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Fractal patterns of fractures in granites
Earth and Planetary Science Letters, 104 (1991) 25-35
25
Elsevier Science Publishers B.V., Amsterdam [DT]
Fractal patterns of fractures in granites B. V e l d e a, j . D u b o i s
b,
D. Moore
c and G. Touchard
~
a D~partement de Gdologie, Ecole Normale Sup~rieure, UR 1316 CNRS, 24 rue Lhomond, 75005 Paris, France b Institut de Physique du Globe Paris, Obseroatoires Volcanologique, 4 pl. Jussieu, 75252 Paris, France c United States Geological Survey, 345 Middlefield Rd., Menlo Park, CA 94025, USA a Laboratoire du M~canique des Fluides, Universitd de Poitiers, Ave. Recteur Pineau, 89022 Poitiers, France
Received October 9, 1990; revised version accepted January 2, 1991
ABSTRACT Fractal measurements using the Cantor's dust method in a linear one-dimensional analysis mode were made on the fracture patterns revealed on two-dimensional, planar surfaces in four granites. This method allows one to conclude that: (1) The fracture systems seen on two-dimensional surfaces in granites are consistent with the part of fractal theory that predicts a repetition of patterns on different scales of observation, self similarity. Fractal analysis gives essentially the same values of D on the scale of kilometres, metres and centimetres (five orders of magnitude) using mapped, surface fracture patterns in a Sierra Nevada granite batholith (Mt. Abbot quadrangle, Calif.). (2) Fractures show the same fractal values at different depths in a given batholith. Mapped fractures (main stage ore veins) at three mining levels (over a 700 m depth interval) of the Boulder batholith, Butte, Mont. show the same fractal values although the fracture disposition appears to be different at different levels. (3) Different sets of fracture planes in a granite batholith, Central France, and in experimental deformation can have different fractal values. In these examples shear and tension modes have the same fractal values while compressional fractures follow a different fractal mode of failure. The composite fracture patterns are also fractal but with a different, median, fractal value compared to the individual values for the fracture plane sets. These observations indicate that the fractal method can possibly be used to distinguish fractures of different origins in a complex system. It is concluded that granites fracture in a fractal manner which can be followed at many scales. It appears that fracture planes of different origins can be characterized using linear fractal analysis.
1. Introduction Fracture distribution and fragmentation of r o c k s h a v e b e e n i d e n t i f i e d as p o w e r law p h e n o m e n a [ 1 - 4 ] w h i c h i n d i c a t e s a f r a c t a l r e l a t i o n [5]. F r a c t a l a n a l y s i s h a s b e e n u s e d m o r e r e c e n t l y to characterize the nature of the pattern of fractures in n a t u r a l r o c k s [6] a n d p a r t i c u l a r l y t h e p a t t e r n s in m a j o r f a u l t s y s t e m s [7-9]. T h e p h y s i c s o f f r a c t u r i n g h a s b e e n to b e c o n s i d e r e d to b e s e l f - s i m i l a r a n d h e n c e f o l l o w t h e rules o f f r a c t a l b e h a v i o r [10,11]. It s h o u l d t h e r e f o r e b e e v i d e n t t h a t t h e use o f a f r a c t a l a p p r o a c h to i d e n t i f y f r a c t u r e p a t t e r n s has g r e a t p o t e n t i a l . H o w e v e r , s e v e r a l r e l a t i o n s w h i c h are a s s u m e d to h o l d in f r a c t a l t h e o r y m u s t b e d e m o n s t r a t e d in t h e a p p l i c a t i o n o f f r a c t a l a n a l ysis to the c o m m o n g e o l o g i c o b s e r v a t i o n o f fractures, e i t h e r m a p s ( t w o - d i m e n s i o n a l r e p r e s e n t a t i o n ) o r drill h o l e d a t a ( e s s e n t i a l l y a o n e - d i m e n 0012-821x/91/$03.50
© 1991 - Elsevier Science Publishers B.V.
sional manifestation). One demonstration of great i m p o r t a n c e is t h a t o f scale. T h e basis o f f r a c t a l a n a l y s i s is t h a t e v e n t s o c c u r in t h e s a m e t y p e s o f d i s t r i b u t i o n o n d i f f e r e n t scales o f o b s e r v a t i o n [12] and most fractal analysis methods use a means of s c a l i n g in o r d e r to d e t e r m i n e the f r a c t a l r e l a t i o n . A n e v e n t o c c u r s as a p o w e r o f t h e d i m e n s i o n o f o b s e r v a t i o n r = x - D w h e r e D is t h e f r a c t a l dimension or number, x the dimension of the measurement and r the occurrence. However the obs e r v a t i o n a l scale u s e d in c u r r e n t s t u d i e s o f fractures in r o c k s u s u a l l y d o e s n o t c h a n g e b y m u c h m o r e t h a n a n o r d e r o f m a g n i t u d e . I f in f a c t t h e f r a c t a l scale r e l a t i o n a p p l i e s to n a t u r a l , g e o l o g i c a l phenomena such a tool should provide a useful p r e d i c t i v e m e c h a n i s m , f o r if o n e c a n use a s m a l l s a m p l e to p r e d i c t t h e b e h a v i o r o f a l a r g e r o n e , o n e c a n e x p l o r e g e o l o g i c a l s p a c e at l o w c o s t s c o m p a r e d to s t u d i e s in t h e past.
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Beyond the use of fractal analysis as a means of describing patterns of occurrence, it would be useful to be able to characterize the various modes of rupture in rocks so that they could be related to the physical phenomena which have created them. The fractal relation could be used as a method of identification. This paper presents some observations on natural rock bodies using fractal analysis as well as rock samples deformed in the laboratory. This is an initial attempt to illustrate possible uses of fractal analysis in the study of fractured rocks.
2. Materials studied The rock type investigated here is that of granite or acid intrusive material. The advantage of using such a rock type is, obviously, its homogeneity which persists on a large scale in nature. Intrusive, batholithic granitic rocks are most often very homogeneous in their physical and chemical properties. Such material should respond in the most direct manner to deformational forces. It has been used frequently in laboratory studies (Westerly granite used by Brace and co-workers) for some years for just these reasons. Our data sets come from four sources, a series of detailed, published studies of fault and joint traces in granites in the Sierra Nevada batholith (Pollard and co-workers) which have been observed at various scales covering a range of four orders of magnitude, maps of ore-filled fracture patterns (vein) from the Butte mining district, Montana which are taken on different topographic levels; a detailed mapping of fractures and alteration veins in a mine in Central France; and finally a series of detailed fracture maps made on granite samples which have been deformed in the laboratory. These data sets are used: (1) to establish the validity of the fundamental fractal relation of scale for fractures; (2) to establish the fractal relation in three-dimensional space; and finally (3) to attempt to understand the relations between the patterns of the different rupture modes caused by a stress field. It is very important to remember that the fractal theory indicates that events are repeated in the same manner on different scales. The problem for a geologist is to know on what scale he has made his observations and whether or not he is compar-
I~. V E L D E E T AL.
ing things which occur on different scales or inadvertently mixing events belonging to different scales. An example of this problem is briefly dealt with in [6] where it was seen that fractures of different widths in a granite were heterogeneous, i.e. some an aggregate of many fractures and others made up of fewer or single fractures on the scale of unaided visual observation. The total of the observed fractures did not indicate a fractal relation but when separated into two groups on a basis of width they showed a fractal relation. It was apparent that the wider fracture zones in fact were assemblages of smaller fractures and the large zones could not be considered on the same scale as those of smaller width. One runs the risk of mixing multi-element units with those which should be considered to be single-element in a given scale of observation. In other words, a given set of fracture maps observed by a geologist might mix events which occur on different scales of observation. We cannot exclude categorically such an effect, especially when using data gathered by others, which is the case in most of the examples treated here. However, we attempt to use the data as it is presented in a fractal analysis; and if problems arise, they must be dealt with in a more detailed treatment of the data presented. It is best to test the data first and analyze its meaning afterwards.
