Fractal Size Distribution Research Articles

A scale-independent approach to fracture intensity and average spacing measurement

Fracture intensity, the number of fractures per unit length along a sample line, is an important attribute of fracture systems that can be problematic to establish in the subsurface. Lack of adequate constraints on fracture intensity may limit the economic exploitation of fractured reservoirs because intensity describes the abundance of fractures potentially available for fluid flow and the probability of encountering fractures in a borehole. Traditional methods of fracture-intensity measurement are inadequate because they ignore the wide spectrum of fracture sizes found in many fracture systems and the consequent scale dependence of fracture intensity. An alternative approach makes use of fracture-size distributions, which allow more meaningful comparisons between different locations and allow microfractures in subsurface samples to be used for fracture-intensity measurement. Comparisons are more meaningful because sampling artifacts can be recognized and avoided, and because common thresholds of fracture size can be enforced for counting in different locations. Additionally, quantification of the fracture-size distribution provides a mechanism for evaluation of uncertainties. Estimates of fracture intensity using this approach for two carbonate beds in the Sierra Madre Oriental, Mexico, illustrate how size-cognizant measurements cast new light on widely accepted interpretation of geologic controls of fracture intensity.

Typical phases of pre-failure damage in granitic rocks under differential compression

Abstract The evolution of pre-failure damage in brittle rock samples subjected to differential compression has been investigated by means of acoustic emission (AE) records. The experimental results show that the damaging process is characterized by three typical phases of microcracking activity: primary, secondary, and nucleation. The primary phase reflects the initial activity of pre-existing microcracks, and is characterized by an increase, with increasing stress, both in event rate and b value. The secondary phase involves subcritical growth of the microcrack population, revealed by an event rate increase and a dramatic decrease of the b value. The nucleation phase corresponds to initiation and accelerated growth of the ultimate macroscopic fracture along one or more incipient fracture planes. During the nucleation phase, the b value decreases rapidly to the global minimum value around 0.5. The temporal variation of b in every phase clearly correlates with grain size of the test sample, hence indicating that a comparatively larger grain size results in a lower b value. In order to investigate the fracture mechanism of each phase, a damage model was tested by employing the constitutive laws of subcritical crack growth of crack populations with a fractal size distribution.

Discrete Element Method Evaluation of Granular Crushing Under Direct Shear Test Conditions

Granular materials forming part of civil engineering structures are subjected to both static and dynamic loads. As a result of these loads, particle crushing occurs. This study focuses on the evolution of crushing in a simulated granular material subjected to a direct shear test. It was found that the discrete element method (DEM) can be used to visualize and understand the evolution of crushing experienced by a granular material during this test. Even though DEM does not normally consider particle breakage, it is possible to simulate crushing by replacing one particle that has failed in tension with a combination of many particles of different sizes. One simulation indicated that crushing did not develop uniformly throughout the sample, and the sample tended to achieve a fractal distribution of particle sizes. The changes in other properties of the simulated granular material such as porosity and shear strength were also recorded, analyzed, and compared to the changes produced during another simulation where crushing was not a predominant feature.

Grain fracture in 3D numerical simulations of granular shear

We present a new method to implement realistic grain fracture in 3D numerical simulations of granular shear. We use a particle based model that includes breakable bonds between individual particles allowing the simulation of fracture of large aggregate grains during shear. Grain fracture simulations produce a comminuted granular material that is texturally comparable to natural and laboratory produced fault gouge. Our model is initially characterized by monodisperse large aggregate grains and gradually evolves toward a fractal distribution of grain sizes with accumulated strain. Comminution rate and survival of large grains is sensitive to applied normal stress. The fractal dimension of the resultant grain size distributions (2.3 ± 0.3 and 2.9 ± 0.5) agree well with observations of natural gouges and theoretical results that predict a fractal dimension of 2.58.

Fractal size and spatial distributions of fault zones: An investigation into the seismic Chelungpu Fault, Taiwan

Abstract The fractality of fault zones of thickness T for the 248‐m‐long core from a borehole penetrating the northern segment of the seismic Chelungpu Fault, Taiwan, was analyzed. The frequency curve of T shows that the fractal dimension is normal when T is smaller than a characteristic thickness Tc, and it becomes abnormally large when T exceeds Tc. The fractal dimensions of size and spatial distributions of T increase as the mean distribution density of T increases, which is inconsistent with the evolution laws for ordinary brittle faults. This discrepancy implies that the thickening rate of T when T is more than Tc is not constant, but a decreasing function of fault displacement. The slow thickening rate is related to the elastohydrodynamic lubrication which was effective on the fault when T exceeds Tc. This slip instability mechanism can explain the large, fast and smooth slip on the northern segment of the Chelungpu Fault during the 1999 Chi‐Chi earthquake.

