## Subcontinuum mass transport of condensed hydrocarbons in nanoporous media

#### Kerstin Falk^{1}, Benoit Coasne^{1}, Roland Pellenq^{1}, Franz-Josef Ulm^{1} & Lyderic Bocquet^{1,w}

## Abstract

Although hydrocarbon production from unconventional reservoirs, the so-called shale gas, has exploded recently, reliable predictions of resource availability and extraction are missing because conventional tools fail to account for their ultra-low permeability and complexity. Here, we use molecular simulation and statistical mechanics to show that continuum description—Darcy’s law—fails to predict transport in shales nanoporous matrix (kerogen). The non-Darcy behaviour arises from strong adsorption in kerogen and the breakdown of hydrodynamics at the nanoscale, which contradict the assumption of viscous ﬂow. Despite this complexity, all permeances collapse on a master curve with an unexpected dependence on alkane length. We rationalize this non-hydrodynamic behaviour using a molecular description capturing the scaling of permeance with alkane length and density. These results, which stress the need for a change of paradigm from classical descriptions to nanoﬂuidic transport, have implications for shale gas but more generally for transport in nanoporous media.

## Introduction

Over the last decade, natural gas recovery from shales has increased worldwide, particularly in the United States, where production rates are skyrocketing—nowadays about 40% of the natural gas produced in the United States, as compared with 1% in 2000 (refs ^{1,2}). Predictions foresee this transformation to continue with part of the attention shifted to shale oil. However, the reliability of these predictions is highly disputed^{3,4} because of large uncertainties over the availability of this resource and large concerns about its environmental impact^{5,6}.

From a scientiﬁc perspective, shale gas and oil are trapped in a complex network of small pores, in particular in organic inclusions (kerogen) with sub-nanometre pore space^{7}. A key characteristic of these unconventional reservoirs is their ultra-low permeability8. Quantitatively, ﬂow rate predictions are classically based on Darcy’s law

stating that the volumetric ﬂuid ﬂux through a porous material depends linearly on the pressure gradient, the inverse of the ﬂuid viscosity η and a material-speciﬁc permeability k. Typically, the permeability scales as the square of the pore diameter and ismeasured in Darcy (1D≈0.987 x10^{-12} m^{2}). Unconventional reservoirs exhibit permeabilities of the order of 10^{-9} D, typically six orders of magnitude smaller than conventional reservoirs8, and in direct line with the nanoporous structures of kerogen^{7}. Considering that kerogen is the hydrocarbon source, which produces the gas and oil through its decomposition, the slow and complex hydrocarbon migration from kerogen to the cracks surface is the rate-limiting step^{9–11}. Such ultra-low permeability raises concerns on the applicability of the Darcy framework itself to account for mass transport in the nanoporous kerogen. Although attempts have been made to palliate for the breakdown of Darcy approach by including slippage in gas ﬂow, via, for example, the Klinkenberg effect^{12,13}, such empirical corrections cannot capture the complex adsorption and transport behaviour of hydrocarbon in ultra-conﬁning porous materials. Such effects must manifest themselves through a complex interplay between apparent viscosity and wettability, as evidenced in recent experiments on nanoconﬁned water^{14}.

^{9}^{10}^{2}

As far as the fundamental question of ﬂuid transport in nanoporous materials is concerned, one expects two major reasons for the breakdown of the Darcy framework. First, strong adsorption effects occurring in nanopores are expected to induce large changes in the phase behaviour of the conﬁned hydrocarbons^{15,16}. The density of the alkane phase inside the nanoporous material is usually much larger than its bulk counterpart and conﬁned hydrocarbons are expected to behave as a condensed phase, at odd with the simple gas picture. This has potentially dramatic consequences for their transport properties^{17–19}.

Second, research in the ﬁeld of ‘nanoﬂuidics’ has demonstrated the breakdown of hydrodynamics at the nanoscale^{20–22}; new phenomena such as slippage, interfacial transport and non-viscous effects appear as the ‘molecular granularity’ of the ﬂuid becomes non-negligible.

