Upload
shone-david
View
242
Download
1
Embed Size (px)
Citation preview
8/9/2019 oil water coning
1/8
A Model of Oil-Water Coning for Two-Dimensional, Areal
eservoir imulation
J.E.CHAPPELEAR
G J. HIRASAKI
MEMBERS SPE IME
ABSTRACT
A model for
oil water
coning
in
a partially
perforated well ha s been developed and
tested by
comparison with numerical
s imulat ions . The
effect
of oil water coning
including
down coning of oil
on field production is
demonstrated
by studying a
small water drive reservoir whose complete
production
data are known.
Th e
coning model
is derived by assuming vert ical
equilibrium and segregated
flow. A necessary
correction
for departure from vertical equilibrium in
the
immediate
neighborhood of
the
w el l i s developed.
The
coning model is suitable for
single well
studies
or for
inclusion
in a
reservoir simulator
for
two dimensional areal studies.
INTRODUCTION
The obj ective of th i s
investigation
of oil -water
coning
was to
develop
tools to evaluate operational
problems for
reservoirs
with bot tom water. Although
any specific question can be answered at least in
principle) by finite-difference simulation, a practical
problem occurs.
Great
detail may be necessary for
a reservoir-wide simulation of problems involving
coning. Two approaches
are possible.
One can use
more accurate finite-difference equations such as
those
derivedby some type
of
Galerkin procedure)
to solve t he p robl em of insuff ic ien t
accuracy. Or
one ca n include
in
hi s simulator a
C
8/9/2019 oil water coning
2/8
8/9/2019 oil water coning
3/8
T BLE 1 -
P R METERS COMMON
TO
LL
T HE T EST CASES
COMPARISON O F CONING MODEL
WITH NUMERICAL SIMULATION
The results
of th e coning model a nd n um er ic al
simulation are s how n as calculated water
cu t
as
a
function
of
t he a ve ra ge w at er
s a tu ra ti on . T h e
case
Value
40.0
1.0
1,000.0
745.0
0.35
0.35
I
r
I
I
I
I
I
SIMULATION
I
I
/ CONING
II
VMODEL I I
/ ) I
FLAT
l I
INTERFACE
T BLE 2 - BASE CASE
BASE
CASE
I
I
v ~
I
WITHOUT
GRAVITY
I
I
I
I
I
I
I
0.6
2
1
0 8
0.7
0.3
0.9
0.4
5
>
u
0.0 ---
.
L 1 L
0.7 0.8 0.9
1 0
AVERAGE WATER
SATURATION
FIG.
2 - BASE CASE.
1 0 r
shown
in Fig 2
is
th e base
case. If
a
flat-interface
model were used, water
production
would have
occurred only
over the range of
saturations
from
0.892 to 0.915. Th e
c a lc u l at i on u s in g Eq .
3 which
is .appropriate
for a
fully perforated well, is also
shown.
Fig 3
shows the.
comparison
fo r
a higher
rate
2,000 B/D). Although th e
vi scous
forces are
higher,
th e
comparison is g ood . B oth
cases
ar e for
a favorable mobility ratio of 0.25. The curve is
closer
to
t h a t p r ed i ct e d without
gravity.
Fig
4
s ho ws t he comparison
for a
more
viscous
oil 5.6 cp). In
this
cas e, th e
rate is always
fa r
above the
critical
rate
and the
mobility ratio is
unfavorable 4.0). Thus, th e water
cut
is quite high.
Notice
t ha t t he v ertic al scale
is different from th e
o
th
e r f ig ur e s.
In Fig.
5 the water
viscosity w as r ed uc ed to
r ep r es e nt g a s- oi l
coning. T he c om pa ri so n b et we en
th e model
an d
th e
simulation
is no t as good in this
ex tremely u n fav o rab le mo b ility -ratio
case.
In Fig 6 th e v al ue o f
kv k
h
w as r ed uc ed to 0. 1
to represent
anisotropy.
Th is retards
water
production.
In
Fig. 7
t he e xt er na l radius ha s been
reduced
to 75
ft.
Fig
8
evaluates th e effect of
a
small grid
block
in
a
l a rg e r e se r vo i r.
