Determination of the absolute CH4 adsorption using simplified local density theory and comparison with the modified Langmuir adsorption model

Accurately determining the adsorbed amount of CH4 on shale is significant for understanding the mechanisms of shale gas storage and shale methane recovery from shale gas reservoirs. Excess CH4 adsorption is measured using the thermogravimetric method. Simplified local density (SLD) theory is applied to calculate the adsorbed CH4 density to obtain the absolute adsorption. Moreover, the modified Langmuir adsorption model is employed to fit the excess adsorption to describe the absolute adsorption. The adsorbed CH4 density from the SLD model is affected by the system pressure and temperature, while such density obtained from the modified Langmuir model is only a function of temperature. Compared to the modified Langmuir model, the SLD model can better capture the adsorbed CH4 density, which allows accurate determination of the absolute CH4 adsorption.


Introduction
Shale gas is one kind of unconventional energy resource, which has become an increasingly important energy in recent years. Shale reservoirs generally exhibit some typical characteristics of extremely low permeability, and heterogeneity. 1 Shale generally contains a large amount of kerogen, which can result in the signicant adsorption of shale gas on the organic shale surface. 2 Accurately measuring the amount of adsorbed shale gas is quite important for the estimation of shale gas storage and the development of shale gas reservoirs.
As is known, CH 4 is a common component existing in shale uid. In shale gas reservoirs, CH 4 is generally stored in three different states, which is claried as free-gas state in nanopores, absorbed-gas state in kerogen, and adsorbed-gas state on pore surface. [3][4][5] It has been found that the adsorbed CH 4 can take 20-85 vol% accounting for the total gas amount. 4 Recently, extensive studies are implemented to measure the CH 4adsorption on shale samples. Volumetric method 6-10 and thermogravimetric (TGA) method 3,11,12 are two main approaches applied for measuring the CH 4 adsorption isotherms on shale. TGA method enables to measure the weight difference as accurate as 1 mg. Thereby, compared to volumetric method, TGA method is more accurate in measuring the amount of adsorbed CH 4 on shale samples.
However, the laboratory measurement only provides the excess adsorption. It has proposed that the measured excess adsorption has possible underestimation of the amount of adsorbed CH 4 . 12 Generally, the measured excess adsorption is transformed to the absolute values, which reects the actual adsorbed amount of CH 4 on shale. 4,12 The density of adsorbed CH 4 is usually employed to make this conversion. Due to the difficulty in measuring such density directly, some constant values are generally used to represent the density of adsorbed CH 4 . For instance, the density of adsorbed CH 4 is suggested to be the liquid CH 4 density at the room boiling point, i.e., 420 kg m À3 . [13][14][15][16][17] However, it is proved that the density of adsorbed CH 4 is strongly affected by temperature, pressure, and pore size. 12,18,19 Most recently, some correlation models, such as the modied Langmuir adsorption model, Dubinin-Radushkevitch equation, Ono-Kondo models, and Supercritical Dubinin-Radushkevitch equation, are widely employed to obtain the absolute adsorption by adjusting the adsorbed CH 4 density. [19][20][21][22] Due to the simplicity for usage, the modied Langmuir adsorption model is extensively used in obtaining the absolute adsorption.
Molecular simulation is also employed to investigate the density of adsorbed CH 4 on shale. By specically considering the uid-pore wall interactions, molecular simulation provides the fundamental mechanisms of CH 4 adsorption in organic pores. Recently,   12 measured the excess adsorption of two hydrocarbon-species, i.e., methane and n-butane, on two typical shale samples; the molecular simulation method was then used to calculate the adsorbed CH 4 density. Such density is then applied to describe the absolute adsorption by correcting the measured excess. Base on their simulation results, they observed that the adsorbed CH 4 density is affected by temperature, pressure, and pore size. Furthermore, Ambrose et al. (2012) 19 also observed the adsorbed CH 4 density changes with temperature, pressure, and pore size. Although molecular simulation could accurately determine the adsorbed phase density, the computation is quite expensive compared to the conventional methods. Simplied local density (SLD) theory specically takes into consideration the uid/pore surface interactions, which can thereby determine the density of adsorbed CH 4 accurately. Compared to molecular simulation method, SLD model signicantly reduces the computational time.
In this study, the excess CH 4 adsorption is measured on the typical shale samples. The modied Langmuir adsorption model and the SLD model are then employed to capture the absolute adsorption based on the excess adsorption. As the previous study, 23 the SLD model captures the absolute adsorption by obtaining the density of adsorbed CH 4 on shale. The modied Langmuir adsorption model describes the absolute adsorption by accurately tting the excess adsorption. The performance of the modied Langmuir adsorption model is then evaluated by comparing with the SLD model. The objectives of this study are: (1) to assess the validity of the widely used modied Langmuir adsorption model in determining the absolute CH 4 adsorption on shale; (2) to propose a practical method, i.e., the SLD model, to determine the absolute CH 4 adsorption. In our SLD model, the carbon-slit pore model is employed to describe the organic pores.

