www.pdfgrip.com

FLUID-INDUCED SEISMICITY

The characterization of fluid-transport properties of rocks is one of the most important, yet difficult, challenges of reservoir geophysics, but is essential for optimal
development of hydrocarbon and geothermal reservoirs. Production of shale oil,
shale gas, heavy oil and geothermal energy, as well as carbon-dioxide sequestration, are relatively recent developments where borehole fluid injection is often
employed to enhance fluid mobility. Unlike active seismic methods, which present
fundamental difficulties for estimating the permeability of rocks, microseismicity induced by fluid injection in boreholes provides the potential to characterize
physical processes related to fluid mobility and hydraulic-fracture growth in rocks.
This book provides a quantitative introduction to the underlying physics,
application, interpretation, and hazard aspects of fluid-induced seismicity with a
particular focus on its spatio-temporal dynamics. It presents many real-data examples of microseismic monitoring of hydraulic fracturing at hydrocarbon fields and
of stimulations of enhanced geothermal systems. The author also covers introductory aspects of linear elasticity and poroelasticity theory, as well as elements of
seismic rock physics and of the mechanics of earthquakes, enabling readers to
develop a comprehensive understanding of the field. Fluid-Induced Seismicity is
a valuable reference for researchers and graduate students working in the fields
of geophysics, geology, geomechanics and petrophysics, and a practical guide for
petroleum geoscientists and engineers working in the energy industry.
SERGE A. SHAPIRO

is Professor of Geophysics at the Freie Universität Berlin,
and since 2004, Director of the PHASE (PHysics and Application of Seismic
Emission) university consortium project. From 2001 to 2008 he was one of the
coordinators of the German Continental Deep Drilling Program. His research interests include seismogenic processes, wave phenomena, exploration seismology and
rock physics. He received the SEG Virgil Kauffman Gold Medal in 2013 for his
work on fluid-induced seismicity and rock physics, and in 2004 was elected a
Fellow of the Institute of Physics.

www.pdfgrip.com


www.pdfgrip.com

FLUID-INDUCED SEISMICITY
S e rg e A . S h a p i r o
Freie Universität Berlin


www.pdfgrip.com

University Printing House, Cambridge CB2 8BS, United Kingdom
Cambridge University Press is part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9780521884570
c Serge A. Shapiro 2015
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2015
Printing in the United Kingdom by TJ International Ltd., Padstow, Cornwall
A catalog record for this publication is available from the British Library
Library of Congress Cataloging in Publication Data
Shapiro, S. A.
Fluid-induced seismicity / Serge A. Shapiro, Earth Science Department, Freie Universität Berlin.
pages cm

Includes bibliographical references and index.
ISBN 978-0-521-88457-0
1. Rock mechanics. 2. Hydraulic fracturing. 3. Induced seismicity. 4. Reservoir-triggered
seismicity. 5. Oil field flooding. I. Title.
QE431.6.M4S53 2015
551.22–dc23
2014043958
ISBN 978-0-521-88457-0 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of URLs for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.


www.pdfgrip.com

Contents

Preface
Acknowledgments
1

2

page ix
xii

Elasticity, seismic events and microseismic monitoring
1.1 Linear elasticity and seismic waves
1.1.1 Strain

1.1.2 Stress
1.1.3 Stress–strain relations
1.1.4 Elastic moduli
1.1.5 Dynamic equations and elastic waves
1.1.6 Point sources of elastic waves
1.1.7 Static equilibrium
1.2 Geomechanics of seismic events
1.2.1 Faults and principal stresses
1.2.2 Friction coefficient
1.2.3 Growth of finite cracks: a sufficient condition
1.2.4 Necessary conditions of crack growth and some results
of fracture mechanics
1.2.5 Earthquake motions on faults
1.3 Elastic wavefields radiated by earthquakes
1.4 Introduction to microseismic monitoring
1.4.1 Detection of seismic events
1.4.2 Seismic multiplets
1.4.3 Location of seismic events
1.4.4 Microseismic reflection imaging

1
1
2
2
4
6
10
12
13
14

15
20
22

Fundamentals of poroelasticity
2.1 Linear stress–strain relations in poroelastic materials
2.2 Linear stress–strain relations in fluid-saturated materials

48
49
55

23
26
30
37
37
38
42
46

v


www.pdfgrip.com

vi

Contents


2.2.1 Independent confining stress and pore pressure
2.2.2 Linear stress–strain relations in undrained media
2.3 Fluid flow and dynamic poroelasticity
2.4 Poroelastic wavefields
2.4.1 Dispersion relations for poroelastic wavefields
2.4.2 Particle motions in poroelastic wavefields
2.4.3 Fluid flow and attenuation of seismic waves
2.4.4 Slow wavefields in the low-frequency range
2.4.5 Elastic P- and S-waves in the low-frequency range
2.5 The quasi-static approximation of poroelasticity
2.6 Sources of fluid mass and forces in poroelastic media
2.6.1 Fluid injection at a point of a poroelastic continuum
2.7 Boundary loading of poroelastic media
2.7.1 Loading of a poroelastic half-space by a fluid reservoir
2.7.2 Fluid injection into a spherical cavity of a poroelastic
continuum
2.8 Stress and pressure coupling for radially symmetric fluid
sources
2.9 Several non-linear effects of poroelastic deformations
2.9.1 Deformation of the pore and fracture space
2.9.2 Stiff and compliant porosities
2.9.3 Stress dependence of elastic properties
2.9.4 Non-linear nature of the Biot–Willis coefficient α
2.9.5 Magnitude of the poroelastic-stress coupling
2.9.6 Effective-stress coefficients
2.9.7 Stress dependence of permeability
2.10 Appendix. Reciprocity-based relationship between
compliances of porous media
3


