Experimental investigation of matrix permeability Department …...Experimental investigation of matrix permeability of gas shales Rob Heller, John Vermylen, and Mark Zoback ABSTRACT

Experimental investigationof matrix permeabilityof gas shalesRob Heller, John Vermylen, and Mark Zoback

ABSTRACT

Predicting long-term production from gas shale reservoirs hasbeen a major challenge for the petroleum industry. To betterunderstand how production profiles are likely to evolve with time,we have conducted laboratory experiments examining the effectsof confining stress and pore pressure on permeability.Experiments were conducted on intact core samples from theBarnett, Eagle Ford, Marcellus, and Montney shale reservoirs.The methodology used to measure permeability allows us to sepa-rate the reduction of permeability with depletion (because of theresultant increase in effective confining stress) and the increasein permeability associated with Knudsen diffusion and molecularslippage (also known as Klinkenberg) effects at very low porepressure. By separating these effects, we are able to estimate therelative contribution of both Darcy and diffusive fluxes to totalflow in depleted reservoirs. Our data show that the effective per-meability of the rock is significantly enhanced at very low porepressures (<1000 psi [<6.9 MPa]) because of the slippageeffects. We use the magnitude of the Klinkenberg effect to esti-mate the effective aperture of the flow paths within the samplesand compare these estimates to scanning electron microscopyimage observations. Our results suggest effective flow pathsto be on the order from tens of nanometers in most samples to100–200 nm, in a relatively high-permeability Eagle Ford sample.Finally, to gain insight on the scale dependence of permeabilitymeasurements, the same core plugs were crushed, and permeabil-ity was again measured at the particle scale using the so-calledGas Research Institute method. The results show much lower per-meability than the intact core samples, with very little correlationto the measurements on the larger scale cores.

Copyright ©2014. The American Association of Petroleum Geologists. All rights reserved.

Manuscript received February 11, 2013; revised manuscript received June 10, 2013; revised manuscriptreceived July 29, 2013; revised manuscript received August 15, 2013; revised manuscript receivedAugust 19, 2013; accepted September 23, 2013.DOI: 10.1306/09231313023

AUTHORS

Rob Heller ∼ Department of Geophysics,Stanford University, Stanford, California94305; [email protected]

Rob Heller earned his Ph.D. in geophysicsfrom Stanford University in 2013. His thesiswas titled “Multiscale Investigation of FluidFlow in Gas Shales”, and his adviser wasDr. Mark Zoback. He earned his M.S. degreefrom Stanford University in environmentalengineering in 2008, and a B.S. degree fromColumbia University in 2007.

John Vermylen ∼ Department ofGeophysics, Stanford University, Stanford,California 94305; [email protected]

John Vermylen earned his Ph.D. ingeophysics from Stanford University in 2011.His thesis was titled Geomechanical Studiesof the Barnett Shale, and his adviser wasMark Zoback. Vermylen earned hisM.S. degree in geophysics from StanfordUniversity in 2008, and his B.S. degree fromPrinceton University in 2005.

Mark Zoback ∼ Department of Geophysics,Stanford University, Stanford, California94305; [email protected]

Mark Zoback is the Benjamin M. PageProfessor of geophysics at Stanford University.His research focuses on in-situ stress, faultmechanics, and reservoir geomechanics withan emphasis of shale gas, tight gas, and tightoil production. He is the author of a textbookentitled Reservoir Geomechanics, published in2007 by Cambridge University Press. He holdsa Ph.D. and an M.S. degree in geophysics fromStanford University.

ACKNOWLEDGEMENTS

We gratefully acknowledge financial supportfrom the Department of Energy (Grant DE-FE-0004731), the Research Partnership to SecureEnergy for America, and the Stanford Rock andBorehole Geophysics consortium. In addition,we thank ConocoPhillips and BP for providingsamples for us to work with. We also thankTerrilyn Olsen for her review of our article.The AAPG Editor thanks Senior AssociateEditor Terrilyn M. Olsen and the followingreviewer for their work on this paper: FarzanJavapour.

AAPG Bulletin, v. 98, no. 5 (May 2014), pp. 975–995 975

INTRODUCTION

A common characteristic of all shale gas reservoirs istheir extremely low intrinsic permeability. To pro-duce gas at any significant rate, the rock must firstbe hydraulically fractured to expose additional sur-face area and provide greater contact with the reser-voir. Although fracture properties (both natural andinduced) may dominate reservoir performance earlyon in the life of a well, it is the matrix properties thatcontrol how a well will perform over longer periodsof time. This study focuses on the fluid-flow proper-ties of the matrix.

Understanding in-situ matrix permeability andhow it evolves with depletion is a major challengeto gas shale reservoir characterization and is essen-tial for accurate production forecasting. As such,many research efforts to date have been aimed at thistopic. Several of these studies have involved meas-uring the impacts of stress on matrix permeability.For example, Bustin et al. (2008) measured the sen-sitivity of permeability to confining pressure at asingle pore pressure. In contrast, Kang et al. (2011)measured sensitivity of permeability to pore pres-sure at a single confining pressure. Kwon et al.(2001) measured intact plug permeability over arange of both pore and confining pressures. Otherefforts have aimed at profiling permeability along awell log or length of core. For example, Yang andAplin (2010) developed a permeability-porosityrelationship for mudstones using a large data setconsisting of both measured permeability valuesand estimates based on clay content and pore-sizedistribution, but did not investigate confining andpore-pressure effects. Clarkson et al. (2012) devel-oped a method for profiling permeability along ashale core sample, which was calibrated using pulsepermeability measurements on plugs. Finally, sev-eral studies have involved efforts to perform rapidpermeability estimation on crushed shale samples(Luffel and Guidry, 1989; Luffel and Hopkins,1993; Egermann et al., 2003; Cui et al., 2009). Thedisadvantages of performing measurements oncrushed samples include potentially eliminatinglarger scale features (such as microcracks) and theinability to perform measurements at reservoir con-ditions or measure how permeability evolves during

production. In this work, we draw upon many ofthese previous studies and methodologies in aneffort to further understand matrix permeability inorganic-rich gas shales and their sensitivity to botheffective confining pressure and pore pressure.

As gas is produced and reservoir pressuredecreases, two different processes alter the per-meability of the rock. These processes can be catego-rized as stress effects and flow regime effects. Stresseffects are relatively straightforward: as pore pressuregoes down, the matrix feels an increase in effectivestress (defined here as the difference between hydro-static confining pressure and pore pressure), whichnarrows pore throats and lowers the permeability ofthe rock. This effect is well known, and variation ofpermeability with effective stress has been studiedfairly extensively for sandstones and carbonates(Warpinski and Teufel, 1992; Al-Wardy andZimmerman, 2004; Ghabezloo et al., 2009). As notedabove, such measurements are much more limitedlyavailable for organic-rich shales.

