Linear Richardson plots from non-fractal data sets

Richardson analysis, one of the principal methods of fractal analysis, is performed by measuring the perimeter of a curve with strides of varying length and constructing a log-log plot of perimeter against stride length. Certain simple geometrical forms can produce linear plots that mimic fractal behavior, and two smooth curves have been discovered that produce linear Richardson plots for strides varying by two orders of magnitude or more. The existence of such curves was not suspected before this study. Richardson analyses that suggest fractal geometry of low dimension or over a limited range of stride length should be checked against the source data for independent evidence of self-similarity.     

First and Second Orders of Suture Complexity in Ammonites: A New Methodological Approach Using Fractal Analysis

The study of septal patterns in ammonoids has been centered on functional and/or constructional issues. Complexly fluted septa have been considered as complementary structures that reinforce the ammonite shell, their frilled sutures possibly manifesting the demand for strength. Ammonitic sutures display features that denote typical fractal behavior, since they can present very long perimeters relative to the contiguous shell areas, and most provide evidence of statistical self-similarity when observed at varying scales of magnification. However, there is a lower limit of scale measurements below which the fractal behavior of the curve no longer holds, and the perimeter length/step size relationship approaches an Euclidean geometry. This paper describes a new methodology that allows the accurate characterization of suture complexity in ammonoids using the technique of fractal analysis (step-line procedure). The proposed methodology helps to fix the position of this cut-off point, allowing for independent estimates of the fractal dimensions of the curve for both large and small measurement scales (i.e., first and second orders of suture complexity). This approach improves the resolution of fractals in the analysis of suture complexity, thus facilitating the potential interpretation of suture patterns in functional/constructional, evolutionary and paleoecological terms.     

Fractal analysis of fracture patterns using the standard box-counting technique: valid and invalid methodologies

On the basis of results of simple box-counting analysis, it has been suggested in several recent publications that natural fracture patterns are fractal. Fractal patterns are characterized by self-similarity of structure on a range of scales and provide straight-line distributions on box-counting plots. New analysis of a fracture pattern that provided the most convincing straight-line box-counting curve previously published, shows that the curve is non-linear and, therefore, that the fracture pattern is not fractal. Non-linear box-counting curves are also characteristic of other natural fracture patterns analysed but spurious linear curves can be obtained if the area analysed extends beyond the mapped area.     

The perimeter-area fractal model and its application to geology

Perimeters and areas of similarly shaped fractal geometries in two-dimensional space are related to one another by power-law relationships. The exponents obtained from these power laws are associated with, but do not necessarily provide, unbiased estimates of the fractal dimensions of the perimeters and areas. The exponent (D_AL) obtained from perimeter-area analysis can be used only as a reliable estimate of the dimension of the perimeter (D_L) if the dimension of the measured area is D_A=2. If D_A<2, then the exponent D_AL=2D_L/D_A>D_L. Similar relations hold true for area and volumes of three-dimensional fractal geometries. The newly derived results are used for characterizing Au associated alteration zones in porphyry systems in the Mitchell-Sulphurets mineral district, northwestern British Columbia.     

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.     

The use of the fractal dimension to quantify the morphology of irregular-shaped particles

For several decades, sedimentologists have had difficulty in obtaining an efficient index of particle form that can be used to specify adequately irregular morphology of sedimentary particles. Mandelbrot has suggested the use of the fractal dimension as a single value estimate of form, in order to characterize morphologically closed loops of an irregular nature. The concept of fractal dimension derives from Richardson's unpublished suggestion that a stable linear relationship appears when the logarithm of the perimeter estimate of an irregular outline is plotted against the logarithm of the unit of measurement (step length). Decreases in step length result in an increase in perimeter by a constant weight (b) for particles whose morphological variations are the same at all measurement scales (self-similarity). The fractal dimension (D) equals 1.0-(b), where b is the slope coefficient of the best-fitting linear regression of the plot. The value of D lies between 1.0 and 2.0, with increasing values of D correlating with increasing irregularity of the outline. In practice, particle outline morphology is not always self-similar, such that two or possibly more fractal elements can occur for many outlines. Two fractal elements reflect the morphological difference between micro-scale edge textural effects (D₁) and macro-scale particle structural effects (D₂) generated by the presence of crenellate-edge morphology (re-entrants). Fractal calibration on a range of regular/irregular particle outline morphologies, plus examination of carbonate beach, pyroclastic and weathered quartz particles indicates that this type of analysis is best suited for morphological characterization of irregular and crenellate particles. In this respect, fractal analysis appears as the complementary analytical technique to harmonic form analysis in order to achieve an adequate specification of all types of particles on a continuum of irregular to regular morphology.     

