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.

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 .

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.

Application of fractal content-gradient method for delineating geochemical anomalies associated with copper occurrences in the Yangla ore field,China

Fractal and multi-fractal content area method finds application in a wide variety of geological,geochemical and geophysical fields.In this study,the fractal content-gradient method was used on1：10,000 scale to delineate geochemical anomalies associated with copper mineralization.Analysis of geochemical data from the Yangla super large Cu-Pb-Zn polymetallic ore district using the fractal content-gradient method,combined with other geological data from this area,indicates that oreprospecting in the ore district should focus on Cu as the main metal and Pb-Zn and Au as the auxiliary metals.The types of deposits include（in chronological order） re-formed sedimentary exhalative（SEDEX）,skarns,porphyries,and hydrothermal vein-type deposits.Three ore-prospecting targets are divided on a S-N basis：（1） the Qulong exploration area,in which the targets are porphyry-type Cu deposits;（2） the Zongya exploration area,where the targets are porphyry-type Cu and hydrothermal vein-type Cu-Pb polymetallic deposits;and（3） the Zarelongma exploration area,characterized mainly skarn-type ＂Yangla-style＂ massive sulfide Cu-Pb deposits.Our study demonstrates that the fractal content-gradient method is convenient,simple,rapid,and direct for delineating geochemical anomalies and for outlining potential exploration targets.

The efficiency and accuracy of probability diagram, spatial statistic and fractal methods in the identification of shear zone gold mineralization: a case study of the Saqqez gold ore district,NW Iran

In this study,geochemical anomaly separation was carried out with methods based on the distribution model,which includes probability diagram(MPD),fractal(concentration-area technique),and U-statistic methods.The main objective is to evaluate the efficiency and accuracy of the methods in separation of anomalies on the shear zone gold mineralization.For this purpose,samples were taken from the secondary lithogeochemical environment(stream sediment samples)on the gold mineralization in Saqqez,NW of Iran.Interpretation of the histograms and diagrams showed that the MPD is capable of identifying two phases of mineralization.The fractal method could separate only one phase of change based on the fractal dimension with high concentration areas of the Au element.The spatial analysis showed two mixed subpopulations after U=0 and another subpopulation with very high U values.The MPD analysis followed spatial analysis,which shows the detail of the variations.Six mineralized zones detected from local geochemical exploration results were used for validating the methods mentioned above.The MPD method was able to identify the anomalous areas higher than 90%,whereas the two other methods identified 60%(maximum)of the anomalous areas.The raw data without any estimation for the concentration was used by the MPD method using aminimum of calculations to determine the threshold values.Therefore,the MPD method is more robust than the other methods.The spatial analysis identified the detail soft hegeological and mineralization events that were affected in the study area.MPD is recommended as the best,and the spatial U-analysis is the next reliable method to be used.The fractal method could show more detail of the events and variations in the area with asymmetrical grid net and a higher density of sampling or at the detailed exploration stage.

The Study of Area-Concentration Fractal Method in Litho-Geochemical Data in Tanurjeh Area, Khorasan Province

Given the scientific progresses as well as the invention of new methods in exploration, it is necessary to conduct some re-investigations in several exploration zones. So, in the present research, geochemical data on Tanurjeh exploration zone, (located in Northern Neishaboor, Khorasane Razavi province) is studied by using some modern statistical methods. Fractal methods are appropriated to study and separate the grades societies in deposits. In this article, litho-geochemical analysis results (ICP) are processed by concentration area fractal method (CA). The distribution diagrams related to the statistical populations are drawn, and anomaly populations of Copper, Gold and Molybdenum are determined besides previous studies (petrography and alteration), the results of statistic methods (CA) and aid presence of the porphyry system in depth.

Analysis of secondary electron emission using the fractal method

Based on the rough surface topography with fractal parameters and the Monte–Carlo simulation method for secondary electron emission properties, we analyze the secondary electron yield(SEY) of a metal with rough surface topography. The results show that when the characteristic length scale of the surface, G, is larger than 1 × 10^(-7), the surface roughness increases with the increasing fractal dimension D. When the surface roughness becomes larger, it is difficult for entered electrons to escape surface. As a result, more electrons are collected and then SEY decreases. When G is less than 1 × 10^(-7),the effect of the surface topography can be ignored, and the SEY almost has no change as the dimension D increases. Then,the multipactor thresholds of a C-band rectangular impedance transfer and an ultrahigh-frequency-band coaxial impedance transfer are predicted by the relationship between the SEY and the fractal parameters. It is verified that for practical microwave devices, the larger the parameter G is, the higher the multipactor threshold is. Also, the larger the value of D,the higher the multipactor threshold.

