Breakdown of Fast Mass Transport of Methane through Calcite

Article pubs.acs.org/JPCC

Breakdown of Fast Mass Transport of Methane through Calcite Nanopores Sen Wang,† Qihong Feng,*,† Farzam Javadpour,‡ and Yong-Biao Yang§ †

School of Petroleum Engineering, China University of Petroleum (East China), Qingdao 266580, China Bureau of Economic Geology, Jackson School of Geosciences, and §McKetta Department of Chemical Engineering, The University of Texas at Austin, Austin, Texas 78712, United States

‡

S Supporting Information *

ABSTRACT: Fast mass transport was identified for gas through nanoscale pores, especially those fabricated from carbon nanotubes and graphene sheets. Even in porous media such as sedimentary rock, it is commonly believed that there exists the well-known Klinkenberg effect due to gas slip. Here, we use molecular simulation to show that the flow enhancement of methane breaks down in calcite nanopores of shale reservoirs. The Klinkenberg effect fails to characterize methane transport through interparticle pores of calcites, and the molecules travel even slower than the prediction of Hagen−Poiseuille equation. The comparison of methane transport in graphene, quartz, and calcite nanopores suggests that this behavior arises from the strong attractive potential and the lack of atomically ultrasmooth surface in calcite, thus leading to the presence of particles sticking at the interface. The Navier−Stokes equation, coupled with a negative slip length and bulk viscosity, can provide a reasonable description of methane flow through calcite nanoslits having apertures greater than 2 nm. Moreover, it is evident that as the pore size increases, the confined methane transforms from a single-file chain to two symmetrical adsorbed layers (extending to ∼0.8 nm), above which a bulk fluid region is present in the central slit. Beyond the theoretical value, the insights gained from this study will advance the exploitation of shale resources and shed light on, more generally, mass transport in nanoporous media.

1. INTRODUCTION Calcite, the rhombohedral polymorph of CaCO3, is a particularly amazing mineral. As a favorable biomaterial, calcite forms solid phases in living beings, for example, teeth, bones, and shells, through the activity of organic compounds produced by organisms.1 Carbonate rocks constituted of calcite serve as hosts for more than 60% of the world petroleum reserves and most aquifers. The interaction of calcite with organic molecules is of broad interest in biological, environmental, and technical disciplines.1−4 Beyond the wide range of implications from enhanced oil recovery to contaminant remediation, such studies potentially contribute to producing biomimetic materials via biomineralization and lowering atmospheric CO2 concentrations through calcite precipitation. Pioneering efforts have primarily focused on the behaviors of organic compounds, for example, ethanol,1,4 carbohydrate,3 and aspartic acid,5 on the calcite−fluid interface. For example, the highly ordered structure of ethanol at the calcite surface, observed by using atomic force microscopy (AFM), X-ray reflectivity (XR), and molecular dynamics (MD), confirms the controlling effect of hydroxyl groups on the adsorption of polysaccharide.1,6,7 Most recently, research on the properties of hydrocarbons confined in calcite nanopores has gained increasing attention in the context of oil and gas production from shale formations. © XXXX American Chemical Society

Although previously regarded only as the source rock of fossil fuels and the exploitation was deemed impractical, shale has been of growing interest from both industry and academia, because the in situ extremely low permeability of shale, ∼6 orders of magnitude lower than the conventional reservoirs caused by its nanoporous structure, can be significantly improved by hydraulic fracturing.8−11 The past decade has seen an explosive growth in shale gas production, which reaches 40% of the total gas yield in the United States, as compared to 2% in 2000.12 Motivated by the large reserves and considerable production rate after fracturing, several countries, including Canada, Germany, China, etc., have started strong programs to develop shale resources.13 This boom has raised the need for the knowledge of hydrocarbon adsorption and transport in the natural composite material. Exploring the properties of hydrocarbons confined in shale nanopores is challenging, not only because of its inherent multiscale and disordered pore structure (typical pore size: 2− 100 nm), but also because the composition of shale matrix is very complex.14−16 X-ray diffraction (XRD) mineralogy analysis Received: June 1, 2016 Revised: June 11, 2016

