Pressure-TransHent Behavior of Horizontal Wells ...

Pressure-TransHent Horizontal Gas F.J.

Wells

Cap

Kuc!tuk,

SPE,

Behavior

or

SPE,

P.A.

and

Without

Aquifer D.J.

SPE,

Goode,

Schlumberger-Doll

With

of

Wllklnsom

SPE,

5LK.M.

and

Thembyimyagem,

●

Research 5!L

Summary. at the

top

and

pressure

New

analytic

bottom

by

solutions

but

differ

tion

and

are

from use

equations

used

most

of

the

and

solutions

are

horizontal to

include

existing

owing

equivalent

existence

criteria

in

Two

real

typsx

of

are

to

the

wellbore

radius

presented

for

time

and

boundmy

wellbore-storage

solutions

correct

presented

planes.

and

skin

use

of

pressure

for

an

anizotropic

the

as

Laplace

l?ansforms

conditions

are

effects.

various

Solutions

are

averaging

flow

to

horizontal

at these

based

on

approximate

reservoir. pericds

for

considered

flow

can

ci%ur

that

welfs

planes,

the

bt

and

reservoirs

the

uniform-flux,

the

New

Iqw’

line-source.

bdiite-umductivity

Pricds

solution,

welfbofe

(regimes)

during

bounded

Laplace-tmnsfofm

are

identified,

a transient

test.

bottom

boundaries

condiand

simple

Entreduetion Determination has

of

aroused

tensive 2.

literature

Most

Ramey5 and

solve

the

commcudy

used.

avemging it

than

requires

the

no

Another

in

of

elliptical-cyliider

however,

are

Using

these

solutions

At

The are

from

most

in for

the

Iimh

of

times are.

times

wells

with

new

welf

are

to use this

Gcmdc new

analytic

the

effects

of

is

accounted of

different

for, resefvoir

flow

from

~

ptit.

pa-

m

Appendix

lent

With

and

Without

Gas

Cap

or

the

A,

we

discuss

constant-rate

the

case

solutions

will

andlor

solutions

without

then

measured

The

basic

be

well

flow

shown

and

with

welfs skin

constant

for

wefl

in

an

intiniteanisotropic

effect?..

These

weflbare

horizontal tal

planes.

directions

The

afe

bounded

boundaries

considered

of to

be

the

so

k

above in

away

that

during

tbe

test.

denoted

by

kx,

general

case

where

pendices

A and

medium

and

assumed types

top

and

but

permeabilities

kz.

mtd three

in the

in

We

below

the

horizonafe

are

afl

and

k, =kv

The

of

constant

compressibility

medium.

Gfavity

boundary

conditicms

flow and

effects

are are

and

of

to

Well

Cqyright

of Pmdeum

86

we!

Sxieiy

in

Appendix

B.

is

drilfed

in

tbe

close

of

p~ure-averaging a &rivation

used

for

time

and

an

by

using

the

an

some we

the

equivalent

the

presis @Ien

correct

anisotmpic pfessnre

ap-

averaging

technique of

actual

dtifer because

condition instead

and

solutions

literature

the

dimensionless

A,

GUI

with be

. .

PD=(kHM141.2@Pi-P@f other

.

equiva-

formation. (in

field

. . . . . . . . . . . . . . . . .

)l,

dimensimdess

units)

by

. . . ...(1)

. . . . . . . . . . . . . . . . . . .(2)

parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(3a)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . (3b)

ZWD=-(ZWILYJ,

the for

Ap-

considered.

time

l+a). domain,

. . . . . . . . . . . . . . . . . . . ..(3c)

constant-rate over

the

drawdown the

instantaneous

dimensionless is

most

pressure

response,

-—.

conveniently

Green’s

given

funcdon

(see

as

PD, a time

Appendix

B):

a sfightly

vismsity

neglected.

the

integral

isnmpic of

fn

is Two

p~(fD)=2~hD~’Dd,Gx(r)Gy(r)Gz(r),

. . . . . . . . . . . . . . ...(4)

3n the where

WOW al S.Mumberw

batom) a comtant-

nut

for

different”in

a transversely

is usually

or

have

duecdom

solutions

consider

so-

by

they

principal the

we

the bottom

the

develop

permeabilities text

k= =kY =kH

fluid Ih2m3ghout

of

the

B,

write

compressible

The

ky,

(top

weff

top

1 This solution

the

Appendix

wgether

andrWD=(rJ2Lti)( seen are

af.