Seismicity and linear diffusion of pore pressure
3.1 Case study: KTB
3.2 Linear relaxation of pore pressure as a triggering mechanism
3.3 Triggering front of seismicity
3.4 Seismicity fronts and poroelastic coupling
3.5 Seismicity and hydraulic anisotropy
3.6 Seismicity in hydraulically heterogeneous media
3.6.1 Eikonal-equation approach
3.6.2 Validity domain of the eikonal-equation approach
3.6.3 Effective-medium approach
3.7 Back front of seismicity

55
59
62
65
67
71
72
73
75
76
80
83
86
87
89
95
100
101

103
106
110
111
113
114
116
118
119
124
127
131
134
137
139
141
143
147


www.pdfgrip.com

Contents

3.8

3.9

Strength of pre-existing fractures
3.8.1 Statistics of rock criticality

3.8.2 Case study: Soultz-sous-Forêts
3.8.3 Case study: Fenton Hill
Spatial density of seismicity

vii

150
150
154
157
158

4

Seismicity induced by non-linear fluid–rock interaction
4.1 Seismicity induced by hydraulic fracturing
4.1.1 Triggering front of hydraulic fracturing
4.1.2 Back front of hydraulic fracturing
4.1.3 Case study: Cotton Valley
4.1.4 Estimating permeability of virgin reservoir rocks
4.1.5 Estimating permeability of hydraulic fractures
4.1.6 Case study: Barnett Shale
4.2 Seismicity induced by non-linear pressure diffusion
4.2.1 Non-linear diffusion and triggering fronts
4.2.2 Triggering fronts and diffusivity estimates
4.3 The model of factorized anisotropy and non-linearity

164
165
165

168
169
176
177
177
180
183
189
196

5

Seismicity rate and magnitudes
5.1 The model of point-like induced events
5.1.1 Event number and event rate during a monotonic
injection
5.1.2 Seismicity rate after termination of an injection
5.1.3 Case studies of magnitude distributions
5.1.4 Seismogenic index
5.1.5 Occurrence probability of events with given magnitudes
5.2 Statistics of large magnitudes
5.2.1 Observations
5.2.2 Statistics of earthquakes with finite rupture surface
5.2.3 Rupture-surface probability and geometry of
stimulated volumes
5.2.4 Distributions of magnitudes and the
Gutenberg–Richter law
5.2.5 Lower and upper bounds for magnitude distributions
5.2.6 Case studies of magnitude distributions and stress
drop

5.2.7 Induced and triggered events
5.2.8 Maximum magnitude and sizes of stimulated volume
5.3 Appendix 1. Probability of discs within a sphere
5.4 Appendix 2. Probability of discs within an ellipsoid

201
202
203
206
208
215
220
224
224
229
232
236
239
243
250
250
253
254


www.pdfgrip.com

viii

Contents


5.5
5.6

Appendix 3. Probability of discs with centers inside a sphere
Appendix 4. Probability notations used in this chapter

References
Index
Color plate section between pages 82 and 83

256
258
259
272


www.pdfgrip.com

Preface

Characterization of fluid-transport properties of rocks is one of the most important,
yet one of most challenging, goals of reservoir geophysics. However, active seismic methods have low sensitivity to rock permeability and mobility of pore fluids.
On the other hand, it would be very attractive to have the possibility of exploring
hydraulic properties of rocks using seismic methods because of their large penetration range and their high resolution. Microseismic monitoring of borehole fluid
injections is exactly the tool that can provide us with such a possibility. Borehole
fluid injections are often applied for stimulation and development of hydrocarbon
and geothermal reservoirs. Production of shale gas and heavy oil as well as CO2
sequestration are relatively recent technological areas that require broad applications of this technology. The fact that fluid injection causes seismicity has been well
established for several decades (see, for example, Pearson, 1981, and Zoback and

Harjes, 1997). Current ongoing research is aimed at quantifying and controlling
this process. Understanding and monitoring of fluid-induced seismicity is necessary for hydraulic characterization of reservoirs and for assessments of reservoir
stimulations.
Fluid-induced seismicity covers a wide range of processes between the two
following limiting cases. In liquid-saturated hard rocks with low to moderate permeability (10−5 –10−2 darcy) and moderate bottom hole injection pressures (as a
rule, less than the minimum absolute value of the principal compressive tectonic
stress) the phenomenon of microseismicity triggering is often caused by the process of linear relaxation of pore-pressure perturbations (Shapiro et al., 2005a,b).
Note that we speak here about the linearity in the sense of corresponding differential equations. In porodynamics this process corresponds to the Frenkel–Biot
slow wave propagation (see Biot, 1962, and a history review by Lopatnikov
and Cheng, 2005, as well as an English translation of Frenkel, 2005). In the
porodynamic low-frequency range (hours or days of fluid-injection duration) this
process reduces to a linear pore-pressure diffusion. Then, the linear pore-pressure
ix