The second phenomenon affecting permeabilityis a flow regime effect and is less intuitive. As dis-cussed in more detail below, transport at low gaspressure and in small pores is a mixture of Darcyflow (in which the flow rate is proportional to thepressure gradient) and diffusion flow (in whichmolecular transport exists along the walls of the flowchannels). This is sometimes referred to as slippageflow, in reference to molecules slipping past oneanother, and was initially studied by Klinkenberg(1941). These processes result in an increase in trans-port relative to pure Darcy flow, which manifestsitself as an apparent increase in the permeability ofthe rock.

In this study, we investigate stress and flowregime effects in the laboratory. An overview of thisstudy is as follows:

1. Using established pulse-decay methods, we mea-sured permeability sensitivity to effective stressin several shale samples over a range of confiningpressures and pore pressures and fit each data setwith an empirical effective stress law.

2. Using a testing methodology that allowed us toisolate stress effects from flow regime (slip flow)effects in the laboratory, we determined the extent

976 Experimental Investigation of Matrix Permeability of Gas Shales

to which slip flow becomes important duringdepletion.

3. We also measure permeability on crushed samplesfrom the same core plugs for comparison.

Finally, using the results from above, we presenta working conceptual model for flow through thatmatrix of gas shales as pressure depletes.

BACKGROUND

Stress-Dependent Permeability

Many physical properties of porous rocks (permeabil-ity, volumetric strain, porosity, etc.) vary as a func-tion of confining pressure and pore pressureaccording to an effective pressure law. This is knownas Terzaghi’s principle, and the Terzaghi effectivestress (sometimes called simple or nominal effectivestress) is defined as follows:

σTerzaghi = CP − PP (1)

where Cp is the confining pressure and Pp is the porepressure. This simple effective stress law (as it willbe referred to hereafter) is frequently modified toinclude a coefficient affecting the pore-pressureterm so that the relative importance of confiningpressure and pore pressure can be described in a sin-gle parameter. When describing how volumetricstrain varies with effective stress, the appropriatecoefficient is defined as the Biot coefficient, α (Nurand Byerlee, 1971).

Many efforts have been made to describe how per-meability depends on effective stress (Zoback andByerlee, 1975; Walls and Nur, 1979; Kwon et al.,2001). The objective of these experiments has been tofit all permeability data to an effective stress law suchthat given a confining pressure and pore-pressure com-bination, the permeability of a rock can be predicted.The effective stress law takes on the following form:

σeff = CP − χPP (2)

where σeff is true effective stress and χ is the effec-tive stress coefficient for permeability. Thus, χdetermines the relative sensitivity of permeability

to changes in confining pressure and pore pressure.Most rocks have been shown to have χ ≤ 1, indicat-ing that the rock is more sensitive to changes in con-fining pressure than pore pressure. However, ahandful of studies of clay-bearing sandstones havefound χ to be greater than 1, implying that changesin pore pressure have a larger impact on permeabil-ity than confining pressure (Zoback and Byerlee,1975; Walls and Nur, 1979). A simple conceptualmodel for this observation, developed by Zobackand Byerlee, 1975, suggests that fluid flows througha pore surrounded by a relatively soft compressiblematrix, whereas the external stresses are supportedby the stiff granular framework of the rock.Therefore, if pore pressure and confining pressurewere to increase by the same amount, for example,the compressible nature of the flow paths relative tothe incompressible frame allows the change in porepressure to have a greater effect on the pore-sizeaperture than the external stress, increasing the over-all permeability of the rock.

In contrast to observations of χ > 1 in high-claysandstones, Kwon et al. (2001) measured χ ∼ 1 invery weak, high-clay Wilcox shale samples fromthe Gulf of Mexico. Since Kwon’s investigation,other studies have measured the sensitivity of gasshale matrix permeability at a single pore pressure(Bustin et al., 2008) or at a single confining pres-sure (Kang et al., 2011). However, there have beenno previous studies of shale matrix permeabilityover a range of effective stresses and porepressures.

Flow Regime Effects on Permeability

Expanding on the introduction above, flow in gasshales (and similarly tight rocks) is described by acombination of transport mechanisms acting at dif-ferent scales (Javadpour et al., 2007; Wang andReed, 2009; Freeman et al., 2011). These includethe following:

1. Surface diffusion of molecules adsorbed to kero-gen and clay surfaces;

2. Knudsen diffusion and slip flow in micro-, meso-,and some macropores; and

3. Darcy flow in macropores.

HELLER ET AL. 977

It is common to denote transitions between thesevarious flow regimes using the dimensionlessKnudsen number, defined as

Kn =λ

dP(3)

where dP is the diameter of the pore and λ is the meanfree path of a molecule moving through it, calculatedas

λ =KBoltzTffiffiffiffiffiffiffiffiffiffiffiffiffiffi2πd2mP

p (4)

where KBoltz is the Boltzmann constant, T is the tem-perature, dm is the molecular diameter, and P is thepressure. The molecular mean free path becomeslarger as pressure (gas density) decreases. The rela-tionship between the Knudsen number, pore width,and pressure is displayed in Figure 1. At very smallKnudsen numbers (Kn < 0.01), the mean free pathof gas molecules is negligible relative to the porewidth, and the continuum assumption (Darcy flow)is valid. At very large Knudsen numbers (Kn > 10),gas molecules collide more frequently with porewalls than with each other, and thus, the continuumassumption becomes invalid, and flow rate and pres-sure drop are no longer accurately predicted usingDarcy’s law. Under these conditions, molecules nolonger move as a viscous fluid but instead move inde-pendently of one another (i.e., gas composition hasno significance), colliding back and forth betweenflow boundaries as they travel through the pore(Knudsen flow). Within the intermediate region(0.01 < Kn < 10) between these two extremes, nei-ther free molecular nor continuum flow models arevalid by themselves, and flow is a combination ofthe two mechanisms. This intermediate zone is fur-ther subdivided into slip and transition flow regimes,with the distinction between the two being the rela-tive amounts of free molecular and Darcy flow con-tributing to total flow. Table 1 summarizes thesefour types of flow regimes and the driving forcesassociated with each.

Flow in gas shales lies mostly within the Darcy,slip, and transition flow regimes. Under these condi-tions, flow has been mostly modeled as a weightedcombination of pure Poiseuille (Darcy) and Knudsenflow. Within the petroleum engineering literature,Klinkenberg (1941) was the first to do this.

Klinkenberg’s primary objective was to develop away of estimating liquid permeability from measure-ments made with gas. In contrast to gas, liquids aresufficiently dense such that the mean free path isalmost always negligible relative to the pore width,and slippage effects are absent. Therefore, to estimateliquid permeability from gas measurements, the mag-nitude of the slippage effect must be quantified andcorrected for. He suggested measuring permeabilityat several pressures and plotting the resulting per-meability measurements versus inverse pressure

(A)

(B)

Figure 1. (A) Dimensionless Knudsen number versus porediameter (nm) for methane at 100°C. Shaded region indicatespressures and pore sizes typical of gas shale reservoirs. Mostof the gas shale reservoirs lie within the slip and transition andDarcy flow regimes given their pore-size distributions and reser-voir pressures. (B) Mean free path of methane and helium ver-sus pressure. The lower the pressure, the lower the gas densityis and the larger the mean free path. The mean free path ofhelium is approximately double that of methane because of itssmaller molecular size. The mean free path of both increasessignificantly at pressures below approximately 500 psi(3.4 MPa).


