00013879_reference for Coreys Formula

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SPE DOE%elaty of U.S. Departmentpetroleum Engineers of Enaroy

SPEIDOE 13879Capillary PressureGas Sands and Permeability Relationships in Tightby J.D. Wells and J.O. Amaefule, etrophyskd ervicesSPE Members

~PYri9h 198S, Society of PetroleumEngineersrhis paperwas presentedatthe SPE/DOE 1985 LowPermeaMtii Gas Reservoirshefdin Osnnr, Colorado,May 19-22, 19S5.The materialissubm tomrractiin by the author. Permissionto copy ia rsstrictsdto an abatmotof notmorethanS00 words.Write SPE, P.O. Box6SS8S6,Rkhardaon, Texss~508S.3836. Telex: 7S0989 SPE DAL.kBSTUCT mercury capillary pressure curves to air-brine or oil-brine data. Almost concurrently with Purcells paper,An improved technique for determination of the Rose and Bruce2 presented a derivation of both absoluteSwanson Petrophysical parameter, (Sb/Pc)A for correla- and relative permeability from capillary pressureting permeability wi h capillary pressure data has been curves.ieveloped. (pc/Sb)152 Burdine3 also presented relationships betweenas a function of Sb, yields a relative permeability and capillary pressure.Jell defined minimum which corresponds to the values ofn k- cwsttennoranhical method.~c ~~~~% f~~= t..=----------r--- The Most recently, correlations between absoluteimproved technique leads to ease of computer deter- permeability and capillary pressure behavior have beennination of the parameter. The relationships developed presented by Thomeer4 and Swanson5 for sandstones andby Swanson for correlating permeability with capillary carbonates from conventional oil and gas reservoirs.pressure work well for samples with pentteabilitiea Also, refinements to the earlier capillary pressure-greater than 10 microdarcies. A new relationship relative permeability equations have been presented bybetween the Swanson parameter and permeability has been Brooka and Corey6. These techniques are very attrac-clevelopedfor tight gas sands which leads to a more tive becauae of their ease of use and because capil-.-....--4.---....=-k~li~y~~di~~i~~ belQv 10 microdarcies.llccuLaLG tJcLutGa*. lary pressure curves, particularly from mercury injec-

tion, are relatively easy to obtain.Mercury capillarypressure data at wetting phasesaturations higher than 50% are approximately a factor However, all of the above correlations and equa-~f 10 greater than centrifuge air-brine capillary tions were developed and/or tested using sandstonepressure for tight gas aands. This contrasts with the samples with permeabilities generally greater thanBore commonly assumed value of 5. Consequently gas- 10 mD. We have extended the Swanson relationship foraater relative permeabilities computed from mercury clean sands to include ssmplea less than 1 microdarcy.,.__...-_- - . 71 . . . . - - A21 W+ l ~esul t i nLItJeCLMU .JALLCH7 y.~~t?tl= In so doing we will present an improved method forpredictions of higher water-gas ratios than the air- calculating the Swanson parameter, (Sb/pc)A,Hg.brine data. Optimistic cleanup times for tight gaswells completed in water-base fluids would, therefore, We have measured the air-brine and mercury injec-be deduced from the mercury injection data. tion capillary curves for several samples and will showhg~ the twQ differ in some typical cases. A newINTRODUCTION relationship is given for permeability versus (Sb/pc)Afrom air-brine capillary pressure. As a final exercise

Tight gas sands are characterized by amsll pore we will ahow how the relative permeability curves, asthroats and crack-like interconnections between pores. calculated from the Brooks-Corey equations, areThese microscopic features result in some characteris- affected by using mercury data instead of air-brinetic macroscopic features such as high capillary data.pressures, low porosity, high irreducible wetting phase References and illustrations at end of paper. from thin-section and X-ray diffraction for the FaJher-.


