A coarse hydraulic fracture sanding fluid and proppant migration constitutive model calculation method

Abstract

The present application provides a kind of rough hydraulic fracture sand slurry, proppant migration constitutive model calculation method, comprising the following steps: (1) based on JRC theory, rough wall surface slope mean square and structure function are calculated, and rough wall surface topography is reconstructed using fast fourier transform;(2) based on the reconstructed rough wall surface, the height of turbulent / non-turbulent interface is calculated;(3) the particle phase constitutive equation considering rough wall surface is established;(4) the sand slurry, proppant migration constitutive equation under different rough wall surface is calculated.The method proposed in the present application further considers the influence of rough wall surface on sand slurry and proppant migration on the basis of Dontsov, and can more truly reflect the sand slurry and proppant migration process during fracturing.

Description

Technical Field

[0001] This invention belongs to the field of reservoir production enhancement and stimulation technology, and in particular relates to a constitutive model calculation method for the migration of rough hydraulic fractured sand-mixed fluid and proppant. Background Technology

[0002] In recent years, with the deepening development of unconventional reservoirs represented by shale gas and tight gas, large-scale volumetric fracturing technology has been widely applied. Due to the strong heterogeneity of reservoirs and the large differences in stress distribution, the fracture walls generated by fracturing usually have a certain degree of roughness. The migration behavior and final placement morphology of proppant and mixed fluid under rough wall conditions will affect post-fracturing productivity. Existing studies often simplify fractures as smooth flat plates, which makes it difficult to accurately reflect the migration characteristics in rough fractures. Existing CFD-DEM studies have shown that rough walls can affect proppant migration by inducing near-wall eddies, increasing velocity gradients and energy losses, and changing the instantaneous velocity distribution of proppant. Therefore, this invention, based on the constitutive model framework of smooth wall mixed fluid and proppant migration established by Dontsov, considers the above-mentioned influence of rough walls and establishes a calculation method for a constitutive model of mixed fluid and proppant migration in rough hydraulic fractures. Summary of the Invention

[0003] To address the shortcomings of existing technologies, this paper further considers the influence of rough walls on proppant migration based on Dontsov's work, and further modifies the constitutive model of sand mixing fluid and proppant migration, establishing a calculation method for the constitutive model of sand mixing fluid and proppant migration in rough hydraulic fractures.

[0004] A method for calculating the constitutive model of the transport of coarse hydraulic fracture mixed sand and proppant includes the following steps: (1) The root mean square slope and structure function of the rough wall were calculated based on JRC theory, and the morphology of the rough wall was reconstructed by fast Fourier transform. (2) Calculate the height of the turbulent / non-turbulent interface based on the reconstructed rough wall; (3) Establish the constitutive equation of the granular phase considering the rough wall surface; (4) Calculate the constitutive equations for the transport of the mixed sand and proppant under different rough wall surfaces.

[0005] Furthermore, step (1) specifically includes the following steps: Based on JRC theory, the root mean square slope of the rough wall is calculated using empirical formulas. Z 2 and structure functions SF : (1); (2); In the formula, JRC represents the joint roughness coefficient of the rough wall surface, which is dimensionless; Based on the above calculation results, a two-dimensional random matrix satisfying a Gaussian distribution is generated. z ( x , y Perform a Fourier transform on the two-dimensional random matrix to obtain... Z ( x , y ), and the autocorrelation function R ( x , y Perform a Fourier transform to obtain the transfer function. H ( x , y ); right Z ( x , y ) H ( x , y Performing an inverse Fourier transform yields the height distribution function of the rough surface. h ( x , y ).

[0006] Step (2) of calculating the height of the turbulent / non-turbulent interface specifically includes the following steps: The mean and standard deviation of the height of the reconstructed rough wall are as follows: (3); (4); In the formula, n Indicates the number of data collection points; h i Indicates the height of any sampling point, in mm; The height of the turbulent / non-turbulent interface is: (5); In the formula, δ represents the height of the turbulent / non-turbulent interface, in mm; The turbulent portion is defined as an ineffective seepage channel, and the ratio of the effective crack width to the true crack width is defined as the crack width roughness factor: (6); In the formula, w Indicates the actual seam width, in mm.

