Predicting Adsorption of Methane and Carbon Dioxide Mixture in Shale Using Simplified Local-Density Model: Implications for Enhanced Gas Recovery and Carbon Dioxide Sequestration

Abstract

Carbon dioxide (CO₂) capture and storage have attracted global focus because CO₂ emissions are responsible for global warming. Recently, injecting CO₂ into shale gas reservoirs is regarded as a promising technique to enhance shale gas (i.e., methane (CH₄)) production while permanently storing CO₂ underground. This study aims to develop a calculation workflow, which is built on the simplified local-density (SLD) model, to predict excess and absolute adsorption isotherms of gas mixture based on single-component adsorption data. Such a calculation workflow was validated by comparing the measured adsorption of CH₄, CO₂, and binary CH₄/CO₂ mixture in shale reported previously in the literature with the predicted results using the calculation workflow. The crucial steps of the calculation workflow are applying the multicomponent SLD model to conduct regression analysis on the measured adsorption isotherm of each component in the gas mixture simultaneously and using the determined key regression parameters to predict the adsorption isotherms of gas mixtures with various feed-gas mole fractions. Through the calculation workflow, the density profiles and mole fractions of the adsorbed gases can be determined, from which the absolute adsorption of the gas mixture is estimated. In addition, the CO₂/CH₄ adsorption selectivity larger than one is observed, illustrating the preferential adsorption of CO₂ over CH₄ on shale, which implies that CO₂ has enormous potential to enhance CH₄ production while sequestering itself in shale. Our findings demonstrate that the proposed calculation workflow depending on the multicomponent SLD model enables us to accurately predict the adsorption of gas mixtures in nanopores based on single-component adsorption results. Following the innovative calculation flow path, we could bypass the experimental difficulties of measuring the multicomponent mole fractions in the gas phase at the equilibrium during the adsorption experiments. This study also provides insight into the CO₂/CH₄ competitive adsorption behavior in nanopores and gives guidance to CO₂-enhanced gas recovery (CO₂-EGR) and CO₂ sequestration in shale formations.

1. Introduction

CO₂ capture, utilization, and storage (CCUS) are effective processes to control and mitigate CO₂ emissions worldwide. In recent years, capturing and storing CO₂ in geological formations, e.g., depleted oil and gas reservoirs, deep saline formations, and coal seams [1,2,3], is regarded as an effective way to reduce the rapid rise in CO₂ emissions from burning fossil fuels. Among the mentioned geological formations, shale gas formations are extensively distributed with abundant original gas-in-place (OGIP), contributing a considerable percentage of unconventional natural gas to the world [4]. In addition, the gas stored in shale belongs to the following three conditions: free gas in pores and natural fractures; adsorbed gas on the surface of organic and clay pore walls; and absorbed (dissolved) gas in organic matters and connate water [5,6]. More importantly, the adsorbed gas in shale may account for up to 85% of the OGIP [7,8], and the adsorption capacity of CO₂ is two to five times the adsorption capacity of CH₄ in shale [9,10,11]. Techniques for injecting CO₂ into oil and gas reservoirs and coal while enhancing oil and gas production from such formations (i.e., CS-EGR, CO₂-EOR, and CO₂-ECBM) have been developed to productively utilize the emitted CO₂ [12,13,14,15]. Compared with CO₂ storage in saline aquifers, CO₂ injection into shale and coal formations would enhance CH₄ production and the revenues could compensate for the expense of CO₂ capture and storage [16]. Therefore, shale gas formations are promising targets for CO₂ sequestration while enhancing gas recovery (CS-EGR).

Recently, investigations of CO₂ injection into organic-rich shales to enhance CH₄ production have been intensively studied via molecular simulation on the microscopic scale and reservoir simulation on the engineering scale [4,16,17,18,19,20,21,22]. Grand canonical Monte Carlo (GCMC) and molecular dynamics (MD) simulation methods are widely employed to investigate competitive adsorption in shale kerogen and clay minerals. Huang et al. [17] investigated the effects of organic type and moisture on CH₄/CO₂ competitive adsorption in shale. The simulation results indicate that the CO₂/CH₄ adsorption capacity and adsorption selectivity increase in a sequence as kerogen IA < IIA < IIIA owing to the fraction of accessible pore volume (i.e., IA, 9.38%; IIA, 13.59%; IIIA, 28.88%) and the CH₄/CO₂ adsorption capacity reduces with the rise of moisture. Hu et al. [23] studied the competitive adsorption behaviors of CO₂/CH₄ mixtures in montmorillonite, illite, and kaolinite. He pointed out that the CO₂ adsorption capacity in the clay minerals decreases in a sequence as montmorillonite > illite > kaolinite. CO₂ molecules are willing to adsorb on the surfaces of montmorillonite and illite nanopores with cation exchange, instead of on the surface of the kaolinite nanopore without cation exchange. In addition, Sun et al. [22] developed a graphene and montmorillonite (graphene-MMT) model to simultaneously study the CH₄/CO₂ adsorption in organic and inorganic compounds in shale. The competitive adsorption behavior close to the MMT surface is much lower than close to the graphene surface, demonstrating an obvious asymmetric CO₂/CH₄ competitive adsorption in actual shale organic-inorganic nanocomposite.

