Evolution Of Fractal Dimension Research Articles

Characterization of shear behavior and damage mechanism of periodic thermal loading sandstone based on NMR technique

We conducted research on the influence of periodic thermal loading on the shear behavior of sandstone through a series of direct shear tests with the aid of acoustic emission techniques. By capturing the evolution in pore structure with nuclear magnetic resonance technology and fractal theory to analyze the damage mechanism. The shear testing reveals that the periodic heating and air cooling sample exhibits a lower shear resistance compared to the continuous heating rock, with more pronounced differences observed in the water cooling samples. The peak shear strength and stiffness are greatly influenced by temperature and thermal loading methods, while the ultimate shear strength remains largely unaffected. Compared to samples with less severe damage, the acoustic emission macrocracking proportion of sandstone with more damage after thermal loading is more significant, corresponding to a smaller b-value. The study of the evolution in pore structure shows that mesopores and macropores exhibit favorable fractal characteristics distinct from micropores. The fractal dimension of mesopores and macropores are observed to be influenced by heating temperature and periodic thermal loading, as evidenced by their negative correlation with the thermal damage variable. After thermal loading, there was a negative correlation between porosity and fractal dimension in mesopores and macropores, indicating that the more severe the damage to the pore structure, the smaller the fractal dimension. The shear strength of periodic thermal loaded sandstone is closely related to the evolution of fractal dimension. The presence of thermal enhancement at 400 °C has been confirmed for the first time through fractal dimension analysis.

Real-time pore structure evolution difference between deep and shallow coal during gas adsorption: A study based on synchrotron radiation small-angle X-ray scattering

Coal seam CO2 sequestration is an important option to address global warming. A better knowledge on coal pore structure evolution during gas adsorption can provide guidance for coal seams CO2 sequestration. However, few investigations on the pore structure evolution differences between the deep and shallow coal were conducted during gas adsorption. In this study, based on the real-time synchrotron radiation small-angle X-ray scattering (SAXS) observation, the average pore diameter and pore surface fractal dimension evolution differences between deep and shallow coal were investigated from the aspects of coal compositions and stress history. Two types of coal deformation (inner-swelling and outer-swelling) coexist during gas adsorption. Coal compositions have significant impact on the dominance of deformation type. The dominance of inner-swelling in deep coal is induced by the higher ash contents, and there is the decrease of average pore diameter during gas adsorption. The impact of stress-history (burial depth) on adsorption-induced deformation is more prominent than that of gas adsorption capacity. In deep coal, the surface fractal dimension evolution presents a negative correlation with the evolution of pore diameters. In shallow coal, the surface fractal dimension evolution presents a Langmuir-type correlation with the adsorption time.

A novel approach for quantitative characterization of aqueous in-situ foam dynamic structure based on fractal theory

In this study, a quantitative characterization method of in-situ foam microstructure is proposed based on fractal theory and the captured two dimension(2D) microstructure image for the study of its fractal characteristic, evolution law, sensitivity of temperature and concentration, as well as the relationship between average particle size and fractal dimension of in-situ foam. The results demonstrate that in-situ aqueous foam in the bulk phase exhibits fractal characteristics with a box-counting fractal dimension ranging from 1.4 to 2.0. Distinct from the “exponential” evolution law of conventional foam, in-situ foam displays the “check mark” evolution law, which can be precisely described by two items with clear physical meaning in our proposed fitting model. Additionally, the in-situ foam's evolutionary characteristic exhibits high adaptability towards temperature and foaming agent concentration. Elevated temperatures accelerate coalescence, resulting in shorter effective times, while higher concentrations of foaming agents prolong the overall stability of in-situ foam. Besides, the variation of average particle size of in-situ foam correspond closely with the evolution of fractal dimension under different temperatures and foaming agent concentrations, providing the potential for predicting changes in microstructure of in-situ foam by fractal dimension.

