John R. Fanchi - Principles of Applied Reservoir Simulation, Second Edition
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Chapter
28
Rock
and
Fluid
Models
The
interaction between reservoir rock
and in
situ
fluids
is
modeled with
relative permeability
and
capillary pressure data. This chapter defines
the
three-
phase
oil
relative permeability model used
in
WINB4D
and its use in
transmissibility
calculations.
It
then presents additional details
of the fluid
property model
after
reviewing
a few
commonly used
thermodynamic
terms.
28.1
Three-Phase Relative Permeability
Relative permeability curves
are
some
of the
most critical data
in the
simulator
because relative permeability curves
can
have
a
significant
impact
on
simulator
performance. Relative permeability curves
are an
important part
of
the
algorithm that
is
used
to
model
the
interaction between reservoir rock
and
fluids.
Unfortunately, relative permeability curves
are
often
among
the
missing
or
poorer quality data.
Relative
permeability data
are
affected
significantly
by
alterations
in
wettability
conditions
in the
core.
Ideally,
the
relative permeability data should
be
measured
in the
laboratory under
the
same conditions
of
wettability that exist
in
the
reservoir.
One
method
of
approaching this ideal
is to use
preserved,
"native state" core samples.
"Native
state"
core samples
are
cores that
are
drilled using crude
oil or
a
special coring
fluid
designed
to
minimize wettability alterations.
The
cores
are
sealed
at the
well site
to
minimize exposure
to
oxygen
or
drying
and
then
preserved
until
ready
to
undergo
flow
testing
in the
laboratory. However, this
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process
is
expensive
and
most relative permeability data
are
obtained
on
restored
state cores
in the
laboratory.
In
principle, three-phase relative permeability should
be
used when oil,
water,
and gas are flowing
simultaneously.
As a
practical matter,
the
difficulty
of
accurately measuring three-phase relative permeabilities
often
makes
their
use
meaningless.
It is
often
sufficient
to
work
with
the
two-phase relative
permeability curves only.
Despite
their
shortcomings,
it may be of
interest
to
perform
a
simulation
using
a set of
three-phase relative permeability curves.
For
this
case,
WINB4D
contains
an
option
for
computing
a
three-phase
oil
relative permeability curve
using
water-oil
and
gas-oil relative permeability curves.
As
with most
calcula-
tions
of
this type,
we
assume:
a.
The
water relative permeability curve
(k^
obtained
for
a
water-oil
system depends only
on
water saturation,
and
b. The gas
relative permeability curve
(k
rg
)
obtained
for a
gas-oil
system depends only
on gas
saturation.
The
validity
of
these assumptions depends
on
such
factors
as
wettability
and
degree
of
consolidation. Given
the
above assumptions,
£
w
and
k
rg
for
water-
oil and
gas-oil systems, respectively,
are
also
valid
for a
water-gas-oil system.
The
three-phase
oil
relative permeability
k
ro3
is
calculated
as
,
(^row
+
^rw)
(krog
+
*rp
,*
_,_
L
\
k^
=
-
1
£-
-
(k
rw
+
k
rg
)
(28J)
*row
where
k
row
- oil
relative permeability
for
water-oil system
k
mg
= oil
relative permeability
for
gas-oil system
k*
row
=
oil
relative permeability
for
water-oil system evaluated
at
the oil
saturation corresponding
to
irreducible water
saturation
Equation
(28.1)
is
based
on the
work
by
Stone
[1973],
and it
corresponds
to
Model
II of
Dietrich
and
Bonder
[1976].
For a
discussion
of
alternative models
of
three-phase
oil
relative permeability,
see
Blunt
[1999].
When
the
three-phase calculation
is
activated,
the
user must
be
sure
the
input
water-oil
and
gas-oil relative
permeability
curves
are
realistic.
For
example,
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if
we
write irreducible water saturation
as
S
wr
,
the
relative permeability constraint
k
mw
(1
-
S
wr
)
=
k
rog
(S
0
+
S
w
=
1.0)
must
be
satisfied since
S
g
= 0 in
both cases.
28.2 Transmissibility
The
simulator
offers
no-flow boundary
conditions,
which lets
you
stop
flow
between
specified
gridblocks
in
chosen directions.
The
no-flow conditions
are
implemented
by
setting
transmissibilities
at
boundary interfaces
to
zero.
The
Transmissibility
Modifications section
in
Chapter
24.3.3
describes
the
directional
conventions
for
transmissibility
in the
model,
Flow
between neighboring blocks
is
treated
as a
series
application
of
Darcy's
law.
A
transmissibility term
at the
interface between
two
blocks
is
defined
using
the
product
of
average values
of
relative permeability
k^
of
phase
0,
absolute permeability
K of
each block
at the
interface,
and
cross-sectional
area
A
c
of
each block
at the
interface, divided
by the
product
of
viscosity
jl
e
and
formation
volume
factor
B%
of the
phase
in
each block.
