Coupling of diffusion-dissolution-deformation on the nature of sorption hysteresis in nanoporous shale kerogen

Summary

Despite being closely relevant in shale gas production, desorption of CH₄ and CO₂ in deformable shale media remains poorly understood. Sorption hysteresis of CH₄ and CO₂ is experimentally observed in kerogen, and the underlying mechanism needs to be unraveled. Here, we develop a microscopic modeling setup to reasonably reproduce the experimentally observed hysteresis using atom-scale simulations. Various previously suggested mechanisms including capillary condensation, chemical interaction, insufficient equilibrium, and surface heterogeneity, are excluded from the hysteresis origins by detailed sorption studies. The observed hysteresis is governed by diffusive mass transfer of dissolved gas in flexible structure, namely, the coupling of diffusion-dissolution-deformation. The worse-connected porous network and the restricted expulsion of aggregated dissolved molecules upon desorption are discussed as the origins of the coupling protocol. These findings significantly enhance the understanding of hysteresis phenomenon in amorphous materials, which, as well as the proposed microscopic modeling, is widely applicable to other soft nanoporous materials.

Graphical abstract

Highlights

• •A novel microscopic modeling setup for sorption hysteresis is developed
• •Various reported hypotheses are excluded from the origins of sorption hysteresis
• •Diffusion of dissolved fluid in flexible kerogen induces the sorption hysteresis
• •Worse connectivity and fluid plug upon desorption govern the sorption hysteresis

Chemistry; Computer science; Materials science

Introduction

Hydrocarbon recovery from shale formations has dramatically transformed the global energy landscape, raising enormous economic benefits while being accompanied by environmental and political concerns.^1,2,3,4 Hydrocarbons in shale are generated from the source called kerogen, whose macromolecular structure is composed of aromatic rings, aliphatic units, and heteroatom groups.⁵ Kerogen is an amorphous organic matter, which possesses the unique advantages of high surface area and flexible structure like other soft nanoporous materials. The large specific surface areas, as well as the abundant nanopores, facilitate the storage of a large number of fluid molecules in kerogen.^6,7,8 Of significant interest to reservoir development is the quantification of various fluid states, which determines the production strategies.⁹ Fluids in kerogen are stored in three states: free state in large pores, adsorbed state on pore surfaces, and dissolved state in kerogen matrix.^10,11 Considerable research has been conducted to characterize the free and adsorbed states as they are commonly believed to be the major fluid states.^12,13,14,15 By contrast, research on the dissolved state is limited, although it is suggested to take a large share.^9,10 The structure flexibility of kerogen results in the strong coupling between fluid distribution and kerogen deformation.¹⁶ The kerogen-fluid interaction can lead to kerogen deformation accompanied by the change of internal porous structure, which feeds back on fluid distribution.¹⁷ The sorption-swelling coupling in kerogen-fluid interaction has been studied by poromechanical theory,^18,19 atomistic simulation,^{20,21,22,23,24,25,26} theoretical model,^26,27,28 and experiment.^29,30 Despite previous efforts, the quantification of fluid states, especially adsorbed and dissolved states in flexible kerogen structure, remains to be unraveled.³¹

Only limited attempts have been performed on desorption from kerogen, although it is well known as the initial process of fluid transport in the shale reservoir. A measurable hysteresis has been experimentally observed for supercritical CH₄ and CO₂ in kerogen by gravimetric methods at both moderate³² and high pressures.³³ The measured sorption/desorption isotherms exhibit a hysteresis loop extending to very low pressures, which departs from the trend induced by capillary condensation for heavier hydrocarbons in mesopores.³⁴ This hysteresis phenomenon has been widely reported in other soft nanoporous materials such as polymers^35,36 and coals.^37,38,39,40 No consensus about the hysteresis origins has been reached so far. Possible explanations including sorption-swelling coupling,⁴¹ changing hydrogen bond networks,³⁵ restricted-access pores,³⁶ reversible structure change,³² dissolution of fluids,³² different fluid locations upon sorption/desorption,³³ as well as chemical interaction, insufficient equilibrium, and surface chemistry heterogeneity⁴² have been suggested. These hypotheses remain to be confirmed. No targeted measures can be expected to improve the shale gas production until the hysteresis origins are well understood. The goal of this work is to clarify the underlying microscopic mechanisms governing the experimentally observed hysteresis for the first time by using a detailed molecular simulation study.

