Beyond the fracture boundary, only elastic deformation occurs due to the wave blast. The elastic deformation in the unit volume of rock δ t is according to the following formula (Cai et al., 2014; Li et al., 2018; Cheng et al., 2022): δ t = 1 2 σ r ε r + σ θ ε θ = σ r 2 2 E m 1 + λ 2 .

Thus, the energy consumed due to the elastic deformation is E t = ∫ r 1 R E 2 π r δ E d r , E t = π 1 + λ 2 p m r m 2 2 E m α − 1 1 − r m R E 2 1 − α ,

where ε r and ε θ are the strain along the radial and tangential directions, respectively, relative to the borehole, m and α are coefficients where α = 2 − v / 1 − v , v = 0.3 , α = 1.57 , E m the Young’s Modulus of coal, 4.6 GPa, r m = b 1 / 2 is the radius of the fracture, λ is a coefficient equivalent to v / 1 − v = 0.43 , R E is the radius at which the elastic deformation starts to occur, R E = 1 + v E m σ t + σ w r m 2 r , 4.0 m, σ t is the tensile strength, and σ w is the initial stress.

The immediate stress induced on coal when air blasting occurs is calculated according to the following equation: p m = ρ v p ⋅ ρ k y v s 2 2 ρ v p + ρ k y v s ,

where ρ is the coal density (1334 kg/m³), v p is the compressive wave velocity (1856 m/s) determined by laboratory testing, ρ k y is the density of the compressed air (83.85 kg/m³), and v s is the compressed air velocity at the releasing nose (243.7 m/s) as determined by measurement.

After the air blast, the fracture propagates to an extent beyond which the deformation remains elastic. Once the blasting wave energy is completely consumed, the fractures at the boundary between the fracture and elastic deformation zone tend to close under the action of load F . For ease of calculation, the friction is ignored and the closure of the fractures is elastic. The energy consumption time is very short (approx. 25  μ s ), and the length of the coal is l (Figure 1).

The probability of damage to the microelement of the coal is assumed to be the damage variable D according to D = ∫ 0 J P x d x = 1 − exp − J / J 0 m .

If the rock microelement strength is J , then σ 1 = σ c , ε 1 = ε c when the peak point gradient is 0, and the relevant microelement strength is J c . When the rock at the closure zone has residual strength, D = 1 ; hence, exp − J / J 0 m = 0 . The strength parameters in the microelement, J , J 0 , and m can be calculated as follows: m = − σ c − 2 μ σ 3 σ c − 2 μ σ 3 − E ε c + h E ε c ln σ c − 2 μ σ 3 − E m ε c + ξ E m ε c h E m ε c , J = σ 1 * + 2 σ 3 * + sin ⁡ φ 3 3 − sin 2 φ σ 1 * − σ 3 * 2 , J 0 = J c J 1 σ c − 2 μ σ 3 − E m σ c + ξ E m ε c σ c − 2 μ σ 3 1 / J 1 ,

where σ 1 * and 2 σ 3 * are the effective stresses of σ 1 and σ 3 , respectively, and φ is the internal friction angle.

The fracture closure coefficient ξ can be calculated according to ξ = 1 − σ r − 2 μ σ r E m ε r ,

where μ is the Poisson's ratio, σ r is the residual strength in MPa, and ε r is the residual strain.

Considering the expansion of the fracture resulting from coal seam gas adsorption in the fracture, σ z is the summation of closure stress in coal fractures σ e (Meng and Li, 2015; Vyacheslav et al., 2016; Peng and Zheng, 2020; Liu et al., 2022) and expansion stress σ p , and the expansion stress due to the coal seam gas adsorption in the fractures is calculated as follows (Xue et al., 2015; Chu et al., 2022): σ p = 2 a R T ρ 1 − 2 v ln b p / 3 V ,

where a and b are adsorption coefficients, p is the compressive stress of the coal seam gas (1.2 MPa), v is the Poisson's ratio, ρ is the coal density (334 kg/m³), R is the molar gas constant (8.31 J/(mol·K)), T is the absolute temperature (297.35 K), and V is the molar volume (22.41 L/mol).

The effective stress σ e and actual stress σ Z are as follows: σ Z = σ e + σ p , σ Z = κ σ e , where κ is the effective stress coefficient according to κ = 1 − K / K s , where K is the water bulk modulus and K s is the coal solid particle bulk modulus. κ is 0.7.

During the blasting, the decay of the blasting wave peak in the coal mass is calculated as follows: p E = p E r m r E 3 .

Assuming the total force on the coal in the elastic zone results only from the blasting wave, p E = F .

Considering the conservation of energy, the energy in the elastic zone is not completely released, with most of the energy converted to kinetic energy, while a minor portion is stored in the elastic zone. E t = E t 1 + Q 1 .

The simplified form is E t ′ = η E t .

The energy from the elastic zone is assumed to contribute to the fracture closure according to E t ′ = η E t = F l ⁡ sin φ 2 cos φ 2 , where η is the energy coefficient, and it equals 1 when all energy is used for fracture closure.

The length of the fracture closure is 1 = η π 1 + λ 2 p m r 0 2 2 E m α − 1 1 − r 0 R E 2 1 − α p m r m r E 3 ⁡ sin φ 2 cos φ 2 .

The effective closure length in the fracture is l y = ξ l .

