1 Introduction

Biological-sedimentary impact on various plants or plants in different positions under a certain geological history causes the differences of coal quality (expressed in ultimate content), coal petrology (expressed in maceral composition) and ash contents (obtained from proximate composition analysis). During the metamorphic evolution, the physicochemical properties of coal experienced a substantial change, resulting into the division of coal into subbituminous coal (\(R_{{\rm o,max}} < 0.5\%\)), low volatile bituminous coal (\(0.5\% < R_{{\rm o,max}} < 0.9\%\)), mid-volatile bituminous coal (\(0.9\% < R_{{\rm o,max}} < 1.6\%\)), High bituminous coal (\(1.6\% < R_{{\rm o,max}} < 2.5\%\)) and anthracite (\(R_{{\rm o,max}}> 2.5\%\)). Physicochemical properties (mainly expressed in proximate composition, maceral composition and ultimate content) variation in coal leads to the adsorption characteristics difference of methane in coal, and many scholars did a lot of research on the influence factors of coal rank and physicochemical properties of coal on gas adsorption, in which the coal rank that can also influence the physicochemical properties is the main controlling factors.

Adsorption of methane in coal is an important research subject in coal mine safety and CBM exploration. Through adsorption isotherm of methane in coal, the critical desorption pressure in coal reservoir can be determined, the coalbed gas reserves can be estimated and the saturation of the CBM can be determined, finally, the extraction rate of coalbed methane can be predicted; The research for adsorption capacity of methane in coal can provide theoretical basis for industrial of CO2-methane displacement; and it is important for the prediction of gas emission quantity and the coal and gas outbursts for studying methane adsorption in coal. In other words, Sorption of gas in coal is an ongoing and extremely important area of research to enhance coalbed methane from coal seams.

Scholars at home and abroad have already done a lot of research work in the coal rank impact on physicochemical properties of coal and coal methane adsorption characteristics (Joubert et al. 1973, 1974; Faiz et al. 1992; Levy et al. 1997; Ceglarska-Stefańska and Brzóska 1998; Laxminarayana and Crosdale 1999; Zhang and Yang 1999; Clarkson and Bustin 2000; Özgen Karacan and Okandan 2000; Fu et al. 2005; Chalmers and Marc Bustin 2007; Faiz et al. 2007a, b; Crosdale et al. 2008; Day et al. 2008; Mares and Moore 2008; Pan et al. 2010; Chen et al. 2011; Pini et al. 2011; Yao et al. 2011; Xiao and Wang 2011; Hao et al. 2012; An et al. 2013). Resulting shows that coal methane adsorption characteristics is affected by many kinds of factors, including coal rank, coal type that can be expressed in terms of many physicochemical properties, e.g. proximate analysis, maceral composition, ultimate analysis, moisture content. In the research of coal rank effect on methane adsorption, most scholars believe that with the increase of coal rank, the adsorption capacity of coal increases monotonously, while Laxminarayana (Laxminarayana and Crosdale 1999, 2002) found a ‘U’ relation presents between coal rank and adsorption capacity. Formerly studies on effect of moisture existence on adsorption capacity have got the following achievements: Kim et al. (2011) studied the adsorption behaviors of CH4 on dry and wet coal and got the conclusion that the sorption capacity of CH4 on anthracite coal was higher than those on bituminous coals at a similar condition, also, they thought that the mutual solubility between the CH4-rich phase and aqueous phase as well as coal swelling should be considered in evaluating the sorption capacity of a wet coal seam. Cai et al. (2013a, b) investigated four coals from Northeast China and found that the data of petrographic, proximate and ultimate analyses, moisture content, ash yield had great effects on CH4 adsorption capacity of coals. Previous studies of coal maceral effect on adsorption capacity showed that vitrinite have the higher adsorption capacity than inertinite, while some researchers hold the opposite view. Maria Mastalerz (2004) and Cai et al. (2013a, b) analyzed eight coals with similar coal rank but varying petrographic for CH4 sorption capacity using a high-pressure adsorption isotherm technique and determined the coal quality and petrographic composition of the coals to study their relationships to the volume of CH4 that could be sorbed into the coal. Previous studies have concluded the qualitative and partly quantitative relationship between physicochemical properties of coal and their adsorption capacity without giving variation scope of adsorption parameters (Unsworth et al. 1989; Yalçin and Durucan 1991; Reich et al. 1992; Clarkson and Bustin 1999; Mastalerz et al. 2009; Charrière et al. 2010; Yao and Liu 2012).

Different from the former research of using Limited quantity of coal samples, focusing on a certain coal mine, or a certain influential factor, data obtained from the National Engineering Research Center of Coal Gas Control (NERC) and the related literatures at home and abroad are collected to discuss the effect of physicochemical properties of coal on methane adsorption capacity by observing the changing rule of physicochemical properties of coal and methane adsorption capacity with coal rank separately. Whereas, physicochemical data are scattered as the macromolecular structure and difference of environmental and metamorphic factors for the formation of a certain coal are complexity. Besides, because of the different research aims in different literatures, not all the data can meet requirements of this paper, then, we did some data transmission work to obtain a Semi-quantitative result and to strengthen the understanding of the factors influencing the adsorption capacity.

2 Methods

For the thorough research on coal metamorphism impact on gas adsorption capacity, this paper collected and summarized parameters of experimental adsorption isotherms, coal maceral, proximate composition and ultimate contents obtained from the National Engineering Research Center of Coal Gas Control (NERC) and related literatures at home and abroad, systematically discussed the coal rank impact on its physicochemical properties and methane adsorption capacity, in which the coal rank was shown in maximum Vitrinite reflectance \(R_{\text{o,max}}\). In this paper, the changing rules of physicochemical properties of coal and methane adsorption capacity with coal rank were observed separately and then the relationships between physicochemical properties of coal and adsorption capacity were obtained.

Proximate analysis of coal (Mahajan and Walker 1971; Bhattacharyya 1972; Reucroft and Patel 1986; Friesen and Mikula 1988; Banerjee 1988, Barker-Read and Radchenko 1989; Yalçin and Durucan 1991; DeGance et al. 1993; Chaback et al. 1996; Clarkson et al. 1997; Bustin and Clarkson 1998; Özgen Karacan and Okandan 2000; Laxminarayana and Crosdale 2002; Wang and Takarada 2003; Bae and Bhatia 2006; Qin et al. 2007; Saghafi et al. 2007; Dutta et al. 2008; Yu et al. 2008a, b, c; Pini et al. 2009; Pone et al. 2009; Battistutta et al. 2010; Gensterblum et al. 2010; He et al. 2010; Pan et al. 2010; Kim et al. 2011; Zheng 2012; Wang 2012; Jiang et al. 2012; Kutchko et al. 2013; An et al. 2013; Hao et al. 2013; Xiao et al. 2016) consists of moisture, ash, volatile, fixed carbon. The widely-used index are: air dried basis (\({\text{ad}}\)), dry basis (\({\text{d}}\)), dry ash-free basis (\({\text{daf}}\)). To avoid the effects of inorganic material of moisture and ash, we applied the index of dry ash-free basis volatile (\(V_{{\rm daf}}\)). Similarly, to avoid the effects of moisture, we applied the index of dry basis ash (\({\text{daf}}\)). The Moisture and Fixed carbon use the air dried basis index, as the index employed by different articles that are different with this paper, we adopted the Eqs. (1)–(5) for transformation:

$$V_{{\rm daf}} = \frac{{100V_{{\rm ad}} }}{{100 - M_{{\rm ad}} - A_{{\rm ad}} }}$$
(1)
$$V_{{\rm daf}} = \frac{{100V_{{\rm d}} }}{{100 - A_{{\rm ad}} }}$$
(2)
$$A_{{\rm d}} = \frac{{100A_{{\rm ad}} }}{{100 - M_{{\rm ad}} }}$$
(3)
$$F_{{\rm cad}} = 100 - M_{{\rm ad}} - A_{{\rm ad}} - V_{{\rm ad}}$$
(4)
$$F_{{\rm cadf}} = 100 - V_{{\rm daf}}.$$
(5)

The maceral (Ceglarska-Stefańska and Brzóska 1998; Crosdale et al. 1998; Laxminarayana and Crosdale 2002; Busch et al. 2003; Mastalerz et al. 2004; Chang et al. 2006; Busch et al. 2006; Qin et al. 2007; Majewska and Ziętek 2007; Siemons et al. 2007; Chang et al. 2008; Day et al. 2008; Mazumder and Wolf 2008; Pone et al. 2009; Majewska et al. 2009; Radliński et al. 2009; Battistutta et al. 2010; Gensterblum et al. 2010; He, et al. 2010; Day et al. 2011; Kim et al. 2011; Jiang et al. 2012; Maphala and Wagner 2012; An et al. 2013; Cai et al. 2013a, b; Lin et al. 2013; Dutka et al. 2013; Xiao et al. 2016) include Vitrinite, Inertinite, Exinite. The common indicators are: moist mineral matter basis, and moist mineral matter-free basis (mmf). As the indexes employed by literatures at home and abroad are not consistent, we adopted the mmf index in this paper and used the Eq. (6) for transformation:

$$X_{{\rm daf}} = \frac{{100X_{{\hbox{mineral}} } }}{100 - Mineral},\quad X = Vitrinite,Inertinite,Exinite .$$
(6)

Ultimate analysis specifies the elements of carbon, hydrogen, nitrogen, oxygen and sulphur (Gan et al. 1972; Joubert et al. 1973; Nandi and Walker 1975; Nelson et al. 1980; Reucroft and Yang and Saunders 1985; Reucroft and Patel 1986; Banerjee 1988; Friesen and Mikula 1988; Giuliani et al. 1991; DeGance et al. 1993; Chaback et al. 1996; Wang and Takarada 2003; Fitzgerald et al. 2005; Chang et al. 2006; Qin et al. 2007; Siemons et al. 2007; Chang et al. 2008; Majewska et al. 2009; Radliński et al. 2009; Pone et al. 2009; Battistutta et al. 2010; Gensterblum et al. 2010; Kim et al. 2011; Hao et al. 2012, 2013; Lin et al. 2013). Widely-used indexes for ultimate analysis are: dry ash-free basis, air dried basis. To avoid moisture effect, we applied dry ash-free basis index in this paper. As the index employed by different articles that are different with this paper, we adopted the Eq. (7) for transformation:

$$X_{{\rm daf}} = \frac{{100X_{{\rm ad}} }}{{100 - A_{{\rm ad}} - M_{{\rm ad}} }},\quad X = C,H,O,N,S.$$
(7)

3 Coal rank effect on adsorption capacity

The following Langmuir adsorption isotherm pattern is used:

$$Q = \frac{abP}{1 + bP}$$
(8)

where, \(Q\) is the adsorption capacity under an equilibrium of \(P\), m3/t, \(a\) is limit monolayer adsorption capacity, m3/t, \(b\) is pressure constant, MP−1.