At the molecular scale, there is no appropriate simulation method at hand that can efficiently reproduce the realistic physical process of adsorption/desorption in flexible kerogen. Grand canonical Monte Carlo (GCMC) fails by virtue of taking into account the structure flexibility of kerogen upon sorption.^43,44 The conventional molecular dynamics (MD) method is inefficient and inaccurate in assessing sorption isotherms by estimating chemical potential from a given number of fluid molecules.^45,46 The hybrid GCMC and MD method (GCMC-MD) neglects the pore connectivity and assumes there is no pore-blocking effect upon sorption.³⁵ While considering the diffusive mass transfer, grand canonical molecular dynamics (GCMD) may not allow the structure flexibility of adsorbent upon sorption without destroying its topological network.⁴⁷

In this work, we develop a microscopic modeling setup that unifies fluid exchange in control regions with GCMC technique and fluid diffusion as well as adsorbent structure flexibility with MD technique (Figure 1A, described in detail in the method details section of STAR Methods). It not only can take the diffusive mass transfer into account but also preserves the topological network of kerogen while allowing its structural flexibility. Also, this hybrid method can effectively differentiate various fluid states upon sorption. Using this modeling, we successfully reproduce the experimentally observed hysteresis.^32,33 Our findings are expected to significantly enhance understanding of the hysteresis phenomenon in amorphous materials, which, as well as the proposed microscopic modeling, is widely applicable to other soft nanoporous materials including natural materials such as wood, plants, and bamboo and synthetic materials such as polymer, foam, and organic membrane.

Figure 1.

Microscopic simulation framework

Microscopic setups of CH₄/CO₂ sorption and desorption in kerogen matrix by the Dual control volume grand canonical molecular dynamics (DCV-GCMD) method (A) and the GCMC-MD method (B). Molecular configuration of kerogen matrix upon CO₂ sorption at 213 atm and 338 K. The yellow dashed lines segregate the diffusion region from the two control volume regions. The yellow arrows denote the transfer of fluids between the diffusion and control regions. The green dashed line represents the raw kerogen matrix before sorption by the GCMC-MD method.

Results and discussion

Sorption hysteresis

The proposed modeling setup for sorption hysteresis in flexible kerogen by the dual control volume grand canonical molecular dynamics (DCV-GCMD) method represents more closely the physical process, whose simulated results are compared with the experimental data³³ in Figure 2. The simulated sorption/desorption isotherms follow a trend identical to the experimental results. The CH₄ isotherm shows an increasing trend similar to the Langmuir type I curve, while the CO₂ isotherm involves a sorption maximum after which the sorption demonstrates a decreasing trend. The interesting phenomenon for decreasing sorption at high pressure has been explained as resulting from the dominance of gas bulk density over its absolute adsorption density in our previous work.¹⁸ The observed deviation between simulations and experiments can be well attributed to the discrepancy in kerogen type⁴³ and maturity,⁴⁴ as well as the limitation in pore sizes of our kerogen matrix.⁷

Figure 2.

Hysteresis of excess sorption

Excess sorption and desorption isotherms in the kerogen matrix for CH₄ (A) and CO₂ (B). The corrected experimental data for total organic carbon in the 333.15 K sample originate from Zhao et al.³³

In addition to sorption isotherms, the simulations display noticeable hysteric behavior comparable to the experimental counterparts, which further validates the simulated results. The hysteresis loops cover the whole pressure range for both CH₄ and CO₂, confirming the observed hysteresis is not caused by capillary condensation. For a typical hysteresis loop from capillary condensation, the desorption branch overlaps the sorption branch at low pressure. The absence of capillary hysteresis can be associated with the fact that the reservoir temperature is much higher than the capillary pseudo-critical temperature for supercritical CH₄/CO₂ in the confined kerogen matrix with the pore size smaller than 1 nm.³⁵

Insufficient equilibrium time is reported as one of the possible origins for observed sorption hysteresis.³⁶ To guide the equilibrium, the total uptake, as well as the temperature, is monitored during the sorption/desorption simulation (see Figure S1). The desorption process takes much longer than the sorption process as expected, and the total uptake converges after about 6 ns of simulation time for both the sorption and desorption processes. This inspection supports the hypothesis that the observed hysteresis in Figure 2 corresponds to an equilibrium state, which is independent of the simulation time.

