This paper was prepared for presentation at the 2017 SPE Western Regional Meeting held in Bakersfield, California, USA, 23 April 2017.

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1 SPE MS Selective Adsorption and Transport Diffusion of CO 2 -CH 4 Binary Mixture in Carbon-Based Organic Nanopores Mohammad Kazemi and Ali Takbiri-Borujeni, West Virginia University Copyright 2017, Society of Petroleum Engineers This paper was prepared for presentation at the 2017 SPE Western Regional Meeting held in Bakersfield, California, USA, 23 April This paper was selected for presentation by an SPE program committee following review of information contained in an abstract submitted by the author(s). Contents of the paper have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper without the written consent of the Society of Petroleum Engineers is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgment of SPE copyright. Abstract Production from shale gas reservoirs provided a unique opportunity for underground storage of the CO 2. Because of the higher affinity of CO 2 to the organic matter of shale compared to CH 4, injecting CO 2 into these resources can cause the adsorbed CH 4 molecules to be desorbed and then replaced by the CO 2 molecules. In this work, we perform molecular dynamics simulations to calculate the Onsager and Fickian diffusion coefficients of CH 4 and CO 2. Onsager coefficients of the pure species increase as its partial pressure increases and then level off at higher pressures. The off-diagonal Onsager coefficients increase as CH 4 pressure increases up to a CO 2 pressure of approximately 100 atm and followed by a reduction in the coefficients. The Onsager diffusion coefficients of CO 2 -CO 2 are the highest compared to CH 4 -CH 4 and CO 2 -CH 4 diffusion coefficients. The Onsager diffusion coefficients for both species are a function of their occupancies. As concentration of a species increases, its Onsager diffusion coefficient increases as well. The results also demonstrated that the off-diagonal terms in the Onsager and Fickian diffusion coefficients matrix have the same order of magnitude as the diagonal terms and therefore cannot be ignored in modeling of CO 2 sequestration in shale. Introduction Carbon capture and sequestration (CCS) in depleted shale gas reservoirs provides an opportunity for underground storage of the CO 2. CO 2 has higher affinity to the organic matter of shale compared to CH 4. Injecting CO 2 into the shale reservoirs can cause the CH 4 molecules in the adsorbed state to be desorbed and and replaced by the CO 2 molecules (enhanced gas recovery-egr). Current understanding of the competitive adsorption/desorption process of CO 2 and CH 4 and its impact on the efficiency of CCS is still insufficient. Furthermore, the transport coefficients describing this process are not well defined. In EGR and CCS processes in shale, the competitive adsorption equilibia is of significant importance. There have been many approaches to determine the adsorption isotherms of pure CO 2 and pure CH 4 to estimate the maximum capacity for CO 2 storage and CH 4 recovery including the volumetric method (Mavor et al., 1990; Nuttall et al., 2005; Heller and Zoback, 2014), manometric method (Khosrokhavar et al., 2014; Yuan et al., 2014), and gravimetric method (Pini et al., 2010). Adsorption of binary CO 2 -CH 4 mixtures has also been measured using gravimetric-chromatographic method (Ottiger et al., 2008). Shale is composed

2 2 SPE MS of organic nanoporous matter called kerogen. Almost half of the total hydrocarbons is assumed to be in stored in kerogen in the adsorbed state (Collell et al., 2014; Wang, 2009). Thus, it is necessary to study the adsorption phenomena from a microscopic point of view. Molecular simulations have been proved to be credible alternatives to experimental measurements of species adsorption in micoroporous materials (Arya et al., 2001). GCMC simulations have been extensively used to study the pure species adsorption in microporous carbon (Billemont et al., 2013), graphite nanoconduits (Kazemi and Takbiri-Borujeni, 2016), and realistic molecular models of coal (Zhao et al., 2016) and shale (Collell et al., 2015) and species mixtures in graphite channels (Kazemi and Takbiri-Borujeni, 2016), microporous carbon (Billemont et al., 2013). There are few studies on species diffusion in EGR process in the literature. There are a few mechanisms, such as inhomogeneous fluid distribution of the species, adsorption selectivity, and apparent viscosity that manifest themselves into different diffusion coefficients, namely self-, corrected (or Maxwell-Stefan), and transport diffusion coefficients. The transport diffusion coefficients of pure CO 2 and pure CH 4 and CO 2 - CH 4 mixtures can be calculated from their trajectories (Sanborn and Snurr, 2000; Zhao et al., 2016). The objective for this work is to shed light on the the diffusion phenomena of the binary mixtures by performing molecular simulations in carbon-based organic nanochannels. Adsorption of pure CH 4 and pure CO 2, CO 2 -CH 4 mixtures, and also adsorption selectivity values of CO 2 over CH 4 are computed by performing GCMC simulations. EMD simulations are also performed to calculate the Onsager transport diffusion coefficient matrix and the Fickian diffusion coefficients via thermodynamic factors at varying pressure and species mole fraction. Diffusion of Binary Mixtures Three different phenomenological approaches are generally used to model the diffusion, namely self-, corrected (or Maxwell-Stefan), and transport diffusion coefficients. In Fick's formulation, the fluxes, (N), are related to the gradients of concentrations, c, with Fick's diffusivity matrix, D T, being the proportionality constant, or in vector notation, where N i is the molecular flux of species i in a mixture (molecules/m 2.s), ρ is the density number (kg/m 3 ), Ci is the concentration, and D T is the matrix of Fick's diffusivities (m 2 /s), which is strongly dependent on mixture composition and loadings. Another popular formulation is Onsager's. In this formulation, the fluxes are casted as a function of gradients of chemical potential, The Onsager matrix, [L], is a symmetric matrix. For a mixture of species i and j, Onsager matrix can be written as, (1) (2) (3) (4) where r l, i (t) is the position of l th molecule of component i at time t. The chemical potential gradient can be related to concentration gradient as,

