Surface Review and Letters, Vol. 10, No. 5 (2003) 751–762
c World Scientific Publishing Company
DIRECT ESTIMATION OF THE FRACTAL
DIMENSIONS OF A FRACTURE SURFACE OF ROCK
H. W. ZHOU∗ and H. XIE
Institute of Rock Mechanics and Fractals,
China University of Mining and Technology,
Xueyuan Road D11, Beijing 100083, P R China
∗
[email protected]
Received 16 February 2003
A direct determination of the fractal dimensions of a fracture surface is essential for a better understanding of its complete topographic characteristics. In this paper, a laser profilometer is employed to
measure the topography of a rock’s fracture surface. With the use of the triangular prism surface area
and projective covering methods, the resultant data set enables us to directly determine the fractal
dimensions of a rock’s fracture surface. Moreover, a new method, referred to as the cubic covering
method, is proposed. The theoretical issues of fractal dimension estimation are also discussed.
Keywords: Fracture surface; fractal dimension; cubic covering method.
1. Introduction
acceptable two-dimensional approaches available for
assessing a fracture surface. Thus, many researchers,
even Mandelbrot,2 suggested that the fractal dimension of a rough surface could be obtained by adding
1.0 to the fractal dimension obtained from a single
profile of that surface. Such an approximation might
be very close to the real fractal dimension of a fracture surface, but from the theoretical viewpoint, it
is unacceptable. Therefore, some new methods have
been developed for direct determination of the real
fractal dimension. The first triangular prism surface area method was proposed by Clarke.3 Muralha4
then used this method to determine the fractal dimensions of 44 joint surfaces of mudstone and sandstone. He found that the fractal dimensions ranged
from 2.00011 to 2.00716. Seven years after Clarke,3
Friel and Pande5 suggested another direct method
for estimating the fractal dimension of a fracture surface of rock and cement paste using scanning electron
microscopy and stereoscopy. This involved measuring the surface area of the true fracture surface over
a base reference plane network. Wang and Diamond6
An insight into the topography of a fracture surface
is important for a better understanding of the fracture mechanism, especially the local failure mechanism. Fracture surfaces usually appear to be rough.
Generally speaking, to characterize a fracture surface roughness is merely a geometrical problem. In
the past few decades, extensive efforts have been devoted to characterization of fracture surfaces of rock.
Current research involves the application of fractal
geometry to the quantification of a fracture surface.1
Fractal dimension, as an index for characterization of a complex degree of natural phenomena,
can also be used to describe the surface topography of a fracture. Many methods, such as the divider, box counting, spectrum and variogram, have
been suggested for estimating the fractal dimensions of a rough profile.2 However, a one-dimensional
analysis provides an incomplete, and even biased,
characterization of a fracture surface. Therefore,
a two-dimensional quantitative description is needed. Nevertheless, there have previously not been
∗ Corresponding
author. Tel.: +86-10-62331286. Fax: +86-10-62331490.
751
752 H. W. Zhou & H. Xie
also employed the stereoscopic SEM method supposed by Friel and Pande5 to evaluate the fractal
characteristics of fracture surfaces of cement pastes
and mortars. They found that fracture surfaces of
cement pastes exhibit two distinct fractal regimes: a
regime of low fractal dimension at low magnification
scales and a significantly higher fractal dimension at
higher magnifications.
Xie and Wang7 proposed another method, called
the projective covering method, which can be regarded as a modification of the triangular prism
surface area method.8 Using the projective covering method, Xie and Wang7 showed that the fractal
dimensions of a fracture surface in sandstone range
from 2.013 to 2.039. Stach et al.9 also applied the
projective covering method to characterize the fracture surfaces of ductile and brittle materials. They
found that the fractal dimensions of fracture surfaces
are equal to 2.0813 and 2.0146 for ductile and brittle
materials, respectively. It should be mentioned that,
in addition, Belem et al.10 assumed that the fracture
surface is a continuum and derivable. Thus, the actual area of a fracture surface can be evaluated with
the integral method.
Recently, Yong et al. indicated that the generalized divider methods, such as the triangular prism
surface area3 and projective covering methods,7 do
not easily identify the difference in two-dimensional
fracture surface roughness because the real surface
area of a two-dimensional joint surface is difficult
to estimate. Thus, Yong et al. extended the
Hurst index applied on the one-dimensional profile to the two-dimensional fracture surface. They
asserted that the Hurst index in two dimensions
could be determined directly by employing the twovariable fractal Brownian motion theory, and thus
could represent the entire surface roughness.
In this research, a laser profilometer is used to
obtain the data set of a fracture surface of sandstone. Using the data set, we determine directly
the fractal dimensions of a fracture surface of rock
by the triangular prism surface area method and the
projective covering method. Aiming at the shortcomings of the triangular prism surface area and
projective covering methods, a new solution called
the cubic covering method is proposed. Moreover,
the theoretical issues of fractal dimension estimation
are discussed.
