**3. Results and Analysis**

#### *3.1. Strain and Strength Characteristics*

Figure 5 compares the stress-strain relationship of specimens under the true-triaxial path, where *ε*<sup>1</sup> and *ε*<sup>3</sup> are the axial strain and radial strain of the specimen, and Δ*σ* is the principal stress difference, i.e., *σ*1–*σ*3. From these diagrams, the stress-strain curves of gas-bearing coal are much the same under the three paths. They all involve five stages: compaction, linear elasticity, plastic deformation, stress decline, and residual stress. Stress path is shown to have a strong impact on the deformation and strength of gas-bearing coal. In Test 1, the specimen strength is the highest, with stress difference of 33.750 MPa. In Test 2, the specimen strength is the second highest, with stress difference of 23.480 MPa. In Test 3, the specimen strength is the lowest, with stress difference of 16.711 MPa. The specimen-bearing capacity is greatly reduced.

At peak strength, *ε*<sup>1</sup> and *ε*<sup>3</sup> are 2.607 and −1.771 in Test 1; 1.301 and −1.354 in Test 2; and 0.793 and −2.186 in Test 3. In Test 1, the axial strain is the largest. In Test 3, the axial strain is the smallest, but the radial strain is the largest. At peak strength, the axial-toradial strain ratios of the specimens are 1.472, 0.961, and 0.363. The gradual reduction in the axial-to-radial strain ratio indicates that, during instability, the axial deformation intensifies and the specimen is more prone to axial failure, especially in Test 3, where the specimen has a strong shear dilatancy. Also, by observing the stress-strain curves, it can be seen that the post-peak curve slope gradually increases. In Test 2 and Test 3, with the unloading of confining pressure, the strain increment gradually increases. When this increase has accumulated to a limit, the bearing capacity declines and the specimen

immediately becomes unstable. This is particularly obvious in Test 3, where the specimen gradually changes from ductile failure to brittle failure.

**Figure 5.** Stress–strain curves of specimens under different test paths.

### *3.2. Permeability Evolution Characteristics*

Gas seepage in the specimens obeys Darcy's law. Permeability is calculated by the following formula [29]:

$$K = \frac{2P\_0'Q\mu L}{A(P\_1^2 - P\_2^2)}\tag{1}$$

where *K* is permeability (m2); *Q* is the gas flow (m3/s); *μ* is the absolute viscosity of the methane; *L* is the specimen length (m); *A* is the effective area of permeability (m2); *P*<sup>0</sup> prime is standard atmosphere; *P*<sup>1</sup> is the inlet pressure (MPa); and *P*<sup>2</sup> is outlet pressure (MPa).

Figure 6 compares the principal stress difference and permeability variations of gasbearing coal as a function of axial strain under different test paths. From the *ε*1–Δ*σ* and *ε*1–*K* curves, as the test goes on, permeability first reduces then increases; the valley deflection of permeability falls before the peak deflection of the *ε*1–Δ*σ* curve in all cases. This is because, during elastic deformation at the beginning of the test, with the loading of external stress, the primary pores and fissures inside the specimen are compressed, which narrows the gas seepage pathway and brings down the permeability. With the loading of *σ*1, the specimen enters plastic deformation. Cracks begin to develop inside and damage is expanded, adding more seepage pathways and stepping up permeability. After that, with the further increase of *σ*1, cracks inside become interconnected, leading to instability failure. The gas seepage pathway is opened and the permeability soars.

**Figure 6.** Variation curves of principal stress difference and permeability with axial strain during test.

Relative permeability (*w*) variations were observed against initial permeability to examine how stress path affects permeability evolution. From the diagram in Figure 7, under Test 1, the specimen permeability variation is modest, with minimum *w* of 0.222, which increases marginally to 0.913 at the end of the test. Under Test 2, *w* is 0.353 minimum and 0.358 maximum. Under Test 3, *w* is 0.370 minimum and 1.405 maximum. Compared with the other paths, the specimen permeability variation is the largest at the end of the test under composite loading/unloading path, proving that the specimen is more badly damaged under this path.