A

DOI: 10.1021/acs.jpcc.6b05511 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

between methane and calcite substrates, gas flow enhancement, a common phenomenon at the nanoscale,31 breaks down in calcite nanopores at reservoir conditions. Continuum hydrodynamics applies at calcite pores of sizes down to 2.0 nm; however, a negative slip length needs to be incorporated. Distinct boundary conditions are essential for CH4 flowing through quartz and graphene pores. Our result indicates that nanoscale fluid transport is strongly dependent on solid materials. Current models for shale fail to predict mass transport in the nanoporous composite material, necessitating further studies on the accurate characterization of shale molecular structure. In addition to providing the fundamental science of fluid transport at the nanoscale, this study can not only improve the accurate prediction of shale permeability and its long-term production performance, but also offers a wide range of implications in the processes of environmental remediation, carbon dioxide sequestration, and gas separation through membranes.11,31,32

shows that shale consists of both organic matter (e.g., kerogen and bitumen) and inorganic substances with various proportions (Figure 1).16 Among these constituents, calcite

2. MODELS AND METHODS Calcite crystallizes in the trigonal system, R3c space group. We first constructed the crystallographic unit cell according to its lattice parameters (a = b = 4.988 Å, c = 17.061 Å, α = β = 90°, γ = 120°).33 The (101̅4) surface then was cleaved from the bulk calcite because it is the most thermodynamically stable crystal plane. Figure 2a shows the computational cell of the

Figure 1. Mineralogical composition of a few shale reservoirs.16

and silica play significant roles in the shale development because their brittleness is favorable for fracture propagation; this is one of the prerequisites for better transport capacity and higher production rate.17 However, most of the existing studies, either experimental or numerical, only concentrated on adsorption and transport of hydrocarbons (mainly methane) in organic media, that is, carbon nanotubes (CNTs),18−22 slitshaped graphitic pores,23−25 and kerogen-like porous carbons.11,26,27 Some researchers are working on the possible chemical bonds between organic phase and other minerals (silica and clay) to reconstruct realistic molecular structure of shale.17,28 Although much progress has been achieved in understanding hydrocarbon behaviors correlated with carbonaceous materials, insight into adsorption and flow of hydrocarbon confined within calcite nanopores is urgently required. A first principle study conducted by Rigo et al.29 revealed that the most energetically favorable locations for a hydrocarbon molecule adsorption on the calcite (101̅4) surface are calcium sites. Xue et al.30 briefly discussed the displacement of dodecane in calcite channels by N2 flooding and observed a slower flow than that in silica pores. Despite its immediate relevance to the applications, there is no thorough investigation on methane adsorption and transport through calcite nanopores. We present a MD study of static properties and pressuredriven flow behaviors of hydrocarbons encapsulated in interparticle pores of calcites, with sizes ranging from 2.0 to 10 nm. We focus on adsorption and dynamics of CH4 interacting with calcite (101̅4) surface, while the results for other alkanes, that is, C3H8 and n-C8H18, together with the properties of CH4 in contact with quartz (101̅0) surface and graphene, which are always employed to represent silica and kerogen in shales, are also reported for comparison. Through equilibrium molecular dynamics (EMD), we calculate the variations of density profile with respect to slit aperture, from which the layering structure of confined alkanes are observed. Then nonequilibrium molecular dynamics (NEMD) enable us to mimic alkane flow driven by a constant pressure gradient and determine velocity profile, shear viscosity, and boundary condition. We find that caused by the strong interactions

Figure 2. (a) Computational cell of methane confined in a 5.38 nm calcite nanopore. (b) Side view of the calcite (101̅4) surface. (c) Sketches of all considered alkanes in our simulations: CH4, C3H8, and n-C8H18. Color code for atoms: green, Ca; red, O; black, C; white, H.

orthorhombic calcite−CH4 system, in which the calcite nanopores in shale are ideally modeled as a slit composed of two calcite (1014̅ ) slabs. The lateral dimensions of each substrate are 32.38 × 29.94 Å2 with a thickness of 12.14 Å in the z direction, corresponding to 240 CaCO3 units (Figure 2b). For the hydrocarbons, three different kinds of alkanes were considered: CH4, C3H8, and n-C8H18 (Figure 2c). We vary the number of alkane molecules in each case to represent the high pressure condition in shale. The effective slit aperture, estimated from the definition of Gibbs dividing surface,34 spans from 2.0 to 10 nm, which is fully consistent with the nanopore sizes in shale.16 We used the potential model recently developed by Xiao et al.35 to describe the calcite substrate. In contrast to the force field of Pavese et al.,36 Xiao et al. treated all of the van der B


