# Comparative Analyses Concerning Triaxial Compressive Yield Criteria of Coal with the Presence of Pore Water

Authors: Rulin Liu; Yanbin Yu; Weimin Cheng; Qingfeng Xu; Haotian Yang; Jialong Shen
Journal: Geofluids
 (2020)

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## Abstract

The least absolute deviation is used as a metric to analyze the applicability of five yield criteria, to describe the yield characteristics of coal based on triaxial compressive strength tests on natural, water-saturated, and seepage coal samples with the presence of pore water. The results show that the strength of coal exhibits nonlinear characteristics with the increase of confining pressure, which the linear Coulomb criterion fails to authentically describe. Although the parabolic Mohr criterion can describe the nonlinearity feature more decently than the linear yield criterion, the fitting error is significant, and the uniaxial compressive strength of coal is overestimated. The Hoek-Brown criterion, quadratic polynomial criterion, and exponential criterion yield decent fitting quality for the coal rock. In particular, the exponential strength criterion can accurately reflect the actual uniaxial compressive strength of the rock. However, the differential principle yield stress for an infinite confining pressure calculated from the exponential strength criterion is lower than the measured value. Furthermore, by employing effective stress principle to analyze the yield criteria for the saturated and seepage coal samples, one can find that the quadratic polynomial criterion and the exponential criterion can also reflect the changes of yield characteristics during the fluid-solid coupling triaxial compression test.


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## Body

## 1. Introduction

In the subsurface mining engineering, hydroelectric power generation, petroleum, and geological engineering applications [1, 2], how to calculate and predict the rock strength is of great significance to the researchers, e.g., engineering designers. The accuracy of the calculation here directly dictates the quality and safety of engineering construction, and then affects the safety of workers. However, the rock strength is closely related to the geological environment. The mechanical properties of the rock also differ in response to the change of rock stress and geological environment. The test of rock strength under various stress states has been a hot research topic for scholars. With the advancement of experimental techniques for testing rock mechanics, the corresponding rock strength evaluation criteria have also been subject to continuous corrections and improvements.Since its advent in 1776, the Coulomb yield criterion has become one of the most widely used yield criteria in geotechnical engineering [3]. In 1900, Mohr extended the Coulomb shear strength criterion to account for the three-dimensional stress state, leading to the M-C criterion [4, 5]. According to the behaviors of fracturing and rock failure and by taking into consideration the triaxial testing data and the rock mass in situ measurement results, Hoek and Brown put forward in 1980 the Hoek-Brown empirical yield criterion [6], which, based on the experimental validation and practical improvements, has been widely accepted by the rock mechanics and engineering community. In addition, many scholars have extended the yield criteria based on experimental measurement and theoretical analyses, giving rise to numerous new criteria, e.g., Drucker-Prager failure criterion [7], Griffith criterion and its associated Murrell three-dimensional extension [8], Mogi yield criterion [9], power function-based yield criterion [10–12], exponential yield criterion [13, 14], and unified strength theory [15].With the continuous application and development of yield criterion in engineering practice, scholars have carried out experiments and analyses on the yield criteria under various stress paths based on different rock types, with a plethora of results published. Among them, Zhao [16] establishes a dynamic Mohr-Coulomb criterion and a dynamic Hoek-Brown criterion for a low loading rate through a series of experiments, e.g., uniaxial compression and triaxial compression. Based on the linear Mogi criterion, Al-Ajmi and Zimmerman [17, 18] established the Mogi-Coulomb criterion by analyzing the relationship between the fitting parameters and the antishearing strength parameters based on linear Mogi criterion. Costamagna and Bruhns [19] establish a nonlinear strength criterion with four complex parameters, which can be transformed into multiple variants based on different selections of parameters. You [20, 21] thoroughly analyze the stress-strain curves of defective rock samples, based on which the impact of confining pressure on the triaxial strength of rock samples is determined, and the yield criteria associated with various rock types are systematically analyzed. Although significant achievements have been made concerning the yield criteria, these criteria are all subject to certain practical limitations with a lack of universal applicability. Therefore, systematic studies are still required for developing yield criteria under different conditions.Many engineering practices have indicated that the rock is not only affected by the stress field in the Earth’s crust but also affected by the presence of water to certain degrees. In some cases, even high osmotic pressure exists in the rock, leading to a complicated water environment that affects the rock. In order to explore the influence of groundwater on rock strength, Ojo and Brook [22] summarize the relevant research results and conclude that the existence of water tends to weaken rock strength, with the increase of humidity compromising the compressive and tensile strengths of rock. Vásárhelyi and Ván [23, 24] test the uniaxial compressive strength and the modulus of elasticity of dry and saturated limestone and find that the strength and modulus of elasticity are reduced by about 34% after the saturation of rock with water. Baud et al. [25] systematically analyze the water’s rock-weakening effect based on mechanical and chemical mechanisms and find that the water’s effect for reducing triaxial compressive strength is attributed to a decrease in the surface energy and the internal friction coefficient of the rock. For the subsurface mining application, Peng et al. [26] study the mechanical properties of deep coal rock samples under five different confining pressure conditions using a triaxial compression test system. Medhurst and Brown [27] conduct a conventional triaxial compression test on a large coal sample using a rock servo tester. The results show that with an increase of confining pressure, the peak strength of the coal sample increases and the expansion deformation decreases. Bell and Jermy [28] analyze the strength and deformation characteristics of coal samples under different stress conditions by performing compressive tensile and permeability tests on coal samples from the East Transvaal coal field. In addition, most of the scholars consider the impact of groundwater environment on the mechanical properties of rocks by conducting related experiments for various rock types. However, the measured results were primarily used to analyze the rock strength and deformation under various stress conditions based on the classical Coulomb yield criterion. In the case of underground mining engineering, due to the joint effects of in situ stress field and seepage field, the coal yield characteristics are more complicated, calling the effectiveness of the classical yield criteria in reflecting the actual rock strength under various stress states into question. Therefore, in-depth investigation in this regard is needed to ensure the reliable safety design of coal mines. To this end, the present paper obtains the conventional triaxial compressive strength of natural and saturated rock as well as the correlation between stress and strain with the presence of pore water seepage under various confining pressures by employing a MTS815 servo controlled rock mechanic testing system. Linear, parabolic and exponential yield criteria are used to characterize the strength of the rock sample subject to various stresses and water environments as well as the applicability of various yield criteria. The results obtained here can serve as a reference for the safety design concerning the operation of underground coal mines.


