RU2787144C2 - Method for determination of exposure time for well after hydraulic fracture, using distribution of creep of hydraulic fracture cracks - Google Patents

Abstract

FIELD: oil and gas industry.

SUBSTANCE: invention relates to the field of development of oil and gas deposits, namely to a method for determination of exposure time of a well after hydraulic fracture, using distribution of creep of hydraulic fracture cracks (hereinafter – HFC). The method includes following stages: obtainment of displacement of hydraulic fracture construction Q, crack height H, a coefficient of loss of hydraulic fracture liquid C, viscosity of hydraulic fracture liquid μ, a rheological index of hydraulic fracture liquid n', a coefficient of viscosity of hydraulic fracture liquid K', Young module of a rock sample on the target horizon E, a Poisson coefficient ν, an elasticity module G, a volumetric module K, and constant of rock formation material D_m, F_m; calculation of a crack length, a crack width, and pressure at different crack places in the end of HFC crack formation; obtainment of the total crack width, the total crack volume, a crack creep width, and a crack creep volume; calculation of J integral and C* integral of a crack vertex at a moment of time j; creation of a graph, and assessment of upper and lower limits of well exposure time, where t_j is a discrete moment of time at the moment of time j, s.

EFFECT: optimization of exposure time of oil and gas wells after HFC, expansion of HFC volume, and efficient increase in extraction.

1 cl, 5 dwg

Description

FIELD OF TECHNOLOGY

The invention relates to the field of oil and gas field development and, in particular, relates to a method for optimizing the soak time of a well after hydraulic fracturing by using the creep expansion of a hydraulic fracture.

BACKGROUND OF THE INVENTION

Due to the continuous development of oil and gas resources, human exploration of unconventional reservoirs such as shale has become a hotspot in the oil field. Currently, hydraulic fracturing is one of the important technical means of developing unconventional oil and gas fields. With the popularization of hydraulic fracturing technology, many hydraulic fracturing construction experiences have shown that in the development of soft rock formations such as shale gas, holding the well after hydraulic fracturing can further expand fractures and increase the penetration distance of fractures, holding the well after hydraulic fracturing for a certain period of time can effectively improve productivity of oil wells. Well soak time has an important influence on the effect of well soak after hydraulic fracturing. If the well soak time is too short, the fracture expansion will be stopped in advance, and the best production increase effect will not be achieved; if the well soak time is too long, it will cause a more severe disturbance of sensitivity, which reduces the productivity of oil wells. Thus, optimization of the well soak time after hydraulic fracturing is of great importance for the development of unconventional oil and gas resources.

Currently, there are many ways to determine the time of exposure of the well after hydraulic fracturing. Some use the appropriate relationship between shale gas well soak time and fracturing fluid rate and initial gas well productivity to optimize the well soak time range at the best performance (Zhang Yin, Li Shien. Analysis of the relationship between shale gas soak time and productivity in shale gas wells Fulin) [J]. Journal of Jianghan Petroleum University of Staff and Workers, 2017, 30 (5): 49-51); some combine the pressure drop inflection point time after a well soak and the relationship between total production and the soak time to determine the soak time (Huang Xiaoqing, Han Yongsheng, Yang Qing et al. Reverse flow law in horizontal well testing of shallow shale gas at the Zhaotong Block Sun Block [J], Xinjiang Petroleum Geology, 2020, (4): 457-463, 470); and some methods use the threshold pressure of hydration and shale fracturing to estimate the soak time of the well to achieve the best stimulation effect (Han Huefen, Yang Bin, Peng Zunliang. Study of water absorption by shale and fracture propagation during well soak after hydraulic fracturing taking Longmaxi formation platform in the Changning Block in the Sichuan Basin as an example [J]. Natural Gas Industry, 2019,39(1): 74-80; Tao, L., Guo, J., Shan, J. A New Shut in Time Optimization Method for Multi-Fractured Horizontal Wells in Shale Gas Reservoirs [J], American Rock Mechanics Association, 2020))

However, none of the above methods for determining the well soak time after hydraulic fracturing does not take into account the propagation of artificial fracture creep in shale reservoirs. This makes the well soak time after fracturing in shale and other soft rock less adaptable and the effects of well soak measures are uneven. Therefore, in order to obtain a more reasonable soak time after hydraulic fracturing in shale reservoirs, a method is needed to estimate the upper and lower limits of the soak time based on the creep propagation of hydraulic fractures.