mmplet-

and

reservoir

far

et

storage and

1 is con.sidezed’to

medium

Daviau

a gas

length

fiD=m(WLfi), ed

dothe

the

rate.

in lVg.

no-

‘Tbam-

Aquifer

horizontal

wellbore-storage combined

downhole

horizontal

for

and

at M

if we

given

tD=0.C602d37k#$pc&; First

wells the

Laplace-rransform

bcnmdaries

cap,

msdify

for

G@xle

because

in

radius

define

the

is discussed

A discussion

welflmre We

Solutions

of

infinite-conductivity

afong

are

important

solutions

the

prewre

as

be

horkmnttd

of

this

tcnmd-

condition by

sec-

of

botuufsry.

develo@ other

the

fefm

the

ccmstant-pressure-tmundazy

one is

methcd

are

and

regimes.

which

for

work

to

may

solutions

presmted

the

such

solution

what

was

(m-flow)

bottum.

in

mobfiw

notation

fonnufaa

the

solutions

and

the

constaht-pressure

flexibility

boundary other

proximate gas

in

from

the

the

these

casJ3

aquifez

we

The

solutions

of

apt

fmm

here, This

the

the

boundaries

presented

sevefal Most

present

different

with

comtanGprewure

at

this

acdve

convenience,

tie but

aquifer

an

model

model,

years,

and

for

or

compared For

no-flow

is at constant

az before

boundmy

high

case), an

14

2 none

is

The

and

few

solution

bottom

paper,

obtain

without effect

of

last

bynayagzm,

to

boundary tup

mcdef.

latter

have

boundaries

no-flow-boundmy

of

mcdel,

pressure

the

bottom.

gas-cap

case

ffow-boundag

ndlow.

the

in the

presented.

A

of

at the is

at the

(the

the

the

solutions

determination of

the

in

work to

and

characteristics

use

forma-

problems.

Ramey3

the

is rhe

it is preferable

the

wellbwe-sturage for

paper

ankotropic

presented

extend and

this

is a no-flow cap

ss

top

been

and

dis-

otier

mobtity

that

for

main.

one

constant-pressure-boundary

During

be-

the

case,

fluid)

model

is at the

have

and

a gas

water

assumes

adapted

approx-

reasons

an

practiced

Clonts

presented

the

in

effects

early

we

horkmntaf

aquifer.

ary

pressure

the

the

the

paper

is

intermediate

These

Solutions

for

and

exact at

elliptical-flow

techniques,

formulas

rameters

radLus

solution.c

for

or

presmtted

wellbore

times.

sufficient

Thambynayagam2

first

lution

solutions

that

at late

the

A.

tie

guarantees

correctly

to

method

is

accurate method.

eq.ivafent

which

new

is more

reservoir

is then

pressure-avemgig

information,

and

Appendix

feature

correct

treated

cap

the

priori’

a

radius,

further

fhe

prefer

which

as

top second

eitier

the

boundary

or

while

(ii

(uni-

the

represent

of to

the

In

Gringar-

bounday-

inner

solution

em

ond

difficult the

equivalent-pressure-point

equivalent-pressufe-puint

cussed

the

an We

wellbore

tion,

infinite-conductivity

either

technique.

cause

of

The

on

both

equation.

condition

a very

condition

tbe

case.

inner-boundary

case,

condbions. pressure,

Ref.

tfam.forms

line-source

poses

in

by

Fourier

ex-

uses

difisivity

finite

An

found

developed

the

sandface)

1+

problem

isotropic

for

a uniform-flux

with

a small

3D used

the

be

fs

wells

yeas.

can

tectuique

the

problem

over

horizontzf

10

wells

intinite-c.mducdvity

problem,

imated

solve

past

horizontal-weU

function

to

pressure

value

the

for

the

hofizontaf

with

ankclropic

Because

behavior over

Tbambymyagam2

the

form

on

dealing Green’s

and

Goode

pressure intefest

survey

work

instantaneous ten

tnmsient

considerable

G.(r)=

lA{erf(l/~)+~[exp(-

I/r)-

l]},

. . . . . . (.5)

