# A Thermal Damage Constitutive Model for Oil Shale Based on Weibull Statistical Theory

Authors: Guijie Zhao; Chen Chen; Huan Yan
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this work, we first studied the thermal damage to typical rocks, assuming that the strength of thermally damaged rock microelements obeys a Weibull distribution and considering the influence of temperature on rock mechanical parameters; under the condition that microelement failure conforms to the Drucker–Prager criterion, the statistical thermal damage constitutive model of rocks after high-temperature exposure was established. On this basis, conventional triaxial compression tests were carried out on oil shale specimens heated to different temperatures, and according to the results of these tests, the relationship between the temperature and parameters in the statistical thermal damage constitutive model was determined, and the thermal damage constitutive model for oil shale was established. The results show that the thermal damage in oil shale increases with the increase of temperature; the damage variable is largest at 700°C, reaching 0.636; from room temperature to 700°C, the elastic modulus and Poisson’s ratio decrease by 62.66% and 64.57%, respectively; the theoretical stress-strain curve obtained from the model is in good agreement with the measured curves; the maximum difference between the two curves before peak strength is only 5 × 10−4; the model accurately reflects the deformation characteristics of oil shale at high temperature. The research results are of practical significance to the underground in situ thermal processing of oil shale.


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## Body

## 1. Introduction

Oil shale resources are abundant and, when converted into shale oil, will amount to more than 400 billion tons, which is approximately three times the proven recoverable reserve of natural crude oil. With the rapid growth of the economy and population, the demand for energy has increased sharply, while conventional oil and gas resources are decreasing. Against the background of this energy crisis, oil shale resources have attracted more attention because of their abundance and future development prospects [1–4].To use the oil shale resources, many methods of exploitation of oil shale have been developed, which can be roughly divided into two categories: surface retorting andin situ thermal processing. In situ thermal processing techniques developed later than surface retorting but have developed rapidly. At present, there are more than ten in situ thermal processing techniques, the core of which relies on heating of the underground oil shale reservoir to produce oil and gas by pyrolysis of kerogen in oil shale and then recovering oil and gas to the surface [5, 6]. In the process of in situ heating of an oil shale reservoir, thermal damage is bound to occur in the oil shale, and the damage will be aggravated continuously. Thermally damaged oil shale will undergo changes in microstructure and macrophysical and mechanical properties. In view of this, it is important to study the thermal damage to oil shale and establish an appropriate constitutive model thereof [7–9].The thermal damage problems of typical rock were first studied. Assuming that the strength of thermally damaged rock microelements obeys a Weibull distribution, considering the influence of temperature on rock mechanical parameters, and defining the thermal damage variable under the condition that microelement failure conforms to the Drucker–Prager criterion, the statistical thermal damage constitutive model of rock after high-temperature exposure was established. Based on the concept of rock yield, the model parameters were determined by the extremum method. On this basis, according to the results of conventional triaxial compression tests of oil shale specimens after heating to different temperatures, the relationship between temperatureT and parameters in the statistical thermal damage constitutive model of oil shale after high-temperature exposure was determined by means of a mathematical fitting method, and the thermal damage constitutive model of oil shale was thus established. The correctness of the model was verified by comparing the theoretical stress-strain curve calculated from the model to the actual curve measured by conventional triaxial tests.


## 2. Damage Model for Rock Based on Weibull Statistical Theory

### 2.1. Statistical Strength Theory of Rock Failure

In 1939, Weibull took the lead in introducing statistical methods into the strength theory of materials to describe the nonuniformity of materials. He believed that the failure probability of materials can be calculated statistically when the stress is constant, although it may be inaccurate to measure the failure strength of materials. Kostak, based on uniaxial compression testing of 320 sandstone specimens, obtained the strength distribution cubic graph by statistical analysis and found that the strength of rock specimens has a certain discreteness, but on the whole, the strength of these discrete rocks has a certain statistical regularity. Tang Chun’an (1993) proposed a hypothesis whereby the strength of the rock microelement satisfies a certain statistical distribution. He proposed that mesoheterogeneity is the fundamental cause of the macro-nonlinearity of quasi-brittle materials and used the statistical damage method to consider the heterogeneity of rock materials and the randomness of defects.Many experts have proposed statistical models to estimate the strength of materials during damage fracture or fatigue failure, among which the most important ones are those following the normal distribution, Weibull distribution, lognormal distribution, and gamma distribution [10–15]. The Weibull distribution statistical model has strong adaptability and fitting ability and thus is the most widely used model. Its probability density function is as follows:(1)fx=mFx−γFm−1exp−x−γFm,x∈γ,∞,0,x∈−∞,γ.Its cumulative probability distribution function is(2)Fx=1−exp−x−γFm,x∈γ,∞,where m is the shape parameter, F is the scale parameter, and γ represents the position parameter.


