Nodal Analysis And Natural Flow

Flow in Production System Compressed Fluids in the Reservoir

Porous Media Perforations Production String Downhole Equipment Restrictions Surface Flowline

Final

Surface Equipment

Destination

Restrictions Mauricio G. Prado – The University of Tulsa

Driving Force for Production

Pr

Reservoir

Individual

Pressure

Components Mommentum

Energy Difference

Mass and

Energy

Energy

Use

balance

Pf

Final Pressure

∆Pc Mauricio G. Prado – The University of Tulsa

Ps

∆Pf ∆Pt

∆Pc ∆Pr Pr

q Mauricio G. Prado – The University of Tulsa

Path of produced fluids

Mauricio G. Prado – The University of Tulsa

Flow in Production System Compressed Fluids in the Reservoir

Porous Media Perforations Production String

Flow in Porous Media

Downhole Equipment Restrictions

Final Destination

Surface Flowline Surface Equipment Restrictions Mauricio G. Prado – The University of Tulsa

Flow in Production System Compressed Fluids in the Reservoir

Porous Media Perforations Production String

Pressure changes in Pipes and Equipment

Downhole Equipment Restrictions Surface Flowline

Production

Surface Equipment

Separator

Restrictions Mauricio G. Prado – The University of Tulsa

Production Flowrate •

For this system to be in equilibrium we must have:

∑ ∆P

c

•

= Pr − Ps

For single phase incompressible fluids, the pressure drop in each of the components is function of the flowrate.

∆Pc = ∆Pc (q ) •

So the equilibrium equation becomes.

∑ ∆P (q) = P − P c

r

s

Mauricio G. Prado – The University of Tulsa

Production Flowrate •

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

∑ ∆P (q) = P − P c

r

qe Mauricio G. Prado – The University of Tulsa

s

Production Flowrate •

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.

∑ ∆P (q) = P − P c

r

s

qe = qe (Components Properties)

Mauricio G. Prado – The University of Tulsa

Components Performance 9Single Phase Incompressible Flow

C-1

C-2

C-3

C-n

∆P1

∆P2

∆P3

∆Pn

∆Pc = ∆Pc (q)

Individual Components

Mauricio G. Prado – The University of Tulsa

Analysis

Production Flowrate •

•

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.

∆Pc = ∆Pc (q, P ) •

So the equilibrium equation becomes.

∑ ∆P (q, P) = P − P c

r

Mauricio G. Prado – The University of Tulsa

s

Production Flowrate •

•

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

q

Pupstream

q

∆Pc = ∆Pc (q, Pupstream )

Pdownstream

∆Pc = ∆Pc (q, Pdownstream )

Mauricio G. Prado – The University of Tulsa

Production Flowrate •

• •

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.

q

Pupstream

q

∆Pc = ∆Pc (q, Pupstream )

Pdownstream

∆Pc = ∆Pc (q, Pdownstream )

Mauricio G. Prado – The University of Tulsa

Components Performance 9Multiphase Flow

P1

P3

P2

C-1

C-2

C-3

C-n

∆P1

∆P2

∆P3

∆Pn

∆Pc (q, P )

Nodal Analysis Individual Components Analysis Mauricio G. Prado – The University of Tulsa

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 Mauricio G. Prado – The University of Tulsa

Nodal Analysis System Composed of

Pr

Interacting Components

∑ ∆P (q, P ) c

Mauricio G. Prado – The University of Tulsa

Ps

Nodal Analysis Node Outflow

Pr

Inflow

Section

Section

∑ ∆P (q, P ) c

Mauricio G. Prado – The University of Tulsa

Ps

Nodal Analysis

Pr

Inflow Section

i node

P

(q) = Pr − ∑ ∆Pc (q, P) IS

Mauricio G. Prado – The University of Tulsa

Nodal Analysis

Pr

i node

P

Inflow Section

i node

P

(q) = Pr − ∑ ∆Pc (q, P) IS

Mauricio G. Prado – The University of Tulsa

Nodal Analysis • The inflow pressure at the node represents the pressure that the inflow section can deliver the flowrate q at the node i node

P

(q) = Pr − ∑ ∆Pc (q, P) IS

Mauricio G. Prado – The University of Tulsa

Nodal Analysis Outflow Section

o node

P

Ps

(q) = Ps + ∑ ∆Pc (q, P) OS

Mauricio G. Prado – The University of Tulsa

Nodal Analysis o node

P

o node

P

Outflow Section

Ps

(q) = Ps + ∑ ∆Pc (q, P) OS

Mauricio G. Prado – The University of Tulsa

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

o node

P

(q) = Ps + ∑ ∆Pc (q, P) OS

Mauricio G. Prado – The University of Tulsa

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 Mauricio G. Prado – The University of Tulsa

Nodal Analysis i node

P

(q) = P

o node

(q)

qe Components performance are included only in the part of the System where the component is located Mauricio G. Prado – The University of Tulsa

Nodal Analysis - Example Production Separator Flowline

Production String

Reservoir

Mauricio G. Prado – The University of Tulsa

Nodal Analysis - Example Production Separator Flowline

Node = Perforations

Production String

Reservoir

Mauricio G. Prado – The University of Tulsa

Nodal Analysis Example - Inflow 5000

Pr

4500 4000

P re ssure (psi)

3500 3000 2500 2000 1500 1000 500 0 0

1000

2000

3000

4000

Flow ra te (bpd)

Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis Example - Inflow 5000 4500

Pr

4000

P re ssure (psi)

3500

∆Pres (q)

3000 2500 2000 1500 1000 500 0 0

1000

2000

3000

4000

Flow ra te (bpd) Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis Example - Inflow 5000 4500

Pr

4000

Pre ssure (psi)

3500

∆Pres (q)

3000 2500 2000 1500

i wf

1000

P

500 0 0

1000

2000

3000

4000

Flowrate (bpd) Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis Example - Outflow 5000 4500 4000

P re ssure (psi)

3500 3000 2500 2000 1500 1000

Psep

500 0 0

1000

2000

3000

4000

Flow ra te (bpd)

Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis Example - Outflow 5000 4500 4000

P re ssure (psi)

3500 3000 2500 2000

∆ Pline (q )

1500 1000

Psep

500 0 0

1000

2000

3000

4000

Flow ra te (bpd)

Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis Example - Outflow 5000 4500 4000

Pre ssu re (p si)

3500 3000 2500 2000

∆ Pline (q )

1500 1000

Psep

500 0 0

1000

2000

3000

4000

F lo wrate (b p d )

Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis Example - Outflow 5000 4500 4000

P re ssure (psi)

3500 3000

∆ Ptubing ( q)

2500 2000 1500

Pwho

1000

Psep

500 0 0

1000

2000

3000

4000

Flow ra te (bpd) Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis Example - Outflow 5000 4500

Pwfo

4000

Pre ssu re (p si)

3500 3000

∆ Ptubing ( q)

2500 2000 1500

Pwho

1000

Psep

500 0 0

1000

2000

3000

4000

F lo wrate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis Example 5000

Pwfo

4500 4000

Pre ssu re (p si)

3500 3000 2500 2000 1500 1000

i

Pwf

500 0 0

1000

2000

3000

4000

F lo wrate (b p d )

Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis Example 5000

Pwfo

4500 4000

Pre ssu re (p si)

3500

Pwf

3000 2500 2000

qe

1500 1000

i

Pwf

500 0 0

1000

2000

3000

4000

F lo wrate (b p d )

Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis - Example Production Separator Flowline

Node = Wellhead Production String

Reservoir

Mauricio G. Prado – The University of Tulsa

Nodal Analysis Example - Wellhead 5000 4500 4000

P re ssure (psi)

