[blasingame] spe 107954

Copyright 2007, Society of Petroleum Engineers

This paper was prepared for presentation at the 2007 SPE Rocky Mountain Oil & GasTechnology Symposium held in Denver, Colorado, U.S.A., 16–18 April 2007.

This paper was selected for presentation by an SPE Program Committee following review ofinformation contained in an abstract submitted by the author(s). Contents of the paper, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, as presented, does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Papers presented atSPE meetings are subject to publication review by Editorial Committees of the Society ofPetroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paperfor commercial purposes without the written consent of the Society of Petroleum Engineers isprohibited. Permission to reproduce in print is restricted to an abstract of not more than300 words; illustrations may not be copied. The abstract must contain conspicuousacknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.Box 833836, Richardson, TX 75083-3836, U.S.A., fax 01-972-952-9435.

AbstractThis work addresses the problem of estimating Klinkenberg-corrected permeability from single-point, steady-statemeasurements on samples from low permeability sands. The"original" problem of predicting the corrected or "liquidequivalent" permeability (i.e., referred to as the Klinkenberg-corrected permeability) has been under investigation since theearly 1940s — in particular, using the application of "gasslippage" theory to petrophysics by Klinkenberg.1

In the first part of our work, the applicability of the Jones-Owens4 and Sampath-Keighin5 correlations for estimating theKlinkenberg-corrected (absolute) permeability from single-point, steady-state measurements is investigated. We alsoprovide an update to these correlations using modern petro-physical data.

In the second part of our work, we propose and validate a new"microflow" model for the evaluation of an equivalent liquidpermeability from gas flow measurements. This work is basedon a more detailed application of similar concepts employedby Klinkenberg. In fact, we can obtain the Klinkenberg resultas an approximate form of our result. Our theoretical "micro-flow" result is given as a rational polynomial in terms of theKnudsen number (the ratio of the mean free path of the gasmolecules to the characteristic flow length (typically theradius of the capillary)).

The following contributions are derived from this work:●Validation and extension of the correlations proposed by Jones-

Owens and Sampath-Keighin for low permeability samples.●Development and validation of a new "microflow" model which

correctly represents gas flow in low permeability core samples.This model is also applied as a correlation for prediction of the

equivalent liquid permeability in much the same fashion as theKlinkenberg model, although our new model is substantiallymore theoretical (and robust) as compared to the Klinkenbergcorrection model.

IntroductionThe gas slippage phenomenon typically occurs in the labora-tory when gas flow experiments are conducted at lowpressures. Gas slippage is defined as the condition where themean free path of the gas molecules is no longer negligiblecompared to the average effective rock pore throat radius —i.e., the gas molecules tend to "slip" on the surfaces of theporous media. This effect yields an overestimation of themeasured gas permeability compared to the true absolutepermeability if it were measured using a liquid.

For flow in tubes, the gas slippage phenomenon has beeninvestigated since the end of the nineteenth century. The firststudy of gas slippage in porous media was conducted byKlinkenberg.1 The Klinkenberg model approximates a linearrelationship between the measured gas permeability and thereciprocal absolute mean core pressure.2 This model has beena consistent basis for the development of methods computingthe absolute liquid permeability of a core sample based on asingle data point — i.e., single-point steady-state permeabilitymeasurement methods.

Subsequent work focused on correlating the parameters of theKlinkenberg model (i.e, the Klinkenberg-corrected permeabil-ity or equivalent liquid permeability (k) and the Klinkenberggas slippage factor (bK). Heid et al3 and Jones and Owens4

proposed two correlations similar in form between bK and k,while Sampath and Keighin5 proposed a different form ofcorrelation using effective porosity () as a third parameter.

At a more theoretical level, the gas slippage phenomenon ispart of a larger area of study — the theory of rarefied gases.This theory has developed substantially in the last fifty years,leading to a better understanding and more accurate modelingof the gas slippage effect. The primary objectives of this workare:

●To compare and evaluate existing correlations for single-point,steady-state measurement of permeability, and

●To develop the concept of a new model for gas slippage in lowpermeability sands.

SPE 107954

Improved Permeability Prediction Relations for Low Permeability SandsF.A. Florence, SPE, Texas A&M U., J.A. Rushing, SPE, Anadarko Petroleum Corp., K.E. Newsham, SPE, Apache Corp.,and T.A. Blasingame, SPE, Texas A&M U.

2 F.A. Florence, J.A. Rushing, K.E. Newsham, and T.A. Blasingame SPE 107594

Overview of Existing CorrelationsThe "Klinkenberg" model was developed to account for thediscrepancies between permeabilities measured with gases andliquids as flowing fluids. This model is based on the argumentdeveloped by Kundt and Warburg6 that, at low pressures, ormore specifically, when the mean free path of the gasmolecules is within two orders of magnitude of the capillaryradius, the velocity of the gas molecules at the surface of theporous medium is non-zero.Klinkenberg proved that there exists an approximately linearrelationship between the measured gas permeability (ka) andthe reciprocal mean pressure ( p/1 ), which is given as:

p

bkk K

a 1 ................................................................ (1)

Where bK is the "gas slippage factor" which is a constant thatrelates the mean free path ( ) of the molecules at the meanpressure ( p ) and the effective pore radius (r).

The bK-term is defined by:

rc

pbK 4

, where 1c .................................................... (2)

In Eq. 1, k∞ is the "equivalent liquid permeability" (which isalso called Klinkenberg-corrected permeability). Klinkenberghas shown that k∞ is the true, absolute permeability of thesample. Derivations of the Klinkenberg relations are given inAppendix A.

Taking the Klinkenberg equation as fact, later work in thisarea focused on the determination of the gas slippage factor.In the late 1940s, Heid et al3 conducted a study including theeffects of pressure, pore size, and the type of porous mediumon permeability. Heid et al determined the gas slippage factor(bK) and the (extrapolated) permeability (k∞) of 11 syntheticcores and 164 natural core samples of representative sandsfrom different producing areas of the United States.

Based on the results of their core sample experiments shownin Fig. 1, Heid et al3 developed a relationship between the gasslippage factor (bK) and the corresponding equivalent liquidpermeability (k∞) which is written as:

39.0)(419.11 kbK ........................................................ (3)

Fig. 1 Heid et al data and correlation (ref. 3).

Additional research in the 1970s focused on the gas slippageeffect because of the emergence of "unconventional gasresources" as an important energy source. In particular, muchof the laboratory research in the 1970s/early 1980s addressedpermeability measurements in tight gas sands — which typi-cally exhibit permeabilities much lower than that of the coresamples Heid et al. studied.

As a case in point, Jones and Owens4 studied the effects of gasslippage for more than one hundred samples from tight gassands. Although the database used by Jones and Owens is notavailable, they did provide a correlation similar in form to theone presented by Heid et al. for the relationship between thegas slippage factor (bK) and the equivalent liquid permeability(k∞). The Jones-Owens formula also confirms that the gasslippage effect is more significant for lower permeabilityrocks. The Jones-Owen result is given as:

33.0)(639.12 kbK ......................................................... (4)

In 1981 (two years after the Jones and Owens work), Sampathand Keighin5 studied 10 core samples from a tight gas sandfield in Uinta County, Utah, and published a formula relatingthe gas slippage factor (bK) to the ratio of Klinkenberg-corrected permeability to effective porosity (i.e., k∞/). TheSampath-Keighin correlation is given by:

53.0851.13

k

bK ...................................................... (5)

The data and the correlation relation developed by Sampath-Keighin are shown in Fig. 2.

Fig. 2 Sampath-Keighin data and correlation (ref. 5).

