Tarek Ahmed. Reservoir engineering handbook

where S

*

o

, S

*

w

, S

*

g

= effective oil, water, and gas saturation, respectively

S

o

, S

w

, S

g

= oil, water and gas saturation, respectively

S

wc

= connate (irreducible) water saturation

1. Wyllie and Gardner Correlation

Wyllie and Gardner (1958) observed that, in some rocks, the relation-

ship between the reciprocal capillary pressure squared (1/P

c

2

) and the

effective water saturation S

*

w

is linear over a wide range of saturation.

Honapour et al. (1988) conveniently tabulated Wyllie and Gardner corre-

lations as shown below:

Drainage Oil-Water Relative Permeabilities

Type of formation k

ro

k

rw

Equation

Unconsolidated sand, well sorted (1 - S

*

w

)(S

*

w

)

3

(5-4)

Unconsolidated sand, poorly sorted (1 - S

*

w

)

2

(1 - S

*

w

1.5

)(S

*

o

)

3.5

(5-5)

Cemented sandstone, oolitic limestone (1 - S

*

o

)

2

(1 - S

w

*

w

2

)(S

*

o

)

4

(5-6)

Drainage Gas-Oil Relative Permeabilities

Type of formation k

ro

k

rg

Equation

Unconsolidated sand, well sorted (S

*

o

)

3

(1 - S

*

o

)

3

(5-7)

Unconsolidated sand, poorly sorted (S

*

o

)

3.5

(1 - S

*

o

)

2

(1 - S

*

o

1.5

) (5-8)

Cemented sandstone, oolitic limestone,

rocks with vugular porosity (S

*

o

)

4

(1 - S

*

o

)

2

(1 - S

*

o

2

) (5-9)

Wyllie and Gardner have also suggested the following two expressions

that can be used when one relative permeability is available:

• Oil-water system

kSk

S

rw w ro

w

=-

-

È

Î

Í

˘

˚

˙

()

*

2

1

(5 -10)

S

g

wc

*

=

-1

(5 - 3)

288 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 288

• Gas-oil system

2. Torcaso and Wyllie Correlation

Torcaso and Wyllie (1958) developed a simple expression to determine

the relative permeability of the oil phase in a gas-oil system. The expres-

sion permits the calculation of k

ro

from the measurements of k

rg

. The

equation has the following form:

The above expression is very useful since k

rg

measurements are easily

made and k

ro

measurements are usually made with difficulty.

3. Pirson’s Correlation

From petrophysical considerations, Pirson (1958) derived generalized

relationships for determining the wetting and nonwetting phase relative

permeability for both imbibition and drainage processes. The generalized

expressions are applied for water-wet rocks.

For the water (wetting) phase

The above expression is valid for both the imbibition and drainage

processes.

For the nonwetting phase

• Imbibition

()k

SS

r nonwetting

wwc

wc nw

=-

-

--

Ê

Ë

Á

ˆ

¯

˜

È

Î

Í

˘

˚

˙

1

2

(5 -14)

kSS

rw w w

=

*3

(5 -13)

kk

S

SS

ro rg

o

oo

=

--

È

Î

Í

˘

˚

˙

()

()(())

*

**

4

22

11

(5 -12)

kSk

S

ro o rg

o

=-

-

È

Î

Í

˘

˚

˙

()

*

1

(5 -11)

Relative Permeability Concepts 289

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 289

• Drainage

where S

nw

= saturation of the nonwetting phase

S

w

= water saturation

S

*

w

= effective water saturation as defined by Equation 5-2

Example 5-1

Generate the drainage relative permeability data for an unconsolidated

well-sorted sand by using the Wyllie and Gardner method. Assume the

following critical saturation values:

S

oc

= 0.3, S

wc

= 0.25, S

gc

= 0.05

Solution

Generate the oil-water relative permeability data by applying Equation

5-4 in conjunction with Equation 5-2, to give:

S

w

k

ro

= (1 - S

*

w

)

3

K

rw

= (S

*

w

)

3

0.25 0.0000 1.000 0.0000

0.30 0.0667 0.813 0.0003

0.35 0.1333 0.651 0.0024

0.40 0.2000 0.512 0.0080

0.45 0.2667 0.394 0.0190

0.50 0.3333 0.296 0.0370

0.60 0.4667 0.152 0.1017

0.70 0.6000 0.064 0.2160

Apply Equation 5-7 in conjunction with Equation 5-1 to generate rela-

tive permeability data for the gas-oil system.

