fracture-distribuition modeling in rock mass using
TRANSCRIPT
FRACTURE-DISTRIBUTION MODELING IN ROCK MASS USING
BOREHOLE DATA AND GEOSTATISTICAL SIMULATION
Katsuaki Koike1, Kazuya Komorida2, and Yuichi Ichikawa3
1: Department of Civil Engineering, Faculty of Engineering, Kumamoto University, 2-39-1, Kurokami,
Kumamoto 860-8555, Japan. e-mail: [email protected]
2: Graduate School of Science and Technology, Kumamoto University, 2-39-1, Kurokami, Kumamoto 860-8555, Japan.
3: PASCO Co. Ltd., 1-1-2, Higashiyama, Meguro-ku, Toyko 153-0043, Japan. e-mail: [email protected]
ABSTRACT- Understanding spatial distribution of rock fractures is significant to various fields in
geosciences. It is, however, difficult to obtain a realistic fracture model because the amount of fracture data
is small and their locations are strongly biased in a study area. Data are usually taken by borehole
investigations and then, spatial correlation structures of fractures among boreholes with different directions
are needed to be clarified. For this problem, we focus on the fracture density along a borehole (number of
fractures per 1-meter interval) and appearance relation of azimuths (strikes and dips) between a fracture pair.
Semivariogram of the density, γ1(h), and cross-semivariograms of the two indicators for two d irections, γ2(h),
are produced. Firstly, fracture density map is produced using the γ1(h) and a sequential gaussian simulation.
Next, a direction of each simulated fracture is assigned using the data set, γ2(h), and ordinary kriging
combined with the principal component analysis. Finally, length of each fracture is defined considering the
distance and difference of azimuths between a fracture pair located closely. This proposed method was
applied to 629 investigation data by several boreholes in a granitic site situated in northeastern Japan. The
size of study area was defined as 60×60 m2. Horizontal distribution of fractures and continuities of them
along the vertical direction were successfully estimated. The most noteworthy features are found in that
high-density zones are located in no data areas and continuous fractures striking along the boreholes are
inferred. In addition, permeability of rock mass was calculated to be about 10-17 m2 from this distribution
model and a permeability tensor analysis.
Key Words : Fracture, Granite, Semivariogram, Stochastic simulation, Permeability tensor
INTRODUCTION
Fractures with diverse origins and different scales from microcracks to faults are generally
developed in rock masses. They were formed attributable to emplacement and cooling of rock
bodies, crustal movements, and regional stress fields at different geologic stages. Understanding spatial
distribution of rock fractures is significant to various fields in geosciences, including hydrogeology for
fracture-affected flow channels, and resource
exploration for vein-type mineral deposits and fluids
in fractured reservoirs (e.g., National Research Council, 1996; Coward et al., 1998; Adler and
Thovert, 1999). In special, characterization of hydraulic properties of rock masses has become
very important recently, because it is closely related to environmental problems.
For this purpose, a general view of fracture distribution in a study area is required to be drawn
using stochastic methods. There are many papers that have tried to clarify scaling laws of fractures
using fractal theory (e.g., Vignes-Adler et al., 1991;
Davy et al., 1992; Dawers et al. 1993; Watterson et
al., 1996; Koike and Kaneko, 1999), but most of them deal with length distribution over wide scales.
In addition to the lengths, locations of fractures should be considered to reveal spatial correlation
structures of fracture attributes. Geostatistical techniques that can incorporate the structures have
also been applied to fracture-distribution modeling (Long and Billaux, 1987; Young, 1987; Chilès,
1988; Gringarten, 1996). It is, however, difficult to obtain a realistic model because the amount of
fracture data is small and their locations are strongly biased in a study area. Fracture attributes that are
applicable to distribution modeling considering
length, appearance pattern, and azimuth of fracture should be firstly identified.
As the first step to a fracture-distribution modeling over various scales, this paper studies the
fracture data obtained by boreholes in granitic rocks. We focus on the fracture density and appearance
relation of azimuths between a fracture pair, and aims at inferring three-dimensional distribution of
joint-sized fractures.
VARIOGRAM ANALYSIS
Borehole Fracture Data The fracture data used in this study were
obtained by eight horizontal boreholes in the
Kamaishi mine, northeastern Japan. The mine is located in an Early Cretaceous dioritic granite
(partly by diorite). The boreholes were dilled over 215-m length in total and toward two perpendicular
Figure 2. Semivariogram of fracture densities, number of
fractures per 1-m interval along each borehole.
