Mathematical Problems in Engineering

Volume 2019, Article ID 4932586, 11 pages

https://doi.org/10.1155/2019/4932586

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

Correspondence should be addressed to Chen Chen; nc.ude.ulj@nehcnehc

Received 2 July 2019; Accepted 23 September 2019; Published 13 October 2019

Academic Editor: Pietro Bia

Copyright © 2019 Guijie Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 and *in 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 temperature *T* 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:

Its cumulative probability distribution function iswhere *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]:where is the stress tensor, denotes the effective stress tensor, is the strain tensor, is the elastic modulus matrix, and is the damage matrix.

Rock has a certain residual strength after failure. Considering the influence of rock residual strength, the damage correction coefficient *δ* was used to modify equation (3) [18]:

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 iswhere *k*_{0} is a constant related to the cohesion and internal friction angle of rock.

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 , which is related to the distribution density of , the damage variable *D* is defined as follows:

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]:where is the internal friction angle; represents the first invariant of the stress tensor; and is the second invariant of the deviatoric stress tensor. The expressions of and are as follows:

In conventional triaxial compression tests on rock specimens, nominal stresses *σ*_{1}, *σ*_{2}, and *σ*_{3} (*σ*_{2} = *σ*_{3}) and strain *ε*_{1} can be measured, and the corresponding effective stresses are , , and (). The formulae for calculating the effective stress are as follows:

The elastic modulus and Poisson’s ratio of rock are *E* and *μ*, respectively. According to Hooke’s law, the following result was obtained:

Formulae (11)–(13) were substituted into formulae (9) and (10), and the following results were obtained:

By substituting formula (14) into formula (7), the strength of rock microelements was acquired as follows:

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

Assuming that the strength *k* 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]: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:

There is a threshold point (*σ*_{D}, *ε*_{D}) in the damage evolution of rock materials. When the stress is less than the threshold point stress *σ*_{D}, the damage of rocks is very small or no damage occurs. At this time, the damage variable can be considered to be zero and

When the stress reaches or exceeds the threshold point stress *σ*_{D}, 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:

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., ) was obtained as follows: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., , 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: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:

#### 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 variable *D*_{T}, the following result can be obtained [23–30]:where *E*_{T} and *E*_{0} are the elastic modulus at temperature *T* and 25°C, respectively.

Weibull parameters of rocks at different temperatures arewhere *A*_{0}, *B*_{0}, *m*_{0}, and *F*_{0} 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:

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:

From equation (23), the elastic modulus of rocks at different temperatures was obtained as follows:

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:

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:

##### 3.2. Calculation of Parameters *A*_{T}, *B*_{T}, *m*_{T}, and *F*_{T}

The values of parameters *A*_{T}, *B*_{T}, *m*_{T}, and *F*_{T} can be calculated by means of the stress and strain (*σ*_{D}, *ε*_{D}) at the damage threshold point and those (*σ*_{c}, *ε*_{c}) 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:

Regarding *m*_{T} and *F*_{T} as unknown, the above equations were solved, and the formulae for calculating parameters *m*_{T} and *F*_{T} were obtained as follows:where

The stress-strain curves of rocks are continuous at the threshold points and

The first derivative of the rock stress-strain curves is continuous at the threshold point and

By combining formulae (36) and (37), we obtainwhere ,

##### 3.3. Physical Meaning of Parameters

Through mathematical analysis of the equation, the physical meaning of parameters *m*, *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 (Figure 1). 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/cm^{3}. 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].