Download Elastic Anisotropy of a Metamorphic Rock Sample of the Canadian

Elastic Anisotropy of a Metamorphic
Rock Sample of the Canadian Shield in
Northeastern Alberta
Judith Chan & Douglas R. Schmitt
Rock Mechanics and Rock
Engineering
ISSN 0723-2632
Volume 48
Number 4
Rock Mech Rock Eng (2015)
48:1369-1385
DOI 10.1007/s00603-014-0664-z
1 23
Your article is protected by copyright and
all rights are held exclusively by SpringerVerlag Wien. This e-offprint is for personal
use only and shall not be self-archived
in electronic repositories. If you wish to
self-archive your article, please use the
accepted manuscript version for posting on
your own website. You may further deposit
the accepted manuscript version in any
repository, provided it is only made publicly
available 12 months after official publication
or later and provided acknowledgement is
given to the original source of publication
and a link is inserted to the published article
on Springer's website. The link must be
accompanied by the following text: "The final
publication is available at link.springer.com”.
1 23
Author's personal copy
Rock Mech Rock Eng (2015) 48:1369–1385
DOI 10.1007/s00603-014-0664-z
ORIGINAL PAPER
Elastic Anisotropy of a Metamorphic Rock Sample
of the Canadian Shield in Northeastern Alberta
Judith Chan • Douglas R. Schmitt
Received: 28 January 2014 / Accepted: 5 October 2014 / Published online: 8 November 2014
Ó Springer-Verlag Wien 2014
Abstract The presence of fractures and textures cause
metamorphic rock masses to be seismically anisotropic.
Neglect of this anisotropy in the processing of field seismic
data causes problems in the final reflection images both in
terms of their quality and in the true positioning of subsurface features. To quantify the degree of seismic
anisotropy in the subsurface, one method is to estimate the
anisotropic parameters from the elastic stiffnesses of a rock
sample. Using the ultrasonic pulse transmission method,
measurements of the compressional and shear wave phase
velocities as a function of confining pressure are used to
calculate the elastic stiffnesses of a metamorphic granite
core sample from the Precambrian basement in northeastern Alberta. Velocities are measured parallel, normal and
oblique to an identified foliation plane of the sample
assumed to be a transversely isotropic medium. The compressional wave velocities are measured to be in the range
of 5,352–6,019 m/s along the foliation plane and
4,752–5,396 m/s normal to the foliation plane over the
range of confining pressures from 0 to 60 MPa. Besides
providing valuable in situ velocity information for the
velocity models, the results also confirm the anisotropic
behavior of the metamorphic rock with the estimated
compressional and shear wave anisotropy valued at 12 and
8 %, respectively. Such degree of seismic anisotropy
should be taken into consideration at the seismic scale
when working with three-dimensional geophysical models
J. Chan (&) D. R. Schmitt
Department of Physics, Institute for Geophysical Research,
University of Alberta, 4-181 CCIS, Edmonton, AB T6G 2E1,
Canada
e-mail: [email protected]
D. R. Schmitt
e-mail: [email protected]
of the Precambrian basement to minimize any out-of-plane
anomalies in the final seismic sections.
Keywords Seismic anisotropy Elastic moduli Ultrasonic pulse transmission Metamorphic rocks Canadian Shield
1 Introduction
Study of the metamorphic cratonic rocks is uncommon in
Alberta because most of the province is blanketed with the
thick sedimentary successions of the Western Canada
Sedimentary Basin (WCSB). Recently, however, there has
been interest in evaluating the potential of geothermal
energy in Alberta as one means towards reducing greenhouse gas emissions. Part of this effort includes studies
within a serendipitously provided borehole-of-opportunity
(Hunt well) drilled to a depth of nearly 2.4 km in which the
lower 1.9 km is drilled through Canadian Shield rock. An
exhaustive geophysical program that included wireline
logging and vertical and walk-away seismic profiling was
carried out to characterize the rock mass in the borehole
with the geothermal gradient recently reported (Majorowicz et al. 2014). A very limited amount of core material
exists, and here we describe measurements of the elastic
anisotropy on one sample of this material.
This paper presents the results of the velocity measurements, elastic constants, and the anisotropic parameters
of the sample based on the assumption that the sample
material has hexagonal, or transversely isotropic (TI),
symmetry. The goal is to quantify the degree of seismic
anisotropy of the rock sample under in situ pressure conditions similar to the in situ stresses. Seismic velocities
were measured in three orientations with respect to a
123
Author's personal copy
1370
visible foliation plane in the sample. The results of these
measurements are intended to assist first in the interpretation of seismic anisotropy measurements in walk-away
borehole seismic measurements and, second, to provide
measures of the general mechanical properties to assist
interpretations of the stress state at depth. A brief theoretical overview of the in situ stress field and the theory of
elasticity are first provided in the next section, followed by
the experimental setup and the results of the study.
2 Background
Seismic anisotropy is a general property of the rocks in the
Earth’s crust that affects propagation velocities, particle
motion polarizations, and amplitudes of the seismic waves.
Anisotropy also further complicates the modeling of such
waves (Vavryčuk and Boušková 2009). Study of anisotropy
offers the opportunities to estimate the stress field (and
fracture orientation), infer the spatial variation of fractures,
predict fluid saturation, and monitor the subsurface pressure changes and fluid pathways in the potential subsurface
reservoirs. The anisotropic behaviors of rocks to small
fractures can be modeled by effective media, whereas
modeling of larger fractures that have an impact on the
reflection, transmission and diffraction features on seismic
data becomes more complicated.
Many crystalline metamorphic formations are geologically heterogeneous in nature due to their compositional
and structural variations. Seismic anisotropy is one aspect
of this heterogeneity and in the case of metamorphic rocks
this anisotropy can be caused by the metamorphic textures
of anisotropic minerals and by cracks oriented in a preferred direction. The latter case may be related to the
regional tectonic stress system that controls the closure and
opening of microfractures on a crustal scale (Crampin
1990). An understanding of the in situ stress fields, confining pressure and their relation to rock mechanics helps to
predict the geometry of induced fractures and the amount
of pressure needed for fracture propagation (Tester et al.
2006).
With the rapid development of incorporating anisotropic
velocity models in seismic methods, it has become
increasingly important to understand the effect of velocity
anisotropy for both seismic imaging and in characterizing
the rock mass. In general, seismic methods serve as good
preliminary imaging of the basement rocks since they offer
coverage over a broad area of interest when deep boreholes
are not commonly available. However, the effect of
anisotropy in seismic imaging can generate problems in the
conventional data processing routine. This includes the
sideslip and smearing of depth-migrated seismic images
leading to the improper positioning of a target structure
123
J. Chan, D. R. Schmitt
(e.g., Alkhalifah and Larner 1994; Vestrum et al. 1999;
Vestrum and Lawton 2010).
Metamorphic rock is commonly assumed to have isotropic, transversely isotropic, or orthorhombic symmetry
based on the degree of foliation and mineral alignment
(e.g., Kern and Wenk 1990; Takanashi et al. 2001; Cholach
et al. 2005). An isotropic rock has no detectable texture and
its physical properties are identical regardless of the
direction through which it is viewed. Foliated (with no
lineation) and layered rocks are considered as transversely
isotropic with a rotational axis of symmetry perpendicular
to the foliation plane. A foliated rock with clear preferential crystallographic alignments of the minerals that are
manifested as a lineation parallel to the foliation plane is
expected to have three orthogonal planes of symmetry, i.e.,
orthorhombic symmetry. By assuming a type of fabric
symmetry and undertaking laboratory measurements of
seismic wave velocities, the correlating elastic constants
with respect to the fabric symmetry can be estimated.
The behaviors of elastic wave velocities and seismic
anisotropy in metamorphic rocks have long been reported
using laboratory measurements and seismic field methods.
Large-scale anisotropy had been found to contaminate
tomographic images with false structures in the upper
mantle when only isotropic velocity models were used
(e.g., Wang and Zhao 2009; Eken et al. 2012). Errors in the
lateral positioning of the subsurface features also arise if an
isotropic earth was assumed in seismic data processing
(e.g., Issac and Lawton 1999; Godfrey et al. 2002; Vestrum
and Lawton 2010).
Johnson and Wenk (1974) measured the physical properties of 110 metamorphic rocks from the Central Alps
under the assumption of orthorhombic symmetry. The
sample was prepared with respect to the visible mesoscopic
fabric directions and elastic wave velocities were measured
using the ultrasonic pulse transmission technique. Strong
anisotropy was observed in the thermal and elastic properties of the samples. It was found that the pattern of
correlations between the various scalar physical properties
became more complex in the presence of small openings
between grains.
