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History matching using the Ensemble Kalman filter
Citation preview
MotivationIntroduction
Papers overviewSummary
Coupling level set methods with the ensembleKalman filter for conditioning geological facies
models to well and production data
David Leonardo Moreno Bedoya
Centre for Integrated Petroleum Research(CIPR)Institute of Mathematics
University of Bergen
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Outline
1 Motivation
2 IntroductionLevel set methodThe ensemble Kalman filterEnKF and level sets - the coupling
3 Papers overviewPapers A - BPaper CPaper DPaper E
4 SummaryDerivation of the analysis scheme
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Outline
1 Motivation
2 IntroductionLevel set methodThe ensemble Kalman filterEnKF and level sets - the coupling
3 Papers overviewPapers A - BPaper CPaper DPaper E
4 SummaryDerivation of the analysis scheme
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Motivation
Continuous model updating using the ensemble Kalmanfilter(EnKF) with emphasis on complex reservoirs
Emphasis on complex, i.e, Non-gaussian reservoirs andnon-linear effects in the EnKFDynamic reservoir characterization of complex reservoirsusing production and static dataSequentially update reservoirs models using real-time data
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Motivation
Continuous model updating using the ensemble Kalmanfilter(EnKF) with emphasis on complex reservoirsEmphasis on complex, i.e, Non-gaussian reservoirs andnon-linear effects in the EnKF
Dynamic reservoir characterization of complex reservoirsusing production and static dataSequentially update reservoirs models using real-time data
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Motivation
Continuous model updating using the ensemble Kalmanfilter(EnKF) with emphasis on complex reservoirsEmphasis on complex, i.e, Non-gaussian reservoirs andnon-linear effects in the EnKFDynamic reservoir characterization of complex reservoirsusing production and static data
Sequentially update reservoirs models using real-time data
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Motivation
Continuous model updating using the ensemble Kalmanfilter(EnKF) with emphasis on complex reservoirsEmphasis on complex, i.e, Non-gaussian reservoirs andnon-linear effects in the EnKFDynamic reservoir characterization of complex reservoirsusing production and static dataSequentially update reservoirs models using real-time data
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Motivation
Facies are defined as distinctive rock units formed undercertain conditions of sedimentation (Reading, 1996)
Different conditions create different rocks with (sometimes)very dissimilar petro-physical propertiesGeological facies models are inherently non-gaussianproblemsThe EnKF requires the prior model for the parameters tobe Gaussian or approximately Gaussian random fields
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Motivation
Facies are defined as distinctive rock units formed undercertain conditions of sedimentation (Reading, 1996)Different conditions create different rocks with (sometimes)very dissimilar petro-physical properties
Geological facies models are inherently non-gaussianproblemsThe EnKF requires the prior model for the parameters tobe Gaussian or approximately Gaussian random fields
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Motivation
Facies are defined as distinctive rock units formed undercertain conditions of sedimentation (Reading, 1996)Different conditions create different rocks with (sometimes)very dissimilar petro-physical propertiesGeological facies models are inherently non-gaussianproblems
The EnKF requires the prior model for the parameters tobe Gaussian or approximately Gaussian random fields
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Motivation
Facies are defined as distinctive rock units formed undercertain conditions of sedimentation (Reading, 1996)Different conditions create different rocks with (sometimes)very dissimilar petro-physical propertiesGeological facies models are inherently non-gaussianproblemsThe EnKF requires the prior model for the parameters tobe Gaussian or approximately Gaussian random fields
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Outline
1 Motivation
2 IntroductionLevel set methodThe ensemble Kalman filterEnKF and level sets - the coupling
3 Papers overviewPapers A - BPaper CPaper DPaper E
4 SummaryDerivation of the analysis scheme
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
An implicit representation of geological faciesFacies Example.
-
-
++
20 40 60 80 100 120
20
40
60
80
100
120
140
1
0.5
0
-0.5
-1
An implicit function defines the domain of study, Ω, as:
ϕ(x) < 0 in Ω−,ϕ(x) = 0 on ∂Ω,ϕ(x) > 0 in Ω+.
(Osher, S. & J. A. Sethian (1988))
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Tracking movementLevel set methods add dynamics to implicit surfaces
Facies Example.
-
-
++
20 40 60 80 100 120
20
40
60
80
100
120
140
1
0.5
0
-0.5
-1
The permeability can be defined as:
K(x) = K1H(ϕ(x)) +K2(1−H(ϕ(x))),
and the level set evolves with:∂ϕ(x)
∂τ= −V(x)|∇ϕ(x)|,
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Tracking movementLevel set methods add dynamics to implicit surfaces
Facies Example.
-
-
++
20 40 60 80 100 120
20
40
60
80
100
120
140
1
0.5
0
-0.5
-1
The permeability can be defined as:
K(x) = K1H(ϕ(x)) +K2(1−H(ϕ(x))),
and the level set evolves with:∂ϕ(x)
∂τ= −V(x)|∇ϕ(x)|,
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Tracking movementLevel set methods add dynamics to implicit surfaces
Facies Example.
-
-
++
20 40 60 80 100 120
20
40
60
80
100
120
140
1
0.5
0
-0.5
-1
The permeability can be defined as:
K(x) = K1H(ϕ(x)) +K2(1−H(ϕ(x))),
and the level set evolves with:∂ϕ(x)
∂τ= −V(x)|∇ϕ(x)|,
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Implicit surfaces as signed distance functions
Facies transformed into signed distance functions
In a signed distance function |∇ϕ(x)| = 1, then:
∂ϕ(x)
∂τ= −V(x).