## 2. Triaxial Strength Test and Yield Criterion

### 2.1. Triaxial Strength Test of Coal

An MTS815 electrohydraulic servo rock test system was applied to three types of coal samples from the Yanzhou Coal Mining Co. Dongtan coal mine [29, 30]. Triaxial compression tests of natural and saturated coal and a triaxial seepage compression test of saturated coal were carried out. The MTS815 electrohydraulic servo rock test system is equipped with three independent servo systems to control the axial pressure, confining pressure, and pore pressure. Test applies a certain axial compression σ1, confining pressure σ3, pore pressure P1, and always maintain P1<σ3; osmotic pressure is produced at the two ends of the test pieces (ΔP=P1‐P2) by reducing the pore pressure at the end of the specimen to P2, so that pore water seepage under the effects of osmotic pressure is through the pore and fracture. When a triaxial seepage compression test of DT coal is carried out, the pore water pressures P1 and P2 are 3.5 MPa and 2 MPa, respectively. The coal sample is fully saturated with water before the seepage test. When a triaxial seepage compression test of BD and YH coal is carried out, the pore water pressures P1 and P2 are 3.8 MPa and 2.3 MPa. Prior to testing, we selected coal samples with no obvious cracks. We then used the ultrasonic velocity measurement method to screen the selected coal samples and used those samples which exhibited similar sonic velocities and waveforms for testing. The results of the triaxial strength test and the data utilized for assessing the yield criteria are shown in Table 1. The principle of triaxial tests is shown in Figure 1.Table 1
Test results of ultimate strength of coal samples.
DT1 natural coal
DT water-saturated nonseeping coal
DT seepage coal
BD2 natural coal [33]
XH3 natural coal [34]
BD seepage coal [34]
YC4 natural coal [32]
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
0
16.078
0
11.094
5
41.267
0
30.16
0
21.07
4
33.78
0
16.5
0
15.68
0
12.181
7
55.242
2
44.38
2
39.13
6
40.21
0
18.13
5
51.659
5
47.617
8
59.113
5
63.04
5
59.04
8
52.24
0
17.87
5
48.682
5
43.675
12
69.864
8
76.89
8
67.99
XH seepage coal [34]
0
18.22
15
82.552
15
69.619
—
—
12
92.51
11
76.77
σ3 (MPa)
σ1max (MPa)
5
47.74
15
81.811
15
65.818
—
—
16
106.07
14
89.91
4
24.32
10
59.13
25
104.548
25
101.212
—
—
20
112.99
17
103.77
6
29.77
20
76.62
25
106.615
25
95.659
—
—
24
123.16
20
113.32
8
33.89
25
97.24
1DT denotes Dongtan coal mine. 2BD denotes Baodian coal mine. 3XH denotes Xinhe coal mine. 4C denotes Yangcun coal mine.Figure 1
Diagram of the tests’ principle.


### 2.2. Mathematical Expression of Yield Criteria

#### 2.2.1. Linear Yield Criterion

Coulomb yield criterion is established by assuming that the rock failure is induced by sheer stress, and the maximum shear stress the rock can withstandτmax is jointly dictated by the cohesion c and the internal friction [31], i.e.,
(1)τmax=c+μσ,where μ is the internal friction coefficient; σ denotes the normal stress. The equation above can be expressed based on the principle stress as follows:
(2)σ1max=2ccosφ1−sinφ+1+sinφ1−sinφσ3,where σ1max denotes the maximum principal stress; σ3 denotes the minimum principal stress; φ denotes the internal friction angle.


#### 2.2.2. Parabolic Yield Criterion

Considering the symmetry, the parabolic Mohr criterion (You 2010) can be written as follows:(3)σ=τmax2aT0−T0,where T0 denotes the uniaxial tensile strength of rock, and a denotes the a parameter related to the compressive strength of the rock R:
(4)a=R21+1+R2.The corresponding principal stress relationship can be expressed as follows:(5)σ1max−σ32=2aT0σ1max+σ3+4aT02−a2T02.The equation above can also be expressed as follows:(6)σ1max−σ32=Aσ1max+σ3+C,where A and C are unknown parameters, which can be determined based on linear regression to finalize the yield criterion expression. Equation (6) can be equivalently written as follows:
(7)σ1max=σ3+σc−σD+4σc−σDσ3+σD2,where σc denotes the uniaxial compressive strength of a complete rock, and σD denotes an unknown parameter to be determined here.


#### 2.2.3. Hoek-Brown Yield Criterion

Based on the experimental testing of rock strength, Hoek and Brown put forward an empirical yield criterion, whose general form [6] is shown below:
(8)σ1max=σ3+σcmσ3σc+sn,where σc denotes the uniaxial compressive strength of a complete rock; m, s, and n are constants dictated by the characteristics of the rock. For a typical complete rock, s=1 and n=0.5.For a complete rock, Equation (8) can be simplified as follows:
(9)σ1max=σ3+σcmσ3σc+10.5.Equation (9) is also referred to as the special Hoek-Brown criterion, which can also be rewritten as follows:
(10)σ1max−σ32=mσcσ3+σc2.The square of the principal stress deviator is linearly correlated with the minimum principal stress, based on which the parametersm and σc can be determined by applying linear regression to the experimental data.


#### 2.2.4. Quadratic Polynomial Criterion

In order to facilitate the regression of experimental data, one can introduce additional unknown parameters to the criterion formula to modify the existing yield criterion. Also, a multitude of options exist for correcting the yield criterion. The general form of the parabolic yield criterion can be rewritten in a quadratic polynomial form [13]:
(11)σ1max−σ32=Aσ1max+Bσ3+C,where A,B,andC are unknown parameters. In the case where A=B, the equation above becomes Equation (6), while a special form of Hoek-Brown criterion is obtained by setting A=0 in the equation above. Equation (11) can be explicitly expressed as follows:
(12)σ1max=σ3+σc−σD+mσcσ3+σD2.


#### 2.2.5. Exponential Yield Criterion

By recognizing that the heterogeneity, cohesive force, and friction cannot coexist in a rock material, You [14, 31] postulates that with an increase of minimum principal stress, the maximum shear stress or the principal stress deviator in rock tends to approach a constant. Therefore, an exponential yield criterion with three unknown parameters is obtained as follows:
(13)σ1max−σ3=Q∞−Q∞−Q0exp−K0−1σ3Q∞−Q0,where Q0 denotes the uniaxial compressive strength, Q∞ denotes the yield differential principle stress, and K0 denotes the impact factor on strength for a zero confining pressure.


### 2.3. Comparative Analysis on Yield Criteria

In this paper, the triaxial compressive test data under different coal conditions are used to compare and analyze the applicability of the following five yield criteria.(1)
Coulomb yield criterion, Equation (2)
(2)
Parabolic Mohr criterion, Equations (6) and (7)
(3)
Hoek-Brown yield criterion, Equations (9) and (10)
(4)
Quadratic polynomial criterion, Equations (11) and (12)
(5)
Exponential yield criterion, Equation (13)


## 2.1. Triaxial Strength Test of Coal

An MTS815 electrohydraulic servo rock test system was applied to three types of coal samples from the Yanzhou Coal Mining Co. Dongtan coal mine [29, 30]. Triaxial compression tests of natural and saturated coal and a triaxial seepage compression test of saturated coal were carried out. The MTS815 electrohydraulic servo rock test system is equipped with three independent servo systems to control the axial pressure, confining pressure, and pore pressure. Test applies a certain axial compression σ1, confining pressure σ3, pore pressure P1, and always maintain P1<σ3; osmotic pressure is produced at the two ends of the test pieces (ΔP=P1‐P2) by reducing the pore pressure at the end of the specimen to P2, so that pore water seepage under the effects of osmotic pressure is through the pore and fracture. When a triaxial seepage compression test of DT coal is carried out, the pore water pressures P1 and P2 are 3.5 MPa and 2 MPa, respectively. The coal sample is fully saturated with water before the seepage test. When a triaxial seepage compression test of BD and YH coal is carried out, the pore water pressures P1 and P2 are 3.8 MPa and 2.3 MPa. Prior to testing, we selected coal samples with no obvious cracks. We then used the ultrasonic velocity measurement method to screen the selected coal samples and used those samples which exhibited similar sonic velocities and waveforms for testing. The results of the triaxial strength test and the data utilized for assessing the yield criteria are shown in Table 1. The principle of triaxial tests is shown in Figure 1.Table 1
Test results of ultimate strength of coal samples.
DT1 natural coal
DT water-saturated nonseeping coal
DT seepage coal
BD2 natural coal [33]
XH3 natural coal [34]
BD seepage coal [34]
YC4 natural coal [32]
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
σ3 (MPa)
σ1max (MPa)
0
16.078
0
11.094
5
41.267
0
30.16
0
21.07
4
33.78
0
16.5
0
15.68
0
12.181
7
55.242
2
44.38
2
39.13
6
40.21
0
18.13
5
51.659
5
47.617
8
59.113
5
63.04
5
59.04
8
52.24
0
17.87
5
48.682
5
43.675
12
69.864
8
76.89
8
67.99
XH seepage coal [34]
0
18.22
15
82.552
15
69.619
—
—
12
92.51
11
76.77
σ3 (MPa)
σ1max (MPa)
5
47.74
15
81.811
15
65.818
—
—
16
106.07
14
89.91
4
24.32
10
59.13
25
104.548
25
101.212
—
—
20
112.99
17
103.77
6
29.77
20
76.62
25
106.615
25
95.659
—
—
24
123.16
20
113.32
8
33.89
25
97.24
1DT denotes Dongtan coal mine. 2BD denotes Baodian coal mine. 3XH denotes Xinhe coal mine. 4C denotes Yangcun coal mine.Figure 1
Diagram of the tests’ principle.


