Nodal Analysis and Natural Flow

Mauricio G. Prado The University of Tulsa

Flow in Production System

Compressed Fluids in the

Reservoir

Porous Media

Perforations

Production String

Downhole Equipment

Restrictions

Surface Flowline

Surface Equipment

Restrictions

Final

Destination


Reservoir

PressureIndividual

Components

Mommentum

Mass and

Energy

balance

Final

Pressure

rP

Driving Force for Production

Energy Difference

Energy

Use

cPfP


Pr

Pr

q

Ps

Pt

Pf

Pc


Path of produced fluids



Compressed Fluids in the Reservoir

Production String

Downhole Equipment

Restrictions

Surface Flowline

Surface Equipment

Restrictions

Final

Destination

Porous Media

Perforations

Flow in Porous Media



Compressed Fluids in the Reservoir

Production

Separator

Porous Media

Perforations

Production String

Downhole Equipment

Restrictions

Surface Flowline

Surface Equipment

Restrictions

Pressure changes in Pipes and Equipment


For this system to be in equilibrium we must have:

Production Flowrate

src PPP = For single phase incompressible fluids, the pressure drop

in each of the components is function of the flowrate.

( )qPP cc = So the equilibrium equation becomes.

src PPqP = )(


We can see that the equilibrium equation is an equation which the independent variable is the flowrate. The flowrate solution for this equation is the equilibrium flowrate of the system

Production Flowrate

eq

src PPqP = )(


We also know that for a certain single phase incompressible fluid, the pressure drop in each component is also function of the properties of the component. For instance the pressure drop in the reservoir is function of the productivity index and pressure drop in pipes is function for instance of pipe diameter, inclination angle and roughness.

Production Flowrate

src PPqP = )()( PropertiesComponentsqq ee =


Components Performance

9Single Phase Incompressible Flow

C-1 C-2 C-3 C-n

1P 2P 3P nP

( )c cP P q = Individual ComponentsAnalysis


For compressible fluids or for multiphase flow, the fluid properties are a strong function of the pressure level in the component.

The pressure drop in each component is then not only function of the flowrate, but also of the a pressure reference on the component.

Production Flowrate

( )PqPP cc ,= So the equilibrium equation becomes.

src PPPqP = ),(


For instance when calculating the pressure available downstream of a pipeline segment, the pressure drop in the segment is function of the flowrates but also of the pressure at the entrance of the pipe segment.

When calculating the pressure required upstream of a pipeline segment, the pressure drop in the segment is function of the flowrates but also of the pressure at the exit of the pipe segment

Production Flowrate

( )upstreamcc PqPP ,=q upstreamP

downstreamPq

( )downstreamcc PqPP ,=


It is obvious then, that the pressure downstream of a component can not be calculate without knowing the behavior of the upstream components.

Also the pressure upstream of a component can not be calculated without knowing the behavior of the downstream components.

The major difference between single and two phase flow problems is that the componenst interact with each other in two phase flow conditions.

Production Flowrate

( )upstreamcc PqPP ,=q upstreamP

downstreamPq

( )downstreamcc PqPP ,=


Components Performance9Multiphase Flow

C-1 C-2 C-3 C-n

1P 2P 3P nP

Individual Components

Analysis

Nodal Analysis),( PqPc

1P 2P 3P


Nodal Analysis Individual components analysis is adequate

when components dont interact with each other. In two phase flow, the pressure drop function not

only of the flowrates but also of the pressure level on the component.

This creates an interdependence between each component.

Individual component analysis is no longer applicable.