3500 3000

Pwhi

2500 2000 1500

o

Pwh

1000 500 0 0

1000

2000

3000

4000

Flow ra te (bpd)

Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis - Example Production Separator Flowline

Node = Separator

Production String

Reservoir

Mauricio G. Prado – The University of Tulsa

Nodal Analysis Example - Separator 5000 4500 4000

P re ssure (psi)

3500 3000

i Psep

2500 2000 1500 1000

Psep

500 0 0

1000

2000

3000

4000

Flow ra te (bpd) Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis - Example Production Separator Flowline

Node = Reservoir Boundary Production String

Reservoir

Mauricio G. Prado – The University of Tulsa

Nodal Analysis Example - Reservoir 5000

Pr

4500 4000

P re ssure (psi)

3500

Pro

3000 2500 2000 1500 1000 500 0 0

1000

2000

3000

4000

Flow ra te (bpd)

Mauricio G. Prado – The University of Tulsa

5000

6000

Nodal Analysis 6000

o

P reservoir

o

P perforations

5000

i

P ressure (psig)

P reservoir 4000

3000

i

P perforations 2000

i

P w ellhead

i

P separator

o

P w ellhead

1000

o

P separator

0 0

1000

2000

3000

4000

Flowrate (bpd) Mauricio G. Prado – The University of Tulsa

5000

6000

Stable and Unstable Conditions

Mauricio G. Prado – The University of Tulsa

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.

Mauricio G. Prado – The University of Tulsa

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. Mauricio G. Prado – The University of Tulsa

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.

Mauricio G. Prado – The University of Tulsa

Pwh

qe

qe Pwf

J

Pr

Stable and Unstable Conditions 5000

Pwfo

4500 4000

Pre ssu re (p si)

3500 3000 2500 2000

P >P i wf

1500

o wf

Pwfi < Pwfo

1000

i

Pwf

500 0 0

1000

2000

3000

4000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

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 ?

Pwh

q < qe

q < qe Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions 5000

Pwfo

4500 4000

Pre ssu re (p si)

3500 3000 2500 2000

P >P

1500

i wf

1000

o wf i

Pwf

500 0 0

1000

2000

3000

4000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

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 !!!) Mauricio G. Prado – The University of Tulsa

Pwh

q < qe

q < qe Pwf

J

Pr

Stable and Unstable Conditions 5000

Pwfo

4500 4000

Pre ssu re (p si)

3500 3000 2500 2000

P >P

1500

i wf

1000

o wf i

Pwf

500 0 0

1000

2000

3000

4000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

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 bottomhole pressure 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. Mauricio G. Prado – The University of Tulsa

Pwh

q < qe

q < qe Pwf

J

Pr

Stable and Unstable Conditions 5000

Pwfo

4500 4000

Pre ssu re (p si)

3500 Time

3000 2500 2000

P >P

1500

i wf

1000

o wf i

Pwf

500 0 0

1000

2000

3000

4000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

Stability Pwh

• A similar analysis can be made when the fluctuations cause the flowrate to be bigger then the equilibrium value.

q > qe

q > qe Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions 5000

Pwfo

4500 4000

Pre ssu re (p si)

3500 3000 2500 2000

P

1500

o wf

1000

i

Pwf

500 0 0

1000

2000

3000

4000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

Stability Pwh •

•

•

•

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 bottomhole pressure 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.

q > qe

q > qe Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions 5000

Pwfo

4500 4000

Pre ssu re (p si)

3500 3000 2500 2000

P

1500

o wf

1000

i

Pwf

500 0 0

1000

2000

3000

4000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

Stable and Unstable Conditions 5000

Pwfo

4500 4000

Pre ssu re (p si)

3500

Time

3000 2500 2000

P

1500

o wf

1000

i

Pwf

500 0 0

1000

2000

3000

4000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

Stability • As a consequence, for the illustrated, the transient between the tubing and the will generate the driving promote equilibrium.

Pwh

situation coupling reservoir force to

q > qe

• For any small disturbances in the system, the reservoir and tubing will interact to accelerate the fluids and bring the production back to steady state stable levels.

q > qe Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions 5000

Pwfo

4500 4000

Pre ssu re (p si)

3500 3000 2500 2000

Stable Production Equilibrium Point

1500 1000

i

Pwf

500 0 0

1000

2000

3000

4000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

Stability Pwh

• In some cases, due to the nature of two phase flow phenomena, two equilbrium points may be possible.

q > qe

q > qe Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions 4000 3500

Pwfo

B 3000

Pre ssu re (p si)

A 2500 2000 1500

?

1000

Stable

i

Pw f

500 0 0

500

1000

1500

2000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

2500

3000

Stability Pwh

• What can you say about the equilibrium conditions for point B ?

q > qe

q > qe Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions 3700

Pwfo

Pre ssu re (p si)

3500

3300

B i

3100

Pwf

2900

Pwfi < Pwfo

2700

Pwfi > Pwfo

2500 0

50

100

150

200

F lo wrate (b p d ) Mauricio G. Prado – The University of Tulsa

250

300

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 bottonhole pressure 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 bottonhole pressure is smaller then the steady state OPR value and this will cause the flowrate in the tubing to decrease. • What will happen ?

Pwh

q < qe

q < qe Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions 3700

Pwfo

Pre ssu re (p si)

3500

3300

B i

3100

Pwf

2900

2700

Pwfi < Pwfo

2500 0

50

100

150

200

F lo wrate (b p d ) Mauricio G. Prado – The University of Tulsa

250

300

Stability Pwh

• When the flowrate is greater then the equilibrium point, the true bottonhole pressure is smaller than the IPR value. • This will cause the reservoir flowrate to increase. • The bottonhole pressure is greater then the OPR value and this will cause the flowrate in the tubing to increase. • What will happen ?

q > qe

q > qe Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions 3700

Pwfo

Pre ssu re (p si)

3500

3300

B i

3100

Pwf

2900

2700

Pwfi > Pwfo

2500 0

50

100

150

200

F lo wrate (b p d ) Mauricio G. Prado – The University of Tulsa

250

300

Stability • • •

Pwh

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.

q < qe

q < qe Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions 3700

Pwfo

Pre ssu re (p si)

3500

3300

B i

3100

Pwf

2900

Unstable Production Equilibrium Point

2700

2500 0

50

100

150

200

F lo wrate (b p d ) Mauricio G. Prado – The University of Tulsa

250

300

Stable and Unstable Conditions 4000 3500

Pwfo

B 3000

Pre ssu re (p si)

A 2500 2000 1500

Unstable

1000

Stable

i

Pw f

500 0 0

500

1000

1500

2000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

2500

3000

Stability • This is a very important phenomena very common in the field. • This well has only one stable flowrate. • A well with this IPR-OPR behavior can not produce the stable higher flowrate without some help. • Point B represents a barrier that the well needs to overcome in order to produce under stable conditions. This is usually accomplished by inducing flow in the well. • If this well dies it can not be put into production simply by opening the choke or valves. • THIS WELL REQUIRES ATTENTION !!! Mauricio G. Prado – The University of Tulsa

Pwh

q < qe

q < qe Pwf

J

Pr

Stability • •

•

Pwh

Several techniques are used in the field to induce flow. Some are good and some …. If the well has an artificial lift installed it can be put into production by using the lift system to help the well overcome the unstable point B. After that if artificial lift is not required you can leave the well under natural flow. If you don’t have an artificial lift system installed you can: – – – –

Rock the well Nitrogen Injection Swab Etc…..

q < qe

q < qe Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions Pwf

Pwfo

Pwfi q Mauricio G. Prado – The University of Tulsa

Stability Pwh

• What about this equilibrium condition ? • Is it stable ? • This is very common for wells that have a high reservoir pressure and a very small productivity index.

q < qe

q < qe Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions Pwf

Pwfo

Pwfi q Mauricio G. Prado – The University of Tulsa

Stability • This well is in fact unsteady (as opposed to steady). • Notice that the IPR and OPR intersect each other in a point that is stable. • Observe however that the pressure difference between the IPR and OPR is very small. • Small fluctuations of variables in the system can cause this well to die. Mauricio G. Prado – The University of Tulsa

Pwh

Pwf

J

Pr

Stable and Unstable Conditions Pwf

Pwfo

Pwfi q Mauricio G. Prado – The University of Tulsa

Stability Pwh

• After the fluctuation is gone, the well is able to produce again.

Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions o Pwf Pwf

Pwfi q Mauricio G. Prado – The University of Tulsa

Stability • Again this is a very common phenomena in the field. • It is called “Heading” • The well will produce in cycles. • Basically this well is screaming to you that it can not produce on a steady basis. • The low productivity index and high reservoir pressure makes this well an ideal candidate for INTERMITTENT lift methods.

Pwh

Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions Pwf Pwfo

Unstable Heading Unsteady

Stable Steady

Pwf Pwfo

i wf

P

q

P wfi

q Mauricio G. Prado – The University of Tulsa

Stability Pwh

• Wait !!!! • That is not all. • High productivity index wells can also be PROBLEMATIC

Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions Pwf

Pwfo

Stable Unsteady - Oscillatory Production

Pwfi

q Mauricio G. Prado – The University of Tulsa

Stable and Unstable Conditions Pwf

Pwfo

Smaller Tubing or choke

Pwfo Pwfo

Stable

Pwfi Smaller tubing or choke and Artificial Lift

q Mauricio G. Prado – The University of Tulsa

Stability • • •

• • •

Pwh

A similar unsteady situation can occur with those wells This is somewhat common for deep water production of high productivity wells. In this case the well flowrate may have huge fluctuations during the day as a function of fluctuations on other variables. The well may not die as in the case of heading, but may have huge fluctuations of flowrate and pressure Summary Nodal analysis a very important tool to design production systems for steady state conditions. It can also help detecting stability problems in wells.

Pwf

J Mauricio G. Prado – The University of Tulsa

Pr

Stable and Unstable Conditions Pwf

Pwfo

Unstable – Does not exist

Pwf

Pwfo

Stable Steady

Stable Unsteady - Heading

Pwfi

Pwfi

q

q

Pwf Pwfo

Pwfi

Stable Unsteady - Oscillatory Production

q Mauricio G. Prado – The University of Tulsa

An Analogy • Lets use a mechanical analogy to get some insight on how important the dynamic equations are for describing the stability of a system. • Lets imagine too magnetic rods facing each other. The force that one rod acts on the other depends on the polarity of the facing tips of the magnets. If they are of equal polarity we have repulsion. If they have different polarities we have attraction. • The force also depends on the distance between the two facing tips of the magnets. The force between the two magnets is inversely proportional to the distance between them. The greater the distance the smaller the magnitude of the force. The smaller the distance the bigger the magnitude of the force

Mauricio G. Prado – The University of Tulsa

An Analogy • • • •

Lets imagine we have two fixed magnets facing each other. The tips of the magnets that are facing each other have different polarities A third magnet is located in between the two fixed magnets Lets analyze the force balance on this third magnet as a function of its relative position in relation to the mid distance between the fixed magnets x

F2

F1

- M1 +

magnet

x

Mauricio G. Prado – The University of Tulsa

-

M2 +

An Analogy -

+

-

+

-

F1

+

-

+

+

F2

F1

F2

x

Force

Force

x

-

-

-1

-0.5

0

0.5

-1

1

-0.5

0

0.5

1

0.5

1

Relative Position

Relative Position

F1+ F2 Resultant Force

Resultant Force

F1+ F2

-1

-0.5

0

0.5

1

-1

Relative Position

-0.5

0 Relative Position

Mauricio G. Prado – The University of Tulsa

+

An Analogy It seems that from the Steady State analysis that an equilibrium condition exists when the third magnet is placed exactly in the mid point between the two fixed magnets This steady state equilibrium point exists independent of the orientation of the third magnet since at that location the resulting force is zero and the steady state force balance is satisfied Do you think this is really possible ?

• •

+

-

-

+

F1+ F2=0

Resultant Force

-

-1

-0.5

+

-

+

+

-

0

0.5

1

-

F1+ F2=0

Resultant Force

•

-1

Relative Position

-0.5

0 Relative Position

Mauricio G. Prado – The University of Tulsa

0.5

1

+

An Analogy •

For both cases a steady state analysis yields an equilibrium point but only the second case is stable In the first case the third magnet never stays in the steady state equilibrium position (mid point between magnets). It always moves to one of the magnets. The steady state equilibrium point for this case is UNSTABLE For the second case, the magnet is capable of staying in the steady state equilibrium position. It is a STABLE point and small disturbances do not affect the final equilibrium position This can only be understood by a analysis of the dynamics of the system

• • •

-

+

-

-

+

+ -

+

-

+

-

-

-

+

Resultant Force

Resultant Force

-

+

+

-1

-0.5

0

0.5

1

-1

Relative Position

-0.5

0 Relative Position

Mauricio G. Prado – The University of Tulsa

0.5

1

+

An Analogy •

For small deviations from the equilibrium point we need to examine the resulting force behavior as a function of the position oscillation In the first case the resulting force will move the magnet in the positive direction increasing the disturbance In the second case the resulting force will move the magnet in the negative direction reducing the disturbance

• •

-

+

+

-

+

-

+

+

-

-

F

F

Resultant Force

Resultant Force

-

-1

-0.5

0

0.5

1

-1

Relative Position

-0.5

0 Relative Position

Mauricio G. Prado – The University of Tulsa

0.5

1

+

An Analogy • This shows that the analysis of the stability has to do not with the steady state force balance, but with the way the net force is related to the magnet position. This has to do with the shape of the force balance around the equilibrium point

-

+

+

-

+

-

+

+

-

-

F

F

Resultant Force

Resultant Force

-

-1

-0.5

0

0.5

1

-1

Relative Position

-0.5

0 Relative Position

Mauricio G. Prado – The University of Tulsa

0.5

1

+

An Analogy • A steady state force balance tells only what the steady state equilibrium points are • The shape of the steady state curve may help us in discerning between stable and unstable points.

a= V=

Transient

dV dt

-

+

+

dx dt

-

-

F

Resultant Force

F ( x) = ma

d 2x F ( x) = m 2 dt Steady State

F ( x) = 0

-1

Mauricio G. Prado – The University of Tulsa

-0.5

0 Relative Position

0.5

1

+

Casing Heading • Unsteady flow conditions occur due to the response of the system to natural fluctuations in some of the variables. • In the previous heading example, we saw how a well can produce intermittently if the wellhead or separator pressure fluctuates. This may occur due to the interference from the production of other wells on a manifold or a common production separator pressure. • Another reason for the heading phenomenon is the natural segregation of gas and liquid at the entrance of the tubing. • This phenomenon is very common as is usually referred as “casing heading”.