(1∕P). Theory predicts a linear trend, with a y-inter-cept (infinite pressure) equal to the liquid permeabil-ity of the rock and the slope reflecting the changingmagnitude of the mean free path with pressure rela-tive to the effective pore width (Knudsen number).The equation he derives is as follows:

ka = k∞

�1þ Kb

P

�(5)

where ka is the apparent permeability (permeabilitymeasured with gas), k∞ is the permeability at infinitepressure (liquid permeability), Kb is the Klinkenbergparameter, and P is the mean flowing pressure acrossa sample. Klinkenberg cared mainly about straight-forwardly obtaining liquid permeability, k∞, whilemeasuring permeability with gas. In the context ofgas shales (with the exception of liquid plays) theparameter Kb is of more importance because it tellsus the extent to which permeability will change as afunction of pressure caused by slippage.

METHODOLOGY

Plug Permeability Measurement

All measurements were made using helium as thetest gas. This was done primarily to avoid the poten-tially significant effects of adsorption and/or associ-ated swelling that might impact permeability

(Cui et al., 2009; Kowalczyk et al., 2010). Samplepreparation included cutting, grinding, drying, andprestressing. Sample plugs were initially cut to alength of approximately 1–1.5 in. (2.5–3.81 cm),and size was cut down if permeability proved toolow for measurements to be completed in a reason-able amount of time. The ends of each plug wereground until flat and subsequently placed in a vac-uum oven at 45°C until constant mass was achievedto remove any water and/or residual hydrocarbonsfrom the core. The temperature was chosen so as tobe above the boiling point of water in near vacuumconditions, but not so high as to remove clay-boundwater and alter the clay properties. Finally, the samplewas loaded into the pressure vessel where it was pre-stressed for a period of 48–72 hr at an effective stress25% greater than that which it would experience dur-ing the experiment. The purpose of this prestressingis to remove any inelastic deformation prior to begin-ning experiments. In addition to this prestressing,measurement repeatability was carefully monitored.

A schematic of the hydrostatic permeability sys-tem used for plug permeability measurements isshown in Figure 2. The sample is wedged betweentwo floating plugs attached to pore lines inside aViton jacket. Confining pressure is controlled man-ually using the high-pressure generator shown on thefar right of Figure 2. Confining pressure is measuredusing a Heise DXD pressure transducer accurate to�0.1% up to 10,000 psi (68.9 MPa). Pore pressureis controlled by a Quizix QX-6000 pump, equippedwith two pistons capable of multiple operating modesincluding constant pressure, constant flow rate, and avariety of paired operating modes. The pump iscapable of reaching a pressure of 6000 psi (41 MPa)

Table 1. Summary of Flow Regimes, Corresponding KnudsenNumber and Driving Force Behind Each Flux*

Flow Regime Knudsen Number Driving Force

Knudsen/freemolecule flow

Kn > 10 Total concentrationgradient andmolecular weight

Transition flow 0.1 < Kn < 10 Mostly Knudsenflow with someDarcy flow

Slip flow 0.01 < Kn < 0.1 Mostly Darcy flowwith someKnudsen flow

Continuum/Darcy/Poiseuille flow

0.01 > Kn Total pressuregradient

*Mineralogy, kerogen analyses, and other sample characteristics for samplesused in this analysis. Pressure and pore sizes typical of most gas shalereservoirs place it within the slip, transition, and Darcy flow regimes.

Figure 2. Simplified schematic of hydrostatic permeabilitysystem. Gas flows from one end of the rock sample to the otherusing the Quizix pump. The entire apparatus is placed inside ofan insulated chamber, and temperature is controlled using aheat lamp with a feedback system.

HELLER ET AL. 979

and measuring pressure to an accuracy of �0.01%.All flow rates were measured via the Quizixsystem, capable of measuring flow rates as low as0.001 mL∕min, accurate to within �0.1% of the setflow rate. Temperature within both the upstream anddownstream pore lines as well as the confining pres-sure lines is measured with thermocouples. To main-tain temperature stability, a custom heat controlsystem was built using a heat lamp and a feedbackalgorithm implemented via LabVIEW™ (laboratoryvirtual instrument engineering workbench). All datawere recorded using LabVIEW as well.

Depending on the permeability of the sample,either a steady-state Darcy flow method or a pressurepulse decay method was employed. Our system iscapable of accurate Darcy flow measurements toapproximately 1 μd for a 1.5-in. (3.81-cm) sample.For samples of lower permeability, equilibrium wasnot reachable in a reasonable amount of time usingsteady-state methods, and the pressure pulseapproach was used instead. Of the six samples dis-cussed in this article (described below), three sam-ples were permeable enough for the Darcy flowmethod (Barnett 27, Eagle Ford 127, and Montney).The pressure pulse method was used for the threeother samples (Barnett 31, Marcellus, and EagleFord 174).

The pressure pulse technique was first developedby Brace et al. (1968) to measure permeability ofgranite. The principle is to create a dynamicallychanging flow across the sample and then fit theresulting pressure curves to a flow model, thus deter-mining permeability. In our case, we stepped thepressure on the upstream side by only 50 psi(0.34 MPa) to minimize the impact of a nonuniformeffective stress (and thus, a nonuniform permeability)along the length of the sample. Although it is typicalto increase the upstream pressure by a certain initialamount and then allow the upstream pressure todecline as the downstream rises, we set up our experi-ments with a constant pressure on the upstream side,creating an effective infinite reservoir volume(Zoback and Byerlee, 1975). This was done primarilyto compensate for any gas leakage on the upstreamside of our system.

An example of the downstream pressure recordedduring a pulse-permeability test is shown in

Figure 3A. When the logarithm of the difference inpressure between the upstream and downstream sidesis plotted versus time (Figure 3B), the decay clearlyfollows a linear trend. This line can be fit by thefollowing equation (after Brace et al., 1968):

ΔPðtÞ = ΔPOe−αt (6)

α =kA

βVdownLμ(7)

where α is the decay exponent (the slope of the lineon a plot of lnðΔPÞ versus time), ΔPðtÞ is the differ-ence in pressure between the upstream and down-stream sides at time t, ΔP0 is the difference inpressure between upstream and downstream at timet = 0, β is the compressibility of the gas, Vdown isthe volume of the downstream reservoir, L is thelength of the sample, A is the cross sectional area of

(A)

(B)

Figure 3. Pulse permeability example, showing the upstreamand downstream pressure curves (A) and the logarithmic decayof the difference in pressure between the two reservoirs (B).


the sample, and μ is the viscosity of the fluid flowingthrough the sample.

