Petroleum Engineering Institute HW UniversityPetroleum Engineering Institute, HW University

W ll T t A l iWell Test Analysis

Chapter 1: Fluid Flow in Porous Media

Dr: M. Jamiolahmady (Jami) Tel: 0131 451 3122Tel: 0131 451 3122Fax: 0131 451 3127Email: jami.ahmady@pet.hw.ac.uk

Flow in Porous MediaFlow in Porous Media

Unsteady-state flow in heterogeneous systems:Three dimensional.Multi (three) phases.Multi (three) forces (viscosity, capillary, gravity, inertial).

However here we focus on radial flow.One dimensional.One phase.One force (viscosity, impact of gravity and capillary for static

pressure distribution discussed in the last three chapters).

Well Testing - Obtained Information - 1Well Testing Obtained Information 1

Pressure behavior.

Average reservoir pressure.

Reservoir properties.Permeability.

Reservoir characterisation.Faults, layering, areal continuity.y g y

Well Testing - Obtained Information - 2Well Testing Obtained Information 2

Well completion efficiency (skin).

Well productivity.PI=q/DP, which stays constant at least for a period of time.PI q/DP, which stays constant at least for a period of time.

Nature of formation fluid.Al l f l b l iAlso samples for lab analysis.

Reservoir temperature.p

Pressure BehaviourPressure Behaviour

Pressure history vs. time during a test leads to:

Determination of average (static) reservoir pressure.

Flow capacity (kh=net pay*permeability).

Skin.

Reservoir discontinuity and limits (fault, …).

OILPROD.RATE

TIME

PLATEAUW.B.T.

TYPICAL OIL PRODUCTION PROFILE

Inj. WellProd. Well

CLUSTER DEVELOPMENT

For an Offshore Fieldthe Target PlateauRate is Typically 10%of RecoverableReserves p.a.Fig 1.1.1

Forties Field4.2 BBOOIPProduced 2.5BBO (Feb 2006)

Minimum Economic Rate for an Offshore Oil Well. . . depends on the following factors:

• Water Depth• Oil Price• Pipeline Tariff• Distance to Existing Facilities• Total Recoverable Reserves. . . each case must be examined in detail and aneconomic assessment made. . . in the early days of the North Sea development afigure of 5000 STB/d was often quoted (no longervalid)

SEPARATOR(1 STAGE)st

GASOILWATER

WELL-HEAD

WELL

RESERVOIR

pwfprps

Reservoir to Separator Flow System

= Flowing Bottom-Hole Pressure= Reservoir Pressure= Separator Pressurepr peppi

pwf

ps

q

Fig 1.1.2

LAMINAR SINGLE-PHASE FLOW IN A POROUS MEDIUMFOR LINEAR HORIZONTAL FLOW A

q

DEFINITION OF THE PERMEABILITY OF A POROUSMEDIUMPERMEABILITY IS AN INTRINSIC ROCK PROPERTY Fig 1.2.1

xqA u k dpdx= = - m

Darcy’s Law

q

Au

k dp

dx= = - m

Definition of the permeability of a porous mediumPermeability is an intrinsic rock property

q in-situ volumetric flow-rateA cross-sectional areau superficial fluid velocityfluid viscosityp pressure or potentialx lengthk permeabilitym

m /smm/sNs/mPamm

32

2

2

. . . single-phase, linearhorizontal flow

Darcy’s Law

B

A

k dp

dxs = - ´ -11271 10 3.

m

Darcy Unitsq : cc/s A : cm x : cm : cp p : atm2 m

k : DarcyOil Field Units

. . . practical unit of permeability : md- the millidarcyq : bbl/day A : ft : cp p : psi x : fts 2 m

k : md

B = Formation volume factor 1 md = 0.986923*10 m-15 2

q

Fig 1.3.1

Laboratory Measurement of the Permeability of Core PlugsCore HolderControlledMeasuredFlowq q

Transducer Cylindrical Core ofCross-sectional AreaA

0p qA k p pL i e k q LA p=

-=1 2e j

mm. . D

Darcy’s Law LPAkQ Δ

= Darcy s Law

Assumptions.

Lμ

Steady state creeping flow.Rock 100% saturated with one fluid.Fluid does not react with the rockFluid does not react with the rock.Rock is homogeneous and isotropic.

UnitsUnits.One Darcy is defined as the permeability which will permit a fluid of one centipoises viscosity to flow at a linear velocity of

ti t d f di t fone centimetre per second for a pressure gradient of one atmosphere per centimetre.

Darcy’s Law LPAkQ Δ

= Darcy s Law

Analogy between Darcy’s law and Ohm’s law .

Lμ

EP ,1k I,Q ArL=R

ce)R(resistanl)E(potentia=I(current) ≈Δ≈≈⇒

rμ

Analogy between Darcy’s law, Fourier’s heat law.

TP ,Kk q,Q LTAK=q '' Δ≈Δ≈

μ≈⇒

Δ

Absolute Permeability DeterminationLPAkQ Δ

=μ

Absolute Permeability Determination

Empirical correlation (e.g. Carman-Kozeny).

Log data.Use of an empirical equation (e.g. Timur) & extending the p q ( g ) g

correlation between measured lab. & log data (porosity & Swi).

Laboratory Measurements.ySteady-state flow of a fluid & Darcy Law with measured Q & ΔP, reservoir conditions preferable otherwise to be corrected.

Well Test analysis.An average (unlike core & log) in-situ (like log) k.

36 f 3D vs2k1Fig 1.3.2

Permeability of Unconsolidated Beds (Sand Packs)

Fixed Bed in Chemical EngineeringFor Laminar Flow:

Carman - Kozeny Equationf = void fractionD = Volume - Surface Mean Particle DiameterVS

(1 )- f 2 k = 1501

kD

kvs= -

36

1

3 2

1

2

ffa f

Shows importance of porosity and grainsize as determinants of permeability

k : permeability k = 150 . . . Kozeny constant : porosityD : Volume - Surface Mean Particle Diameter = 6(1 - )/aa : specific surface area of bed (wetted surface / unit volume)

1vs

ff

Carman - Kozeny Equation

Berg CorrelationInvestigated the permeability of well sorted detrital rocks withporosities down to 10%maximum value for granular aggregatesk : permeability (Darcy) : porosityMD : Weight median grain sizePD : phi percentile deviation - measure of sorting

f

a

Berg : Trans Gulf Coast Assoc of Geol Soc 20, 303 (1970)