Laboratory velocity measurements of rock samples were
also used to assist in the interpretation of in situ seismic
velocity data. Kern and Schenk (1988) investigated the
relationship of velocity anisotropy with the chemical and
mineralogical characteristics of metamorphic rocks from
the Serre Mountains of southern Italy. Using the structural
and stratigraphic field data, and by determining the pressure and temperature derivatives of the compressional and
shear wave velocities, they generated a model seismic
profile that showed the velocity variation as a result of
mineralogical change with depth. It was found that the
degree of anisotropy was controlled by the elastic
Author's personal copy
Elastic Anisotropy of a Metamorphic Rock Sample
anisotropy of mineral constituents as well as by the modal
concentration, distribution, and preferred orientation of
mineral constituents at high confining pressure. Furthermore, Poisson’s ratio was found to be more sensitive to
lithology than the compressional (P-) and shear (S-) wave
velocities alone.
Kern et al. (2002) examined the physical properties of
metamorphic rocks from the Sulu terrain, China under
ultrahigh pressure (up to 600 MPa). Chemical composition
and metamorphic grade were known to contribute to the
change in mineralogy, and this was evident in the relationship between P-wave velocities and Poisson’s ratio.
The measured in situ velocities of P- and S-waves
increased with density and metamorphic grade of the rocks.
They were compared with the near-intrinsic velocities from
seismic measurements (refraction and reflection) which led
to the conjecture that fractures play an important role in the
in situ seismic properties of the seismic crustal profile.
Reflectivity in the seismic profile was suggested to be due
to the shear deformation-induced (lattice preferred orientation-related) seismic anisotropy.
Vilhelm et al. (2010) compared the velocities determined in the laboratory to those measured in the surface
seismic records in the Ivrea zone in northwestern Italy.
This zone is composed of strongly metamorphic rocks
including an ultra-mafic peridotite massif that was selected
for rock sampling. It was found that the laboratory and
field-scale velocities differ significantly up to a scale of
41 %. The discrepancy can be due to the frequency of the
applied seismic signals, density of parallel cracks, and
fracture stiffness observed in the outcrops that were
otherwise undetected by the smaller scale rock samples.
This study demonstrated that it can be difficult to directly
transfer laboratory-measured velocities to a field-scale
study.
Cholach et al. (2005) measured the P- and S-wave
speeds in three orthogonal directions and with three
orthogonal polarizations on a suite of 35 metavolcanics and
metasediments from an exposed mid-crustal shear zone in
the Canadian Shield near Flin Flon, Manitoba. These
samples visually displayed a range of textures indicative of
isotropic, transversely isotropic, and orthorhombic symmetries; and both strong shear wave splitting and high
values of anisotropy were observed. Interestingly, averaging of the elastic properties of this suite suggested that the
larger formation would be close to having transverse isotropic symmetry.
In situ quantification of seismic anisotropy can be done
using multi-offset and multi-azimuths vertical seismic
profiling (VSP). Rabbel (1994) observed the presence of
shear wave birefringence in the near-offset VSP of the
KTB borehole in Oberpfalz, Germany. Velocity for direct
compressional and split shear waves was picked from the
1371
near-offset VSP. The polarization direction of the faster
shear wave was found to coincide with the steeply dipping
rock formation of dominantly biotite gneiss. Under the
assumption of hexagonal-type symmetry, the average
anisotropy of the gneiss was estimated using the leastsquares fit method with known seismic velocities and
angles of wave propagation. Moreover, the final reported
anisotropy was found to agree with the laboratory measurements performed on the rock samples from the
borehole.
Significant seismic anisotropy was observed in the Outokumpu scientific borehole in Finland using walk-away
VSP. Schijns et al. (2012) found that the reported anisotropy from the VSP data could not be fully explained alone
by the intrinsic rock anisotropy measured in the laboratory
(Kern et al. 2008). As such, forward modeling of the VSP
results was undertaken to predict an overall anisotropy of
schist by incorporating the intrinsic material properties
obtained from core measurements and the oriented crack
anisotropy. The modeling results revealed that the phase
velocities in an intrinsically orthorhombic medium containing a set of aligned cracks would be consistent with the
in situ P-wave velocities from the walk-away VSP.
Most of the Canadian Shield Craton is made up of
Precambrian igneous and high-grade metamorphic rock.
The Canadian Shield is exposed in the northeastern part of
Alberta where the crystalline basement can be traced
unambiguously into the subsurface dipping gently southwest at 4–5 m/km and overlain by the Phanerozoic successions of the WCSB. Recent research interests in
characterizing the basement rocks in northern Alberta arise
in investigating the feasibility of geothermal energy
development in Alberta. Existing geothermal gradient
studies suggested that deep drilling into the metamorphic
Canadian Shield is required in northeastern Alberta to
reach the target temperature for higher enthalpy geothermal
exploitation (Majorowicz et al. 2012).
In Alberta, there were over 489,000 boreholes drilled in
Western Canada until 2011 (Canadian Association of
Petroleum Producers (CAPP) 2013). However, boreholes
drilled deep into the Precambrian basement in Alberta are
extremely rare because these low porosity and permeability
rocks are not considered prospective for hydrocarbons. As
such, our understanding of the basement rocks, let alone
their seismic anisotropy, remains limited. However, commercial interests seeking to test novel hypotheses of
hydrocarbon generation drilled a deep borehole (hereafter
referred to as Hunt well after its owner) through nearly
2,000 m of metamorphic cratonic rocks in NE Alberta
(Chan 2013). This borehole extends deep into the basement
rocks and thus provides a unique opportunity for
researchers to study the in situ physical properties of the
metamorphic rocks. Currently, the Hunt well is believed to
123
Author's personal copy
1372
J. Chan, D. R. Schmitt
be the deepest borehole drilled into the metamorphic
Canadian Shield in Alberta.
Source
O
*
x
Horizontal Position
3 Theoretical Background
According to the theory of linearized elasticity, a linear
relation (Eq. 1) between stress and strain can be applied to
characterize the material properties at a particular point by
the generalized Hooke’s Law. In an elastic medium, the
stress–strain relationship is related in the form of the
generalized Hooke’s Law:
rij ¼ Cijkl ekl
ð1Þ
where rij, ekl denote the variables of stress and strain as
second-rank tensors, respectively, Cijkl are the components
of the elastic stiffness tensor C (or elastic constants) as
fourth-rank tensor with 81 components, and i, j, k, l = 1, 2,
3 indicates one of the three orthogonal axes. Einstein
summation convention applies here for each instance of a
repeated index on the same side of the equation (Auld
1990). Equation (1) can be further reduced from four
indices ijkl down to two indices mn by introducing the
Voigt notation (Nye 1985):
Cijkl ¼ Cmn ði; j; k; l ¼ 1; 2; 3; m; n ¼ 1; . . .; 6Þ
ð2Þ
which then simplify the generalized Hooke’s Law into to a
matrix equation:
rI ¼ cIJ eJ
ð3Þ
where rI and eJ are the 6 9 1 vectors containing independent components of stress and strain tensors, respectively (Auld 1990). Each CIJ is one of the components of a
6 9 6 symmetric matrix with 21 independent elastic stiffnesses that is symmetric about the diagonal:
32 3
2 3 2
e1
C11 C12 C13 C14 C15 C16
r1
6 r2 7 6 C12 C22 C23 C24 C25 C26 76 e2 7
76 7
6 7 6
6 r3 7 6 C13 C23 C33 C34 C35 C36 76 e3 7
76 7
6 7¼6
6 r4 7 6 C14 C24 C34 C44 C45 C46 76 e4 7 ð4Þ
76 7
6 7 6
4 r5 5 4 C15 C25 C35 C45 C55 C56 54 e5 5
r6
C16 C26 C36 C46 C56 C66
e6
where the indices I and J are related to the ij in according to
the cyclical notation I, J = 1, 2, 3, 4, 5, 6 when ij or
kl = 11, 22, 33, 13 or 31, 23 or 32, 12 or 21, respectively.