www.pdfgrip.com

x

Preface

diffusion defines features of the rate of spatial growth, geometry and density of
clouds of microearthquake hypocenters (Shapiro et al., 2002, 2003, 2005a,b; Parotidis et al., 2004). In some cases, spontaneously triggered natural seismicity, like
earthquake swarms, also shows similar diffusion-like signatures (Parotidis et al.,
2003, 2004, 2005; Hainzl et al., 2012; Shelly et al., 2013).
Another extreme case is a strong non-linear fluid–solid interaction related to the
hydraulic fracturing of sediments like a tight sandstone or a shale with extremely
low permeability (10−9 –10−5 darcy). In this case a fluid injection leads to a
strong enhancement of the permeability. Propagation of a hydraulic fracture is
accompanied by opening of a new fracture volume, fracturing fluid loss and its

infiltration into reservoir rocks, as well as diffusion of the injection pressure into
the pore space of surrounding formations and inside the hydraulic fracture (Economides and Nolte, 2003). Some of these processes can be seen from features of
spatio-temporal distributions of the induced microseismicity (Shapiro et al., 2006b;
Fischer et al., 2008; Dinske et al., 2010). The initial stage of fracture volume
opening as well as the back front of induced seismicity (propagating after termination of the fluid injection) can be observed. Evaluation of spatio-temporal
dynamics of induced microseismicity can help to estimate physical characteristics
of hydraulic fractures, e.g. penetration rate of the fracture, its permeability as well
as the permeability of the reservoir rock. Therefore, understanding and monitoring of fluid-induced seismicity by hydraulic fracturing can be useful for describing
hydrocarbon and geothermal reservoirs and for estimating the results of hydraulic
fracturing.
Seismicity induced by borehole fluid injections is a central topic of this book. It
describes physical fundamentals of interpretation of fluid-induced seismicity. The
first two chapters of the book provide readers with an introduction to the theoretical background of concepts and approaches useful for understanding fluid-induced
seismicity. An application-interested reader can probably skip these two chapters
and just go directly to Section 1.4 and then Chapters 3–5, using Chapters 1 and 2
mainly as reference material.
In Chapter 1 the book starts with a brief introduction to the theory of elasticity and seismic-wave propagation. This chapter also includes elements of
fracture mechanics and of the geomechanics of faulting. Then there is an introductory description of earthquake sources of the seismic wavefield. Finally, the
chapter contains a brief schematic description of methodical approaches of microseismic monitoring. Many important processing-related methodical aspects of
microseismic monitoring remain outside of the scope of this book.
Chapter 2 provides a detailed introduction to the theory of poroelasticity. The
main physical phenomena responsible for fluid-induced seismicity and discussed in
this book in detail are fluid filtration and pore-pressure relaxation. They are closely


www.pdfgrip.com

Preface

xi


related to slow waves in porous fluid-saturated materials. The dynamics of slow
wavefields is the focus of this chapter. The chapter also includes a discussion of
some non-linear effects related to deformations of the pore space. They are relevant
for characterizing poroelastic coupling and for formulating models of the pressuredependent permeability. Such models will be used for the considerations of nonlinear pressure diffusion in subsequent chapters. The topic of thermo-poroelastic
interaction is not discussed in the book.
In Chapters 3–5 of this book we describe the main quantitative features of different types of fluid-induced microseismicity. Different properties of induced seismicity related to reservoir characterization and hydraulic fracturing are addressed,
along with the magnitude distribution of seismicity induced by borehole fluid injections. Evidently, this is an important question closely related to seismic hazard
of injection sites. Many corresponding aspects of the book are also applicable to
induced tectonic seismicity.
This book attempts to contribute to further elaboration of the seismicity-based
reservoir characterization approach (see also Shapiro, 2008).


www.pdfgrip.com

Acknowledgments

This book contains results of research funded by different institutions in different time periods. This includes the German Federal Ministry for the Environment,
Nature Conservation and Nuclear Safety (BMU), and the German Federal Ministry of Education and Research (BMBF) supported the section of Geophysics
of the Freie Universität Berlin for their projects MAGS and MEPRORISK. This
also includes the Deutsche ForschungsGemeinschaft (DFG) in whose Heisenberg
research program I started to work, in 1997, with interpretation of microseismic
monitoring of fluid injection at the German KTB. At that time I worked at the
Geophysical Institute of the Karlsruhe University and also spent a short period
at the GeoForschungsZentrum Potsdam. I continued this research at the Geological School of Nancy, France, in close cooperation with Jean-Jacques Royer and
Pascal Audigane, where our work was significantly supported by the GOCAD
consortium project led by Professor Jean-Laurent Mallet. In 1999 I moved to the
Freie Universität Berlin. Many of the results reported here were then obtained in
our common work with Elmar Rothert, Jan Rindschwentner, Miltiadis Parotidis,