3. Experimental method The present study uses only one approach of fractal analysis applied to the problem of describing irregular events. Our approach is that of the method of Cantor's dust [13] used by Smalley et al. [14] for problems similar to ours. The Cantor's dust method is applied using a linear analysis of the pattern of linear events (fractures) revealed on a two-dimensional surface (fracture map). The linear approach has been seen to be very useful in time series. Since many fracture patterns are the result of sets of planar events which are commonly revealed in their intersection with a plane of observation (the surface of the earth) as lines, the geometry of this situation presents a series of essentially parallel lines to the observer which have different length and spacings. The line sets of different orientation are assumed to represent different failure causes and modes; tension, shear,
FRACTAL PATTERNS OF FRACTURES IN GRANITES
etc. Given this geometry, it is obviously useful to orient the analysis method orthogonally to these events (sets of parallel lines). Hence a linear analysis perpendicular to the lines of fracture intersection with the analysis plane is appropriate for such a study. It has already been demonstrated that such an analysis can be used to describe fracture patterns revealed on a two-dimensional surface [6]. The advantage of a linear analysis is that it can be oriented in different azimuths of the twodimensional fracture array in order to take into account the orientation differences in the fracture pattern of each set of fracture planes, i.e., shear, tension and compression. The Cantor's dust method gives a fractal number which indicates the probability of finding an event in the next segment of analysis according to the relation S = 1 - P and P = x 1-D where S is the probability of finding an event in the next interval of x distance and P is the probability of not finding an event in that interval. D is the fractal number or dimension. The significance of the value of D is unknown in terms of rock mechanics. It is, at the moment, just a convenient way of comparing one analysis set with another. The method is simple; a linear analysis line is established, divided into segments and the number of events is recorded as a function of the number of intervals which contain them [see 9]. The length of the analysis interval is changed on the analysis line and the operation is repeated. A plot of log x (the dimension of the analysis interval) against log p (the proportion of the intervals along the analysis line which contain at least one event) will give a straight line when the relation is a power law (fractal). If the relation is stochastic or Gaussian in distribution, the points will describe a curve. Two limiting conditions must be established in the Cantor's dust analysis method: (1) a determination of the interval which is large enough so that all contain an event (where p = 1) which gives a log-log slope of zero. Segments of this size and larger must not be included in the analysis as they have no meaning; and (2) the interval which is so small that each event is found in only one interval must also be avoided. Further division of the analysis segment lengths gives a log-log plot of
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Fig. 1. Illustration of the fundamental relations of the log p log x plots of the Cantor's Dust method of fractal analysis where p is the proportion of the measurement units which contain an event and x is the length of the measurement unit. Three slopes are apparent with two critical points indicated as Xmax and Xmin. Xmax indicates the point where the length of the analysis unit (see text) is sufficiently large to always contain an event. Increasing the measurement unit engenders a slope on the plot of O. Xmin indicates the point where each event is found in only one analysis unit and further decrease in the size of the analysis unit creates a constant slope of 1. The useful zone is between Xmax and Xroin.
constant slope also, which is 1. These critical measurement lengths are called here Xmax and S m i n respectively. Any further decrease or increase of the length of the analysis unit, beyond g m i n and Xm,x, will only lead to lines of constant slope on the log-log plot which are artifacts of the analysis method. These relations are given in Fig. 1. If these precautions are not respected, one will develop a graphical analysis of three lines. Such a situation can give an overall disposition of the data which can be mistaken for a curved instead of linear relations which would indicate nonfractal, stochastic relations. In practice, it has been observed that at the point of change between the useful part of the curve and the value of Xmax where the slope of the relations is zero, there is not an abrupt change in slope but the values show a curve. The p values above 0.8-0.9 frequently, but not always, seem not to follow the theoretical fractal relation. This has been noted but not explained in [14]. Therefore these points, between p = 0.8 and p = 1.0 have been omitted from the regression line used to fit the other data. Therefore the value of Xmax is estimated from the intersection of the regression curve and the log x ordinate for p = 1.0.
28
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Also, we have found that the linear relation of the analysis points (usually 5 or 6 in number) should give a correlation coefficient, R, of 0.97 or more. If not, the measurements should be reconsidered. The fractal analysis using the Cantor's dust m e t h o d then gives a slope of the log-log relation which is one minus the fractal n u m b e r or fractal dimension ( D = 1 - slope). In general, a pattern of segregated events, in bursts, will give a slope towards one and a D of near zero. Events that are more evenly spaced tend towards a slope of zero and a D of one (Fig. 2). The special case of perfectly periodically spaced events gives the following relations: any measurement interval of less than the distance between the regularly spaced lines will be below S m i n a s each event or line will occur in only one measurement interval. The slope of values below S m i n is one. W h e n the measurement interval X is greater than half the distance between the lines, Xmax is reached, i.e. all intervals contain an event and p = 1. The slope of values above Xm~, is zero. In the case of a perfectly ordered pattern (Fig. 2a), gma x will be confused with Xmin and the system is unmeasurable by the method used. As the pattern is slightly distorted (Fig. 2b), either by missing lines or by slightly irregular spacings between them, the slope of the fractal relation will b e c o m e apparent, near zero, between g m a x and Xmi.. The length of the interval of measurable fractality will increase as the pattern becomes more segregated or unevenly spaced. In cases where the events
ET
AL.
occur in bursts or clusters with large irregular spaces between them, the slope of the log-log relation will a p p r o a c h one and the fractal n u m b e r will approach zero. In the instance of a very nearly perfect array of fractures, one can easily imagine that each of the heavy fracture lines on the pattern is in reality a group of small, well spaced lines (Fig. 2c). This would be a highly ordered, but fractal relation operating on m a n y scales of observation. Each event is in fact a cluster of events which are revealed as one changes scale. The slope of this ordered, clustered array on a C a n t o r ' s dust analysis will be near one. If one sees only the ordered clusters of lines which make up the larger lines the slope will be 1 and the D value zero. Such a system will not be measurable even though it is ideally fractal. In the case of segregation of the events into groups in an irregular m a n n e r (Fig. 2d) the fractal value increases and the useful measurement zone increases also. This is the most frequent situation for fracture patterns. Thus far the cases investigated by the present authors have not involved such ordered, either clustered or unclustered, patterns. The fractal D values are used for comparative purposes.
4. Examples analyzed F o u r sets of data are used to establish three crucial points in the justification of the application of fractal analysis to fractures in natural rocks. The first point dealt with is the p r o b l e m of scale, repetition of the same p h e n o m e n o n on different scales of observation. Use has been m a d e of the published work of Pollard and co-workers [15-18] all of which is based on the initial largescale m a p p i n g work of L o c k w o o d and L y d o n [19]. The sum of the information is a series of very detailed maps of fault, joint or fracture separations in rocks in the Sierra N e v a d a batholith in east central California. The data sets were derived from maps made from kilometre-scale air photos, maps m a d e from metre-scale g r o u n d observations, and centimetre-scale maps. All of the data comes f r o m the same area in the Bear Creek region on the Muir trail. Fractal, C a n t o r ' s dust analysis has been made of the published m a p s from this area. The second problem is a d e m o n s t r a t i o n of the continuity of a fracture pattern at different posi-
29
F R A C T A L PATTERNS O F F R A C T U R E S IN G R A N I T E S
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Fig. 3. Fracture patterns taken from the studies in the Upper Bear Creek portion of the Mount Abbot area of the Sierra Nevada mountains, Calif. (a) The fracture or joint pattern of the Upper Bear Creek area on a kilometre scale (adapted from [19]). The area analyzed is about 33 km indicated by the dotted line in the figure. Linear measurement direction was on a diagonal in the area indicated, which is perpendicular to the major strike direction of the fractures. Localization of the two sub-areas is indicated by K C and TF. (b) The Kipp camp outcrop of the Upper Bear Creek area showing the fracture pattern mapped on a metric scale [17]. (c) The Trail Fork outcrop from the above reference. (d) A detail of the Trail Fork outcrop of the Upper Bear Creek area [15]. Arrow indicates the direction in which the linear fractal analysis was made. This was assumed to be perpendicular to the major strike direction of the joint fracture field.
TABLE 1 Sierra Nevada Mt. Abbot batholith fractures
D R Xmax Surface Intersections
Upper Bear Creek area
Trail Fork area
Kip Camp
Trail Fork
0.28 0.993 8×102 m 33 km 112
0.35 0.987 17X10 -1 m 17 m 2 115
0.38 0.991 7 x 1 0 -1 m 32 m2 136
0.36 0.994 7X10 -2 m 200 cm 2 135
30
B. VELDE E T AL.
tions in the earth's crust, an approach to the problem of three-dimensionality. The level maps of the Butte mining district (very kindly supplied by G. Burns and S. Czehura, Montana Resources Co.) are from three depths, the 1800, 2800 and 3800 ft levels in the old mining complex. These are the mined veins established as main stage mineralization, which is a secondary event in the alteration sequence of the area [see 20,21]. The importance of these veins is that they were all open to percolation at the same time and represented the open fracture system of the rocks at a geological instant. They do not by any means represent the totality of the fractures observed in these rocks. The third and fourth data set are observations of complex crack systems in granite which represent the shear, tension and compressional mode of deformation. The natural example is from a 200 m deep mine in the French Massif Central ( F a n a y / Augrres, Haute Vienne, France) where a detailed set of maps was made in a mine tunnel of 100 m length. The complementary data set to this natural example is that of an experimental deformation study of cracks formed in a granite subjected to slip-stick deformation under high confining pressure (3 kbar). These data sets give a certain breadth to the scope of investigation concerning the validity of using the Cantor's dust method to describe the fracture patterns in granites.