CumWt%CumWt%

phi

k MD e PDs= ´ - -51 10 6 5 1 2 1 385. . .f e j

Dphi

= FHGIKJ

1

2

Log Resistivity & PorosityLog Resistivity & Porosity

Formation Resistivity factor. RyRo resistivity of water saturated rock.Rw resistivity of water in the pores.

w

o

RRFr =

Fr can be related to porosity by an empirical correlation.a and m are constants.– Archie, carbonates, a=1, m=2.– Humble, sandstone, a=0.62, m=2.15.

maFrφ

=

Log Resistivity & SaturationLog Resistivity & Saturation

Resistivity of a rock saturated with hydrocarbon and y ywater is greater than that of a rock saturated with water Rt>Ro.

n values range from 1 7 to 2 2n

oRS/1

⎟⎟⎞

⎜⎜⎛

=n values range from 1.7 to 2.2.t

ow R

S ⎟⎟⎠

⎜⎜⎝

=

2/12/1

⎟⎟⎠

⎞⎜⎜⎝

⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

t

w

t

ow R

FrRRRSFor n=2

A

S

B

w irrC= f

, kSw irr

1 22 25

100/.

,

= f

kc

w S

w

RR

c

w cementation saturation onent

Fa

SFR

R

w irr

w

w

t irr

h h

m wn w

t

1 24

2

2

2

3 75

2 2

223 465 188

/

,

,.

log .

exp

= FHGIKJ

= - +

FHGIKJ +

LNMM

OQPP

= + -=

= =

f

fr r

f

Permeability from Open-hole Log DataTimur S . . . Grain size indicatorw,irr

Modified Form:

Coates and Dumanoir

k

MSLSea Bed

Over-Burden

Formation

Overburden at Depth = pc= total weight of rock plus sea waterpsi

Fluid pore pressure = pp psi

(obtained from density log)

(measured by WFT)

Contact forcebetween particles= grain pressure= psipg

Force Balance: p = p + pc p g

Basic Rock Mechanics

NetEffectiveStressFig1.3.3

543210 0 2000 6000 10000 14000

p = 14500 psigp varyingcp }

p = 0, p varyingp c

BEREAf= 20%

SANDSTONES

LIMESTONES

ff fPA A= 100 - 3.37 (p - p )C p 0.3

f

fp

f f

fpa

PA A= 100 - 0.432(p - p )C p 0.42

= POROSITY FROM CORE ANALYSIS= POROSITY IN RESERVOIR AT PRESSURE pppC = OVERBURDEN PRESSURE (~ 1 psi/ft)pp = PORE PRESSURE

Grain Pressure, p p (psi)c p-

V Vpa p-Vpa(%)

Effect of Pressure on Pore Volume

Fig 1.3.4

kokko

pc (bar)

1.0

0.9

0.8

0.7200 600 1000

BA = 0.4 mdBE = 52 mdMA = 737 mdSP = 944 mdFA = 817 md

BA BE

FAMA SP

p = 20 barpNormalised Permeability versus Confining Pressure

Fig1.3.5

6.1 BillionbblOIP Overburden Stress 9000 psi6 km OverburdenDanian Chalk (Paleocene)Cretaceous Chalk

TightZone400 ft600 ft

Seabed

f = 0.42

Initial Reservoir Pressure = 7000 psi (Pore Pressure)Initial Net Effective Stress = 2000 psi

End of 1985 : Pore Pressure = 4000 psiNet Effective Stress = 5000 psi Fig1.5.5

NoteVeryHighPorosity

Ekofisk Reservoir3

50403020100 200 400 0 km

Satellite Measurements of PlatformSubsidence Since March 1985

Time (days)

Subsidenc

e(cm)

40 cm/yr

Predicted Subsidence (1986) underThen Current Depletion Policy - 6m

1985 Ekofisk SubsidenceSeabed Map (cm)

Bathymetric SurveySubsidence of Seabed intoan Elliptical BowlFig 1.5.6

050100200250

AC

B

Reservoir originally thought to be oil wetIn 1987 all six of the steel jacket structureswere raised by cutting, installing flangesand jacking up the platforms and insertingextensions of 6m

Ekofisk ReservoirVery little pressure maintenance. . . naturally fractured reservoir with solutiongas drive and some gas re-injection

5000 4000 3000 2000303540455055

f(%)

Reservoir Pressure (psia)

Valhall Porosity versus Reservoir PressureElastic Plastic

Yield Point Pressure

Fig 1.5.7After Cook and Jewel

Valhall Reservoir

Cook and Jewel:“Simulation of a North Sea Field Experiencing SignificantCompaction Drive”, SPE Res. Eng., 11,(1), 48-53, Feb 1996

• Half the oil produced from Valhall is a direct result ofthe rock compressibility mechanism• In the crest rock compressibilities can be as high as150´10-6 psi-1• On the reservoir crest a measured PTA permeabilityof 120 md was corrected to an original value of300 md• Final set of compaction curves shown in Fig 1.5.7• Original rock curves had to be multiplied by a factorof 1.5• Net stress exposure had hardened the rock samples

Flow RegimesFlow Regimes

Steady-state, P=f(r), qr=constant.Strong aquifer support or injection wells.

Semi steady state P=f(r t) but ΔP/dt=constantSemi-steady state, P=f(r,t) but ΔP/dt=constant.Closed no flow outer boundary.

Unsteady-state transient, P=f(r, t).Transient well test data.

reREGION OF AREALRADIAL FLOW

PRODUCING WELLOBSERVATION WELL

ACCESSIBLEFROM PLT

WELL SHUT-IN

ACCESSIBLE FROMRFT DATApepeq

pw f re

Reservoir Pressure Distribution

Fig 1.4.1

Radial Flow Single Well Model

hrw re

kf

Fig 1.4.2

q

Radial FlowSituation

Model Cylindrical Reservoirwith Central Well

rw

rw

re

re

r

rurur

q

qqpe

hpe

u =r q2prh k dpdrm Fig 1.4.3

Steady-State RadialIncompressible Flow

= =uq B

hr

k dp

drrs

2p m

q B

kh

dr

rdps

r

r

p

p

w

e

w

emp2

=

\ - =p pq B

khrre w

s e

w

mp2

ln

pp pq B

kh

r

rrD

e w

s

e

wDe= - = =m

p2

ln ln

Steady-State Radial Darcy (Creeping) FlowDarcy's Law

Separating the Variables and Integrating:

