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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Seismic reservoir characterization aims to build 3-dimensional models of rock and fluid properties, including elastic and petrophysical variables, to describe and monitor the state of the subsurface for hydrocarbon exploration and production and for CO₂ sequestration. Rock physics modeling and seismic wave propagation theory provide a set of physical equations to predict the seismic response of subsurface rocks based on their elastic and petrophysical properties. However, the rock and fluid properties are generally unknown and surface geophysical measurements are often the only available data to constrain reservoir models far away from well control. Therefore, reservoir properties are generally estimated from geophysical data as a solution of an inverse problem, by combining rock physics and seismic models with inverse theory and geostatistical methods, in the context of the geological modeling of the subsurface. A probabilistic approach to the inverse problem provides the probability distribution of rock and fluid properties given the measured geophysical data and allows quantifying the uncertainty of the predicted results. The reservoir characterization problem includes both discrete properties, such as facies or rock types, and continuous properties, such as porosity, mineral volumes, fluid saturations, seismic velocities and density.
Seismic Reservoir Modeling: Theory, Examples and Algorithms presents the main concepts and methods of seismic reservoir characterization. The book presents an overview of rock physics models that link the petrophysical properties to the elastic properties in porous rocks and a review of the most common geostatistical methods to interpolate and simulate multiple realizations of subsurface properties conditioned on a limited number of direct and indirect measurements based on spatial correlation models. The core of the book focuses on Bayesian inverse methods for the prediction of elastic petrophysical properties from seismic data using analytical and numerical statistical methods. The authors present basic and advanced methodologies of the current state of the art in seismic reservoir characterization and illustrate them through expository examples as well as real data applications to hydrocarbon reservoirs and CO₂ sequestration studies.
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Veröffentlichungsjahr: 2021
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Table of Contents
Cover
Title Page
Copyright Page
Dedication Page
Preface
Acknowledgments
1 Review of Probability and Statistics
1.1 Introduction to Probability and Statistics
1.2 Probability
1.3 Statistics
1.4 Probability Distributions
1.5 Functions of Random Variable
1.6 Inverse Theory
1.7 Bayesian Inversion
2 Rock Physics Models
2.1 Rock Physics Relations
2.2 Effective Media
2.3 Critical Porosity Concept
2.4 Granular Media Models
2.5 Inclusion Models
2.6 Gassmann's Equations and Fluid Substitution
2.7 Other Rock Physics Relations
2.8 Application
3 Geostatistics for Continuous Properties
3.1 Introduction to Spatial Correlation
3.2 Spatial Correlation Functions
3.3 Spatial Interpolation
3.4 Kriging
3.5 Sequential Simulations
3.6 Other Simulation Methods
3.7 Application
4 Geostatistics for Discrete Properties
4.1 Indicator Kriging
4.2 Sequential Indicator Simulation
4.3 Truncated Gaussian Simulation
4.4 Markov Chain Models
4.5 Multiple‐Point Statistics
4.6 Application
5 Seismic and Petrophysical Inversion
5.1 Seismic Modeling
5.2 Bayesian Inversion
5.3 Bayesian Linearized AVO Inversion
5.4 Bayesian Rock Physics Inversion
5.5 Uncertainty Propagation
5.6 Geostatistical Inversion
5.7 Other Stochastic Methods
6 Seismic Facies Inversion
6.1 Bayesian Classification
6.2 Bayesian Markov Chain Gaussian Mixture Inversion
6.3 Multimodal Markov Chain Monte Carlo Inversion
6.4 Probability Perturbation Method
6.5 Other Stochastic Methods
7 Integrated Methods
7.1 Sources of Uncertainty
7.2 Time‐Lapse Seismic Inversion
7.3 Electromagnetic Inversion
7.4 History Matching
7.5 Value of Information
8 Case Studies
8.1 Hydrocarbon Reservoir Studies
8.2 CO
2
Sequestration Study
Appendix: MATLAB Codes
A.1 Rock Physics Modeling
A.2 Geostatistical Modeling
A.3 Inverse Modeling
A.4 Facies Modeling
References
Index
End User License Agreement
List of Tables
Chapter 2
Table 2.1 Density, bulk modulus, and shear modulus of quartz, clay, calcite, ...
