1992_ss-kr from log - pc and k distribution

8/12/2019 1992_SS-Kr From Log - Pc and K Distribution

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SPWLA 33rd Annual Logging Symposium, June 14-17, 1

DETERMINATION OF RELATIVE PERMEABILITY CURVESIN TIGHT GAS SANDS USING LOG DATA

Adel Ibrahim, Zaki Bassiouni, and Robert DesbrandesDepartment of Petroleum Engineering

Louisiana State UniversityBaton Rouge, LA 70803

AbstractThis paper presents an approach to relative permeability determination integrating well

log and petrophysical data. The proposed technique is based on matching a log-derived watersaturation profile in the transition zone to empirically determined capillary pressure typecurves. The technique is particularly useful in tight formations where the transition zoneextends over a large interval. The match yields an estimation of absolute permeability, thefree water level, and a capillary pressure curve specific to the formation studied. Thecapillary pressure data thus derived is used to generate relative* permeability curves using amodified Purcells equation. Field examples are presented to illustrate the application of theproposed technique to tight gas-bearing sands.

IntroductionLaboratory measurements of relative permeability characteristics of tight sand cores

are very complex and time consuming. (1*2*3*4) technique that can be used to extrapolateexisting core data to cases where such data is absent or not representative of in-situconditions is of interest.596 Such a technique has been developed and is based on using logdata to derive a water saturation versus depth profile in the transition zone of the formationof interest. The log-derived water saturation distribution is then correlated to generalizedcapillary pressure curves typical of the formation studied. This curve matching yields, bycomparison, a capillary pressure curve specific to the formation of interest. The capillarypressure type curves are generated from already available core data and other petrophysicalinformation. Relative permeability curves are subsequently generated using correlations basedon Purcells model.

The formulation of the proposed technique for tight gas-bearing sand formationsrequires the development of:

1) A capillary pressure (PC) to water saturation (S,) empirical relationshiprepresentative of tight sands;2) A water saturation to relative permeability relationship based on the previouslydeveloped PJS, relationship;


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3) Generalized capillary pressure type curves for tight sands; and4) A normalization method to reduce the number of variables affecting the log-derived water saturation profile.The core and log data used to develop and test the proposed relationships are

representative of the Cotton Valley, Travis Peak and Falher formations. Petrophysical data,including absolute permeability, porosity, and capillary pressure, available for nine coresamples are listed in Table 1. The absolute permeability and porosity of the cores rangedfrom 13.6 to 260 pd and from 7.9 to 12.5%, respectively. -

Capillary Pressure-Water Saturation Relationship in Tight SandsExamining the petrophysical data of available tight sand core samples suggested that

the capillary pressure-water saturation relationship can be approximated over most of thesaturation range by a linear trend. Thus the P,/S, relationship for tight sands can beexpressed empirically by:

where:PC as, (1)

P, = capillary pressure, psis, = water saturation, fraction, anda and b are coefficients reflecting the formation pore size distribution.Fig. 1 shows the PO/S, plot for three core samples representative of the Cotton Valley,

Travis Peak, and Falher formations. The values of the coefficients a and b of Eq. 1 for thesethree and the other six tight sand cores are listed in Table 1.

The empirical relationship of Eq. 1 is used to simplify the evaluation of the integralin Purcells equation relating relative permeability to water saturation.

Capillary Pressure-Relative Permeability Relationshipa. Drainage RegimePurcell derived a theoretical expression relating relative permeability to capillary

pressure and water saturation. (7) His derivation is based on the analogy between Darcysempirical law for sand packs and Poiseuilles formula that models the reservoir rock with abundle of capillary tubes of the same length but different diameters. To simulate theprobability of the interconnection of pores, Wyllie and Grander modified Purcells model,cutting the bundle of tubes into a large number of thin slices and reassembling them in arandom way. (8) This modification in the model accounts for the tortuosity and results in two

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equations expressing the relative permeability for the wetting and non-wetting phases underdrainage conditions:

d wSJPC2KTWt CO2 v;d dS,JC2WIwhere:

Kw is the relative permeability to the wetting phase,Kmwt is the relative permeability to the non-wetting phase,SW is the irreducible water saturation,PC is the capillary pressure, andS, is the effective water saturation defined as:

s; = w - wi1 - swi

(2)

(3)

Wyllie and Granders equations are selected because they yield values in goodagreement with experimentally determined relative permeabilities.

b. Imbi bition RegimeNaar and Henderson developed a relative permeability model for imbibition

conditions.9~0 Their model accounts for the entrapment of the non-wetting phase during theimbibition process of the wetting phase. The authors relate the drainage and imbibitionsaturation for equal values of non-wetting relative permeability as:

SW(Imb.) = s W(a) - R G g.)where:

SW(Imb.)SW(@)is the effective water saturation for imbibition, andis the effective water saturation for drainage

Both are expressed by Eq. 4.

