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OutlineIntroduction to Reservoir Engineering

Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques

Case StudyResults

ConclusionQuestions

Parameter Estimation in Reservoir EngineeringModels via Data Assimilation Techniques

Mariya V. Krymskaya

TU Delft

July 16, 2007



Case StudyResults

ConclusionQuestions

Introduction to Reservoir Engineering

Two-Phase Water-Oil Fluid Flow Model

Kalman Filtering TechniquesEnsemble Kalman Filter (EnKF)Iterative Ensemble Kalman Filter (IEnKF)

Case StudyState Vector FeasibilityRe-scaling state vectorExperimental Setup

ResultsEnKFIEnKF

Conclusion



Case StudyResults

ConclusionQuestions

Structure of Reservoir EngineeringReservoir Engineering

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**VVVVVVVVVVVVVVVVV

Production Engineering Reservoir Simulation

approaches

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yyyy

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yyyy

y

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Analogical

Experimental

Mathematical



Case StudyResults

ConclusionQuestions

A Reservoir

General information> 40, 000 oil fields in the world300 m to 10 km below the surface2− 500 million years old

Ghawar oil fieldthe biggest among discoveredLocation: Saudi ArabiaRecovery: since 1951Size: 280× 30 kmAge: 320 million years oldProduction: 5 million barrels(800, 000 m3) of oil per day



Case StudyResults

ConclusionQuestions

Oil Recovery

I Primary20% extracted

I Secondary (water flooding)25% to 35% extracted

I Tertiary50% left



Case StudyResults

ConclusionQuestions

Reservoir Properties

I Rock propertiesI PorosityI (Absolute) permeabilityI Rock compressibility

I Fluid properties

I Fluid compressibilityI Fluid densityI Fluid viscosity

I Fluid-rock properties

I Fluid saturationI Capillary pressureI Relative permeability (Corey-type model)



Case StudyResults

ConclusionQuestions



I Fluid propertiesI Fluid compressibilityI Fluid densityI Fluid viscosity

I Fluid-rock properties

I Fluid saturationI Capillary pressureI Relative permeability (Corey-type model)



Case StudyResults

ConclusionQuestions



I Fluid propertiesI Fluid compressibilityI Fluid densityI Fluid viscosity

I Fluid-rock propertiesI Fluid saturationI Capillary pressureI Relative permeability (Corey-type model)



Case StudyResults

ConclusionQuestions

Two-Phase Water-Oil Fluid Flow Model

I Mass balance equation for each phase

I Darcy’s law for each phase

I Capillary pressure equation

I Relative permeability equations(Corey-type model)

I Equations of state

I Initial / boundary conditions

I Well model



Case StudyResults

ConclusionQuestions

Model Discretization

E(X) X − A(X) X − B(X) U = 0↑ ↑ ↑ ↑ ↑

accumulationmatrix

systemmatrix

statevector

inputmatrix

inputvector

⇓

X =

[pS

]



Case StudyResults

ConclusionQuestions

History Matching Process

Historical Match

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--ZZZZZZZZZZZZZZZZZZZZZZZZZZZZ

Manual Automaticapproaches

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��

Traditional

Kalman Filtering



Case StudyResults

ConclusionQuestions

Data Assimilation Problem StatementEnsemble Kalman Filter (EnKF)Iterative Ensemble Kalman Filter (IEnKF)

Data Assimilation Problem Statement

I SystemXk+1 = F (Xk ,Uk ,m) + Wk ,Zk+1 = MXk + Vk

I Uncertainties

X0 ∼ N (X0,P0) − uncertain initial state,Wk ∼ N (0,Q) − model noise,Vk ∼ N (0,R) − measurement noise,

I Independency assumption

X0⊥Wk⊥Vk

I State conditional pdf

(Xk |Z1, . . . ,Zl) ∼ N (mean, cov)



Case StudyResults

ConclusionQuestions


Ensemble Kalman Filter (EnKF)

Initial conditionX0, P0

��

DataZk

��Ensemblegeneration

(ξi )0, X0, P0

//Forwardmodel

integration

//Forecastedensembleand state

(ξi )fk , Xf

k , Pfk

// Dataassimilation

��Analyzed ensemble

and state(ξi )

ak , Xa

k , Pak

@A

OO



Case StudyResults

ConclusionQuestions


Parameter Estimation via EnKF

Augmented state vector

X =

[pS

]V X =

log k} = mpS

}= Y

d



Case StudyResults

ConclusionQuestions


Iterative Ensemble Kalman Filter (IEnKF)

[m0, Y0]T,

P0

��_ _ _ _ _ _ _�

�

�

�_ _ _ _ _ _ _

Initial conditionX0, P0

// EnKF //

_ _ _ _ _ _ _ _�

�

�

�

�

�_ _ _ _ _ _ _ _

Analyzed ensembleand state

(ξi )ak , Xa

k ,Pak

//Estimated

model parameterma

k

EDma

koo



Case StudyResults

ConclusionQuestions

State Vector FeasibilityRe-scaling state vectorExperimental Setup

State Vector Feasibility

Ensemblegeneration

// Forward modelintegration

//

Yak−1��

Dataassimilation

// Confirmationstep

mak

ssg g g g g g g g g g g g g g g g g g g g g

mlgf��

New staticparameters

//

Re-initializedensembleand state(ξi )

ak−1,

Xak−1, Pa

k−1

//Forward model

integrationfrom k − 1 to k

//

Confirmedensembleand state

(ξi )ck ,

Xck , Pc

k

OO��



Case StudyResults

ConclusionQuestions


Re-scaling state vector

X =

log kpSY

,

AB

=

10−7

10

10−7

103

⇒ AMLBL



Case StudyResults

ConclusionQuestions



Kalman gain

K =1

N − 1

(B−1B

)