The Christoffel equation is an eigenvalue problem in
which known elastic constants can be used to calculate the
phase velocities of the elastic waves. Detailed derivations
need not be repeated here but can be found for example in
the monograph published by Musgrave (1970). The equations for linking velocities and elastic constants simplify
123
d
3.1 Elasticity
Vertical Position
D
P
Pl
an
e
W
av
ef
ro
nt
Wave Front Surface
z
Fig. 1 The relationship between ray (group, D) and phase velocities
(d). The axis of symmetry is along the z direction. Image adapted
from Daley and Hron (1979) and Kebaili and Schmitt (1997)
substantially when wave behavior is discussed within a
plane of symmetry and along the principal axis (e.g.,
Brugger 1965; Fedorov 1968; Musgrave 1970; Aki and
Richards 1980; Crampin 1981; Nye 1985).
3.2 Anisotropy
Metamorphic rocks are more heterogeneous in nature as
presence of foliations and lineations result in strong
intrinsic anisotropy due to the preferential alignment of
minerals, textural–structural features such as bedding and
foliation, and pores and cracks (e.g., Crampin 1981;
Mainprice and Nicolas 1989; Kern and Wenk 1990).
An important point to note when dealing with wave
propagation in anisotropic media is that care must be taken
in the definition of the wave speed itself. Figure 1 shows
the shape of a wavefront in the x–z plane at time t after the
source of the elastic wave was activated at position O. An
observer at point P on the wavefront, a straight distance
D from point O, would calculate a group velocity
Vray = D/t under the assumption that he was in the far field
from a point source. However, this observer cannot know
from his single vantage point whether or not the arrival he
sees lies along the front of a plane wave that would have
conveniently passed through the origin O at t = 0. This
plane wave front is tangent to locus of the point source
wave front at the observation point P; and the phase
velocity of this plane wave is instead given by Vphase = d/t.
In an isotropic medium d = D but once the material
becomes anisotropic this is no longer generally true. Ray
(group) velocity is the velocity of energy propagation along
a ray path at propagation angle /, whereas phase velocity
is the velocity of a mono-frequency plane wave at propagation angle h. These two velocities are equivalent in an
Author's personal copy
Elastic Anisotropy of a Metamorphic Rock Sample
Fig. 2 Symmetry of a transversely isotropic medium where elastic
properties are rotationally symmetric about the z-axis. Arrows of the
same color indicate waves propagating at the same speed while their
directions indicate the wave’s polarization
isotropic material, but while linked to one another still need
to be considered separately for an anisotropic material to
properly relate the velocities to elastic constants. This is an
important point as determining the CIJ directly from the
wave speeds requires knowledge of the phase (plane wave)
velocities.
This must be considered in the experimental design
particularly with regards to the dimensions of the transmitting and receiving transducers, the distance of propagation between them, and the degree of ‘side-slip’ of the
beam due to the preferred path of the wave’s energy flow
through the medium (Dellinger and Vernik 1994; Vestrum
1994; Kebaili and Schmitt 1997; Mah and Schmitt 2001).
Modeling for more anisotropic shales (Meléndez Martı́nez
2014) demonstrates that plane (or group) wave speeds are
observed in the current experiments.
1373
plane wave propagation direction relative to the axis of
symmetry. P, SV and SH indicate the polarization vectors
in the compressional, vertical (i.e., parallel to the rotational
axis of symmetry z) and horizontally polarized shear
directions, respectively. For example, VSH90 refers to the
propagating S-wave that is horizontally polarized with
respect to the symmetry axis (i.e., at 90° from the symmetry axis). The Hunt well sample is presumably a TI
sample with the foliation plane as the plane of symmetry.
In TI media, VP0 is different from VP90 and VS0 is also
different from VSH90. The S-waves propagating parallel to
the symmetry axis do not split and do not need separate
polarization labels as related by VSV0 = VSH0. Another
special characteristic of TI media is that VSH0 = VSV90.
This form of symmetry is similar to a crystal that could be
described in the Voigt-form stiffness matrix C (Musgrave
1970):
3
2
C11 2C66 C13
0
0
0
C11
6 C11 2C66
C11
C13
0
0
0 7
7
6
6
C13
C13
C33
0
0
0 7
7
CIJ ¼ 6
6
0
0
0 C44
0
0 7
7
6
4
0
0
0
0 C44
0 5
0
0
0
0
0 C66
ð5Þ
The expressions for deriving the elastic constants from
phase velocities were developed by Brugger (1965), Daley
and Hron (1977) and Thomsen (1986) as:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
C33 þ C44 þ ðC11 C33 Þ sin2 h þ DðhÞ
VP ð h Þ ¼
ð6Þ
2q
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
C33 þ C44 þ ðC11 C33 Þ sin2 h DðhÞ
VSV ðhÞ ¼
2q
ð7Þ
3.3 Transverse Isotropy
Many rocks have structure with hexagonal symmetry and
elastically will be transversely isotropic in which the
elastic properties are the same in a plane that is perpendicular to a symmetry axis (i.e., z-axis, Fig. 2) but different parallel to the axis. To avoid confusion, it must be
pointed out that this differs from the sixfold optical
symmetry observed in the minerals. Transverse isotropy
requires five independent elastic constants that are
dependent on the angle between the propagation direction
and the symmetry axis.
Many low- to medium-grade metamorphic rocks often
have well-developed bedding or foliation planes and
behave as transversely isotropic (TI) elastic solids (Gupta
2011). Figure 2 illustrates the symmetry of a TI medium.
The subscripts 0 and 90 correspond to the angle h of the
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
C66 sin2 h þ C44 cos2 h
VSH ðhÞ ¼
q
ð8Þ
where q is the density of the rock and h is the angle
between the direction of wave propagation and the sample’s rotational axis of symmetry. D(h) is denoted by:
h
DðhÞ ðC33 C44 Þ2 þ2 2ðC13 þ C44 Þ2 ðC33 C44 Þ
i
ðC11 þ C33 2C44 Þ sin2 h
1=2
h
i
2
2
4
þ ðC11 þ C33 2C44 Þ 4ðC13 þ C44 Þ sin h
ð9Þ
Note that the wave front surface for the VSH mode will
always take an elliptical form according to Eq. (8).
123
Author's personal copy
1374
J. Chan, D. R. Schmitt
As there are five independent elastic coefficients a
minimum of five different measured velocities are needed
to calculate the stiffness constants of the material. For a TI
material these equations simplify significantly in the symmetry directions where the P- and S-waves propagate as
pure modes, i.e., modes in which the longitudinal and shear
particle motions are parallel and perpendicular, respectively, to the direction of the wave propagation with:
2
C11 ¼ qVP90
ð10Þ
2
C33 ¼ qVP0
ð11Þ
2
C44 ¼ qVSH0
ð12Þ
2
C66 ¼ qVSH90
ð13Þ
C12 ¼ C11 2C66
ð14Þ
where VP90, VP0, VSH0, and VSH90 are P- and S-wave
velocities in different directions as indicated in Fig. 2. SH
refers to the shear wave with particle motions perpendicular to the z-axis and is the component vibrating along
the foliation plane. The remaining elastic stiffness C13
requires that an additional wave speed to be measured at
an angle offset from the symmetry axis. Conveniently
taking the velocity measurements at 45° from the symmetry axis, C13 can be obtained using the following
equation:
C13 ¼ C44
h
i1=2
2
2
þ C11 þ C44 2qVp45
C33 þ C44 2qVp45
c¼
C66 C44 VSH90 VSH0
¼
2C44
VSH0
ðC13 þ C44 Þ2 ðC33 C44 Þ2
2C33 ð
C33 C44 Þ VP45
VP90
¼4
1 1
VP0
VP0
ð18Þ
d¼
ð19Þ
where e is a measure of the P-wave anisotropy, c is a
measure of SH-wave anisotropy, and d is an expression
related to the anellipticity of the P-wave wavefront. e, c, d
are dimensionless and have values smaller than 0.5, but
frequently much smaller (Sheriff 2002). These three
parameters are useful in quantifying anisotropy when
simple inspection of the elastic moduli do not present
obvious indications of anisotropy.
It is worth noting that all of the studies on the anisotropy
of metamorphic rocks described earlier made measurements in directions aligned with visible foliations and lineations. However, as examination of the above equations
demonstrates, the full set of elastic stiffnesses requires that
wave speeds must also be measured at angles away from
these directions and to our knowledge this is very rarely
accomplished (e.g., Sano et al. 1992; Pros et al. 1998,
2003; Takemura and Oda 2005; Sarout et al. 2007; Nara
et al. 2011). Such information, however, is crucial to a full
understanding of the wave behavior and for this reason in
this study we take care to specially machine our sample to
allow for measurements perpendicular, parallel, and at 45°
from the symmetry axis.
