4 ipr undersaturated 15

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Chair of Petroleum & Geothermal Energy Recovery

Petroleum Engineering

Summer Course 2015

Inflow Performance

Relationship (IPR)

Clemens Langbauer petrowiki.org


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Beggs: Production Optimization using Nodal Analysis

Petroleum Production System


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Petroleum Production System


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Reservoir Types


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Reservoir Types

Undersaturated Oil Reservoirs

Reservoir pressure (pe, p) and well flowing pressure (pw ) are above thebubble point pb.

Oil and water are present and flowing

Saturated Oil Reservoirs / Multi-phase Reservoirs

pw and / or the reservoir pressure is / are below the bubble point.

Oil, water and gas are present and flowing

Gas Reservoirs

Gas is present and flowing (little amount of water)


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Undersaturated Oil Reservoirs


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Darcy‘s Law

L.P.Drake: Fundmentals of Reservoir Engineering


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Darcy‘s Law


Generalized form: (frictionless & laminar flow)

u

.

d

d Φ

d

+g.(zA z)

u … Velocity (m/s, ft/s) k … Permeability (m², D)

ρ … Density (kg/m³, lb/ft³) μ … Viscosity (Pas, cp)

Φ … Potential energy per unit mass l … Length (m, ft)q … Flow rate (m³/s, bbl/day) A … Cross-section (m², ft²)

Incompressible & horizontal flow: Φ

u

.

d

d q

A

.

d

d

Radial form: q

r.

d

d


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Diffusivity Equation


Basic differential equation for radial flow in a porous media

- Homogeneous rock properties and isotropic permeability

- Fully radial flow (entire reservoir thickness covered)

- Formation is completely saturated with one fluid

- Reservoir of cylindrical shape


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Diffusivity Equation


Radial flow of any single fluid in a porous media:

r

ϕcρ

Linearization for incompressible fluids:


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Flow Regimes


Flow regimes describe the outer boundary pressure decline over time:

Steady-State Flow:

=. 0

Constant pressure boundary

Pseudo-(Semi-) Steady-State Flow:

=. const.

No flow boundary

Transient Flow: =.

f (log t)

Infinite acting

so far no boundary response - until pressure wave hits boundary


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Steady State Solution

Steady State applies after the transient period and assumes a constant

pressure over time at the outer boundary.

p

t =.

0


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Steady State

SI – Units (m³/s) Field – Units (bbl/day)

Relationship p(r),r p pw ln

+ S p pw ,. ln

+ S

Outer boundary

pressure pe pe pw

ln

+ S pe pw ,.

ln

+ S

Average reservoir

pressure p

p pw

ln

,.

+ S p pw ,.

ln

,.

+ S

C f & G

C f & G


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Pseudo- (Semi-) Steady State Solution

Pseudo Steady State applies after the transient period and assumes no

fluid flow at the outer boundary and a constant pressure drop over time.

p

t =.

const.

Ch i f P t l & G th l E R



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Pseudo Steady State


Relationshipp(r),r

p pw ln

²

+ S p pw ,. ln

²

+ S

Outer boundary

pressure pe pe pw

ln

,.

+ S pe pw ,.

ln

,.

+ S

Av. reservoir

pressure p p pw

ln,.

+ S p pw ,.

ln,.

+ S




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Transient Solution

The reservoir is acting infinite – at the beginning of production and time

dependent!

p

t =.

f (log t)




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Transient Solution


pi pw

ln

² + 2. S pi pw

,

log

² 3,23 + 0,87. S

re … Outer boundary radius (m, ft)rw … Wellbore radius (m, ft)

pe … Outer boundary pressure (Pa, psi)pi … Initial reservoir pressure (Pa, psi)pw … Well flowing pressure (Pa, psi)q … Production rate under reservoir conditions (m³/s, STB/d)

μ … Viscosity of the fluid (Pas, cp)h … Height of the reservoir (m, ft)

k … Total permeability (m², mD)

Φ … Porosity (-)

c … Compressibility (Pa-1, psi-1) γ … Euler constant = 1,78t … Time (s, h)

S … Skin factor (-)




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Economides: Petroleum Production System

Surface Rates from IPR

Conversion of the volumes under reservoir conditions to surface

conditions.

B V

V q

q

B … Formation volume factor (-)V … Volume of fluid under reservoir conditions (m³, bbl, …) Vs … Volume of fluid under stock tank conditions (m³, bbl, …) q … Flow rate under reservoir conditions (m³/s, bbl/day, …) qs … Flow rate under stock tank conditions (m³/s, bbl/day, …)




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Surface Rates

Example

IPR rate calculation

Calculate the surface production rate (here p = 0 MPa) and the

quantities at a pressure of 5 MPa for the following reservoir under steady

state conditions.

p = 20 MPapw = 10 MPaμ = 0,003 Pas

k = 100 mDh = 10 m

re = 100 mrw = 0,15 mS = 1

Chair of Petrole m & Geothermal Energ Reco er

Chair of Petrole m & Geothermal Energ Reco er


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Surface Rates


Homework

Calculate the production rate for steady state , pseudo steady state and

transient fluid flow for the well with the following parameters:

pe = pi = 17 MPa μ = 1,62. 10-3 Paspw = 10 MPa B = 1,1

re = 100 m h = 10 m

rw = 6,538” = 83 mm k = 100 mD S = 0

Φ = 0,2 c = 1,3. 10-5 Pa-1 γ = 1,78 t = 30 days = 2592000 s




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Productivity Index J or PI: J

−

Expression describes the deliverability of the wellbore

Constant for undersaturated reservoirs

Maximizing the PI is a major objective

Inflow Performance Relationship




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Productivity Index

Example

Productivity index

The average reservoir pressure of well XY in a depth of 2000 m is 21

MPa. The test point of this undersaturated reservoir was at 18 MPa and

generated 250 m³/day.