DimensionlessPressure

rw

h

re

pwpe

q

rw re Fig 1.4.4

Steady-State Radial Creeping Flow

p = rD Dln

65

3210 1 100 200 300 400r / rw

PRESSURE PROFILE IN THE VICINITY OF A WELL

Fig 1.4.5

p =Dp p- wq Bs o om2 khp

4

Steady-State, Radial, Single-Phase Flow

SEPARATOR(1 STAGE)st

GASOILWATER

WELL-HEAD

WELL

RESERVOIR

pwfprps

Reservoir to Separator Flow System

= Flowing Bottom-Hole Pressure= Reservoir Pressure= Separator Pressurepr peppi

pwf

ps

q

Fig 1.1.2

- Radial steady-state flow modelProductivity Index, PINo Skin J

q

p psss

e w

= -b g

bbl/day/psi

Since:q B

khr

r

p pse

w

e w= ´ --1127 10 23.

ln

pm

b g FieldUnitsthen

Jkh

Brr

ssee

w

= ´ -1127 10 23.

ln

pm

Hence well productivity index depends strongly on- - mPermeability- thickness product,in-situ oil viscosity, kh

Well Productivity Index

Straight Line Inflow Performance Relation (IPR)qs = Jsss(pe - pwf)

. . . Definition of P.I.i.e. p p J qwf e sss s= - 1pepwf

qsIPR

slope - 1Jsss

. . . equation of astraight line

Determination of Well Operating ConditionsDetermination of Well Operating Conditions

In oil wells under laminar (creeping, Darcy) flow:qs=Jss(Pr-Pwf):

Overall pressure difference for vertical liftOverall pressure difference for vertical lift performance is not linear:

Pwf-Ps=fVLP(qs)wf VLP(q )

Solving the above two equations simultaneously for & P k i J d f f ti ifiqs & Pwf, knowing Jss and fVLP functions, specifies

the well operating conditions.

p pe w- DRAWDOWN1J(p )e 1

(p )e 2(p )e 3

Relation Between Three Key Variables: q , p and ps w e

VLP IPRProduction Rate,(STbbl/D) qs

FBHP

(psi)pw

slope = -

Well Inflow Performance Diagram

Fig1.4.6

Due to Gilbert

qs

Bottom-Ho

lePressu

re

Oil Production RateMatch vertical lift performance (VLP) to inflow performancerelation (IPR) i.e. find q from nodal analysiss

Well Performance Diagramprpwfps

p pr wf-

p pwf s-Dp

OperatingPoint

Drawdown

Lift TotalIPR VLP

Fig 1.1.3

p(r) = p +wq m2 k h ln rrwp Fig 1.4.7

pep

pwrw re

p(r)

Average Pressure in SS Radial Flow

r pq

kh

r

rww

b g = + mp2

ln

p ppdV

V

p r rhdr

r hwr

r

e

w

e

= + =b g2

2

p

p

p pq

kh

r

rwe

w

= + -LNM

OQP

mp2

1

2ln p p

q

khe - = 1

2 2

mp

Jq

p p

kh

Brr

sss

w e

w

= - = ´-L

NMOQP

-1127 10 2

12

3.

ln

pm

Average Reservoir Pressure in SS FlowPressure Distribution

VolumeAveragedPressure

STbbl/day/psi

i.e.

p

qt0

WELL PRODUCED ATCONSTANT RATE

CLOSED (NO FLOW)OUTER BOUNDARY

tTRANSIENTINFINITE-ACTINGPERIOD

SEMI-STEADY-STATErw reFig 1.5.1Transient Pressure Behaviour of a Single Wellat the Centre of a Closed Reservoir

r

PressureinReservoir

t1t2t3

rerw r

PRESSU

RE

Fig 1.5.2

Semi-Steady-State Depletion of a Circular Reservoir with a Central Well

dpdt = constant . . . all r

In SSS Pressure Profiles Retain the Same Shape

q

q

rerw r

PRESSU

RE

p w f

p

AT SSS dp dpdt dt Fig 1.5.3

p p r rdrrr

r

ewe

= z e j22pp

Average Pressure in SSS Radial Flow

V

V

p= 1 ¶

¶

DV qdt cVdp= =

dp

dt

q

cV

q B

c r hs

e

= - = - p f2

Compressibility of a Liquid. . . fractional change in volumeper unit change in pressure

ReservoirMaterialBalance Volume producedin time interval dt Expansion of theLiquid in the reservoir

. . . simplest possible form of the material balance equation

c

. . . a more sophisticated analysis shows that c should be replacedby the total system compressibility c where:t

c c S S c ct w wc wc o f= + - +1b gc . . . water compressibil ity c . . . oil compressibil ityc . . . formation (pore volume) compressibilityS . . . connate water saturationw ofwc

cV

V

pfp

p= - 1 ¶¶

Allows for the presence of connate water and formation compactionLatter term is significant in unconsolidated formations

Total System Compressibility

Definition ofRockCompressibility

ppi slope ,

Linear Pressure Decline in Primary Depletion

0 Time,tReservoirLimit Test

m q Bc r hst e* = - p f2

Fig 1.5.4

urr rwre

q

Fig 1.5.10

No Flow Across External BoundarySlightly Compressible FlowOil Production at Central Well isSustained by Expansion of Fluidin Place

Semi-Steady-State (SSS) Flow

u q rhr k dpdrr = = -e j2p m

Flow Distribution

Closed System

rw reqr qr

q

rqr

rw rer0

q

Mechanism of Semi-Steady-State Depletion

Fig1.5.11q cV dpdtr r re= - ®

cVdp

dtc r r h

dp

dtr r r ee= - = - -® p f2 2c h

q c r hdp

dte= - p f2 \ = - = -q

q

r r

r

r

rr e

e e

2 2

2

2

21

- = =uq

rh

k dp

drrr

2p m

Darcy's Law qr0rw

q

r re

qqr qr

rw rerq

qr

r

rkh dp

drre

= -FHGIKJ =1

22

2

pm

122

2-FHGIKJ =r

r

dr

r

kh

qdp

e

pm

122

2-FHGIKJ =r

r

dr

r

kh

qdp

er

r

p

p

w

e

w

epm

Hence on substitution:

which on separating the variables becomes:

Integration gives:

q

kh

r

r

r

r

p pe

w

w

e

e w=- +F

HGIKJ

-2

1

2 2

2

2

pm ln

b g

qkh

rr

rr

rr

p p

w e

w

e

w=- +F

HGIKJ

-2

2 2

2

2

2

2

pm ln

b g

qkh

r

r

p pe

w

e w=-F

HGIKJ

-2

1

2

pm ln

b g

The analytical solution to this is:

and the pressure at any radius r is given by the equivalent formula

For r >> re w

q

p r dV

V

h p r r dr

r r hr

r

e w

w

e

= = -z zb g b g

c h2

2 2

p f

p f

p pq

kh

r

r

p p pq

kh

we

w

e

= + -LNM

OQP

= - =

mp

mp

2

3

4

2

1

4

ln

D

Volume Average Reservoir Pressure in SSS Flowq

ppw

pe

rw rep

pDqr

qr

q

q

r

re

rw

rw= p p- w ln rrw r2

r22re2

re2q2p km h1

SS FLOWSSS FLOW= 400

Dimensionless Pressure Profile and Flow Distribution in SSS Flow1.00.80.60.40.20 1 100 200 300 400

6543210

pD

Fig 1.5.12

q

p p

q

B p pssss

w w

= - = -b g

qkh p p

r

r

w

e

w

= --F

HGIKJ

2

3

4

pmb g

ln

Jkh

Brr

SSS

e

w

=-F

HGIKJ

2

34

pm ln

Jkh

Brr

SSS

e

w

= ´-F

HGIKJ

-11271 10 2

34

3.

ln

pm

Well Productivity Index in a Bounded (Closed) Drainage Area

The SSS well inflow equation is:hence

or in field units:

J

Semi-Steady-State Steady-StateSemi Steady State, Steady State

At Semi-Steady-State (SSS) conditions, pressure gradient with time is constant, no flow closed boundary.

P*=Pe, c=0.5P*=P c=0 75 PPkh2Q w

* −π=P =Pave, c=0.75.

Pave , volumetric average pressure.Pe , external boundary pressure.

crrLn

Q

w

e −⎟⎠

⎞⎜⎝

⎛μ=

At Steady-State (SS) conditions, no variation in P & saturation with time, constant external pressure.

P*=Pe, c=0P*=Pave, c=0.5.

Productivity Index JProductivity Index, J

PI is the ratio of production to the pressure drawdown in the drainage area of a well.

SS, P*=Pe, c=0, P*=Pave , c=0.5.SSS P*=P c=0 5 P*=P c=0 75SSS, P =Pe, c=0.5, P =Pave, c=0.75.

1kh2QPkh2 πΔπ

crrLn

1kh2P-P

QJc

rrLn

Pkh2Q

w

e

w

e −⎟⎠

⎞⎜⎝

⎛μπ

==−⎟⎠

⎞⎜⎝

⎛Δ

μπ

= ∗

Dimensionless Productivity Index JDDimensionless Productivity Index, JD

JD does not include the impact of reservoir thickness, fluid and rock properties.

Only depends on drainage area & rw.crLn

1kh2

JJe

D

−⎟⎞

⎜⎛

=πμ

=

Non-redial, improvement, damage expressed by

cr

Lnw

⎟⎠

⎜⎝

p g p yskin factor.

SrL

1JSrL

Pkh2Qe

De ⎟

⎞⎜⎛

=⎟⎞

⎜⎛

Δμπ

=Sc

rLnSc

rLn

w

e

w

e +−⎟⎠

⎞⎜⎝

⎛+−⎟

⎠

⎞⎜⎝

⎛μ

p1 p3V1

p4 V4V3

p2 V2q1

q4q2

q3

VIRTUAL NO-FLOW BOUNDARIES Fig 1.5.8

Concept of Drainage Areas and Virtual No-Flow Boundaries. . . Multiwell Reservoirs

DietzDrainageAreas

q

qVi

i

i

= åp p p p1 2 3 4= = =

Under semi-steady-state (SSS) conditions the reservoir pore volumedrained by a well is proportional to that well's production rate i.e.V = total reservoircompartmentvolume

V determined by planimeteringat joint SSSassumes a communicating systemdue to Dietzreal no-flow boundaries such as sealing faults must berespected before assigning drainage areas

i

V

p1p3

V1

p4 V4

V3p2 V2

q1q4

q2 q3

REAL NO-FLOW BOUNDARIES SUCH AS SEALING FAULTS MUST BERESPECTED BEFORE ASSIGNING VIRTUAL DRAINAGE AREAS

Fig 1.5.9

Effect of Real No-Flow Boundaries on the Assignment of Drainage Areas

Physical No-Flow Boundaries e.g. Faults

Real no-flow boundaries such as sealing faults must berespected before assigning virtual drainage areas

Fig 1.6.1

Deviation from radial flowin non-symmetric drainagecaused by well proximityto a physical boundary

Radial streamlines in acircular drainage areawith a central well

Generalised Form of the SSSInflow Equation

pq

khrrw

e

w

- = -FHG

IKJ

mp2

34

ln

p pq

kh

r

r ewe

w

- = mp

pp2

1

2

2

2 3 2ln

/

Generalised Form of the SSS Inflow Equation- note the longer length of flow paths and the bunchingof streamlines with a non-central well- areal flow convergence effect

Dietz Shape FactorsThe basic radial flow equation for SSS is:

which can be written alternatively as:

The natural log term can be rearranged as:

p

4

4

56 32

4

31 62

2

3 2 2 2 2

pp g

r

e r

A

r

A

re

w w w/ . .

= =

p pq

kh

A

C rwA w

- = mp g2

1

2

42

ln

where A = Area of drainage region= 1.781 . . . exponential of Euler's constantC = Dietz shape factorgA

For a circular region with a central well C = 31.62. . . the maximum value which C can takeAA

The generalised inflow equation takes the form:

4

kh

BA

C r

SSS

A w

= FHG

IKJ

4

42

pm gln

SSS Well Productivity Index

For non-symmetric drainage areas and well locationsC < 31.62Aand the PI is smaller than that of a well in the centre of a circleDietz evaluated C for a wide variety of shapes and well positionsA

e.g. rectangle C = 4.514AEspecially important in long narrow reservoirs e.g. channel sandsand when well is close to a fault

J

C27.6t0.2A

DAsss

C12.98t0.7A

DAsss

C4.51t0.6A

DAsss

C3.34t0.7A

DAsss

C21.9t0.4A

DAsss

C0.098t0.9A

DAsss

C30.88t0.1A

DAsss

C31.6t0.1A

DAsss

1/33 4

7/8

1 1 1

111

2 2 2

222

C = 21.8 t = 0.3A DAsss C = 10.8 t = 0.4A DAsss C = 2.08 t = 1.7A DAsss

C = 4.51 t = 1.5A DAsss

C = 5.38 t = 0.8A DAsss C = 2.70 t = 0.8A DAsss C = 0.23 t = 4.0A DAsss

C = 3.15 t = 0.4A DAsss C = 0.58 t = 2.0A DAsss

7/87/8

1 1 14 4 4

Fig 1.6.2Selection of Dietz Shape Factors

Norwegian (Whitson and Golan) Form of the Inflow Equationp p qkh rr Sw e

w A- = - +FHG

IKJ

mp2 34ln

Areal flow convergence contributionto the total skin factorwhere: S CA A= +12 4 34ln p

g

• Total skin factor is a vehicle for allowing for alldeviations from ideal radial flow• No formation damage contribution to skin inthis formulation