堆石料缩尺方法的分形特性及缩尺效应研究

采用6种不同缩尺方法将同一条现场级配曲线缩制成不同的数值模拟级配曲线,建立以不同粒径范围内颗粒数为测量数的分形模型,研究了颗粒级配分布的分形特性;基于颗粒流方法,生成与级配曲线相对应的6组数值试样进行双轴压缩试验,研究了缩尺方法对数值试样宏观力学特性及细观力学响应的影响;并讨论了颗粒级配分布的分形特性与数值试样力学特性之间的关系。结果表明：采用不同方法缩尺后数值试样的颗粒级配分布具有分形特性,分维数D为1.463～1.783;随着缩尺方法相似比尺M的增大,数值试样中细颗粒数量增多,粗细颗粒的充填关系得到改善,力学特性逐步提高;颗粒级配分布分维数D与数值试样力学特性指标之间存在较好的线性关系。     

Fractal analysis of magnetite grains — implications for interpreting deformation mechanism

In the present study, the grain size (d) and shape of 225 magnetite grains, that crystallized at T>600°C in a syntectonic granite (Godhra Granite, India) are evaluated and implications of data to decipher deformation mechanism of magnetite are discussed. Fractal (ruler) dimension (D) analysis of magnetite grains is performed and it is demonstrated that they show fractal behaviour. Smaller magnetite grains tend to be more serrated than the larger ones, which is manifested in the higher fractal (ruler) dimension (D) of the former. Assuming a natural strain rate ranging between 10⁻¹⁰ s⁻¹ and 10⁻¹⁴ s⁻¹, the grain size data fall dominantly in the dislocation creep field of the existing deformation mechanism map of magnetite for 630°C. However, SEM-EBSD studies reveal that subgrains are absent in the magnetite grains and they did not undergo dislocation creep. Thus it is inferred that the shape of magnetite grains was not controlled by dislocation creep. It is concluded that the higher serration and increased fractal dimension of finer magnetite grains implies the importance of diffusion creep as an important deformation mechanism at high-T for magnetite in polymineralic rocks.     

Investigation of fractal distribution law for the trace number of random and grouped fractures in a geological mass

A fractal study method of the number of geological mass fractures is introduced in detail in this paper. Three main aspects of the problem were studied: (1) The random distribution of fractures in a geological mass was in good agreement with the fractal law. The size scale of the studied geological mass ranged from 2400 m to 1 mm for the length of each side, and the geological mass samples were taken from 13 coal areas in China. (2) The geological mass fractures were evidently directional and anisotropic, having originated from tectonic movement. Observation and statistics for the data from the Xuangang, Fenxi and Dongshan coal areas in Shanxi, China, demonstrated that the fracture distribution of each group, classified by the strike of the strata, still follow the fractal law, even though the fractal dimension varies to a certain extent with different strikes. (3) The sedimentary strata containing the coal seams, as a geological mass, underwent almost similar tectonic movements in their geological history. The mechanical experiments on geological mass samples from Fenxi and Jiexiu in Shanxi demonstrated that the fractal dimension of the number of fractures in the same strata is in good power function with the product of strength and elastic modulus. The larger the product of the strength of the elastic modulus is, the larger is the fractal dimension, and vice versa.     

虎跳峡龙蟠右岸土石混合体粒度分形特征研究

应用分维理论对虎跳峡龙蟠右岸分布的土石混合体粒度分布的分维规律进行了研究分析,建立了平均粒径与相应的分维数之间的定量关系模型。通过研究表明,土石混合体具有良好的统计自相似性,由于其本身为级配不良土,在分维曲线上表现为双重分形分布,这种特殊的分维分布与土石混合体的成因及形成过程有关。     

A Nonstratified Model for Natural Fractal Curves

The stratified structure of Koch curves limits their value as models of nonstratified curves typical in nature, which may nevertheless have statistically self-similar character. Richardson plots for basic and randomized Koch curves are marked by strong, scale-periodic disruptions, and compare poorly with the smoothly sloping, compact Richardson plots derived for some natural, irregular curves. These disruptions relate to the heirarchies of dominant element sizes created in the discrete steps of Koch curve construction, with element vertices acting as attractors for divider steps. We have developed a curve-generation algorithm that adds individual bend elements to a line segment in a nonstratified manner, explicitly following the self-similar limit relation of curve length (L) to added line segment length (r). Values that may be specified include curve fractal dimension (D) and bend element shape (angle of inclination for the added segments). The procedure allows combinations of fractal dimension and element shape, which encompass the range of D values encountered for natural fractal curves. The nonstratified construction method provides useful standard curves for testing and norming of geometric analysis methods commonly applied to complex natural features. In a preliminary test of a multisampling divider method algorithm, the resulting Richardson plots consistently diverge from linear form, and are likely to lead to an overapproximation of D value.     

