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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Outline
Outline

  • Motivation

  • Background: Alternative implementation

  • Examples

    • Simple 1-D linear model

    • PUNQS3

  • Further work


Coupled enkf motivation
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
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
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
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 example1
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


Localization by hierarchical ensemble filter simple example 40 runs

EnKF

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

Results from 40 runs (ens. mean)

Average of 40 runs

Hierarchical

True

True


Localization by hierarchical ensemble filter simple example 40 runs1

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
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
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
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
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


Comparison ordinary enkf compared to 5x40 members with hierarchical ensemble filter forecasted fopt1

EnKF hierarchical ensemble filter – Forecasted FOPT

Hierarchical

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


Quality of the solutions history matching
Quality of the solutions: History matching hierarchical ensemble filter – Forecasted FOPT

  • Evaluate the estimated fields by rerunning from time zero

  • 117 measurements

  • Objective function:

EnKF

Hierarchical


Quality of solutions estimated porosity
Quality of solutions: Estimated porosity hierarchical ensemble filter – Forecasted FOPT

  • Compare quality of solutions with following measure:

  • Hierarchical more robust

EnKF

Hierarchical


Quality of solutions estimated log perm
Quality of solutions: Estimated log-perm hierarchical ensemble filter – Forecasted FOPT

Log-permZ

Log-permX

EnKF

Hierarchical


Quality of solutions estimated dynamic quantities
Quality of solutions: Estimated dynamic quantities hierarchical ensemble filter – Forecasted FOPT

Pressure

Gas-oil ratio

EnKF

Hierarchical


Quality of solutions estimated saturations
Quality of solutions: Estimated saturations hierarchical ensemble filter – Forecasted FOPT

Water saturation

Gas saturation

EnKF

Hierarchical


Comparison of mean of final estimates the concept

EnKF hierarchical ensemble filter – Forecasted FOPT

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
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
Porosity layer 5 – 10 runs – final time runs: Porosity layer 1 – final time

Hierarchical filter

Ordinary EnKF


Log permx layer 1 final time
Log-Permx - layer 1 – final time runs: Porosity layer 1 – final time

Hierarchical filter

Ordinary EnKF


Water saturation layer 2 final time
Water saturation - layer 2 – final time runs: Porosity layer 1 – final time

Hierarchical filter

Ordinary EnKF


Gas saturation layer 3 final time
Gas saturation - layer 3 – final time runs: Porosity layer 1 – final time

Hierarchical filter

Ordinary EnKF


Pressure layer 2 final time
Pressure - layer 2 – final time runs: Porosity layer 1 – final time

Hierarchical filter

Ordinary EnKF


Conclusion of punqs3 study
Conclusion of PUNQS3 study runs: Porosity layer 1 – final time

  • 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
Conclusions & suggestions for further work runs: Porosity layer 1 – final time

  • 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
Acknowledgment runs: Porosity layer 1 – final time

  • 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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