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 Method for Calculating Fractal Dimension of Amicrons and Fractal Simulation of Boundary Micro-Topography

A simple and practical method to calculate the fractal dimension （FD） of amicron＇s projective surface profile based on fractal theory is proposed. Taking AI（OH）3 material particles as an example, the scanning electron microscope （SEM） photos of particles were processed using an image.processing software （IPS） Photoshop. Taking the pixel as a fixed yardstick with the enlargement of the size of the particle image, the box-dimension and circumference-area （C-S） methods were used to calculate the FD of the surface profile of the particle. The FD of 1.2623 of the classic Koch curve is obtained, which approximates the true value of 1.2628. The complexities of the object＇s boundary and surface micro-topography are simulated successfully by a generator method.

Classification of forearm action surface EMG signals based on fractal dimension

Surface electromyogram （EMG） signals were identified by fractal dimension.Two patterns of surface EMG signals were acquired from 30 healthy volunteers＇ right forearm flexor respectively in the process of forearm supination （FS） and forearm pronation （FP）.After the raw action surface EMG （ASEMG） signal was decomposed into several sub-signals with wavelet packet transform （WPT）,five fractal dimensions were respectively calculated from the raw signal and four sub-signals by the method based on fuzzy self-similarity.The results show that calculated from the sub-signal in the band 0 to 125 Hz,the fractal dimensions of FS ASEMG signals and FP ASEMG signals distributed in two different regions,and its error rate based on Bayes decision was no more than 2.26%.Therefore,the fractal dimension is an appropriate feature by which an FS ASEMG signal is distinguished from an FP ASEMG signal.

Fractal Features of Urban Morphology and Simulation of Urban Boundary

Using fractal dimension to reflect and simulate urban morphology are two applications of fractal theory in city geography. As the only consistent description of a fractal, fractal dimension plays an important role in describing the basic features of fractals. Just like other fractals, our cities have similar characteristics. Fractal dimension to some extent is regarded as an indicator of urban expansion, and it may change with urban morphology in different time and space. Based on the Geographic Information System (GIS), taking Wuhan city as a test area, the fractal dimensions of different land use were calculated, and a linear regression equation was established to analyze the relationship between fractal dimension and residential areas. Then the author used fractal dimension to simulate the urban boundary which is an important part of urban mor-phology. A mid-point subdivision fractal generator is needed in the simulation process, and the shape of the generator is determined by fractal dimension. According to the fractal theory, fractal boundaries in different scales have self-similarity and they have the same fractal dimensions. Based on this fact, the simulation method the author used could quantitatively keep the similarity of configuration of the urban boundaries.

Damage Characteristics Analysis and Fractal Study of Shale With Prefabricated Fractures under Thermal-mechanical Coupling

To study the damage and failure of shale with different fracture inclination angles under uniaxial compression loading,in this work,RFPA2D-Thermal,a two-dimensional real failure process analysis software,was used for numerical simulation.Numerical simulation results show that quartz in shale mainly affects the tensile and compressive strength of shale by increasing rock brittleness.The coupling of temperature and pressure will cause lateral and volume destruction of shale,which enables the shale body to be more easily broken.Fracture inclination is the key factor affecting shale damage patterns.The failure mode of shale with low-and high-angle fractures is mainly shear failure,and the compressive strength does not vary with crack inclination.The damage mode of obliquely intersecting fractured shale is slip damage along the fracture face,the compressive strength decreases and then increases with the fracture inclination,and a minimum value exists.The acoustic emission simulation results of the damage process effectively reflect the accumulated internal damage and macroscopic crack appearance until fracture instability when the prefabricated fractured shale is subjected to uniaxial compressive loading.The crack inclinations of 0°and 120℃ corresponds to the most complex"N"shape damage mode.The crack inclinations of 30°and 60°,and the damage mode is an inverted"λ"shape.

