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WELL TESTING

( 1 ) PRODUCTI VI TY TESTS( 1 ) PRODUCTI VI TY TESTSto determine the performance of a well, i.e. the flow rates that can beproduced under an imposed pressure drop

Oil=>Productivity Index (PI ) Gas=>W ell de l i verab i l i t y

( 2 ) DRAW DOW N TESTS

early transient stateearly transient state (k , S)

transient statetransient state -- single rate (k , S or S)- two rates (k , S, D)

pseudopseudo--steady statesteady state (k , Vp )

(RESERVOI R LI MI T TESTRESERVOI R LI MI T TEST)

to evaluate the average parameters of the reservoir rock, and to assessthe degree of damage or stimulation in the vicinity of the wellbore

FLOW REGI ME

(3) BUILDUP TESTSto evaluate the average permeability of the reservoir rock, and averagereservoir pressure (k , )p

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Deliverability Tests for Gas Wells

Back pressure test

qsc

p

SPURGO

tT TTT

qsc1

qsc2

qsc3

qsc4

pwf1

pwf2

pwf3

pwf4

T > ts

p

CLEAN-UP

qSC

q SC1

q SC2

q SC3

q SC4

T TTT

T>ts

t

p

pp w f1

p w f2

p w f3

p w f4

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Empirical relationship ofEmpirical relationship ofRowlinsRowlins-- SchellardtSchellardt

0.5

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+

= 'S2

rc

A25.2ln

T

p

kh2

Tq)p(m)p(m

2wAsc

scscw

Rigorous equation for gas flow under pseudopseudo--steady statesteady stateconditions:

a

b

SCq

)p(m

SCq

sc

sc

w bqaq

)p(m)p(m+=

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Isochronal test

qsc

p

SPURGO

tT2T TT

qsc1

qsc2

qsc3qsc4

pwf1

pwf2

pwf3

pwf4

2T 2T

EXTENDED FLOW

p

CLEAN-UP

qSC

qSC1

qSC4

qSC3

qSC2

T T T2T2T 2T

EXTENDEDFLOW

p

p

pw f 1

pw f 4

pw f3

pw f2

t

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Empirical relationship ofEmpirical relationship ofRowlinsRowlins-- SchellardtSchellardt

2h4h8hpseudopseudo--steady statesteady state

)pplog(2wf

2

scqlog

n2wf

p2

pCscq

=

1/n

1/n

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Modified isochronal test

n2w f

2

sc ppCq

====For the first producing rate:For the first producing rate:

(((( ))))n2w f2w ssc ppCq ====For the rates following the first:For the rates following the first:

qsc

p

SPURGO

tT TT

qsc1qsc2

qsc3qsc4

pwf1

pwf2

pwf3pwf4

EXTENDED FLOW

T T T

pws4

pws3pws2p

t

CLEAN-UP

qSC

qSC1

qSC4qSC3

qSC2

T

EXTENDEDFLOW

p

ppw f1

pw f4

pw f3

pw f 2

T T T T T

pw s2 pw s3

pw s4

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Productivity tests for oil wells

qoST

p

SPURGO

t

qoST

pwf

T > ts

p

T

CLEAN-UP

qOSTq OST

T

T>ts

tp

p

p w f

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wf

STo

pp

qPI

=

Stabilized Productivity Index (for single phase flow only)Stabilized Productivity Index (for single phase flow only)

PI

Flusso bifase

qoST

pwf

p

Flusso monofase

psat

Single phase flowSingle phase flow

psat

pw f

p

qost

PI

Two phase flowTwo phase flow

Productivity tests for oil wells

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Drawdown tests for oil wells

qoST

p

t

p(rw,t)

p

q

p

qOSTq

pp(rw,t)

t

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The reservoir is assumed as an infinite acting system(radial geometry)

Hy po th esi s o f cons tan t f l ow ra t e q Hy po th esi s o f cons tan t f l ow ra t e q

=

= D

w

iw t,1r

rP

kh2

qp)t,r(p

=

=

=

trc

kt

pq

kh2t,1rrP

2w

D

D

w

Early transient stateEarly transient state:

1 0 0t D

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+

=

+

=

=

tlogrc

klogtlog

plogq

kh2logt,1

r

rPlog

2w

D

Dw

On a l og - l og g raph )t(ft,1

r

rP)t(fp DD

w

=

==

Match point

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tD

= D

w

t,1

r

rP

t

p

Match point

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t

p

tD

= D

w t,1r

r

P

* M* M

Match point

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Any point M chosen in the superposition area of the two diagramshas two coordinate sets, one for each graph. The coordinatevalues, read on the corresponding axes, allow the calculation ofthe constants by which the two curves are translated and,

therefore, the evaluation of the parameters of the reservoir rockparameters of the reservoir rock:

MD

M2w

M

M

D

w

tt

rkc

p

t,1rrP

2

qk h

====

====

====

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Wellbore Storage

surface flowrate

sandface flowrate

drawdown

q

time

qwh

qsf< qwh

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Wellbore Storage

wcVpVC =

=

tqV = tC

qp =

The w e l lbo re st o rage coe f f i c ien t depends on The w e l lbo re st o rage coe f f i c ien t depends on ::

! fluid compressibility

! wellbore dimension

and

! formation permeability: low permeability formations induce morerelevant wellbore storage effects

p is a linear function of t, and it will be demonstrated that on a

dimensionless log-log graph the slope is equal to unity.

2w

Dhrc2

CC

=

Dimensionless wellbore storage coefficientDimensionless wellbore storage coefficient

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Type-Curves of Agarwal et al.(1970)

The t y pe The t y pe --cu r ves ob t a ined by cu r ves ob t a ined by A g a r w a l A g a r w a l e t a l.e t a l.

== DD

w

D C,S,t,1r

rPfP

describe the pressure decline at the wellbore in dimensionless terms asa function of wellbore storage and skin factor.

Again, the match point method allows the evaluation of kh (and c):

MD

M

2

w

M

MD

t

t

r

kc

p

P

2

qkh

=

=

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PD

tD

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Transient stateTransient state:

DD

D

C)S5.36 0(t

1 0 0t

++++

Pressure data can be assumed free from wellbore storage effects

( )S2t25.2lnkh4

qp)t,r(p Diw +

=

++

+

= S281.0

rc

klntln

kh4

qp)t,r(p

2wt

iw

Transient state

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m

ln t

p(rw,t)

Transient State

p

ln t

p

p(rw, t)

m

k h4

qm

====

( )S2t25.2lnkh4

qp)t,r(p Diw +

=

S

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Transient State

== 0.9

rc

kln

m

p)h1t,r(p

2

1S

2wt

iw

At t= 1 hour

++

== S20.9

rc

kln

kh4

qp)h1t,r(p

2

wt

iw

Determination of the skin factor SDetermination of the skin factor S

R i Li it T t

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Reservoir Limit Test

PseudoPseudo--steady state conditionssteady state conditions

++

= S2t4

rc

A25.2ln

kh4

qp)t,r(p

AD2wA

iw

*mc

qAhV

t

p ==Pore volume drained by the well:Pore volume drained by the well:

m*

t

p(rw,t)p(rw,t)

t

tAhc

qS2rc

A25.2lnkh4

qp)t,r(pt

2wA

iw

+

=

Ahcq*m

t ====

Di t Sh F t

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===

m

)0t,r(p)h1t,r(pexp

*m

m

3600

4c wwA

Dietz Shape Factor cA

It is obtained by combination of the analysis of the drowdown test rununder both transient conditions [from which m and p(rw,t) at t=1h areknown] and pseudo-steady conditions [from which m* and p(rw,t) att=0 are known].

+

==

++

==

S2rc

A25.2ln

kh4

qp)0t,r(p

S20.9rc

klnkh4

qp)h1t,r(p

2

wA

iw

2wt

iw

The calculated value for cA can then be compared with the Dietz shapefactors presented for a variety of different geometrical configurations

to infer the shape of the drainage area and the well location within thedrainage area.

D d T t f G W ll

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Drawdown Tests for Gas Wells

In the case of gas wells single rate drawdown tests can be performed toevaluate the parameters of the producing formationparameters of the producing formationjust as for oil wellsprovided that the appropriate gas flow equations are used forinterpretation.

)'S281.0rc

klnt(ln

T

p

kh2

Tq)p(m)p(m

2wtsc

scsciw ++

+

=

Transient state flow equationTransient state flow equation

PseudoPseudo--steady state flow equationsteady state flow equation

+

+

= 'S2t

Ac

k4

rc

A25.2ln

T

p

kh2

Tq)p(m)p(m

t2wAsc

scsciw

However, t w o sing le ra t e i soch r ona l tests are usually performed ingas well testing. In fact, interpretation of the late transient pressure

data allows the determination of the apparent skin factor only, but twoflow rates are needed to determine both S and D .

