Characterization of the Structure of River-Bed Gravels Using Two-Dimensional Fractal Analysis

This paper is concerned with the application of fractal analysis to understand the structure of water-worked gravel-bed river surfaces. High resolution digital elevation models, acquired using digital photogrammetric methods, allowed the application of two-dimensional fractal methods. Previous gravel-bed river studies have been based upon sampled profiles and hence one-dimensional fractal characterisation. After basic testing that bed elevation increments are Gaussian, the paper uses two-dimensional variogram surfaces to derive directionally dependent estimates of fractal dimension. The results identify mixed fractal behavior with two characteristic fractal bands, one associated with the subgrain scale and one associated with the grain scale. The subgrain scale characteristics were isotropic and sensitive to decisions made during the data collection process. Thus, it was difficult to differentiate whether these characteristics were real facets of the surfaces studied. The second band was anisotropic and not sensitive to data collection issues. Fractal dimensions were greater in the downstream direction than in other directions suggesting that the effects of water working are to alter the level of surface organisation, by increasing surface irregularity and hence roughness. This is an important observation as it means that water-worked surfaces may have a distinct anisotropic signal, revealed when using a fractal type analysis.     

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.     

JRC分形估测方法的实用性

基于分形几何的码尺法分维数与岩石节理粗糙度系数的物理意义剖析，认为Ｄ－ＪＲＣ之间不存在必然的相关性．分析标准轮廓曲线的分维数，发现其分维数差级微小，难以实行粗糙度系数分级．根据实测资料阐述了岩石节理表面轮廓曲线的“自相似”是统计意义而不是绝对的，它要求ＪＲＣ分形估测应统计求取，而过繁的分维数测量步骤削弱了ＪＲＣ的分形统计估测的可行性．建立在实测资料统计分析基础上的ＪＲＣ尺寸效应分形模型ＪＲＣｎ＝ＪＲＣ０（Ｌｎ／Ｌ０）－Ｄ客观而真实地刻画了粗糙度系数随取样长度增大而降低的规律，其中，ＪＲＣ尺寸效应分维数（Ｄ）具明确的物理意义，它描述了ＪＲＣ随结构面规模增大而降低的衰减速率．最后，运用ＪＲＣ尺寸效应分维数（Ｄ）探讨了岩石节理粗糙度系数尺寸效应的各向异性规律．     

确定田间土壤水力传导率的分形方法

采用了一种简便易行的分形方法来间接估计水力传导率函数,即根据土壤粒径分布曲线确定孔隙表面分形维数,将其作为分形水力传导率模型的参数来预测整个压力水头范围内的水力传导率.利用UNSODA数据库中217个样本的实测资料对模型进行检验,结果表明分形方法预测的水力传导率其精度明显高于根据水分特征曲线利用统计孔径分布模型的估计结果.     

龙马溪组页岩三维缝网重构及分形分析

传统上研究含气页岩宏观缝网特征的方法多是基于小样品开展,通过观察小样品的裂缝展布特征来了解页岩的裂缝发育的宏观特性。但是这类方法因尺度过小,缺乏代表性,不能获取连续面上的裂缝特征,难以分析裂缝发育和地应力方向之间的关系。为获取更详细的大尺度含气页岩的宏观裂缝特征,本文从重庆市涪陵地区石柱县采样获取2m×3m×0.7m完整大尺度岩块,并采用有序标号切割来获取30cm3规格岩样,在此基础上提取小岩样的表面裂缝并按照大尺度岩样切割的相对顺序构建三维裂缝网络。分析观测大尺度的三维裂缝网络可以发现,含气页岩的裂缝分布具有明显的规律性:水平主应力方向上的裂缝发育度普遍高于最小水平主应力方向;垂直方向上的裂缝与最小水平主应力的夹角为-62°。分维研究表明所有不同面上的裂缝均符合分维特性,说明小尺度裂缝特征和大尺度裂缝特征具有相似的规律性。此外,研究还表明最大水平主应力上的裂缝密度和分形维数最大,缝网结构最为复杂,最小水平主应力方向缝网结构次之,垂直主应力方向的裂缝结构最为简单;水平应力方向上主要以层理裂缝为主,垂直方向上以剪切缝为主,且与最小水平主应力的夹角为-62°。在掌握宏观裂缝分布特征基础上,研究可为页岩显微观测实验提供相应的理论依据。     