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Facies as signed distance functions
∂ϕ(x)
∂τ+ S(ϕo) (|∇ϕ(x)| − 1) = 0︸ ︷︷ ︸,
Where (Osher & Fedkiw, 2003; Mitchell, 2007)
S(ϕo) =
−1 iff x ∈ Ω−
1 iff x ∈ Ω+.
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
View of the implicit interface
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
1D view (cross section)
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Transform into a signed distance function
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Signed distance and corresponding cross-section
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
The ensemble Kalman filter
The ensemble Kalman filter(EnKF) is introduced by GeirEvensen(1994)Designed to address the problems related to the EKF(unbounded error growth for the covariance, calculation ofgradients)Monte carlo method for sequential Bayesian inversionA prior or forecast ensemble of reservoir models ispropagated in time to assimilate data resulting into aposterior or analyzed ensemble of reservoir models
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
The ensemble Kalman filter
prior
t0
time
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
The ensemble Kalman filter
prior
d1|t
1
time
t0
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
The ensemble Kalman filter
prior
d1|t
1
G( . ,t0
t1)
time
t0
f
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
The ensemble Kalman filter
Aprior
d1|t
1
aG( . ,t0
t1)
time
t0
f
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
The ensemble Kalman filter
A
A
prior
d1|t
1
a
f
A
a
f
A
f forecastaa analyzed state (after filtering)
A analysis scheme of the EnKF
G(.) transforms the analyzed state into the next forecast
G( . ,t0
t1)
time
dk
observations at time k
d2|t
2d
3|t
3dn|tn
G( . ,t1
t2)
G( . ,t2
t3)
G( . ,tn-1
tn)
...
...
...
...
Aa
f
...
...
...
dn-1
|tn-1
t0
f
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
The analysis scheme
For the discrete form
ψf = ψt + pf ,d = Hψt + ε,
where, ψf , is a model forecast, d, is a measurement of ψt ,H a linear operator that extracts the measurements out of ψt
(Evensen 2007), and
pf = 0,ε = 0,
pfεT = 0.
pf(pf)T = Cfψψ,
εεT = Cεε,
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
The analysis scheme
The analyzed(corrected) state is updated as a linear combina-tion of the forecast and the residual between the measured andthe simulated data.
ψa = ψf + Ke(d− Hψf ),Caψψ = (I− KeH)Cf
ψψ,
Ke = CfψψHT(HCf
ψψHT + Cεε)−1,
In the Kalman filter
Cfψψ = (ψf − ψt)(ψf − ψt)T ,
Caψψ = (ψa − ψt)(ψa − ψt)T ,
In the ensemble Kalman filter
(Ceψψ)f = (ψf − ψf )(ψf − ψf )T ,
(Ceψψ)a = (ψa − ψa)(ψa − ψa)T .
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Level set and EnKF
Facies Example.
-
-
++
20 40 60 80 100 120
20
40
60
80
100
120
140
1
0.5
0
-0.5
-1
ϕ(x)=⇒
∂ϕ(x)∂τ
=−V(x)
=⇒
Traditional EnKF Proposed Methodmodel parameters
K(x);φ(x)
V(x)
dependent dynamic variables
pi ; si
pi ; si
simulated data
wct; · · · ; bhp
wct; · · · ; bhp
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Level set and EnKF
Facies Example.
-
-
++
20 40 60 80 100 120
20
40
60
80
100
120
140
1
0.5
0
-0.5
-1
ϕ(x)=⇒
∂ϕ(x)∂τ
=−V(x)
=⇒
Traditional EnKF Proposed Methodmodel parameters
K(x);φ(x)
V(x)
dependent dynamic variables
pi ; si
pi ; si
simulated data
wct; · · · ; bhp
wct; · · · ; bhp
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Level set and EnKF
Facies Example.
-
-
++
20 40 60 80 100 120
20
40
60
80
100
120
140
1
0.5
0
-0.5
-1
ϕ(x)=⇒
∂ϕ(x)∂τ
=−V(x)
=⇒
Traditional EnKF Proposed Methodmodel parameters
K(x);φ(x)
V(x)
dependent dynamic variables
pi ; si
pi ; si
simulated data
wct; · · · ; bhp
wct; · · · ; bhp
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Level set and EnKF
Facies Example.
-
-
++
20 40 60 80 100 120
20
40
60
80
100
120
140
1
0.5
0
-0.5
-1
ϕ(x)=⇒
∂ϕ(x)∂τ
=−V(x)
=⇒
Traditional EnKF Proposed Methodmodel parameters
K(x);φ(x)
V(x)
dependent dynamic variables
pi ; si
pi ; si
simulated data
wct; · · · ; bhp
wct; · · · ; bhp
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling
Level set and EnKF
Facies Example.
-
-
++
20 40 60 80 100 120
20
40
60
80
100
120
140
1
0.5
0
-0.5
-1
ϕ(x)=⇒
∂ϕ(x)∂τ
=−V(x)
=⇒
Traditional EnKF Proposed Methodmodel parameters
K(x);φ(x)
V(x)
dependent dynamic variables
pi ; si
pi ; si
simulated data
wct; · · · ; bhp
wct; · · · ; bhp
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Outline
1 Motivation
2 IntroductionLevel set methodThe ensemble Kalman filterEnKF and level sets - the coupling
3 Papers overviewPapers A - BPaper CPaper DPaper E
4 SummaryDerivation of the analysis scheme
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Papers A - B
A−→ Stochastic Facies Modelling Using the Level setMethod (Moreno & Aanonsen)Proceeding at the 2007 EAGE Petroleum GeostatisticsConference. Cascais - Portugal.
B−→ Continuous Facies Updating Using the EnsembleKalman Filter and the Level Set Method (Moreno &Aanonsen)Submitted to Mathematical Geosciences, Jan 2009.
