Diversity of channeling flow in heterogeneous
aperture distribution inferred from integrated
experimental‐numerical analysis on flow
through shear fracture in granite
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Watanabe Noriaki, Hirano Nobuo, Tsuchiya
Noriyoshi
Journal of Geophysical Research
114
B04208
2009
http://hdl.handle.net/10097/52105
doi: 10.1029/2008JB005959
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, B04208, doi:10.1029/2008JB005959, 2009
Diversity of channeling flow in heterogeneous aperture distribution
inferred from integrated experimental-numerical analysis on flow
through shear fracture in granite
Noriaki Watanabe,1 Nobuo Hirano,1 and Noriyoshi Tsuchiya1
Received 28 July 2008; revised 29 December 2008; accepted 9 March 2009; published 28 April 2009.
[1] Shear (Mode II) fractures with shear displacements of 1 and 5 mm were generated by
direct shear on granite under normal stresses of 1, 20, and 60 MPa. Fracture surface
mapping showed that the surface roughnesses of the shear fractures decreased with
increasing shear displacement and normal stress and were smaller than those of tensile
fractures reported in our previous study. Fluid flow experiments on the shear fractures
provided fracture permeabilities at a wide range of confining pressures of 10–100 MPa.
Nonmonotonic permeability was usually observed to decrease with increasing confining
pressure. However, the permeability changes were different between the shear fractures
generated at the normal stress of 20 and 60 MPa. In addition, obvious permeability
changes with shear displacement were observed for 60 MPa, whereas no significant
difference was observed for 20 MPa. Comparing the shear fractures with the tensile
fractures having shear displacements revealed clear differences, even for equivalent shear
displacements. Numerical models that were constructed using the data of the fracture
surface mapping by matching their permeabilities with the experimentally evaluated
fracture permeabilities revealed the development of preferential flow paths, i.e.,
channeling flows, for the shear fractures, providing a diversity of channeling flow in
heterogeneous aperture distributions of rock fractures in the Earth’s crust.
Citation: Watanabe, N., N. Hirano, and N. Tsuchiya (2009), Diversity of channeling flow in heterogeneous aperture distribution
inferred from integrated experimental-numerical analysis on flow through shear fracture in granite, J. Geophys. Res., 114, B04208,
doi:10.1029/2008JB005959.
1. Introduction
[2] Fluid flows through rock fractures in the Earth’s crust
have been a subject of interest for some time because rock
fractures usually have much greater permeability than the
rock matrix and are therefore recognized as the predominant
pathways of resources or hazardous materials, e.g., groundwater, geothermal fluids, hydrocarbons, and radioactive
wastes. The fluid flow and transport properties of rock
fractures have been investigated with respect to the geological disposal of high-level radioactive waste. As a result, our
understanding of the subsurface flow system has been
greatly improved and has been applied to the development
of geothermal and oil/gas fractured reservoirs. Recently, the
prediction of flow and transport phenomena through rock
fractures based on precise modeling of the flow system in a
fractured rock mass with natural heterogeneities has become
increasingly important because recent environmental and
energy problems require urgent solutions using underground
space based on the safe and effective development of
reservoirs. A modeling with a natural heterogeneity of a
1
Graduate School of Environmental Studies, Tohoku University,
Sendai, Miyagi, Japan.
Copyright 2009 by the American Geophysical Union.
0148-0227/09/2008JB005959$09.00
fracture network has been established by the Discrete
Fracture Network (DFN) modeling technique [Min et al.,
2004]. In DFN modeling, rock fractures are described by
smooth parallel plates, although, in nature, rock fractures have
heterogeneous aperture distributions due to rough surfaces.
Therefore simple description requires an ideal fracture aperture (hydraulic aperture), rather than a physical fracture
aperture obtained by field observations, to predict fracture
permeabilities. However, detailed analyses have suggested
that DFN modeling requires further improvement of the
heterogeneous aperture distributions and the resulting flow
paths within them. This requires a good understanding of the
relationships among the aperture distributions, flow paths,
hydraulic apertures, and geological conditions, which include
a wide range of confining pressures at depths of up to several
thousand meters. In addition to engineering applications, such
detailed analyses appear to be essential to improve our
understanding of some important Earth processes, including
those of diagenesis, strength gain on faults, evolving pressure
solution, development of reservoir seals, and evolution of
flow pathways [Durham et al., 2001; Morrow et al., 2001;
Yasuhara and Elsworth, 2004].
[3] Rock fractures are characterized as having a threedimensional distribution of local apertures formed by rough
surfaces that, depending on the confining pressure, are in
partial contact with each other with, presumably, preferen-
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tial flow paths (i.e., the occurrence of channeling flow
[Brown, 1987; Pyrak-Nolte et al., 1988; Tsang and Tsang,
1989; Durham, 1997; Brown et al., 1998; Bruderer-Weng
et al., 2004]). In general, fracture permeability changes
not only with confining pressure, but also with shear
displacement [Raven and Gale, 1985; Esaki et al., 1999;
Chen et al., 2000; Gutierrez et al., 2000; Plouraboué et al.,
2000; Olsson and Barton, 2001]. The changes indicate
that the aperture distributions and the resulting flow paths
are affected by both confining pressure and shear displacement. Fluid flows through numerically generated aperture
distributions have therefore been studied extensively and
channeling flows have been observed [Unger and Mase,
1993; Yang et al., 1993; Pyrak-Nolte and Morris, 2000;
Lespinasse and Sausse, 2000; Sausse, 2002; Koyama et al.,
2006; Matsuki et al., 2006]. In numerical studies, the
relationships between numerical observations and geological conditions, particularly with respect to confining pressures, have not been accurately clarified due to difficulties
associated with demonstrating model robustness through
experimental observations [Cheng et al., 2004] for a rock
fracture under a confining pressure. This failure to incorporate experimental observations into numerical techniques
means that numerical observations, such as observations of
channeling flow, can only rarely be applied to rock fractures
under actual geological conditions. On the other hand,
casting techniques [Hakami and Larsson, 1996; Kostakis
et al., 2003; Konzuk and Kueper, 2004; Yasuhara et al.,
2006], X-ray CT techniques [Bertels et al., 2001; Polak et
al., 2004], and the coupled casting X-ray CT technique
[Pyrak-Nolte et al., 1997; Montemagno and Pyrak-Nolte,
1999] have been used to characterize the aperture distributions of rock fractures experimentally. However, applying
these techniques to the determination of the aperture distributions under a varying confining pressure or a wide
range confining pressures is technically difficult [Gertsch,
1995; Fredrich et al., 1995; Montoto et al., 1995].