L 2

L1a 2 a1Well

Well in General Position in a Rectangular Drainage Area

b2b1

yx

a < b2 2a < b1 1

x aL y aLD D= =22

11 Fig 1.6.3

a

L

bb

L

cc

W

D

D

D

=

=

=

A WL=

pp p kh

q

A

C rwDw

A w

= - =b g 2 1

2

42

pm gln

pA

Wa b

Wr cwD D D

w D

= -FHGIKJ +2 1

3 22

pp pln

sinb g

ln lnsin

CA

W

A

Wa bA D D= LNM

OQP - -FHG

IKJ - ¢16 4 1

3

2 2

2 2

p p g

Rectangular Drainage Area SSS Flow

Result due to Yaxley based on linear flow theory

i.e.For a 5:1 rectangle (central well) C = 2.359 = 0.5772A g

a bc Wa

L 2

L1a 2 b2a = 0 = y1 DTotallyOffsetWell

Fig1.6.4

Three Well ClusterFig 1.6.5

WellSpacingL c

WellClustersReservoirLimit

Fig 1.6.5a

1 42 3

Approximate Drainage Areas for a Four Well Cluster Fig 1.6.5b

Overall Block

FiveWellCluster

Virtual No-FlowBoundaries Fig 1.6.6

Triangular or Wedge Shaped Drainage Area

+ (r , )o oq

q qo

Well

rero

AREA = r e2 q2

AfterYaxley Fig 1.6.7

Dietz Shape Factor for a Well in a Wedge-Shaped Reservoir

. . . Due to Yaxley

C Arr rAeo

oo

=

-FHG

IKJ + F

HGIKJ

L

N

MMMM

O

Q

PPPP

44 34 2 2g pq

qp p q

qexp ln ln sin

Intersection Angle q = 60° q = 90°Well Distance from Apex

ro

Dietz Shape Factor, CA

Dietz Shape Factor, CA

1 9.52895´10-27 7.1163´10-16

5 2.3264´10-18 1.77907´10-10

10 3.4304´10-15 1.1386´10-8

20 3.51275´10-12 7.2871´10-7

50 3.3500´10-8 1.77907´10-4

100 3.4304´10-5 1.1386´10-2

200 3.51275´10-2 0.7287300 2.0256 8.3004350 9.46298 20.9305400 35.9706 46.6372

Table 6.1 Dietz Shape Factors for a Well on the Bisectorof Intersecting Faults (re = 1000, rw = 1)• Yaxley formula is valid provided ro < 1/3 re• Use default value of 31.62 if formula predicts alarger value

Fig 1.7.1

pepwpwf

Dpsrw rs

ideal profilealtered profile

For a variety of reasons there is often an annular region of alteredpermeability around the wellboreSince most of the pressure drop in radial flow occurs within the regionfrom r to 100r near wellbore permeability alteration is very importantw w

Formation damage

Near Wellbore Altered Zone

k . . . Altered Zone Permeability r . . . Extent of Alterationp . . . Actual Bottom-hole Pressure p . . . Incremental Pressure Drops swf sD

Dps pwfpw

rw rs

ks peIdeal Pressure ProfileBased on HomogeneousPermeability, k

Actual Pressure ProfileSteepened by ReducedPermeability, k , inAltered Zones

Near Wellbore Altered Zone

Fig 1.7.1

p . . . Mud Hydrostatic Pressure p p . . . Excess Formation Pressure"Supercharging"p p . . . Mud Overbalance p . . . Sandface Pressurep . . . Formation Pressurem sf f

m f sff

-

-

pmDpmc

psfpf

ql

Time

StaticFiltrationDynamicFiltrationSpurtLoss

Mud Filtrate Invasion

Fig 1.7.2

1-Sor

Swc

FLUSHEDZONEql

t1 t2

rw r (t )i 1 r (t )i 2r

Sw

Low Permeability Formation

Qe = Cumulative Fluid Loss Per Unit Height

Saturation Profiles

Fig 1.7.3

r QS Si eor wc= - -pf 1b g

Piston-like displacement with the creation of a flushed zone atresidual oil saturationr (t)i . . . depth of invasion of mud filtrateDepends on porosity and cumulative fluid (mud filtrate) injected

rQS Si

l

or wc

= - -pf 1b gQ = cumulative fluid loss per unit height of formationl

Often is synonomous with - the extent of the altered zoner ri s

Mud fluid loss rate depends on the overbalance and thefiltration properties of the drilling mud

DUE TO VAN EVERDINGEN AND HURST

SK IN

pw pwpw f

pw f

PRESSURE PROFILEIN THE FORMATIONBASED ON UNALTEREDPERMEABILITY k

D

D

Dps

ps

psDpsS q2p km h

INCREMENTAL SKIN PRESSURE DROP(POSITIVE FOR DAMAGE) Fig 1.7.6S . . . Dimensionless Skin Factor

Skin Factor ConceptDue to van Everdingen and Hurst

= Incremental skin pressure drop(Positive for Damage)

Negative Skin Effect. . . Due to Near Wellbore Permeability Improvement i.e. Stimulation

Homogeneous MediumPrediction

PossibleActualProfile

Dps pwpwf. . . is a negative quantityDps

Region ofIncreasedPermeability

"Skin"

S pqkhs= D

mp2

Fig 1.7.7

Deliberate Well Stimulat ionacidisinghydraulic fracturingHigh Shot Density PerforationThermal Fracturing of Injection WellsWell Deviation"Geoskin"

--

High PermeabilityLens Straddlingthe WellboreFormation ofPermeability, k

Reasons for Negative Skin

Fig 1.12.1

pwf

pe

Dpsrerwreff r r ew eff w S

, = -

Effective Well Radius

Alternative Way of Characterising Near Wellbore AlterationParticulary Useful for Negative Skin Situations e.g. fractures

pwf rwreffDps

re

pe

EffectiveWell Radius

pr

rS

r

rDe

w

e

w eff

= + =ln ln,

\ = -S r rw w effln ln ,

r r or Sr

rw eff wS w

w eff,

,

= =-e ln

krs

rw

h

re

ks

Fig 1.7.8k = Bulk Formation Permeabilityk = Altered Zone Permeabilityr = Radius of Altered Zoness

