04/19/23 ENGI6324: A-2-1 1

Radial flow equation

t

p

k

c

r

pr

rr r

t

1

0 err

r

r

pk

Outer boundary: CR Inner boundary:

wrro r

pr

B

khq )(

2

wrrw trpp ),(

CP Inner boundary:

Initial: pi

04/19/23 ENGI6324: A-2-1 2

Dimensionless Radial(cylindrical source)

D

D

D

DD

DD t

p

r

pr

rr

1

0),(0

DtDDD rtp

11

DrD

DD r

pr

Inner boundaryCR CP

initial

Outer boundary: no flow 0

eDD rrD

DD r

pr

11 DrDp

04/19/23 ENGI6324: A-2-1 3

Intuitive Concept of Productivity Index

boundary

re

wfaveei pp

qJ

,,

Production Rate Drawdown (Driving Force)

well

04/19/23 ENGI6324: A-2-1 4

Pseudo-steady State IPR: Impact of Reservoir Pressure

AOF

04/19/23 ENGI6324: A-2-1 5

Irregular Drainage Area and/or LocationDietz shape factor

2

4ln

21

2.141wArCe

AB

khJ

CA = 31.6

CA = 30.9

...78107.1

...577216.0

e

Euler’s constant

04/19/23 ENGI6324: A-2-1 6

How to be smart?

DJB

khJ

2

DJB

khJ

2.141

)472.0

ln(

1

43

)ln(

1

w

e

w

eDpss

rr

rr

J

Pss: p is defined with average pressure

04/19/23 ENGI6324: A-2-1 7

A/acre 40 120

re /ft

JDpss =

75.0)/ln( we rr

Assume rw = 0.3 ft

How to be smart?

04/19/23 ENGI6324: A-2-1 8

Pseudosteady-State Performance with Skin (Undersaturated Oil)

]472.0

[ln2.141

)(

sr

rB

ppkhq

w

e

wf

There is –3/4

Average reservoir press

04/19/23 ENGI6324: A-2-1 9

Single Phase Pss IPRProductivity Index and Skin

pssJm

1

pAverage reservoir pressure (NOT average between reservoir and wellbore !!!)

Slope: 1/JNOT J !

sr

rJ

w

eDpss

43

)ln(

1

04/19/23 ENGI6324: A-2-1 10

Effective Wellbore Radiuss

ww err

'472.0

ln2.141

)(

]472.0

[ln2.141

)(

w

e

wf

w

e

wf

rr

B

ppkh

sr

rB

ppkhq

E.g. Steady-state

Definition of r’w

04/19/23 ENGI6324: A-2-1 11

Effect of Stimulation Represented as a Folds of

Increase in PI

postw

e

prew

e

pre

post

sr

r

sr

r

J

J

472.0ln

472.0ln

Spot acidizing: 0carbonate acidizing: -2hydraulic fracturing: -5

04/19/23 ENGI6324: A-2-1 12

Skin Components•damage •penetration+slant•perforation•pseudo (e.g. non-Darcy, condensate)•plus reservoir shape and well location:(later: stimulation)

Elements of Skin Effect(Accounting for well

geometry, perforation, etc)

CApseudopcd ssssss

ACA C

s62.31

ln5.0

The easiest (will be

always plus)

the “original”

04/19/23 ENGI6324: A-2-1 13

Boundary-dominated State

No flow boundaryConstant wellbore pressure,

time elapsed is enough to stabilize the "shape" of the pressure distribution

wfave pp

qJ

)(

wfave ppJq )(

43

ln

12

w

e

rrB

khJ

04/19/23 ENGI6324: A-2-1 14

Steady-state

Constant outer pressure

Constant wellbore pressure*,time elapsed is enough to stabilize

the "shape" of the pressure distribution

wfave pp

qJ

)( wfave ppJq )(

21

ln

12

w

e

rrB

khJ

*or constant rate

wfeess ppJq ,

w

eess

r

rB

khJ

ln

12,

04/19/23 ENGI6324: A-2-1 15

Steady-State Performance with Skin (Undersaturated Oil)

][ln2.141

)(

sr

rB

ppkhq

w

e

wfe

There is no –3/4

Press at outer boundary

04/19/23 ENGI6324: A-2-1 16

Theory

Akif Ibragimov and Peter Valkó

http://www.isc.tamu.edu/iscpubs/0005.pdf

04/19/23 ENGI6324: A-2-1 17

Productivity Index • Intuitive:

– for a given reservoir-well geometry, the ratio of production rate to some pressure difference between the reservoir and the well is basically independent from production history or even from actual operating conditions, once the well production is "stabilized"

