John R. Fanchi - Principles of Applied Reservoir Simulation, Second Edition
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22
Principles
of
Applied
Reservoir Simulation
atmosphere.
Laboratory
fluids
should
also
be at
reservoir
conditions
to
obtain
the
most reliable measurements
of
wettability.
Based
on
laboratory tests, most
known
reservoirs have
intermediate
wettability
and are
preferentially
water wet.
3.2
Capillary
Pressure
Capillary
pressure
is the
pressure
difference
across
the
curved
interface
formed by two
immiscible
fluids in a
small capillary tube:
PC
=
P
m
-
P
w
(3.4)
where
P
c
capillary pressure [psi]
P
nw
pressure
in
nonwetting
phase [psi]
P
w
pressure
in
wetting
phase
[psi]
Capillary Pressure Theory
Equilibrium between
fluid
phases
in a
capillary tube
is
satisfied
by the
relationships/ores
up
=
force down. These forces
are
expressed
in
terms
of the
radius
r of the
capillary tube,
the
contact angle
6, and the
interfacial tension
0,
The
forces
are
given
by
force
up -
IFT
acting around perimeter
of
capillary tube
=
O
cos 0
x
2Kr
and
force down
=
density gradient difference
x
cross-sectional
area
x
height
h of
capillary
rise in
tube
The
density gradient
F
is the
weight
of the fluid per
unit length
per
unit cross-
sectional area.
For
example,
the
density gradient
of
water
T
w
is
approximately
0.433
psia/ft
at
standard conditions.
If we
assume
an
air-water system,
the
force
down
is
force
down
=
(T
w
-
T
air
)Ttr
2
h
where
the
cross-sectional
area
of the
capillary tube
is
Tlr
2
.
Capillary pressure
P. is
defined
as the
force/unit area, thus
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P
c
=
force
up
I
Tlr
2
=
force
down
/
Tlr
2
.
Capillary
Pressure
and
Pore
Radius
Expressing
capillary pressure
in
terms
offeree
up per
unit area gives:
n
27crocos0
2ocos0
PC
=
_
-
-
=
-----
(3>5)
71
r
4
r
where
r
pore radius [cm]
a
interfacial
(or
surface) tension
[mN/m
or
dynes/cm]
6
contact angle [degrees]
Equation
(3.5) shows that
an
increase
in
pore radius will cause
a
reduction
in
capillary
pressure while
a
decrease
in
IFT
will
cause
a
decrease
in
capillary
pressure.
Equivalent Height
Expressing
P
c
in
terms
of
force
down leads
to the
expression
PC
=
-
2_-
=
h(
r
w
-
T
air
)
(3.6)
Tir
2
where
h
height
of
capillary rise [ft]
P
c
capillary pressure [psi]
T
w
water,
or
wetting
phase,
density gradient
[psi/ft]
r
ai>
air,
or
nonwetting
phase, density gradient [psi/ft]
Solving
for h
yields
the
defining relationship between capillary pressure
and
equivalent
height, namely
The
equivalent height provides
an
estimate
of the
height
of the
transition zone
between immiscible phases.
A
more precise
definition
of
transition zone
is
given
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Principles
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Applied Reservoir Simulation
in
the
following
section.
Equivalent
height
is
inversely
proportional
to the
difference
in
densities between
two
immiscible phases.
The
relatively
large
density
difference
between
gas and
liquid results
in a
smaller transition
zone
height
than
the
relatively
small
difference
between
two
liquid
phase
densities.
Oil-Water
Capillary Pressure
Oil
is the
nonwetting phase
in a
water-wet reservoir.
Capillary
pressure
for
an
oil-water
system
is
PC,*
=
p
o
-
p
w
(3-8)
where
P
0
pressure
in the oil
phase [psi]
P
w
pressure
in the
water phase [psi]
Capillary pressure increases with height above
the
oil-water contact (OWC)
as
water saturation decreases,
Gas-Oil Capillary Pressure
In
gas-oil systems,
gas
usually behaves
as the
nonwetting phase
and oil
is
the
wetting phase. Capillary pressure between
oil and gas in
such
a
system
is
P
= P - P
(39}
ego
g
o
\J.y)
where
P
g
pressure
in the gas
phase [psi]
P
0
pressure
in the oil
phase [psi]
Capillary
pressure
increases
with height above
the
gas-oil
contact (GOC)
as gas
saturation decreases.
3.3
Mobility
A
measure
of the
ability
of a
fluid
to
move through interconnected pore
space
is the
concept
of
mobility.
It is
defined
here
for
single phase
and
multiphase
flow.
