Cantor's Dust Method Research Articles

A random Cantor set as a measure of the statistical homogeneity of fractured rock masses

This study addresses the applicability of the exponent of a power-law scaling model, as an index of the statistical homogeneity of fractured rock masses. We systematically examine the capability of a random Cantor set, Dc, to quantify the combined effect of fracture density and size distribution on fracture networks. Total 129 two-dimensional fracture configurations were generated, based on different combinations of fracture size distribution and fracture density. The Dc for each generated fracture network system was calculated using the random Cantor dust method. We found that the standard deviation of trace length has no effect on the calculated Dc, and the estimated Dc increases with the total density and/or mean trace length. To explore the field applicability of this study, we investigated the statistical homogeneity of a fractured rock mass of the Upper Devonian Grosmont Formation in Alberta, Canada. The results clearly show that the methodologies given in this study are capable of determining the statistical homogeneity of fractured rock masses.

Quantification of Microcrack Anisotropy in Quartzite – A Comparison between Experimentally Undeformed and Deformed Samples

Abstract In this paper, microcrack patterns in a quartzite are quantified using fractal geometry based methods. Since the quartzite does not show a mesoscopic foliation, the fabric was recognized using anisotropy of magnetic susceptibility (AMS) analysis. Microcracks were investigated in thin sections prepared along the three principal planes of the AMS ellipsoid. Point load tests were performed on cores drilled parallel as well as perpendicular to the magnetic foliation. After experimental deformation, thin sections were prepared in two orientations – (a) parallel to the plane of failure (i.e., parallel to the direction of loading), (b) perpendicular to the plane of failure (i.e., perpendicular to the direction of loading), and microcrack patterns in these sections were investigated. The box-counting method of fractal analysis was first applied to microcracks traced from SEM images from each thin section of the experimentally undeformed as well as deformed samples to establish the fractal nature of the microcrack pattern. It was found that in thin sections perpendicular to the direction of loading, the box (fractal) dimension tends to marginally increase. This is inferred as a manifestation of the increase in complexity of the pattern. The software AMOCADO, which is based on the modified Cantor Dust method of fractal analysis, was applied to microcrack pattern from each thin section in order to quantify the pattern anisotropy. It is noted that the anisotropy significantly reduces in sections perpendicular to the loading direction. SEM data are presented to demonstrate that this reduction in anisotropy is on account of generation and/or growth of new cracks in random orientations. It is envisaged that the approach adopted in this investigation maybe useful in rock mechanics and mineral-resource applications in future.

Does Species Evolution Follow Scale Laws? First Applications of the Scale Relativity Theory to Fossil and Living-beings

We have demonstrated, using the Cantor dust method, that the statistical distribution of appearance and disappearance of rodents species (Arvicolid rodent radiation in Europe) follows power laws strengthening the evidence for a fractal structure set. Self-similar laws have been used as model for the description of a huge number of biological systems. With Nottale we have shown that log-periodic behaviors of acceleration or deceleration can be applied to branching macroevolution, to the time sequences of major evolutionary leaps (global life tree, sauropod and theropod dinosaurs postural structures, North American fossil equids, rodents, primates and echinoderms clades and human ontogeny). The Scale-Relativity Theory has others biological applications from linear with fractal behavior to non-linear and from classical mechanics to quantum mechanics.

Quantification of flow patterns in sheared tonalite crystal-melt mush: Application of fractal-geometry methods

Fractal-geometry-based analysis techniques offer simple and efficient ways for analyzing magmatic fabrics that are otherwise difficult to describe quantitatively. This study shows an application of two different methods on flow patterns observed in a syntectonic magmatic body injected into the lower crust. XZ and YZ rock cuts are scanned, and the scans are automatically transferred to binary patterns of mafic and felsic minerals. These are analyzed by boxcounting as well as the modified Cantor-dust method. Box-counting leads to characterization of the entire patterns, proves their fractality in two different scale ranges, and yields information about magma mingling and grain-aggregate forming processes. The modified Cantor-dust method quantifies the anisotropy of pattern complexity and represents a potentially powerful method for determination of shear sense during magmatic flow. Both methods represent useful tools specifically for analyzing diffuse magmatic fabrics and for connecting field-related studies with analyses on the microscale.

