Study on Fractal Characteristics of Mineral Particles in Undisturbed Loess and Lime-Treated Loess

In order to explore the fractal characteristics of particle size distribution (PSD) of various minerals in loess and lime-treated loess, the Q4 undisturbed loess and lime-treated loess were studied. From the perspective of multi-scaled microstructure, the internal characteristics of loess were observed and the regularity statistics were carried out from a macroscopic view. Fractal theory was used to quantitatively study the distribution of mineral particles in undisturbed loess and lime-treated loess. It was found that the skeleton particles of undisturbed loess were obvious and the structure of soil was loose. While that of lime-treated loess decreased, the fine particles were connected with each other, and the structure of soil changed from loose to dense. The three mineral particles in the undisturbed loess and lime-treated loess did not accord with the single fractal distribution characteristics, but the total particles had fractal characteristics. The percentage content of the mineral particles in the soil varied greatly with the particle size. In addition, the non-uniform degrees of mineral particles in the two soils from large to small were carbonate minerals of lime-treated loess, carbonate minerals of undisturbed loess, quartz minerals of lime-treated loess, feldspar mineral of lime-treated loess, feldspar mineral of the undisturbed loess, and the quartz mineral of the undisturbed loess. This paper provided a basis for the future study of the different soil mechanical properties of undisturbed loess and lime-treated loess.

Some Notes on Granular Mixtures with Finite, Discrete Fractal Distribution

Why fractal distribution is so frequent? It is true that fractal dimension is always less than 3? Why fractal dimension of 2.5 to 2.9 seems to be steady-state or stable? Why the fractal distributions are the limit distributions of the degradation path? Is there an ultimate distribution? It is shown that the finite fractal grain size distributions occurring in the nature are identical to the optimal grading curves of the grading entropy theory and, the fractal dimension n varies between –¥ and ¥. It is shown that the fractal dimensions 2.2–2.9 may be situated in the transitional stability zone, verifying the internal stability criterion of the grading entropy theory. Micro computed tomography (μCT) images and DEM (distinct element method) studies are presented to show the link between stable microstructure and internal stability. On the other hand, it is shown that the optimal grading curves are mean position grading curves that can be used to represent all possible grading curves.

Possible Effects of the Fractal Distribution of Relic Wormholes

We discuss the possibility that the distribution of relic wormholes may possess fractal properties. Relic wormholes and their fractal distributions are predicted in a natural way by lattice quantum gravity models. This provides a new approach to some long standing problems. That is the nature of dark matter phenomena, the origin of Faber-Jackson and Tully-Fisher relations and the observed deficit of baryons. We derive corrections to the Newton’s potential caused by the presence of relic wormholes and show that the analysis of dark matter distribution in galaxies allows us to fix the parameters of the fractal distribution of wormholes.

Predicting Electrokinetic Coupling and Electrical Conductivity in Fractured Media Using a Fractal Distribution of Tortuous Capillary Fractures

Electrokinetics methods have attracted increasing interest to characterize hydrogeological processes in geological media, especially in complex hydrosystems such as fractured formations. In this work, we conceptualize fractured media as a bunch of parallel capillary fractures following the fractal size distribution. This conceptualization permits to obtain analytical models for both the electrical conductivity and the electrokinetic coupling in water saturated fractured media. We explore two different approaches to express the electrokinetic coupling. First, we express the streaming potential coupling coefficient as a function of the zeta potential and then we obtain the effective charge density in terms of macroscopic hydraulic and electrokinetic parameters of porous media. We show that when the surface electrical conductivity is negligible, the proposed models reduces to the previously proposed one based on a bundle of cylindrical capillaries. This model opens up a wide range of applications to monitor the water flow in fractured media.

Mathematical models for calculating the quantitative characteristics of the optimal quantization of information

Various additional mathematical aspects related to solving the problem of optimal information quantization in the sense of filling are considered, such as control of quantum elements, accounting for errors of quantum elements, determining the amount of information during quantization, and determining the numerical values of fractals of distributions represented as a sequential fractal distribution. The purpose of the article is to consider additional questions based on a specific "heavy" probability distribution the normal distribution. The considered questions are made in order to facilitate the solution of applied problems for researchers dealing with the problem of information quantization.

Carbonate Acidising Calculation Model Coupled with Dual-Fractal Wormhole

As recognized as the most economical and effective measure to increase carbonate oil and gas well production, matrix acidising is widely used. The main feature of acidising and the key influence factor of increasing production is what kind of acid etched wormholes can be formed. The actual etched wormholes grow in disorder and randomly, that's why it is extremely difficult to describe by classical mathematical methods. Due to the lack of a quantitative calculation model for the growth law of acid-etched wormholes, penetration depth, competition and distribution patterns among different etched wormholes, an effective method for acidising parameters optimization cannot be formed. The stimulated production of different wells vary greatly. In order to establish the corresponding quantitative calculation model for geometric size of acid etched wormholes, three-dimensional competitive distribution of different wormholes, and production prediction, also to achieve the quantitative optimization of treatment parameters for different wells and improve oil and gas production, firstly, we designed and completed indoor core etched experiments and CT scanning technology to obtain the true three-dimensional morphology of linear acid etched wormholes. Besides, the radial wormholes in 14 cubic feet of super large cores were proven to meet the requirement of fractal. The fractal dimensions of these two wormhole types were also obtained. Furthermore, a quantitative calculation model for the wormhole length expansion was established. Secondly, according to the mathematical model of extending competition among different acid etched wormholes, the fractal distribution law of the length of wormholes in the vertical direction is obtained. Combined with the fractal wormhole length calculation model, a method to solve the dual fractal model by calculating the maximum wormhole length is given. Finally, the classic acidised production rate calculation model was revised. The influence of the three-dimensional expansion length and distribution of wormholes on the skin factor was considered in detail. The sensitivity of key acidising parameters, such as acid strength and pumping rate, was also analyzed. The results show that both the linear wormholes obtained from conventional cores and the radial wormholes obtained from super-large cores can be described by fractal geometry. The fractal dimension corresponding to the optimal pumping rate is 1.46-1.63. Considering the dual-fractal distribution of acid-etched wormholes, the skin factor is larger than that of the conventional equalization model. This is mainly due to the fact that the equalization model only uses the maximum length of the wormholes. This also explains why wells or layers with negative skin factors can still increase production rate after uniform acidising. At the same time, for a specific layer, there is a better acid strength and pumping rate. With these parameters, the acid consumption and predicted production rate are better, which provides a theoretical basis for the quantitative optimization of acidising treatment parameters.

Cell-to-cell Mathematical Modeling of Arrhythmia Phenomena in the Heart

With an aperiodic, self-similar distribution of two-dimensional arrangement of atrial cells, it is possible to simulate such phenomena as Fibrillation, Fluttering, and a sequence of Fibrillation-Fluttering. The topology of a network of cells may facilitate the initiation and development of arrhythmias such as Fluttering and Fibrillation. Using a GPU parallel architecture, two basic cell topologies were considered in this simulation, an aperiodic, fractal distribution of connections among 462 cells, and a chessboard-like geometry of 60×60 and 600×600 cells. With a complex set of initial conditions, it is possible to produce tissue behavior that may be identified with arrhythmias. Finally, we found several sets of initial conditions that show how a mesh of cells may exhibit Fibrillation that evolves into Fluttering.