D a do n Tests fo Gas Wells

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Drawdown Tests for Gas Wells

qsc1

p

t

p(rw,t)

p

q

qsc2

T T2T

q

p

p

qSC1

qSC2

p(rw,t)

T T2T

t

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ln t

m(pw)

m1m2

m p( )

ln t

m(pw)

m(p)

=

sc

sc2sc2

Tp

kh2Tqm

=

sc

sc1sc1

T

p

kh2

Tqm

Fort = 1 h

=+

=+

=

=

0.9

rc

kln

m

)p(m)p(m

2

1DqS

0.9rc

klnm

)p(m)p(m21DqS

2

wt2

i2h1tw

2sc

2wt1

i1h1tw

1sc

Buildup Tests for Oil Wells

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Buildup Tests for Oil Wells

The well is shut in, i.e. q=0, after producing at a constant flow rate for a certain time.Pressure variations occur under transient conditions.

p(rw,t)

qOST

tp+ttp

p

If the well has been produced at a constant flow rate then t is the production timeprior to shut-in. If the well has been produced at different flow rates then theeffective production time t eq is to be used

STo

p

eqqNt ====

t=0

cumulative volume of oil since beginning of production

final flow rate

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)S2t25.2(lnkh4

qp)t,r(p D

*w +

=

Flow equation which extend the flow equation derived for an infinite acting system(radial geometry) to the case of a finite, irregularly-shaped drainage area:

i* pp =

If the well is new:

Fkh4

qpp *

+=

If the well has been produced:

The equation applies when the flow rate q is constant. The superposition theoremis applied:

+q

-q

tp+ttp

++++

= S281.0)ttln(

rc

kln

kh4

qp)t,r(p p2

wt

*w

+++

+ S281.0tln

rc

kln

kh4

q2wt

Horner Plot

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t

ttln

kh4

qp)t,r(p

p*w

+

=

"" Horne r p lo t Ho rne r p lo t

k h4

qm

====

pw

t

ttln

p

+

m

p*

"" Bu i l dup equa t i on o r Ho rne r equa t i on Bu i l dup equa t i on o r Ho rne r equa t i on

Horner Plot

mFpp * =

The pressure p*, from which the volume averaged reservoir pressure p can be

determined, is evaluated at an infinite shut-in time t :

Afterflow

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surface flowrate

sandface flowrate

build-up

q

time

qwh=0

qsf>0

Afterflow

Horner Plot

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The solution is derived with respect to an ideal well, and in addition S does not appearin the buildup equation. However, the skin effects as well as the after flow effects dueto shutting the well in at the surface rather than downhole have to be taken intoaccount.

HORNER PLOTHORNER PLOT

t

ttln

p

+

pw

m

After flowAfter flow

Skin effectSkin effect

*

p

Horner Plot

Determination of the Skin Factor

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atat t=0, i.e.at the production time tt=0, i.e.at the production time t

)S2t25.2(lnkh4

qp)0t,r(p D

*w +

==

+++

== S281.0rc

klntlnkh4

qp)0t,r(p 2

wt

iw

atat t=1 hourt=1 hour

)3600lnt(lnkh4qp)h1t,r(p*w

==

h1tifttt =+

+++

=

==3600lnS281.0

rc

kln

kh4

qpp

2

wt

h1tw0tw

= == 9rc

kln

m

pp

2

1S

2

wt

0twh1tw

Oil Well

Buildup Tests for Gas Wells

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Buildup Tests for Gas Wells

)'S2t25.2(lnTp

kh2Tq)p(m)p(m D

sc

scsc*w +

=

+

=

t

ttlnT

p

kh2

Tq)p(m)p(m p

sc

scsc*w

"" Bu i l dup equa t i on o r Ho rne r equa t i on Bu i l dup equa t i on o r Ho rne r equa t i on

t

ttln

p

+

"" Horn er Plo t Horn er Plo t

m(pw)

m

sc

scsc

T

p

k h2

Tqm

====m(p*)


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= == 0.9

rckln

m)p(m)p(m

21'S

2wt

0twh1tw

Gas Well


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Type Curves of Agarwal et al.(1970)

The typeThe type--curves obtained bycurves obtained by AgarwalAgarwalet alet al..

== DD

w

D C,S,t,1r

rPfP

represent solutions to the drawdown equation to describe the pressure decline at thewellbore, expressed in dimensionless terms, as a function of wellbore storage effectsand skin factor.

However, the use of the type-curves to interpret also buildup tests could be justifiedprovided that the producing time prior to well shut-in was sufficiently large so that

the rate of pressure decline could be assumed negligible during the shut-in period. Ifthis assumption was satisfied the buildup curve would be a mirror image of thedrawdown curve, and as such analyzed.

