CN116877067A - Method for predicting hydraulic fracturing generated cracks and swept area fluid pressure - Google Patents

Method for predicting hydraulic fracturing generated cracks and swept area fluid pressure Download PDF

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CN116877067A
CN116877067A CN202310878940.XA CN202310878940A CN116877067A CN 116877067 A CN116877067 A CN 116877067A CN 202310878940 A CN202310878940 A CN 202310878940A CN 116877067 A CN116877067 A CN 116877067A
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fluid
hydraulic fracturing
fracture
pressure
hydraulic
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CN116877067B (en
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贺培
周建坤
王锦喜
黄永葵
蒙春
余华文
李梦来
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Chongqing Huadi Zihuan Technology Co ltd
Chongqing Institute of Geology and Mineral Resources
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Chongqing Huadi Zihuan Technology Co ltd
Chongqing Institute of Geology and Mineral Resources
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    • EFIXED CONSTRUCTIONS
    • E21EARTH DRILLING; MINING
    • E21BEARTH DRILLING, e.g. DEEP DRILLING; OBTAINING OIL, GAS, WATER, SOLUBLE OR MELTABLE MATERIALS OR A SLURRY OF MINERALS FROM WELLS
    • E21B49/00Testing the nature of borehole walls; Formation testing; Methods or apparatus for obtaining samples of soil or well fluids, specially adapted to earth drilling or wells
    • EFIXED CONSTRUCTIONS
    • E21EARTH DRILLING; MINING
    • E21BEARTH DRILLING, e.g. DEEP DRILLING; OBTAINING OIL, GAS, WATER, SOLUBLE OR MELTABLE MATERIALS OR A SLURRY OF MINERALS FROM WELLS
    • E21B43/00Methods or apparatus for obtaining oil, gas, water, soluble or meltable materials or a slurry of minerals from wells
    • E21B43/25Methods for stimulating production
    • E21B43/26Methods for stimulating production by forming crevices or fractures
    • EFIXED CONSTRUCTIONS
    • E21EARTH DRILLING; MINING
    • E21BEARTH DRILLING, e.g. DEEP DRILLING; OBTAINING OIL, GAS, WATER, SOLUBLE OR MELTABLE MATERIALS OR A SLURRY OF MINERALS FROM WELLS
    • E21B43/00Methods or apparatus for obtaining oil, gas, water, soluble or meltable materials or a slurry of minerals from wells
    • E21B43/25Methods for stimulating production
    • E21B43/26Methods for stimulating production by forming crevices or fractures
    • E21B43/267Methods for stimulating production by forming crevices or fractures reinforcing fractures by propping
    • EFIXED CONSTRUCTIONS
    • E21EARTH DRILLING; MINING
    • E21BEARTH DRILLING, e.g. DEEP DRILLING; OBTAINING OIL, GAS, WATER, SOLUBLE OR MELTABLE MATERIALS OR A SLURRY OF MINERALS FROM WELLS
    • E21B47/00Survey of boreholes or wells
    • E21B47/06Measuring temperature or pressure
    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10Complex mathematical operations

Abstract

The application provides a method for predicting fluid pressure of a hydraulic fracturing generated crack and swept area, which is based on the fact that the fluid pressure of the hydraulic fracturing fluid in the whole process can be dynamically predicted from the beginning of pumping fluid from the ground to the consumption of a circulation channel and perforation, then to the generation of the crack and swept area, seamless connection is realized in each link, the progressive accurate calculation is carried out layer by layer, the whole process of dynamically predicting the hydraulic fracturing pressure change is realized, the accuracy of predicting the fluid pressure of the hydraulic fracturing generated crack and swept area is improved, and the safety of hydraulic fracturing construction is ensured. And the hydraulic fracturing construction is optimized by realizing real-time regulation of the pressure of the ground pumping fluid through back-thrust calculation.

Description

Method for predicting hydraulic fracturing generated cracks and swept area fluid pressure
Technical Field
The application relates to the technical field of hydraulic fracturing, in particular to a method for predicting hydraulic fracturing generated cracks and swept area fluid pressure.
