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Subcontinuum mass transport  of condensed hydrocarbons in nanoporous  media

Kerstin Falk1, Benoit Coasne1, Roland Pellenq1, Franz-Josef Ulm1  & Lyderic Bocquet1,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 disputed3,4 because of large uncertainties over the availability of this   resource  and   large  concerns  about   its  environmental impact5,6.

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  space7. 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 

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 m2).  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 kerogen7. 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 step9–11. 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 effect12,13, 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 water14.

 

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 gradient-dependent permeability of kerogen. They predict a power  law for  the  decline  of  the  production  rate,  which  is compatible with early-life 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 Darcy-like gas flow bewteen these fracture  planes,  they  predict  a  crossover from  an  early-time algebraic decay to an exponential decline at long time. Although such   macroscale  modellings   capture   some   specificities  of the  gas  recovery,  in  particular  the   long-time  decay,  they, however, point to the lack of knowledge on small-scale behaviours, and in particular on the role of the adsorption and desorption  processes, as  well as  non-Darcy  multiphase  flow. Further  research  is  needed  to  improve  the—so far  limited— scientific understanding2.

 

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  hydrocarbons15,16.  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 properties17–19. Second, research  in  the  field of ‘nanofluidics’

has demonstrated the breakdown of hydrodynamics at the nanoscale20–22;  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,  dis- ordered 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 dis- order)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 N2  and CO2  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).

(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 a

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  carbon28  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}).the  values of  the permeance K are in agreement with equilibrium calculation based on Green-Kubo

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.

Flow of different n-alkanes in nanoporous  carbon under an external driving force-gradient(P): methane  (black), propane (blue), hexane (green)

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 temperature15,16. 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 isotherms29.

 

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:

Subcontinuum mass transport  of condensed hydrocarbons in nanoporous media. Kerstin Falk, Benoit Coasne, Roland Pellenq, Franz-Josef Ulm 

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 times27:

at small k and long times. When confined inside a solid matrix, a viscous  relaxation of the form exp[-(γ0 + k2ν)t]  may be expected, with γ0

 the kinematic viscosity). When confined inside a solid matrix, a viscous  relaxation of the form exp[-(γ0 + k2ν)t]  may be expected, with γ0  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  [jz(k,t)jz(-k,0)]equ= Aexp(-akt) -B exp(-bkt), 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 occur30. 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.

a) Permeability ηxK versus loading Γ (Γ= ρ=ρn∞), showing the breakdown of the hydrodynamic prediction for the permeance: methane (black), propan

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 n0) (with n0 = 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 (ratioof 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+n0) with n0≈2 for all alkanes provides an excellen

where f(Γ) is a simple function of the loading Γ. More specifically, we find that K(Γ)= f(Γ)/(n+n0) with n0≈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 FDT27. In this fluctuation of the total momentum via a Green–Kubo equation:

where D0 is the collective diffusion coefficient, N the number of alkane molecules, V the volume of the matrix, respectively, and q0=N-1∑lvz(l)(t

where D0  is the collective diffusion coefficient, N the number of alkane molecules, V the volume of the matrix, respectively, and q0=N-1lvz(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 D0 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 Ds by cross-correlation terms  (vz(i)tvz(j)(0) with i≠j) of the form

Owing to its collective nature, D0 differs from the molecular self-diffusivity Ds by cross-correlation terms  (vz(i)tvz(j)(0) with i≠j) of the

In all studied systems, the difference between D0  and Ds  was found  to  be small in  most  conditions. This is highlighted in Fig. 4a showing that  D0≈Ds, 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).

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

(a) Comparison of the self (Ds) and collective (D0) diffusion. Inset: Scaling of the permeance K with the self-diffusion. (b) Rescaled diffusion

Figure 4 | Mass transport and diffusion: towards  a free volume theory. (a) Comparison of the self (Ds) and collective (D0) 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 Vfree /V0. 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 Ds 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 Dsbulk = kBT/(4pZR0) for a particle with an effective diameter close to its longitudinal cross-section 2R0 ≈σCH2; see Supplementary Fig. 6, where R0~Dsxη 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   Ds~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 measurements31. 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. 5a32. This would require further investigation of the  chain  dynamics  using,  for  example,  recently  developed diffusion maps33.