膨胀土湿干胀缩裂隙演化及其定量分析

膨胀土是一种分布广泛且具有明显胀缩性的裂隙土,其裂纹萌生及裂隙扩展会受到诸多因素的影响。此试验利用基于MATLAB开发的图像处理与裂隙定量分析程序,研究了膨胀土裂隙开展的尺寸和温度效应。制备25个不同初始状态的膨胀土泥浆试样,利用拍照台对其失水收缩过程中试样表面裂隙演化过程以及对应含水率进行记录。运用裂隙定量分析程序,分别对两类效应单独作用、耦合作用下的膨胀土裂隙度 、长径比C、裂隙平均宽度 、分形维数值等指标进行定量化分析发现：膨胀土裂隙的分形维数值具有较好的稳定性,其指标仅受试样厚度轻微影响,分形维数与含水率ω的关系呈近似对数曲线;在失水收缩过程中膨胀土最终裂隙指标主要受到试样厚度的影响,表面尺寸对裂隙长径比、宽度最终值也有一定影响,温度影响主要体现在促使裂隙更早、更快的发育并稳定,对最终指标影响相对不明显,影响显著性顺序为：厚度>表面尺寸>温度;此外还对膨胀土失水收缩开裂过程机制进行分析,进一步研究膨胀土的裂隙发育以及两类效应对其裂隙扩展的影响规律。     

Fractal Reconstruction with Unorganized Geochemical Data

Fractal geometry is receiving increased attention as a model for many natural phenomena. In this paper, we address two issues in dealing with unorganized 3-D measured data using fractal geometry: extraction of fractal characters and fractal surface (geochemical landscapes) reconstruction. We start from a set of randomly measured 3-D data on a plane. After classification and concentration of the input data, we present methods for estimating the fractal dimension and texture deviation as fractal characters, and then an improved subdivision scheme is developed to reconstruct fractal surface on the basis of the extracted fractal characters. We demonstrate both characterization and reconstruction with irregularly measured geochemical data from 1767 stream sediment samples in the middle district (450km² in area) of Zhejiang, China.     

Estimating fractal dimension of profiles: A comparison of methods

This paper examines the characteristics of four different methods of estimating the fractal dimension of profiles. The semi-variogram, roughness-length, and two spectral methods are compared using synthetic 1024-point profiles generated by three methods, and using two profiles derived from a gridded DEM and two profiles from a laser-scanned soil surface. The analysis concentrates on the Hurst exponent H,which is linearly related to fractal dimension D,and considers both the accuracy and the variability of the estimates of H.The estimation methods are found to be quite consistent for Hnear 0.5, but the semivariogram method appears to be biased for Happroaching 0 and 1, and the roughness-length method for Happroaching 0. The roughness-length or the maximum entropy spectral methods are recommended as the most suitable methods for estimating the fractal dimension of topographic profiles. The fractal model fitted the soil surface data at fine scales but not at broad scales, and did not appear to fit the DEM profiles well at any scale.     

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.     

Natural fracture profiles,fractal dimension and joint roughness coefficients

Summary Thirteen natural rock profiles (Barton and Choubey, 1977) are analyzed for their fractal properties. Most of the profiles were found to approximate fractal curves but some also showed features of specific wavelengths and amplitudes superimposed on fractal characteristics. The profiles showed fractal dimensions from 1.1 to 1.5 covering a range of selfsimilar and self-affine curves. The analysis results suggest a negative correlation between fractal dimension,D, and amplitude,A. Joint roughness coefficients (JRC) show a positive correlation with amplitude,A, and a negative correlation with fractal dimension,D. A numerical model of fracture closure is used to investigate the effects of different profile characteristics (D, A and sample size) on the nature of dilation and contact area, using the natural profiles and synthetic fractional Brownian motion profiles. Smooth profiles (low JRC, highD, lowA) display many small contact regions whereas rough fractures (high JRC, lowD, highA) display few large contact areas. The agreement with published experimental data supports the suggested correlations between JRC and the fractal parameters,A andD. It is suggested that observed scale effects in JRC and joint dilation can be explained by small differential strain discontinuities across fractures, which originate at the time of fracture formation.     

Fractal and geostatistical methods for modeling of a fracture network

The modeling of fracture networks is useful for fluid flow and rock mechanics studies. About 6600 fracture traces were recorded on drifts of a uranium mine in a granite massif. The traces have an extension of 0.20–20 m. The network was studied by fractal and by geostatistical methods but can be considered neither as a fractal with a constant dimension nor a set of purely randomly located fractures. Two kinds of generalization of conventional models can still provide more flexibility for the characterization of the network: (a) a nonscaling fractal model with variable similarity dimension (for a 2-D network of traces, the dimension varying from 2 for the 10-m scale to 1 for the centimeter scale, (b) a parent-daughter model with a regionalized density; the geostatistical study allows a 3-D model to be established where: fractures are assumed to be discs; fractures are grouped in clusters or swarms; and fracturation density is regionalized (with two ranges at about 30 and 300 m). The fractal model is easy to fit and to simulate along a line, but 2-D and 3-D simulations are more difficult. The geostatistical model is more complex, but easy to simulate, even in 3-D.This paper was presented at Emerging Concepts, MGUS-87 Conference, Redwood City, California, 13–15 April 1987.     

