Fractals and Dynamic Systems in Geoscience

The frequency—magnitude statistics of earthquakes have long been known to satisfy the Gutenberg—Richter relation; it is easy to show that this relation is fractal with D=1.8–2. Since it is generally accepted that individual faults have characteristic earthquakes, it follows that the number—size statistics of faults are also fractal. Limited field studies indicate that the surface exposures have a two—dimensional fractal dimension of D ~ 1.6. (The seismic and fault fractal dimensions need not be equal since the repeat time of earthquakes can also have a power—law dependence on scale.) Fragments have a fractal number—size relation with D ~ 2.5 under a wide range of conditions. One model for tectonic deformation is comminution; such a scale—invariant model appears to be consistent with the fractal correlations discussed above. Seismicity has many of the characteristic features of “self—organized criticality”. Energy (strain) is continuously added to the crust and is lost in discrete events (earthquakes) that have a fractal frequency—size distribution. It appears reasonable to hypothesize that the continental crust is everywhere in this critical state (similar to the critical stress associated with perfect plasticity). Evidence for this comes from the distribution of intraplate earthquake and from the occurrence of induced seismicity almost anywhere an artificial reservoir is filled.

Fragmentation over a wide range of size scales has been discussed in previous papers. A variety of statistical power—law relations have often been used successfully to describe scaling laws of the size distribution and shape (surface roughness) of fragments, indicating that fragmentation is a scale invariant process. But it has been assumed that fracture shapes do not have any effect on the fractal dimension of fragment size distributions. New scaling laws of the fracturing energy for rock fragmentation can be derived from concepts of fractal geometry and the Griffith energy balance. These laws show that the two fractal dimensions describing size distributions and roughness in the shape of rock fragments increase as the fracturing energy increases. A relation between the two kinds of fractal dimensions indicates that the fractal dimension of roughness is the mean value of the fractal dimension of fracture—size distribution and the Euclidean space dimension of specimen volume. This relation indicates a constraint of fractal geometry for fragmentation and is expected to be a powerful tool for the fractography analysis in the fields of tectonics and seismicity.

A numerical simulation method is used to predict connectivity of fractured rock masses. Hiere is a threshold of fracture density, below which fractures are poorly connected. Where fracture density is at or above the threshold, there is a continuous fracture cluster (i.e. the largest cluster) throughout the fractured rock mass. Fractal dimension, D_f, is used to describe quantitatively the connectivity and compactness of the largest fracture cluster in the fractured rock mass and increases with fracture density. The critical fractal dimension, D_fc, describes the geometry of the largest fracture cluster at the threshold of fracture density, and has a rather constant value of 1.22 to 1.38 for wide variations in the distribution of size and orientations of the fractures.Simulation of biaxial compressive tests of fractured rock masses has been carried out using a numerical method, UDEC (Universal Distinct Element Code). The deformation of fractured rocks increases greatly with fractal dimension and is mainly created by the shear displacements and openings along fractures. A link between fracture density and deformability of a fractured rock mass is established through the fractal dimension.

The spatiotemporal evolution of damage in stochastic rocks is studied in terms of the multi—scale approach with the fractal tree used as an analogue of a discretization scheme for the region under study. Such an approach, based on the fractal theory, allows quantitative information on spatial (morphology of fracture surface) and temporal scaling of the failure process to be obtained. The numerical simulation for different types of stochasticity has shown that the fractal dimension of the fractured zone is the invariant characteristic of the failure process and depends totally on the stochastic properties of material. The analysis of the load vs time—to—fracture relation for a vast interval of loads proved the scale—invariance of the fracture process. The multifractal properties of the load distribution near fractures are studied.

Generation and propagation of macroscopic joints in brittle rocks depend on the evolution of defects and their interaction with cracks. The proposed approach accounts for damage accumulation and joint—damage interaction in terms of local stress intensity factors (SIF). The numerical simulation of the joint’s development in the 2D region is carried out for the various kinds of the rock stochasticity. It is shown that the fractal dimension of the crack front is the invariant of the joints’ propagation process: the fractal dimension increases with the uniformity of the material properties distribution.

The present chapter investigates the applicability of fractal geometry in fractography. Samples at different times during excavation have been studied to examine fractal characteristics of fracture surfaces. The fractal dimensions of the fracture surfaces were determined by using a vertical section method. Fracture toughness has been obtained by using the three point SENB method. Results show the fracture surfaces to be fractal, but no correlation between the measured quantities has been noticed.

