Analysis
of
Pressure
and
Transient
F.
Ku?uk,
L.
Ayestaran,
Simultaneously
Sandface
Well
SPE,
Schl.mbc%er
SPE,
Measured
Flow
Rate
in
Testing
Well
@
Services
Schl.mberger
Well
5R!5
Services
lal’71
SUrmnary
New
well
test
eliinate
methods
and
interpretation
wellbore
use
fluence
function)
Jkmdface
can
flow
storage-free
the
and
to 6stimate
system
synthetic
tion
also
are
that
(unit
response
Darcy
r“te
saturation
canbc
in-
parameter
The
testing
(radial
multi-
estimation
and
and
to tbe
must
+
imate
beginning
measured
rate
from
.comstant
rate
skin,
at?
a
convolu-
period.
The
ternal
ing
to
effect
end
of
of
of
and
average
its
ampIe,
the
and
divided
the
into
sake
the
of
be
and typecurve
capacity
(k/z),
The
behavior
of
according
pxessuze.
These
and
ex-
of a well.
these
for
the
test
to which
periods
the
to deterskin,
and
influence
of
time
these
over
can
be
acting
tions,
defined
f9.33
sww
1985
.f
P.t,.h+
The
combmed
effects
of
and
pseudoskin
(which
include
EWW.IS
the
and
as
and
in pressure
wellbme
par-
tink
coefficient
fluid
higher
than
for
is
of the
geater
is
ticukrly
during
weflbore
storage
ones,
during
the
the
of
may
the
not
alone
the
be
‘wellbore
drawdown,
be
the
correct.
makes
the
com-
10 or even
of the
the
fluid
piessure
depti,
can
that
shut-
of the
Since
the
period.
tests,
compresiibJl-
compressibility
during
coefficient
thus,
tie
in the
@e test
buildup
compressibility
assumption
buildup,
by these
under
for
wellhead
use
Eafkmgher
(density)
wells.
at the
1979,
namely
presented
developed
rapidly;
the
as welf
In
that
5 and
curves
to
applica-
1.
earlier
constant
fgnction
constant
Ref.
compressibility
a
the
introduced
type-curves
were
fluid
than
T@,
in
the
type
producing
behavior
methods,
found
pamicularly
ve~
behave
theories,
4 McKinley,
remains
time,
been
The
new
others,
con-
intinite-
heterogeneous
type-curve
be
as
behavior.
radial
have
the
incrkases
wellbore
the @ttOm.
All
tests
reservoir
reservoirs
problems.
et al.,
~e
reservoir
can-
period.
than
many
ty is usually
the. middle-
buildup
the
of the
of the
types.
early
each
bottom-
sometimes
tie
early-time
can
anmdus
the
the
or
approaches
3 Agar.val
During
ex-
on log-
of
during
storage-free
most
the
of these
that
start
of
pammetetiation
tubing
For
from
effects
approach
ez al. 2 introduced
and
periods
the
may
the
elaborations
assumption
in
cap
” most
references,
au~ors,
skin.
boundary
line.
effect
these
drawdown
because
Kemch6
is
used
and
outer
straight
of
semilog
during
some
and
flow
are
pressure
period,
to homogenize
&pe-cume
different
100
skin,
may
all.
during
WY”
initial
tbk
radial
a curve-flattening
gas
the
at
period
pr&ibfity
+nmge
also
techniques
particularly,
tend
place
in the
phenomenon
are
porosity
period,
semilog
separation
words,
Rhmey,
Dur-
of
recognized
techniques
@mage
gas
pressure
plots.
app~,ed
Gringarten
wellbore,
shows
Thus,
today
and
the
andtor
other
as
each
influx
The
stat
of a wg?fl progresses
convenience,
periods
Period.
REBRU.4RY
cap
the
overcome
,
EarIy-T@
mwmt
gas
period.
be
thii
During
distort
homogeneously
effects),
may
the
data,
water
takes
effects.
bebavior
must
rate
infinite-acting
also
by
period.
semilog
kh
is impossible;
In
should
complexities,
pressure.
~
is affecting
follows
storage,
to
Homer
ducted
ra&d
well
storage
storage
otier
not
compressible
radial
dud
this
During
Period.
start
Furtbeirnore,
or at a zero
boundaries
transient
on transient
For
the
the
and
formation
effects
approiimatc
of
~pacity
dtition
formation
phenomena
the
pressure-time
aPPEcatiOn
Of SeMilog
mme
formation
flow
time.
Period.
Late-Time
time
reservoir
during
(weflbore
reach
rate
approx-
estimating
ptmneabili-
sandface
wellbore
affeqt
aualysis
Tfds
period,
rim-wellbore
boundaries
phenomena
tbk
be
The
single
Because
storage
geometry,
the.
for
at the
outer
the
the
a
pressure
string
@e
gener~,
wellbore
rate
can
and{or
even
During
production
the
In
obtain
is condu~ed.
a long
before
calcuk
sufficient
rate
engineering.
to
formation
conkant
in the
it (alces
been
test
flow
measurements.
period.
if a buildup
fluid
to
initial
infinite-acting
produce
time
pressure
has
and
sandface
of reservoir
over
constant
parameter
ty,
measured
nOn-
caused
dominate
during
Conventionally,
Sometimes
with
wellbore)
pressure
to determine
presented.
testing
the
fractures,
reduction
stratification
wellbore
es@blished.
log
traced
The
aciduing,
permeability
around
affect
Introduction
Welj
and
Middfe-Time
to iden-
is under
deconvolution
perforation,
flow,
of
data.
analyzed
that
penetration,
behavior.
or
deconvolution
Convolution
for
tkd.
formation
pressure
pattern)
reservoir
examples
that
new
flow
shown
from
behavior
flow
These
sandfuce
wellbore
its parameters.
for
is
obtained
and
(reservoir
methods
few
It
effects
be
rate
formation
ti&
rite)
Ma.
storage
presented
effects.
measured
pr~ure
without
are
(aftertlow)
simultaneously
weffbore
behavior
methods
storage
fluid
at
storage
and
pal-
A variable
application
323
of type-curve
tion
of
methods
variable
wellbore
“geometry
~tching
process.
tiY
not
cume
of
of
can
purpose
of
@ transient
work
flow
pressure
and
to
study
storage
the
we
(drawdown
use
curves
or
the
Theoretical
regard
During
the
developed
use
interpretation
of
(buildup
case)
me
the
case).
not
riew.
Hurst7
tie
sandface
To
oti
were
rate
they
approximate
rate
in transient
knowledge,
the
flow
flow
first
to calc~ate
van
to
estimate
and
wellbore
pressure.
the
stidface
testing
flow
use
the
sandface
buildup
test
attempt
flow
rate
Homer
stmight
to use
data
in
method
line
weU
can
earlier
km
be
type-
irdcates.
by
do
the
4 or 5 decades,
transierit
many
fluid,
theorem
flow
multiple-rate
superposition
solutions
through
(Dahamel’s
for
have
porous
been
media.
tieorem)
has
been
time-dependent
boundary
con-
boundary
conditions.
For
time-independent
the
testing
is a special
application
theorem.
classic
paper
Everdingen
wellbore
sandface
To
rate
fo?
successful
the
@e
used
data
sandface
a semifog
solutions
from
the
van
and
last
@derive
fn their
is
Everdingens
the
first
that
1 I%-cycle
for
ditions
of
of
of
superposition
used
Background
use
17 have
Developments
example,
The
the
al.
pressure
the
showed
to ‘&ach
use
of
with
explore
afterflow
been
measurements
modilied
type-
has
They
et
with
data.
sense
$1 the
with
of
sbnuftaneous
Furthermore,
of a well
use
pressure
in a broad
deconvolution
beliavior
wellbore
and
is
rate
weU”testing.
of convolution
or
this
rate
direct
afone
Metier
This
testing.
the
by the
flow
sandface
with
efiiated
&ta
recently,
measurements
amafysis.
storage.
inherent
be
flow
with
type-curve
pre.wure
wellbore
More
combina-
storage
the
wellbore
changing
sandface
The
wellbore
complicates
problems
methods
measured
impossible.
constant
Moreover,
the
measured
‘fhe
even
timber
indicate
Some
afmost
or
~d
pressure
on unsteady-state
Hurst’s
for
flow
presented
the
a continuously
problems,
dimensionless
varying
flow
rate
as
this,
forr+nda
(b)
PwD(b)=mJ(f9Pso
. . . . . . . . . . . . . . . . . . . . . ...(1)
qsf=(’-’-”o$
+
where
the
13 is a positive
constant,
voir
cm
parameters.
volution
an
~,
expression
for
storage
method
to determine
measuring
buiklup
the
(vimiible)
these
only
is
constant
assuined.
techniques
direct
solution
or
alternative
This
Hutchison
PwD(tD)
unit
from
g fie
the
field
yeys.
and
van
tests
afso
of
of the
pwD
data
for
to combine
also
production
1s also
constant-rote
used
solution
and
*1%-PM$l!
de#s
0.0002637kt
is
~D =
qipctr$
inner
psD
12 and
Coates
g the
The
=
variable
rate
pressure
pressure
rate
extended
and
PD(tD)
=
well
proc-
to obtain
~e
with
variable
measurements
data
steady-state
sf&
skin
without
for
wellbore
effects,
factor,
to
et al.