34rvlo-, Englnws

GY(7)=l/2~,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...(6)

SPE

Formation

Evatuadon,

bfarcb

1991

TABLE

1-RESERVOIR

PARAMETERS

FOR

EXAMPLES

Examples f

3

d

100

100

100

200

500

5:0

500

5:0

100

100

100

40

Jo 200

-L

kx, md k,, md L%, ft h, ft

Fig.

l-Horizonfai.well

1+2

hD

~

_n2=27

[

COS~COS—

“=*

for

—

hD

hD

‘Xp

h;

the

nc.flow

upper-bounday

case,

-..

0.0dA72

is

define

o.oole37

caze.

at

+rWD.

form

of

F@)

ZD =ZWD tbe

5.6?665

...

7eE9

0.566s!3

0.00197

upper-boundary evaluated

a function

O.&7

. . ..— 0.1 .

0.00289

and

Y2)uzD

solution

fn

is

0.00!

both

given

EqE.

in

530

7 and

Appendii

by

~~2(u)

~2m

o

cos—

hD

. . . . . . . . . . . . . . . . . .. . . . . . . . . ...(9)

“du—

(n-%)mzw~

COS—

“:,

0.8ii43

O.d

bplacedomain

We

F@)=J (n-

::

~:o~-

0.72Sd9

pressure

Tbe B.

. GZ(T)=;

20 0.62240

c.mztant-pressure

the

1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...0) for

the

8,

5

-

5

i46

ImQl

GZ(7)=—

--

10

model.

1and

100

Zw, ft hD .2.,. ;:;

,

9

hD

m j7zWD Then,

pD(sD)=~

f7(sD)+2

‘D [

-(n–r%)zrz, Xexp

. .

.

. .

F(tj)cosJzcos—

~ j=

,

hD

I

hD

1

. . . . . . . . . . . ...(8) . . . . . . . . . . . . . . . . . . . . . . . . . . .

h;

~ ~

● O

~:o++++

~o

.

. . (10)

●

0+

~oflci. o

●.

0+

~o O*

0 O* ~ 0.”

-..

0+ 0

10-~

00+ 0°

$

0

0 0+

0

e 0°

0

elliptical

0

0

(uniform

horizontal

●

+

pressure)

(pressure

averaging)

q~=loo r ~D

10-2

I

I

10-6

= 0.00385

I

I

10-4

.[

10-2

T 10°

tD

Fig.

SPE

Fomadon

Evduadon,

2-Comparison

March

19P1

of

dlmenslonlzz.s

pmssurw

from

an

elliptical

and

a hotfzonfzd

well.

87

2-

2s.5

-. 5 G. .

FICJ.

3–First

early.time

radial

flow

(after

Ref.

2).

I EX.4 0.1-

where

$j

=SD

+ ( j~/hD)2

for

the

no-flow-boundary

GISe,

md ~.4

2bD(sD)

(j–

=;

j~,

YJuzD

(j-

F(gj)cOs—

,

hD

[

hD

‘D

1

Fig. 4-Dimensionless

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(11)

through where

~j

case.

Again,

=$D

sionless

j–

+ [( tie

M)r/hD]

value

faplace

2 for

of zD

variable,

the

is

SD=@+L~S/0.0fD2637kH.

at ZWD +rWD.

defined

seen

from

The

above

equations,

(12)

tion

terms

on

well

ends,

while

the

dmensimdess

the

use

tie

line-source

of

hD,

tD,

approximation,

Z.D,

Eqs.

and rwD.

Because

when

large

tDlr~D

>25.

This

an

10,

11

4,

important

to

satisfy

this

the

and

limitation

are

flow

given

those

in

1),

it

condition.

in tie

y-z

6.

Ref.

tlom

Table

may

For

plan%

Fig. 1

when

take

more

smaller

transient

than

tD,

the

several

true

wlutions

2 compares

in

Ref.

solution

tD

c

0.001.

for

Also,

at

pD

6

for

values

ends

of the

spreading

from

of

from

Appendx

the

formalism

flow

period

a large not

Eq.

