Measurement and modeling of the adsorption isotherms of CH4 and C2H6 on shale samples

CH4 and C2H6 are two common components in shale gas. Adsorption isotherms of CH4, C2H6, and their binary mixtures on shale samples are significant for understanding the fundamental mechanisms of shale gas storage and the recovery of shale resources from shale reservoirs. In this study, the thermogravimetric method is applied to obtain the adsorption isotherms of CH4, C2H6 and their binary mixtures on two typical shale core samples. A simplified local density theory/Peng–Robinson equation of state (SLD-PR EOS) model is then applied to calculate the adsorption of CH4 and C2H6 on shale, and the efficiency of the SLD-PR EOS model is thus evaluated. The results show that C2H6 exhibits a higher adsorption capacity than CH4 on shale samples, indicating the greater affinity of C2H6 to organic shale. As the molar fraction of C2H6 increases in the CH4/C2H6 mixtures, the adsorption capacity of the gas mixtures increases, indicating the preferential adsorption of C2H6 on shale. Based on the predicted results from the SLD-PR EOS model, a reasonable agreement has been achieved with the measured adsorption isotherms of CH4 and C2H6, validating the reliability of the SLD-PR EOS model for predicting adsorption isotherms of CH4 and C2H6 on shale samples. In addition, the SLD-PR EOS model is more accurate in predicting the adsorption of CH4 on shale than that of C2H6. This study is expected to inspire a new strategy for predicting the adsorption of hydrocarbons on shale and to provide a basic understanding of competitive adsorption of gas mixtures in shale reservoirs.


Introduction
Shale gas has been widely accepted as an important energy resource in recent years. Shale gas reservoirs possess some unique characteristics, such as extremely low permeability, rendering shale gas quite difficult to recover from such reservoirs. Compared to the conventional reservoirs, shale reservoirs generally have some unique characteristics, such as high organic content, which leads to more adsorption of hydrocarbons on shale. 1 Understanding the adsorption behavior of shale gas is signicant for estimating shale gas-in-place and the fundamental mechanisms of shale gas recovery. 2 In recent years, the adsorption behavior of shale hydrocarbons has been extensively investigated. CH 4 is the most abundant gas component in shale gas, which has been paid signicant attention in the previous studies. [3][4][5][6][7][8] Thermogravimetric analysis and the volumetric method are two commonly used approaches for measuring the adsorption isotherms of CH 4 on shale. [9][10][11][12][13][14] It is found that the thermogravimetric analysis method can measure the weight difference down to 1 mg; thereby, the thermogravimetric analysis method is more accurate than the volumetric method in determining the amount of adsorption on shale. Besides CH 4 , C 2 H 6 also takes an important proportion in shale gas. CH 4 and C 2 H 6 generally show different adsorption capacity on shale rocks, the so-called competitive adsorption. 15 Using thermogravimetric analysis method, Wang et al. 15 measured the sorption isotherms of CH 4 and C 2 H 6 on shale and reveal the competitive adsorption behavior between both components on shale samples. However, their measurements were only conducted at temperature up to 333.15 K, which is not practical to shale reservoir conditions. Furthermore, to the best of our knowledge, studies regarding to C 2 H 6 adsorption on shale is still scarce.
Extensive mathematical adsorption models have been proposed to match the adsorption of shale hydrocarbons on shale samples, including Langmuir model, the Brunauer-Emmett-Teller (BET) model, Dubinin-Astakhov (D-A), and Dubinin-Radushkevich (D-R) models. 16,17 Compared to the Langmuir model and Brunauer-Emmett-Teller (BET) model, Dubinin-Astakhov (D-A) and Dubinin-Radushkevich (D-R) models are more accurate because they specically consider the heterogeneous and hierarchical structures of shale cores. 18,19 However, these aforementioned adsorption models are only a mathematical matching process without any physical meaning. Simplied local density (SLD) theory has been recently proposed to model the adsorption of hydrocarbons on shale; such model is more accurate than the conventional models due to its consideration of the pore surface-uid interactions. 20 In addition, most of the modeling works are conducted only for the CH 4 adsorption, with less studies performed for C 2 H 6 . Here, one of the main motivations behind our efforts is to validate the SLD-PR EOS model in describing adsorption of light hydrocarbons, i.e., CH 4 and C 2 H 6 , on shale samples.
In this paper, the excess adsorption isotherms of CH 4 and C 2 H 6 and their binary gas mixtures are measured on two typical shale core samples using the thermogravimetric method. The adsorption of gas mixtures is compared with the adsorption of pure gases to reveal the occurrence of competitive adsorption under the shale reservoir conditions. The SLD-PR EOS model is then applied to predict the adsorption of gases on the shale samples and the effectiveness of the SLD-PR EOS model is then evaluated. The main objectives in this study are to understand the mechanisms of adsorption behavior of CH 4 and C 2 H 6 on shale samples and to evaluate the validity of SLD-PR EOS model in describing adsorption behavior of CH 4 and C 2 H 6 on shale. As a comprehensive study on gas adsorption behavior, the SLD-PR EOS model is the rst time to be applied to model the adsorption isotherms of C 2 H 6 .

