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Geir Nævdal and Brice Vallès. Coupled EnKF. Outline. Motivation Background: Alternative implementation Examples Simple 1-D linear model PUNQS3 Further work. Coupled EnKF – motivation. Lorentzen et. al., 2005, SPE96375 Problem with consistency between repeated runs

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Coupled EnKF

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#### Presentation Transcript

Geir Nævdal and Brice Vallès

### Outline

• Motivation

• Background: Alternative implementation

• Examples

• Simple 1-D linear model

• PUNQS3

• Further work

### Coupled EnKF – motivation

• 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

### Reminder: SPE96375

• 10 initial ensembles

• Generated with same distribution

• Kolmogorov-Smirnov test on posterior distributions

• Posterior distributions are not coming from same distribution

Example: FOPT

### Hierarchical ensemble filter (by Anderson)

• 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

### Localization by Hierarchical EnKF – simple 1-D example

• 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)

### Localization by Hierarchical EnKF – simple 1-D example

• 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

### Localization by Hierarchical ensemble filter – simple example (40 runs)

Results from 40 runs (ens. mean)

Average of 40 runs

Hierarchical

True

True

EnKF

### Localization by Hierarchical ensemble filter – simple example (40 runs)

Results from 40 runs (ens. mean)

Standard deviation of mean of 40 runs

Hierarchical

True

True

### PUNQS3

• 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

### PUNQS3 – production history and forecasting

• 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.

### Investigation

• 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

### Comparison: Ordinary EnKF compared to 5x40 members with hierarchical ensemble filter – Forecasted FOPT

• 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

### Quality of the solutions: History matching

• Evaluate the estimated fields by rerunning from time zero

• 117 measurements

• Objective function:

EnKF

Hierarchical

### Quality of solutions: Estimated porosity

• 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

### Comparison of mean of final estimates – the concept:

Results from 40 runs (ens. mean)

Standard deviation of mean of 40 runs

Hierarchical

True

True

### Comparison of std. deviation of the mean estimate for the 10 runs: Porosity layer 1 – final time

Hierarchical filter

Ordinary EnKF

### Porosity layer 5 – 10 runs – final time

Hierarchical filter

Ordinary EnKF

### Log-Permx - layer 1 – final time

Hierarchical filter

Ordinary EnKF

### Water saturation - layer 2 – final time

Hierarchical filter

Ordinary EnKF

### Gas saturation - layer 3 – final time

Hierarchical filter

Ordinary EnKF

### Pressure - layer 2 – final time

Hierarchical filter

Ordinary EnKF

### Conclusion of PUNQS3 study

• 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

### Conclusions & suggestions for further work

• 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

### Acknowledgment

• 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