ENI_WTA_3

7/29/2019 ENI_WTA_3

http://slidepdf.com/reader/full/eniwta3 1/93

RESERVOI RPROPERTI ES

REAL SYSTEMREAL SYSTEM

Q

t

FI ELD I NPUTFI ELD I NPUT

Q

t

MODEL I NPUTMODEL I NPUT

p

t

MODEL OUTPUTMODEL OUTPUT

p

t

RESERVOI RRESERVOI R

RESPONSERESPONSE

MODELPARAMETERS

MATHEMATI CAL MODELMATHEMATI CAL MODEL

MMAA

TT

CC

HH

CALIBRATION OF MODEL PARAMETERSCALIBRATION OF MODEL PARAMETERS

Interpretation

7/29/2019 ENI_WTA_3


Model

MODEL: simplified version of the real system, by which the behaviorof the real system can be approximately but yet representativelysimulated

• MASS BALAN CE EQUATI ONS:MASS BALAN CE EQUATI ONS:for the considered extensive quantities

• FLOW EQUATI ONS:FLOW EQUATI ONS: relate theextensive quantities to the significant statevariables of the problem

• STATE EQUATI ONS:STATE EQUATI ONS: define thebehavior of the components of the system

• I N I TI AL AND BOUNDARYI NI TI AL AND BOUNDARY

CONDI TI ONS:CONDI TI ONS: must be defined after thedomain geometry has been established

MATHEMATI CAL M ODELMATHEMATI CAL M ODEL

THEORETI CAL MODELTHEORETI CAL MODEL Set of simplifyingSet of simplifyingassumptionsassumptions

7/29/2019 ENI_WTA_3


REALREALSYSTEMSYSTEM MATHEMATI CALMATHEMATI CAL

MODELMODEL

SOLUTI ONS OFSOLUTI ONS OF

MATHEMATI CAL MODELMATHEMATI CAL MODEL

NUM ERI CAL M ETHODNUM ERI CAL M ETHODNecessary in the case of:• non linearity of the equations

which constitute the model• complexity of the boundary

conditions, etc.

ANALYTI CAL METHODANALYTI CAL METHODPreferable for the ease of applicability of the solutions

MODEL COEFFI CI ENTSMODEL COEFFI CI ENTS:the transport coefficients of theconsidered extensive quantities

Model solutions

7/29/2019 ENI_WTA_3


Flow Problem

Number of phasesNumber of phases•single phase (kass)•multiphase (krel)

Nature of fluidsNature of fluids•compressible (liquid)•very compressible (gas)

GeometryGeometry•monodimensional•bidimensional (radial flow)•tridimentional

Hydraulic regimeHydraulic regime

•steady state flow•pseudo-steady state flow•transient flow

7/29/2019 ENI_WTA_3


t

)()v(

∂

φρ∂−=ρ∇

CONTI NUI TY EQUATI ONCONTI NUI TY EQUATI ON

STATESTATE EQUATI ONSEQUATI ONS

zRT

Mp=ρ

)pp(c

00e −−−−==== ρρLiquid

Real Gas

FLOW EQUATI ONSFLOW EQUATI ONS(gravity effects are neglected)

Turbulent flow (Turbulent flow (ForchheimerForchheimer))

pkv ∇µ

−=

Laminar flow (Darcy)Laminar flow (Darcy)

2vvk

p βρ+µ

=∇−

Basic equations

7/29/2019 ENI_WTA_3


Diffusivity Equation

MONOPHASI C FLOW OF A SLI GHTLY COMPRESSI BLE FLUI D ( LI QUI D)THROUGH A HOMOGENEOUS AND I SOTROPI C POROUS MEDI UM.PRESSURE GRADI ENTS ARE SMALL AN D DA RCY’S LAW APPLI ES

ηη DIFFUSIVITY CONSTANTDIFFUSIVITY CONSTANT

t

p1

t

p

k

cp t2

∂

∂

η=

∂

∂φµ=∇

φµ

=η

tc

k

MONOPHASIC FLOW MONOPHASIC FLOW : in the case of oil flow the water saturation is

equal to the irreducible value Swi, and the pressure is always greaterthan the bubble point pressure; in the case of water flow the oilsaturation is equal to the residual value Sor

7/29/2019 ENI_WTA_3


Pressure Profile

! Each time the well production is modified (rate change) apressure disturbance starts to propagate in the reservoir

!The DIFFUSIVITY EQUATION DIFFUSIVITY EQUATION describes how the pressuredisturbance evolves within the reservoir

Q

7/29/2019 ENI_WTA_3


Infinite Acting Radial Flow

(I.A.R.F.)HYPOTESIS: constant thickness of the producing formation,

and wellbore open to production across the entire formation

thickness.Therefore, the fluid flow is horizontal.

7/29/2019 ENI_WTA_3


Radial Flow Equations

t

p

k

c

r

p

r

1

r

p t

2

2

∂

∂φµ=

∂

∂+

∂

∂

!transient flow

!pseudo-steady state flow

dt

dp

k

c

dr

dp

r

1

dr

pd t2

2 φµ=+

!steady state flow

0drdp

r1

drpd2

2=+

7/29/2019 ENI_WTA_3


Average Pressure

MEASUREMENT OF AVERAGE PRESSUREMEASUREMENT OF AVERAGE PRESSURE

•• TheoreticallyTheoretically: the average pressure could be measured in the wellbore

under static conditions if the well (or the field) had been shut in for an

infinitely long time so as to allow the reservoir pressure to reach

equilibrium.