a known
qD
15
(tD)
qi(b)
=
!hf(tD)fq,
q,
=
dqD(tD)/dtD
,
=
reference
flow
pressare
,
constant
technique
as
techniques
rate
and
pressure
case
a constant-rate
deconvolution
buldup
constant-rate
storage
inffnence
Bostic
sandface
dimensiodws
to use
data
(the
testing.
history
first
the
the
‘decon-
pressure
behavior
deconvolution
from
and
the
PD+L$,
er
influence
is cafled’
perhaps
tecbni@e
a variable
They
an
diffusivity
aquifers.
function
=
constant-rote
function)
determ.inin
formation’in
a dwonvolution
history.
by
can
so far
the
to the
et af.,
14 were
constant-rate
from
obtained
rate
“convolution.”
influence
Poolen
deconvolution
solution
be
where
The
the-dependent
solutiom
for
30
S =
and
fiction)
can
+~fZD(7h’D&r~)dC . . .. . . . . . ..(zb)
”
the
2a
on
test.
response
11 Katz
methods
~ectly
compute
to E.q.
as
constant-rate
mentioned
is calfed
.%kora,
last
is measured
words,
with
process
and
of determining
Jargon
parts
‘qD(tD)psD@)
well
done
drawdown
ivork
to obtain
13 presented
volition.
form
by
by
gas’
been
of multirate
the
(superimposed)
equation.
324
type
the
has
the
each
In other
influence
condition
function
(2a)
.
ivellbore.
to
sequential
pressure
1. AU
@~dary
surface)
and
obtained
the
.
- ~)dr.
a
pressure
in
afso
dining
during
problem.
(the
convolved
Pascal
a variable
An
tegration
g presented’
approach
tests
to thk
in Ref.
the
used
with
were
level
basically
transient
related
found
the
liquid
of work
rate
are
dratidowns;
ew
the
Gladfelter
con-
presented
al.
(f~
res?r-
the
Hursts
kh from
&ta
aad
and
jtDqD‘(T)p@
0“
0
However,
af.
formation
amount
multirate
with
et
that
tests.
A considerable
be
in
the
well
pressure
Gladfefter
afterflow
rise
stated
formula
and
weUbore
the
applied
authors
from
above
Everdingen7
The
the
10
determined
effect.
ikita.
These
tie
van
wellbore
aftertlow
be
Using
integd,
Ramey
constant.
a single
test.
to obtain
(measured
of a drawdown
should
a
q~t)
=
test.
variable
be
qD(2Li)
=
replaced
by
flow
stabilized
then
q,
q~,
rate
(flowmeter
, and
sandface
dheUSiOdeSS
JOURNAL
the
is available,
sandface
readings)
at the
rate—if
rate
OF
PETROLEUM
rate.
TECHNOLOGY
in-
TABLE
1.0.
l—HOMOGENEOUS
RESERVOIR
AND
FLUID
ROCK
DATA
8, bbl/STS
0,, psi-’
1.0
,..5
1,000
ccl
h.
100
ft
40
2,666,94
3,000
0.35
10,000
0.76
x10-4
p, Cp
0.8
0.2
.6
Time,
shut-m
Fig.
I—Dimensionless
tiel
AI,
sandface
Ibore
storage
hrs
flow
and
rates
exponential
for
decfi
constant
ne
The
cases.
purpose
mine
traditionslfy
dimensionless
quantity
formation
of
the
a unit
is known
thiid
the
imer
rate
boundmy
is
the
effect
This
considered
be
treated
for
the
its goverrdng
the
of the
for
many
linesrity
of the
2a
2b
and
systems
dmwdown
and
For
form
can
be
as
F@.
long
as
if the
with
m
The
us to use
Eqs.
the
fluid
initial
idkd
[pD(0)=O],
2a
to be
the
3a,
Taking
and
~~D
be
0
(3b)
–~)dr.
s
3a
the
sfso
can
be
expressed
Thu3,
we
tion
intograf
tion
as
Eqs:
. . .’ . . . . . . . .
(3c)
snd
first
tid
3a
3C we
known
kind
aod
Akhough
tion
ly
it
can
csn
be
fected
this
by
FEBRUARY
the
just
paper,
fluid
1985
““
a
is
hzve
of the
and
above
we
3a
of
the
constant-rate
integral
p,D(tD)
(sandface)
3a
and
is
case.
solving
for
(4)
only
by
io
this
P,D
the
Eqs.
3a,
case
solution
flow
rste
can
3c.
allows
csn
be
for
be
superposithsn
Laplace
us to express
forms.
the
expressed
the
tmnsthe
Furtherof
constant-
convolution
is constant,
=S.
convolu-
The
a solution
the
3C will
p,D(o)
more
The
different
case
storage
and
and
of the
notliig
qD (tD).
msny
or comtsnt-pre-wure
wellbore
3b
intepal
in
that
form
is
aod
(tD)
3b
in mind
operational
convolution
kemef
Eqs.
in
keeping
one
given
variable.
P,D
integral.
dmensiordes:
as4,1s
dp ~,D (tD)
q~(tD)=l–c~—,
. . . . . . .
. . . .. . . . . . (5)
dtD
physical-
the
pressure
assume
below
the
that
the
for
gauge
As
a
special
wellbore
cao
However,
perforations.
will
fluid
u,!dess
sms.lf
‘q,f(t)fqr.
In
welfbore
perforations.
lx
qD(tD)
case.
flowmeter
will
where
solu-
and
by the
the
through
volume
If
ls@.
equation
previously,
of Eq.
of
4,
integral
If the
exists.
used.
a constant-storage
flowmeter
flowmeter
placed
PO
below
is measured
the
integral
be affecti
will
q~(rD)
be
mathematically
of
volume
the
must
psD(tD),
of the
type.
that
effect,
below
since
throughout
a Volterra
solution
the
that
3b
convolution
afways
rate
volume
wells,
ly
a
p$D
ssndface
as
stomge
occupies
the
Eq.
then
the
be
pmctice,
it is assumed
it is assumed
without
that
3c,
constant,
Eq.
of
sandface
is
m
convolution
.0
qD(2D)
stated
transform
Esnsform
thmrem
rate
~’DqA(iD–T)P@(r)dT.
pressure,
. . . . . ..8. . . . . . . . . . . . . . . . . .
Laplace
the
same
more,
fn
is
Laplace
form
Eq.
(including
methods.
ssndface
transform
pressure
conventional
deconvolution
As
be reduced.
with
constant-flow-rate
‘s)
~:D
=,
convolution
0
Furthermore,
will
2b
. . . . . . . . . . . .(3a)
fDqD(T)p,’D(tD
the
Lsplace
($)=
The
=SgD(tD)+
3c.
out,
yields
where
q~(7)p@(tD–T)dT
by the
for
as
PWD(fD)=f
the
as
problem
increase,
rate
interpretation
and
sre
uni-
‘D
PWD(tD)=
3b,
the
problem
the
problem,
determined
solution
~@(s)*
snd
Eqs.
hss
test
inverse
inverse
et al. 2 pointed
flow
enhance
in the
is known
measurements
invers.k
sandface
will
of the
of
dsts
problem
Gringarten
range
of the
weU
AS m
reser-
conditions
constant
tie
measured
Thk
solution
As
snd
measurement
to both
in the
be applied
unique.
combining
in Eqs.
snd
Tbe
nonuniqueness
csm be ap-
anisotropic,
2b cm
tests
distribution
expressed
allows
2a and
buifdup
2b
problems.
layered,
a reservoir
pressure
2a and
equation
fractored,
phase.
Eqs.
engineering
diffusivity
for
is single
known.
that
reservoir
heterogeneous
voir
is not
from
wellhead.
problem.
type-curve)
be emphasized
psmrneters
at the
number
Thus,
kind..
It shoufd
plied
inverse
if the
condition
condition
tie
and
usuafly
here
solution
boundsry
boundary
a
dimensionless
will
condition
case.
homogeneous
the
effect
from
skin
production
as
skin
different
pressure,
as part
of
the
as stated
by
and deter-
et al.,
wellbore
Although
of the test welf interpretation,
z is to identify
the system
Gringarten
be
written
most
usual-
p,D
is
not
directly
Vhmughout
f.ncsion
Eq.
for
from
4,
the
Eqs.
the
dimensionless
constant-storage
4 and
case
5 as
($)
(6)
(s)=
1+
af-
of
solution
~sD
~wD
Therefore,
flowmeter.
application
pressure
lhls
paper,
CDS2F,D(S)’
the
runclian
““””’”’””””””’”””””
?(.9
w
be
Mod
the
Wlace
tram
form.!
tha
F(t).
325
As
k&
,/’,
,,,.”..,
storage
ciple,
Eq.
a much
~..
systems.
ly during
the
Iy
except
In
, ~.,
the
10“
n..,
M,
hr.
for
decline
constant
~$D (s)
is the
witbout
wellbore
storage
sandface
pressure
effect
but
for
of the
simifa
to Eq.
equation
6, and
Hurst
18 presented
Aganval
and
Kufmk
and
weflbore
pressure
the
same
other
can
be
dimensionless
(for
partially
for
in Eq.
obtained
the
for
radial
same
systems.
penetrated
snd
5 and
pressure
tle
wells)
.