A,

flow

the the

at

upper

the and

ends

of

the

term@

well)

and

implies

an

results

earlier

fmm

end

to

than

the

corresponding

the

pressure-point

ex-

100.

high-anisotropy

Note

case

solutions

time

the

deviate

from

contributions

of

horizontal

well

start

affecting

the

rti

the fmt

radial

increases

and

flow the

pericd

conditions

>25

the

tD41

(required

(required

for

be

for

satisfied

the

pseudosteady

effects

of

flow the

simultaneously.

welf

near

Note

ends

the

that

to

well)

the

be

and

first

neglgible)

effects

of

the

inaccurate

each

flow

kH/kv,

ratio, because

with

that

is

anisotropy

develop

flow 10

other

end

from

pressure

develop

earlier

in

the

pressure-averaging

formalism

than

afusual

pressure-pint

methcd

(see

Appmdix

A).

Tbe

effect

of

tirst

radial

beyond flow

the

in

effects

result

is

elliptic

from

kH/kv=

for

this

the

which

d~cussed

the

line

sec-

behavior

in tbe tD >0.1,

flow

radd

well ter

As

from

second

for

cannot horizontal-well

resuft

vahd

because

tD/r~D are

line in the

pressure-averaging

may elliptical

fmt those

boundaries.

For onds

. .

pmssio+

(kH/kvB

anisotropy

is

1

we first

only

Examples

pressure (representing

is a function

for

dimen-

. . . . . . . . . . . . . . . . . . . . . . ..

the

weUbore

derivative

as

lower As at the

pressure

5.

constant-PreSSIIrebWIIdaIY

is evaluated SD,

@

~.,

W%@

Cos—

period

wiU

most

definitely

end

when

at the

wellbore

(and,

as

the

the

nearest

behavior. boundary

is felt

explained

above,

may

end

eadier). Perfods

Flow

ffit The in

duration

Refs. flow

may

develop

which

causes

of

tbe

occur

during ZWD,

Esrly-Time

pressure can

pressure and

cylindrical.

period

becomes

derivatives radiaf

period,

welf

first

radial

of

tbe

arc

in

time,

an

infinite P&ml

dimensionless

and is

can

pressure

as

equivalent The

be

clearly

to

a fulfy

from by

(hD –Z ~D)2 }, provided

min{z~D,

in

4)

Eq.

that

As

welfs

Fig.

3.

is

rD/r3.

bifity,

of

fD 25,

we length

the

this

penetrating,

if

where to

ing

behavior

4

flow

-20g;(&+&)l

flow

indicate

the

radial

(14)

can

Period.

asymptotic

obtained given

shown

first

ef-

1.

elliptic-cylindrical

4 (Fig.

reservoir.

Pfow

the

values

Table

horizontal

=

for as

syn-

that Tbe

in

Pi –Pwf

equation

written

of

the five

wells.

be

vertical

pericds

for

the

radial,

which

flow

pattern

the

before

given

the

may

regimes

Using

Pirst Radial flow

some

in

time

exists,

units

permeability,

flow

horizontal

detaif

some

flow

slowly

various

in

examine

testing

of

3,

more

is observed.

Periods.

1,

or

vertical

parameters

approximately

flow

of

the

fust

discussed

propagate

testing other

After

Examples

the

to

is closely one

smafl

hours

show

very

for

vertical

tD
2.5, curacy The

perind

will

not

appear

for

wells

with

a gas

cap

or

this

approximation

frill

form

height

pressure

of

may becomes

an

Alternatively,

if

the be

solution

estimated equal

‘h= O.Ol&tcbJ@c,.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(22) flow

the

formation

weUbore

sz=-1’’l’g~g[?(’+g)’i’(?)l”

This

and

to

from the

tie

sm%cient must

time

steady-state

be

acused.

at which

the

vnlue

. . . . . . . . . . . . . . . . . . . . . . . . . . ..(28) h is known,

Eq.

28 can

be

used

m

estimate

to

obtain

kv.

aquifer. Downhole Late-Time sure a third

8PE

Radial

frnnt rdlal

Fmmadon

will

Flow.

become flow

pattern

Evatuatinn,

After

a sufficiently

approximately will

March

long

radial

develop

t991

(see

in Fig.

time,

the

x-y

plane

the 4).