Furthermore, reservoir simulation models are employed to evaluate the feasibility of CS-EGR and optimize the operation processes on the reservoir scale. Yu et al. [19] compared CO₂ flooding and CO₂ huff-n-puff in Barnett shale using a numerical reservoir simulation model coupled with multiple hydraulic fractures and multicomponent Langmuir isotherms. Jiang et al. [20] developed a fully coupled multi-continuum and multicomponent simulator associated with the extended Langmuir and embedded discrete fractures models to evaluate the CO₂/N₂ injection strategy in shale gas reservoirs. Zhang et al. [21] integrated the hybrid discrete fracture network (DFN), multiple interacting continua (MINC), and generalized Ono-Kondo (OK) lattice model to characterize the multicomponent gas flowing through the multiscale pores in shale matrix and fracture networks.

In addition, experimental measurements on competitive adsorption between CO₂ and CH₄ in nanoporous shale were carried out in several studies [12,24,25,26,27,28]. Currently, there are two methods to evaluate the adsorption amount and selectivity of the CO₂ and CH₄ gas mixture in shale. One is the static volumetric adsorption experimental setup coupled with the gas chromatograph [24,25,26], and another one is the dynamic gas displacement setup coupled with the low-field nuclear magnetic resonance (NMR) [27,28]. The preferential adsorption of CO₂ over CH₄ in shale is observed in experimental measurements, which demonstrates that the injection of CO₂ into shale gas reservoirs would be beneficial to CH₄ production and, meanwhile, the injected CO₂ could be permanently stored in the shale matrix.

In previous studies, adsorption models for pure gas (single component) adsorption were intensively studied [29,30,31,32,33,34,35]. Meanwhile several adsorption models for a gas mixture were also proposed to characterize the CH₄/CO₂ competitive adsorption isotherms. The extended Langmuir model [36,37], ideal adsorbed solution theory [37,38], extended multicomponent Brunauer–Emmett–Teller (BET) model [39], multicomponent SLD model [40], and multicomponent OK lattice model [41] were widely used to describe the adsorption behaviors of gas mixtures in porous media. In general, such developed models are used to evaluate the measured competitive adsorption isotherms, and majority of the models require the component mole fraction in the gas phase at the equilibrium and the density of adsorbed gas mixture [42]. The multicomponent mole fractions in the gas phase at the equilibrium are determined using the gas chromatograph, which is the most complicated part for the experimental measurements of competitive adsorption. Fortunately, the multicomponent SLD model may enable us to predict the competitive adsorption behavior in shale without knowing the component mole fraction in the gas phase at the equilibrium.

The SLD model, which was first introduced by Rangarajan et al. [32], has been utilized to characterize pure-gas adsorption in porous media [43,44]. It assumes that all the pores associated with the porous media exist in the form of a rectangular slit with a constant slit width. In addition, the slit pore wall consists of the spherical carbon molecules, and the spherical gas molecules within the slit pore will interact with slit walls. For the SLD model, the fluid-fluid interactions between gas molecules will be described by the Peng–Robinson equation of state (PR-EOS) [45], while the fluid-solid interactions will be depicted on the basis of the 10-4 Lennard-Jones potential [46]. The advantages of the SLD model include: (1) it simulates both the fluid-fluid and fluid-solid interactions in the slit pore; and (2) it is able to predict the excess and absolute adsorption isotherms and gas density profile in the slit pore through rapid computation.

Although molecular simulation can precisely depict the multicomponent gas adsorption on the microscopic scale and experimental measurements can determine the multicomponent gas adsorption on core samples, the high computation costs of molecular simulation and the complex procedures and various devices of experiments are obstacles to achieving a fast prediction and evaluation for the competitive adsorption in porous media. Thus, the purpose of this study is to propose an iterative calculation workflow based on the multicomponent SLD model to determine the excess and absolute adsorption of a gas mixture on shale relying on the measured single-component adsorption data, which could considerably decrease the costs of computation and measurements.

