GEOPHYSICS, VOL. 75, NO. 3 共MAY-JUNE 2010兲; P. N51–N63, 10 FIGS.
10.1190/1.3420734
Plastic yielding as a frequency and amplitude independent mechanism
of seismic wave attenuation
Victoria M. Yarushina1 and Yuri Y. Podladchikov1
Gurevich and Lopatnikov, 1995; Pride et al., 2004; Carcione and Picotti, 2006; Quintal et al., 2009兲, scattering 共Kuster and Toksöz,
1974; Frankel and Clayton, 1986兲, or matrix anelasticity 共Mindlin
and Deresiewicz, 1953; Knopoff and MacDonald, 1958; Walsh,
1966; Hagin and Zoback, 2004兲 have been proposed 共see e.g.,
Johnston et al., 1979; Bourbié et al., 1987兲.
Although dissipation mechanisms associated with a characteristic
time scale are important for interpreting the frequency dependence
of attenuation 共Aki, 1980; Sams et al., 1997; Sothcott et al., 2000兲,
there is a considerable amount of data indicating weaker-than-expected dependence of attenuation on frequency 共Knopoff, 1964;
Gordon and Davis, 1968; McKavanagh and Stacey, 1974; Tittman,
1977; Clark et al., 1980; Murphy, 1982; Winkler and Nur, 1982;
O’Hara, 1985兲. Several alternative explanations for frequency-independent attenuation were proposed, e.g., hysteresis with discrete
memory in nonlinear elastic materials 共McCall and Guyer, 1994兲,
frictional dissipation 共Mindlin and Deresiewicz, 1953; Knopoff and
MacDonald, 1960; Walsh, 1966兲, grain contact adhesion hysteresis
共Sharma and Tutuncu, 1994; Aleshin and Van Den Abeele, 2007兲,
scattering in the fractal earth 共Van der Baan, 2002兲, and a superposition of intrinsic attenuation mechanisms, each with an individual
resonance 共Debye兲 peak 共Liu et al., 1976兲. The idea of frictional sliding on crack surfaces and grain boundaries was particularly popular
due to the works of Walsh 共1966兲 and Mindlin and Deresiewicz
共1953兲. Walsh 共1966兲 considers a very thin 2D elliptic crack with
contacting surfaces sliding relative to each other during the passage
of the wave and Mindlin and Deresiewicz 共1953兲 assume two identical elastic homogeneous spherical particles in contact when subjected to external loads. Although these and similar frictional models
provide attenuation that is independent of frequency, they also predict some features that contradict early experiments 共Mavko, 1979;
Winkler et al., 1979; Winkler and Nur, 1982兲. First, the specific attenuation factor 1 / Q that is predicted by these models is proportional to strain amplitude although experimental attenuations are independent of the signal amplitude. Second, for typical strain amplitudes of seismic waves and for reasonable microcrack dimensions,
the computed slip across crack faces is less than the interatomic
spacing; therefore, frictional losses at small strains are negligible
ABSTRACT
We have developed a mathematical formulation of two
mechanisms of compressional wave attenuation, which can
occur within the solid rock frame prestressed up to its yield
stress in part of its volume. Energy losses are attributed to two
distinct processes: irreversible plastic yielding and formation
of radial microfractures around microscopic cavities. Smallamplitude waves propagating through the rocks prestressed
at their yield point would cause nonelastic strain to avoid
building local stresses above the yield limit and attenuate
some fraction of their energy per every loading cycle. New
mechanisms of microscale yielding and microfracturing give
rise to frequency-independent attenuation due to rate-independence of plasticity formulation. Quality factor Q predicted by the model is independent of strain amplitude for small
strains and decreases with increasing amplitude for large
strains. We found that attenuation can be high even for small
seismic strains 共10ⳮ9 – 10ⳮ5兲. Thus, Q ⳱ 12. . . 20 is achieved
at effective pressures greater than twice the yield strength of
the solid matrix for the plastic yielding mechanism and
at overpressures exceeding half tensile strength for microfracturing.
INTRODUCTION
The attenuation of elastic waves has received a good deal of attention from the experimental 共e.g., Toksöz et al., 1979; Castagna et al.,
1985; Klimentos and McCann, 1990; Best et al., 1994兲 and the modeling points of view 共Johnston et al., 1979; Carcione et al., 1988;
Chapman et al., 2006兲. Several seismological observations on the attenuation of seismic waves have been recorded 共Aki, 1980; Fehler et
al., 1992; Beresnev et al., 1995; Dasios et al., 2001兲. Based on experimental and in situ data, several alternative explanations relating attenuation to the global or local flow of saturating fluids 共Biot, 1956;
Palmer and Traviolia, 1980; Stoll, 1989; Dvorkin et al., 1994;
Manuscript received by the Editor 1 April 2008; revised manuscript received 29 December 2009; published online 20 May 2010.
1
University of Oslo, Physics of Geological Processes, Oslo, Norway. E-mail: v.m.yarushina@matnat.uio.no; iouri.podladtchikov@matnat.uio.no.
© 2010 Society of Exploration Geophysicists. All rights reserved.
N51
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N52
Yarushina and Podladchikov
in rock samples 共Savage, 1969兲. In addition, it was shown that frictional attenuation results in nonlinear wave propagation. This was
in contradiction with early data indicating that at seismic strains
共⬍10ⳮ6兲 attenuation is dominated by linear processes because there
was no transfer of energy between frequency components of the signal. Although frictional losses are still accepted to be considerable
even at strains on the order of 10ⳮ6 – 10ⳮ7, for unconsolidated marine sediments 共Prasad and Meissner, 1992; Buckingham, 1997兲
friction is considered only as a secondary loss mechanism in rocks.
Recent observations reported important nonlinear effects. In a
great variety of laboratory experiments over large intervals of stress,
strain, and frequency, rocks exhibit a resonance frequency shift, dependence of wave velocity and attenuation on strain amplitude, nonlinear stress-strain relation, and stress-strain hysteresis 共Beresnev
and Nikolaev, 1988; Johnson et al., 1996; Johnson and Rasolofosaon, 1996; Mashinskii, 2006; Nazarov et al., 2007兲. These and
other nonlinear phenomena can be seen even at very low strains
共10ⳮ9兲. Seismological data from the recordings of large earthquakes
agree with laboratory measurements and support the idea of nonlinear hysteretic ground behavior 共see Beresnev and Wen 关1996兴 for a
review兲.
Usually, a common belief is that the intrinsic anelasticity of minerals is small and therefore totally neglected. The observed stressstrain hysteresis indicates that no unique dependence exists between
stress and strain, which casts doubt on the elastic nature of the process. Cusped hysteresis loops reported by Beresnev and Wen 共1996兲
and Kadish et al. 共1996兲 allow us to suspect the presence of permanent time independent 共i.e., plastic兲 deformation to which such
cusped loops are peculiar 共Kachanov, 2004; Figure 1兲. In the experiments on bending prismatic bars of dry sedimentary rocks from
Western Siberia with a stepwise increasing load, Mashinsky 共1994兲
shows that at strains as small as 10ⳮ6 – 10ⳮ3 a permanent deformation is achieved. This deformation was proven to be plastic because
it appeared immediately after unloading and did not change over
a)
b)
c)
d)
Figure 1. Hysteresis loops for different types of rheology: 共a兲 Maxwell viscoelastic solid, 共b兲 Kelvin-Voight solid, 共c兲 elastic–perfectly
plastic solid, and 共d兲 elastic solid with microdamage. Maxwell and
Kelvin-Voight viscoelastic solids exhibit smooth hysteresis loops.
