SPE 16394 PA Applications of Convolution and deconvolution to transient well tests
Transcript of SPE 16394 PA Applications of Convolution and deconvolution to transient well tests
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8/17/2019 SPE 16394 PA Applications of Convolution and deconvolution to transient well tests
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pplications of
Convolution
and
Deconvolution to
Transient
Well Tests
F.J. Kuchuk SPE
Schiumberger-Doll
Research
Summary This paper presents the application of convolution and deconvolution interpretation methods. Two well-test field exam
ples, interpreted with these methods, suggest that the downhole flow rate is crucial for system identification and parameter estimation
and that the wellbore volume below the pressure gauge and flowmeter must be taken into account. A new generalized rate-convolution
method is presented to obtain the reservoir pressure. This new method gives better results than both the
Homer
and modified
Homer
methods. A new formula also is presented to determine the vertical permeability for partially penetrated wells.
Introduction
Transient well testing is a measurement of the output (observation)
of the system response to a given input. Control of the input, which
has traditionally been a constant flow rate or pressure at the well
head, is as important as the output measurement to obtain system
parameters. Control of the input has been a difficult problem for
well testing, with the exception of buildup tests at late times.
It has been recognized in the last decade that the measurement
of
the input signal (usually flow rate) at the sandface, along with
the output (usually pressure), is needed to ~ u c e wellbore-storage
effects and to account for rate variations. Furthermore, downhole
flow measurements are necessary to determine producing zones to
estimate permeability and skin from well-test data.
Well-test interpretation is the process of obtaining information
(reservoir parameters) from measurements (output) by use
of
the
input signal, all other pertinent data available for the system, such
as geological and well-log data, and the past production history.
For most well-test-interpretation problems, system identification
(diagnosis) and estimation of its parameters are done sequentially.
Since the early 1930 s,1 many interpretation techniques have
been developed to estimate reservoir parameters from measured
pressure and flow-rate data. The objective of this paper is to ana
lyze measured downhole pressure and flow-rate data from two
different wells with conventional and recently developed interpre
tation techniques.
Mathematical
Preliminaries
The relationship between flow-rate and pressure signals across the
sandface (in the wellbore) can be described as a convolution
operation
1 4
:
Apw t) = J qSjD T)Ap ~ j t - T ) d T , I)
o
where .lpw=wellbore pressure drop and qsjD=normalized sand
face flow rate,
qsjlq
where
qsj
=sandface flow rate and q,=a
reference flow rate. For Eq. I, the initial pressure of the forma
tion is assumed to be constant, uniform, and the same as the initial
pressure of the wellbore . .lp
j t )
in Eq. I is defined as
5
A p ~ j t ) = A p f t ) + A p l ) t ) , 2)
where
o t)
is
the Dirac delta function.
Apj(t)
and
Aps
are the pres
sure drops across the formation and the skin region, respectively,
for a constant flow rate q,. The Laplace transform
of
Eq. I can
be written as
. lPw s)=sijsfD s).lPsf s). . (3)
For most well tests, the tool (including pressure gauge and flow
meter) is located just above the perforations. However, they could
also be located at any point in the wellbore, including the well
head. Like the distinction between the surface and downhole flow
rates, a difference also exists between the sandface flow rate,
qSj'
and the flow rate at the tool location (measured flow rate, qm) be
cause of storage. This difference can be expressed as
4
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7
Copyright 199 Society of Petroleum Engineers
SPE Fonnation Evaluation, December 1990
qSj(t)-qm(t)= C dPw dt) (4)
where C is the wellbore-storage coefficient caused by the wellbore
volume below the tool. In the first formulation
of
the wellbore
storage effect on the sandface flow rate by van Everdingen and
Hurst,4 qm
is assumed
to
be constant. The substitution
of
Eq.
4
into Eq. I gives the wellbore pressure in terms of the measured
flow rate and the wellbore storage for a given formation response:
r
[
C
dpw]
.lPw t) = J
q m D T ) + - -
A p ~ j t - T ) d T , (5)
o
q dT
and its Laplace transform is
.lPw s) =sijmD S)[
fljisf(s) ]
1
CI
q)s2 flji
sf
s)
(6)
where qmD=measured normalized flow rate, qm1q,. Note that
if
there is no additional volume between the sandface and the tool,
Eqs. 5 and 6 reduce to Eqs. I and 3, respectively. Note also that
the term given within brackets in Eq. 6 is the well-known constant
rate solution, Apwj' with the wellbore-storage and skin effects. 4 7
If Apw is the wellbore pressure (measured or computed),
.lPwj
must be the response of the system, which includes the storage
volume below the measurement point. Thus Eq. 6 can also
be
written
in terms
of .lPwf
in the time domain:
.lPw t)
=
J
qmD
T).lp
'wj(t-T)dT
7)
o
For some well-test conditions, the relationship between the sand
face and measured flow rates can be expressed as
8
•
9
qsj(t)=qm(t)[l-exp(-at»),
(8)
where a*O and is constant. Substituting Eq. 8 into Eq. I yields
.lPw t)
=
JqmD T)[I-exp -at»)Ap
j t - T ) d T
.
(9)
o
The Laplace transform of Eq. 9 can be written
.lPw s)=s[ij mD s) - i jmD
(s+a)).lpsj(s). .
(10)
As Eq. 8 shows,
if
qm t) is constant, Eq. 9 will become the solu
tion for the exponential-wellbore-flow-rate case presented by van
Everdingen
8
and Hurst. 9 The Laplace transform
of
Eq. 9 for the
same case, qmD =
I,
can be written 1
. lpw(s)=a.lpsf (s)/(a+s) .
11)
Eq.
II
will be used later to analyze one of the field examples.
The above equations for the wellbore pressure (output) provide
a general framework for the solution of time-dependent internal
boundary conditions (input). They also permit the constant-wellbore
storage or exponential-wellbore-flow-rate solutions to be used as
a kernel (influence
or
unit response). Thus, in this formulation,
the wellbore volume between the measurement point and the sand
face can be included as an additional wellbore storage. The addi
tional wellbore-storage volume below the tool is usually more
significant for horizontal wells and wells with fractures and rat holes.