This

presand pericd

Because cal

and

storage

Pressure the

early-time

horizontal effects

Response interpretation

penneabiiitks, by

either

measming

is

important

it is desimble the

dowmbole

to

reduce tlow

verd-

wellborerate.

or

using

89

101

z

“++

pwD

A

for

no-flow

‘. :6+

10-1

T

CD=8.94X10-4

+ + . .

o

p’ti

for

no-flow

x

pm

for

constant

pressure

+

p’w

for

constant

pressure

+ + . . {

*

10-2

I 10-*

I

10°

I

fl

102

103

&;

a downhole ed

to

shut-in

the

sandface, mcdel

td.

Furthermore,

wellbore

volume

the

assumption

usual

is LWterjustitied.

in is

used,

below

the

volume same

and

the

last

formation

meabilily

early

flow

rate

volume

and below

anisotropy

to

wellbore

the

measurement

the

becomes and

the

well-lmown

skiII%l

wellbore-pressure

tlom

a verdcal

well

. . . . . . . ..(34)

tbe

and

its

Iaplace

transform

is

~rPsD(sD)

pressure

in

terms

resulting point

is

of

the

from

given

the

.

AD(SD)=

measured

-CD-]

wellbore

by

Pi~(r~-7)d7>

now

The

. . . . (29) 31)

CD

is

defined

by

convolution

provides

by

a constant

reference

flow

~d

skin

q~

h

giv~

In%

29,

given

by

is

bI

tie

te~

Of

derivative

the

m~ur~

Of the

flOW

cO*t-m@

~~

the

Laplace

domain

q.

revme

by wi~

.. ... . ... . ... . .... . ... ..(30) Eq.

for

horizontal

@SD(SD)=@D(SD)+

wells

The

PD

is given

coefficient

by

I?qs.

+cDs;pm(sD) (hD/2)S,

4,

10,

...

pnint.

If

93

qm

is

with msasured

and

1

Elg.

transform

or

35

of

@q.

dmedependedt

should

be

wellbore

used

if

Eq.

and

skin

1 from

Eq.

CbIcoLey

Eq. 35

al.

34,

pressure,

the

effects 34

and

takes

Eq.

was

with

first

the

nu-

Samaniego-V.

however,

becase

p.D,

34

is

10

much

of

more

ID =0.

when

the

In

a given

addition,

computed

5 presents

pWD,

for

involution

a log-log

and

its

as

from

E+

plot

of

derivatives

PWD is always

time, shown 34 the

by

could

be

as

11 the high

dmensiotdess

with

respect

computed

Kuchuk,

as

wellbore

to

In

tD

for

error 1%. pres-

Exnmple

. . . . . . . . . . . . . . . . . . . .(32) 11.

The

dimensionless

storage

Fig.

ap

. . . . . . . . . . . . . . . . . . . . . . ..(33)

the wellbore at

st of

from

from

words,

from

and

5 shows

that

volume

wellhead

below and

is

the constant,

measuring @.

i?. masked 29

the

nc-flow

Fig. (ii by

5 also

this the

ZWD =0.12649)

a very

pressure

effect.

bounday C is defined

Laplace

The

Daviau mcdel

than

other

dimensionless

where

34

pWD

and

q..

solution

wellbore-storage by

ofpWD

time in

pWD

in

is

CD=5.615C/2r@c~~,

with

computation

1 0ZD=0.63240 and

its

the

measured.

wsllbore-stomge

sure, where

or

Q.

is not

computed

starting

, . . . . . . . . . ...”.(31) 1

29)

volume,

rate

29 becomes PsD(sD)

[

wellbore

for

conditions.

numerically

VW,

bwD(SD)=sD@M(5D)

(l?+

framework

flowrate

computer In

total

constant

rate

merical

Pb(O=Pb(O+(w2)w).

the

the

integral

a genersl

downhole q,.

from

internal-boundary

pWD is nondimensionaliied

. . . . . . . . . . . . (35)

. . . . . . .. . . .

cDs&%D(sD)

1+

is nondmensionalized

qJq~

storage

this in

effective

the

PwD(tD)=~*D[l-CD-]p~(tD-r)dr,

where

where

with

shut-

volume

resulting

for reduces

storage

Pw~(t~)=~’D[qti(r)

solution

O:

=.

wellbore the

or

wellbore

effects

those

limit-

and

measurement

considerable

.stornge than

the

times

dimensionless

ke

is

gauge

constant-storage

downbole

still

longer

because

at

The

will

storage

downhole

a single-phase,

when

there

toOl,%10

typically

the

of

Even

however,

if wellbore

between

case

with

pronounced or

its

shows it is

derivative that

closer

tim

pressure,

and

effect

tbe

of

wellbore

effect.