Relates Skin Factor, Sto the IntrinsicProperties of theAltered Zone

Hawkins Equation

Dpq

k h

r

r

q

kh

r

rss

s

w

s

w

= -mp

mp2 2

ln ln

\ = -FHGIKJ

LNM

OQP

Dpq

kh

k

k

r

rss

s

w

mp2

1 ln

Sp

q

kh

k

k

r

rs

s

s

w

= = -FHGIKJ

LNM

OQP

Dm

p2

1 ln

Dp = additional pressure drop over the altered zones

Actual Pressure Dropover Altered Zone Pressure Drop that wouldhave occurred if thePermeability was unaltered

Hawkins Equation (Open-Hole)

1 20 40 60 80 10002468

S

pD

rDk s= k2

Bulk FormationPermeability, k

Damaged Zone

Addition of Incremental Skin Pressure Drop toHomogeneous Radial Flow Prediction

Fig1.7.9

REGION OF FREEGAS SATURATIONWHEREk =kk (s )o ro g

pb

pw frw r

Fig 1.7.10

Gas Block Around an Oil Well where BHP is Below the Bubble Point

- one of the main reasons for pressure maintenanceby water injection

Direct Application of the Hawkins Equation

PARTIAL PENETRATION

PARTIAL COMPLETION

VERTICAL FLOW CONVERGENCEADDITIONAL PRESSURE DROPREQUIRED

Fig 1.9.1

Flow Occurs acrossBedding Planes HenceVertical Permeabilityis Important

Deviation from Pure Radial Flow Due to Limited Entry

OIL BEARINGRESERVOIR

GAS

WATER HIGH WORWELL

HIGH GORWELLMain Reason ForDeliberate LimitedEntry is to AvoidConing

OG CWOC

Fig 1.10.1

WaterandGasConing

Format ion o f Wa ter ConeWhen We ll i s Under la inby Wat er

q

q

OOWC

Fig 1.10.2

Pressure Distribution in Oil Phaseis Little Affected by Presence ofStatic Cone

Partially Penetrating Well

qc

ho pw pehapp + hw w apr p + ghe o apr

Water Hydrostatic Equilibrium

Fig1.10.3

pw pepw - pe -r rg a pg h 0 apgh

Cri t ica l Rate For Gas Free Produc ti on

GAS HYDROSTATIC EQUILIBRIUM

hphap

ho

Fig 1.10.4

Fig 1.9.5

FlowConvergenceintoGroups ofPerforations Form ofLimitedEntrySkinPluggedPerforation

htheor.p

he f f .p

PRESUMED SITUATION

ACTUAL SITUATIONFLOW CONVERGENCE ZONEALTERED ZONE

EFFECTIVE PERFORATEDINTERVAL REVEALEDBY PRODUCTIONLOGGING

DR AWDOWN I SI NC REA SED AN D P . I .R E DU CED BYAD D I T I O NAL F LOWCONV ERG EN CEFig 1.9.8

Effective Penetration Ratio

Top or Bottom Central General Positionh = Formation Heighth = Perforated intervalh = Height of a SymmetryElementps

h h hhshs hs

hphp hphp

Brons and Marting Parameters

Geometry of Limited Entry

Fig1.9.2b hh h kk hrp D vsw= =

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

30252015105

Sp

b

hD = 10000

20100

5

1000

Fig 1.9.3= Penetration Ratio

BronsandMartingCorrelation

Limited EntryGeometricSkin

b

h kk hrD z w=

b hhp=

0

Based on Homogeneous Theoryi.e. uniform horizontal and verticalpermeabilitiesUncertainty in the Value of the PenetrationRatio, bLack of Knowledge of the Effective orAverage (macroscopic) Vertical PermeabilityDoes not Allow for Layering

Limitations of the Brons and Marting Correlation

FLOWCONVERGENCEZONE

FORMATIONPRESSURE PROFILE

NO DAMAGENO CONVERGENCECONVERGENCE WITHNO DAMAGErD

SpSdSa

pD

Fig 1.9.4

Combined Effects of Partial Completion and a Thin Altered Zone

Superposition of Skin EffectsDue to Partial Completionand Damage

Al teredZone

Limited Entry with Formation Damage

Note Augmented Velocitythrough the Damaged Region

hkhp

ks

rw rs Flow Convergence Zone Fig1.9.6

Fig 1.9.6

Limited Entry with Formation Damage

Flow convergence zoneAlteredzone

For no altered zonei.e. k = ka

rw rs

Incremental Pressure Drop over the Altered Zone

Note that h is used in this formulationFluid velocity through the altered zone is controlled by q/hpp

hpks

p pq

kh

r

rSe wf

e

wp- = +F

HGIKJ

mp2

ln

Dpq

k h

r

r

q

kh

r

rds p

s

w p

s

w

= -mp

mp2 2

ln ln

Dpq

kh

h

h

k

k

r

rdp s

s

w

= -FHGIKJ

LNM

OQP

mp2

1 ln

or Dpq

kh

S

k

k

r

r

bd

ds

s

w

mp2

1

= =-FHGIKJ ln

The quantity: k

k

r

rs

s

w

-FHGIKJ1 ln is the intrinsic or true skin factor

Thus: SS

bdtr=

Formula due toRowlandandJones and Watts

characteristic of the altered zone denoted Str

Rearranging thisequation gives:

SS

bdtr=

The effect of damage is enhanced by a limited entry due to theincreased velocity through the altered regionTotal Apparent SkinThe inflow equation including damage effect and geometric skinbecomes:

p pq

kh

r

rS

S

be wfe

wp

tr- = + +FHG

IKJ

mp2

ln

The total skin effect is written as:S S S

S

bSa d p

trp= + = +

Skin Factor for a Damaged Well with a Limited Entry

Limited Entry with Damage (contd)

Formula due toRowland andJones and Watts

4p khm

S = S + S =a d p S t rb + Sp

ETR MTR LTR p*pws

0

SLOPE , m =

ln t +p DDtt

Fig 1.9.7Jones & WattsEquation

Total Apparent Skin from the Intercept of a Horner Plot

S fromBrons & MartingCorrelationp

Hence St r

straight line segment MTRExtrapolatedPressure("Intercept")

h a

a

0-2-4-6

102 103 104

Sswp

hrw

30o45o60o=75o

15o

Fig 1.11.1

Due toCinco &Miller

Deviated Wells

Effect of Deviation i.e. Flow Divergence Expressed as AnotherComponent of the Skin Factor viz. Sswp

h a

For a Completely Perforated Well:

0 < <75a o Due to Cinco and Miller

Deviated (Slant) Wells

S h rswp w= -FHGIKJ - FHG

IKJ

FHGIKJ

a a41 56 1002 06 1 865. . logh

rw

> 40

i.e. b > 1

Fig 1.11.2hcos a

where:b hh

hhp= = =cos cosa

a1

S S S Sb Sa d swp tr swp= + = + S pqk h

kk rrtr ss

sw= = -FHG

IKJ

Dm

p a21

cosln

Combination of Deviated Welland Thin Damaged Zone

Replace S by S in Preceding Formulaed cS S Sc damage perforation= +

- not possible to decompose this term into individual, additive contributionse.g. well with limited entry:

qkh p p

rr

S

e wf

e

wa

= -+F

HGIKJ

2pmb g

ln

where: S S Sa c p= +

S = Combined Skin Effect forAlteration and PerforationS = Combined True Skin Factorc

c,trS

S

bcc tr= ,

For a Perforated Well

h

rex f

Fig 1.12.2xf

Vertical Fracture ofLimited Radial Extent

xf = Fracture Half-Length

Double Wing Fracture

Vertically Fractured Well

h rex fxf

Vertical Fracture of Limited Radial Extent x . . . Fracture Half LengthFracture Height Equal to Formation ThicknessInfinite Conductivity Fracture

f

For Steady-State Flow:Prats, M. SPEJ June 1961 p105

Vertically Fractured Well

rx

providedr

xw efff e

f, = >

22

Fig9.3.6bPseudo-RadialF low

InnerQuasi-SSRegion Radius ofInvestigation

Dotted Lines - FiniteWellbore RadiusRadial FlowRegion ofTransientRadialPressurePropagation

Fig8.3.6b

Fractured Well SSS Productivity IndexJ

kh

Br

r

sss

e

w eff

=-F

HGIKJ

2

34

pm ln

,

rx

w efff

, =2

orJ

kh

Brr

Ssss

e

wpr

=- +F

HGIKJ

2

34

pm ln

Sr

xprw

f

= ln2

This pseudoradialskin is negative upto about -5.5Pseudoradialskin factorNote that any skin on the well before fracturing is bypassed

pq

kh

r

rSe wf

e

w

- = +FHG

IKJ

mp2

ln

pp p

qkh

r SDee wf

De= - = +mp2

ln

Jq

p p

kh

Brr

SSS

s

e wf e

w

= - =+F

HGIKJ

2pm ln

Inflow Equations Including Skin EffectSteady-State Radial Flow

. . . skin factor is added to pure radial flow term i.e. ln(r /r )e w

Steady-State Productivity IndexSkin is important if S is comparable to ln(r /r ) which istypically of the order of 7 - 8Hence skin factors greater than about 3 are seriously reducing PI

e w

p

pq

kh

r

rSwf

e

w

- = - +FHG

IKJ

mp2

3

4ln

pp p

qkh

r SDwf

De= - = - +mp2

3

4ln

Jq

p p

kh

Brr

SSSS

s

wf e

w

= - =- +F

HGIKJ

2

34

pm ln

Semi-Steady-State (SSS) Radial Flow. . . based on average pressure of the drainage area

or

SSS ProductivityIndex

p

J =SSS 2p

2B 1 ln

PERMEABILITY - THICKNESS PRODUCT

OILVISCOSITYWELLSPACING

DRAINAGEAREA SHAPE

WELLBOREDAMAGE

WELLDIAMETER

kh

1

m S

432

4gACa rw2

56

Well Productivity Depends on: . . . using Dietz shape factor

SSSPI

Generalised Formulation

Fig1.8.1

prpwf

VLP

dqs dqs

High kh Well(Tubing Control)

Low kh Well(Formation Control)

qsIPR+S IPR-S

Skin Removal Workover on Well Performance Diagram

Fig 1.8.2

CONFINEDPRODUCERS

PRODUCTIONWELL

INJECTIONWELL

7062575451

504946433830

Steady-State, HomogeneousEqui-Pressure Contours andStreamlines in the Quadrantof a Five-Spot Element

Five SpotFlooding Pattern

Fig 1.8.3

k h2 2

k h3 3

pepeq 1q2q 3

pw

Strat i fied Reservoir

Fig1.13.1

Steady-State, Single-Phase FlowNon-Communicating, Homogeneous LayersLayer Skin Factors ZeroCommon External Pressure, peq

k hrr

p pii i

e

w

e w= -2pm lnb g

\ = = -=

=å åq q

k h

rr

p pii

N i ii

N

e

w

e w1

1

2p

m lnb g

since p , p , r and are common to all layerse w e m

Layered System BehaviourIndividual Layer Rate

Summation to give total flow:

q

Fig 1.13.2

peq1q2q3

qk h1 1

k h2 2

k h3 3

Layer 1Layer 2Layer 3

S1

S2

S3

q = qS i

pw

Common WellborePressure, pw

Reservoir Communication

Layered System

Fig 1.15.2

LAYER i

LAYER N k hN N

k hi i

k h1 1

Fig1.13.3

Perfect Layered System

Arithmetic Average PermeabilityApplicable to Perfect Layered or Stratified Systems with a CommonPressure on Each Flow Face

Each Individual Layer is Homogeneous and of Constant Thickness

kk h

h

k h

h

i ii

N

ii

N

i ii

N

= ==

=

=åå

å1

1

1

k2k3

k i+1k ik i - 1

kNkN-1kN-2Fig1.13.5

Geometric Mean Permeability

k , i = 1 . . . N Randomly Distributedi

p1 p2

Applicable to Systems with Random Permeability Distributions

Biased to Lower Permeabilities Usually Used with Cut-offk k k k k ki N N

N= ´ ´ ´ ´ ´ ´-1 2 1

1... ...

