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THE
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September 2007 Vol. 26, N
o. 9Pages 91081-1232
Fractures
September 2007, Vol. 26, No. 9
Special Section: Special Section:
Fractures
THE SOCIETY OF EXPLORATION GEOPHYSICISTSThe international
society of applied geophysicsISSN 1070-485X
2609_c1_cover1.qxd 10/9/07 10:56 AM Page 1
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F ractures in rock cause seismicanisotropy because they are
orientedmechanical discontinuities that mayoccur in sets with
preferred orienta-tions. However, anisotropy from ori-entation can
be altered or masked bygradients in stress and fluids.
Suchgradients in stress occur naturally inthe Earth or are induced
throughanthropogenic activities. For instance,stress increases with
depth in the Earthbecause of lithostatic forces, and in agiven
region, local gradients in hori-zontal stresses can arise through
thetectonic history of the region (e.g.,folding, faulting, etc.).
In addition, gradients in fluid dis-tributions as well as pore
pressures occur when fluids areeither injected or withdrawn from a
rock formation.
Gradients in stress are particularly significant for frac-tures
because fractures are weakly coupled through thepoints of contact
between the two fracture surfaces. Thismeans that small physical
modifications of the fracture cancause large changes in the
mechanical, seismic, and hydraulicresponse of a fracture system.
For instance, even lowamounts of fluid saturation at the level of
only 10% can dou-ble seismic transmission across a fracture. This
occursbecause the fluid provides a transmission medium
aroundexisting asperities, couple asperities, or it couples
asperitiesacross the fracture aperture. For such a weak-coupling
sys-tem, the fracture geometry plays an essential role. The
geom-etry defines not only the seismic properties of the
fracture,but also the fluid flow through the fracture, which can
behighly sensitive to stress. For example, the flow pathsthrough
the fracture are composed of the apertures and spa-tial
distribution of regions of contacts that are sensitive tochanges in
stress. A factor of two decrease in aperture froman increase in
stress on a fracture can lead to an order ofmagnitude decrease in
the volumetric flow rate.
To understand these stress-induced changes on the frac-ture
geometry, a useful quantity is fracture-specific stiff-ness.
Stiffness relates the deformation of a fracture (closingapertures
and increasing asperity contacts) to the stress.Stiff fractures
displace less, while compliant fractures dis-place more.
Theoretically, a fracture can be represented bya displacement
discontinuity, i.e., stresses across the frac-ture are continuous
but the displacements are not. The dis-continuity in displacement
is inversely proportional to thestiffness of the fracture. From
experimental and numericalstudies it has been shown that there is a
general correlationof fracture geometry and fracture stiffness:
compliant frac-tures tend to have larger apertures and fewer or
smallerregions of contact than fractures that are stiffer. It is
impor-tant to keep in mind that this correlation is statistical
andnot deterministic. Specific counter-examples certainly existthat
show the opposite trend, but these special cases do notalter the
average connection between fracture geometry andfracture
stiffness.
The displacement-discontinuity theory has been used toderive
plane-wave transmission and reflection coefficientsfor fractured
media, group time delays, compressional-modeand Rayleigh-mode
interface waves, guided Love waves,and dispersion relationships.
Many of these studies haveassumed that fracture stiffness is
uniform along a fractureand uniform among fractures within a set.
But, as describedearlier, because stress gradients commonly exist
or areinduced in the Earth, one must consider what effects
stressgradients have on seismic wave propagation in fracturedmedia.
Gradients in stress cause gradients in fracture-spe-cific stiffness
and ultimately affect any interpretation of seis-mic
anisotropy.
If fracture-specific stiffness varies along a fracture, thetime
delay of a seismic wave also varies along the fractureplane. For
example, consider a fracture with a linearly vary-ing
fracture-specific stiffness (Figure 1a) caused by a gradi-ent in
stress on the fracture. A plane wave propagatingupward from below
is refracted because the time-delay isa function of
fracture-specific stiffness. A low fracture stiff-ness produces a
larger time delay than a region of the frac-ture with high
stiffness. While this refraction certainlyviolates Snell’s law, it
does not violate Fermat’s principle ofthe path of least time.
An interesting example of the effect of stress gradienton
seismic wave propagation across a fracture was studiedby Oliger et
al. (2003). They showed that a single plane frac-ture with an
axially symmetric stress distribution (radial gra-dient in stress)
behaves as a seismic lens that focuses seismicenergy to a beam
‘‘waist’’ at a focal plane. A schematic ofthis concept is shown in
Figure 1b. The fracture in Figure1b has a stiffness that increases
quadratically away from thecenter. The low stiffness in the center
of the fracture delaysthe waves relative to the stiffness farther
out along the frac-ture plane. For a wave propagating upwards from
below,the wave is focused by the fracture. The location of
thereceiver in front of, at, or behind the focal plane
determineswhether a converging, planar, or diverging wave front
isobserved. Their work demonstrated that a 2D planar frac-ture with
a stress gradient (contrasted with three-dimen-sional geologic
structures such as basins and domes) canfocus seismic waves.