SPE

without exists

with

the to

wellbore-storage

boundary

and

difference

Thus, or

Formation

its

a gas

without the

cap.

between the

lower

than

the gasc-

nc-flow

the

gas

cap)

identification derivative,

Evahmtion,

of appears

March

1991

~cuk

when

ence to

of be

wellbore

a nc-flow

known

sient

stotage

dominates.

boundmy

it3 dktance

for

a unique

independently

weU-te3t

Jn Ois

and

situation,

from

the

tie

interpretation

of

ft =

pres-

wellbofe

have the

ij

tran-

data.

Conclusions New of

solutiofM

Lsplace

tal

either

a no-flow

of

Although

we our

because the

of

the

various

flow

~ui~~ent

flow

in

by

two

the

top

=

dummy

=

porosity,

condi-

H

= horizontal

bottom

j

second

m

sys-

other

the

those

pressure well

Tbi3

of

for

horizontal

=

fiti~

.

ma~~

for

vetdcal

in a reasonable

sjbe

= time

to feel

far

Snbe

=

time

to

neaf

=

steady

in

k

flow

to

the

we

be

well

used

in

by

be

and

are

identified, for

a tmmsient

test.

weUs

at unknown the

new

fmm

and

flow

me

simple

periods

Ahfmgh

of

to

can

flowing

~

~mdmate

and

obtain

With

and

regime

“

=

Laplace

image

plzne

*

=

pressure

point

coordinate

‘ =

radial

wellh-m

not

also

that

solutions,

A

multiple

increase

time

large

(automafed

It is

lyze

weU-test

data the

2.

.

onnDlexitv

is

because

of

of

in

to

develop

principle,

cutves

use

for

nonlinear

different

Iafge

be

of

4.

to

of

the

6.

anaFor

7.

solutions S.

work.

1987)

Cloti, Wells

With

1986:

SPE

Ozkan.

a

=

D.

b = horizontal =

total

C

=

wellbore-storage

G

equivalent

axis

Ct

F(13)

of

of

equivalent

cnmpressibifity,

= fiction

defined fimction

by

weU,

eU@tical

atm

-1

ft

[m]

well,

ft

[m]

RB-psi’1

[ms

.Pa-l]

9

f!.q.

on

Wells,,,

pap,

thickness,

Kllchuk,

fi [m]

k = permesbUity, K.

md

= mcdified

K,2

[m2]

Bessel

= second

Richardson,

function

integral

of

zero

order

12.

= weU =

Fo”ljer

~~fO~

vfiable

n

=

Fourier

tisfo~

v~able

p

= pre3sure,

[m] 13.

dia, 14.

Ap g

psi

=

pressure

=

VOhUIJetfiC

Pa]

drop,

VeJue

Media M.:

Grin@rtm,

A. C., Solution

(April

Pa]

rate,

B/D

1975)

=

cOnstit

r

=

~i”~,

s

=

dimensionless

S= t

=

time,

=

integration

=

Cartesian

~

SPE

reference [m]

flow

rate,

B/D

H.J.

1970)

at the

DenveJ,

to

Wetf Test “All

blV.%ti-

Trans.,

of WdtLmre of

Flow. AJME,

Storage

Vertically

SPE

fnter-

Liquid

279-%

1977

Oct.

Transient 15671,

Jr.:

in Unsteady

Behavior

presented

Function

Simplifies

Rarmy,

. F. : ,’Effect

Pressure

p~

Curves

Effect

(Sept.

—..

Systems,s,

and

Fmdured

Annual

Technical

9-12.

Re$wre

available

from

solutions

SPE

With

Book

Order

‘well.

Dept.,

(1986). and

Hammond,

Problems

(19990)

P.:

‘*A Pemn’bafion

in Pressure 5,

and

140-48;

Ramey, a Partially Trans.,

Medmd

for Mid

Testins,-,

Tmn.vxm

60P-36.

