honarpour, d., koederitz, l. and harvey, a. h. - relative permeability of petroleum reservoir

8/20/2019 Honarpour, D., Koederitz, L. and Harvey, A. H. - Relative Permeability of Petroleum Reservoir

1/141

Relative

Permeability

of

Petroleum

Reservoirs

Authors

Mehdi

Honarpour

Associate

Professor

f Petroleum

Engineering

Department f

Petroleum

Engineering

MontanaCollegeof Mineral ScienceandTechnology

Butte,

Montana

Leonard

Koederitz

Professor

f Petroleum

Engineering

Department

f

PetroleumEngineering

University of

Missouri

Rolla. Missouri

A.

Herbert

Harvey

Chairman

Department f

Petroleum

Engineering

University

of Missouri

Rolla, Missouri

@frc')

CRC

Press,

nc.

Boca

Raton,

Florida


2/141

PREFACE

In 1856

Henry P. Darcy determined

hat the

rate of

flow of water through a

sand ilter

could be

described y the equation

h , - h .

q : K A

- L

where

q

represents

he rate at

which water

flows downward

through a

vertical sand

pack

with cross-sectional

rea

A

and

ength

L; the terms

h,

and

h, represent

ydrostatic

eadsat

the

nlet and outlet,

respectively,

f the sand

ilter, and

K is a constant.

Darcy's experiments

were confined to

the flow

of water through

sand

packs which were 1007o

aturatedwith

water.

Later

investigators etermined

hat Darcy's law could be modified to describe he flow

of

fluids other than

water, and

that the

proportionality onstant

K

could

be replaced

by k/

p,

where k is a

property

of the

porous

material

permeability)

and

p

is a

property

of the

fluid

(viscosity).

With this

modification,

Darcy's

aw may be

written n

a

more

general

orm

AS

k

l-

dz

dPl

u ' : * L P g o s - d s l

where

S

v

Distance

n direction

of flow,

which is taken as

positive

Volume of

flux across

unit area

of the

porous

medium n unit time

along

flow

path

S

Vertical coordinate,

which is taken

as

positive

downward

Density of the

fluid

Gravitational

acceleration

Pressure radientalong S at the point to which v. refers

The

volumetric

lux

v. may be further

defined

as

q/A, where

q

is the volumetric

low

rate

and A

is the average

ross-sectional

rea

perpendicularo the

lines of flow.

It can

be shown

that the

permeability erm

which appears

n Darcy's

law has units

of

lengthsquared.

A

porous

material

has a

permeabilityof

I D when a single-phase

luid with

a

viscosityof

I cP completely

saturates

he

pore space f the

mediumand

will flow through

it under

viscous

flow at the

rate of

I

cm3/sec/cm2

ross-sectional

rea

under

a

pressure

gradientof 1 atm/cm. It is important o note the requirement hat the flowing fluid must

completely

saturate

he

porous

medium.

Since his

condition

s

seldom

met

n

a

hydrocarbon

reservoir,

t is evident

hat

further

modification

of Darcy's

law is needed

f the

law is to be

applied o

the flow

of fluids

in

an

oil or

gas

reservoir.

A

more useful

form of

Darcy's law can

be obtained

f we assurne hat

a

rock which

contains

more than

one

fluid has an effective

permeability o each

luid

phase

and

that the

effective

permeability

o

each

luid is a

function of

its

percentage aturation.

he effective

permeabilityof a

rock

to

a fluid

with which

it is 1007.o

aturated

s equal

to the absolute

permeabilityof the

rock.

Effective

permeability o each

fluid

phase

s considered

o be

independent f the other fluid phases nd the phases re consideredo be immiscible.

Z

p

g

D

dP

dS


3/141

V o . : T ( 0 . , * K - * )

V* . : * (o - ' 13 - t )

V o , : H ( o - r # - k )

where

he

subscripts

,

g,

and

w represent

il,

gas' and

water,

respectively'

Note

that

k,,,'

k.", and

k,*

are

he

relative

permeabilities

o

the

hree

luid

phases

t

he

respective

aturations

of the

phases

within

the

rock'

Darcy's

law

is the

basis

or

almost

all

calculations

f

fluid

flow

within

a

hydrocarbon

reservoir.

n

order

to

use

the

law,

it

is

necessary

o

determine

he

relative

permeability

of

the

reservoir

ock

to

each

of

the

fluid

phases;

his

determination

must

be

made

hroughout

the

range

of

fluid

saturations

hat

will be

encountered.

he

problems

nvolved

n

measuring

and predicting elativepermeabilityhavebeenstudiedby many investigators. summary

of

the

major

results

of

this

research

s

presented

n

the

following

chapters'


4/141

THE AUTHORS

Dr. Mehdi

"Matt"

Honarpour

is

an

associate

rofessor

of

petroleum

engineeringat

the

MontanaCollege

of Mineral Science

and

Technology,

Butte, Montana.

Dr. Honarpour

obtained is B.S., M.S., and Ph.D. in petroleum ngineeringrom the Universityof Mis-

souri-Rolla.

He hasauthored

many

publications

n

the

areaof reservoir ngineering

nd

core

analysis.

Dr. Honarpour

has

worked

as

reservoirengineer,

esearch ngineer,consultant,

and teacher

or the

past

15

years. He is a

memberof several

professional rganizations,

including he

Societyof

Petroleum

Engineers f

AIME, the

honorarysocietyof Sigma

Xi,

Pi

Epsilon Tau and

Phi Kappa

Phi.

Leonard

F. Koederitz

is a

Professor

of Petroleum

Engineeringat

the University

of

Missour i-Rol la.

ereceivedB.S.,

.S. , andPh.D.

egrees

romtheUniversi tyof

issour i-

Rolla. Dr. Koederitzhasworked

or Atlantic-Richfie ld

nd

previouslyservedas Department

Chairman

at Rolla.

He has authored

or

co-authored

everal echnical

publicationsand two

texts

related o

reservoirengineering.

A. Herbert Harvey

receivedB.S. and

M.S. degrees rom Colorado School

of Mines

and a Ph.D. degree rom the University

of Oklahoma.

He has authoredor co-authored

numerous

echnical

ublications

n topics

elated o the

production

f

petroleum.

Dr. Harvey

is

Chairman

of both the Missouri Oil

and

Gas

Council and the

PetroleumEngineering

Department t the University of

Missouri-Rolla.


5/141

ACKNOWLEDGMENT

The

authorswish

to acknowledge

he Societyof Petroleum

Engineers nd

the American

Petroleum

nstitute

or granting

permission

o use heir

publications.

Special hanksare due

J. Joseph

of Flopetrol

Johnston

and

A.

Manjnath of Reservoir nc.

for their

contributions

and

reviews

hroughout

he writing of

this book.


6/141

TABLE

OF CONTENTS

Chapter

I

Measurement

of

Rock

Relative

Permeability

.

I. Introduction. .

il.

Steady-State

ethods

.

.

A.

Penn-State

Method

B.

Single-Sample

Dynamic

Method

C.

Stationary

Fluid

Methods

D.

Hassler

Method.

E.

Hafford

Method

F.

Dispersed

FeedMethod

.

I

I

1

I

2

4

4

5

5

6

8

9

10

t 2

II I .

IV.

V .

VI .

Unsteady-

tate

Methods

Capillary PressureMethods

Centrifuge

Methods

Calculation

from

Field Data

.

References.

. . .

Chapter

2

Two-Phase

Relative

Permeabil ity

......

15

I .

I n t r o d u c t i o n . . .

. . . . . . . 1 5

II.

Rapoport

nd

Leas

..

'

15

I I I . G a t e s , L i e t z , a n d F u l c h e r . . .

. . . . . 1 6

IV.

Fa t t ,

Dykst ra ,

nd

Burd ine .

. . . . . .

16

V.

Wyl l ie, Sprangler,

nd

Gardner.

. . . . .

' .

19

VI.

Timmerman,

Corey,

andJohnson

.. . . . .20

VII.

Wahl, Torcaso,

and

Wyllie

VIII.

Brooksand

Corey

. . . .27

XIIX.

Wyllie, Gardner,

and

Torcaso

. .

.... .

.29

X.