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Papers A - B
A−→ Stochastic Facies Modelling Using the Level setMethod (Moreno & Aanonsen)Proceeding at the 2007 EAGE Petroleum GeostatisticsConference. Cascais - Portugal.
B−→ Continuous Facies Updating Using the EnsembleKalman Filter and the Level Set Method (Moreno &Aanonsen)Submitted to Mathematical Geosciences, Jan 2009.
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Introduce a new the methodology for the history matchingof geological facies
Present a coupling between the level set method and theEnKF for continuous model updatingModel geological facies through the level set methodPerform dynamic reservoir characterization based onproduction dataTest different forms for evolving the interfaces based on thelevel set equation(Convective equation, equation of movement into itsnormal direction , and linear non-linear combinations of thesigned distance function)
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Introduce a new the methodology for the history matchingof geological faciesPresent a coupling between the level set method and theEnKF for continuous model updating
Model geological facies through the level set methodPerform dynamic reservoir characterization based onproduction dataTest different forms for evolving the interfaces based on thelevel set equation(Convective equation, equation of movement into itsnormal direction , and linear non-linear combinations of thesigned distance function)
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Introduce a new the methodology for the history matchingof geological faciesPresent a coupling between the level set method and theEnKF for continuous model updatingModel geological facies through the level set method
Perform dynamic reservoir characterization based onproduction dataTest different forms for evolving the interfaces based on thelevel set equation(Convective equation, equation of movement into itsnormal direction , and linear non-linear combinations of thesigned distance function)
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Introduce a new the methodology for the history matchingof geological faciesPresent a coupling between the level set method and theEnKF for continuous model updatingModel geological facies through the level set methodPerform dynamic reservoir characterization based onproduction data
Test different forms for evolving the interfaces based on thelevel set equation(Convective equation, equation of movement into itsnormal direction , and linear non-linear combinations of thesigned distance function)
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Introduce a new the methodology for the history matchingof geological faciesPresent a coupling between the level set method and theEnKF for continuous model updatingModel geological facies through the level set methodPerform dynamic reservoir characterization based onproduction dataTest different forms for evolving the interfaces based on thelevel set equation(Convective equation, equation of movement into itsnormal direction , and linear non-linear combinations of thesigned distance function)
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Procedure
Use of equations developed in the level set community to adddynamics to the interfaces
Hamilton-Jacobi equationsLevel set methods evolve ϕ according to
∂ϕ
∂τ+H(∇ϕ) = 0,
where H can be a function of both space and time anddepends at most to the first derivative
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Experiments based on the different level set equations
∂ϕ(x)
∂τ= −−→v · ∇ϕ(x), (1)
∂ϕ(x)
∂τ= −V(x)|∇ϕ(x)|, (2)
ϕ(x) = ϕsigned(x) + ∆τV(x) |(V≈N (0,Cx )), (3)ϕ(x) = ϕsigned(x)× V(x) |(V≈N (1,Cx )), (4)
(Osher & Sethian, 1988; Osher & Fedkiw, 2003)
Exp. 1 Exp. 2 Exp. 3 Exp. 4True case Eq. 1 Eq. 1 Eq. 1 Eq. 3Prior ensemble Eq. 1 Eq. 3 Eq. 4 Eq. 3
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Exp. one. True(level set); Prior(level set)True case, sand/shale ratio 62
10 20 30 40 50
10
20
30
40
50
Prior - time 1, member no 13, sand/shale ratio 57
10 20 30 40 50
10
20
30
40
50Final - time 61, member no 13, sand/shale ratio 61
10 20 30 40 50
10
20
30
40
50
10 20 30 40 50 60200
250
300
350
400Bottom hole pressure for injector 1.
priorHistoryPosterior
10 20 30 40 50 60200
250
300
350
400Bottom hole pressure for injector 2.
priorHistoryPosterior
Prior - time 1, member no 37, sand/shale ratio 57
10 20 30 40 50
10
20
30
40
50Final - time 61, member no 37, sand/shale ratio 60
10 20 30 40 50
10
20
30
40
50
10 20 30 40 50 600
0.2
0.4
0.6
0.8
1Water cut for producer 1.
priorHistoryPosterior
10 20 30 40 50 600
0.2
0.4
0.6
0.8
1Water cut for producer 2.
priorHistoryPosterior
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Exp. three. True(level set); Prior(non-linear eq. 4)True case, sand/shale ratio 62
10 20 30 40 50
10
20
30
40
50
Prior - time 1, member no 25, sand/shale ratio 52
10 20 30 40 50
10
20
30
40
50
10 20 30 40 50 60200
250
300
350
400Bottom hole pressure for injector I1.
priorHistoryPosterior
10 20 30 40 50 60200
250
300
350
400Bottom hole pressure for injector I2.
priorHistoryPosterior
Prior - time 1, member no 50, sand/shale ratio 51
10 20 30 40 50
10
20
30
40
50Final - time 61, member no 50, sand/shale ratio 64
10 20 30 40 50
10
20
30
40
50
10 20 30 40 50 600
0.2
0.4
0.6
0.8
1Water cut for producer P1.
priorHistoryPosterior
10 20 30 40 50 600
0.2
0.4
0.6
0.8
1Water cut for producer P2.
priorHistoryPosterior
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
WBHP results for experiment three
10 20 30 40 50 60200
250
300
350
400Bottom hole pressure for injector I1.
priorHistoryPosterior
10 20 30 40 50 60200
250
300
350
400Bottom hole pressure for injector I2.
priorHistoryPosterior
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Results for experiment fourBase Case.
I1
P1 P2
P3
P4
20 40 60 80 100 120
20
40
60
80
100
120
140True Case.