Materials
The gases, i.e., CH 4 and C 2 H 6 , used in this study have the purities of 99.90 wt% and 99.95 wt%, respectively. Thus, the uncertainty in the measurements is not caused from the impurity of gases. Two typical shale samples are retrieved from the depth of 1356 and 1437 m in the Longmaxi formation in the Sichuan Basin of China, where the reservoir temperature is approximately 343.15 K. In order to avoid the moisture in the air, the shale core samples are crushed into small particles and sealed in the zip-locked bags.

Characterization of shale core samples
In this work, the two shale samples are characterized to obtain the total organic carbon (TOC) and pore size distribution. TOC content is measured by a combustion elemental tester. First, H 2 SO 4 is added into the shale particles to form a solution; O 2 is then used to sparged the solution to remove the purgeable inorganic and organic carbon. The non-purgeable organic carbon is formed by CO 2 in a combustion tube, which is then detected and used for the calculation of TOC content. The measured TOC content for each shale core sample is shown in Table 1.
To obtain the specic surface area and pore size distribution of the shale core samples, N 2 adsorption/desorption tests are adopted. The gas sorption analyzer (Quantachrome, USA) is used for conducting the measurements by measuring the N 2 adsorption/desorption at 77.0 K. The specic surface area is computed with the BET equation. 16 The BET surface area for each shale sample are obtained with an accuracy of AE0.5%. The results of the BET surface area for each shale sample are shown in Table 1. Fig. 1 presents the measured pore size distribution for the two shale core samples. As shown in this gure, the two shale cores possess pores with the pore size falling in the nanoscale range. In addition, the dominant pore size for the two shale cores are 4.23 and 3.00 nm, respectively.

Measurements of adsorption isotherms
Adsorption isotherms of CH 4 and C 2 H 6 are measured by using an Intelligent Gravimetric Analyser (Nanjing Haohai Science Instruments and Apparatuses Limited Company, China). Fig. 2 presents the schematic diagram of the experimental setup for measuring adsorption isotherms of CH 4 and C 2 H 6 . Before the isothermal measurements, the shale core particles are placed at 385.15 K and vacuumed for 12 hours for dehydration. The Gravimetric Analyser employs the thermogravimetric analysis approach for the measurement; this means the adsorption amount is obtained by calculating the weight change of shale sample. The mass of the empty sample container, m c , and its volume, v c , are rst measured at the experimental temperature. The mass of shale sample, m s , and the sample volume, v s , are then measured by placing the shale sample into the adsorption chamber. Then, the adsorption chamber is lled with the adsorbent gas, i.e., CH 4 and C 2 H 6 aer vacuuming the sample chamber for 12 h at the experimental temperature. The pressure in the sample chamber increases gradually to the experimental value. Then, the apparent weight, Dm, is measured at the given pressure and temperature until it reached stabilization, 2 where m a represents the mass of gas adsorbed on shale; v a represents the adsorbed gas volume; r is the gas density in bulk; m c and v c represent the mass of the empty sample container and its volume, respectively; m s and v s represent the mass of shale sample and the sample volume, respectively.  The adsorbed mass can be calculated by, Then the excess adsorbed mass, m e , can be calculated as,

