The strength and fractal dimension characteristics of alum–kaolin flocs

Flocs generated by various shear forces exhibit different characteristics of size, strength and structure. These properties were investigated by employing a continuous optical monitoring and a microscope with CCD camera to directly monitor aggregation under six different shear intensities. The floc structure was characterized by the fractal dimension. The results showed that the flocculation index (FI) decreased from 1.16 at 20 rpm to 0.25 at 250 rpm and the floc size decreased from 550 μm to 150 μm, meantime, the FI value showed a good correlation with floc size. In order to determine the floc strength, two methods were used. One was the strength factor, ranging from 18.3% to 62.5%, calculated from FI curve, and the other was a theoretical value between 0.005 N/m² and 0.240 N/m², estimated by calculation. The floc strength increased with the G value in both cases. Furthermore, the fractal dimension increased with G and its value was between 1.30 and 1.63. The relation between fractal dimension and strength was also obtained.     

Application of fractal models to characterization of vertical distribution of geochemical element concentration

Characterization of the vertical distribution of geochemical element concentration is essential for economic planning in the mining industry. 10 mineralized boreholes and 1 non-mineralized borehole from the Qulong copper deposit, Tibet, western China, were collected to identify the vertical distribution properties of Cu values using fractal models. The vertical distribution of Cu values in mineralized and non-mineralized boreholes shows a positive skewed distribution in the former and multimodal distribution in the latter. The results obtained by the box counting method show that the vertical distributions of Cu values in mineralized and non-mineralized boreholes exhibit self-similarity with box dimensions ranging from 1.28 to 1.37. The box dimensions of mineralized boreholes are greater than that of Cu values in the non-mineralized borehole, indicating that the mineralization makes the distribution of Cu values more irregular. The power-law frequency analysis reveals that Cu values in mineralized boreholes are bifractal. The two portions of the plot define a crossover point at 0.33%, for Cu values less than and greater than 0.33, fractal dimensions range from 0.1 to 0.65, in non-mineralized rocks, and range from 2.71 to 5.79, in the mineralized rocks. Hurst exponents for mineralized boreholes occur at 0.8, which are greater than 0.5, indicating that Qulong copper deposit has a good continuity of mineralization.     

Geomorphometry and fractal dimension of a riverine badland in Maharashtra

One of the most serious limitations in studying the surface morphometry of a badland landscape is the nonavailability of a very fine resolution data which is essential for such types of studies. Local relief of most of the badlands in India and also from other parts of the world exhibit limited relief amplitude, often within a few meters. The paper reports a case study carried out in a riverine badland formed along the Western Deccan Trap Region. An attempt has been made in the present paper to extract the morphometric variables of the landscape from the DEMs derived from a high resolution field generated data, because the accuracy of the DEM derived values are dependent on the pixel resolution of the DEM from which they are generated. The size of the pixel resolution should be fixed differently for different landscapes depending on the landscape process in the area. The local relief of the area is around 10 m and for such types of landscapes the topographical maps and also the web-available DEMs are of very coarse resolutions which are not suitable for the analysis. Therefore two well defined tributary catchments were chosen from the area under investigation and theodolite surveys were carried out, contours were generated with 10 cm interval, DEMs were derived by using Arc GIS software. SRTM (Shuttle Radar Topography Mission) 90 m resolution data were utilized to generate DEM for the whole basin. Hypsometric and the drainage basin parameters were extracted from these data by using the same software. Fractal dimension of the whole basin and the sample basins were also obtained for the same data. The morphometric data generated were used to understand the geomorphic processes operating in the area.     

The fractal dimension of pores in sedimentary rocks and its influence on permeability

Perimeter-area power-law relationship of pores in five sedimentary rocks are estimated from scanning electron micrographs of thin sections. These relationships for the pores of four sandstones were found to lie between 1.43 and 1.49, while that of an Indiana limestone was found to be 1.67. We show how the perimeter-area power-law relationship of pores, along with a pore-size distribution, can be used to estimate the hydraulic permeability. A discussion is given of how the fractal dimension of the pore perimeter derived by Mandelbrot for islands whose boundaries are fractal: P = ε^DA^D/2, where ε is some constant that depends on the length of the measuring grid size and D is the fractal dimension of the pore perimeter, influences permeability.     

Investigation of the effect of aggregate shape and surface roughness on the slake durability index using the fractal dimension approach

Argillaceous rocks cover about one thirds of the earth's surface. The major engineering problems encountered with weak- to medium-strength argillaceous rocks could be slaking, erosion, slope stability, settlement, and reduction in strength. One of the key properties for classifying and determining the behavior of such rocks is the slake durability. The concept of slake durability index (SDI) has been the subject of numerous researches in which a number of factors affecting the numerical value of SDI were investigated. In this regard, this paper approaches the matter by evaluating the effects of overall shape and surface roughness of the testing material on the outcome of slake durability indices.

