Ensemble Kalman Filtering in Distance-Kernel...

Introduction EnKF Overview Proposed Methodology Example Summary Future works

Ensemble Kalman Filteringin Distance-Kernel Space

Kwangwon Park and Jef Caers

Department of Energy Resources EngineeringStanford University

May 8, 2008

K. Park and J. Caers SCRF 21st ANNUAL MEETING (May 8-9, 2008)


In the previous presentation (Caers, 2008),How to generate new realizations with a given set of realizations in distance-kernel space

A new set of realizations which satisfy geologic informationand static data has been generated in distance-kernel space.



In this presentation, ...Objectives

Generate multiple realizations satisfying dynamic data

Preserve geologic information

Joint conditioning to static and dynamic data

Simultaneous generation of multiple realizations

Efficient optimization for multiple solutions

Find multiple x’s such that o(x) = do(x): result calculated by forward solution

d: dynamic data observed from a field

ex) 4D seismic, BHP, OPR, WPR, WCT,...

x = o−1(d) is a highly nonlinear inverse problem



Outline

1 Introduction

2 EnKF Overview

3 Proposed Methodology

4 ExamplePrior and posterior densityEnsemble size reduction

5 Summary

6 Future works



’Dynamic data conditioning’ = Optimization problemImpossible to evaluate ϕ−1(d) explicitly

Minimize the difference between the observed and the calculated

xopt = argminx

∆ (1)

∆(x; d) = ‖d− o(x)‖ (2)

However, if x represents a reservoir model,

1 (xopt) consistent with geologic information and static data

2 (x) high dimensional

3 (∆) multiple local minima and non-unique solution

4 (o(x)) time-consuming forward simulation

5 (argminx ∆) One optimal solution per one optimization



’Dynamic data conditioning’ = Optimization problemImpossible to evaluate ϕ−1(d) explicitly

Minimize the difference between the observed and the calculated

xopt = argminx

∆ (1)

∆(x; d) = ‖d− o(x)‖ (2)

However, if x represents a reservoir model,

1 (xopt) consistent with geologic information and static data

2 (x) high dimensional

3 (∆) multiple local minima and non-unique solution

4 (o(x)) time-consuming forward simulation

5 (argminx ∆) One optimal solution per one optimization



’Dynamic data conditioning’ = Filtering problemEnsemble Kalman Filtering (EnKF; Evensen, 1994)



Outline

1 Introduction

2 EnKF Overview



5 Summary

6 Future works



Ensemble Kalman Filtering OverviewEnKF updates a state vector

State vector contains dynamic variables as well as staticmodel parameters

zn =(

xp(x; tn)

)(3)

where,

n: time index

x: static model parameters(permeability, porosity, ... at each gridblock)

p(x; t): simulated dynamic variables at time = t(pressure, saturation, ... at each gridblock)



Ensemble Kalman Filtering OverviewKalman Gain G

Update multiple state vectors based on the prediction error

z+ = z− + G( d− o(z−)︸︷︷︸prediction error

) (4)

Kalman Gain G depends on

Cd (data observation error covariance)

Co (simulated output error covariance)

Czo (relationship between state vector and its output)

Kalman Gain GG = Czo (Co + Cd)−1



Ensemble Kalman Filtering OverviewRecursive nature

First step (n = 0): Generate an initial ensemble

zj0 (j = 1, ..., L) (5)

Second step (n = 0 to 1): Simulate the ensemble to the timeof first data (d1) (forecast step: reservoir simulation)

z0→ z−1 (6)

Third step (n = 1): Correct the ensemble based on the dataat n = 1 (assimilation step: conditioning)

z+1 = z−1 + G(d1 − o(z−1 )) (7)

Recursive nature:z+

1 → z−2 → z+2 → z−3 → z+

3 → z−4 → z+4 → z−5 → ...



Ensemble Kalman Filtering Simple ExampleMeasurements



Ensemble Kalman Filtering Simple ExampleForecast: reservoir simulation



Ensemble Kalman Filtering Simple ExampleAssimilation: condition to dynamic data



Ensemble Kalman Filtering Simple ExampleForecast: reservoir simulation



Ensemble Kalman Filtering Simple ExampleAssimilation: condition to dynamic data



Ensemble Kalman Filtering Simple ExampleRecursive nature



Ensemble Kalman Filtering Pros and Conshow to overcome the ’Cons’

Filtering multiple realizations based on the prediction error

z+ = z− + G(d− o(z−)) (8)

Pros

1 Simultaneous update multiple realizations

2 Real-time update whenever new data available

3 One forward simulation per one conditional realization

Cons

1 Only works with Gaussian realizations

2 High-dimensional filtering problem

3 Sometimes physically unrealistic output

4 Needs sufficiently large ensemble

Ensemble Kalman filtering in distance-kernel space!



Ensemble Kalman Filtering Pros and Conshow to overcome the ’Cons’

Filtering multiple realizations based on the prediction error

z+ = z− + G(d− o(z−)) (8)

Pros

1 Simultaneous update multiple realizations

2 Real-time update whenever new data available

3 One forward simulation per one conditional realization

Cons

1 Only works with Gaussian realizations

2 High-dimensional filtering problem

3 Sometimes physically unrealistic output

4 Needs sufficiently large ensemble

Ensemble Kalman filtering in distance-kernel space!



