Average Water Saturation From Capillary Pressure Data

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SPWLA TWELFTH

ANNUAL

LOGGING

SYMPOSIUM,

MAY 2-5,

AVERAGE WATER SATURATION FROM CAPILLARY PRESSURE DATA

L. Paul Westbrook and W. John Lee Mississippi State University ABSTRACT

This paper proposes a method for estimating average water saturation in stratified reservoirs. This method requires capillary pressure data and an estimate of average permeability in the reservoir. In the past, there has been some question as to how this average permeability should be determined. Therefore, computer studies, using field capillary pressure data, were performed on hypothetical stratified reservoirs to determine a satisfactory method of estimating average permeability to determine average water saturation. Use of geometric average permeability proved to yield reliable average water saturation estimates; use of arithmetic average permeability frequently led to unacceptably large errors.

INTRODUCTION

Average water saturation of a reservoir is frequently determined from capillary pressure data. This is a convenient method for homogeneous reservoirs, but in a stratified reservoir, an accurate value of average water saturation is more difficult to determine. The difficulty arises because water saturation depends on permeability; for a reservoir in which permeability varies from point to point, water saturation is a complex function of height in the reservoir. Average water saturation in a stratified reservoir could be estimated rather easily from capillary pressure data if an average permeability could be determined for the reservoir. If, in addition to making calculations easier, use of this average permeability results in reasonably accurate values of average water saturation, we would have a technique of practical value to the formation evaluation engineer. At least two methods of computing average permeability in stratified reservoirs can be considered: arithmetic average permeability and geometric average permeability. Both of these averaging methods have been used by engineers to estimate the average water saturation for oil reservoirs. We found no clear evidence in the literature supporting either of these methods. Amyx, Bass and Whitingl suggested that average permeability obtained with the geometric average method will yield more accurate values of average water saturation, but they offeretino evidence based on field capillary pressure data to support their suggestion. Therefore, the objective of this work was to use field data to determine which of the two proposed methods of averaging permeability will yield a value of average water saturation closer to true average water saturation.

-1-

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MAY 2-5, 1971

A computer program was written to calculate and compare true average water saturation with average water saturations obtained from arithmetic average permeability and geometric average permeability. These computer studies verified that the geometric average permeability should be used to find average water saturation in stratified reservoirs.

CAPILLARY PRESSURE

Water saturations calculated from capillary pressure data can be very accurate. D. L. Luffel and R. V. Randa112 have -compared water saturations calculated from capillary pressure data with those obtained by coring with an oil base mud and a pressure core barrel. Figure 1 gives depth plotted against water saturation determined by these two methods. Each point represents a depth in an oil well with a certain water saturation. Note the excellent agreement when comparing the directly determined water saturations with those calculated from the air-kerosene capillary pressure curve of extracted cores. Also note the complicated shape of the resulting water distribution curve.

A

---”

.

---

1

1

1

I

-r-l’ =

-—._ —~

-n I

1

1

I [

1

~Me@wredDirectly by Vacuum Distillation of Oil-CutCores 3

o FIG. I

20

‘-0

Calculated from Gas-Oil Capiilory PressureCurves

r

40 Water SoturotionPercent

Capillary pressure data for reservoir cores are usually reported in the form of a graph similar to Figure 2. These capillary pressure curves were obtained from cores with different permeability but from the same reservoir. These data give water saturation as a function of capillary pressure. Capillary pressure is not useful as such for studying water saturation

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MAY 2-5,

in an oil reservoir; therefore, capillary pressure is usually converted to height above free water level in the following manner:*

144 Pc H=

(1) PW-P ~

Free water level is defined as the point in the reservoir where capillary pressure is zero. Figure 2 gives water saturation plotted against both capillary pressure and height above free water level. For the conversion from capillary press~re to height above free water-level in Figure 2, it was assumed that the density of oil was 53.6 lb/ft3 and the density of water was 68 lb/ft3.

