Table Of ContentList of Contributors
Besuelle, Pierre Laboratoire 3S, BP 53, 38041 Grenoble Cedex 09, France.
[email protected]
Bouteca, Maurice Institut Fran9ais du Petrole, 1 et 4 avenue de Bois-Preau,
92852 Rueil-Malmaison Cedex, France,
cepm @ francenet.fr
David, Christian Departement des Sciences de la Terre, Universite de Cergy-
Pontoise, 8 Le Campus Bat. I, 95031 Cergy-Pontoise Cedex, France.
[email protected]
Dormieux, Luc CERMMO, ENPC, 6-8 Avenue Blaise Pascal, Champs sur Mame,
77455, Mame la Vallee Cedex 2, France,
dormieux @ cermmo-omp.enpc.fr
Dresen, Georg GeoForschungZentrum Potsdam, Telegrafenberg D 425, D-14473
Potsdam, F.R.G.
dre @ gfz-potsdam.de
Gueguen, Yves Ecole Normale Superieure, ENS, 24 rue Lhomond,
75231 Paris Cedex 05, France.
[email protected]
Kumpel, Hans-Joachim GGA Institut, Stillweg 2, D-30655 Hannover, F.R.G.
kuempel @ gga-hannover.de
Lehner, Florian Am Pfaffenbiihel 9, A 5201, Seekirchen, Austria.
[email protected]
Leroy, Yves LMS, Ecole Polytechnique, 91128 Palaiseau Cedex, France,
leroyy @ 1ms .polytechnique.fr
VII
viii List of Contributors
Main, Ian Department of Geology and Geophysics, University of Edinburgh, West
Mains Road, Edinburgh EH9 3JW, U.K.
[email protected]
Menendez, Beatriz Departement des Sciences de la Terre, Universite de Cergy-
Pontoise, 8 Le Campus Bat. I, 95031 Cergy-Pontoise Cedex, France.
[email protected]
Rudnicki, John W. Department of Civil Engineering, Northwestern University,
Evanston, Illinois, 60201, U.S.A.
[email protected]
Wong, Teng-fong State University of New York, Department of Geosciences,
Stony Brook, NY 11794-2100, U.S.A.
tfwong @ notes.cc.sunysb.edu
Zimmerman, Robert Imperial College, Department of Earth Science and
Engineering, London SW7 2BP, U.K.
[email protected]
Foreword
Fluids permeate the pores and cracks of crustal rocks and often have a significant
effect on rock deformation and failure under stress. The fluids of interest may, in
different circumstances, be water, carbon dioxide, or hydrocarbons. Straining of
rock alters the pore volume and thus changes pore pressure in the fluid. Unless
the pressure changes are spatially uniform, the resulting pore pressure gradients
induce a Darcy flow towards a new fluid equilibrium state. However, any Darcy
flow for which the fluid flux divergence alters the pore volume also induces a
strain field in the host medium. Hence, solid deformation and fluid transport are
generally coupled processes. Proper elastic constitutive descriptions show that the
strain components, along with the fluid mass content in the pore space, must be
regarded as functions of the stress components, pore pressure, and temperature.
Criteria for nonelastic, frictional deformation and failure under compression can
often be formulated in terms of the effective stress tensor, defined as the total
stress with each diagonal component augmented by the pore pressure. That is,
shear strength depends on the differences between compressive normal stresses
and the pore pressure.
As examples of coupled effects in fluid-saturated rocks, we may note that well
water levels respond to the transient stressing of the crust by tidal forces, as well as
by earthquakes, and even by passing trains. Buildings on saturated soils of low per
meability slowly settle as the sediments compact and fluid is squeezed out. Land
slides on slopes of poorly consolidated saturated materials can quickly increase
pore pressure and transform into yet more destructive debris flows. For related
reasons, earthquake straining can liquefy shallow sediments. Fluid-rich materials
of the ocean floor are thrust deep into subduction zones, where pore compaction
and mineral dehydration set up a complex regime of fluid pressurization and trans
port which accompanies, and surely interacts with, the ongoing deformation and
faulting there. On long time scales, fluids allow stress-driven creep in the middle
crust by pressure solution. On very short time scales, thermal pressurization of
pore fluid by frictional heating may provide a major mechanism of weakening
during earthquake slip. The sinking of a drill hole into the crust, for extraction of
petroleum resources, requires attention to the mechanics of fluid-saturated rocks at
all stages, to prevent wall collapse, to avoid catastrophic blowouts from overpres-
surized formations, and to hydraulically crack from the wall to enhance oil inflow
IX
X Foreword
from the reservoir. Conversely, in planning for waste repositories, one rather strives
to obstruct or divert groundwater inflow, so as to stably isolate storage zones.
