Fractals

Fractal represents a special feature of nature and functional objects. However, fractal based computing can be applied to many research domains because of its fixed property resisted deformation, variable parameters and many unpredictable changes. Theoretical research and practical application of fractal based computing have been hotspots for 30 years and will be continued. There are many pending issues awaiting solutions in this domain, thus this thematic issue containing 14 papers publishes the state-of-the-art developments in theorem and application of fractal based computing, including mathematical analysis and novel engineering applications. The topics contain fractal and multifractal features in application and solution of nonlinear odes and equation.

Magnetic resonance imaging (MRI) is the most sensitive method to detect chronic nervous system diseases such as multiple sclerosis (MS). In this paper, Brownian motion Hölder regularity functions (polynomial, periodic (sine), exponential) for 2D image, such as multifractal methods were applied to MR brain images, aiming to easily identify distressed regions, in MS patients. With these regions, we have proposed an MS classification based on the multifractal method by using the Self-Organizing Map (SOM) algorithm. Thus, we obtained a cluster analysis by identifying pixels from distressed regions in MR images through multifractal methods and by diagnosing subgroups of MS patients through artificial neural networks.

In the present paper, a family of the special functions via the celebrated Mittag–Leffler function defined on the Cantor sets is investigated. The nonlinear local fractional ODEs (NLFODEs) are presented by following the rules of local fractional derivative (LFD). The exact solutions for these problems are also discussed with the aid of the non-differentiable charts on Cantor sets. The obtained results are important for describing the characteristics of the fractal special functions.

This paper presents a preliminary work to evaluate the influence of slag and superplasticizer on the early-age hydration of cement pastes by an innovative non-contact impedance measurement, heat evolution method, compressive strength and setting time tests. Besides, the cumulative pore volume obtained from modulus and phase of impedance in different hydration sections is taken to continuously correlate the cumulative heat releasing of cement pastes via the fractal analysis. Retarded phenomena and mechanism of hydration in cement pastes incorporated with slag and superplasticizer are studied, respectively. It is found that the compressive strength and setting time have a good linear relation with the slag amount in blended cement pastes.

Fractal encoding method becomes an effective image compression method because of its high compression ratio and short decompressing time. But one problem of known fractal compression method is its high computational complexity and consequent long compressing time. To address this issue, in this paper, distance clustering in high dimensional sphere surface is applied to speed up the fractal compression method. Firstly, as a preprocessing strategy, an image is divided into blocks, which are mapped on high dimensional sphere surface. Secondly, a novel image matching method is presented based on distance clustering on high dimensional sphere surface. Then, the correctness and effectiveness properties of the mentioned method are analyzed. Finally, experimental results validate the positive performance gain of the method.

This paper presents a preliminary work to evaluate the electrical impedance of three-dimensional pore fractal networks embedded in a cube. The construction and structural parameters of pore and electrical networks are illustrated in detailed. The total impedance response (modulus and phase) of networks is primarily associated with structural parameter of network, pore structure parameter of cube, conductivity of pore and solid phases. The impedance response of cube with the evolution of fractal network is analyzed comprehensively in this work. Besides, the influence of conductivity of pore phase caused by different types of electrolytes, electrical frequencies and conductivity of solid phase on the total impedance response is systematically studied.

The new Boussinesq-type model in a fractal domain is derived based on the formulation of the local fractional derivative. The novel traveling wave transform of the non-differentiable type is adopted to convert the local fractional Boussinesq equation into a nonlinear local fractional ODE. The exact traveling wave solution is also obtained with aid of the non-differentiable graph. The proposed method, involving the fractal special functions, is efficient for finding the exact solutions of the nonlinear PDEs in fractal domains.

According to hydraulic-fracturing practices conducted in shale reservoirs, effective stimulated reservoir volume (ESRV) significantly affects the production of hydraulic fractured well. Therefore, estimating ESRV is an important prerequisite for confirming the success of hydraulic fracturing and predicting the production of hydraulic fracturing wells in shale reservoirs. However, ESRV calculation remains a longstanding challenge in hydraulic-fracturing operation. In considering fractal characteristics of the fracture network in stimulated reservoir volume (SRV), this paper introduces a fractal random-fracture-network algorithm for converting the microseismic data into fractal geometry. Five key parameters, including bifurcation direction, generating length ($d$), deviation angle ($\alpha$), iteration times ($N$) and generating rules, are proposed to quantitatively characterize fracture geometry. Furthermore, we introduce an orthogonal-fractures coupled dual-porosity-media representation elementary volume (REV) flow model to predict the volumetric flux of gas in shale reservoirs. On the basis of the migration of adsorbed gas in porous kerogen of REV with different fracture spaces, an ESRV criterion for shale reservoirs with SRV is proposed. Eventually, combining the ESRV criterion and fractal characteristic of a fracture network, we propose a new approach for evaluating ESRV in shale reservoirs. The approach has been used in the Eagle Ford shale gas reservoir, and results show that the fracture space has a measurable influence on migration of adsorbed gas. The fracture network can contribute to enhancement of the absorbed gas recovery ratio when the fracture space is less than 0.2 m. ESRV is evaluated in this paper, and results indicate that the ESRV accounts for 27.87% of the total SRV in shale gas reservoirs. This work is important and timely for evaluating fracturing effect and predicting production of hydraulic fracturing wells in shale reservoirs.