Materials
The CH 4 used in this work has a purity of 99.95 mol%. The two shale samples used are obtained from Longmaxi formation. Before experiment, the shale samples are sealed to avoid the moisture.

Characterization of the shale samples
(1) N 2 adsorption/desorption tests. In this study, N 2 adsorption/desorption tests are conducted to characterize pore size distribution of both shale samples. The Gas Adsorption Analyzer (Quantachrome, America) is employed for this characterization. By analyzing the adsorption data measured at 77.0 K, we can obtain the pore size distribution as well as the specic surface area.
(2) TOC measurement. To obtain the TOC content of both shale samples, an elemental analyzer is employed. The organic carbon in shale is rst formed by CO 2 ; a non-dispersive infrared analyzer is then applied to measure the total molar amount of CO 2 .
(3) Scanning election microscopy (SEM). In this work, the Hitachi SEM setup is applied to obtain the surface morphology of both shale samples. Before the SEM scanning, argon ion is used to polish the shale surface. Then, the shale surface is then covered by a golden lm to improve the conductivity. The shale samples are scanned at a voltage of 20.0 kV.

Measurement of the excess adsorption
In this adsorption experiment, we measure the excess CH 4 adsorption with a thermalgravimetric (TGA) analyzer at the temperatures of 303.15, 345.15, and 387.15 K, and pressures as high as 15.0 MPa. With the TGA method, the measured excess adsorption uptake can be expressed as, 24 where M ex represents the excess adsorption uptake; M ad represents the adsorbed uptake, which is recognized as the uptake of the absolute adsorption; r represents the CH 4 density in bulk; and V ad represents the adsorbed volume of CH 4 . We nd that the measured excess adsorption is smaller than the adsorbed CH 4 adsorption on shale.
The adsorbed volume of CH 4 can be given by, Substituting eqn (2) into (1), we can obtain the expression for absolute adsorption, which represents the actual adsorption uptake of CH 4 on shale, To conrm the reliability and reproductivity of the measured data, we repeat each test twice, and it is found that the maximum deviation is always smaller than AE3.76% between two measuring runs.

Modied Langmuir adsorption model
As shown in eqn (1), the adsorbed CH 4 density is important to obtain the absolute adsorption. Recently, three categories of conversion methods have been proposed to represent this quantity. One approach is to predetermine the adsorbed CH 4 density as a constant value, which generally ranges between 0.373 g cm À3 (ref. 19 and 25) and 0.423 g cm À3 . 3,26,27 Our previous study has proved that this method is unphysically reasonable considering that the adsorbed CH 4 density is generally inuenced by system pressure, temperature, and pore size. 12 Another approach is to determine this value by tting a modied equation to the measured excess isotherm by adjusting the adsorbed CH 4 density. [26][27][28] Due to its simplicity and low computational cost, the modied Langmuir adsorption model is widely used for tting excess isotherms and then calculating the absolute isotherms. 26,27 This model is based on the assumption that CH 4 generally exhibits monolayer adsorption on carbon surface, 29,30 which can be expressed as, 27 where V L represents the maximum adsorbed amount of CH 4 ; p L represents the Langmuir pressure; p represents the system (4) is the standard Langmuir equation. r ad is initially determined by tting eqn (4) to the directly measured excess adsorption. According to eqn (4), the absolute adsorption uptake is then calculated as, 27

