Calculation method for long-term diversion capacity of deep shale fractures based on rock creep effect

Abstract

The present invention discloses a calculation method for the long-term conductivity of deep shale fractures based on the rock creep effect, including: calculating the initial fracture width, the elastic embedding depth of proppants, the deformation of proppants, and the change in fracture width based on Hertz contact theory, establishing a fracture width change model that varies with time, and performing Laplace transform to calculate the change in fracture width in the Laplace domain; performing inverse Laplace transform on the change in fracture width in the Laplace domain to obtain the change in fracture width considering the rock creep effect and the change in proppant creep embedding over time; calculating the long-term conductivity of fractures considering the creep effect of deep shale. The present invention proposes an accurate and effective analysis model to quantify the variation law of the long-term conductivity of fractures, reveals the influence of the creep effect on the long-term conductivity of fractures, and helps to achieve high fracture conductivity of unconventional oil and gas resources.

Description

Technical Field

[0001] The present invention belongs to the field of oil and gas well engineering, and particularly relates to a calculation method for the long-term conductivity of deep shale fractures based on the rock creep effect. Background Art

[0002] Deep shale gas reservoirs are characterized by deep burial, high temperature, large formation stress, enhanced rock rheology, etc. These characteristics lead to a sharp decrease in fracture permeability and conductivity due to their strong stress sensitivity, serious insufficiency in fracture support effect, and a sharp decline in cumulative production. Through actual production data analysis and physical simulation experiments, it is shown that fracture closure is the main reason for the rapid decline in the production of fractured wells. The enhanced rheological characteristics of rocks further deepen the embedding of proppants. After the stimulation process is completed, the fluid pressure in the fracture rapidly decreases, the effective closure stress acting on the fracture surface increases, and phenomena such as proppant compaction, deformation, embedding, and rock creep occur, resulting in a significant reduction in fracture permeability and fracture width (Jiang, 2018; Ahamed et al., 2021).

[0003] Currently, the research methods for long-term fracture conductivity mainly include experimental tests and theoretical models. Wen et al. (2007) used a long-term fracture conductivity test instrument to study the influence of closure stress on the loss of fracture conductivity and established a relationship for the long-term fracture conductivity under the action of closure stress. Lee et al. (2010) studied the influence of proppant diagenesis on permeability and compared the model results of Yasuhara (Yasuhara., et al., 2003). Hou et al. (2017) experimentally tested the influence of heterogeneous proppant filling and temperature changes on fracture conductivity. In terms of theoretical research, some scholars proposed an empirical model for the embedding depth of proppants based on the test results of proppant and rock physical properties, explaining the relationship between the embedding depth and closure stress (Volk et al., 1981; Chen et al., 2017). However, the empirical model cannot elaborate on the stress-sensitive characteristics therein and cannot understand the interaction mechanisms between particles and fractures, and between particles and particles (Neto et al., 2015). Guo et al. (2017) combined the Hertz contact theory to establish a stress-sensitive proppant embedding depth model, considered the influence of particle elastic deformation on the fracture width change, and predicted the fracture conductivity based on the porosity and permeability of the filled proppants.

[0004] In summary, current experimental studies have clarified the adverse effects of the rheological characteristics of deep shale on the long-term fracture conductivity, but the relevant theories are insufficient. There is no established mechanism for the interaction between proppants and fracture surfaces during the hydraulic fracture closure process considering the influence of the creep effect of deep shale, and there is a lack of a reliable long-term conductivity prediction method.

[0005] Therefore, there is an urgent need to establish a calculation method for the long-term conductivity of deep shale fractures based on the rock creep effect, which will help improve the stimulation effect of deep shale, reduce the production decline rate, extend the stable production period, and further enhance the development potential of deep shale reservoirs. Summary of the Invention

[0006] In order to overcome the problems in the prior art, the present invention provides a calculation method for the long-term conductivity of deep shale fractures based on the rock creep effect. This method can conveniently measure the influence of the creep effect of deep shale on the long-term conductivity of fractures, and provides some insights for the parameter design and optimization of fracturing execution.

