The effect of adsorption and Knudsen diffusion on thesteady-state permeability of microporous rocks

Adam M. Allan1 and Gary Mavko1

ABSTRACT

Microporous rocks are being increasingly researched as novelexploration and development technologies facilitate productionof the reserves confined in the low-permeability reservoir. Theability to numerically estimate effective permeability is pivotalto characterizing the production capability of microporous reser-voirs. In this study, a novel methodology is presented for esti-mating the steady-state effective permeability from FIB-SEMvolumes. We quantify the effect of a static adsorbed monolayerand Knudsen diffusion on effective permeability as a function ofpore pressure to better model production of microporous rockvolumes. The adsorbed layer is incorporated by generating aneffective pore geometry with a pore pressure-dependent layer ofimmobile voxels at the fluid-solid interface. Due to the steady-state nature of this study, surface diffusion within the adsorbedlayer and topological variations of the layer within pores areneglected, potentially resulting in underestimation of effective

permeability over extended production time periods. Knudsendiffusion and gas slippage is incorporated through computationof an apparent permeability that accounts for the rarefaction ofthe pore fluid. We determine that at syn-production porepressures, permeability varies significantly as a function ofthe phase of the pore fluid. Simulation of methane transportin micropores indicates that, in the supercritical regime, the ef-fect of Knudsen diffusion relative to adsorption is significantlyreduced resulting in effective permeability values up to 10 nano-darcies (9.87 × 10−21 m2) less or 40% lower than the continuumprediction. Contrastingly, at subcritical pore pressures, the effec-tive permeability is significantly greater than the continuum pre-diction due to rarefaction of the gas and the onset of Knudsendiffusion. For example, at 1 MPa, the effective permeability ofthe kerogen body is five times the continuum prediction. Thisstudy demonstrates the importance of, and provides a novel meth-odology for, incorporating noncontinuum effects in the estima-tion of the transport properties of microporous rocks.

INTRODUCTION

The decline in production from conventional natural gas re-sources over the past 15 years has created a significant increasein unconventional resource demand (EIA, 2010a) and research.The United States Energy Information Administration (EIA) reportsthat 30% of primary energy consumed in the United States in 2010was supplied by natural gas. Gas shale reservoirs, in particular, areprojected to supply 13.6 TCF of natural gas in 2035, as compared tothe 5.0 TCF produced in 2010, which would constitute 49% of totalannual dry gas production in the United States in 2035 (EIA, 2012).As of 2009, 60.6 TCF of total gas shale resources in the UnitedStates are believed to be proven — 21.3% of all economicallyrecoverable natural gas reserves in the United States (EIA,

2010b). As a result, it is important that a fundamental understandingof transport properties in this expansive resource is developed.Flow in porous media has, until recently, been modeled under the

assumption that the fluid is continuous and viscous, with negligibleflow of molecules adjacent to the pore wall. This continuum ap-proach is, however, only valid when the mean free path of thegas, λ, is much smaller than the channel width, H. This stipulationcan be quantified by the dimensionless Knudsen number Kn: theratio of the mean free path of the gas to the width of the porethe gas is contained by

Kn ¼ λ

H: (1)

Manuscript received by the Editor 16 August 2012; revised manuscript received 29 October 2012; published online 31 January 2013.1Stanford University, Stanford Rock Physics Laboratory. E-mail: amallan@stanford.edu; mavko@stanford.edu.

© 2013 Society of Exploration Geophysicists. All rights reserved.

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GEOPHYSICS, VOL. 78, NO. 2 (MARCH-APRIL 2013); P. D75–D83, 11 FIGS.10.1190/GEO2012-0334.1

The continuum approach, and use of the no-slip Navier-Stokesequations, is only valid for small values of the Knudsen number(Kn < 0.001) (Schaaf and Chambre, 1961). In confined porespaces, as the pore width approaches the mean free path of thegas, the gas rarefies, resulting in an increase in the frequency ofmolecule-wall collisions with respect to intermolecular collisions,and the velocity of the molecules adjacent to the pore wall is nolonger negligible. The slip flow regime is applicable when Kn issmall (<0.1) but not negligible. In the transition flow regime,molecule-wall collisions and intermolecular collisions are bothimportant. Flow regimes corresponding to Kn > 0.001, character-ized by nonnegligible velocities at the wall, have been shownto increase the mass transfer rate (Arkilic et al., 1997). Faithfulprediction of mass transfer rates is requisite for correct estimationof permeability from numerical simulation. For clarity, the flowregimes highlighted here are illustrated in Figure 1.It is now routinely reported that most of porosity in gas shale is