3 SPE MS 3 (5) where Q i is the fractional occupancy of component i, c i,sat is saturated molar concentration of adsorbed species, f i is the fugacity of component i, and Γ ij is the matrix of thermodynamic factors. If the binary mixture adsorption isotherms can be modeled using multicomponent Langmuir isotherm, then the elements of matrix of thermodynamic factors can be written as, Therefore, the matrix of Onsager diffusion coefficients can be converted to those of Fickian as, (6) (7) While Fick's and Onsager formulations are phenomenological approaches, the Maxwell-Stefan formulation bal-ances diffusive and drag forces. Diffusion of species i is described as, (8) where is the Maxwell-Stefan diffusivity coefficient of component i and are the binary exchange Maxwell-Stefan diffusivities. For single component systems, is called the corrected diffusivity. It is important to emphasize that all aforementioned formulations of are strictly equivalent. Simulation Method The computational methodology are described in (Kazemi and Takbiri-Borujeni, 2016). In order to perform GCMC and EMD simulations, a 2 nm graphite channel is created with rough surfaces. The MD simulation time integration of equation of motion are performed using Verlet velocity algorithm. Pressure inside the channel is kept constant using adequate number of GCMC insertion and deletion. The inserted molecules are assigned based on specified reservoir temperature (350 K). The driving force for gas molecules movement are the pressure or chemical potential difference between the two reservoirs (L and H). The wall and fluid temperatures are kept constant at 350 K in NVT (constant number of molecules, constant volume and constant temperature) ensemble. All the simulations are performed using Large-scale Atomic/ Molecular Massively Parallel Simulator (LAMMPS) (Plimpton, 1995) and Visual Molecular Dynamics (VMD) (Humphrey et al., 1996) is used for visualization. Results and Discussion Single-Component Adsorption Isotherms In order to determine adsorption isotherms of pure CH 4 and CO 2, grand canonical Monte Carlo (GCMC) simulations are performed. Absolute adsorption isotherms are determined at 350 K for a pressure range of 10 to 330 atm. Density profiles of CH 4 and CO 2 in the channel are shown in Fig. 1. At 10 atm (blue lines), two sets of density peaks are observed; the first peaks (approximately at z = 7 and 33 A), represents the gas molecules

4 4 SPE MS adsorbed to the two-layered graphite sections of the channel (sections of the channel where the carbon atoms are taken out). The second peaks (approximately at z = 12.5 and 28.5 A) demonstrate the density of gas molecules adsorbed to the three-layered sections of the surface, where no carbon atoms are removed from the surface. As pressure increases, third layers of adsorbed molecules are formed. Furthermore, as pressure increases, values of density at the first, second, and third peaks and also the density at the middle of the channel increase. Figure 1 Density profiles of CO 2 and CH 4 across the channel (a) CO 2 and (b) CH 4. The density peaks for CO 2 are highest at the second layer of molecules (approximately at z = 12.5 and 28.5 A) demonstrating the density of gas molecules adsorbed to the three-layered sections of the surface (Fig. 1a). At pressures higher than 90 atm, formation of three CO 2 layers is observed. Similar to CO 2, density peaks for CH 4 are highest at the second layer of molecules (Fig. 1b). For CH 4, formation of the third layer occurs at higher pressures compared to that of CO 2 and once it forms, the difference between the density of the third layer with that in the second layer is less significant compared to CO 2. For instance, the ratios of the densities of third layer to density in middle of channel for CO 2 and CH 4 at 330 atm are 1.4 and 1.1, respectively. Compared to CH 4, the density profiles for CO 2 show a multilayered structure. This is in agreement with the BET observations. The absolute and excess loading quantities (moles of adsorbed molecules per solid mass in terms of mol/kg)at 350 K are demonstrated in Fig. 2. The isotherms display a steep increase in adsorption at low relative pressures followed by a plateau region at higher relative pressure, which is believed to be caused by the completion of a monolayer of adsorbed gas. The absolute loadings of both gases increase as pressure increases. For the pressure range tested, absolute adsorption values for CO 2 are higher than those for CH 4.