Land,

Wyl l ie ,

Rose,

P i rson ,

nd

Boatman. . .

. . . . . .

30

XI.

Knopp,

Honarpour

et al.,

and

Hirasaki

. . .

. . .37

R e f e r e n c e s . . . . .

. . . . . . . . 4 1

Chapter

3

Factors

Affecting

Two-Phase

Relative

Permeability

.... 45

I .


. . . . . . . 4 5

il.

Two-Phase

Relative

Permeabil ity

urves

....45

n. Effects

f Saturat ion

tates

. . . . . .49

IV.

Effectsof

Rock Properties

....

... 50

V. Def in i t ion

nd Causes

f

Wettabi l i ty.

. . . . . . . .54

V I .

De te rmina t i ono fWet tab i l i t y . . . .

. . . . . . . 5 8

A. Contact

Angle Method

... 58

B .

Imb ib i t i onMe thod .

. . . . . . . 6 0

C .

Bureau f

M i n e s

Method

. . . . . . . 6 3

D. Cap i l l a r imet r i c

e thod . . .

. . . . . .63

E.

Fract iona lSur faceAreaMethod. .

. . . .64

F.

Dye

Adsorp t ion

ethod

' . . . . . .

.64

G.

DropTest

Method.

. . . . .64

H .

Me thods f

B o b e k t

a l .

. . . . . . . . 6 4

I.

Magnetic

Relaxation

Method

...64

J. ResidualSaturationMethods .. .65

27


7/141

K.

Pe r m e a b i l i t y

e t h o d . . .

. . . . . . . 6 5

L. Connate

Water-Permeabi l i ty

ethod

. . . . . . . 66

M.

Relat ive ermeabi l i ty

ethod

.. .

. . . . 66

N.

Relat ive

ermeabi l i ty

ummation

ethod

.. . . . . . .61

O.

Relat ive

ermeabi l i ty

at io

Method

.. . . . . . .67

P. Water f lood ethod . . . . . . . 68

a.

Capil lary

Pressure

ethod

....

.

68

R.

Resist iv i ty

ndex

Method

. . . . . . .

68

VII.

Factors

nfluencing

Wettability

Evaluation

.. . 68

VIII.

Wettability

Influence

on

Multiphase

Flow

. . .72

I X .

E f f e c t s f Sa t u r a t i o n

i s t o r y . . . .

. . . . . . ' 7 4

X.

Effectsof Overburden

ressure

.

...

' . . 78

K)(I .

Effects

f Porosity

nd

Permeabi l i ty. . .

. . . . . .79

XII.

Effects

of Temperature.

. .. .82

XIII.

Effects

of Interfacial

Tensionand

Density

. . .82

XIV. Ef fec ts f V iscos i ty . . ; . . . . . . ' 83

XV. Effects

of

Init ial

Wetting-Phase

aturation

... 89

XVI.

Effects

of an

Immobile

Third

Phase

. . 90

XVII.

Effects

of Other

Factors

. . .92

R e f e r e n c e s . . . . .

. . - . . . . . 9 7

Chapter

4

Three-Phase

Relative

Permeability

... f 03

I .


. . . . . . 1 0 3

i l .

D r a i n a g e R e l a t i v e Pe r m e a b i l i t y . . .

. . ' . 1 0 4

A. Leverett

andLewis ... ' . . 104

B. Corey,

Rathjens,

Henderson,

nd

Wyll ie

.. 105

C .

R e i d .

. .

1 0 7

D .

Sn e l l .

. .

l 0 g

E.

Donaldson

nd

Dean

..

. . I l0

F .

S a r e m

. . . . . . . 1 1 3

G.

Sara f

nd

Fat t

. . . . . I 15

H .

W y l l i e a n d G a r d n e r . . .

. ' l l 5

m.

Imbibit ion

Relat ive

ermeabi l i ty. . .

. . .117

A .

C a u d l e , s l o b o d , a n d B r o w n s c o m b e

. . . . . . . 1 1 7

B .

N a a r n d

W y g a l . . . . .

. . . .

1 7

C .

L a n d .

. . 1 2 0

D . S c h n e i d e r a n d O w e n s . . . .

. . . . . 1 2 3

E .

Sp r o n s e n

. ' . . 1 2 3

IV.

Probabil i ty

Models

. .123

V. ExperimentalConfirmation

.. . . .126

U\/I . LaboratoryApparatus.. .

. .127

VII.

PracticalConsiderations

or Laboratory

Tests

....

' 132

V I I I . C o m p a r i s o n o f M o d e l s

. . . ' 1 3 3

R e f e r e n c e s " " ' " " " ' 1 3 4

Appendix

Sy m b o l s .

. . . . . . .

1 3 7


8/141

Chapter

MEASUREMENT OF

ROCK RELATIVE PERMEABILITY

I.

INTRODUCTION

The

relative

peffneability

of a

rock

to each

luid

phase

can be

measured n

a core

sample

by either

"steady-state"

or

"unsteady-state"

methods. n the

steady-state ethod, a fixed

ratio of fluids is forced through he test sampleuntil saturation nd

pressure

quilibria are

established.

Numerous

echniqueshave been successfully mployed o obtain a uniform

saturation.

The

primary

concern n designing he experiment

s

to eliminateor reduce he

saturation

radient

which is

caused

y capillary

pressure

ffects

at the outflow boundary

of

the core. Steady-state ethodsare

preferred

o unsteady-state ethods y some nvestigators

for rocksof intermediatewettability,' althoughsomedifficulty hasbeen eported n applying

the

Hassler

steady-state ethod o this type

of rock.2

ln

the capillary

pressure

method,only the nonwetting

hase

s injected nto

the coreduring

the test. This fluid displaces he

wetting

phase

and the

saturations

f both

fluids

change

throughout he test. Unsteady-stateechniques

are

now employed or most laboratory

meas-

urementsof

relative

permeability.3

Some

of the more commonly used

laboratory methods

for measuring elative

perrneability

are

describedbelow.

II. STEADY-STATE

METHODS

A. Penn-State Method

This steady-statemethod

for measuring

elative

perrneability

was designedby

Morse

et

al.a and

ater modified by Osobaet aI.,5

Henderson nd

Yuster,6

Caudle

et a1.,7 nd Geffen

et al.8 The

version of the apparatus

which was describedby Geffen

et al., is illustrated by

Figure

l. In

order

to reduce end effects

due to capi llary

forces, the sample o be tested s

mounted between wo

rock sampleswhich

are similar to the test

sample. This

arrangement

also

promotes

thorough

mixing of the

two fluid

phases

before they enter the test sample.

The laboratory

procedure s

begun

by saturating he

sample with one fluid

phase

(such

as

water) and adjusting he flow rate of this phase hrough he sampleuntil a predetermined

pressure radient

s obtained. njection of

a second

phase

such

as

a

gas)

s then begun at

a

low rate and flow of the first

phase

s reducedslightly

so that the

pressure

ifferential

across he

system emainsconstant.

After an equilibriumcondition

s reached, he two flow

rates

are

recordedand the

percentage

aturationof each

phase

within the test sample

s

determined y removing he test sample

rom the assernbly nd

weighing t. This

procedure

introduces

a

possible

sourceof experimental rror,

since a small amount

of fluid may be

lost because f

gas

expansionand

evaporation.One authority

ecommendshat the core be

wgighedunderoil, eliminating

he

problem

of obtaining he

same

amount

of liquid film on

the

surfaceof the core for each

weighing.3

The estimation

of water saturation y measuring lectric

esistivity s a

faster

procedure

than

weighing the core. However, the accuracy

of saturations btained

by

a

resistivity

measurements

questionable,

ince esistivitycan be

nfluenced y fluid distributionas

well

as fluid saturations. he four-electrode ssembly

which is illustratedby Figure

I was

used

to investigate

water saturation istributionand o determine

when low

equilibrium

hasbeen

attained.Other methods

which have beenused or in situ determination

f fluid saturation

in cores nclude

measurement

f electric

capacitance, uclear

magnetic esonance, eutron

scattering,

X-ray

absorption,

gamma-ray

absorption,

volumetric

balance,

vacuum distilla-

tion, and microwave echniques.