To independently test the effects of effectivestress, confining pressure, and pore pressure on per-meability, it is necessary to measure the permeabilityover a wide range of Pp and Cp values. An exampleof the procedure is shown in Figure 4. At each of sev-eral pore-pressure steps, the sample is subjectedto incrementally increasing confining pressures,and permeability is measured at each increment.Confining pressure is then decreased in the sameincrements, and permeability is again measured tomonitor for repeatability. Following this sequence,pore pressure is increased, and the process is repeatedup until the last pore pressure of interest (typically4000 psi [27.6 MPa]).

To calculate the effective stress parameterfor permeability, we use the ratio of slope methodfollowing Kwon et al. (2001) using the followingequation:

χ =�−

∂ log k∕∂PP

∂ log k∕∂CP

�(8)

The change in permeability with change in porepressure at a given confining pressure is divided bythe change in permeability with change in confiningpressure at a given pore pressure. This ratio definesthe relative sensitivity of permeability to changes inpore and confining pressures. The use of the

logarithm of permeability is based purely on empiri-cal fitting of the permeability decline as effectivestress increases.

Although it is common to terminate measure-ments once ΔPðtÞ reaches approximately 30%ofΔP0 (one-third the way to equilibrium), we allowedeach step to proceed nearly to equilibrium to aid inleak rate characterization (see the Discussion sectionfor further elaboration on leak detection). On average,this took approximately 24 hr. A full suite of confin-ing pressure and pore-pressure combinations (asshown in Figure 4) took approximately 30 days tocomplete for a single sample.

A methodology for separating stress and flowregime effects on permeability will be presented inthe next section using our experimental data.However, once this is accomplished, we can createstandard Klinkenberg-style plots that can be used toestimate the effective aperture of the flow paths(Soeder, 1988). Suppose we assume that thetotal mass flow is the sum of the viscous flow plussome empirical constant times the Knudsen flow(Klinkenberg made a similar assumption). In addi-tion, we assume the pore geometry to be slit-shapedbased on observations of scanning electron micros-copy (SEM) images. By modeling the viscous fluxusing the Poiseuille equation and combining this withthe Knudsen flow and Klinkenberg equations, we canderive the following:

w =16cμKb

�2RTπM

�(9)

where w is slit width, c is an empirical constant, μis the viscosity, R is the gas constant, T is the temper-ature,M is the molar mass, and Kb is the Klinkenbergconstant. This provides a direct linkage between theempirical Klinkenberg constant and the effectivepore aperture (full derivation presented inAppendix 1).

Crushed Permeability Measurements

Shale permeability is frequently measured on crushedshale sample as opposed to plugs because of thelength of time required to make accurate measure-ments on such low-permeability rock (Luffel and

Figure 4. Confining pressure and pore-pressure steps duringpermeability testing. A long hold for equilibrium exists after eachpore-pressure change. Tests are conducted with increasing effec-tive stress and decreasing effective stress to monitor for hystere-sis affects. A typical suite of measurements as shown in thisfigure takes approximately 30 days to complete.

HELLER ET AL. 981

Guidry, 1989; Luffel and Hopkins, 1993; Egermannet al., 2003). However, many benefits of measuringpermeability on intact core plugs exist as opposed tocrushed core, including straightforward and estab-lished methods for interpreting data, sampling a large(and perhaps more representative) volume of rock,and the ability to test the influence of confining stresson permeability. Given our experimental objectives,we chose to perform our experiments on intact coreplugs. However, crushed permeability measurementswere additionally performed on several of the sam-ples for comparison.

Once plug permeability measurements were com-plete, several core samples were subsequently crushedto a particle size of 1000–2000 μm and again placed ina vacuum oven at 45°C until constant mass wasobserved, implying that moisture had been removed.All crushed permeabilities measured were made onthese chips inside a dual-chamber pressure apparatus.A schematic of this simple setup is shown in Figure 5.

The setup consists of a reference cell and samplecell, placed inside of a temperature-controlled oil bathcapable of maintaining a constant temperature to�0.05°C. The volumes of both cells are adjustableto minimize dead volume and allow measurementsto be made on varying amounts of sample withoutsacrificing sensitivity. The crushed sample wasplaced inside the sample cell, and the entire systemwas vacuumed to remove air. The valve separatingthe reference and sample cells was then closed, andthe reference cell filled with helium to a pressureof about 150 psi (1 MPa). After pressure and

temperature stabilization was reached, this gas wasallowed to expand into the sample cell. The dead vol-ume within the lines and between the particles isfilled immediately, and the pressure subsequentlydecays as gas permeates the shale particles.Equilibrium was reached after approximately 10 minor less. The pressure vs. time data leading up to equi-librium can then be used to estimate permeability, asfirst outlined by the Gas Research Institute (Luffeland Guidry, 1989) and later elaborated on by otherauthors (Luffel and Hopkins, 1993; Egermann et al.,2003; Profice et al., 2011). In this study, we interpretthe pressure decay curves in the means outlined byCui et al. (2009).

SAMPLE DESCRIPTION

In this article, results will be presented from twoBarnett, two Eagle Ford, one Marcellus, and oneMontney shale sample. The Barnett Shale is aMississippian-age shale located in the Forth WorthBasin in north central Texas. The Eagle Ford Shaleis a Late Cretaceous-age shale located in southTexas. The Montney Shale is an Early Triassic-ageshale, located within the Western CanadaSedimentary Basin in British Columbia and Alberta.

With the exception of the Marcellus plug, allsamples were oriented horizontally (sample axis ori-ented parallel to the surface of the Earth) such thatthe direction of flow is parallel to the bedding. TheMarcellus was a vertical sample (sample axis orientedperpendicular to the surface of the earth); thus, flowwas perpendicular to bedding. The mineralogy ofthe samples is shown in Figure 6, and a table withmineralogy and other sample information is includedin Appendix 2. Mineralogy varies significantly fromsample to sample, particularly the amount of quartz,carbonate, and clay. Both Barnett samples containeda high amount of quartz, whereas the Eagle Ford sam-ples were dominated by carbonates and the Marcellusby clay. The Montney sample comprises a relativelyeven mixture of quartz, carbonate, clay, and feldspar.

In addition to mineralogy, three of the samplescontained visible anomalies worth noting. BothBarnett 27 and Eagle Ford 127 samples contained cal-cite streaks that extended across the width and length

Figure 5. Schematic of system used for crushed permeabilitymeasurements. Both the reference and sample cells are placedinside of a temperature-controlled oil bath for stability. The vol-umes of both cells are adjustable using aluminum spacers toaccommodate varying sample sizes without sacrificing measure-ment sensitivity.


of the samples. The Montney sample contained a veryminor crack (visible only when wet) oriented alongits axis. Both the calcite streaks and crack enhancedpermeability significantly, as will be described in theresults section.

RESULTS

Plug Permeability Measurements

For all six samples studied, permeability wasmeasured at pore pressures from 1000 to 4000 psi(6.9 to 27.6 MPa), confining pressures from 2000 to8000 psi (13.8 to 55 MPa), and simple effectivestresses ranging from 1000 to 4000 psi (6.9 to27.6 MPa). These data were fit to an effective stresslaw. Of those six samples, three were chosen for addi-tional permeability measurements at pore pressures

below 1000 psi (6.9 MPa). The results for the highpore-pressure data (all six samples) will be presentedfirst, followed by a more in-depth look at the rocksfor which low-pressure permeability measurementswere also made.

