mscrem3

SYZ1 Petroleum Reservoir Monitoring and Testing

MSc REM

Reservoir Evaluation and Management

Wellbore Storage, Radial Derivatives and Type Curve Analysis


Well Test Analysis Summary


ln

t

tt +p

L T R M T R E T R

Semilog StraightLine of Ideal Slope

m = 4khq

StimulatedConstant Pressure

Boundary Effector

Closed System

Faults

WellboreStorage and Skin

Fig 3.1.1a

Horner Pressure Buildup Graphs


Flow periodAnalysis

Model BestMatch

DataPreparation

ContextInput

STOP

Yes

No

Reasonable

?

Well Test Analysis Algorithm

Log-LogDerivativeDiagnostic

SpecialistPlots

NonlinearRegression

Fig 3.1.1b


High Resolution Gauges

Preprocessors with Filters

Derivative Computation Methods"Flopetrol Algorithm"

Spectrum of Well Documented Basic Model Responses

Stehfest Algorithm for Numerical Laplace Transform Inversion

Advent of PC with Interactive Graphics

Development of Methodology


New Time Functions and Derivatives

Deconvolution

Efficient Computation Methods

Multiwell Capability

Combined Solution Forms

Reservoir and Boundary Modelling

Horizontal Well Models

Variable Wellbore Storage

Handling of Multiphase Flow

Recent Improvements


Better Multiphase Flow Capability

Clarify Effects of Spatial Heterogeneity

Possible Extension to Reservoir

Characterisation

Analysis of Production Data

Downhole Flow Measurement

Sophist icated Help Systems

in Software

Future Needs


Limitations of Well Test Interpretation

Problem of nonuniqueness in model response andparameter estimation

Inability to demonstrate the presence of layering Calculated skin factor, S, cannot be decomposed

into a depth of damage, ra , and an alteredpermeability, ka

Poor quality of rate measurement Failure of error estimates in nonlinear regression

procedures

Testing time too short in tight reservoirs


Early Time Pressure ResponseAnd Wellbore Storage


Physical Reasons for ETR

Radial Composite - Injection Well Near Wellbore Region belowSaturation Pressure: Bubble Point - Gas Block Dew Point - Liquid Dropout

or


Reservoir atPressure

pi

WellFlowing

at Surface

q s

t = 0

t > 0

p = pw s i

p < pw f i

q s f

Wellbore of Volume VFilled with a Liquid of

Compressibility c

Pressure Transducer

Fig 3.1.2

Wellbore Storage Effect


Well is considered to be a tank of volume, V ,fi lled with a fluid of compressibility, c

Capacity , C = cVsCs = Wellbore storage coefficient

Wellbore

CapacityCs

Pressurep

w

qs f

qsqs

qs f

t

FLOW

Surface Rate , , is assumed constantand

the Wellbore Storage Coefficient, , is alsoassumed constant

qs

Cs

Simplified Model of Liquid Wellbore Storage


CapacityCs

Pressurep

w

Fig 3.1.3

FLOW

t

qs

qs

qs f

qs f

Wellbore Storage Coefficient, C = cVs

q = Sandface Flow-Rate

q = Surface Flow-Ratesf

s

0

Constant Surface Rate Drawdown

qsf

qs

Well-bore

P L T


GAS

OIL

WELL

HIGHLY NONIDEAL

WELLBORE STORAGE

SITUATION

WELLBORE

PHASE

REDISTRIBUTION

IN BUILDUP

* HIGH GOR

* LOW WELLHEAD PRESSURE

ANNULARFLOW

FROTHFLOW

SLUG FLOW

BUBBLE FLOW

SINGLE PHASE

Fig 3.1.4


Rate of

Input Output = Accumulation

q q B C d pd tsf s sw =

where

q r kh prsfw

r rw

==

2

p p q kh S pw wfsf

s = =2 Hence

2

r kh pr q B C

pt

w

r rs s

r rw w= = =

Wellbore Storage Inner Boundary Condition

Wellbore Material Balance

- Darcy Law


At very early time the flow, , f rom the

formation is negligible and all the surface

production is sustained by the expansion

of the wellbore f luid

qsf

Thus q 0s f

Wellbore Storage (WBS) Dominated Flow Regimeq

s

Capacity

PressureC

p

s

wf

d pd t

q BC

wf s

s=

Plot of versus time is linear with a slope

-q B/C

pw f

s s

Hence actual wellbore storage coefficient , ,may be determined

Cs

Early Time Behaviour of a Well with Storage


Wellbore Storage (WBS) Dominated Flow

Time, t

Time, t

pwf

pi

slope = Cs

q Bs

WBS Dom.WBS Affected

qs f

qs

0

qsf negligibleWBS Dominated

High Sampling RateGauge often Necessary

to See this Regime

CartesianGraph

Flow-Rate


0 1 20

1

2

Log p

Log t

pM

t M

Determination of C from a Log-Log Plot

of p - p versus ts

i w f

Choose anymatch point onunit-slope l ine

Line of unit slope

WBSdom.