/b g

Fig1.15.5

. . . Plot of Frequency versus Log(k)Fig 1.13.4

Probability Distribution Functions (PDF)orCore Permeability Frequency Distributions

Frequ-ency

May exhibit a Bell Shape i.e. distribution may be log normalMedian Value of a Log Normal Distribution is the Geometric Mean

Log k

Fig1.15.4

WestSeminoleSan AndresUnit(Exxon)Well Falloff Core kh Core khTest Arithmetic Geometrickh Mean Mean(md.ft) (md.ft) (md.ft)305W306W307W609W610W611W707W

109410085331306944599889

9107326371008446467868

242312355265193197335Harpole SPE 8274 Fall Mtg. Las Vegas 1979

Comparison of Pressure Transient khwith kh Derived from Core Data

Geometric Average

. . . Random Distribution of PermeabilityArithmetic Average

. . . Horizontal FlowPerfect Layering

Permeability Averages

k kGi

N

i

N

= FH IK=P

1

1/

kk

NA

ii

N

= =å

1

. . . Vertical FlowPerfect Layeringk and k are Upper and Lower LimitsRespectively of Average PermeabilityA H

. . . In Series Vertical FlowHarmonic Average

k Nk

Hii

N=

=å 11

Gas Permeability - 1Gas Permeability 1

Klinkenburgh, non-zero velocity at pore walls.

Slippage of gas molecules along the solid grain whenalong the solid grain when the pores diameter is in the range of the gas free path.A function of pressure poreA function of pressure, pore size and gas type (smaller the molecules, larger effect). Liquid permeability

0 reciprocal mean pressure0 reciprocal mean pressuremean pressure infinity

Pmkk LG +=

Gas Permeability - 2Gas Permeability 2

Compressibility.Use of Boyl’s law (P1V1=P2V2=>ΔP2/(2Pb) instead of ΔP).

2Inertia, Non-Darcy, Nonlinear, ΔP=(μ/k)V+βρV2.Low viscosity gas dictates higher velocities for same ΔP.If Darcy law is used k decreases at higher velocitiesIf Darcy law is used, k decreases at higher velocities.

WELL A, Upper Jurassic

10

10

10

10

10

10 10 10 10 10

3

2

1

0

-1

-1 0 1 2 3

Air Permeability, K [md]a

BrinePermeabili tyK [md]b

TYPICAL RELATIONSHIP

BETWEEN AIR PERMEABILITY

AND IN-SITU BRINE

PERMEABILITY

(After Juhasz)Fig 1.15.6

Single-Phase Creeping/Inertial FlowSingle-Phase Creeping/Inertial Flow

Darcy Equation.

vkdx

dP μ=−

Forchheimer Equation for dry gas.

2vvkdx

dPβρ+

μ=−

dpdr k

u ur r= +m br 2

Reynold's Number for Porous Mediainertial term is important only if

is comparable tobr mu /kr

Re .= = ³brm

brm

u

k

k ur r 0 1

only ever t rue near the wellboreratio of inertial to viscous forces

Forcheimer Equation

Single-Phase Non-Darcy Flow

2βρvvdP+

μForcheimer Equation.

Single Phase Non Darcy Flow

βρvvkdx

+μ

=q

C ti l ti & I l h ti flConvective acceleration & deceleration of fluid particles

Irregular, chaotic flow of fluid, Turbulent

Laboratory, Field β Measurements or calculated from correlations.

Laminar Creeping (Darcy) Flow

Reynolds Experiment

Laminar Flow

Flow in a Synthetic Porous Medium (Micromodel)

E. Sketne

Re D vp p= <rm 2100 D

DpL av= Hagen-PoiseuilleEquation

Re D vp p= £rm 1 D

DpL av=

DDpL av bv= + 2

Darcy’sLaw

Fig 1.2.2

Single-Phase β MeasurementsSingle Phase β Measurements

β is a fundamental rock property.β p p y

Core laboratory.

Available correlations.

Field, open hole with homogenous porous medium.

– Field β with any non-uniformity in flow, different from lab data.

Laboratory β MeasurementLaboratory β Measurement

Core flow at incremental flow rates.

1W)AP(PM 22

21W +β=−Real gas law & Forchiemer Eq.

kAWL2zRT+β

μ=

μz & μ=f(P), negligible.

i l f 1β is slope of y vs. x.k1xY +β=

Laboratory β MeasurementLaboratory β Measurement

2.7E+12Clashach Core, Swi=0%, k=553 mD

2 3E+12

2.5E+12

y = 1.035E+08x + 1.699E+122.1E+12

2.3E+12

Y /m

-2

1.7E+12

1.9E+12

0 2000 4000 6000 8000 10000

β /m-1

0 2000 4000 6000 8000 10000x /m-1

β From Correlationsβ From Correlations

There are numerous correlations in the literature.

First correlation, Janicek and Katz (1955)k in md and β in (1/cm). 4345

8

k1082.1φ×

=β

Most widely used, Geertsma (1974).k in m2 and β in (1/ft). 5.50.5k

0.005φ

=βk in m and β in (1/ft). k φ

Katz and Firoozabadi another popular one.,k in mD and β in (1/ft).

201.1

1010*33.2k

=β

Field β MeasurementField β Measurement

scT DQSS +=High velocity an additional skin.

Variable rate test, essential.

scT DQSS +g y

Stabilized or Transient.

– e.g., Isochronal test or Step rate transient.⎟⎟⎞

⎜⎜⎛⎟⎟⎞

⎜⎜⎛ PkMD scwβ

⎟⎟⎠

⎜⎜⎝ π⎟⎟⎠

⎜⎜⎝ μ

=2RThr

Dsc

sc

w

wβ

D from slope of Q vs. ST.

β from Dβ from D.

Field β Open Hole or Perforated WellField β, Open Hole or Perforated WellscT DQSS +=

80Clashach, Swi=0, 90 degree phasing

Lp

/inβ*1E-8

/m-1

(from Slope)

S

(intercept)50

60

70

(ST)

36912

Lp /inch

)

369

12.7653.9702.044

1.870.600.2920

30

40

Tot

al S

kin 15

Open HoleLi (3)

1215

1.3380.952

-0.03-0.06

OH 1.012 0.02-10

0

10

0 0 5 1 1 5 2 2 5 3 3 5

h=1 ft,4 SPF, Swi=0, β(core)=1.035E8 m-1

0 0.5 1 1.5 2 2.5 3 3.5Q /MMSCFD

IPR

pwf

qsDpND

pe

OperatingPointpwhslope = A-

Dp = BqND s2

Quadratic IPR for an Oil Well Exhibiting Non-Darcy Flow

Well Performance Diagram

VLP

Fig 1.16.2

rs

Ideal Pressure Profile(No damage or non-Darcy flow)

DpsDDpsND

DamagedRegionks k(unaltered formationpermeability)

Pressure Profile in Damaged Regionwith no non-Darcy flowPressure Profile in Damaged Regionincluding non-Darcy effect

Influence of Damaged Zone Including Non-Darcy Flow

rw Fig 1.16.3