How to analyze and identify the characteristics of radiation sources and estimate the threat level by means of detecting, intercepting and locating has been the central issue of electronic support in the electronic warfare, and communication signal recognition is one of the key points to solve this issue. Aiming at accurately extracting the individual characteristics of the radiation source for the increasingly complex communication electromagnetic environment, a novel feature extraction algorithm for individual characteristics of the communication radiation source based on the fractal complexity of the signal is proposed. According to the complexity of the received signal and the situation of environmental noise, use the fractal dimension characteristics of different complexity to depict the subtle characteristics of the signal to establish the characteristic database, and then identify different broadcasting station by gray relation theory system. The simulation results demonstrate that the algorithm can achieve recognition rate of 94% even in the environment with SNR of −10dB, and this provides an important theoretical basis for the accurate identification of the subtle features of the signal at low SNR in the field of information confrontation.

The granularity distribution of powder is shown to have the fractal characters. Based on the fractal theory, three algebraic expressions for the fractal dimension, the cumulative probability and the mass distribution of particles were obtained. Three expressions are the functions of the concentration of particles, the diameter of particles, the maximum and the minimum particle diameters. No additional empirical constant is introduced in these expressions. These fractal models contain less empirical constants than the conventional correlations. The fractal behavior about coal particles was discussed in detail. Simulations were also performed by fractal Monte Carlo technique. The proposed technique may provide us a new approach to analyze the granularity distribution of powder.

Multiple sclerosis (MS) is a severe brain disease. Early detection can provide timely treatment. Fractal dimension can provide statistical index of pattern changes with scale at a given brain image. In this study, our team used susceptibility weighted imaging technique to obtain 676 MS slices and 880 healthy slices. We used synthetic minority oversampling technique to process the unbalanced dataset. Then, we used Canny edge detector to extract distinguishing edges. The Minkowski–Bouligand dimension was a fractal dimension estimation method and used to extract features from edges. Single hidden layer neural network was used as the classifier. Finally, we proposed a three-segment representation biogeography-based optimization to train the classifier. Our method achieved a sensitivity of 97.78±1.29%, a specificity of 97.82±1.60% and an accuracy of 97.80±1.40%. The proposed method is superior to seven state-of-the-art methods in terms of sensitivity and accuracy.

Because of the low permeability, multi-stage hydraulic fractured horizontal wells (MHFHWs) occupy a dominant position among production wells in tight gas reservoir. However, net present value (NPV) estimation method for MHFHW in tight gas reservoirs often ignores the effect of heterogeneity in microscopic pore structure. Apart from that, a new fractal model is presented for NPV of MHFHW, based on the fractal expressions of formation parameters. First, with the aid of apparent permeability model, a pseudo pressure expression considering both reservoir fractal features and slippage effect is derived, contributing to establish the productivity model. Secondly, economic assessment method is built based on the fractal productivity model, in order to obtain the NPV of MHFHW. Thirdly, the type curves are illustrated and the influences of different fractal parameters are discussed. The pore fractal dimensions $D_f$ and the capillary tortuosity fractal dimensions $D_T$ have significant effects on the NPV of an MHFHW. Finally, the proposed model in this paper provides a new methodology for analyzing and predicting the NPV of an MHFHW and may be conducive to a better understanding of the optimal design of MHFHW.

In this study, the period-$n$ steady regions of McMullen family M sets are discussed. The calculation methods of the numbers of period-$n$ steady regions are presented and the center points and boundaries of period-1 are studied by complex dynamical system theory. In addition, the problem about the free critical points is discussed. The experimental results show that the free critical points do not influence the distributions of period-$n$ steady regions when $m=d$, which is proved theoretically.

X-ray computed tomography (CT) scanning method is the most accurate method to construct digital core, which can reflect the microscopic pore structure of real cores; therefore, it is widely used and researched by experts and scholars all over the world. However, there are few reports about how to select the CT scan core samples at present, and the current practice is to make CT scan samples by visually observing rocks or core columns to select a region that is considered representative or interesting, which can lead to a large difference between the selected sample and the whole rock and a digital core that cannot represent the real rock as a whole. In order to construct the digital cores that can reflect the whole rock structure and reservoir properties, combining with fractal theory, a scientific and reasonable method was proposed to select representative rock samples for digital core modeling. First of all, a core column is scanned by X-ray CT at a certain resolution and CT gray scale images are obtained and stored in the order of scan. Secondly, the fractal dimension (FD) of each image is calculated by box-counting method, and the calculated porosity of each image is achieved by the existing formula. Then, according to the size of the digital core to be constructed, the CT gray scale images are grouped, and the average FD and the average porosity of each combination are calculated by the derived equations. Finally, based on the proposed criteria the best image combination is selected and the preferred sample is determined accordingly. At the same time, a facile experiment was conducted to test the effectiveness of this method. The experimental results show that there are some errors between the subjectively selected cores and the long core in terms of permeability and porosity, and the petrophysical parameters of the core selected by the proposed method are close to those of the long core; as a consequence, the validity of this method was verified and it is feasible and practical to select the representative rock samples for digital core modeling by this method.

As natural fractures show statistically fractal scaling laws, fractal geometry has been proposed and applied to model the fracture geometry and to study the hydraulic properties of fractured porous media. In this paper, a fractal dual-porosity model is developed to study the single-phase fluid flow through fractured porous media. An analytical expression for effective permeability of fractured porous media is derived, which depends on the fractal dimension and fracture aperture. The effect of fractal dimensions for fracture aperture distribution and tortuosity, the ratio of minimum to maximum fracture apertures and fracture fraction on the effective permeability have been discussed. In addition, a power law relationship between the effective permeability and fracture fraction is proposed to predict the equivalent hydraulic properties of fractured porous media. Compared with empirical formulas for effective permeability, the present fractal dual-porosity model can capture the statistical characteristics of fractures and shed light on the transport mechanism of fractured porous media.