## 2.2. Mathematical Expression of Yield Criteria

### 2.2.1. Linear Yield Criterion

Coulomb yield criterion is established by assuming that the rock failure is induced by sheer stress, and the maximum shear stress the rock can withstandτmax is jointly dictated by the cohesion c and the internal friction [31], i.e.,
(1)τmax=c+μσ,where μ is the internal friction coefficient; σ denotes the normal stress. The equation above can be expressed based on the principle stress as follows:
(2)σ1max=2ccosφ1−sinφ+1+sinφ1−sinφσ3,where σ1max denotes the maximum principal stress; σ3 denotes the minimum principal stress; φ denotes the internal friction angle.


### 2.2.2. Parabolic Yield Criterion

Considering the symmetry, the parabolic Mohr criterion (You 2010) can be written as follows:(3)σ=τmax2aT0−T0,where T0 denotes the uniaxial tensile strength of rock, and a denotes the a parameter related to the compressive strength of the rock R:
(4)a=R21+1+R2.The corresponding principal stress relationship can be expressed as follows:(5)σ1max−σ32=2aT0σ1max+σ3+4aT02−a2T02.The equation above can also be expressed as follows:(6)σ1max−σ32=Aσ1max+σ3+C,where A and C are unknown parameters, which can be determined based on linear regression to finalize the yield criterion expression. Equation (6) can be equivalently written as follows:
(7)σ1max=σ3+σc−σD+4σc−σDσ3+σD2,where σc denotes the uniaxial compressive strength of a complete rock, and σD denotes an unknown parameter to be determined here.


### 2.2.3. Hoek-Brown Yield Criterion

Based on the experimental testing of rock strength, Hoek and Brown put forward an empirical yield criterion, whose general form [6] is shown below:
(8)σ1max=σ3+σcmσ3σc+sn,where σc denotes the uniaxial compressive strength of a complete rock; m, s, and n are constants dictated by the characteristics of the rock. For a typical complete rock, s=1 and n=0.5.For a complete rock, Equation (8) can be simplified as follows:
(9)σ1max=σ3+σcmσ3σc+10.5.Equation (9) is also referred to as the special Hoek-Brown criterion, which can also be rewritten as follows:
(10)σ1max−σ32=mσcσ3+σc2.The square of the principal stress deviator is linearly correlated with the minimum principal stress, based on which the parametersm and σc can be determined by applying linear regression to the experimental data.


### 2.2.4. Quadratic Polynomial Criterion

In order to facilitate the regression of experimental data, one can introduce additional unknown parameters to the criterion formula to modify the existing yield criterion. Also, a multitude of options exist for correcting the yield criterion. The general form of the parabolic yield criterion can be rewritten in a quadratic polynomial form [13]:
(11)σ1max−σ32=Aσ1max+Bσ3+C,where A,B,andC are unknown parameters. In the case where A=B, the equation above becomes Equation (6), while a special form of Hoek-Brown criterion is obtained by setting A=0 in the equation above. Equation (11) can be explicitly expressed as follows:
(12)σ1max=σ3+σc−σD+mσcσ3+σD2.


### 2.2.5. Exponential Yield Criterion

By recognizing that the heterogeneity, cohesive force, and friction cannot coexist in a rock material, You [14, 31] postulates that with an increase of minimum principal stress, the maximum shear stress or the principal stress deviator in rock tends to approach a constant. Therefore, an exponential yield criterion with three unknown parameters is obtained as follows:
(13)σ1max−σ3=Q∞−Q∞−Q0exp−K0−1σ3Q∞−Q0,where Q0 denotes the uniaxial compressive strength, Q∞ denotes the yield differential principle stress, and K0 denotes the impact factor on strength for a zero confining pressure.


## 2.2.1. Linear Yield Criterion

Coulomb yield criterion is established by assuming that the rock failure is induced by sheer stress, and the maximum shear stress the rock can withstandτmax is jointly dictated by the cohesion c and the internal friction [31], i.e.,
(1)τmax=c+μσ,where μ is the internal friction coefficient; σ denotes the normal stress. The equation above can be expressed based on the principle stress as follows:
(2)σ1max=2ccosφ1−sinφ+1+sinφ1−sinφσ3,where σ1max denotes the maximum principal stress; σ3 denotes the minimum principal stress; φ denotes the internal friction angle.


## 2.2.2. Parabolic Yield Criterion

Considering the symmetry, the parabolic Mohr criterion (You 2010) can be written as follows:(3)σ=τmax2aT0−T0,where T0 denotes the uniaxial tensile strength of rock, and a denotes the a parameter related to the compressive strength of the rock R:
(4)a=R21+1+R2.The corresponding principal stress relationship can be expressed as follows:(5)σ1max−σ32=2aT0σ1max+σ3+4aT02−a2T02.The equation above can also be expressed as follows:(6)σ1max−σ32=Aσ1max+σ3+C,where A and C are unknown parameters, which can be determined based on linear regression to finalize the yield criterion expression. Equation (6) can be equivalently written as follows:
(7)σ1max=σ3+σc−σD+4σc−σDσ3+σD2,where σc denotes the uniaxial compressive strength of a complete rock, and σD denotes an unknown parameter to be determined here.


## 2.2.3. Hoek-Brown Yield Criterion

Based on the experimental testing of rock strength, Hoek and Brown put forward an empirical yield criterion, whose general form [6] is shown below:
(8)σ1max=σ3+σcmσ3σc+sn,where σc denotes the uniaxial compressive strength of a complete rock; m, s, and n are constants dictated by the characteristics of the rock. For a typical complete rock, s=1 and n=0.5.For a complete rock, Equation (8) can be simplified as follows:
(9)σ1max=σ3+σcmσ3σc+10.5.Equation (9) is also referred to as the special Hoek-Brown criterion, which can also be rewritten as follows:
(10)σ1max−σ32=mσcσ3+σc2.The square of the principal stress deviator is linearly correlated with the minimum principal stress, based on which the parametersm and σc can be determined by applying linear regression to the experimental data.