1000 000 600 500 400 300 200

I

1 11

11

II

1 1

1 \

1

1

Im -

z ●. ~ 4 2 %

,~~ 80

f: 40 30 20

OL o

I I t0203040508C

I

I

I

I

I 1 70808000°

I

I

10

WATER SATURATION , !4

F!OURE 2 — CORRELATION OF WATER SAT UR4T10N WITH HEIGHT ABOVLZ FREE WATER LEVEL ANO CAPILLARY PRESSURE AT V&RlOUS PERME&OILiTI ES.

0

10

20304050307083 WATER

SOKIO

SATURATION, %

Fl@URE 3 — CORi+ELAT(ON OF WATER $ANRATION WITH PERMEABILITY FOR VARICIJS MEIQHTS ABOVE FREE WATER LEVEL,

If the data in Figure 2 are replotted on semi-log paper with parameters of height above free water level, straight lines result: for example, see Figure 3. Capillary pressure data plotted as in Figure 3 are more convenient to work with.

*symbols are defined in the table of nomenclature

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MAY 2-5, 1971

220 I

Ooc

I

I

II

I

I tl

200

20

WC

I

00

I

I

I

I

I

I

10

20

30

40

I

L

I

50

60

1-

I 1

70

I

IoiO

I

I

I

BO

SO

,003Z10

WATER SATURATt ON , %

FlOURE 4-

CORRELATION OF WATER SATURATION WITH MEIQHT ABOVE FREE WATER LEvEL AND SIJ8SEA CEPTM AT TME AVERAOE RESERVOIR PERMEAC.ILITY.

h “average” water distribution curve’for a stratified reservoir can be obtained easily from Figure 3, given a value of average permeability for the reservoir. Figure 4 is an example of this water distribution curve for an average permeability of 90.6 md as obtained from Figure 3. Figure 4 can be treated as the water distribution curve for the reservoir. This assumes that the reservoir is homogeneous, with permeability equal to the average permeability. Average water saturation can be obtained from Figure 4 since

I Ew

Woc

Sw dH GOC

=

(2) WOC - GOC

This integral is normally evaluated by graphical integration; an example problem using this method is given at the end of the paper.

COMPUTER SOLUTION

A computer program was written to compute true and approximate water saturations (based on average reservoir permeability) for stratified

-4-

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reservoirs. These calculations were performed by assigning a thickness and a permeability to each layer of the reservoir. Capillary pressure data published by Wright and Wooddy3 were used to relate water saturation to permeability and height above free water level. True average water saturation for a reservoir was calculated with Eq. 3: ~wlhl +EW2

h2 + . . .

Fwt =

(3) h1+h2+

. . .

Arithmetic average permeability and geometric average permeability were then calculated for the reservoir. Arithmetic average permeability was calculated with Eq. 4: hlkl+h2k2+ Za

. . .

(4)

= hl+hl+

. . .

Geometric average permeability was calculated with Eq. 5.

hllogkl+hzlogkz+. log Eg

= h1+h2+

. .

(5)

. . .

Values of average water saturation were determined using each of these two average permeabilities. Each average water saturation was compared with true average water saturation and a percentage error was calculated. Sixty-six test cases were studied. The number of layers and the permeability of each layer were varied from case to case. Reservoirs had as few as two layers and as many as five layers. The layers had permeability values as low as five md and as high as 900 md. Some reservoirs had the higher permeability on top and others had the higher permeability on the bottom.

PRESENTATION OF RESULTS

Table 1 gives the results obtained in this study. Each hypothetical reservoir is denoted by a case number. The layers in each reservoir are numbered from bottom to top; i.e. , layer one is closest to the wateroil contact.