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2 CAPILLARY PRESSURE AND PERMEABILITY RELATIONSHIPS IN TIGHT GAS SANDS SPE 1387samples and the average bulk mineralogy for the Travis wherePeak samples7. This data ia shown in Table 1, Theprincipal clay in the Falher samples is authegenic Pc = mercuryjair capillary pressure, psiakaolinite with small amounts of illite occasionallyobserved. Travis Peak clays consist of mostly illite pd = mercury/air extrapolated displacementbut with a significant chlorite content. pressure

We have measured gas permeability on plug sampleswhich were either 1 inch (2.54 cm) or 2 inches (5.08 cm) s~ = mercury saturation fraction bulk volumein diameter and 1 inch (2.54 cm) to 3 inches (7.62 cm)long. Mercury injection and air-brine capillary pres-

Sb@ = mercury saturation at infinite pressuresure was measured on 1 inch diameter plug samples of C2 = curve shape factor which is reflective ofabout 20 cm3 volume. Core plugs were cut, trimmed, andvacuum dried at 60C. However some samples containing pore geometrytraces of residual fluids were also extracted withchloroform-methanol before drying and testing. Each sample can then be characterized by the parauvDm3T f muT T TIvl@lTO~l~~ meters, Pd, Sbm, and C2. However,DA.u..-...- -----.. the procedure forgetting these parameters iiF.lGl.W2S@~Chill~ the PC - S~Capillary Pressure curve of interest with a curve in the family ofcomputed capillary pressure curves with defined Pd,

Mercury injection capillary pressure has been Sbcc.,nd C2.~ea~=red to ~o,~o~ Psi (~07 &lPa)on all samples. Thismethod was described by Purcelll. In addition air- Because of the somewhat tedious nature of thebrine capillary pressure data were measured on 20 of Thomeer procedure, Swanaon5 later developed an alter-the samples. These measurements were made in a high- ~nativeprocedure for obtaining a correlation parameterspeed centrifuge capable of producing capillary \(Sb/Pc)Abetween UIerCU~Y h~e~~iea e~~i~h~~ ~~*SSC~epressures to about 1,000 psi (69 MPa). This procedure landpermeability data. Swansons procedure alsois discussed by Hassler and Brunner8. requires that a log-log plot of mercury injection:capillarypressure, Pc versus mercury saturationGas Permeability Measurement (% bulk volume), Sb be made and a tangent drawn at the~pointof intersection of the hyperbola with a 45 line

The permeability of the samples was measured using {passingthrough the origin of the hyperbola axis.a transient flow method with nitrogen as the mobile lFigure 1 illustrates this procedure and shows that thefluid. This method is described in detail elsewhere, :pointof tangency, A, on the curve is peculiar to eachso only a brief description of the experimental setup sample and defines the point where the ratio (Sb/Pc) iwill be given here. The system is an automated, five- maximum.sample apparatus in which confining stress and porepressure are independently controlled. This allows Though this procedure is an improvement overmeasurement of permeability at in-situ stress condi- Thomeers, it is also time consuming and requires thetions which is of utmost importance when working with construction of two perpendicular lines that intersecttight gas sands. Permeability is measured by introduc- the Pc - Sb curve in order to define (Sb/pc)A.ing a small step change in pore pressure to one end of Attempting to generate (Sb/pc)A automatically using athe sample and monitoring the r$sultant pressure computer routine is made difficult by the aforemen-gradient versus time with a differential transducer. tioned problems. We have developed a less laboriousSince one end of the sample is connected to a small procedure for ease of computer generation of the para-known volume and the other end is connected to a large, meter (Sb/pc)A. l%is new technique is an extension ofessentially infinite volume, the pressure pulse decay both the Thomeer and Swanson procedures.can be used to calculate the gas flow rate versus time=cd he~cei the permeability can be obtained. This As discussed by Thomeer4, if the hyperbolicmethod has the advantages of speed and accuracy over Equation 1 is differentiate~ with respect to Fc siitithe more usual steady-state gas flow method. All is eliminated through combination of the resultingmeasurements were made at pore pressures around differential equation with Equation 1, the following2,000 psi (13.8 MPa) so they will be equivalent to iinear equation results:Klinkenberg-corrected and liquid permeabilities.DATA ANALYSIS AND INTERPRETATION 10g Sb - 10g Sb = c [(pc&,) (dS@pc)]l2 (2Determination of Swanson Parameter (Sb/pc)A for Corre-lating Capillary Pressure With Permeability We have observed that a plot of (pc/Sb)l2 verSuSlog Sb results in a well defined minimum. This minimu

