John R. Fanchi - Principles of Applied Reservoir Simulation, Second Edition
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32
Principles
of
Applied
Reservoir Simulation
mass
entering
the
block
-
mass leaving
the
block
=
accumulation
of
mass
in the
block.
If
the
block
has
length
A*,
width
Ay,
and
depth
Az,
then
we can
write
the
mass
entering
the
block
in a
time interval
A/
as
|
(',/
)
AvAz+
(j
}
AxAz
+
(j
}
AjcAv
A/
=
Mass
in
(4,1)
I
'<
x
'
x \ y I
v
\
z
I
:
"
\
where
we
have generalized
to
allow
flux in the y and z
directions
as
well,
The
notation
(J
x
)
x
denotes
the x
direction
flux
at
location*,
with analogous meanings
for
the
remaining terms,
Corresponding
to
mass entering
is a
term
for
mass exiting which
has the
form
lW*.*,*y*
z
+
(^W
A
*
A
*
+
(^)
2+
A,
A
*
A
^
A
'
(4.2)
+
gAxAyAzA/
=
Mass
out
We
have added
a
source/sink term
q
which represents mass
flow
into (source)
or out of
(sink)
a
well.
A
producer
is
represented
by q > 0, and an
injector
by
q<0.
Accumulation
of
mass
in the
block
is the
change
in
concentration
of
phase
<!
(C
4
)
in the
block over
the
time interval
A?.
If the
concentration
C
t
is
defined
as
the
total mass
of
phase
0
(oil, water,
or
gas)
in the
entire
reservoir
block
divided
by the
block volume, then
the
accumulation term becomes
[(^X+A*
~
(C^JAjtAyAz
=
Mass
accumulation
(43)
Using
Eqs.
(4.1)
through (4.2)
in the
mass
conservation equality
Mass
in -
Mass
out =
Mass accumulation
gives
[(/^AyAz
H-
(J^AjcAz
+
0/
Z
)
Z
A*A7]A?
-
lW
x
.***y*
z
+
(^W
A
*
A
*
+
W
z
^
z
&x&y]&t
(4.4)
-
^AxAyAzA/
=
[(C
(
)^
Ar
-
(C
f
),]A*A;pAz
Dividing
Eq.
(4.4)
by
AxAyAzA/
and
rearranging gives
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33
A*
A*
A
7
=
Az
Ar
In
the
limit
as
A*,
Aj,
Az, and
A?
go to
zero,
Eq.
(4.5) becomes
the
continuity
equation
dJ
x
dJ
v
dJ
z
dC,
y
(A
&}
\*T,Oj
dx
dy dz dt
The
oil, water,
and
gas
phases each
satisfy
a
mass conservation equation having
the
form of Eq.
(4.6).
4.2
Flow Equations
for
Three-Phase
Flow
The flow
equations
for an
oil, water,
and gas
system
are
determined
by
specifying
the fluxes and
concentrations
of the
conservation equations
for
each
of
the
three phases.
A flux in a
given direction
can be
written
as the
density
of
the
fluid
times
its
velocity
in the
given direction. Letting
the
subscripts
o,
w,
and
g
denote oil, water,
and
gas, respectively,
the fluxes
become:
D
B
(A,
-
-^
(4.8)
g
B
B
B
(
where
R
so
and
R
sw
are gas
solubilities;
B
0
,
B
w
,
and
B
g
are
formation volume
factors;
the
subscript
sc
denotes standard conditions (usually 60°F
and
14.7
psia
in
oilfield
units);
and p
denotes
fluid
densities.
The
velocities
v
are
assumed
to
be
Darcy velocities
and
their
x
components
are
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34
Principles
of
Applied
Reservoir
Simulation
®
P -
'"_-
(A 1
A\
a
r
o
...
(4,10)
a^
w
w
_a_
dx
P
-
W
/A
1 1
\
r
w
,,>
(4.11
j
(4.12)
where
g
is the
acceleration
of
gravity
in ft/sec
2
,
andg
c
is
32.174
ft/sec
2
(WINB4D
assumes
g -
g
c
).
These equations should
be
valid
for
describing
fluid flow in
porous media even
if
g
and
g
c
change, such
as on the
Moon, Mars,
or
the
space
shuttle.
