively extracted, noise interference can be reduced, and the accuracy of signal processing can be improved.

Second, the method of the present disclosure uses local signal analysis technology. The rising and falling segments of the water hammer wave signal are processed separately, effectively avoiding the influence of signal interference and distortion. Furthermore, the introduction of processing methods such as linear trend removal further improves the accuracy of signal analysis.

Third, the method of the present disclosure has strong real-time performance and adaptability. Through real-time acquisition and analysis of water hammer wave signals, the information of the extension length of the main fracture can be obtained in time, providing an important basis for fracturing process optimization. This method is suitable for various formation conditions and fracturing processes and has broad application prospects.

BRIEF DESCRIPTION OF THE DRAWINGS

To describe the technical solutions in the embodiments of the present disclosure more clearly, the following briefly describes the drawings required for describing the embodiments or the prior art. Apparently, the drawings in the following description show merely some of the embodiments of the present disclosure, and those of ordinary skill in the art may still derive other drawings from these drawings without creative efforts.

FIG. 1 is a flowchart of a method for calculating an extension length of a main fracture in hydraulic fracturing;

FIG. 2 is a schematic structural diagram of a three-dimensional well fracture model in hydraulic fracturing with a main fracture length of 40 meters in a first embodiment of the method for calculating an extension length of a main fracture in hydraulic fracturing;

FIG. 3 is a waveform diagram of a water hammer wave signal obtained after pump shutdown during hydraulic fracturing in a well fracture model with a main fracture length of 40 meters in the first embodiment of the method for calculating an extension length of a main fracture in hydraulic fracturing;

FIG. 4 is a cepstrum curve of a water hammer wave based on cepstrum analysis in a well fracture model with a main fracture length of 40 meters in the first embodiment of the method for calculating an extension length of a main fracture in hydraulic fracturing;

FIG. 5A shows a local signal of the water hammer wave before trend removal in a well fracture model with a main fracture length of 40 meters, and FIG. 5B shows a local signal of the water hammer wave after trend removal in a well fracture model with a main fracture length of 40 meters;

FIG. 6 is a diagram of a frequency spectrum of the local signal of the water hammer wave in an ascending phase or a descending phase in a well fracture model with a main fracture length of 40 meters in the first embodiment of the method for calculating an extension length of a main fracture in hydraulic fracturing;

FIG. 7 is a waveform diagram of a water hammer wave signal generated after pump shutdown during hydraulic fracturing in a well fracture model with a main fracture length of 50 meters in the first embodiment of the method for calculating an extension length of a main fracture in hydraulic fracturing; and

FIG. 8 is a diagram of a frequency spectrum of the local signal of the water hammer wave in an ascending phase or a descending phase in a well fracture model with a main fracture length of 50 meters in the first embodiment of the method for calculating an extension length of a main fracture in hydraulic fracturing.

DETAILED DESCRIPTION OF THE EMBODIMENTS

To make the objectives, technical solutions, and advantages of the embodiments of the present disclosure clearer, the technical solutions in the embodiments of the present disclosure are clearly and completely described below with reference to the accompanying drawings in the embodiments of the present disclosure.

FIG. 1, FIG. 2, FIG. 3, FIG. 4, FIGS. 5A-5B, FIG. 6, FIG. 7, and FIG. 8 illustrate a first embodiment of a method for calculating an extension length of a main fracture in hydraulic fracturing provided by the present disclosure. In this embodiment, the method for calculating an extension length of a main fracture in hydraulic fracturing includes the following steps:

• • acquiring a water hammer wave signal at a wellhead after pump shutdown during hydraulic fracturing;
  • performing cepstrum analysis on the water hammer wave signal to obtain an average velocity of a water hammer wave in a well;
  • performing spectral analysis on the water hammer wave signal to obtain a propagation time of the water hammer wave in a main fracture of the hydraulic fracturing; and
  • calculating an extension length of the main fracture based on the average velocity of the water hammer wave in the well and the propagation time of the water hammer wave in the main fracture of the hydraulic fracturing.