(5)

R is the residual saturation of the non-wetting phase which is empirically related toporosity by:()

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SPWLA 33rd MU~ Logging Symposium, June 14-17, 1992

where 41 s the porosity.R = 0.617 - 1.2841 (6)

The relativeconditions using:

permeability of the wetting phase is calculated under imbibition

@*3cT2K, = - K dS

where:0 is the interfacial tension,

is the absolute permeability, andis the reduced porosity defined as:

W=@(l -SW> (8)

The Naar and Henderson models are selected because they predict values that are ingood agreement with steady-state measurements of core relative permeabilities.Relative Permeability Water Saturation Relationships for Tight Sands

a. Drainage RegimeSubstituting the empirical expression of P, given by Eq. 1 into Eqs. 2 and 3 yields

integrals that can be solved analytically. The final expressions for the wetting and non-wetting phase are:

K, = Si2 9)

where:

b. Imbibition RegimeSubstituting Eq. 1 in Eq. 6 and integrating analytically results in:

(10)

(11)

(12)


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Capillary Pressure Type Curves for Tight SandsLeverett pioneered the generalizations of capillary pressure data based on experimental

study of sand pack co1umns(2S3*4).Leveretts generalization is based on a dimensionlessfactor known as the J-function. The J-function is defined as:

where:J 13)

J = Leveretts J-function (dimensionless)P, = Capillary pressure (dyne/cm2)CT = Interfacial tension (dyne/cm)K = Permeability (cm2)cb = Porosity (fraction)To include the wettability effect, the contact angle (e) has been incorporated into theabove equation.

J PC K= (14)0 cos 8 Q,Fig. 2 shows the values of the J-function calculated using the capillary pressure data

available for tight sand core samples. In these calculations values of 74 dynes/cm and 90were used for o and 8 respectively. When theJ-function is plotted versus water saturationon log-log scale, a linear trend emerges as shown in Fig. 3. A best fit of the data results in:

where: J=? S (15)a = 0.039, andP = 2.308Eq. 14 is of the same form as Eq. 1, as should be expected.Combining Eqs. 1 and 15 gives:

A practical presentation of type curves is in terms of height above the free water level,h-, which is related to the capillary pressure and water and gas densities by:

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PC 0, -P,)h,, 17)where g is the acceleration of gravity.

Combining Eqs. 16 and 17 and using practical units yields:h_= 10.66

s,PP/P,) K1 T(18)

where:h,, = height above free water level, ft.=; = Interfacial tension, dynes/cmContact angle, degreess, = Water saturation, fractionc&P = coefficients-; 1 absolute permeability, mdporosity, fractionpw,pg= water and gas densities, g/cm3Fig. 4 shows an example of type curves generated using Eq. 18 and the following

parameters:a = 0.039P = 2.308=; = 74 dyne/cm90, ando = 0.15Each curve is for a specific absolute permeability value. Each curve is characterized

by an irreducible water saturation value derived from the correlation of Fig. 5 obtained forstudied tight sand core samples. Each core is also characterized by the value of coefficienta in Eq. 1. Using the approximation of b = p, a is calculated from:

a = a 0 cos 8

rTWater Saturation Profile (1%Water saturation values are calculated from log data using known interpretationtechniques. These techniques take into account the effect of fluid properties and shalinesson different log measurements used in the interpretation. The generated water saturationprofile (S, vs. depth) is seldom smooth since it reflects variations in petrophysical properties.

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Variations in porosity and permeability follow from variations in pore characteristics. Tofacilitate curve matching the log-derived water saturation profile is normalized to the averageporosity in the transition zone, &VP

The normalized water saturation, S,, at any level is defined as the water saturationvalue calculated from the measured formation resistivity, R,, and average porosity, 41~~.Using Archies equation:

where:Rw RvJR, = - = -

OrnSwn cp:,s:

R, is the formation water resistivitym is the cementation exponentn is the saturation exponent

solving for S,:

and if m is assumed equal to n:

(21)

(22)The normalized water saturation profile still reflects variations in permeability, i.e.

pore size. These variations are shown schematically in Fig. 6. According to this figure,based on a capillary tube model, the J-function vs. saturation profile is subdivided intosegments reflecting zones of different permeabilities. Each of these segments must bematched separately, as illustrated later by field examples.