In a pioneering paper, Thomeer4 showed that when (pc/Sb)l/2 ia a unique petrophysical parameter fOr ameasured values of mercury capillary pressure, Pc and given sample. l%is parameter, which we will call YHg,mercury saturation (fraction bulk volume), Sb, for is related to the Swanson parameter by yHg =various rock samples were plotted on a log-log plot, [( Sb/pc)A ]-1/2.7 1

Figure 2 shows a typicala family of curves are obtained that can be adequately plot of (P; b) 12 versus log Sb for some of therepresented by the equation of a rectangular hyperbola samples presented in Table 2.(KY =

Notice the well defineda) as shown below: minimum (YH ) for all the samples.%

vdUeS of (Sb/pc)determined y the yHg technique were found to corre-spond exactly with values determined by the Swanson(log Pc - log Pd) (log Sb - log SW) = -C2 (1) method. The main advantage of this procedure is the

-


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SPE 13879 J. D. Walls & J. O. Amaefule 3eaae with which YHg can be determined even in samples p= Hg UHg COS9HRwith ill-defined curvatures. No constructions or nomo- . (51pc air-brine I?a-bCOSea-bgraphs are needed because YHg can be easily picked froma tabular presentation of (Pc/Sb)l/2 and Sb data. wherePermeability - (Sb/p~)A Relationship for Tight Gss Sands OHg = 480 dynes/cm

Swanaon developed several correlations for predic- U = 70 dynes/cma-bting permeability from mercury injection capillarypressure data. Tinesecorrelations wers bzsed cm the ~ = 14QCpower law least squares fit of gas permeability, kwith the parameter (Sb/pc)A. The samplea used in fhese e:b = ocorrelations were of medium to high permeabilityvalues. As a matter of fact, Swanson pointed out thegreater deviation between predicted and observed valuea Using the above values, Purcell concluded that theof permeability using his correlations especially in ratio of pc Hf 0 c a-b was about 5. He showedlow-permeability regions. several examp es of mercury and air-brine curvesplotted for samples ranging from 23 mD to 1,150 mD and

We have, therefore, reviewed the Swanson relation- with,porosities from 16% to 26%. Figure 4 ia anship to check its applicability in tight gas sands. example of these curves where the Pc Hg scale isFigure 3 shows the absolute permeability of our samples expanded by X5. His conclusion was that the curvesplotted againSt (Sb/pc)A. As mentioned earlier, our were sufficiently similar to allow the uae of mercurygas permeabilities were measured at in-situ conditions and air-brine data interchangeably.of overburden atress ranging from 5,000 psi (21 MPa) to6,000 psi (28 MPa) and pore pressures of 2,000 psi Following a similar scaling procedure we have(138 MPa). These values wouid, therefore, be cOm@a- -~-.+a.l-----4-_k*~rIn~e~.trif~ge~urveS and mercurypLu.1.Guu. uA.-..A..-rable to Klinkenberg-corrected gas permeabilities or injection curves for four samples. Figures 5 throughequivalent liquid permeabilities at overburden stress. 8 are typical of the tight gas sanda we tested. Thecurves for these samples have several.features in

The permeability values predicted using Swansons common. At high wettfng phase saturations, the air-empirical relationship at 1,000 psi (6.9 MPa) confining brine capillary pressure curves are considerably lowerstress for clean sandstones are shown in Figure 3 aa than those for mercury, even with the factor of fivedashed lines and are defined by: scaling difference. Different threshold entry pres-sures and calculated pore throat radii at high Sw,

will therefore result from the two methods.k@D) = 431 (Sb/pc)2109 (3) Mercury and air-brine capillary pressures appearto be proportional by a factor of five in the region ofwhere Sb is percent bulk saturation of mercury and Pc lower Sw and higher capillary pressure, and tend tois in psi. give similar residual wetting phase saturations, SW.The solid line is the least square fit to our Reexamining Purcells data, we find that his lower

data set (25 samples) defined by: permeability samples behave in the same manner as ours.Figure k for a 23 mD sample shows the same &isagreemeiitat high Sw and close match at low ~ as the four tightkg(mD) = 30.5 (Sb/pc)A,Hg1.56 (4) gas samples.

