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 Istanbul Karachi Kolkata Kuala Lumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris S5o Paulo Shanghai Singapore Taipei Tokyo Toronto Warsaw 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. QE48.8.M34 2001 550'.1'13—dc21 00-047894
I. Title.
gOcad is a trade mark of the Association Scientifique pour la Geologic et ses Applications.
9 8 7 6 5 4 3 2 1 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 severe 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 mathematical 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 modeling 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 preparation 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
[email protected] 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 Interpolation 1.4 Examples of applications 1.5 Conclusions
1 1 5 12 19 26
2 Cellular Partitions 2.1 Introduction 2.2 Elements of topology 2.3 Cellular partition of an n-manifold object 2.4 Generalized Maps 2.5 Implementing GMap-based models 2.6 Conclusions
27 27 28 36 57 81 93
3 Tessellations 3.1 Introduction 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 Implicit surfaces 3.7 Conclusions
97 97 98 109 122 132 134 137
4 Discrete Smooth Interpolation 4.1 Introduction 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
139 139 147 153 160 170 174 182 190 193 196
5 Elements of Differential Geometry 5.1 Parametric curves 5.2 Parametric surfaces
199 199 203
vn
CONTENTS
vii 5.3 5.4 5.5 5.6 5.7
Curvature of curves drawn on a surface Miscellaneous Discrete modeling Examples of applications to geology Conclusions
211 218 226 233 244
6 Piecewise Linear Triangulated Surfaces 6.1 Introduction 6.2 Basic DSI constraints 6.3 Modeling a faulted surface 6.4 Continuity through faults 6.5 Global parameterization 6.6 Modifying the topology 6.7 Conclusions
245 245 256 272 278 287 305 315
7 Curvilinear Triangulated Surfaces 7.1 Introduction 7.2 Building a smooth curvilinear triangle 7.3 Gregory G1 patchwork 7.4 Recursive subdivisions 7.5 Conclusions
317 317 318 336 350 369
8 Elements of Structural Geology 8.1 Geometry of faults and horizons 8.2 Modeling stratified media 8.3 Merging seismic data with well data 8.4 Deformation analysis 8.5 Unfolding a horizon 8.6 Unfolding a stack of layers 8.7 Conclusions
371 371 379 390 392 405 418 441
9 Stochastic Modeling 9.1 Simulation versus interpolation 9.2 Probabilities in a nutshell 9.3 Random Functions 9.4 Random Fourier Series 9.5 Uniform Random Functions and P-fields 9.6 Stochastic simulators 9.7 Kriging-based methods 9.8 Blending-based method 9.9 Assessing geometric uncertainties 9.10 Conclusions
443 443 445 459 474 488 492 502 519 528 532
10 Discrete Smooth Partition 10.1 Introduction 10.2 The probabilistic approach 10.3 Structural constraints 10.4 Moving-Centers-based methods 10.5 A tutorial example 10.6 Conclusions
533 533 538 548 559 572 579
Index
595
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GEOMODELING
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Chapter
1 Discrete Modeling for Natural Objects This introductory chapter presents a discrete approach specially designed for modeling the geometry and the properties of natural objects such as those encountered in biology and geology. Contrary to classical Computer-Aided Design methods based on continuous (polynomial) functions, the proposed approach is based on a discretization of the objects close to the finite element techniques used for solving partial differential equations. Each object is modeled as a set of interconnected nodes having both the geometry and the physical properties of the objects, and the Discrete Smooth Interpolation method is used for fitting the geometry and the properties to complex data. Each datum is turned into a linear constraint and some constraints related to typical information encountered in numerical geology are presented.
