Fractal characteristics of rock discontinuities

Engineering Geology, 34 (1993) 1-9 Elsevier Science Publishers B.V., Amsterdam

l

Fractal characteristics of rock discontinuities Amitava Ghosh a and Jaak J.K. Daemen b aCenterfor Nuclear Waste Regulatory Analyses Southwest Research Institute, San Antonio, TX 78228-5166, USA bMining Engineering Department, Mackay School of Mines University of Nevada, Reno, NV 89557-0139, USA (Received March 11, 1992; revised version accepted February 18, 1993)

ABSTRACT The discontinuities in a rock mass usually control its overall behavior. Rock mass deformability, stability of underground excavations, and flow of fluid depend significantly on the intensity, the degree of interconnection, and the characteristics of the fracture network present. We apply the theory of fractal geometry to describe the rock fracture network. Three parameters characterize the discontinuities visible in the exposed face of the investigated rock mass. The first parameter measures the complexity of the network formed by the individual traces of the discontinuities. The intensity and the interconnectivity of the discontinuities are characterized by the fracture density and the block density respectively. We use data from four faces of a copper mine in Arizona. All three parameters show fractal characteristics over the range investigated with coefficientsof determination better than 0.99. The fractal structure of these parameters suggests that the rock fracturing process may be a scale-independent phenomenon.

Introduction Rock is rarely a continuous medium. Rock masses contain m a n y natural breaks such as faults, joints, bedding planes and schistosity planes. Understanding the geometry of the fracture network in rock is helpful for characterizing the flow of fluid through and the mechanical behavior of rock masses. The effect of fractures on fluid flow is important for underground waste disposal, for estimating groundwater flow into mines and tunnels, for predicting production of petroleum from naturally fractured reservoirs, oil shale and tar sand deposits. Intensity, degree of interconnection, and characteristics of a fracture network are important for determining the stress history of a rock mass (Olson and Pollard, 1988), rock mass deformability, and the stability of underground excavations ( G o o d m a n and Shi, 1985), and flow of fluid through fractures (Witherspoon and Long, 1987). "From an engineering point of view, a knowl-

edge of the type and intensity of the rock defects may be much more important than the type of rock which will be encountered" (Terzaghi, 1946). It is difficult to obtain detailed fracture data, especially 0013-7952/93/$06.00

subsurface - - from boreholes, outcrops, or remote geophysical techniques. Modeling fluid flow in fractured rock is a relatively simpler problem than defining the geometry of the fractures in the rock mass and their interconnectedness (Witherspoon and Long, 1987). Generally, only the dominant sets are identified in field mapping. The characteristics of the joint sets, such as, dip, dip direction, linear extent, etc., are measured and the statistical averages are determined. The average values are used to characterize the rock mass. Additionally, m a n y studies take into account the statistical distribution fitted to each parameter (Call et al., 1976; Priest and Hudson, 1981; Hudson and Priest, 1979, 1983; Baecher, 1983; Einstein and Baecher, 1983; Dershowitz and Schrauf, 1987; Dershowitz and Einstein, 1988; Kulatilake, 1988). The fractures are assumed planar. In case of complicated geology, the rock unit is divided into structural domains having statistical homogeneity (Miller, 1983; M a h t a b and Yegulap, 1984; Kulatilake, 1988). In this study, data have been obtained from four rock faces in a mine using photographs of

© 1993 - - Elsevier Science Publishers B.V. All rights reserved.

2

A. G H O S H A N D J . J . K D A E M E N

the faces. Photographic mapping of rock joints has considerable advantages in terms of safety, speed, and efficiency (e.g., Franklin and Dusseault, 1989, p. 50; Hunt, 1984, p. 259), particularly in open pit bench faces which may be exposed only briefly, and may only be accessible for a very short time. Three parameters are used to characterize the intensity and the degree of interconnection of the discontinuities visible in these photographs. The theory of fractal geometry (Mandelbrot, 1982) is used to study the self-similarity of the fracture network. The fractal nature of these parameters can be incorporated into characterizing the rock mass discontinuities. Exploiting the fractal behavior, the information provided by the discontinuities is combined into a single value of each parameter.