Our curves all show that the ratio of five forwith a correlation coefficient, R2 = 0.93. mercury and air-brine data as stated by Purcell is notapplicable to these tight sands. The reason for thisAlthough a least square fit of the points produces discrepancy is not completely clear, but of the foura line with considerably different slope and intercept variables in Equation 5, the contact angle for the air-from Swansons, it is noteworthy that Swansons line water system would be the most likely suspect. Ourpasses through our higher permeability data with almost data shows that the ratio of Pc H /Pc a-b is actuallyequal number of points above and below it. Therefore, closer to 10 at high ~. $hen th s scaling factor isif our data set were added to Swanaona, the equation inserted into Equation 5 and the equation solved forof the line would not be greatly affected. Note that es-b, a value of about 60 resulta. Whether this valuethe two lines intersect at about 0.02 mD. The is representative of the larger body of tight sands or

relationship in Equation 4 will, however, provide a ia dependent upon sample preparation, etc. will have tobetter estimate of permeability for tight gas sands be established by further testing.(k< O.ol@).Many other reasons may be advanced to explain whyComparison of Centrifuge Air-Brine Versus Mercury the mercury capillary pressure data may not be directlyInjection Capillary Pressure Data For Tight Gas Sands proportional to the air-brine data. These includesample size conaiderationa and rates of mercury injec-The mercury injection method for obtaining capil- tion aa discussed by Wardlaw and TaylorlO. Since theIary pressure curves waa int educed to the petroleum1 air-brine capillary pressure data are not simplyengineering field by Purcell . He observed that the related to the mercury injection data by a factor ofpressure required to force mercury into the evacuated five for tight gas sands, it is essential to uae thepores of rock could be related to the air-brine capil- mercury injection data with caution.lary pressure by taking tnto account the differences

in contact angle and surface tension. This relation With this in mind we calculated Y for each samplewas expressed as: from the air-brine centrifuge data to check whether a*r-=


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4. CAPILLARY PRESSURE AND PERMEABILITY RELATIONSHIPS IN TIGHT GAS SANDS SPE 1387relationship can also be established between ya-b andpermeability. An example of the behavior of (Pc/Sb)l/2?ersus Sb for air-brine ia shown for four samples inPigure 9. Aa with the mercury capillary pressure data,+- -4*L +ho minim~rn which is defined aait is V-EiYe=sy .- ~-b- ...------~a-b. Using ya-b to calculate the air-brine Swanaon~arameter, we plotted in-situ permeability, k, veraua(Sb/p=)A ~-b for the samples which had air-brine dataavailabl~ (Figure 10). A least squares best fit tothis data (19 samplea) ia given by:

k(mD) = 1.22 (Sb/pc)A,a-b1.61 (6)

aith a correlation coefficient (R2) of 0.91. Here, Sbia non-wetting phaae saturation percent bulk volume andPc ia in psi. ~is data is listed in Table 2.Sas-Water Relative Permeability Calculations Forright Gas Sands

Relative permeability data for tight gaa sands areextremely difficult to obtain by the conventionalsteady state or dynamic displacement techniques. Ingas reservoir engineering calculations where these dataare needed as input parameter for hydraulic fracturingpredictions or completion strategies, gaa-waterrelative permeabilities have been computed from capil-lary pressure curves. The computational techniquefrequently employed is based on the Brooka and Corey

1A[1km = S$

and

&w=[l-g]2

where $ = Sw - ~

kmkrnw =

Brooks

1 - Sw

t(7)

(8)

wetting phase relative permeabilitynon-wetting phaae relative permeability

and Corey observed that a log-log plot ofS$ against Pc results in a straight line with a slope Awhich is characteristic of the pore structure. pd, thep$r~ ~try pressure is obtained from the intercept at-w A.

Mercury injection capillary pressure data havetraditionally been used for the above computations

c

mder the assumption that the data can be scaled toeservoir behavior. In view of the differences:urrently observed between mercury and air-brine capil.ary pressure data for tight gas sands, we have com-mted gaa-water relative permeabilities to understand:he resultant effect of the aforementioned diff~i~iie~m gas recovery.

Log-log plots of G versus pc for our mercurY andLir-brine capillary pressure data were made as sug-;eated by Brooks and Corey. The lesst square fit for:ypical tight gaa sample is given in Table 3. Thecorresponding (krg/kN) ratios for both techniques arellSO presented in this table. The relative gas andraterpermeabilities computed from mercury and air-mine capillary pressure data for this sample are showin Figure 11.