For the purpose, different types of rocks (marl, clayey limestone, tuff, sandstone, weathered granite) were broken into chunks and were intentionally shaped as angular, subangular, and rounded and tested for slake durability. Before testing the aggregate pieces of each rock type, their surface roughness was determined by using the fractal dimension. Despite the variation of final values of SDI test results (values of I_d), the rounded aggregate groups plot relatively in a narrow range, but a greater scatter was obtained for the angular and subangular aggregate groups. The best results can be obtained when using the well rounded samples having the lowest fractal values. An attempt was made to analytically link the surface roughness with the I_d parameter and an empirical relationship was proposed. A chart for various fractal values of surface roughness to use as a guide for slake durability tests is also proposed. The method proposed herein becomes efficient when well rounded aggregates are not available. In such condition, the approximate fractal value for the surface roughness profile of the testing aggregates could be obtained from the proposed chart and be plugged into the empirical relation to obtain the corrected I_d value. The results presented herein represent the particular rock types used in this study and care should be taken when applying these methods to different type of rocks.     

On the practice of estimating fractal dimension

Coastlines epitomize deterministic fractals and fractal (Hausdorff-Besicovitch) dimensions; a divider [compass] method can be used to calculate fractal dimensions for these features. Noise models are used to develop another notion of fractals, a stochastic one. Spectral and variogram methods are used to estimate fractal dimensions for stochastic fractals. When estimating fractal dimension, the objective of the analysis must be consistent with the method chosen for fractal dimension calculation. Spectal and variogram methods yield fractal dimensions which indicate the similarity of the feature under study to noise (e.g., Brownian noise). A divider measurement method yields a fractal dimension which is a measure of complexity of shape.     

Enhancing PIV image and fractal descriptor for velocity and shear stresses propagation around a circular pier

In this study,the fractal dimensions of velocity fluctuations and the Reynolds shear stresses propagation for flow around a circular bridge pier are presented.In the study reported herein,the fractal dimension of velocity fluctuations(u′,v′,w′) and the Reynolds shear stresses(u′v′ and u′w′) of flow around a bridge pier were computed using a Fractal Interpolation Function(FIF) algorithm.The velocity fluctuations of flow along a horizontal plane above the bed were measured using Acoustic Doppler Velocity meter(ADV)and Particle Image Velocimetry(P1V).The PIV is a powerful technique which enables us to attain high resolution spatial and temporal information of turbulent flow using instantaneous time snapshots.In this study,PIV was used for detection of high resolution fractal scaling around a bridge pier.The results showed that the fractal dimension of flow fluctuated significantly in the longitudinal and transverse directions in the vicinity of the pier.It was also found that the fractal dimension of velocity fluctuations and shear stresses increased rapidly at vicinity of pier at downstream whereas it remained approximately unchanged far downstream of the pier.The higher value of fractal dimension was found at a distance equal to one times of the pier diameter in the back of the pier.Furthermore,the average fractal dimension for the streamwise and transverse velocity fluctuations decreased from the centreline to the side wall of the flume.Finally,the results from ADV measurement were consistent with the result from PIV,therefore,the ADV enables to detect turbulent characteristics of flow around a circular bridge pier.     

基于地质建造的土壤地质调查及应用前景分析——以大凉山区西昌市为例

为探索适合中—高山区的土壤地质调查方法,选择兼具高寒山区、干热峡谷和中山宽谷盆地等不同地貌单元,以及沉积建造、岩浆建造和变质建造等不同地质建造类型的大凉山区西昌市进行试点研究。本次工作基于地质建造,将西昌市成土母质划分为7大类14个类,土壤地质单元分为31个类,并探讨了他们的互馈机制。研究认为,地质建造是土壤的物质基础,大体控制了土壤的平面分带,而海拔等地理因素则控制土壤的垂向分带。本项研究可为地方政府的规划布局提供数据支持和科学依据,也为高山区的土壤地质调查和生态地质调查提供了思路与范例。     

波形长度分形维数在结构振动诊断中的应用

在Hausdorff 维数和分配维数的基础上,采用拓扑映射技术,提出了振动波形长度分形维数的定义和计算公式,并对其数学严密性进行了证明.结合工程实例探讨了基于振动波形长度分形维数计算的结构振动诊断技术:模型增长法和滑动窗法.该技术对于波形异常检测表现出精度高、抗噪能力强及自动快捷等优势,构成了一条分形理论用于结构振动诊断的有效技术路线,具有良好的应用价值.     