Simplied local density (SLD) model
The SLD model is originally proposed by Rangarajan et al. (1995), 31 which is generally applied to describe gas adsorption on adsorbate surface over a wide pressure/temperature range. The SLD model specically considers the uid-pore surface and uid-uid interactions, which can accurately describe the gas adsorption on pore surface. 32 The main assumptions proposed for the SLD model are summarized as below, (1) Near the pore surface, the chemical potential at any point is equal to the chemical potential in bulk; (2) At any point, the chemical potential at equilibrium is equal to the summation of potentials due to uid-pore surface and uid-uid interactions; (3) The uid-pore surface potentials at any point do not correlate the total number of molecules around this point.
When adsorption reaches equilibrium, the gas chemical potential at position z is calculated as the summation of the potentials due to uid-pore surface and uid-uid interactions; it is regarded as the bulk chemical potential.
where the subscript "ff" represents uid-uid interactions, "fs" represents uid-pore surface interactions, and "bulk" represents the gas in bulk.
The bulk chemical potential of gas can be given as a function of fugacity by, where f bulk represents the fugacity of gas in bulk, f 0 represents the fugacity at a reference state. The chemical potential due to uid-uid interaction is expressed as, where f ff (z) represents the gas fugacity at position z; f 0 represents the fugacity at the same reference state as that in eqn (7). The gas chemical potential in nanopores due to the uidpore wall interaction is expressed as, 31 where J fs (z) and J fs (L À z) represent the uid-pore surface interactions from both walls of a pore; L represents the pore size; N A represents Avogadro number. The Lee's partially integrated 10-4 Lennard-Jones potential 33 is applied to represent the uid-pore surface interaction, where r atoms is the solid-atom density, 38.2 atoms per nm 2 ; 34 3 fs is the interaction parameter between uid and pore surface; s fs is the uid-pore surface molecular diameter, which can be calculated by s fs ¼ (s ff + s ss )/2, where s ff represent the molecular diameter of CH 4 , while s ss represents the carbon-interplanar distance. As for graphite, s ss is taken as0.355 nm; z' is the dummy coordinate, z' ¼ z + s ss /2.
Substituting eqn (8)-(10) into eqn (6), we can obtain the following, where k is the Boltzmann constant, 1.38 Â 10 À23 J K À1 ; and T is the absolute temperature. The Peng-Robinson equation of state (PR-EOS) is employed to take into consideration the uid-uid interactions, which can be expressed as a function of density (r) as, where The term a(T) in eqn (13) can be expressed as. 35 aðTÞ ¼ exp where A, B, C, and D represent the correlation parameters with the values xed at 2.0, 0.8145, 0.508, and À0.0467, respectively. As for CH 4 , the value of the acentric factor (u), the critical pressure (P c ), the critical temperature (T c ), and the molecular diameter are 0.0113, 4.6 MPa, 190.56 K, and 0.3758 nm, respectively. Applying the PR-EOS, the gas fugacity in bulk is expressed as, where P represents the pressure of gas in bulk. The fugacity of CH 4 taking into consideration the uid-uid interactions is expressed as, where the term a ads (z) correlates with the position in pores and the dimensionless pore size L/s ff . 36 Chen et al. (1997) 36 proposed the equations for calculating the term a ads (z). r(z)represents the gas density in pores, which is a function of position in pores. It has been found that the covolume parameter b in eqn (17) affects the adsorbed CH 4 density near the pore surface. 30 In order to consider the repulsive interactions of the adsorbed CH 4 , Fitzgerald (2005) 37 modied this term to improve the predictive capacity of CH 4 on pore surface. It is expressed as, 37 where b ads is the modied covolume; L b represents the empirical correction, which is usually ranges from À0.4 to 0.0 for shale gas. 32 In this work, this value is xed at À0.20 for CH 4 . Eqn (17) is then rewritten as, ln f ff ðzÞ P ¼ b ads rðzÞ 1 À b ads rðzÞ À a ads ðzÞrðzÞ PT À 1 þ 2b ads rðzÞ À b ads 2 r 2 ðzÞ Á Àln P RTrðzÞ À Pb ads RT ! À a ads ðzÞ 2 ffiffi ffi 2 p b ads RT ln The density prole of CH 4 in a pore is calculated by combining eqn (6) through (19). The excess adsorption is expressed as, where n ex represents the excess adsorption; A is the surface area.
As for the integration of s ff /2, the lower limit is the center of CH 4 adsorbed on pore surface, while the upper limit L À (s ff /2) represents the center of CH 4 molecule adsorbed on pore surface. The average density (r ave ) of the adsorbed CH 4 in nanopores is calculated by, where W is the width of the adsorbed phase of CH 4 .

