a theoretical analysis of pore size distribution effects

Research ArticleA Theoretical Analysis of Pore Size Distribution Effects onShale Apparent Permeability

Shouceng Tian Wenxi Ren Gensheng Li Ruiyue Yang and Tianyu Wang

State Key Laboratory of Petroleum Resources and Engineering China University of Petroleum Beijing 102249 China

Correspondence should be addressed to Shouceng Tian tscsydxcupeducn

Received 19 July 2017 Accepted 27 September 2017 Published 26 November 2017

Academic Editor Zhenhua Rui

Copyright copy 2017 Shouceng Tian et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Apparent permeability is an important input parameter in the simulation of shale gas production Most apparent permeabilitymodels assume a single pore size In this study we develop a theoretical model for quantifying the effect of pore size distributionon shale apparent permeability The model accounts for the nonuniform distribution of pore sizes the rarefaction effect and gascharacteristics The model is validated against available experimental data Theoretical calculations show that the larger the poreradius the larger the apparent permeability Moreover the apparent permeability increases with an increase in the width of poresize distribution with this effect being much more pronounced at low pressure than at high pressure

1 Introduction

The unconventional gas production boom in North Americahas attracted increasing interest internationally [1] Uncon-ventional gas production has been the topic ofmuch researchbut its accurate prediction is still a challenge [2ndash6] especiallyfor shale gas The simulation of shale gas extraction requiresinformation about the apparent permeability of shale whichdeviates its intrinsic permeability due to the complex fluiddynamics in tight porous media Gas transport in shaleinvolves different transport mechanisms including viscousflow slip flow Knudsen diffusion and surface diffusion[7ndash9] A variety of studies have been done on studyingthese differentmechanisms [10ndash14] However most of studiesconsider shale media as a bundle of uniform capillarieswith an effective pore radius 119903e The determination of 119903e isnot a trivial work Some researchers employ the mercuryinjection method to determine 119903e For conventional rocksWinland [15] proposed using the pore size at 35 mercurysaturation as 119903e Pittman [16] extended Winlandrsquos work andpointed out that the pore size at 25 mercury saturationcontrols the permeability For tight gas sands Rezaee et al[17] indicated that the pore size at 10 mercury saturationis a good permeability predictor For shale media very highpressure would be required for mercury to enter small pores

around 36 nm [18] At such high pressures the deformationof the rock and the possibility destruction of pore structurewill affect the measurement [18] For determination of 119903e amore sophisticated method is the Effective Medium Approx-imation (EMA) Ghanbarian and Javadpour [19] combinedEMA with the Javadpour model [7] to estimate the apparentpermeability of shale Their model is very sensitive to 119903e thatis determined by EMA As seen from the above literaturereview the determination of 119903e is still an open issue

Shale media possesses a complex pore architecture [20]Thus the uniform capillary model does not adequatelyaccount for the heterogeneity of shale Civan [21] indicatedthat the apparent permeability of tight rock is directly relatedto pore size distribution (PSD) But limited work has studiedthe effect of PSD on gas transport in shale Villazon et al [22]employed a log-normal density function to characterize thePSD of shale and proposed an apparent permeability modelThemodel is developed based on the HagenndashPoiseuille equa-tion and the general slip boundary condition [23] Thus themodel only considers viscous flow and slip flow But at lowpore pressure Knudsen diffusion dominates gas transportThe fractal capillary model has received much attention overseveral decades [24ndash27] Recently some researchers used thefractal theory to describe the PSD of shale and developed aseries of apparent permeabilitymodels [28ndash30]Thesemodels

HindawiGeofluidsVolume 2017 Article ID 7492328 9 pageshttpsdoiorg10115520177492328

2 Geofluids

assume that the PSD of shale can be modeled as a power lawdistribution Apart from a power law distribution the PSDof shale can be described by a nonsymmetrical distributionsuch as a gamma distribution [31ndash34]

Fewer studies have been carried out to theoreticallyinvestigate the effect of PSD on shale apparent permeabilityIn this article an apparent permeability model that accountsfor the nonuniformdistribution of pore sizes is developed andvalidated with experimental results The model can describeunique flow behaviors in shale and can be used to study theeffect of PSD on shale apparent permeability

2 Model Development

When the characteristic size of pore channels is comparableto or smaller than the gas mean free path the moleculendashwallcollisions predominate over the intermolecular collisionswhich is known as the rarefaction effect At this conditionthe NavierndashStokes equations break down [37] Thus theapplicability of the conventional Darcy law is invalid Shaleis referred to as extremely tight porous media with pore sizesin the nanoscale [38] Due to the rarefaction effect shale gastransport is a complex process involving viscous flow slipflow and Knudsen diffusion [12 14] The Knudsen numberquantifies the rarefaction of fluid The Knudsen numberfor producing shale gas reservoirs is less than unity whichcorresponds to the Darcy flow slip flow and early transition-flow regimes [39] In this work we consider slip flow as a partof Knudsen diffusion [40] Moreover we do not consider thesurface diffusion in this work For a single capillary themolarflow rate for viscous flow 119869v is given by

119869v = 11988812058711990348120583Δ1199011205911198710 (1)

119888 = 119901119885119877119879 (2)

where 119888 is the molar density molsdotmndash3 120583 is the viscosityPasdots 119901 is the pore pressure Pa 119877 is the ideal gas constantJsdotmolndash1sdotKndash1 119879 is the temperature K 119885 is the compressibilityfactor 120593 is the porosity 120591 is the tortuosity 1198710 is the length ofcapillary m and 119903 is the pore radius m

The compressibility factor can be estimated using theSoaveminusBenedictminusWebbminusRubin equation of state (SBWR-EOS) due to its accuracy at reservoir condition [41] Gasviscosity data are taken from the NIST database [42]

For Knudsen diffusion the molar flow rate 119869K is [43]

119869K = 1205871199032119863KΔ1198881205911198710 = 1205871199032119863K119877119879Δ(119901119885) 1

1205911198710= 1205871199032119863K119877119879

119901119885 ( 1

119901 minus 1119885119889119885119889119901 ) Δ119901

1205911198710 (3)

119863K = 23119903radic

8119885119877119879120587119872 (4)

where119872 is gas molar weight kgsdotmolndash1

The molar flow rate 119869 in a single capillary is [40 44]

119869 = 119869v + 119869K= 119888 [12058711990348120583 + 2

31205871199033radic8119885119877119879120587119872 (1

119901 minus 1119885119889119885119889119901 )] Δ119901

1205911198710 (5)

Since shale is assumed to be composed of parallel capil-laries with different sizes the total molar flow rate throughunit area 119869T can be obtained using the following integral