Although based on a reduced number of samples, the Sam-path-Keighin correlation (Eq. 5) is interesting at a theoreticallevel since it is noted in the literature7 that the square root ofthe Klinkenberg-corrected permeability/porosity ratio is con-sidered to be a "characteristic length" (recall that permeabilityhas the dimension of length-squared). Eq. 2 shows that the

SPE 107954 Improved Permeability Prediction Relations for Low Permeability Sands 3

Klinkenberg gas slippage factor, bK, is inversely proportionalto the capillary radius — where the capillary radius is relatedto the square root of k∞/by:

/10886.8 6

kr ...................................................... (6)

Eq. 6 is derived analytically in Appendix B — we note that ris the capillary radius (expressed in centimeters) and k∞ is theKlinkenberg-corrected permeability (in md). Assuming stan-dard conditions, it is possible to derive a general but rigorous"square-root" k∞/correlation of the form:

5.0

kbK ....................................................................... (7)

The "-term" in Eq. 7 is a parameter that depends on the typeof gas used in the core flow experiment — we present Eq. 7computed for different gases on various data correlation plotslater in this paper.

Core Data Used in Current ResearchThe core data used in our work consists of data gathered in theliterature and from industry sources. The "literature database"is comprised of data from Heid et al and Sampath andKeighin. The "industry database" is composed of two datasetsobtained from the Lower Cotton Valley sandstones in NorthLouisiana.8,9 The first "Cotton Valley" dataset includes 12core samples; the second has 18 core samples. We note thatboth datasets utilized multi-point, steady-state permeabilitymeasurements. For each sample, we have effective porosity aswell as several gas permeabilities (ka) measured at variousmean core pressures ( p ). Klinkenberg-corrected permeability(k) and gas slippage factor (bK) are estimated using theclassical Klinkenberg plotting method.

We also obtained "older" datasets from a Gas ResearchInstitute (GRI) project conducted in the 1980s and early 1990s

in the Travis Peak (East Texas) and Frontier (Wyoming)formations (refs. 10-12). We should note that all of the GRIdatasets were derived from unsteady-state permeabilitymeasurements; however, the new correlations and modelsderived in our work are based on the multi-point, steady-statedata. We have included the unsteady-state data for (visual)reference only — the data are not used in the correlations.

Comparison of Core Data and CorrelationsFig. 3 presents a comparison of available core data with theJones and Owens and Heid et al correlations in the bK versusk format. In Fig. 4 we present a comparison of the availabledata with the Sampath-Keighin correlation and our generalizedsquare-root model given by Eq. 7 (bK versus k/format plot).Figs. 3 and 4 serve to highlight trends in the data — as well asto establish existence of a significant vertical spread in thedata sets — which can lead to errors in permeability estimatesfor low permeability reservoirs.

Next, we compare the Jones-Owens correlation, the Sampath-Keighin correlation, and our new proposed square-rootcorrelation with the data from the Lower Cotton Valleyformation (or modern "standard" database). The comparisonof the Heid et al and Jones-Owens correlations with the(Cotton Valley) steady-state data is shown on Fig 5 (bK vs.k). The Sampath-Keighin correlation and our proposed"square-root" correlation are compared to the steady state datain Fig 6 (bK vs. k/).

For both of the Cotton Valley datasets, the steady-statepermeability measurements were performed using nitrogen asthe flowing fluid — and we note that the nitrogen curve onFig. 6 overestimates the gas slippage factor (bK), whereas thecarbon dioxide curve provides a good fit. Regardless of the"intercept" issue, our model matches the data trends well.

Fig. 3 Comparison of the "bK vs. k" correlations by Heid et al (ref. 3) and Jones-Owen (ref. 4) with variousfield and literature data, acquired with unsteady-state (USS) or steady-state (SS) techniques.


Fig. 4 Comparison of the "bK vs. k/" correlations by Sampath and Keighin (ref. 5) and this work with various fieldand literature data, acquired with unsteady-state (USS) or steady-state (SS) techniques.

Fig. 5 Comparison of the steady-state data with the Jones-Owens and Heid et al correlations. The Heid et alcorrelation seems well adapted for these datasets,whereas the Jones-Owens correlation generallyunderestimates the gas slippage factor, especially forthe first Cotton Valley sample set.

Fig. 6 Comparison of the Cotton Valley steady-state data withthe Sampath-Keighin and proposed square-root correla-tions — although measurements were performed withnitrogen as the flowing gas, we note the square-rootcorrelation for carbon dioxide gives the best match (thebest fit is actually obtained for a -value of 34).


In Figs 7 and 8 we present the computed equivalent liquidpermeability plotted against the extrapolated Klinkenberg-corrected permeability on log-log scales for both "CottonValley" datasets. We should note that the "Klinkenberg-corrected permeability" refers only to the value of permeabil-ity actually extrapolated from a Klinkenberg plot (k∞measured),while the permeability values computed from correlations (k∞computed) are referred as "equivalent liquid permeabilities."However, we will use the same symbol (k∞) for either case.

Fig. 7 Computed equivalent liquid permeability versusextrapolated Klinkenberg-corrected permeability, for theLower Cotton Valley Sample No. 1 — the Jones-Owensand Sampath-Keighin correlations overestimate k∞ andthe theoretical square-root model for nitrogen under-estimates k∞.

Fig. 8 Computed equivalent liquid permeability versusextrapolated Klinkenberg-corrected permeability, for theLower Cotton Valley Sample No. 2 — the Jones-Owensand Sampath-Keighin correlations provide fair resultsfor k∞ > 1 d. The theoretical square-root model fornitrogen underestimates k∞, but performs well for lowpermeability samples.

In Figs. 9 and 10 we present the average absolute relativeerrors for the computed equivalent liquid permeability plottedagainst the extrapolated Klinkenberg-corrected permeability.This format helps us to assess the accuracy of the correlationon a "point-by-point" basis for a particular dataset.

For Cotton Valley Sample No. 1, the Jones-Owens and Sam-path-Keighin models generally overestimate the permeability.In both cases (Cotton Valley samples 1 and 2), the theoreticalsquare-root model (nitrogen curve) underestimates theKlinkenberg-corrected permeability — but the overall per-formance of the square-root relation is comparable to the othermethods.

Fig. 9 Average absolute relative errors for equivalent liquidpermeability versus measured Klinkenberg-correctedpermeability, for Lower Cotton Valley Sample No. 1.Errors are generally less than 35 percent.

Fig. 10 Average absolute relative errors for equivalent liquidpermeability versus measured Klinkenberg-correctedpermeability, for Lower Cotton Valley Sample No. 2.Errors are generally less than 30 percent, although thereare numerous outliers which exhibit greater than 50percent error.

As shown in Figs. 9 and 10, the three models for k∞exhibit afairly high absolute relative error (>50 percent for some cases)— however; the "clustering" for both datasets suggests that


most samples exhibit less than 30 percent error. Given thedata and the relatively approximate nature of the correlations,we consider the performance to be good, perhaps very good.While it is difficult to discriminate the "best" of the threecorrelations based on our results, we believe the Sampath-Keighin correlation and the generalized square-rootcorrelations are more "consistent" with the theory — and assuch, should be favored for this application.

Similar to previous work, the objective of our research is toestimate the bK-term accurately, and then utilize this estimatein Eq. 1 to yield the Klinkenberg-corrected permeability. Ageneral procedure for "single-point data" is to use any givenpair of ka and values to estimate bK, and then to use the bKand p values to estimate k∞. In our case, we will use a model(i.e., correlation) for bK and substitute this result into Eq. 1 toestimate k∞.