基于分形理论研究土壤结构及其水分特征关系

为定量获得土壤结构对其水力性质的指示作用,室内实验选用华北平原子牙河流域原状土样为研究对象,用张力计法和激光粒度分析仪分别测定土壤水分特征曲线和样品粒度分布,基于分形理论计算土壤粒度分布的分形维数,采用实验测定与模型验证相结合的方法对水分特征曲线进行分析.结果表明,土壤颗粒粒度分布在[10 μm,50 μm]区间内的分段分维值是表征土壤粒度累积分布显著上升段特征的关键参数,与0~80 kPa吸力范围内的土壤水分特征曲线幂函数模型拟合参数(a、b)有极显著相关关系.研究区内土壤水分特征曲线以分形形式表达的幂函数模型为:θ=100.78×(3-D)S(D-3)/3,利用土壤结构分形特征能够有效指示其水力性质.      

分形技术在攀西地区铂族元素异常查证中的应用

区域地球化学异常的形成往往是复杂的地质历史演化的综合结果,它表现出确定性的一面,又同时具有随机性的一面.分形理论作为非线性科学的一个分支,是研究自然界空间结构复杂性的一门学科,在本研究区域中,应用分形技术中的元素含量等值线面积模型来确定研究区铂族元素是否存在分形结构,并圈定出异常区域.处理结果表明该方法较传统方法效果更好.     

基于不同分形模型的冻融黄土孔隙特征研究

为了得到冻融作用对黄土孔径分布的影响规律, 以重塑黄土为研究对象, 利用压汞法测试经历不同冻融循环次数后黄土样品的孔隙特征, 采用3种分形模型对冻融作用后的黄土微观孔隙结构进行定量表征和对比研究。结果表明： 未经冻融作用的黄土孔隙分布曲线呈单峰分布, 经历冻融作用的黄土孔隙分布曲线呈双峰甚至多峰分布。冻融作用对黄土中孔径分布在0.1 ~ 10 μm范围内的孔隙影响较大。前10次冻融作用使黄土孔隙率增加, 特别是经历6次冻融作用后, 与未经历冻融作用的黄土相比孔隙率增大约18.8％。随着冻融作用的继续, 黄土孔隙率减小且趋于稳定。经历不同冻融循环次数后的黄土孔隙分布均具有良好的统计分形特性。基于热力学模型和毛细管压力曲线法表征黄土孔隙结构时, 黄土孔隙呈现显著的分形特性, 可在整个孔径尺度范围内给出唯一且合理的分形维数。基于Menger海绵模型表征的经历冻融作用后黄土孔隙分形特征呈现多尺度分形, 在不同的尺度范围内, 有不同的分形维数。结合分形理论可知冻融作用改变了黄土孔隙均匀性及复杂程度。     

On Fractal Dimensions of China's Coastlines

The fractal dimensions of China's coastlines are preliminarily discussed on the basis of GIS in this paper. Some significant conclusions are drawn. The fractal relationship between the length of China's continental coastline L and the yardstick r is log L = 3.99 – 0.16 log r on the scale 1:2,500,000 map. The fractal relationship between the length of Jiangsu province's coastline and the yardstick may be established as log L = 2.82 – 0.10 log r on the scale 1:50,000 map. Using the divider method, the fractal dimension of China's continental coastline is 1.16, Taiwan Island is 1.04, etc. The fractal dimensions of coastlines of the Bohai sea, the Yellow sea, the East China sea, and the South China sea tend to increase from north to south, indicating that the complexity of China's coastlines increases toward low latitudes. The substantial components of coast, biological function, and climate from north to south result in a change in fractal dimensions along the coasts of China. The fractal dimension of a coastline is different from the average fractal dimension of all its parts. The more parts of a coastline, the larger the difference between the fractal dimension of the original coastline and the average fractal dimension of all its parts. Faults control the basic trends and fractal dimensions of coastlines as a whole in the studied areas of Taiwan Island and Changle-Lufeng. The more the controlling effect of the faults, the smaller the fractal dimension of the coastline. The less the controlling effect of the faults, the larger the fractal dimension of the coastline. The results indicate that the faults control not only the basic trend of a coastline but also the complexity in the studied areas of Taiwan Island and Changle-Lufeng.     