Mauricio G. Prado – The University of Tulsa

Casing Heading • Lets imagine we have the well producing with a certain dynamic liquid level in the annulus. • The casing is closed and we have no packer in the well • The well is producing a very small liquid flowrate from the reservoir • At the intake of the tubing a natural segregation of the phases will force part of the gas to bypass the tubing and flow upwards in the annulus. • Since gas is moving upwards in the annulus and being pressurized in the top part of the casing, the liquid level is slowly being lowered by forcing casing liquid into the tubing Mauricio G. Prado – The University of Tulsa

qltubing q

qlcasing qg ql

casing g

q gtubing

Casing Heading • The GLR in the tubing is smaller than the production GLR since part of the gas is bypassing the tubing intake and the liquid flowrate is being increased by the liquid being produced from the annulus. • This process continues, with gas pushing the liquid level down • A significant amount of gas is being stored in the upper part of the casing

qltubing q

qlcasing qg ql

Mauricio G. Prado – The University of Tulsa

casing g

q gtubing

Casing Heading • •

• •

• •

Eventually the liquid level is going to reach the tubing intake. At this point, no more gas can be stored in the annulus and gas is going to start to enter the tubing. The liquid flowrate now in the tubing is only the liquid flowrate from the reservoir The GLR inside the tubing is now considerably greater than the production GLR since some extra gas is being provided by the casing This reduces the bottomhole flowing pressure allowing the well to increase the flowrate We will see an increase then in the surface flowrate

qltubing q

qg ql

Mauricio G. Prado – The University of Tulsa

casing g

q gtubing

Casing Heading • Eventually the gas from the casing is dissipated and the GLR in the tubing is reduced. This limits the flowrate that can be lifted by the tubing • The reservoir is still producing a high flowrate and part of the liquid production start to be accumulated in the annulus • The bottomhole flowing pressure will increase with time since the dynamic liquid level is being raised • Gas start to bypass the tubing intake reducing the GLR in the tubing further reducing the liquid flowrate in the tubing and increasing the liquid storage in the annulus Mauricio G. Prado – The University of Tulsa

qltubing q

qlcasing qg ql

casing g

q gtubing

Casing Heading • The flowrate from the reservoir is still bigger than the flowrate from the tubing and the liquid level continue to rise. • The bottom hole flowing pressure continues to increase reducing the flowrate coming from the reservoir • Gas continues to bypass the tubing intake being accumulated in the annulus and also contributing for the increase in the bottomhole flowing pressure

qltubing q

qlcasing qg ql

Mauricio G. Prado – The University of Tulsa

casing g

q gtubing

Casing Heading • Eventually the increase in bottomhole flowing pressure creates a balance between the tubing liquid outflow capacity and the reservoir inflow capacity. • The liquid level stabilizes and the flowrates in the tubing and from the reservoir are equal. • Gas continues to bypass the tubing intake and to accumulate in the top portion of the casing. • The process then restarts when the gas accumulated at the top starts to push the liquid level down. Mauricio G. Prado – The University of Tulsa

qltubing q

qg ql

casing g

q gtubing

Two Phase Flow Phenomena • Lets now summarize those results and obtain a “model” for two phase flow OPR. • This is NOT an accurate model. • Its purpose is just to illustrate some of the phenomena characteristic of two phase flow. • We will neglect the mass transfer between the phases. • We will assume gas never goes in or out of solution. • We will also assume the gas to be “incompressible” for the sake of simplicity. • We will neglect transient effects on the mass balance equation (no fluid segregation is possible) • We still need a correlation for the slip velocity so that we can develop a “transient” momentum balance equation.

Mauricio G. Prado – The University of Tulsa

Transient Two Phase Flow • We will use a very simplified version of the drift flux model for two phase flow. • The drift flux model consists of two mass balance equations (one for each phase), a mixture momentum balance equation and a closure equation for the slip velocity or void fraction. • We are interested in very short transients. We will assume that the void fraction does not change with time or position and we are interested only in the behavior of the liquid flowrate and pressure with time. • As a consequence our model only needs the momentum balance equation and a closure equation for the void fraciton • Lets start by examining the steady state momentum balance equation and slip closure relationship. Mauricio G. Prado – The University of Tulsa

Total Pressure Gradient •

The Steady State OPR is given by:

ft bbl/d

psi

Pwfo = Pwh + 0.433 γ m sin( β ) L + 1.1471 10 −5 f mmoody

γ m q m qm d

5 p

L in

Mauricio G. Prado – The University of Tulsa

Moody Friction Factor • For the multiphase fricton factor we will use Moody friction factor given by:

f moody

⎡ 8 12 ⎤ 1 ⎛ ⎞ ⎥ = 8 ⎢⎜ ⎟ + 3 ⎢⎝ Re ⎠ ⎥ 2 ( ) A + B ⎣ ⎦

⎡ ⎤ ⎢ ⎥ 1 ⎥ A = ⎢2.457 ln 0.9 ⎢ ⎛ 7 ⎞ ⎛ ε ⎞⎥ ⎜ ⎟ + 0.27⎜ ⎟ ⎥ ⎢ ⎝ Re ⎠ ⎝ d ⎠⎦ ⎣

1 12

16

16

⎛ 37530 ⎞ B=⎜ ⎟ ⎝ Re ⎠

Mauricio G. Prado – The University of Tulsa

Summary of Single Phase Flow in Pipes •

The Reynolds Number is given by: bbl/d

Re = 92.2

γ m qm µm d p

in

cp

•

The mixture flowrate is given by:

qm = ql + q g qm = ql + ql VLR bbl/d

bbl/bbl

qm = ql (1 + VLR ) Mauricio G. Prado – The University of Tulsa

Frictional Term • The mixture properties are defined as: ρ m = ρl α l + ρ g α g µ m = µl α l + µ g α g lb/ft3

γm =

ρl α l + ρ g α g 62.4

Mauricio G. Prado – The University of Tulsa

Frictional Term • The gas phase fraction is given by the closure equation:

αg =

VLR V A 1 + VLR + ∞ p ql

VLR

αg =

1 + VLR +

2 π V∞ d p

4

ql

bbl/bbl ft/s

αg =

VLR 1 + VLR + 83.93

V∞ d p2 ql

in

bpd Mauricio G. Prado – The University of Tulsa

Frictional Term • The gas phase fraction is given by: V∞ V∞ = 8

24 Cd = Re ∞

rd ( ρ l − ρ g ) g sin( β ) 3 Cd ρ l

Re ∞ =

2 rd V∞ ρ l

µl

2 ( ρ l − ρ g )rd V∞ = 9 µl ft/s

Cd = 12

2

g sin( β ) in

lb/ft3

V∞ = 73.94

( ρ l − ρ g )rd

µl

2

sin( β ) cp

Mauricio G. Prado – The University of Tulsa

µl

rd V∞ ρ l

Frictional Term • For the Steady state OPR we have: Pwh

P (q ) = Pwh + ∆Pg (q ) + ∆Pf (q ) o wf

Pwh

∆Pg + ∆Pf

• If the pressure changes from the steady state equilibrium value, the system must accelerate ∆Pg + ∆Pf

q

q

∑ F = ma

(P

wf

− Pwh − ∆Pg (q) − ∆Pf (q ) )Ap = ma Ap

Pwf − Pwh − ∆Pg (q ) − ∆Pf (q) =

ma Ap Pwf

o wf

P

Mauricio G. Prado – The University of Tulsa

Frictional Term • If the pressure changes from the steady state equilibrium value, the system must accelerate Pwh ∆Pg + ∆Pf

q

Ap

L

Pwf − Pwh − ∆Pg (q ) − ∆Pf (q ) =

m = L Ap ρ m

a=

1 dqm Ap dt

Pwf − Pwh − ∆Pg (q ) − ∆Pf (q ) =

Pwf Mauricio G. Prado – The University of Tulsa

ma Ap

L ρ m dqm Ap dt

Frictional Term • For the transient OPR we obtain: bpd psi

ft

⎛ γq q ⎞ L dq ⎜ Pwf (t ) − Pwh (t ) − 0.433γ m sin( β ) L − 1.1471 10−5 f Moody m 5 m L ⎟ = 1.6046 10− 4 γ m 2 m ⎜ ⎟ dt d t dt ⎝ ⎠ in

• To simulate a Standing Valve we have: Pwf (t ) = 1.6046 10 − 4 γ m

γ m qm qm L dqm −5 γ β + P ( t ) + 0 . 433 sin ( ) L + 1 . 1471 10 f L + ∆Psv wh m Moody d t2 dt d t5 ⎧0 if q > 0 ∆Psv = ⎨ ⎩ BigNumber q if

q<0

Mauricio G. Prado – The University of Tulsa

s

Instability Analysis – The Phenomena • We need to determine for each time step what is the value of the flowrate that makes the inflow pressure equals to the transient outflow pressure.