**Figure 7.** Relative permeability change.

#### **4. Numerical Simulation of Crack Characteristics**

In order to examine how cracks in gas-bearing coal evolve under triaxial stress paths, numerical simulations were performed with PFC software to see how crack-count changes in the specimens during loading.

### *4.1. Model Construction*

The model was sized with the same dimensions as the laboratory specimens, i.e., φ50 mm × 100 mm. In the computational model, the minimum particle radius was 0.25 mm, with a particle size ratio of 1.66. A total of 3665 particle samples were generated.

Figure 8 shows the initial and boundary conditions used for simulation. First, the model was loaded with biaxial compression of axial and confining pressure to 6 MPa by the servo mechanism. Then a high-pressure zone with pressure P was set at the top of the model. The fluid field pressure at the bottom of the model was fixed to 0.1 MPa to indicate connecting to atmospheric pressure.

**Figure 8.** Model, initial conditions and boundary conditions.

#### *4.2. Determination of Coal Meso-Mechanical Parameters*

First, coal compression and tension simulations were made with PFC software to correlate the meso- and macro-mechanical parameters of coal.

Data regression yielded a correlation coefficient of 0.992 (R2 = 0.992) between elastic modulus and meso-mechanical parameter, 0.998 (R2 = 0.998) between Poisson's ratio and meso-mechanical parameter, 0.997 (R<sup>2</sup> = 0.997) between compressive strength and meso-mechanical parameter, and 0.995 (R2 = 0.995) between tensile strength and mesomechanical parameter. The empirical equations between these coefficients are shown in Equations (2)–(5).

$$\frac{E}{E\_c} = a\_1 + b\_1 \ln(\frac{k\_n}{k\_s}) \tag{2}$$

where *E* is the elastic modulus, GPa; *Ec* is the Young's modulus, GPa; *kn*/*ks* is the stiffness ratio; *a*<sup>1</sup> = 1.652, and *b*<sup>1</sup> = −0.395.

$$
\mu = a\_2 + b\_2 \ln(\frac{k\_n}{k\_s}) \tag{3}
$$

where *μ* is the Poisson's ratio; *a*<sup>2</sup> = 0.111; *b*<sup>2</sup> = 0.209.

$$\frac{\sigma\_{\mathfrak{C}}}{\overline{\sigma}} = \begin{cases} a\_3 \left(\frac{\mathfrak{v}}{\sigma}\right)^2 + b\_3 \frac{\mathfrak{v}}{\sigma}, & 0 < \frac{\mathfrak{v}}{\mathfrak{C}} \le 1 \\\ c\_1 & \text{, } 1 < \frac{\mathfrak{v}}{\sigma} \end{cases} \tag{4}$$

where *σ<sup>c</sup>* is the compressive strength, MPa; *σ* is the parallel connection normal strength, MPa; *τ* is the parallel connection tangential strength, MPa; *a*<sup>3</sup> = −0.965; *b*<sup>3</sup> = 2.292; and *c*<sup>1</sup> = 1.327.

$$\frac{\sigma\_l}{\overline{\sigma}} = \begin{cases} a\_4 \left(\frac{\overline{\pi}}{\overline{\sigma}}\right)^2 + b\_4 \frac{\overline{\pi}}{\overline{\sigma}}, & 0 < \frac{\overline{\pi}}{\overline{\sigma}} < 1\\ c\_2 & , \ 1 < \frac{\overline{\pi}}{\overline{\sigma}} \end{cases} \tag{5}$$

where *σ<sup>t</sup>* is the tensile strength, MPa; *a*<sup>4</sup> = −0.174; *b*<sup>4</sup> = 0.463; *c*<sup>2</sup> = 0.289. Based on Equations (2)–(5) and the data in Table 1, the coal mesoscopic parameters required for simulation can be retrieved as shown in Table 2.

**Table 1.** Macro-mechanical characteristics of the moulded coal specimen.


**Table 2.** Mesoscopic mechanical parameters of the DEM numerical model.


It has been demonstrated that among all fluid parameters, residual pore size *a*<sup>0</sup> and fluid viscosity *μ* make the greatest difference to permeation pressure–stress coupling during permeation. After repeated simulation, fluid computational parameters were worked out, as presented in Table 3.