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Figure 3. (a) Mass density profiles of CH4 molecules in a 3.44 nm calcite nanopore (386 K, 39.5 MPa). The inset shows the number density distributions of carbon and hydrogen atoms. (b) Two-dimensional RDFs of carbon atoms in selected slabs of this slit. The locations of each slab with respect to the density profile are indicated by the gray stripes in the inset.

acceleration along the x direction was assigned to the carbon atoms of all of the alkane molecules.21,34 We tuned the magnitude of the external force to ensure a linear response.21 The temperature thermostat was coupled only to the velocity component in the y direction.34 Steady state was always reached in less than 2 ns. The simulation steps used to sample the transport properties varied from 16 to 60 ns, depending on the driving force and pore size.

Waals interactions with Lennard-Jones (LJ) potentials, which not only eases the derivation of cross-term interactions between calcite and other materials, but also improves the computational efficiency by replacing exponential Buckingham functions with faster LJ models. This force field has been employed to understand the adsorption of simple biomolecules on calcite surfaces,35,37 as well as the interactions between carbohydrate and calcite in brine.3 To be consistent with the convention applied in the force field development, we adopted the atomic parameters from the OPLS-AA force field38 to characterize the alkane molecules, and calculated the pairwise LJ interactions within a cutoff distance of 1.0 nm using the geometric-average mixing rules.35 Potential parameters are summarized in Table S1. Throughout our simulations, the solid substrates were restrained to their lattice constants.3 To ensure that this treatment does not introduce erroneous results, we performed simulations in which the vibrational dynamics of the calcite substrates is fully included, while maintaining other conditions unchanged. Negligible deviations are observed in density and velocity profiles of confined methane obtained by using rigid and flexible frameworks of calcite (Figure S1). A periodic boundary condition was imposed only in the x and y directions to prevent the interactions between the computational cell and its image.34 We used the particle−particle particle-mesh (PPPM) technique, which has been modified for 2D-slab by inserting a vacuum space beyond the solid, to calculate the long-range Coulomb interactions.39 All of the MD simulations were conducted using LAMMPS40 (integration time step: 1 fs). A Nosé−Hoover thermostat with a coupling constant of 0.1 ps was used to maintain a constant temperature. For each case, a bulk fluid region consisting of an arbitrary number of alkane molecules was defined between the solid substrates. To control the pressure, we adjusted the basal spacing by treating the upper substrate as a piston while keeping the lower one frozen.21 If the equilibrium distance was achieved, the pore size was measured and the locations of both slabs were fixed in the subsequent simulations. Each system then was equilibrated for 2.0 ns in the NVT ensemble via EMD simulations, and the trajectories were collected in the following 6.0 ns production run, from which the static properties of alkanes confined in calcite nanopores were explored. Finally, we used nonequilibrium molecular dynamics to study the transport behaviors of hydrocarbons though slit-shaped calcite nanopores. To mimic the pressure-driven flow in shale, a constant

3. RESULTS AND DISCUSSION 3.1. Molecular Structure. Previous studies suggested that methane confined in CNTs,22,41 graphene slits,24 and silica pores42 exhibits a layered structure. We observed similar behavior in calcite nanopores. The mass density profile of methane as well as the independent number densities of carbon and hydrogen atoms are present in Figure 3a for a 3.44 nm slit (T = 386 K, P = 39.5 MPa). The symmetric density profile exhibits intense fluctuations near the solid walls and forms two dense layers: one directly absorbed onto the calcite surface and the other one, located closer to the central plane, having a lower density. The thickness of each adsorbed layer (∼3.98 Å), determined by measuring the spacing between two successive crests, is close to the molecular diameter of methane (3.8 Å). Farther away from the substrate, caused by the decreasing attractive potentials, the spatial distributions of CH4 molecules become disordered, leading to a constant density of 0.177 g/ cm3. The consistency between this value and the bulk density at the same condition, 0.182 g/cm3,43 suggests that the confined methane is bulk-like in the central slit. The peak intensities of the hydrogen atoms approximate to 4 times those of the carbon atoms, because the molar ratio of carbon to hydrogen atoms is 1:4. We calculated the two-dimensional radial distribution functions (RDFs) of carbon atoms by dividing the pore space into several slabs, each of which is parallel to the calcite substrate (thickness: 1 Å):34 g (r ) =