Overall, hydrocarbon transport in the multiscale and disordered nanoporosity of kerogen remains essentially not understood. Keeping in mind that large parts of the total amount of hydrocarbons is trapped in this nanoporosity, and that the overall permeability of the formation will be limited by the lowest permeability in the ﬂuid path, there is a strong need for a reliable theoretical framework of hydrocarbon transport in nanoporous matrix, with the ultimate goal of obtaining more reliable predictions, towards a more efﬁcient and environmentally safe exploitation technology.

Here, we present an in-depth theoretical study of n-alkane transport in a kerogen-like nanoporous matrix, which aims at proposing such a new theoretical framework. By relying on statistical mechanics molecular simulations that capture the interplay between adsorption and transport as well as the breakdown of hydrodynamics at the nanoscale, our approach does not require assuming any ﬂow type (Darcy, diffusive, Knudsen and so on). We ﬁrst show that the continuum description—the so-called Darcy’s law—dramatically fails to describe transport within a molecular model of nanoporous kerogen. Such a failure of the conventional description is shown to be due to the non-viscous nature of the ﬂow in such complex media, which arises from strong alkane adsorption. Nevertheless, despite the intrinsic complexity of such heterogeneous, disordered media, all permeances are shown to follow an unexpected yet simple scaling with the alkane length. To account for the scaling of permeance with alkane length and ﬂuid density, we propose a molecular description in which transport arises from a combination of slip-like friction of the hydrocarbons with the matrix and a free volume term. This model provides an analytical expression for the permeance, which allows to rationalize hydrocarbon transport in kerogen and quantitatively describe the permeance for all alkanes, at all densities.

## Results

Alkane transport in kerogen. Figure 1a,b shows the nanoporous structure used for this study; a disordered porous carbon, obtained using an atom-scale reconstruction technique, which can be seen as a reasonable molecular model of kerogen as it captures its main features (pore size, density, chemical composition including sp2/sp3 hybridization ratio, morphological disorder)^{23–26}. The pore size distribution of the numerical sample considered here spans from a few Å to ~15 Å, which is fully consistent with the pore sizes probed by N_{2} and CO_{2} adsorption in kerogen (see Supplementary Fig. 1 for a comparison with available experimental pore size distributions).

We investigated both hydrocarbon adsorption and transport in this molecular model of kerogen using conﬁgurational biased grand-canonical Monte Carlo and molecular dynamics simulations. Details about the models and simulations can be found in the Supplementary Discussion and Methods; see also Supplementary Table 1. This will serve as the basis of a theoretical scaling model of transport, based on the analysis of the ﬂuctuations of microscopic variables via the ﬂuctuation dissipation theorem (FDT)^{27}. Such a bottom-up approach will allow us to assess ﬂuid transport in ultra-low permeable materials on the relevant microscopic scale (Fig. 1).

**Figure 1 | Hydrocarbons in kerogen-like nanoporous carbon under reservoir conditions.** (a) System setup: n-alkanes adsorbed in a porous carbon matrix (volume (5 nm)^{3}); (b) zoom on one dodecane molecule (red) with its neighbours and the surrounding carbon structure; (c) adsorption isotherms of methane (black), propane (blue), hexane (green), nonane (yellow) and dodecane (red), normalized by the maximum density ρ1 reached at high pressures; the mass density ρ1 increases slightly with the alkane length (see Supplementary Methods and Supplementary Fig. 2). Because of the small pore sizes (~1 nm), the system is dominated by ﬂuid/solid interfaces, and the ﬂuid is in a supercritical phase, that is, no gas–liquid phase transition occurs. Inset: bulk phase diagrams for comparison.