SUMMARY OF THE INVENTION

The purpose of the present invention is to provide a method for determining the dwell time of a well after hydraulic fracturing using fracture creep extension, which is used to optimize the dwell time of oil and gas wells after hydraulic fracturing, expand the volume of hydraulic fracturing, effectively improve productivity, it overcomes the shortcomings and defects of the existing technology, and has broad prospects. market applications.

To achieve the above technical goals, the present invention uses the following technical solutions.

In the present invention, the J-integral and the C*-integral are used as the crack propagation criterion. The J-integral and the C*-integral play an important role in studying the related issues of crack propagation of creep. The J-integral is time dependent and is a criterion for the delayed initiation of creep cracks. Over time, the J-integral will increase cumulatively and creep cracks will initiate crack formation after a period of inactivity. C* is commonly used to characterize the distribution of stress and strain at the tip of a creep crack. It has a strong correlation with the crack tip propagation rate and is the basis for evaluating the strength of creep crack propagation.

The method for determining the well soak time after hydraulic fracturing using the creep propagation of hydraulic fractures includes the following steps in succession.

(1) Fracture construction displacement Q, fracture height H, fracturing fluid loss coefficient C, fracturing fluid viscosity μ, fracturing fluid rheological index n', fracturing fluid viscosity coefficient K', Young's modulus E of the rock sample at the target horizon are obtained , Poisson's ratio v, modulus of elasticity G, bulk modulus K and constant of the rock material D _m , Ф _m .

(2) Calculate the fracture length, fracture width and pressure at different locations of the fracture at the end of hydraulic fracture formation, the method is as follows: using the two-dimensional PKN fracture propagation model, calculate the fracture length L at the end of hydraulic fracture formation by the following formula (Wang Hongsun, Zhang Shicheng, Numerical Design Method for Hydraulic Fracturing [M], Beijing: Petroleum Industry Press, 1998);

where L - fracture length at the end of hydraulic fracture formation, m;

Q - construction displacement, m ³ /min;

H - fracture height, m;

C - hydraulic fracturing fluid loss factor, m/min ^0.5 ;

t - time from the beginning of hydraulic fracturing construction to the end of formation of hydraulic fracturing fractures, min.

The following formula is used to calculate the fracture width w _x=0 at the fracture opening at the end of hydraulic fracturing (Li Yingchuan, Oil Production Engineering [M]. Petroleum Industry Press, 2009):

- ширина трещины в отверстии трещины в конце образования трещины ГРП, м;where

- width of the fracture in the fracture opening at the end of the formation of the hydraulic fracture, m;

x - coordinates of various positions along the length of the crack, m, if we take the opening of the crack as the origin of coordinates;

μ is the viscosity of the fracturing fluid, mPa⋅s;

E - Young's modulus of the rock sample on the target horizon, MPa;

ν - Poisson's ratio of the rock sample at the target horizon, dimensionless.

The following formula is used to calculate the fracture width w _x at various positions in the fracture length direction at the end of hydraulic fracture formation (Wang Hongxun, Zhang Shicheng. Numerical calculation method for hydraulic fracturing design [M]. Beijing: Petroleum Industry Press, 1998):

, чтобы получить давление Рх в различных положениях по длине трещины:Based on obtaining the fracture length and fracture width at the end of the fracture formation, hydraulic fracturing calculates the pressure gradient in the fracture

to get the pressure Px at various positions along the length of the crack:

where n' is the rheological index of the fracturing fluid, a number less than 1, in a fracturing fluid made from cross-linked guar gum n'=0.5;

K' - viscosity coefficient of the fracturing fluid, mPa⋅s, in the fracturing fluid made of cross-linked guar gum K'=0.55;

q(x) - flow rate in different positions along the length of the crack in the crack, m ³ /s;

σ _h - minimum horizontal principal stress, Pa.