[4] Recently, we analyzed water flows within aperture
distributions of the granite tensile fractures for a range of
shear displacements of up to 10 mm at confining pressures
of 10 – 100 MPa by numerical modeling in fluid flow
experiments [Watanabe et al., 2008]. In a previous study,
we showed the development of preferential flow paths in
aperture distributions, which suggest that the concept of
channeling flow is applicable for rock fractures under a
wide range of confining pressures and that three-dimensional preferential flow paths can exist in a subsurface fracture
network, as indicated by experimental [Carmeliet et al.,
2004; Johnson et al., 2006] and field studies [Cacas et al.,
1990; Abelin et al., 1991; Dverstorp et al., 1992; Rowland
et al., 2008]. In order to understand the preferential flow
path and its significance to the prediction of subsurface flow
and transport phenomena, channeling flow should be carefully investigated as a function of geological condition
(shear displacement, confining pressure, etc.) because significant differences in flow paths can arise due to heterogeneities that exist within the aperture distributions.
However, no detailed analysis on fluid flows through shear
(Mode II or III) fractures can be found in previous studies,
despite the fact that shear fractures are more likely to be
generated under the subsurface compressive stress fields in
nature [Rao et al., 2003] and can be generated even by
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artificial hydraulic fracture [Lockner and Byerlee, 1977;
Cornet et al., 2007]. This is based on the hypothesis that a
shear fracture can be replaced by a tensile fracture with a
shear displacement because generating shear fractures is
much more challenging than generating tensile fractures.
Here there is an open question as to whether tensile
fractures with shear displacements are actually equivalent
to shear fractures.
[5] The objective of the present study is to provide a
fundamental understanding of fluid flow through shear
fractures in granite by using an integrated experimental and
numerical analysis to examine differences between shear
fractures and tensile fractures with shear displacements.
Since shear fractures can be generated under various stress
conditions in nature, shear fractures were generated using
direct shear technique on granite under various stress conditions. The roughnesses of the fracture surfaces were measured using Laser scanning equipment. A fluid flow
experiment at confining pressures of up to 100 MPa was
performed. In addition, the aperture distributions and flow
paths were analyzed using numerical models constructed
from the surface mapping data of a fracture with the experimentally evaluated fracture permeability. The experimental
and numerical results were compared with our previous
experimental and numerical results for the granite tensile
fractures with shear displacements [Watanabe et al., 2008].
2. Methods
2.1. Sample Preparation
[6] Direct shear experiments were performed on granite
cores (diameter: 95 mm, length: 200 mm) in order to create
granite samples that contained a single shear (Mode II)
fracture with final dimensions of 95 mm 150 mm. The
direct shear was performed using the Multipurpose Testing
machine for Rock Mass (MTRM) at the Obayashi Corporation Technical Research Institute, Tokyo, Japan. The test
machine was used to subject a fracture to different shear
displacements (1 and 5 mm) along prescribed shear planes
in the cores for various normal stresses (1, 20, and 60 MPa).
In principle, two samples were prepared with respect to each
condition (combination of shear displacement and normal
stress) for the fracture surface mapping (section 2.2) and for
the fluid flow experiment (section 2.3). This was because
debris associated with the fracturing made it impossible to
return a fracture to its original state once the fracture was
opened for surface mapping.
[7] A granite core obtained from Inada medium-grained
granite (Ibaraki, Japan) was placed in the MTRM with a jig
and roller bearings (Figure 1). The jig was used to constrain
the shear plane in the core, and the roller bearings were used
to reduce the friction between the jig and the loading platens.
Vertical and horizontal loads of 5 kN were used to hold the
components. A displacement transducer was attached to the
jig to monitor the shear displacement along the shear plane.
The horizontal load was increased so that the prescribed
normal stress was applied to the shear plane. The vertical
displacement was applied to the jig at a rate of 0.2 mm/min
while maintaining the normal stress and was stopped when
the core was fractured and the prescribed shear displacement
was achieved. The fractured core was retrieved from the
MTRM carefully so that the shear displacement did not
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Figure 1. Schematic diagram of the direct shear on a granite core using the Multipurpose Testing
machine for Rock Mass (MTRM) at the Obayashi Corporation Technical Research Institute, Tokyo, Japan.
change. The sample that contained the single fracture generated along the shear plane was prepared by cutting both
ends of the core to a prescribed length.
2.2. Fracture Surface Mapping
[8] One of the two samples prepared for a given condition
was used in the fracture surface mapping for data acquisition
and analysis of fracture surface geometry. For convenience, the
samples were separated into upper and lower fracture surfaces.