N (r , r + Δr ) n2D2πr Δr

(1)

Here, N(r,r+Δr) is the number of carbon atoms in this slab with a distance [r,r+Δr] to the center particle, and n2D denotes the two-dimensional number density. Figure 3b shows the RDFs of carbon atoms in different slabs of the 3.44 nm slit. Slabs “a” and “b” are centered at 1.59 and 5.57 Å from the C


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Figure 4. Density profiles and snapshots of methane in calcite slits with different apertures. The green slabs represent the calcite substrates. (a) d = 4.3 Å, a single-file chain; (b) d = 6.5 Å, one symmetrical adsorbed layers; (c) d = 11 Å, two primary layers in the near-wall region and a central layer; (d) d = 21.9 Å, a bulk fluid region forms in the central slit.

surface, corresponding to the two density maxima (Figure 3b, inset). Two additional slabs are randomly selected in the constant-density region. The RDFs exhibit short-range positional order of methane and tend to be 1 at longer distance. For comparison, the result for bulk methane under the same condition is superimposed. It is evident that the RDFs of the adsorbed layers are greater than that of the bulk value, which manifests the densification of methane molecules in the near-wall region due to the confinement effect. The identical RDFs of the other slabs and the bulk fluid suggest the presence of bulk methane at the slit center. To probe the effect of pore size on the variation of layering structure, we illustrate, in Figure 4, the density profiles of methane in calcite slits with apertures d ranging from 4.3 to 21.9 Å. The corresponding simulation snapshots are also included to show the adsorption structures. Because of the superimposed interaction potentials from the two solid surfaces, a single-file chain is present in a 4.3-Å calcite slit, reaching the highest peak density (Figure 4a). For a larger pore, the attractive potentials exerted by the two substrates begin to divide; thus, one symmetrical adsorbed layer with lower density peaks appears in the wall vicinity (Figure 4b). When the slab separation further increases to 11 Å, the calcite slit creates another space in the central region, which allows for the formation of another layer (Figure 4c). In comparison with the primary adsorbed layers that reside nearest to the walls, the peak of the central layer is much lower, manifesting that the adsorption sites at the calcite surfaces are favorably occupied by methane, then the pore center. Two secondary adsorbed layers, besides the primary peaks, exist in a slit having an aperture greater than ∼16 Å (Figure 4d). A region with density identical to that of bulk fluid forms in the central pore, because the interactions from the pore walls have been completely separated into two independent potential systems and the methane molecules near the slit center are beyond the range of the surface confinement. 3.2. Adsorption Isotherm. We use the total gas content and excess adsorption to characterize the storage amount of

CH4 in calcite nanopores. The total adsorption density is defined as the entire quantity of gas confined in the slit divided by the pore volume.22,24 As shown in Figure 5a, the total adsorption increases monotonically with the pressure and asymptotically converges to a saturation state. The bulk density computed from our simulation agrees well with the values reported by the National Institute of Standards and Technology (NIST).43 Because of the adsorption, the total content is greater than the bulk density at the same pressure, indicating that the calcite nanopore is capable of storing more methane than the bulk fluid. Subtracting the bulk density from the total amount provides the excess adsorption density attributed to the confinement.22−24 When the pressure becomes larger, the quantity of excess adsorption first increases up to a maximum value and then decreases (Figure 5a). During methane molecules quickly occupying the adsorption sites distributed on the calcite surface, the additional pressure exerted by the substrates renders the total adsorption greater than that of the bulk density, leading to a positive excess adsorption. As pressure continually increases, the available adsorption sites are gradually filled up, and the space remaining for accommodating adsorbed methane is insufficient, from which point the total adsorption rises slower than the bulk density, causing the reduction of excess adsorption.22,23 The nonmonotonic correlation between excess adsorption and pressure manifests the existence of an optimum pressure for maximum adsorption. A similar feature is also observed in the excess adsorption isotherms of methane confined in organic and silica pores fabricated by smooth graphene and fully hydroxylated quartz (101̅0) crystallographic plane (Figure 5b). These three types of minerals can represent the components of a typical shale sample. We obtained these results from independent MD simulations by using OPLS-AA and CLAYFF force fields, respectively. The molecular models and detailed potential parameters can be found in refs 44 and 45. At the same pressure, graphitic pores show a larger excess adsorption quantity than calcite pores, and the silica pores are the lowest, D