The diffusion region possesses a box length of around 45 × 45 × 30 Å along the x, y, and z directions, respectively, which is larger than the reported critical size of 25 Å that can reproduce the major structural and thermodynamic features of kerogen.^48,49 In our previous work,^43,44 we have validated that this kind of kerogen size can obtain reasonable physical density, thermodynamic property, and gas sorption comparable to experimental data. To examine the possible effect of diffusion region size, as well as pore size, we extend the diffusion region to be 40 Å along the z direction, which contains some larger pores (see Figure S2). Specifically, the diffusion region is extended after the first sorption/desorption run, and a second hysteresis scan simulation is then performed on the extended diffusion region, where the sorption is reversed at a lower pressure compared with the first hysteresis scan. In Figure 2B, the hysteresis loop over the extended diffusion region is comparable to the original results, which is indicative of a negligible effect of diffusion region size and pore size. Also, the hysteresis loops based on the two different runs resemble each other, demonstrating that the simulation is reproducible and the structure change of the flexible kerogen matrix upon sorption/desorption is reversible, which is accordant with the previous experimental observations.³² The reversible structure change of the kerogen matrix is further confirmed as analogous results are observed when reversing the sorption branch at an intermediate pressure. The documented experimental data^32,33 exhibit a consistent reversible structure of kerogen as our simulations, indicating that chemical interaction can be excluded from the origins of sorption hysteresis since it can lead to irreversible structural changes.

To examine the role of diffusive mass transfer, we have simulated the sorption/desorption isotherms by the DCV-GCMD and GCMC-MD methods. Also, the two methods are performed in both the frozen and flexible kerogen matrices to test the effect of structural deformation. For the GCMC-MD method, the effect of deformation magnitude of kerogen structure is studied by manipulating the stress condition on the kerogen matrix, wherein the zero confining stress corresponds to the maximum deformation and the zero effective stress leads to a smaller deformation. The sorption/desorption isotherms from various simulations are presented in Figure 2. The GCMC-MD simulations on both frozen and flexible kerogen structures fail to capture the hysteresis behavior, and the deformation magnitude of kerogen structure shows no effect on hysteresis for this method. The findings suggest that the sorption thermodynamics alone and its coupling with structure deformation are not sufficient to account for the CH₄/CO₂ sorption hysteresis. The absence of sorption hysteresis for the GCMC-MD simulations guides us to turn to the DCV-GCMD method involving diffusive mass transfer, which represents more closely the physical process in the sorption experiment. The sorption hysteresis comparable to experimental results is observed for the DCV-GCMD simulation in flexible kerogen structure, revealing that diffusive mass transfer is a necessary ingredient for sorption hysteresis. However, the sorption/desorption isotherms in frozen structure by the DCV-GCMD simulation are found to be hysteresis free. This observation demonstrates that the coupling between diffusive mass transfer and kerogen structure deformation is required to achieve sorption hysteresis.

Fluid states

Fluid states in realistic kerogen are divided into free, adsorbed, and dissolved states. In our microscopic model (Figure 1A), the free state mainly exists in the control region, the adsorbed state aggregates on the external surface of the kerogen matrix, and the dissolved state is confined in the diffusion region. The total uptake in the flexible diffusion region presents similar hysteresis behavior (Figure 3) as the excess sorption in Figure 2. The adsorbed and dissolved states are quantified from the total uptake to analyze their contributions to the hysteric behavior. The adsorption and dissolution isotherms in the flexible diffusion region upon sorption and desorption are presented in Figure 4. For both CH₄ and CO₂, the adsorption is not hysteric, while the dissolution exhibits a pronounced hysteresis loop. The observed sorption hysteresis is governed by fluid dissolution, and fluid adsorption exerts a negligible effect given its reversibility. Figure S3 shows the hysteresis scan of adsorbed and dissolved states in the frozen diffusion region. In addition to the adsorbed state, the dissolved state becomes hysteresis free when kerogen deformation is restricted. Consequently, the underlying microscopic mechanisms for CH₄/CO₂ sorption hysteresis in kerogen nanopores rely on the coupling of fluid diffusion, fluid dissolution, and kerogen deformation.

Figure 3.

Hysteresis of total uptake

Hysteresis scans of total uptake in kerogen diffusion region at 338 K for CH₄ (A) and CO₂ (B).