In this procedure, great care was taken not to disturb the
original surface geometry, where the debris associated with the
fracturing was left on the surfaces because debris was present
on the samples in the fluid flow experiment. The asperity
heights zx,y were measured for the upper and lower fracture
surfaces in a 250-mm square grid system using laser scanning
equipment having a resolution of 10 mm along the z axis
and a positioning accuracy of 20 mm in the x-y plane,
providing 380 600 data points for each 95 mm 150 mm
fracture surface. The data was used as input data for the
numerical model described in section 2.4 and for roughness
analyses to evaluate the influence of the condition of fracture
generation on the fracture surface roughness.
[9] The fractal nature is usually observed for surfaces of
natural rock fractures [Power and Durham, 1997; Tsuchiya
et al., 1994; Brown, 1995; Tsuchiya and Nakatsuka, 1995,
1996; Lee and Bruhn, 1996]. In the present study, the fractal
nature of the artificial fractures was examined using the
spectral method [Power and Durham, 1997], where the
relationship between the logarithms of power spectral
density and spatial frequency was linear for a fractal
surface, and the slope of the line provides the fractal
dimension. In addition, the roughness coefficient (Z2) and
the tortuosity (t) were evaluated for the surfaces [Sausse,
2002]:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
N 2 u 1 X
ziþ1 zi 2
;
Z2 ¼ t
Dx
N 1 i¼0
t¼
ffi
XN 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
ð
Þ
ð
Þ
z
z
þ
Dx
iþ1
i
i¼0
L
ð1Þ
;
ð2Þ
where zi is the asperity height, Dx is the distance between
adjacent data points, L is the direct linear length, and N is the
number of the data points, with respect to a two-dimensional
(x-z or y-z) surface profile. The roughness coefficient is
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Figure 2. Schematic diagram of the fluid flow experiment on a fractured granite sample for the
permeability measurement using the experimental system developed using the Rubber-Confining
Pressure Vessel (R-CPV) [Hirano et al., 2005].
defined as the root mean square of local slopes, which
provides a measure of the microscopic (grid scale) roughness. The tortuosity is defined as the ratio of the length of the
profile line to the direct linear length, which provides a
measure of macroscopic (sample scale) roughness.
2.3. Fluid Flow Experiments
[10] The second of the two samples prepared for a given
condition was used in the fluid flow experiment to evaluate
the permeability of the generated fracture for a range of
confining pressures. The experiment was performed using
the Rubber-Confining Pressure Vessel (R-CPV) experimental system [Hirano et al., 2005]. The permeability was
calculated based on the relationships between the flow rates
through the fracture and differential pressures between the
ends of the fractures at confining pressures ranging from
10 to 100 MPa (equations (3) and (4) in the next section).
The experimentally obtained permeability was also used in
the numerical model (described in the next section) to
determine fluid flow through the measured aperture distribution of a fracture for a range of confining pressures. A
comparison of the measured permeability to the numerically
determined permeability was required.
[11] A sample with a stainless steel jacket (thickness:
2 mm) was placed in the R-CPV (Figure 2). The jacket was
used to prevent the sample from failure in the R-CPV
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because preliminary tests revealed that samples prepared by
direct shear usually failed, even at relative low confining
pressures of from 10 to 20 MPa. The sample failure is
hypothesized to originate in microscopic cracks generated
in the rock matrix. The jacket is attached to the rock with
silicone sealant. Note that the jacket had slits filled with
Neoprene1 strips (width: 2 mm, thickness: 2 mm) to ensure
that the jacket did not affect the fracture normal displacement without short-circuiting the flow. The R-CPV with the
sample was placed in the load machine (maximum load:
250 kN). The prescribed confining pressures (normal stress)
were applied to the sample (fracture) by compressing the
Neoprene1 sleeve (thickness: 5 mm) of the sample as per
Pascal’s principle [Bloomfield, 2006]. Distilled water (at
room temperature) was pumped into the R-CPV at a
constant flow rate (maximum flow rate: 0.3 cm3/s). The
fluid flowed from the bottom to the top of the sample and
out from the R-CPV at ambient pressure. Small pressure
gradients were applied so that pore pressures were kept
smaller than the applied confining pressure by at least one
order of magnitude. Therefore in the present study, the term
‘‘confining pressure’’ refers to the effective confining pressure. The flow rate through the fracture was measured by
weighing the effluent using an electronic balance. The
differential pressure between both ends of the fracture was
measured using a differential pressure gauge.
[12] A fluid flow experiment on tensile fracture of granite
performed using the same experimental system revealed that
the relationships between the flow rates and the various
pressures were linear at the confining pressures of 10–
100 MPa [Watanabe et al., 2008], indicating that the
permeability of the fractures could be calculated based on
the cubic law assumption. Therefore the permeability (k) of
the fractures generated in the present study will be evaluated
using the following equation when the hypothesis is confirmed [Witherspoon et al., 1980; Tsang and Witherspoon,
1981; Zimmerman and Bodvarsson, 1996]:
k¼
e2h
;
12
ð3Þ
where eh is the hydraulic aperture, which can be calculated
from the following equation:
eh ¼
1
12mLQ =3
;
W DP
ð4Þ
where Q is the flow rate, DP is the differential pressure, m is
the viscosity of water, and L and W are, respectively, the
apparent lengths of the fracture in the directions parallel and
perpendicular to the macroscopic flow direction.
2.4. Numerical Modeling
[13] The data obtained from the surface mapping and the
fluid flow experiments were used in the numerical simulation of the water flow through the aperture distribution of
the fracture in the samples. The numerical modeling technique used in the present study is based on the work of
Watanabe et al. [2008]. This technique determines aperture
distributions and water flows within the structures at given
confining pressures by matching the permeability of the
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numerical model constructed from the surface mapping data
of a fracture with the experimentally evaluated fracture
permeability for a given confining pressure of the same
fracture or an equivalent fracture generated under the same
condition.