Theory of fractai geometry "A f r a c t a l is a shape made o f parts similar to the whole in some way" (Mandelbrot, 1987, as quoted in Feder, 1988). The fractal dimension quantitatively measures the ruggedness of an object (Mandelbrot, 1982; Jullien and Botet, 1987; Feder, 1988; Vicsek, 1989). Shapes in Euclidian space, for example, a straight line, a plane surface, or a sphere, are smooth. Real objects are far from ideal ones in Euclidian space. A fractal object can have a non-integer dimension that exceeds the topological dimension in the Euclidian space. The fractal shapes are nonanalytic at any point. This means the shapes are not differentiable and a tangent cannot be drawn at any point on the entire object (Pietronero, 1988). If an object may be divided into N(R) = b subintervals of dimension R = l/b, then the dimension D of the object is defined as (Mandelbrot, 1982):

or: N(R) = R - ~

(2)

This means that an object may be subdivided into N parts each of which has been scaled down from the original by a factor R < I (Voss et al., 1983). Equation 2 is valid for an object which is selfsimilar, meaning that the structure of the object is scale invariant. A part of the object, when enlarged, is identical to the object (Mandelbrot, 1982; Turcotte, 1986; Jullien and Botet, 1987). A mathematical fractal object, derived using rigid, deterministic, iterative rules, possesses the true scale invariant property (Mandelbrot, 1982; Voss et al., 1983; Jullien and Botet, 1987; Feder, 1988; Vicsek, 1989). A natural object is rarely, if ever, exactly self-similar. Many shapes found in nature are only approximately self-similar. A small portion of the body looks alike, but it is not exactly the same as the scaled version of the total body. The selfsimilarity property is only valid in the average and, therefore, natural objects are statistically selfsimilar (Mandelbrot, 1982; Voss et al., 1983; Jullien and Botet, 1987; Feder, 1988; Vicsek, 1989). Usually upper and lower limits exist for the scale over which the system shows the self-similar property. If the number of squares with a characteristic size ¢ > L (some specific value) required to cover the entire discontinuity network N(~ > L) varies as (Voss et al., 1983; Turcotte, 1986; Jullien and Botet, 1987): N(¢ > L) ~ L - o

(3)

then D is the fractal dimension of the discontinuity network. This relationship can have a wide range of applications in fracturing of rock masses by explosives, size reduction of fragments in crushers, and in many other geomechanics areas.

Methodology log [N(R)] Dlog [l/R]

(1)

which is known as the Hausdorff-Besicovitch dimension, after two mathematicians who developed it. Equation 1 can be written as: log [N(R)] = D log [l/R]

Information on the discontinuities present in a rock mass has been collected in an open pit mine in Arizona. Available line surveys of the bench face are not sufficient to map the discontinuities for this study as only major structures are mapped routinely due to their importance for slope stability analysis. Such information is not detailed enough

FRACTAL CHARACTERISTICS OF ROCK DISCONTINUITIES

for the present study. Photographs of four bench faces have been taken to gather discontinuity information in greater detail. Each photograph contains a graduated tape laid across the face. All the visible discontinuities are traced from the photographs. It is assumed that the areas photographed are representative of the total bench. Precambrian pinal schist and dacite are the common rock types in the area of the mine where this study has been carried out. The rock is heavily jointed. Schmidt plots of the joints show twelve to nineteen central tendencies. Dip direction varies from about $45°W to East. Dip varies from almost vertical to about 30 ° from the horizontal. Three photographs (16A17, 17A18 and 18A19) of the first face, one photograph (9) of the second face, one photograph (25) of the third face, and one photograph (1) of the fourth face are analyzed. The discontinuity trace map obtained from each photograph is shown in Figs. la-g. Areas, where no information is available because of dust or other materials, are shown as dark patches in Fig. 1. Initially only very prominent discontinuities in photograph 17A18 are traced (Fig. lc). All the analyses are carried out using these traces. Subsequently, all the discontinuities visible in photograph 17A18 have been traced (Fig. lb). Both maps of photograph 17A18 are used in this study. Three parameters characterize the pre-existing discontinuities visible in each photograph. The first parameter measures the complexity of the network formed by the individual traces of the discontinuities. The intensity and the interconnectivity of the discontinuities are characterized by the fracture density and the block density respectively, following La Pointe (1988). The discontinuities visible on the face can be considered as space-filling curves. The fractal dimension of the fracture network has been determined using the box-counting method (Voss, 1988). The number of squares, N(R), of size R needed to cover all the discontinuities using different grid sizes R can be used to determine the fractal dimension. The fractal dimension of the discontinuity network should be 1 < D < 2. D equal to 1 gives a single straight line. D equal to 2 gives the entire plane, which means the discontinuity traces cover

