Table Of ContentFRACTALS IN
SOIL SCIENCE
E d ite d b y
Philippe Baveye
Jean-Yves Parlange
Bobby A. Stewart
First published 2000 by CRC
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© 1998 by Taylor & Francis
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Preface
“Fractals everywhere!” This title of a successful book on fractals, published a
few years ago, echoes probably very well the current views of many soil
scientists. Every new issue of just about every soil science, or soil science
related, periodical has a number of articles dealing in some fashion with the
application of fractal geometry. This tidal wave shows no sign yet of abatement.
Claims continue to be made that fractals and fractal geometry will allow the
solution of unresolved questions, like the elusive and difficult quantification of
soil structure. At the same time, however, critics accuse fractal enthusiasts of
succumbing to yet another fad and forecast that, when the dust settles, fractals
will prove of little value in describing the intricate geometry of soils and the
complex processes that occur in them.
Tidal waves periodically sweep the shores of soil science research.
Seeing that “universal” frameworks like the thermodynamics of irreversible
processes, the theory of dissipative structures, catastrophe theory, percolation
theory, chaos theory, the theory of self-organized criticality, to name only a few,
were applied to systems ranging from biological organisms, to the history of
human institutions (churches, governments), all the way to the description of the
large-scale structure of the universe, some soil scientists were encouraged to try
to apply them as well to soils and soil processes. This “jumping on the
bandwagon” often is associated with an initial neglect of the intrinsic limitations
of the theory being applied, a tendency to exaggerate and misrepresent its
potential, followed by subsequent disappointment, if not even to a backlash and
eventual condemnation by all but a minority of “die-hards”. This pattern is
perhaps clearest in soil science in the case of the use of the thermodynamics of
irreversible processes.
Is the use of fractals in soil science another one of these fads? Certainly
the claims made for it initially were formidable: nothing less than the
encapsulation of the whole geometry of soils into a single parameter, the so-
called fractal dimension. The starry-eyed predictions that, with this single
dimension, one would be able to describe faithfully all the dynamical processes
occurring in soils are now usually smiled at. Yet, along the way, fractal
geometry has managed to reveal aspects of soils that were largely ignored
before. As a result, the “fractal movement” is still going strong in soil science,
some 16 years after its inception, unlike any other tidal wave in the past. It
seems therefore appropriate to inquire about its current status and about the
direction toward which it is heading.
The present book was designed to provide detailed and comprehensive
information on the current status of the application of fractal geometry in soil
science, and on prospects for its future use. Because applications of fractals in
soil science are both relatively recent and in constant evolution, there
occasionally are sharp differences among the viewpoints expressed in the various
chapters. In order for the book to reflect accurately existing trends, nothing has
been done to resolve these divergences of views nor to present a unified view.
At the same time, with the inclusion of an introductory chapter, in which the
theoretical foundations of fractal geometry are reviewed and illustrated, a special
effort has been made to propose a self-contained coverage of the field, suitable
as a textbook for an upper-level undergraduate or graduate course.
As a final word, we wish to thank the authors and reviewers for their
dedication in producing the chapters contained in this book and Susan Fox (CRC
Press LLC) for her thorough copyediting of the manuscript. Without their
professionalism and commitment, this volume would not have been possible.
P. Baveye
J.-Y. Parlange
B.A. Stewart
Contributors
P. Baveye, Laboratory of Environmental Geophysics, Bradfield Hall, Cornell
University, Ithaca, New York 14853-1901. [[email protected]]
F. Bartoli, Centre de Pédologie Biologique, UPR 6831 du CNRS, associé à
l’Université Henri Poincaré, 54501 Vandoeuvre-les-Nancy, France.
C. W. Boast, Department of Natural Resources and Environmental Sciences,
University of Illinois at Urbana-Champaign, 1102 South Goodwin Avenue,
Urbana, Illinois 61801 [[email protected]]
G. M. Berntson, Department of Organismic and Evolutionary Biology, Biological
Laboratories, Harvard University, 16 Divinity Avenue, Cambridge,
Massachusetts 02138. [[email protected]]
G. K. Bowman, Civil Engineering Department, Auburn University, Auburn,
Alabama 36849.