In field units

p q BC tM s s M= 24

q ... STB/D

p ... psi

t ... hr

C ... bbl/psi

s

M

M

s

Hence compute Cs

C Chc rDs

t w= 561462 2

. and


d pd t C i e p

tC

D

D DD

D

D= =1 . .

. . . s i n c e p = 0 w h e n t = 0D D

C Ch c rDs

t w= 2 2Here

Alternatively

log p = log t - log CD D D

10

1

0.110 2 10 3 10 4

pD

tD

C = 10D2 103 104 105

Log-Log

D iagnost ic

In Terms of Dimensionless Variables


104

10 510 3

10

1

C = 10D2

0.1

Lines ofUnit Slope

103

10 4 10 5

t D

t Dp

D

pD CD

=

Fig 3.1.6

Straight Lines of Unit Slope on a Dimensionless Log-Log Plot


Log

p

t-1 0 1 2

0

1

2

[ t ]M

[ p]M

Match Point

Line of Unit Slope

Fig 3.1.7

Determination of C from a Log-Log Plot of p versus ts

WBSd om .


10

1

0.1102 10

810610 4 tD

pD

C =

10

D

5

C =

10

D

4

C =

10D

3

201050

SC = 0D

C =

10

D

2

1 log cycles12

Ramey Log-Log Type Curve

Fig 3.1.8


10

1

0.1102 10

8106

104 tD

pD

C =

10

D

4

C =

10D

3

201050

SC = 0D

C =

10

D

2

Ramey Log-Log Type Curve

1000.1 1.0 100.01

t (hr)

0.1

1.0

10

p (psi)

Fig 3.2.1bType Curve Matching Process

Log-Log Data Plot


100

pD

0.1 104

10 t DCD

Ideal Wellbore Storage Log-Log Type Curve

CDe2SParameter =

CRD

Fig 3.2.3Earlougher-Gringarten Form


10

1

0.1

p(psi)

0.01 0.1 1 10

Fig 3.2.4at (hr)CRD

Measured Data on a Log - Log Plot

CompatibleScale to

Type Curve

p = p p (t)i wf


100

pD

0.1 104

10

CDe2S

Parameter

CRD

Type Curve Matching by Overlay of Measured Data

0.01 0.1 1 10t (hr)

Log - Log Plot of Measured Data Gringarten

Type Curve

p(psi)

10

1

100 Fig3.2.4b


Pressure Match . . . Field Units

[ ] [ ]p k h pqD M M= 28871 .[ ]

[ ]khq p

p md ftD M

M= 88722

..

Time Match

[ ] [ ]tC k h tCDD M Ms= 0 000295. [ ][ ]C k h ttC bbl psi C Ch c rs MDD M D st w= =

0 00295 561462 2

. / .

Parameter Match

[ ] [ ]C e S C eCD S M DS

M

D

221

2=

ln

Type Curve Match Evaluation


Classification of Wellbore Response

WellboreStorage

SemilogStraight

Line

Late TimeBoundary

Effects

Slope

Log tFig 3.3.1

p

0


Ln p

Ln t

Ln t

pp =

d( p)d( t)ln

Fig 3.3.2

. . . Local slope ofsemilog graph

Tangents to Curve(Obtained by Numerical

Differentiation)

Plateau


1001

100

10

1

p

t

10

10.5

0.1

0.01

pD

0.01 1 100t /CD D Fig 3.3.9

Derivative Plateau Match

Equivalent to Fitting Semilog Straight Line Slope


Id e a l We llb o re Sto ra g e a nd Skin

0.10.1

1

10

100

1 10 100 103 104

pD

pD

tD CDFig 3.3.10

0.5

Parameter = CDe2S

After Bourdet


10

0.01 0.1 1 10 100

pp

(psi)

t (hr)

Pressure and Logarithmic DerivativeTime Record

on TC Compatible Scale

Fig 3.3.11


0.10.1

1

10

100

1 10 100 103 104

pD

pD

tD CD

Fig 3.3.12

0.5

Parameter = CDe2S

100

10

0.01 0.1 1 10 100

pp

t (hr)1


pBU

x

x

x

x p ( t)

w f

e x

p

t

Desuperposition

Time Scale Functioning

t

t e

Fig 3.4.8Build-up Analysis


0.1

1

10

pBU

(psi)

0.001 100.1Fig 3.4.2te

te

(hr)

=t tp

t + tp

Log-Log Plot Based on Agarwal Equivalent Drawdown Time


1.0

10

100

0.1 1 10

0.1 1

100 103

10310 1000.01

0.1

1

10

pD

pBU

(psi)

t /CD D

te(hr)

+

MatchPoint

CDe2S

Match of Buildup to Drawdown Type CurveUsing Agarwal Equivalent Time, t e

Fig 3.4.3


t t tt tt

tt

ep

p

p

p= + = +1

Hence

t t te p asDrawdown response can only be

defined up to

Must be careful to use correct

t

t

p

p

Limit of Agarwal Equivalent Time,te


t Tt T T

p

p

=

3

3 2Short Shut-in

to Run Gauge

Flow Periodto Re-establish

Conditions

T1 T20 T3

Improper Selection of tp

q

pw

Time, t

FinalBuildup

Fig 3.4.7


Fig 3.5.1a

0.01 0.1 1.0t e (hr)10

100

1000

pp

(psi)

Equivalent Drawdown Time

Log - Log Diagnostic Plot

Unit Slope Line Nonideal Wellbore Storage

p

C = 0.015 bbl/psis


100.1 1 10

1000

100

pp

(psi)

t e

Type Curve Overlay

(log - Log)

Nonideal Wellbore Storage

IdealT. C.