1 presents
flOW raE
6 in terms
of
as
. . . . . . . . . .
7 as a function
oir
and
of real
fluid
tlom
this
face
tkw
storage
figure,
for
rate
cwe.
of qD(tD)
time
properties
wdculated
for
given
the
from
faster
and
as a function
a buildup
test
using
1. As
can
than
case,
the
Agarv.m121
of tD
Eqs.
in Table
exponential&clime
declines
IZamey
for
the
the
of
lack
pcesentd
various
tegral
given
dingen,
inF.qs.’3a
7
wellbore
Hurst,
g
pressure
and
and
of
3C was
skin
by
using
the
and
Ea..
a finite
pressure
1 and
wellbore
for
1 and
solution
4 for
the
afso
radius,
tie
can
written
be
exponential
~wD(.,=y —.
i3, is
giVeII
by van
in-
u can
as
the
dimensionless
326
constant
used
for
to determine
p
~,
(At)
shown
to be
data
prc%ented
and
by
thk
straight
in Fig.
flow
the
of
and
formation
semilog
data,
rate.
rate
can
s&tiOns,
simultaneously
pressure
pressure
Fur-
also
following
deconvolution
rate
the
flow
the
rate
is used
early-time
sandface
In
flow
by
of the
the
from
usuafly
sandface
anzfysis.
abO~e
stems
sandface
supported
the
tbe
2 is that
analysis
analyses
flow
frO.m
analysis
and
and
case.
made
the
fluid
figure,
a semilog
type-curve
At,
vs.
reservoir
data
and
will
be
parameters.
the
Ever-
source
rate
from
decline
Eq.
Eqs.
case
(8)
M
This
the
case.
volved
qD
with
of
function
ly is used
for
the
also
can
tests
the
chosen
pa
with
the
of
for
measured
the
the
cOn-
can
be
conthe
to
modify
the
cOnvOlutiOn
log
analysis
for
pressure
fractured
using
approximaCCJMMOR-
tests.
sandface
P.D
in oilfield
line-
or
press~e
muldrate
psD
a
by
the
wellbore
approtition
in measured
3c)
example,
rate
used
rate)
tkougb
For
and
with
10g
3a
anafysis
function
flow
a
~
case
variable
be
cO~d
vertical
pwD.
tion
tbep,D
tb~On
approach
assume
conductivity
(Eqs.
(superposition)
for
psD
(sandface
or
simplest
is to
constant-flow-rate
The
integril
function
USing
Eve@gen7
V~
for
stant-pressure
time
as
3a.
The
integral
infinite
etc.,
convolution
Method.
convolution
solution,
solution,
wellbore
directly
in
source
the
line
the
The
same
of buildup
flow
in
units
rate
f%.
can
or
data.
3a!
fie
be
ex-
as
k,
be determined
wellbore
par-
different
(Superposition)
solving
pressed
such
been
wells,
8 for
and
of
point
Multirate
function
as
where
A8
of measured
to obtain
change
(3=m#yic,rw210.000264
use
sandface
Continuous
,., ,.,. .,, ,,, ,. .,, . . . . . . . .
constant,
dif-
Eq.
approaches
type-mu-w
semilog
drawdown
exponential
the
for
cases
plot
measure
convolution
technique
‘fhe
from
developed
about
the
to
the
Convolution
calculating
dimensionless
sandface
be
1.
quantitative
the
values
solution.
For
has
curves
constant-wellbore-storage
answers
For
shown
for
storage
case
information
quabtive
storage
convolution
resented
*J
Ramey
analysis
constant-wellborealso
application
Of
same
storage
8 by using
of the
Thus,
measured
samd-
can
Table
buildup
limitation
the
reservbe seen
in
the
principle
to
important
con-
time
at tie
as fractured
curves
7 and
important
and
improve
1 and
constants:
Another
and
pressure
wellbore
semilog
Eqs.
it is necessary
of qD
changing
clOsing
of type
wellbore
than
most
analysis.
(7)
. .
also
decline
earlier
behavior.
(s)1
vafues
affect
such
skin
given
thermore,
Fig.
fifi~
Em in early-time
Type
a
from
The
for
St 1 + CDS2ESD
recognize
tie
will
etc.
presents
exponential
line
1
~~(s)=
it is a gradual
W?.
2
parameters
dimensionless
sandface
Eqs.
to
because
a constant
decreasing
discussion
from
condnuoLls-
difficult
geometries,
of O and
calctiated
6.
formation
decrease
change
redistribution.
construction
7 for
wellbore
Fig.
reservoirs),
the
dimensionless
directly
thus
the
of Eq.
graduafly
equatiOn
the
fractured
expression
as
hand,
the
19 {for
Kirwan20
the
on
form
Mmaniego
presented
an
et al. 4 presented
as an integrodifferentid
Cmco-Ley
and
and
work
penetration,
skin
Everdingen
also
toward
vafues
including
skin.
Van
valve
sfow-
rapid
and
directed
storage
is
it slows
time.
cases.
dimemionIess
case
it
then
up vay
of the
wilf
phase
effdcts
more
boilds
Because
storage
of
wi~
increwes
tests;
and
Furthermore,
welffread
tial
constant-rate
case
change.
ferent
the
wellbore
cases,
solutions
where
period.
the
storage
tinuous
buildup
period
re@ily
in prin-
cO~~tiOn
pressure
early-time
as
However,
~ediate
the
transition
in the
estimated
CD.
more
fact,
semilog
many
wellbore
Most
pressures
exponential
the
of pressure,
~.z
2—Shut-in
the
during
be
constant,
In
during
down
shut-to
Fig.
9 has
real
rapidly
!.!
i3 cannot
weflbore
the
-...O.W9
10 noted,
Ramey
as the
from
storage
like
welfbore
constant.
CD.
pressure
Note
that
data,
(3 is
a
Af)&t)
=m@(,)@og(t-r)
+~]d~,
. . . . . . . .
0
JOURNAL
OF
PETROLEUM
TECHNOLOGY
(9)”
As
where
noted
gested
Ap ~f)
‘pi
‘p
earlier,
here
wellbore
tit),
the
will
storage,
bottom
the
the
The
method
muhirate
method.
al. 9 method
12.
effect.
by
Gladfelter
existence
al. 9 further
improve
the
Gladfelter
method
of one
it will
not
of
mukirate
but
et
does
disadvantage
the
of the
it
of the
between
meter,
storage
is that
to three
sug-
effect
volume
flow
However,
The
over
the
test
the
is a finite
and
suggested
Eq.
variable-rate
completely
there
well
wellbore
simplifies
pxsibfity
continuous
eliminate
since
of
minimize
‘=”’’’+’”g(*)-’’’”
the
not
the
et
here
different
is a
straight
lines.
and
162.6
@q
~.—
kh
Motiled
Homer
rate
&ta,
Meunier
the
Homer
Eq.
9 can
be
rewritten
be
as
for
reduced
on
sandface
rate
The
Letus approximate
sum,
which
the
integral
in Eq.
9 by the
Riemamr
yields
Horner
is appealiig
of
AP.jc(/n)
(tn)
the
the
sandface
measured
face
tion
t.
is the
has
been
analysis.
Eq.
measured
presented
10 gives
(variable-rate)
well
side
vs.
will
the
yield
b.
the
Riemarm
the
first
sum
a straight
For
of
line
buildup
the
with
tests,
form
of the
11,
Eq.
10.
right
side
a slope
The
as
be
used
of
the
Eqs.
ten
on
in
for
the
this
tests
the
and
straight
during
the
unless
the
fluid
of the
semilog
line.
is that
the
semilog
as long
wellbore,
effect
ex-
method
method
of view,
at
ease
the
Homer
Homer
zero
Horner
buildup
and
skin,
semilog
point
rhe
of the
kh,
the
becomes
transient
3 decades.
generality,
Homer
as
the
in the
the
sand-
wellbore
decaying
straight
sandline
wifI
section.
integral
buildup
Horner
last
analysis
from
be zero
convolution
for
popular
tie
estimated
Thus,
rate
section,
given
by
Eq.
3b
can
be
writ-
as
to
PDS(~pD
left
10 or
an
be
flow
the
increases
will
investigated
muhirate
can
m and
face
be
A plot
of
should
qD(f)
equa-
techniques,
in Eq.
in
This
multiple-rate
integration
used
given
term
point.
for
condnuous
Any
integral
time
elsewhere
equation.
as the
evaluate
(discrete)
In this
if the
modified
over
of
rate
is incompressible.
where
ex-
ratio
most
reliability
a theoretical
never
detailed
dme
and
simplicity,
of the
pressure
rate
conventional
available.
the
required
flow
Flom
rate-convolved
the
Homer
Homer
for
its
slope
assumption
period.
the
oil industry
used
for
The
on
17 gave
is perhaps
pressure
The
al.
are
by the
application.
depends
qD
of
of tie
method
trapolated
=
test
test
the
can
examined.
practiced
The
that
line
of
measurements
rate-
showed
using
the
of
“the
straight
by
to modify
buildup
named
instead
how
sandface
a modification
semilog
et
be
measured
” They
of the
Metier
will
testing
,~m.
function.
start
mture
the
they
ratio.
fundamental
methods
b=
time
the
function
time
planations
where
which
consi&rably
time
Homer
Using
17 presented
al.
ratio,
buildup
required
buildup
the
et
time
convolved
time
Method.