9/141

RelativePermeabilin of

Petroleum

Reservoirs

El-ectrodes

Outl-et

Differential

Pressure

Taps

Inlet

Inlet

FIGURE

l. Three-section ore assembly.8

After fluid

saturation n the core has been determined, he Penn-State

pparatus s reas-

sembled,a new equilibrium

condition

s

established t a higher flow rate for

the second

phase,

and

fluid

saturationsare determinedas

previously

described.This

procedure

s re-

peated

sequentially

at

higher

saturationsof the second

phase

until the complete relative

permeability

curve

has

been established.

The Penn-State

method can be

used o

measure elative

permeability

at either increasing

or decreasing aturations

f the wetting

phase

and t can be applied

o both

liquid-liquid

and

gas-liquid

systems.The direction

of

saturation

hangeused

n

the laboratoryshould cor-

respond o field conditions.

Good capillary contactbetween he test sample

and the adjacent

downstream core is

essential

for

accurate

measurements

nd temperaturemust be held

constantduring the test. The

time

required or

a test to

reach

an equilibrium condition may

be I day or more.3

B.

Single-Sample Dynamic Method

This technique for

steady-statemeasurement f

relative

permeability

was developed

by

Richardson

t al.,e Josendal

t

al.,ro

and

Loomis and Crowell.ttThe

apparatus nd exper-

imental

procedure

differ from those

used

with the Penn-State echnique

primarily

in the

handling of

end effects. Rather han using a test sample

mounted

between wo core samples

(as

llustrated

by

Figure

1), the two fluid

phases

re

njected

simultaneously hrough a

single

core. End effects are minimized

by using

relatively high flow rates,

so the region of high

wetting-phase

aturationat the outlet faceof the core s small. The theorywhich was

presented

by Richardsonet al. for describing

the

saturationdistribution within

the core

may

be de-

veloped

as

follows. From Darcy's law, the

flow of two

phases

hrougha horizontal inear

systemcan be

described y the equations

-dP*,

:

Q*,

F* ,dL

k*,

A

( l )

and

,n

Q.

Fr"

dL

- d P n :

= i ^

Q )

where he subscriptswt

and

n

denote he

wetting

and

nonwetting

phases,

espectively.From

the definition of capillary

pressure,

P", it follows

that


10/141

1 . 0

o

a

0

5 1 0

1 5

2 0

2 5

D i s t a n c e

f r o m O u t f l o w

F a c e ,

c f f i

FIGURE 2.

Comparison

of saturation

gradients

at low

flow rate.e

d P . : d P . - d P * ,

These hree equations

may be

combined to

obtain

qP.

:

/Q*,

Fr,*,

9"U=\

/

o

dL

\

k*,

kn

//

where dP"/dL is the capillary

pressure

gradient

within the core. Since

dP. : dP. ds*,

dL

dS*, dL

it is

evident

that

(3)

(4)

(s)

(6)

S*,

dL

|

/Q*,

Fr*,

Q"p.\

I

: A \

k *

-

L "

/ o p . r u s *

Richardsonet

al. concluded

from experimental

evidence

hat the nonwetting

phase

sat-

uration at the dischargeend of the core was at the equilibrium value, (i.e., the saturation

at

which the

phase

becomes

mobile).

With this

boundary

condition,

Equation 6 can

be

integrated

graphically

to

yield

the

distribution

of wetting

phase

saturation

hroughout

the

core.

If the

flow rate

is sufficiently

high,

the calculation

indicates hat

this saturation

s

virtually constant

rom the

inlet

face to a

region a

few centimeters

rom the

outlet.

Within

this

region he

wetting

phasesaturation

ncreaseso the equilibrium

valueat the

outlet

ace.

Both

calculations

and experimental

evidence

show that

the region

of high

wetting-phase

saturation

at

the discharge

end

of the core

is

larger at low

flow rates than

at high

rates.

Figure

2 illustrates

the saturation

distribution

for a

low flow rate and

Figure 3

shows the

distribution at a higher rate.

\o

\.o

> { ^

/

-i-

-o-

T h e o r e t i c a l

s a t u r a t i o n

g r a d i e n t

f n f o w f a c e 1 >


11/141

Relative

Permeability

of

Petroleum

Reservoirs

1 . 0

\ o

t

I

-o-o- -o--o- :- --

:

-

J

t

T h e o r e t i c a l

s a t u r a t i o n

g r a d i e n t

I n f o w

a c "

a > l

o

5

1 0

1 5

2 0

2 5

D i s t a n c e

f r o m

O u t f l o w

F a c e ,

c t r l

FIGURE

3.

Comparison

of saturation

gradients

at high

flow

rate.e

Although

the

flow

rate

must

be

high

enough

o

control

capillary

pressure

ffects

at

the

discharge

nd

of the

core,

excessive

ates

must

be

avoided.

Problems

which

can

occur

at

very

high

rates

nclude

nonlaminar

low.

C.

Stationary

Fluid

Methods

Leas

et al.12

escribed

technique

or

measuring

ermeability

o

gas

with

the

iquid phase

held

stationary

within

the

core

by

capillary

orces.

Very

low gur

flo*

rates

must

be

used,

so

the

iquid

is not

displaced

uring

the

test.This

technique

was

modified

slightly

by

Osoba

et

al.,s

who

held

the

iquid phase

tationary

within

the

core

by

means

f

barriers

which

were

permeable

o

gas

but not

to the

liquid.

Rapoport

and

Leasr3

mployed

a

similar

technique

using

semipermeable

arriers

which

held

the gas phase

stationary

while

allowing

the

liquid

phase

o

flow.

Corey

et

al.ra

extended

he stationary

luid

method

o

a

three-phar.

yri..

by

using

barriers

which

were

permeable

o water

but impermeableo oil and gas.Osobaet

al.

observed

hat

relative permeability

to

gas

determined

by

the

stationary

iquid

method

was

in

good

agreement

with

values

measured

by

other

techniques

or

some

of

the

cases

which

were

examined.

However,

they

found

that


o

gas

determined

by

the

stationary

iquid

technique

was

generally

ower

than

by

other

methods

n

the

region

of

equilibrium

gas

saturation.

This

situation

resulted

n

an

equilibrium

gas

saturation

value

which

was

higher

than

obtained

by

the

other

methods

used

(Penn-Siate,

Single-Sample

Dynamic,

and

Hassler).

Saraf

and

McCaffery

consider

he

stationary

luid

methods

o be

unrealistic,

since

all mobile

fluids

are

not

permitted

o flow

simultaneously

uring

the

test.2

D. Hassler Method

This

is

a steady-state

method

or


measurement

hich

was

described

by

Hasslerrs

n 1944.

The

technique

was

later

studied

and

modified

by

Gates

and

Lietz,16

Brownscombe

et

?1.,"

Osoba

et

al.,s

and

Josendal

et

al.ro

The

laboratory

apparatus

s

illustrated

by

Figure

4.

Semipermeable

membranes

are

installed

at each

end of

the

Hassler

test

assembly.

These

membranes

eep

he

two

fluid

phases

eparated

t the

inlet

and

outlet

of

the

core,

but

allow

both

phases

o

flow

simultaneously

hrough

he

core.

The pressure

o

a


12/141

F L O W M E T E R

FIGURE

4.

Two-phase relative

permeability apparatus.r5

in each

luid

phase

s measured

eparatelyhrough

a semipermeable

arrier.

By

adjusting

the flow

rate of the

nonwetting

phase,

he

pressure

radients

n the

two

phases

an be

made

equal, equalizing

he

capillary

pressures

t the

inlet and outlet

of the core.

This

procedure

is designed o

provide

a

uniform saturation

hroughout

he

length of the core, even

at low

flow

rates, and thus

eliminate the

capillary end

effect.

The technique

works well under

conditions

where he

porousmedium s strongly

wet

by one

of the fluids, but

somedifficulty

has been

reported

n using the

procedureunder conditions

of

intermediate

wettability.2'r8

The

Hasslermethod s not widely usedat this time, since he data can be obtainedmore

rapidly

with other

aboratory

echniques.

E.

Hafford

Method

This steady-state

echnique

was described

by Richardson

et al.e In this

method the non-

wetting

phase

s injected directly

into the

sample and the

wetting

phase

s

injected through

a disc

which is impermeable

o the

nonwetting

phase.