•• I n p r a ct i c e:I n p r a ct i c e: the average pressure can be determined from buildup tests.

AVERAGE PRESSURE AVERAGE PRESSURE : the representative reservoir pressureat which the pressure dependent parameters in the materialbalance equations should be evaluated.

∫=VpdV

V

1p

VOLUME AVERAGEDVOLUME AVERAGED

RESERVOI R PRESSURERESERVOI R PRESSURE

7/29/2019 ENI_WTA_3


Steady State Flow

w

e

we

rrln

ppkh2q

−

µ

π=

MUSKAT EQUATION

eerrconstpp

t,rtp

===

∀∀=∂∂ 0

Initial and boundary conditions:

The pressure p(r) at a distance r from the well can be evaluated as follows:

w

wr

rln

kh2

qp)r(p

π

µ+=

0dr

dp

r

1

dr

pd2

2

=+Radial Flow Equations

7/29/2019 ENI_WTA_3


rw re

h

p

r

k

pe

p w

If there is a damaged zone in the vicinity of the wellbore characterizedby a reduced permeability k’ with respect to the formation permeabilityk - due, for instance, to invasion of drilling or completion mud into theformation - an additional pressure drop is observed at the wellbore.

Skin Effect

SKI N EFFECT SKI N EFFECT : permeability damage in the vicinity of thewellbore.

r'

r’=0.5 - 1 m

k'p ’w

7/29/2019 ENI_WTA_3


Skin Factor

ww'r

r 'r ln

kh2qppπ

µ+=k ’ = k

w

w'r

r

'r ln

h'k2

q'pp

π

µ+=k ’ < k

wSww

r

'r ln1'k

k

kh2

qp'pp

−π

µ=∆=−

Additional pressure drop in the wellboreAdditional pressure drop in the wellbore: :

k ’ < k S > 0 permeability reduction (damage)

k ’ = k S = 0 no change in permeability

k ’ > k S < 0 permeability enhancement (stimulation)

SKI N FACTORSKI N FACTOR:wr

'rln1

'k

kS

−−−−====

7/29/2019 ENI_WTA_3


Sw

w pr

r ln

kh2

qp)r (p ∆+

π

µ+=

+

π

µ+= S

r

r ln

kh2

qp)r (p

ww

S

Additional pressure drop

wS

r

'r ln1

'k

k

kh2

qp

−π

µ=∆

7/29/2019 ENI_WTA_3


COMPLETI ON FACTORCOMPLETI ON FACTOR

)0S(PI

)0S(PICF

ltheoretica

real

=

≠=

real

S

real

Sreal

real

ltheoretica

pp1

ppp

ppCF

∆∆−=

∆∆−∆=

∆∆=

PRODUCTI VI TY I NDEXPRODUCTI VI TY I NDEX

p

qPI STo

∆

=

Sr

r

ln

1

B

kh2PI

w

eo

+

µ

π=

Productivity Index

7/29/2019 ENI_WTA_3


Partial Penetration and Partial Perforation

If the wellbore is partially penetrating in the reservoir, or the wellbore is open toproduction over a restricted interval of the reservoir, there is a distortion of theradial flow pattern close to the well (the flow lines are not parallel in the vicinity of the wellbore) giving rise to an ADDITIONAL PRESSURE DRAWDOWN.ADDITIONAL PRESSURE DRAWDOWN.

This pressure drop is generally accounted for by including the effect of partial

penetration as an additional skin factor. It can not be measured, but it can bequantitatively evaluated by numerical models that take the wellbore completioncharacteristics into account.

The skin factor S determined by well tests comprises both the component due to a

change in permeability Sk, and the component due to partial penetration SB:

S =S = SSkk ++ SSBB

7/29/2019 ENI_WTA_3


Pseudo-Steady State Flow



e

ei

rrr

p

rconstdt

dpt

rrppt

==

∂

∂

∀=>

===

0

0

0

Solution in terms of average pressure:

S4

3

r

rln

ppkh2q

w

e

w

+−

−

µ

π=

w

e

w

e

r

r472.0ln

4

3

r

rln =−

Pseudo-steady state drainage radius: 0.472 re=rds

dt

dp

k

c

dr

dp

r

1

dr

pd t2

2 φµ=+

7/29/2019 ENI_WTA_3


Transient Flow


π

µ−= D

wi t,

r

r P

kh2

qp)t,r (p

Solution:

c

Dt

tt =Dimensionless time

k

rct

2wt

c

φµ=Characteristic time


==

∂

∂

==

>

∀==

w

ei

i

rrconstr

p

rrpp

t

rppt

0

0

CONSTANT TERMINAL RATECONSTANT TERMINAL RATE

t

p

k

c

r

p

r

1

r

p t2

2

∂

∂φµ=

∂

∂+

∂

∂

7/29/2019 ENI_WTA_3


Function P(r/rw,tD)