Figure 1. Arrangement of boreholes and segments that
represents the locations and strikes of the fractures.
directions, N15°W and N75°E, at almost the same
level, 250 m above the sea. Orientations (strikes and dips), in addition to kinds of interstitial minerals,
width of alteration halo, and fracture width were measured for 629 fractures, by the examination of
the borehole TV images and drilling cores. Figure 1 shows the arrangement of boreholes and segments
that represent the locations and strikes of the fractures.
Semivariograms Clarifying whether the spatial distributions of fracture attributes are perfectly random or have
certain spatial correlation structures is the most fundamental process for fracture-distribution
modeling. By producing experimental semi-
variograms of several attributes, it was found that the widths of alteration halos have almost a pure
Figure 3. Semivariograms of first principal values for the
indicator sets of four strike sectors.
N
KDH-1
KM-2
KDK-1
KDH-3
KDT-1
KD
A-8
KD
A-1
0
KDT-2
NN
KDH-1
KM-2
KDK-1
KDH-3
KDT-1
KD
A-8
KD
A-1
0
KDT-2
NN
15 m15 m
0 1 2Distance between data h (×10 m)
2
3
4
5
6
7
Sem
ivar
iogr
am γ
(h)
0 1 2Distance between data h (×10 m)
2
3
4
5
6
7
2
3
4
5
6
7
Sem
ivar
iogr
am γ
(h)
Sem
ivar
iogr
am γ
(h)
Distance between data h (×10 m)
1.1
1.2
1.3
1.4
1.5
1.6
0 1 2 3 4
Sem
ivar
iogr
am γ
(h)
Distance between data h (×10 m)
1.1
1.2
1.3
1.4
1.5
1.6
1.1
1.2
1.3
1.4
1.5
1.6
0 1 2 3 40 1 2 3 4
0 2 4 6 8 10Distance between data h (×104 m)
9
10
11
12
13
14
15
Sem
ivar
iogr
am γ
(h)
0 2 4 6 8 100 2 4 6 8 10Distance between data h (×104 m)
9
10
11
12
13
14
15
9
10
11
12
13
14
15
Sem
ivar
iogr
am γ
(h)
NN
10 km
nugget effect, whereas the widths of interstitial minerals contain a weak correlation structure. As
compared to these attributes, fracture densities, which are defined by the number of fractures per
1-m interval along each borehole, showed a more clear feature. It can be demonstrated by the
experimental semivariogram in Figure 2. A spherical model drawn by the broken line is
applicable to approximate the semivariogram. Another attribute considered here is the spatial
relation of fracture orientations which means directional similarity between a fracture pair. We did
not use directly the difference of strike angles or dip
angles between two fractures, but attached great importance to the distribution relation of strike or
dip data classified into several groups. This idea was proposed to consider continuit ies of the same
fracture set and corporate conjugate-pattern features into fracture-distribution modeling. For this purpose,
the four directional sectors for the EW, NW-SE, NS, and NE-SW are defined, and each fracture strike is
transformed into an indictor set (I1, I2, I3, I4) that represents presence (Ii=1) or absence (Ii=0) of strike
in the (EW, NW-SE, NS, NE-SW). For example, N45°W strike defines the set as (0, 1, 0, 0 ).
This set requires four semivariograms for the same sector and six cross-semivariograms for the
different sector pairs. To reduce calculation amount,
the principal component analysis was adopted and the set data were transformed into four principal
values. Since the fourth principal values become constant for the above indicator set, only three
Figure 4. Lineaments around the Kamaishi mine
extracted from the Landsat TM band 4 image using the
Segment Tracing Algorithm (STA).
experimental semivariograms are needed for the
first to third principal values. Figure 3 shows the experimental semivariogram of the first principal
values. While the values are scattered, the semivariogram can be approximated by an
exponential model (the broken line in Fig. 3). The same model is applicable to the semivariograms of
second and third principal values.
