CN115774089A - Establishment Method of Statistical Damage Constitutive Model of Coal and Rock Under Gas Pressure - Google Patents

Abstract

The invention discloses a method for establishing a coal rock statistical damage constitutive model under the action of gas pressure, which specifically comprises the following steps: by considering gas pressure, introducing an effective stress principle, defining an effective stress tensor under stress gas seepage coupling, setting coal rock infinitesimal strength to obey weibull distribution, and establishing a coal rock statistical damage constitutive model under the action of the gas pressure by taking a Hoek-Brown rule as a damage rule of the infinitesimal strength; and introducing a damage correction coefficient defined by the residual strength and the peak strength, and establishing a coal rock statistical damage constitutive model under the action of the corrected gas pressure. The model established by the invention can accurately reflect the constitutive relation of the coal rock under the combined action of three-dimensional stress and gas pressure, reveals the influence of the gas pressure on damage variables, and provides certain theoretical guidance for the exploitation of a coal rock reservoir.

Description

Establishment Method of Statistical Damage Constitutive Model of Coal and Rock Under Gas Pressure

technical field

The invention belongs to the technical field of rock mass engineering, and in particular relates to a method for establishing a statistical damage constitutive model of coal and rock under the action of gas pressure.

Background technique

With the exploitation and utilization of deep resources, a large number of deep rock engineering have emerged, such as coal mining, coal bed methane extraction and underground space construction, etc., and various disasters are often accompanied in the construction process. Especially in coal mining and coal bed methane extraction, dynamic disasters such as mine pressure, coal and gas outburst often occur, which pose a serious threat to the safe production of coal mines. Therefore, an in-depth understanding of the influence of gas pressure on the mechanical behavior of coal rocks and the damage and fracture of coal rocks is of great significance for the rational design and safe production of deep mining gas extraction projects.

At present, there are few studies on the constitutive model considering the influence of gas pressure on the mechanical properties of coal and rock, and the mechanical characteristics of the post-peak residual stress stage are often ignored when constructing the constitutive model. The statistical damage constitutive model, as one of the most widely used models at present, can more accurately describe the damage evolution characteristics of rocks, and can better reflect the mechanical mechanism of coal rock damage under the action of gas pressure. Therefore, it is necessary to design a method for establishing a statistical damage constitutive model of coal and rock under the action of gas pressure.

Contents of the invention

The purpose of the present invention is to provide a method for establishing a statistical damage constitutive model of coal rock under the action of gas pressure. The mining provides certain theoretical guidance.

The technical scheme adopted in the present invention is a method for establishing a statistical damage constitutive model of coal and rock under the action of gas pressure, which is specifically implemented according to the following steps:

如式(2)所示：Step 1: By considering the gas pressure and introducing the effective stress principle, define the effective stress tensor under the coupling of stress gas seepage

As shown in formula (2):

In the formula: b is the Biot coefficient, b=1; p _a is the pore gas pressure; δ _ij is the Kronecker symbol; D is the damage variable;

Step 2: Based on the effective stress tensor under the coupling of stress and gas seepage, the strength of coal-rock micro-elements obeys the Weibull distribution, and the Hoek-Brown criterion is used as the failure criterion of micro-element strength to establish the coal-rock statistics under gas pressure Damage constitutive model;

Step 3: Based on the established coal-rock statistical damage constitutive model under the action of gas pressure, the damage correction coefficient defined by the residual strength and peak strength is introduced to establish the corrected coal-rock statistical damage constitutive model under the action of gas pressure.

