An integrated numerical simulation method for multi-cluster hydraulic fracture propagation and proppant migration

Abstract

The present invention discloses a method for integrated numerical simulation of multi-cluster hydraulic fracture expansion and proppant migration, comprising the following steps: S1: defining a numerical simulation region and inputting basic parameters; S2: establishing a wellbore fracturing fluid flow control equation, a fracture internal fracturing fluid flow control equation and a hydraulic fracture deformation control equation, and coupling and solving to obtain fracture width, fluid flow rate and fluid pressure; S3: judging whether the fracture is extended: if extended, adding a new displacement discontinuous unit at a position adjacent to the existing fracture tip unit; S4: placing a proppant particle package in the numerical simulation region at a predetermined time node, and establishing a fracture internal proppant migration control equation; S5: calculating various force terms on the particles, thereby updating the particle position; S6: calculating the fluid-particle interaction term; S7: repeating steps S2-S6 until the preset simulation time is reached. The present invention can realize the integrated numerical simulation of hydraulic fracture expansion and proppant migration in the process of multi-cluster fracturing of horizontal wells.

Description

Technical Field

[0001] The invention relates to the technical field of hydraulic fracturing in oil and gas field development engineering, and in particular to a numerical simulation method for integrating multi-cluster hydraulic fracture expansion and proppant migration. Background Art

[0002] Horizontal well multi-cluster fracturing technology is one of the main technologies for developing unconventional oil and gas reservoirs such as shale and tight sandstone. At present, horizontal well multi-cluster fracturing technology has the significant characteristics of a large number of perforation clusters and a small cluster spacing. The small cluster spacing will produce a more obvious stress shadow effect, resulting in the competition and expansion of hydraulic fractures, making it impossible for hydraulic fractures to achieve balanced expansion. In addition, the stress shadow effect will also make the proppant migration process in hydraulic fracturing operations more complicated. Therefore, establishing an integrated numerical method that can simulate the expansion and proppant migration of horizontal well multi-cluster hydraulic fractures has obvious practical significance for optimizing the fracturing parameters of unconventional oil and gas wells and improving the production capacity of oil and gas wells. Summary of the invention

[0003] In view of the above problems, the present invention aims to provide an integrated numerical simulation method for multi-cluster hydraulic fracture expansion and proppant migration.

[0004] The technical solution of the present invention is as follows:

[0005] A method for integrated numerical simulation of multi-cluster hydraulic fracture extension and proppant migration includes the following steps:

[0006] S1: Define the numerical simulation area and input basic parameters;

[0007] S2: Establish the wellbore fracturing fluid flow control equation considering the effects of perforation friction and wellbore friction, the fracture internal fracturing fluid flow control equation considering the proppant migration effect, and the hydraulic fracture deformation control equation based on the planar three-dimensional displacement discontinuity method, and solve them in a coupled manner to obtain the fracture width, fluid flow rate, and fluid pressure;

[0008] S3: Determine whether the crack is expanding: if it is expanding, add a new displacement discontinuity unit adjacent to the existing crack tip unit; if it is not expanding, keep the crack length unchanged;

[0009] S4: placing proppant particle packages into the numerical simulation area at a predetermined time node and establishing a control equation for proppant migration inside the fracture;

[0010] S5: Calculate various forces acting on the particles, thereby updating the particle positions;

[0011] S6: Calculate fluid-particle interaction terms;

[0012] S7: Repeat steps S2-S6 until the preset simulation time is reached.

[0013] Preferably, in step S2, the wellbore fracturing fluid flow control equation considering the influence of perforation friction and wellbore friction along the way includes:

[0014] Fluid mass conservation equation:

[0015]

[0016] Where: Q t is the total flow rate of fracturing fluid injected into the fracturing stage, m 3 / s; i is the crack index; N f is the total number of cracks in the fracturing section; Q f,i is the fluid inflow of the ith hydraulic fracture, m 3 / s;

[0017] Fluid pressure continuity equation:

[0018]

[0019] Where: p o is the reference point pressure, Pa; is the inlet pressure of the ith hydraulic fracture, Pa; is the perforation friction pressure drop of the i-th hydraulic fracture, Pa; is the friction pressure drop along the wellbore from the reference point to the location of the i-th hydraulic fracture, Pa.