### 2.2. Damage Variable and Establishment of the Damage Constitutive Relationship

At present, there are two main ways in which to establish rock constitutive relationships by using damage theory: one is to establish a relationship [16] according to the principle of energy equivalence before and after damage; the other relies on the principle of strain equivalence before and after deformation. These two approaches establish the constitutive relationship of rock materials from different perspectives, but the mechanical concept of the latter is clearer. Therefore, in this research, the latter was used to establish the damage constitutive relationship of rock.According to Lemaitre’s equivalent strain hypothesis, the following results were obtained [17]:(3)σ∗=σ1−D=Eε1−D,where σ is the stress tensor, σ∗ denotes the effective stress tensor, ε is the strain tensor, E is the elastic modulus matrix, and D is the damage matrix.Rock has a certain residual strength after failure. Considering the influence of rock residual strength, the damage correction coefficientδ was used to modify equation (3) [18]:(4)σ∗=σ1−δD=Eε1−δD.The key to establishing the damage constitutive relationship of rock is to determine the damage variable. In previous studies, the strength of rock microelements is usually expressed by rock axial principal strain, but this method cannot reflect the influence of the complex stress state on the strength of rock microelements; therefore, considering the failure model and criterion of rocks, others have proposed a new expression of damage variables reflecting the strength of rock microelements [19–22]. It is assumed that the general failure criterion for rocks is(5)fσ∗−k0=0,where k0 is a constant related to the cohesion and internal friction angle of rock.f
σ
∗ reflects the criticality of rock microelement failure and can be used as the strength of a rock microelement. Assuming that the probability of rock microelement failure is Ffσ∗, which is related to the distribution density of fσ∗, the damage variable D is defined as follows:(6)D=∫0fσ∗Fxdx.Formula (6) shows that the key to establishing the damage model for the entire process of rock deformation and failure lies in the determination of rock microelement strength and its failure probability.


### 2.3. Determination of Rock Microelement Strength

At present, there are many forms of failure criteria that can be used in rock mechanics research. Among them, the Drucker–Prager criterion has a simple parametric form that is applicable to rock media and is thus widely used. Therefore, the strength of a rock microelement based on the Drucker–Prager criterion was determined as follows [12, 16]:(7)fσ=k=α0I1+J2,(8)α0=sinφ9+3sin2φ,where φ is the internal friction angle; I1 represents the first invariant of the stress tensor; and J2 is the second invariant of the deviatoric stress tensor. The expressions of I1 and J2 are as follows:(9)I1=σ1∗+σ2∗+σ3∗,(10)J2=16σ1∗−σ2∗2+σ2∗−σ3∗2+σ3∗−σ1∗2.In conventional triaxial compression tests on rock specimens, nominal stressesσ1, σ2, and σ3 (σ2 = σ3) and strain ε1 can be measured, and the corresponding effective stresses are σ1∗, σ2∗, and σ3∗ (σ2∗=σ3∗). The formulae for calculating the effective stress are as follows:(11)σ1∗=σ11−δD,(12)σ2∗=σ3∗=σ31−δD.The elastic modulus and Poisson’s ratio of rock areE and μ, respectively. According to Hooke’s law, the following result was obtained:(13)ε1=σ1∗−2μσ3∗E.Formulae (11)–(13) were substituted into formulae (9) and (10), and the following results were obtained:(14)I1=σ1+2σ3Eε1σ1−2μσ3,J2=σ1−σ3Eε13σ1−2μσ3.By substituting formula (14) into formula (7), the strength of rock microelements was acquired as follows:(15)fσ=Eε1σ1−2μσ3α0+33σ1+2α0−33σ3.


### 2.4. Damage Model of Rock Based on Weibull Statistical Theory

Assuming that the strengthk of rock microelements obeys a Weibull distribution, the probability density function of rocks can be obtained from the Weibull statistical model introduced earlier as follows [16]:(16)fk=mFkFm−1exp−kFm.where k is the distribution variable of the Weibull distribution and m and F are Weibull distribution parameters. By substituting formula (16) into formula (6), the damage variable D was calculated as follows:(17)D=1−exp−kFm.There is a threshold point (σD, εD) in the damage evolution of rock materials. When the stress is less than the threshold point stress σD, the damage of rocks is very small or no damage occurs. At this time, the damage variable can be considered to be zero and(18)D=0,σ1<σD.When the stress reaches or exceeds the threshold point stressσD, the damage variable can be calculated according to equation (17). The damage variable expression of rock-like materials under arbitrary stress state can be expressed as follows:(19)D=0,σ1<σD,1−exp−kFm,σ1≥σD.By substituting formulae (11), (12), and (17) into formula (13), the damage constitutive equation of rocks based on the Weibull distribution (after the damage threshold point, i.e., σ1≥σD) was obtained as follows:(20)σ1=Eε11−δ+δexp−kFm−2μσ3,σ1≥σD,where the parameter m reflects the shape of the function; the parameter F reflects the scale characteristics of the function and corresponds to the homogeneity and average strength of rock materials; and parameter δ reflects the residual strength of rocks.Before reaching the damage threshold, i.e.,σ1<σD, the constitutive relation of rock materials can be obtained by fitting the polynomial function relation. Considering that the strain of rocks is zero when the stress is zero, the relationship can be expressed as follows:(21)σ1=Aε1ε1+B,σ1<σD,where A and B are coefficients.By combining formulae (20) and (21), a rock damage model based on Weibull statistical theory under an arbitrary stress state was obtained as follows:(22)σ1=Aε1ε1+B,σ1<σD,Eε11−δ+δexp−kFm−2μσ3,σ1≥σD.


## 2.1. Statistical Strength Theory of Rock Failure

In 1939, Weibull took the lead in introducing statistical methods into the strength theory of materials to describe the nonuniformity of materials. He believed that the failure probability of materials can be calculated statistically when the stress is constant, although it may be inaccurate to measure the failure strength of materials. Kostak, based on uniaxial compression testing of 320 sandstone specimens, obtained the strength distribution cubic graph by statistical analysis and found that the strength of rock specimens has a certain discreteness, but on the whole, the strength of these discrete rocks has a certain statistical regularity. Tang Chun’an (1993) proposed a hypothesis whereby the strength of the rock microelement satisfies a certain statistical distribution. He proposed that mesoheterogeneity is the fundamental cause of the macro-nonlinearity of quasi-brittle materials and used the statistical damage method to consider the heterogeneity of rock materials and the randomness of defects.Many experts have proposed statistical models to estimate the strength of materials during damage fracture or fatigue failure, among which the most important ones are those following the normal distribution, Weibull distribution, lognormal distribution, and gamma distribution [10–15]. The Weibull distribution statistical model has strong adaptability and fitting ability and thus is the most widely used model. Its probability density function is as follows:(1)fx=mFx−γFm−1exp−x−γFm,x∈γ,∞,0,x∈−∞,γ.Its cumulative probability distribution function is(2)Fx=1−exp−x−γFm,x∈γ,∞,where m is the shape parameter, F is the scale parameter, and γ represents the position parameter.


