A new method to determine varying adsorbed density based on Gibbs isotherm of supercritical gas adsorption

Abstract

In the calculation of the absolute adsorption of supercritical gas adsorbed on the microporous materials, most existing methods regard the adsorbed density as a constant, which is very unreasonable. In this study, an extended pressure point method combined with Langmuir adsorption model is proposed in which the varying adsorbed density under different pressures is considered at the same time. The utility of the proposed method to correlate accurately the experimental data for supercritical gas adsorption system is demonstrated by high-pressure methane adsorption measurements on two groups of shale samples. Taking advantage of the proposed method, we can obtain the adsorbed density and the adsorbed volume corresponding to different pressures. Compared with the conventional methods under the assumption of fixed and parameterized adsorbed density, the proposed method yields better fitting results with the experimental data. Our work should provide important fundamental understandings and insights into the supercritical gas adsorption system.

Keywords

Supercritical gas adsorption extended pressure point method adsorbed density adsorbed volume Langmuir adsorption model

Introduction

Supercritical gas adsorption on the microporous materials is the physicochemical basis of many important engineering processes and industrial applications, such as the hydrogen storage on activated carbon, sequestration of carbon dioxide in coalbed, gas-in-place calculation of shale (Ge et al., 2016; Moellmer et al., 2011; Pini, 2014; Qi et al., 2017; Tang and Ripepi, 2017; Tang et al., 2017). It occurs through a physical interaction between the gas molecules and the adsorbent surface, thus creating a region whose density is different from that of the homogeneous bulk phase (Pini, 2014). The design of the abovementioned applications relies greatly on the availability of high-pressure adsorption data as well as reliable adsorption models (Tang et al., 2017). Although significant advances have been made in the accuracy of the common adsorption measuring techniques (e.g. gravimetric, volumetric methods) (Broom and Webb, 2017; Gensterblum et al., 2009), the realistic mechanism of supercritical gas adsorption is still far from being fully understood. Adsorption experiment can only provide Gibbs excess adsorption (m_ex), the difference between the absolute adsorption (m_abs) and the amount that would be present in the same volume at the density of the gas in the bulk phase (Do and Do, 2003; Pini, 2014). In adsorption system, gas molecules are affected by two kinds of forces: the force between the gas molecules and the force of the adsorbent surface exerting on the gas molecules (Fitzgerald et al., 2003). When the compound force system reaches equilibrium, there seems to be a layer of invisible film at a distance from the adsorbent surface. If the gas molecules outside the membrane move toward the adsorbent surface, they will rebind rather than enter the membrane (Staudt et al., 1993). The quantity of gas molecules between the membrane and the adsorbent surface is the absolute adsorption. As we know, the difference between the Gibbs excess adsorption and the absolute adsorption becomes larger and larger with the increase of equilibrium pressure (Merkel et al., 2015). Meanwhile, all adsorption models are initially developed to determine the absolute adsorption (Zhou and Zhou, 2009). Therefore, the conversion of the experimentally determined Gibbs excess adsorption to the absolute adsorption is crucial to the interpretation of the adsorption system (Zhou et al., 2018). The relationship between the Gibbs excess adsorption and the absolute adsorption is given as (Hu et al., 2018; Tian et al., 2016)

m_{a b s} = m_{e x} \cdot (\frac{ρ_{a d} (T, P)}{ρ_{a d} (T, P) - ρ (T, P)})

(1)