The central

portion

of the semiperme-

able

disc is

isolated from the

remainder of the

disc by a small

metal sleeve, as

illustrated

by

Figure 5.

The central

portion

of the disc

is used to measure

he

pressure

n the

wetting

fluid at the inlet of the sample.The nonwetting luid is injecteddirectly into the sampleand

its

pressure

s measured hrough

a standard

pressure

ap

machined nto the

Lucite@sur-

rounding the sample.

The

pressure

difference between

he

wetting and the nonwetting

fluid

is a

measureof the

capillary

pressure

n the

sample at the

inflow end. The design

of the

Hafford apparatus

acilitates nvestigation

of

boundary

effects at the

influx

end

of the core.

The outflow boundary effect

is minimized by using

a high flow

rate.

F.

Dispersed

Feed Method

This is a steady-state

method

for measuring

elative

permeability

which was designed

by

Richardsonet al.e The technique s similar to the Hafford and single-sample ynamic meth-


13/141

Relative

Permeabilin

of PetroleumReservoirs

G A S

I

G A S

P R E S S U R E

G A U G E

P R E S S U R E

G A S

M E T E R

O I L B U R E T T E

FIGURE

5.

Hafford relative permeability

apparatus.e

ods.

In

the dispersed

eed

method,

the wetting

fluid

enters

he test sample

by first passing

through

a

dispersing

section,

which

is made

of a

porous

material

similar

to the test sample.

This

material

does not

contain

a

device or measuring

he input

pressure

f the wetting phase

as does

he Hafford

apparatus.

he

dispersing

ection

distributes

he wetting

luid

so

that

it

enters

he test sample

more

or less

uniformly

over the inlet

face.

The

nonwetting

phase

s

introduced

into radial grooves

which

are machined

nto

the

outlet face

of the

dispersing

section,

at the

unction

between

he

dispersingmaterial

and

he testsample.

Pressure radients

used for

the

tests are high

enough

so the boundary

effect at

the outlet

face

of the

core is

not

significant.

III.

UNSiuoo"-STATE

METHoDS

Unsteady-state

elative

permeability

measurements

an

be made

more rapidly

than

steady-

state measurements,

ut the mathematical

analysis

of

the unsteady-state rocedure

s

more

difficult. The

theory

developed

by Buckley and Leverettre

and extended

by

Welge2o

s

generally

used or

the measurement

f


under

unsteady-state

onditions.

The

mathematical

basis for interpretation

of the

test data

may be

summarized

as follows:

Leverett2r

combined

Darcy's

law

with a definition

of

capillary

pressure

n differential

form

to obtain

f*z

' * ; h ( * - e A p s i n o )

( 7 1

r +

I n . &

k*

Fo

where

f*,

is

the fraction

water

in

the outlet stream;

q,

is

the superficial

velocity

of total fluid

leaving

the

core;

0 is

the angle

between

direction x

and horizontal;

and

Ap is

the density

P R E S S U R E


14/141

.(#)

,(a

7

difference between displacing and displaced

fluids. For

the case of horizontal flow

and

negligible capillary

pressure,

Welge2o

howed hat

Equation

7

implies

S*.u,

-

S*z

:

f.r,

Q*

where he

subscript denotes he

outletend of the core,S*.ou

s

the averagewatersaturation;

and

Q*

is the cumulative

water njected,

measuredn

pore

volumes.

Since

Q*

and S*.,ucan

be

measured xperimentally,

",

(fraction

oil in the outlet stream)

can be determined rom

the

slopeof a

plot

of

Q*

as

a function of S*,ou.

By

definition

l , z : q , , / ( q , , * q * )

By combining his

equationwith Darcy's

law, it can be shown hat

I

f , , r :

'

t l O t

I1.,/

K..,

t

*

tr/.,*

Since

p"

and

pw

are known, the relative

permeability

ratio k.o/k.*

can be determined rom

Equation 10. A

similar expression an be derived or the caseof

gas

displacingoil.

The work

of

Welge was

extendedby Johnsonet a1.22

o

obtain

a technique

(sometimes

called he JBN method) or calculating ndividual

phase

elative

permeabilities

rom

unsteady-

state est data. The

equations

which were

derived are

k..

:

(8)

(e)

f,,,

and

k . o :

l t o o , , ,

t.z

ttr.

where I,, the ?elative nject ivity, is defined as

(

)

(12)

(

3 )

I , :

injectivity

initial

injectivity

(q*,/Ap)

(q*,/Ap)

at startof

injection

A

graphical

technique

for solving Equations 1l and 12 is i llustrated in Reference L3..

Relationships describing relative

permeabilities

n a

gas-oil

system may be obtained

by

replacing

he subscript

w"

with

"g"

in Equations I,12, and 13.

In designingexperiments o

determine elative

permeability

by the unsteady-state

ethod,

it

is

necessarv

hat:

The

pressure radient

be

large

enough o

minimize

capillary

pressure

ffects.

The

pressure

differential across he core be sufficiently small compared with

total

operating

pressure

o that compressibility ffectsare

nsignificant.

The core be homogeneous.

The driving force and fluid

properties

be held constantduring

the test.2

l .

2 .

3 .

4 .


15/141

Relative

Permeabilin of

Petroleum

Reservoirs

Laboratory

equipment

s

available or making the unsteady-state

measurements

nder sim-

ulated

eservoir

conditions.2a

In

addition to the JBN method, several alternative echniques or determining relative

permeability

rom

unsteady-stateest data

have

been

proposed.

Saraf and McCaffery2

de-

veloped

a

procedure

or obtaining elative

permeability

urves rom

two

parameters

eter-

mined by least squares it of oil recovery and pressuredata. The technique s believed to

be superior to the JBN method for heterogeneous arbonatecores. Jones and Roszelle25

developed a

graphical

technique for evaluation

of individual

phase

relative

permeabilities

from

displacementexperimentaldata which are

linearly scalable.

Chavent et al. described

a

method for

determining two-phase

elative

permeability

and capillary

pressure

rom

two

sets of displacement

experiments,

one set conductedat a

high

flow rate and the other at a

rate representative

f reservoir conditions.

The

theory

of Welge was

extendedby Sarem o

describe elative

permeabilities

n a systemcontaining hree luid

phases.

Sarememployed

a

simplifying

assumption

hat the

relative

permeability

o each

phase

depends nly on its

own saturation,

nd he

validity

of this assumption

particularly

with respect o the

oil

phase)

hasbeen

questioned.2

Unsteady-state elative

permeability

measurements

re

frequently

used to determine

he

ratios k*/ko,

ks/k", and kr/k*. The ratio k*/k" is used o

predict

the

performance

of reservoirs

which

are

produced

by waterflood

or

natural water

drive;

kr/k"

is employed o

estimate he

production

which will be

obtained

rom recovery

processes

where

oil is displaced

by

gas,

such as

gas

injection or solution

gas

drive. An important use of

the

ratio k*/k*

is in the

prediction

of

performance

of natural

gas

storage

wells,

where

gas

s injected nto

an aquifier.

The ratios

k*/ko, kg/ko,

and

kr/k*

are usually

measured n

a system

which

containsonly

the

two fluids for which

the relative

permeability

ratio is to be determined. t is

believed that

the connatewater n the reservoirmay have an influenceon kg/k.,, xpecially n sandstones

which

contain

hydratableclay mineralsand

in low

permeability

ock. For these ypes of

reservoirs t may

be advisable

o measure

*/k., n

cores

which

containan

immobile water

saturation.2a

IV. CAPILLARYPRESSURE ETHODS

The

techniqueswhich are

used

or

calculating

elative

permeability

rom capillary

pressure

data were

developed

or

drainagesituations,

where a

nonwettingphase

gas)

displacesa

wetting phase

oil

or water). Therefore

use

of the techniques

s

generally

imited

to

gas-oil

or gas-water ystems,where the reservoir s producedby a drainageprocess.Although it

is

possible

o calculate

elative

permeabilities

n a water-oil system rom capillary

pressure

data, accuracyof this technique s

uncertain;

he displacement f oil by

water

in a water-

wet rock

is an imbibition

process

ather han a drainage

rocess.