Two Phase Flow Simulation of Fractal Oil Reservoir Based on Meshless Method

The reservoir is the networked rock skeleton of an oil and gas trap,as well as the generic term for the fluid contained within pore fractures and karst caves.Heterogeneity and a complex internal pore structure characterize the reservoir rock.By introducing the fractal permeability formula,this paper establishes a fractal mathematical model of oil-water two-phase flow in an oil reservoir with heterogeneity characteristics and numerically solves the mathematical model using the weighted least squares meshless method.Additionally,the method’s correctness is verified by comparison to the exact solution.The numerical results demonstrate that the fractal oil-water two-phase flow mathematical model developed using the meshless method is capable of more accurately and efficiently describing the flow characteristics of the oil-water two-phase migration process.In comparison to the conventional numerical model,this method achieves a greater degree of convergence and stability.This paper examines the effect of varying the initial viscosity of the oil,the initial formation pressure,and the production and injection ratios on daily oil production per well,water cut in the block,and accumulated oil in the block.For 10 and 60 cp initial crude oil viscosities,the water cut can be 0.62 and 0.80,with 3100 and 1900 m^(3)cumulative oil production.Initial pressures have little effect on production.In this case,the daily oil production of well PRO1 is 1.7 m^(3)at 7 and 10 MPa initial pressure.Block cumulative oil production is 3465.4 and 2149.9m^(3)when the production injection ratio is 1.4 and 0.8.The two-phase meshless method described in this paper is essential for a rational and effective study of production dynamics patterns in complex reservoirs and the development of reservoir simulations of oil-water flow in heterogeneous reservoirs.

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).

Сomputational Experience Optimization of Colors in Complex Fractal Images in Carpet Design Using the Simplex Method

This article proposes a new approach based on linear programming optimization to solve the problem of determining the color of a complex fractal carpet pattern.The principle is aimed at finding suitable dyes for mixing and their exact concentrations,which,when applied correctly,gives the desired color.The objective function and all constraints of the model are expressed linearly according to the solution variables.Carpet design has become an emerging technological field known for its creativity,science and technology.Many carpet design concepts have been analyzed in terms of color,contrast,brightness,as well as other mathematical concepts such as geometric changes and formulas.These concepts represent a common process in the carpet industry.This article discusses the use of complex fractal images in carpet design and simplex optimization in color selection.

Research and experimental analysis of precision degradation method based on the ball screw mechanism

As a key transmission component in computer numerical control(CNC) machine tools,the ball screw mechanism(BSM) is usually investigated under working load conditions. Its accuracy degradation process is relatively long,which is not conducive to the design and development of new products. In this paper,the normal wear depth of the BSM nut raceway is calculated under the variable speed operation condition using the fractal wear analysis method and the BSM’s accelerated degradation proportional wear model. Parameters of the acceleration degradation model of the double-nut preloaded ball screw pair are calculated based on the physical simulation results. The accelerated degradation test platform of the BSM is designed and manufactured to calculate the raceway wear model when the lubrication condition is broken under the variable-speed inertial load and the boundary lubrication condition under the uniform speed state. Three load forces and two samples are selected for the accelerated degradation test of the BSM. The measured friction torque of the BSM is employed as the evaluation index of the accuracy degradation test. In addition,the life cycle of the accuracy retention is accurately calculated by employing the parameters of the physical simulation model of the BSM. The calculations mentioned above can be used to estimate BSM’s accuracy performance degradation law under normal operating conditions. The application of the proposed model provides a new research method for researching the precision retention of the BSM.

基于孔隙分形特征的水泥基毛细吸力预测模型

为深入理解水泥基毛细吸力与含水量的关系,采用分形几何学方法和毛细管理论提出了考虑孔隙分形特征的水泥基毛细吸力预测模型。该模型通过建立分形模型参数与试样配合比的拟合关系,实现了不同配合比水泥基毛细吸力的快速预测。基于相对湿度法开展了水泥基孔径分布测量验证试验,结果表明:不同配合比(水、灰、砂)水泥基试样孔隙均具有明显分形特性,可用Menger海绵体分形模型较好描述;毛细吸力模型预测结果与验证试验测试结果的偏差主要集中在±20%以内,模型可以较为准确地预测水泥基毛细吸力与含水量的关系。

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。

Long memory of price-volume correlation in metal futures market based on fractal features

An empirical test on long memory between price and trading volume of China metals futures market was given with MF-DCCA method. The empirical results show that long memory feature with a certain period exists in price-volume correlation and a fittther proof was given by analyzing the source of multifractal feature. The empirical results suggest that it is of important practical significance to bring the fractal market theory and other nonlinear theory into the analysis and explanation of the behavior in metal futures market.