where ρ(T, P) and ρ_ad(T, P) are the density of the bulk phase and the adsorbed phase under specific temperature and pressure. According to equation (1), we can see that the Gibbs excess adsorption is positive when the adsorbed density is greater than the bulk density, while it is negative when the adsorbed density is smaller than the bulk density. If the adsorbed density is known a priori, the absolute adsorption can be calculated based on the experimental Gibbs excess adsorption. However, there are not yet any unambiguous ways to determine the adsorbed density (Hao et al., 2014; Tang et al., 2016). In practice, some assumptions have to be made to determine the adsorbed density. Typically, these assumptions can be divided into two categories. The first category is to treat the adsorbed density as a known constant. Dubinin (1960) assumed that the adsorbed phase was in the accumulation of compressed gas molecules. Ozawa et al. (1976) considered the adsorbed phase as a superheated liquid with a density-dependent upon the thermal expansion of the liquid. Tsai et al. (1985) argued that the adsorbed density is equivalent to the critical density of the supercritical gas. Later, Dreisbach et al. (1999) suggested using the saturated liquid density at the normal boiling point. However, for supercritical gas adsorption, it is uncertain to identify whether the adsorbed phase is in the form of a liquid or a gas state (Zhou et al., 2001). Thus, applying the physical properties of the gas to calculate the adsorbed density may result in unreliable conclusions. In addition, when the Gibbs isotherm is plotted as a function of the bulk density, the Gibbs excess adsorption will be reduced linearly under sufficiently high bulk density. The linearity in this region can be explained by assuming that no additional adsorption is occurring (Fitzgerald et al., 2003). By linearly fitting the data in this region, we can obtain the adsorbed density when the Gibbs excess adsorption is zero (Gensterblum et al., 2009; Gensterblum et al., 2010; Humayun and Tomasko, 2000). Note that the adsorbed density gained by using the above graphical estimate method (GEM) can only represent the condition of saturated adsorption. The second category is to treat the adsorbed density as an unknown constant. The adsorbed density is added as a parameter to the adsorption model, and the Gibbs excess adsorption is directly fitted to obtain a so-called adsorbed density (Gasparik et al., 2014; Li et al., 2017; Lutynski and González González Miguel, 2016). In a sense, the adsorbed density obtained by the abovementioned parameterized method (PM) is actually a mean density for the global experimental pressure.

It should be mentioned that all these assumptions are sometimes unreliable in that they all treat the adsorbed density as a constant independent of the pressure of the bulk phase. Nowadays, scientists have applied molecular dynamics (MD) simulations and grand canonical Monte Carlo (GCMC) simulations to study supercritical gas adsorption system (Jiang and Lin, 2018; Mosher et al., 2013; Song et al., 2018; Xiong et al., 2017; Zhang et al., 2018). In MD or GCMC simulations, the pores in microporous materials are often simplified to be slit pores, circle pores, triangle pores, square pores, etc. The gas molecular density distribution under different pressures can be calculated from MD or GCMC simulations (Xiong et al., 2016). It is shown that the gas density distribution exhibits a gradual oscillation from the adsorbent surface to the pore center because of the enhanced interactions between the gas molecules and the adsorbent surface atoms. Through the density distribution diagram under different pressures, we can find that it is unreasonable to assume that the adsorbed density does not change as the pressure increases (Jiang and Lin, 2018). However, the heterogeneous properties of microporous adsorbent (Cai et al., 2010) cannot be reasonably portrayed by the simplified, homogeneous pore structures of MD or GCMC simulations. Therefore, in order to accurately describe the supercritical gas adsorption, an appropriate and robust method of evaluating adsorbed density is necessary to obtain the absolute adsorption from the experimental Gibbs isotherm.

In this work, a method that combines extended pressure point method (EPPM) and Langmuir adsorption model is proposed to determine the varying adsorbed density under different pressures for the supercritical gas adsorption. The high-pressure methane adsorptions on two groups of shale samples from the Changning block and Fuling block were measured at a pressure range up to 25 MPa by gravimetric method. The proposed method can capture the characteristics of the adsorbed density variations under different pressure points instead of treating the adsorbed density as a constant. Additionally, the adsorbed volume can be obtained based on the calculated adsorbed density. It is found that the newly developed method provides a more accurate estimate of the adsorbed density than the GEM and the PM.

Methodology and experiments

Langmuir adsorption model

The Langmuir adsorption model is one of the most popular models to describe the physical process of gas adsorption, which is represented as (Do and Do, 2003)

V = V_{L} \frac{P}{P + P_{L}}

(2)

where V is the absolute adsorption under equilibrium pressure P and V_L and P_L are the Langmuir volume and the Langmuir pressure, respectively. V_L represents the maximum adsorption capacity and mainly reflects the properties of the adsorption system under the saturated condition. Therefore, in the adsorption experiment, V_L is mainly determined by the sufficiently high-pressure data points. P_L is defined as the pressure corresponding to one-half Langmuir volume and reflects the speed of the rising stage in the adsorption isotherm. P_L is an important parameter reflecting the gas desorption process (Hu et al., 2018) and mainly reflected by the relatively low-pressure data points. In addition, the adsorption heat and standard entropy for the adsorption system are related to P_L under different temperatures (Gasparik et al., 2014).