The present invention is also characterized in that,

In step 2, specifically:

The damage variable D of coal and rock can be expressed as the ratio of the number of damaged units to the number of undamaged material units, as shown in formula (4):

In the formula: N is the total number of microelements, and the number of microelements that are damaged in any interval [F, F+dF] is Np(F)dF;

Assuming that the micro-element strength of coal and rock obeys the Weibull random distribution, the probability density distribution function p(F) is shown in formula (5):

In the formula: a, η are Weibull distribution parameters;

Substituting (5) into (4), the damage variable D can be obtained, as shown in formula (6):

The Hoek-Brown strength criterion is used to describe the micro-element strength F of coal rock, as shown in formula (7):

分别为最大、中间、最小的有效主应力；In the formula: σ _c is the uniaxial compressive strength of intact coal rock; m, s are constants related to the characteristics of coal rock; θ is 30°;

are the maximum, middle, and minimum effective principal stresses, respectively;

Substituting formula (2) into formula (7), the micro-element strength F of coal rock can be expressed as formula (8);

The axial stress-strain relationship can be expressed as formula (9):

为最大有效主应变；E为弹性模量；ν为泊松比；In the formula:

is the maximum effective principal strain; E is the modulus of elasticity; ν is Poisson's ratio;

将式(2)代入式(9)，得到应力-瓦斯压力耦合作用下的轴向应力-应变的关系如式(11)所示：Let the axial strain

Substituting Equation (2) into Equation (9), the axial stress-strain relationship under the stress-gas pressure coupling is obtained as shown in Equation (11):

σ ₁ =(1-2ν)p _a +2νσ ₃ +Eε ₁ (1-D) (11);

Substituting Equation (11) into Equation (8), the coal-rock microelement strength F represented by nominal stress is obtained, as shown in Equation (12):

The axial deviatoric stress σ _1t is the difference between the axial stress σ ₁ and the confining pressure σ ₃ , as shown in formula (13):

σ _1t = σ ₁ -σ ₃ (13);

The initial strain ε ₀ before axial stress is applied, as shown in formula (14):

The real axial strain ε ₁ is the sum of the test measured strain ε _1t and the initial strain ε ₀ , as shown in formula (15):

ε ₁ =ε ₀ +ε _1t (15);

Substituting (6), (13), (14) and (15) into formulas (11) and (12) can obtain the coal-rock statistical damage constitutive model under gas action, as shown in formula (16):

In step 3, specifically:

According to the deformation characteristics of coal rock, the damage correction coefficient k defined by the residual strength and peak strength is introduced, as shown in formula (17):

Where: σ _r is the residual strength; σ _p is the peak strength;

表示为式(18)：The effective stress tensor corrected by the damage correction factor

Expressed as formula (18):

From this, the revised statistical damage model of coal and rock under the action of gas pressure can be established as formula (19):

The beneficial effect of the present invention is,

1. By considering the damage effect of gas pressure on coal and rock, the principle of effective stress is introduced, and the statistical damage constitutive model is used to describe the constitutive relationship of coal and rock under the action of gas pressure. Based on the damage correction coefficient reflecting the residual strength and peak strength, the The constitutive model is corrected so that the established model can accurately reflect the constitutive relationship of coal and rock under the joint action of three-dimensional force and gas pressure;

2. Carry out triaxial tests on coal and rock under different gas pressures, and use the peak point method and curve fitting method to determine the model parameter values according to the obtained test data, and compare and analyze them with the test curves to verify the correctness of the constitutive model sex and superiority.

3. By analyzing the influence of model parameters and damage correction coefficients on the theoretical curve shape characteristics, the physical significance of model parameters to gas-containing coal and rock is clarified, which reflects the applicability of the statistical damage constitutive model of coal and rock under the action of gas pressure. The safety analysis of actual engineering of gas-bearing coal and rock has good reference value.