The
transmissibility
to
each phase
is
determined using
a
harmonic average calculation
of the
product
of
absolute permeability
and
cross-sectional area
at the
interface between
neighboring
blocks.
An
arithmetic average
of
phase viscosities
and
formation
volume
factors
is
used.
The
average relative permeability
is
determined using
an
upstream weighted averaging technique.
The
resulting Darcy transmissibility
is
A'
-"•
Air
rt(
upstream)
B
u
2(KA
C
and the finite
difference
transmissibility
A
tii
.
]/2
for
phase
<!
between block
/
-1
and
block
/
used
in the
simulator
is
+
*-t
;
I/O
•*•••
a
;..
i
where
they,
k
indices
are
suppressed
and the
spatial
differences
are
Ax
—
x
(
—
x
f
_
}
,
Ax =
X
M
-
x
i
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Similar
definitions
of
transmissibility
apply
in all
three coordinate directions.
28.3
Terminology
and
General Comments
Some
of the
fluid
property terms that
are
most
frequently
used
in
black
oil
simulation
are
defined here.
Density
Density
is
defined
as the
mass
of a
substance divided
by the
volume
it
occupies.
The
density
of a fluid
depends
on the
pressure,
temperature,
and
composition
of the fluid.
Composition
The
composition
of a fluid
depends
on
whether
the fluid
consists
of
a
pure
component, such
as
water
or
methane,
or is a
mixture.
For
example, petroleum
and
in
situ water
are
mixtures. Petroleum
is a
mixture
of
hydrocarbon com-
pounds,
and in
situ water usually contains dissolved solids,
such
as
salt,
and
may
contain dissolved
gases
such
as
methane
and
carbon dioxide.
The
composition
of a fluid is a
list
of the
components contained
in the fluid.
The
relative amount
of
each component
in a
mixture
is
defined
as the
concentration
of the
component. Concentration
may be
expressed
in a
variety
of
units, such
as
volume fraction, weight
fraction, or
molar fraction.
It is
important
to
know
the
units
associated
with
the
composition.
If
the
concentration
units
are not
clearly expressed
in
a fluid
report, they should
be
determined
before
use in
calculations.
It is
common
to find
composition expressed
in
mole
fractions.
The
symbols
{*„>>„
z
t
}
are
often
used
to
denote
the
mole
fraction
of
component
i in the oil
phase,
gas
phase,
and
wellstream
respectively.
The
equilibrium
K
value
is a
measure
of the
amount
of
component
i in
the gas
phase relative
to the oil
phase.
It is
defined
as the
ratio
K,
=
y./x
t
If
component
i
exists entirely
in the oil
phase,
then^,
is 0 and
K
{
is 0.
Conversely,
if
component
i
exists
entirely
in the gas
phase, then
x
t
is 0 and
K
f
approaches
infinity.
Thus,
the
equilibrium
K
value
for
component
/
may
range
from
0
to
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infinity.
It
should
be
noted that these concepts apply
to
both hydrocarbon
components
and any
other
distinct
molecular species, such
as
carbon dioxide
and
nitrogen.
Pressure
The
average pressure
on a
surface
is the
total normal
force
applied
to the
surface
divided
by the
area
of the
surface.
The
normal
force
is the
component
of
the
force that
is
acting perpendicularly
to the
surface.
Consider
a fluid in the
pore space
of a
rock.
The
pressure
at any
point
in
the
fluid
is
equal
in all
directions.
If the
fluid
is at
rest
in the
pore space,
the
pressure
is
equal
at all
points
in the fluid at the
same
depth.
Pascal's
law
says
that
pressure applied
to an
enclosed
fluid
will
be
transmitted without
a
change
in
magnitude
to
every point
of the fluid and to the
walls
of the
container.
Temperature
Temperature
is a
measure
of the
average kinetic energy
of a
system.
Several temperature scales
are in
use.
The
most commonly used temperature
scales
are the
Fahrenheit
and
Celsius
scales.
The
relationship
between
these
scales
is
where
T
c
and
T
F
are
temperatures
in
degrees Celsius
and
degrees Fahrenheit
respectively.
Applications
of
equations
of
state require
the use of
absolute temperature
scales.
Absolute temperature
may be
expressed
in
terms
of
degrees Kelvin
or
degrees Rankine.
The
Kelvin scale
is
related
to the
Celsius scale
by
T
K
=
T
c
+ 273
where
T
K
is
temperature
in
degrees Kelvin.
The
Rankine scale
is
related
to the
Fahrenheit
scale
by
T
R
=
T
F
+ 460
where
T
R
is
temperature
in
degrees Rankine.