## 2.2.4. Quadratic Polynomial Criterion

In order to facilitate the regression of experimental data, one can introduce additional unknown parameters to the criterion formula to modify the existing yield criterion. Also, a multitude of options exist for correcting the yield criterion. The general form of the parabolic yield criterion can be rewritten in a quadratic polynomial form [13]:
(11)σ1max−σ32=Aσ1max+Bσ3+C,where A,B,andC are unknown parameters. In the case where A=B, the equation above becomes Equation (6), while a special form of Hoek-Brown criterion is obtained by setting A=0 in the equation above. Equation (11) can be explicitly expressed as follows:
(12)σ1max=σ3+σc−σD+mσcσ3+σD2.


## 2.2.5. Exponential Yield Criterion

By recognizing that the heterogeneity, cohesive force, and friction cannot coexist in a rock material, You [14, 31] postulates that with an increase of minimum principal stress, the maximum shear stress or the principal stress deviator in rock tends to approach a constant. Therefore, an exponential yield criterion with three unknown parameters is obtained as follows:
(13)σ1max−σ3=Q∞−Q∞−Q0exp−K0−1σ3Q∞−Q0,where Q0 denotes the uniaxial compressive strength, Q∞ denotes the yield differential principle stress, and K0 denotes the impact factor on strength for a zero confining pressure.


## 2.3. Comparative Analysis on Yield Criteria

In this paper, the triaxial compressive test data under different coal conditions are used to compare and analyze the applicability of the following five yield criteria.(1)
Coulomb yield criterion, Equation (2)
(2)
Parabolic Mohr criterion, Equations (6) and (7)
(3)
Hoek-Brown yield criterion, Equations (9) and (10)
(4)
Quadratic polynomial criterion, Equations (11) and (12)
(5)
Exponential yield criterion, Equation (13)


## 3. Comparative Analyses of Yield Criteria

### 3.1. Metrics for Assessing the Regression Quality of Yield Criteria

The least square method is used to determine the unknown parameters in the formula, which can minimize the summation over the square of error between the model prediction and measurement data.(14)δ1=∑σs−fσ32.You [13] conducts regression on the experimental data by using two different methods, i.e., minimizing the sum of square of error and minimizing the sum of absolute error. The study indicates that the latter approach can highlight the outliers in the measurement data and ensure that the measured points are distributed over both sides of the fitted curve in a balanced manner while staying as close as possible to the fitted curve.
(15)δ2=∑absσs−fσ3.For this reason, the present study is aimed at minimizing the sum of absolute error so as to determine the unknown parameters in the formula and usesδ2m to indicate the average fitting error:
(16)δ2m=∑absσs−fσ3N=δ2N,where N denotes the number of data points.


### 3.2. Comparative Analyses of Coal Strength Criteria

The present study applies curve fitting to the data of triaxial compression test on the coal rock samples as well as the triaxial compression test data published by Su et al. [32] and Yang et al. [33, 34] (as shown in Table 1), based on Coulomb criterion, parabolic Mohr criterion, Hoek-Brown criterion, quadratic polynomial criterion, and exponential strength criterion, respectively, with the fitted parameters shown in Tables 2 and 3, and the fitted curves associated with various yield criteria under the same coal rock condition shown in Figure 2. To facilitate the comparison, the fitting errors corresponding to the five yield criteria studied here are shown with a column diagram in Figure 3. It can be seen that for the natural water-bearing coal samples, the linear Coulomb yield criterion exhibits a large fitting error, while the parabolic Mohr criterion yields a lower fitting error compared with the linear Coulomb yield criterion for all tested cases except the XH natural coal sample. However, the uniaxial compressive strength of a complete rock σc obtained based on Equation (7) is significantly higher than that of the measured data. In other words, the calibrated parabolic Mohr criterion tends to overestimate the uniaxial compressive strength of the rock. For the Hoek-Brown criterion, quadratic polynomial criterion, and exponential strength criterion, the corresponding fitting errors are all less than those associated with the linear yield criterion and the parabolic Mohr criterion. Except that the quadratic polynomial criterion fails to decently fit the BD and XH natural coal samples, the fitted uniaxial strength of coal rock derived from the three criteria above all agree well with experimental data, and the performances of Hoek-Brown and exponential yield criteria are particularly satisfactory.Table 2
Test results concerning ultimate strength of coal samples.
Category
σc (MPa)
Coulomb criterion
Parabolic Mohr criterion
Hoek-Brown criterion
Quadratic polynomial criterion
c (MPa)
φ (°)
δ2m (MPa)
σc (MPa)
σD (MPa)
δ2m (MPa)
m
σc (MPa)
δ2m (MPa)
m
σc (MPa)
σD (MPa)
δ2m (MPa)
DT natural
15.879
10.05
29.13
6.35
26.7
0.0001688
4.53
17.44
16.08
2.42
10.29
16.39
2.17e−06
0.88
BD natural
30.16
12.71
32.55
5.56
37.87
2.457e−14
4.47
14.93
30.16
2.55
8.338
30.16
8.61
1.66
XH natural
21.07
7.454
38.15
2.98
29.75
0.006519
6.22
19.61
21.07
1.86
17.51
21.07
14.95
1.80
YC natural
17.68
5.096
31.31
3.74
20.32
2.651e−08
3.00
10.79
18.13
2.30
5.484
18.01
0.0002742
1.90
DT water-saturated
11.638
3.382
34.02
5.25
20.37
1.291e−07
5.31
15.91
12.18
3.27
10.89
12.45
6.427e−08
2.64
DT seepage
10.17
29.36
2.27
23.23
8.244e−05
2.72
9.546
24.44
2.16
35.34
6.406
0.05028
2.16
BD seepage
3.348
40.08
1.24
15.15
0.00016
2.67
1859
0.1197
1.48
1608
26.19
1.248e+04
3.27
XH seepage
4.84
24.23
0.29
8.959
9.518e−07
0.25
5.197
12.56
0.22
6.578
6.873
1.673e−12
0.14Table 3
Fitted results of exponential yield criterion.
Category
σc (MPa)
Exponential yield criterion
K0
Q0 (MPa)
Q∞ (MPa)
δ2m (MPa)
DT natural
15.879
8.059
16.08
84.83
1.41
BD natural
30.16
7.62
30.16
110.1
0.49
XH natural
21.07
5.876
26.72
145.9
2.40
YC natural
17.68
5.075
18.13
89.1
2.36
DT water-saturated
11.638
7.892
12.18
74.61
3.82
DT seepage
14.01
1.052e−08
63.22
1.11
BD seepage
1.279
26.96
26.28
0.05
XH seepage
6.771
5.569
29.27
0.00097Figure 2
Triaxial compressive strength of coal samples and fitted curves of five yield criteria. (a) DT Natural coal. (b) BD Natural coal. (c) XH Natural coal. (d) YC Natural coal. (e) DT Water-saturated coal. (f) DT Seepage coal. (g) BD Seepage coal. (h) XH Seepage coal.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure 3
Mean fitting deviations of five yield criteria. (a) DT Natural coal. (b) BD Natural coal. (c) XH Natural coal. (d) YC Natural coal. (e) DT Water-saturated coal.(f) DT Seepage coal. (g) BD Seepage coal. (h) XH Seepage coal.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)For water-saturated coal and compressive seepage coal, the linear yield criterion and the parabolic Mohr criterion still yield large errors in fitting the measured data. The cohesive force of the natural coal derived from the Coulomb criterion is smaller than those of all the other coal samples, e.g., water-saturated and seepage coal samples, except for the DT coal sample. The internal friction angle of the natural coal sample is smaller than those of all the water-saturated and seepage coal samples, except for the XH water-seeping coal with well-developed fractures and pore space [34], whose internal friction angle is smaller than that of the DT water-saturated coal and slightly larger than that of the DT natural coal. Due to the complex distribution of fine joints, fractures, and pore space in the coal, the linear yield criterion cannot authentically reflect the actual variation of coal strength, and thus, the applicability of the internal friction angle computed from the linear criterion to describing the mechanical characteristics of the rock needs further experimental validation.In addition to the BD seepage coal test, the Hoek-Brown criterion, the quadratic polynomial criterion, and the exponential yield criterion can also decently fit the experimental strength data of the water-saturated and water-seeping rock samples, with the exponential yield criterion giving the best result, as shown in Figure3. However, considering the mechanical interpretation underlying the exponential yield criterion, the DT seepage uniaxial compressive strength Q0 derived from the exponential yield criterion shows a local minimum value, which is inconsistent with the reality. In addition, the fitting parameters of the exponential yield criterion, Q0, all give low fitting errors, implying its ability to truly reflect the uniaxial strength of the coal rock. It is noteworthy that, for both the natural coal rock and the water-saturated water-seeping coal rock, the yield differential principle stress, Q∞, derived from the exponential yield criterion is lower than the measured value. Intuitively speaking, however, the exponential yield criterion exhibits a major advantage in fitting the strength data of the water-seeping coal. Therefore, it is necessary to conduct further experimental validation concerning the applicability of the exponential yield criterion to analyzing the coal rock strength.