(3) The total crack width, total crack volume and crack creep width and crack creep volume are obtained, and the process is as follows:

The pressure in the fracture is discretized in space, the fracture is divided into n segments along the length of the fracture, and the pressure on the k-th segment is designated as p _k ; then time discretization is performed, and the pressure in the fracture will be equivalent to a series of rectangular pressure pulses with equal time intervals. The pressure of the kth fracture segment at time j is equal to p _k,j , and the width w _k,j of the kth fracture segment at time j is calculated by the following formula: (Palmer, ID A Model of the Hydraulic Fracturing Process for Elongated Vertical Fractures and Comparisons of Results With Other Models [C] SPE/DOE Low Permeability Gas Reservoirs Symposium, 1985)

where p _k,j is the pressure of the k-th segment of the crack at the time], MPa;

B, D - intermediate function;

N is the Laplace inversion constant, usually equal to 10;

l is the Laplace inversion constant, the value of which is determined by N;

- коэффициент Пуассона в пространстве Лапласа;

- Poisson's ratio in Laplace space;

- модуль Юнга в пространстве Лапласа, МПа;

- Young's modulus in Laplace space, MPa;

t _j - discrete time point at time j, s;

a _j - crack half-length at time j, m;

b _k - distance from the left end of the k-th fracture segment to the fracture opening, m;

c _k - distance between the right end of the k-th fracture segment and the fracture opening, m,

, безразмерная,s - Laplace variable,

, dimensionless,

z - variable of the analytical function, m.

при

представляет собой общую ширину трещины. Полагают поперечное сечение трещины как эллипс, длину трещины L и получают общий объем трещины V_t; для момента времени j, если p_k,1 известно, вычисленное

при

- ширина ползучести трещины, и объем ползучести трещины V_c получают тем же методом.Taking the pressure p _x in different positions of the fracture at the end of the hydraulic fracture formation as the pressure p _{k,1 of} the k-th fracture segment at time 1, substitute it into formulas (5)-(7), then calculated

at

is the total crack width. Assume the cross section of the crack as an ellipse, the length of the crack L and get the total volume of the crack V _t ; for time j, if p _k,1 is known, the computed

at

is the width of the crack creep, and the creep volume of the crack V _c is obtained by the same method.

(4) Calculate the J-integral and C*-integral of the crack tip at time j, the process is as follows:

The total crack volume is equal to the sum of the elastic crack volume and the crack creep volume. The elastic crack width w _e is calculated based on the elastic crack volume V _e =V _t -V _c , and the pressure p _e under conditions of elastic crack width is calculated by the following formula: (Gao Qing. Fracture Mechanics Engineering [M]. Chongqing University Press, 1986 )

where b _n is the distance between the left end of the crack tip and the crack opening, m.

By putting p _e in the following formula, the stress intensity factor K _I at the crack tip at time j (i.e., the position of the farthest distance between the crack and the crack opening) can be obtained:

Include K _I in the following formula to obtain the stress and displacement of the crack tip at time j:

where σ _xx is the stress in the x direction (along the crack length) of the crack tip, MPa;

σ _yy - stress in the y direction (perpendicular to the crack length) of the crack tip, MPa;

σ _xy - shear stress at the crack tip, MPa;

X - displacement of the crack tip in the x direction (along the length of the crack), m;

Y - displacement of the crack tip in the y direction (perpendicular to the crack length), m;

θ - polar angle of the polar coordinate system installed at the crack tip, rad;

r is the polar diameter of the polar coordinate system installed at the crack tip, m.

Calculate the J-integral J _j at the crack tip at time j:(Lee, HS et al. General time-dependent C(t) and J(t) estimation equations for elastic-plastic-creep fracture mechanics analysis[J]. Fatigue & Fracture of Engineering Materials & Structures, 2016.39(9): 1092-1104):

where J _j - J-integral at the crack tip at time j, MJ/m ² (or MPa.m);

W is the strain energy density at the crack tip, MJ/m ³ (or MPa);

ε _ij - deformation at the crack tip, where all indices ij represent three types xx, yy and xy;

σ _ij - stress at the crack tip, MPa;

T _ij - thrust force at the crack tip, MPa;

Г - any loop from the lower surface of the crack tip to the upper surface, m;

n - unit normal vector perpendicular to the integral loop at the crack tip, dimensionless.