Cusped loops are specific to elastoplastic materials and microdamaged elastic solids. The hysteresis in elastoplastic solids shows permanent deformation upon unloading whereas elastic hysteresis upon
unloading returns strains to zero.
time. Moreover, elastic and plastic strains in the specimens were almost equal when the total strain was within the range of 10ⳮ6 – 10ⳮ5.
The presence of permanent deformation in rocks is not surprising.
The high temperatures and presence of fluids reduce its strength
共Nur and Byerlee, 1971; Byerlee, 1978兲. Furthermore, the heterogeneities, e.g., pores and cracks, are well-known stress concentrators.
So, the real stresses around the cavities, grain boundaries, etc., can
be much higher than in the rock in general and can lead to plastic
yielding or fracturing around structural imperfections. If local
stresses are already at the critical state, the passing seismic wave of
even the smallest amplitude will only add to the existing load and
will lead to further plastic flow or to the development of microfractures that radiate from the inclusions.
In this paper, we propose two new frequency-independent mechanisms of intrinsic energy losses in prestressed porous media. Attenuation is due to dissipative processes, i.e., permanent plastic deformation 共failure兲, at the heterogeneities scale. Unlike viscous deformation, which is also permanent but rate dependent, plastic deformation is rate independent and occurs only when stresses reach a certain
yield criterion. An increment of plastic strain is related to a stress increment only; there is no characteristic time scale in the model. This
deformation is illustrated by a mass-spring system sliding on a rough
surface 共Kachanov, 2004, p. 454–455兲 and in this sense our model
can be considered as a modification of a frictional sliding mechanism. We consider two distinct failure processes 共microscale yielding and microfracturing兲 around cavities in the prestressed rock matrix as a potential source of energy losses. We estimate attenuation in
terms of the quality factor Q. In our model, energy loss per cycle of
harmonic oscillation is independent of the time scale of oscillations
and therefore Q is exactly independent of frequency 共Knopoff,
1964兲. Porous rock can be fully saturated, partially saturated, or dry.
The proposed mechanism is nonlinear; however, its nonlinearity is
weak and the attenuation is amplitude independent at low seismic
strains.
The organization of the body of this paper is as follows: first we
describe the mathematical model for P-wave attenuation around
cavities and state the boundary-value problem for a representative
volume element 共RVE兲 for porous rock. Then we give a rigorous
mathematical analysis of deformation of the RVE during one cycle
of harmonic oscillation 共plastic loading, elastic unloading, and subsequent reloading兲 caused by a passing seismic wave for two distinct
failure modes: microscale yielding and microfracturing. In the “effective behavior of porous media” section, the response of a single
pore is generalized and macroscopic properties of a porous rock are
established. Details of the numerical computations are presented in
the “numerical verification” section. The section on “results and discussion” presents the main new results of the paper, including the behavior of the quality factor with changing parameters and discussion
on the relevance of the new mechanisms. This is followed by a section in which our conclusions are summarized.
MODEL FOR P-WAVE ATTENUATION
AROUND CAVITIES
We base our analysis on the effective media theory approach
共Christensen, 1979; Nemat-Nasser and Hori, 1999兲 and model porous rock as a distribution of cylindrical or spherical pores embedded in a solid matrix 共Figure 2兲. The study of a typical response of a
single cavity gives an estimation of the bulk response of a porous
rock. It was shown previously that a similar spherical model gives a
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Plastic yielding as attenuation mechanism
good fit to experimental data on compaction of porous carbonate
rocks and tuff 共Carroll and Holt, 1972; Baud et al., 2000; Vajdova et
al., 2004兲. Microstructural observations have shown that inelastic
compaction in limestone is associated with pore collapse that seems
to initiate from stress concentrations at the surface of an equant pore,
which induces a ring of localized damage in its periphery 共Zhu et al.,
in press兲.
We consider a single cylindrical or spherical cavity of radius R in a
solid deformable matrix that is subjected to a confining far-field
pressure P⬁ at the remote boundary 共Figure 2兲. The cavity wall is
subjected to pore pressure p. We do not consider the origin of this
pressure. At fully saturated conditions, p is the fluid pressure. When
gas is present in the fluid phase, p gives the pressure within the fluid
and the gas. Because we consider the low-frequency regime here,
these two pressures are equilibrated. Dry conditions are reproduced
when p ⳱ 0. The behavior of the solid in general is controlled by the
effective pressure ⌬P ⳱ P⬁ ⳮ p, which is prescribed to vary in time
as a periodic function to imitate a seismic P-wave. The cavity can
contract or dilate depending on whether the confining pressure is
higher than the fluid pressure or the fluid pressure is higher than the
confining pressure. We measure the stresses as positive in tension
and pressures as positive in compression. Here ⌬P ⬎ 0 for contracting pore space and ⌬P ⬍ 0 for the expanding cavity. In the case of
the cylindrical cavity, plane strain conditions are fulfilled. If no voids
or imperfections are present at smaller scales, which can weaken the
solid matrix, then it can be considered incompressible with a Poisson
ratio of 0.5. By considering the elastoplastic pore collapse, Carroll
and Holt 共1972兲 show that the effect of elastic compressibility on the
overall compaction of the porous rock is quite small. Therefore, the
assumption of matrix incompressibility seems to be justified.
␴ ␪ ⳱ ⳮp0 ⳮ ⌬P0
⳵␴ r
⳵r
Ⳮk
␴rⳮ␴␪
⳱ 0,
r
共1兲
the incompressibility condition
⳵u
u
Ⳮ k ⳱ 0,
r
⳵r
冉冉 冊
1 R0
k r
kⳭ1
冊
Ⳮ1 ,
共4b兲
where R0 is the initial cavity radius.
PLASTIC YIELDING AROUND THE CAVITY
Initial elastoplastic state
Note that the elastic stress state given by equation 4a and 4b is not
hydrostatic. It has a nonzero deviatoric component ␴ r ⳮ ␴ ␪
⳱ ⌬P0共R0 / r兲kⳭ1共1 Ⳮ 1 / k兲. At an elevated initial effective pressure
⌬P0 ⳱ P⬁0 ⳮ p0, it can exceed the yield stress. In the case of cylindrical and spherical pore geometry, the Tresca and von Mises yield criteria reduce to the form
␴ r ⳮ ␴ ␪ ⳱ 2Y ␰ ,
共5兲
with Y being the yield limit of solid matrix for pure shear for the cylindrical pore and half of the yield limit for simple tension for the
spherical one, ␰ ⳱ sgn共⌬P兲. The cavity is contracted when the pressure of the host rock exceeds the pore pressure 共⌬P ⬎ 0兲, which can
be the case for the connected porosity when fluid can be expelled
from the pore without raising the pore pressure when a wave passes.