The
multiphase
flow
definition
is
based
on the
concept
of
relative permeability, which
is
presented next.
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Relative
Permeability
The
general definition
of
relative permeability
is
k
r
=
-^
(3.10)
k
abs
where
k
r
relative permeability between
0 and
1
,
k
eff
effective
permeability [md]
k
abs
absolute permeability [md]
Fluid
phase relative permeabilities
for
oil, water
and gas
phases, respectively,
are
k
ro
-kjk
t
k^kjk
t
k
r
^kjk
(3.11)
where
k$
is the
effective
permeability
of
phase
(!,
&
ri
is the
relative permeability
of
phase
d,
and k is
absolute permeability.
Mobility
Fluid
phase mobility
is
defined
as the
ratio
of
effective
phase
permeability
to
phase viscosity. Mobility
for
oil, water
and gas
phases respectively
are
V
—
•
^
=
—
•
V-
(3.12)
U
LI
U,
~0
r*W
'g
where
|l
f
is the
viscosity
of
phase
1
Relative mobility
is
defined
as
relative
permeability divided
by
viscosity [Dake,
1978].
Absolute permeability
is not
a
factor
in the
definition
of
relative mobility.
Mobility Ratio
Mobility ratio
is
defined
as the
mobility
of the
displacing
fluid
"k
D
behind
the
front
divided
by the
mobility
of the
displaced
fluid
K
d
ahead
of the front,
thus
(3.13)
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Principles
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An
example
of
mobility
ratio
is the
mobility
ratio
of
water
to oil for a
waterflood:
In
this case, relative permeability
to
water
is
evaluated
at
residual
oil
saturation
S
or
,
and
relative permeability
to
oil is
evaluated
at
connate water saturation
S
wc
.
Notice
that absolute permeability factors
out of the
expression
for
mobility
ratio.
Consequently,
mobility ratio
can be
calculated using either mobilities
or
relative
mobilities.
3.4
Fractional Flow
The
fractional
flow of
water
is the
ratio
of
water production rate
to
total
production
rate.
In the
case
of an
oil-water system,
the
fractional
flow of
water
is
given
by
/,
=
~
=
-^—
(3.15)
*r
<?
w
+
<lo
where
f
w
fractional
flow of
water
q
w
water volumetric
flow
rate
[RB]
q
0
oil
volumetric
flow
rate [RB]
q
t
total
volumetric
flow
rate [RB]
Notice that
the flow
rates
are
expressed
in
terms
of
reservoir volumes.
The
fractional
flow
of
oil/,
and the
fractional
flow of
water
are
related
byf
w
=
1
-f
0
for
an
oil-water system. Based
on the
definition
of
fractional
flow, we see
that
fractional
flow
should
be a
value between
0 and
1.
Simplified
Fractional Flow
Equation
A
simplified fractional
flow
equation
is
obtained
by
replacing
flow
rates
with
Darcy's
Law in the
definition
of
fractional
flow. If we
neglect capillary
pressure
and
gravity
for
simplicity,
we
obtain
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/.
-
-
-
(3.16)
kkAdP
klA
dP
where
^
is
cross-sectional
area
and
/^
is the
pressure
of
phase
L
Since capillary
pressure
is
neglected,
we
have
the
equality
of
phase pressures
P
w
=
P
0
so
that
Equation
(3.17)
can be
expressed
in
terms
of
mobilities
as
1
1
^£
¥z
i+jjl
(
3;l8
)
k.
rw
The
construction
of Eq.
(3.18)
is
based
on the
following simplifying assump-
tions:
Darcy's
Law
adequately describes
flow
rate,
and
capillary pressure
and
gravity
are
negligible. Given these assumptions,
we can
calculate/,,
at
reservoir
conditions.
Fractional Flow
Equation
with Gravity
Gravity
can be
included
in the
fractional
flow
equation
as
follows. First,
let
us
consider
the
two-phase
flow of oil and
water
in a
tilted linear system.
Darcy's
Law
including capillary pressure
and
gravity
effects
for
linear
flow is
kkA
ro
^
dx
•
u
^
where
(3.19)
P
0
gsina
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Principles
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Applied
Reservoir Simulation
a dip
angle
of
formation
g
gravitational constant
If
we
differentiate
capillary pressure
for a
water-wet system with respect
to
position
x
along
the
dipping bed,
we find
dx
dx dx
Combining
Eqs.
(3.19)
and
(3.20) gives
Akk
ru
M»
where
we
have used
q
t
=
q
0
+
q
w
.