To validate the spatial correlation structures clarified in the two semivariograms, satellite image-
derived lineaments were investigated as another fracture-related geologic element. Figure 4 depicts
Sem
ivar
iogr
am γ
(h)
Distance between data h (×103 m)
0.8
1.0
1.2
1.4
1.6
0 1 2 3 4 5 6
Sem
ivar
iogr
am γ
(h)
Distance between data h (×103 m)
0.8
1.0
1.2
1.4
1.6
0.8
1.0
1.2
1.4
1.6
0 1 2 3 4 5 60 1 2 3 4 5 6
Figure 5. Semivariograms of A, the densities expressing number of lineament centers in
unit mesh; and B, the first principal values for indicator sets of four strike sectors.
A B
Kamaishi mine
Figure 6. Semivariogram of the indicator-transformed dip
direction data. Indicators are 1 for northern dip and 0
for southern dip.
the lineaments around the Kamaishi mine extracted from the Landsat TM band 4 image. The Segment
Tracing Algorithm (STA: Koike et al., 1995) was adopted to extract the lineaments. Using the centers
of lineaments, two experimental semivariograms of the densities expressing number of centers in unit
mesh and the first principal values for the indicator sets of strike sectors were produced as shown in
Figure 5. Since they can be approximated by the same semivariogram models as the case of borehole
fractures, the semivariograms shown in Figures 2
and 3 are considered to allow to estimate fracture distribution in the study area.
As for the fracture dips, only the dip directions indicated a spatial correlation structure. The dip
B
data were simply transformed into indicator data, 1 for northward dip and 0 for southward dip. The
experimental semivariogram of these indicators is shown in Figure 6, which can be approximated by a
spherical model.
TWO-DIMENSIONAL MODELING OF FRACTURE DISTRIBUTION
Production of Fracture-Density Map
The fracture density was chosen as an important key to sow the seeds of fractures which
are bases on simulating fracture distribution. It is
necessary to extend the one-dimensional fracture densities along the boreholes to two-dimensional
densities over the study area. The study area, defined as 60×60 m2 in size, was superimposed on a
grid with 1×1 m2 meshes. Calculation of fracture density in each mesh is the first step for the seeds, to
which the semivariogram of fracture densities (Fig. 2) is available.
We used ordinary kriging (OK) for the calculation, but could not obtain an appropriate
result as shown in Figure 7A. It is obvious that the densities are strongly affected by the nearest data
and there are many linear, unnatural boundaries of densities. To overcome this problem, a sequential
gaussian simulation (SGS: Journel, 1989) was
examined. It consists of transformation of the data into normal scores, determination of random path,
Monte-Carlo simulation using kriging variance at each grid point to draw an estimation value, addition
Number of
fractures in 1 m2
NN
Figure 7. Comparison of two fracture-density distributions produced by A, OK; and B, SGS.
0.18
0.2
0.22
0.24
0.26
0.28
0 10 20 30 40 500 1 2 3 4 50.18
0.20
0.22
0.24
0.26
0.28
Distance between data h (×10 m)
Sem
ivar
iogr
am γ
(h)
0.18
0.2
0.22
0.24
0.26
0.28
0 10 20 30 40 500 1 2 3 4 50.18
0.20
0.22
0.24
0.26
0.28
Distance between data h (×10 m)
Sem
ivar
iogr
am γ
(h)
15 m
A
15 m
N
15 m15 m
NN
NN
A B
15 m
of the value to the data set, and repetition of the calculation. The validness of SGS can be confirmed
from the result (Fig. 7B) in that high-density zones are estimated in no data areas in the northwest and
unnatural boundaries are not formed. In addition, the density distribution shows large spatial variation
as compared to that produced through OK.
Fracture-Distribution Modeling A method of fracture-distribution modeling
proposed here utilizes the fracture-density map and consists of the following five steps. (1) In each mesh, fracture centers are generated by
the same number as the fracture density estimated
for the mesh and their locations are given using random numbers.
(2) The strike data expressed in the above indicator
set are transformed into the principal values. OK uses these values and interpolates them at each
fracture center localized by the step (1). The interpolated principal values are then converted to
values in the original coordinate system corresponding to the indicator set. The strike
sector with the highest value is chosen as the estimated fracture direction. For example, a
calculated set (0.8, 0.3, 0.1, 0.4) assumes the strike to be EW.
(3) With respect to dip direction, OK interpolates the indicator-transformed dip directions of the
measured fractures at each fracture center. The interpolated value above 0.5 assumes the dip
direction of fracture to be northward, otherwise southward.