J _IC is the threshold value of rock creep propagation, it can be measured experimentally.

If J _j <J _IC , then this does not satisfy the creep propagation condition, and the crack length at this time remains unchanged, a _j+1 =a _j , p _k,j-1 = _pe , steps (3), (4) repeat to get the J-integral J _j+1 ; at the crack tip at time j+1;

вершины трещины в момент времени j:If J _j ≥J _IC , the creep propagation condition is reached, and the С*-integral is calculated

crack tip at time j:

- С*-интеграл вершины трещины в момент времени j, МДж/(м²⋅с);where

- C*-integral of the crack tip at time j, MJ/(m ² s);

- скорость изменения плотности энергии деформации в вершине трещины, МДж/(м³⋅с),

- the rate of change in the strain energy density at the crack tip, MJ / (m ³ s),

- скорость изменения смещения вершины трещины в направлении х, м/с;

- rate of change in the displacement of the crack tip in the x direction, m/s;

:Calculate the half-length of the crack a _j+1 at time j+1 from the propagation velocity of the crack tip creep

:

- скорость распространения ползучести вершины трещины, м/с;where

- velocity of propagation of crack tip creep, m/s;

D _m - rock material constant, m ³ /MJ;

f _m - rock material constant, dimensionless.

при условиях полудлины трещины

на основе упругого объема трещины V_e, таким образом, получают давление в трещине при условиях упругой ширины трещины; так как ползучесть породы не влияет напрямую на давление в трещине, поэтому давление p_k,j+1 в k-ом сегменте трещины в момент времени j+1 вычисляют по следующей формуле:Get the elastic crack width

under crack half-length conditions

based on the elastic fracture volume V _e , the pressure in the fracture is thus obtained under the conditions of the elastic fracture width; since the creep of the rock does not directly affect the pressure in the fracture, therefore, the pressure p _k,j+1 in the k-th segment of the fracture at time j+1 is calculated using the following formula:

и

этапы (3) и (4) повторяют, чтобы вычислить J-интеграл J_j+1 и С*-интеграл

вершины трещины в момент времени j+1.After receiving

and

steps (3) and (4) are repeated to calculate the J-integral of J _j+1 and the C*-integral

crack tip at time j+1.

и оценивают верхний и нижний пределы времени выдержки скважины, процесс выглядит следующим образом:(5) Create plot dJ _j /dt _j -t _j and plot

and estimate the upper and lower limits of the well soak time, the process is as follows:

When the curve dJ _j /dt _j -t _j gradually becomes flat and the rate of change of the J-integral remains stable for more than 7 hours, it can be concluded that it has entered the steady state creep stage, and this value is the lower limit of the well soak time; when the C*-integral value is lower than 0.015, the fracture creep propagation power is insufficient and the fracture propagation stops, this is the upper limit of the well soak time.

Existing methods for determining well dwell time are mainly based on field experience and construction parameters of adjacent wells, so they are poorly applicable and have a big difference when applied in different blocks and different wells. The method of determining the well soak time in the present invention is based on the basic rock parameters and fracture design parameters to obtain the J-integral and C*-integral by calculation, thereby estimating the upper and lower limits of the well soak time after hydraulic fracturing. Compared with existing technologies, the present invention considers the geology and fracturing parameters of the well, and also allows you to calculate and design the optimal well soak time after hydraulic fracturing for various wells.

DESCRIPTION OF GRAPHICS

In FIG. 1 shows a plot of dJ _j /dt _j -t _j for wells Y1, Y2 and Y3 by calculation.

для скважин Y1, Y2 и Y3 посредством расчета соответственно.In FIG. 2, fig. 3, fig. 4 presented graphs

for wells Y1, Y2 and Y3 by calculation respectively.

In FIG. 5 shows a comparison of instantaneous gas production in wells Y1, Y2 and Y3.

SPECIFIC IMPLEMENTATION OPTIONS

The present invention will be further explained below based on figures and examples so that those skilled in the art can understand the present invention. However, it should be clear that the present invention is not limited to the scope of specific implementation options. For those skilled in the art, as long as various changes are within the spirit and scope of the present invention as defined by the appended claims, all of them are reserved.

