Topic: Analysis of Inflow Performance Relationships for Niger Delta Oil Wells
Student name: Keren Ebiokpo Paul
Objective: The objective of this project is to establish the Inflow Performance
Relationship (IPR) that best describes Niger delta Oil wells.
Brief description: The Inflow Performance curve ( a Cartesian plot of bottom-hole
flowing pressure versus surface flow rate) is one of the diagnostic tools used by
Petroleum engineers to evaluate the performance of a flowing well. The plot is used to
determine whether any well under consideration is performing as expected or not. If it
is not, then remedial action may be necessary. The equation that describes this curve is
the Inflow Performance Relationship (IPR). This equation can be determined both
theoretically and empirically.
Theoretical Inflow Performance Relationship
Accounting for the pressure dependence of oil viscosity, oil formation volume factor,
and oil relative permeability, the theoretical inflow equation for oil in a radial system
under pseudo-steady flow is:
q o 
k h
  re 
141.2  ln 

  rw
P


 
 0.75  s  P

R
k ro
 o B o
dp
wf
Where s is the skin factor.
Evaluating the integral above is difficult because unlike oil viscosity and formation
volume factor, relative permeability has no direct dependence on pressure. However, it
k ro
 B
has been shown that below the bubble point, a cartesian plot of o o versus Pressure
results in a line that can be assumed linear. This line starts from the origin and ends at
 k ro 


  o B o  P R
. Thus the integral is easily evaluated by calculating the trapezoidal area
under the straight line. The value of this integral is easily shown to be:
2
2
P R  P wf
2 P R
 k ro 

  o B o  P R

Introducing the above into the oil flow rate gives:
 P R2  P wf2  k ro  
k h
q o 


 


B
  re 
  2 P R
 o o  P R 
141.2  ln 
 0.75  s  

  rw

The inflow performance curve can then be generated from this equation by first
calculating qo using the equation above and plotting Pwf against qo.
Inflow performance curve calculation from single rate data
For oil wells, multiple rate data are not always available. Therefore, this equation can
be used to calculate inflow performance curve. First, the equation is simplified to the
form:
qo  C * Area
Kh
C
 r 

141.2 ln  e   0.75  s 
  rw 

Where
The procedure is as follows:
 K ro 


  o Bo  PR
(1) Plot the single point
versus PR. Then draw a straight line from this point
to the origin.
(2) Find the slope of this line and therefore the equation of the line.
(3) Knowing the equation of the line, the value of at any other pressure such as at Pwf
 K ro 



B
 o o  Pwf
can be calculated as
(4) Then calculate the area of the trapezoid between Pwf and PR.
(5) Use the calculated area and the known single value of qo to calculate C as
C=qo/Area.
(6) Having calculated C, we can now calculate any other value of q corresponding to
any Pwf by altering Pwf, calculating the new area and calculating qo as qo = C * Area.
The example calculation below illustrates the procedure.
EXAMPLE INFLOW PERFORMANCE CURVE FROM SINGLE RATE FLOW TEST DATA
The drawdown test for a partially depleted undersaturated oil reservoir stabilized at 310 STB/D
at a flowing bottom-hole pressuer of 715 psia. The average reservoir pressure = 2830 psia. Given
the PVT, gas saturation, and relative permeability data below, calculate and plot the IPR curve
for this well . Use P wf values of 0,250,500,750,1000,1250,1500,1750,2000,2250,2500 psia.
q o  310
Pwf  715
STB/D
skin  4.7
Bo  1.35
Sg  0.12
Psia
RVB/STB
PR  2830
psia
 o  0.52
cp
Kro  0.8
SOLUTION
( 1) The plot of
Kro
 o Bo
 1.14
Kro
 o Bo
versus Pwf
is approximately a straight line starting at the origin.
at P R = 2830 psia
Since there is only one data point, the plot would look like this:
Kro
o Bo
1.14
Area
Pwf =715
The slope of the line is:
m 
P
1

PR=2830
Kro
PR  o Bo
4
m  4.027  10
Since the intercept = 0, The equation of the straight line is:
That is, the equation of the line is F(P) = 0.0004027*P
F( P)  m P  0
Therefore, when P = P wf = 715, F(P wf) = 0.0004027 * 715 = 0.288
F PR  1.14
Therefore ,
F Pwf  0.288
The area of the trapezoid between P R and P wf is then:
Area Pwf  0.5  F PR  F Pwf    PR  Pwf
Area Pwf  1509.6
Since the inflow performance equation can be represented as: q = C * Area
C 
Then
qo
Area Pwf
C  0.205
Now use the different values of P wf to calculate the corresponding values of q and plot
j  1  12
p wf 
j
0
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
area p wf  0.5  F PR  F p wf    PR  p wf
Q p wf  C area p wf
1500
1750
2000
2250
2500
2750
p wf 
j
 
area p wf
 
Q p wf
j
j
0
1.613·10 3
331.137
250
1.6·10 3
328.553
500
1.562·10 3
320.801
750
1.499·10 3
307.88
1000
1.411·10 3
289.791
1250
1.298·10 3
266.534
1500
1.16·10 3
238.108
1750
995.923
204.515
2000
807.164
165.753
2250
593.237
121.822
2500
354.142
72.724
2750
89.879
18.457
3000
2000
pwf
j
1000
0
0
50
100
150