EPPM

In order to determine the varying adsorbed density under different pressures, this paper proposes the EPPM. Its basic idea is based on the concept of the point subgroup. The so-called point subgroup is to use a few points with a small pressure increment as a subgroup, as illustrated in Figure 1. Considering the pressure increment is small for the pressure points at the same subgroup, their adsorbed densities are assumed to be approximately equal. The procedures of the EPPM are provided below:

Figure 1.

Schematic diagram of the EPPM. The red region indicates that the adsorption has reached saturation while the black region indicates that the adsorption is unsaturated.

Determining the normal pressure points distribution based on the conventional Gibbs isotherm and each pressure point is extended to form a subgroup according to a certain increment. The increment should be small enough to ensure the assumption that the adsorbed densities are constant in each point subgroup.

Obtaining the Gibbs excess adsorption: The methods of obtaining Gibbs excess adsorption at all pressure points can be divided into two kinds. The first is to use all pressure points as the experimental points to obtain the Gibbs excess adsorption, which is referred to as experimental expansion method (EEM). The second is to get the Gibbs excess adsorption of normal pressure distribution through adsorption experiment, and then get the Gibbs excess adsorption of other pressure points in each subgroup by the interpolation method, which is called fitting expansion method (FEM).

Determining the adsorbed density ρ_sat under the saturated condition: According to the GEM, the adsorbed density at several points or point subgroups in the saturated state is obtained.

Obtaining the model parameters and adsorbed density of each unsaturated subgroup by using cyclic fitting: The purpose of the cyclic fitting is to make the objective function smaller, such as the least-squares residual minimization algorithm, which is defined as

r e s i d u a l = {\sum_{i} \sum_{j} (m_{i j, c a l} - m_{i j, c i r})}^{2}

(3)

where i represents the index of point subgroups, j represents the index of pressure points in each subgroup, cal represents the absolute adsorption calculated by circulatory Langmuir volume and Langmuir pressure, cir represents the absolute adsorption calculated by the experimentally determined Gibbs excess adsorption and circulatory adsorbed density according to equation (1). Through the above process, we can get the adsorbed density under different pressure points and more accurate Langmuir parameters. Once this is done, the adsorbed volume can be calculated directly

V_{a d} (T, P) = \frac{m_{a b s} (T, P)}{ρ_{a d} (T, P)}

(4)

Experiments

Shale gas is one of the most promising unconventional natural gas resources, and huge reserves are located around the southwestern of China. In southwestern China, Changning block in Sichuan and Fuling block in Chongqing are two representative shale gas production blocks. These two blocks have different material compositions and production properties. So we selected these two shale samples for experimental study.

Each shale sample was crushed and sieved through a 60–80 mesh metal sifter to obtain the desired power size and placed in a drying oven at 60°C for more than 48 h to dehydrate. Prior to starting the adsorption experiments, the samples were dried again at 60°C in a vacuum for 24 h to remove residual free water.

The high-pressure methane adsorption experiments were conducted using the magnetic suspension balance ISOSORP-HP (Rubotherm). Compared to the volumetric adsorption equipment, the advantage of the gravitational adsorption equipment is that it will not accumulate relative error in the procedure of increasing adsorption pressure (Zhou and Zhou, 2009). The precision of the magnetic suspension balance is 0.01 mg and the pressure precision is 0.01 MPa. For all samples, the experimental temperature was 60°C and the pressure was up to 25 MPa. The detailed experimental procedures for the magnetic suspension balance equipment have been well documented in a series of publications (Pan et al., 2016; Yang et al., 2014; Zhou et al., 2018).

During the experiment, the whole pressure segment was divided into 16 subgroups and each subgroup contained about five pressure points. The adsorption equilibrium measurements were performed for a sufficient length of time to ensure that there was no significant variation in weight and pressure.