• Math: Key concept is invariance– From time– From production rate or pressure

difference

04/19/23 ENGI6324: A-2-1 18

3 basic flow regimes with invariance properties

• Steady-State: Constant pressure outer

boundary and constant pressure (or flowrate)

at the well

• Pseudo-Steady State: No flow outer

boundary and constant flowrate at the well

• Boundary-Dominated State: No flow outer

boundary and constant pressure at the well

04/19/23 ENGI6324: A-2-1 19

Unified View• Driving force is average reservoir

pressure minus wellbore pressure

• We are looking for an initial distribution of pressure PROVIDING time invariance at once

• In the case of Pseudo-steady and Boundary-dominated states time invariance applies only to PI and some characteristics of the pressure distribution ("shape")

04/19/23 ENGI6324: A-2-1 20

Circular drainage area

reD

1

04/19/23 ENGI6324: A-2-1 21

kck kD

0/1 kck

PIcJ JD

B

hkcJ

02

weD rrR / 1DWS2

12 D

DG

RV

Table 1. Dimensionless variables

Definition Circular Drainage Area, Constant-rate

Circular Drainage Area, Constant -pressure

)( ppcp refpD

QB

hkcp

02

wrefp pp

c

1

qcq qD

qcQ

1 qc

)(2 0 wref

o

pphk

B

QcQ qD see above see above

xcx xD

wr rc /1 (x becomes r)

tct tD

20

wtt

rc

kc

kck kD

0/1 kck

PIcJ JD

B

hkcJ

02

weD rrR / 1DWS2

12 D

DG

RV

Table 1. Dimensionless variables

Definition Circular Drainage Area, Constant-rate

Circular Drainage Area, Constant -pressure

)( ppcp refpD

QB

hkcp

02

wrefp pp

c

1

qcq qD

qcQ

1 qc

)(2 0 wref

o

pphk

B

QcQ qD see above see above

xcx xD

wr rc /1 (x becomes r)

tct tD

20

wtt

rc

kc

Table 1. Dimensionless variables

Definition Circular Drainage Area, Constant-rate

Circular Drainage Area, Constant -pressure

)( ppcp refpD

QB

hkcp

02

wrefp pp

c

1

qcq qD

qcQ

1 qc

)(2 0 wref

o

pphk

B

QcQ qD see above see above

xcx xD

wr rc /1 (x becomes r)

tct tD

20

wtt

rc

kc

04/19/23 ENGI6324: A-2-1 22

Pseudo-steady State

Def 1 (CR PI)

Def 2 (PSS)

Def: Auxiliary Problem 1 (sol: pD1 with zero average)

GDDWDDDcr tptp

J)]([)]([

1

constant remains DcrJ

DWDGDD SV

pxkDiv/

1))(( 1

BonpD 01

Wonp

S DDW 11

04/19/23 ENGI6324: A-2-1 23

11 )()( cxpxp DDDDi WD

DpsDDcr pJtJ

][

1)(

1

Specific Results:

22

442

2

2

2

2

1 )1(4

)ln(4321)ln(

1)1(2)(

D

DDDDD

D

D

D

DDD R

RRRRr

R

R

R

rrp

)ln(4341

)1(4

)(

1442

22

11 DDDD

D

rDDDps RRRR

R

rpJ

D

43

)ln(

1

D

Dps

RJ

General Results:

04/19/23 ENGI6324: A-2-1 24

Boundary-dominated

W

DWDDD

DD dStxp

tQ ),()(

Def 1 (CP PI)

Def 2 (BD)

Def: Auxiliary Problem 3

Def: Auxiliary Problem 4 (zero-zero transient)

GDDDw

DDDcp tpp

tQJ

)]([

)(

constant remains DcpJ

04/19/23 ENGI6324: A-2-1 25

121)( mDDi cxp

DGDbdDDcp VJtJ 1)(

)]()()()([1)( 1'

01

01

01'02 zYr

R

zJr

R

zYzJcrp D

DD

DDDi

2

221 2

1

D

DDbd R

RzJ

0)()()()( '000

'0 zY

R

zJ

R

zYzJ

DD

Specific Results:

General Results:

04/19/23 ENGI6324: A-2-1 26

Table 2. Comparison of Dimensionless Productivity Indices(Circular domain)

Ratio of drainage radius to wellbore

radius, RD

Standard approximation,

JDst

Pseudo-steady state, JDps

Boundary-dominated, JDbd

10 0.644087 0.627240 0.601888

100 0.259392 0.259330 0.256797

1,000 0.162397 0.162396 0.161765

10,000 0.118199 0.118199 0.117955

100,000 0.092912 0.092912 0.092794

Table 2. Comparison of Dimensionless Productivity Indices(Circular domain)