4.1. Sierra Nevada batholith The Mt. Abbot quadrangle in central eastern California was mapped using air photos to determine the jointing pattern in the area which was largely glaciated giving a good exposure of the bedrock. A portion of this area was mapped in detail on metre scale and finally a map of a small area between two faults which was fractured on centimetre scale has been used. The four fault patterns are given in Fig. 3, where the differences in the roughly parallel sets of joint and shear planes are seen to be rather similar in nature. In fact, the linear fractal analysis of these two-dimensional arrays of fracture patterns shows that they are all very similar in their characteristics of distribution when analyzed perpendicular to the strike of the fault planes. Table 1 gives the relations of each data set.
The values of S m a x a r e those of the length of the unit of measure which will always contain a fault event (proportion of events = 1). This is a sort of measure of the scale of the faulting event which reflects the mode of observation, i.e., the scale at which the events were distinguished. In the case of the Sierra Nevada granite fractures, the Xmax values vary by a factor of 10 5, which is a reasonable demonstration of one of the basic tenets of fractal analysis, that of scale. It can be assumed from this demonstration that the scale of the observation in this range does not matter, the fractal type of the fracture pattern will be repeated up to the point where the stresses on the rock or its response are no longer the same. An example of such a limit might be seen at the scale of individual mineral grains. Here the physical properties of each grain, being different in each species present, will deform the stress field which will create different, very local stress fields according to the competences of the different minerals. This will modify the response of the material and create new, local fracture patterns in each grain according to its mineral species and perhaps orientation. Then, microscopic fracture analysis will not give the same results as those on the centimetric scale. An example of this effect is given in [6].
4.2. Three-dimensional aspect of fractal analysis The ore vein system of the Butte mining district in the Boulder batholith, Montana was used as an example of the applicability of fractal analysis to a three-dimensional problem. The mine maps of three levels were used in a fractal analysis similar to that of the Sierra Nevada granite. These mine maps represent horizontal, two-dimensional plane surfaces at different depths in the batholith. The veins mapped then are the intersections of the veins with these three two-dimensional planes. As in the case of the fractures in the Sierra Nevada batholith, the joint pattern is subparallel, they are
TABLE 2 Butte mining district main stage veins Level 3800' D 0.244 R 0.988 Xmax 90 m Intersections 80-115
2800' 0.268 0.986 105 m
1800' 0.244 0.984 110 m
FRACTAL
PA'ITERNS OF FRACTURES
31
IN GRANITES
also subvertical. In most cases the veins strike N W with minor N E striking components which become more distinct in the upper level. Figure 3 shows the general aspect of the vein pattern at the three levels, which cover a vertical distance near 700 m. The horizontal surface area of each level is about 7 x 105 m 2. Results of the fractal analysis are given in Table 2. Although there is a slight increase in the maxim u m distance necessary to include a vein (Xmax), the fractal ( D ) values do not change significantly from level to level. In moving up in the batholith, one has a visual impression of a stronger NE-striking vein component in the overall fracture patterns. Also the average length of the veins as mapped decreases. However, despite these changes in the fracture pattern, the fractal values are remarkably constant. Therefore, even though the components change somewhat, the fractal nature (probability of finding a fracture event in a linear analysis length of a given distance) does not vary. These results indicate that the fractal method can be used to follow patterns of fractures over significant distances in the third dimension. The fracturing process which the veins follow is the response to unchanging geological conditions of stress, as would be expected seeing the patterns,
and as such it can be followed by the Cantor's dust method.
4.3. Complex fracture sets The first data sets described above involve only single-mode joints or fracture patterns. They are essentially parallel cracks. However, it is important to be able to describe the fractures in the same rocks which are due to different types of failure: shear, tension, and compression. Two sets of fractured granites were investigated in order to observe the importance of the different fracture modes on fractal analysis. The first is a well controlled experiment of deformation in the laboratory using Barre granite (Barre, Vt.) which was deformed under a confining pressure of 3 kbar with a strain rate of 1 0 - 5 / s , or 6.35 x 10 -4 m m / s (see [22] and [23] for details). Eleven stickslip cycles were made in the experiment to deform the 65 m m long by 25 m m diameter sample. Kinked micas were observed in every 4 m m segment of the observed surface of the sample. The diagonal central portion of the sample was strongly brecciated and not used in the fractal analysis procedure. The crack patterns in the non-breccia zone were greatly accentuated in number by the stick-slip cycles. Samples observed after stresses of
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Fig. 4. Maps of the main stage veins of the Butte Mt. Boulder batholith deposit. Three levels are shown. Arrows indicate the direction in which the linear fractal analysis was taken.
32
B. V E L D E E T A L .
TABLE 3 Barre granite, 3 kbar confiningpressure All fractures
Fracture type Tension D 0.29 R 0.985 X,~ax 5.0 mm Intersections > 1000 in each example
Shear
Compression
(I ol)
(II ol)
0.26 0.990 4.7 mm
0.13 0.991 7.6 mm
0.32 0.989 2.5 mm
0.32 0.978 2.8 mm
90% failure showed many fewer cracks than in those the stick-slip experiments. This suggests that the full fracture field was produced after failure of the sample. Observation of the cracks produced by the deformation was made by point counting on petrographic thin sections. Figure 4 shows the fracture map of the sample as seen in thin section. The surface is 24 × 34 mm in dimension. The shaded diagonal area is that of the breccia. The overall fracture pattern was divided into three types: that perpendicular to the long axis of the sample and thus perpendicular to the applied stress (sigma 1); that vertical on the figure and thus parallel to the principal stress; and finally all of the fractures which remain, forming a general 25-60 ° angle with respect to the major stress force. The secondary stresses, confining pressure, are isotropic. The linear fractal analysis of these individual fracture sets and analysis of the total fracture set taken perpendicular and parallel to the deformation direction of ol are given in Table 3. Several remarks can be made. The most interesting observation is that the compressional fractures (due to a decompression of the ol stress and of tension, mode I type) have a different pattern (D value) than those of shear and tension which are very similar. The compressional Xmax value is significantly different from that of the tension and shear fractures. The conclusions to be drawn from this experiment are that the fracture plane sets of the dynamic ruptures (tensional and shear) are spaced with the same probability of occurrence, even though their maximum interspacing (Xm~x) is not identical. The secondary, compressional fracture set (mode I tensional release), due to a relaxation of the compressive, densifying forces, does not follow the same pattern.
4.4. Complex fracture set in a natural granite A set of fractures apparently similar to those of the experiment can be observed in the former mine at F a n n a y / A u g e r s , Drpt. Haute Vienne, France [see 24, 25]. The assembled, mapped fractures in the 100 m long tunnel are complex but can be divided into three categories; vertical, (90 + 15 o ), and horizontal (0 + 10 o ). Other fractures range in bisection angle with these two sets from 25 to 45 o. In general, it is assumed that the major tectonic events in the area occurred in a horizontal direction, defining the O1 direction. There could have been several orientations to successive stress fields. The disposition of the ensemble and the individual fracture families is given in Fig. 5 for a 10 m distance in the 100 m measured in the tunnel. As a first approximation, we assume that the oblique angle faults are of shear origin, and probably the horizontal fractures are of tensional origin originating during the application of the stress field. The vertical fractures could then be considered to be due to a relaxation of the compression, densification event and be of tensional, type I mode. Each set of fractures could in fact be composed of multiple series of planes created by compressional events of different orientation. Table 4 shows the derived fractal numbers for the data sets. It is apparent in this series of determinations that the oblique (shear) and horizontal (tensional) fractures are distributed in the same manner although their frequency (Xmax) is different, as in the case of the experimental deformation. The fractal value of the combined fracture pattern in the direction of the length of the tunnel (horizontal) is 0.40 and Xmax is 0.6 m. The fractures due to the compressional-densification forces give a different fractal number. There is a striking resemblance between the
FRACTAL
PATTERNS OF FRACTURES
33
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//!' ?,~ l! 'J/,~,,' , ..-l,' ~?/z//X.;(f 7 X,sJIT,,'~-' ~:./-,
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l,
" ','
, I~ //lli,.' / // t
//t
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, i/
'
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~, -<+. ~/I//,jli'f l','~l, '71 , i~ '<. -
tension
;'
~ ,,, ,
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i(21
compression
Fig. 5. (a) Fracture map of the Barre granite deformed in the laboratory at 3 kbar confining pressure, el is the deformational axis applied to the sample. Central diagonal area (shown without fractures) was highly brecciated and could not be analyzed. (b) Shear fractures of the sample. (c) Tension fractures of the sample. (d) Compressional fractures of the sample. Arrows indicate the analysis directions.
