radial flow chap1v1

1

DownholeMemory

SurfaceRecording

Casing

Tubing

Testing Valve(operated by

annulus pressure)

Packer(set by weight on

string)

PressureTransducer

Tailpipe

Figure 2.1.1

Drillstem

Testing

Assembly

Fig 1.1.1

Well Test Surface Hardware

Choke

Gas

Oil

Surface Choke providesRate Control

Orifice PlateFlow Measurement

qo

Q

Test Rate Limited bySeparator Capacity

Fig 2.1.1b

Test Separator

Fig 1.1.1b

2

InitialFlow

In i t i alS hu t in

Afterflow

FinalShutin

Final Flow

Time

Prod.Rate

Initial Res. Pressure

Drawdown BuildupBHP

Time Figure 2.1.2

Dual Flow - Dual Shutin Test

Fig 1.1.2

Fig 2.1.3

4 k hπ

q Bsµ

Transient Well Testing

Buildup Analysis - Horner (Theis) Plot

Semilog Analysisln

t + tp ∆∆t

BuildupAffected

byWellboreStorage Intercept

givesskin

factorS

Affectedby

Boundaries

ETR MTR LTR

slope = −

p*pws

Nomenclature due to

D. Pozzi (EPR)

Fig 1.1.30

3

alteredzone

pw

pw f

∆ps

UnalteredPermeability

k

rw rs re

wellboreradius

altered zoneradius

externalradius

∆p = Incremental Skin Pressure Drops

Fig2.1.4a

ks

Near Wellbore Altered Zone Fig 1.1.4a

Fig 2.1.4b

n = Number of perforations per foot

l = Length of penetration

= Phase angle

s

p

θ rw rs

ks

lp

Altered Zone

k

Combined SkinDamage + Perforation

Perforated Completion

S f n l k r ks p s s= , , , , ,θe j

Fig 1.1.4b

4

E T R M T R L T R

slope

p*

q µ4 khπ

m = −

l n t + tsi a ∆∆t

p

pws

∆pMBH

Fig

2.1.5Horner

Plot

0

Determination of Average Pressure

Fig 1.1.5

WELL PRESSURESTARTS TO BEAFFECTED BYBOUNDARIES

MTR LTR

TRANSIENTI.A. FLOW TRANSITION

LATETRANSIENT

SEMI-STEADY-STATEFLOW

pi

pwf

0 TIME

SCHEMATIC PLOT OF PRESSURE DECLINE AT

PRODUCING WELL

CONSTANT RATE WELL BOUNDED RESERVOIR

Figure 2.1.6

d pd t

q Bc r h

s

t e

= −φ π 2

Cartesian Plot Flow Regimes Fig 1.1.6

5

MTR LTR

lnt + tp ∆∆t

depletion

buildupdrawdown

HornerPlot

pws

p*p**

Closed "tank" ofpore volume, V

Figure 2.1.7

pi

0 Time, t q

Detection of Depletion

Fig 1.1.7

Skinre

k

Homogeneous Finite Reservoir

d d

Well Image

No FlowBoundary

k1

k2

r1

Composite InfiniteReservoir

d2

d1

Single Linear Fault Multiple Faults

µ2

µ1

θ

Some Well Test Models

Fig 1.1.8

6

T(t=0)

x

x

Heated Bar

Thermocouples in thermowells

Transient HeatConduction Equation

Inner B.C.

i.e.

. . . specified gradient(2 kind BC)nd

Linear Flow

t

Penetration Depth

Initial Temp.

T

FaceTemp.

Dynamic Temperature Distribution

Fig 2.1.9

∂∂ ρ

∂∂

Tt

kC

Txp

=2

2

qA

k Tx

x

= −=

∂∂

0

∂∂Tx

qkA

x=

= −0

Constant heat flux

Fig 1.1.9

h

re

rw

qWell in theCentre of a

CircularReservoir

RadialFlow

Fig 2.2.1

Model Reservoir

Fig 1.2.1

7

ln r

t

p(r ,t)

pi

Well-Bore

Trans ien t Deve lopment o fthe Format ion Pressure

Dis t r ibu t i on

Fig 2.2.2Fig 1.2.2q

104 105 106 107 108 109

10

1

10-1

10-2

10-1 1 10 102 103 104

pD

t /rD D2

Figure 2.2.3Single Well in an Infinite Reservoir (No Skin)