3.1 Coal rank effect on adsorption constant \(a_{{\rm daf}}\)

To avoid the effects of moisture, ash on methane adsorption capacity, we adopted the dry ash-free basis index to analyze the limit monolayer adsorption capacity \(a\), expressed in \(a_{{\rm daf}}\). As the index employed by different articles that are different with this paper, we adopted the Eq. (9) for transformation:

$$a_{{\rm daf}} = \frac{{100a_{{\rm ad}} }}{{100 - M_{{\rm ad}} - A_{{\rm ad}} }}.$$
(9)

The statistical analysis of the relationship between limit monolayer adsorption capacity (\(a_{{\rm daf}}\)) and vitrinite reflectance \(R_{{\rm o,max}}\) was shown in Fig. 1.

Fig. 1
figure 1

Value of adsorption constant \(a_{{\rm daf}}\) versus vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: NERC-290, Chinese (Zhang and Yang 1999; Fu et al. 2002; Zhang et al. 2004; Fu et al. 2005; Su et al. 2005; Su et al. 2006; Chen et al. 2008; Fu et al. 2008; Zhang and Sang 2008; Wang et al. 2008; Yu et al. 2008a; Chen et al. 2008; Jiang 2009; Zhang and Sang 2009; Yang et al. 2009; Ren 2010; Tian et al. 2010; Zhang et al. 2011; Zhang Jun-fan and Liang Wen-qing 2011; Zheng 2012; Gao 2012; Ma et al. 2014; Zhang et al. 2014)-173; English(Joubert et al. 1973; Yalçin and Durucan 1991; Beamish and O’Donnell 1992; DeGance et al. 1993; Chaback et al. 1996; Bustin and Clarkson 1998; Nodzeński 1998; Faiz et al. 2007a, b; Yu et al. 2008a, b, c; Xiao and Wang 2011)-103)

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Adsorption isotherm results (Fig. 1) indicate that Langmuir volumes \(a_{{\rm daf}}\) follows a upwards, discrete, second-order polynomial trends with increasing coal rank, which is consistent with the conclusion of Levy et al. 1997. Figure 1 shows that the range of \(a_{{\rm daf}}\) for Lignite/subbituminous is 13–34 m3/t, for low volatile bituminous is 8–48 m3/t, for middle volatile bituminous is 4–52 m3/t, for high volatile bituminous is 6–60 m3/t and for semi-anthracite/anthracite is 16–50 m3/t. Besides, upper and lower limits exists in the relationship between \(a_{{\rm daf}}\) and \(R_{{\rm o,max}}\), the upper and lower curves are separately represented in the equations of \(a_{{\rm daf,max}} = 13.4R_{{\rm o,max}} - 25.01R_{{\rm o,max}} + 57.72\) and \(a_{{\rm daf,min}} = 1.014R_{{\rm o,max}}^{2} - 0.014R_{{\rm o,max}} + 7.18\), the corresponding average curve is: \(a_{{\rm daf,avr}} = 9.3135R_{{\rm o,max}}^{2} - 22.2865R_{{\rm o,max}} + 32.673\), from the corresponding average curve, we can see that the \(a_{{\rm daf}}\) decrease as the coal rank increase in the range of \(R_{{\rm o,max}} = 0.4\% - 1.196\%\), while in the range of \(R_{{\rm o,max}} = 1.196\%\,{-}\,7\%\), the \(a_{{\rm daf}}\) increase as the coal rank increase. The reason is that the porosity of the low rank coal is larger, resulting in the stronger adsorption capacity. With the increase of buried depth, adsorption capacity decrease with the compaction of larger pores because of the increasing overburden pressure. With the enhancement of maturity, adsorption capacity increase again with the increasing pore volume and specific area because of the increasing reservoir pressure coefficient.

3.2 Coal rank effect on adsorption constant \(b\)

To analyze the relationship between Langmuir constant \(b\) and vitrinite reflectance \(R_{{\rm o,max}}\), we did the Fig. 2.

Fig. 2
figure 2

Value of adsorption constant \(b\) versus vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: NERC-290, Chinese (Zhang and Yang 1999; Su et al. 2006; Ma et al. 2008; Yang et al. 2009; Tian et al. 2010; Zheng 2012; Gao 2012; Ma et al. 2014; Pan and Xu 2015; Xu and Pan 2015)-217, English (Joubert et al. 1973; Yalçin and Durucan 1991; DeGance et al. 1993; Chaback et al. 1996; Bustin and Clarkson 1998; Nodzeński 1998; Faiz et al. 2007a, b; Yu et al. 2008a, b, c)-153)

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Figure 2 indicates that constant b follows a downward, discrete, second-order polynomial trends with vitrinite reflectance \(R_{{\rm o,max}}\), which is not related to coal type. Constant b for the Lignite/subbituminous is between 0 and 0.9 MPa−1, for low volatile bituminous is between 0.15 and 1.8 MPa−1, for middle volatile bituminous is between 0.25 and 2 MPa−1, for high volatile bituminous is between 0.55 and 2.1 MPa−1 and for semi-anthracite/anthracite is between 0.56 and 2.25 MPa−1. The variation range is between the upper and lower curves represented in the equations of \(b_{{\rm max}} = - 0.16R_{{\rm o,max}}^{2} + 0.74467R_{{\rm o,max}} + 0.899\) and \(b_{{\rm min}} = - 0.2687R_{{\rm o,max}}^{2} + 1.321R_{{\rm o,max}} - 0.6754\), and the corresponding average curve is: \(b_{{\rm avr}} = - 0.43019R_{{\rm o,max}}^{2} + 0.90056R_{{\rm o,max}} + 0.86969\), from the corresponding average curve, we can see that the constant \(b\) increase as the coal rank increase in the range of \(R_{{\rm o,max}} = 0.4\% {-}1.05\%\); while in the range of \(R_{{\rm o,max}} = 1.05\% {-}7\%\), The constant \(b\) decreases as the coal rank increases. And from the upper curve, we can see that the constant b reaches max at \(R_{{\rm o,max}}\) = 1.765MP−1.

3.3 Coal rank effect on adsorption \(Q_{{\rm daf}}\)

The Eq. (10) was employed to obtain dry ash-free basis gas content \(Q_{{\rm daf}}\):

$$Q_{{\rm daf}} = \frac{{a_{{\rm daf}} bP}}{1 + bP}$$
(10)

Set the equilibrium pressure in Eq. (10) as 1 MPa, we can obtain the corresponding gas content \(Q_{{\rm daf}}\). Figure 3 shows the relationship between \(Q_{{\rm daf}}\) at 1 MPa and the vitrinite reflectance \(R_{{\rm o,max}}\).

Fig. 3
figure 3

Value of adsorption constant \(Q_{{\rm daf}}\) versus vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: NERC-290, Chinese (Yu et al. 2005; Zhang 2007; Zheng 2012; Zou et al. 2013; Liu et al. 2015; Yue et al. 2016)-217, English(Joubert et al. 1973; Yalçin and Durucan 1991; DeGance et al. 1993; Chaback et al. 1996; Bustin and Clarkson 1998; Nodzeński 1998; Yu et al. 2008a, b, c)-153)

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Figure 3 shows that \(Q_{{\rm daf}}\) for the Lignite/subbituminous is between 0 and 8 m3/t, for low volatile bituminous is between 3 and 24 m3/t, for middle volatile bituminous is between 5 and 29 m3/t, for high volatile bituminous is between 6 and 36 m3/t and for semi-anthracite/anthracite is between 8 and 30 m3/t. The variation range is between the upper and lower curves represented in the equations of \(Q_{{\rm daf,max}} = - 4.9549R_{{\rm o,max}} ^{2} + 13.263R_{{\rm o,max}} + 27.6\) and \(Q_{{\rm daf,min}} = 2.847R_{{\rm o,max}}^{2} - 1.84R_{{\rm o,max}} + 4.97\).

3.4 Coal rank effect on gas content of raw coal

Taken into account the factors influence the actual gas content, we applied the Eq. (11) (Chen et al. 2010; Gao 2012) for transformation:

$$Q = Q_{{\rm daf}} \frac{{100 - A_{{\rm ad}} - M_{{\rm ad}} }}{100}\frac{1}{{1 + 0.31M_{{\rm ad}} }}$$
(11)

where, \(A_{{\rm ad}}\) is the air dried basis ash, %; \(M_{{\rm ad}}\) is the air dried basis moisture, %; \(Q_{{\rm daf}}\) is the dry ash-free basis gas content, m3/t; \(Q\) is the gas content of raw coal, m3/t.