2. Experimental Procedure and
Acquisition of the Data Set
In the present study, a large size sandstone sample was selected from an underground coalmine, the
Myslowice Mine of the Upper Silesian Coal Basin in
Poland. The size of the sample was 180 × 180 ×
118 mm. A fresh, unfilled fracture surface was produced in the sample using the Brazilian method.
Topographical measurement of the fracture surface
was carried out with a 3D laser profilometer. This
is a noncontact device which enables one to obtain
data files of x, y and z coordinates (Fig. 1). The laser
profilometer consists of a laser probe mounted on a
coordinate-measuring machine. The probe provides
an accuracy of ±7 µm, a resolution of 7.5 µm and
an elevation range of 30 mm. It can move automatically over the sample using a preprogrammed path to
measure the topography of any portion of the fracture surface. A personal computer then performs
data collection and processing. The collected data
set of a fracture surface topography consists of both
coordinates and corresponding heights of the object
surface.
The overall dimension of the scanning field was
160 mm × 160 mm. The fracture surface of the defined area was digitized using a 0.25 mm sampling
interval. In this case, the total number of data points
should be 641 × 641 (641 points for each profile and
641 profiles for the entire surface). The height information at each point was then transferred to the
computer and an isometric view of the entire scanning field was reconstructed with imaging software
(as shown in Fig. 2).
co-ordinate measuring machine
monitor
laser probe
sample
Fig.1 1. Photo
Laser
Figure
ofprofilometer.
laser profilometer
Y, mm
Direct Estimation of the Fractal Dimensions of a Fracture Surface of Rock 753
Y, mm
Z, mm
O
X, mm
Fig. 2. Surface topography of a rock fracture measured by the laser profilometer (scanning interval = 0.25 mm; total
number of measured points = 641 × 641).
Figure 2 The surface topography of a rock fracture measured by the laser profilometer
(scanning interval =0.25mm, total number of measured points =641 ×641)
3. Estimation of Fractal Dimensions
by Previously Reported Methods
3.1.
The triangular prism surface
area method
S2
S1
fracture surface
S3
S4
h(i+1, j)
h0
h(i, j)
In 1986, Clarke proposed a method for calculat14
(i, j)
ing the fractal dimension of a fracture surface. The
h(i+1, j+1) (i+1, j)
h(i, j+1)
fundamental principle of the method is that if the
δ
heights of all points on the fracture surface above a
base reference plane
base reference plane can be established, its true sur(i, j+1)
δ
(i+1, j+1)
face area can be measured. For example, suppose
that there is a regular square grid, which has a scale
Fig. 3. Schematic view of the triangular prism surface
of δ (as shown in Fig. 2), below a fracture surface.
area method (after Clarke3 ).
Figure
3
Schematic
view of the triangular prism surface area method (after Clarke, 1986)
Each intersection point of a regular square grid corresponds to a point on the fracture surface and an
elevation. Thus, four points of every square grid
With this method the approximate area is estimated by calculating the area of four triangles. The
correspond to four points on a fracture surface. In
elevation at the center of each grid cell is determined
this case, the surface area of the fracture surface can
by linear interpolation (average) of the four heights
be determined. However, it is almost impossible to
of the adjacent points (after Clarke3 ):
make an exact calculation of the true area of the
fracture surface within the grid cell shown in Fig. 3,
1
because, usually, the four points considered seldom
h0 = [h(i, j)+h(i, j +1)+h(i+1, j)+h(i+1, j +1)] ,
4
lie on the same plane. One possible solution is to
(1)
estimate the approximate value of the true area of
where h0 , h(i, j), h(i + 1, j), h(i, j + 1) and h(i + 1,
the fracture surface.
+ 1)and
areδas
shown in Fig. (b)
3.
N=100 and δ =16mm
=32mm
(a)jN=25
(i, j+1)
δ
(i+1, j+1)
S
3
Then the area of one of triangles is determined by
The corresponding areas S , S and S of the
other
three triangles are then calculated. Thus, the
h(i+1, j)
p
754 H. W. Zhou & H. Xie
1
S4
2
S1 =
l1 (l1 − a1 )(l1 − b1 )(l1 − c1 ) ,
where l1 = 21 (a1 + b1 + c1 ),
h0 Figure
p
3
, j)
a1 =
(i, j)
(2)
3
4
approximate real area of a fracture surface in a given
grid cell with a scale of δ × δ is given by
S =
S + S + S + S prism
.