contributed by nanometer-scale pores in the organic content of therock (e.g., Loucks et al., 2009; Sondergeld et al., 2010; Curtis et al.,2011). These nanometer-scale pores support flow characterized byKnudsen numbers in the slip and transition regimes, subsequently,noncontinuum flow physics are nonnegligible; additionally, thesenanopores have significant internal surface areas which can resultin substantial adsorption of a high-density, liquid-like phase to thewalls of the pores. The high density of the adsorbed phase createsthe potential for significant storage of gas, and the effect of storageon gas-in-place (GIP) calculations has been widely studied (e.g.,Bustin et al., 2008; Cui et al., 2009; Ambrose et al., 2012); however,the effect of an adsorbed layer of gas on sample permeabilityhas been less widely researched, as noted in Sakhaee-Pour andBryant (2011).Several recent studies propose that the permeability of micropor-

ous reservoir rocks will be greater than the continuum prediction atreservoir temperature and pore pressure due to rarefaction of thepore-filling gas. However, much of the recent research has beenconducted on simplified pore geometries, e.g., single, or bundlesof, cylindrical pores. For instance, Knudsen diffusion and gas slip-page have been studied with a modified dusty gas model (DGM) for2D pipes (Javadpour et al., 2007; Freeman et al., 2009; Javadpour,2009), a single cylindrical nanopore (Swami and Settari, 2012), anda pack of spherical grains (Shabro et al., 2011a, 2011b); whereas theapparent permeability expression of Florence et al. (2007) derivedfrom Karniadakis et al. (2005) has been applied to a single cylind-rical pore (Xiong et al., 2012) and a network and a bundle ofcylindrical pipes by Sakhaee-Pour and Bryant (2011) and Michelet al. (2011), respectively, to determine the effective permeabilityof microporous geometries as a function of pore pressure and

channel width. Additionally, Fathi et al. (2012) develop a novel Lat-tice-Boltzmann model (LBM) capable of directly modeling Knud-sen diffusion, gas slippage, and molecular streaming in 2D tubes.Other authors add that the presence of an adsorbed gas phase may

decrease in situ permeability values at high pore pressure and causethe sustained production rates that are characteristic of shale gasreservoirs as discussed by Valkó and Lee (2010). Adsorption is ac-counted for by applying Langmuir adsorption theory dynamicallyas an additional source/sink term in flow PDEs (Shabro et al.,2011a, 2011b; Xiong et al., 2012) and as a Langmuir-slip modifiedLattice Boltzman model (LBM-Ls: Fathi and Akkutlu, 2011).Sakhaee-Pour and Bryant (2011) represent adsorption through asimplified, static process in which a uniform coating of the interiorwalls of cylindrical pores thickens linearly with pore pressure.However, even with the advent of advanced imaging techniquessuch as FIB-SEM, TEM, and CLSM, and the subsequent abilityto image nanometer-scale heterogeneity in shale samples as demon-strated by Loucks et al. (2009), Kale et al. (2010), Sondergeld et al.(2010), Curtis et al. (2011), and others, only the finite-differencediffusive-advective (FDDA) method of Shabro et al. (2009) enablesthe use of complex, realistic pore geometries extracted from FIB-SEM or AFM experiments.This study provides an additional methodology to apply these

new imaging techniques to complex pore geometries to quantifythe magnitude of the effect of nanometer-scale, noncontinuum phe-nomena, e.g., adsorption and Knudsen diffusion, on microporousreservoir rock transport properties. Additionally, this work appearsto be the first to explicitly address the implications of the neglig-ibility of diffusive mass transfer mechanisms in the supercriticalregime, as identified in the life sciences (Patil et al., 2006), to flowin microporous earth materials. To reach this objective, we simulatecontinuum fluid flow of methane through a sample kerogen porenetwork with an existing continuum regime LBM simulator(Keehm, 2003) with static, discretely modeled, adsorbed pseudo-molecules at the resolution of a methane molecule (0.38 nm), suchthat each voxel is approximately the size of a methane molecule. Inthis manner, we set criteria governing where in the pore spaceadsorption occurs and better represent the heterogeneity of transportthrough the tortuous pore space. The rarefaction of the pore-fillinggas is accounted for by computing an apparent permeability that is afunction of the continuum permeability and the average Knudsennumber in the pore space as presented by Florence et al. (2007).The average Knudsen number is calculated as a flux-weighted aver-age to account primarily for the pores that support flow, thus ac-counting for isolated porosity and nonpercolating branches ofthe pore network. In this manner, we provide a novel workflowfor the estimation of in situ, steady-state permeability variationsas a function of pore pressure for any arbitrary pore geometry acrossa complete range of pore pressures.

METHODS

Pore geometry generation

In anticipation of the acquisition of FIB-SEM volumes of kero-gen bodies from a gas shale reservoir, our flow-simulation metho-dology is tested on a geostatistically rendered proxy kerogen body.A 2.9 × 3.6 μm SEM image (Figure 2) captured and provided byArjun Kohli (Stanford University) is thresholded and rendered inthe third dimension by sequential indicator simulation (Journeland Gomez-Hernandez, 1993). To fully resolve and model the

0.001 0.1 10KnudsenNumber, Kn

Flow regime Contin

uum

Slip Trans

ition

Free

mole

cular

Figure 1. A schematic representation of the extents of each of theKnudsen flow regimes discussed in the text. The flow of gas in mi-croporous reservoir rocks is generally confined to the slip and tran-sition flow regimes characterized by a Knudsen number between10−3 and 10.