[0007] Step (3) specifically includes the following steps: The constitutive relation of the granular phase considering the influence of rough walls is as follows: (7); (8); In the formula, p Represents the normal stress in the granular phase, in MPa; t Represents the shear stress in the granular phase, in MPa; This represents the frictional stress in the granular phase, expressed in MPa. This represents the particle phase collision stress, expressed in MPa. The normal stress of granular phase friction is: (9); The normal stress of particle collision is: (10); The granular phase frictional shear stress is: (11); The particle collision shear stress is: (12); Therefore, the constitutive relation of the granular phase considering the influence of rough walls can be further expressed as: (13); (14); in, (15); (16); (17); In the formula, φ represents the proppant concentration, which is dimensionless; φ m Indicates the maximum permissible proppant concentration, φ m =0.585; μ f The viscosity of the fracturing fluid is expressed in mPa·s; v p The proppant particle velocity is represented in m / s; e represents the impact rebound coefficient, which is dimensionless; ρ s This indicates the density of the proppant particles, in kg / m³. 3 η s (φ) represents the shear viscosity coefficient, which is dimensionless; μ1, μ2 and I0 are empirical coefficients, which are dimensionless.

[0008] Step (4) specifically includes the following steps: The equation for the conservation of granular phase momentum is: ; Substituting equation (14) into equation (18), we get: (19); (20); In the formula, ; The velocity, expressed in m / s, represents the velocity under conditions of no boundary slip. ft It is the shear roughness factor, which is dimensionless; The velocity expressed is given in m / s under rough wall conditions. The normal stress of the granular phase is further expressed as: (twenty two); (twenty three); (twenty four); According to equations (22)-(24), the proppant concentration at any point within the crack is expressed as: (25); make , Then equations (22) and (25) can be simplified to: (26); (27); For functions and The limits at low and high normalized proppant concentrations are as follows: ; For functions and The limits at low and high normalized proppant concentrations are as follows: (29); Combining equations (23) and (26), the velocity distribution function is further expressed as: ; The velocities of the particulate phase, fluid phase, and mixing solution are as follows: (31); Integrating equation (31), we obtain the unit cross-sectional area flow rates of the mixing fluid and proppant: (32); in: (33); The limits and fitting formulas of equation (33) at low and high normalized proppant concentrations are as follows: (34); (35); The sand plugging function is used to characterize the critical width at which proppant enters the crack: (36); In the formula, N = 3; a Indicates the diameter of the proppant, in mm; ; H Represents the Heaviside step function; Considering the size effect between proppant particle size and crack width, equation (33) is further expressed as: (37); The limits and fitting formulas of equation (37) at low and high normalized proppant concentrations are as follows: (38); (39); Based on the above relationships, the transport equations for the mixing fluid and proppant are expressed as follows: (40); Compared with the prior art, the present invention has the following beneficial effects: This invention introduces the influence of crack wall roughness on the basis of the Dontsov model, which can more reasonably characterize the migration characteristics of sand-mixing fluid and proppant in rough hydraulic cracks and reduce the deviation caused by using a smooth plate model to characterize actual rough cracks. This invention incorporates the influence of crack wall roughness on effective flow channels, flow resistance, and particle migration into the constitutive relation of the particle phase and the migration equation of the mixed sand and proppant. It can characterize the convective migration and gravity settlement behavior of proppant under different crack roughness conditions, and provide a model basis for the simulation of proppant migration and placement in rough cracks. The calculation method established in this invention is applicable to different conditions such as fracture roughness, fracture width, and proppant particle size. It can be used to simulate proppant migration under conditions of single rough fractures and coexistence of rough fractures with natural fractures, joints, and bedding. It can provide a calculation basis for the analysis of proppant migration distance, settlement characteristics, and placement patterns during hydraulic fracturing, and provide a reference for the selection of proppant particle size, optimization of construction parameters, and evaluation of proppant placement in fractures in unconventional reservoirs. Attached Figure Description

[0009] Figure 1 This is a flowchart of the present invention; Figure 2 For Example 1, the seam width roughness factor fw Plotting the relationship between the curve and JRC fitting. Figure 3 Shear roughness factor for Example 1 ft Plotting the relationship between the curve and JRC fitting. Figure 4 The flow function of sand-carrying fluid under different JRC values ​​in Example 1 and Relationship curve diagram; Figure 5 Propionage diffusion function under different JRC values ​​in Example 1 and Relationship curve diagram; Figure 6 The proppant gravity settling function under different JRC values ​​in Example 1 and Relationship curve graph. Detailed Implementation

[0010] The present invention will be further described below with reference to specific embodiments, but this does not constitute any limitation on the invention. The basic parameters used in the calculation are shown in Table 1: Table 1. Propionate Properties and Hydraulic Fracture Parameters .

[0011] Example 1: To compare the calculation results of Dontsov and the present invention, w / a Taking a 6,40 / 70 mesh proppant as an example, the values ​​are detailed in Table 1; obtained through computer simulation. and The curve showing the fitting relationship with JRC is as follows: Figure 2 and Figure 3 As shown; calculations were performed for different JRC values. , and and Relationship curves, such as Figure 4 , Figure 5 and Figure 6 As shown, it can be seen that with the increase of hydraulic fracture roughness, the flow function of the proppant-carrying fluid, the convection-diffusion function of the proppant, and the gravity settling function of the proppant all decrease. That is, the greater the roughness of the hydraulic fracture wall, the greater the resistance to the migration of the proppant and the proppant.