DecreasingWBS

Fig 3.5.1b

D P

C = 3873D

t = 4.0415 hrp(hr)


1200

2730

pw s

(psia)

25

Cartesian Plot of Build-up

tElapsed time ,Fig 3.5.1cNonideal WBS

(hr)


LTRETR

WBSA

WBSD

Derivative Data

Liquid Solution based on WBSDWellbore Storage Coefficient

(US Construction)

Area of Zero or Low Weightingin Sum of Squares

p

tWeighting of Data Points in Objective Function

Log-Log Diagnostic Plot

Fig 3.5.2


BubbleSlip

Velocityvs

GasCushion

LiquidColumn

Steady-StateFlowing Condition

Segregated

Phases

Drift Flux ModelFig3.5.7


t0

0p

B U

pB U

pH

p

p

- ve

Gas Phase Redistribution or "Humping"

= p - p (t )w s w f p

=d td p

B U

Cartesian Plot

Fig 3.5.8


slope =q

4 khpw s

ln t + tp t

Apparent ParallelStraight Lines

Fig 3.5.9

p

te

DerivativeFingerprint

U S DP

MTR

Nonideal WBS Having Appearence of Dual Porosity


Compressibility of Dry Gas as a Function of Pressure

0

5

10

15

20

25

0 1000 2000 3000 4000 5000 6000

Pressure (psia)

C

o

m

p

r

e

s

s

i

b

i

l

i

t

y

(

1

/

p

s

i

*

1

0

4

)

Gamma = 0.65T = 200 deg F

Fig 3.5.10


p

p

t e

t e

Increasing StorageC > 0

Decreasing StorageC < 0

IdealBehaviour

Cs

IdealBehaviour

Cs

IdealBehaviour

C

IdealBehaviour

C

U SC

U SC

Hump due toC

Log -Log Plot

Fig 3.5.11


1600

1400

1400

1200

1200

1000

1000

800

800

0 1 2 3 4 5 6

0 5 10 15 20

Time (hours)

Time (hours)

With Down-hole Shut-in

Without Down-hole Shut-in

P

r

e

s

s

u

r

e

(

p

s

i

a

)

P

r

e

s

s

u

r

e

(

p

s

i

a

)

Annular Communication orLeaking Gas Lift Mandrels

Fig 3.5.19


1200

1000

800

6000 2 4 6 8 10

0 2 4 6 8 10 12

1400

1200

1000

800

600



Time (hours)

Time (hours)

P

r

e

s

s

u

r

e

(

p

s

i

a

)

P

r

e

s

s

u

r

e

(

p

s

i

a

)

Phase Redistribution Combinedwith Annular Communication

Fig 3.5.20


Liquid Fall-back Causing Humping



1400

1400

1200

1200

1000

1000

800

800

600

600

0 1 2 3 4 5 6 7

0 2 4 6 8 10 12 14

Time (hours)

Time (hours)

P

r

e

s

s

u

r

e

(

p

s

i

a

)

P

r

e

s

s

u

r

e

(

p

s

i

a

)


10

20

20

30

30

40

40

50

50

2

1

0

50000

45000

40000

35000

Gas Rate

Q

(sm /d*10 )3 -6

Time, (hr)t

pw

(kPa)

Test Sequence for Well A

BHP

2 BUnd2 DDnd

After Larsen et al

After Larsen et al

t

Fig 3.6.1


2 4 6 8 10

120

100

80

60

Hyperbolic Function for Skin Behaviour During First Drawdown in Well A

Fig 3.6.3

a = 10b = 0.16c = 67.0

Flowing Time, (hr)t

SkinFactor

S

After Larsen et al


2 4 6 8 1075

80

85

a = 6.5b = 0.55c = 76.0

Hyperbolic Function for Skin BehaviourDuring Second Drawdown in Well A

After Larsen et al

SkinFactor

S

Flowing Time, (hr)t Fig 3.6.4


0

20

20

40

40

60

60

80

80

100

100

120

120

1000

500

0

40500

40000

39500

OilRate

qs

(sm /d)3

Time, (hr)t

BHPpw

(kPa)

Test Sequence for Well B Fig 3.6.5

t


40

35

30

250 2 4 6 8 10

SkinFactor

S

Hyperbolic Function for Skin Behaviour During First Drawdown of Well B

a = 32.0b = 1.90c = 25.5

After Larsen et al



20

21

22

0 2 4 6 8 10

SkinFactor

S

Hyperbolic Function for Skin Behaviour in Second Drawdown of Well B

After Larsen et al

a = 6.0b = 2.40c = 19.5


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