119869T = 119899t int119903max

119903min

119869119891 (119903) 119889119903

= 119899t119888 int119903max

119903min

[12058711990348120583 + 231205871199033radic

8119885119877119879120587119872 (1

119901 minus 1119885119889119885119889119901 )]

sdot Δ1199011205911198710119891 (119903) 119889119903

(6)

119899t = 120593int119903max

119903min1205871199032119891 (119903) 119889119903 (7)

where 119899t is the total number of capillaries per unit areamndash2 119903max is the radius of the maximum capillary m 119903minis the radius of the minimum capillary m and 119891(119903) isthe probability density function for capillaries (pores) Inthis work we take 119903min as the gas molecular radius Thevalue of 119903max is taken as 200 nm [45] Moreover followingMoghaddamand Jamiolahmady [33] we assume that the PSDof shale can be modeled by a gamma distribution

119891 (119903) = 120573120572119903120572minus1119890minus120573119903Γ (120572) (8)

with a mean 119903m = 120572120573 and variance 1205902 = 1205721205732 where119903 is the pore radius m 120572 is the shape parameter 120573 is therate parameter mndash1 119903m is the mean pore radius m andΓ(120572) is the gamma function The parameters 120572 and 120573 canbe any value greater than zero Note that (8) is limited toa relatively narrow monomodal distribution of pore sizesThe pore size distributions described by the gamma densityfunction are shown in Figure 1 for several values of 120572 and 120573As shown in Figure 1(a) the most likely pore radius (the poreradius corresponding to the maximum value of the gammadensity function) is smaller than 119903m Moreover the differencebetween 119903m and most likely pore radius increases with anincrease in the width of the distribution (see Figure 1(a) solidlines) This behavior is due to the asymmetry in the gammadistribution which is characterized by a relatively long tail atlarge 119903 For most porous materials the PSD is not symmetricbut skewed [46]

Based on the Darcy equation the total molar flow ratethrough unit area is

119869T = 119888119896a120583Δ1199011198710 (9)

Comparing (6) and (9) the apparent permeability isobtained as

Geofluids 3

Table 1 Fitted parameters of the apparent permeability model

Sample Permeating gas 120593 [-] 120591 [-] 120572 [-] 120573 [-]Eagle Ford shale samplea Helium 55d 31d 533d 013dXX86b Helium 72c 73d 218d 096dXX88b Helium 27c 27d 312d 095daExperimental data from Aljamaan et al [35] bExperimental data fromMathur et al [36] cThe value is taken fromMathur et al [36] dFitting parameter

= 3 = 1 (rm = 3 ＨＧ 2 = 3nＧ2) = 6 = 2 (rm = 3 ＨＧ 2 = 15nＧ2) = 12 = 4 (rm = 3 ＨＧ 2 = 075nＧ2) = 4 = 2 (rm = 2 ＨＧ 2 = 1nＧ2) = 8 = 812 (rm = 2828 ＨＧ 2 = 1nＧ2) = 16 = 4 (rm = 4 ＨＧ 2 = 1nＧ2)

2 4 6 8 10 12 140Pore radius (nm)

00

01

02

03

04

05

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)


00

02

04

06

08

10

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

3 6 9 12 150Pore radius (nm)

(b)

Figure 1 Representative plots of the gamma distribution (a) probability density function (b) cumulative distribution function

119896a = 120593120591int119903max

119903min[12058711990348 + (23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903

int119903max

119903min1205871199032119891 (119903) 119889119903 (10)

3 Results and Discussion

31 Model Validation with Experimental Data To eliminateor minimize the effect of gas adsorptiondesorption shalepermeability tests with helium are used to validate our model(see (10)) For fitting the model to the experimental data the120591 120572 and 120573 are treated as fitting parameters 120593 is also treatedas a fitting parameter unless its value is knownThe databaseused in this work consists of experiments from Aljamaan etal [35] and Mathur et al [36]

Aljamaan et al [35] carried out permeability measure-ments on two shale samples from Barnett and Eagle FordThey conducted the tests with helium methane nitrogenand carbon dioxide at temperature 29695 K Note that thepermeability measurements with sorbing gases (methanenitrogen and carbon dioxide) are not used to validate ourmodel Figure 2 shows the comparison between the model

results and the experimental data for the Eagle Ford shalesample Table 1 lists the fitted parameters 119903min for helium is026 nmAs can be seen fromFigure 2 ourmodel captures thetrends of the experimental data very well Figure 2 also showsthat the apparent permeability is high initially drops steeplywith increasing pore pressure and approaches a constantvalue at high pore pressure (gt5MPa) This is because atlow pore pressure the dominant transport mechanism isKnudsen diffusion which enhances gas transport in shale[40] With increasing pore pressure the effect of Knudsendiffusion vanishes which results in the decrease of the appar-ent permeability Furthermore according to (10) at high porepressure the apparent permeability of shale converges to itsintrinsic permeability and is not sensitive to pore pressure

The present model is also compared with the experi-mental data from Mathur et al [36] This dataset includestwo shale samples (XX86 and XX88) from the Wolfcamp

4 Geofluids

1 2 3 4 5 60Pore pressure (MPa)

Aljamaan et al (helium)Present model

05

10152025303540455055

Appa

rent

per

mea

bilit

y (u

D)

Figure 2 Comparison of the model results with the experimentaldata of Aljamaan et al [35] The solid line is calculated from (10)The parameter set is presented in Table 1

formation These measurements are performed under steadystate at temperature 29815 K Helium and nitrogen are usedas permeating fluids Once again we only use the heliummeasurements to validate the present model Figure 3 showsthat the present model matches the experimental data wellError bars for the experimental data of Aljamaan et al [35]

and Mathur et al [36] are omitted because experimentaluncertainties are not reported in those studies The fittedparameters are listed in Table 1 The 120591 values fall withinthe range of 23minus119 reported by Katsube et al [47] More-over the 119903m values range from 227 nm to 41 nm which isconsistent with the findings of Javadpour et al [38] Theyfound that pore radiuses in the range of 2 nm to 100 nmare the dominant flow channels in shale Figures 3 and 2show that the permeability values of the Eagle Ford shalesample are significantly higher than those of the Wolfcampshale samples Intuitively one should expect permeability tovary from one sample to another Moreover we note thatin Mathur et alrsquos [36] experiment the measurements areconducted on ldquoas-receivedrdquo samples But in Aljamaan et alrsquos[35] experiment the measurements are conducted on driedsamples Ghanizadeh et al [48] indicated that permeabilityvaluesmeasured on dried samples are higher than thosemea-sured on ldquoas-receivedrdquo samples which is due to the presenceof pore fluid (boundadsorbed water and hydrocarbon) inldquoas-receivedrdquo samples [48] The boundadsorbed water andhydrocarbon partially block pore channels which results inthe decrease of permeability [49]