As an alternative, one could simply establish a correlation ofk∞ = f(ka) — this process was proposed by Jones and Owensbased on the gas permeability (ka) estimated at 100 psig. Thiscorrelation is given as:

md1μd0.1for,10 )0825.0log067.1log398.0( 2

a

kk kk aa

........................................................................................... (8)The purpose of our work is to establish a theoretically robustcorrelation of ),...,,( akpfk — so we will not pursue aresult of the form of Eq. 8.

Klinkenberg Model — RevisitedWe begin our discussion of the Klinkenberg concept by firstreturning to the fundamentals — we will utilize the samemeasurements as with the correlations presented in theprevious section (i.e., ka, p , , and backpressure (if used)) —but we now consider the rigorous mechanisms of flow at themicroscopic (and smaller) scales. As a first step, we acceptthe Klinkenberg approximation as a "first order" type ofestimate — Eq. 1, repeated below for clarity:

p

bkk K

a 1 ................................................................ (1)

We "define" k∞ as the true, equivalent liquid permeability ofthe system, but we will also use estimates of k∞ estimatedusing the Klinkenberg correction as a standard to correlateagainst. That is, we will use the Klinkenberg-corrected per-meability (k∞) as reported (or calculated) from our datasources as the reference permeability. Obviously, we wouldprefer to use a more rigorous estimate, but for the presentstudy, we must prove the concept model (i.e., the "microflow"model presented in this section) against some standard — andwe believe that the Klinkenberg-corrected estimates areappropriate for that purpose.

In Fig. 11, we present the Klinkenberg-corrected (equivalentliquid) permeability (k∞) plotted against the measured gaspermeability (ka) on log-log coordinates as a "correlation" toensure that, at least directionally, Eq. 1 is valid.

Fig. 11 Klinkenberg-corrected (extrapolated) permeability (k∞)versus measured gas permeability (ka).

As a primer for the development of our microflow model, weconsider the phenomenon of gas slippage at a fundamentallevel — as a phenomenon which occurs as a subset of a muchlarger area of study known as the theory of rarefied gases.

With the development of the common interest for aeronauticsand aerospace, as well as more recently, the development ofMEMS (Micro Electro-Mechanical Systems), the flow ofgases at very low pressure — or, more aptly, the flow of gasmolecules with a high mean free path (the mean free pathbeing inversely proportional to the pressure) has beenthoroughly studied in the last half century.

In the pursuit of a new microflow model, it then seems naturalto revisit the Klinkenberg model, if for no other reason than toconfirm that our new microflow model reverts to theKlinkenberg model under certain conditions (which it does).

The flow regime for a gas flowing in a micro-channel istypically determined by the value of the Knudsen number (Kn)— which is defined as:13

charlKn

......................................................................... (9)

In Eq. 9, is the mean free path of the gas molecules (i.e., theaverage distance (length) between two consecutive molecularinteractions) and lchar is the characteristic length of the flowgeometry (e.g., channel height, pipe radius).

For our purposes, the Knudsen number is difficult to definerigorously for a porous medium — but for the sake ofargument, we presume that some "characteristic length" can beestimated for a porous medium. More generally, we assumethat we can define the Knudsen number based on the pro-perties of the porous medium.


The classical definition of the mean free path from thermo-dynamics is:

)],([1

2/),( TpMRT

pTp ......................... (10)

where:p = mean pressure (absolute)

T = absolute temperature = viscosity of the gas at T and pR = universal gas constantM = molecular weight of the gas

This particular definition (Eq. 10) is based on the kinetictheory for a perfect gas.14

In addition to the definitions above, we also must considervarious other flow regimes as follows (see Fig. 12):

●Continuum Flow Regime: For Kn < 0.01, the mean freepath of the gas molecules is negligible compared to thecharacteristic dimension of the flow geometry (i.e., the lchar-parameter). In this case the continuum hypothesis of fluidmechanics is applicable (i.e., the system is described by theNavier-Stokes equations).

●"Slip-Flow" Regime: For 0.01 < Kn < 0.1, the mean freepath is no longer negligible, and the slippage phenomenonappears in the "Knudsen" layer (layer of gas moleculesimmediately adjacent to the wall).

●"Transition" Regime: For 0.1 < Kn < 10.●Free Molecular Flow Regime: For Kn > 10, the flow is

dominated by diffusive effects.

Fig. 12 Limits of the different flow regimes, as a function ofcharacteristic length lchar, and reciprocal mean freepath normalized (1 atm, 300K). Lines defining variousflow regimes are based on flow of air at isothermalconditions (Modified from ref. 13).

Karniadakis and Beskok13 developed a unified model for gasflow in micro-tubes (see Appendix C) where this model isvalid over the entire range of flow regimes.

The volumetric gas flowrate (q) flowing through a capillary ofradius r and length L is given by:13

Lp

KnKn

KnKnr

q

14

1))(1(4

8

........................... (11)

The (Kn)-term is defined by Karniadakis and Beskok13 as:

4.01-2

4tan15

128)( KnKn

............................................. (12)

Using the same capillary model as Klinkenberg,1 we derivedthe following "microflow" model which relates the measuredpermeability to gas (ka), the equivalent liquid permeability(k), and the Knudsen number (Kn):

Kn

KnKnKnkka 1

41])(1[ ..................................... (13)

Eq. 13 is rigorously derived and should be valid for all lowpressure/low velocity flow regimes which exist for gas flow inporous media. We have not investigated the application of Eq.13 for high pressure and/or high velocity flow.

Correlation of the Knudsen Number with Porosity,Permeability and Mean PressureThe Knudsen number (Kn), unlike the mean pressure ( p ),cannot be measured by direct laboratory measurements. Ourobjective is to design a model for estimating the equivalentliquid permeability (k∞) using single-point, steady-state mea-surements (i.e., using a single pair of ka and p values). Toremedy this issue, we propose a "pseudo" Knudsen number(Knp) — which is defined as a function of the "typically"measured parameters (i.e., p , , ka).

Assuming that the core flow experiments are conducted atisothermal conditions (i.e. there is no temperature gradient inthe core) and that variations in viscosity are negligible overthe range of pressures considered, then the Knudsen number(defined by Eq. 9) is inversely proportional to both the meanpressure ( p ) and the characteristic flow geometry length(lchar). Typically the pore throat radius (defined by Eq. 6 as afunction of the equivalent liquid permeability (k∞) and theporosity ()) will be used to estimate lchar.

Substituting Eq. 12 into Eq. 13 yields

p

pppa Kn

KnKnKnkk

1

414tan

15

1281 4.01-

2........................................................................................ (14)

where Knp is the "pseudo" Knudsen number. In order toutilize Eq. 14, we must assume that the mean pressures,porosities, and the gas and equivalent liquid permeabilities areavailable. To solve for the "pseudo" Knudsen number (Knp)for a particular case, we then rearrange Eq. 14 into thefollowing "root solution" form:

01

414tan

15

1281 4.01-

2

p

pppa Kn

KnKnKnkk

........................................................................................ (15)


Applying the Pseudo-Knudsen Number ConceptTo demonstrate an application of the "pseudo" Knudsennumber (Knp), we only use the Lower Cotton Valley data (No.2) (ref. 9), from which we derive two separate correlations forKnp (see Appendix D). Our first correlation relates Knp to thereciprocal mean pressure, the porosity, and the gas per-meability as follows:

352.0598.014.0 ap k

pKn ............................................ (16)

As implied, Eq. 16 was derived for the specific case of theCotton Valley No. 2 data, in a process where Knp (Eq. 15) wasestimated using known data for ka, p , and k∞. After Knp wasobtained as the root of Eq. 15, we then correlated these Knp

values with the ka, p , and data to yield Eq. 16.