375
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4500
4000
2500
drawdown (solid)
buildup
(SYDlbols)
2000 -L - - F '= - ,-_--==_ .....___
.... .__ ..
0.01
0.1
10
t lme,hr
18000
14400
10BOO e:
7200
3600
III
oj
i
Fig.
1 Pressure
and flow rate
for
Well A drawdown and build
up
tests.
nterpretation
Methods
In this section, we briefly discuss the convolution, nonlinear-least
squares-estimation, and deconvolution methods, which will be used
to analyze the well-test examples.
Convolution. Here, logarithmic and generalized rate convolutions,
as well as modified Homer methods, are discussed.
The conventional multirate
ll
-
4
(Ref.
14
gives more literature
on the subject) and logarithmic (sandface-rate) convolution
10 15 16
methods are the same
if
the Riemann sum is used for the integra
tion
of
the convolution integral given by Eq.
1. For
both methods,
one also can use other numerical integration techniques. For the
multirate case, however, it does not make any difference which
integration technique is used because the number
of
the measured
rate data is small for a large time span, making the integration
timestep large. On the other hand, for the sandface-rate convolu
tion, the flow rate can be measured every second. Thus, a variety
of
numerical methods
lO
14-20
can
be
used to integrate Eq. 1.
In terms
of
testing procedure, flow rates for a multirate test are
measured at the surface, while pressure is measured at the sand
face. In other words, a multirate test basically consists
of
sequen
tial constant-rate drawdowns during which only transient downhole
pressure is continuously measured and flow rates usually are meas
ured intermediately. During each drawdown, the flow rate has to
become constant rapidly; otherwise, the wellbore storage will
strongly affect pressure measurements. Thus,
if
the flow rates fluc
tuate rapidly, the test cannot be analyzed with the multirate proce
dure. For this situation, one has to use a nonlinear least-squares
estimation (automated type curve) with the model given by Eq. 5
if
the wellbore storage is constant. Pressure and flow-rate meas
urements in the same time span and at the same wellbore location
close to the sandface will minimize problems associated with mul
tirate testing.
Ideally, we would like to know the sandface flow rate to inter
pret the measured wellbore pressure.
If
wellbore flow rate is not
measured, other indirect met"ods exist to determine the sandface
flow rate. The first method is to measure the movement
of
the
gas/liquid interface with an acoustic device.
21-23
The second ap
proach is to apply the mass-balance principle to the wellbore
volume.
24
25
The third method is to determine the sandface flow
rate from the measured wellbore pressure
26
-
28
with Eq.
4,
provid
ed that
qm is
constant
or
zero and that the wellbore storage remains
constant for the duration of the test.
The logarithmic convolution can be obtained from Eq.
1 by use
of
the logarithmic approximation for
t:.Pj
as
ll
12
(oilfield units)
Jw(t)= :.pw(t)/qmD(t)=m[jlct(t,qmD)+b], (12)
where
w
is the "reciprocal PI"29-31 or "rate-normalized pres
sure,
10 15 16 ftct(t,qmD)=[I/qmD t ) l l M ~ r ) log
(t-r)dr=log-
376
12050
1 2 ~ 0 0
12150
12200
12250
production
rate
profile.
ID
3500 7000 10500 14000
Fig,
2 Productlon
profile
for
Well A,
arithmic convolution time,
m= 162.6qpJkh,
and
b=log(k/p.ctrJ)
-3.2275+0.87S.
For radial flow, a linear plot
of w
vs. hct should yield a straight
line with a slope m and an intercept mb from which permeability
and skin can be estimated.
The logarithmic convolution method is simple and easy to use
and
is
similar to semilog methods in many respects. It performs
reasonably well for a fully penetrated well in a homogeneous reser
voir with negligible wellbore storage between the tool and sand
face. Thus a diagnostic logarithmic convolution derivative
27
32
may help determine whether the use
of
a radial model is valid for
the convolution interpretation.
Other convolution techniques can be developed for different flow
geometries as a diagnostic tool. Next, we consider use of
the gen
eralized rate-convolution method to estimate the reservoir pressure
and to verify the model.
For convenience, let us assume that a well is produced at a nor
malized rate
of qmD
until shut-in (or another drawdown). At any
time after shut-in, Eq. 7 can be partitioned as
Pw(t)=Pi-
JpqmD(r) :.p'wj(t-r)dr- JqmD
(r) :.p'wj (t-r)dr,
o
13)
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o
0.2
0.4
0.6 0.8
penetration ratio.
b
Fig.
4 Dlmenslonless
time
of
the start
of
the pseudoradlal
flow period for the estimation of vertical permeability.
In general, the measured pressure (as in Eq.
19),
measured flow
rate,49 and/or any processed form
of
the measured pressure, such
as a derivative, can be used to match the response
of
the system.
In the nonlinear least-squares estimation with rate, the response
of
a selected model is convolved with the measured downhole flow
rate, as in Eq.
1,
to obtain reservoir parameters.
Deconvolution.
The deconvolution method
lO
,17-19,29-31,50-53
is
the determination
of
the constant rate/pressure behavior
of
the sys
tem (unit response
of
influence function) from measured pressure
(output) and flow rate (input). In other words, deconvolution com
putes the pressure behavior
of
a well/reservoir system as
if
the well
was producing at a constant rate with or without constant-wellbore
storage or exponential-flow-rate effects. As discussed above, if the
sandface flow rate differs from the measured flow rate, flpd will
include the effect of the wellbore volume below the rate
measurement point. Once
flpd
is computed, conventional interpre
tation methods, including type-curve matching, can be used to iden
tify the well/reservoir system and to estimate its parameters. The
idea
of
deconvolution is simple
if
it is considered as a solution
of
the integral equation given by Eq. 1. In other words, for a given
set
of
pressure,
flpw,
and flow-rate,
qsjD
measurements, decon
volution is the process
of
computing t:..Psj flpd if the measured
flow rate is used) from
Eq.
1. Using the Riemann sum for the in
tegration
of
Eq. 1, one can write a simple deconvolution formula:
t:..Pw)n
-E;:/
qmD)n-i(flpd)i
(flpd)n = ,
(20)
(qmDh
where n=I,2,3
N
m
.
