Integrating Production and Seismic Data

into Gaussian and Pluri-Gaussian

Models with EnKF(S)

Yong Zhao

Yudou Wang

Gaoming Li

Al Reynolds

EnKF Workshop: Voss June 2008

Sequential Data Assimilation (Ensemble Kalman Filter)

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EnKF Analysis (Bayesian Updating and Sampling)

Critical Assumptions:1. Predictions of state vectors are Gaussian;2. Covariances can be represented by ensemble members;3. Gaussian noise in data; 4. Predicted data are a linear function of the state vector.

Or, with data augmented state vector1. Predictions of augmented vector are Gaussian;2. Gaussian noise in data;3. Covariances can be represented by ensemble members.

Potential Problems in EnKF

1. Each analyzed vector of model parameters is a linear combination of

initial ensemble.

2. Difficult to match large data sets, e.g., seismic data.

3. Non-Gaussianity.

4. Strong non-linearity.

5. Poor knowledge of measurement errors.

6. Modeling of modeling errors.

7. Sampling errors due to finite ensemble size.

8. Inconsistency: updated pressure and saturations are inconsistent with

the updated models (statistically different from those obtained by

simulating from time zero)

Rescaling for Different Types of Data

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Assimilating production data:

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Assimilating with rescaled data:

Better conditioned

Truncation of Singular Values, PUNQ, Est. Contact Depths

Truncated at 0.9999 Rescaled

Channel Model

2-D case2-D case– 100 X 100 grid, 100 X 100 grid, – 4 producers and 1 injector are located in the channel facies4 producers and 1 injector are located in the channel facies– 360 days of production with BHP and WCT measurements360 days of production with BHP and WCT measurements– 300 days of prediction300 days of prediction– 100 ensemble members100 ensemble members

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Facies En20

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True facies

EnKF Predictions

Prior prediction Prediction from EnKF Rerun from time zero

Normal Score Transform

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Before Analysis

After Analysis

Prediction Domain Analysis Domain

Normal Score Transform

Standard EnKF Global Transform Local Transform

Predictions From Transforms

No Transform Global LocalEnKF

Rerun

HIEnKF Method

If model changes significantly, updated primary field may be more inconsistent with the updated model.

When the change of model is significant, rerun from zero; otherwise, we use the EnKS.

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Model changes significantly EnKF

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EnKF HIEnKF

True

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EnKF vs. HIEnKF

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psi)

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Three-Facies Model

3-D case– 50 X 50X3 grid, – 4 producers and 1 injector– Total rate constraint for each well– Hard data: observed facies in well gridblocks– 360 days of production with BHP and WCT measurements (monthly)– 300 days of prediction– Seismic data (at time zero and 300 days)– 100 ensemble members– Fixed porosity and permeability

Permeability (11md, 100md, 528md); Porosity (0.06, 0.13, 0.21)

Layer 1 Layer 2 Layer 3

Assimilating Dynamic Data While Satisfying Hard Data, SPE 113990

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EnKF Predictions

Prior prediction Prediction from EnKF state Rerun from time zero

Acoustic Impedance

t = 0

Match seismic data at the time they are measuredMatch seismic data at the time they are measured

Prod-1

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t = 300days

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Matching Seismic Data:Local Analysis of EnKF

Local analysis:– Analyzed models are not constrained to the

sub-space spanned by the initial ensemble

– Undesired roughness can be introduced into the analyzed models

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Projection Method for Local Analysis

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A large ensemble with 1200 realizations of model

that honors the hard data (M0)– Use the first 200 eigenvectors

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Assimilate Seismic Data- Local Analysis (2 seismic + prod)

True Facies

No projectionEn20

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With projectionEn20

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Assimilate Seismic Data- Local Analysis With Projection

First Seismic OnlyContinue EnKF for production data

Rerun from time zero

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1st seismic

2nd seismic

1st seismic

2nd seismic

Structure Map of PUNQ-S3

grid.

Fault, gas cap, strong aquifer.

52819

Data: BHP GOR WCTMatch to 4032 days

Estimate the Depths of Fluid Contacts with EnKS

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Introduction to HIEnKS Method

If model changes significantly, updated primary field may be more inconsistent with the updated model.

Only use HIEnKS when the change of model is significant. otherwise, we use the EnKS.

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Model changes significantly EnKS

Examples

Example A:• Prior mean of OWC shifted up 20 feet

• Prior mean of GOC shifted down 20 feet.

True GOC

True OWC

Prior Mean of GOC

Prior Mean of OWC

Example A, prior oil column too thin

True GOC

True OWC

Prior Mean of GOC

Prior Mean of OWC

Example B, prior contact depths too deep

Example B:• Prior mean of OWC shifted down 20 feet

• Prior mean of GOC shifted down 20 feet.

20ft

20ft

20ft

20ft

STD: 20 ft

Comparison of Estimates of Fluid Contacts Example A Example B

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nK

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En

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HIE

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Consistency of Prediction, Example A

EnKS HIEnKS

EnKS: Future predictions poor, inconsistent. HIEnKS: Data matches good, consistent.

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Consistency of Prediction, Example A

EnKS

EnKS: Assimilation good, prediction poor, inconsistent. HIEnKS: Assimilation/Prediction good, roughly consistent.

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HIEnKS

Rock Property Fields- 4th, 5th layersT

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Vertical Permeability

HIE

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Horizontal Permeability

Comments

Iteration can improve reliability of data match,

predictions and consistency between parameters

and dynamical variables but is expensive.

Scaling can be critical if SVD is used.

EnKF combined with pluri-Gaussian gives

reasonable results (3D - rock properties – hard

data).

Comments

Pluri-Gaussian inappropriate for fluvial systems

– Cosine transforms, MRFs, KPCA?

Seismic: local analysis with projection seems

feasible but is currently ad hoc.