[14] The aperture distribution was modeled by the twodimensional distribution of local apertures that was calculated using the data from the surface mapping (250-mm
square grid system), with local apertures represented by
vertical separations between opposite fracture surfaces. In
the determination of aperture distributions and resulting
water flows, a model with at least a single contact point
was initially constructed. The model was then modified to
simulate the normal deformation of a rock fracture to match
the permeability of the model with the fracture permeability
from the fluid flow experiment. In the simulation, all local
apertures were reduced uniformly, and overlapped asperities
were eliminated (set to be zero local apertures), precluding
the formation of local apertures [Brown, 1987; Power and
Durham, 1997; van Genabeek and Rothman, 1999; Matsuki
et al., 2006].
[15] The fluid flow through an aperture distribution was
simulated using the Reynolds equation for a steady state
laminar flow in a two-dimensional (x-y) field of a viscous fluid
(water) [Brown, 1987; Mourzenko et al., 1995; Zimmerman
and Bodvarsson, 1996; Ge, 1997; Oron and Berkowitz, 1998;
Yeo et al., 1998; Lespinasse and Sausse, 2000; Pyrak-Nolte
and Morris, 2000; Sausse, 2002; Brush and Thomson, 2003;
Konzuk and Kueper, 2004]:
3
3
@
e @P
@
e @P
þ
¼ 0;
@x 12m @x
@y 12m @y
ð5Þ
where e is the local aperture, m is the water viscosity, and P is
the water pressure. Since the viscosity was assumed to be
constant, this parameter can be removed from equation (5).
Using the Deterministic/Stochastic Crack network modeler
(D/SC) [Tezuka and Watanabe, 2000], linear equations derived
from a finite difference form of equation (5) were solved under
boundary conditions equivalent to those in the fluid flow
experiments (Figure 3). Once the flow field is calculated, the
volumetric flow rate Q in equation (4) can be obtained.
Therefore the permeability of the aperture distribution can be
calculated using equations (3) and (4). Note that the zero local
apertures were replaced with significantly small nonzero local
apertures of 1 mm in the D/SC. This was done primarily
because the pressures at the zero local apertures cannot be
defined, but also zero permeability should be invalid, even for
the contacting asperities. The permeability of the contacting
asperities should not be much smaller than that of the rock
matrix (1019 –1018 m2). However, the permeability of the
contacting asperities may be much smaller than that of
noncontacting points. The local aperture of 1 mm, which
corresponds to a local permeability of 8.3 1014 m2, was
selected for contacting asperities because this generally
allowed results that were not significantly different compared
to those obtained by a much smaller local aperture of 1 103 mm, which corresponds to a local permeability of 8.3 1020 m2, to be obtained in reasonable calculation times of
less than a day. The resolution of the surface mapping in the
present study provides a relatively large number of calculation
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Figure 3. Boundary conditions in the fluid flow simulation for a model of an aperture distribution of
a rock fracture.
points. Therefore a relatively long calculation time is required
for convergence in an iterative solution of the D/SC. A model
with a wider range of local apertures, associated with smaller
local apertures at contacting asperities, generally requires a
longer calculation time. Comparing several results by
substituting contacting asperities with local apertures of 1 and
1 103 mm, the differences in hydraulic apertures were less
than 5%, depending on the percentage of contacting asperities
among the total data points, which was as high as
approximately 50%, and no significant differences were
observed in flow rate distributions. In addition, the calculation
times were usually several hours when the local aperture was
1 mm, which is reasonable for permeability matching because
the matching procedure involved a number of calculations.
However, the local aperture of 1 mm precludes numerical
models with permeabilities smaller than the local permeability
of 8.3 1014 m2. The numerical models were therefore
constructed only when the experimentally evaluated permeability was greater than the local permeability.
[16] The aperture distributions determined from the
numerical modeling were evaluated by means of the geostatistical analysis of size and spatial distributions of the local
apertures (histogram and semivariogram). The obtained flow
paths were evaluated with respect to the development of
preferential flow paths (channeling flow). The area of the
preferential flow paths was calculated when channeling flow
was observed.
3. Results
3.1. Generated Fracture
[17] In the direct shear on the granite cores, single shear
(Mode II) fractures could be generated under specific
conditions. At the lowest normal stress, however, only
one core with a single fracture with a shear displacement
of 1 mm could be obtained due to the difficulty of single
fracture generation (occurrences of multiple fractures).
Therefore this sample was used in the fluid flow experiment. Figure 4a shows the core that fractured at a shear
displacement of 5 mm and a normal stress of 60 MPa. The
single fracture with the given shear displacement was
clearly observed along the core long axis, i.e., the shear
plane. The shear stress calculated for the shear plane
changed with shear displacement, as is typically observed
in direct shear on rocks or rocks containing a single fracture
without shear displacement [Bernaix, 1969; Rosso, 1976;
Barton et al., 1985]. The shear stress increased to the peak
shear stress of approximately 130 MPa with increasing
shear displacement and then decreased to the constant
residual shear stress (Figure 4b). Therefore the observed
peak shear stresses indicated the shear stress at failure.
Similar results were observed under all given conditions,
and the relationships between the shear stresses at fracture
and the given normal stresses from all of the cores was
indicated by a unique linear curve (Figure 5), indicating that
Mode II fracturing occurred along the shear plane in all of
the cores.