34
B. V E L D E E T AL.
(a)
<
5. Discussion and conclusions
50 m
60 m
Shear
(c)
/
Compression
,
Tension
Fig. 6. Fracture maps for l0 m of the 100 mapped in the Fanay/Aug~res mine tunnel [24,25]. (a) Total fracture pattern; (b) shear fractures; (c) compressional fractures; (d) tensional fractures. Analysis direction was horizontal in (a)-(c) and vertical in (d).
results from the naturally deformed, multi-plane fracture system and the sample deformed in the laboratory. Both show similar fractal patterns ( D values) for the shear and tension sets of fracture planes while the compressional set is of a different organization. However, the compressional fracture fractal number is greater than the others in the Fanay tunnel and less than the others in the experimentally deformed granite. We have no explanation for this at present.
TABLE 4 Fractures in granite Fanay/Augrres, France Vertical D 0.51 R 0.986 Xma, 2.7 m Intersections 200-400
Horizontal 0.19 0.986 2.7 m
Oblique 0.21 0.994 4.8 m
It is apparent from the measurements made that the fractal relation of scale holds in the fracture patterns in the fracture maps of the granites investigated. The Sierra N e v a d a data show that change of scale by 10 5 does not change the fractal number concerning the disposition of the fractures and joints in these granites. The demonstration of continuity in depth was shown to hold in the mineralized veins of the Butte, Montana Boulder batholith. The fractal pattern holds over a depth range of 700 m. Thus linear fractal analysis of the expression of fracture planes on a two-dimensional surface (fracture map) shows that arrays of fractures are similar at different depths in the crust even though they change slightly towards the surface. These two examples of fractures in granite indicate that the initial premises of homogeneity and isotropic deformation forces are valid on a significant scale and that the linear fractal analysis method is capable of revealing these consistencies. The investigation of the complex fracture patterns which were produced by the one set of stresses, shows that the different failure modes, shear, tension and compressional relaxation, can give different fractal relations, two modes being almost identical in the examples presented and the third different from the other two modes. This indicates that the origin of the different modes follows some different laws. An interesting point in these sets of observations, in both the natural granite and the laboratory deformation, is that the overall fracture pattern, i.e. using all of the fractures, remains fractal although composed of different fractal sets. The fractality of the total fracture patterns can be deduced from the arguments of [5] for the general fragmentation process. Another interesting observation is that all of the cases of the fracture patterns produced in granites from different tectonic environments show nearly the same fractal number, D near 0.3 for the overall fracture patterns. This suggests perhaps that the stresses applied to the granites were similar (undoubtedly with all forces positive and a significant confining pressure) or that the mechanical properties of the granites do not vary greatly from one set of constraints to another. The initial assumption that granites are a gen-
FRACTAL PATTERNSOF FRACTURESIN GRANITES
erally homogeneous material responding in a regular manner to stress sollicitations seems to be reasonably valid.
35
15
References 16 1 C.J. Alirgre, J-L. LeMouel and A. Provost, Scaling rules in rock fracture and possible implications for earthquake prediction, Nature 297, 47-49, 1982. 2 D.E. Grady and M.E. Kipp, Dynamic rock fragmentation, in: B.E. Atkinson, ed., Fracture Mechanics of Rocks, pp. 429-475, Academic Press, London, 6021 pp., 1987. 3 C.G. Jacquin and M. Adler, Fractal geological structures, Acta Stereol. 6, 821-826, 1987. 4 C.A. Barton and M.D. Zoback, Self-similar distributions of macroscopic fractures at depth in crystalline rock in the Cajon Pass scientific drillhole, Proc. Int. Conf. Rock Joints: Scale Effects in Rock Masses, Loen, Norway, in press, 1990. 5 D.L. Turcotte, Fractals and fragmentation, J. Geophys. Res. 91, 1921-1926, 1986. 6 B. Velde, J. Dubois, G. Touchard and A. Badri, Fractal analysis of fractures in rocks: the Cantor's Dust method, Tectonophysics 179, 345-352, 1990. 7 P.G. Okubo and K. Aki, Fractal geometry in the San Andreas fault system, J. Geophys. Res. 92, 345-355, 1987. 8 C.G. Sammis, R.H. Osborne, J.L. Anderson, B. Mavonwe and P. White, Self-similar cataclasis in the formation of fault gouge, Pure Appl. Geophys. 124, 53-77, 1986. 9 C.A. Aviles, C.H. Scholz and J. Boatwright, Fractal analysis applied to characteristic segments of the San Andreas fault, J. Geophys. Res. 92, 331-344, 1987. 10 S.J. Gibowitcz, Physics of fracturing and seismic energy release/a review, Pure Appl. Geophys. 124, 612-658, 1986. 1l G. King, The accommodation of large strains in the upper lithosphere of the earth and other solids by self-similar fault systems/the geometrical origin of b-value, Pure Appl. Geophys. 121, 815, 1983. 12 B.B. Mandelbrot, The Fractal Geometry of Nature, 468 pp., Freeman, New York, N.Y., 1982. 13 G. Cantor, Uber die Ausdehnung eines Satzes aus der Trigonometrischen Reihen, Math. Ann. 5, 123-132, 1872. 14 R.E. Smalley, L-L. Chatelain, D.U Turcotte and R. Prevot,
17
18
19
20
21
22
23
24
25
A fractal approach to clustering of earthquakes: applications to the seismicity of the New Hebrides, Bull. Seismol. Soc. Am. 77, 1368-1381, 1987. P. Segall and D.D. Pollard, Nucleation and growth of strike-slip faults in granite, J. Geophys. Res. 88, 555-568, 1983. P. Segall and D.D. Pollard, Joint formation in granitic rock of the Sierra Nevada, Geol. Soc. Am. Bull. 94, 563-575, 1983. R.K. Davies and D.D. Pollard, Relations between leftlateral strike-slip faults and right-lateral monoclinal kink bands in granodiorite, Mt. Abbot quadrangle, Central Sierra Nevada, CA, Pure Appl. Geophys. 124, 177-201, 1986. S.J. Martel, D.D. Pollard and P. Segall, Development of simple strike-slip fault zones, Mount Abbot quadrangle, Sierra Nevada, California, Geol. Soc. Am. Bull. 100, 14511465, 1988. J.P. Lockwood and P.A. Lydon, Geologic map of the Mount Abbot quadrangle, Central Sierra Nevada, CA, GQ1155, U.S. Geol. Surv., Reston, Va., 1975. R.H. Sales and C. Meyer, Results for preliminary studies of vein formation at Butte, Montana, Econ. Geol. 44, 465-484, 1949. C. Meyer, E.P. Shea and C.C. Goddard, Ore deposits at Butte Montana in Ore Deposits of the United States, Am. Inst. Mining Metall. Pet. Eng. pp. 1373-1416, 1968. D.E. Moore, R. Summers and J.D. Byrlee, Deformation of granite under triaxial stress friction tests, Proc. Int. Conf. Mech. Jointed Faulted Rocks, Vienna, 1990. D.E. Moore, R. Summers and J.D. Byrlee, Faults, fractures and other deformation features produced during loading of granite in triaxial friction tests, U.S. Geol. Surv. Open File rep. 90, 1990. D. Billaux, J-P. Chiles, K. Hestir and J. Long, Three-dimensional statistical modelling of a fractured rock mass-an example from Fanay-Augrres Mine, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 26, 281-299, 1989. D. Lassagne, Essai de charactrrisation du milieu fractur6 en massif granitique, Rep. BRGM 83SGN576, in: Etude des possibilitrs de stockage dans les formations grologiques (granite), Rapp, Srmestriel, Jan. 1987-Juin 1987, CEA SPD 88/06, Fontenay aux Roses, 92, 1988.
