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USMS019389 Absol:lteand Relative Permeability Studies of Gas/WaterFlowL. Zawisza, Inst of Drilling Teuhnolqgy

*WW 1- =W OtpoWaum Engln4afa~lsm@@q-WWW~@~mE@-tid~andpoeaibhpubikationnanSPEjoumaL Thematerialhwbjectocwreotiobyh authof(s).ermissionooopyisWstktadtoanabatractofotmorethanW&&K&n ~, SPE SookOrckrCIaP:Libraryechnician,.O.Box 833936* W833836 U.S.A.Telex730989SPEDAL.

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UNWJLMlWG111989SPE -1-PUBLICATIONS

Absolute and{ Relative Permeability Studies of Gas - %ater FIQ.

Ludwik 7awiszaInstitute of Drilling Techrlology and Petroleum EngineeringUniversity of Mining and Metallurgy .Cracow, Al. Mickiewicza 30 tPoland

summary.

Absolute and relative permeability estimation through direlog responses iS still out of reach. Yet permeability is a kparameter among those controlling reservoir qualities.

Studies were conducted by the Author on the interpretatioof lab data i.e. porosity, irreducible water saturatio.distribution of pore size, internai surface area per unit pvolume, capillary pressure, in their relationship with absoluand relative per...=bilities. These studies have resulted inelaboration of the physical equations for absolute and relatipermeability calculations of which the empirical calibratifor any geological formation can readily be worked out.computing equations include log - derived terms only and canintegrated in any computer processed log analysis pregram.

1. Introduction .

The estimation of the properties of reservoir rocks,

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, S?E 1938-ii?-

usually connected with the determination of such parametersporosity, absolute permeability, saturation (totalsaturation>, irreducible water saturation and relpermeabilities, The knowledge of these parameters is necesfor the calculation of multiple - phase fiows thrmghporous medl um. Laboratory measurements , of reservoirparameters are usually labor consuming and arduous. Iconnected especially With the estimation of absopermeability and reletive permeabilities. For this reasothis paper there have been considered the possibilitiesevaluating these parameters i.e. absolute permeabilityrelative permeabillties on the basia of physicalproperties using modelllng considerations, exemplified byAutochtonous Miocene of the Carpathian Foredeep, Poland.

As a result of the overdone researches thereconstructed some prediction models thanks to which itpossible to conduct a continuous digital prognosis Of:absopermeability and relative permeabllities from log - deporosity and irreducible water saturation.

2, Determination of absolute rock permeability coefficient

2. 1. A review of the existing methods of a quantitadetermination of the absolute rock permeability

Over the year, a variety of relationships have

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-3-

developed for estimating absolute permeability. These inclurelationships proposed by Tlxiera, Pirsonz, Wyl1ie and RosTimur4, Coates and Dunanoi rs, Raymer6, Rai~?.-Clemenceau7,others.

One of such methods is the estimation of absolupermeability on the basis of the resistivity gradient by socalled Tixier method~. This method bases on the relations ampermeability, capi11 ary pressure and difference betweenresistivity of the formation water and this of the hydrocarb. The method of the resistiviSradient may be SXC1 US1vel1y appl led in the water - hydrocarbtransition zone.

In many cases Lhere exi 8+,9 a relation betweenpermeabi 1ity al]d other physical properties of reservoir roci. e. relationships between absolute permeabi 1ity and log.derl ved porosity and irreducible water saturation values,this correl atlon comes from the empirical dependence validdefinite geological formations, for which tests were carrout .

S. J, Plrson2 gave an equation for the determl nati on ofabsol u>= permeability coefficient k in the following form:

k s+= (+-]2+,, ,,,,.., ....o ............ci.

where~ . o.6a ... .,,.,, ..........,,...,,* ,0 ..,, .,, .~

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4-h

water saturation ~ and F is the formation factor d~faccoraing to the Humble equation .

M. R. J. Wyllle and W. D. Roses presented the empirrelation referring to the medium of grain sands, i.e.

k = c +=/ s )2, . . .. . . . . . . . .. . . . . .. . . . . . . . .. . . . . .vi,where constant c depends on the density; of the hydrocarbonsassumes values from c = 250 for oil having an average densto c = 79 for dry gas.