+AtD)=%D(ArD)
+PD(t@
+AtD)
11
intercept
replaced
-~[1-qD(T)@;(MD-~)d~,
. . . . . . . ..(12)
by
o
1
(At)
–qD
and
–p~At=O)
The
(rate
major
just
minimized.
The
ventional
The
should
be
should
drawdown
The
equal
or
second
is the
buildup
surface
step
change
for
rate
change
from
the
PEBRUARY
conventional
1985
one
over
wellbore
the
effect
other
has
not
been
mtddrate
in the
the
storage-free
of the
words,
con-
effects
used
from
end
the
storage
conventional
taken
the
the
constant-surface
with
rate
the
test
storage
con-
effect
sandface
rate
change
rate
conven!iond
cannot
be
Neglecting
to another
multirate
q
is
rate
for
each
multirate
test
in,
the
by
van
the
tie
test
will
analysis.
taken
the
affect
as a step
continuous
the
result
and
curve
that
exponentially,
Hurst7
after
in
shut-
as suggested
so that
Eq.
1 becomes
‘-&,
(13)
e~ple,
a
shut-in.
let us assume
decliies
a is determined
For
available,
square.s
rate
Everdingen6
=e
before
rate.
of simplicity,
afterflow
where
data.
constant
sake
qD(At)
the
sandface.
and
For
period.
problem
rate
of
Thus,
the
variable-rate
pressure
be
In
to
~, (At)
where
storage
measured
determined.
=p
perforation)
the
for
should
period.
be
the
wellbore
wellbore
Ap ~, (At)
11.
continuous
tit
literature
malysis
in ftite-acting
of
testis
in the
by
9 through
above
multirate
discussed
tests.
Eqs.
advantage
measure
vention
are
AP ~f(At)
in
Eq.
fitting
from
measured
if
sandface
13
can
of Eq.
be
rate
sandface
determined
13 to the
flow
measurements
sandface
by
rate
are
the
least-
rate
data
327
Fig.
5—OimenslonIess
times
modified
Horner
for
the
semilog
start
of
straight
the
lines
Horner
as
a
and
function
of 0.
tremely
as
long
At
[[tP
time
Fig.
&
3-Modified
Modified
Horner
dec~ne
Horner
and
sandface
Time
Horner
Functions
plots
for
rect
term
in Fig.
term
straight
in
of the
start
of
the
the
correct
ror
criterion
slopes.
4—Functions
for
Horner
semilog
the
start
of
straight
the
Horner
and
Homer
Horner
for the
start
as
tie
time
Homer
2ctPD
–AtD[(rF
+-In
4+EXJ3AtD)+2S]-
we
and
B,
m
1
@
be
for
the
never
can
define
yields
an
computed
the
error
criterion
straight
line.
formula
line
[–in
er-
Homer
approximate
‘btiD
stint
Theoretically
straight
1/AtD
the
it will
line
semfiog
semilog
+At)D]
for
line.
at which
in Appendix
shut-
large.
line,
straight
other
aftertlow
formula
straight
of a Homer
In
the
formula
correct
to the
Homer
At.
by
approximate
modified
tines.
cor-
maximum
becomes
However,
the
start
of the
At
semi30g
hne.
large
straight
sem-ilog
we. take.
developed
at
to the
caused
an
Homer
as the
As
modified
sefiog
between
Then,
compaed
and
g an approximate
straight
is satisfied
plot,
the
large
negligible
the
log
Homer
Horner
compaed
to determine
speaking,
the
approaches
correction
determining
interesting
Thus,
identical
large
the
modified
14 that
but
is very
the
almost
almost
Before
Fig.
are
Eq.
smaller,
line.
then
then
becomes
well.
(ZP +Ar)/At
lines
from
becomes
3, asymptotically
if tp is veiy
time,
as
straight
l/2.303aAt,
words,
It is obvious
303cYAi
decreases
semilog
If the
exponential
rate,
1/2.
+Ar)/At]
as it is seen
Horner
period.
increases,
is given
by
p –27
1
=0,
. . . . . ..
.(15)
where
Substitution
of the
and
Eq.
13
for
tion
are
given
qD
in
exponential
in
Eq.
integral
12
Appendix
(the
A)
solution
details
of
forp~
the
deriva-
yields
an
pi-pw’’’)=~[’”g(%)+al
The
constant
a in Eq.
parameter.
&me
ratio,
tilog
liie
The
term
(tP +At)/At.
{log[(tP
with
a correct
in Fig.
3 (approximately
straight
328
—-—
and
liie
the
Homer
Brighamz
on
the
considered
will
This
straight
one-half
Homer
that
plot
of
afterflow
(At)
yield
line
vs.
slOpe
Anm~~~
starts
much
as
shown
cycle
earlier).
In
correct
semilog
after
=
an
fact,
ex-
of
correct
Horner
semi-log
straight
lie,
a ~ght
line,
is obtained
%.,
Horner
straight
the
criterion,
(14)
the
pw~
wilJ
“setnilog
found
an
modify
plot
+ l/2.303aAt}
slope.
than
be
A semilog
+At)/&]
earlier
Chen
14 can
l/2.303cYAt
error
=
slope
of
line,
tpD
For
of
dimensiOIdess
=
a given
e (relative
of
15
zeros
the
computed
Eq.
Homer
Homer
semilog
stmaight
and
with
semilog
JoURNAL
producing
error),
respect
stmight
OF
6,
time.
skin,
to
AtD
line.
PETROLEUM
and
producing
time,
will
give
It
interesting
is
the
TECHNOLOGY
sw
to
observe
is
that
the
a function
A simple
Homer
start
of
formula
semifog
as
4,
S=0.0,
E41.
15 has
S, &d
e given
early-time
The
time
for
be
tion
time
previously.
The
first
upper
curve
start
of
15,
15 slso
can
crpde
for
approximation
Eq.
15 cm
an optin@
tain
accuracy
ticularly,
production
For
dimensiordess
line
is a very
a function
been
line
As
for
of
the
observed
constant-we~bore-storage
for
13. However,
the
func-
the design
usefuf
is
is limited
by
tie,
a ve~
tests
effect
Eq.
line.
measured
pressure
data.
mukirate,
by
also
using
can
the
be
same
derived
principle
If p i.iapproximated
ten
from
drawdown
given
by
I/CD,
Eq.
As
g
well
(16)
values
deed
for
the
The
tie
Eq.
stat
weflbore
Homer
stmight
Appendm
e[2@(tPAt)D
yield
semilog
remains
time
semilog
in
will
of Homer
storage
dimensionless
given
17
the
start
partially
be
(details
also
can
+(tP+At)D]
be
of
be
writ-
line
a very
in the
tie
dimensionless
semilog
Homer
the
motied
test.
as two
paral!el
a fractured
straight
vslues
of Eq.
tPD=’106,
The
for
the
For
time
for
the
lime
as
-4>
root
sw
a function
of
S=0.0,
wid
in Fig.
5 presents
start
of the
modified
Horner
modified
function
large
rPD=106,
Homer
of
8.
Fig.
semilog
stmigbt
(rD).
is
to
wellbore
geomet~
tion
deconvolution
tPD,
@.
time
for
There
time
straight
line
c= O.01.
The
we
18 cm
chamcteristic
the
early-time
without
cahdated
or
pressure
from
P
the
should
of
of
the
F@
WD (tD)
VS. tD , such
information
3a
and
2$ liim,
about
idendi5cation
a given
is not
knowledge
more
the
complex
that
the
linear
well
of the
because
and
system
often
we
do
equation.
fluid
flow
dh%sivity
given
genersl,
about
problem
differhisl
methods
in-
reservoir.
governing
by the
wi13
wellbore
identification
In general,
repeating
of
well,
of formation
be
in-
plot
graph
these
Of PsD(t~)
to system
is described
For
either
p,~(r~)
conventional
the
a Horner
deconvolution
provide
skin
fractured
formation
the
is much
fmow
the
conventional
behaviors.
a log-log
must
the
and
our
testing.
on
durtig
merely
will
approach
is also
5iMP~i~
several
3c.
following
in the
equation
formafor
the
next.
tion,
the
methods
These
for
metiods
tie
deconvolution
will
be
of Eqs.
discussed
in
the
section.
Lineariza
a
~
are
through
directly
@ne,
on
graphs
etc.,
it uses
effect
used
damage
well
behavior
For
ae
of
dominates
stiffi-
identilcation
However,
calculate
seve~
and
system
reservoirs
effect
wells
kh
part
have
slope
on
types
analyzed.
a vefically
limes
pressure
sphericsl,
give
skin.
producing
the
4 presents
dimensionless
and
(in-
in
reservok
being
the
part
usually
for
3C
3a
semilog
decon-
functions
must
both
resemoir.
storage
tlqough
.(18)
6=0.01.