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nanopores.46−49 All of these processes will lead to a fast mass transport of methane, and their contributions depend on the Knudsen number (Kn), that is, the ratio of mean free path to the pore size.46−49 However, this conclusion was adopted from methane flow in graphitic nanopores and does not differentiate kerogen with inorganic minerals. Considering the different potentials between methane and these substrates, the flow mechanisms in these mineral nanopores should be distinct; therefore, the use of this supposition to characterize gas flow in shale and predict the long-term production might be incorrect. In this section, we will assess the validity of widely used gas transport models in calcite nanopores of shale. In Figure 6a, we show the steady-state velocity profile of methane driven by an external force of F = 25 × 10−4 kcal/ (mol·Å) in a 3.44 nm calcite nanoslit (T = 386 K, P = 39.5 MPa). In contrast to the interfacial slip reported for methane flow through CNT,50−52 graphene slit,53,54 sandstone,55,56 and porous alumina disc,57 the streaming velocity of methane at the calcite surface approximates to be zero; that is, there exists almost no slip. To analyze this phenomenon in detail, we first estimate the shear viscosity of methane confined in calcite nanopores. The parabolic velocity profile present in the slit center confirms the relevance of continuum hydrodynamics.34,58 Thus, according to the Navier−Stokes equation, the viscosity of confined methane can be obtained from the curvature: η=−

nF d2v dz 2

( )

(2)

where n is the number density of the particles driven by the external force F. We made a parabolic fit to the simulated velocities in the constant-density region. Within the accuracy of estimates, the resulting viscosity, 24.158 ± 0.224 μPa s (v = −0.2696 × 1020z2 + 72.29), is in good agreement with the experimental value of the bulk phase at the same condition, 24.178 μPa s,43 indicating that the fluid in the pore center is free from the confinement effect of the calcite surface. Figure 6a also implies that the result vfit can fairly characterize the whole velocity profile vMD, even though only the central portion is employed for the fit. Figure 6b compares the simulated velocity profile of methane in calcite nanopores with the predictions of two theoretical models: Karniadakis et al.’s formula59 and Hagen−Poiseuille equation. On the basis of a general slip boundary condition, Karniadakis et al. proposed a unified model that predicts the volumetric flux and velocity profile through simple geometries in the entire flow regime. For a slit having an aperture of d, the flow rate is

Figure 5. (a) Isotherms for the total and excess adsorption of methane in a 3.44 nm calcite slit and the bulk densities at 386 K. (b) Comparison of the excess adsorption isotherms between calcite, quartz, and graphene (d ≈ 3.44 nm). (c) Density profiles of methane confined in different mineral pores having the same aperture (T = 386 K, P ≈ 18.6 MPa).

indicating that the hydrocarbon adsorption capacity decreases in the order graphene, calcite, and quartz (Figure 5b). The comparison of methane density profiles across different types of slit-shaped mineral pores further confirms this conclusion. Figure 5c shows there is only a monolayer of methane adsorbed on the silica surface. However, two dense layers form at the graphene and calcite surface, and the peak densities of the adsorbed layers in a carbonaceous slit are much greater than that of calcite, which suggests that the attractive potential between calcite and methane is stronger than the silica pores, but weaker than the graphene pores. 3.3. Transport Behavior. We now turn to the pressuredriven flow behavior of methane through calcite nanopores. It is commonly believed that slip flow, Knudsen diffusion, surface diffusion, etc., are the fundamental mechanisms that should be taken into account to describe gas transport through shale

q=−

⎛ d 3 dp 6Kn ⎞⎟ (1 + αKn)⎜1 + ⎝ 12η dx 1 − bKn ⎠

(3)

where the general slip coefficient b is −1,59 and the parameter α varies as a function of Kn: α = α0

2 tan−1(α1Kn β ) π

(4)

In this equation, α0 is the constant asymptotic value 64/15π corresponding to the free-molecular regime (Kn → ∞). α1 and β are obtained from numerical and experimental data, and the resulting values are 4.0 and 0.4, respectively.59 The velocity profile normalized with the average value U̅ gives E


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Figure 6. (a) Velocity and density profiles of CH4 in a 3.44 nm calcite pore. vMD is obtained from NEMD simulation with a driving force equal to 25 × 10−4 kcal/(mol·Å). vfit is a parabolic fitting to the simulated velocities in the constant-density region. (b) Comparison of the simulated velocity profile with classical mathematical models for fluid flow in nanochannel. The purple dashed line (vKarniadakis) and magenta solid line (vPoiseuille) are the velocity profile predicted from Karniadakis et al.’s model and Hagen−Poiseuille equation, respectively.