A new tool is necessary Nodal Analysis


Nodal Analysis

System Composed of

Interacting Components

rP sP

( ) PqPc ,


Nodal Analysis

sPrP InflowSectionOutflow

Section

Node

( ) PqPc ,


Nodal Analysis

rP InflowSection

),()( PqPPqPIS

crinode =


Nodal Analysis

rP InflowSection

inodeP

),()( PqPPqPIS

crinode =


Nodal Analysis

The inflow pressure at the node represents the pressure that the inflow section can deliver the flowrate q at the node

),()( PqPPqPIS

crinode =


Nodal Analysis

sPOutflow

Section

),()( PqPPqPOS

csonode +=


Nodal Analysis

sPOutflow

SectiononodeP

),()( PqPPqPOS

csonode +=


Nodal Analysis

The outflow pressure at the node represents the pressure that the outflow section requires to produce the flowrate q up to the separator

),()( PqPPqPOS

csonode +=


Nodal Analysis

The equilibrium point is the point at which the inflow section is capable of delivering the flowrate at a pressure enough for the outflow section to flow the fluids up to the separator


Nodal Analysis

( ) ( )i onode nodeP q P q=

eqComponents performance are included

only in the part of the System where the component is located


Nodal Analysis - Example

Production String

Production Separator

Reservoir

Flowline



Production String


Reservoir

Flowline

Node = Perforations


Nodal Analysis Example - Inflow

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flow rate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

rP



0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flow ra te (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

rP

( )resP q



0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

rP

( )resP q

iwfP


Nodal Analysis Example - Outflow

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flow ra te (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

sepP



0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flow ra te (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

sepP

( )lineP q



0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

sepP

( )lineP q



0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flow ra te (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

sepP

owhP

( )tubingP q



0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

sepP

owhP

( )tubingP q

owfP


Nodal Analysis Example

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

iwfP

owfP


Nodal Analysis Example

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

iwfP

owfP

eq

wfP



Production String


Reservoir

Flowline

Node = Wellhead


Nodal Analysis Example - Wellhead

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flow ra te (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

iwhP

owhP



Production String


Reservoir

Flowline

Node = Separator


Nodal Analysis Example - Separator

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flow ra te (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

isepP

sepP



Production String


Reservoir

Flowline

Node = Reservoir Boundary


Nodal Analysis Example - Reservoir

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flow ra te (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

orP

rP


Nodal Analysis

0

1000

2000

3000

4000

5000

6000

0 1000 2000 3000 4000 5000 6000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

g

)

Preservoiri

Preservoiro

Pperforationsi

Pperforationso

Pw ellheadi

Pw ellheadoPseparatori

Pseparatoro


Stable and Unstable Conditions


Stability Generally speaking, mechanical equilibrium is defined as a condition

where the summation of forces acting on a body equal to zero. This means that a body in equilibrium has no acceleration.

The equilibrium can be stable, unstable or indifferent. Stable equilibrium is a condition where after a small disturbance, the

system will return to the original equilibrium position Unstable equilibrium is a condition where after a small disturbance, the

system will move away from the original equilibrium position Indifferent equilibrium is a condition where after a small disturbance,

the system will not move since the points around the original equilibrium condition are also equilibrium points.


Stability For a well, we understand equilibrium as the steady state

condition. The equilibrium flowrate is a flowrate where the IPR and OPR

meet. This equilibrium can also be stable, unstable or indifferent. The nodal analysis is a very powerful tool to determine steady

state equilibrium conditions. We can clearly see that determination of stability conditions

requires an analysis of the behavior of the system after a disturbance.

This analysis requires determination of the performance of the system for points that are not in equilibrium and as a consequence are NOT in steady state.

This is a transient problem and the steady state nodal analysis tool is very limited of fully describing the phenomena.

Nonetheless, an unsteady analysis of the problem can lead us to stability criteria that may be used to check the stability of the equilibrium flowrate determined by Nodal Analysis.


Stability Imagine that we have a closed completion

system as shown. During transient conditions that normally

occur after a disturbance, mass and momentum balance equations are still valid.

In order to investigate the stability, lets examine the case of a single phase incompressible fluid being produced.

If we assume that the fluid is incompressible, the flowrate coming from the reservoir needs to be equal to the flowrate going into the tubing string.

The dynamic Inflow bottonhole flowing pressure needs to be equal to the dynamic outflow bottonhole flowing pressure.