Q pwf
200
j

250
300
350
Inflow performance calculation from multiple rate data
Note that this equation can be simplified to the form:
 k ro 
k h  

  o B o  P R
2
2
q o 
  re 
141.2  ln 
  0.75 
  rw

s   2 P R


 P R  P wf


2
2
Which can be expressed as: C PR  Pwf

where C is a constant.
Alternatively, the oil flow rate can be expressed as:
 k ro 
 P R
  o B o  P R
k h  
 P R2  P wf2 

q o 

  re 
  P R2 
141.2  ln 
  0.75  s   2
  rw

Which can be simplified to the form:
2

P wf 

q o  C 1 
2 

PR 

For the more general case (where Pwf is below the bubble point and PR is above),
 k ro 
1


  o B o  is linear with pressure below the bubble point whereas  o B o is constant with
pressure above the bubble point. Therefore, evaluating the integral means calculating
the area under the combined linear and constant sections. The resulting equation for
the flow rate of oil becomes:
1 

 k ro 

k h  
k h  




  o B o  P b
  o B o  P b
2
2
q o 
  P R  P wf  
 P b  P wf 
  re 

 141.2  ln r e   0.75  s   2 P

141.2  ln 
 0.75  s 

 r 

b


r
  w


  w



Theoretical IPR Analysis Procedure
I. For any fixed average reservoir pressure PR, calculate the oil saturation
corresponding to PR from material balance calculations.
II. Read the value of (Kro)PR at the calculated oil saturation corresponding to PR
III. Read the values of (oBo)PR corresponding to PR
IV. Calculate qo for varying values of Pwf and plot Pwf against qo to give the IPR curve.
Empirical Inflow Performance Relationships
The most common empirical IPR equations are: (a) Vogel’s equation, (b) the backpressure equation (Bureau of Mines equation), and (c) normalized back-pressure
equation. All three methods will be used to analyze the field data and plotted.
Analysis Procedure
 Make a field IPR Plot (Pwf versus qsc) from the field-measured multi-rate well
test data. Determine the maximum oil rate (qo)max graphically.
 Make a Vogel IPR plot as follows: Using qo from one data point, and PR from
the data (at qo = 0), calculate (qo)max from the equation:
qo
q o.max 

2

 P wf 
 P wf  
 1  0.2 

0.8


P 

 PR 
 R
Then, knowing (qo)max, calculate the values of qo at corresponding values of
Pwf and plot Pwf versus qsc.
Make a Back Pressure Equation IPR plot: This is based on the empirical


2 n
2
equation q  C P R  P wf attributed to the Bureau of Mines where C and n
are constants to be determined using the field data. If the field data can be

P
2
P
2
 , a log-
2
log( q)  log( C)  n log P R  P wf
represented by this equation, then since
2
wf
log plot of R
versus qo should give a straight line from which, n = the
slope and C = the intercept. Note that the slope can be obtained simply by
measuring the rise and run with a ruler and taking the ratio. Alternatively,
n 

2

2
log P R  P wf
2

2

2
 log P R  P wf
1
log ( q ) 2  log ( q) 1
slope =
Having obtained C and n, calculate the values of qo at corresponding values



2 n
2
q  C P R  P wf
of Pwf using the equation
and plot Pwf versus qsc. If the loglog plot did not give a straight line, then this method is not valid.
Make a normalized back pressure equation IPR plot: This method is based
on the normalization of the Bureau of Mines empirical equation by (qo)max.
n
q
The resulting equation is:
q o.max
  P wf  2
  1  

  P R   . This implies that:
   nlogP
log( q)  log q o.max  n log P R
2
2
R

2
 P wf
Therefore, just like in the Back pressure equation method, a log-log plot of
2
2
P R  P wf
versus qo should give a straight line from which, n = the slope.
 


log( q) 
 n logwhich
PR q
P wf
However, in this case, the intercept
= logq o.max  n log P R from
omax
can be calculated. Having obtained the values of qomax and n, calculate the
2
2
2
values of qo at corresponding values of Pwf using the equation:
n
q
q o.max
  P wf  2
  1  

  P R   and plot Pwf versus q.
The theoretical (above bubble point only) and the three empirical curves will be
compared with the field curve in order to establish which of them is best for Niger delta
oil wells.
Data needed from SPDC:
(a) Oil PVT data (particularly viscosity and formation volume factor).
(b) Relative permeability data (Oil-water, and oil-gas)
(c) Field measured multi-rate well test data
(d) Existing correlations for oil viscosity and oil formation volume factor.
At the end, you would have generated graphs that look like this:
Pwf
PRo
PR1
qo
qomax