Simplified local density/Peng-Robinson equation of state (SLD-PR EOS) model
The SLD-PR EOS model 21 is applied to describe the CH 4 and C 2 H 6 adsorption on both shale core samples, while carbon-slit pores are employed to simulate the organic pores appeared in the shale samples. The SLD-PR EOS model can accurately calculate the uid adsorption in nanopores by considering the uid-uid and uid-solid surface interactions. Within the framework of the SLD-PR EOS model, the equation of state of CH 4 and C 2 H 6 employed the local-density approximation in obtaining the congurational energy of the adsorbed CH 4 and C 2 H 6 . It is noted that the adsorbed CH 4 and C 2 H 6 distribute in-homogeneously in nanopores. 22 Compared to molecular simulations, SLD/PR-EOS model considerably decreases the cost of computation. Generally, three main assumptions are used in the SLD-PR EOS model, 22 (1) Chemical potential of uid at any point in nanopores is identical to the bulk chemical potential near the solid surface; (2) Chemical potential of uid in nanopores is the summation of uid-uid and uid-surface potentials at adsorption equilibrium; (3) Chemical potential from uid-surface at any point is not inuenced by molecules around this point.
At adsorption equilibrium, the chemical potential of CH 4 and C 2 H 6 at the position z is calculated by the potential summation due to the uid-uid and uid-surface interactions; it equals to the chemical potential of CH 4 and C 2 H 6 in bulk.
where the subscript "ff" is the uid-uid interactions, "fs" is uid-surface interactions, and "bulk" represents bulk CH 4 and C 2 H 6 . The bulk chemical potential of CH 4 and C 2 H 6 is expressed as a function of fugacity, where f bulk represents the bulk fugacity of CH 4 and C 2 H 6 , f 0 represents fugacity at a reference state, m bulk represents chemical potential in bulk; m 0 represents chemical potential at a reference state; T represents temperature; R represents ideal gas constant. The chemical potential of CH 4 and C 2 H 6 in nanopore from the CH 4 -CH 4 and C 2 H 6 -C 2 H 6 interactions is calculated as, where f ff (z) represents the fugacity of CH 4 and C 2 H 6 at the position z; f 0 represents fugacity at the same reference state as that in eqn (5). The chemical potential of CH 4 and C 2 H 6 in nanopore from the CH 4 -solid surface interaction is calculated as, 21 where J fs (z) and J fs (L À z) are interactions from the CH 4 -solid and C 2 H 6 -solid surface of a carbon-slit pore with a pore size of where r atoms is the solid-atom density, 38.2 atoms per nm 2 ; 24 3 fs is the parameter from the CH 4 -solid surface and C 2 H 6 -solid surface interactions; s fs is the molecular diameters of CH 4 and C 2 H 6 , which is computed by s fs ¼ (s ff + s ss )/2, where s ff and s ss represent molecular diameters of CH 4 and C 2 H 6 and the carbon-interplanar distance, respectively. The value of s ss is 0.355 nm for graphite; z 0 represents the dummy coordinate, which is calculated as z 0 ¼ z + s ss /2. Substituting eqn (6)-(8) into eqn (4), the criterion for adsorption equilibrium is expressed as, where k represents Boltzmann's constant, 1.38 Â 10 À23 J K À1 ; T represents absolute temperature. The PR-EOS is used to calculate CH 4 -CH 4 and C 2 H 6 -C 2 H 6 interactions. The PR EOS can be given as a function of density (r), where The a(T) term in eqn (11) is given with the following expression. 25 where A, B, C, and D are correlation parameters, 2.0, 0.8145, 0.508, and À0.0467, respectively. The values of acentric factor (u), the critical pressure (P c ), the critical temperature (T c ), and the molecular diameter for CH 4 are 0.0113, 4.6 MPa, 190.56 K, and 0.3758 nm, respectively. Acentric factor (u), the critical pressure (P c ), the critical temperature (T c ), and the molecular diameter for C 2 H 6 are 0.0990, 4.9 MPa, 305.32 K, and 0.4000 nm, respectively. In the PR-EOS, the fugacity of bulk CH 4 and C 2 H 6 are calculated as, where P is the bulk pressure. With a similar analogy, fugacity of the adsorbate due to the CH 4 -CH 4 and C 2 H 6 -C 2 H 6 interactions is expressed as, ln f ff ðzÞ P ¼ brðzÞ 1 À brðzÞ À a ads ðzÞrðzÞ PTð1 þ 2brðzÞ À b 2 r 2 ðzÞÞ À ln P RTrðzÞ where a ads (z) is related with the position in the nanopore and the dimensionless pore width L/s ff . 26 a ads (z) is obtained from Chen et al. (1997). 26 r(z) correlates with the position in carbonslit pores, which represents the in situ gas density in nanopores.
In the PR EOS, covolume parameter b affects the local density of adsorbed CH 4 and C 2 H 6 . 22 To improve the predictive capacity of pure CH 4 and C 2 H 6 on carbon surface, Fitzgerald (2005) 27 modied the covolume parameter b. To consider the repulsive interactions of the adsorbed CH 4 and C 2 H 6 at high pressure conditions, covolume parameter b is modied as, 27 where b ads is the modied covolume; L b is the empirical correction for shale gases, ranging from À0.4 to 0.0. 22 In our model, L b is set as À0.20 for CH 4 and C 2 H 6 . As a result, eqn (15) is expressed 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 À ln P RTrðzÞ À Pb ads RT ! À a ads ðzÞ 2 ffiffi ffi 2 p b ads RT ln Density distribution of CH 4 and C 2 H 6 in nanopores can then be calculated by combining eqn (4) through (17). Within the SLD/PR-EOS model, the excess adsorption of CH 4 and C 2 H 6 is calculated as, where n ex represents the excess CH 4 and C 2 H 6 adsorption, which is calculated in moles per unit mass of adsorbent; A represents the total surface area of adsorbed CH 4 and C 2 H 6 on carbon surface. The lower limit of integration s ff /2 is the center of the sphere-shaped CH 4 and C 2 H 6 molecules adsorbed on the pore surface, while the upper limit of integration L À (s ff /2) is the center of CH 4 and C 2 H 6 molecules adsorbed on the pore surface of the other wall. The average density (r ave ) of CH 4 and C 2 H 6 in nanopores is expressed as, where W is the pore size of nanopore. The SLD model applies the equation of state for CH 4 and C 2 H 6 , which has been simplied with a local-density approximation in obtaining the conguration energy of the adsorbed CH 4 and C 2 H 6 . The local-density approximation simplied the calculation for the long-range physical interactions, which is the difference from the conventional molecular simulation methods. This simplication renders the SLD model more efficient than the molecular simulation methods in calculating the conned uid properties in nanopores, while it could be less accurate in describing some more complex molecules compared to the molecular simulation methods.