Similar expressions
can be
written
for
the>>
and z
components.
The
phase
mobility
A
e
is
defined
as the
ratio
of the
relative permeability
to
flow of the
phase divided
by its
viscosity, thus
A,
=
V»*i
(4.13)
The
phase densities
are
related
to
formation volume factors
and gas
solubilities
by
Po
=
—
[Po*c
+
^oPff*J'
(4.14)
(4.15)
(4.16)
Besides
fluxes, we
also need
concentrations.
These
are
given
by
C
o
=
4»Po,A/*
0
»
(4.17)
(4.18)
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c
c
c
so
£„
B,
(4.19)
where
<j>
is the
porosity
and
S
t
is the
saturation
of
phase
fi. The
saturations
satisfy
the
constraint
O
4.
O
4.
P — 1
//i
OA\
5
o
+
^w
^g
~
l
(4,20)
Combining
Eqs. (4.6), (4.7) through (4.9),
and
(4.17)
through
(4.19)
gives
a
mass
conservation equation
for
each phase:
Oil
dx
_a_
dz
£££
v
Z0
(4.21)
a
f
A
5
'
Water
\
dx
a
xw
p
wsc
-V.
yw
'w
/
dz
B.
(4.22)
--Up
~1
q
»~
dt(
W
*
C
B
0
)
Gas
"
I
8&c
-.
,
so"gsc
SY/*
gsc
I
av
I
B B B
i
*
\
K
O
W
I
(4.23)
O
'gSC
SO^gSC
SW'gSC
I
a_
_
^
_
^^
_
z
^
^
O2T
JD
jD
/>
1
a/
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Principles
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Reservoir Simulation
The
densities
at
standard conditions
are
constants
and can be
divided
out of the
above
equations. This reduces
the
equations
to the
following
form:
Oil
d
*
B
Water
/
\
d
v,
yw
I
u
zw
9*
M
d
y(
B
»)
dz
(
B
»s
<?w
Pw.c
dt
Gas
-
A
a*
By
(
B
g
B
ft
w
\
(4.26)
Zff"
JO
JW
o
-i.
11
J_
ti
(
c c
c
N
-JL
+
R
__2.
+
R
_JL
^
"^
^
w
,
4.3
Flow Equations
in
Vector Notation
Equations
(4.10)
through
(4.16),
(4.20),
and
(4.24) through (4.26)
are the
basic
fluid
flow
equations which
are
numerically
solved
in a
black
oil
simulator.
A
glance
at
Eqs. (4.24) through (4.26) illustrates
the
computational complexity
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of
the
basic
three-dimensional, three-phase black
oil
simulator equations.
Equivalent
but
much simpler appearing
forms
of
the
flow
equations
are
presented
in
terms
of
vector operators
as
- V
(427)
q
™
3
-
- •
(
4,28)
and
- V
Pgsc
dt
(4.29)
+1?
*
"
}
_
-
v =
—-v
x
+
—v
+
—-v
z
.
(430)
where
the
symbol
V • v for the
divergence
of the
velocity vector
is
shorthand
for
the
expression
d
d d
—
v +
—v
+ —
dx
dy
y
dz
A
review
of
vector analysis
can be
found
in
many references, such
as
Kreyszig
[1999]
and
Fanchi
[2000].
Exercises
Exercise
4.1
Suppose
the
unit
of
density
p
osc
is
mass
per
volume
at
standard
conditions,
and the
unit
of
Darcy velocity
is
length
per
time.
Use
dimensional
analysis
to
determine
the
unit
of
flux
in Eq.
(4.7).
Exercise
4.2 The
densities
in
Eqs.
(4.14)
and
(4.15) include
gas
dissolution.
Rewrite Eqs. (4.19), (4.23),
and
(4.29)
for a
system with
no gas
dissolved
in
either
the oil or
water phases.
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Principles
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Reservoir Simulation
Exercise
4.3 Run
EXAM3.DAT
and
record
the
time, pressure,
oil
rate, water
rate,
gas
rate,
and GOR at the end of the
run.
These values
are
obtained
from
the
one
line
timestep
summary
file
WTEMP.TSS (also
see
Chapter 26.2).
Is gas
significant
in
this model?