 

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, Ds  can be rescaled as the inverse of the alkane length, Ds~1/n. Furthermore,  the  rescaled diffusion nxDs 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 diffusion34. 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, fv =-ξ0vi, on top of internal forces (ξ0  is  the  friction  coefficient for  a  single monomer). Therefore, the total external force acting on an alkane molecule scales as FT= ξ0ivi= nxξ0v, with its centre of mass velocity v= n-1∑ivi  and its mobility μT= (nxη0) -1. Accordingly, the self-diffusion coefficient should scale as

Each monomer i of the alkane experiences a slip-like friction force from the matrix, fv =-ξ0vi, on top of internal forces (ξ0  is  the  friction

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 approach35,36, which relies on the probability to find a void space larger than a critical volume vcrit  next to the diffusing molecule. One can therefore write

This effect can be quantified by a free volume approach, which relies on the probability to find a void space larger than a critical volume vcrit

where vcrit  is the minimum size of a void that allows the molecule to move into it, whereas N is the number of alkane molecules and Vfree  the accessible free volume. Estimating vcrit as the size of one alkane molecule valkane  and using Vfree=V0-Nxvalkane  (with free  the bare free volume in the matrix), equation (9) can be recast as:

equation (9) can be recast as. with Ds proportional to 1/n. The free volume fraction Vfree=Vfree is calculatedindependently for a given adsorbed

with Ds proportional to 1/n. The free volume fraction Vfree =Vfree is calculatedindependently for a given adsorbed amount Γ, see Supplementary Information. As shown in Fig. 4b, equation (10) describes very well the dependence of Ds  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).

Left: In bulk, molecular diffusion is well described by the hydrodynamic Stokes–Einstein relation Ds=μkBT=kB/(4πZη0) (with slip boundary conditi

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 Ds=μkBT=kB/(4πZη0) (with slip boundary conditions). The effective particle diameter 2R0  is consistent  with σCH2 —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 Vfree/V0free the free volume fraction and we used the relationship Nvalkane/Vfree=1-Vfree /V0 α a is a numerical constant. 

with Vfree/V0free the free volume fraction and we used the relationship Nvalkane/Vfree=1-Vfree /V0 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 n0  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. V0free and Vfree  at a given adsorbed amount N can be estimated  from  the  adsorption  isotherm.  α  and Ds(0), which describe the dynamics of the confined alkanes, can be assessed from   diffusion  experiments  such  as  Quasi-Elastic  Neutron Scattering.

 Rescaled permeance  K x  (n+n0) for different n-alkanes as a function of the free volume fraction Vfree =V0  (with n0 = 2). The colour code is

Figure 6 | Mass transport of alkanes  in nanoporous matrix. Rescaled permeance  K x  (n+n0) for different n-alkanes as a function of the free volume fraction Vfree =V0  (with n0 = 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 Kx(n+ n0)= K0/(1-Vfree/V0)xexp(-αV0free/Vfree), with K0=2,38x10-15m2Pa-1s-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 Vfree =V01-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, Vfree/ V0free=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 an

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  quantity37.  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 pro- jects 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 Falk1, Benoit Coasne1, Roland Pellenq1, Franz-Josef Ulm1  & Lyderic Bocquet1,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: coasne@mit.edu) or to L.B. (email: lyderic.bocquet@ens.fr).

 

 

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

 

Supplementary  Information accompanies this paper at http://www.nature.com/ naturecommunications

 

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

 

Reprints  and permission  information is available online at http://npg.nature.com/

reprintsandpermissions/

 

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).