Fractal Features and Computer Simulation of Fan-Shaped Sedimentary Bodies in Faulted Basins

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Classification of surface EMG signal with fractal dimension

Surface EMG (electromyography) signal is a complex nonlinear signal with low signal to noise ratio (SNR). This paper is aimed at identifying different patterns of surface EMG signals according to fractal dimension. Two patterns of surface EMG signals are respectively acquired from the right forearm flexor of 30 healthy volunteers during right forearm supination (FS) or forearm pronation (FP). After the high frequency noise is filtered from surface EMG signal by a low-pass filter, fractal di-mension is calculated from the filtered surface EMG signal. The results showed that the fractal dimensions of filtered FS surface EMG signals and those of filtered FP surface EMG signals distribute in two different regions, so the fractal dimensions can rep-resent different patterns of surface EMG signals.

One-Dimensional Fractal Curves and Fractal Dimension Determination

Various methods for evaluating the fractal curves were reviewed and simulated on computer. It is shown box-counting and power spectrum methods generally give poor measuring results, while yard and variation methods could obtain good results. However,owing to multiple influencing factors, further study needs to be done before widespread application of variation method. In order to improve the measuring accuracy of yard method, a new method has been propoed to measure the fractal dimension by changing the instrumental resolutions.

COMPUTER SIMULATION OF PHASE SEPARATION IN CaO-MgO-Fe_2O_3-Al_2O_3-SiO_2 GLASS

The fractal and spinodal decomposition models combined Monte Carlo method were used in computer simulation of phase separation from CaO MgO Fe 2O 3 Al 2O 3 SiO 2 glass. The fractal dimension was applied to quantitative analysis. The mechanism of phase separation was investigated. The results showed that the fractal dimensions and simulated micrographs of phase separation were in good agreement with the experimental results. And the mechanism of phase separation was spinodal decomposition.

A numerical simulation of seepage structure surface and its feasibility

According to Cubic law and incompressible fluid law of mass conservation, the seepage character of the fracture surface was simulated with the simulation method of fractal theory and random Brown function. Furthermore, the permeability coefficient of the single fracture was obtained. In order to test the stability of the method, 500 simulations were conducted on each different fractal dimension. The simulated permeability coefficient was analyzed in probability density distribution and probability cumulative distribution statistics. Statistics showed that the discrete degree of the permeability coefficient increases with the increase of the fractal dimension. And the calculation result has better stability when the fractal dimension value is relatively small. According to the Bayes theory, the characteristic index of the permeability coefficient on fractal dimension P（Dfi｜ Ri） is established. The index, P（Dfi｜ Ri）, shows that when the simulated permeability coefficient is relatively large, it can clearly represent the fractal dimension of the structure surface, the probability is 82%. The calculated results of the characteristic index verify the feasibility of the method.

3-D fracture network dynamic simulation based on error analysis in rock mass of dam foundation

Accurate 3-D fracture network model for rock mass in dam foundation is of vital importance for stability,grouting and seepage analysis of dam foundation.With the aim of reducing deviation between fracture network model and measured data,a 3-D fracture network dynamic modeling method based on error analysis was proposed.Firstly,errors of four fracture volume density estimation methods(proposed by ODA,KULATILAKE,MAULDON,and SONG)and that of four fracture size estimation methods(proposed by EINSTEIN,SONG and TONON)were respectively compared,and the optimal methods were determined.Additionally,error index representing the deviation between fracture network model and measured data was established with integrated use of fractal dimension and relative absolute error(RAE).On this basis,the downhill simplex method was used to build the dynamic modeling method,which takes the minimum of error index as objective function and dynamically adjusts the fracture density and size parameters to correct the error index.Finally,the 3-D fracture network model could be obtained which meets the requirements.The proposed method was applied for 3-D fractures simulation in Miao Wei hydropower project in China for feasibility verification and the error index reduced from 2.618 to 0.337.

Influence of water coupling coefficient on the blasting effect of red sandstone specimens

