Improved sampling

Geir Nævdal and Kristin M. Flornes*

EnKF with improved sampling of the initial ensembleGoal: Improve the performance of the EnKF

without increasing the ensemble size

• Different resampling techniques for the initial ensemble have been proposed

• In this work we have looked more closely at the effect of using Geir Evensen's resampling scheme (2004)

Outline• Evensen’s resampling scheme

• Does resampling preserve the variogram?

• Description of test case

• Results

• Conclusion

Evensen’s improved sampling schemeAim

Introduce a maximum rank and conditioning of the ensemble matrix for a given ensemble size.

Based on ideas from Singular Evolution Interpolated Kalman (SEIK) Filters

(Pham, 2001).

Evensen’s improved sampling schemeTo generate an ensemble of size N do the following: Generate a large ensemble of size β*N. Do a SVD of the ensemble matrix A and retain only

the N largest singular values.

Create a new ensemble of size N based on these singular values.

TVUA

Does Evensen’s resampling scheme preserve the variogram?• Is the variogram the same for a resampled

ensemble of N members as for the large initial ensemble with β*N members?

• We used the analytical covariance matrix in 1-D for Gaussian, spherical and exponential variogram to study theeffect of removing

singular values

Does resampling preserve the

variogram? • For the Gaussian model the elements in the covariance matrix

are

• Relationship between variogram and covariance

2

23exp

a

ipi

ii pp 0

Effect of retaining only ½ of the singular values

Gaussian Spherical Exponential

Analytical model variogram

Evensen’s algorithm

Evensen’s algorithm

Evensen’s algorithm

Effect of retaining 1/8 of the singular values

Gaussian Spherical Exponential

This shows that the variograms will be influenced by resampling if we truncate a large portion of the singular values, leading to smoother fields.

Evensen’s algorithm

Analytical model variogram

Evensen’s algorithm Evensen’s algorithm

Test Case - Description• Synthetic 2D case (50 X 50)• 3 producers (corners),1 injector (in the

middle)• Fields are generated using sgsim2

– Spherical variogram– Variogram range: 10 grid cells

• Static variables: PORO and PERMX• PERMY=PERMX• Measurement errors

– OPR, WPR, GPR: 10 % – BHP: 1%

True static fields

Injector in the middle, producers in the upper left and right corners and lower left corner

PERMX PORO

Test Case - Resampling setupStarted with 500 ensemble members.

Resampled down to 100 members

1. 100 random ensemble members • Used the 100 first of the 500 ensemble

members

2. Generate 100 ensemble members using Evensen’s improved sampling scheme

Example of initial porosity fields I

Ordinary ensemble member

Ensemble member generated fromresampling

Example of initial porosity fields II

Ordinary ensemble member

Ensemble member generated fromresampling

Example of initial porosity fields III

Ordinary ensemble member

Ensemble member generated fromresampling

Effect of resampling on initial

fields • The resampled ensemble members are smoother

• This is an effect of removing singular values

• Permeability fields are generated independently from porosity fields, and also resampled independently

Injection pressure Forecast AnalyzedMeasurement

Data for PROD1Forecast AnalyzedMeasurement

Data for PROD2Forecast AnalyzedMeasurement

Data for PROD3Forecast AnalyzedMeasurement

Performance measures• For a field m we use the formula

Compared results of 30 runs

e mN

i

N

j

truej

ij

me

mmNN

mR1 1

21)(

Effect on estimated water saturation

Effect on estimated pressure

Effect on estimated porosity

Effect on estimated permeability

Summary and conclusion• The dynamic variables are better

estimated using ordinary ensembles compared to resampled– Holds in particular for water saturation

• Small effects on static variables

• No argument for resampling

Acknowledgement This work was done with financial

support from the ROAW project, funded by the Research Council of Norway (PETROMAKS) and industrial sponsors. Licenses to the Eclipse simulator were provided by Schlumberger.

Literature• Evensen (Ocean Dynamics 2004)

“Sampling strategies and square root analysis schemes for the EnKF”

• Zafari et. al., SPE95750 (RMS measure)

• G. Nævdal, K.M. Flornes SPE118729 “Using ensemble Kalman filter with improved sampling of the initial ensemble” Submitted