1.1
Introduction
In this introduction, we would like to emphasize the need for a new breed of Computer-Aided Design (CAD) specially dedicated to the modeling of natural objects such as those encountered in biology and geology. In a nutshell, we could say that a user of a traditional CAD software creates nice curves, surfaces and volumes without any constraint, while a user of a CAD software dedicated to the modeling of natural objects has to respect constraints induced by the observed data. All of these traditional CAD software programs are based on parametric methods such as those introduced by Bezier in the early 1970s. The goal of the mathematicians who designed these methods (e.g., Barnhill 1
2
CHAPTER 1. DISCRETE MODELING FOR NATURAL OBJECTS
Figure 1.1 An example of complex data to take into account when modeling geological horizon. (Data courtesy of Unocal) [13]; Bezier [27]; de Boor [54, 55]; Farin [77]; . . . ) was simply to propose tools for modeling nice curves and surfaces interactively and not to respect complex data such as those encountered in geology. For example, if we consider the geological horizon (=surface) shown in figure (1.1), we can say that the data available are complex for the following reasons: • The data are heterogeneous because we have to account simultaneously for: — the locations of the intersections of the well-curves with the surface to be modeled (see pages 184 and 263), — the slope of the tangent plane to the surface when this slope has been measured at the intersections of the wells with the surface (see page 267), — seismic data corresponding to cross sections of the surface (see page 264), — structural information consisting of 3D fault-throw vectors (see page 273), — geometric information specifying that the surface should respect a given partial differential equation (see page 267), and — physical properties (seismic velocities, reflectivities,...) attached to the surface and observed directly or indirectly at some locations (see page 256).
1.1. INTRODUCTION
3
• The data are more or less reliable; for example, well data are more reliable than seismic data, which in turn are more reliable than structural data. • The data are irregularly distributed and are generally strongly clustered on lines and surfaces. In practice, this clustering generates numerical instabilities in most of the numerical methods used for interpolating the data. Another major drawback with traditional CAD methods is that they are designed for modeling the geometry of objects and not to take into account the physical properties attached to these objects. In geology, there is a strong need to model the geometry and the properties simultaneously since there are many cases where they are linked: for example, the geometry and the seismic velocity of geological layers are interdependent. Mathematical methods used in traditional CAD have not been designed for addressing such complex data and it is too optimistic to think that these methods could be adapted to account for all of the data available, while respecting their complexity [148]. In fact, these methods were initially designed for the needs of the car industry [27] and we are in search of a specific class of mathematical modeling methods specially designed to meet the needs of the geosciences. The success of polynomial models used in traditional CAD [166] comes from the fact that polynomials generate aesthetic curves and surfaces and this is of paramount importance in the car industry [27]. However, in the geosciences, our primary concern is more to respect the constraints induced by the data than to produce nice-looking objects. It is generally admitted that discretized problems are simpler to solve than those based on continuous representations and this is why, in this chapter, we propose to abandon the polynomial models used in traditional CAD in favor of a discrete approach close to the "finite elements" technique used for solving partial differential equations. Despite the fact that they are less well known than parametric methods, discrete modeling methods have been formulated and implemented in varying ways for over seventy years. Examples include: Whittaker [232]; Horton [107]; Bergthorsson and Doos [21]; Weaver [227]; Arthur [9]; Harder and Desmarais [102]; Briggs [34]; Akima [4]; Sibson [202]; Mallet [147, 149, 150]; Overveld [170]. The goal of this chapter is to propose a generic formulation of discrete modeling which generalizes most of these methods. The approach presented in this chapter was specially designed for modeling natural objects and is potentially able to account for any series of (linear) constraints corresponding to the influence of the data on the model. Each of these constraints can be weighted by a "certainty factor" used for specifying its importance relative to the other constraints. This is particularly interesting in the geosciences where, due to sampling errors, it may happen that some constraints become contradictory. For example, if the projection of two seismic cross sections in the (x, y) horizontal plane meet at some point P(x, y), it
4
CHAPTER 1. DISCRETE MODELING FOR NATURAL
OBJECTS
Figure 1.2 Examples of geomodeling applications: (A) Reconstruction of a distorted and broken Neanderthal skull (metric scale). (B) Modeling the internal organs of an embryo (millimetre scale). (C) Modeling the structure of "Hoes" in a mud resulting from the treatment of wasted water (micrometric scale). (D) Modeling geological structures (kilometric scale). (E) One more example of natural object. is almost certain that these two seismic cross sections do not, in actual fact, intersect in the (x,y,z) 3D space (see figure (1.1)), and it is important to be able to weight each of them with a given certainty factor proportional to the quality of the data.