⎧ γ m q m qm L dqm −4 −5 γ γ β P t P t sin L f L + ∆Psv ( ) 1 . 6046 10 ( ) 0 . 433 ( ) 1 . 1471 10 = + + + ⎪ wf m wh m Moody ⎪ d t2 dt d t5 ⎨ ⎪ P (t ) = P − ql r ⎪⎩ wf J

⎧0 if q > 0 ∆Psv = ⎨ ⎩ BigNumber q if

q<0

Mauricio G. Prado – The University of Tulsa

Example

Mauricio G. Prado – The University of Tulsa

Example •

Naturally Flowing Well – Stable - Opening : 7000

1.6046 10 − 4 γ m

L dqm d t2 dt

6000 Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

Pressure (psi)

5000

4000

3000

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 6000 psi 1 bpd/psi 1 bbl/bbl 300 psi

2000

1000

0 0

1000

2000

3000

4000

Flowrate (bpd) Mauricio G. Prado – The University of Tulsa

5000

6000

7000

Example Naturally Flowing Well – Stable - Opening : 7000

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

6000

Flowrate (bpd)

5000

4000

8000

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 6000 psi 1 bpd/psi 1 bbl/bbl 300 psi

7000

6000

5000

4000 3000 3000 2000 2000 1000

1000

0

0 0

5

10

15 Time (s)

Mauricio G. Prado – The University of Tulsa

20

25

Pressure (psi)

•

Example •

Naturally Flowing Well – Stable - Opening : Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

7000

6000

Pressure (psi)

5000

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 6000 psi 1 bpd/psi 1 bbl/bbl 300 psi

4000

3000 Bottomhole

2000 Surface

1000

0 0

5

10

15 Time (s)

Mauricio G. Prado – The University of Tulsa

20

25

Example Naturally Flowing Well – Stable Well – Surface Pressure Fluctuations Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

600

Surface Pressure (psi)

500

400

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 6000 psi 1 bpd/psi 1 bbl/bbl 300 psi

3900

3850

3800

300

3750

200

3700

100

3650

0

3600 0

5

10

15 Time (s)

Mauricio G. Prado – The University of Tulsa

20

25

Bottomhole Pressure (psi)

•

Example Naturally Flowing Well – Stable Well – Surface Pressure Fluctuations 7000

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

6000

Flowrate (bpd)

5000

4000

8000

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 6000 psi 1 bpd/psi 1 bbl/bbl 300 psi

7000

6000

5000

4000 3000 3000 2000 2000 1000

1000

0

0 0

5

10

15 Time (s)

Mauricio G. Prado – The University of Tulsa

20

25

Pressure (psi)

•

Example •

Naturally Flowing Well – Stable - Opening: 7000

1.6046 10 − 4 γ m 6000

L dqm d t2 dt

Pressure (psi)

5000

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

4000

3000

2000

1000

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

0 0

500

1000

1500

2000

2500

3000

Flowrate (bpd) Mauricio G. Prado – The University of Tulsa

3500

4000

4500

Example •

Naturally Flowing Well – Stable - Opening : 7000

5000 4500

6000 4000 3500 Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

4000

3000

2000

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

3000 2500 2000 1500 1000

1000 500 0

0 0

5

10

15 Time (s)

Mauricio G. Prado – The University of Tulsa

20

25

Pressure (psi)

Flowrate (bpd)

5000

Example •

Naturally Flowing Well – Stable - Opening:

•

When this well is shut down at the surface, the required pressure is higher than the reservoir pressure. That means that the liquid hidrostatic load kills the well. When we open the choke to put this well into production, the well is incapable of reaching the natural equilibrium flowrate. Observe that indeed there are two equilibrium flowrates at 854 bpd and at 100 bpd. Lets examine what happens with this well when when the surface pressure is set to 300 psi. Lets examine the cases when the initial flowrate before the disturbance on the surface pressure is:

• • •

– – – –

99 bpd 101 bpd 400 bpd 2000 bpd

Mauricio G. Prado – The University of Tulsa

Example Naturally Flowing Well – Opening – Inital Flowrate 99 bpd : 120

4020

4000

100

80

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

60

40

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

3960

3940

3920

20

3900

0

3880 0

5

10

15 Time (s)

Mauricio G. Prado – The University of Tulsa

20

25

Pressure (psi)

3980

Flowrate (bpd)

•

Example •

Naturally Flowing Well – Stable - Opening: 7000

1.6046 10 − 4 γ m 6000

dq <0 dt

5000

Pressure (psi)

L dqm d t2 dt

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

4000

3000

2000

1000

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

0 0

500

1000

1500

2000

2500

3000

Flowrate (bpd) Mauricio G. Prado – The University of Tulsa

3500

4000

4500

Example 900

4500

800

4000

700

3500

600

3000 Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

500 400 300 200

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

2500 2000 1500 1000

100

500

0

0 0

5

10

15 Time (s)

Mauricio G. Prado – The University of Tulsa

20

25

Pressure (psi)

Naturally Flowing Well – Opening – Inital Flowrate 101 bpd :

Flowrate (bpd)

•

Example •

Naturally Flowing Well – Stable - Opening: 7000

1.6046 10 − 4 γ m 6000

dq >0 dt

5000

Pressure (psi)

L dqm d t2 dt

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

4000

3000

2000

1000

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

0 0

500

1000

1500

2000

2500

3000

Flowrate (bpd) Mauricio G. Prado – The University of Tulsa

3500

4000

4500

Example Naturally Flowing Well – Opening – Inital Flowrate 400 bpd : 3700

900 800

3600 700

500 400 300

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

3500

3400

3300

200 3200 100 0

3100 0

5

10

15 Time (s)

Mauricio G. Prado – The University of Tulsa

20

25

Pressure (psi)

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

600 Flowrate (bpd)

•

Example •

Naturally Flowing Well – Stable - Opening: 7000

1.6046 10 − 4 γ m 6000

dq >0 dt

5000

Pressure (psi)

L dqm d t2 dt

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

4000

3000

2000

1000

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

0 0

500

1000

1500

2000

2500

3000

Flowrate (bpd) Mauricio G. Prado – The University of Tulsa

3500

4000

4500

Example Naturally Flowing Well – Opening – Inital Flowrate 2000 bpd : 2500

3500

3000

1500

1000

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

2500

2000

1500

1000 500 500

0

0 0

5

10

15 Time (s)

Mauricio G. Prado – The University of Tulsa

20

25

Pressure (psi)

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

2000

Flowrate (bpd)

•

Example •

Naturally Flowing Well – Stable - Opening: 7000

1.6046 10 − 4 γ m 6000

dq <0 dt

5000

Pressure (psi)

L dqm d t2 dt

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

4000

3000

2000

1000

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

0 0

500

1000

1500

2000

2500

3000

Flowrate (bpd) Mauricio G. Prado – The University of Tulsa

3500

4000

4500

Example •

Naturally Flowing Well – Stable Point – Surface Pressure Fluctuations 3200

600

3180 500

Surface Pressure (psi)