S

SS

S

w

wwc

wc

*

=

-

-1

() ( ) ( )

**.

.

kSSS

r nonwetting w w w

=- -

[]

11

025

05

(5 -15)

290 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 290

S

g

S

o

= 1 - S

g

- S

wc

k

ro

= (S

*

o

)

3

k

rg

= (1 - S

*

o

)

3

0.05 0.70 0.933 0.813 —

0.10 0.65 0.867 0.651 0.002

0.20 0.55 0.733 0.394 0.019

0.30 0.45 0.600 0.216 0.064

0.40 0.35 0.467 0.102 0.152

0.50 0.25 0.333 0.037 0.296

0.60 0.15 0.200 0.008 0.512

0.70 0.05 0.067 0.000 0.813

Example 5-2

Resolve Example 5-1 by using Pirson’s correlation for the water-oil

system.

Solution

S

w

0.25 0.0000 0.000 1.000

0.30 0.0667 0.007 0.793

0.35 0.1333 0.016 0.695

0.40 0.2000 0.029 0.608

0.45 0.2667 0.047 0.528

0.50 0.3333 0.072 0.454

0.60 0.4667 0.148 0.320

0.70 0.6000 0.266 0.205

4. Corey’s Method

Corey (1954) proposed a simple mathematical expression for generat-

ing the relative permeability data of the gas-oil system. The approxima-

tion is good for drainage processes, i.e., gas-displacing oil.

kS

ro g

=-()

*

1

4

(5 -16)

kSSS

ro w w w

=- -

È

Î

Í

˘

˚

˙

()()

**.

.

11

025

05

kSS

rw w w

=

*3

S

SS

1

S

*

w

wwc

wc

=

-

S

o

wc

*

=

-1

Relative Permeability Concepts 291

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 291

where the effective gas saturation S

*

g

is defined in Equation 5-3.

Example 5-3

Use Corey’s approximation to generate the gas-oil relative permeabili-

ty for a formation with a connate water saturation of 0.25.

Solution

S

g

k

ro

= (1 - S*

g

)

4

k

rg

= (S

*

g

)

3

(2 - S

*

g

)

0.05 0.0667 0.759 0.001

0.10 0.1333 0.564 0.004

0.20 0.2667 0.289 0.033

0.30 0.4000 0.130 0.102

0.40 0.5333 0.047 0.222

0.50 0.6667 0.012 0.395

0.60 0.8000 0.002 0.614

0.70 0.9333 0.000 0.867

5. Relative Permeability from Capillary Pressure Data

Rose and Bruce (1949) showed that capillary pressure p

c

is a measure

of the fundamental characteristics of the formation and could also be

used to predict the relative permeabilities. Based on the concepts of tor-

tuosity, Wyllie and Gardner (1958) developed the following mathemati-

cal expression for determining the drainage water-oil relative permeabili-

ty from capillary pressure data:

k

S

dS p

ro

w

wc

S

w

wc

S

wc

=

-

Ê

Ë

Á

ˆ

¯

˜

Ú

1

2

1

2

1

()

(5 -19)

k

SS

S

dS p

rw

wwc

wc

S

wc

S

w

wc

S

wc

=

-

Ê

Ë

Á

ˆ

¯

˜

Ú

1

2

1

(5 -18)

S

g

wc

*

=

-1

kS S

rg g g

=-()( )

**

2 (5-17)

292 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 292

Wyllie and Gardner also presented two expressions for generating the

oil and gas relative permeabilities in the presence of the connate water

saturation. The authors considered the connate water as part of the rock

matrix to give:

where S

gc

= critical gas saturation

S

wc

= connate water saturation

S

or

= residual oil saturation

Example 5-4

The laboratory capillary pressure curve for a water-oil system between

the connate water saturation and a water saturation of 100% is represent-

ed by the following linear equation:

P

c

= 22 - 20 S

w

The connate water saturation is 30%. Using Wyllie and Gardner meth-

ods, generate the relative permeability data for the oil-water system.