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TABLE L RESULTS OF COMPUTER STUDIES ON STRATIFIED RESERVOIRS

Average

Average ~ermeabilities Case

1 2 3 h 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

of

K

K

layers, K

1

2

15 200 15 200 900 10 10 10 15 15 15 75 75 75 200 200 200 300 300 300 10 10 10 10 10 21.5 21.5 21.5 21.5 21.5 21.5 100 100 100 100 100 100 100 225 225 225 225 225 225 900 900 900 900 900 900 900 5 10 500 800 900 20 40 500 600 700 40 50 300 400 500

200 15 200 900 15 5 200 200 75 300 300 15 200 300 15 15 75 75 200 200 21.5 21.5 21.5 900 900 100 100 100 100 225 225 10 21.5 21.5 225 225 900 900 10 10 10 21.5 100 900 10 10 21.5 100 LOO 225 225 900 5 10 500 800 700 20 30 500 600 500 40 50 300 400

3

900 15 200 20 600 900 300 74 200 300 15 200 75 300 15 200 15 75 100 100 225 21.5 225 10 225 225 900 10 100 900 10 225 10 900 10 225 21.5 900 900 100 900 10 21.5 21.5 100 21.5 21.5 10 21.5 800 900 5 10 500 600 700 20 30 500 400 500 40 50 300

K 4

100 900 600 200 200 75 200 300 15 300 75 300 15 75 15 225 900 100 100 21.5 900 10 900 225 900 900 225 225 10 900 10 21.5 10 100 21.5 Loo 10 10 21.5 100 225 225 10 225 21.5 100 500 800 900 5 10 500 600 700 20 30 300 400 500 40 50

Water —

Permeability

mcl

K 5

s

i

R a

g

55 55 140 140 140 18 181 145 91 91 91 91 91 91 91 91 91 91 91 91 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 85 112 112 112 112 112 166 166 166 166 166 164 164 164 164 164

108 108 370 370 370 34 428 383 148 148 148 148 148 148 148 148 148 148 148 148 251 900 225 251 251 900 225 251 100 251 225 251 251 900 251 10 10 251 100 251 251 10 21.5 251 251 900 251 900 21.5 251 21.5 251 225 251 21.5 251 900 251 100 251 21.5 25L 251 900 21.5 251 100 251 225 251 100 251 10 251 225 251 10 251 100 251 10 251 10 433 500 433 800 433 900 433 433 5 30 370 370 500 370 600 370 700 20 370 50 258 300 258 400 258 258 500 4f.1 258

-6-

Saturation —

s Wt

47.1 45.6 37.5 35.6 35.7 59.0 34.8 37.1 41.7 41.7 41.5 41.2 41.2 40.6 41.0 40.7 40.9 40.2 40,5 40.2 42.8 42.6 42.7 42.2 40.7 42.3 42.2 41.9 41.8 42.1 41.8 41.7 42.2 42.0 41.5 41.2 41.5 41.1 42.o 41.5 41.4 41.8 41.1 41.2 41.4 41.3 40.9 41.1 40.8 41.0 40.7 39.0 39.8 38.9 38.2 37.4 34.7 35.3 34.6 34.2 33.6 34.7 35.1 34.7 34.4 33.9

wa

40.4 40.4 27.0 27.0 27.0 53.0 25.4 26.6 37.0 37.0 37.0 37.0 37.0 37.0 37.0 37.0 37.0 37.0 37.0 37.0 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31.2 31,2 25.1 25.1 25.1 25.1 25.1 27.0 27.0 27.0 27.0 27.0 39.0 39.0 39,0 39.0 39.0

