Natural rock joint roughness quantification through fractal techniques

Accurate quantification of roughness is important in modeling hydro-mechanical behavior of rock joints. A highly refined variogram technique was used to investigate possible existence of anisotropy in natural rock joint roughness. Investigated natural rock joints showed randomly varying roughness anisotropy with the direction. A scale dependant fractal parameter, K _v, seems to play a prominent role than the fractal dimension, D _r1d, with respect to quantification of roughness of natural rock joints. Because the roughness varies randomly, it is impossible to predict the roughness variation of rock joint surfaces from measurements made in only two perpendicular directions on a particular sample. The parameter D _r1d × K _v seems to capture the overall roughness characteristics of natural rock joints well. The one-dimensional modified divider technique was extended to two dimensions to quantify the two-dimensional roughness of rock joints. The developed technique was validated by applying to a generated fractional Brownian surface with fractal dimension equal to 2.5. It was found that the calculated fractal parameters quantify the rock joint roughness well. A new technique is introduced to study the effect of scale on two-dimensional roughness variability and anisotropy. The roughness anisotropy and variability reduced with increasing scale.     

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.     

一种新的岩石节理面三维粗糙度分形描述方法

研究并提出一种新的岩石节理面三维粗糙度分形描述方法。首先,基于激光扫描数据将节理表面离散成三角网,并建立与剪切方向相关的三维均方根抵抗角的计算方法。其次,运用分形数学理论,提出一种新的基于三维均方根抵抗角的节理面粗糙度分形描述方法。最后,采用新方法对天然玄武岩节理和花岗岩张拉型节理的粗糙特性进行分析。研究结果表明,提出的新方法能够较全面地反映节理面的三维几何形貌信息,并能描述节理粗糙度的各向异性特性。研究成果为进一步建立岩石节理面的三维剪切强度公式和剪切本构理论奠定了基础。     

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.     

Requirements for accurate estimation of fractal parameters for self-affine roughness profiles using the line scaling method

Summary A new concept of feature size range of a roughness profile is introduced in the paper. It is shown that this feature size range plays an important role in estimating the fractal dimension,D, accurately using the divider method. Discussions are given to indicate the difficulty of using both the divider and the box methods in estimatingD accurately for self-affine profiles. The line scaling method's capability in quantifying roughness of natural rock joint profiles, which may be self-affine, is explored. Fractional Brownian profiles (self-affine profiles) with and without global trends were generated using known values ofD, input standard deviation, , and global trend angles. For different values of the input parameter of the line scaling method (step sizea ₀),D and another associated fractal parameterC were calculated for the aforementioned profiles. Suitable ranges fora ₀ were estimated to obtain computedD within ±10% of theD used for the generation. Minimum and maximum feature sizes of the profiles were defined and calculated. The feature size range was found to increase with increasingD and , in addition to being dependent on the total horizontal length of the profile and the total number of data points in the profile. The suitable range fora ₀ was found to depend on bothD and , and then, in turn, on the feature size range, indicating the importance of calculating feature size range for roughness profiles to obtain accurate estimates for the fractal parameters. Procedures are given to estimate the suitablea ₀ range for a given natural rock joint profile to use with the line scaling method in estimating fractal parameters within ±10% error. Results indicate the importance of removal of global trends of roughness profiles to obtain accurate estimates for the fractal parameters. The parametersC andD are recommended to use with the line scaling method in quantifying stationary roughness. In addition, one or more parameters should be used to quantify the non-stationary part of roughness, if it exists. The estimatedC was found to depend on bothD and and seems to have potential to capture the scale effect of roughness profiles.     

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.     

利用毛管压力曲线分形分维方法研究流动单元

利用取心井铸体薄片获得的图像资料和毛管压力曲线,通过图像分形几何学方法以分维数的形式定量地表征出了复杂的微观孔隙喉道结构特征,发现能够很好地划分和评价孔隙岩石中油、气、水的渗流差异,可以用于储层微观流动单元表征。文中阐述了岩石微观孔隙喉道结构分形的理论基础、计算方法和应用于表征流动单元的依据。建立了中国西部砾岩低渗透油藏微观孔隙喉道分维数与孔隙度、渗透率之间计算图版,据此在油藏中利用常规测井资料获得的孔隙度、渗透率参数计算微观孔隙喉道分维数,开展全油藏流动单元划分与评价,取得了良好的效果。研究结果表明,利用毛管压力曲线分形分维方法研究储层微观流动单元是一种很有效的途径。     

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.     