Robert Patzig, Inna Edelman, Nicolas Delepine and Volker Rath. Corresponding research works were funded to a significant extent by the Wave Inversion
Technology (WIT) university consortium project led by Professor Peter Hubral
of Karlsruhe University.
Starting in 2005, the research reported here was to a large extent funded by
the PHysics and Application of Seismic Emission (PHASE) university consortium project at the Freie Universität Berlin. Susanne Rentsch, Carsten Dinske,
Jörn Kummerow, Stefan Buske, Erik Saenger, Stefan Lüth, Cornelius Langenbruch, Oliver Krüger, Nicolas Hummel, Anton Reshetnikov, Antonia Oelke, Radim
Ciz, Maximilian Scholze, Changpeng Yu, Sibylle Mayr and Karsten Stürmer contributed strongly to the work performed in this project. Corresponding results are
of especial importance for this book. I express my sincere thanks to all these
colleagues and friends, and I would like to thank the sponsors of the PHASE
xii


www.pdfgrip.com

Acknowledgments

xiii

consortium project and of two other consortium projects mentioned above as well
as the DFG, the BMU and the BMBF for their generous support of my work related
to this book.
I am also indebted to colleagues and institutions who helped me to access different microseismic data sets used here. These are Hans-Peter Harjes (Bochum
University),1 André Gérard and Roy Baria (SOCOMINE), Andrew Jupe (EGS
Energy), Michael Fehler and James Rutledge (LANL), Shawn Maxwell (Pinnacle
Technology), Kenneth Mahrer (USBR), Ted Urbancic, Adam Baig and Andreas
Wuesterfeld (ESG), Hideshi Kaieda (Central Research Institute of Electric Power
Industry, Tohoku), Takatoshi Ito (Institute of Fluid Science, Tohoku), Günter
Asch (GFZ-Potsdam), Martin Karrenbach (P-GSI), Ulrich Schanz and Markus
Häring (Geothermal Explorers), Sergey Stanchits and Georg Dresen (Deutsches
GeoForschungsZentrum, GFZ-Potsdam).

I am deeply grateful to colleagues who provided me with their comments on
the book manuscript. These are Boris Gurevich from Curtin University and CSIRO
(Perth) and Robert Zimmerman from Imperial College London (the first two chapters). These are also my colleagues from the Freie Universität Berlin: Carsten
Dinske (Chapter 4), Jörn Kummerow (various sections), Cornelius Langenbruch
(Chapter 3), Oliver Krüger (Chapter 5). Of course, I have sole responsibility for
the complete book content.
I sincerely acknowledge the Society of Exploration Geophysicists (SEG) for
extending permission to use materials from a series of publications coauthored by
me in Geophysics and The Leading Edge.

1 Here, and in the following, the affiliations are given for the time periods during which the access to the data

was made possible.


www.pdfgrip.com


www.pdfgrip.com

1
Elasticity, seismic events and microseismic monitoring

By “seismic events” we understand earthquakes of any size. There exists a broad
scientific literature on earthquakes and on the processing of seismologic data. We
refer readers interested in a detailed description of these subjects to corresponding books (see, for example, Lay and Wallace, 1995, and Shearer, 2009). We
start this book with an introductory review of the theory of linear elasticity and
of the mechanics of seismic events. The aim of this chapter is to describe classical fundamentals of the working frame necessary for our consideration of induced
seismicity. We conclude this chapter with a short introduction to methodical aspects
of the microseismic monitoring.

1.1 Linear elasticity and seismic waves
Deformations of a solid body are motions under which its shape and (or) its size
change. Formally, deformations can be described by a field of a displacement
vector u(r). This vector is a function of a location r of any point of the body in an
initial reference state (e.g., the so-called unstrained configuration; see, for example, Segall, 2010). Initially we accept here the so-called Lagrangian formulation,
i.e. we observe motions of a given particle of the body.
However, the field of displacements describes not only deformations of the body
but also its possible rigid motions without changes of its shape and its size, such as
translations and/or rotations.
In contrast to rigid motions, under deformations, distances (some or any)
between particles of the body change. Therefore, to describe deformations, a
mathematical function of the displacement field is used that excludes rigid motions
of a solid and describes changes of distances between its particles only. This function is the strain tensor , which is a second-rank tensor with nine components i j .
Here the indices i and j can accept any of values 1, 2 and 3 denoting the coordinate
directions of a Cartesian coordinate system in which the vectors u and r have been
defined.
1


www.pdfgrip.com

2

Elasticity, seismic events and microseismic monitoring

1.1.1 Strain
In the case of small deformations (i.e. where absolute values of all spatial derivatives of any components of the vector u(r) are much smaller than 1) the strain
tensor has the form of a 3 × 3 symmetric matrix with the following components:
ij


=

1
2

∂u i
∂u j
+
∂x j
∂ xi

.

(1.1)

This form of the strain tensor describes deformations within a small vicinity of a
given location. This form remains the same also by consideration of small deformations in the Eulerian formulation (see Segall, 2010), where instead of a motion
of a given particle of the body (i.e. the Lagrangian approach) rather a motion at
a given coordinate location (i.e. at a given point of the space) is considered. In
this book we accept the small-deformation approximation and do not distinguish
between the Lagrangian and Eulerian approaches.
Strains i j can be arbitrary (small) numbers. However, because of their definition
(1.1) they cannot be arbitrarily distributed in space. Spatial derivatives of strains
must be constrained by the following compatibility equations (see Segall, 2010):
∂ 2 kl
∂ 2 ik
∂ 2 jl
∂2 i j
+
=

+
.
∂ x k ∂ xl
∂ xi ∂ x j
∂ x j ∂ xl
∂ xk ∂ xi

(1.2)

Deformations of a body results from applications of loads to it. Deformations
that will disappear completely if the loads are released are called elastic. Bodies
that can have elastic deformations are called elastic bodies.