As a preferred technical solution of the present disclosure, the step of performing cepstrum analysis on the water hammer wave signal to obtain the average velocity of the water hammer wave in the well specifically includes:

• • converting the water hammer wave signal into a complex cepstrum, with a conversion expression as follows:

x ˆ ( t ) = ℱ - 1 [ log [ ❘ "\[LeftBracketingBar]" ℱ [ x ⁡ ( t ) ] ❘ "\[RightBracketingBar]" ] ] ;

• • where {circumflex over (x)}(t) is the complex cepstrum of the water hammer wave signal; ⁻¹[·] denotes an inverse Fourier transform; [·] denotes a Fourier transform; x(t) is a time-domain signal of the water hammer wave;
  • calculating a minimum period between positive pulse extremum points in a curve of the complex cepstrum and taking the minimum period as a time T for the water hammer wave to propagate from the wellhead to the bridge plug along the wellbore; and
  • calculating an average velocity P of the water hammer wave in the well based on a distance d and the time T for the water hammer wave to propagate from the wellhead to the bridge plug along the wellbore, with an expression as follows:

v ¯ = d T ;

As a preferred technical solution of the present disclosure, the spectral analysis specifically includes:

• • performing linear trend removal on a local signal x_k(t) of the water hammer wave during an ascending phase or a descending phase of the water hammer wave along a wellbore;

{ y k ( t ) = x k ( t ) - ( at + b ) a = ( n ⁢ ∑ i = 1 n t i ⁢ x k ( t i ) - ∑ i = 1 n t i ⁢ ∑ i = 1 n x k ( t i ) ) ( n ⁢ ∑ i = 1 n t i 2 - ( ∑ i = 1 n t i ) 2 ) b = ( ∑ i = 1 n x k ( t i ) - a ⁢ ∑ i = 1 n t i ) n ;

• • where y_k(t) is data after linear trend removal of a k-th local signal of the water hammer wave in the ascending phase or descending phase; x_k (t) is the k-th local signal of the water hammer wave in the ascending phase or descending phase; t is a time series of the local signal; n is a length of the time series;
  • performing a Fourier transform on the detrended data to obtain a frequency spectrum, with a calculation expression as follows:

y k ( ω ) = ❘ "\[LeftBracketingBar]" ℱ [ y k ( t ) ] ❘ "\[RightBracketingBar]" ;

• • where y_k (ω) is the frequency spectrum of the k-th local signal of the water hammer wave in the ascending phase or the descending phase;
  • extracting a fundamental frequency corresponding to maximum value points in the frequency spectrum of the local signal of the water hammer wave as a response frequency ω′ in the main fracture, and calculating the propagation time t′ of the water hammer wave in the main fracture, with a calculation expression as follows:

t ′ = 1 2 ⁢ ω ′ .

Below is a specific application scenario of this embodiment. In this application scenario, a case study of three-dimensional hydraulic fracturing was established, with the scenario setup as shown in FIG. 2. The specific configuration details are as follows: inner diameter of the pipe: 0.2 m; distance from the bridge plug to the wellhead along the wellbore: approximately 3000 m; pipe wall thickness: 7.72 mm; Young's modulus of the pipe: 206 GPa; Darcy friction factor of the pipe wall: 0.028. At a position 2900 m from the wellhead along the wellbore, there is a perforation with an inner diameter of 0.015 m, connected to the main fracture. The main fracture is equivalent to a rectangular fracture network having a length of 40 m, a width of 1 cm, and a height of 10 m. Fluid parameters are as follows: initial flow rate: 0.27 m³/s; initial pressure: 600 atm.

After instantaneous pump shutdown in the fracturing well model, a pressure change curve at the wellhead is output at a sampling frequency of 1000 Hz as the water hammer wave signal recorded by the high-frequency pressure gauge. The recorded water hammer wave signal is shown in FIG. 3.

Cepstrum analysis is performed on the water hammer wave signal in FIG. 3.

