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