Draw the IPR curve!

Calculate the PI-Index and the AOF!

Calculate the skin factor!

μ = 1,62. 10-3 PasB = 1,1

re = 100 m h = 10 m rw = 6,538” = 83 mm k = 100 mD




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Dietz Shape Factor

Up to now only cylindrical reservoirs were considered – the shape of the

reservoir has a significant influence on the IPR.

Dietz Shape Factor: It considers the irregularity of the drainage area.

CA … Dietz shape factor A … Drainage area (m², ft²)

Irregular Drainage Pattern


Steady state

p pw

ln

A

,.²+ S p pw

,.

ln

A

,.²+ S

Pseudo steady state

p pw

ln

A

²+ S p pw

,.

ln

A

²+ S




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Dietz Shape Factor




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Dietz Shape Factor

Example

Dietz shape factor

Compare the surface production rates under steady state conditions for

the two different drainage patterns and the given reservoir data. Draw

the IPR curves and calculate the productivity index.

B = 1,1 A = 570 acre = 24829200 ft²

μ = 1,5 cP S = 0k = 9 mD h = 35 ft

rw = 0,3281 ft γ = 1,78p = 3000 psi pw = 1200 psi

21




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Separation between oil and water production

e.g. steady state solution:

k … Relative permeability oil (-)kw … Relative permeability water (-)

Relative Permeabilities

q 2πkkh p pw μB ln

0,61.rerw

+ S

qw 2πkkwh p pw

μwBw ln0,61.re

rw + S




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Relative Permeabilities

Example

Water and Oil Production

Calculate the oil and water production rate under steady state

conditions. The water saturation is 55 %.

p = 15 MPa pw = 10 MPare = 100 m rw = 83 mmh = 10 m k = 100 mD

S = 3

μi = 1,62. 10-3 Pas Bi = 1,12μwe = 0,8. 10-3 Pas Bwe = 1,04




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The drainage shape for horizontal wells is ellipsoidal. One must

distinguish between horizontal and vertical permeability.

Horizontal Wells




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Horizontal Wells

Steady state flow in the horizontal plane and pseudo steady state flow in

the vertical plane are assumed:

a

0,5 + 0,25 +

, ,

for

≤ 0,9 . reH

Ii kH

k

q k

H

.h.∆p

141,2. Bμ lna + a²

L2 ²

L2

+Iih

L ln Iih

rw(Ii + 1)




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Example

IPR Horizontal Wells

Calculate the production rate for the given horizontal well.

kH = 10 mD k = 3 mD reH = 500 ft

pi = 5000 psi pw = 2750 psih = 55 ft

B = 1,15μ = 1,55 cP

rw = 0,351 ft




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Homework

Deadline: 25.08.2015 09:00

The following examples should be calculated using the following

undersaturated reservoir properties:




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Homework

1) Infinite – acting oil well:

Develop a production rate profile (q – t diagram) for 1 year assuming

that no boundary effects emerge. Do this in increments of 2 months and

use a pw of 3500 psi.

2) Steady state production:

Assume that the well has a drainage area equal to 640 acre (re = 2980ft)and is producing at steady state with an outer boundary (constant)

pressure equal to 5651 psi. Calculate the steady-state production rate ifthe pw is equal to 4500 psi. Use a skin factor of 10. Describe twomechanisms to increase the flow rate by 50%. (Show calculations)




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Homework

3) No-flow boundary reservoir:

What would be the average reservoir pressure if the outer boundary

pressure is 6000 psi, the pw is 3000 psi, the drainage area 640 acresand the well radius is 0,328 ft ? What would be the ratio of the flow rates

before and after the average reservoir pressure drops by 1000 psi?

Assume skin = 0.

4) Irregular well positioning:

Assume that two wells each drain 640 acre. Furthermore, assume that

p =5651 psi (same as pi) and that S = 0. The pw in both wells is 3500psi. Well A is placed at the center of a square, whereas well B is at thecenter of the upper right quadrant of a square drainage shape. Calculate

the production rates from the two wells for pseudo steady state!




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Homework

5) Transient IPR:

Construct transient IPR curves for 1, 6 and 24 months. Assume zero

skin!

6) Steady state IPR:

Assume that the initial reservoir pressure of the well is also the constant

pressure of the outer boundary. Draw IPR curves for skin effects equal to

0, 5, 10 and 50. Use a drainage radius of 2980 ft (A = 640 acre).

7) Pseudo steady state IPR:

Calculate the IPR curves for zero skin effect but for average reservoir

pressures in increments of 500 psi from initial 5651 psi to 3500 psi.

drainage radius of 2980 ft

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