bayesian inversion methods for seismic reservoir characterization ...

BAYESIAN INVERSION METHODS FOR SEISMIC RESERVOIR CHARACTERIZATION AND TIME-LAPSE STUDIES

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF GEOPHYSICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Dario Grana August 2013

c Copyright by Dario Grana 2013

All Rights Reserved

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I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

(Gary Mavko) Principal Adviser


(Tapan Mukerji)


(Jack Dvorkin)

Approved for the University Committee on Graduate Studies

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Abstract This dissertation addresses mathematical methodologies for seismic reservoir characterization and time-lapse studies. Generally the main goal of reservoir modeling is to provide 3-dimensional models of the main properties in the reservoir in order to perform fluid flow simulations. These properties generally include rock properties, such as porosity and lithology; fluid properties, such as water and hydrocarbon saturations; and dynamic properties, such as pressure and permeability. None of these properties can be directly measured in the subsurface, therefore reservoir properties must be estimated from other measurements. In petroleum geophysics we generally have two kinds of measured data: well log data and seismic data. Well log data contain high resolution information about elastic and petrophysical properties, but they can only sample few locations of the reservoir. On the other side, seismic data cover the whole reservoir but the resolution is lower than well log data. Electromagnetic data are sometimes acquired in addition to seismic data to improve the reservoir description but the resolution is still limited. In order to obtain suitable models of the reservoir, we have to combine these two sources of information, wells and seismic, and integrate physical relations (rock physics and seismic modeling) with mathematical methodologies (inverse theory and probability and statistics). In particular by using a Bayesian approach to seismic and rock physics inversion we aim to obtain reservoir models of rock and fluid properties and the associated uncertainty. Since the resolution and the quality of seismic data are generally not ideal, uncertainty quantification plays a key role in reservoir modeling. This thesis includes three innovative methodologies for seismic reservoir characterization: the first method is a Bayesian inversion methodology suitable for reservoirs iv

in exploration phases with a limited number of wells, the second method is a Bayesian sampling methodology that can provide multiple reservoir models honoring the given seismic dataset, the third one is a stochastic inversion methodology that provides high-detailed models suitable for reservoirs with a large number of wells. The key innovation in all these methods is the use of new statistical tools to describe the multimodal behavior of rock and properties in the reservoir and the direct integration of the rock physics model. The main principle of these methodologies is then extended to time-lapse studies to invert time-lapse seismic data and improve the reservoir description in terms of changes in rock and dynamic properties. The novelty of this method is the simultaneous inversion of the pre-production base seismic survey and repeated monitor surveys. This dissertation contributes to both deterministic and statistical seismic-based reservoir characterization. Complementary, I investigated velocity-pressure transforms to determine analytical physical models to describe the pressure effect on elastic properties and integrate these models in time-lapse reservoir studies. Finally I also developed a statistical methodology to integrate rock physics models in formation evaluation analysis and log-facies classification. All the proposed probabilistic reservoir-characterization techniques can predict reservoir models with multiple properties (static and dynamic) and the associated uncertainty. Multiple models can then be derived to run multiple scenarios and the corresponding risk analysis. All the methodologies were tested using synthetic data and applied to real case datasets. In the future, these methodologies could be integrated with history matching techniques to develop statistical methodologies for seismic history matching and improve reservoir description and monitoring by simultaneously matching seismic data and production data.

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Acknowledgements First, I want to express my thankfulness to my advisers: Gary Mavko for his support, trust and for the enjoyable conversations; Jack Dvorkin for all the help during my time here; and Tapan Mukerji for his collaboration which allowed me to expand my knowledge in many different domains of geophysics and reservoir engineering. I am really grateful to Gary, Jack and Tapan for the outstanding help and friendship, and for their helpful comments and challenging questions. It was an honor and privilege to have them all as advisers. Many thanks go to the SRB program and all its sponsors for the funding and support during these three years. Thanks to this generous support, I have been allowed to choose this research topic and complete my Ph.D. at Stanford. A special thank goes to all the SRB members: Tiziana for her constant support as a scientist and as a friend; Amos for the interesting conversations; Adam, Amrita and Anthony for sharing the office; and all the other students and postdocs that I have met over my time at Stanford: Piyapa, Danica, Adam Tew, Nishank, Yu, Sabrina, Ammar, Yuki, Chisato, Humberto, Prianka, Sissel, Fabian, Stephanie and Jane. Thanks to all the staff members in Geophysics, particularly Fuad, Nancy and Stephanie. It was a pleasure to work with all of them. There are a lot of people that I would like to thank for the time we spent together: Kevin, who was clearly born in the wrong country, for sharing a lot of enjoyable time in my unofficial office, Randi and Sarah who were wonderful unofficial officemates, Denys and Adam who always have funny story to tell, Jason, Andreas and Gader for the afternoon study sessions, Amrita for being such a good listener, Shahar and Ksenia for the international students solidarity. A great thank goes to the volleyball vi

group and all the volleyball players and friends I have met: Alec, Guillaume, Sean, Alice, Jack, Ryan, Alex, Anthony, Shandor, Kaipo, the chinese team and many others. Finally, thanks to Maurizio for his friendship. The most important acknowledgement goes to my mom. She is the one without whom nothing of what I did in my life would have been possible. Thanks to the rest of my family especially to my nieces with whom I return to be a ten year old kid. A special thank goes to Ernesto Della Rossa, an amazing mathematician, great scientist and wonderful person. I want to thank my friend Valentina a special person for me who makes our friendship unique. I finally want to thank all my friends from Italy, who I have constantly been in touch with, at least through my ”fairy tales from the west coast”. There are other many people that would deserve many acknowledgements, and there is not enough space here to thank all of them, but the nice thing is that if I close my eyes I have a memory for each of you guys, and when I reopen my eyes, most of you are still around me, and this is a great satisfaction. These three years in California have been an amazing experience.

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Contents Abstract

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Acknowledgements

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1 Introduction

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1.1 Motivation and objectives . . . . . . . . . . . . . . . . . . . . . . . .

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1.2 Chapter description . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.3 Inversion methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.4 Bayesian approach . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Probabilistic reservoir properties estimation

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2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3 Probabilistic approach to petrophysical inversion . . . . . . . . . . . .

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2.3.1

Statistical rock physics modeling . . . . . . . . . . . . . . . .

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2.3.2

Probabilistic upscaling . . . . . . . . . . . . . . . . . . . . . .

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2.3.3

From seismic to petrophysics inversion . . . . . . . . . . . . .

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2.4 Real case application . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.4.1

Methodology implementation . . . . . . . . . . . . . . . . . .

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2.4.2

Data application . . . . . . . . . . . . . . . . . . . . . . . . .

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2.4.3

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Sequential Gaussian mixture simulation

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3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.3 Theory: Linearized Gaussian Mixture Inversion . . . . . . . . . . . .

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3.4 Theory: Sequential approach . . . . . . . . . . . . . . . . . . . . . . .

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3.4.1

Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.5 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Stochastic inversion for facies modeling

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4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.3.1

Geostatistical methods . . . . . . . . . . . . . . . . . . . . . .

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4.3.2

Geophysical forward model . . . . . . . . . . . . . . . . . . . .

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4.3.3

Stochastic optimization algorithm . . . . . . . . . . . . . . . .

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4.3.4

Secondary information . . . . . . . . . . . . . . . . . . . . . .

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4.4 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.4.1

Synthetic case . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.4.2

Real case . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5 Statistical methods for log evaluation

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5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3.1

Quantitative Log Interpretation . . . . . . . . . . . . . . . . . 117

5.3.2

Rock physics modeling . . . . . . . . . . . . . . . . . . . . . . 126

5.3.3

Log-facies classification . . . . . . . . . . . . . . . . . . . . . . 130

5.4 Application: first example . . . . . . . . . . . . . . . . . . . . . . . . 133 5.5 Application: second example . . . . . . . . . . . . . . . . . . . . . . . 146 ix

5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 6 Pressure dependence of elastic properties

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6.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.3 Physical model and application to lab data . . . . . . . . . . . . . . . 161 6.4 Application to log data and sensitivity . . . . . . . . . . . . . . . . . 168 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7 Bayesian time-lapse inversion

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7.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 7.3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7.4 Application: synthetic case . . . . . . . . . . . . . . . . . . . . . . . . 178 7.5 Application: real case . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 8 Final remarks

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References

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List of Tables 2.1 Rock physics model parameters: density ρ, bulk modulus K and shear modulus µ of the matrix components. . . . . . . . . . . . . . . . . . .

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2.2 Correlation coefficients between estimated petrophysical properties and real data at well A location. . . . . . . . . . . . . . . . . . . . . . . .

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2.3 Contingency analysis of petrophysical properties estimation at well A location (f = Absolute frequency; R = Reconstruction rate; r = Recognition rate; E = Estimation index). . . . . . . . . . . . . . . . . . . .

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4.1 Confusion matrix of the reference case (T stands for true facies, C stands for classified facies). . . . . . . . . . . . . . . . . . . . . . . . .

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4.2 One-way sensitivity analysis of the synthetic inversion test: the first column show the different cases, in the second column we report the main diagonal of the confusion matrices of the different cases, in the third column we show the average of the elements of the main diagonal (sum of the trace of the matrix normalized by the number of facies). .

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4.3 Mean values of petro-elastic properties in the different facies. Values of porosity and clay content have been estimated from well log data, elastic properties values have been computed by rock physics model. .

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4.4 Confusion matrix of the inversion results obtained by stochastic inversion at well 2 location (T stands for true facies, C stands for classified facies). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.1 Standard deviations associated to log measurements. . . . . . . . . . 140

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List of Figures 2.1 Flowchart of the probabilistic petrophysical properties estimation. . .

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2.2 Petrophysical curves derived from log interpretation of well A. From left to right: P-impedance (well log in blue and rock physics model predictions in red), S-impedance (well log in blue and rock physics model predictions in red), effective porosity, clay content, water saturation and litho-fluid classification (oil sand in yellow, water sand in brown, shale in green) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.3 Well log data distribution and litho-fluid classification of well A. (top) P-impedance versus effective porosity color coded by litho-fluid class. (bottom) S-impedance versus P-impedance color coded by litho-fluid class (oil sand in yellow, water sand in brown, shale in green). . . . .

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2.4 Rock physics model calibration on well A: velocity at wet conditions (obtained performing a fluid substitution on well log velocities) versus effective porosity, color coded by clay content. The curves are from the stiff-sand model for a mixture of wet clay and sand, with clay content equal to (from bottom to top) 0.9, 0.7, 0.5, 0.3 and 0.1. . . . . . . . .

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2.5 Prior distribution of petrophysical variables. (top) 2D marginal distribution of effective porosity and clay content with the associated 1D marginal distributions. (bottom) 2D marginal distribution of clay content and water saturation with the associated 1D marginal distributions. The black crosses represent petrophysical data of well A, the background color is the joint probability. . . . . . . . . . . . . . . . .

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2.6 Experimental and fitted variograms of effective porosity for the three litho-fluid classes. From left to right: variogram of effective porosity in shale, water sand and oil sand (color lines are the experimental variograms and dotted lines are the fitted variograms). . . . . . . . .

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2.7 Petrophysical properties estimation at well A location: effective porosity, clay content and water saturation probability distributions obtained with GMM. (top) Petrophysical properties estimation conditioned by high resolution impedances, extracted from P (R|mf ). (bottom) Petrophysical properties estimation conditioned by upscaled data, extracted from P (R|mc). The background color is the conditional probability. Black lines are the actual petrophysical curves, red dotted lines represent P10, median and P90. . . . . . . . . . . . . . . . . . .

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2.8 Petrophysical properties estimation conditioned by synthetic seismic data at well A location: effective porosity, clay content and water saturation probability distributions extracted from P (R|Sz ) computed with GMM (top) and KDE (bottom). The background color is the conditional probability. Black lines are the actual petrophysical curves, red dotted lines represent P10, median and P90. . . . . . . . . . . . .

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2.9 Litho-fluid probabilistic classification conditioned by synthetic seismic at well A location. (top) From left to right: probabilities of lithofluid classes P (πz |Sz ) based on petrophysical inversion, MAP of the

probability and actual litho-fluid classes. (bottom) Some realizations obtained with a Markov chain approach (oil sand in yellow, water sand in brown, shale in green). . . . . . . . . . . . . . . . . . . . . . . . . .

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2.10 2D seismic sections passing through well A (on the right) and well B (on the left): (top) angle stack 20o ; (bottom) angle stack 44o . . . . . .

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2.11 Prior model and posterior distributions at well locations. From left to right: P-impedance of well A, S-impedance of well A, P-impedance of well B, S-impedance of well B. Blue curves are the actual logs, green curves represent the upscaled data, black curves are the prior model, red curves represent the inverted values. Dotted lines represent the P10 and P90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.12 2D sections of inverted impedances: (top) inverted P-impedance; (bottom) inverted S-impedance. . . . . . . . . . . . . . . . . . . . . . . .

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2.13 Probability distributions of effective porosity at wells locations (well A on the right, well B on the left) and at an intermediate location between the two wells. Black lines are the actual effective porosity curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.14 Estimation of effective porosity (top), clay content (middle), and water saturation (bottom) in the 2D section obtained from the mode of the posterior distributions.

. . . . . . . . . . . . . . . . . . . . . . . . .

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2.15 Litho-fluid probabilistic classification conditioned by real seismic at well A location. From top left to bottom right: probabilities of lithofluid classes P (πz |Sz ) based on petrophysical inversion, six realizations obtained with a Markov chain approach, and actual litho-fluid classi-

fication (oil sand in yellow, water sand in brown, shale in green). . . .

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2.16 Estimation of effective porosity along two lines extracted from the 3D volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.17 Isoprobability surface of 70% probability of oil sand litho-fluid class. The background slices represent two 2D sections of probability of oil sand occurrence.

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3.1 Conditional realizations of porosity and reservoir facies obtained by SGMixSim. The prior distribution of porosity and the hard data values are shown on top. The second and third rows show three realizations of porosity and facies (grey is shale, yellow is sand). The fourth row shows the posterior distribution of facies and the ensemble average of 100 realizations of facies and porosity. The last row shows the comparison of SGMixSim results with and without post-processing. . . . . . . . .

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3.2 Linearized sequential inversion with Gaussian Mixture models for the estimation of porosity map from acoustic impedance values. On top we show the true porosity map and the acoustic impedance map; on the bottom we show the inverted porosity and the estimated facies map. 63 3.3 Sequential Gaussian Mixture inversion of seismic data (ensemble of 50 realizations). From left to right: acoustic impedance logs and seismograms (actual model in red, realization 1 in blue, inverted realizations in grey, dashed line represents low frequency model), inverted facies profile corresponding to realization 1, maximum a posteriori of 50 inverted facies profiles and actual facies classification (sand in yellow, shale in grey). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3.4 Application of linearized sequential inversion with Gaussian Mixture models to a reservoir layer. The conditioning data is P-wave velocity (top left). Two realizations of porosity and facies are shown: realization 1 corresponds to a prior proportion of 30% of sand, realization 2 corresponds to 40% of sand. The histograms of the conditioning data and the posterior distribution of porosity (realization 2) are shown for comparison.

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4.1 Workflow of stochastic inversion.

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. . . . . . . . . . . . . . . . . . . .

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4.2 Flowchart of PPM algorithm. . . . . . . . . . . . . . . . . . . . . . .

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4.3 Synthetic well log dataset (well A), from left to right: effective porosity, volume of clay, water saturation, P-wave and S-wave velocity, density and facies profile (green represents shale, brown represents silty-sand, and yellow represents sand). . . . . . . . . . . . . . . . . . . . . . . .

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4.4 (Top) Rock physics crossplots of well dataset: clay content versus effective porosity (top left), P-wave velocity versus effective porosity (top right), S-wave velocity versus P-wave velocity (mid left), and VP /VS ratio versus P-impedance (mid right), color coded by facies classification (green represents shale, brown represents silty-sand, and yellow represents sand). (Bottom) Joint probability of petrophysical properties distribution: conditional probability contours color coded by facies (bottom left), and joint probability surface (bottom right). . . . . . .

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4.5 Variograms of porosity estimated at the well location, from top to bottom: variogram of porosity in shale, silty-sand, and sand. . . . . .

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4.6 Stochastic inversion results at well location, from left to right: actual facies classification, initial realization, and partial results of the optimization loop after 3, 10 and 25 iterations classification (green represents shale, brown represents silty-sand, and yellow represents sand). The last result (right plot) is the optimized model according to the fixed tolerance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.7 Synthetic seismograms (red) corresponding to the optimized model of Figure 4.6 compared to input seismic traces (black). From left to right: near, mid and far stack, corresponding to the incident angles of 12o , 24o , and 36o. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.8 Set of 10 different realizations obtained by stochastic inversion. We show 10 optimized models (obtained from 10 different runs) compared to the actual classification (green represents shale, brown represents silty-sand, and yellow represents sand). . . . . . . . . . . . . . . . . .

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4.9 Real case application: well log dataset from well 2 (calibration well). From left to right: P-wave and S-wave velocity, effective porosity, clay content, water saturation, and actual facies classification (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4.10 Rock physics model: (left) P-wave velocity versus effective porosity; (right) S-wave velocity versus effective porosity, color coded by clay content. Black lines represent constant-cement sand model for different clay contents (from top to bottom: 0%, 25%, 50%, 75%, and 100%). .

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4.11 Marginal probability density functions of effective porosity and clay content conditioned by facies classification. The pdfs of petrophysical properties are used to distribute rock properties within the reservoir model at each iteration of stochastic inversion (shale in green, siltysand in brown, stiff sand in light brown, soft sand in yellow). . . . . .

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4.12 Inversion results at well 2 location with synthetic seismic data, from left to right: actual facies classification, upscaled facies profile, seismic facies probability, initial model, optimized model after 50 iterations (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.13 Multiple realizations with different tolerance conditions: (top) 25 simulations obtained with a small tolerance; (bottom) 25 simulations obtained with a larger tolerance (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow). For each set of simulations we plot the ensemble average (e-type) and we compare the results with the upscaled facies classification at well 2 location. . . . . . . . . . . . 103 4.14 Inversion results at well 2 location with real seismic data, from left to right: actual facies classification, upscaled facies profile, seismic facies probability, initial model, optimized model after 50 iterations (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow).104

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4.15 (Top) Convergence plot of stochastic inversion results with and without low frequency information (blue and red symbols respectively) as a function of iteration number. (Bottom) Boxplots of 25 runs consisting of 50 iterations with and without low frequency information (left and right respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.16 Inversion results at well 3 (left plots) and well 5 (right plots) locations. We compare the actual classification with the optimized model obtained by stochastic inversion (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow).

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4.17 2D seismic section passing through wells 2 and 3: (top) near angle stack corresponding to 8o , (bottom) far angle stack corresponding to 26o . The black line represents the top horizon, in time domain, corresponding to the interpreted top of the reservoir. . . . . . . . . . . . . 107 4.18 Inversion results along the 2D section shown in Figure 4.17. On the left we show the optimized model of reservoir facies (top left), porosity (mid left) and P-wave velocity (bottom left), obtained by stochastic inversion. On the right we show the corresponding synthetic seismic sections, near (top right) and far (top left) and the maximum a posteriori (MAP) of seismic facies probability (converted in depth and mapped in the geocellular grid) used as secondary information in the inversion and obtained by multi-step inversion (shale in green, siltysand in brown, stiff sand in light brown, soft sand in yellow). . . . . . 108 5.1 Flowchart of the probabilistic petrophysical properties estimation. . . 118 5.2 Schematic representation of petro-elastic uncertainty estimation and log-facies classification through Monte Carlo simulations of petrophysical and elastic properties. . . . . . . . . . . . . . . . . . . . . . . . . 123 5.3 Well log dataset interval of well A, from left to right: P-wave and S-wave velocity; density; neutron porosity; gamma ray. . . . . . . . . 134

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5.4 Petrophysical curves performed in Quantitative Log Interpretation, from left to right: porosity (total porosity in blue, effective porosity in red); volumetric fraction curves (clay in green, quartz in yellow, and silt in brown); water saturation; and finally the cumulative volumetric display (shale in orange, silt in brown, quartz in yellow, water in blue, and oil in green; red dashed line represents 1 minus porosity and it separates solid and fluid phase). . . . . . . . . . . . . . . . . . . . . . 135 5.5 Petrophysical outputs comparison (red curves represent our results; blue curves represent standard commercial software outputs). . . . . . 136 5.6 Calibration of the rock physics model, from left to right P-wave and Swave velocity, and VP /VS ratio (black curves represent the actual sonic log, blue dashed curves represent the predicted rock physics model). The rock physics model has been calibrated in wet condition and then applied to the well scenario by Gassmann fluid substitution. . . . . . 137 5.7 Dendogram associated to log-facies classification. A dendrogram consists of many U-shaped lines connecting objects in a hierarchical tree. The stem of each U represents the distance between the two objects being connected. Red clusters refer to the connecting histories of the three recognized facies: low concentration turbidite (LCT), mid concentration turbidite (MCT), high concentration turbidite (HCT).

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5.8 Log-facies classification performed at well location: LCT in green, MCT in brown, and HCT in yellow. Log-facies classification is derived by using petrophysical curves (porosity and clay content) and velocity data (VP /VS ratio). On the right we show two crossplots in petrophysical (top right) and petroelastic (bottom right) domain, color coded by facies classification. . . . . . . . . . . . . . . . . . . . . . . . 139 5.9 Set of 100 realizations of petrophysical curves (gray curves), from left to right: effective porosity, volume of clay, and volume of quartz (volume of silt is computed by difference 1 minus the sum of effective porosity, clay, and quartz). The pointwise median curve (P50) is displayed in red.141

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5.10 Set of 100 realizations of elastic curves (gray curves), from left to right: P-wave and S-wave velocity. The pointwise median curve (P50) is displayed in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.11 Posterior probability density functions of porosity (left side) and Pwave velocity (right side). For two depth locations z1 and z2 we also show the histogram of the simulated values. . . . . . . . . . . . . . . 143 5.12 Set of 100 realizations of facies (top left) at well location and estimated most likely facies profile (top right). On the bottom 5 realizations are shown: LCT in green, MCT in brown, and HCT in yellow. . . . . . . 144 5.13 Posterior probability distribution of facies estimated by Monte Carlo simulation (left), most likely facies profile (middle) and associated entropy function (right): LCT in green, MCT in brown, and HCT in yellow. On the right we show two crossplots in petrophysical (top right) and petroelastic (bottom right) domain, color coded by the associated entropy given by the probabilistic facies classification. . . . . . . . . . 145 5.14 Well log dataset and preliminary petrophysical curves at well location, from left to right: P-wave and S-wave velocity; density; porosity (total porosity in blue, effective porosity in red); volumetric fraction curves (clay in green, quartz in yellow, and muscovite in brown); and water saturation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.15 Preliminary facies classification: a set of 8 facies has been identified in this reservoir: 1) marine silty-shale; 2) pro-delta; 3) flood plain; 4) mouth bar; 5) distributary channel; 6) crevasse splay; 7) tidal deltaic lobes; 8) tight (left). Grouped histogram of effective porosity (top right) and clay content (bottom right) as a function of facies classification.148 5.16 Rock physics crossplots: (left) P-wave velocity versus effective porosity; (right) S-wave velocity versus effective porosity color coded by volume of clay. Black lines represent stiff sand model for different clay contents (from top to bottom: 10%, 20%, 30%, 40%, 50% and 60%). . . . . . . 149

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5.17 Set of 50 realizations of petrophysical and elastic curves (gray curves), from left to right: effective porosity, volume of clay, and volume of quartz (volume of silt is computed by difference 1 minus the sum of effective porosity, clay, and quartz), P-wave velocity and S-wave velocity predictions. The pointwise median curve (P50) is displayed in red. The dashed blue line represents sonic log data. . . . . . . . . . . . . . 150 5.18 Set of 50 realizations of facies (left) at well location, etype (ensemble average) and estimated most likely facies profile (top right). Color codes are the same as in Figure 5.15. . . . . . . . . . . . . . . . . . . 151 5.19 Facies probability estimated from ensemble and extracted statistics, from left to right: probability of facies (first three plots); entropy, maximum a posteriori of facies distribution and new facies classification performed by Markov chain integrated approach. Color codes are the same as in Figure 5.15. . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5.20 Comparison of two different classifications. On the left (first two plots) we show the 8-facies classification: maximum a posteriori of facies and Markov chain classification (color codes are the same as in Figure 5.15). On the right we show the simplified 3-facies classification (namely seismic facies classification, last three plots): posterior probability distribution, entropy, and maximum a posteriori of facies (shale in green, silty-sand in brown, sand in yellow). . . . . . . . . . . . . . . . . . . . 153 6.1 Han’s dataset: dry-rock bulk modulus versus effective pressure color coded by porosity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.2 Han’s dataset: dry-rock bulk modulus versus effective pressure. The color represents porosity (left) and a linear combination of porosity and clay content(right). . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.3 Eberhart-Phillips relation: on the left each sample has been fitted independently, on the right all samples have been fitted all together. . . 163 6.4 MacBeth relation (bulk modulus): each sample has been fitted independently. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

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6.5 Sensitivity analysis on fitted parameters on MacBeth relation: (top) fitted parameters versus porosity; (bottom) fitted parameters versus linear combination of porosity and clay content. . . . . . . . . . . . . 165 6.6 Self similarity concept: (left) measured bulk moduli versus a linear combination of porosity and clay content color coded by effective pressure; (right) bulk modulus at 50 MPa versus a linear combination of porosity and clay content and linear fitting.

. . . . . . . . . . . . . . 166

6.7 Modified MacBeth relation (bulk modulus). . . . . . . . . . . . . . . 166 6.8 Modified MacBeth relation (shear modulus). . . . . . . . . . . . . . . 167 6.9 Modified MacBeth relation (set of five samples): bulk modulus (left) and shear modulus (right). . . . . . . . . . . . . . . . . . . . . . . . . 167 6.10 Well log data: from left to right P-wave velocity, S-wave velocity, density (well log in blue and rock physics model in red), and volumetric fractions (volume of quartz in yellow, clay in green, silt in black, effective porosity in red, and water saturation in blue). . . . . . . . . . . . 169 6.11 P-wave velocities computed through the rock physics model in 8 different scenarios (in-situ condition in red, and production scenarios in black). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.12 S-wave velocities computed through the rock physics model in 8 different production scenarios (in-situ condition in red, and production scenarios in black). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 6.13 Density computed through the rock physics model in 8 different production scenarios (in-situ condition in red, and production scenarios in black). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.14 Synthetic seismograms in 8 different production scenarios (short offset). 171 6.15 Synthetic seismograms in 8 different production scenarios (long offset). 172

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7.1 Combined Bayesian inversion results for elastic properties and elastic properties relative changes for a three-layer wedge model: estimated P-impedance relative change, S-impedance relative change and P-impedance base model (black curves represent the prior model, blue curves the actual model, solid red curves represent the mean values and dotted red curves percentiles represent P10 and P90). . . . . . . 179 7.2 Combined Bayesian inversion results for elastic properties and elastic properties relative changes for a real well log dataset: estimated posterior distributions of P-impedance relative change, S-impedance relative change and P-impedance base model (black curves represent the actual model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.3 Highly informative prior distribution for pressure and saturation changes.181 7.4 Combined Bayesian inversion results for elastic properties and elastic properties relative changes for a real well log dataset : estimated posterior distributions of P-impedance relative change, S-impedance relative change and P-impedance base model (black curves represent the actual model).

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

7.5 Poorly informative prior distribution for pressure and saturation changes.182 7.6 Combined Bayesian inversion results for elastic properties and elastic properties relative changes for a real well log dataset assuming a poorly informative prior distribution: estimated posterior distributions of water saturation change, effective pressure change and porosity base model (black curves represent the actual model. . . . . . . . . . . . . 182 7.7 Fluid saturations and effective pressure before (top) and after production (mid). Bottom left, effective porosity; bottom right, net-to-gross.

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7.8 Base (left) and repeated (right) seismic survey (perfect data): from top to bottom near (10o ), mid (20o ), and far (30o ).

. . . . . . . . . . 184

7.9 Time-lapse seismic differences: from top to bottom near (10o), mid (20o ), and far (30o ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

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7.10 Mean values of inverted elastic properties and elastic properties relative changes: from top to bottom P-impedance relative change, Simpedance relative change, and P-impedance estimation from base survey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.11 Mean values of inverted elastic properties and elastic properties absolute changes: from top to bottom P-impedance absolute change, S-impedance absolute change, and P-impedance estimation from base survey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.12 Mean values of inverted reservoir properties and dynamic properties relative changes: from top to bottom gas saturation change, oil saturation change, effective pressure change, and porosity estimation. . . . 189 7.13 Examples of point-wise posterior distributions of reservoir properties and dynamic properties relative changes from top-left to bottom-right gas saturation change, oil saturation change, effective pressure change, and porosity.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

7.14 Main structure of the geocellular model (blue, top layer in red) and seismic survey geometry (black). . . . . . . . . . . . . . . . . . . . . . 191 7.15 Active cells of the geocellular model (blue, top layer in red). Seismic survey geometry is shown in black for comparison. . . . . . . . . . . . 191 7.16 Well log data (well E2): from left to right P-wave velocity, S-wave velocity, density, clay content, porosity and water saturation. . . . . . 192 7.17 Well log data (well E2): crossplot of P-wave velocity versus porosity (left) and S-wave velocity versus porosity (right) color coded by clay content. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 7.18 Seismic data differences in 2003: from left to right, near, mid and far.

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7.19 Seismic data differences in 2006: from left to right, near, mid and far.

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7.20 Predicted gas saturation in 2003. . . . . . . . . . . . . . . . . . . . . 194 7.21 Predicted gas saturation in 2006. . . . . . . . . . . . . . . . . . . . . 194 7.22 Predicted fluid pressure in 2003. . . . . . . . . . . . . . . . . . . . . . 195 7.23 Predicted fluid pressure in 2006. . . . . . . . . . . . . . . . . . . . . . 195

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Chapter 1 Introduction 1.1

Motivation and objectives

The main goal of this thesis is to provide mathematical methodologies to model rock properties in the subsurface. When we want to create a subsurface model we generally cannot measure the rock properties we are interested in, at least not exhaustively in space, but we can measure other properties that are related, directly or indirectly, to the properties of interest. If we know the physical relations that link the measured properties (data) with the properties we want to estimate (model ), we can estimate the model from the data using a mathematical tool known as inverse theory. The set of physical relations that relates the model to the data is called forward physical model, and the problem of estimating the model from the data is called inverse problem. Most of the modeling problems in earth sciences and geological engineering are inverse problems. For example in reservoir modeling we generally want to estimate a 3D model of porosity and hydrocarbon saturation, but we have only few wells where we measure these properties directly, however we can measure seismic data that are related to the rock properties through seismic and rock physics models. Similarly, in near surface geophysics we want to estimate the spatial distribution of water saturation but we measure electromagnetic data or resistivity data; in mining engineering we are interested in mineral grades but the only source of information available away from the well locations is seismic. These are all different examples of 1

CHAPTER 1. INTRODUCTION

2

geophysical inverse problems, where we measure some data d, we know the forward model f that links the model m to the data d and we want to estimate the model m as a solution of an inverse problem. Most of the methods presented in this thesis can be applied to different domains, however I will focus on seismic reservoir characterization problems. In seismic reservoir characterization we are generally interested in modeling the porosity of the rocks, the fluid saturations (gas, oil, and water) and lithological information (for example, sand content and clay content). However these properties cannot be directly measured in the subsurface away from the well locations. The main source of information is given by seismic data: traveltime and seismic amplitudes of elastic waves. Seismic data provide information about the elastic contrasts at the interface between adjacent layers in the subsurface: the amplitudes depend on the elastic properties (velocities and densities of the upper and lower layer) and the elastic properties depend on the rock and fluid properties. Estimating reservoir properties given seismic data is clearly a non-trivial inverse problem. This problem is generally split in two sub-problems: in the first step, generally called elastic inversion, we estimate elastic attributes (such as, P- and S-wave velocities, or P- and S-impedances) from seismic data, whereas in the second step, called petrophysical property estimation, we estimate rock properties from elastic attributes. Most part of geophysical research focuses on the first subproblem. Seismic data are very complex to acquire since tools are generally located on the surface and the target can be few kilometers deep in the subsurface. Moreover seismic is also difficult to be processed and interpreted as well, since the data are uncertain and noisy. Elastic attributes obtained as a solution of the first part of the inverse problem, can be used by geologists to qualitatively or quantitatively interpret lithologies and fluids presence in the subsurface. This interpretation is clearly uncertain especially when few well data are available in the area. Physical relations between elastic properties and rock properties are one of the main matters of interest in rock physics. Generally, the forward model is well known, but the solution of the inverse problem is not unique, which calls for probabilistic approaches. By combining the results of elastic inversion and petrophysical property estimation we aim to build 3-dimensional models of the reservoir in


3

terms of rock properties: porosity, lithology, fluid saturation, and possibly effective pressure. These models can be subsequently used by reservoir engineer as input for the reservoir fluid flow simulator. So far in literature, few complete methodologies for solving the corresponding inverse problem have been proposed in order to obtain reservoir models of rock properties from seismic data. The inverse problem is quite complex and in the past reservoir engineers preferred to use statistical relations and simulations to populate their models rather than estimated models obtained as a solution of the inverse problem. The most challenging aspects of the inverse problem are: quality of the seismic data, non-uniqueness of the solution, uncertainty of the measurements, approximations in the physical models, and changes of scale and domain from seismic data to the reservoir model. The solution of an inverse problem should not be limited to a deterministic estimation of the solution which is supposed to be optimal in terms of best-fitting of the given dataset, but it should also include the evaluation of the uncertainty of the inversion results. Uncertainty evaluation necessarily requires a probabilistic approach to inverse problems, in which the inversion results include the most likely model and the associated uncertainty, or in other words the estimation of the posterior probability of the inverted parameters. Furthermore, in seismic reservoir characterization, the number of variables that we want to estimate from seismic data is generally greater than the number of input variables: for example in an oil clastic reservoir we generally want to estimate porosity, clay and sand content, water and hydrocarbon saturation, but we generally have a limited number of angle stacks in the seismic dataset (generally three partial stacks, representing near, mid, and far offset) and the solution could be not unique. In addition to that, seismic data are generally noisy and even in optimistic conditions the signal to noise ratio is quite low. In other words the inverse problem is underdetermined with noisy data, and the deterministic approach is generally not suitable. The natural choice for many geophysical inverse problems is the Bayesian framework, where we combine geological prior knowledge with the information contained in measured data (Tarantola (2005)). In the following chapters we are going to present some methodologies to solve this inverse problem in different situations (exploration stage with few wells or development stage with more


4

wells).

1.2

Chapter description

In Chapter 1, I introduce the inverse problem and review the main contributions in literature. In this chapter I also provide the mathematical background of probability and statistics of the Bayesian approach. In Chapter 2 we present a probabilistic approach to reservoir properties estimation based on Bayes rule and Gaussian mixture models. The method aims to estimate at each location of the seismic grid the posterior probability of reservoir properties conditioned by elastic properties. The Bayesian inversion accounts for different sources of uncertainty: noise, scale and resolution of seismic data, approximations of the physical models, natural variability and heterogeneity of the rocks. The final reservoir model is then obtained by taking a statistical estimator of the posterior probability. In Chapter 3 we introduce a sequential approach for Bayesian linear inverse problems in the Gaussian mixture case. The goal of this method is to provide multiple high-detailed models in terms of reservoir properties. The method is based on the same Bayesian approach presented in Chapter 2, but instead of summarizing the probabilistic information in a single statistical estimator at each grid location, the aim of this method is to sample from the pointwise posterior distributions, or in other words to draw a random sample according to the estimated posterior distribution. Random sampling in a 3-dimensional space is not straightforward: in fact, if we sample independently at two adjacent locations (i.e., we draw a random sample for the first location and then the second random sample for the second location independently of the value we drew at the first location), we may obtain a very high value at the first location and a very low value at the second location, or vice versa. Such independent random sampling ignores the spatial continuity expected in the variations of the property, according to depositional and sedimentological laws. In the sequential Bayesian approach, I account for a spatial statistics model to describe the spatial correlation of the rock properties within the reservoir. In Chapter 4 we propose a stochastic inversion methodology based on the probability perturbation method to obtain high detailed reservoir models that match the


5

input seismic dataset. This method is based on a stochastic optimization technique, therefore the computational cost is generally higher than the methods presented in Chapters 2 and 3, however it allows us to include more complicated spatial statistics models and inter-dependence relations between rock properties. In Chapter 5 we explore a Monte Carlo approach for log evaluation to integrate formation evaluation analysis and rock physics modeling and to quantify the uncertainty in well log data. In Chapters 6 and 7 we extend the Bayesian inversion method proposed in Chapter 2 to time-lapse seismic data to estimate pressure and saturation changes during reservoir production from time-lapse seismic data. In Chapter 6 we introduce a new rock physics model to relate velocity changes to effective pressure changes whereas in Chapter 7 we present the Bayesian inversion workflow. The novelty of this method is the quantitative use of rock physics models in the inversion workflow and the simultaneous inversion of the base survey and the seismic differences (between repeated seismic surveys and base survey). The simultaneous inversion allows us to account for the correlation between rock properties in the static model (pre-production model) and changes in dynamic properties (during production).