Solution

Step 1. Integrate the capillary pressure equation, to give:

I

dS

S

ba

w

a

b

=

-

(

)

=

-

È

Î

Í

˘

˚

˙

-

È

Î

Í

˘

˚

˙

Ú

22 20

1

440 400

1

440 400

2

k

SS

dS p

rg

oor

ggc

oc

S

o

oc

=-

-

Ê

Ë

Á

ˆ

¯

˜

Ú

1

2

1

2

0

1

(5 - 21)

k

SS

S

dS p

ro

oor

or

oc

S

o

oc

=

-

Ê

Ë

Á

ˆ

¯

˜

Ú

1

2

0

2

0

1

(5 - 20)

Relative Permeability Concepts 293

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 293

Step 2. Evaluate the above integral at the following limits:

Step 3. Construct the following working table:

k

rw

k

ro

S

w

Equation 5-18 Equation 5-19

0.3 0.0000 1.0000

0.4 0.0004 0.7195

0.5 0.0039 0.4858

0.6 0.0157 0.2985

0.7 0.0466 0.1574

6. Relative Permeability from Analytical Equations

Analytical representations for individual-phase relative permeabilities

are commonly used in numerical simulators. The most frequently used

functional forms for expressing the relative-permeability and capillary-

pressure data are given below:

Oil-Water Systems:

kk

SS

ro ro S

wc

w orw

wc orw

n

o

=

--

È

Î

Í

˘

˚

˙

()

1

(5 - 22)

•.

dS

S

w

S

w

22 20

0 025

1

440 400

2

1

-

(

)

=-

-

È

Î

Í

˘

˚

˙

Ú

•.

.

dS

S

w

S

w

22 20

1

440 400

0 00313

2

3

-

(

)

=

-

È

Î

Í

˘

˚

˙

Ú

•

() ( .)

.

dS

S

w

22 20

1

440 400 1

1

440 400 0 3

0 02188

2

03

1

-

(

)

=

-

È

Î

Í

˘

˚

˙

=

Ú

294 Reservoir Engineering Handbook

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 294

Gas-Oil Systems:

with

S

lc

= S

wc

+ S

org

where S

lc

= total critical liquid saturation

(k

ro

)

S

wc

= oil relative permeability at connate water saturation

(k

ro

)

S

gc

= oil relative permeability at critical gas saturation

S

orw

= residual oil saturation in the water-oil system

S

org

= residual oil saturation in the gas-oil system

S

gc

= critical gas saturation

(k

rw

)

S

orw

= water relative permeability at the residual oil

saturation

n

o

, n

w

, n

g

, n

go

= exponents on relative permeability curves

p

cwo

= capillary pressure of water-oil systems

pp

SS

cgo c S

lc

ggc

lc gc

n

pg

=

-

--

È

Î

Í

˘

˚

˙

()

1

(5 - 27)

kk

SS

rg rg S

wc

ggc

lc gc

n

g

=

-

--

È

Î

Í

˘

˚

˙

()

1

(5 - 26)

kk

SS

ro ro S

gc

glc

gc lc

n

go

=

--

È

Î

Í

˘

˚

˙

()

1

(5 - 25)

pp

SS

cwo c S

wc

w orw

wc orw

n

p

=

--

Ê

Ë

Á

ˆ

¯

˜

()

1

(5 - 24)

kk

SS

rw rw S

orw

wwc

wc orw

n

w

=

-

--

È

Î

Í

˘

˚

˙

()

1

(5 - 23)

Relative Permeability Concepts 295

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 295

(p

c

)

S

wc

= capillary pressure at connate water saturation

n

p

= exponent of the capillary pressure curve for the oil-

water system

p

cgo

= capillary pressure of gas-oil system

n

p

g

= exponent of the capillary pressure curve in gas-oil

system

(p

c

)

S

lc

= capillary pressure at critical liquid saturation.

The exponents and coefficients of Equations 5-22 through 5-26 are

usually determined by the least-squares method to match the experimen-

tal or field relative permeability and capillary pressure data.

Figures 5-4 and 5-5 schematically illustrate the key critical saturations

and the corresponding relative permeability values that are used in Equa-

tions 5-22 through 5-27.

296 Reservoir Engineering Handbook

Figure 5-4. Water-oil relative permeability curves.

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 296

Example 5-5

Using the analytical expressions of Equations 5-22–5-27, generate the

relative permeability and capillary pressure data. The following informa-

tion on the water-oil and gas-oil systems is available:

S

wc

= 0.25 S

orw

= 0.35 S

gc

= 0.05 S

org

= .23

(k

ro

)

S

wc

= 0.85 (k

rw

)

S

orw

= 0.4 (P

c

)

S

wc

= 20 psi

(k

ro

)

S

gc

= 0.60 (k

rg

)

S

wc

= 0.95

n

o

= 0.9 n

w

= 1.5 n

p

= 0.71

n

go

= 1.2 n

g

= 0.6 (p

c

)

S

lc

= 30 psi

n

pg

= 0.51

Relative Permeability Concepts 297

Figure 5-5. Gas-oil relative permeability curves.

Reservoir Eng Hndbk Ch 05 2001-10-24 09:51 Page 297

Tarek Ahmed. Reservoir engineering handbook

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