Adsorption and transport of linear alkanes—methane, propane, hexane, nonane and dodecane—in the kerogen-like nanoporous carbon^{28} shown in Fig. 1 were investigated under temperature and pressure relevant to shale reservoir conditions6 (T = 423 K and P≤100 MPa). Here, we present the key results from an extensive investigation of adsorption, diffusion and steady-state ﬂow under constant pressure gradients. As shown in Fig. 2a, the mean ﬂuid ﬂow velocity q in the matrix depends linearly on the pressure gradient dzP for all considered n-alkanes and static pressures,

vz(l) is the velocity of molecule l, lA{1;N}). We emphasize that this linear relation is in no way imposed, but is a result of the simulations. In other words, no nonlinear effects occur. Note that we checked that the values of the permeance K are in full agreement with equilibrium calculation based on Green-Kubo relationship, see Supplementary Fig. 5a. This demonstrates that the linear relationship obtained here pertains to the small pressure drops relevant to experimental conditions. However, the proportionality factor K—called permeance to make a clear distinction from the permeability k~Kxη usually deﬁned by Darcy’s law (equation (1))—depends on the ﬂuid type and the thermodynamic conditions as shown in Fig. 2b.

**Figure 2 |Alkane transport in kerogen-like nanoporous carbons. ** Flow of different n-alkanes in nanoporous carbon under an external driving force -gradient(P): methane (black), propane (blue), hexane (green), nonane (yellow) and dodecane (red). (a) Linear response of the mean ﬂow velocity to the pressure gradient (T = 423 K, P=25 MPa, dashed lines: linear ﬁts); (b) permeance K =-q/gradient(P) as a function of the thermodynamic equilibrium pressure P. Values of the permeance K are in full agreement with equilibrium calculation based on Green–Kubo relationship, see Supplementary Fig. 5a. This demonstrates that the linear relationship obtained here pertains to small pressure drops relevant to experimental conditions.

### Non-Darcy behavior and transport scaling law

When using Darcy’s law, it is implicitly assumed that the permeability k is an intrinsic material property, that is, k~Kxη is a constant depending only on the geometry of the porous matrix. Figure 3a shows that this expectation dramatically fails for hydrocarbon transport in kerogen as k is found to depend on both the ﬂuid type and adsorbed amount. A ﬁrst reason for this failure of the classical porous-media-ﬂow description can be found in the adsorption behaviour. As seen from the form of the adsorption isotherms in Fig. 1c, owing to the severe conﬁnement in small nanopores such as in kerogen, the conﬁned alkanes are in a state that drastically differs from their bulk counterpart at the same pressure and temperature^{15,16}. In particular, comparison with the bulk phase shows that longer alkanes are in a condensed liquid like phase under conﬁnement while they are in a gaseous phase in bulk. As a result, the use of the bulk viscosity in this case is clearly inappropriate to calculate ﬂow properties in the nanopores. In an attempt to extend Darcy’s law to hydrocarbon transport in nanoporous media, we compared its predictions against the data density of the conﬁned phase (Supplementary Fig. 3). As shown in the inset in Fig.3a, Darcy’s law with such corrected viscosities also fails to describe the permeabilities observed in the molecular simulations.

We emphasize that such a pure dynamical effect cannot be accounted for by the so-called ‘Darken factor’, which describes the thermodynamic effect of adsorption on transport by correcting local density gradients using local adsorption isotherms^{29}.

To further assess the magnitude of the hydrodynamic break-down, many insights are provided by the molecular dynamics. An interesting probe of the dynamical processes is the transverse momentum ﬂuctuations, deﬁned in Fourier space:

in Fig. 3 when using the bulk viscosity of the alkanes at the (with x and z two perpendicular directions). In a viscous ﬂuid, transverse momentum relaxes via momentum diffusion and its correlation should exhibit a ‘universal’ exponential decay at small k and long times^{27}:

is the kinematic viscosity). When conﬁned inside a solid matrix, a viscous relaxation of the form exp[-(γ_{0} + k^{2}ν)t] may be expected, with γ_{0} steming from the Darcy friction of the liquid with the solid matrix. In strong contrast, we ﬁnd a very different behaviour for the transverse momentum ﬂuctuation of the conﬁned alkanes, with a correlation in the form of a double exponential [j_{z}(k,t)j_{z}(-k,0)]_{equ}= Aexp(-a_{k}t) -B exp(-b_{k}t), and a complex dependence of the decay coefﬁcients ak and bk on the wave vector k (Supplementary Fig. 4). These features demonstrate the violation of the hydrodynamic relaxation for all explored k scales.