ð15Þ
or
C13
4 Experimental Procedure
31=2
2 2
2
2
4q2 Vp45
VSV45
ðC11 C33 Þ2
7
6
¼ C44 þ 4
5
4
ð16Þ
Equation (16) is an alternative expression to Eq. (15)
proposed by Hemsing (2007) to estimate C13 using a
combination of VP45 and VSV45 to reduce the degree of
uncertainty. SV refers to shear waves whose polarization
lies in the vertical plane (i.e., the plane containing the
material’s axis of symmetry).
To relate the elastic constants (C11, C13, C33, C44 and
C66) to P- and S-wave velocities measured parallel and
perpendicular to the symmetry axis, Thomsen (1986) suggested three parameters for quantifying the weak anisotropic behavior of TI materials:
e¼
C11 C33 VP90 VP0
¼
2C33
VP0
123
ð17Þ
4.1 Sample Preparation
Limited core samples from the Hunt well were available
from the Core Research Centre of the Alberta Energy
Regulator
at
two
different
depth
intervals:
1656.5–1,657.8 m and 2,347.5–2,364.3 m in the 2.4 km
deep borehole (AOC GRANITE 7-32-89-10) located in NE
Alberta (latitude 56°450 N, longitude 111°330 W). A rock
sample was selected from a depth of 2,350.3 to 2,350.5 m.
It is a foliated granite composed mainly of quartz and
feldspars. Other minerals present in the rock are biotite,
garnet, pyroxene, hematite, magnetite, ilmenite, titanite,
zircon, monazite and apatite (Walsh 2013; G. Njiekak,
personal communication, January 23, 2014). Some samples
also display schlieren textures. Thin sections were made to
provide microscopic analysis of the textural features in the
sample. Aggregates of elongated quartz and the presence of
feldspar bands provide evidence of the preferentially
Author's personal copy
Elastic Anisotropy of a Metamorphic Rock Sample
Fig. 3 Thin sections taken under plane-polarized (left) and crosspolarized (right) lights. The preferential orientation of minerals and
microcracks are indicated by the yellow arrows. Two evident zones of
1375
microcracks are annotated by the letters A and B for a comparative
interpretation between the photographs
Fig. 4 Photographs of the
metamorphic rock sample
a before and b, c after sample
preparation. Due to limited
cores available, the sample has
to be carefully cut along the
dotted blue lines with respect to
the identifiable foliation plane to
obtain a sample with the largest
dimension possible from the
core
123
Author's personal copy
1376
oriented fabric displayed by the study rock. Furthermore,
microcracks are seen parallel to the felsic bands (Fig. 3).
P- and S-wave velocities were measured in three different directions: parallel, perpendicular and at 45° to the
foliation plane. The sample was cut, ground flat and parallel on opposing surfaces, on a diamond grinding disk in
three different angles relative to the apparent direction of
the foliation plane to ensure good surface contacts for the
transducers. This minimized the errors in velocity measurements and improved pulse transmission from one surface of the sample to another (Meléndez Martı́nez and
Schmitt 2013). The final sample size was 8.15 cm
(length) 9 7.00 cm (width) 9 4.50 cm (height) with the
45° surfaces 3.80 cm apart (Fig. 4). Error for each measured dimension is estimated to be 0.05 cm. After grinding,
the sample was placed in the oven to vacuum dry at 45 °C
for over 12 h as a precautionary measure to ensure that the
sample is fully dry.
Bulk density of the sample was measured in the laboratory using a mercury (Hg) injection porosimeter and a
bulk density value of 2.62 g/cm3 was reported at
0.0036 MPa (i.e., at low pressure before any Hg could
intrude the sample). A bulk density geophysical log (i.e.,
Compton scattering) was also available and this provided a
density of 2.65 g/cm3 at 2,350.40 m, which closely agrees
with the density value measured in the laboratory. A grain
density of 2.66 g/cm3 was also reported from the Hg porosimeter and reveals a low porosity of 1.45 %.
The main component for transmitting and receiving
elastic waves in the laboratory are the transducers. For this
experiment, ultrasonic transducers composed of longitudinally and transversely polarized piezoelectric ceramics and
copper foil electrodes are used. The transmitting transducer
converts the electrical pulse to a mechanical signal, which
is transmitted through the rock. The receiving transducer
changes the wave back into an electrical pulse, which is
then amplified and displayed on the oscilloscope screen.
The type of vibration generated is determined by the
polarization of the piezoelectric ceramic. Longitudinal
wave (P-wave) PZT ceramics were in the form of circular
discs with a diameter of 20 mm and the transverse wave
(S-wave) ceramics were square plate with a length of
15 mm. Both were made from APC’s 851 material (APC
International Ltd. 2011). The piezoelectric material is a
ceramic made from lead zirconate titanate with a resonant
frequency operating at 1 MHz. This material is capable of
producing an electrical potential with applied stress such as
a mechanical vibration.
Six pairs of piezoelectric transducers were mounted on
the aforementioned surfaces relative to the foliation plane
to generate and record the P- and S-waves (Fig. 5). Thin
copper strips were bonded beneath and on top of the
transducers as electrodes using CircuitWorksÒ CW2400
123
J. Chan, D. R. Schmitt
Fig. 5 Schematic diagram showing the placement of six pairs of
transducers on the sample. The subscripts indicate the direction with
respect to the axis of symmetry and the dashed lines represent the
direction of wave propagation. Image adapted from Wong et al.
(2008)
Fig. 6 Schematic diagram of the experimental setup
conductive epoxy. The copper foil attached to the underside of the transducer acted as a ground for the transducers.
Insulated electrical wires were soldered to the electrodes,
and the entire sample was then sealed in a flexible urethane
compound (FlexaneÒ 80 Putty) to protect the sample from
the hydraulic oil inside the pressure vessel that is used as
Author's personal copy
Elastic Anisotropy of a Metamorphic Rock Sample
1377
the confining pressure medium. This assemblage was then
placed in the pressure vessel. A voltage is applied to the
transmitting transducer, which then sends a frequency
pulse through the sample and is detected by the transducer
on the opposite ends.
4.2 Data Acquisition
The ultrasonic pulse transmission technique was used to
study the dependence of seismic velocity under varying
confining pressure condition with a QuizixTM Q5000 pump
system. The transmitted signal of the elastic waves was
generated using a JSR-PR35 pulse generator that was
connected to the electrical wires of the transmitting transducers and the propagated signal was recorded by a digital
oscilloscope developed by National Instrument (Fig. 6).
In situ peak vertical stress for the sample is determined
based on the density of the overburden and the depth in
which it was extracted from. Vertical stress (Sv) is often
assumed to be equivalent to the overburden weight by
integrating the densities of overlying rocks according to the
following equation:
Sv ð hÞ ¼ Zh
qðhÞgdh
ð20Þ
0
where h is the overburden depth, q is the average density
value and g is the acceleration of gravity. The vertical
stresses expected in the Hunt well were calculated by
integrating the borehole density log. The negative value
arises under the assumption that both g and h are positive
numbers with the compressional stress increasing with
depth (Schmitt et al. 2012). The in situ peak pressure for
the sample was found to be close to 60 MPa.
The travel times were recorded in 3 MPa increments
between 0 and 15 MPa, and in 5 MPa increments between
15 and 60 MPa during both the compression (pressurization) and decompression (depressurization) cycles conducted at room temperature. The maximum confining
pressure of 60 MPa was selected to recreate the in situ
pressure conditions similar to those at depth. After reaching
the target confining pressure, waveforms were recorded
after a waiting period (5 min) to allow the pressure condition to stabilize in the pressure vessel. This step was also
taken to ensure the opening or closing of any cracks or
pores in the sample. The travel time and amplitude of the
wave were recorded at each pressure interval and each
waveform trace was constructed from at least 300 progressively stacked records to minimize the random noise
effects (Fig. 7). The recorded travel time across the sample
in a direction normal to the wave front is defined as the
phase travel time. Phase velocities are obtained using the
known distance that the wave has traveled normal to the
plane wave front and the measured phase travel time
(Vestrum 1994). Travel time was interpreted from the first
extremum value of the waveform on each stacked record
(Molyneux and Schmitt 1999). It was also observed that the
signal amplitudes were stronger with pressure due to the
combined effects of the closing of microcracks, influence
of pressure on the stiffness of rock forming minerals and
the improved surface contacts.