(3)
Schematic view of the
triangular
surface ar
i,j
1
2
3
4
[h(i, j) − h(i, j + 1)]2 + δ 2 ,
The total area of the fracture surface is
p
b1 = [h(i, j) − h0 ]2 + 0.5δ 2 ,
h(i+1, j+1)
(i+1, j)
N (δ)
S(δ) =
p
c1 = [h(i, j + 1) − h0 ]2 + 0.5δ 2 .
X
(4)
Si,j ,
i,j=1
δ
base reference plane
δ
(i+1, j+1)
e triangular prism surface area method (after Clarke, 1986)
mm
(a) N = 25 and δ = 32 mm
(a) N=25 and δ =32mm
(b)
N=
(d)
N=
(b) N = 100 and δ = 16 mm
(b)
Fig. 4.
N=100 and δ =16mm
First four steps of estimation of real surface areas.
(c) N=400 and δ =8mm
Figure 4
First four-steps of estimations of re
mm
mm
(a) N=25 and δ =32mm
N=
(b)
Direct Estimation of the Fractal Dimensions of a Fracture Surface of Rock 755
(b)
N=100 and δ =16mm
(c) N = 400 and δ = 8 mm
(c) N=400 and δ =8mm
Figure 4
N=
(d)
First four-steps of estimations of re
(d) N = 1600 and δ = 4 mm
(d)
N=1600 and δ =4mm
Fig. 4.
where N (δ) is the total number of grid cells with
scale δ × δ needed to cover the fracture surface.
If the scale δ is changed, different values of S(δ)
may result. Obviously, the measured real area of
the fracture surface depends on δ. The less dense
the network of points whose height has been established, the less accurately the individual cells
are mirrored, and the smaller the measured surface
area. Conversely, the smaller the scale δ, the larger
(Continued )
the measured surface area. In fractal geometry, the
measured surface area is related to the scale δ by
our-steps of estimations of real surface areas
S(δ) ∼ δ 2−D ,
(5)
where D is the self-similar fractal dimension of a
fracture surface.
The data set of topography of a rock fracture
surface shown in Fig. 2 was used to make a direct
determination of the fractal dimension of a fracture
15
756 H. W. Zhou & H. Xie
Table 1.
Relation between S(δ) and δ estimated by the triangular prism surface area method.
δ (mm)
32
16
8.0
4.0
2.0
1.25
1.0
0.5
0.25
N (δ)
5×5
10 × 10
20 × 20
40 × 40
80 × 80
128 × 128
160 × 160
320 × 320
640 × 640
Total points
6×6
11 × 11
21 × 21
41 × 41
81 × 81
129 × 129
161 × 161
321 × 321
641 × 641
S(δ) (mm2 )
25633.9
25680.6
25770.6
25912.8
26120.1
26360.7
26488.3
27230.1
28575.0
2
d)], mm
log[S(
log[S(δ)],
mm
2
log[S(δ)],
log[S(d)],mm
mm
2 2
10.26
surface. The first stage consists in choosing a scale
10.26
data in Table 1
of grid cells equal to 32 mm. We may estimate the
10.24
data in Table 1
linear fi
10.24
total area of a fracture surface using Eqs. (1)–(4).
linear fit
10.22
The next stage involves adjusting the scale of grid
10.22
cells downward to equal 16 mm. A total surface area
10.20
10.20
slope = -0.019
corresponding to scale 16 mm may be counted again.
slope = -0.019
Then the count is repeated by decreasing the size of
10.18
10.18
grid cells (the process is shown in Fig. 4 and results
10.16
are given in Table 1).
10.16
Based on Table 1, the relation between total sur10.14
10.14
-1
0
1
2
3
4
face area S(δ) and grid cell size δ can be plotted in
-1
0
1
2
3
4
log(d), mm
a log–log way (as shown in Fig. 5). Figure 5(a) indilog(δ), mm
cates that the log–log relation between the total real
(a)
(a)
area of the fracture surface and the grid scale does
(a)
not appear to be linear, i.e. the rock fracture surface
10.26
does not show strict self-similar fractal behavior on
10.26
data in Table 1
the scale ranging from 0.25 mm to 32 mm. However,
10.24
data
linearinfitTable 1
10.24
it is evident that the data points fall into two dislinear fi
10.22
slope=-0.05
tinct linear regions with different slopes, as shown
10.22
slope=-0.05
in Fig. 5(a). In this case, a dual-segment linearity
10.20
10.20
should be performed. The correlation coefficients
10.18
are equal to 0.985 within the region of 0.25–2.0 mm,
10.18
slope=-0.008
and 0.968 within the range of 2.0–32 mm. One may
slope=-0.008
10.16
find that the log–log slope on the scale ranging from
10.16
0.25 mm to 2.0 mm is higher than that of the scale
10.14
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
10.14-1.5 -1.0 -0.5
variation from 2.0 mm to 32 mm. Substituting the
-1.5 -1.0 -0.5
0.0
0.5 log(δ),
1.0
1.5
mm 2.0 2.5 3.0 3.5 4.0
slopes in Fig. 5(b) into Eq. (5), we may conclude that
log(d), mm
(b)
the rock fracture surface has fractal dimensions equal
(b)
to 2.05 on the scale from 0.25 mm to 2.0 mm and
(b)
plots
S(δ) and
estimatedprism
by the
δ of
Figure 5 Log-log Fig.
plot 5.of SLog–log
(δ ) and
estimated
byδtriangular
surface area metho
2.008 on the scale from 2.0 mm to 32 mm.
