Quantification of Natural Fracture Surfaces Using Fractal Geometry

The purpose of this paper is to present an extensive evaluation of the methods to calculate the fractal dimension of natural fracture surfaces. Three methods; variogram analysis (VA), power spectral density (PSD), and roughness-length method (RMS) are applied to 2-D surface data (PSD) and 1-D profiles (VA and RMS) extracted from the surface data of 54 mm diameter crystallized limestone samples. Surface topography of the samples is quantified through a newly designed fully automated device. Before the application, self-affinity of the surface roughness and the applicability of these methods are validated using synthetically generated fractal surfaces. Fractal dimension values of the profiles are obtained as between 1 and 1.5 with a few exceptions. VA and RMS methods yield consistent fractal dimensions while the PSD values are lower than those of the other two methods. In terms of practical applicability, the VA is found more convenient than other two methods because there still exists shortcomings with the PSD and RMS methods due to difficulties in the mathematical analysis of the plots whose slopes are used in the computation of fractal dimension. However, it is observed that the data of limited size fracture surfaces are convenient for fractal analysis and the results are promising for further applications if the fracture surface size is restricted like cores recovered from deep boreholes.     

New techniques in rock mass classification: application to welded tuffs at the Nevada Yucca Mountain

Summary Many rock mass classification systems exist to assist the engineer in assessing the rock support requirements for underground design. On-going research in this area is directed at attempting to utilize the fractal dimension and the acoustic emission response of the tuffs at the Nevada Yucca Mountain to further aid in rock mass classification. Acoustic emission response is shown to be correlated with the porosity of the sample. Engineering behaviour of the rock varies dramatically with porosity; events and peak amplitude offer a means to distinguish between fracture porosity and pore porosity and consequently the engineering behaviour of the rock. Fractal dimension is used to characterize the roughness of fracture surfaces. Two fractal dimension calculation methods, one based on the semi-variogram for the surface and the other based on the use of dividers, are applied for this purpose. The divider method is shown to resolve deviation from a straight line; the semi-variogram method is shown to identify statistical similarity to various types of noise.Nomenclature D fractal dimension - AE acoustic emission - b b-value determined from log(frequency) against log(amplitude) plots - (h) semi-variogram function - h lag distance for semi-variogram function - H an exponent term related to fractal dimension asD=2 –H     

Quantification of Aperture and Relations Between Aperture,Normal Stress and Fluid Flow for Natural Single Rock Fractures

Accurate quantification of rock fracture aperture is important in investigating hydro-mechanical properties of rock fractures. Liquefied wood’s metal was used successfully to determine the spatial distribution of aperture with normal stress for natural single rock fractures. A modified 3D box counting method is developed and applied to quantify the spatial variation of rock fracture aperture with normal stress. New functional relations are developed for the following list: (a) Aperture fractal dimension versus effective normal stress; (b) Aperture fractal dimension versus mean aperture; (c) Fluid flow rate per unit hydraulic gradient per unit width versus mean aperture; (d) Fluid flow rate per unit hydraulic gradient per unit width versus aperture fractal dimension. The aperture fractal dimension was found to be a better parameter than mean aperture to correlate to fluid flow rate of natural single rock fractures. A highly refined variogram technique is used to investigate possible existence of aperture anisotropy. It was observed that the scale dependent fractal parameter, K _v, plays a more prominent role than the fractal dimension, D _a1d, on determining the anisotropy pattern of aperture data. A combined factor that represents both D _a1d and K _v, D _a1d × K _v, is suggested to capture the aperture anisotropy.     

基于SEM图像的既有地基红黏土颗粒微观结构及分形特征研究

采用IPP（Image-Pro Plus）图像分析软件对贵州省贵阳市某工厂三种既有地基红黏土SEM图像的信息进行提取和处理,定性描述和定量分析土体的微观结构,并引入分形理论分析SEM图像,提出在IPP软件中获取颗粒三维分形维数的计算方法。结果表明:（1）抗剪强度参数随微观颗粒数、颗粒形态比的增大而增大,随颗粒平均面积的增大而减小;（2）颗粒分布及形态均具有明显的分形特征,分形维数介于2~3之间,抗剪强度参数均随颗粒分布及形态分维值的增大而增大;（3）与构建三维模型的方法相比,利用IPP软件计算土颗粒三维分形维数的方法具有可行性,简单易操作,结果可靠。      

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.     