Although

capillary

pressure

echniques

re

not usually he

preferred

methods or

generating

relative

permeability

data,

the

methods

are useful

for

obtaining

gas-oil

or

gas-water

elative

permeabilities

when rock samples

are too

small for flow tests

but

large

enough or mercury

injection. The

techniquesare also useful in rock which has such ow

permeability

hat

flow

testsare impractical

and for instanceswhere capillary

pressure

ata have

been

measured

ut

a sample

of the

rock is

not available or measuring elative

permeability.

Another

use

which

has been

suggested

or

the capillary

pressure

echniques s in

estimating

kr/k"

ratios for

retrograde

gas

condensate eservoirs,

where oil

saturation ncreases

as

pressure

decreases,

with

an

initial

oil saturationwhich may be as low as zero. The capillary

pressure

methods

are recommended

or this situation because he conventionalunsteady-state

est

is not

de-

signed or very

low oil saturations.

Data obtained

by

mercury njection are customarily

used when relative

permeability

s

estimated y the capillary

pressure

echnique.The core s

evacuated

nd

mercury

which

is


16/141

9

the

nonwetting

phase)

s injected

n

measuredncrements

t

increasing

ressures. pprox-

imately

20 data

points

are

obtained

n a typical

aboratory est

designedo

yield

the complete

capillary

pressure urve,

which

is required

or calculating

elative

permeability

y the meth-

ods described

elow.

Several

nvestigators

ave developed

quations

or estimating

elative

permeability rom

capillarypressure ata. Purcell2e resentedhe equations

f s * i

l,

dS/pi

k.* ,

:

f l

t

dS/Pi

I'

ds/p!

J S o

i

k . n * , :

f l

J,

dS/pi

(

4 )

and

(

5 )

where

the

subscripts

wt

and

nwt denote he

wetting and

nonwetting

phases,

espectively,

and

n has a

value

of

2.0. Fatt and

Dykstra3o eveloped

imilar equations

with

n

equal

to

3 . 0 .

A slightly different

esult

s

obtained

by combining

he equations

eveloped y

Burdine3l

with

the

work of Purcell.2e

he resultsare

(

6 )

(

7 )

where

S,

is the total liquid

saturation.

V.

CENTRIFUGE

METHODS

Centrifuge

echniques

or

measuring

elative

permeability

nvolve

monitoring

liquids

pro-

duced

from

rock samples

which were

initially saturated

niformly

with one

or two

phases.

Liquids arecollected

n transparent

ubesconnected

o the rock

sampleholdersand

production

is monitored hroughout

he test.

Mathematical

echniques

or deriving

relative

permeability

data

from these

measurements

re described

n References

26, 27, and

28.

Although the centrifugemethodshavenot beenwidely used, heydo offer someadvantages

over alternative

echniques.

The centrifuge

methodsare

substantially

aster han the

steady-

state echniques

and they apparently

are

not subject o

the viscous

ingering

problems

which

sometimes

nterfere

with the unsteady-state

easurements.

n

the other

hand, the centrifuge

methods are

subject to

capillary

end effect

problems

and they

do not

provide

a

means or

determining

relative

permeability to the

invading

phase.

O'Mera

and

Lease28 escribe

an automated

entrifuge

which employs a

photodiode

array

in

conjunction

with a

microcomputer

o

image and

dentify

liquids

produced

during

the test.


17/141

t0

Relative

Permeabiliy

of Petroleum Reservoirs

C A M E R

C E N T R I F U G E

L I Q U I D P R O D U C T I O N

T R O B E

S P E E D

D I S K

FIGURE

6. Automated

centrifuge system.28

Stroboscopic

ights

are located

below

the

rotating

tubes

and movement

of fluid interfaces

is monitored

by

the transmitted

ight.

Fluid collection

tubes

are square

n cross

section,

since

a cylindrical

tube would

act as

a

lens

and concentrate

he light

in a narrow

band

along

the major

axis of

the tube. A

schematic

diagram of

the apparatus

s shown

by Figure

6.

VI. CALCULATIONFROM FIELD DATA

It is

possible

o calculate

elative

permeability

ratios

directly

from field

data.23In

making

the

computation

t is

necessary

o recognize

hat

part

of the

gas

which is

produced

at the

surface

was dissolved

within

the

liquid phase

n

the reservoir.

Thus;

(produced

as)

:

(free

gas)

*

(solution

as)

(18)

If

we

consider

he

flow

of free

gas

n

the reservoir , Darcy's

law

for

a radial

systemmay

be

written

9g.fr""

:

k h P . P

? . 0 9 - E - e

- w

FrB,

ln

(r./r*)

(

9 )

C O M P U T E R

o

z

LIJ

o

o

uJ

LIJ

o-

a)

o

U'

IJJ

tr

o

o

J

:

C O N T R O L L E R

S P E E D

S E T

P O I N T


18/141

l l

FIGURE

7. Calculation

of

gas-oil

relative

permeability values rom

production

data.

Similarly,

the rate of

oil flow

in

the

same system

s

where r* is the well radius and r" is the radiusof the externalboundaryof the area

drained

by the

well.

B"

and

B, are

the oil and

gas

formation

volume

factors, respectively.

The ratio

of free

gas

to oil

is obtained

by

dividing

Equation

19 by

Equation

20. lt

we

express

Ro,

cumulative

gas/oil

ratio and

R,, solution

gasioil ratio, in terms

of standard

cubic

foot

per

stock tank

barrel,

Equation

l8 implies

R o :

s . 6 t s l u * ' *

* .

Ko

ltrs

be

Thus, the

relative

permeability

ratio

is

given

by

(20)

(22)

(2t)

k"

ko

S . :

( t -

t o o , )

* , t -

s * )

_

(Ro

R.)&- ! !

5 .615

B.

F .

The oil

saturation

which corresponds

o this

relative

permeability atio may be determined

from a material

balance.

f

we

assume

here

s no

water influx, no

water

production,

no

fluid

injection,

and

no

gas

cap, the

material

balance

equation

may be

written

where minor effects

such

as change

n reservoir

pore

volume have been

assumed

egligible.

In Equation

23 the symbol

N denotes

nitial

stock tank barrels

of oil

in

place;

No is number

of

stock tank

barrelsof oil

produced;

and B",

is the ratio of the

oil volume at

initial reservoir

conditions

to oil

volume at s tandard

conditions.

If total

liquid saturation

n the

reservoir

s expressed s

(23)

s , : s * + ( r - s * ) ( \ } )

( * )

(24)

then the

relative

permeability

curve

may be

obtainedby

plotting

kr/k" from Equation 22 as

a function

of S,-

rom Equation

24. Figure 7

illustrates a convenient

ormat for tabulating

the data.

The curve

is

prepared

by

plotting

column 9

as a flnction

of column 6 on semilog

paper,

with

k/k"

on the

logarithmic

scale.

The

technique

s usefuleven

f only a

few high-

liquid-saturation

data

points

can be

plotted.

These

kr/k" values can be used to

verify the

accuracyof

relative

permeability

predicted

by empirical

or laboratory

echniques.

Poor

agreementbetween

relative

permeability determined

from

production

data and

from

laboratory

experiments

s not uncommon.

The causes

of these

discrepancies

may include

the following:


19/141

t2

Relative Permeability

of Petroleum Reservoirs

l.

The core

on which relative

permeability

s measured

may not be representative

f the

reservoir n regard

to such factors as fluid distributions,

secondary

porosity,

etc.

2. The

technique

customarily

used o computerelative

permeability

rom

field data

does

not

allow for

the

pressure

and saturation

gradients

which

are

present

n

the reservoir,

nor does

t allow for

the fact that wells may

be

producing

from

several strata which

are at various stagesof depletion.

3. The

usual

technique or calculating relative

permeability

rom field

data assumes

hat

Ro

at any

pressure

s constant

hroughout the oil

zone.

This assumption

can lead to

computational

errors if

gravitational

effects

within

the reservoir

are significant.

When relative

permeability

to water is computed from

field data, a common

source of

elror is

the

production

of water from

some source other than the hydrocarbon

reservoir.