Note that the relative permeability calculated3 ..4,.-.....41s- p~~~~,u~~.c~rve~rom t-hemerCUry iiiJeCLWLl ~-p--+-.=.eada to predictions of higher water-gas ratios at any~aturation. In turn, these higher ratios lead toexpectations of faster cleanup times for gas wells:ompleted or fractured with water baae fluids. The:entrifugeair-brine capillary preseure datawould,:herefore,be recommended for gas recovery predictions.n tight gas sands.CONCLUSIONS

An improved method for finding the Swansonparamet$$O(Sb/pc)A has been presented.pc/Sb l aS a fUnCtiOn Of Sb haS b~ea fOiifto yield a minimum value that corresponds tothe value of Pc and Sb at A given by Swanaongraphical method.The least square fit to our data given byk(mD) = 30.5 (Sb/pc)A,Hg1.56 IS preferable fotight gaa sanda with permeabilities below0.01 MD.sbfpc)A a-b fr~ centrifuge air-brinecapillar~ preaaure also correlated stronglywith the gas permeability as given byk(mD) = 1.22 (Sb/pc)A,a-b161.Mercury capillary preasurea at higher wettingphase aaturationa (~ > 50%) are approximatela factor of 10 greater than the centrifugeair-brine capillary pressures for tight gassands. This contrasts with zhe iaazeemmofiljused value of 5. Air-brine and mercury dataindicate very similar residual wetting phasesaturations, Sm.Gas-water relative permeabilities computeds---~Luu mercury i=j~~ti~~ capillary preaaure datwould lead to predictions of higher water-gasratios than air-brine data. ---J- .*4A -la nWpkLUAnkAL ~4=a.up times for gas wells completed in water basfluids would therefore result from the mercurinjection data.

}


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.

SPE 13879 J. Il.Walls & J. O. Amaefule 5NOMENCLATURE

k=Pc =pb =s~ =

s=Y

snw =

Swr=s; =

A=

Y=XJBSCRIPTS

Hg =a-b =

permeabilitycapillary pressureentry pressuresaturation of non-wetting phase in percentof bulk volumewetting phase saturation, percent porevolumenon-wetting phase saturation, percent porevolumeresidual wetting%-SWl-swr

point on the logwhich a 45 line

@@b)mercury

phase saturation

air-brine

2 minimum

p= versus k g Sbbecomes tangent curve at

ACKNOWLEDGMENTSThe authora thank Canadian Hunter Exploration, Ltd

:orproviding samples for this study and the GasLesearch Institute who provided samples and funding aalart of GRI contract no. 5083-211-0313. Wealsoxpress our grstitude to Lynne Welte for manuscriptreparation and Terri Knight, Stephen Carter, andlalph Perez for data plotting. Penniasion to publish:hiswork was granted by Canadian Hunter Exploration,td; the Gas Research Institute; and Littonetrophysical Services.ETRIC

111

CONVERSION FACTORSinch = 2.54 cmpsi = 0.006897 MPaDarcy = 0.9869 pm2

i-

references1.

2.

2d.

4.

5.

6.

7.

8.

9.

10.

Purcell, W.R., Capillary pressures-theirmeasurement using mercury and the calculationof permeability therefrom, PetroleumTransactions, AIME, February 1949, pp. 39-48.Rose, Walter and W.A. Bruce, Evaluation ofcapillary character fn petroleum reservoirrOCk, ?~tr~leiifiT~~IN3aCtiO?W, AIMEj MS~ 1949,pp. 127-142.~Jr


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TABLE 1.AVERAGEBULS MINERALOGYOF SAMPLESSTUDIED ( )

Falher FormationQuartz 37.8Qwrtz o.G. 12.5Ca:[cfte 0.9

Travis Peak Formation56.B20.B0.0 TABLE 2.POROSITY, PERMEABILITY,AND SWANSONPARAMETERFORMSRCUKYAND AIR-BRINEDATA001omite 11.2

Feldspar 0.4Chert 16.8Heavy Minerals 5.0

3.00.40.6

PorosityIn-SituGas Permk(pd)

Swan eonParameter(sb/pc)A,g

5.516 X 10::. .433x 10-36.595X 10-37.365 X 10-32.037 X 10-35.641 X 10-31.970x 10-34.137x 10-32.832 X 10-31.970x 10-35.909x 10-3.4.875 10-34.833 X 10_34.s33x lo_31.300 x 10-31.660x 10-36.S00 X 10-35.600X 10-25.900x 1 12.570X 10-31.700 x 10-42.480 X 10-42.170 X 10-33.690x 10-23.020 X 10_51.540x 10-34.438 X 10-11.627X 10-35.550x 10----------