Figure 4 shows the relationship between \(Q\) at 1 MPa and the vitrinite reflectance \(R_{{\rm o,max}}\):

Fig. 4
figure 4

Value of adsorption constant \(Q\) versus vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: NERC-290, Chinese (Yu et al. 2008a; Ma et al. 2008; Jiang 2009; Tian et al. 2010; Zhang et al. 2011; Zheng 2012; Gao 2012; Ma et al. 2014; Xu and Pan 2015)-117, English(Joubert et al. 1973; Yalçin and Durucan 1991; DeGance et al. 1993; Chaback et al. 1996; Nodzeński 1998; Yu et al. 2008a, b, c)-153)

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From Fig. 4, it can be concluded that the \(Q\) increases with the coal rank. The formula between \(Q\) and \(R_{{\rm o,max}}\) was shown in Eq. (12):

$$Q = k_{{\rm Q}} R_{{\rm o,max}} + b_{{\rm Q}}$$
(12)

where, \(Q\) is the gas content at the equilibrium pressure of 1 MPa; \(k_{{\rm Q}}\) and \(b_{{\rm Q}}\) are the coefficients in Eq. (12), and \(k_{{\rm Q}} > 0\).

Figure 4 shows that \(Q\) for the Lignite/subbituminous is between 0 and 5 m3/t, for low volatile bituminous is between 2 and 12 m3/t, for middle volatile bituminous is between 1 and 13 m3/t, for high volatile bituminous is between 1 and 15 m3/t and for semi-anthracite/anthracite is between 3 and 15 m3/t. The variation range is between the upper and lower curves represented in the equations of \(Q_{\rm{max} } = 4.49R_{{\rm o,max}} + 8.9386\) and \(Q_{\rm{min} } = 3.9266R_{{\rm o,max}} - 2.545\). Comparing with Fig. 3, it can be seen that the value of \(Q\) is smaller than \(Q_{{\rm daf}}\) because of the Physicochemical difference.

4 Analysis of factors effect on adsorption capacity of coal

4.1 Proximate composition effect on adsorption capacity

Proximate analysis is usually used to define the chemical composition of coals. With proximate analysis, the substances of moisture, ash, volatile matter and fixed carbon in the coal content are determined as weight percent. In the four indexes, moisture and ash represents the inorganic material, while volatile and fixed carbon represents the organic material.

4.1.1 Volatile content effect on adsorption capacity

Commonly used indexes for Volatile are: dry air basis volatile (\(V_{{\rm ad}}\)), dry basis volatile (\(V_{{\rm d}}\)), dry ash-free basis volatile (\(V_{{\rm daf}}\)) and Received basis volatile (\(V_{{\rm ar}}\)). To avoid the effect of moisture, ash on volatile matter, the volatile matter was determined by using dry ash-free basis on weight percent. Figure 5 shows the relationship between volatile (\(V_{{\rm daf}}\)) and vitrinite reflectance (\(R_{{\rm o,max}}\)).

Fig. 5
figure 5

Variation of \(V_{{\rm daf}}\) with vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: NERC-189, Chinese (Yu et al. 2004; Chang et al. 2006; Qin et al. 2007; Chang et al. 2008; Chen et al. 2010; Hao et al. 2012; Zheng 2012; Jiang et al. 2012; Gao 2012; Lin et al. 2013; Li et al. 2013; Xiao et al. 2016)-135, English, (Mahajan and Walker 1971; Joubert et al. 1973; Yalçin and Durucan 1991; DeGance et al. 1993; Chaback et al. 1996; Nodzeński 1998; Wang 2012)-171)

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Figure 5 shows the relationship between Dry ash-free basis volatile \(V_{{\rm daf}}\) and Vitrinite reflectance \(R_{{\rm o,max}}\) expressed in Eq. (13):

$$V_{{\rm daf}} = k_{{\rm vdaf}} R_{{\rm o,max}}^{{b_{{\rm vdaf}} }}$$
(13)

where, \(V_{{\rm daf}}\) is the dry ash-free basis volatile; \(k_{{\rm vdaf}}\), \(b_{{\rm vdaf}}\) are the coefficients in Eq. (13), and \(k_{{\rm vdaf}} > 0\), \(b_{{\rm vdaf}} < 0\).

The first and order derivative are separately shown in Eqs. (14) and (15):

$$V_{{\rm daf}}^{\prime } = k_{{\rm vdaf}} b_{{\rm vdaf}} R_{{\rm o,max}}^{{b_{{\rm vdaf}} - 1}}$$
(14)
$$V_{{\rm daf}}^{\prime } = k_{{\rm vdaf}} b_{{\rm vdaf}} (b_{{\rm vdaf}} - 1)R_{{\rm o,max}}^{{b_{{\rm vdaf}} - 2}}$$
(15)

As \(k_{{\rm vdaf}} > 0\), \(b_{{\rm vdaf}} <\,0\), the first derivative (Eq. 14) of Eq. (13) is always less than zero, while the second derivative (Eq. 15) is always greater than zero. According to the properties of function, it can be concluded that Eq. (13) exhibits depression, power-decreasing type. In conclusion, volatile decrease with the increase of coal rank, and compared with the higher coal rank, the volatile at the low metamorphic grade has a much more significantly decrease. The reason is that Volatile matter are mainly small molecular compound broke from unsaturated group in the side-chain of fat and the oxygen-containing functional groups that decreases with the coal rank.

Figure 5 shows that Dry ash-free basis volatile \(V_{{\rm daf}}\) for the Lignite/subbituminous is larger than 30%, for low volatile bituminous is between 25% and 58%, for middle volatile bituminous is between 10% and 38%, for high volatile bituminous is between 5% and 20% and for semi-anthracite/anthracite is smaller than 10%. The variation range is between the upper and lower curves represented in the equations of \(V_{{\rm daf, max}} = 42.39R_{{\rm o, max} }^{ - 1.183}\) and \(V_{{\rm daf,min} } = 16.39R_{{\rm o,max} }^{ - 0.756}\), and the corresponding average curve is \(V_{{{{\rm daf}},{{\rm avr}}}} = 29.63R_{{\rm o,max} }^{ - 1.05}\). For low metamorphic bituminous coal, the vitrinite reflectance changes smaller with scattered volatile \(V_{{\rm daf}}\) content falling in 25%–58%;

Features for the mid volatile bituminous coal are with the wide range of volatile matter content falling in 10%–38% because of the complex generation environment and constitution of coal rock; volatile matter content of the high volatile bituminous coal falls in 5%–20%; while the volatile matter content in anthracite is the lowest, less than 10%.

Combing Eqs. (12) and (13), the quantitative formula of Eq. (16) can be obtained:

$$Q = k_{{\rm Q}} \left( {\frac{{V_{{\rm daf}} }}{{k_{{\rm vdaf}} }}} \right)^{{\frac{1}{{b_{{\rm vdaf}} }}}} + b_{{\rm Q}}$$
(16)
$$Q^{{\prime }} = \frac{{k_{{\rm Q}} }}{{b_{{\rm vdaf}} k_{{\rm vdaf}}^{{\frac{1}{{b_{{\rm vdaf}} }}}} }}V_{{\rm daf}}^{{\frac{1}{{b_{{\rm vdaf}} }} - 1}}$$
(17)
$$Q^{\prime \prime } = \frac{{k_{Q} \left( {\frac{1}{{b_{{\rm vdaf}} }} - 1} \right)}}{{b_{{\rm vdaf}} k_{{\rm vdaf}}^{{\frac{1}{{b_{{\rm vdaf}} }}}} }}V_{{\rm daf}}^{{\frac{1}{{b_{{\rm vdaf}} }} - 2}}$$
(18)

As \(\frac{1}{{b_{{\rm vdaf}} }} < 0\), \(k_{{\rm Q}} > 0\), \(k_{{\rm vdaf}} > 0\), so \(k_{{\rm vdaf}}^{{\frac{1}{{b_{{\rm vdaf}} }}}} > 0\),the first derivative (Eq. 17) of Eq. (16) is always less than zero, while the second derivative (Eq. 18) is always greater than zero, thus, it can be concluded that Eq. (16) exhibits depression, power-decreasing type, it can be concluded that gas content in raw coal decreases with the increase of dry ash-free basis volatile \(V_{{\rm daf}}\), and the gas content in raw coal at the low metamorphic grade has a much more significantly decrease.

4.1.2 Moister content effect on adsorption capacity

Moisture in coal generally divided into inherent moisture and external moisture. Inherent moisture existing in the underdeveloped pore is from the plant turned to coal; external moisture existing in the well-developed pore and crack is form the process of mining and transport. The moisture was determined using air dried basis on weight percent, expressed in \(M_{{\rm ad}}\) with the value equal to inherent moisture. Figure 6 shows the relationship between air dried basis moisture (\(M_{{\rm ad}}\)) and vitrinite reflectance (\(R_{{\rm o,max}}\)).