U U* = = U̅

−

2

( dy )

1 6

( ) +( ) +

1 4

+

2 − σv σv

2 − σv σv

Kn 1 − bKn

Kn 1 − bKn

(5)

For methane, the tangential momentum accommodation coefficient σv is ∼1.04.60 Equations 3−5 constitute the fundamental description of Karniadakis et al.’s model, which has been widely used to determine gas permeability through shale systems.47,49 We calculate the mean free path of the confined molecules by61 λ=

16η 5ρ 2πRT

(6)

where R is the universal gas constant Rg (8314 J/(K·kmol)) divided by the molar mass in kg/kmol, that is, R = Rg/M, η is the viscosity, Pa·s, and ρ is the fluid density, kg/m3. Sometimes the factor 16/5 is replaced by π. The estimated Knudsen number, ∼0.1098, lies between the slip (0.01 < Kn < 0.1) and transition flow (0.1 < Kn < 10) regimes. We then use Karniadakis et al.’s model to predict the average streaming velocity and the distribution of local velocity. As indicated by Figure 6b, large derivations are observed between the simulated results and the theoretical model. The formula of Karniadakis et al. predicts a tremendous slip at the calcite−CH4 interface, whereas no obvious slip is present in the NEMD simulation, which suggests that such a model may fail to capture the nature of methane transport in calcite nanopores and overestimate the shale permeability. Also included in Figure 6b is the prediction of the Hagen−Poiseuille equation (slip length Ls = 0). In contrast to almost all of the transport phenomena reported for methane through nanopores and the well-known Klinkenberg effect that states that due to the gas slip the measured permeability of gas through a rock sample is greater than its absolute (liquid) permeability,55 methane flow in calcite nanopores behaves even slower than the no-slip Poiseuille equation. In Figure 7, we present the steady-state velocity profiles of methane through different mineral pores. The effective slit apertures of graphene, silica, and calcite nanopores are 3.24,

Figure 7. Comparison of velocity profiles for CH4 transport in graphene, quartz, and calcite nanopores. The scatter points represent the streaming velocities obtained from MD simulations, and the solid lines are parabolic fits to the simulation data vMD.

3.36, and 3.44 nm, respectively. Because of the perfectly ordered structure and the ultrasmooth atomic surface, fluids always exhibit a fast mass transport through CNTs and graphene pores, and the velocities exceed values computed from the continuum hydrodynamics by 1−3 orders of magnitude.21,32,54 Therefore, to highlight the effect of mineral types on the transport behavior of confined hydrocarbons, we conducted the simulations under different driving forces. Although the external field exerted on the methane molecules (F = 5 × 10−4 kcal/(mol·Å)) in a graphene pore is only 1/5 of that applied to the other systems, the streaming velocity in the carbonaceous slit far exceeds that in quartz and calcite nanopores, and the profile tends to a plug instead of the parabolic shape. This conclusion implies that gas transport through nanoscale pores constituted from different materials might not be characterized by a general theoretical model. We attribute the underlying mechanism for the distinct hydrodynamic behaviors to the inherent characteristics of fluid−solid interaction. Both the density profiles and the adsorption isotherms shown in Figure 5 indicate that the attractive potential between methane and these minerals decrease in the F


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Figure 8. (a) Velocity profiles of methane through calcite nanopores with different sizes. The same force, F = 25 × 10−4 kcal/(mol·Å), is applied in all cases. The scatter points indicate the streaming velocities obtained from MD simulations; the solid lines are the parabolic fits. (b) Effect of slit aperture on the estimated slip length Ls and viscosity ηMD. The dashed line stands for the viscosity of bulk fluid, ηbulk.43