[0007] The technical solution provided by the present invention to solve the above technical problems is: A calculation method for the long-term conductivity of deep shale fractures based on the rock creep effect, including the following steps:

[0008] Step S10: Obtain reservoir parameters, fracture parameters, and proppant parameters;

[0009] Step S20: Calculate the change in the width of the supported fracture, the deformation of the proppant, and the elastic embedding depth of the proppant based on the reservoir parameters, fracture parameters, proppant parameters, and Hertz contact theory;

[0010] Step S30: Describe the creep deformation of deep shale through the elastic embedding depth of the proppant in combination with Maxwell's viscoelastic model, establish a model for the change in fracture width over time, and perform Laplace transform to calculate the change in fracture width in the Laplace domain;

[0011] Step S40: Substitute the transformation relationship of the elastic parameters in the time domain and the Laplace domain and the operator function in the Maxwell viscoelastic model into the change in fracture width in the Laplace domain, and perform inverse Laplace transform to obtain the change in fracture width considering the rock creep effect and the change in the creep embedding of the proppant over time;

[0012] Step S50: According to the calculation results of steps S20 and S40, obtain the effective width of the supported fracture, and combine with the fracture permeability calculated by the Kozeny-Carman porous medium permeability model to calculate the long-term conductivity of the fracture considering the creep effect of deep shale.

[0013] Furthermore, the reservoir geological parameters include elastic modulus, Poisson's ratio, viscoelastic coefficient, and in-situ stress, the fracture parameters include fracture length and fracture width, and the proppant parameters include elastic modulus, Poisson's ratio, sand concentration, particle size, and apparent density.

[0014] A further technical solution is that the calculation formula in step S20 is as follows:

[0015]

[0016]

[0017]

[0018]

[0019] In the formula: ω ini is the initial slot width, in mm; α is the change in slot width, in mm; δ is the deformation of the proppant, in mm; d e is the elastic embedding depth of the proppant, in mm; n is the number of proppant placement layers; R is the diameter of the proppant, in mm; λ is the distance coefficient, dimensionless; p e is the effective closure stress, in MPa; E1 is the elastic modulus of the proppant, in MPa; E2 is the elastic modulus of the reservoir, in MPa; v1 and v2 are the Poisson's ratios of the proppant and the reservoir, dimensionless.

[0020] A further technical solution is that the crack width change model varying with time in step S30 is as follows:

[0021]

[0022] In the formula: α1(t) is the change in crack width with time, dimensionless; λ is the distance coefficient, dimensionless; p e is the effective closure stress, in MPa; E1 is the elastic modulus of the proppant, in MPa; E2 is the elastic modulus of the reservoir, in MPa; v1 and v2 are the Poisson's ratios of the proppant and the reservoir, dimensionless.

[0023] A further technical solution is that the calculation formula for the change in crack width in the Laplace domain in step S30 is as follows:

[0024]

[0025] In the formula: α(s) is the change in crack width in the Laplace domain, dimensionless; E2(s) is the elastic modulus of the rock in the Laplace domain, dimensionless; v2(s) is the Poisson's ratio of the rock in the Laplace domain, dimensionless; is the distance coefficient, dimensionless; p e is the effective closure stress, in MPa; E1 is the elastic modulus of the proppant, in MPa; v1 is the Poisson's ratio of the proppant, dimensionless; p e (s) is the effective closure stress in the Laplace domain, dimensionless.

[0026] A further technical solution is that the calculation formula for the change in fracture width considering rock creep in step S40 is as follows:

[0027]

[0028] In the formula: α2(t) is the change in fracture width under rock creep, in mm; η2 is the reservoir viscoelastic coefficient, in GPa·s; t is the creep time, in day.

[0029] A further technical solution is that the calculation formula for the change in proppant creep embedment over time in step S40 is as follows:

[0030]

[0031] In the formula: d e (t) is the proppant creep embedment depth, in mm; η2 is the reservoir viscoelastic coefficient, in GPa·s; t is the creep time, in day; λ is the distance coefficient, dimensionless; p e is the effective closure stress, in MPa; E1 is the proppant elastic modulus, in MPa; E2 is the reservoir elastic modulus, in MPa; v1 and v2 are the proppant Poisson's ratio and reservoir Poisson's ratio, dimensionless; R is the diameter of the proppant, in mm.