I1
P1 P2
P3
P4
20 40 60 80 100 120
20
40
60
80
100
120
140
10 20 30 40 50 60
250
300
350
400
450
500
550
600Bottom Hole Pressure - Injector 1
PriorPosteriorHistoryBase Case
10 20 30 40 50 600
1000
2000
3000
4000
5000
6000Well Oil Production Rate - Producer 1.
PriorPosteriorHistoryBase Case
Permeability member 62 - Prior.
I1
P1 P2
P3
P4
20 40 60 80 100 120
20
40
60
80
100
120
140Permeability member 62 - Posterior.
I1
P1 P2
P3
P4
20 40 60 80 100 120
20
40
60
80
100
120
14010 20 30 40 50 60
0
1000
2000
3000
4000
5000
6000Well Oil Production Rate - Producer 2.
PriorPosteriorHistoryBase Case
10 20 30 40 50 600
2000
4000
6000
8000
10000Well Oil Production Rate - Producer 3.
PriorPosteriorHistoryBase Case
Permeability member 66 - Prior.
I1
P1 P2
P3
P4
20 40 60 80 100 120
20
40
60
80
100
120
140Permeability member 66 - Posterior.
I1
P1 P2
P3
P4
20 40 60 80 100 120
20
40
60
80
100
120
14010 20 30 40 50 60
0
0.2
0.4
0.6
0.8
1Well Water Cut - Producer 1
PriorPosteriorHistoryBase Case
10 20 30 40 50 600
0.2
0.4
0.6
0.8
1Well Water Cut - Producer 3
PriorPosteriorHistoryBase Case
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
WOPR results for experiment four
10 20 30 40 50 600
1000
2000
3000
4000
5000
6000Well Oil Production Rate - Producer 2.
PriorPosteriorHistoryBase Case
10 20 30 40 50 600
2000
4000
6000
8000
10000Well Oil Production Rate - Producer 3.
PriorPosteriorHistoryBase Case
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Two synthetic reservoirs with bimodal petro-physicalregions were studied
The methodology was effective for the modelling andupdating reservoirs containing faciesDynamic data (production data) seems to have enoughinformation for a good reconstruction of the topologiesThe models are sensitive to the prior models (a collapsetowards a single model is evident)The coupling of the level set method with the EnKF seemsto be a good alternative for the modelling and updating offacies using production data
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Two synthetic reservoirs with bimodal petro-physicalregions were studiedThe methodology was effective for the modelling andupdating reservoirs containing facies
Dynamic data (production data) seems to have enoughinformation for a good reconstruction of the topologiesThe models are sensitive to the prior models (a collapsetowards a single model is evident)The coupling of the level set method with the EnKF seemsto be a good alternative for the modelling and updating offacies using production data
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Two synthetic reservoirs with bimodal petro-physicalregions were studiedThe methodology was effective for the modelling andupdating reservoirs containing faciesDynamic data (production data) seems to have enoughinformation for a good reconstruction of the topologies
The models are sensitive to the prior models (a collapsetowards a single model is evident)The coupling of the level set method with the EnKF seemsto be a good alternative for the modelling and updating offacies using production data
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Two synthetic reservoirs with bimodal petro-physicalregions were studiedThe methodology was effective for the modelling andupdating reservoirs containing faciesDynamic data (production data) seems to have enoughinformation for a good reconstruction of the topologiesThe models are sensitive to the prior models (a collapsetowards a single model is evident)
The coupling of the level set method with the EnKF seemsto be a good alternative for the modelling and updating offacies using production data
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Two synthetic reservoirs with bimodal petro-physicalregions were studiedThe methodology was effective for the modelling andupdating reservoirs containing faciesDynamic data (production data) seems to have enoughinformation for a good reconstruction of the topologiesThe models are sensitive to the prior models (a collapsetowards a single model is evident)The coupling of the level set method with the EnKF seemsto be a good alternative for the modelling and updating offacies using production data
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Paper C
Channel facies estimation based on Gaussianperturbations in the EnKF(Moreno, Aanonsen, Evensen & Skjervheim)11th European Conference on the Mathematics of OilRecovery. Bergen - Norway. 8 - 11 September 2008
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Test the viability of the methodology for a 3D two faciessemi-synthetic reservoir of the North sea (the Osebergfield)
Introduce different approaches for the parametrization offacies models (level set - shapiro filter)Compare experiments where the true model is generatedwith a different method as that used for the priorrealizationsInvestigate the ease of implementation of the methodology
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Test the viability of the methodology for a 3D two faciessemi-synthetic reservoir of the North sea (the Osebergfield)Introduce different approaches for the parametrization offacies models (level set - shapiro filter)
Compare experiments where the true model is generatedwith a different method as that used for the priorrealizationsInvestigate the ease of implementation of the methodology
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Test the viability of the methodology for a 3D two faciessemi-synthetic reservoir of the North sea (the Osebergfield)Introduce different approaches for the parametrization offacies models (level set - shapiro filter)Compare experiments where the true model is generatedwith a different method as that used for the priorrealizations
Investigate the ease of implementation of the methodology
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Test the viability of the methodology for a 3D two faciessemi-synthetic reservoir of the North sea (the Osebergfield)Introduce different approaches for the parametrization offacies models (level set - shapiro filter)Compare experiments where the true model is generatedwith a different method as that used for the priorrealizationsInvestigate the ease of implementation of the methodology
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Methodology - the Shapiro filter
Generates a φinit by smoothing the initial indicator functionThe initial surface is then given by (Shapiro 1970, 1975):
φinit(xi) = S(φ0(xi)) +2n∑
k=0
(−1)n+k−1(2n)!
22nk !(2n − k)!S(φ0(xi+n−k )),
Where, n = 8 is chosen (Evensen, 1994)For more than one-dimensional problems the filter isapplied on a dimension by dimension fashion
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Perturbations to implicit surfaces
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Cross-section of the perturbations
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Base case and true models, layer 9
Base Case Shapiro Level Set
Well 1
Well 2
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Posterior realizations
Case 1 Case 2
Case 3 Case 4
Figure: Posterior realization 1, layer 9.