7/29/2019 ENI_WTA_3


η−−=

t4

rE

2

1t,

r

rP

2

iD

w

At very short times, for :20r

r

w

≥

At the well radius rw, if :100tD ≥

≥=

= D

w

D

w

t,20rrPt,1

rrP

Therefore, if k

rc25t

2t φµ

≥

kt25.2

rclnt4

rE

2t

2

i

φµ=

η−

Also, when x ≤ 0.01 )x781.1ln()x(Ei =−

η

−−=

=

t4

r E

2

1t,1

r

r P

2

iD

w

Now some developments…

7/29/2019 ENI_WTA_3


)t25.2(lnkh4

qp)t,r (p Diw

π

µ−=

φµπ

µ−=

2wt

iwr c

kt25.2ln

kh4

qp)t,r (p

Solution at the well radius rw , for tD>100

2wtc

Dr c

kt

t

tt

φµ==Dimensionless time

…and here’s the solution

φµ−

π

µ−=

kt25.2

r cln

2

1

kh2

qp)t,r (p

2wt

iw

)S2t25.2(lnkh4

q

p)t,r (p Diw +π

µ

−=

7/29/2019 ENI_WTA_3


Tim e to reach pseud oTim e to reach pseud o -- st eady st a test eady s ta t e tt ss

i2et

sise p

rc

kt25.2ln

kh4

qp)t,r(p =

φµπ

µ−=

krc

25.21t

2

ets φµ=

Dra inage o r i n vest i ga t i on r ad ius rD ra inage o r i n vest i ga t i on r ad ius r dd

φµ=

2wtw

d

r c

kt25.2ln

2

1

r

r ln

t5.1tc

k5.1r

td η=

φµ=

Investigation radius

W h

7/29/2019 ENI_WTA_3


Water encroachment

t

p

k

c

r

p

r

1

r

p t

2

2

∂

∂φµ

=∂

∂

+∂

∂

Tr ansien t f l owTr ansien t f l ow

VanVan Everd ingenEverd ingen --

Hurs t Mode lHurs t Mode l

Car terCar ter -- TracyTracy

Mode lMode l

rw rw

V E di H t l ti

7/29/2019 ENI_WTA_3


Van Everdingen-Hurst solution

∞≠=

∂∂

∞==

=

=∆−==

>

∀==

w

e

w

eie

e

wiww

i

rr0

rp

r

rp)t,r(p

rr

tcospp)t,r(prr

0t

rpp0t


Van Everdingen-HurstSOLUTI ONSSOLUTI ONS

to diffusivity equation

CONSTANT TERMINAL RATE :Rate of water influx=const for ∆t

" calculation of pressure drop

CONSTANT TERMINAL PRESSURE:boundary pressure drop=const for ∆t

" calculation of water influx rate

V E di H t if

7/29/2019 ENI_WTA_3


Van Everdingen-Hurst aquifer

Dimensionless water influxDimensionless water influx

w

eD

rr,tQ

Dimensionless timeDimensionless time 2wtw

D r c

kt

t φµ=

Dimensionless radiusDimensionless radius

=

w

eD

r

r r

Cumulative Water influxCumulative Water influx

∆=

w

eDwe

rr,tQpBW

where B: water influx constantB: water influx constant hr cB wt

2

2 πφ=hf rcB wt

22 πφ=360

θ=f

F ti Q( / t )

7/29/2019 ENI_WTA_3


Dim ens ion less Tim e, t D

W a t e r I

n f l u x ,

Q

Function Q(re /rw,tD)

Carter Tracy aquifer

7/29/2019 ENI_WTA_3


Carter-Tracy aquifer

The CarterThe Carter--Tracy solution is not an exact solution to theTracy solution is not an exact solution to thediffusivity equation, it is an approximationdiffusivity equation, it is an approximation

( ) ( ) ( ) ( )[ ]( ) ( )

( ) ( ) ( )

−

−∆−+=

−

−−−

nD1nDnD

nD1nen1nDnD1nene

'ptp

'pWpBttWW

Where : B=Van Everdingen-Hurst water influx constant

n=current time stepn-1=previous time step

∆pn=total pressure drop, pi-pn

pD=dimensionless pressurep’ D=dimensionless pressure derivative

CONSTANT W ATER I NFLUX RATE

ove r each f i n i t e t im e in t e r va l .

Condition: Condition:

Gas Flow

7/29/2019 ENI_WTA_3


Gas Flow

Therefore, the Forchheimer equation should be used instead of the Darcy

equation. However, if a quadratic expression for the fluid velocity is substitutedinto the continuity equation a non linear, partial differential equation is obtained,which can not be solved analytically.

In general t u r b u l e n t f l o w c o n d i t i o n st u r b u l e n t f l o w c o n d i t i o n s have to be assumed near thewellbore when the reservoir fluid is gas due to its low viscosity

Lam ina r f l owLam ina r f l ow for Qg<50000 Sm3 /d approximately

Tur bu len t f l owTu r bu len t f l ow for Qg>100000 Sm3 /d

11 ST ST APPROACH APPROACH : Da rcy equa t i on fo r l am ina r f l ow

An equation is obtained, the solution of which can not be applied in the vicinity of thewellbore without restrictions.