Thus, the topic of this book is of major interest to a wide spectrum of earth
scientists and engineers. That is all the more so because of the effectiveness of
the presentation. The editors and their contributing authors, together constituting
some of the most notable scientists on the mechanics of fluid-saturated rocks, have
assembled a volume of remarkable soundness and consistency of level. Attaining
that consistency was likely aided by the attractions of Paris, making it easy for
several of the authors to accept invitations for collaborative visits at Ecole Normale
Superieure during the period of gestation.
The book has a decided focus, as should be the case for any account of this
broad field, at such relatively short length, yet elevated conceptual level. That focus
is on the fundamental mechanics of elastic, elastic-plastic and creep deformation of
fluid-saturated rocks, and on the closely related topics of their deformation local
ization, faulting and cracking, and fluid transport through them. Correspondingly,
appUcations in geology, hydrology, resource extraction, and civil and environmen
tal engineering receive relatively less attention, but several are discussed, and the
fundamentals here provide an essential underpinning to all.
Poroelastic and elastic-plastic theory is a mainstay of the subject. It inte
grates the Darcy law for pore fluid transport with the mechanics of deformable
solids, in a manner first proposed by Terzaghi in 1923 for describing the one-
dimensional consolidation of saturated clay soils. Later, poroelasticity was given
a three-dimensional generalization in a rigorous continuum mechanics framework
in a series of papers by Biot, starting in 1941 and continuing into the 1970s. One
would be hard pressed to identify a better short introduction to poroelasticity than
what is to be found in the opening chapter here, written by the editors. It is one
that I would advise students to use as a starting point.
The understanding of compactive deformation, damage, and failure in porous
rocks has been an area of major progress in the field over the last few years, includ
ing macroscopic elastic-plastic characterizations and microscopic descriptions of
cracking processes. That is well acknowledged in the subsequent group of four
chapters here, having among the authors some of the major contributors to that
progress. One also finds in that group a thorough and rigorous treatment of creep
by pressure solution, and a broad ranging, accessible survey of the mechanics of
deformation localization and fault formation.
The closing two chapters focus on poroelastic properties and phenomena over
a range of scales, including large field scales. That includes deformation responses
to pumping at wells, subsidence, and fracture compliance, as well as fluid transport
with its strong stress and scale dependence in extensively cracked and fractured
rock masses. The latter exposes the Achilles heel of current formulations of the
continuum mechanics of fluid-saturated rocks (already noted in the opening para
graph of the book): Nature often provides no clear scale separation, particularly in
the description of transport. Thus, poroelastic predictions involving fluid motion,
dependent on an ill-characterized permeability or hydraulic diffusivity, must often
Foreword xi
be interpreted with some indecisiveness, in a manner attentive to the scale and local
setting of the intended application. Let us hope that a new generation of workers,
attracted to the subject in part by the effectiveness of this volume, will develop
some novel inroads there.
James R. Rice
Cambridge, MA USA
December 2003
Preface
Major progress has been achieved in recent years in the field of mechanics of
fluid-saturated rocks, at the border between geology and mechanics. This field is
of direct interest to those who are concerned with upper-crust mechanics and fluid
movements, the most important fluids being oil and water. This book aims to be
an up-to-date, comprehensive presentation of the recent important developments.