Results and discussion

The measured Gibbs isotherms are presented in the unit of cm³/g rock under STP conditions and shown in Figure 2 using bulk density as the x coordinate. The experimental results show that the Gibbs excess adsorption increases to a maximum value at the bulk density of approximately 0.024 cm³/g. After that, the Gibbs excess adsorption begins to slightly decrease and eventually decreases linearly and turns into negative with further increasing bulk density. This phenomenon is also reported in previous studies (Lutynski and González González Miguel, 2016; Pini, 2014; Tang et al., 2016). It is caused by the fact that the Gibbs excess adsorption is calculated on the basis of neglecting the buoyancy of the adsorbed phase. By use of the GEM, we obtained the saturated adsorbed density: the saturated adsorbed density of Changning sample is 0.100 g/cm³, while that of Fuling sample is 0.098 g/cm³. These two values are far smaller than the liquid density of methane at its boiling point (0.373 g/cm³) and the van der Waals density (0.423 g/cm³) (Bi et al., 2017; Jiang and Lin, 2018; Zhou et al., 2018), but are close to the result obtained by Song et al. (2018) Besides, in order to compare the results of the EEM and the FEM, we expand the experimental data to obtain Figure 3 by interpolation method based on the center point of each point subgroup.

Figure 2.

Measured Gibbs isotherms m_ex of methane on Changning sample and Fuling sample plotted against bulk density at 60°C. The linear equations of m_ex are fitted within the quasi-linear range of the last three point subgroups.

Figure 3.

The Gibbs isotherms of methane on Changning sample and Fuling sample obtained by interpolation method.

To compare with the existing adsorbed density treatment methods, the experimental results are also fitted by the GEM and the PM. The fitting parameters are shown in Table 1, and the calculated absolute adsorptions are shown in Figure 4. In Figure 4, the scatter points represent the absolute adsorption calculated by the cyclic fitting process and the solid lines represent the fitting curves using the fitting parameters (V_L and P_L) listed in Table 1. Compared with the GEM and PM, the least-squares residual calculated by the EPPM is about an order of magnitude lower, which indicates that the EPPM is more accurate to fit the adsorption data. The calculated Langmuir volumes obtained by these three methods are almost similar, but the calculated Langmuir pressures are very different. This kind of phenomenon can also be shown very well in Figure 4. The rising tendency of the EPPM curves is more intense than that of the curves calculated by the other two methods while the horizontal parts are almost consistent. For the two samples, the GEM overestimates the Langmuir pressure by up to 100% and PM up to 70%. This is because that the difference of the adsorbed density under low pressures is not taken into account in the GEM and the PM. It results in the undercompensation of the calculated absolute adsorption. In addition, the results obtained from the EEM and FEM are almost similar. This consistency is very meaningful for the application of the EPPM in that we can directly analyze the experimental adsorption data based on the normal pressure point distribution. This will greatly reduce the workload of the experiment.

Figure 4.

Comparison of the absolute adsorption of methane on (a) Changning sample and (b) Fuling sample calculated using different methods. EEM: experimental expansion method; EPPM: extended pressure point method; FEM: fitting expansion method; GEM: graphical estimate method; PM: parameterized method.

Table 1.

Comparison of the fitting results for Changning sample and Fuling sample using different methods.

	Changning sample		Fuling sample
	EEM	FEM	EEM	FEM
Lv_EPPM(cm³/g)	6.11	6.06	5.52	5.52
Lp_EPPM (MPa)	1.48	1.42	2.13	2.22
residual_EPPM	0.493993	0.147046	0.906230	0.743218
Lv _GEM (cm³/g)	6.26	6.20	5.58	5.71
Lp _GEM (MPa)	2.88	2.82	4.56	4.77
residual_GEM	3.146668	2.755558	9.537248	10.395333
Lv _PM (cm³/g)	5.96	5.92	5.03	5.08
Lp _PM (MPa)	2.56	2.51	3.58	3.65
residual_PM	2.591465	2.395807	2.617281	2.554605

EEM: experimental expansion method; EPPM: extended pressure point method; FEM: fitting expansion method; GEM: graphical estimate method; PM: parameterized method.