1.1.2 Stress
Elastic bodies resist their elastic deformations by means of elastic forces. Elastic
forces in a solid body are analogous to a pressure in an ideal fluid. They occur due
to mutual interactions of elastically deformed parts of the body. These interactions
in turn take place on surfaces where the parts of the body are contacting each other
(see also Landau and Lifshitz, 1987).
Let us consider an elementary part of a body under deformation (see Figure 1.1).
Other parts of the body act by means of elastic forces onto this elementary part over
its surface S. Let us consider a differentially small element of this surface at its
arbitrary point r. Such a surface element can be approximated by a differentially
small part of a plane of area d S tangential to S at point r with a unit normal n
directed outside this part of the surface. Owing to elastic deformations an elastic
force dF(r, n) (also called a stress force) acts on the plane element with the normal
n. The following limit defines a traction vector:


www.pdfgrip.com


1.1 Linear elasticity and seismic waves
x3

3

n
dS
dF(r,n)

S
r
x2

x1

Figure 1.1 A sketch for defining a traction.

τ (r, n) = lim

d S→0

dF(r, n)
.
dS

(1.3)

Note that the traction has the same physical units as a pressure in a fluid (e.g. Pa
in the SI system). Note also that the traction is a function of a location r and of an

orientation of the normal n.
Let us consider three plane elements parallel to coordinate planes at a given
location. We assume also that their normals point in the positive directions of
coordinate axes, which are perpendicular to the plane elements. Therefore, the corresponding three normals coincide with the unit basis vectors xˆ 1 , xˆ 2 and xˆ 3 of the
Cartesian coordinate system under consideration. Tractions acting on these plane
elements are τ (r, xˆ 1 ), τ (r, xˆ 2 ) and τ (r, xˆ 3 ), respectively. A 3 × 3 matrix composed
of nine coordinate components of these tractions defines the stress tensor, σ . Its
element σi j denotes the ith component of the traction acting on the surface with
the normal xˆ j :
σi j = τi (xˆ j ).

(1.4)

Let us consider a differentially small elastic body under an elastic strain and
assume for all deformation processes enough time to bring parts of this body into
an equilibrium state. From the equilibrium conditions for the rotational moments
(torques) of elastic forces it follows that the stress tensor is symmetric:
σi j = σ ji .

(1.5)


www.pdfgrip.com

4

Elasticity, seismic events and microseismic monitoring

Note that if the body torques are negligible (which is usually the case) this relation
is valid even in the case of the presence of rotational motions. This is because

of the fact that, in the limit of a small elementary volume, the inertial forces are
decreasing faster than the elastic force torques (see Auld, 1990, volume 1, section
2, for more details).
Similarly, a consideration of forces (elastic forces, body forces and inertial
forces) acting on a volume element in the limit of its vanishing volume shows
that elastic forces applied to the surface of such a volume must be in balance (see
Auld, 1990, volume 1, section 2, for more details). It then follows that a traction
τ (r, n) acting on an arbitrarily oriented plane surface element can be computed by
using the stress tensor:
τi (n) = σi j n j .

(1.6)

Note that here and generally in this book (if not specially mentioned) we accept
the agreement on summation on repeated indices, e.g. ai bi = a1 b1 + a2 b2 + a3 b3 .
Definition (1.4) of the stress tensor corresponds to a common continuum
mechanics sign convention that tensile stresses are positive and compressive
stresses are negative (see, for example, a thin elementary volume and tractions
acting on its outer surface with normals pointing outside this volume; Figure 1.2).
1.1.3 Stress–strain relations
The strain-tensor and stress-tensor notations give a general form of an observational fact, known as Hooke’s law, that small elastic deformations are proportional
to elastic forces:
x3
dB
n(B–dB)
τ(B–dB)

n(B)
B


τ(B)

x2

x1

Figure 1.2 A sketch illustrating positiveness of tensile stresses. Indeed,
equation (1.6) requires that the components σ22 in the both points, B and B − d B
must be positive. Note that the point B is shown as a dot on the right-hand side
of the disc. The point denoted as B − d B is not seen. It is on the left-hand side of
the disc; d B denotes the width of the disc.


www.pdfgrip.com

1.1 Linear elasticity and seismic waves
ij

= Si jkl σkl ,

5

(1.7)

where the fourth-rank tensor S, with components Si jkl , is the tensor of elastic compliances. Note that their physical units are inverse to the unit of stress: 1/Pa. Owing
to the symmetry of the strain and stress tensors, the tensor of elastic compliances
has the following symmetries:
Si jkl = S jikl = Si jlk .

(1.8)


Another fourth-rank tensor C, with components Ci jkl , called the tensor of elastic
stiffnesses, yields an alternative formulation of Hooke’s law:
σi j = Ci jkl

kl .