## 2.2. Damage Variable and Establishment of the Damage Constitutive Relationship

At present, there are two main ways in which to establish rock constitutive relationships by using damage theory: one is to establish a relationship [16] according to the principle of energy equivalence before and after damage; the other relies on the principle of strain equivalence before and after deformation. These two approaches establish the constitutive relationship of rock materials from different perspectives, but the mechanical concept of the latter is clearer. Therefore, in this research, the latter was used to establish the damage constitutive relationship of rock.According to Lemaitre’s equivalent strain hypothesis, the following results were obtained [17]:(3)σ∗=σ1−D=Eε1−D,where σ is the stress tensor, σ∗ denotes the effective stress tensor, ε is the strain tensor, E is the elastic modulus matrix, and D is the damage matrix.Rock has a certain residual strength after failure. Considering the influence of rock residual strength, the damage correction coefficientδ was used to modify equation (3) [18]:(4)σ∗=σ1−δD=Eε1−δD.The key to establishing the damage constitutive relationship of rock is to determine the damage variable. In previous studies, the strength of rock microelements is usually expressed by rock axial principal strain, but this method cannot reflect the influence of the complex stress state on the strength of rock microelements; therefore, considering the failure model and criterion of rocks, others have proposed a new expression of damage variables reflecting the strength of rock microelements [19–22]. It is assumed that the general failure criterion for rocks is(5)fσ∗−k0=0,where k0 is a constant related to the cohesion and internal friction angle of rock.f
σ
∗ reflects the criticality of rock microelement failure and can be used as the strength of a rock microelement. Assuming that the probability of rock microelement failure is Ffσ∗, which is related to the distribution density of fσ∗, the damage variable D is defined as follows:(6)D=∫0fσ∗Fxdx.Formula (6) shows that the key to establishing the damage model for the entire process of rock deformation and failure lies in the determination of rock microelement strength and its failure probability.


## 2.3. Determination of Rock Microelement Strength

At present, there are many forms of failure criteria that can be used in rock mechanics research. Among them, the Drucker–Prager criterion has a simple parametric form that is applicable to rock media and is thus widely used. Therefore, the strength of a rock microelement based on the Drucker–Prager criterion was determined as follows [12, 16]:(7)fσ=k=α0I1+J2,(8)α0=sinφ9+3sin2φ,where φ is the internal friction angle; I1 represents the first invariant of the stress tensor; and J2 is the second invariant of the deviatoric stress tensor. The expressions of I1 and J2 are as follows:(9)I1=σ1∗+σ2∗+σ3∗,(10)J2=16σ1∗−σ2∗2+σ2∗−σ3∗2+σ3∗−σ1∗2.In conventional triaxial compression tests on rock specimens, nominal stressesσ1, σ2, and σ3 (σ2 = σ3) and strain ε1 can be measured, and the corresponding effective stresses are σ1∗, σ2∗, and σ3∗ (σ2∗=σ3∗). The formulae for calculating the effective stress are as follows:(11)σ1∗=σ11−δD,(12)σ2∗=σ3∗=σ31−δD.The elastic modulus and Poisson’s ratio of rock areE and μ, respectively. According to Hooke’s law, the following result was obtained:(13)ε1=σ1∗−2μσ3∗E.Formulae (11)–(13) were substituted into formulae (9) and (10), and the following results were obtained:(14)I1=σ1+2σ3Eε1σ1−2μσ3,J2=σ1−σ3Eε13σ1−2μσ3.By substituting formula (14) into formula (7), the strength of rock microelements was acquired as follows:(15)fσ=Eε1σ1−2μσ3α0+33σ1+2α0−33σ3.


## 2.4. Damage Model of Rock Based on Weibull Statistical Theory

Assuming that the strengthk of rock microelements obeys a Weibull distribution, the probability density function of rocks can be obtained from the Weibull statistical model introduced earlier as follows [16]:(16)fk=mFkFm−1exp−kFm.where k is the distribution variable of the Weibull distribution and m and F are Weibull distribution parameters. By substituting formula (16) into formula (6), the damage variable D was calculated as follows:(17)D=1−exp−kFm.There is a threshold point (σD, εD) in the damage evolution of rock materials. When the stress is less than the threshold point stress σD, the damage of rocks is very small or no damage occurs. At this time, the damage variable can be considered to be zero and(18)D=0,σ1<σD.When the stress reaches or exceeds the threshold point stressσD, the damage variable can be calculated according to equation (17). The damage variable expression of rock-like materials under arbitrary stress state can be expressed as follows:(19)D=0,σ1<σD,1−exp−kFm,σ1≥σD.By substituting formulae (11), (12), and (17) into formula (13), the damage constitutive equation of rocks based on the Weibull distribution (after the damage threshold point, i.e., σ1≥σD) was obtained as follows:(20)σ1=Eε11−δ+δexp−kFm−2μσ3,σ1≥σD,where the parameter m reflects the shape of the function; the parameter F reflects the scale characteristics of the function and corresponds to the homogeneity and average strength of rock materials; and parameter δ reflects the residual strength of rocks.Before reaching the damage threshold, i.e.,σ1<σD, the constitutive relation of rock materials can be obtained by fitting the polynomial function relation. Considering that the strain of rocks is zero when the stress is zero, the relationship can be expressed as follows:(21)σ1=Aε1ε1+B,σ1<σD,where A and B are coefficients.By combining formulae (20) and (21), a rock damage model based on Weibull statistical theory under an arbitrary stress state was obtained as follows:(22)σ1=Aε1ε1+B,σ1<σD,Eε11−δ+δexp−kFm−2μσ3,σ1≥σD.


