A novel box-counting method for quantitative fractal analysis of threedimensional pore characteristics in sandstone

Fractal theory offers a powerful tool for the precise description and quantification of the complex pore structures in reservoir rocks,crucial for understanding the storage and migration characteristics of media within these rocks.Faced with the challenge of calculating the three-dimensional fractal dimensions of rock porosity,this study proposes an innovative computational process that directly calculates the three-dimensional fractal dimensions from a geometric perspective.By employing a composite denoising approach that integrates Fourier transform(FT)and wavelet transform(WT),coupled with multimodal pore extraction techniques such as threshold segmentation,top-hat transformation,and membrane enhancement,we successfully crafted accurate digital rock models.The improved box-counting method was then applied to analyze the voxel data of these digital rocks,accurately calculating the fractal dimensions of the rock pore distribution.Further numerical simulations of permeability experiments were conducted to explore the physical correlations between the rock pore fractal dimensions,porosity,and absolute permeability.The results reveal that rocks with higher fractal dimensions exhibit more complex pore connectivity pathways and a wider,more uneven pore distribution,suggesting that the ideal rock samples should possess lower fractal dimensions and higher effective porosity rates to achieve optimal fluid transmission properties.The methodology and conclusions of this study provide new tools and insights for the quantitative analysis of complex pores in rocks and contribute to the exploration of the fractal transport properties of media within rocks.

A Successive Shift Box-Counting Method for Calculating Fractal Dimensions and Its Application in Identification of Faults

Fractal dimensions of a terrain quantitatively describe the self-organizedstructure of the terrain geometry. However, the local topographic variation cannot be illustrated bythe conventional box-counting method. This paper proposes a successive shift box-counting method,in which the studied object is divided into small sub-objects that are composed of a series of gridsaccording to its characteristic scaling. The terrain fractal dimensions in the grids are calculatedwith the successive shift box-counting method and the scattered points with values of fractaldimensions are obtained. The present research shows that the planar variation of fractal dimensionsis well consistent with fault traces and geological boundaries.

A Statistical Method for Determining the Fractal Dimension of Time Series

In this paper,we present a new method for determining the fractal dimension of time series and the algorithm of H index(Hurst index).

Relevance between abutment pressure and fractal dimension of crack network induced by mining

Based on the geological conditions of coal mining face No.15-14120 at No.8 mine of Pingdingshan coal mining group,the real-time evolution of coal-roof crack network with working face advancing was collected with the help of intrinsically safe borehole video instrument.And according to the geology of this working face,a discrete element model was calculated by UDEC.Combining in situ experimental data with numerical results,the relationship between the fractal dimension of boreholes'wall and the distribution of advanced abutment pressure was studied under the condition of mining advance.The results show that the variation tendency of fractal dimension and the abutment pressure has the same characteristic value.The distance between working face and the peak value of the abutment pressure has a slight increasing trend with the advancing of mining-face.When the working face is set as the original point,the trend of fractal dimension from the far place to the origin can be divided into three phases:constant,steady increasing and constant.And the turning points of these phases are the max-influencing distance(50 m)and peak value(15 m)of abutment pressure.

Running-in Test and Fractal Methodology for Worn Surface Topography Characterization

Studying and understanding of the surface topography variation are the basis for analyzing tribological problems,and characterization of worn surface is necessary.Fractal geometry offers a more accurate description for surface roughness that topographic surfaces are statistically self-similar and can be quantitatively evaluated by fractal parameters.The change regularity of worn surface topography is one of the most important aspects of running-in study.However,the existing research normally adopts only one friction matching pair to explore the surface topography change,which interrupts the running-in wear process and makes the experimental result lack authenticity and objectivity.In this paper,to investigate the change regularity of surface topography during the real running-in process,a series of running-in tests by changing friction pairs under the same operating conditions are conducted on UMT-II Universal Multifunction Tester.The surface profile data are acquired by MiaoXAM2.5X-50X Ultrahigh Precision Surface 3D Profiler and analyzed using fractal dimension D,scale coefficient C and characteristic roughness Ra ＊based on root mean square（RMS） method.The characterization effects of the three parameters are discussed and compared.The results obtained show that there exists remarkable fractal feature of surface topography during running-in process,both D and Ra ＊increase gradually,while C decreases slowly as the wear-in process goes on,and all parameters tend to be stable when the wear process steps into the normal wear process.Ra ＊illustrates higher sensitivity for rough surface characterization compared with the other two parameters.In addition,the running-in test carried with a set of identical surface properties is more scientific and reasonable than the traditional one.The proposed research further indicates that the fractal method can quantitatively measure the rough surface,which also provides an evidence for running-in process identification and tribology design.