LLTMT

(AA−1

)

(1

N − 1MLLTMT + R

)−1

(A−1A

)= B−1 1

N − 1(BL) (AML)T

(1

N − 1AML (AML)T + ARA

)−1

︸︷︷︸K1

A

Ensemble update

(ξi )ak = (ξi )

fk + K

(Z−M(ξi )

fk + Vi

)= (ξi )

fk + B−1K1A

(Z−M(ξi )

fk + Vi

)= (ξi )

fk + B−1

(K1

(AZ− AM(ξi )

fk + AVi

))



Case StudyResults

ConclusionQuestions



Kalman gain

K =1

N − 1

(B−1B

)LLTMT

(AA−1

) (1

N − 1MLLTMT + R

)−1 (A−1A

)

= B−1 1

N − 1(BL) (AML)T

(1


)−1

︸︷︷︸K1

A

Ensemble update

(ξi )ak = (ξi )

fk + K

(Z−M(ξi )

fk + Vi

)= (ξi )

fk + B−1K1A

(Z−M(ξi )

fk + Vi

)= (ξi )

fk + B−1

(K1

(AZ− AM(ξi )

fk + AVi

))



Case StudyResults

ConclusionQuestions



Kalman gain

K =1

N − 1

(B−1B

)LLTMT

(AA−1

) (1

N − 1MLLTMT + R

)−1 (A−1A

)= B−1 1

N − 1(BL) (AML)T

(1


)−1

︸︷︷︸K1

A

Ensemble update

(ξi )ak = (ξi )

fk + K

(Z−M(ξi )

fk + Vi

)= (ξi )

fk + B−1K1A

(Z−M(ξi )

fk + Vi

)= (ξi )

fk + B−1

(K1

(AZ− AM(ξi )

fk + AVi

))



Case StudyResults

ConclusionQuestions


Experimental setup

Twin experimentInitialization

True permeability field (log(m2))

5 10 15 20

2

4

6

8

10

12

14

16

18

20

−32

−31.5

−31

−30.5

−30

−29.5

−29

−28.5

−28Mean of permeability fields ensemble (log(m2))

5 10 15 20

2

4

6

8

10

12

14

16

18

20−29.1

−29

−28.9

−28.8

−28.7

−28.6

−28.5



Case StudyResults

ConclusionQuestions

EnKFIEnKF

RMS Error in Model Parameter

0 100 200 300 400 500 6000.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (days)

RM

S e

rror

(lo

g(m

2 ))

Ensemble meanEnsemble members



Case StudyResults

ConclusionQuestions

EnKFIEnKF

Estimated Permeability Field

True permeability field (log(m2))

5 10 15 20

2

4

6

8

10

12

14

16

18

20

−32

−31.5

−31

−30.5

−30

−29.5

−29

−28.5

−28

Mean of initial ensemble (log(m2))

5 10 15 20

5

10

15

20−32

−31

−30

−29

−28Variance of initial ensemble ((log(m2))2)

5 10 15 20

5

10

15

200

0.2

0.4

0.6

0.8

1

Variance of estimated ensemble ((log(m2))2)

5 10 15 20

5

10

15

200

0.2

0.4

0.6

0.8

1

Estimated permeability field (log(m2))

5 10 15 20

5

10

15

20−32

−31

−30

−29

−28



Case StudyResults

ConclusionQuestions

EnKFIEnKF

Forecasted Reservoir Performance

0 500 1000 15000.2

0.4

0.6

0.8

1

Time t ( days)

Wat

er s

atur

atio

n S

w a

t wel

l (

1,1)

(−

)

0 500 1000 15000.2

0.4

0.6

0.8

1

Time t ( days)

Wat

er s

atur

atio

n S

w a

t wel

l (

21,1

)(−

)

0 500 1000 15000.2

0.4

0.6

0.8

1

Time t ( days)

Wat

er s

atur

atio

n S

w a

t wel

l (

1,21

)(−

)

0 500 1000 15000.2

0.4

0.6

0.8

1

Time t ( days)

Wat

er s

atur

atio

n S

w a

t wel

l(2

1,21

)(−

)



Case StudyResults

ConclusionQuestions

EnKFIEnKF


0 100 200 300 400 500 6000.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (days)

RM

S e

rror

(lo

g(m

2 ))


0 100 200 300 400 500 6000.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (days)

RM

S e

rror

(lo

g(m

2 ))




Case StudyResults

ConclusionQuestions

EnKFIEnKF


0 100 200 300 400 5000.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Time (days)

RM

S e

rror

(lo

g(m

2 ))


0 100 200 300 400 5000.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Time (days)

RM

S e

rror

(lo

g(m

2 ))




Case StudyResults

ConclusionQuestions

EnKFIEnKF

Estimated Permeability Field


5 10 15 20

5

10

15

20−32

−31

−30

−29


5 10 15 20

5

10

15

200

0.2

0.4

0.6

0.8

1


5 10 15 20

5

10

15

200

0.2

0.4

0.6

0.8

1


5 10 15 20

5

10

15

20−32

−31

−30

−29

−28


5 10 15 20

5

10

15

20−32

−31

−30

−29


5 10 15 20

5

10

15

200

0.2

0.4

0.6

0.8

1


5 10 15 20

5

10

15

200

0.2

0.4

0.6

0.8

1


5 10 15 20

5

10

15

20−32

−31

−30

−29

−28



Case StudyResults

ConclusionQuestions

Conclusion

I Model calibration is essential

I EnKF provides reasonable parameter estimation

I There are cases at which IEnKF is superior to EnKF

I Further investigations on IEnKF sensitivities are required



Case StudyResults

ConclusionQuestions

Questions

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