Tissue phantom for optical diagnostics based on a suspension of microspheres with a fractal size distribution

We demonstrate experimentally the possibility of reproducing the phase function, absorption, and scattering coefficients of a real biological tissue (adult brain white matter and liver) using a suspension of polystyrene microspheres with a fractal size distribution. The design of a light scattering goniometer with a cylindrical cell in air is discussed, and phase function measurements using the device are described. The scattering coefficient is measured using transmission spectrophotometry and the absorption and reduced scattering coefficients are measured using a time-resolved method. A good match between real tissue and phantom parameters is demonstrated.

Analysis of Crushing of Granular Material Under Isotropic and Biaxial Stress Conditions

ABSTRACT Granular materials forming part of natural slopes, embankments, subgrades of foundations and pavement structures are subjected to both static and dynamic loads during their engineering lives. As a result of these loads, particle crushing occurs. The present study focuses on the evolution of crushing in a simulated granular material subjected to different combinations of biaxial stresses. It was found that the Discrete Element Method (DEM) can be used to visualize and understand the evolution of crushing experienced by a granular material under these conditions. Even though DEM does not normally consider particle breakage, it is possible to simulate crushing by replacing one particle that has failed in tension with a combination of many particles of different sizes. The results from the developed simulations indicated that the samples tended to achieve a fractal distribution of particle sizes, although at the end of the simulations these distributions were still being dominated by the size of the original particles. Changes in other properties of the simulated granular material such as the void ratio and the shear strength were also recorded and analyzed. It was found that the internal friction angle decreased as a result of particle crushing.

Calculation of Broadband Time Histories of Ground Motion, Part II: Kinematic and Dynamic Modeling Using Theoretical Green's Functions and Comparison with the 1994 Northridge Earthquake

In the evolution of methods for calculating synthetic time histories of ground motion for postulated earthquakes, kinematic source models have dominated to date because of their ease of application. Dynamic models, however, which incorporate a physical relationship between important faulting parameters of stress drop, slip, rupture velocity, and rise time, are becoming more accessible. This article compares a class of kinematic models based on the summation of a fractal distribution of subevent sizes with a dynamic model based on the slip-weakening friction law. Kinematic modeling is done for the frequency band 0.2 to 10.0. Hz, dynamic models are calculated from 0.2 to 2.0. Hz. The strong motion data set for the 1994 Northridge earthquake is used to evaluate and compare the synthetic time histories. Source models are propagated to the far field by convolution with 1D and 3D theoretical Green’s functions. In addition, the kinematic model is used to evaluate the importance of propagation path effects: velocity structure, scattering, and nonlinearity. At present, the kinematic model gives a better broadband fit to the Northridge ground motion than the simple slip-weakening dynamic model. In general, the dynamic model overpredicts rise times and produces insufficient shorter-period energy. Within the context of the slip-weakening model, the Northridge ground motion requires a short slip-weakening distance, on the order of 0.15 m or less. A more complex dynamic model including rate weakening or one that allows shorter rise times near the hypocenter may fit the data better.

Fractal model for size effect on ice failure strength

A significant ‘size effect’ is observed in ice failure strength: the ice strength is not independent of ice fragment size, but decreases with increasing size. The fractal approach to the ice mechanics with a multiscale microstructure provides an elegant and unified explanation of the size effect on ice failure strength. Ice fragments are found to have a fractal size distribution, with a fractal dimension ( D) close to 2.50 over a remarkably wide range of ice fragment size. The size effect on ice strength can be well predicted using the fractal dimension D=2.50 of the ice fragment size distribution.

Macroscopic permeability of three-dimensional fracture networks with power-law size distribution.

Fracture network permeability is investigated numerically by using a three-dimensional model of plane polygons uniformly distributed in space with sizes following a power-law distribution. Each network is triangulated via an advancing front technique, and the flow equations are solved in order to obtain detailed pressure and velocity fields. The macroscopic permeability is determined on a scale which significantly exceeds the size of the largest fractures. The influence of the parameters of the fracture size distribution--the power-law exponent and the minimal fracture radius--on the macroscopic permeability is analyzed. Eventually, a general expression is proposed, which is the product of a dimensional measure of the network density, weighted by the individual fracture conductivities, and of a fairly universal function of a dimensionless network density, which accounts for the influences of the fracture shapes and of the parameters of their size distribution. Two analytical formulas are proposed which successfully fit the numerical data over a wide range of network densities.

