John R. Fanchi - Principles of Applied Reservoir Simulation, Second Edition
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TEAM LinG - Live, Informative, Non-cost and Genuine!
PartV
Technical Supplements
TEAM LinG - Live, Informative, Non-cost and Genuine!
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TEAM LinG - Live, Informative, Non-cost and Genuine!
Chapter
27
Simulator
Formulation
WINB4D
is an
implicit pressure-explicit saturation
finite
difference
simulator.
It can
simulate isothermal Darcy
flow in up to
three dimensions.
Reservoir
fluids are
described
by up to
three
fluid
phases (oil, gas,
and
water),
whose physical properties
are
functions
of
pressure only. Solution
gas may be
present
in
both
the oil and
water phases.
27.1 Equations
The
black
oil
simulator mass conservation equations
for the
oil-,
water-
and
gas-phases
are
derived
in
Chapter
4.
They
can
be
succinctly written
in
vector
notation
as
follows:
Oil
a
B
0
Pose
dt
Water
4>
-r-l
(27.1)
B
o
285
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286
Principles
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Applied
Reservoir Simulation
Gas
R
-V •
V
j?
2 SO
-*
„-„„,*?„
-4.
.
m
._
LJ
_
Li
.
L
,
ij
1
K
'
R
0
°
-D
15
^
^
a
|
'
a,
j
g
+
^o
5
g
5
0
v
-
v
^
(27.3)
Letting
the
subscript
/
denote
o
(oil),
w
(water),
and
g
(gas),
the
symbols
in
Eqs.
(27.1)
to
(27.3) have
the
following
definitions:
B
t
=
formation
volume factor
of
phase
i
q
i
=
mass
flow
rate
per
unit reservoir volume
of
phase
/
R
so
=
solubility
of gas in oil
R
sw
=
solubility
of gas in
water
5,
=
saturation
of
phase
i
v,
=
Darcy's
velocity
of
phase
i
Pise
~
density
of
phase
i at
standard conditions
<j)
=
porosity
Three additional equations
-
called auxiliary equations
- are
employed
when solving
the
preceding
fluid
flow
equations. They
are the
saturation
constraint
S
0
+
S
w
+
S
g
=l
(27.4)
and
the
capillary pressure relationships
P
c
OW
(S
w
)
=
P
0
-
P
w
(27.5)
P
cgo
(S
g
)
=
P
8
'
P
0
(27.6)
where
P,
is the
pressure
of
phase
i,
P
cow
is the
oil-water capillary pressure,
and
P
cgo
is the
gas-oil capillary pressure.
Darcy's velocity
for
phase
/
is
(27.7)
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where
K
is
a
permeability tensor that
is
usually assumed
to
be
diagonalized along
its
principal
axes,
k
ri
is
relative permeability
and
|l,
is
viscosity
of
phase
i.
The
phase
potentials
^
t
are
given
as
functions
of
depth
z by
o
z o z
o
z
A
=p
_.L£__
$
= P
-P
--L2L.
d>
=p +p
_£*_
(2781
0
°
144'
w
°
cow
144'
*
°
cgo
144
l
j
Phase densities
are
calculated
from
input
PVT
data
as
P
0
=
-^-[P
0
,c
+
^
0
PgJ'
P
w
=
"-[P
0
,c
+
^wP
g
J'
P^-f^
(27.9)
5
o
5
w
5
g
Expressions
for
rock
and
phase compressibilities
are
1
84)
1
dB
g
£.
—
~
f\
=
2.
^
4>
5
p
*
Bd
P
(27.10)
C*.
~ ~
_L^-
B
*
8R
>»
"
dP
B
dp
These equations
are
discretized
and
solved numerically
in
WINB4D.
The
procedure
for
solving
these
equations
is
outlined
in
Chapter
32.
27.2
Coordinate Orientation
The
WINB4D
reservoir model assumes
a
block-centered grid with
the
axes
aligned
i
ising
the
right-handed coordinate system illustrated
ir
K
*
y
\
I
1
z
i
Figure
27-1.
Figure
27-1.
Coordinate system.
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Principles
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Reservoir Simulation
The top
layer
(K
= 1) is
shown.
The
second layer
(K = 2) is
below
the K
=
1
layer,
and so on. The top of
each
gridblock
may
vary
from one
block
to
another.
This
allows
the
model
to
perform calculations using grid representations
of
reservoirs ranging
from
flat
layer cake models
to
dipping structures such
as
anticlines
and
domes.
An
anticlinal structure
is
shown
in
Figure 27-2.
D
-6200-60001
D-6400-6200
•
-6600-6400
•-6800-6600
North
-
South
Figure
27-2.
Depth
to top of
anticlinal structure.
27.3
Petrophysical
Model
Monitoring
changes
in the
seismic characteristics
of a
reservoir
as the
reservoir
is
produced
is the
basis
of
time-lapse
(4D)
seismic monitoring
[Anderson,
etal.,
1995;
He,
etal.,
1996; Fanchi,
etal.