S=0.0,
p,D
penetrated
indicate.
storage
Tbis
not
will
the
Thus,
This
Homer
of dimensionless
curve
of j3 for
16 will
modified
continuous
Thus,
is an important
system
fully
a one-half
and
minimized
qD
the
for
we.ffs
weflbore
but
of Eq.
the
early-time
as
and
engineer
straight
of
resemoir
of the
lower
of
weak
second
18 as a function
“13=10
w a fUIItiWI
sw
tie
to
from
parameters
reservok
wells,
or’afterflow”
system.
are
by
4+2S)=0.
case,
to obtain
etc.)
important
data
It is worth
As
such
functions.
Thus,
identifying
dicate
be
-b’2AtD2[(tP+AZ)Dl
&7-27+fII
improves
to very
geometries
curves
identification
.e-8~D(–1”
be
has
anafysis
composite,
penetrated
pressure
the
if in-
the
derivations
given
It must
restdts
formation
underestimated.
period.
during
of
explore
to
(afterflow)
applied
used
about
if w
storage
optimistic
straight
constant
for
line
B)
very
be
The
information
bebaviors
. . . . . . . . . . . . . . . . . . . . . . . . . . ..(17)
previously,
used
reliable
methods,
wellbore
testing.
wellbore
noted
19.
section..
stance,
as
AfD=:CD.
rate
Horner
and
layered,
for
B.
can
the
hours
analysis.
and
be
otier
will
(fractured,
will
solutions
16 then
me
prerequisitepD
functions)
becomes
in Appendix
accurate
The
methls
for
16 and
parameters,
Horner
flow
it cannot
Deteminin
Par-
be simPlifi~
. . . . . . . . . . . . . . .. . . . . . . . . . . . ..
16
the
modified
require
Eqs.
analysis.
the
anafysis,
fluid
modified
time.
sandface
Homer
though
of the
required
from
functions
on
the
semilog
tie
l+q.
testing
pD
to obtain
modified
since
to
AfD=~.
that
example,
facilities.
15 can
the
and
of afterflow
recognized
cient
a cer-
tests,
production
tPD,
on
and
stat
time
be seen
formation
qD
the
the
Deconvolution
Eq.
of buildup
driflstem
saved
simple
following
case
straight
for
be
the
for
of
and
to achieve
semilog
be
on
Very
of
for
the
Depending
fluence
“B.
Homer
producing
as can
volution
stwt
Chen
UCD
line,
the
as
case.
the
straight
2.
production
by
Homer
be
Cm
the
wesk
is half
Even
6=0.1.
time
line
the
Fig.
straight
and
of a producingdme
15 will
time
large
from
semilog
useful
value
for
easily
words,
the
6,
from
Homer
for
be quite
Eq.
of tPD,
resufts
required
stmight
,
Fig.
dimensionless
had
lICD
for
from
values
zero
seen
seen
time
Homer
could
of
9
tPD = 10
for
constaht-wellbore-storage
be used
substituting
it is
5 presents
S-might
tie
15 because
in Fig.
the
This
for
of the
values
the
the
start
inkinite-acting
It is basically
IS.
be
other
line
radial
the
semilog
skin.
and
can
the
be
tPD = 10 6, S=0.0,
Eq.
Binghamzz
by
for
of P for
of
can
(zeros)
from
Homer
As
roots
which
from
of the
time
results
me
seen
4 presents
zero
period..l%e
Eq.
Fig..
period,
second
a function
from
E= O.01.
straight
expected.
for
dimensionless
and
two
as
derived
tie
function.
of
Homer
time,
be
straight
a” function”
(3=10-4,
semilog
cannot
a transcendental
15
of the
the”producing
tion
of the
Convolution
“convolution
by
discretized
using
equation,”
the
limaiization
Integral.
Eq.
3c,
niethod.
In this
will
Eq.
be
sec-
solved
3C can
be
as
n
AtD=:,
. . . . .... . . . . . . . . . . . . . . . . . . . ..
(19)
PWZND”+l)=
4@3
The
modified
one-half
FEBRUARY
cycle
1985
Horner
earfier
semilog
than
the
straight
Homer
line
star@
straight
z
i-o
~“mlti%+l
~Dt
-~)
at least
fine.
fn
“ps~(~)d~.
. . . . . . . . . . . . . . . . . . . ..(20)
329
,6,0=
:
‘“:”’’’””:;:L:L----
“-”-”
:,200
il’’’’”F”’-’F
. . .. . .“..,. !.
i
‘.”.,
!,.
ah”..!.
s,..,.!.
:
m,
,
-
./”/
+
.,
.
./
.0,
“
~’”:~,~l
-”
--.
.-.
0
,
..
,o-
10
,,!!
n.,,
slut-l.
At,
formation
method
and
Fig.
hrs
pressure
wellbore”
drop
pressure
using
.!,
d
b
0“
7—Calculated
formation
method
6—Calculated
‘m.,
SW-i-
,
,0-’
1,.’
,0-
~~~~~~~~~
,o-
Fig.
/“
i
:
and
pressure
wellbore
drop
Pre?sure
using
Hamming
drop.
linearization
drop.
where
“sum”
is equal
to
n—1
By
using
the
trapezoid
rule
for
integration
in
Eq.
20,
,2
n
PwD(tDfI+l)=
X
bSD(tDt+I)Qi(tDn+I
‘tDi+l)
‘qD(tDn+I
i=!l
and
+P.D(tDi)q~(2Dm+
Eq.
I –2Di)l(tDi+I
givea
21
coefficient
a system
inairiix
triangular
niatri~
tie
right
lower
can
be
solved
For
and
field
(p w,
tively;
data,
qD
(1
straight
can
21
for
if
the
formula.
However,
the
osciffatory
result.%
very
early
times,
oscillates.
the
the
rule,
that
direct
the
a stible
be
If tie
to
slupe
of
plot.
conventiowf
used
in Fig.
values
of
integmfs
for
in Eq.
in
Eq.
20
can
be
rate
data,
Eq.
21,
semilog
effect.
(pW,
–P
at
~)f
Laplace
RI.
pSD(tD)=
also
the
linearization
ing
for
p,D
can
will
r@t
approximated
he
in w
in Eq.
directly.
20,
and
solv-
yields
\
fZD(tDn+
,.
33U
Fig.
straight
hours)
same
the
Decunvulntiun.
@tb
figure
we~bOre
The
–7)d~,
K(T)PWD(tD
is tbe
it~rage
convolution
of
. . . . . . . . . . . . (25)
.,....
(26)
($)
can
data.
to
qD(tD)
gD(tD)
real
pute
.
. . (23)
power
data
space
in
and
types
in Laplace
must
integrate
Eq.
We
25
have
Eq.
OF
qD
tried
of
PETROLEUM
can
rational
26.
Thus,
data.
(tD)
m comp,D(tD).
be used
(tD)
to
functions,
Expmentid
qD
to
transform
be easy
determine
functions.
representafiori
and
for
it will
to
uf
consuming
space
used
Laplace
equation
@e
functiuns
ex~nentid
good
the
with
be
of approximation
and
a
be
is obtained,
qD (tD).
series,
give
from
a curve-fitted
accordance
functions
K(tD)
or
it would
an approximation
A few
eifher
data
However,
all the
it back
cumputed
be
of
JOURNAL
.—
(At=O.05
in the
includes
in
a
“1
tions
I ‘tD.)
time
given
yields
lr(z~)=c-1
— . .................
M
approximate
p,D(tDn+,A)=~”~(f~+l)–s~
At,
vs.
data,
as
integrated
resuhs
but
q&D),
where
Once
22
for
for
showed
r~i
Eq.
,,.,+)f
test
curve
which
Transform
aPProhtion
integration
used
23.
dif-
‘D
integral
. . . . . . ..(22)
of
in Eq.
A iinite
inclu~g
the
scheme
He
‘PSD(~Di+%)~1q~(2Dn+l-7)dT.
side
needed
of the
4 yields
invert
of the
–p
early
lower
–p;),
in-
derivative
known.
be
and
lost.
@ ~,
a veiy
The
be
a stable
convolution
Eq.
6,
qD(tD)
right
is not
buildup,
kom
slope.
of
of
gives
the
the
can
be
synthetic
starting
23
for
must
also
plot
Eq.
q~(tD),
q;(t)
would
radi+
a correct
plot
7,
Furthermore,
appromixation
K(rD)
Substitution
by
scheme
20.
for
accufacy
for
line
Fig.
flow
transforms
The
from
integration
s?D
intigral
given
. . . . . . . . . . .. . . .. . . . . . ...(24)
seen
given
The
7
skin
owiflation.
The
is
,D
~
integration
usually
be
ference
Skin
in
@w.
mefiods,
as
can
sandface
fur
flOW ~S
the
seen
integrul~.
As
some
used
method
(discussed
of convolution
+fferenc=
drop).
qD
results
ofp
—
However,
.’ .. .
‘%wolntion
integration
wj)
and
be
can
value
nonosdlatory
respec-
–pw)f
log’Af
me
divergent
sumzested
~f
method)
vs.
from
As
(pi
from
–pM)f
..”