2. SLD Model

2.1. SLD Model for Single Component Gas

For the SLD model, the equilibrium chemical potential is equivalent to the bulk fluid potential, which is composed of the fluid-fluid and fluid-solid interaction potentials [32].

$$\mu \left(z\right)={\mu }_{ff}\left(z\right)+{\mu }_{fs}\left(z\right)={\mu }_{bulk}$$

(1)

where µ(z) is the chemical potential at the position z, µ_bulk is the chemical potential of bulk fluid, and the “bulk”, “ff”, and “fs” as subscripts indicate the bulk fluid, fluid-fluid, and fluid-solid interactions, respectively. The chemical potential of the bulk fluid is a function of fugacity expressed as [32]:

$${\mu }_{bulk}={\mu }_0\left(T\right)+RT\ln \left(\frac{f_{bulk}}{f_0}\right)$$

(2)

where the subscript “0” denotes an arbitrary reference state and f_bulk refers to the fugacity of the bulk fluid. In a similar way, the chemical potential of a fluid-fluid interaction is a function of fugacity written as [32]:

$${\mu }_{ff}\left(z\right)={\mu }_0\left(T\right)+RT\ln \left(\frac{f_{ff}\left(z\right)}{f_0}\right)$$

(3)

where f_ff(z) is the fugacity of the fluid-fluid interaction at the position z.

In addition, the chemical potential of a fluid-solid interaction is calculated by [47]:

$${\mu }_{fs}\left(z\right)=N_A\left[{\Psi }^{fs}\left(z\right)+{\Psi }^{fs}\left(L-z\right)\right]$$

(4)

where N_A is Avogadro’s number, and ψ_fs(z) and ψ_fs(L − z) are potential energy functions demonstrating a fluid molecule at position z interacting with slit walls at the distance of z and L − z, respectively. The fluid-solid interaction is determined based on the Lee’s partially integrated 10 − 4 Lennard-Jones potential [46]:

$${\Psi }^{fs}\left(z\right)=4\pi {\rho }_{atoms}{\epsilon }_{fs}{\sigma }_{fs}^2\left[\frac{{\sigma }_{fs}^{10}}{5{\left(z^{\prime }\right)}^{10}}-\frac{1}{2}{\displaystyle \sum }_{i=1}^4\frac{{\sigma }_{fs}^4}{{\left(z^{\prime }+\left(i-1\right){\sigma }_{ss}\right)}^4}\right]$$

(5)

where ρ_atoms is the density of solid atoms, which is equal to 38.2 atoms/nm², ε_fs is the fluid-solid interaction energy parameter, and σ_fs is the mean value of σ_ff and σ_ss, calculated by σ_fs = (σ_ff + σ_ss)/2. The σ_ff and σ_ss are the adsorbate diameter and the carbon interplanar distance, respectively. The value of the carbon interplanar distance is set as the value for graphite (i.e., σ_ss = 0.335 nm). z′ is the dummy coordinate defined as z′ = z + σ_ss/2. Substituting Equations (2)–(4) into Equation (1), the adsorption equilibrium is written as:

$$f_{ff}\left(z\right)=f_{bulk}·\exp \left(-\frac{{\Psi }^{fs}\left(z\right)+{\Psi }^{fs}\left(L-z\right)}{kT}\right)$$

(6)

where k is Boltzmann’s constant (i.e., k = 1.38 × 10⁻²³ J/K). The fluid-fluid interaction is characterized by PR-EOS. Herein, the PR-EOS is rewritten as a function of gas density (ρ) as follows [45]:

$$\frac{p}{\rho RT}=\frac{1}{\left(1-\rho b\right)}-\frac{a\left(T\right)\rho }{RT\left[1+\left(1-\sqrt{2}\right)\rho b\right]\left[1+\left(1+\sqrt{2}\right)\rho b\right]}$$

(7)

in which:

$$a\left(T\right)=\frac{0.457535\alpha \left(T\right)R^2{T}_c^2}{p_c}$$

(8)

$$b=\frac{0.077796R{T}_c}{p_c}$$

(9)

The term α(T) In Equation (8) is given as [48]:

$$\alpha \left(T\right)=\exp \left[\left(A+B{T}_r\right)(1-T_r^{C+D\omega +E{\omega }^2}\right]$$

(10)

where T_r = T/T_c, ω is the gas acentric factor, P_c is the critical pressure, T_c is the critical temperature, and parameters A, B, C, D, and E are assigned as 2.0, 0.8145, 0.134, 0.508 and −0.0467, respectively. Furthermore, the fugacity of the bulk fluid can be determined based on Equation (7) as:

$$\ln \frac{f_{bulk}}{p}=\frac{b\rho }{1-b\rho }-\frac{a\left(T\right)\rho }{RT\left(1+2b\rho -b^2{\rho }^2\right)}-\ln \left(\frac{p}{RT\rho }-\frac{pb}{RT}\right)-\frac{a\left(T\right)}{2\sqrt{2}bRT}\ln \left(\frac{1+\left(1+\sqrt{2}\right)\rho b}{1+\left(1-\sqrt{2}\right)\rho b}\right)$$