Application MethodologyCapillary pressure type curves are constructed in terms of height above the free water

level versus water saturation. The curves are constructed using Eq. 17, with specific fluidproperties and an average porosity, for different permeability values.

The log-derived water saturation profile is normalized to the same value of averageporosity used in generating the type curves. The normalized water saturation profile is laidover the type curves and shifted vertically, i.e. along the hFwL axis, to obtain an acceptablematch.

The matched capillary pressure curve defines the following parameters:

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1) absolute permeability, K.2) irreducible water saturation, &.3) depth of the free water level, at h,, = 0; and4) the parameter a of Eq. 1.

The defined parameters are then used to generate relative permeability curves using Eqns. 5,9, 10 and 12.

Stratified formations yield a segmented water saturation. Each segment is matchedseparately to generate a capillary pressure and relative permeability characteristics specificto that segment. Only the free water level depth is the same for all segments.Field Example 1

The well logs obtained in a tight gas-bearing sand formation in a well in Colorado areshown in Fig. 7. Fig. 8 shows the porosity, water saturation, and normalized water saturationprofiles. The normalization to the average porosity of 7.6% results in a smoother saturationprofile, which gives the reasonable match illustrated in Fig. 9. The match, which shows thatthe permeability varies within a narrow range, yields the following match parameters:

Average Permeability = 0.08 mdIrreducible Water Saturation = 17%a = 13 psiFree water level depth = 7,922 ft

The determination of the free water level by visual inspection of the logs is difficult due tothe increase in formation shaliness with depth.Field Example 2

The second field example represents a tight gas-bearing sand in Colorado. Fig. 10shows the well log data, Fig. 11 shows the porosity and saturation profiles, and Fig. 12 showsthe best match obtained. Unlike Example 1, the transition zone of this formation seems tobe composed of three strata of distinct properties sharing a common free water level at adepth of 8,130 ft.

The match ammeters for the three strata are:Strata A Strata B

K, md 0.3 0.08swi, % 6 15a, psi 5.6 10.9

Strata C0.05

1913.8

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SPWLA 33rd Annual Logging Symposium, June 14-17,

The decrease of permeability with depth is consistent with the increase in shaliness.

Field Example 3Figs. 13, 14, and 15 show the data and interpretation for a tight gas-bearing sand inWyoming. The match show three strata x, y and z sharing a common free water level at a

depth of 9,162 ft.The match parameters for the three strata are:

I Strata x I Strata y I Strata zK, md 1 0.05 1 0.1 1 0.08Swir 6 25 20 22a, psi 21.7 15.4 18.3

An example of the relative permeability curves which can be constructed using thematch parameters is shown in Fig. 16 representing strata y.

Cores available for interval Y show permeabilities ranging from 0.08 to 0.16 md andaveraging 0.108 md, which agree extremely well with the match permeability. Because thesand is reasonably clean the Tixier method, based on the resistivity gradient, can be used toestimate the permeability of the transition zone. (16) The Tixier method results in an averagepermeability of 0.05 md for the three strata. This value is also in agreement with the matchvalues.

The saturation profile in the top interval of the sand is complex and cannot beanalyzed using the proposed technique.Conclusions

A capillary pressure-water saturation empirical relationship is developed for tight sandformations. This relationship is used to adapt available relative permeability models to thecase of tight sands.

The capillary pressure-water saturation relationship is used in conjunction with the J-function to develop generalized capillary pressure curves typical of tight sand formations.When matched to the capillary pressure type curve, a log-derived normalized water

saturation profile yields reasonable estimates of the absolute permeability, irreducible watersaturation, the free water level and relative permeability characteristics.

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PWLA 33rd Annual Logging Symposium, June 14-17, 1992

LimitationsThe development of the type curves is based on petrophysical parameters derived from

curve fitting of core data. The parameters used in this study are the average of nine coresrepresenting Cotton Valley, Travis Peak and Falher tight sand formations. The addition ofmore core data when available should improve the statistical representation of the matchresults.

The petrophysical models discussed in this paper should only be used in the analysisof other tight formations if a petrophysical similarity is demonstrated.

Although the water saturation profile is normalized to an average porosity, the use ofthe proposed technique should be limited to cases where porosity variations fall within areasonable range.

AcknowledgementsSupport for this work was provided by the Gas Research Institute under contract no.

