Flow experiments were conducted on six cylindrical coal samples with single fractures and four cubic coal samples with complex fracture networks under both confining and true triaxial stress loading conditions, respectively. These samples were obtained from the Xichenzhuang coal mine in Jincheng City, Shanxi Province, China. The results of the industrial analysis for the coal samples are shown in Table 1. The mineralogical compositions of the coal sample were detected via X-ray diffraction (D8 Advance, State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University). The results revealed that the selected coal samples mainly consisted of 64.96% kaolinite, 16.22% illite, 9.11% quartz, 5.71% dolomite, and 4% pyrite and calcite. Additionally, the selected coal sample exhibited a density of 1.31 g/cm³.

All cylindrical coal samples were cut from a single coal block without macro fractures and had a volume of π × 25 × 25 × 50 mm³. Moreover, all cubic coal samples were cut from the same coal block and had a volume of 50 × 50 × 100 mm³ (Figure 1). The Brazilian split method was used to generate single fractures in all cylindrical coal samples. The surfaces of all samples were carefully polished to minimize the surface roughness (<0.01 mm), which helps to reduce the end effects. Furthermore, all samples were dried for 24 h at 78°C to prevent any residual liquid from influencing the flow experiments.

The flow experiments were conducted using the experimental system (RTX-3000, GCTS Company, the USA) for high-temperature and high-stress rock mechanics under both confining and true triaxial stress loading conditions (Figure 2). The system was mainly used to assess the mechanical properties and flow characteristics of rock, concrete, and coal under complex stress conditions. The testing system exhibited a maximum axial stress capacity of 3000 kN, a maximum horizontal stress capacity of 200 MPa, a maximum confining stress capacity of 200 MPa, a maximum pore fluid pressure of 200 MPa, and a maximum temperature of 200°C. Further details about the experimental apparatus are available in other sources (Peng et al., 2019; 2020).

Before the flow test, the surfaces of both cubic and cylindrical coal samples were uniformly coated with silicone rubber and then wrapped with heat shrink tubing. Subsequently, the coal sample was installed into the loading cell, and the silicone oil was injected into the loading cell. Afterward, the initial stress condition was sequentially loaded onto the coal samples. For flow experiments conducted under true triaxial stress loading, the initial values for the three principal stresses (σ₁, σ₂, and σ₃) and confining stress (σ_c) were set at 9, 6, 8, and 4 MPa, respectively. For flow experiments conducted under confining stress loading, the initial confining stress (σ_c) was set at 1.5 MPa. Finally, the flow experiments conducted under different stress conditions were conducted, and the loading paths of confining stress and true triaxial stress are shown in Figures 3A, B, respectively. During all flow experiments, the pore pressure was carefully maintained at a level lower than the confining stress to prevent the leakage of N₂ from the contact area between the coal sample and heat shrink tubing.

The detailed experimental steps for confining stress loading are as follows: 1) N₂ was injected into the coal sample until the pressure at both the inlet and outlet of the experimental setup reached 1 MPa. 2) The outlet valve was opened to reduce the pressure at the outlet to 0 MPa. Subsequently, the outlet valve was closed to allow N₂ flow through the coal sample, and the change in the pressure difference between the inlet and outlet of the apparatus was recorded under the initial stress condition. 3) The confining stress was increased (Figure 3A), and steps a) and b) were repeated to record the change in pressure difference between the inlet and outlet under different stress conditions.

The detailed experimental steps for true triaxial loading are as follows: 1) To explore the influence of principal stress σ₁ on the permeability of the coal samples, a) N₂ was injected into the coal sample until the pressure at both the inlet and outlet of the experimental setup reached 3 MPa. b) The flow experiment was conducted, and the change in pressure difference between the inlet and outlet was recorded under the initial stress condition before achieving equilibrium. c) The σ₂ and σ₃ values were set at 6 and 8 MPa, respectively. The σ₁ was then gradually increased in intervals of 2–17 MPa, and steps a) and b) were repeated for each increment. 2) To explore the influence of principal stress σ₂ on the permeability of fractured coal samples, a) σ₁ and σ₃ were maintained at 17 and 8 MPa, respectively. b) σ₂ was gradually increased in intervals of 2–14 MPa. The flow experiment was conducted, and the change in the pressure difference between the inlet and outlet of the experimental setup was recorded under each stress condition. 3) To explore the influence of principal stress σ₃ on the permeability of fractured coal samples, a) σ₁ and σ₂ were maintained at 17 and 14 MPa, respectively. b) σ₃ was increased in an interval of 2–16 MPa. The flow experiment was conducted, and the change in the pressure difference between the inlet and outlet of the experimental setup was recorded under each stress condition.