(11)

Similarly, the fugacity of the adsorbate is calculated by:

$$\begin{gathered} \ln \frac{f_{ff}\left(z\right)}{p}=\frac{b_{ads}\rho \left(z\right)}{1-b_{ads}\rho \left(z\right)}-\frac{a_{ads}\left(z\right)\rho \left(z\right)}{RT\left(1+2b_{ads}\rho \left(z\right)-b_{ads}^2{\rho }^2\left(z\right)\right)}-\ln \left(\frac{p}{RT\rho \left(z\right)}-\frac{p{b}_{ads}}{RT}\right) \\ -\frac{a_{ads}\left(z\right)}{2\sqrt{2}{b}_{ads}RT}\ln \left(\frac{1+\left(1+\sqrt{2}\right)\rho \left(z\right)b_{ads}}{1+\left(1-\sqrt{2}\right)\rho \left(z\right)b_{ads}}\right) \end{gathered}$$

(12)

$$b_{ads}=b\left(1+A_b\right)$$

(13)

where the functions for a_ads(z) are derived by Chen et al. [47], the ρ(z) in Equation (12) is the gas density profile, and A_b in Equation (13) is the covolume correction factor. Finally, the excess adsorption isotherm is written as:

$$n^{Excess}=\frac{A}{2}{{\displaystyle \int }}_{\frac{3{\sigma }_{ff}}{8}}^{L-\frac{3{\sigma }_{ff}}{8}}\left(\rho \left(z\right)-{\rho }_{bulk}\right)dz$$

(14)

where parameter A denotes the specific surface area (in m²/g) and $n^{Excess}$ denotes the excess adsorption amount (in mmol/g). Equation (14) can be integrated numerically with the Simpson’s rule. In general, four regression parameters including the specific surface area, A (in m²/g), fluid-solid interaction energy, ε_fs/k (in Kelvin), slit width, L (in nm), and covolume correction factor, A_b, will be determined during the regression process of fitting the single component gas adsorption by use of the SLD model. We noticed that ε_fs/k is the fluid-solid interaction energy divided by the Boltzmann’s constant (k).

2.2. SLD Model for Multicomponent Gas

For multicomponent gas, the functions for fluid-fluid and fluid-solid interactions and adsorption equilibrium mentioned previously will be applied to each component in the gas mixture. Thus, the function of fluid-solid interaction for component i in the gas mixture is given as:

$${\Psi }_i^{fs}\left(z\right)=4\pi {\rho }_{atoms}{\epsilon }_{fs,i}{\sigma }_{fs,i}^2\left[\frac{{\sigma }_{fs,i}^{10}}{5{\left(z^{\prime }\right)}^{10}}-\frac{1}{2}{\displaystyle \sum }_{i=1}^4\frac{{\sigma }_{fs,i}^4}{{\left(z^{\prime }+\left(i-1\right){\sigma }_{ss}\right)}^4}\right]$$

(15)

Then, the adsorption equilibrium function for each component i is written as:

$$f_i^{ff}\left(z\right)=f_i^{bulk}·\exp \left(-\frac{{\Psi }_i^{fs}\left(z\right)+{\Psi }_i^{fs}\left(L-z\right)}{kT}\right)$$

(16)

Similarly, the fugacity of component i in bulk phase can calculated by:

$$\ln \frac{f_i^{bulk}}{y_{i}p}=\frac{b_i}{b}\left(\frac{p}{RT\rho }-1\right)-\ln \left(\frac{p}{RT\rho }-\frac{pb}{RT}\right)-\frac{a}{2\sqrt{2}bRT}\left(\frac{b_i}{b}-\frac{2{{\displaystyle \sum }}_j{y}_j{a}_{ij}}{a}\right)\ln \left(\frac{1+\left(1+\sqrt{2}\right)\rho b}{1+\left(1-\sqrt{2}\right)\rho b}\right)$$

(17)

$$a={\displaystyle \sum }_i{\displaystyle \sum }_j{y}_i{y}_j{\left(a_{bulk}\right)}_{ij}$$