The complex cepstrum of the water hammer wave signal is calculated, with results shown in FIG. 4.

A minimum period between positive pulse extremum points the complex cepstrum curve in FIG. 4 is calculated, and is taken as the time T=2.289 s for the water hammer wave to propagate from the wellhead to the bridge plug along the wellbore.

The average velocity

v ¯ = d T = 3 ⁢ 0 ⁢ 0 ⁢ 0 2 . 2 ⁢ 8 ⁢ 9 ≈ 1 310.6 m / s

of the water hammer wave in the well is calculated based on the distance d=3000 m and the time T for the water hammer wave to propagate from the wellhead to the bridge plug along the wellbore.

Spectral analysis is performed on the water hammer wave signal in FIG. 3.

Linear trend removal is performed on a local signal of the water hammer wave during the ascending phase or the descending phase of the water hammer wave along the wellbore, as shown in FIGS. 5A-5B.

A Fourier transform is performed on the detrended data to obtain a frequency spectrum, as shown in FIG. 6.

The fundamental frequency corresponding to the maximum value points in the frequency spectrum of the local signal of the water hammer wave as is taken as the response frequency ω′=18.03 Hz in the main fracture, and the propagation time

t ′ = 1 2 ⁢ ω ′ = 1 ( 2 × 18.03 ) ≈ 0.0277 s

of the water hammer wave in the main fracture is calculated.

The extension length 1=v·t′=1310.6×0.0277≈36.3 m of the main fracture is calculated based on the average velocity of the water hammer wave in the well and the propagation time of the water hammer wave in the main fracture of the hydraulic fracturing. The calculated value shows minimal error compared to the actual value and can be effectively used for fracturing effect evaluation.

Below is another application scenario of this embodiment. In this application scenario, a hydraulic fracturing case study was established, with the following specific configuration details: inner diameter of the pipe: 0.2 m; distance from the bridge plug to the wellhead along the wellbore: approximately 3000 m; pipe wall thickness: 7.72 mm; Young's modulus of the pipe: 206 GPa; Darcy friction factor of the pipe wall: 0.028. At a position 2900 m from the wellhead along the wellbore, there is a perforation with an inner diameter of 0.015 m, connected to the main fracture. The main fracture is equivalent to a rectangular fracture network having a length of 50 m, a width of 1 cm, and a height of 10 m. Fluid parameters are as follows: initial flow rate: 0.27 m³/s; initial pressure: 600 atm.

After instantaneous pump shutdown in the fracturing well model, a pressure change curve at the wellhead is output at a sampling frequency of 1000 Hz as the water hammer wave signal recorded by the high-frequency pressure gauge. The recorded water hammer wave signal is shown in FIG. 7.

Cepstrum analysis is performed on the water hammer wave signal in FIG. 7.

The complex cepstrum of the water hammer wave signal is calculated.

A minimum period between positive pulse extremum points in the complex cepstrum curve is calculated, and is taken as the time T=2.289 s for the water hammer wave to propagate from the wellhead to the bridge plug along the wellbore.

The average velocity

v _ = d T = 3000 2.289 ≈ 1310.6 m / s

of the water hammer wave in the well is calculated based on the distance d=3000 m and the time T for the water hammer wave to propagate from the wellhead to the bridge plug along the wellbore.

Spectral analysis is performed on the water hammer wave signal in FIG. 7.

Linear trend removal is performed on a local signal of the water hammer wave during the ascending phase or the descending phase of the water hammer wave along the wellbore.

A Fourier transform is performed on the detrended data to obtain a frequency spectrum, as shown in FIG. 8.

The fundamental frequency corresponding to the maximum value points in the frequency spectrum of the local signal of the water hammer wave as is taken as the response frequency ω′=15.03 Hz in the main fracture, and the propagation time

t ′ = 1 2 ⁢ ω ′ = 1 ( 2 × 15.03 ) ≈ 0.0333 s

of the water hammer wave in the main fracture is calculated.