32 Gas Transport Dynamics in Shale The flow regimes forproducing shale gas reservoirs include continuum flow slipflow and transition flow [40]Thepresentmodel incorporatesviscous flow and Knudsen diffusion to cover these flowregimes To enunciate the gas transport dynamics in shalewe define the permeability ratios for viscous flow 119896rv andKnudsen diffusion 119896rk

119896rv =int119903max

119903min(12058711990348) 119891 (119903) 119889119903

int119903max

119903min[12058711990348 + (23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (11)

119896rk =int119903max

119903min[(23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903

int119903max

119903min[12058711990348 + (23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (12)

Figure 4 shows the permeability ratios 119896rv and 119896rk as afunction of pore pressure The results shown in Figure 4 areobtained with 120572 = 533 and 120573 = 013 The other parametersare listed in Table 2 As can be seen from Figure 4 thepermeability ratio 119896rv increaseswith increasing pore pressureand the permeability ratio 119896rk decreases with increasing porepressure which is consistent with the molecular simulationresults of Hari et al [50] This behavior is due to the fact thatwhen pore pressure is increased the gas molecules collidewith each other more frequently than with the pore walland thus viscous flow contributes more to the total flowFigure 4 also shows that when 0MPa lt 119901 lt 6MPa 119896rvincreases steeply while for 119901 gt 6MPa 119896rv increases slowlySpecifically 119896rv increases sharply from 0327 to 0709 whenthe pore pressure increases from 1 to 5MPa Meanwhile119896rk decreases sharply from 0673 to 0291 When the porepressure increases from 10 to 50MPa 119896rv shows a minorincrease from 0832 to 0967 and 119896rk decreases smoothly

from 0168 to 0033 This is because that according to (11)with increasing pore pressure 119896rv gradually converges to 1Thus at high pore pressure (gt30MPa) 119896rv is not sensitiveto pore pressure Moreover at high pore pressure (gt30MPa)Knudsen diffusion is negligible Thus at this condition 119896rk isalso not sensitive to pore pressure

33 Effect of Pore Size Distribution on Shale Apparent Perme-ability Figure 5(a) shows how themean value of distributionaffects shale apparent permeability The results shown inFigure 5(a) were obtained with 1205721 = 4 1205731 = 2 1205722 = 8 1205732 =812 1205723 = 16 and 1205733 = 4 The corresponding values of 119903m and1205902 are given in the legend of Figure 5(a)The other parametersare listed in Table 2 As shown in Figure 5(a) the apparentpermeability increases with an increase in 119903m Specificallyat the pore pressure of 12MPa the apparent permeabilityincreases from 0771 to 1152 uD when 119903m increases from 2to 4 nm This behavior is due to the fact that the apparent

Geofluids 5

04 06 08 10 12 14 16 18Reciprocal of pore pressure (－０minus1)

02

04

06

08

10

12

14

16

Appa

rent

per

mea

bilit

y (u

D)

Mathur et al (XX88)Present model (XX88)


Figure 3 Comparison of the model results with the experimentaldata of Mathur et al [36] The solid and dashed lines are calculatedfrom (10) The parameter set is presented in Table 1

permeability is positively associated with pore radius Thelarger the pore radius the larger the apparent permeabilityFigure 5(a) also shows that the effect of 119903m on the apparentpermeability is much larger at low pore pressure than at highpore pressure Specifically the difference between 119896a3 and 119896a1decreases from 0742 to 0092 uD when the pore pressureincreases from 06 to 6MPa At low pore pressure thedifference between 119896a3 and 119896a1 is contributed by both viscousflow and Knudsen diffusion But at high pore pressureKnudsen diffusion could be ignored Thus the increase inpore pressure decreases the effect of 119903119898 on the apparentpermeability

Figure 5(b) shows how the variance of distribution affectsthe apparent permeability The results shown in Figure 5(b)were obtained with 1205721 = 3 1205731 = 1 1205722 = 6 1205732 = 2 1205723 = 12 and1205733 = 4The corresponding values of 119903m and 1205902 are given in thelegend of Figure 5(b) The other parameters are presented inTable 2 For the three considered distributions the apparentpermeability increases with an increase in 1205902 At the pore

50 5020 35 6010 4025 554515 30Pore pressure (MPa)

00

02

04

06

08

10

Perm

eabi

lity

ratio

Knudsen diffusionViscous flow

Figure 4 Permeability ratio as a function of pore pressureThe solidline is calculated from (11) and the dash dot line is calculated from(12) The parameter set is presented in Table 2

pressure of 12MPa the apparent permeability increases from0863 to 1315 uD when 1205902 increases from 075 to 3 nm2 Thisresult is attributed to the width of the PSD being representedby1205902The gammadistribution is a positive distributionThuswith increasing 1205902 the PSD presents a lower pick at smallpore sizes but a longer tail at large pore sizes as shown inFigure 1(a) Moreover the molar flow rates for viscous flow119869v and for Knudsen diffusion 119869K have a strong dependenceon pore radius 119869v is proportional to 1199034 (see (1)) and 119869K isproportional to 1199033 (see (3)) respectively Thus in the caseof the same 119903m increasing 1205902 results in the increase of theapparent permeability Figure 5(b) also shows the effect of 1205902on the apparent permeability is more pronounced under lowpore pressure than under high pore pressure This is becauseat high pore pressure Knudsen diffusion is negligible

The effect of PSD on gas transport in shale can also beseen from the behavior of the fractional flow rate 119891q which isdefined as

119891q =int1199031015840119903min

[12058711990348120583 + (23) 1205871199033radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903int119903max

119903min[12058711990348120583 + (23) 1205871199033radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (13)

In (13) the denominator characterizes the total flow rateand the numerator characterizes the flow rate contributedby pores with radius 119903 le 1199031015840 Figure 6(a) shows the effectof 119903m on 119891q The results shown in Figure 6(a) are obtainedwith 1205721 = 4 1205731 = 2 1205722 = 8 1205732 = 812 1205723 = 16 and 1205733 =4 The corresponding values of mean and variance are givenin the legend of Figure 6(a) The other parameters are listedin Table 2 As can be seen from Figure 6(a) in the case of

the same 1205902 increasing 119903m shifts the fractional flow curvetoward the right which means that the majority of the flowoccurs through large pores Specifically at the pore pressureof 10MPa 119891q1 for the pore radius of 3 nm is 364 At thesame pore pressure 119891q3 for the pore radius of 3 nm is only33 which means that more than 90 of the total flow wascontributed by pores with radius 119903 ge 3 nm Moreover at theconditions of pore pressure and pore radius of 10MPa and