 

This  work  is  licensed under  a  Creative Commons  Attribution  4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

 stating that the volumetric fluid flux through a porous material depends linearly on the pressure gradient, the inverse of the fluid viscosity η 
(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 a
vz(l)  is the velocity of molecule l,lA{1;N}).the  values of  the permeance K are in agreement with equilibrium calculation based on Green-Kubo
Flow of different n-alkanes in nanoporous  carbon under an external driving force-gradient(P): methane  (black), propane (blue), hexane (green)
Subcontinuum mass transport  of condensed hydrocarbons in nanoporous media. Kerstin Falk, Benoit Coasne, Roland Pellenq, Franz-Josef Ulm 
at small k and long times. When confined inside a solid matrix, a viscous  relaxation of the form exp[-(γ0 + k2ν)t]  may be expected, with γ0
a) Permeability ηxK versus loading Γ (Γ= ρ=ρn∞), showing the breakdown of the hydrodynamic prediction for the permeance: methane (black), propan
where f(Γ) is a simple function of the loading Γ. More specifically, we find that K(Γ)= f(Γ)/(n+n0) with n0≈2 for all alkanes provides an excellen
where D0 is the collective diffusion coefficient, N the number of alkane molecules, V the volume of the matrix, respectively, and q0=N-1∑lvz(l)(t
Owing to its collective nature, D0 differs from the molecular self-diffusivity Ds by cross-correlation terms  (vz(i)tvz(j)(0) with i≠j) of the
which captures the main behaviour of the permeance K (see the inset of Fig. 4a).
(a) Comparison of the self (Ds) and collective (D0) diffusion. Inset: Scaling of the permeance K with the self-diffusion. (b) Rescaled diffusion
Each monomer i of the alkane experiences a slip-like friction force from the matrix, fv =-ξ0vi, on top of internal forces (ξ0  is  the  friction
This effect can be quantified by a free volume approach, which relies on the probability to find a void space larger than a critical volume vcrit
equation (9) can be recast as. with Ds proportional to 1/n. The free volume fraction Vfree=Vfree is calculatedindependently for a given adsorbed
Left: In bulk, molecular diffusion is well described by the hydrodynamic Stokes–Einstein relation Ds=μkBT=kB/(4πZη0) (with slip boundary conditi
with Vfree/V0free the free volume fraction and we used the relationship Nvalkane/Vfree=1-Vfree /V0 α a is a numerical constant. 
 Rescaled permeance  K x  (n+n0) for different n-alkanes as a function of the free volume fraction Vfree =V0  (with n0 = 2). The colour code is
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 an
 stating that the volumetric fluid flux through a porous material depends linearly on the pressure gradient, the inverse of the fluid viscosity η 
(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 a
vz(l)  is the velocity of molecule l,lA{1;N}).the  values of  the permeance K are in agreement with equilibrium calculation based on Green-Kubo
Flow of different n-alkanes in nanoporous  carbon under an external driving force-gradient(P): methane  (black), propane (blue), hexane (green)
Subcontinuum mass transport  of condensed hydrocarbons in nanoporous media. Kerstin Falk, Benoit Coasne, Roland Pellenq, Franz-Josef Ulm 
at small k and long times. When confined inside a solid matrix, a viscous  relaxation of the form exp[-(γ0 + k2ν)t]  may be expected, with γ0
a) Permeability ηxK versus loading Γ (Γ= ρ=ρn∞), showing the breakdown of the hydrodynamic prediction for the permeance: methane (black), propan
where f(Γ) is a simple function of the loading Γ. More specifically, we find that K(Γ)= f(Γ)/(n+n0) with n0≈2 for all alkanes provides an excellen
where D0 is the collective diffusion coefficient, N the number of alkane molecules, V the volume of the matrix, respectively, and q0=N-1∑lvz(l)(t
Owing to its collective nature, D0 differs from the molecular self-diffusivity Ds by cross-correlation terms  (vz(i)tvz(j)(0) with i≠j) of the
which captures the main behaviour of the permeance K (see the inset of Fig. 4a).
(a) Comparison of the self (Ds) and collective (D0) diffusion. Inset: Scaling of the permeance K with the self-diffusion. (b) Rescaled diffusion