Fractal Method for Statistical Analysis Geological Data

This paper establishes the phase space in the light of spacial series data , discusses the fractal structure of geological data in terms of correlated functions and studies the chaos of these data . In addition , it introduces the R/S analysis for time series analysis into spacial series to calculate the structural fractal dimensions of ranges and standard deviation for spacial series data -and to establish the fractal dimension matrix and the procedures in plotting the fractal dimension anomaly diagram with vector distances of fractal dimension . At last , it has examples of its application .

Realizing the Box-counting Method for Calculating Fractal Dimension of Urban Form Based on Remote Sensing Image

In the research of fractal cities, the fractal dimension is very important. It is used to describe the fractal character of the city. The authors have designed two approaches to calculate the fractal dimension by the box-counting method through an example of Beijing, which are called the vector method and the grid method, respectively. The former calculates the fractal dimension through an intersecting analysis in ArcView; and the latter is carried out by programming in Matlab. They are compared from three aspects： the calculating process, the limits in use, and the results. As a result, the conclusion is made that there are merits and faults on both methods, and they should be chosen to use properly in practical situation.

Evaluation of dimension of fractal time series with the least square method

Properties of fractional Brownian motions （fBms） have been investigated by researchers in different fields, e.g. statistics, hydrology, biology, finance, and public transportation, which has helped us better understand many complex time series observed in nature [1-4]. The Hurst exponent H （0 〈 H 〈 1） is the most important parameter characterizing any given time series F（t）, where t represents the time steps, and the fractal dimension D is determined via the relation D = 2 - H.

Research on nonlinear R／S method and its application in earthquake prediction

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Fractal analysis of granular ore media based on computed tomography image processing

九组矿石样品的代表性的图象被 X 光检查获得计算断层摄影术(CT ) 扫描仪。基于 CT 图象分析，稳固的矩阵的分数维的尺寸，毛孔空格和每件样品的矩阵 / 毛孔接口被使用框数测量方法。有粒子尺寸，孔，和渗出物系数的三种分数维的尺寸的关联被调查。结果证明为这些样品的所有图象，矩阵阶段有最高的尺寸，由矩阵毛孔接口的毛孔阶段，和尺寸列在后面有最小的价值；当毛孔阶段的尺寸与孔断然并且线性地联系时，矩阵阶段和矩阵毛孔接口的尺寸否定地并且线性地与孔被相关；矩阵毛孔接口的分数维的尺寸与渗出物系数否定地并且线性地联系。矩阵 / 毛孔接口的更大的分数维的尺寸为解决方案流动显示更不规则的复杂隧道，导致低渗透。

From Fractal Geometry to Fractal Analysis

Euclidian geometry pertained only to the artificial realities of the first, second and third dimensions. Fractal geometry is a new branch of mathematics that proves useful in representing natural phenomena whose dimensions (fractal dimensions) are non-integer values. Fractal geometry was conceived in the 1970s, and mainly developed by Benoit Mandelbrot. In fractal geometry fractals are normally the results of an iterative or recursive construction using corresponding algorithm. Fractal analysis is a nontraditional mathematical and experimental method derived from Mandelbrot’s Fractal Geometry of Nature, Euclidean geometry and calculus. The main aims of the present study are: 1) to address the dimensional imbalances in some texts on fractal geometry, proving that logarithm of a physical quantity (e.g. length of a segment) is senseless;2) to define the modified capacity dimension, calculate its value for Koch fractal set and show that such definition satisfies basic demands of physics, before all the dimensional balance;and 3) to calculate theoretically the fractal dimension of a circle of unit radius. A quantitative determination of the similarity using the set of Koch fractals is carried out. An important result is the relationship between the modified capacity dimension and fractal dimension obtained using the log-log method. The text includes some important modifications and advances in fractal theory. It is important to notice that these modifications and quantifications do not affect already known facts in fractal geometry and fractal analysis.