The pore dilates if ⌬P ⬍ 0, i.e., pore pressure pushes the cavity wall
more than the host rock tends to squeeze it. The latter can happen in
the case of occluded porosity when a passing wave raises pore pressure. Our modeling accounts for pore compaction and pore dilation
modes. The parameter ␰ ⳱ 1 if the pore space is compacting and ␰
⳱ ⳮ1 if the pore space is dilating. In either case, the yield criterion
from equation 5 will first be reached when the initial effective pressure ⌬P0 ⳱ P⬁0 ⳮ p0 reaches the critical value
INITIAL ELASTIC STATE AROUND THE CAVITY
If before the initial arrival of the first wave the undisturbed effective pressure ⌬P0 ⳱ P⬁0 ⳮ p0 is small, then the deformation of the
solid matrix around the pore will be purely elastic and governed by
the equilibrium equation
N53
⌬Pcr ⳱
2k
Y␰ ,
kⳭ1
共6a兲
2k
Y.
kⳭ1
共6b兲
or equivalently,
兩⌬Pcr兩 ⳱
We assume that the absolute value of the initial undisturbed effective
pressure ⌬P0 exceeds 兩⌬Pcr兩 and the initial state of the rock is elastoplastic. In this case, a plastic region spreads into the shell surround-
共2兲
and Hooke’s law 共Muskhelishvili, 1953兲
u 2Ⳮk
共␴ ␪ ⳮ ␴ r兲,
⳱
r 12k␮
共3兲
where ␴ r and ␴ ␪ are the radial and hoop stresses, u is the radial displacement, and ␮ is the shear modulus of the solid matrix. Parameter
k indicates the geometry of the pore, with k ⳱ 1 for the cylindrical
pore and k ⳱ 2 for the spherical pore. The solution to equations 1–3
gives an initial elastic distribution of stresses around the cavity,
␴ r ⳱ ⳮp0 Ⳮ ⌬P0
冉冉 冊
R0
r
kⳭ1
冊
ⳮ1 ,
共4a兲
Figure 2. The cylindrical 共spherical兲 model of the representative volume element in the porous material. The cavity wall of radius R is
subjected to a uniform fluid pressure p and a pressure P⬁ at a great
distance from the cavity. At a sufficiently high pressure difference,
plastic regions can increase in volume.
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N54
Yarushina and Podladchikov
ing the pore 共Figure 2兲. For symmetry reasons, the plastic boundary
is a cylindrical 共spherical兲 surface. Its initial radius is denoted by c0.
Due to incompressibility of the solid matrix expressed in equation
2, the radial displacement in the shell is
u ⳱ A0 /rk,
␴ r ⳱ ⳮp0 ⳮ 2kY ␰ ln共r/R0兲
␴ ␪ ⳱ ⳮp0 ⳮ 2Y ␰ 共1 Ⳮ k ln共r/R0兲兲
冎
共R0 ⱕ r ⱕ c0兲.
共8兲
Stresses in the elastic region can be found by solving equations 1–3
together with the remote boundary condition
再
␴ r ⳱ ⳮP⬁0 Ⳮ B0 /rkⳭ1
␴ ␪ ⳱ ⳮP⬁0 ⳮ B0 /共krkⳭ1兲
冎
共r ⱖ c0兲.
冉
冊
RkⳭ1
k Ⳮ 1 ⌬P0
Y␰
0
exp
ⳮ1 ,
共k Ⳮ 1兲 ␮
2k Y ␰
冉
冊
RkⳭ1
k Ⳮ 1 ⌬P0
B0 ⳱ 0 2kY ␰ exp
ⳮ1 ,
kⳭ1
2k Y ␰
where an upper dot stands for the material time derivative 共ȧ
⳱ ⳵a / ⳵t Ⳮ v · ⳵a / ⳵r兲. The initial distribution of stresses in the elastic
region is given by equation 9 at the remote boundary ␴ r ⳱ ⳮP⬁.
Solving equilibrium equation 1 and Hooke’s law 共equation 15兲 for
␴ r and ␴ ␪ with a velocity taken in the form of equation 13 yields
冦
␴ r ⳱ ⳮP⬁ Ⳮ
␴ ␪ ⳱ ⳮP⬁ ⳮ
RkⳭ1
2k
0
rkⳭ1 k Ⳮ 1
RkⳭ1
0
2
rkⳭ1 k Ⳮ 1
冉
冉
Y ␰ exp
Y ␰ exp
k Ⳮ 1 ⌬P0
2k
Y␰
k Ⳮ 1 ⌬P0
2k
Y␰
冊
冊
ⳮ1 ⳮ
ⳮ1 Ⳮ
B
rkⳭ1
B
krkⳭ1
冧
共10a兲
共10b兲
冊
1
⌬P0
c0 ⳱ R0 exp
ⳮ
.
2kY ␰
kⳭ1
冦
␴ r ⳱ ⳮP⬁ Ⳮ
2 ckⳭ1
␴ ␪ ⳱ ⳮP ⳮ
Y␰
k Ⳮ 1 rkⳭ1
⬁
冉
共11兲
v⳱ⳮ
The P-wave disturbs the medium and causes the redistribution of
stresses. Its first phase can add to the load or cause unloading from
the current stress state. Consider the wave that first increases the absolute value of the effective pressure by changing the pore and outside pressures 共Figure 3兲. It provokes the further plastic flow regardless of the wave amplitude. The plastic region grows to the radius c
⬎ c0, dissipating the energy of the wave. We follow the incremental
solution procedure 共Hill, 1950兲 to evaluate changes in the rock
caused by the wave. The radial velocity v in the volume satisfies the
incompressibility equation
⳵v
2k ckⳭ1
Y␰
k Ⳮ 1 rkⳭ1
c ⳱ R exp
Loading
v
Ⳮk ⳱0
r
⳵r
共12兲
v ⳱ A/rk .
共13兲
共r ⱖ c兲.
共16a兲
Unknown parameters A and B and the current elastoplastic radius c
are found from the continuity of stresses at the elastoplastic boundary, leading to
and the initial radius of the elastoplastic boundary
冉
共15兲
共9兲
From the continuity of stresses at the elastoplastic interface, we find
parameters
A0 ⳱ ⳮ
v 2Ⳮk
⳱
共␴˙ ␪ ⳮ ␴˙ r兲,
r 12k␮
共7兲
where A0 is a parameter to be determined. Stresses in the plastic region are defined by the equilibrium equation 1, the yield criterion 5,
and the boundary condition at the radius of the hole
再
where we accounted for the fact that the pore pressure at the current
cavity radius R equals p. The increments of stresses in the elastic region satisfy Hooke’s law
冧
共r ⱖ c兲,
冊
1
⌬P
ⳮ
,
2kY ␰
kⳭ1
冉
共16b兲
共17兲
冊
1 d⌬P
ckⳭ1 Y ␰ dR
Ⳮ
.
k
r ␮ R dt
2k dt
共18兲
Although stresses given by equation 16b are already in the final
form, the velocity can still be resolved further. Because v ⳱ dR / dt at
the cavity wall, the equation for the evolution of the hole radius can
be written as
and is of the form
Parameter A is to be determined. From the equilibrium equation 1
and the yield criterion 5, we find stresses in the plastic region
再
␴ r ⳱ ⳮp ⳮ 2kY ␰ ln共r/R兲
␴ ␪ ⳱ ⳮp ⳮ 2Y ␰ 共1 Ⳮ k ln共r/R兲兲
冎
共R ⱕ r ⱕ c兲,
共14兲
Figure 3. A passing pressure wave causes an increase in the absolute
value of the initial pressure up to ⌬Pu and starts to decline up to the
value ⌬Pr, which can be above the reverse plastic flow point.
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Plastic yielding as attenuation mechanism
冉
1
dR
⌬P
R d⌬P
exp
ⳮ
⳱ⳮ
dt
2kY ␰ dt
2kY ␰
kⳭ1
冉 冉
exp
冊 冊
冊冒
is greatest on the internal surface as it follows from equation 21a and
21b, reverse plastic flow will initiate from the hole rim when the effective pressure reaches the critical value
1
⌬P
␮
ⳮ
Ⳮ
.