1.3

Inversion methods

Seismic signals depend on the elastic properties of the media through which they propagate. Elastic properties of a porous medium depend on intrinsic rock properties (lithology, porosity, etc.), fluid pressures and fluid saturations. Different inversion methodologies have been proposed for reservoir properties characterization to infer rock and fluid properties from seismic signals. Generally, rock properties (porosity for example) are better-resolved than pressures and saturations. Inverse theory is a very wide topic of mathematics which includes analytical and numerical approaches to inverse problems. Many excellent books on geophysical inverse theory have been published, the most related to exploration geophysics being Tarantola (2005) and Oliver (2008). However the final goal of this thesis is not to give an exhaustive overview of the mathematical methods to solve inverse problems, but to describe new methodologies to infer reservoir properties that include characteristics of the


6

rocks (lithology, fluid, porosity). For this purpose, Doyen (2007) is a good reference which contains the description of many mathematical tools used in this thesis. The forward models used in this thesis are extensively described in Mavko et al. (2009) and Avseth et al. (2001), whereas the geostatistical methods are presented in Doyen (2007), Deutsch and Journel (1992), and Goovaerts (1997). There are different methods to solve inverse problems in reservoir characterization including deterministic and probabilistic methods, optimization and sampling techniques. Following the classification proposed by Bosch et al. (2010), the available methods can be divided in two main families: 1) hierarchical or multi-step approaches (e.g., Grana and Della Rossa (2010)), and 2) simultaneous stochastic workflows (e.g., González et al. (2008)). In the hierarchical approach, first seismic data are inverted, deterministically or stochastically, into elastic properties; then rock physics models transform those elastic properties to the reservoir properties of interest. The simultaneous workflow aims to estimate simultaneously elastic parameters and the reservoir properties, guaranteeing consistency between these properties and seismic data. Stochastic sampling methods and geostatistical algorithms allow representing the natural variability and heterogeneity and incorporating subseismic scales of heterogeneities but they generally require higher computational times compared to sequential approaches. In hierarchical inversion, seismic elastic inversion is performed on seismic data to arrive at volumes of P- and S-wave impedances, deterministic or probabilistic (for example Bayesian elastic inversion); then, a rock physics model is established at a well to link the elastic properties to porosity, lithology, and fluid saturation; following these two steps, the rock physics model is used to arrive at the spatial distributions of rock properties from seismic-derived elastic properties; independently, well data are used (via, e.g., cluster analysis) to derive a facies classification based on the reservoir properties rather than elastic properties; finally, this classification is fed into the probabilistic volume of rock properties obtained in the previous step. If a probabilistic approach is applied at each step of the inversion process (Grana and Della Rossa (2010)), the result is a set of volumes of facies probabilities. The natural approach for all these steps is the Bayesian approach in order to provide full posterior


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distributions and quantify the uncertainty. These computations are fast, but the resolution of the resulting most probable facies volume is the same as of the seismic data. Stochastic inversion (also called geostatistical inversion) was introduced in the 90s by Bortoli et al. (1992) and Haas and Dubrule (1994). In stochastic simultaneous inversion, high-resolution models of subsurface properties (facies and rock properties) are generated; then relevant rock physics transforms are applied to these volumes to generate the corresponding volumes of the elastic properties; synthetic seismic traces are computed on these volumes; finally, the so-obtained synthetic seismic is compared to real seismic to evaluate the mismatch and an optimization algorithm (deterministic or stochastic) is applied to determine the most likely reservoir model. This approach provides a high-resolution model of the subsurface (which still is verified by the lowresolution seismic data) but is computationally expensive. Some hybrid methods have been presented by first using the output of the first technique, namely the spatial probability of rock properties, as a secondary information for a high-resolution geostatistical simulation of the facies with spatial variograms a priori derived at the well(s) or anticipated geological occurrences verified on analogous reservoirs. The aim of these methods is to integrate the flexibility of Bayesian inversion of seismic data with advanced geostatistical techniques for detailed reservoir characterization. Traditionally reservoir models are used to describe static reservoir properties such as porosity and lithology and possibly fluid saturations. With the emerge of the concept of time-lapse (or 4D) earth models, we can integrate static and dynamic reservoir properties in the reservoir modeling workflow. These models are driven by the need to integrate time-lapse seismic and reservoir fluid flow simulation predictions in a unique framework to improve reservoir monitoring and production forecast.

1.4

Bayesian approach

The Bayesian approach is a probabilistic method that allows us to combine a priori information about a model and data measurements. Suppose that we are interested

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in some property of a rock that we cannot measure in the subsurface, for example porosity, and that for the same rock we can measure another property, for example P-wave velocity. From geological, sedimentological or depositional information, geologists can formulate hypotheses about the porosity of the rock: for example if the rock is a sandstone, then they can suppose that the porosity is between 10% and 40%. This information is called prior information and it summarizes the prior knowledge that we have before looking at the data. In addition to this information, we can generally measure some properties that are physically related to the property we want to estimate, and we can generally establish a physical model between these two properties. In our example if we can measure velocity and we know the rock physics model for that rock, then we can infer some information about porosity: for instance if the velocity is high, then the porosity will be low since porosity and velocity are anti-correlated. This information is called likelihood function, since it links the data to the model. Both information, the prior information about the rock and the rock physics likelihood function are uncertain. If we can express these two information in probability distributions, Bayes’ rule allows integrating these two information in a single probabilistic information called posterior distribution. Given an event A, for which we have some prior information, and another event B that is somehow related to A, then the posterior probability P (A|B) of the event A given the outcome of the event B is given by: P (A|B) =

P (B|A)P (A) P (B)

(1.1)

where P (A) is the prior probability of A, P (B) is the probability of B, and P (B|A) is the conditional probability of B|A. Intuitively, we can think about Bayes’ rule as a method to reduce the uncertainty of our prior information when new relevant data are available. Suppose for example that we know that a rock sample is a sandstone, and from geological information we know that its porosity is generally between 10% and 40%. Velocity measurements provide additional data to improve the estimation of porosity. Suppose that in our example we measured a relative high velocity, which means that the rock is quite stiff and as a consequence porosity is quite low. These qualitative relations can be


9

translated into physical-mathematical models which are used to build the likelihood function (the conditional probability P (B|A)). The product of the two probabilities allows us to reduce the uncertainty in the prior information. A nice feature of the Bayesian approach is that when the prior distribution and the likelihood function are Gaussian, then the posterior distribution is again Gaussian, and the solution of the inverse problem can be analytically derived (see Chapter 3).

Chapter 2 Probabilistic reservoir properties estimation 2.1

Abstract

We propose a joint estimation of petrophysical properties which combines statistical rock physics and Bayesian seismic inversion. Since elastic attributes are correlated with petrophysical variables (effective porosity, clay content, and water saturation) and this physical link is associated with uncertainties, then the petrophysical properties estimation from seismic data can be seen as a Bayesian inversion problem. The purpose of this work is to present a strategy for estimating the probability distributions of petrophysical parameters and litho-fluid classes from seismic. The estimation of reservoir properties and the associated uncertainty is performed in three steps: linearized seismic inversion to estimate the probabilities of elastic parameters, probabilistic upscaling to include the scale changes effect, and petrophysical inversion to estimate the probabilities of petrophysical variables and litho-fluid classes. Rock physics equations provide the link between reservoir properties and velocities, while linearized seismic modeling connects velocities and density to seismic amplitude. We adopt a full Bayesian approach to propagate uncertainty from seismic to petrophysics in an integrated framework which takes into account different sources of uncertainty:

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CHAPTER 2. PROBABILISTIC RESERVOIR PROPERTIES ESTIMATION

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the heterogeneity of the real data, the approximation of physical models, measurement errors and scale changes. The methodology has been tested, as a feasibility step, on real well data and synthetic seismic to show the reliable propagation of the uncertainty through the three different steps and to compare two different statistical approaches, parametric and non parametric. The application to a real reservoir study, including two wells data and partially stacked seismic volumes, has provided as main result the probability densities of petrophysical properties and litho-fluid classes and it demonstrates the applicability of the proposed inversion methodology.

2.2

Introduction

In reservoir characterization studies constrained by seismic data, statistical rock physics is normally used to combine statistical techniques with physical equations to generate different petroelastic scenarios. The goal of statistical rock physics is to predict the probability of petrophysical variables when velocities (or impedances) and density are assigned, and to capture the heterogeneity and complexity of the rocks and the uncertainty associated with theoretical relations. The use of statistics in rock physics is becoming more and more frequent: in the typical statistical rock physics workflow (Avseth et al. (2005) and Doyen (2007)), deterministic models are firstly established in order to build physical relations between elastic properties and reservoir attributes; then, probabilistic petroelastic transformations are determined, combining these relations with Monte Carlo simulations, to include the uncertainty associated to the real data (measurement errors and natural heterogeneity of the rocks) and to the degree of accuracy of the model itself. The traditional Bayesian framework (Tarantola (2005)) used for uncertainty evaluation in elastic inversion (Buland and Omre (2003)) has recently been adopted for problems of litho-fluid prediction from seismic data (Larsen et al. (2006), Gunning and Glinsky (2007), and Buland et al. (2008)). Statistical rock physics has been introduced in Mukerji et al. (2001a) and Eidsvik et al. (2004a) to estimate reservoir parameters from prestack seismic data and to evaluate the associated uncertainty. Petrophysical seismic inversion methods based


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on statistical relations between elastic and rock properties have been presented in Bornard et al. (2005) and Coleou et al. (2005). Subsequently, stochastic rock physics models have been used in Bachrach (2006) for a joint estimation of porosity and saturation and in Sengupta and Bachrach (2007) for pay volume uncertainty evaluation, while Spikes et al. (2008) developed a probabilistic seismic inversion to constrain reservoir properties estimation with well data and seismic. To infer litho-fluid classification from seismic data, Larsen et al. (2006) proposed an integrated litho-fluid inversion methodology based on a Markov chain model, while Gallop (2006) presented an approach based on mixture distributions for facies estimations. Geostatistical methods in seismic inversion have been introduced in Doyen (1988), Bortoli et al. (1992), and Haas and Dubrule (1994) and good reviews can be found in Dubrule (2003) and Doyen (2007). New approaches integrating advanced geostatistical techniques have recently been proposed in González et al. (2008), and Bosch et al. (2009). We present a method to integrate statistical rock physics and Bayesian elastic inversion to compute the probability distributions of the petrophysical properties. Similar approaches have already been presented, with some assumptions and limitations about the form of the probability distributions, the size of the data and the type of dependencies considered. By means of more general parametric distributions, such as Gaussian Mixture Models (GMMs) (Hastie and Tibshirani (1996)), or non parametric statistical techniques such as Kernel Density Estimation (KDE) (Silverman (1986)), some limitations can be overcome by our approach. In addition we take into account the upscaling problem (Lake and Srinivasan (2004)), in order to face the limited resolution and the greater uncertainty of seismic data compared to well log data and we integrate this step within the probabilistic inversion framework. The workflow we propose (Figure 2.1) can be summarized as follows: 1. Rock physics model calibration: a rock physics model is established using well log data to predict elastic attributes (velocities or impedances) from petrophysical properties. 2. Linearized Bayesian seismic inversion: we estimate elastic properties from partially stacked seismic angle gathers. 3. Conditional probabilities estimation: we calculate the conditional probabilities of


13

petrophysical variables and litho-fluid classes in a multiproperties, multiscale model. The methodology includes the upscaling effect within an integrated probabilistic approach.

Prestack seismic data

Well data & Rock Physics Elastic attributes m (Ip, Is)

Conditional estimations

Petrophysics R (ĭ, C, sw)

Litho fluid class

Figure 2.1: Flowchart of the probabilistic petrophysical properties estimation. The rock physics model is a set of equations which transforms petrophysical variables in elastic attributes. The rock physics model type depends on the reservoir rocks we deal with: the set of equations can be a simple regression on well data or a more complex physical model (Mavko et al. (2009)). Once the rock physics model has been calibrated on well logs, we can apply the model to situations not sampled by log data and generate different scenarios by means of Monte Carlo simulations. This approach is used to explore, for example, all possible ranges of porosity, saturation and clay content and to simulate the corresponding acoustic and elastic responses. The proposed methodology propagates the uncertainty from seismic data to petrophysics combining three conditional probabilities. The first one is the probability of elastic properties given seismic data obtained by a Bayesian approach to elastic inversion (Buland and Omre (2003)). The second one is the probability of elastic attributes at fine scale (high resolution) when the coarse scale values are known. The third one is the probability of petrophysical properties conditioned by elastic attributes obtained


14

by integrating the rock physics model equations with Monte Carlo simulations and generating different geological scenarios. We applied the methodology to a clastic reservoir in the North Sea, where two wells and four partial stacked seismic angle gathers are available. As a feasibility step we tested the methodology on a calibration well using a synthetic seismic trace and we compared two different approaches: GMMs and KDE. We then applied the methodology to the whole seismic volume and we obtained trace-by-trace the probability density functions of the petrophysical variables and litho-fluid classes.

2.3

Probabilistic approach to petrophysical inversion

The target of this section is to illustrate the probabilistic formulation of the joint petrophysical inversion. We describe the derivation of the posterior probabilities of petrophysical properties and litho-fluid classes, conditioned by seismic, using different attributes (elastic, petrophysical, and categorical) and integrating data coming from different sources (high resolution well data, coarse resolution data and seismic data). The methodology is divided in three steps: 1) statistical rock physics modeling, 2) upscaling and 3) petrophysical inversion from seismic data. In the following we will use m to indicate the acoustic and elastic properties, typically impedances Ip and Is (m = [Ip Is ]T ), and R to represent the petrophysical data, typically effective porosity, water saturation and clay content (R = [φ sw C]T ).

2.3.1

Statistical rock physics modeling

One of the important aspects of statistical rock physics is that it combines physical models with statistics to account for situations not seen in the well data. If all the variables are considered as random vectors, the rock physics model can be written as: m = fRP M (R) + ε

(2.1)


15

where fRP M represents the rock physics model and ε is the random error that describes the degree of accuracy of the model. For the prior distribution, which is the same at any depth, we assume a multivariate Gaussian Mixture (GM), a linear combination of Gaussian distributions, with a fixed number of components Nc : P (R) =

Nc X

αk N(R; µkR , ΣkR ),

(2.2)

k=1

where N indicates the multi-Gaussian distribution of the vector R with mean µkR and covariance matrix ΣkR for all k = 1, ..., Nc and αk are the weights of the linear P c combination (with N k=1 αk = 1).

This choice is motivated by two reasons. The first one is that this formulation

allows us to model each litho-fluid class detectable from a petrophysical point of view as a single Gaussian component of the mixture. The second one is the analytical convenience of this approach since the analytical results valid for Gaussian distributions can be extended also to Gaussian Mixtures. The number of components initially used in our tests is three, because the litho-fluid classification we consider consists of shale, oil sand and water sand. The approach we propose for the estimation of the conditional probability P (R|m) is semi-analytical: we generate a set of Ns samples from the prior distribution P (R), we apply the rock physics model fRP M , we estimate the joint distribution assuming a Gaussian Mixture distribution and we analytically derive the conditional distribution. From the prior distribution, different scenarios can be generated by Monte Carlo simulations: the petrophysical variables can be sampled from the prior distribution and the elastic response can be computed by the rock physics model fRP M ; in the pseudo logs generation, we model the vertical correlation of petrophysical properties by means of a vertical variogram in order to obtain pseudo logs of elastic variables with a realistic vertical correlation and subsequently perform upscaling on elastic variables, in the probabilistic upscaling step. We also made the additional assumption that the error ε in Eq. 2.1 is Gaussian with zero mean and covariance Σε that can be estimated from well log data. With

16


this assumption we can state that P (m|R) = N(m; µ(R), Σε )

(2.3)

where the mean µ(R) = fRP M (R) and the covariance matrix Σε can be assumed independent of R and related only to ε . This model allows us to account for the uncertainty associated to the rock physics model predictions by means of Monte Carlo simulations and conditional probabilities estimations. In fact we can generate a set of Ns samples {Ri }i=1,...,Ns from the petrophysical prior P (R); then we compute the response of the rock physics model

µ(Ri ) = fRP M (Ri ), for all i = 1, ..., Ns and we generate Ns samples {mi }i=1,...,Ns from

the normal distributions N(m; µ(Ri ), Σ).

The joint distribution P (m, R) can be estimated from the Ns samples If the joint distribution is a Gaussian Mixture P (m, R) =

Nc X

πk N([m, R]T ; µk[m,R] , Σk[m,R]),

"

mi Ri

#

.

i=1,...,Ns

(2.4)

k=1

then the conditional distribution P (R|m) is again a Gaussian Mixture. If the rock physics model fRP M was linear, the joint distribution could be analytically derived from the prior; but in general fRP M is not linear and the joint distribution P (m, R) can be obtained from the Monte Carlo samples. The technique adopted to estimate the parameters of the Gaussian components and the weights of the mixture is Expectation-Maximization (EM) algorithm (Hastie and Tibshirani (1996)). We point out that these weights can be interpreted as the indicator probability of the discrete random variable that represents the litho-fluid class. As a consequence, the conditional distribution P (R|m) is a Gaussian Mixture P (R|m) =

Nc X

λk N(m; µkR|m , ΣkR|m )

(2.5)

k=1

and we can analytically compute its parameters; in particular the means and the covariance matrices of the mixture components are given by −1 m − µkm µkR|m = µkR + ΣkR,m Σkm,m

(2.6)


ΣkR|m = ΣkR,R − ΣkR,m Σkm,m

−1

Σkm,R

17

(2.7)

for each given m. The assumption that both petrophysical and elastic variables are distributed as a Gaussian Mixture is compatible with the hypothesis of GM distribution for the prior model and it is reasonable if the rock physics model is not too far from linear. However, if these assumptions are not in agreement with well log data, a non parametric approach for the conditional probability estimation P (R|m) should be adopted. In this case, we propose to estimate the joint distribution P (m, R) by applying Kernel density estimation on the Monte Carlo samples in a multidimensional domain. Kernel density estimation is a non parametric technique that allows us to estimate the probability distribution by fitting a base function, the kernel function, at each data point including only those observations close to it. The joint probability can be expressed as the sum of the contributions of the same kernel function centered at each data point location (see Silverman (1986)); for example in 2D, if m = [Ip ] and R = [φ]: Ns X Ip − Ipi 1 φ − φi P (m, R) = P (Ip , φ) = K K . Ns hp hφ i=1 hp hφ

(2.8)

where K is the kernel function, Ipi , φi i=1,...,N are the data samples, and hp and hφ s

are the scaling lengths (also called kernel widths). The kernel function K is a nonnegative symmetric function; in this work we used Epanechnikov kernel as in Doyen (2007): K(x) =

(

3 (1 4

0

− x2 ) x ∈ [−1, 1] otherwise.

(2.9)

The scaling lengths, for each variable, control how far we incorporate observations close to data points and they have to be assessed using training data. In the current workflow, we estimate the joint distribution by KDE on a multidimensional grid, then the conditional distribution P (R|m) can be numerically evaluated by definition: P (R|m) = R

P (m, R) , P (m, R)dR

(2.10)


18

that corresponds to a normalization of the joint probability P (m, R) for each given m. Statistical formulation We recall here the analytical results for conditional distributions of Gaussian Mixtures, extending the results valid in the Gaussian case. If the joint distribution is a Gaussian Mixture of Nc components, we indicate the joint probability as P (m, R) =

Nc X

πk N(y; µky , Σky )

(2.11)

k=1

with mean and covariance of each component given by " # " # µkm Σkm,m Σkm,R k k µy = , Σy = ; µkR ΣkR,m ΣkR,R where y =

"

m

(2.12)

#

. R Then the conditional distribution P (R|m) is again a Gaussian Mixture (see for

example Dovera and Della Rossa (2011)) P (R|m) =

Nc X

λk N(R; µkR|m , ΣkR|m )

(2.13)

k=1

where λk are the weights of the conditional distribution: πk N(m; µkm , Σkm ) λk (m) = PNc , l , Σl ) π N(m; µ l m m l=1

(2.14)

and the mean and the covariance of each component of the conditional distribution can be analytically derived as follows: µkR|m = µkR + ΣkR,m Σkm,m

−1

ΣkR|m = ΣkR,R − ΣkR,m Σkm,m for each given m.

m − µkm −1

Σkm,R ,

(2.15) (2.16)


2.3.2

19

Probabilistic upscaling

The described methodology does not explicitly consider the difference in scale and domain of the available data due to different sources of information. In fact the typical domain of a rock physics model is depth with the resolution of well logs, whereas the inverted seismic attributes are obtained in time domain with a lower resolution. The objective of this section is to define a step in the methodology that accounts for these differences and that is consistent with the general probabilistic approach we are proposing. In general there are two main issues to take into account in the scale reconciling problems: the computation of physically equivalent measures at different scales and the correct propagation of the uncertainty from one scale to another. The upscaling, both of petrophysical and elastic properties, is complicated by the presence of spatial and vertical correlation in the heterogeneities distributions (see for example Lake and Srinivasan (2004)), but in agreement with the choice of our petrophysical inversion model we concentrate on the problem of coherently transforming different measures from one resolution to another and of estimating the corresponding changes in probability distribution. The approach we adopted to face the first problem for elastic properties is Backus averaging (Backus (1962)), while we tackle the second issue by estimating the conditional distribution of elastic parameters at high resolution scale (fine scale) given the corresponding data at low resolution scale (coarse scale). The starting point for the probabilistic scale change is the rock physics model with the associated uncertainty defined as in Eq. 2.1. If mf represents the fine scale (log scale) vector of the elastic parameters and mc the corresponding coarse scale (seismic) data, we indicate the change of scale with mc = g(mf ), where the function g represents Backus averaging for velocities and linear average for density. Backus upscaling is used to determine the effective transversely isotropic elastic constants of layered media for long wavelength seismic waves. In order to integrate the upscaling problem in our probabilistic framework we propose the following scheme. If mf is conditionally distributed with probability P (mf |R), the problem of the estimation of the distribution conditioned by mc can


n o be solved by generating a set of Ns samples mfi

i=1,...,Ns

20

according to P (mf |R) by

Monte Carlo simulation, andnapplying o the upscaling transformation g, so that we f c obtain a set of joint samples (mi , mi ) that can be used to estimate, with i=1,...,Ns

Gaussian models for example, the conditional distribution P (mf |mc ).

The conditional distribution of the petrophysical parameters given coarse scale

elastic data P (R|mc ) can be obtained combining P (R|mf ) with P (mf |mc ) by means of Chapman-Kolmogorov equation (Papoulis (1984)): Z c P (R|m ) = P (R|mf )P (mf |mc )dmf ,

(2.17)

Rn

where n is the dimension of mf (n = 2 in the case of Ip and Is ). This expression is the probabilistic model that includes the uncertainties due to rock physics (P (R|mf )) and to upscaling (P (mf |mc )).

2.3.3

From seismic to petrophysics inversion

In order to obtain the posterior distribution of elastic parameters from seismic, we use a linearized AVO inversion technique, in a Bayesian framework. The inversion method adopted here assumes an isotropic and elastic medium and it combines the convolutional model with Aki-Richards linearized approximation of Zoeppritz equations valid for vertical weak contrasts, as in Buland and Omre (2003). If S refers to seismic data, the elastic model can be expressed as: S = Gℓ + e,

(2.18)

where G is the forward linearized operator including both convolution and weak contrasts Aki-Richards approximation, ℓ is the vector of the logarithms of the whole trace of the elastic parameters and e is a Gaussian error term with zero mean and covariance Σe . We also assume that ℓ is distributed according to a multivariate Gaussian prior ℓ ∼ N(ℓ; µℓ , Σℓ ).

Under these hypothesis it can be shown (Buland and Omre (2003)) that P (ℓ|S)

is again Gaussian: P (ℓ|S) = N(ℓ; µℓ|S , Σℓ|S )

(2.19)


21

where µℓ|S = µℓ + ΣTS,ℓ (ΣS )−1 (S − µS )

Σℓ|S = Σℓ − ΣTS,ℓ (ΣS )−1 ΣS,ℓ

(2.20) (2.21)

with µS = Gµℓ , ΣS = GΣℓ GT +Σe and ΣS,ℓ = GΣℓ is the cross-covariance between the vector of parameters ℓ and seismic data S. Once the conditional distribution P (ℓ|S) is known, also the lognormal distribution at each vertical position z, P (mc |Sz ), can be derived (assuming a depth conversion

of seismic inversion results). The final step to compute the probability of the petrophysical variables conditioned by seismic data P (R|Sz ) including the upscaling effect (Eq. 2.17), can be written as: P (R|Sz ) =

Z

Rn

P (R|mc)P (mc |Sz )dmc .

(2.22)

By means of Eq. 2.22 we finally obtain the posterior probabilities of the petrophysical properties. We point out that even though Backus averaging is applied to upscale elastic properties and estimate elastic properties at coarse scale, anisotropic effects are not accounted for in the proposed seismic inversion method. We can also introduce a further step to apply the same methodology in the discrete domain, in litho-fluid classes classification studies for example. Formally we compute the probability P (πz |Sz ) =

Z

P (πz |R)P (R|Sz )dR

Rn

(2.23)

where πz is the generic litho-fluid class at vertical position z, n is the dimension of R (n = 3, if R = [φ sw C]T ) and P (πz |R) is the rock physics likelihood function.

In order to generate different realizations of litho-fluid classes conditioned by

seismic, including vertical correlation to model the vertical continuity of litho-fluid classes, we can combine the posterior probability (Eq. 2.23) with a Markov chain prior model (as in Larsen et al. (2006)): Z Z P (πz |Sz ) = P (πz |R)P (R|Sz )dR ∝ Rn

∝

P (R|πz )P (πz )P (R|Sz )dR

Rn

Y z

P (πz |πz−1 )

Z

Rn

P (R|πz )P (R|Sz )dR

(2.24)


22

where z indicates depth and the probability P (πz |πz−1 ) can be obtained from the

downward Markov chain transition matrix of litho-fluid classes estimated on actual well log classification (with the assumption that P (πz1 ) = P (πz1 |πz0 ) for notational convenience).

2.4 2.4.1

Real case application Methodology implementation

The methodology application is described for an oil saturated clastic reservoir but it can be adapted to different saturation and lithology reservoir conditions, with the choice of a suitable rock physics model. First of all a rock physics model is calibrated at well location using velocity logs and petrophysical curves obtained in formation evaluation analysis. The rock physics model can be written in the following formulation [Vp , Vs , ρ] = fRP M (φ, sw, C) + ε,

(2.25)

where Vp and Vs are respectively P and S-waves velocities, ρ is the density, φ is the effective porosity, sw is the water saturation, C is the clay content and ε is the error that represents the difference between model predictions and real data. The function fRP M can be an empirical relation or a theoretical set of equations such as granular media models or effective media models (see Mavko et al. (2009)). Secondly we estimate elastic attributes from seismic data: we use a reformulation of the approximation of Zoeppritz equations by Aki-Richards (Aki and Richards (1980)) in terms of impedances, and we jointly estimate P and S-impedances and density following the Bayesian approach presented in Buland and Omre (2003). In terms of impedances the reflection coefficient RP P as a function of the reflection angle θ becomes: ! 2 ¯s 2 1 ∆Ip ∆I I ∆ρ 1 I¯s 1 s RP P (θ) ∼ − + 2 2 sin2 θ (2.26) − 4 2 sin2 θ ¯ + 2 2 ¯ 2 cos θ Ip 2 2 cos θ ρ¯ Is I¯p I¯p where I¯p , I¯s , and ρ¯ are respectively the averages of impedances and density over the reflecting interface; and ∆Ip , ∆Is , and ∆ρ are the corresponding contrasts. With


23

realistic noise levels, the inversion cannot retrieve reliable information about density (as in Buland and Omre (2003)), for this reason in our real case application we do not use density in the petrophysical inversion workflow. Finally we calculate the conditional probabilities of petrophysical variables and litho-fluid classes conditioned by seismic following the methodology described in the theory section: a).

We assume a prior distribution of the petrophysical variables: in our case

P (φ, sw, C) is assumed as a trivariate GMMs to take into account the observed correlations between variables in each litho-fluid class; in this case the prior is the same at any vertical position. b). We generate pseudo logs of petrophysical properties from the prior distribution with a realistic vertical correlation, in two steps. We firstly create litho-fluid classes profiles, for example using a first order Markov chain downward model (Larsen et al. (2006)), and then we generate in each litho-fluid class petrophysical properties vertically correlated using a variogram estimated on well data. c). We apply the rock physics model fRP M to the petrophysical pseudo logs to obtain the corresponding elastic attributes and we add a random error ε (Eq. 2.25); then we compute fine scale impedances. d). Using the random samples generated in step b) and c), we estimate the joint probability P (Ipf , Isf , φ, sw, C) and we derive the conditional probability of petrophysical properties conditioned by impedances P (φ, sw, C|Ipf , Isf ) at fine scale. e). We upscale the elastic properties applying sequential Backus averaging on a running window whose length is found by estimating the wavelength from the seismic bandwidth and the average velocity. We then compute the conditional probabilities at coarse scale: P (φ, sw, C|Ipc, Isc )

=

Z

R2

P (φ, sw, C|Ipf , Isf )P (Ipf , Isf |Ipc , Isc )dIpf dIsf .

(2.27)

f). This last conditional probability is then combined by means of Chapman-Kolmogorov


24

equation (Papoulis (1984)) with the probability of elastic properties coming from linearized Bayesian inversion P (Ipc , Isc |Sz ), to obtain the posterior probability of petrophysical properties:

P (φ, sw, C |Sz ) =

Z

R2

P (φ, sw, C|Ipc, Isc )P (Ipc , Isc |Sz )dIpc dIsc .

(2.28)

at each vertical position z. g). Finally we estimate the probability of litho-fluid classes conditioned by seismic: Z P (πz |Sz ) = P (πz |φ, sw, C)P (φ, sw, C |Sz )dφdswdC (2.29) R3

and we integrate it with a Markov chain model by means of Eq. 2.24. In the case of Gaussian Mixture assumption, the joint distribution P (Ipf , Isf , φ, sw, C) is estimated using Expectation-Maximization algorithm (if the rock physics model was linear, the joint distribution could be analytically derived from the prior). EM is an iterative algorithm which allows us to find maximum likelihood estimates of parameters in probabilistic models in the presence of missing data. EM is a two steps method: Expectation step computes an expectation of the log-likelihood respect to the current estimate of the distribution, Maximization step maximizes the expected log-likelihood found in the previous step. The algorithm converges to the optimal solution in a number of steps which depends on different factors, such as the distribution shape and data dimensions (see Hastie and Tibshirani (1996)). It is used here to estimate the parameters (means and covariance matrices) and the weights of the Gaussian components of the mixture for the joint distribution in the case of multimodality of the data. Once the weights and the parameters of the joint distribution are known, the conditional distribution is analytically derived using Eqs. 2.6 and 2.7. In alternative we propose to use Kernel density estimation which allows us to estimate on a multidimensional grid a probability density function. In this case we apply Kernel density estimation to estimate the joint distribution P (Ipf , Isf , φ, sw, C), extending Eq. 2.8 in 5D domain. In our implementation we use the same kernel function, Epanechnikov kernel, for the five different variables and a specific scaling length for each variable. The critical point of this approach is the calibration of the scaling lengths: the higher they are, the farther from the data points are the


25

observations included in the distribution. The choice of the scaling lengths depends on the number of data points and the spread of the distribution (Doyen (2007)). Once the joint distribution is estimated, we compute the conditional distribution at fine scale (Eq. 2.10) by normalizing the joint distribution at each given (Ipf , Isf ). In both cases, parametric and non parametric, the following steps are similarly performed: a Gaussian model is assumed for the upscaling step, while a lognormal distribution is used for seismic inversion. These two probabilities are combined with fine scale probabilities by means of Chapman-Kolmogorov equation (Eqs. 2.27 and 2.28).

2060

2080

2100

Depth (m)

2120

2140

2160

2180

2200

2220

2240 5000 9000 13000 3000 5000 7000 Ip (m/s g/cm3)

Is (m/s g/cm3)

0 0.2 0.4 Effective porosity

0 0.4 0.8 Clay content

0 0.5 1 Litho−fluid classes Water saturation

Figure 2.2: Petrophysical curves derived from log interpretation of well A. From left to right: P-impedance (well log in blue and rock physics model predictions in red), S-impedance (well log in blue and rock physics model predictions in red), effective porosity, clay content, water saturation and litho-fluid classification (oil sand in yellow, water sand in brown, shale in green)


2.4.2

26

Data application

The methodology has been applied to an oil saturated clastic reservoir in the North Sea, using angle-stack seismic data and well log data coming from two wells of the field: well A (relative coordinates: x=1530, y=450) used for model calibration and well B (x=120, y=1000) used for methodology validation. The input data of the rock physics model are the petrophysical curves obtained in formation evaluation analysis: effective porosity, clay content and water saturation. We focus our attention on a specific reservoir level (Figure 2.2), where we can identify three litho-fluid classes: Oil Sand, Water Sand and Shale. The litho-facies have been obtained by means of log-facies classification, on the basis of the petrophysical curves and the available sedimentological information. The litho-fluid discrimination still remains visible both in petro-elastic domain and in impedances domain (Figure 2.3). The adopted rock physics model is the stiff sand model based on Hertz-Mindlin contact theory. The model was previously calibrated on well A (Figure 2.4) and then used on well B data; for the calibration, we performed a fluid substitution on velocities of well A to obtain the corresponding velocities in wet conditions and we determined the model parameters in order to obtain a good match with well data. The critical porosity used is 0.4 and the coordination number is 7, while the effective pressure in the reservoir is 70 MPa. For the solid phase we used a matrix model made by two components: sand (mostly made of quartz) and wet clay (mostly made of illite), with the parameters indicated in Table 2.1. The matrix parameters have been selected on the basis of the available mineralogical information about the rock composition and for the good match between model predictions and well log data (Figure 2.2). The choice of considering clay as a mixture of mineral and clay bound water is coherent with the choice of using effective porosity. The stiff sand model was selected on the basis of the available geological information and because it is appropriate to describe a well consolidated sand. In shale the effective porosity is near to zero, so that the rock physics model reduces to the computation of velocities and density of a matrix made of wet clay, by means of Voigt-Reuss-Hill average and we obtain a good approximation of the velocities in shale.