Description of drawings

Fig. 1 is a schematic flow chart of a method for establishing a coal-rock statistical damage constitutive model under the action of a gas pressure of the present invention;

Figure 2 is a comparison chart (1) of the theoretical curve and test curve of the damage constitutive model of coal rock when the gas pressure is 1 MPa;

Figure 3 is a comparison chart (2) of the theoretical curve and test curve of the damage constitutive model of coal rock when the gas pressure is 1 MPa;

Figure 4 is a comparison chart (1) of the theoretical curve and test curve of the damage constitutive model of coal rock when the gas pressure is 2 MPa;

Figure 5 is a comparison chart (2) of the theoretical curve and test curve of the damage constitutive model of coal rock when the gas pressure is 2 MPa;

Figure 6 is a comparison chart (1) of the theoretical curve and test curve of the damage constitutive model of coal rock when the gas pressure is 3 MPa;

Figure 7 is a comparison chart (2) of the theoretical curve and test curve of the damage constitutive model of coal rock when the gas pressure is 3 MPa;

Figure 8 is a comparison chart (1) of the theoretical curve and test curve of the damage constitutive model of coal rock when the gas pressure is 5 MPa;

Figure 9 is a comparison chart (2) of the theoretical curve and test curve of the damage constitutive model of coal rock when the gas pressure is 5 MPa;

Fig. 10 is a diagram showing the influence of the damage correction coefficient k on the full stress-strain curve of coal;

Fig. 11 is a diagram showing the influence of model parameter a on the total stress-strain curve of coal;

Fig. 12 is the figure of influence of model parameter η on the full stress-strain curve of coal rock;

Figure 13 is a diagram of the evolution law of the damage variable D of coal and rock under different gas pressures.

Detailed ways

The present invention will be described in detail below in conjunction with the accompanying drawings and specific embodiments.

A method for establishing a coal-rock statistical damage constitutive model under the action of gas pressure of the present invention, as shown in Figure 1, is specifically implemented according to the following steps:

如式(1)所示：Step 1: Within the framework of poroelasticity theory, according to the Lemaitre strain equivalence theory, the effective stress tensor of the undamaged part can be obtained

As shown in formula (1):

如式(2)所示：By considering the gas pressure, the effective stress principle is introduced to define the effective stress tensor under the coupling of stress gas seepage

As shown in formula (2):

In the formula: b is the Biot coefficient, b=1; p _a is the pore gas pressure; δ _ij is the Kronecker symbol; D is the damage variable.

Step 2: Based on the effective stress tensor under the coupling of stress and gas seepage, the strength of coal-rock micro-elements obeys the Weibull distribution, and the Hoek-Brown criterion is used as the failure criterion of micro-element strength to establish the coal-rock statistics under gas pressure Damage constitutive model; specifically:

The micro-element strength criterion of coal rock can be expressed as formula (3):

为煤岩微元强度；B为材料参数。In the formula:

is the micro-element strength of coal rock; B is the material parameter.

假定p(F)为微元体的概率密度分布函数，N为总微元体数目，在任意区间[F,F+dF]内产生破坏的微元数目为Np(F)dF，煤岩的损伤变量D可以表示为受损破坏的单元数与无损时材料单元数的比值，如式(4)所示：make

Assume that p(F) is the probability density distribution function of microelements, N is the total number of microelements, and the number of microelements that are damaged in any interval [F, F+dF] is Np(F)dF. The damage variable D can be expressed as the ratio of the number of damaged elements to the number of undamaged material elements, as shown in formula (4):

Assuming that the micro-element strength of coal and rock obeys the Weibull random distribution, the probability density distribution function p(F) is shown in formula (5):

In the formula: a, η are Weibull distribution parameters.

Substituting formula (5) into formula (4), the damage variable D can be obtained, as shown in formula (6):

The Hoek-Brown strength criterion is used to describe the micro-element strength F of coal rock, as shown in formula (7):

分别为最大、中间、最小的有效主应力。其中：In the formula: σ _c is the uniaxial compressive strength of intact coal rock; m, s are constants related to the characteristics of coal rock; θ is Rhodes angle;

are the maximum, intermediate, and minimum effective principal stresses, respectively. in:

In the triaxial test, σ ₁ >σ ₂ =σ ₃ , and θ is 30° at this time. Substituting formula (2) into formula (7), according to the relationship between effective stress and nominal stress, the micro-element strength F of coal rock can be expressed as formula (8);

Assume that the stress-strain relationship of coal and rock obeys the generalized Hooke's law, and the axial stress-strain relationship is formula (9);

为最大有效主应变；E为弹性模量；ν为泊松比。In the formula:

is the maximum effective principal strain; E is the modulus of elasticity; ν is Poisson's ratio.