Towards automated pattern quantification: Time-efficient assessment of anisotropy of 2D patterns with AMOCADO

The software tool AMOCADO is presented here. It is designed for quantifying pattern anisotropy. This automated method, based on a modified Cantor-dust method, is applied to various artificial and natural 2D patterns. The results illustrate different applications for the software tool. The automation, results in a high data density and in a precision which is manually not achievable and represents a qualitative improvement of the analysis of complex patterns. Specifically, the possible high angular resolution of 1∘ or less, forms the basis for more precise investigations of pattern details which potentially comprises more complex information about the pattern-anisotropy-forming processes, their superposition and interaction.

Anisotropy quantification: the application of fractal geometry methods on tectonic fracture patterns of a Hercynian fault zone in NW Sardinia

Three different fractal geometry methods for pattern analysis are applied on a quartz-filled late-Hercynian fracture zone in NW Sardinia. Fragment–size distribution analysis and the box-counting method offer the possibility to (i) detect self-similarity of patterns and, therefore, argue for a specific pattern forming process, and (ii) allow the quantification of the patterns and their comparison with other patterns from similar natural and artificial environments. The failure of these methods in analyzing pattern anisotropies can be overcome by the Cantor-dust method, direction-related and based on the analysis of 1-D distribution of material in relation to its 2-D orientation. With the aid of fractal-dimension orientation diagrams (DOD), a specific parameter, the azimuthal anisotropy of fractal dimension (AAD), can be determined. It quantifies the pattern anisotropy and, consequently, provides the basis for analyzing the pattern-forming processes. In addition, the Cantor-dust method shows the existence of two different pattern-forming processes.

The alluvial events in the last two centuries at Sarno, southern Italy: their classification and power-law time-occurrence

In past and recent years, the town of Sarno, Southern Italy, about 17 km E from the Vesuvio volcano, has been devastated by alluvial events with abrupt slippings of the volcanoclastic mantle covering the surrounding mountains. The succession of these calamities has been accurately reconstructed since 1794 on the basis of old documents and tentatively classified according to an index of strength that takes into account both the rainfall infiltrated into the fractured rock in the 30 days before the event and the rain falling during the event, apart from other descriptive terms characterizing the event. The catalogue has been verified to be complete only for events classified at least as weak. The application of the Cantor dust method to such a catalogue has allowed the identification of a scale-invariance in the time clustering of the events, even if limited to the interval between 8 and 64 months. An important result is the evidence of a significant decrease in the time clustering of the events after 1854, immediately after the Little Ice Age (ca 1430–1850), characterized by a significant decrease of the time clustering of rainy days in Italy, related to the coherent decrease of zonal atmospheric circulation over the Mediterranean area. Moreover, the application of the rank-ordering statistics to the repose-times between successive events, classified at least as moderate and strong, has provided further significant evidence for time-scale-invariances. The reality of different scale-invariance laws in the time-occurrence of alluvial events, even if indicative of the critical state of the pyroclastic cover of the mountains surrounding Sarno, might be very helpful for the assessment and reduction of the hazard of future disasters. For example, on the basis of the results of the rank-ordering statistics, the most probable occurrence of an alluvial event at Sarno, classified at least as strong is predicted to occur within the 2007–2017 interval.

The Time Clustering of Floodings in Venice and the Cantor Dust Method

The catalogue of sea-floodings in Venice, accurately reconstructed for a period of 12 centuries (interval: 872–1996) has been analysed according to the Cantor Dust method. This provides a means of testing whether clustering in time is a scale-invariant process: if the fraction R of the intervals of length t containing flooding events is related to the time interval by: R∼t (1−D) , then the fractal clustering is occurring with fractal dimension D (0<D<1). The main result is the evidence for a gradual increase of the fractal clustering starting from 1914, when the soil subsidence of the lagoon basin determines an increase in its hydrodynamic response to the marine forcing with a gradual increase of flooding occurrences.