These

possible

sources

of extraneouswater include

casing eaks, fractures

hat extend from

the hydrocarbon

zone into

an aquifer,

etc.

REFERENCES

l.

Gorinik, B. and Roebuck,

J.

F.,

Formation Evaluation

through

Extensive

Use of

Core Analysis,

Core

Laborator ies,

nc.,

Dallas,Tex.,

1979.

2.

Saraf, D. N.

and McCaffery,

F.

G.,

Two-

and

Three-Phase

elative

Permeabilit ies: Review,

Petroleum

Recovery

nstituteReport

#81-8,

Calgary, Alberta,

Canada,1982.

3. Mungan,

N., Petroleum

Consultants

td.,

personal

ommunication,1982.

4. Morse,

R. A.,

Terwill iger,

P. L.,

and Yuster, S. T., Relative

permeability

measurements

n small

samples,Oil GasJ. , 46, 109, 1947.

5. Osoba,

J.

S., Richardson,

J.

G., Kerver,

J.

K., Hafford,

J.

A.,

and

Blair,

P. M., Laboratory

elative

permeability

measurements,

rans.

AIME, 192, 47, 1951.

6.

Henderson,

J.

H.

and Yuster, S.T.,

Relat ive

ermeabil i ty

tudy,World

Oil,3,139, 1948.

7. Caudle,

B. H.,

Slobod, R. L.,

and Brownscombe, E.

R. W., Further

developmentsn

the laboratory

determination

f relative

permeability,

Trans. AIME,

192, 145,

1951.

8.

Geffen, T.

M., Owens,

W. W., Parrish,

D. R., and Morse, R.

A., Experimental

nvestigation f factors

affecting laboratory

relative

permeability

Teasurements,

Trans. AIME,

192,

99,

1951.

9. Richardson,

J.

G., Kerver,

J.

K.,

Hafford,

J.

A.,

and Osoba,

J.

S.,

Laboratory

etermination

f relative

permeability,

Trans.

AIME, 195,

187, 1952.

10.

Josendal,

V. A.,

Sandiford, B.

B., and Wilson,

J.

W., Improved multiphase

low

studiesemploying

radioactive

tracers,

Trans. AIME,

195, 65, 1952.

I l. Loomis, A.

G. and

Crowell,

D.

C., RelativePermeability

Studies:

Gas-Oiland Water-Oil

Systems,

U.S.

Bureau

of Mines Bulletin

BarHeuillr,

Okla., 1962,599.

12.

Leas, W.

J., Jenks,

L.

H., and Russell,

Charles D., Relativepermeability

o

gas,

Trans. AIME,

189,

65, 9s0.

13.

Rapoport, L.

A. and Leas,

W.

J.,

Relative

permeability

o liquid

in liquid-gas

systems,Trans.

AIME,

1 9 2 ,

3 , l 9 5 l .

14.

Corey, A. T.,

Rathjens,

C. H., Henderson,

J.

H., and Wyllie,

M. R.

J.,

Three-phaseelativeperme-

abil i ty,

J.

Pet.

Technol. ,Nov.,

63, 1956.

15.

Hassler,

G. L., U.S. Patent

,345,935,

1944.

16.

Gates,

J.

I. and Leitz,

W. T., Relative permeabilities

of

California cores

by the capillary-pressure

method,

Drilling

and Production

Practices,

American Petroleum

nstitute, Washington,

D.C. 1950,

285.

17.

Brownscombe,

E. R.,

Slobod, R. L., and Caudle, B. H., Laboratory determinationof relative perrne-

ab i l i t y ,O i l

GasJ . ,48 ,98 ,

1950 .

18.

Rose,

W., Some

problems

n

applying he Hassler elativepermeability

method,

.

Pet.

Technol.,

8, I l6l,

1980 .

19. Buckley,

S.

E.

and Leverett,

M.

C.,

Mechanism

f fluid displacement

n sands,

Trans.AIME,

146,107,

1942.

20 .

Welge 'H.J . rAs imp l i f iedmethod fo rcomput ingrecoverybygasorwate rd r ive ,Trans .A|ME,

95 ,91 ,

1952.

21. Leverett,

M.

C., Capillary

behavior n

porous

solids,

Trans. AIME,

142, 152, 1941.


20/141

13

22.

Johnson,

E. F., Bossler,

D. P.,

and Naumann,

V.

O., Calculationof relative permeability

rom

dis-

placement

xperiments,

rans. AIME,

216,310, 1959.

23.

Crichlow, H. B.,

Ed., Modern Reservoir

Engineering

A

SimulationApproaclr,

Prentice-Hall,

Englewood

Cliffs,

1977,

chap.7.

24.

SpecialCore Analysis,

Core Laboratories,

nc., Dallas,

1976.

25.

Jones,

S.

C. and R oszelle, W.

O., Graphical

techniques or

determining

elative permeability

ro m

displacement xperiments, . Pet. Technol.,5, 807, 1978.

26.

Slobod, R. L.,

Chambers, A.,

and Prehn, W. L.,

Use of

centrifuge or

determining

connatewater,

residual

oil, and capillary

pressure

urvesof small core

samples,

Trans.AIME,

192,

127, 1952.

27

Yan Spronsen,

E., Three-phase

elative

permeability

measurements

sing the

Centrifuge

Method,

Paper

SPE/DOE

10688

presented

t the Third

Joint Symposium,

Tulsa,

Okla., 1982.

28.

O'Mera, D.

J., Jr.

and Lease,W.

O.,

Multiphase

elative

permeability

measurements

singan automated

centrifuge,

Paper

SPE

12128presented

t the SPE 58th

Annual Technical

Conference

nd Exhibition,

San

Franc isco .1983 .

29. Purcell,

W. R.,

Capillary

pressures

their measurement

singmercury

and he

calculation

f

permeability

therefrom,

Trans. AIME,

186, 39. 1949.

30. Fatt, I.

and Dyksta, H .,,Relative permeability

tudies,Trans.

AIME,

192,41,

1951.

31. Burdine, N. T., RelativePermeabilityCalculations rom PoreSize DistributionData, Trans.AIME, lg8,

7 t , 1 9 5 3 .


21/141

l 5

Chapter

2

TWO-PHASE

RELATIVE

PERMEABILITY

I. INTRODUCTION

Direct

experimental

measurement

o determine

elative

permeabilityof

porous ock has

long

been

recorded

n

petroleum elated

iterature.

However,

empirical

methods or deter-

mining

relative

permeability

are

becoming

more

widely used,

particularlywith the

advent

of digital

reservoir

simulators.

The

general

shape

of the

relative

permeabilitycurves

may

be approximated

y the

following

equations:

.*

:

A(S*)';

k..,

:

B(l

-

S*)"';

where

A,

B. n. and

m are

constants.

Most

relative

permeability

mathematical

models

may

be classified

under

one

of

four

categories:

Capillary

models

-

Are

based

on the

assumption

hat a

porousmedium

consists

of

a

bundle

of capillary

ubes

of

various

diameters

ith a

fluid

path ength

onger han

he

sample.

Capillary

models

gnore

the

interconnected

atureof

porousmedia and

frequently

do

not

provide realistic

esults.

Statistical

models

-

Are also

based

on the

modeling

of

porousmedia by a

bundle

of

capillary

ubes

with various

diameters

istributed

andomly.

The

modelsmay

be described

as

being

divided

into a

large

number

of

thin

slices

by

planes

perpendicularo the

axes

of

the tubes.

The slices

are

imagined

to

be

rearranged

nd

reassembled

andomly.

Again,

statistical

models

have the

same

deficiency

of

not being able

to

model the

interconnected

nature

of

porousmedia.

Empirical

models

-

Are

based

on

proposed mpirical

relationships

escribing

experi-

mentally

determined

elative

permeabilities

nd

n

general, ave

provi{ed

he

most

successful

approximations.

Netwoik

models

-

Are

frequently

based

on the

modeling

of fluid

flow in

porousmedia

using a

network

of electric

resistors

s

an analog

computer.

