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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
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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