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

-

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


2/10

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)



3/10


4/10

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

SPE Formation Evaluation, December 1990


5/10

llpw

1000

1...,-

7

\' dpw/dlntH

1

. . : : ~ _ : P j t s u p

'


6/10

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,



7/10

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


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


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



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


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.

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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.


Documents

SPE 16394 PA Applications of Convolution and deconvolution to transient well tests