[0020] Preferably, the perforation friction pressure drop is calculated by the following formula:

[0021]

[0022] Where: f is the density of the fracturing fluid, lbs / gal; n p,i is the number of perforations corresponding to the ith hydraulic fracture; d p,i is the perforation diameter corresponding to the ith hydraulic fracture, in; K d is an empirical constant;

[0023] The friction pressure drop along the wellbore is calculated by the following formula:

[0024]

[0025] Q f,1 =Q t (7)

[0026] Where: C cf is the friction coefficient along the wellbore, Pa·s / m 4 ;xj is the coordinate of the jth crack, m; x j-1 is the coordinate of the j+1th crack, m; Q f,j is the fluid inflow of the jth hydraulic fracture, m 3 / s; μ f is the viscosity of the fracturing fluid, Pa·s; D is the horizontal wellbore diameter, m; Q t is the total injection rate, m 3 / s;Q f,k is the fluid inflow of the kth hydraulic fracture, m 3 / s;Q f,1 is the fluid inflow of the first hydraulic fracture, m 3 / s.

[0027] Preferably, in step S2, the flow control equation of the fracturing fluid inside the fracture considering the proppant migration effect includes:

[0028] The mass conservation equation:

[0029]

[0030] Where: w is the width of the hydraulic fracture, m; θ f is the fluid volume fraction, m 3 / m 3 ; t is time, s; u f is the fracturing fluid flow rate, m / s; C L is the fracturing fluid loss coefficient, m / s 0.5 ; τ(x) is the time it takes for the fracturing fluid to reach a certain position (x), s;

[0031] Momentum conservation equation:

[0032]

[0033] Where: μ f is the fracturing fluid viscosity, Pa·s; is the gradient operator; p is the fluid pressure, Pa; F fp is the fluid-particle interaction term, Pa / m.

[0034] Preferably, in step S2, the hydraulic fracture deformation control equation based on the plane three-dimensional displacement discontinuity method is:

[0035]

[0036] Where: σ is the normal stress of the crack unit, Pa; τ is the tangential stress of the crack unit, Pa; A ij is the influence coefficient of crack unit j on crack unit i, Pa / m; D x , D y , Dz are the displacement discontinuities in the x, y, and z directions respectively; the subscript o represents the in-situ geostress.

[0037] Preferably, in step S3, when judging whether the crack is expanding, the judgment is made based on the Type I stress intensity factor of the crack tip unit; if the Type I stress intensity factor of the crack tip unit is greater than or equal to the fracture toughness of the rock, the crack expands; if the Type I stress intensity factor of the crack tip unit is less than the fracture toughness of the rock, the crack does not expand.

[0038] Preferably, the type I stress intensity factor of the seam tip unit is calculated by the following formula:

[0039]

[0040] Where: K Ι is the type I stress intensity factor, Pa·m 1 / 2 ; E is Young's modulus, Pa; v is Poisson's ratio; d is the crack tip unit length, m; D n is the normal displacement discontinuity of the crack tip unit, m.

[0041] Preferably, in step S4, the control equation for proppant migration inside the fracture is:

[0042]

[0043] Where: a p is the particle acceleration, m / s 2 ;u p is the particle velocity, m / s; t is the time, s; D p is the drag force coefficient, 1 / s; u f is the fracturing fluid velocity, m / s; ρ p is the particle density, kg / m 3 ρ f is the density of fracturing fluid, kg / m 3 ; g is the acceleration due to gravity, m / s 2 ; is the gradient operator; p is the fracturing fluid pressure, Pa; θ p is the particle volume fraction, m 3 / m 3 ; τ p is the particle stress, Pa; p . is the average velocity of all particles around a certain particle, m / s; τ D is the collision relaxation time, s.

[0044] Preferably, in step S5, the particle positions are updated based on the forward Euler method.

[0045] ​Preferably, in step S6, the fluid-particle interaction term is calculated by the following formula:

[0046]

[0047] Where: F fp is the fluid-particle interaction term, Pa / m; f is the particle phase probability distribution function; V p is the volume of the proppant particle package, m 3 ρ p is the particle density, kg / m 3 ;D p is the drag force coefficient, 1 / s; u f is the fracturing fluid flow rate, m / s; u p is the particle velocity, m / s; is the gradient operator; p is the fracturing fluid pressure, Pa.

