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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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Structure of Reservoir EngineeringReservoir Engineering
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Production Engineering Reservoir Simulation
approaches
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Analogical
Experimental
Mathematical
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Oil Recovery
I Primary20% extracted
I Secondary (water flooding)25% to 35% extracted
I Tertiary50% left
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Oil Recovery
I Primary20% extracted
I Secondary (water flooding)25% to 35% extracted
I Tertiary50% left
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Oil Recovery
I Primary20% extracted
I Secondary (water flooding)25% to 35% extracted
I Tertiary50% left
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Oil Recovery
I Primary20% extracted
I Secondary (water flooding)25% to 35% extracted
I Tertiary50% left
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Reservoir Properties
I Rock propertiesI PorosityI (Absolute) permeabilityI Rock compressibility
I Fluid propertiesI Fluid compressibilityI Fluid densityI Fluid viscosity
I Fluid-rock properties
I Fluid saturationI Capillary pressureI Relative permeability (Corey-type model)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Reservoir Properties
I Rock propertiesI PorosityI (Absolute) permeabilityI Rock compressibility
I Fluid propertiesI Fluid compressibilityI Fluid densityI Fluid viscosity
I Fluid-rock propertiesI Fluid saturationI Capillary pressureI Relative permeability (Corey-type model)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Reservoir Properties
I Rock propertiesI PorosityI (Absolute) permeabilityI Rock compressibility
I Fluid propertiesI Fluid compressibilityI Fluid densityI Fluid viscosity
I Fluid-rock propertiesI Fluid saturationI Capillary pressureI Relative permeability (Corey-type model)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Reservoir Properties
I Rock propertiesI PorosityI (Absolute) permeabilityI Rock compressibility
I Fluid propertiesI Fluid compressibilityI Fluid densityI Fluid viscosity
I Fluid-rock propertiesI Fluid saturationI Capillary pressureI Relative permeability (Corey-type model)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Reservoir Properties
I Rock propertiesI PorosityI (Absolute) permeabilityI Rock compressibility
I Fluid propertiesI Fluid compressibilityI Fluid densityI Fluid viscosity
I Fluid-rock propertiesI Fluid saturationI Capillary pressureI Relative permeability (Corey-type model)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Reservoir Properties
I Rock propertiesI PorosityI (Absolute) permeabilityI Rock compressibility
I Fluid propertiesI Fluid compressibilityI Fluid densityI Fluid viscosity
I Fluid-rock propertiesI Fluid saturationI Capillary pressureI Relative permeability (Corey-type model)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Model Discretization
E(X) X − A(X) X − B(X) U = 0↑ ↑ ↑ ↑ ↑
accumulationmatrix
systemmatrix
statevector
inputmatrix
inputvector
⇓
X =
[pS
]
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
History Matching Process
Historical Match
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Manual Automaticapproaches
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Traditional
Kalman Filtering
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Data Assimilation Problem StatementEnsemble Kalman Filter (EnKF)Iterative Ensemble Kalman Filter (IEnKF)
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Data Assimilation Problem StatementEnsemble Kalman Filter (EnKF)Iterative Ensemble Kalman Filter (IEnKF)
Parameter Estimation via EnKF
Augmented state vector
X =
[pS
]V X =
log k} = mpS
}= Y
d
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Data Assimilation Problem StatementEnsemble Kalman Filter (EnKF)Iterative Ensemble Kalman Filter (IEnKF)
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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���
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
State Vector FeasibilityRe-scaling state vectorExperimental Setup
Re-scaling state vector
X =
log kpSY
,
AB
=
10−7
10
10−7
103
⇒ AMLBL
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
State Vector FeasibilityRe-scaling state vectorExperimental Setup
Re-scaling state vector
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
))
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
State Vector FeasibilityRe-scaling state vectorExperimental Setup
Re-scaling state vector
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
))
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
State Vector FeasibilityRe-scaling state vectorExperimental Setup
Re-scaling state vector
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
))
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
State Vector FeasibilityRe-scaling state vectorExperimental Setup
Re-scaling state vector
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
))
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
State Vector FeasibilityRe-scaling state vectorExperimental Setup
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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
)(−
)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
EnKFIEnKF
RMS Error in Model Parameter
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 ))
Ensemble meanEnsemble members
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 ))
Ensemble meanEnsemble members
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
EnKFIEnKF
Estimated Permeability Field
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
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
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
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