基于分形理论的蚀变异常遥感解译——以广东阳山县为例

围岩蚀变异常是热液型矿床的重要找矿标志,运用遥感技术解译提取围岩蚀变信息是辅助地质找矿的重要手段。本文采用主成分分析法提取广东阳山地区Landsat-5的TM遥感影像中铁染和泥化蚀变信息,对影像中所提取的异常主成分进行“像元-面积”划分并统计像元个数,再将异常阈值进行分形非线性划分,得到更为精准的分级方法。门限法是基于突变点选择阈值进行等级划分,在提取蚀变异常的结果中往往会包含大量的背景信息,利用分形理论确定临界值进行分级减少了这些信息的影响。本次研究通过门限法和分形方法两种划分方法对研究区的蚀变异常进行对比,最后根据研究区蚀变异常信息的分形特征,利用求和方法对研究区的铁染和泥化蚀变异常信息进行分级。结果表明,相对于门限法,在遥感蚀变信息提取中分形方法加强了阈值选取的准确度,并发现研究区泥化蚀变与铁染蚀变主要分布于北东部及北部区域,其它区域小面积离散分布,这可为矿产勘查工作提供更好的技术支撑。

Fractal Analysis of Spatial Fault Distribution in the Southern South China Sea

Fractal geometry can effectively quantify naturally nonlinear or much complicated phenomena, and fractal dimension is the quantitative parameter. In this paper, we applied fractal geometry to analyze spatial fault distribution of the southern South China Sea. Calculation results indicate that the spatial fault distribution of this area have remarkable fractal features in a non-scale interval of 25 - 250 km, with fractal dimensions of 1.660 1, 1.387 5, and 1.269 3 for all faults, NE-trending faults and NW-trending faults, respectively. These dimensions （capacity dimensions）, obtained by box-counting method, represent well the characteristics of spatial fault distribution. Displayed by isoline-maps of fractal dimensions, faults in the study area arrange symmetrically along the spreading axis of the Southwest Sub-basin, and density of the NE-trending faults supplements to that of the NW-trending faults, revealing a property of mutual restriction between these two groups of faults. At last, by correlating the fractal features of spatial fault distribution with the evolution of the South China Sea and the distribution of oil or gas basins, we tried to reveal their relationships.

基于多季相分形特征的Landsat 8 OLI影像耕地信息提取方法

利用遥感技术快速准确地提取耕地信息是耕地保护的关键环节。以山东省商河县为例,提出了一种基于多季相分形特征的Landsat 8 OLI影像耕地信息提取方法。首先采用毯子覆盖法计算多季相遥感影像每个像元的上分形信号和下分形信号,对比分析耕地和其他土地利用类型的分形特征,选取上分形信号的第3尺度作为特征尺度,提取商河县耕地空间分布特征;其次采用同时期的土地利用矢量数据、Esri land cover数据和统计数据进行耕地信息提取精度评价;最后分别设置多季相分形提取与单季相分形提取、现有土地利用数据产品的对比实验,并基于点位匹配度和面积匹配度进行评价。结果表明:多季相数据更能反映农作物生长的复杂性,有助于提高耕地信息的提取精度;不同土地利用类型在不同分形尺度的信号值各不相同,分形特征可以在不同尺度上清晰地刻画出不同土地利用类型的分异性;基于矢量数据和Esri land cover数据评价的多季相分形特征耕地提取点位匹配度为87.13%和89.83%,面积匹配度为99.73%和97.91%,均比单季相分形提取结果精度高;综合考虑点位匹配度、面积匹配度和空间分布特征,研发方法能有效区分耕地和其他土地利用类型,提取结果更优,且与统计数据有更高的一致性。该方法可准确提取耕地信息,为耕地的动态监测和损害评估提供技术支撑。

基于分形特征和阈值分析的图像边缘检测

针对传统的边缘检测算法对噪声敏感且伪边缘较多,提出一种基于分形特征和阈值分析的图像边缘检测方法。该方法利用改进的毯覆盖算法计算出图像的分形特征,将图像的灰度分布映射到分形维数空间上,再根据提取的分形特征图进行阈值分析,获得高低两个阈值,将像素值分为非边缘、弱边缘和强边缘3类,再对弱边缘像素进一步加以判断。实验结果与其他算法相比较表明,该算法检测出来的图像边缘伪边缘和噪声最少。