6 Geofluids

0

1

2

3

4Ap

pare

nt p

erm

eabi

lity

(uD

)


ka1 (rm = 2 ＨＧ 2 = 1nＧ2)ka2 (rm = 2828 ＨＧ 2 = 1nＧ2)ka3 (rm = 4 ＨＧ 2 = 1nＧ2)

(a)

0

1

2

3

4

Appa

rent

per

mea

bilit

y (u

D)


ka1 (rm = 3 ＨＧ 2 = 3nＧ2)ka2 (rm = 3 ＨＧ 2 = 15 nＧ2)ka3 (rm = 3 ＨＧ 2 = 075nＧ2)

(b)

Figure 5 Apparent permeability as a function of pore pressure The lines are calculated from (10) The parameter set is presented in Table 2(a) The effect of 119903m on the apparent permeability (b) The effect of 1205902 on the apparent permeability

5 nm 119891q1 is equal to 851 119891q2 is equal to 815 and 119891q3is equal to 595 This is because in the case of the same1205902 increasing 119903m means that more large pores are presentThus themain contribution to the total flow rate is from theselarge pores Figure 6(a) also shows the effect of pore pressureon 119891q As can be seen from Figure 6(a) decreasing the porepressure shifts the fractional flow curve toward the leftWhenthe pore pressure is 10MPa and the pore radius is 4 nm 119891q1 =657119891q2 = 549 and119891q3 = 239When the pore pressureis 001MPa and the pore radius is 4 nm 119891q1 = 687 119891q2 =577 and 119891q3 = 258 This is because Knudsen diffusiondominates gas transport at lowpore pressure which enhancesgas flowThus at the same pore radius119891q at low pore pressureis larger than that at high pore pressure

Figure 6(b) shows the sensitivity of 119891q to the varianceof PSD 1205902 The results shown in Figure 6(b) were obtainedwith 1205721 = 3 1205731 = 1 1205722 = 6 1205732 = 2 1205723 = 12 and 1205733 = 4 Thecorresponding values of 119903m and 1205902 are given in the legend ofFigure 6(b) As shown in Figure 6(b) in the case of the same119903m increasing 1205902 shifts the fractional flow curve toward theright Specifically at the pore pressure of 10MPa 119891q1 for thepore radius of 4 nm is 182 At the same pore pressure 119891q3for the pore radius of 4 nm is 61This behavior is explainedby the fact that decreasing 1205902 improves the uniformity of thePSD namely the width of the PSD decreases with decreasing1205902 see Figure 1(a) (solid lines) Figure 1(a) (solid lines) alsoshows that with decreasing 1205902 the most likely pore radiusapproaches themean pore radiusWhen 1205902 equals 0 the poresize is uniform and equal to 119903m At this condition119891q0 shows astep increase from 0 to 1 at 119903m as shown in Figure 6(b) Thusin the case of the same 119903m with decreasing 1205902 119891q increasesmore steeply at 119903m Figure 6(b) also shows that 119891q at low porepressure is larger than that at high pore pressure It is worthnoting that the difference between119891q at low pore pressure and

Table 2 Parameters for the sensitivity analysis

Permeating gas 119879 [K] 120593 [-] 120591 [-] 120572 [-] 120573 [mminus1]Helium 29695 55 31 3sim16 013sim4

that at high pore pressure increases with increasing 1205902This isexplained by the fact that a larger 1205902 corresponds to a higherquantity of extremely small pores (119903 le 1 nm) as shown inFigure 1(b) (solid lines) At low pore pressure the smaller thepore radius the stronger themoleculeminuswall collisions and thestronger the rarefaction effect which enhances gas transport

4 Conclusion

In this paper we have studied the effect of PSD on shaleapparent permeability We have derived an apparent perme-ability model with the assumption that shale media consistsof a bundle of tortuous capillaries with a gamma distributionThe apparent permeability model matches experimental datawell Although the model is not completely realistic it allowsus to understand the effect of PSD on shale apparent perme-ability Moreover the proposedmodel possesses an analyticalform and can be easily incorporated into existing reservoirsimulators without large code changeThemajor findings areas follows In the case of the same variance of PSD increasing119903m increases the apparent permeability In the case of thesame 119903m the apparent permeability increases with an increasein the width of PSD Moreover the apparent permeability ismore sensitive to PSD at low pore pressure than at high porepressure because Knudsen diffusion enhances gas flow at lowpore pressure and is negligible at high pore pressure

It would be interesting to generalize the present workby considering the effect of surface diffusion and stresssensitivity This will be done in future research

Geofluids 7

00

02

04

06

08

10

Frac

tiona

l flow

rate

2 4 6 8 10 12 14 16 180Pore radius (nm)

fq1 (rm = 2 ＨＧ 2 = 1nＧ2 10 MPa)fq2 (rm = 2828 ＨＧ 2 = 1nＧ2 10 MPa)fq3 (rm = 4 ＨＧ 2 = 1nＧ2 10 MPa)f

q1 (rm = 2 ＨＧ 2 = 1nＧ2 001 MPa)f

q2 (rm = 2828 ＨＧ 2 = 1nＧ2 001 MPa)f

q3 (rm = 4 ＨＧ 2 = 1nＧ2 001 MPa)

(a)

00

02

04

06

08

10

Frac

tiona

l flow

rate

3 6 9 12 15 18 210Pore radius (nm)

fq1 (rm = 3 ＨＧ 2 = 3nＧ2 10 MPa)fq2 (rm = 3 ＨＧ 2 = 15 nＧ2 10 MPa)fq3 (rm = 3 ＨＧ 2 = 075nＧ2 10 MPa)f

q1 (rm = 3 ＨＧ 2 = 3nＧ2 001 MPa)f

q2 (rm = 3 ＨＧ 2 = 15 nＧ2 001 MPa)f

q3 (rm = 3 ＨＧ 2 = 075nＧ2 001 MPa)fq0 (rm = 3 ＨＧ 2 = 0nＧ2 10 MPa)

(b)

Figure 6 Fractional flow rate as a function of pore radius The lines are calculated from (13) The parameter set is presented in Table 2 (a)The effect of 119903m on the fractional flow rate (b) The effect of 1205902 on the fractional flow rate

Nomenclature

119903e Effective pore radius m119869v Molar flow rate for viscous flow molsdotsndash1119888 Molar density molsdotmndash3

120583 Viscosity Pasdots119901 Pore pressure Pa119877 Ideal gas constant Jsdotmolndash1sdotKndash1

119879 Temperature K119885 Compressibility factor120593 Porosity120591 Tortuosity1198710 Length of capillary m119903 Pore radius m119869K Molar flow rate for Knudsen diffusionmolsdotsndash1119872 Gas molar weight kgsdotmolndash1119869 Molar flow rate in a single capillary molsdotsndash1119869T Total molar flow rate molsdotsndash1119899t Total number of capillaries per unit area mndash2