Focusing of seismic waves by fractures
Fracture anisotropy: The role of fracture-stiffness
gradientsLAURA J. PYRAK-NOLTE, Purdue University, West Lafayette,
Indiana, USA
1124 THE LEADING EDGE SEPTEMBER 2007
Figure 1. A fracture with (a) a linearly varying
fracture-specific stiffness (high on the left to lowon the right)
and (b) a radially varying fracture-specific stiffness (low in the
center and increasesquadratically radially outward).
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should be considered in the interpretation of seismic datafrom
fractured strata with heterogeneous stress distributions.
Another example of the effect of stress gradients on
theinterpretation of seismic data from a set of parallel
fracturescan be found in Hildyard and Young (2002). They
showedthrough a comparison of data and numerical modeling
thatknowledge of the state of stress in fractured systems is keyfor
accurate interpretation of seismic data. In an attempt toreproduce,
numerically, wave attenuation for shear andcompressional waves
propagated parallel or perpendicularto a set of parallel fractures,
they found that stress distrib-
ution was of critical importance. Based on an analysis of
thestress distribution in the experiment, the
fracture-specificstiffness could not be assumed to be either the
same amongthe fractures in the set, nor uniform along any single
frac-ture within the set (Figure 2). In their analysis, the stress
var-ied from 9 MPa to 30 MPa across the fracture set, and alongthe
central fracture the stress varied from 3 MPa near theedges to 9
MPa in the center. By using a stress-dependentfracture-specific
stiffness, they were able to match the lab-oratory data.
While stress gradients can be eliminated or created inthe
laboratory through experimental design, stress distrib-utions in
the subsurface are set by lithostatic pressures,changes in pore
pressure, as well as the local tectonic set-ting. Because wave
attenuation and signal delay stronglydepend on the stiffness of the
fracture, gradients in stresscertainly have significant
consequences for the seismic inter-pretation of fracture properties
in fractured reservoirs wherethe local stress field is perturbed
during production.
Another strong fracture anisotropy arises when seismicwaves are
propagated parallel to fracture sets. Even whenthe fracture spacing
is smaller than a wavelength, strongwaveguiding can occur when the
fracture-specific stiffnessvaries along the set. Figure 3 shows a
set of parallel frac-tures with a fracture spacing much smaller
than a wave-length where the fracture stiffness is high in the
centralfractures but decreases with distance from the center.
Thisspatial variation in fracture-specific stiffness can
producewaveguiding (as shown in Figure 3). Because of the
gradi-ents in stiffness, wave transmission across the
fracturesdecreases and the waves become internally reflected
withina group of fractures producing a wave guide. If the gradi-ent
in stiffness were similar to what is known in optics as aGRIN
(graded index), the wave could snake though the lay-ers.
SEPTEMBER 2007 THE LEADING EDGE 1125
Figure 2. The stress distribution (for stress component normal
to thefracture plane) in a set of parallel fractures under bi-axial
loading(after Hildyard and Young, 2002).
Figure 3. A gradient in fracture stiffness (red=high stiffness
decreasingto black=low fracture stiffness) among fractures in a set
where thefracture spacing is much smaller than a wavelength.
Figure 4. Images of the acoustic wavefront propagated through
(a) anintact sample F0, (b) fracture sample F3 in the dry
condition, and (c)fracture sample F3 in the saturated condition.
The fractures in theimages for sample F3 are oriented parallel to
the time axis and arespaced 3 mm apart (i.e., approximately 20
fractures occur between 0and 60 mm). All of the wavefront images
have a common time axis.The color scale on the right of each image
represents the amplitude involts. Note that the color scale is
different for images (a), (b), and (c).
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In a fracture system, energy confinement that spans sev-eral
fractures is possible either with a stress-induced distri-bution of
fracture-specific stiffness as shown in Figure 3 orwith an uneven
fluid saturation of fractures within a sys-tem of fractures. Water
saturation increases both the nor-mal and shear fracture-specific
stiffnesses. As describedearlier, a fracture can be viewed as being
weakly coupledthrough the points of contact between the two
surfaces. Theaddition of a fluid to this weakly coupled system
signifi-cantly enhances the fracture stiffness. For normal
stiffness,the coupling is enhanced through the bulk modulus of
thefluid filling the fracture (e.g., water relative to air). For
shearstiffness, the amount of enhancement is a function of
thegeometry of the fracture. For example, if the fracture sur-faces
are rough and interlocking, the void shapes mightenhance shear-wave
transmission (i.e., like a wet clutch). Forsmooth parallel
surfaces, little enhancement in shear-wavetransmission would be
expected because shear waves decayin fluids. Thus, if a set of
fractures were not uniformly sat-urated, a variation of both normal
and shear fracture-spe-cific stiffness can occur.