This study investigates the impact of different water coupling coefficients on the blasting effect of red sandstone.The analysis is based on the theories of detonation wave and elastic wave,focusing on the variation in wall pressure of the blasting holes.Using DDNP explosive as the explosive load,blasting tests were conducted on red sandstone specimens with four different water coupling coefficients:1.20,1.33,1.50,and 2.00.The study examines the morphologies of the rock specimens after blasting under these different water coupling coefficients.Additionally,the fractal dimensions of the surface cracks resulting from the blasting were calculated to provide a quantitative evaluation of the extent of rock damage.CT scanning and 3D reconstruction were performed on the post-blasting specimens to visually depict the extent of damage and fractures within the rock.Additionally,the volume fractal dimension and damage degree of the post-blasting specimens are calculated.The findings are then combined with numerical simulation to facilitate auxiliary analysis.The results demonstrate that an increase in the water coupling coefficient leads to a reduction in the peak pressure on the hole wall and the crushing zone,enabling more of the explosion energy to be utilized for crack propagation following the explosion.The specimens exhibited distinct failure patterns,resulting in corresponding changes in fractal dimensions.The simulated pore wall pressure–time curve validated the derived theoretical results,whereas the stress cloud map and explosion energy-time curve demonstrated the buffering effect of the water medium.As the water coupling coefficient increases,the buffering effect of the water medium becomes increasingly prominent.

爆炸荷载作用下深部煤体损伤特征试验研究

为研究深部地应力影响下爆破后煤体的损伤情况及裂纹扩展规律,建立了地应力下煤体损伤模型和爆破孔环向应力分布规律。通过相似模拟试验和数值模拟与分形维数相结合的方法,探讨了地应力作用下煤体爆破后裂纹扩展规律。理论分析表明,炮孔受地应力的影响会发生二次应力分布;在双向等压条件下,裂纹沿偏离主应力方向45°发展,在双向异压条件下,裂纹朝与主应力角度为θ(θ=arctan(σ_(x)/σ_(y)))方向发展。对比四组相似模拟试验:相对于无地应力条件下,3组对照试验各测点的应力大小和动载作用时间均减少,证明地应力对煤体爆破后的裂纹扩展起抑制作用;煤体裂纹会朝最大主应力方向扩展,最小地应力方向裂纹长度会随着应力差的增大而减小。通过ANSYS/LS-DYNA软件建立了三维煤体模型对不同地应力条件下进行了爆破模拟,模拟结果表明:平行于最大主应力方向的裂纹发育受到促进作用,而垂直于最大主应力方向的裂纹则会受到抑制。同时,通过线性拟合得到损伤量w和分形维数D之间的函数关系,建立了w=5.4381D10.45381的煤体损伤预测模型。

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。

计算机模拟研究油脂结晶网络的显微分形

以计算机编程模拟生成不同形状、尺寸、面积分数和分布模式的晶体显微图像,计算计盒维数(D_(B))、颗粒计数维数(D_(F))和分形布朗维数(D_(FBM)),并采用非参数回归分析和多重线性回归分析观察各个因素对各种分形维数的影响。结果表明:当晶体形状一定时,D_(B)随晶体尺寸和面积分数的增大而增大,且晶体的D_(B)主要由其晶体面积分数所决定;晶体的D_(FBM)主要由其尺寸所决定,晶体的面积分数对其影响较小。均匀分布条件下的D_(B)和D_(FBM)一般小于随机分布,聚集分布条件下的D_(B)和D_(FBM)一般大于随机分布。晶体的形状、尺寸、面积分数和分布模式对D_(F)均无显著影响。在油脂结晶的显微分形维数中,D_(B)可以用于表征晶体面积分数的变化,D_(FBM)可以用于表征晶体尺寸的变化。

石英含量及方解石脉倾角对页岩破坏特征影响数值模拟

应用数值分析软件RFPA2D-DIP对不同石英含量及方解石脉倾角页岩试样在单轴压缩下的力学特征及裂缝扩展规律进行研究,分析了石英含量与方解石脉倾角α对试样抗压强度、应力-应变曲线、破裂模式和分形维数的影响。结果表明,在石英含量相同的情况下,α=0°、α=15°和α=90°时,抗压强度较高;α从15°至30°时,试样的抗压强度大幅降低;当α在30°~75°时,受到的滑移作用较明显,试样的抗压强度较低。α=0°时,试样的抗压强度随着石英含量的增加而增加;但将α增加到30°后,相同倾角下试样的抗压强度随石英含量增加而减小。α=0°、α=15°、α=30°和α=90°的试样破坏模式较为复杂,对应的分形维数较大,便于裂缝在试样中扩展,促进复杂缝网的形成。

