Adaptive Observer Design for the Bottomhole Pressure of a ...

Adaptive Observer Design for the BottomholePressure of a Managed Pressure Drilling SystemCDC - Cancun

Øyvind Nistad Stamnes, Jing Zhou,Glenn-Ole Kaasa, Ole Morten AamoDepartment of Engineering Cybernetics

10 Dec. 2008

www.ntnu.no Ø. Stamnes, Adaptive Observer Design for Bottomhole Pressure 1/15

Outline

— Drilling 101— System Model— Observer Design— Simulation Results

Pressure

Sea bed

Conventional, pump offConventional, pump on

Outline

— Drilling 101

— System Model— Observer Design— Simulation Results

Pressure

Sea bed

Outline

— Drilling 101— System Model

— Observer Design— Simulation Results

Pressure

Sea bed

Outline

— Drilling 101— System Model— Observer Design

— Simulation Results

Pressure

Sea bed

Conventional, pump off

Outline

— Drilling 101— System Model— Observer Design

— Simulation Results

Pressure

Sea bed

Outline

— Drilling 101— System Model— Observer Design— Simulation Results

Pressure

Sea bed

Drilling 101

— Conventional Drilling— Managed Pressure Drilling (MPD)

Pann = Pfric + Phydro

+ Pchoke

Pann = Annular pressurePfric = Friction pressure

Phydro = Hydrostatic pressure

Pchoke = Pressure upstream choke

Drilling 101

— Conventional Drilling

— Managed Pressure Drilling (MPD)

+ Pchoke

Pressure

Conventional, pump onConventional, pump off

Drilling 101

— Conventional Drilling

— Managed Pressure Drilling (MPD)

+ Pchoke

Pressure

Conventional, pump onConventional, pump off

Drilling 101

Pann = Pfric + Phydro + Pchoke

Phydro = Hydrostatic pressurePchoke = Pressure upstream choke

Pressure

Conventional, pump onConventional, pump offMPD, pump off

Pchoke

Drilling 101

Pann = Pfric + Phydro + Pchoke

Phydro = Hydrostatic pressurePchoke = Pressure upstream choke

Pressure

Conventional, pump onConventional, pump offMPD, pump off

Pchoke

Model- Main Assumptions

— 1. phase, effect of gas in wellincluded in density and effectivebulk modulus

— Rigid flow in momentum balance— Turbulent flow regime— 1-dimensjonal flow— Isothermal conditions

— Rigid flow in momentum balance

— Turbulent flow regime— 1-dimensjonal flow— Isothermal conditions

— Rigid flow in momentum balance— Turbulent flow regime

— 1-dimensjonal flow— Isothermal conditions

— Rigid flow in momentum balance— Turbulent flow regime— 1-dimensjonal flow

— Isothermal conditions

βapc + Va = qbit + qback + qres − qchoke

βdpp = qpump − qbit

Mqbit = pp − pc − (Fd + Fa)|qbit |qbit

+ (ρd − ρa)ghbit

pbit =

{pc + Maqbit + Fa|qbit |qbit + ρaghbitpp + Md qbit − Fd |qbit |qbit + ρdghbit

p=[bar], V=[m3], q=[

β=[bar], h=[m], g=9.81[m

F=friction factor, ρ=[

M = Md + Ma

Md = ρd∫ lbit

Ad (x)dx ,

Ma = ρa∫ lw

Aa(x)dx

pbit =

p=[bar], V=[m3], q=[

β=[bar], h=[m], g=9.81[m

M = Md + Ma

Md = ρd∫ lbit

Ad (x)dx ,

Ma = ρa∫ lw

Aa(x)dx

pbit =

p=[bar], V=[m3], q=[

β=[bar], h=[m], g=9.81[m

M = Md + Ma

Md = ρd∫ lbit

Ad (x)dx ,

Ma = ρa∫ lw

Aa(x)dx

pbit =

p=[bar], V=[m3], q=[

β=[bar], h=[m], g=9.81[m

M = Md + Ma

Md = ρd∫ lbit

Ad (x)dx ,

Ma = ρa∫ lw

Aa(x)dx

pbit =

p=[bar], V=[m3], q=[

β=[bar], h=[m], g=9.81[m

M = Md + Ma

Md = ρd∫ lbit

Ad (x)dx ,

Ma = ρa∫ lw

Aa(x)dx

Model- Verification

0 500 1000 1500240

pbitsim

0 500 1000 1500−50

Figure: Model fitted to WeMod

0 0.5 1 1.5 2 2.550

pbitpppbitf it

ppf it

0 0.5 1 1.5 2 2.50

pc(barg)

up( liters )

Figure: Model fitted to data

Model- Verification

0 500 1000 1500240

pbitsim

0 500 1000 1500−50

0 0.5 1 1.5 2 2.550

pbitpppbitf it

ppf it

0 0.5 1 1.5 2 2.50

pc(barg)

up( liters )

Model- Verification

0 500 1000 1500240

pbitsim

0 500 1000 1500−50

0 0.5 1 1.5 2 2.550

pbitpppbitf it

ppf it

0 0.5 1 1.5 2 2.50

pc(barg)

up( liters )