Note that the above deconvolution formula is recursive. In other
words, ( f lpdh,
(t:..Pdh··· t:..Pd)n-1
(all previously computed
values) are needed to compute
(flpd)n
Small perturbations in the
flow-rate measurements (errors) result in large perturbations in the
solutions
(flpd)
computed from Eq. 20 because the solution
of
the
integral equation given by
Eq.
I is ill-posed.
19
It
is
well known
that measurements in general, no matter how carefully acquired,
have errors. Thus, the constrained deconvolution method, 19 which
minimizes the instability problem caused by measurement errors,
will be used to analyze the examples.
/i-Deconvolution. For
the exponential-wellbore-flow-rate case
(the sandface rate is approximated by
Eq.
8), the deconvolved pres
sure,
flpd,
from Eq.
11
can be written
lO
1 dilPw(t)
f lpd t =- +t:..Pw t) .
(21)
ex dt
This technique makes it quite simple to compute flpd from the
measured downhole pressure, its derivative, and ex which is ob-
378
7000
5250
Q
III
4i
l
3500
1750
..
0
. f····
0
1000
2000 3000
4000
5000
dp/dt
psilhr
Fig.
5 Flow
rate as a function of the derivative of pressure
with respect
to
time.
tained from the measured downhole rate
if
the wellbore flow rate
varies exponentially.
Gas Wells.
One
of
the well-test examples to be analyzed is from
a gas well. A brie f discussion
of
pseudovariables, which will be
used for the interpretation
of
the gas well-test data, is given here
for convenience. The real gas potential (pseudopressure) given by
AI-Hussainy et al
54
is
modified by Meunier et at 55 as
J l iZi r
p
P
1/;N p)=2-J dp
(22)
Pi
h
J.l. p)z p)
Although
1 ;N
is called normalized pseudopressure, we call it
pseudopressure,
1/;.
Unlike the unit (psi2/cp)
of
the real gas poten
tial, it has the unit of pressure. The pseudovariables given in Eq.
22 partially linearize the diffusivity equation. 56 They are, how
ever, sufficient for the pressure and permeability range
of
our well
test problems.
ield
Example
The objective
of
the interpretation
of
the following tests is not to
produce numbers from each analysis. Instead, we demonstrate cer
tain salient features
of
each technique and compare them with con
verttional techniques. The well-test examples given are well-run field
experiments compared with well tests we usually encounter. In many
instances, the infinite-acting radial flow does not occur during a
well test. Cost
or
operational restrictions can make it impractical
to carry out a test of sufficient duration to attain radial flow. In
these circumstances, convolution and deconvolution techniques may
be the only approach available for the interpretation
of
short tests.
For example, well-test interpretation for saturated reservoirs is often
confounded by the presence
of
a gas cap, which often creates at
least two well-known interpretation problems: the allowance
of
a
large standoff to inhibit gas coning can lead to very low penetra
tion ratios, and
if
a well is in direct communication with a gas cap,
the infinite-acting radial-flow period will never occur.
Well
A: A
Partially Penetrated Well.
This is a deep well in a
thick reservoir and has an
1
OOO-ft rathole below the producing,
zones. The geological, log, and core data suggest that the forma-'
tion is mildly layered; i.e. the properties
of
each zone are not ex
pected to be very different. After a 2-day shut-in period, the tool
was lowered and stationed at thy top
of
the formation, and the down
hole pressure was recorded for about 30 minutes. The well was
then put on production with the expectation that the production rate
would stabilize at a constant rate
of
15,000 BID. Within a 20-minute
period, a significant drop in the downhole pressure was noticed.
In fact, the pressure fell below the bubblepoint pressure.
To
avoid
two-phase flow in the wellbore and formation, the production rate
was decreased (Fig.
1).
After
7
hours
of
recording the downhole
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llpw
1000
1...,-
7
\' dpw/dlntH
1
. . : : ~ _ : P j t s u p
'
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1
r
0.01 0.1
1
tbne hr
10
Fig.
8-Comparlson of
derivatives.
dp./dlnt
100
Buildup Test. Fig. 1 also presents the buildup pressure and af
tertlow rate during the buildup test, which was started after about
19 hours of production. As Fig. 1 shows, the measurable after
flow rate period is short 40 minutes). The missing sandface rate
data could be computed with Eq. 3 as discussed above. As shown
in Fig.
5
however, the wellbore-storage coefficient, C, which is
from the whole wellbore-storage volume and represents the slope
of the linear plot of the sandface rate vs. dp/dt see Eq. 4), is not
constant. For buildup tests, when the wellbore-storage coefficient
becomes constant, a plot of qm t) vs. dp/dt should yield a straight
line passing through the origin. Fig. 5 shows that the common
method of obtaining C for the sandface-flow-rate estimation from
the wellbore volume and the compressibility of the wellbore fluid
would not be reliable for this test because
of changing wellbore
storage.
The log-log plots
of
the derivatives
of
the wellbore pressure with
respect to the
Homer
superposition time, dpw/d In tH), and the
multirate superposition time,
dpw/dtsup'
with the flow rate meas
ured during the drawdown test) shown in Fig. 6, indicate that after
the wellbore-storage effect, the system slowly approaches a possi
ble radial-flow period. The plot at the upper right shows that the
Homer semilog straight line is not fully developed. This could be
a result of the effect of the short producing time because the mul
tirate superposition indicates a radial-flow period. As explained
above, the time for the start
of
the radial-flow period from the
derivative
of
the superposition plot and Eq. 23 can be used to esti
mate kv= 11.4 md. This value compares favorably with the
kv
ob
tained from the spherical derivative plot of the drawdown
deconvolved pressure. The horizontal permeability and skin com
puted from the same plot are given in Table 1.
The convolution,
dJ
w/d lcp and deconvolution, dpd1d In tH),
derivatives do not show any diagnostic features Fig. 6). On the
other hand, as in the drawdown case, the derivative of the decon
volution pressure with respect to the spherical time function,
dpd1d spt,
indicates a short hemispherical flow period. The system,
at least, is changing from a hemispherically dominated flow to a
radially dominated flow. Thus the buildup behavior
of
the system
is similar to the drawdown behavior.