5 Results
To investigate the anisotropic properties of the metamorphic core sample from the Hunt well, P- and S-wave phase
velocities were measured at a confining pressure of up to
60 MPa in three different directions with respect to the
visible foliation plane during both the pressurization and
depressurization cycles of the measurements. Arrival times
for P- and S-waves were then interpreted from the first
extremum of the stacked waveforms. At 0–3 MPa, the
waveforms for S-waves for both the pressurization and
depressurization cycles were relatively noisier compared to
all other waveforms at higher confining pressures. P- and
S-wave velocities were thereafter calculated using the
measured distances between the pulsing and receiving
surfaces of the sample which were plotted as a function of
confining pressure in Fig. 8. Numerous sources of error
need to be considered when calculating the velocities from
the measurements. They include the travel time delay
through the piezoelectric ceramic transducers (Meléndez
Martı́nez 2014), the uncertainty in the complete parallelism
of the sample surfaces (0.05 cm) and the accuracy of the
travel time picks from the recorded waveforms (0.01 ls).
At low confining pressures, the anisotropic character of
pore space geometry was suggested to have an influence on
seismic anisotropy (e.g., Kern and Schenk 1988). The
initial increase in velocity with pressure is often explained
by the closing of microcracks and pores in the rock sample.
In other words, the closing of cracks under pressure would
increase the elastic stiffness which consequently leads to
the rapid increase of velocities. Microcracks could be
originated from sources such as stress relief during coring
and are present in many rocks. The small changes in the
anisotropic parameters of the sample show that there is
minute stress dependence with the cracks closing at below
9 MPa. As such, the anisotropic behavior of the sample is
mainly attributed to its intrinsic properties. At high confining pressures, velocities in all three directions increase
linearly at a similar rate as a result of the intrinsic effect of
the mineral components (e.g., Birch 1961; Kern and
Schenk 1988). Figure 8 shows that the velocity gradient is
significantly higher at lower confining pressure (0–9 MPa).
The velocity gradient gradually decreases as the pressure
123
Author's personal copy
1378
J. Chan, D. R. Schmitt
Fig. 7 Waveforms of P- and
S-waves propagating a, b along
the foliation plane, c, d normal
to foliation plane, and e, f at 45°
with respect to the foliation
plane. The lengths of the travel
paths are 0.070, 0.045 and
0.038 m, respectively
increases which reflects the intrinsic effects of pressure on
the mineralogical velocities of the rock sample. The fastest
P-wave velocities were measured along the foliation plane
from 5,352 to 6,019 m/s. Velocities for P-waves measured
across and oblique to the foliation plane are quite similar at
4,752–5,396 m/s and 4,810–5,337 m/s, respectively.
Figure 9a is useful in demonstrating whether the sample
is an isotropic medium. If a sample is isotropic, it is
expected that C11 = C33 and C44 = C66. This is not the
case for this sample which demonstrates the anisotropic
behavior of the rock. As the cracks closed under the
influence of increasing pressure, the velocity gradients of
the propagative waves also decreased as they encountered
less low-rigidity material (air) and cracks. This led to the
increase in the elastic stiffnesses that eventually approached the constant linear level when all the cracks are closed
at high confining pressure. It is also possible that the
123
stiffnesses of the sample can be affected by other minerals
in smaller proportions.
Figure 9b shows the change in the anisotropic parameters under pressure. The three anisotropic parameters
remained relatively stable beyond a confining pressure of
9 MPa. A sudden sharp decrease of S-wave anisotropy at
low confining pressures could be due to the erroneous
travel time picks in the noisy waveforms as mentioned
earlier at low confining pressures. The anisotropic parameters e and c were obtained from the P- and S-wave
velocities (Eqs. 17, 18) in the axial directions and have
reported values of 12 and 8 %, respectively. These values
indicate that the sample is considered as moderately
anisotropic (Table 1) as suggested by Thomsen (1986)
when the anisotropic parameters exceed 10 %. In between
0 and 3 MPa, c and d experience greater changes that could
be attributed to the uncertainties in the first extremum
Author's personal copy
Elastic Anisotropy of a Metamorphic Rock Sample
(a) P-Wave Velocities
5800
5600
5400
Velocity (m/s)
Fig. 8 a P- and b S-wave
velocities as a function of
confining pressure between 0
and 60 MPa for the
pressurization (solid line) and
depressurization (dashed line)
cycle. The hysteresis effect is
observed in both the
pressurization and
depressurization cycles
1379
5200
5000
4800
V P90
V P45
4600
V P0
4400
0
10
20
30
40
50
60
70
Confining Pressure (MPa)
(b) S-Wave Velocities
3100
3000
2900
Velocity (m/s)
2800
2700
2600
2500
VSH90
2400
VSH45
2300
2200
VSH0
0
10
20
30
40
50
60
70
Confining Pressure (MPa)
picking of the S-wave waveforms. The third anisotropic
parameter d contains off-diagonal elastic stiffnesses and is
related to the short-spread normal moveout velocities in TI
media with a reported value of 13 %.
6 Discussion
P-wave velocity anisotropy of the most common metamorphic rocks is typically almost orthorhombic or transversely isotropic (Barberini et al. 2007). The anisotropy in
rocks is primarily due to the preferred orientation of the
constituent minerals, textural–structural features, pores,
and cracks. As the confining pressure increases, the effect
of microcracks on anisotropy is expected to diminish, and
the remaining anisotropy is more likely to result from
layering and preferred orientation of minerals, i.e., intrinsic
anisotropy.
For most types of symmetry, Backus (1970) reported
that the phase and group velocities are identical in magnitude along the principal axes in the case of weak
anisotropy. The propagating velocity as a function of
pressure is primarily dependent upon mineralogy and
porosity (Gupta 2011). The effects of cracks and pores in
the anisotropic nature of the rocks should be expected to
decrease as the applied confining pressure progressively
123
Author's personal copy
1380
J. Chan, D. R. Schmitt
Fig. 9 Display of a elastic
constants and b anisotropic
parameters as a function of
confining pressure
(a) Elastic Constants
120
C11
C33
C44
C66
C13
100
(GPa)
80
60
40
20
0
0
10
20
30
40
50
60
50
60
Confining Pressure (MPa)
(b) Anisotropic Parameters
0.5
Anisotropic Parameters
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
0
10
20
30
40
Confining Pressure (MPa)
increases along with the closing of the cracks which in turn
would result in the increase of P- and S-wave velocities
(Walsh 1965).
In this study, velocity measurements obtained during the
initial pressurization cycle are slightly lower than the
velocities measured during depressurization. This effect is
called hysteresis in which the physical property that has
been changed does not return to its original state after the
cause of the change has been removed (Gardner et al.
1965). The cracks that were closed during the pressurization cycle would be expected to open at a lower pressure
123
than the original closing pressure associated with the frictional adjustment of crack faces and grain boundaries.
Alternatively, it is also possible that some cracks or pore
spaces do not re-open during the depressurization cycle as
a result of the irreversible change in grain contacts and
reduction of porosity in the rock sample (Ji et al. 2007).
The hysteresis effect is generally smaller if sufficient time
is allowed for data measurements at each confining pressure step.
P-wave velocities parallel to the foliation plane are
greater than those in the perpendicular and oblique
Author's personal copy
Elastic Anisotropy of a Metamorphic Rock Sample
1381
directions. The high gradients in P- and S-wave velocities
at low confining pressures suggest that velocities are largely controlled by the microcracks, as observed in Fig. 8.
Thomsen parameters also experience similar sharp gradient
with the decrease in values at lower confining pressures. At
higher pressures, layering and the preferred orientation of
minerals are expected to have greater effects on the
anisotropic behavior of the rock as opposed to the cracks.