Numerical simulation of wave propagation in media with discrete distributions of fractures: effects of fracture sizes and spatial distributions

SUMMARY We model seismic wave propagation in media with discrete distributions of fractures using the pseudospectral method. The implementation of fractures with a vanishing width in the 2-D finite-difference grids is done using an effective medium theory (that is, the Coates and Schoenberg method). Fractures are treated as highly compliant interfaces inside a solid rock mass. For the physical representation of the fractures the concept of linear slip deformation or the displacement discontinuity method is used. According to this model, the effective compliance of a rock mass with one or several fracture sets can be found as the sum of the compliances of the host (background) rock and those of all the fractures. To first order, the background rock and fracture parameters can be related to the effective anisotropic coefficients, which govern the influence of anisotropy on various seismic signatures. We test the validity of the method and examine the accuracy of the synthetic seismograms by a comparison with theoretical ray traveltimes. We present three numerical examples to show the effects of different fracture distributions. The first example shows that different spatial distributions of the same fractures produce different wavefield characteristics. The second example examines the effects of variation of fracture scale length (size) compared with the wavelength. The final example examines the case of fractures with a power-law (fractal) distribution of sizes and shows how that affects the wavefield propagation in fractured rock. We conclude that characterization of fractured rock based on the concept of seismic anisotropy using effective medium theories must be used with caution. Scale length and the spatial distributions of fractures, which are not properly treated in such theories, have a strong influence on the characteristics of wave propagation.

Fractal Size Distributions of Wildfires in Hierarchical Landscapes: Natura Facit Saltus?

Fractal Size Distributions of Wildfires in Hierarchical Landscapes: Natura Facit Saltus?

Fractal Principle of Mineral Deposit Size Forecasting and Its Implication for Gold Resource Potential Evaluation in China

AbstractMineralization distributions are very heterogeneous in nature. As large or superlarge mineral deposits are quite rare whether in time or in space, it is difficult to detect all the largest mineral deposits in a region in a limited period of time owing to the restriction of technology and exploration degrees—this is called “not all discovered”. However, all discovered large, especially superlarge, mineral deposits generally have a complete census in the geological literatures. On the other hand, not all discovered small mineral deposits are recorded in the geological literatures because for economic reasons people have not much interest in them—this is called “not all recorded”. This practice often results in the observation truncations, that is, the data points near the two ends in an observable population, which is obtained by fitting a power law size‐frequency distribution to discovered mineral deposits in a given region, show concave‐down departure from the correlation line fitted. The authors suggest that the size and number of undiscovered deposits may be forecast by fitting a fractal size distribution to discovered mineral deposit sizes between the upper and lower truncation observations and then extrapolating the scale‐independent area to deposit sizes larger than the upper truncation limit. Based on the statistical results obtained by the fractal size‐frequency distributions of 394 discovered gold deposits with sizes greater than 2 t Au in China and 83 known gold deposits with sizes over 0.3 t Au in the Jiaodong area of China, the authors forecast according to the present commercial standards for gold ores that the total resources of undiscovered gold deposits ranging in tonnage from 50 to 2000 t Au are more than 4500 t in China, and that in the Jiaodong area of China the total resources of potential gold deposits with sizes in the range of 30 to 650 t Au are about 700 t.

Hydrology of subarctic Canadian shield: bedrock upland

Runoff generation processes in the Canadian Shield were investigated to reconcile the considerable variability of runoff ratios noted by previous studies. Four bedrock plots near Yellowknife, Northwest Territories, Canada were studied between May 1999 and September 2000. Surface runoff increases with rainfall and snowmelt intensity because higher values of the latter can exceed fracture infiltration capacities. The size distribution of fractures affects infiltration while the fracture network controls the direction of surface flow. Frost has no effect on infiltration since the fractures are seldom ice filled. Storms of long duration and intermittent rainfall provide intervals between rainfalls during which evaporation can reduce the water available to runoff. Available storage in bedrock depressions, both empty and soil filled, retard surface flow and lengthen residence time, in turn enhancing evaporation and infiltration at the expense of runoff. The further downslope available storage is on the slope the more control it has on plot runoff magnitude. These field results show that rainfall–runoff relationships for bedrock slopes in the Canadian Shield should be considered in terms of microtopography, bedrock and soil characteristics and climate.

Derivation of parent fracture population statistics from trace length measurements of fractal fracture populations

This note describes how to estimate the three-dimensional fracture size distribution from trace length statistics.

Statistical Characteristics of Xenoliths in the Antioch Kimberlite Pipe, Marshall County, Northeastern Kansas

Geometrical characteristics of xenoliths in the Antioch kimberlite pipe have been considered in statistical terms. A method of conversion of 2D intersections to 3D dimensions was used. It has been shown that the Rosin-Rammler distribution of mass leads to the Weibull distribution of sizes, whereas a fractal distribution of sizes can be expressed as the Pareto distribution. Lognormal, Weibull, and Pareto distributions have been tested as model distributions. The Pareto distribution could be the most appropriate model for the distribution of xenoliths. This conclusion is in agreement with the general concept that the xenoliths formed as a result of an underground explosion without additional breakage occurring during magma transport. The final distribution may be shifted from the initial model as a result of processes of redistribution and sorting of xenoliths in liquid-crystalline flows.