Furthermore, Agarwal has empirically found that by plotting the buildup pressuredata:

)0t,r(p)t,r(p ww =

on a log-log graph versus an equivalent expression of time, defined as follows:

tt

ttt

p

p

e

+

=

the type-curve analysis can be made without the requirement of a long drawdownperiod.

Type-Curves

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Type Curves

== DD

w

D C,S,t,1rrPfPThe Agarwal type curves:

can be written as a function of: tC

kh2

C

t

D

D

=

trc

kt

2wt

D

=

2wt

Dhrc2

CC

=

Infinite acting radial flow equation for fluid flux occurring in a homogeneous porousmedium under transient conditions:

( )S2t25.2lnkh4

qp)t,r(p Diw +

=

( )S2t25.2ln21P DD +=

obtained from

( )

++= S2D

D

DD eCln81.0

Ctln

21P

By substitution:

Type-Curves Plot

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Type Curves Plot

D

D

D C

t

P =

(i.e., unit slope)

0.1

1

10

100

0.1 1 10 100 1000 10000

tC

D

D

PD C eDS2 10000

0.1

PD CDe2S

10000

0.1

tD

CD

When pure wellbore storage flow occurs:

Pressure Derivative Method

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According to this method, which is largely based on the solutions obtained by

Agarwal et al., the type-curves are redrawn in terms of the derivative of thedimensionless pressure PD:

)C/t(d

dP'P

DD

DD =

The dimensionless pressure values are plotted on a log-log graph as follows:

)C/t(f)C/t('P DDDDD =

The drawdown test interpretation is carried out by matching the field pressure data

p(rw,t), expressed in terms of pressure derivative:

dt

)pp(d'p wfi

=

)t(ft'p =and plotted on a log-log graph as follows:

with the type-curves of the derivative of the dimensionless pressure.

In the case of a buildup test the field time-pressure data p(rw,t) are plotted on a log-log graph as follows:

)t(ft

ttt'p =

+

Type-Curve Derivatives

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yp

=

D

D

D'D

C

td

dPP

PURE W ELLBORE STORAGEPURE W ELLBORE STORAGE

1

Ctd

dPP

D

D

D'D =

=

I NFI NI TE ACTI NG RADI AL FLOW ( I .A.R.F. )I NFI NI TE ACTI NG RADI AL FLOW ( I .A.R.F. )FOR A H OMOGENEOUS POROUS MEDI UMFOR A H OMOGENEOUS POROUS MEDI UM

=

=

D

D

D

D

D'D

Ct

5.0

Ctd

dPP

Type-Curves and Type-Curve Derivatives

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yp yp

10-210-3 10-1 1 10

t (hr)

103

102

101

104

p(psia)&

p

WellboreStorage

Radial Flow

CDe2S

1030

1012

106

10-1

102

Real data vs type-curves

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t (hr)

p(psia)&

p

103

102

101

100

10-1

10-2 10-1 1 10 102 103 104103

LogLog pp

D er i va t i veD er i va t i ve

104

yp

Superposition real data & type-curves

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103

102

101

10-210-3 10-1 1 10 101

104

t (hr)

p(psia

)&

p

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Interpretation Result

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CCg ives 2wt

Dhrc2

CC

=

Cu r ve Mat chCu r ve Mat ch S2DeC

SkinSkin

=

D

S2

DC

eCln2

1S

Pr essur e Mat chPr essur e Mat ch k hk hpqB

kh4Po

D

=

Tim e Mat ch Tim e Mat ch CCtCkh2Ct DD

=

Drawdown Test Analysis

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Pressure

L

og

p

Log t

Pr essu r e vs t im e

Pressu r e de r i va t i ve vs t im e

Log t

Buildup Test Analysis

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Log t

Pressur

e

Log

p

Pr e ssu r e v s Ho r n er t i m e

Pr essu r e de r i va t i ve vs t im e

Interpretation with the Pressure Derivative Method

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The derivatives of the type-curves exhibit a linear dependency on time ofpressure data points. Linearity can occur according to different yet

characteristic slopes which are a function of flow geometry:

FLOW TYPEFLOW TYPE PRESSUREPRESSURE SLOPESLOPE

linearlinear

bilinearbilinear

radialradial

sphericalspherical

(hemispherical)(hemispherical)

)t(fp =

)t(fp 4=

)t(lnfp =

=

t

1fp

1/ 21/ 2

1/ 41/ 4

--1/ 21/ 2

horizontalhorizontal

straight linestraight line

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ENI_WTA_4