The gas adsorption constant of coal characterizes the adsorption capacity of coal to methane, reflecting the maximum gas adsorption capacity and coal quality characteristics (Lei, 2017). The purpose of determining the gas adsorption constant in the laboratory is to calculate the gas content in coal seam, which is an indirect method to determine the gas content in coal seam. At present, the roadways in a vast majority of mines are located along the coal seam, so the indirect method is widely used to determine the gas content in coal seams (Zhao and Jia, 2019; Plaksin and Kozyreva, 2021). High pressure volumetric method is used to determine adsorption constant (7 gas adsorption capacities corresponding to 7 equilibrium pressures with approximate average distribution in the range of 0∼5 MPa are measured), and the HCA-1 high pressure capacity method (Chen et al., 2012; Zhang et al., 2015; Zhou et al., 2019) is used to determine the experimental determination process of gas adsorption device as shown in Figure 1.

Ninety coal samples were collected from 90 coal mines in 13 provinces, including Guizhou, Jiangxi, Anhui, Xinjiang, Sichuan, Liaoning, Shaanxi, Henan, Shanxi, Jilin, Qinghai, Yunnan and Inner Mongolia. According to the national standards such as GB/T 482-2008 “Methods for Coal Seam Samples”, GB/T 474-2008 “Coal Sample Preparation Methods”, GB/T 477-2008 “Coal Sample Sieving Test Methods” and other national standards, about 5 kg powder mixed coal samples taken from freshly exposed coal seams, after being marked with the mine name and sampling location on the tightly sealed packages, were sent to the laboratory for numbering, registration, sampling, drying, crushing, sieving, etc., before they were prepared to be coal samples of different particle sizes for inspection.

According to the coal industry standards and national standards such as MT/T 752-1997 “Method for Determination of Methane Adsorption of Coal”, GB/T 212-2008 “Method for Industrial Analysis of Coal”, GB/T 217-2008 “Method for Determining the True Relative Density”, and GB/T 6949-2008 “Method for Determination of Apparent Relative Density of Coal”, 90 coal samples were measured in the laboratory using HCA-1 high pressure volumetric gas adsorption device, industrial analysis instrument, density meter and other instruments and equipment. More specifically, the coal quality indexes and gas basic parameters such as moisture M _ad , ash A _ad , volatile V _daf , true density TRD, apparent density ARD, porosity F, atmospheric adsorption capacity Q 0 , gas adsorption constant a, b, initial velocity index of gas emission Δ p and firmness coefficient of coal f were measured in the laboratory. The measured results of the 90 coal samples are shown in Table 1.

From Table 1, the variation ranges of the parameters of 90 coal samples measured are as follows: M _ad is 0.48–9.31%, A _ad is 2.71–70.73%, V _daf is 5.33–55.91%, TRD is 1.33–2.27 g.cm⁻³, ARD is 1.28–2.11 g.cm⁻³, F is 2.19–13.04%, Q 0 is 1.4558–7.4993 cm³.g⁻¹, a is 12.6231–38.2783 cm³.g⁻¹, b is 0.6361–1.76294 MPa⁻¹, Δ p is 4–43 mmHg, and f is 0.15–1.80. From the distribution characteristics of the measured data, the selected coal samples are universal and extensive.

In general, multivariate nonlinear regression is defined as follows: based on the fact that the nonlinear function has multiple derivatives in the independent variable range, the nonlinear function relationship between multivariate independent variables and dependent variables is established by transforming the nonlinear relationship into a generalized linear relationship and carrying out regression analysis by means of mathematical statistics. The general nonlinear regression model can be expressed as follows: y = f x , c + ε (1) In the equation, x represents observable random independent variables; c represents parameter vectors to be estimated; y represents independent observation variables; ε represents random variables. Eq. 1 is often used to solve the estimated value of the parameters using the least square method to minimize the sum of squares of the residual. The sum of squares function of residual error and its first derivative are: S c = ∑ i = 1 n ε i 2 = ∑ i = 1 n y i − f X i , c 2 d S d c = 2 y i − f X i , c − d f X i , c d c = 0 (2)

The c of the above equation can be solved by calculating the Equation 2, and the global minimum can be estimated.

Taking gas adsorption constants a and b as dependent variables, 9 factors such as M _ad , A _ad , V _daf , TRD, ARD, F, Q 0 , Δ p , and f as independent variables, and 90 coal mine sample data in Table 1 as samples, the curve equation which is most suitable for gas adsorption constants a, b and related parameters is determined by using chart construction program and curve estimation in SPSS data software. After eliminating the independent variable whose fitting coefficient R 2 is less than 0.300, the curve estimation results of a and two independent variables Q 0 , Δ p as well as b and two independent variables V _daf , ARD are obtained are shown in Table 2.

