history matching of geological facies, data assimilation, complex

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



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




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




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




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




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




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




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




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




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




Level set methodThe ensemble Kalman filterEnKF and level sets - the coupling

Outline

1 Motivation








An implicit representation of geological faciesFacies Example.

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20 40 60 80 100 120

20

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1

0.5

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-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))





Tracking movementLevel set methods add dynamics to implicit surfaces

Facies Example.

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20 40 60 80 100 120

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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)|,





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).





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+.





View of the implicit interface





1D view (cross section)





Transform into a signed distance function





Signed distance and corresponding cross-section





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






prior

t0

time






prior

d1|t

1

time

t0






prior

d1|t

1

G( . ,t0

t1)

time

t0

f






Aprior

d1|t

1

aG( . ,t0

t1)

time

t0

f






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





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,





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 .





Level set and EnKF

Facies Example.

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1

0.5

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-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




Papers A - BPaper CPaper DPaper E

Outline

1 Motivation








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.





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)





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)





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)





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)





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)





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





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





Exp. one. True(level set); Prior(level set)True case, sand/shale ratio 62

10 20 30 40 50

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30

40

50

Prior - time 1, member no 13, sand/shale ratio 57

10 20 30 40 50

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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.



10 20 30 40 50

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40


10 20 30 40 50

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10 20 30 40 50 600

0.2

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1Water cut for producer 1.


10 20 30 40 50 600

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1Water cut for producer 2.






Exp. three. True(level set); Prior(non-linear eq. 4)True case, sand/shale ratio 62

10 20 30 40 50

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50


10 20 30 40 50

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10 20 30 40 50 60200

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10 20 30 40 50 60200

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10 20 30 40 50

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1Water cut for producer P1.


10 20 30 40 50 600

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1Water cut for producer P2.






WBHP results for experiment three

10 20 30 40 50 60200

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10 20 30 40 50 60200

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Results for experiment fourBase Case.

I1

P1 P2

P3

P4

20 40 60 80 100 120

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40

60

80

100

120

140True Case.

I1

P1 P2

P3

P4

20 40 60 80 100 120

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10 20 30 40 50 60

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PriorPosteriorHistoryBase Case

10 20 30 40 50 600

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5000

6000Well Oil Production Rate - Producer 1.


Permeability member 62 - Prior.

I1

P1 P2

P3

P4

20 40 60 80 100 120

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40

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100

120

140Permeability member 62 - Posterior.

I1

P1 P2

P3

P4

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100

120

14010 20 30 40 50 60

0

1000

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10 20 30 40 50 600

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Permeability member 66 - Prior.

I1

P1 P2

P3

P4

20 40 60 80 100 120

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40

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120


I1

P1 P2

P3

P4

20 40 60 80 100 120

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14010 20 30 40 50 60

0

0.2

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1Well Water Cut - Producer 1


10 20 30 40 50 600

0.2

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0.6

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1Well Water Cut - Producer 3






WOPR results for experiment four

10 20 30 40 50 600

1000

2000

3000

4000

5000



10 20 30 40 50 600

2000

4000

6000

8000







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






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






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






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






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





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





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





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





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





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





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





Perturbations to implicit surfaces





Cross-section of the perturbations





Base case and true models, layer 9

Base Case Shapiro Level Set

Well 1

Well 2





Posterior realizations

Case 1 Case 2

Case 3 Case 4

Figure: Posterior realization 1, layer 9.





Case 2: EnKF(level set); true model(Shapiro)

Jan95 Jan00 Jan050

0.1

0.2

0.3

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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


Figure: Simulated well gas-oil ratio using 20 realizations from priorand posterior ensemble.





Case 4: EnKF(Shapiro); true model(Shapiro)

Jan95 Jan00 Jan050

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0.7

Time (days)

WW

CT

(%

)

WELL 2: WWCT


Jan000

500

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1500

2000

2500

3000

Time (days)

WG

OR

(S

M3/

DA

Y)

WELL 2: WGOR


Figure: Simulated well water cut using 20 realizations from prior andposterior ensemble.






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






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






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






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






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






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





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





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





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





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





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





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





Methodology - several level set functions

N Level sets can represent up to 2N subdomains

Figure: Level set functions and facies types.





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.





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


I1

P1

P2

P3

P4

P5

P6

P7

P8

5 10 15 20 25

5

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15

20


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


I1

P1

P2

P3

P4

P5

P6

P7

P8

5 10 15 20 25

5

10

15

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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





Results with 1000 ensemble members


I1

P1

P2

P3

P4

P5

P6

P7

P8

5 10 15 20 25

5

10

15

20


I1

P1

P2

P3

P4

P5

P6

P7

P8

5 10 15 20 25

5

10

15

20

25





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


Jul02 Jan050

500

1000

1500

Time (days)

WO

PR

(B

ars)

P8: WOPR






Incorporating geological information into the EnKF


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





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





Ass. of facies and proportions(without localization)

20 40 60 800

500

1000

1500Well Oil Production Rate No loc - Producer 8





Ass. of facies and proportions(with localization)- 1 Ite.

20 40 60 800

500

1000

1500Well Oil Production Rate Ite. 1 - Producer 8





Ass. of facies and proportions(with localization)- 3 Ite.

20 40 60 800

500

1000

1500Well Oil Production Rate Ite. 3 - Producer 8





How is the localization Applied

5 10 15 20 25

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15

20

25

0

0.1

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0.9

1





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


I1

P1

P2

P3

P4

P5

P6

P7

P8

5 10 15 20 25

5

10

15

20


I1

P1

P2

P3

P4

P5

P6

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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






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






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






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






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





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





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





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





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





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





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.





History matching results

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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






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






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






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






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




Derivation of the analysis scheme

Outline

1 Motivation








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






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






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






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






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






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





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






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






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






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






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





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)





Thank you





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





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





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,






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






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 )






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






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)
