Use of Model Error With the EnKF

Centre for Integrated Petroleum Research

University of Bergen, Norway

Use of Model Error With the EnKF

S. I. Aanonsen

Voss Workshop 18 – 20 June 2008



Motivation

• An unkown model error is present in all real problems

• Originally, the Kalman filter and the EnKF was developed to update state variables for models with error

• Within the petroleum applications focus has mainly been on EnKF as a history-matching tool. Model error is normally not included(?) – However, real-time reservoir management also requires

short term predictions based on a very accurate data match



Questions

• Is it possible to obtain good short term predictions and better estimates of unknown parameters by including a model error term?

• Can a model error be estimated as a part of the EnKF updating process?



Model Problem

• Single-phase flow:

on0,0)0(

0,],0[],0[

)())((

n

ptp

tLLx

xQpxKt

p



True permeability field Fine grid (20x20)



Estimated mean permeabilityFine Scale EnKF. No model error

Log-normal Prior model:

Mean log(K) = 2 (100mD)

SD [log(K)] = 0.7

Spherical variogram with correlation length = L/4

Ensemble size: 40



EnKF update and ensemble predictions Well pressures vs time

Measurement error: 0.3



Rerun from time zero Well pressures vs time



EnKF update and ensemble predictionsCoarse grid (10x10). No model error

Same prior model as in fine-scale EnKF, except that the correlation length is L/2



Mean estimated permeability coarse modelNo model error



Including model error

• Alternative 1. Uncorrelated error (in time):

,...2,1,)( 11 ncwugu nan

fn

where

,...2,1,0,),0(~ nCNw wn

Cw has the same correlation structure as the prior permeability field

and same variance as the measurement error. c = 1.

No error added to the permeability.



Including model error

• Alternative 2. Correlated error (from Geir Evensen’s book). Model error is added to the state vector and updated “as usual”.

Forecast step:

fn

an

fn

nan

fn

cqugu

wqq

)(

)1(

1

11

where

1

),0(~

,...2,1,0,),0(~

0

w

wn

CNq

nCNw



EnKF update and ensemble predictionsCoarse grid (10x10). Uncorrelated model error



EnKF update and ensemble predictionsCoarse grid (10x10). Correlated model error, = 1



Mean estimated permeability coarse model With model error

Uncorrelated error Correlated error, = 1



Rerun from time zeroCoarse model. No model error



Rerun from time zeroCorrelated model error



Rerun from time zeroUncorrelated model error



Conclusions

• A typical model error may result in EnKF not being able to match data, ensemble collapse and poor predictions and parameter estimates.

• Adding an “unknown” model error may improve all of this and provide reasonably good predictions, at least short term.

Use of Model Error With the EnKF

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