Microscopic analysis of the thin sections of the rock
sample is used to verify the interpretations with the presence of microcracks and aggregates of elongated quartz
oriented in the same direction (Fig. 3). P and S-wave
velocities at 45° are unexpectedly measured to be lower
Table 1 Thomsen parameters obtained from the rock sample during
the pressurization cycle
Pressure (MPa)
e
c
d
0
0.13 ± 0.06
0.27 ± 0.06
0.18 ± 0.02
3
0.12 ± 0.06
0.20 ± 0.05
0.24 ± 0.03
6
0.12 ± 0.06
0.08 ± 0.04
0.16 ± 0.02
9
0.12 ± 0.06
0.08 ± 0.04
0.13 ± 0.02
12
0.12 ± 0.06
0.08 ± 0.04
0.13 ± 0.02
15
0.13 ± 0.06
0.08 ± 0.04
0.14 ± 0.02
20
25
0.13 ± 0.06
0.12 ± 0.06
0.08 ± 0.04
0.07 ± 0.04
0.14 ± 0.02
0.12 ± 0.02
30
0.12 ± 0.06
0.08 ± 0.04
0.12 ± 0.02
35
0.12 ± 0.06
0.08 ± 0.04
0.13 ± 0.02
40
0.13 ± 0.06
0.08 ± 0.04
0.13 ± 0.02
45
0.13 ± 0.06
0.07 ± 0.04
0.14 ± 0.02
50
0.13 ± 0.06
0.08 ± 0.04
0.14 ± 0.02
55
0.13 ± 0.06
0.07 ± 0.04
0.13 ± 0.02
60
0.12 ± 0.06
0.07 ± 0.04
0.13 ± 0.02
Table 2 Elastic constants
estimated from the recorded
waveforms during the
pressurization cycle
Pressure (MPa)
C11 (GPa)
than the velocities measured normal to the foliation plane.
One possible explanation for this is the heterogeneity of the
sample that could not be identified from the surface of the
selected sample for this experiment.
Five independent elastic stiffnesses C11, C13, C33, C44
and C66 are calculated from the phase velocities according
to Eq. 10 through Eq. 15 (Table 2) and are plotted in
Fig. 9a. Elastic constants gradually increase with the
increase in confining pressure up to 60 MPa. The calculated C11 and C33 are dependent on the P-wave velocities
whereas C44 and C66 are dependent on the S-wave velocities. The results indicate that C11 [ C33 [ C13 [ C66 [
C44. Through error propagations, the uncertainties for C11,
C33, C66 and C44 are expected to be lower compared to that
for C13 due to the buildup of uncertainties in the equation
(Eq. 15). An interpretation of Fig. 9a also reveals that the
hysteresis effect is also noticeable in the elastic constants
similar to the observation of the velocity measurements.
In the case of a weakly TI medium, the dependence of
Thomsen parameters (e, c, d) on the phase velocities is
illustrated using Thomsen (1986) approximations (Fig. 10).
One of Thomsen’s (1986) phase velocity equations
Vp(h) = VP0 (1 ? d sin2 h cos2 h ? e sin4 h) demonstrates d
as the dominant factor which controls the anisotropic features of most situations in exploration geophysics. Changes
in d relative to e illustrate the anellipticity of the propagating
P-wave wavefront and controls whether the wavefront is
concave (positive anellipticity, d \ e), elliptical (d = e), or
convex (negative anellipticity, d [ e). An important note to
mention is that the original elastic equations for the P and SV
curves do not actually follow the ellipse or circle in Fig. 10 as
they are illustrated using Thomsen approximations.
The above demonstrates that we have been able to calculate a complete set of elastic stiffnesses for this rock
C33 (GPa)
C44 (GPa)
C66 (GPa)
C13 (GPa)
0
75.1 ± 2.9
59.2 ± 3.1
15.3 ± 0.7
23.6 ± 0.8
38.1 ± 4.4
3
79.7 ± 3.2
64.3 ± 3.5
17.3 ± 0.8
24.1 ± 0.8
43.3 ± 4.8
6
86.9 ± 3.5
70.3 ± 3.9
21.4 ± 1.0
24.8 ± 0.8
37.7 ± 5.0
9
89.4 ± 3.6
71.8 ± 4.0
21.6 ± 1.0
25.0 ± 0.8
36.9 ± 5.0
12
89.5 ± 3.6
72.1 ± 4.0
21.7 ± 1.0
25.1 ± 0.8
37.4 ± 5.1
15
90.7 ± 3.7
72.3 ± 4.0
21.8 ± 1.0
25.1 ± 0.8
38.1 ± 5.1
20
91.7 ± 3.7
72.5 ± 4.0
21.9 ± 1.0
25.4 ± 0.8
38.1 ± 5.1
25
92.0 ± 3.8
73.9 ± 4.1
22.1 ± 1.0
25.4 ± 0.8
38.0 ± 5.2
30
92.3 ± 3.8
74.0 ± 4.1
22.3 ± 1.0
25.7 ± 0.8
37.6 ± 5.2
35
92.6 ± 3.8
74.2 ± 4.1
22.4 ± 1.0
25.9 ± 0.9
38.4 ± 5.2
40
94.0 ± 3.9
74.6 ± 4.1
22.5 ± 1.0
26.0 ± 0.9
38.5 ± 5.3
45
94.0 ± 3.9
74.9 ± 4.2
22.7 ± 1.0
26.0 ± 0.9
39.0 ± 5.3
50
94.7 ± 3.9
75.3 ± 4.2
22.8 ± 1.0
26.2 ± 0.9
39.2 ± 5.3
55
60
94.8 ± 3.9
95.0 ± 3.9
75.8 ± 4.2
76.4 ± 4.3
22.9 ± 1.0
23.1 ± 1.0
26.4 ± 0.9
26.5 ± 0.9
39.2 ± 5.3
39.6 ± 5.4
123
Author's personal copy
1382
J. Chan, D. R. Schmitt
2.0 (Bourbie et al. 1987). From the dipole shear sonic
imager log recorded for the Hunt well, the Vp/Vs ratio has
an average value of 1.71 between 1,012 and 1,865 m
interval (Chan 2013). Current laboratory measurements
report averaged Vp/Vs ratios in the range of 1.82–1.90 along
the three directional variations at 2,350 m depth which
reveals an increase in Vp/Vs with depth. At this point, it is
unclear whether the increase in Vp/Vs could be attributed to
any geological significance.
7 Conclusions
Fig. 10 Illustration of the anisotropic parameter behavior of e, c, and
d of a weak transversely isotropic medium. The ellipsoids of the
longitudinal (dark grey) and SH polarization shear waves (light grey)
are included for references
sample using the observed wave speeds under the
assumption that the material is transversely isotropic.
However, in light of the illustration in Fig. 10 it is
important to return to the observed wave speeds in Fig. 8
where VP45 \ VP90 and VSH45 \ VSH90. While under certain circumstances the former is allowed, the latter case is
not. There are a number of possible reasons for this discrepancy that include heterogeneity of the sample, a misalignment of the true intrinsic transversely isotropic
symmetry axes from those chosen visually based on the
foliation, or the sample having a lower intrinsic symmetry.
Our observations here serve as a caution with regards to
determining the elastic stiffnesses from velocity measurements assumptions that the visible foliation and lineation,
as is assumed in nearly all measurements of the anisotropy
of metamorphic rocks, provide the true reference orientations for the intrinsic anisotropy.
Besides analyzing the velocity measurements and elastic
stiffnesses, the ratio of P- and S-wave velocities (Vp/Vs) can
be used as an indicator of the presence of fluids and fractures that are controlled mainly by rock type and pore fluid.
Existing literature has reported that such increase suggests
greater fracture density, higher metamorphic grade, larger
fluid content and/or changes in mineral composition (e.g.,
Sanders et al. 1995; Sibbit 1995; Kern et al. 2002; Boness
and Zoback 2004; Newman et al. 2008). In the crystalline
rock environment, Vp/Vs values generally range from 1.6 to
123
Seismic anisotropy may be caused by the presence of
fractures and anisotropic minerals oriented in a preferred
direction. To determine the elastic properties and to
quantify the directional dependence of a metamorphic rock
over a range of confining pressures, compressional and
shear wave velocities were measured using the ultrasonic
pulse transmission technique on a core sample from the
Hunt well, Alberta, Canada. Laboratory measurements are
performed up to a confining pressure of 60 MPa and the
recorded velocities are presented in graphs. Under the
assumption that the symmetry of this sample is transversely
isotropic, velocities were measured parallel, perpendicular,
and at 45° with respect to the visually selected foliation
plane. The graphical results reveal a gradual increase in
velocity as a function of confining pressure which reflects
the closing of microcracks. The hysteresis effect is
observed in both the velocity measurements and the
anisotropic parameter plots during the pressurization and
depressurization cycles.