27

shale water sand oil sand

12000

P−impedance (m/s g/cm3)

11000 10000 9000 8000 7000 6000 0

0.05

0.1

0.15 0.2 Effective porosity

0.25

0.3

0.35

7000

S−impedance (m/s g/cm3)

6500 6000 5500 5000 4500 4000 3500 3000 6000

7000

8000

9000

10000

11000

12000

P−impedance (m/s g/cm3)

Figure 2.3: Well log data distribution and litho-fluid classification of well A. (top) P-impedance versus effective porosity color coded by litho-fluid class. (bottom) Simpedance versus P-impedance color coded by litho-fluid class (oil sand in yellow, water sand in brown, shale in green). We describe here the implementation of the inversion methodology and its application to the data (we recall that R = [φ sw C]T and m = [Ip Is ]T ). We assume that the prior distribution P (R) is a Gaussian Mixture (Figure 2.5) which weights are the actual proportions of litho-fluid classes. In particular we assume a Gaussian Mixture distribution with truncations for φ and C, while a Gaussian Mixture Score transformation (extension of the Normal Score transformation) is applied to water saturation sw. The simulation and the inversion are conducted from the Gaussian Mixture scores, and at the end of the simulation the results are back-transformed to recover the actual saturation values. If we assume a large variability within each


wet clay sand

ρ (g/cm3 ) 2.5 2.7

K (GP a) 20 33

28

µ (GP a) 8 36

Table 2.1: Rock physics model parameters: density ρ, bulk modulus K and shear modulus µ of the matrix components. litho-fluid class (large covariance matrices) we can also generate samples that are not present in well data (Figure 2.5); the advantage of this assumption is that it allows us to simulate the petroelastic properties of different scenarios. Clay content

5500

0.7

C=0.1 5000

0.6 4500 C=0.3

Vp (m/s)

0.5

C=0.5

4000

0.4

C=0.7

0.3

3500 C=0.9

0.2

3000 0.1

2500

0

0.05

0.1


0.25

0.3

0.35

Figure 2.4: Rock physics model calibration on well A: velocity at wet conditions (obtained performing a fluid substitution on well log velocities) versus effective porosity, color coded by clay content. The curves are from the stiff-sand model for a mixture of wet clay and sand, with clay content equal to (from bottom to top) 0.9, 0.7, 0.5, 0.3 and 0.1. We then generate pseudo petrophysical curves in two steps. We firstly generate litho-fluid classes profiles by means of a first order Markov chain downward model using two transition matrices P1 and P2 honoring well A proportions and transition


0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

x 10

29

−3

Clay content

Clay content

2.5

0.4 0.3

2

1.5 0.4 1

0.3

0.2

0.2

0.1

0.1

0.5

0 0.06

0 0.04 0.02 Probability

0

0.05

0.1

0.15 0.2 0.25 Effective porosity

0.3

0.35

0.4 Probability

0.05

0.1

0.15 0.2 0.25 Effective porosity

0.3

0.35

0.4

Probability

0 0.02 0.04 0.06

−3

x 10 3.5

1

1

0.9

0.9

0.8

0.8

0.7

0.7

2.5

0.6

2

Water saturation

Water saturation

3

0.6 0.5 0.4

0.5 1.5 0.4

0.3

0.3

1

0.2

0.2

0.5

0.1

0.1

0.06

0.04 0.02 Probability

0

0

0.1

0.2

0.3

0.4 0.5 Clay content

0.6

0.7

0.8 Probability

0

0.1

0.2

0.3


0.6

0.7

0.8

Probability

0 0.02 0.04 0.06

Figure 2.5: Prior distribution of petrophysical variables. (top) 2D marginal distribution of effective porosity and clay content with the associated 1D marginal distributions. (bottom) 2D marginal distribution of clay content and water saturation with the associated 1D marginal distributions. The black crosses represent petrophysical data of well A, the background color is the joint probability.


30

probabilities respectively above and under the oil-water contact (at 2182 m):     0.95 0.03 0.02 0.95 0.03 0.02      P2 =  0.06 0.94 0  (2.30) P1 =  0.20 0.80 0     0 0.19 0.81 0.01 0.01 0.98 where rows correspond to shale, water sand and oil sand at generic depth z, while columns refer to shale, water sand and oil sand at depth z−1 (the downward transition from water sand to oil sand is impossible in both cases). If the well information is not representative of reservoir conditions, litho-fluid classes proportions and as a consequence the transition matrices can be modified. Then, for each litho-fluid class, we can estimate a variogram to model the vertical correlation of the petrophysical properties (the results for porosity are presented in Figure 2.6, as an example), and we generate pseudo logs of petrophysical properties from the prior with a realistic vertical correlation. −3

−3

Shale

x 10

5

Water Sand

x 10

5

4.5

4

4

4

3.5

3.5

3.5

3 2.5 2 1.5

3 2.5 2 1.5

2 1.5

1

1

0.5

0.5

0 2

4 Distance (m)

6

8

Oil Sand

3

1

0

−3

2.5

0.5 0

x 10

4.5

Variogram of porosity

4.5



5

0 0

2

4 Distance (m)

6

8

0

2

4 Distance (m)

6

8

Figure 2.6: Experimental and fitted variograms of effective porosity for the three litho-fluid classes. From left to right: variogram of effective porosity in shale, water sand and oil sand (color lines are the experimental variograms and dotted lines are the fitted variograms). We then apply the rock physics model (fRP M ) to obtain the corresponding pseudo logs of velocities and impedances. The error ε (see Eq. 2.1) added to the elastic variables, Ip and Is , computed with the rock physics model, is distributed as a bivariate Gaussian distribution, and its parameters (the elements of covariance matrix) are estimated from the difference between real data and rock physics model predictions


31

on well A logs (σp = 530 and σs = 280). Using the pseudo logs generated by means of the rock physics model we perform the upscaling using sequential Backus averaging in a running window of about 12.5 meters (the estimated wavelength is 125 m, the operator length is obtained as wavelength/10 as in Avseth et al. (2005)) and we estimate the conditional probabilities at coarse scale (Eq. 2.27). Finally the linearized AVO inversion technique is used to jointly estimate the posterior distribution of P and S-impedance and density. We estimated the four wavelets independently for each available angle gather, while the trend for the prior model has been obtained from well logs by filtering impedances logs to a high-cut value of 4 Hz and by interpolating these logs along the interpreted horizons. The probabilistic inversion approach is based on the convolutional model and Aki-Richard linearized approximation of Zoeppritz equations in the limit of vertical weak contrasts. The elastic parameters derived from seismic inversion are characterized by a logGaussian random field. The posterior probabilities of petrophysical properties and litho-fluid classes are obtained by means of Eqs. 2.28 and 2.29 and the results are shown in the next section. Stiff sand model The stiff sand model is based on Hertz Mindlin grain contact theory (see Mavko et al. (2009)). This model provides estimations for the bulk (KHM ) and shear moduli (µHM ) of a dry rock assuming that the sand frame is a dense random pack of identical spherical grains subject to an effective pressure P with a given porosity φ0 and an average number of contacts per grain n (coordination number): s n2 (1 − φ0 )2 µ2mat P KHM = 3 18π 2 (1 − ν)2 µHM

5 − 4ν = 10 − 5ν

s 3

3n2 (1 − φ0 )2 µ2mat P 2π 2 (1 − ν)2

(2.31)

(2.32)

where ν is the grain Poisson’s ratio and µmat is the matrix shear modulus. The matrix elastic moduli are obtained by Voigt-Reuss-Hill averages for a matrix


32

made of two components, wet clay (mixture of clay and clay bound water) and sand: ! 1−φ 1 CKc + (1 − C)Ks + C (2.33) Kmat = 1−C 2 1−φ + Kc Ks ! 1 Cµc + (1 − C)µs 1−φ µmat = + C (2.34) 2 1−φ + 1−C µc µs

where C is the volume of wet clay, φ is the effective porosity, Kc , µc , Ks , and µs are

respectively the bulk and shear moduli of wet clay and sand. For effective porosity values between zero and the critical porosity φ0 , this model connects the solid phase elastic moduli Kmat and µmat respectively with the elastic moduli KHM and µHM of the dry rock at porosity φ0 , by interpolating these two end members at the intermediate effective porosity values by means of the modified Hashin-Strikman upper bound: −1 φ/φ0 1 − φ/φ0 4 Kdry = (2.35) − − µmat 4 4 3 KHM + 3 µmat Kmat + 3 µmat −1 1 − φ/φ0 φ/φ0 1 − µdry = − ξµmat , (2.36) 1 1 6 µHM + 6 ξµmat µmat + 6 ξµmat 9Kmat + 8µmat . ξ= Kmat + 2µmat Gassmann’s equations are used for calculating the effect of fluid on velocities using the matrix and the fluid properties (see Dvorkin et al. (2007) for the use of effective porosity in Gassmann): Ksat = Kdry +

1−

φ Kf l

+

Kdry Kmat

1−φ Kmat

µsat = µdry .

2

−

Kdry 2 Kmat

(2.37) (2.38)

From the rock saturated elastic moduli we finally obtain velocities as s Ksat + 34 µsat (2.39) Vp = ρ r µsat Vs = . (2.40) ρ where ρ is the density of the saturated rock, estimated as a weighted linear average.


2.4.3

33

Results

We firstly applied the methodology using well log data coming from well A and a synthetic seismic trace to verify the applicability and the validity of the method. Through this feasibility step we compare two different statistical approaches and we demonstrate the coherent propagation of the uncertainty through the three steps of the methodology. Following the approach presented in the previous sections we will show the results of the petrophysical properties estimation in three different conditions: at fine scale, at coarse scale and conditioned by seismic data. In the first step we take into account only the uncertainty related to the rock physics model at fine scale, without considering the uncertainty associated to coarse scale and to seismic. In order to estimate the conditional distribution P (R|mf ), the EM algorithm has been applied assuming three mixture components (one component for each litho-fluid class) combined with the analytical expression of Gaussian Mixtures. In Figure 2.7 (top) we display the marginal conditional probabilities of effective porosity, clay content and water saturation extracted from P (R|mf ) at fine scale. As the rock physics model is accurate, the uncertainty propagated to petrophysics is quite small and the petrophysical properties estimation honors the actual curves of effective porosity, clay content and water saturation derived from log interpretation. In Figure 2.7 (bottom) we show the results of the probability estimation (Eq. 2.27) conditioned by the upscaled impedances obtained by applying sequential Backus averaging to rock physics model predictions: the probabilistic upscaling step allows us to take into account the uncertainty associated to the scale change. The comparison between Figure 2.7 (top) and Figure 2.7 (bottom) clarifies the impact of coarse resolution on uncertainty, which is, as expected, larger in the second case (P (R|mc )), especially for water saturation. Finally we combined the results of the statistical rock physics model with seismic inversion performed with the Bayesian approach, by means of Eq. 2.28, in order to obtain an estimation of the petrophysical properties conditioned by seismic data P (R|Sz ). As a feasibility step we applied the methodology using synthetic seismic data with a signal to noise ratio (SNR) equal to 5. The conditional distributions


34

Probability 0.4

Conditional probability distributions (fine scale) 2060

0.35

2080

0.3

2100

Depth (m)

2120

0.25

2140 0.2 2160 0.15 2180 0.1

2200 2220

0.05

2240 0

0.2 Effective porosity

0.4

0

0.2 0.4 0.6 Clay content

0.8

0

0.5 Water saturation

1

0

Probability 0.25

Conditional probability distributions (coarse scale) 2060 2080

0.2 2100 2120 Depth (m)

0.15 2140 2160 0.1 2180 2200 0.05 2220 2240 0

0.2 Effective porosity

0.4

0

0.2 0.4 0.6 Clay content

0.8

0

0.5 Water saturation

1

0

Figure 2.7: Petrophysical properties estimation at well A location: effective porosity, clay content and water saturation probability distributions obtained with GMM. (top) Petrophysical properties estimation conditioned by high resolution impedances, extracted from P (R|mf ). (bottom) Petrophysical properties estimation conditioned by upscaled data, extracted from P (R|mc). The background color is the conditional probability. Black lines are the actual petrophysical curves, red dotted lines represent P10, median and P90.


35

(Figure 2.8, top) show the multimodality of the petrophysical data and the increase of uncertainty in particular in thin layers sequences. In the case of multimodal data, the median is not a good estimator, but we can observe that the probability distributions still capture the bimodality of petrophysical properties. We compare now the previous results at seismic scale obtained with Gaussian Mixture Models with the results obtained with Kernel density approach (Figure 2.8). The two results are quite similar, in fact in both cases the petrophysical inversion can capture the bimodality of each variable. The top of the reservoir is characterized by a thick high porosity oil sand layer and it is well detected in both cases. In this application Gaussian Mixtures are an appropriate solution and they provide a good result because the litho-fluid classification of well data (which identifies the components of the mixture) allows for a good discrimination of petrophysical and elastic properties; also the approach based on KDE provides a good estimation of the posterior probability as the Kernel density estimation recognizes the multimodality of the data if the scaling lengths are correctly chosen. With respect to the GMM approach, the non parametric approach is more computationally demanding and it requires the tuning of the scaling lengths parameters. Even though the linear correlation coefficients cannot be used as a full quantitative measure of the inversion quality in the case of multimodal distributions, we tried to evaluate the quality of the match between the inversion results and real data computing the correlation coefficients between the estimated petrophysical properties and the actual curves (Table 2.2). The analysis of the correlation coefficients confirms what we observed from the probability densities, in particular how the scale change affects the uncertainty. The methodology has also been applied in a discrete domain, for litho-fluid classification based on seismic data: from the probability distributions of petrophysical properties we predicted litho-fluid classes (Eq. 2.29) at well A location and we used the resulting posterior probabilities to generate multiple realizations of litho-fluid classes vertical sequences. The rock physics likelihood P (πz |R) has been estimated

using the petrophysical curves and the litho-fluid classification, assuming a Gaussian distribution for each litho-fluid class.


Probability distributions conditioned by synthetic seismic (GMM)

36

Probability 0.25

2060 2080 0.2 2100

Depth (m)

2120 0.15 2140 2160 0.1 2180 2200 0.05 2220 2240 0


0

0.2 0.4 0.6 0.8 Clay content

0

0.5 1 Water saturation

Probability distributions conditioned by synthetic seismic (KDE)

0

Probability 0.25

2060 2080 0.2 2100

Depth (m)

2120 0.15 2140 2160 0.1 2180 2200 0.05 2220 2240 0


0


0

0.5 1 Water saturation

0

Figure 2.8: Petrophysical properties estimation conditioned by synthetic seismic data at well A location: effective porosity, clay content and water saturation probability distributions extracted from P (R|Sz ) computed with GMM (top) and KDE (bottom). The background color is the conditional probability. Black lines are the actual petrophysical curves, red dotted lines represent P10, median and P90.


Correlation coefficient of effective porosity

Correlation coefficient of clay content

Correlation coefficient of water saturation

0.95

0.89

0.87

0.55

0.67

0.63

0.58

0.55

0.64

0.55

0.60

0.69

Conditioned by fine scale data Conditioned by coarse scale data Conditioned by synthetic seismic (GMM) Conditioned by synthetic seismic (KDE)

37

Table 2.2: Correlation coefficients between estimated petrophysical properties and real data at well A location. The results of the classification conditioned by seismic are shown in Figure 2.9 (top): we can observe a high probability value for the oil sand class reflecting the thick high porosity sand layer at the top of the reservoir. It is important to observe that, even though the maximum a posteriori (MAP) of litho-fluid classes probabilities is not a good estimator in this case, the fluctuations of the probability curves have a good match with the actual litho-fluid classes profile and they can be used as a prior probability for multiple realizations. Integrating the probability of litho-fluid classes conditioned by seismic with the probability obtained from the transition matrices (Eq. 2.24), we can generate several realizations of litho-fluid classes profiles at well location (Figure 2.9, bottom). We used contingency analysis (Table 2.3) in order to evaluate the misclassification errors, comparing the maximum a posteriori of the probability P (πz |Sz ) with the actual classification. In the contingency table we computed the absolute frequencies, the reconstruction rate, the recognition rate and the estimation index. The reconstruction rate is obtained by normalizing the frequency table per row, while the recognition rate is obtained by normalizing the frequency table per column. The reconstruction rate represents the percentage of the samples belonging to a litho-fluid class (actual) which are classified in that class (predicted). The recognition rate represents the percentage of the samples classified in a litho-fluid class (predicted) that actually belong to that class (actual). The information concerning under/overestimation can be inferred from the estimation index which is defined as the difference between the reconstruction rate and the recognition rate. A negative estimation index in the


38

2060 2080 2100

Depth (m)

2120 2140 2160 2180 2200 2220 2240 0

0.5 LFC probabilities

1

MAP of LFC probabilities

Actual LFC

2060 2080 2100

Depth (m)

2120 2140 2160 2180 2200 2220 2240 Simulation 1

Simulation 2

Simulation 3

Simulation 4

Simulation 5

Actual LFC

Figure 2.9: Litho-fluid probabilistic classification conditioned by synthetic seismic at well A location. (top) From left to right: probabilities of litho-fluid classes P (πz |Sz ) based on petrophysical inversion, MAP of the probability and actual lithofluid classes. (bottom) Some realizations obtained with a Markov chain approach (oil sand in yellow, water sand in brown, shale in green).


Shale

Water sand

Oil sand

Predicted shale f = 472 R = 69% r = 69.2% E = −0.2% f = 114 R = 35.5% r = 16.7% E = 18.8% f = 96 R = 19.4% r = 14.1% E = 5.3%

Predicted water sand f = 104 R = 15.2% r = 40% E = −24.8% f = 106 R = 33% r = 40.8% E = −7.8% f = 50 R = 10.1% r = 19.2% E = −9.1%

39

Predicted oil sand f = 108 R = 15.8% r = 19.4% E = −3.6% f = 101 R = 31.5% r = 18.1% E = 13.4% f = 349 R = 70.5% r = 62.5% E = 8%

Table 2.3: Contingency analysis of petrophysical properties estimation at well A location (f = Absolute frequency; R = Reconstruction rate; r = Recognition rate; E = Estimation index). main diagonal indicates underestimation, while a positive estimation index indicates overestimation; the off-diagonal terms describe in which class the samples are misclassified. In our case, oil sand could be reconstructed with probability 70.5% by the inversion algorithm, and recognized with probability 62.5%, thus sand is overestimated (estimation index 8%). The actual oil sand samples not detected by the inversion are mostly classified in shale (96 samples, reconstruction rate 19.4%). The recognition rates of predicted oil sand tell us that some shale samples (108) and water sand samples (101) are classified in oil sand, which is the reason of the overestimation of oil sand. Similarly, the negative estimation index for water sand (−7.8%) in the main diagonal of the contingency table shows an underestimation of water sand. In some cases we cannot discriminate water sand from actual shale and oil sand from actual water sand (relatively high estimation index of predicted water sand in actual shale and of predicted oil sand in actual water sand). This result can be justified by the rock physics template (Figure 2.3) where we can note the overlaps between those classes. The misclassification between oil sand and shale is mainly due to the upscaling effect on thin layers. Finally we applied the methodology to the whole reservoir level using real seismic data in order to obtain 3D volumes of petrophysical properties with the associated uncertainty. First of all we performed a Bayesian inversion on a small 3D volume


40

including well A used for rock physics model calibration and well B used for methodology validation. The seismic volume (4 angle gathers available) contains about 10000 traces in a time window corresponding to a depth interval of approximately 250 m. In Figure 2.10 we display two seismic sections (related to the partial angle stacks 20o and 44o ), passing through the two wells. In Figure 2.11 we show the prior model used for inversion and the inverted values with the associated uncertainties at well locations, while the corresponding inverted impedances sections, Ip and Is , estimated by Bayesian inversion, are displayed in Figure 2.12. Well B

Well A

2050

Depth (m)

2100

2150

2200

2250

300

600

900

1200 Distance (m)

1500

1800

2100

2400

300

600

900

1200 Distance (m)

1500

1800

2100

2400

2050

Depth (m)

2100

2150

2200

2250

Figure 2.10: 2D seismic sections passing through well A (on the right) and well B (on the left): (top) angle stack 20o ; (bottom) angle stack 44o . The final result of the study is the posterior probability of petrophysical properties on the whole 3D volume. Figure 2.13 shows the probability distributions of effective porosity at three different locations along the 2D section passing through the wells. The comparison between the actual effective porosity curves and the estimated probabilities gives evidence that at the top of the reservoir the estimation is more accurate than in the lower part. In Figure 2.14 we display the maximum a posteriori of the


well A

well A

well B

41

well B Prior model P10 prior P90 prior Well log Upscaled log Inverted log P10 inversion P90 inversion

2060

2080

2100

Depth (m)

2120

2140

2160

2180

2200

2220

2240 5000

9000 Ip (m/s g/cm3)

13000

3000

5000

Is (m/s g/cm3)

7000

5000

9000 Ip (m/s g/cm3)

13000

3000

5000

7000

Is (m/s g/cm3)

Figure 2.11: Prior model and posterior distributions at well locations. From left to right: P-impedance of well A, S-impedance of well A, P-impedance of well B, S-impedance of well B. Blue curves are the actual logs, green curves represent the upscaled data, black curves are the prior model, red curves represent the inverted values. Dotted lines represent the P10 and P90. posterior probabilities of effective porosity, clay content and water saturation. In the upper part of the section we can clearly detect the overcap clay and the top of the reservoir characterized by a high porosity sand filled by oil; in the lower part the thin layers observed in the well logs are not detected and the uncertainty associated to the inverted properties increases. This is mainly due to the quality of the seismic data, which is higher at the top (SNR ≃ 3) and very poor at the bottom (SNR ≃ 1), as we

can observe in the seismic sections (Figure 2.10).

We also performed a litho-fluid classification based on seismic data; in Figure 2.15 we show the curves of the conditional probabilities of litho fluid classes at well A


Well B

Ip (m/s g/cm3)

Well A

2050

12000 11000

2100 Depth (m)

42

10000 2150 9000 2200

8000 7000

2250

6000 300

600

900

1200 1500 Distance (m)

1800

2100

2400

Is (m/s g/cm3) 7000

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Depth (m)

2100

6000 5500

2150

5000 4500

2200

4000 3500

2250

3000 300

600

900

1200 1500 Distance (m)

1800

2100

2400

Figure 2.12: 2D sections of inverted impedances: (top) inverted P-impedance; (bottom) inverted S-impedance. location conditioned by the corresponding seismic trace and some realizations obtained integrating the posterior probability of litho-fluid classes with the Markov chain model: the quality of Markov chain realizations is acceptable at the top of the reservoir and quite poor at the bottom where the SNR of seismic data is very low. The results on the 3D volume, for example for effective porosity, are shown in Figure 2.16 by extracting from the estimated effective porosity volume a crossline passing for well B and an inline for well A. Finally, in Figure 2.17 we propose a 3D visualization of the hydrocarbon sands probability: the oil sand probability cube has been thresholded to reveal the areas where the probability of oil sand litho-fluid class occurrence is greater than 0.7.


43

Figure 2.13: Probability distributions of effective porosity at wells locations (well A on the right, well B on the left) and at an intermediate location between the two wells. Black lines are the actual effective porosity curves.


Well A

Well B

44

Effective porosity

2050 0.2

Depth (m)

2100 0.15 2150 0.1

2200

0.05

2250 300

600

900

1200 Distance (m)

1500

1800

2100

2400

Depth (m)

Clay content 2050

0.7

2100

0.6 0.5

2150 0.4 2200

0.3 0.2

2250 300

600

900

1200 Distance (m)

1500

1800

2100

2400

0.1

Water saturation 2050 0.9

Depth (m)

2100

0.8 0.7

2150

0.6 0.5

2200

0.4 2250

0.3 300

600

900

1200 Distance (m)

1500

1800

2100

2400

Figure 2.14: Estimation of effective porosity (top), clay content (middle), and water saturation (bottom) in the 2D section obtained from the mode of the posterior distributions.


45

2060 2080 2100

Depth (m)

2120 2140 2160 2180 2200 2220 2240 0

0.5 1 LFC probabilities

Simulation 1

Simulation 2

Simulation 3

Simulation 5

Simulation 6

Actual LFC

2060 2080 2100

Depth (m)

2120 2140 2160 2180 2200 2220 2240 Simulation 4

Figure 2.15: Litho-fluid probabilistic classification conditioned by real seismic at well A location. From top left to bottom right: probabilities of litho-fluid classes P (πz |Sz ) based on petrophysical inversion, six realizations obtained with a Markov chain approach, and actual litho-fluid classification (oil sand in yellow, water sand in brown, shale in green).


46

Figure 2.16: Estimation of effective porosity along two lines extracted from the 3D volume.

Figure 2.17: Isoprobability surface of 70% probability of oil sand litho-fluid class. The background slices represent two 2D sections of probability of oil sand occurrence.


2.5

47

Discussion

The feasibility test based on synthetic seismic shows the propagation of the different sources of uncertainty through the different steps and the applicability of the methodology with both the proposed statistical approaches. The real case application, integrating well data and real seismic, pointed out that the results are quite satisfactory as long as the quality of seismic is acceptable. In particular the use of Gaussian Mixture seems to be a valid approach for the classification of petrophysical and categorical parameters, which can be applied to real cases with reduced computational time. The application of the rock physics model is not computationally demanding whereas the estimation of the conditional probability in a Bayesian framework can be quite hard to obtain because it requires the estimation of the joint distribution in a space of high dimensions. Gaussian Mixture Models are a suitable solution because of their analytical convenience, especially when the distributions of petrophysical and elastic attributes describe different litho-fluid classes features. The non parametric alternative, Kernel density estimation, is more computationally demanding because it requires the numerical evaluation of a joint probability on a multidimensional domain. A more efficient method to tackle the multidimensional extension of KDE is based on Fast Fourier Transform (FFT). In fact KDE can also be seen as a convolution, so that we can reduce the computational time by realizing the convolution by means of FFT (Buland et al. (2008)). However, one of the most critical point for the estimation of probabilities by means of Kernel density is the choice of the scaling lengths parameters, which have to be determined through different trials. The main simplification we adopted in our approach is the overlooking of the spatial correlation of petrophysical variables for the estimation of the conditional distributions, in order to reduce the dimension of the probability space. However we take into account the vertical correlation in seismic inversion by including a vertical correlation in the prior covariance matrix of the vector of elastic parameters and in the prior covariance matrix of the error on seismic amplitudes for each angle gather (Buland and Omre (2003)); whereas the spatial correlation is not explicitly accounted for,


48

because we adopt a trace-by-trace inversion approach to jointly estimate impedances and densities from seismic. We remark that the lateral continuity of our results is mainly related to the imaging, in fact part of the lateral correlation of seismic data is imposed by the migration operator which is a spatial filter whose correlation length is associated with the Fresnel zone. In order to perform the final step from continuous petrophysical variables to lithofluid classes modeling, Gaussian Mixture Models seem to be an appropriate approach as they can express the multimodal features of the petrophysical variables in the different litho-fluid classes. The integration of more advanced geostatistical techniques could be a significant improvement in order to use the probabilistic information related to litho-fluid classes to generate multiple realizations for reservoir characterization.

2.6

Conclusions

The presented methodology aims to propagate the uncertainty from seismic to petrophysical properties, including the effect of scale change, the seismic noise error and the degree of approximation of the physical models. Statistical rock physics, combined with the probabilitic approach adopted for seismic inversion, is proposed in order to quantify the uncertainty. The main results of the methodology are the probability distributions of the estimated petrophysical parameters, that can be used to assess the reliability of reservoir properties estimation. In order to obtain the posterior distribution of the petrophysical properties, we point out that one of the key point of our methodology is the use of Gaussian Mixture Models and the identification of the weights of the mixture as the indicator probability of the litho-fluid classes. Even though the considered uncertainty factors do not cover all the possible sources, the 1D feasibility test shows that the main effects due to scale changes and seismic noise are taken into account and that these two factors can explain an important part of the uncertainty. In the application case, the method works better in the upper layers, where the signal to noise ratio is high, rather than in the lower layers, where the signal to noise


49

is low. In conclusion, where the signal to noise is acceptable, the probabilistic petrophysical evaluation on the real case shows the applicability of the methodology and that the reliability of the seismic data is coherently propagated to the petrophysical properties prediction. The proposed methodology can be applied to all reservoirs where elastic characterization of petrophysical properties is possible and where the physical link can be described by a suitable rock physics model.

Chapter 3 Sequential Gaussian mixture simulation 3.1

Abstract

We present here a method for generating realizations of the posterior probability density function of a Gaussian Mixture linear inverse problem in the combined discretecontinuous case. This task is achieved by extending the sequential simulations method to the mixed discrete-continuous problem. The sequential approach allows us to generate a Gaussian Mixture random field that honors the covariance functions of the continuous property and the available observed data. The traditional inverse theory results, well known for the Gaussian case, are first summarized for Gaussian Mixture models: in particular the analytical expression for means, covariance matrices, and weights of the conditional probability density function are derived. However, the computation of the weights of the conditional distribution requires the evaluation of the probability density function values of a multivariate Gaussian distribution, at each conditioning point. As an alternative solution of the Bayesian inverse Gaussian Mixture problem, we then introduce the sequential approach to inverse problems and extend it to the Gaussian Mixture case. The main novelty compared to the approach presented in Chapter 2 is that in this method we sample from the posterior

50

CHAPTER 3. SEQUENTIAL GAUSSIAN MIXTURE SIMULATION

51

pdfs rather than just taking a statistical estimator, which allows us to obtain multiple realizations all honoring the initial conditioning measurements. The Sequential Gaussian Mixture Simulation (SGMixSim) approach is presented as a particular case of the linear inverse Gaussian Mixture problem, where the linear operator is the identity. Similar to the Gaussian case, in Sequential Gaussian Mixture Simulation the means and the covariance matrices of the conditional distribution at a given point correspond to the kriging estimate, component by component, of the mixture. Furthermore, Sequential Gaussian Mixture Simulation can be conditioned by secondary information to account for non-stationarity. Examples of applications with synthetic and real data, are presented in the reservoir modeling domain where realizations of facies distribution and reservoir properties, such as porosity or net-to-gross, are obtained using Sequential Gaussian Mixture Simulation approach. In these examples, reservoir properties are assumed to be distributed as a Gaussian Mixture model. In particular, reservoir properties are Gaussian within each facies, and the weights of the mixture are identified with the point-wise probability of the facies.

3.2

Introduction

Inverse problems are common in many different domains such as physics, engineering, and earth sciences. In general, solving an inverse problem consists of estimating the model parameters given a set of observed data. The operator that links the model and the data can be linear or non-linear. In the linear case, estimation techniques generally provide smoothed solutions. Kriging, for example, provides the best estimate of the model in the least-squares sense. Simple kriging is in fact identical to a linear Gaussian inverse problem where the linear operator is the identity, with the estimation of posterior mean and covariance matrices with direct observations of the model space. Monte Carlo methods can be applied as well to solve inverse problems Mosegaard and Tarantola (1995) in a Bayesian framework to sample from the posterior; but standard sampling methodologies can be inefficient in practical applications. Sequential simulations have been introduced in geostatistics to generate high resolution models and provide a number


52

of realizations of the posterior probability function honoring both prior information and the observed values. Deutsch and Journel (1992) and Goovaerts (1997) give detailed descriptions of kriging and sequential simulation methods. Hansen et al. (2006) proposes a methodology that applies sequential simulations to linear Gaussian inverse problems to incorporate the prior information on the model and honor the observed data. We propose here to extend the approach of Hansen et al. (2006) to the Gaussian Mixture case. Gaussian Mixture models are convex combinations of Gaussian components that can be used to describe the multi-modal behavior of the model and the data. Sung (2004), for instance, introduces Gaussian Mixture distributions in multivariate nonlinear regression modeling; while Hastie and Tibshirani (1996) proposes a mixture discriminant analysis as an extension of linear discriminant analysis by using Gaussian Mixtures and Expectation-Maximization algorithm Hastie et al. (2002). Gaussian Mixture models are common in statistics (see, for example, Hasselblad (1966) and Dempster et al. (1977)) and they have been used in different domains: digital signal processing Reynolds (2000) and Gilardi et al. (2002), engineering Alspach and Sorenson (1972), geophysics Grana and Della Rossa (2010), and reservoir history matching Dovera and Della Rossa (2011). In this chapter we first present the extension of the traditional results valid in the Gaussian case to the Gaussian Mixture case; we then propose the sequential approach to linear inverse problems under the assumption of Gaussian Mixture distribution; and we finally show some examples of applications in reservoir modeling. If the linear operator is the identity, then the methodology provides an extension of the traditional Sequential Gaussian Simulation (SGSim, see Deutsch and Journel (1992), and Goovaerts (1997)) to a new methodology that we call Sequential Gaussian Mixture Simulation (SGMixSim). The applications we propose refer to mixed discrete-continuous problems of reservoir modeling and they provide, as main result, sets of models of reservoir facies and porosity. The key point of the application is that we identify the weights of the Gaussian Mixture describing the continuous random variable (porosity) with the probability of the reservoir facies (discrete variable).


3.3

53

Theory: Linearized Gaussian Mixture Inversion

In this section we provide the main propositions of linear inverse problems with Gaussian Mixtures (GMs). We first recap the well-known analytical result for posterior distributions of linear inverse problems with Gaussian prior; then we extend the result to the Gaussian Mixtures case. In the Gaussian case, the solution of the linear inverse problem is well-known Tarantola (2005). If m is a random vector Gaussian distributed, m ∼ N(µm , Cm ), with mean µm and covariance Cm ; and G is a linear operator that transforms the

model m into the observable data d d = Gm + ε,

(3.1)

where ε is a random vector that represents an error with Gaussian distribution N(0, Cε ) independent of the model m; then the posterior conditional distribution of m|d is Gaussian with mean and covariance given by µm|d = µm + Cm GT (GCm GT + Cε )−1 (d − Gµm )

Cm|d = Cm − Cm GT (GCm GT + Cε )−1 GCm .

(3.2) (3.3)

This result is based on two well known properties of the Gaussian distributions: (A) the linear transform of a Gaussian distribution is again Gaussian; (B ) if the joint distribution (m, d) is Gaussian, then the conditional distribution m|d is again Gaussian. These two properties can be extended to the Gaussian Mixtures case. We assume that x is a random vector distributed according to a Gaussian Mixture with P Nc k k Nc components, f (x) = k=1 πk N(x; µx , Cx ), where πk are the weights and the

distributions N(x; µkx , Cxk ) represent the Gaussian components with means µkx and covariances Cxk evaluated in x. By applying property (A) to the Gaussian compo-

nents of the mixture, we can conclude that, if L is a linear operator, then y = Lx


54

is distributed according to a Gaussian Mixture. Moreover, the pdf of y is given by P c k k T f (y) = N k=1 πk N(y; Lµx , LCx L ). As a matter of fact, by using the definition of characteristic function, we can write T

Φy (t) = E(et y ) = E(et

T Lx

) = E(e(L

T t)T x

) = Φx (LT t)

(3.4)

where t ∈ ℜN , y ∈ ℜN , x ∈ ℜM and L : ℜM → ℜN . As x is a Gaussian Mixture,

then the characteristic function Φx (s) is a linear combination of the characteristic functions of the Gaussian components; then Φy (t) =

Nc X

πk Φxk (LT t)

(3.5)

k=1

where Φxk is the characteristic function of the Gaussian component k. Property (A) applied to each Gaussian component allows to conclude that also y is distributed according to a Gaussian Mixture and the pdf is given by f (y) =

Nc X

πk N(y, Lµkx , LCkx LT ) .

(3.6)

k=1

Similarly we can extend property (B ) to conditional Gaussian Mixture distributions. The well-known result of the conditional multivariate Gaussian distribution has already been extended to multivariate Gaussian Mixture models (see, for example, Alspach and Sorenson (1972)). In particular, if (x1 , x2 ) is a random vector whose joint distribution is a Gaussian Mixture f (x1 , x2 ) =

Nc X

πk fk (x1 , x2 ),

(3.7)

k=1

where fk are the Gaussian densities, then the conditional distribution of x2 |x1 is

again a Gaussian Mixture

f (x2 |x1 ) =

Nc X k=1

λk fk (x2 |x1 ),

(3.8)

and its parameters (weights, means, and covariance matrices) can be analytically derived. The coefficients λk are given by πk fk (x1 ) λk = PNc , π f (x ) ℓ ℓ 1 ℓ=1

(3.9)


55

where fk (x1 ) = N(x1 ; µkx1 , Ckx1 ); and the means and the covariance matrices are −1 k (3.10) x1 − µkx1 µkx2 |x1 = µkx2 + Ckx2 ,x1 Ckx1 −1 T Ckx2 |x1 = Ckx2 − Ckx2 ,x1 Ckx1 Ckx2 ,x1 , (3.11) where Ckx2 ,x1 is the cross-covariance matrix.

In fact, the general conditional distribution can be written by definition as: f (x2 |x1 ) =

f (x1 ,x2 ) . f (x1 )

We then explicitly write the numerator and the denominator by evaluating

the single Gaussian components of the mixture. If x = (x1 , x2 ) is distributed according to a Gaussian Mixture distribution, then the single components of the joint pdf are Gaussian with means and covariances given by " # " # µkx1 Ckx1 ,x1 Ckx1 ,x2 k k µx = , Cx = µkx2 Ckx2 ,x1 Ckx2 ,x2

(3.12)

where x1 ∈ ℜN1 and x2 ∈ ℜN2 . Now the property (B) of Gaussian distributions

applied to each single component of the mixture allows us to write the conditional distributions for each component as Gaussian fk (x2 |x1 ) with means and covariances

given by Eqs. 3.3 and 3.3. If the joint density of each component is written as the product fk (x1 , x2 ) = fk (x2 |x1 )fk (x1 ), then the Gaussian Mixture joint pdf of (x1 , x2 ) is

f (x1 , x2 ) =

Nc X k=1

πk fk (x2 |x1 )fk (x1 ) .