F/c o w of Honweneow Book Co. fn~., New

for

Transient

H,J.

Ffuid$ Ymk

Jr. : “An

Pmeuafing AIME,

Pom!As Me-

Jbnwh

City

(1937).

Approximate Lim-,%mrce

Lntinite

Ccm-

SPEl

Welt;,

-—..

239.

tramform

A-TheoretlcaI

Conslderatlons

[m3k] We

Laplace

SPEI

vzriable

use

dimensionless

XD=XLLA,

variables

(written

in

consistent

units).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

(A-la)

skin

u xsY,z

t(t)

ft

Skin

, SPEI

SPE

omen’s

[rn3/s]

Appendix %

Jle

McGraw-Hill

ductivity

psi

flow

D.J.

Porous

Muskat,

and

EU@d

of the 1465-70.

1983). R., and

%emmliied pa~r

and

__..

of Type

Set

Exhibiticm,

TX

Wilkinson,

in

L

ft

kind,

Boundary

m

length,

second

K.

of

F.J.:

Storage,”

he

“Application 19S0)

New

6752

of Source

2S7.

in Reservoirs,,,

.4._,

SammiegwV,

SPE

Use

..1Trarsieet~zww:

(Aug.

Tran.si.mt

and

‘The

pres-

Tmns. , A3ME,

Problems

i-m.,

Storage

and

the

Conference 11.

h = reservoir

H.

Dau@ge

2-4.

?= . ..-. —..

A.:

Treatment,,

249. 10. CirI~Ley,

Jr.:

April

at the

‘TJoriwntd-WeU

1989) ’567-7%

W. E.,

“A

of WeUbere

S. D.:

for

p~”ti

, AfMF

401-l@

et al.:

and

SPSFE

.batysis

15116

Oakland,

Josbi.

HJ.

Trans.

JPT

S SPE

MedrIg,

worM Oil Way R.G. , A1-H”sminy,

1. AnrJylical

ma-I]

coefficient,

=

Green’s

eUiptical

paper

and

RauEy,

B@am,

Drawdown Media,.,

‘Tressure-Tramient

Unsteady-Flow

1979)

D.

gation

axis

and

md KumaI, Analysis:,

Bourdet,

JJr.:

(&c,

285-96

F. and

9. Asam’’al, vwticd

SPEti

‘Tressure

2JW —..-, -—.

R..

in SolviLg

Tkb. ferewe

SPEFE

Wells,,,

~.. H.J.

Regioml

Rae.havan.

(Dec.

Horizontal

in Ankotmpic

Drainhole,,,

California

1973)

Wells

Trans., Ramey,

Hmizmual

SPEJ

for

R. K. M.:

of Hm-immat

and

E..

Kucuk.

Analysis

Tbambymyagam,

683-97;

M.D.

~Ysis.” Nomenclature

md

Amlysis

(Oct.

psmmetefs.

implementation

P.A.

Funcfims

horizontalestima-

‘sPressure

71&24.

sure Analysis,’; 5. Grinsarten, A.C.

pre3-

however,

number

used

effects.

least-squares

matc~g),

the

this

can skin

computer

festure

a constant-pressur~ solutions and

type-curve

efficient

inqmrtant

to

the

Ciooie,

(D&

builduv

weUbore-storage

possible,

better

of

and

et al.:

1988)

Buildup

.

the

hours

=

Euler’s

=

Difac

Formadon

YD=G(YILS),

[seconds] vzfiable ccofdinatm,

andzD=w(zlLfi). n

delta

Evaluation,

function

March

andhD=~(h/Lti).

1991

. . . . . . . . . . . . . . . . . . . . . . . . . ..(A-lb) . . . . . . . . . . . . . . . . . . . . . . . . . ..(A-lc)

[m] ZWD=~(ZW/LH)

conwant=O.577215

.

derivative

F.

(Dec.

boundaries

drawdown

witbout form

type

paramefms.

to

witb”~d

pressurederivative

reason,

amdied

include it

metbcds

an

be

. .