Claims

1. A method for calculating the constitutive model of the transport of coarse hydraulic fracture mixed sand and proppant, characterized in that, Includes the following steps: S10: Calculate the root mean square slope and structure function of the rough wall based on JRC theory, and reconstruct the rough wall morphology using Fast Fourier Transform. The steps include: S11: Calculate the root mean square slope of rough wall surface using empirical formulas based on JRC theory. Z 2 and structure functions SF : （1）； （2）； In the formula, JRC represents the joint roughness coefficient of the rough wall surface, which is dimensionless; S12: Generate a two-dimensional random matrix that satisfies a Gaussian distribution based on the calculation results of step S11. z ( x , y Perform a Fourier transform on the two-dimensional random matrix to obtain... Z ( x , y ), and the autocorrelation function R ( x , y Perform a Fourier transform to obtain the transfer function. H ( x , y ); S13: Yes Z ( x , y ) H ( x , y Perform an inverse Fourier transform to obtain the height distribution function of the rough surface. h ( x , y ); S20: Calculate the height of the turbulent / non-turbulent interface based on the reconstructed rough wall surface. Step S20 includes: S21: The mean and standard deviation of the height of the reconstructed rough wall surface are as follows: （3）； ； In the formula, n Indicates the number of data collection points; h i Indicates the height of any sampling point, in mm; S22: Height of the turbulent / non-turbulent interface: （5）； In the formula, δ represents the height of the turbulent / non-turbulent interface, in mm; S23: Treat the turbulent portion as an ineffective seepage channel, and define the ratio of the effective crack width to the true crack width as the crack width roughness factor: （6）； In the formula, w Indicates the actual seam width, in mm; S30: Establish constitutive relations for the granular phase considering rough walls: The constitutive relation for the granular phase considering the rough wall surface is as follows: （7）； （8）； In the formula, p τ represents the normal stress in the granular phase, MPa; τ represents the shear stress in the granular phase, MPa. This represents the frictional stress in the granular phase, expressed in MPa. This represents the particle phase collision stress, expressed in MPa. The normal stress of granular phase friction is: （9）； The normal stress of particle collision is: （10）； The granular phase frictional shear stress is: （11）； The particle collision shear stress is: （12）； The constitutive relation of the granular phase considering the rough wall surface can be further expressed as: （13）； （14）； in, （15）； （16）； （17）； In the formula, φ represents the proppant concentration, which is dimensionless; φ m Indicates the maximum permissible proppant concentration, φ m = 0.585; μ f The viscosity of the fracturing fluid is expressed in mPa·s. v p This indicates the velocity of the proppant particles, in m / s; e ρ represents the collision rebound coefficient, which is dimensionless; s This indicates the density of the proppant particles, in kg / m³. 3 η s (φ) represents the shear viscosity coefficient, which is dimensionless; μ1, μ2, and I 0 These are empirical coefficients, dimensionless; S40: Determine the transport constitutive equations of the mixed sand and proppant under different rough wall conditions. Step S40 includes: S41: The equation for the conservation of momentum in the granular phase is: ； Substituting equation (14) into equation (18), we get: （19）； （20）； ； In the formula, ; f represents the velocity under no-slip boundary conditions, in m / s. τ It is the shear roughness factor, which is dimensionless; The velocity expressed is given in m / s under rough wall conditions. S42: The normal stress of the granular phase is expressed as: （22）； （23）； （24）； According to equations (22)-(24), the proppant concentration at any point within the crack is expressed as: （25）； make , Then equations (22) and (25) simplify to: ； （27）； For functions and The limits at low and high normalized proppant concentrations are: ； For functions and The limits at low and high normalized proppant concentrations are: （29）； Combining equations (23) and (26), we obtain the velocity distribution function: ； S43: The velocities of the particulate phase, fluid phase, and mixing solution are respectively: （31）； Integrating equation (31), we obtain the unit cross-sectional area flow rates of the mixed sand and proppant: （32）； in: ； The limits and fitting formulas of equation (33) at low and high normalized proppant concentrations are as follows: （34）； （35）； The sand-plugging function is used to characterize the limiting width of proppant penetration into the crack. The sand-plugging function is: （36）； In the formula, N = 3; a Indicates the diameter of the proppant, in mm; ; H Represents the Heaviside step function; Further considering the size effect between proppant particle size and crack width, equation (33) is expressed as: （37）； The limits and fitting formulas of equation (37) at low and high normalized proppant concentrations are as follows: （38）； （39）； The migration equations for the mixing fluid and proppant are as follows:

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