Reservoir Modeling with EnKF: A sensitivity study

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Controlled CO2 | Diversified fuels | Fuel-efficient vehicles | Clean refining | Extended reserves

Reservoir Modeling with EnKF: A sensitivity study

L. Heidari (IFP), V. Gervais (IFP), M. Le Ravalec (IFP), H. Wackernagel (ENSMP)Ecole Des Mines de Paris (ENSMP)Institut Français du Pétrole (IFP)

4th Ensemble Kalman Filter Workshop, 22-24 June 2009 - Bergen , Norway

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Overview Background

Reservoir Study and History Matching Kalman Filters Ensemble Kalman Filter – EnKF

Application 2D synthetic case study Sensitivity Tests

Conclusions & perspectives

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Overview Background

Reservoir Study and History Matching Kalman Filters Ensemble Kalman Filter – EnKF

Application 2D synthetic case study Sensitivity Tests

Conclusions & perspectives

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Reservoir Study

?=U

ps c

a lin

g

(Ph

i , K

)

Geologicalmodel

(facies, porosity,permeability)

Flu

id F

low

S

imu

lati

on

Reservoirmodel

100

150

200

250

0 500 1000 1500 2000 2500 3000

Time (day)

BH

FP (b

ar)

0

100

200

300

400

0 500 1000 1500 2000 2500 3000

Time (day)

GO

R

0.0

0.1

0.2

0.3

0.4

0 500 1000 1500 2000 2500 3000

Time (day)

WC

UT

Observed Production

data

100

150

200

250

0 500 1000 1500 2000 2500 3000

Time (day)

BH

FP (b

ar)

0.0

0.1

0.2

0.3

0.4

0 500 1000 1500 2000 2500 3000

Time (day)

WC

UT

0

100

200

300

400

0 500 1000 1500 2000 2500 3000

Time (day)

GO

R

Simulated production data

Aim Constrain reservoir models to production history by

adjusting parameters: porosity, permeability, etc.

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Sequential Methods

Initial state

Timet0 t1

Measurements

Forecasted state (t1)

Timet0 t1

Dynamical model

Corrected state (t1)

Assimilation method

Forecasted state (t2)

Dynamical model t2

t2

Measurements

Correctedstate (t2)

Assimilation method

…

dfa21 AA

Kalman Filter

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Overview Background

Reservoir Study and History Matching Kalman Filters Ensemble Kalman Filter – EnKF

Application 2D synthetic case study Sensitivity Tests

Conclusions & perspectives

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Assumptions

Linear dynamic model

Gaussian probability distribution function

First Order Markov processes

Independent and uncorrelated model and measurement errors

Kalman Filter estimate is based on

Forecast Estimate

Measurements

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Kalman filter

dfa21 AA

)t()t( 1 ka

kf F

td H

Dynamic Model Operator

Measurement Operator

True State

Measurement error

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Kalman filter estimate, a , minimizes

Kalman filter estimate, a :

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Kalman filter analysis

fTfTffa HdCHHCHC 1

Kalman gain Simulated data

)()()()()(J 11 HdCHdC TffTf

( )( ) : Forecast error covariance matrixf t f t f TC

Distance between and f

matrix covarianceerror tsMeasuremen : TεεC

Distance between and d

dfa21 AA

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System Dynamics Linear

Non linear

Slightly Non Linear Models Extended Kalman Filter Taylor expansion of nonlinear model operator F

Highly Non Linear Models Ensemble Kalman Filter Ensemble representation of error covariance matrix

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Kalman filter forecast

)()( ),()( 11T

ka

kf

ka

kf FtFCtCtFt

Model operator ?)( ,)()( 1 k

fk

ak

f tCtFt

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Overview Background

Reservoir Study and History Matching Kalman Filters Ensemble Kalman Filter – EnKF

Application 2D synthetic case study Sensitivity Tests

Conclusions & perspectives

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Ensemble representation of error covariances for Model State Measurements

Forecast The time evolution of uncertainty is obtained by integrating

each model state forward in time

Analysis Kalman linear analysis scheme applied to each model state

with the sample covariance

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Ensemble Kalman filters (EnKF)