The model
had
a
radius of
88
Parameter
h
wi
ft
kv k
h
qL
BID
e ft
flo
cp
flw
cp
5)
Value
0.20
0.25
0.25
1. 0
750.0
0.987
1.372
0.4844
0.299
0.33
40.0
0. 0
5. 0
Parameters
Swc
Sot
k
rw
at
Sot
kt o at
Swc
k
hl
md
B
WI
RB STB
8
RB STB
w psi f t
Po
psi f t
W I
ft
hoi
ft
h
ct
ft
h
Cb 9
ft
SELECTION O F T E S T PARAMETERS
The t est parameters were selected from
th e
values thought to
be
typical
o f o ff sh or e
Loui si ana
w at er -c on in g p ro bl em s
descr i bed by
Miller an d
Rogers.
9
Th e
v alu es o f
parameters common
in
al l
th e
test
c a s e s
are shown in
Tabl e
1. The base
case from which th e
val ues
of the parameters were
changed for
subsequent
test
c a s e s is shown
in
Tabl e
2. Th e values of the mobility-thickness ratio,
N
mt
an d
th e
critical
rate,
qe change
throughout
each case as
th e
a v er ag e s a tu ra ti on c h an g es .
COMP
ARISON
WITH
SIMULATIONS
DESCRIPTION
O F
T E S T PROCEDURES
For the simulation runs
descr i bed
here a
semieoimplicit black-oil
reservoir
simulator
descr i bed
by Chappelear and Rogers
8
wa s
used. The
g eo me tr ic al c on fi gu ra ti on m od el ed i n t he
tests
is
shown in Fig. 1. An a qu if er w as
at t ached
at the
e x te r na l r a di u s
to
r ep la ce t he
p r od u ce d l i qu i d,
an d
th e
reservoir was
produced
a t
a
c on st an t g ro ss
liquid rate. The water cu t was calculated.
as
a
function of
time until 98-percent water cut was
obtained.
These
simulated
values ar e
compared
with
coning
model prediction s to d et er mi ne t he
validity
of the model.
The
same
r s u ~
holds
i f
h
eb
an d
h
et
ar e small
compared with
h
o
. Eq . 5
i l lustrates
that, for
this
case, the water cut
is
a linear function of qe/qt
When qt
=
qe
that i s ,
qc/ qt
=
1 t he w at er
cu t
is
equal
to z ero. In th e l im it o f qt much larger
than
qc
qe/qt
= 0) ,
t he w ate r
cu t
is just a function of th e
mobility-thickness
ratio,
a s d is cu s se d
previously.
of
th e
mobility-thickness
ratio.
If h
eb
h
et
small per f or at ed i nt er val) an d
u n it m ob il i ty r a ti o ), Eq . 1
r educes
to
APRIL, 1976
67
8/9/2019 oil water coning
4/8
1
qL
=
2000 I
A
II
.9
I
~
I
I
tl
0.8
{
I
I
0.7
IWIT OUT
GRAVITY
0.6
I /
I
I
SIMULATION
I
u
I
t::
0.5
/
w
I
/
CONING
VMODEL
I
I
I
0.4
I
I
0.3
I
/
I
I
I
.2
r
I
F l T ~
1
I
INTERFACE
I
I
I
0.0
0 7
0.8
0.9
1
AVERAGE
WATER
SATURATION
FIG.
3 GROSS
PRODUCTION RATE 2 000 ID
ft to
represent
th e volume of a
single
grid b lock .
The s imulat ion was conducte d with th e external
radius at 745 ft
and
th e ave rage s atu ra tio n was
evaluated
in the region within a radius of
88
ft .
Both water
and
oi l ar e flowing into this region.
Thus
the model
that ha s
only
water
influx at th e
external radius is a good
representation
for a
sys tem that
has
both water and
oi l
influx.
In
ig
9 we
have
a
smaller
initial water
layer
and total thickness. Thus th e critical rate is much
smaller
than
for the bas e c as e.
The
coning
model also can be
used
to
compute
oi l production with
time if
it is
assumed
that only
water
encroaches at
th e external
radius. The
cumulative oi l production
vs
time for the base case
is compared
with th e
numerical s imulation
in Fig.
10.
After
20
years
the difference
in
the cumulat ive
oi l production between
the
numerical s imulat ion and
the coning model is 6 percent of the cumulative oi l
production or
0.4 percent of the cumulat ive gros s
fluid
production.