(18)

$$b={\displaystyle \sum }_i{y}_i{b}_i$$

(19)

$${\displaystyle \sum }_i{y}_i=1$$

(20)

where y_i indicates the mole fraction of component i in bulk phase at the equilibrium, (a_bulk)_ij in Equation (18) is calculated by Equation (8), and b_i in Equation (19) is calculated by Equation (9). Moreover, the fugacity of component i in adsorbed phase can calculated by:

$$\begin{array}{ll}\ln \frac{f_i^{ads}\left(z\right)}{x_{i}p}=& \frac{2{{\displaystyle \sum }}_j{x}_j\left(z\right)b_{ij}-b}{b}\left(\frac{p}{RT\rho \left(z\right)}-1\right)-\ln \left(\frac{p}{RT\rho \left(z\right)}-\frac{pb}{RT}\right)\\ & +\frac{a\left(z\right)}{2\sqrt{2}bRT}\left(\frac{2{{\displaystyle \sum }}_j{x}_j\left(z\right)b_{ij}-b}{b}-\frac{2{{\displaystyle \sum }}_j{x}_j\left(z\right)a_{ij}\left(z\right)}{a\left(z\right)}\right)\ln \left(\frac{1+\left(1+\sqrt{2}\right)\rho \left(z\right)b}{1+\left(1-\sqrt{2}\right)\rho \left(z\right)b}\right)\end{array}$$

(21)

$$a\left(z\right)={\displaystyle \sum }_i{\displaystyle \sum }_j{x}_i\left(z\right)x_j\left(z\right){\left(a_{ads}\right)}_{ij}$$

(22)

$$b={\displaystyle \sum }_i{\displaystyle \sum }_j{x}_i\left(z\right)x_j\left(z\right){\left(b_{ads}\right)}_{ij}$$

(23)

$${\left(a_{ads}\right)}_{ij}=\sqrt{{\left(a_{ads}\right)}_i{\left(a_{ads}\right)}_j}\left(1-C_{ij}\right)$$

(24)

$${\left(b_{ads}\right)}_{ij}=\frac{b_i\left(1+A_{b,i}\right)+b_j\left(1+A_{b,j}\right)}{2}$$

(25)

$${\displaystyle \sum }_i{x}_i\left(z\right)=1$$

(26)

where x_i indicates the mole fraction of component i in adsorbed phase at the equilibrium and C_ij is the binary interaction parameters (BIPs) between gas components. The (a_ads)_ij in Equation (22) is calculated using the functions derived by Chen et al. [47] and the (b_ads)_ij in Equation (23) is calculated based on Equation (13). Finally, the excess adsorption isotherm for gas mixture can be written as [40]:

$$n_i^{Excess}=\frac{A}{2}{{\displaystyle \int }}_{\frac{3{\sigma }_{ff,i}}{8}}^{L-\frac{3{\sigma }_{ff,i}}{8}}\left(\rho \left(z\right)x_i\left(z\right)-{\rho }_{bulk}{y}_i\right)dz$$

(27)

For pure gas, the x_i and y_i are equal to 1. Moreover, the absolute adsorption isotherm for gas mixture can be calculated by:

$$n_i^{Abs}=A{{\displaystyle \int }}_{\frac{3{\sigma }_{ff,i}}{8}}^{z_{cutoff}}\left(\rho \left(z\right)x_i\left(z\right)\right)dz$$

(28)

where $n_i^{Abs}$ is the absolute adsorption amount for component i in the gas mixture (in mmol/g) and z_cutoff is the cutoff point. Beyond such point, the density of component i in the adsorbed phase is nearly identical to the density of component i in the bulk phase. In this study, the z_cutoff is determined when the difference between the densities of adsorbed and free gases is less than 5%.

3. Calculation Workflow for Adsorption of Multicomponent Gas

In this study, we developed an iterative calculation workflow to determine the competitive adsorption of the CH₄/CO₂ binary mixture in shale based on the pure CH₄ and CO₂ adsorption results. Such a calculation workflow is able to forecast the adsorption isotherms of multicomponent gas without demanding the experimental measurements for the adsorption of gas mixture on porous media, which significantly reduces the complexity, time, and cost of experiments.

The main idea of the developed calculation workflow is to utilize the SLD model for a gas mixture to evaluate the measured adsorption isotherm of each component in the gas mixture (i.e., pure gas) simultaneously, and then use the determined ε_fs,i and A_b,i for each gas component coupled with the L and A corresponding to the slit pore structure to predict the adsorption isotherms of gas mixture with various feed-gas mole fractions. The calculation workflow for predicting the adsorption of the CH₄/CO₂ binary mixture is presented in Figure 1 in detail.