## 3.1. Metrics for Assessing the Regression Quality of Yield Criteria

The least square method is used to determine the unknown parameters in the formula, which can minimize the summation over the square of error between the model prediction and measurement data.(14)δ1=∑σs−fσ32.You [13] conducts regression on the experimental data by using two different methods, i.e., minimizing the sum of square of error and minimizing the sum of absolute error. The study indicates that the latter approach can highlight the outliers in the measurement data and ensure that the measured points are distributed over both sides of the fitted curve in a balanced manner while staying as close as possible to the fitted curve.
(15)δ2=∑absσs−fσ3.For this reason, the present study is aimed at minimizing the sum of absolute error so as to determine the unknown parameters in the formula and usesδ2m to indicate the average fitting error:
(16)δ2m=∑absσs−fσ3N=δ2N,where N denotes the number of data points.


## 3.2. Comparative Analyses of Coal Strength Criteria

The present study applies curve fitting to the data of triaxial compression test on the coal rock samples as well as the triaxial compression test data published by Su et al. [32] and Yang et al. [33, 34] (as shown in Table 1), based on Coulomb criterion, parabolic Mohr criterion, Hoek-Brown criterion, quadratic polynomial criterion, and exponential strength criterion, respectively, with the fitted parameters shown in Tables 2 and 3, and the fitted curves associated with various yield criteria under the same coal rock condition shown in Figure 2. To facilitate the comparison, the fitting errors corresponding to the five yield criteria studied here are shown with a column diagram in Figure 3. It can be seen that for the natural water-bearing coal samples, the linear Coulomb yield criterion exhibits a large fitting error, while the parabolic Mohr criterion yields a lower fitting error compared with the linear Coulomb yield criterion for all tested cases except the XH natural coal sample. However, the uniaxial compressive strength of a complete rock σc obtained based on Equation (7) is significantly higher than that of the measured data. In other words, the calibrated parabolic Mohr criterion tends to overestimate the uniaxial compressive strength of the rock. For the Hoek-Brown criterion, quadratic polynomial criterion, and exponential strength criterion, the corresponding fitting errors are all less than those associated with the linear yield criterion and the parabolic Mohr criterion. Except that the quadratic polynomial criterion fails to decently fit the BD and XH natural coal samples, the fitted uniaxial strength of coal rock derived from the three criteria above all agree well with experimental data, and the performances of Hoek-Brown and exponential yield criteria are particularly satisfactory.Table 2
Test results concerning ultimate strength of coal samples.
Category
σc (MPa)
Coulomb criterion
Parabolic Mohr criterion
Hoek-Brown criterion
Quadratic polynomial criterion
c (MPa)
φ (°)
δ2m (MPa)
σc (MPa)
σD (MPa)
δ2m (MPa)
m
σc (MPa)
δ2m (MPa)
m
σc (MPa)
σD (MPa)
δ2m (MPa)
DT natural
15.879
10.05
29.13
6.35
26.7
0.0001688
4.53
17.44
16.08
2.42
10.29
16.39
2.17e−06
0.88
BD natural
30.16
12.71
32.55
5.56
37.87
2.457e−14
4.47
14.93
30.16
2.55
8.338
30.16
8.61
1.66
XH natural
21.07
7.454
38.15
2.98
29.75
0.006519
6.22
19.61
21.07
1.86
17.51
21.07
14.95
1.80
YC natural
17.68
5.096
31.31
3.74
20.32
2.651e−08
3.00
10.79
18.13
2.30
5.484
18.01
0.0002742
1.90
DT water-saturated
11.638
3.382
34.02
5.25
20.37
1.291e−07
5.31
15.91
12.18
3.27
10.89
12.45
6.427e−08
2.64
DT seepage
10.17
29.36
2.27
23.23
8.244e−05
2.72
9.546
24.44
2.16
35.34
6.406
0.05028
2.16
BD seepage
3.348
40.08
1.24
15.15
0.00016
2.67
1859
0.1197
1.48
1608
26.19
1.248e+04
3.27
XH seepage
4.84
24.23
0.29
8.959
9.518e−07
0.25
5.197
12.56
0.22
6.578
6.873
1.673e−12
0.14Table 3
Fitted results of exponential yield criterion.
Category
σc (MPa)
Exponential yield criterion
K0
Q0 (MPa)
Q∞ (MPa)
δ2m (MPa)
DT natural
15.879
8.059
16.08
84.83
1.41
BD natural
30.16
7.62
30.16
110.1
0.49
XH natural
21.07
5.876
26.72
145.9
2.40
YC natural
17.68
5.075
18.13
89.1
2.36
DT water-saturated
11.638
7.892
12.18
74.61
3.82
DT seepage
14.01
1.052e−08
63.22
1.11
BD seepage
1.279
26.96
26.28
0.05
XH seepage
6.771
5.569
29.27
0.00097Figure 2
Triaxial compressive strength of coal samples and fitted curves of five yield criteria. (a) DT Natural coal. (b) BD Natural coal. (c) XH Natural coal. (d) YC Natural coal. (e) DT Water-saturated coal. (f) DT Seepage coal. (g) BD Seepage coal. (h) XH Seepage coal.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)Figure 3
Mean fitting deviations of five yield criteria. (a) DT Natural coal. (b) BD Natural coal. (c) XH Natural coal. (d) YC Natural coal. (e) DT Water-saturated coal.(f) DT Seepage coal. (g) BD Seepage coal. (h) XH Seepage coal.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)For water-saturated coal and compressive seepage coal, the linear yield criterion and the parabolic Mohr criterion still yield large errors in fitting the measured data. The cohesive force of the natural coal derived from the Coulomb criterion is smaller than those of all the other coal samples, e.g., water-saturated and seepage coal samples, except for the DT coal sample. The internal friction angle of the natural coal sample is smaller than those of all the water-saturated and seepage coal samples, except for the XH water-seeping coal with well-developed fractures and pore space [34], whose internal friction angle is smaller than that of the DT water-saturated coal and slightly larger than that of the DT natural coal. Due to the complex distribution of fine joints, fractures, and pore space in the coal, the linear yield criterion cannot authentically reflect the actual variation of coal strength, and thus, the applicability of the internal friction angle computed from the linear criterion to describing the mechanical characteristics of the rock needs further experimental validation.In addition to the BD seepage coal test, the Hoek-Brown criterion, the quadratic polynomial criterion, and the exponential yield criterion can also decently fit the experimental strength data of the water-saturated and water-seeping rock samples, with the exponential yield criterion giving the best result, as shown in Figure3. However, considering the mechanical interpretation underlying the exponential yield criterion, the DT seepage uniaxial compressive strength Q0 derived from the exponential yield criterion shows a local minimum value, which is inconsistent with the reality. In addition, the fitting parameters of the exponential yield criterion, Q0, all give low fitting errors, implying its ability to truly reflect the uniaxial strength of the coal rock. It is noteworthy that, for both the natural coal rock and the water-saturated water-seeping coal rock, the yield differential principle stress, Q∞, derived from the exponential yield criterion is lower than the measured value. Intuitively speaking, however, the exponential yield criterion exhibits a major advantage in fitting the strength data of the water-seeping coal. Therefore, it is necessary to conduct further experimental validation concerning the applicability of the exponential yield criterion to analyzing the coal rock strength.


