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

2

Outline

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

Depth

Pressure

Sea bed

Conventional, pump offConventional, pump on


2

Outline

— Drilling 101

— System Model— Observer Design— Simulation Results

Depth

Pressure

Sea bed



2

Outline

— Drilling 101— System Model

— Observer Design— Simulation Results

Depth

Pressure

Sea bed



2

Outline

— Drilling 101— System Model— Observer Design

— Simulation Results

Depth

Pressure

Sea bed

Conventional, pump off


2

Outline

— Drilling 101— System Model— Observer Design

— Simulation Results

Depth

Pressure

Sea bed



2

Outline

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

Depth

Pressure

Sea bed



3

Drilling 101

— Conventional Drilling— Managed Pressure Drilling (MPD)

Pann = Pfric + Phydro

+ Pchoke

Pann = Annular pressurePfric = Friction pressure

Phydro = Hydrostatic pressure

Pchoke = Pressure upstream choke


3

Drilling 101

— Conventional Drilling

— Managed Pressure Drilling (MPD)


+ Pchoke




Depth

Pressure

Conventional, pump onConventional, pump off


3

Drilling 101

— Conventional Drilling

— Managed Pressure Drilling (MPD)


+ Pchoke




Depth

Pressure

Conventional, pump onConventional, pump off


3

Drilling 101


Pann = Pfric + Phydro + Pchoke


Phydro = Hydrostatic pressurePchoke = Pressure upstream choke

Depth

Pressure

Conventional, pump onConventional, pump offMPD, pump off

Pchoke


3

Drilling 101


Pann = Pfric + Phydro + Pchoke


Phydro = Hydrostatic pressurePchoke = Pressure upstream choke

Depth

Pressure

Conventional, pump onConventional, pump offMPD, pump off

Pchoke


4

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


4





4



— Rigid flow in momentum balance

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


4



— Rigid flow in momentum balance— Turbulent flow regime

— 1-dimensjonal flow— Isothermal conditions


4



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

— Isothermal conditions


4





5

Model

Va

βapc + Va = qbit + qback + qres − qchoke

Vd

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=[

m3

s

]

β=[bar], h=[m], g=9.81[m

s2

]

F=friction factor, ρ=[

kgm3

]

M = Md + Ma

Md = ρd∫ lbit

01

Ad (x)dx ,

Ma = ρa∫ lw

01

Aa(x)dx


5

Model

Va


Vd




pbit =


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

m3

s

]

β=[bar], h=[m], g=9.81[m

s2

]


kgm3

]

M = Md + Ma

Md = ρd∫ lbit

01

Ad (x)dx ,

Ma = ρa∫ lw

01

Aa(x)dx


5

Model

Va


Vd




pbit =


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

m3

s

]

β=[bar], h=[m], g=9.81[m

s2

]


kgm3

]

M = Md + Ma

Md = ρd∫ lbit

01

Ad (x)dx ,

Ma = ρa∫ lw

01

Aa(x)dx


5

Model

Va


Vd




pbit =


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

m3

s

]

β=[bar], h=[m], g=9.81[m

s2

]


kgm3

]

M = Md + Ma

Md = ρd∫ lbit

01

Ad (x)dx ,

Ma = ρa∫ lw

01

Aa(x)dx


5

Model

Va


Vd




pbit =


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

m3

s

]

β=[bar], h=[m], g=9.81[m

s2

]


kgm3

]

M = Md + Ma

Md = ρd∫ lbit

01

Ad (x)dx ,

Ma = ρa∫ lw

01

Aa(x)dx


6

Model- Verification

0 500 1000 1500240

260

280

300

320

340

s

barg

pbit

pbitsim

0 500 1000 1500−50

0

50

100

150

s

barg

pp

pc

ppsim

pcsim

Figure: Model fitted to WeMod

0 0.5 1 1.5 2 2.550

100

150

200

250

barg

pbitpppbitf it

ppf it

0 0.5 1 1.5 2 2.50

10

20

30

hr

pc(barg)

up( liters )