## 3. Thermal Damage Model of Rock

### 3.1. Thermal Damage Evolution and Constitutive Equations

The room temperature is set to 25°C. When the initial damage to rock materials is neglected, no thermal damage occurs at room temperature. If the elastic modulus is used to define the thermal damage variableDT, the following result can be obtained [23–30]:(23)DT=1−ETE0,where ET and E0 are the elastic modulus at temperature T and 25°C, respectively.Weibull parameters of rocks at different temperatures are(24)AT=A01−DT,(25)BT=B01−DT,(26)mT=m01−DT,(27)FT=F01−DT,where A0, B0, m0, and F0 are the Weibull function parameters of rock materials at room temperature.By substituting equation (7) into the damage evolution equation (19) of rock, the following results were obtained:(28)Dσ,T=0,σ1<σD,1−exp−α0I1+J2FTmT,σ1≥σD.The macroscopic elastic modulus changes obviously after the rock has been affected by a higher temperature. At this time, the change in elastic modulus can be considered to express the thermal damage effect of rocks. By substituting formula (23) into formula (28), the damage evolution equation of rock materials at different temperatures could be obtained as follows:(29)Dσ,T=0,σ1<σD,1−exp−α0I1+J2F0ET/E0m0ET/E0,σ1≥σD.From equation (23), the elastic modulus of rocks at different temperatures was obtained as follows:(30)E=ET=ET=E01−DT.By substituting equation (30) into equation (20), the thermal damage constitutive equation of rock materials based on conventional triaxial compression tests (after the damage threshold) could be obtained as follows:(31)σ1=E01−DTε11−δ+δexp−α0I1+J2FTmT+2μσ3.By combining the constitutive equations before and after the damage threshold, i.e., formula (21) and formula (31), the thermal damage constitutive equation of rock materials based on conventional triaxial compression tests could be obtained as follows:(32)σ1=ATε1ε1+BT,0≤ε≤εD,ETε11−δ+δexp−α0I1+J2FTmT+2μσ3,ε>εD.


### 3.2. Calculation of ParametersAT, BT, mT, and FT

The values of parametersAT, BT, mT, and FT can be calculated by means of the stress and strain (σD, εD) at the damage threshold point and those (σc, εc) at the peak point on the rock stress-strain curve as measured by conventional triaxial compression tests. At the peak, the stress and strain satisfy the constitutive equation, and the first derivative of the stress-strain curve is zero at that point. The following results were obtained:(33)σε=εc=σc,∂σ∂εε=εc=0.RegardingmT and FT as unknown, the above equations were solved, and the formulae for calculating parameters mT and FT were obtained as follows:(34)mT=−fcσc−2μσ3/Eεc−α0+3/3Eε−10α0+53/3Eεc⋅α0σc+2σ3+3/3σc−σ3/σc−2μσ3⋅lnσc−2μσ3+δ−1Eεc/Eεcδ⋅1σc−2μσ3/Eεc+δ−1,(35)FT=fclnσc−2μσ3+δ−1Eεc/Eεcδ1/mT,where fc=Eεcα0σc+2σ3−3/3σc−σ3/σc−2μσ3The stress-strain curves of rocks are continuous at the threshold points and(36)ATε12+ATBTε1=Eε11−δ+δexp−fFTmT+2μσ3.The first derivative of the rock stress-strain curves is continuous at the threshold point and(37)∂σ1∂ε1ε1=εD=2Aε1+AB.By combining formulae (36) and (37), we obtain(38)AT=dεD−2μσ3εD2,(39)BT=cεD+dεD2dεD−2μσ3−2εD,where c=EεD1−δ+δexp−fD/Fm,(40)d=EεDδ⋅exp−fDFm⋅−mfDm−1Fm⋅α02E2εD+2σ3EεD−2μσ3−EεD+2σ3E2εDEεD−2μσ32+2E2εD−Eσ33EεD−2μσ3−3E2εDEεD−σ33EεD−2μσ32,fD=EεDα0σD+2σ3−3/3σD−σ3σD−2μσ3.


### 3.3. Physical Meaning of Parameters

Through mathematical analysis of the equation, the physical meaning of parametersm, F, A, B, and δ is obtained: m reflects the morphological characteristics of the stress-strain curve, and the smaller m is, the shallower the stress-strain curve is and the tougher the rock is; F represents the average strength of rocks, and when F increases, the peak strength of rocks increases accordingly; A represents the concavity in the compacting stage, and the larger A is, the more notable the upwards concavity is; B is the initial elastic modulus of rocks; and δ reflects the residual strength thereof (the larger δ is, the lower the residual strength is).


