John R. Fanchi - Principles of Applied Reservoir Simulation, Second Edition

42

Principles

of

Applied

Reservoir

Simulation

dx\

-'

jfl

,<,K

^c

I

(5,12)

*

i

The

derivative

(dxidi)

Sw

is the

velocity

of the

moving plane

S

w

,

and

(dfJdS

w

)

Sw

is

the

slope

of the

fractional

flow

curve.

The

integral

of the

frontal

advance

equation

gives

r

-

W

<

(

dL

\

s

-~7*

\7s~l

(5J3)

\

'

$„

where

x

Sw

distance traveled

by a

particular

S

w

contour [ft]

Wj

cumulative water

injected

[cu ft]

(df

w

/dsj\

slope

of

fractional

flow

curve

Water Saturation Profile

A

plot

of

S

w

versus distance using

Eq.

(5.13)

and

typical fractional

flow

curves leads

to the

physically impossible situation

of

multiple values

of

S

w

at

a

given location.

A

discontinuity

in

S

w

at a

cutoff location

x

c

is

needed

to

make

the

water saturation

distribution

single valued

and to

provide

a

material balance

for

wetting

fluids. The

procedure

is

described

by

Collins

[

1961

] and

summarized

below.

5.2

Welge's Method

In

1952,

Welge

published

an

approach that

is

widely

used

to

perform

the

Buckley-Leverett

frontal

advance calculation.

Welge's

approach

is

best

demonstrated using

a

plot

off

w

vs

S

w

(Figure 5-2).

A

line

is

drawn

from

the

water saturation

S

w

before

the

waterflood

-

irreducible water saturation

S

wirr

- and

tangent

to a

point

on

thef

w

curve.

The

resulting tangent line

is

called

the

breakthrough tangent,

or

slope.

It is

illustrated

in

Figure 5-2. Water saturation

at the flood

front

S^is

the

point

of

tangency

on

thef

w

curve. Fractional

flow of

water

at the flood

front

is/^and

occurs

at the

point

of

tangency

S^

on

the/

w

curve.

In

Figure 5-2,

S

wf

\s

65%

and/

w/

is

95%

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Average

water saturation behind

the flood

front

S

wbt

is the

intercept

of the

main

tangent line with

the

upper limiting line

where/,,

=

1.0.

In

Figure

5-2,

average

5^

is

67%.

0 0.2 0.4 0.6 0.8 1.0

Figure 5-2. Welge's Method

In

summary, when water reaches

the

producer,

Welge's

approach gives

the

following results:

•

Water saturation

at the

producing well

is

S

wf

•

Average water saturation behind

the front is

S

wbt

•

Producing water

cut at

reservoir conditions

isf

wf

Other

useful

information about

the

waterflood

can be

obtained

from

Welge's

approach.

The

time

to

water breakthrough

at the

producer

is

.

LA*

"

9,

(4W)

V

(5J4)

where

q

t

injection rate

(df

w

I

dS^\

slope

of

main tangent line

V

'

S

wf

L

linear distance

from

injection well

to

production well

Cumulative

water

injected

is

given

by

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Principles

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Applied

Reservoir Simulation

Qi

=

where

Q

i

is the

cumulative pore volume

of

injected

water.

The

slope

of

the

water

fractional

flow

curve with respect

to

water saturation evaluated

at the

water

saturation

at

breakthrough gives

Q

(

at

breakthrough.

Effects

of

Capillary Pressure

and

Gravity

In

the

absence

of

capillary pressure

and

gravity

effects,

the flood front

propagates

as a

"sharp" step

function,

or

piston-like displacement.

The

presence

of

capillary pressure leads

to the

imbibition

of

water ahead

of the front.

This

causes

a

change

in the

behavior

of

produced

fluid

ratios. Rather than

an

abrupt

increase

in WOR

associated with piston-like displacement,

the WOR

will

increase gradually

as the

leading edge

of the

mobile water reaches

the

well

and

is

produced.

In

addition,

the WOR

will begin

to

increase sooner than

it

would

have

in the

absence

of

capillary pressure.

By

contrast, gravity causes high

S

w

values

to lag

behind

the front. The

result

is a

smeared

or

"dispersed"

flood front.

5.3

Miscible Displacement

Buckley-Leverett

theory treats

the

displacement

of one fluid by

another

under

immiscible, piston-like conditions.

An

immiscible displacement occurs

when

the

displaced

and

displacing

fluids do not

mix.

The

result

is a

readily

discernible interface between

the two fluids. In a

miscible displacement,

the

fluids

mix

and

the

interfacial

tension approaches zero

at

the

interface.

A

miscible

displacement system

is

described

by a

convection-dispersion (C-D) equation.

As

an

illustration, consider

the

one-dimensional

C-D

equation

for

the

concentra-

tion

C of the

displacing

fluid:

n

d

2

C

BC

dC

D

v

= —

f5

16\

dx

2

dx dt

IJ

'

10J

We

assume here that dispersion

D and

velocity

v are

real,

scalar constants.

The

diffusion

term

has the

Fickian

form

D'd

2

C/dx

2

and the

convection term

is

vdC/dx.

When

the

diffusion

term

is

much larger than

the

convection

term,

the

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vdC/dx.

When

the

diffusion

term

is

much larger than

the

convection

term,

the

C-D

equation

behaves

like

the

heat conduction equation, which

is a

parabolic

partial

differential

equation (PDE).

If the

diffusion

term

is

much smaller than

the

convection term,

the C-D

equation behaves

like

a first-order

hyperbolic PDE.

The C-D

equation

is

especially valuable

for

studying numerical solutions

of

fluid flow

equations because

the C-D

equation

can be

solved

analytically

and

the

C-D

equation

may be

used

to

examine

two

important classes

of

PDEs

(parabolic

and

hyperbolic).

