Statistical self-affinity, fractal dimension, and geologic interpretation

Properties of statistical self-affinity are explored and explained. Semi-variogram analysis can be used for identifying statistical self-affine behavior. This is, however, not the only method available for such an analysis. Some error and interpretation is involved; therefore, estimating the Hurst dimension (and from this the fractal dimension) from the semi-variogram can be misleading. Simulations based on statistical self-affine properties are alternatively used to develop an empirical approach to assessing statistical self-affine behavior. Analyzing simulations using semi-variograms, and comparing these semi-variograms to those from actual data, offers an alternate and perhaps superior approach to the understanding of the statistical self-affine properties of a geologic phenomenon. This empirical approach offers a method of reverse modeling for verifying estimates of Hurst dimension from semi-variograms.     

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.     

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.     

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.     

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.     

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。     

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.     

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。     

泥石流源区砾石土分形特征及其与渗透系数关系试验研究

以取自泥石流易发区的182个砾石土土样为基础,进行室内颗分试验,通过分形理论计算各土样的分维值,经计算发现,泥石流源区砾石土以一重分形为主,一重分形的土样占样本总数的88.46%,一重分形土样的分维值介于2.250～2.798之间;以此数据为基础配置土样,通过自制、可控的常水头试验装置进行渗透试验。试验结果表明,渗透系数k与分维值D之间有极显著的相关性,且在干密度为1.8 g/cm3时相关性最好。通过多元回归分析发现,不同密度条件下,k与D之间均有较好的幂函数关系;相同的分维值条件下土样的渗透系数随密度的增大呈减小的趋势,分维值在2.450～2.600之间时,样本的干密度 与渗透系数k之间的幂函数关系较为明显。试验结果可以为泥石流启动的临界雨量研究提供理论基础,提高已有预报模型的普适性及精度。     

高温硅酸盐熔体粘度与网络分数维值的相关性研究

对高温硅酸盐熔体粘度的估算一直是国际地学界热点问题之一,本文在研究了高温硅酸盐熔体网络分数维值的基础上,建立了估算熔体粘度的新模式 (简称 FD模式 ) ,阐明了熔体粘度值除了与温度成反比外,还与分子网络介观尺度的自相似比 ri 和分数维值 D相关,即与分子网络 (纳米尺度以上 )中的桥氧数 Ni成正比 (Ni∝ ) , 与单位硅氧四面体中的非桥氧数成反比.经对 4个硅酸盐系列高温熔体的粘度测定证实, FD模式的理论计算值与实测值吻合,且优于现今国际通用的 VTF模式.为探索岩浆迁移演化规律和完善新型低维材料的性能提供了新的理论依据.     

Box fractal dimension as a measure of statistical homogeneity of jointed rock masses

A written computer programme to estimate the box fractal dimension (D_B) is verified by estimating D_B of the triadic Koch curve for which the theoretical D is known. The influence of a number of input parameters of the box-counting method on the accuracy of estimated D_B is evaluated using the same Koch curve. The employed size range of the applied box networks was found to be the parameter which has the strongest influence on the accuracy of estimated D_B. This indicated the importance of finding the range of self-similarity or self-affinity for the object considered to select the proper range for the box sizes and, in turn, to obtain accurate estimates of D_B. By calculating D_B for different block sizes sampled from three generated two-dimensional joint patterns, it is shown that D_B can capture the combined effect of joint-size distribution and joint density on the statistical homogeneity of rock masses. The spatial variation of D_B along a 350 m stretch of a tunnel in the shiplock area of the Three Gorges dam site is computed using the joint data mapped on the walls and the roof of the tunnel. This spatial variation of D_B is used, along with the visual geological evaluation of the joint trace maps of the tunnel, in making decisions about statistical homogeneity of the rock mass around the tunnel. The results obtained on statistically homogeneous regions were found to be quite similar to the results obtained from a previous statistical homogeneity investigation which incorporated the effect of number of joint sets and their orientation distributions, but not the spatial variation of D_B. It is recommended that the spatial variation of D_B is used, along with the results of other methods such as contingency table analysis and equal area plots, which incorporate the effect of joint orientation distribution, in addition to the geology of the site, in determining the statistically homogeneous regions of jointed rock masses.     

承德地区煤矿分布特征及其受断裂控制的分形研究

运用Fry方法对承德地区已知煤矿点的空间分布形式做了分析,认为矿点整体走向和主要断裂带的走向一致。选择出露矿点较多的A地区,计算其矿点和断裂的分维值,将其与全区相比较,认为承德地区断裂带主要控制了煤层的走向,而煤的形成则受构造、古地理、古气候等因素的综合影响。     

多孔材料等效导热系数与分形维数关系的数值模拟研究

采用有限元方法模拟了多孔材料的导热系数与孔隙率和分形维数之间的非线性关系。有限元模型中的固体骨架和孔隙根据孔隙率的大小随机生成,模型中的材料参数和单元属性用ANSYS中的APDL参数化语言赋值。根据有限元随机模拟断面的热流密度分布和稳态热传导傅立叶定律,计算在不同孔隙率条件下的等效导热系数。研究表明,多孔材料模型等效导热系数随着孔隙的增加而线性减小。在孔隙率为常数的条件下,模型等效导热系数随着模型比例因子的增加而呈指数增加。等效导热系数随着孔隙分形维数的增加而减小,随着固体颗粒分形维数的增加而增加。     

基于岩体结构面统计与分维数的岩体质量评价及其在矿山工程中的应用

以三山岛金矿-510中段和-555中段为例,基于结构面实测数据,利用Matlab对结构面特征进行统计模型,进而实现了岩体裂隙网络Monte-Carlo模拟。然后结合分形理论确定了岩体质量的分维值。研究表明,-510中段和-555中段F3以北的岩体稳定性要好于F3以南的岩体稳定性,从定性和定量的角度对岩体质量进行了分析,为矿山的安全生产提供了一定的依据。     

三轴压缩煤岩三维裂隙CT图像重构及体分形维研究

通过对煤岩试件进行常规三轴力学试验,并对破坏后的煤岩试件进行CT扫描,之后将处理过的CT图片导入Mimics10.01进行三维重构,得到了煤岩试件内部裂隙空间的分布情况。另外,利用自编的Matlab程序对煤岩试件的体分形维进行了初步分析。结果表明：煤岩试件在三轴加载条件下主要发生压剪破坏;且围压不同,煤岩试件表现出了特定的力学规律;利用Mimics10.01重构得到的煤岩三维裂隙空间分布情况显示,随着围压的增大,煤岩试件剪切破坏面的连通率和体密度有增大的趋势;体分形维计算的结果显示,煤岩内部破坏越严重,体分形维值越小。