## 3.1. Thermal Damage Evolution and Constitutive Equations

The room temperature is set to 25°C. When the initial damage to rock materials is neglected, no thermal damage occurs at room temperature. If the elastic modulus is used to define the thermal damage variableDT, the following result can be obtained [23–30]:(23)DT=1−ETE0,where ET and E0 are the elastic modulus at temperature T and 25°C, respectively.Weibull parameters of rocks at different temperatures are(24)AT=A01−DT,(25)BT=B01−DT,(26)mT=m01−DT,(27)FT=F01−DT,where A0, B0, m0, and F0 are the Weibull function parameters of rock materials at room temperature.By substituting equation (7) into the damage evolution equation (19) of rock, the following results were obtained:(28)Dσ,T=0,σ1<σD,1−exp−α0I1+J2FTmT,σ1≥σD.The macroscopic elastic modulus changes obviously after the rock has been affected by a higher temperature. At this time, the change in elastic modulus can be considered to express the thermal damage effect of rocks. By substituting formula (23) into formula (28), the damage evolution equation of rock materials at different temperatures could be obtained as follows:(29)Dσ,T=0,σ1<σD,1−exp−α0I1+J2F0ET/E0m0ET/E0,σ1≥σD.From equation (23), the elastic modulus of rocks at different temperatures was obtained as follows:(30)E=ET=ET=E01−DT.By substituting equation (30) into equation (20), the thermal damage constitutive equation of rock materials based on conventional triaxial compression tests (after the damage threshold) could be obtained as follows:(31)σ1=E01−DTε11−δ+δexp−α0I1+J2FTmT+2μσ3.By combining the constitutive equations before and after the damage threshold, i.e., formula (21) and formula (31), the thermal damage constitutive equation of rock materials based on conventional triaxial compression tests could be obtained as follows:(32)σ1=ATε1ε1+BT,0≤ε≤εD,ETε11−δ+δexp−α0I1+J2FTmT+2μσ3,ε>εD.


## 3.2. Calculation of ParametersAT, BT, mT, and FT

The values of parametersAT, BT, mT, and FT can be calculated by means of the stress and strain (σD, εD) at the damage threshold point and those (σc, εc) at the peak point on the rock stress-strain curve as measured by conventional triaxial compression tests. At the peak, the stress and strain satisfy the constitutive equation, and the first derivative of the stress-strain curve is zero at that point. The following results were obtained:(33)σε=εc=σc,∂σ∂εε=εc=0.RegardingmT and FT as unknown, the above equations were solved, and the formulae for calculating parameters mT and FT were obtained as follows:(34)mT=−fcσc−2μσ3/Eεc−α0+3/3Eε−10α0+53/3Eεc⋅α0σc+2σ3+3/3σc−σ3/σc−2μσ3⋅lnσc−2μσ3+δ−1Eεc/Eεcδ⋅1σc−2μσ3/Eεc+δ−1,(35)FT=fclnσc−2μσ3+δ−1Eεc/Eεcδ1/mT,where fc=Eεcα0σc+2σ3−3/3σc−σ3/σc−2μσ3The stress-strain curves of rocks are continuous at the threshold points and(36)ATε12+ATBTε1=Eε11−δ+δexp−fFTmT+2μσ3.The first derivative of the rock stress-strain curves is continuous at the threshold point and(37)∂σ1∂ε1ε1=εD=2Aε1+AB.By combining formulae (36) and (37), we obtain(38)AT=dεD−2μσ3εD2,(39)BT=cεD+dεD2dεD−2μσ3−2εD,where c=EεD1−δ+δexp−fD/Fm,(40)d=EεDδ⋅exp−fDFm⋅−mfDm−1Fm⋅α02E2εD+2σ3EεD−2μσ3−EεD+2σ3E2εDEεD−2μσ32+2E2εD−Eσ33EεD−2μσ3−3E2εDEεD−σ33EεD−2μσ32,fD=EεDα0σD+2σ3−3/3σD−σ3σD−2μσ3.


## 3.3. Physical Meaning of Parameters

Through mathematical analysis of the equation, the physical meaning of parametersm, F, A, B, and δ is obtained: m reflects the morphological characteristics of the stress-strain curve, and the smaller m is, the shallower the stress-strain curve is and the tougher the rock is; F represents the average strength of rocks, and when F increases, the peak strength of rocks increases accordingly; A represents the concavity in the compacting stage, and the larger A is, the more notable the upwards concavity is; B is the initial elastic modulus of rocks; and δ reflects the residual strength thereof (the larger δ is, the lower the residual strength is).


## 4. Thermal Damage Tests on Oil Shale Specimens

The oil shale used in this study was sampled from Yong’an Township, Nong’an County, Jilin Province, China (Figure1). The rocks are gray-black, the horizontal bedding planes are well developed, the rocks contain a large number of crustal fossils, and the density of rocks ranges from 1.92 to 2.11 g/cm3. To simulate the heating of oil shale during in situ thermal processing more accurately, thermogravimetric analysis was carried out to determine the temperature range required for heating, and then the oil shale specimens were heated by special equipment [31].Figure 1
Location map of the sampling site in China.


### 4.1. Thermogravimetric Analysis of Oil Shale

The thermogravimetric curve (Figure2) of oil shale specimens can be measured by a thermogravimetric analyzer (Figure 3). The thermogravimetric curve was used to analyze the possible temperature range of oil shale during in situ thermal processing, and the temperature range of oil shale thermal damage tests in the present work was determined accordingly. Through analysis of Figure 2, the following conclusions can be drawn: the pyrolysis of kerogen, which produces shale oil (gas) and combustible gas, occurs between 400°C and 500°C. If the kerogen is to be fully pyrolyzed, only the oil shale is heated to the upper temperature bounding the high-temperature weight-loss stage; therefore, in this experiment, the temperature range from room temperature to 700°C is chosen to study thermal damage to oil shale specimens. The possible temperatures of oil shale in all in situ thermal processing modes are basically in this range (room temperature to 700°C) thus ensuring that the research results are representative.Figure 2
Thermogravimetric curve.Figure 3
Thermogravimetric analyzer.