Results and discussion
In this subsection, characterization results of the shale samples are rst presented. Then, we show the absolute CH 4 adsorption calculated from the modied Langmuir adsorption model. SLD model is then employed to obtain the adsorbed CH 4 density in pores. Using the calculated adsorbed CH 4 density, the measured excess adsorption is then corrected to obtain the absolute adsorption. Finally, we evaluate the performance of the modied Langmuir adsorption model by comparing with SLD model. Table 1 shows the measured TOC content and specic surface area of both shale samples. We nd that the TOC content in shale sample-1 is higher than that in shale sample-2. However, the specic surface area of shale sample-1 is lower than that of shale sample-2. High TOC content indicates high content of kerogen in shale, which contributes to the specic surface area of shale samples. However, for given shale sample, the specic surface area also correlates with the clay content, heterogeneity, and pore size distribution etc. Fig. 1 shows the measured pore size distribution of both shale samples. Pores in both shale samples are generally in nano-scale locating in the range of 1-100 nm. The dominant pore sizes for the two shale samples are 4.35 nm and 3.12 nm, respectively. Fig. 2 presents the scanned SEM digital images for the shale samples. The X-ray spectroscopy analysis is conducted on the locations of A and B on both shale samples. We observe a high content of carbon element residing in the two points, indicting kerogen. As shown in Fig. 2, we can also observe a bunch of pores present in kerogen, which is recognized as a unique characteristic of kerogen in shale. The modied Langmuir adsorption model is employed to t the excess adsorption by adjusting the adsorbed CH 4 density (r ad ). We can observe that a perfect matching has been achieved between the measured results and the predicted values from the modied Langmuir adsorption model. At 303.15 K, the excess adsorption of CH 4 is enhanced as pressure increases. The excess adsorption reaches the maximum at around 8.0 MPa on the two shale samples. However, as pressure further increases, the measured excess adsorption decreases. Tian et al. (2017) 2 attributed this behavior to the much higher CH 4 density at the center of organic pores at higher pressure conditions. The excess adsorption is then corrected to absolute adsorption using eqn (5) from the modied Langmuir adsorption model, as shown in Fig. 3. The absolute adsorption is clearly affected by the system temperature and pressure; specically, it decreases with increasing temperature but increases as pressure increases. Moreover, compared to the excess adsorption, the absolute adsorption is higher, especially at higher pressure conditions, which agrees well with the previous ndings. 2,12 It suggests the amount of adsorbed CH 4 on the organic shale is underestimated by the excess adsorption. In addition, the absolute CH 4 adsorption varies for different shale samples at the same testing pressure/temperature conditions. Besides of system pressure and temperature, CH 4 adsorption is expected to be also inuenced by mineral contents, heterogeneity, specic surface area, and total organic carbon content etc. Fig. 1 The measured pore size distribution of (a) shale sample-1, and (b) shale sample-2. Fig. 2 The SEM digital images of (a) shale sample-1, (b) shale sample-2.

