CN115774089A - Method for establishing coal rock statistical damage constitutive model under action of gas pressure - Google Patents
Method for establishing coal rock statistical damage constitutive model under action of gas pressure Download PDFInfo
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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
Technical Field
The invention belongs to the technical field of rock mass engineering, and particularly relates to a method for establishing a coal rock statistical damage constitutive model under the action of gas pressure.
Background
With the exploitation and utilization of deep resources, a large number of deep rock projects such as coal mining, coal bed gas extraction, underground space construction and the like appear, and various disasters are often accompanied in the construction process. Particularly, in coal mining and coal bed gas extraction, dynamic disasters such as mine pressure, coal and gas outburst and the like can be caused frequently, and serious threats are caused to the safety production of coal mines. Therefore, the deep understanding of the mechanical behavior influence of the gas pressure on the coal rock and the damage and the fracture of the coal rock has important significance for the reasonable design and the safe production of deep mining gas production engineering.
At present, the study of a constitutive model considering the influence of gas pressure on the mechanical properties of coal rocks is less, and the mechanical characteristics of a post-peak residual stress stage are often ignored when the constitutive model is constructed. The statistical damage constitutive model is one of the most widely applied 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 coal rock statistical damage constitutive model under the action of gas pressure.
Disclosure of Invention
The invention aims to provide a method for establishing a coal rock statistical damage constitutive model under the action of gas pressure, the established model can accurately reflect the constitutive relation of coal rock under the combined action of three-dimensional stress and the gas pressure, and certain theoretical guidance is provided for mining of a coal rock reservoir.
The invention adopts the technical scheme that the method for establishing the coal rock statistical damage constitutive model under the action of gas pressure is implemented according to the following steps:
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; delta ij Is a kronecker symbol; d is a damage variable;
step 2: setting the strength of a coal rock infinitesimal body to obey weibull distribution based on the effective stress tensor under stress gas seepage coupling, and establishing a coal rock statistical damage constitutive model under the action of gas pressure by taking a Hoek-Brown criterion as a failure criterion of the infinitesimal body strength;
and 3, 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.
The present invention is also characterized in that,
in the step 2, the method specifically comprises the following steps:
the damage variable D of the coal rock can be expressed as the ratio of the number of damaged units to the number of units of the material when the damage is not damaged, as shown in the formula (4):
in the formula: n is the total number of the 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):
and describing the coal rock infinitesimal strength F by adopting a Hoek-Brown strength rule, wherein the formula (7) is as follows:
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 stress-gas pressure coupling effect, which is 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 (6), (13), (14) and (15) into the formulas (11) and (12) to obtain a coal rock statistical damage constitutive model under the action of gas, as shown in the formula (16):
in step 3, the method specifically comprises the following steps:
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):
the beneficial effect of the invention is that,
1. by considering the damage effect of gas pressure on coal rocks, introducing an effective stress principle, describing the constitutive relation of the coal rocks under the action of the gas pressure by adopting a statistical damage constitutive model, and correcting the constitutive model based on a damage correction coefficient reflecting residual strength and peak strength, the established model can accurately reflect the constitutive relation of the coal rocks under the combined action of three-dimensional stress and the gas pressure;
2. and performing triaxial tests of the coal rock under the action of different gas pressures, respectively determining model parameter values by adopting a peak point method and a curve fitting method according to the obtained test data, and performing comparative analysis on the model parameter values and the test curves to verify the correctness and superiority of the constitutive model.
3. By analyzing the influence rule of the model parameters and the damage correction coefficient on the morphological characteristics of the theoretical curve, the physical significance of the model parameters on the gas-containing coal rock is determined, the applicability of the coal rock statistical damage constitutive model under the action of gas pressure is reflected, and the method has a better reference value on the safety analysis of the actual engineering of the gas-containing coal rock.
Drawings
FIG. 1 is a schematic flow chart of a method for building a coal rock statistical damage constitutive model under the action of gas pressure according to the invention;
FIG. 2 is a comparison graph (I) of a theoretical curve and a test curve of a damage constitutive model of coal rock under the condition that the gas pressure is 1 MPa;
FIG. 3 is a comparison graph (II) of a theoretical curve and a test curve of a damage constitutive model of coal rock under the condition that the gas pressure is 1 MPa;
FIG. 4 is a comparison graph (I) of a theoretical curve and a test curve of a damage constitutive model of coal rock at a gas pressure of 2 MPa;
FIG. 5 is a comparison graph (II) of a theoretical curve and a test curve of a damage constitutive model of coal rock under the condition that the gas pressure is 2 MPa;
FIG. 6 is a comparison graph (I) of a theoretical curve and a test curve of a damage constitutive model of coal rock at a gas pressure of 3 MPa;
FIG. 7 is a graph (II) comparing a theoretical curve and a test curve of a damage constitutive model of coal rock at a gas pressure of 3 MPa;
FIG. 8 is a comparison graph (I) of a theoretical curve and a test curve of a damage constitutive model of coal rock at a gas pressure of 5 MPa;
FIG. 9 is a graph (II) comparing a theoretical curve and a test curve of a damage constitutive model of coal rock at a gas pressure of 5 MPa;
FIG. 10 is a graph of the effect of the damage correction factor k on the coal petrography full stress-strain curve;
FIG. 11 is a graph of the effect of model parameter a on the coal petrography full stress-strain curve;
FIG. 12 is a graph of the effect of model parameter η on the coal petrography total stress-strain curve;
FIG. 13 is a diagram of the evolution law of the damage variable D of the coal rock under the action of different gas pressures.