### 4.2. Heating Tests on Oil Shale Specimens

According to prevailing requirements, the oil shale blocks were processed into a cylinder with a diameter of 50 mm and a height of 100 mm perpendicular to the bedding direction (Figure4). They were heated by an MXG1200-80 tubular furnace. The experiment was carried out in eight groups. The heating temperatures of the groups were set to 25°C (room temperature), 100°C, 200°C, 300°C, 400°C, 500°C, 600°C, and 700°C, respectively. To ensure uniformity of heating, heating was carried out at a rate of 5°C/min. After reaching the predetermined temperature, the temperature was maintained for 4 h, and then, specimens were naturally cooled to room temperature in the furnace tube. The heated oil shale specimens are as shown in Figure 5, and they were used in the following conventional triaxial compression tests.Figure 4
Typical oil shale specimens.Figure 5
Heated (700°C) specimens.


### 4.3. Conventional Triaxial Compression Tests on Thermally Damaged Oil Shale Specimens

For thermally damaged oil shale after heating, the conventional triaxial compression tests were carried out with a TAW-2000 triaxial rock-testing machine. Load control mode, a loading rate of 0.2 kN/s, and a confining pressure of 5 MPa were adopted; the axial deformations of the specimens were measured by a 10 mm extensometer, and the radial deformations were measured by a 5 mm extensometer. The axial force was measured by a pressure sensor with a range of 2000 kN, while confining pressure was measured by a pressure sensor with a range of 100 MPa. Three specimens were selected for parallel tests at the same temperature. Figure6 shows the triaxial rock-testing machine selected for these tests, and Figure 7 shows an oil shale specimen after compressive failure.Figure 6
TAW-2000 triaxial rock-testing machine.Figure 7
Compressive failure.Figure8 shows the stress-strain curves of eight groups of oil shale specimens in conventional triaxial compression tests (σ3 = 5 MPa).Figure 8
Stress-strain curves.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Through the analysis of Figure8, it can be found that the peak strength and slope of the elastic stage decrease with the increase of heating temperature. In addition, the compacting stage of the stress-strain curve becomes longer than that before 400°C when the temperature exceeds 400°C. The results indicate that the thermal damage in oil shale specimens increases with increasing temperature.


## 4.1. Thermogravimetric Analysis of Oil Shale

The thermogravimetric curve (Figure2) of oil shale specimens can be measured by a thermogravimetric analyzer (Figure 3). The thermogravimetric curve was used to analyze the possible temperature range of oil shale during in situ thermal processing, and the temperature range of oil shale thermal damage tests in the present work was determined accordingly. Through analysis of Figure 2, the following conclusions can be drawn: the pyrolysis of kerogen, which produces shale oil (gas) and combustible gas, occurs between 400°C and 500°C. If the kerogen is to be fully pyrolyzed, only the oil shale is heated to the upper temperature bounding the high-temperature weight-loss stage; therefore, in this experiment, the temperature range from room temperature to 700°C is chosen to study thermal damage to oil shale specimens. The possible temperatures of oil shale in all in situ thermal processing modes are basically in this range (room temperature to 700°C) thus ensuring that the research results are representative.Figure 2
Thermogravimetric curve.Figure 3
Thermogravimetric analyzer.


## 4.2. Heating Tests on Oil Shale Specimens

According to prevailing requirements, the oil shale blocks were processed into a cylinder with a diameter of 50 mm and a height of 100 mm perpendicular to the bedding direction (Figure4). They were heated by an MXG1200-80 tubular furnace. The experiment was carried out in eight groups. The heating temperatures of the groups were set to 25°C (room temperature), 100°C, 200°C, 300°C, 400°C, 500°C, 600°C, and 700°C, respectively. To ensure uniformity of heating, heating was carried out at a rate of 5°C/min. After reaching the predetermined temperature, the temperature was maintained for 4 h, and then, specimens were naturally cooled to room temperature in the furnace tube. The heated oil shale specimens are as shown in Figure 5, and they were used in the following conventional triaxial compression tests.Figure 4
Typical oil shale specimens.Figure 5
Heated (700°C) specimens.


## 4.3. Conventional Triaxial Compression Tests on Thermally Damaged Oil Shale Specimens

For thermally damaged oil shale after heating, the conventional triaxial compression tests were carried out with a TAW-2000 triaxial rock-testing machine. Load control mode, a loading rate of 0.2 kN/s, and a confining pressure of 5 MPa were adopted; the axial deformations of the specimens were measured by a 10 mm extensometer, and the radial deformations were measured by a 5 mm extensometer. The axial force was measured by a pressure sensor with a range of 2000 kN, while confining pressure was measured by a pressure sensor with a range of 100 MPa. Three specimens were selected for parallel tests at the same temperature. Figure6 shows the triaxial rock-testing machine selected for these tests, and Figure 7 shows an oil shale specimen after compressive failure.Figure 6
TAW-2000 triaxial rock-testing machine.Figure 7
Compressive failure.Figure8 shows the stress-strain curves of eight groups of oil shale specimens in conventional triaxial compression tests (σ3 = 5 MPa).Figure 8
Stress-strain curves.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Through the analysis of Figure8, it can be found that the peak strength and slope of the elastic stage decrease with the increase of heating temperature. In addition, the compacting stage of the stress-strain curve becomes longer than that before 400°C when the temperature exceeds 400°C. The results indicate that the thermal damage in oil shale specimens increases with increasing temperature.