Observer Design - Assumption

— Measure top-side pressures andflow through main pump

— measure/know the geometry of thewell

— all parameters except friction anddensity in annulus known

— qres = 0— qbit > 0, in reality qbit ≥ 0

— qres = 0

— qbit > 0, in reality qbit ≥ 0

Observer Design - ModelSame model, unkown friction and density

pp = −a1qbit + b1up

qbit = a2(pp − pc)− θ1|qbit |qbit + θ2v3

pbit = pc + Maqbit + (Mθ1 − Fd )q2bit + (ρd −

Mgθ2)hbit

a1 =βd

Vd, b1 = a1

a2 =1M, θ1 =

Fa + Fd

θ2 =(ρd − ρa)g

M, v3 = hbit

Mgθ2)hbit

a1 =βd

Vd, b1 = a1

a2 =1M, θ1 =

Fa + Fd

θ2 =(ρd − ρa)g

M, v3 = hbit

Mgθ2)hbit

a1 =βd

Vd, b1 = a1

a2 =1M, θ1 =

Fa + Fd

θ2 =(ρd − ρa)g

M, v3 = hbit

Mgθ2)hbit

a1 =βd

Vd, b1 = a1

a2 =1M, θ1 =

Fa + Fd

θ2 =(ρd − ρa)g

M, v3 = hbit

Mgθ2)hbit

a1 =βd

Vd, b1 = a1

a2 =1M, θ1 =

Fa + Fd

θ2 =(ρd − ρa)g

M, v3 = hbit

Observer Design - TransformationDefine ξ = qbit + l1pp. l1 is a tuning gain.

Dynamics for ξ

ξ = −l1a1qbit − θ1|qbit |qbit + θ2v3 + a2(pp − pc) + l1b1up

A state estimator for qbit

˙ξ = −l1a1qbit − θ1|qbit |qbit + θ2v3 + a2(pp − pc) + l1b1up

qbit = ξ − l1pp

Error dynamics for ξ = qbit − qbit

˙ξ = −l1a1ξ − θ1(|qbit |qbit − |qbit |qbit ) + θTφ(qbit , v3)

φ(qbit , v3) =[−|qbit |qbit , v3

Observer Design - TransformationDefine ξ = qbit + l1pp. l1 is a tuning gain. Dynamics for ξ

qbit = ξ − l1pp

Observer Design - Lyapunov Analysis

Lyapunov type analysis

— V = 12 ξ

2 + 12 θ

T Γ−1θ

— V ≤ −l1a1ξ2

— for the choice ˙θ = −Γφξ

For which we can conclude— |V | ≤ V0 which gives |ξ| < c1 og |θ| < c2

— Using Barbalat’s lemma we get limt→∞ ξ = limt→∞ qbit = 0

— We also get limt→∞ θTφ = limt→∞

(−θ1qbit + θ2v3

— qbit → 0 og θTφ→ 0 enables us to get an estimate pbit → pbit

Lyapunov type analysis— V = 1

2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2

(−θ1qbit + θ2v3

2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2

(−θ1qbit + θ2v3

2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2

(−θ1qbit + θ2v3

2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2

For which we can conclude

— |V | ≤ V0 which gives |ξ| < c1 og |θ| < c2

(−θ1qbit + θ2v3

2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2

(−θ1qbit + θ2v3

2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2

(−θ1qbit + θ2v3

2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2

(−θ1qbit + θ2v3

2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2

(−θ1qbit + θ2v3

Observer Design - Adaptive lawProblem: ˙

θ = − ˙θ = Γφξ

Define: σ = θ + η(qbit , v3) and differentiate w.r.t. time

σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

Let an estimate θ be:

θ = σ − η(qbit )

˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

Observe that σ = θ og ˙θ = ˙σ = l1a1∂η∂qbit

qbit .

Solve pde: l1a1∂η∂qbit

= −Γφξ

θ = − ˙θ = ΓφξDefine: σ = θ + η(qbit , v3) and differentiate w.r.t. time

σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

qbit .

= −Γφξ

σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

qbit .

= −Γφξ

σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

qbit .

= −Γφξ

σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

qbit .

= −Γφξ

σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

qbit .

= −Γφξ

σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

qbit .

= −Γφξ

σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂v3v3

qbit .

= −Γφξ

Simulation - WeMod

0 10 20 30 40 50 60240

0 10 20 30 40 50 600

50010001500

0 10 20 30 40 50 60

0 10 20 30 40 50 600.01

Initial conditions:qbit (t0) = up(t0), ρa(t0) = 2ρa,Fa(t0) = 3Fa, t0 = 50s

Simulation - Data

0 0.5 1 1.5 2 2.550

pbitpppbitf it

ppf it

0 0.5 1 1.5 2 2.50

pc(barg)

up( liters )

0 0.5 1 1.5 2 2.5215

0 0.5 1 1.5 2 2.50

Initial conditions:qbit (t0) = up(t0), ρa(t0) = 1.2ρa,Fa(t0) = 1.5Fa

Conclusion

— A simple model for estimation has been presented

— An observer has been developed— Adapts to unknown friction and density— Low complexity— Good simulation results— Tested on data from North Sea well

Conclusion

— A simple model for estimation has been presented— An observer has been developed

— Adapts to unknown friction and density— Low complexity— Good simulation results— Tested on data from North Sea well

Conclusion

— A simple model for estimation has been presented— An observer has been developed— Adapts to unknown friction and density

— Low complexity— Good simulation results— Tested on data from North Sea well

Conclusion

— A simple model for estimation has been presented— An observer has been developed— Adapts to unknown friction and density— Low complexity

— Good simulation results— Tested on data from North Sea well

Conclusion

— A simple model for estimation has been presented— An observer has been developed— Adapts to unknown friction and density— Low complexity— Good simulation results

— Tested on data from North Sea well

Conclusion

— A simple model for estimation has been presented— An observer has been developed— Adapts to unknown friction and density— Low complexity— Good simulation results— Tested on data from North Sea well

Thank you!

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