In the experiment, the flow was assumed to be isothermal and adhered to Darcy’s law. The transient pulse method was used to measure the coal permeability under different loading conditions. The coal permeability was calculated based on the pressure difference between the inlet and outlet of the experimental setup under a specific stress condition, as Eq. 1 (Lin and Kovscek, 2014; Zhou et al., 2020). k = μ ς V ln Δ P i / Δ P f / 2 Δ t A / L (1) where k represents the permeability of the coal sample, μ denotes the viscosity of N₂, ϛ indicates the volume compressibility of N₂, and V signifies the reference volume (0.0025 m³). ΔP _i and ΔP _f represent the initial and final pressure differences between the inlet and outlet, respectively. Δt denotes the duration of the N₂ flow, A indicates the crossing area of the coal sample (0.025 m²), and L denotes the length of the coal sample (0.1 m).

The internal images of all coal samples were obtained using the medical X-ray computerized tomography scanning system (SOMATOM Scope). The internal fractures were reconstructed using image processing technology such as a non-local means filter (Buades et al., 2008) and watershed segmentation (Jones et al., 2007) (Figure 4). The Avizo image processing tool was used to measure crucial fracture structure parameters, such as porosity ϕ _f , fracture aperture a, the azimuth and dip angle of fracture θ ₁ and θ ₂ , and the maximum length of fracture branch (l ^max ), from the reconstructed coal sample fractures. During the flow experiments, the coal sample was oriented to align the flow direction parallel to the fracture surface. Therefore, the azimuth angle of fracture θ ₁ was 0°. The tortuosity of fracture τ, the fractal dimension of fracture D _f , tortuosity fractal dimension of fracture D _Tf are defined as follows.

The tortuosity of fracture τ is defined as the ratio of the fracture actual length L _t to the cell unit length L ₀ (Ghanbarian et al., 2013). τ = L t / L 0 (2)

Yu and Cheng and Miao et al. concluded that the cumulative number N of fractures in natural rock, with a fracture length L greater than or equal to ≥l, follows the fractal scale relationship of Eq. 3 (Yu and Cheng, 2002; Miao et al., 2015a): N L ≥ l = l max / l D f (3) where D _f represents the fractal dimension of fracture, with 1 < D _f < 2 in two dimensions and 2 < D _f < 3 in three dimensions. Differentiating Eq. 2 with respect to l yields the following Eq. 4. d N l = − D f l max D f l − D f + 1 d l (4)

As the ratio of minimum to maximum fracture length (l ^min /l ^{max r} ) in natural rock was less than 0.01, the relationship between fractal dimension D _f , l ^min /l ^max , and porosity ϕ can be described using the fractal theory of Eq. 5 (Yu and Li, 2001). ϕ f = l min / l max d E − D f (5) where dE represents the number of dimensions of the Euclidean space, with dE = 2 for two dimensions and dE = 3 for three dimensions.

Liu et al. (2016) and Li B. et al. (2016) indicated a relationship between the characteristic length of fracture L ₀ , actual length L _t , fracture aperture a, and tortuosity fractal dimension D _Tf , expressed as follows: L t = a 1 − D T f L t 0 D T f = L 0 cos ⁡ θ 2 D T f a 1 − D T f (6) where a = βl ⁿ , β represents the proportionality coefficient, and its value generally ranges from 10^–4 to 10^–1. For self-similar fracture structures, n is equal to 1 (Klimczak et al., 2010; Torabi and Berg, 2011).

