1. Dario Grana, Tapan Mukerji
Annual Meeting 2013
Stanford Center for Reservoir Forecasting
Bayesian inversion of time-lapse seismic data for porosity, pressure and saturation changes
Dario Grana, Tapan Mukerji

2. Introduction
In time-lapse studies we aim to estimate pressure and saturation changes
Changes in dynamic properties can be measured from well/lab data. The main source of information are time-lapse seismic data.
Time-lapse seismic data
Seismic dataset records information about the elastic contrasts in the subsurface. Seismic attributes depend on rock properties.
Dynamic property changes
Inverse problem
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3. Motivation
Inverted data can be used in seismic history matching to improve the reservoir description
The probabilistic approach allows to quantify the uncertainty in predicted data
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4. Introduction
In pressure-saturation estimation the physical model is not linear and the dynamic property changes are not normally distributed.
Changes in dynamic properties are not independent of initial rock properties (static reservoir model)
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5. Bayesian inversion
We propose a Bayesian approach for the simultaneous estimation of porosity jointly with pressure and saturation changes from time-lapse seismic data.
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6. Bayesian inversion
We propose a Bayesian approach for the simultaneous estimation of porosity jointly with pressure and saturation changes from time-lapse seismic data.
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7. Bayesian inversion
We propose a Bayesian approach for the simultaneous estimation of porosity jointly with pressure and saturation changes from time-lapse seismic data.
Seismic data, Changes in seismic data
Elastic properties, Changes in elastic properties
(velocities or impedances)
Reservoir properties (porosity)
Changes in dynamic properties
(saturation and pressure)
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8. Bayesian inversion
Seismic data S depend on reservoir properties R through elastic properties m
We can split the inverse problem into two sub-problems:
seismic linearized modeling
rock physics model
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9. Reflection coefficients
Physical model
Seismic forward model:
Wavelet convolution
Linearized Aki-Richards approximation of Zoeppritz equations
Wavelet
Reflection coefficients
Seismogram
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10. P-wave velocity versus effective porosity
Physical model
Rock physics forward model:
Granular media models (Hertz-Mindlin contact theory)
Gassmann’s equations
Velocity-pressure relations
P-wave velocity versus effective porosity
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11. Outline Introduction Theory and Inversion workflow Application
Conclusions
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12. Inversion workflow We first estimate
Buland and Omre, 2003 Buland and El Ouair, 2006
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13. Inversion workflow We first estimate We then estimate
Buland and Omre, 2003 Buland and El Ouair, 2006
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14. Inversion workflow We first estimate We then estimate
We combine and using Chapman-Kolmogorov equation
Buland and Omre, 2003 Buland and El Ouair, 2006 1 2
Grana and Della Rossa , 2010
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15. Inversion workflow
Seismic data
Elastic properties
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16. Rock physics likelihood function
Inversion workflow
Seismic data
Rock physics likelihood function
Elastic properties
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17. Rock physics likelihood function
Inversion workflow
Seismic data
Rock physics likelihood function
Elastic properties
Reservoir properties
(Chapman-Kolmogorov)
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18. Simultaneous Bayesian 4D inversion
Combined Inverse problem
We estimate the posterior distribution
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19. Reservoir property estimation
Using statistical rock physics we estimate the likelihood
We use non-parametric pdfs and we estimate them using
Kernel Density Estimation (KDE)
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20. Outline Introduction Theory and Inversion workflow Application
Conclusions
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21. 2D application: synthetic model
Porosity
injector
Net-to-gross
producer
Pressure
Saturation
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22. 2D application: synthetic model
Saturation
2012
2017
Pressure
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23. Assumptions Porosity does not change in time (no compaction effect).
Initial pressure and saturation (pre-production) are known.
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24. 2D application: seismic model
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25. Synthetic seismic surveys
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26. Time-lapse seismic differences
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27. Rock physics likelihood
Different scenarios:
Pressure decreases – Saturation insitu
Pressure increases – Saturation insitu
Pressure insitu – Water replaced oil
Pressure insitu – Gas replaced oil
Pressure decreases – Water replaced oil
Pressure decreases – Gas replaced oil
Pressure increases – Water replaced oil
Pressure increases – Gas replaced oil
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28. Rock physics likelihood
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29. Elastic property changes%
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30. Reservoir property changes
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31. Point-wise posterior probabilities
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32. Application: Norne
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33. Application: Norne
Seismic differences ( )
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34. Application: Norne
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35. Application: Norne
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36. Outline Introduction Theory and Inversion workflow Application
Conclusions
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37. Conclusions
We presented a full Bayesian methodology to estimate reservoir properties and their changes from seismic data
The method allows to assess the uncertainty in the estimation of reservoir properties
Inverted data can be used in seismic history matching to improve the reservoir description
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38. Acknowledgments Prof. Tapan Mukerji and Prof. Jef Caers
Stanford University, SRB and SCRF for supporting my research
Thank you for the attention
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39. Backup
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40. Rock physics model Log of porosity, saturation, Mineral parameters
mineralogical fractions,
Fluid properties
Mineral parameters
Mixing laws
Voigt-Reuss-Hill
Mineral fractions
Matrix Moduli
Estimated Vp, Vs
and Density
RP Model
Nur, Hertz-Mindlin, Kuster-Toksoz,…
Saturation,
Fluid properties
Batzle Wang
Gassmann
MacBeth’s relation
Dry Rock Moduli
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41. MacBeth modified
We assume that
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42. Bayesian 3D inversion
Seismic Inverse problem
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43. Bayesian 3D inversion Seismic Inverse problem
If the prior distribution of m is Gaussian
If the model G is linear
Then the posterior distribution is Gaussian
Buland and Omre (2003)
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44. Bayesian 4D inversion
Time-lapse Inverse problem
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45. Bayesian 4D inversion Time-lapse Inverse problem
If the prior distribution of Δm is Gaussian
If the model G is linear
Then the posterior distribution is Gaussian
Buland and El Ouair (2006)
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46. Base seismic and time-lapse inversion
We then compute the relative impedance change as ΔIP = 1 - IPrep/IPbase
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47. Base seismic and time-lapse inversion
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48. Non-parametric pdfs: KDE
Clay
Vs (m/s)
Sand
Shale
Vp (m/s)
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49. Application: Norne Calibration well: well E2
Three angle stacks: near, mid, far
Base survey + seismic differences
( ; ; )
Goal: estimate dynamic property changes in 2003, 2004, and 2006
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50. Application: Norne
Well E2
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51. Application: Norne
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52. Application: Norne
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53. Application to Norne
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54. Application: Norne
Prediction of gas saturation pressure: 2003 and 2006
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55. Application: Norne Prediction of fluid pressure: 2003 and 2006
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