Error —

s

E w

47.8 27.8 37.6 37.6 37.6 60.0 34.8 37.2 b2.3 h2.3 42.3 62.3 42.3 42.3 42.3 42.3 42.3 h2.3 42.3 62.3 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 43.0 40.0 40.0 40.0 40.0 40.0 35.7 35.7 35.7 35.7 35.7 35.8 35.8 35.8 35.8 35.8

a

14.1 11.4 27.9 24.2 24.3 10.1 26.9 28.2 11.3 11.2 10.8 10.2 10.2 8.9 9.7 9.2 9.6 8.0 8.6 7.9 27.0 26.8 26.9 26.1 23.3 26.2 26.1 25.4 25,3 25.9 25.3 25.1 25.9 25.6 24.8 24.3 24.8 24.1 25.6 24.8 24.6 25.2 24.o 24.2 24.6 24.4 23.7 24.0 23.5 23.8 23.3 35.8 37.1 35.5 34.3 32.9 22.2 23.5 21.9 20.9 19.6 10.9 12.0 10.9 10.1 8.9

E —

g

1.4 4.7 0.2 5.5 5.3 1.7 0.1 0.3 1.4 1.5 2.0 2.5 2.6 4.1 3.2 3.3 3.4 5.1 4.5 5.2 0.7 1.0 0.8 1.9 5.7 1.7 1.9 2.8 3.0 2.1 3.0 3.3 2.1 2.5 3.7 4.4 3.6 4.6 2.6 3.6 3.9 3.0 4.7 4.4 4.0 4.1 5.2 4.7 5.4 5.1 5.7 2.4 0.3 2.8 4.7 6.9 2.9 1.2 3.2 4.6 6.3 3.2 2.0 3.2 4.2 5.5

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INTERPRETATION OF RESULTS

Table 1 shows that the average water saturation obtained using arithmetic average permeability varied from 7.9%, in case 20, to 37.1%, in case 53. Error in average water saturation obtained using geometric average permeability varied from 0.1%, in case 7, to 6.9%, in case 56. Table 1 also shows that reservoirs which have high permeability layers near the water-oil contact, overlain by layers of low permeability, yield larger errors in average water saturation calculated using geometric average permeability. Cases 20, 51, and 56 are examples of this. Cases 8, 22, and 53 in Table 1 show the large errors caused in the average water saturation calculated using arithmetic average permeability when very low permeability layers are near the water-oil contact. However, in every case, geometric average permeability yielded average water saturation closer to true average water saturation than did arithmetic average permeability.

CONCLUSIONS

Geometric averaging yields permeabilities which can be used to obtain more accurate estimates of average water saturation than arithmetic averaging. If a relatively small error can be tolerated in average water saturation, then the geometric average permeability can be used with confidence to determine average water saturation. In any case, if left with a choice between arithmetic average permeability and geometric average permeability, the choice should be geometric average permeability.

APPLICATION

Consider a stratified reservoir which has layers with the following permeabilities:

subsea depth, feet

3000 3000 3040 3100 3143 3200

permeability, md

Gas-oil contact

-

10 200

3040 3100 3143 3200

50

500 Water-oil contact

’i’

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MAY 2-5, 1971

Assume that the capillary pressure data in Figure 2 applies to this reservoir and that core data show that the free water level is ten feet below the water-oil contact. The average water saturation of the reservoir is determined as follows: 1.

Determine geometric average permeability:

log Zg = 40 log 10 + 60 log 200 + 43 log 50 + 57 log 500

40+60+43+57 ~g = 90.6 2.

Replot the capillary pressure data in Figure 2 as logarithm of permeability against water saturation with parameters of height above free water level. Results are given in Figure 3.

3. Enter Figure 3 with ~

= 90.6 and determine water saturation and height values that correspon% to this average permeability.

H, feet

Swg%

61.5 51 47 44 42 37.5

10 20 30 40 60 170

Plot these Swg vs. H values as in Figure 4.

8

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4. Determine average water saturation from Figure 4 by solving Eq. 2 graphically. The area under the curve between the water-oil contact and the gas-oil contact, shown as the shaded area in Figure 5, is divided by the thickness between the water-oil contact and the gasoil contact to obtain the average water saturation.

