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.

Combining Fractals and Box-Counting Dimension

In this paper, the box-counting dimension is used to derive an explicit formula for the dimension of a fractal constructed using several contractions or by combining fractals. This dimension agrees with the Hausdorff dimension in the particular case when the scales factors considered are all the same. A more general sufficient condition for the box-counting dimension and the Hausdorff dimension to be the same is given. It is also shown that the dimension of the fractal obtained by combining two fractals is the weighted average of the dimensions of the two fractals.

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.

Fractal characterization and frequency properties of near-fault ground motions

This study explores the irregularity and complexity of strong earthquake ground motions from the perspective of fractal geometry, and constructs a relation with the frequency content of the ground motions. The box-counting fractal dimensions and five representative period parameters of near-fault ground motions from the Chi-Chi and Northridge earthquakes are calculated and compared. Numerical results indicate that the acceleration and velocity time histories of ground motions present the statistical fractal property, and the dominant pulses of near-fault ground motions have a significant influence on their box dimensions and periods. Further, the average box dimension of near-fault impulsive ground motions is smaller, and their irregular degree of wave forms is lower. Moreover, the box dimensions of ground motions reflect their frequency properties to a large extent, and can be regarded as an alternative indicator to represent their frequency content. Finally, the box dimension D of the acceleration histories shows a considerably negative correlation with the mean period T. Meanwhile, the box dimension of the velocity histories Dye is negatively correlated with the characteristic period T and improved characteristic period Tgi.

Fractal analysis of thermal conduction of loose coal

For deep mining engineering, heat transfer of coal mass is a vital factor in the thermal environment of coal mines. In order to study the thermal conduction mechanism, we obtained gray images of coal mass microstructure by scanning samples with a digital microscope. With the use of Matlab, these gray images were transformed into binary images, which were then transformed into a corresponding matrix consisting only of the values 0 and 1. According to the calculation method of box-counting dimension, we calculated the fractal dimension of the loose coal to be approximately 1.86. The thermal conductivity expressions of loose coal were derived based on the simulation method of thermal resistance. We calculated the thermal conductivity of loose coal by using a fractal model and compared the calculated values with our experimental data. The results show that the test data show an encourag-ing agreement with the calculated values. Hence fractal theory is a feasible method for studying thermal conductivity of loose coal.

红枫分形维数中的物理机制

采用Matlab程序计算与实验测量相结合的方法,以重庆理工大学校园内随机选取的3棵红枫为研究对象,分别测量并计算了植物整体和部分的三维空间分形维度,得出该种植物分形维度与阳光等物理因素之间的关系,以及植物生长过程中的维度变化规律。研究结果表明:植物的维度大小与其生长的物理环境具有逻辑关系。这些关系有助于深入研究高等植物的生长规律,进而为人工裁剪和控制的自然原理提供理论支持。

Fractals in Spatial Patterns of Vegetation Formations

The spatial distribution patterns of species are always scale-dependent and spatially self-similar in ecological systems. In this work, vegetation distribution data collected from the vegetation map of the Xigaze region was analyzed using a box-counting method. The power law of the box-counting dimension （DB） across a range of scales （5-160 km） confirms the fractal patterns for most vegetation formations, while the fluctuations of the scale-specific DB among the different abundance groups indicate limitations of fractal coherence. The fractal method is shown to be a useful tool for measuring the distribution patterns of vegetation formations across scales, which provides important information for both species and habitat conservation, especially in landscape management.

Fractal Studies on Primary Phase Morphology of TA15 Titanium Alloy after Hot Compressive Deformation

The goal of this study is to establish relationships between the hot compression deformation behaviors and the fractal dimension of primary phase morphology of TA15 titanium alloy using the analytical methods of metallurgical microscope and transmission electron microscope coupled with box-counting dimension method. The hot compression deformation behaviors vary with decreasing fractal dimension owing to the change of microstructure caused by different parameters of the hot compressive deformation.The results indicate that TA15 alloy shows dynamic recrystallization characteristics at deformation temperature lower than 850℃while fractal dimension exhibits a moderate decreasing trend with the temperature increasing,and shows dynamic recovery characteristics at deformation temperature higher than 850℃while fractal dimension reduces rapidly with the temperature increasing.The fractal dimension displays non-linear relationship with fraction of primary phase and with aspect ratio of primary phase.

Visualization of collision and aggregation behavior of particles simulating movement of inclusions in molten steel

Inclusions with sizes less than 1 μm in molten steel are difficult to float up to the molten steel and slag interface owing to their slow terminal velocity. Thus, increasing the size of inclusion is essential for accelerating the removal of inclusions. Polystyrene particles simulating inclusions in molten steel were quantified by direct observation of the particle collision behavior in a turbulent flow in a water model. The box-counting fractal dimension of particles was calculated by processing the binary images of aggregated particles. The fractal dimension of the outer contours of the single plastic particles was smaller than that of the aggregated particles. The fractal dimension was varied from 1.14 to 1.35. When two or more monomer particles collide, the aggregates are separated more easily, as the temperature increases from 40 to 80 ℃. The aggregated particles were loose and easy to separate in the high-temperature aqueous solution. The effect of temperature on the surface tension of liquid and the interracial tension of solid and liquid is obvious. The particles are wetting in the water solution at a temperature more than 60 ℃. The relationship between the velocity of the particles and the fractal dimension of the solid particles with the equivalent diameter was discussed.