(3.13)

Similarly, the denominator of the full conditional pdf can be written by using the definition of marginal density as Z Z f (x1 ) = f (x1 , x2 )dx2 = ℜN2

=

Nc X k=1

πk

ℜN2

Z

fk (x1 , x2 )dx2 ℜN2

Nc X

!

πk fk (x1 , x2 ) dx2 =

k=1

=

Nc X

πk fk (x1 )

(3.14)

k=1

where fk (x1 ) is the marginal density of the k th Gaussian component. If we explicitly write the numerator and the denominator of the conditional distribution f (x2 |x1 ),

we obtain

f (x2 |x1 ) =

P Nc

k=1 πk fk (x2 |x1 )fk (x1 ) PNc ℓ=1 πℓ fℓ (x1 )

(3.15)

56


If we set the new coefficients λk as in Eq. 3.3, then the full form of conditional pdf can be written as in Eq. 3.3 and the so obtained pdf is again a Gaussian Mixture. By combining these propositions, the main result of linear inverse problems with Gaussian Mixture can be derived. Theorem 3.3.1. Let m be a random vector distributed according to a Gaussian MixP c k k k ture m ∼ N k=1 πk N(µm , Cm ), with Nc components and with means µm , covariances Ckm , and weights πk , for k = 1, . . . , Nc . Let G : ℜM → ℜN be a linear operator, and

ε a Gaussian random vector independent of m with 0 mean and covariance Cε , such that d = Gm + ε, with d ∈ ℜN , m ∈ ℜM , ε ∈ ℜN , then the posterior conditional

distribution m|d is a Gaussian Mixture.

Moreover, the posterior means and covariances of the components are given by µkm|d = µkm + Ckm GT (GCkm GT + Cε )−1 (d − Gµkm )

Ckm|d = Ckm − Ckm GT (GCkm GT + Cε )−1 GCkm ,

(3.16) (3.17)

where µkm and Ckm , are respectively the prior mean and covariance of the k th Gaussian component of m. The posterior coefficients λk of the mixture are given by πk fk (d) λ k = P Nc , ℓ=1 πℓ fℓ (d)

(3.18)

where the Gaussian densities fk (d) have means µkd = Gµkm and covariances Ckd = GCkm GT + Cε .

3.4

Theory: Sequential approach

Based on the results presented in the previous section, we introduce here the sequential approach to linearized inversion in the Gaussian Mixture case. We first recap the main result for the Gaussian case Hansen et al. (2006). The solution of the linear inverse problem with the sequential approach requires some additional notation. Let mi represent the ith element of the random vector m, and let ms represent a known subvector of m. This notation will generally be used


57

to describe the neighborhood of mi in the context of sequential simulations. Finally we assume that the measured data d are known having been obtained as a linear transformation of m according to some linear operator G. Theorem 3.4.1. Let m be a random vector, Gaussian distributed, m ∼ N(µm , Cm )

with mean µm and covariance Cm . Let G be a linear operator between the model m and the random data vector d such that d = Gm + ε, with ε a random error vector independent of m with 0 mean and covariance Cε . Let ms be the subvector with direct observations of the model m, and mi the ith element of m. Then the conditional distribution of mi |(ms , d) is again Gaussian.

Moreover, if the subvector ms is extracted from the full random vector m with the linear operator A such that ms = Am, where the ith element is mi = Ai m, with Ai again linear, then the mean and variance of the posterior conditional distribution are: " # i h −1 ms − Aµm C(ms ,d) µmi |(ms ,d) = µmi + Ai Cm AT Ai Cm GT (3.19) d − Gµm # " i h T AC A −1 m i 2 2 , (3.20) C(ms ,d) σm = σm − Ai Cm AT Ai Cm GT i |(ms ,d) i GCm ATi 2 where µmi = Ai µm , σm = Ai Cm ATi , and i " # ACm AT ACm GT C(ms ,d) = . GCm AT GCm GT + Cε

(3.21)

To clarify the statement we give the explicit form of the operators Ai and A. In particular, Ai is written as Ai =

h

0 0 ... 1 ... 0

i

,

(3.22)

with the one in the ith column. If the subvector ms has size n, ms = {mi1 , mi2 , . . . , min }, and m has size M; then the operator A  0 0   0 ...  A= . .  .. ..  0 1

is given by

... 1 ... 0 1 .. .

0 ... .. .. . .

... 0

0

0 .. . 0



   ,  

(3.23)

58


where A has dimensions n×M and the ones are in the i1 , i2 , . . . , in columns. Theorem 3.4.1 can be proved using the properties (A) and (B ) described in Section 3.3 (see Hansen et al. (2006)). Then, by using Theorem 3.3, we extend the result to the Gaussian Mixture case. Theorem 3.4.2. Let m be a random vector distributed according to a Gaussian MixP c k k k ture, m ∼ N k=1 πk N(µm , Cm ), with Nc components and with means µm , covariances Ckm , and weights πk , for k = 1, . . . , Nc . Let G a linear operator such that d = Gm+ε,

with ε a random error vector independent of m with 0 mean and covariance Cε . Let ms be the subvector with direct observations of the model m, and mi the ith element of m. Then the conditional distribution of mi |(ms , d) is again a Gaussian Mixture.

Moreover, the means and variances of the components of the posterior conditional distribution are: µkmi |(ms ,d) = µkmi + 2 (k) σmi |(ms ,d)

=

2 (k) σm i

−

2 (k)

where µkmi = Ai µkm , σmi

h

Ai Ckm AT Ai Ckm GT h

Ai Ckm AT

i

Ai Ckm GT

Ck(ms ,d) i

−1

Ck(ms ,d)

"

ms − Aµkm

d − Gµkm # " −1 ACkm ATi GCkm ATi

#

(3.24)

,(3.25)

= Ai Ckm ATi , and

Ck(ms ,d) =

"

ACkm AT

ACkm GT

GCkm AT GCkm GT + Cε

#

.

(3.26)

The posterior coefficients of the mixture are given by πk fk (ms , d) λk = PNc , ℓ=1 πℓ fℓ (ms , d)

where the Gaussian components fk (ms , d) have means # " k Aµ m µk(ms ,d) = Gµkm and covariances Ck(ms ,d) .

(3.27)

(3.28)


59

In the case where the linear operator is the identity, the associated inverse problem reduces to the estimation of a Gaussian Mixture model with direct observations of the model space at given locations. In other words, if the linear operator is the identity, the theorem provides an extension of the traditional Sequential Gaussian Simulation (SGSim) to the Gaussian Mixture case. We call this methodology Sequential Gaussian Mixture Simulation (SGMixSim), and we show some applications in the next section.

3.4.1

Proof

First of all we observe that the joint vector (m, ε) is distributed according to a Gaussian Mixture. If we consider the following linear operator   Ak 0    B=  A 0  G 1N

(3.29)

then, we observe that





" # mk   m  md  = B ·   ε s

(3.30)

and the joint vector (mk , md , s) is distributed according to a Gaussian Mixture. The statement follows by assuming x1 = (md , s) and x2 = mk , and deriving the conditional distribution x2 |x1 .

3.5

Application

We describe here some examples of applications with synthetic and real data, in the context of reservoir modeling. First, we present the results of the estimation of a Gaussian Mixture model with direct observations of the model space as a special case of Theorem 3.4 (SGMixSim). In our example, the continuous property is the porosity of a reservoir, and the discrete variable represents the corresponding reservoir facies, namely shale and sand. This means that we identify the weights of the mixture components with the facies probabilities. The input parameters are then the prior


60

distribution of porosity and a variogram model for each component of the mixture. The prior is a Gaussian Mixture model with two components and its parameters are the weights, the means, and the covariance matrices of the Gaussian components. We assume facies prior probabilities equal to 0.4 and 0.6 respectively, and for simplicity we assume the same variogram model (spherical and isotropic) with the same parameters for both. We then simulate a 2D map of facies and porosity according to the proposed methodology (Figure 3.1). The simulation grid is 70x70 and the variogram range of porosity is 4 grid blocks in both directions. The simulation can be performed with or without conditioning hard data; in the example of Figure 3.1, we introduced four porosity values at four locations that are used to condition the simulations, and we generated a set of 100 conditional realizations (Figure 3.1). When hard data are assigned, the weights of the mixture components are determined by evaluating the prior Gaussian components at the hard data location and discrete property values are determined by selecting the most likely component. As we previously mentioned, the methodology is similar to Hansen et al. (2006), but the use of Gaussian Mixture models allows us to describe the multi-modality of the data and to simulate at the same time both the continuous and the discrete variable. SGMixSim requires a spatial model of the continuous variable, but not a spatial model of the underlying discrete variable: the spatial distribution of the discrete variable only depends on the conditional weights of the mixture (Eq. 3.27). However, if the mixture components have very different probabilities and very different variances (i.e. when there are relatively low probable components with relatively high variances), the simulations may not accurately reproduce the global statistics. If we assume, for instance, two components with prior probabilities equal to 0.2 and 0.8, and we assume at the same time that the variance of the first component is much bigger than the variance of the second one, then the prior proportions may not be honored. This problem is intrinsic to the sequential simulation approach, but it is emphasized in case of multi-modal data. For large datasets or for reasons of stationarity, we often use a moving searching neighborhood to take into account only the points closest to the location being simulated Goovaerts (1997). If we use a global searching neighborhood (i.e. the whole grid) the computational time, for large datasets, could significantly


61

increase. In the localized sequential algorithm, the neighborhood is selected according to a fixed geometry (for example, ellipsoids centered on the location to be estimated) and the conditioning data are extracted by the linear operator (Theorem 3.4) within the neighborhood. When no hard data are present in the searching neighborhood and the sample value is drawn from the prior distribution, the algorithm could generate isolated points within the simulation grid. For example, a point drawn from the first component could be surrounded by data, subsequently simulated, belonging to the second component, or viceversa. This problem is particularly relevant in the case of multi-modal data especially in the initial steps of the sequential simulation (in other words when only few values have been previously simulated) and when the searching neighborhood is small. To avoid isolated points in the simulated grid, a post-processing step has been included (Figure 3.1). The simulation path is first revisited, and the local conditional probabilities are re-evaluated at all the grid cells where the sample value was drawn from the prior distribution. Then we draw again the component from the weights of the re-evaluated conditional probability. Finally, we introduce a kriging correction of the continuous property values that had low probabilities in the neighborhood. Next, we show two applications of linearized sequential inversion with Gaussian Mixture models obtained by applying Theorem 3.4. The first example is a rock physics inverse problem dealing with the inversion of acoustic impedance in terms of porosity. The methodology application is illustrated by using a 2D grid representing a synthetic system of reservoir channels (Figure 3.2). In this example we made the same assumptions about the prior distribution as in the previous example. As in traditional sequential simulation approaches, the spatial continuity of the inverted data depends on the range of the variogram and the size of the searching neighborhood; however, Figure 3.2 clearly shows the multi-modality of the inverted data. Gaussian Mixture models can describe not only the multi-modality of the data, but they can better honor the data correlation within each facies. The second example is the acoustic inversion of seismic amplitudes in terms of acoustic impedance. In this case, in addition to the usual input parameters (prior distribution and variogram models), we have to specify a low frequency model of


62

Figure 3.1: Conditional realizations of porosity and reservoir facies obtained by SGMixSim. The prior distribution of porosity and the hard data values are shown on top. The second and third rows show three realizations of porosity and facies (grey is shale, yellow is sand). The fourth row shows the posterior distribution of facies and the ensemble average of 100 realizations of facies and porosity. The last row shows the comparison of SGMixSim results with and without post-processing.


63

impedance, since seismic amplitudes only provide relative information about elastic contrasts and the absolute value of impedance must be computed by combining the estimated relative changes with the low frequency model (often called prior model in seismic modeling). Once again, the discrete variable is identified with the reservoir facies classification. In this case shales are characterized by high impedance values, and sand by low impedances. The results are shown in Figure 3.3. We observe that even though we used a very smoothed low frequency model, the inverted impedance log has a good match with the actual data (Figure 3.3), and the prediction of the discrete variable is satisfactory compared to the actual facies classification performed at the well. In particular, if we perform 50 realizations and we compute the maximum a posteriori of the ensemble of inverted facies profiles, we perfectly match the actual classification (Figure 3.3). However, the quality of the results depends on the separability of the Gaussian components in the continuous property domain.

Figure 3.2: Linearized sequential inversion with Gaussian Mixture models for the estimation of porosity map from acoustic impedance values. On top we show the true porosity map and the acoustic impedance map; on the bottom we show the inverted porosity and the estimated facies map.


64

Finally we applied the Gaussian Mixture linearized sequential inversion to a layer map extracted from a 3D geophysical model of a clastic reservoir located in the North Sea (Figure 3.4). The application has been performed on a map of P-wave velocity corresponding to the top horizon of the reservoir. The parameters of the variogram models have been assumed from existing reservoir studies in the same area. In Figure 3.4 we show the map of the conditioning velocity and the corresponding histogram, two realizations of porosity and facies, and the histogram of the posterior distribution of porosity derived from the second realization. The two realizations have been performed using different prior proportions: 30% of sand in the first realization and 40% in the second one. Both realizations honor the expected proportions, the multi-modality of the data, and the correlations with the conditioning data within each facies. 1000 1020 1040

Actual data Prior model Realization 1 Inverted data

Time (ms)

1060 1080 1100 1120 1140 1160 1180 4000

8000 12000 P−impedance 3 (m/s g/cm )

−200 0 200 Seismic amplitudes

Facies Maximum a posteriori (Realization 1) of facies

Actual facies classification

Figure 3.3: Sequential Gaussian Mixture inversion of seismic data (ensemble of 50 realizations). From left to right: acoustic impedance logs and seismograms (actual model in red, realization 1 in blue, inverted realizations in grey, dashed line represents low frequency model), inverted facies profile corresponding to realization 1, maximum a posteriori of 50 inverted facies profiles and actual facies classification (sand in yellow, shale in grey).


65

Figure 3.4: Application of linearized sequential inversion with Gaussian Mixture models to a reservoir layer. The conditioning data is P-wave velocity (top left). Two realizations of porosity and facies are shown: realization 1 corresponds to a prior proportion of 30% of sand, realization 2 corresponds to 40% of sand. The histograms of the conditioning data and the posterior distribution of porosity (realization 2) are shown for comparison.

3.6

Conclusion

In this chapter, we proposed a methodology to simultaneously simulate both continuous and discrete properties by using Gaussian Mixture models. The method is based on the sequential approach to Gaussian Mixture linear inverse problem, and it can


66

be seen as an extension of sequential simulations to multi-modal data. Thanks to the sequential approach used for the inversion, the method is generally quite efficient from the computational point of view to solve multi-modal linear inverse problems and it is applied here to reservoir modeling and seismic reservoir characterization. We presented four different applications: conditional simulations of porosity and facies, porosity-impedance inversion, acoustic inversion of seismic data, and inversion of seismic velocities in terms of porosity. The proposed examples show that we can generate actual samples from the posterior distribution, consistent with the prior information and the assigned data observations. Using the sequential approach, we can generate a large number of samples from the posterior distribution, which in fact are all solutions to the Gaussian Mixture linear problem.

Chapter 4 Stochastic inversion for facies modeling 4.1

Abstract

The main objective of this work is to present a new stochastic methodology for seismic reservoir characterization that combines advanced geostatistical methods with traditional geophysical models, in order to provide fine-scaled reservoir models of facies and reservoir properties, such as porosity, and net-to-gross. The methodology we propose is a stochastic inversion where we simultaneously obtain earth models of facies, rock properties, and elastic attributes. It is based on an iterative process where we generate a set of models of reservoir properties by using sequential simulations, calculate the corresponding elastic attributes through rock physics relations, compute synthetic seismograms and, finally, compare these synthetic results with the real seismic amplitudes. The optimization is performed through a stochastic technique, the probability perturbation method, that perturbs the probability distribution used to generate the initial realization and allows obtaining a facies model consistent with all available data through a relatively small number of iterations. The probability perturbation approach is based on a probabilistic method called Tau model, which provides an analytical representation to combine single probabilistic information into a joint conditional probability. The advantages of probability perturbation method 67

CHAPTER 4. STOCHASTIC INVERSION FOR FACIES MODELING

68

are that it transforms a 3D multiparameter optimization problem into a set of 1D optimization problems and it allows us to include several probabilistic information through the Tau model. The method has been tested on a synthetic case where we generated a set of pseudo-logs and the corresponding synthetic seismograms. We then applied the method to a real well profile, and finally extended it to a 2D seismic section. The application to the real reservoir study includes data from three wells and partially stacked near and far seismic sections, and provided as a main result the set of optimized models of facies, and of the relevant petrophysical properties, to be used as initial static reservoir models for fluid flow reservoir simulations.

4.2

Introduction

One of the aims of reservoir modeling is to describe the spatial variability of reservoir properties: facies, and the corresponding petrophysical properties, such as porosity, permeability, net-to-gross, and fluid saturation. The estimation of reservoir properties from seismic data is a complex underdetermined non-linear inverse problem. Several techniques, both deterministic and probabilistic, have been developed to solve the problem and estimate the optimal reservoir model Bosch et al. (2010) to be used as initial model in fluid flow simulations. We can classify all the existing methodologies in two categories: 1) multi-step inversion methods and 2) stochastic inversion approaches. In multi-step inversion methods, the problem of estimating reservoir properties from seismic data is split into two or more sub-problems: generally elastic properties are first derived from partial stacked seismic data by elastic inversion; then facies are pointwise classified from the resulting volumes of elastic attributes by statistical techniques, such as, for example, discriminant analysis, neural networks, or Bayesian classification (see Avseth et al. (2001), and Mukerji et al. (2001a)). If a Bayesian elastic inversion (Buland and Omre (2003)) is performed, we obtain in the first step a set of volumes of probability of elastic properties which can be used with a suitable likelihood function to classify seismic facies through the Bayesian approach (Doyen (2007)). In more recent approaches, reservoir properties such as porosity and clay


69

content are estimated from inverted seismic velocities (Grana et al. (2009)) and facies distribution can be subsequently derived from the reservoir properties volume. Similarly, in Grana and Della Rossa (2010), a three-step probabilistic approach based on Gaussian mixture models is introduced to estimate the probability of seismic elastic attributes, reservoir properties, and litho-fluid classes or facies. The traditional Bayesian framework (Tarantola (2005)) has been also adopted for problems of litho-fluid prediction from seismic data, as presented in Buland et al. (2008). The probabilistic approach allows us to correctly propagate the uncertainty associated with input data and physical model approximations to the posterior probability of reservoir properties. For non-Gaussian posterior distributions, different estimators can be used to obtain the most likely model, such as mean, median or maximum a posteriori. However if the inversion does not include any sampling method of the posterior distribution, the resolution of the estimated properties is the same as that of the input conditioning data (seismic amplitudes) and the final volumes of facies and reservoir properties are representative of a coarser scale than the characteristic scale of reservoir dynamic models. As a consequence, these methodologies require the integration with geostatistical methods to include seismic inversion results into reservoir models (e.g., Mukerji et al. (2001a)). The most common strategy (Doyen (2007)) is to perform sequential simulations to generate high resolution facies models, by conditioning the simulation with the ”coarse-scaled” volume of facies estimated from seismic. Facies models can be generated by either two-point (sequential indicator simulation, see e.g. Deutsch and Journel (1992)) or multi-point geostatistics (single normal equation simulation, see e.g. Remy (2009)). Both methods allow one to include secondary information derived from seismic data to condition the simulations. The corresponding models of continuous reservoir properties are generated by sequential Gaussian simulation, conditioned by the facies model. Other methods have been recently proposed, mainly in reservoir history matching, including geomechanical models to condition reservoir simulations (Wilschut et al. (2011)). On the other hand, stochastic inversion approaches are generally based on the iterative application of a forward model and the inversion step is performed using


70

deterministic or stochastic optimization techniques: in particular models of subsurface properties (facies and rock properties) are generated; then suitable rock physics transforms are applied to generate the corresponding volumes of the elastic properties; finally synthetic seismic volumes are computed and compared to real seismic data to evaluate the mismatch. The initial models are usually generated using previously mentioned geostatistical techniques (sequential indicator simulation or multipoint geostatistics, and sequential Gaussian simulation) to create fine-scaled reservoir models (González et al. (2008)). The final model is found by applying a suitable optimization method. The main limitation of stochastic inversion techniques is that the optimization step, in real applications, can be computationally expensive. The optimization cannot be applied independently point-by-point, because the objective function depends on seismic data which represent coarse scale information reflecting contrast between sub-surfaces. Moreover, as the solution of the inverse problem could have local mimima, the final model could depend on the initial model. Different initial models could lead to different optimized models with the same seismic response especially when layers thinner than the seismic resolution are included in the reservoir model. Different optimization methods can be used. In González et al. (2008), the optimization is deterministic and it is performed trace-by-trace, and the optimized profile at the current trace is then used to condition the following simulations. Other methods have been introduced: for example, Bosch et al. (2009) propose an iterative optimization based on Newton’s method to simultaneously update the multi-property model. Another family of stochastic inversion approaches is based on Markov chain Monte Carlo methods (Eidsvik et al. (2004a), Larsen et al. (2006), Gunning and Glinsky (2007), Rimstad and Omre (2010), Ulvmoen and Omre (2010), Hansen et al. (2012)). Several geostatistical methods have been proposed to generate ensembles of reservoir property realizations: two-point geostatistics (for example, sequential indicator and sequential Gaussian simulations, or pluri-Gaussian methods, Doyen (2007)) and multi-point geostatistics methods (González et al. (2008)) are the most common. Some of these methods have been combined with optimization techniques (such as simulated annealing, genetic algorithms, gradual deformation, probability


71

perturbation method and ensemble Kalman filter) to obtain optimal models of reservoir properties. These methods have been applied to elastic inversion or simultaneous inversion of elastic and reservoir properties and facies (see Sams et al. (1999), Merletti and Torres-Verdin (2006), and Sams et al. (2011)). We propose here a new approach that aims at estimating fine-scaled reservoir models in a stochastic inversion by combining geostatistical methods, such as sequential simulations (Deutsch and Journel (1992)) and a stochastic optimization technique called probability perturbation method (PPM, Caers and Hoffman (2006)), with classical geophysical methods, such as seismic convolution and rock physics models (Mavko et al. (2009). Our methodology is mainly aimed to determine the optimal facies model for the reservoir. In our approach, we use sequential indicator simulation (SISim, Journel and Gomez-Hernandez (1989)) to generate facies models, and the probability perturbation method to perturb the probability used in SISim. At each optimization step, a new facies model is generated; reservoir properties, in particular porosity and clay content (or net-to-gross), are then simulated by sequential Gaussian simulation conditioned by the facies distribution; elastic properties are subsequently calculated by applying a rock physics model and converted in the corresponding time domain; and finally the synthetic seismic response is computed with a traditional convolutional model (Figure 4.1). The optimization objective function is the 2-norm of the difference between the synthetic seismic and the real seismic data. A similar approach has been presented in González et al. (2008), with the target being the direct inversion of facies with the integration of rock physics models and multipoint geostatistics. However, in their method, at each iteration of the optimization, the perturbation of the facies model is performed directly on the realization, whereas in our approach we perturb the underlying probability distribution used to generate the model. We introduce the probability perturbation method to obtain the optimal model in a reasonable number of iterations. In our methodology we also account for non-stationarity by introducing an additional probability distribution that can be derived from different sources (seismic or AVO attributes, for example). We first apply the stochastic inversion to a synthetic case with the objective of reconstructing the actual facies classification, in order to test the validity of the method.


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Figure 4.1: Workflow of stochastic inversion. As the proposed methodology includes different methods and models and several parameters have to be calibrated or assumed from prior knowledge or information from nearby fields, we propose a sensitivity analysis that investigates their effect on the corresponding estimations. The method is then applied to a well log profile and to a 2D seismic section of a real seismic reservoir characterization study in the North Sea (offshore Norway). Several studies have been published on a number of nearby fields in the North Sea (Avseth et al. (2001), Mukerji et al. (2001a), and Avseth et al. (2005)). In this example we integrated into the methodology a further probability derived from seismic data by means of a traditional Bayesian approach, to speed up the convergence and account for non-stationarity.


4.3

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Methodology

The inversion methodology we propose attempts to directly integrate the petroelastic model and facies classification into the seismic inversion workflow. The flowchart of the method is shown in Figure 4.1. In the following subsections we will describe each step of the method and the techniques that are used. The methodology is presented for a clastic reservoir, but it can be adapted to different lithology reservoir conditions, with the choice of a reliable rock physics model and a suitable facies classification.

4.3.1

Geostatistical methods

The application of geostatistics to reservoir modeling aims to integrate data from various sources (well, seismic and production data) into a consistent model to describe the rock properties of the reservoir and their spatial continuity. Sequential simulations are geostatistical methods that can be used to generate realizations of a probability density function of either discrete or continuous properties. These methods are based on various stochastic algorithms and they are applied in reservoir modeling to generate different realizations of reservoir properties. This procedure produces high-resolution simulations of the property we are interested in, by sequentially visiting the grid cells of a 1D, 2D or 3D space, along a random path. In each cell, the simulated value is drawn from the local conditional distribution, which depends on the prior distribution and on the previously simulated values in the neighborhood of the given cell. This procedure is repeated for all the cells of the grid. The methods nowadays available can be divided into two big categories: two-point geostatistics and multi-point geostatistics. Two-point geostatistics algorithms are in general faster as they only account for the correlation between two spatial locations at a time; the spatial continuity of the property distribution being ensured by variogram models. On the other hand, multi-point geostatistics takes into account the correlation between multiple spatial points but as it is very complex to analytically treat the associated conditional probability, the multi-point statistics are inferred from a training image generated for example by unconditional Boolean modeling. In our approach we use two-point geostatistics algorithms, but if a suitable training image is available, with


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size at least larger than the reservoir size, then multi-point geostatistics could be easily integrated. The two most common algorithms in two-point geostatistics are: sequential indicator simulation, SISim, and sequential Gaussian simulation, SGSim (see Deutsch and Journel (1992) or Goovaerts (1997)). Sequential indicator simulation deals with discrete random variables, for example facies in reservoir modeling, while sequential Gaussian simulation deals with continuous random variables, for example porosity. In our approach, facies are first simulated by SISim possibly with a secondary conditioning data derived from seismic (seismic facies probability, for example); then porosity is simulated by using SGSim. In particular, the simulation of porosity for each facies is performed independently of the simulations for other facies; each simulation is performed over the whole 3D grid, then the simulations are re-assembled into the final simulated porosity realization according to the facies classification. To grid-cells not belonging to the reservoir layer, a constant value of porosity (equal or close to 0) is assigned. Finally other reservoir properties (for example, net-to-gross, irreducible water saturation, and even permeability if necessary) are subsequently simulated by sequential Gaussian co-simulation (CoSGSim) with porosity distribution or previously simulated properties, as secondary information. In CoSGSim, a continuous variable is simulated by accounting for the correlation with another variable: this is the case for example of porosity and net-to-gross or porosity and permeability. At the end of this step, we obtain a reservoir model of facies, porosity φ, and net-to-gross ntg. In our approach, net-to-gross is converted into clay content (vclay ) by assuming vclay = 1 − ntg. Fluid saturation distributions (water, oil and gas saturations, namely sw, so, sg) could be simulated as well, but in order to obtain realistic

litho-fluid models we prefer to impose in the model the oil- (or gas-) water contact and the gas-oil contact (if present) and assume a constant distribution of the fluid within the so-obtained fluid layers. To increase the realism of the model, we could however simulate the irreducible water saturation through sequential Gaussian simulation or co-simulation or deterministically distribute it by assuming empirical relations with other properties such as porosity or permeability. One of the favorable features of sequential simulation is the ability to incorporate different types of conditioning data.


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However, if A is the unknown property and B and C are the conditioning data (for example hard data B and soft data C), then in many cases the analytical expression of the posterior probability P (A|B, C) can be difficult to obtain. Journel (2002) has proposed an efficient method to integrate secondary data (soft data C) in the probability model P (A|B) to get the posterior probability P (A|B, C). To combine P (A|B) and P (A|C) into P (A|B, C), Journel (2002) proposes the following expression: 1 1+x

(4.1)

τ1 c τ2 b a a

(4.2)

P (A|B, C) = where

x = a

and a=

1 − P (A) , P (A)

a=

1 − P (A|B) , P (A|B)

c=

1 − P (A|C) P (A|C)

(4.3)

P (A) is the prior distribution of the unknown property. The ratios a, b, and c can be interpreted as the distance to an event occurring. For example the ratio a is the distance to the event A occurring, prior to knowing the information associated with B and C: if P (A) = 1, then a = 0 and A is certain to occur. The parameters τ1 and τ2 account for the redundancy for each set of conditioning data B and C (Krishnan (2008)). Setting τ1 = τ2 = 1 is equivalent to assuming a form of conditional independence between P (B|A) and P (C|A) expressed in terms of permanence of ratio (Eq. 4.2). In other words we assume that the incremental contribution of data event C to knowledge of A is the same after or before knowing B; this assumption is however less restrictive than assuming independence between data B and C. The parameter τ2 can be modified to tune the contribution of the conditioning data C: if τ2 > 1 then the influence of C is increased (in our context, this could be the case where C is crosswell seismic where the resolution is higher than common surface seismic); if 0 < τ2 < 1 then the influence of C is decreased (this could be the case where the quality of the seismic is not optimal and the low resolution of seismic could obscure facies transitions). In our work we assume τ1 = τ2 = 1, but a preliminary sensitivity analysis at the well location is necessary to investigate the effect of these parameters. Several possible definitions are proposed for the information content


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measure related to Tau-parameters (Liu (2003)), but the determination of the optimal parameter values is still object of research. Tau-model formulation allows avoiding the computation of the probabilities P (B|C) and P (C|A, B) or P (B|A, C) that appear in the exact decomposition of P (A|B, C) and that are generally more difficult to calculate. In our application the unknown property A is the facies classification within the reservoir, B are the hard data and C is the seismic information.

4.3.2

Geophysical forward model

In order to compute the seismic response of the earth models generated by sequential simulations, we first calculate the elastic properties, such as velocities or impedances, within the reservoir model and subsequently compute the corresponding seismic signature. We observe that in many real applications, the geocellular grid (corner-point grid) used to model petrophysical and dynamic properties in the reservoir does not coincide with the seismic grid. In particular geocellular grid cells are usually larger than the bin size of the seismic survey, which calls for a downscaling of the grid (Castro et al. (2009)). Furthermore, the velocity models must be converted from depth domain to time domain, in order to perform seismic convolution and obtain synthetic seismic volumes. Depth-to-time conversion necessarily requires an accurate background velocity model, which is consistent with the seismic processing steps performed on the real seismic dataset. Elastic properties are usually computed through a rock physics model. This model is a set of equations that transforms petrophysical variables, typically porosity, mineralogy (clay and sand content, for example, in clastic reservoirs), and fluid saturations, into elastic properties, such as P-wave and S-wave velocities and density (or, as in many practical applications, P- and S-impedance). The rock physics model type depends on the reservoir rocks we are dealing with: the set of equations can be a simple regression on well data or a more complex physical model (Mavko et al. (2009)). Generally the model is first calibrated on well logs, where both petrophysical and elastic properties are available: in fact most of the models traditionally used contain one or


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more parameters (for example critical porosity and coordination number for granular media models, Mavko et al. (2009)) that should be determined from core analysis or estimated by comparing well logs and rock physics model predictions. Once the rock physics model has been calibrated on well logs, the model is applied, point-by-point, to the volumes of reservoir properties generated by sequential simulations. In the case of a clastic reservoir, these properties are generally porosity φ, clay content vclay , and fluid saturations sw, so, and sg. Clay content is usually computed from net-to-gross as vclay = 1 − ntg; this assumption does not account for mineralogical texture related

to laminated and dispersed shale that can influence the elastic properties of the rock.

However if more accurate petrophysical relations linking net-to-gross to lithological properties are available, they can be included in the forward model. The rock physics forward model can be described as follows: first we estimate the elastic properties of the solid phase, i.e. bulk and shear modulus of the matrix, Kmat and µmat , and density ρmat , by using solid phase mixing laws (Voigt-Reuss-Hill average or HashinShtrikmann bounds); then we compute the elastic properties of the fluid phase, i.e. bulk modulus Kf l and density ρf l , by using fluid mixing laws (Reuss average or Brie’s law); dry rock properties are then computed from solid phase properties by using one of the available theories in literature (for example, granular media or inclusion models) to obtain dry rock bulk and shear moduli, Kdry and µdry ; finally the saturated rock properties, Kdat and µsat , are calculated by Gassmann’s equations (Mavko et al. (2009)). Density of the saturated rock is computed as a linear combination of matrix density ρmat and fluid density ρf l weighted by their respective volume fractions ρ = φρf l + (1 − φ)ρmat

(4.4)

and P-wave and S-wave velocities are calculated by definition as function of saturated elastic properties, Kdat and µsat , and density ρ: s Ksat + 4/3µsat VP = ρ VS =

r

µsat ρ

(4.5)

(4.6)


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The result of this first step of the forward modeling is a set of volumes of elastic attributes: typically, P-wave and S-wave velocity, VP and VS , and density ρ, or Pand S-impedances IP and IS . The rock physics model can be facies dependent, such as in the field study we propose in the Application section. Data are subsequently re-sampled in the seismic grid and converted from depth to time. In our workflow, we downscaled the data using a subsampling technique, but more advanced downscaling methods could be used (Castro et al. (2009)). Depth-to-time conversion can be performed by applying a velocity model obtained by collocated cokriging using the stacking velocity volume used for the processing of seismic amplitudes and sonic well-logs filtered at low frequency. The second phase of the forward modeling step is the computation of the synthetic seismic signature. If partial stacked seismic data (dobs (θ)) are available, then we compute the corresponding angle stacks, otherwise if only post stack data are available we compute only the zero angle seismic traces. We describe here the partial stack case, as the post stack can be seen as a particular case of this application. Synthetic seismic traces (dsynth (θ)) are computed here by seismic convolution: the forward modeling is based on a convolutional model and Zoeppritz equations (Aki and Richards (1980)). Specifically, at each trace, the synthetic seismogram is computed by convolving the wavelets (estimated from the real dataset) with the reflection coefficients series: dsynth (t, θ) = wsynth (t, θ) ∗ RP P (t, θ)

(4.7)

where t it the travel-time, wsynth (t, θ) is the vector of the angle-dependent wavelets and RP P (t, θ) the vector of reflection coefficients.

Seismic reflection coefficients

RP P (t, θ) depend on the angle and the material properties of the subsurface: an isotropic, elastic medium is completely described by P-wave and S-wave velocity and density. For angles smaller than the critical angle of the seismic dataset, we can alternatively use a linearized weak contrast approximation of Zoeppritz equations (Aki and Richards (1980)). We point out that in order to perform the convolution, we estimate the wavelets independently for each available angle gather. Both forward models, seismic convolution and rock physics model, lead to underdetermined inverse problems. In our


79

approach, in order to solve these problems we adopt a stochastic optimization method in the inversion.

4.3.3

Stochastic optimization algorithm

The forward model results are included in an optimization loop to find the optimal model of facies (F). We perform a seismic-driven stochastic optimization where the objective function is the 2-norm of the difference between synthetic seismic data (dsynth ) and the real seismic amplitudes (dobs ). The stochastic optimization algorithm we used in our methodology is based on the probability perturbation method (Caers and Hoffman (2006)) and Tau model (Journel (2002) and Krishnan (2008)). In our approach the target proportions and the variogram models are assumed to be assigned, but they could be made stochastic and optimized simultaneously. However preliminary sensitivity tests in 1D showed that the convergence of the algorithm could be more than 10 times slower. We describe the methodology for a generic reservoir with NF facies. In particular the categorical variable F can assume NF possible values fk (for k = 1, , NF ). The facies value at a given location is coded using a set of indicator variables i(u, fk ) ( 1 if fk occurs at u i(u, fk ) = (4.8) 0 otherwise where u = (x, y, z) denotes a generic spatial location, corresponding to a grid cell in the reservoir grid. We first select a random seed and determine a random path of simulation; then we generate an initial realization of facies by using SISim and according to the selected variogram. We then simulate porosity by using SGSim, cosimulate other rock properties by using co-SGSim and apply the forward model to compute elastic properties and synthetic seismic data. This realization honors the hard data (for example the facies profiles at the well locations) but it does not necessarily match the seismic data. The initial realization is then perturbed: in the probability perturbation method rather than perturbing the initial realization directly, we propose a perturbation of the probability model used to generate the realization. We denote the underlying


80

probability model of SISim as P (Fk |G) where G indicates the hard data (well data

or previously simulated values). We remind that, in SISim, the probability of facies given hard data G is obtained at each location by indicator kriging.