3.2. Surface Roughness
[18] The results of the surface roughness analyses of the
shear fractures generated in the present study and those of the
tensile fractures generated in our previous study [Watanabe et
al., 2008] are summarized in Table 1. Roughness analyses
were performed on the surface profiles with respect to the
directions parallel to and perpendicular to the shear displacements in the case of the shear fractures. The fractal nature of
the fractures generated under each condition was confirmed,
indicating that the shear fractures were an acceptable substitute for natural rock fractures. The fractal dimensions were
1.36– 1.53, with greater values for the parallel direction. The
roughness coefficients (grid-scale roughness) and the tortuosities (sample-scale roughness) tended to be smaller for
higher normal stresses, greater shear displacement, and the
parallel direction.
[19] The range of fractal dimensions of the shear fractures
was similar to that of the tensile fractures of granite in our
previous study (Table 1). The fractal dimension showed no
significant difference between the shear and tensile fractures.
However, the roughness coefficients and the tortuosities
showed clear differences between the shear and tensile
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Figure 4. Fracture generation in a granite core by direct shear under the condition of a shear
displacement of 5 mm and a normal stress of 60 MPa. (a) The single fracture generation in the shear plane
and (b) the relationship between shear stress and shear displacement measured for the shear plane.
fractures. Figure 6 shows the relationships between the
tortuosities and the roughness coefficients, where the plotted
values are the averages of the values for the two orthogonal
directions. Figure 6 shows that the roughnesses in both the
grid and sample scales of the shear fractures were out of the
range obtained for the tensile fractures. The differences in
the roughness coefficient and the tortuosity between the
tensile fractures and the shear fractures generated at a normal
stress of 20 MPa were comparable to those between the
natural fractures in sandstone and granite [Sausse, 2002].
3.3. Permeability
[20] For fluid flow experiments, the linear relationships
between the flow rates and the differential pressure were
confirmed for each given confining pressure for all of the
samples. The permeability was therefore calculated using
equations (3) and (4). In general, although the permeabilities of the shear fractures decreased with increasing confining pressure, the permeabilities at 100 MPa remained
much greater than those of the granite matrix permeability
of 1019 – 1018 m2 (Figure 7a), as observed in our previous
study for the tensile fractures with different shear displacements of 0 – 10 mm [Watanabe et al., 2008]. This showed
that it was not likely for a fracture in granite to get complete
closure associated with plastic deformation at contacting
asperities [Yoshioka and Scholz, 1989a, 1989b].
[21] Permeability changes with confining pressure or
shear displacement were remarkably different between the
shear fractures generated at the lower normal stresses of
20 MPa and the higher stress of 60 MPa (Figure 7a).
Although, for all of the generated fractures, the permeabilities decreased up to a confining pressure of 50 MPa and
Figure 5. Relationship between shear stress and normal
stress in the shear plane at failure. The solid line shows the
liner regression by the least squares method.
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Table 1. Fractal Dimensions, Roughness Coefficients, and Tortuosities of the Shear Fractures Generated in the Present Study and the
Tensile Fractures Generated in Our Previous Study [Watanabe et al., 2008]a
Condition at Generation
Fracture Type
Shear fracture
Normal
Shear
Stress (MPa) Displacement (mm)
20
60
Tensile fracture
-
1
5
1
5
-
Fractal Dimension
Roughness Coefficient
Tortuosity
Parallel (-)
Perpendicular (-)
Parallel (-)
Perpendicular (-)
Parallel (-)
Perpendicular (-)
1.50
1.51
1.51
1.53
1.40 – 1.59b
1.47
1.45
1.50
1.36
0.71
0.63
0.56
0.50
0.81 – 0.90b
0.74
0.71
0.60
0.55
1.13
1.12
1.10
1.09
1.15 – 1.18b
1.15
1.16
1.12
1.11
a
Parallel and perpendicular denote directions to the shear displacement in the analyses.
b
Omnidirection.
then became approximately constant, the additional permeability decrease at the confining pressures of 70– 100 MPa
was observed only in the case of the fractures generated at
higher normal stress. The difference between the fractures
generated at the different normal stresses was also observed
in the permeability change with the shear displacement.
A notable difference (enhancement) was observed with
increasing shear displacement only in the case of the shear
fractures generated at the higher normal stress.
[22] The results for the shear fractures were clearly
different from those for the tensile fractures with equivalent shear displacements obtained in our previous study
[Watanabe et al., 2008, Figure 7b]. For the tensile fractures,
permeabilities became systematically greater with increasing shear displacement. In addition, the permeabilities
became less stress dependent when the shear displacement
was greater than 3 mm. However, even in the case of the
shear fractures generated at the low normal stresses of 1 and
20 MPa, such results were not observed for the shear
fractures.
3.4. Aperture Distribution and Flow Path
[23] For the numerical models constructed from the surface
mapping data of fractures with the experimentally evaluated
fracture permeability, the histograms of the aperture distributions were always characterized by the number of contact
points (local apertures of 1 mm) and the skewed distributions
of noncontact points (local apertures of >1 mm) with long
tails, i.e., lognormal-like distributions, as shown in Figure 8.
The histograms were therefore evaluated by contact areas as
percentages of contact points, geometric means, and geometric standard deviations of local apertures, as well as the
histograms of the tensile fractures in our previous study
[Watanabe et al., 2008]. In addition, the aperture distributions were evaluated by semivariograms. In the case of the
aperture distributions of the shear fractures generated at the
lower normal stress of 20 MPa, the semivariograms with no
clear sills (long spatial correlation exceeding the sample
scale) did not provide the spatial correlation lengths, i.e.,
ranges (Figure 9a). On the other hand, the semivariograms
of the aperture distributions of the shear fractures generated
at the higher normal stress of 60 MPa had clear sills,
providing the spatial correlation lengths by good fitting
with the exponential model (Figure 9b), as well as the
semivariograms of the tensile fractures in our previous
study [Watanabe et al., 2008]. The results in the geostatistical analyses are summarized in Table 2 with the values
for the tensile fractures.