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.     

地表的分形测量及其大地构造学意义

以湖北红安地区为例,采用投影覆盖法(projectivecoveringmethod)对地表进行了二维分形测量,结果表明,地表面积具有双分形(bifractal)关系,即具有小尺度的结构分形(texturalfractal)和大尺度的构造分形(structuralfractal),分叉点(breakpoint)的尺度为3610m,分维值都在2～3之间且结构分维值大于其构造分维值。可见,地表形态具有分形性质,分维值可以指示地表形态的复杂程度。构造分维值可作为构造活动强度的一个指标,可为大地构造单元的划分提供定量依据。复杂地表形态主要是由构造活动(内营力作用)和各种复杂表生地质作用(外营力作用)引起的,前者主要控制大尺度的地形起伏,后者则塑造小尺度的地表形态。地表分维值可以指示地表的发育成熟度,该地区小尺度的结构分维值大于大尺度的构造分维值表明其处于地表形态的发育晚期。此外,地表的分形尺度可以来用确定构造活动尺度,从而指导构造地质与找矿勘探研究。     

岩石破裂带的分维及变化特征

用盒维数方法估计了受压岩石表面断裂带的分维值,研究了分维值随载荷、结构(平板伏试件合圆孔或与加载方向成不同角度的割缝)、岩性(大理岩、灰岩、砂岩)和粒度的变化特征,分析了统计自相似性成立的范围与岩性和粒度的关系。研究结果表明。在微裂纹发育未连通为宏观断裂带阶段,分维值随载荷上升而上升,当宏观断裂带形成之后,分维值保持为常数,且随岩性不同而不同。对于所测三种岩石,自相似性成立的尺度范围的上限为颗粒尺寸的Zyljlo倍。最后,结合岩石破裂过程的能量耗散机理,讨论了岩石分维断裂的物理意义。     

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.     

基于分维数的土体裂隙表征单元体估算

由于干湿循环、剪切或其它作用,自然界土壤表面发育着大量裂隙。裂隙相互连接形成网络,裂隙网络是一个随机无序的系统,常规统计方法描述裂隙的分布特征存在很大的难度。网状土体裂隙一般具有分形特征,可以用分维数来进行描述。用数码相机拍摄土体表面的裂隙,将其转换成灰度图像,并结合分形理论,建立了摄影法测定土体裂隙分维数的方法,得到了土体裂隙的分维数。根据表征单元体的物理意义,建立了基于分维数估算表征单元体的方法,并对实例进行了分析。研究表明：分维数随裂隙密度,裂隙宽度和次一级裂隙发育程度的增大而增大;表征单元体与裂隙的密度均匀性和隙宽均匀性相关,基于分维数估算表征单元体的方法简便可行。     

2017 - 2018年冬季乌梁素海湖冰表面裂缝形态分析

在2017 - 2018年冬季使用无人机对乌梁素海湖冰的冰面裂缝进行了航拍, 采用改进的自适应阈值分割方法进行图像二值化处理, 提取了冰裂缝的密度和分形维数。分析结果显示： 在固定区域的冰裂缝分形维数在1.35 ~ 1.50间变化, 冰裂缝数量随时间增加, 其分形维数也随之线性增大; 在冰生长期间, 冰厚度与冰裂缝分形维数也呈现出明显的线性关系（相关系数R²=0.75）。冰面不同区域的冰裂缝密度和分形维数在0.017 ~ 0.079、 1.38 ~ 1.64间变化, 且两者之间存在显著的对数相关性（相关系数R²>0.96）, 不同日期航拍数据拟合结果近似相等, 说明冰裂缝密度越大, 对应的分形维数越大。作为表征冰面形态特征的一种物理指标, 建立得到的冰裂缝分形维数与冰厚、 裂缝密度的相关关系, 对未来利用冰面裂缝形态监测冰层的生消过程可提供科学的参考。     

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.     

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.     