Therefore,
δ ) and
δ estimated
Figure
5 Log-log triangular
plot of S (prism
by triangular
prism surface area metho
surface
area method:
(a) one-segment
(a) one-segmental linearity; (b) two-segmental linearity
linearity,
(b) two-segment
linearity.
the fracture surface of the present research does not
(a)
one-segmental
linearity;
(b) two-segmental linearity
have a universal fractal dimension on all scales. In
addition, the result of Fig. 5(b) suggests the follow-(i,j)
(i,j)
(i+1,j)of the four heights
(i+1,j)
of the ad(i,j)
ing tendency: the smaller the value of δ, the greater(i,j) by linear interpolation
(i+1,j)
(i+1,j)
projective
covering
plane
jacent points (as shown in Fig. 3). Thus, the point
the fractal dimension.
projective covering plane
at the center
(i+1,j+1) of a grid cell is not usually on the(i+1,j+1)
(i,j+1)
(i,j+1)
(i+1,j+1)
rough surface, and might even
h(i,j)
h(i,j)from the true(i+1,j+1)
(i,j+1)
(i,j+1)be far
h(i+1,j)
h(i+1,j)
3.2. The projective covering method h(i,j)
fracture surfac
h(i,j) S1Because ofS2
rough surface. This h(i+1,j)
may lead to an error.
h(i+1,j)
fracture surfac
S1
S2
S1 fracture surfac S2
7
3
proposed a new direct measureWith the triangular prism surface area method, the S1 this, XieSand Wang
δ
δ
2
fracture
surfac
ment method,
referred
to as
the projective
covering
elevation at the center of each grid cell ish(i,j+1)
determined
δ
h(i,j+1)
δ
h(i+1,j+1)
h(i+1,j+1)
h(i,j+1)
δ
δ
h(i+1,j+1)
(a)
Figure 6
Figure 6
h(i,j+1)
δ
δ
h(i+1,j+1)
(b)
(a) view of projective covering method: (a) Case
(b) 1 and (b) Case 2
Schematic
Schematic view of projective covering method: (a) Case 1 and (b) Case 2
(a)
(a)one-segmental
one-segmentallinearity;
linearity;(b)
(b)two-segmental
two-segmentallinearity
linearity
Direct Estimation of the Fractal Dimensions of a Fracture Surface of Rock 757
(i,j)
(i,j)
(i,j)
(i,j)
(i+1,j)
(i+1,j)
(i+1,j)
(i+1,j)
projectivecovering
coveringplane
plane
projective
(i+1,j+1)
(i+1,j+1)
(i,j+1)
(i,j+1)
h(i,j)
h(i,j)
h(i,j+1)
h(i,j+1)
S2S2
δδ
h(i,j)
h(i,j)
h(i+1,j)
h(i+1,j)
fracturesurfac
surfac
fracture
S1S1
(i+1,j+1)
(i+1,j+1)
(i,j+1)
(i,j+1)
δδ
h(i+1,j+1)
h(i+1,j+1)
S1S1
fracturesurfac
surfac
fracture
h(i,j+1)
h(i,j+1)
h(i+1,j)
h(i+1,j)
S2S2
δδ
(a)
δδ
h(i+1,j+1)
h(i+1,j+1)
(b)
(a)
(a)
(b)
(b)
Fig. 6. Schematic view of the projective covering method: (a) Case 1, (b) Case 2.
Figure
Figure 66 Schematic
Schematicview
viewofofprojective
projectivecovering
coveringmethod:
method:(a)
(a)Case
Case11and
and(b)
(b)Case
Case22
Table 2.
δ (mm)
S(δ) (mm2 )
Relation between S(δ) and δ estimated by the projective covering method.
16
8.0
4.0
2.0
1.25
1.0
0.5
Case 1
25637.3
25698.0
25797.1
25956.8
26184.2
26459.1
26605.9
27464.4
28953.5
Case 2
25637.3
25698.0
25797.1
25956.7
26183.9
26458.4
26605.3
27462.5
28950.0
method, to determine the fractal dimension of a fracture surface. With this method, box covering was
considered in the projective plane of the fracture
surface and its covered areas were estimated within
the mapping space between the projective plane and
the real fracture surface (as shown in Fig. 6). Simply said, the real area surrounded by four points
on the fracture surface is approximated by two triangles. In this case, every point for calculation of
the approximate area can be assured to be on the
fracture surface.