PetrophysicalParameter(Sb/pc)A,-b

--------------------------

SampleNumber1234567891011121314151617181920212223242526272829303132333435

10.301.41S.68.1efritalGrainsRock Fragments 7.5OrganicMaterial 0.0Clay andClayMatrix 2.1

2.61.66.1S.1

6.97.75,55.377.241.237.112.o47.24.06.04.66.610.08.46.97.65.16.09.49.4B.515.01.54.04.38.09.82.8

11.415.97.7

2.3S3.B9O,rOsity* 5.8 4.B574.9011.8014.4014.002.000.5619.8013.60238.001,870.003.200.140.022.20

*Porositymeasuredby HellumBoylesLaw method.-- -21.19x 10-21.49x 10-25.65 X 10-24.66 X 10-16.94 X 1003.10x 10-26.09 X 10_32.13 X 10_31.00x 10-22.87 X 10-14.25 X 10--

3.77x 10;21.42 X 10-23.77x 10-22.21x 10-15.44x 1002.14x 10-21.37x 10-22.24 X 10-21.70x 10

TASLE3.CAPILLARYPIRESSURSRELATIVEPERMSABILITT ANOSATURATIONRS2A.TIONSHIPSOR A TTFICALTIGHTGAS SAND 1 0.0218.706,400.004.70

Sample 17kgi - 19.8pd; d - 9.4

6.011.5 1.505,930.0067,900.000.492.001.80

Irreduciblecapillary WettingPhase ReducedPressure Saturation,Siw SaturationTechnique 4

14.58.47.5COrrelationCmefficient~z

EntryPressurePd (p6i)

312.2

31.0

.A

1.4992

1.1060

5.6MercuryInjectionData 30 5,491.80PC-1-4992 0.9895

0.9808CentrifugeAir-BrineData 30 44.64Pc-l.1060

u (MercuryG Injection)- 8,194.oe--l5.167 ;R2 - 0.9985kg (centrifuge~; Air-Brine)- 15,296.3e-15.570~;R2 - 0.9990


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Ffg.l-MannerofdefiningSwansonparameterfromthecapllleryreeeureurve(Ref.).

A Io

L A30 _ \

25 : -- 2

Z* 7-...

k ~t

FIs. 2-(PJSL.)W WAS~(bulkduration) mercurymrourFalhef eempfee.A-tc14

//

/

//

I 1 1 I I 1 I~. ,-1

I

(),?CAWE 13879


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5

4

9

2

1

0

-;-, a TevAL w *GI Oux n mFig. 4-Air -brine and mercury oapillary pressurecurvesora 23-mD sarnpk (Ref. 1 ).

0Xg oxx o

Xa8

I I I Ii 20 40 m a

UETTING PHASE SATURATION COFIo . S-Air-briw andmercw capil lw p fusum CUIWS for 3aI II@ 17 (9. 4% -W,19.8 MD).

1

m

o00

1000,

nox n

n1 0s ox oII o sx 1 0K . Al-I= CIP ~ a0- IERMYCA?IURV ?-3=- X 11

s (*1 t I I 0

i 30 40 a m i=

UETTINGHASE SATURATI~ (%)Fig. &Air.brine and mercury cApillatY PH~ CUIWS for *m@e 15 (5.1 ~ ~r~i~.-,. m,L.u p,.

5

A

3

2

1

0

090000

000

x 0

5

t I I 1

i 20 40 50 30 100WETTING PHA3E SATURATION (X)

~. 7-AiFbrinendmercuryapillatyf0a9ureurvesorSample18(9.4 W~W,13.6PD).


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1

9

8

.4

.2

0

X0

x1 0

x a s

x1 1 I Iaoa~m

SRI)E SATURATION ~Fig. O-Air-brine andmercury capillary preaeure Cuww for Sam@ 25 (9.~~ W@tiS

100 IQ).

Iu

4

I I


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r , 1 ,

~ , , , , II* , , , ,4 .

0\r xI 1 1 I

00013879_reference for Coreys Formula

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