Final Interpretotion and Discussion. So far, we have been con
cerned mainly with the system-identification problem. At this point,
we have observed from both tests 1) changing wellbore storage,
2) partial penetration effects, 3) no apparent outer-boundary ef
fects, and 4) a fully developed radial-flow period owing to the en
tire formation. Moreover, the buildup test without the drawdown
flow-rate measurements for the superposition) could have given
a misleading interpretation. For this buildup test, the parameters
obtained from the superposition derivative Fig. 6) are assumed to
be more accurate than those from other techniques because the
radial-flow period is well-defined and the vertical permeability com
pares well with that from the drawdown deconvolved pressure.
380
4500
4300
~
.,
ill
'
4100
.;
I
III
l 3900
'
3700
3500
100
Horner
modified Horner
generalized rate convolution
10· 10
8
t ime hr
'
10
12
Fig.
9-Horner
modified Horner, and generallzed-rate
convolution plots
for
Well A.
These parameters will be used as initial guesses for the nonlinear
estimation, which will be carried out next.
The nonlinear estimation method type-curve matching with rate)
is applied to the drawdown test to improve the results obtained previ
ously. In this estimation, the effect
of
wellbore storage on well
bore pressure is included. In other words, the mathematical model
will be Eq. 1 where qsj is given by Eq. 4 as a function
of
both
the measured wellbore flow rate and an unknown wellbore-storage
coefficient caused by the wellbore volume below the flow-rate
measurement point).
It
is assumed that the wellbore-storage coeffi
cient from this additional wellbore volume is constant. This assump
tion is reasonable because the wellbore pressure was kept above
the bubblepoint pressure, with the exception
of
a short time during
the drawdown. In general, the variation of the wellbore storage is
a result
of
two-phase flow in the tubing from the wellbore to the
wellhead. The reservoir model, I1p the impulse response
of
the
system) in Eq. 1 is the derivative of PD in dimensionless form
given by Eq. A-I57 plus the damage skin
S.
The horizontal and
vertical permeabilities, skin, and wellbore-storage coefficient will
be estimated by the nonlinear estimation procedure with the known
formation thickness and penetration ratio. The thickness
of
the open
interval is directly determined from the production profiles. The
formation thickness is detemtined from the geological and openhole
log data. Although possible in principle, the estimation
of b
is more
difficult than the estimation of other parameters. Thus, we will at
tempt to estimate b only if we do not obtain a satisfactory match
with its present value of 0.49.
Fig. 7 shows a good match between the measured and computed
pressures as log-log and semilog plots. As stated above, the deriva
tives are not included because they were noisy as a result
of
the
flow-rate variations. Table 1 gives the final estimates obtained
from this match. C=0.OO56 bbl/psi, which yields
CD
5.6146
C 27rf >c
t
hrJ)=48,
compares well with the additional wellbore
volume below the tool. hwD [the dimensionless wellbore length,
hWD
= hw1rwh/kH1kv] is calculated as 830 from the estimated
kH=
110.0 md and
kv=
10.6 md.
Now that we know the model and its parameters, let us compute
the derivatives
of
the wellborepressure for this partially penetrat
ed well
b=0.49
and
h
wD
=830) with
CD =48
and
S=4.8)
and
without wellbore-storage and skin effects. These derivatives are
compared with the derivative
of
the deconvolved pressure computed
from the drawdown data. This comparison is important because
both the convolution nonlinear estimation or logarithmic) and
deconvolution, and their derivatives, may be affected to a certain
degree by the different smoothing processes. This comparison is
shown in Fig. 8 [see dpsjld spt for
CD =0]
which indicates that
we do not have a hemispherical flow regime. The derivative of the
sandface pressure without the wellbore-storage effect, dpsjld spt,
indicates a long transition period, which results from the partial
penetration effect before the flow becomes pseudoradial. Of course,
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4500
3800
1
,;
3100
t400
1Il
1:1
1700
.
pseudopressure
o
measured
rate
compu t ed rate
~
1
/
l 0 0 0 - - - - - - - . - - - - - - - r - - - - - ~ ~ - - - - ~ - - - L
0.001 0.01
0.1
tlme,hr
10
2500
2000
1500
i
1000
=
500
Fig
10 Pseudopressure
and afterflow rate
for
Well
B.
there is a single point in this curve that would have the correct
hemispherical slope. In other words, the length
of
the open inter
val
is
too large compared with the distance to the lower no-flow
boundary to have a well-defined hemispherical flow period. The
curve of I dp4s
/dfs
pt
)
I in Fig. 8 (the spherical derivative
of
the
wellbore pressure including the effects of
D
=48 and S=4.8) has
a minimum; this is also true for the curves
of
I
dp25
/d
fs
p
t)
I
(with
D =25
and
S=4.8)
and
I
dp lOo/dfspt
I
(with
D
= 100 and
S=4.8)
at different times. The spherical derivative of the deconvolved pres
sure probably becomes flat for a short time period because
of
wellbore-storage effect. It must then be by coincidence that the
derivative at this flattening period becomes approximately equal
to the hemispherical slope. A low
or
high value of the wellbore
storage would yield an inaccurate hemispherical slope. In general,
the hemispherical slope obtained from this flattening period will
be inaccurate. Nevertheless, the spherical derivative
of
the decon
volved pressure exhibits the true characteristics of a partially
penetrated well.
Fig. 9 presents the Homer, modified Homer, and generalized
rate-convolution plots where time functions are defined as
(t
p
+l1t)/l1t
for the Homer, 10/mHt (Eq. 18) for the modified
Homer, and 10Irct (Eq. 16) for the generalized rate convolution.
As can be seen from Eq.
18,fmHt
is a function
of
the skin,
St,
and
diffusivity constant, . . We therefore use the final estimates with
a total skin of 15.9
(St=Slb+S
p
)
where Sp=6.0 (from Ref. 14
for
b=0.49
and
h
wD
=830).
Strictly speaking, the application
of
the modified
Homer
method is not valid because the well is par
tially penetrated. The generalized-rate-convolution
time,frct,
is ob
tained fromPD given by Eq. A-I of Ref. 57 and the final estimates
of
C, S,
kH
and
kv.