The first chapter provides the appropriate theoretical framework to deal with the
various mechanical behaviours of fluid-saturated rocks. The following two chap
ters address a key domain. Compaction of porous rocks is a process that is now
well understood because of recent experimental and theoretical results. Two dis
tinct types of compaction are known to play a major role: mechanical compaction
and chemical compaction. In the first case, the fluid effect is mainly an effective
pressure effect, whereas in the second case, it is a pressure solution effect. In the
field of damage and rock physical properties, fundamental research and specific
experimental investigations on rocks in geologically appropriate conditions allow
us to understand how damage modifies the different properties, and in particular,
elastic properties and permeability. At a certain level of deformation, however,
localization takes place. This is a process of great importance in geology because
it concerns the birth of faults and has major implications for fluid flow. The con
nection between fluid flow and deformation is covered by the last two chapters,
which address complementary issues: fluid transport in deforming rocks and fluid
flow in fractured rocks.
Potential readers are expected to be advanced undergraduate students, grad
uate students, scientists, and professionals in the fields of geology, geophysics,
rock mechanics and physics, petroleum engineering, geological engineering, civil
and environmental engineering, and hydrogeology. Important issues such as sub
sidence, geological fault formation, earthquake faulting, hydraulic fracturing, and
transport of fluids are examined throughout the book, as are natural and direct
applications.
XIII
xiv Preface
This book would not have existed without a series of three Euroconferences
that took place in 1998,1999, and 2000, on rock physics and mechanics in geology.
These Euroconferences played an important role in the maturation process of some
key ideas in that field, involving researchers from universities, the oil industry, and
underground nuclear waste storage.
Yves Gueguen and Maurice Bouteca
Paris, 2003
Chapter 1 •
Fundamentals of
Poromechanics
Yves Gueguen^
Luc Dormieux^
Maurice Bouteca^
1.1 Introduction
The mechanical behavior of the Earth's crust is often modeled as that of a porous,
fluid-saturated medium. Crustal rocks, as are many other solids, are porous and
fluid saturated down to at least 10 km depth. Because they are made of minerals and
open pores, they show an internal structure. Classical continuum mechanics de
scribe such a medium as an idealized continuum model where all defined mechani
cal quantities are averaged over spatial and temporal scales that are large compared
with those of the microscale process, but small compared with those of the investi
gated phenomenon. We follow this type of approach in this chapter's presentation
of the classical macroscopic theories of porous rock deformation. Such a separation
of scales is a necessary condition for developing a macroscopic formulation.
A complementary point of view is that of mixture theory. In that approach,
solids with empty pore spaces can be treated relatively easily because all the com
ponents have the same motion when the solid is deformed. However, if the porous
solid is filled with liquid, the solid and liquid constituents have different motion,
and so the description of the mechanical behavior is more difficult. Interactions are
taking place between the constituents. A convenient way to approximately solve
that problem is to idealize the saturated rock as a mixture of two components that
would fill the total space shaped by the porous solid. This is the model of mixture
theory where each component occupies the total volume of space simultaneously
^Ecole Normale Superieure, ENS, 24 rue Lhomond, 75231 Paris Cedex 05, France.
[email protected]
^CERMMO, ENPC, 6-8 Avenue Blaise Pascal, Champs sur Mame, 77455, Mame la Vallee
Cedex 2, France, [email protected]
^Institut Fran9ais du Petrole, 1 et 4 avenue de Bois-Preau, 92852 Rueil-Malmaison Cedex, France,
cepm @ francenet.fr
1
2 Chapter 1 Fundamentals of Poromechanics
with the others. The assumptions of the theory of mixtures are not completely
vaHd for fluid-saturated rocks, because the solid and fluid phases are not miscible
phases. This theory yields, however, a possible framework for the macroscopic
treatment of liquid-saturated porous solids. An additional assumption is that only
the fluid phase is allowed to leave the total space defined by the porous body. The
pores are assumed to be statistically distributed and the porosity value fixes the
ratio of the pore volume to the total porous body volume. Introducing the above
assumptions means that the microscale should be taken into consideration. For
that reason, the microscopic approach is developed in this chapter along with the
classical macroscopic theory. Although this is unusual, we believe that it gives a
better insight into rock behavior.
This chapter examines the most important mechanical types of behavior that
can be observed for porous crustal rocks: small reversible deformation (poroelas-
ticity), large irreversible deformation (poroplasticity), and rupture. Some comple
mentary results on each of these are given in other chapters when appropriate.