INSTALLATION IN A
RESERVOIR
SIMULATOR
Using th e con ing
model oil-water
coning within
a
grid
b lock can
be s imula ted in
a two-dimensional
areal or a three-dimensional noncommunicating
layer
system. The condi tions for
vertical
equilibrium
and
segregated flow should
be
satisfied
everywhere
1
w l T ~ o U T T
00
0 9
I GRAVITY
/
I
98
ISIMULATION /
0 8
I
96
0 7
I
/
94
SIMULATION
0 6
I
/
0 92
I
:::
U
u
~ O N I N
0 5
I
UJ
I
MODEL
I
fLo
5 6 CP
0 4
0 88
I
0 86
II
0 3
FLAT
INTERFACE
I
0 2
84
L T ~ I
J
I
I
INTERFACE I
I
I
82
1
I
w = 0 022
I
I
I
0 80
0 0
7
0 8 0 9
1
0 7
0 8
0 9
1 0
AVERAGE WATER SATURATION
AVERAGE WATER
SATURATION
FIG.
4
OIL
VISCOSITY 5.6
CPo
FIG.
5 WATER VISCOSITY 0.022 CPo
68
SOCIETY OF PETROLEUM ENGINEERS JOURNAL
8/9/2019 oil water coning
5/8
.except near t he wel l.
The con in g model s
exp re ss ed a s an
equation
that
relates the w ~ t r cut, f
w
the average o l
column
t h i ~ n s s
h
o
and th e
total
rate, qt The
value of h
o
s
determined from
th e grid-block
saturation.
The
reservoir and
fluid
properties
appear as parameters in th e
equation.
If
t he value
of
t he wat er cu t
s
obtained
from
th e
coning model
using average oil-column th ickness a t
t he p reviou s time step,
instability
can
occur. We
avoid
this
problem
by making the grid-block
produc
tivity index impl ic it in saturation, as recommended
by Spivak and Coats.
lO
The der ivat ive requi red
s
obtained
numerically.
Even though t he coning model was developed for
oil-water coning,
some
consideration
must be
made
for gas
production. In
our case,
any
gas
production
was r ep re sent ed by th e flat-interface mode. The
f ract ion of the per fo ra ted interva l covered
by oi l
as
computed
by th e coning model was reduced by the
frac t ion covered
by
gas.
FIELD EXAMPLE
The o il -wa te r con ing model has been
installed
in
a reservoir
simulator.
Shown here are some results
from t he s imulat ion
of
a small reservo ir ,
Eugene
Island
Block 18-N
Sand.
As shown in Fig. 11, the
reservoir s a simple
domal
structure
developed
by
three
wells.
T he entire history of the
field
is ava ilab le . The
over-al l f ie ld performance was charac teri zed by a
l ong per iod o f
water-free oi l production.
Then, after
water
breakthrough, each well produced
a
substantial
portion about 50 percent of ts total oi l reserves
at
a
steadi ly increasing water cut. T his behavior
contrasts
with other deeper reservoirs in
this
field,
which experience very ear ly water breakthrough.
Fo r
i l lust rat ive purposes,
a
very coarse,
two-
dimensional,
areal grid was adopted,
with individual
blocks of 781
x
812
ft. There are
no
intervening
grid blocks between t he p roduci ng wells. In the
simulation, a
stable
cone was formed. After
substantial
oi l
depletion, when th e
average
water
c on ta ct h as ris en
significantly, th e con ing model
e t e r ~ i n e s
that
water p roduct ion should
begin.
An
east-west cross-section through
t he f ie ld
Fig.
12
s hows th e
calculated average
oil-water
con ta ct a s
of
Jan.
1967. A
permeability value
of
10,000
md
was
used
for
this
t r ia l h istory-matching
run.
Although
the ave rage
contact
was
some
2 ft
below the bottom
perforation
of Wells
17 and 18
bo th a re p roduci ng
at
a
substantial
water cut .
The
breakthrough
t m s for water w ere determined_.
to be extremely sensitive to
the hor izontal
permeability. T he large difference in
calculated
performance
of
the individual wells
for two
1 0r------r------------.,..---------,
1 0
.8 0.9
V R G W T R
S TUR TION
FIG.
