Geir n vdal and brice vall s
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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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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

  • 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

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


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


Quality of solutions: Estimated log-perm

Log-permZ

Log-permX

EnKF

Hierarchical


Quality of solutions: Estimated dynamic quantities

Pressure

Gas-oil ratio

EnKF

Hierarchical


Quality of solutions: Estimated saturations

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


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