R/S method for evaluation of pollutant time series in environmental quality assessment

The significance of the fluctuation and randomness of the time series of each pollutant in environmental quality assessment is described for the first time in this paper. A comparative study was made of three different computing methods： the same starting point method, the striding averaging method, and the stagger phase averaging method. All of them can be used to calculate the Hurst index, which quantifies fluctuation and randomness. This study used real water quality data from Shazhu monitoring station on Taihu Lake in Wuxi, Jiangsu Province. The results show that, of the three methods, the stagger phase averaging method is best for calculating the Hurst index of a pollutant time series from the perspective of statistical regularity.

云南永胜古滑坡堰塞湖沉积物粒度多重分形特征及其指标适用研究

本文以云南永胜县下院滑坡堰塞湖沉积物为研究对象,依据粒度测试结果采用传统粒度指标与分形理论相结合的方式对其沉积环境进行系统化研究,探讨各传统指标在堰塞湖沉积解析中的适用性,探索多重分形理论在沉积学中的应用价值。研究结果表明:①堰塞湖沉积物粒度统计参数明显不同于其他环境下的沉积物,具有河湖相沉积独特的分布曲线形式、分选状态和分形特征。优势粒级在分形计算中起主导地位,全局分形维数与分选系数有着良好的对应关系。不同的分形维数对应着不同的堰塞湖沉积条件,分形维数在堰塞湖粒度解析中有着很好的运用前景;②多重分形结果显示2种不同时期的堰塞湖沉积物均以高聚集度组分为主体,表明粒径分布集中,反映出堰塞湖沉积过程中水动力的稳定性,沉积来源的唯一性,进一步证明了该堰塞湖曾长期存在。细粒组沉积粒径分布范围较窄,分布相对集中,优势粒组多但单个含量低,尾端含量低,内部分异较大;粗粒组分布呈局部集中整体分散的趋势,高聚集度粒组数量少但含量高,低聚集度粒组数量多但单个含量低,呈现中间高,四周低的特点。③多重分形分析表明Δα、Δf在沉积物粒度分析中能够解析粒组内部分布特征,具有传统指标不可替代的作用,D_(0)和D_(1)两者结合可作为沉积环境解析的替代指标,与其他q阶多重分形联合可进一步作为堰塞湖解析指标,而D_(1)/D_(0)和D2则存在一定局限。

冲刷作用下砂岩宏观力学特性及微观结构

水流环境下库岸边坡岩石劣化会严重危害坡体稳定性,如何精细化揭示动水冲刷作用下岩石宏观力学特性及微观结构对库区灾害防治具有重要意义。为模拟库岸边坡所处的水流环境,设计了一套岩石抗冲刷试验装置。以砂岩为研究对象,考虑自然、静水浸泡和动水冲刷3种工况。结合室内试验、数字图像相关方法、扫描电镜和分形理论,从宏观力学参数、应变场分布特征、微观结构等角度系统探讨了不同水环境对岩石力学特性的影响。研究结果表明:(1)动水冲刷后的岩石抗压强度和弹性模量软化系数比饱水试件分别降低了0.07和0.06;(2)应变场分异速率-轴向应变曲线在峰值应力前出现拐点P,与断裂过程区的集聚相关,可作为岩石破裂的前兆信号;相较于饱水作用,冲刷作用使得破裂前兆点出现变早,这是因为冲刷作用加剧了岩石内部缺陷的发育,更易形成裂纹;(3)相较于饱水试件,动水冲刷后砂岩内部矿物颗粒和胶结物流失程度加大,微观结构劣化更为显著,这种变化在扫描电镜图像的分形维数计算结果中得到了印证,冲刷试件的分形维数最大。

片岩隧道围岩稳定性与断面轮廓分形维数关系

在薄层状片岩隧道中,隧道超欠挖对于围岩稳定性有一定的影响。通过Python编程,采用盒维数法计算断面轮廓分形维数的方法,对某隧道10个超欠挖断面进行分形维数的计算。并通过数值模拟的方法,计算了薄层状片岩隧道在超欠挖状态下拱顶的应力集中系数和最大位移影响系数,作为评价围岩稳定性的指标。结合分形理论建立的断面轮廓分形维数与薄层状片岩隧道围岩稳定性关联模型,发现分形维数和围岩稳定性的最佳函数关系为线性关系,回归系数为0.87;而超欠挖百分比与二者并没有直接的线性关系。研究此种围岩条件下隧道超欠挖与围岩受力状况之间的规律关系,对指导薄层状片岩中隧道的开挖有现实意义。