After compressed air blasting, fractures form, leading to an increase in the gas flow channels. In contrast, the fractures close to the elastic zone tend to close, resulting in decreased gas flow channels. Therefore, the borehole for air blasting should be drilled within the fracture propagation zone. In addition, if the borehole height is more than the height of the fracture closure zone, the amount of coal seam drainage decreases greatly.

The effective drainage zone is assumed to be as far as L from the borehole according to L = a ∗ + b 1 + b 2 − l y , where a * is the blasting radius (0.094 m), b 1 is the thickness of the loose area, and b 2 is the thickness of the fracture propagation.

The diameter of the loosened area can be calculated as follows (Qian, 2010): b 1 = 1 − 2 v 1 + v 3 1 − v ρ c p 2 σ ∗ 1 / 3 1 2 0.61 E z 1 / 3 ρ c p σ ∗ 1 / 9 .

The diameter of the fracture propagation is calculated as follows (Qian, 2010): b 2 = ρ c p 2 4 1 + sin ⁡ θ 1 − sin ⁡ θ σ 0 1 / 3 1 2 0.61 E z 1 / 3 ρ c p σ ∗ 1 / 9 .

As such, L = 0.094 + 1 − 2 v 1 + v 3 1 − v ρ c p 2 σ ∗ 1 / 3 1 2 0.61 E z 1 / 3 ρ c p σ ∗ 1 / 9 + ρ c p 2 4 1 + sin ⁡ θ 1 − sin ⁡ θ σ 0 1 / 3 1 2 0.61 E z 1 / 3 ρ c p σ ∗ 1 / 9 − ξ η π 1 + λ 2 p m r 0 2 2 E m α − 1 1 − r 0 R E 2 1 − α p m r m r E 3 ⁡ sin φ 2 cos φ 2 , where v is Poisson's ratio (0.3), ρ is the coal density (1334 kg/m³), and c p is the compressive stress velocity in the coal mass (1856 m/s). η is 0.75, E Z is the energy resulting from the blast, σ * is the compressive strength of the coal (28.63 MPa), σ 0 is the tensile strength (0.462 MPa), θ = π / 18 , and φ = 2 π / 45 . Substituting all these parameters into the equation mentioned previously leads to L = 3.8 m .

The field testing was conducted in the transport roadway in a single coal mine. The compressed air blasting was initiated in a coal seam with a thickness of 3.6 m. The stress from the coal seam gas was 1.6 MPa, and the gas intake was around 6.11 m³/t. The seam was weak and the gas permeability was 0.0111 m²/MPa²·d. The roadway was reinforced by cable bolts and steel mesh.

A 0.5-m-deep borehole was drilled into the seam, followed by the installation of the blasting detonation. When the stress reached 60 MPa, the compressed air was released. Two to three blasts were performed in each borehole. Eventually, one drainage borehole and 12 blasting boreholes were formed. In addition, adjacent to the blasting boreholes, another 14 boreholes were drilled for observation (Figure 2). The observation boreholes were drilled 0.5 m deep into the seam with a drill bit 94 mm in diameter. All these boreholes were sealed thereafter. Blasting boreholes were drilled similarly. As soon as the compressed air blasting occurred, the drainage borehole was connected to the gas drainage system. All gas drainage parameters were the same in each borehole. The variations in coal seam gas drainage volume during the drainage were measured. The gas drainage volumes measured in each observation borehole were used to aid in identifying the locations of fracture closure.

As shown in Table 1 and Figure 3, blasting in 12 blasting boreholes showed increased gas flow in the observation boreholes. The observation boreholes within 2 m of the blasting holes had the largest gas flow. The gas flow decreased from the observation borehole 1 m (borehole 13) to that 1.5 m (borehole 16) from the blasting borehole. The flow then increased to the observation boreholes 1.8 m (boreholes 20, 21, 24, and 25) away from the blasting borehole before decreasing again to the observation boreholes 2 m (borehole 14) and 2.5 m (boreholes 19, 22, 23, and 26) away from the blasting borehole. The gas flow increase peaked at the observation borehole 3 m (borehole 15) from the blasting borehole and reached its minimum in the observation borehole 4 m (borehole 18) from the blasting borehole. The gas flow increase was more obvious in the zone within 3 m from the blasting borehole. The increasing rate of gas intake peaked in the zone 1.8–2.5 m from the blasting borehole. The increasing rate of gas flow peaked in the zone 1.5 m from the blasting borehole. The rate was as high as 55.56%, and the gas drainage was also enhanced most significantly in this area. The gas flow increase tended to decrease more than 2.5 m from the blasting borehole. The increasing rate reached its minimum in the zone 4 m from the blasting borehole, where it was close to the fracture closure zone.

After the air blasting, the rate of gas intake tended to increase and then decrease with distance from the blasting borehole. When approaching the fracture closure zone, the increasing rate reached its minimum. The increasing rate of the gas intake on the left-hand side of the blasting borehole was greater than that on the right-hand side. This might be attributed to the orientation of the in situ major principal stress. The observation boreholes above the blasting borehole (boreholes 19–22) had higher gas intake increase rates than those below the blasting borehole (boreholes 23–26). Although the induced stress level at the top and bottom of the blasting borehole by the blasting wave tended to be the same, the induced failure differed, with tensile and compressive failures dominating the top and bottom, respectively. As the tensile strength is always much less than the compressive strength, the influential area resulting from air blasting at the top would be much larger than that at the bottom.