[0048] The beneficial effects of the present invention are:

[0049] The present invention respectively adopts a planar three-dimensional displacement discontinuity method and a mixed Euler-Lagrangian framework with high simulation accuracy to describe the process of fracture expansion and proppant migration, and realizes an integrated numerical simulation method of hydraulic fracture expansion and proppant migration in the process of horizontal well segmented multi-cluster fracturing. It can study the influence of key parameters such as non-uniform geostress, perforation parameters, and cluster spacing, and can be used for the design and optimization of hydraulic fracturing operations on an engineering scale, thereby improving the fracturing transformation effect of oil and gas wells. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0051] Figure 1 It is a schematic diagram of the calculation flow of the integrated numerical simulation method of multi-cluster hydraulic fracture extension and proppant migration of the present invention;

[0052] Figure 2 A schematic diagram of particle volume fraction distribution results at the end of simulation in a specific embodiment;

[0053] Figure 3 It is a schematic diagram of the hydraulic fracture width distribution results at the end of the simulation in a specific embodiment. DETAILED DESCRIPTION

[0054] The present invention is further described below in conjunction with the accompanying drawings and embodiments. It should be noted that, in the absence of conflict, the embodiments in this application and the technical features in the embodiments can be combined with each other. It should be noted that, unless otherwise specified, all technical and scientific terms used in this application have the same meanings as those generally understood by those of ordinary skill in the art to which this application belongs. The words "including" or "comprising" and the like used in the disclosure of the present invention mean that the elements or objects appearing before the word cover the elements or objects listed after the word and their equivalents, without excluding other elements or objects.

[0055] like Figure 1 As shown, the present invention provides a method for integrated numerical simulation of multi-cluster hydraulic fracture extension and proppant migration, comprising the following steps:

[0056] S1: Define the numerical simulation area and input basic parameters.

[0057] In a specific embodiment, the basic parameters include geomechanical parameters (Young's modulus, Poisson's ratio, fracture toughness, etc.), fracturing construction parameters (fracturing fluid density, proppant diameter, etc.) and completion parameters (cluster spacing, single cluster perforation number, etc.).

[0058] S2: Establish the wellbore fracturing fluid flow control equation considering the effects of perforation friction and friction along the wellbore, the fracturing fluid flow control equation inside the fracture considering the proppant migration effect, and the hydraulic fracture deformation control equation based on the planar three-dimensional displacement discontinuity method, and solve them in a coupled manner to obtain the fracture width, fluid flow rate and fluid pressure.

[0059] In a specific embodiment, the wellbore fracturing fluid flow control equation considering the influence of perforation friction and wellbore friction along the way includes:

[0060] Fluid mass conservation equation:

[0061]

[0062] Where: Q t is the total flow rate of fracturing fluid injected into the fracturing stage, m 3 / s; i is the crack index; N f is the total number of cracks in the fracturing section; Q f,i is the fluid inflow of the ith hydraulic fracture, m 3 / s;

[0063] Fluid pressure continuity equation:

[0064]

[0065] Where: p o is the reference point pressure, Pa; is the inlet pressure of the ith hydraulic fracture, Pa; is the perforation friction pressure drop of the i-th hydraulic fracture, Pa; is the friction pressure drop along the wellbore from the reference point to the location of the i-th hydraulic fracture, Pa.

[0066] It should be noted that when the influence of perforation friction and wellbore friction is not considered, and are all equal to zero.

[0067] Optionally, the perforation friction pressure drop is calculated by the following formula:

[0068]

[0069] Where: f is the density of the fracturing fluid, lbs / gal; n p,i is the number of perforations corresponding to the ith hydraulic fracture; d p,i is the perforation diameter corresponding to the ith hydraulic fracture, in; K d is an empirical constant;

[0070] The friction pressure drop along the wellbore is calculated by the following formula:

[0071]

[0072] Q f,1 =Q t (7)

[0073] Where: C cf is the friction coefficient along the wellbore, Pa·s / m 4 ;x j is the coordinate of the jth crack, m; x j-1 is the coordinate of the j+1th crack, m; Q f,j is the fluid inflow of the jth hydraulic fracture, m 3 / s; μ f is the viscosity of the fracturing fluid, Pa·s; D is the horizontal wellbore diameter, m; Q t is the total injection rate, m 3 / s;Q f,k is the fluid inflow of the kth hydraulic fracture, m 3 / s;Q f,1 is the fluid inflow of the first hydraulic fracture, m 3 / s.