## 4. Comparative Analyses of Yield Criteria with the Presence of Pore Water in Coal

In 1936, Terzaghi proposed the principle of effective stress [35]. According to the principle of effective stress, the total stress in the soil is equal to the sum of the effective stress and the pore water pressure, and only the change of effective stress will cause the change of strength. In the field of geotechnical engineering, the complex groundwater environment and the huge engineering risks associated with it have prompted engineering professionals to introduce the principle of effective stress into rock mechanics for rock strength analysis. To further analyze the effect of pore pressure on the mechanical properties of coal, the effective stress principle and the aforementioned test results are used to evaluate the applicability of the Coulomb criterion, the parabolic Mohr criterion, the Hoek-Brown criterion, the quadratic polynomial criterion, and the exponential yield criterion to computing the strength of water-saturated and water-seeping rock samples. Herein, the maximum axial stress σ1max and the confining pressure σ3 corresponding to the coal rock solid skeleton under the pore pressure are expressed as the effective maximum axial stress σ1max′ and the effective confining pressure σ3′.


### 4.1. Comparative Analyses of Yield Criteria in Water-Saturated Nonseeping Coal

During triaxial compression tests on the water-saturated rock sample, the water present within the fractures and pore spaces between solid particles in the solid skeleton of the coal rock is subject to a combination of axial, confining, and pore pressure [36, 37]. Existing research has shown that under the physical processes of water lubrication, softening, argillization, drying and wetting, freezing and thawing, the chemical processes of dissolution, hydration, hydrolysis, acidification, and oxidation, and the mechanical effects of pore water pressure, the mechanical property parameters of rock, such as strength, all decrease in a groundwater environment [38–40]. Although the structure and composition of coal rock is different from other rocks, studies have shown that the strength limit of coal rock is also related to its degree of water saturation and pore pressure [41, 42]. During the saturated nonseeping compression test, the pore pressure enclosed in the solid skeleton also increased in response to the compression of the solid skeleton, and the resulting pore pressure is related to the volumetric stress of the coal rock skeleton. Therefore, the pore pressure inside the coal rock should be ζσ1+2σ3, where ζ denotes the coefficient related to the porosity and its distribution within the coal rock, with the value ranging between 0 and 1. As a result, the effective strength associated with the saturated nonpermeating coal rocks can be calculated as follows:
(17)σ1max′=σ1max−ζσ1max+2σ3σ3′=σ3−ζσ1max+2σ3.Using the five yield criteria above, the experimental data tied to the effective strength of DT saturated coal under different conditions are fitted, and the average fitting errors are analyzed, as shown in Table4. For the sake of conciseness, this paper only presents the fitted parameters and the corresponding fitting errors of the effective coal strength for ζ being 0.01, 0.05, and 0.08. In order to intuitively illustrate the performance of each criterion in fitting the measurement data of saturated coal rock, the fitting errors of various yield criteria are shown in a column diagram (Figure 4), with Figure 5 showing the corresponding fitted curves of the five yield criteria for ζ=0.05.Table 4
Triaxial strength fitting results of five effective yield criteria for saturated coal based on the least absolute deviation.
Category
ζ
Coulomb criterion
Parabolic Mohr criterion
Hoek-Brown criterion
c (MPa)
φ (°)
δ2m (MPa)
σc (MPa)
σD (MPa)
δ2m (MPa)
m
σc (MPa)
δ2m (MPa)
DT water-saturated
0.01
3.403
34.99
5.47
21.53
0.0005366
5.59
16.89
12.18
3.44
0.05
3.522
39.58
6.68
23.83
2.09e−05
6.33
23.23
12.18
4.63
0.08
4.458
42.97
8.23
27.13
1.573e−07
8.07
33.1
12.18
6.97
Category
ζ
Quadratic polynomial criteria
ζ
Exponential yield criterion
m
σc (MPa)
σD (MPa)
δ2m (MPa)
K0
Q0 (MPa)
Q∞ (MPa)
δ2m (MPa)
DT water-saturated
0.01
10.5
13.31
1.073e−06
2.94
0.01
8.758
12.18
73.85
3.99
0.05
8.424
18.34
0.0001455
4.66
0.05
16.17
12.18
71.28
5.08
0.08
21.03
12.18
3.368e−08
5.73
0.08
59.29
12.18
55.24
6.91Figure 4
Mean fitting deviations of yield criteria for saturated DT coal samples.Figure 5
Triaxial compressive strength of water-saturated coal samples and fitted curves of five yield criteria.It can be seen that the Coulomb criterion and the parabolic Mohr criterion yield large fitting errors, and the parabolic Mohr criterion tends to overestimate the uniaxial compressive strength of saturated coal rock. In addition, it is found from the Coulomb criterion-based fitting data under different pore pressure conditions that the internal friction angle and cohesion of saturated coal increase with the pore pressure. By contrast, the Hoek-Brown criterion, the quadratic polynomial criterion, and the exponential yield criterion are superior to the Coulomb criterion and the parabolic Mohr criterion in terms of the fitting quality of effective saturated rock strength. For a variety of pore pressures, the fitting errors of the three methods are comparable. All three methods can decently fit the uniaxial strength data of saturated coal rock.Among the three yield criteria with small fitting errors, the Hoek-Brown criterion is regarded as a special case of the second-degree polynomial rule [14] (with A=0 in Equation (11)). Figure 6 shows the fitted curves of quadratic polynomial criterion and exponential yield criterion under different pore pressures. It can be seen in the figure that the fitting parameter m, as a measure of the rock hardness, increases with the pore pressure, that is, the rock skeleton gets reinforced under the influence of the pore pressure. Besides, from the fitting quality of the exponential yield criterion, one can conclude that with an increase of pore pressure, the yield differential principal stress Q∞ of the saturated coal rock gradually decreases, i.e., the increase of pore pressure compromises the yield stress of coal rock. Nonetheless, the yield differential principal stress Q∞ fitted from the exponential yield criterion is still lower than the experimentally measured value. This deviation, however, is acceptable from the perspective of engineering construction safety.Figure 6
Triaxial compressive strength of water-saturated coal samples and the fitted curves of two yield criteria under different pore pressures.


