Liquid Productivity Index

•The productivity index for an oil well is defined as

•This simple expression relates the flow rate to the driving force with all else being a constant distinct to the flow system.

•This relation is a function of bottom hole pressures only, which removes the constraints of well and surface equipment.

o

e wf

qJ

p p

Liquid Productivity Index

• J can be constructed from well test data where flowing bottom hole pressures are known.

• J can also be constructed from any flow model previously discussed.

o

e wf

qJ

p p

IPR for liquid flow

• The steady state radial equation for oil can be used to construct J

• If the reservoir conditions are either above the bubble point or at a pressure where fluid properties and relative permeability are not changing much, J is a constant.

141.2 ln

o o

e wf eo o

w

q k hJ

p p rB s

r

Liquid Inflow Performance (IPR)

•If J is a constant, it can be used to create an inflow performance relation.

•The inflow performance relation (IPR) is a description of what the reservoir can deliver to the well at different well bore flowing pressures.

•In the next example two flow tests are used to construct the IPR by simultaneously solving

STBO/day

psio

e wf

qJ

p p

Oil IPR Example

• Oil test one of 1000 STB/Day at Pwf=4400 psi.• Oil test two of 2000 STB/Day at Pwf=4150 psi• J=4 STBO/Day/Psi

0500

100015002000250030003500400045005000

0 5000 10000 15000 20000

Oil Rate (STB/Day)

Pw

f (p

si)

Future IPR for liquid flow

• A ratio of the equation for J allows for the calculation of a future J at a different pressure.

141.2 ln

141.2 ln

/

/

of

eof of

wf

opp

eop op

w

o o o ff p

o o o p

k h

rB s

rJ

k hJ

rB s

r

k BJ J

k B

Future IPR for liquid flow

• The J equation is good only for single phase flow. As such the fluid properties will not be changing much.

• Therefore, one should find that –

• So, future IPR curves have identical slopes, but start at different reservoir pressures.

/ constant

/

o o o ff p

o o o p

k BJ J

k B

Future IPR Curves

Single Phase Oil

0

1000

2000

3000

4000

5000

0 5000 10000 15000 20000

q (STBO/day)

Pw

f (p

si)

original Pe

Pe =3000

IPR for gas flow

• A similar equation for steady state flow of gas is:

• This is the steady state deliverability equation for gas and is the more common way of plotting gas well performance.

2 2

1424 ln

g g

e wf eg

w

q k h

p p rzT s

r

Gas Deliverability Example

100

1000

10000

100000

1000 10000 100000

Gas Flow Rate (MScf/Day)

(Pe^

2-P

wf^

2)/1

000

(psi

/100

0)

IPR for gas flow

• Plotted as Pwf vs q (like we use for liquid), the line will no longer be straight.

• This is because of the pressure squared term and because gas properties change with pressure.

• The gas properties are evaluated at the average pressure.

2 2 2

MScf/Day

( ) psi1424 ln

g g

e wf eg

w

q k h

p p rzT s

r

Gas IPR Example

0.00

1000.00

2000.00

3000.00

4000.00

5000.00

0 10000 20000 30000 40000 50000

Gas Flow Rate (MScf/Day)

Pw

f (p

si)

Future IPR for gas flow

• Future performance can be predicted if permeability, temperature, and skin can be considered a constant.

• Using the current well test data the unknowns are solved for as a group and used to predict performance at a different reservoir pressure in the future.

2 21424

( )ln

g gg

e wfe

w

k h qz

p prT s

r

2 2( )

1424 ln

gf g

e wf f eg f

w

q k h

p p rz T s

r

Two Phase Flow In The Reservoir Below the Bubble Point

Two phase flow in a reservoir below the bubble point can be described using the radial form of Darcy’s Law.

The oil flow rate is now a quadratic function of pressure.

The permeability to the flowing fluids will be unknown in most cases.

Recognizing the quadratic nature of flow rate, Vogel presented an empirical relation

based on the study of a number of wells flowing below the bubble point.

2 2

2 2

141.2 ln 1424 ln

141.2 1 ln 1424 1 ln

in oilfield units

g e wfo e wf

o g

e eo o g

w w

g o

g e wfo e wf

o

e eo o g

w w

k h p pk h p pq q

r rB s zT s

r r

q q GOR

k h p pk h p pq

r rGOR B s GOR zT s

r r

.

IPR Below the Bubble Point

• In 1968 Vogel introduced an empirical IPR for oil reservoirs that are below the bubble point.

• It is convenient because it requires no more data to use than two valid production tests where the oil flow rate and flowing BHP are known.

• Then the maximum oil rate and average reservoir pressure can be computed and the IPR curve developed.

2

max

1 0.2 0.8wf wfo

o

p pq

q p p

Single Phase vs 2 Phase IPR

010002000300040005000

0 5000 10000 15000 20000

q (STBO/day)

Pw

f (p

si)

J based

Vogel's

• Future two phase flow IPR can be constructed in a manner similar to the single phase case IF we reference the reservoir conditions to an average pressure.

• See the supplemental notes on Vogel’s equation for this development

* max

*

max

*

as approaches , then the limit of is

(1.8)

1.8

is important because at the limit where ,

all the reservoir properties are constant at the values.

wf

o

o

wf

p p J

qJ

p

J pq

J p p

p

• Using the same concept as for single phase flow to ratio future to present conditions:

• Where is the limiting value at the future average pressure.

* *

2

max

2*

/ ( )

/ ( )

and placing back in Vogel's Relation

1 .2 .8

1 .2 .81.8

o o o ff p

o o o p

wf wfof o f

f f wf wfof

k BJ J

k B

p pq q

p p

J p p pq

p p

*fJ

IPR Utility

• The Inflow Performance Relation allows the prediction of flow rate from the reservoir versus drawdown.

• It can provide a snap shot of performance as the reservoir pressure falls.

• This is useful as a guide to production prediction and equipment design. Picture from Hurd Enterprises.

Tubing Size Selection

• To produce this well at the highest possible rate requires a choice of tubing size.

• If the surface separator pressure is set at 1100 psi and this well is 10,000’ deep, the vertical lift performance for several common tubing sizes can be developed from the mechanical energy balance.

VLP

0

2000

4000

6000

8000

10000

12000

0 20 40 60 80

Gas Rate (MMcf/Day)

Pw

f (p

si) 3 1/2" tubing

2 7/8" tubing

2 3/8" tubing

Tubing Size with Well Deliverability

• If the IPR graph for this well is superimposed on the VLP graph we create a picture of what the reservoir can deliver to the bottom on the well and what each tubing size can then transport to the surface with the required surface pressure to enter the separator.

VLP & IPR

0

2000

4000

6000

8000

10000

12000

0 20 40 60 80 100

Gas Rate (MMcf/Day)

Pw

f (p

si) 3 1/2" tubing

2 7/8" tubing

2 3/8" tubing

IPR

Additional Reading

• This lecture covers text sections 2-3, 2-6, 3-4, 3-5, 3-6, 4-3, and Chapter 8.

• Also see the lecture supplement on Vogel’s equation.