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Intensive
and
Extensive
Properties
Pressure, temperature,
and
density
are
examples
of
intensive properties.
An
intensive property
is a
fluid
property which
is
independent
of the
amount
of
material.
For
example,
if a
cubic cell
of gas in an
equilibrium state
is
divided
into
two
halves
by a
vertical partition,
the gas in
each half
of
the
cell should have
the
same pressure
and
temperature.
By
contrast,
the
mass
and
volume
in
each
half
will
be one
half
of the
original mass
and
volume. Mass
and
volume
are
examples
of
extensive properties.
An
extensive property
is a
property that
depends
on the
amount
of
material.
Compressibility
If
the
surface
of an
object
is
subjected
to an
external
force,
the
resulting
pressure applied
to the
object
can
change
the
volume
of
the
object. Compressibil-
ity
is a
measure
of the
volume change resulting
from the
applied pressure.
The
fractional
volume change
AF/Fof
an
object
may be
estimated
from
V
where
c is the
compressibility
of the
object,
AP
is the
pressure
applied,
and the
minus sign implies that
an
increase (decrease)
in
applied pressure results
in a
decrease (increase)
in the
volume
of the
object.
Formation Volume Factor
Formation volume
factor
is
defined
as the
volume occupied
by a fluid
phase
at
reservoir conditions divided
by the
volume occupied
by the fluid
phase
at
standard conditions.
The fluid
phase volume
may
change substantially
as
pressure
and
temperature change.
Ordinarily
the
volume
of a fluid
with constant composition will increase
as
the
applied pressure
and
temperature decrease.
The
behavior
of
petroleum
is
made more complex because
it is a
mixture
and can
experience
a
change
in
composition
as
temperature
and
pressure change.
For
example,
a
barrel
of oil
at
reservoir
conditions
(relatively high pressure
and
temperature) will shrink
as
the
barrel
is
brought
to the
surface (relatively
low
pressure
and
temperature).
The
shrinkage
is
associated with
the
release
of
solution
gas as the
pressure
and
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temperature
of
the
oil
decline
from
reservoir
to
surface conditions. Consequently,
measurements
of the
change
in
volume
as a
function
of
pressure
are
desirable,
especially
for the oil
phase.
The
determination
of gas
formation
volume
factor
provides
an
interesting
contrast
to the
determination
of oil
formation
volume factor.
Gas
formation
volume
factor
is
often
determined with reasonable accuracy using
the
real
gas
equation
of
state
PF=
ZnRT where
n is the
number
of
moles
of gas in
volume
Fat
pressure
P and
temperature
T. The gas
compressibility factor
Z
equals
one
if
the gas is an
ideal gas.
For
real gases,
Z
*
1 for
most
pressures
and
tempera-
tures,
Specific Gravity
Specific
gravity
is
defined
as the
density
of a fluid
divided
by a
reference
density.
Gas
specific gravity
is
calculated
at
standard
conditions
using
air
density
as the
reference density.
The
specific gravity
of gas is
defined
by
g
M
a
(air)
29
where
M
a
is
apparent molecular weight. Apparent molecular weight
is
calculated
as the
mole
fraction
weighted average
M
=
Yy.M.
a
L-t
J
i
i
i=\
where
N
c
is the
number
of
components,^,
is the
mole
fraction of
component
/,
and
M
{
is the
molecular weight
of
component
i.
Oil
specific gravity
is
calculated
at
standard conditions using
fresh
water
density
as the
reference
density.
Oils
are
often
characterized
by
specifying their
API
gravity, which
is
related
to oil
specify
gravity
y
0
at
standard temperature
and
pressure
by the
equation
To
Heavy oils
are
oils with
a
relatively large
J
0
and a
relatively
low API
gravity.
Heavy
oils typically
do not
contain much
gas in
solution.
By
contrast, light oils
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have
a
relatively
small
y
o
and a
correspondingly large
API
gravity.
Light
oils
typically
contain
a
large amount
of gas in
solution.
Gas-Liquid
Ratio
The
gas-liquid ratio
is
defined
as
the
volume
of gas
divided
by the
volume
of
liquid,
usually
oil or
water.
The gas
volume
and
liquid volume should
be
expressed
at the
same temperature
and
pressure.
Viscosity
The
coefficient
of
viscosity
is a
measure
of
resistance
to flow of the fluid.
In
general, gases have
a
lower viscosity than liquids.
The
inverse
of
viscosity
is
called
fluidity
[McCain,
1990].
Thus,
a fluid
with
a
large viscosity
has a low
fluidity.
The
relationship
between viscosity
and
shear rate defines
the
rheology
of
the fluid. If fluid
viscosity
is
independent
of flow
rate,
the fluid is
referred
to
as a
Newtonian
fluid. If fluid
viscosity depends
on flow
rate,
the fluid is
considered
a
non-Newtonian
fluid.
