An example of flow based covariance localisation in the EnKF

Dan Kuznetsov, Dick KachumaGRC, Total E&P UK

2

Talk

Simple 2D example

Streamline-based localisation

Real field example

Conclusions

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Covariance in the EnKF

Update equation

is the key element

is estimated from the ensemble

Estimation from the ensemble of a practical size may lead to a spurious correlation between model parameters and predictions

fDfa HψdCHHCHCψψ

1TT

C

N

iiiN 1

T

1

1ψψψψC

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2D synthetic example

Single well

51x51x1

Permeability is a Gaussian field

Conditioned at the well location

Other parameters are constant

BHP is the observation data

Ensemble of 100 realisations

5 EnKF assimilation steps

5

Initial ensemble

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BHP

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Consider data along the line

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Correlation coefficient and pressure along the line

Step 1, 100 members

Pressure

Corr Coef

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Correlation coefficient for bigger ensemble

Step 1, 1000 members

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Correlation coefficient and pressure along the line

Step 1, 100 members

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Covariance localisation

Covariance matrix estimated from the ensemble is modified in order to make it decreasing at some distance from the observation point

Multiplied by 1 within some zone and by 0 outside or multiplied by some coefficients derived, for example from streamline sensitivities

Distance based localisation Houtekamer and Mitchell 1998, Hamill et al 2001, Skjervheim et al 2006, … Truncation of the covariance for production data may not be physically based since

the well influence zone may have non trivial shape

Streamline based localisation Aroyyo-Negrete et al 2006, Devegowda et al 2007 Covariance is truncated with respect to the flow path and/or streamline based

sensitivities

dd CC ~

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SL based localisation for production data

Limit covariance to the zone that has been swept by the produced liquid

Reverse problem: if a producer had been an injector the swept zone would be limited by a front of the injected liquid

Propagation of the liquid front along streamlines can be described in terms of Time Of Flight

To find the localisation zone: Trace the streamlines based on the velocity field Calculate time of flight Limit drainage and injection zones accordingly to the current model time

dv

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SL based localisation

Trace streamline -> Select a zone -> Update within the zone

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Pressure TOF

Asymptotic solution to the pressure diffusivity equation Kulkarni, Datta-Gupta and Vasco, 2000

is the velocity of the pressure front

Pressure Time Of Flight

Pressure TOF is related to the observed time as

in 2D and in 3D

Track the pressure front and set covariance to 0 outside the influence zone

1 ptc

k

d

p

4

2t

6

2t

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Localisation within the drainage zone. Step 1

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Localisation within the drainage zone. Step 5

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Correlation coefficient. Step 1

Pressure TOF

Pressure

Corr Coef

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Correlation coefficient. Step 1 after truncation

Pressure TOF

Pressure

Corr Coef

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Correlation coefficient. Step 5

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Correlation coefficient. Step 5 after truncation

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Match with and without localisation

No localisation SL TOF localisation

Initial

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Permeability along the line. Step 5Initial

SL TOF localisation

1000 realisation

No localisation

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Real field case example

70x49x34, 69290 active cells

Modified parameters:porosity,permeability,aquifer strength

NTG is different for each realisation but not updated

Initial ensemble for the fault permeability was generated separately and included into Kx and Ky

3 producers, 2 water injectors

900 days of production

Data to match: WBHP, WOPR, WWCT

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Algorithm for reservoir simulation

Forward step with a conventional finite-difference reservoir simulator

Streamline tracing using separate routine (Texas A&M)

Compute TOF and Pressure TOF Practically, for a full field reservoir simulation a full drainage zone is used for

localisation of the covariance with bottom hole pressure

Stack influence zones for each well and observation for the current time

EnKF update with modified covariance

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Perm X, realisation 1, Lay 8

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Drainage zones, lay 8

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Drainage zones, lay 8

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Drainage zones, lay 8

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Drainage zones, lay 8

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Drainage zones, lay 8

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Drainage zones, lay 8

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Drainage zones, lay 8

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Drainage zones, lay 8

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Drainage zones, lay 8

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Drainage zones, lay 8

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Covariance: perm X, lay 8 and OPR in P1

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Covariance: perm X, lay 8 and OPR in P1

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Covariance: perm X, lay 8 and OPR in P1

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Covariance: perm X, lay 8 and OPR in P1

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Covariance: perm X, lay 8 and OPR in P1

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Covariance: perm X, lay 8 and OPR in P1

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Covariance: perm X, lay 8 and OPR in P1

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Covariance: perm X, lay 8 and OPR in P1

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Covariance: perm X, lay 8 and OPR in P1

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Covariance: perm X, lay 8 and OPR in P1

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BHP Well 1Initial

No Localisation SL TOF Localisation

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BHP Well 2Initial

No Localisation SL TOF Localisation

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BHP Well 3

Initial

No Localisation SL TOF Localisation

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WOPR Well 1Initial

No Localisation SL TOF Localisation

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Initial

No Localisation SL TOF Localisation

WOPR Well 2

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WOPR Well 3Initial

No Localisation SL TOF Localisation

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WWCT Well 1Initial

No Localisation SL TOF Localisation

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Initial

No Localisation SL TOF Localisation

WWCT Well 2

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Initial

No Localisation SL TOF Localisation

WWCT Well 3

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Permeability, lay 4. No localisation

Initial No localisation

P3

P2

P1

I2

I1

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Permeability, lay 4. SL TOF based localisation

P3

P2

P1

I2

I1

Initial SL TOF localisation

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Estimation of permeability in cell 1

cell 1

No Localisation SL TOF Localisation

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Estimation of permeability in cell 2

cell 2

No Localisation SL TOF Localisation

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Summary

Significant covariance disturbance happens outside well influence zone

Streamline based covariance localisation helps to reduce spurious correlation and decrease non-data based perturbation of the model parameters

It’s not necessary to use a SL simulator when applying a SL based covariance localisation

SL based localisation does not necessarily improve the match;although it’s case dependant

Initial realisations are less modified when localisation is used