Figure 4.

Contribution of occurrence state to hysteresis

CH₄/CO₂ adsorption (A) and dissolution (B) in flexible kerogen matrix during the sorption and desorption paths at 338 K by the DCV-GCMD method.

Nanostructure deformation

The porous network of kerogen constantly changes along with gas sorption due to kerogen flexibility. Figures 5A and 5B show kerogen porosity as a function of pressure and gas loading in the dissolved region. For the DCV-GCMD simulation, the kerogen porosity changing with pressure displays similar hysteric behavior as the isotherms of total uptake in Figure 3. However, the hysteresis loops become insignificant as the porosity is plotted as a function of gas loading. The porosity converges into an approximately linear correlation with gas loading, confirming the kerogen structure is reversible, which also indicates that the expanded kerogen porosity is governed by the gas amount dissolved in the diffusion region. This finding is further supported by the hysteresis-free kerogen porosity in the GCMC-MD simulation, which is accordant with the overlapping isotherms in Figure 3. The GCMC-MD simulation shows a resembling correlation (kerogen porosity vs. gas loading) with the DCV-GCMD simulation, suggesting the correlation is determined by inherent kerogen properties like crosslinked structures and aromatic units.

Figure 5.

Coupling of diffusion-dissolution-deformation during CH₄/CO₂ sorption/desorption at 338 K

(A and B) Accessible porosity in kerogen diffusion region probed by the dissolved fluid as functions of pressure (A) and dissolved fluid loading (B).

(C) Distribution of accessible pores (colored in purple) in kerogen diffusion region for CO₂ sorption/desorption at 45 atm and CH₄ at 72 atm.

(D) Schematic representation of the diffusion-dissolution-deformation coupling mechanism for CH₄/CO₂ sorption hysteresis.

With all these findings into consideration, we are confident that the observed CH₄/CO₂ sorption hysteresis is caused by the coupling of fluid diffusion, fluid dissolution, and kerogen deformation. This conclusion guides us straightforward to the question, what are the underlying microscopic mechanisms for the coupling protocol to induce the hysteresis? At this point, we are interested if there are some topological and molecular reasons for the hysteresis. The amorphous kerogen matrix possesses a highly irregular topology, in which diffusive mass transfer follows a process close to creeping. The creeping process leads to a constant change of porous network, which results in different pore size distributions (PSDs) at the same pressure upon sorption and desorption (see Figure S4). Figure 5C shows the pore distributions in the kerogen matrix upon sorption/desorption. Compared with the sorption path, more isolated pores are observed in the desorption path, corresponding to worse connectivity. These isolated pores are formed as the elastic response of the kerogen structure reduces or even closes some narrow throats connected to the expanded pores during the creeping process.³⁶ The response is reasonably stronger upon desorption since the kerogen matrix has more expanded pores (Figure 5C). Consequently, the worse-connected porous network upon desorption could make significant contributions to the hysteresis.

Closely associated with the changing porous structure is the restricted expulsion of aggregated dissolved molecules in expanded pores upon desorption. Figure S5 shows that the dissolved molecules are dispersed in the diffusion region upon sorption, while some dissolved molecules are aggregated in the expanded pores upon desorption. The aggregated molecules can block the narrow throats connected to the expanded pores, and the passage of such narrow throats is expected to be activated at lower pressure. This hypothesis can be interpreted by the thermodynamics of aggregated molecules, specifically, the solvation pressure. The solvation pressure of aggregated molecules in expanded pores upon desorption can be strongly negative,³⁶ causing the contraction of the narrow passage, which makes it passable only at lower external pressure. By contrast, the solvation pressure in smaller pores upon sorption can be less negative or even positive,^48,50 leading to less contraction or elastic expansion, which allows for easier passage. Therefore, the restricted expulsion of aggregated dissolved molecules in expanded pores upon desorption can also contribute to the observed hysteresis.

Figure 5D schematically summarizes the underlying microscopic mechanisms for hysteresis. The free molecules in large pores and the adsorbed molecules on the external surface of the kerogen matrix are reversible. The dissolved molecules diffuse into the interior of the kerogen matrix following a creeping process, which increases the porosity and creates some expanded pores. The stronger elastic response related to the expanded pores can induce a worse-connected porous network upon desorption, producing some isolated molecules. Also, some aggregated molecules can block the narrow throats connected to the expanded pores, inducing the restricted expulsion of these molecules.