In Fig. 6(a), the area of one of the triangles is
given by
p
S1 = l1 (l1 − a1 )(l1 − b1 )(l1 − c1 ) ,
where l1 = 12 (a1 + b1 + c1 ),
p
a1 = [h(i, j) − h(i, j + 1)]2 + δ 2 ,
q
b1 = [h(i, j + 1) − h( i + 1, j + 1)]2 + δ 2 ,
c1 =
1616
0.25
32
p
[h(i, j) − h(i + 1, j + 1)]2 + 2δ 2 .
Similarly, the area S2 can be easily determined.
Then, the total surface area of the fracture surface
within a projective covering cell with a scale of δ × δ
is given by
Si,j = S1 + S2 .
(6)
The total area of a fracture surface may be estimated using Eq. (4). With the calculation process
shown in Fig. 4, we may get the relation between
the total surface area S(δ) and the grid cell scale
δ (Table 2), as well as the log–log relation between
them (as shown in Fig. 7). It should be noted there
are two ways to divide the field surrounded by the
four points on the fracture surface into two triangles. Thus, we will have two results corresponding
to Case 1 and Case 2 in Fig. 6, respectively. Both
are given in Table 2 and Fig. 7.
Table 2 and Fig. 7 show that there are two ways
to divide the field surrounded by four points on
the fracture surface into two triangles (as shown
in Fig. 6). However, their results are almost the
same. Thus, one may divide the field surrounded
by four points into two triangles, according to either Case 1 or Case 2. In addition, the data
points fall into two straight segments. The linear
regression analysis shows that the correlation coefficients within the region 0.25–2.0 mm and the region 2.0–32 mm are equal to 0.987 and 0.972, respectively. Substituting the slopes in Fig. 7 into
Eq. (5), we may conclude that the rock fracture
surface has a fractal dimension of 2.056 on a scale
ranging from 0.25 to 2.0 mm, and 2.008 on a scale
ranging from 2.0 to 32 mm. The fractal dimensions
10.28
data in Table 2
data infitTable 1
linear
linear fit
10.24
10.24
slope=-0.056
slope=-0.05
2
log[S(δ)],
log[S(mm
d)], mm
2
10.26
10.26
10.22
10.22
δ
10.20
10.20
10.18
10.18
slope=-0.008
slope=-0.008
10.16
10.16
cover grid
10.14
10.14 -1.5
-1.5
-1.0
-1.0
-0.5
-0.5
0.0
0.0
0.5
0.5
2.0
2.5
3.0
3.5
4.0
1.0 d),1.5
log(
mm2.0
log(δ), mm
1.0
1.5
2.5
3.0
3.5
4.0
irregular curve
(a)
(a)
(b)
Fig. 8. view
Schematic
view of the
box-counting
method
usedcurve by squar
Figure 8 Schematic
of box-counting
method
used to cover
irregular
cover
irregular
with etsquare
grid (after Zhou
method
Log-log plot10.28
of S (δ ) and δ estimated by triangular prism surfacetoarea
al, 2003)
grid curve
(after Zhou
et al.12 ).
(a) one-segmental linearity; (b) two-segmental linearity
data in Table 2
linear fit
2
10.26
log[S(δ)], mm
gure 5
758 H. W. Zhou & H. Xie(a)
covering units, units number Ni,j
δ
10.24
3
rough method
surface 7
area method
and the projective covering
(i+1,j)
10.22
Z
have to estimate the approximate value of the fracprojective covering plane
Y calculations will
10.20
ture surface area. Such approximate
(i+1,j+1)
(i+1,j+1)
certainly
result
in
error.
In
this
case,
to find an ac,j+1)
(i,j+1)
10.18
h(i,j)
h(i,j)
slope=-0.008
curateh(i+1,j)
calculation is a way for an accurate estimah(i+1,j)
fracture surfac
S1
S2
10.16
tion of the fractal dimension of a fracture. In fractal
S1
S2
fracture surfac
2
10.14
δ
δ
geometry,
the box-counting method (i.e. the cover-1.5 -1.0 -0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
h(i,j+1)
,j+1)
h(i+1,j+1)
h(i+1,j+1)
ing
method)
is a widely accepted approach to fracO
δ
δ
log(δ), mm
tal measurement. Just as a two-dimensional square
(b)
can be used to cover an irregular curve (indicated in
(a)
(b)
(b)
Fig.
812 2), a three-dimensional cube can be used to
Fig.
7.
Log–log
plots
of
S(δ)
and
δ
estimated
by
the
Figure 6 Schematic view of projective covering method: (a) Case 1 and (b)
Case
an irregular surface.
Case 1, (b) Case 2.
