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Bifurcation Analysis in Geomechanics

by

I.VARDOULAKIS
Department of Engineering Science
National Technical University of Athens
Greece
and
J.SULEM
Centre d’Enseignement et de Recherche en Mécanique
des Sols
Ecole Nationale des Ponts et Chaussées/LCPC
France

Published by
Blackie Academic & Professional, an imprint of Chapman & Hall,
Wester Cleddens Road, Bishopbriggs, Glasgow G64 2NZ
This edition published in the Taylor & Francis e-Library, 2005.

What is the use of bifurcation analysis in petroleum engineering rock mechanics?
This question no doubt will be asked by engineers working in this area with generally
rather practical and application oriented portfolios.
Is there any use at all, is it not just an academic subject, at best useful for wellcontrolled
conditions such as those dealt with in structural engineering of surface
facilities?
Are subsurface rock conditions with inherent heterogeneity, anisotropy and
layering not so uncertain that highly sophisticated numerical techniques with very
precise determination of failure are out of balance?
Many sceptics, amongst whom I have found myself for some time, look upon bifurcation
analysis in that way. Until recently there was definitely no strong business pull from
petroleum engineering to develop bifurcation analysis in geomechanics because no
operating company was really asking for improved production performance. Application
of bifurcation analysis to petroleum engineering was clearly a technology push; a
technique that was available and was (and is) waiting for people seeing opportunities to
apply it.
Once these opportunities were found, this fine technology appears to be a lot more
beneficial for our ‘rough’ business than initially anticipated. For instance, it allows us to
solve the very important question of scale dependency of our rock mechanical tests,
such as whether a hollow cylinder with a 8 mm hole can be representative for an 8½′′
wellbore. Transition between different borehole failure and stabilisation mechanisms can
now be understood, which significantly increases the scope for barefoot completions and
exclusion or postponement of sand control.
Suddenly a technology that was initially regarded as rather academic contributes to
millions of dollars savings.
This book gives an excellent overview and in-depth treatment of all aspects of
bifurcation analysis in geomechanics. This is sophisticated and powerful technology,
which can be exploited not only in petroleum engineering rock conditions but in many
other geomechanical applications.
Cor Kenter

This book by Vardoulakis and Sulem is an outstanding contribution to the important
field of geomechanics and will also be useful to the many branches of engineering and
applied science, particularly those dealing with mechanical behavior.
Following a self-contained Introduction, chapter 2 introduces the basic concepts from
continuum mechanics and thermodynamics before the incremental formulation of
standard continuum mechanics, particularly useful for stability and bifurcation
problems, is presented in chapter 3. This formalism is employed in chapter 4 to analyze
in detail buckling and interface instabilities, including multilayered and cracked media
which offer a natural stimulus for introducing the concept of Crosserat continuum.
Chapter 5 provides an updated contribution to the mechanics of water-saturated
media. It takes Biot’s original contribution to a new level of understanding, including
important physical phenomena such as grain crushing and non-Darcean flow. It uses
the essentials of mixture formalism but achieves a great economy of postulates and
mathematical generalizations with no direct physical interpretation.
In the same spirit, chapters 6 to 8 provide self-contained reviews of the current
plasticity theories and discuss in a systematic and thorough manner the problems of
bifurcation and shear-band formation. The effects of non-coaxiality, confining pressure
and grain characteristics are explained. The benchmark problems of the triaxial
compression test, the biaxial compression test and the cavity inflation test are analyzed
in detail as examples of loss of material stability and progressive failure.
The stage has now been set for the introduction of higher order continuum models to
describe correctly bifurcation and periodic phenomena in granular materials. Standard
theories of soil plasticity do not contain an internal length scale, and thus features such
as thickness and spacing of shear bands, as well as the stress-strain response in the
softening regime, cannot be addressed. As a result, finite element calculations do not
converge and related boundary value problems are ill-posed. The authors were among
the first to show that Cosserat and higher order gradient plasticity models can resolve
these difficulties.
Self-contained accounts of Cosserat plasticity and second-grade plasticity are given in
chapters 9 and 10, respectively. Bifurcation and shear-band analyses are performed and
the necessity of higher order terms in removing ill-posedeness in the softening regime
with simultaneous determination of shear-band characteristics (not captured by
standard continuum theory) is documented.
The book concludes with an account of stability of undrained deformations. This
problem is complex and very little understood despite its practical importance. The
authors demonstrate that fundamental work can intelligibly be applied to solve
outstanding geomechanics problems with extended engineering implications. Careful
laboratory tests are used to calibrate the theoretical models.
This well-organized volume is a substantial and valuable reference for civil engineers
and geologists, as well as mechanical engineers and metal physicists. It should be of
value to any junior or senior researcher with a keen interest in the stability of
deformation.
Elias C.Aifantis
Center for Mechanics of Materials and Instabilities
Michigan Technological University
and Aristotle University of Thessaloniki

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