Adsorbed CH 4 density in nanopores
Using SLD model, we investigate the CH 4 distribution in the 4.35 nm and 3.12 nm pores. Note that the most probable pore sizes of shale samples-1 and -2 are 4.35 nm and 3.12 nm, respectively. Based on the previous studies, it has been found that CH 4 is single-layered adsorption in organic pores. 2,12 As is known that the molecular diameter of CH 4 is about 0.37 nm, previous works generally used 0.37 nm as the phase width of the adsorbed CH 4 . In our work, we also take 0.37 nm as the phase width of the adsorbed CH 4 in nanopores. The average density of the adsorbed phase is calculated with r ave ¼ Ð b a r ads ðzÞdz=z ab (where r ave represents the averaged adsorbed phase density of CH 4 ; r ads represents the in situ density in the adsorbed phase of CH 4 ; and z ab represents the phase width of the adsorbed CH 4 ). Fig. 4 shows the calculated density of the adsorbed CH 4 in the 4.35 nm and 3.12 nm pores at the experimental temperature/ pressure conditions. We observe the adsorbed CH 4 density is related with the experimental temperature and pressure. Specically, the adsorbed CH 4 density increases with increasing pressure but decreases as temperature increases. We observe that such density varies in the two different pores. Therefore, we may expect that the density of the adsorbed CH 4 in is affected by temperature, pressure, and pore size. The previous works that employed constant values to represent the density of adsorbed CH 4 is not physically reasonable. [13][14][15][16][17] 3.4 Absolute adsorption isotherms of CH 4 from SLD model In this work, two key parameters, i.e., uid-pore surface interaction energy (3 fs /k) and covolume correction parameter (A b ), are adjusted in the SLD model. These parameters are obtained by adjusting these parameters to t the measured excess adsorption. Table 2 shows the adjusted parameters in the SLD model   for both shale samples. We observe that the covolume correction parameter is in the range of À0.3-0.3, which has a good agreement with the previous studies. 34,38,39 Fig. 5 shows the measured excess adsorption and the calculated absolute CH 4 adsorption on both shale samples from the SLD model. We nd that the SLD model can properly represent the excess CH 4 adsorption. Moreover, the converted absolute adsorption is also greater than the measured excess, especially at high pressure conditions, which is similar to the observation from the modi-ed Langmuir adsorption model.

Evaluation of the modied Langmuir adsorption model
It has been proved that SLD model can reasonably capture the adsorbed CH 4 density and can thus accurately describe the absolute adsorption isotherms. In Fig. 6, the absolute adsorption isotherms calculated from SLD model are compared with those obtained from the modied Langmuir adsorption model. The performance of the modied Langmuir adsorption model is then evaluated in describing the absolute adsorption. We observe that the absolute adsorption obtained from SLD model are always higher than those obtained from the modied Langmuir adsorption model. The modied Langmuir adsorption model describes the absolute adsorption isotherm with constant density values representing the adsorbed CH 4 density at a given temperature (see Fig. 3). However, based on the results calculated from SLD model, the density of adsorbed CH 4 is related with the temperature, pressure, and pore size. Thereby, the widely used modied Langmuir adsorption model underestimates the actual adsorption and is not reasonable in obtaining the absolute CH 4 adsorption on organic shale samples.

Conclusions
In this paper, the excess CH 4 adsorption is measured on two shale core samples. We then use the modied Langmuir adsorption model and SLD model to t the excess adsorption and then describe the absolute CH 4 adsorption on the shale core samples. SLD model considers the uid/pore surface interactions, which can thereby capture the density of adsorbed CH 4 in nanopores. This study evaluates the performance of the modied Langmuir adsorption model in describing absolute adsorption of CH 4 on organic carbon surface, and more importantly, it raises a more efficient approach (i.e., SLD theory) than the sophisticated molecular simulation tools in determining the absolute adsorption. The detailed conclusions can be summarized as follows: Based on the simulation results from SLD model, the density of adsorbed CH 4 is affected by temperature, pressure, and pore size. It highlights the importance for accurately determining the adsorbed CH 4 density in obtaining the absolute CH 4 absorption; It is found that the corrected absolute adsorption is greater than the excess CH 4 adsorption on shale, especially at high pressures. It indicates that the measured excess CH 4 adsorption shows underestimation of the amount of adsorbed CH 4 on shale; Compared to the SLD model, the absolute adsorption obtained from the modied Langmuir adsorption model is always smaller than that obtained from the SLD model. It suggests that the absolute adsorption obtained from the modied Langmuir adsorption model underestimates the actual adsorbed CH 4 .
This study may inspire us new tools in determining the absolute adsorption uptake of CH 4 on shale samples, which is practical in estimating the shale gas storage in shale gas reservoirs. The SLD model is more efficient in calculating the adsorbed CH 4 density on shale than the molecular simulation methods. However, besides CH 4 , some heavier hydrocarbon components may also appear in shale uids. Therefore, in the future works, the excess adsorption is suggested to be measured for the heavier hydrocarbon species and the SLD model recommended to calculate the adsorbed density for the heavier hydrocarbons on shale samples.

Conflicts of interest
There are no conicts to declare.