[0032] A further technical solution is that the calculation formula for the effective width of the propped fracture in step S50 is as follows:

[0033] d total =d e +d e (t)

[0034] ω=ω ini -2d total

[0035] In the formula: d total is the total embedment depth of the proppant, in mm; ω is the effective width of the propped fracture, in mm; ω ini is the initial fracture width, in mm; d e (t) is the proppant creep embedment depth, in mm; d e is the elastic embedment depth of the proppant, in mm.

[0036] A further technical solution is that the calculation formula for the long-term conductivity of fractures considering the creep effect of deep shale in step S50 is as follows:

[0037]

[0038]

[0039]

[0040]

[0041] Where: F cd_e is the long-term fracture conductivity under deep shale creep embedding, μm 2 ·cm; δ is the deformation of the proppant, mm; ω ini is the initial fracture width, mm; r is the pore throat radius of the propped fracture, μm; τ is the tortuosity of the propped fracture, dimensionless; φ e is the porosity of the propped fracture, %; φ0 is the fracture porosity in the initial state, %; r0 is the pore throat radius of the fracture in the initial state, μm; τ0 is the tortuosity of the fracture in the initial state, dimensionless.

[0042] The present invention has the following beneficial effects: Compared with the existing methods, the present invention reveals the influence of deep shale creep effect on the long-term fracture conductivity, expounds the mechanism of proppant embedding in fracturing, and provides some insights for the effective development of unconventional oil and gas. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 is a schematic flow chart of the present invention;

[0044] Figure 2 is a schematic diagram of the proppant embedding process;

[0045] Figure 3 is a schematic diagram of the fracture pore throat structure after proppant embedding of the present invention;

[0046] Figure 4 is a schematic diagram of the comparison between the model data and the experimental results of the present invention;

[0047] Figure 5 is the result of the long-term fracture conductivity of deep shale fractures in a well block of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0048] The present invention will be further described in detail below with reference to the embodiments and the accompanying drawings.

[0049] To make the objectives, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Apparently, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art without making creative efforts based on the embodiments of the present invention belong to the scope of protection of the present invention. Therefore, the detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the present invention claimed, but merely represents selected embodiments of the present invention.

[0050] AsFigure 1 As shown in Figure 1 , the long-term diversion capacity calculation method of deep shale fractures based on the rock creep effect of the present invention includes the following steps:

[0051] S1. Obtain reservoir geological parameters (elastic modulus, Poisson's ratio, viscoelastic coefficient, in-situ stress), fracture parameters (fracture length, fracture width), and proppant parameters (elastic modulus, Poisson's ratio, sand placement concentration, particle size, apparent density);

[0052] S2. Based on the obtained parameters and Hertz contact theory, calculate the change in the width of the supported fracture, the deformation of the proppant, and the elastic embedding depth of the proppant;

[0053]

[0054]

[0055]

[0056]

[0057] In the formula: ω ini is the initial fracture width, in mm; α is the change in fracture width, in mm; δ is the deformation of the proppant, in mm; d e is the elastic embedding depth of the proppant, in mm; n is the number of proppant placement layers; R is the diameter of the proppant, in mm; λ is the distance coefficient, dimensionless; p e is the effective closure stress, in MPa; E1 is the elastic modulus of the proppant, in MPa; E2 is the elastic modulus of the reservoir, in MPa; v1 and v2 are the Poisson's ratios of the proppant and the reservoir, dimensionless;

[0058] S3. Through the elastic embedding depth of the proppant calculated in S2, combine the Maxwell viscoelastic model to describe the creep deformation of deep shale, establish a fracture width change model that changes with time, and perform Laplace transform to calculate the change in fracture width in the Laplace domain;

[0059] Among them, the change in fracture width with time α1(t) can be written as:

[0060]

[0061] According to the approximate relationship between p e (t) and its Laplace transform p e (s):

[0062]

[0063] When t approaches 0 + , the effective closure stress p in the Laplace domain e(s) can be written as:

[0064]

[0065] where: p e (s) is the closed pressure in the Laplace domain; p e (0) is the initial value of p e at the initial time in the time domain; s is the complex variable in the Laplace domain;