Energy origin

The coupling of fluid dissolution, fluid diffusion, and kerogen deformation is controlled by the nonbonded interaction between kerogen and fluids; the energy change upon sorption/desorption is studied to provide a better understanding of the hysteresis. Figure 6 shows the interaction energy plotted as a function of pressure and the specific interaction energy changing with sorption amount. The interaction energy exhibits noticeable hysteresis loops with pressure, analogous to the trend in Figure 3. Nevertheless, the hysteresis disappears as the specific interaction energy is correlated with the sorption amount. The observation indicates that the hysteresis in Figure 6A is due to the extra gas molecules in the desorption branch. The overlapping specific interaction energy suggests the hysteresis may not be attributed to the different gas locations related to kerogen surface chemistry in the sorption and desorption paths. The magnitude of specific interaction energy shows a slightly decreasing trend with increasing sorption amount, which implies gas molecules preferentially occupy the higher energy sites upon sorption and are prone to be stripped from the lower energy sites upon desorption. Compared with the electrostatic interaction, the van der Waals (VDW) interaction is observed to play a dominant role in the nonbonded energy between fluids and kerogen.

Figure 6.

Interaction energy between flexible kerogen and fluids upon sorption/desorption at 338 K by the DCV-GCMD method

(A) Kerogen-fluid interaction energy as a function of pressure.

(B) Kerogen-fluid-specific interaction energy (defined as interaction energy divided by fluid sorption) as a function of fluid sorption.

Based on the findings above, several practical implications relevant to the context of this study can be drawn: during the late stage of shale gas depletion, the gas recovery rate is generally low. Our study reveals that this is closely related to the deteriorated connectivity of the kerogen pore network and sorption hysteresis. This low recovery phenomenon may be mitigated through CO₂ injection, primarily via two mechanisms: (1) reopening clogged throats to restore pore network connectivity and (2) displacing shale gas trapped in isolated pores, thereby significantly alleviating sorption hysteresis.

Conclusions

The microscopic mechanisms for sorption hysteresis of CH₄ and CO₂ in shale kerogen are clarified by molecular simulations with proposed modeling setups. The experimentally observed hysteresis is reasonably reproduced by our devised DCV-GCMD simulations, which eliminate the size and pore effect and allow for sufficient equilibrium time. The kerogen structure is reversible upon sorption/desorption, and the hysteresis shows good reproducibility. The capillary condensation, chemical interaction, and insufficient equilibrium could be excluded from the hysteresis origins. Sorption thermodynamic alone and even its coupling with structural deformation could not induce the hysteresis. The free and adsorbed molecules are reversible upon sorption/desorption. Evidence is provided that the observed hysteresis is governed by the coupling of fluid diffusion, fluid dissolution, and kerogen deformation. The underlying mechanisms for the coupling protocol can be attributed to the worse-connected porous network and the restricted expulsion of aggregated dissolved molecules upon desorption. The energy analysis reveals that the hysteresis is not linked to the different gas locations due to kerogen surface heterogeneity. Gas molecules preferentially occupy the higher energy sites upon sorption and are prone to be stripped from the lower energy sites upon desorption. The nonbonded interaction between gas and kerogen is dominated by the VDW interaction rather than the electrostatic interaction.

Limitations of the study

This study, while elucidating the microscopic mechanism of gas sorption hysteresis in kerogen, has certain limitations. First, the newly revealed mechanism remains qualitative, and its quantitative impact on shale gas recovery has yet to be evaluated. Future work can incorporate key parameters into multi-scale simulations or integrate them with productivity prediction models to enable quantitative assessment. Second, the current research focuses solely on single-component static adsorption and fails to capture the dynamic competitive adsorption and displacement effects during actual CO₂ injection. Subsequent studies can develop physical models that incorporate injection processes to investigate the interplay between competitive gas adsorption and pore evolution.

Resource availability

Lead contact

Requests for further information and resources should be directed to and will be fulfilled by the lead contact, Liang Huang (huangliang@cdut.edu.cn).

Materials availability

This study did not generate new unique reagents.