We propose a simple model suitable for the computer simulation of fracture phenomena in brittle materials (rocks, ceramics). The real object of interest —polycrystalline sample — is replaced by a mathematical object—lattice of binary variables. Every variable represents a bond between two structural units of the sample and the values of this variable correspond to the two possible states of the bond (intact, broken). So any situation (crack inside the sample) can be described by the distribution of suitable values of variables on a model lattice and the energy of the real state can be associated with a given distribution on the basis of the physical considerations.For illustration we performed Monte Carlo simulation on a simple model (two—dimensional square lattice of variables and the energy determined according to Griffith’s theory). We obtained the shape of the crack and the fractal dimension of the fracture pattern as functions of temperature.

The brittle widening of rocks during regional extension follows a power—law relationship and, therefore, is scale—invariant. The development of extensional veins during this process is best described by the Cantor—dust model with a fractal dimension 0.955 < D < 0.970. D is independent of the rock types, their anisotropics, the amount of widening, the vein type and the vein thickness. The model implies that successively thinner veins are formed reflecting the increase in irregularity of the total stress field with time.

Displacement on faults can be described using a power—law distribution with exponent D. To justify using a power—law, the relationship needs to be proved over several orders of magnitude. Offshore, faults with displacement >20m can be measured within Triassic rocks from seismic reflection profiles. Around the Moray Firth there is limited exposure of Triassic rocks which are continuous with the offshore basin. Faulted sections from Triassic sandstones were measured, with fault displacements varying from 1 to 500 mm. These seismic and field data are limited in both size and scale range.Computer simulations were run where a population with a known power—law was sampled to see how the measured distributions matched the original. The samples gave a variable and biased power—law exponent. Strategies were designed to find the underlying value from these samples. The simulations were also used to estimate the confidence intervals for the D—values calculated.A combined graph of the field and seismic data showed that a single power—law with a D—value of—0.8 can describe the fault population in this area.

The relaxation function φ (ξ) for high temperature flow of rocks can be represented by φ (ξ) ∝ ξ ^γ where ξ is the temperature reduced time. This shows the relaxation behavior has a temporal fractal property called a long time tail. This relaxation function can be regarded as the result of the superposition of standard exponential decay functions with different mean lifes, τ. Here, a universal relaxation function of rocks is introduced and the relaxation of high temperature flow of rocks is discussed from the viewpoints of fractal concepts.

Geologic structures are repetitive in a quasi—periodic or erratic manner. The geometries of these structures are also manifestations of the mechanical behaviour of a deforming rock mass. Can we therefore obtain information on the dynamics of a complex geologic system from an examination of the geometry of geologic structures? We examine this question from the point of view of non—linear dynamics.We investigate the variability in space of the velocity of growth of crenulations in a model rock mass which is undergoing a (numerical) simple shearing deformation. That is, we follow the distribution and evolution of the velocity in space, rather than in time. We investigate the thesis that the behaviour of this one variable reflects the presence of all other variables participating in the dynamics and, by use of increasing multiples of a fixed space lag, discretize the system, and unfold the system’s dynamics into a multidimensional phase space. The trajectories within this phase space of the system converge to a subspace which is the geometrical attractor for the system. We infer from this that our deforming model rock can be described by a set of deterministic laws. The dimension of this attractor is about 2.5; that is, the system may be completely represented by a fractal attractor. Further, this fractal attractor embeds in a phase space of three so that at least three variables must be considered in the description of the underlying dynamics. These are the variables involved in the three independent differential equations of the numerical model: the stress equations of motion, the yield criterion and the flow rule. We conclude that geological systems may be successfully modelled on the basis of such a system of equations, and analysed using the concepts of fractal geometry.This new application of nonlinear dynamics to the spatially erratic structures of deformed rocks results in an improved understanding of rock deformation behaviour and in an improved prediction of the distributions of structurally—controlled phenomena.

Two sources of non-linearity in the initiation and continuation of rupture processes are the build-up of critical stresses and their propagation along inhomogeneous rupture zones including the underlying non-seismogenic crust. The most important sources for a non-linear build-up of critical stresses seem to be dilatancy processes, transfer of stresses from nearby foreshocks, and possibly interactions with the lower crust. Propagation of rupture processes are dependent on inhomogeneities of the dynamic friction, asperities, fault gouge, geometry, and possibly complex interaction with the non-seismogenic lower crust. Models based on chaos theory are shortly reviewed. They can explain several statistically observed phenomena. A deterministic approach to earthquake prediction requires a dense network as well as short-and long-time monitoring of any deformation in the neighbourhood of the suspected rupture area. In accordance with weakly chaotic systems a limited prediction might be possible.