Network

models

are

probably

the best

ools

for understanding

luid

flow

in

porousmedia'r'aa

The hydrodynamic

aws

generallybear

ittle use

n the

solution

of

problemsconcerning

single-phase

luid

flow

through

porous

media,

et alone

multiphase

luid

flow,

due to

the

complexity of the porous system.One of the early attempts o relateseveral aboratory-

measured

arameterso

rock

permeability

was he

Kozeny-Carmen

quation.2

his equation

expresses

he

permeability

of a

porousmaterial

as a

function

of the

productof the

effective

path ength

of the

flowing

fluid and

he

mean

hydraulic

adius

of the

channels

hrough

which

the fluid

flows.

Purcell3

ormulated

an

equation

for the

permeability

of a

porous system

n terms

of

the

porosityand

capillary

pressure esaturation

urve

of that

system

by

simply

considering

he

porousmedium

as

a bundle

of

capillary

ubes

of

varying sizes.

Several

authorsa-r6

dapted

he

relations

developed

y Kozeny-Carmen

nd Purcell

o

the

computation f relativepermeability.Theyall proposedmodelson thebasisof theassumption

that

a

porous medium

consists

of a bundle

of capillaries

n order to

apply

Darcy's

and

Poiseuille's

equations

n their

derivations.

They used

he

tortuosity

concept

or texture

pa-

rameters

o

take

nto account

he

tortuous

path

of

the flow

channels

s

opposed

o the

concept

of capillary

ubes.

They tried

to

determine

ortuosity

empirically

n order

o

obtain

a close

approximation

of

experimental

data.

II. RAPOPORT

ND LEAS

Rapoport

nd

Lease

resentedwo

equations

or

relative

permeability

o

the

wetting

phase.


22/141

16 Relative

Permeabilin

of PetroleumReservoirs

These

equations

were

basedon surfaceenergy elationships

nd the Kozeny-Carmen

qua-

tion.

The

equations

werepresented

sdefining

imits for wetting-phase

elative

permeability.

The

maximum

and

minimum

wetting-phaseelative

permeability

presented

y Rapoport

and Leas are

k.*,(max) ( l )

P. dS

f s *

Jr*,

'ot

,['*'

(tj) T#)'

and

.['*'

P.

dS

k,*,(min)

(ti

-

j;

) '

fs- fS*,

I

P . d s + R . a s

J r ' J r

(2)

where

S- represents

he minimum

rreducible

aturation f

the wetting

phase

rom

a drainage

capillary

pressure

urve, expressed

s a fraction;S*, represents

he saturation

f the wetting

phase

or which

the

wetting-phase

elative

permeability

s

evaluated,

xpressed

sa

fraction;

P. representshe drainage apillarypressure xpressedn psi and

S

representshe porosity

expressed

s a fraction.

III.

GATES. LIETZ. AND

FULCHER

Gatesand Lietzs

developed

he ollowing

expression ased

n Purcell's

model or wetting-

phase

elativepermeability:

t . _K.*r -

Fulcher

et al.,ashave

nvestigated

he influence

of

capillary number

ratio

of viscous

o

capillary

orces)

on two-phase

il-water relativepermeability

urves.

IV. FATT,

DYKSTRA,

AND BURDINE

Fatt

and Dykstrarr

developed

an expression

or relativepermeability

ollowing

the

basic

methodof Purcell or calculating he permeabilityof a porousmedium.They considered

lithology

factor

(a

correction

for

deviation

of the

path

length

from

the length

of

the

porous

medium)

to be a function

of

saturation.

They

assumed

hat

the radius

of the path

of

the

conductingpores

was

related

o the lithology

factor,

tr,

by the

equation:

ru

I $

(3 )

(4)

\ : -

ro


23/141

L7

Table I

CALCULATION OF

WETTING.PHASERELATIVE

PERMEABILITY BASED

ON

THE FATT AND

DYKSTRA EQUATION

Area from 0

S*,

Vo

P", cm Hg l/P"'],

(cm

Hg)-t to

S*,

in.2

k.*,, Vo

100 4.0 0.0156

9 0 4 . 5 0 . 0 1 1 0

80 5.0 0.0080

'70

5.5 0.0060

60 6.0 0.0046

s0 6.7 0.0033

40 7.s 0.0024

30 8.7 0.00 5

20 13.0 0.0005

'

7.88/11.25 100 70.0.

"

5.54111.25 l0O 49.2.

n . 2 5

7 . 8 8

5 .54

3 .80

2.49

t . 5 0

0.75

0.30

0.20

100.0

70.0,

49.2b

33 .8

2 2 . 1

1 3 . 3

6 . 1

2 . 7

0 .4

where r represents

he radius of a

pore,

a and b representmaterialconstants,

and }, is a

function of saturation.

The equation or the wetting-phaseelative

permeability,

.*,, reported

y

Fatt

andDykstra

is

f t* '

ds

t -

,

J n

P 2 ( l

+

b )

K.*,

:

l.r

dS

Jo

P2(

*

b)

agreementwith

observed

data when b

:

(5)

r/r,

reducing

att and Dykstra found

good

Equation

5 to

They stated hat their

equation

it

their own data as well

as the data of

Gatesand

Lietz

more

accurately han other

proposed

models.

The

procedure

or

the calculation of relative

permeability

from

capillary

pressure

data is

illustrated

by Table I and

the

results

are shown in Figures

I

and 2.

Burdine'3

reported

equations or computing relative

perrneability

or

both the wetting

and

nonwetting

phases.

His equations an be shown

o

reduce

o a form similar

to thosedeveloped

by

Purcell. Burdine's

contribution is

principally

useful in handling

tortuosity.

Defining the

tortuosity

factor

for a

pore

as L when the

porous

medium is saturated

with

only one

fluid

and using the symbol tr*, for

the

wetting-phase

ortuosity

factor when

two

phases

are

present,

a tortuosi ty ratio can be defined

as

ft*' ds

Jo

P:

TF

(6)

T

tr.*,

;

(7)


24/141

l8

Relative

Permeabilitv

of Petroleum

Reservoirs

9

I

| 7

P o l

(cm

Hg)

6

5

4

3

2

I

oo'

lo 20

40 50

60

70

80

S w +

FIGURE

1. Capillary

pressure

s a

function

of

water saturation.

/'*'

{^,*,)' ds/(\)' (P.)'

kr*,

/ '0r,1^;'1r.y'

fS*'

t

ds/(P.)r

k.*t

:

(tr.*.) '

r l

t

ds/(p")l

In a similar

fashion,

the

relative

permeability o

the

nonwetting

phase

can

utilizing

a

nonwetting-phase

ortuosity

atio,

tr,,*,,

then

Burdine

has

shown

that

( 9 )

be expressed

( 0)

( 8 )

If tr

is

a

constant

or the

porous

medium

and

tr,*t dependsonly

on

the

final saturation,

hen

f l

I

dst1e.)'

^

J S * t

k .n * , : ( t r rn * , ) '

J"

ds/(P.)2

S * , -

S -

Arwt

-

( l

)

1 - S -


25/141

l9

r60

r50

r40

r30

t20

l l

roo

90

70

60

50

40

30

20

t o

o5

lo 20 30 40 50 60 70

Sw

-+

il,;yul}:

Reciprocal

f

(capillary

ressure)rs

a function

f

water

where

S- represents

he

minimum

wetting-phase

aturation

rom

a capillary-pressure

urve.

The relative

perrneability

is assumed

o approach

zero

at this

saturation.

The

nonwetting

phase ortuosity

can be

approximated

by

\ - ^ . . , . :

.

S n * t - - S '

(12 )

r n w t

l - s * - s "

where

S.

is the

equilibrium

saturation

o the

nonwetting

phase.

The

expression

or the

wetting

phase

Equation

9)

fit the

data

presentedmuch

better

han

the expression

or

the

nonwetting

phase

Equation

10).

V. WYLLIE,

SPRANGLER,

ND

GARDNER

Wyllie and Spranglertz reported equationssimilar to those presentedby Burdine for

computing

oil

and

gas

relative

permeability.

Their

equations

can

be expressed

as

follows:

I t

Pc3

|

(CmHq i3

fs"

k,,,:

iil'

J

os"rp;

/'

or",rl

(

3 )


26/141

I

E

Relative

Permeabilin

of Petroleum

Reservoirs

o

A

I

a

WYLLIE

ond

SPANGLER

GATES

ond

LIETZ

i l | | t l

BEREA

NO.4

FIGURE 3. Reciprocalof (capil larypressure)r s a functionof saturationor normalized

data . rT

k,*

r

r+"

)' !Y

S*,/

/ '

or",r3

where

S- represents

he

lowest

oil

saturation

at which

the

gas

phase

s

discontinuous:

S-

:

( l

-

S".).