As shown in Figure 1, the excess adsorption isotherms of each gas component (pure gas) in a gas mixture are the only required experimental measurements. The first key step is to simultaneously perform curve fitting on the measured excess adsorption of each gas component in the gas mixture using a multicomponent SLD model (Equations (15)–(27)) by setting the mole fraction of component i in the gas phase at the equilibrium very close to one (e.g., y_i = 0.999). The second key step is to set the binary interaction parameters equal to zero (C_ij = 0) and find the suitable y_i via iteration. The iteration is controlled by the component mass balance equation given as:

$$z_i=\frac{n_i^{Excess}+{\rho }_{bulk}{V}_{slit}{y}_i}{n_t^{Excess}+{\rho }_{bulk}{V}_{slit}}$$

(29)

where z_i is the feed-gas mole fraction, V_slit is the specific volume of slit, and $n_t^{Excess}$ is the sum of excess adsorption of each component in gas mixture. It is noted that the V_slit can be obtained through fitting the pure gas adsorption by an SLD model for single component gas (i.e., Equations (4)–(14)). Herein, the V_slit used in the calculation is $169.2\times {10}^{-9}$ m³/g. If the y_i calculated by Equation (29) does not match the assumed y_i to initiate the calculation, a new set of y_i will be assumed to start another trial. Such an iterative process will end when the difference between the calculated y_i and assumed y_i is less than 0.002 in this study. The physical properties of CH₄ and CO₂ used in the SLD model are shown in

Table 1. Physical properties of CH₄ and CO₂.

Gas Molecule	Critical Pressure (MPa)	Critical Temperature (K)	Molecular Diameter (nm)	Acentric Factor
CH4	4.599	190.56	0.373	0.0113
CO2	7.377	304.13	0.3648	0.225

.

4. Results and Discussion

4.1. Validation of Developed Calculation Workflow

In this study, the results of experimentally measured pure CH₄ and CO₂ adsorption and CH₄/CO₂ competitive adsorption in Yanchang shale from the Ordos Basin in China reported by Qin et al. [49] are employed as the references to validate the developed calculation workflow. The adsorption isotherms of CH₄, CO₂, and an equimolar CH₄/CO₂ mixture (i.e., the mole fractions of the feed CH₄ and CO₂ are identical, expressed as CH₄/CO₂ 50/50 in the following figures) measured at 313.15 K from Qin et al. [49] are displayed in Figure 2. Herein, we examine the practicability and reliability of the developed calculation workflow by comparing two cases. For case one, we use all the data determined via the experimental measurements to perform regression analysis on the measured competitive adsorption of the CH₄/CO₂ binary mixture in shale by the SLD model for multicomponent gas. For case two, we employ the SLD model for multicomponent gas to match such measured CH₄/CO₂ competitive adsorption isotherms following the calculation workflow merely based on the measured pure CH₄ and CO₂ adsorption isotherms. The main differences between the two cases are the demands of the component mole fraction in the gas phase at the equilibrium (y_i) and the binary interaction parameters (C_ij). The y_i is usually determined using a gas chromatograph during experimental measurements, which is the most complex part in the adsorption experiments with multicomponent gas. In addition, the C_ij is the curve-fitting parameter obtained during the regression process. It is noted that y_i and C_ij are the primary obstacles to estimating the adsorption isotherms of a gas mixture depending on the corresponding single-component adsorption results.

For both cases, the first step is to simultaneously fit the adsorption isotherms of pure CH₄ and pure CO₂ with the SLD model for multicomponent gas, from which the ε_fs,i and A_b,i for both CH₄ and CO₂ coupled with the L and A can be determined. The curve-fitting results are demonstrated in Figure 2a, and the regression parameters and average absolute percentage deviation (%AAD) are listed in

Table 2. Regression parameters and %AAD for pure gas adsorption.

Gas Component	L (nm)	εfs/kB (K)	A (m2/g)	Ab	%AAD
CH4	8.6	37.81	16	0.1	3.76
CO2	8.6	72.25	16	−0.01	4.63

. Herein, the %AAD used to reveal the curve-fitting quality is calculated by:

$$\%AAD=\frac{1}{N}{\displaystyle \sum }_{i=1}^N\left|\frac{n_{EXP}-n_{SLD}}{n_{EXP}}\right|\times 100$$

(30)

where N is the amount of the total data point, n_EXP is the adsorption amount determined via the experiments, and n_SLD is the adsorption amount calculated by the SLD model. From Figure 2a, it can be seen that the SLD model for multicomponent gas can properly depict the measured pure CH₄ and CO₂ adsorption isotherms. Next, the y_i measured by the gas chromatograph at each testing pressure and the regression parameters shown in Table 2 are set as inputs for the SLD model to match the measured CH₄/CO₂ competitive adsorption in shale. The C_ij is the only curve-fitting parameter during the regression process. The values of y_i are from Qin et al. [49] and the results of regression analysis are presented in Figure 2b. The optimal curve-fitting results associated with C_ij = −0.27 correspond to the %AAD = 21.06 for CH₄ (red dashed line) and %AAD = 13.33 for CO₂ (green dashed line). It was found that the CO₂ adsorption amount obtained with the SLD model (green dashed line) is a little bit larger than the experimental data (purple triangles), but the trend of the CO₂ adsorption isotherm is nearly identical to that of the measured data points. Similarly, the adsorption amount of CH₄ calculated by the SLD model (red dashed line) is larger than the experimental data (blue circles) at p < 3 MPa, but the calculated adsorption amount matches well with the experimental data when p > 3 MPa. The main reasons resulting in the imperfect match may be attributed to (1) the inaccuracy of measured y_i, (2) the deviations that existed between the measured adsorption of pure gas and such adsorption calculated by the SLD model (Figure 2a), (3) the inaccuracy of determined x_i, and (4) the convergence of regression analysis. Although the deviations reveal that the SLD model for multicomponent gas cannot perfectly describe the CH₄/CO₂ competitive adsorption results on shale, the %AADs for the CH₄/CO₂ binary mixtures share a similar degree of accuracy with the %AADs for pure gases reported in the literature. Mohammad et al. summarized that the %AADs for evaluating the CO₂ adsorption in coals using the SLD model are in the range of 10 to 12 [43]. Additionally, Chareonsuppanimit et al. pointed out the prediction of adsorption in shales associated with the %AADs for CH₄ lying between 2 and 12 and the %AADs for CO₂ lying between 4 and 14 by use of the SLD model [44].

