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 RelationshipAccounting 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 ok h

141.2 lnr e

r w

0.75 s

P wf

P R

pk ro

o B o

dk

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 has been shown that below the bubble point, a cartesian plot of

k ro

o B 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:

P R2 P wf

2

2 P R

k ro

o B o

P R

Introducing the above into the oil flow rate gives:

q ok h

141.2 lnr e

r w

0.75 s

P R2 P wf

2

2 P R

k ro

o B o

P R

k

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:

AreaCoq *

Where

sr

r

KhC

w

e 75.0ln2.141

The procedure is as follows:

(1) (1) Plot the single point RPoo

ro

B

K

versus PR. Then draw a straight line from this point to the origin.

(2) (2) Find the slope of this line and therefore the equation of the line.(3) (3) Knowing the equation of the line, the value of at any other pressure such as

at Pwf can be calculated as wfPoo

ro

B

K

(4) (4) Then calculate the area of the trapezoid between Pwf and PR.

(5) (5) Use the calculated area and the known single value of qo to calculate C as C=qo/Area.

(6) (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.

Since there is only one data point, the plot would look like this:

at PR = 2830 psiaKro

o Bo1.14

is approximately a straight line starting at the origin. versus Pwf

Kro

o BoThe plot of1( )

SOLUTION

Kro 0.8Sg 0.12

cpo 0.52RVB/STBBo 1.35skin 4.7

psiaPR 2830PsiaPwf 715STB/Dqo 310

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 Pwf values of 0,250,500,750,1000,1250,1500,1750,2000,2250,2500 psia.

EXAMPLE INFLOW PERFORMANCE CURVE FROM SINGLE RATE FLOW TEST DATA

oo

ro

B

K

1.14 Pwf =715 P PR=2830

Area

The slope of the line is: m1

PR

Kro

o Bo

m 4.027 10 4

Since the intercept = 0, The equation of the straight line is: F P( ) m P 0

That is, the equation of the line is F(P) = 0.0004027*P

Therefore, when P = Pwf = 715, F(Pwf) = 0.0004027 * 715 = 0.288

Therefore , F PR 1.14 F Pwf 0.288

The area of the trapezoid between PR and Pwf 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

Then Cqo

Area Pwf C 0.205

Now use the different values of Pwf to calculate the corresponding values of q and plot

j 1 12

pwfj

025050075010001250150017502000225025002750

area pwf 0.5 F PR F pwf PR pwf

Q pwf C area pwf

pwfj

025050075010001250150017502000225025002750

pwfj

0

250

500

750

1000

1250

1500

1750

2000

2250

2500

2750

area pwfj

1.613·10 3

1.6·10 3

1.562·10 3

1.499·10 3

1.411·10 3

1.298·10 3

1.16·10 3

995.923

807.164

593.237

354.142

89.879

Q pwfj

331.137

328.553

320.801

307.88

289.791

266.534

238.108

204.515

165.753

121.822

72.724

18.457

0 50 100 150 200 250 300 3500

1000

2000

3000

pwf j

Q pwf j

Inflow performance calculation from multiple rate data

Note that this equation can be simplified to the form:

q o

k hk ro

o B o

P R

141.2 lnr e

r w

0.75 s

2 P R

P R2 P wf

2

k

Which can be expressed as: C PR2 Pwf

2 where C is a constant.

Alternatively, the oil flow rate can be expressed as:

q o

k hk ro

o B o

P R

P R

141.2 lnr e

r w

0.75 s

2

P R2 P wf

2

P R2

k

Which can be simplified to the form:

q o C 1P wf

2

P R2

C

For the more general case (where Pwf is below the bubble point and PR is above), k ro

o B o

is linear with pressure below the bubble point whereas

1

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:

q o

k h1

o B o

P b

141.2 lnr e

r w

0.75 s

P R P wf

k hk ro

o B o

P b

141.2 lnr e

r w

0.75 s

2 P b

P b2 P wf

2

k

Theoretical IPR Analysis ProcedureI. I. For any fixed average reservoir pressure PR, calculate the oil saturation

corresponding to PR from material balance calculations.II. II. Read the value of (Kro)PR at the calculated oil saturation corresponding to PR

III. III. Read the values of (oBo)PR corresponding to PR

IV. IV. Calculate qo for varying values of Pwf and plot Pwf against qo to give the IPR curve.

Empirical Inflow Performance RelationshipsThe most common empirical IPR equations are: (a) Vogel’s equation, (b) the back-pressure 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:

q o.max

q o

1 0.2P wf

P R

0.8P wf

P R

2

q o

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 equation q C P R2 P wf

2 n P R 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 represented by this equation, then since

log q( ) log C( ) n log P R2 P wf

2 P wf , a log-log plot of P R2 P wf

2 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, slope =

nlog P R

2 P wf2 2 log P R

2 P wf2 1

log q( )2 log q( )1

P R

Having obtained C and n, calculate the values of qo at corresponding

values of Pwf using the equation q C P R2 P wf

2 n P R and plot Pwf versus qsc.

If the log-log 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. The resulting equation is:

q

q o.max

1P wf

P R

2

n

P wf

. This implies

that: log q( ) log q o.max n log P R2 n log P R

2 P wf2 P wf

Therefore, just like in the Back pressure equation method, a log-log plot

of P R2 P wf

2 versus qo should give a straight line from which, n = the

slope. However, in this case, the intercept = log q( ) log q o.max n log P R2 n log P R

2 P wf2 P wffrom

which qomax can be calculated. Having obtained the values of qomax and n, calculate the values of qo at corresponding values of Pwf using the

equation:

q

q o.max

1P wf

P R

2

n

P wf

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) (a) Oil PVT data (particularly viscosity and formation volume factor).(b) (b) Relative permeability data (Oil-water, and oil-gas)(c) (c) Field measured multi-rate well test data(d) (d) Existing correlations for oil viscosity and oil formation volume factor.

At the end, you would have generated graphs that look like this:

PR1

PRoqomax

qo

Pwf