[0066] Then, when there is viscoelastic deformation in the deep shale reservoir, the change in fracture width α(s) in the Laplace domain can be expressed as:

[0067]

[0068] In the formula: α(s) is the change in fracture width in the Laplace domain; E2(s) is the elastic modulus of the rock in the Laplace domain; v2(s) is the Poisson's ratio of the rock in the Laplace domain;

[0069] S4. Substitute the transformation relationship between the elastic parameters in the time domain and the Laplace domain and the operator function in the Maxwell viscoelastic model into the change in fracture width in the Laplace domain, and perform the inverse Laplace transform to obtain the change in fracture width considering the rock creep effect and the change in support creep embedding over time;

[0070] Among them, for the change in fracture width in the Laplace domain, according to the conversion relationship between the elastic parameters in the time domain and the Laplace domain, the elastic parameters E2(s) and v2(s) are:

[0071]

[0072]

[0073] Then:

[0074]

[0075] Considering the instantaneous deformation of the formation rock under the closed stress, select the Maxwell viscoelastic model, and the change in fracture width in the Laplace domain can be obtained:

[0076]

[0077] where:

[0078]

[0079] After simplification, it can be obtained:

[0080]

[0081] Taking the inverse Laplace transform on both sides, we get the following:

[0082]

[0083] Where:

[0084]

[0085] Finally, the change in fracture width α2(t) considering rock creep can be obtained as follows:

[0086]

[0087] In the formula: α2(t) is the change in fracture width considering rock creep, in mm; η2 is the reservoir viscoelastic coefficient, in GPa·s; t is the creep time, in day;

[0088] At this time, the change in the creep embedding depth of the proppant with time is:

[0089]

[0090] In the formula: d e (t) is the creep embedding depth of the proppant, in mm;

[0091] S5. According to the creep embedding depth of the proppant calculated in S4, the total embedding depth d of the proppant is obtained total And the effective width ω of the propped fracture is:

[0092] d total = d e + d e (t)

[0093] ω = ω ini - 2d total

[0094] In the formula: d total is the total embedding depth of the proppant, in mm; ω is the effective width of the propped fracture, in mm;

[0095] Combined with the initial state, the fracture porosity, pore throat radius, and tortuosity can be expressed as:

[0096]

[0097]

[0098]

[0099] Where: φ0 is the fracture porosity in the initial state, %; r0 is the fracture pore throat radius in the initial state, μm; τ0 is the fracture tortuosity in the initial state, dimensionless; H is the fracture height, m; L is the half-length of the fracture, m; N t is the total number of particles, dimensionless;

[0100] As the creep time increases, the proppant embeds into the fracture surface, and the fracture porosity φ e can be expressed as:

[0101]

[0102] Where: φ e is the propped fracture porosity, %;

[0103] When the proppant is arranged in multiple layers, the pore throat radius is a function of the proppant embedment depth. Regarding the proppant filling as a capillary model, the pore throat is divided into two parts. One part is the pore throat on the fracture surface, and the other part is the pore throat in the fracture flow channel. When the volume of the proppant embedded in the fracture wall can be regarded as a spherical segment with a height of d total , and a bottom radius of a, the embedding volume V of a single particle can be obtained e :

[0104]

[0105] Where: V e is the embedding volume of the proppant particle, mm 3 .

[0106] When the proppant embedment depth is less than the particle radius ( Figure 3 a), the effective radius r1 of the flow channel on the fracture wall after proppant embedding and the effective radius r2 of the flow channel inside the fracture after proppant embedding are:

[0107]

[0108]

[0109] Among them:

[0110]

[0111]

[0112]

[0113]

[0114] When n = 3m - 2, A = 2m - 2, B = 0, D = 0, S = 2m - 2; when n = 3m - 1, A = 2m - 1, B = 1, D = 1, S = 2m - 2; when n = 3m, A = 2m, B = 1, D = 1, S = 2m - 1;

[0115] Where: r1 and r2 are the effective radii of the flow channels on the fracture wall and inside the fracture respectively, in μm; H is the fracture height, in m; L is the half-length of the fracture, in m; N t is the total number of particles, dimensionless; N tc is the number of proppant particles in contact with the fracture surface, N C is the number of pore throats on the fracture wall, dimensionless; N I is the number of internal pore channels in the fracture, dimensionless;