TeCC

Best-guess

Tffff

e

ff CC

fTfTfaj

aj HdCHHCHC

1

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Advantages of EnKF for History Matching

Only one fluid-flow simulation over the whole production history per model of the ensemble

Real-time assimilation of data, for any data frequency

Uncertainty quantification

Practical implementation

Can be plugged with any fluid-flow simulator Suitable for parallel computation Relatively low computational cost

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Overview Background

Reservoir Study and History Matching Kalman Filters Ensemble Kalman Filter – EnKF

Application 2D synthetic case study Sensitivity Tests

Conclusions & perspectives

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Application : 2D synthetic case

P P

P P

I

Reservoir Size : 500 × 500 × 1 m Grid : 50× 50 × 1

Petrophysical Properties porosity = cte , Kv/Kh = cte Reference ln(Kh): spherical variogram

Fluid Flow oil-water, IFP simulator: PumaFlow Water Injection

Well plan 5 wells: 4 producers and 1 injector

Production data Bottom hole Pressure

(BHP) Surface Oil Rate

(QOS) Water cut

(WCT)

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Reference data

P1

P2 P3

P4

Inj

Ref ln(Kh)

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Assimilation with EnKF

Ensemble of 50 models State-parameter vector

ln( )

k

k

Kh

P

WSat

QOS

BHP

WCUT

Pressure

Water saturation

Surface oil rate

Bottom hole pressure

Water cut

Log of permeability Static parameter

Dynamical state

Simulated measurements

One value pergrid block

One value per well

17 assimilations, Once every month

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Evolution of ln(Kh) mean and variance

1st assimilation 4th assimilation 10th assimilation 17th assimilationInitial Model

Me

an

Va

rian

ce

ln(Kh) reference model

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Simulated answer and prediction – Prod3

Initialensemble

Finalensemble

Ensemblemembers

Ensemblemean Reference

Prediction

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Root Mean Square Error - RMS

Total number of grids

Ensemble Mean value for each

grid

Reference value at each grid

Number of time steps

Number of data per time step

Reference model Measurements

Simulated data for ensemble member

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Overview Background

Reservoir Study and History Matching Kalman Filters Ensemble Kalman Filter – EnKF

Application 2D synthetic case study Sensitivity Tests

Conclusions & perspectives

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Effect of Initial Ensemble 10 Ensembles of size 50 Prediction of cumulative oil production after 5.5

years

True Oil Cumulative Oil Production

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Effect of Assimilation Time Interval

Long Assimilation Interval

Base Case

Short Assimilation Interval

2 Months

Time

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Effect of Ensemble Size - 1

Reference ln(Kh)

Init

ial

Fin

alF

inal

Init

ial

Mea

nV

aria

nce

Ensemble Size

50 100 200 500

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Effect of Ensemble Size - 2Ensemblemembers

Ensemblemean Reference

50 Members

100 Members

200 Members

500 Members

BHP QOS WCT

Prediction

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Measurement Uncertainty

High Case : 10%Base Case : 5%Low case : 1%Initial Model

Me

an

Va

rian

ce

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Overview Background

Reservoir Study and History Matching Kalman Filters

Ensemble Kalman Filter Theory Implementation

Application 2D synthetic case study Sensitivity Tests

Conclusions & perspectives

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Main challenges Finite number of models used to represent uncertainty

Quality of initial sampling Variability Preservation

Linear analysis for a non linear model Physical inconsistency between parameters and state after

analysis

Non Gaussian distributions Multimodal distributions are not well preserved (e.g.

channel and barrier permeability distribution, facies distributions)

Large amount of data 4D seismic

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Conclusions and perspectives EnKF application to a 2D model and Sensitivity studies

Several ensembles of the same size for a sound uncertainty quantification

Smaller assimilation time interval : Better Match Lower Variance

Larger ensemble size : Better match for production data Variance preservation Larger simulation and computational cost

Larger Measurement Uncertainty Variance Preservation Less efficient for matching production data

Perspectives Facies History Matching Real field case study

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Thank you for your attention !Any Questions ?