A method for predicting the conductivity of multi-size proppant piles under pressure breakage.

Abstract

This invention discloses a method for predicting the conductivity of multi-size proppant stacks considering compressive breakage, comprising the following steps: establishing a slot width prediction model considering the effects of proppant embedding and proppant stack deformation; establishing a porosity prediction model using the slot width deformation prediction model; obtaining the particle size distribution of multi-size proppant stacks after breakage by combining the relationship between the particle size distribution of a single-size proppant group and the work input per unit volume with the gradation transfer matrix; establishing a permeability prediction model based on the multi-size KC equation; and combining the slot width prediction model and the permeability prediction model to obtain the conductivity of multi-size proppant stacks considering compressive breakage. The method for predicting the conductivity of multi-layer multi-size proppant stacks provided by this invention fully considers the influence of particle embedding and breakage evolution on the conductivity of multi-size proppant groups, and is more consistent with actual conditions.

Description

Technical Field

[0001] This invention relates to the field of hydraulic fracturing technology, and in particular to a method for predicting the conductivity of multi-size proppant stacks under pressure fracturing. Background Technology

[0002] Deep unconventional gas reservoirs have low permeability and low or no natural production capacity, necessitating fracturing technology for effective development. High in-situ stress and enhanced rock plasticity in deep reservoirs lead to high fracture closure pressure, resulting in severe proppant breakage and embedding. This causes a rapid decline in fracture conductivity, making it difficult to maintain long-term conductivity and severely limiting the effectiveness of fracturing in these reservoirs. This poses a significant challenge to the establishment and stabilization of production in unconventional gas reservoirs.

[0003] To evaluate fracturing effectiveness, predict oil and gas well productivity, and further optimize hydraulic fracturing design for better economic benefits, conductivity prediction is crucial. In conductivity prediction, permeability is a critical parameter. Permeability can be calculated from proppant particle size. However, during hydraulic fracturing, due to the use of proppant with combined particle sizes and proppant breakage, the proppant filling the fracture is not composed of particles of uniform size. Instead, it typically consists of particles within a certain range of screen sizes.

[0004] Different sizes of proppant affect the permeability of propped fractures. However, existing technologies generally use average particle size or weighted average particle size instead of the actual particle size for permeability prediction, which introduces a certain error compared to the actual results. These factors contribute to the errors in the current technology for predicting the conductivity of multi-size proppant stacks. Therefore, how to provide a permeability prediction method that considers the effects of multi-size proppant stacking to obtain more realistic prediction results is an urgent technical problem to be solved in the field of fracture conductivity prediction. Summary of the Invention

[0005] In view of this, the present invention provides a method for predicting the conductivity of multi-size proppant stacks that take into account compression and breakage, in order to overcome the shortcomings of the prior art.

[0006] The specific technical solution of this invention is as follows:

[0007] A method for predicting the conductivity of multi-size proppant piles considering compressive breakage, characterized by comprising the following steps:

[0008] (1) Considering the effects of proppant embedding and proppant stack deformation, a joint width prediction model is established;

[0009] (2) Establish a porosity prediction model using the aforementioned seam width deformation prediction model;

[0010] (3) By combining the relationship between the particle size distribution after crushing a single particle size group and the input work per unit volume with the gradation transfer matrix, the particle size distribution after crushing a multi-particle size group can be obtained.

[0011] (4) Establish a permeability prediction model based on the multi-particle-size KC equation;

[0012] (5) Combine the gap width prediction model in step (1) and the permeability prediction model in step (4) to obtain the flow conduction capacity of multi-size proppant stack considering pressure breakage.

[0013] Furthermore, the seam width prediction model in step (1) is as follows:

[0014] w f =w0-δ1-δ2 (1)

[0015] Among them, w f The crack width, in mm, is calculated to account for the proppant embedding effect and the proppant stack deformation effect; w0 is the initial crack width, in mm; δ1 is the crack surface deformation, in mm; and δ2 is the proppant stack deformation, in mm.

[0016] Furthermore, the method for obtaining the surface deformation of the crack includes:

[0017]

[0018] Where, d s ν1 is the equivalent particle size of the multi-particle-size stack; σ is the closure pressure; ν1 is the Poisson's ratio of the proppant particles; ν2 is the Poisson's ratio of the rock slab; E1 is the Young's modulus of the proppant particles; E2 is the Young's modulus of the rock slab.