Woc

SwgdH GOC

[

Fwg

=

=

7960

200 Woc - Goc

Fwg

=

39.8%

The same problem can be solved using Eq. 3 to calculate the true average water saturation. For the purpose of illustration and comparison the same problem is solved using Eq. 3 as follows: 1.

0

Eq. 3 is best solved in steps because average water saturation for each layer requires graphical integration of the capillary pressure curve for that layer. Also, because water saturation depends on height above free water level, the average water saturation of each layer must be determined at the layer’s position with respect to the water-oil contact. Therefore, it is convenient to number the layers from bottom to top in the reservoir, which will place laYer one nearest the water-oil contact. Figure 6 shows how the average water saturation should be obtained for each layer in the reservoir from capillary pressure data.

10

20

30

40 wATm

50

60

70

m

90

,m WATER

MATURATION,%

FIQURE 3 — GRAPHICAL INTEGRATION OF THE WATE8 DISTRIBUTION CURVE AT AvERAGE RESERVOIR PERMEABILITY,

FIGIRE

9

3ATuRATWM, %

6 — CORnELATlON OF WATER SATURATION WITH MEIQHT ABOVE FREE WATER LEVEL SMOWING ORAPHICAL lNTEQRAnON OF THE WATER DISTRIBUTION CURVE FM EACH LAwIR IN A RESJZRVO!R

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1971

For layer one, the average water saturation is:

67 SwdH J 10 Fwl

=

=

1588 57

67 - 10 Fwl

=

27.8%

Similarly for layers two, three, and four:

3.

ZW2

=

46.8%

:W3

=

30.5%

FW4

=

57.5%

After the individual average water saturations have been determined for each layer, Eq. 3 is used directly

Zwt . 127.8)(57) + (46.8)(43)+ (30.5)(60)+ (57.5)(40) 57+43+60+40

FWt= 7750 200

Zwt = 38.8% The value of w~ter saturation obtained using permeability~ Swg = 39.8%, is within 2.5% of saturation, Swt_= 38.8%. It is obvious that in determining Swt would become very tedious strata; however, the calculations leading to by an increase in the number of strata.

1

the the the ~or S Wg

geometrically averaged true average water calculations involved a reservoir with many would not be complicated

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NOMENCLATURE

error in average water saturation, %

WC

=

gas-oil contact

h=

thickness of individual bed, ft

H=

height above free water level, ft

k=

absolute permeability of individual bed, md

E=

average absolute permeability for reservoir, md capillary pressure, psi water saturation, % average water saturation, %

Woc

=

water-oil contact density of fluid, lb/ft3

SUBSCRIPTS

123 99s...

a=

=

individual beds numbered from bottom to top

quantity obtained using arithmetic average permeability

g

=

quantity obtained using geometric average permeability

o

=

oil

t

=

true value

w=

water

MAY 2-5, 19

I

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MAY 2-5, 1971 1 I

REFERENCES

1.

Amyx, J. W., Engineering”,

2.

Luffel, D. L., and Randall, R. V. :“Core Handling and Measurement Techniques for Obtaining Reliable Reservoir Characteristics”, Formation Evaluation Symposium, AIME, Houston, Texas, 1960, 1-21.

3.

Wright, H. T., Jr. and Woody, L. D., Jr. : “Formation Evaluation of the Borregos and Seeligson Field, Brooks and Jim Wells County, Texas”, Formation Evaluation Symposium, AIME, New Orleans, Louisiana, October 1955.

Bass, D. M., McGraw-Hill

Jr. and Whiting, R. L. : “Petroleun~ Reservoir Book Company, Inc., New York (1960) 546.

ABOUT THE AUTHORS

Paul Westbrook is a Petroleum Engineer with Shell Oil Company in New Orleans, Louisiana. He received his B.S. degree in Petroleum Engineering from Mississippi State University in 1970. (Please refer to Paper E for the biography John Lee.) and photo of the co-author, !IJ.

L. Paul Westbrook

-12-

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