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Case 2: EnKF(level set); true model(Shapiro)
Jan95 Jan00 Jan050
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (days)
WW
CT
(%
)
WELL 2: WWCT
HistoryPriorPosterior
Jan000
500
1000
1500
2000
2500
3000
Time (days)
WG
OR
(S
M3/
DA
Y)
WELL 2: WGOR
HistoryPriorPosterior
Figure: Simulated well gas-oil ratio using 20 realizations from priorand posterior ensemble.
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Case 4: EnKF(Shapiro); true model(Shapiro)
Jan95 Jan00 Jan050
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Time (days)
WW
CT
(%
)
WELL 2: WWCT
HistoryPriorPosterior
Jan000
500
1000
1500
2000
2500
3000
Time (days)
WG
OR
(S
M3/
DA
Y)
WELL 2: WGOR
HistoryPriorPosterior
Figure: Simulated well water cut using 20 realizations from prior andposterior ensemble.
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Two different approaches for parameterization of the faciesmodels in terms of Gaussian perturbations of an existing"best guess" model were presented
The methods were tested on a 3D model inspired by a realNorth Sea fluvial reservoirA large variation of realistic facies model realizations maybe generated from Gaussian random fields andconditioned to production data using the EnKF
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Two different approaches for parameterization of the faciesmodels in terms of Gaussian perturbations of an existing"best guess" model were presentedThe methods were tested on a 3D model inspired by a realNorth Sea fluvial reservoir
A large variation of realistic facies model realizations maybe generated from Gaussian random fields andconditioned to production data using the EnKF
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Two different approaches for parameterization of the faciesmodels in terms of Gaussian perturbations of an existing"best guess" model were presentedThe methods were tested on a 3D model inspired by a realNorth Sea fluvial reservoirA large variation of realistic facies model realizations maybe generated from Gaussian random fields andconditioned to production data using the EnKF
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
The updating procedure performs reasonably well alsowhen the true model is not a realization from the samestatistical model
The posterior realizations to a large degree reflects theprior modelThe novelty of the paper lies in the inclusion of additionalparameterization for the facies, its extension to a 3D fortwo facies, and its test on a North sea base case reservoir
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
The updating procedure performs reasonably well alsowhen the true model is not a realization from the samestatistical modelThe posterior realizations to a large degree reflects theprior model
The novelty of the paper lies in the inclusion of additionalparameterization for the facies, its extension to a 3D fortwo facies, and its test on a North sea base case reservoir
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
The updating procedure performs reasonably well alsowhen the true model is not a realization from the samestatistical modelThe posterior realizations to a large degree reflects theprior modelThe novelty of the paper lies in the inclusion of additionalparameterization for the facies, its extension to a 3D fortwo facies, and its test on a North sea base case reservoir
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Paper D
Conditioning geological facies to production and welldata using the ensemble Kalman filter and the level setmethod: a study on ensemble size and localization(Moreno & Aanonsen)Preprint form, to be submitted to the SPE Journal
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Extension of the methodology to more than two faciesmodels (four facies)
Investigate models where the base case is not conditionedinitially to the petro-physical properties at the position ofthe wellsIncorporate prior geological information into the models(Bayesian framework)Address the collapse of the members observed in previousworkIllustrate advantages of distance dependent localizationschemes for assimilation of petro-physical properties
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Extension of the methodology to more than two faciesmodels (four facies)Investigate models where the base case is not conditionedinitially to the petro-physical properties at the position ofthe wells
Incorporate prior geological information into the models(Bayesian framework)Address the collapse of the members observed in previousworkIllustrate advantages of distance dependent localizationschemes for assimilation of petro-physical properties
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Extension of the methodology to more than two faciesmodels (four facies)Investigate models where the base case is not conditionedinitially to the petro-physical properties at the position ofthe wellsIncorporate prior geological information into the models(Bayesian framework)
Address the collapse of the members observed in previousworkIllustrate advantages of distance dependent localizationschemes for assimilation of petro-physical properties
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Extension of the methodology to more than two faciesmodels (four facies)Investigate models where the base case is not conditionedinitially to the petro-physical properties at the position ofthe wellsIncorporate prior geological information into the models(Bayesian framework)Address the collapse of the members observed in previouswork
Illustrate advantages of distance dependent localizationschemes for assimilation of petro-physical properties
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Extension of the methodology to more than two faciesmodels (four facies)Investigate models where the base case is not conditionedinitially to the petro-physical properties at the position ofthe wellsIncorporate prior geological information into the models(Bayesian framework)Address the collapse of the members observed in previousworkIllustrate advantages of distance dependent localizationschemes for assimilation of petro-physical properties
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Methodology - several level set functions
N Level sets can represent up to 2N subdomains
Figure: Level set functions and facies types.
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Two level sets required
(a) Level set one (b) Level set two
Figure: Level sets for the base case facies map. Here, the blue zonesare negative ϕi(x) < 0|x ∈ Ω−; i = 1,2 and the red ones positiveϕi(x) > 0|x ∈ Ω+; i = 1,2.
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Results with different ensemble sizes
100 200 1000Base Case
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25
4
3
2
1
2D Four facies - True Case.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
252D Four facies - True Case.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
252D Four facies - True Case.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25
Prior Realization No. 62.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25Permeability member 62 - Posterior.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25
Permeability member 62 - Posterior.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25
Permeability member 62 - Posterior.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Results with 1000 ensemble members
2D Four facies - True Case.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25Permeability member 62 - Posterior.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
History matching(WBHP) 100,200 and 1000 ens.