Geomodeling: A definition By contrast with traditional CAD methods devoted to the modeling of manufactured objects and to emphasize the specific nature of the modeling of geological objects, the following definition of the notion of geomodeling is proposed: Geomodeling consists of the set of all the mathematical methods allowing to model in an unified way the topology, the geometry and the physical properties of geological objects while taking into account any type of data related to these objects. In fact as illustrated in figures (1.2), (1.16), or (6.23), most of the methods developed for modeling geological objects are also potentially applicable to the modeling of natural objects as those encountered, for example, in biology, medicine, paleoanthropology and environmental sciences. This book mainly focuses on geological applications, and particular attention is paid to discrete modeling methods and geostatistical methods frequently used in the realm of geosciences. However, to point out the possible
1.2. DISCRETE MODELING
5
Figure 1.3 Neighborhood N(a} of node a is composed of a plus all nodes of its orbit N°(a) = {0i,02, • • •}• This figure corresponds to the special case where N(o) is denned as the set of nodes that can be reached in at most s(a) = I step from the node a. use of these methods in other fields of science, a few non-geological examples of applications will also be given.
1.2
Discrete modeling
This section presents in a general and abstract way the notion of a discrete model, which can be used for approximating complex natural objects encountered, for example, in the geosciences.
Discrete topological model £7(f2, N) As shown in figures (1.3) and (1.5), the main idea of a discrete model is that the topology of any object can be approximated by a graph ^(0, N) where: • 0 is the set of all the nodes of the graph, each of these nodes being identified with its rank order:
N is an application from into a subset of such that
can be reached in at most steps from
CHAPTER 1. DISCRETE MODELING FOR NATURAL
OBJECTS
Figure 1.4 An example of topologically ambiguous object defined by nodes connectivities only. where s(a] > 0 is a given function of the node a. In practice, to minimize the complexity of the model, we suggest choosing s(a) = I but, from a mathematical point of view, this is absolutely not mandatory. We assume also that the graph £(Q,/V) is symmetrical and this implies that N ( - ) is such that 0 e N(a)
a £ JV(0)
The subset N(a) is called "neighborhood" of a and it must be noted that TV (a) contains the node a and all its surrounding nodes (see figure (1.3)): N(a)
=
{a, ft, &,...}
From a mathematical point of view [24, 15], we can say that N ( - ) defines a discrete topology on Jl in the sense that
As suggested in figure (1.3), the "orbit" 7V°(a:) of a node a is defined as the set of nodes {/?i, 02> • • •} different from a and belonging to N(a}:
Topological ambiguity It is important to note that, in general, the notion of discrete topological model defined above only approximates the topology of real objects. Figure (1.4) gives an example that clearly shows that the nodes connectivities defined by the neighborhood operator N(a) may have several interpretations in terms of solid objects. In other words, the notion of discrete topological model introduced above is topologically ambiguous: the neighborhood operator N(a) is not sufficient to describe the topology of real objects. The whole point of chapter (2) is precisely to provide the additional information needed for removing such an ambiguity.
1.2. DISCRETE
MODELING
7
Figure 1.5 Examples of objects approximated by discrete model: (A) Faulted triangulated geological surface. (B) Set of adjacent tetrahedra filling a geological layer. (C) Polygonal curves. (D) Faulted curvilinear grid filling a geological layer.
Examples of discrete topological models Potentially, the notion of discrete topological model presented in the previous section can be used for approximating the topology of any geological object. For example and as suggested in figure (1.5), it is possible to model: • a geological horizon or a fault (surface) as a set of adjacent triangles (figure (1.5)-A), • a geological body (solid) as a set of adjacent tetrahedra (figure (1.5)-B), • a geological cross section (curve) as a set of adjacent segments (figure (1.5)-C), and • a geological layer (solid) as a regular curvilinear or rectilinear grid (figures (1.5)-D and (1.6)-B). In these models, the vertices of the triangles, tetrahedra, and segments correspond to the nodes of &(£l, N), while the edges define (partly) the topology. The introduction of a discontinuity between two adjacent nodes a and (3 is straightforward, and, as suggested in figure (1.6), this can be achieved simply by removing the edge (a, /?) from the network.