The plots given in Fig. 9 are a convenient
way to display and compare the Homer, modified Homer, and gen
eralized rate convolution together. The generalized-rate-convolution
plot, which is a semilog plot of Pw vs. frct yields a straight line
with a slope
m
(although it was known) and an intercept
p*
(the
initial or extrapolated pressure). The slope slightly increases after
fret
= 100 (
<
1 hour) possibly because the partially penetrated well
model may not be not exact because all perforated zones are com
bined as a single-zone model and the afterflow rate could not be
measured at late times during the buildup.
Fig. 9 exaggerates the early-time data; in fact, the time interval
between 0 and 1 hour
is
about
14
log cycles, and it
is
onl) two
log cycles for the time interval between 1 and 24 hours. Like other
semilog plots, it is unfortunate that this type
of
display relies on
the plotting scale. Of course, we could have looked at the deriva
tives of these plots, as we did for the Homer plot. They may not
be useful for the determination of the initial pressure, however,
which
is
the main objective
of
this type
of
plotting. Fig. 9 also
presents the late-time enlargement. The extrapolated pressure,
p*,
obtained from the generalized rate-convolution curve,
is
4,496 psi,
which
is
1 psi higher than the initial pressure before the drawdown
test. Note that both the Homer and modified Homer methods de
pend on the existence
of
a storage-free, infinite-acting radial flow
SPE Fonnation Evaluation, December 1990
11000
I
I I 100
~
=0.11, 1- =0.017 cp,
c
t
=0.OOO31 psi -I r
w=0.365 ft, h= 120 ft,
tp
=567 hours, pseu
dopressure,
1 Iw
at
tp
= 1,221.0 psi, and production rate,
q=2,450.0
BID.
To have better flowmeter response, the continuous produc
tion logging tool, which was located just above the tubing shoes
during the test, was used. This well-test example was selected be
cause
of
its interesting features.
For this gas well, the measured pressure data are converted into
pseudopressure, defined by Eq. 22. The computed pseudopressure
is treated as a pressure data set of an equivalent liquid case (see
Ref. 55).
Fig. 10 presents the measured pseudopressure and flow rate. In
Fig. 10, the afterflow rate
is
measurable for a few hours, after which
the rate becomes too small to be measured. We notice that the down
hole flow rate can be approximated by an exponential function as
qC(t)=2,450e-
5
.
3t
,
24)
381
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8/17/2019 SPE 16394 PA Applications of Convolution and deconvolution to transient well tests
8/10
measured (symbols)
computed (solid)
1 0 ~ ~ ~ ~ ~ ~ r - - r - r ~ ~ , - - ~ ~ ~ ~ ~ ~ ~
0 01 0.1
1
tlme.hr
10
Fig. 12-Comparlson
of
measured and computed pseudopres
sures and derivatives.
where the exponential constant a=5 3
is
determined from the meas
ured flow rate. The constant 2,450 BID is the flow rate before shut
in. Fig. 10 also presents the computed (from Eq. 24) flow rates.
Fig. 10 shows that the exponential decline given in Eq. 24 approx
imates the measured flow rate well up to 1 hour. The flow rate
computed from Eq. 24 is much smaller than the actual values be
cause the flow rate declines very slowly after 1 hour (Fig. 10).
Fig. 11 presents the derivatives of the pseudopressure and nor
malized pseudopressure with respect to different time functions.
These derivatives indicate that the wellbore pseudopressure
is
heav
ily dominated by the wellbore storage and that the system is possi
bly becoming an infinite-acting radial flow after 10 hours (first
diagnostic observation). The convolution and deconvolution deriva
tives may not be accurate after 1 hour because the flow rate meas
urem nts or their extrapolation is unreliable. In general, when the
flow rate is undermeasured (less than its true value)
or
underesti
mated, its effect will appear as a wellbore storage provided that
the surface flow rate does not fluctuate rapidly. This is apparent
in convolution and deconvolution plots in Fig. 11. Thus, these
derivatives do not indicate any feature
of
the system earlier than
the Homer derivative. The semilog slope of an infinite-acting radial
flow period from Fig.
11
is 228 psi/cycle, which gives
k=0 25
md
and S=l l l
The derivative
of
the deconvolution pseudopressure, with respect
to the spherical time function,
dl/ld1dfspt,
is also included in Fig.
11
to show whether the pseudopressure might be affected by lost
or plugged perforations.
It
is known that this well is fully perforat
ed. The spherical derivative also indicates the pronounced effect
of the wellbore storage and possibly the beginning
of
an infinite
acting radial flow period.
As Fig.
11
shows, with the exception of very few data points
at the beginning, the deconvolved pseudopressures from the con
strained deconvolution 19 and {3-deconvolution (Eq. 21) methods
give identical results. The advantage of the {3-deconvolution method
is that it is easy to compute.
It
can be continued even af ter the flow
meter data become unreliable below the flowmeter threshold value,
with the assumption that the downhole flow rate declines exponen
tially during the test. As stated above, this assumption did not work
for this test.
Fig. 12 shows the match of the derivative of the deconvolved
pseudopressure (the constant-rate behavior of the system includ
ing the effect of the additional volume) with the constant-welibOre
storage type curves for a fully penetrated well in an infinite reser
voir. The parameters obtained from derivatives are used as initial
guesses for this matching. The estimates obtained from this type
curve matching are k=0 26 md, S=11.8, and
C=O.OI
bbl/psi
(CD
= 16). This computed C value is slightly higher than that ob
tained from the 180-ft wellbore volume below the tool. These pa
rameters compare well with those from derivatives.
Another nonlinear estimation
is
performed with a fully penetrat
ed well in an infinite radial reservoir for the verification and im-
382
4500
' )
.-
3700
~ .....
.
Ul
.
s::Io
.
;
,.
2900
' \
II
III
e
'\
s::Io
2100
Horner
·······\.
modified
Homer
generalized
rate
convolution
1300
10 1000
lOS
10
7
10"
lOll
1013
time, hr
Fig.
13-Horner
modified
Horner, and generalized-rate-
convolution plots for Well B.
provement of the above estimates. The mathematical model, ilp.£
used in Eq.
19
is given by Eq. 7, where
qmD
is the normalized
measured flow rate.