When the gas enters the pores of the coal rock body, the coal rock body absorbs the gas and causes swelling. So the axial stress-strain relationship can be expressed as formula (10):

In the formula: ε _p is the expansion strain; the coal adsorption expansion strain is related to the gas pressure, and the gas pressure P is constant, so the adsorption expansion strain is considered to remain unchanged.

将式(2)代入式(9)，得到应力-瓦斯压力耦合作用下的轴向应力-应变的关系如式(11)所示：Considering the coordinated deformation of coal and rock mass, the axial strain can be obtained

Substituting Equation (2) into Equation (9), the axial stress-strain relationship under the stress-gas pressure coupling is obtained as shown in Equation (11):

σ ₁ =(1-2ν)p _a +2νσ ₃ +Eε ₁ (1-D)(11);

Substituting Equation (11) into Equation (8), the coal-rock microelement strength F represented by nominal stress is obtained, as shown in Equation (12):

The axial deviatoric stress σ _1t recorded in the test is actually the difference between the axial stress σ ₁ and the confining pressure σ ₃ , as shown in formula (13):

σ _1t = σ ₁ -σ ₃ (13);

The initial strain ε ₀ before axial stress is applied, as shown in formula (14):

The real axial strain ε ₁ is the sum of the test measured strain ε _1t and the initial strain ε ₀ , as shown in formula (15):

ε ₁ =ε ₀ +ε _1t (15);

Substituting Equation (6), Equation (13), Equation (14) and Equation (15) into Equation (11) and Equation (12), the constitutive model of coal and rock statistical damage under gas action can be obtained, as shown in Equation (16). Show:

Step 3: Based on the established coal-rock statistical damage constitutive model under the action of gas pressure, introduce the damage correction coefficient defined by the residual strength and peak strength, and establish the corrected coal-rock statistical damage constitutive model under the action of gas pressure;

According to the deformation characteristics of coal rock, the damage correction coefficient k defined by the residual strength and peak strength is introduced, as shown in formula (17):

Where: σ _r is the residual strength; σ _p is the peak strength.

表示为式(18)：The effective stress tensor corrected by the damage correction factor

Expressed as formula (18):

From this, the revised statistical damage model of coal and rock under the action of gas pressure can be established as formula (19):

in:

The model verification and parameter analysis are carried out through the triaxial test data of coal and rock under different gas pressures. Specifically:

Carry out the triaxial test of coal and rock under the action of gas pressure, set the confining pressure constant at 10MPa, and the gas pressure at 1, 2, 3 and 5MPa respectively, obtain the test data and test curves, and obtain the mechanical properties of coal samples under different gas pressures. Parameter values, as shown in Table 1;

Table 1 shows the mechanical parameter values of coal samples under different gas pressures

According to the above parameter values in the test data, the peak point method and the curve fitting method are used to determine the model parameter values respectively. Based on the above model (Eq. The derivative of the constitutive model at the and peak points is 0. The curve fitting method uses the least squares principle to perform nonlinear fitting on the test data to solve the model parameters. The parameter values of the coal-rock statistical damage constitutive model under different gas pressures are as follows: As shown in Table 2;

Table 2 shows the parameter values of the coal-rock statistical damage constitutive model under different gas pressures