It suggests that non-local effects and memory effects as described in generalized hydrodynamics with Mori-Zwanzig memory functions may occur^{30}. This result shows unambiguously that alkane transport in disordered nanoporous materials such as kerogen cannot be accounted for, at any length scale explored, by a hydrodynamic description.

**Figure 3 | Breakdown of Darcy law and permeance master curve for alkane transport.** (a) Permeability ηxK versus loading Γ (Γ= ρ=ρn^{∞}), showing the breakdown of the hydrodynamic prediction for the permeance: methane (black), propane (blue), hexane (green), nonane (yellow) and dodecane (red). The viscosity is that of the bulk hydrocarbon at the corresponding pressure and temperature. Inset: Same plot with the bulk viscosity replaced by the bulk viscosity calculated at the relevant pore density, ρpores. For comparison, the dashed lines give the permeability of a cylindrical pore with diameter equal to the mean size of the matrix pore-size distribution. (b) Permeance master curve: Kx (n x n_{0}) (with n_{0} = 2) versus loading (same symbols as in a). This demonstrates that K~1/n with n the alkane length. The dashed line is a guide to the eye.

### Scaling law and nanoﬂuidic transport

The failure of the hydrodynamic approach under extreme conﬁnement therefore calls for alternative frameworks of alkane transport in kerogen. A lead is suggested in Fig. 3b where it is shown that, in spite of this complexity, permeances K for all alkanes can be collapsed onto a single master curve as a function of loading complexity, permeances K for all alkanes can be collapsed onto a single master curve as a function of loading Γ=ρn (P)=ρ^{∞}n (ratio of the alkane density to its value at very large pressure, where the adsorbed amount reaches a plateau. The maximum density ρn^{∞} , which was obtained from a Langmuir ﬁt of the adsorption isotherms shown in Supplementary Fig. 2, slightly depends on the alkane length n. The permeance K is found to scale as the inverse of the alkane length (number of carbon atoms n):

where f(Γ) is a simple function of the loading Γ. More speciﬁcally, we ﬁnd that K(Γ)= f(Γ)/(n+n_{0}) with n_{0}≈2 for all alkanes provides an excellent rescaling.

With the aim to propose a molecular model of alkane transport in disordered nanoporous materials such as kerogen, we make use of the intimate links between dissipation and ﬂuctuation of microscopic quantities, as described by the FDT^{27}. In this ﬂuctuation of the total momentum via a Green–Kubo equation:

where D_{0} is the collective diffusion coefﬁcient, N the number of alkane molecules, V the volume of the matrix, respectively, and q_{0}=N^{-1}∑_{l}v_{z}^{(l)}(t) the ﬂuctuating centre-of-mass velocity of the ﬂuid with respect to the frozen matrix. As expected from the FDT described in equation (5), the collective diffusivity D_{0} for the different conﬁned alkanes—computed using equilibrium molecular dynamics (MD) of the q-autocorrelation function—is in full agreement with the permeances K estimated using non- equilibrium MD, in which the ﬂow is induced by a pressure gradient (Supplementary Fig. 5a).

This results further conﬁrms that hydrocarbon transport in kerogen is in the linear regime over the entire range of pressure gradients considered. Owing to its collective nature, D0 differs from the molecular self-diffusivity D_{s } by cross-correlation terms (v_{z}^{(i)}tv_{z}^{(j)}(0) with i≠j) of the form

In all studied systems, the difference between D_{0} and D_{s} was found to be small in most conditions. This is highlighted in Fig. 4a showing that D_{0}≈D_{s}, despite some differences for the shortest alkanes. Consequently, we can relate the permeance K to the mobility of single molecules as

which captures the main behaviour of the permeance K (see the inset of Fig. 4a).

