1. Classification: Internal Status: Draft Using the EnKF for combined state and parameter estimation Geir Evensen

2. 2 Outline Reservoir modelling and simulation History matching problem and uncertainty prediction Ensemble Kalman filter (EnKF) Field case example

3. 3 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

4. 4 The geological model Geological 3D model Structural framework (Seismic data) Depositional model Rock properties distribution Lithology: facies, porosity and permeability Depth of fluid contacts and fluid properties

5. 5 Production data Time (days) Oil flow rate (m3/day) History matching reservoir models Traditional parameter estimation Find parameter-set that gives best match to 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 only simulation model Simulation model

6. 6 History matching and uncertainty prediction HistoryPrediction Initial uncertainty Predicted uncertainty Reduced initial uncertainty Reduced predicted uncertainty

7. 7 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

8. 8 General formulation Find posterior pdf of state and parameters given measurements and model with prior error statistics Combined parameter and state estimation problem Bayesian formulation

9. 9 Bayes’ theorem Gaussian priors Markov model Independent data Quadratic cost-function Sequential processing of measurements Sequence of inverse problems p(x|d)~p(x)p(d|x) Minimization/Sampling ”Ignore model errors” Solve only for parameters? Ensemble methods

10. 10 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)

11. 11 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

12. 12 EnKF can update geo-realizations Geo-model Geo-realizations Simulation realizations EnKFLog data RFT/PLT data Production rates 4D seismics

13. 13 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

14. 14 Initial ensemble uncertainty span Oil production rate Water cut Measurements Initial ensemble

15. 15 OPR WCT Measurements Initial ensemble EnKF updated ensemble Posterior prediction and uncertainty span Oil production rate Water cut

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17. 17 Porosity layer 19 (UT), prior and posterior Initial EnKF updated

18. 18 Porosity standard deviation layer 19, prior and posterior Initial EnKF updated

19. 19 Improved estimate of initial WOC depth 2907± 5m 2890 ± 2m

20. 20 Fault transmissibility estimation

21. 21 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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25. 25 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

26. 26 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

27. 27 Operational ocean prediction system TOPAZ system: 27 000 000 unknowns 148 000 weekly observations 100 ensemble members Local analysis