oil water coning

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


2/8


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


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


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


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



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


8/8

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


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