Part 10 Buildup Followed CR Drawdown 3slides on 1page

8/13/2019 Part 10 Buildup Followed CR Drawdown 3slides on 1page

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Well Test Analysis September 20

Chapter 10

Buildup AnalysisFollowing A Constant-Rate

Drawdown

Problems with Drawdown Tests

It is difficult to produce a well at astrictly constant rate

Even small variations in rate distort thepressure response

There is one rate that is easy to maintain A flow rate of zero.

A buildup test is conducted by shuttingin a producing well and measuring theresulting pressure response.

Buildup

Usually more common to run and analyzepressure buildup tests.

Well is produced at constant or variable rate

for some time, then it is shut in (i.e.,produced at zero rate)

The increasing wellbore pressure isrecorded as a function of shut-in time.

Buildup data are analyzed to giveestimates of the same parameters that areavailable from pressure drawdown data.


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Buildup Test - Rate History

t0

- q

0 tp + t

q

tp t0

q

Buildup Test - Pressure Response

ttp

t0

tp + t

Pi

pi

Model Buildup Pressure Response


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Buildup Superposition

Pressure behavior determined by a simpleapplication of the principle ofsuperposition

It is simply the superposition of twoconstant-rate drawdown solutions:

( ) ( ) ( )tpqttqptpp upuwsi ++=

-First term: continuous production at rate q-Second term: production at rate q starting at tp (t= 0)

Buildup Superposition

For log-log analysis purposes, we may use adifferent from of the buildup superpositionequation:

( )[ ] ( )[ ] ( ) ( ) ( ) ( )[ ]tpqttqptqptpptpp upupuwsipwfi ++=

Pressure During Buildup

Buildup equation valid for all reservoir flowregimes and also account for wellborestorage.

For transient infinite-acting radial flow where

Buildup pressure given by

( ) ( )

+

+= 0.87s3.23

rc

klogtlog

kh

162.6Btp

2

wt

u

( ) ( ) ( ) ( )

+=

++=

t

ttlog

kh

162.6qB

tpqttqptpp

p

upuwsi

Horners Equation


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Horner Equation and Analysis

++++====

t

ttlog

kh

qB6.162pp p

10iws

y = mx + b

Buildup TestStraight Line Analogy

hm

qB6.162k

====

1t

tt@bp p

i ====

++++====

Horner Plot

infinite shut-in

p*pi


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Skin Factor from Horner Analysis

Note that, unlike the semilog analysis fordrawdown data, the skin factor is notdirectly obtainable from a Horner plot underinfinite acting radial flow.

Horner Plot

This does not meanthat the skin doesnot affect thepressure in thebuildup --- it does.

It simply has noeffect on thesemilog straight linethat we use foranalysis.

Shut-in

Pressure,psi

kh

qBm o

=

6.162

1

+

t

ttplog

Increasing shut-in time

Horner Line

WellboreStorage

Skin Factor

To obtain the skin factor, we use the lastdrawdown data point, pwf(tp)=pws(t=0).

From semilog straight line just before

shut-in

Subtract Horner straight line

( )

+

+= 0.87s3.23

rc

klogtlog

kh

162.6qB)(tpp

2

wt

ppwfi

( )

+=

t

ttlog

kh

162.6qBtpp

p

wsi


6/15


Determination of Skin

Resulting Equation is the Argawal Equivalenttime semilog equation

To obtain skin, we evaluate this expression at ashut-in time of 1 hour, and call the value as p1hr

Solve for skin gives:

( )

+

+

+= 0.87s3.23

rcklog

ttttlog

kh162.6qB)(tptp

2

wtp

p

pwfws

+

+

+= 0.87s3.23

rc

klog

t

1tlog

kh

162.6qB)(tpp

2

wtp

p

pwf1hr

( )

+

++

= 3.23

t

1tlog

rc

klog

m

tpp1.151s

p

p

2

wt

pwf1hr

Notes on Horner Plot

The value of p1hr is obtained by extrapolatingthe semilog straight line to a time of 1 hr. It isnot the actual value of shut-in pressure at1hr.

If the reservoir is infinite-acting, extrapolatingthe Horner semilog straight line to a Hornertime ratio of 1 will yield an estimate of theinitial reservoir pressure, pi

( )

+=

t

ttlog

kh

162.6qBtpp

p

wsi

Horner Analysis for Closed Systems

Time

Pressure

re


7/15


Bounded Systemspi=5000

4993)5( ==tp

1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 1E+1 1E+24880

4920

4960

5000

0.2 saat retim

0.5 saat retim

1 saat retim

5 saat retim

10 saat retim

20 saat retimBottomh

olepressure,pwf,psi

Total time, t, hour

Radial flow

period

4985)10( ==tp

4930)20( ==tp

qsc = 54 STB/D

re = 500 ft

0.2 h prod.

0.5

1.0

5.0

10.0

20.0

re

Average Reservoir Pressure

In any closed system, average reservoirpressure can be expressed by the followingequation:

If flow rate is constant, then

=t

0pt

i dtqBVc

5.615pp

Cumulative production(STB)

tVc

5.615qBpp

pt

i=

hAVp =

Bounded Systems-Horner Analysis

If the reservoir is notinfinite acting at thetime of shut-in, theHorner straight line will

extrapolate to a falsepressure p*.

Matthews-Brons-Hazebroek correlatedaverage reservoirpressure with Hornerfalse pressure. MBHanalysis will bediscussed later.

