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Coupled EnKF

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Geir Nævdal and Brice Vallès

- Motivation
- Background: Alternative implementation
- Examples
- Simple 1-D linear model
- PUNQS3

- Further work

- Lorentzen et. al., 2005, SPE96375
- Problem with consistency between repeated runs

- Thulin et. al., in preparation, previous talk + ECMOR XI
- Set of independent EnKFs to estimate Monte-Carlo uncertainty

- Anderson, Physica D, vol. 230, 2007
- “Hierarchical ensemble filter” to calculate localization
- Use several independent EnKF, calculate a reduction in Kalman gain based on statistics

- 10 initial ensembles
- Generated with same distribution

- Kolmogorov-Smirnov test on posterior distributions
- Posterior distributions are not coming from same distribution

Example: FOPT

- Split the ensemble in several sub-ensembles
- Run each sub-ensemble using different Kalman gain matrices
- Modify each of the Kalman gain matrices
- multiplied with factor

- Initial guess:
- Zero mean
- Gaussian variogram
- Correlation length: 5
- Standard deviation: 1

- Measurement: y=2 at x=26
- Measurement uncertainty: 2

- Kalman filter gives updated mean (and covariance)

- Initial guess:
- Zero mean
- Gaussian variogram
- Correlation length: 5
- Standard deviation: 1

- Measurement: y=2 at x=26
- Measurement uncertainty: 2

- Kalman filter gives updated mean (and covariance)
- Compare
- EnKF with 100 ens. members
- Hierarchical ensemble filter with 5x20 members

EnKF

Results from 40 runs (ens. mean)

Average of 40 runs

Hierarchical

True

True

EnKF

Results from 40 runs (ens. mean)

Standard deviation of mean of 40 runs

Hierarchical

True

True

- The PUNQ-S3 is a small-size synthetic 3-D reservoir engineering model.
- The reservoir consists of 19 x 28 x5 gridblocks, where 1761 are active.
- Equal 180 meter sides in x- and y-directions.
- Reservoir is bounded by a fault in east and south.
- Reservoir is bounded by an aquifer in west and north.
- New webpage: http://www3.imperial.ac.uk/earthscienceandengineering/research/perm/punq-s3model

- First 8 years: history matching phase.
- 1 year of well testing,
- 3 year shut-in period, and
- 4 years of production.

- Next 8.5 years: forecasting phase.
- During history matching phase:
- wells are controlled by using history target rates for oil.

- During forecasting phase:
- wells are controlled using target oil rate of 150 scm/day.
- Minimum bottom hole pressure of 120 bar.
- If gas/oil ratio is greater than 200, a cutback factor of 0.75 is used.

- Initial ensemble generated based on description on old PUNQS3 webpage
- Permeability and porosity are estimated
- Comparing result of forecasts
- Ordinary EnKF versus hierarchical ensemble filter (200 members vs. 5 x 40 members)
- Arguing for 40 members in each batch:
- For PUNQS3 Gu & Oliver found reasonable history match with 40 members
- For field case, Bianco et. al. found reasonable history match with 50 members

- 10 initial ensembles used in both cases
- Compare forecasted FOPT from final estimates
- Figure shows maximum, mean, and minimum of cdfs for FOPT
- There is generally less deviation in the results from hierarchical ensemble filter

EnKF

Hierarchical

EnKF

Hierarchical

- Evaluate the estimated fields by rerunning from time zero
- 117 measurements
- Objective function:

EnKF

Hierarchical

- Compare quality of solutions with following measure:
- Hierarchical more robust

EnKF

Hierarchical

Log-permZ

Log-permX

EnKF

Hierarchical

Pressure

Gas-oil ratio

EnKF

Hierarchical

Water saturation

Gas saturation

EnKF

Hierarchical

EnKF

Results from 40 runs (ens. mean)

Standard deviation of mean of 40 runs

Hierarchical

True

True

Hierarchical filter

Ordinary EnKF

Hierarchical filter

Ordinary EnKF

Hierarchical filter

Ordinary EnKF

Hierarchical filter

Ordinary EnKF

Hierarchical filter

Ordinary EnKF

Hierarchical filter

Ordinary EnKF

- Slightly better history matches with EnKF compared to hierarchical ensemble filter
- Hierarchical ensemble filter seems to be more robust and have less variations in repeated runs
- Computation time is of same order for the two approaches
- PUNQS3 forecasts do not differ to much

- Hierarchical ensemble filter
- Gives the opportunity to estimate Monte-Carlo uncertainty
- Seems to be more robust
- Have computation time as ordinary EnKF

- Other approaches for localization could be evaluated
- Datta-Gupta and coworkers based on streamlines
- Approaches based on Schur product
- …

- Evaluate hierarchical ensemble filter on more challenging examples
- Evaluate different partitions than 5 x 40 members

- This work has been done with financial support from Research Council of Norway (PETROMAKS) and industrial partners
- Licenses for Eclipse have been provided by Schlumberger