岩溶洼地建库背景下水-气耦合试验相似材料研制

在岩溶洼地建库研究中,为研制出满足水-气耦合模拟试验所需的相似材料,以砂子、普通水泥、石膏为原料,设计了4组不同粒径级配的试验组。通过岩石低渗测试技术和核磁共振分析技术,分析了4组不同颗粒级配相似材料的渗透率、孔隙分布及逾渗特征。研究结果表明:在分形维数Df=1.68~2.31范围内,相似材料的渗透率随颗粒级配分形维数增加而线性增大。4组试样中,分形维数越大,则材料内孔隙总量越多,孔径分布越集中,对应的主控孔隙尺寸降低。随着分形维数的增大,灰度阈值升高,逾渗阈值降低,材料越容易发生逾渗。基于实测渗透率,建立了相似材料的渗透率计算模型,计算值与实测值吻合度较高。研究成果成功运用于岩溶洼地模型试验并取得显著效果,为进一步研究水-气作用提供了合适的试验材料。

Application Study of Fractal Theory in Mechanical Transmission

Mechanical transmissions are applied widely in various electrical and mechanical products, but some qualities of some high-end products can＇t meet people＇s demand, and need to be improved with some new methods or theories. The fractal theory is a new mathematic tool, which provides a new approach for the further study in the area of the mechanical transmission, and helps to solve some problems. The basic contents of the fractal theory are introduced firstly, especially the two important concepts, the self-similar fractal and the fractal dimension. Then, the deferent application of the fractal theory in this area are given to display how to further the study and improve some important characteristics of the mechanical transmission, such as contact surfaces, manufacturing precise, friction and wear, stiffness, strength, dynamics, fault diagnosis, etc. Finally, the problems of the fractal theory and its application are discussed, and some weaknesses, such as the calculation capacity of the fractal theory is not strong, are pointed out. Some new solutions are suggested, such as combining the fractal theory with the fuzzy theory, the chaos theory and so on. The new application fields of the fractal theory in the area of the mechanical transmission are proposed.

Investigation on jointed rock strength based on fractal theory

Strength of discontinuities with complex structure is an important topic in rock engineering.A large number of studies have shown that fractal is applicable in the description of this discontinuity.Using fractal interpolation method for the generation of rock joints,numerical experiments of shear tests of the jointed rock mass model were carried out using FLAC^(3D).The test results show that the real rock joints can be simulated by fractal curves obtained by fractal interpolation.The fractal dimension is an important factor for the characterization of jointed rock mass;test results show that the fractal dimension of rock joints can be related to the equivalent cohesion strength and shear strength of the rock mass.When the fractal dimension of the joint surface is less than critical dimension Dc 1.404,the cohesion strength and shear strength of the rock mass increase as the fractal dimension increases;for larger fractal dimensions,all mechanical parameters decrease as the fractal dimension increases.Joint surfaces with different degrees of roughness were obtained by the fractal interpolation method.Three types of failure modes were observed in the tests:climbing slip failure,climbing gnawing fracture,and non-climbing gnawing fracture.

On the Fractal Design in Human Brain and Nervous Tissue

Digital imaging techniques have enabled to gain insight into complex structure-functional processes involved in the neo-cortex maturation and in brain development, already recognized in anatomical and histological preparations. Despite such a refined technical progress most diagnostic records sound still elusive and unreliable because of use of conventional morphometric approaches based on a unique scale of measure, inadequate for investigating irregular cellular components and structures which shape nervous and brain tissues. Instead, these could be efficiently analyzed by adopting principles and methodologies derived from the Fractal Geometry. Through his masterpiece, The Fractal Geometry of Nature [1], Benoît Mandelbrot has provided a novel epistemological framework for interpreting the real life and the natural world as they are, preventing whatever approximation or subjective sight. Founded upon a body of well-defined laws and coherent principles, the Fractal Geometry is a powerful tool for recognizing and quantitatively describing a good many kinds of complex shapes, living forms, organized patterns, and morphologic features long range correlated with a broad network of functional interactions and metabolic processes that contribute to building up adaptive responses making life sustainable. Scale free dynamics characterized biological systems which develop through the iteration of single generators on different scales thus preserving proper self-similar traits. In the last decades several studies have contributed to showing how relevant may be the recognition of fractal properties for a better understanding of brain and nervous tissues either in healthy conditions or in altered and pathological states.