The elastic constants and Thomsen’s anisotropic
parameters are calculated to estimate the degree of seismic
anisotropy; e and c are reported to be 12 and 8 %,
respectively, which indicate the anisotropic behavior of the
metamorphic rock sample. The results can be explained by
the preferential alignment of minerals and microcracks. It
was also observed that there is little variation in Thomsen
parameters for confining pressures beyond 9 MPa after the
closure of microcracks. As the confining pressure increases, the alignment of minerals (i.e., intrinsic effect) and
cracks become the dominant factors in the anisotropic
behavior of the rocks with the lessening effect of the microcracks. However, it might be possible to observe the
decreasing gradient of anisotropic parameters as a result of
the complete closure of microcracks and the preferred
alignment of mineral combination if higher confining
pressure is applied to the rock sample.
The overall results of the laboratory measurements
confirm the anisotropic behavior of the metamorphic rock
and could be useful in the construction of velocity models
in the seismic processing workflow. This could help to
Author's personal copy
Elastic Anisotropy of a Metamorphic Rock Sample
remove the anomalies caused by the directional dependence in the seismic velocities and further enhance the
imaging of seismic reflection profiles. An alternative
approach for identifying seismic anisotropy is the analysis
of shear wave birefringence. This was not performed on the
rock sample that was made available at the time but should
be considered in the future measurements of other samples.
The comparison of seismic velocities between laboratory
measurements and field seismic data is complicated by the
frequency ranges of the respective data sets due to dispersion effects. A single rock sample is used to showcase
the presence of anisotropy in the crystalline basement, but
it is not a representation of the entire crystalline basement
rock section. However, the result of this data does provide
an indication of the seismic intrinsic anisotropy in the
crystalline basement rocks of the study borehole, which
suggests that the use of an isotropic velocity model in
conventional seismic processing workflow might cause
imaging problems as a result of seismic anisotropy.
Seismic anisotropy has also been analyzed using walkaway VSP in the Hunt well (Chan 2013). Furthermore,
recent acquisition of the sonic scanner in the borehole is
also expected to provide and verify the results reported in
this paper. It is expected that an integration of the laboratory measurements (this paper), walk-away VSP and sonic
scanner interpretations can extend the seismic anisotropy
analysis from a single depth interval to other depth levels in
the borehole.
Acknowledgments The authors would like to thank Lucas Duerksen and Randolf Kofman for their assistance in setting up the laboratory equipment in the Rock Physics Laboratory of the Experimental
Geophysics Group (EGG) at the University of Alberta, and Gautier
Njiekak for his assistance in analyzing the thin sections. Core samples
were generously on loan from the Alberta Core Research Centre in
Calgary. This research was sponsored by funding from Natural Sciences and Engineering Research Council of Canada (NSERC), the
Canada Research Chairs Program, and the Geothermal Theme of the
Helmholtz-Alberta Initiative (HAI). We would also like to thank our
reviewer for insightful and observant feedback.
References
Aki K, Richards PG (1980) Quantitative seismology, theory and
methods, vol 1. W. H Freeman and Company, San Francisco
Alkhalifah T, Larner K (1994) Migration error in transversely
isotropic media. Geophysics 59(9):1405–1418. doi:10.1190/1.
1443698
APC International Ltd. (2011) Physical and piezoelectric properties of
APC materials. http://www.americanpiezo.com/apc-materials/
piezoelectric-properties.html
Auld BA (1990) Acoustic fields and waves in solids, 2nd edn. Kreiger
Publishing Company, Malaba
Backus G (1970) A geometrical picture of anisotropic elastic tensors.
Rev Geophys Space Ge 8(3):633–671. doi:10.1029/Rg008i003
p00633
1383
Barberini V, Burlini L, Zappone A (2007) Elastic properties, fabric
and seismic anisotropy of amphibolites and their contribution to
the lower crust reflectivity. Tectonophysics 445(3–4):227–244.
doi:10.1016/j.tecto.2007.08.017
Birch F (1961) The velocity of compressional waves in rocks to 10
kilobars (part II). J Geophys Res 66:2199–2224
Boness NL, Zoback MD (2004) Stress-induced seismic velocity
anisotropy and physical properties in the SAFOD Pilot Hole in
Parkfield, CA. Geophys Res Lett 31(15):L15S17. doi:10.1029/
2003gl019020
Bourbie T, Coussy O, Zinszner B (1987) Acoustics of porous media.
Gulf Publishing Company, Houston
Brugger K (1965) Pure modes for elastic waves in crystals. J Appl
Phys 36(3):759. doi:10.1063/1.1714215
Canadian Association of Petroleum Producers (CAPP) (2013) Statistical handbook for Canada’s upstream petroleum industry, vol
2013–9999. Technical Report. Canadian Association of Petroleum Producers, Calgary, Alberta, Canada
Chan J (2013) Subsurface geophysical characterization of the
crystalline Canadian Shield in Northeastern Alberta: Implications for Geothermal Development. University of Alberta,
Edmonton. http://hdl.handle.net/10402/era.32882
Cholach PY, Molyneux JB, Schmitt DR (2005) Flin Flon Belt seismic
anisotropy: elastic symmetry, heterogeneity, and shear-wave
splitting. Can J Earth Sci 42(4):533–554. doi:10.1139/E04-094
Crampin S (1981) A review of a wave motion in anisotropic and
cracked elastic media. Wave Motion 3:343–391
Crampin S (1990) Alignment of near surface inclusions and
appropriate crack geometries for hot dry rock experiments.
Geophys Prospect 38:621–631
Daley PF, Hron F (1977) Reflection and transmission coefficients for
transversely isotropic media. Bull Seismol Soc Am
67(3):661–675
Daley PF, Hron F (1979) Reflection and transmission coefficients for
seismic-waves in ellipsoidally anisotropic media. Geophysics
44(1):27–38. doi:10.1190/1.1440920
Dellinger J, Vernik L (1994) Do travel times in pulse-transmission
experiments yield anisotropic group or phase velocities? Geophysics 11:1774–1779
Eken T, Plomerova J, Vecsey L, Babuska V, Roberts R, Shomali H,
Bodvarsson R (2012) Effects of seismic anisotropy on P-velocity
tomography of the Baltic Shield. Geophys J Int 188(2):600–612.
doi:10.1111/j.1365-246X.2011.05280.x
Fedorov FI (1968) Theory of elastic waves in crystals. Plenum Press,
New York
Gardner GHF, Wyllie MRJ, Droschak DM (1965) Hysteresis in the
velocity-pressure characteristics of rocks. Geophysics 30:111–116
Godfrey NJ, Christensen NI, Okaya DA (2002) The effect of crustal
anisotropy on reflector depth and velocity determination from
wide-angle seismic data: a synthetic example based on South
Island, New Zealand. Tectonophysics 355(1–4):145–161. doi:10.
1016/S0040-1951(02)00138-5
Gupta HK (2011) Encyclopedia of Solid Earth Physics, vol Volume
1., Encyclopedia of Earth Science SeriesSpringer, The
Netherlands
Hemsing DB (2007) Laboratory determination of seismic anisotropy
in sedimentary rock from the Western Canadian Sedimentary
Basin. University of Alberta, Edmonton
Issac JH, Lawton DC (1999) Image mispositioning due to dipping TI
media: a physical seismic modeling study. Geophysics
64(4):1230–1238
Ji S, Wang Q, Marcotte D, Salisbury MH, Xu Z (2007) P wave
velocities, anisotropy and hysteresis in ultrahigh-pressure metamorphic rocks as a function of confining pressure. J Geophys Res
112(B09204):1–24. doi:10.1029/2006JB004867
123
Author's personal copy
1384
Johnson LR, Wenk HR (1974) Anisotropy of physical properties in
metamorphic rocks. Tectonophysics 23(1–2):79–98. doi:10.
1016/0040-1951(74)90112-7
Kebaili A, Schmitt DR (1997) Ultrasonic anisotropic phase velocity
determination with the Radon transformation. J Acoust Soc Am
101(6):3278–3286. doi:10.1121/1.418344
Kern H, Schenk V (1988) A model of velocity structure beneath
Calabria, southern Italy, based on laboratory data. Earth Planet
Sci Lett 87:325–337
Kern H, Wenk HR (1990) Fabric-related velocity anisotropy and
shear-wave splitting in rocks from the Santa Rosa Mylonite
zone. California. J Geophys Res Solid 95(B7):11213–11223.
doi:10.1029/Jb095ib07p11213
Kern H, Jin ZM, Gao S, Popp T, Xu ZQ (2002) Physical properties of
ultrahigh-pressure metamorphic rocks from the Sulu terrain,
eastern central China: implications for the seismic structure at
the Donghai (CCSD) drilling site. Tectonophysics
354(3–4):315–330. doi:10.1016/S0040-1951(02)00339-6
Kern H, Ivankina TI, Nikitin AN, Lokajicek T, Pros Z (2008) The
effect of oriented microcracks and crystallographic and shape
preferred orientation on bulk elastic anisotropy of a foliated
biotite
gneiss
from
Outokumpu.