The characteristic parameters of single fractures and complex fracture networks are shown in Table 2. These parameters were used to predict the coal permeability.

The permeability evolution of single-fractured coal under different effective confining stresses is shown in Figure 5A. The findings indicated a negative power relationship (i.e., SD model) between fracture permeability and effective confining stress, and the fitting parameters are shown in Table 3. This suggests that the SD model effectively described the permeability evolution of single-fractured coal under confining stress loading. At low confining stress levels, the fracture underwent minimal compression, resulting in its highest permeability. As the confining stress increased, the fracture underwent further compression, leading to a gradual increase in the fluid flow resistance and a corresponding decrease in fracture permeability.

According to the cubic law, which approximates the fluid flow inside a single fracture as a flow between two plates, the fracture aperture can be calculated, as illustrated in Figure 5B. The results revealed that during the first three loading steps, the fracture aperture decreased by ∼54.63%, 51.79%, 53.72%, 52.19%, 57.20%, and 57.83% of the total closure for SC1, SC2, SC3, SC4, SC5, and SC6 samples, respectively. Correspondingly, the permeability decreased by ∼43.49%, 38.06%, 45.80%, 39.52%, 40.81%, and 42.12% for samples SC1, SC2, SC3, SC4, SC5, and SC6, respectively. However, during the final three loading steps, the fracture aperture decreased by ∼19.26%, 20%, 22.49%, 23.08%, 18.26%, and 19.72% of the total closure for SC1, SC2, SC3, SC4, SC5, and SC6 samples, respectively. Moreover, permeability decreased by ∼22.91%, 22.15%, 24.29%, 26.93%, 23.93%, and 25.49% for samples SC1, SC2, SC3, SC4, SC5, and SC6, respectively. The results indicated that as the effective stress increased, fracture closure initially occurred more rapidly, slowed down, and then eventually tended to stabilize. This pattern suggests that with increasing confining stress, the permeability first rapidly decreased, followed by a gradual decrease, and eventually reached a constant value. Therefore, this phenomenon elucidates why the fractured coal exhibited a higher sensitivity to stress in permeability compared with intact coal.

Figure 6 illustrates the correlation between true triaxial stresses and the permeability of fractured coal. The findings indicated a consistent decrease in the permeability of fractured coal rock with increasing principal stresses (σ₁, σ₂, and σ₃) across the three loading steps. Additionally, the loading of each principal stress σ₁, σ₂, and σ₃ had different effects on fracture permeability. Particularly, the permeability of the ST1 sample decreased by ∼43.08%, 14.84%, and 42.08% during the entire loading process. Throughout the entire loading process, the permeability of the ST2 sample decreased by ∼65.74%, 14.29%, and 19.97%. Similarly, the permeability reductions for the ST3 sample were ∼34.03%, 55.85%, and 10.12%, while those for the ST4 sample were ∼35.97%, 46.51%, and 17.52% during the entire loading process. The permeability of ST1 was significantly sensitive to σ₁ and σ₃, while the permeability of ST2 was mainly sensitive to σ₁. However, the permeability of ST3 and ST4 was influenced by both σ₁ and σ₂.

This sensitivity can be attributed to the complex fracture networks within the cubic sample, which exhibited varying crossing angles with each principal stress (Figure 7). The macro fractures within the ST1 sample exhibited a crossing angle of 0° with σ₂, while the crossing angle between the fracture surface and σ₁ and σ₃ was equal. As σ₁ or σ₃ increased, the macro fractures within ST1 underwent compression, leading to a decrease in the permeability of the coal sample. Similar mechanisms were applied to elucidate the permeability evolution of TS2, TS3, and TS4 during the true triaxial stress loading (Figures 7B–D). Thus, the interaction between the dip and azimuth angles of the fractures and the true triaxial stress influenced the permeability of the fractured coal, resulting in anisotropic fluid flow within the coal. This suggests that the dip and azimuth angles of the fracture surfaces should be considered when developing a permeability model for coal rock with complex fracture networks.