At the following step of the optimization we propose a new probability, P (Fk |dobs ),

as a linear combination of the indicator model i0 (u, fk ) associated to the initial realization and the prior probability P (Fk ) of the facies: P (Fk |dobs ) = (1 − r)i0 (u, fk ) + rP (Fk )

(4.9)

where r is the scalar deformation parameter to be optimized between 0 and 1 (Caers and Hoffman (2006)). In our approach, we used a uniform discretization of the interval [0, 1] with spacing 0.1, which means that we evaluated the probability in Eq. 4.9 for 11 values of the parameter r. For each facies, the probability in Eq. 4.9 is a function of spatial location u, but for simplicity of notation we omitted the spatial dependency. At each spatial location u, we now have to combine P (Fk |dobs obtained from

Eq. 4.9, with the prior probability P (Fk ) and the probability P (Fk |G) obtained from indicator kriging conditioned to hard data, to obtain the probability P (Fk |G, dobs ).

This is done by using the Tau-model (Journel (2002)): τ1 1 c τ2 b P (Fk |G, dobs ) = , x=a 1+x a a

(4.10)

where τ1 and τ2 are the Tau-model parameters, and a, b, and c, are obtained by Eq. 4.3 with A = Fk , B = G, and C = dobs . In the case τ1 = τ2 = 1 (i.e, in case of conditional independence) Eq. 4.10 simplifies as follows: P (Fk |G, dobs ) =

a , a + bc

(4.11)

We sample from the distribution P (Fk |G, dobs ) to generate a new facies model

ir (u, fk ) and we repeat the above described reservoir modeling, by simulating rock properties, computing elastic attributes and synthetic seismic data dsynth (θ, r). For each facies, the probability of Eq. 4.10 depends on the scalar parameter r, in other words Eq. 4.3 provides a set of distributions, and the forward model result is a set of

81


models that depends on the deformation parameter r. For each model we calculate the objective function O(r) =

Nθ X k=1

ωθk ||dobs (θk ) − dsynth (θk , r)||2

(4.12)

where Nθ is the number of angle stacks and the ωθk are the weights assigned to the different angle stacks based on the quality of the seismic dataset. For example we could choose the weights directly proportional to the signal-to-noise ratio of the angle stacks. We finally perform a 1D optimization on the deformation parameter r. However the search space provided by the set of distributions in Eq. 4.9 is too limited because it is obtained as linear deformation of two realizations. Thus we introduce another optimization loop where we change the random seed and the optimal realization ir (u, fk ) obtained at the previous step replaces the initial realization i0 (u, fk ). The optimization step is indeed performed within two nested loops. In the outer loop, we change the random seed until a good match between the synthetic seismic traces of the trial model and the real seismic traces is achieved. At each step we perform a 1D optimization (inner loop) on the deformation parameter r of the probability perturbation method to obtain the parameter that minimizes the error between the synthetic and real seismic data. If the error of the new model is less than the error of the previous model, we accept the new model and we set i0 (u, fk ) = ir (u, fk ) otherwise we change the random seed, and we repeat the previously described steps. We iterate this procedure until the error is less than a fixed tolerance value T , which can be selected depending on the quality of seismic data, for example in terms of signal-to-noise ratio. The basic structure of the algorithm (Figure 4.2) can be described as follows: 1. Select a random seed, generate an initial realization of facies (namely i0 (u, fk )) using SISim; simulate rock properties and apply the geophysical forward model; 2. Perform a seismic-driven stochastic optimization using the probability perturbation method:


82

(a) in the outer loop change the random seed and iterate to obtain a good match, i.e O(r) < T ; (b) in the inner loop perform a 1D optimization to obtain the optimal deformation parameter r. In this inner loop, we propose a new probability P (Fk |dobs ), obtained as a

linear combination of the realization i0 (u, fk ) and the prior probability P (Fk )

of the facies (Eq. 4.9), we compute the conditional probability P (Fk |G, dobs )

by using Tau-model (Eq. 4.10), we generate a new facies model ir (u, fk ), apply the forward model and evaluate the objective function of the objective function of Equation 4.12. If O(r) < T , then we stop the algorithm, otherwise we set ir (u, fk ) = i0 (u, fk ) and repeat the procedure (steps 2a and 2b), with a different random seed.

Figure 4.2: Flowchart of PPM algorithm.


83

Two-point geostatistics Sequential Gaussian simulation (SGSim) is a geostatistical method that allows us to generate several realizations of a continuous property which honor: 1) the hard data (if any) used to condition the simulation, 2) the target distribution (if the distribution of the property is not already a standard Gaussian pdf) and 3) the variogram model which describes the spatial continuity of the property. In SGSim we sequentially visit the grid cells along a random path. At each cell, we simulate a value by sampling from a Gaussian distribution with mean equal to the kriging estimate at that location and variance equal to the kriging variance. Kriging estimate and variance at the given location are computed by solving the kriging system (Goovaerts (1997)). The procedure is repeated for all the cells in the grid. As in many practical applications, when the target distribution is not Gaussian, we usually apply a preliminary normal score transformation, perform the simulation and back-transform the results at the end. Sequential Gaussian simulation provides higher detailed map of the simulated property compared to the corresponding smoothed kriging map. Multiple simulations are generated by changing the random seed. Sequential indicator simulation can be seen as a generalization of SGSim. It is based on the concept of indicator variable, i.e. a binary variable, which is the indicator of occurrence of an event. SISim is generally applied to simulate discrete properties. The probability of a certain cell assuming a certain value of a discrete property, given the set of neighboring values, is calculated by indicator kriging. In indicator kriging we estimate the probability of a certain categorical event at a given location as a weighted linear combination of the indicator data falling within the searching neighborhood. As in traditional kriging the weights are obtained by solving the linear system of kriging equations which accounts for the indicator spatial covariance model (Goovaerts (1997)). The methodology relies on the result that the expected value of a binary indicator is the probability of the corresponding categorical event occurring; kriging, as a least square error estimation method, allows calculating this probability. At each generic location xn of the random path we compute the indicator kriging


84

probability pIK fk of the generic categorical event fk : pIK fk

= πfk +

n−1 X i=1

wi (i(xi , fk ) − πfk )

(4.13)

where πfk is the prior proportion of the categorical event fk , i(xi , fk )i=1,,n−1 are the indicators of the previously simulated values at the locations xi ; and wi are the kriging weights. By computing this distribution at each location and sampling from it, SISim allows us to perform simulations of discrete variables such as facies or litho fluid-classes.

4.3.4

Secondary information

To speed up the convergence we can include a further probability term in the Tau model, for example, the probability of facies obtained by inverted seismic attributes, P ∗ (Fk |S), where P ∗ represents the pointwise probability of facies conditioned by a

set of seismic attributes S at seismic scale, i.e. at low resolution. The probability of facies at seismic scale can be obtained by using different methods and it can be conditioned by different data, for example seismic impedances, P ∗ (Fk |IP , IS ), or seismic

amplitudes, P ∗ (Fk |dobs ), as in Grana and Della Rossa (2010), or AVO properties R0

and Gr , P ∗ (Fk |R0, Gr ) (see Mukerji et al. (2001a)). This step allows us to account

for low resolution secondary information, i.e. the probability of facies conditioned by

seismic, which improves the convergence speed and accounts for non-stationarity of the data. The star symbol in the following will indicate the probability of facies at seismic scale used as a low resolution trend to condition stochastic inversion simulations. This probability can be integrated into the workflow in different ways: in our method we used the Tau-model, by modifying Eq. 4.10 as follows: τ1 τ3 1 c τ2 d b P (Fk |G, S, dobs ) = , x=a 1+x a a a

(4.14)

where

1 − P ∗ (Fk |, S) (4.15) P ∗ (Fk |, S) However we point out that other methods could be adopted, such as, collocated d=

cokriging or Bayesian updating in sequential indicator simulations (Doyen (2007)).


85

Different sources of information can be used to obtain low resolution estimation of facies distribution, i.e. to derive the probability of facies at seismic scale, P ∗ (Fk |S).

We adopt here a probabilistic approach to seismic facies classification consisting of three main steps: seismic inversion to recover elastic attributes from seismic amplitudes; estimation of petrophysical properties from elastic attributes; and facies classification to classify seismic facies from petrophysical properties (Grana and Della Rossa (2010)). A full Bayesian approach has been adopted, based on the integration of the probabilities obtained from Bayesian seismic inversion and statistical rock physics model. First of all, a Bayesian seismic inversion is performed (Buland and Omre

(2003)) to obtain the probabilities of impedances from seismic amplitudes (step a). Then a probabilistic characterization of petrophysical properties is applied (Grana and Della Rossa (2010)) to estimate the probability of porosity, and clay content (step b) by integrating the statistical rock physics model with the probabilities of impedances obtained from Bayesian elastic inversion. Finally a probabilistic facies characterization is performed (Grana and Della Rossa (2010)): the estimation of facies probabilities P ∗ (Fk |S) conditioned by seismic attributes (step c) is obtained

combining petrophysical properties probabilities (step b), log-facies classification, and

seismic information from Bayesian elastic inversion (step a). The final results of this probabilistic multi-step approach are the probability volumes P ∗ (Fk |S) of seismic

litho-facies, that are used in stochastic inversion as additional information in the Tau model to condition the geostatistical simulations and account for non-stationarity. In general P ∗ (Fk |S) can be obtained from any set of seismic attributes, such as

inverted impedances, AVO attributes, full waveform inversion properties; however

the weight of this information, i.e. the exponent τ3 in the Tau model (Eq. 4.14), should be tuned after a sensitivity analysis, as all these properties are derived from the same seismic dataset which appears in the optimization objective function. In our application we tested the following set of parameters 0.5, 1, 2.5. By assuming a high exponent τ3 , we increase the convergence speed but we tend to disregard the prior information related to the spatial continuity model described by the variogram, in particular we generally obtain a model with a resolution closer to the seismic one. This result can be explained by the fact that we are accounting seismic information in


86

two different terms, c and d, with a relatively high weight in Eq. 4.14. According to the theory, the parameters of the Tau-model measure the additional contribution of the probabilistic information they are associated with. Since the parameter t3 is associated to the seismic-derived probability, and this information is already accounted by the probability term P ∗ (Fk |dobs ), we tend to exclude values greater than or equal

to 1.

4.4

Application

The application of stochastic inversion is first presented in a one-dimensional synthetic case to show the different steps of the method and verify its reliability. Then we applied the methodology on a real reservoir study in the North Sea where a complete dataset, including a set of well log data coming from three wells and partial stacked seismic data, is available.

4.4.1

Synthetic case

For the synthetic test, we built a set of pseudo-logs to mimic a realistic depositional turbiditic system along a well profile (namely well A). This synthetic case models a clastic reservoir filled by oil: the top of the reservoir is supposed to be positioned at around 2000m depth at the well location and the oil-water contact is fixed at 2100m. We focus on a depth interval of 200m, by assuming a thick overcap of clay on top of the reservoir. We created a synthetic facies sequence and a set of pseudo-logs mimicking the behavior of rock and elastic properties associated to the facies profile; and we finally generated synthetic seismic traces at three different angle stacks: 12o , 24o and 36o. Three facies are defined: sand, silty-sand, and shale. We first modeled the synthetic stratigraphic sequence in the well by a first-order Markov chain and then distributed the corresponding rock properties. Markov chains are a statistical tool that has been used in geophysics to simulate facies sequences to capture the main features of the depositional process (Krumbein and Dacey (1969)). Markov chains are based on a set of conditional probabilities that


87

describe the dependency of the facies value at a given location with the facies values at the locations above (upward chain) or the locations below (downward chain). The chain is said to be first-order, if the transition from one facies to another depends only on the immediately preceding facies. The conditional probability of the transitions are the elements of the so-called transition matrix P, where the generic element Pij represents the probability of a transition from the facies i located above the interface to the facies j located below. In our example, we estimated the input parameters, i.e. prior proportions and transition probabilities, from a real well dataset. The transition matrix is estimated by counting the number of transitions in the facies classification at the well:



 P= 

0.9

0.05 0.05



 0.93 0.07   0.05 0 0.95 0

(4.16)

Rows correspond to shale, silty-sand, and sand at the generic depth location z, and columns correspond to shale, silty-sand, and sand at the generic depth location z − 1. In other words, in our facies profile, we never have a shale on top of a silty-sand

or a silty-sand on top of a sand. The terms on the diagonal of the transition matrix are related to the thickness of the layers: in fact the higher are the numbers on the diagonal, the higher is the probability to observe no transition (i.e. high probability that a facies has a transition to itself), and as a consequence the thicker will be the layer. We define the first sample of the well profile as shale in order to have a shale layer above the top of the reservoir. At the next step the facies value is sampled from the conditional probability P (Fi |Fi−1 ), and we iterate the sampling till the bottom

of the interval (Figure 4.3).

Through this method we generated a facies profile which is assumed to be the true model of this synthetic example. The facies proportions in this well profile are 0.28, 0.34, and 0.38 respectively for sand, silty-sand and shale. We then generated pseudo-logs of rock properties, namely porosity and clay content (Figure 4.3). The pseudo-logs must be vertically correlated within each facies and at the same time they must be correlated between each other, as porosity in general almost linearly depends on the clay content of the rock. In our dataset, we created the pseudo-logs


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Figure 4.3: Synthetic well log dataset (well A), from left to right: effective porosity, volume of clay, water saturation, P-wave and S-wave velocity, density and facies profile (green represents shale, brown represents silty-sand, and yellow represents sand). of porosity and clay content by sampling from three bivariate Gaussian distributions, one for each facies. We assumed a correlation of 0.8 for every facies and we included a vertical correlation by multiplying (Kronecker product) the covariance matrices of each distribution by a spatial covariance matrix obtained from a 1D (vertical) exponential variogram with correlation range 7.5 m (for every facies). The simulated logs are finally re-assembled into the final pseudo-logs according to the facies classification profile. Then we computed the corresponding pseudo-logs of density and elastic properties, P-wave and S-wave velocity, by means of the soft sand model (Mavko et al. (2009)). Rock physics models are generally good approximations of the elastic behavior of rocks, but these relations cannot account for the heterogeneity and the natural variability of the rocks in the subsurface. We then added a random error, vertically correlated (the correlation range is 1 m), to mimic a more realistic behavior similar to measured well log data. Traditional rock physics crossplots and the estimated probability distributions are shown in Figure 4.4. Finally we computed the synthetic seismograms corresponding to three angle stacks: 12o , 24o and 36o by using three Ricker wavelets with three different center frequencies 30, 25, and 20 Hz PP reflection coefficients have been computed with Aki-Richards approximation. In the inversion methodology, we assume that the rock physics model is known (but


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Figure 4.4: (Top) Rock physics crossplots of well dataset: clay content versus effective porosity (top left), P-wave velocity versus effective porosity (top right), S-wave velocity versus P-wave velocity (mid left), and VP /VS ratio versus P-impedance (mid right), color coded by facies classification (green represents shale, brown represents silty-sand, and yellow represents sand). (Bottom) Joint probability of petrophysical properties distribution: conditional probability contours color coded by facies (bottom left), and joint probability surface (bottom right).


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the error is unknown) and the three wavelets corresponding to the three angle stacks are known as well. We finally assume that the facies proportions of the true model are known and correspond to the real proportions of the well log profile. The variograms of facies and rock properties used in sequential simulations and the cross-correlation between rock properties have been estimated from the pseudo-logs (Figure 4.5). For all the three facies we assumed a Gaussian model with the following correlation ranges: 3, 10, and 4, respectively for shale, silty-sand, and sand.

Figure 4.5: Variograms of porosity estimated at the well location, from top to bottom: variogram of porosity in shale, silty-sand, and sand. We show the results of the stochastic inversion methodology applied to the synthetic well A, assuming perfect signal-to-noise ratio, in Figures 4.6 and 4.7. The facies profile classified by our approach has a good match with the actual classification; good results are obtained after only 25 iterations, which correspond to 275


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evaluations of the forward model (since each iteration requires 11 forward model evaluations to locate the optimal parameter r). The more we proceed with the iterations, the lower is the acceptance rate: in fact even if we perform 100 iterations the improvement compared to the result obtained after 25 steps is quite small. In other words the stochastic optimization used in this approach quickly reaches a sufficiently small neighborhood of the minimum, then the convergence to the exact minimum becomes slower. We point out that since seismic data are generally noisy, we do not want to match perfectly the data, but only match the data within a certain tolerance (T ). The convergence can be sped up, by introducing secondary information describing the probability of facies at seismic frequency.

Figure 4.6: Stochastic inversion results at well location, from left to right: actual facies classification, initial realization, and partial results of the optimization loop after 3, 10 and 25 iterations classification (green represents shale, brown represents silty-sand, and yellow represents sand). The last result (right plot) is the optimized model according to the fixed tolerance. We then applied the methodology several times by using different random seeds: the result of the optimization after a fixed number of iterations will be statistically similar but every time different in the details: in Figure 4.8, we show the variability


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Figure 4.7: Synthetic seismograms (red) corresponding to the optimized model of Figure 4.6 compared to input seismic traces (black). From left to right: near, mid and far stack, corresponding to the incident angles of 12o, 24o , and 36o .


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between the solutions we obtained by plotting 10 of 25 different runs; this variability mainly depends on the tolerance T we fixed to satisfy the convergence criterion (O(r) < T ).

Figure 4.8: Set of 10 different realizations obtained by stochastic inversion. We show 10 optimized models (obtained from 10 different runs) compared to the actual classification (green represents shale, brown represents silty-sand, and yellow represents sand). Finally we used this synthetic example to perform a sensitivity analysis on different parameters (for each case, one parameter is changed at the time): 1. Signal to noise ratio of seismic data: SNR=5 (good quality seismic) or SNR=2.5; 2. Rock physics model: known (we use the same rock physics model used for the model generation) or unknown;


C shale C silty-sand C sand

T shale 0.81 0.11 0.07

T silty-sand 0.06 0.74 0.05

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T sand 0.13 0.15 0.88

Table 4.1: Confusion matrix of the reference case (T stands for true facies, C stands for classified facies). 3. Wavelets: known (we use the same wavelets used for the synthetic seismic generation) or estimated from real seismic data; 4. Variograms: real variogram or a different one with wrong correlation ranges; 5. Prior proportions of facies: real proportions or different ones; 6. Number of angle stacks: post stack seismic or partial stack seismic (2 or 3 angles); 7. Fluid effect: oil water contact known or ignored. The results of the different cases have been quantitatively compared by computing for each case the corresponding confusion matrix associated to the facies classification. The confusion matrix is a tool used in supervised learning to visualize the quality of the classification: in our application, each column of the matrix represents the percentage of samples in a predicted facies, whereas each row represents the percentage of samples in the actual facies. The confusion matrix of the reference case shown in Figure 4.6 (right plot) is summarized in Table 4.1. For all the three facies we obtain a satisfactory reconstruction rate. The results of the sensitivity tests are collected in Table 4.2, where we report the main diagonal of the confusion matrix and the percentage of correctly classified samples (normalized sum of the trace of the confusion matrix). Even though these statistics are not exhaustive to evaluate the quality of the inversion, this sensitivity analysis confirms that the number of angle stacks and the quality of the seismic data are the major sources of uncertainty in seismic reservoir characterization studies. The rock physics model is essential in this methodology; however the degree of accuracy of


True model Reference case Low signal-to-noise (SNR=2.5) Rock physics model (stiff sand) Wavelets Variogram (long range) Variogram (short range) Biased prior proportions Number of angle stacks (Nθ = 1) Number of angle stacks (Nθ = 2) Fluid effect

Diagonal confusion matrix 1.00, 1.00, 1.00 0.81, 0.74, 0.88 0.76, 0.72, 0.73 0.92, 0.69, 0.78 0.79, 0.83, 0.77 0.71, 0.48, 0.79 0.76, 0.59, 0.69 0.93, 0.52, 0.66 0.55, 0.33, 0.68 0.84, 0.49, 0.75 0.80, 0.76, 0.77

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Percentage of identified samples 100 % 81.0 % 73.6 % 79.6 % 79.6 % 66.0 % 68.0 % 70.3 % 52.0 % 69.3 % 77.6 %

Table 4.2: One-way sensitivity analysis of the synthetic inversion test: the first column show the different cases, in the second column we report the main diagonal of the confusion matrices of the different cases, in the third column we show the average of the elements of the main diagonal (sum of the trace of the matrix normalized by the number of facies). the model is usually quite high as the model can be calibrated at the well locations by using well data. Finally, in our tests, we observed that even if we underestimate the correlation range of the variogram, we still obtain good results in the optimization; conversely if we overestimate the range, then the optimization model cannot reproduce the correct thickness of the sediments, but it tends to create thicker layers. However we point out that these results are not completely general and they depend on the parameters we chose. For example, if we use the wrong rock physics model but the predictions are close enough to the observed data values (for example, a multi-linear regression), or if we ignore the fluid effect but the velocity in hydrocarbon sand is close to the velocity in brine sand, then the difference between the inversion results with the correct parameters and the ones with wrong assumptions could be small. Similarly the results of the sensitivity analysis on variogram parameters and prior proportions could be worse, if we introduce a more significant bias in the parameters.

4.4.2

Real case

As a final step, we applied the methodology to a real seismic reservoir characterization study, in the North Sea (offshore Norway). It is a deepwater clastic reservoir made


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of sand and shale and filled by oil. Four seismic-scale sedimentary litho-facies can be identified: soft sand and stiff sand (both filled by oil in the upper part of the reservoir), silty-sand with clay dispersed in it, and shale. The reservoir is located at approximately 2150m; the oil-water contact was measured at 2190m at the well locations and it is supposed to be known in the inversion. The data available are partial stacked seismic data (near stack corresponding to 8o , and far stack to 26o), a set of horizons for the reservoir level we are interested in, and 5 well log datasets. Well 2 has been used as calibration well as it contains the main acquired logs: P-wave velocity, S-wave velocity, gamma ray, density, neutron porosity and resistivity. Data of well 2 and well 3 are used to condition the simulations along the 2D line used to test the methodology in 2D. Finally well 5, that is located outside the seismic survey, is used as an additional test. First, the method has been tested at well 2 location to predict the facies distribution from seismic data. The litho-facies classification has been performed using sedimentological information, core analysis, and a clustering technique applied to petrophysical curves: effective porosity, volume of clay and volume of sand (Figure 4.9). A rock physics model has been calibrated at the well location: the more suitable model for this scenario is a constant-cement sand model (Avseth et al. (2005)). This model is a combination of the contact-cement model and friable sand model (Mavko et al. (2009)), where we assumed critical porosity equal 0.4 and coordination number 9 (Figure 4.10); however we point out that in some wells (wells 2 and 5) we can identify a relatively thick layer of soft sand at the top of the reservoir, therefore in this facies a soft sand model has been applied to explain the low velocity values measured at the well locations in the corresponding intervals. From well data we can also infer the marginal distribution of petrophysical properties conditioned by the facies classification (Figure 4.11): the estimated pdfs are used in the forward model to generate the simulated realizations at each iteration of the stochastic inversion. We assumed for simplicity, Gaussian distributions, but SGSim does not require that the prior distribution is Gaussian and other distributions could be used. The descriptive statistics of the different properties are shown in Table 4.3. The results of the 1D application with a perfect synthetic seismic trace (SNR=8)


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Figure 4.9: Real case application: well log dataset from well 2 (calibration well). From left to right: P-wave and S-wave velocity, effective porosity, clay content, water saturation, and actual facies classification (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow). are shown in Figure 4.12, where we reconstruct the litho-facies classification at well 2 location. In this test we assumed that the wavelets were known, in other words the wavelets used in the inversion were the same as those used in the forward modeling performed to generate the synthetic seismic trace. The actual facies profile is severely non-stationary and the estimation of variogram parameters was not reliable; we then assumed the same exponential model for all the facies with correlation range equal to 2.5m. We notice that even if we start from an initial model with a short vertical correlation range, we obtain a good result in the inversion: after 50 iterations (right plot) the error between the input seismic and the synthetic seismic generated from the optimized model is lower than the fixed tolerance T ′ , and the match between the optimized facies profile and the actual classification is satisfactory. The tolerance T ′ is fixed such that the ratio between the variance of the signal and the variance of the residuals approximates the SNR: in this example with perfect synthetic seismic we stop the convergence when the ratio approximates 10. The upscaled classification


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Figure 4.10: Rock physics model: (left) P-wave velocity versus effective porosity; (right) S-wave velocity versus effective porosity, color coded by clay content. Black lines represent constant-cement sand model for different clay contents (from top to bottom: 0%, 25%, 50%, 75%, and 100%). (see Stright et al. (2009)) over a support of 1m is provided for comparison. As already pointed out the tolerance value used to stop the inversion process depends on a number of factors, in particular the quality of seismic data (signal-to-noise and resolution): the more reliable is the seismic dataset, the smaller can be the tolerance. In most of the cases, we do not want to perfectly match the seismic data, but we prefer to obtain a model (or a set of models) that match the data within a fixed tolerance. In Figure 4.13 we show two different sets of inverted models: on top we show 25

Porosity Volume of clay VP (m/s) VS (m/s) Density (g/cm3 )

Shale 17% 47% 2591 1112 2.26

Silty-sand 24% 35% 2872 1292 2.18

Stiff sand 28% 28% 2820 1313 2.13

Soft sand 30% 14% 2581 1198 2.12

Table 4.3: Mean values of petro-elastic properties in the different facies. Values of porosity and clay content have been estimated from well log data, elastic properties values have been computed by rock physics model.


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Figure 4.11: Marginal probability density functions of effective porosity and clay content conditioned by facies classification. The pdfs of petrophysical properties are used to distribute rock properties within the reservoir model at each iteration of stochastic inversion (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow). inverted models obtained by imposing a small tolerance T ′ (on average 51 iterations are needed to reach the required accuracy); on bottom we show 25 inverted models obtained with a larger tolerance equal to 1.2T ′ (33 iterations required on average). By decreasing the tolerance value, we improve the match between the input seismic and the synthetic seismic of the generated models, but we increase the computational time to reach the convergence condition. The different variability within the two sets of realizations is shown by the e-types of the two ensembles (Figure 4.13). The e-type is the ensemble average of the set of models and it is a continuous variable. We then performed the same inversion exercise with the real collocated seismic


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Figure 4.12: Inversion results at well 2 location with synthetic seismic data, from left to right: actual facies classification, upscaled facies profile, seismic facies probability, initial model, optimized model after 50 iterations (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow). trace (Figure 4.14). The quality of the inversion is worse than the previous case but it is still satisfactory as pointed out by the confusion matrix reported in Table 4.4, where we obtain high recognition indexes for all the facies. In both applications, with synthetic and real seismic, we used as secondary information a low resolution probability estimated at the well location. In this application we assumed that the parameters of the Tau-model are: τ1 = τ2 = 1 and τ3 = 0.5. The use of this probability is necessary to account for the non-stationarity of the actual facies classification. In Figure 4.15 we show the convergence of the methodology with and without secondary information and we show the boxplots of the normalized error of 25 optimization runs, each of them consisting of 50 iterations. We notice that the convergence is much slower if no secondary information is used, and the average error is generally higher.


C shale C silty-sand C stiff sand C soft sand

T shale 0.95 0.09 0.07 0

T silty-sand 0.05 0.81 0.08 0.10

T stiff sand 0 0.10 0.72 0

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T soft sand 0.00 0 0.13 0.90

Table 4.4: Confusion matrix of the inversion results obtained by stochastic inversion at well 2 location (T stands for true facies, C stands for classified facies). We then applied the methodology to two other wells (wells 3 and 5), by using the same parameters for variograms, prior distributions of properties and forward models calibrated at well 2. Well 3 does not present soft sand in the classification, whereas the scenario in well 5 is similar to the calibration well. In both cases the inversion results are good (Figure 4.16), even though we notice that we cannot reproduce the thin layers below a certain thickness. We could actually obtain a set of models with the same vertical correlation observed at the well by using a shorter correlation range and a larger tolerance T and/or a smaller value of the Tau-model parameter τ3 (Eq. 4.14) related to the low resolution information. In fact if we assign a high weight to seismic data (low tolerance T and/or high parameter τ3 ) we tend to match the seismic data with higher accuracy and we cannot recover thin layers under the seismic resolution. If a larger tolerance is fixed or a lower weight is assigned to seismic data, we could recreate thin layers according to the input variograms; however, in this application we retain that some of the layers visible in the actual profile are due to clustering artifacts in the log-facies classification process. As conclusion of this study we perform the stochastic inversion in terms of facies of a 2D seismic section passing through wells 2 and 3. The near and far stacks are shown in Figure 4.17 and the top horizon (in time domain) of the reservoir is superimposed. The seismically derived probability of facies, P ∗ (Fk |S) has been computed following

the approach proposed in Grana and Della Rossa (2010). The maximum a posteriori of the so-estimated seismic facies probability converted in depth within the geocellular grid, is shown, as a reference, in Figure 4.18 (bottom right). The velocity model used for the time-to-depth conversion has been obtained by applying a kriging method to the 2D section by using filtered sonic logs (at a frequency of 4 Hz).


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The initial model has been generated using SISim: the 2D variograms of the facies have been estimated using information from previous studies on this field and nearby fields of the same area, and the prior information assumed by averaging the facies proportions at the well locations. The 2D variograms describe the spatial continuity model of the facies. For each facies we assumed an exponential model with the following parameters: the lateral correlation range is 1000m for soft sand, 2000m for stiff sand, 2500m for silty-sand and 800m for reservoir shale; the azimuth is 0o for all the facies. The optimization is performed for all traces simultaneously. The main result of this study is the optimized facies model (Figure 4.18, top): this result honors the prior information and the spatial continuity of the data; furthermore, the optimized realization honors the seismic data. Here we show the results after 10 iterations, corresponding to 110 2D simulations, in terms of facies, porosity, P-wave velocity, and the corresponding synthetic seismic (Figure 4.18). For simplicity, the porosity of the non-reservoir shale has been set constant equal to 0.05. As expected the areas with higher variability are the sequences of silty-sand and stiff sand in the lower part of the reservoir. As this application points out, the main advantage of stochastic techniques is that the estimated facies model has a higher resolution than the model obtained from the maximum a posteriori of the probability of facies directly inferred from seismic, and it can be directly used as initial model in fluid flow simulation, without the integration of additional geostatistical methods.


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Figure 4.13: Multiple realizations with different tolerance conditions: (top) 25 simulations obtained with a small tolerance; (bottom) 25 simulations obtained with a larger tolerance (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow). For each set of simulations we plot the ensemble average (e-type) and we compare the results with the upscaled facies classification at well 2 location.


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Figure 4.14: Inversion results at well 2 location with real seismic data, from left to right: actual facies classification, upscaled facies profile, seismic facies probability, initial model, optimized model after 50 iterations (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow).


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Figure 4.15: (Top) Convergence plot of stochastic inversion results with and without low frequency information (blue and red symbols respectively) as a function of iteration number. (Bottom) Boxplots of 25 runs consisting of 50 iterations with and without low frequency information (left and right respectively).


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Figure 4.16: Inversion results at well 3 (left plots) and well 5 (right plots) locations. We compare the actual classification with the optimized model obtained by stochastic inversion (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow).


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Figure 4.17: 2D seismic section passing through wells 2 and 3: (top) near angle stack corresponding to 8o , (bottom) far angle stack corresponding to 26o . The black line represents the top horizon, in time domain, corresponding to the interpreted top of the reservoir.


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Figure 4.18: Inversion results along the 2D section shown in Figure 4.17. On the left we show the optimized model of reservoir facies (top left), porosity (mid left) and P-wave velocity (bottom left), obtained by stochastic inversion. On the right we show the corresponding synthetic seismic sections, near (top right) and far (top left) and the maximum a posteriori (MAP) of seismic facies probability (converted in depth and mapped in the geocellular grid) used as secondary information in the inversion and obtained by multi-step inversion (shale in green, silty-sand in brown, stiff sand in light brown, soft sand in yellow).


4.5

109

Discussion

In this chapter we summarized the main results obtained by applying a new methodology for inversion of seismic data in terms of facies. Most of the geostatistical methodologies published so far, require large computational times to be performed for real reservoir studies. The introduction of the probability perturbation method allows us to reduce the computational time compared to other stochastic optimization methods. The probability perturbation method incorporates the Tau model, which is a probabilistic model to account for information coming from different sources. By using the Tau model, we can integrate well and seismic data and we could potentially extend the methodology to include other information such as EM data or production data. In our approach we used two-point geostatistics methods, to obtain models of facies and petrophysical properties; however the presented stochastic inversion could be applied also with multi-point based priors, if a suitable training image is available. Multi-point geostatistics could be necessary in litho-fluid inversion, to avoid nonphysical scenarios such as brine sand on top of oil sand. In our study we assumed that the oil water contact was known. We point out that in both cases, it is important to include secondary information in the methodology, specifically in the Tau model: this information represents the probability of facies at seismic scale, or in a broad sense, the low resolution probability of facies. This additional step has two goals: the first one is to speed up the convergence as shown in our application, the other one is to account for nonstationarity. Two-point and multi-point geostatistics are based on the assumption of stationarity, but in most of the cases this assumption is not completely satisfied. In our real case application for example, silty-sand and stiff sand approximately have a stationary behavior in the reservoir layer, but this is not true for soft sand, which appears only at the top of the reservoir. The scenario is more complex if we consider the shale layers at the top and at the bottom of the reservoir. The layers bordering the reservoir can be neglected in reservoir modeling if a reliable set of horizons is provided; however in seismic reservoir characterization the elastic contrasts at the


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top and at the bottom of the reservoir have a key role in seismic interpretation and inversion. The introduction of the probability of seismic facies, even though with a low resolution, allows us to account for the non-stationarity observable in the vertical stratigraphy. Moreover, we observed in the proposed real field application, that the integration of the secondary probability information reduces the number of iterations by a factor of 5 (assuming τ3 = 0.5). We point out that secondary information could be integrated in different ways: in addition to the Tau model used in our research, we could include this additional probability data in sequential simulations by using cokriging or Bayesian updating; and it could be derived from different sources of information, such as full waveform inversion, AVO attributes, or EM methods. Non-stationarity is also visible in petrophysical and elastic properties and it can be recognized by the presence of trends as a function of depth. To overcome this issue, we introduced in the real case application a deterministic trend of porosity and elastic properties to integrate/correct the rock physics model which does not explicitly account for depth-dependency. The trend observable in porosity could be due to different mechanisms and also to changes in lithology; for example we suspect that the layer of shale at the bottom of the reservoir has a different mineralogy than the overcap clay, even though the amount of clay recognized by means of gamma ray log is approximately the same. In general the depth trend is not linear (see Rimstad and Omre (2010)) and several models are available, but in a relatively small depth interval it can be approximated by a linear regression. We observe that in the forward modeling we used a convolutional model, which is a Born single scattering approximation of the first order. However, other techniques could be used such as the reflectivity methods (Kennett algorithm) or 2-D Born filtering. Similarly different rock physics models could be introduced to link elastic and petrophysical properties. A limitation of this methodology is the presence of tuneable parameters, such as the Tau model exponents, variogram parameters, and the optimization tolerance. The determination of the Tau-model weights is still a subject of research and a methodology to quantitatively express the data-dependency or the dependency of the probabilistic information, through the Tau-model exponents, is still missing. Especially when redundant information is incorporated in the Tau-model,


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it is recommended to perform a preliminary sensitivity analysis on the parameters of the Tau model, possibly at the well location where a facies classification is generally provided by formation evaluation studies. In our study we used τ1 = τ2 = 1 and τ3 = 0.5 after some synthetic tests performed at the well location to investigate the effect of the Tau model parameters, but the final choice is partially a subjective decision. The optimization tolerance T significantly influences the variability of the set of models of multiple realizations: the smaller is the tolerance the smaller is the variability within the set of models obtained from different runs. This choice is related to how well we want to match the data. This parameter depends on several conditions related to the seismic dataset of the reservoir study and it should be assessed by trial and error. A similar conclusion can be drawn for variogram parameters: in this case an assessment of the parameters is harder, especially for the correlation ranges in the lateral directions, due to the lack of multiple wells distributed spatially. We have seen that, in the 1D application, the key is to select the appropriate parameters to reconcile the expected vertical variogram and the match with real seismic data; but this assessment cannot be done in a 2D or 3D application unless a very large number of wells are available. The assumptions related to the choice of correlation ranges and anisotropic parameters of the variogram model are crucial; in real reservoir studies this choice can only rely on prior geological information of the field or nearby fields. We observed that solutions are better, with finer scale features, when the variogram range is underestimated. As pointed out by an anonymous reviewer, this behavior may be quite general. By giving a short range we specify a prior information outside the seismic bandwidth and the inversion process can adapt better the solution to the specified prior (i.e. it has more freedom to arrange the thin layers in a way that gives a good seismic response). On the contrary if we specified a long range, the prior may be less complementary to the seismic frequency content. In the case of a wrong prior (too large thickness linked to an overestimation of the ranges) it may be then more difficult for the inversion algorithm to adapt the (thick) layers to give the assigned seismic response. Seismic data could be used to estimate these properties, but the lateral continuity of seismic is generally affected by the migration operator applied to data, which could lead to an overestimation of the lateral ranges of the


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variograms. One possible future development is to use a multi-parameter approach in the probability perturbation method (MP-PPM), where we optimize more than one parameter simultaneously. One particular property that could be made stochastic and perturbed in this manner is the facies proportion, especially when few wells are available in the field.