Fig. 6
figure 6

Variation of \(M_{{\rm ad}}\) with vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: NERC-189, Chinese (Yu et al. 2004; Chang et al. 2006; Qin et al. 2007; Chang et al. 2008; Hao et al. 2012; Zheng 2012; Jiang et al. 2012; Gao 2012; Lin et al. 2013; Li et al. 2013; Xiao et al. 2016)-169, English(Allardice and Evans 1971; Mahajan and Walker 1971; Bhattacharyya 1972; Joubert et al. 1973; Reucroft and Patel 1986; Banerjee 1988; Barker-Read and Radchenko 1989; Yalçin and Durucan 1991; Ciembroniewicz and Marecka 1993; DeGance et al. 1993; Chaback et al. 1996; Clarkson et al. 1997; Crosdale et al. 1998; Bustin and Clarkson 1998; Friesen and Mikula 1988; Özgen Karacan and Okandan 2000; Laxminarayana and Crosdale 2002; Busch et al. 2003; Wang and Takarada 2003; Weishauptová et al. 2004; Mastalerz et al. 2004; Fitzgerald et al. 2005; Hildenbrand et al. 2006; Bae and Bhatia 2006; Dai et al. 2006; Busch et al. 2006; Faiz et al. 2007a, b; Saghafi et al. 2007; Majewska and Ziętek 2007; Faiz et al. 2007a, b; Astashov et al. 2008; Yu et al. 2008a, b, c; Zhang 2008; Dutta et al. 2008; Day et al. 2008; Pini, et al. 2009; Kędzior 2009; Pone et al. 2009; Majewska et al. 2009; Majewska et al. 2010; Battistutta et al. 2010; Gensterblum et al. 2010; Pan et al. 2010; He, et al. 2010; Jin et al. 2010; Kim et al. 2011; Day et al. 2011; Wang 2012; An et al. 2013; Dutka et al. 2013; Hao et al. 2013; Kutchko et al. 2013)-258)

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Figure 6 show that moisture in coal decreases with the coal rank, and the functional relationship between inherent moisture \(M_{{\rm ad}}\) and vitrinite reflectance \(R_{{\rm o,max}}\) is shown in Eq. (19):

$$M_{{\rm ad}} = k_{{\rm mad}} \exp (R_{{\rm o,max}} /c_{{\rm mad}} ) + b_{{\rm mad}}$$
(19)

where \(M_{{\rm ad}}\) is air dried basis inherent moisture, %; \(k_{{\rm mad}}\), \(c_{{\rm mad}}\), \(b_{{\rm mad}}\) are coefficients of Eq. (19), and \(c_{{\rm mad}} < 0\), \(k_{{\rm mad}} > 0\).

The first and order derivative are separately shown in Eqs. (20) and (21):

$$M_{{\rm ad}}^{{\prime }} = \frac{{k_{{\rm mad}} }}{{c_{{\rm mad}} }}\exp \left( {R_{{\rm o,max}} /c_{{\rm mad}} } \right)$$
(20)
$$M_{{\rm ad}}^{{\prime \prime }} = \frac{{k_{{\rm mad}} }}{{c_{{\rm mad}}^{2} }}\exp \left( {R_{{\rm o,max}} /c_{{\rm mad}} } \right)$$
(21)

As \(c_{{\rm mad}} < 0\), \(k_{{\rm mad}} > 0\), the first derivative (Eq. 20) of Eq. (19) is always less than zero, while the second derivative (Eq. 21) is always greater than zero, thus, it can be concluded that Eq. (19) exhibits depression, exponent-decreasing type, it can be concluded that inherent moisture decreases with the coal rank and has a much more significantly decrease at the low metamorphic grade. From Fig. 6, it can be seen that the maximum of moisture are generally less than the values in the curves of \(M_{{\rm ad,max} } = 2.217 + 83.84\exp \left( {R_{{\rm o,max}} / - 0.64} \right)\). The ranges of inherent moisture in subbituminous coal are from 0% to 55%, in low volatile bituminous coal with lots of pores are from 0% to 35%, in mid volatile bituminous coal with lots of hydrophobic fused ring structure in organic structure are from 0% to 30%, in high volatile bituminous coal are from 0% to 8%, in anthracite are from 0% to 10%. Besides, at the range of 3.2%–4.2% and 5.2%–5.9% had an increasing zone, and the corresponding inherent moisture are separately 0%–10% and 0%–9%, the possibly reason maybe the existence of the aphanitic structure.

Reasons for the changing trend of inherent moisture are summarized below: (1) adsorption character of pore surface on moisture, the higher internal surface area is the high moisture absorbed. (2) The more Polarity oxygen-containing groups in coal, the larger inherent moisture content. (3) The surface area and Polarity oxygen-containing groups decrease with the increase of coal rank, resulting in the decease of inherent moisture.

Combing Eqs. (12) and (19), Eq. (22) can be obtained:

$$Q = k_{{\rm Q}} c_{{\rm mad}} \ln \left( {\frac{{M_{{\rm ad}} - b_{{\rm mad}} }}{{k_{{\rm mad}} }}} \right) + b_{{\rm Q}}$$
(22)
$$Q^{'} = \frac{{k_{{\rm Q}} c_{{\rm mad}} }}{{M_{{\rm ad}} - b_{{\rm mad}} }}$$
(23)
$$Q^{''} = \frac{{ - k_{{\rm Q}} c_{{\rm mad}} }}{{\left( {M_{{\rm ad}} - b_{{\rm mad}} } \right)^{2} }}$$
(24)

As \(k_{Q} > 0\), \(c_{{\rm mad}} < 0\), \(k_{{\rm mad}} > 0\), the first derivative (Eq. 23) of Eq. (22) is always less than zero, while the second derivative (Eq. 24) is always greater than zero, thus, it can be concluded that Eq. (22) exhibits depression, logarithmic-decreasing type, that is the gas content \(Q\) in raw coal has a negative relationship with the inherent moisture \(M_{{\rm ad}}\). Also, it can be concluded from Eq. (22) that gas content \(Q\) in raw coal has a much more significantly decrease at the low inherent moisture grade. The reasons for the effects of moisture content on adsorption of coal are generally as follow: (1) Effective adsorption sites at the pore surface are limited, the more the moisture is, the more adsorption sites can be occupied, then the less adsorption sites can be used for methane. (2) The preferential adsorption of moisture results in the decreased gas content. (3) Due to the adsorption of methane in micro-pore with high capillary pressure and strong self-priming capacity, the moisture in coal can restrict the adsorption of methane. (4) Existing of moisture can cause matrix shrinkage of coal resulting in the decrease of gas content.

4.1.3 Ash content effect on adsorption capacity

Ash is the remaining residue after full combustion of coal, and almost all of it comes from minerals. Usually, the ash content is proportional to the mineral content. Ash is not an inherent component of coal but a very different matter from minerals in coal. First, the ash content is lower than corresponding mineral content, and then their composition is changed, Minerals at high temperature after decomposition, oxidation, combining chemical reaction can turn into ash. Because of the changed moisture in the air dried coal with the change of air humidity, the ash content of the coal sample changed. But in terms of absolute dry coal, the ash content is the same. Thus, dry basis ash \(A_{{\rm d}}\) is used in the actual. Figure 7 shows the relationship between dry basis ash (\(A_{{\rm d}}\)) and vitrinite reflectance (\(R_{{\rm o,max}}\)).

Fig. 7
figure 7

Variation of \(A_{{\rm d}}\) with vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: NERC-189, Chinese (Yu et al. 2004; Chang et al. 2006; Qin et al. 2007; Chang et al. 2008; Hao et al. 2012; Zheng 2012; Gao 2012; Jiang et al. 2012; Li et al. 2013; Lin et al. 2013; Xiao et al. 2016)-268, English, (Mahajan and Walker 1971; Joubert et al. 1973; Yalçin and Durucan 1991; DeGance et al. 1993; Chaback et al. 1996; Bustin and Clarkson 1998; Nodzeński 1998; Hildenbrand et al. 2006; Faiz et al. 2007a, b; Faiz et al. 2007; Yu et al. 2008a, b, c; Gensterblum et al. 2010; Taraba 2011; Wang 2012)-297)

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Figure 7 shows that the ash content value is more dispersed. This is because the ash is affected largely by the geological evolution conditions; Ash showed a decreasing trend with the increase metamorphism degree. The maximum ash value does not exceed the curve shown in this formula: \(A_{{\rm d,max}} = - 8.657R_{{\rm o,max}} + 68.435\). The ranges of ash content in subbituminous coal are from 0% to 70%, in low volatile bituminous coal are from 0% to 62%, in mid volatile bituminous coal are from 0% to 59%, in high volatile bituminous coal are from 0% to 49%, in anthracite are from 0% to 43%.

Existence of ash in coal has adverse effects on coal adsorption capacity. The effect mechanism of ash on coal adsorption is mainly: (1) the first is based on the principle of similar compatibility, the properties of methane is similar to the organic matter with weak polarity and better adsorption character, but ash is mostly inorganic with weak adsorption capacities, the presence of ash reduced the adsorption content of methane in coal; (2) the presence of silica, iron, calcium and magnesium and other group, especially the presence of metal elements resulting in the basal cementation tendency of rock, and the increasing cements will block the coal pore and fissure, reducing the porosity of coal, resulting in the decreasing of adsorption capacity of methane in coal. The comprehensive function of the two aspects reduces the ability of organic matter to adsorb methane in coal, In view of this, in the evaluation of ash effect on adsorption of coal, authors should consider not only the impact of ash content, but also consider the state of ash adsorption sites.

4.1.4 Fixed carbon content effect on adsorption capacity

Carbon Fixed is the residue after subtracting the ash from the coke residue determined after the measurement of volatile in coal, In fact, it is the pyrolysis product produced by the organic matter under certain heating conditions, and it is a part of the coke residue. Carbon Fixed is mainly composed of carbon elements, some of the hydrogen, sulfur and a small amount of oxygen and nitrogen elements. In proximate analysis, Fixed carbon always expressed in \(Fc_{{\rm ad}}\), Fig. 8 shows the relationship between Fixed carbon (\(Fc_{{\rm ad}}\)) and Vitrinite reflectance (\(R_{{\rm o,max}}\)).