Table 2.2 Density and bulk modulus of water, oil, and gas. The ranges of valu...
Chapter 6
Table 6.1 Absolute frequencies of the facies classification.
Table 6.2 Reconstruction rate of the facies classification.
Table 6.3 Recognition rate of the facies classification.
Table 6.4 Estimation index of the facies classification.
List of Illustrations
Chapter 1
Figure 1.1 Bar chart of the probability mass function of a discrete random v...
Figure 1.2 Graphical interpretation of the probability
P
(2 <
X
≤ 3) of a con...
Figure 1.3 Comparison of statistical estimators: mean (square), median (circ...
Figure 1.4 Multivariate probability density functions: bivariate joint distr...
Figure 1.5 Bayes' theorem: the posterior probability is proportional to the ...
Figure 1.6 Examples of different correlations of the joint distribution of t...
Figure 1.7 Uniform probability density function in the interval [1, 3].
Figure 1.8 Standard Gaussian probability density function with 0 mean and va...
Figure 1.9 Log‐Gaussian probability density function associated with the sta...
Figure 1.10 Univariate Gaussian mixture probability density function, with t...
Figure 1.11 Bivariate Gaussian mixture probability density function estimate...
Figure 1.12 Beta probability density function in the interval [0, 1]: the so...
Chapter 2
Figure 2.1 Han's dataset: P‐wave velocity versus porosity color coded by cla...
Figure 2.2 Wyllie and Raymer's equations as a function of porosity for clay ...
Figure 2.3 Voigt, Reuss, and Voigt–Reuss–Hill averages for the bulk and shea...
Figure 2.4 Hashin–Shtrikman elastic bounds for the bulk and shear moduli of ...
Figure 2.5 Fluid mixing laws for a mixture of water and gas: the dashed line...
Figure 2.6 Soft sand and stiff sand models for the bulk and shear moduli of ...
Figure 2.7 Inclusion models for the bulk and shear moduli of a porous rock m...
Figure 2.8 Application of Gassmann's equations and fluid substitution to a s...
Figure 2.9 P‐wave and S‐wave velocity predictions as a function of porosity ...
Figure 2.10 P‐wave and S‐wave velocity predictions as a function of porosity...
Figure 2.11 P‐wave and S‐wave velocity variations as a function of effective...
Figure 2.12 Well logs for the Weber Sandstone formation, from left to right:...
Figure 2.13 Stiff sand rock physics model for the Weber Sandstone formation:...
Figure 2.14 Well logs for the Madison Limestone formation, from left to righ...
Figure 2.15 Berryman's inclusion rock physics model for the Madison Limeston...
Chapter 3
Figure 3.1 Three different porosity profiles: profile 1 shows the original d...
Figure 3.2 Histograms of porosity profiles in Figure 3.1, showing a similar ...
Figure 3.3 Cross‐plots of pairs of porosity samples measured at gradually in...
Figure 3.4 Experimental vertical correlation functions for porosity profiles...
Figure 3.5 Example of experimental vertical correlation function and corresp...
Figure 3.6 Spatial covariance functions and corresponding variogram models w...
Figure 3.7 Ellipse representing the azimuth‐dependent correlation length and...
Figure 3.8 Schematic representations of four data configurations. The dots r...
Figure 3.9 Spatial configuration and covariance model used for the kriging s...
Figure 3.10 Elevation and temperature maps of Yellowstone National Park. The...
Figure 3.11 Simple kriging estimate maps of elevation and corresponding krig...
Figure 3.12 Cokriging estimate maps of elevation and corresponding cokriging...
Figure 3.13 Cokriging estimate maps of elevation and corresponding cokriging...
Figure 3.14 Schematic representation of sequential Gaussian simulation for a...
Figure 3.15 Three random sequential Gaussian simulation realizations and the...