Figure: Model fitted to data


6

Model- Verification

0 500 1000 1500240

260

280

300

320

340

s

barg

pbit

pbitsim

0 500 1000 1500−50

0

50

100

150

s

barg

pp

pc

ppsim

pcsim


0 0.5 1 1.5 2 2.550

100

150

200

250

barg

pbitpppbitf it

ppf it

0 0.5 1 1.5 2 2.50

10

20

30

hr

pc(barg)

up( liters )



6

Model- Verification

0 500 1000 1500240

260

280

300

320

340

s

barg

pbit

pbitsim

0 500 1000 1500−50

0

50

100

150

s

barg

pp

pc

ppsim

pcsim


0 0.5 1 1.5 2 2.550

100

150

200

250

barg

pbitpppbitf it

ppf it

0 0.5 1 1.5 2 2.50

10

20

30

hr

pc(barg)

up( liters )



7

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


7







7







7





— qres = 0

— qbit > 0, in reality qbit ≥ 0


7







8

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

M

θ2 =(ρd − ρa)g

M, v3 = hbit


8





Mgθ2)hbit

a1 =βd

Vd, b1 = a1

a2 =1M, θ1 =

Fa + Fd

M

θ2 =(ρd − ρa)g

M, v3 = hbit


8





Mgθ2)hbit

a1 =βd

Vd, b1 = a1

a2 =1M, θ1 =

Fa + Fd

M

θ2 =(ρd − ρa)g

M, v3 = hbit


8





Mgθ2)hbit

a1 =βd

Vd, b1 = a1

a2 =1M, θ1 =

Fa + Fd

M

θ2 =(ρd − ρa)g

M, v3 = hbit


8





Mgθ2)hbit

a1 =βd

Vd, b1 = a1

a2 =1M, θ1 =

Fa + Fd

M

θ2 =(ρd − ρa)g

M, v3 = hbit


9

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

]T


9

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




qbit = ξ − l1pp




]T


9





qbit = ξ − l1pp




]T


9





qbit = ξ − l1pp




]T


9





qbit = ξ − l1pp




]T


9





qbit = ξ − l1pp




]T


10

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

)= 0

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


10


Lyapunov type analysis— V = 1

2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2





(−θ1qbit + θ2v3

)= 0



10



2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2





(−θ1qbit + θ2v3

)= 0



10



2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2





(−θ1qbit + θ2v3

)= 0



10



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

)= 0



10



2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2





(−θ1qbit + θ2v3

)= 0



10



2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2





(−θ1qbit + θ2v3

)= 0



10



2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2





(−θ1qbit + θ2v3

)= 0



10



2 ξ2 + 1

2 θT Γ−1θ

— V ≤ −l1a1ξ2





(−θ1qbit + θ2v3

)= 0



11

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)) +

∂η

∂v3v3

Let an estimate θ be:

θ = σ − η(qbit )

˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3

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

qbit .

Solve pde: l1a1∂η∂qbit

= −Γφξ


11


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

σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3



˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3


qbit .


= −Γφξ


11



σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3



˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3


qbit .


= −Γφξ


11



σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3



˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3


qbit .


= −Γφξ


11



σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3



˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3


qbit .


= −Γφξ


11



σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3



˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3


qbit .


= −Γφξ


11



σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3



˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3


qbit .


= −Γφξ


11



σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3



˙σ =∂η

∂qbit(

˙ξ1 − l1(−a1qbit + b1up)) +

∂η

∂v3v3


qbit .


= −Γφξ


12

Simulation - WeMod

0 10 20 30 40 50 60240

260

280

300

320

[bar

]

pbit

pbit

0 10 20 30 40 50 600

50

100

150

[min]

[bar

]

pc

pp

0 10 20 30 40 50 600

50010001500

qbit

qbit

0 10 20 30 40 50 60

246

x 104

Fa

Fa

0 10 20 30 40 50 600.01

0.015

[min]

ρa

ρa

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


13

Simulation - Data

0 0.5 1 1.5 2 2.550

100

150

200

250

barg

pbitpppbitf it

ppf it

0 0.5 1 1.5 2 2.50

10

20

30

hr

pc(barg)

up( liters )

0 0.5 1 1.5 2 2.5215

220

225

230

235

240

245

barg

pbit

pbit

0 0.5 1 1.5 2 2.50

0.005

0.01

hr

ρa

Fa

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


14

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


14

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


14

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


14

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


14

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


14

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!


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

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