The extension length l=v·t′=1310.6×0.0333≈43.6 m of the main fracture is calculated based on the average velocity of the water hammer wave in the well and the propagation time of the water hammer wave in the main fracture of the hydraulic fracturing. The calculated value closely matches the actual value and can be effectively used for fracturing effect evaluation.

A second embodiment of the method for calculating an extension length of a main fracture in hydraulic fracturing provided by the present disclosure is described below. In this embodiment, a horizontal well fracturing operation in a shale gas block is used to illustrate the specific implementation process of the present disclosure. The horizontal well has a depth of 3500 m, with a horizontal section length of 1500 m; the target formation is shale reservoir, with a formation temperature of 95° C., a formation pressure of 35 MPa, and a minimum horizontal principal stress of 45 MPa.

This fracturing operation used slickwater fracturing fluid with a designed injection volume of 1200 m³, a displacement rate of 12 m³/min, and a sand filling volume fraction of 15%. During fracturing, staged fracturing was performed at 11 perforation clusters, with each perforation cluster having a length of 0.8 m, a perforation density of 16 holes per meter, and an inter-cluster spacing of 15 m. The bridge plug was installed at 3200 m from the wellhead to facilitate staged fracturing.

Throughout the fracturing process, a high-frequency pressure gauge was employed at the wellhead to collect pressure data at a sampling rate of 1000 Hz, with data acquisition continuing for 120 s after pump shutdown. The pressure measurement range of the high-frequency pressure gauge was 0 to 100 MPa, with an accuracy of 0.01 MPa. A fiber-optic temperature monitoring system was used to measure wellbore temperature with a precision of 0.1° C.

Water hammer wave signal recording began immediately after pump shutdown, with the acquired raw pressure data shown in Table 1.

TABLE 1
Wellhead pressure data after pump shutdown (partial)

	Time (s)	Pressure (MPa)

	0.000	45.263
	0.001	45.258
	0.002	45.251
	0.003	45.245
	0.004	45.238
	0.005	45.232
	0.006	45.225
	0.007	45.219
	0.008	45.212
	0.009	45.206
	0.010	45.199

Complex cepstrum analysis was first performed on the acquired pressure data. Through steps of Fourier transform, modulo operation, logarithmic operation and inverse Fourier transform, the complex cepstrum of the water hammer wave signal was obtained. By analyzing the complex cepstrum curve, positive pulse extremum points were identified, with the minimum period between adjacent extremum points calculated as 4.267 s, which represents the round-trip propagation time of the water hammer wave from the wellhead to the bridge plug.

Based on the actual distance (3200 m) from the wellhead to the bridge plug and the propagation time (4.267 s), the preliminary calculated average velocity of the water hammer wave in the well was 1498.24 m/s. Considering the influence of pressure and temperature, correction calculations are required. During the pump shutdown process, the monitored wellbore pressure variation was 2.5 MPa, and the temperature variation was 1.2° C. With a selected pressure correction coefficient of 0.003 and temperature correction coefficient of 0.0003, the corrected average velocity of the water hammer wave is 1505.83 m/s.

During local signal analysis on the water hammer wave signal, a typical rising segment with a duration of 0.5 s containing 500 data points was selected. First, linear trend removal was performed, yielding linear trend coefficients a=−0.0856 and b=45.263. A noise correction factor of 0.2 and signal noise standard deviation of 0.005 MPa were selected to correct the detrended data.

Spectral analysis was conducted on the corrected local signal. A Hanning window function was applied for weighting, followed by discrete Fourier transform. In the spectral analysis, a spectral correction factor of 0.03 and frequency attenuation factor of 0.003 were selected. Through peak detection, dominant peaks in the spectrum were identified, with the maximum peak corresponding to a frequency of 23.15 Hz, which was taken as the response frequency in the main fracture.

The propagation time of the water hammer wave in the main fracture was calculated based on the response frequency. Considering variations in fracturing fluid properties, measured viscosity change was 15 Pa s and density change was 50 kg/m³. With the selected viscosity correction factor of 0.0003 and density correction factor of 0.00003, the calculated propagation time of the water hammer wave in the main fracture was 0.0216 s.