[24] As a representative result, Figure 10 shows fluid flow
in the aperture distribution at a confining pressure of 10 MPa
for the case of a shear fracture with a shear displacement of
1 mm generated at a normal stress of 60 MPa. As shown in
the histograms (Figure 8b), the aperture distribution had
strong heterogeneity due to the significant number of contact
points, which were represented by local apertures of 1 mm,
and the wide range of local apertures of up to several
millimeters (Figure 10a). Figure 10b shows the fluid flow
for the aperture distribution in Figure 10a. The fluid flow
was represented as the spatial distribution of the ratios of a
local flow rate to the total flow rate, indicating the relative
contributions of each point to the total flow rate, i.e., the
permeability of the fracture. Ratios of greater than 0.01
(1% of the total flow rate) are shown in color, while smaller
ratios are shown in black as negligible values. When the
colored and black regions are uniformly distributed, no
preferential flow path is developed. Therefore Figure 10b
clearly shows the development of preferential flow paths,
i.e., channeling flow, as remarkable localization of the
colored regions. A composite image of the aperture distributions and water flow describes the channeling flow more
clearly (Figure 10c). The preferential flow paths were
usually observed in the noncontact regions of the aperture
distribution, despite the nonzero permeability at the contacting asperities. Although the contacting asperities had a local
Figure 6. Relationships between tortuosities and roughness coefficients of the shear fractures generated in the
present study and of the tensile fractures generated in our
previous study [Watanabe et al., 2008].
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Figure 7. Changes in permeability with confining pressure for shear fractures generated under various
conditions (various shear displacements at various normal stresses). (a) Shear fractures and (b) comparison
between the shear fractures generated at a normal stress of 20 MPa and tensile fractures with equivalent
shear displacement in Watanabe et al. [2008].
aperture of 1 mm, the no-flow condition can essentially be
assumed at these points. In addition, the noncontact area can
be divided into flow and stagnant areas, where the flow and
stagnant areas are the areas in which flowing and stagnant
fluids exist, respectively. Such fluid flow was usually
observed in the aperture distributions at confining pressures
of up to 100 MPa with respect to the various shear fractures
generated in the present study (Figure 11). Table 3 lists the
results of evaluation of the flow area (total areas of preferential flow paths), the ratios of the flow area to the noncontact
area, the hydraulic apertures, the ratios of hydraulic apertures
to geometric mean of the local apertures, and the permeabilities. Note that hydraulic apertures and corresponding
permeabilities are identical to the experimental values. The
values for the tensile fractures are also listed in Table 3 for
comparison.
[25] As observed in the semivariograms, the models
showed obvious differences when the shear fractures were
generated at the different normal stresses of 20 MPa and
60 MPa. In addition, the models for the shear fractures
generated at the lower normal stress of 20 MPa were similar,
despite the different shear displacements (Figure 11a),
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Figure 8. Histograms of the local apertures of the aperture distributions at a confining pressure of
10 MPa for the shear fractures. Shear fractures generated at normal stresses of (a) 20 MPa and (b) 60 MPa.
whereas the models for the shear fractures generated at the
higher normal stress of 60 MPa differed between the different
shear displacements (Figure 11b). For the shear fractures
generated at the smaller normal stress, the values in Tables 2
and 3 were similar, regardless of the shear displacement.
However, for the shear fractures generated at the higher
normal stress, the spatial correlation lengths and anisotropy
of the spatial correlation lengths were greater for the smaller
shear displacement (Table 2). In addition, the ratios of the
hydraulic apertures to the geometric means of the local
apertures were smaller for the smaller shear displacement
(Table 3).
[26] The shear fractures generated at the lowest normal
stress of 1 MPa may be relevant, as compared to a tensile
fracture with a shear displacement. However, numerical
results for 1 MPa were not obtained. Therefore the models
of the shear fractures generated at the lower stress of 20 MPa
were compared with those of the tensile fractures with
equivalent shear displacements (1 mm and 5 mm) in our
previous study [Watanabe et al., 2008]. Although no
numerical result was obtained for the fractures generated
at the lowest normal stress of 1 MPa, the similar permeabilities indicate that there was no significant difference
between the shear fractures generated at 1 MPa and those
generated at 20 MPa. In comparison, at the same shear
displacements, the major difference in the aperture distributions between the shear and tensile fractures with shear
displacements was the spatial correlation lengths, although
other differences were observed. For example, the spatial
correlation lengths of the tensile fractures with shear displacements were smaller (did not exceed the sample size)
and had clear anisotropy with greater values for the direction perpendicular to the shear (Table 2). On the other hand,
at the same confining pressures, the major difference was
the change in the geometric means of the local apertures
with the shear displacement. In the case of the tensile
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Figure 9. Semivariograms of the local apertures of the aperture distribution at a confining pressure of
10 MPa for the shear fractures. (a) Shear fracture with a shear displacement of 1 mm generated at a
normal stress of 20 MPa and (b) shear fracture with a shear displacement of 5 mm generated at a normal
stress of 60 MPa, wherein the dashed lines are the fits by the exponential model.
fractures, the geometric means increased dramatically with
increasing shear displacement (Table 2). The ratios of the
flow areas to the noncontact areas also tended to be smaller
for the tensile fractures (Table 3).