At this point, Eq. 16 can be substituted into Eq. 15 — and thisresult can be solved directly for the equivalent liquidpermeability (k∞). Obviously, k∞ data must be available to"calibrate" Eq. 15 (i.e., to estimate Knp). However, we believethat the calibration of Eq. 16 may (in future work) be reducedto the determination of the intercept term (the ka exponent isapproximately -0.5 and the exponent may be imposed as 0.5— similar to the "square-root" correlations).

As an alternative, we can follow the exact same calibrationprocedure, but this time substitute k∞ for ka in Eq. 16 (i.e., theKnp = f(ka, p ,) correlation). For this case, we obtain:

3897.0553.0162.2

kp

Kn p .......................................... (17)

The equivalent liquid permeability values computed using Eq.15 (based on the values of Knp from Eq. 16 or 17) arecorrelated with the "given" Klinkenberg-corrected permea-bility data for this case in Fig. 13.

We recognize that Klinkenberg-corrected permeability datamay not always be available in practice — and the process ofsolving simultaneously for the pseudo-Knudsen number (Knp)and the equivalent liquid permeability (k∞) will require animplicit formulation (an analog in the field of phase behaviorwould be an equation-of-state (EOS) which is implicit in fluiddensity).

In such a case, (i.e., an implicit relation) we would have todetermine coefficients for the Knp = f(k∞, p , ) simul-taneously with the estimation of k∞using Eq. 15. Writing thecorrelation for the pseudo-Knudsen number in general form,we have:

211

0aa

p kp

aKn ........................................................ (18)

where Eq. 18 provides a "correlation" for the variables in thisproblem — and this relation allows us, in theory, to solve Eq.15 for the equivalent liquid permeability (k∞) in an implicitfashion.

Substituting Eq. 18 into Eq. 15 and rearranging yields:

1/11

41

114tan

15128

1

21

2121

0

0

4.0

01-

2

aa

aaaaa

kp

a

kp

akp

akk

............................................................................................ (19)

Fig. 13 Computed (Klinkenberg) equivalent liquid permeability(k∞) versus Klinkenberg-corrected (extrapolated)permeability (k∞). (pseudo-Knudsen number approach,implicit and explicit relations for Knp)

The "trick" is to obtain the "calibration" coefficients (a0, a1,and a2) in Eq. 19. As noted earlier, Eq. 19 can be formulatedas an "equation-of-state" and the coefficients (a0, a1, and a2)can be tuned using non-linear regression — provided thatthere are sufficient data measurements, and that such measure-ments are taken for samples from the same depositionalsequence.

We must note that we do not in any manner propose that Eq.18 is "universal" (i.e., one set of coefficients (a0, a1, and a2)does not apply to all possible data cases, Eq. 18 must becalibrated for each dataset). For this case, we did formulateEq. 19 as a "fully implicit" solution using non-linear regres-sion and we achieved following results:

a0 = 0.93a1 = -0.49a2 = 0.13

The results of our "fully implicit" correlation are shown inFig. 14.


Fig. 14 Computed (Klinkenberg) equivalent liquid permeability(k∞) versus Klinkenberg-corrected (extrapolated)permeability (k∞). (fully implicit formulation for thepseudo-Knudsen number).

Summary and ConclusionsSummaryThe development of the microflow model for permeabilityprediction in low permeability gas sands offers an alternativeto the classical Klinkenberg model, which is the long-timereference for measuring permeability. Although the Klinken-berg model is the universally accepted standard, we believe itmay yield poor results for very or ultra low permeability coresamples.

Our new model has been validated using field data from a lowpermeability reservoir case, but this "proof" is by no meansexhaustive — additional validation is warranted.

Conclusions1. The Jones-Owens, Sampath-Keighin, and our "Square-Root"

correlations may be used satisfactorily for single point, steady-state measurements as mechanisms to estimate the equivalentliquid permeability. The Sampath-Keighin and Square-Rootcorrelations should be preferred based on the theoreticalformulations for these models.

2. The microflow model presented in this work is promisingas it provides a second-order correction for gas slippage(beyond the "first-order" Klinkenberg formulation. The useof such a correction should be especially relevant for low toultra low permeability core samples.

Recommendations/CommentWe note that the theoretical square-root correlation (AppendixB) has a limited accuracy in this work. However, the square-root model validates use of the permeability/porosity ratio inmodeling the gas slippage factor and serves to connect theclassical Klinkenberg model with our new "microflow model."Upon further validation, we believe that our microflow model

will provide significant improvement for the estimation ofequivalent liquid permeability from typical steady-state (gas)permeability measurements.

NomenclatureVariables

b = General slip coefficientbK = Klinkenberg gas slippage factor, psi

c = Proportionality constantKn = Knudsen number, dimensionless

Knp = Pseudo-Knudsen number, dimensionlesska = Apparent gas permeability, md

k = Equivalent liquid permeability or Klinkenberg-corrected permeability, md

lchar = Characteristic length of the flow geometry, cmp = Mean core pressure, psiap = Differential pressure, psi

q = Gas flowrate, cc/secr = Effective pore radius, cm = Rarefaction coefficient parameter, dimensionless = Gas viscosity, cp. = Mean free path of the gas molecules, cm = Porosity, fraction

References1. Klinkenberg, L.J.: "The Permeability of Porous Media to Liquid

and Gases," paper presented at the API 11th Mid Year Meeting,Tulsa, Oklahoma (May 1941); in API Drilling and ProductionPractice (1941), 200-213

2. RP40, Recommended Practices for Core Analysis, 2nd edition,API, Washington, DC (1998).

3. Heid, J.G., et al: "Study of the Permeability of Rocks toHomogeneous Fluids," in API Drilling and Production Practice(1950), 230-246

4. Jones, F.O. and Owens, W.W.: "A laboratory Study of LowPermeability Gas Sands," paper SPE 7551 presented at the 1979SPE Symposium on Low-Permeability Gas Reservoirs, May 20-22, 1979, Denver, Colorado.

5. Sampath, K. and Keighin, C.W.: "Factors Affecting Gas Slippagein Tight Sandstones," paper SPE 9872 presented at the 1981SPE/DOE Low Permeability Symposium, May 27-29, 1981,Denver, Colorado.

6. Kundt, A. and Warburg, E.: "Über Reibung und Wärmelei-tungverdünnter Gase," Poggendorfs Annalen der Physik und Chemie(1875), 155, 337.

7. Jones, S.C.: "Using the Inertial Coefficient, , to CharacterizeHeterogeneity in Reservoir Rock," paper SPE 16949 presented atthe 1987 SPE Annual Technical Conference and Exhibition,September 27-30, 1987, Dallas, Texas

8. Lower Cotton Valley Formation Core Report — AnadarkoPetroleum Corp., (2005)

9. Lower Cotton Valley Formation Core Report — AnadarkoPetroleum Corp., (2006)

10. Travis Peak Formation Core Report —Well Howell No. 5, S.A.Holditch, (1986)

11. Travis Peak Formation Core Report — Well S.F.E. No. 2, S.A.Holditch, (1987).

12. Frontier Formation Core Report — Well S.F.E. 4-24, S.A.Holditch, (1991).


13. Karniadakis, G.E. and Beskok, A.: Micro-flows, Fundamentalsand Simulation, Springer-Verlag, New-York (2002).

14. Loeb, L.B.: The Kinetic Theory of Gases, second edition,McGraw-Hill Co. Inc., New York City (1934).

15. NIST Chemistry Webbook, NIST, http:\\webbook.nist.gov.

Appendix A: Derivation of Klinkenberg Equation

This appendix presents our derivation of the Klinkenbergequation (ref. 1). Note that our derivation uses componentsthat Klinkenberg did not employ originally (i.e., elements ofthe literature for the kinetic theory of gases (ref. 14)).