3140 400 3120 300

3100 Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

200

100

3080

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

3060 3040 3020

0

3000 0

5

10

15 Time (s)

Mauricio G. Prado – The University of Tulsa

20

25

Bottomhole Pressure (psi)

3160

Example •

Naturally Flowing Well – Stable Point – Surface Pressure Fluctuations 4000

1000 900

3500

800 3000

2500

600 Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure

500 400 300 200

10000 ft 1.995” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4000 psi 1 bpd/psi 1 bbl/bbl 300 psi

2000

1500

1000

500

100 0

0 0

5

10

15 Time (s)

Mauricio G. Prado – The University of Tulsa

20

25

Pressure (psi)

Flowrate (bpd)

700

Example • It is clear that this well has two equilibrium points, but the first one at 100 bpd is UNSTABLE. • When shut down, this well is not capable of producing a stable equilibrium flowrate once opened to production • If by some means we can put this well to produce a certain flowrate slightly higher than the first equilibrium point (100 bpd) the well will naturally increase its production till the second stable equilibrium point of 854 bpd. • This is a well that needs a “kick-off” • Why is the first equilibrium point unstable ? • Can we obtain a criteria for stability ?

Mauricio G. Prado – The University of Tulsa

Instability and Cyclic Production • Unstable equilibrium point and cyclic production are two different phenomena. • The first one, the well has an equilibrium point, but any disturbances (they always exist) will make this equilibrium point unstable and the well will seek a more stable condition (either die or find a new stable equilibrium point). • The cyclic production (heading) occurs when a well has a stable equilibrium point, but oscillations in some varialbe cause this equilibrium point to have dramatic changes. The well may produce for a period of time, then dies, then comes back into production. This is not an unstable condition. Mauricio G. Prado – The University of Tulsa

Example •

Naturally Flowing Well – Opening – High Reservoir Pressure : 7000

6000

Pressure (psi)

5000

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

4000

3000

2000

1000

10000 ft 1.000” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 5000 psi 0.08 bpd/psi 0.4 bbl/bbl 300 psi 76 bpd

0 0

100

200

300

400

500

600

700

Flowrate (bpd)

Mauricio G. Prado – The University of Tulsa

800

900

1000

Example •

Naturally Flowing Well – Opening – High Reservoir Pressure : 350

300

Pressure (psi)

250

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

200

150

100

50

10000 ft 1.000” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 5000 psi 0.08 bpd/psi 0.4 bbl/bbl 300 psi 76 bpd

0 0

50

100

150 Time (s)

Mauricio G. Prado – The University of Tulsa

200

250

Example •

Naturally Flowing Well – Opening – High Reservoir Pressure : 90

6000

80 5000 70 4000

50

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

40 30 20 10

10000 ft 1.000” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 5000 psi 0.08 bpd/psi 0.4 bbl/bbl 300 psi 76 bpd

3000

2000

1000

0

0 0

50

100

150 Time (s)

Mauricio G. Prado – The University of Tulsa

200

250

Pressure (psi)

Flowrate (bpd)

60

Instability and Cyclic Production • This well showed some cyclic production, but was reasonably stable. • Notice that the IPR and OPR intercept to the left of the minimun. • The equilibrium flowrate is stable but will oscillate according to the variation of the operating conditions. • What will happen when the reservoir pressure decline ?

Mauricio G. Prado – The University of Tulsa

Example •

Naturally Flowing Well – Opening – Low Reservoir Pressure : 4700

4600

7000 Pressure (psi)

4500

6000

4400

4300

4200

4100

4000

Pressure (psi)

5000

0

10

20

30

40

50

60

70

Flowrate (bpd)

4000

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

3000

2000

1000

10000 ft 1.000” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4620 psi 0.08 bpd/psi 0.4 bbl/bbl 300 psi 25 bpd

0 0

100

200

300

400

500

600

700

Flowrate (bpd) Mauricio G. Prado – The University of Tulsa

800

900

1000

Example •

Naturally Flowing Well – Opening – Low Reservoir Pressure : 350

300

Pressure (psi)

250 Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

200

150

100

10000 ft 1.000” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4620 psi 0.08 bpd/psi 0.4 bbl/bbl 300 psi 25 bpd

50

0 0

50

100

150 Time (s)

Mauricio G. Prado – The University of Tulsa

200

250

Example •

Naturally Flowing Well – Opening – Low Reservoir Pressure : 35

4700 4650

30 4600 4550

25

20

4450 4400

15

4350 10

4300 4250

5 4200 0 0

50

100

150 Time (s)

Mauricio G. Prado – The University of Tulsa

200

4150 250

Pressure (psi)

Flowrate (bpd)

4500

Instability and Cyclic Production • • • •

This well is clearly heading. This is not an unstable problem It is a cyclic stable production. Can occur in several situations. Usually due to the proximity of the OPR and IPR. Any disturbance in the operational parameters can kill the well. • There are several solutions but clearly the problem of this well is productivity. So this is usually a natural candidate for a stimulation job. • The other solutions may be: – If the well is on gas lift, increase the injection GLR – Reduce tubing size (workover or velocity string) • Reduce tubing size and increase GLR Mauricio G. Prado – The University of Tulsa

Example •

Naturally Flowing Well – Gas Lift : 7000

6000 Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

Pressure (psi)

5000

4000

3000

10000 ft 1.0” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4620 psi 0.08 bpd/psi 1.0 bbl/bbl 300 psi 102 bpd

2000

1000

0 0

100

200

300

400

500

600

700

Flowrate (bpd)

Mauricio G. Prado – The University of Tulsa

800

900

1000

Example Naturally Flowing Well – Opening –Gas Lift : 350

300

250 Wellhead Pressure (psi)

•

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

200

150

100

10000 ft 1.0” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4620 psi 0.08 bpd/psi 1.0 bbl/bbl 300 psi 102 bpd

50

0 0

50

100

150 Time (s)

Mauricio G. Prado – The University of Tulsa

200

250

Example Naturally Flowing Well – Opening –Gas Lift :

120

5000 4500

100 4000 3500 3000

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

60

40

20

10000 ft 1.0” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4620 psi 0.08 bpd/psi 1.0 bbl/bbl 300 psi 102 bpd

2500 2000 1500 1000 500

0 0

50

100

150 Time (s)

Mauricio G. Prado – The University of Tulsa

200

0 250

Pressure (psi)

80 Flowrate (bpd)

•

Instability and Cyclic Production •

The Gas Lift was able to eliminate the heading and increase the flowrate to a steady value of 102 bpd. If the productivity index remains the same this solution may work untill the reservoir pressure reduces to 4200 psi 7000

6000

5000

Pressure (psi)

•

4000

3000

2000

1000

0 0

100

200

300

400

500

600

700

Flowrate (bpd)

Mauricio G. Prado – The University of Tulsa

800

900

1000

Example •

Naturally Flowing Well – Velocity String :

7000

6000

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

Pressure (psi)

5000

4000

3000

10000 ft 0.6” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4620 psi 0.08 bpd/psi 0.4 bbl/bbl 300 psi 45 bpd

2000

1000

0 0

100

200

300

400

500

600

700

800

Flowrate (bpd)

Mauricio G. Prado – The University of Tulsa

900

1000

Example Naturally Flowing Well – Opening –Velocity String : 350

300

250 Wellhead Pressure (psi)

•

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

200

150

100

10000 ft 0.6” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4620 psi 0.08 bpd/psi 0.4 bbl/bbl 300 psi 45 bpd

50

0 0

50

100

150 Time (s)

Mauricio G. Prado – The University of Tulsa

200

250

Example •

Naturally Flowing Well – Opening –Velocity String :

4700

60

4600 50

Flowrate (bpd)