Detailed Description
The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments.
The invention relates to a method for establishing a coal rock statistical damage constitutive model under the action of gas pressure, which is specifically implemented according to the following steps as shown in figure 1:
step 1: in the framework of porous elastic theory, according to Lemailre strain equivalence theory, the effective stress tensor of the undamaged part can be obtained
As shown in formula (1):
by taking into account gasPressure, effective stress principle is introduced, and 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; delta ij Is a kronecker symbol; d is the lesion variable.
Step 2: setting the strength of a coal rock infinitesimal body to obey weibull distribution based on the effective stress tensor under stress gas seepage coupling, and establishing a coal rock statistical damage constitutive model under the action of gas pressure by taking a Hoek-Brown criterion as a failure criterion of the infinitesimal body strength; the method specifically comprises the following steps:
the coal petrography infinitesimal intensity criterion can be expressed as formula (3):
in the formula:
the strength of coal rock infinitesimal; and B is a material parameter.
Order to
Assuming that p (F) is probability density distribution function of the infinitesimal body, N is total infinitesimal body number, and in any interval [ F, F + dF]The number of infinitesimal elements generating damage in the coal rock is Np (F) dF, and the damage variable D of the coal rock can be expressed as the ratio of the number of damaged units to the number of units of the material without damage, as shown in formula (4):
if the infinitesimal strength of the coal rock obeys Weibull random distribution, the probability density distribution function p (F) is shown as formula (5):
in the formula: a, eta are Weibull distribution parameters.
Substitution of formula (5) for formula (4) yields a damage variable D, as shown in formula (6):
and describing the coal rock infinitesimal strength F by adopting a Hoek-Brown strength rule, wherein the formula (7) is as follows:
in the formula: sigma c Uniaxial compressive strength of intact coal rock; m and s are constants related to coal rock characteristics; theta is a rod angle;
maximum, intermediate, minimum effective principal stress, respectively. Wherein:
in the triaxial test, σ 1 >σ 2 ＝σ 3 In this case, θ is 30 °. Substituting the formula (2) into the formula (7), and representing the coal rock infinitesimal strength F as a formula (8) according to the relation between the effective stress and the nominal stress;
assuming that the stress-strain relation of the coal rock obeys the generalized Hooke's law, and the axial stress-strain relation is an equation (9);
in the formula:
is the maximum effective principal strain; e is the modulus of elasticity; ν is the poisson ratio.
When the gas enters the pores of the coal rock body, the coal rock body absorbs the gas to cause expansion. The stress-strain relationship in the axial direction can be expressed by the following equation (10):
in the formula: epsilon p Is an expansion strain; the coal body adsorption expansion strain is related to the gas pressure, and the gas pressure P is constant, so it is considered that the adsorption expansion strain remains unchanged.