Permeability as a function of simple effectivestress (Cp − Pp) is shown for all six samples inFigure 7 at different pore pressures. Several featuresare worth pointing out. First, note the scale differencefor the Barnett 27, Eagle Ford 127, and Montneysamples (microdarcy as opposed to nanodarcy) andrecall that each of these relatively high-permeabilitycore plugs contained an anomalous throughgoing fea-ture. Both the Barnett 27 and Eagle Ford 127 samplescontained a carbonate layer extending throughthe core plugs. We believe these carbonate layers arehighly permeable relative to the rest of the shalematrix, resulting in a larger than typical plug per-meability value in these two samples. In addition,recall that the Montney sample contained whatappeared to be a very minor crack oriented along itsaxis, which seems to have enhanced permeability.Finally, notice that for each sample, permeabilityvaries as a function of simple effective stress(Cp − Pp) in a similar way, allowing for a singlefunction to be fit to the whole data set.

The data for each rock were fit to a unique per-meability effective stress law as described previously.These results are presented in Figure 8. In all cases, χwas found to be less than 1, indicating that the rocksare more sensitive to changes in confining pressurethan changes in pore pressure. Noting that permeabil-ity as a function of modified effective stress forms atrend enables us to attribute all permeability variationobserved (for Pp > 1000 psi [6.9 MPa] thus far) toeffective stress effects.

As previously mentioned, three of the six sampleswere chosen for further characterization at lower porepressures (Pp < 1000 psi [6.9 MPa]). A typical per-meability sample (Eagle Ford 174), a high-permeability sample (Eagle Ford 127), and a verticalsample (Marcellus) were selected for variation.Permeability versus effective stress is plotted on theleft in Figure 9 using the same effective stress law(same χ) previously found to fit the high-pressuredata. It is apparent in this figure that the measurementsmade at low pore pressure deviate from the trend fit by

Figure 6. Mineralogy of samples included in this study (A).Ternary diagram representation of sample composition (B).Mineralogy varies significantly from sample to sample, particu-larly the amount of quartz, carbonate, and clay. Both Barnettsamples contained a high amount of quartz, whereas the EagleFord samples were dominated by carbonates and theMarcellus by clay. The Montney sample comprised a relativelyeven mixture of quartz, carbonate, clay, and feldspar.

HELLER ET AL. 983

the high pore-pressure data. We attribute this per-meability enhancement (relative to what one wouldpredict given the effective stress) to slippage (i.e.,Klinkenberg) effects, and we proceed to investigatehow permeability varies as a function of pore pressurebecause of non-stress-related (Klinkenberg) effects.We select several values of true effective stress (indi-cated in Figure 9 by the black dashed lines) and createKlinkenberg plots for each by selecting points ofintersection with the pore-pressure trend. The resultsare shown in the right half of Figure 9.

Several aspects of these Klinkenberg plots areworth pointing out. First, note the increase in theKlinkenberg constant (Kb) with increasing effectivestress for each rock. This increase in Kb essentiallyrepresents an increase in gas slippage with increasingeffective stress. This observation is intuitively reason-able, given that one would expect an increase ineffective stress to narrow flow paths, thereby increas-ing the Knudsen number and the contribution fromslip flow. It is also worthwhile to compare the magni-tude of Kb between samples. Both the Eagle Ford 174

Figure 7. Permeability versus simpleeffective stress (Cp − Pp). Note that per-meability units for samples Barnett 27,Eagle Ford 127, and Montney samples(right column) are in microdarcys; allother samples are in the nanodarcy range.


and Marcellus samples are characterized as having asimilar permeability, as well as similar contributionsfrom slip flow as evidenced by the similar Kb values.This contrasts with the higher permeability EagleFord 127 sample, which has a lower Kb and, thus, asmaller overall contribution of slip flow to total flow.The overall contribution of slip flow to total flow inthese samples will be further considered in the discus-sion section.

With Klinkenberg parameters established foreach sample, we are now able to use equation 9 to cal-culate the effective size of the flow paths. Moreover,

because we have also determined how Kb varies witheffective stress for each rock, we can estimate howthe effective size of the flow paths varies with effec-tive stress as well. These results are presented inFigure 10. Our calculations suggest that the effectivepore size varies moderately with effective stress(about 5% in the case of Eagle Ford 127, factor of 2in the case of Marcellus, and a factor of about 5 inthe case of Eagle Ford 174). Although many assump-tions went into these estimates (slit-shaped pores inparticular), the results seem plausible based on theapertures of pore sizes commonly observed in SEM

×

×

× ×

×

×

Figure 8. Permeability versus modifiedeffective stress after data were fit to theeffect stress law for each sample. In allcases, χ was found to be less than 1, indi-cating that the rocks are more sensitive tochanges in confining pressure thanchanges in pore pressure. Noting that per-meability as a function of modified effec-tive stress forms a trend enables us toattribute all permeability variationobserved (for Pp > 1000 psi [6.9 MPa]thus far) to effective stress effects.

HELLER ET AL. 985

photos in the shale gas literature (Loucks et al., 2009;Curtis et al., 2010).


Of the six core plugs, four were chosen for crushedpermeability measurements at particle sizes of1000–2000 μm. As previously described, the experi-ments were performed by expanding gas from a refer-ence cell into a separate chamber containing the shaleparticles. Fractional uptake (defined as the fraction of

gas initially within the reference cell that has enteredthe shale particles) is plotted as a function of time inFigure 11. Permeability is approximately propor-tional to the amount of time required to reach equilib-rium (when all intraparticle porosity has been filled).The magnitude of the fractional uptake (height ofcurve on the y-axis) at equilibrium is simply depen-dent on the porosity within the particles.

The data displayed in the fractional uptake versustime curves were interpreted following Cui et al.(2009), and lower and upper bounds on permeability

×

×

×

Figure 9. Left: permeability versusmodified effective stress with low pore-pressure data. Note the deviation from thehigh pore-pressure trend by the low pore-pressure points. Black dashed lines indi-cate effective stress isolines used for gen-erating Klinkenberg plots by selecting theirintersection with the permeability trendfor several pore pressures (Pp = 250, 500,and 750 psi [1.7, 3.4, and 5.1 MPa]). Right:standard Klinkenberg plots (permeabilityversus 1∕Pp). Permeability points wereselected for true effective stress values of2000, 3000, and 4000 psi (13.8, 20.7, and27.6 MPa). Notice the increase in Kb value(slope divided by intercept) with increas-ing effective stress.


were estimated. The results are displayed inFigure 12. We calculate the permeability at this scaleto be within the narrow range of about 1 to 10 nd.Because these measurements were made on the samesamples as the plug measurements, we directly com-pare the calculated permeability values using thetwo different approaches and sample sizes inFigure 13. Note the large difference between per-meability at the two different scales, with permeabil-ity at the plug scale being much higher than that at

the crushed particle scale. This suggests that muchless heterogeneity exists at smaller scales than atlarger scales, where microcracks and other features(such as carbonate layers) could enhance permeabil-ity. Moreover, we observe no systematic trendbetween plug and crushed permeability measure-ments. A discrepancy in permeability between thetwo different scales and lack of an obvious trendrelating the two were also reported by Sinha etal. (2012).