The steady state nodal analysis assume that the fluids are not accelerating and the flow is steady state.

rP

eq

whP

wfP

eq

J


0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

iwfP

owfP


owf

iwf PP > owfiwf PP


Stability Imagine that during a transient

phenomena (for instance due to fluctuations on wellhead pressure) the flowrate in the system becomes smaller than the equilibrium steady state value.

If you observe on the steady state nodal analysis graph you will see that for this condition, the inflow pressure is higher than the outflow pressure

How is this possible ? What is the bottomhole flowing pressure during the transient that follows a disturbance ?

rP

eqq


Stability The solution is as follows. During the transient disturbance,

the true bottonhole flowing pressure is between the steady state inflow and the outflow values.

This difference in pressure (true values compared to the steady state values is going to cause the fluids to accelerate !!! (changes in time transient !!!)

rP

eqq


Stability For the reservoir, since the true

bottomhole pressure is smaller than the steady state value, the flowrate is going to increase.

For the tubing, since the bottomholepressure is greater than the steady state value, the system will also accelerate and the flowrate is going to increase as well.

This transient coupling between reservoir and system is going to promote an increase with time of the flowrate through the system. rP

eqq

Time


Stability A similar analysis can be made

when the fluctuations cause the flowrate to be bigger then the equilibrium value.

rP

eqq >

whP

wfPJ

eqq >


0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

iwfP

owfP


owf

iwf PP


Stability

In this case the true bottonhole flowing pressure is again in between the steady state values for the IPR and OPR.

For the reservoir, since the true bottomholepressure is greater than the steady state value, the flowrate is going to decrease.

For the tubing, since the bottomhole pressure is smaller than the steady state value, the system will also accelerate and the flowrate is going to decrease.

This transient coupling between reservoir and system is going to promote an decrease with time of the flowrate through the system.

rP

eqq >

whP

wfPJ

eqq >


0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

iwfP

owfP


owf

iwf PP


0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

iwfP

owfP


owf

iwf PP

whP

wfPJ

eqq >


0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 1000 2000 3000 4000 5000 6000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

iwfP

owfP


Stable Production Equilibrium Point


Stability In some cases, due to the nature of two

phase flow phenomena, two equilbriumpoints may be possible.

rP

eqq >

whP

wfPJ

eqq >



0

500

1000

1500

2000

2500

3000

3500

4000

0 500 1000 1500 2000 2500 3000

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

iw fP

owfP

A

B

Stable?


Stability What can you say about the equilibrium

conditions for point B ?

rP

eqq >

whP

wfPJ

eqq >



2500

2700

2900

3100

3300

3500

3700

0 50 100 150 200 250 300

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

iwfP

owfP

B

owf

iwf PP < owfiwf PP >


Stability Again, during the disturbance, the true

bottonhole flowing pressure is between the steady state IPR and OPR values.

When the flowrate is smaller then the equilibrium point, the true bottonholepressure is greater than the steady state IPR value.

This will cause the reservoir flowrate to decrease.

When the flowrate is smaller then the equilibrium point, the true bottonholepressure is smaller then the steady state OPR value and this will cause the flowrate in the tubing to decrease.

What will happen ?rP

whP

wfPJ

eqq

eqq >



2500

2700

2900

3100

3300

3500

3700

0 50 100 150 200 250 300

Flowrate (bpd)

P

r

e

s

s

u

r

e

(

p

s

i

)

iwfP

owfP

B

owf

iwf PP >


Stability Point B is an unstable operating point. If the flowrate is suddenly decreased from

the equilibrium point, the well will die. If the flowrate is suddenly increased from

the equilibrium point, the well is going to produce the next stable flowrate value.

rP

whP

wfPJ

eqq

Date post:	28-Aug-2015
Category:	Documents
Upload:	ipbrendamv
View:	29 times
Download:	5 times

Nodal Analysis and Natural Flow

Documents