Jean-Laurent Mallet. Geomodeling
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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
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in
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Copyright
©
2002
by
Oxford University Press, Inc.
Published
by
Oxford University
Press,
Inc.
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York,
New
York
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is a
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No
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may be
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or
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or by any
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or
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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.
987654321
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
1.1
Introduction
1
1.2
Discrete modeling
5
1.3
Interpolation
12
1.4
Examples
of
applications
19
1.5
Conclusions
26
2
Cellular Partitions
27
2.1
Introduction
27
2.2
Elements
of
topology
28
2.3
Cellular partition
of an
n-manifold object
36
2.4
Generalized Maps
57
2.5
Implementing GMap-based models
81
2.6
Conclusions
93
3
Tessellations
97
3.1
Introduction
97
3.2
Delaunay's tessellation
98
3.3
Non-Delaunay
triangulated
surfaces
109
3.4
Notion
of a
regular n-grid
122
3.5
Notion
of an
irregular n-grid
132
3.6
Implicit surfaces
134
3.7
Conclusions
137
4
Discrete Smooth Interpolation
139
4.1
Introduction
139
4.2
The DSI
problem
147
4.3
Uniqueness
of the DSI
solution
153
4.4
The
local
DSI
equation
160
4.5
Accounting
for
hard constraints
170
4.6
Accelerating
the
convergence
174
4.7
The
fuzzy
Control-Point paradigm
182
4.8
The
fuzzy
Control-Node paradigm
190
4.9
From
a
discrete
to a
continuous model
193
4.10 Conclusions
196
5
Elements
of
Differential
Geometry
199
5.1
Parametric
curves
199
5.2
Parametric
surfaces
203
vn
vii
5.3
Curvature
of
curves drawn
on a
surface
211
5.4
Miscellaneous
218
5.5
Discrete modeling
226
5.6
Examples
of
applications
to
geology
233
5.7
Conclusions
244
6
Piecewise
Linear
Triangulated
Surfaces
245
6.1
Introduction
245
6.2
Basic
DSI
constraints
256
6.3
Modeling
a
faulted surface
272
6.4
Continuity through faults
278
6.5
Global parameterization
287
6.6
Modifying
the
topology
305
6.7
Conclusions
315
7
Curvilinear
Triangulated
Surfaces
317
7.1
Introduction
317
7.2
Building
a
smooth curvilinear triangle
318
7.3
Gregory
G
1
patchwork
336
7.4
Recursive subdivisions
350
7.5
Conclusions
369
8
Elements
of
Structural
Geology
371
8.1
Geometry
of
faults
and
horizons
371
8.2
Modeling stratified media
379
8.3
Merging seismic
data
with
well
data
390
8.4
Deformation analysis
392
8.5
Unfolding
a
horizon
405
8.6
Unfolding
a
stack
of
layers
418
8.7
Conclusions
441
9
Stochastic
Modeling
443
9.1
Simulation versus interpolation
443
9.2
Probabilities
in a
nutshell
445
9.3
Random Functions
459
9.4
Random Fourier Series
474
9.5
Uniform
Random Functions
and
P-fields
488
9.6
Stochastic simulators
492
9.7
Kriging-based methods
502
9.8
Blending-based method
519
9.9
Assessing geometric
uncertainties
528
9.10 Conclusions
532
10
Discrete
Smooth
Partition
533
10.1 Introduction
533
10.2
The
probabilistic
approach
538
10.3 Structural constraints
548
10.4 Moving-Centers-based methods
559
10.5
A
tutorial example
572
10.6 Conclusions
579
Index
595
CONTENTS
Pour
examiner
la
verite,
il est
besoin
une
fois
dans
sa vie de
mettre
toutes
choses
en
doute
autant
qu'il
se
pent...
...
Ceux
qui
cherchent
le
droit
chemin
de la
verite
ne
doivent s'occuper
d'aucun objet dont
Us ne
puissent avoir
une
certitude
egale
a
cells
des
demonstrations
de
I'arithmetique
et de la
geometric.
ffene
Descartes
(Discours
de la
methode)