Unlike the above deconvolved pseudopressure
matching, at each iteration during this nonlinear estimation, the
constant-rate solution with the wellbore-storage and skin effects for
the fully penetrated well model is convolved with the flow rate,
as in Eq. 7. Thus, the nonlinear estimation with rate data requires
more computation time than does the deconvolved pseudopressure,
from which the effect of
the flow rate variations are eliminated.
It
is therefore desirable for the nonlinear estimation with rate to
have the initial guesses as close as possible to the final solution.
Thus, deconvolution not only indicates diagnostic features of the
system, but also provides satisfactory estimates. Both nonlinear es
timations should be carried out, however, because of the smooth
ing properties of convolution and the ill-posed nature of
deconvolution.
Fig. 12 shows an excellent match between measured and com
puted pseudopressures and their derivatives. The estimates obtained
from this match are k=0 26 md, S= 12.0, and C=O OI bbl/psi.
The above analysis, including the diagnostic and estimation step,
has produced a model with parameters except the reservoir pseu
dopressure. The model fits the observed behavior
of
the system
very well. To complete the interpretation of this buildup test, we
not only have to estimate the reservoir pseudopressure (extrapo
lated or initial), but also have to know its effect on other estimates
because, for the convolution, deconvolution, and nonlinear esti
mation procedure, we have used measured
t...jIw=l/Iw fJ.t)-l/Iw fJ.t
=0 , where l/Iw fJ.t=;O) is the flowing pseudopressure before shut
in and not the initial pseudopressure. In other words, the draw
down solutions are used. This aspect of the problem can be solved
accurately if we use Eq. 16. Unfortunately, it may become a for
midable task computationally. Thus the generalized rate-convolution
technique is used to estimate the reservoir pseudopressure.
Fig. 13 presents the Homer, modified Homer, and generalized
rate-convolution plots where time functions are defined as tp
+
fJ.t)/fJ.t for the Homer, 1 ImHJ for the modified Homer, and
10lrct
for the generalized rate convolution. It is convenient
to
display and
compare all of them together. Note that both the Homer and modi
fied Homer methods depend on the existence of a storage-free,
infinite-acting radial flow period. On the other hand, the general
ized rate-convolution method can give the extrapolated pseudopres
sure at any test time. Fig. 13 also presents the late-time enlargement.
As Fig.
13
shows, each curve extrapolates
to
a different pseudopres
sure,
l/I*,
as
4,765.4,4,772.2,
and 4,778 psi for generalized rate
convolution, modified Homer, and Homer, respectively. The ex
trapolated pseudopressure obtained from the generalized rate con
volution should be the most accurate one.
onclusions
In this paper we applied convolution and deconvolution interpreta
tion methods to two well tests.
It
is
clear from the interpretation
SPE Formation Evaluation, December 1990
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8/17/2019 SPE 16394 PA Applications of Convolution and deconvolution to transient well tests
9/10
of
these well-test examples that the downhole flow rate
is
crucial
for system identification and parameter estimation. Both measured
downhole pressure and flow rate, however, also can be affected
by the wellbore volume below the pressure gauge and flowmeter.
Thus, this must be taken into account for the interpretation.
t is shown that the deconvolved pressure and its derivative are
an effective system identification tool and also can provide initial
estimates for nonlinear estimation. Without diagnostic steps, rely
ing solely on nonlinear estimation may lead to an erroneous model
and estimates.
A new interpretation method, called generalized rate convolu
tion,
is
introduced to obtain reseI'voir pressure and the final
verification
of
the model and its estimated parameters.
t
is shown
that this new method works better than the Horner and modified
Horner methods.
fj-deconvolution provides a simple technique for obtaining de
convolved pressure that can be used for system identification and
parameter estimation, if the flow rate varies exponentially.
A new method
is
presented to determine the vertical permeabil
ity for partially penetrated wells. The method uses the onset of the
radial flow period, if it evolves during the test.
t is
shown that an integrated interpretation approach reduces a
possible inaccurate interpretation and harmonizes features
of
the
system with the well-test data.
Nomenclature
b
=
penetration ratio or intercept
e
= total system compressibility, psi - 1
C
=
wellbore-storage constant, bbllpsi
f
= time function
h
= formation thickness,
ft
J
= positive scalar objective function for minimization
J
w
= reciprocal PI or rate-normalized pressure
k =
permeability, rod
m
= slope
N
=
number of measured data points
p
= pressure, psi
q = flow rate, BID
r = radius,
ft
s
= Laplace image space variable
S
=
damage skin
t =
time, hours
W = positive weight factor
x = parameter vector
0/
= positive constant
[
= Dirac delta function
1 = pressure diffusivity,
ft
2
/hr
J I
=
oil viscosity, cp
T
= dummy integration variable
c/>
=
system porosity
1 ; = pseudopressure, psi
Subscripts
d
= deconvolved
D = dimensionless
f
=
formation
H = horizontal
H = Horner time
i = initial
let
= logarithmic convolution time
m t
= modified Horner time
N = normalized
p = perforated
r
= reference
ret = rate convolution time
s
= skin
sf = sandface
sl
=
sernilog
spt = spherical time
SPE Formation Evaluation, December 1990
sup
=
superposition time
V = vertical
w = well, wellbore, or perforated
w = wellbore flowing
Superscripts
e = model or computed
m
= measured
- =
Laplace transform of
I
= derivative
*
= interpreted
cknowledgments
I thank Schlumberger-Doll Research for permission to publish this
paper and Christine Ehlig-Econornides
of Schlumberger Well Serv
ices for providing helpful discussions.
References
1 Schilthuis, R.J. and Hurst, W.: Variations in Reservoir Pressure in
the East Texas Field, Trans. AIME (1935) 114, 164-76.
2. Muskat, M.:
The
Flow of Compressible Fluids Through Porous Me
dia and Some Problems in Heat Conduction, Physics (March 1934).
3. Muskat, M.: The Flow
of
Homogeneous Fluids Through Porous Me-
dia
McGraw-Hill Book Co., New York City (1937).
4. van Everdingen, A.F. and Hurst, W.: Application of the Laplace
Transformation
to
Flow Problems in Reservoirs, Trans. AlME (1949)
186, 305-24.
5. Kuchuk,
FJ.
and Wilkinson, D.: Transient Pressure Behavior
of
Com
mingled Reservoirs, paper SPE 18125 presented at the 1988 SPEAn
nual Technical Conference and Exhibition, Houston, Oct. 2-5.