Outline

1 Introduction

2 EnKF Overview



5 Summary

6 Future works



Ensemble Kalman Filtering in Distance-Kernel Spaceovercomes the ’Cons’ of the conventional EnKF

Do KL expansion of z into y in distance-kernel space!

z =(

xp(x)

)→(

y)

(9)

y always standard Gaussian → (non-Gaussian x applicable!)

y’s are assigned to all x’s by KL expansionEnKF updates y’s rather than x’sFind new x’s from the new y’s through the pre-image problem.

y much shorter than x → (low dimensionality!)

x’s are of 106 to 108 dimensiony’s are of 10 to 103 dimensionFast and stable filtering



Ensemble Kalman Filtering in Distance-Kernel Spaceovercomes the ’Cons’ of the conventional EnKF

Do KL expansion of z into y in distance-kernel space!

z =(

xp(x)

)→(

y)

(10)

Single y represents both x and p(x) →(physically realistic and consistent update!)

Solve the pre-image problem for x’s from the new y’sSolve the pre-image problem for p(x)’s from the same y’sUpdate x and p(x) in a consistent wayConventional EnKF updates x and p(x) independently.

The size of ensemble reduced by kernel-kmean clustering →(increased efficiency!)



Procedure proposedexplained in the previous presentation (Caers, 2008)


Introduction EnKF Overview Proposed Methodology Example Summary Future worksPrior and posterior density Ensemble size reduction

Outline

1 Introduction

2 EnKF Overview



5 Summary

6 Future works



Given informationDetermine reservoir permeability with given information

Geology: Spherical variogram (NE50, r1 = 200 ft, r2 = 50 ft)Static data: 150 md at well locations (5,5) and (28,28)Dynamic data: qw/qt measured every 2 months (10% noise)



Ensemble generationin realization space



Ensemble generation300 SGSIM realizations conditioned to hard data (150 md at (5,5) and (28,28))



Ensemble generationInitial mean and variance

Figure: Initial mean Figure: Initial variance



Ensemble generationWatercut curves for initial 300 realizations



Distancefrom realization space to distance space



Connectivity distance btw any two realizationsConnectivity distance: difference in analytical watercut curves

Figure: On the y-axis is the dissimilarity in dynamic data of any tworealization. On the x-axis the connectivity distance between any tworealizations.



Multidimensional Scalingfrom distance space to MDS space



Multidimensional Scaling2D MDS space based on connectivity distance



Ensemble Kalman Filteringin parameterization space



EnKFForecast to t = 260 days

Figure: All the simulated watercut match the data before breakthrough(no update!)




Figure: Most of the realizations have lower watercut values than themeasurement.



EnKFAssimilation at t = 520 days

Figure: Realizations move from disconnected region to connected region.




Figure: Realizations move from disconnected region to connected region.




Figure: Most of the realizations show lower watercut values than thedata.




Figure: Realizations move toward a particular point.




Figure: Realizations move toward a particular point.



EnKFFinal realizations



EnKFFinal mean and variance

Figure: Final mean Figure: Final variance



EnKFPrediction with final 300 realizations

Figure: Final Figure: Initial

Final realizations match the data within the error level (10%).

For 300 history-matched models, exactly 300 forwardsimulations required.



Outline

1 Introduction

2 EnKF Overview



5 Summary

6 Future works



Analyze probability densityPrior probability density of unconditional realizations



Analyze probability densityPosterior probability density of conditional realizations to static data only



Analyze probability densityPosterior probability density of conditional realizations to static and dynamic data



Outline

1 Introduction

2 EnKF Overview



5 Summary

6 Future works



Reduce the size of ensembleby selecting a few realizations which represent ensemble statistics (Scheidt, 2008)

Figure: Select 30 realizations by kernel k-mean clustering and applyEnKF to the 30 realizations.



Reduce the size of ensembleby selecting a few realizations which represent ensemble statistics (Scheidt, 2008)

Figure: Select 30 realizations by kernel k-mean clustering and applyEnKF to the 30 realizations.



Prior to EnKFEnsemble statistics reproduction

Figure: 30 realizations Figure: 300 realizations

30 realizations selected by kernel k-mean clustering representthe ensemble (300 realizations) statistics (p10, p50, and p90).



Prior to EnKFEnsemble statistics reproduction

Figure: 30 realizations Figure: Comparison

30 realizations selected by kernel k-mean clustering representthe ensemble (300 realizations) statistics (p10, p50, and p90).



After EnKFPrediction with final 30 realizations

Figure: 30 realizations Figure: 300 realizations

30 realizations represent the ensemble (300 reals) statistics.To get 30 history-matched models, only need 30 forwardsimulations.



Outline

1 Introduction

2 EnKF Overview



5 Summary

6 Future works



Summary

Objective

Generate multiple realizations satisfying all available static and dy-namic data

EnKF in distance-kernel space

updates multiple realizations to dynamic data simultaneously.

is applied to any type of geologic model including Gaussian.

is boosted when the size of ensemble reduced by selection ofkernel k-mean clustering.

MDS space is very useful to analyze the optimization processas well as the probability density.



Outline

1 Introduction

2 EnKF Overview



5 Summary

6 Future works



Future works

Implementation

Extend to non-Gaussian case (e.g. multiple-point Geostatistics)

Application

Apply to 3D real case (Brugge data set)


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