Estimation of saturation and pressure changes from time-lapse seismic data and model re-

parameterization for seismic history matching.

Dario GranaDepartment of Geology and Geophysics University of Wyoming

IOR workshop, 28 April 2016

Seismic reservoir characterization

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• Seismic data S depend on reservoir properties R through elastic properties m

• We can split the inverse problem into two sub-problems:

•

• ff

gg

)(

)(

mR

Sm

))(( SR gf

seismic linearized modeling

rock physics model

Forward model

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Seismic forward model:Wavelet convolution

Linearized Aki-Richards approximation of Zoeppritz equations

Wavelet Reflection coefficients Seismogram

),()( mhrPP

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Bayesian seismic inversion

• The seismic response of a sequence of layers can be then written as:

• At a given time t, the reflections coefficients can be approximated by the three term Shuey approximation

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Bayesian seismic inversion

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Bayesian seismic inversion

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Bayesian seismic inversion

Forward model

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Rock physics forward model:

Granular media models (Hertz-Mindlin contact theory)

P-wave velocity versus effective porosity

sw

cfV

V

RPMS

P

Bayesian petrophysical inversion

Elastic properties

Seismic data)( Sm|P

Gaussian mixture

likelihood

)|( mRP

Rock properties

)( SR|P

(Chapman-Kolmogorov)

Grana and Della Rossa, 2010

Buland and Omre, 2003

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Time-lapse inversion

• In time-lapse reservoir modeling we aim to model reservoir property changes from repeated seismic surveys.

Inverse problem

Time-lapse seismic dataReservoir property changes

(saturation and pressure)

Grana and Mukerji, 2014

Challenges

• Elastic changes introduce time-shifts in the repeated seismic surveys (either we invert amplitudes and travel time or we correct for time-shifts and invert for amplitudes)

• The link between dynamic properties and elastic properties is not direct. Need a RPM to transform pressure and saturation into elastic properties.

• Mapping of data between different grid representations corresponding to the flow simulation, earth model and seismic grid frameworks.

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Example

• Time-lapse seismic changes

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Increase Pp by 100 MPa Before After Difference

Time-

aligned

difference

(Courtesy of G. Mavko)

Methods

• Method 1: to simultaneously invert base and repeated seismic surveys

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(Courtesy of P. Doyen)

Methods

• Method 2: to invert seismic changes (difference in amplitudes)

* We cannot compute the difference directly because horizons of the repeated surveys are not aligned with the base survey. We must apply a warping method first.

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Methods

• 2-step approach

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1. We first estimate

2. We then estimate

3. We combine and using Chapman-Kolmogorov equation

( | ) ( | ) ( )P P P m S S m m

( | ) ( | ) ( )P P P R m m R R

Buland and El Ouair, 2006

( | )P m S ( | )P R m

( | ) ( | ) ( | )m

P P P d

R S R m m S mGrana and Mukerji, 2014

Seismic model

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Seismic forward model:Wavelet convolution

Linearized Aki-Richards approximation of Zoeppritz equations

Wavelet Reflection coefficients Seismogram

),()( mhrPP

MacBeth equation

( )

1 K

dry P

P

K

KK P

A e

Methods for seismic history matching

• Option1: to match production data and time-lapse seismic data

• Option 2: to match production data and inverted seismic velocities (or impedances)

• Option 3: to match production data and inverted pressure and saturation (or a re-parameterization)

(plus several options for the parameterization and for the inverse method)

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Re-parameterized seismic history matching

Bayesian time-lapse inversion Re-parameterization

Seismic history matching

(ES-MDA)

Synthetic example: waterflooding

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Synthetic reservoir model (modified from Panzeri et al., 2014)

Synthetic example: waterflooding

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Synthetic example

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• Porosity constant and known

• History matching of 6 years; forecast of 4 years

• Inverted seismic data every 6 months for 6 years (13 surveys)

• Ensemble Smoother MDA (4 assimilations)

Synthetic example

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Synthetic example

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Synthetic example

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Synthetic example

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Prior model

Updated model

Synthetic example: comparison

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Acknowledgements

Thanks for your attention

• Thanks to Geir for the invitation

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Methodology

• Bayesian time-lapse inversion for pressure and saturation changes

• Re-parameterization of saturation snapshots through POD-DEIM

• History matching of production data and re-parameterized inverted seismic data through ES-MDA

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Re-parameterization

• If we match the seismic data pointwise, we need a large ensemble to avoid ensemble collapse

• We want to avoid to compute CDD

• Issues with resolution, noise, quality of seismic data

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Proper Orthogonal Decomposition

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Discrete Empirical Interpolation Method

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Discrete Empirical Interpolation Method

Chaturantabut and Sorensen, 2010

• The projection basis is given by POD

• The interpolation indices are given by DEIM

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Example 1

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Example 1

Other parameterization methods: wavelet, DCT, waterfront, spatial PCA, ...