This chapter concerns the study of the temporal evolution of seismicity in a particular region, through a time repeated fractal analysis of the spatial distribution of the seismic events. The technique is applied to the NE—Italy Friuli seismic region, where a local seismic network has been active since 1977 (OGS 1977–1981, 1982–1990). Fractal dimensions have been calculated at fixed time intervals of thirty days. The spectrum of the time series thus obtained reveals that in this region the seismicity is characterized by a sort of periodicity, with periods that vary from one year to about four years, superimposed onto a longer period term.The comparison of these results with the geodetic measurements in the same region suggests a ‘periodic’ variation of the stress field as the cause of both phenomena, and confirms the usefulness of this kind of fractal analysis in the study of the seismic process.

A cellular automaton for seismological stick—slip processes is developed. Reducing complexity to a minimum, a binary description is obtained for a Poincaré section defined by the states of rest. This description allows us to compare the results with the binary number—theoretical automata by Wolfram. It is found that the Poincaré section can be approximated by stochastic modifications of Wolfram’s automata, and a classification into the four automata classes proposed by Wolfram is possible. In aperiodic regimes, the spatial measure entropy reaches a constant minimum after transients. Gutenberg—Richter statistics yield large, resp. small, slopes for large, resp. small seismic moments, as well as a “characteristic earthquake”, in agreement with seismological and geological data.

In the mid 1980’s, fractal research into earthquakes developed in China and became an active branch. This chapter gives a review of the general situation of fractal research into earthquakes in China, on the important topics which have been investigated and on the problems which still exist and should be solved.The main topics of fractal research into earthquakes in China are as follows: (a) It is testified by much evidence that the temporal, spatial and magnitude distributions of the earthquakes are self—similar on some levels and scales, and are fractal in the statistical sense. (b) Before and after a strong earthquake, the fractal structures of the medium—small earthquakes around the epicenter region are complex: some show deepen dimension while others show rising dimension. (c) The fault systems on the Chinese continent have fractal structures. In general, the thrust structure has a lower fractal dimension than the tensional. (d) It is also found that the tectonic topography, which is related to neotectonic and earthquake activity, also has fractal structures, such as the channels, ridges, and so on. (e) Fractal research into earthquakes has led to the development of a new discipline in seismology and new ideas in geology.Some problems should still be solved: (a) There is still a lack of scientific demonstration as to how many statistical samples are enough for fractal analysis of earthquakes. (b) There is no unified quantitative standard to determine the upper and lower limits of the scaling range. (c) The parameters Dq and f (a) should be studied intensively.

Recent models of the earth’s tidal potential include more than 103 individual harmonic constituents (Tamura 1987, Xi Qinwen 1987, 1989), enough to screen these data for the existence of self—similarities. It is found that the number of constituents surpassing a certain amplitude level shows a fractal distribution over at least four orders of magnitude. Its fractal dimension is close to 0.5. Accordingly, a refined tidal model which is complete should include about 10 times as much harmonic constituents as a coarser model, when the smallest amplitude of the refined model is 100 times smaller than that of the coarse one. Whether this feature masks an inherent structure in the spatial distribution or in the dynamics of celestial bodies governing the tidal forces is yet unknown.

On the base of a one—degree grid of elevation values lengths of different topographic isolines were computed for seven grades of resolution. It was ascertained that elevation contours have fractal dimensions significantly greater than unity at the scales from 100 to 2000 km. Some additional considerations extend these limits to 5 km and 20000 km respectively. Average value of fractal dimension of global relief as a whole is stated to be 1.37. To test this estimation the topography of the equatorial belt was examined with the help of the method of variance which provided the value of 1.38. An original method of dispersion counter—scaling was also applied and proved the existence of spatial variations in self—similar properties of regional topography.

Fractal geometry is the creation of Mandelbrot, who published his first book on the subject in English in 1977 (Mandelbrot 1977). Fractal geometry is the geometry of rugged systems, that is objects having “non—smooth” boundaries and non—Euclidean shapes. From a study of rugged systems fractal geometers have derived many descriptive parameters for describing fractured systems such as broken rock and the random space filling structures of sedimentary systems such as sandstone and other porous systems. The term fractal geometry was coined by Mandelbrot from a Latin word meaning fractured. In the 15 years since the publication of Mandelbrot’s book, fractal geometry has found many applications in the mining industry.