Example

Taking as an example three adjacent shale gas wells (Y1, Y2 and Y3) in a shale gas block in southern Sichuan, the method described in the present invention is used to determine the upper and lower limits of the well soak time after fracturing each well and compare them with the actual construction time to analyze the effect of well soak time on productivity.

Step 1. Fracture construction displacement Q, fracture height H, fracturing fluid loss coefficient C, fracturing fluid viscosity μ, fracturing fluid rheological index n', fracturing fluid viscosity coefficient K', Young's modulus E, Poisson's ratio v are obtained, modulus of elasticity G, bulk modulus K and rock material constants D _m and f _m .

Step 2. In combination with the step 1 parameter, using formulas (1)~(4), calculate the fracture length L, fracture width w _x and pressure p _x at each fracture position at the end of hydraulic fracture formation.

и общий объем трещины V_t. В то же время можно получить ширину ползучести трещины

и объем ползучести трещины V_c в первый момент. Поскольку значение V_t остается неизменным в каждый момент времени, после того, как V_t вычисляют в первый момент, общий объем трещины V_t не вычисляют в другие моменты времени.Step 3. Spatio-temporal pressure dispersion p _x is performed at different fracture positions at the end of hydraulic fracture formation and pressure p _k , _{1 of} the k-th fracture segment at the first moment is obtained, which is substituted into formula (5)-(7) to obtain full crack width

and total fracture volume V _t . At the same time, it is possible to obtain the width of the crack creep

and the crack volume V _c at the first moment. Since the value of V _t remains the same at each time point, after V _t is calculated at the first moment, the total fracture volume V _t is not calculated at other times.

Step 4. Using the total crack volume V _t and the crack creep volume V _c , calculate the elastic crack volume V _e and the elastic crack width w _e at the first moment. Pe can be calculated by formula (8), and then the stress and displacement at the crack tip can be calculated by formulas (9)-(11).

Substitute the stress and displacement of the crack tip into formulas (12)-(15) and calculate the J-integral J1 at the first moment.

When J ₁ <J _IC (for shale reservoir J _IC =0.325) the fracture does not creep and does not propagate, and the fracture half-length a ₂ = a ₁ at the second moment is obtained, and the pressure in the fracture at the second moment p _k,2 = p _e , repeat steps (3) and (4), calculate the J-integral at the crack tip at the next moment.

в первый момент вычисляют по формуле (16). Используют формулу (17), чтобы получить длину трещины а₂ во второй момент. Упругую ширину трещины w_a2 при условиях полудлины трещины а₂ вычисляют по упругому объему V_e, а затем подставляют в формулу (18) для расчета давления p_k,2 во втором моменте. После получения а₂ и p_k,2 повторяют этапы (3) и (4), чтобы вычислить J-интеграл и С*-интеграл вершины трещины в следующий момент.At J≥J _IC (J _IC =0.325 for shale reservoirs), the fracture creeps and expands, and the С*-integral

at the first moment is calculated by formula (16). Formula (17) is used to obtain the crack length a ₂ at the second moment. The elastic width of the crack w _a2 under the conditions of the half-length of the crack a ₂ is calculated from the elastic volume V _e , and then substituted into formula (18) to calculate the pressure p _k,2 in the second moment. After obtaining a ₂ and p _k,2 , steps (3) and (4) are repeated to calculate the J-integral and C*-integral of the crack tip at the next moment.

(фиг. 2, фиг. 3, фиг. 4). Когда кривая dJ_j/dt_j-tj постепенно становится пологой и скорость изменения J-интеграла остается стабильной в течение более 7 часов, можно сделать вывод, что это вошло в стадию установившейся ползучести, и данное значение является нижним пределом времени выдержки скважины; когда значение С*-интеграла ниже 0,015, мощность распространения ползучести трещины недостаточна, и распространение трещины прекращается, это верхний предел времени выдержки скважины.Step 5. Create a graph dJ _j /dt _j -tj (Fig. 1) and a graph

(Fig. 2, Fig. 3, Fig. 4). When the curve dJ _j /dt _j -tj gradually becomes flat and the rate of change of the J-integral remains stable for more than 7 hours, it can be concluded that this has entered the steady-state creep stage, and this value is the lower limit of the well soak time; when the C*-integral value is lower than 0.015, the fracture creep propagation power is insufficient and the fracture propagation stops, this is the upper limit of the well soak time.