Furthermore, for case two, the C_ij is set as 0 and y_i is determined through the iterative calculation workflow (Figure 1). The curve-fitting results obtained via the developed calculation workflow are illustrated in Figure 2b as well. The %AADs for CH₄ (orange dashed line) and CO₂ (blue dashed line) are 35.27 and 5.23, respectively. As shown in Figure 2b, the CO₂ adsorption isotherm calculated by the SLD model following the calculation workflow (blue dashed line) approximately fits in with the experimental data (purple triangles). In contrast, the CH₄ adsorption amount calculated by the SLD model following the calculation workflow (orange dashed line) partially deviates from the experimental data, especially at p < 3 MPa. Compared with case one, case two provides a better prediction for CO₂ adsorption but a worse prediction for CH₄ adsorption. Overall, the calculation methods in both cases, to some extent, are able to forecast the gas mixture adsorption isotherms. Following the proposed calculation workflow, the SLD model for multicomponent gas might be used to predict competitive adsorption of a binary gas mixture in shale based merely on the single-component adsorption. After determining the ε_fs,i and A_b,i for each gas component in a gas mixture coupled with the L and A for a slit-pore structure, the adsorption isotherms of a gas mixture for any feed-gas mole fractions can be calculated, which in turn greatly reduces the complexity, time, and cost of experimental measurements.

4.2. Prediction of CH₄/CO₂ Competitive Adsorption

The absolute and excess adsorption isotherms of CH₄/CO₂ mixtures with different feed-gas mole fractions can be predicted following the developed calculation workflow once the regression parameters of the SLD model are obtained. It is noted that the following Figure 3, Figure 4, Figure 5 and Figure 6 are generated using the SLD model with the four regression parameters listed in Table 2. Figure 3 shows the excess adsorption isotherms of 75/25 and 25/75 CH₄/CO₂ gas mixtures in shale at 313.15 K. The corresponding adsorption isotherms measured by Qin et al. [49] are set as comparisons. In spite of the deviations for CH₄ adsorption isotherms similar to those shown in Figure 2b, acceptable agreement for the CO₂ adsorption isotherms between the multicomponent SLD model (calculation workflow) and the experimental results is reached, which further proves the practicability of the calculation workflow to some extent. Moreover, Figure 4 presents the absolute adsorption isotherms of CH₄ and CO₂ in 75/25, 50/50, and 25/75 CH₄/CO₂ gas mixtures in shale core samples at 313.15 K. The absolute adsorption isotherms shown in Figure 4 are calculated by Equation (28) with the regression parameters from Table 2. It is found that the CH₄ adsorption amount increases with the feed-gas mole fraction (CH₄/CO₂), while the CO₂ adsorption amount decreases with the feed-gas mole fraction. The CO₂ adsorption amounts are considerably larger than the CH₄ adsorption amounts for the 75/25 and 50/50 feed-gas mole fractions. However, the CO₂ adsorption amount is lower than the CH₄ adsorption amount for the 25/75 feed-gas mole fraction. As a result, there will be a threshold of a feed-gas mole fraction, above which the CO₂ adsorption amount would be larger than the CH₄ adsorption amount. Finding such a threshold can provide guidance for CO₂-EGR and CO₂ sequestration operations in shale gas reservoirs.

In addition, after determining the regression parameters (Table 2), we can predict the excess adsorption isotherms of CH₄/CO₂ mixtures at different temperatures following the developed calculation workflow. Figure 5 exhibits the excess adsorption isotherms of each component in a CH₄/CO₂ (50/50) gas mixture in shale at temperatures of 313.15, 333.15, and 353.15 K, respectively. It is found that the excess adsorption amounts for both the CH₄ and CO₂ decrease with the increase in temperature, which are consistent with the observations in previous studies [50,51,52]. Moreover, we can conclude from Figure 5 that CO₂ sequestration during the CO₂-EGR process would benefit from the lower temperature, since the effect of temperature on the CO₂ adsorption amount is more distinct.