2 2 n d n d APPROACH: APPROACH: Fo r c h h ei m e r e q u at i o n f o r t u r b u l e n t f l o w

The Forchheimer equation is integrated to obtain an equation valid under steady state

conditions only. The assumption of steady state flow can be removed if the drainageradius is inserted in place of the reservoir external radius, and the appropriatepressure is used .

Pseudo Pressure Function m(p)

7/29/2019 ENI_WTA_3


Pseudo-Pressure Function m(p)

∫ µ= p

p 0

dpzp2)p(m

m(p) is a function of: • pressure

• temperature• gas composition

m(p)

p

1st Approach

7/29/2019 ENI_WTA_3


1st Approach

Flow Equation: LAMINAR flow assumption

1)z

p

µ

tp1

tp

kcp t2

∂∂

η=

∂∂φµ=∇

= CONSTANT= CONSTANT (p > 3000(p > 3000 psipsi))

2) zµ

t

p1

t

p

k

cp

22t22

∂

∂

η=

∂

∂φµ=∇

= CONSTANT= CONSTANT (p < 2000(p < 2000 psipsi))

RIGOROUS SOLUTIONRIGOROUS SOLUTION

(no simplifying assumptions)(no simplifying assumptions)

t)p(m1

t)p(m

kc)p(m t2

∂∂

η=

∂∂φµ=∇

3)

SOLVI NG EQUATI ONSOLVI NG EQUATI ON

tpzpkcrprzprr1t

∂∂φ=

∂∂µ∂∂

2nd Approach

7/29/2019 ENI_WTA_3


2nd Approach

Flow Equation: TURBULENT flow assumption2vv

kdr

dpβρ+

µ=

++

π=−

scw

e

scsc

sc

weDqS

r

rlnq

T

p

kh

T)p(m)p(m

),k( φβ=β

Turbulence factorφ

k

β

It is assumed that there exists a damaged zone in the vicinity of the

wellbore characterized by k’ and β’ different from the formation values.Furthermore, an average viscosity value is assumed.

Apparent Skin

7/29/2019 ENI_WTA_3


S’

qSC

S

D

−β+

−β

µπ=

ewsc

sc

r

1

'r

1

'r

1

r

1

T

p

Rh2

kMD

DD “ n o n“ n o n -- Dar cy f l ow coe f f i ci en t ”Dar cy f l ow coe f f i ci en t ”

APPARENT SKI N S’APPARENT SKI N S’

S’=S+DqSC

Apparent Skin

Flow Equations

7/29/2019 ENI_WTA_3


Flow Equations

PSEUDOPSEUDO-- STEADY STATE FLOW EQUATI ONSTEADY STATE FLOW EQUATI ON

++

π=− sc

w

esc

sc

scw DqS

r

r472.0lnq

T

p

kh

T)p(m)p(m

sde r r 472.0 =

pp e →

TRAN SI ENT FLOW EQUATI ONTRAN SI ENT FLOW EQUATI ON

Under transient conditions D=f(tD) but a constant value is attained when rd>> r’

)Dq2S2t25.2(lnqT

p

kh2

T)p(m)p(m scDsc

sc

scwi ++

π=−

φµ=

2wtw

d

rc

kt25.2ln

2

1

r

rln

ie pp →

Flow Equations for Reservoir Engineering

7/29/2019 ENI_WTA_3


If a well is asymmetrically located in a bounded, irregularly shapeddrainage area the flow equations derived in the case of a porous mediumof infinite extent can still be applied provided that the effects of thepermeability limits which bound the drainage area are taken into account.

Flow Equations for Reservoir Engineering

METHOD OF I MAGE W ELLSMETHOD OF I MAGE W ELLSA virtual, hydraulically equivalent situation is assumed to simulate theadditional pressure drops due to the reflection of the pressuredisturbances on the permeability limits (no-flow boundaries).

d/2

P P'

k,φ

d/2

k,φ

Liquid Flow Equations

7/29/2019 ENI_WTA_3



For 20rd

w

≥

η−−=

t4dE

21t,

rdP

2

iD

w

η−−π

µ

−= t4

d

Et25.2lnkh4

q

p)t,r(p

2

iDiw

+

π

µ−= D

w

Diw t,r

dPt25.2ln

2

1

kh2

qp)t,r(pLinear boundary:

For an asymmetrically well located in a bounded, irregularly shaped drainage area aninfinite number of images has to be accounted for, and the influence of each of the

images depends on the distance from the producing well:

η−−

π

µ−= ∑

∞

=1 j

2 j

iDiwt4

dEt25.2ln

kh4

qp)t,r(p

0t

dif 0

t4

dE

2

i→

∞→→

η−

Superposition in space

F function

7/29/2019 ENI_WTA_3


Dimensionless time with respect to the drainage area:

A

rt

Ac

ktt

2w

D

t

AD =φµ

=

FF fun ct i on o r MBHMBH f u n ct i on ( Mat t h e w s, Br o ns, Hazeb r oek )