Equation of A. Timur4 is another moditicatlon of Eq. 3k=e5el.02+4*4/sw:. ...........................

This relatiOn is set ~?r clean compact sa~]dstone witaverage porosity @ = 15 td 20% .

The Coates and Dumanoir relationships is

O.k 300 + Zvk = T b ........................w vi tirrWhei e

[logRv Rt +2.2]2WE ~ 3.75 -tp+ 2 . ... ...,, .....

In the above equations, RV is the formation water resistlvR~ is the rock resistiv]ty, Rtbrr is the rock In the above Eqs. 20, 21, and 22 values C, a, and b aco wtants which should be empirically determined,

As a reeult of the applied statistical analysis of the daton , Swi and k coefficients determined in lab conditionsth-re were obtained the following relations valid for t*Miocene of Lhe Carpathian Fwedeep:,

I:

11:

III:

k 4 f33e.3et$2.4~s 0,93 , ...00.....*. .O*viCoefficient of multiple correlation T?= 0.75

Coefficient of multip~e correlation R = O.@O

( )2.20k = 15 023.49 [1 - Swi] * .*....., .*.

Correlation coefficient r = 0.?9It fdlowe from tke comparison that the best approxlmatie

can be obt~ined when using the second ana third modedescribed by Eqs, 84 and &?5. Modele I, II, and III wepresented in seperate nomograms . In Eqs. ~24 ar,~ 25, ~ and Swt coefficients are expressed in a fractiof uniL, wheras k in md. The absolute permeability coefficlmtmay be defined by means of Eqs. 24 or ~5 ana the nomograms frFigs, ~ or 3. Including the correction factor c defined by

. L, Raymere $nto the models 1,makes the parameters of those

II and III

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1o

3. Ntermination of rslative permeabili~.ywatsr k and gas krv r~ti

3. 1. A review of the existing models d~scrlbj,ng the relatpermeability distribut~.ms

In the paper there was anal ysed the relation betweenrelatiive permeability and the porosity, absolute permeabl 1irreducible saturation w~.th the wetting phase (bour,d water)the capillary pressure distributions, These conmideraticoncerned the two - phase m~tem water - gas, because the tphase, i. e, oil 1s practically absent in the Miocene ofCarpathlan Foredeep. Ar result of this analy8is therebuilt two classes of models describing the flow of flthrough the porous medium, namely the class of capillary mo

.and the class of geoktrical mcrdels.To the class of capillary models one may include

following models, classified according to the degree ofcomplsxitye9:i) model of parallel pipes with idenblcal radius describedthe tollowing equationa:

krv(II) ,,, ,,, , ,, 0 ,,, ,,, , ,, , .

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3>

Distrlbubion function t5cr)l/Pe*cs>model of pipes with radius

=i-k ..co**, o*ca7)rvux}

has the density equal

defined by the determininfunction ~rw(XIXoJ sj dS/P 2(1+8}e j: dS\~e]g o 0

for e =~,5and

{v dS/(PQ)2k m [TJS ~ ,O. ,......icso)so)rw IIIb) j dS/(Pe]2 o

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$ dS/[PG]krg(IIXb) (T,,] :Vf dS/[P=]2o

B .s . ,,, .,0. , ,, , cm)

where :s - s, mtnT=Jrw vi -s v,mtn

(1-s)-srrg = y - S,m,ti]whare Trv and T~g play the role of the porous mediumtortuo=ity ratios r-spectlvely for the water and for thegas.In the class of geometrical models there

empirical modelsj $. e. modelm proposedBoaLmm*a, The Wyllie mpirical model for

were analyued someby Wyllieio andthe sandstone

mudstone rocks which corresponds to the ,lithological type orocks in the Carpathian Foredeep has the following formio:

k . [S*] ,*.. s,, ,*.,,** ,** ,. .,* ,,, *,,,,,Cw>rv(w}krg(w)ql-s*]2(i-s *2] , ,.OO,,,,,,.O,,,,,,, (s3)

and the Boatman. uationwts for gas - water flow are~k = (S*]%: , )rwm} 9..***n...*** ,, 9 ,,,.,,.*,

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properties ofrocks and fluids and has the following form:P

[)0 5J(SW] = + +-

.

a given rock type may be shown inin the form of J function.many Authors@iOi2,for a given

the fo

1ithe:o

where d is the interf acialFunction J enables the

tension.normal ization of the measurements

capi 1lar y pressure to be conducted by taking into accountproperties of rocks and the fluids inside theme-lo. As a rof this normalization al1 the data on tihe capi 11ary presdistribution forone curve, i. e.