自然水系的计算机处理与分维计算

自然水系的分维值能够反映某一地区一定的地质构造和地壳运动性质,但目前的分维算法大都采用人工完成。本文通过VB编程实现了自然水系的计算机处理与分维计算,并以中国三大水系为例,求得黄河、长江、珠江水系的分维值以及各水系主河道的分维值。该方法快速精确,并能适用于各种线性构造。     

Fractal dimension of profiles and surfaces using fuzzy morphological coverings

In this paper, we propose a new method, based on fuzzy morphology coverings, to estimate the fractal dimension of profiles and surfaces. This method is geometrically intuitive and simple to implement. Algorithmically, the method fits a covering to the frames or blocks of the profile or surface using fuzzy morphology. Varying the dimension of the frame or block, estimates of the length or area covered are then used to find the fractal dimension. Validation of the proposed method is performed by comparing its results with known fractal dimensions of mathematical profiles. The method is used to obtain the fractal dimension of rock profiles and surfaces.     

Uncertainty in fractal dimension estimated from power spectra and variograms

The reliability of using fractal dimension (D) as a quantitative parameter to describe geological variables is dependent mainly on the accuracy of estimated D values from observed data. Two widely used methods for the estimation of fractal dimensions are based on fitting a fractal model to experimental variograms or power-spectra on a log-log plot. The purpose of this paper is to study the uncertainty in the fractal dimension estimated by these two methods. The results indicate that both spectrum and variogram methods result in biased estimates of the D value. Fractal dimension calculated by these two methods for the same data will be different unless the bias is properly corrected. The spectral method results in overestimated D values. The variogram method has a critical fractal dimension, below which overestimation occurs and above which underestimation occurs. On the bases of 36,000 simulated realizations we propose empirical formulae to correct for biases in the spectral and variogram estimated fractal dimension. Pitfalls in estimating fractal dimension from data contaminated by white noise or data having several fractal components have been identified and illustrated by simulated examples.     

河流形态的分维及与洪水关系的探讨--以长江中下游为例

河流水系形态特征可以通过河流的分形特征来反映,分形维数则是河流分形特征的量化表示,其与河流洪水之间存在着一定的关系。以长江中下游为例,利用网格覆盖法计算出长江中下游河流分维,分析了长江中下游河流的分形特性,并在此基础上结合长江中下游洪水分析不同水系特征下洪水的特点。研究结果表明,一般来说河道分维越大、河网分维越小,洪水发生可能性则越高。     

岩石破裂过程中的声发射b值及分形特征研究

应用声发射及其定位技术,通过单轴受压岩石破坏声发射试验,对岩石破裂过程中的声发射b值和空间分布分形维值随不同应力水平的变化趋势进行了研究。研究结果表明：声发射分形维值D和b值反映了岩石破坏过程中微裂纹的初始和扩展;在小尺度微裂纹所占比例较高的加载初期,分形维值和b值在较高的水平波动变化,部分岩石试件分形维值和b值呈现升高现象;随着载荷的增加,岩石内部微裂纹的空间分布由无序向有序转变,大尺度裂纹所占比例增加,声发射定位事件出现群集现象,分形维值和b值开始较快速下降并在岩石失稳破坏时达到最低值。在岩石破坏过程中,声发射分形维值和b值的变化趋势相近。由于实际应用时,分形维值和b值的最小值（临界点）难以确定,故可将2个参数相结合,以分形维值D和b值较快速下降作为前兆特征,以提高现场岩体稳定性监测的准确性。     

Fractal analysis of the roughness and size distribution of granular materials

The roughness, i.e. general shape and surface irregularity, of particulate soil is an important characteristic that affects the mass behavior of the soil. Characterization of roughness has typically been limited to visual comparison of particles to standard charts, although other more quantitative methods such as Fourier analysis have also been used. Particle size distribution is another important mass-behavioral characteristic of granular soils, and similar to roughness, is defined within limited boundaries. Fractal geometry can be applied to irregular or fragmented patterns such as roughness and grain size distribution to provide quantifying and unique numerical values. This paper presents an evaluation of the applicability of fractal dimensioning techniques to the quantification of both physical particle roughness and grain size distribution of granular soil. The divider and the area-perimeter fractal dimensioning techniques are used to quantify roughness of planar profiles of individual sand grains. The characterization of the size distribution of granular material using fractal geometry is evaluated through Korcak's fragmentation theory. As shown herein, both the divider and the area-perimeter fractal dimensioning techniques are useful in characterizing soil particle roughness, and the results confirm the importance of differentiating between textural and structural aspects of roughness. Fractal geometry can also be used to quantify the size distribution of granular soils with relatively well-graded size distributions.     