5089-211-1842, CER subcontract no. PO233-0275-S. The authors wish to acknowledge thetechnical support of the members and staff of the LSU petroleum engineering department.Special thanks to Brenda Macon, who generously assisted in editing the manuscript.References1. Jones, F. 0. and W. W. Owens, 1980. A Laboratory Study of Low-Permeability Gas

Sands. Journal of Perrol eum Technol ogy (September): 1631-1640.2. Luffel, D. L. and W. E. Howard, 1988. Reliability of Laboratory Measurement of

Porosity in Tight Gas Sands. SPE Format ion Evaluati on (December): 705-7 10.3. Wells, J. S. and J. 0. Amaefule, 1985. Capillary Pressure and Permeability

Relationships in Tight Gas Sands, SPE/DOE Paper 13879. Presented at theSPE/DOE Joint Symposium on Low Permeability Gas Reservoirs, May 19-22, Denver,co.

4. Ward, J. S. and N. R. Morrow, 1985. Capillary Pressure and Gas RelativePermeabilities of Low-Permeability Sandstone, SPE Formarion Evaluation(September): 345-356.

5. Raymer, L. L. and P. M. Freeman, 1984. In-Situ Determination of CapillaryPressure Pore Throat Size and Distribution and Permeability from Wireline Data.Presented at the Society of Professional Well Log Analysts 25th Annual LoggingSymposium, June lo- 13.

-lO-


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6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

Alger, R. P., D. L. Luffel, and R. B. Truman, 1987. New Unified Method ofIntegrating Core Capillary Pressure Data with Well Logs, SPE Paper 16793.Presented at the 62nd Annual Technical Conference and Exhibition of the Society ofPetroleum Engineers, September 27-30, Dallas, TX.Purcell, W. R., 1949. Capillary Pressures: Their Measurement Using Mercury andthe Calculation of Permeability Therefrom. Pet rol eum Tr ansact ions of AI M E 186:39 48.Wyllie, M. R. and G. H. F. Grander, 1958. The Generalized Kozeny-CarmanEquation. Wor ld O i l (April): 210-227.Naar, J. and J. H. Henderson, 1961. An Imbibition Model: Its Application to FlowBehavior and the Prediction of Oil Recovery. Societ y of Pet rol eum Engi neersJournal (June): 61-70.Naar, J. and R. J. Wygal, 1961. Three-Phase Imbibition Relative Permeability.Societ y of Pet rol eum Engi neers Journal (December): 254-258.Katz, Donald L., Max W. Legatski, M. Rasin Tek, L. Gorring, and R. L. Nielsen,1966. How Water Displaces Gas from Porous Media. The Oi l and Gas Journal(January 10): 55-60.Leverett, M. C., 1941. Capillary Behavior in Porous Solids. PetroleumTransacti ons of A IM E 142: 152 169.Brown, H. W., 1951. Capillary Pressure Investigations. Petroleum Transactionsof A IM E 192: 67 74.Heseldine, G. M.: 1974. A Method of Averaging Capillary Pressure Curves.Presented at the Society of Professional Well Log Analysts 15th Annual LoggingSymposium, June 2-5.Rose, W. and W. A. Bruce, 1949. Evaluation of Capillary Character in PetroleumReservoir Rock. Pet rol eum Transact ions of AIME 186: 127-142.Tixier, M. P., 1949. Evaluation of Permeability from Electric-Log ResistivityGradients, O i l and Gas Journal (June): 113-122.

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Formation core K cud)Cotton ValleyCotton Valley

5A 80 12.0 32.0 35.200 2.30410B 194 8.2 10.5 9.030 2.167

Cotton Valley 7A 97 12.5 23.0 35.694 2.132Cotton Valley 8A 65 8.8 30.0 11.929 3.400Cotton Valley IllAl 49Cotton ValleyCotton Valley

12A 5113A 30

Travis PeakTravis Peak

17A 260 9.9 10.5 2.574 2.647NAl 100 9.8 16.0 4.264 1.468

Falher SSFalher SS

NA2 19.8 9.4 30.0 19.648 2.623NA3 13.6 9.4 32.0 26.156 2.385

Q, (%I vi (%I a (psi) b

I,TABLE 1. Petrophysical data from tight sand core samples used in this study.

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Fig. 1.

1 I-2 -1 0

Ln (SW)

Logarithmic P, vs. S, plot for three core samples representative of theCotton Valley, Travis Peak, and Falher tight sands.