(1.9)

From this equation it is clear that the tensor of elastic stiffnesses also has the
symmetry:
Ci jkl = C jikl = Ci jlk .

(1.10)

Both the tensor of elastic stiffnesses and the tensor of elastic compliances are
physical characteristics of a given elastic body.
Often both forms of Hooke’s law (1.7) and (1.9) are written symbolically as (see
Auld, 1990):
= S : σ, σ = C : .

(1.11)

Here the double-dot (or double scalar) products denote summations over pairs of
repeating indices in (1.7) and (1.9), respectively.
A deformed elastic body possesses an elastic strain energy. At zero strain this
energy is equal to zero. With increasing strain by an increment d kl due to the stress
σkl , the volumetric density of this energy (energy per unit volume) must increase
by the increment d E = σkl d kl (see Landau and Lifshitz, 1987). The tensor of
elastic stiffnesses can then be used to define the density of the elastic strain energy

(by integration of the increment d E) as a positive quadratic function of non-zero
strains:
1
1
1
(1.12)
E = Ci jkl i j kl = σkl kl = Skli j σkl σi j ,
2
2
2
where in the two last expressions the two forms of Hooke’s law (1.7) and (1.9) have
been used. The product i j kl remains unchanged if the index pair i j is replaced by
kl and kl is replaced by i j, respectively. Thus, the tensor of elastic stiffnesses as
well as the the tensor of compliances must also have the following symmetry:
Ci jkl = Ckli j , Si jkl = Skli j .

(1.13)


www.pdfgrip.com

6

Elasticity, seismic events and microseismic monitoring

Symmetries (1.8), (1.10) and (1.13) of the stiffness and compliance tensors
reduce the number of their independent components. From 81 possible components of a tensor (tensors’ indices can be equal to 1, 2 or 3) only 21 components
are mutually independent. These components are also called elastic moduli or
elastic constants (the latter notation neglects such effects as pressure dependence
and temperature dependence of these quantities). The requirement that the elastic

strain energy must be a positive-definite quadratic form of arbitrary strain/stress
components (called also the stability condition) provides additional restrictions on
allowed values of elastic moduli.

1.1.4 Elastic moduli
The tensors Ci jkl and Si jkl are inverse to each other so that (see Cheng, 1997):
1
(1.14)
Ci jkl Sklmn = (δim δ jn + δin δ jm ),
2
where quantity δkl is the so-called Kronecker matrix, with components δkl = 1, for
k = l, and δi j = 0 in other cases.
The tensors of stiffnesses and compliances can be expressed in convenient matrix
forms by using their 21 independent components, respectively. For this, one uses
the so-called contracted notation (or the Voigt notations). Let us introduce capital
indices (e.g. I, J, etc.), which can take values 1, 2, 3, 4, 5 and 6. The following
correspondence between the capital indices and the pairs of the usual indices (i j)
is assigned: 1 → 11, 2 → 22, 3 → 33, 4 → 23, 5 → 13, and 6 → 12. In these
notations Hooke’s law has the following forms (Jaeger et al., 2007; Auld, 1990):
⎤ ⎡
⎤⎡ ⎤
⎡
s11 s12 s13 s14 s15 s16
σ11
11
⎥ ⎢
⎥⎢ ⎥
⎢
⎢ 22 ⎥ ⎢s12 s22 s23 s24 s25 s26 ⎥ ⎢σ22 ⎥
⎥ ⎢

⎥⎢ ⎥
⎢
⎢ 33 ⎥ ⎢s13 s23 s33 s34 s35 s36 ⎥ ⎢σ33 ⎥
(1.15)
⎥=⎢
⎥⎢ ⎥,
⎢
⎢2 23 ⎥ ⎢s14 s24 s34 s44 s45 s46 ⎥ ⎢σ23 ⎥
⎥ ⎢
⎥⎢ ⎥
⎢
⎣2 13 ⎦ ⎣s15 s25 s35 s45 s55 s56 ⎦ ⎣σ13 ⎦
2 12
s16 s26 s36 s46 s56 s66
σ12
⎤⎡
⎤
⎡ ⎤ ⎡
σ11
c11 c12 c13 c14 c15 c16
11
⎥⎢
⎥
⎢σ ⎥ ⎢c
⎢ 22 ⎥ ⎢ 12 c22 c23 c24 c25 c26 ⎥ ⎢ 22 ⎥
⎥⎢
⎥
⎢ ⎥ ⎢
⎢σ33 ⎥ ⎢c13 c23 c33 c34 c35 c36 ⎥ ⎢ 33 ⎥
(1.16)