Figure 3.16 Three random sequential Gaussian simulation realizations and the...
Figure 3.17 Comparison of simple kriging and sequential Gaussian simulation ...
Figure 3.18 Three random sequential Gaussian co‐simulation realizations and ...
Figure 3.19 Application of geostatistical sequential simulations to reservoi...
Figure 3.20 Application of geostatistical sequential simulations to reservoi...
Chapter 4
Figure 4.1 Peak–valley classification map of elevation data in the Yellowsto...
Figure 4.2 Maps of indicator kriging probability of peaks and corresponding ...
Figure 4.3 Schematic representation of sequential indicator simulation for a...
Figure 4.4 Three random sequential indicator simulation realizations and the...
Figure 4.5 Three random sequential indicator co‐simulation realizations and ...
Figure 4.6 Three random truncated Gaussian simulation realizations and the a...
Figure 4.7 Facies simulations obtained using a stationary first‐order Markov...
Figure 4.8 Schematic representation of ENESim and SNESim algorithms for a tw...
Figure 4.9 Training image and three multiple‐point statistics random realiza...
Figure 4.10 Application of sequential Gaussian mixture simulation to reservo...
Chapter 5
Figure 5.1 Well log data from a borehole in the Norwegian Sea: P‐wave veloci...
Figure 5.2 Synthetic seismic data for three partial angle stacks: near (12°)...
Figure 5.3 Bayesian linearized AVO inversion results: posterior distribution...
Figure 5.4 Examples of Gaussian mixture models: a bivariate joint distributi...
Figure 5.5 Petrophysical curves computed from well log data in Figure 5.1, m...
Figure 5.6 Bayesian linearized rock physics inversion results: posterior dis...
Figure 5.7 Bayesian non‐linear and non‐parametric rock physics inversion res...
Figure 5.8 Uncertainty propagation (Gaussian mixture case): posterior distri...
Figure 5.9 Uncertainty propagation (non‐parametric case): posterior distribu...
Figure 5.10 Prior ensemble of petrophysical properties: porosity, clay volum...
Figure 5.11 Posterior ensemble of petrophysical properties: porosity, clay v...
Figure 5.12 Posterior ensemble of elastic properties: P‐wave velocity, S‐wav...
Chapter 6
Figure 6.1 Likelihood function of P‐wave velocity in sand and shale in Examp...
Figure 6.2 Contour plots of multivariate Gaussian likelihood functions of P‐...
Figure 6.3 Contour plots of multivariate non‐parametric likelihood functions...
Figure 6.4 Bayesian facies classification along a borehole dataset, from lef...
Figure 6.5 Bayesian Markov chain Gaussian mixture seismic inversion for faci...
Figure 6.6 Multimodal McMC seismic inversion for facies, P‐ and S‐impedance,...
Chapter 8
Figure 8.1 Rock physics relations and prior model: rock physics calibration ...
Figure 8.2 Posterior distributions of petrophysical properties and litho‐flu...
Figure 8.3 Partially stacked seismic data and corresponding petrophysical pr...
Figure 8.4 Predictions of porosity and oil sand facies probability in the th...
Figure 8.5 Posterior ensemble of petrophysical properties: porosity, clay vo...
Figure 8.6 Partially stacked seismic data and corresponding petrophysical pr...
Figure 8.7 Predicted reservoir model of petro‐elastic properties and facies ...
Figure 8.8 Stochastic realization of the reservoir model of petro‐elastic pr...
Figure 8.9 Isosurface of high-porosity carbonate probability higher than 0.6...
Figure 8.10 Structural and reservoir model of the Johansen formation and sim...
Figure 8.11 Inversion results for CO
2
saturation: (top) reference model of C...
Guide
Cover Page
Title Page
Copyright Page
Dedication Page
Preface
Acknowledgments
Table of Contents
Begin Reading
Appendix: MATLAB Codes
References
Index
Wiley End User License Agreement
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Seismic Reservoir Modeling
Theory, Examples, and Algorithms
Dario Grana
University of Wyoming
Laramie, WY, USA
Tapan Mukerji
Stanford University
Stanford, CA, USA
Philippe Doyen
Independent Consultant
London, UK
This edition first published 2021© 2021 John Wiley & Sons Ltd
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The right of Dario Grana, Tapan Mukerji, and Philippe Doyen to be identified as the authors of this work has been asserted in accordance with law.