On the Yielding and Plastic Compression of Sand.

This paper presents an analysis of the yielding and plastic hardening of uniformly-graded samples of a silica sand subjected to one-dimensional normal compression. Single grains of silica sand have been compressed diametrically between flat platens to measure indirectly tensile strength. Approximately 30 grains were tested for each of the following nominal particle sizes: 0.5mm, 1mm and 2mm diameter. It was found that the data could be described by the Weibull statistics of brittle ceramics, and the Weibull modulus could be taken to be about 3. 1. Uniform aggregates of the same sand were then compacted to maximum density and subjected to one-dimensional compression. The initial particle size distributions were 0.3-0.6mm, 0.6-1.18mm and 1.18-2mm, and aggregates were subjected to stresses of up to 100MPa. All particles were initially of similar shape, and hence the initial voids ratios of the aggregates at maximum density were approximately equal. The yield stress was defined to be the point of maximum curvature on a plot of voids ratio against the logarithm of effective stress, and found to increase with decreasing particle size, and to be approximately proportional to the tensile strength of the constituent grains. However, the plastic compressibility index was found to be approximately constant and independent of the initial grading, and a fractal distribution of particle sizes appeared to evolve under increasing stress. There is evidence to suggest that the aggregates evolve towards a fractal dimension of 2.5 under high stresses.

Micropore size analysis in hydrated cement paste by NMR

Abstract We present a time evolution of 1 H spin-lattice relaxation rates in the laboratory (1/ T 1 ) and in the rotating (1/ T 1ρ ) frame of a synthetic cement paste. The typical results found for both rates allow us to follow the main hydration stages of the cement paste and the refinement of its microporosity. In particular the texturation of the porosity and the structuration of the surface of the material are evidenced on two model cement pastes. An interpretation in terms of fractal size distribution is considered as well as the effect of the curing temperature.

Multifractal Characterization of Soil Particle‐Size Distributions

A particle‐size distribution (PSD) constitutes a fundamental soil property correlated to many other soil properties. Accurate representations of PSDs are, therefore, needed for soil characterization and prediction purposes. A power‐law dependence of particle mass on particle diameter has been used to model soil PSDs, and such power‐law dependence has been interpreted as being the result of a fractal distribution of particle sizes characterized with a single fractal dimension. However, recent studies have shown that a single fractal dimension is not sufficient to characterize a distribution for the entire range of particle sizes. The objective of this study was to apply multifractal techniques to characterize contrasting PSDs and to identify multifractal parameters potentially useful for classification and prediction. The multifractal spectra of 30 PSDs covering a wide range of soil textural classes were analyzed. Parameters calculated from each multifractal spectrum were: (i) the Hausdorff dimension, f(α); (ii) the singularities of strength, α; (iii) the generalized fractal dimension, Dq; and (iv) their conjugate parameter the mass exponent, τ (q), calculated in the range of moment orders (q) of between −10 and +10 taken at 0.5 lag increments. Multifractal scaling was evident by an increase in the difference between the capacity D0 and the entropy D1 dimensions for soils with more than 10% clay content. Soils with <10% clay content exhibited single scaling. Our results indicate that multifractal parameters are promising descriptors of PSDs. Differences in scaling properties of PSDs should be considered in future studies.

Numerical Simulation of the Hijiori Geothermal Reservoir Hydraulic Stimulation(Ceramics & Rocks 2)

In this paper an improved three-dimensional simulation model (FRACSIM-3D) is presented. It incorporates the mechanical rock-fluid interaction during hydraulic stimulation in more rigorous way than the existing two-dimensional model FRACSIM-2D and quantitatively predicts the 3D reservoir growth behavior. In this model the pre-existing fractures are generated stochastically. The fractal size distribution of the penny-shaped fractures with random orientation is assumed. The fluid flow through the discrete fracture network is simulated by mapping the connectivity of the fractured rock to a regular cubic grid under condition of constant injection pressure. The fluid flow is approximated by the Darcy law equation. Pressure distribution within the fractured rock is used for calculating the growth of the cracks apertures and shear displacements. This requires several iterations; solving the complete flow model simulating the fracture system anew each time and recalculating the shear displacement using the inner fracture pressure from the previous iteration. The results of this numerical solution for hydraulic stimulation are in good agreement with the existing field data collected at the Hijiori hot-dry-rock (HDR) reservoir.