Two
types
of
viscosity
may be
specified: dynamic viscosity
p,
and
kinematic
viscosity
V.
They
are
related
by the
expression
[I
=
p
v
where
p is the
density
of
the fluid.
Dynamic viscosity
|i
is
used
in
Darcy's
law to
calculate
the
rate
of fluid
movement
fluid flow in
porous media. Typically,
the
unit
of
dynamic
viscosity
Ji
is
centipoise.
If fluid
density
p has the
unit
of
g/cc,
then
kinematic
viscosity
v has the
unit
of
centistoke. Thus,
1
centistoke equals
1
centipoise divided
by 1
g/cc.
Reservoir
fluid
properties (PVT data) include
fluid
viscosities,
densities,
formation
volume factors,
gas
solubilities, etc. These data
are
usually obtained
by
laboratory analyses applied
to fluid
samples taken
from
the
reservoir. They
are
sketched
in
ChapterB.
Differential
to
Flash Conversion
Laboratory
reservoir
fluid
analyses generally
provide
data
from
both
a
differential
liberation experiment
and a flash
experiment approximating
field
separator conditions.
The
differential
and flash
liberation data
can be
signifi-
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cantly
different
for
some oils.
The
actual behavior
of the
production
process
is
some
combination
of the
differential
and
flash
processes.
The
assumption
normally
made
in
preparing
PVT
data
for use in a
black
oil
simulator
is
that
the
differential
liberation data represent
the
process occurring
in the
reservoir
and
the
flash
data represent production
to
stock tank conditions. Thus,
for use in the
simulator,
the
differential
liberation data should
be
corrected
to
flash
values
at
field
separation
conditions. This procedure
is
described
in the
literature
[Arnyx,
et
al,
1960;
Moses,
1986]
and is
summarized below.
Physical
property data obtained
from
a
testing laboratory
for a
black
oil
system will generally
be a
differential liberation study
coupled
with
a
separator
study. Most reservoir simulators require that these data
be
converted
to
flash
form
so
that
the
effects
of the
surface separation
facility
are
included. Conversion
of
the
data
is
restricted
to oil
formation volume factor
and
solution
gas-oil
ratio
data.
If the
separator
B
0
and
R
so
are
known,
the
conversion equations are:
"odbp
and
B
ofbp
"odbp
where subscripts
are
defined
as:
d
=
differential liberation data
/
=
flash
data
bp
-
bubble point
28.4
Extrapolating
Saturated
Curves
Guidelines
for
extrapolating
PVT
data
to
pressures
above
the
measured
saturation pressure
are
presented below.
i.
The
B
g
versus pressure curve
is
strongly non-linear
and an
extrapolation
of
this curve
to
small
B
g
values
at
high pressures
can
result
in
errors.
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most natural gases,
the
relationship
\!B
g
versus pressure
will
be
very
nearly
linear, especially
at
moderate
to
high
pressures. Plotting
l/B
g
versus
pressure
and
extrapolating
to
PMAX should provide more realistic
values
of
B
g
at
higher pressures. Interpolating
B
g
using
l/B
g
versus
pressure substantially improves material balance.
2.
Once
the
B
g
versus
P
curve
is
fixed,
R
so
versus
P and
B
0
versus
P
curves
must
be
extrapolated
so as to
avoid
a
negative
oil
compressibility
being
calculated over
any
pressure increment.
To
ensure that negative
oil
compressibilities will
not be
calculated
by the
program,
the
following
test
should
be
used.
For any
pressure increment
P,
to
P
2
,
where
P
2
>
P
{
,
the
following
relationship should hold:
0
*
-
(B
02
-
B
ol
)
+
v
;
.
5.615
where
the
units
of
B
0
,
B
g
,
and
R
so
are
RB/STB,
RCF/SCF,
and
SCF/ STB,
respectively. Note that this test applies only
to the
saturated
oil PVT
data.
3
.
The
above concepts also apply
to the
water
PVT
data. However,
for
most
simulations,
it can be
assumed that
R
sw
=
0.0,
thus
-
AB
W
/B
W
AP
approxi-
mates water compressibility.
28.5
Gas PVT
Correlation Option
Basic
Gas
Properties
Following
Govier
[1975],
real
gas
Z-factors
are
computed using
the
Dranchuk,
et
al.
[1974]
representation
of the
Standing-Katz
Z-factor charts
[1942].
This representation employs
the
Benedict-
Webb-Rubin
[1940]
eight-
parameter equation
of
state
to
express
the
Z-factor
as a
function
of
pseudo-
critical temperature
T
r
and
pseudo-critical pressure
P
r
,
thus
Z =
Z(P
r
,
T
r
)
(28.2)
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