7 EXTERNAL RADIUS 75FT.
0 1
0 2
O O____ o
.. ::;......a. L..-L.. . . . . . .I . .--
0 7
0 4
0.3
0 7
0 6
1 0
r .
I
re
=
75
FEET /iJ I
I
/ /
/
I
/ / /
\
~ WITHOUT I
I GRAVITY I :I
I
CONING I I
M O D E l l
i I
I I
I I
I
L L FLAT
I
Ii
INTERFACE
I
I I
I
I
0.8
0 9
1 0
.8 0.9
V R G W T R S TUR T ON
kv/kh
0 1
/ J
I
I
II
I ~
~ T O U T
/
I
GRAVITY / :
I
ONING
/
MODEL
1
:
I
I :
I
I
1
II
I I FLAT I
I l iNT R FACE I
I
SIMULATION i I
I I
I
a
.O L
.l -- --- ---- - J
0.7
0 1
0 4
0 2
0.3
0.7
0 6
0.8
0 9
0 5
I-
~
FIG. 6
VERTICAL-TO-HORIZONTAL
PERMEABILITY
RATIO , 0 .1 .
APRIL, 1976
69
8/9/2019 oil water coning
6/8
1 .0 . . . . . . . . . .
20
6
2
TIME, YR
.AT
SIMULATION
.. .. 1
CONING
MODEL
,II'
,,
,
GROSS FLUIDS
I
hwi = 10 FT
/
.9
I
0.8
I
I
I
I
0.7
I
I
,
0.6
I
4
I
IWITHOUT O
I
:::>
,
u
I
0 5
I GRAVITY /
U
~
I \ /
I
I
0 4
I
I
I
0 3
V
I
SIMULATION
I
I
.2
7 FLATA
I
/ INTERFACE i
0 1
I
r CONING
I
MODEL I
1200
co
:E
:-800
o
we
made
a
f inal run
with a
uniform
permeability
of
10,000
md
and
used
the
flat-interface
well
modeL
With this option, th e
oil-water
contact is
assumed
to be per fect ly hor izonta l
within
a grid block
containing a
well.
The
water
cut
is
calculated by
comparing
that
level with the well s
perforated
interval. In our simulation, a
broad
((uplift
of
water
is
created because
of
th e
oi l
withdrawals.
But the calculated water breakthrough understand
ably
occurs
much later
in
the f ield history (Fig. 13).
However, a surprising
and
interesting
event
was
observed
in
the simulation. The producing wells
abruptly shut in before producing all the
oi l
that we
had spe ci fie d. This is because the completion
AVERAGE WATER SATURATION
FIG.
9 -
INITIAL WATER-ZONE TIIICKNESS,
10 FT .
O
0 L - - - - - - ~ - L - _ - 1 . . .
__
L
I L ____I
0 3
0.4 0.5 0.6 0.7
0 8
FIG. 10 - PREDICTION OF CUMULATIVE
OIL
PRODUCTION.
1.0
r ~
1.0
re =
88 FT
I
INFLUX AT
I
7 FT
/ /
/
II
I / 1
/
v
WITHOUT
I
GRAVITY I
I
C O N I N G
MODEL ---1 I
I
I
I :
I
I
II
I :
I
I I
I L l
FLAT
I ; rNTERFACE
0.0 I L _ L _ I L _ ~
_ ___I___L l _ ___I
0 7
0 7
0 2
0 4
0 1
0 6
0.3
0.9
0 8
t -
:)
U
0::: 0.5
w
t -
0.8 0.9
AVERAGE
WATER
SATURATION
FIG.
8
MODEL
EXTERNAL
RADIUS
AT
88
FT;
EXTERNAL RADIUS OF SIMULATION AT
745
FT AND
AVERAGE
SATURATION EVALUATED TO 88 FT .
assumptions
( 4, 000 and 10,000 md) can be clearly
seen in Figs. 13 through 15e There is no t a
single
permeability value
that
wil l permi t
an exact
fi t
for
all three
wells.
But
additional
((fine
tuning
of
breakthrough
times
in
i nd iv idual well s
could be
obtained
by varying
the
horizontal permeability
slightly in
the
coning model for
each
well.
From
the
cal cu la ted r esul ts
for a
range of
permeabilities,
it
is understandable
that
deeper
measures
with comparable
geometry
could
easily
exhi it
much earlier
water-breakthrough times
At
deeper measures, permeabil it ies
a re p robably still
((very high., but they only
need
to be somewha t
lower than that in the subj
ect
N sand.