Gold deposits in two study areas in the Zimbabwe Archaean craton have fractal spatial distributions over a length scale from 2.5 to 25 km. The number of squares of side d necessary to cover every deposit is proportional to a power of the length of the square, and the number of deposits within a circle of radius r is proportional to a power of the radius. The fractal dimension, given by the exponent of the length scale in each method, is approximately 1. The fractal relations for both study areas are very similar. The distributions of deposits are interpreted as the result of hydrothermal mineralization by fractal fluid systems focussed in deformation zones.

Self-organization phenomena leading to the formation of patterned mineral fabrics are discussed in connection with electric field effects frequently occurring in the lithosphere. The different internal sources of electric potentials which may intensify ionic fluxes over long time scales are compiled. Model experiments with electrolysis in quartz sand basins between iron electrodes lead to the formation of Liesegang—like precipitate bands of iron hydroxides. Amplification of inhomogenities in the electric field or the capillary transport results in undulatory shapes of precipitation bands. Local breakthrough phenomena of ionic transport are observed leading to “boudinage” patterns along a horizontal precipitation band or “breccia”-like fabrics on top of vertical breakthroughs. Ripening effects within the primarily formed bands due to competitive particle growth lead to speckled or nodular patterns.Several morphological similarities between these experimental findings and mineral fabrics are pointed out. Furthermore, the role of other non—electric effects, like the presence of diffusion barriers and capillar front instabilities (“Runge” pictures) in the evolution of geological patterns, is discussed.

The formation of periodic band structures involving certain structural defects (lateral interruptions and branching of bands) through selforganization in colloidal media is demonstrated both by classical Liesegang experiments and by numerical simulation of a competitive particle growth model. Fragmentation or branching of bands can be explained as consequence of an instability during the precipitation process itself (Ostwald ripening) by excluding of any external mechanical influences. The detailed morphological similarity of these patterns with periodic bands in some mineral samples (zebra rock) suggests that the mineral pattern formation was in the same manner governed by self-organized precipitation processes.

A variety of objects growing in lithified or unlithified environments exhibit self-organization. These include single crystals and polycrystalline or amorphous solids such as agates, geodes, concretions and orbicular granites. Here we discuss mechanisms for their genesis and quantitative reaction-transport-(and possibly)mechanical (RTM) equations that can account for many of their most spectacular characteristics.The main two classes of phenomena are internal compositional or textural patterns and morphological patterns. These phenomena can occur under a wide range of conditions of geochemical change: igneous, metasomatism, metamorphosis, contact metamorphosis, diagenesis and weathering. Conditions of change, and more precisely of strong displacement from thermodynamic equilibrium, are required for self-organization.Mechanisms of self-organization fall into two classes. Growth instabilities occur during the growth of the inclusion from components in the host medium. In contrast to these “open system” phenomena, there are mechanisms which rely only on material in the inclusion (already formed) in its pre-self-organized state in the matrix. Both types of models are considered, contrasted and formulated as quantitative RTM models.Self-organization dynamics of four types are examined. In the transport-limited, open case the inclusion grows from an initial nucleus or seed as a low or zero porosity body with a feedback arising from the growth rate mediated by the state of the surface of the body and that of the host medium near the body surface. Another open case involves growth of isolated crystals in a nucleation zone from mobile components derived outside the original zone; growth of these crystals may be mediated by their surface composition and that of the host near their surface and may proceed to the removal of host grains within their interior by force of crystallization effects. A third mechanism involves the replacement of an initial inclusion phase by a late phase such that minimal mass exchange between the inclusion and host medium is required. Finally, growtn of the body may be under open or closed conditions but involves Ostwald ripening.

Manganese oxide patterns known as “pyrolusite dendrites” are explained as the result of the mineral record of flow structures in porous media. This interpretation is supported by a) fractal characterization, b) Mn profiles across the mineral pattern and the matrix rocks, c) structures reminiscent of flow pattern observed at the scale of the grain size of the matrix rocks, d) the lack of long-range order of the manganese oxide particles and e) the existence of clays and quartz grains of colloidal size intimately linked to the manganese particles. This interpretation also explains other fractal and non-fractal patterns accompanying the beautiful treelike fractal forms associated with these manganese oxide patterns.

Based on a continuum model of nucleation and growth in two and three dimensions we present results for grain size and cluster size distributions for different nucleation and growth mechanisms. The percolative properties of these systems are determined with respect to percolation threshold and appropriate exponents for volume, surface, and hull of the clusters. The universal behaviour of these systems is confirmed. The present results are applicable for the description of crystallization processes in igneous rocks as well as for understanding the structure of pore spaces in sedimentary rocks.