Finally, the extension length of the main fracture was calculated. During calculation, considering formation parameter effects, measured rock elastic modulus change was 5 GPa and horizontal in-situ stress change was 3 MPa. With the selected elastic modulus correction factor of 0.0003 and horizontal stress correction factor of 0.003, the final calculated extension length of the main fracture was 32.56 m.

To verify reliability of the calculation result, microseismic monitoring and production testing were conducted after fracturing. Microseismic monitoring showed that the hypocenter distribution range basically matched the calculated extension length of the main fracture, with an error within 10%. Production test data indicated that post-fracturing gas production reached 35,000 m³/day, meeting expected stimulation results.

This embodiment demonstrates the following advantages of the proposed method: First, the method employs a high-frequency pressure gauge for data acquisition, featuring simple equipment, convenient operation, and low cost. Secondly, by combining complex cepstrum analysis with spectral analysis, the method effectively extracts characteristic information from the water hammer wave signal. Thirdly, by considering multiple influencing factors including pressure, temperature, fluid properties and formation parameters, the method improves calculation accuracy. Fourthly, calculation results show good consistency with other monitoring methods, verifying the reliability of the method.

In practical applications, the method of the present disclosure allows parameter adjustments based on specific working conditions. For example, appropriate correction factors can be selected according to formation conditions and fracturing process characteristics. For different reservoir types, the elastic modulus correction factor can be adjusted based on rock mechanical properties. For different fracturing fluid systems, viscosity and density correction factors can be adjusted according to fluid properties. This flexibility provides strong adaptability.

This embodiment also shows that the method can be applied not only to the evaluation of individual fracturing stages but also to the fracturing effect evaluation of the entire horizontal well. Analysis of multiple fracturing stages can produce fracture network distribution characteristics, providing basis for fracturing process optimization. Meanwhile, the real-time feature of the method enables dynamic monitoring during fracturing operations to promptly identify and resolve issues.

From the perspective of engineering application, the implementation cost of the method is significantly lower than traditional methods like microseismic monitoring. In this embodiment, main equipment investment only included high-frequency pressure gauges and data acquisition system, totaling approximately 100,000 RMB, whereas microseismic monitoring systems typically cost over 1 million RMB. Additionally, the data processing of the method can be automated, greatly reducing labor costs.

During operations, the calculation results of the method can guide fracturing parameter adjustments in real time. For example, if a main fracture extension length of a fracturing segment significantly deviates from a design value, operation parameters like injection volume, displacement, or proppant concentration of subsequent stages can be promptly adjusted. This real-time optimization capability is significant for improving fracturing effectiveness.

In summary, this embodiment comprehensively demonstrates the application process of the proposed method in horizontal well fracturing of shale gas. Through detailed data acquisition, processing and analysis, a reliable extension length evaluation result of the main fracture is obtained. Implementation shows that the method features simple operation, low cost and reliable accuracy, providing a practical evaluation tool for hydraulic fracturing engineering.

A third embodiment of the method for calculating an extension length of a main fracture in hydraulic fracturing provided by the present disclosure is described below. In this embodiment, a vertical well fracturing stimulation in a loose sandstone reservoir is taken as an example to illustrate the specific implementation process of the present disclosure. The well depth is 2,800 meters, and the target layer is a loose sandstone reservoir with a porosity of 25%, permeability of 85×10{circumflex over ( )}(−3) μm², formation temperature of 85° C., formation pressure of 28 MPa, minimum horizontal principal stress of 38 MPa, Young's modulus of 15 GPa, and Poisson's ratio of 0.25.

This fracturing operation adopted a low-damage fracturing fluid system with a viscosity of 35 mPa s and a density of 1.15 g/cm³. The designed total injection volume of the fracturing fluid was 800 m³, with a pumping rate of 8 m³/min. The proppant selected was 40/70-mesh ceramic sand, with a designed maximum proppant concentration of 500 kg/m³. The bridge plug was installed at 2,650 meters from the wellhead, and the target segment length of the fracturing operation was 50 meters.