119903max Radius of maximum capillary m119903min Radius of minimum capillary m120572 Shape parameter120573 Rate parameter mndash1

119903m Mean pore radius m1205902 Variance m2119896rv Permeability ratio for viscous flow119896rk Permeability ratio for Knudsen diffusion119891q Fractional flow rate

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China (nos 51210006 U1562212 and51234006)

References

[1] M Jakob and J Hilaire ldquoUnburnable fossil-fuel reservesrdquoNature vol 517 no 7533 pp 150ndash152 2015

[2] Y He S Cheng S Li et al ldquoA semianalytical methodology todiagnose the locations of underperforming hydraulic fracturesthrough pressure-transient analysis in tight gas reservoirrdquo SPEJournal vol 22 no 03 pp 0924ndash0939 2016

[3] W Ren G Li S Tian M Sheng and L Geng ldquoAnalytical mod-elling of hysteretic constitutive relations governing spontaneousimbibition of fracturing fluid in shalerdquo Journal of Natural GasScience and Engineering vol 34 pp 925ndash933 2016

[4] R Yang Z Huang G Li et al ldquoAn innovative approach tomodel two-phase flowback of shale gas wells with complexfracture networksrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition ATCE 2016 are September 2016

[5] R Yang Z Huang W Yu et al ldquoA comprehensive model forreal gas transport in shale formations with complex non-planar

8 Geofluids

fracture networksrdquo Scientific Reports vol 6 Article ID 36673pp 1ndash13 2016

[6] W Yu K Wu K Sepehrnoori and W Xu ldquoA comprehensivemodel for simulation of gas transport in shale formation withcomplex hydraulic-fracture geometryrdquo SPE Reservoir Evalua-tion and Engineering vol 20 no 3 pp 547ndash561 2016

[7] F Javadpour ldquoNanopores and apparent permeability of gasflow in mudrocks (shales and siltstone)rdquo Journal of CanadianPetroleum Technology vol 48 no 8 pp 16ndash21 2009

[8] K Wu X Li C Wang Z Chen and W Yu ldquoA model for gastransport in microfractures of shale and tight gas reservoirsrdquoAIChE Journal vol 61 no 6 pp 2079ndash2088 2015

[9] K Wu Z Chen X Li et al ldquoFlow behavior of gas confined innanoporous shale at high pressure real gas effectrdquo Fuel vol 205pp 173ndash183 2017

[10] H Darabi A Ettehad F Javadpour and K Sepehrnoori ldquoGasflow in ultra-tight shale stratardquo Journal of Fluid Mechanics vol710 pp 641ndash658 2012

[11] M Kazemi and A Takbiri-Borujeni ldquoAn analytical model forshale gas permeabilityrdquo International Journal of Coal Geologyvol 146 pp 188ndash197 2015

[12] K Wu X Li C Wang W Yu and Z Chen ldquoModel for surfacediffusion of adsorbed gas in nanopores of shale gas reservoirsrdquoIndustrial amp Engineering Chemistry Research vol 54 no 12 pp3225ndash3236 2015

[13] H Singh and F Javadpour ldquoLangmuir slip-Langmuir sorptionpermeability model of shalerdquo Fuel vol 164 pp 28ndash37 2016

[14] W Ren G Li S Tian M Sheng and L Geng ldquoAdsorption andsurface diffusion of supercritical methane in shalerdquo Industrialand Engineering Chemistry Research vol 56 no 12 pp 3446ndash3455 2017

[15] S Kolodzie ldquoAnalysis of pore throat size and use of theWaxman-Smits equation to determine OOIP in Spindle FieldColoradordquo Society of Petroleum Engineers of AIME 1980

[16] E D Pittman ldquoRelationship of porosity and permeability tovarious parameters derived from mercury injection-capillarypressure curves for sandstonerdquo The American Association ofPetroleum Geologists Bulletin vol 76 no 2 pp 191ndash198 1992

[17] R Rezaee A Saeedi and B Clennell ldquoTight gas sands perme-ability estimation frommercury injection capillary pressure andnuclear magnetic resonance datardquo Journal of Petroleum Scienceand Engineering vol 88-89 pp 92ndash99 2012

[18] U Kuila and M Prasad ldquoSpecific surface area and pore-sizedistribution in clays and shalesrdquoGeophysical Prospecting vol 61no 2 pp 341ndash362 2013

[19] B Ghanbarian and F Javadpour ldquoUpscaling pore pressure-dependent gas permeability in shalesrdquo Journal of GeophysicalResearch Solid Earth vol 122 no 4 pp 2541ndash2552 2017

[20] H E King A P R Eberle C CWalters C E Kliewer D Ertasand C Huynh ldquoPore architecture and connectivity in gas shalerdquoEnergy amp Fuels vol 29 no 3 pp 1375ndash1390 2015

[21] F Civan ldquoA triple-mechanism fractal model with hydraulicdispersion for gas permeation in tight reservoirsrdquo inProceedingsof the SPE International Petroleum Conference and Exhibition2002

[22] M Villazon R F Sigal F Civan and D Devegowda ldquoParamet-ric Investigation of Shale Gas Production Considering Nano-Scale Pore Size Distribution Formation Factor and Non-DarcyFlow Mechanismsrdquo in Proceedings of the SPE Annual TechnicalConference and Exhibition Denver Colorado USA

[23] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquo NanoscaleandMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999

[24] J Cai B Yu M Zou and L Luo ldquoFractal characterization ofspontaneous co-current imbibition in porous mediardquo Energy ampFuels vol 24 no 3 pp 1860ndash1867 2010

[25] J Cai B Yu M Zou and M Mei ldquoFractal analysis ofinvasion depth of extraneous fluids in porous mediardquo ChemicalEngineering Science vol 65 no 18 pp 5178ndash5186 2010

[26] J Cai X Hu D C Standnes and L You ldquoAn analyticalmodel for spontaneous imbibition in fractal porous mediaincluding gravityrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 414 pp 228ndash233 2012

[27] R Liu Y Jiang B Li and L Yu ldquoEstimating permeabilityof porous media based on modified HagenndashPoiseuille flow intortuous capillaries with variable lengthsrdquo Microfluidics andNanofluidics vol 20 no 8 pp 1ndash13 2016

[28] Y Yuan N G Doonechaly and S Rahman ldquoAn analyticalmodel of apparent gas permeability for tight porous mediardquoTransport in Porous Media vol 111 no 1 pp 193ndash214 2016