[0074] It should be noted that, in the above embodiments, and Q f,iThe units are psi (pound force / square inch) and bpm (barrel / minute). When solving equations (1) and (2) simultaneously, their units need to be converted to Pa and m. 3 / s.

[0075] In a specific embodiment, the flow control equation of the fracturing fluid inside the fracture considering the proppant migration effect includes:

[0076] The mass conservation equation:

[0077]

[0078] Where: w is the width of the hydraulic fracture, m; θ f is the fluid volume fraction, m 3 / m 3 ; t is time, s; u f is the fracturing fluid flow rate, m / s; C L is the fracturing fluid loss coefficient, m / s 0 .5; τ(x) is the time it takes for the fracturing fluid to reach a certain position (x), s;

[0079] Momentum conservation equation:

[0080]

[0081] Where: μ f is the fracturing fluid viscosity, Pa·s; is the gradient operator; p is the fluid pressure, Pa; F fp is the fluid-particle interaction term, Pa / m.

[0082] In a specific embodiment, the hydraulic fracture deformation control equation based on the plane three-dimensional displacement discontinuity method is:

[0083]

[0084] Where: σ is the normal stress of the crack unit, Pa; τ is the tangential stress of the crack unit, Pa; A ij is the influence coefficient of crack unit j on crack unit i, Pa / m; D x , D y , D z are the displacement discontinuities in the x, y, and z directions respectively; the subscript o represents the in-situ geostress.

[0085] S3: Determine whether the crack is expanding: if it is expanding, add a new displacement discontinuity unit adjacent to the existing crack tip unit; if it is not expanding, keep the crack length unchanged.

[0086] In a specific embodiment, when judging whether a crack expands, the judgment is made based on the Type I stress intensity factor of the crack tip unit; if the Type I stress intensity factor of the crack tip unit is greater than or equal to the rock fracture toughness, the crack expands; if the Type I stress intensity factor of the crack tip unit is less than the rock fracture toughness, the crack does not expand.

[0087] Optionally, the type I stress intensity factor of the seam tip unit is calculated by the following formula:

[0088]

[0089] Where: K Ι is the type I stress intensity factor, Pa·m 1 / 2 ; E is Young's modulus, Pa; v is Poisson's ratio; d is the crack tip unit length, m; D n is the normal displacement discontinuity of the crack tip unit, m.

[0090] It should be noted that judging whether a crack has expanded is a prior art. In addition to the method used in the above embodiment, other prior art methods for judging whether a crack has expanded can also be applied to the present invention.

[0091] S4: placing proppant particle packages into the numerical simulation area at a predetermined time node and establishing a proppant migration control equation inside the fracture.

[0092] In a specific embodiment, the control equation for proppant migration inside the fracture is:

[0093]

[0094] Where: a p is the particle acceleration, m / s 2 ;u p is the particle velocity, m / s; t is the time, s; D p is the drag force coefficient, 1 / s; u f is the fracturing fluid velocity, m / s; ρ p is the particle density, kg / m 3 ρ f is the density of fracturing fluid, kg / m 3 ; g is the acceleration due to gravity, m / s 2 ; is the gradient operator; p is the fracturing fluid pressure, Pa; θ p is the particle volume fraction, m 3 / m 3 ; τ p is the particle stress, Pa; p 〉 is the average velocity of all particles around a certain particle, m / s; τ D is the collision relaxation time, s. ​

[0095] In the above embodiment, by adopting the multiphase particle grid method to establish the control equation of proppant migration inside the fracture, it is possible to flexibly and conveniently realize the simulation of more complex proppant injection conditions such as variable density and variable particle size.

[0096] S5: Calculate various forces acting on the particles and update the particle positions.

[0097] In a specific embodiment, the particle positions are updated based on the forward Euler method.

[0098] S6: Calculate the fluid-particle interaction term.

[0099] In a specific embodiment, the fluid-particle interaction term is calculated by the following formula:

[0100]

[0101] Where: F fp is the fluid-particle interaction term, Pa / m; f is the particle phase probability distribution function; V p is the volume of the proppant particle package, m 3 ρ p is the particle density, kg / m 3 ;D p is the drag force coefficient, 1 / s; u f is the fracturing fluid flow rate, m / s; u p is the particle velocity, m / s; is the gradient operator; p is the fracturing fluid pressure, Pa.