6. Agarwal, G.R., AI-Hussainy, R., and Ramey, H.J. Jr.:
An
Investi
gation
of
Wellbore Storage and Skin Effect in Unsteady Liquid Flow:
I. Analytical Treatment , SPEJ (Sept. 1970) 279-90; Trans. AIME,
249.
7. McKinley, R.M.: Wellbore Transmissibility from Afterflow
Dominated Pressure Buildup
Data,
JPT July 1971) 863-72; Trans.
AIME,251.
8.
van
Everdingen, A.F.: The Skin Effect and Its Influence on the Produc
tive Capacity
of
a
Well,
Trans. AIME (1953) 198, 171-76.
9. Hurst, William: Establishment of the Skin Effect and Its Impediment
to Fluid Flow into a Well Bore, Pet. Eng. (Oct. 1953) A-6-A-16.
10. Kueuk, F. and Ayestaran, L.: Analysis of Simultaneously Measured
Pressure and Sandface Flow Rate in Transient Well Testing,
JPT
(Feb.
1985) 323-34.
II. Russell, D.G.: Determination
of
Formation Characteristics From Two
Rate Flow Tests, JPT (Dec. 1963) 1347-55; Trans. AIME, 228.
12 Odeh, A.S. and Jones, L.G.: Pressure Drawdown Analysis, Variable
Rate Case, JPT (Aug. 1965) 960-64; Trans. AIME, 234.
13
Matthews, C.S. and Russell, D.G.: Pressure Buildup and Flow Tests
in Wells
Monograph Series, SPE, Richardson, TX (1967) 1.
14 Earlougher, R C. Jr.: Advances in Well Test Analysis Monograph Ser
ies, SPE, Richardson, TX (1975) 5.
15. Meunier,
D.,
Wittmann, MJ., and Stewart, G.: Interpre tation
of
Pres
sure Buildup Test Using
In-Situ Measurement of Afterflow, JPT (Jan.
1985) 143-52.
16 Fetkovich, M.J. and Vienot, M.E.:
Rate
Normalization
of
Buildup
Pressure
By
Using Afterflow
Data, JPT
(Dec. 1984) 2211-24.
17. Thompson, L.G., Jones, J.R., and Reynolds, A.C.: Analysis of Pres
sure Buildup Data Influenced by Wellbore Phase Redistribution,
SPEFE
(Oct. 1986) 435-52.
18. Thompson, L.G. and Reynolds, A.C.: Analysis
of
Variable-Rate Well
Test Pressure Data Using Duhamel's Princip le, SPEFE (Oct. 1986)
453-69.
19. Kuchuk, FJ., Carter, R.G., and Ayestaran, L.: Deconvolution
of
Wellbore Pressure and Flow Rate, SPEFE (March 1990) 53-59.
20. McEdwards, D.G.: Multiwell Variable-Rate Well Test Analysis,
SPEI (Aug. 1981) 444-46.
21. Godbey, 1.K. and Dimon, C.A.: The Automatic Liquid Level Moni
tor for Pumping Wells, JPT Aug. 1977) 1019-24.
22. Podio, A.L., Tarrillion, M.J., and Roberts, E.T.: Laborato ry Work
Improves Calculations, Oil Gas J (Aug. 25, 1980) 137-46.
23. Hasan, A.R. and Kabir, C.S.: Determining Bottomhole Pressures in
Pumping Wells, SPEI (Dec. 1985) 823-38.
24. Hasan, A.R. and Kabir, C.S.: Application of Mass Balance in Pumping
Well Analysis,
JPT
(May 1982) 1002-10.
383
-
8/17/2019 SPE 16394 PA Applications of Convolution and deconvolution to transient well tests
10/10
I
uthor
FIIot I Kuchuk
is a senior scientist and
program leader at Schiumberger-Doll Re-
search Center In Ridgefield, CT. He
researches fluid dynamics in porous me·
dla and performs reservoir testing.
Kuchuk holds an
MS
degree from the
Technical
U of
Istanbul and
MS
and
PhD
degrees from Stanford U., all In petrole
um engineering.
25. Simmons, J.F.: Convolution Analysis
of
Surge Pressure Data,
JPT
(Jan. 1990) 74-83.
26. Ramey, H.J. Jr. and Agarwal, R.G.:
Annulus
Unloading Rates as
Influenced by Wellbore Storage and Skin
Effect, SPEJ
(Oct. 1972)
453-62;
Trans. AIME, 253.
27. Bourdet, D. and Alagoa, A.:
New
Method Enhances Well Test In
terpretation, World Oil (Sept. 1984).
28. Westaway, P.J., EI Shafei,
1.,
and Wittmann, M.J. : A Combined Per
forating and Well-Testing
System,
paper SPE 14686 presented at the
1985 SPE Production Technology Symposium, Lubbock, Nov. 11-12.
29. Gladfelter, R.E., Tracy, G.W., and Wilsey, L.E.: Selecting Wells
Which Will Respond to Production-Stimulation Treatment,
Drill.
Prod. Prac.
API, Dallas (1955) 117-29.
30. Ramey, H.J. Jr.: Verification of the Gladfelter, Tracey, and Wilsey
Concept for Wellbore Storage Dominated Transient Pressures During
Production,
J. Cdn. Pet. Tech.
(April-June 1976)
84-85.
31. Winestock, A.G. and Colpitts, G.P. : Advances in Estimating Gas Well
Deliverability, J. Cdn. Pet. Tech. (July-Sept. 1965) 111-19.
32. Ehlig-Economides, C. et al.: Evaluation
of
Single-Layer Transients
in a Multilayered
System,
paper SPE 15860 presented at the 1986
European Offshore Petroleum Conference, London, Oct. 20-22.
33. Bourdet, D. et af.: A New Set
of
Type Curves Simplifies Well Test
Analysis,
World Oil
(May 1983).
34. Bourdet,
D.,
Ayoub, J.A., and Pirard, Y.M.: Use of Pressure Deriva
tive in Well-Test Interpretation, SPEFE (June 1989) 293-302; Trans.
AIME,287.