Ratio of drainage radius to wellbore

radius, RD

Standard approximation,

JDst

Pseudo-steady state, JDps

Boundary-dominated, JDbd

10 0.644087 0.627240 0.601888

100 0.259392 0.259330 0.256797

1,000 0.162397 0.162396 0.161765

10,000 0.118199 0.118199 0.117955

100,000 0.092912 0.092912 0.092794

Table 2. Comparison of Dimensionless Productivity Indices(Circular domain)

Ratio of drainage radius to wellbore

radius, RD

Standard approximation,

JDst

Pseudo-steady state, JDps

Boundary-dominated, JDbd

10 0.644087 0.627240 0.601888

100 0.259392 0.259330 0.256797

1,000 0.162397 0.162396 0.161765

10,000 0.118199 0.118199 0.117955

100,000 0.092912 0.092912 0.092794

Table 2. Comparison of Dimensionless Productivity Indices(Circular domain)

Ratio of drainage radius to wellbore

radius, RD

Standard approximation,

JDst

Pseudo-steady state, JDps

Boundary-dominated, JDbd

10 0.644087 0.627240 0.601888

100 0.259392 0.259330 0.256797

1,000 0.162397 0.162396 0.161765

10,000 0.118199 0.118199 0.117955

100,000 0.092912 0.092912 0.092794

Table 2. Comparison of Dimensionless Productivity Indices(Circular domain)

Table 2. Comparison of Dimensionless Productivity Indices(Circular domain)

Ratio of drainage radius to wellbore

radius, RD

Ratio of drainage radius to wellbore

radius, RD

Standard approximation,

JDst

Standard approximation,

JDst

Pseudo-steady state, JDps

Pseudo-steady state, JDps

Boundary-dominated, JDbd

Boundary-dominated, JDbd

1010 0.6440870.644087 0.6272400.627240 0.6018880.601888

100100 0.2593920.259392 0.2593300.259330 0.2567970.256797

1,0001,000 0.1623970.162397 0.1623960.162396 0.1617650.161765

10,00010,000 0.1181990.118199 0.1181990.118199 0.1179550.117955

100,000100,000 0.0929120.092912 0.0929120.092912 0.0927940.092794

Table 2. Comparison of Dimensionless Productivity Indices(Circular domain)

Ratio of drainage radius to wellbore

radius, RD

Standard approximation,

JDst

Pseudo-steady state, JDps

Boundary-dominated, JDbd

10 0.644087 0.627240 0.601888

100 0.259392 0.259330 0.256797

1,000 0.162397 0.162396 0.161765

10,000 0.118199 0.118199 0.117955

100,000 0.092912 0.092912 0.092794

Table 2. Comparison of Dimensionless Productivity Indices(Circular domain)

Ratio of drainage radius to wellbore

radius, RD

Standard approximation,

JDst

Pseudo-steady state, JDps

Boundary-dominated, JDbd

10 0.644087 0.627240 0.601888

100 0.259392 0.259330 0.256797

1,000 0.162397 0.162396 0.161765

10,000 0.118199 0.118199 0.117955

100,000 0.092912 0.092912 0.092794

Table 2. Comparison of Dimensionless Productivity Indices(Circular domain)

Table 2. Comparison of Dimensionless Productivity Indices(Circular domain)

Ratio of drainage radius to wellbore

radius, RD

Ratio of drainage radius to wellbore

radius, RD

Standard approximation,

JDst

Standard approximation,

JDst

Pseudo-steady state, JDps

Pseudo-steady state, JDps

Boundary-dominated, JDbd

Boundary-dominated, JDbd

1010 0.6440870.644087 0.6272400.627240 0.6018880.601888

100100 0.2593920.259392 0.2593300.259330 0.2567970.256797

1,0001,000 0.1623970.162397 0.1623960.162396 0.1617650.161765

10,00010,000 0.1181990.118199 0.1181990.118199 0.1179550.117955

100,000100,000 0.0929120.092912 0.0929120.092912 0.0927940.092794

04/19/23 ENGI6324: A-2-1 27

R = 1000PPSS = -6.15776809 - 5.0000050e-7 r2 + ln(r)

PBD = 1.227462 J0(0.000568798 r ) + 0.254100Y0(0.000568798 r) - 1

When both average pressures (0) and wellbore pressures (-1) are the same

04/19/23 ENGI6324: A-2-1 28

PSS

0 200 400 600 800 1000r

- 1

- 0.8

- 0.6

- 0.4

- 0.2

0

SS

PpR=1000

04/19/23 ENGI6324: A-2-1 29

BD

0 200 400 600 800 1000r

- 1

- 0.8

- 0.6

- 0.4

- 0.2

0

DBp

R=1000

04/19/23 ENGI6324: A-2-1 30

Difference: PPSS-PBD

0 200 400 600 800 1000r

- 0.001

- 0.0005

0

0.0005

0.001

0.0015

0.002

0.0025

Lp

Difference, R=1000