D76 Allan and Mavko

adsorption process (discussed fully in the following subsection), the3D volume is scaled with a discrete cosine transform (DCT) suchthat the grid spacing, dx, is equal to the molecular diameter ofmethane (dx ¼ 0.38 nm). The proxy kerogen body displayed inFigure 3 is 167 nm (440 voxels) in each dimension and has a totalporosity of 17.2%. To handle the computational expense of the 4403

voxel pore geometry, numerical simulation is conducted on theRock Physics Laboratory cluster of the Stanford Center for Com-putational Earth and Environmental Science (CEES). An existingD3Q19 continuum LBM code (Keehm, 2003) is used to computethe intrinsic continuum prediction permeability of the sample. Thecontinuum permeability of the sample, k∞, for dx ¼ 0.38 nm is25.2 nanodarcies (2.49 × 10−20 m2).

Simulation methodology

Sorption

Adsorption, as described by Langmuir adsorption theory, is theadhesion of gas molecules to the surface of a solid resulting in ahigh-density, liquid-like gas film that is distinct from the bulk

gas phase. The adsorptive potential of a material is quantified bythe amount of adsorption at many relative pressures (the ratio ofthe present gas pressure to the saturation pressure) and constanttemperature as an isotherm. The adsorption isotherm is a functionof the pore size distribution, composition of the pore-filling gas (ormixture of gases), and the attraction energy between the surface(sorbent) and pore-filling gas (adsorbate) (Langmuir, 1916; Bru-nauer et al., 1938). To satisfy the tenets of the Langmuir adsorptionmodel in simulation, the following assumptions must be main-tained: (1) the adsorbed molecules are immobile, (2) all adsorptionsites are equivalent, (3) each adsorption site can hold at most onemolecule, and (4) no interactions exist between the adsorbed mo-lecules (Langmuir, 1916). Adsorption will affect permeability intwo manners: (1) the pore width will become geometrically nar-rower by 2n molecular diameters where n is the number of layersadsorbed, and (2) the adsorbed layer changes the nature of interac-tions of molecule and wall through the tangential momentumaccommodation coefficient (TMAC). Effect (1) will be of little con-cern in larger pores; however, in narrow (e.g., 10 nm) pores, areduction in width by twice the molecular diameter of methane(0.38 nm) will have a non-negligible effect on the permeabilityof the pore. Effect (2) is addressed more completely in the followingsubsection; however, in simulations we hold TMAC constant withvalue 1.0, e.g., all molecule-surface interactions are fully diffusive.It is trivial to note that desorption is the reverse physical processfrom adsorption, in which adsorbed molecules reenter the bulk, freegas from the surface layer. Given that the reservoir pore pressure,generally, drops as hydrocarbons are produced, it is more sensiblefor the remainder of the discussion herein to be presented as des-orption in microporous rocks.The assumptions of the Langmuir sorption model, especially the

statistical equivalence of adsorption sites, result in a homogeneouspore space in which desorption may occur simultaneously in allpores regardless of the behavior of the bulk gas in each pore.Because the study geometry is derived from a physical kerogenimage, the geometry is characterized by a tortuous pore networkwith isolated porosity and pores that percolate the sample. Thesepercolating pores provide additional heterogeneity as they may varywidely in terms of their average pore throat width and the fraction ofthe total flux passing through each pore. Assuming constant influx

Figure 3. Geostatistically rendered 3D kerogen body used for flow simulations. White voxels are solid matrix, wheras black voxels are pore.The porosity of the volume is 17.2% and the continuum permeability is 25.2 nanodarcies (2.49 × 10−20 m2). The flow-driving pressure gra-dient is applied in the x-direction. Orientation (a) shows the outlet face of the cube at x ¼ 440, whereas orientation (b) shows the inlet face ofthe cube at x ¼ 0.

Figure 2. Secondary electron SEM image of a kerogen body in anorganic-rich shale plug imaged and provided by A. Kohli (StanfordUniversity). The width of each pixel is 5.56 nm. Black pixels in-dicate void space. Upon thresholding, the porosity of the kerogenbody is estimated to be 4.6%.

Permeability of microporous rocks D77

conditions, the variable fraction of the total flux passing througheach pore will cause some pores to experience more net desorptionthan others. In other words, molecules desorbing from the walls of ahigh flux pore have a greater likelihood of being advected out of thesystem in the free gas before the molecule can readsorb within thesame pore. Contrarily, in a pore that supports very little flux, there isa higher likelihood that desorbed molecules will readsorb at anotheradsorption site within the same pore — in this case, no net des-orption has occurred. To capture such an effect in the simulation,molecules are desorbed initially from high flux pores before net des-orption occurs in the negligibly low flux pores.It is important to bear in mind that the continuum LBM simulator

used in this study is steady-state, e.g., adsorption/desorption doesnot occur dynamically during the simulation; as a result, the ad-sorbed layer configuration must be specified before flow simulationis executed. The permeability estimated through LBM simulation is,then, the continuum permeability of the effective pore geometrycomposed of the original pore network and the layer of adsorbedmolecules that the authors refer to as kads. The methodology forgenerating the effective pore geometry via addition of an adsorbedlayer is presented in Figure 4.LBM simulation is initially performed on the unaltered pore geo-

metry to obtain the flux through each voxel in the pore space. Eachdiscrete, continuous pore is then grouped into sets based on the frac-tion of the total mean flux supported by that pore as represented inequation 2, where χi is the flux fraction in pore i, and q̄i is the meanflux in pore i:

χi ¼q̄iPiq̄i. (2)

The demarcation of each set boundary is arbitrary and user-specified, e.g., all pores with χi > 0.4 are “high flux,” set one pores.The list of all possible adsorption sites is generated by searching forall pore voxels with a face-connected solid voxel. The list of adsorp-tion sites is then sorted by set number to enable easy implementa-tion of preferential desorption; additionally, the adsorption siteswithin each set are randomized so that the sites are statisticallyequivalent in accordance with the Langmuir model.The number of molecules adsorbed at a specified pore pressure

must then be computed. In the study of Sakhaee-Pour and Bryant(2011), an isotherm for methane and shale is described to calculate

the amount of adsorption at a given pore pressure, their model as-sumes a linear increase of adsorption with pore pressure that reachesa double-layer of bound molecules at 28 MPa. The dissertation ofVermylen (2011) documents methane adsorption isotherms mea-sured on crushed Barnett shale samples at room temperature andpore pressures up to ∼10 MPa. The results of Vermylen indicateisotherms demonstrating Langmuir behavior, e.g., a monolayerof adsorbed methane molecules with the isotherms exhibiting a non-linear relationship with pore pressure. The isotherm of sample31vcde from Vermylen (2011) is sampled and fit with a logarithmiccurve in Figure 5 and used hereafter as a reference for the amount ofmethane adsorbed at a given pore pressure. As documented inFigure 4, the isotherm-prescribed number of pore voxels is thenconverted to solid voxels resulting in a new effective pore geometryaccounting for the true rock and the static adsorbed layer. Note that,by representing the adsorbed molecules as solid voxels in the effec-tive pore geometry, the remaining assumptions in the Langmuirmodel (e.g, immobile, noninteracting molecules that occupy onlyone adsorption site) are honored in the simulation.In this study, the sample geometry is updated with adsorbed

molecules at 14 discrete pore pressures between 40 MPa and atmo-spheric pressure (0.101 MPa), representing production induced-depletion from in situ conditions. These updated geometries arethen used as inputs to LBM simulation and the resultant permeabil-ity values are reported as the continuum permeability in the pre-sence of adsorbed gas, kads.

Knudsen diffusion

The permeability of conventional reservoirs has long been accu-rately predicted by Darcy’s Law (equation 3), a phenomenologicalrelationship between the flux through the porous medium (in ms−1)q to the permeability k, through the viscosity μ, and pressure gra-dient, ∇P (Darcy, 1856)

q ¼ −kμ

∇P. (3)

5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

Pore pressure (MPa)

Frac

tion

of m

onol

ayer

ads

orbe

d

Figure 5. Adsorption isotherm used to determine the amount ofadsorbed gas at a given pore pressure. The black curve is a loga-rithmic fit to data points extracted from an experimental isothermfor methane adsorption on a crushed Barnett shale sample measuredby Vermylen (2011) (open circles).

Start

Run LBM simulationon unaltered geometry

Group pores by fractionof flux supported

Generate list of possible adsorption sites by group

Identify number of adsorbedmolecules at specified Pp

from experimental isotherm

Convert pore voxelsto solid voxels in model

Run LBM simulation onupdated geometry

Compute kads from resultantflux field

End

Figure 4. A flow chart demonstrating the work flow to be imple-mented to account for the presence of an adsorbed layer in steady-state, continuum permeability estimation.

D78 Allan and Mavko

The formulation in equation 3 is, however, only valid in the con-tinuum regime. As a result, an apparent permeability kapp must bedefined by considering the effect of rarefaction-induced Knudsendiffusion of the pore-filling gas for use in noncontinuum flowregimes.The “unified flow model” (UFM) of Karniadakis et al. (2005) is a

second-order inKnmodel of rarefied gas flow that has been demon-strated to be valid across the full range of Kn by comparison tomolecular dynamics simulations. The reader is directed to chapter4 of Karniadakis et al. (2005) for a complete derivation of the mod-el. Subsequent work by Florence et al. (2007) derived an apparentpermeability kapp (equation 4) that accounts for Knudsen diffusionin nanopores. Note that the rarefaction term ½1þ αðKnÞKn� inequation 4 accounts for the transition between the slip flow and freemolecular flow regimes and is expanded in equation 5 as defined byFlorence et al. (2007). In equation 4, k0 is a continuum permeabilityvalue that may be selected a priori, computed from flow simulation,or taken as kads as calculated in the previous work flow (Figure 4)

kapp ¼ k0½1þ αðKnÞKn��1þ 4Kn

1þ Kn

�(4)