Substituting the obtained parameter values into equations (16) and (19) to obtain the corrected damage constitutive model theoretical curve and the uncorrected damage constitutive model theoretical curve, and compare and analyze them with the test curves to verify the coal-rock statistical The rationality of the damage constitutive model, as shown in Figures 2 and 3, is the comparison between the theoretical curve and the test curve of the damage constitutive model of coal rock when the gas pressure is 1MPa; Comparison chart of damage constitutive model theoretical curve and test curve when the pressure is 2MPa; as shown in Figure 6 and 7, it is a comparison chart of damage constitutive model theoretical curve and test curve when the gas pressure of coal rock is 3MPa; Figure 8 and 9 Shown is the comparison chart between the theoretical curve and the test curve of the damage constitutive model of coal rock when the gas pressure is 5 MPa. It can be seen from the figure that in the pre-peak stage, the four theoretical curves and the test curve are well fitted, and the relationship between the theoretical curves The fitting conditions between them are highly coincident, but the theoretical curve using the peak point method has the best fitting situation at the peak point. In the post-peak stage, the corrected model curve has the best fit with the test curve, which can reflect the stress drop phenomenon and softening characteristics of the rock, while the uncorrected model curve has a poor fit with the experimental curve, which cannot reflect the rock’s stress drop and softening characteristics. residual strength. Generally speaking, the corrected coal-rock statistical damage constitutive model curves under different gas pressures can better reflect the real mechanical characteristics of coal rocks. Curve fit is best.

Based on the residual strength and peak strength, the damage correction coefficient k (Equation 17) can be obtained. In order to explore the influence of the correction coefficient on the coal-rock constitutive relationship, keeping other parameters unchanged, the damage correction coefficient is used as a variable to analyze the effect of the correction coefficient on the coal-rock The influence of theoretical curve shape characteristics, the results are shown in Figure 10. The change of the damage correction factor has no effect on the pre-peak stage of the stress-strain curve of coal rock, but in the post-peak stage, with the increase of the damage correction factor, the residual strength of the coal rock gradually decreases. When the damage correction factor is 1 At that time, the model will not reflect the residual strength of coal rock. The damage correction coefficient can reflect the post-peak softening characteristics of coal rock, which further improves the accuracy and applicability of the statistical damage constitutive model of coal rock.

The strength of coal and rock microelements obeys the Weibull distribution, which contains two model parameters a and η. In order to explore the physical meaning of the model parameters and make the model more widely applicable, keep other parameters unchanged, and take a and η as variables respectively , analyze the influence of these two parameters on the total stress-strain curve of coal and rock, and the results are shown in Fig. 11 and Fig. 12. As a gets larger, the stress drop rate of the coal rock after the peak point is faster, that is, the brittleness of the coal rock becomes more and more obvious. Therefore, the parameter a mainly reflects the brittleness characteristics of the coal rock and the internal micro-element strength of the coal rock material. distribution concentration. With the increase of parameter η, the strength of coal rock is getting bigger and bigger, which shows that parameter η reflects the size of the average strength of coal rock macroscopic statistics. Combined with the analysis in Table 2, it can be seen that with the increase of gas pressure, the model parameters a and η show a decreasing trend, which indicates that the brittleness and strength of coal rocks decrease.

The growth of damage variables is closely related to the deformation and failure of coal rocks. The relationship between damage variables and strains of coal rocks under different gas pressures is shown in Figure 13. The damage degree of coal rock is almost 0 in the elastic deformation stage, but as the strain increases, the damage degree of coal rock will begin to increase at a certain moment, and then the damage degree will increase sharply until the coal rock Destruction; under different gas pressures, the damage rate of coal rocks is not the same when the degree of damage increases sharply, and the maximum damage evolution rate of coal rocks gradually decreases with the increase of gas pressure. When the gas pressure is low, coal rocks are more prone to brittle failure.

