metric ensemble kalman filter: application to the brugge...

Metric Ensemble Kalman Filter: Application to the Brugge Synthetic Data

Kwangwon Park and Jef CaersStanford Center for Reservoir Forecasting

Energy Resources EngineeringStanford University

Apr 30, 2009

In Metric Space Modeling, 4th presentation

• Modeling Uncertainty in Metric Space, Jef Caers– Defining a Random Function From a Given Set of Model

Realizations, Celine Scheidt– Bootstrap Confidence Intervals for Reservoir Model Selection

Methods, Celine Scheidt– Stochastic Simulation of Patterns by Means of Distance-Based

Pattern Modeling, Mehrdad Honarkhah– The Metric Ensemble Kalman Filter (mEnKF): Application to The

Brugge Synthetic Data, Kwangwon Park– Direct Construction and History Matching Ensembles of Coarse

Flow Models, Celine Scheidt

Objectives

• Generate multiple realizations satisfying all data– Preserve geologic information– Joint conditioning to static and dynamic data– Simultaneous generation of multiple realizations

• One solution: Ensemble Kalman Filtering– Ensemble approach– Non-iterative algorithm– Real-time update

Ensemble Kalman Filtering OverviewEvensen, 1994

d: time-varying nonlinear data

p(t; x): dynamic variables

x: spatial variables

G: Kalman filterprediction error

d – g ( x )

z-: prior state vector z+: posterior state vector

Ensemble Kalman Filtering OverviewKalman Gain G

z+ = z- + G ( d - g ( x ) )

prediction error

G = Czg (Cg + Cd)-1

Data-data covariance

Output-output covariance

State-output covariance

Kalman Gain

Updated Initial Update

Ensemble Kalman Filtering Limitations

• Formulated for Gaussian field (Gaussianity issue)

• Large scale filtering problem (Stability issue)

• Sometimes physically unrealistic outpu(Consistency issue)

z+ = z- + G ( d - g ( x ) )

Ensemble Kalman Filtering in Metric SpacePark et al., 2008

z+ = z- + G ( d - g ( x ) )

y+ = y- + G ( d - g ( x ) )

Distance calculationMulti-dimensional scalingKernel KL expansion

featurespace

Model expansion

Φ

21/KKVΦ Λ=

ector)Gaussian v standard a is (L

1 with )(

:expansionLoeve-Karhunen

y

ybbx KVΦ ==ϕ

MDSUsing K

• Although z is non-Gaussian, y always Gaussian. (Non-Gaussian model applicable)

• y much shorter than z(Fast and more stable filtering)

• Single y represents both x and p(t; x)(Physically realistic and consistent update)

y+ = y- + G ( d - g ( x ) )

Ensemble Kalman Filtering in Metric Space

Brugge Field Synthetic Data SetPeters et al., 2009 (SPE119094)

• Benchmark project– Test optimization and history matching methods

• Given information– Reservoir geometry

(high-resolution model: 20 million gridblocks)(flow simulation model: 60,048 gridblocks)

– 104 initial models (NTG, PERMx, PERMy, PERMz, PORO, …)– 10-year production history (WBHP, WOPR, WWPR)

Waterflooding in Brugge FieldWells and oil saturation

Oil saturation10 injectors20 producers

10-year Production Historysimulated from a synthetic reservoir

d2: difficult dataOil Production Water Production

20 Producers10 years

Bottom Hole Pressure

Assess consistency of models and dataWatercut curves for initial models

W01 W02 W03 W04 W05

W06 W07 W08 W09 W10

W11 W12 W13 W14 W15

W16 W17 W18 W19 W20

Assess consistency of models and data (zoom)Initial model not represent the data at all

W05 W10

W16W15

How to check the prior with the given data quantitatively? Projection from metric space with MDS

Clearly Wrong priorneed to modify prior set

New problem set up:Choose one of the initial model and get new data

W01 W02 W03 W04 W05

W06 W07 W08 W09 W10

W11 W12 W13 W14 W15

W16 W17 W18 W19 W20

New datastill far from the mean of the initial watercut curves

W05 W14

W17 W19

Facies-based models are chosenNTGs, PERMx, PERMy, PERMz, PORO

Initial real 1

Initial real 2Initial real 3

Initial real 4

Projection from metric space with MDSD = difference in well watercut curves (exact distance)

truth

Update using metric EnKFProjection from metric space with MDS

Conventional EnKF update vector: length(z) = 60048 * 7Metric EnKF: length(y) = 65

One step update, not sequentially (it’s more stable)

Final modelsWatercut curves for final models

W01 W02 W03 W04 W05

W06 W07 W08 W09 W10

W11 W12 W13 W14 W15

W16 W17 W18 W19 W20

Final match (zoom)watercut curves for final models

W05 W14

W17 W19

initial

final

initial

final

initial

finalinitial

final

Final 65 modelsStill facies modelsNTGs, PERMx, PERMy, PERMz, PORO

Final real 1

Final real 3

Final real 4

Final real 2

Initial and Final modelsE-type and conditional variance

Initial etype

Initial c.v.

Final etype

Final c.v.

Final modelsMatching watercut data,Prediction of WOPR

W01 W02 W03 W04 W05

W06 W07 W08 W09 W10

W11 W12 W13 W14 W15

W16 W17 W18 W19 W20

W05 W14

W17 W19

initial

final

initial

final

initialfinal

initial

final

Final match (zoom)well oil production curves for final models

Final modelsMatching watercut data,Prediction of WBHP

W01 W02 W03 W04 W05

W06 W07 W08 W09 W10

W11 W12 W13 W14 W15

W16 W17 W18 W19 W20

initial

W05 W14

W17 W19

final initial

final

initial

final

initial

final

Final match (zoom)well bottom hole pressure for final models

Summary

Metric Ensemble Kalman Filter

• Successfully applied to multi-well large reservoir • Applicable to any type of spatial continuity model• Stable and consistent filtering

– Simultaneous update of all the variables (PERM, PORO,…)

• Efficiently generate multiple conditional models.

• Discussion– Sensitive to prior model– EnKF (estimation) has limitations for uncertainty quantification

Any question about Metric EnKF?

• Thanks!

• wonii@stanford.edu