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Classification: Internal Status: Draft

Using the EnKF for combined state and

parameter estimationGeir Evensen

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Outline

• Reservoir modelling and simulation

• History matching problem and uncertainty prediction

• Ensemble Kalman filter (EnKF)

• Field case example

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Reservoir Geophysics and Fast Model Updating

• Business challenge

 – To reduce uncertainty in reserves and

production targets

• Project goal

 – Provide continuously updated and

integrated models with reduced and

quantified uncertainty

•  Activities

 – Seismic acquisition and imaging

 – 4D quantitative analysis

 – Integrated use of 4D seismic data

 – Well based reservoir monitoring

 – Model uncertainty and updating

 – Integrated IOR work processes

http://sp-st10.statoil.com/sites/d7ca533a-20dd-4ae1-af1d-613bc17828c6/Document%20library/Forms/DispForm.aspx?ID=42&RootFolder=%2Fsites%2Fd7ca533a%2D20dd%2D4ae1%2Daf1d%2D613bc17828c6%2FDocument%20libraryhttp://sp-st10.statoil.com/sites/d7ca533a-20dd-4ae1-af1d-613bc17828c6/Document%20library/Forms/DispForm.aspx?ID=42&RootFolder=%2Fsites%2Fd7ca533a%2D20dd%2D4ae1%2Daf1d%2D613bc17828c6%2FDocument%20library

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The geological model

Log(K)

Phie

Geological 3D model

Structural framework

(Seismic data)

Depositional model

Rock properties distribution

Lithology: facies, porosity and permeability

Depth of fluid contacts and fluid properties

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Production data

Time (days)   O   i   l   f   l  o  w  r  a   t  e   (  m   3   /   d  a  y   )

History matching reservoir models

• Traditional parameter estimation

• Find parameter-set that gives best matchto data

 – Production and seismic data

• Definition of quadratic cost function

 – Perfect model assumption

• Minimization of cost function

 – Adjoints, gradients, genetic

algorithms, ensemble methods

• Traditional workflow updates onlysimulation model

Simulation model

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History matching and uncertainty prediction

History Prediction

Initial uncertainty Predicteduncertainty

Reduced initialuncertainty

Reducedpredicteduncertainty

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Assisted history matching

• Parameterization

• Definition of cost function

• Minimization/sampling

• High-dimensional problem

• Highly nonlinear problem

• Model errors ignored

• Multiple local minima

• Hard to solve

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General formulation

Find posterior pdf of state and parameters givenmeasurements and model with prior error statistics

Combined parameter and state estimation problem

Bayesian formulation

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Bayesian formulation

Bayes’ theorem

Gaussian priors Markov model

Independent data

Quadratic cost-function Sequential processing ofmeasurements

Sequence of inverse

problems

p(x|d)~p(x)p(d|x)

Minimization/Sampling”Ignore model errors”

Solve only for parameters? Ensemble methods

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History matching and uncertainty prediction

EnKF procedure

Todays posterior is tomorows prior 

p(x|d1) ~ p(x) p(d1|x)p(x|d1,d2) ~ p(x|d1) p(d2|x)

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Ensemble Kalman Filter

• Sequential Monte Carlo method

• Representation of error statistics by an ensemble of model states

 – Mean and covariance

• Evolution of error statistics by ensemble integrations

 – Stochastic model equation

•  Assimilation of measurements using a variance minimizing update

 – Sequential updating of both model state and static parameters

 – Model state and parameters converge towards true values

 – Information accumulates and uncertainty is reduced at each update

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EnKF can update geo-realizations

Geo-model

Geo-realizations

Simulation realizations

E nK F 

Log data

RFT/PLT data

Production rates

4D seismics

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Oseberg Sør reservoir model

• Dimensions:

• Field 3 km x 7 km, 300m thick• Cells size 100 x 100m, z variable

•  60 ‘000 active cells

• Complex reservoir• Heterogeneous flow properties

• Many faults, poorly known properties

• Fluid contacts poorly known

• Parameters to estimate• Porosity and permeability fields

• Depth of fluid contacts

• Fault properties

• Relperm parameterization

• Condition initial ensemble on production data• 4 producers, 1 water injector

• 6 years of production history

Permeability field

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Initial ensemble uncertainty span

Oil production rate

Water cut

Measurements

Initial ensemble

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OPRWCT

Measurements

Initial ensemble

EnKF updated ensemble

Posterior prediction and uncertainty span

Oil production rate

Water cut

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Oil Water relative permeability

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Water Saturation

Krw Initial mean

Krow Initial mean

Krw Updated

Krow Updated

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Porosity layer 19 (UT), prior and posterior 

Initial EnKF updated

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Porosity standard deviation layer 19, prior and posterior 

Initial EnKF updated

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Improved estimate of initial WOC depth

2907±

5m

2890 ± 2m

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Fault transmissibility estimation

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• Grane reservoir 

 – Grid consists of 90x168x20 grid cells

 – Homogenous/high permeability

 – Unclear vertical communication

 – Poorly known initial contacts

• Parameters to estimate

 – PORO and PERM

 – MULTZ

 – WOC & GOC

 – RELPERM

• Conditioning on production

 – 3 years production history, 19 wells

 – OPR, WCT, GOR

Real time prediction of oil production using the EnKF

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Conclusions

EnKF can efficiently history match complex reservoir models

General tool for parameter and/or state estimation.

Practically no limitation on parameter space.

Problem with local minima avoided.

Workflow and EnKF method allow for:

Consistency in model chain.

Estimates with quantified uncertainty.

Real time and sequential updating of models.

Updated ensemble provides future prediction with uncertainty estimates

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Issues and future challenges

• EnKF with general facies models

 – Involves non-Gaussian variables

• Pluri-Gaussian representation

• Kernel methods

• EnKF for estimating structural

parameters like faults and surfaces

 – Changes model grid

• Conditioning geo-models

 – Consistent links between geo- and

simulation model

• Operational workflow / best practice

 – Generally applicable

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Operational ocean prediction system

TOPAZ system:

27 000 000 unknowns

148 000 weekly observations

100 ensemble members

Local analysis