Gas Flow Through a Straight Capillary. We consider acapillary tube of radius (R0) and length (L) with its axiscoincident with the x-axis. Gas is flowing through thecapillary as a result of a pressure drop (p = (p1 - p2)) acrossthe length of the capillary tube. Assume that the velocity ofthe flowing gas (v) is a function only of the distance r from thex-axis. Consider now a cylindrical shell of length (L) betweenthe cylinders of radii r and r+r (see Fig. A.1). The force (F1)acting on the cross-section of this shell is the normal pressuredue to the flowing gas. In derivative form, this force is:

rprdF 21 ................................................................ (A-1)

Assuming steady-state fluid motion, this force is balanced bythe viscous drag exerted by the flowing gas on the outer andthe inner surfaces (along the x-axis) of the cylindrical shell.The viscous drag is defined by:

drdv

SF .................................................................... (A-2)

where:Coefficient of viscosity of the gas.S = Surface considered (2rL for the inner surface,

2(r+r)L for the outer surface).

The gas velocity reaches its maximum on the axis of the cylin-der (i.e., r=0) and decreases radially towards the wall to avelocity of zero — therefore the velocity gradient dv/dr isnegative. The viscous drag exerted on the inner surfaces is:

drdv

LrdF 2Surfaceinner ............................................. (A-3)

The component of the viscous drag on the outer surface of thecylindrical shell, where the gas velocity is lower, is:

dr

rdrdv

vdrrLdF

)(2SurfaceOuter ................... (A-4)

The resulting viscous drag exerted on the entire shell is givenby:

ceInnerSurfaceOuterSurfa2 dFdFdF

The equilibrium yields dF1 = dF2, hence:

dr

rdrdv

vdrrL

drdv

Lrrpr

)(222 ............... (A-5)

Expanding the second term on the right-hand side of Eq. A-5

yields:

2

22

2

2)(2

)()(2

dr

vdr

drdv

dr

vdrr

drdv

rL

dr

rdrdv

vdrrL

...................................................................................... (A-6)In Eq. A-6, the term r2(d2v/dr2) is negligible compared to theother terms — neglecting this term and substituting Eq. A-6into Eq. A-5 yields:

)(2

22

2

2

drdv

dr

vdrr

drdv

rL

drdv

Lrrpr

...................................................................................... (A-7)Expanding and simplifying Eq. A-7 gives:

drdv

dr

vdrLpr

2

2 .................................................. (A-8)

Dividing Eq. A-8 by (-rL) yields the following differentialequation:

Lp

drdv

rdr

vd

1

2

2........................................................ (A-9)

A particular solution of Eq. A-9 (i.e., a non-linear differentialequation) is:

20 4

rLp

Av

........................................................... (A-10)

where A0 is a constant fixed by the boundary conditions. Twodifferent boundary conditions can be used — slip-flow at thewall or no slip-flow at the wall.

If no slip-flow is assumed at the wall, the condition is v = 0

for r = R0; this imposes 200 4

RLp

A

. The velocity solution

to Eq. A-10 is then:

)(4

220 rR

Lp

v

...................................................... (A-11)

The gas flowrate (qg) is obtained by integrating the gas flow-rate through the considered cylinder shell section (v(2rdr))over the cross-section of the capillary tube, which yields thePoiseuille equation:

40

220

0

0 8)(

42 R

Lp

drrRLp

rqR

g

................ (A-12)

If the gas slippage is considered at the wall, the condition canbe written v = v0 for r= R0 (where v0 is non-zero). Kundt andWarburg (ref. 6) proved that the velocity at the wall (v0) isproportional to the velocity gradient at the wall (

0/ Rrdrdv ),as given by:


00 / Rrdrdvcv ................................................... (A-13)

In Eq. A-13, c is a constant with a value slightly less than 1 (asgiven by Kundt and Warburg) and is the average mean freepath of the gas molecules (i.e., the mean free path of the gasmolecules at the mean pressure p (see ref. 14)), defined by

2/21 ppp ). The velocity gradient at the wall is obtainedby differentiating Eq. A-10:

rLp

drdv

2

............................................................... (A-14)

Applying the condition given by Eq. A-13 to Eq. A-14 yields:

00 2R

Lp

cv

............................................................ (A-15)

Using Eq. A-12 in Eq. A-9, at r = R0, and rearranging yields:

02

00 24

RcRLp

A

............................................... (A-16)

Substituting Eq. A-16 for A0 in Eq. A-10 gives:

022

0 24

RcrRLp

v

.......................................... (A-17)

As previously done for Eq. A-12, the gas flowrate (qg) isobtained by integrating v(2rdr) over the whole radius of thecapillary:

0

20

20

0

40

022

00

0

41

18

41

8

24

Rc

Lp

RR

Rc

RLp

drRcrRLp

qR

g

..................................................................................... (A-18)Re-writing Eq. A-18 in terms of superficial velocity (vl):

0

20

20

tube

41

18 R

cLpR

R

q

Aq

v

l

lg

..................................................................................... (A-19)

Gas Flow Through an Idealized Porous Medium. The"Klinkenberg" idealized porous medium is composed of solidmaterial through which the capillaries are oriented randomlyand have all the same radius, r. The direction of flow isparallel to one of the planes of the cube, and let there be Ncapillaries. The system for Klinkenberg's idealized porousmedium is shown in Fig. A-1.

The liquid flow rate (ql) through the capillary is given byPoiseuille's law as: (3-dimensional flow, so Klinkenberg used1/3 of the total flow for a particular direction).

Lp

RR

Nql

1

831 2

02

0

( 20R = Atube) ................... (A-20)

Rewriting Eq. A-20 in terms of superficial velocity (vl):

LpR

N

Aq

v ll

183

1 20

tube

( 20R = Atube) .............................. (A-21)

Fig. A.1 System schematic for a tube (Loeb (ref. 14)).

Darcy's law gives:1

)( 2Lp

Lkq ll

(L2 = Acube (i.e., rock sample))..... (A-22)

Rewriting Eq. A-22 in terms of superficial velocity (vl):

1cube

Lp

k

Aq

v

l

ll

(L2 = Acube (i.e., rock sample))..... (A-23)

where kl is the permeability to liquid of the porous medium —which is also the absolute permeability of the porous medium(provided that the saturation is 100 percent — i.e., single-phase flow). Equating Eqs. A-19 and A-20 and simplifyingyields:

831 2

0RNkl .............................................................. (A-24)

If we assume that that the flowing fluid is a gas, with a gasslippage condition at the wall (v0 ≠ 0), the gas flowrate (qg) isgiven by Eq. A-18:

0

20

20

41

18 R

cLpR

R

qv g

g

................................ (A-19)

Substituting83

1 20R

Nkl into Eq. A-19 for yields:

0

41

1Rc

Lp

kv lg

................................................ (A-25)

For the same gas flow rate, Darcy's law gives:

Lp

kv ag

1

.............................................................. (A-26)

Where ka is the apparent permeability to gas. Setting Eq. A-25 equal to Eq. A-26 and simplifying, yields:

0

41

Rc

kk la ......................................................... (A-27)

As the mean free path () is proportional to the reciprocalmean pressure ( p ), we can define:


pb

rc K4 ................................................................... (A-28)

where bK is a coefficient of proportionality, known as theKlinkenberg gas slippage factor. Substituting Eq. A-28 for

rc /4 in Eq. A-24 gives the Klinkenberg equation:

)definitionourfor(1

kk

pb

kk lK

la ................... (A-26)

Appendix B: Derivation of a Theoretical Square-Root Correlation

Theoretical Capillary Radius. Considering the flow of aliquid through a capillary tube of inner radius r and length L,the fluid flow rate (q) is given by Poiseuille’s law as:

42

2 18

18

rLp

Lpr

rq

....................................... (B-1)

where is the viscosity of the fluid and p is the pressuredrop across the length of the tube. Considering a cylinder ofidealized porous medium with a radius R0 and a length L,composed of n identical capillary tubes (such as describedabove), with the same orientation — parallel to the axis of thecylinder. We note that for this discussion (as opposed to ourprevious work in this section), that R0 is now defined as theouter radius of the bulk core sample. By definition, theporosity of a porous medium is given by:

20

2

20

2

VolumeBulkVolumeVoid

R

rn

LR

Lrn

................................. (B-2)

The total volumetric flow rate (qtot) of a fluid flowing throughthis cylinder is defined by multiplying Eq. B-1 by the numberof tubes, n. This gives:

422 1

8)(

18

rLp

nLp

rr

nqtot

............................ (B-3)

For flow in a porous medium, Darcy's law is defined as:

Lp

Ak

qtot

................................................................ (B-4)

In Eq. B-4, k is the absolute permeability of the porousmedium, and A is the cross-sectional area of the cylinder.Since A = R0

2, Eq. B-4 yields:

Lp

Rk

qtot

)( 20

.......................................................... (B-5)

Equating Eqs. B-3 and B-5, and rearranging yields:

20

4

8Rk

rn ................................................................... (B-6)

Rearranging Eq. B-6 gives:

krR

rn 82

20

2

Substituting Eq. B-2 into the above result, we have:

krkrR

rn 88 22

20

2

........................................ (B-7)

Or, solving Eq. B-7 for the "equivalent capillary" radius, r, weobtain:

/22 kr ............................................................. (B-8)

In Eq. B-8, the "units" of permeability depend on the unitssystem. For consistency, we must apply a conversion factor,C0, as a multiplier (e.g., when k is in md and r is in cm — C0 =3.141510-6 md/cm , recall that 1 Darcy = 9.8692310-9

cm2). The general form of Eq. B-8 with the units conversionfactor (C0) is:

/22 0 kCr .................................................. (B-9)

Using C0 = 3.141510-6, Eq. B-9 becomes:

/10886.8 6 kr .................................................. (B-10)

Theoretical Square-Root Correlation. To correct for theeffects of gas slippage in permeability measurements, Klin-kenberg (ref. 1) derived an approximate linear relationshipbetween the measured gas permeability (ka) and the reciprocalmean pressure ( p ). This result is given as:

p

bkk K

a 1 ......................................................... (B-11)

In Eq. B-11, k is the Klinkenberg-corrected permeability andbK is the Klinkenberg gas slippage factor. The gas slippagefactor (bK) is a "constant" which relates the mean free path ofthe gas molecules () at the mean (absolute) pressure ( p )and the effective pore-throat radius (r), as given by:

pb

rc K4 ................................................................... (B-12)

In Eq. B-12, c is a constant very close to unity (ref. 6). Wewill consider c = 1 for the remainder of this derivation.Rearranging Eq. B-12 yields:

pr

bK 4 ................................................................... (B-13)

Substituting r in Eq. B-13 with Eq. B-10 gives:5.0

610886.8

4

k

pbK .................................... (B-14)

In Eq. B-14, the mean free path of the gas molecules isdefined by: (ref. 14)

)],([1

2/),( TpMRT

pTp ...................... (B-15)

In this definition, R is the universal gas constant, M is themolecular weight of the gas and is the viscosity of the gas atthe corresponding thermodynamic state (i.e., is a function ofthe absolute pressure, p , absolute temperature, T, andcomposition of the gas). For clarity, the subsequent formulaeare expressed in S.I. units, unless otherwise specified.


The square-root correlation developed in this work conformsto the following assumptions:

●The temperature in the core sample is uniform duringsteady-state flow (assumed: T = 298 K).

●The gas used for the experiments is nitrogen (M =28.01348 kg/kg-mole).

●Assuming that, in the range of pressure applied in thelaboratory, nitrogen behaves as an ideal gas, the variationof the nitrogen viscosity with pressure is negligible; hence

)(),( atm1,22TpT NN .

Nitrogen viscosity as a function of temperature ( )(atm1,2TN ,

expressed in Pa.s) is computed using Sutherland's equation:(ref. 2)

75.1

atm1, 10102

85.13)(

2

T

TTN ................................... (B-16)

The value of the universal gas constant, R = 8,314 J/K/kg-mole. The product p in Eq. B-15 can be calculated usingEq. B-15 and B-16:

00664.010102298

29885.1328.01348

298831412/ 7

5.1

pp

p

........................................................................................ (B-17)To obtain the radius, r, in meters (recall that the base unit ofthe mean free path is the meter), Eq. B-10 (expressed intraditional units) becomes;

/10886.8 8 kr (B-18)

Using the result from Eq. B-17 and Eq. B-18 in Eq. B-14yields:

5.05

5.0

8109885.2

10886.8

00664.04

kk

bK ........ (B-19)

Where bK is expressed in Pa. Converting bK to psi yields:5.0

345.43

k

bK .................................................... (B-20)

The significance of Eq. B-20 is that this result establishes thebasis for the bK versus /k correlation given by Sampathand Keighin (ref. 5). The exponent (-0.5) is establishedrigorously from theory as shown above.

The intercept (43.345) is based on the following assumptions:●ambient temperature (T = 298 K) and●ambient pressure (1 atm).

However, these assumptions are probably not significantcompared to the assumption of nitrogen as the reference gas.In Table B.1 we summarize our estimates of the interceptcoefficient for hydrogen, helium, air, nitrogen, and carbondioxide. We have also shown the intercept of the square-rootmodel plotted against molecular weight for hydrogen, helium,air, nitrogen, and carbon dioxide in Fig. B.1.

Table B.1 Intercept for Eq. B-20, various gases..4

Flowing Gas

MolecularWeight

(kg/kg-mole)

Gas Viscosity at1atm and 298 K

(Pa.s)

Eq. B-20Intercept

(psi)hydrogen 2.0159 8.845x10-6 80.236helium 4.0026 1.985x10-5 127.802air 28.9586 1.842x10-5 44.106nitrogen 28.01348 1.781x10-5 43.345carbon dioxide 44.0095 1.503x10-5 29.181

These estimates are based on the fluid properties of ref.2 andthe NIST correlations (ref.15) as appropriate.

Fig. B.1 Intercept of the square root model versus molecularweight of the gas used in the measurements — theviscosity of the different gases except H2 is similar(see Table B.1), hence the p-term (Eq. B-17) variesapproximately with the reciprocal square-root of themolecular weight.

Appendix C: A Rigorous Microflow Model Applied tothe Problem of Gas Flow through Porous Media

Unified Flow Model Gas Flow in Pipes. Karniadakis andBeskok (ref. 13) developed a unified model that predicts volu-metric and mass flow rates for gas flow in channels and pipesover the entire Knudsen regime (i.e., all flow regimes). TheKarniadakis-Beskok "microflow" model is given (withoutderivation) as:

14

1])(1[1

84

bKnKn

KnKnLp

lq char

.................(C-1)

where:q Volumetric flow rate in the conduit, cc/seclcharCharacteristic length of the flow geometry (e.g.,

channel height, pipe radius), cmL Length of conduit, cmp Pressure drop across the length of the conduit, atm = Gas viscosity at temperature and pressure, cpb = Dimensionless slip coefficient, (b is defined as -1)Kn) = Dimensionless term in the rarefaction coefficient

In Eq. C-1, the Knudsen number (Kn) is defined by:


charlKn

..................................................................... (C-2)

where is the mean free path of the gas molecules (i.e., theaverage distance (length) between two consecutive molecularinteractions).