[0116] Then the calculation formula for the effective radius r0 of the fracture flow channel is:

[0117]

[0118] And when the embedding depth is greater than the particle radius ( Figure 3 b), then:

[0119]

[0120] Where: r0 is the effective radius of the flow channel in the fracture, in μm;

[0121] Considering the influence of the contact deformation between the proppant particles, the expression of the pore throat radius should be corrected as:

[0122]

[0123] Where: r is the pore throat radius of the propped fracture, in μm;

[0124] The pore tortuosity satisfies the following formula:

[0125]

[0126] Where: τ is the tortuosity of the propped fracture, dimensionless;

[0127] Finally, the long-term conductivity of the fracture considering the creep effect of deep shale is:

[0128]

[0129] Where: F cd_e is the long-term conductivity of the fracture under the creep embedding of deep shale, in μm 2 ·cm.

[0130] Example 1

[0131] Taking the reservoir geological parameters of a deep shale fracturing well in China as an example, using the basic fracturing parameter table shown in Table 1, the relationship between the long-term fracture conductivity and time under the reservoir creep effect can be calculated.

[0132] Carry out the calculation of the embodiment according to the steps described above, output the model calculation results, and compare the results under the same experimental test conditions (as Figure 4 shown), and use the parameters in Table 1 to draw the curves of the long-term fracture conductivity under different elastic moduli of proppants and viscoelastic coefficients of reservoir rocks (as Figure 5 shown).

[0133] Table 1 Basic Fracturing Parameters of a Deep Shale Reservoir

[0134] Parameter Name Value Unit Reservoir Elastic Modulus 35.506 GPa Reservoir Poisson's Ratio 0.2136 Dimensionless Reservoir Viscoelastic Coefficient 2.375e5 GPa·s Reservoir Vertical Stress 149.7 MPa Maximum Horizontal Principal Stress 137.7 MPa Minimum Horizontal Principal Stress 126 MPa Effective Closure Stress 0-80 MPa Fracture Half-Length 0.175 m Fracture Height 0.036 m Proppant Granules 40 / 70 Mesh Average Particle Size 0.283 mm Apparent Density of Proppant 1.45 <![CDATA[g / cm 3 > Proppant Elastic Modulus 28.87 GPa Poisson's Ratio 0.143 Dimensionless Sand Placement Concentration 2.5 <![CDATA[kg / m 2 > Creep Time 80 Days

[0135] As mentioned above, it is not a restriction on the present invention in any form. Although the present invention has been disclosed through the above embodiments, it is not intended to limit the present invention. Any person skilled in the art can make some changes or modifications to equivalent embodiments with equivalent changes within the scope of the technical solution of the present invention by using the technical content disclosed above. However, any simple modification, equivalent change and modification made to the above embodiments based on 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 calculation method for the long-term diversion capacity of deep shale fractures based on the rock creep effect, characterized in that, It includes the following steps: Step S10: Obtain reservoir parameters, fracture parameters, and proppant parameters; Step S20: Calculate the change in propped fracture width, proppant deformation, and elastic embedment depth of proppant based on the reservoir parameters, fracture parameters, proppant parameters, and Hertz contact theory; where: ω ini is the initial slot width, in mm; α is the change in slot width, in mm; δ is the deformation of the proppant, in mm; d e is the elastic embedding depth of the proppant, in mm; n is the number of proppant placement layers; R is the diameter of the proppant, in mm; λ is the distance coefficient, dimensionless; p e is the effective closure stress, in MPa; E1 is the elastic modulus of the proppant, in MPa; E2 is the elastic modulus of the reservoir, in MPa; v1 and v2 are the Poisson's ratios of the proppant and the reservoir, dimensionless; Step S30: Describe the creep deformation of deep shale by combining the elastic embedment depth of the proppant with Maxwell's viscoelastic model, establish a model for the change in fracture width over time, perform Laplace transform, and calculate the change in fracture width in the Laplace domain; In the formula: α1(t) is the change in fracture width over time, dimensionless; λ is the distance coefficient, dimensionless; p e is the effective closing stress, MPa; E1 is the elastic modulus of the proppant, in MPa; E2 is the elastic modulus of the reservoir, in MPa; v1 and v2 are the Poisson's ratios of the proppant and the reservoir, dimensionless; Step S40: Substitute the transformation relationship between the elastic parameters in the time domain and the Laplace domain and the operator function in the Maxwell viscoelastic model into the change in fracture width in the Laplace domain, and perform inverse Laplace transform to obtain the change in fracture width considering the rock creep effect and the change in proppant creep embedment over time; Step S50: According to the calculation results of steps S20 and S40, obtain the effective width of the propped fracture, and combine with the fracture permeability calculated by the Kozeny-Carman porous medium permeability model to calculate the long-term fracture conductivity considering the creep effect of deep shale; 2. The long-term diversion capacity calculation method of deep shale fractures based on the rock creep effect according to claim 1, wherein The reservoir geological parameters include elastic modulus, Poisson's ratio, viscoelastic coefficient, and in-situ stress. The fracture parameters include fracture length and fracture width. The proppant parameters include elastic modulus, Poisson's ratio, sand placement concentration, particle size, and apparent density.