[29] A Behrang and A Kantzas ldquoA hybrid methodology to predictgas permeability in nanoscale organic materials a combinationof fractal theory kinetic theory of gases and Boltzmann trans-port equationrdquo Fuel vol 188 pp 239ndash245 2017

[30] L Geng G Li S TianM ShengW Ren and P Zitha ldquoA fractalmodel for real gas transport in porous shalerdquo AIChE Journalvol 63 no 4 pp 1430ndash1440 2017

[31] R Zhang Z Ning F Yang X Wang H Zhao and Q WangldquoImpacts of nanopore structure and elastic properties on stress-dependent permeability of gas shalesrdquo Journal of Natural GasScience and Engineering vol 26 pp 1663ndash1672 2015

[32] E A Letham and R M Bustin ldquoKlinkenberg gas slippage mea-surements as a means for shale pore structure characterizationrdquoGeofluids vol 16 no 2 pp 264ndash278 2016

[33] R N Moghaddam and M Jamiolahmady ldquoSlip flow in porousmediardquo Fuel vol 173 pp 298ndash310 2016

[34] R N Moghaddam and M Jamiolahmady ldquoFluid transport inshale gas reservoirs simultaneous effects of stress and slippageon matrix permeabilityrdquo International Journal of Coal Geologyvol 163 pp 87ndash99 2016

[35] H Aljamaan K Alnoaimi and A R Kovscek ldquoIn-DepthExperimental Investigation of Shale Physical and TransportPropertiesrdquo in Proceedings of the Unconventional ResourcesTechnology Conference pp 1120ndash1129 Denver Colorado USA

[36] A Mathur C H Sondergeld and C S Rai ldquoComparisonof steady-state and transient methods for measuring shalepermeabilityrdquo in Proceedings of the SPE Low Perm Symposium2016 May 2016

[37] N G Hadjiconstantinou ldquoThe limits of Navier-Stokes the-ory and kinetic extensions for describing small-scale gaseoushydrodynamicsrdquo Physics of Fluids vol 18 no 11 Article ID111301 pp 1ndash19 2006

[38] F Javadpour D Fisher and M Unsworth ldquoNanoscale gasflow in shale gas sedimentsrdquo Journal of Canadian PetroleumTechnology vol 46 no 10 pp 55ndash61 2007

[39] S Sinha E M Braun M D Determan et al ldquoSteady-statepermeability measurements on intact shale samples at reservoirconditions - effect of stress temperature pressure and type ofgasrdquo inProceedings of the 18thMiddle East Oil andGas Show andConference 2013 Transforming the Energy Future MEOS rsquo13 pp956ndash970 bhr March 2013

Geofluids 9

[40] W Ren G Li S Tian M Sheng and X Fan ldquoAn analyticalmodel for real gas flow in shale nanopores with non-circularcross-sectionrdquo AIChE Journal vol 62 no 8 pp 2893ndash29012016

[41] G S Soave ldquoAn effective modification of the Benedict-Webb-Rubin equation of staterdquo Fluid Phase Equilibria vol 164 no 2pp 157ndash172 1999

[42] E Lemmon M Huber and M McLinden ldquoREFPROP Ref-erence fluid thermodynamic and transport propertiesrdquo NISTstandard reference database Version 90 2010

[43] K Wu Z Chen and X Li ldquoReal gas transport throughnanopores of varying cross-section type and shape in shale gasreservoirsrdquo Chemical Engineering Journal vol 281 pp 813ndash8252015

[44] T Veltzke and JThoming ldquoAn analytically predictive model formoderately rarefied gas flowrdquo Journal of Fluid Mechanics vol698 pp 406ndash422 2012

[45] R G Loucks R M Reed S C Ruppel and D M JarvieldquoMorphology genesis and distribution of nanometer-scalepores in siliceousmudstones of themississippian barnett shalerdquoJournal of Sedimentary Research vol 79 no 12 pp 848ndash8612009

[46] J Ghassemzadeh M Hashemi L Sartor and M Sahimi ldquoPorenetwork simulation of imbibition into paper during coating Imodel developmentrdquo AIChE Journal vol 47 no 3 pp 519ndash5352001

[47] T J Katsube B S Mudford and M E Best ldquoPetrophysicalcharacteristics of shales from the Scotian shelfrdquoGeophysics vol56 no 10 pp 1681ndash1689 1991

[48] A Ghanizadeh S Bhowmik O Haeri-Ardakani H Sanei andC R Clarkson ldquoA comparison of shale permeability coeffi-cients derived using multiple non-steady-state measurementtechniques examples from the Duvernay Formation Alberta(Canada)rdquo Fuel vol 140 pp 371ndash387 2015

[49] A Ghanizadeh M Gasparik A Amann-Hildenbrand Y Gen-sterblum and B M Krooss ldquoExperimental study of fluidtransport processes in the matrix system of the europeanorganic-rich shales I scandinavian alum shalerdquo Marine andPetroleum Geology vol 51 pp 79ndash99 2014

[50] P D S Hari S K Prabha and S P Sathian ldquoThe effect ofcharacteristic length on mean free path for confined gasesrdquoPhysica A Statistical Mechanics and its Applications vol 437Article ID 16137 pp 68ndash74 2015

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of

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Journal ofPetroleum Engineering


GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in


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Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Geological ResearchJournal of


Geology Advances in

2 Geofluids

assume that the PSD of shale can be modeled as a power lawdistribution Apart from a power law distribution the PSDof shale can be described by a nonsymmetrical distributionsuch as a gamma distribution [31ndash34]

Fewer studies have been carried out to theoreticallyinvestigate the effect of PSD on shale apparent permeabilityIn this article an apparent permeability model that accountsfor the nonuniformdistribution of pore sizes is developed andvalidated with experimental results The model can describeunique flow behaviors in shale and can be used to study theeffect of PSD on shale apparent permeability

2 Model Development

When the characteristic size of pore channels is comparableto or smaller than the gas mean free path the moleculendashwallcollisions predominate over the intermolecular collisionswhich is known as the rarefaction effect At this conditionthe NavierndashStokes equations break down [37] Thus theapplicability of the conventional Darcy law is invalid Shaleis referred to as extremely tight porous media with pore sizesin the nanoscale [38] Due to the rarefaction effect shale gastransport is a complex process involving viscous flow slipflow and Knudsen diffusion [12 14] The Knudsen numberquantifies the rarefaction of fluid The Knudsen numberfor producing shale gas reservoirs is less than unity whichcorresponds to the Darcy flow slip flow and early transition-flow regimes [39] In this work we consider slip flow as a partof Knudsen diffusion [40] Moreover we do not consider thesurface diffusion in this work For a single capillary themolarflow rate for viscous flow 119869v is given by

119869v = 11988812058711990348120583Δ1199011205911198710 (1)