Hamminz23
in
fp i –p
tests,
ch.
iugher-urder
yield
aze
equations
by
must
case
cal~ulated
in Eq. 20.
evaluation
buildup
fist
PsD(tD%)=:;;;;
tegral
press~e
pressure
trapezoidal
Furthermore,
SiniPson
hy
~eld
21
gives
replaced
be determined
(p ~,
of
replaced
b~dup
be “determined
The
elements
system
and
for
kh can
line
also
be
–qD)
Eq.
in
formation
factor
be
drawduwn
‘tDi+I)l,
the
is a lower
substitution.
sho~d
~D
equations.
nonzero
‘b)
. . (21)
equations
The
~bich
we
s@dface
effecm
(formation
and
replace
its
forward
should
Q%
–P&,
without
storage
the
p
p,D
drawdown
all
of
mat@
‘by
for
algebmic
system
is,
the
easily
-PJ
and
radial,
that
of
‘2Di)/2.
of linear
of this
psD(bi+fi)[qD(tDrI+l
func:
data.
TECHNOLOGY
Ex-
—
TABLE
2—FRACTURED
RESERVOIR
ANO
FLUID
TABLE
ROCK
3—COMPARISON
COMPUTED
DATA
AND
PRE?SURE
~,
100
400
md
4,216.94
psia
STB/0
q,
0.35
10
hours
10,000
,.-5
k
(spherical
~,
matrix
used)
0.8
Cp
0.2
+,
0.06
.
ponentiel
functions
expressed
as
approximation
qD(tD)=cleB1tD
where
i=
After
ing
the
+c2e82tD
1,2.
131.93
131.96
0.04
133.57
133.78
qD
be
can
+...cne8~tD,
.:.
(27)
Lapktce
Ci
xnd
from
L3i
transformation
resulting
qD(tD)
to Eq.
expression
98.32
0.06
134.66
134.86
115.30
0.08
136.47
135.64
124.37
0.10
136.12
136.25
129.44
0.20
138.27
138.27
137.03
0.30
139.62
139.55
138.87
0,40
140.62
140.53
139.83
0.50
141.42
141.33
140.85
0.60
142.10
142.02
141.59
0.70
142,68
142.62
142.24
0.80
143.20
143.15
142.81
0.90
143.66
143.63
142.32
1.00
144.08
144;07
143.78
2.00
146.94
146.98
146.84
3,00
148.!36
148,70
148.60
4.00
149.92
149.92
149.85
5.00
150.88
160.86
150.80
6.00
151.66
151.63
151.59
7.00
152.%
152.29
152.24
8.00
152.85
152.85
152,81
9.00
153.33
153.35
153.32
in Eq.
data,
27 and
26
applyrepresents
substituting
yields
“..
the
behavior
apPrOfimatiOn
case,
the
such
tbatzz
was
natural
,
>sD(tD)=
x
bv
not
its inverse’
P~D(tD)
We
Eq.
have
fonction
has
from
qD
if
the
be
vw
be
cti
For
be
to
tramfOrma-
inverse
“28 should
is possible.
25
seen
thxt
24
Eq.
trxnsform
fi
works
Sneddo”.
&fist,
divided
each
by”p
~vision
and
by
m
,.
..,.
Hurst’sg
gjven
exponential
. . . . . . . . . . . . .. . . . . .
and
a fmctured
the
cm
is the
be
simplest
wriiten
p,D
power
the
(tD).
10g
In tis
series
of in tD
. (29)
for
the
third
form
directly
of Eq.
from
27.
Eq.
For
this
case,
P$D
fluid
and
of
effect.
with
As
seen
curve-fit
mm=?’”’”;’
‘+PwD(tD).
. . .
(30)
. . . . . . .
xlmost
q
Table
3,
The
2, while
vxlues
of
in
Table
Ap,f
without
3 presents
effect.
the
relative
3 presents
approxima-
solution
dMerences
and
from
pam.meteis
in Table
the
column
storage
approximation
identicel.
given
amlyticel
fourth
wellbore
fluid
curve-fit
3 presents
have
to weIfbore
obtained
of Table
from
date
Table
from
from
and
.;,
We
31,
were
column
Apti
reservok
The
the
&f.
that
of
paramters,
problem.
resewoir
second
of
directly
storzge
&~,
3 as
2. The
the
number
n
pxmneters,
these
polynomial,
for
column
yields
obtxining
(synthetic)
values
the
20
unknown
opdmiiation
data
reservoir
in Table
Eq.
of
In fact,
rxte
cxkdated
tion
. . . . . . . . ..(31)
. . . . . .
into
a fotirtb-degree
pressure
Everdngen’s7
approximated
number
an unconstrained
appEed
ofp,
differentiated.
31
“m
,c2 .:.c~)T.
becomes
calculated
which
be the
tD)i-l.
of Eq.
and
.=(cI
until
as
qD=l_e-@D
to approximate
will
C@ I
Substitution
d~~s
earlier,
{=1.
.—,
equations
SugEeSM
noted
..!:.’: ””””.”.:.:”’: ’(2. !)4
“
. .. . .
,.,
tion
used
As
p,D.
m
~
“’”””
AS
of
choice
1.”
i?(y)=
65.33
.,.n.
determining
the
for
*
0.02
3>000
rw
R
tp,
FRACTURED
.-
~
At
1,000
h,Dft
pti,
A
,..5
psi-l
k,,
FOR
RESERVOIR
1.0
bbl/STB
B,
OF
ANALYTICAL
in
Ap,f
anxlydcel
error
from
the
solutions
decreases
are
for
lerge
times.
From
Eq.
cm
qD
be
(tD
30
it is very
simple
approximated
)
data
ponential
ds.o
to calculate
suc.csss~y
can
fun~om
be
for
,by
approximated
p,D
if qD
.,,using
,:.Eq.
by
a selective
dat6
29..
piecewise
interval
for
C0nc1u8i0nx
ex-
Eq.
26.
1. The
used
flow
the
Curve-fit
and
App:oximmtioiis’o
fp,D”.
3:..c~wb&approximated
tion~
by
approximations.
copld
be power
tions,
or exponential
depends
FEBRUARY
series,
how
on
1985
These
continued
functions.
well
~ctionel
functions,
The
the
p~~(tD)
choosing
success
approximation
in Eqs.
suitable
3a
func-
approximations
rational
of this
funcmethod
function
for
rate
convolution
integml
the
of
analysis
and
pressure.
conventional
2.
Wellbore
The
Horner
storage
face
flow
The
rate
last
one-half
data
modified
cycle
in the
analysis
to
obtain
earfier
is
wellbore
is ve~
effects
Homer
is modified
Horner
theorem)
varying
similxr
to
meth0d3.
(afterflow)
degree
an~ysis
liie.
This
mukirxte
to a significant
(superposition
continuously
can
semi-log
by using
a correct
measured
semilog
semilog
straight
tlmn
conventional
the
be
present
strxight
line
Iiue.
sxndstraight
steits”xt
Horner
331
straight
stat
line.
Approximate
of the
limes
as
time,
and
puted
modified
a function
the
presented
Homer
sandface
error
formation
cakulated
rate
for
semilog
decline,
between
pressure
ffom
pressure
and
volution
the
the
the
ffow
techniques
are
and
At
com-
(atlertlow)
thei’more,,
W&bore
from
the
rate
this
new
to compute
and
be
the
format-
wellbore
geometries
ca
pressure.
Fur-
can
be used
to analyze
to detefmine
and
ffactured
pute
the
‘IW
computed
of synthetic
rekervoir
formation
data
shows
pressure
reveak
and
fractured
from
tiiat
from
pressure
homogeneous
a homogeneous
it is possible
a
=
O .000264kLWpc,r;
#
=
a positive
thank
0.57722
p
=
viscosity,
f
...
=
rate-preasufe
=
hours
Euler’s
cp
constant
pa-s]
convolved
~ =
dwmny
+
=
porosi~,
fraction
-
=
LaPlace
transform
tbe”beghing
the
of the
characteristic
test.
of
integration
t@&
the
integral
aid
A.C.
Wellbore
~Y@,”
Tedmicaf
in
for
acknowledge
Catala
during
H.J.
Services
the
Ramey
Jr.
Appendm
the
of
initial
time
function
variable
of
Ranmy,
4.
H.J.
of
this
providing
A&u-waJ,
uid
work.
We
a solution
for
5.
A.
Ffmw
6.
McKinley,
R. M.:
Dominat&J
Pressure
Twns.
Ead.mgk,
B
=
oil
fofmation
[res
volume
m3/stock
factor,
tank
1974)
RBISTB
7.
m3]
van
=
system
total
compressibility,
psi-1
ma-l]
C
=
8.
wellbore
,storage
coefficient,
,9.
[m3/kPa]
CD
h
=
wellbore
=
formation
, Am
MIV.:
Fluid
constant,
tilckness,
dimensionless
ft
K
p:
=
fofmation
=
kernel
=
deferential
permeability,
of
the
md
convolution
integraf
11,
PDS
pressure
=
shut-in
=
fOfnJatiOn
pi
=
initial
PSD
=
PD
PD(tD)
drop,
pressure,
pressure,
+S
psi
H.J.