4. Discussion and Conclusions
[27] Experimental results on fluid flow through shear
fractures generated in granite were presented. Fluid flow
through fractures is important in numerous crustal processes. Laboratory studies on fluid flow through various types
of fractures provide important constraints in many geological and geotechnical problems. Since generating shear
fractures is much more challenging than generating tensile
fractures, the present study provides valuable experimental
data on the fluid flow properties of shear fractures.
[28] In the present study, direct shear experiments were
performed on granite cores in order to create granite
samples that contained a single Mode II fracture. The linear
relationship between the shear stress at failure and normal
stress indicated that all of the prepared samples contained
Mode II fractures (Figure 5). Due to the debris associated
with fracture, a number of samples were required for the
fracture surface mapping and the fluid flow experiments.
Although two samples were generally prepared for each
condition at fracture generation, the sample preparation
required considerable effort, particularly due to multiple
fractures, which resulted in an insufficient number of
samples for quantitative discussion on the reproducibility
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Table 2. Contact Areas, Mean Apertures, Standard Deviations, and Spatial Correlation Lengths of the Aperture Distributions at
Confining Pressures of 10, 60, and 100 MPa for the Shear Fractures in the Present Study and the Tensile Fractures in Our Previous Study
[Watanabe et al., 2008]a
Spatial
Correlation Length
Condition at Generation
Fracture Type
Shear fracture
Normal
Stress (MPa)
20
Shear
Displacement (mm)
1
5
60
1
5
Tensile fracture
-
0b
1b
3b
5b
10b
Confining
Pressure (MPa)
Contact
Area (%)
Mean
Aperture (mm)
Standard
Deviation (-)
Parallel
(mm)
Perpendicular
(mm)
10
60
100
10
60
100
10
60
80
10
60
100
10
60
100
10
60
100
10
60
100
10
60
100
10
60
100
37
55
56
43
61
61
47
47
51
36
50
65
56
62
63
31
42
66
35
37
37
41
42
42
34
34
34
186
67.2
65.8
199
84.4
67.9
202
194
146
278
183
60.8
70.9
65.3
64.5
196
171
75.2
229
219
219
437
398
398
891
871
871
5.4
13.1
13.2
7.8
17.6
23.2
5.6
5.8
7.5
4.8
8.2
27
2.9
2.9
2.9
2.3
2.3
3.0
2.4
2.4
2.4
2.3
2.5
2.5
2.1
2.1
2.1
24
24
24
11
11
11
3
3
3
12
12
13
13
13
13
13
13
13
25
25
25
42
42
42
10
9
9
2
2
2
18
18
18
28
28
28
34
33
33
35
35
35
a
The mean apertures and standard deviation are geometric means and geometric standard deviations of local apertures at noncontact points, respectively.
Parallel and perpendicular denote directions to the shear displacement in the analyses.
b
Given after fracture generation.
of the experimental results. However, the fracture surface
mapping and the fluid flow experiments on different samples prepared under the same conditions provided consistent
results for the fracture surface roughness and fracture
permeability, which implied that the fracture generation
and resulting fluid flow properties were reproducible. The
roughness coefficient and tortuosity of the fracture surface
changed systematically with shear displacement and normal
stress, which supports the reproductively of the fracture
generation (Figure 6). The surface roughness was significantly different between the shear fractures generated at
normal stresses of 20 MPa and 60 MPa. The difference
between natural fractures in sandstone and granite was
comparable, whereas the fluid flow properties were reported
to differ for natural fractures in different rocks [Sausse,
2002]. Since the influences of microcracks and/or debris
formation associated with the fracture generation at different
stresses are expected to differ for permeability and surface
roughness, the different results obtained in the fluid flow
experiments were reasonable for shear fractures generated
at different normal stresses (Figure 7). Similarly, the difference in the results for the shear and tensile fractures
with equivalent shear displacements was also reasonable.
Although further experiments are required for quantitative
discussion, the fluid flow properties of shear fractures are
thought to depend on the conditions of fracture generation
and are essentially different from those of tensile fractures
with shear displacements.
[29] In the present study, numerical models were also
constructed using the data of fracture surface mapping by
matching the numerical and experimental fracture permeabilities to determine the aperture distributions at the
confining pressures of up to 100 MPa in the experiments
and fluid flow within the apertures (Figure 11). The numerical results provided findings that were fundamentally
consistent with the experimental results. An especially
valuable contribution of the numerical results to understanding the fluid flow through rock fractures was the observation of the developments of preferential flow paths, i.e.,
channeling flow, in the heterogeneous aperture distributions. Although it remains unclear as to whether the
channeling flow was likely for shear fractures, the present
study demonstrated the channeling flow for both shear
fractures at a wide range of confining pressures and tensile
fractures. Fluid flow through rock fractures should take
place essentially within a limited region, with a diversity
associated with fracture types.