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To our loved families
Preface
Geophysical data are commonly acquired to understand the structure of the subsurface for a variety of applications, such as energy resource exploration, natural hazards, groundwater, and surface processes. Seismic reservoir modeling is an interdisciplinary field that integrates physics, mathematics, geology, statistics, and computer science. The goal of seismic reservoir characterization studies is to predict three‐dimensional models of rock and fluid properties in the subsurface based on the available geophysical data, such as borehole measurements and surface geophysical surveys. Mathematical methods are required to solve the so‐called inverse problem in which the properties of interest are estimated from their geophysical response. The solution to this problem is generally non‐unique owing to the data errors, their limited resolution, the physics approximations, and the natural variability and heterogeneity of porous rocks. Multiple model realizations of rock and fluid properties, constrained by geophysical measurements and prior geological information, can be generated by combining statistical methods and computer science algorithms.
The main goal of this book is to bring together in one place basic and advanced methodologies of the current state of the art in seismic reservoir characterization. This work finds inspiration in the book Seismic Reservoir Characterization by Philippe Doyen. For the rock physics part, it strongly relies on The Rock Physics Handbook by Gary Mavko, Tapan Mukerji, and Jack Dvorkin, whereas for the geostatistics part, it relies on Geostatistical Reservoir Modeling by Michael Pyrcz and Clayton Deutsch, and Multiple‐Point Geostatistics by Gregoire Mariethoz and Jef Caers. Unlike other textbooks on seismic reservoir modeling, this book offers a detailed description of the mathematical–physical methods used for quantitative seismic interpretation. Indeed, it focuses on mathematical methods for the estimation of petrophysical properties, such as porosity, mineral volumes, and fluid saturations from geophysical data and attributes. Owing to the non‐uniqueness of the solution, we present a set of probabilistic methods that aim to estimate the most likely model as well as the uncertainty associated with the predictions. The model uncertainty can be quantified using probability distributions, confidence intervals, or a set of model realizations.
Chapter 1 reviews the main mathematical and statistical concepts used in the methods proposed in this book. We first present a review of probability and statistics to familiarize the reader with the concept of probability distribution. We then introduce the notion of a mathematical inverse problem, where the model variables are estimated from a set of measurements, based on the physics operator that links the model variables to the measured data. The inverse problem is formulated in a Bayesian setting.
Chapter 2 includes an overview of rock physics models. A rock physics model is defined as one or multiple equations that compute the elastic response (P‐wave and S‐wave velocity and density) of a porous rock given its petrophysical properties (porosity, mineral volumes, and fluid saturations). Empirical and theoretical models are presented with examples in different geological environments.
Chapters 3 and 4 focus on geostatistical methods for spatial interpolation and simulation of multiple realizations of subsurface properties. Geostatistics is a branch of spatial statistics that aims to analyze and predict properties associated with spatiotemporal phenomena in geosciences. Spatial statistics notions and algorithms are commonly used in geoscience to mimic the spatial and temporal continuity of geological processes. Chapter 3 describes methods for the interpolation and stochastic simulation of continuous properties, such as petrophysical and elastic variables, whereas Chapter 4 extends these algorithms to discrete properties, such as rock types and facies. An example of application of geostatistical methods to elevation and temperature maps in the Yellowstone National Park is presented to illustrate the algorithms and compare the predictions with the exact measurements.