4.6

Conclusion

We presented a new methodology for facies and reservoir properties modeling which combines traditional geophysical models, such as rock physics and seismic forward modeling, with geostatistical methods. The proposed approach is a stochastic optimization based on probability perturbation method and Tau model. The use of sequential simulations allows us to generate fine-scaled models, while the probability perturbation method guarantees that the optimized model matches the real data within a fixed tolerance. The main advantage of this technique is that it provides high resolution models of facies and the associated properties in a moderate amount of computational time. The method is fast, especially if secondary information is provided. This secondary information can be obtained from seismic attributes through different techniques: if the secondary information is not taken into account, the convergence could be quite slow, especially in complex sequences of thin layers, and the assumption of stationarity of geostatistical methods cannot be satisfactorily overcome. Different probabilistic information from different sources and at different scales can be integrated into the methodology thanks to the use of the Tau model. The application to the real well data shows that the methodology can be applied to complex reservoirs with good results.

Chapter 5 Statistical methods for log evaluation 5.1

Abstract

Formation evaluation analysis, rock physics models and log-facies classification, are powerful tools to link the physical properties measured at wells with petrophysical, elastic and seismic properties. However this link can be affected by several sources of uncertainty. In this chapter, we propose a complete statistical workflow for obtaining petrophysical properties at the well location and the corresponding log-facies classification. This methodology is based on traditional formation evaluation models and cluster analysis techniques but it introduces a full Monte Carlo approach in order to account for uncertainty evaluation. The workflow includes rock physics models in log-facies classification to preserve the link between petrophysical properties, elastic properties and facies. The use of rock physics model predictions guarantees obtaining a consistent set of well log data that can be used to calibrate the usual physical models used in seismic reservoir characterization and to condition reservoir models. The final output is the set of petrophysical curves with the associated uncertainty, the profile of the facies probabilities and the entropy, or degree of confusion, related to the most probable facies profile. The full statistical approach allows us to propagate the uncertainty from data measured at well location to the estimated petrophysical 113

CHAPTER 5. STATISTICAL METHODS FOR LOG EVALUATION

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curves and facies profiles. We apply the proposed methodology to two different well log studies to show its applicability, the advantages of the new integrated approach and the value of uncertainty analysis.

5.2

Introduction

In reservoir modeling and seismic reservoir characterization most of the physical models used are calibrated at well location. However well log data do not provide direct measurements of the reservoir properties we account for in reservoir models: the desired reservoir properties are generally obtained from the measured well logs through formation evaluation analysis. This process contains several sources of uncertainty. Similarly log facies classification can be severely affected by the uncertainty of petrophysical curves performed in formation evaluation. Furthermore, in some cases, the classification could be not linked to seismically derived attributes if rock physics modeling results are not included in the classification methodology. The first goal of this work is to present a new methodology for log-facies classification based on both petrophysical and acoustic/elastic properties in order to link log-facies to seismic inverted attributes. The second added value of the workflow is the introduction of Monte Carlo simulations to generate several realizations of petrophysical and elastic curves to obtain different log-facies profiles which are used to infer facies probabilities and facies uncertainty. The first step of the methodology is formation evaluation analysis. The petrophysical evaluation of subsurface formations requires the combined efforts of log measurements and core data together with a quantitative log interpretation model. The main results obtained from the petrophysical interpretation of well logs are the volumes of certain formation components (solid matrix and fluids) at each data level which combine the measurements provided by several tools such as well log resistivity, acoustic, density, neutron, nuclear magnetic resonance, fluid sampling, coring, and imaging (for details we refer the reader to Darling (2005), and Ellis et al. (2007)). The standard approach consists of optimizing simultaneously equations described by one or more interpretation models. Formation evaluation is done by solving the so-called inverse


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problem, in which well log measurements and response parameters are used together to compute volumetric results. An improved approach has been recently introduced by Heidari et al. (2010). However, there is always some uncertainty in the inferred parameters obtained by the inversion process since errors in any of the measurements lead to errors in the final values (see Fylling (2002), Verga et al. (2002), Kennedy et al. (2010), and Viberti (2010)). Moreover, other sources of uncertainties in the determination of the volume fractions come from the theoretical models used in the interpretation and the input parameters characterizing them. Hence a statistical quantitative log interpretation is performed in order to obtain reliable results and to assess the uncertainty for the characterization. Rock physics modeling represents the second step. Rock physics is mainly used to establish the link between petrophysical parameters and acoustic/elastic properties. In our approach, the uncertainty from formation evaluation is propagated through the well-calibrated rock physics model in order to provide a full probabilistic petroelastic model. This chapter only covers the application of the basic models used in rock physics and the statistical approach. For those who are interested in in-depth knowledge of rock physics, we refer to Bourbie et al. (1988), Nur and Wang (1989), Wang and Nur (1992), Wang and Nur (2000), Avseth et al. (2005) and Mavko et al. (2009). Statistical rock physics has been introduced by Mukerji et al. (2001a) and it is essentially based on Monte Carlo simulations. An extensive explanation is included in Doyen (2007). Finally a log-facies classification is performed at each depth location of the well. Reservoir facies are usually defined from sedimentological information and core analysis. This classification must then be consistent with the one performed on well log data in order to link petrophysical properties with reservoir facies. Traditionally, log-facies classification is based on multivariate techniques (in particular cluster analysis, see Kaufman and Rousseeuw (1990)) introduced to automatically identify common features within well log data and computed curves (generally petrophysical properties). In this work, we also account for elastic properties in order to guarantee a clear discriminability in the petro-elastic domain, through rock physics modeling. This issue is of key importance for the following step of seismic facies classification


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performed on seismic data to obtain reliable reservoir models. In this chapter we do not focus on the classification of the seismic data but on the initial well log analysis and log-facies classification. The probabilistic petro-elastic model yields a statistical log-facies classification. Thus the probability of occurrence of different facies at the well is obtained. The concept of entropy (Shannon (1948)) is then exploited in order to quantify the uncertainty related to the discrete random variable describing the facies distribution along the well. At each step of the workflow, namely formation evaluation analysis, rock physics modeling, and log-facies classification, we account for the sources of uncertainty associated with measured data, numerical models and natural heterogeneity in order to provide for each estimated curve, the corresponding confidence interval at each depth location along the well. The crucial added value is the assessment and propagation of the uncertainty at each step in order to finally provide a reliable classification fully based on the amount of information characterizing the entire workflow. In general, formation evaluation, rock physics models and log-facies analysis are commonly implemented neglecting the uncertainty involved in these processes and this fact can lead to an erroneous final interpretation. This is exactly the pitfall the probabilistic petro-elastic classification tries to avoid by quantifying the reliability of each step and of the final result. Moreover, the discriminated facies are automatically characterized from the petrophysical and acoustic/elastic point of view, which is a key factor in data integration in seismic reservoir characterization. Eidsvik et al. (2004b) presented a probabilistic log-facies formulation using hidden Markov models to get the posterior probability distributions of the log facies accounting for uncertainties in the log data and rock physics model, but did not account for the uncertainties in the formation evaluation analysis. Differently from previously published works in formation evaluation analysis and log-facies classification, the uncertainty propagation problem is faced by using a full Monte Carlo approach for each step of the method. Monte Carlo methods allow us to numerically estimate the posterior probability density functions (PDFs) of the variables we are interested in: in formation evaluation we estimate the posterior


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PDFs by propagating, point by point, the uncertainty from acquired well logs; in rock physics modeling we estimate the posterior PDFs by propagating the uncertainty from the previously obtained petrophysical curves; finally both PDFs are combined in log-facies classification where we sample from these distributions and we combine the results to statistically classify the facies, point by point, and estimate the associated uncertainty. The so obtained uncertainty represents the uncertainty associated to the degree of approximation of the physical models used in the method and the uncertainty related to error measurements. However random errors are also taken into account to represent the natural variability and heterogeneity of the rocks. The whole methodology is first illustrated by applying all the steps to the same well log dataset of an offshore West Africa field. We then perform the same study on another real case, located in the North Sea, where a complex sedimentological facies classification has been identified.

5.3

Methodology

The methodology can be divided into three parts: 1. Quantitative Log Interpretation (QLI); 2. Rock Physics Model (RPM) computation; 3. Log-Facies Classification (LFC). The flowchart of the methodology is shown in Figure 5.1. In this specific case it refers to a clastic environment, but it can be applied to different scenarios if suitable log interpretation and rock physics models are available. In the following we analyze in detail these three steps and the uncertainty propagation through them.

5.3.1

Quantitative Log Interpretation

The petrophysical characterization of subsurface formations is an inverse problem and generally requires an integrated approach. Well log measurements and core data are


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Figure 5.1: Flowchart of the probabilistic petrophysical properties estimation. combined together with a Quantitative Log Interpretation (QLI) model. QLI, also known as formation evaluation, provides the volume fractions of interest (solid matrix and fluids) at each depth level by solving the corresponding inverse problem, in which well log measurements and response parameters are reconciled with the theoretical interpretation models. In particular, the inverse problem is based on an opportune cost function that expresses the distance between the observed measurements and the predictions of the model chosen to describe the system. The final aim is to minimize the cost function and determine the solution (i.e. the volume fractions of the formation components) that optimally mimics the observables. Theoretical response-equations in QLI come from a simplified description of the physics involved in the well log instruments and are considered together with several constraints. These are typically used in order to impose geological and/or petrophysical prior knowledge to avoid physically meaningless results and define the domain space of the problem. In details, QLI can be divided into three different phases: • system parameterization, defining the set of parameters (unknown volumes)


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which characterizes the formation; • direct modeling which involves the laws (linear and/or non-linear) able to generate the synthetic values for the observables, once the parameters describing the model are fixed; • inverse modeling, playing its role in volumetric quantification. System parametrization and direct modeling are quite straightforward; even so, it is worth mentioning a few points of discussion that need to be taken into account to properly define the domain space. Mathematically, at a given depth x , the vector d(x) represents the observational log data (e.g. gamma ray, neutron, density and so on). QLI model is determined by the components of a vector p(x) containing the depth-dependent unknown volumetric fractions, such as porosity φ, volume of clay vclay , water saturation sw and so on. For simplification purposes, from now on the dependency upon the spatial position x is disregarded and it is assumed that all the recorded log measurements are sampled with a constant depth step. Formally we can write: p = [φ, vclay , sw , ...]

(5.1)

Once a well is drilled into the formation, to maintain the pressure balance, some mud fluid is pumped into the borehole. Often, drilling is done at over-balanced condition and mud has a pressure slightly higher than the formation pore pressure. This pressure gradient induces mud filtrate to seep into the porous rock system. Hence, well log measurements are affected by, at least, the existing step invasion profile due to the circulating fluid. The mud invasion divides the formation in two separated parts: the so-called flushed zone (usually labeled XO) and the unflushed one, or deep zone (DE). This splits the set of unknowns into two subsets, and the vector p can be written as: p = (pXO , pDE ).

(5.2)

Shallow reading log measurements are assumed to respond only to volumes of formation components in the first zone (pXO ). Similarly, deep reading logs only see


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the unflushed zone pDE ). Some tools, which have a medium depth of investigation, are assumed to be influenced by both zones, and their response equations contain terms for all formation components, regardless of the zone. On the other hand, physically-based assumptions constrain and reduce the domain space of the unknowns. For instance, the standard normalized-to-unity investigated total volume fractions (both in shallow and deep zone) need to be achieved. Moreover, it is common to assume lateral continuity: all the solid formation components extend infinitely from the borehole at zero degrees dip and porosity in the flushed and undisturbed zone is the same, regardless of the type of fluids filling the pore space. So, the sum of the fluid volumes in the flushed zone is equal to the sum of fluid volumes in the unflushed zone. In a more compact form, all of the constraints can be collected in the function . Synthetic log data are obtained by applying the direct modeling operator for a given fixed values of the model parameters u(p): Synthetic log data s are obtained by applying the direct modeling operator fQLI (p) for a given fixed values of the model parameters p : s = fQLI (p).

(5.3)

The forward model is a set of equations, linear or non-linear, which describes the link between volumetric curves (p) and petrophysical properties measurable at the well (d). Inverse modeling tries to determine the components of the parameter vector p, using the observational data d. The optimization problem involves a cost function to be minimized in order to reduce the discrepancy e between the observed data and the synthetic data generated by the direct modeling: e = s − d = fQLI (p) − d.

(5.4)

The definition of the cost function introduces the concept of a weighted 2-norm for the vector e. As a matter of fact, since the physical observables and measurements come from several well log tools and involve absolute values which are in general very different (and have different units), we have to normalize the 2-norm in order to treat all the quantities at the same level. This normalization mainly depends on


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typical tool accuracy. Moreover an additional term, a multiplier, is incorporated in order to weight the different tools with respect to their influence in the final answer and it is based on log analyst experience. The understanding of how a tool can affect the results and the knowledge of tool physics is required to select the multiplier values. Finally, normalizations and weight multiplier factors can be organized into a weighting matrix W. So, the cost function reads 1 T j(p) = e We . 2 u(p)

(5.5)

which is a scalar and it can be represented as a function of the unknowns collected in the vector p. The aim of optimization is to determine the optimal solution p∗ that minimizes the above cost function, under the set of constraints u(p) representing the solution space for the problem. The so-obtained vector p∗ represents the output and solution of the standard QLI. We now set the scenario regarding the uncertainty analysis and evaluation for the QLI methodology. As a matter of fact, there is always some degree of uncertainty in the inferred volumes p∗ obtained by the inversion process since each piece of information introduced in system parametrization and direct modeling can be affected by several errors. Quantification and propagation of these uncertainties throughout the inversion problem is here considered and assessed together with appropriate methods. The main sources of uncertainty in QLI come from well log measurements, heterogeneity of rock system, possible thin layered intervals, invasion effects, lithological and/or textural assumptions, environmental correction issues, simplified interpretation models and input parameters. A complete analysis of the above uncertainties would involve a full knowledge of the individual contributions and such detailed preliminary evaluation is rarely, if ever, available. Thus, the practical quantification problem simplifies the classification of sources into three macro-groups: methodological, systematic and random errors (see Theys (1991), Theys (1994), and Theys (1997)). Methodological errors are mainly due to an incorrect choice of interpretation models and related parameters. In general, model selection is one of the most critical steps in a reliable QLI and, for sure, it is mandatory to keep in mind all of the simplifying hypotheses of the real system it intends to describe. Since an inaccurate model


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selection could introduce significant errors to the interpretation, at this particular stage core measurement calibration and validation are fundamental. Petrophysical experience plays a role as well. Potential errors associated to an incorrect choice of the theoretical models are difficult to quantify and their magnitude may vary significantly. This is to state that methodological errors, once assessed, should be mainly used as a sort of guidelines on the applicability limits and constraints underlying each model. Systematic errors are defined as reproducible inaccuracies of the measurement mainly due to imprecision of the instrumental system, processing of the data, environmental conditions and so on. Systematic errors cannot be removed by repeated runs, thus they need to be recognized and corrected before any calculation. Random errors are those that need to be accounted for and propagated throughout the QLI process. In fact, random errors are mostly associated to the physics of the well log measurement system and cannot be corrected because they cannot be reproduced. Statistical variations in count rates or signal noise are examples of random errors. Also the uncertainty associated to the parameters used to correct the measurements for any environmental effect has an important impact in the final petrophysical characterization and must be accounted for. Given the broad spectrum of random uncertainty, a statistical treatment of the problem is suitable. Despite this apparent need, most formation evaluations in practice are based on a single deterministic description that can reproduce with a certain quality and confidence the well observations. The single deterministic model and its description make the adopted model rather unsuitable for uncertainty assessment. On the other hand, the statistical approach combines the deterministic QLI with a Monte Carlo Simulation (MCS) in order to build a probabilistic framework to study the natural variability of the results (Figure 5.2). In order to avoid confusion, in what follows deterministic QLI means with no MCS implemented. We start from a number of uncertain inputs representing log measurements. Uncertainty in each input variable is represented by a probability density function (PDF)


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Figure 5.2: Schematic representation of petro-elastic uncertainty estimation and logfacies classification through Monte Carlo simulations of petrophysical and elastic properties. whose range of values correspond to the nominal uncertainty provided with the measurement. In particular, Gaussian distributions are used to represent the random vari˜ ∼ N(d, Σd ) and describe the uncertainty of acquired data: the measured value able d

d represents the mean, the standard deviation is equal to the instrumental/processing

error (collected in the covariance matrix Σd ), and N is a compact notation for the normal distribution. This set of distributions, estimated at each depth location along the well profile, provides the bandwidth of tolerance for the response variables of the forward model. Model uncertainties can be added to the forward modeling fQLI by means of a random noise εQLI , distributed with zero mean and a prior covariance matrix, so that the forward model becomes: s = fQLI (p) + εQLI .

(5.6)


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In most of the cases the forward model can be represented as: s = [gr, nt, dn, rs] = fQLI (φ, vmin , sw ) + εQLI .

(5.7)

where φ is porosity, vmin is the vector of the volumes of mineral fractions, sw is water saturation, gr is Gamma ray, nt is neutron porosity, dn is the acquired density and rs is resistivity, and it allows computing the volumetric fractions (porosity, volume fractions of mineral, and saturations) from the acquired logs. In our approach we include also velocity data to guarantee the consistency with rock physics modeling step that follows. ˜ at each The simulation is performed by randomly sampling from the PDFs of d data level the values for each input log required for the interpretation model (see also Viberti (2010)). Suitable vertical correlated and conditioned Monte Carlo sampling can be implemented in a straightforward way in order to perform a geological driven sampling and minimize the probability of occurrence of physically meaningless scenarios. Once a set of values has been sampled, the QLI cost function is optimized and the results are stored. When a statistically representative number of realizations have been drawn the results can be sorted and histograms created to approximate the local ˜ PDFs of the uncertain output p ˜ ˜ = F (d) p

(5.8)

where F = (fQLI + εQLI )−1 . Thus, the single vector p∗ obtained in the deterministic QLI is now enhanced to ˜. account for the uncertainty of p Once this goal is achieved, all inferences can be obtained from the posterior PDFs by computing statistics relative to individual parameters. Since the final result is presented as probability density functions, this tool provides a unified framework for volume estimates and for the uncertainty associated with them. Formation evaluation analysis equations We recap here the basic equations for formation evaluation analysis (Darling (2005), and Ellis et al. (2007)). Starting from well log measurements, formation evaluation


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computes volumetric curves of the corresponding formation components. Assuming a unitary investigated total volume then the following relation holds: N X

vh = 1

(5.9)

h=1

where 0 < vh < 1 is the volumetric fraction of the single hth component and N is the total number of minerals and fluids defining the modelled formation. Fundamental in dealing with response equations is the knowledge of how clay is handled and the concept of wet versus dry clay. Here, we take the clay as wet so the volume fraction, vclay , is intended to consider this phase composed of dry clay and associated bound water. As already mentioned, well log measurements could be affected by the existing step invasion profile due to the circulating mud fluid. The mud invasion divides the formation in two separated parts: 1) the flushed zones (XO) and 2) the unflushed deep zone (DE). Well logs tools have different depths of investigation and can be influenced only by one zone (shallow reading log tools and deep reading logs) or by both zones; then their response can contain terms for only one or for all formation components. Mathematically this is taken into account by a special factor, 0 < Ψi < 1, called the invasion factor, which controls how much influence for each log i (such as neutron porosity or density) comes from the XO zone. The remaining influence (1 − Ψi ), comes from the DE zone. In well log interpretation processes it is

common to assume a lateral homogeneity and as a consequence to assume that the sum of the fluid volumes in the flushed zone is equal to the sum of fluid volumes in the unflushed zone:

Nf luid

X k=1

(vXOk − vDE k ) = 0

(5.10)

where the sum only considers the fluid involved. Other constraints involve the feature of the mud used. For example, when oil based mud is used the invading fluid is hydrocarbon. Therefore, we must impose a constraint that limits the volume of water in the flushed zone to be less than or equal to the volume of water in the undisturbed zone on the basis of the normal response equations. The reverse is true for water based mud.


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Finally it is also possible to limit the sum of the fluid volumes to be less than a particular critical porosity. This critical porosity is the value which marks the limit between a saturated rock and a suspension and it will be relevant also for the rock physics model. In general, a response equation is a mathematical description of how a given measurement mi (where i is the generic index to identify a well log type) varies with respect to each formation components. The simplest linear response equations are of the form mi (v) =

NX solid j=1

Nf luid

Nf luid

m ˜ j vj + (1 − Ψi )

X

m ˜ XOk vXOk + Ψi

k=1

X

m ˜ DE k vDE k

(5.11)

k=1

where m ˜ j, m ˜ XOk , m ˜ DE k are the response parameters for the corresponding formation components. This form is typical for neutron, density, and sonic logs. As gamma ray log only considers solid components, then it is not affected by invasion and the response equation does not involve the invasion factor. On the other hand, the resistivity/conductivity model used is non-linear and in our formation we consider two different log measurements: one related to the flushed zone (shallow reading log) and the other in the undisturbed zone (deep reading). The response equations are the same for both situations, but the parameters involved depends on the investigated zone. For conductivity (C), we consider the theoretical form of the Indonesian equation (Poupon and Leveaux (1971)) since it is widely used in shaly sand formations: √ n2 √ p vw Cw 12 (m+ mφ2 ) (1− 12 vclay ) C= Cclay vclay + √ φ (5.12) a φ where Cclay and Cw are the specific conductivities of clay and water respectively, vw

is the volume of water and the parameters m, m2 , n and a come from empirical observations.

5.3.2

Rock physics modeling

A Rock Physics Model (RPM) is represented by a set of equations that transform petrophysical variables to acoustic/elastic variables. RPM can be a simple regression

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on well data or a more complex physical model characterized by a number of parameters that need to be estimated, such as elastic moduli of matrix and fluid components, critical porosity, aspect ratio, and/or coordination number (Mavko et al. (2009)). Generally, the standard workflow starts from the model calibration from well log data, or the estimation of the physical parameters involved, in order to obtain a good match between RPM predictions and well log measurements. The model type depends on the geological environment and the parameter estimation is given by the solution of an inverse problem. In the traditional rock physics workflow, the generic RPM can be written in the following concise form: r = fRP M (p).

(5.13)

where the vector p represents the petrophysical input obtained from QLI and fRP M comprises the set of equations defining the model. The output vector r is the vector containing acoustic and elastic properties, typically P-wave velocity (VP ), S-wave velocity (VS ) and density (ρ): r = (VP , VS , ρ)

(5.14)

In literature, there are many different RPMs and they can be classified in three big classes (see Avseth et al. (2005), and Mavko et al. (2009)): • empirical models: in this case RPM usually is a simple regression using elastic moduli or directly velocities;

• granular media models: these are based on Hertz-Mindlin contact theory which assumes that the rock is represented by a random pack of spherical grains;

• inclusions models: these models describe the rocks as a sequence of inclusions (typically ellipsoidal) until the desired pore fraction is achieved.

The various models differ for the calculation of the dry rock properties. In fact, the usual scheme of a RPM starts from the computation of fluid and solid phase (matrix) properties. Then, dry rock properties, i.e. the properties of the solid phase


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with its own porosity, are calculated with equations depending on the RPM chosen. Finally, the effect of fluids using Gassmann equation is included. In general, once the RPM has been calibrated to well log data, to evaluate the fluid effect, different saturation scenarios could be considered: a ”brine” scenario where the hydrocarbon column in the reservoir is substituted by water and a ”full hydrocarbon” scenario where hydrocarbon saturation is constantly increased up to one minus irreducible water saturation. As described, our aim is to assess the uncertainty in RPM by propagating it from QLI probabilistic output. From the previous QLI analysis we have a number of realizations of the petrophysical variables representing basic volumetric properties of ˜ . Thus, fRP M the formation: uncertainty is represented by the obtained PDFs of p links the uncertain QLI inputs to multiple RPM output variables. In the end, the output PDFs of elastic properties ˜r are generated through a MCS (Figure 5.2, top), extending the deterministic RPM to the probabilistic case: ˜r = G(˜ p)

(5.15)

where G = (fRP M + εRP M )−1 . This means that the random errors of the first step (QLI) are propagated through RPM. In order to account for the uncertainty associated with the RPM approximation, a random noise εRP M can be added to the fRP M + direct modeling operator. MCS still provides a simple way to propagate all uncertainties in the RPMs since the latter are in general non-linear (except for linear empirical models). As we previously mentioned, we will restrict our description to a clastic reservoir, without losing generality. For high-porosity clastic reservoirs, common rock physics models are the so-called granular media models (see Mavko et al. (2009)). These models are based on Hertz-Mindlin contact theory (see Dvorkin et al. (1994), Dvorkin and Nur (1996), and Gal et al. (1998)) which describes a rock as a random pack of spherical grains. In the reservoir layer, in addition to input petrophysical curves, i.e. effective porosity, volumes of mineral components, and saturations, the applied rock physics model requires the following input data: fluid and matrix properties, reservoir pressure and temperature. Fluid parameters and reservoir condition


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information are generally available from PVT analysis and well tests measurements, whereas solid phase properties are inferred from core mineralogical analysis. In reservoir characterized by low porosity rocks, inclusion models (e.g. see Avseth et al. (2005), and Mavko et al. (2009)) are more appropriate. Soft sand model Granular media models are based on Hertz Mindlin contact theory (Mavko et al. (2009)) and, they provide estimates for the bulk (KHM ) and shear moduli (µHM ) of the dry rock assuming that the frame is a dense random pack of identical spherical grains subject to an effective pressure P with a given critical porosity φ0 and an average number of contacts per grain n. Critical porosity is the value which marks the porosity limit between a saturated rock and a suspension. We recap here the set of equations of soft sand model used in the application case. First Hertz Mindlin equations are given by: KHM =

µHM

P [n(1 − φ0 )µmat ]2 18[π(1 − ν)]2

5 − 4ν = 5(2 − ν)

3P [n(1 − φ0 )µmat ]2 2[π(1 − ν)]2

where ν is the grain Poisson’s ratio, namely ν=

31

(5.16) 31

3Kmat − 2µmat 2(3Kmat + µmat )

(5.17)

(5.18)

and Kmat and µmat are the solid phase (zero porosity) elastic moduli. For porosity values ranging between zero and the critical porosity, the soft sand rock physics model connects the grain elastic moduli Kmat and µmat with the elastic moduli KHM and µHM of the dry rock at critical porosity. This is done by interpolating these two end members in the intermediate porosity values by means of a heuristic modified Hashin-Shtrikman lower bound: −1 φ/φ0 1 − φ/φ0 Kdry = + − 4/3µHM KHM + 4/3µHM Kmat + 4/3µHM

(5.19)


µdry =

µHM

1 − φ/φ0 φ/φ0 + + 1/6ξµHM µmat 1/6ξµHM

−1

− 1/6ξµHM

130

(5.20)

9KHM + 8µHM (5.21) KHM + 2µHM On the other hand, the stiff-sand model connects the two end-members by with ξ=

the modified Hashin-Shtrikman upper bound (Mavko et al. (2009)). Gassmann’s equation is then used for calculating the effect of fluid: 2 Kdry 1 − Kmat Ksat = Kdry + φ K − K 2dry + K1−φ Kf l mat

(5.22)

mat

µsat = µdry

(5.23)

where Kf l the fluid bulk modulus, and Ksat and µsat are the rock saturated moduli. Fluid properties can be obtained by means of Batzle-Wang formulas (Batzle and Wang (1962)), a set of empirical equations which allows calculating bulk moduli and densities of fluid components starting from fluid analysis parameters (such as gas gravity, oil gravity, gas-oil-ratio, temperature and pressure). Finally, velocities are obtained as follows VP =

s

Ksat + 4/3µsat ρ r µsat VS = ρ

(5.24) (5.25)

where ρ is the fluid saturated bulk density.

5.3.3

Log-facies classification

The goal of this section is to present a new methodology for Log-Facies Classification (LFC) and the related uncertainty evaluation. The classification of facies at well location is a key important step in reservoir modeling: the static reservoir model essentially consists of stochastic simulations which are driven by log-facies classification at the well locations. We propose here a classification based on both petrophysical and acoustic/elastic properties in order to link log-facies to seismic inverted attributes which are often


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used as soft conditioning data in reservoir simulation. However, different sources of uncertainty affect high resolution log-facies classification. We have classified the associated uncertainty in two main groups: the uncertainty related to petrophysical curves obtained in QLI and the uncertainty associated to elastic properties (velocities or impedances) recovered by RPM. We finally couple the so obtained results by Monte Carlo simulations to generate several realizations of log-facies profiles which are used to infer facies uncertainty from the probabilistic analysis previously performed (Figure 5.2, bottom). Here, we first present the standard approach in order to set the basis of the statistical methodology. LFC usually combines a multivariate statistics technique (cluster analysis) and interpretation of prior sedimentological information. Cluster analysis is a well-known technique that helps to group objects according to their mutual similarity. Petrophysical and elastic logs can be statistically processed to find clusters, by using, for example,an unsupervised hierarchical clustering algorithm. This algorithm groups the log data based on their statistical similarity into hierarchically ordered set of clusters. During the characterization phase, an initial number of clusters is selected. The identified clusters are compared to quantitative and qualitative information derived from cores (routine and special core analyses, lithological, sedimentological and petrographical descriptions). The integration of information from different sources (logs, core measurements, lithology, sedimentology, etc) is a key step in the characterization of the different clusters. In particular, a suitable training set t is selected from QLI and RPM outputs (p∗ and r respectively) and LFC is run including the data all along the well, providing a depth-dependent, discrete, class vector, collecting the results of the classification. In detail, the log-facies vector gives the classified litho-classes at a given depth value. Eventually, some of the clusters are grouped to identify and classify log-facies for use in the 3D geological model. In the conventional workflow, LFC does not consider the uncertainty associated to input curves (both petrophysical and acoustic/elastic ones). In order to evaluate the uncertainty related to LFC, we use the Monte Carlo approach. From the previous steps of the methodology (QLI and RPM), we have obtained PDFs of petrophysical and acoustic/elastic curves (˜ p and ˜r) and these define

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the probabilistic training set ˜t. In particular, from this set of PDFs collected in ˜t, N realizations can be sampled and then used to perform N LFC profiles through MCSs, and estimate the posterior probability on facies classification, in addition to the most probable facies profile: in other words, cluster analysis is applied to the different QLI and RPM realizations to obtain the related scenarios of log-facies. Then, from the sogenerated profiles we can count, at each depth location, the frequency of occurrence of each log-facies, and infer a depth-by-depth posterior probability distribution. The results are collected in a vector c whose entries are the probability of each classified log-facies, so that if have a set of M possible discriminated facies, their probabilities of occurrence are c = (c1 , ..., cM ). Sedimentological information can be further integrated by multiplying the probability vector by the confusion matrix obtained from the sedimentological classification (e.g. the matrix accounting for the degree of confusion among the different facies with those provided by sedimentologists). We finally observe that, from vector c, for instance, the most likely facies scenario can be estimated. We can take a further step in order to quantify the uncertainties associated to the probabilistic LFC result. In fact, the most probable scenario, as the output of a probabilistic workflow, should be coupled with a confidence value showing the amount of information brought by the discrete classification. For this reason the concept of entropy (Shannon (1948)) can be exploited. This function is a statistical parameter able to provide some numerical insights on the intrinsic variability of the discrete variable studied with respect to all the outcomes of the classification. The scalar information entropy (see Shannon (1948), or Mavko and Mukerji (1998), and Mukerji et al. (2001b)) associated to the probability vector c is defined as follows: T

h = −c logM (c) = −

M X

ci logM (ci ),

(5.26)

i=1

where the logarithm is computed in base M (number of facies) since we have classified M different facies; this restricts and normalizes the entropy to the range [0, 1]. As a simple example, assume that M = 3 and at a particular point of the depth profile, the three facies are equally probable: c = ( 31 , 13 , 31 ). Hence, the entropy is


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h = 1, namely the maximum entropy value and this means that uncertainty is at its maximum. On the other hand, if c = (1, 0, 0) we get h = 0, giving a minimum entropy scenario and no uncertainty. In-between situations are characterized by entropy values that quantify the degree of disorder brought by the classification: 0 < h < 1. Entropy measure of uncertainty goes beyond measures such as variance and covariance, and can be used for categorical variables (e.g. facies). The information provided by entropy analysis on LFC, comprising uncertainty from QLI and RPM, will be discussed in two real examples.

5.4

Application: first example

The methodology has been applied to a real dataset from an exploration well (here called A), offshore West Africa. Well A is vertical and drilled in a deep offshore clastic reservoir made by soft sandstones of a turbidite channel complex. The prospect is a typical combined structural-stratigraphic trap. It is composed by a quite large turbidite sandstone channel which drapes over the north flank of a large salt diapir. Sparse shale beds are present and analysis of core samples revealed that the clay is a mixture of illite and kaolinite (respectively 70% and 30%). The reservoir fluid is oil. A comprehensive set of wireline logs (standard and high resolution) has been acquired to provide a reliable formation evaluation: gamma ray (gr), resistivity (rs), sonic (sn), density (dn), neutron (nt) and nuclear magnetic resonance (nmr) logs. Globally, the quality of the acquired data is very good (Figure 5.3). Following our notation the depth-dependent data vector is: d = (gr, rs, sn, dn, nt).

(5.27)

Deterministic QLI via a straightforward fQLI resulted in the generation of mineralogical volumes (vsand , vsilt , and vclay ), effective porosity φ, and water saturation sw : p = (vsand , vsilt , vclay , φ, sw ).