SOURCES OF ERROR

Any investigation of permeability in such low per-meability and complex formations would not be com-plete without discussing some of the potential sourcesof error. We begin this section by discussing generalsources of error relevant to both plug and crushedmeasurements and follow that by addressing addi-tional challenges specific to each approach.

General Error Discussion

An important consideration with regard to both plugand crushed permeability experiments is that we haveperformed all flow measurements using helium asopposed to methane. As previously discussed, with asmaller molecular diameter (helium has a diameterof 0.25 nm, whereas methane is 0.38 nm), heliumhas a larger mean free path compared to methane ata given pressure. Thus, the contribution of slip flowis enhanced relative to methane. In addition, as anadsorbing gas, the flow of methane may be impededby the presence of an adsorbed layer. Such a complexcombination of physics poses a major experimentalchallenge, particularly in the context of materials withflow paths as intricate as gas shales.

Another potentially significant detail worth dis-cussing is the presence of residual oil in our samples.As mentioned, our samples were dried in a vacuumoven at 45°C prior to all experiments. Althougheffective at removing residual water, these conditionsare insufficient to remove heavier hydrocarbons thatmay be trapped within the samples. Despite onlyone of our samples being conclusively within thegas window, very little reported oil production from

2000 2500 3000 3500 40000

20

40

60

80

100

120

140

Effective Stress (psi)

Effe

ctiv

e P

ore

Wid

th (

nm)

Eagle Ford 127MarcellusEagle Ford 174

Figure 10. Effective pore width versus true effective stress, asestimated for the Klinkenberg parameter (Kb). Our calculationssuggest that the effective pore size varies moderately with effec-tive stress. Despite having to make several assumptions to arriveat these estimates, the results seem plausible based on the aper-tures of pore sizes commonly observed in SEM photos in the gasshale literature.

Figure 11. Fractional uptake (defined as the fraction of gasinitially within the reference cell that has entered the shale par-ticles) versus time. In this study, we interpret these pressuredecay curves as outlined by Cui et al. (2009). Permeability isapproximately proportional to the amount of time required toreach equilibrium.

HELLER ET AL. 987

the wells exists in which these samples were col-lected. Therefore, we can safely assume that the rocksamples were not heavily saturated in oil. In addition,any hydrocarbons that were present are likely to berelatively immobile and simply acting as part of therock. Thus, for a given sample, residual hydrocarbonswould impose the same influence for all measure-ments and, therefore, not impact our results in anysignificant way.

Plug Permeability Measurements

The rate of flow of gas through the shale plugs isextremely low. Thus, minute system leaks can be animportant challenge in these experiments. We wereable to overcome this issue in several ways. First,leaks were minimized by keeping the number of fit-tings and valves to a minimum. All connections werecarefully adjusted and monitored for leaks prior toexperiments, including numerous pressure tests. The“leakier” side of the system was always chosen as

dnd

ndnd

ndnd

ndnd

Figure 12. Lower and upper crushed permeability bounds (indicated by dashed lines bounding data) calculated using the methodoutlined by Cui et al. (2009). We calculate the permeability at this scale to be within the narrow range of about 1 to 10 nd. A compari-son of crushed and intact permeability values is made in Figure 13.

Figure 13. Crushed permeability versus intact plug per-meability. Error bars indicate range of crushed permeability esti-mate. Plug permeability at zero effective stress was estimated byextrapolating the permeability-effective stress trends shown inFigure 8 to zero effective stress. We observe a narrow range ofpermeability at the particle scale (approximately 1–10 nd)regardless of the magnitude at the plug scale. In addition, wedo not identify any obvious trend relating permeability at thetwo different scales.


the upstream side and, thus, negated given our con-stant upstream pressure boundary condition. Typicalleak rates from the downstream volume were lessthan 10−8 mol∕second. Note the relatively largedownstream volume we have chosen (∼15 mL)relative to what may be typical for these experiments.Using a larger volume helps minimize the signifi-cance of leak rates on the pressure signal, albeit atthe expense of longer equilibration times.

Given that setting up an experiment involvesreconnecting the pore lines each time, it would notbe valid to assume a leak rate measured in a calibra-tion run to be the same leak rate occurring duringour experiments. However, we believe we can safelyassume a lower bound on leak rate to be our best cal-ibration run and, therefore, correct all pressure datafor this minimum rate. In Figure 14, we present anexample of the data before and after the minimum-leak-rate correction, as well as curves demonstratingbest-guess-leak-corrected and overcorrected data.Comparing these curves at early times (1–3 hr) andestimating permeability from each, it is apparent thatleaks have a minimal impact on the data such thatregardless of whether the permeability is estimatedfrom the raw data, minimum-leak-corrected, best-guess-leak-corrected or overcorrected data, the

permeability estimate lies in a narrow range (9.4 to11.7 nd). If permeability were estimated from latertime data, however, the results would be quite differ-ent. Therefore, we chose to estimate all permeabilityfrom early time data (1–3 hr) so that the impact ofleaks and leak corrections is minimized.

As discussed, great efforts were taken to mini-mize sources of error to the pressure vs. time data.Nevertheless, our data are not without error, androom for improvement exists. In Table 2, we summa-rize what we believe to be some of the key technicalchallenges standing in the way of higher quality data,and the approaches that we have already taken or areconsidering for future experiments.

In addition to minimizing error in the pressure vs.time data, several studies have suggested alternativeapproaches for analyzing these curves. In particular,authors have examined higher order models thatmight more accurately describe the combination offlow regimes acting in these rocks (Roy et al., 2003;Javadpour, 2009). Darabi et al. (2012) modified equa-tion 6 to include slip flow and Knudsen diffusion andused a data-matching approach to arrive at a final per-meability estimate. Future works might compare theresults achieved using such sophisticated approacheswith those attained through more traditional methods.

Figure 14. Left: pressure in the upstream and downstream volume of our system versus time during a permeability measurement. Inaddition to the raw data, minimum-leak-rate-corrected data, best-guess-leak-rate-corrected data and leak-overcorrected data areshown as well. Estimating permeability from early time data (1–3 hr) minimizes the impact of leaks and leak corrections, such thatregardless of the leak correction approach, permeability lies within a narrow and acceptable range.