1999].
Changes
in
seismic
characteristics
are
determined
in
WINB4D
by
calculating seismic attributes
as
a
function
of
time.
The
seismic attributes calculated
in
WINB4D
are
defined
below.
(
ompressional
and
Shear Velocities
Seismic
compressional
velocity
and
shear velocity
are
calculated
from
the
expressions
[Scho'n,
1996; McQuillin,
et
al.,
1984]:
'*
+ ty*
"
3__
(27.11)
P
B
and
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V
-
r
s
(27.12)
where
V
p
—
compressional velocity
V
s
=
shear velocity
K*
=
effective
bulk modulus
fl*
=
effective
shear modulus
p
B
=
effective
bulk density
=
(!-<J))p
Ma
+4^
p
ma
=
density
of
grains (solid matrix material)
p,
= fluid
density
=
p
0
S
0
+p
w
S
w
+p
g
S
g
<j>
=
porosity
Gassman
[1951]
derived
an
expression
for K*
from
the
theory
of
elasticity
of
porous media [Schon, 1996; McQuillin,
et
al.,
1984]:
K
K
B
+
i
K
(27
where
K
B
-
bulk modulus
of
empty reservoir, that
is, dry
rock
or
porous matrix material
K
G
=
bulk modulus
of
grains (solid matrix material)
K
F
=
bulk modulus
of
fluid =
IIc
f
c
f
=
fluid
compressibility
=
c
0
S
0
+
c
w
S
w
+
c
g
S
g
The
grain modulus
K
G
equals
the
bulk modulus
K
B
when porosity equals
zero. Figure 27-3 shows that bulk modulus
and
shear modulus
are
linear
functions
of
porosity
for
quartz sandstone [Murphy,
et
al.,
1993]
for
porosity
less than 35%.
The
WINB4D
user
must enter data that cannot
be
calculated
from
traditional black
oil
simulator input data.
In
particular,
the
user must enter
K
B
,
K
G
,\i*,
and
p
ma
.
The
references give values that
may be
used
if the
data
are not
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Principles
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Reservoir Simulation
available
from
well logs
such
as
shear wave logging tools
or
laboratory measure-
ments
of
parameters such
as
acoustic velocities
or the dry
frame
Poisson's
ratio.
Bulk
and
Shear
Moduli
for
Quartz
Sand
Correlation
Valid
for
Porosity
< 35%
0.00 0.05 0.10
0.15
0.20 0.25 0.30 0.35
Porosity
Bulk
(psia)
-»-
Shear
(psia)~*
Bulk (GPa)
-"-
Shear (GPa)
Figure 27-3. Correlation
for
bulk
and
shear moduli.
Acoustic
Impedance
and
Reflection Coefficients
Acoustic impedance
Z is
defined
as
Z
=
9
B
V
P
(27.14)
The
reflection
coefficient
RC at the
interface between
two
layers with acoustic
impedances
Z,
and
Z
2
is
given
by
Z
2
- Z,
RC
=
——
(27.15)
Z,
2
+
Z,j
The
transmission coefficient
TC is
2Z.
TC
=
z
+
z
(27J6)
or
TC = 1 - RC.
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27.4 Material Balance
Material
balance
is one
measure
of the
numerical stability
and
accuracy
of
a
simulator.
The
WINB4D
material balance calculation
at
time
t is
given
by
FIP
Material Balance
=
:
(27
17)
OFIP
-
Prod
+
Inj
l
where
FIP
=
fluid in
place
at
time
/
OFIP
=
original
fluid in
place
Prod
=
cumulative
fluid
produced
at
time
/
Inj
=
cumulative
fluid
injected
at
time
t
Based
on
this definition, material balance should equal
one in an
idealized
calculation. Actual simulator material balance
may not
equal one.
Material
balance error reported
by
WINB4D
is
calculated using
the
formula
{
FTP
1
—
1 \ x
100% (27.18)
OFIP
-
Prod
+ Inj J
Material
balance
can be a
sensitive indicator
of
error.
Material balance error
is
greatest
in
WINB4D
when
a
gridblock
undergoes
a
phase transition,
for
example,
when
a
gridblock
passes
from
single phase
oil to
two-phase
oil and gas
during
a
timestep.
Material balance errors
can
be
corrected
by
adding
or
subtracting enough
fluid
to
reestablish
an
exact material balance
[Nolen
and
Berry,
1973;
Spillette,
et
al.,
1986],
This material balance correction technique
is
equivalent
to
adding
a
source/sink
term
to
the
mass conservation equations
for
every gridblock. These
terms
are not
included
in the
WINB4D
formulation.
The
exercises
in
Parts
I and
II
show that
the
uncorrected
formulation
can be
used with good accuracy
in
many
practical situations.
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