(5.28)

In well A, invasion effects were negligible so that pXO = pDE . In Figure 5.4, the results of our local minimization, p∗ , in other words the final set of petrophysical


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Figure 5.3: Well log dataset interval of well A, from left to right: P-wave and S-wave velocity; density; neutron porosity; gamma ray. curves performed in Quantitative Log interpretation, are shown. In Figure 5.5, we give an example of comparison between our result and a standard commercial software output, which shows a satisfactory match between the two estimated set of data. A rock physics model is then used to establish the link between petrophysical parameters p∗ and acoustic properties r. For this reservoir study, we used the soft sand (uncemented) model, based on Hertz-Mindlin contact theory (Dvorkin and Nur (1996)). Petro-elastic property uncertainty assessment and facies classification has been performed in the lower reservoir whose top is located at approximately 2510m. In the reservoir layer the applied RPM requires the following input data from QLI: effective porosity, volumes of mineral components (sand, silt and clay) and fluid saturations. The parameters include effective pressure of 35 MPa, a critical porosity of 0.4 while the coordination number is 9. In this particular case effective pressure has been assumed constant within the reservoir but in general pressure effects on velocities should be taken into account. Several models have been developed to account for pressure effect on velocities (e.g. Eberhart-Phillips et al. (1989)) or on dry rock


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Figure 5.4: Petrophysical curves performed in Quantitative Log Interpretation, from left to right: porosity (total porosity in blue, effective porosity in red); volumetric fraction curves (clay in green, quartz in yellow, and silt in brown); water saturation; and finally the cumulative volumetric display (shale in orange, silt in brown, quartz in yellow, water in blue, and oil in green; red dashed line represents 1 minus porosity and it separates solid and fluid phase). elastic moduli (MacBeth (2004)); however we point out that these methods must be calibrated (typically using lab measurements at different pressure regimes) to determine the empirical parameters. The rock physics model is calibrated at well location, by comparing the velocity and density estimated from the elastic parameters with the velocity and density from the recorded logs in the borehole. The calibration is performed in wet condition (meaning that a preliminary fluid substitution is performed on well logs) in order to avoid the effect of fluid in rock parameters calibration. Focusing on the depth interval including the lower reservoir layer (2480m - 2600m), the results can be seen in Figure 5.6 . The calibration is performed with a trial and error method; optimization techniques could be used as well but these methods do not guarantee that the optimized parameters still preserve their physical meaning. Rock


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Figure 5.5: Petrophysical outputs comparison (red curves represent our results; blue curves represent standard commercial software outputs). physics model calibration, in many practical applications, can be quite complex as the physical-mathematical model cannot account for heterogeneity and natural variability of the rock. In our application the model overpredicts sonic log velocities at 2520m depth and does not match the high peak of P-wave velocity at 2550m depth. The mismatch at 2520m is due to lack of accuracy of the model, while the velocity peak at 2550m could be a measurement error or represent a different lithology. The lack of accuracy of the rock physics model in some small depth intervals should be partially compensated by the variability introduced in the following Monte Carlo simulations. Traditional log-facies characterization is achieved through cluster analysis, carried out using key-curves (defining the training set t) selected to be effective porosity, clay content and VP /VS ratio: t=

VP φ, vclay , VS

.

(5.29)

In particular, VP /VS ratio refers to brine condition to avoid any fluid effect due to hydrocarbon presence. In this case, a preliminary analysis showed that VP /VS ratio


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Figure 5.6: Calibration of the rock physics model, from left to right P-wave and Swave velocity, and VP /VS ratio (black curves represent the actual sonic log, blue dashed curves represent the predicted rock physics model). The rock physics model has been calibrated in wet condition and then applied to the well scenario by Gassmann fluid substitution. and clay content have a good correlation; however the training set could include only VP , or VS (or both of them), if these logs are identified as good lithological indicators for the reservoir (see second example). The training set t was statistically processed by means of a hierarchical agglomerative clustering algorithm with Ward’s minimum variance linkage method (Ward (1963)). Hierarchical algorithms find successive clusters using previously established clusters (i.e. the resultant classification has an increasing number of nested classes). Thus, given a data set consisting of N0 objects, agglomerative clustering methods


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generate outputs where objects are gradually partitioned. Ward’s minimum variance linkage method makes use of squared Euclidean distances to define the dissimilarity among clusters. The most common and practical way to visualize the results is through a plot called dendrogram: this allows reconstructing the merging history of the objects of the studied data set from the beginning (each object forms a cluster of its own) to the end of the clustering process (all objects are in the same cluster). The dendrogram for our case application is plotted in Figure 5.7.

Figure 5.7: Dendogram associated to log-facies classification. A dendrogram consists of many U-shaped lines connecting objects in a hierarchical tree. The stem of each U represents the distance between the two objects being connected. Red clusters refer to the connecting histories of the three recognized facies: low concentration turbidite (LCT), mid concentration turbidite (MCT), high concentration turbidite (HCT). In traditional log-facies analysis, the number of the classes (facies) that can be identified from sedimentological information is often higher than the number of classes of interest in reservoir models: the reason is that the quality of the data at the well location, and the indirect measured data far away from the well (seismic and


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electromagnetic data) do not allow us to distinguish sedimentological features with the same accuracy. Based on the dendrogram that shows a very stable clustering (Figure 5.7) and on core analysis calibration, a 3-facies classification, based on sand concentration, is determined (Figure 5.8) and it consists of: • low concentration turbidite (LCT) facies in green; • mid concentration turbidite (MCT) facies in brown; • high concentration turbidite (HCT) facies in yellow.

Figure 5.8: Log-facies classification performed at well location: LCT in green, MCT in brown, and HCT in yellow. Log-facies classification is derived by using petrophysical curves (porosity and clay content) and velocity data (VP /VS ratio). On the right we show two crossplots in petrophysical (top right) and petroelastic (bottom right) domain, color coded by facies classification. The proposed facies classification is based on the percentage of quartz and clay in the facies as derived from interpreted log curves. This classification is simplified compared to sedimentological models but it is strongly consistent with the facies classification of the static reservoir model where porosity and net-to-gross are distributed


Log Neutron Density Gamma Ray Resistivity Sonic

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Standard Deviation 7% 0.015 g/cm3 5% 10% 7%

Table 5.1: Standard deviations associated to log measurements. following distributions estimated from well logs in each facies. Actually facies LCT could be subdivided into 2 sub-classes, namely low and very low concentration turbidite (which includes non-reservoir shale), as thin interbeddings of hemipelagic shales have been observed in the well and as two clusters can be identified in the dendogram. However these two classes are not distinguishable from the petro-elastic point of view within the reservoir. MC simulations are then exploited to provide the uncertainty propagation and evaluation. The starting point are the normal PDFs associated to the set of wireline ˜ used in QLI with with standard deviations which are typical of the log logs (d) measurements and of the processing of raw data: Table 5.1 collects common standard deviations (Viberti (2010)). In this example a sensitivity analysis shows that 100 realizations are enough to provide a stable solution and reliable results. Realizations of input curves sampled ˜ provide the set of cost functions to be minimized and the related set of from d ˜ . In Figure 5.9 we plot the different realizations and petrophysical curves defining p the median of the petrophysical sets of curves. From the previous QLI analysis we have a number of realizations of the petrophysical variables representing basic properties of the formation such as porosity and mineral volumetric contents. Thus, we can generate a set of realizations of elastic properties logs through a Monte Carlo simulation by applying the rock physics model to the set of petrophysical curves realizations from quantitative log interpretation. In the MC simulation we add a random error to account for the uncertainty associated ˜ and the previously calibrated rock physics model, the to RPM. rom the PDFs of p set of acoustic and elastic curves and the corresponding PDFs of velocities ˜r are then


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Figure 5.9: Set of 100 realizations of petrophysical curves (gray curves), from left to right: effective porosity, volume of clay, and volume of quartz (volume of silt is computed by difference 1 minus the sum of effective porosity, clay, and quartz). The pointwise median curve (P50) is displayed in red. computed (Figure 5.10). In Figure 5.11 we show the full probability distributions of porosity and P-wave velocity, and as an example at two given depth locations we extract the corresponding histograms. On the base of the set of 100 realizations of effective porosity, volume of clay and VP /VS ratio, we can obtain 100 training sets sorted in ˜t to be used to generate 100 profiles of facies. In fact, from the previous steps of the methodology, we have obtained 100 realizations of petrophysical curves (QLI results, Figure 5.10) and 100 realizations of rock physics curves (RPM results, Figure 5.10). We now perform 100 log-facies


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Figure 5.10: Set of 100 realizations of elastic curves (gray curves), from left to right: P-wave and S-wave velocity. The pointwise median curve (P50) is displayed in red. classifications through MCSs (Figure 5.12, top), and estimate the posterior probability on facies classification, in addition to the most probable facies profile. In particular cluster analysis is applied to the 100 realizations to obtain 100 profiles of log-facies; then the frequency of the facies, at each depth location of the profile, provides the posterior probability of facies occurrence and the most likely facies estimation. In Figure 5.12 (bottom), we show, as an example, 5 realizations of facies obtained from Monte Carlo simulations: the thick high porosity sand layer is well identified in all the profiles, whereas the thin layers at the top and at the bottom of the reservoir are more uncertain. This means that in the upper and lower part of the reservoir the log-facies classification can vary as a function of RPM and QLI input parameters in a traditional deterministic workflow and the underestimation of the uncertainty could


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Figure 5.11: Posterior probability density functions of porosity (left side) and P-wave velocity (right side). For two depth locations z1 and z2 we also show the histogram of the simulated values. lead to a misclassification of the facies. From the so-generated 100 profiles we can count, point by point along the profile, the frequency of occurrence of each log-facies and infer the posterior probability distribution c, by normalizing, at each data level, the frequencies by the total number of realizations (100). In Figure 5.13 we show the probability of log-facies obtained from petro-elastic Monte Carlo realizations and the final most probable classification. We notice that in the mid part of the profile the probability of having a HCT facies is very high and the uncertainty is small, whereas in the thin layers zones the uncertainty is much higher. This is also reflected in the entropy curve that we plot together with the most likely facies profile (Figure 5.13) in order to quantify the uncertainty in the discrete probabilistic scenario so provided. Entropy is computed in base 3 since we have discriminated 3 different facies, in order to get values between 0 (no uncertainty) and 1 (maximum uncertainty).


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Figure 5.12: Set of 100 realizations of facies (top left) at well location and estimated most likely facies profile (top right). On the bottom 5 realizations are shown: LCT in green, MCT in brown, and HCT in yellow. To interpret entropy results we comment here on the surface described by the variation of the probabilities related to the three discriminated facies. Since the three probabilities sum to 1, we can take any two as independent variables to map the behavior of entropy for different values of the facies probabilities. Specifically: h(cHCT , cLCT ) = −[cHCT log3 (cHCT )+cLCT log3 (cLCT )+(1−cHCT −cLCT ) log3 (1−cHCT −cLCT )] (5.30)

with cHCT + cLCT ≤ 1, where cHCT is the probability of obtaining HCT at a given

depth, and cLCT is the probability associated with LCT; while cM CT = (1 − cHCT −

cLCT ). As previously explained the three probabilities are collected in the vector


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Figure 5.13: Posterior probability distribution of facies estimated by Monte Carlo simulation (left), most likely facies profile (middle) and associated entropy function (right): LCT in green, MCT in brown, and HCT in yellow. On the right we show two crossplots in petrophysical (top right) and petroelastic (bottom right) domain, color coded by the associated entropy given by the probabilistic facies classification. c = (cLCT , cM CT , cHCT ). When the three facies are equiprobable, we have maximum entropy, whereas when the probability of a given facies is close to 1, then the entropy tends to 0. The two crossplots on the right of Figure 5.13 help in clarifying the information provided by entropy analysis. With Figure 5.8 in mind (right), it can be easily seen that the transition areas between the different facies are characterized by high entropy (yellow clouds in Figure 5.13) as expected: this fact mainly affects the generalized high entropy associated to MCTs. On the other hand, the extreme LCT and HCT facies show low entropy (blue clouds).


5.5

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Application: second example

The second case study is in the North Sea: the reservoir is located at approximately 1750m, has an average thickness of about 80m, and it is part of a complex fluviodeltaic system with sequences of sandstone and shale. Sandstones and shales layers are relatively thin compared to the seismic resolution. Several works have been published on a number of nearby fields in the North Sea (Mukerji et al. (2001a), Mukerji et al. (2001b), and Avseth et al. (2005)). A complete set of well logs is available for one well of the area. The well is slightly deviated and passes through the main reservoir layer filled with oil. The set of sonic logs, density, and a set of preliminary petrophysical curves performed with commercial software are shown in Figure 5.14. We point out that three main mineralogical fractions have been identified: clay (mainly illite), muscovite, and quartz. Porosity is relatively high in clean sand in the upper part of the reservoir. The quality of the sonic log is generally good except for few data samples where some high peaks in P-wave velocity are recorded while no changes are recorded by S-wave velocity, generating unrealistic values of Poisson ratio and VP /VS ratio. Furthermore a preliminary facies classification has been performed based on sedimentological information, accurate depositional models, and core sample analysis. A set of 8 facies has been identified in this reservoir: 1) marine silty-shale; 2) prodelta; 3) flood plain; 4) mouth bar; 5) distributary channel; 6) crevasse splay; 7) tidal deltaic lobes; 8) tight. The distribution of porosity and clay content color coded by facies classification (Figure 5.15) shows that some of these sedimentilogical facies cannot be discriminated by petrophysical properties: for example marine silty-shale, pro-delta and flood plain have similar distributions in terms of porosity and clay content (Figure 5.15). As a consequence the elastic response of some facies is approximately the same. For this reason, we applied the proposed methodology with two different classifications: first we used the 8-facies sedimentological definition; then we used a simplified 3-facies classification, namely sand, silty-sand (mixed), and shale, to classify facies that can be recognized at seismic scale. As in the previous example we show the different steps of the methodology. First


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Figure 5.14: Well log dataset and preliminary petrophysical curves at well location, from left to right: P-wave and S-wave velocity; density; porosity (total porosity in blue, effective porosity in red); volumetric fraction curves (clay in green, quartz in yellow, and muscovite in brown); and water saturation. of all, a rock physics model is calibrated at the well location: in this case we applied a stiff sand model (Mavko et al. (2009)). Both soft sand and stiff sand models belong to the group of granular media models and are based on Hertz-Mindlin contact theory: soft sand model extrapolates elastic property values to low porosities by using modified Hashin-Shtrikman lower bound; stiff sand model uses modified HashinShtrikman upper bound, resulting in an increase of elastic properties values compared to soft sand model. Similarly to the previous case, the calibration is performed in wet conditions by comparing sonic logs data and rock physics model predictions (Figure 5.16). The model parameters are: effective pressure P = 27MP a, critical porosity of φ0 = 0.42, and coordination number n = 9. The overestimation of S-wave velocity in stiff sand model is well known (Mavko et al. (2009)), for this reason we applied a factor of 3/4 to reduce S-wave velocity predictions and match the data: this correction is purely heuristic but it is often applied in real cases (a physical explanation


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Figure 5.15: Preliminary facies classification: a set of 8 facies has been identified in this reservoir: 1) marine silty-shale; 2) pro-delta; 3) flood plain; 4) mouth bar; 5) distributary channel; 6) crevasse splay; 7) tidal deltaic lobes; 8) tight (left). Grouped histogram of effective porosity (top right) and clay content (bottom right) as a function of facies classification. of this reducing factor for shear wave can be found in Bachrach and Avseth (2008)). Another method to achieve a good fit is to reduce the values of shear moduli of mineralogical fractions but this could lead to unrealistic values for this mineralogy and would require a recalibration of P-wave velocity. The direct comparison of rock physics model predictions and sonic logs is shown in Figure 5.17: generally we have a good agreement except for the data samples with high peaks in P-wave velocity. A slight overestimation of velocity in the upper shaly layer is observable, but this could be due to a different mineralogical composition of clay in the overcap layer on top of the reservoir. We then apply the uncertainty propagation methodology to petrophysical properties and rock physics elastic attributes. We generate a set of 50 realizations of effective porosity, volume of clay and volume of quartz (volume of muscovite is computed by difference to guarantee that the sum of the mineralogical fraction is 100%); we then compute the corresponding set of P-wave and S-wave velocities and density


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(Figure 5.17). The parameters used for the probabilistic formation evaluation analysis are the same used in the previous case and summarized in Table 5.1.

Figure 5.16: Rock physics crossplots: (left) P-wave velocity versus effective porosity; (right) S-wave velocity versus effective porosity color coded by volume of clay. Black lines represent stiff sand model for different clay contents (from top to bottom: 10%, 20%, 30%, 40%, 50% and 60%). Rock physics diagnostics (Figure 5.15) showed that, for this specific case, the use of S-wave velocity could improve facies classification in the combined petro-elastic domain. For this reason we decided to use in this case an extended dataset made by effective porosity, volume of clay, volume of quartz, P-wave, and S-wave velocities. We then performed the facies classification with 8 sedimentological facies. In such application we expect more variability and higher entropy compared to the previous example as the number of facies is higher and some facies cannot be discriminated from the petrophysical and/or elastic point of view. The full set of 50 realizations is shown in Figure 5.18: by comparing all the realizations we can detect some similar features (such as layers made by facies 1, 2, and 3, and layers made by facies 4, and 5 in the upper part) but as expected it is hard to discriminate between facies with similar petro-elastic properties. The extracted statistics, in particular the etype (ensemble average) and the maximum a posteriori (MAP) of facies, help to interpret the results (Figure 5.18). The etype is the ensemble average of the set of models and it is a continuous variable; the maximum a posteriori represents the facies that maximizes the posterior probability estimated from the ensemble of realizations as


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Figure 5.17: Set of 50 realizations of petrophysical and elastic curves (gray curves), from left to right: effective porosity, volume of clay, and volume of quartz (volume of silt is computed by difference 1 minus the sum of effective porosity, clay, and quartz), P-wave velocity and S-wave velocity predictions. The pointwise median curve (P50) is displayed in red. The dashed blue line represents sonic log data. the facies frequency divided by the total number of realizations (Figure 5.19). In the upper part of the profile we can recognize the shaly overcap made by facies 1, 2, and 3 and the thicker layer of the main reservoir made by facies 4 and 5. In the lower part the variability in the classification is higher, which results in a sequence of thin layers. However if we look at the whole set of realizations we observe that even in the upper part of the reservoir the variability is quite high; this is confirmed by the entropy profile (Figure 5.19). Differently from the previous case the overall entropy is generally high: this can be due to the large number of facies in the classification but also to the vertical heterogeneity of petrophysical and elastic properties (Figure 5.17). To improve the facies classification at the well location, we introduced here a new approach based on Markov chain methods (Krumbein and Dacey (1969)). Markov chains are a statistical tool that has been used in geophysics to simulate facies sequences to capture the main features of the depositional process. Markov chains are


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Figure 5.18: Set of 50 realizations of facies (left) at well location, etype (ensemble average) and estimated most likely facies profile (top right). Color codes are the same as in Figure 5.15. based on a set of conditional probabilities that describe the dependency of the facies value at a given location with the facies values at the locations above (upward chain) or the locations below (downward chain). The chain is said to be first-order, if the transition from one facies to another depends only on the immediately preceding facies. The transition matrix is built by assuming (or estimating from other wells in the field or nearby fields) prior proportions and transition probabilities. The matrix in our case is 8 by 8 elements: the terms on the diagonal of the transition matrix are related to the thickness of the layers. The higher are the numbers on the diagonal, the higher is the probability to observe no transition, and as a consequence the thicker will be the layer. In our application the posterior probability of facies estimated from logs of petroelastic properties is combined with the probabilities of the transition matrix (for mathematical details, see Grana and Della Rossa (2010)) by a simple integration, and facies values are sampled from the resulting distribution. In Figure 5.19 (right) we show one realization of facies obtained by this technique: the use of Markov chain allows us to account for vertical continuity and produces a more realistic classification in cases where a high number of facies is identified. However the main limitation of


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Figure 5.19: Facies probability estimated from ensemble and extracted statistics, from left to right: probability of facies (first three plots); entropy, maximum a posteriori of facies distribution and new facies classification performed by Markov chain integrated approach. Color codes are the same as in Figure 5.15. such a choice is due to the assumptions required to assemble the transition matrix. In our case, for example, we do not have any information about the distribution and thickness of facies 7 and 8 and the results we obtained only depend on the information assumed from the depositional and sedimentological model. We finally repeated the same application with a simplified classification: we assumed 3 facies in the reservoir that we called for simplicity shale, silty-sand, and sand. The reservoir characterization study performed on this field had shown that these facies can be discriminated in the petro-elastic domain and represent the facies classification recognizable at seismic scale. Figure 5.20 shows the comparison of the results obtained with the different classification. The methodology clearly works better when a limited number of facies is identified (also by comparing the entropy profiles of Figures 5.19 and 5.20); however we notice that there is a good agreement between the two classifications as facies 1, 2, and 3 (in 8 facies definition) correspond to shaly layers, and facies 4, and 5 (in 8 facies definition) correspond to sandy layers.


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Figure 5.20: Comparison of two different classifications. On the left (first two plots) we show the 8-facies classification: maximum a posteriori of facies and Markov chain classification (color codes are the same as in Figure 5.15). On the right we show the simplified 3-facies classification (namely seismic facies classification, last three plots): posterior probability distribution, entropy, and maximum a posteriori of facies (shale in green, silty-sand in brown, sand in yellow).

5.6

Discussion

The proposed methodology classifies log-facies at well location by integrating three different disciplines: formation evaluation, rock physics modeling, and cluster analysis, by means of a statistical approach, that allows us to understand how plausible the obtained interpretation is or, in other words, how large the uncertainties are in the obtained solution. The assessment of uncertainty in geophysical inverse problems has been addressed in several works (Jackson (1972), Jackson (1979), and Tarantola (2005)) but a standard application to well log analysis is in general neglected. Classical deterministic workflows in formation evaluation analysis and, in particular, log-facies classification, do not allow us to propagate the uncertainty in measured data to the property estimates in a robust and comprehensive way. Statistical methods,


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on the other hand, can incorporate all available information on the studied system (observational data, theoretical predictions and prior knowledge) and this information can be represented by probability distribution functions. According to this, the solution is not unique, but rather it consists of a posterior probability distribution function over the model domain space which describes the probability of a given model being the closest to the true one. The aim of this chapter is to enhance the classical approach by introducing Monte Carlo simulations providing a proper way to account for a statistical view. Since each step of the methodology strongly depends on the previous one, a robust uncertainty analysis and classification from the very beginning (initial acquired dataset) is a mandatory requirement for a safe and trustworthy probability distribution definition. The input uncertainties represent the critical point of the methodology. Formation evaluation is the aforementioned starting point and so it drives the remaining modeling. As we have already pointed out, the prior characterization of input uncertainties for quantitative log interpretation relies on separating the different sources of errors associated with tool measurements, environmental corrections, pre-processing computations and model parameterizations. Systematic errors need to be corrected since their propagation is meaningless. Random errors may not be neglected when the number of measurements to be averaged is low and/or the measurement precision is very poor. Monte Carlo simulations try to address this uncertainty issue but a robust and straightforward way to account for all the sources of uncertainty is still lacking. However, other open problems exist and two of the major ones are the possible correlation existing between the uncertainty affecting the different log acquisitions and the various reference volumes of investigation involved (vertical and lateral resolution). The assignment of probability distribution functions to log measurements assumes a fixed and certain value of the given nominal depth. A more robust approach should also consider the vertical uncertainty related to the recorded depth of any log measurements. An important point of discussion is also related to the constrained number of facies in the probabilistic workflow. In particular the methodology quantifies and


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propagates the uncertainty through log-facies classification once the number of facies has been a priori fixed. Monte Carlo simulations are applied to the multivariate statistical technique chosen to discriminate a given number of log-facies. Different petrophysical and elastic scenarios in the probabilistic training set could be better described by a number of facies which is higher or lower with respect to the a priori fixed one. A generalization of the methodology in order to take into account this issue could also help for a more rigorous characterization. The application of the entire workflow to real datasets seems to confirm its feasibility and reliability. Log-facies probability and entropy evaluation quantify the uncertainty in the classification and represent a probabilistic approach from the very beginning of borehole data interpretation for seismic reservoir characterization. It is worth mentioning that even more complex systems need to be tested and studied in order to finally state all the potentiality of the methodology here developed. It is also worth mentioning the choice of Monte Carlo simulations for accounting the uncertainty propagation problem. First of all, Monte Carlo techniques calculate the probability density of any functions of random variables and do not require any a priori assumption on the type of distribution for the results. In fact, quantitative log interpretation and rock physics modeling can be highly non-linear and the uncertainty propagation problem cannot be solved analytically. Under this light, Monte Carlo simulations provide a simple means by which uncertainties in inputs can be translated into uncertainties in the calculated output properties. Moreover, this methodology is very flexible, allowing different interpretation models to be built and uncertainties tested in a robust way. Correlated as well as independent parameters can be handled and all the possible constraints defining the problem can be taken into account in a straightforward way. The downside to Monte Carlo simulations is that a large number of simulations are typically required for meaningful statistics to be developed. So, it is fundamental to understand the number of samplings needed in order to get stable results. To conclude, by Monte Carlo simulations we have a powerful tool to propagate uncertainties in the different steps of the methodology: the technique can be applied to different studies, independently of the physical models (defining formation evaluation and rock physics interpretation) and mathematical techniques


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(for log-facies classification) chosen for the specific case. As described in the application sections, in this work we performed log-facies classification following a hierarchical agglomerative clustering algorithm with a Ward’s minimum variance linkage method. Other methods can be used to classify facies, such as linear discriminant analysis, partitioning cluster algorithms or neural networks. In particular, in petrophysics neural networks are used in unsupervised learning mode. The different classification algorithms were previously investigated in order to rank the obtained results, undertaking tests on known synthetic data sets of variable complexity. The objective was to understand which methods are the most effective in generating a log-facies classification suitable for reservoir characterization purposes. In particular, the proposed and here used hierarchical agglomerative Ward’s technique has been evaluated by far the best, as it correctly classified most of the synthetic data sets. K-means clustering, unsupervised neural networks, centroid linkage methods and unweighted average (UPGMA) with Euclidean distance algorithms showed reasonably good classification rates, even though they were not suitable for arbitraryshaped data clouds. Moreover, no significant advantages were observed when Manhattan or Mahalanobis distance were used in substitution for the Euclidean distance in the UPGMA algorithm. Other clustering algorithms (e.g. single linkage, complete linkage, weighted average) have been proved unsuccessful since they were characterized by high misclassification rates. Given the above considerations on the ranking and availability of the different classification algorithms, we here focused on Ward’s hierarchical agglomerative method. However, the overall methodology works with any appropriate classification algorithm. As previously mentioned, the main advantage of the presented workflow is that the Monte Carlo simulations allow us to extend the training dataset and to propagate the uncertainty from the input measured data to the facies classification. Uncertainty evaluation is important from a qualitative point of view to assess the reliability of the estimated properties, but it can be used in quantitative modeling by using the full probability distributions of the property. In reservoir modeling, many geostatistical simulations methods (for example sequential indicator simulation and sequential Gaussian simulation) require input prior distributions of the properties we


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want to model, i.e. facies proportions and the prior distributions of rock properties (such as porosity and net-to-gross). These distributions are generally assumed from prior geological knowledge of the field or nearby fields; but in our case the distributions can be extracted directly from Monte Carlo simulations. Nevertheless we point out that the PDFs derived from Monte Carlo simulations are depth dependent; then stationary conditions should be first verified before extrapolating marginal distributions. In a practical reservoir modeling workflow (see for example Doyen (2007)), we first generate reservoir facies models with sequential indicator simulations (or multipoint geostatistics if a suitable training image is available) and then generate porosity and other reservoir property models with sequential Gaussian simulation where the prior distributions of reservoir properties are facies-dependent and are estimated from the marginal distributions derived from Monte Carlo simulations in each facies. An important point of our methodology is the integration of rock physics modeling. The main advantage of this step is the link between facies classification and elastic properties, which is provided by the model calibrated at the well location. In fact, this link guarantees that facies classification estimated from seismic amplitudes in seismic characterization studies is consistent with log-facies classification at well location. Instead of rock physics predictions, sonic logs could be used as well but the correlation between sonic logs and petrophysical curves is not always optimal in well log analysis. Furthermore, resolution of sonic logs is generally lower than the resolution of well logs used in formation evaluation analysis. If the sonic logs are reliable and correlated to petrophysical logs, the lower resolution of sonic logs would reduce the entropy but at the same time increase the uncertainty in the output estimation. On the other side rock physics models are applied to a set of petrophysical curves with similar resolution, which results in a complete consistent petro-elastic dataset. The drawback of such a choice is that the rock physics model approximation could be not accurate in some interval layers, as pointed out in our applications, but this lack of accuracy is partially compensated by Monte Carlo multiple realizations. We finally point out that in our methodology we tried to include different sources of uncertainty: log measurements, rock physics model approximations, heterogeneity and natural variability of the rocks. However other factors could influence formation


158

evaluation analysis, rock physics modeling, and log-facies classification, such as for example anisotropy, laminar and/or dipping layering, and well deviation. These issues should be carefully evaluated in each case and a sensitivity analysis should be performed to verify if they can be neglected or not.

5.7

Conclusion

The proposed probabilistic log-facies classification workflow is based on the integration of three different disciplines: quantitative log interpretation, rock physics modeling, and multivariate statistical techniques (cluster analysis) for log-facies classification., Furthermore, this workflow focuses on the uncertainty propagation through the three different steps, and it is based on Monte Carlo simulation method. The application of the proposed methodology allows to generate several log-facies profiles at well location and to estimate the most probable facies classification and the associated uncertainty via entropy computation. The so obtained classification includes both petrophysical properties and acoustic/elastic attributes to directly link log-facies to seismic velocities used in facies simulation constrained by seismic information. This workflow represents a rigorous starting point to propagate the well facies classification and related uncertainty to the entire reservoir modeling. These obtained uncertainties are more realistic inputs to reservoir modeling, rather than taking log data as fixed and certain (so-called ”hard data” in geostatistics). The method has been illustrated on real well log datasets in clastic environments, but it can be applied to different scenarios.

Chapter 6 Pressure dependence of elastic properties 6.1

Abstract

In time-lapse studies, the knowledge of the saturation and pressure effects on elastic properties is a key factor. The relation between saturation changes and velocity variations is well known in rock physics and at seismic frequency it can be satisfactorily described by Gassmann’s equations. The pressure effect still requires deeper investigations in order to be included in rock physics models for 4D studies. Theoretical models of velocity-pressure relations often do not match lab measurements, or contain empirical constants or theoretical parameters that are difficult to calibrate or do not have a precise physical meaning. In this chapter, I present a new model to describe the pressure sensitivity of elastic moduli for sandstones. This relation is then integrated with a complete rock physics model to describe the relation between rock properties (porosity and clay content) and dynamic conditions (saturation and pressure) and elastic properties. Our new model is calibrated with lab measurements of room-dry sand samples over a wide range of pressure variations and then applied to well data to simulate different production scenarios. The complete rock physics model is then used in time-lapse inversion to predict the spatial distribution of dynamic property changes in the reservoir within a Bayesian framework. 159

CHAPTER 6. PRESSURE DEPENDENCE OF ELASTIC PROPERTIES

6.2

160

Introduction

Isotropic effective pressure is defined as the difference between the overburden pressure, which is the integral of density from the surface to the depth of interest times the acceleration due to gravity, and the pore pressure (Terzaghi (1943)). The effective pressure, Pe , is taken as the difference between the lithostatic overburden pressure Po and the hydrostatic pressure Pp (see for example, Mindlin (1949), Christensen and Wang (1985)): Pe = Po − Pp

(6.1)

In traditional workflows, effective pressure - velocity transformations are calibrated by using direct formation pressure measurements at the well, or by fitting lab measurements. The effective pressure - velocity model can be a simple polynomial or exponential relation or a more complex equation that also contains other rock properties, such as porosity and clay content. Eberhart proposed a more complex empirical model to relate rock properties and isotropic effective pressure to elastic attributes using the Han’s dataset (Han (1986)). More recent papers contain variants of those equations (Dvorkin et al. (1996); Dutta (2002); Sayers et al. (2003)). In Gutierrez et al. (2006) different models are, suggested the most appropriate relationships, and discussed the calibration of the adjustable parameters, which appears to be critical. All these papers contain relations calibrated on dry-rock velocities. MacBeth (2004) proposed an analogous equation to link dry rock elastic moduli to effective pressure with an exponential equation by fitting a set of lab measurements conducted on sand and shaly sand dry samples. Multi-property relations are taken into account in the model proposed by Saul and Lumley (2012). These two relations can be directly integrated in rock physics models used in reservoir characterization to describe dry rock elastic moduli variations as a function of effective pressure, the effect of fluid being then modeled by Gassmann’s equation. However the empirical parameters depend on porosity and mineralogy of the rock under study.


6.3

161

Physical model and application to lab data

The goal of this chapter is to find a reliable relation to link dry rock elastic moduli to the effective pressure to be used in time-lapse studies. Most of the relations published in literature so far are empirical and contain some parameters that must be fitted by using lab measurements, but these empirical constants change as a function of porosity and mineralogy. Laboratory measurements show that velocities, as well as the elastic moduli, generally increase with effective pressure and the relation is generally non-linear (Han (1986)). A relevant laboratory dataset of dry-rock velocity data is Han’s dataset, from which we selected a subset of 32 samples whose clay content varies between 5% and 25% (Figure 6.1). In the low effective pressure range, we observe a significant increase in velocity, whereas for higher pressure values, velocity changes tend to become smaller and exhibit an asymptotic behavior. Therefore, the relations between pressure and velocity and pressure and elastic moduli can be expressed by an exponential function that reaches the asymptotic value when pressure becomes large. However if we observe the measured data, we notice that even though each sample has a similar exponential behavior, the slope of the curve, the initial value and the asymptotic value are generally not the same. If we color code the data by porosity, we observe that the curves tend to move downward as porosity increases: as a matter of fact, if the clay content is approximately constant, generally the softer is the rock, the lower is the velocity; if the clay content changes, the relation is more complex, but still follows the general trend. The slope and the asymptotic value depend on different parameters, such as initial porosity, lithology and number of cracks in the rock. A similar behavior can be observed for the bulk modulus, as well as for S-wave velocity and the shear modulus. For density, the behavior is generally more linear and it reflects the porosity reduction due to increasing effective pressure. In Han’s case no relevant variations in porosity have been observed and density was assumed to be constant. Multi-linear regression can be used to link elastic property changes to effective

162


35

0.3

Dry Bulk Modulus (GPa)

30

0.25

25 0.2 20 0.15 15

0.1

10

5

0

10

20 30 Effective pressure (MPa)

40

50

0.05

Figure 6.1: Han’s dataset: dry-rock bulk modulus versus effective pressure color coded by porosity. pressure, porosity and clay content, but these relations are generally difficult to calibrate due to the lack of exhaustive datasets. Alternatively we can reduce the number of variables by introducing the concept of self-similarity (Dvorkin (2007)). In particular Dvorkin (2007) proposed to use a single variable given by the linear combination of porosity and clay content: φ + avclay , where a is an empirical constant (Figure 6.2). The petrophysical-properties dependence of velocity becomes more clear thanks to the self-similarity concept. 35

0.3

25 0.2 20 0.15 15 0.1

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5

30

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0

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30

Porosity+0.3Clay 0.35

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25 0.25 20 0.2 15 0.15

10

5

0

10


40

50

0.1

Figure 6.2: Han’s dataset: dry-rock bulk modulus versus effective pressure. The color represents porosity (left) and a linear combination of porosity and clay content(right).

