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.
1 Review of Probability and Statistics
Statistics and probability notions and methods are commonly used in geophysics studies to describe the uncertainty in the data, model variables, and model predictions. Statistics and probability are two branches of mathematics that are often used together in applied science to estimate parameters and predict the most probable outcome of a physical model as well as its uncertainty. Statistical methods aim to build numerical models for variables