[0102] S7: Repeat steps S2-S6 until the preset simulation time is reached.

[0103] In a specific embodiment, the numerical simulation method for integrated numerical simulation of multi-cluster hydraulic fracture extension and proppant migration of the present invention is used for numerical simulation. In this embodiment, the basic parameters input are shown in Table 1:

[0104] Table 1 Basic parameters

[0105] Fracturing fluid viscosity 1mPa·s Young's modulus of rock 35GPa Fracturing fluid density <![CDATA[1000kg / m 3 ]]> Rock Poisson's ratio 0.25 Proppant density <![CDATA[2650kg / m 3 ]]> Rock fracture toughness <![CDATA[2MPa·m 0.5 ]]> Proppant particle size 0.6mm Pre-liquid time 600s Proppant injection concentration 7.5％ Injection displacement <![CDATA[0.01m 3 / s]]> Filtration coefficient <![CDATA[0m / s 0.5 ]]> Crack spacing 25m

[0106] The particle volume fraction distribution at the end of the numerical simulation of this embodiment is as follows Figure 2 As shown in Figure 2, the hydraulic fracture width distribution is as follows: Figure 3 As shown. Figure 2-3 It can be seen that due to the existence of stress shadow effect, the three hydraulic fractures show competitive expansion. The fluid and sand inflow of the middle hydraulic fracture are the lowest.

[0107] The above description is only a preferred embodiment of the present invention and does not limit the present invention in any form. Although the present invention has been disclosed as a preferred embodiment as above, it is not used to limit the present invention. Any technician familiar with this profession can make some changes or modify the technical contents disclosed above into equivalent embodiments without departing from the scope of the technical solution of the present invention. However, any simple modification, equivalent change and modification made to the above embodiments according to the technical essence of the present invention without departing from the content of the technical solution of the present invention still fall within the scope of the technical solution of the present invention.

Claims

1. A method for integrated numerical simulation of multi-cluster hydraulic fracture expansion and proppant migration, characterized in that: The following steps are involved: S1: Define the numerical simulation area and input basic parameters; S2: Establish the wellbore fracturing fluid flow control equation considering the effects of perforation friction and wellbore friction, the fracture internal fracturing fluid flow control equation considering the proppant migration effect, and the hydraulic fracture deformation control equation based on the planar three-dimensional displacement discontinuity method, and solve them in a coupled manner to obtain the fracture width, fluid flow rate, and fluid pressure; The flow control equation of the fracturing fluid inside the fracture considering the proppant migration effect includes: The mass conservation equation: Where: w is the width of the hydraulic fracture, m; θ f is the fluid volume fraction, m 3 / m 3 ; t is time, s; u f is the fracturing fluid flow rate, m / s; C L is the fracturing fluid loss coefficient, m / s 0.5 ; τ(x) is the time it takes for the fracturing fluid to reach a certain position (x), s; Momentum conservation equation: Where: μ f is the fracturing fluid viscosity, Pa·s; is the gradient operator; p is the fluid pressure, Pa; F fp is the fluid-particle interaction term, Pa / m; S3: Determine whether the crack is expanding: if it is expanding, add a new displacement discontinuity unit adjacent to the existing crack tip unit; if it is not expanding, keep the crack length unchanged; S4: placing proppant particle packages into the numerical simulation area at a predetermined time node and establishing a proppant migration control equation inside the fracture; S5: Calculate various forces acting on the particles, thereby updating the particle positions; S6: Calculate fluid-particle interaction terms; S7: Repeat steps S2-S6 until the preset simulation time is reached.

2. The integrated numerical simulation method for multi-cluster hydraulic fracture expansion and proppant migration according to claim 1, characterized in that: In step S2, the wellbore fracturing fluid flow control equation considering the influence of perforation friction and wellbore friction along the way includes: Fluid mass conservation equation: Where: Q t is the total flow rate of fracturing fluid injected into the fracturing stage, m 3 / s; i is the crack index; N f is the total number of cracks in the fracturing section; Q f,i is the fluid inflow of the ith hydraulic fracture, m 3 / s; Fluid pressure continuity equation: Where: p o is the reference point pressure, Pa; is the inlet pressure of the ith hydraulic fracture, Pa; is the perforation friction pressure drop of the i-th hydraulic fracture, Pa; is the friction pressure drop along the wellbore from the reference point to the location of the i-th hydraulic fracture, Pa.