The adsorbed density obtained by cyclic fitting is shown in Figure 5. The EPPM curves show that the adsorbed density is a function of the equilibrium pressure, while the GEM curves and the PM curves are horizontal lines independent of the equilibrium pressure. Under low pressures, the adsorbed density obtained by EPPM is bigger than the bulk density, but the term ρ_ad/(ρ_ad − ρ) in equation (1) should not be approximated to 1. At present , this problem is always ignored in most of the studies (Mukherjee and Misra, 2018; Zhou et al., 2018). Researchers usually think that there is no obvious difference between the Gibbs excess adsorption and the absolute adsorption when the pressure is low, so it is not necessary to analyze or correct the adsorbed density under low pressures (Murata and Kaneko, 2000; Zhao et al., 2018). Judging from our results, this kind of comment is obviously incorrect. As the pressure increases, the increasing trend of the adsorbed density gradually slows down and finally, it is lower than that of the bulk density. This result explains the emergence of a negative Gibbs excess adsorption and it should be considered in conjunction with the force field in the adsorption system. Under low pressures, due to the attraction force of the adsorbent surface exerting on the gas molecules, the gas molecules near the adsorbent surface enrichment. But with the increase of the bulk density, the attraction force limits the further increase of the adsorbed density, and the adsorbed density finally becomes stable at a certain limit. Although many researchers have confirmed that the Gibbs excess adsorption will become negative under sufficiently high pressure, they do not consider it from the point of view that the adsorbed density is gradually changing with pressure (Lutynski and González González Miguel, 2016; Ross and Marc Bustin, 2007). Again, the curve of the adsorbed density obtained by the EEM and the FEM is almost consistent with each other.

Figure 5.

Comparison of the adsorbed density of methane on (a) Changning sample and (b) Fuling sample calculated using different methods. EEM: experimental expansion method; EPPM: extended pressure point method; FEM: fitting expansion method; GEM: graphical estimate method; PM: parameterized method.

After the absolute adsorption and the adsorbed density are determined, the adsorbed volume calculated according to equation (4) is shown in Figure 6. It can be seen that when the pressure is higher than 0.8 MPa, the adsorbed volume gradually decreases with the increase of pressure. As a comparison, the adsorbed volume converted by a constant adsorbed density is a linear function of the absolute adsorption. This behavior can be explained by the force field of the gas molecules in the adsorption system. From a molecular perspective, the adsorbed volume is the near-wall space where significant molecule–solid interactions exist (Mohammad et al., 2009). The force of the adsorbent surface exerting on the gas molecules at a specific location is constant, but the force between gas molecules increases with the increase of bulk density in the experimental pressure range. As the bulk density increases, regions originally dominated by the force between adsorbent surface and gas molecules transform to be dominated by intermolecular interactions. This means that the membrane to demarcate the adsorbed phase and bulk phase moves toward the adsorbent surface with the increase of bulk density and finally becomes stable at a certain distance from the adsorbent surface. Note that whether the adsorbed volume is gradually increasing with increasing pressure needs further investigation when the pressure is lower than around 0.8 MPa. This is probably due to the measuring precision of the pressure sensor in the adsorption equipment.

Figure 6.

Comparison of the adsorbed volume of methane on (a) Changning sample and (b) Fuling sample calculated using different methods. EEM: experimental expansion method; EPPM: extended pressure point method; FEM: fitting expansion method; GEM: graphical estimate method; PM: parameterized method.

Conclusions

With the introduction of the EPPM, the varying adsorbed density under different pressures can be calculated straightforwardly according to the Gibbs excess adsorption. The experimental data of two groups of shale-methane adsorption system were employed to examine the effectiveness of the proposed method. The significant conclusions can be summarized as follows:

The adsorbed density shows a pressure-dependent tendency. It increases with the bulk density, then slows down and gradually reaches saturation under sufficiently high pressures. The bulk density increases with pressure and eventually exceeds the adsorbed density, which leads to the negative Gibbs excess adsorption under sufficiently high pressures.

The adsorbed volume decreases gradually with pressure and eventually stabilizes. A reasonable explanation is given from the perspective of the force field of the gas molecules.

The results obtained by the EEM and the FEM are consistent, depicting that the EPPM can be applied to the normal pressure point distribution adsorption experiment.

It should be mentioned that we employ the conventional Langmuir adsorption model in this study. In fact, adopting different adsorption models may make some difference. This point needs to be further evaluated in the following work.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was financially supported by CAS Strategic Priority Research Program (Grant No. XDB10030402), CNPC-CAS Strategic Cooperation Research Program (Grant No. 2015A-4812), and National Science and Technology Major Project (Grant No. 2017ZX05009005-002).