**Figure 4 | Mass transport and diffusion: towards a free volume theory. **(a) Comparison of the self (D_{s}) and collective (D_{0}) diffusion. Inset: Scaling of the permeance K with the self-diffusion. (b) Rescaled diffusion coefﬁcient Dsxn, with the alkane length n, versus the free volume fraction V_{free} /V_{0}. The latter is calculated independently for a given adsorbed amount G, see Supplementary Information. The dashed line is the prediction of the free volume theory, Ds xn proportional to V0=Vfree, see text. Inset: bare data for the self-diffusion coefﬁcient for the various alkanes. The colour code is the same as Fig. 1: methane (black), propane (blue), hexane (green), nonane (yellow) and dodecane (red).

To proceed further, one needs to provide a molecular description of the self-diffusion of the dense alkane phase in the kerogen matrix. First, we compared the scaling of the self diffusion coefﬁcient D_{s} with the chain length n for bulk and framework, the permeance K is expressed in terms of the conﬁned alkanes. For the bulk ﬂuid, we ﬁnd that the diffusion of a linear alkane molecule is well described by the Stokes–Einstein relation with slip boundary conditions D_{sbulk}= k_{B}T/(4pZR_{0}) for a particle with an effective diameter close to its longitudinal cross-section 2R_{0} ≈σ_{CH2}; see Supplementary Fig. 6, where R_{0}~D_{s}xη is shown to be independent of the alkane density and length (under typical shale reservoir conditions T=423 K and P=25 MPa, the self-diffusivity and viscosity in the bulk liquid scale roughly as D_{s}~n 0.7 and Z~n0.7, respectively, and the n-dependence of the self-diffusion coefﬁcient and the viscosity compensate each other).

The fact that the hydrodynamic molecular sizes of alkanes are independent of their length n is in agreement with experimental measurements^{31}. However, we emphasize that the origin of this behaviour is far from trivial. It can be actually accounted for by the slippage of the continuum alkane ﬂuid on an individual alkane molecule along its length, as illustrated in Fig. 5a^{32}. This would require further investigation of the chain dynamics using, for example, recently developed diffusion maps^{33}.

Coming back to the molecular diffusion of an alkane chain in the amorphous carbon matrix, we ﬁnd a very different picture, as sketched in Fig. 5b. First, as shown in Fig. 4b, we ﬁnd that, like the permeance, D_{s} can be rescaled as the inverse of the alkane length, D_{s}~1/n. Furthermore, the rescaled diffusion nxD_{s }is found to be a generic function of the free volume accessible to the alkane molecules (Fig. 4b). Qualitatively, this scaling behaviour can be attributed to two effects. First, the strong molecular interaction of the alkanes with the carbon matrix leads to a large ﬂuid-wall friction (in contrast to the low liquid–liquid friction in the bulk). This suggests a description in the spirit of the Rouse model for polymer diffusion^{34}. Consider a single alkane molecule in the matrix. Each monomer i of the alkane experiences a slip-like friction force from the matrix, Each monomer i of the alkane experiences a slip-like friction force from the matrix, f_{v}=-ξ_{0}v_{i}, on top of internal forces (ξ_{0} is the friction coefﬁcient for a single monomer). Therefore, the total external force acting on an alkane molecule scales as FT= ξ_{0}∑_{i}v_{i}= nxξ_{0}v, with its centre of mass velocity v= n-1∑_{i}v_{i} and its mobility μ_{T}= (nxη_{0})^{ -1}. Accordingly, the self-diffusion coefﬁcient should scale as

where the superscript (0) stands for a single molecule.