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。     

Water quality index and fractal dimension analysis of water parameters

Statistical analysis of water quality parameters were analyzed at Harike Lake on the confluence of Beas and Sutlej rivers of Punjab (India). Mean, median, mode, standard deviation, kurtosis, skewness, coefficient of variation, regression lines, correlation coefficient, Hurst exponent, fractal dimension and predictability index were estimated for each water parameter. Monthly variation of water quality index using month-wise and parameter-wise value of quality rating and actual value present in water sample was calculated and compared with World Health Organization/Environmental Protection Agency standard value of these parameters. It was observed that Brownian time series behavior exists of potential of hydrogen with total dissolved solids, hardness, alkalinity, sulfate, chloride and conductance parameters; biochemical oxygen demand with total dissolved solids, hardness, alkalinity, sulfate, chloride, conductance and calcium parameters; dissolved oxygen with total dissolved solids, hardness, alkalinity, sulfate, chloride, conductance and calcium parameters; ferrous with total dissolved solids, hardness, alkalinity, sulfate, conductance and calcium parameters; chromium with total dissolved solids, hardness, alkalinity, sulfate, chloride, conductance and zinc parameters; zinc with total dissolved solids, hardness, sulfate, chloride, conductance and calcium parameters; fluoride with total dissolved solids, hardness, alkalinity, sulfate, chloride and conductance parameters; nitrate with total dissolved solids, sulfate and conductance parameters; nitrite with potential of hydrogen, total dissolved solids, hardness, alkalinity, sulfate, chloride, conductance and calcium parameters. Also, using water quality index, it was observed that water of the lake was severely contaminated and became unfit for drinking and industrial use.     

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。     

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.     

粗粒土的分数阶应变率及其与分形维度的关系

在波浪荷载、潮汐作用下砂土等粗粒土常常经受长期动力变形。运用分数阶微积分理论,分析了5种不同粗粒土在不同加载条件下的累积变形特性及粗粒土的分数阶应变率,传统的整数阶应变率随着加载次数的变化而变化,而粗粒土的分数阶应变率在同一加载条件下保持为常数。通过粗粒土颗粒破碎的分形理论,尝试建立分数阶应变率与土颗粒分布的分形维度之间的关系,分析土体分形维度对分数阶应变率大小的影响,发现随着分形维度的增加,分数阶应变率的数值降低。     

基于分维和相关性的自动初至拾取技术及应用

初至拾取是近地表反演的基础数据,采集的原始数据呈指数增长,尤其对于复杂山地采集的低信噪比单炮地震记录,越来越需要具有较高精度的自动初至拾取技术来完成巨大的工作量。这里采用了空变炮内和炮间的初至拾取时窗,实现了基于分维和相关性相结合的初至自动拾取技术。首先根据分形维的变化特征来判别地震波的初至走时;然后再根据拾取的是波峰、波谷,或过零点进行细微的调整,而对于山地采集的低信噪比单炮资料,还不能调整为最优;最后,在此基础上,将相关性技术应用于局部的初至最优化。实际资料的初至拾取结果表明：该方法能有效地改善初至拾取的精度。使用拾取的初至时间进行层析反演,并计算层析静校正量,可用于后续的处理。     

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.     

Determination of stiffness and other joint properties from roughness measurements

Summary The measurement of surface profiles is presented as a useful and simple approach to classifying statistically the essential features of rock joints. After introducing the reader to some existing analytical joint contact models for normal loading, a discrete numerical technique is developed. Using this technique the mechanical behaviour of a number of different slate joints is examined. The functional relationships between nominal stress, stiffness, true contact area and initial aperture are shown, for this class of joints, to be surprisingly simple. Experimental evidence is used to substantiate the numerical results. From the point of view of in-situ joint stiffness and hydraulic conductivity, numerical predictions seem feasible provided the degree of mating at some known stress level can be determined.     

Assessment of fractal dimension and geometrical characteristics of the landslides identified in North of Tehran,Iran

The aim of the presented study is to assess the fractal dimension (D) and the geometrical characteristics (length and width) of the landslides identified in North of Tehran, Iran. At first, the landslide locations (528 landslides) were identified by interpretation of aerial photographs, satellite images and field surveys, and then to calculate the fractal dimension (D), we used the computer programming named as FRACEK. In the next step, geometrical characteristics of each landslide such as length (L) and width (W) were calculated by ArcGIS software. The landslide polygons were digitized from the mentioned landslide inventory map and rotated based on movement direction. The fractal dimension for all landslides varied between 1.665 and 1.968. Subsequently, the relationship between the length/width ratios and theirs fractal D values for 528 landslides was calculated. The results showed that correlation coefficients (R), which are different regression models such as exponential, linear, logarithmic, polynomial, and power, between D and L/W ratio are relatively high, respectively (0.75, 0.75, 0.76, 0.78, and 0.75). It can be concluded that the fractal dimension values and geometry characteristics of landslides would be useful indices for the management of hazardous areas, susceptible slopes, land use planning, and landslide hazard mitigation.     

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。