## 5. Establishment and Verification of the Thermal Damage Constitutive Model for Oil Shale

### 5.1. Determination of Elastic Moduli, Poisson’s Ratios, and Thermal Damage Variables

According to the stress-strain curves of oil shale specimens at different temperatures (Figure7), the elastic moduli of each specimen were calculated by choosing the elastic stage before the peak strength, and then Poisson’s ratios of oil shale specimens were calculated by the ratio of radial strain to axial strain (Table 1).Table 1
Elastic moduli and Poisson’s ratios of oil shale specimens at different temperatures.
Temperature (°C)
Elastic modulus (GPa)
Poisson’s ratio
1
2
3
Average
25
18.649
17.375
18.932
18.319
0.460
100
11.992
12.400
13.973
12.788
0.469
200
12.319
12.856
13.044
12.740
0.434
300
10.585
11.004
11.134
10.908
0.427
400
9.795
9.161
9.601
9.519
0.276
500
8.800
8.407
8.885
8.697
0.272
600
8.333
7.644
8.658
8.212
0.195
700
7.181
6.785
6.556
6.841
0.163Based on the elastic moduli in Table1, the thermal damage variable DT of oil shale specimens at different temperatures was calculated by using equation (19). The results are listed in Table 2.Table 2
Thermal damage variables of oil shale specimens at different temperatures.
Temperature (°C)
25
100
200
300
400
500
600
700
D
T
0
0.302
0.305
0.405
0.480
0.525
0.546
0.636


### 5.2. Determination of Model ParametersAT, BT, mT, and FT

According to the conventional triaxial compression test results, the specific values ofmT and FT at different temperatures were calculated by using formulae (34) and (35), respectively. Among them, the value of δ is 0.95, the stress and strain at the peak point (σc, εc) of the stress-strain curve are the average of stress and strain at the peak point of three specimens, the internal friction angle is 30°, and the calculated value of α0 is 0.16. The calculated values of parameters AT, BT, mT, and FT at different temperatures are as listed in Table 3. The variations of constitutive equation parameters AT, BT, mT, and FT with temperature are shown in Figure 9.Table 3
Constitutive model parameters at different temperatures.
Temperature (°C)
25
100
200
300
400
500
700
A
T (×106)
−17.86
−13.38
−8.65
−7.07
−4.47
−6.11
−0.23
B
T (×10−3)
−1.61
−1.66
−2.18
−2.31
−2.60
−2.20
−9.06
m
T
3.33
6.43
4.95
6.59
7.09
6.68
5.42
F
T
39.42
44.54
47.62
52.37
51.00
44.83
39.71Figure 9
Variations of constitutive equation parameters with temperature.By fitting the data in Table3, the mathematical relationships between model parameters AT, BT, mT, and FT and temperature T were obtained:(41)AT=−21.9×107×exp−T343.735+2.8×106,R2=0.989,BT=−3.285×10−4×expT217.9−1.128×10−3,R2=0.977,mT=−3.141×10−8T3+1.207×10−5T2+0.0103T+2.962,R2=0.95,FT=1.26×10−7T3−2.4×10−4T2+0.11T+36.2,R2=0.85.


### 5.3. Thermal Damage Constitutive Model for Oil Shale

The values of, or mathematical expressions for, the parameters required in the model have been determined, and the thermal damage constitutive equation of oil shale can be obtained as follows:(42)σ1=ATε1ε1+BT,0≤ε≤εD,ETε11−δ+δexp−α0I1+J2FTmT+2μσ3,ε>εD,where AT=−2.28×107×exp−T/353.75+3.36×106; BT=−3.48×10−4×expT/192.4−8.19×10−4; mT=1.946×10−5T2+0.00757T+4.251; FT=2.4×10−7T3−3.95×10−4T2+0.134T+33.5; I1 is the first invariant of the stress tensor, I1=σ1+2σ3Eε1/σ1−2μσ3; J2 is the second invariant of the deviatoric stress tensor; J2=σ1−σ3Eε1/3σ1−2μσ3; α0=sinφ/9+3sin2φ; φ is the internal friction angle; and δ reflects the residual strength (the larger δ is, the lower the residual strength is).


### 5.4. Verification of the Thermal Damage Constitutive Model for Oil Shale

The theoretical stress-strain curve for oil shale at 600°C was calculated by the constitutive equation established above, and the theoretical curve was compared with the measured curves to verify the correctness of the model.It can be found from Figure10 that the theoretical curve is in good agreement with the measured curves, especially before the peak point. The hypothesis and factors deemed negligible in the process of establishing the governing equation and parameter determination have little influence on the correctness of the equation. The constitutive model accurately reflects the deformation characteristics of oil shale at high temperatures.Figure 10
Comparison of theoretical and measured stress-strain curves at 600°C.


## 5.1. Determination of Elastic Moduli, Poisson’s Ratios, and Thermal Damage Variables

According to the stress-strain curves of oil shale specimens at different temperatures (Figure7), the elastic moduli of each specimen were calculated by choosing the elastic stage before the peak strength, and then Poisson’s ratios of oil shale specimens were calculated by the ratio of radial strain to axial strain (Table 1).Table 1
Elastic moduli and Poisson’s ratios of oil shale specimens at different temperatures.
Temperature (°C)
Elastic modulus (GPa)
Poisson’s ratio
1
2
3
Average
25
18.649
17.375
18.932
18.319
0.460
100
11.992
12.400
13.973
12.788
0.469
200
12.319
12.856
13.044
12.740
0.434
300
10.585
11.004
11.134
10.908
0.427
400
9.795
9.161
9.601
9.519
0.276
500
8.800
8.407
8.885
8.697
0.272
600
8.333
7.644
8.658
8.212
0.195
700
7.181
6.785
6.556
6.841
0.163Based on the elastic moduli in Table1, the thermal damage variable DT of oil shale specimens at different temperatures was calculated by using equation (19). The results are listed in Table 2.Table 2
Thermal damage variables of oil shale specimens at different temperatures.
Temperature (°C)
25
100
200
300
400
500
600
700
D
T
0
0.302
0.305
0.405
0.480
0.525
0.546
0.636