3

the whole face area. Therefore, 1 and 2 form lower and upper bounds for D. If the discontinuity network follows self-similar behavior in a twodimensional cut (bench face), then the fractal dimension of the discontinuity network in three dimensions is ( D + 1) with 2 < ( D + 1)<3. Fracture intensity or fracture density is defined as the number of fractures per unit square area of the bench face. A square grid 8.9 cm x 8.9 cm (3.5 in x 3.5 in) is laid over the trace. The number of joints in each square of the grid is counted. A grid of the same size but with more yet smaller squares, is used in the next calculation. Again, the number of joints in each small square is counted. This process is repeated with grids with finer and finer squares. As the square grid does not cover the whole photograph (size 8.9 cm x 13.0 cm or 3.5 in x 5.2 in), the same procedure is carried out for both sides of the photo. There is some overlap at the middle between the two grids. Fracture interconnectivity is defined as the number of blocks produced by the joints per unit square area of the face. The same grids used for the fracture density calculations are used in the interconnectivity analysis also. The procedure for counting the blocks is the same. Fracture density, as well as fracture interconnectivity, can be represented as a rugged surface in X - Y coordinates associated with the bench (where X is horizontal and Y is vertical). The Z coordinate at any point is proportional to the number of fractures or number of blocks per unit area. The area of this rugged surface is estimated by summing the height of equal size small squares required to cover the whole surface. As the size of each square decreases, more peaks and troughs of the surface are taken into account. As a result, the area of the surface increases due to better estimation. This method, devised by Clarke (1986), is similar to the divider method described by Mandelbrot (1982) to determine the length of a rugged line. The calculated fracture density and the block density in any region of the photograph increase as the size of the unit square used for counting is increased. Therefore, the calculated densities need to be normalized by the maximum density obtained at any square of the grid to represent them in a common scale. These normalized densities are used in the fractal analysis.

4

A. G H O S H A N I ) J . J . K DAEMI~N

mm

100

200

300

mm

l,,i,,h,i,,l,,h,h,i,,l,,i,,l,,i,,l Fig. l(a). Joint trace map from Photograph 16A17

100

200

300

l,,l,,l,,i,,l,,J,,l,,+,,l,,r,,l,,hl} Blast 1.

Fig. l(b). Joint trace map from Photograph 17A18, Blast l, with all visible joints.

7-'-

I00

ram

200

300

mm

I00

200

300

l,,,,,h,,,,l,,h,l,,,,,l,,,,,h,,,,l

I,,I,,I,,I,,I,,a,,h,J,,I,, h,l,,Iz~l

Fig. l(c). Joint trace map from Photograph 17A18, Blast 1, with only major joints traced.

Fig. l(d). Joint trace map from Photograph 18A19 - - Blast 1.

rnm

100

206

300

l,..,,i,,,,,l,..,.i,.,.,l,,,,~l,.i.l

Fig. ICe). Joint trace map from Photograph 9 - - Blast 2.

mm 300 hldli,hhl

Fig. l(f). Joint trace map from Photograph 25 - - Blast 3.

FRACTAL

CHARACTERISTICS

OF

ROCK

DISCONTINUITIES

5

10 4

,

,

,

, ,,,,t

'

'

'

'

....

i

q

i

i

i LI,,I

2

3

4 3

2

Z

mm

100

200

10 3

• • •

300

la,l,hl,,,hhf,uhhl

Fig. l(g). Joint trace map from Photograph 1 - - Blast 4.

10

2

Photo 9 Photo 25 Photo 1 i

i

2

3

I0 0

IIIILI

I0!

10 2

Figures 2 and 3 show the N(R) versus R plots of all six photographs. Results from the analysis of the prominent joints only in photograph 17A18 are also shown in Fig. 2. Best-fit lines, plotted as a solid line in each graph, have a coefficient of determination r 2 greater than 0.99 in all cases. This suggests that the discontinuity network visible in the photographs is a fractal, i.e., the discontinu4

4 3

10a

Fractal dimension of the discontinuity Photograph number

P h o t o 16A 17 P h o t o 1 7A / P h o t o 1 7A 18 ( C o a r s e ) P h o t o 18A19

• ,

100

,

~ ,,,

10 3

ity network in the rock mass shows statistically self-similar behavior. Therefore, the network visible on a large scale is statistically similar to the network on a small scale. The fractal dimensions calculated from each photograph are given in Table 1. When all the visible discontinuities in photograph 17A18 are considered, the fractal dimension of the discontinuity network increases. This is due to more discontinuities filling the same area of the face covered in the photograph. Nevertheless, both traced maps of photograph TABLE l

lo2

,

3

Fig. 3. Number of squares N(R) of size R required to cover all the discontinuities in Photographs 9, 25, 1.