,s,~,,”m=

develop

ditlcuk. of

(drawdown)

indicators

infinite-acting

may

interpretation

Lsplacs-donmin

well

is

this

existence

the

can

tion

this

that

makes

weUs

the

solutions

tbwe

sure

fact

pressure

occur

late-time

the

weU

supsr3Rfpt3

w

3.

solutions

boundary

Stiti

=

x,yhz

equations

that

the

is analogous

potential

tbm

horizontal

boundary, to

the

dktamxs

boundary

afcor-

interpretation.

These tests

the time

and

feel

gives

the

the

=

wf

ex.

ends

identified

w

using

to

radius

near

addition,

by

lJelieved12

wellbore

which Jn

presented

wells,

ratio

state

V = vertical

the

weU

than

SS

other in

at the

rather

technique

radius

testing

anisotropy

steady

was

like

appearing

response

length

vanishing

way

solution,

from

the

of

are

course

to reach

= dimensionless

(the

the

= time

1. Daviau,

of

s]

fraction

D

the

pressure while

line-source

(regimes)

criteria

period

pericd

variable

r d

cbp

hodzon-

and fn

at constant aquifer)

form

bounday

the

condition).

active

regimes.

pericds

the

flow

and

formation.

existence

dufing

both

somewhat

weUbo=

amsotmprc New

~a.

vtiable

before.

point.

indkation

e

in

Two

case,

was

an

along

liit

the

or

estimate

the

fects

and

we

times

bottom.

first

uniform-flux

pressure

SU

and

time

bounded

(no-flow

differ

pressure

equivalent

a better

the

real

welk

boundaries

a3

use

literature

top the

cap

results

averaging

at

the

a gas

boundary,

authors,

act

the

impemmable

one bad

in

horizontal

3iI

were

either

an

at

considered.

boundaries

presented

for

planes

are

case,

an

transforms

boundary

tem

cp

dmy

Subscripts

analytic

tions

viscosity,

=

..” . . . . . . . . . . . . . . . . . . . . . . . ..

(A-24

. . . . . . . . . . . . . . . . . . . . . . . . ..(A-2b)

91

tD=%t&’PCjL;

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-3a)

. .

amisD=@c&&lkx

. . . . . . . .

.

.

.

.

comepseudmteady,

and

s&tdonhas

xa,ndpre

unifortnfbz

variables,

mation

with

tbe.diftiion

unit

equation

diffusion

is that

of

an

isotropic

for-

coefficient:

it

clearly

not

taneous

at~

.,

ax;

by:

is

mrrext

true.

zlso

This

Green’s

is

line

at

most

fiction

uniform-flux

‘=%%ti.

very

early

.ssure.Hen

times,

when

the

true

is

very

ceitisreasonableto

(A-3b) Zswmetbzt

fiI these

akoat

intermediate

eazily

by

(dmemionkss

source

in

an

times.

seen

the

instan-

respw)

for

a finite

pulse

infinite

This

examining

medium

– r~~

. . . . . . .. . . . . . .. . . . . . . . ..(A-4) G(xD,tD)=~G,(xD,tD)exp—

az;

.

. . . . . . . . .

.

(A-9)

4tD For

drawdown

sure

at conmnt

fieldpD

a dimensionless

preswhere

(see

Appendw

(

Gx(xD,rD)=~

. . . . . . . . ..(A-5)

;pD(xD.yDJD>tD~wD.hD).

The

boun~’

a very we

condition

that

the

diffkxdt

use

dit%culty best

is

to

of

it

how

to

to

be

line

write

decoupled

from

in

that

sandface.

Thus,

the

this

The

the

the

k

long

well transverse

the

TMS achml

but

longitudinal

x

that is just

simply how

like

a fully

in

tme

da

(1 –x32,

the with

its

z directions

1M%

the

ly

direction.

Direction.

original

circufar

h

well

the

above

occupies

transformed

coordinates

tbeelliptiwl

the

but

@t. the

in

This

until

is not

the

that

This

the

effect

weU

felt

is

metbcd

in

which

or-

nearest of

not

devi-

sudden-

fite

in

is automadcaly

we kmk

of

end onset

be

weU

cor-

to times

the

should

solu-

if

continue

correc-tbe

solution

en&.

to

the the

dinwnsimdess

physically

realization

we12

and

character,

wifl

time

penetrating as

in

solution

source

%,

Because

exponential

the

pressure-averaging

~J

2

length

accounted

Gx

becomes

‘dzDGz(xD>fD)

2-,

–l