A high-frequency pressure gauge was installed at the wellhead with a sampling frequency of 2,000 Hz, measurement accuracy of 0.01 MPa, and a pressure measurement range of 0 to 80 MPa. A high-precision temperature sensor with an accuracy of 0.1° C. was also installed. The fracturing fluid storage tank was equipped with a densitometer and viscometer to monitor real-time changes in fracturing fluid properties.

After pump shutdown, the raw data of the water hammer wave signal were recorded for complex cepstrum analysis. First, a Fourier transform was first performed on the acquired pressure signal x(t):

X ⁡ ( ω ) = F [ x ⁡ ( t ) ] = ∫ - ∞ ∞ x ⁡ ( t ) ⁢ e - j ⁢ ω ⁢ t ⁢ dt .

Then, an amplitude spectrum of the signal was calculated:

❘ "\[LeftBracketingBar]" X ⁡ ( ω ) ❘ "\[RightBracketingBar]" = Re [ X ⁡ ( ω ) ] 2 + Im [ X ⁡ ( ω ) ] 2 .

A natural logarithm was applied to the amplitude spectrum:

Y ⁡ ( ω ) = ln ⁢ ❘ "\[LeftBracketingBar]" X ⁡ ( ω ) ❘ "\[RightBracketingBar]" .

Finally, an inverse Fourier transform was performed to obtain the complex cepstrum:

x ⁡ ( t ) = F - 1 [ Y ⁡ ( ω ) ] = 1 2 ⁢ π ⁢ ∫ - ∞ ∞ Y ⁡ ( ω ) ⁢ e j ⁢ ω ⁢ t ⁢ d ⁢ ω .

By analyzing the complex cepstrum curve, positive pulse extremum points were extracted, and the time differences between adjacent extremum points are shown in Table 2.

TABLE 2
Time differences between positive pulse
extremum points in complex cepstrum

	No.	Time Difference (s)

	1	3.852
	2	3.865
	3	3.849
	4	3.858
	5	3.861

The minimum value of 3.849 seconds was taken as the round-trip propagation time T of the water hammer wave in the wellbore. Based on the distance from the wellhead to the bridge plug d=2650 m, the preliminary average velocity of the water hammer wave in the well was calculated:

v _ 0 = d T = 2650 3.849 = 688.49 m / s .

Considering the effects of pressure and temperature, the measured wellbore pressure change in the wellbore was Δp=3.2 MPa, and the temperature change was ΔT_f=1.8° C. A pressure correction factor α=0.004 and a temperature correction factor δ=0.0004 were selected, and the corrected average velocity of the water hammer wave was:

v=v₀+αΔp+βΔT_f=688.49+0.004×3.2×10⁶+0.0004×1.8=701.33m/s

Local analysis was performed on the water hammer wave signal, selecting a data segment within 0.5 seconds after pump shutdown. Let the original data be x_k(t). First, the linear trend coefficients were calculated:

a = n ⁢ ∑ i = 1 n t i ⁢ x k ( t i ) - ∑ i = 1 n t i ⁢ ∑ i = 1 n x k ( t i ) n ⁢ ∑ i = 1 n t i 2 - ( ∑ i = 1 n t i ) 2 = - 0.0923 ; b = ∑ i = 1 n x k ( t i ) - a ⁢ ∑ i = 1 n t i n = 38.562 .

A noise correction factor γ=0.25 was selected, and the signal noise standard deviation was calculated: σ=0.006 MPa. Trend removal and noise correction were performed on the data:

y k ( t ) = x k ( t ) - ( at + b ) + γ ⁢ σ n .