### 4.2. Comparative Analyses of Yield Criteria in Water-Saturated Seeping Coal

The internal pores of coal samples are not always connected, so the internal pore water pressure is not necessarily equal to the water pressure exerted by the end face. Therefore, the pore water pressure in coal samples should beδP, where δ is the coefficient related to the porosity of coal and its connected state, and its value is related to the porosity of coal and its connected state. In addition, due to the existence of osmotic pressure differences in coal, pore water pressure is a function of pore distribution coordinate, but it is difficult to accurately determine its distribution state during the test. Therefore, two extreme assumptions were adopted to simplify estimation of the range of effective stress values on coal under conditions of saturated drainage.
(1)
Assuming that the coal sample is absolutely dense, i.e., the porosityϕ is 0 (completely impermeable), the water pressure applied to the specimen section is equivalent to the axial stress superimposed on the specimen section. Therefore, the effective stress on the coal is as follows:
(18)σ1max′=σ1max+Pσ3′=σ3.
(2)
Assuming that the porosity of the coal sample is large enough to make its structure similar to that of soil, the effective stress on the coal is as follows:
(19)σ1max′=σ1maxσ3′=σ3−P.Due to the complex structure of pores and fractures in coal, it is impossible to specifically calculate the effective stress on the internal skeleton of coal. However, from formula (18) and formula (19), it can be seen that the effective stress on seepage coal is somewhere between the two. Therefore, based on the above two extreme hypotheses, the maximum value of effective stress is selected for analysis by fitting data of the effective strength of seepage coal. Table 5 shows the fitting parameters and average fitting deviation of different strength criteria of seepage coal.Table 5
Triaxial strength fitting results of five types of effective strength criteria for coal during a seepage process based on the least absolute.
Category
ϕ
Coulomb criterion
Parabolic Mohr criterion
Hoek-Brown criterion
c (MPa)
φ (°)
δ2m (MPa)
σc (MPa)
σD (MPa)
δ2m (MPa)
m
σc (MPa)
δ2m (MPa)
DT seepage
0
12.14
29.36
2.27
27.42
3.412e−05
1.69
5.085
40.16
2.09
ϕ±
11.29
29.36
2.28
24.72
0.001393
2.68
5.649
36.14
2.09
BD seepage
0
6.546
40.08
1.24
27.85
5.867
1.87
8.983
27.97
1.54
ϕ±
5.284
40.08
1.24
18.55
6.171e−06
2.01
12.48
20.14
1.54
XH seepage
0
6.55
24.23
0.29
20.05
10.46
0.17
3.24
20.14
0.22
ϕ±
5.875
24.23
0.29
16.79
7.194
1.58
3.72
17.55
0.22
Category
ϕ
Quadratic polynomial criterion
ϕ
Exponential yield criterion
m
σc (MPa)
σD (MPa)
δ2m (MPa)
K0
Q0 (MPa)
Q∞ (MPa)
δ2m (MPa)
DT seepage
0
2.801
30.84
0.001214
1.65
0
14.28
20.68
60.05
0.17
ϕ±
4.342
24.96
4.045e−05
1.85
ϕ±
19.71
2.559e−09
60.64
0.23
BD seepage
0
1.026e+04
29
4.099e+04
0.94
0
4.617
28.12
1.377e+04
1.25
ϕ±
1.316e+04
22.93
4.133e+04
1.13
ϕ±
1.715
28.03
26.28
0.0054
XH seepage
0
0.7782
19.68
2.075
0.00065
0
3.288
19.87
29.27
0.0026
ϕ±
1.94
13.62
1.056e−13
0.089
ϕ±
4.296
15.73
29.27
0.0012It can be seen from Figure7 that the Hoek-Brown criterion, quadratic polynomial criterion, and exponential yield criterion are better than the Coulomb criterion and parabolic Mohr criterion, and especially the exponential yield criterion. Figure 8 shows the fitting curve of the DT seepage coal quadratic polynomial criterion and the exponential yield criterion, from which we can see that the exponential yield criterion has a good fitting deviation for the effective strength of seepage coal. However, due to the limited amount of coal seepage test data, the exponential yield criterion for the partial seepage test coal data, the fitting parameters of uniaxial compression strength Q0, and the ultimate principal stress difference Q∞ are not accurate. Therefore, to apply the exponential yield criterion to the effective strength of seepage coal, further seepage tests are needed.Figure 7
Mean fitting deviations of strength criteria for seepage coal samples.Figure 8
Triaxial compression strengths of DT coal samples and fitting curves of two strength criteria under seepage pressures.


## 4.1. Comparative Analyses of Yield Criteria in Water-Saturated Nonseeping Coal

During triaxial compression tests on the water-saturated rock sample, the water present within the fractures and pore spaces between solid particles in the solid skeleton of the coal rock is subject to a combination of axial, confining, and pore pressure [36, 37]. Existing research has shown that under the physical processes of water lubrication, softening, argillization, drying and wetting, freezing and thawing, the chemical processes of dissolution, hydration, hydrolysis, acidification, and oxidation, and the mechanical effects of pore water pressure, the mechanical property parameters of rock, such as strength, all decrease in a groundwater environment [38–40]. Although the structure and composition of coal rock is different from other rocks, studies have shown that the strength limit of coal rock is also related to its degree of water saturation and pore pressure [41, 42]. During the saturated nonseeping compression test, the pore pressure enclosed in the solid skeleton also increased in response to the compression of the solid skeleton, and the resulting pore pressure is related to the volumetric stress of the coal rock skeleton. Therefore, the pore pressure inside the coal rock should be ζσ1+2σ3, where ζ denotes the coefficient related to the porosity and its distribution within the coal rock, with the value ranging between 0 and 1. As a result, the effective strength associated with the saturated nonpermeating coal rocks can be calculated as follows:
(17)σ1max′=σ1max−ζσ1max+2σ3σ3′=σ3−ζσ1max+2σ3.Using the five yield criteria above, the experimental data tied to the effective strength of DT saturated coal under different conditions are fitted, and the average fitting errors are analyzed, as shown in Table4. For the sake of conciseness, this paper only presents the fitted parameters and the corresponding fitting errors of the effective coal strength for ζ being 0.01, 0.05, and 0.08. In order to intuitively illustrate the performance of each criterion in fitting the measurement data of saturated coal rock, the fitting errors of various yield criteria are shown in a column diagram (Figure 4), with Figure 5 showing the corresponding fitted curves of the five yield criteria for ζ=0.05.Table 4
Triaxial strength fitting results of five effective yield criteria for saturated coal based on the least absolute deviation.
Category
ζ
Coulomb criterion
Parabolic Mohr criterion
Hoek-Brown criterion
c (MPa)
φ (°)
δ2m (MPa)
σc (MPa)
σD (MPa)
δ2m (MPa)
m
σc (MPa)
δ2m (MPa)
DT water-saturated
0.01
3.403
34.99
5.47
21.53
0.0005366
5.59
16.89
12.18
3.44
0.05
3.522
39.58
6.68
23.83
2.09e−05
6.33
23.23
12.18
4.63
0.08
4.458
42.97
8.23
27.13
1.573e−07
8.07
33.1
12.18
6.97
Category
ζ
Quadratic polynomial criteria
ζ
Exponential yield criterion
m
σc (MPa)
σD (MPa)
δ2m (MPa)
K0
Q0 (MPa)
Q∞ (MPa)
δ2m (MPa)
DT water-saturated
0.01
10.5
13.31
1.073e−06
2.94
0.01
8.758
12.18
73.85
3.99
0.05
8.424
18.34
0.0001455
4.66
0.05
16.17
12.18
71.28
5.08
0.08
21.03
12.18
3.368e−08
5.73
0.08
59.29
12.18
55.24
6.91Figure 4
Mean fitting deviations of yield criteria for saturated DT coal samples.Figure 5
Triaxial compressive strength of water-saturated coal samples and fitted curves of five yield criteria.It can be seen that the Coulomb criterion and the parabolic Mohr criterion yield large fitting errors, and the parabolic Mohr criterion tends to overestimate the uniaxial compressive strength of saturated coal rock. In addition, it is found from the Coulomb criterion-based fitting data under different pore pressure conditions that the internal friction angle and cohesion of saturated coal increase with the pore pressure. By contrast, the Hoek-Brown criterion, the quadratic polynomial criterion, and the exponential yield criterion are superior to the Coulomb criterion and the parabolic Mohr criterion in terms of the fitting quality of effective saturated rock strength. For a variety of pore pressures, the fitting errors of the three methods are comparable. All three methods can decently fit the uniaxial strength data of saturated coal rock.Among the three yield criteria with small fitting errors, the Hoek-Brown criterion is regarded as a special case of the second-degree polynomial rule [14] (with A=0 in Equation (11)). Figure 6 shows the fitted curves of quadratic polynomial criterion and exponential yield criterion under different pore pressures. It can be seen in the figure that the fitting parameter m, as a measure of the rock hardness, increases with the pore pressure, that is, the rock skeleton gets reinforced under the influence of the pore pressure. Besides, from the fitting quality of the exponential yield criterion, one can conclude that with an increase of pore pressure, the yield differential principal stress Q∞ of the saturated coal rock gradually decreases, i.e., the increase of pore pressure compromises the yield stress of coal rock. Nonetheless, the yield differential principal stress Q∞ fitted from the exponential yield criterion is still lower than the experimentally measured value. This deviation, however, is acceptable from the perspective of engineering construction safety.Figure 6
Triaxial compressive strength of water-saturated coal samples and the fitted curves of two yield criteria under different pore pressures.