(24.158 μPa·s) nanopores approximate to that of the bulk fluid (24.178 μPa·s). Falk et al. suggest that the friction coefficient, λ, of fluids on the interface serves as a mathematical description for the origin of slip:21

order graphene, calcite, and quartz. Therefore, in comparison with calcite, hydrocarbon is less strongly adsorbed by the silica surface and travels much faster through the pore. Although the adsorption capacity of methane on the graphene substrate is greater than the inorganic materials, the atomically smooth nature of the surface leads to an ultralow friction and a large slip.21 For the velocity profiles shown in Figure 7, the slip-modified Poiseuille equation can be utilized to characterize the flow behaviors:34,44,58 v (z ) = −

⎞ nF ⎛ 2 d 2 − dLs⎟ ⎜z − 2η ⎝ 4 ⎠

λ=

η Ls

(8)

Thus, we calculate the friction coefficients between methane and these minerals by using eq 8. The smaller λ for graphene pores (2.48 × 103 Ns/m3) explains the molecular origin of fast mass transport. Unfortunately, caused by the negative slip length on calcite surfaces, the friction coefficient is negative, −30.05 × 103 Ns/m3, which implies that its definition may not apply to the sticking boundary condition. The friction coefficient for quartz−methane systems, 16.13 × 103 Ns/m3, is 6.5 times greater than that measured for gas through graphene slits. Therefore, a parabolic velocity profile is present when there is methane flow in silica nanopores, in contrast to the plug flow expected for large slip. 3.4. Sensitivity Analysis. Figure 8a shows the variation of velocity profiles of confined methane as a function of slit aperture. In all cases, the pressure gradient (nF), that is, the product of number density and external field, was maintained constant. We suggest, in Figure 4, that if the pore size is greater than ∼1.6 nm, two symmetric monolayers appear nearest to the solid surfaces, and a bulk fluid region is present in the central slit. When we vary the pore size from 2.08 to 10.0 nm, the simulated velocity profiles are indeed parabolic, and slight deviations are observed in the near-wall region. We made parabolic fits to the velocity profiles and estimated the viscosities. The resulting values are 24.40, 24.16, 24.33, 24.68, and 24.67 μPa·s for the increasing pore sizes. As the slit apertures increases, methane molecules travel much faster; however, the streaming velocities are still lower than that predicted by the no-slip Poiseuille equation because some particles may stick at the surface and lead to a negative slip. We also observe that wider slits exhibit larger slip length as compared to the smaller pores; for example, the slip lengths for d = 2.08 and 10.0 nm are −0.83 and −0.64 nm, respectively (Figure 8b). This conclusion confirms that the Hagen− Poiseuille equation (Ls = 0) tends to be valid in a larger pore.

(7)

The slip length Ls, which quantifies the velocity jump at the interface, is defined as an adjustment of the slit aperture to the location where the fluid velocity vanishes.20,58 If Ls is set equal to zero, the Hagen−Poiseuille equation is recovered. Therefore, a second-order polynomial fit to the simulated velocity profile provides the information on slip length and viscosity of confined fluid. To avoid the uncertainty (a relative error up to 100%) caused by an unconstrained parabolic fit to a plug flow,58 we made a fit to the velocity profile of methane in graphene pores under the constraint of bulk viscosity, from which a slip length of ∼9.75 nm was estimated. For the silica and calcite nanopores, we conducted unconstrained parabolic fits to the velocity profiles and determine the slip length as well as the viscosity, because the velocity difference between the central slit and the interface is very large.44 Although both of the two velocity distributions exhibit a parabolic shape, the slip length for methane transport through quartz slits is positive (i.e., 1.51 nm), indicating an enhanced flow, which contradicts the negative slip (−0.804 nm) in calcite nanopores. The flow pattern of methane through calcite suggests that besides the no-slip and slip boundary conditions, there exists a third case, sometimes termed stick slip;62−64 that is, a few molecules adsorbed nearest to the solid boundaries are immobilized. The phenomenon does not always appear at the nanoscale, whereas experimental and numerical studies have confirmed the possibility and its presence depends on the fluid−solid attractive potential, the strength of external force, and the thermodynamic state.62−64 The entire viscosities of methane confined in quartz (24.35 μPa·s) and calcite G


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Figure 9. (a) Velocity profiles of methane driven by varying external forces. The symbols correspond to the values derived from NEMD simulations, and the solid lines are obtained from fitting scatters to a parabolic profile. (b) Effect of driving force on the estimated slip length Ls and viscosity ηMD. The dashed line stands for the viscosity of bulk fluid, ηbulk.43

Figure 10. Comparison of velocity (a) and density (b) profiles for CH4 (●), C3H8 (■), and n-C8H18 (▲) confined in calcite nanopores.

potentials between longer carbon chains and calcite substrate. The preferential adsorption of heavier hydrocarbons relative to light alkanes has been confirmed by both experimental and numerical studies.25,65 Therefore, the fluid−solid friction coefficient is larger for a longer hydrocarbon, thus leading to a smaller slip length.