In this paper, the fractal structure model, fractal dimensional calculation and the expressions of quasicrystal lattice with fivefold or eightfold symmetry have been discussed. In a fivefold symmetry quasicrystal, the foundational cell is an icosahedron, and the enlargement coefficient is l+(√5+l)/2, and D = 2.6652. In an eightfold symmetry quasicrystal, the foundational cell is a hexakaidecahedron, and the enlargement coefficient is 1+ √2, and D = 2.7206. The fractal structure model has many advantages over the Penrose model. These quasicrystal fractal lattices patterns drawn by my mapping program are close to the high-resolution electron microscopic image of real quasicrystal.

Octahedral coordinated aluminum is released by the attack of protons from clay minerals like montmorillonite. Apart from other effects, this leads to an alteration of the form of the surface of the clay particles. We used fractal geometry to describe this change in form. The effective surface fractal dimension (D) was determined before and after acid treatment by using the dependency: number of yardsticks necessary to cover the surface versus size of the yardstick area. We used n-alkanols with different cross sectional areas to determine this dependency. This resulted in a D: 2.55 for the untreated and a D: 2.85 for the acid treated clay.

Porous media in geologic systems are usually inhomogeneous with respect to porosity and permeability. In addition, these parameters may evolve with time because of the chemical interaction between solids and pore fluid.We will illustrate how fluxes evolve in strongly inhomogeneous media and how different flow patterns and diagenetic changes depend on the porosity and permeability distribution.Finally we attempt to characterize anisotropic media by simple scaling rules and power laws as well as to predict the average effective transport coefficients. It is shown that the anisotropic media can be transformed into equivalent isotropic media with respect to the conductivity tensor.

It is shown that SEM/EDX imaging is a powerful tool for detecting the spatial distribution of different elements and minerals in recent bioactive sediments. The evaluation of these distribution patterns by multifractal analysis provides a quantitative assessment of the characteristic inhomogeneities.The combination of optical and multifractal techniques leads to the discrimination of intricate structures, the description of element — element correlations, and the identification of specific processes: e.g. the formation of (i) microbially induced calcite or (ii) authigenic clay—minerals indicated by an evident local correlation between aluminium and silicium.

Fractal dimensions of oxygen isotope (δ18O) data obtained from deep—sea records were estimated. The globally distributed data reflect climate variability during the late Pleistocene.All fractal dimensions fall into the range of 1<D<2. However, the records do not show self—affinity over the entire range of investigated time scales, instead two significantly different fractal dimensions can be observed. Ulis result contrasts with previous suggestions that time series obtained from climate proxies might possess self—affinity with a single fractal dimension over a range of 10 to 105 years.The estimated fractal dimensions of the (δ18O) records for time scales between 3 and ~20 ka fall into a narrow range, with an average value of D = 1.51 ± 0.06.Estimated fractal dimensions for longer time scales may be biased by the presence of periodic components in the time series. The limited range of time scales which can be described by a single fractal dimension raises questions about the applicability of simple fractal theory to this kind of data.

A new method of estimation of fractal properties of self—affine sets is suggested which allows the analysis of scale invariance of many natural processes, objects, and phenomena. The method is based on scale tracing of the values of two different kinds of dispersions of the set — relative to the smallest and to the largest possible scales. They are called internal and external dispersions respectively. Some case histories of its use are discussed showing that the “counter—scaling” method provides reliable estimates of fractal dimensions for various geophysical sets.

The Grassberger—Procaccia algorithm, which allows one to estimate the correlation dimension of a chaotic system, was investigated on oxygen isotope data from the deep—sea sediment core ODP Site 659 from the eastern North Atlantic. These data provide uniform coverage of the last 5.2 Ma and show an average spacing of 4.5 ka (1170 points). They are assumed to reflect global ice volume as a proxy for global climate.This paper focusses on methodical aspects. One important point consists in the use of nonlinear fits instead of linear fits after a logarithm transformation. The latter overestimates the correlation dimension. An advantage is that the comparable large amount of original data allowed the number of (equidistant) working—data points (generated by linear interpolation) to be held equal to the number of original data points in order to strive for small statistical dependences between the working—points.The Pliocene and Pleistocene climatic signatures can clearly be distinguished in the reconstructed space of climatic states.The behavior of the estimated correlation dimension on the embedding dimension does not indicate the existence of a low—dimensional attractor. The behavior on a variable time lag does not indicate a definite lower boundary for the time lag above 4.5 ka. The behavior on the fit region shows a strong dependence on the upper boundary for the region, whereas the dependence on the lower boundary is weak.