We use a value of -1 for the general slip coefficient (b) asrecommended by Karniadakis and Beskok. The role of therarefaction coefficient [1+Kn) Kn] is to account for thetransition between the "slip-flow" regime (for which Klin-kenberg model was developed) and the "free molecular flow"regime. In the "slip-flow" regime (i.e., 0.01 < Kn < 0.1), therarefaction coefficient is equal to 1 (i.e., = 0); in the "freemolecular flow" regime (i.e., Kn > 10), the volumetric flow-rate is independent of the Knudsen number and the parametertends toward a constant value (for Kn ).

For reference, the "Knudsen" flow regimes are defined asfollows below and are illustrated graphically in Fig. 12 in thetext. .

●Continuum Flow Regime: For Kn < 0.01, the mean freepath of the gas molecules is negligible compared to thecharacteristic dimension of the flow geometry (i.e., the lchar-parameter). In this case the continuum hypothesis of fluidmechanics is applicable (i.e., the system is described by theNavier-Stokes equations).

●"Slip-Flow" Regime: For 0.01 < Kn < 0.1, the mean freepath is no longer negligible, and the slippage phenomenonappears in the "Knudsen" layer (layer of gas moleculesimmediately adjacent to the wall)

●"Transition" Regime: For 0.1 < Kn < 10.●Free Molecular Flow Regime: For Kn > 10, the flow is

dominated by diffusive effects.

The variation of the -parameter as a function of Kn is repre-sented using: (see ref. 13)

tan2

)( 211-

0

cKncKn

.......................................... (C-3)

where 0, c1 and c2 are constants. The values for the constantsc1 and c2 are respectively 4.0 and 0.4, and the parameter 0 isgiven by:

1564

41

1364

0

b

................................................... (C-4)

Validation of the Microflow Model Concept. In order tovalidate the applicability of the concept of the unified flowmodel (Eq. C-1) for the problem of steady-state gas flowthrough a porous medium, we computed the Knudsen numberfor a given data set of 11 core samples tested at various meanpressures.

The mean free path of gas molecules is defined by:

)],([1

/2),( TpMRT

pTp ........................ (C-5)

The mean free path () is computed using the following (all

terms in S.I. units):●The core flow experiments were performed at a constant

room temperature (T = 298 K).●The gas used for the experiments was nitrogen (M =

28.01348 lb/lbmole).●The nitrogen viscosity was computed using Sutherland's

equation (ref. 2):

3623

atm1,22

5.1atm1,2

105052.11005452.1

123688.012474.0)(),(102

85.13)(

pp

pTpTT

TT

NN

N

.................................................................................................(C-6)The characteristic length of the conduit (lchar) is required inorder to estimate the required Knudsen numbers. The"typical" characteristic length is estimated from capillary pres-sure data or an equivalent single capillary concept — andthese "lengths" are typically the capillary radii. We use thisexercise simply to evaluate one definition over another —conceptually, there is no perfect definition because we areinvestigating flow in porous media, not uniform capillaries.

The definitions we consider for lchar are:●The computed average pore throat radius estimated from

capillary pressure data (these data are given in the variousreports from which the sample data were extracted).

●The theoretical "capillary radius" given by (see AppendixB):

/10886.8 6

klchar ...................................................(C-7)

The minimum and maximum computed Knudsen numbers(based on mean pressures) for each sample are presented inTable C-1. Since most of the Knudsen numbers are greaterthan 0.1, the "Transition" Regime (0.1 < Kn < 10) is thedominant type of flow regime — therefore, the rarefactioncoefficient defined in the previous section plays a major rolefor modeling the volumetric flow rate. In concept, this obser-vation validates the application of the microflow model for(steady-state) gas flow in porous media.

Table C.1 Minimum and maximum Knudsen numbers corres-ponding to the models used to represent thecharacteristic length lchar.

4

Kn Estimated from theAverage Pore Throat Radius

Kn Estimated fromEq. C-7

Sample ID min Kn max Kn min Kn max Kn1 0.15 0.34 0.64 1.512 0.15 0.36 0.32 0.773 0.43 0.97 0.40 0.914 0.06 0.15 0.72 1.675 0.15 0.34 0.12 0.276 0.37 0.72 0.41 0.807 0.15 0.34 0.22 0.508 0.05 0.09 0.16 0.269 0.15 0.33 0.28 0.63

10 0.18 0.34 0.28 0.5212 0.18 0.34 0.31 0.57


A Rigorous Microflow Model for Gas Flow in an IdealizedPorous Medium. The "microflow" model is defined in theprevious section as:

14

1])(1[1

84

bKnKn

KnKnLp

lq char

................ (C-1)

Poiseuille's law for fluid flow in a pipe (or tube) is given by:

Lp

lq char

1

84

......................................................... (C-8)

while Darcy's law for fluid flow in a porous media is:

Lp

Akq

1

core ............................................................ (C-9)

In Eq. C-9, A is the cross-sectional area of the porous medium(perpendicular to the direction of the flow). Following theprocedure given by Klinkenberg (ref. 1), we can equatePoiseuille's and Darcy's laws to yield an expression for thepermeability (k). This procedure requires us to equate Eqs. C-8 and C-9, which yields:

core

4

8 Al

k char ................................................................ (C-10)

Substituting Eq. C-10 into Eq. C-1, we have:

14

1])(1[core

bKnKn

KnKnLp

Ak

q

.................. (C-11)

where:k = Permeability, Dq Volumetric flow rate in the conduit, cc/seclchar The characteristic length of the flow geometry (e.g.,

channel height, pipe radius), cmL Length of conduit, cmp Pressure drop across the length of the conduit, atm = Gas viscosity at temperature and pressure, cpb = Dimensionless slip coefficient (b is defined as -1)Kn) = Dimensionless rarefaction coefficientKn = Knudsen number, dimensionless

Multiplying through Eq. C-11 byp

LA and using b=-1, we

obtain:

14

1])(1[

bKnKn

KnKnkp

LA

q ...................... (C-12)

where we note that the left-hand-side of Eq. C-12 is simply the"gas" permeability (ka) as defined by Darcy's law (i.e., the"uncorrected" permeability). Making this reduction, we haveour base form:

14

1])(1[

Kn

KnKnKnkka ............................... (C-13)

Eq. C-13 provides an independent relation between the ap-parent gas permeability (ka), the slip-corrected permeability(or Klinkenberg-corrected permeability) (k), and theKnudsen number (Kn). We now need to finalize Eq. C-13 bysubstitution of the relations for Kn). We substitute Eq. C-3into Eq. C-1 (and assume that c1 = 4.0 and c2 = 0.4 in Eq. C-

3), which yields a direct relation for Kn) of the form of:

4tan15

128)( 4.01-

2KnKn

.........................................(C-14)

We now substitute Eq. C-14 into Eq. C-13 to yield our formal(or complete) result for this work:

14

14tan15

1281 4.01-

2

Kn

KnKnKnkka

.....................................................................................(C-15)

Klinkenberg Model as a Simplified Microflow Model forSlip-Flow Regime. We can prove "mathematically" (usingrigorous assumptions) that the model developed byKlinkenberg is actually a simplification of the microflowmodel for the case of the slip-flow regime (0.01 < Kn < 0.1).Recalling the Klinkenberg equation:

p

bkk K

a 1 ..........................................................(C-16)

where:

k = Klinkenberg-corrected permeability, DkaApparent gas permeability, DbKKlinkenberg gas slippage factor, atmp Mean core pressure, atm

The Klinkenberg gas slippage factor (bK) is defined byKlinkenberg as a coefficient of proportionality:

pb

rc K4 ....................................................................(C-17)

where c is a constant close to 1 (see ref. 6) and r is theeffective pore throat radius (in our case r is the characteristiclength lchar).