The velocity profiles of methane subject to various external fields in a 3.44 nm calcite slit are shown in Figure 9a. As expected, with the increment of driving force, the streaming velocity increases while keeping a parabolic profile. The resulting viscosities, 24.35 ± 0.22 μPa·s, approximate to that of bulk phase. For the slip length, we found an increasing tendency with respect to the driving force (Figure 9b), which suggests that the sticking particles can be swept away by the strong force.62 We compare the steady-state velocity profiles of CH4, C3H8, and n-C8H18 through a calcite nanopore in Figure 10a (T = 386 K, P = 39.5 MPa). An external force of 25 × 10−4 kcal/(mol·Å) is exerted on all carbon atoms of the confined alkanes. The qualitative features of the profiles of C3H8 and n-C8H18 are akin to that shown for CH4. We calculate their viscosities from the curvature of velocities in the central slit. The results, 95.66 and 339.03 μPa·s, are consistent with those of bulk fluids. All of these hydrocarbons transport slower than the prediction of Hagen−Poiseuille equation because of the negative slip lengths. The computed Ls are −0.8682 and −0.959 nm for confined C3H8 and n-C8H18, respectively, which suggest the slip length decreases monotonically with increasing carbon chains. As elaborated in Figure 10b, the number of adsorbed layers and the density peaks of longer hydrocarbons are much greater than those of shorter alkanes, revealing the strong attractive

4. CONCLUSIONS To summarize, we demonstrated, using molecular dynamics, the breakdown of fast gas transport through calcite nanopores, and hence the Klinkenberg effect may fail to characterize mass transfer of methane in nanoporous materials constituted of calcite. Our results manifest that under pressure-driven flow, methane molecules confined in calcite nanopores manifest a parabolic velocity profile but transport even slower than that predicted by Hagen−Poiseuille equation, which contradicts with plug flow between graphene sheets and slight enhancement in quartz slits. The contrastive nature from flow through carbonaceous and silica slits arises from the atomically rough surface and the strong attractive interactions leading to particles sticking at the interface. The Navier−Stokes equation applies even in a slit with apertures down to 2 nm; however, a correction to the slip boundary condition, that is, a negative interfacial slip length of ∼ −1 nm, should be incorporated. We also reveal that calcite surfaces carry as many as two dense H


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symmetric monolayers of methane; therefore, the decaying behavior of the density profile, perturbed by the substrate, extends to ∼8 Å, above which a region with density identical to that of bulk phase appears in the slit center. With the increase of pore size, the molecular structure of confined methane changes from a single-file chain to the coexistence of a bulkfluid region and two adsorbed layers. Because of the distinct attractive potentials, adsorption capacity of methane decreases in the order graphene, calcite, and quartz. The slip length increased with the increment of slip aperture and driving force, but for longer hydrocarbons, this parameter decreases monotonically. We suggest that nanoscale flow patterns, that is, slip, no-slip, and sticking, strongly depend on the adsorbed layers−solid interactions and the nature of substrate surface. Current models for shale may fail to predict the transport of methane; mineral types should be accounted for in the construction of shale pore network and flow modeling. More generally, this study may also show implications for mass transport through nanoporous materials, for example, cell membranes, catalyst, micro/nano energy conversion machines, etc.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.6b05511. Force field parameters and the result comparison between rigid and flexible frameworks of calcite (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported partly by the National Program for Fundamental Research and Development of China (973 Program 2014CB239005), the Program for Changjiang Scholars and Innovative Research Team in University (IRT1294), the Outstanding Doctoral Dissertation Training Program of China University of Petroleum (LW140201A), the Graduate School Innovation Program of China University of Petroleum (YCX2014009), and the NanoGeosciences lab at the Bureau of Economic Geology, Jackson School of Geosciences, The University of Texas at Austin.

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Breakdown of Fast Mass Transport of Methane through Calcite

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