2kY ␰
kⳭ1
Y␰
共19兲
Substitution of equation 19 into equation 18 yields the final result for
velocity
v⳱ⳮ
1 ckⳭ1 d⌬P
2kY ␰ rk dt
冒冉
冊
ckⳭ1
␮
.
kⳭ1 Ⳮ
R
Y␰
共20兲
Unloading
As the wave approaches its crest, the effective pressure reaches its
extreme value ⌬Pu 共Figure 3兲 and starts to decrease, giving rise to
elastic unloading. From this moment, stresses and velocity in the entire shell are governed by equilibrium equation 1, incompressibility
condition equation 12, and elastic constitutive equation 15. Confining and fluid pressures are prescribed at the external and internal
boundary as before 共Figure 2兲. The solution to these equations gives
the following increments of stresses
d␴ r ⳱ ⳮdP⬁ Ⳮ d⌬P ·
d␴ ␪ ⳱ ⳮdP⬁ ⳮ d⌬P ·
and velocity
v⳱
冉 冊冉
kⳭ1 R
2k␮ r
k
共⌬P ⳮ ⌬Pu兲
冉冊
冉冊
R
r
kⳭ1
1 R
k r
共21a兲
,
kⳭ1
共21b兲
,
冊
R d⌬P
dR
ⳮ
,
dt
k Ⳮ 1 dt
共21c兲
where ⌬Pu is the effective pressure at the onset of unloading. The residual stresses in the shell can be obtained by subtracting equation
21a and 21b from equations 14 and 16b. Because v ⳱ dR / dt at the
cavity wall, the equation for the evolution of the hole radius during
unloading can be written as
d⌬P
dR
R
.
⳱ⳮ
dt
共k Ⳮ 1兲共⌬P ⳮ ⌬Pu兲 Ⳮ 2k␮ dt
共22兲
Combining equations 21c and 22, one can write
v⳱ⳮ
N55
1
RkⳭ1
d⌬P
. 共23兲
k
共k Ⳮ 1兲r ⌬P ⳮ ⌬Pu Ⳮ 2k␮ /共k Ⳮ 1兲 dt
Reloading
Elastic unloading will continue until the stresses reach the yield
criterion again and the reverse plastic flow starts or until the wave
reaches its trough and turns to its final phase, leading to an increase
in the absolute value of the effective pressure 共Figure 3兲. In the latter
case, stresses might not reach the yield criterion again and unloading
will be purely elastic. The yield criterion for the reverse plastic flow
is ␴ r ⳮ ␴ ␪ ⳱ ⳮ2Y ␰ . Because the numerical magnitude of ␴ r ⳮ ␴ ␪
⌬Prf ⳱ ⌬Pu ⳮ 2Y ␰
2k
.
kⳭ1
共24兲
Whether stresses reach the critical value during unloading or not
depends on the material parameters and the wave amplitude defined
through ⌬Pu. For waves with very small amplitude, the critical pressure for reverse flow might not be reached. The scenario when the
plastic flow will develop at both wave peaks provides more energy
losses than that of the plastic flow occurring only during the first
phase of the wave. We consider the worst possible situation and assume that the stresses during unloading did not reach the critical value given by equation 24 and, after passing a trough, began reloading
elastically again. Stresses, velocity, and cavity radius during elastic
reloading are determined by equations 21a–21c, 22, and 23 until the
effective pressure reaches its initial value ⌬P0 and the wave is gone.
MICROFRACTURING AROUND THE CAVITY
Rocks forming the earth have composite or heterogeneous microstructures with a variety of pre-existing or stress-induced defects in
the form of voids, microcracks, and weak interfaces. These imperfections cause localized stress amplification, which can be high
enough to lead to microfracturing around larger defects if slightly
disturbed. It has been observed that the microfracturing process can
be correlated to the progressive failure of a circular opening in brittle
rock 共e.g., Suknev et al., 2003; Guéguen and Boutéca, 2004兲. Microfractures due to overpressures extending from intraskeletal pores
are observed in reservoirs 共Márquez and Mountjoy, 1996兲. Seismic
waves, when traveling through the critically stressed formations,
could cause brittle fracturing around spherical or cylindrical pores in
rocks. The earth is a dynamic system. After a seismic event, a number of mechanisms can anneal the induced microscopic fractures
around defects and cavities and reduce their number, particularly in
high-temperature environments. In addition, the stress state in the
earth’s interior is always slightly changing due to a number of processes, e.g., fluid migration, metamorphic reactions, seismicity, tectonic processes, etc. There are infinitely many virtual stress states in
rock and seismic waves that never have repeating paths. One seismic
wave cannot open all cracks; therefore, every new seismic event
could possibly generate a new set of fractures.
Even in well-controlled experiments, cracks are still forming after
a number of loading cycles. It is known from damage mechanics
共Chaboche, 1988兲 that crack propagation is either stable or unstable.
Results from damage-control tests involving cyclic loads exceeding
the crack-damage threshold show that subsequent to the first damage
increment, very little new cracking occurred when under stable conditions; crack growth can be stopped by controlling the applied load.
It appears that with each subsequent damage increment, new cracks
and existing cracks initiate and propagate 共Eberhardt et al., 1999兲.
This would seem to imply that with each damage increment, a crack
population of new and existing cracks develops and grows. In addition, the population of cracks is being changed by high stress with
each opening or extension so that new microfractures can be generated by passing elastic waves, producing acoustic emission events
reported in experiments on cyclic loading 共Holcomb, 1981兲. Some
researchers report that the width of the hysteresis loops changes very
little for several cycles during cyclic loading experiments 共Son-
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N56
Yarushina and Podladchikov
dergeld and Estey, 1981兲. Soils and rocks show that all subsequent
unload-reload cycles result in a gradual accumulation of permanent
strain and excess pore pressure 共for undrained cases; Wood, 1990;
Yu et al., 2007兲. The memory of a rock of all the previous deformation results in a softening behavior and a loss of cohesion 共Cox and
Meredith, 1993; Gatelier et al., 2002兲, which would make it easier to
deform rocks irreversibly even by small-amplitude seismic waves.
Microdamage on a grain scale leads to a permanent rate-independent
deformation in rocks; it is a major mechanism of brittle plastic failure.
再
␴ r ⳱ Y ⳮ p0 ⳮ Y共R0 /r兲k
␴ ␪ ⳱ Y ⳮ p0
冎
共R0 ⱕ r ⱕ c0兲.
共27兲
As before, the subscript 0 denotes initial undisturbed values of parameters. In the elastic region, one has to satisfy the stress balance
equation 1, the incompressibility condition equation 2, and Hooke’s
law 共equation 3兲. The solution to these equations is in the form of
equation 9 as before with
B0 ⳱ ⳮYRk0c0
k
.
kⳭ1
共28兲
The initial radius of the elastoplastic interface is
Initial state
When ⌬P ⬍ 0, i.e., fluid pressure locally exceeds the confining
pressure, the pore space undergoes isotropic expansion and tensile
microcracks might develop around major pores 共Figure 4兲. According to Griffith’s failure criterion, mode I cracks will generate in the
vicinity of the cylindrical or spherical pore when tensile hoop stress
␴ ␪ reaches the critical value. For fluid-saturated rocks, the effective
stress concept suggests that the criterion for failure takes the form
共Paterson and Wong, 2005, p. 163; Jaeger et al., 2007, p. 99–100兲
␴␪ Ⳮp⳱Y.
共25兲
Through substitution of the elastic stresses defined by equation 4a
and 4b into failure criterion equation 25, we find that tensile cracks
will be generated if the effective pressure reaches the threshold value
⌬Pc ⳱ ⳮ
k
Y.
kⳭ1
共26兲
In the failure region adjacent to the cavity, equilibrium stresses are
fully defined by stress balance, i.e., equation 1, failure criterion
equation 25, and the boundary condition at the pore radius, resulting
in
ck0 ⳱
Rk0
Y
.
k Ⳮ 1 ⌬P0 Ⳮ Y
共29兲
Loading
When a wave comes and disturbs the initial equilibrium, further
failure occurs. Radial cracks propagate further into the rock, causing
the growth of an initial failure region. From stress-balance equation
1 and failure criterion equation 25, together with the boundary condition at the cavity wall 共兩 ␴ r兩r⳱R ⳱ ⳮp兲, one finds that
再
␴ r ⳱ Y ⳮ p ⳮ Y共R/r兲k
␴␪ ⳱Y ⳮp
冎
共R ⱕ r ⱕ c兲.
