Abstract
Although hydrocarbon production from unconventional reservoirs, the socalled shale gas, has exploded recently, reliable predictions of resource availability and extraction are missing because conventional tools fail to account for their ultralow 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 nonDarcy 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 nonhydrodynamic 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 subnanometre pore space^{7}. A key characteristic of these unconventional reservoirs is their ultralow permeability^{8}. 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 materialspecific permeability k. Typically, the permeability scales as the square of the pore diameter and is measured in Darcy (1D≃0.987 × 10^{−12} m^{2}). Unconventional reservoirs exhibit permeabilities of the order of 10^{−9} D, typically six orders of magnitude smaller than conventional reservoirs^{8}, 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 ratelimiting step^{9,10,11}. Such ultralow 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 ultraconfining 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}.
At a more global scale, some recent works aimed at explaining the specific longtime production rates of shales beyond traditional reservoir modelling. Monteiro et al.^{9} suggested a hydrodynamic model of gas flow in nanoporous media by introducing a pressure gradientdependent permeability of kerogen. They predict a power law for the decline of the production rate, which is compatible with earlylife data for several major US shale plays. In the same line, Patzek et al.^{10} proposed a simplified model of shale reservoirs, which are made up of parallel equidistant fracture planes. Assuming Darcylike gas flow bewteen these fracture planes, they predict a crossover from an earlytime algebraic decay to an exponential decline at long time. Although such macroscale modellings capture some specificities of the gas recovery, in particular the longtime decay, they, however, point to the lack of knowledge on smallscale behaviours, and in particular on the role of the adsorption and desorption processes, as well as nonDarcy multiphase flow. Further research is needed to improve the—so far limited—scientific understanding^{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^{17,18,19}. Second, research in the field of ‘nanofluidics’ has demonstrated the breakdown of hydrodynamics at the nanoscale^{20,21,22}; new phenomena such as slippage, interfacial transport and nonviscous effects appear as the ‘molecular granularity’ of the fluid becomes nonnegligible. 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 indepth theoretical study of nalkane transport in a kerogenlike 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 socalled 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 nonviscous 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 sliplike 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 atomscale 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 sp^{2}/sp^{3} hybridization ratio, morphological disorder)^{23,24,25,26}. 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 grandcanonical 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 bottomup approach will allow us to assess fluid transport in ultralow permeable materials on the relevant microscopic scale (Fig. 1).
Adsorption and transport of linear alkanes—methane, propane, hexane, nonane and dodecane—in the kerogenlike nanoporous carbon^{28} shown in Fig. 1 were investigated under temperature and pressure relevant to shale reservoir conditions (T=423 K and P≤100 MPa). Here, we present the key results from an extensive investigation of adsorption, diffusion and steadystate 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 ∂_{z}P for all considered nalkanes and static pressures,
( is the velocity of molecule l, l∈{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 GreenKubo 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∼K × η usually defined by Darcy’s law (equation (1))—depends on the fluid type and the thermodynamic conditions as shown in Fig. 2b.
NonDarcy behaviour 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∼K × η 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 porousmediaflow 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 liquidlike 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 in Fig. 3 when using the bulk viscosity of the alkanes at the 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 socalled ‘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 breakdown, many insights are provided by the molecular dynamics. An interesting probe of the dynamical processes is the transverse momentum fluctuations, defined in Fourier space:
(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}: 〈j_{z}(k,t)j_{z}(−k,0)〉_{equ}∼exp(−k^{2}vt) (ν=η/ρ 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}=A exp(−a_{k}t)−B exp(−b_{k}t), and a complex dependence of the decay coefficients a_{k} and b_{k} 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 nonlocal effects and memory effects as described in generalized hydrodynamics with MoriZwanzig 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.
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 (ratio of the alkane density to its value at very large pressure, where the adsorbed amount reaches a plateau. The maximum density , 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 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 framework, the permeance K is expressed in terms of the 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 the fluctuating centreofmass 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 qautocorrelation function—is in full agreement with the permeances K estimated using nonequilibrium 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, D_{0} differs from the molecular selfdiffusivity D_{s} by crosscorrelation terms ( 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).
To proceed further, one needs to provide a molecular description of the selfdiffusion of the dense alkane phase in the kerogen matrix. First, we compared the scaling of the selfdiffusion coefficient D_{s} with the chain length n for bulk and 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 for a particle with an effective diameter close to its longitudinal crosssection ; see Supplementary Fig. 6, where R_{0}∼D_{s} × η is shown to be independent of the alkane density and length (under typical shale reservoir conditions T=423 K and P=25 MPa, the selfdiffusivity and viscosity in the bulk liquid scale roughly as D_{s}∼n^{−0.7} and η∼n^{0.7}, respectively, and the ndependence of the selfdiffusion 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 n × D_{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 fluidwall 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 sliplike 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 F_{T}=ξ_{0}∑_{i}v_{i}=n × ξ_{0}v, with its centre of mass velocity v=n^{−1}∑_{i}v_{i} and its mobility μ_{T}=(n × ξ_{0})^{−1}. Accordingly, the selfdiffusion 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 v_{crit} as the size of one alkane molecule v_{alkane} and using (with the bare free volume in the matrix), equation (9) can be recast as:
with D_{s}^{(0)} ∝ 1/n. The free volume fraction 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).
Altogether, these results show that the bulk and confined diffusion of alkanes follow very different mechanisms. As illustrated in Fig. 5, the molecular selfdiffusion 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 the free volume fraction and we used the relationship ; α 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. 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 QuasiElastic Neutron Scattering.
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 relationship between the free volume fraction and the loading Γ, that is, . 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 ultralow 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 nonviscous 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 fluidspecific prediction of hydrocarbon transport properties in ultralow permeable media such as kerogen. Once integrated into a bottomup model of fluid transport in multiscale porous materials (using, for example, wellestablished homogeneization techniques), this can be the starting point for the development of a new generation of unconventional reservoir simulators. More generally, it proposed a useful framework for mass transport of dense fluids in nanoporous materials, which is pertinent to questions relevant to catalysis, energy storage and so on. Beyond the immediate practical implications, the presented results about the exotic transport in porous materials also raises new challenging fundamental questions. In particular, the crossover between hydrodynamic to nonhydrodynamic 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 wellgrounded molecular basis to adress these questions.
Additional information
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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Acknowledgements
This work has been carried out within the framework of the french ‘Investissements d’Avenir’ program (projects ICoME2 Labex, ANR11LABX0053 and A*MIDEX projects ANR11IDEX000102). K.F., B.C., F.J.U. and R.P. also acknowledge funding from Royal Dutch Shell and Schlumberger through the MIT XShale Hub.
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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.
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Supplementary Figures 17, Supplementary Table 1, Supplementary Discussion, Supplementary Methods and Supplementary References (PDF 267 kb)
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Falk, K., Coasne, B., Pellenq, R. et al. Subcontinuum mass transport of condensed hydrocarbons in nanoporous media. Nat Commun 6, 6949 (2015). https://doi.org/10.1038/ncomms7949
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