基于离散元法的砂石混合体直剪试验结果分析

砂石混合体由力学性质以及结构相差极大的材料组成,其组成的重塑地层易发生塌陷等问题,因此对砂石混合体力学特性的研究具有重要的工程意义。砾石形状是砂砾石力学特性研究的重要属性参数,但采用规则图形对砾石进行描述不能反映出其真实的力学性质,采用数字图像处理技术构建的砾石数据库能反映砾石真实形状并可对特定形状参数进行具体分析。由于砂石混合体的粒径分布较广,采用特征粒径等无法描述整体粒度分布,故本文结合分形理论构建砂石混合体的二重分形结构模型,通过粒度分维值反演出完整的级配分布曲线。考虑到砂石混合体离散型的特点,采用离散元软件进行直剪试验数值模拟并对细观结构进行分析,研究结果表明,砂石混合体一般具有2个粒度分维值:砂粒度分维值和砾石粒度分维值,砂、砾石粒度分维值越接近,抗剪强度和内摩擦角越大;当两者相等时,砂石混合体具有一重分维,此时均一性最好,抗剪强度和内摩擦角最大;轴向系数是形容砾石形状的一个重要参数,随着轴向系数的增加,砾石显示出明显的条状性,在直剪试验中抗转动能力增强、周围接触数量增加,导致抗剪强度和内摩擦角不断增加。     

泥石流堆积物粒度分布特征影响因素分析及分形维数预测

泥石流堆积物作为泥石流发育最终的产物,含有大量与泥石流发生过程和发育特征相关的信息,能够反映泥石流灾害程度和活动强度。研究表明,泥石流堆积物颗粒具有明显的自相似性和无标度区间,运用分形理论,计算泥石流堆积物颗粒分布的分维数。分析分维数与主沟长度、泥砂补给段长度比、主沟平均比降、流域最大相对高差和松散物源量的关系,结果表明分维数与各因素之间存在较强的非线性响应关系。以乌东德库区泥石流实测数据为例,以上述的5个因素作为输入单元,建立了泥石流堆积物分维数支持向量机预测模型,并对分维数进行了预测,其预测结果的最大误差为1.25%,说明预测值与实测值吻合度较高。综合表明支持向量机预测模型能够较好地模拟和泛化数据,是一种行之有效的泥石流堆积物分形维数预测方法,可用于不具备筛析条件的泥石流堆积物粒度分布特征的预测与研究,进而可为研究泥石流的形成机理、类型、危险度和堆积物的形成演化特征及物理力学性质提供一个新思路。     

Scanline Method to Determine the Fractal Nature of 2-D Fracture Networks

A previous method proposed to measure the fractal dimension of pore spaces is adapted and modified for 2-D fracture networks. The method relies on scanning a 2-D fracture network through successive straight lines from top to bottom and measuring the distance between two fractures. The fractal dimension is then obtained using the log–log plot of the feature size and the number of features for this particular size at different magnifications. It is shown in this study that the method proposed to measure the fractal dimension of porous structures can be applicable to 2-D fracture networks with some modifications after testing it on synthetic and natural fracture patterns. The method is simplified to be useful for practical applications in the fractal analysis of fracture networks. The results reveal that, on the basis of the direction of scanning lines, different fractal behavior and dimensions can be obtained indicating that 2-D fracture networks possess multifractal character. This approach takes into account the effect of fracture orientation on the fractal behavior and anisotropic nature of fracture networks as well as the fracture density, length, and spatial distribution.     

分形渗透模型在饱和冻土中的应用

冻土中的渗透系数对于评估冻土工程中的水,热和溶质迁移至关重要。以往研究表明,渗透系数主要依赖孔隙结构,经常被描述为孔径大小和孔隙率,但是这两个参数并不能充分地表征孔隙结构。为加强对孔隙结构的描述,引用分形理论研究了冻土中的渗透系数。基于非均匀毛细管束模型和分形理论,提出了饱和冻土中渗透系数的分形模型,并提出通过土体冻结特征曲线获取冻土中孔径分布的理论方法。为了验证分形模型的有效性,对已有实验数据进行分析。分析表明,分形渗透系数模型是毛细管分维、最大孔径、黏度和迂曲度的函数,孔径分布变化是导致冻土渗透系数变化的根本原因。通过对比,计算值与实测值吻合较好。结果表明分形模型可以较好的预测冻土中的渗透系数,研究结果可为冻土渗透机理研究提供参考。     

稳定井流直线图解法确定含水层参数

对于潜水井流,利用Dupuit公式计算的参数往往相差很大,难以直接选用。对Dupuit公式进行线性化,建立降深(s)与距离(r)的s-lnr直线关系,利用直线斜率、直线在x轴上的截距,可以直接求得唯一的渗透系数K和影响半径R。利用松花江河谷的承压水井抽水试验资料和洮儿河扇形地27个潜水抽水井及其观测孔的抽水试验资料,应用直线图解法分析计算,得到唯一的含水层参数T、K和R。将计算结果与直接利用Dupuit公式所计算的结果相比较,前者的求参效果较好。