Results and discussion
4.1 Adsorption isotherms of CH 4 and C 2 H 6 on shale samples Fig. 3-6 present the measured adsorption isotherms of CH 4 and C 2 H 6 on the two typical shale samples. It is observed that adsorption of CH 4 and C 2 H 6 is expected to be inuenced by the system pressure and temperature; specically, adsorption of CH 4 and C 2 H 6 increases as pressure increases but decreases as temperature increases. As for the same shale sample, C 2 H 6 adsorption is signicantly higher than that of CH 4 at the same temperature and pressure conditions, indicating the more affinity of C 2 H 6 to the organic shale. Compared with the shale sample #1, adsorption of CH 4 and C 2 H 4 on the shale sample #2 is much higher. Based on the characterization results for the two shale samples, the specic surface area and the total organic carbon content of shale sample #2 is signicantly higher than that of the shale sample #1. The adsorption capacity of hydrocarbons on solid surface correlates with the physical properties of solid, such as surface area, and mineralcomposition heterogeneity etc. 27 Possibly, it is the main reason why the adsorption of CH 4 and C 2 H 6 on the shale sample #2 is stronger than that on the shale sample #1.

Competitive adsorption of CH 4 and C 2 H 6 on shale samples
The adsorption isotherms of the binary gas mixtures of CH 4 -C 2 H 6 are measured on the two shale samples. In this work, four different gas compositions, i.e., 60.20-39.80 mol%, 53.25-46.75 mol%, 82.35-17.65 mol%, and 63.12-36.88 mol% for CH 4 -C 2 H 6 mixtures, are selected. The manner for isotherm Fig. 3 The measured excess adsorption of CH 4 on the shale sample #1. Fig. 4 The measured excess adsorption of C 2 H 6 on the shale sample #1.  measurements are conducted similarly to that adopted for the pure components. Fig. 7 and 8 show the measured adsorption isotherms for these gas mixtures. As for the two shale samples, the total excess adsorption of CH 4 -C 2 H 6 mixtures increases as the molar concentration of C 2 H 6 increases in the gas mixtures. It is possibly caused by the competitive adsorption between CH 4 and C 2 H 6 on the organic shale surface; C 2 H 6 exhibits the preferential adsorption over CH 4 on shale surface, resulting in a higher adsorption than CH 4 but lower than C 2 H 6 . Additionally, we observe the maximum excess adsorption loading at about 130 bar for the four gas mixtures, while it tends to decrease beyond this pressure. However, this behavior is an exception for the pure gas adsorption isotherm under the studied conditions. The adsorption difference between gas mixtures and pure gases may be resulted from the interactions between two hydrocarbon species in the adsorption phase as well as in the free-gas phase.