[0019] Furthermore, the method for obtaining the proppant stack deformation includes:

[0020]

[0021] Among them, E a The apparent Young's modulus of the proppant pile.

[0022] Furthermore, the porosity prediction model includes:

[0023]

[0024] in Porosity.

[0025] Furthermore, the permeability prediction model for the multi-particle-size KC equation in step (4) is as follows:

[0026]

[0027] Furthermore, k fτ is the permeability, τ is the proppant stack tortuosity, d is the proppant particle size, H is the total specific surface area of ​​the two types of particles, and i is the serial number.

[0028] Furthermore, the method for obtaining the flow conductivity of the multi-size proppant stack in step (5) is as follows:

[0029] D f =w f k f (27)

[0030] In the formula, D f This enhances the flow conductivity of multi-size proppant stacks.

[0031] On the other hand, the present invention provides a system for predicting the conductivity of multi-size proppant piles under pressure breakage, comprising:

[0032] The gap width prediction module is used to establish a gap width prediction model that takes into account the effects of proppant embedding and proppant stack deformation.

[0033] The porosity prediction module is used to establish predicted porosity based on the seam width deformation prediction model.

[0034] The multi-size proppant particle size distribution prediction module is used to predict the particle size distribution after multi-size proppant crushing by combining the relationship between the particle size distribution after single-size proppant crushing and the input work per unit volume, and the gradation transfer matrix.

[0035] The permeability prediction module is used to predict the permeability of multi-particle-size KC equations;

[0036] The flow capacity prediction module is used in conjunction with the slot width prediction module and the permeability prediction module to obtain the flow capacity of multi-size proppant stacks that take into account pressure breakage.

[0037] Compared with the prior art, the beneficial effects of the present invention are as follows: The present invention provides a method and system for predicting the conductivity of multi-size proppant stacks under pressure breakage. It considers the changes in porosity and gradation curve under proppant breakage conditions, and proposes a method for predicting the conductivity of multi-layer multi-size proppant stacks based on the modified KC equation. This method fully considers the influence of the particle breakage evolution law of multi-size proppant stacks on the crack conductivity, and is more consistent with the actual situation. Attached Figure Description

[0038] Figure 1 This is a flowchart of the method steps of the present invention. Detailed Implementation

[0039] The details of the present invention can be more clearly understood by referring to the accompanying drawings and the description of specific embodiments. However, the specific embodiments of the present invention described herein are for illustrative purposes only and should not be construed as limiting the invention in any way. Under the teachings of this invention, those skilled in the art can conceive of any possible modifications based on the invention, and these should all be considered to fall within the scope of the invention.

[0040] like Figure 1 As shown, this invention provides a method for predicting the conductivity of multi-size proppant piles considering compression breakage, comprising the following steps:

[0041] (1) Considering the effects of proppant embedding and proppant stack deformation, a joint width prediction model is established;

[0042] Fracture conductivity refers to the ability of a proppant-filled fracture to allow fluid to pass through it under reservoir stress. It is generally expressed as the permeability (Kp) of the proppant zone. f ) and support joint width (w f The product of (K) f *w f The width w is represented by ) . Under the action of closure pressure, proppant embedding and breakage will both lead to a decrease in the gap width w. f Therefore, in order to better match the actual situation, the effects of proppant embedding and proppant stack deformation need to be considered when calculating the joint width.

[0043] In one embodiment, since both the proppant embedding effect and the proppant stacking deformation effect lead to a reduction in crack width, the crack width can be expressed as:

[0044] w f =w0-δ1-δ2 (1)

[0045] Among them, w f The crack width, in mm, is calculated to account for the proppant embedding effect and the proppant stack deformation effect; w0 is the initial crack width, in mm; δ1 is the crack surface deformation, in mm; and δ2 is the proppant stack deformation, in mm.

[0046] It should be noted that those skilled in the art can select other methods or approaches in the prior art to obtain the parameters involved in this invention. For example, those skilled in the art can also use other methods in the prior art to predict the embedding of the proppant. The embodiments involved in this invention should not be construed as limiting the scope of protection of this patent.