Fractal and length analysis of fractures during brittle to ductile changes

Linear Cantor's dust type fractal analysis of spacial distribution and fracture length analysis has been made on the fracture sets obtained in four deformation experiments performed at different confining pressures (0.5 to 5 kbar) using the Barre Granite. The fractures, mapped using petrographic microscope methods, were subsequently classed into those of shear, tension, and relaxation or decompression origin. Fracture lengths were found to have a distribution similar to a log normal frequency distribution for all sets. The difference between the mean lengths for the fractures of different orientations in a given experiment decreases with pressure, ie. the fracture lengths for all orientations (stress origins) tend to be more homogeneous as confining pressure increases. The fracture sets were also subjected to the linear Cantor's dust method of analysis. These analyses show that the fractal dimension for all of the fractures in a sample increases slightly as confining pressure increases. The difference in fractal dimension of individual fracture sets in a sample (shear, tension, decompression) tends to decrease with increased confining pressure. In the experiments with the lowest confining pressure, one finds the greatest contrasts in both fracture length and fractal dimension for fractures of different origin. These analyses show that one can describe the differences in fracture style in fracture fields using fractal measurements and length measurements. This follows the change from anisotropic stress fields to those more homogeneous through an increase of confining pressure. This shows that one can measure the change in fracture style in going from brittle rupture toward plastic deformation.

Geometrical and fractal analysis of a three-dimensional hydrothermal vein network in a fractured granite

Abstract A three-dimensional study of a fracture network has been performed on a 0.66-m 3 block of fine-grained granite from La Peyratte (Deux-Sevres, France). The block was sawn in nine parallel plates. The fractures associated with altered haloes were mapped for each plate using a photographic method. The nine fracture sets were subjected to an analysis of the percolated network based on a measurement of the width of altered haloes developed around fractures. Hydrothermal alteration is created by circulation of a certain amount of fluid flow in the rock. We assume that the alteration zone width is proportional to fluid mass flow. The width of alteration zones will give a good idea of the path preferentially followed by fluids. The relation between fracture length and width of the alteration zone shows that the total fracture network consists in fact of two sorts. The largest fractures (surface >0.1 m 2 ) create a loose and poorly interconnected pattern in the block. However, they have developed wide alteration zones (> 3 mm) and have then had a great role in fluid conduction. They represent only 10% of the fractures but are responsible for 47% of alteration (in volume). Approximately 80% of these fractures with wide alteration zones are oriented in the same direction. This main network is linked to a more dense, highly interconnected system made of shorter fractures with narrower altered zones. Fractal analysis has also been performed on pictures of the fracture patterns using the linear Cantor's dust method. Fractal dimension values ( D ) indicate an anisotropy of fracture distribution on each plane. Moreover, when the planes are compared to one another in succession, variations of D values can be noticed indicating an inhomogeneity in the third dimension. The study of such a three-dimensional fracture set shows that fluids follow preferential paths within a fracture network and helps the understanding of fracture distribution.

Fractal analysis of fractures in rocks: the cantor's dust method—reply

Fractal analysis of fractures in rocks: the cantor's dust method—reply

Fractal analysis of fractures in rocks: the Cantor's Dust method : Velde, B; Dubois, J; Touchard, G; Badri, A TectonophysicsV179, N3/4, July 1990, P345–352

Fractal analysis of fractures in rocks: the Cantor's Dust method : Velde, B; Dubois, J; Touchard, G; Badri, A TectonophysicsV179, N3/4, July 1990, P345–352

Fractal patterns of fractures in granites

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.

Fractal analysis of fractures in rocks: the Cantor's Dust method

The Cantor's Dust method of fractal analysis has been used to describe the regularity of fracture patterns in geological materials seen on a two-dimensional surface. This linear method of analysis is well adapted to a determination of two-dimensional phenomena. The determination of the fractal dimension, D, indicates that it varies regularly as a function of the orientation of the analysis direction. This is taken to indicate that there is a relation, unexplained as yet, between the regularity of the fracture pattern and the forces which produced it. The use of the Cantor's Dust method on naturally fractured materials ranging in dimension from 3 mm to 45 m in maximum dimension indicates the fractal nature of the fractures found in natural materials. The analysis method can also be used to identify local variations in the stress field which caused failure, such as the fractures in a feldspar compared to those in the host granite, etc. The method appears to be a sensitive descriptive measure of the failure patterns in rocks.