In sum, without requiring extensive laboratory measurements, the developed calculation workflow is capable of predicting absolute and excess adsorption isotherms of CH₄/CO₂ mixtures with any feed-gas mole fractions at various temperatures. It is also an optimal method to determine the threshold of the feed-gas mole fraction, which enables us to optimize CO₂ injection amounts and rates for the CO₂-EGR operation in shale gas reservoirs.

4.3. CO₂/CH₄ Adsorption Selectivity

Adsorption selectivity is a critical parameter to reflect the feature of CH₄/CO₂ competitive adsorption. The adsorption priority of CO₂ against CH₄ in shale can be expressed by:

$$S_{C{O}_2/C{H}_4}=\frac{x_{C{O}_2}/x_{C{O}_2}}{y_{C{O}_2}/y_{C{O}_2}}=\frac{n_{C{O}_2}^{Abs}/n_{C{H}_4}^{Abs}}{y_{C{O}_2}/y_{C{O}_2}}$$

(31)

where x and y refer to the mole fractions of CH₄ and CO₂ in adsorbed and free gases, respectively, and S_CO2/CH4 is the adsorption selectivity of CO₂ against CH₄. It is noted that if the adsorption selectivity is larger than 1, CO₂ will be preferentially adsorbed [23,53]. In addition, the larger selectivity signifies the more intense adsorption affinity of CO₂ to shale rock. Herein, Figure 6 demonstrates the adsorption selectivity of 75/25, 50/50, and 25/75 CH₄/CO₂ gas mixtures determined via the developed calculation workflow. The adsorption selectivity ranges approximately from 2.5 to 5, indicating the preferential adsorption of CO₂ over CH₄ on shale. Moreover, it is observed that the adsorption selectivity decreases with the pressure because the high-energy adsorption sites are easily occupied by CO₂ at low pressures, whereas CO₂ and CH₄ will start to take up the low-energy adsorption sites through the competition when the high-energy adsorption sites are fully occupied at high pressures, leading to the decline of the selectivity at elevated pressures [17]. Figure 6 also shows the increase of adsorption selectivity with the feed-gas mole fraction, which implies that the higher mole fraction of CO₂ in feed-gas will weaken the selectivity of CO₂ over CH₄, although the higher mole fraction of CO₂ in feed-gas will lead to more CO₂ adsorption amounts in shale as shown in Figure 4. This finding is consistent with the results of adsorption selectivity reported by Wei et al. [54]. Therefore, it is significant to select a proper mole fraction for feed-gas to keep the high adsorption capacity and selectivity of CO₂ over CH₄ for CO₂-EGR and CO₂ sequestration processes.

5. Conclusions

In this work, an iterative calculation workflow based on the multicomponent SLD model is developed to predict and characterize competitive adsorption of a gas mixture in shale. Such a calculation workflow depends not on the competitive adsorption measurements that demand the multicomponent mole fractions in the gas phase at the equilibrium but on the single-component adsorption results, which significantly lower the complexity, time, and cost of the gas adsorption experiments. The main findings and conclusions in this study are summarized as follows:

• The developed iterative calculation workflow is able to characterize and predict the excess and absolute adsorption of a gas mixture with various feed-gas mole fractions through manipulating the multicomponent SLD model to implement regression analysis on the measured single-component adsorption results simultaneously, thus bypassing the demands of the mole fraction of each component in the gas phase at the equilibrium in experiments and the binary interaction parameters obtained from the curve-fitting process.
• The adsorption capacity of the gas component in a gas mixture is proportional to the initial feed-gas mole fraction. Accordingly, there will be a threshold of feed-gas mole fraction, above which the CO₂ adsorption capacity would be larger than the CH₄ adsorption capacity. Such a threshold is critical for CO₂-EGR and CO₂ sequestration operations in shale gas reservoirs.
• In this study, adsorption selectivity (S_CO2/CH4) ranges approximately from 2.5 to 5, indicating the preferential adsorption of CO₂ over CH₄ in shale. The adsorption selectivity increases with the feed-gas mole fraction, demonstrating that the higher mole fraction of CO₂ in feed-gas will weaken the selectivity of CO₂ over CH₄, despite that the higher mole fraction of CO₂ in feed-gas will lead to the larger CO₂ adsorption capacity. As a result, a proper mole fraction for feed-gas should be determined to ensure the high adsorption capacity and selectivity of CO₂ over CH₄ in the process of CO₂-EGR and CO₂ sequestration.