土壤水分特性曲线的分形模拟

应用Menger海绵结构模型,推导出包含有分形维数的土壤水分特征曲线的解析模型,该模型与Brook Corey (1964)及Campbell (1974)经验模型的结构相似。通过对不同地区所采集的10种土壤样本利用压力薄膜仪实测得到水分特性曲线资料反求得到相应的分形维数,分析了分形维数与土壤质地、结构之间的关系,结果表明分形维数随土壤粘粒含量的增大而增大,随砂粒含量的增大而减小。同时对土壤水分特性曲线模型的分形维数与基于质量的土壤颗粒分布分形维数进行了比较,结果表明,两者十分接近,而且具有良好的线性相关关系。根据上述关系,利用易测得的土壤粒径分形维数结合所推导的解析模型,对土壤水分特性曲线进行了预测,预测值与实测值具有良好的一致性。     

岩石碎裂作用的分形尺度(英文)

各种各样的统计幂定律关系经常成功地被用来描述碎屑大小分布和断口形状 (不平整度 )的分级规律 ,表明碎裂作用是一种尺度不变的作用过程。一条新的有关岩石碎裂作用破裂能量的尺度律 ,可以由分形几何以及Griffith能量平衡的概念推导出来 ,而且它与先前三条关于大小缩减或Hall Petch关系的理论是相一致的。从材料强度的观点来看 ,断口形状的分形维数是形状因子和岩石Weibull均质系数的函数。在任何一次采集中 ,如果碎屑都是致密压实的 ,那么大小分布和碎屑不平整度的分形维数是相同的。然而有些地壳碎屑在三维体积上已不再是致密压实的 ,破裂的地壳可被当作分形的多孔物质来处理。在此情况下 ,地壳碎屑形状的分形容量与地壳断口大小分布的分形维数相关 ,可以预期 ,在大地构造及地震强度的分形分析中 ,地壳断口大小分布的分形维数可作为断裂制约条件之一。     

利用FE-SEM、HIP、N2吸附实验表征生物气化煤系有机岩储层微观孔隙结构演化

微生物降解前后的煤系有机岩（煤岩和泥页岩）储层微观孔隙结构的变化对生物成气和成藏过程具有重要的意义.利用场发射扫描电子显微镜、高压压汞仪、孔比表面积孔隙度分析仪以及分形维数理论对厌氧微生物降解前后的煤系有机岩样品储层孔隙结构演化进行分析,根据孔隙结构特征并结合微生物生态学特征,将生物气化煤系有机岩的孔隙结构类型分为3类,即孔隙直径大于5 μm的微米孔,孔隙直径介于5 μm~100 nm的微纳孔,以及孔隙直径小于100 nm大于2 nm的纳米孔.微生物作用后的煤岩与泥页岩的微米孔孔容增加,微纳孔和纳米孔孔容减小,孔隙比表面积降低,平均孔隙直径增大.分形维数对比结果表明受微生物作用的煤岩与泥页岩样品的面分形维数（D₁）和孔隙结构分形维数（D₂）均降低,微生物作用使得有机岩孔隙表面变的光滑,孔隙结构变得简单,有利于游离气的运移和富集.      

Characterization and reconstruction of a rock fracture surface by geostatistics

It is well understood that, in studying the mechanical and hydromechanical behaviour of rock joints, their morphology must be taken into account. A geostatistical approach has been developed for characterizing the morphology of fracture surfaces at a decimetre scale. This allows the analysis of the spatial variability of elevations, and their first and second derivatives, with the intention of producing a model that gives a numerical three‐dimensional (3D) representation of the lower and upper surfaces of the fracture. Two samples (I and II) located close together were cored across a natural fracture. The experimental data are the elevations recorded along profiles (using recording steps of 0.5 and 0.02 mm, respectively, for the samples I and II). The goal of this study is to model the surface topography of sample I, so getting estimates for elevations at each node of a square grid whose mesh size will be, for mechanical purposes, no larger than the recording step. Since the fracture surface within the sample core is not strictly horizontal, geostatistical methods are applied to residuals of elevations of sample I. Further, since structural information is necessary at very low scale, theoretical models of variograms of elevations, first and second derivatives are fitted using data of both that sample I and sample II. The geostatistical reconstructions are computed using kriging and conditional simulation methods. In order to validate these reconstructions, variograms and distributions of experimental data are compared with variograms and distributions of the fitted data. Copyright © 2002 John Wiley & Sons, Ltd.     