5 SPE MS 5 Figure 2 Absolute and excess adsorption isotherms of CH 4 and CO 2 The excess loading of CH 4 increases with pressure to reach a maximum value at an optimum pressure and then slightly decreases. The behavior of CH 4 isotherms is similar to Langmuir isotherm for the pressure range tested. This can be confirmed by looking at the density profiles of CH 4 in Fig. 1b. CH 4 densities in the third peaks (which is the second layer from non-defective surfaces) are not significantly high compared with the densities on the wall (first and second peaks). Similar to CH 4, the CO 2 absolute loading increases and saturates around 300 atm. Binary Adsorption isotherms Binary adsorption isotherms of CH 4 and CO 2 in the channel are determined using GCMC simulations. The simulations are performed according to grid of partial pressures of gases, shown in Fig. 3, to obtain a wide range of possible gas compositions. In order to determine the thermodynamic correction factors according to Eq. 6, the adsorption isotherm data are fitted to Langmuir multicomponent adsorption isotherms as (9) (10) Figure 3 Grid of partial pressure pairs for the CO 2 -CH 4 mixture used in GCMC simulations.

6 6 SPE MS where c 1 and c 2 is the concentration of methane and carbon dioxide in the channel and P 1 and P 2 are the bulk partial pressure of these species, respectively. Here and throughout the remainder of the paper, the subscripts 1 and 2 for concentrations, partial pressures, etc. refer to CH 4 and CO 2, respectively. Coefficients obtained for the best fit are listed in Table 1. Table 1 Calculated parameters for the binary adsorption isotherms (Eqs. 9 and 10) Parameter Value A A A B B B The fitted values are compared with the computed ones in Fig. 4. The straight line shows that the fitted values are close to the simulation results. The fitted data are within 7 and 13% of simulation results. Figure 4 Comparison between the simulation results and fits using Eqs. 9 and 10 for the binary mixture adsorption for CH 4 (red open circles) and CO 2 (filled circles). The line denotes exact agreement. Selectivity GCMC simulations are performed to compare the tendency of different components to be adsorbed to the channel wall, by computing the adsorption selectivity of CO 2 over CH 4 at different pressures. Selectivity of CO 2 over CH 4 can be determines as, where S is selectivity, x is the molar fraction of the gas in the adsorbed phase, and y represents the gas molar fraction in the bulk gas phase. Selectivity of one demonstrates a case in which both types of molecules have similar affinities to the solid walls. For each simulation, CH 4 pressure is kept constant and the same as the pressure of CO 2. The simulations are performed for a pressure range of 100 to 200 atm at 350 K. (11)

7 SPE MS 7 Computed values of the adsorption selectivity of CO 2 over CH 4 are greater than one for the pressures tested, which demonstrates that the CO 2 molecules have higher tendency to be adsorbed to the wall than CH 4 molecules (Fig. 5). As pressure increases, the selectivity values decrease and tend to reach one. At lower pressures, CO 2 molecules easily move within the channel and therefore, there is a higher probability that they reach the carbon molecules at the wall. This has important implications in processes involving injection of CO 2 in depleted shale reservoirs. Based on these results, in these systems, as pressures increases, storage of CO 2 in carbon-based channels in presence of CH 4 becomes less efficient. Figure 5 Selectivity of CO 2 over CH 4 at different pressures. Binary Diffusion Coefficients Equilibrium molecular dynamics (EMD) simulations are performed to determine the Onsager and Fickian diffusion coefficients at 350 K. The pressures for both species range from 20 to 170 atmthe Onsager diffusion coefficients are determined using Eq. 4 and Fickian coefficients are calculated using Eq. 7. The components of Onsager and Fickian diffusion matrices are plotted against the species partial pressures in Figs. 6 and 8. The subscript 1 refers to CH 4 and 2 refers to CO 2. Due to the symmetric nature of the Onsager diffusion coefficient matrix, L 12 = L 21. The matrix of Fickian diffusion coefficients, on the other hand, is not symmetric and therefore D 12 D 21.