∑∞

=

η−+π=

1 j

2 j

iADt4

dEt4F

F depends on:

•Time (tDA)•Shape of the drainage area

•Well location within the drainage area

F function

F function or MBH function

7/29/2019 ENI_WTA_3


F function or MBH function

Late transient conditions

7/29/2019 ENI_WTA_3


)Ft4t25.2(lnkh4

q

p)t,r(p ADDiw −π+π

µ

−=

)Ft4(kh4

qpp

ADi* −π

π

µ−=

The influence of the boundaries of the drainage area is felt

D*

w t25.2lnkh4

qp)t,r(p

π

µ−=

An equivalent pressure value p*

, which has no physical meaning,is defined:

Late transient conditions

Pseudo-steady state conditions

7/29/2019 ENI_WTA_3


At the time ts when the pseudo-steady state is reached, the FF functionstabilizes and becomes linear. In dimensionless terms:

ADA tclnF =sADAD tt >when

DI ETZ SHAPE FACTORDI ETZ SHAPE FACTOR ccAA: depends on the shape of the drainage

area, and on the well location within the drainage area

π+

π

µ−=

AD2wA

iw t4

rc

A25.2ln

kh4

qp)t,r(p

kh4

qt4pp

ADiπ

µπ=− 2

wA

wrc

A25.2ln

kh4

qp)t,r(p

π

µ−=

Furthermore: Fkh4

qpp *

π

µ+=

In the case of pseudo-steady state the average pressure is used:

p

V

V

1

c ∆

∆

= φ=− cAh

qtpp i

Pseudo steady state conditions

Dietz shape factor cA

7/29/2019 ENI_WTA_3


Dietz shape factor cA

Pressure decline at the wellbore in time

7/29/2019 ENI_WTA_3


LATE TRANSIENT CONDI TIONS LATE TRANSIENT CONDI TIONS (one or more boundaries are felt)

EARLY TRANSIENT CONDI TIONS EARLY TRANSIENT CONDI TIONS (radial flow)

PSEUDO PSEUDO - - STEADY STATE CONDITIONS STEADY STATE CONDI TIONS (all boundaries are felt)

t

P(rw, t)

( 1 )( 3 )

( 2 )


7/29/2019 ENI_WTA_3


( 1 ) General equation

)S2Ft4t25.2(lnkh4

qp)t,r(p

ADDiw +−π+π

µ−=

+

π

µ−= S2

rc

A25.2ln

kh4

qp)t,r(p

2wA

w

( 3 ) Equation valid under pseudopseudo--steady state conditionssteady state conditions

)S2t25.2(lnkh4

qp)t,r(p Diw +π

µ−=

( 2 ) Equation valid under early transient conditionsearly transient conditions only, when noboundaries are felt, also called infinite acting state (radial flow)

q q

Gas Flow Equations

7/29/2019 ENI_WTA_3


t)p(m1)p(m2

∂∂

η=∇tp1p2

∂∂η=∇Liquids Liquids Gas Gas

Analogy with the liquid flow: the solutions to the two equationsin dimensionless terms are the same, only the proportionality

groups are different.

ADtP

kh4

qp

π

µ=∆

ADsc

scsc

tP

Tp

kh2Tq)p(m

π=∆

Ft4t25.2lnt

PADD

AD−π+=

2wA

AD rc

A25.2ln

tP =

DAD

t25.2lnt

P =

Always validAlways valid

PseudoPseudo--steady state conditionssteady state conditions

Early transient conditionsEarly transient conditions

Pressure decline at the wellbore in time

7/29/2019 ENI_WTA_3


PSEUDO PSEUDO - - STEADY STATE CONDITIONS STEADY STATE CONDI TIONS (all boundaries are felt)

LATE TRANSIENT CONDI TIONS LATE TRANSIENT CONDI TIONS (one or more boundaries are felt)

EARLY TRANSIENT CONDI TIONS EARLY TRANSIENT CONDI TIONS (radial flow)

t

m(pw)

( 1 )( 3 )

( 2 )

Gas Flow Equations

7/29/2019 ENI_WTA_3


)'S2t25.2(lnT

p

kh2

Tq)p(m)p(m D

sc

scscwi +

π=−

)'S2Ft4t25.2(lnT

p

kh2

Tq)p(m)p(m

ADD

sc

scscwi +−π+

π=−

+

π=− 'S2

rc

A25.2ln

T

p

kh2

Tq)p(m)p(m

2wAsc

scscw

( 1 ) General equation

( 2 ) Equation valid under early transient conditionsearly transient conditions only, when noboundaries are felt, also called infinite acting state (radial flow)

( 3 ) Equation valid under pseudopseudo--steady state conditionssteady state conditions