In
an
earlier
study,
it
was found t ha t r educing
th e
effec tive ver ti ca l
permeability (even to very
low
levels) had
relatively
little
effect
in del ay ing
th e water
breakthrough. It
is
important
to note tha t,
in these calcu la tions, a much smaller (500 md)
horizontal pe rmeability was us ed. Thus, the
resulting water cone was no t s tabl e; t ha t
is , the
critical rate was
less
than the actual per-well,
gross fluid-production rates. We
attribute
the long
period
of
water-free
oi l
production to
high
in-situ
permeabi l it ies rather
than
to subtle impediments to
vertical
flow
of
water.
To illustrate the
necessity for
the
coning
model,
70
SOCIETY OF PETROLEUM ENGINEERS JOURNAL
8/9/2019 oil water coning
7/8
956
W
FIG. 11 STRUCTURAL CONTOURS
OF EXAMPLE RESERVOIR.
interval
of a
well
does not quite reach the
to p of
i ts
g rid b lo ck . T he f la t i nt er fa ce w ell moqel
no t
only
neglects
upward water c;oning
bu t
also
precludes
any
down coning of oil. Fortunately th e
analyt ical expression that
the con ing
model does
represent
th e latter effect. So in our
earlier
runs
no such shut
In
occurred.
As seen
here
this
k =10 000 md
EACH WELL P RODUCI NG A BOUT
6 BID
OIL
@
JAN.
1967
1
}
.
WATER
WI
~ ~
RGINAL
OIL
WATER
CONTACT
o FIELD DATA
SIMULATOR
CONCLUSIONS
T WATER CUT
100.
1.
The
agreem ent of
more
detailed
numerical
simulations with th e
simple
coning model is good.
2. A
steady state model
should be adequate
for
lllost
studies
where some
deviation
from actual
performance during a
brief
i ni ti al t ransien t
period
c an be t ol er at ed .
incidental capability of
the
coning model ca n
be
quite important.
EAST
9 7
ELL NO. 8
WEST
9570
9550
9540
:: 9560
9580
o
9590
..L . L... . - - - L - I
l
L _
o
3 4 5
GRID
BLOCK
NUMBER
FIG. 14
WATER CUT
PERFORMANCE
OF WE LL
18.
FIG. 12 EAST WEST
SECTION OF RESERVOIR
SHOW-
ING OR IGIN AL OIL WATER CONTACT
AND GRID-
BLOCK AVERAGE OIL WATER CONTACT IN JAN.
1967.
1964
1966
YEAR
o
o
o
o I
/
/
I
I
I
I
I
I
I
/
10 000 . . /
/
/
/
WELL 9
/
10 000 md=k ...
o
F IELD DATA
-
SIMULATOR
- - SIMULATOR
NO
CONING
WATER
CUT
100
WELL 7
o
o
o
o
10 000 md
=k
o
4000 ...
o F IELD DATA
- SIMULATOR
r -WATER CUT
100
1964
1966
1968
YEAR
1970
FIG.
13 WATER CUT PERFORMANCE OF WELL 17.
1964
1966
1968
YEAR
1970
FIG. 15 WATER CUT
PERFORMANCE
OF
WELL 19. DASHED LINE IS PERFORMANCE
WITH THE FLAT INTERFACE
MODEL.
APRIL 1976
7
8/9/2019 oil water coning
8/8
3 The effect of vertical
flow
resistance IS
significant. It has
been
included
in the coning
model
by
th e
use
of an
effective radius.
4
Anisotropy
can retard vertical flow.
Under
extremely
anisotropic condi tions, the basic
assump
tion of
vertical
equ il ibr ium wil l not
be j us ti fi ed .
Nevertheless,
the model
was
shown
to
be
adequate
for
moderate
k
v
h
2
0.1) anisotropy.
5
Installation in a reserv oir simulator h as
subs tan tial ly increased our ability
to represent
well
performance
in
field-wide
simulations o f wat er
dri
ve reservoirs.