Each monomer i of the alkane experiences a slip-like friction force from the matrix, fv =-ξ0vi, on top of internal forces (ξ0  is  the  friction
This effect can be quantified by a free volume approach, which relies on the probability to find a void space larger than a critical volume vcrit
equation (9) can be recast as. with Ds proportional to 1/n. The free volume fraction Vfree=Vfree is calculatedindependently for a given adsorbed
Left: In bulk, molecular diffusion is well described by the hydrodynamic Stokes–Einstein relation Ds=μkBT=kB/(4πZη0) (with slip boundary conditi
with Vfree/V0free the free volume fraction and we used the relationship Nvalkane/Vfree=1-Vfree /V0 α a is a numerical constant. 
 Rescaled permeance  K x  (n+n0) for different n-alkanes as a function of the free volume fraction Vfree =V0  (with n0 = 2). The colour code is
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 an
 stating that the volumetric fluid flux through a porous material depends linearly on the pressure gradient, the inverse of the fluid viscosity η 
(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 a
vz(l)  is the velocity of molecule l,lA{1;N}).the  values of  the permeance K are in agreement with equilibrium calculation based on Green-Kubo
Flow of different n-alkanes in nanoporous  carbon under an external driving force-gradient(P): methane  (black), propane (blue), hexane (green)
Subcontinuum mass transport  of condensed hydrocarbons in nanoporous media. Kerstin Falk, Benoit Coasne, Roland Pellenq, Franz-Josef Ulm 
at small k and long times. When confined inside a solid matrix, a viscous  relaxation of the form exp[-(γ0 + k2ν)t]  may be expected, with γ0
a) Permeability ηxK versus loading Γ (Γ= ρ=ρn∞), showing the breakdown of the hydrodynamic prediction for the permeance: methane (black), propan
where f(Γ) is a simple function of the loading Γ. More specifically, we find that K(Γ)= f(Γ)/(n+n0) with n0≈2 for all alkanes provides an excellen
where D0 is the collective diffusion coefficient, N the number of alkane molecules, V the volume of the matrix, respectively, and q0=N-1∑lvz(l)(t
Owing to its collective nature, D0 differs from the molecular self-diffusivity Ds by cross-correlation terms  (vz(i)tvz(j)(0) with i≠j) of the
which captures the main behaviour of the permeance K (see the inset of Fig. 4a).
(a) Comparison of the self (Ds) and collective (D0) diffusion. Inset: Scaling of the permeance K with the self-diffusion. (b) Rescaled diffusion
Each monomer i of the alkane experiences a slip-like friction force from the matrix, fv =-ξ0vi, on top of internal forces (ξ0  is  the  friction
This effect can be quantified by a free volume approach, which relies on the probability to find a void space larger than a critical volume vcrit
equation (9) can be recast as. with Ds proportional to 1/n. The free volume fraction Vfree=Vfree is calculatedindependently for a given adsorbed
Left: In bulk, molecular diffusion is well described by the hydrodynamic Stokes–Einstein relation Ds=μkBT=kB/(4πZη0) (with slip boundary conditi
with Vfree/V0free the free volume fraction and we used the relationship Nvalkane/Vfree=1-Vfree /V0 α a is a numerical constant. 
 Rescaled permeance  K x  (n+n0) for different n-alkanes as a function of the free volume fraction Vfree =V0  (with n0 = 2). The colour code is
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 an
 stating that the volumetric fluid flux through a porous material depends linearly on the pressure gradient, the inverse of the fluid viscosity η 
(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 a
vz(l)  is the velocity of molecule l,lA{1;N}).the  values of  the permeance K are in agreement with equilibrium calculation based on Green-Kubo
Flow of different n-alkanes in nanoporous  carbon under an external driving force-gradient(P): methane  (black), propane (blue), hexane (green)
Subcontinuum mass transport  of condensed hydrocarbons in nanoporous media. Kerstin Falk, Benoit Coasne, Roland Pellenq, Franz-Josef Ulm 
at small k and long times. When confined inside a solid matrix, a viscous  relaxation of the form exp[-(γ0 + k2ν)t]  may be expected, with γ0