Notion of discrete model .M n (n,7V, (p, C) The discrete topological model £?(i7, N) introduced in the previous sections does not take into account the properties of the objects. As shown in figure (1.7), such properties are modeled as a series denned on the set 0, consisting of the nodes of a graph g(£l,N).
In practice, three components of (p(ot), noted {(px(a), ipy(a), tpz(a)}, correspond to the location of the node a € O in the 3D space, while the other components correspond to physical properties. By definition, the notion of discrete model .M n (f2, N, ,C) consists of a triplet composed of Q(Q,N), the functions :
Each constraint c e C is assumed to be linear and to have one of the following three general forms where {^(a)} and bc are given coefficients denning the constraint c:
1.2. DISCRETE MODELING
9
By definition, we will say that • C~ is the set of "soft" equality constraints that have to be honored in a least square sense, • C= is the set of "hard" equality constraints that have to be strictly honored, and • C> is the set of "hard" inequality constraints that have to be strictly honored.
It should be noted that the notion of "soft inequality" constraint does not make sense and is thus not defined.
Simple and cross constraints Let us consider a constraint c E C^:
Depending on the nature of the coefficients {A"(a)} associated with c, we will say that c is either a "simple" constraint or a "cross" constraint: • If the coefficients {^(a)} are such that
then c will be called a "simple" constraint. As you can see, a simple constraint involves only one component tpVc of "(ai)
=
A"(a 0 ,ai)
and this information can easily be turned into a constraint c 6 C~ or c e C= such that
In figures (1.1) and (1.6)-A, this constraint can be used to specify that the coordinates (px(a0), tpy(a0), ipz(o>0)) and (px(ai), (py(ai), ipz(ai}) of two nodes located on both sides of a fault should be equal to a given throw vector (Ax(ao,ai), Ay(a0,ai), A^o^cei))- For more details on a generalization of such a constraint, see page 258. To constrain the point {(f>x(UQ), (py (aQ}, (pz(ao)} to be located on a given plane P(p, ri) containing the point p = (px,py ,pz) and orthogonal to the vector n = (nx,ny,nz), we should have (
and this relationship can easily be turned into a constraint c G C~ or c G C~ such that
It is important to note that, contrary to the two previous constraints, this constraint is a "cross" constraint merging several components of (p.
1.2. DISCRETE MODELING
11
In figure (1.1), this constraint can be used for specifying that a node ao should be located in a given tangent plane. For more details on such a constraint, see page 265. • If {v?1(a0), V^^o), • • -, implies a slight modification in the main loop of this algorithm.
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CHAPTER 1. DISCRETE MODELING FOR NATURAL
OBJECTS
Generalization of the DSI method In practical applications, the function (p of the discrete model Ai n (0, TV, \a] together with the degree of violation of the constraints p(f\c) suffices:
In the above definition of the roughness R(y>\a), rv is a given positive weighting coefficient allowing the contribution of the function "(•) to the local roughness R((p\ot) to be modulated. If the functions {^{-)} have approximately the same magnitude, then it is possible, for example, to choose all the coefficients {r"} equal to 1. The equation allowing the value (pv(oi) in the iterative DSI algorithm to be updated remains almost unchanged in its form (see equation (4.46)) and becomes: The only noticeable difference in the monocomponent case is a slight modification of the term Tv(a\(f>) in the case of cross constraints involving several components of (p (see page 163). When there is no such cross constraint attached to the node a, then the above formulas are absolutely identical to the ones in the monocomponent case.
Implementation of the DSI method In spite of its apparent complexity, the local DSI equation is very simple to program and the very heart of the iterative algorithm introduced on page 17 can be implemented in about 30 lines of source code. The real programming difficulty comes from the discrete model .A/ln(O, N,