Jul02 Jan050
500
1000
1500
Time (days)
WO
PR
(B
ars)
P8: WOPR
HistoryPriorPosteriormean
Jul02 Jan050
500
1000
1500
Time (days)
WO
PR
(B
ars)
P8: WOPR
HistoryPriorPosteriormean
Jul02 Jan050
500
1000
1500
Time (days)
WO
PR
(B
ars)
P8: WOPR
HistoryPriorPosteriormean
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Incorporating geological information into the EnKF
2D Four facies - True Case.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25
I1 P1 P2 P3 P4 P5 P6 P7 P81 1 3 4 4 3 1 4 1
%F1 %F2 %F3 %F437.12 14.72 9.44 38.72
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conditioning to the petrophysics with the EnKF
Proposed methodmodel parameters
V(x)
simulated data
facies type well I1facies type well P1
...facies type well P8
sand/shale ratio facies 1...
sand/shale ratio facies 4
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Ass. of facies and proportions(without localization)
20 40 60 800
500
1000
1500Well Oil Production Rate No loc - Producer 8
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Ass. of facies and proportions(with localization)- 1 Ite.
20 40 60 800
500
1000
1500Well Oil Production Rate Ite. 1 - Producer 8
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Ass. of facies and proportions(with localization)- 3 Ite.
20 40 60 800
500
1000
1500Well Oil Production Rate Ite. 3 - Producer 8
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
How is the localization Applied
5 10 15 20 25
5
10
15
20
25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Results with 100 ensemble members
No loc. Loc. 1 ite. Loc. 3 ite.Base Case
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25
4
3
2
1
2D Four facies - True Case.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
252D Four facies - True Case.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
252D Four facies - True Case.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25
Prior Realization No. 62.
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25Posterior , realization no 62
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25Posterior , realization no 62
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25Posterior , realization no 62
I1
P1
P2
P3
P4
P5
P6
P7
P8
5 10 15 20 25
5
10
15
20
25
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Extended the methodology to four facies systems andmade a study on the effect of increasing the ensemble size
Studied the adding of geological features to the EnKFapplied to the four facies modelSuccessfully applied distance dependent localization ofstatic hard dat (petro-physical) for conditioning faciesmodelsSuccessfully history matched reservoir models with fourfaciesThe results opens a door for the inclusion of other"geological" parameters
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Extended the methodology to four facies systems andmade a study on the effect of increasing the ensemble sizeStudied the adding of geological features to the EnKFapplied to the four facies model
Successfully applied distance dependent localization ofstatic hard dat (petro-physical) for conditioning faciesmodelsSuccessfully history matched reservoir models with fourfaciesThe results opens a door for the inclusion of other"geological" parameters
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Extended the methodology to four facies systems andmade a study on the effect of increasing the ensemble sizeStudied the adding of geological features to the EnKFapplied to the four facies modelSuccessfully applied distance dependent localization ofstatic hard dat (petro-physical) for conditioning faciesmodels
Successfully history matched reservoir models with fourfaciesThe results opens a door for the inclusion of other"geological" parameters
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Extended the methodology to four facies systems andmade a study on the effect of increasing the ensemble sizeStudied the adding of geological features to the EnKFapplied to the four facies modelSuccessfully applied distance dependent localization ofstatic hard dat (petro-physical) for conditioning faciesmodelsSuccessfully history matched reservoir models with fourfacies
The results opens a door for the inclusion of other"geological" parameters
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
Extended the methodology to four facies systems andmade a study on the effect of increasing the ensemble sizeStudied the adding of geological features to the EnKFapplied to the four facies modelSuccessfully applied distance dependent localization ofstatic hard dat (petro-physical) for conditioning faciesmodelsSuccessfully history matched reservoir models with fourfaciesThe results opens a door for the inclusion of other"geological" parameters
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Paper E
Dynamic reservoir characterisation of a 3D Geocellularmodel using the ensemble Kalman filter(Moreno, Aanonsen, Carlsson and Howell)Preprint form, to be submitted to the AAPG Journal
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Extension of the methodology to a 3D four facies model
Use an outcrop-like model built using advanced lidartechniques and well logs from the Woodside Canyon inUtahStudy the effect of the increase of the ensemble sizes inthe modelCondition the model to production data using the EnKF
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Extension of the methodology to a 3D four facies modelUse an outcrop-like model built using advanced lidartechniques and well logs from the Woodside Canyon inUtah
Study the effect of the increase of the ensemble sizes inthe modelCondition the model to production data using the EnKF
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Extension of the methodology to a 3D four facies modelUse an outcrop-like model built using advanced lidartechniques and well logs from the Woodside Canyon inUtahStudy the effect of the increase of the ensemble sizes inthe model
Condition the model to production data using the EnKF
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Motivation
Extension of the methodology to a 3D four facies modelUse an outcrop-like model built using advanced lidartechniques and well logs from the Woodside Canyon inUtahStudy the effect of the increase of the ensemble sizes inthe modelCondition the model to production data using the EnKF
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Outcrop of the Woodside canyon
Figure: Cross view of the geocellular five facies 3D-model with atopographic surface from the Woodside Canyon. The height of themodeled unit is 50 m.