Data and code availability

• •All data reported in this paper will be shared by the lead contact upon request.
• •This paper does not report original code.
• •Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Acknowledgments

This work was supported by the National Key R&D Program of China (grant no. 2023YFF0614100) and the National Natural Science Foundation of China (grant no. 52204031, 52574030). Computer time was provided by the computing clusters of the College of Chemistry at the University of California, Berkeley. We appreciate the support of the China Scholarship Council (CSC no. 201806440095). We would like to thank Prof. Clayton J. Radke of the University of California, Berkeley and Prof. Abbas Firoozabadi, Dr. Tianhao Wu, Dr. Stéphane Tesson, and Prof. Armando Gama Goicochea of the Reservoir Engineering Research Institute for discussion.

Author contributions

Investigation, data curation, formal analysis, validation, conceptualization, funding acquisition, supervision, writing – original draft preparation, L.H.; funding acquisition, supervision, writing – review & editing, H.Z.; data curation, visualization, writing – review & editing, Q.C. and Z.X.; validation, writing – review & editing, Q.Y., X.F., and B.T.; writing – review & editing, Z.Y., Z.Q., and S.A.

Declaration of interests

L.H. has a patent related to this work.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Deposited data
Density data under bulk conditions	NIST Chemistry WebBook	RRID:SCR_026225
Software and algorithms
Molecular dynamics codes	LAMMPS	RRID:SCR_015240
Molecular visualization program	VMD	RRID:SCR_001820
Pore characterization program	CSIRO PorosityPlus	https://doi.org/10.1088/2515-7639/aada5f

Method details

Modeling setup

A microscopic setup incorporating the GCMC and MD techniques is developed to reproduce more closely the physical process of gas sorption in kerogen, similar to the process that happened in the sorption experiment. This approach is inspired by the Dual Control Volume Grand Canonical Molecular Dynamics (DCV-GCMD) method used in fluid flow simulation.⁵¹ This framework innovatively incorporates dynamic fluid transport processes governed by molecular diffusion into sorption hysteresis study, which has posed significant challenges for conventional methodologies employed in previous research. A diffusion region made of kerogen matrix is sandwiched by two control volume regions, together constituting the microscopic model (see Figure 1A). The two control regions are served as fluid sources and sinks, in which fluid insertion, deletion, and random walk are allowed to maintain the given chemical potential by the GCMC algorithm. Fluid transfer in the diffusion region is governed by the MD principle. Exchange of fluid molecules between the control regions and the diffusion region is realized either by random walk in GCMC simulation or via thermodynamic walk in MD simulation. Asymmetric design is adopted for the two control regions. The left control region contains no adsorbent, corresponding to an external bulk reservoir, while the right control region is created as a heterogeneous cave enclosed by kerogen matrix and quartz, which is aimed to mimic the realistic interplay between fluid and inorganic-organic nanocomposite at reservoir conditions. These simulation settings effectively reproduce the diffusion transport process of CO₂ within the micro- and nanoscale spaces of shale under actual conditions. The diffusion region is targeted for the study of gas sorption hysteresis, and the kerogen matrix within it can be either flexible or fixed. To prevent the flexible kerogen matrix from being destroyed, the adjacent kerogen matrix in the right control region is always kept fixed, which can be interpreted as the consolidation effect in realistic kerogen. Recently, a nailing technique has been proposed to achieve a similar purpose by fixing a small part of kerogen atoms.²² The graphene and quartz walls define the boundaries of the two control regions, which are also fixed during the whole simulation. In addition to the DCV-GCMD algorithm performed on the generated microscopic setup, the reported GCMC-MD algorithm³⁵ integrating GCMC and MD techniques is implemented on the molecular model of bulk kerogen matrix (see Figure 1B). This hybrid strategy can probe gas sorption behavior using GCMC simulation and allow for the coupling deformation of the kerogen matrix using MD simulation simultaneously. Details of this approach can be found in the work of Chen et al.³⁵ We adopt this method to analyze the role of sorption thermodynamic and kerogen deformation on sorption hysteresis.