Now, one should take into account density effects by considering that a molecule is able to diffuse provided it ﬁnds a free cavity around it. This effect can be quantiﬁed by a free volume approach^{35,36}, which relies on the probability to ﬁnd a void space larger than a critical volume v_{crit} next to the diffusing molecule. One can therefore write

where v_{crit} is the minimum size of a void that allows the molecule to move into it, whereas N is the number of alkane molecules and V_{free} the accessible free volume. Estimating vcrit as the size of one alkane molecule valkane and using V_{free}=V_{0}-Nx_{valkane} (with free the bare free volume in the matrix), equation (9) can be recast as:

with D_{s} proportional to 1/n. The free volume fraction V_{free}=V_{free} is calculated independently for a given adsorbed amount Γ, see Supplementary Information. As shown in Fig. 4b, equation (10) describes very well the dependence of D_{s} on the alkane chain length and free volume, and therefore conﬁrms the validity of our description (the calculation of the free volume in the simulations is described in the Supplementary Methods).

**Figure 5 | Diffusion mechanisms for bulk alkanes and alkanes conﬁned in the porous carbon matrix. **Left: In bulk, molecular diffusion is well described by the hydrodynamic Stokes–Einstein relation D_{s}=μk_{B}T=k_{B}/(4πZη_{0}) (with slip boundary conditions). The effective particle diameter 2R_{0} is consistent with σ_{CH2} —independently of the alkane length—because the diffusive motion is mostly in longitudinal direction. Right: In contrast, in the nanopores, movement of alkane molecules is dominated by friction on the carbon matrix, corrected for the free volume accessible to the molecule (green sphere). The total friction force is a sum of the forces between the individual monomers with the pore wall—therefore scaling linearly with the alkane length. v stands for the molecule velocity.

Altogether, these results show that the bulk and conﬁned diffusion of alkanes follow very different mechanisms. As illustrated in Fig. 5, the molecular self-diffusion in bulk alkanes involves molecular motion that is mainly a translational movement in the longitudinal direction, subject to very little friction with the surrounding ﬂuid molecules. In contrast, the diffusion of the conﬁned alkanes stems merely from the friction of the molecule against the matrix, corrected for the free volume accessible to the molecule under motion.

Coming back to alkane transport, one therefore predicts that the permeance K behaves as

with V_{free}/V_{0free }the free volume fraction and we used the relationship N_{valkane}/V_{free}=1-V_{free} /V_{0 } α a is a numerical constant. Figure 6 shows that the prediction in equation (11) is in excellent agreement with the MD results for the permeance K for all alkanes at various densities. We allow for a shift n_{0} in the alkane length dependence, as our arguments provide merely the generic scaling behaviour. All the parameters needed in the derivation of equation (10) can be determined from simple experiments. V_{0free }and V_{free} at a given adsorbed amount N can be estimated from the adsorption isotherm. α and D_{s}(0), which describe the dynamics of the conﬁned alkanes, can be assessed from diffusion experiments such as Quasi-Elastic Neutron Scattering.

**Figure 6 | Mass transport of alkanes in nanoporous matrix.** Rescaled permeance Kx(n+n_{0}) for different n-alkanes as a function of the free volume fraction V_{free}=V_{0} (with n_{0}=2). The colour code is the same as Fig. 1: methane (black), propane (blue), hexane (green), nonane (yellow) and dodecane (red). The dashed line is the prediction in equation (11), written here as K_{x}(n+n_{0})= K_{0}/(1-V_{free}/V_{0})xexp(-αV^{0}_{free}/V_{free}), with K_{0}=2,38x10^{-15}m^{2}Pa^{-1}s^{-1}, α=0,23, obtained from a best ﬁt. Inset: Permeance K versus the loading Inset: Permeance K versus the loading Γ=ρ/ρn_{∞} for various alkanes. The dashed lines correspond to the previous prediction in terms of the loading, using the relationship V_{free} =V^{0}_{1}-βΓ, with β=0.6, , see Supplementary Fig. 7.

