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Well Inflow Performance

1

Well Inflow Performance
Session Objectives
At the end of the session, you will be able to:
- explain the flow regimes governing the
reservoir well inflow and recognise the
relationships for stabilized flow.
- recall and do simple calculations on the well
inflow performance.
- recall what is “Skin Effect” and its effects on
the Well Inflow Quality.
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Well Performance
The performance of a Well is governed by the behaviour
of two basic factors:
 the capacity of a reservoir to pass fluids against
down-hole conditions (the Inflow Performance)
 the ability of the produced fluids to flow through the
well conduit to surface (Vertical or Outflow

Performance)
– The two factors are closely linked, because the final
condition of the inflow performance, is the starting
point of the vertical flow performance.

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Well Inflow Performance
•
•
•
•
•
•
•

Types of flow models
Radial flow
Productivity Index (PI)
Straight-line IPR
Vogel’s IPR and generalised IPR
Gas well IPR
The “Skin Effect” concept

4

Possible Flow Models for Stabilized Well Inflow

LINEAR

RADIAL
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Pressure drawdown in the Wellbore
Well

re

rw

Pe

¯
P

¯
P

PDD
Pwf
Pwf
6

Darcy’s Law
• In 1856, while performing experiments for the design of
filter beds for water purification, Henry Darcy proposed an
equation relating apparent fluid velocity to pressure drop
across the filter bed.
• Darcy’s experiments involved only one fluid, water, and the
sand filter was completely saturated with water. Therefore
no effects of fluid properties or saturations were involved.
• Darcy’s sand filters were of constant cross-sectional areas,
so the equation did not account for changes in velocity
with location.
• Although the experiments were performed with flow in the
downward vertical direction, the expression is also valid for
horizontal flow.

7

Darcy’s Law
L

Area A

q

q

p1

p

k dp
v
 dx

kAdp
q  vA  
dx

p2

 = fluid viscosity
k = permeability

8

The Basic Partial Differential Equation for the radial
flow of any single phase fluid in a porous medium

The Radial Diffusivity Equation

1  k p
p
(  r r )  c
r r
t
pressure : radius : time
A non-linear equation because of:
- dependence of pressure on density, compressibility and viscosity
- analytical solutions are obtainable under various boundary and
initial conditions to describe Well Testing and Stabilized Well Inflow.
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The Radial Flow Model
for a single phase fluid in a porous medium
Constant Terminal Rate Solution
qB

qB
qB

qB

qB

qB
- reservoir is homogeneous in all rock properties
- reservoir is isotropic in permeability
- producing well is completed across the entire formation thickness,
thus ensuring fully Radial Flow
- formation is completely saturated with a single fluid
- the fluid compressibility is small and constant
10

Reservoir Flow Regimes
1) Infinite acting or Transient state
2) Semi-steady state or Steady State

2 Stages

re

r3
r2
r1

t2=60 days
t1=20 days
t3=200 days
t4=400 days

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Infinite acting or Transient State Flow Regime
•

re

•

r3
r2

t2=60 days

r1

t1=20 days
t3=200 days

•

•

When a well is put on production, the
pressure disturbance caused by the
well transmits outward radially. Fluid
away from the well-bore experiences
the pressure gradient and it begins to
move towards the well-bore.
The rate of pressure propagation
outward depends upon reservoir
permeability.
As time progresses, this pressure
disturbance reaches the “drainage
boundary”
The time period required to reach the
boundary of a circular area is called
“infinite acting time”

t4=400
days

12

Semi-Steady State Flow Regime
re

r3
r2

t2=60 days

r1

t1=20 days
t3=200 days
t4=400
days

• Once the pressure
disturbance has reached the
boundary, then as time
progresses, and as more and
more fluids are withdrawn
from the reservoir, the
average reservoir pressure
begins to decrease over time.
• This occurs in a reservoir in
which there is no flow across
the well’s drainage boundary
- the no-flow boundary
condition

 This is called a Semi-steady state or
pseudo-steady state flow regime

13

Steady State Flow Regime

re

r3
r2

t2=60 days

r1

t1=20 days
t3=200 days
t4=400
days

• However if the condition
at the well’s drainage
boundary is such that
there is flow across the
boundary - the fluidinflux boundary
condition, then we have
fixed (constant) pressure
at the boundary
• This is called the Steady
State flow regime

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Stabilized Inflow Equation - Steady state solution
qB = constant
p
0
t