Figure 7 Logprojective
-log plots covering
of S (δ ) method:
and δ (a)
estimated
by projective coveringcover
method
There exists a regular square grid on the plane
(a) Case 1; (b) Case 2
XOY (Fig. 16
9). In each grid cell with scale δ, four
estimated either by the triangular prism surface area
intersection points correspond to four heights of a
method or by the projective covering method are esX h(i, j + 1), h(i + 1, j) and
fracture surface: h(i, j),
sentially the same.
h(i + 1, j + 1) (where 1 ≤ i, j ≤ n − 1, n being the
Figure
9 Schematic
viewpoints
of cubiconcovering
method
total
number
of sampling
each individual
4. Estimation of the Fractal Dimension
profile on a fracture surface). If we use a cube with
by Cubic Covering, a New Method
scale δ to cover the fracture surface, the maximum
difference among h(i, j), h(i, j + 1), h(i + 1, j) and
It is well known that it is almost impossible to make
h(i + 1, j + 1) will determine the number of cubes
an exact calculation of the actual surface area of a
needed to cover the irregular surface area within
fracture surface within the grid cell shown in Figs. 3
the scale δ, i.e. the number Ni,j of cubes needed to
and 6. Usually, the four points considered seldom
cover the fracture surface in the field of the (i, j)th
lie on a plane. Both the triangular prism surface
grid unit on the reference plane XOY is given by
(i,j)
(i,j)
slope=-0.056
(i+1,j)
17
1
[max(h(i, j), h(i, j + 1), h(i + 1, j), h(i + 1, j + 1))
δ
− min(h(i, j), h(i, j + 1), h(i + 1, j), h(i + 1, j + 1))] + 1 ,
Ni,j = INT
where INT denotes the integrating function.
(7)
18
Figure 8
Schematic view of box-counting method used to cover irregular curve by squar
grid (after Zhou et al, 2003)
Direct Estimation of the Fractal Dimensions of a Fracture Surface of Rock 759
covering units, units number Ni,j
δ
rough surface
Z
Y
O
X
Fig. 9.
Figure 9
Table 3.
Schematic view of the cubic covering method.
Schematic view of cubic covering method
Relation between N (δ) and δ estimated by the cubic covering method.
δ (mm)
32
16
8.0
4.0
2.0
1.25
1.0
0.5
0.25
N (δ)
25
100
400
1600
6400
16397
25672
106223
452606
Then the total number of cubes needed to cover
the whole fracture surface is
N (δ) =
n−1
X
Ni,j .
(8)
i,j=1
Changing the scale δ, we may get different values of
N (δ). The total number N (δ) of cubes depends upon
the sampling interval δ used. If the fracture surface
appears to be a fractal, the relation between N (δ)
and δ is
N (δ) ∼ δ −D ,
(9)
where D is the fractal dimension of the fracture
surface.
The data set of the topography of the rock
fracture surface shown in Fig. 2 was used to estimate the fractal dimension of a rock fracture surface.
Based on Eqs. (7) and (8), the relation between the
total numbers N (δ) of cubes needed to cover18the
whole fracture surface and the sizes δ of cubes is
given in Table 3. In Fig. 10, giving the log–log
plot of N (δ) and δ, the data points also fall into
two straight segments. Their correlation coefficients
of linear regression analysis are 0.9998 on the scale
ranging from 0.25 to 2.0 mm, and 1.00 on the scale
ranging from 2.0 to 32 mm, respectively. Substituting the slopes in Fig. 10 into Eq. (9), one may assert
that, on the scale of 0.25–2.0 mm, the fractal dimension of the fracture surface is 2.062, while on the scale
of 2.0–32 mm, the fractal dimension is exactly equal
to 2.0. It can be concluded, therefore, that at the
lower measurement resolution from 2.0 to 32 mm, the
fracture surface does not exhibit fractal behavior at
all. Only at the higher measurement resolution from
760 H. W. Zhou & H. Xie
14
log[N(δ)]
mentioned above are quite low. One of the reasons
is that all the fractal measurements are
12
slope=-2.062
data in Table 3
self-similar measurements. This also happens with
linear fit
one-dimensional fractal measurement. For example,
10
the joint roughness coefficient (JRC) is the most
8
attractive parameter for the description of rock
slope=-2.000
joint surface in rock engineering. Barton15 proposed
6
10 standard JRC profiles; their values have been
assigned between 0 and 20 in steps of two, starting
4
from the smoothest to the roughest. The self-similar
fractal dimensions of the roughest JRC profile are
2
-1.5 -1.0 -0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
usually smaller than 1.02.16,17 The sampling interval
log(δ), mm
in the present research is not small enough. This also
causes lower fractal dimensions.