25
Elsevier Science Publishers B.V., Amsterdam [DT]
Fractal patterns of fractures in granites B. V e l d e a, j . D u b o i s
b,
D. Moore
c and G. Touchard
~
a D~partement de Gdologie, Ecole Normale Sup~rieure, UR 1316 CNRS, 24 rue Lhomond, 75005 Paris, France b Institut de Physique du Globe Paris, Obseroatoires Volcanologique, 4 pl. Jussieu, 75252 Paris, France c United States Geological Survey, 345 Middlefield Rd., Menlo Park, CA 94025, USA a Laboratoire du M~canique des Fluides, Universitd de Poitiers, Ave. Recteur Pineau, 89022 Poitiers, France
Received October 9, 1990; revised version accepted January 2, 1991
ABSTRACT Fractal measurements using the Cantor's dust method in a linear one-dimensional analysis mode were made on the fracture patterns revealed on two-dimensional, planar surfaces in four granites. This method allows one to conclude that: (1) The fracture systems seen on two-dimensional surfaces in granites are consistent with the part of fractal theory that predicts a repetition of patterns on different scales of observation, self similarity. Fractal analysis gives essentially the same values of D on the scale of kilometres, metres and centimetres (five orders of magnitude) using mapped, surface fracture patterns in a Sierra Nevada granite batholith (Mt. Abbot quadrangle, Calif.). (2) Fractures show the same fractal values at different depths in a given batholith. Mapped fractures (main stage ore veins) at three mining levels (over a 700 m depth interval) of the Boulder batholith, Butte, Mont. show the same fractal values although the fracture disposition appears to be different at different levels. (3) Different sets of fracture planes in a granite batholith, Central France, and in experimental deformation can have different fractal values. In these examples shear and tension modes have the same fractal values while compressional fractures follow a different fractal mode of failure. The composite fracture patterns are also fractal but with a different, median, fractal value compared to the individual values for the fracture plane sets. These observations indicate that the fractal method can possibly be used to distinguish fractures of different origins in a complex system. It is concluded that granites fracture in a fractal manner which can be followed at many scales. It appears that fracture planes of different origins can be characterized using linear fractal analysis.
1. Introduction Fracture distribution and fragmentation of r o c k s h a v e b e e n i d e n t i f i e d as p o w e r law p h e n o m e n a [ 1 - 4 ] w h i c h i n d i c a t e s a f r a c t a l r e l a t i o n [5]. F r a c t a l a n a l y s i s h a s b e e n u s e d m o r e r e c e n t l y to characterize the nature of the pattern of fractures in n a t u r a l r o c k s [6] a n d p a r t i c u l a r l y t h e p a t t e r n s in m a j o r f a u l t s y s t e m s [7-9]. T h e p h y s i c s o f f r a c t u r i n g h a s b e e n to b e c o n s i d e r e d to b e s e l f - s i m i l a r a n d h e n c e f o l l o w t h e rules o f f r a c t a l b e h a v i o r [10,11]. It s h o u l d t h e r e f o r e b e e v i d e n t t h a t t h e use o f a f r a c t a l a p p r o a c h to i d e n t i f y f r a c t u r e p a t t e r n s has g r e a t p o t e n t i a l . H o w e v e r , s e v e r a l r e l a t i o n s w h i c h are a s s u m e d to h o l d in f r a c t a l t h e o r y m u s t b e d e m o n s t r a t e d in t h e a p p l i c a t i o n o f f r a c t a l a n a l ysis to the c o m m o n g e o l o g i c o b s e r v a t i o n o f fractures, e i t h e r m a p s ( t w o - d i m e n s i o n a l r e p r e s e n t a t i o n ) o r drill h o l e d a t a ( e s s e n t i a l l y a o n e - d i m e n 0012-821x/91/$03.50
© 1991 - Elsevier Science Publishers B.V.
sional manifestation). One demonstration of great i m p o r t a n c e is t h a t o f scale. T h e basis o f f r a c t a l a n a l y s i s is t h a t e v e n t s o c c u r in t h e s a m e t y p e s o f d i s t r i b u t i o n o n d i f f e r e n t scales o f o b s e r v a t i o n [12] and most fractal analysis methods use a means of s c a l i n g in o r d e r to d e t e r m i n e the f r a c t a l r e l a t i o n . A n e v e n t o c c u r s as a p o w e r o f t h e d i m e n s i o n o f o b s e r v a t i o n r = x - D w h e r e D is t h e f r a c t a l dimension or number, x the dimension of the measurement and r the occurrence. However the obs e r v a t i o n a l scale u s e d in c u r r e n t s t u d i e s o f fractures in r o c k s u s u a l l y d o e s n o t c h a n g e b y m u c h m o r e t h a n a n o r d e r o f m a g n i t u d e . I f in f a c t t h e f r a c t a l scale r e l a t i o n a p p l i e s to n a t u r a l , g e o l o g i c a l phenomena such a tool should provide a useful p r e d i c t i v e m e c h a n i s m , f o r if o n e c a n use a s m a l l s a m p l e to p r e d i c t t h e b e h a v i o r o f a l a r g e r o n e , o n e c a n e x p l o r e g e o l o g i c a l s p a c e at l o w c o s t s c o m p a r e d to s t u d i e s in t h e past.
26
Beyond the use of fractal analysis as a means of describing patterns of occurrence, it would be useful to be able to characterize the various modes of rupture in rocks so that they could be related to the physical phenomena which have created them. The fractal relation could be used as a method of identification. This paper presents some observations on natural rock bodies using fractal analysis as well as rock samples deformed in the laboratory. This is an initial attempt to illustrate possible uses of fractal analysis in the study of fractured rocks.
2. Materials studied The rock type investigated here is that of granite or acid intrusive material. The advantage of using such a rock type is, obviously, its homogeneity which persists on a large scale in nature. Intrusive, batholithic granitic rocks are most often very homogeneous in their physical and chemical properties. Such material should respond in the most direct manner to deformational forces. It has been used frequently in laboratory studies (Westerly granite used by Brace and co-workers) for some years for just these reasons. Our data sets come from four sources, a series of detailed, published studies of fault and joint traces in granites in the Sierra Nevada batholith (Pollard and co-workers) which have been observed at various scales covering a range of four orders of magnitude, maps of ore-filled fracture patterns (vein) from the Butte mining district, Montana which are taken on different topographic levels; a detailed mapping of fractures and alteration veins in a mine in Central France; and finally a series of detailed fracture maps made on granite samples which have been deformed in the laboratory. These data sets are used: (1) to establish the validity of the fundamental fractal relation of scale for fractures; (2) to establish the fractal relation in three-dimensional space; and finally (3) to attempt to understand the relations between the patterns of the different rupture modes caused by a stress field. It is very important to remember that the fractal theory indicates that events are repeated in the same manner on different scales. The problem for a geologist is to know on what scale he has made his observations and whether or not he is compar-
I~. V E L D E E T AL.
ing things which occur on different scales or inadvertently mixing events belonging to different scales. An example of this problem is briefly dealt with in [6] where it was seen that fractures of different widths in a granite were heterogeneous, i.e. some an aggregate of many fractures and others made up of fewer or single fractures on the scale of unaided visual observation. The total of the observed fractures did not indicate a fractal relation but when separated into two groups on a basis of width they showed a fractal relation. It was apparent that the wider fracture zones in fact were assemblages of smaller fractures and the large zones could not be considered on the same scale as those of smaller width. One runs the risk of mixing multi-element units with those which should be considered to be single-element in a given scale of observation. In other words, a given set of fracture maps observed by a geologist might mix events which occur on different scales of observation. We cannot exclude categorically such an effect, especially when using data gathered by others, which is the case in most of the examples treated here. However, we attempt to use the data as it is presented in a fractal analysis; and if problems arise, they must be dealt with in a more detailed treatment of the data presented. It is best to test the data first and analyze its meaning afterwards.
