Cap or. Aquifer. First we discuss the basic solutions for horizontal welfs for the ...... knit of vanisbingly small wellbore radius. We prefer this method for several.
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