Axial strain can be obtained by considering the coordinated deformation of coal and rock masses
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 offset stress sigma recorded in the test 1t In fact the axial stress sigma 1 And confining pressure σ 3 As shown in equation (13):
σ 1t ＝σ 1 -σ 3 (13)；
initial strain epsilon 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 formula (6), the formula (13), the formula (14) and the formula (15) into the formula (11) and the formula (12) can obtain a coal rock statistical damage constitutive model under the action of gas, as shown in the formula (16):
and step 3: based on the established coal rock statistical damage constitutive model under the action of the gas pressure, introducing a damage correction coefficient defined by residual strength and peak strength, and establishing the corrected coal rock statistical damage constitutive model under the action of the gas pressure;
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 The 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):
wherein:
and performing model verification and parameter analysis through triaxial test data of the coal rock under the action of different gas pressures. The method specifically comprises the following steps:
performing a triaxial test of the coal rock under the action of gas pressure, setting the ambient pressure constant to be 10MPa, and setting the gas pressure to be 1,2,3 and 5MPa respectively, obtaining test data and a test curve, and obtaining mechanical parameter values of the coal sample under different gas pressures, wherein the values are shown in Table 1;
table 1 shows the values of the mechanical parameters of the coal samples at different gas pressures
Respectively determining model parameter values by adopting a peak point method and a curve fitting method according to the parameter values in the test data, calculating the model parameters by using the peak point method based on the model (formula 19) and meeting 2 conditions, namely the two sides of a model equation at the peak point are equal and the derivative of the constitutive model at the peak point is 0, performing nonlinear fitting on the test data by using a least square principle to solve the model parameters by using the curve fitting method, wherein the parameter values of the constitutive model statistically damaged by coal and rock under different gas pressures are shown in a table 2;
table 2 shows the parameter values of the coal rock statistical damage constitutive model under different gas pressures
Respectively substituting the obtained parameter values into the formulas (16) and (19) to obtain a corrected damage constitutive model theoretical curve and an uncorrected damage constitutive model theoretical curve, comparing and analyzing the corrected damage constitutive model theoretical curve and a test curve, and verifying the rationality of the coal rock statistical damage constitutive model, wherein the corrected damage constitutive model theoretical curve and the uncorrected damage constitutive model theoretical curve are comparison graphs of the damage constitutive model theoretical curve and the test curve of the coal rock when the gas pressure is 1MPa as shown in figures 2 and 3; as shown in fig. 4 and 5, the graph is a comparison graph of a theoretical curve and a test curve of the damage constitutive model of the coal rock under the gas pressure of 2 MPa; as shown in fig. 6 and 7, the graph is a comparison graph of a theoretical curve and a test curve of the damage constitutive model of the coal rock when the gas pressure is 3 MPa; as shown in fig. 8 and 9, which are graphs comparing theoretical curves and test curves of the damage constitutive model of coal rock at a gas pressure of 5MPa, it can be seen that, in the pre-peak stage, four kinds of theoretical curves and test curves are well fitted, and the fitting conditions between the theoretical curves are highly overlapped, but the fitting condition of the theoretical curve at the peak point by the peak point method is the best. In the post-peak stage, the fitting degree of the corrected model curve and the test curve is the best, the stress drop phenomenon and the softening characteristic of the rock can be reflected, and the fitting of the uncorrected model curve and the test curve is poor, so that the residual strength of the rock cannot be reflected. In general, the corrected coal rock statistical damage constitutive model curve under different gas pressures can reflect the real stress characteristics of the coal rock better, wherein the corrected coal rock statistical damage constitutive model curve adopting the curve fitting method is best fitted with the experimental curve.
Based on the residual intensity and the peak intensity, a damage correction coefficient k (formula 17) can be obtained, in order to explore the influence of the correction coefficient on the coal rock constitutive relation, other parameters are kept unchanged, the damage correction coefficient is used as a variable, the influence of the correction coefficient on the coal rock theoretical curve morphological characteristics is analyzed, and the result is shown in fig. 10. The change of the damage correction coefficient has no influence on the pre-peak stage of the stress-strain curve of the coal rock, while in the post-peak stage, the residual strength of the coal rock is gradually reduced along with the increase of the damage correction coefficient, and when the damage correction coefficient is 1, the model cannot reflect the residual strength of the coal rock. The damage correction coefficient can reflect the softening characteristics of the coal rock after the peak, and the accuracy and the applicability of the coal rock statistical damage constitutive model are further improved.
The coal rock infinitesimal body strength is subject to weibull distribution, comprises two model parameters of a and eta, and is used for analyzing the influence of the two parameters on a coal rock total stress-strain curve by respectively taking the a and the eta as variables in order to explore the physical significance of the model parameters and enable the model to have wider applicability and keep other parameters unchanged, and the results are shown in fig. 11 and fig. 12. With the increasing a, the stress drop rate of the coal rock after the peak point is faster, namely the brittleness of the coal rock is more and more obvious, so the parameter a mainly reflects the brittleness characteristic of the coal rock and the concentration degree of the infinitesimal strength distribution in the coal rock material. The intensity of the coal petrography is larger and larger along with the increase of the parameter eta, which shows that the parameter eta reflects the size of the macroscopic statistical average intensity of the coal petrography. The analysis combined with the table 2 can obtain that the model parameters a and eta are in a decreasing trend along with the increase of the gas pressure, and represent the brittleness and the strength reduction of the coal rock.
The growth of the damage variable is closely related to the deformation damage of the coal rock, and the relationship of the damage variable-strain of the coal rock under different gas pressures is shown in fig. 13. The damage degree of the coal rock is almost 0 in the elastic deformation stage, but the damage degree of the coal rock begins to increase at a certain moment along with the continuous increase of the strain amount, and the damage degree rapidly increases immediately until the coal rock is damaged; under the action of different gas pressures, the rates of the damage degree of the coal rock are different when the damage degree of the coal rock is increased sharply, and the maximum damage evolution rate of the coal rock is gradually reduced along with the increase of the gas pressure. When the gas pressure is lower, the coal rock is more easily subjected 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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| CN116611265B (en) * | 2023-07-18 | 2023-09-22 | 北京建筑大学 | Method and device for predicting stress and strain of deep anisotropic rock |
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