35. Theis, C.V.: The Relation Between the Lowering of the Piezometric
Surface and the Rate and Duration
of
Discharge
of
Well Using Ground
Water
Storage,
Trans. AGU (1937) 519-24.
36. Papadopulos,
I.S.
and Cooper, H.H. Jr.: Drawdown in a Well
of
Large
Diameter,
Water Resources Res. (1967)
3,
No.1, 241-44.
37. Ramey, H.J. Jr.: Short-Time Well Test Data Interpretation in the Pres
ence of Skin Effect and Wellbore Storage, JPT (Jan. 1970) 97-104;
Trans. AIME, 249.
38. Earlougher, R.C. Jr. and Kersch, K.M.: Analysis
of
Short-Time Tran
sient Test Data by Type-Curve Matchin g,
JPT
(July 1974) 793-800 ;
Trans. AIME, 257.
39. Gringarten,
A.C.
et af.: A Comparison Between Different Skin and
Wellbore Storage Type-Curves for Early-Time Transient Analysis,
paper SPE 8205 presented at the 1979 SPE Annual Technical Confer
ence and Exhibition, Las Vegas, Sept.
23-26.
40. Earlougher, R.C. Jr. and Kersch,
K.M.:
Field Examples of Auto
matic Transient Test Analysis,
JPT Oct.
1972) 1271-77.
41. Dixon,
T.N.
et
aI.:
Reliability
of
Reservoir Parameters From History
Matched Drill Stem Tests, paper SPE 4282 presented at the 1973 SPE
Symposium on Numerical Simulation of Reservoir Performance,
Houston, Jan. 11-12.
42. Panmanabhan, L. and Woo,
P.T.:
A
New Approach to Parameter
Estimation in Well Testing, paper SPE 5741 presented at the 1976
SPE
Symposium on Reservoir Simulation, Los Angeles, Feb. 1-2.
384
43. Welty, D.H. and Miller, W.C. : Automated History Matching
of
Well
Tests,
paper SPE 7695 presented at the 1979 SPE Symposium on
Reservoir Simulation, Denver, Jan. 31-Feb. 2.
44. Rosa, A.J. and Home, R.N.: Automated Type-Curve Matching in
Well Test Analysis by Using Laplace Space Determination ofParame
ter Gradie nts, paper SPE 12131 presented at the 1983 SPE Annual
Technical Conference and Exhibition, San Francisco, Sept.
5-8.
45. Barua,
J.,
Kucuk,
F.,
and Gomez-Angulo, J.: Application of Com
puters in the Analysis
of
Well Tests From Fractured Reservoirs, paper
SPE 13662 presented at the 1985 SPE California Regional Meeting,
Bakersfield, March 27-29.
46. Barua, J.
et al.:
Improved Estimation Algorithms for Automated Type-
Curve Analysis
of
Well Tests,
SPEFE
(March 1988) 186-96;
Trans.
AIME,285.
47. Guillot, A.Y. and Home, R.N.: Using Simultaneous Downhole Flow
Rate and Pressure Measurements To Improve Analysis
of
Well Tests,
SPEFE (June 1986) 217-26.
48. Kucuk,F., Karakas,
M.,
and Ayestaran, L.: Well Testing and Anal
ysis Techniques for Layered Reservoirs, SPEFE (Aug. 1986) 342-54.
49. Shah, P.C. et al.: Estimation of the Permeabilities and Skin Factors
in
Layered Reservoirs With Downhole
Rate
and Pressure Data, SPEFE
(Sept. 1988) 555-66.
50. Coats, K.H. et af.: Determination
of
Aquifer Influence Functions From
Field
Data, JPT
(Dec. 1964) 1417-24;
Trans.
AIME, 231.
51. Jargon, J.R. and van Poollen, H.K.:
Unit
Response Function From
Varying-Rate Data, JPT (Aug. 1965) 965-69; Trans. AIME, 234.
52. Bostic, J.N., Agarwal,
R.G.,
and Carter, R.D.: Combined Analysis
of Post racturing Performance and Pressure Buildup Data for Evaluat
ing an MHF Gas
Well, JPT Oct.
1980) 1711-19.
53. Pascal, H.: Advanc es in Evaluating Gas Well Deliverability Using
Variable Rate Tests Under Non-Darcy Flow, paper SPE 9841 presented
at the 1981 SPE/DOE Low Permeability Gas Recovery Symposium,
Denver, May 27-29.
54. Al-Hussainy, R., Ramey, H.J.
Jr.,
and Crawford, P.B.:
The
Flow
of
Real Gases Through Porous
Media, JPT
(May 1966)
624-36;
Trans.
AIME, 237.
55. Meunier, D.F., Kabir, C.S., and Wittmann, M.J.:
Gas
Well Test Anal
ysis: Use
of
Normalized Pseudovariables,
SPEFE (Dec.
1987) 629-36.
56. Lee, W.J. and Holditch, S.A. : Application of Pseudo ime to Buildup
Test Analysis
of
Low-Permeability Gas Wells With Long-Duration Well
bore Storage Distortion,
JPT Dec.
1982) 2877-87.
57. Kuchuk, F.J.:
New
Methods for Estimating Parameters of Low Per
meability Reservo irs, paper SPE 16394 presented at the 1987
SPEIDOE Low Permeability Reservoirs Symposium, Denver, May
18-19.
58. Raghavan, R. and Clark, K.K.: Verti cal Permeability From Limited
Entry Flow Tests in Thick Formations, SPEJ (Feb. 1975) 65-73;
Trans. AIME, 259.
5
etric
Conversion Factors
·bbl
x
1.589873
E-Ol
m
3
cp
x
1.0*
E-03
Pa's
ft
x
3.048*
E-Ol
m
md
x
9.869233
E-04
p
2
psi
x
6.894757
E OO kPa
psi I
x
1.450377 E-Ol
kPa-
1
• Conversion factor is exact.
SPEFE
Original SPE manuscript received for review May 18, 1987. Paper accepted for publica·
tion March 28, 1990. Revised manuscri pt received Jan. 19, 1990. Paper SPE 16394) first
presented at the 1987 SPEIDOE Low Permeability Reservoirs Symposium held in Denver,
May 18-19.
SPE Formation Evaluation, December 1990