3. The integrated numerical simulation method for multi-cluster hydraulic fracture expansion and proppant migration according to claim 2 is characterized in that: The perforation friction pressure drop is calculated by the following formula: Where: f is the density of the fracturing fluid, lbs / gal; n p,i is the number of perforations corresponding to the ith hydraulic fracture; d p,i is the perforation diameter corresponding to the ith hydraulic fracture, in; K d is an empirical constant; The friction pressure drop along the wellbore is calculated by the following formula: Q f,1 =Q t (7) Where: C cf is the friction coefficient along the wellbore, Pa·s / m 4 ;x j is the coordinate of the jth crack, m; x j-1 is the coordinate of the j+1th crack, m; Q f,j is the fluid inflow of the jth hydraulic fracture, m 3 / s; μ f is the viscosity of the fracturing fluid, Pa·s; D is the horizontal wellbore diameter, m; Q t is the total injection rate, m 3 / s;Q f,k is the fluid inflow of the kth hydraulic fracture, m 3 / s;Q f,1 is the fluid inflow of the first hydraulic fracture, m 3 / s.

4. The integrated numerical simulation method for multi-cluster hydraulic fracture extension and proppant migration according to claim 1, characterized in that: In step S2, the hydraulic fracture deformation control equation based on the plane three-dimensional displacement discontinuity method is: Where: σ is the normal stress of the crack unit, Pa; τ is the tangential stress of the crack unit, Pa; A ij is the influence coefficient of crack unit j on crack unit i, Pa / m; D x , D y , D z are the displacement discontinuities in the x, y, and z directions respectively; the subscript o represents the in-situ geostress.

5. The integrated numerical simulation method for multi-cluster hydraulic fracture expansion and proppant migration according to claim 1, characterized in that: In step S3, when judging whether the crack is expanding, the judgment is made according to the type I stress intensity factor of the crack tip unit; if the type I stress intensity factor of the crack tip unit is greater than or equal to the rock fracture toughness, the crack expands; if the type I stress intensity factor of the crack tip unit is less than the rock fracture toughness, the crack does not expand.

6. The integrated numerical simulation method for multi-cluster hydraulic fracture expansion and proppant migration according to claim 5, characterized in that: The type I stress intensity factor of the seam tip unit is calculated by the following formula: Where: K Ι is the type I stress intensity factor, Pa·m 1 / 2 ; E is Young's modulus, Pa; v is Poisson's ratio; d is the crack tip unit length, m; D n is the normal displacement discontinuity of the crack tip unit, m.

7. The integrated numerical simulation method for multi-cluster hydraulic fracture expansion and proppant migration according to claim 1, characterized in that: In step S4, the control equation for proppant migration inside the fracture is: Where: a p is the particle acceleration, m / s 2 ;u p is the particle velocity, m / s; t is the time, s; D p is the drag force coefficient, 1 / s; u f is the fracturing fluid velocity, m / s; ρ p is the particle density, kg / m 3 ; ρ f is the density of fracturing fluid, kg / m 3 ; g is the acceleration due to gravity, m / s 2 ; is the gradient operator; p is the fracturing fluid pressure, Pa; θ p is the particle volume fraction, m 3 / m 3 ; τ p is the particle stress, Pa; p > is the average velocity of all particles around a particle, m / s; τ D is the collision relaxation time, s.​ 8. The integrated numerical simulation method for multi-cluster hydraulic fracture extension and proppant migration according to claim 1, characterized in that: In step S5, the particle positions are updated based on the forward Euler method.

9. The integrated numerical simulation method for multi-cluster hydraulic fracture expansion and proppant migration according to claim 1, characterized in that: In step S6, the fluid-particle interaction term is calculated by the following formula: Where: F fp is the fluid-particle interaction term, Pa / m; f is the particle phase probability distribution function; V p is the volume of the proppant particle package, m 3 ; ρ p is the particle density, kg / m 3 ;D p is the drag force coefficient, 1 / s; u f is the fracturing fluid flow rate, m / s; u p is the particle velocity, m / s; is the gradient operator; p is the fracturing fluid pressure, Pa.

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