HELLER ET AL. 989


Permeability experiments on crushed samples aremuch simpler and faster to perform, but along withthis convenience carry many challenges from a dataanalysis standpoint. Estimates are very sensitive toboth porosity and system volume measurements. Wehave mostly diminished this source of uncertaintyusing a carefully designed system with adjustablevoid volumes for optimal sensitivity. We haveassumed that after initial gas expansion and fillingof void volume, the pressure change was smallenough to neglect compressibility effects. We havealso assumed all particles to be spherical in shape ashas been done by previous investigators. In addition,we emphasize that we report bounds on crushed-sample permeability as opposed to a single-value esti-mate. The resolution and sampling frequency of ourpressure transducers limited our ability to more pre-cisely constrain the shape of the pressure decay curve.Finally, despite recognizing its potential significanceto flow at such low pressures, we have neglectedKlinkenberg slippage effects in our analysis. Untilmeasurement and data analysis standards are estab-lished for crushed permeability, adding complexities

like the Klinkenberg effect provide little benefit.Other authors have cited the need for establishingsuch standards as well (Sondergeld et al., 2010;Sinha et al., 2012).

DISCUSSION

A summary of all experiments performed and datapresented thus far is presented in Table 3. We firstdeveloped permeability effective stress laws for sixsamples from four different shale plays. The magni-tude of the Klinkenberg effect was measured for threeof these six samples by making permeability mea-surements at relatively low pressures and separatingthe stress effects from slippage effects using a graphi-cal method. Based on the magnitude of theKlinkenberg effect, we were able to make pore widthestimates. Finally, four of the six samples werecrushed for further permeability characterization atthe particle scale.

All of our intact plug permeability measurementslay within a similar range of those presented in the lit-erature by other authors. In the case of the BarnettShale, we measured one sample to have a

Table 2. Experimental Challenges, Actions that Have Been Taken to Address Those Challenges as Well as Potential FutureImprovements

Technical Challenge Action Taken Level of Influence Potential Improvements

Helium leaking out ofupstream volume

Infinite upstream volume boundarycondition (imposed using QX pump)renders leaks irrelevant onupstream side.

None N/A

Helium leaking out ofdownstream volume and/or through jacket

Leak rate low; corrected for lowestobserved. Relatively largedownstream volume reducessignificance of leak rate.

Low Switch from helium to argon gas formeasurements. Further reduceconnections on downstream side.

Helium bypassing sample Control experiment with steel plug inplace of sample showed noevidence of this.

None Repeat control experiment regularlyto monitor for issues related tosample bypass.

Confining stress fluctuationswith temperatureimpacting the shape of theΔPðtÞ curve

Pressure data analyzed whereconfining pressure and temperatureare stable.

Moderate Thermal stability of system has beenimproved. An electronic pump couldbe installed to ensure confiningpressure stability.

Nonuniform fluid propertiesacross length of coresample

Minimized using smaller pulses at lowpressures, where fluid propertiesvary more significantly with pressure.

Dependent on gas Other approaches for data analysis(example, numerical simulations ofpressure vs. time data).


permeability in the range of 60 to 160 nd dependingon effective stress, whereas another contained a car-bonate streak and ranged from about 0.8 to 1.8 μd.Kang et al. (2011) similarly measured the permeabil-ity of a Barnett plug to be in the range 10 to 100 nd,whereas Wang and Reed (2009) measured a plug per-meability in the range of 1 to 10 μd. We measured per-meability of two Eagle Ford samples: one with apermeability in the range of 5 to 40 nd and anotherwith a carbonate streak in the range of 6 to 12 μd.Mullen (2010) described the permeability of theEagle Ford as ranging from 1 to 800 nd. TheMontney sample we characterized turned out to bevery permeable (8 to 14 μd), although this was likelycaused by the presence of a crack. However, note thatother studies of Canadian shale samples have mea-sured permeability in a similar range of 1 to 10 μd(Clarkson et al., 2012). Finally, we measured verticalpermeability in the Marcellus to range from 20to 60 nd.

Direct comparison of our crushed permeabilityresults with other studies is difficult because ofdifferences in methodology and particle size. Forexample, Sinha et al. (2012) sent the same set of foursamples to two different vendors for crushedpermeability measurement. One vendor calculatedpermeability to be 10 to 20 nd, whereas another ven-dor estimated permeability to be between 100 and1000 nd. Cluff et al. (2007) presented approximately20 Barnett shale crushed permeability measurementswith an average value of about 0.1 nd, but rangingto as low as 1 pd. Guidry et al. (1995) shares resultsof about 20 “black shale” crushed permeability mea-surements, ranging in magnitude between 1 and1000 nd. An order-of-magnitude comparison of our

results with these studies places our estimates approx-imately in the middle of the range.

Our permeability effective stress experimentsindicate that the matrix permeability of gas shales islikely more sensitive to changes in pore pressure thanchanges in confining stress, as indicated from ourobservation of χ < 1 in all of our samples. Given theseresults, the model proposed by Zoback and Byerlee(1975), in which a compressible pore space existswithin a stiff granular framework, must be modifiedfor shales. Kwon et al. (2001) similarly observed χ <1 in their measurements of the Wilcox shale and pro-posed a model in which clays form a connected matrixand are thus similarly affected by confining pressureand pore pressure (this model is shown in Figure 15).This work seems to confirm this hypothesis.However, given our estimation of pore widths andobservation of pores of a similar width within bothclay and kerogen in SEM images, we believe con-nected flow paths to exist in kerogen as well. From aproduction perspective, the implications of χ < 1 arepositive because decreasing the pore pressure by agiven amount will reduce the permeability of the shaleto a lesser amount. Nevertheless, the effective stress isstill increasing, and one can expect the capacity of thematrix to allow Darcy flow to decline with pore pres-sure during production.

Although effective stress effects narrow the aper-ture of flow paths and inhibit fluid flow, slip flowcould offer a mechanism of transport to offset thereduced Darcy flux. In other words, as pressuredecreases during production, competing processesacting on the rock matrix exist, one acting to decreaseflow capacity (pore compression) and one acting toincrease it (slip flow). Because we have measured

Table 3. Summary of Experiments Performed and Data Presented in This Work

SamplePp Range(psi)

Cp Range(psi) χ

Plug Permeability atσeff = 0 (nd)

CrushedPermeability

(nd) Comments

Barnett 27 1000–4000 1000–4000 0.82 1200 10.5 Carbonate streakBarnett 31 1000–4000 1000–4000 0.68 100 8.0Eagle Ford 127 250–4000 1000–4000 0.60 9000 2.0 Carbonate streakEagle Ford 174 250–4000 1000–4000 0.40 15 N/AMarcellus 250–3000 1000–4000 0.15 35 3.5Montney 1000–4000 1000–3000 0.85 11,000 N/A Small crack

HELLER ET AL. 991

the magnitude of slip flow occurring in our samplesas a function of effective stress, we can estimatehow the total contribution to flow from each transportmechanism will evolve during production. Theseresults are presented in Figure 16, in which the contri-bution of diffusive flux (slip flow) relative to Darcyflux is plotted as a function of pore pressure andeffective stress. In the two lower permeability sam-ples (Eagle Ford 174, and Marcellus), the ratiobecomes greater than one near a pore pressureof 400 psi (2.8 MPa), indicating an important

contribution from slip flow at such conditions. In thecase of the more permeable Eagle Ford 127 sample,the contribution from slip flow is less significant,although still not negligible, at pore pressures lessthan 400 psi (2.8 MPa).