References

Jiang

et al . (2017) Ono–Kondo model for supercritical shale gas storage: A case study of Silurian Longmaxi shale in southeast Chongqing, China. Energy & Fuels 31(3): 2755–2764.

Broom

Webb

(2017) Pitfalls in the characterisation of the hydrogen sorption properties of materials. International Journal of Hydrogen Energy 42(49): 29320–29343.

Cai

Zou

et al . (2010) Fractal analysis of surface roughness of particles in porous media. Chinese Physics Letters 27(2): 024705.

(2003) Adsorption of supercritical fluids in non-porous and porous carbons: Analysis of adsorbed phase volume and density. Carbon 41(9): 1777–1791.

Dreisbach

Staudt

Keller

(1999) High pressure adsorption data of methane, nitrogen, carbon dioxide and their binary and ternary mixtures on activated carbon. Adsorption 5(3): 215–227.

Dubinin

(1960) The potential theory of adsorption of gases and vapors for adsorbents with energetically nonuniform surfaces. Chemical Reviews 60(2): 235–241.

Fitzgerald

Sudibandriyo

Pan

et al . (2003) Modeling the adsorption of pure gases on coals with the SLD model. Carbon 41(12): 2203–2216.

Gasparik

Bertier

Gensterblum

et al . (2014) Geological controls on the methane storage capacity in organic-rich shales. International Journal of Coal Geology 123: 34–51.

Fan

Cao

et al . (2016) Investigation of organic related pores in unconventional reservoir and its quantitative evaluation. Energy & Fuels 30(6): 4699–4709.

10.

Gensterblum

van Hemert

Billemont

et al . (2009) European inter-laboratory comparison of high pressure CO2 sorption isotherms. I: Activated carbon. Carbon 47(13): 2958–2969.

11.

Gensterblum

van Hemert

Billemont

et al . (2010) European inter-laboratory comparison of high pressure CO₂ sorption isotherms II: Natural coals. International Journal of Coal Geology 84(2): 115–124.

12.

Hao

Chu

Jiang

et al . (2014) Methane adsorption characteristics on coal surface above critical temperature through Dubinin–Astakhov model and Langmuir model. Colloids and Surfaces A: Physicochemical and Engineering Aspects 444: 104–113.

13.

Hao

Guo

et al . (2018) Investigation of methane sorption of overmature Wufeng-Longmaxi shale in the Jiaoshiba area, Eastern Sichuan Basin, China. Marine and Petroleum Geology 91: 251–261.

14.

Humayun

Tomasko

(2000) High‐resolution adsorption isotherms of supercritical carbon dioxide on activated carbon. AIChE Journal 46(10): 2065–2075.

15.

Jiang

Lin

(2018) Molecular dynamics investigation of conversion methods for excess adsorption amount of shale gas. Journal of Natural Gas Science and Engineering 49: 241–249.

16.

Tian

Xiao

et al . (2017) Geochemical characterization and methane adsorption capacity of overmature organic-rich Lower Cambrian shales in northeast Guizhou region, southwest China. Marine and Petroleum Geology 86: 858–873.

17.

Lutynski

González González Miguel

(2016) Characteristics of carbon dioxide sorption in coal and gas shale – The effect of particle size. Journal of Natural Gas Science and Engineering 28: 558–565.

18.

Merkel

Gensterblum

Krooss

et al . (2015) Competitive sorption of CH₄, CO₂ and H₂O on natural coals of different rank. International Journal of Coal Geology 150–151: 181–192.

19.

Moellmer

Moeller

Dreisbach

et al . (2011) High pressure adsorption of hydrogen, nitrogen, carbon dioxide and methane on the metal–organic framework HKUST-1. Microporous and Mesoporous Materials 138(1–3): 140–148.

20.

Mohammad

Fitzgerald

Robinson

et al . (2009) Experimental uncertainties in volumetric methods for measuring equilibrium adsorption. Energy & Fuels 23(5): 2810–2820.

21.

Mosher

Liu

et al . (2013) Molecular simulation of methane adsorption in micro- and mesoporous carbons with applications to coal and gas shale systems. International Journal of Coal Geology 109–110: 36–44.