119888 = 119901119885119877119879 (2)

where 119888 is the molar density molsdotmndash3 120583 is the viscosityPasdots 119901 is the pore pressure Pa 119877 is the ideal gas constantJsdotmolndash1sdotKndash1 119879 is the temperature K 119885 is the compressibilityfactor 120593 is the porosity 120591 is the tortuosity 1198710 is the length ofcapillary m and 119903 is the pore radius m

The compressibility factor can be estimated using theSoaveminusBenedictminusWebbminusRubin equation of state (SBWR-EOS) due to its accuracy at reservoir condition [41] Gasviscosity data are taken from the NIST database [42]

For Knudsen diffusion the molar flow rate 119869K is [43]

119869K = 1205871199032119863KΔ1198881205911198710 = 1205871199032119863K119877119879Δ(119901119885) 1

1205911198710= 1205871199032119863K119877119879

119901119885 ( 1

119901 minus 1119885119889119885119889119901 ) Δ119901

1205911198710 (3)

119863K = 23119903radic

8119885119877119879120587119872 (4)

where119872 is gas molar weight kgsdotmolndash1

The molar flow rate 119869 in a single capillary is [40 44]

119869 = 119869v + 119869K= 119888 [12058711990348120583 + 2

31205871199033radic8119885119877119879120587119872 (1

119901 minus 1119885119889119885119889119901 )] Δ119901

1205911198710 (5)

Since shale is assumed to be composed of parallel capil-laries with different sizes the total molar flow rate throughunit area 119869T can be obtained using the following integral

119869T = 119899t int119903max

119903min

119869119891 (119903) 119889119903

= 119899t119888 int119903max

119903min

[12058711990348120583 + 231205871199033radic

8119885119877119879120587119872 (1

119901 minus 1119885119889119885119889119901 )]

sdot Δ1199011205911198710119891 (119903) 119889119903

(6)

119899t = 120593int119903max

119903min1205871199032119891 (119903) 119889119903 (7)

where 119899t is the total number of capillaries per unit areamndash2 119903max is the radius of the maximum capillary m 119903minis the radius of the minimum capillary m and 119891(119903) isthe probability density function for capillaries (pores) Inthis work we take 119903min as the gas molecular radius Thevalue of 119903max is taken as 200 nm [45] Moreover followingMoghaddamand Jamiolahmady [33] we assume that the PSDof shale can be modeled by a gamma distribution

119891 (119903) = 120573120572119903120572minus1119890minus120573119903Γ (120572) (8)

with a mean 119903m = 120572120573 and variance 1205902 = 1205721205732 where119903 is the pore radius m 120572 is the shape parameter 120573 is therate parameter mndash1 119903m is the mean pore radius m andΓ(120572) is the gamma function The parameters 120572 and 120573 canbe any value greater than zero Note that (8) is limited toa relatively narrow monomodal distribution of pore sizesThe pore size distributions described by the gamma densityfunction are shown in Figure 1 for several values of 120572 and 120573As shown in Figure 1(a) the most likely pore radius (the poreradius corresponding to the maximum value of the gammadensity function) is smaller than 119903m Moreover the differencebetween 119903m and most likely pore radius increases with anincrease in the width of the distribution (see Figure 1(a) solidlines) This behavior is due to the asymmetry in the gammadistribution which is characterized by a relatively long tail atlarge 119903 For most porous materials the PSD is not symmetricbut skewed [46]

Based on the Darcy equation the total molar flow ratethrough unit area is

119869T = 119888119896a120583Δ1199011198710 (9)

Comparing (6) and (9) the apparent permeability isobtained as

Geofluids 3




2 4 6 8 10 12 140Pore radius (nm)

00

01

02

03

04

05

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)


00

02

04

06

08

10

Cum

ulat

ive d

istrib

utio

n fu

nctio

n


(b)


119896a = 120593120591int119903max

119903min[12058711990348 + (23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903

int119903max

119903min1205871199032119891 (119903) 119889119903 (10)






4 Geofluids



05

10152025303540455055

Appa

rent

per

mea

bilit

y (u

D)





119896rv =int119903max

119903min(12058711990348) 119891 (119903) 119889119903

int119903max

119903min[12058711990348 + (23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (11)

119896rk =int119903max

119903min[(23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903

int119903max

119903min[12058711990348 + (23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (12)




Geofluids 5


02

04

06

08

10

12

14

16

Appa

rent

per

mea

bilit

y (u

D)







00

02

04

06

08

10

Perm

eabi

lity

ratio





119891q =int1199031015840119903min


119903min[12058711990348120583 + (23) 1205871199033radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (13)



6 Geofluids

0

1

2

3

4Ap

pare

nt p

erm

eabi

lity

(uD

)



(a)

0

1

2

3

4

Appa

rent

per

mea

bilit

y (u

D)



(b)







4 Conclusion



Geofluids 7

00

02

04

06

08

10

Frac

tiona

l flow

rate

2 4 6 8 10 12 14 16 180Pore radius (nm)


q1 (rm = 2 ＨＧ 2 = 1nＧ2 001 MPa)f

q2 (rm = 2828 ＨＧ 2 = 1nＧ2 001 MPa)f

q3 (rm = 4 ＨＧ 2 = 1nＧ2 001 MPa)

(a)

00

02

04

06

08

10

Frac

tiona

l flow

rate

3 6 9 12 15 18 210Pore radius (nm)


q1 (rm = 3 ＨＧ 2 = 3nＧ2 001 MPa)f

q2 (rm = 3 ＨＧ 2 = 15 nＧ2 001 MPa)f


(b)


Nomenclature








Acknowledgments


References






8 Geofluids




































Geofluids 9



















Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 3




2 4 6 8 10 12 140Pore radius (nm)

00

01

02

03

04

05

Cum

ulat

ive d

istrib

utio

n fu

nctio

n

(a)


00

02

04

06

08

10

Cum

ulat

ive d

istrib

utio

n fu

nctio

n


(b)


119896a = 120593120591int119903max

119903min[12058711990348 + (23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903

int119903max

119903min1205871199032119891 (119903) 119889119903 (10)






4 Geofluids



05

10152025303540455055

Appa

rent

per

mea

bilit

y (u

D)





119896rv =int119903max

119903min(12058711990348) 119891 (119903) 119889119903

int119903max

119903min[12058711990348 + (23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (11)

119896rk =int119903max

119903min[(23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903

int119903max

119903min[12058711990348 + (23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (12)




Geofluids 5


02

04

06

08

10

12

14

16

Appa

rent

per

mea

bilit

y (u

D)







00

02

04

06

08

10

Perm

eabi

lity

ratio





119891q =int1199031015840119903min


119903min[12058711990348120583 + (23) 1205871199033radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (13)



