John R. Fanchi - Principles of Applied Reservoir Simulation, Second Edition
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1.2
Principles
of
Applied
Reservoir
Simulation
where
N
original
oil in
place [STB]
<f)
reservoir porosity [fraction]
A
reservoir area [acres]
h
0
net
thickness
of oil
zone
[feet]
S
oi
initial reservoir
oil
saturation
[fraction]
B
oi
initial
oil
formation
volume
factor
[RB/STB]
Associated
gas,
or gas in
solution,
is the
product
of
solution gas-oil ratio
R
so
and
original
oil in
place
N.
Original
free
gas in
place
for a gas
reservoir
is
given
by
775844^
S .
G
=
^_£.
(22)
gi
where
G
original
free
gas in
place [SCF]
h
g
net
thickness
of gas
zone
[feet]
Sg
initial reservoir
gas
saturation [fraction]
B
gi
initial
gas
formation volume factor
[RB/SCF]
Equation
(2.2)
is
often
expressed
in
terms
of
initial water saturation
S
wi
by
writing
S
gi
= 1 -
S
wi
.
Initial water saturation
is
usually determined
by
well
log
or
core analysis.
2.2
Material
Balance
The law of
conservation
of
mass
is the
basis
of
material balance calcula-
tions. Material balance
is
an
accounting
of
material entering
or
leaving
a
system.
The
calculation
treats
the
reservoir
as a
large
tank
of
material
and
uses
quantities
that
can be
measured
to
determine
the
amount
of a
material
that cannot
be
directly
measured. Measurable quantities include cumulative
fluid
production
volumes
for
oil, water,
and gas
phases;
accurate reservoir
pressures;
and fluid
property data
from
samples
of
produced
fluids.
Material balance calculations
may be
used
for
several purposes. They
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3
provide
an
independent method
of
estimating
the
volume
of
oil, water
and gas
in
a
reservoir
for
comparison with volumetric estimates.
The
magnitude
of
various
factors
in the
material balance equation indicates
the
relative contribution
of
different
drive mechanisms
at
work
in the
reservoir. Material balance
can be
used
to
predict
future
reservoir performance
and aid in
estimating
cumulative
recovery
efficiency.
More discussion
of
these topics
can be
found
in
references
such
as
Dake
[1978]
and
Craft,
et
al.
[1991],
The
form
of the
material balance equation
depends
on
whether
the
reservoir
is
predominately
an oil
reservoir
or a gas
reservoir. Each
of
these cases
is
considered separately.
Oil
Reservoir Material Balance
The
general material
balance
equation
for an oil
reservoir
is the
Schilthuis
equation
[1961]
expressed
in a
form
given
by
Guerrero
[1966]:
(2.3)
-TV
R B
-{W+W.-W)B
p
so g \ e
i
p
J
w
All
of
these terms
are
defined
in the
Nomenclature
at the end of
this chapter.
The
unit
of
each quantity
is
presented
in
square brackets
in the
Nomenclature.
The
physical significance
of the
terms
in Eq.
(2.3)
can be
displayed
by first
defining
the
terms
B
-B .
D
_
"
l-S
wlg
B
twi
(2.4)
B
ti
cAP.
«
/
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Principles
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Reservoir Simulation
Substituting
Eq.
(2.4)
in Eq.
(2.3) gives
the
general material balance equation
in
the
form
(2,5)
The
terms
in
Eq.
(2.5)
have
a
physical significance.
The
terms
on
the
right
hand
side
of Eq.
(2.3) represent
fluid
production
and
injection, while
the
terms
on
the
left
hand side represent volume changes.
The
significance
of the
terms
Is
summarized
in
Table
2-1.
Table
2-1
Physical
Significance
of
Material Balance Terms
Term
ND
0
ND
,o
MA.