Background
Hydraulic fracturing is a technology for improving the rock by injecting a large amount of water into the rock and generating cracks on the rock by means of hydraulic energy so as to improve the permeability of the rock. During hydraulic fracturing, a large amount of water is injected into the rock from the ground through a shaft by a pump truck, the whole process realizes rock reconstruction by taking the water as an energy carrier, and one important evaluation parameter of hydraulic energy is fluid pressure. The magnitude of the fluid pressure directly influences whether the rock can be fractured or not, whether the fractured fracture can continuously extend and expand or not, whether the fracture can be triggered to activate or not, and the like. More directly, it is an important parameter in evaluating the ability and effectiveness of hydraulic fracturing modification of rock, and relates to the success or failure of the hydraulic fracturing engineering.
At present, some documents at home and abroad report fluid pressure prediction methods and techniques in hydraulic fracturing implementation, and some patents also have some hydraulic fracturing fluid pressure prediction methods. Patent CN106326591B grants a method and a device for obtaining a pressure field of a fracturing fluid in a fracture in a hydraulic fracturing process, which include obtaining a length and a width of a formation in a solution area, a fracture length, a formation parameter, a construction parameter and a fluid pressure parameter, obtaining a rock displacement field in the fracture, obtaining a fracture width at each calculation node in the fracture according to the rock displacement field, and obtaining a predicted fluid pressure field in the fracture. Patent CN109522579B grants a method for predicting fracture pressure of fracturing construction of a horizontal well, which comprises counting fracture construction parameters of a fractured well of each horizon in a target area, determining a median value of the fracture pressure of the construction of a certain horizon in each horizon, a fracture pressure gradient, a relation between the fracture pressure of the construction and the vertical depth of a reservoir, a relation between the fracture pressure of the construction and the clay content of the reservoir, establishing a fracture pressure difference of the construction of the horizontal well, and calculating the fracture pressure of the construction at a target point of the horizontal well. The literature 'calculation and analysis of friction resistance of liquid CO2 fracturing construction pipe column' provides a friction resistance calculation plate in a hydraulic fracturing pipe column, and a fitting calculation formula easy to apply in engineering is obtained. The literature radial well hydraulic fracturing friction factor and calculation formula establishes a radial well guar gum fracturing fluid friction loss calculation relation by carrying out regression fitting on 322 groups of experimental data, wherein the radial well hydraulic fracturing fluid friction loss calculation relation considers the bore hole inner diameter, the displacement, the fracturing fluid viscosity, the propping agent particle size and the sand ratio. In addition, some patents and literature disclose hydraulic fracturing fluid pressure calculation methods.
Although the above has been published with respect to methods for predicting hydraulic fracture formation fractures and swept zone fluid pressures. The method has the advantages, but is mainly limited to calculating the hydraulic fracturing fluid pressure in the pipe column channel, and has the problem that consideration is incomplete and dynamic changing fluid pressure of a generated fracture and a swept area in the hydraulic fracturing process can not be obtained.
Disclosure of Invention
Aiming at the defects existing in the prior art, the application provides a method for predicting the fluid pressure of a hydraulic fracturing generated crack and a swept area, which aims to solve the problems that the hydraulic fracturing fluid pressure in a pipe column channel is mainly limited to be calculated in the prior art, consideration is incomplete, and the dynamic change fluid pressure of the generated crack and the swept area in the hydraulic fracturing process can not be obtained. Technical problems of (2).
The application provides a method for predicting hydraulic fracturing generated cracks and swept area fluid pressure, which comprises the following steps:
step S1, determining the pressure, viscosity and displacement of ground hydraulic fracturing injection fluid, the vertical depth, well depth and inner diameter of a well completion sleeve of a fracturing well according to a hydraulic fracturing construction object, and perforating to form the size and number of holes;
step S2, determining the fluid pressure of hydraulic fracturing fluid at the fracturing perforation positions;
step S3, determining the pressure loss of hydraulic fracturing fluid passing through the perforation holes;
and S4, determining the fluid pressure of a neutralization wave area of the hydraulic fracturing generated fracture.