Fig. 8
figure 8

Variation of \(Fc_{{\rm ad}}\) with vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: NERC-189, Chinese (Yu et al. 2004; Chang et al. 2006; Qin et al. 2007; Chang et al. 2008; Hao et al. 2012; Zheng 2012; Jiang et al. 2012; Gao 2012; Li et al. 2013; Lin et al. 2013)-266, English (Mahajan and Walker 1971; Bhattacharyya 1972; Joubert et al. 1973; Joubert et al. 1974; Yang and Saunders 1985; Banerjee 1988; Friesen and Mikula 1988; Barker-Read and Radchenko 1989; Yalçin and Durucan 1991; DeGance et al. 1993; Chaback et al. 1996; Clarkson et al. 1997; Crosdale et al. 1998; Bustin and Clarkson 1998; Özgen Karacan and Okandan 2000; Laxminarayana and Crosdale 2002; Wang and Takarada 2003; Weishauptová et al. 2004; Fitzgerald et al. 2005; Bae and Bhatia 2006; Dai et al. 2006; Saghafi et al. 2007; Majewska and Ziętek 2007; Faiz et al. 2007a, b; Siemons et al. 2007; Yu et al. 2008a, b, c; Zhang 2008; Dutta et al. 2008; Day et al. 2008; Majewska et al. 2009; Kędzior 2009; Pini et al. 2009; Pone et al. 2009; Battistutta et al. 2010; Pan et al. 2010; Gensterblum et al. 2010; He et al. 2010; Kim et al. 2011; Day et al. 2011; Wang 2012; Dutka et al. 2013; An et al. 2013; Hao et al. 2013; Kutchko et al. 2013)-268)

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Figure 8 shows the relationship between air dried basis Fixed carbon and vitrinite reflectance:

$$Fc_{{\rm ad}} = k_{{\rm Fcad}} \exp (c_{{\rm Fcad}} R_{{\rm o,max}} ) + b_{{\rm Fcad}}$$
(25)

where, \(Fc_{{\rm ad}}\) is air dried basis fixed carbon, %; \(k_{{\rm Fcad}}\), \(c_{{\rm Fcad}}\), \(b_{{\rm Fcad}}\) are coefficients of Eq. (25),and \(c_{{\rm Fcad}} < 0\), \(k_{{\rm Fcad}} < 0\).

The first and order derivative are separately shown in Eqs. (26) and (27):

$$Fc_{{\rm ad}}^{\prime } = \frac{{k_{{\rm Fcad}} }}{{c_{{\rm Fcad}} }}\exp \left( {R_{{\rm o,max}} c_{{\rm Fcad}} } \right)$$
(26)
$$Fc_{{\rm ad}}^{''} = \frac{{k_{{\rm Fcad}} }}{{c_{{\rm Fcad}}^{2} }}\exp (R_{{\rm o,max}} c_{{\rm Fcad}} )$$
(27)

As \(c_{{\rm Fcad}} < 0\), \(k_{{\rm Fcad}} < 0\), the first derivative (Eq. 26) of Eq. (25) is always larger than zero, while the second derivative (Eq. 27) is always less than zero, thus, it can be concluded that Eq. (25) exhibits up-convex, power-decreasing type, it can be concluded that Fixed carbon increases with the increase of coal rank, and at low metamorphic stage, Fixed carbon content increases greatly.

Figure 8 shows that with the increase of coal rank, the fixed carbon content exists the upper and lower curves expressed in the following equations:\(Fc_{{\rm ad,max}} = 92.3433 - 88.918\exp ( - 1.7628R_{{\rm o,max}} )\) and \(Fc_{{\rm ad,min}} = 73.01726 - 273.61127\exp ( - 0.9814R_{{\rm o,max}} )\), The corresponding average curve is as follow: \(Fc_{{\rm ad,avr}} = 82.57758 - 162.60685\exp ( - 1.03532R_{{\rm o,max}} )\),besides, The ranges of fixed carbon content in subbituminous coal are from 2% to 56%, in low volatile bituminous coal are from 18% to 70%, in mid volatile bituminous coal are from 18% to 85%, in high volatile bituminous coal are from 21% to 92%, in anthracite are from 50% to 95%.

Combing Eqs. (25) and (12):

$$Q = \frac{{k_{{\rm Q}} }}{{c_{{\rm Fcad}} }}\ln \left( {\frac{{Fc_{{\rm ad}} - b_{{\rm Fcad}} }}{{k_{{\rm Fcad}} }}} \right) + b_{{\rm Q}}$$
(28)
$$Q^{\prime } = \frac{{k_{{\rm Q}} /c_{{\rm Fcad}} }}{{Fc_{{\rm ad}} - b_{{\rm Fcad}} }}$$
(29)
$$Q^{\prime \prime } = \frac{{ - k_{{\rm Q}} /c_{{\rm Fcad}} }}{{\left( {Fc_{{\rm ad}} - b_{{\rm Fcad}} } \right)^{2} }}$$
(30)

As \(k_{{\rm Q}} > 0\), \(c_{{\rm Fcad}} < 0\), \(k_{{\rm Fcad}} < 0\),the first derivative (Eq. 29) of Eq. (28) is always greater than zero, while the second derivative (Eq. 30) is always greater than zero, thus, it can be concluded that Eq. (28) exhibits depression, logarithmic-increasing type. The increase of carbon Fixed content has a reinforcing effect on the adsorption capacity of raw coal, and in the low carbon Fixed content stage, the amount adsorbed gas in coal decreases significantly.

The research shows that with the increase of carbon fixed content, the specific surface area and pore volume increases, The surface morphology of coal shows that the particles become small, the size is uniform, the arrangement is more orderly, the space for the adsorption of methane in coal is increasing, and finally the coal methane adsorption capacity is enhanced.

4.2 Maceral composition effect on adsorption capacity of raw coal

The maceral types in coals are mainly vitrinite, inertinite, rare exinite and little mineral matter. Differences of maceral groups in composition and structure resulted in the different characteristics, and these characteristics directly decides the macroscopic properties of coal. Branch of various group experienced different changes, resulting in the difference of chemical composition, molecular structure and pore. Gelation and wire charring effect is different, resulting in different degree of plant tissue preservation. In the process of coal metamorphism, the difference of hydrocarbons, volatile matter content resulted in the different pore growth degree. Thus, there is difference in the adsorption capacity of macerals.

It is generally believed that adsorption capacity of exinite is the lowest, and as the main components of the organic maceral, the vitrinite and inertinite have the strong adsorption capacity with the increase in the metamorphism degree, the two show the shift tendency, indicating that inertinite gradually evolved into vitrinite. Therefore, in this paper, we mainly study the relationship between vitrinite and inertinite with the metamorphism degree, in order to get the effect of the two matters on adsorption capacity.

4.2.1 Vitrinite content effect on adsorption capacity

Vitrinite is the main maceral in coal, and is the jelly-like substance formed from the wood cellulose of plant tissue under the gel conditions. According to the intact degree of cell structure and cell forms, size and other characteristics,the most common vitrinite components are divided into anthraxylon, homocollinite and degradinite. Figure 9 represents the relationship between vitrinite and vitrinite reflectance (\(R_{{\rm o,max}}\)).

Fig. 9
figure 9

Variation of vitrinite content with vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: NERC-61, Chinese (Yu et al. 2004; Chang et al. 2006; Qin et al. 2007; Chang et al. 2008; Jiang et al. 2012; Lin et al. 2013; Xu and Pan 2015; Xiao et al. 2016)-159, English(Gan et al. 1972; Nelson et al. 1980; Giuliani et al. 1991; Crosdale et al. 1998; Marecka and Mianowski 1998; Bustin and Clarkson 1998; Laxminarayana and Crosdale 2002; Busch et al. 2003; Mastalerz et al. 2004; Hildenbrand et al. 2006; Busch et al. 2006; Siemons et al. 2007; Busch et al. 2007; Faiz et al. 2007a, b; Majewska and Ziętek 2007; Yu et al. 2008a, b, c; Astashov et al. 2008; Day et al. 2008; Mazumder and Wolf 2008; Pone et al. 2009; Majewska et al. 2009; Radliński et al. 2009; Battistutta et al. 2010; Gensterblum et al. 2010; He et al. 2010; Kim et al. 2011; Day et al. 2011; Le Gal et al. 2012; Maphala and Wagner 2012; Sakurovs 2012; An et al. 2013; Cai et al. 2013a, b; Dutka et al. 2013)-292)

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Figure 9 shows that, the relationship between vitrinite and vitrinite reflectance is expressed as Eq. (31)

$$Vitrinite = k_{{\rm vitr}} c_{{\rm vitr}}^{{R_{{\rm o,max}} }} + b_{{\rm vitr}}$$
(31)

where \(k_{{\rm vitr}}\), \(c_{{\rm vitr}}\), \(b_{{\rm vitr}}\) are the coefficients in Eq. (31) and \(k_{{\rm vitr}} < 0\), \(c_{{\rm iner}} < 1\).

The first and order derivative are separately shown in Eqs. (32) and (33):

$$Vitrinite^{{\prime }} = k_{{\rm vitr}} \ln c_{{\rm vitr}} c_{{\rm vitr}}^{{R_{{\rm o,max}} }}$$
(32)
$$Vitrinite^{{\prime \prime }} = k_{{\rm vitr}} \ln^{2} c_{{\rm vitr}} c_{{\rm vitr}}^{{R_{{\rm o,max}} }}$$
(33)

As \(k_{{\rm vitr}} < 0\), \(c_{{\rm iner}} < 1\), the first derivative (Eq. 32) of Eq. (31) is always greater than zero, while the second derivative (Eq. 33) is always less than zero, thus, it can be concluded that Eq. (31) exhibits up-convex, exponent-increasing type. It can be concluded that vitrinite content increases with the coal rank and at low metamorphic stage, vitrinite content increases greatly. The reason is: with the increase of metamorphism degree, exinite continuously generate hydrocarbon with a decreasing rate; in the stage of low metamorphic degree, exinite experience decarboxylation and the generation of oil process; the mid metamorphic stage experiences the transformation of gaseous hydrocarbons process. Therefore, in the stage of low metamorphic degree, is commonly seen, to mid metamorphic stage degree, the number of exinite reduced resulting in the increase of vitrinite content.