+
Arw)
ND
r
#A
*AA
GA
GptPgc
Gfi,'
WJB
W
wp
w
*A
Physical
Significance
Change
in
volume
of
initial
oil and
associated
gas
Change
in
volume
of
free
gas
Change
in
volume
of
initial connate water
Change
in
formation
pore
volume
Cumulative
oil
production
Cumulative
gas
produced
in
solution with
oil
Cumulative solution
gas
produced
as
evolved
gas
Cumulative
gas cap gas
production
Cumulative
gas
injection
Cumulative water
influx
Cumulative water injection
Cumulative
water production
Equation
(2.3)
is
considered
a
general material balance equation because
it
can be
applied
to an oil
reservoir with
a
gas cap and an
aquifer.
The
derivation
of the
material balance equation
is
based
on
several assumptions:
the
system
is
in
pressure
equilibrium;
the
system
is
isothermal; available
fluid
property data
are
representative
of
reservoir
fluids; the
reservoir
has a
constant volume;
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production
data
is
reliable;
and
gravity
segregation
of
phases
can be
neglected.
A
discussion
of the
relative importance
of
drive mechanisms obtained
from
Eq,
(2.3)
is
presented
in
Chapter
8.
Gas
Reservoir Material Balance
The
general material balance equation
for a gas
reservoir
can be
derived
from
Eq.
(2.3)
by first
recognizing
the
relationship
GB
gi
=
NmB
H
(2,6)
defines
original
gas in
place
G.
Substituting
Eq.
(2.6) into
Eq.
(2.3) gives
the
general
material
balance equation
N(B,
-
B,.
'
B
gi
S
wlg
(
B»
-
B\
(
NB
ti
GB
gi
}
(2.7)
'
—
__
_j.
^
ur
I-
S.__(
5.,
J
1-
5.....
1-
S...J
f
*A
+ fas,
+
G^
-
G,jy]
-
]V
pJ
R
so
5
g
Equation
(2.7)
is
further
simplified
by
recognizing that
the
material balance
for
a gas
reservoir
does
not
include
oil in
place
so
that
N - 0 and
N
p
= 0. The
resulting material balance equation
is
B
~
B
-
GB
Water compressibility
and
formation compressibility
are
relatively small
compared
to gas
compressibility. Consequently,
Eq.
(2.8)
is
often
written
in the
simplified
form
(
B - B }
GB
^^
=
B
«
-
G
<
B
*'
-
(
w
-
+
w
<
-
W
^
B
-
(2
-
9)
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Principles
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2.3
Decline Curve Analysis
Arps
[1945] studied
the
relationship between
flow
rate
and
time
for
producing
wells. Assuming constant
flowing
pressure,
he
found
the
relationship;
^
=-«?""
(2-10)
where
a and n are
empirically determined
constants.
The
empirical constant
n
ranges
from
0 to
1,
Solutions
to Eq.
(2.10)
show
the
expected decline
in flow
rate
as the
production time
increases.
A
fit
of an
equation
of the
form
of Eq.
(2.10)
to flow
rate
data
is
referred
to as
decline curve
analysis.
Three
decline
curves
have
been
identified
based
on the
value
of
n.
The
Exponential Decline curve corresponds
to n
—
0. It has the
solution
q=q
t
e'
at
(2.11)
where
q
i
is
initial rate
and a is a
factor that
is
determined
by
fitting
Eq.
(2.11)
to
well
or field
data.
The
Hyperbolic Decline curve corresponds
to a
value
of n in the
range
0 < n <
1.
The
rate solution
has the
form
q~
n
= nat
+
q:
n
(2.12)
where
q
i
is
initial rate
and a is a
factor that
is
determined
by fitting Eq.
(2.12)
to
well
or field
data.
The
Harmonic
Decline
curve
corresponds
to n -
1.
The
rate
solution
is
equivalent
to Eq.
(2.12)
with
n =
1,
thus
q~
l
=
nat
+
q:
1
(2.13)
where
q
l
is
initial
rate
and a is a
factor that
is
determined
by fitting Eq.
(2.13)
to
well
or field
data.