Accordl ng torock type, the Leverett J function depends only on saturatB~Xmg on the data on , k, P=, there were listed the valueJ f unctim defined by Eq. 36 in the function of saturationthe wetting phase .for 106 Miocene core samples

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In the above equation 2.08x10-9 was defined by thethe so - calledpressure*, u is thethe rock ~haliness,

In Fig. 5 thereC44, 52, 67 and

aver age normal ized initialcc~efficient which, among others~

Authorcapilla

depefidsand which varies between 1 and 35..were presented J curves for four sampl

One may observe almost01>. an idagreeability of theoretical curvesempirical curves determined fromappropriate value of a).

described by Eq. 37 andEq . 36 Cafter assuming

It is not..able that J cur- occurs within the rasaturation of rocks with the wetting phase, i. e. water, 1.between S =.Vthe Leverettfollows fromsaturation,

iandS =S ~i

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~ 19389-16 -

As a result of the tests conducted for all 106 samples itbeen stated that for S = S the value of the Leverettv vifunction is constant and equals to:

J (s.] =a.oe. ...................................

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P = o *,3 J(SV) . ..........................

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WE 19589-ie -

and having analytically calculated the integrals in tequations ~ we get the tormulae for relative permeabil;Capillary models II and III) in the following forms:

Capillary model II:k:v;xx,=S%+2? ...............................v t503

k=l i+za-s . . . . .rg(II) ..............w .........krg(IIIb)= [T) [1 -Sv+q . ................rg C55>

3. 3. Construction of relative permeability and ppermeability curves on the basis of stand

measurements on physical rock properties

Futher experimentstheoretical model empi

were to lead to the construction @rically precised, which w~~ld enable

to defil,n the relative permeabillt.y on the basis ofparameters as: porosity, absolute permeability and irreduciwater saturation, so standard and easy measurements of physi

-.

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.

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rock properties.For this reason ~ the description of the relati

permeability distribution, given by individual models

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-ao -

~9.[SW]k:_. . . . . . . . . . . . . . . . . . . . . . . . . .. 57) = T*r9In the above formulae k*~W and k* denote the relatirg

permeability curves for water and gas, determined on the basof the capillary model II defined by Eqs. 26 and 27 or Eqs.and 51, best. fitting the lab data, T~W(SW], r~gcSW] - ratiosporous medium tortuosity for water and gas, respectively,the function of wetting phase

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-21-.

Finally, Eqs. 60 and 61 assume the following form:kw = ;sswa*20 , ............................. i= a resultmodel II. In Eqs. 61

was obtainedcapillary modof an emp~rlca

*and 62 k--- ak* denote curves of relative permeabillties for water and grgConstructed on the basis of capillary modelfitting* the relative permeability laboratory.relative permeability curves obtained from Eqsi 61with the laboratory data were presented in Figs.

II), bedata. T

and 6~ alo7 - 10. T

maximal error of the model IIIc in relation to the lab daequals to 4% . In Figs. 7 - 10 there were presented also tWyllie and Boatman empirical models described by Eqs. 32,and 34, 35. All the models show good agreeability:

In order to calculate the phase permeabilities for water agas, i. e. kw and ks , appropriate coefficients krv andshould be multiplied by the coefficient of the absolupermeability k:

k k k, .*....*.* *,*,,.*.. ..*..*.*, ,,.O, ,s,, ,v rv

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k k k, ..........* ..*.*,. ............ .,...0

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= .19369

.