Besides, Fig. 9 shows that the minimum value of vitrinite content are generally larger than the curve expressed in the following equal: \(Vitrinite_{\hbox{min} } = 76.14 - 16996 \cdot 0.01063^{{R_{{\rm o,max}} }}\), and the corresponding curve is \(Vitrinite_{{\rm avr}} = 88.07 - 8498 \times 0.01063^{{R_{{\rm o,max}} }}\). The ranges of vitrinite content in subbituminous coal are from 11% to 97%, in low volatile bituminous coal are from 5% to 99%, in mid volatile bituminous coal are from 5% to 97%, in high volatile bituminous coal are from 8% to 97%, in anthracite are from 62% to 97%.

Combing Eqs. (31) and (12), the following Eq. (34) can be obtained:

$$Q = \frac{{k_{{\rm Q}} }}{{\ln c_{{\rm vitr}} }}\ln \left( {\frac{{Vitrinite - b_{{\rm vitr}} }}{{k_{{\rm vitr}} }}} \right) + b_{{\rm Q}}$$
(34)
$$Q^{'} = \frac{{k_{{\rm Q}} }}{{\ln c_{{\rm vitr}} }}\frac{1}{{Vitrinite - b_{{\rm vitr}} }}$$
(35)
$$Q^{''} = \frac{{ - k_{{\rm Q}} }}{{\ln c_{{\rm vitr}} }}\frac{1}{{\left( {Vitrinite - b_{{\rm vitr}} } \right)^{2} }}$$
(36)

As \(k_{{\rm Q}} > 0\), \(c_{{\rm vitr}} < 1\), \(\ln c_{{\rm iner}} < 0\), so \(\frac{{k_{{\rm Q}} }}{{\ln c_{{\rm iner}} }} < 0\), and also because \(k_{{\rm vitr}} < 0\), the first derivative (Eq. 35) of Eq. (34) is always greater than zero, while the second derivative (Eq. 36) is always greater than zero, thus, it can be concluded that Eq. (34) exhibits depression, logarithmic-increasing type. It can be concluded that there is a positive relationship between gas content and vitrinite content and at low vitrinite content stage, gas content increases slowly. This is because of the loose structure of coal sample with the higher vitrinite content, the less water absorption, the formation of the gel like network structure. Besides, with the increase of metamorphism degree, there are more volatile matter produced in vitrinite, resulting in the increase in the number of micropores, the increase of the surface area and the adsorption capacity of coal.

4.2.2 Inertinite content effect on adsorption capacity

Inertinite is one of the common macerals in coal, and its content is lower than that of vitrinite in coal, It is made of wood carbonization products of fibrous tissue in the peat swamps, under oxidizing conditions, plant body experience a greater degree of aromatic condensation due to the loss of oxidized atoms resulting in the dehydrogenation and dehydration, The most common inertinite components are fusinite, fusovitrite and macrinite. According to the structure and morphology of cell, Inertinite is divided into fusinite, half fusinite, fungus, secretion, coarse grain, particles, debris of inertinite, etc. Figure 10 shows the relationship between inertinite and Vitrinite reflectance (\(R_{{\rm o,max}}\)).

Fig. 10
figure 10

Variation of Intertinite content with vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: NERC-64, Chinese (Ceglarska-Stefańska and Brzóska 1998; Yu et al. 2004; Chang et al. 2006; Qin et al. 2007; Faiz et al. 2007a, b; Chang et al. 2008; Jiang et al. 2012; Lin et al. 2013; Xiao et al. 2016)-233, English(Giuliani et al. 1991; Crosdale et al. 1998; Bustin and Clarkson 1998; Marecka and Mianowski 1998; Laxminarayana and Crosdale 2002; Busch et al. 2003; Mastalerz et al. 2004; Busch et al. 2006; Hildenbrand et al. 2006; Busch et al. 2007; Majewska and Ziętek 2007; Siemons et al. 2007; Astashov et al. 2008; Day et al. 2008; Yu et al. 2008a, b; Mazumder and Wolf 2008; Pone et al. 2009; Radliński et al. 2009; Majewska et al. 2009; Gensterblum et al. 2010; Battistutta et al. 2010; He et al. 2010; Day et al. 2011; Kim et al. 2011; Maphala and Wagner 2012; Sakurovs 2012; An et al. 2013; Cai et al. 2013a, b; Dutka et al. 2013)-169)

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Figure 10 shows that the relationship between inertinite and vitrinite reflectance can be expressed in Eq. (37)

$$Inertinite = k_{{\rm iner}} c_{{\rm iner}}^{{R_{{\rm o,max}} }} + b_{{\rm iner}}$$
(37)

where \(k_{{\rm iner}}\), \(c_{{\rm iner}}\), \(b_{{\rm iner}}\) are the coefficients in Eq. (37) and \(k_{{\rm iner}} > 0\), \(c_{{\rm iner}} < 1\).

The first and order derivative are separately shown in Eqs. (38) and (39):

$$Inertinite = k_{{\rm iner}} \ln c_{{\rm iner}} c_{{\rm iner}}^{{R_{{\rm o,max}} }}$$
(38)
$$Inertinite{''} = k_{{\rm iner}} \ln^{2} c_{{\rm iner}} c_{{\rm iner}}^{{R_{{\rm o,max}} }}$$
(39)

As \(k_{{\rm iner}} > 0\), \(c_{{\rm iner}} < 1\),the first derivative (Eq. 38) of Eq. (37) is always less than zero, while the second derivative (Eq. 39) is always greater than zero, thus, it can be concluded that Eq. (37) exhibits depression, exponent-decreasing type, that is the relationship between inertinite and vitrinite reflectance presents negative exponential correlation. Inertinite is negative exponential decline with the increase of metamorphism degree, and at the low metamorphism degree inertinite decreases largely.

Figure 10 shows that the data for the relationship between inertinite and vitrinite reflectance are more dispersed, and its maximum value did not exceed the upper curve with the following equation: \(Inertinite_{\hbox{max} } = 18.557 + 706.1 \cdot 0.11058^{{R_{{\rm o,max}} }}\), and the corresponding average curve is: \(Inertinite_{{\rm avr}} = 9.2785 + 353.05 \times 0.11058^{{R_{{\rm o,max}} }}\). The ranges of inertinite content in subbituminous coal are from 0% to 78%, in low volatile bituminous coal are from 0% to 82%, in mid volatile bituminous coal are from 0 to 80 in high volatile bituminous coal are from 0% to 62%, in anthracite are from 0% to 37%.

Combing Eqs. (37) and (12), the Eq. (40) can be obtained:

$$Q = \frac{{k_{{\rm Q}} }}{{\ln c_{{\rm iner}} }}\ln \left( {\frac{{Inertinite - b_{{\rm iner}} }}{{k_{{\rm iner}} }}} \right) + b_{{\rm Q}}$$
(40)
$$Q^{'} = \frac{{k_{{\rm Q}} }}{{\ln c_{{\rm iner}} }}\frac{1}{{Inertinite - b_{{\rm iner}} }}$$
(41)
$$Q^{''} = \frac{{ - k_{{\rm Q}} }}{{\ln c_{{\rm iner}} }}\frac{1}{{\left( {Inertinite - b_{{\rm iner}} } \right)^{2} }}$$
(42)

As \(k_{{\rm Q}} > 0\), \(c_{{\rm iner}} < 1\), so \(\ln c_{{\rm iner}} < 0\), the first derivative (Eq. 41) of Eq. (40) is always less than zero, while the second derivative (Eq. 42) is always greater than zero, thus, it can be concluded that Eq. (40) exhibits depression logarithmic-decreasing type. That is, the gas content has a negative relationship with inertinite content, and at the low inertinite content stage, gas content decreases largely. The reason is that: inertinite has the greatest proportion of pores, a smaller surface area, and has a lot of internal oxygen functional groups and, therefore, has a weak adsorption capacity.

4.3 Ultimate analysis effect on adsorption capacity

Coal contains a lot of elements, but as the coal organic matter composition, the mainly five elements is carbon, hydrogen, oxygen, nitrogen, sulfur. These elements content are related to the formation of coal type, coal and rock composition and the degree of coalification. The different elements show not only the degree of coal metamorphism, but also reflect the coal nature. Carbon, hydrogen, and oxygen of Coal are mainly aromatic structure, aliphatic and alicyclic structure, the sum occupies more than 95%, and thus play major role in the adsorption ability of coal, and so, this paper mainly studies the three elements effect on the coal adsorption abilities.

4.3.1 Carbon content effect on adsorption capacity

Carbon is an important component of organic matter in coal. It is the main element of hex-acyclic-ring in the coal structure. A small amount of carbon is present in carbonate and carbon dioxide, and the content of carbon in coal is higher than that in any other element. Therefore, the metamorphism degree of coal is often referred to as carbonization degree. In order to reduce the impact of ash, moisture and other content, we use ash-free basis Dry indicator, Fig. 11 shows the relationship between carbon and vitrinite reflectance (\(R_{{\rm o,max}}\)).

Fig. 11
figure 11

Variation of \(C\) content with vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: Chinese (Ceglarska-Stefańska and Brzóska 1998; Chang et al. 2006; Qin et al. 2007; Chang et al.; Lin et al. 2013)-169, English(Allardice and Evans 1971; Bhattacharyya 1972; Gan et al. 1972; Joubert et al. 1973, 1974; Nandi and Walker 1975; Nelson et al. 1980; Yang and Saunders 1985; Reucroft and Patel 1986; Banerjee 1988; Friesen and Mikula 1988; Giuliani et al. 1991; DeGance et al. 1993; Ciembroniewicz and Marecka 1993; Milewska-Duda et al. 1994; Chaback et al. 1996; Martyniuk and Wiȩckowska 1997; Xu et al. 1997; Marecka and Mianowski 1998; Nodzeński 1998; Wang and Takarada 2003; Fitzgerald et al. 2005; Bae and Bhatia 2006; Dai et al. 2006; Siemons et al. 2007; Busch et al. 2007; Majewska and Ziętek 2007; Day et al. 2008; Majewska et al. 2009; Pone et al. 2009; Kędzior 2009; Radliński et al. 2009; Battistutta et al. 2010; Majewska et al. 2010; Gensterblum et al. 2010; He et al. 2010; Taraba 2011; Kim et al. 2011; Sakurovs 2012; Hao et al. 2013)-277)

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Figure 11 shows that the relationship between carbon content and vitrinite reflectance can be expressed in Eq. (23)

$$C_{{\rm daf}} = k_{{\rm C}} \ln (R_{{\rm o,max}} + b_{{\rm C}} ) + d_{{\rm C}}$$
(43)

where \(k_{{\rm C}}\), \(b_{{\rm C}}\), \(d_{{\rm C}}\) are the coefficients in Eq. (43) and \(k_{{\rm C}} > 0\).