共30兲
The plastic flow rule can be used to find a pressure-expansion relation for the shell. We assume that the total strain increment in the failure region is the sum of the elastic and irreversible plastic strain increments, i.e., de ⳱ dee Ⳮ de p. For the total strain increments, one
has 共Hill, 1950兲
der ⳱
⳵v
⳵r
dt,
v
de␪ ⳱ dt.
r
共31兲
The associated flow rule gives
derp ⳱ d␭
⳵F
⳵␴ r
,
de␪p ⳱ d␭
⳵F
⳵␴ ␪
,
共32兲
where d␭ is a positive increment of the plastic multiplier and F
⳱ ␴ ␪ Ⳮ p ⳮ Y is a plastic potential. Elastic strain increments are
given by Hooke’s law
dere ⳱
2Ⳮk
共d␴ r ⳮ d␴ ␪ 兲,
12␮
共33a兲
de␪e ⳱
2Ⳮk
共d␴ ␪ ⳮ d␴ r兲.
12k␮
共33b兲
Rearranging equations 31–33, we obtain a differential equation for
velocity
⳵v
Figure 4. The cylindrical 共spherical兲 model of tensile brittle failure
around the pore. At sufficiently high isotropic tensile pressure P⬁, at
a great distance from the cavity, or at a pore pressure that is significantly higher than the confining pressure P⬁, mode I cracks form in
the vicinity of the pore.
⳵r
⳱
2Ⳮk
共␴˙ r ⳮ ␴˙ ␪ 兲,
12␮
共34兲
where the stresses are known from the solution given by equation 30.
Integrating and neglecting the convective part of the time derivatives
of stresses, we find after some rearrangements that
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Plastic yielding as attenuation mechanism
v⳱ⳮ
Y dR
ln r Ⳮ A 共R ⱕ r ⱕ c兲
4␮ dt
共35a兲
v⳱ⳮ
N57
c2 2Y共R ⳮ r兲 Ⳮ 3␮r d⌬P
2␮r YR Ⳮ c共6␮ ⳮ 4Y兲 dt
共R ⱕ r ⱕ c兲,
共41a兲
for the cylindrical cavity and
v⳱
2Y R dR
Ⳮ A 共R ⱕ r ⱕ c兲
3␮ r dt
for the spherical cavity. Parameter A is to be determined.
In the elastic region, we solve the force balance and incompressibility equations 1 and 12 and Hooke’s law, i.e., equation 15, together
with the remote boundary condition. Demanding the continuity of
the stresses and velocities at the interface r ⳱ c between the failure
and elastic domains, we obtain
冦
␴ r ⳱ ⳮP⬁ ⳮ
Rkc kY
rkⳭ1 k Ⳮ 1
R kc Y
␴ ␪ ⳱ ⳮP Ⳮ kⳭ1
r
kⳭ1
⬁
冧
共r ⱖ c兲,
1 ckⳭ1 d⌬P
Y Rkⳮ1c dR
ⳮ
2␮ rk dt
2k␮ rk dt
ck ⳱
Y
Rk
.
k Ⳮ 1 ⌬P Ⳮ Y
共r ⱖ c兲, 共36b兲
共36c兲
共R ⱕ r ⱕ c兲
共37a兲
around the cylindrical cavity and
v⳱
c d⌬P
Y R 4c ⳮ r dR
ⳮ
6␮ c r dt
4␮ dt
共R ⱕ r ⱕ c兲 共37b兲
2c
d⌬P
dR
⳱ⳮ
dt
共ln共R/c兲 ⳮ 2兲Y Ⳮ 4␮ dt
Achieving its crest point at an effective pressure ⌬Pu, the wave
causes the unloading of the media. As for the ductile plastic yielding,
we assume that unloading is purely elastic and is totally described by
equations 21a–21c, 22, and 23.
EFFECTIVE BEHAVIOR OF POROUS MEDIA
The next step is to determine the effective response of porous aggregate with random distribution of voids. The macroscopic stresses
in the porous material are given in terms of the surface data by
Nemat-Nasser and Hori 共1999, p. 27–38兲,
¯ ij ⬅
␴
3
d⌬P
dR
c2
⳱ⳮ
2 共6␮ ⳮ 4Y兲c Ⳮ YR dt
dt
for the cylindrical case and
1
V
V
冕
tix jdS,
共42兲
⳵V
where ti are surface tractions, xi are coordinates on the boundary, and
V is the overall volume of the representative sample bounded by a
surface ⳵V. For the case at hand, the average stresses in the porous
¯ r ⳱ ⳮP⬁,␴
¯ ␪ ⳱ ⳮP⬁. The average mean stress is ␴
¯m
material are ␴
⬁
⳱ ⳮP ; the average effective pressure is ⌬P ⳱ P⬁ ⳮ p.
The porosity of a porous aggregate equals the ratio of the volume
of pores V p to the total volume of the porous material V, namely
␸ ⳱ V p /V.
共43兲
From equation 43, we also have that
d␸ dV p dV
⳱
ⳮ .
␸
Vp
V
共44兲
d␸
dV p
⳱
␸ 共1 ⳮ ␸ 兲
Vp
共45兲
d␸
dV
⳱ .
␸ 共1 ⳮ ␸ 兲 V p
共46兲
共R ⱕ r ⱕ c兲,
共r ⱖ c兲
␴ ijdV ⳱
共39兲
共40a兲
4␮ ⳮ Y ln共c/R兲 d⌬P
c2
2r␮ Y共ln共c/R兲 Ⳮ 2兲 ⳮ 4␮ dt
冕
The total volume V is occupied by the pore space V p and solid volume Vs so that V ⳱ Vs Ⳮ V p. If rock mineral grains composing the
matrix are assumed to be incompressible then dVs ⳱ 0 and
dV ⳱ dV p. Thus, equation 44 takes one of the two forms:
for spherical geometry. Substitution of equations 38 and 39 simplifies the equations for velocities and gives
v⳱
1
V
共38兲
for cylindrical geometry and
4␮ ⳮ Y ln共r/R兲 d⌬P
c
2␮ Y共ln共c/R兲 Ⳮ 2兲 ⳮ 4␮ dt
共41b兲
Unloading and reloading
around the spherical cavity. From equation 37a and 37b, we find that
the velocity of the cavity wall is
v⳱
共r ⱖ c兲
for the spherical case.
共36a兲
Because v must be continuous across the plastic boundary, we find
from equations 35b and 36b that in the plastic region velocity has the
following form:
c d⌬P
Y
dR
共ln共c/r兲 Ⳮ 2兲 ⳮ
4␮
dt
2␮ dt
c3 2YR Ⳮ c共3␮ ⳮ 2Y兲 d⌬P
2r2␮ YR Ⳮ c共6␮ ⳮ 4Y兲 dt
General considerations
v⳱
v⳱
v⳱ⳮ
共35b兲
共40b兲
or
Equations 45 and 46 are equivalent. The particular choice of one
or the other equation depends only on computational convenience.
These equations give a description of porosity evolution in a porous
material in terms of pore-space compressibility. In poroelastic or poroelastoplastic materials, the relative change of pore volume would
depend on the material properties of solid rock grains, porosity, and
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N58
Yarushina and Podladchikov
d␸
applied effective pressure. For the axisymmetric model considered
here, porosity can be expressed in terms of internal and external radii
because the pore and total volumes are defined as
2Ⳮk
Vp ⳱
␲ RkⳭ1,
3
2Ⳮk
kⳭ1
V⳱
␲ Rout
,
3
exp
⳵t
Ⳮ ⵜ 共共1 ⳮ ␸ 兲vs兲 ⳱ 0,
␸˙
.