Pressure (p)

p  pe

for all r and t

at r = re

qB
pwf
rw
General relationship
between p and r
Expressed in terms of
p = pe at r = re

Expressed in terms of
average reservoir
pressure

r

p  pwf

re

141.2qB

{ln rr }
w
kh

141.2qB
re
pe  pwf 
{ln r }
w
kh
141.2qB
r
pR  pwf 
{ln re  12}
w
kh

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Stabilized Inflow Equation - Semi-steady state solution
qB = constant
 p
 constant for all r and t
t

Pressure(p)

 p
0
r

pwf

at r = re

No-flow
boundary

rw

r

re
2
141.2qB
r
r

{ln r  2 }
w
2 re
kh

General relationship
between p and r

p  pwf

Expressed in terms of
p = pe at r = re

pe  pwf 

Expressed in terms of
average reservoir
pressure

pR  pwf

141.2qB
r
{ln re  12}
w
kh
141.2qB
r

{ln re  34}
w
kh

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Summary of Stabilized Inflow Equations
Semi-Steady State
General
relationship
between
p and r

Expressed in
terms of:
p = pe at r = re

Expressed in
terms of
Average reservoir
pressure

p  pwf

141.2qB r r 2

{ln r  2 }
w 2re
kh

pe  pwf

141.2qB
r

{ln re  12}
w
kh

pR  pwf

141.2qB re 3

{ln r  4}
w
kh

Steady State

p  pwf

pe  pwf

141.2qB

{ln rr }
w
kh

141.2qB re

{ln r }
w
kh

pR  pwf 

141.2qB
r
{ln re  12}
w
kh

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Radius of Drainage (re) for various Well Spacings
Drainage Radius (re)
Well Spacing
m2
per well

20,234
40,468
80,937
161,874
323,748
647,497
1,294,994
2,589,988

acres
per well

(5)
(10)
(20)
(40)
(80)
(160)
(320)
(640)

Radius m (ft)
circular area to
equal well
spacing
80
113
161
227
321
454
642
908

(263)
(372)
(527)
(745)
(1053)
(1489)
(2106)
(2979)

- m (ft)

Square
spacing
71
101
142
201
284
402
569
804

(233)
(330)
(466)
(660)
(933)
(1320)
(1867)
(2640)

Triangular
spacing
76
108
153
216
306
432
611
865

(250)
(355)
(501)
(709)
(1003)
(1418)
(2006)
(2837)

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Well Inflow Performance Indicator

The Productivity Index (PI) Concept
J = is the production rate divided by the drawdown

qo
7.08*103 koh
J
)
(pR  pwf oBo ln( re rw )
the inflow equation is :



qo  J ( pR  pwf )
qo
pwf  p R 
J

if J remains constant with drawdown, once a value of J is obtained from
one production test, or calculated, it can be used to predict future
inflow performance. The plot will be linear with slope = - 1/J
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Straight-Line Inflow Performance Relationship (IPR)
Valid for single phase liquid flow
(k, , Bo = constant)

pR
p wf

p



(Maximum

pwf1

inflow potential - MIP,
Absolute Open Flow, AOF)

q omax
q1

q

qo1
qo1
qo max
qo1  qo 2
PI  J 

 tan 

p p  pwf 1
p
pwf 2  pwf 1
20

Two Phase Flow in a Reservoir
• Bubblepoint pressure (pb)
• Pressure at which first bubble of gas is released
from reservoir oils

21

Inflow Performance Relationship for
initially undersaturated reservoir

Pressure

pR

Straight Line IPR

Pb

Curved IPR

- Two-phase flow
- Fluid properties
- Relative permeability
- Turbulence

Flow Rate

22

Vogel’s Method for two-phase flow reservoirs
• Used a mathematical reservoir model to calculate the IPR for
oil wells, producing from several hypothetical saturated
reservoirs. with widely differing oil characteristics, relative
permeabilities, and well spacing.
• After plotting dimensionless IPR curves for all cases
considered, Vogel proposed an empirical relationship for
saturated , dissolved-gas-drive reservoirs.

qo
qo(max )

pwf
pwf 2
 1  0. 2(
)  0. 8(
)
pR
pR
23

Vogel IPR for saturated oil wells

24

Multiphase Flow
• Vogel’s Behavior
• IPR Curve - Vogel plotted the data using the
following dimensionless variables

p wf
p

and

q
qmax
25

IPR of undersaturated oil well producing
at pwfbelow the bubble point

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Combination Single Phase Liquid and
Two Phases Flow

q
STB / D / psi 
J
p  pwf
+

 q  
 pwf

  1  0.2 
 qmax  
 p


 pwf
  0.8 

 p

2


 
 