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
History matching results
100 200 500
Jan96 Jan98 Jan00 Jan02
315
320
325
330
335
Time (days)
WB
HP
(B
AR
S)
I1: WBHP
HistoryBase CasePriorPosteriormean
Jan96 Jan98 Jan00 Jan02
315
320
325
330
335
Time (days)
WB
HP
(B
AR
S)
I1: WBHP
HistoryBase CasePriorPosteriormean
Jan96 Jan98 Jan00 Jan02
315
320
325
330
335
Time (days)
WB
HP
(B
AR
S)
I1: WBHP
HistoryBase CasePriorPosteriormean
Jan96 Jan98 Jan00 Jan02315
320
325
330
335
340
345
350
Time (days)
WB
HP
(B
AR
S)
I2: WBHP
HistoryBase CasePriorPosteriormean
Jan96 Jan98 Jan00 Jan02315
320
325
330
335
340
345
350
Time (days)
WB
HP
(B
AR
S)
I2: WBHP
HistoryBase CasePriorPosteriormean
Jan96 Jan98 Jan00 Jan02315
320
325
330
335
340
345
350
Time (days)
WB
HP
(B
AR
S)
I2: WBHP
HistoryBase CasePriorPosteriormean
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
History matching of outcrop 3D models, analogs toNorth-sea reservoirs is possible with the methodology
Petro-physical conditioning is very important for the historymatchingInclusion of prior information that accounts for a lot of theuncertainty in the reservoir modelsGood history matching but deficient reconstruction of thefacies(highly non-linear problem)Extension of the methodology to 3D cases for multiplefacies
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
History matching of outcrop 3D models, analogs toNorth-sea reservoirs is possible with the methodologyPetro-physical conditioning is very important for the historymatching
Inclusion of prior information that accounts for a lot of theuncertainty in the reservoir modelsGood history matching but deficient reconstruction of thefacies(highly non-linear problem)Extension of the methodology to 3D cases for multiplefacies
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
History matching of outcrop 3D models, analogs toNorth-sea reservoirs is possible with the methodologyPetro-physical conditioning is very important for the historymatchingInclusion of prior information that accounts for a lot of theuncertainty in the reservoir models
Good history matching but deficient reconstruction of thefacies(highly non-linear problem)Extension of the methodology to 3D cases for multiplefacies
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
History matching of outcrop 3D models, analogs toNorth-sea reservoirs is possible with the methodologyPetro-physical conditioning is very important for the historymatchingInclusion of prior information that accounts for a lot of theuncertainty in the reservoir modelsGood history matching but deficient reconstruction of thefacies(highly non-linear problem)
Extension of the methodology to 3D cases for multiplefacies
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
History matching of outcrop 3D models, analogs toNorth-sea reservoirs is possible with the methodologyPetro-physical conditioning is very important for the historymatchingInclusion of prior information that accounts for a lot of theuncertainty in the reservoir modelsGood history matching but deficient reconstruction of thefacies(highly non-linear problem)Extension of the methodology to 3D cases for multiplefacies
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Papers A - BPaper CPaper DPaper E
Conclusions & contribution
History matching of outcrop 3D models, analogs toNorth-sea reservoirs is possible with the methodologyPetro-physical conditioning is very important for the historymatchingInclusion of prior information that accounts for a lot of theuncertainty in the reservoir modelsGood history matching but deficient reconstruction of thefacies(highly non-linear problem)Extension of the methodology to 3D cases for multiplefacies
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Outline
1 Motivation
2 IntroductionLevel set methodThe ensemble Kalman filterEnKF and level sets - the coupling
3 Papers overviewPapers A - BPaper CPaper DPaper E
4 SummaryDerivation of the analysis scheme
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Summary - advantages
Introduce a new methodology based on a couplingbetween the level set method and the EnKF
Dynamic reservoir characterization using production andwell data with the EnKFMethodology completely automatic (No adjoints orgradients on the data are necessary)History matching of 2D,3D reservoirs containing faciessystems (synthetic, Outcrop and a real north sea field)The level set method has proven once more to be aversatile tool for facies characterizationIt is possible to include additional constraints(geological)into the problem
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Summary - advantages
Introduce a new methodology based on a couplingbetween the level set method and the EnKFDynamic reservoir characterization using production andwell data with the EnKF
Methodology completely automatic (No adjoints orgradients on the data are necessary)History matching of 2D,3D reservoirs containing faciessystems (synthetic, Outcrop and a real north sea field)The level set method has proven once more to be aversatile tool for facies characterizationIt is possible to include additional constraints(geological)into the problem
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Summary - advantages
Introduce a new methodology based on a couplingbetween the level set method and the EnKFDynamic reservoir characterization using production andwell data with the EnKFMethodology completely automatic (No adjoints orgradients on the data are necessary)
History matching of 2D,3D reservoirs containing faciessystems (synthetic, Outcrop and a real north sea field)The level set method has proven once more to be aversatile tool for facies characterizationIt is possible to include additional constraints(geological)into the problem
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Summary - advantages
Introduce a new methodology based on a couplingbetween the level set method and the EnKFDynamic reservoir characterization using production andwell data with the EnKFMethodology completely automatic (No adjoints orgradients on the data are necessary)History matching of 2D,3D reservoirs containing faciessystems (synthetic, Outcrop and a real north sea field)
The level set method has proven once more to be aversatile tool for facies characterizationIt is possible to include additional constraints(geological)into the problem
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Summary - advantages
Introduce a new methodology based on a couplingbetween the level set method and the EnKFDynamic reservoir characterization using production andwell data with the EnKFMethodology completely automatic (No adjoints orgradients on the data are necessary)History matching of 2D,3D reservoirs containing faciessystems (synthetic, Outcrop and a real north sea field)The level set method has proven once more to be aversatile tool for facies characterization
It is possible to include additional constraints(geological)into the problem
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Summary - advantages
Introduce a new methodology based on a couplingbetween the level set method and the EnKFDynamic reservoir characterization using production andwell data with the EnKFMethodology completely automatic (No adjoints orgradients on the data are necessary)History matching of 2D,3D reservoirs containing faciessystems (synthetic, Outcrop and a real north sea field)The level set method has proven once more to be aversatile tool for facies characterizationIt is possible to include additional constraints(geological)into the problem
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Summary - disadvantages
When the level set equation is used, extra time is requiredto solve the solution of the problem
Very sensitive to the prior model (Perhaps, the base caseshould also evolve?)Evident collapse of the members of the ensemble (largerensemble sizes!, localization?)Highly non-linear problem (multiple solutions)Dynamic conditioning of the reservoirs with the EnKFmight disturb initial uncertainties about the petro-physics
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Summary - disadvantages
When the level set equation is used, extra time is requiredto solve the solution of the problemVery sensitive to the prior model (Perhaps, the base caseshould also evolve?)