1ⳮ␸
共49兲
共50兲
共51兲
The logarithmic volumetric strain of the matrix can be obtained by
the integration of the last equation
冉
冊
1 ⳮ ␸0
ēv ⳱ ln
,
1ⳮ␸
冉
␸ ⳱ ␸0
共1 ⳮ ␸ 0兲exp
k Ⳮ 1 ⌬P0
k Ⳮ 1 ⌬P
2k Y ␰
2k
冊
Y␰
冊
冉
共53兲
d⌬P.
ⳮ 1 Ⳮ ␮ /Y ␰
ⳮ 1 Ⳮ ␸ 0 exp
k Ⳮ 1 ⌬P0
2k
Y␰
冊
.
ⳮ 1 Ⳮ ␮ /Y ␰
共54兲
Combining equations 51 and 53 gives the desired equation for volumetric strain rate
where vs is the solid matrix velocity and ␸ is the porosity. Defining
the volumetric strain rate of the matrix ¯␧v as the divergence of the
matrix velocity and introducing the material time derivative of porosity in equation 50, we obtain
¯␧v ⳱ ⵜ vs ⳱
2k ␮ Ⳮ Y ␰ ckⳭ1 /RkⳭ1
共48兲
As before, equations 48 and 49 are two alternative forms of the porosity equation. Equation 48 explicitly accounts only for the changes
in porosity due to contraction-expansion of a cylindrical 共spherical兲
void whereas equation 49 accounts for all possible porosity changes
due to void growth and developing microfractures. To obtain the final form of the pressure-expansion relation, one needs to specify radii as functions of external load.
The effective strain rate of a porous rock can be related to porosity
changes in a porous rock with an incompressible solid. The conservation of solid mass requires
⳵共1 ⳮ ␸ 兲
冉
共47兲
and
Rk dRout
d␸
⳱ 共k Ⳮ 1兲 outkⳭ1 .
␸ 共1 ⳮ ␸ 兲
R
ckⳭ1 /RkⳭ1
kⳭ1
We integrate equation 53 and obtain
where Rout is the external radius of the representative volume. Substitution of equation 47 into 45 and 46 leads accordingly to
d␸
dR
⳱ 共k Ⳮ 1兲
␸ 共1 ⳮ ␸ 兲
R
␸ 共1 ⳮ ␸ 兲
⳱ⳮ
¯␧v ⳱ ⳮ␸
ckⳭ1 /RkⳭ1
kⳭ1
d⌬P
.
kⳭ1 kⳭ1
2k ␮ Ⳮ Y ␰ c /R
dt
共55兲
Loading: microfracturing
When microfracturing occurs during the active plastic loading,
i.e., the first cycle of deformation, changes of the pore volume are influenced by the growth of the cylindrical 共spherical兲 void and by the
initiation of new microfractures. Therefore, we use equation 49 to
obtain the pore dilation relation. Noting that RkoutdRout / dt
⳱ 兩r · v兩r⳱Rout and using equations 40b and 41b, we obtain
4␮ ⳮ Y ln c/R
d␸
c2
⳱ⳮ 2
d⌬P
␸ 共1 ⳮ ␸ 兲
␮R 4␮ ⳮ Y共ln c/R Ⳮ 2兲
共56兲
for cylindrical voids and
3 c3 2Y Ⳮ 共3␮ ⳮ 2Y兲c/R
d␸
⳱ⳮ
d⌬P 共57兲
␸ 共1 ⳮ ␸ 兲
2␮ R3 Y Ⳮ 2共3␮ ⳮ 2Y兲c/R
for spherical voids. The elastoplastic radius c is defined by equation
36c. Substitution of equations 56 and 57 into equation 51 gives the
effective strain rate of a porous rock with cylindrical voids
¯␧v ⳱ ⳮ␸
4␮ ⳮ Y ln c/R d⌬P
c2
2
␮R 4␮ ⳮ Y共ln c/R Ⳮ 2兲 dt
共58兲
and the effective strain rate of a rock with spherical holes
共52兲
where ␸ 0 is the initial porosity corresponding to an undisturbed state.
In the next sections, we are interested in quantitatively predicting the
amount by which the porosity and total volumetric strain in rock are
altered because of a passing seismic P-wave. One cycle of loading
consists of plastic loading and subsequent elastic unloading-reloading modes. Calculations are based on the averaging procedure outlined above.
Loading: plastic yielding
The behavior of a small volume of material containing a single
void with localized plastic yielding is described by the analytical solution given by equations 12–15, 16a, 16b, and 17–20. Substitution
of equation 19 for dR / dt into equation 48 gives the following porosity equation:
¯␧v ⳱ ⳮ␸
3 c3 2Y Ⳮ 共3␮ ⳮ 2Y兲c/R d⌬P
.
2␮ R3 Y Ⳮ 2共3␮ ⳮ 2Y兲c/R dt
共59兲
Unloading and reloading
Substitution of equation 22 into equation 48 gives the porosity
equation for unloading and subsequent elastic reloading in porous
media with microscale yielding and microfracturing
d␸
d⌬P
⳱ⳮ
.
␸ 共1 ⳮ ␸ 兲
⌬P ⳮ ⌬Pu Ⳮ 2k␮ /共k Ⳮ 1兲
共60兲
Integration gives the finite form of porosity-pressure relation
␸⳱
2k␮␸ u
.
2k␮ Ⳮ 共k Ⳮ 1兲共⌬P ⳮ ⌬Pu兲共1 ⳮ ␸ u兲
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共61兲
Plastic yielding as attenuation mechanism
2␲
冕
From equation 51, the volumetric strain rate during unloading will
be
␸
d⌬P
¯␧v ⳱ ⳮ
.
⌬P ⳮ⌬Pu Ⳮ 2k␮ /共k Ⳮ 1兲 dt
N59
⌬E ⳱ ⳮ ⌬Pdev ⳱ ⳮ
共62兲
冕
⌬P␧vdt.
共64兲
0
Strain induced by a seismic wave in all considered cases of loading
and unloading can be obtained by time integration of obtained volumetric strain rates.
One cycle of harmonic loading is imitated by a sinusoidal perturbation of effective pressure of frequency ␻ , ⌬P ⳱ ⌬P0 Ⳮ 共⌬Pu
ⳮ ⌬P0兲sin共2␲ t / ␻ 兲, taking for a period of t ⳱ 0..2␲ 共Figure 3兲. The
maximum stored strain energy is given by
NUMERICAL VERIFICATION
E ⳱ max共ⳮ ⌬P · ev兲.
Analytical solutions for a single void considered above are obtained under several assumptions. First, we assumed that the rockmineral grains composing the matrix are incompressible. Second,
plastic flow occurs only during the first phase of harmonic loading
and the possibility of the reverse plastic flow during unloading is ignored. Finally, we assumed that the effect of the void is not felt on the
outside boundary. To investigate the behavior of the system beyond
these limits, a finite-element numerical code for the von Mises type
of plasticity was developed.