Optionally, the determining the fluid pressure of the hydraulic fracturing fluid at the fracturing perforation location comprises:
fluid pressure P at the frac perforation location 3 The method comprises the following steps: p (P) 3 =P 1 +P 2 -ΔP 1 Wherein P is 1 Injection of fluid pressure, P, for hydraulic fracturing 2 Fluid column pressure, ΔP, of fluid injected for hydraulic fracturing 1 A fluid column flow pressure loss for hydraulic fracturing injection;
hydraulic fracturing injection fluid column pressure P 2 The method comprises the following steps:
wherein ρ is 1 Is the density and p of hydraulic fracturing fluid 2 Is the density, rho of propping agent body in hydraulic fracturing fluid 3 The density of propping agent is considered, delta is the volume ratio of propping agent to hydraulic fracturing fluid, g is gravity acceleration, and h is the vertical depth from the ground at the position of the fracturing perforation;
hydraulic fracturing injection fluid string flow pressure loss Δp 1 The method comprises the following steps:
where Q is hydraulic injection fluid displacement, H is well depth at the fracture perforation location, d is hydraulic fracture well completion casing inner diameter, a is injection fluid power law index, and a is injection fluid consistency coefficient.
Optionally, the determining the pressure loss of the hydraulic fracturing fluid through the perforation comprises:
the hydraulic fracturing fluid is passed through the perforation holes and has pressure loss delta P 2 The method comprises the following steps:
wherein d 1 The perforation diameter is expressed as:d 0 is the initial theoretical diameter of the perforation, B is the number of perforation, < >>The perforation displacement correction coefficient is expressed as:
optionally, the determining the fluid pressure of the hydraulic fracture generation fracture neutralization sweep region includes:
the fluid pressure in the hydraulic fracturing generation fracture is as follows:
wherein p is 4 (x, y, t) is the fluid pressure in the hydraulic fracture generation fracture, φ (ζ, η) is the fluid pressure of the hydraulic fracturing fluid just passing through the perforations: phi (ζ, eta) =P 3 -ΔP 2 The method comprises the steps of carrying out a first treatment on the surface of the ψ (ζ, η) is the first order partial differentiation of φ (ζ, η); f (xi) , η, τ) is equal to P 3 -ΔP 2 A related integrated damping variable;is the propagation wave velocity of hydraulic fracturing fluid, beta 1 Is the fluid compression coefficient, ρ is the fluid density: />Lambda is the integrated damping coefficient of the hydraulic fracturing fluid flowing in the fractureThe method comprises the steps of carrying out a first treatment on the surface of the Omega is the integral region, ζ and η are integral variables +.>
Optionally, the determining the fluid pressure of the hydraulic fracture generation fracture neutralization sweep region further comprises:
the hydraulic fracturing sweep zone fluid pressures fall into two categories: the first type of hydraulic fracturing sweep zone fluid pressure is a zone where the matrix permeability is zero at the non-fracture location, and the second type of hydraulic fracturing sweep zone fluid pressure is a zone where the matrix permeability is non-zero at the non-fracture location;
the first type of hydraulic fracturing sweep and zone fluid pressure is:
wherein beta is 2 Is the compression coefficient of matrix skeleton, beta 3 Is the compression coefficient of matrix particles, omega 4 Is the fracture fluid pressure conjugate coefficient omega 5 Is the conjugate coefficient of the matrix fluid pressure, phi is the porosity of the matrix in the affected area, p' 4 (x, y, t) is fracture fluid pressure closest to the matrix;
the second type of hydraulic fracturing sweep and zone fluid pressure is:where μ is the fluid viscosity and k is the matrix permeability.
Compared with the prior art, the application has the following beneficial effects:
the hydraulic fracturing fluid pressure dynamic prediction method based on the hydraulic fracturing fluid can start pumping fluid from the ground, start pumping fluid to a circulation channel and perforation loss, then start generating cracks and sweep areas, and can realize dynamic hydraulic fracturing fluid pressure prediction in the whole process, seamless connection is realized in each link, accurate calculation is performed layer by layer, the whole process of dynamic hydraulic fracturing pressure change prediction is realized, accuracy of hydraulic fracturing generation cracks and sweep area fluid pressure prediction is improved, and hydraulic fracturing construction safety is guaranteed.
Drawings
The accompanying drawings, which are incorporated in and constitute a part of this specification, illustrate embodiments consistent with the application and together with the description, serve to explain the principles of the application.