Comparing the contribution of slip flow to totalflow as a function of effective pressure, one observesan apparent relationship between this ratio and effec-tive stress. In the case of the two Eagle Ford samples,increasing effective stress at a given pore pressureresults in an increase in the relative contribution of

Figure 15. Left: simplifiedpore models developed byZoback and Byerlee (1975)sandstone with strong stress-supporting frame (black) andhigh-compressibility clays(gray) lining pores (white).Right: Kwon et al. (2001) modelwhere clay forms a connectedmatrix in which flow pathsreside. Given our observationof χ < 1 in all samples, wesupport the Kwon model (onthe right) for shales, in whicheffective pore widths are simi-larly affected by changes inconfining and pore pressure.

Figure 16. Contribution of slip flow rel-ative to Darcy flux as a function of porepressure and effective stress. Note in thetwo lower permeability samples (EagleFord 174 and Marcellus), the ratiobecomes greater than 1 at about 400 psi(2.8 MPa). In the case of the more per-meable Eagle Ford 127 sample, the con-tribution from slip flow is less significantbut not negligible at pore pressures lessthan approximately 400 psi (2.8 MPa).


slip flow (likely for narrowing pores, thereby bothdecreasing Darcy flux and increasing Knudsen flow).In the case of the Eagle Ford 174 sample, increasingeffective stress from 3000 to 4000 psi (20.7 to27.6 MPa) seems to drastically impact this ratio forreasons that are unclear. Recall that this was the leastpermeable sample of the group. Finally, notice thatthe Marcellus vertical sample does not behave in thesame way as the two Eagle Ford samples in responseto increasing effective stress. More specifically, therelative contribution of slip flow to total flow appearsto remain rather constant.

Note that the magnitude of slip flow will be lesswith methane as the flowing gas relative to helium.Given that the mean free path of helium is roughly dou-ble that of methane at the pressures studied in theseexperiments, it is likely that the magnitude of the slip-page effect for methane would be roughly half whatwe measured for helium. Although this implies less ofa contribution to total flow than shown in Figure 16, itdoes not obviate the conclusion that slip flow may bean important contributor to total flow at low pressures.Moreover, note that in the case of methane, the poten-tial for surface diffusion exists (Tien, 1994). Despitebeing an entirely different and separately occurringtransport mechanism, surface diffusion would superfi-cially be the same factor as slip flow. That is, it wouldallow an enhancement of fluid flow at low pressures.This additional flow mechanism may partially offsetthe lesser amount of slip flow in the case of methane,resulting in a similar cumulative importance of all dif-fusive fluxes at low pore pressure.

CONCLUSIONS

We have conducted laboratory experiments on gasshales examining the effects of confining stress andpore pressure on matrix permeability. Our results showthe matrix permeability of gas shales to be more sensi-tive to changes in confining pressure than changes inpore pressure. In addition, data suggest that the effec-tive permeability of the rock is significantly enhancedat very low pore pressures (<500 psi [<3.4 MPa])because of slippage effects. By separating these effects,we are able tomake standard Klinkenberg permeabilityplots and calculate the effective size of the flow pathswithin the sample and how those sizes might vary with

effective stress. Our estimates are on the order of tensof nanometers, in general agreement with SEM imageobservations of pores within kerogen. Finally, we cal-culate the relative contribution of slip flow to Darcyflow as a function of pore pressure and effective stress.We estimate a significant contribution from slip flow tototal flow at pore pressures less than about 400 psi(2.8 MPa), perhaps exceeding that of Darcy flow intighter, lower permeability rocks. We hypothesize thatthe increasing importance of slip flow at low pore pres-sures may help explain the relatively long and flat pro-duction tails observed in some shale plays (e.g., Valkoand Lee, 2010).

APPENDIX 1: DERIVATION OF EFFECTIVEPORE SIZE AS A FUNCTION OFKLINKENBERG CONSTANT

Suppose we assume that the total flow is the sum of the viscousflow plus some empirical constant times the Knudsen flow(Klinkenberg made a similar assumption). We also assume aslit-shaped pore geometry. If we model the viscous flow usingthe Poiseuille equation, we can write the following:

Q =w4

12μLΔPþ c

43P

�2RTπM

�1∕2 w3

LΔP (10)

where Q is the total volumetric flow rate (m3∕s), the first term isthe viscous flow rate, the second is the Knudsen flow rate, c isan empirical constant, and w is the slit width. If we combine thiswith Darcy’s Law,

Q =kAμL

ΔP (11)

and then solve for permeability, we get

k =w4

12Aþ c

43μw3

PA

�2RTπM

�1∕2

(12)

which we can then rewrite as

k =w4

12A

�1þ 16cμ

wP

�2RTπM

�1∕2

�(13)

Equation A4 has the same form as the Klinkenberg equation(equation 5). Thus, this simple model yields the following rela-tion for the Klinkenberg constant, Kb:

Kb =16cμw

�2RTπM

�1∕2

(14)

which can easily be solved for w,

w =16cμKb

�2RTπM

�1∕2

(15)

providing us with a direct linkage between the empiricalKlinkenberg constant and pore aperture.

HELLER ET AL. 993

APPENDIX 2: SAMPLE PROPERTIES

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Barnett 27 Barnett 31Eagle Ford

127Eagle Ford

174 Marcellus Montney

Depth (ft) 8629.2 8640.8 12,771.35 12,845.25 6300 7614.52Depth (m) 2630.2 2633.7 3893.70 3916.23 1920.73 2321.50Plug orientation Horizontal Horizontal Horizontal Horizontal Vertical HorizontalLength (in.) 0.95 0.64 1.50 0.27 0.32 1.23Length (cm) 2.4 1.61 3.81 0.69 0.81 3.12

TOC (%) 3.8 5.3 1.81 4.4 1.17 2.04Quartz (%) 56.7 51.3 7 16.4 38 42.3Plagioclase/feldspar (%) 3.8 4 4 1.9 6 11.9Calcite (%) 7.7 0 80 47.5 1 8.1Dolomite (%) 1.4 0.4 1 0 1 9.9Pyrite (%) 1.8 1.7 1 6.7 1 3.5Apatite (%) 1 0 2 0.6 1 0Total clay (%) 23.8 37.4 5 22.4 52 24.1

S1 (mg HC/g rock) 4.8 4.4 2.88 N/A 2.06 N/AS2 (mg HC/g rock) 4.5 6.1 1.36 N/A 5.15 N/AS3 (mg HC/g rock) 0.3 0.3 0.54 N/A 0.31 N/ATmax (°C) 448 452 466 N/A 369 N/AHI (mg HC/g TOC) 119 115 75 N/A 439 N/AOI (mg CO2∕g TOC) 8 6 30 N/A 26 N/A


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Experimental investigation of matrix permeability Department …...Experimental investigation of matrix permeability of gas shales Rob Heller, John Vermylen, and Mark Zoback ABSTRACT