基于关联维数的地质灾害空间分布特征分析

通过分形理论中关联维数及其动态变化来揭示黑水河流域地质灾害的时空分布特征,并探讨了影响区内地质灾害空间分布的相关因素。研究表明:区内不同规模地质灾害点系统空间分布具有分形特征,崩塌灾害点的分布最为集中,滑坡次之,泥石流则最为分散;近十年来区内每年所发生灾害的空间分布集中程度变化剧烈,呈现强集中、弱分散的特点;近五年来每年所发生的灾害呈逐渐分散趋势,但灾害的累积却有逐渐集中的趋势;影响区内地质灾害空间分布异同的主要因素为地形坡度、地层岩性、河流、降雨、人类工程活动、高程、断裂。     

Relationship between the fractal dimension of orthopyroxene distribution and the temperature in mantle xenoliths

As rock textures reflect the physical conditions and the mechanisms of formation of the rocks, new approaches are used for improving texture analyses, both qualitatively and quantitatively. Pioneer work has recently boosted interest in fractal analysis for quantifying and correlating patterns. Fractal‐like patterns relate to a degree of multiscale organization, and fractal dimensions (FD) and their potential variations can be used to infer the physical conditions of rock formation at various scales of observation. Here, we characterize quantitatively the shape and distribution of orthopyroxene grains in ultramafic xenoliths in terms of FD and their relation with temperature of equilibration. Fractal analysis has been applied to several populations of mantle xenoliths: 7 xenoliths collected in the vicinity of Pico Santa Isabel on Bioko Island, an alkaline basaltic volcano in oceanic domain (Gulf of Guinea, Equatorial Atlantic), 9 samples from Sangilen, in the Agardag alkaline lamprophyre dyke (Russia), and 11 samples form Śnieżnik (Lutynia, Poland), in the continental domain. Fractal analysis has been conducted to characterize the degree of complexity of the petrographic textures: it is indeed known that large values of FD are associated to more complex textures. The correlation here observed between the orthopyroxene fractal dimension and the temperature of equilibration suggests that FD captures a significant textural feature directly related to the temperature (i.e. generated by a temperature‐controlled process). The significant difference in the FD–T correlation observed for the continental and oceanic mantle domains suggests that the mechanical and rheological behaviour is distinct in the oceanic and continental lithospheres. These first promising results should be confirmed by analysing other mantle suites of rocks in different geodynamic settings. Copyright © 2015 John Wiley & Sons, Ltd.     

中国花岗岩类化学元素组成的分维数特征

将各种化学元素的丰度集合视为一组丰度序列,首次将分维数(盒维数)应用于研究中国花岗岩类的化学元素组成及演化特征。原理上,分形的遍在性源于随机性的遍在性,分维数的取值可表征丰度序列的内部结构性及演化的方向性,为矿产资源勘查提供重要信息;实践上,以观测元素相容性序列为基础分析其地球化学行为,采用盒维数以不失观测变量的整体性。详细探讨了分维数与成岩序列上的演化趋势或分异程度与研究对象地球化学特征间的相互关系及作用机制,据以解释为何在组分、时间与空间上同属花岗岩类却呈现出截然相异的成矿专属性与含矿性;确立了从频数特征角度探讨化学元素组成研究的新思路。研究结果表明：中国花岗岩类化学元素在丰度序列演化中均存有某种形式的结构性演化,即在组成、时间及空间上的有序结构,传统的(对数)正态分布模型难以描述其频数规律。然而,累加(差)的频数却是服从分形分布的,分维数与聚类分析的结果十分接近,且其递减序列可作为研究体系演化程度的判据,具有相当重要的地球化学意义。     

紫色泥岩水热条件下崩解过程的分维特性

以马头山组（K2m）、妥甸组（J3t）和下禄丰组（J1l）自然风干的紫色泥岩为研究对象,分别进行干湿、冷热和干湿冷热3种处理,并运用分维方法分析不同处理下紫色泥岩的崩解特性,发现分维数能比较准确地表征紫色泥岩的崩解过程：分维数越大,岩石破碎程度越高;分维数越小,岩石破碎程度越低。3种紫色泥岩在不同处理方式下崩解的分维数大小都是干湿冷热 干湿 冷热,可见,水热共同作用对母岩崩解的影响明显大于仅有水或热的作用。相同处理方式下3种泥岩的分维数大小为J3t J1l K2m,且在干湿和干湿冷热条件下J3t的崩解分维数与K2m和J1l的差异显著,这主要是由这3种泥岩的物理特性、元素组成以及其风化成度所决定的。     

基于孔隙分形维数的土壤大孔隙流水力特征参数研究

根据数字图像制备分析技术得到大孔隙数目、大小和形状等特征数据,用分形理论计算孔隙分形维数,并对两类不同质地土壤的孔隙分形维数进行分析比较,应用基于孔隙分形维数的大孔隙分形模型,建立了土壤水分特征曲线和非饱和导水率的预测模型。结果表明,孔隙分形维数D_v可定量描述不同质地土壤结构,其模型预测精度优于颗粒大小分布估计的分形模型,可用于实际问题的研究。