Decline curves
are fit to
actual data
by
plotting
the
logarithm
of
observed
rates versus time
t. The
semi-log plot yields
the
following equation
for
exponential
decline:
tnq=tnq
i
-at
(2.14)
Equation
(2.14)
has the
form
y - mx + b for a
straight line with slope
m and
intercept
b. In the
case
of
exponential decline, time
/
corresponds
to the
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independent
variable*,
$n
q
corresponds
to the
dependent
variable
y,
$n
q
i
is the
intercept
b, and -a is the
slope
m of the
straight
line.
Cumulative
production
for
decline curve analysis
is the
integral
of the
rate
from
the
initial rate
q
t
at t - 0
to the
rate
q at
time
t. For
example,
the
cumulative production
for
the
exponential
decline
case
is
N
f
<*
(2
J5)
The
decline factor
a is for the
exponential decline case
and is
found
by re-
arranging
Eq.
(2.
11),
thus
fl=--ln
—
(2.16)
t
Exercises
Exercise
2.1
Copy
file
EXAM1
.DAT
to file
WTEMP.DAT
and
run
WINB4D.
What
are the
volumes
of
initial
fluids in
place
in the
model? Hint: Open
the run
output
file
WTEMP.ROF
to find
initial
fluids in
place.
Exercise
2.2
Derive
the
material balance equation
for a
system with
no gas cap
beginning
with Eqs. (2.3)
and
(2.4).
Exercise
2.3 Use Eq.
(2.9)
to
show that
the
material balance equation
for a
depletion
drive
gas
reservoir
is
"
(P/Z).
where
G is
original
free
gas in
place,
G^
is
cumulative
free
gas
produced,
P is
reservoir pressure,
and Z is the
real
gas
compressibility factor. Subscript
t
indicates
that
the
ratio
P/Z
should
be
calculated
at the
time
/
that corresponds
to
G
pc
,
and
subscript
/
indicates that
the
ratio
P/Z
should
be
calculated
at the
initial
time.
The
units
of
G
P
and
G
must agree
for the
equation
to be
consistent.
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Exercise
2.4
Derive
Eq.
(2.15)
for the
exponential
decline
case
by
using
Eq.
(2.11)
as the
integrand
and
performing
the
integration.
Nomenclature
for
Equation
(2.3)
B
c
f
G
G,
m
N
R
yw
R,
s.
'wig
WIO
w
w
AP
gas
formation volume
factor
(FVF)
[RB/SCF]
gas cap FVF
[RB/SCF]
injected
gas FVF
[RB/SCF]
oil
FVF
[RB/STB]
B
0
+
(R
si
-
R
so
)B
g
=
composite
oil FVF
[RB/STB]
B
w
+
(R
swi
-
R
sw
)B
g
=
composite water
FVF
[RB/STB]
formation
(rock) compressibility
[1/psia]
initial
gas in
place [SCF]
cumulative
gas
injected [SCF]
cumulative
gas cap gas
produced [SCF]
cumulative solution
gas
produced
as
evolved
gas
[SCF]
ratio
of gas
reservoir volume
to oil
reservoir volume
initial
oil in
place [STB]
cumulative
oil
produced [STB]
solution gas-oil ratio [SCF/STB]
initial solution gas-oil ratio [SCF/STB]
solution gas-water ratio
[SCF/STB]
initial solution gas-water ratio [SCF/STB]
gas
saturation
[frac.]
oil
saturation
[frac.]
water saturation
[frac.]
initial water saturation
[frac.]
initial water saturation
in gas cap
[frac.]
initial
water saturation
in oil
zone
[frac.]
cumulative
water
influx
[STB]
cumulative
water injected [STB]
cumulative
water produced [STB]
P,'P
=
reservoir pressure change [psia]
initial
reservoir pressure [psia]
reservoir pressure corresponding
to
cumulative
fluid
times [psia]
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Chapter
3
Multiphase Flow Concepts
This chapter summarizes
the
basic concepts
of
multiphase
flow
including
interfacial
tension,
wettability,
and
contact angle.
These
concepts lead naturally
to a
discussion
of
capillary pressure, mobility,
and
fractional
flow.
3.1
Basic Concepts
Some basic concepts
must
be
introduced
as
prerequisites
for
understanding
capillary pressure.