Exponential Integral Solution

Fig 1.2.3

8

∆ps

pw

pwf

"SKIN"

tPRESSURE PROFILEIN THE FORMATION

rw

NEGATIVE SKINFACTOR

i.e. STIMULATION

−∆p s

pwf

pw

k > ka

PRESSURE PROFILEIN FORMATION

RIGOROUS SKINCONCEPT PROFILE

t

STIMULATED ZONE

Figure 2.2.4

DimensionlessSkin

S pq B

kh

s

s

=∆

µπ2

Positive Skin Factor

i.e. Damage

Fig 1.2.4

C R Dpi

0

qpw f

TIME, t0

CARTESIAN PLOT

INTERCEPT

SLOPE, m

ln t0

pw f

SEMILOG PLOT

Figure 2.2.5

= −qµ

4 khπ

p (t=1)wf

Rate Schedule

Ideal (CSFR) Drawdown

Fig 1.2.5

9

104

105 5x105

106

2 x10 6

t = 3x10D6

0

1

2

3

4

5

6

7

8

91 200 400 600 800 1000

rD

pD

r =10De3

Figure2.2.6

Dimensionless Pressure Distributions in Radial Flow

SSS

I A

Well in aClosed

Reservoir

Fig 1.2.6

HENCE E (x) IS DENOTED -Ei(-x)1

1

2

3

0.4 1.0 1.6

E (x)1

-Ei(-x)

0

x Fig 2.2.7

Ei x eu

u

x

e j = −−

−

∞z

Exponential Integral Function

Fig 1.2.7

10

103 5x103 t =10D4

1 100 200rD

PRESSUREDISTURBANCE

FRONT Fig 2.2.8

Radius of Influence

p = 0.1D

Fig 1.2.8

102

q

ACTIVEWELL

OBSERVATIONWELL

MINIMUM OBSERVABLE pDEPENDS ON GAUGE RESOLUTION

∆

rD

"ARBITRARY"CRITERION

Ei SOLUTION

pi

pwo

OBSWELL

PRESSURE

0 t Fig 2.2.9

p Ei rtDD

D

=FHGIKJ

12 4

2

p p khqD = =

∆ 2 0 1πµ

.