Chapter 5 summarizes the developments in seismic and petrophysical inverse problems of the past two decades. The chapter includes three main topics: Bayesian linearized AVO inversion, Bayesian rock physics inversion, and geostatistical inversion. Bayesian linearized AVO inversion is an elegant and efficient algorithm proposed by Arild Buland and Henning Omre for the prediction of elastic properties from measured seismic data. Bayesian rock physics inversion refers to a set of probabilistic methods for the prediction of petrophysical properties from elastic attributes, based on different statistical assumptions for the distribution of the model variables and different linear or non‐linear rock physics models. Geostatistical inversion methods include multiple stochastic algorithms that aim to predict petrophysical properties from measured seismic data, by iteratively perturbing and updating an initial realization or a set of realizations. All the methodologies are illustrated through one‐dimensional examples based on borehole data.
Chapter 6 extends the Bayesian inversion methodology to facies classification and to joint inversion of facies and petrophysical properties from seismic data. This chapter discusses traditional Bayesian facies classification methods based on seismic data and seismically derived attributes, and introduces recent advances in stochastic sampling methods for the joint prediction of facies and reservoir properties, integrating Bayesian inverse theory and geostatistical algorithms.
Chapter 7 introduces additional sources of uncertainty associated with data processing, natural heterogeneity, and geological interpretation, and elaborates on the integration of seismic reservoir characterization methods in other domains of reservoir modeling. The chapter discusses the application of seismic and petrophysical inversion methods to time‐lapse geophysical data, the use of different geophysical datasets, such as electromagnetic data, and the assimilation of seismic data in fluid flow simulation for updating the reservoir model in monitoring studies. These topics introduce recent research challenges and directions. Probabilistic model predictions are also used in decision‐making studies associated with the value of information of geophysical data.
Chapter 8 presents several case studies previously published in peer‐reviewed journals. The applications include two hydrocarbon reservoirs in the Norwegian Sea, a carbonate field offshore Brazil, and a CO2 sequestration study offshore Norway.
The Appendix contains the description of the Seismic Reservoir Modeling (SeReM) MATLAB package including codes for rock physics, geostatistics, inversion, and facies modeling. The Matlab package SeReM and the Python version SeReMpy are available at the following link: https://seismicreservoirmodeling.github.io/SeReM/.
Acknowledgments
We completed our education as PhD students of the Stanford Rock Physics and Borehole Geophysics Project at Stanford University, under the supervision of Professor Amos Nur and Professor Gary Mavko. Amos and Gary continuously promoted innovations in theoretical and experimental rock physics for geophysical studies. We have also collaborated with the Stanford Center for Reservoir Forecasting group founded by Professor André Journel, who pioneered innovations in geostatistics theory and applications. Amos, Gary, and André's knowledge and mentorship have contributed to our professional growth as geoscientists.
We acknowledge our colleagues Jack Dvorkin, Henning Omre, Jo Eidsvik, Jef Caers, Per Avseth, Patrick Connolly, Brian Russell, Subhashis Mallick, Ezequiel González, Mita Sengupta, Lucy MacGregor, Alessandro Amato del Monte, and Leonardo Azevedo, as well as our collaborators in academia and industry for the constructive discussions and their help throughout these years. A special acknowledgment goes to Leandro de Figueiredo and Mingliang Liu for their help with the examples and Ernesto Della Rossa for the meticulous review.
The examples included in this book have been made possible by the availability of data and open source software. We would like to thank Equinor (operator of the Norne field) and its license partners Eni and Petoro for the release of the Norne data and the Center for Integrated Operations at NTNU for cooperation and coordination of the Norne case. We also thank SINTEF for providing the MATLAB Reservoir Simulation Toolbox. We thank the Society of Exploration Geophysicists for the permission to include some examples originally published in Geophysics.
We acknowledge the School of Energy Resources of the University of Wyoming and the Nielson Energy Fellowship, the sponsors of the Stanford Center for Earth Resources Forecasting, the Stanford Rock Physics and Borehole Geophysics Project, and the Dean of the Stanford School of Earth, Energy, and Environmental Sciences, Professor Steve Graham, for their continued support.
We want to thank all our co‐authors of previous publications, especially the MSc and PhD students that contributed to the recent developments in our research. This book would not have been possible without the contribution of numerous students, colleagues, and friends who have constantly inspired and motivated us. Finally, we thank our families for their love, encouragement, and support.