[0047] In one embodiment of the present invention, the method for obtaining the crack surface deformation and the proppant pile deformation is as follows: The crack surface deformation is obtained by the average proppant embedment depth:

[0048]

[0049] Where d s ν1 is the equivalent particle size of the multi-particle-size stack; σ is the closure pressure; ν1 is the Poisson's ratio of the proppant particles; ν2 is the Poisson's ratio of the rock slab; E1 is the Young's modulus of the proppant particles; E2 is the Young's modulus of the rock slab.

[0050] δ2 is the deformation of the proppant pile, which is obtained by the following formula:

[0051]

[0052] Where E a The apparent Young's modulus of the proppant pile was obtained through experimental testing.

[0053] When we substitute equations (2)-(3) into equation (1), we get:

[0054]

[0055] (2) Establish a porosity prediction model using the seam width deformation prediction model.

[0056] Changes in crack width will cause changes in the porosity of sand-filled cracks. Therefore, a porosity prediction model for sand-filled cracks can be obtained based on the changes in crack width. Let A be the area of ​​the crack surface perpendicular to the crack width. Then the porosity can be obtained by equation (5).

[0057]

[0058] in Porosity is used to calculate permeability using the KC equation after obtaining the porosity.

[0059] (3) Obtain the particle size distribution after crushing of multiple particle size groups.

[0060] For the particle size distribution of the broken proppant, those skilled in the art can use image processing methods to draw its gradation curve, or they can use the transfer matrix method described in detail in this specification to obtain it.

[0061] In one embodiment of the present invention, a method for obtaining the particle size distribution of a multi-size proppant after crushing using a transfer matrix method is explained in detail. Since the particle size distribution after crushing is related to its fractal dimension, and the crushing rate of a single-size proppant under pressure is related to the work per unit volume, the relationship between unit volume, particle size range, and fractal dimension can be derived, thereby establishing the relationship between the particle size distribution after crushing a single-size proppant and the work per unit volume input. After obtaining the relationship between the particle size distribution after crushing a single-size proppant and the work per unit volume input, the particle size distribution of the multi-size proppant after crushing can be calculated by combining the gradation transfer matrix.

[0062] Assume that the post-compression proppant multi-size group consists of n single-size groups, specifically (d0-d1, d1-d2, ... d n-1 -d n The fractal dimensions of each particle size group after crushing are D1, D2, ... D. n The relative breakage rates correspond to B in sequence. r1 * B r2 * ,...B rn * .

[0063] Taking the i-th particle size group (d) i-1 -d i Taking ) as an example, the derivation is performed. In the limiting state, the fractal dimension D is taken as 2.6, and its B pi * With B ti * The calculation formula is:

[0064]

[0065]

[0066] Where d is the proppant particle size, D is the fractal dimension, i is the index, and B is the number of the proppant particles. p * B is the initial fracture potential of the proppant. t * This refers to the amount of proppant broken.

[0067] Relative breakage rate B r * The expression:

[0068]

[0069] In the formula, B r * The relative breakage rate of the proppant.

[0070] Solving the above equation, we can obtain the fractal dimension D. i The expression is:

[0071]

[0072] Substituting the above equation into the work input per unit volume W in Relationship:

[0073] D i =G(k i W in ,d i-1 ,di (10)

[0074] The work input per unit volume can also be expressed as the product of the closed stress and the displacement. The above formula can be transformed into:

[0075] D i =G(k i Δwσ,d i-1 ,d i (11)

[0076] The breakage condition of any single particle size group in the multi-particle size group can be obtained as shown in the following formula:

[0077]

[0078] In the formula: P i-x For particle size group i in a multi-size group, the particle size after crushing is smaller than particle size d. x The ratio of the mass of particle size group i to the total mass of particle size group i.