中阶煤的孔隙特征研究

明确煤体的孔隙结构对煤层气的开采及瓦斯突出危险性评价有重要意义。为了解中阶煤孔隙特征,首先通过扫描电子显微镜测试煤层表面,后采用低温液氮吸附法和压汞法分析孔隙结构,并对测得实验数据进行绘图,研究其分形特征。结果表明:钱家营焦煤(QJY)、宁夏1/3焦煤(NX)、东欢坨气煤(DHT)三种煤样的吸附等温曲线均属于Ⅱ类吸附曲线,且呈倒"S"型。东欢坨煤的比表面积最大,宁夏煤次之,钱家营煤的比表面积较小,煤的变质程度与比表面积呈现明显的负相关性。孔径在≥50 nm的孔含量与平均孔径呈正相关;3个煤样中孔径≤10 nm的孔容占比均最大,煤孔隙主要是由微孔组成。孔隙发育情况:DHT>NX>QJY;钱家营煤与东欢坨煤的分形维数都呈现出吸附孔D _(2)>渗流孔D 1的特征,其微小孔的孔隙结构比较复杂,且无规则性。宁夏煤吸附孔的分形维数D_(2)=2.890,在2~3之间,微小孔具有分形特征。

基于分形理论的传统商肆建筑的形态对比分析

分形理论作为非线性科学研究中的主要内容之一,为建筑形态方面的研究提供了全新的思路。该文以内蒙古呼和浩特地区和山西祁县的调研资料为基础,以分形几何理论为依据,选取两地具有代表性的传统商肆建筑作为研究对象,运用计盒维数法结合FracLab软件进行计算,对两地商肆的平面、立面和装饰部分的形态特征进行对比分析,发现两地商肆建筑的形态上具有相似性与差异性,探索数据背后造成建筑形态差异的原因,以期为两地商肆建筑的文脉传承和发展提供参考。

基于结构函数分形表征的地铁钢轨波磨评价方法

钢轨波磨问题出现至今已有百年历史,但仍未得到有效解决,在现有钢轨打磨标准中提出的钢轨波磨评价指标普遍存在无法从多尺度对钢轨波磨粗糙情况进行评价的问题,难以与波磨对列车运行动力学性能的影响对标。通过对钢轨波磨轨面不平顺数据进行分形分析,改进并提出适用于钢轨波磨轨面不平顺的分形指标,以此来量化评估钢轨波磨对列车运行动力学性能的影响。研究表明,与轮轨力波动指标拟合效果最优的钢轨波磨评价指标为修正后的分形维数+尺度系数,并建立三者之间的映射关系经验公式,可用于指导基于钢轨波磨的钢轨打磨策略。

基于三维体视分析技术对金属断口的分形研究

金属断口包含金属材料的性质、性能及服役过程的重要信息,是金属断裂定性分析、定量表征的关键证据。研究证明分形维数与材料断裂参数密切相关,但传统二维图像分析难以全面揭示断口特征。为了更深入了解断裂机制和材料性能,三维形貌技术逐渐成为研究热点。在分析仪器中三维体视显微镜可以提供更便捷、更准确的断口信息,进而通过计算分形维数推测材料断裂性质。其中计盒维数分析方法具有易于实现和理解、应用范围广、稳定性好、可解释性强等优势。利用三维体视显微镜和MATLAB程序获取断口三维形貌并计算分形维数。实验表明,拉伸断口抗拉强度和屈服强度与分形维数呈负相关,冲击断口的冲击值与分形维数呈正相关。

基于分形理论的闪络放电痕迹分析方法研究

绝缘子连续闪络会导致绝缘子表面出现碳化痕迹,导致绝缘劣化。对绝缘子表面碳迹的研究有助于解绝缘子长期运行时的闪络特性。文中通过对绝缘子进行直流闪络实验,获取连续放电后的碳化痕迹图像。基于分形理论,采用改进的差分盒计数法(improved differential box-counting method,IDBCM)对这些图像进行分析,对于边缘被切割尺寸框覆盖的矩形图像,在计数过程中提出加权项,所提出的方法可以估计绝缘子表面的放电传播特性和劣化特性。由于考虑了碳化痕迹图像的颜色深度,IDBCM在图像预处理和评估绝缘子劣化性能方面表现出了比传统盒计数方法更大的优势。该图像分析方法在处理碳化痕迹时具有通用性,可为闪络电压提供附加信息,包括闪络时间、电压大小、工况等。为今后研究放电机理及相应的绝缘子劣化问题提供了一种方法。