⎥⎢
⎥.
⎢ ⎥=⎢
⎢σ23 ⎥ ⎢c14 c24 c34 c44 c45 c46 ⎥ ⎢2 23 ⎥
⎥⎢
⎥
⎢ ⎥ ⎢
⎣σ13 ⎦ ⎣c15 c25 c35 c45 c55 c56 ⎦ ⎣2 13 ⎦
σ12
c16 c26 c36 c46 c56 c66
2 12
In these two equations the contracted notation is used in the two symmetric 6 × 6
matrices of components s I K and c I K , where I corresponds to a pair of normal


www.pdfgrip.com

1.1 Linear elasticity and seismic waves

7

indices, e.g. i j, and K corresponds to another their pair, e.g. kl. It is clear that
these matrices are inverse to each other, i.e. their matrix product gives a 6 × 6
unit matrix. Their components are also called elastic compliances and elastic stiffnesses, respectively. The relations between the contracted-notation stiffness matrix
components and corresponding components of the fourth-rank tensor of elastic
stiffnesses is simple: c I K = Ci jkl . This correspondence for the compliances is a
bit more complicated: s I K = Si jkl if I, K = 1, 2, 3, s I K = 2Si jkl if I = 1, 2, 3 and
K = 4, 5, 6, and s I K = 4Si jkl if I, K = 4, 5, 6.
The higher the physical symmetry of the elastic medium, the smaller is the number of non-vanishing independent elastic constants. For mineral crystals, different
symmetries are of relevance (see Auld, 1990, for a comprehensive description). In

the most general case of triclinic crystals the elastic properties are characterized by
21 independent compliances (or, equivalently, 21 independent stiffnesses). This situation corresponds to equations (1.15) and (1.16), respectively. If the medium has a
single symmetry plane (the monoclinic symmetry) then the number of independent
constants will be reduced to 13 (for example, if we assume the x y coordinate plane
as the plane of symmetry, this will result in the invariant coordinate transformation
z → −z and thus, all elastic constants with odd numbers of index 3 must be equal
to zero). This situation corresponds, for example, to a layered medium with a single
system of plane cracks oblique to the lamination plane.
One of most relevant symmetries for rocks is the orthorhombic one. It can be
applied to describe different geological situations, like rocks with three mutually perpendicular systems of cracks or horizontally layered rocks permeated by
a single system of aligned vertical fractures. An orthorhombic medium has three
mutually perpendicular symmetry planes. This means that in such a medium under
corresponding coordinate transformations (reflections across symmetry planes) the
tensors of elastic constants must remain unchanged. In a coordinate system with
axes normal to the symmetry planes it follows that all components Ci jkl and Si jkl
with odd numbers of any index must be equal to zero. This leads to the following
forms of the compliance and stiffness matrices, respectively:
⎤ ⎡
⎤
⎡
0
0
0
0
s11 s12 s13 0
c11 c12 c13 0
⎢s
⎢
0
0⎥

0
0⎥
⎥ ⎢c12 c22 c23 0
⎥
⎢ 12 s22 s23 0
⎥ ⎢
⎥
⎢
s
s
0
0
0
c
c
0
0
0
s
c
⎥
⎥
⎢ 13 23 33
⎢ 13 23 33
⎥; ⎢
⎥ . (1.17)
⎢
0⎥ ⎢0
0⎥
0

0 s44 0
0
0 c44 0
⎢0
⎥ ⎢
⎥
⎢
⎣0
0
0
0 s55 0 ⎦ ⎣ 0
0
0
0 c55 0 ⎦
0
0
0
0
0 s66
0
0
0
0
0 c66
We see that nine independent constants are enough to completely describe the
elastic properties of an orthorhombic medium. The compliances can be obtained
from stiffnesses by the matrix inversion and vice versa. In the case of an arbitrary


www.pdfgrip.com


8

Elasticity, seismic events and microseismic monitoring

coordinate orientation, three additional constants (corresponding to three rotational
angles) are required.
A useful and geologically relevant subset of orthorhombic symmetry is
transverse isotropy. Layered sedimentary rocks can frequently be described by
this symmetry. The plane of lamination is then the symmetry plane. If one of
the coordinate planes coincides with the symmetry plane, then a coordinate axis
normal to the symmetry plane will be an axis of an arbitrary-angle rotational
symmetry. This symmetry results in four additional relations between the elastic
constants, reducing the number of independent ones to five. If the symmetry axis
coincides with the direction of the axis x3 , then in equations (1.17) additional relations will be (Auld, 1990): s22 = s11 , s23 = s13 , s55 = s44 and s66 = 2(s11 − s12 ).
Correspondingly, c22 = c11 , c23 = c13 , c55 = c44 and c66 = (c11 − c12 )/2.
Finally, in the case of an elastic isotropic medium (all coordinate axes are
arbitrary-angle rotational symmetry axes and any plane is a plane of symmetry),
two constants remain independent only: s22 = s33 = s11 , s23 = s13 = s12 ,
s66 = s55 = s44 and s44 = 2(s11 − s12 ). Correspondingly, c22 = c33 = c11 ,
c23 = c13 = c12 , c66 = c55 = c44 and c44 = (c11 − c12 )/2. The independent elastic
stiffnesses are usually denoted as the elastic moduli λ and μ, so that c44 = μ and
c12 = λ. Inverting the matrix ci j we obtain compliances of an isotropic medium:
s11 =

λ+μ
λ
1
, s12 = −
, s44 = .

μ(3λ + 2μ)
2μ(3λ + 2μ)
μ

(1.18)

Let us consider a volumetric strain (dilatation) of an elementary volume V of an
arbitrary anisotropic elastic medium:
≡

dV
.
V

(1.19)

We can choose such an elementary volume to be a cuboid with side lengths l x , l y
and l z . Thus we see that
=

dl x
dl y
dl z
d(l x l y l z )
=
+
+
=
lx l y lz
lx

ly
lz

11

+

22

+

33 .