WELL TESTING

7/29/2019 ENI_WTA_3


( 1 ) PRODUCTI VI TY TESTS( 1 ) PRODUCTI VI TY TESTS

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

- k , Vp

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

Deliverability Tests for Gas Wells

7/29/2019 ENI_WTA_3


Back pressure test

qsc

p

SPURGO

tT TTT

qsc1

qsc2

qsc3

qsc4

pwf 1

pwf 2

pwf 3

pwf 4

T > ts

p

CLEAN-UP

qSC

q SC1

q SC2

qSC3

q SC4

T TTT

T>ts

t

p

pp w f1

p w f2

p w f3

p w f4

7/29/2019 ENI_WTA_3


Empirical relationship of Empirical relationship of

Rowlins Rowlins

- -

Schellardt Schellardt

0.5<n<1

Turbu len tf low

Lamina rf low

AOFAOF ( Abso l u te Open Fl ow ) :( Abso l u te Open Fl ow ) : maximum producing rate which

could be theoretically obtained for ∆pmax,i.e. when pwf is zero

n2wf

2sc ppCq

−=

logqscAOFAOF

logp2

)pplog(2

wf

2

−

logC

1/n

7/29/2019 ENI_WTA_3


+

π=− 'S2

rc

A25.2ln

T

p

kh2

Tq)p(m)p(m

2wAsc

scscw

Rigorous equation for gas flow under pseudopseudo--steady statesteady state

conditions:

a

b

SCq)p(m∆

SCq

sc

sc

w bqaq

)p(m)p(m+=

−


7/29/2019 ENI_WTA_3


Isochronal test

qsc

p

SPURGO

tT2T TT

qsc1

qsc2

qsc3

qsc4

pwf 1

pwf 2

pwf 3

pwf 4

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

7/29/2019 ENI_WTA_3


Empirical relationship of Empirical relationship of Rowlins Rowlins - - Schellardt Schellardt

2h4h8hpseudopseudo--steady statesteady state)pplog(

2wf

2−

scqlog

n2wf

p2

pCscq

−=

1/n

1/n


7/29/2019 ENI_WTA_3


Modified isochronal test

n2w f

2

sc ppCq

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

(((( ))))n2w f

2w ssc ppCq −−−−====For the rates following the first:For the rates following the first:

qsc

p

SPURGO

tT TT

qsc1

qsc2

qsc3

qsc4

pwf 1

pwf 2

pwf 3

pwf 4

EXTENDED FLOW

T T T

pws4pws3

pws2p

t

CLEAN-

UP

qSC

qSC1

qSC4

qSC3

qSC2

T

EXTENDED

FLOW

p

ppw f1

pw f4

pw f3

pw f 2

T T T T T

pw s2 pw s3

pw s4

Productivity tests for oil wells

7/29/2019 ENI_WTA_3


qoST

p

SPURGO

t

qoST

pwf

T > ts

p

T

CLEAN-UP

qOST

qOST

T

T>ts

t

p

p

p w f

Productivity tests for oil wells

7/29/2019 ENI_WTA_3


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

pFlusso monofase

psat

Single phase flowSingle phase flow

psat

pw f

p

qost

PI

Two phase flowTwo phase flow

Drawdown tests for oil wells

7/29/2019 ENI_WTA_3


qoST

p

t

p(rw,t)

p

q

p

qOSTq

p

p(rw,t)

t

7/29/2019 ENI_WTA_3


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

=

πµ−= D

w

iw t,1rrP

kh2qp)t,r(p

φµ=

∆µ

π

=

=

trc

kt

pq

kh2

t,1r

r

P

2w

D

Dw

Early transient stateEarly transient state:

1 0 0t D <<<<

Match point

7/29/2019 ENI_WTA_3


+φµ

=

∆+µ

π=

=

tlogr c

klogtlog

plogq

kh2logt,1r r Plog

2w

D

Dw

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

rP)t(f p DD

w

=

=⇔=∆

Match point

7/29/2019 ENI_WTA_3


tD

= D

w

t,1r

rP

t

∆p

Match point

7/29/2019 ENI_WTA_3


t

∆p

tD

= D

w

t,1r

rP

* M* M

7/29/2019 ENI_WTA_3


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 of the constants by which the two curves are translated and,

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

MD

M

2w

M

MD

w

t

t

r

k

c

p

t,1r

r

P

2

qk h

µµµµ====φφφφ

∆∆∆∆

====

ππππ

µµµµ====

Wellbore Storage

7/29/2019 ENI_WTA_3


surface flowrate

sandface flowrate

drawdown

q

time

qwh

qsf < qwh

Wellbore Storage

7/29/2019 ENI_WTA_3


wcVp

V

C =∆

∆

=

tqV ∆=∆ tCqp ∆=∆

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 dimensionand

# 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

D hrc2

C

C φπ=

Dimensionless wellbore storage coefficient Dimensionless wellbore storage coefficient

Type-Curves of Agarwal et al.(1970)

7/29/2019 ENI_WTA_3


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

rPf P

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 cφ):