163


The two relations that better describe the exponential behavior are: EberhartPhillips and MacBeth relations. Eberhart-Phillips’s relation links velocity and pressure and it contains porosity and clay content terms: √ VP = α − βφ − γ vclay + δ(Pe − e−εPe )

(6.2)

where α, β, γ, δ, ε are empirical parameters. This equation can be divided into two √ terms: the first term α − βφ − γ vclay does not depend on pressure, the second is the pressure-dependent exponential component. In other words the slope of the curves

obtained from this model is assumed to be the same and independent of porosity and lithology, and the curves differ only for the changes in the first term, that shifts the curve upward and downward depending on porosity and clay content. If we fit each sample independently (Figure 6.3, left), the fitted parameters will depend on the specific sample; on the other hand, if we fit all the samples simultaneously, the slope of the curves does not change: only the initial point and the asymptotic value are shifted (Figure 6.3, right). If we fit all the data simultaneously we obtain the following values α = 5.77, β = 6.94, γ = 1.73, δ = 0.446, ε = 16.7 as in Eberhart-Phillips et al. (1989). 5

0.3

4 0.2 3.5 0.15 3

0.1

2.5

2

0

10


40

50

0.05

0.3

4.5

0.25 P−wave velocity (km/s)

P−wave velocity (km/s)

4.5

5

0.25

4 0.2 3.5 0.15 3

0.1

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2

0

10


40

50

0.05

Figure 6.3: Eberhart-Phillips relation: on the left each sample has been fitted independently, on the right all samples have been fitted all together. MacBeth’s relation links dry rock bulk modulus and effective pressure Kdry (Pe ) =

K∞ 1 + AK e−Pe /PK

(6.3)


164

where K ∞ , AK , and PK are empirical parameters: K ∞ represents the asymptotic value, whereas AK and PK are related to the slope. For the physical derivation of Eq. 6.3 we refer the reader to MacBeth (2004). When we fit this relation independently for each sample (Figure 6.4), we obtain different values for the empirical parameters: if we plot these values versus porosity φ or a linear combination of porosity and clay content such as φ + avclay , a being for instance equal to 0.3, we observe a clear dependence for K ∞ and AK (Figure 6.5). Porosity 0.3

35

Dry bulk modulus (GPa)

30

0.25

25 0.2 20 0.15 15

0.1

10

5

0

10


40

50

0.05

Figure 6.4: MacBeth relation (bulk modulus): each sample has been fitted independently. We also observe that K ∞ and AK are not independent. As a matter of fact, if K0 is the measured value at a given effective pressure P0 , then AK can be expressed as a function of K ∞ and K0 : K∞ , K0 = Kdry (Pe = P0 ) (6.4) K ∞ − K0 To modify the Eq. 6.3 to include petrophysical-properties dependence, I used here AK =

the concept of self similarity to express K ∞ as a linear function of φ + avclay (with a = 0.3, Figure 6.6) and introduced Eq. 6.4. The modified equation then becomes: Kdry (Pe ) =

K∞ , K∞ (P0 −Pe )/PK 1 + K∞ e −K0

K ∞ = λ1 (φ + avclay ) + λ2

(6.5)


16

30

25

14

Parameter AK

20 15 10

15 10

0

0.1

0.2 0.3 Porosity

5

0.4

12 10 8

0

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0.2 0.3 Porosity

6

0.4

35

30

16

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25

14

Parameter AK

∞

Parameter K

20

25 20 15

Parameter PK

Parameter K

25

Parameter PK

30

∞

35

20 15 10

10 0.1

0.2 0.3 0.4 Porosity+0.3Clay

0.5

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5 0.1

0

0.1

0.2 0.3 Porosity

0.4


0.5

12 10 8


0.5

6 0.1

Figure 6.5: Sensitivity analysis on fitted parameters on MacBeth relation: (top) fitted parameters versus porosity; (bottom) fitted parameters versus linear combination of porosity and clay content. where λ1 and λ2 are empirical parameters that must be fitted using lab measurements. When we apply this equation to Han’s dataset, the results are quite satisfactory (Figure 6.7). Similar results have been obtained for the shear modulus (Figure 6.8). The modified equation for the shear modulus is: µdry (Pe ) =

µ∞ , 1 + µ∞ −µ0 e(P0 −Pe )/Pµ µ∞

µ∞ = λ3 (φ + avclay ) + λ4

(6.6)

where µ∞ is a linear function of φ + avclay , Pµ , λ3 and λ4 are empirical parameters, µ0 is the shear modulus at the initial pressure P0 .

45

Pressure (MPa) 50

45

40

45

40

35

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35

25

30

20

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20

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15

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0.2 0.3 Porosity+0.3Clay

0.4

166

measurements P=50MPa Linear regression K=−95.9(φ+0.3 vclay)+41.6

35 Dry Bulk Modulus (GPa)



30 25 20 15

0

0.1

0.2 0.3 Porosity+0.3Clay

0.4

Figure 6.6: Self similarity concept: (left) measured bulk moduli versus a linear combination of porosity and clay content color coded by effective pressure; (right) bulk modulus at 50 MPa versus a linear combination of porosity and clay content and linear fitting.

Porosity 0.3

35


30

0.25

25 0.2 20 0.15 15

0.1

10

5

0

10


40

50

0.05

Figure 6.7: Modified MacBeth relation (bulk modulus).

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Porosity 0.3 25

Dry shear modulus (GPa)

0.25 20 0.2 15 0.15 10 0.1 5 0

10


40

50

0.05

Figure 6.8: Modified MacBeth relation (shear modulus).

Porosity 0.3

30

Porosity 0.3

22

0.25

20

0.2

15

0.15

10

0.1

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18 Dry shear modulus (GPa)


20 25

16 0.2 14 12 0.15 10 8

0.1

6 5

0

10


40

50

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4

0

10


40

50

0.05

Figure 6.9: Modified MacBeth relation (set of five samples): bulk modulus (left) and shear modulus (right).


168

To clarify the application we show the results by only using 5 samples (Figure 6.9). In the next section we show how to apply the new equation in a complete rock physics workflow. As shown in the previous chapters, in seismic reservoir characterization we want to establish a set of rock physics relations to link reservoir properties with elastic properties. We now extend these relations to time-lapse studies. The input data of the rock physics model are: effective porosity, clay content, fluid saturations and effective pressure. We first compute the elastic moduli of the rock frame (matrix) using Voigt-Reuss-Hill average, we then compute the elastic moduli of the dry rock using a suitable rock physics model (empirical relations, granular media models, or inclusion models), we then apply the pressure effect using the modified MacBeth’s equations (Eqs. 6.5 and 6.6), and we finally model the fluid effect using Gassmann’s equations to obtain the elastic moduli of the saturated rock under assigned pressure conditions.

6.4

Application to log data and sensitivity

In this section we want to explore the combined effect of pressure and saturation changes on well log data and the corresponding synthetic seismograms. We consider a complete set of well log data from an oil field in the North Sea. Sonic logs and petrophysical curves from the formation evaluation analysis are shown in Figure 6.10. Insitu reservoir conditions are: effective pressure 20 MPa and gas saturation given by the saturation curve. We then design 8 different production scenarios mimicking different production mechanism, such as gas injection, water injection and depletion: 1. Effective pressure decreases (Pe = 10 MP a) and saturation is the initial saturation; 2. Effective pressure increases (Pe = 25 MP a) and saturation is the initial saturation; 3. Effective pressure equals Pe = 20 MP a and rock is water-saturated;


169

4. Effective pressure equals Pe = 20 MP a and rock is gas-saturated; 5. Effective pressure decreases (Pe = 10 MP a) and rock is water-saturated; 6. Effective pressure decreases (Pe = 10 MP a) and rock is gas-saturated; 7. Effective pressure increases (Pe = 25 MP a) and rock is water-saturated; 8. Effective pressure increases (Pe = 25 MP a) and rock is gas-saturated.

Porosity Water saturation

Quartz Clay Silt

Well log Rock physics model

2020

2040

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2060

2080

2100

2120

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3000 4000 5000 P−wave velocity (m/s)

1500 2000 2500 S−wave velocity (m/s)

2.2

2.4

2.6

Density (g/cm3)

2.8

0

0.5 1 Mineral fractions (v/v)

0

0.5 Pore fluid (v/v)

1

Figure 6.10: Well log data: from left to right P-wave velocity, S-wave velocity, density (well log in blue and rock physics model in red), and volumetric fractions (volume of quartz in yellow, clay in green, silt in black, effective porosity in red, and water saturation in blue). We first applied a rock physics model (stiff sand model) and the performed the fluid and pressure substitution on rock physics model prediction curves (Figure 6.10). The effects on elastic properties are shown in Figures 6.11 (P-wave velocity), 6.12 (S-wave velocity), and 6.13 (density). The effects on seismic properties are computed using a convolutional model and are shown in Figures 6.14 (short offsets), and 6.15 (long offsets).

170


P=25 MPa sw insitu

P=20 MPa (insitu) sw=1

P=20 MPa (insitu) sw=0 (gas)

P=10 MPa sw=1

P=10 MPa sw=0 (gas)

P=25 MPa sw=1

P=25 MPa sw=0 (gas)

2500 3500 4500 V (m/s)

2500 3500 4500 V (m/s)

2500 3500 4500 V (m/s)

2500 3500 4500 V (m/s)

2500 3500 4500 V (m/s)

2500 3500 4500 V (m/s)

2500 3500 4500 V (m/s)

P=10 MPa sw insitu

2020

2040

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2100

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2140

2160

2500

3500 4500 V (m/s) p

p

p

p

p

p

p

p

Figure 6.11: P-wave velocities computed through the rock physics model in 8 different scenarios (in-situ condition in red, and production scenarios in black).

P=10 MPa sw insitu

P=25 MPa sw insitu



P=10 MPa sw=1

P=10 MPa sw=0 (gas)

P=25 MPa sw=1

P=25 MPa sw=0 (gas)

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2000 3000 1000 2000 3000 1000 2000 3000 1000 2000 3000 1000 2000 3000 1000 2000 3000 1000 2000 3000 V (m/s) V (m/s) V (m/s) V (m/s) V (m/s) V (m/s) V (m/s) s

s

s

s

s

s

s

Figure 6.12: S-wave velocities computed through the rock physics model in 8 different production scenarios (in-situ condition in red, and production scenarios in black).

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P=25 MPa sw insitu

P=10 MPa sw insitu



2

2

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P=25 MPa sw=0 (gas)

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2

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Rho (g/cm3)

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Figure 6.13: Density computed through the rock physics model in 8 different production scenarios (in-situ condition in red, and production scenarios in black).

Seismograms

Seismograms

Seismograms

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Seismograms

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0

25 50 75 100 0 Offset (m)

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0

25 50 75 100 Offset (m)

0

25 50 75 100 Offset (m)

Figure 6.14: Synthetic seismograms in 8 different production scenarios (short offset).


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Figure 6.15: Synthetic seismograms in 8 different production scenarios (long offset).

6.5

Conclusion

The proposed new model describes the behavior of bulk and shear moduli over a wide range of effective pressures by including the porosity and lithology effects through the self-similarity concept. The new model is semi-empirical, it assumes funiform fluid distribution, and it must be fitted using laboratory measurements of rock samples in dry conditions. The model equation is represented by an exponential relation that shows the increase of dry elastic (bulk and shear) moduli as a function of effective pressure. The fluid effect can be then introduced using the traditional Gassmann’s equations. The new model also improves the accuracy of fluid substitution results because the dry-rock moduli is a key input parameter to Gassmanns fluid substitution equations. Our new velocity-pressure can be used for improved prediction of rock properties as a function of effective pressure and for time-lapse inversion and interpretation.

Chapter 7 Bayesian time-lapse inversion 7.1

Abstract

Traditionally static reservoir models are obtained as a solution of an inverse problem where reservoir properties, such as porosity and lithology, are estimated from seismic data. With the emergence of time-lapse reservoir models, we can integrate static and dynamic reservoir properties in the seismic reservoir characterization workflow. Here, we propose a methodology to jointly estimate rock properties, such as porosity, and dynamic property changes, such as pressure and saturation changes, from time-lapse seismic data. The methodology is based on a full Bayesian approach to seismic inversion and can be divided into two steps. First we estimate the conditional probability of elastic property and their relative changes, then we estimate the posterior probability of rock properties and dynamic property changes. We applied the proposed methodology to a synthetic reservoir study where we created synthetic seismic survey for a real dynamic reservoir model including pre-production and production scenarios. The final result is then a set of point-wise probability distributions that allow us to predict the most probable reservoir models at each time step and to evaluate the associated uncertainty.

173

CHAPTER 7. BAYESIAN TIME-LAPSE INVERSION

7.2

174

Introduction

Seismic data, repeatedly acquired at multiple times, monitor the changes in seismic response during the production phase. Reservoir models are driven by the need to include these time-lapse seismic conditioning data in the reservoir fluid flow simulator, in order to assess the predictions of dynamic property changes (in particular hydrocarbon saturations and pressure changes), compared to the pre-production static model and initial conditions (see Landrø (2001), and Landrø et al. (2003)). Due to the low resolution and degree of accuracy of seismic data, uncertainty quantification in pressure and saturation changes plays a key role, which calls for a probabilistic approach to time-lapse seismic inversion (see Buland and El Ouair (2006), and Trani et al. (2011)). Mostly, time-lapse seismic data has been used as a reservoir monitoring tool. If the repeatability of the seismic data is sufficiently good, time-lapse seismic data will contain small uncertainty and can be used to determine changes in the earth that are related to reservoir production. For to this reason, time-lapse seismic data have recently been used as an additional set of dynamic data for history matching in reservoir applications (see for example Aanonsen et al. (2003) and Dong and Oliver (2005)). A lot of efforts have been done in the direction of integration of production data and time-lapse seismic data (see for example Huang et al. (1997), Kretz et al. (2004) and Landa and Horne (1997)). Direct inversion methods based on first order or second order approximations of the analytical expression of pressure and saturation changes as a function of reflectivity coefficients have been presented as well (see Landrø (2001), Landrø et al. (2003), Dadashpour et al. (2008) and Trani et al. (2011)). In this work we propose a Bayesian inversion to estimate the posterior distribution of pressure and saturation changes from time-lapse seismic data. The problem is split into two sub-problems: the estimation of relative changes in elastic properties from seismic differences and the estimation of saturation and pressure changes from elastic property changes. First, the posterior distribution of relative changes in elastic properties is estimated from time-lapse partial-stacked seismic data using the Bayesian time-lapse inversion proposed by Buland and El Ouair (2006)), that combines the


175

convolutional model with Aki-Richards linearized approximation of Zoeppritz equations (see Buland and Omre (2003)). Then the likelihood of saturation and pressure changes given relative elastic property changes is estimated using rock physics models and kernel density estimation methods. Finally the two probabilities are combined to estimate the posterior distribution of reservoir property changes given time-lapse seismic data. In order to improve reservoir property predictions, we propose a simultaneous Bayesian inversion of the pre-production seismic survey and time-lapse seismic differences, to account for the correlation between static properties and changes in dynamic properties. The methodology has been applied to a real reservoir model with synthetic seismic data. The seismic inversion for elastic properties is performed combining the approaches presented in Buland and Omre (2003) and Buland and El Ouair (2006), whereas we perform the inversion for reservoir properties by using rock physics models and non-parametric probabilistic techniques. Time-lapse seismic data are corrected for time-shifts due to reservoir production, however this information could be made probabilistic (Avseth et al. (2012)) and integrated in the inversion workflow.

7.3

Methodology

In seismic reservoir characterization, we aim to solve the inverse problem of assessing a model m given the data Sbase Sbase = Gm + es

(7.1)

where Sbase represents the base seismic survey, G is the forward linear model, m is the elastic parameter, e.g. the logarithm of elastic impedances IP and IS , and es is a random error. The linear operator G can be written as G = WAD where W is the matrix containing the wavelets, A represents the reflectivity coefficients computed by Aki-Richards approximation and D is a differential matrix (as in Buland and Omre (2003)). In time-lapse studies we aim to solve the inverse problem of assessing ∆m given ∆S with ∆S = G∆m + ǫs

(7.2)

176


where ∆S is the difference between the repeated seismic survey and the base seismic survey ∆S = Srep −Sbase , G is the same forward linear model as in Eq. 7.1, ∆m is the model parameter consisting of the logarithm of relative changes in elastic properties,

and ǫs is a random error. The variable ∆m can be written as (Buland and El Ouair (2006)): 

∆m = 

rep IP ln I base Prep IS ln I base S



.

(7.3)

Once we have estimated the elastic properties, we then aim to estimate the static rock properties R, such as porosity, from the model m as a solution of another inverse problem m = fRP M (R) + er

(7.4)

and the changes in reservoir properties ∆D, such as saturation and pressure changes, from the model ∆m as ∆m = gRP M (∆D) + ǫr

(7.5)

where the functions fRP M and gRP M are rock physics models and er and ǫr are random errors. Different petro-elastic models fRP M , e.g. empirical relations, granular media models, or inclusion models (Mavko et al. (2009)) can be used, and these are generally calibrated with well data. The rock physics model gRP M relating elastic property changes to pressure and saturation changes generally combines an empirical equation with analytical models. In our work we use Gassmann’s equation for the saturation effect (Mavko et al. (2009)), whereas for the pressure effect we use a modified version of MacBeth’s relation (Mavko et al. (2009)) where we include the effect of initial reservoir properties R. By combining the formulations in Buland and Omre (2003) and Buland and El Ouair (2006) we can write: " # Sbase ∆S

and

"

R ∆m

=

#

"

G 0 0 G

= FRP M

#" "

m ∆m m

∆m

#

#!

+

"

es εs

#

+ eRP M ,

,

(7.6)

(7.7)

177


where FRP M is the rock physics model that include fRP M and gRP M and eRP M is the error associated to it. We propose here a Bayesian approach to solve simultaneously the combined inverse problem (Eqs. 7.6 and 7.7). For the first step (seismic inversion, Eq. 7.6), we assume a Gaussian prior distribution for the joint variable y = [m, ∆m]T . We then estimate the likelihood function using a linearized seismic forward model consisting of the convolution between the wavelet and the Aki-Richards linearized approximation of Zoeppritz equations (see Buland and Omre (2003), and Buland and El Ouair (2006)). Finally, we estimate the posterior distribution of the joint vector y|x, where x = [Sbase , ∆S]T using Bayes’ rule (Tarantola (2005)): " # " #! " # " #! m Sbase Sbase m P (y|x) = P | ∝P | P ∆m ∆S ∆S ∆m

"

m ∆m

#! (7.8)

T

In particular, if we assume that the joint vector y = [m, ∆m] is prior distributed according to a Gaussian model: # " m ∼N y= ∆m

"

µm µψ

# " ,

Σm

0

0

Σψ

#!

(7.9)

then the posterior distribution y|x where x = [Sbase , ∆S]T is Gaussian: y|x ∼ N µy|x , Σy|x

(7.10)

and the expressions of the conditional mean µy|x and variance Σy|x is analytical. For the second step (rock physics inversion, Eq. 7.7), if R represents the rock properties in the reservoir, such as porosity, and ∆D represents the changes in dynamic properties in the reservoir, such as pressure and saturation changes, then from rock physics theory (Mavko et al. (2009)) we know that the joint distribution of w = [R, ∆D]T depends on y, i.e. on the initial elastic properties m and their relative changes ∆m. Using Gassmann’s equation applied to well log data and the modified MacBeth’s relation fitted on core measurements describing different geological and


178

production scenarios, we can estimate the likelihood function P (w|y): " # " #! " # " #! " #! R m m R R P (w|y) = P | ∝P | P ∆D ∆m ∆m ∆D ∆D (7.11) The conditional distribution in Equation 7.11 is generally non-Gaussian and cannot be described by parametric distributions. For this reason we estimated the likelihood by using kernel density estimation (Silverman (1986)) where the kernel function is the Epanechnikov kernel. From the first step (seismic inversion) we obtained the posterior probability of elastic properties and their relative changes conditioned by seismic data and their differences P (y|x), whereas from the second step (rock physics) we obtained the posterior probability of reservoir properties and their changes conditioned by elastic properties P (w|y). Now we combine the two probabilistic information using Chapman-Kolmogorov equation (Papoulis (1984)): # " " #! Z Sbase R | P = P (w|x) = P (w|y) P (y|x)dy ∆D ∆S Rn

(7.12)

In order to compute the integral in Eq. 7.12, we use uniform discretizations of the model spaces of x, y, and w, evaluate the probability at each point of the discretized space, and compute the matrix product of the corresponding probability matrices. From Eq. 7.12,, we can estimate the most likely model (by taking a point-wise statistical estimator, such as the mean or the median), or simulate different realizations using geostatistical methods, such as probability field simulations.

7.4

Application: synthetic case

We tested the methodology using three different synthetic models: 1) a well log dataset representing a three layer model, 2) a real well log dataset with synthetic seismic, and 3) a 2D section of a real reservoir model with synthetic seismic. In the first example we simply invert for elastic properties and their relative changes the synthetic partial-stacked seismograms of a three layer model. The geological model represents a thick sand layer embedded in two shaly layers. Both rock and

179


elastic properties are assumed to be constant in each layer and there is no error on the seismogram. We then apply the Bayesian inversion proposed in the previous sections where we combine the approaches presented by Buland and Omre (2003) and Buland and El Ouair (2006). The elastic model is given by P-impedances, P-impedance relative change, and S-impedance relative change. The prior model is the Gaussian distribution in Eq. 7.9 where we assume that the mean value of P-impedance is a low frequency model of the actual log of P-impedance (filtered to 4 Hz), the mean value of the relative changes is equal to 0 everywhere, and the correlation between elastic properties and their changes is 0. In Figure 7.1 we show the actual data and the results of the inversion: the mean values and the percentiles P10 and P90 of the point-wise posterior distributions. P−impedance relative change 2000

S−impedance relative change

P−impedance

Estimated model True model Prior model

2020

Depth (m)

2040

2060

2080

2100

2120 0.8 1 1.2 1.4 3 P−impedance (m/s g/cm )

0.9 1 1.1 1.2 3 S−impedance (m/s g/cm )

4000 6000 8000 3 P−impedance (m/s g/cm )

Figure 7.1: Combined Bayesian inversion results for elastic properties and elastic properties relative changes for a three-layer wedge model: estimated P-impedance relative change, S-impedance relative change and P-impedance base model (black curves represent the prior model, blue curves the actual model, solid red curves represent the mean values and dotted red curves percentiles represent P10 and P90). In the second example, we use a real set of well logs coming from a well dataset in the North Sea. At the well location we created synthetic partial-stacked seismograms

180


at different times to mimic realistic time-lapse data. Time-lapse seismic data are obtained from pseudo-logs mimicking production scenarios as presented in Chapter 6. In the following we present a production scenario where effective pressure increases due water injection and oil production. The inversion is performed for elastic properties (impedances and relative changes) and reservoir properties (porosity and saturation and pressure changes). The full posterior distribution is show as background color in Figure 7.2. We then performed the inversion of elastic properties in terms of rock properties (porosity and changes in dynamic properties (saturation and pressure) for two different cases: in the first case we used a highly informative prior distribution for pressure and saturation changes (Figure 7.3) estimated from the pseudo-wells used to create synthetic seismic; in the second case we use a poorly informative prior distribution for pressure and saturation changes to include other production scenarios in addition to the real one. P−impedance relative change 2000

S−impedance relative change

P−impedance (base survey) Probability 0.3

2020 0.25

2040

Depth (m)

2060

0.2

2080 2100

0.15

2120 0.1

2140 2160

0.05

2180 2200 0.9 1 1.1 3 P−impedance (m/s g/cm )

1 1.1 3 S−impedance (m/s g/cm )

6000 8000 10000 12000 3 P−impedance (m/s g/cm )

0

Figure 7.2: Combined Bayesian inversion results for elastic properties and elastic properties relative changes for a real well log dataset: estimated posterior distributions of P-impedance relative change, S-impedance relative change and P-impedance base model (black curves represent the actual model. The results obtained for the first case are shown in Figure 7.4. For the second case we used a uniform distribution for pressure, a bimodal distribution for porosity and a

181


non-parametric distribution for saturation (Figure 7.5). The corresponding results are shown in Figure 7.6. The comparison of Figures 7.4 and 7.6 show that the uncertainty in the posterior distribution generally increases when the prior distribution is less informative. Histrogram of water saturation changes 500

Histrogram of effective pressure changes 700

Histrogram of effective porosity 250

600 400

200

Frequency

500 300

150

400 300

200

100

200 100

50 100

0 0

0.5 Water saturation (v/v)

1

0 0


6

0 0

0.1 0.2 0.3 Effective porosity (v/v)

0.4

Figure 7.3: Highly informative prior distribution for pressure and saturation changes.

Water saturation change 2000

Effective pressure change

Predicted Measured

2020

Depth (m)

Effective porosity

Probability 0.2 0.18

2040

0.16

2060

0.14

2080

0.12

2100

0.1

2120

0.08

2140

0.06

2160

0.04

2180

0.02

2200 0

0.5 1 Water saturation (v/v)

0 5 10 Effective pressure (MPa)

0 0.2 0.4 Effective porosity (v/v)

0

Figure 7.4: Combined Bayesian inversion results for elastic properties and elastic properties relative changes for a real well log dataset : estimated posterior distributions of P-impedance relative change, S-impedance relative change and P-impedance base model (black curves represent the actual model).

182

Frequency


Histrogram of water saturation changes 2500

Histrogram of effective pressure changes 2500 2500

2000

2000

2000

1500

1500

1500

1000

1000

1000

500

500

500

0 0

0.5 Water saturation (v/v)

1

0 0

2 4 6 Effective pressure (MPa)

Histogram of effective porosity

0 0

0.1 0.2 0.3 Effective porosity (v/v)

Figure 7.5: Poorly informative prior distribution for pressure and saturation changes.

Water saturation change

Effective pressure change

Effective porosity

2000 Predicted Measured

2020 2040 2060

Depth (m)

2080 2100 2120 2140 2160 2180 2200 0

0.2 0.4 0.6 0.8 Water saturation (v/v)

0 2 4 6 Effective pressure (MPa)

0

0.2 0.4 Effective porosity (v/v)

Figure 7.6: Combined Bayesian inversion results for elastic properties and elastic properties relative changes for a real well log dataset assuming a poorly informative prior distribution: estimated posterior distributions of water saturation change, effective pressure change and porosity base model (black curves represent the actual model.


183

The methodology has then been tested using a real reservoir model with synthetic time-lapse seismic data. The input reservoir model represents a clastic reservoir with sand and shale. The high-porosity sand layer at the top of the reservoir is filled by gas, whereas the mid section of shaly sand contains oil. The production mechanism consists of injecting water in the lower part of the reservoir and producing hydrocarbon in the upper part. From the reservoir fluid flow simulator we extracted a 2D section (Figure 7.7) at two different time steps: 2012 (pre-production) and 2017 (after 4 years of production).

Figure 7.7: Fluid saturations and effective pressure before (top) and after production (mid). Bottom left, effective porosity; bottom right, net-to-gross.

184


For both time steps we computed the corresponding elastic properties using a rock physics model and synthetic partial stacked seismic data with three different angle stacks: near (10o), mid (20o ), and far (30o ) using a convolutional model. The velocity and density models used to calculate the seismic dataset were computed using a rock physics model that combines: stiff sand model, Gassmann fluid substitution and modified MacBeth’s relation. We finally applied a correction to seismic data to account for time-shifts using a warping technique (Figure 7.8). Time-lapse differences are shown in Figure 7.9.

Time (ms)

Base survey Near (10o) 1100

1100

1150

1150

1200

1200

1250

1250

Time (ms)

50

100 Mid (20o)

150

1100

1100

1150

1150

1200

1200

1250

1250 50

Time (ms)

Repeated survey Near (10o)

100 Far (30o)

150

1100

1100

1150

1150

1200

1200

1250

1250 50

100 Trace number

150

50

100 Mid (20o)

150

50

100 Far (30o)

150

50

100 Trace number

150

Figure 7.8: Base (left) and repeated (right) seismic survey (perfect data): from top to bottom near (10o ), mid (20o ), and far (30o ).

185


Repeated survey − Base survey Near (10o)

Time (ms)

1100 1150 1200 1250 20

40

60

80

100

120

140

160

100

120

140

160

80 100 Trace number

120

140

160

o

Mid (20 )

Time (ms)

1100 1150 1200 1250 20

40

60

80 o

Far (30 )

Time (ms)

1100 1150 1200 1250 20

40

60

Figure 7.9: Time-lapse seismic differences: from top to bottom near (10o ), mid (20o ), and far (30o ).


186

In this application the model parameter m is P-impedance, ∆m is the variable given in Eq.

7.3, R is effective porosity, and ∆D represents changes in effective

pressure, oil saturation and gas saturation. To build the prior model of [m, ∆m]T we filtered the true impedance model (m) using a maximum frequency of 4 Hz, while for ∆m we assumed a prior mean of 0 everywhere. The prior correlation between m and ∆m is 0. The mean values of the marginal distributions of the point-wise posterior probabilities of elastic properties and their relative changes (1 mrep /mbase ) are shown in Figure 7.10. The corresponding absolute changes assuming that the initial model is correct, are shown in Figure 7.11. The rock physics likelihood has been estimated using well log data and core measurements and combining Monte Carlo simulations and rock physics models (Gassmann fluid substitution and modified MacBeth’s relation). In particular the pressure-velocity relation has been established using lab measurements at different effective pressure conditions (Chapter 6). In Figure 7.12, we display the mean of the point-wise posterior distributions of porosity, and saturation and pressure changes (Eq. 7.12). The high porosity layer at the top of the reservoir and the changes in fluid contacts are well detected by the inversion results, whereas pressure changes are slightly underestimated close to the well locations. Since the methodology is a full Bayesian inversion, for each point in the seismic grid we can evaluate the posterior distribution of each estimated property. In Figure 7.13, we show the posterior distributions of the predicted parameters at a given location in the seismic grid. The uncertainty in saturation changes is larger than for other properties since the fluid effect on elastic properties is generally less detectable. The posterior distribution of porosity is bimodal since it reflects the overall behavior of porosity in a mixture of sand and shale.


187

Figure 7.10: Mean values of inverted elastic properties and elastic properties relative changes: from top to bottom P-impedance relative change, S-impedance relative change, and P-impedance estimation from base survey.


188

Figure 7.11: Mean values of inverted elastic properties and elastic properties absolute changes: from top to bottom P-impedance absolute change, S-impedance absolute change, and P-impedance estimation from base survey.


189

Figure 7.12: Mean values of inverted reservoir properties and dynamic properties relative changes: from top to bottom gas saturation change, oil saturation change, effective pressure change, and porosity estimation.

190


Posterior probability 0.2

0.15

0.15

Probability

Probability

Posterior probability 0.2

0.1 0.05 0 0

0.5 Water saturation change

0.1 0.05 0 0

1

0.2

0.15

0.15

0.1 0.05 0

1

Posterior probability

0.2 Probability

Probability

Posterior probability

0.5 Oil saturation change

0.1 0.05

−5 −2.5 0 2.5 5 Pressure change (MPa)

0 0

0.1

0.2 Porosity

0.3

Figure 7.13: Examples of point-wise posterior distributions of reservoir properties and dynamic properties relative changes from top-left to bottom-right gas saturation change, oil saturation change, effective pressure change, and porosity.

7.5

Application: real case

We applied the inversion methodology to the Norne dataset, a real case study in the Barentz Sea. It is an oil field and oil production started in 1997. Sea depth in the area is about 380 m and the main reservoir is located at a deep of 2500-2700 m. Norne field consists of two oil compartments; Norne main structure (Norne C, D and E segments), which contains 97% of the oil in place, and the north-east segment (Norne Gsegment). Total hydrocarbon column is 135 m which contains 110 m oil and 25 m gas. The Norne main structure is relative flat. The reservoir is subdivided into four different formations. Hydrocarbons in this reservoir are located in the Lower to Middle Jurassic sandstones. The reservoir sandstones are dominated by fine-grained and well to very well stored arenites. Most of the sandstones are good reservoir rocks. The porosity is in the range of 25-30%. The average reservoir thickness is 200 m.


191

Figure 7.14: Main structure of the geocellular model (blue, top layer in red) and seismic survey geometry (black).

Figure 7.15: Active cells of the geocellular model (blue, top layer in red). Seismic survey geometry is shown in black for comparison.

192


The dataset includes well log data, a base seismic survey (2001) and three repeated seismic surveys in 2003, 2004 and 2006. Rock physics and petrophysics of the Norne main field are based on data from two exploration wells: 6608/10-2 (well E2 in the following) and 6608/10-3 (well F3 in the following). The geocellular model and the dynamic model are available as well. Seismic acquisition geometry and the main structure of the irregular reservoir grid are shown in Figure 7.14. In Figure 7.15 we only show the active cells of the reservoir grid (cells that contribute to fluid flow). Well logs are shown in Figure 7.16, whereas the rock physics diagnostics are shown in Figure 7.17. The applied rock physics model is the contact cement model (see Avseth et al. (2001)). Time-laspe seismic data are shown in Figures 7.18 and 7.19.

2750

Depth (m)

2800

2850

2900

Porosity Water saturation

2950

2000 4000 6000 P−wave velocity (m/s)

1000 2000 3000 S−wave velocity (m/s)

1

2

3 3

Density (g/cm )

0

0.5 1 Clay content (v/v)

0 0.5 1 Porosity and saturation (v/v)

Figure 7.16: Well log data (well E2): from left to right P-wave velocity, S-wave velocity, density, clay content, porosity and water saturation.

193


color = clay content

color = clay content

4600

3000 0.9

0.9

0.8

0.8

4400

0.7

4000

0.6

3800

0.5

3600

0.4

3400

0.3

3200

0.2

3000

0.1

2800

0

0.1

0.2 Porosity (v/v)

0.3

0.4

0.7 S−wave velocity (m/s)

P−wave velocity (m/s)

4200

2500

0.6 0.5 0.4

2000

0.3 0.2 0.1

1500

0

0.1

0.2 Porosity (v/v)

0.3

0.4

Figure 7.17: Well log data (well E2): crossplot of P-wave velocity versus porosity (left) and S-wave velocity versus porosity (right) color coded by clay content.

Figure 7.18: Seismic data differences in 2003: from left to right, near, mid and far.

Figure 7.19: Seismic data differences in 2006: from left to right, near, mid and far.


Figure 7.20: Predicted gas saturation in 2003.

Figure 7.21: Predicted gas saturation in 2006.

194


Figure 7.22: Predicted fluid pressure in 2003.

Figure 7.23: Predicted fluid pressure in 2006.

195


196

For this application we did not invert simultaneously for static and dynamic properties but only for dynamic property changes since the initial reservoir model was already provided in the dataset and obtained with a similar technique. We then performed the Bayesian time-lapse inversion of time-lapse seismic differences for changes in pressure and saturation. The inversion was performed in time-domain on a regular grid. We then converted the data from time to depth using the velocity model provided with the dataset and then we mapped (and upscaled) the regular seismic grid into the geocellular model using 3-dimensional interpolation. We assumed that the initial model for pressure and saturation was known and we added the predicted changes to obtain estimations of dynamic properties at the corresponding time-steps (2003, 2004, and 2006). To avoid non-physical results, such as saturations greater than 1, we applied cut-off values to preserve physical ranges. In Figures 7.20 and 7.21 we show the predictions for gas saturation in 2003 and 2006 where we observe a decrease of gas saturation in the upper part of the reservoir and a shift in the gas-oil contact. In Figures 7.22 and 7.23 we show the corresponding predictions for pore pressure, where we observe two different pressure behaviors in two different regions of the reservoir.

7.6

Conclusion

We presented a Bayesian inversion approach for assessing reservoir properties and their changes from time-lapse data. The simultaneous inversion of base seismic survey and tile-lapse seismic differences accounts for the correlation between static properties (e.g. porosity, net-to-gross) and changes in dynamic properties (saturations and pressure), and the full Bayesian approach allows us to quantify the associated uncertainty. This approach is different from the usual one where 3D seismic data is used to first characterize static reservoir properties and then time-lapse seismic data is used to estimate saturation and pressure changes. Here, instead we make use of the rock physics relations between pressure/saturation changes and porosity/lithology, to jointly invert for static and dynamic properties. The use of non-parametric distributions allows us to describe the non-Gaussian behavior of dynamic properties, in


197

particular saturations. The mean, median or most probable values of the estimated distributions of reservoir properties can be used in seismic history matching to improve reservoir model predictions and assess their uncertainty. The application to a real reservoir model with synthetic data showed promising results.

Chapter 8 Final remarks The main achievement of this thesis consists of a set of probabilistic methodologies to estimate rock properties in the subsurface. All the methods presented in this thesis combine seismic inversion with rock physics modeling and geostatistics. Rock physics models allow us to study the relations between rock properties and elastic attributes, specifically the physical models that link porosity, lithology and fluid properties with velocities of elastic waves. However, spatial variations and inter-dependence between different attributes are complex and cannot be described using simple deterministic functions. Geostatistics provides a framework to represent complex reservoir heterogeneities and model complicated relationships between reservoir attributes and is particularly useful when we have incomplete data information about the reservoir. The first Bayesian inversion method presented in Chapter 2 is particularly suitable in exploration phases where few data are available, whereas the inversion methods presented in Chapters 3 and 4 can be applied in advanced stages where the goal of the study is to obtain high detailed reservoir models. Similar methodologies have been applied in the past to elastic inversion problems, the introduction of rock physics modeling and statistical relations allows us to link the results of elastic inversion with static and dynamic reservoir models used in fluid flow simulators. Another statistical methodology is presented in Chapter 5 to account for the uncertainty of well log data and integrate formation evaluation results with rock physics relations. Finally, for a

198

CHAPTER 8. FINAL REMARKS

199

full integration of geophysics and reservoir engineering, a Bayesian inversion of timelapse seismic data is presented in Chapter 7, in order to estimate not only the initial static reservoir model, but also changes in dynamic properties, such as pressure and saturation. This goal is achieved by integrating seismic inversion techniques with a rock physics model describing pressure-velocity relations in different reservoir facies as shown in Chapter 6. Most of the methods presented in this thesis have been applied to clastic reservoirs. A possible research direction for the future is the application of these methodologies to unconventional reservoirs, in particular CO2 sequestration problems. A second important future development focuses on the integration of the inversion results in the reservoir modeling workflow. Time-lapse inversion results could be integrated in seismic history matching to improve the reservoir description and reduce the uncertainty. A third innovative research topic is the integration of other types of data and disciplines, such as electromagnetic data and geomechanical models.

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