Spectral analysis was performed on the corrected data. First, windowing was applied using a Hamming window:

w ⁡ ( n ) = 0.54 - 0.46 cos ⁢ ( 2 ⁢ π ⁢ n N - 1 ) , N = 0 , 1 , ⋯ , N - 1 ;

where N is the number of data points. Discrete Fourier transform was performed on the windowed data:

Y k ( ω ) = ∑ n = 0 N - 1 y k ( n ) ⁢ w ⁡ ( n ) ⁢ e - j ⁢ ω ⁢ n .

A spectral correction factor λ=0.035 and a frequency attenuation factor μ=0.0035 were selected, and the corrected spectrum was:

Y k ′ ( ω ) = ❘ "\[LeftBracketingBar]" Y k ( ω ) ❘ "\[RightBracketingBar]" + λ ⁢ e - μω .

Through peak detection, the dominant peaks in the spectrum were identified, as shown in Table 3.

TABLE 3
Dominant peaks in the spectrum

No.	Frequency (Hz)	Amplitude

1	18.25	0.856
2	27.63	0.723
3	35.42	0.645
4	42.18	0.589
5	51.36	0.482

The frequency corresponding to the maximum amplitude, 18.25 Hz, was selected as the response frequency ω′ in the main fracture.

During calculation of the propagation time of the water hammer wave in the main fracture, changes in fluid properties were considered. The measured viscosity change of the fracturing fluid was Δη=12 Pa·s, and the density change was Δp=45 kg/m³. A fluid viscosity correction factor δ=0.00035 and a density correction factor θ=0.000035 were selected, and the propagation time was calculated:

t ′ = 1 2 ⁢ ω′ + δΔη + θΔρ = 0.0274 + 0.00035 × 12 + 0.000035 × 45 = 0.0316 s .

Finally, the extension length of the main fracture was calculated. The measured rock elastic modulus change was ΔE=4.5 GPa, and the horizontal in-situ stress change was Δσ_h=2.8 MPa. An elastic modulus correction factor φ=0.00035 and a horizontal stress correction factor ψ=0.0035 were selected, and the extension length of the main fracture was calculated:

l = v _ ⁢ t ′ + ϕΔ ⁢ E + ψΔσ h = 701.33 × 0.0316 + 0.00035 × 4.5 × 10 9 + 0.0035 × 2.8 × 10 6 = 22.16 m .

To verify the accuracy of the calculation results, the following validation was performed:

(1) Acoustic logging verification: After fracturing, acoustic logging was conducted, and the measured extension range of the main fracture was approximately 21 to 24 meters, which is consistent with the calculation results.

(2) Production performance verification: After fracturing stimulation, the production data of the well are shown in Table 4.

TABLE 4
Comparison of production data before and after fracturing

		Before	After
	Parameter	Fracturing	Fracturing

	Daily Oil Production (tons)	2.5	15.8
	Water Cut (%)	12.5	15.2
	Dynamic Fluid Level (m)	1850	1620
	Bottomhole Flowing Pressure (MPa)	18.5	16.8

The production data indicate that the fracturing stimulation achieved good results, matching the calculated extension length of the main fracture.

(3) Fracturing fluid flowback verification: Within 48 hours after fracturing, the flowback rate of the fracturing fluid reached 65%, and flowback rate of the proppant carryover was less than 0.5%, indicating the formation of a stable propped fracture, which is consistent with the calculation results.

The specific implementation process of this embodiment demonstrates that:

Signal processing method is effective: Complex cepstrum analysis can accurately extract the propagation time of the water hammer wave in the wellbore, and spectral analysis reliably identifies the response frequency in the main fracture.

The application of this embodiment shows that the method can be extended to similar oil and gas well fracturing projects. Whether for vertical well fracturing or horizontal well fracturing, the calculation results exhibit high reliability. The extension length information of the main fracture obtained through this method can guide subsequent fracturing parameter optimization and production plan formulation.

This method can not only evaluate the effectiveness of a single fracturing operation but also be used to optimize the design of staged fracturing. By monitoring the extension length of the main fracture of each fracturing stage in real time, fracturing parameters can be dynamically adjusted to improve overall stimulation effectiveness. This optimization capability is of great significance for enhancing the economic efficiency and effectiveness of fracturing operations.