An example of waveguiding induced in a set of paral-lel
fractures caused by uneven saturation within the set offractures is
given by Xian et al. (2001). They used wavefrontimaging on aluminum
samples with multiple parallel syn-thetic fractures to examine the
effect of stress and satura-
tion on wave propagation in amedium containing a set of
par-allel fractures. Aluminum sam-ples were used, so all
effectsobserved are the result of thefactures and not of the
back-ground medium. Figure 4shows the acoustic wavefrontrecorded
for an intact referencesample, F0, and a fracture sam-ple, F3. The
fracture sample wascomposed of 28 fractures with afracture spacing
of 3 mm, a spac-ing that was roughly equal to aquarter of a
wavelength at a fre-quency of 0.5 MHz. The intactsample exhibited a
uniformwavefront (Figure 4a) becauseof the isotropy of the
aluminum.On the other hand, for the dryfracture sample in Figure
4b, thefirst-arriving wavefront had asignificantly smaller
amplitudethan the intact sample, and thewaveform was mostly
confinedto within the central region span-ning several fractures.
There isa systematic trend in confine-ment as a function of
frequency:the high-frequency componentsof the signal were the
moststrongly confined, while thedominant low-frequency energywas
more weakly confined andhad an earlier first arrival thanthe
high-frequency componentsof the signal.
An anomalous behavioroccurred when the fractureswere exposed to
water with the
purpose of saturating the fractures. Because of trapped airin
the fractures, the water saturation was incomplete.Furthermore,
there were gradients in the fracture saturationacross the set,
which enhanced wave confinement rather thansuppressing it. In the
saturated condition (Figure 4c), thewavefront for the fracture
sample was more strongly con-fined than the wavefront in the dry
condition and was muchlarger in amplitude. This apparently
anomalous behaviorarose because the dominant low-frequency energy
was con-fined by two fractures distant from the center of the
sam-ple, approximately at locations of 20 mm and 40 mm. Thecentral
fractures exhibited a larger value of stiffness thanthe fractures
farther from the center, which confined thewave. These data
illustrate that stiffness gradients in a setof parallel fractures
can induce waveguiding even when thefracture spacing is much
smaller than a wavelength.Interpretation of seismic data from
fractured media wouldrequire both a knowledge of the local stress
distribution aswell as possible gradients in saturation.
Certainly, the opposite effect is more usually the case, inwhich
saturation with water would be expected to decreaseconfinement
anisotropy. As an example of this, Xian (2001)observed that a
combination of stress and saturation couldmask the presence of
fractures. He used an aluminum frac-ture sample that had seven
fractures with a fracture spacingequal to a wavelength (12 mm for a
frequency of 0.5 MHz).
1126 THE LEADING EDGE SEPTEMBER 2007
Figure 5. Acoustic wave front images are shown for fracture
sample F12: (a) dry at 1.4 MPa, (b) dry at7.0 MPa, (c)
water-saturated at 1.4 MPa, and (d) water-saturated 7.0 MPa. The
images on the left showthe spatial distribution of energy for each
condition taken at 63.5 microseconds. The images on the
rightrepresent a 20-microsecond window of the wavefront as a
function of horizontal position (taken at avertical position of 30
mm in the images on the left). The color scale associated with each
image repre-sents the amplitude in volts. Please note that the
color scale is different for each image.
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In Figure 5 (left), the fractures were oriented parallel to
thevertical axis and were located at horizontal positions of
approx-imately 6 mm, 18 mm, 30 mm, 42 mm, and 54 mm, and sim-ilarly
for the right side of the figure.
The wavefronts for the dry fracture sample comparedat low and
high stresses (Figures 5a and b) show that increas-ing pressure on
the sample decreased the confinementanisotropy. At low stress, the
acoustic wavefront in the frac-ture sample was strongly confined to
the central layer. Forthis sample, the source was located at about
26 mm on thehorizontal axis and about 30 mm on the vertical axis.
Theacoustic wavefront was delayed by the fractures and spreadout
faster within the central confining layer than across thefractures.