Our numerical approach is based on the incremental first-order
forward Euler solution strategy with correction of the drift from the
yield surface and use of continuum-tangent moduli. The algorithm is
combined with Galerkin’s finite-element method in its weak form
and implemented using the 2D MATLAB-based finite-element
method. We used an adaptive moving grid with a 2D four-node isoparametric element for plane strain. Due to radial symmetry, an analysis needs only to account for a quarter-cell. The mesh thickens as r2
at the vicinity of the hole. The applied load is partitioned in several
small incremental loads that are repeatedly applied to the internal
and external boundaries. After stresses reached the prescribed value
imitating the initial prestress in a porous rock, the cycle of loadingunloading-reloading is performed during which applied effective
pressure 共defined as the difference between outside and inside pressures兲 repeats the path shown on Figure 3. The analytical solution for
a cylindrical cavity described previously was used as a benchmark
for the code. During each cycle of harmonic loading, porosity evolution is computed as a squared ratio of internal and external radii, i.e.,
␸ ⳱ 共R / Rout兲2. Average strain and strain rate are calculated using
equations 51 and 52. Loading cycles were repeated for different values of initial porosity, initial prestress, and wave amplitude. The
comparison of numerical results for different Poisson ratios with analytical incompressible solution shows that compressibility of the
solid matrix has little impact on porosity evolution during quasi-static loading as was already noted by Carroll and Holt 共1972兲. The effect of reverse plastic flow during unloading and finite radius of external boundary on results for attenuation will be discussed in the
next section.
RESULTS AND DISCUSSION
As a measure of attenuation in porous media, we choose the specific attenuation factor 1 / Q defined as
2␲ /Q ⳱ ⌬E/E,
共63兲
where ⌬E is the amount of energy dissipated per stress cycle and E is
strain energy stored in the unit volume when the strain is a maximum
共Knopoff, 1964兲. The dissipated energy can be found by integrating
the deformation work over the whole cycle of harmonic loading:
共65兲
As discussed earlier, one stress cycle consists of active loading, unloading, reloading, and final unloading; therefore, equation 64 is decomposed into four separate integrals over quarter periods:
␲
␲ /2
⌬E ⳱ ⳮ
冕
兩⌬P␧v兩loadingdt ⳮ
0
冕
␲ /2
3␲ /2
ⳮ
冕
␲
兩⌬P␧v兩unloadingdt
2␲
兩⌬P␧v兩reloadingdt ⳮ
冕
兩⌬P␧v兩unloadingdt.
3␲ /2
In each of the four integrals, one of equations 55 and 59 or 62 for volumetric strain rate during different deformation modes was substituted. Volumetric strain ev from equation 65 is found by time integration of equations 55 and 59 or 62. Here Q is further computed as a
function of the following four dimensionless parameters: ratio of
initial effective pressure to the yield stress 共⌬P0 / Y兲, strain amplitude e0, porosity, and ratio of the shear modulus to the yield stress of
the solid frame 共 ␮ / Y兲. In computations, ⌬P0 / Y was varied from one
to six, e0 from 10ⳮ9 to 10ⳮ3, and ␸ was changing in the 10%–50%
range.
The dependence of Q on ␮ / Y and porosity is relatively unimportant within the porosity range investigated. The contouring of Q versus the remaining two parameters for the case of microscopic plastic
yielding around cylindrical and spherical pores is shown in Figures
5a and 6a for ␮ / Y ⳱ 33 and 10% of initial porosity. Figures 5b and
6b show Q as a function of normalized prestress ⌬P0 / Y, initial porosity ␸ , and wave amplitude e0. The abscissa parameter on both
plots is chosen in such a way that 3D data for Q at various ⌬P0 / Y, ␸ ,
and e0 would fit a single curve. At strains below 10ⳮ5, the dependence of quality factor on amplitude and porosity disappears and effective pressure is a dominant factor influencing attenuation. However, at higher strains 1 / Q almost linearly depends on e0 so that prestress and strain amplitude additively contribute to Q. The porosity
effect is very low at small seismic strains but increases at larger
strains. Attenuation increases 共and Q decreases兲 with an increase of
initial prestress given in terms of normalized effective pressure
⌬P0 / Y. Figures 5b and 6b show that the quality factor asymptotically approaches a minimum value of approximately 12. However, this
limiting value can be a consequence of the limited range of initial
prestress 共⌬P0 / Y ranges from one to six兲 and strain amplitude used
in computations presented in Figures 5 and 6 and of the assumption
on dilute void distribution. Lower values of Q can be obtained if the
model of a cylinder 共sphere兲 of finite radius would be considered.
Such a model would allow for the full plastic pore collapse at some
values of initial prestress and wave amplitude and therefore would
provide Q ⳱ 2␲ corresponding to the total energy absorption.
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N60
Yarushina and Podladchikov
Figures 7 and 8 demonstrate energy losses in porous media due to
the opening of mode I microfractures around the major cylindrical or
spherical pores for ␮ / Y ⳱ 33 and 10% of initial porosity. They correspond to a case of “expanding” porosity, when fluid pressure locally exceeds pressure in a solid frame, which can happen in fluid-bearing rocks. The quality factor Q increases with increasing isotropic
pressure P⬁0 and decreases with an increase in pore pressure p0. The
lowest Q that can be achieved with this mechanism is 2␲ . This value
corresponds to situations when rock is in a fully plastic state and all
the energy that was imparted to the rock frame was dissipated. It is
the lowest possible theoretical value for Q defined as a ratio of peak
elastic energy to the energy dissipated per stress cycle 共Barton,
2007兲.
Figure 9 shows the comparison of analytical and numerical results
for the specific attenuation factor obtained for microscopic von
Mises yielding. The slight discrepancy of results is caused by the
a)
b)
–
–
–
a)
b)
–
–1.2
–
–1.3
–
–1.4
–
–
–1.5
–
–1.6
–
–1.7
–
–1.8
–
–1.9
–
–
–
–
–
–2.0
–
–
–
–2.1
–2.2
–
–
–
–
Figure 5. Quality factor as a function of the initial prestress, strain
amplitude, and porosity in the model of attenuation due to microscale yielding around cylindrical pores. 共a兲 Contour plot for Q as a
function of strain amplitude e0 and normalized prestress ⌬P0 at ␸
⳱ 10%. 共b兲 The data collapse of Q as a function of a single expression ⌬P0 / Y Ⳮ 200共1 ⳮ 2␸ 兲e0.
a)
b)
Figure 7. Quality factor as a function of the initial prestress, strain
amplitude, and porosity in the model of attenuation due to microfracturing around cylindrical pores. 共a兲 Contour plot for Q as a function of strain amplitude e0 and normalized prestress ⌬P0 at ␸ ⳱ 10%.
共b兲 The data collapse of Q as a function of a single expression
⌬P0 / Y ⳮ 110共1 ⳮ ␸ 兲2.5e0. The quality factor Q increases with increasing isotropic confining pressure P⬁0 and decreases with an increase in pore pressure p.
a)
b)
–
–
–
–1.2
–
–1.3
–
–1.4
–
–1.5
–
–1.6
–
–
–
–1.7
–1.8
–1.9
–2.0
–
–
–
–
–
–2.1
–
–
–
–
–
–2.2
–
–
–
–
–
–
–
–
Figure 6. Quality factor as a function of the initial prestress, strain
amplitude, and porosity in the model of attenuation due to microscale yielding around spherical pores. 共a兲 Contour plot for Q as a
function of strain amplitude e0 and normalized prestress ⌬P0 at ␸
⳱ 10%. 共b兲 The data collapse of Q as a function of a single expression ⌬P0 / Y Ⳮ 250共1 ⳮ 2␸ 兲e0.
Figure 8. Quality factor as a function of the initial prestress, strain
amplitude, and porosity in the model of attenuation due to microfracturing around spherical pores. 共a兲 Contour plot for Q as a function of strain amplitude e0 and normalized prestress ⌬P0 at ␸ ⳱ 10%.
共b兲 The data collapse of Q as a function of a single expression
⌬P0 / Y ⳮ 170共1 ⳮ ␸ 兲2.5e0. The quality factor Q increases with increasing isotropic confining pressure P⬁0 and decreases with an increase in pore pressure p.
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Plastic yielding as attenuation mechanism
presence of reverse plastic flow during unloading and the effect of a
pore on the finite external radius of the shell at higher porosities. It
can be noted that these factors tend to increase attenuation. The dependence of the attenuation factor on ␮ / Y calculated according to
the model of plastic yielding around cylindrical cavity is presented
in Figure 10. In the case of spherical pore geometry and microfracturing, the same trend is observed.