In order to more clearly illustrate the embodiments of the application or the technical solutions of the prior art, the drawings which are used in the description of the embodiments or the prior art will be briefly described, and it will be obvious to a person skilled in the art that other drawings can be obtained from these drawings without inventive effort.
FIG. 1 is a schematic flow chart of the present application;
FIG. 2 is a schematic illustration of the fluid pressure distribution at various locations during hydraulic fracturing in accordance with the present application;
FIG. 3 is a schematic illustration of injection fluid parameters according to a first and a second embodiment of the present application;
FIG. 4 is a schematic diagram of fluid pressure and pressure loss obtained by calculation in accordance with the first and second embodiments of the present application;
FIG. 5 is a schematic diagram of a predicted hydraulic fracture generation fracture and sweep area fluid pressure in accordance with an embodiment of the present application;
FIG. 6 is a schematic diagram of example two predicted hydraulic fracture generation fractures and swept zone fluid pressures.
Reference numerals illustrate: 1. hydraulic fracturing to generate cracks; 2. hydraulic fracturing of the swept area; 3. perforation holes; 4. example hydraulic fracturing to generate fracture pressure; 5. examples hydraulic fracturing wave and zone pressure.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present application more apparent, the technical solutions of the embodiments of the present application will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present application, and it is apparent that the described embodiments are some embodiments of the present application, but not all embodiments of the present application. All other embodiments, which can be made by those skilled in the art based on the embodiments of the application without making any inventive effort, are intended to be within the scope of the application. The functional units of the same reference numerals in the examples of the present application have the same and similar structures and functions.
Referring to fig. 1, the present application provides a method of predicting hydraulic fracture generation fracture and swept zone fluid pressure comprising:
step S1, determining the pressure, viscosity and displacement of ground hydraulic fracturing injection fluid, the vertical depth, well depth and inner diameter of a well completion sleeve of a fracturing well according to a hydraulic fracturing construction object, and perforating to form the size and number of holes;
step S2, determining the fluid pressure of hydraulic fracturing fluid at the fracturing perforation positions;
step S3, determining the pressure loss of hydraulic fracturing fluid passing through the perforation holes;
and S4, determining the fluid pressure of a neutralization wave area of the hydraulic fracturing generated fracture.
Referring to fig. 2, the actual hydraulic fracturing is performed in real time, and the surface fluid passes through the fracturing pump set to obtain the hydraulic fracturing surface injection fluid pressure P 1 Then flows to the perforation position through the injection fluid pipe column, and the pressure loss delta P occurs in the flow of the hydraulic fracturing injection fluid pipe column in the process 1 At the same time, fluid is injected into the ground to increase the liquid column pressure P 2 . In addition, pressure loss ΔP occurs when liquid passes through perforations 3 2 Then flows into the rock to crack the rock and generate hydraulic fracture crack 1, and generates fluid pressure p in the crack 4 (x, y, t). At the same time, the fluid pressure in the fracture diffuses toward the matrix region, generating a swept region pressure p 5 (x,y,t)。
Embodiment one, referring to fig. 2, step S1 is performed on a shale gas fracturing well. The well-hanging depth is 4000m, the inside diameter of the well completion casing is 115.02mm, the viscosity of the fracturing fluid is 60 mPa.s, the density of the fracturing fluid is 1.05g/cm3, the volume density of propping agent (sand) is 1.57g/cm3, the apparent density of propping agent (sand) is 3.48/cm3, the perforation position and the well depth are 5600m, and the fluid injection pressure, the displacement and the sand concentration (propping agent occupies the volume ratio of hydraulic fracturing fluid) are shown in figure 3. Perforation diameter 12mm,60 degree spiral phase layout, single section length 80m, perforation quantity 90, matrix permeability is close to 0, matrix porosity 1.5%.
Step S2, adopting a formulaObtaining the pressure P of the liquid column of the injected fluid 2 41.2MPa.
Using the formulaObtaining the flow pressure loss delta P of the injection fluid pipe column 1 As shown in fig. 4.
Step S3, adopting a formulaObtaining the pressure loss delta P of hydraulic fracturing fluid passing through perforation holes 2 As shown in fig. 4.