-a3-

universal for an arbitrary type of rock and may be used for thconstruction of a prediction model giving the possibility ofvery precise determination of the capillary pressure on thbasis of the standard and easy measurements of the values osuch propertiesand irreducible

Nomenclature

J=krc

k=rvk =wP=P=cr =R=

as porosity, absolute permeability, saturatiowater saturation.

constantsformation factor, fractionLevurett functionabsolute permeability, mdeffective

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SPE19389-a4

sp =

Sw =Swi =

s =v,min

fw.

T=rgT=rw

equj Iibruin gas saturation, fractioninternal sUrf ace area per unit pore volume, 1A [1/ftwater saturation, fractionirreducible water saturation, fractionminimum water saturation from a capillary curvefractiontextural parameter, fractioncoefficient fractioncontact angle, dmgreeswater viscoeityD Pa**

(radl[Cpl

density, kgm* flbfitalinterfacia tension, N~ [dynezcml tortuesity of the porous medium, fractiongas tortuoslty ratio. fre.cionwater tortuosity ratio, fractionporosity, fraction

References

i. Tlxler, M,P,: Evaluation of Permeability from Electrlc LResistivity Gradients,Oil and Gas ;, CJune 1S49) 48, li3lea

~. Plrson, S,J. :HandbookFormation Evaluation,

of Well Log Analysis for Oil and GPrentice - Hall Inc., Engl ewo

cliffs, New York

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4.

6*

6,

7,

e.

9.

10.

11.

-a5-

the Physical Characteristics of Reservoir Rock ioElectrical Log Data, J. Pet, Tech.

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S 19389-2e -.

1300,12, Anderson} W,G,: Wettabili ty Literature Survey - Part 4:

Effects of Wettabllity on Capillary Pressure, J. Pet.Tech,

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- 19389-a7-

Figures

Fig. 1 - Nomogram for determination of absolute permeability.on the basis of porosity ~, irreducible wasaturation SWL and Eq. 23.

Fig. 2 - Nomogram for determination of absolute permeabilityon the basis of porosity @, irreducible wasaturation S and Eq. 24.VL

Fig. 3 - Nomogram for det~rmination of absolute permeabilityon the basis c: porosity $, irreducible wa5atur?At10n SWb and Eq. 26.

Fig. 4 - A **J function diagram. All samples.Fig. 5 - A J i%~-ictioniagram. Samples No. 44, 52, 67, 10Fig. 6 - Graph representing relation between coefficient

and irreducible water saturation SWiFig. 7 - Comparison of observed and calculated relat

permeabilities. Curves of relatiye permeabilitywater krv and gas k obtained from capilla;y mr9IIIc C61), C62), Wyllie equation C322, C33)Boatman equation C34), C35). Sample No, 32.

Fig. e - Comparison of observed and calculated relatpermabilities. Curves of relative permeabilitywaber krw and gas k obtained from capillary mrgIIIc CG1>, C62>, Wyllie equation C32J, C33)Boatman equation C,34), C36>. Sample No. 81.

Fig. S - Comparison of observed and calculated relatpermeabilities. Curves of relative permeability

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-28-

= 19389

water k and gas kPv obtained from capillary morgIIIc

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.

c

u

G

No

%

IRREDUCIBLE WATER SATURATION, Swi VXo)

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-

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WE 19589

z-gt-4s100

WATER SATURATION, SW (VQ)

19389

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30 0

25 0 ~.

2040 a=

1/15.0 /

/1/ v~A10.0 A //5.0

t0.0: 0 20 40 60 80 iio

IRREDUCIBLE WATER SATURATION, Swi [Yol

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.

.

RELATIVE PERMEABILITY - [~0]M mo 0 ms o 0 so

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.,.-.,... . ..

I 00

80

60

40

20

.

1\\1

\

Swix 23.S0%

//.

0 20 40 60 80 100WATER SATURATION, SW [~0]

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100

w>i=aid 20a

n

- 19s89

Swi = 41,30a= 7,81

o 20 40 60 80 100WATER SATURATION, Sw [? el

\\ L

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.

100

o

S~i8 75.05%

1 1 1 I I I 10 20 40 60 00 100WATER SATURATION, SW [%]

,