40

30

20

10000 ft 0.6” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4620 psi 0.08 bpd/psi 0.4 bbl/bbl 300 psi 45 bpd

4400

4300

4200

4100

10 4000

0 0

50

100

150 Time (s)

Mauricio G. Prado – The University of Tulsa

200

3900 250

Pressure (psi)

4500 Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

Instability and Cyclic Production •

The velocity string or reduction of tubing size was able to eliminate the heading and increase the flowrate to a steady value of 45 bpd. If the productivity index remains the same this solution may work untill the reservoir pressure reduces to 4385 psi 7000

6000

5000

Pressure (psi)

•

4000

3000

2000

1000

0 0

100

200

300

400

500

600

700

Flowrate (bpd)

Mauricio G. Prado – The University of Tulsa

800

900

1000

Example Naturally Flowing Well – Opening – Gas Lift + Velocity String : 7000

6000 Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

5000

Pressure (psi)

•

4000

3000

2000

10000 ft 0.6” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4620 psi 0.08 bpd/psi 1 bbl/bbl 300 psi 76 bpd

1000

0 0

100

200

300

400

500

600

700

Flowrate (bpd)

Mauricio G. Prado – The University of Tulsa

800

900

1000

Example Naturally Flowing Well – Opening – Gas Lift + Velocity String : 350

300

250 Wellhead Pressure (psi)

•

Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

200

150

100

50

10000 ft 0.6” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4620 psi 0.08 bpd/psi 1 bbl/bbl 300 psi 76 bpd

0 0

50

100

150 Time (s)

Mauricio G. Prado – The University of Tulsa

200

250

Example •

Naturally Flowing Well – Opening – Gas Lift + Velocity String :

90

5000

80

4500 4000

70

3500 Well Depth Tubing Roughness Liquid Viscosity Liquid Density Gas Viscosity Gas Density Interf. Charact. Length Reservoir Pressure Productivity Index VLR Wellhead Pressure Equilibrium Flowrate

50 40 30 20

10000 ft 0.6” 0.00015 ft 1 cp 62.4 lb/ft3 0.01 cp 6.24 lb/ft3 0.015576 in 4620 psi 0.08 bpd/psi 1 bbl/bbl 300 psi 76 bpd

3000 2500 2000 1500 1000

10

500

0

0 0

50

100

150 Time (s)

Mauricio G. Prado – The University of Tulsa

200

250

Pressure (psi)

Flowrate (bpd)

60

Instability and Cyclic Production •

The velocity string or reduction of tubing size and the gas lift were able to eliminate the heading and increase the flowrate to a steady value of 76 bpd. If the productivity index remains the same this solution may work untill the reservoir pressure reduces to 3800 psi 7000

6000

5000

Pressure (psi)

•

4000

3000

2000

1000

0 0

100

200

300

400

500

600

700

Flowrate (bpd)

Mauricio G. Prado – The University of Tulsa

800

900

1000

Instability Analysis – The Phenomena • Lets examine if it is possible to obtain an analytical criteria for the stability of this problem. • We start first with a small disturbance that moves the system away from the equilibrium point. • The following equations are then valid for this transient problem after the disturbance. • This is valid also for two phase flow

⎛ γ qq ⎞ L dq −4 ⎜ Pwf (t ) − Pwh − 0.433γ L − 1.1471 10 −5 f Moody ⎟ L = 1 . 6046 10 γ ⎜ d t5 ⎟⎠ d t2 dt ⎝

⎛ rr ⎞ ⎛ rr ⎞ qµ − 7 γ dq ⎜ ⎟ Pwf (t ) − Pr + ln⎜ ⎟ = −1.3929 10 ln⎜⎜ ⎟⎟ h dt ⎝ rw ⎠ 0.00708 k h ⎝ rw ⎠ Mauricio G. Prado – The University of Tulsa

Stable and Unstable Conditions 5000 4500

Pwfo

Pwfo

4000

Pwf (t )

Pre ssu re (p si)

3500

Pwfi

3000

qe

2500

q(t )

2000 1500 1000

i

Pwf

500 0 0

1000

2000

3000

4000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

Instability Analysis – The Phenomena • But for the steady state problem we have: P = Pwh + 0.433γ L + 1.1471 10 f Moody −5

o wf

γ qq d

5 t

L

⎛ rr ⎞ qµ P = Pr − ln⎜⎜ ⎟⎟ 0.00708 k h ⎝ rw ⎠ i wf

• Then we have a relationship between the steady state and transient bottomhole pressures and the fluid accelerations:

Pwf (t ) − Pwfo = 1.6046 10 − 4 γ

Pwf (t ) − Pwfi = −1.3929 10−7

L dq d t2 dt

γ dq ⎛ rr ⎞ ln⎜⎜ ⎟⎟ h dt ⎝ rw ⎠

Mauricio G. Prado – The University of Tulsa

Stable and Unstable Conditions 5000 4500

Pwfo

Pwfo

4000

1.6046 10 − 4 γ

Pre ssu re (p si)

3500

L dq d t2 dt

Pwf (t )

Pwfe

− 1.3929 10−7

3000

γ dq ⎛ rr ⎞ h dt

ln⎜⎜ ⎟⎟ ⎝ rw ⎠

qe

2500

q

2000

Pwfi

1500 1000

i

Pwf

500 0 0

1000

2000

3000

4000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

Instability Analysis – The Phenomena • The steady state outflow and inflow pressure are not linear funcitons of the flowate

L dq Pwf (t ) − P = 1.6046 10 γ 2 d t dt −4

o wf

Pwf (t ) − P = −1.3929 10 i wf

−7

γ dq ⎛ rr ⎞ ln⎜⎜ ⎟⎟ h dt ⎝ rw ⎠

• We have a better chance of solving this system if somehow we can find a linear approximation for the IPR and OPR pressures. • This can be done by a 1st order Taylor expansion around the equilibrium point.

Mauricio G. Prado – The University of Tulsa

Instability Analysis – The Phenomena • The steady state IPR and OPR can be linearized around the equilibrium point for a small disturbance as: Pwfo = Pwfe +

Pwfi = Pwfe +

dPwfo dq

(q − qe ) + .... qe

dPwfi dq

(q − qe ) + ..... qe

Mauricio G. Prado – The University of Tulsa

Stable and Unstable Conditions 5000 4500

Pwfo

Pwfo

dPwfo

4000

Pre ssu re (p si)

3500

dq

Pwfe

dPwfi

3000

dq

(q − qe )

(q − qe )

qe

2500

q

2000

Pwfi

1500 1000

i

Pwf

500 0 0

1000

2000

3000

4000

F lo w rate (b p d ) Mauricio G. Prado – The University of Tulsa

5000

6000

Instability Analysis – The Phenomena • Then we have:

Pwf (t ) − Pwfe −

Pwf (t ) − Pwfe −

dPwfo dq

(q − qe ) = 1.6046 10 − 4 γ qe

dPwfi dq

(q − qe ) = −1.3929 10−7 qe

Mauricio G. Prado – The University of Tulsa

L dq d t2 dt

γ dq ⎛ rr ⎞ ln⎜⎜ ⎟⎟ h dt ⎝ rw ⎠

Instability Analysis – The Phenomena • We can then obtain the following ODE: − 1.3929 10