Shut-inPressure,psi

kh

qBm o

=

6.162

1

+

t

ttplog

Increasing shut-in time

Horner Line

WellboreStorage

p*

p

ReservoirAveragePressure

False Pressure

Initial Pressure


8/15


Buildup MDH Analysis

The buildup pressure drop can be written as

Miller-Dyes-Hutchinson approximated

MDH buildup equation:

( ) ( ) ( ) ( ) ( )[ ]pupuupwfwsws tpttptpqtptpp ++=

pupu tpttp +

( ) ( ) ( )tqptptptp upwfwsws ==

Significance

MDH equation states that for smallshut-in times, buildup pressure changewill duplicate the correspondingdrawdown pressure drop at the samevalue of time.

True regardless of well or reservoirgeometry.

Applies for wellbore storage problem

For radial flow

( ) ( ) ( )

+

+= 0.87s3.23

rcklogtlog

kh162.6qBtptp 2

wt

pwfws

MDH Plot

kh

162.6qBm=

+

= 3.23

rc

klog

m

p1.151s

2

wt

1hr

pwfws,1hr1hr tppp =


9/15


Buildup versus Drawdown

We saw that for small shut-in times, builduppressure change data will duplicate

drawdown data.All analysis techniques that worked for

drawdown will work on buildup data;wellbore storage, radial flow, type curveanalysis

Buildup data will not enter pseudosteadystate.

Shut-in pressure will approach averagereservoir pressure before the time topseudosteady state flow.

Buildup vs Drawdown Curves

Vertical well in closed system

Pressure change

derivative

Argarwal Equivalent Time

Most modern well test analysis packagesand commercial software use anequivalent shut-in time to analyzebuildup data

First proposed by Argarwal for analyzingbuildup semilog data following asemilog drawdown period.

Can be applied to any well/reservoirgeometry

Usually extends the range of buildupdata that can be analyzed.


10/15


Argarwal Equivalent Time

This time transformation is an attempt toconvert buildup pressure signal p

ws

=pws(t)-pwf(tp) to an equivalent drawdownpressure signal change so that buildup datacan be analyzed by using drawdown analysismethods (straight line and log-log analyses):

( ) ( ) ( )eupwfwsws tqptptptp ==

tt

ttt

p

p

e+

=

Derivative Analysis for Buildup

For log-log diagnostic purposes, we differentiatebuildup pressure change with respect toAgarwals equivalent time:

Then, plot pws vs te together with dpws/dlntevs t.

( ) ( )[ ]

( )e

pwfws

wstdln

tptpdp

=

Notes on Use of te Plotting buildup pressure change and its derivative using

Agarwal equivalent time (or multi-rate Agarwal equivalentor superposition time function to be discussed) willcorrelate quite well with the drawdown solutions if we

have long producing times and infinite acting radial f low. However, the use of equivalent time:

Over corrects if tp is small compared to buildupduration,

Reduces a long buildup into a short equivalentdrawdown because te approaches to tp as t getslarger,

Distorts the flow regimes and transition periods whenboundaries affect the buildup response.


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31

Buildup vs Drawdown Curves

Pressure change

Derivative

Producing time effects distort flow regimes!

m

2m

Well near a sealing fault

Well In A Channel

0.01

0.1

1

10

100

1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09

Dimensionless time

Dimensionlesspre

ssure

Well In A Channel ReservoirDrawdown Type Curve


12/15


0.01

0.1

1

10

100

1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09

Dimensionless shutin time

Dimensionlesspressure

tpD=108

tpD=107

tpD=106

tpD=105

Drawdown

Channel Reservoir - Buildup ResponseDerivative With Respect To Shutin Time

0.01

0.1

1

10

100

1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09

Dimensionless equivalent time

Dimensionlesspressure

tpD=105

tpD=106

tpD=107

tpD=108

Drawdown

Channel Reservoir - Buildup Response

Derivative With Respect To Equivalent Time

0.01

0.1

1

10

100

1E+03 1E+04 1E+05 1E+06 1E+07 1E+08 1E+09

Dimensionless time function

Dimensionlesspre

ssure

tpD=105

tpD=106

tpD=107

tpD=108

Drawdown

Channel Reservoir - Buildup Response

Derivative WRT Equivalent Time vsShutinTime


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37

Horner Type Straight LineAnalysis For Other Flow Regimes

For flow regimes different from radial, we

can perform Horner type analysis basedon the superposition time that isappropriate for the flow regime identified.

For linear flow

Bi-linear flow

Spherical flow

tttt ps +=

44 tttt ps +=

tttt

p

s+

= 11

38

Spherical Flow Example

Spherical flow period 8-20 sec

Start of radial flow 30 sec

39

Spherical Flow Horner Plot

tttt

p

s+

= 11

( ) 3/13/2

1856)/(t

s

s c

m

qk

=

slope

cm3/sec

vhsvhs kkkkkk == 3/23/12 )(


14/15


40

Radial Flow Horner Plot

=

r

hm

qhk 36.88)/(

cm3/sec

slope

41

General Approach-Convolution

Buildup is in fact a variable rate flow problemand cannot rigourosuly correlated withdrawdown. Thus, the general approachmodeling any variable rate flow problem shoulduse Duhamels equation:

( )mod0

( , ) ( , )

t

i ucp t p q p t d =

r r

Flow rate history Unit-rate solution

42

Non-Linear RegressionAnalysis

Modern approach uses non-linear regressionanalysis to estimate parameters by historymatching obseerved pressures and accountingfor variable flow rate history. The commonmethod is to use weighted-least squaresapproach.


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43

Weighted Least-Squares

( )

=

=

obsN

i o

imodiobs tpp

J1

2

, ,)(

r

r

( ) ( )

==

+

=

2,1,

1

2

2,

,2

1

2

1,

,1 ,,)(obsobs N

i o

imodiobs

N

i o

imodiobs tpptpp

J

rr

r

44

Computed Aided Approach

45

Computed Aided Approach

field data

Calculated model

Part 10 Buildup Followed CR Drawdown 3slides on 1page

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