αðKnÞ ¼ 128

15π2tan−1½4Kn0.4�. (5)

There are two intrinsic assumptions in the derivation of equa-tion 4: first, that all fluid-surface interactions are purely diffusive,i.e., σV ¼ 1 where σV is the tangential momentum accommodationcoefficient (TMAC), and, secondly, that the pores are long channels.Laboratory experiments by Grauv et al. (2009) find that, for argon,nitrogen, and helium impinging on a silicon surface, TMAC variesbetween bounds of 0.848� 0.008 and 0.956� 0.005 at atmo-spheric conditions. Additional experimental measurements for hy-drogen, helium, and air impingement on aluminum, iron, andbronze surfaces by Seidl and Steinheil (1974) and Lord (1976) fallwithin the same range as those reported by Grauv et al. (2009). As aresult, there appears to be a physical basis for assuming mostly dif-fusive molecule-pore wall reflections. The assumption that σV ¼ 1

will, when Knudsen diffusion contributes substantially to masstransfer, significantly underpredict the effective permeability ifσV ≪ 1 in situ. Subsequently, future work will investigate the va-lidity of the purely diffusive reflection assumption for characteristicgeologic materials (e.g., quartz, illite, and carbon) at reservoir tem-perature-pressure conditions with and without an adsorbed layerthrough molecular dynamics simulations to quantify the effect ofTMAC on effective permeability estimation.The assumption that pores are long channels is proven invalid for

the sample under study through inspection of the geometry of thepercolating pores (Figure 6). The highly tortuous, complex poregeometry requires that the long channel assumption be correctedfor. Because the correction for Knudsen diffusion in equation 4is a function of only the intrinsic, continuum permeability andKn, we assume that by computing the continuum permeability fromthe original pore geometry via LBM, we need only accurately es-timate the average pore width, and thus Kn, to minimize error in theestimation of gas permeability by UFM. The implementation of thisgeometrical correction is surmised in the flow chart in Figure 7. Theminimum pore width is calculated for each discrete percolating porein each 2D cross section of the volume cut perpendicular to the flow

direction. There are, thus, Nav minimum pore width values, whereNav is the product of the number of discrete percolating pores ineach cross section and the number of cross sections cut orthogonalto the flow. To account for the tortuosity of the pore space, eachminimum pore width value is computed as the 3D minimum Eu-clidean distance across the pore between any two solid voxels.To account for the heterogeneity of flow between different pores,each minimum pore width value is weighted by the fraction ofthe total mean flux passing through the corresponding 2D poreslice. The total flux-averaged pore width is then the sum of eachweighted minimum pore width value as documented in equation 6,where fAPW is the flux-averaged pore width, di is the individualminimum pore width value, and the ratio involving q̄i and q̄j isas in equation 2. In this manner, the pores that most heavily con-tribute to the determination of the average pore width, and thusKnudsen number, are the percolating pores that support the greatestfraction of flow

fAPW ¼XNav

i¼1

di

�q̄iPNavj¼1 q̄j

�. (6)

The overestimation of the slip magnitude due to the inclusion ofnonpercolating pores and narrow pores that support minimal flowin the average Knudsen number calculation is shown to exceed 10%for much of the range of pore pressures under consideration(Figure 8). The total in situ effective permeability of the samplektot is then estimated by applying equation 4 with k0 ¼ kads forthe appropriate average Knudsen number, as computed throughequation 1 for the calculated flux-weighted average pore widthfAPW .

Consideration of critical pressure

The critical pressure and temperature of methane at the conti-nuum-scale have been determined as 4.60 MPa and 190 K, respec-tively (Lemmon et al., 2012). It has been noted by Michel et al.(2012) that the critical properties of hydrocarbons vary in pores withradii less than 10 nm using the empirical relationships of Sapmanee

Figure 6. Isosurface plot of the percolating pore network indicatingthe tortuous pore structure of the geostatistically rendered rockgeometry. The flow-driving pressure gradient is applied in thex-direction from x ¼ 0 to x ¼ 440.

Permeability of microporous rocks D79

(2011): For example, in a 5 nm pore the critical properties become5.822 MPa and 185.20 K, respectively. However, because the aver-age pore width of the sample in this study (15 nm) is 200% greaterthan the widest radius empirical relation in Sapmanee (2011), thevariability in critical properties is neglected with little loss of effi-cacy in the tool here presented. Subsequently, a substantial set of thepore pressures simulated, i.e., all those greater than 4.60 MPa, cor-respond to supercritical methane. It is, therefore, of critical impor-tance that we account for the effects of the supercritical phasetransition in simulation.Previous work in the literature indicates that the effect of the