The
concepts
are
interfacial tension, wettability,
and
contact
angle. They
are
defined
here.
Interfacial Tension
On
all
interfaces between solids
and fluids, and
between immiscible
fluids,
there
is a
surface
free
energy resulting
from
electrical forces. These forces cause
the
surface
of a
liquid
to
occupy
the
smallest possible area
and act
like
a
membrane. Interfacial tension
(IFT)
refers
to the
tension between liquids
at a
liquid/liquid
interface. Surface tension refers
to the
tension between
fluids at
a
gas/liquid interface.
Interfacial
tension
is
energy
per
unit
of
surface area,
or
force
per
unit
length. Interfacial
tension
is
often
abbreviated
as
IFT.
The
units
of IFT are
typically
expressed
in
milli-Newtons/meter
or the
equivalent dynes/cm.
The
value
of IFT
depends
on the
composition
of the two fluids at the
interface
between phases. Table
3-1
lists
a few
examples:
19
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Principles
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Table
3-1
Examples
of
Interfacial Tension
Fluid
Pair
Air-Brine
Oil-Brine
Gas-Oil
IFT
Range
(niN/m
or
dyne/cm)
72-100
15-40
35-65
Interfacial
tension (IFT)
can be
estimated using
the
Macleod-Sugden
correlation.
The
Weinaug-Katz variation
of the
Macleod-Sugden correlation
is
N
e
'M
L
y,
(3.1)
where
o
1
chi
M
L
My
Pi
PK
interfacial tension [dyne/cm]
parachor
of
component
i
[(dynes/cm)
1/4
/(g/cm
3
)]
molecular weight
of
liquid phase
molecular weight
of
vapor phase
liquid
phase
density
[g/cm
3
]
vapor
phase density
[g/cm
3
]
x
:
mole
fraction of
component
/
in
liquid phase
y
t
mole
fraction of
component
i in
vapor phase
Parachors
are
empirical parameters.
The
parachor
of
component
i can be
estimated using
the
molecular weight
M
i
of
component
i and the
empirical
regression equation
P
chi
=
10.0
+2.92
M
l
(3.2)
This procedure works reasonably well
for
molecular weights ranging
from
100
to
500.
A
more accurate procedure
for a
wider range
of
molecular weights
is
given
by
Fanchi
[1990].
Wettability
Wettability
is the
ability
of a fluid
phase
to
preferentially
wet a
solid
surface
in the
presence
of a
second immiscible phase.
The
wetting,
or
wettability,
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condition
in a
rock/fluid
system depends
on
IFT.
Changing
the
type
of
rock
or
fluid
can
change
IFT
and, hence,
wettability
of the
system. Adding
a
chemical
such
as
surfactant, polymer, corrosion inhibitor,
or
scale inhibitor
can
alter
wettability.
Contact Angle
Wettability
is
measured
by
contact angle. Contact angle
is
always
measured
through
the
more dense phase. Contact angle
is
related
to
interfacial
energies
by
(3.3)
os
ws
a
ow
cos0
where
o
os
interfacial energy between
oil and
solid [dyne/cm]
o
m
interfacial energy between water
and
solid [dyne/cm]
o
ow
interfacial energy,
or
IFT, between
oil and
water [dyne/cm]
0
contact angle
at
oil-water-solid interface measured through
the
water
phase
[degrees]
Examples
of
contact angle
are
presented
in
Table
3-2 for
different
wetting
conditions.
Table
3-2
Examples
of
Contact Angle
Wetting
Condition
Strongly
Water-
wet
Moderately Water-wet
Neutrally
Wet
Moderately Oil-wet
Strongly
Oil-wet
Contact
Angle,
degrees
0-30
30-75
75-105
105-150
150-180
Wettability
is
usually measured
in the
laboratory. Several factors
can
affect
laboratory measurements
of
wettability. Wettability
can be
changed
by
contact
of the
core
during coring with drilling fluids
or
fluids
on the rig floor,
and
contact
of the
core during core handling with oxygen
and/or
water
from
the
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