Solution of the first and the second derivative of Eq. (43) are shown in (44) and (45):

$$C_{{\rm daf}}^{{\prime }} = \frac{{k_{{\rm C}} }}{{R_{{\rm o,max}} + b_{{\rm C}} }}$$
(44)
$$C_{{\rm daf}}^{{\prime \prime }} = \frac{{ - k_{{\rm C}} }}{{\left( {R_{{\rm o,max}} + b_{{\rm C}} } \right)^{2} }}$$
(45)

As \(k_{{\rm C}} > 0\), the first derivative (Eq. 44) of Eq. (43) is always greater than zero, while the second derivative (Eq. 45) is always less than zero, thus, it can be concluded that Eq. (43) exhibits up-convex, logarithmic-increasing type. That is, the content of carbon in coal increases with the increase of metamorphism degree. In the stage of low metamorphic grade, the increase of carbon content is larger. The reason is that: with the increase of coal rank, the microstructure of coal tends to be regular. Figure 11 shows that the data for the relationship between carbon content and vitrinite reflectance are limited in the upper and downward curves: \(C_{{\rm daf,max}} = 89.83 + 5.16\ln (R_{{\rm o,max}} - 0.215)\) and \(C_{{\rm daf,min}} = 83.62 + 5.788\ln (R_{{\rm o,max}} - 0.522)\), The corresponding average curve is: \(C_{{\rm daf,avr}} = 87.21 + 5.216\ln (R_{{\rm o,max}} - 0.47)\), The ranges of carbon content in subbituminous coal are from 0% to 87%, in low volatile bituminous coal are from 64% to 90.5%, in mid volatile bituminous coal are from 67% to 94%, in high volatile bituminous coal are from 82% to 95%, in anthracite are from 87% to 99%. The reason is that at the low metamorphism degree, due to the shallow bury, compaction and maturity is not high, so that the carbon content of coal is less.

Combing Eqs. (43) and (12), The Eq. (46) can be obtained:

$$Q = k_{{\rm Q}} \left[ {exp\left( {\frac{{C_{{\rm daf}} - d_{{\rm C} } }}{{k_{{\rm C}} }}} \right) - b_{{\rm C} } } \right] + b_{{\rm Q} }$$
(46)
$$Q^{'} = \frac{{k_{{\rm Q}} }}{{k_{{\rm C}} }}exp\left( {\frac{{C_{{\rm daf}} - d_{{\rm C}} }}{{k_{{\rm C}} }}} \right)$$
(47)
$$Q^{''} = \frac{{k_{{\rm Q}} }}{{k_{{\rm C}}^{2} }}exp\left( {\frac{{C_{{\rm daf}} - d_{{\rm C}} }}{{k_{{\rm C}} }}} \right)$$
(48)

As \(k_{{\rm Q}} > 0\), \(k_{{\rm C}} > 0\), the first and second derivatives (Eqs. 47 and 48) of Eq. (46) are always greater than zero, thus, it can be concluded that Eq. (46) exhibits depression, exponent -increasing type, That is, under the condition of the same temperature, adsorption capacity of coal increases with the increase of carbon content, and at the low carbon content stage, the gas content increases slowly. This is due to the increase of carbon content with the increase of coal rank, and the microstructure of coal tends to be regular. With the increase of \(R_{{\rm o,max}}\), the total porosity of coal increases, especially for small pores. The pore specific surface area of the coal is increasing, the adsorption capacity of coal is increasing, and the adsorption ability of methane is enhanced.

4.3.2 Hydrogen content effect on adsorption capacity

The importance of Hydrogen in coal is second only to the Carbon element, because of its minimum atomic mass, the number of atoms and Carbon elements are in the same magnitude order, and even more than Carbon elements. Hydrogen is also an important element in the composition of coal macromolecular backbone and side-chain, compared with the carbon element, hydrogen has greater reaction capacity, hydrogen elements content is different in different types of coal. Figure 12 shows the relationship between Hydrogen content and vitrinite reflectance (\(R_{{\rm o,max}}\)).

Fig. 12
figure 12

Variation of \(H\) content with vitrinite reflectance \(R_{{\rm o,max}}\). (Data size: Chinese (Chang et al. 2006; Qin et al. 2007; Chang et al. 2008; Lin et al. 2013)-152, English(Allardice and Evans 1971; Bhattacharyya 1972; Gan et al. 1972; Joubert et al. 1973, 1974; Nandi and Walker 1975; Yang and Saunders 1985; Reucroft and Patel 1986; Banerjee 1988; Friesen and Mikula 1988; Giuliani et al. 1991; DeGance et al. 1993; Chaback et al. 1996; Marecka and Mianowski 1998; Martyniuk and Wiȩ̧ckowska 1997; Wang and Takarada 2003; Fitzgerald et al. 2005; Dai et al. 2006; Bae and Bhatia 2006; Siemons et al. 2007; Majewska and Ziętek 2007; Day et al. 2008; Pone et al. 2009; Radliński et al. 2009; Majewska et al. 2009; Kędzior 2009; Majewska et al. 2010; Battistutta et al. 2010; Gensterblum et al. 2010; He, et al. 2010; Kim et al. 2011; Taraba 2011; Sakurovs 2012; Hao et al. 2013)-220)

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Hydrogen content are closely related to coal rank. Figure 12 shows that the relationship between Hydrogen content and vitrinite reflectance can be expressed by Eq. (49).

$$H_{{\rm daf}} = k_{{\rm H}} \ln (R_{{\rm o,max}} + b_{{\rm H}} ) + d_{{\rm H}}$$
(49)

where \(k_{{\rm H}}\), \(b_{{\rm H}}\), \(d_{{\rm H}}\) are the coefficients in Eq. (49),and \(k_{{\rm H}} < 0\).

The solution of the first and second derivative of Eq. (49) are shown in Eqs. (50) and (51):

$$H_{{\rm daf}}^{{\prime }} = \frac{{k_{{\rm H}} }}{{R_{{\rm o,max}} + b_{{\rm H}} }}$$
(50)
$$H_{{\rm daf}}^{{\prime \prime }} = \frac{{ - k_{{\rm H}} }}{{\left( {R_{{\rm o,max}} + b_{{\rm H}} } \right)^{2} }}$$
(51)

As \(k_{{\rm H}} < 0\), the first derivative (Eq. 50) of Eq. (49) is always less than zero, while the second derivative (Eq. 51) is always greater than zero, thus, it can be concluded that Eq. (49) exhibits depression, logarithmic-decreasing type, That is the hydrogen content decreases with the increases of coal rank, and at the low coal rank, The hydrogen content decreases greatly.

Figure 12 shows that the data of the relationship between hydrogen content and coal rank has the limits of the upper and downward curves expressed in the following Equations: \(H_{{\rm daf,max}} = 7.87474 - 3.26191\ln (R_{{\rm o,max}} + 0.50856)\); \(H_{{{\rm daf} ,min}} = 4.27974 - 1.77558\ln (R_{{{\rm o} ,max}} + 0.62858)\), the equation for the corresponding average curve is: \(H_{{\rm daf,avr}} = 7.50833 - 3.16908\ln (R_{{\rm o,max}} + 1.19543)\), The ranges of hydrogen content in subbituminous coal are from 2.7% to 8.4%, in low volatile bituminous coal are from 2.5% to 6.9%, in mid volatile bituminous coal are from 2.5% to 7%, in high volatile bituminous coal are from 2% to 6.2%, in anthracite are from 0.3% to 5.3%. The reason for this change is the formation of low metamorphic coal from the Low hydrogen containing organisms.

Combing Eqs. (49) and (12), the Eq. (52) can be obtained:

$$Q = k_{{\rm Q}} \left[ {exp\left( {\frac{{H_{{\rm daf}} - d_{{\rm H}} }}{{k_{{\rm H}} }}} \right) - b_{{\rm H}} } \right] + b_{{\rm Q}}$$
(52)
$$Q^{'} = \frac{{k_{{\rm Q}} }}{{k_{{\rm H}} }}exp\left( {\frac{{H_{{\rm daf}} - d_{{\rm H}} }}{{k_{{\rm H}} }}} \right)$$
(53)
$$Q^{''} = \frac{{k_{{\rm Q}} }}{{k_{{\rm H}}^{2} }}exp\left( {\frac{{H_{{\rm daf}} - d_{{\rm H}} }}{{k_{{\rm H}} }}} \right)$$
(54)

As \(k_{{\rm Q}} > 0\), \(k_{{\rm H}} < 0\), the first derivative (Eq. 53) of Eq. (52) is always less than zero, while the second derivative (Eq. 54) is always greater than zero, thus, it can be concluded that Eq. (52) exhibits depression, exponent-decreasing type, thus, the gas content dropped exponentially with the Dry ash-free basis hydrogen content. The reason is that coal methyl and methylene chain increases the methane adsorption trap in coal, enhances adsorption ability of methane on coal surface. And with the increase in metamorphic grade, side chain of coal decreased, resulting in the hydrogen element content. Contribution of hydrogen to the overall coal adsorption capacity is negative.