a) Permeability ηxK versus loading Γ (Γ= ρ=ρn∞), showing the breakdown of the hydrodynamic prediction for the permeance: methane (black), propan
where f(Γ) is a simple function of the loading Γ. More specifically, we find that K(Γ)= f(Γ)/(n+n0) with n0≈2 for all alkanes provides an excellen
where D0 is the collective diffusion coefficient, N the number of alkane molecules, V the volume of the matrix, respectively, and q0=N-1∑lvz(l)(t
Owing to its collective nature, D0 differs from the molecular self-diffusivity Ds by cross-correlation terms  (vz(i)tvz(j)(0) with i≠j) of the
which captures the main behaviour of the permeance K (see the inset of Fig. 4a).
(a) Comparison of the self (Ds) and collective (D0) diffusion. Inset: Scaling of the permeance K with the self-diffusion. (b) Rescaled diffusion
Each monomer i of the alkane experiences a slip-like friction force from the matrix, fv =-ξ0vi, on top of internal forces (ξ0  is  the  friction
This effect can be quantified by a free volume approach, which relies on the probability to find a void space larger than a critical volume vcrit
equation (9) can be recast as. with Ds proportional to 1/n. The free volume fraction Vfree=Vfree is calculatedindependently for a given adsorbed
Left: In bulk, molecular diffusion is well described by the hydrodynamic Stokes–Einstein relation Ds=μkBT=kB/(4πZη0) (with slip boundary conditi
with Vfree/V0free the free volume fraction and we used the relationship Nvalkane/Vfree=1-Vfree /V0 α a is a numerical constant. 
 Rescaled permeance  K x  (n+n0) for different n-alkanes as a function of the free volume fraction Vfree =V0  (with n0 = 2). The colour code is
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 an
 stating that the volumetric fluid flux through a porous material depends linearly on the pressure gradient, the inverse of the fluid viscosity η 
(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 a
vz(l)  is the velocity of molecule l,lA{1;N}).the  values of  the permeance K are in agreement with equilibrium calculation based on Green-Kubo
Flow of different n-alkanes in nanoporous  carbon under an external driving force-gradient(P): methane  (black), propane (blue), hexane (green)
Subcontinuum mass transport  of condensed hydrocarbons in nanoporous media. Kerstin Falk, Benoit Coasne, Roland Pellenq, Franz-Josef Ulm 
at small k and long times. When confined inside a solid matrix, a viscous  relaxation of the form exp[-(γ0 + k2ν)t]  may be expected, with γ0
a) Permeability ηxK versus loading Γ (Γ= ρ=ρn∞), showing the breakdown of the hydrodynamic prediction for the permeance: methane (black), propan
where f(Γ) is a simple function of the loading Γ. More specifically, we find that K(Γ)= f(Γ)/(n+n0) with n0≈2 for all alkanes provides an excellen
where D0 is the collective diffusion coefficient, N the number of alkane molecules, V the volume of the matrix, respectively, and q0=N-1∑lvz(l)(t
Owing to its collective nature, D0 differs from the molecular self-diffusivity Ds by cross-correlation terms  (vz(i)tvz(j)(0) with i≠j) of the
which captures the main behaviour of the permeance K (see the inset of Fig. 4a).
(a) Comparison of the self (Ds) and collective (D0) diffusion. Inset: Scaling of the permeance K with the self-diffusion. (b) Rescaled diffusion
Each monomer i of the alkane experiences a slip-like friction force from the matrix, fv =-ξ0vi, on top of internal forces (ξ0  is  the  friction
This effect can be quantified by a free volume approach, which relies on the probability to find a void space larger than a critical volume vcrit
equation (9) can be recast as. with Ds proportional to 1/n. The free volume fraction Vfree=Vfree is calculatedindependently for a given adsorbed
Left: In bulk, molecular diffusion is well described by the hydrodynamic Stokes–Einstein relation Ds=μkBT=kB/(4πZη0) (with slip boundary conditi
with Vfree/V0free the free volume fraction and we used the relationship Nvalkane/Vfree=1-Vfree /V0 α a is a numerical constant. 
 Rescaled permeance  K x  (n+n0) for different n-alkanes as a function of the free volume fraction Vfree =V0  (with n0 = 2). The colour code is
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 an

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