Fig. 10. Log–log plots of N (δ) and δ estimated by the
N (δ ) and δ estimated by cubic covering method
Figure 10 Log
-logcovering
plots ofmethod.
cubic
A significant work of the present research is that
a new method for direct estimation of the fractal dimension of a fracture surface, referred to as the cubic
0.25 to 2.0 mm does the fracture surface appear to
covering method, is proposed. Both the triangular
be a fractal.
prism surface area3 and projective covering methods7
cannot avoid the problem of approximate estimation
Y, mm
mm
5. Discussion X,and
Conclusions
of the real area surrounded by four points on the
5.1. Comparisons
of fractal dimensions
fracture surface, because the four points considered
(a)
(b)
estimated by the above methods
seldom lie on a plane. Such approximate calculations
will certainly result in error. However, the cubic
The fractal dimensions of the rock
Y fracture surface
covering method can assure that every step is acshown in Figs. 5(b), 7 and 8, estimated by the
curate. Therefore, it can be regarded as a reliable
methods mentioned above, exhibit, to some extent,
method
for direct determination of the fractal dimensimilar behavior. All results indicate that the data
sion
of
a
fracture surface. Moreover, Fig. 10 shows
Z
points in the log–log plot do not fall into a straight
that the fractal dimension of a fracture surface is
line, but two distinct straight segments. Thus, there
exactly
equal to 2.0 on the scale larger than 2.0 mm.
is no universal fractal dimension on all scales. The
Only
if
the
measurement scale is smaller than 2.0 mm
O
fractal dimensions on smaller scales are usually
does
the
fracture
surface exhibit fractal behavior.
higher than those on larger scales (similar results
13
were obtained by Fardin et al. ). At a higher measurement resolution, the influence of the structure
5.2. Theoretical approach to the
of the fracture surface on smaller scales, such as
fracture surface
pores and grain clusters, on fractal characteristics
Xie and Wang7 discussed the relation between the
comes into play. In other words, the fractal dimenX
fractal dimension of the fracture surface and the
sion reflects the finer details
(c) of the fracture surface
fractal
dimension of individual profiles on the fractexture. It implies the following tendency: the higher
ture surface. One might have a desire to construct
the measurement
resolution,
higher
the production
fractal
Figure
11 Generation
of fractal the
surface
by star
a fractal surface using two fractal profiles. Usually,
dimension.
the(c)calculated
fractalwith two
(a) a profile in plane
ZOX; (c) aMathematically,
profile in plane ZOY;
a surface generated
profiles
three ways — the special Cartesian product, fracdimension will be close to the real fractal dimension
tional Brownian motion surface and star product
of a fracture surface only if the measurement scale
fractal surface18 — can be used to approach the
δ → 0. However, the sampling interval cannot be
19 surface.
natural fractal
reduced infinitely, and thus the fractal surface as a
Generally speaking, a fractal surface is a special
kind of natural physical fractal is quite different from
set
in three-dimensional Euclidean space R3 . The
the mathematical fractal (see also Ref. 14).
natural fracture surface can be regarded as a binary
In addition, the fractal dimensions of the
function z = f (x, y). Suppose that A is a fractal
fracture surface estimated by the different methods
Z, mm
6
4
2
6
4
2
0
0
0
20
40
60
80
100
0
20
40
60
80
100
14
14
14
Direct Estimation of the Fractal Dimensions of a Fracture Surface of Rock 761
slope=-2.062
slope=-2.062
profile in plane ZOX,
A
:
x
∈
[c, d], z =
12
slope=-2.062
g(x)}, while B is an interval in R, B = [a, b]; then a
10
simple fractal surface F can
10 be produced using the
Cartesian product: 10
F = A×B
8
log[N(
d)]d)]
log[N(
log[N(d)]
12
12 {(x, z)
=
8
8
data in Table 3
(x, y); (2) for adata
given in
(x,Table
y), (h, 3
k) ∈ R2 , the probabilfit 3
datalinear
in Table
ity distributionlinear
function
fit of increment BH (x + h, y +
linear
fit
k) − BH (x, y) is a Gaussian, or normal distribution
with a zero average increment and a variance of the
increment of (h2 + K 2 )H , i.e.
slope=-2.000
slope=-2.000
slope=-2.000
= {(x, y, z) : x ∈ [c, d], y 6∈ [a, b], z = g(x)} . (10)
the6
P (BH (x + h, y + k) − BH (x, y) ≤ z)
6
This means that
fractal surface F can be ob4 fractal profile A along
tained by the movement of the
4 fractal dimension of a
the straight line B. Thus
the
4
19
simple Cartesian product can
2 be given by
-1.5
-1.0
dim(A ×
2 B) =2 1 + dim A .
-0.5
0.0
0.5
(11)
= (2π)−1/2 (h2 + k 2 )−H/2
×
1.0
1.5
2.0
2.5
Z
z
−r2
2
2 H
3.5 2(h
4.0 + k )
exp
−∞
3.0
(12)
dr .