3. Experimental method The present study uses only one approach of fractal analysis applied to the problem of describing irregular events. Our approach is that of the method of Cantor's dust [13] used by Smalley et al. [14] for problems similar to ours. The Cantor's dust method is applied using a linear analysis of the pattern of linear events (fractures) revealed on a two-dimensional surface (fracture map). The linear approach has been seen to be very useful in time series. Since many fracture patterns are the result of sets of planar events which are commonly revealed in their intersection with a plane of observation (the surface of the earth) as lines, the geometry of this situation presents a series of essentially parallel lines to the observer which have different length and spacings. The line sets of different orientation are assumed to represent different failure causes and modes; tension, shear,
FRACTAL PATTERNS OF FRACTURES IN GRANITES
etc. Given this geometry, it is obviously useful to orient the analysis method orthogonally to these events (sets of parallel lines). Hence a linear analysis perpendicular to the lines of fracture intersection with the analysis plane is appropriate for such a study. It has already been demonstrated that such an analysis can be used to describe fracture patterns revealed on a two-dimensional surface [6]. The advantage of a linear analysis is that it can be oriented in different azimuths of the twodimensional fracture array in order to take into account the orientation differences in the fracture pattern of each set of fracture planes, i.e., shear, tension and compression. The Cantor's dust method gives a fractal number which indicates the probability of finding an event in the next segment of analysis according to the relation S = 1 - P and P = x 1-D where S is the probability of finding an event in the next interval of x distance and P is the probability of not finding an event in that interval. D is the fractal number or dimension. The significance of the value of D is unknown in terms of rock mechanics. It is, at the moment, just a convenient way of comparing one analysis set with another. The method is simple; a linear analysis line is established, divided into segments and the number of events is recorded as a function of the number of intervals which contain them [see 9]. The length of the analysis interval is changed on the analysis line and the operation is repeated. A plot of log x (the dimension of the analysis interval) against log p (the proportion of the intervals along the analysis line which contain at least one event) will give a straight line when the relation is a power law (fractal). If the relation is stochastic or Gaussian in distribution, the points will describe a curve. Two limiting conditions must be established in the Cantor's dust analysis method: (1) a determination of the interval which is large enough so that all contain an event (where p = 1) which gives a log-log slope of zero. Segments of this size and larger must not be included in the analysis as they have no meaning; and (2) the interval which is so small that each event is found in only one interval must also be avoided. Further division of the analysis segment lengths gives a log-log plot of
2"7
IOg x O-
T
~ ~
,~'f
usef ul
zone
X min
o
Fig. 1. Illustration of the fundamental relations of the log p log x plots of the Cantor's Dust method of fractal analysis where p is the proportion of the measurement units which contain an event and x is the length of the measurement unit. Three slopes are apparent with two critical points indicated as Xmax and Xmin. Xmax indicates the point where the length of the analysis unit (see text) is sufficiently large to always contain an event. Increasing the measurement unit engenders a slope on the plot of O. Xmin indicates the point where each event is found in only one analysis unit and further decrease in the size of the analysis unit creates a constant slope of 1. The useful zone is between Xmax and Xroin.
constant slope also, which is 1. These critical measurement lengths are called here Xmax and S m i n respectively. Any further decrease or increase of the length of the analysis unit, beyond g m i n and Xm,x, will only lead to lines of constant slope on the log-log plot which are artifacts of the analysis method. These relations are given in Fig. 1. If these precautions are not respected, one will develop a graphical analysis of three lines. Such a situation can give an overall disposition of the data which can be mistaken for a curved instead of linear relations which would indicate nonfractal, stochastic relations. In practice, it has been observed that at the point of change between the useful part of the curve and the value of Xmax where the slope of the relations is zero, there is not an abrupt change in slope but the values show a curve. The p values above 0.8-0.9 frequently, but not always, seem not to follow the theoretical fractal relation. This has been noted but not explained in [14]. Therefore these points, between p = 0.8 and p = 1.0 have been omitted from the regression line used to fit the other data. Therefore the value of Xmax is estimated from the intersection of the regression curve and the log x ordinate for p = 1.0.
28
B. VELDE
C])
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!
i
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i
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l
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Fig. 2. Indications of different linear patterns. A perfectly ordered array (a) where Xmax = Xmin and no useful slope can be determined on a log-log plot. A slope of near zero will be found on a plot as in Fig. 1 due to a slight deformation of the array (a) where the spacings are slightly changed (b). A slope tending towards one occurs when events are highly grouped, in clusters (c). Intermediate cases (d) give intermediate values of D and slope on a log-log plot.
Also, we have found that the linear relation of the analysis points (usually 5 or 6 in number) should give a correlation coefficient, R, of 0.97 or more. If not, the measurements should be reconsidered. The fractal analysis using the Cantor's dust m e t h o d then gives a slope of the log-log relation which is one minus the fractal n u m b e r or fractal dimension ( D = 1 - slope). In general, a pattern of segregated events, in bursts, will give a slope towards one and a D of near zero. Events that are more evenly spaced tend towards a slope of zero and a D of one (Fig. 2). The special case of perfectly periodically spaced events gives the following relations: any measurement interval of less than the distance between the regularly spaced lines will be below S m i n a s each event or line will occur in only one measurement interval. The slope of values below S m i n is one. W h e n the measurement interval X is greater than half the distance between the lines, Xmax is reached, i.e. all intervals contain an event and p = 1. The slope of values above Xm~, is zero. In the case of a perfectly ordered pattern (Fig. 2a), gma x will be confused with Xmin and the system is unmeasurable by the method used. As the pattern is slightly distorted (Fig. 2b), either by missing lines or by slightly irregular spacings between them, the slope of the fractal relation will b e c o m e apparent, near zero, between g m a x and Xmi.. The length of the interval of measurable fractality will increase as the pattern becomes more segregated or unevenly spaced. In cases where the events
ET
AL.
occur in bursts or clusters with large irregular spaces between them, the slope of the log-log relation will a p p r o a c h one and the fractal n u m b e r will approach zero. In the instance of a very nearly perfect array of fractures, one can easily imagine that each of the heavy fracture lines on the pattern is in reality a group of small, well spaced lines (Fig. 2c). This would be a highly ordered, but fractal relation operating on m a n y scales of observation. Each event is in fact a cluster of events which are revealed as one changes scale. The slope of this ordered, clustered array on a C a n t o r ' s dust analysis will be near one. If one sees only the ordered clusters of lines which make up the larger lines the slope will be 1 and the D value zero. Such a system will not be measurable even though it is ideally fractal. In the case of segregation of the events into groups in an irregular m a n n e r (Fig. 2d) the fractal value increases and the useful measurement zone increases also. This is the most frequent situation for fracture patterns. Thus far the cases investigated by the present authors have not involved such ordered, either clustered or unclustered, patterns. The fractal D values are used for comparative purposes.
4. Examples analyzed F o u r sets of data are used to establish three crucial points in the justification of the application of fractal analysis to fractures in natural rocks. The first point dealt with is the p r o b l e m of scale, repetition of the same p h e n o m e n o n on different scales of observation. Use has been m a d e of the published work of Pollard and co-workers [15-18] all of which is based on the initial largescale m a p p i n g work of L o c k w o o d and L y d o n [19]. The sum of the information is a series of very detailed maps of fault, joint or fracture separations in rocks in the Sierra N e v a d a batholith in east central California. The data sets were derived from maps made from kilometre-scale air photos, maps m a d e from metre-scale g r o u n d observations, and centimetre-scale maps. All of the data comes f r o m the same area in the Bear Creek region on the Muir trail. Fractal, C a n t o r ' s dust analysis has been made of the published m a p s from this area. The second problem is a d e m o n s t r a t i o n of the continuity of a fracture pattern at different posi-
29
F R A C T A L PATTERNS O F F R A C T U R E S IN G R A N I T E S
(b) ~
~.
0 2m
(a)
//~"
~
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/~
/
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TF ::
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0"
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Fig. 3. Fracture patterns taken from the studies in the Upper Bear Creek portion of the Mount Abbot area of the Sierra Nevada mountains, Calif. (a) The fracture or joint pattern of the Upper Bear Creek area on a kilometre scale (adapted from [19]). The area analyzed is about 33 km indicated by the dotted line in the figure. Linear measurement direction was on a diagonal in the area indicated, which is perpendicular to the major strike direction of the fractures. Localization of the two sub-areas is indicated by K C and TF. (b) The Kipp camp outcrop of the Upper Bear Creek area showing the fracture pattern mapped on a metric scale [17]. (c) The Trail Fork outcrop from the above reference. (d) A detail of the Trail Fork outcrop of the Upper Bear Creek area [15]. Arrow indicates the direction in which the linear fractal analysis was made. This was assumed to be perpendicular to the major strike direction of the joint fracture field.
TABLE 1 Sierra Nevada Mt. Abbot batholith fractures
D R Xmax Surface Intersections
Upper Bear Creek area
Trail Fork area
Kip Camp
Trail Fork
0.28 0.993 8×102 m 33 km 112
0.35 0.987 17X10 -1 m 17 m 2 115
0.38 0.991 7 x 1 0 -1 m 32 m2 136
0.36 0.994 7X10 -2 m 200 cm 2 135
30
B. VELDE E T AL.
tions in the earth's crust, an approach to the problem of three-dimensionality. The level maps of the Butte mining district (very kindly supplied by G. Burns and S. Czehura, Montana Resources Co.) are from three depths, the 1800, 2800 and 3800 ft levels in the old mining complex. These are the mined veins established as main stage mineralization, which is a secondary event in the alteration sequence of the area [see 20,21]. The importance of these veins is that they were all open to percolation at the same time and represented the open fracture system of the rocks at a geological instant. They do not by any means represent the totality of the fractures observed in these rocks. The third and fourth data set are observations of complex crack systems in granite which represent the shear, tension and compressional mode of deformation. The natural example is from a 200 m deep mine in the French Massif Central ( F a n a y / Augrres, Haute Vienne, France) where a detailed set of maps was made in a mine tunnel of 100 m length. The complementary data set to this natural example is that of an experimental deformation study of cracks formed in a granite subjected to slip-stick deformation under high confining pressure (3 kbar). These data sets give a certain breadth to the scope of investigation concerning the validity of using the Cantor's dust method to describe the fracture patterns in granites.