Depth of

Investigation

Fig 1.2.9

11

1

2

3

4

5

6

pD

0 1 2 3 4 5 6 7ln rD

Radiusof

Drainage

rD

Fig 2.2.10

I.-A. TransientPressure Profile

at t = 10D

5

Steady-StatePressure Profile

for Same p (1,t )D D

rDd rD i

r = classical depth of investigation

D i

Fig 1.2.10

t = 0.3De

r = 1000De

4

2

0

t = 10D t = 25D

pwD

ln tD

1 10 102 103 104 105 106 0 2 x 103 4 x 103 6 x 103 8 x 103 10 x 103

0

1

2

3

4

5

pwD

PRODUCTION

SHUT-IN0

q

0 tD

0 2 4 6 8 10 12 14

6

8

10

Cartesian Graph

tD tD

Semilog Plot

Pressure Drawdown at the Wellbore Fig 1.2.11

12

rD = 1.0

2.0

1.2

20

EXPONENTIAL INTEGRALSOLUTION

10-2 10-1 1 10 102 10310-2

10-1

1

10

pD

t /rD2D

Finite Wellbore Radius Solution

Finite Wellbore Radius (FWR) Solution

Fig 1.2.12

pp

e

pw f

StabilisedPressure

Distribution

SSS Depletion Wel l in Cent re o f a

Closed Circu lar Reservoi r

rerw

Fig 2.2.13Por e Vo lume = hAφ

qs

Fig 1.2.13

13

TIME, t

Pressure Drawdown Testing

RATEq

0

0

SHUT-IN

PRODUCING

TIME, t

p = pws i

0 Fig2.3.1

Bottom Hole

Pressurepwf Fig

1.3.1

0

pt=1

ln tNOTE : ln t = 0 corresponds to t = 1

Fig 2.3.2

Deviation from straight linecaused by damage andwellbore storage effects

slope, m = − 4 khπq Bs µ

Drawdown Semilog Plot

Bottom Hole

Pressurepwf

Fig 1.3.2

14

•o

Well 1q

1

Well 2q

2

r1

r2

Well 3

Observation Well

Active Well

Three Well System Figure 2.4.1

Principle of Superposition

Fig 1.4.1

q2

q

t T1

q - q2 1

Two-Rate Flow Schedule

Fig 2.4.2

Inject ion wellrate q q

2 1−

Production wellrate q

1

Superposition of RatesFig

1.4.2

q1

0

15

Total Response

ExtrapolatedPressure

Principle of Superposition

Injection Wellat Rate q - q2 1

Well atRate q1

0 T1 t

0

p pi w−

∆pDD

∆pDD

Fig 1.4.3

RATE

t p ∆

∆

t

t

FLOWING

SHUT-IN

t p

BHP

pws

p ( t=0)wf∆

Figure2.5.1

Schematic Flow-Rate and Pressure Behaviour for an Ideal Buildup

q

Fig 1.5.1a

16

pws

ln t + tp ∆∆ t

p*

Deviation from StraightLine caused by

Afterflow and Skin

0

slope, m = −4 π k hq Bs

µ

Semilog (Horner) Plot for a Buildup

Fig 2.5.1bFig 1.5.1b

rD0

5

1 200

q

0tpD

∆ t D

t = 10p D

4

Pressure Build-Up in a Reservoir

Figure 2.5.2

t = 10D4

∆

10

50

200

2 10 3

psD

Fig 1.5.2

17

0

40 7

ln ∆∆

tD

tDt +pD0

5

ps D

∆ t D

10 103

××10 3 5 10 3

t = 10p D4

Pressure Build-Up

at Wellbore

Semilog

Cartesian

DimensionlessResponse

(Horner)

Fig 1.5.3

psD

t + tp ∆∆t

135

2

6

∆tD

t = 10pD

4

∆t < 10D

Ei Function notRepresented by

Log Approximation

Fig 2.5.4

pD

Dimensionless Build-up Semilog (Horner) Plot

Fig 1.5.4

18

t + tp ∆∆t

Effect of Afterflow on a Horner Plot

Data Affectedby Wellbore

Storage

Correct SemilogStraight Line

pws

0

p*

slope m

Fig 2.5.5

ETR MTR

Fig 1.5.5

1

1000

0.001 100t

∆p(psi)

UnitSlope

Da ta o f Co rr ec tS emi log S lo pe

(hr)Fig 2.5.6

MTR

Log - Log Diagnostic Plot for Afterflow

Fig 1.5.6

19

Logt + tp ∆∆t

xx

xx

x x x x x x x

p*p1 hr

slope m

0

Determination of p on the Horner Plot1hr

Fig 2.5.7

MTRStraight

Line

∆t = 1 hr

S p pm

tt

kc r

wf hr p

p t w

=−

++

− +LNMM

OQPP11513

13 22751

2. .log log

φµ

pws

Fig 1.5.7

Determine and

very accurately

p ( t=0 t( t=0)wf ∆ ∆)

+

t( t=0)∆p ( t=0)

wf∆

End ofDrawdown

Buildup

∆t

Stabilise flow-rate before shutin

q

Q = cumulative volume

Flow-Rate

∆t

Shutin

Afterflow∆ − ∆p = p p ( t=0)BU ws wf

∆ − ∆t = t t( t=0) t =pQq

Test Precautions

Fig 1.5.8

20

Time

Pressure Flow period

ReservoirDisturbance

Shut-in periodReservoirRecovery

Horner Plot

Largem

Smallm

Samepr

pws

Log t + tp ∆∆t

Small ReservoirDisturbance

Low flow-rateSmall viscosityHigh permeability

Large ReservoirDisturbance

High flow-rateViscous fluidLow permeability

Permeability of Reservoir Rock from a DST

After Matthews and RussellLarge mSmall m

Scribed Tin Chart from Amerada Gauge

Fig 1.5.9

3424

3423

3422

pws

8 7 6 5 4

Horner Plot

Early Piper Well(HP Gauge) slope

m = 0.7465 psi−

kh = 1.067*10 md.ftS = 3.08

6

q = 11750 bbl/d

B = 1.28 = 0.75 cpr = 0.362 ft = 0.237

c = 1.234*10 psi

s

w

t

µ

φ-5 -1

(psia)

Fig 2.5.10 lnt t

tp + ∆

∆

3425

Fig 1.5.10

21

0tp

∆t

Pressure Build-Up in a Reservoir

∆ t 1

∆ t 2

∆ t 3

∆ t 4

∆ t 5

p (t ,r )r p p1

p (t ,r )r p p2

p (t ,r )r p p3

p (t ,r )r p p4

p (t ,r )r p p5

pr

rrw

Reservoir pressure distributionat moment of shut-in, p (t )r p

Peaceman ProbeRadius Concept

Fig 2.5.11

p pw r r rw

==

Fig 1.5.11

pwf

pi

Time, t

tp2 tp1tp1 tp3

tp3

tp4

Transient Productivity Index, Jt

p (t )wf p3

pws

or

J is strongly time dependentt

pwf

J qp p tt

s

i wf p

=− d i

Fig 1.5.12