−7

γ dq ⎛ rr ⎞ h dt

dPwfi

ln⎜⎜ ⎟⎟ + P + dq ⎝ rw ⎠ e wf

dPwfo L dq e (q − qe ) = 1.6046 10 γ 2 + Pwf + d t dt dq −4

qe

( q − qe ) qe

• Then we have: − 1.3929 10

dPwfi dq

−7

⎛ rr ⎞ dq dPwfi ln⎜ ⎟ + h ⎜⎝ rw ⎟⎠ dt dq

γ

(q − qe ) − qe

o L dq dPwf (q − qe ) = 1.6046 10 γ 2 + d t dt dq −4

qe

(q − qe ) qe

⎡ L γ ⎛ r ⎞⎤ dq (q − qe ) = ⎢1.6046 10 − 4 γ 2 + 1.3929 10 −7 ln⎜⎜ r ⎟⎟⎥ dq q dt h ⎝ rw ⎠⎦ dt ⎣ e

dPwfo

⎡ dPwfi dPwfo ⎤ ⎡ ⎛ rr ⎞⎤ dq L −4 −7 γ ⎜⎜ ⎟⎟⎥ − − = + γ q q ln ( ) 1 . 6046 10 1 . 3929 10 ⎢ ⎥ ⎢ e 2 dq ⎥⎦ dt h ⎝ rw ⎠⎦ dt ⎣ ⎣⎢ dq qe Mauricio G. Prado – The University of Tulsa

Instability Analysis – The Phenomena • We can then obtain the following ODE: ⎡ dPwfi dPwfo ⎤ − ⎢ ⎥ dq dq ⎥⎦ q ⎣⎢ e

dq = dt ( q − qe ) ⎡ ⎛ rr ⎞⎤ L −4 −7 γ ⎜⎜ ⎟⎟⎥ + γ ln 1 . 6046 10 1 . 3929 10 ⎢ 2 dt h ⎝ rw ⎠⎦ ⎣

• Since for small disturbances the term multiplying dt is a constant: dq = m dt (q − qe ) ln(q − qe ) = m t + c (q − qe ) = Ae m t

m=

⎡ dPwfi dPwfo ⎤ − ⎢ ⎥ dq dq ⎥⎦ q ⎣⎢ e ⎡ ⎛ rr ⎞⎤ L −4 −7 γ ⎜⎜ ⎟⎟⎥ ln γ 1 . 6046 10 1 . 3929 10 + ⎢ 2 dt h ⎝ rw ⎠⎦ ⎣

Mauricio G. Prado – The University of Tulsa

Instability Analysis – The Phenomena • The flowrate changes with time is then: (q − qe ) = Ae m t

• Then the solution for the flowrate is: q = qe + Ae m t

Mauricio G. Prado – The University of Tulsa

Instability Analysis – The Phenomena • For the flowrate to return to equilibrium as time increases we must have

q = qe + Ae m t

m<0

• Then the stability criteria becomes:

m=

⎡ dPwfi dPwfo ⎤ − ⎢ ⎥ dq dq ⎥⎦ q ⎢⎣ e ⎡ ⎛ rr ⎞⎤ L −4 −7 γ ln⎜⎜ ⎟⎟⎥ ⎢1.6046 10 γ 2 + 1.3929 10 dt h ⎝ rw ⎠⎦ ⎣

Mauricio G. Prado – The University of Tulsa

<0

Instability Analysis – The Phenomena • Finally the stability criteria is: dPwfi dq

− qe

dPwfo dq

> qe

dPwfo

<0

dq

qe

dPwfi dq

qe

dPwfi − dPwfo dq

<0 qe

Mauricio G. Prado – The University of Tulsa

An Analogy •

The stability criteria is related with the shape of the IPR and OPR. It is related to the slope of the difference of the IPR and OPR pressures with the flowrate. The same way as the stability for the magnets was related to the slope of function describing the behavior of the resultant force with the third magnet position The analogy of the resultant force is the difference in the IPR and OPR pressures. The analogy for the magnet position is the flowrate.

• •

-

+

+

-

+

-

+

+

-

-

F

F

Resultant Force

Resultant Force

-

-1

-0.5

0

0.5

1

-1

Relative Position

-0.5

0 Relative Position

Mauricio G. Prado – The University of Tulsa

0.5

1

+

Stable and Unstable Conditions Pwf

Pwfo

Unstable – Does not exist

Pwf dPwfo dq

−

dPwfi dq

dPwfo o wf

dq

P >0

dq

<0

Stable Steady

Stable Unsteady - Heading

dPwfo

i wf

P

Pwfi

−

dPwfi

dq

q

−

dPwfi

q

Pwf Pwfo

Pwfi

Stable Unsteady - Oscillatory Production dPwfo dq

−

dPwfi dq

>0

Mauricio G. Prado – The University of Tulsa

q

dq

>0

Instability Analysis – The Phenomena • How fast does the system reacts to a small disturbance ? dq = m dt (q − qe )

m=

⎡ dPwfi dPwfo ⎤ − ⎢ ⎥ dq dq ⎥⎦ q ⎣⎢ e ⎡ ⎛ rr ⎞⎤ L −4 −7 γ ⎜⎜ ⎟⎟⎥ ln γ 1 . 6046 10 1 . 3929 10 + ⎢ 2 dt h ⎝ rw ⎠⎦ ⎣

dq = m (q − qe ) dt

⎡ dPwfi dPwfo ⎤ − ⎢ ⎥ (q − qe ) dq ⎥⎦ ⎢⎣ dq qe

dq = dt ⎡ ⎛ rr ⎞⎤ L −4 −7 γ ln⎜⎜ ⎟⎟⎥ ⎢1.6046 10 γ 2 + 1.3929 10 dt h ⎝ rw ⎠⎦ ⎣ Mauricio G. Prado – The University of Tulsa

Instability Analysis – The Phenomena • How fast does the system reacts to a small disturbance ? ⎡ dPwfi dPwfo ⎤ − ⎢ ⎥ (q − qe ) dq ⎥⎦ ⎢⎣ dq qe

dq = dt ⎡ ⎛ rr ⎞⎤ L −4 −7 γ ln⎜⎜ ⎟⎟⎥ ⎢1.6046 10 γ 2 + 1.3929 10 dt h ⎝ rw ⎠⎦ ⎣

Pwfo = Pwfe +

dPwfo dq

(q − qe )

Pwfi = Pwfe +

qe

dPwfi dq

(q − qe ) qe

Pwfi − Pwfo dq = dt ⎡ ⎛ rr ⎞⎤ L −4 −7 γ ln⎜⎜ ⎟⎟⎥ ⎢1.6046 10 γ 2 + 1.3929 10 dt h ⎝ rw ⎠⎦ ⎣ Mauricio G. Prado – The University of Tulsa

Instability Analysis – The Phenomena • How fast does the system reacts to a small disturbance ? Pwfi − Pwfo dq = dt ⎡ ⎛ rr ⎞⎤ L −4 −7 γ ln⎜⎜ ⎟⎟⎥ ⎢1.6046 10 γ 2 + 1.3929 10 dt h ⎝ rw ⎠⎦ ⎣

(

dq is proportional to Pwfi − Pwfo dt

Mauricio G. Prado – The University of Tulsa

)

Stable and Unstable Conditions • How fast is the reaction ? Pwf

Pwfo

(

dq ∝ Pwfi − Pwfo dt

)

Unstable – Does not exist

Pwf

Pwfo Stable Steady

Stable Unsteady - Heading

Pwfi

Pwfi Pwf

q

q Pwfo

Pwfi

Stable Unsteady - Oscillatory Production

q Mauricio G. Prado – The University of Tulsa

Stable and Unstable Conditions Pwf Pwfo

Stable Criteria II

Unstable Criteria I Unstable Criteria II

Pwf Pwfo

i wf

P

q

P wfi

q Mauricio G. Prado – The University of Tulsa

Nodal can only be done upwind or downwind must be interactive Stability Surging and Flow Inducing Well Control – Choke Choke for Stability Mauricio G. Prado – The University of Tulsa