2

Z

,

2

R(mm)

Results

10

i

,

;

. . . . .

, 101

~

,

,

. . . .

102

R(mm) Fig. 2. Number of squares N(R) of size R required to cover all the discontinuities in Photographs 16A17, 17Al8 and 18A19. 17Al 8 (coarse) refers to the case when only the major discontinuities of Photograph 17A 18 are considered.

Blast 1 16A17 17A18 17A18 18A19 Blast 2 9 Blast 3 25 Blast 4 1

• Coarse map.

Fractal dimension

1.83 1.92 1.34 1,50 1,72 1.58 1.69

6

A GHOSH

17A18 indicate fractal behavior of the discontinuity network. Figures 4 and 5 show the variation of the fracture density and the block density calculated from photograph 17A 18. Figure 4 shows the result when all the fractures of photograph 17A18 are included in the analysis. Results with only major fractures are shown in Figure 5. The solid lines represent the least-square line fitted through the calculated points. The slope of the best-fit line defines the fractal dimension. The variations of fracture density and of block density with grid size are straight lines with r2 value greater than 0.99 in all cases. Therefore, both fracture density and block density are fractal, at least within the range investigated. Fractal dimensions of the block density and the fracture density are given in Table 2. Results from the coarse map of photograph 17A18 are also included. Left and right sides of each photograph

AND J.JK

L)AIiM[:N

show almost identical fractal dimensions for the block and the fracture densities from one photograph to another. Therefore, the apparent difference among the fracturing characteristics of the rock mass seen in different photographs, even between the left and the right sides of the same photograph, is only due to the scale effect. Statistically the degree of fracturing seen in the heavily fragmented part of the face is the same as that in the relatively less fragmented part if the former is enlarged to the appropriate scale. Again, the fracturing characteristics produced by the prominent joints in photograph 17A 18 are statistically similar when all the fractures are taken into account. Conclusions The fractal nature of discontinuity networks suggests that the discontinuity traces are not truly

BLOCK DENSITY Right

Left Side 8.0

,

,

l

8.0

6.0

c~ v

v

Z; 4 . 0

z 4.0

2.0

2:.0

0.0

i

0.0

i

,

1.0

i

2.0

i

0.0

3.0

Side

=/

8.0

,

i

1.0

0.0

Ln (i/R)

Ln

J

I

2.0

(l/R)

FRACTURE DENSITY Left Side 8.0

l

6.0 v

z

/

4.0 2.0

0.0

i 0.0

Right B.0

,

D=2.Sy

6.0

/

v

z 4.0 ,.d 2.0

I 1.0

i

Ln (I/R)

i

0.0

I 2.0

3.0

Side

I

0.0

l

D=2.U

i

1.0

i

I

2.0

3.

Ln (:/R)

Fig. 4. Variations of the fracture density and the block density with grid size in Photograph 17A18 - - Blast 1. All the visible fractures are included in this analysis. Both fracture and block densities are fractal.

FRACTAL CHARACTERISTICS

OF ROCK DISCONTINUITIES

7

BLOCK DENSITY Right Side

Left Side u

8.0

u

8.0

6.0

i

l

s =

.

v Z;

z

4.0

2.0

4.0

2.0

i

0.0 0.0

I

i

1.0

I 2.0

0.0

i

0.0

3.0

~ (l/R)

I 1.0

,

i 2.0

~(1/R)

FRACTURE DENSITY

807/ Right Side

Left S i d e S.0

z e~

i

S.0

u

4.0

v

p,l

i

l

1.0

2.0

4.0

,.d 2.0

2.0

0o0

i

0.0

I

i

1.0

I

0.0

0.0

2.0

t~ ( 1/R)

Ln ( l / R )

Fig. 5. Variations of the fracture density and the block density with grid size in Photograph 17A18 - - Blast 1. Only the major fractures are included in this analysis. Both fracture and block densities show fractal behavior. Calculated fractal dimensions are almost the same (maximum difference is 0.18) as those in Fig. 4, where all visible fractures are included.