To

solve

the C-D

equation,

we

must

specify

two

boundary conditions

and an

initial condition.

The two

boundary conditions

are

needed because

the C-D

equation

is

second-order

in the

space derivative.

The

initial

condition satisfies

the

need

for a

boundary condition

in

time associated

with

the first-order

derivative

in

time.

The

boundary conditions

for the

miscibie

displacement

process

are

that

the

initial

concentration

of

displacing

fluid is

equal

to one at the

inlet

(x

=

0), and

zero

for all

other values

of

x. The

mathematical

expressions

for

these boundary conditions

are

concentration C(0,

t) =

I

at the

inlet, concentration

C(°°,

f)

= 0 at the

edge

of the

linear system

for all

times

t

greater than

the

initial time

t = 0, and the

initial

condition C(x,

0) = 0 for all

values

of

x

greater than

0.

The

propagation

of the

miscibie displacement

front

is

calculated

by

solving

the C-D

equation.

The

analytical solution

of the

one-dimensional

C-D

equation

is

[Peaceman,

1977]

C(jc,

0

=

-U

erfc

2

l

JC

- Vt

X

+ Vt

2

]

/Dt

where

the

complementary error

function

erfc(y)

is

defined

as

(5.17)

2

r

2

erfc(j)

= 1 - — je

z

rfz

(5,18)

V*

o

Abramowitz

and

Stegun

[

1972]

have presented

an

accurate numerical algorithm

for

calculating

the

complementary error

function

erfcO/).

A

comparison

of the

analytical

solution

of the C-D

equation with numerical solutions

is

given

in

Fanchi

[2000].

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46

Principles

of

Applied

Reservoir Simulation

Exercises

Exercise

5.1

Consider

an

oil-water

system

in

which

oil

viscosity

is

0.64

cp and

water

viscosity

is 0.5 cp. Oil

relative permeability

(k

row

)

and

water relative

permeability

(Ar

w

)

are

given

in the

following table

as a

function

of

water

saturation

(S

w

).

Complete

the

table

by

using

the

viscosity

and

relative permeabil-

ity

information

to

calculate

oil

mobility

(A,

0

),

water mobility

(A.

w

),

total mobility

(1,),

water fractional

flow

(/"„,),

and oil

fractional

flow

(£,).

Total mobility

is the

sum

of

oil

mobility

and

water mobility. Assume absolute permeability

is 100

md.

s

w

-

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.80

*n.

0.000

0.005

0.010

0.017

0.023

0.034

0.045

0.064

0.083

0.120

^

row

1.000

0.590

0.320

0.180

0.080

0.030

0.010

0.001

0.000

^

k

*,

L

Exercise

5.2

Plot

X

0

,

X

w

,

and A, in

Exercise

5.1

as a

function

of

S

w

.

What

is the

mobility

ratio

of the

oil-water system? Hint:

See Eq.

(3.14).

Exercise

5.3

Piotf

0

and/,,

in

Exercise

5.1

as a

function

of

S

w

.

Use the

plot

of

f

w

versus

S

w

and

Welge's

method

to

determine water saturation

at

the

producing

well,

average water saturation behind

the front, and

producing water

cut at

reservoir conditions.

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Exercise

5.4 Run

EXAM3.DAT

and

plot water saturation

as a

function

of

distance between wells

at

the

midpoint

of

the

ran and

at

the end of

the

ran.

Hint:

water

saturation

is

reported

in the run

output

file

WTEMP.ROF.

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Chapter

6

Frontal

Stability

The

stability

of a flood

front

can

influence

the

efficiency

of fluid

displacement.

A front is

stable

if it

retains

the

shape

of the

interface between

displaced

and

displacing

fluids as the

front

moves through

the

medium.

An

analysis

of

frontal

stability

is

presented

in

this chapter

in

terms

of a

specific

example

-

the

advance

of

a

water-oil displacement

front in the

absence

of

gravity

and in the

presence

of

gravity.

The

stability

of the

front

is

then studied using

linear

stability analysis.

6.1

Frontal Advance Neglecting Gravity

The

displacement

of one

phase

by

another

may be

analytically studied

if

a

linear, homogenous porous medium

is

assumed.

Let us first

consider

the

displacement

of oil by

water

in a

horizontal porous medium

of

length

L. We

assume

piston-like displacement

of a

front

located

atx

f

.

Application

of

Darcy's

law

and the

continuity equation leads

to a

pressure distribution described

by

Poisson's equation.

The

absence

of

sources

or

sinks

in the

medium reduces

Poisson's

equation

to the

Laplace equation

for the

water phase

pressure:

£3

D

=

0,

0<x<x

r

(6.1)

dx

The

corresponding

equation

for oil

phase pressure

is

48

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49

3

f

= 0,

x

f

<

x < L

(62)

Equations

(6.1)

and

(6.2) apply

to

those parts

of the

medium containing water

and

oil

respectively.

They assume that

the fluids are

incompressible,

and

that

the

oil-water interface

is a

piston-like displacement

in the

^-direction.

The

piston-

like

displacement assumption implies

a

discontinuous change

from

mobile

oil

to

mobile water

at the

displacement

front.

This concept

differs

from the

Buckley-

Leverett

analysis

presented

in

Chapter

5.

Buckley-Leverett

theory with

Welge's

method shows

the

existence

of a

transition zone

as

saturations grade

from

mobile

oil

to

mobile water.

The

saturation

profile

at the

interface between

the

immiscible

phases

depends

on the

fractional

flow

characteristics

of the

system.

The

present

method

has

less structure

in the

saturation profile,

but is

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