4.1. Sierra Nevada batholith The Mt. Abbot quadrangle in central eastern California was mapped using air photos to determine the jointing pattern in the area which was largely glaciated giving a good exposure of the bedrock. A portion of this area was mapped in detail on metre scale and finally a map of a small area between two faults which was fractured on centimetre scale has been used. The four fault patterns are given in Fig. 3, where the differences in the roughly parallel sets of joint and shear planes are seen to be rather similar in nature. In fact, the linear fractal analysis of these two-dimensional arrays of fracture patterns shows that they are all very similar in their characteristics of distribution when analyzed perpendicular to the strike of the fault planes. Table 1 gives the relations of each data set.
The values of S m a x a r e those of the length of the unit of measure which will always contain a fault event (proportion of events = 1). This is a sort of measure of the scale of the faulting event which reflects the mode of observation, i.e., the scale at which the events were distinguished. In the case of the Sierra Nevada granite fractures, the Xmax values vary by a factor of 10 5, which is a reasonable demonstration of one of the basic tenets of fractal analysis, that of scale. It can be assumed from this demonstration that the scale of the observation in this range does not matter, the fractal type of the fracture pattern will be repeated up to the point where the stresses on the rock or its response are no longer the same. An example of such a limit might be seen at the scale of individual mineral grains. Here the physical properties of each grain, being different in each species present, will deform the stress field which will create different, very local stress fields according to the competences of the different minerals. This will modify the response of the material and create new, local fracture patterns in each grain according to its mineral species and perhaps orientation. Then, microscopic fracture analysis will not give the same results as those on the centimetric scale. An example of this effect is given in [6].
4.2. Three-dimensional aspect of fractal analysis The ore vein system of the Butte mining district in the Boulder batholith, Montana was used as an example of the applicability of fractal analysis to a three-dimensional problem. The mine maps of three levels were used in a fractal analysis similar to that of the Sierra Nevada granite. These mine maps represent horizontal, two-dimensional plane surfaces at different depths in the batholith. The veins mapped then are the intersections of the veins with these three two-dimensional planes. As in the case of the fractures in the Sierra Nevada batholith, the joint pattern is subparallel, they are
TABLE 2 Butte mining district main stage veins Level 3800' D 0.244 R 0.988 Xmax 90 m Intersections 80-115
2800' 0.268 0.986 105 m
1800' 0.244 0.984 110 m
FRACTAL
PA'ITERNS OF FRACTURES
31
IN GRANITES
also subvertical. In most cases the veins strike N W with minor N E striking components which become more distinct in the upper level. Figure 3 shows the general aspect of the vein pattern at the three levels, which cover a vertical distance near 700 m. The horizontal surface area of each level is about 7 x 105 m 2. Results of the fractal analysis are given in Table 2. Although there is a slight increase in the maxim u m distance necessary to include a vein (Xmax), the fractal ( D ) values do not change significantly from level to level. In moving up in the batholith, one has a visual impression of a stronger NE-striking vein component in the overall fracture patterns. Also the average length of the veins as mapped decreases. However, despite these changes in the fracture pattern, the fractal values are remarkably constant. Therefore, even though the components change somewhat, the fractal nature (probability of finding a fracture event in a linear analysis length of a given distance) does not vary. These results indicate that the fractal method can be used to follow patterns of fractures over significant distances in the third dimension. The fracturing process which the veins follow is the response to unchanging geological conditions of stress, as would be expected seeing the patterns,
and as such it can be followed by the Cantor's dust method.
4.3. Complex fracture sets The first data sets described above involve only single-mode joints or fracture patterns. They are essentially parallel cracks. However, it is important to be able to describe the fractures in the same rocks which are due to different types of failure: shear, tension, and compression. Two sets of fractured granites were investigated in order to observe the importance of the different fracture modes on fractal analysis. The first is a well controlled experiment of deformation in the laboratory using Barre granite (Barre, Vt.) which was deformed under a confining pressure of 3 kbar with a strain rate of 1 0 - 5 / s , or 6.35 x 10 -4 m m / s (see [22] and [23] for details). Eleven stickslip cycles were made in the experiment to deform the 65 m m long by 25 m m diameter sample. Kinked micas were observed in every 4 m m segment of the observed surface of the sample. The diagonal central portion of the sample was strongly brecciated and not used in the fractal analysis procedure. The crack patterns in the non-breccia zone were greatly accentuated in number by the stick-slip cycles. Samples observed after stresses of
/
j---~__7--~j
1800'
2800'
lOOm
f
f
3800'
Fig. 4. Maps of the main stage veins of the Butte Mt. Boulder batholith deposit. Three levels are shown. Arrows indicate the direction in which the linear fractal analysis was taken.
32
B. V E L D E E T A L .
TABLE 3 Barre granite, 3 kbar confiningpressure All fractures
Fracture type Tension D 0.29 R 0.985 X,~ax 5.0 mm Intersections > 1000 in each example
Shear
Compression
(I ol)
(II ol)
0.26 0.990 4.7 mm
0.13 0.991 7.6 mm
0.32 0.989 2.5 mm
0.32 0.978 2.8 mm
90% failure showed many fewer cracks than in those the stick-slip experiments. This suggests that the full fracture field was produced after failure of the sample. Observation of the cracks produced by the deformation was made by point counting on petrographic thin sections. Figure 4 shows the fracture map of the sample as seen in thin section. The surface is 24 × 34 mm in dimension. The shaded diagonal area is that of the breccia. The overall fracture pattern was divided into three types: that perpendicular to the long axis of the sample and thus perpendicular to the applied stress (sigma 1); that vertical on the figure and thus parallel to the principal stress; and finally all of the fractures which remain, forming a general 25-60 ° angle with respect to the major stress force. The secondary stresses, confining pressure, are isotropic. The linear fractal analysis of these individual fracture sets and analysis of the total fracture set taken perpendicular and parallel to the deformation direction of ol are given in Table 3. Several remarks can be made. The most interesting observation is that the compressional fractures (due to a decompression of the ol stress and of tension, mode I type) have a different pattern (D value) than those of shear and tension which are very similar. The compressional Xmax value is significantly different from that of the tension and shear fractures. The conclusions to be drawn from this experiment are that the fracture plane sets of the dynamic ruptures (tensional and shear) are spaced with the same probability of occurrence, even though their maximum interspacing (Xm~x) is not identical. The secondary, compressional fracture set (mode I tensional release), due to a relaxation of the compressive, densifying forces, does not follow the same pattern.
4.4. Complex fracture set in a natural granite A set of fractures apparently similar to those of the experiment can be observed in the former mine at F a n n a y / A u g e r s , Drpt. Haute Vienne, France [see 24, 25]. The assembled, mapped fractures in the 100 m long tunnel are complex but can be divided into three categories; vertical, (90 + 15 o ), and horizontal (0 + 10 o ). Other fractures range in bisection angle with these two sets from 25 to 45 o. In general, it is assumed that the major tectonic events in the area occurred in a horizontal direction, defining the O1 direction. There could have been several orientations to successive stress fields. The disposition of the ensemble and the individual fracture families is given in Fig. 5 for a 10 m distance in the 100 m measured in the tunnel. As a first approximation, we assume that the oblique angle faults are of shear origin, and probably the horizontal fractures are of tensional origin originating during the application of the stress field. The vertical fractures could then be considered to be due to a relaxation of the compression, densification event and be of tensional, type I mode. Each set of fractures could in fact be composed of multiple series of planes created by compressional events of different orientation. Table 4 shows the derived fractal numbers for the data sets. It is apparent in this series of determinations that the oblique (shear) and horizontal (tensional) fractures are distributed in the same manner although their frequency (Xmax) is different, as in the case of the experimental deformation. The fractal value of the combined fracture pattern in the direction of the length of the tunnel (horizontal) is 0.40 and Xmax is 0.6 m. The fractures due to the compressional-densification forces give a different fractal number. There is a striking resemblance between the
FRACTAL
PATTERNS OF FRACTURES
33
IN GRANITES
(a)
0
2
I
4ram
Oi
(k~-~
~
~
i
\-.,
\-z
/, )/-..~; y . -~ /
"
, II ' t' '~ 4 I) ~ ~", ~q,,,,,
,;
>~' , ~ ' 7 - ' y / /
r....