H. W.G. Booltink, Department of Soil Science and Geology, Agricultural
University, Wageningen, The Netherlands.
J. W. Crawford, Soil-Plant Dynamics Group, Cellular and Environmental
Physiology Department, Scottish Crop Research Institute, Invergowrie,
Dundee DD2 5DA, Scotland, [[email protected]]
S. Cr es tana, Centro Nacional de pesquisa e Desenvolvimento de Instrumentaçâo
Agropecuâria (CNPDIA, Empresa Brasileira de Pesquisa Agropecuâria
(EMBRAPA), Rua XV de Novembre 1452, Caixa Postal 741, Centro-CEP
13560-970, Sâo Paulo, Brazil. [[email protected]].
M Dubuit, Centre Interrégional de Resources Informatiques de Lorraine
(CIRIL), Château du Montet, 54500 Vandoeuvre-les-Nancy, France.
P. Dutartre, Centre de Pédologie Biologique, UPR 6831 du CNRS, associé à
l’Université Henri Poincaré, 54501 Vandoeuvre-les-Nancy, France.
C. Fuentes, Instituto Mexicano de Tecnologia del Agua (IMTA), Paseo
cuauhnahuac No 8532, Col. Progreso, 62550 Jiutepec, Morelos, Mexico.
V. Gomendy, Centre de Pédologie Biologique, UPR 6831 du CNRS, associé à
l’Université Henri Poincaré, 54501 Vandoeuvre -les-Nancy, France.
R. Hatano, Faculty of Agriculture, Hokkaido University, Kita 9, Niski 9, Kita-
Ku, Sapporo 060, Japan. [[email protected]]
R. Haver kamp, Laboratoire d’Etude des Transferts en Hydrologie et
Environnement (LTHE)-CNRS URA 1512, INPG, B.P. 53, 38041 Grenoble
Cédex, France. [[email protected]]
T. A. Hewett, Petroleum Engineering Department, Stanford University, Stanford,
California 94305.
V.J. Homer, Lenoir City, Tennessee 37771.
C.-h. Huang, National Soil Erosion Research Laboratory, 1196 Soil Building,
Purdue University, West Lafayette, Indiana 47907-1196.
J. Lynch, Department of Horticulture, 101 Tyson Building, Pennsylvania State
University, University Park, Pennsylvania 16802. [[email protected]]
F.J. Molz, Civil Engineering Department, Auburn University, Auburn, Alabama
36849. [[email protected]]
S. Niquet, Centre Interrégional de Resources Informatiques de Lorraine (CIRIL),
Château du Montet, 54500 Vandoeuvre-les-Nancy, France.
J.-Y. Parlange, Department of Agricultural and Biological Engineering, Riley-
Robb Hall, Comell University, Ithaca, New York 14853-1901.
[[email protected]]
E. Perrier, Laboratoire d’informatique Appliquée, Centre ORSTOM Bondy, 70-
74 Route d’Aulnay, 93143 Bondy Cédex, France.
[[email protected]]
A.N. Posadas, Centro Nacional de Pesquisa e Desenvolvimento de
Instrumentaçâo Agropecuâria (EMBRAPA), Rua XV de Novembre 1452,
Caixa Postal 741, Centro -CEP 13560-970, Sâo Paulo, Brazil.
M. Rieu, Laboratorie d’Hyrophysique des Sols, Centre ORSTOM Bondy, 70-74
Route d’Aulnay, 93143 Bondy Cédex, France.
S. Snapp, Rockefeller Foundation, PO 30721, Lilongwe 3, Malawi, SE Africa.
H. Van Damme, Centre de Recherche sur la Matière Divisée, CNRS and
Université d’Orléans, 45071 Orléans Cédex 2, France. [hvd@cnrs-
orleans.fr]
M. Vauclin, Laboratoire d’Etude des Transferts en Hydrologie et Environment
(LTHE)-CNRS URA 1512, INPG, B.P. 53, 38041 Grenoble Cédex, France.