MD

M

2w

M

MD

t

t

r

kc

p

P

2

qkh

µ=φ

∆π

µ=


7/29/2019 ENI_WTA_3


PD

tD

Transient state

7/29/2019 ENI_WTA_3


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 +

π

µ−=

++

φµ+

π

µ−= S281.0

rc

klntln

kh4

qp)t,r(p

2wt

iw

Transient State

7/29/2019 ENI_WTA_3


m

ln t

p(rw,t)

p

ln t

p

p(rw, t)

m

k h4

qm

ππππ

µµµµ−−−−====

( )S2t25.2lnkh4

qp)t,r(p Diw +

π

µ−=

Transient State

7/29/2019 ENI_WTA_3


−

φµ−

−== 0.9

rc

kln

m

p)h1t,r(p

2

1S

2wt

iw

At ∆t= 1 hour

++

φµπ

µ−== S20.9

rc

kln

kh4

qp)h1t,r(p

2wt

iw

Determination of the skin factor S Determination of the skin factor S

Reservoir Limit Test

7/29/2019 ENI_WTA_3


PseudoPseudo--steady state conditionssteady state conditions

+π+

π

µ−= S2t4

rc

A25.2ln

kh4

qp)t,r(p

AD2wA

iw

*mc

qAhV

t

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

m*

t

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

t

m *

tAhc qS2rcA25.2lnkh4qp)t,r(pt

2wA

iwφ−

+πµ−=

Ahc

q*m

t φφφφ−−−−====

Dietz Shape Factor cA

7/29/2019 ENI_WTA_3


=−=π=

m

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

*m

m

3600

4c ww

A

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].

+

π

µ−==

++

φµπµ−==

S2rc

A25.2ln

kh4

qp)0t,r(p

S20.9rc

klnkh4

qp)h1t,r(p

2wA

iw

2wt

iw

The calculated value for cA can then be compared with the Dietz shapefactors presented for a variety of different geometrical configurationsto infer the shape of the drainage area and the well location within the

drainage area.

Drawdown Tests for Gas Wells

I th f ll i l t d d t t b f d t

7/29/2019 ENI_WTA_3


In the case of gas wells single rate drawdown tests can be performed toevaluate the parameters of the producing formation parameters of the producing formation just 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 ++

φµ+

π−=

Transient state flow equationTransient state flow equation

PseudoPseudo--steady state flow equationsteady state flow equation

+

φµπ+

π−= '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 pressuredata allows the determination of the apparent skin factor only, but two

flow rates are needed to determine both S and D .

Drawdown Tests for Gas Wells

7/29/2019 ENI_WTA_3


qsc1

p

t

p(rw,t)

p

q

qsc2

T T2T

q

p

p

qSC1

qSC2

p(rw,t)

T T2T

t

7/29/2019 ENI_WTA_3


m2

ln t

m(pw)

m1

m p( )

ln t

m(pw)

m(p)

π−=sc

sc2sc

2 T

p

kh2

Tq

m

π−=

sc

sc1sc1

Tp

kh2Tqm

For ∆t = 1 h

−

φµ−

−=+

−φµ−

−=+

=

=

0.9rc

kln

m

)p(m)p(m

2

1DqS

0.9rc

klnm

)p(m)p(m

2

1DqS

2wt2

i2h1tw

2sc

2wt1

i1h1t

w

1sc

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.

7/29/2019 ENI_WTA_3


u , q 0, a p odu g a a o a o a o a a

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

eqq

N

t ====

∆t=0

cumulative volume of oil since beginning of production

final flow rate

Flow equation which extend the flow equation derived for an infinite acting system

7/29/2019 ENI_WTA_3


)S2t25.2(lnkh4

qp)t,r(p D

*w +

π

µ−=

q q g y

(radial geometry) to the case of a finite, irregularly-shaped drainage area:

i* pp =

If the well is new:F

kh4

qpp *

π

µ+=

If the well has been produced:

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

-q

tp+∆t

tp

++∆++

φµπµ−= S281.0)ttln(

rckln

kh4qp)t,r(p p2

wt

*w

++∆+

φµπµ+ S281.0tln

rckln

kh4q

2wt

Horner Plot

7/29/2019 ENI_WTA_3


t

ttln

kh4

qp)t,r(p

p*w

∆

∆+

π

µ−=

!! Horne r p lo t Ho rne r p lo t

k h4

qm

ππππ

µµµµ====

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

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

7/29/2019 ENI_WTA_3


surface flowrate

sandface flowrate

build-up

q

time

qwh=0

qsf >0

Horner Plot

7/29/2019 ENI_WTA_3


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 PLOT HORNER PLOT

t

ttln

p

∆

∆+

pw

m

After flowAfter flow

Skin effectSkin effect

*p

Determination of the Skin Factor

••atat ∆∆t=0, i.e.at the production time tt=0, i.e.at the production time t Oil Well

7/29/2019 ENI_WTA_3


)S2t25.2(lnkh4

qp)0t,r(p D

*w +

π

µ−==∆

++

φµ+

π

µ−==∆ S281.0

rc

klntln

kh4

qp)0t,r(p

2wtiw

••atat ∆∆t=1 hourt=1 hour

)3600lnt(lnkh4

qp)h1t,r(p *

w−

π

µ−==∆

h1tif ttt =∆≈+∆

+++

φµπ

µ−=−

=∆=∆3600lnS281.0

rc

kln

kh4

qpp

2

wt

h1tw0tw

−

φµ−−= =∆=∆ 9

rckln

mpp

21S

2

wt

0twh1tw

Buildup Tests for Gas Wells

pTq scsc*

7/29/2019 ENI_WTA_3


)'S2t25.2(lnT

p

kh2

q

)p(m)p(m D

sc

scscw +π−=

∆∆+

π−=

tttln

Tp

kh2Tq)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

ttlnp

∆

∆+

!! Horn er Plo t Horn er Plo t

m(pw)

m

sc

scsc

T

p

k h2

Tqm

ππππ====m(p*)

Determination of the Skin Factor

Gas Well

7/29/2019 ENI_WTA_3


−

φµ

−−

= =∆=∆ 0.9

rc

kln

m

)p(m)p(m

2

1'S

2

wt

0twh1tw


The type The type - - curves obtained by curves obtained by Agarwal Agarwal et al et al ..