Step S4, adopting a formulaObtaining fluid pressure p in hydraulic fracturing generated cracks 4 (x, y, t). Because the shale matrix permeability is almost zero, the hydraulic fracturing sweep zone fluid pressure formula of the first class is adopted +.>Obtaining hydraulic fracturing wave and regional fluid pressure p 5 (x, y, t), the calculation result is shown in fig. 5.
Wherein the first type of hydraulic fracturing sweep zone fluid pressure is a zone of zero (approaching zero) matrix permeability at the non-fracture location.
In a second embodiment, referring to fig. 2, step S1 is performed on a shale gas fracturing well. The well-hanging depth is 4000m, the inside diameter of the well completion casing is 115.02mm, the viscosity of the fracturing fluid is 60 mPa.s, the density of the fracturing fluid is 1.05g/cm3, the volume density of propping agent (sand) is 1.57g/cm3, the apparent density of propping agent (sand) is 3.48/cm3, the perforation position and the well depth are 5600m, and the fluid injection pressure, the displacement and the sand concentration (propping agent occupies the volume ratio of hydraulic fracturing fluid) are shown in figure 3. Perforation diameter 12mm,60 degree spiral phase layout, single section length 80m, perforation number 90, matrix permeability 10mD, matrix porosity 5.8%.
Step S2, adopting a formulaObtaining the pressure P of the liquid column of the injected fluid 2 41.2MPa.
Using the formulaObtaining the flow pressure loss delta P of the injection fluid pipe column 1 As shown in fig. 4.
Step S3, adopting a formulaObtaining the pressure loss delta P of hydraulic fracturing fluid passing through perforation holes 2 As shown in fig. 4.
Step S4, adopting a formulaObtaining fluid pressure p in hydraulic fracturing generated cracks 4 (x, y, t). Sandstone matrix permeability 10mD and matrix porosity 9.8%, so the second type hydraulic fracturing sweep area fluid pressure formula is adopted +.>Obtaining hydraulic fracturing wave and regional fluid pressure p 5 (x, y, t), the calculation result is shown in fig. 6.
Based on the first embodiment and the second embodiment, it can be seen that the method for predicting the fluid pressure of the hydraulic fracturing generation cracks and the swept area is used for realizing the prediction of the fluid pressure of the hydraulic fracturing fluid in the whole process from the start of pumping fluid to the consumption of a circulation channel and a perforation and then to the generation of the cracks and the swept area, and performing the accurate calculation of the progressive layer by layer on each link in the hydraulic fracturing process. The method not only realizes the whole process of dynamically predicting the pressure change during hydraulic fracturing, but also improves the accuracy of predicting the hydraulic fracturing generated cracks and the fluid pressure of the swept area, ensures the safety of hydraulic fracturing construction, and has important significance for the production increase of oil gas exploitation and the safety production of geothermal resource development.
Meanwhile, the hydraulic fracturing fluid pressure prediction of different types of rocks is realized based on the prediction of the fluid pressure of the swept area considering different matrix permeabilities except the cracks (different rock is reflected by different matrix permeabilities). Based on the predicted fluid pressure changes of the generated cracks and the swept areas in the hydraulic fracturing rock, the real-time adjustment of the fluid pressure of the ground pumping can be realized through back-thrust calculation, the hydraulic fracturing construction is optimized, and the expected purpose is achieved.
It should be noted that in this document, relational terms such as "first" and "second" and the like are used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Moreover, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. Without further limitation, an element defined by the phrase "comprising one … …" does not exclude the presence of other like elements in a process, method, article, or apparatus that comprises the element.
The foregoing is only a specific embodiment of the application to enable those skilled in the art to understand or practice the application. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the application. Thus, the present application is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims (5)

1. A method of predicting hydraulic fracture generation fracture and swept zone fluid pressure comprising:
step S1, determining the pressure, viscosity and displacement of ground hydraulic fracturing injection fluid, the vertical depth, well depth and inner diameter of a well completion sleeve of a fracturing well according to a hydraulic fracturing construction object, and perforating to form the size and number of holes;
step S2, determining the fluid pressure of hydraulic fracturing fluid at the fracturing perforation positions;
step S3, determining the pressure loss of hydraulic fracturing fluid passing through the perforation holes;
and S4, determining the fluid pressure of a neutralization wave area of the hydraulic fracturing generated fracture.