From FIG. Figure 1 shows that well Y1 entered steady state creep mode after 73 hours of well soaking, and the lower soak time limit for this well was 73 hours, well Y2 went into steady state creep mode after 68 hours soak time, the lower soak time limit for this well was 68 hours , well Y3 entered steady state creep after 85 hours of well soak time, the lower soak time limit for this well was 85 hours.

для скважин Y1, Y2 и Y3 соответственно. Когда значение С* падает до 0,015, соответствующее время для скважин Y1, Y2 и Y3 составило 173,83 ч, 193,50 ч и 185,00 ч, соответственно. Это верхние пределы времени выдержки для трех скважин.In FIG. 2, fig. 3, fig. 4 shows the curves

for wells Y1, Y2 and Y3, respectively. When the C* value drops to 0.015, the corresponding times for wells Y1, Y2, and Y3 were 173.83 hours, 193.50 hours, and 185.00 hours, respectively. These are the upper soak time limits for the three wells.

According to the construction parameters of the well soak and reverse flow, the soak time of wells Y1, Y2 and Y3 was 144 hours, 48 hours and 264 hours, respectively. By comparison, it can be concluded that the actual soak time of wells Y2 and Y3 is not within the upper and lower limits of the calculated soak time of the well, and only the construction time of the soak well Y1 is within the upper and lower limits of the soak time calculated according to the present invention.

In FIG. 5 shows a comparison of instantaneous reverse gas production in wells Y1, Y2 and Y3. Analyzing FIG. 5, it can be seen that the Y1 reflux well achieved a gas production rate that is better than the other two wells and could be maintained at 36×10 ⁴ m ³ /day. In well Y2, the soak time was too short and the well was shut down before the creep expansion of the fracture was complete, so the increase in productivity was not apparent. Since the soak time in well Y3 exceeded the upper limit, the advantage of fracture creep propagation was lost due to damage to the reservoir sensitivity, which reduced the productivity of the oil well. This example demonstrates that the method of determining the dwell time of a well after hydraulic fracturing in the present invention is reasonable and can provide guidance for the efficient development of oil and gas resources in unconventional reservoirs.

Claims (64)

A method for determining the well holding time after hydraulic fracturing using the propagation of hydraulic fracture creep (HF), which sequentially includes the following steps: (1) Obtaining the fracture construction offset Q, fracture height H, fracture fluid loss coefficient C, fracture fluid viscosity μ, fracture fluid rheology index n', fracture fluid viscosity coefficient K', Young's modulus E of the rock sample at the target horizon , Poisson's ratio ν, elastic modulus G, bulk modulus K and constants of the rock material D _m , Ф _m ; (2) calculation of fracture length, fracture width, and pressure at various fracture locations at the end of hydraulic fracture formation; (3) obtaining total crack width, total crack volume, crack creep width and crack creep volume, (4) calculation of the J-integral and C*-integral of the crack tip at time j; (5) create a graph

and graphics

and estimation of the upper and lower limits of the well soak time, where t _j is a discrete point in time at time j, s, wherein: in step (2), the fracture length, fracture width, and pressure at various fracture locations at the end of hydraulic fracture formation are calculated, and the method is as follows: calculate the fracture length L at the end of the hydraulic fracture formation according to the following formula:

where t is the time from the start of hydraulic fracturing construction to the end of hydraulic fracturing cracks, min; calculate the width of the fracture W _x=0 in the opening of the fracture at the end of the formation of the hydraulic fracture according to the following formula:

where x - coordinates of various positions along the length of the crack, m, if we take the opening of the crack as the origin; calculate the fracture width W _x at various positions in the direction along the fracture length at the end of the hydraulic fracture creation according to the following formula:

calculating a pressure gradient across the fracture to obtain pressure p _x at various positions along the length of the fracture; in step (3) get total crack width, total crack volume and crack creep width and crack creep volume, this process is as follows: the fracture is divided into n segments along the length of the fracture, and the pressure in the fracture is equivalent to a series of rectangular impulse pressures with equal time intervals, then the pressure of the k-th segment of the fracture at time j is equal to p _k,j , and the width W _k,j of the k-th crack segment at time j; taking the pressure p _x in different positions of the fracture at the end of the hydraulic fracture formation as the pressure p _{k,1 of} the k-th fracture segment at time 1, calculated earlier,

at

is the total crack width; calculating the cross section of the crack as an ellipse, the length of the crack L, get the total volume of the crack V _t ; computed

at

the crack creep width and the crack creep volume V _c are obtained in the same way, while t ₁ ^- - time point, which is infinitely approaching t ₁ from the left side, s; t _j ⁺ - moment of time, which infinitely approaches t _j from the right side, s; at step (4), the J-integral and C*-integral of the crack tip at time j are calculated, this process is as follows: calculate the elastic volume of the crack V _e =V _t -V _c , the elastic width of the crack w _e , and the pressure p _e under the conditions of the elastic width of the crack is calculated by the following formula:

where b _n is the distance between the left end of the crack tip and the crack opening, m;

- crack half-length at time j, m; the position of the furthest distance between the crack and the crack opening at time j is determined by the following formula, that is, the stress intensity factor K _I at the crack tip:

, where K is the bulk modulus, MPa; thus, one can obtain the stress and displacement of the crack tip at time j:

where σ _xx - stress in the x direction of the crack tip, MPa; σ _yy - stress in the y direction of the crack tip, MPa; σ _xy - shear stress at the crack tip, MPa; X - displacement of the crack tip in the x direction, m; Y - displacement of the crack tip in the y direction, m; θ - polar angle of the polar coordinate system installed at the crack tip, rad; r is the polar diameter of the polar coordinate system installed at the crack tip, m; calculate the J-integral J _j at the crack tip at time j:

where W is the strain energy density at the crack tip, MJ/m ³ ; ε _ij - deformation at the crack tip, where all indices ij represent three types xx, yy and xy; σ _ij - stress at the crack tip, MPa; T _ij - thrust force at the crack tip, MPa; Г - any loop from the lower surface of the crack tip to the upper surface, m; n - unit normal vector perpendicular to the integral loop at the crack tip, dimensionless; if J _j <J _IC , J _IC is the creep propagation threshold of the rock, then this does not satisfy the creep propagation condition, and the crack length at this time remains unchanged, a _j+I = a _j , p _k,j+1 = p _e , steps (3), (4) are repeated to obtain the J-integral J _j+1 ; at the crack tip at time j+1; if J _j ≥J _IC , the creep propagation condition is reached and the C*-integral is calculated

crack tip at time j:

where

- the rate of change in the strain energy density at the crack tip, MJ / (m ³ s),

- rate of change in the displacement of the crack tip in the x direction, m/s, calculate the half-length of the crack a _j+1 at time j+1 from the propagation velocity of the crack tip creep:

where D _m - rock material constant, m ³ /MJ; φ _m - rock material constant, dimensionless; obtain the elastic width of the fracture under the conditions of the fracture half-length a _j+1 based on the elastic volume of the fracture, the rock creep does not directly affect the pressure in the fracture, so the pressure p _k,j+1 in the kth fracture segment at time j+1 is calculated from the following formula:

where

- elastic crack width at crack half-length component a _j+1 , m; b _k - distance from the left end of the k -th fracture segment to the fracture opening, m; steps (3) and (4) are repeated to obtain the J-integral

and C*-integral

crack tip at time j+1; at step (5) draw a graph

and schedule

and estimate the upper and lower limits of the well soak time, this process is as follows: when the curve

gradually becomes flat and the rate of change of the J-integral remains stable for more than 7 hours, it can be concluded that this has entered the stage of steady creep, and this value is the lower limit of the well holding time, when the value of C*-integral is below 0.015, the propagation power If the fracture creep is insufficient, and the fracture propagation stops, this is the upper limit of the well soak time.

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