8 8 SPE MS Figure 6 Onsager diffusion coefficients of CH 4 -CO 2 mixtures in the channel; (a) L 11 (b) L 12 (c) L 22. Figure 7 Onsager diffusion coefficients as a function of gas occupancy.

9 SPE MS 9 Figure 8 Fickian diffusion coefficients of CH 4 -CO 2 mixtures. Onsager coefficients for each species, namely, L 11 and L 22, increase as the species pressure increases and level off at higher pressures (Fig. 6). For instance at 170 atm CH 4 pressure, the CO 2 -CO 2 Onsager coefficients, L 22, sharply increase from 7.45 x 10-7 to 1.22 x 10-6 as CO 2 pressure increases from 20 to 140 atm above which the coefficients gradually decrease (Fig. 6c). Similarly, at CO 2 pressure of 170 atm, Onsager coefficients for CH 4 -CH 4, L 11, increase from 6.60 x 10-9 to 1.69 x 10-7 as CH 4 pressure changes from 20 to 140 atmfollowed by a reduction in coefficients to 4.38 x 10-7 at 170 atm. The off-diagonal Onsager

10 10 SPE MS coefficients (CH 4 - CO 2 ), L 12, increase as CH 4 pressure increases up to CO 2 pressure of approximately 100 atm and then decrease at higher pressures (Fig. 6b). The CO 2 -CO 2 Onsager diffusion coefficients are higher than CH 4 -CO 2 and CH 4 -CH 4 diffusion coefficients. In order to understand the diffusion behavior of the species mixture, the calculated Onsager coefficients are plotted against the species occupancy (Fig. 7). As CH 4 occupancy, θ 1, increases, its Onsager diffusion coefficients increase. The CO 2 -CO 2 diffusion coefficients do not change significantly with CH 4 occupancy although at higher CH 4 occupancy, they tend to decrease. The CH 4 -CO 2 Onsager diffusion coefficients do not change significantly as the species occupancies increases. Similarly, as CO 2 occupancy increases, its Onsager coefficients, L 22, increase as well and CH 4 -CH 4 coefficients decrease. Fickian diffusion coefficients of the pure species, namely, D 11 and D 22, increase as the species pressure increases and then level off at higher pressures (Fig. 8). Similar to Onsager diffusion coefficients, species Fickian diffusion coefficients increase as their partial pressure increases. However, at high pressures (> 100atm), the diffusion coefficients decrease. The CH 4 -CO 2 and CO 2 -CH 4 Fickian diffusion coefficients are not equal. The values of off-diagonal terms in the Onsager and Fickian diffusion coefficient matrices are in the same orders of magniytude as the diagonal terms and cannot be neglected. Therefore, the modeling techniques should account for the off-diagonal terms in order to accurately model the CO 2 sequestration in shale. Conclusions Molecular dynamics simulations are performed to study the transport of CH 4 -CO 2 mixtures in carbon-based organic channels. A comparison of adsorptions of CH 4 and CO 2 indicates that the CO 2 have higher affinity to organic walls than CH 4. For CH 4, formation of the third layer occurs at higher pressures and once it forms, the difference between the density of the third layer with that in the second layer is less significant compared to CO 2. The adsorption selectivity values of CO 2 over CH 4 are larger than one for all pressures tested, which indicates that the CO 2 has a higher tendency to be adsorbed on the channel walls compared to CH 4. For CO 2, a behavior similar to BET adsorption is observed, while for CH 4, the adsorption isotherm is similar to Langmuir isotherm. Onsager coefficients for each species, CH 4 -CH 4 and CO 2 -CO 2, increase as the partial gas pressure increases and levels off at higher pressures. The off-diagonal Onsager coefficients (L 12 ) increase as CH 4 pressure increases up to a CO 2 pressure of approximately 100 atm and starts to reduce after then. The Onsager diffusion coefficients of CO 2, CO 2 -CO 2, are the highest compared to CH 4 -CH 4 and CH 4 -CO 2 diffusion coefficients. The Onsager diffusion coefficient of each species is a function of their occupancies. As the occupancy of species increases, their Onsager diffusion coefficient increases. Similar to Onsager diffusion coefficients, species Fickian diffusion coefficients increase as their partial pressure increases. The results also demonstrated that the off-diagonal terms in the Onsager and Fickian diffusion coefficients matrix have the same order of magnitude as the diagonal terms and therefore cannot be ignored in modeling of CO 2 sequestration in shale. Acknowledgements Authors acknowledge use of Super Computing Systems (Spruce Knob and Mountaineer) at West Virginia University, which are funded in part by the National Science Foundation EPSCoR Research Infrastructure Improvement Cooperative Agreement # , the state of West Virginia (WVEPSCoR via the Higher Education Policy Commission) and West Virginia University.

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