3. The long-term conductivity calculation method for deep shale fractures based on the rock creep effect according to claim 1, characterized in that The calculation formula for the change in fracture width in the Laplace domain in step S30 is: where: α(s) is the change in fracture width in the Laplace domain, dimensionless; E2(s) is the elastic modulus of the rock in the Laplace domain, dimensionless; p e (s) is the effective closure stress in the Laplace domain, dimensionless; v2(s) is the Poisson's ratio of the rock in the Laplace domain, dimensionless; is the distance coefficient, dimensionless; p e is the effective closure stress, MPa; E1 is the elastic modulus of the proppant, in MPa; v1 is the Poisson's ratio of the proppant, dimensionless.

4. The long-term diversion capacity calculation method for deep shale fractures based on the rock creep effect according to claim 1, wherein The calculation formula for the change in fracture width considering rock creep in step S40 is: In the formula: α2(t) is the change in fracture width under rock creep, in mm; η2 is the viscoelastic coefficient of the reservoir, in GPa·s; t is the creep time, in days.

5. The long-term diversion capacity calculation method for deep shale fractures based on the rock creep effect according to claim 4, wherein The calculation formula for the change in proppant creep embedment over time in step S40 is: where: d e (t) is the proppant creep embedment depth, in mm; η2 is the reservoir viscoelasticity coefficient, in GPa·s; t is the creep time, in day; λ is the distance coefficient, dimensionless; p e is the effective closure stress, in MPa; E1 is the elastic modulus of the proppant, in MPa; E2 is the elastic modulus of the reservoir, in MPa; v1 and v2 are the Poisson's ratios of the proppant and the reservoir, dimensionless; R is the diameter of the proppant, in mm.

6. The long-term diversion capacity calculation method for deep shale fractures based on the rock creep effect according to claim 5, characterized in that, The calculation formula for the effective width of the propped fracture in step S50 is: d total = d e + d e (t) ω = ω ini -2d total Where: d total is the total embedding depth of the proppant, in mm; ω is the effective width of the propped fracture, in mm; ω ini is the initial fracture width, in mm; d e (t) is the creep embedding depth of the proppant, in mm; d e is the elastic embedding depth of the proppant, in mm.

7. The long-term diversion capacity calculation method for deep shale fractures based on the rock creep effect according to claim 6, characterized in that, The calculation formula for the long-term fracture conductivity considering the creep effect of deep shale in step S50 is: Where: F cd_e is the long-term conductivity of the fracture under deep shale creep embedding, μm 2 ·cm; δ is the deformation of the proppant, mm; ω ini is the initial fracture width, mm; r is the pore throat radius of the propped fracture, μm; τ is the tortuosity of the propped fracture, dimensionless; φ e is the porosity of the propped fracture, %; φ0 is the porosity of the fracture in the initial state, %; r0 is the fracture pore throat radius in the initial state, in μm; τ0 is the fracture tortuosity in the initial state, dimensionless.

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