SLD-PR EOS model for representing the adsorption of CH 4 and C 2 H 6
The SLD-PR EOS model is applied to predict the adsorption of CH 4 and C 2 H 6 on shale samples, which are then applied to match the measured adsorption data. Specically, two key parameters, i.e., uid-pore surface interaction energy (3 fs /k) and covolume correction parameter (A b ), are adjusted in the SLD-PR EOS model to t the measured excess adsorption. Table 2 shows the adjusted parameters in the SLD-PR EOS model. Fig. 9-12 present the comparison results between the measured excess adsorption and the predicted excess adsorption of CH 4 and C 2 H 6 from the SLD-PR EOS model. We observe that the SLD-PR EOS model can reasonably represent the measured excess CH 4 and C 2 H 6 adsorption on the two shale samples. In addition, compared to CH 4 , we observe that the SLD-PR EOS model is less accurate for predicting C 2 H 6 adsorption.
Based on the comparison results, the absolute relative error of the calculated adsorption of CH 4 and C 2 H 6 are calculated from the measured excess adsorption. The absolute relative error is calculated as, where RE represents the absolute relative error, %; Ad c represents the calculated adsorption on shale surface, mmol g; Ad m represents the measured excess adsorption on shale surface, mmol g À1 .    Fig. 13 presents the calculated absolute relative error for CH 4 and C 2 H 6 at various pressure conditions. We observe a higher absolute relative error at lower pressures for both CH 4 and C 2 H 6 , which, specially, can be as high as 40% for C 2 H 6 , while the absolute relative error decreases as pressure increases. It suggests the SLD-PR EOS model is not accurate in predicting adsorption of CH 4 and C 2 H 6 on shale samples at low pressure conditions. In addition, compared with CH 4 , a much higher absolute relative error is observed for C 2 H 6 , indicating that the SLD-PR EOS model may not be suitable for the prediction of the adsorption of heavier hydrocarbon species.

Conclusions
In this work, the excess adsorption isotherms of CH 4 , C 2 H 6 and their binary gas mixtures are measured on two typical shale core samples using thermogravimetric method. The adsorption of gas mixtures is compared with that of pure gases to reveal the     behavior of competitive adsorption under the shale reservoir conditions. The SLD-PR EOS model is then applied for predicting the adsorption of CH 4 and C 2 H 6 on both shale samples to evaluate its efficiency in predicting the adsorption of shale hydrocarbons. The detailed conclusions can be drawn as below: () C 2 H 6 has higher adsorption capacity than CH 4 on the two shale samples under the same conditions; it suggests the more affinity of C 2 H 6 on the organic shale; () As observed from the measured adsorption isotherms of CH 4 -C 2 H 6 mixtures, as the molar fraction of C 2 H 6 in CH 4 -C 2 H 6 mixtures increases, adsorption of the gas mixture increases, indicating the preferential adsorption of C 2 H 6 on shale.
() Based on the predicted results from the SLD-PR EOS model, a reasonable agreement has been achieved with the measured adsorption isotherms, indicating the accuracy of the SLD-PR EOS model in predicting the gas adsorption on shale samples. In addition, compared with C 2 H 4 , the SLD-PR EOS model is more accurate in predicting adsorption of CH 4 on shale.
This study proposes the SLD-PR EOS model for the prediction of gas adsorption on shale samples; in addition, it may provide a basic understanding of the competitive adsorption of hydrocarbons in shale reservoirs. To our knowledge, the adsorption measurements of gas mixtures on typical shale samples are presented for the rst time. However, future works should be supplemented to our study. Besides CH 4 and C 2 H 6 , some other heavier hydrocarbons, such as nC 3 H 8 , nC 4 H 10 , may also be an important component in shale gas. Thereby, future works are suggested to measure the adsorption/desorption isotherms of the heavier hydrocarbons on shale. In addition, in our work, we measure the adsorption of C 2 H 6 at pressures as high as 60 bar based on the saturated vapor pressure of C 2 H 6 at given temperature. New experimental setups should be designed to achieve the adsorption measurement at pressures as close as the shale reservoir conditions.

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