NOMENCLATURE
B
formation volume factor,
RB/STB
f
w
water cu t
g accelerat ion of
gravity,
1/144 Ibf/ lb
m
h
height,
ft
hcb height o f complet ion bottom
from
top of
formation,
it
h
height of completion from top of
formation, ft
h
o
average oil-column
thickness,
ft
k
permeability,
md
k
r
relative
permeability
M mobili ty and formulation volume factor ra tio
Nm
t mobili ty/thickness ratio
p pressure, psi
capillary pressure, psi
q rate,
STB/D
qc critical rate,
STB/D
qt total
gro ss
fluid rate,
STB/D
r =
r adiu s,
ft
re
=
drainage
radius,
ft
r
w
= well
radius, it
r effective radius,
ft
S saturation
So r
residual
oi l
saturation
S
wc connate
water
saturation
t time days
u flux, t lday
z vertical
coordinate,
ft
porosity
p
density,
lb /
cu
ft
f viscosity, cp
SUBSCRIPTS
e
=
external
f = final
72
h horizontal
initial
o
oi l
total
thickness
w
water
v vertical
SUPERSCRIPTS
o relative
permeability
a t re si du al s atu ra tio n
of
other
fluid
vertical average
over the
reservoir thicknes s,
except
for h
o
and
In 7; these averages are
taken
radially as
de fi ned i n
Eqs.
A 56
and A 57
REFERENCES
1
Scho1s,
R
s : An Empir ical Formula for t he Cr it ica l
Oi l Production Rate , Erdoel
Erdgas
Z (Jan.
1972)
Vol. 88,
No.1 ,
6-11.
2
Hawthorne,
R
G.:
Estimating
th e
Effect
of
Produc
tion Rate and Tub ing I nt ake Dep th in Water /Oi l and
Gas/Oil Ratios , paper SPE 2748 presented at th e
SPE-AIME
40th
California g i o n ~ l Meeting,
San
Francisco, Nov. 6-7, 1969.
3 Muskat,
M :
Phys ical Pr incip les of
Oi l
Production
McGraw-Hill
Book Co. ,
Inc., New York (1949).
4 Meyer,
H 1 and Garder,
A
0 : :
Mechanics
of
Two
Immiscible
Fluids
in
Porous Media, ] Appl. P hys.
(1954)
Vol.
25, No. 11, 1400-1406.
5 Bournaze1, C., and Jeanson,
B.:
Fa s t
Water-Coning
Evaluation
Method, paper
SPE
3628
presented at
th e SPE-AIME
46th
Annual
Fall
Meeting, New
Orleans,
La.,
Oct.
3 -6 , 1 971.
6
Chappelear, J.
E.,
an d
Hirasaki, G
J. : I
A
Model 0
f
Oil-Water
Coning
for
2-D
Areal
Reservoir Simulation,
paper SPE 4980 presented at th e SPE-AIME
49th
Annual
Fall
Meeting , Hous ton , Oct .
6-9 , 1974.
7 Coats, K H., Demps ey , J. R. , and Henderson, J. H :
The Use o f Vert ica l Equilibrium
in
Two-Dimensional
Simulation of Three-Dimensional
Reservoir
Perform
ance ,
Soc. Pet. Eng ] (March 1971); Trans.
AIME
Vol. 25l.
8 Chappelear, J E and
Rogers,
W L.: Some
Practical Considerations
i n the C on st ru ct io n of a
Semi-Implicit
Simulator,
Soc.
Pet.
Eng.
]
(June
1974)
216
220.
9
Miller,
R
T., an d
Rogers,
W
L. :
Performance of
Oi l Wells in Bottom
Water-Drive Reservoirs,
H
paper
SPE 4633 presented at the SPE -A IME 48th Annua l
Fall
NIeeting, Las Vegas, Nev. , Sept. 30-0ct. 3
1973.
10 Spivak,
A.,
and
Coats,
K
H.:
t lNumerical
Simulation
of Coning Usi ng . Impl ici t Product ion
Terms,
Soc.
Pet. Eng
] . (Sept.
1970) 257 -267 ;
Trans. AIME
Vol. 249.
11
Matthews, C. S. , an d Russel,
D G :
Pressure Buildup
an d Flow Tests in Wells Monograph
Seri es , Soc ie ty
of Petroleum
Engineers
of AIME
Dallas (1967) Vol. l
SOCIETY OF PETROLEUM ENGINEERS JOURNAL