Ensemble-based DA for multidisciplinary models: An example from the oil industryGeir Evensen

Data assimilation problem

Inverse problems:

• Combined state and parameter estimation

• Parameter estimation

• State estimation

Dynamical systems:

• Nonlinear chaotic and unstable dynamics.

• Weakly nonlinear and stable dynamics.

Ensemble data-assimilation methods:

• Ensemble Kalman Filter

• Ensemble Smoother

Applications:

• Ocean

• Atmosphere

• Oil reservoir

• Soil models

• Ground water

• Climate

Bottlenecks in reservoir modelling

• Very complex and time consuming to build and maintain models.

• No proper tools for uncertainty modelling.

• Model updating is nearly impossible.

• Robust decisions cannot be made based on a single reality.

Geological modelling

Dynamicalsimulation

Model Conditioning

Production Optimization

Fast Model Update

Automated modelling and simulation process.

Representation of uncertainties using multiple realizations.

Flexible model updating using ensemble methods.

Uncertain input parameters

Simulated response

Automated reservoir-modeling

workflow

FMU concept

Model conditioning

MeasurementsEnsemble Smoother

Improved model with quantified

uncertainty

Uncertain input parameters

Simulated response

Automated reservoir-modeling

workflow

FMU concept

Correlations

Model conditioning

MeasurementsEnsemble Smoother

Improved model with quantified

uncertainty

Uncertain input parameters

Simulated response

Automated reservoir-modeling

workflow

FMU concept

Correlations

Model conditioning

MeasurementsEnsemble Smoother

Improved model with quantified

uncertainty

Assumptions:

• Major uncertainty is in the parametrization.

• No stochastic model errors.

• For a parameterization the solution is given.

ES example – initial ensemble

ES example – Analyzed ensemble

ES example – Parameter estimation

Ensemble Smoothers

• Handles arbitrary model workflows.

• No restarts.

• Updates model and model solution.

• Easier measurement-error

modelling.

• Conceptually simple.

• ES – Ensemble Smoother.

• IES – Iterative ES.

• ES MDA – Multiple Data

Assimilation.

Ensemble Smoother Issues

• Large data sets m >> N

• Measurements with correlated

errors

• Stable inversion of C=SS’ + EE’

• Efficient subspace inversion

• ES – Nonlinearity.

• IES – Convergence.

• ES MDA – Number of steps and

choice of weights.

› Need to invert

› May be singular or poorly conditioned.

› Pseudo inversion:

Stable inversion

Eig(1)/eig(5)= O(10^5)

Truncation at 90% retains one eigenvalue out of five.

Truncation at 99.9% retains four eigenvalues out of five.

EnKF with pseudo inversion of C

EnKF: Truncation at 99% retains about 40 eigenvalues.

EnSQRT: Truncation at 99% retains about 40 eigenvalues.

Example with 200 measurements

EnKF: Truncation at 99% retains about 40 eigenvalues.

EnSQRT: Truncation at 99% retains about 40 eigenvalues.

Example with 200 measurements

› Compute inversion in N-dim ensemble space rather than the m-dim meas. space.

› SVD of S

› Pseudo inverse of S

Ensemble-subspace inversion

Ensemble analysis schemes

Non-diagonal C_ee and clustered data19

Example

› Data assimilation in oil reservoirs:• Parameter-estimation problem.• Weakly nonlinear and stable dynamics --> ES• ES allows for handling complex workflows as black-box model.• Operational ensemble workflow for uncertainty modelling and model updating.

› Necessary to deal with the rank of C.• Pseudo inversion is key.• A diagonal C_ee may improve conditioning but does not add more “information.”• Analysis with full C_ee as efficient but no need to actually construct the matrix.• Easier and faster to sample measurement perturbations E.• Represent the measurement errors within the ensemble subspace!

› Stable and efficient analysis schemes that handle non-diagonal C_ee are available.• Details in Evensen (2009), Chapter 14.

Summary