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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
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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
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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
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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
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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
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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 ) )
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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
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• 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
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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)
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Waterflooding in Brugge FieldWells and oil saturation
Oil saturation10 injectors20 producers
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10-year Production Historysimulated from a synthetic reservoir
d2: difficult dataOil Production Water Production
20 Producers10 years
Bottom Hole Pressure
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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
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Assess consistency of models and data (zoom)Initial model not represent the data at all
W05 W10
W16W15
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How to check the prior with the given data quantitatively? Projection from metric space with MDS
Clearly Wrong priorneed to modify prior set
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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
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New datastill far from the mean of the initial watercut curves
W05 W14
W17 W19
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Facies-based models are chosenNTGs, PERMx, PERMy, PERMz, PORO
Initial real 1
Initial real 2Initial real 3
Initial real 4
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Projection from metric space with MDSD = difference in well watercut curves (exact distance)
truth
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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)
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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
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Final match (zoom)watercut curves for final models
W05 W14
W17 W19
initial
final
initial
final
initial
finalinitial
final
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Final 65 modelsStill facies modelsNTGs, PERMx, PERMy, PERMz, PORO
Final real 1
Final real 3
Final real 4
Final real 2
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Initial and Final modelsE-type and conditional variance
Initial etype
Initial c.v.
Final etype
Final c.v.
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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
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W05 W14
W17 W19
initial
final
initial
final
initialfinal
initial
final
Final match (zoom)well oil production curves for final models
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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
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initial
W05 W14
W17 W19
final initial
final
initial
final
initial
final
Final match (zoom)well bottom hole pressure for final models
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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