From Table 2, for a and b, five commonly used unary curve equations such as linear, logarithmic, quadratic, power and index are established respectively. By comparing the coefficient of determination R 2 and significance level Sig of the five curves, it is known that the adjustment of conic R 2 = 0.717 fitted by a and Q 0 is the largest, and the significance level S i g . = 0.482 > 0.05. This shows that the quadratic curve is not significant, while the adjustment of logarithmic curve equation R 2 = 0.711 is larger. The significance level S i g . is 0.000 < 0.05. Therefore, a logarithmic curve equation is established for a and Q 0 . The adjustment of quadratic curve equation R 2 = 0.523 fitted by a and Δ p is the largest, significance level S i g . = 0.003 < 0.05. Therefore, a quadratic curve equation is established for a and Δ p , and the adjustment of logarithmic curve equation fitted by b and V _daf is 0.698, significance level S i g . = 0.000 < 0.05. Therefore, a logarithmic curve equation is established for b and V _daf . The adjustment of quadratic curve equation R 2 = 0.378 fitted by b and ARD is the largest, and the significance level is 0.000 < 0.05. Therefore, a quadratic curve equation is established for b and ARD.

Let the gas adsorption constant a be the equal of the dependent variable f i i = 1,2 and b the dependent variable f i i = 3,4 , the atmospheric adsorption capacity Q 0 be the equal of x 1 , the initial velocity of gas emission Δ p be the equal of x 2 , the volatile matter of coal V _daf be the equal of x 3 and the apparent density of coal ARD be the equal of x 4 . The nonlinear regression was carried out by using SPSS data analysis software, and the nonlinear regression Equation 3 was obtained after 9, 2, 5, and 8 iterations respectively. f 1 = 7.850 + 16.292 ⁡ ln ⁡ x 1 f 2 = 11.378 + 1.151 x 2 − 0.015 x 2 2 f 3 = 2.238 − 0.383 ⁡ ln ⁡ x 3 f 4 = − 7.641 + 10.621 x 4 − 3.145 x 4 2 (3)

The gas adsorption constants a and b are taken as dependent variables and f ₁ , f ₂ , f ₃ and f ₄ as independent variables respectively, and the linear regression model of Equation 4 is obtained. a = − 3.576 + 0.7861 f 1 + 0.348 f 2 b = − 0.208 + 0.851 f 3 + 0.351 f 4 (4)

After testing, the decision coefficients R ² of a and b are 0.737 and 0.728 respectively, indicating that f ₁ and f ₂ can explain 73.7% of the changes of the dependent variable a and 72.8% of the changes of the dependent variable b, as well as high goodness of fit of regression. The significance level (the probability value corresponding to t statistics) is S i g . = 0.002 < 0.05, S i g . = 0.029 < 0.05 respectively, which suggests that there is a significant correlation between independent variables and dependent variables. Durbin-Watson values of 1.617 and 2.034 are close to 2, respectively, which suggests that the sequences are independent of each other and there is no autocorrelation. The maximum variance expansion factor (VIF) is 2.112 and 1.483 respectively, which is less than 5, which suggests that there is no strong multicollinearity among independent variables.

The multiple nonlinear regression models of a and b are obtained by putting f ₁ , f ₂ , f ₃ , and f ₄ into Equation 4, as shown in Eq. 5. a = 6.5536 + 12.8055 ⁡ ln ⁡ Q 0 + 0.4005 Δ p − 0.0052 Δ p 2 b = − 0.9855 − 0.3259 ⁡ ln ⁡ V d a f + 3.728 A R D − 1.1039 A R D 2 (5)

Atmospheric adsorption capacity, also known as residual gas capacity, refers to the maximum gas capacity of coal adsorption under atmospheric pressure. In the laboratory, the degassing method is used to make the coal sample in the state of negative pressure, and then make the coal sample adsorb methane gas at atmospheric pressure to reach adsorption saturation. Atmospheric adsorption is an important part of the prediction of gas emission from mining face, and it is also one of the prevention and control indexes of coal and gas outburst (Wu et al., 2011; Lu et al., 2022). Both an and a characterize the adsorption and desorption performance of coal, the difference between them is that the adsorption pressure is different, the adsorption pressure is standard atmospheric pressure, and the adsorption pressure of an is the limit.