Energy confinement within the central layer wasalso observed in the
image of the propagating wavefront(Figure 5a on the right). At this
low stress, the low fracture-specific stiffness prevents any
significant energy transmis-sion across the fractures. The arrival
time of the compressionalmode wavefront is approximately the same
as for the intactsample (Figure 4a) while the wavefronts outside
the centralwaveguide are delayed and attenuated by each fracture
thatthe wavefront crosses.
On the other hand, at high stress (Figure 5b), theincreased
fracture-specific stiffness allows greater trans-mission of energy
across the fractures, less delay of thewavefront, and less energy
confinement in the central layerbetween fractures. The nonsymmetry
in the amplitude inthe image of the acoustic wavefront (Figures 5a
and b) alsoindicates that the fracture-specific stiffness for each
fracture,and even for different locations on the same fracture,
werenot equal. For example, Figure 5b shows that the fractureat a
horizontal position of 30 mm had a different stiffnessthan the
fracture at a horizontal position of 18 mm basedon the asymmetry of
the wavefront. Thus, symmetric posi-tions in the wavefront relative
to the central layer do nothave the same energy.
Homogeneous saturation of the fractures with water, com-bined
with high stress, removes the strong effect of fractureson the
acoustic wavefront and makes the fractures nearlyinvisible. By
comparing Figures 5a and 5c (or Figures 5b and5d) for the same
confining pressure, more energy is propa-gated across the saturated
fractures than across the dry frac-tures. Increasing the stiffness
of a fracture further increasesthe transmission of energy across
the fracture and reduces theamount of energy that is internally
reflected into a guidedmode. At a confining pressure of 7.0 MPa for
the saturatedcondition (Figure 5d), the first-arriving wavefront is
nearly uni-form and the presence of fractures is only observed in
the laterarrivals. The wavefront exhibited a slight ellipticity
(Figure5d on the left), although the high stiffnesses of the
fracturesenabled the wavefront to spread out almost as if there
wereno fractures. The energy distribution was observed to beroughly
symmetric and only a slight delay in the wavefrontoccurred as it
propagated across the fractures. This demon-strates that seismic
anisotropy caused by parallel sets of frac-tures can be masked by
saturating the fractures with a liquid.Hence, application of both
stress and fluid saturation (homo-geneously) essentially erases the
effects of the fractures on apropagating wavefront.
In summary, I have shown that the ability to interpretfracture
properties from seismic data is linked to spatial vari-ations in
fracture-specific stiffness. Several examples of theeffect of
gradients in fracture-specific stiffness on seismicwave propagation
were presented to convey a sense that gra-dients in
fracture-specific stiffness play an important rolein seismic
anisotropy caused by fractures. Fracture-specificstiffness is
intimately linked to the many different length
scales associated with fracture geometry (apertures,
spatialcorrelations, contact area, etc.) and to how these
lengthscales are altered through physical processes. The
lengthscales associated with the examples given in this paper
aremuch smaller than those encountered in the field. However,an
understanding of length scales associated with fracturesand
fracture sets relative to seismic length scales is relevantbecause
it determines when a fractured medium can betreated as an effective
medium, or when a discrete fractureapproach is necessary. In
addition, it is important to recog-nize that physical processes
(such as stress, fluids, etc.) caneither homogenize a fracture
stiffness, or it can induce greatercomplexity within a fracture or
among fractures within aset. Therefore, a system that initially may
be represented asan effective medium may fail to be so as
alterations to a frac-tured reservoir proceed, and vice versa. A
fundamentalunderstanding of how gradients in fracture-specific
stiffnessalter the seismic response of a fractured medium will
helpimprove the interpretation of seismic data and to
recognizedeviations from homogeneity.
Suggested reading. “Modeling seismic waves around under-ground
openings in fractured rock” by Hildyard and Young(Pure and Applied
Geophysics, 2002). “Focusing of seismic wavesby a single fracture”
by Oliger et al. (Geophysical Research Letters,2003). “Transmission
of seismic waves across single naturalfractures” by Pyrak-Nolte et
al. (Journal of Geophysical Research,1990). “Single fractures under
normal stress: The relationbetween fracture-specific stiffness and
fluid flow” by Pyrak-Nolte and Morris (International Journal of
Rock Mechanics andMining Sciences, 2000). “Compressional waves
guided betweenparallel fractures” by Xian et al. (International
Journal of RockMechanics and Mining Sciences, 2001). “Wavefront
imaging ofenergy confinement by multiple parallel fractures” by
Xian(MS thesis, Purdue University, 2001). TLE
Acknowledgments: The author acknowledges support of this work by
theGeosciences Research Program, Office of Basic Energy Sciences,
U.S.Department of Energy (DE-FG02-97ER14785 08).
Corresponding author: [email protected]
SEPTEMBER 2007 THE LEADING EDGE 1127