[0079] Based on the Markov chain model, by selecting i quasi-boundary particle sizes, the gradation evolution model established through the gradation transition matrix can more comprehensively reflect the breakage of particles in different size groups:

[0080] g T (n)=g T (0)P n (13)

[0081] Where: g T (0)=(G1(0),G2(0),...,G n (0)) and g T (n)=(G1(n),G2(n),...,G n (n) represents the mass percentage of each single particle size group in the multi-particle size group of the soil before crushing and after n crushing cycles. Generally, particle crushing refers to the crushing situation after one test, i.e., n=1. Grading transition matrix P n Represented as:

[0082]

[0083] In the formula: S ii P represents the probability that particle size group i is not broken. ij S1 represents the mass percentage of particle size group i broken down into particle size group j. The i-th row represents the breakage status of particle size group i within a multi-particle-size group. The j-th column represents the breakage status of a single particle size group larger than particle size group j within the multi-particle-size group, and the particle size group j itself not being broken. Since the smallest particle size group remains within that particle size group after breakage, S1 is 1. The sample satisfies the law of conservation of mass before and after breakage.

[0084]

[0085] By solving formula (14), the particle composition gradation curve of the multi-size group after one or n crushings can be calculated.

[0086] (4) Establish a permeability prediction model based on the multi-particle-size KC equation.

[0087] The Kozeny-Carman equation (KC equation) is a formula in the art that expresses the relationship between rock permeability, porosity, and rock specific surface area. However, the existing KC equation assumes single-size particles, making it no longer applicable for predicting the permeability and conductivity of multi-size proppant after fragmentation, and therefore requires modification.

[0088] When considering rock masses with uniform particle size distribution, the specific surface area S can be expressed as:

[0089]

[0090] Substituting equation (10) into equation (9) yields the KC equation based on the uniform particle size distribution.

[0091]

[0092] In the formula, τ represents the tortuosity of the proppant pile.

[0093] For a particle pack composed of different particle sizes, the particle sizes are arranged from smallest to largest as follows: d1 < d2 < ... < d i <…<d n The specific surface area S is:

[0094]

[0095] The total specific surface area of ​​the two particle sizes is defined as:

[0096]

[0097] Substituting equation (19) into equation (18), we get:

[0098]

[0099] In the formula m i For particle size d i The number of proppant particles; H i It is the total specific surface area of ​​the two types of particles.

[0100] Therefore, for particle size groups composed of multi-size proppants, the particle number can be correlated with the gradation curves obtained through image processing, and the particle size group (d0-d1, d1-d2, ... d) can be considered. n-1 -dn The corresponding normalized quality fractions are (χ0-χ1, χ1-χ2, ... χ). n-1 -χ n )

[0101] For particle size group (d i -d i+1 Assuming the particles are round before and after crushing, the number of particles can be calculated using the law of conservation of mass:

[0102]

[0103]

[0104] In the formula, M i For particle size d i The actual mass of the proppant particles; ρ is the density of the proppant particles.

[0105] Therefore, the specific surface area for two particle size groups can be expressed as:

[0106]

[0107] In the formula, Δχ i For proppant particle diameter from (d i-1 -d i The mass fraction of ) is determined by the gradation curve.

[0108] Actual particles are often irregular and rough, with some shape characteristics, and there is actually some cementation or overlap between particles, which leads to a certain error in the obtained specific surface area. Therefore, considering the shape factor, formula (20) can be corrected.

[0109]

[0110] In the formula, ξ represents the surface roughness of the proppant particles.

[0111] Substituting equation (24) into equation (17), we obtain the modified KC equation:

[0112]

[0113] Therefore, the equivalent single particle size of a particle stack can be expressed by the following formula.

[0114]

[0115] The permeability of the proppant after breakage can be calculated using formulas (25) and (23).

[0116] (5) Combine the gap width prediction model in step (1) and the permeability prediction model in step (4) to obtain the flow conduction capacity of multi-size proppant stack considering pressure breakage.

[0117] D f =w f k f (27)

[0118] In the formula, D f This enhances the flow conductivity of multi-size proppant stacks.

[0119] Correspondingly, the calculation formula (27) can also be written in the following calculation mode.