supercritical phase transition will be to inhibit diffusive mass trans-fer while preserving the continued adsorption of molecules from thebulk phase under the assumptions of this methodology. Patil et al.(2006) have demonstrated experimentally by permeating fluidsthrough nanopores in inorganic membranes that Knudsen diffusion(and other diffusive flow mechanisms) becomes a negligible masstransport mechanism when the pore-filling fluid is supercritical.Subsequently, for pore pressures greater than the critical pressure(4.60 MPa), equation 4 is not implemented and slip velocitiesare neglected. Additionally, because above a pore pressure of4.60 MPa Knudsen diffusion is negligible, the potential effectivepermeability underprediction due to the purely diffusive reflectionassumption, e.g., TMAC ¼ 1, will only be observed at lower porepressures. It should also be noted that this study pertains to puremethane as the pore-filling fluid, in situ fluid mixtures, e.g.,methane, ethane, carbon dioxide, and hydrogen sulfide, will gener-ally have higher critical pressures than that of pure methane. Theseelevated critical pressures indicate that Knudsen diffusion mayaffect in situ permeability values at greater pore pressures than ispredicted by this work.It has been noted by Zhou et al. (2002), among others, that super-

critical adsorption will continue at pore pressures greater than thecritical pressure for nitrogen and methane adsorbing onto silica gel.Zhou et al. (2002) do add the caveats that they detect a transition in

adsorption mechanism across the critical transition (a change in theisotherm structure) and that an upper limit exists for supercriticalabsorption. The experimental results of Vermylen (2011) are col-lected up to ∼10 MPa, in excess of the critical pressure of methane,but show no deviation from a type I (Langmuir) isotherm upon be-coming supercritical. This result may be representative of the dif-fering interactions of methane with shale versus nitrogen andmethane with silica gel, or may indicate that additional higher porepressure data points are required for complete representation of theexperimental isotherm. Given the geometry of the Langmuir iso-therm purported by Vermylen (2011), a maximum adsorption statecan be observed in the supercritical regime (Figure 5) fulfilling thesecond caveat of Zhou et al. (2002). Subsequently, in this study, theprocess of accounting for adsorption discussed in Figure 4 isnot modified in the supercritical regime. By not modifying thesupercritical adsorption methodology, we are able to honor theexperimental isotherm geometry of Vermylen (2011) which charac-terizes the corresponding lithofacies to this study (gas shale) whilepreserving the upper limit of supercritical adsorption identified byZhou et al. (2002) until further laboratory characterization of super-critical adsorption in shale is completed.

RESULTS

The flow of methane was simulated through the proxy kerogenbody at pore pressures ranging from atmospheric (0.101 MPa) to40 MPa. Figure 9 shows the change in permeability as a functionof pore pressure due to adsorption and Knudsen diffusion in thecase of honoring the supercritical transition and the case of gaseousmethane at all pore pressures, respectively. Adsorption is seen toreduce the permeability of the body at all pore pressures greaterthan atmospheric pressure where it is assumed that no adsorptionoccurs. The effect of adsorption is as great as 40% or 10 nanodarcies(9.87 × 10−21 m2) at high pore pressure. Knudsen diffusion of agaseous phase that does not become supercritical in the pressurerange investigated, increases permeability across the entire range(albeit negligibly at high pore pressure). If the supercritical transi-tion of methane is honored, Knudsen diffusion has no effect on per-meability at pore pressures greater than the critical pressure in

Start

Run LBM simulationon unaltered geometry

End

Include adsorbedlayer?

k0 = kads

(see Figure 4)Compute k0 from

LBM flux fieldYesNo

Compute average Knudsennumber at specified Pp

Compute ktot by applyingEqn. 4

Compute averagepore width fromeffective pore

geometry

Compute averagepore width from

original poregeometry

Figure 7. A flow chart demonstrating the work flow to be imple-mented to account for Knudsen diffusion in steady-state, continuumpermeability estimation. This work flow is readily amenable to ad-sorbing and nonadsorbing pore walls.

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

45

Pore pressure (MPa)

% e

rror

in s

lip m

agni

tude

Figure 8. Error in the slip magnitude resulting from not flux-averaging the pore widths of the study geometry (Figure 3). Theerror becomes increasingly more significant as the pore-fillinggas rarefies (decreasing pore pressure) and the elevated Knudsennumber increases the contribution of the smallest isolated pores.

D80 Allan and Mavko

keeping with the findings of Patil et al. (2006), but can result in afive fold increase in permeability at 1 MPa.Figure 10 shows the combined effect of adsorption and Knudsen

diffusion for both cases illustrated in Figure 9. In both cases, thetotal permeability decreases with increasing pore pressure.When the critical transition of methane is neglected, the total per-meability is observed to be less than the continuum permeability(2.49 × 10−20 m2) for pore pressures greater than ∼12 MPa. Whenthe pore-filling methane is allowed to become supercritical, the totalpermeability remains less than the continuum permeability untilpore pressure decreases to the critical pressure (4.6 MPa) asopposed to at 12 MPa in the previous case.