(1.20)

Let us further assume that this dilatation is a result of a hydrostatic stress,
σkl = − pδkl , applied to the medium, where p is the pressure loading the medium.
A general relation between the dilatation and the pressure can be obtained by taking a double-dot product (the scalar product) of Hooke’s law (1.7) with the δi j (i.e.
multiplying the both sides with δi j and summing up over repeating indices):
= −Siikk p.

(1.21)


www.pdfgrip.com

1.1 Linear elasticity and seismic waves

9


The proportionality coefficient here is a bulk compressibility C mt of the elastic
material:
C mt ≡ Siikk = S1111 + S2222 + S3333 + 2(S1122 + S1133 + S2233 )
= s11 + s22 + s33 + 2(s12 + s13 + s23 ).

(1.22)

It follows from (1.19)–(1.21) that the bulk compressibility of a sample has the
following relation to its bulk density ρ:
C mt = −

dV
d(1/ρ)
1 dρ
=−
=
.
V dp
(1/ρ)dp
ρ dp

(1.23)

In the case of an isotropic elastic material we obtain (see equations (1.22) and
(1.18)) C mt = 3s11 + 6s12 = 1/(λ + 2μ/3). Therefore,
K = λ + 2μ/3

(1.24)

is a bulk modulus describing the stiffness of the material to volumetric

deformations.
The following representation of the stiffness tensor of an isotropic medium is
useful (Aki and Richards, 2002):
Ci jkl = λδi j δkl + μ(δik δ jl + δil δ jk ).

(1.25)

In the same terms, Hooke’s law for isotropic elastic media can be written in the
following form:
σi j = λδi j + 2μ

ij.

(1.26)

From this equation it follows that μ is a shear modulus of the material, describing
its stiffness to shear deformations (under which i = j). It follows also that under
uniaxial stress conditions (for example σ33 = 0 and σ11 = σ22 = 0) the ratio ν of
the transverse strain to the longitudinal strain, − 11 / 33 , is equal to
ν=

λ
.
2(λ + μ)

(1.27)

This quantity is called Poisson’s ratio. For an isotropic elastic solid the stability
condition requires that both bulk and shear moduli must be positive. For Poisson’s
ratio this yields the restriction −1 ≤ ν ≤ 0.5. For realistic rocks this coefficient is

positive. Its upper limit of 0.5 corresponds to fluids. Frequently, its values for stiff
tight isotropic rocks are close to 0.25 (corresponding to λ ≈ μ).
All elastic moduli introduced above will usually be assumed to be isothermal
ones, if static deformations or processes being very slow in respect to the thermal
diffusion are considered. In this book we consider processes that are faster than the
temperature equilibration (e.g. wave propagation and pore-pressure equilibration).
We will assume that these processes are approximately adiabatic. Thus, we assume


www.pdfgrip.com

10

Elasticity, seismic events and microseismic monitoring

that the elastic moduli introduced above are adiabatic. Note that the adiabatic and
isothermal moduli of hard materials (e.g. rocks) differ by a small amount (see also
Landau and Lifshitz, 1987).
In this book we will frequently assume that the elastic properties of the medium
are isotropic. This simplifying assumption is often too rough for problems of
seismic event location and imaging (which are not the main subject of our
consideration). For such problems velocity models should take into account seismic anisotropy at least in the weak anisotropy approximation (Thomsen, 1986;
Tsvankin, 2005; Grechka, 2009). For describing dominant effects responsible for
the triggering of induced microseismicity the assumption of elastic isotropy seems
to be adequate at least as the first approximation. For such effects hydraulic
anisotropy of rocks is much more important. Elastic anisotropy in rocks is usually
below 10% and seldom exceeds 30%, in respect to the velocity contrast between
the slowest and fastest wave propagation directions. In shale the elastic anisotropy
can be even higher. However, usually it is much smaller than a possible anisotropy
of the hydraulic permeability, which can reach several orders of magnitude.


1.1.5 Dynamic equations and elastic waves
By an elastic deformation, a transfer of an elastic solid from one equilibrium state
to another equilibrium state occurs by means of propagation of elastic waves.
Elastic waves in rocks in the frequency range between 10−3 and 104 Hz are usually referred to as seismic waves. Resulting elastic forces acting on an elementary
volume of the elastic medium define its acceleration vector. Owing to Hooke’s
law and the definition of the strain tensor, the second Newtonian law (i.e. the
momentum conservation) takes the form of the following dynamic equation (Lamé
equation):
∂u k
∂ 2ui
∂
Ci jkl
=ρ 2 .
∂x j
∂ xl
∂t

(1.28)

This equation describes the propagation of elastic waves in the most general case
of a heterogeneous anisotropic elastic medium. Note that this is a system of three
equations for three unknown components of the displacement vector. A planewave analysis (see also our later discussion of poroelastic waves) is instructive
for investigating modes of propagation of elastic perturbations.
Let us consider the case of a homogeneous arbitrary anisotropic elastic medium.
Then equation (1.28) simplifies to:
Ci jkl

∂ 2uk
∂ 2ui

=ρ 2 .
∂ x j xl
∂t

(1.29)