Figures 5–8 demonstrate that the attenuation factor 1 / Q is strongly strain-amplitude dependent at large strains 共⬎10ⳮ5 for ␮ / Y ⳱ 33兲
and independent of the wave amplitude at smaller strains, as is observed experimentally 共Winkler et al., 1979; Murphy, 1982兲. The
strain at which attenuation loses its amplitude dependence is controlled by the ratio of the yield stress to the shear modulus.
For both mechanisms considered in the paper, attenuation is negligible if the initial prestress 共measured as initial normalized effective
pressure ⌬P0 / Y兲 is small. However, at larger initial effective pressures, larger values of attenuation are predicted while preserving
strain amplitude independence for the small strain amplitudes. In
other words, increasing initial prestress leads to progressing failure
within the rock, thus causing an increase in attenuation. This is consistent with laboratory measurements of attenuation in increasingly
damaged rocks where attenuation caused by increasing fracturing of
the samples was shown to increase 共Q decreases兲 with increasing
uniaxial pressure 共Wulff et al., 1999兲. However, most of the experiments on dry or saturated rocks show that attenuation decreases 共Q
increases兲 with increasing confining pressure 共i.e., Johnston et al.,
1979; Bourbié et al., 1987兲, which is most probably caused by the
closing of thin cracks and minor pores. This is not captured by our
simple model having effective pressure as a single parameter quantifying the entire stress state, which cannot reproduce the opposite effects of confining pressure and differential stress on the yielding of a
rock sample. In this paper, we therefore do not aim to predict the precise experimental dependence of attenuation on pressure, porosity,
effect of partial saturation, or other physical parameters. Our main
Figure 9. Analytical 共solid line兲 and numerical 共dotted line兲 calculations for the quality factor Q. Analytical calculations are performed
for the incompressible solid matrix while numerical calculations on
this graph are for v ⳱ 0.3. In numerical computations, initial normalized prestress varies in 0.4 increments over the range of one to five,
initial porosity varies from 10% to 50% in increments of 10%, and
wave amplitude changes from 10ⳮ9 to 10ⳮ3 in ten logarithmic increments.
N61
goal is to demonstrate that a nonlinear mechanism can cause attenuation of small-amplitude seismic waves independent of the wave amplitude.
To summarize, the essence of our result is that for local effective
pressures, lower than or slightly higher than the yield stress, the attenuation due to plastic yielding around cavities is negligible indeed.
Attenuation curves presented in Figures 5b and 6b show that for
⌬P0 ⬎ 2Y the quality factor Q lies within the range of 12–18, typical
for reservoir rocks 共Klimentos, 1995; Dasgupta and Clark, 1998;
Korneev et al., 2004兲. Observed at hydrocarbon-saturated zones, Q
⳱ 50– 100 共Dasgupta and Clark, 1998; Dasios et al., 2001兲 can be
achieved at ⌬P0 ⳱ 1.5Y. Microfracturing would cause Q ⳱ 12. . . 20
at effective pressures exceeding half tensile strength 共Figures 7 and
8兲.
If we assume that confining pressure is lithostatic 共which would
mean that there are no other heterogeneities in a rock other than
spherical or cylindrical pores that would cause local stress amplification兲 and the fluid pressure is hydrostatic, then the estimation of
the critical depth at which plastic yielding becomes important can be
obtained from the inequality ⌬P0 ⳱ gz共 ␳ s ⳮ ␳ f 兲 ⬎ 2Y. Substitution
of typical values of the yield stress Y ⳱ 20 MPa, density of the fluid
␳ f ⳱ 103 kg/ m3, and density of the rock ␳ s ⳱ 2 · 103 kg/ m3 give the
critical depth of 4 km for Q ⳱ 12. . . 15 and 3 km for Q ⳱ 50. These
depth restrictions seem high and exclude typical reservoir depth.
However, they provide only the upper bound on the depth estimates.
Indeed, if similar assumptions on depth dependence of effective
pressure and the tens of MPa strength level would be used for the
prediction of the porosity with depth, the sediments would preserve
their near-surface porosity up to the same 1 – 4-km depth range because the conditions for the plastic pore collapse would not be
reached and elastic deformation cannot significantly reduce porosity. The effects of pore interaction 共Tvergaard, 1990兲, pores of different shapes 共such as elliptical cracks of different aspect ratios兲, and a
nonhydrostatic stress state 共Baud et al., 2000; Vajdova et al., 2004;
Yarushina and Podladchikov, 2007兲 would reduce the critical depth
necessary for the initiation of inelastic compaction and increase the
volume fraction of the rock frame at plastic yield and therefore further decrease Q.
Figure 10. Dependence of the minimum value of the quality factor Q
on shear modulus ␮ and yield stress Y.
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N62
Yarushina and Podladchikov
In the paper, we considered two processes that might happen in
porous rocks: compaction and dilation of pore space. The first one is
associated with confining pressure exceeding fluid pressure, whereas the second one takes place when fluid pressure exceeds local confining pressure. Pore-scale yielding might happen in compaction and
dilation modes and indeed there is experimental evidence for pore
collapse happening in compacting sedimentary rocks 共Baud et al.,
2000兲. Pore-scale plastic yielding is much more easily triggered in
fluid-saturated rocks with elevated fluid pressures and therefore
might be more pertinent to fluid-filled formations such as reservoirs
in which inelastic deformation and failure are manifested by surface
subsidence, well failure, and induced seismicity 共i.e., Boutéca et al.,
1996; Segall, 1989兲.
Microfracturing around pores in some sense is a process similar to
hydrofracturing and this mechanism would be significant in rocks
with high fluid overpressures close to the lithostatic pressure. There
is field and experimental evidence that this type of fracturing happens in reservoir rocks 共Márquez and Mountjoy, 1996; Zhu et al., in
press; Applod and Nunn, 2002; Zoback, 2007兲.
CONCLUSION
In this paper, the model of seismic wave attenuation due to porescale plastic yielding and microfracturing in prestressed porous materials is introduced and studied. Modeling is concerned with the
components of P-wave loss that occur within the rock framework;
there will be additional losses arising from the interaction between
the solid and the pore fluid, which are not considered here. The relative importance of solid frame inelasticity over viscous flow mechanisms is yet to be determined.
The effects of strain amplitude, effective pressure, and porosity
have been investigated. We showed that the amplitude dependence
of attenuation disappears at strains below 10ⳮ5 共at ␮ / Y ⳱ 33兲,
whereas at larger strains there is a clear increase in attenuation in
agreement with experimental data. The effect of the initial porosity
and the ratio of the shear modulus to the yield stress on attenuation is
rather weak. The effective pressure in the prestressed media is the
main factor controlling the level of attenuation 共Figures 5–9兲. At low
effective pressures, attenuation due to pore-scale inelasticity is indeed negligible. However, if the local effective pressure is amplified,
the effect of plastic yielding on attenuation becomes more pronounced and asymptotically approaches a minimum value of approximately 12 in the model of microscale yielding and 2␲ in the
model of microfracturing.
The theory does not aim to predict the attenuation in dry or saturated rocks under “normal” pressure conditions 共negligible differential stress兲 for small-strain seismic waves 共although it might predict a
component of that attenuation兲, but the mechanisms are very relevant for waves propagating in rocks under stress near or above their
yield point.
ACKNOWLEDGMENTS
We are grateful to Clive McCann, Harro Schmeling, and anonymous reviewers whose constructive comments improved this paper.
This work was supported by the Research Council of Norway; additional financial support was from Spectraseis AG and ETHZ.
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