, , . ~,, ,,~,- , ". 5 : 1 / ; ~ ; , ~ --\ ,'2 .">."/ ~ \~-i -'-t"¢Ms"," ~ j .~/~'] ~is"."-< ) " [7,,\7 f,$"- ' ,"--~
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,
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,
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s
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, 'f,fT,.,
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"l
//!' ?,~ l! 'J/,~,,' , ..-l,' ~?/z//X.;(f 7 X,sJIT,,'~-' ~:./-,
'l,', ',/
(c), y,,
" L . /,
,~.~,
l,
" ','
, I~ //lli,.' / // t
//t
I'
, i/
'
y~ ,I
//,,/,~lf,~It,
~, -<+. ~/I//,jli'f l','~l, '71 , i~ '<. -
tension
;'
~ ,,, ,
,I,"] I
i(21
compression
Fig. 5. (a) Fracture map of the Barre granite deformed in the laboratory at 3 kbar confining pressure, el is the deformational axis applied to the sample. Central diagonal area (shown without fractures) was highly brecciated and could not be analyzed. (b) Shear fractures of the sample. (c) Tension fractures of the sample. (d) Compressional fractures of the sample. Arrows indicate the analysis directions.
34
B. V E L D E E T AL.
(a)
<
5. Discussion and conclusions
50 m
60 m
Shear
(c)
/
Compression
,
Tension
Fig. 6. Fracture maps for l0 m of the 100 mapped in the Fanay/Aug~res mine tunnel [24,25]. (a) Total fracture pattern; (b) shear fractures; (c) compressional fractures; (d) tensional fractures. Analysis direction was horizontal in (a)-(c) and vertical in (d).
results from the naturally deformed, multi-plane fracture system and the sample deformed in the laboratory. Both show similar fractal patterns ( D values) for the shear and tension sets of fracture planes while the compressional set is of a different organization. However, the compressional fracture fractal number is greater than the others in the Fanay tunnel and less than the others in the experimentally deformed granite. We have no explanation for this at present.
TABLE 4 Fractures in granite Fanay/Augrres, France Vertical D 0.51 R 0.986 Xma, 2.7 m Intersections 200-400
Horizontal 0.19 0.986 2.7 m
Oblique 0.21 0.994 4.8 m
It is apparent from the measurements made that the fractal relation of scale holds in the fracture patterns in the fracture maps of the granites investigated. The Sierra N e v a d a data show that change of scale by 10 5 does not change the fractal number concerning the disposition of the fractures and joints in these granites. The demonstration of continuity in depth was shown to hold in the mineralized veins of the Butte, Montana Boulder batholith. The fractal pattern holds over a depth range of 700 m. Thus linear fractal analysis of the expression of fracture planes on a two-dimensional surface (fracture map) shows that arrays of fractures are similar at different depths in the crust even though they change slightly towards the surface. These two examples of fractures in granite indicate that the initial premises of homogeneity and isotropic deformation forces are valid on a significant scale and that the linear fractal analysis method is capable of revealing these consistencies. The investigation of the complex fracture patterns which were produced by the one set of stresses, shows that the different failure modes, shear, tension and compressional relaxation, can give different fractal relations, two modes being almost identical in the examples presented and the third different from the other two modes. This indicates that the origin of the different modes follows some different laws. An interesting point in these sets of observations, in both the natural granite and the laboratory deformation, is that the overall fracture pattern, i.e. using all of the fractures, remains fractal although composed of different fractal sets. The fractality of the total fracture patterns can be deduced from the arguments of [5] for the general fragmentation process. Another interesting observation is that all of the cases of the fracture patterns produced in granites from different tectonic environments show nearly the same fractal number, D near 0.3 for the overall fracture patterns. This suggests perhaps that the stresses applied to the granites were similar (undoubtedly with all forces positive and a significant confining pressure) or that the mechanical properties of the granites do not vary greatly from one set of constraints to another. The initial assumption that granites are a gen-
FRACTAL PATTERNSOF FRACTURESIN GRANITES
erally homogeneous material responding in a regular manner to stress sollicitations seems to be reasonably valid.
35
15
References 16 1 C.J. Alirgre, J-L. LeMouel and A. Provost, Scaling rules in rock fracture and possible implications for earthquake prediction, Nature 297, 47-49, 1982. 2 D.E. Grady and M.E. Kipp, Dynamic rock fragmentation, in: B.E. Atkinson, ed., Fracture Mechanics of Rocks, pp. 429-475, Academic Press, London, 6021 pp., 1987. 3 C.G. Jacquin and M. Adler, Fractal geological structures, Acta Stereol. 6, 821-826, 1987. 4 C.A. Barton and M.D. Zoback, Self-similar distributions of macroscopic fractures at depth in crystalline rock in the Cajon Pass scientific drillhole, Proc. Int. Conf. Rock Joints: Scale Effects in Rock Masses, Loen, Norway, in press, 1990. 5 D.L. Turcotte, Fractals and fragmentation, J. Geophys. Res. 91, 1921-1926, 1986. 6 B. Velde, J. Dubois, G. Touchard and A. Badri, Fractal analysis of fractures in rocks: the Cantor's Dust method, Tectonophysics 179, 345-352, 1990. 7 P.G. Okubo and K. Aki, Fractal geometry in the San Andreas fault system, J. Geophys. Res. 92, 345-355, 1987. 8 C.G. Sammis, R.H. Osborne, J.L. Anderson, B. Mavonwe and P. White, Self-similar cataclasis in the formation of fault gouge, Pure Appl. Geophys. 124, 53-77, 1986. 9 C.A. Aviles, C.H. Scholz and J. Boatwright, Fractal analysis applied to characteristic segments of the San Andreas fault, J. Geophys. Res. 92, 331-344, 1987. 10 S.J. Gibowitcz, Physics of fracturing and seismic energy release/a review, Pure Appl. Geophys. 124, 612-658, 1986. 1l G. King, The accommodation of large strains in the upper lithosphere of the earth and other solids by self-similar fault systems/the geometrical origin of b-value, Pure Appl. Geophys. 121, 815, 1983. 12 B.B. Mandelbrot, The Fractal Geometry of Nature, 468 pp., Freeman, New York, N.Y., 1982. 13 G. Cantor, Uber die Ausdehnung eines Satzes aus der Trigonometrischen Reihen, Math. Ann. 5, 123-132, 1872. 14 R.E. Smalley, L-L. Chatelain, D.U Turcotte and R. Prevot,
17
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A fractal approach to clustering of earthquakes: applications to the seismicity of the New Hebrides, Bull. Seismol. Soc. Am. 77, 1368-1381, 1987. P. Segall and D.D. Pollard, Nucleation and growth of strike-slip faults in granite, J. Geophys. Res. 88, 555-568, 1983. P. Segall and D.D. Pollard, Joint formation in granitic rock of the Sierra Nevada, Geol. Soc. Am. Bull. 94, 563-575, 1983. R.K. Davies and D.D. Pollard, Relations between leftlateral strike-slip faults and right-lateral monoclinal kink bands in granodiorite, Mt. Abbot quadrangle, Central Sierra Nevada, CA, Pure Appl. Geophys. 124, 177-201, 1986. S.J. Martel, D.D. Pollard and P. Segall, Development of simple strike-slip fault zones, Mount Abbot quadrangle, Sierra Nevada, California, Geol. Soc. Am. Bull. 100, 14511465, 1988. J.P. Lockwood and P.A. Lydon, Geologic map of the Mount Abbot quadrangle, Central Sierra Nevada, CA, GQ1155, U.S. Geol. Surv., Reston, Va., 1975. R.H. Sales and C. Meyer, Results for preliminary studies of vein formation at Butte, Montana, Econ. Geol. 44, 465-484, 1949. C. Meyer, E.P. Shea and C.C. Goddard, Ore deposits at Butte Montana in Ore Deposits of the United States, Am. Inst. Mining Metall. Pet. Eng. pp. 1373-1416, 1968. D.E. Moore, R. Summers and J.D. Byrlee, Deformation of granite under triaxial stress friction tests, Proc. Int. Conf. Mech. Jointed Faulted Rocks, Vienna, 1990. D.E. Moore, R. Summers and J.D. Byrlee, Faults, fractures and other deformation features produced during loading of granite in triaxial friction tests, U.S. Geol. Surv. Open File rep. 90, 1990. D. Billaux, J-P. Chiles, K. Hestir and J. Long, Three-dimensional statistical modelling of a fractured rock mass-an example from Fanay-Augrres Mine, Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 26, 281-299, 1989. D. Lassagne, Essai de charactrrisation du milieu fractur6 en massif granitique, Rep. BRGM 83SGN576, in: Etude des possibilitrs de stockage dans les formations grologiques (granite), Rapp, Srmestriel, Jan. 1987-Juin 1987, CEA SPD 88/06, Fontenay aux Roses, 92, 1988.