In summary, this embodiment, through an oil well fracturing case study, details the implementation process and effectiveness verification of the method of the present disclosure. The results show that the method has the advantages of high reliability, simple operation, and low cost, providing a practical evaluation tool for oil and gas well fracturing engineering. At the same time, this embodiment also provides valuable references for broader application of the method.

The technical principle of the present disclosure is based on the propagation characteristics of water hammer waves in the wellbore and fractures. After fracturing pump shutdown, a water hammer wave is generated in the wellbore due to sudden pressure changes, and this wave propagates back and forth in the wellbore and the main fracture. By analyzing the propagation characteristics of the water hammer wave signal, information about the extension length of the main fracture can be obtained.

The table below lists the formula symbols used in the present disclosure and their meanings:

TABLE 5
Formula symbols used in the present disclosure and their meanings

{circumflex over (x)}(t): Complex cepstrum of the water	F−1[•]: Inverse Fourier transform operator
hammer wave signal
F[•]: Fourier transform operator	x(t): Time-domain signal of the water hammer
	wave
t: Time	ln: Natural logarithm function
ν: Average velocity	d: Distance of the water
of the water hammer	hammer wave travelling
wave in the well	from the wellhead to the bridge plug along the
	wellbore
T: Time for the water hammer wave to	yk(t): Data of the k-th local signal of the water
travel from the wellhead to the bridge	hammer wave after linear trend removal
plug along the wellbore
Δp: Wellbore pressure change	β: Temperature correction factor
xk(t): K-th local signal of the water	yk(ω): Frequency spectrum of the k-th
hammer wave	local signal of the water hammer wave
n: Length of the time sequence	γ: Noise correction factor
ω: Angular frequency	t′: Propagation time of the water hammer wave in
	the main fracture
ω′: Response frequency in the main	λ: Spectral correction factor
fracture
Δη: Fracturing fluid viscosity change	δ: Fluid viscosity correction factor
Δρ: Fracturing fluid density change	θ: Density correction factor
φ: Elastic modulus correction factor	l: Extension length of the main fracture
ψ: Horizontal stress correction factor	ΔE: Rock elastic modulus change
α: Pressure correction factor	Δσh: Horizontal in-situ stress change
σn: Standard deviation of signal noise	ΔTf: Wellbore fluid temperature change
μ: Frequency attenuation factor

Specifically, the present disclosure first uses complex cepstrum analysis to process the water hammer wave signal. The complex cepstrum is the inverse Fourier transform of the logarithm of the signal spectrum and has the advantage of extracting periodic components. Through complex cepstrum analysis, the round-trip propagation time of the water hammer wave in the wellbore can be accurately obtained. The ratio of the time to the wellbore length can be used to calculate the average propagation velocity of the water hammer wave in the well.

For spectral analysis, the present disclosure processes the local signal of the water hammer wave. By removing linear trends, the influence of signal non-stationarity can be eliminated. Spectral analysis can reveal the response characteristics of the water hammer wave in the main fracture; particularly, by extracting the fundamental frequency of the spectrum, the propagation time of the water hammer wave in the main fracture can be calculated.

The scientific nature of this method is reflected in two aspects: On one hand, it is based on reliable physical principles, namely the propagation characteristics of water hammer waves; on the other hand, it employs advanced signal processing techniques, including complex cepstrum analysis and spectral analysis. By combining physical principles with signal processing technology, a comprehensive evaluation system for the extension length of the main fracture is established.

The technical solution of the present disclosure is logical. Starting from signal acquisition, it proceeds through signal processing and parameter extraction steps to ultimately determine the extension length of the main fracture. Each step has solid theoretical foundations and practice basis, forming a complete technical chain. This method not only accounts for the complexity of physical processes but also considers the feasibility of engineering applications, thereby effectively addressing the technical challenge of evaluating an extension length of a main fracture.