DISCUSSION

In this study, we present trends in the estimated effective perme-ability of a digitally rendered microporous solid as a function ofpore pressure. It is determined that the effects of adsorption andKnudsen diffusion are opposite with regard to the Darcy-predictedpermeability. We see in Figure 9 that adsorption causes the perme-ability to be less than the continuum, Darcy prediction at all porepressures whereas the onset of Knudsen diffusion results in the per-meability exceeding the Darcy prediction. At supercritical porepressures, Knudsen diffusion has been demonstrated to be negligi-ble (Patil et al., 2006); subsequently, the in situ permeability is re-duced below the continuum prediction by the presence of theadsorbed layer. Contrarily, at subcritical pore pressures, the effectof Knudsen diffusion in the rarefied pore-filling gas outweighs theeffect of partial adsorption resulting in a permeability value signif-icantly greater than the continuum prediction (Figure 10). It hasbeen previously noted by Sakhaee-Pour and Bryant (2011) andXiong et al. (2012) that increasing pore pressure causes an increasein gas density resulting in decreasing Knudsen diffusion and sub-sequent overestimation of in situ permeability if adsorption isneglected. However, due to the omission of the critical phase transi-tion, this effect is underestimated as a nonnegligible contributionfrom Knudsen diffusion is preserved resulting in overestimatedin situ permeability values even in the presence of an adsorbed

layer. This overestimation of the contribution of Knudsen diffusionis also evident above the critical pressure in the work of Shabro et al.(2011b). This overestimation of total permeability is illustrated inFigure 10, in which we neglect the supercritical transition, docu-menting that the trend of decreasing permeability with increasingpore pressure remains; however, the effect of adsorption will over-come Knudsen diffusion at a higher pore pressure. The pore pres-sure at which adsorption dominates diffusion is a function of theadsorption isotherm, the average pore width of the sample, andthe pore-filling gas chemistry and will, therefore, change from sam-ple to sample; however, in this case study, total permeability is over-estimated by at least 10% for all pore pressures above the criticalpressure (4.60 MPa).The effect of neglecting adsorption and Knudsen diffusion on in

situ permeability estimation is presented as a function of pore pres-sure in Figure 11. In both cases, gaseous phase at all pore pressuresand including the phase transition, the error in permeability estima-tion attains a maximum at atmospheric pore pressure when Knudsendiffusion causes the greatest underestimation of the in situ perme-ability by continuum Darcy theory. As pore pressure increases, thesuppression of Knudsen diffusion causes the misfit in permeabilityestimation to decrease until the point at which adsorption offsets theeffect of diffusion. Further increases in pore pressure, and theassociated increase in adsorption causes the Darcy prediction toincreasingly overestimate the permeability of the sample. Formuch of the range of pore pressures simulated in this study,the error in permeability estimation exceeds eight nanodarcies(7.90 × 10−21 m2) or 30% of the Darcy prediction. The significanterror in permeability estimation is proof in point of the need toaccount for wall effects such as adsorption and Knudsen diffusionwhen estimating the permeability of microporous rocks.The permeability trends presented in this paper are simulated in a

complex, tortuous pore network composed of four discrete, perco-lating pore branches. Most of previous publications concerning gasflow in geologic microporous media are validated on idealized geo-metries such as bundles of cylindrical tubes (e.g., Javadpour, 2009:Sakhaee-Pout and Bryant, 2011: Xiong et al., 2012, and others).

0 5 10 15 20 25 30 35 40101

102

103

104

Pore pressure (MPa)

Per

mea

bilit

y (n

D)

Adsorption onlySlip only (No phase transition)Slip only (Phase transition)

Critical pressure

Figure 9. Variation of permeability as a function of pore pressureresulting from the presence of an adsorbed layer (dot-dashed layer),Knudsen diffusion of a gaseous phase (dotted curve), and Knudsendiffusion of a fluid that undergoes a supercritical transition (solidcurve). The black arrows outside the graph indicate the value ofthe continuum permeability of the sample (25 nD).

0 5 10 15 20 25 30 35 4010

1

102

103

104

Pore pressure (MPa)

Tota

l per

mea

bilit

y (n

D)

No phase transitionPhase transition

Critical pressure

Figure 10. Variation of the total permeability, accounting foradsorption and Knudsen diffusion, of the kerogen body as a func-tion of pore pressure. For comparison, two curves are presented: thepermeability to a gaseous phase (dashed curve), and the permeabil-ity to a fluid phase that undergoes a supercritical transition (solidcurve). The black arrows outside the graph indicate the value of thecontinuum permeability of the sample (25 nD).