If we
write
the
density difference
as
Ap
=
P
w
-
P
0
,
(3.22)
collect terms,
and
simplify
we
obtain
q
t
li
BP
~t
>o
, cow
Ak\ k
k
ro
rw
/
Akk dx
sina
(3.23)
Rearranging
and
collecting terms gives
the
fractional flow
to
water
f
w
in
conventional oilfield units:
1+0.001127
Akk
dp
0.433(Y
~Yjsina
(3.24)
k
u
rw
PO
A
cross-sectional
area
of flow
system
[ft
2
]
k
absolute permeability
[md]
k
ro
relative permeability
to oil
k
m
relative permeability
to
water
|J,
0
oil
viscosity [cp]
\l
w
water viscosity [cp]
P
cow
oil-water capillary pressure [psi]
=
P
0
-
P
w
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x
direction
of
linear
flow
[ft]
a dip
angle
of
formation
[degrees]
Y
0
oil
specific gravity (water
=1)
j
w
water specific gravity (water
=
1
)
The
general expression
forf
w
includes
all
three terms governing immiscible
displacement,
namely
the
viscous term
(k
ro
/k
rw
)
(|i
w
/
(1
0
)
,
the
capillary
pressure
term
d
P
co
J$x,
and the
gravity term
(Y
W
~Y
0
)
sin
a,
It
is
interesting
to
note that
the
capillary pressure
and
gravity terms
are
multiplied
by
II
q
t
in Eq.
(3.24). Most
waterfloods
have sufficiently high
flow
rates
that capillary pressure
and
gravity
effects
can be
neglected, leaving
the
simplified
expression:
f
a
_
L
__
,
^
k
ro
V«
(3.25)
••--
••
Equation
(3.25)
is in
agreement with
Eq.
(3.18),
as it
should
be.
Gas
Fractional Flow
A
similar analysis
can be
performed
to
determine
the
fractional
flow of
gasjC
The
result
for a
gas-oil system
is
f _-
Akk
1+0.001127
U
Q
'
dP
—
^
-
0.433
(Y
~Y
0
dx
°
t
}
(3.26)
l
+
£
k
u.
rg
~o
where
k
rg
relative permeability
to gas
M-g
gas
viscosity [cp]
P
cgo
gas-oil
capillary
pressure
=
P
g
-P
0
[psi]
Y
g
gas
specific gravity [water
=
1]
q
g
gas
volumetric
flow
rate
[RB/D]
q
t
'
total volumetric
flow
rate
=
q
0
+
q
g
[RB/D]
Immiscible displacement
of oil by gas is
analogous
to
water displacing
oil
with
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Principles
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Applied
Reservoir Simulation
the
water terms replaced
by gas
terms.
In
general,
the
gravity term
in^
should
not
be
neglected unless
q,'
is
very high because
of the
specific
gravity
difference
between
gas and
oil.
Exercises
Exercise
3.1
Estimate
the
parachors
for
butane
and
decane.
Exercise
3.2
Derive
the
relationship between
the
equivalent height
of a
transition
zone
and
pore radius
by
using
Eq.
(3.5)
to
eliminate
capillary
pressure
from
Eq.
(3.7).
Exercise
3.3
Suppose
kJiS^
~
k
ro
(S
wc
)
in Eq.
(3.14)
and
water
viscosity
is 1
cp.
Plot
M
w
0
versus
oil
viscosity
for oil
viscosity ranging
from
0.1
cp to
100
cp.
Exercise
3.4
Derive
Eq.
(3.21)
from
Eqs.
(3.19)
and
(3.20).
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Chapter
4
Derivation
of the
Flow
Equations
Many
derivations
of the
oil, water,
and gas fluid flow
equations exist
in
the
literature
[for
example,
see
Crichlow,
1977;
Peaceman,
1977].
Consequently,
only
a
brief
discussion
will
be
presented
here.
It
closely
follows
the
presentation
originally
published
in
Fanchi,
et
al.
[1982].
4.1
Conservation
of
Mass
We
begin
by
considering
the flow of fluid
into
and
out of a
single reservoir
block
(Figure
4-1).
Let the
symbol
J
denote
fluid flux.
Flux
is
defined
as the
rate
A
/
j,
—
**
i
i
7
J
X+AX
y
^^-
fmnm—^nm^^.
j£
T
z
Figure
4-1. Reservoir block:
the
coordinate
convention
follows
Sawyer
and
Mercer
[1978].
of
flow of
mass
per
unit cross-sectional area normal
to the
direction
of flow,
which
is the x
direction
in the
present
case.
Assume
fluid flows
into
the
block
atx
(J
x
)
and out of the
block
at
;c
+
A
x
(J
x
+&x)-
By
conservation
of
mass,
we
have
the
equality:
31
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