## 4.2. Comparative Analyses of Yield Criteria in Water-Saturated Seeping Coal

The internal pores of coal samples are not always connected, so the internal pore water pressure is not necessarily equal to the water pressure exerted by the end face. Therefore, the pore water pressure in coal samples should beδP, where δ is the coefficient related to the porosity of coal and its connected state, and its value is related to the porosity of coal and its connected state. In addition, due to the existence of osmotic pressure differences in coal, pore water pressure is a function of pore distribution coordinate, but it is difficult to accurately determine its distribution state during the test. Therefore, two extreme assumptions were adopted to simplify estimation of the range of effective stress values on coal under conditions of saturated drainage.
(1)
Assuming that the coal sample is absolutely dense, i.e., the porosityϕ is 0 (completely impermeable), the water pressure applied to the specimen section is equivalent to the axial stress superimposed on the specimen section. Therefore, the effective stress on the coal is as follows:
(18)σ1max′=σ1max+Pσ3′=σ3.
(2)
Assuming that the porosity of the coal sample is large enough to make its structure similar to that of soil, the effective stress on the coal is as follows:
(19)σ1max′=σ1maxσ3′=σ3−P.Due to the complex structure of pores and fractures in coal, it is impossible to specifically calculate the effective stress on the internal skeleton of coal. However, from formula (18) and formula (19), it can be seen that the effective stress on seepage coal is somewhere between the two. Therefore, based on the above two extreme hypotheses, the maximum value of effective stress is selected for analysis by fitting data of the effective strength of seepage coal. Table 5 shows the fitting parameters and average fitting deviation of different strength criteria of seepage coal.Table 5
Triaxial strength fitting results of five types of effective strength criteria for coal during a seepage process based on the least absolute.
Category
ϕ
Coulomb criterion
Parabolic Mohr criterion
Hoek-Brown criterion
c (MPa)
φ (°)
δ2m (MPa)
σc (MPa)
σD (MPa)
δ2m (MPa)
m
σc (MPa)
δ2m (MPa)
DT seepage
0
12.14
29.36
2.27
27.42
3.412e−05
1.69
5.085
40.16
2.09
ϕ±
11.29
29.36
2.28
24.72
0.001393
2.68
5.649
36.14
2.09
BD seepage
0
6.546
40.08
1.24
27.85
5.867
1.87
8.983
27.97
1.54
ϕ±
5.284
40.08
1.24
18.55
6.171e−06
2.01
12.48
20.14
1.54
XH seepage
0
6.55
24.23
0.29
20.05
10.46
0.17
3.24
20.14
0.22
ϕ±
5.875
24.23
0.29
16.79
7.194
1.58
3.72
17.55
0.22
Category
ϕ
Quadratic polynomial criterion
ϕ
Exponential yield criterion
m
σc (MPa)
σD (MPa)
δ2m (MPa)
K0
Q0 (MPa)
Q∞ (MPa)
δ2m (MPa)
DT seepage
0
2.801
30.84
0.001214
1.65
0
14.28
20.68
60.05
0.17
ϕ±
4.342
24.96
4.045e−05
1.85
ϕ±
19.71
2.559e−09
60.64
0.23
BD seepage
0
1.026e+04
29
4.099e+04
0.94
0
4.617
28.12
1.377e+04
1.25
ϕ±
1.316e+04
22.93
4.133e+04
1.13
ϕ±
1.715
28.03
26.28
0.0054
XH seepage
0
0.7782
19.68
2.075
0.00065
0
3.288
19.87
29.27
0.0026
ϕ±
1.94
13.62
1.056e−13
0.089
ϕ±
4.296
15.73
29.27
0.0012It can be seen from Figure7 that the Hoek-Brown criterion, quadratic polynomial criterion, and exponential yield criterion are better than the Coulomb criterion and parabolic Mohr criterion, and especially the exponential yield criterion. Figure 8 shows the fitting curve of the DT seepage coal quadratic polynomial criterion and the exponential yield criterion, from which we can see that the exponential yield criterion has a good fitting deviation for the effective strength of seepage coal. However, due to the limited amount of coal seepage test data, the exponential yield criterion for the partial seepage test coal data, the fitting parameters of uniaxial compression strength Q0, and the ultimate principal stress difference Q∞ are not accurate. Therefore, to apply the exponential yield criterion to the effective strength of seepage coal, further seepage tests are needed.Figure 7
Mean fitting deviations of strength criteria for seepage coal samples.Figure 8
Triaxial compression strengths of DT coal samples and fitting curves of two strength criteria under seepage pressures.


## 5. Conclusion

(1)
With an increase of confining pressure, the coal rock strength shows a nonlinear characteristic. The linear Coulomb yield criterion cannot describe this behavior well, showing a large fitting error. It is therefore necessary to conduct experimental validation when it comes to using its internal friction angle and cohesion to describe the variation of the rock’s mechanical properties
(2)
Although the parabolic Mohr yield criterion can reflect to certain extent the nonlinear characteristics of coal rock strength, its fitting error is too large, and the fitted parameters are inconsistent with the reality
(3)
The Hoek-Brown criterion, quadratic polynomial criterion, and exponential yield criterion all exhibit decent fitting quality, i.e., they can all reflect the characteristics of coal strength under natural and fluid-structure interactions. The exponential yield criterion, in particular, can truly reflect the uniaxial compressive strength of coal. Although the yield strength of the rock derived from the exponential yield criterion is less than the measured value, such conservative approach is favored for ensuring engineering safetyAlthough the test and analysis results presented in this paper provide some insights into the applicability of various yield criteria to characterize the yield variation (especially with the presence of fluid-solid interaction), the size and self-consistency of experimental dataset is limited; thereby, the conclusions presented in this paper are preliminary. As such, the applicability of various yield criteria still requires further experimental analyses.



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*Source: 4670812-2020-02-03.xml*