Permeability of microporous rocks D81

Although Shabro et al. (2011b) do investigate a complex poregeometry, exclusion of the phase transition results in overestimationof permeability at high pore pressure. By incorporating voxel-basedadsorption modeling and the apparent permeability correction ofFlorence et al. (2007) with a flux-weighted average pore width,we are able to estimate the effect of adsorption and Knudsen diffu-sion on permeability in any arbitrary geometry while accounting forthe negligibility of diffusion in the supercritical regime. The abilityto estimate the effective permeability of any arbitrary microporousgeometry coupled with the ability to easily parallelize LBM, makethe methodology documented in this paper a powerful, cost-efficient tool for more accurately estimating the steady-state perme-ability of nearly impermeable rock samples.Although the present methodology removes the need to make

geometrical assumptions about the pore space, there remain intrin-sic limitations in our simulation of gas flow in confined pores. Wemaintain the assumption of Florence et al. (2007) that all gas-surface reflections are diffusive, e.g., σV ¼ 1, which will underes-timate the effective permeability for systems in which specularreflections dominate. Future work will be aimed at addressingthe applicability of this assumption and any error associated withits implementation. Additionally, in applying the apparent perme-ability of Florence et al. (2007), we assume that by computing a 3D,flux-weighted average pore width and derivative average Knudsennumber we are able to account for the complex, tortuous nature ofthe pore system. Due to the pseudomolecular grid spacing appliedin this model to capture individual adsorbed molecules, we are lim-ited with this methodology to the investigation of rock volumes onthe order of a few hundreds of nanometers. However, the study ofnonadsorbing systems may be conducted with the same Knudsendiffusion methodology as discussed in this work at a coarser reso-lution enabling the investigation of larger rock volumes. The rockvolume limitation requires than an additional technique such asFEM or FD modeling, such as in Shabro et al. (2009), be usedto investigate the effective permeability of core-scale and largershale samples. Furthermore, it is important to remember that thismethodology is a steady-state effective permeability estimator and

that no dynamic effects, e.g., time-dependent adsorption/desorptionor surface diffusion, are considered in the LBM flow simulation.We also assume in implementing the supercritical transition that

the phase change occurs instantaneously at the critical pressure.This phase change assumption is the source of the discontinuityin the in situ permeability plots (Figures 9, 10, and 11). Future studywill account for a more physical, continuous phase transition in ourmethodology. Finally, we assume constant effective pressure suchthat no mechanical compaction of the rock occurs with changes inpore pressure which will overestimate in situ permeability at lowpore pressure for a significantly compliant composite.In addition to addressing the limitations of the current methodol-

ogy, future investigations will attempt to physically validate theresults presented in this paper through comparison to laboratorymeasurements. We will measure permeability values on a suite ofshale samples before estimating permeability values using ourmethodology on FIB-SEM volumes of various constitutive compo-nents, e.g., clay matrix, kerogen bodies, clastic-rich zones, forcomparison. Future work will also attempt to address outstandingquestions regarding the communication of flow between kerogenbodies and the manner in which shale reservoirs produce.

CONCLUSION

We present a methodology that simulates steady-state fluid flowthrough tortuous pore networks with a static adsorbed layer thatexperiences variable Knudsen diffusion as a function of pore pres-sure. The new methodology uses a segmented binary volume as adigital representation of a microporous rock volume and combinessteady-state, continuum Lattice Boltzmann simulation with voxel-based modeling of the adsorbed layer and the Unified Flow modelto account for the necessary wall effects, e.g., adsorption and Knud-sen diffusion. We investigate the effect of pore pressure at constanteffective pressure on the effective permeability of microporousrocks to gaseous and supercritical pore-filling methane. The effec-tive permeability of microporous rock volumes is shown to behighly variable with respect to pore pressure due to changes in thedominant wall effect as the phase of the pore-filling fluid changesbetween sub- and supercritical. The main advantage of this meth-odology is the ability to estimate the steady-state effective perme-ability of any arbitrary microporous geometry across the sub- andsupercritical regimes and for all flow regimes. This work is limited,however, to studying volumes on the order of a few hundreds ofnanometers due to computational cost. Additionally, by studyingthe steady-state permeability to pure methane, surface diffusionin the adsorbed layer, time-dependent realignment of the adsorbedlayer, and the elevated critical pressure of gas mixtures are ne-glected which may result in underestimation of the effective perme-ability. It is important to remember that, although the example inthis study is restricted to a kerogen body, this methodology canbe used to estimate the effective permeability of any microporousfacies, e.g., tight sands or micritic carbonates.

ACKNOWLEDGMENTS

The authors acknowledge the support of the Stanford Rock Phy-sics and Borehole Geophysics Project and its industrial affiliates. A.M. Allan is sponsored by The William R. and Sara Hart KimballStanford Graduate Fellowship Fund. The authors thank Arjun Kohlifor the SEM image, and Fabian Krzikalla, M. D. Zoback, and J.Wilcox for their discussions.

0 5 10 15 20 25 30 35 40−20

0

20

40

60

80

100

Pore pressure (MPa)

Mis

fit fr

om c

ontin

uum

val

ue (

nD)

No phase transitionPhase transitionNo misfit

Critical pressure

Figure 11. The difference of the total permeability and the conti-nuum permeability as a function of pore pressure. For comparison,two curves are presented corresponding to a gaseous phase (dot-dashed curve), and to a fluid phase that undergoes a supercriticaltransition (solid curve). The horizontal dotted line marks that thetotal permeability is equal to the continuum permeability.

D82 Allan and Mavko

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