(4) Strike and dip angles at each fracture center are stochastically drawn by combining the
cumulative distribution functions of those angles in the selected strike sector and dip direction with
Monte-Carlo method. (5) Connection of fracture centers considering both
the center distance and directional difference is carried out at the final step. If an arbitrary
fracture center (A) can find out another center (B)
whose differences of strike and dip angles from the A are under 10° and 20°, respectively, the two
centers are connectable. Then, the B is used as a start point for the next search. Isolated fracture
that has no connectable center is eliminated.
Figure 8A depicts the result of the fracture-
distribution modeling. To clarify distribution characteristics in detail, only fractures estimated
longer than 10 m are extracted as shown in Figure 8B. The conspicuous directions of continuous
fractures (N20°-40°E and N20°-50°W) agree with those of the measured fractures. Usefulness of the
proposed method can be proved in that it can estimate the ENE-WSW striking fractures almost
parallel to the boreholes, which are hardly appeared in the boreholes, in the northern area and continuous
Figure 8. A, Result of t wo-dimensional fracture- distribution modeling; and
B, distribution of extracted fractures estimated longer than 10 m.
B A
C
fractures in the zones without data. Another
interesting feature is existence of the zones that contain the concatenated fractures passing through
the study area.
VALIDATION OF THE PROPOSED METHOD
The proposed method of fracture-distribution
modeling must be validated whether it allows to estimate practical distribution and characterize
hydraulic properties of the rock mass. For this purpose, a rock specimen over which fracture
distribution is entirely observed is desirable. We prepared a thin-section of the Inada granite with
1.5×1.5-cm2 size. The Inada granite situated in central Japan is known to contain many microcracks.
A modified version of the STA suitable for the microcrack analysis was used to extract linear
features from the digital image of thin-section. This
non-filtering technique can specify the portions in an image where the brightness largely change
attributable to cracks and trace the portions along
the cracks. Figure 9A shows the extracted 3559 linear
features that partly correspond to microcracks. On the image, virtual lines were set by modeling after
the arrangement of boreholes in the study area (Fig. 1), and the intersections of the lines and linear
features (Fig. 9B) were used to reconstruct the linear-feature distribution. Unit mesh size for
calculating the linear-feature density was defined as
0.25×0.25 mm2 so that the number of meshes, 60×60, is equal to that used for the fracture-density
map in Figure 7. As seen in the final result (Fig. 9C), the
reconstructed pattern is similar to the linear-feature distribution in that it highlights three dominant
directions, about 30°, 120°, and 160° counter- clockwise from the x-axis. The validness of the
proposed method is demonstrated because these dominant continuous line features are estimated in
the zones with no data.
EXTENSION TO THREE-DIMENSIONAL FRACTURE- DISTRIBUTION MODELING
For more practical modeling of fracture distribution, it is necessary to extend the proposed
Figure 9. A, Linear features extracted from an
Inada granite thin-section, using a modified
version of the STA; B, intersections of the
supposed lines and linear features; and C,
estimated linear-feature distribution.
5 mm x
y
method to be available in the three-dimensional
space. Information on vertical continuities of the fractures cannot be obtained because the boreholes
are located at almost the same level. Therefore, an assumption is needed for a three-dimensional
modeling. We assumed that the fractures observed at the 250-m level may be continuous only within the
±5-m depth range. The fracture locations at the 245 to 255-m
levels were given at 1-m depth interval by extending the observed fractures along their strikes and dips.
At each level, virtual boreholes were set on the same locations at the 250-m level, and the
intersection points of the virtual boreholes and the
extended fractures were used for calculating the fracture densities. Two-dimensional distribution
modelings were at first executed by the above- mentioned method and then, fracture planes were
constructed by connecting simulated fractures at adjacent levels based on the similarity of strike and
dip and the nearness of locations. For example, the distributions conjectured at the 248 and 252-m
levels are shown in Figure 10, from which different patterns can be found with the level.
The modeling result is visualized in Figure 11, which represents the locations of fracture planes
longer than 5 m along the vertical direction. The constructed planes are slightly undulate, but can be
considered to steeply dip at larger than 70°. Figure
12 shows the azimuthal frequencies of the planes using the lower hemisphere projection of the
Schmidt’s net. There are two sets of the simulated continuous fracture planes, the most remarkable
NW-SE striking set and the NNE-SSW striking set subordinate to it.