Jr.:
drawdown,
pressure
PW
=
bOttotiole
flowing
P~.
=
bottomhole
shut-in
=
stabflkid
qD
=
s@face
qr
=
reference
q
includlng
dmensi0n3ess
=
WD
14.
dmensicmless
psi
ate,
psi
flow
qR
=
resewoti
q~
=
smdface
flow
flow
~w
–
—
wellbore
radius,
s
=
Laplme
S
=
skin
m~~fO~
16,
rate,
H.tchimm.
rate,
B/D
BID
raw,
B/IJ
ft
X&
/d]
time,
time,
=
production
Oct.
et al.:
From
FieJd
Data,’-
AJME,
231.
Jargon,
1.R.
and
(JuJY
Tech
on
1953)
a“d
the
171-76;
Its
kopediment
1953)
L.E,
Well%
,’, Drill.
117-29.
and
Wdl&me
Storage
Wells,,,
J.
Effects
Pet,
Tech.
234.
. ‘A Generalized
1959)
169-77:
kd.Joms,’
of Gas
“i
1962
AfME.
216.
Gene&ized
Reservoirs
at
Water-Drive
Tram.:
S. C,:
presented
B6-B16.
: ‘ ‘Selecting
Tresxnent
of Gas
Mcdel
S.bjec4
SPE
ta Water
Anmd
Meeting,
fnfluence
Functions
7-10.
“Determimtion
J.
van
J.N.
et af.;
mance
and
Pressure
Well,,,
J.
PascaJ,
H.:
Pet.
PcalJen,
‘y J.
of Aquifer
Tech.
Pet.
(Dec.
H.K.:.
Pet.
1964)
“Unit
1417-24
ResFon%
1965)
(Aug.
Tech.
Pet.
Dam
Tram.,
Funcdon
965-69;
From
“Trans.,
tie
1981
under
Evaluating
Perfor-
an
MHF
Gas
1711-19.
in EvaJuatinE
Tests
of Pmtfracmrin’g
for
1980)
(Oct.
Rate
at
Analysis
BtdJdup
Tech
May
Gas
No”-Darcy
SPEIDOE
Well
Deliverability
Flow,’
LOW
3 paper
Perrneabili@
Us-
SPE
9841
Symposium,
27-29.
D.,
of pressure
J.
‘, Combined
‘ ‘Advanm
Vtible
Withna@
Buildup
“TecJa,
Test
(Jan.
M.J.,
Using
19S51
ed
Steward,
IreSim
G.:
Measnrmnert
‘Tnterpretatio.
of Afterglow,”
143-52.
vfiable
format&
19,
dimensionless
Naturally
1982
Orleans,
~d
sanxmieso,
Fractured
SPE
Sept.
in R&voirs,x,
Tram.,
.&E
AnnuaJ
F.:
Reservoirs,,
Te&dcaI
‘aPressure
s paper
Co”
femnca
AmlY$is
Transient
SPE
and
11026
presented
Exhibition,
26-29.
JOURNAL
.
Pmble&
305-24.
H.
CkO.bY,
at the
hours
to FlrJw
1S6,
332
—
1971)
Infken.x
(Oct.
Wikq,
V.J,:
(Jui
42S,
Dam.
Bred.,
(1949)
time,
Its
Effect
, AIME,
Pa’fomce
K,H.
for
=
Pet.
(June
E.g.
(1955)
Sikora.
M.R.,
SPE
.@eles,
17., Meunier,
[~3/d]
[m]
hops
tp
Afterflow(July
of Shofl-Time
J.
and
Skin
Drawdown
Tech.
the
papr
Denver,
factor
t~
1970)
Tech.
Tech.
,7 Pet.
Flow
Trcms.
and
Tek,
Predicting
Trans
t =
T.S.
presented
[m3/d]
[m3
(Sept.
J.
-“
Effect
the
DaJlas
and
9 J. Pet.
D.i.,
of
“NomDaIcy
Coats,
ing
dimensimdezs
rate,
3n-
tiE,-234.
15.
STB/D
m3/d]
flow
[!-@a]
~]
“AD
Liq-
“AnaJysis
x,J, Pet.
G. W. , and
223-33;
V&nz.R?@
pressure,
pressure,
constant
[stock-tank
1965)
Los
pressure
Jr.:
from
Pa
to Fmducticm3dmulation
Build-Up
Drive,>>
13.
S!dn
Bore,
, API,
fi!lC.
Prod.
Ranw,
for
[kPa]
fofmation
=
skin,
dimensionless
dimensionless
J.
257.
‘“i%
Tracy,
Res.mmd
Am fvsii.,
12.
(Jan.
in Unsteady
EW
Matching,,>
,-&JME,
a WdJ
into
V?ii
(Feb.
of pD
A%
K. M.:
of a Well,
R. E.,
in Pressure
k
H.J.
Effect
TransmissibiShy
Data,”
Kersch,
‘ ‘Establishmerd
Gladfelwr,
and
10.
[m]
Rarmy,
Skin
L%.,
TVWCIUW
A. F.:
Flow
Which
storage
in e.
Tech.
.-.. -, 1. QR
..
Tram.
Hurst,
t.
bbllpsi
and
bv
CPpacity
J. Pet.
251.
Tr&.
EverdimgeII,
Prcducdve
ct
Jr.
793-SC@
and
and
Buifdup
Data
Inteqxctation
Storage,,,
R.,
‘YWeUbore
R.C.
Dm
AIInuaJ
23-26.
249.
, AIME,
Test
Test
Treatment,!,
, AJME,
Skin
Transient
249.
Storage
1. AmJMcaJ
Transient
Well
AJME.,
Different
Early-Time
at tie
1979
SpE
Las Vegak,
Sept.
WelJbore
AJ-Hussainy,
Tram,
S63-7%
Nomencfatnre
Tram,
of Wellbore
279-90;
for
8205
p=nted
and
Exhibition,
and
Monograph
Analysis,
Bctwee.
Cm’ves
“ShorGTime
R. G.,
Test
5.
; ‘A Cmnpmism
Effyt
97-10%
well
(1977),
Type
SpE
. . . .
Jr.:
in
TX
er .1.:
of S!dn
vesdgatim
Techniques
Advances
Storage
pawr
Confer
1970)
en~u~gement
Servic&
stage
for
its peffnission
JI.:
Richardson,
Gri”gamen,
and
reservoirs.
We!
Gerard
Scblumberger
alio
2.
behavior
time
OdS paper
help
R.C.
SPE,
Serk,
3.
Scblumberger
pub3ish
P
time,
dimensionless
constant
.y =
1. Earlo.gher,
to com-
Acknowledgment
and
testing
time,
References
4. Deconvolution
o
shut-in
formation
parameters.
We
running
=
storage
reserv@r
pressure
t@e
decon-
fofmation
methods
formation
can
production
transpose
=
ArD
wellbore
Some
without
computed
conventional
computed
data.
function)
effects.
function)
dimensionless
=
T =
production
of measured
introduced
(iiuence
be identified
of
tPD
straight
correct
(influence
deconvolution
sandface
pressure
this
of
relative
afe
and
s30pes.
3. The
ion
formulas
Homer
OF
PETROLEUM
TECHNOLOGY
New
20.
K@uk,
F.
TYCX
CUweS
and
pms.nmd
21,
at fie
hfarcb
23-25,
Kamey,
H.J.
22,
Chin,
1983
H.K.
Storase
SpE
and
skin
and
Wells,
c~.~
Wellbore
”
paper
Region~
A&val,
R,G.
Meeting.
: ‘ ‘Annulus
WOW.
and
Skin
W,E.:’
‘Pressure
where
StoraSe
SPE
11676
venti~,
Unloading
y=
O.5772
Rates
Effect,’+
.%c.
Buildup
for a Well
Pet.
Ew.
and
453-62.
and
and
“New
Penetrated
by W.lhm
1972)
(Oct.
P. A.:
PardallY
Jr.
as Influenced
J.
Kirwan,
for
Brigham,
Skin
in a Closed
Square,,,
J,
Pet.
Tech.
with
(Jan.
Et(aAt)=
197S)
\8M:du.
14146.
23.
Hammm
g, R. W.:
McGwv-HiU,
24.
%eAdcm,
York
25.
I, M.:
City
Stehf..st,
&
H.:
City
APPENDIX
Sckvu&s
anAEn@zeers,
375-77.
Trmqfom,
McGmw-3fiU,
New
the
Neglecting
and
207–14.
“Ntuncricaf
of
(1973)
of huq’ml
Use
for
Mefiods
York
(1972)
numications
fnversion
ACM
(Jan.
of Lapk.ce
1970)
13,
Transforms,”
No,
1,
13
the
for
exponential
qD
in
integral
Eq.
term
are
.i,
obtained
the
van
Everdingen7
as
368.
solution
for
–%e-@MD
pD
[–III
j3-2-y+fn
4
. . . . . . . . . . . . . . . . . . . . ..(A-7)
+Ei(@tD)].