TABLE 2 Fractal dimensions of the block density and the fracture density Photograph number

Fracture density

Block density

Left

Right

Left

Right

2.80 2.83 2.67 2.76

2.77 2.79 2.61 2.72

2.77 2.80 2.74 2.73

2.76 2.83 2.67 2.78

2.73

2.72

2.77

2.71

2.71

**

2.70

**

2.83

2.78

2.81

2.82

Blast 1

16A17 17A18 17A18' 18A19 Blast 2

9 Blast 3

25 Blast 4

1

* Coarse map. ** Only left side of the photograph has been measured.

planar. They show nonlinear features down to a scale much smaller than that used in regular line mapping. Surfaces of natural joints and faults have shown fractal character rather than linear features. Brown and Scholz (1985) and Scholz and Aviles (1986) have observed fractal surfaces of rock joints from microscopic scales (10 -5 m) to large field scales (105 m). Joint surfaces near Libby Dam, Montana, also show fractal characteristics (Carr and Warriner, 1987) on a scale practically of interest for dam foundation and slope stability. The San Andreas main fault (Aviles et al., 1987), San Andreas and associated faults (Okubo and Aki, 1987), and faults at Yucca Mountain, Nevada (Barton and Larsen, 1985), show fractal structure on a scale of many meters. Line mapping, even if it is carried out along several parallel lines on the rock exposure, gives the information only at the intersection points of the discontinuity and the line along which the measurements are carried out. As

8

a result, line mapping ignores the self-similar characteristics of the discontinuities. This fractal characteristic should be kept in mind when using the results of line mapping. The fractal behavior of rock discontinuity networks is potentially useful for many applications concerning rock mass behavior, such as, rock mass classification, support design, fluid flow, etc. The fractal nature of discontinuities suggests a not entirely random distribution of the individual discontinuity planes. It suggests a relationship among the different discontinuities especially in terms of length, location and orientation. This relationship should depend upon the history of loading acting on the rock mass and on the subsequent fracturing process. Structural geologists are investigating rock fracture formation using fractal analysis (e.g., Davy et al., 1990; Sornette et al., 1990), and it seems probable that an improved insight into the fractal aspects of rock fracture genesis will enhance our engineering applications of describing structural features using the fractal theory. When simulating the rock mass discontinuity network using statistical procedures this relationship should be preserved. Results from Figs. 2-5 show an order in the apparently disordered distribution of the discontinuities in the rock mass. It is not known why the fracture network, the fracture density, and the block density of the rock mass, like many other natural objects, show fractal behavior. One logical explanation is that the perception of the physical phenomenon and the associated scale of measurement are due to the limitations of the human mind rather than the underlying facts of nature (Stewart, 1989). Although nature operates simultaneously on all scales, the human mind associates a scale to understand the natural phenomena. The fracture network, the fracture density, and the block density are different characteristics or parameters representing the same fracture pattern. As a result, there must be some relationship among the three parameters describing the same fracture formation process in the rock mass. As the number of fractures per unit area (fracture density) increases, the number of blocks in the same area (block density) also may increase, depending on the degree of interconnection among the fractures.

A. G H O S H A N D J.J.K I ) A E M E N

The degree of interconnection or the intricacy of the fracture network is characterized by the fractal dimension of the fractal network. The fractal structure of the three parameters indicates that the rock fracturing process may be a scale-independent phenomenon. Formation and subsequent propagation of a crack is a scale invariant process with self-similar structure. We postulate that the presence of pre-existing microflaws and the heterogeneous material properties at the microlevel make the structure a stochastic fractal by incorporating some randomness in the growth process. If the fracture formation in a rock mass, subjected to a given loading history, is a selfsimilar process then the information obtained about the fracture pattern at any scale would be statistically similar to that at any other scale. This has significant importance as it implies that information about large scale behavior can be obtained from small scale observations. Fractal behavior has been observed in other rock fracture processes - - shear of rock at different scales (Tchalenko, 1970), shear deformation of a strikeslip fault (Rispoli, 1981), continental deformation and laboratory simulations thereof (Davy et al., 1990; Sornette et al., 1990), microfractures ahead of a crack tip (Ghosh, 1990, pp. 449-453 ), and microfracture formation in uniaxial compression (Hirata et al., 1987). Fractal structure of the fracture system is not limited only to the joints in a face. The fault planes at this mine also show fractal structure (Ghosh, 1990). At the same level in which these blasts are carried out, the fracture density of the fault planes has a fractal dimension of 2.40. The fault network has a fractal dimension of 1.14.

Acknowledgement We wish to thank Dr. Dirk van Zyl for allowing us to use the data for this study. Data gathering was supported through NSF Grant No. CEE-8417990 to Dr. van Zyl at the University of Arizona.

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