6 Geofluids

0

1

2

3

4Ap

pare

nt p

erm

eabi

lity

(uD

)



(a)

0

1

2

3

4

Appa

rent

per

mea

bilit

y (u

D)



(b)







4 Conclusion



Geofluids 7

00

02

04

06

08

10

Frac

tiona

l flow

rate

2 4 6 8 10 12 14 16 180Pore radius (nm)


q1 (rm = 2 ＨＧ 2 = 1nＧ2 001 MPa)f

q2 (rm = 2828 ＨＧ 2 = 1nＧ2 001 MPa)f

q3 (rm = 4 ＨＧ 2 = 1nＧ2 001 MPa)

(a)

00

02

04

06

08

10

Frac

tiona

l flow

rate

3 6 9 12 15 18 210Pore radius (nm)


q1 (rm = 3 ＨＧ 2 = 3nＧ2 001 MPa)f

q2 (rm = 3 ＨＧ 2 = 15 nＧ2 001 MPa)f


(b)


Nomenclature








Acknowledgments


References






8 Geofluids




































Geofluids 9



















Mining


Journal of









Journal of




Advances in











Geology Advances in

4 Geofluids



05

10152025303540455055

Appa

rent

per

mea

bilit

y (u

D)





119896rv =int119903max

119903min(12058711990348) 119891 (119903) 119889119903

int119903max

119903min[12058711990348 + (23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (11)

119896rk =int119903max

119903min[(23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903

int119903max

119903min[12058711990348 + (23) 1205871199033120583radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (12)




Geofluids 5


02

04

06

08

10

12

14

16

Appa

rent

per

mea

bilit

y (u

D)







00

02

04

06

08

10

Perm

eabi

lity

ratio





119891q =int1199031015840119903min


119903min[12058711990348120583 + (23) 1205871199033radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (13)



6 Geofluids

0

1

2

3

4Ap

pare

nt p

erm

eabi

lity

(uD

)



(a)

0

1

2

3

4

Appa

rent

per

mea

bilit

y (u

D)



(b)







4 Conclusion



Geofluids 7

00

02

04

06

08

10

Frac

tiona

l flow

rate

2 4 6 8 10 12 14 16 180Pore radius (nm)


q1 (rm = 2 ＨＧ 2 = 1nＧ2 001 MPa)f

q2 (rm = 2828 ＨＧ 2 = 1nＧ2 001 MPa)f

q3 (rm = 4 ＨＧ 2 = 1nＧ2 001 MPa)

(a)

00

02

04

06

08

10

Frac

tiona

l flow

rate

3 6 9 12 15 18 210Pore radius (nm)


q1 (rm = 3 ＨＧ 2 = 3nＧ2 001 MPa)f

q2 (rm = 3 ＨＧ 2 = 15 nＧ2 001 MPa)f


(b)


Nomenclature








Acknowledgments


References






8 Geofluids




































Geofluids 9



















Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 5


02

04

06

08

10

12

14

16

Appa

rent

per

mea

bilit

y (u

D)







00

02

04

06

08

10

Perm

eabi

lity

ratio





119891q =int1199031015840119903min


119903min[12058711990348120583 + (23) 1205871199033radic8119885119877119879120587119872(1119901 minus (1119885) (119889119885119889119901))] 119891 (119903) 119889119903 (13)



6 Geofluids

0

1

2

3

4Ap

pare

nt p

erm

eabi

lity

(uD

)



(a)

0

1

2

3

4

Appa

rent

per

mea

bilit

y (u

D)



(b)







4 Conclusion



Geofluids 7

00

02

04

06

08

10

Frac

tiona

l flow

rate

2 4 6 8 10 12 14 16 180Pore radius (nm)


q1 (rm = 2 ＨＧ 2 = 1nＧ2 001 MPa)f

q2 (rm = 2828 ＨＧ 2 = 1nＧ2 001 MPa)f

q3 (rm = 4 ＨＧ 2 = 1nＧ2 001 MPa)

(a)

00

02

04

06

08

10

Frac

tiona

l flow

rate

3 6 9 12 15 18 210Pore radius (nm)


q1 (rm = 3 ＨＧ 2 = 3nＧ2 001 MPa)f

q2 (rm = 3 ＨＧ 2 = 15 nＧ2 001 MPa)f


(b)


Nomenclature








Acknowledgments


References






8 Geofluids




































Geofluids 9



















Mining


Journal of









Journal of




Advances in











Geology Advances in

6 Geofluids

0

1

2

3

4Ap

pare

nt p

erm

eabi

lity

(uD

)



(a)

0

1

2

3

4

Appa

rent

per

mea

bilit

y (u

D)



(b)







4 Conclusion



Geofluids 7

00

02

04

06

08

10

Frac

tiona

l flow

rate

2 4 6 8 10 12 14 16 180Pore radius (nm)


q1 (rm = 2 ＨＧ 2 = 1nＧ2 001 MPa)f

q2 (rm = 2828 ＨＧ 2 = 1nＧ2 001 MPa)f

q3 (rm = 4 ＨＧ 2 = 1nＧ2 001 MPa)

(a)

00

02

04

06

08

10

Frac

tiona

l flow

rate

3 6 9 12 15 18 210Pore radius (nm)


q1 (rm = 3 ＨＧ 2 = 3nＧ2 001 MPa)f

q2 (rm = 3 ＨＧ 2 = 15 nＧ2 001 MPa)f


(b)


Nomenclature








Acknowledgments


References






8 Geofluids




































Geofluids 9



















Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 7

00

02

04

06

08

10

Frac

tiona

l flow

rate

2 4 6 8 10 12 14 16 180Pore radius (nm)


q1 (rm = 2 ＨＧ 2 = 1nＧ2 001 MPa)f

q2 (rm = 2828 ＨＧ 2 = 1nＧ2 001 MPa)f

q3 (rm = 4 ＨＧ 2 = 1nＧ2 001 MPa)

(a)

00

02

04

06

08

10

Frac

tiona

l flow

rate

3 6 9 12 15 18 210Pore radius (nm)


q1 (rm = 3 ＨＧ 2 = 3nＧ2 001 MPa)f

q2 (rm = 3 ＨＧ 2 = 15 nＧ2 001 MPa)f


(b)


Nomenclature








Acknowledgments


References






8 Geofluids




































Geofluids 9



















Mining


Journal of









Journal of




Advances in











Geology Advances in

8 Geofluids




































Geofluids 9



















Mining


Journal of









Journal of




Advances in











Geology Advances in

Geofluids 9



















Mining


Journal of









Journal of




Advances in











Geology Advances in








Mining


Journal of









Journal of




Advances in











Geology Advances in

a theoretical analysis of pore size distribution effects

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