**Table 3.** Computational parameters of fluid.

Figure 9 compares the numerical simulation results with the test results of gas-bearing coal under the stress path of triaxial loading. From this diagram, at the end of the loading path of axial pressure, an oblique shear crack appears in the specimen. The stress-strain curve and failure form from numerical simulation almost entirely agrees with the laboratory test results. This suggests that our numerical simulation model and meso-mechanical parameters are appropriate enough for subsequent meso-mechanical simulation.

**Figure 9.** Comparison of physical test and numerical simulation results.

#### *4.3. Crack Number Characteristic Analysis*

To observe crack development inside the specimen under different paths, a cracks count–axial strain plot shown in Figure 10 was drawn. The cracks count inside the specimen changes in much the same way. As axial stress increases, cracks count first increases slowly and then increases quickly, especially near the site of peak stress. Finally, the rate of increase slows down and gradually stabilizes. The evolution of tension and shear-induced cracks count is not much different: it first increases and then stabilizes. However, as tensile strength is greater than shear strength in coal particles, there are many more tensile cracks than shear cracks.

Figure 11 compares the cracks counts at the end of simulation under different paths. In Test 1, the total model cracks count is 4.36 × <sup>10</sup>3, including 3.72 × 103 tensile cracks, accounting for 85.29% of total cracks; and 0.64 × 103 shear cracks, accounting for 14.71% of total cracks. In Test 2, the model cracks count is 7.29 × <sup>10</sup>3, including 6.05 × <sup>10</sup><sup>3</sup> cracks, accounting for 82.91% of total cracks; and 1.25 × 103 shear cracks, accounting for 17.09% of total cracks. In Test 3, the model cracks count is 5.65 × <sup>10</sup>3, including 4.46 × 103 tensile cracks, accounting for 78.93% of total cracks; and 1.19 × <sup>10</sup><sup>3</sup> shear cracks, accounting for 21.07% of total cracks. By cracks count, Test 2 > Test 3 > Test 1; by cracks proportion, from Test 1 to Test 3, the proportion of shear cracks gradually increases, suggesting that both tensile cracks and shear cracks are present at the time of failure. For this reason, the model displays a composite tensile–shear failure, although the specimens are more prone to shear failure under loading and composite loading/unloading paths.

**Figure 10.** Relationship between the number of cracks and axial strain. (**a**) Test 1, (**b**) Test 2, (**c**) Test 3.

**Figure 11.** Characteristics of the number of cracks under different paths.

## **5. Conclusions**

In Test 1, the specimen strength is the highest, with Δ*σ* = 33.750 MPa. In Test 2, the specimen strength is the second highest, with Δ*σ* = 33.750 MPa. In Test 3, the specimen strength is the lowest, with Δ*σ* = 23.480 MPa. The specimen-bearing capacity is much lower under loading and composite loading/unloading paths.

Under the three paths, as the test goes on, permeability first reduces then increases; the valley deflection of permeability falls before the peak deflections of the ε1–Δ*σ* curve in all cases. Compared with the other paths, the specimen-permeability variation is the largest at the end of Test 3, proving that the specimens are more badly damaged in Test 3.

The total cracks count is the largest in Test 2. From cracks proportion, from Test 1 to Test 3, the proportion of shear cracks in the model gradually increases, suggesting that both tensile and shear cracks are present at the time of macroscopic failure. For this reason, the model displays a composite tensile–shear failure, although the specimens are more prone to shear failure under loading and composite loading/unloading paths.

**Author Contributions:** Conceptualization, H.S. (Haibo Sun), B.Z., Z.S., B.S. and H.S. (Hongyu Song); Writing—original draft, H.S. (Haibo Sun); Funding acquisition, B.Z.; Formal analysis, H.S. (Haibo Sun), B.Z. and Z.S.; Software, H.S. (Haibo Sun), B.S.; Data curation, H.S. (Haibo Sun), B.Z. and H.S. (Hongyu Song). All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** All relevant data presented in the article are according to institutional requirements and, as such, are not available online. However, all data used in this manuscript can be made available upon request to the authors.

**Conflicts of Interest:** The authors declare no conflict of interest.