Finally, the permeance as described by equation (11) can be recast to describe the dependency on the loading Γ, which is an experimentally accessible quantity. One expects a linear relation- ship between the free volume fraction and the loading Γ, that is, V_{free}/ V_{0free}=1-βΓ. This is conﬁrmed by our simulations, see Supplementary Fig. 7, providing the value β= 0.60. Accordingly, one obtains the following prediction for the mass transport permeance K in terms of alkane length n and adsorbed amount Γ:

As shown in the inset of Fig. 6, this single expression provides a very good description of the permeance K for all alkanes, at all densities and does conﬁrm the relevance of the underlying microscopic description.

This expression takes into account both the strong adsorption of the alkane in the microporous kerogen, via the dependence on loading Γ, as well as the speciﬁc nanoﬂuidic transport of this dense alkane phase in the disordered matrix. It provides an explicit prediction for the permeance K, which quantitatively captures hydrocarbon transport in the nanoporous kerogen matrix, in spite of the breakdown of Darcy’s law. Furthermore, our prediction allows rationalizing the 1/n rescaling of the permeance, as found in Fig. 3b. Altogether our prediction, equation (12) therefore establishes a framework able to describe quantitatively hydrocarbon transport in ultra-low permeable materials.

## Discussion

We demonstrated that hydrodynamics and, hence, Darcy’s law fail to describe hydrocarbon transport in nanoporous media because of strong molecular adsorption leading to non-viscous ﬂow. As an alternative to the continuum Darcy’s description, we propose a microscopic description for the permeance K derived from the theoretical framework of statistical mechanics, which culminates in a quantitative prediction of the permeance as a function of alkane length n and adsorbed amount Γ, an experimentally accessible quantity^{37}. This relation offers a valuable tool for the ﬂuid-speciﬁc prediction of hydrocarbon transport properties in ultra-low permeable media such as kerogen. Once integrated into a bottom-up model of ﬂuid transport in multiscale porous materials (using, for example, well-established homogeneization techniques), this can be the starting implications, the presented results about the exotic transport in porous materials also raises new challenging fundamental questions. In particular, the cross-over between hydrodynamic to non-hydrodynamic transport in disordered nanoporous media calls for a shift of paradigm as conventional approaches—based on percolation, porosity and tortuosity concepts—^{38,39} do rely on continuum descriptions. The present work offers a well-grounded molecular basis to adress these questions.

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#### Acknowledgements

This work has been carried out within the framework of the french ‘Investissements d’Avenir’ program (projects ICoME2 Labex, ANR-11-LABX-0053 and A*MIDEX projects ANR-11-IDEX-0001-02). K.F., B.C., F.-J.U. and R.P. also acknowledge funding from Royal Dutch Shell and Schlumberger through the MIT X-Shale Hub.

Authors: Kerstin Falk^{1}, Benoit Coasne^{1}, Roland Pellenq^{1}, Franz-Josef Ulm^{1} & Lyderic Bocquet^{1,w}

^{1}Department of Civil and Environmental Engineering and MultiScale Material Science for Energy and Environment UMI 3466 CNRS-MIT, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA. ^{w}Present address: LPS, UMR CNRS 8550, Ecole Normale Supe´rieure, 24 rue Lhomond, 75005 Paris, France. Correspondence and requests for materials should be addressed to B.C. (email: [email protected]) or to L.B. (email: [email protected]).

#### Author contributions

L.B., B.C., R.P. and F.-J.U. designed the work. K.F. performed the simulations. K.F., L.B. and B.C. analysed the data and wrote the manuscript.

#### Additional information

Competing ﬁnancial interests: The authors declare no competing ﬁnancial interests.

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How to cite this article: Falk, K. et al. Subcontinuum mass transport of condensed hydrocarbons in nanoporous media. Nat. Commun. 6:6949 doi: 10.1038/ncomms7949 (2015).

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