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GEOMODELING

APPLIED GEOSTATISTICS SERIES General Editor Andre G. Journel Jean-Laurent Mallet, Geomodeling Pierre Goovaerts, Geostatistics for Natural Resources Evaluation Clayton V. Deutsch and Andre G. Journel, GSLIB: Geostatistical Software Library and User's Guide, Second Edition Edward H. Isaak and R. Mohan Srivastava, An Introduction to Applied Geostatistics

GEOMODELING Jean-Laurent Mallet OXFORD UNIVERSITY PRESS 2002

OXPORD UNIVERSITY PRESS Oxford New York Athens Auckland Bangkok Bogotd Buenos Aires Cape Town Chennai Dar es Salaam Delhi Florence Hong Kong Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris S5o Paulo Shanghai Singapore Taipei Tokyo Toronto Warsaw Istanbul Karachi and associated companies in Berlin Ibadan Copyright © 2002 by Oxford University Press, Inc. Published by Oxford University Press, Inc. 198 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Mallet, Jean Laurent. Geomodeling / Jean-Laurent Mallet. p. cm. Includes bibliographical references and index. ISBN 0-19-514460-0 1. Geology—Computer simulation. 2. Computer-aided design. I. Title. QE48.8.M34 2001 550'.1'13—dc21 00-047894 gOcad is a trade mark of the Association Scientifique pour la Geologic et ses Applications. 9 8 7 6 5 4 3 21 Printed in the United States of America on acid-free paper

Preface During the 1980s, it became clear that, in spite of their success in modeling simple surfaces, classic automatic mapping systems would never be able to model complex geological surfaces and, more generally, complex geological volumes affected by se- vere tectonic events with overturned folds, salt domes, and reverse faults. At the same time, experience using traditional Computer-Aided Design software developed for the car industry brought out its inability to accommodate the complex data encountered in the geosciences. For this reason, within the framework of the gOcad research project, in 1989 I proposed a completely different strategy involving the discrete modeling of natural objects. In this discrete approach, the geometry of any object is defined by a finite set of nodes (points) in the 3D space, while its topology is modeled by links bridging these nodes. For example, if the object to be modeled is composed of surfaces, then the links can be arranged in such a way that the mesh so defined generates triangular facets. These facets can be interpolated locally by flat triangles or, if need be, by curvilinear triangles. It is not difficult to imagine how this strategy can be extended to the modeling of curves and volumes. In practice, such a discrete approach is of no interest without a powerful math- ematical tool able to interpolate the location (x, y, z) of the nodes defining the geometry of the objects in the 3D space. For this reason, I proposed a new method, called Discrete Smooth Interpolation (DSI), which today is at the heart of the gOcad research project. This new interpolation method was specially designed for model- ing natural objects, while taking into account a wide range of complex and more or less precise data. In fact, adopting a new mathematical core for a Computer-Aided Design system has huge consequences that render inadequate most of the existing tools developed for traditional systems. The new research fields thus opened up resulted in the launching of the gOcad research project in the fall of 1989. After more than a decade of research, the tools developed within the framework of the gOcad project are now well honed and widely used in the oil and gas industry for modeling complex geological structures in the subsurface. At the same time, some exciting applications have come to light in very different fields such as, for example, medicine, anthropology, and the environmental sciences. This book presents some of the more important methods that have constituted the backbone of the gOcad project from its early days to the present. v

VI Acknowledgments I am pleased to acknowledge the many companies and universities around the world that have supported and actively participated in the gOcad project. GOcad was, and still is, a tremendous adventure, not only for me, but also for all the brave "gocadians" who have agreed to accompany me in this exciting research. There is no doubt that gOcad would never have become so popular if it were not for the enthusiastic and outstanding contribution of the students and senior researchers around the world who decided to participate in this adventure; without further ado, I want to thank them collectively for the great work they have done. My particular thanks go to all those who generously helped me in the prepa- ration and reviewing of this book, including Yves Bertrand, Amy Cheng, Richard Cognot, Joel Conraud, Stephane Conreaux, Jean-Claude Dulac, Pierre Goovaerts, Andre Haas, Andre Journel, Bruno Levy, Pascal Lienhardt, Heather Ludden, John Ludden, Olivier Mariez, Isabelle Moretti, Jarek Rossignac, Jean-Jacques Royer, Arben Shtuka, Chuck Sword. I am also grateful to all the many friends around the world who provided useful critiques in the preparation of this manuscript. Thanks also go to the companies Chevron, Elf, T-Surf and Unocal who provided data used for building some models of the subsurface that illustrate this book. Finally, I must acknowledge the ASGA organization who agreed to manage the day to day administration of the gOcad project from the very beginning to the present. I would like also to thank the Kluwer Publishing Company for the written permission to use and reproduce in this book some of my text and figures previously published in the Journal of Mathematical Geology (articles [150] and [151]). Finally, I express my deepest thanks to Oxford University Press which has agreed to publish this book. These acknowledgments would be incomplete without mentioning the essential role played by my wife, Danielle. For the last thirty years, her agreement to accept total responsibility for the smooth running of the family has freed me to devote myself one-hundred percent to my research. Without her, neither gOcad nor this book would ever have been possible. Jean-Laurent Mallet Jean-Laurent.Mallet@ensg.inpl-nancy.fr Institut National Polytechnique de Lorraine Ecole Nationale Superieure de Geologie/CRPG/Loria Nancy, July 2001

Contents 1 Discrete Modeling for Natural Objects 1.1 Introduction 1.2 Discrete modeling 1.3 1.4 Examples of applications 1.5 Conclusions Interpolation 2 Cellular Partitions Introduction 2.1 2.2 Elements of topology 2.3 Cellular partition of an n-manifold object 2.4 Generalized Maps 2.5 2.6 Conclusions Implementing GMap-based models 3 Tessellations Introduction 3.1 3.2 Delaunay's tessellation 3.3 Non-Delaunay triangulated surfaces 3.4 Notion of a regular n-grid 3.5 Notion of an irregular n-grid 3.6 3.7 Conclusions Implicit surfaces 4 Discrete Smooth Interpolation Introduction 4.1 4.2 The DSI problem 4.3 Uniqueness of the DSI solution 4.4 The local DSI equation 4.5 Accounting for hard constraints 4.6 Accelerating the convergence 4.7 The fuzzy Control-Point paradigm 4.8 The fuzzy Control-Node paradigm 4.9 From a discrete to a continuous model 4.10 Conclusions 5 Elements of Differential Geometry 5.1 Parametric curves 5.2 Parametric surfaces vn 1 1 5 12 19 26 27 27 28 36 57 81 93 97 97 98 109 122 132 134 137 139 139 147 153 160 170 174 182 190 193 196 199 199 203

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