Simulation details

To construct the microscopic model for the DCV-GCMD simulation, a loose kerogen matrix is first built by randomly placing 15 reported type I-A units (C₂₅₁H₃₈₅O₁₃N₇S₃)⁵² into a large cuboid box (45×45×400 Å along the x, y, and z-direction). A suite of dummy particles with different LJ diameters (30 Å for 3 particle and 20 Å for 4 particles) are then inserted into the loose kerogen matrix. The box dimension of the mixed system is extended along the z-direction to integrate the quartz and the graphene, forming the initial microscopic model. The initial system is thereafter condensed through a rigorous annealing procedure (see Table S1). In this procedure, the graphene is served as a piston, and the pressure in the NPT ensemble (constant number of molecules, pressure, and temperature) is converted to the external force on the graphene along the z-direction. The box dimensions are fixed along the x and y-direction. Subsequently, the graphene is moved to generate the left control region, while the dummy particles are removed to generate the right control region. The condensed kerogen matrix is further relaxed in the NVT ensemble (constant number of molecules, box volume, and temperature) for 1000 ps to eliminate the stress concentration and create the final microscopic model for gas sorption. In parallel, the microscopic model of the kerogen matrix for the GCMC-MD simulation is condensed from 9 reported type I-A units by using our previous annealing process.^43,44

The generated two microscopic models are utilized for gas sorption simulation by the DCV-GCMD method and the GCMC-MD method, respectively. Gas sorption is conducted in the grand canonical ensemble for both of the two methods, wherein gas in the bulk kerogen matrix has the same chemical potential and temperature with an imaginary external bulk reservoir for the GCMC-MD simulation, while gas in the diffusion region possesses the equivalent chemical potential and temperature with the two control regions at equilibrium state for the DCV-GCMD simulation. Chemical potential is required as an input parameter in the sorption simulation. In this study, we perform additional GCMC-MD simulation for gas in bulk phase to obtain the correlation between pressure and chemical potential (see Figure S6A). Sorption/desorption isotherms are required to study the hysteresis loops. The sorption branch is yielded by performing simulation at a sequence of increasing chemical potential, while the desorption branch is yielded by recovering the chemical potential starting from the last point in the sorption branch. Information regarding the relevant simulation schemes for this work can be found in Table S2.

The simulations are conducted using the Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS).⁵³ The atomic types and point charges of kerogen are parameterized with the CVFF force field,⁵⁴ which has been employed to describe kerogen interaction with gas molecules.¹⁸ CH₄ molecule is represented with the TraPPE-UA model,⁵⁵ while CO₂ molecule is treated by the TraPPE-EH model,⁵⁶ as these force fields have been extensively validated in prior studies and reliably reproduce the physical and chemical properties of CH₄ and CO₂. Nonbonded interactions between atomic pairs are calculated by the Lennard-Jones (LJ) 12-6 function and the unscreened Coulomb potential. Lorentz-Berthelot combining rules⁵⁷ are adopted for unlike atoms. The cutoff distance for short-range pairwise interaction is 14 Å, and the long-range electrostatic interaction is described by the Ewald summation.^58,59 Periodic boundary conditions are used in the three directions of space. The temperature at 338 K is controlled by the Nosé-Hoover thermostat,⁶⁰ and the timestep is set as 0.1 fs. The sorption simulation at each chemical potential has a total of 4000 cycles, each including 2500 GCMC exchanges after every 10000 MD steps. The simulation details are validated by the comparable gas densities from the simulation with the PR-EOS predictions and NIST data (see Figures S6B and S6C). During the simulation, the total uptake and temperature are monitored to guide the equilibrium (see Figure S1).

Computational analysis

Fluid uptake, fluid states, porous properties, and energy terms are computed based on post-processing of the model trajectories. Total uptake is directly obtained by counting gas molecules in the diffusion region for the DCV-GCMD simulation or the bulk kerogen model for the GCMC-MD simulation. In the DCV-GCMD modeling setup, total uptake is divided into the dissolved state in the diffusion region and the adsorbed state on the interface between the left region and the diffusion region. The adsorbed state is defined as the portion of the density profile in the z-direction of the left control region where the density exceeds that of the free state. To be comparable with experimental data, total uptake is also converted into excess sorption by subtracting the corresponding free state at the same chemical potential. Pore structures including porosity and PSD are characterized using the Metropolis Monte Carlo integration method based on geometry determination.⁶¹ The CSIRO PorosityPlus code⁶² is adopted to calculate the porous properties using probe diameters corresponding to the studied gas molecules. The interaction energy is obtained by subtracting the energy of isolated components from the total energy of the mixtures.

Quantification and statistical analysis

Origin software was used for statistical analysis. The computational results are averaged among 10 structure configurations. All statistical were expressed as mean ± standard deviation (SD), and were shown in figures.

Published: October 29, 2025