22.

Mukherjee

Misra

(2018) A review of experimental research on enhanced coal bed methane (ECBM) recovery via CO₂ sequestration. Earth-Science Reviews 179: 392–410.

23.

Murata

Kaneko

(2000) Determination of the interface between gas and adsorbed phases in high pressure gas adsorption. Adsorption Science and Technology, pp.314–318.

24.

Ozawa

Kusumi

Ogino

(1976) Physical adsorption of gases at high pressure. IV. An improvement of the Dubinin—Astakhov adsorption equation. Journal of Colloid and Interface Science 56(1): 83–91.

25.

Pan

Xiao

Tian

et al . (2016) Geological models of gas in place of the Longmaxi shale in Southeast Chongqing, South China. Marine and Petroleum Geology 73: 433–444.

26.

Pini

(2014) Interpretation of net and excess adsorption isotherms in microporous adsorbents. Microporous and Mesoporous Materials 18: 740–752.

27.

Jia

et al . (2017) Nanoporous structure and gas occurrence of organic-rich shales. Journal of Nanoscience and Nanotechnology 17(9): 6942–6950.

28.

Ross

DJK

Marc Bustin

(2007) Impact of mass balance calculations on adsorption capacities in microporous shale gas reservoirs. Fuel 86(17–18): 2696–2706.

29.

Song

Yao

et al . (2018) Grand canonical Monte Carlo simulations of pore structure influence on methane adsorption in micro-porous carbons with applications to coal and shale systems. Fuel 215: 196–203.

30.

Staudt

Sailer

Tomalla

et al . (1993) A note on gravimetric measurements of gas‐adsorption equilibria. Berichte Der Bunsengesellschaft Für Physikalische Chemie 97(1): 98–105.

31.

Tang

Ripepi

(2017) High pressure supercritical carbon dioxide adsorption in coal: Adsorption model and thermodynamic characteristics. Journal of CO₂ Utilization 18: 189–197.

32.

Tang

Ripepi

Luxbacher

et al . (2017) Adsorption models for methane in shales: Review, comparison, and application. Energy & Fuels 31(10): 10787–10801.

33.

Tang

Ripepi

Stadie

et al . (2016) A dual-site Langmuir equation for accurate estimation of high pressure deep shale gas resources. Fuel 185: 10–17.

34.

Tian

Zhang

et al . (2016) Characterization of methane adsorption on overmature Lower Silurian–Upper Ordovician shales in Sichuan Basin, southwest China: Experimental results and geological implications. International Journal of Coal Geology 156: 36–49.

35.

Tsai

Chen

Cen

et al . (1985) Adsorption of gas mixture on activated carbon. Carbon 23(2): 167–173.

36.

Xiong

Liu

et al . (2016) Molecular simulation of methane adsorption in slit-like quartz pores. RSC Advances 6(112): 110808–110819.

37.

Xiong

Liu

Liang

et al . (2017) Adsorption behavior of methane on kaolinite. Industrial & Engineering Chemistry Research 56(21): 6229–6238.

38.

Yang

Ribeiro

et al . (2014) Adsorption equilibrium and kinetics of methane and nitrogen on carbon molecular sieve. Industrial & Engineering Chemistry Research 53(43): 16840–16850.

39.

Zhang

Kang

(2018) Monte Carlo simulations of methane adsorption on kaolinite as a function of pore size. Journal of Natural Gas Science and Engineering 49: 410–416.

40.

Zhao

Ning

et al . (2018) Molecular simulation of methane adsorption on type II kerogen with the impact of water content. Journal of Petroleum Science and Engineering 161: 302–310.

41.

Zhou

Bai

et al . (2001) Determination of the adsorbed phase volume and its application in isotherm modeling for the adsorption of supercritical nitrogen on activated carbon. Journal of Colloid and Interface Science 239(1): 33–38.

42.

Zhou

Xue

Ning

et al . (2018) Experimental study of supercritical methane adsorption in Longmaxi shale: Insights into the density of adsorbed methane. Fuel 211: 140–148.

43.

Zhou

(2009) Fundamentals of high pressure adsorption. Langmuir 25(23): 13461–13466.