[30] The numerical results may provide important information regarding the diversity of channeling flow of rock
fractures in the range of confining pressures considered in
the present study. Since the contact area may account for
approximately 30% to 70% of the fracture plane, depending
on the confining pressure (Table 2), the area in which fluids
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Figure 10. Relationship between a numerically determined aperture distribution and resulting water
flow in the structure for a shear fracture. (a) The aperture distribution at 10 MPa for a shear fracture with a
shear displacement of 1 mm (95 150 mm) generated at a normal stress of 60 MPa, (b) the water flow in
the structure, and (c) a composite image showing the development of the preferential flow paths
(occurrence of the channeling flow) in the aperture distribution. The preferential flow paths consist of the
points at which the ratios of the local flow rate to the total flow rate are greater than 0.01 in the image of
water flow.
can easily flow (i.e., noncontact area) may be approximately
70% at maximum. However, aperture distributions that are
far from uniform should cause fluid flow only along
preferential (least resistant) flow paths, which account for
a smaller area than the noncontact area. As a result, the
noncontact area is divided into flow and stagnant areas. The
flow area in particular may be relevant to the flow wetted
surface (FWS), which is significant for radionuclide transport analyses in fractured rock masses [Rasmuson and
Neretnieks, 1986; Crawford et al., 2003; Neretnieks,
2006]. The flow area may be only approximately 5% to
20% of the fracture plane, in other words, only approximately 10% to 30% of the noncontact area may contribute
to fluid flow (Table 3). Although the flow area seemed
surprisingly small, such a small flow area has previously
been indicated by field investigations at depths of up to
600 m in the Stripa mine in Sweden, where more than 90%
of the water flow was thought to take place in channels,
accounting for approximately 5% to 20% of the fracture
plane depending on the confining pressure [Abelin et al.,
1985]. A small flow area is therefore likely to occur in
nature, which would emphasize the significance of prediction of the channeling flow in many geological and geotechnical problems. The numerical results presented herein
may provide useful information for the prediction of the
channeling flow. Since the relationships among contact/
flow/stagnant areas and the permeability were roughly
linear (Figure 12), it may be possible to predict the
channeling flow simply by a permeability measurement.
[31] Although the numerical results provided findings of
significant interest, a thorough investigation should be
conducted in the future because of the limitations of the
fracture deformation model employed in the present study.
The present model, which uses a uniform reduction of local
apertures, does not take into account any deformation of a
fracture surface as a function of stress, even for contacting
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Figure 11. Changes in the water flows in the aperture distributions with confining pressure for shear
fractures generated under various conditions. Shear fractures (95 150 mm) generated at normal stresses
of (a) 20 MPa and (b) 60 MPa.
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Table 3. Flow Areas, Ratios of Flow Area to Noncontact Area, Hydraulic Apertures, Ratios of Hydraulic Aperture to Mean Aperture,
and Permeabilities at Confining Pressures of 10, 60, and 100 MPa for the Shear Fractures in the Present Study and the Tensile Fractures in
our Previous Study [Watanabe et al., 2008]a
Condition at
Fracture Generation
Fracture Type
Shear fracture
Normal
stress
(MPa)
20
Shear
Displacement
(mm)
1
5
60
1
5
Tensile fracture
-
0b
1b
3b
5b
10b
Confining
Pressure
(MPa)
Flow
Area (%)
Flow
Area/Noncontact
Area (-)
Hydraulic
Aperture (mm)
Hydraulic
Aperture/Mean
Aperture (-)
10
60
100
10
60
100
10
60
80
10
60
100
10
60
100
10
60
100
10
60
100
10
60
100
10
60
100
17
11
11
13
12
12
13
13
13
14
12
10
5
8
8
12
10
8
13
12
12
14
13
13
13
13
13
0.27
0.24
0.25
0.23
0.31
0.31
0.25
0.25
0.27
0.22
0.24
0.29
0.11
0.21
0.22
0.17
0.17
0.24
0.20
0.19
0.19
0.24
0.22
0.22
0.20
0.20
0.20
140
17
7.5
184
18
15
8.5
4.3
2.9
183
15
2.1
9.1
3.1
2.8
192
100
11
194
170
170
202
184
184
223
196
196
0.75
0.25
0.11
0.92
0.21
0.22
0.04
0.02
0.02
0.66
0.08
0.03
0.13
0.05
0.04
0.98
0.58
0.15
0.85
0.78
0.78
0.46
0.46
0.46
0.25
0.23
0.23
Permeability
(m2)
1.6
2.4
4.7
2.8
2.7
1.9
6.0
1.5
7.0
2.8
1.9
3.8
6.9
8.0
6.5
3.1
8.3
1.0
3.1
2.4
2.4
3.4
2.8
2.8
4.1
3.2
3.2
a
Where the mean apertures are geometric means of local apertures at noncontact points.
Given after fracture generation.
b
Figure 12. Relationships between the contact/flow/stagnant areas and the permeability of various types
of fractures generated in granite. Drawn using data in Tables 2 and 3. The solid lines are intended to
clarify the linear relationships between the areas and the permeability.
15 of 17
109
1011
1012
109
1011
1011
1012
1012
1013
109
1011
1013
1012
1013
1013
109
1010
1011
109
109
109
109
109
109
109
109
109
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WATANABE ET AL.: DIVERSITY OF CHANNELING FLOW
asperities [Gangi, 1978]. Therefore the model that uses a
uniform reduction in local apertures cannot be used to
clarify the aperture distribution or resulting flow paths as
a function of applied confining pressure. In future studies,
we hope to implement more realistic models [Pyrak-Nolte et
al., 1988; Pyrak-Nolte and Morris, 2000] to examine
fracture deformation as a function of confining pressure.
[32] Acknowledgments. The authors would like to thank Drs. Kazuhiko
Tezuka and Tetsuya Tamagawa (Research Center of Japan Petroleum Exploration Co., Ltd.), Dr. Kimio Watanabe (RichStone Limited), and Dr. Kenichiro
Suzuki (Technical Research Institute of Obayashi Corporation) for their assistance with the sample preparations and numerical simulations.
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N. Hirano, N. Tsuchiya, and N. Watanabe, Graduate School of Environmental Studies, Tohoku University, 6-6-20 Aramaki-aza-Aoba, Aoba-ku,
Sendai, Miyagi 980-8579, Japan. ([email protected])
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