[[email protected]]
H. Vivier, Laboratoire des Sciences du Génie Chimique (LSGC), UPR 6811 du
CNRS, ENSIC, BP 451, 54001 Nancy Cédex, France.
I. Young, Soil-Plant Dynamics Group, Cellular and Environmental Physiology
Department, Scottish Crop Research Institute, Invergowrie, Dundee DD2
5DA, Scotland. [[email protected]]
Contents
Fractal Geometry, Fragmentation Processes and the Physics of
Scale-Invariance: An Introduction..........................................................................1
P. Baveye and C. W. Boast
Structural Hierarchy and Molecular Accessibility in Clayey Aggregates... 55
H. Van Damme
Fractal Probes of Humic Aggregation: Scattering Techniques for Fractal
Dimension Determinations......................................................................................75
V. Homer
Fractal Geometry and the Description of Plant Root Systems:
Current Perspectives and Future Applications..............................................113
G. Berntson, J. Lynch and S. Snapp
Quantification of Soil Microtopography and Surface Roughness.............153
C.-h. Huang
Fractal Models of Fragmented and Aggregated Soils.....................................169
M Rieu and E. Perrier
Fractals and Soil Structure...................................................................................203
F. Bartoli, P. Dutartre, V. Gomendy, S. Niquet, M. Dubuit and H. Vivier
The Interactions between Soil Structure and Microbial Dynamics............233
J. W. Crawford and I.M. Young
Using Fractal Dimensions of Stained Flow Patterns in Clay Soils to
Predict Bypass Flow...............................................................................................261
R. Hatano and H. W.G. Booltink
2-D and 3-D Fingering in Unsaturated Soils Investigated by Fractal
Analysis, Invasion Percolation Modeling and Non-Destructive Image
Processing................................................................................................................293
S. Crestana andA.N.D. Posadas
Soil-Water Conductivity of a Fractal Soil.......................................................333
C. Fuentes, M. Vauclin, J.-Y. Parlange and R. Haverkamp
A Pseudo-Fractal Model for Hydraulic Property Distributions
in Porous Media.....................................................................................................341
F.J. Molz, T.A. Hewett and G.K. Bowman
Index 373
Q
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Fractal Geometry, Fragmentation
Processes and the Physics of
Scale-Invariance: An Introduction
P. Baveye and C.W. Boast
Contents
I. Introduction...................................................................................................2
II. A Gallery of Mathematical Monsters.........................................................3
A. Cantor’s Set and the Devil’s Staircase.....................................................3
B. The Peano-Hilbert Plane-Filling Curves.................................................7
C. The Triadic von Koch Curve...................................................................9
D. The Sierpinski Gasket and Carpet.........................................................10
E. The Menger Sponge...............................................................................11
F. Bolzano-Weierstrass-Like Functions .....................................................12
G. Random Monsters and Fractional Brownian Motion ...........................13
III. Hausdorff Dimension and Alternatives.....................................................18
A. Hausdorff Measure and Dimension.......................................................18
B. Similarity Dimension.............................................................................22
C. Box-Counting Dimension.......................................................................23
D. Divider Dimension.................................................................................26
E. Pointwise Dimension and Holder Exponent.........................................26
F. Correlation Dimension...........................................................................27
G. An Arsenal of Dimensions.....................................................................27
IV. Fractals.......................................................................................................28
A. Physical Motivation ...............................................................................28
B. “Definition” of Fractals .........................................................................29
C. “Natural” versus Mathematical Fractals.................................................32
V. Is “Power Law” Equivalent to “Fractal”? .................................................34
A. Power Law or Paretian Distribution andF ractals..................................34
B. Fragmentation Fractals...........................................................................37
VI. The Physics of Fractals.............................................................................40
A. Going Beyond Fractal Dimensions .......................................................40
B. Illustration: Diffusion-Limited Aggregation (DLA)............................41
ISBN 1-56670-105-8
© 1998 byCRC Press LLC 1