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Chapter
5
Fluid
Displacement
Fluid
displacement processes require contact between
the
displacing
fluid
and
the
displaced
fluid. The
movement
of the
interface between displacing
and
displaced
fluids and the
breakthrough time associated
with
the
production
of
injected
fluids
at
producing wells
are
indicators
of
sweep
efficiency.
This chapter
shows
how
to
calculate such indicators using
two
analytical
techniques:
Buckley-
Leverett
theory with
Welge's
method
for
immiscible
fluid
displacement,
and
solution
of the
convection-dispersion equation
for
miscible
fluid
displacement.
5.1
Buckley-Leverett
Theory
One of the
simplest
and
most widely used methods
of
estimating
the
advance
of a fluid
displacement
front in an
immiscible
displacement
process
is
the
Buckley-Leverett method. Buckley-Leverett Theory
[1942]
estimates
the
rate
at
which
an
injected water bank moves through
a
porous medium.
The
approach uses
fractional
flow
theory
and is
based
on the
following assumptions:
•
Flow
is
linear
and
horizontal
•
Water
is
injected
into
an oil
reservoir
•
Oil and
water
are
both incompressible
• Oil and
water
are
immiscible
•
Gravity
and
capillary pressure
effects
are
negligible
The
following
analysis
can be
found
in a
variety
of
sources, such
as
Collins
[1961],
Dake [1978], Wilhite [1986],
and
Craft,
et
al.
[1991].
39
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Principles
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Applied
Reservoir Simulation
Frontal
advance theory
is an
application
of the law of
conservation
of
mass.
Flow through
a
small volume element (Figure
5-1)
with length
Ax and
/
/
Porous Material
(e.g. Rock)
v
/
A
Figure 5-1. Flow Geometry
cross-sectional
area
A can be
expressed
in
terms
of
total
flow
rate
q
t
as
9*
=
QO
+
<?
w
(s.i)
where
q
denotes volumetric
flow
rate
at
reservoir conditions
and the
subscripts
{o,
w,
t}
refer
to
oil, water,
and
total rate,
respectively.
The
rate
of
water entering
the
element
on the
left
hand side (LHS)
is
q
t
f
w
=
entering
LHS
(5.2)
for
a
fractional
flow to
water
f
w
.
The
rate
of
water leaving
the
element
on the
right
hand
side
(RHS)
is
q
t
(f
w
+
A/
w
)
=
leaving
RHS
(53)
The
change
in
water
flow
rate across
the
element
is
found
by
balancing mass
for
an
immiscible, incompressible
system,
thus
rate change
=
water entering
-
water leaving
(54)
The
change
in
water saturation
per
unit time
is the
rate change
in Eq.
(5.4)
divided
by the
pore volume
of the
element, thus
Ax
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In the
limit
as
A/
->
0 and Ax
-*
0, we
pass
to the
differential
form
of Eq.
(5.5)
for
the
water
phase:
dt
-4(j)
dx
A
similar equation applies
to the oil
phase:
(5.7)
dt
A$
dx
Since/,
depends only
on
S
w
,
we can
write
the
derivative
of
fractional
flow as
"a7
=
Ist
"a7
(5
'
8)
Substituting
dfjdx
into
dSJdx
yields
dS
w
-q
t
df
w
dt A dS dx
(5,9)
It
is not
possible
to
solve
for the
general distribution
of
water saturation
S^x,
t)
in
most realistic cases because
of
the
nonlinearity
of
the
problem.
For
example,
water fractional
flow is
usually
a
nonlinear
function
of
water saturation.
It is
therefore
necessary
to
consider
a
simplified approach
to
solving
Eq.
(5.9).
We
begin
by
considering
the
total differential
of
S
w
(x,
t):
w
w
QX
w
dt
dx dt dt
(5.10)
Equation
(5.10)
can be
simplified
by
choosing
x to
coincide
with
a
surface
of
fixed
S
w
so
that
dSJdt
=
0 and
(
dS.}
dx\
( dt
)
_____
I —
_
\
/
ff
1 1
\
.J
/ \
(5.H)
dx
Substituting Eqs. (5.8)
and
(5.9) into
Eq.
(5.
II)
gives
the
Buckley-Leverett
frontal
advance equation:
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