Evident collapse of the members of the ensemble (largerensemble sizes!, localization?)Highly non-linear problem (multiple solutions)Dynamic conditioning of the reservoirs with the EnKFmight disturb initial uncertainties about the petro-physics
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Summary - disadvantages
When the level set equation is used, extra time is requiredto solve the solution of the problemVery sensitive to the prior model (Perhaps, the base caseshould also evolve?)Evident collapse of the members of the ensemble (largerensemble sizes!, localization?)
Highly non-linear problem (multiple solutions)Dynamic conditioning of the reservoirs with the EnKFmight disturb initial uncertainties about the petro-physics
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Summary - disadvantages
When the level set equation is used, extra time is requiredto solve the solution of the problemVery sensitive to the prior model (Perhaps, the base caseshould also evolve?)Evident collapse of the members of the ensemble (largerensemble sizes!, localization?)Highly non-linear problem (multiple solutions)
Dynamic conditioning of the reservoirs with the EnKFmight disturb initial uncertainties about the petro-physics
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Summary - disadvantages
When the level set equation is used, extra time is requiredto solve the solution of the problemVery sensitive to the prior model (Perhaps, the base caseshould also evolve?)Evident collapse of the members of the ensemble (largerensemble sizes!, localization?)Highly non-linear problem (multiple solutions)Dynamic conditioning of the reservoirs with the EnKFmight disturb initial uncertainties about the petro-physics
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Acknowledgments
Thanks to: Sigurd I. Aanonsen, Magne Espedal, GeirEvensen, Geir Nædval and Dean S. OliverPetromaks project "Continuous model updating using theEnKF with emphasis on complex reservoirs"Centre for Integrated Petroleum Research(CIPR)University of BergenInternational Research Institute of Stavanger(IRIS)
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Thank you
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Gibou, F., Fedkiw, R. Caflisch, R. and Osher S., "A LevelSet Approach for the Numerical Simulation of DendriticGrowth", J. Sci. Comput. 19, 183-199 (2003)Osher, S. & J. A. Sethian (1988), "Fronts propagating withcurvature-dependent speed: Algorithms based onHamilton-Jacobi formulations", J. Comput. Phys. 79:12U49.Reading, H. G. (Ed.), (1996), Sedimentary Environmentsand Facies. Blackwell Scientific Publications.Shapiro, R. Smoothing, filtering, and boundary effects Rev.Geophys. Space Physics, 1970, Vol 8, pp 359-387
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Evensen, G, (1994), Sequential data assimilation with anon-linear quasi-geostrophic model using Monte Carlomethods to forecast error statistics. J Geophys. Res.10(99):143-162Evensen, G, (2007), Data Assimilation, The ensembleKalman filter. Springer, New YorkToolbox of Level Set Methods, its source, and itsdocumentation are Copyright 2007 by Ian M. Mitchell
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Derivation of the Analysis scheme for the scalar case
Given two different estimates of the true case ψt
ψf = ψt + pf ,d = ψt + ε,
ψf may be a model forecast or a first-guess estimate and d is ameasurement if ψt . pf denotes the unknown error in the forecastand ε is the unknown measurement error. The problem accountsto find an improved analyzed estimate ψa of ψt .
pf = 0,ε = 0,
pfεT = 0
(pf)2 = Cfψψ,
(ε)2 = Cεε,
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Derivation of the Analysis scheme for the scalar case
We seek for a linear estimator for the analyzed state
ψa = ψt + pa = α1ψf + α2d ,
where pa = 0 and (pa)2 = Caψψ (unbiased).
inserting previous estimates
ψt + pa = α1(ψt + pf ) + α2(ψ
t + ε),
the expectation of this equation leads to
ψt = α1ψt + α2ψ
t = (α1 + α2)ψt ,
therefore α1 + α2 = 1
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Derivation of the Analysis scheme for the scalar case
And a linear unbiased estimator for ψt is given as
ψa = (1− α2)ψf + α2d ,
= ψf + α2(d − ψf ).
an expression for the error in the analysis can be achieved in thesame form leading to
pa = pf + α2(ε− pf )
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Derivation of the Analysis scheme for the scalar case
Calculating the variance of pa
(pa)2 = Caψψ = (pf + α2(ε− pf ))2,
= Cfψψ − 2α2Cf
ψψ + α22(Cεε + Cf
ψψ).
Taking the derivative of this expression with respect to α2 andmaking it equal to 0, that is, finding the minimum variance
dCaψψ
dα2= 2Cf
ψψ + 2α2(Cεε + Cfψψ) = 0
David Moreno History matching of geological facies with the EnKF
MotivationIntroduction
Papers overviewSummary
Derivation of the analysis scheme
Derivation of the Analysis scheme for the scalar case
and solving for α2
α2 =Cfψψ
(Cεε + Cfψψ)
and the analyzed estimate becomes
ψa = ψf +Cfψψ
(Cεε + Cfψψ)
(d − ψf )
similarly, for the error variance of the analyzed estimate can becalculated as
Caψψ = Cf
ψψ
(1−
Cfψψ
Cεε + Cfψψ
)(Evensen, 2007)
David Moreno History matching of geological facies with the EnKF
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