## 5.2. Determination of Model ParametersAT, BT, mT, and FT

According to the conventional triaxial compression test results, the specific values ofmT and FT at different temperatures were calculated by using formulae (34) and (35), respectively. Among them, the value of δ is 0.95, the stress and strain at the peak point (σc, εc) of the stress-strain curve are the average of stress and strain at the peak point of three specimens, the internal friction angle is 30°, and the calculated value of α0 is 0.16. The calculated values of parameters AT, BT, mT, and FT at different temperatures are as listed in Table 3. The variations of constitutive equation parameters AT, BT, mT, and FT with temperature are shown in Figure 9.Table 3
Constitutive model parameters at different temperatures.
Temperature (°C)
25
100
200
300
400
500
700
A
T (×106)
−17.86
−13.38
−8.65
−7.07
−4.47
−6.11
−0.23
B
T (×10−3)
−1.61
−1.66
−2.18
−2.31
−2.60
−2.20
−9.06
m
T
3.33
6.43
4.95
6.59
7.09
6.68
5.42
F
T
39.42
44.54
47.62
52.37
51.00
44.83
39.71Figure 9
Variations of constitutive equation parameters with temperature.By fitting the data in Table3, the mathematical relationships between model parameters AT, BT, mT, and FT and temperature T were obtained:(41)AT=−21.9×107×exp−T343.735+2.8×106,R2=0.989,BT=−3.285×10−4×expT217.9−1.128×10−3,R2=0.977,mT=−3.141×10−8T3+1.207×10−5T2+0.0103T+2.962,R2=0.95,FT=1.26×10−7T3−2.4×10−4T2+0.11T+36.2,R2=0.85.


## 5.3. Thermal Damage Constitutive Model for Oil Shale

The values of, or mathematical expressions for, the parameters required in the model have been determined, and the thermal damage constitutive equation of oil shale can be obtained as follows:(42)σ1=ATε1ε1+BT,0≤ε≤εD,ETε11−δ+δexp−α0I1+J2FTmT+2μσ3,ε>εD,where AT=−2.28×107×exp−T/353.75+3.36×106; BT=−3.48×10−4×expT/192.4−8.19×10−4; mT=1.946×10−5T2+0.00757T+4.251; FT=2.4×10−7T3−3.95×10−4T2+0.134T+33.5; I1 is the first invariant of the stress tensor, I1=σ1+2σ3Eε1/σ1−2μσ3; J2 is the second invariant of the deviatoric stress tensor; J2=σ1−σ3Eε1/3σ1−2μσ3; α0=sinφ/9+3sin2φ; φ is the internal friction angle; and δ reflects the residual strength (the larger δ is, the lower the residual strength is).


## 5.4. Verification of the Thermal Damage Constitutive Model for Oil Shale

The theoretical stress-strain curve for oil shale at 600°C was calculated by the constitutive equation established above, and the theoretical curve was compared with the measured curves to verify the correctness of the model.It can be found from Figure10 that the theoretical curve is in good agreement with the measured curves, especially before the peak point. The hypothesis and factors deemed negligible in the process of establishing the governing equation and parameter determination have little influence on the correctness of the equation. The constitutive model accurately reflects the deformation characteristics of oil shale at high temperatures.Figure 10
Comparison of theoretical and measured stress-strain curves at 600°C.


## 6. Conclusions

(1)
Assuming that the strength of thermally damaged rock microelements obeys a Weibull distribution, considering the influence of temperature on rock mechanical parameters, defining the thermal damage variable by elastic modulus, and under the condition that microelement failure conforms to the Drucker–Prager criterion, the statistical thermal damage constitutive model of rock after high-temperature exposure was established. The constitutive model is suitable for typical rocks including oil shale, and the effect of temperature on the stress-strain relationship is considered. The mathematical expressions of parametersAT, BT, mT, and FT in the model do not include the unconventional rock mechanics parameters, and the specific values can be determined according to the conventional triaxial compression test results.
(2)
The pyrolysis of kerogen in oil shale occurs mainly in the temperature range from room temperature to 700°C, of which 400–500°C is the most concentrated. From room temperature to 700°C, with the increase of temperature, the elastic modulus of oil shale decreases gradually from 18.319 GPa to 6.841 GPa, which is a decrease of 62.66%; Poisson’s ratio of oil shale decreases gradually in the whole from 0.460 to 0.163, which is a decrease of 64.57%. With the increase of temperature, the thermal damage of oil shale is aggravated, and the damage variable increases continuously. At 700°C, the damage variable is the largest, reaching 0.636.
(3)
According to the results of the triaxial compression test of oil shale, the values of model parameters in the statistical thermal damage constitutive model of rock at different temperatures were calculated, the relationship between model parameters and temperatureT was obtained by a mathematical fitting method, and the thermal damage constitutive model for oil shale was established. The parameters AT, BT, mT, and FT of the model are no longer related to the rock mechanics parameters but only to the temperature T, which is convenient for engineering application.
(4)
Comparing the theoretical stress-strain curve calculated by the model with those measured by conventional triaxial tests at 600°C, the theoretical strain values are slightly smaller than the measured strain values, but the difference between them is not large; the maximum difference before the peak strength is only 5 × 10−4, and the difference does not increase with the increase of strain. This result shows that the theoretical results are in good agreement with the conventional test results and that the model accurately reflects the deformation characteristics of oil shale at high temperature.



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*Source: 4932586-2019-10-13.xml*