Parameter Estimation in Reservoir Engineering Models via ...
Transcript of Parameter Estimation in Reservoir Engineering Models via ...
![Page 1: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/1.jpg)
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
![Page 2: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/2.jpg)
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
![Page 3: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/3.jpg)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Structure of Reservoir EngineeringReservoir Engineering
sshhhhhhhhhhhhhhhhhh
**VVVVVVVVVVVVVVVVV
Production Engineering Reservoir Simulation
approaches
tthhhhhhhhhhhhhhhhh
||yyyy
yyyy
yyyy
yyyy
yyyy
y
��
Analogical
Experimental
Mathematical
![Page 4: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/4.jpg)
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
![Page 5: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/5.jpg)
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
![Page 6: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/6.jpg)
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
![Page 7: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/7.jpg)
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
![Page 8: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/8.jpg)
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
![Page 9: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/9.jpg)
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)
![Page 10: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/10.jpg)
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)
![Page 11: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/11.jpg)
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)
![Page 12: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/12.jpg)
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)
![Page 13: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/13.jpg)
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)
![Page 14: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/14.jpg)
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)
![Page 15: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/15.jpg)
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)
![Page 16: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/16.jpg)
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
![Page 17: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/17.jpg)
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
![Page 18: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/18.jpg)
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
![Page 19: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/19.jpg)
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
]
![Page 20: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/20.jpg)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
History Matching Process
Historical Match
ssfffffffffffffffffff
--ZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
Manual Automaticapproaches
ttiiiiiiiiiiiiiii
��
Traditional
Kalman Filtering
![Page 21: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/21.jpg)
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)
![Page 22: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/22.jpg)
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
![Page 23: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/23.jpg)
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
![Page 24: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/24.jpg)
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
![Page 25: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/25.jpg)
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���
![Page 26: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/26.jpg)
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
![Page 27: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/27.jpg)
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
))
![Page 28: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/28.jpg)
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
))
![Page 29: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/29.jpg)
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
))
![Page 30: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/30.jpg)
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
))
![Page 31: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/31.jpg)
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
![Page 32: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/32.jpg)
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
![Page 33: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/33.jpg)
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
![Page 34: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/34.jpg)
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
)(−
)
![Page 35: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/35.jpg)
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
![Page 36: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/36.jpg)
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
![Page 37: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/37.jpg)
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
![Page 38: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/38.jpg)
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
![Page 39: Parameter Estimation in Reservoir Engineering Models via ...](https://reader030.fdocuments.net/reader030/viewer/2022020705/61fbc9ec0b657423a8479766/html5/thumbnails/39.jpg)
OutlineIntroduction to Reservoir Engineering
Two-Phase Water-Oil Fluid Flow ModelKalman Filtering Techniques
Case StudyResults
ConclusionQuestions
Questions