APPLICATION TO CHARACTERIZATION
OF HYDRAULIC PROPERTIES
The three-dimensional fracture-distribution
model can be linked to the permeability tensor analysis to estimate permeability of the rock mass.
According to Oda et al. (1987), the permeability tensor, kij of degree three is expressed by
( )
( )∫ ∫ ∫∞ ∞
Ω
Ω=
−=
0 0
32 ,,4
drdtdtrnEtr jiij
ijijkkij
nnP
PPk
πρ
δλ (1)
where δij: delta function, λ: a constant expressing the continuity of fractures, r: diameter of fracture, t:
hydraulic aperture of fracture, n: normal vector of
fracture plane，E(n, r, t): probabilistic density func- tion with respect to n, r, and t, ρ: volumetric
fracture density, and Ω: solid angle. We defined the λ as 1/12 and supposed that the t is 10-6 times the r.
The r was determined by converting a fracture plane into a disc with the equivalent area.
Discretization of Eq. (1) was carried out and three principal values (k1, k2, k3) and directions of
the principal axes were calculated as shown in Table
NN
Figure 10. Fracture distributions estimated at the 248 and 252-m levels. 15 m
252-m level 248-m level
Perspective view from the east side.
Perspective view from the bottom side.
Figure 12. Azimuthal frequencies of the planes using the
lower hemisphere projection of the Schmidt’s net.
1. The principal axes generally correspond to the
two dominant strikes for the k1 and k2 and the dip direction of the predominant fractures for the k3.
The magnitude of principal values cannot be validated because there is no measured permeability
data in the study area. However, hydraulic well tests were executed in other areas in the Kamaishi mine
where the fractures are much developed as
Perspective view from the top side.
Figure 11. Result of three-dimensional
fracture-distribution modeling, which
represents the locations of simulated
fracture planes longer than 5 m along
the vertical direction.
Table 1. Three principal values and directions of principal
axes calculated for the permeability tensor of Eq. (1).
compared to the present study area. Considering the
data, condition of the rock mass, and the permeability data reported for hard granitic rocks in
many areas (e.g., Brace, 1980; Brace, 1984; Clauser, 1992), the first principal value , 10-17 m2 can be
regarded as a plausible magnitude.
CONCLUSION
By analyzing the borehole fracture data in the Kamaishi mine, spatial correlations were clarified
on the three attributes of fractures, fracture densities along the boreholes, appearance relations of fracture
strikes, and dip directions with respect to northward or southward dip. The experimental semivariograms
Directions of principal axes Principal values (m2)
Azimuth Dip
k1 1.0 × 10-17 N68 ° W 14°
k2 4.3 × 10-18 N20 ° E
4°
k3 2.4 × 10-18 N83 ° W
76°
N
10 m
10 m
5 m
yz
xN
10 m
N
10 m
10 m
5 m
yz
x
NN
10 m10 m
10 m
5 m
yz
xN
10 m
N
10 m
xy
z
10 m
10 m
5 m
N
xy
z
10 m
10 m
5 m
xy
z
xy
z
xy
z
10 m10 m
10 m
10 m
5 m5 m
NN
5 m
8 m +
6 - 7 m
5 m
8 m +
6 - 7 m
S
E W
N
0
2.3
4.7
7.0
9.4+
Frequency (%)
of these attributes could be approximated by spherical, exponential, and spherical models,
respectively. The SGS was proved to be effective for estimating fracture densities in the sparse data
zones. To consider the azimuthal correlation of fracture pairs, we transformed the strike and dip
data into the indicators. Applying the principal component analysis and ordinary krig ing to the
indicators, two- and three-dimensional models of fracture distributions, which incorporate both the
azimuthal and positional information of the fracture data, could be constructed.
The three-dimensional model was effectively
linked to the permeability tensor analysis for estimating the permeability and principal axes. Two
characteristics were detected by it in that the permeability along the first principal axis is about
10-17 m2 and the major axes correspond to the dominant azimuths of simulated fracture planes. The
estimated permeability is plausible by considering the condition of rock mass and the hydraulic test
data obtained at other sites in the mine.
Acknowledg ments: The authors wish to express their
grateful thanks to the Japan Nuclear Cycle Development
Institute for providing the in situ measurement results in
the Kamaishi mine.
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