Figure 2 shows the overall variation trend of a under different Q 0 and different Δ p . With the increase of Q 0 , a shows an obvious increasing trend. And when Q 0 increases to a certain extent, the increasing amplitude of a decreases gradually and tends to reach the limit state. This is because the adsorption capacity of coal is determined by the specific surface area of coal. Under the same conditions, the larger the Q 0 , the stronger the adsorption capacity of coal. The gas adsorption constant a is an index to measure the gas adsorption capacity of coal. The larger the specific surface area is, the stronger the adsorption capacity is, the greater the a is. Under the same conditions, the adsorption capacity of coal increases with the increase of gas pressure, that is, the greater the Q 0 , the greater the a. When Q 0 increases to a certain extent, the specific surface area of coal decreases gradually with the increase of adsorption capacity, the ability of coal to adsorb methane weakens gradually, the growth rate of adsorption gas gradually slows down, and a gradually reaches the limit.

The initial gas emission velocity of coal represents the gas release capacity of coal at the moment of pressure relief. In Figure 2, with the increase of Δ p , a shows the trend of quadratic function from low to high to low, and most coal samples increase with the increase of Δ p before the extreme point. This is because the size of Δ p first depends on the strength of coal adsorption capacity, and under the same gas pressure, the coal with larger specific surface area has a larger amount of gas adsorbed. Under the condition of the same gas pressure, the gas released by the coal with good adsorption performance is larger than that released by the coal with relatively poor adsorption performance (Tang, 2014; Saghafi, 2017; Si et al., 2021b). That is to say, the greater the initial velocity Δ p of the gas released by the coal body in the same period of time, the stronger the adsorption performance of the coal body under the same conditions, and the greater the a. However, after the extreme point, very few kinds of coal decrease with the increase of Δ p , which may be due to the fact that the coal sample is relatively dry and the water content is very low during the preparation of a few coal samples, the particle size of the prepared coal sample may be more concentrated near 0.25 mm, the particle size of the coal sample with a certain mass is relatively larger, the total pore volume is also relatively large, and the gas migration channel in coal is unobstructed. The initial amount of gas desorption is also relatively large in the same period of time (Zhao and Niu, 2022). However, when determining the gas adsorption constant a, the quantity of this coal sample will decrease with the increase of particle size at a certain mass, and the total specific surface area can be considered to maintain invariable. Therefore, a will not increase with the increase of coal sample particle size, and the adsorption saturation time may be prolonged due to the increase of coal sample particle size. This explains why a decreases with the increase of Δ p after the extreme point.

According to the multivariate nonlinear regression equation5, a curved surface fitting diagram of the influence of atmospheric pressure adsorption capacity Q 0 and gas emission initial velocity on gas adsorption constant a is established, as shown in Figure 3. The influence of Q 0 on a shows an upward curve, that is, when Δ p is in the range of 4–38 mmHg, a increases significantly with the increase of Δ p . when Δ p is in the range of 39–43 mmHg, a shows a decreasing trend with the increase of Q 0 , but the decreasing trend is not obvious. The three-dimensional curved surface reflects the evolution process from trough to peak, that is, the change process of specific surface area of coal and gas pressure.

The gas adsorption constant b characterizes the speed of coal adsorption and desorption of gas. The overall change law of b under different volatile matter V _daf and apparent density ARD can be seen from Figure 4. With the increase of volatile matter V _daf , b attenuates as a logarithmic function, and when V _daf increases to a certain extent, b decreases to a minimum. This is because V _daf characterizes the metamorphic degree of coal, and the larger the V _daf , the lower the metamorphic degree of coal, the weaker the adsorption capacity of coal, the slower the adsorption speed, and the longer the time to reach adsorption saturation, the smaller the b. On the contrary, the smaller the V _daf , the stronger the adsorption capacity of coal, the faster the adsorption speed, and the shorter the time to reach the adsorption saturation state, the greater the b. With the increase of apparent density ARD, b shows the trend of quadratic function from low to high and from high to low. This is because ARD characterizes the pore volume of coal. When ARD increases, the pore volume of coal increases, the pore size of coal increases, the gas escape channel becomes shorter, the resistance to gas desorption decreases, and the desorption rate increases, b increases accordingly. When ARD increases to a certain extent, some types of coal may change due to higher degree of coal metamorphism, the arrangement of coal molecular structure changes from disorder arrangement to neat arrangement, the pore volume decreases rapidly, the gas desorption path lengthens with higher resistance, b decreases accordingly.

According to the multivariate nonlinear regression model (5), a surface fitting diagram of the effect of coal volatile matter V _daf and coal apparent density ARD on gas adsorption constant b is established, as shown in Figure 5. The impact trend of V _daf on b is in a shape of downward curve, that is, when ARD is in the range of 1.28–1.69 g/cm³, the metamorphic degree of coal decreases with the increase of V _daf , and the b value increases significantly with the increase of V _daf . when ARD is in the range of 1.70–2.11 g/cm³, with the increase of V _daf , the metamorphic degree of coal decreases significantly. The three-dimensional curved surface reflects the influence of V _daf - ARD correlation on the evolution of b from peak to valley, that is, the pore volume of coal changes from high to low and low to high.