Some Distinctions Between Self-Similar and Self-Affine Estimates of Fractal Dimension with Case History

Compass, power-spectral, and roughness-length estimates of fractal dimension are widely used to evaluate the fractal characteristics of geological and geophysical variables. These techniques reveal self-similar or self-affine fractal characteristics and are uniquely suited for certain analysis. Compass measurements establish the self-similarity of profile and can be used to classify profiles based on variations of profile length with scale. Power spectral and roughness-length methods provide scale-invariant self-affine measures of relief variation and are useful in the classification of profiles based on relative variation of profile relief with scale. Profile magnification can be employed to reduce differences between the compass and power-spectral dimensions; however, the process of magnification invalidates estimates of profile length or shortening made from the results. The power-spectral estimate of fractal dimension is invariant to magnification, but is generally subject to significant error from edge effects and nonstationarity. The roughness-length estimate is also invariant to magnification and in addition is less sensitive to edge effects and nonstationarity. Analysis of structural cross sections using these methods highlight differences between self-similar and self-affine evaluations. Shortening estimates can be made from the compass walk analysis that includes shortening contributions from predicted small-scale structure. Roughness-length analysis reveals systematic structural changes that, however, cannot be easily related to strain. Power-spectral analysis failed to extract useful structural information from the sections.     

一种描述结构面剖面线粗糙度的新方法

采用法向矢量单位圆描述结构面剖面线粗糙度,从各微分段的角度关系阐述粗糙度,进而将二维问题转化为一维问题处理并提出"角度粗糙度"的概念.考虑到各微分段的实际长度对粗糙度的贡献,采用加权均值与加权方差定量描述角度粗糙度;角度粗糙度越大表明该剖面越粗糙.对规则剖面线与不规则剖面线采用"角度粗糙度"进行描述,所得结果跟已有的剖...     

测井数据滑动窗关联维计算和沉积学分析

应用滑动窗关联维计算方法对鲁北济阳坳陷某钻井剖面石炭系—二叠系陆表海沉积层段GR测井响应数据进行了分形维数计算,结果表明海侵体系域关联维数一般较小,且维数曲线振荡变化较弱,而高水位体系域则相反。这与海侵体系域形成时海平面上升的主控因素压制或掩盖了其他因素的显现有关,同时也证明测井序列分形分析能从另一个新的视角观察识别测井序列中所包含的地质信息。随着分形方法应用研究的不断开展,必将从测井数据中挖掘出更多的有用信息,指导油气等资源勘探与开发,丰富及完善测井地质学理论。相信将来分形会成为地质数据分析的标准工具。     

石油储量分布的分形特征及其预测

以准噶尔盆地为例，应用分形理论及方法分析了石油储量分布特征，探讨了分形维数的地质含义。分形维数的大小可以作为分析含油气盆地石油勘探潜力的一种指标，分形维数越大，表明盆地的石油勘探潜力越大，并与盆地的勘探历程相一致，在此基础上进行了储量预测。从分形方法预测的探明储量和已发现的探明储量可以地而推测盆地总资源量，与应用聚集系数法计算的结果基本吻合，从而为资源评价提供了一种新方法，为勘探部署和决策开辟了一条新途径。     

Fractal analysis of bathymetry and gravity profiles across the Chagos-Laccadive Ridge and the Carlsberg Ridge in the Indian Ocean

Gravity and bathymetry data have been extensively used to infer the thermo-mechanical evolution of different segments of the oceanic lithosphere. It is now understood that magmatic fluid processes involved in the accretion of oceanic crust are spatially complex and episodic. The nature of these processes which are in general nonlinear, can be described using fractal analysis of marine geophysical data. Fractal analysis has been carried out for gravity and bathymetry profiles over the aseismic Chagos-Laccadive Ridge and the spreading Carlsberg Ridge. The Iterated Function Systems (IFS) have been used to generate synthetic profiles of known dimension (D) and these are compared with the observed profiles. The D for the data sets are in the range of 1–1.5. The D for gravity profiles is less than those of bathymetry and the D for gravity and bathymetry over spreading ridge is higher than the aseismic ridge. The low fractal dimension indicates that the processes generating them are of low dimensional dynamical systems.