An investigation into the effects of block size distribution function on the strength of bimrocks based on large-scale laboratory tests

Considering the geomechanical characteristics of rocks which are a mixture of rock blocks embedded in fine grain matrix can be challenging due to their complex structure and difficulty in preparing representative in situ or laboratory samples. In the literature, this group of rocks is called block in matrix Rocks (bimrocks). In this paper, the effects of volumetric block proportion (VBP), maximum block size, and distribution function on the compressive strength and failure patterns of these rocks were considered by executing the unconfined compressive test on large-scale synthetic samples. The compressive strength of bimrocks was found to have a direct relation with VBP and maximum block size and an indirect one with fractal dimension in accordance with multivariable regression analysis. The fracture patterns of remolded samples were changed from an inter-granular facture to a matrix and between-granular one by the reducing the VBP and increasing the fractal dimension. In addition, the failure mechanism was spalling for the low values of VBP and the high values of fractal dimension and it was changed to shear failure by increasing VBP and decreasing fractal dimension.     

钙质砂一维蠕变分形破碎特性宏微观试验研究

钙质砂是远洋地区港口、机场和民用建筑等构筑物的天然地基材料。通过钙质砂一维压缩蠕变试验和微观结构测试,发现了蠕变前后表面孔隙面积减小且呈分散分布的规律以及试验过程中试样瞬时变形、快速变形和衰减变形特征与粒径的高度相关性;利用基于分形理论改进的相对颗粒破碎率和质量分形维数描述了蠕变前后颗粒破碎程度,得到了分形维数和蠕变与时间的衰减形态曲线关系以及宏观质量分形维数和微观表面分形维数的线性关系,并在此基础上对单一粒径组钙质砂蠕变过程中的分形破碎行为进行了多尺度分析和宏微观跨尺度关联性研究,获得了蠕变过程中颗粒破碎发展以及微观孔隙变化规律,证明了钙质砂蠕变过程中的颗粒重组排列、破碎和研磨行为,揭示了钙质砂蠕变机制。     

大理岩断裂能与细观结构几何特征相关性

岩石在细观尺度上是由矿物颗粒胶结而成,除细观结构的力学特性外,其几何特征（颗粒界面网络结构和颗粒粒度分布）也对宏观断裂能具有一定的影响。为了探索这种几何特征与大理岩断裂能之间的相关性,从而进一步揭示岩石细观断裂机制,首先通过三点弯曲试验测定大理岩的断裂能,然后在试验后的三点弯试件断口处取岩石切片,进行电镜（SEM）扫描以获取岩石细观结构图像。利用图像处理技术获取岩石细观颗粒的界面网络图。通过对颗粒界面网络图分析,发现颗粒界面网络拓扑结构与颗粒粒度分布具有较强的分形特征。将二者分形维数与试件宏观断裂能相对比,发现它们之间具有很强的相关性。界面网络分形维数与宏观断裂能具有很强的二次相关性,而粒度分布维数与宏观断裂能之间具有较强的线性相关性。宏观断裂能随着两种分形维数的增大而增大。该研究结果揭示了宏观断裂能与岩石细观结构几何特征的相关性,加深了对宏观断裂能细观机制的理解。     

南海东部69柱粒度分维指标与气候代用指标的分形比较及其意义

南海69柱粒度数据分析表明,晚更新世以来,粒度在0.2~11μm的标度范围内具有分形特征,粒度分维值DS为1.72~2.02,平均1.84;不同时期粒度分维值明显不同,氧同位素1期(冰后期)为1.88,氧同位素2期(冰期)为1.81,温暖的冰后期粒度分维值高且变化幅度大,寒冷的冰期粒度分维值低且变化幅度小,表明粒度分维值DS对沉积物的形成环境演化具有良好的指示意义。运用R/S分析法,对69柱沉积物的粒度分维值DS、UK37法估算的表层海水古温度、浮游有孔虫的δ18O、δ13C及其转换函数计算的冬季表层海水古温度、夏季表层海水古温度、季节性温差、沉积物SiO₂、CaCO₃含量9项指标进行尺度重整分析计算,得到其时间序列分维值D_H分别为1.2855、1.1712、1.2659、1.4244、1.2719、1.2214、1.2979、1.1366、1.1609。虽然这些气候指标是用不同物理单位测量的,但运用分形分析可以对它们进行数学比较,粒度分维值DS与季节性温差二者的时间序列分维值D_H最接近,从分形理论方面进一步证实粒度分维值D_S的变化主要反映气候的冷暖变化。综合粒度分维值DS及其他气候代用指标分析,识别出南海东部全新世以来的4次突发事件。