2. A method of predicting hydraulic fracturing generated fracture and swept zone fluid pressures as claimed in claim 1, wherein said determining fluid pressure of hydraulic fracturing fluid at the location of fracturing perforations comprises:
fluid pressure P at the frac perforation location 3 The method comprises the following steps: p (P) 3 =P 1 +P 2 -ΔP 1 Wherein P is 1 Injection of fluid pressure, P, for hydraulic fracturing 2 Fluid column pressure, ΔP, of fluid injected for hydraulic fracturing 1 A fluid column flow pressure loss for hydraulic fracturing injection;
hydraulic fracturing injection fluid column pressure P 2 The method comprises the following steps:
wherein ρ is 1 Is the density and p of hydraulic fracturing fluid 2 Is the density, rho of propping agent body in hydraulic fracturing fluid 3 The density of propping agent is considered, delta is the volume ratio of propping agent to hydraulic fracturing fluid, g is gravity acceleration, and h is the vertical depth from the ground at the position of the fracturing perforation;
hydraulic fracturing injection fluid string flow pressure loss Δp 1 The method comprises the following steps:
where Q is hydraulic injection fluid displacement, H is well depth at the fracture perforation location, d is hydraulic fracture well completion casing inner diameter, a is injection fluid power law index, and a is injection fluid consistency coefficient.
3. A method of predicting hydraulic fracture generation fracture and sweep zone fluid pressures as recited in claim 1, wherein said determining the pressure loss of hydraulic fracturing fluid through the perforations comprises:
the hydraulic fracturing fluid is passed through the perforation holes and has pressure loss delta P 2 The method comprises the following steps:
wherein d 1 The perforation diameter is expressed as:d 0 is the initial theoretical diameter of the perforation, B is the number of perforation, < >>The perforation displacement correction coefficient is expressed as:
4. a method of predicting hydraulic fracture-generating fracture and swept zone fluid pressures as claimed in claim 1, wherein said determining hydraulic fracture-generating fracture-neutralizing swept zone fluid pressures comprises:
the fluid pressure in the hydraulic fracturing generation fracture is as follows:
wherein p is 4 (x, y, t) is the fluid pressure in the hydraulic fracture generation fracture, φ (ζ, η) is the fluid pressure of the hydraulic fracturing fluid just passing through the perforations: phi (ζ, eta) =P 3 -ΔP 2 The method comprises the steps of carrying out a first treatment on the surface of the ψ (ζ, η) is the first order partial differentiation of φ (ζ, η); f (ζ, η, τ) is equal to P 3 -ΔP 2 A related integrated damping variable;is the propagation wave velocity of hydraulic fracturing fluid, beta 1 Is the fluid compression coefficient, ρ is the fluid density: />Lambda is the integrated damping coefficient of the hydraulic fracturing fluid flowing in the fracture; omega is the integral region, ζ and η are integral variables +.>
5. The method of predicting hydraulic fracture-generating fracture and swept zone fluid pressures of claim 1, wherein the determining hydraulic fracture-generating fracture-neutralizing swept zone fluid pressures further comprises:
the hydraulic fracturing sweep zone fluid pressures fall into two categories: the first type of hydraulic fracturing sweep zone fluid pressure is a zone where the matrix permeability is zero at the non-fracture location, and the second type of hydraulic fracturing sweep zone fluid pressure is a zone where the matrix permeability is non-zero at the non-fracture location;
the first type of hydraulic fracturing sweep and zone fluid pressure is:
wherein beta is 2 Is the compression coefficient of matrix skeleton, beta 3 Is a matrix particleGrain compression coefficient omega 4 Is the fracture fluid pressure conjugate coefficient omega 5 Is the conjugate coefficient of the matrix fluid pressure, phi is the porosity of the matrix in the affected area, p' 4 (x, y, t) is fracture fluid pressure closest to the matrix;
the second type of hydraulic fracturing sweep and zone fluid pressure is:where μ is the fluid viscosity and k is the matrix permeability.
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