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Multiphase Flow
• Combination Darcy/Vogel
p

Pressure

pb

pwf

J pb

qb

qmax

1.8

O
O

Rate

q

28

Multiphase Flow

• Mathematical relationship between Vogel
(qmax) and Darcy (AOF)

qmax

J  Pb
 qb 
1.8

29

Multiphase Flow
• How to find qmax:
for q  qb , Darcy' s law applies : q  Jp  p wf 


2




p
p
for q  qb then : q  qb  qmax  qb  1  0.2 wf  0.8  wf  
pb

 pb  



qmax  qb 

J pb
1 .8
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Well Inflow Quality Indicator (WIQI)

Pressure

J ideal Jcalculated 

p

3

7.08*10 koh
oBo ln( re rw )

q1
q1
J actual  J measured  
p p  pwf 1
Pwf1

S



q1

q
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Well Inflow Quality Indicator (WIQI)
WIQI = Damage Ratio (DR)
Actual stabilised production rate

WIQI =

Ideal production rate

=

J actual
J ideal

( the Ideal production rate is derived by excluding any avoidable inflow damage)

WIQI (max) = 1
WIQI < 1  positive Skin!
32

Flow Efficiency (FE)

Pr  P´wf
FE 
Pr  Pwf
Where:
P´wf =
Pwf
=
Pr
=

The equivalent undamaged flowing pressure
actual flowing pressure
static reservoir pressure
33

Flow Efficiency (FE)

34

Flow Efficiency (FE)

35

Gas IPR

36

Actual well inflow performance
- A well rarely exhibits the ideal flow conditions, thus
far considered in the ideal well model
- Actual flow into a well is affected by:
• changes in permeability near the wellbore

• changes in radial flow geometry
• breakdown of Darcy’s law at high flow rates
- To adjust the radial flow equation for these
deviations, the skin-factor approach has been
chosen
37

The “Skin” effect
(van Everdingen & Hurst)
Skin is a wellbore phenomenon, that causes
an additional pressure drop in the nearwellbore region:

(p)skin



141.2q o Bo
ko h

S

38

Positive Skin

39

The Stabilized Inflow Equations
incorporating “Skin”
General (p, r ) relationship

Semi-steady state flow
in terms of Average
reservoir pressure

Steady state flow in
terms of Average
reservoir pressure

p  pwf

141.2q o Bo
r

(ln  S )
ko h
rw

pR  pwf

141.2qo Bo re 3

(ln   S )
ko h
rw 4

pR  pwf

141.2q o Bo
re 1

(ln   S )
ko h
rw 2

40

Straight-line IPR and skin

41

Straight-line IPR and reservoir pressure

42

Actual IPR vs ideal model IPR

43

Three or Four Points Test

•

Fetkovich proposed that flow after flow or isochronal test as used on

gas wells could also be used on oil wells

q  J ' o(Pr  Pwf )
2

2 n

q  C (Pr  Pwf )
2

•

2 n

These equations are straight lines on log log with J’o and C
representing the intercept on the q axis (where Pr2-Pwf2=1 and n =
1/slope)
44

Three or Four Points Test

•

Jones, Blount, and Glaze suggest that radial flow for both oil and gas

could be represented to show wether near wellbore restriction exist

7.08 X 10 k h  p  pwf 
3

q

  re 

 o Bo  ln    0.75  s  a' q 
  rw 

45

Three or Four Points Test
 9.08 x10 13  Bo2   2
 Bo ln re / rw  0.75  s 
xq
pr  pwf  
x q  

3
2

7.08 x10 k h
4

h
r


p
w



b

a

pr  pwf  Bo ln re / rw  0.75  s   9.08 x10 13  Bo2  
xq

 

3
2

q
7.08 x10 k h
4 hp rw

 


46

Future IPR
• Future production rate
• Determine when a well is to be placed on
artificial lift
• Rate acceleration projects and comparing
artificial lift methods

50

51

Future IPR
• Fetkovich procedure



 Pr2  2
 Pr2  Pwf 2
qo  J ' o1
 Pr1 



n

From a three or four point flow test it is posible
to predict IPR curves at other static reservoir
pressures
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