0 40 60

Water saturation, percent

Fig. 2. Leveretts J-function for tight sands core samples from Cotton Valley,Travis Peak, and Falher formations.


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4i

Y - 3.2482 . 2.3080~ RA2 0.9082

2

43 2 1 0

Ln (SW)Fig. 3. Loganthmic plot of the J-function data displayed in Fig. 2.

X

:a

200

100

0 -Id 60 Itaa

Fig. 4. Example of capillary pressure type curves


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2

CE2&

1

E:

0 I 0.1 0.2 0.3SWI

Fig. 5. Correlailon betweenJ-

K and S,, for tight sand core samples from Cotton5

Valley, Traws Peak, and Father formations

Fig. 7. Well logs forfield example 1


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HighK

HlghK

F.W.L. .-1-11--1-

F.W.L.

\

\\

\\ \

\\ \

\\ \ \

0.0 Watrr srturatlon 100

Water srturatlon

4

Fig. 6. Schematic showing variations in water saturation profile due to changes inpore size and permeability.


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771,

773c

77 50

7770DeplhFL)

7790

7e:o.

1)-

SPWLA 33rd Annual Logging Symposium, June 14-1Water Salurarlon. Frachon

Fig. 8. Water saturation profilesfor field example 1.

Fig, IO. Well log data foifield example 2.


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240

230

220

210

200

190

18C

17c

= 16x

T 1600,&i 141Bg 13ctz 12c8m 11CEP2 1O

9(

at

7(

6(

54

4(

3(

2l

1t

t

I-

,-

I-

,-

)-

)-

)-

,-

I-

)-

I-

)-

I-

)-

I-

I-

I-

I-

1-r0

- - 7700

- - 7710

- . 7720

-- 77x)

L. 7740

- . 7790 Depth- - 7800 f t

- - 7810

- . 7820

- - 7630

t- 7660

Water saturation, %

Fig. 9. Matching normalized water saturation profile to capillary pressure typecurves for field example 1.


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\

+

+

\-

cl _...

-8080Depth

-8090

-8100

-8110

-8120

Fig. 12.

40 60Water saturation,

Best match of normalized water saturation tofor field example 2.

80 100

capillary pressure type curves


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SPWLA 33rd AMU~ Logging Symposium, June 14-17, 1992

i

-.... .-.. . . ._.

s.:.

._. . ..+-...-.....-... .__(.

/.

Fig. 11. Water saturation profllesfor field example 2.

-. _-_- -

>i.iCE 1 /:Y....,.

Fig. 13. Well logs forfield example 3.

I -- .. i rl -1 .I i l--i UI


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SPWLA 33rd AMU~ Logging Symposium, June 14-17, 1

9080

9090

Depth(ft)

9100

9110

9120

9130

------AI

i;iiIIi1

-_-.; Pi -____---sily

i

l o 15.02 porosity., / ,

0.2 0.4 0.6 08 1.0

Water saturation, fraction

Fig. 14. Water saturation profile for field example 3


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1.0

0.8:38z

0.0

DeDthmm 17 ftl

- kmw dr)

0.4 0.6

Water saturation, fraction0.8

Fig. 16. Calculated relative permeability curves for strata y of example 3.


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Adel Ibrahim is a Ph.D. candidate in the department of petroleum engineering atLouisiana State University in Baton Rouge, Louisiana. He earned both his B. S. 1984) andM. S. 1988) degrees in petroleum engineering from Suez Canal University in Suez, Egypt.His research interests include formation evaluation, reservoir engineering, and well logging.

Robert Desbrandes holds the position of Distinguished Professor in the petroleumengineering department at Louisiana State University in Baton Rouge, Louisiana. He teachesand performs research in petrophysics, well logging, formation evaluation, MWD/LWD, andproduction logging. Previously, he taught two years at the University of Houston and 24years at the French Petroleum Institute. Prior to his teaching career, Dr. Desbrandes workedfor Schlumberger for 11 years in South America and in Houston. He earned his B. S. degreein mechanical engineering at Arts et Metiers 1944) and M. S. 1962) and Dr. SC. 1965)degrees in physics at the University of Lyon, France. He is the author of The Encyclopediaof Well Logging

Zaki Bassiouni is a professor and chairman of the petroleum engineering departmentat Louisiana State University, Baton Rouge, Louisiana. He holds a B. S. degree in petroleumengineering from Cairo University, a diploma in geophysics from the French PetroleumInstitute, and a Ph.D. degree from the University of Lille, France.

1992_ss-kr from log - pc and k distribution

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