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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
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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
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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'
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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.
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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.
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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 .
I n t r o d u c t i o n . . .
. . . . . . . 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
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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 .
I n t r o d u c t i o n . . .
. . . . . . 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
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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.
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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
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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.
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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
relative permeability
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
relative permeability
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
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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-
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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
relative permeability
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
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.(#)
,(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 .
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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
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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.
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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
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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:
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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.
-
8/20/2019 Honarpour, D., Koederitz, L. and Harvey, A. H. - Relative Permeability of Petroleum Reservoir
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 .
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8/20/2019 Honarpour, D., Koederitz, L. and Harvey, A. H. - Relative Permeability of Petroleum Reservoir
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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.
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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
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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)
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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 -
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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 )
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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.
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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
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8/20/2019 Honarpour, D., Koederitz, L. and Harvey, A. H. - Relative Permeability of Petroleum Reservoir
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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
relative permeability
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
relative permeability
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 .
-
8/20/2019 Honarpour, D., Koederitz, L. and Harvey, A. H. - Relative Permeability of Petroleum Reservoir
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.