d), mm
0.5 log(
1.0
1.5 2.5
2.0 3.0
2.5 3.5
3.0 4.0
3.5
4.0
1.0
1.5
2.0
Fractional Brownian motion surface can be used
Then {(x, y, z) : (x, y) ∈ R2 , z = BH (x, y)} is called
d), mm
log(d),log(
mm
to simulate the morphology of the fracture surface.
a Brownian surface with the exponent of H. It can be
δ estimated
bythe
cubic
covering
method
Figure
10 inLog
-log
plots
(δ1,) ifand strictly
For any
real number
the
range
of 0of
< HN<
proved that
Hausdorff
dimension
and box
(1) with
probability
1,
the
two-dimensional
Browndimension
of
the
Brownian
surface
with
the
exponent
δ estimated
Figure
Logplots
-log plots
(δ ) δandestimated
by cubic
covering
method
by cubic
covering
method
Figure
10 10
Log-log
of Nof(δ )N and
ian function BH (x, y) is a continuous function for any
of H are equal to 3 − H.19
-1.5
-1.5
-1.0
-1.0
-0.5
-0.5
0.0
0.0
0.5
6
2
0
2
2
Z, mm
Z, mm
4
4
4
4
6
6
0
20
40
60
80
0
0
100
X, mm
0
0
Z, mm
Z, mm
Z, mm
Z, mm
6
20
20
40
40
60
60
(a)
80
80
100
6
2
4
0
6
4
2
20
0
0
40
60
80
100
Y, mm
0
0
100
20
20
20
40
40
X, mm X, mm
60
60
(b)
(a)
80
80
100
100
Y, mm Y, mm
(b)
(a) (a)
(b) (b)
Y
Y
Y
Z
Z
Z
O
O
O
X
(c)
(c)
X
Fig. 11. Generation of a fractal surface by star production: X
(a) a profile in plane ZOX, (b) a profile in plane ZOY,
Figure
Generation (c)
of fractal
(c) a surface generated
with two11
profiles.
(c) surface by star production
(a) a profile in plane ZOX; (c) a profile in plane ZOY; (c) a surface generated with two profiles
Figure
11 11
Generation
of fractal
surface
by star
Figure
Generation
of fractal
surface
byproduction
star production
(a) a profile in plane ZOX; (c) a profile in plane ZOY; (c) a surface generated with two profiles
(a) a profile in plane ZOX; (c) a profile in plane ZOY; (c) a surface generated with two profiles
19
19
19
762 H. W. Zhou & H. Xie
The fractal dimension of a naturally developed
rough surface is assumed to be the fractal dimension
3 − H of the Brownian motion surface. Nevertheless, the Brownian motion is a completely random
process, while the fracture surface is not completely
stochastic. Therefore the simulation of natural surfaces using Brownian motion may cause an error. For
this reason, the fractal dimension obtained by 3 − H
usually exceeds the actual fractal dimension.
The third way of theoretical construction of a
fractal surface is referred to as the star product of
two fractal profiles.18 Suppose that A and B are
the fractal profiles in plane ZOX and plane ZOY respectively, i.e. A = {(x, z) : x ∈ [a, b], z = g(x)},
B = {(y, z) : y ∈ [a, b], z = h(x)} (indicated in
Fig. 11). The star product F ∗ of A and B is defined
as the movement of A along B or the movement of
B along A:
F ∗ = A ∗ B = B ∗ A = {(x, y, z) : (x, y) ∈ [a, b]
× [c, d], z = g(x) + h(y) − g(a)} .
(13)
It has been proved that18
dim F ∗ = 1 + max(dim A, dim B) .
(14)
A natural fractal surface can be approached by
the special Cartesian product and Brownian motion
surface, or rather the star product of two fractal
individual profiles. It is indicated that the fractal
dimension of a star product of two fractal sets
is closer in approximation to the fractal dimension of a rock fracture surface than the Cartesian
product. Of course, the star product fractal surface
is more regular than the natural fracture surface,
so its dimension should be less than the dimension
of the fracture surface. In other words, the fractal
dimension of a rock fracture surface falls between
the fractal dimension of the star production surface
and the sum of fractal dimensions estimated along
two individual perpendicular directions:
dim F ∗ < D < Dx + Dy ,
(15)
where Dx and Dy are the average values of fractal
dimensions estimated in the x and y directions,
respectively.
Acknowledgments
This work is supported by the National Natural
Science Foundation (50074032 and 50221402) and
the 973 Program (2002CB412701). The financial
support is gratefully acknowledged. The experiment
was done when Zhou visited the Rock Mechanics
Laboratory of the Technical University of Silesia,
Poland. He wishes to thank Dr M. A. Kwasniewski
for his kind help.
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