Claims (5)

1. The method for establishing the coal rock statistical damage constitutive model under the action of gas pressure is characterized by comprising the following steps of:

step 1: by considering gas pressure and introducing effective stress principle, the effective stress tensor under stress gas seepage coupling is defined

As shown in formula (2):

in the formula: b is the Biot coefficient, b =1; p is a radical of _a Pore gas pressure;δ _ij is a kronecker symbol; d is a damage variable;

step 2: based on the effective stress tensor under stress gas seepage coupling, setting the coal rock infinitesimal strength to obey weibull distribution, and establishing a coal rock statistical damage constitutive model under the action of gas pressure by taking a Hoek-Brown rule as a damage rule of the infinitesimal strength;

and step 3: and based on the established coal rock statistical damage constitutive model under the action of the gas pressure, introducing a damage correction coefficient defined by the residual strength and the peak strength, and establishing the corrected coal rock statistical damage constitutive model under the action of the gas pressure.

2. The method for establishing the coal rock statistical damage constitutive model under the action of the gas pressure as recited in claim 1, wherein in the step 2, the method specifically comprises:

the damage variable D of the coal rock can be expressed as a ratio of the number of damaged units to the number of material units when the coal rock is not damaged, as shown in equation (4):

in the formula: n is the total number of microelements, and the number of the microelements which are destroyed in any interval [ F, F + dF ] is Np (F) dF;

if the coal rock infinitesimal strength obeys Weibull random distribution, a probability density distribution function p (F) is shown as formula (5):

in the formula: a, eta are Weibull distribution parameters;

substituting (5) into (4) can obtain a damage variable D, as shown in formula (6):

the Hoek-Brown strength criterion is adopted to describe the coal rock infinitesimal strength F, as shown in formula (7):

in the formula: sigma _c Uniaxial compressive strength of intact coal rock; m and s are constants related to coal rock characteristics; theta is 30 degrees;

maximum, intermediate, minimum effective principal stress, respectively;

substituting the formula (2) into the formula (7), and expressing the coal rock infinitesimal strength F as a formula (8);

the stress-strain relationship in the axial direction can be expressed by the following formula (9):

in the formula:

is the maximum effective principal strain; e is the modulus of elasticity; ν is the poisson ratio;

strain in axial direction

Substituting the formula (2) into the formula (9) to obtain the relation of axial stress-strain under the coupling action of stress-gas pressure as shown in the formula (11):

σ ₁ ＝(1-2ν)p _a +2νσ ₃ +Eε ₁ (1-D)(11)；

substituting the formula (11) into the formula (8) to obtain the coal rock infinitesimal strength F represented by the nominal stress, as shown in the formula (12):

axial bias stress sigma _1t For axial stress σ ₁ And confining pressure σ ₃ The difference of (d) is shown in equation (13):

σ _1t ＝σ ₁ -σ ₃ (13)；

initial strain before application of axial stress ₀ As shown in formula (14): :

true axial strain epsilon ₁ Measuring strain values epsilon for the test _1t With initial strain epsilon ₀ And (3) the sum is represented by formula (15):

ε ₁ ＝ε ₀ +ε _1t (15)；

substituting the (6), (13), (14) and (15) into the formulas (11) and (12) can obtain a coal rock statistical damage constitutive model under the action of gas.

3. The method for establishing the coal petrography statistical damage constitutive model under the action of the gas pressure as claimed in claim 2, wherein the coal petrography statistical damage constitutive model under the action of the gas is represented by formula (16):

4. the method for establishing the coal rock statistical damage constitutive model under the action of the gas pressure as claimed in claim 2, wherein in the step 3, specifically:

according to the deformation characteristics of the coal rock, introducing a damage correction coefficient k defined by residual strength and peak strength, as shown in formula (17):

in the formula: sigma _r Is the residual strength; sigma _p Peak intensity;

effective stress tensor corrected by damage correction factor

Represented by formula (18):

therefore, a coal rock statistical damage model under the action of the corrected gas pressure can be established as a formula (19):

5. the method for building a coal petrography statistical damage constitutive model under the action of gas pressure as claimed in claim 4, wherein in formula (19),

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