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.

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.

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.

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.

Abstraction and operator characterization of fractal ladder viscoelastic hyper-cell for ligaments and tendons

This study investigates the viscoelastic behavior of soft bio-fibres in association with their fractal structures. A spring-dashpot fractal network with the self-similar topology, named the -type fractal ladder hyper-cell (FLHC), is abstracted from the micro/nano-structure of ligaments and tendons (LTs). Its constitutive operator is derived by the Heaviside operational calculus, which is of intrinsic fractional order. In terms of this operator, the long-term viscoelastic relaxation of bio-fibres arising from the fractal ladder topology is expounded. In addition, the fractional-order viscoelastic constitutive equation is obtained based on the FLHC of LTs, and its results are consistent with those of available human knee and spinal LT relaxation experiments. Results on the constitutive equation of FLHCs are formulated into two propositions. The multidisciplinary invariance and implications from the fractal ladder pattern of bio-fibres are also discussed.

Adaptability of n-γdiscrimination and filtering methods based on plastic scintillation

Neutrons have been extensively used in many fields,such as nuclear physics,biology,geology,medical science,and national defense,owing to their unique penetration characteristics.Gamma rays are usually accompanied by the detection of neutrons.The capability to discriminate neutrons from gamma rays is important for evaluating plastic scintillator neutron detectors because similar pulse shapes are generated from both forms of radiation in the detection system.The pulse signals measured by plastic scintillators contain noise,which decreases the accuracy of n-y discrimination.To improve the performance of n-y discrimination,the noise of the pulse signals should be filtered before the n-y discrimination process.In this study,the influences of the Fourier transform,wavelet transform,moving-average filter,and Kalman algorithm on the charge comparison method,fractal spectrum method,and back-propagation neural network methods were studied.It was found that the Fourier transform filtering algorithm exhibits better adaptability to the charge comparison method than others,with an increasing accuracy of 6.87%compared to that without the filtering process.Meanwhile,the Kalman filter offers an improvement of 3.04%over the fractal spectrum method,and the adaptability of the moving-average filter in backpropagation neural network discrimination is better than that in other methods,with an increase in 8.48%.The Kalman filtering algorithm has a significant impact on the peak value of the pulse,reaching 4.49%,and it has an insignificant impact on the energy resolution of the spectrum measurement after discrimination.

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

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Structural Control on the Salmas Geothermal Region, Northwest Iran, from Fractal Analysis and Paleostress Data

The Salmas geothermal field is located in NW Iran. Subduction of Neo-Tethys oceanic crust beneath the Iranian microcontinent caused to propagation of the magmatic-Arc. Fractures and faults in the convergent zone have created path-ways for the circulation of geothermal fluid. Fracture concentration in the Salmas geothermal field has been characterized using of the fractal method and creation of a fracture density map that shows the highest concentration in the central part of the study area. The permeability of fractures has been evaluated by analyzing their orientation in respect to the paleostress axes. Also, the fractal analyzing result indicates the maximum fractal dimension（1.96） is around the thermal spring outlet. Paleostress analyzing revealed that in the central part of the study area, σ1 axes orientation is S90°W/10° and the σ2 dip is near to the vertical in this stress field, where strike slip faults can be propagated. In the SE part near the recharge of the thermal springs, the σ3 plunge increases to 70？ and σ1 orientation is N15°E/20°, in this local tectonic regime thrust fault developed. Fractures have an important role in the circulation of fluid and the fractal dimension increases near the thermal springs in the Salmas geothermal field. Regarding the paleostress data fracture with N-S direction such as the F1 fault zone（parallel to the σ1 axes）, a suitable pathway for deep circulation of geothermal fluid flow has been created.

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.