Tectonophysics
457(3–4):143–149. doi:10.1016/j.tecto.2008.06.015
Mah M, Schmitt DR (2001) Near point-source longitudinal and
transverse mode ultrasonic arrays for material characterization.
IEEE Ultrason Ferroelectr Freq Control 48:691–698
Mah M, Schmitt DR (2003) Determination of the complete elastic
stiffnesses from ultrasonic phase velocity measurements. J Geophys Res 108:11. doi:10.1029/2001JB001586
Mainprice D, Nicolas A (1989) Development of shape and lattice
preferred orientations—application to the seismic anisotropy of
the lower crust. J Struct Geol 11(1–2):175–189. doi:10.1016/
0191-8141(89)90042-4
Majorowicz J, Gosnold W, Gray A, Safanda J, Klenner R, Unsworth
M (2012) Implications of post-glacial warming for Northern
Alberta heat flow—correcting for the underestimate of the
geothermal potential. GRC Trans 36:693–698
Majorowicz J, Chan J, Crowell J, Gosnold W, Heaman LM, Kück J,
Nieuwenhuis G, Schmitt DR, Walsh N, Unsworth M, Weides S
(2014) The first deep heat flow determination in crystalline
basement rocks beneath the Western Canadian Sedimentary
Basin. Geophys J Int 197(2):731–747. doi:10.1093/gji/ggu065
Meléndez Martı́nez J (2014) Elastic properties of sedimentary rocks.
PhD University of Alberta, Edmonton
Meléndez Martı́nez J, Schmitt DR (2013) Anisotropic elastic moduli
of carbonates and evaporites from the Weyburn-Midale reservoir
and seal rocks. Geophys Prospect 61(2):363–379. doi:10.1111/
1365-2478.12032
Molyneux JB, Schmitt DR (1999) First-break timing: arrival onset
times by direct correlation. Geophysics 64(5):1492–1501.
doi:10.1190/1.1444653
Musgrave MJP (1970) Crystal acoustic. Holden-Day, San Francisco
Nara Y, Kato H, Yoneda T, Kaneko K (2011) Determination of threedimensional microcrack distribution and principal axes for
granite using a polyhedral specimen. Int J Rock Mech Min Sci
48(2):316–335. doi:10.1016/j.ijrmms.2010.08.009
Newman GA, Gasperikova E, Hoversten GM, Wannamaker PE
(2008) Three-dimensional magnetotelluric characterization of
the Coso geothermal field. Geothermics 37(4):369–399. doi:10.
1016/j.geothermics.2008.02.006
Nye JF (1985) Physical properties of crystals. Oxford University
Press, London
Pros Z, Lokajiček T, Klı́ma K (1998) Laboratory approach to the
study of elastic anisotropy on rock samples. Pure appl Geophys
151:619–629
123
J. Chan, D. R. Schmitt
Pros Z, Lokajiček T, Přikryl R, Klı́ma K (2003) Direct measurement
of 3-D elastic anisotropy on rocks from the Ivrea Zone (Southern
Alps, NW Italy). Tectonophysics 370:31–47
Rabbel W (1994) Seismic anisotropy at the continental deep drilling
site (Germany). Tectonophysics 232(1–4):329–341. doi:10.1016/
0040-1951(94)90094-9
Sanders CO, Ponko SC, Nixon LD, Schwartz EA (1995) Seismological evidence for magmatic and hydrothermal structure in longvalley Caldera from local earthquake attenuation and velocity
tomography. J Geophys Res-Sol Ea 100(B5):8311–8326. doi:10.
1029/95jb00152
Sano O, Kudo Y, Mizuta Y (1992) Experimental determination of
elastic constants of Oshima granite, Barre granite, and Chelmsford granite. J Geophys Res 97(B3):3367–3379. doi:10.1029/
91JB02934
Sarout J, Molez L, Guéguen Y, Hoteit N (2007) Shale dynamic
properties and anisotropy under triaxial loading: experimental
and theoretical investigations. Phys Chem Earth A/B/C
32(8–14):896–906. doi:10.1016/j.pce.2006.01.007
Schijns H, Schmitt DR, Heikkinen PJ, Kukkonen IT (2012) Seismic
anisotropy in the crystalline upper crust: observations and
modelling from the Outokumpu scientific borehole, Finland.
Geophys J Int 189(1):541–553. doi:10.1111/j.1365-246X.2012.
05358.x
Schmitt DR, Currie CA, Zhang L (2012) Crustal stress determination
from boreholes and rock cores: fundamental principles. Tectonophysics 580:1–26. doi:10.1016/j.tecto.2012.08.029
Sheriff RE (2002) Encyclopedic Dictionary of Applied Geophysics,
vol 13. Geophysical Reference Series, 4th edn. Society of
Exploration Geophysicists
Sibbit AM (1995) Quantifying porosity and estimating permeability
from well logs in fractured basement reservoirs. In: PetroVietnam, Ho Chi Minh City, Vietnam, Mar 1–3, 1995 1995. Society
of Petroleum Engineers, pp 1–8. doi:10.2118/30157-MS
Takanashi M, Nishizawa O, Kanagawa K, Yasunaga K (2001)
Laboratory measurements of elastic anisotropy parameters for
the exposed crustal rocks from the Hidaka Metamorphic Belt,
Central Hokkaido, Japan. Geophys J Int 145(1):33–47
Takemura T, Oda M (2005) Changes in crack density and wave
velocity in association with crack growth in triaxial tests of Inada
granite. J Geophys Res 110:B05401. doi:10.1029/2004JB003395
Tester JW, Anderson BJ, Batchelor AS, Blackwell DD, DiPippo R,
Drake EM, Garnish J, Livesay B, Moore MC, Nichols K, Petty S,
Toksoz MN, Veatch RWJ (2006) The future of geothermal energy:
Impact of enhanced geothermal systems (EGS) on the United
States in the 21st Century. Massachusetts Institute of Technology
Thomsen L (1986) Weak Elastic-Anisotropy. Geophysics
51(10):1954–1966. doi:10.1190/1.1442051
Vavryčuk V, Boušková A (2009) S-wave splitting from records of
local micro-earthquakes in West Bohemia/Vogtland: an indicator of complex crustal anisotropy. Stud Geophys Geod
52(4):631–650. doi:10.1007/s11200-008-0041-z
Vestrum RW (1994) Group and phase-velocity inversions for the
general anisotropic stiffness tensor. University of Calgary,
Calgary
Vestrum R, Lawton D (2010) Reflection point sideslip and smear in
imaging below dipping anisotropic media. Geophys Prospect
58(4):541–548. doi:10.1111/j.1365-2478.2009.00849.x
Vestrum RW, Lawton DC, Schmid R (1999) Imaging structures
below dipping TI media. Geophysics 64(4):1239–1246. doi:10.
1190/1.1444630
Vilhelm J, Rudajev V, Zivor R, Lokajicek T, Pros Z (2010) Influence
of crack distribution of rocks on P-wave velocity anisotropy––a
laboratory and field scale study. Geophys Prospect
58(6):1099–1110. doi:10.1111/j.1365-2478.2010.00875.x
Author's personal copy
Elastic Anisotropy of a Metamorphic Rock Sample
Walsh JB (1965) The effects of cracks on the compressibility of rock.
J Geophys Res 70:381–389
Walsh NJ (2013) Geochemistry and geochronology of the Precambrian basement domains in the vicinity of Fort McMurray,
Alberta: a geothermal perspective. University of Alberta,
Edmonton
Wang J, Zhao DP (2009) P-wave anisotropic tomography of the crust
and upper mantle under Hokkaido, Japan. Tectonophysics
469(1–4):137–149. doi:10.1016/j.tecto.2009.02.005
1385
Wong RCK, Schmitt DR, Collis D, Gautam R (2008) Inherent
transversely isotropic elastic parameters of over-consolidated
shale measured by ultrasonic waves and their comparison with
static and acoustic in situ log measurements. J Geophys Eng
5(1):103–117. doi:10.1088/1742-2132/5/1/011
123