The above equations for oil and gas relative permeabilitiesmay be evaluated when

a

reliable

drainage

capillary pressure

curve

of

the

porous

medium

is

available,

so

that

a

plot

of

llP"2

as a function

of

oil

saturation

an be

constructed.

Obviously,

reliable

values

of

S-

and

So.

are

also needed

or

the oil

and

gas

relative

permeability

evaluation.

Figure

3 shows

some

examples

of

llP.2

vs.

saturation

urves.rT

Wyllie

and

GardnerrT

developed

equations

or

oil

and

gas

relative permeabilities

n

the

presence

of

an ineducible

water

saturation,

with

the water

considered

as

part

of the

rock

matrix:

f t ' ds*

k,.:(H),

+*

.s;

Jr*,

Pi

f '

ds*

k,,

(*)'

f*

'6)

Jr*,

Pi

where Sl representsotal liquid saturation.Note that theseequations

may

be

applied

only

when

the water

saturation

s

at the

irreducible

evel.

VI.

TIMMERMAN,

COREY,

AND

JOHNSON

Timmermanr8

uggests

he following

equations

ased

on

the

water-oil

drainage

apillary

pressure,

or

the

calculation

of low

values

of

water-oil

relative

permeability.

( t 4 l

o.


27/141

2l

Wetting-Phase

rainage

Process:

k.o

:

S.

k.*

:

S*

Wetting-Phasembibition Process:

kro

:

So

Injection Curve

Injection

Curve

Trap-Hysteresis

urve

Injection

Curve

k.o

:

So

(20)

Coreyre

combined

he work

of Purcell3

and

Burdiner3

nto a

form that has considerable

utility and

s widely

accepted

or its simplicity.

It requires

imited

nput data

since

esidual

saturation

s the

only

parameter eeded o develop

a

setof relative

permeability urves)and

it is fairly accurate

or consolidated

porous

media

with

intergranular

porosity.

Corey's

equations

are often

used

or calculation

of

relative

permeability n reservoirs

ubject o

a

drainage rocess r externalgasdrive. His methodof calculationwas derived rom capillary

pressure

oncepts

nd he

fact that

for certain

cases,

/P"2

s

approximately

linear

unction

of the effective

saturation

ver

a considerable

angeof saturations;

.e. llP"2

:

C

[(S"

-

S".)/(1

-

S",)]

where C

is a constant

and

S"

is an oil saturation

reater

han S.,,.

On the

basis

of this observation

nd

he

indingsof

Burdiner3

oncerning

he nature

of the ortuosity-

saturation

unction, the

following

expressions

ere derived:

fl'"H.1"

LTFI

InjectionCurve

InjectionCurve

Injection

Curve

lnjection

Curve

( t7)

f[Hl"

LrFl

(

8 )

[l'"H"

LTFj

[[H]"

Lrsl

( l e )

(2r)

(22)

(23)

o :

k,o

[ S '

-

S ' * l o

L

-

s *J


28/141

22

Relative Permeability

of Petroleum

Reservoirs

where

S'- s the

total liquid

saturation

nd equal

o

(l

-

Sr);

S-

is

the lowest

oil saturation

(fraction)

at which

the gas phase

s discontinuous;

nd

Sr* is

the residual

iquid

saturation

expressed

s

a

fraction.

Corey

and Rathjens2o

tudied

he

effect of

permeability

ariation

n

porous

media

on the

value

of the

S- factor

in

Corey's equations.

They confirmed

hat S,,, s

essentially

qual

o

unity for uniform and isotropicporous media;however, valuesof S,, were found to be

greater

han

unity when

there was

stratification erpendicular

o the direction

of flow

and

less

han unity in

the

presence

f stratification arallel

o the

directionof

flow. They

also

concluded

hat oil relative permeabilities

were

less sensitive

o stratification

han

the

gas

relativepermeabil i t ies.

The gas-oil


equation s

often used

for testing,

extrapolation,

and

smoothing

xperimental

ata. t is

alsoa

convenient xpression

hat

may be

used n computer

simulation

of reservoir

performance.

Corey's

gas-oil


atio

equation

can be solved

f only

two

points

on

the k,r/k,., s.

S* curve are

available.However,

he algebraic

olution

of the k,g/k..,

quation

when two pointsare available s very tediousand the graphicalsolution hat Corey offers

in his

original

paper

equires

engthygraphical

onstruction

swell

asnumerical

omputation.

Johnson2r

as offered

a

greatly

simplified

and

useful method or

determination

f Corey's

constant.

Johnson onstructed

hree

plots

by

assuming

alues

of Sr*,

S,,, and k.s/k..,

y

calculating

the

gas

saturation,

1

-

S,_),

sing

Corey's equations.The

calculation

was

carriedout for

various

Sr*

and S- combinations

nd for k.s/k,o alues

of

l0 to 0.1,

1.0

to 0.01, and

0. I

to 0.001. Johnson's

graphs

may be

used o

plot

a

more

completek.g/k,,,

urve based

on

limited

experimental

ata. The

spanof

the experimental

atadetermines

which

of the

three

figures

should

be selected.

The suggested rocedureor k.g/k., alculation,based n Corey's equation, s as follows:

l.

Plot

the experimental

.r/k,"

vs.

S, on semilog

paper

with

k,*/k,oon

the logarithmic

scale.

2.

From the experimental

ata

determine

he

gas

saturation

t

k.r/k,o

equal o 10.0

and

0.1,

1.0 and

0.01,

or 0.1 and

0.001.

The

isted

ai rs

of

values

orrespond

o Figures

4,5,

and 6 of

Johnson's

ata, respectively,

nd

the range

of the

experimental

ata

dictates

which

figure is

to be

employed.Note

that if

the data do not

span

he entire

permeability

atio

interval

of 10.0

o

1.0,

Figure4

may not

be employed

irst; instead

Figure

5 with

the k,*/k.o

nterval

of 1.0

to 0.01 or Figure

6

with

the k.*/k,.,

nterval

of

0.10 to 0.001may be used irst . )

Enter

the

appropriate

Figure

(4,5,

or 6)

using the gas

saturations

orresponding

o

the

pair

of k.r/k.o

values

selected

n step2.

Pick

a unique

S.* and

S- at the intersection

f the

gas

saturation alues;

nterpolate

if necessary.

5. Using

these

S.*

and

S-

values

and

employing

the two

other figures

of Johnson,

determine wo

more gas

saturation

alues

and the

k,*/k," ratio

indicated

on

the axes

of each igure.

6.

Add

these

points

o the

experimental

lot

for obtaining

he relativepermeability

atio

over

the

region

of interest.

This

procedure

rovides

alues

f

gas

saturation

t

k.*/k.o

atios

of

10.0, 1.0,

0.10,

0.01,

and

0.001, which

are

sufficient

o

plot

an expanded

.s/k.o

urve.

It

should be noted

that if

the

data cover

a

wide

range

of

permeability

atios,

multiple

determinations

f

Sr* and

S- can

be made. f

the calculated

alues

differ from

the

exper-

imental

data,

he discrepancy

ndicates

hat there s

no single

Corey curve which

will

fit

al l

3 .

4 .


29/141

23

o

t l

I

o)

J

I

o)

U)

20

S n

%

k r g

/ k r o

=

0 . 1

O

FIGURE 4.

Corey equation

constants.2l

the

points;

an averageof the values or

each constantshould yield

a better

curve fit.

Figure

7 illustrates he graphical echniqueof Johnson.

Corey's equations or

drainageoil and

gas

relative

permeabilities

nd

the

gas-oil

elative

permeability

atio in

the simplest orm

are as

follows:

and

they

are related through

I

k.o

:

(s".)o

k . r : ( l - S " . ) 2 x ( l - S 3 " )

k.. k.

honarpour, d., koederitz, l. and harvey, a. h. - relative permeability of petroleum reservoir

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