Dynamic effects are clearly out of the scope of this book, which focuses on quasi-
static behavior.
1.21 Poroelasticity
Poroelasticity theory accounts reasonably well for small deformations of a fluid-
saturated porous soHd. It is an extension of elasticity theory to the precise situation
we are interested in: that of a porous rock submitted to a small reversible strain. Re
versibility is a major assumption because it allows us to develop the theory within
the framework of classical thermodynamics. The extension to the theory of elastic
behavior of a solid medium is that the fluid phase is taken into account, and this
implies that two additional parameters are required to describe the thermodynamic
state of the fluid: its pressure and its volume (or mass). Two possible descriptions
are very usefuht he drained description, where the fluid pressure is the appropriate
thermodynamic variable, and the undrained description, where the mass content
is the appropriate thermodynamic variable. The fluid is viscous and compress
ible. The isothermal theory of poroelasticity was first presented by Biot (1941,
1955, 1956, 1957, 1972), and later reformulated by Rice and Cleary (1976) and
Coussy (1991). Nonisothermal effects were later considered by Palciauskas and
Domenico (1982) and McTigue (1986). Nonlinear poroelasticity was introduced
by Biot (1973). Different reviews of poroelastic theory have been published by
Detoumay and Cheng (1993), Zimmerman (2000), Rudnicki (2002). Wang (2000)
has presented a monograph on the theory of linear poroelasticity with applications
to geomechanics and hydrogeology.
1.2.1 Linear Isothermal Poroelasticity
We follow here the classical sign convention of elasticity: compressive stresses
are considered to be negative for the solid rock but fluid pressure is positive. We
1.2 Poroelasticity 3
consider the linear quasi-static isothermal theory, and assume that the rock at a
macroscopic scale can be viewed as isotropic and homogeneous. Let P be the
mean pressure, P = —l/3akk, where a/y is a component of the stress tensor,
defined as the measure of total force per unit area of an element of porous rock.
Let p be the fluid pore pressure, which is the equilibrium fluid pressure inside
the connected and saturated pores: p can be understood as the pressure on an
imagined fluid reservoir that would equilibrate an element of rock to which it is
connected from either giving off or receiving fluid from the reservoir. We assume
that all pores are connected. In addition, let m be the fluid mass content per unit
volume in the reference state. Fluid mass density p is defined locally as the mass
density of fluid in the equilibrating reservoir. The apparent fluid volume fraction
is V = m/p. Because we are considering a saturated rock, VQ = OQ, where OQ
is the initial porosity of a given porous rock volume VQ. In the deformed state,
however, the volume VQ is transformed into V so that v — VQ is not identical to
O — <I>o because O = Vp/V, where Vp is the pore volume in the rock volume
V: V = Vp/Vo = ^(V/Vo). In general the fluid is compressible and its density
depends on p: p = p{p). Only isothermal deformations are considered in this
section. Linear poroelasticity is not restricted to fluids of low compressibility. The
drained description is convenient to deal with highly compressible fluids. Two types
of deformation will be considered: drained deformation refers to deformations
at constant fluid pore pressure p, undrained deformation refers to deformations
at constant fluid mass content m. Strain refers to the relative displacement of
solid points in the solid phase. The components of the strain tensor are 6/y =
\/2{djUi + 9/wy), where ui is the displacement vector component. Stress, strain,
and fluid pressure will be defined as small perturbations with respect to a given
equilibrium state, so that body forces are ignored. The rock has an apparent elastic
bulk modulus for drained conditions K
1
(1.1)
~K v\dP),
Effective Stress Concept
The concept of effective stress is of great importance to poroelasticity. As an
introductory step, let us consider first the case of effective pressure. The effective
pressure Pe is defined by
P^ = P-bp, (1.2)
where b is the Biot coefficient, which will be computed as follows. The elastic
deformation of a rock sample submitted to both an isotropic pressure P and a pore
pressure p can be obtained by superimposing two states of equilibrium (Figure 1.1).
The first one corresponds to a pressure P — p applied to the external surface of the
rock and a zero fluid pressure within the pores. The second one is that of the same
rock sample submitted to a pressure p on both the external surface of the rock