4.3.3 Oxygen content effect on adsorption capacity

Oxygen is one of the main elements of coal and mainly consists of carboxyl (\(- {{\rm COOH}}\)), hydroxyl (\(- {{\rm OH}} -\)), carbonyl, methoxyl (\(- {{\rm OCH}}_{ 3} -\)), and ether groups (–C–O–C –), and some of the oxygen combing with carbon skeleton into a heterocyclic ring (White 1909). The total amount and shape of oxygen in coal have a direct influence on the coal properties. Figure 13 shows the relationship between Oxygen content and vitrinite reflectance (\(R_{{\rm o,max}}\)).

Fig. 13
figure 13

Variation of \(O\) content with vitrinite reflectance \(R_{{\rm o,max}}\). (Data size:Chinese (Chang et al. 2006; Qin et al. 2007; Chang et al. 2008; Lin et al. 2013)-145, English(Allardice and Evans 1971; Gan et al. 1972; Joubert et al. 1973, 1974; Nandi and Walker 1975; Nelson et al. 1980; Yang and Saunders 1985; Reucroft and Patel 1986; Banerjee 1988; Friesen and Mikula 1988; DeGance et al. 1993; Chaback et al. 1996; Wang and Takarada 2003; Fitzgerald et al. 2005; Dai et al. 2006; Bae and Bhatia 2006; Siemons et al. 2007; Majewska et al. 2009; Pone et al. 2009; Radliński et al. 2009; Battistutta et al. 2010; Gensterblum et al. 2010; He et al. 2010; Kim et al. 2011; Hao et al. 2013)-254)

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Figure 13 shows the relationship between oxygen content and vitrinite reflectance:

$$O_{{\rm daf}} = k_{{\rm O}} b_{{\rm O}}^{{R_{{\rm o,max}} }} + d_{{\rm O}}$$
(55)

where \(k_{{\rm O}}\), \(b_{{\rm O}}\), \(d_{{\rm O}}\) are the coefficients in Eq. (55), and \(k_{{\rm O}} > 0\), \(b_{{\rm O}} < 1\).

The solutions of the first and second derivative for the Eq. (55) are shown in Eqs. (56) and (57):

$$Q_{{\rm daf}}^{{\prime }} = k_{{\rm O}} \ln b_{{\rm O}} b_{{\rm O}}^{{R_{{\rm o,max}} }}$$
(56)
$$O_{{\rm daf}}^{{\prime \prime }} = k_{{\rm O}} \ln^{2} b_{{\rm O}} b_{{\rm O}}^{{R_{{\rm o,max}} }}$$
(57)

As \(k_{{\rm O}} > 0\), the first derivative (Eq. 56) of Eq. (55) is always less than zero, while the second derivative (Eq. 57) is always greater than zero, thus, it can be concluded that Eq. (55) exhibits depression, exponent-decreasing type, that is,the oxygen content decreases with the increase of coal rank,and at the low metamorphic degree, the oxygen declines greatly. Reasons for this change is increases of oxygen content decreases the caking property of coal and increased the hydrogen bound, so side chain of the six carbon ring in young coal is more, and most of side chain polar groups are oxygen containing functional groups. Side chain will decrease with the coal rank, and finally resulting in the decrease of oxygen content.

Figure 13 shows that the data for the relationship between oxygen content and vitrinite reflectance exists upper and downward limits expressed in the equations: \(O_{{{\rm daf} ,max}} = 4.98 + 67.19 \cdot 0.1655^{{R_{{\rm o,max}} }}\), \(O_{daf,\hbox{min} } = 0.866\,+\,30.482 \cdot 0.047^{{R_{{\rm o,max}} }}\), the corresponding average curve is expressed in the following equation: \(O_{{\rm daf,avr}} = 2.943\,+\,45.555 \cdot 0.138^{{R_{{\rm o,max}} }}\), from the dashed line in Fig. 13, it can be seen that: The ranges of oxygen content in subbituminous coal are from 1% to 29%, in low volatile bituminous coal are from 0.9% to 26%, in mid volatile bituminous coal are from 0.8% to 11.5%, in high volatile bituminous coal are from 0% to 11%, in anthracite are from 0% to 5.5%.

Combing the Eqs. (55) and (12), the Eq. (58) can be obtained:

$$Q = \frac{{k_{{\rm Q}} }}{{\ln b_{{\rm O}} }}\left[ {\ln \left( {O_{{\rm daf}} - d_{{\rm O}} } \right) - \ln k_{{\rm O}} } \right] + b_{{\rm Q}}$$
(58)
$$Q^{{\prime }} = \frac{{k_{{\rm Q}} }}{{\ln b_{{\rm O}} }}\frac{1}{{O_{{\rm daf}} - d_{{\rm O}} }}$$
(59)
$$Q^{'\prime \prime } = \frac{{ - k_{{\rm Q}} }}{{\ln b_{O} }}\frac{1}{{\left( {O_{{\rm daf}} - d_{O} } \right)^{2} }}$$
(60)

As \(k_{{\rm Q}} > 0\), \(b_{{\rm O}} < 1\), \(\ln b_{{\rm O}} < 0\), so \(\frac{{k_{{\rm Q}} }}{{\ln b_{{\rm O}} }} < 0\), the first derivative (Eq. 59) of Eq. (58) is always less than zero, while the second derivative (Eq. 60) is always greater than zero, thus, it can be concluded that Eq. (58) exhibits depression, logarithmic-decreasing type, that is the oxygen has a negative effect on gas adsorption capacity, and at the low oxygen content stage, the gas content decreases greatly. The reason is that the oxygen element exists in the form of oxygen functional groups in coal, And the oxygen functional groups of carboxyl, hydroxyl and aldehyde reduced the adsorption wells of methane, weakened the adsorption capacity of methane. Therefore, the reduction of oxygen functional groups in coal increases the adsorption capacity of coal. It was observed that there was, in general, a positive correlation between the methane saturated adsorption capacity and the micropore volume of coals while a negative correlation between methane saturated adsorption capacity and the Ototal/Ctotal. Coal with a higher amount of oxygen surface groups, and consequently with a less hydrophobic character, had lower methane adsorption capacity.

4.4 Comprehensive analysis of physicochemical properties of coal effect on methane adsorption

The reason for the fixed carbon content increases with the coal rank is: In the process of coal metamorphism, the carbonization trend is obviously shown, that is, the low rank contains five kinds of elements of C, H, O, N, S, While, the anthracite basically only contain one element of carbon. The reason is that: the chemical components and the structure of coal changes in the process of coal metamorphic, the performance for this is the rot colonization material constantly aggregate, hydrophilic oxygen group such as hydroxyl and carboxyl lost large of water, from low metamorphic degree of coal with variety functional groups structure, to anthracite containing condensation aromatic nuclear of structure, hydrophilic functional groups constantly reduced, carbon content constantly increases, resulting in the weaken of hydrophilic constantly and the adsorption of methane constantly enhanced. At the low metamorphic degree stage, the oxygen and hydrogen content in vitrinite are all greater than in inertinite,which means that the oxygen-containing groups of hydroxyl and carboxyl in vitrinite at this stage is higher, resulting in the higher hydrophilia and weakened gas adsorption capacity. Thus, at low metamorphic degree stage, the gas adsorption capacity of vitrinite is weaker than that of inertinite vitrinite, the reason is that with the increase of coal rank, hydrophilic group in vitrinite fall off, and the gas adsorption capacity can be strengthen.

5 Conclusions

  1. 1.

    Physicochemical properties of coal and the methane adsorption capacity had a well-regulated change with coal rank. Limit adsorption capacity expressed in \(a_{{\rm daf}}\) decreases firstly and then increases with the increase of coal rank, but Adsorption constant of \(b\) falling in the range of 0.15–2.25 MP−1 present the opposite changing law. Dry ash-free basis gas content of \(Q_{{\rm daf}}\) increases with increasing metamorphism of coal, and the gas content of raw coal (\(Q\))increases linearity with the coal rank.

  2. 2.

    Fixed carbon, vitrinite, carbon element present the positive interrelated relationship to maximum vitrinite reflectance \(R_{{\rm o,max}}\), and \(R_{{\rm o,max}} \sim Fc_{{\rm ad}}\), \(R_{{\rm o,max}} \sim Vitrinite\) exhibits power-decreasing type, \(R_{{\rm o,max}} \sim C_{{\rm daf}}\) exhibits Logarithm function type. The volatile, inherent moisture, inertinite, oxygen content, hydrogen content present the negative interrelated relationship to maximum vitrinite reflectance \(R_{{\rm o,max}}\),and \(R_{{\rm o,max}} \sim V_{daf}\) exhibits power-decreasing type, \(R_{{\rm o,max}} \sim M_{{\rm ad}}\), \(R_{{\rm o,max}} \sim Inertinite\), \(R_{{\rm o,max}} \sim O_{daf}\) exhibit exponential function type, and \(R_{{\rm o,max}} \sim H_{daf}\) exhibit Logarithm function type. Furthermore, we can derive fixed carbon and vitrinite present a positive correlation with adsorption capacity of \(Q\), and \(Fc_{ad} \sim Q\) and \(Vitrinite \sim Q\) are Logarithm function, while \(C_{{\rm daf}} \sim Q\) exhibits exponential function type. Besides, volatile,vinherent moisture, inertinite, oxygen content, hydrogen content of coal present a negative correlation with adsorption capacity of \(Q\), and \(V_{{\rm daf}} \sim Q\) exhibits a power function type; \(M_{{\rm ad}} \sim Q\), \(Inertinite \sim Q\) and \(O_{{\rm daf}} \sim Q\) exhibit Logarithm function type, while \(H_{{\rm daf}} \sim Q\) is an exponential function.

  3. 3.

    The essence of the effect of coal rank on methane adsorption capacity is the changing of Physicochemical Properties of Coal.