Applying Eq. C-17 to Eq. C-16 yields:

r

ckka

41 .........................................................(C-18)

Recall the base form of our microflow model (Eq. C-13):

14

1])(1[

Kn

KnKnKnkka ................................(C-13)

We assume that we are in a slip-flow regime, then by defini-tion, (Kn) = 0 (see ref. 13). Hence Eq. C-13 becomes:

14

1

KnKn

kka .....................................................(C-19)

Also, for 0.01 < Kn < 0.1, we have:

)(11

1KnoKn

Kn

.................................................(C-20)

Hence:

4

)(4414 22

Kn

KnoKnKnKn

Kn

......................................(C-21)

By definition of the Knudsen number (Eq. C-2):


charlKn

..................................................................... (C-2)

Eq. C-19 now yields:

char

a

lk

Knkk

41

41

.................................................... (C-22)

Eq. C-22 is very similar (and almost identical) to Eq. C-18,provided that the constant c in Eq. C-18 is equal to one and themean free path of the gas molecules used in the microflowmodel () is defined as the average mean free path of the gasmolecules ( in the Klinkenberg model, evaluated at themean core pressure). We can consider that the Klinkenbergmodel is an approximation of the microflow model.

Appendix D: Correlations for the "Pseudo-Knudsen"Number — Application to a Tight Gas Example(North Louisiana, USA)

Determination of the Pseudo-Knudsen Numbers. This dataset (ref. 9) includes 18 core samples where the Klinkenberganalysis (ref. 1) was performed in order to estimate the liquidequivalent permeability (k). The measured mean pressure,the permeability to gas, and the porosity are given for eachsample — and using the mean pressure and permeability togas data, a Klinkenberg plot is constructed, and the gasslippage factor (bK) and the Klinkenberg-correctedpermeability (k) are estimated.

As noted, the porosity of the sample is also given, but noreference conditions were given, so we assumed 300 K (80Deg F) and 1 atm. The data summary is given in Table D.1.

Table D.1 Summary of the core data (given data and com-puted results).

4

SampleID

k(md)

bK

(psi)

(fraction)1-8 0.0001 565.3 0.023

2-10 0.1322 14.2 0.0632-22 0.0168 32.5 0.0563-8 0.0439 32.8 0.078

3-48 0.0035 83.6 0.071-1 0.00007 753.5 0.0291-5 0.00004 801.0 0.021-7 0.00003 771.0 0.0172-2 0.0004 165.1 0.0362-7 0.0124 26.8 0.062-8 0.0025 77.1 0.051

2-12 0.0025 215.1 0.0552-28 0.0052 58.6 0.0373-4 0.0002 303.1 0.043-6 0.0999 26.4 0.066

3-10 0.0067 84.8 0.0713-36 0.0168 39.2 0.0573-55 0.0013 183.7 0.066

Since we cannot compute the Knudsen number explicitly, thegoal of this step is to evaluate a "pseudo Knudsen" number(Knp) using the relation below: (see Appendix C and ref. 13)

014

14tan15

1281 4.01-

2

Kn

KnKnKnkka

...................................................................................... (D-1)

It is easily proven that for any pair of values for ka and k, Eq.D-1 has a unique positive solution: we refer to this solution asthe "pseudo Knudsen" number.

Correlation of the Pseudo-Knudsen Number with Pres-sure, Permeability and Porosity. We have developed twocorrelations to relate the "pseudo-Knudsen" numbers (Knp)and the available data (ka, k∞, , p). The first correlation givenbelow relates the "pseudo-Knudsen" number (Knp) to thereciprocal mean pressure, the porosity (), and the gaspermeability (ka):

352.0598.014.0 ap k

pKn .......................................... (D-2)

Applying Eq. D-2 to Eq. D-1 yields:

352.0598.0

352.0598.0

352.0598.04.0352.0598.012

14.01

14.04

1

14.0]

14.0[4tan

15

1281

a

a

aaa

kp

kp

kp

kp

kk

...................................................................................... (D-3)The equivalent liquid permeability can then be easilycomputed by:

352.0598.0

352.0598.0

352.0598.04.0352.0598.012

14.01

14.04

1

14.0]

14.0[4tan

15

1281/

a

a

aaa

kp

kp

kp

kp

kk

...................................................................................... (D-4)For the second correlation we replaced the gas permeabilitywith the equivalent liquid permeability:

3897.0553.0162.2

kp

Kn p ........................................ (D-5)

Substitution of Eq. D-5 into Eq. D-1 yields:

3897.0553.0

3897.0553.0

3897.0553.04.03897.0553.012

162.21

162.24

1

162.2]

162.2[4tan

15

1281

kp

kp

kp

kp

kka

...................................................................................... (D-6)Rearranging Eq. D-6 yields:


01

62.21

162.24

1

162.2]

162.2[4tan

15

1281

3897.0553.0

3897.0553.0

3897.0553.04.03897.0553.012

kp

kp

kp

kp

kka

....................................................................................... (D-7)Eq. D-7 is an implicit relation where k∞ is solved as a root ofthis relation. Fig. D.1 presents the "pseudo-Knudsen" num-bers obtained with Eqs. D-2 and D-5 plotted against the refer-ence "pseudo-Knudsen" numbers (recall that we defined thereference "pseudo-Knudsen" number as solution of Eq. D-1).

Fig. D.1 Correlated "pseudo-Knudsen" number (Eq. D-2 andEq. D-5) versus reference "pseudo-Knudsen" number(solution of Eq. D-1) — the implicit model (Eq. D-5)achieves a better estimation of Knp than the explicitmodel (Eq. D-2).

In Fig. D.2 we present the equivalent liquid permeabilitiescomputed using Eqs. D-4 and D-7 plotted against thereference Klinkenberg-corrected permeability. Figs. D.3 andD.4 show the behavior of the following function of k∞ for agiven data point (e.g., Sample 1-8, ka = 0.0012699 md, p =45.27 psig, = 0.023).

Fig. D.2 Computed equivalent liquid permeability versusreference Klinkenberg-corrected permeability (extra-polated from a Klinkenberg plot (ka versus reciprocalmean pressure)) — both models present reasonablygood results.

Fig. D.3 Plot of residuals computed with Eq. D-8 againstequivalent liquid permeability (k∞) for an examplepoint from our data set — Sample 1-8, ka = 0.0012699md, p = 45.27 psig, = 0.023. The Klinkenberg-corrected permeability for this sample is k∞ =0.000125 md. The root solution for this case yields k∞= 0.00018 md, which implies an absolute relative errorof 30.56 percent (compared to the "Klinkenberg"permeability).


Fig. D.4 Plot of absolute value of residuals computed with Eq.D-8 against equivalent liquid permeability (k∞) for thesame point as in Fig. D.3 — Sample 1-8, ka =0.0012699 md, p = 45.27 psig, = 0.023. The Klin-kenberg-corrected permeability for this sample is k∞=0.000125 md. The root solution for this case yields k∞= 0.00018 md, which implies an absolute relative errorof 30.56 percent (compared to the "Klinkenberg"permeability).

[blasingame] spe 107954

Documents