[0120]

[0121] This invention treats the proppant as a porous medium, uses the apparent Young's modulus of the proppant to describe the deformation of the proppant pile, considers the proppant embedding effect, and establishes a crack width prediction model considering the influence of the proppant embedding effect and the proppant pile deformation effect. Furthermore, a porosity prediction model for sand-filled cracks is obtained based on the change in crack width. Since the breakage of a single particle size group is related to its fractal dimension, and the breakage rate of a single particle size group under pressure is related to the work per unit volume, there is a functional relationship between the unit volume, particle size range, and fractal dimension of the single particle size group proppant. Therefore, a relationship between the particle size distribution after the breakage of a single particle size group and the input work per unit volume is established. The particle size distribution after the breakage of a multi-particle size group is obtained by combining the gradation transition matrix. Addressing the drawback of requiring a single particle size for calculating the permeability of multi-particle size proppant using the traditional KC equation, a modified method is proposed. The permeability after the breakage of a multi-particle size group proppant is calculated by combining the gradation curve of the multi-particle size group proppant. Finally, multiplying the crack width by the permeability can predict the conductivity of the multi-particle size group after pressure breakage.

[0122] This invention fully considers the influence of the fracture width and permeability caused by the particle breakage evolution law of multi-size proppant groups, and proposes a method for predicting the conductivity of multi-size proppant piles under pressure breakage, providing relevant theoretical support for the optimization of fracture parameters and construction parameters.

[0123] Although specific embodiments of the present invention have been described in detail with reference to the accompanying drawings, this should not be construed as limiting the scope of protection of this patent. Various modifications and variations that can be made by those skilled in the art without inventive effort within the scope described in the claims still fall within the scope of protection of this patent.

Claims

1. A method for predicting the conductivity of multi-size proppant piles considering compressive breakage, characterized in that, Includes the following steps: (1) Considering the effects of proppant embedding and proppant stack deformation, a joint width prediction model is established; The seam width prediction model in step (1) is as follows: w f =w0-δ1-δ2 (1) Among them, w f The crack width, in mm, is calculated to account for the proppant embedding effect and the proppant stack deformation effect; w0 is the initial crack width, in mm; δ1 is the crack surface deformation, in mm; δ2 is the proppant stack deformation, in mm. The surface deformation of the crack is obtained by measuring the average embedment depth of the proppant. The method for obtaining the surface deformation of the crack is as follows: Where, d s ν1 is the equivalent particle size of the multi-particle-size stack; σ is the closure pressure; ν1 is the Poisson's ratio of the proppant particles; ν2 is the Poisson's ratio of the rock slab; E1 is the Young's modulus of the proppant particles; E2 is the Young's modulus of the rock slab. The method for obtaining the proppant stack deformation includes: Among them, E a σ is the apparent Young's modulus of the proppant pile; σ is the closure pressure. (2) Establish a porosity prediction model using the aforementioned seam width prediction model; The porosity prediction model is as follows: in Porosity; (3) By combining the relationship between the particle size distribution after crushing a single particle size group and the input work per unit volume with the gradation transfer matrix, the particle size distribution after crushing a multi-particle size group can be obtained. (4) Establish a permeability prediction model based on the multi-particle-size KC equation; The permeability prediction model for the multi-particle-size KC equation in step (4) is as follows: Where k f τ is the permeability, ξ is the proppant pile tortuosity, ξ is the surface roughness of the proppant particles, d is the proppant particle size, H is the total specific surface area of ​​the two types of particles, and n is the serial number. (5) Combine the gap width prediction model in step (1) and the permeability prediction model in step (4) to obtain the flow conduction capacity of multi-size proppant stack considering pressure breakage.

2. The method for predicting the conductivity of a multi-size proppant stack considering compression breakage as described in claim 1, wherein the method for obtaining the conductivity of the multi-size proppant stack in step (5) is as follows: D f =w f k f In the formula, D f This enhances the flow conductivity of multi-size proppant stacks.

3. A system for predicting the conductivity of multi-size proppant piles under pressure breakage, used to implement the conductivity prediction method according to any one of claims 1-2, comprising: The gap width prediction module is used to establish a gap width prediction model that takes into account the effects of proppant embedding and proppant stack deformation. The porosity prediction module is used to establish a porosity prediction model based on the seam width prediction model. The multi-size proppant particle size distribution prediction module is used to predict the particle size distribution after multi-size proppant crushing by combining the relationship between the particle size distribution after single-size proppant crushing and the input work per unit volume, and the gradation transfer matrix. The permeability prediction module is used to predict the permeability of multi-particle-size KC equations; The flow capacity prediction module is used in conjunction with the slot width prediction module and the permeability prediction module to obtain the flow capacity of multi-size proppant stacks that take into account pressure breakage.

Images