7/29/2019 ENI_WTA_3


== DD

w

D C,S,t,1r

rPf P

represent solutions to the drawdown equation to describe the pressure decline at the

wellbore, 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 thatthe rate of pressure decline could be assumed negligible during the shut-in period. If

this 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(pww

=∆−

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

r

7/29/2019 ENI_WTA_3


== DDw

D C,S,t,1rPf PThe Agarwal type curves:

can be written as a function of: t

C

kh2

C

t

D

D

µ

π=

trc

kt

2wt

Dφµ

=

2wt

Dhrc2

CC

φπ=

Infinite acting radial flow equation for fluid flux occurring in a homogeneous porous

medium under transient conditions:

( )S2t25.2lnkh4

qp)t,r(p Diw +

π

µ−=

( )S2t25.2ln2

1P DD +=

obtained from

( )

++=

S2

D

D

DD eCln81.0C

t

ln2

1

P

By substitution:

Type-Curves Plot

100

7/29/2019 ENI_WTA_3


D

DD

C

tP =

(i.e., unit slope)

0.1

1

10

0.1 1 10 100 1000 10000t

CD

D

PD C eDS2 10000

0. 1

PD CDe2S10000

0. 1

tD

CD

When pure wellbore storage flow occurs:

Pressure Derivative Method

According to this method, which is largely based on the solutions obtained byAgarwal et al., the type-curves are redrawn in terms of the derivative of the

7/29/2019 ENI_WTA_3


g , ypdimensionless 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 datap(rw,t), expressed in terms of pressure derivative:

dt

)pp(d'p wf i −

=∆

)t(f t'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(f t

ttt'p ∆=∆+∆∆

Type-Curve Derivatives

D' dP

7/29/2019 ENI_WTA_3


=

D

DD

C

td

P

PURE W ELLBORE STORAGE PURE W ELLBORE STORAGE

1

C

t

d

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 UM FOR A H OMOGENEOUS POROUS MEDI UM

=

=

D

D

D

D

D'D

C

t

5.0

C

td

dPP

Type-Curves and Type-Curve Derivatives

7/29/2019 ENI_WTA_3


10-210-3 10-1 1 10

∆∆∆∆t (hr)

103

102

101

104

∆ ∆∆ ∆ p ( p s

i a ) &

∆ ∆∆ ∆ p ’

WellboreStorage

Radial Flow

CDe2S

1030

1012

106

10-1

102

104

Real data vs type-curves

7/29/2019 ENI_WTA_3


∆∆∆∆t (hr)

∆ ∆∆ ∆ p ( p s i a ) &

∆ ∆∆ ∆ 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

Superposition real data & type-curves

7/29/2019 ENI_WTA_3


103

102

101

10-210-3 10-1 1 10 101

104

∆∆∆∆t (hr)

∆ ∆∆ ∆ p ( p s i

a ) &

∆ ∆∆ ∆ p ’

7/29/2019 ENI_WTA_3


Interpretation Result

Pr ess r e Mat chPr ess r e Mat ch k hk hB

kh4

PD ∆

π

7/29/2019 ENI_WTA_3


• C C g ives 2wt

Dhrc2

CC πφ=

•• Cu r ve Mat ch Cu r ve Mat ch

S2

DeC

•• Skin Skin

= D

S2D

C

eC

ln2

1

S

• Pr essur e Mat ch Pr essur e Mat ch k h k h pqBPo

D ∆µ=

••Tim e Mat ch Tim e Mat ch C C tC

kh2

Ct

D

D

∆µ

π

=

Drawdown Test Analysis

7/29/2019 ENI_WTA_3


Pressure

L o g ∆ ∆∆ ∆ 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

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

Buildup Test Analysis

7/29/2019 ENI_WTA_3


Log t

P r e s s u

r e

L o g ∆ ∆∆ ∆ 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

The derivatives of the type-curves exhibit a linear dependency on time of

pressure data points Linearity can occur according to different yet

7/29/2019 ENI_WTA_3


pressure data points. Linearity can occur according to different yetcharacteristic slopes which are a function of flow geometry:

FLOW TYPE FLOW TYPE PRESSURE PRESSURE SLOPE SLOPE

linear linear

bilinear bilinear

radial radial

spherical spherical

(hemispherical)(hemispherical)

)t(f p =

)t(f p 4=

)t(lnf p =

=

t

1f p

1/ 2 1/ 2

1/ 4 1/ 4

- - 1/ 2 1/ 2

horizontal horizontal

straight line straight line

Documents

ENI_WTA_3