12 yields
The
1
pSD[(tP+&)D]’
imaginary
forms
&(AtD)=
of
Eq.
the
Hursts
Com-
&OriIhUl
A
Substitution
and
Numerical
NW
-m
‘~~i
1
‘—
[
4(tP
+At)D
of
becomes
0
When
AtDs
@tD)
becomes
smaller
dr
.—+SqD(AtD),
2(AtD
-,)
. . . . . . . . . . . .. . ..
A-I
were
the
Stehfest
The
two
difference.
of the
log
given
in
integrafs
yields
z
using
between
1%.
Substitution
integral
com-
technique
exponential
the
A-7
difference
than
increases.
the
for
the
less
as AtD
A-7
f3kf.
from
inversion
30,
for
and
from
computed
trsnsform
approximations
Eqs.
:(&D)
vafues
Laplace
A-3.
vslues
integral
the
munericaf
-&]
of
with
Eq.
.%E,(.&)-j””e-Bexp[
values
psred
(in pmcticd
units)
(A-l)
~-pw(~)=~[l”g(%)+~(A’)l,
-(A-8)
where
~D=e
Thus,
tie
The
two
integral
forms
of gD
in Eq.
A-1
unfortunately,
present
van
the
Rafney’s
of
derivation
of
the
be
used
and
method
readily
Ramey
Ramey*
wss
integraf
interchangeably.
be integrated
integration.
that
integration
transform
wifl
cannot
Everdingen7
&tds
a heurirtic
where
. . . .. . . . . . . . . . . . . . . ..(A-2)
—d==-LmrD.
edited
is given
given
of his
below.
in
Eq.
10 dld
+~.
10
*KO(J).
The
long-time
approximation
for
(v
Koc~)=–lny+y.
Substitution
. .
The
of
Eq.
. . . . ..
long-time
approximation
substituting
limiting
times.
For
j3-27+ln
zero.
term
The
Eq.
A-3
g(At)
of ci+tain
lsrger
values
4+2$
e ‘aNEi(aAt)
e-a&Ei(aAt)=~,
in
Eq.
can
be
maybe
terms
obtained
in Eq.’ A-7
of
At,
the
term
A-9
approaches
when
yields
approxinsted
as
. . . . . . . . . . . . . . . . . ..(A-1O)
&+l.
Substituting
in
for
fbims
(A-3)
. . . . . . . . . . . . . . . ..(A4)
A4
Ei(aAf)
. . . . . . . ... . . . . . . . . . . . . . . . . . . . ..(A-9)
for
long
e ‘@(-bI
is
KO(~
4+
LapIace
is
by
Z(S)=
f3-27+fn
me
paper.
The
A-1
‘aAr[–fn
not
provided
out
i(At)=*e
and,
~(At)
Eq.
A-10
in
Eq.
A-9
yields
=~
(A-11)
2.3026dr”
““”””’”””””’’”””””””
1
j(s)
= –—
(h
s–in
4+2T).
. . . . . . . . ..(A-5)
2(s+.0)
The
inverse
Further
in
Laplace
transform
of
Eq.
[–in
&~i-2y+fn
A-5
the
yields
APPENDIX
4
ten
The
@ro)=
–fie-@tiD
+Ei(6AtD)],
‘Ranw
H.J,
FEBRUARY
Jr.:
P+rsmal
19S5
substitution
main
of Eq.
A-11
in Eq.
A-8
gives
Eq.
14
text.
dimensionless
B
form
of Eqs.
A-8
and
A-9
cm
be writ-
as
. . . . . . . . . . . . . . . . . . . . ..(A-6)
eommunkallon
S$anfwd
U.,
Stanford,
CA
(Mm+l
1984).
PD(A’)=05[”[-1+’(A’D)]
w
333
from
where
Eqs.
A-1
~(AtD)=
%.
‘~ND
+2,g.
The
[–In
4+
of
+At)D]/AfD}
Eq.
B-1
A-1
terms
and
of
A-2
the
as
follows.
modified
Let
Homer
us
rewrite
Eq.
time
Ei@AtD)
f’D@@=”4w%31”
. . . . . . .. . . . . . . . . . . . . . . . . . ...03-2)
differentiation
In{[(tp
(3-2y+ln
in
with
respect
1
+—
26AtD
to
yields
I
+%$(AtD),
,. (B-7)
. .
where
‘com=0.5++[g(Ad],
..................@-3)
$(AtD)=!4e-5ti.c
lA$(AtD)
where.
of
mcom
the
ln[(f,n
is
the
Homer
semifog
+At)D/AtD]
Let
us &fine
~=
I
computed
dimensionless
straight
line
slope
and
x
not
equals
a relative
error
=Ij
for
tie
computed
slope
...... .... .........
in Eq.
included
relative
.
as
(–fn
in
error
straight
pressed
as
is the
modified
.
(as
4+2S).
ordy
that
is
Thus,
modified
Homer
.(B-S)
term
time.
computed
in the
.:
remaining
Horner
of the
line
df(AtD)
—...-
~.
. (B-I)
A-7
the
of slope
semilog
fl-2y+ln
case)
the
Homer
can
be
ex-
. . . . . (E-9)
. . . . . .. . . . .. . . . .. . .
Ax
where
where
mcom>
‘=1”[(%%]++
0.5.
Differentiation
Substitution
~=
of
Eq.
df(AtD)
~~
—..
Ax
BzI
in
Eq.
B-3
yields
tion
. .. . . . . ... .. . . . . . . . . . . . . . . . .
of
of Eq.
the
result
the
&rivative
substituting
of
in Eq.
B-5
$(AtD)
with
respect
to x
Eq.
yields
o,
[–1.
A3e-~&D
6–2y+ln
B-11
gives
tPD,
and
.1
+.Ei((3AtD)+2S]-~
gives
the
334
“tie
semilog
13-2-y+ln
modified
4+2S).
. . . . . . . . . . . .(B-11)
dimensionless
semiIog
stiaigbt
time
for
line
as a fiction
the
start
of the
of
e,
4
Conversion
dimensionless
straight
line
. . . . . . . . . ...@-@
dme
as
for
the
a function
start
of
1.589873
E–01
=
Cp
x
1 .“*
E–03
=
the
Homer
dimensionless
semilog
time
straight
line
for
the
can
Pas
ft
X
3.048*
E–01
=
m
x
6.894757
E+OO
=
kPa
found
of
Pew
14,
term.
rmnusuiDt
accaPted
19%.
JP’f
Iaclor is exact
.Gmversl.m
fPD,
start
be
m3
of the
e, 0,
0ri@n=4
for
Factors
X
psi
S.
A formula
substhw
skin.
bbl
and
the
Homer
S1 Metric
B-5
to x and
[
2tpD
Eq.
respect
yields
+M)DI
~=
Homer
with
B-9
and
modified
AtD[(tP
B-8
Eq.
(B-5)
.
.(–in
Taking
in
‘@-’o)
and
rec.stved
for
PaPer
(SPE
mhlbitlo.
in the
P.MWO.
121771
held
Satiety
M.,ch
first
in
of Pe,,%um
11.
Prem.ted
I ss,.
at
S..
Franckm
JOURNAL
OF
Engineers
~~vi,ed
the
19S3
0~.
~s.
PETROLEUM
ollb
manuscript
SPE
Annual
Od,
,ece~ved
T@ch”icnl
~CHNOLOGY
5,
? 9S3.
‘e*
Con.
4aoo.o
[email protected]
I
LEQEND
horner
_ mo ,dKde l!w!!,e~
4280.0
4970.0
—.
—
.
—
—
.
4ZSOm0
\
4260.0
Y
‘\
4240,0
4aao,o
\\
\
\.
4220,0
—
\
‘\
4210s0
T
4aoo.o
lb’
HORNER & MODIFIED
Fig, 3-Modlfled
Horner and Horner plots
decllne sandfaco rate,
)2/77
—
.
1800.0
1700,0
.
II Ill
II
1600,0
1600,0
P
1400.0
Iaoo.o
1
TTll 111
1200s0
“
LEGEND
G wdlbOre DKM3SUWi
0
_m!M?lm.EE
!E!M!!u
1100.0
.
1000.0
1’
Fig, 4-Calculated
formation
pressure
dropusingIkwkatkm
method
and wellbore pressure drop,
1A/7,7
1600,0
?Emh
[
n
m
II’
I
0
—
0
—
1400,0
1s00.0
1200,0
/
/
1100,0
100 .0
1--1
10-8
10’
TIME, HOUR8
Fig, 5-Calculated
formation pressure drop using Hamming method and wellbore pressure drop,
10R
.
aooooo
2800.0
.
—
——
LEQEND
o formation pre8sun3
LY?s!!!l ore
, ixessurp_
—
2600.0
6
z
a
w“
e
g
—
—
S400.O
i
@@”—
)
r
>
—
—
—
—
—
—
1600.0
—
—
—
1600.0
—
—
—
1400.0
—
—
—
1200.0
—
—
1000.Q
—
2200,0
$
a
an
z
T
t=
s
x
@
2000.0
——
~
d
w
a
—-
.
-
.
I
Ii)-’
Fig, 6-Calculated
formation pressure drop using polynomial approximation
fractured reservoir,
and welibore pressure for a