Improvement to the Fractal Adsorption Isotherms of Porous Surfaces

The theoretical and experimental Adsorption Isotherms (AI) are essential for the study of surface area and porous structure in solid materials. Also, the fractal dimension characterizes the roughness of a surface which means that a smooth surface has dimension two, whereas that of a highly rugged surface approaches to three. So, although there are several methods to evaluate fractal adsorption isotherms the Fripiat proposal can be considered as the generalization of the Brunauer-Emmett-Teller (BET) formulation for homogeneous surfaces because it contains, as particular case, the classical BET relationship for fractal dimension D=2. However, in the fitting of experimental Nitrogen adsorption data with the Fripiat approach is compulsory to use the BET classical formula for the calculation of the C constant and the monolayer volume. Thus, in order to overcome such constraint, in this work is presented an alternative treatment of fractal adsorption isotherms as applied to porous surfaces. The proposal is based on a relationship between the physical adsorption in homogeneous and inhomogeneous surfaces followed to the assumption that the emerging series are well correlated with polynomials. i.e. we propose an improved formulation to the fractal dimension which is solved by lineal regression avoiding the use of classical parameters. As expected, our proposition contains as particular cases the BET and Fripiat’s formulas. As an example of the usefulness of our proposal, we consider the experimental data of isothermal adsorption obtained with Al2O3 that were synthesized via sol-gel at various temperatures; the results are in good agreement with the fractal dimension evolution and with the microporous and mesoporous structure of the materials under consideration.

Fractal Characteristic Evolution of Coastal Settlement Land Use: A Case of Xiamen, China

Coastal settlements in urban areas show certain degrees of spatial complexity. Understanding the evolution law of fractal settlements is practically important for marine engineering and urban planning. In this paper, we investigate the fractal evolution of coastal settlement land use based on fractal theory. The fractal dimensions of the land uses for three typically coastal settlements in Xiamen city, China, are obtained to quantify their spatial complexity. The results reveal the fractal characteristics and regional differences of the coastal settlements. Furthermore, nonlinear modeling is applied to describe the fractal dimension evolution of the coastal settlement land uses from 2000 to 2018. Three settlements in rapid urbanization show different nonlinear evolution equations of the fractal dimension due to their different land uses. This study might provide a theoretical basis for understanding the fractal characteristic evolution of coastal settlements in urban areas and show its potential application in urban geography.

Evolution of fractal dimension in pulsed laser deposited MoO3 film with ablation time and annealing temperature

Evolution of fractal dimension in pulsed laser deposited MoO3 film with ablation time and annealing temperature

Analysis of Damages in Layered Surrounding Rocks Induced by Blasting During Tunnel Construction

A layered rock mass is a special type of geological body. The existence of a bedding surface may lead to a poor cutting effect (over/under-excavation of the surrounding rock), falling of blocks, or collapse, thereby affecting most constructions in areas with such rocks. Given the lack of a proper quantitative analysis method for surrounding rock damages, the construction process of layered surrounding rock tunnels becomes difficult. To address these problems, three types of cut blasting models with single, double, and four holes are studied in this paper. With this, the LS-DYNA program is used to analyze the behaviors of stress wave propagation, crack propagation, and fracture modes, as well as fracture mechanisms of mudstone, sandstone, and layered rock. Using the image processing technology and fractal theory, the fractal dimension change trend and progressive damage evolution behavior of the three types of rocks under different cut blasting conditions are determined. Also determined is the corresponding relationship between the fractal dimension and the rock damage degree. The results indicate that crack initiation, propagation, bifurcation, and fractal dimension evolution are more closely related to the phenomenon where the compression wave is ahead of the tension wave, and the[Formula: see text]incompatible deformation of the bedding under single-hole blasting. Under double-hole and four-hole blasting, the phenomena, such as spalling, bedding crack penetration, and fracture connection between the explosive holes are caused mainly by the effects of stress concentration, reflected tension waves, and stress wave superposition. Moreover, under different blasting conditions, the rocks exhibit a similar progressive damage process, i.e. a rapid increase at first, then a slow rise, and finally a stabilization phase. The dynamic damage degree of the rock exhibits a linear increasing trend under different blasting holes. The study results provide a useful reference for blasting scheme design and optimization of underground engineering projects.

Optimization of the shale gas reservoir fracture parameters based on the fully coupled gas flow and effective stress model

AbstractFracture parameters are important for oil and gas production. In this paper, we establish a fully coupled gas flow and effective stress model for the single well multi‐stage hydraulic fracturing shale gas reservoir and analyze the optimal fracture parameters of this model. Firstly, the reservoir consists of single‐porosity medium zone, dual‐porosity medium zone, and hydraulic fracture zone. Then, on the basis of the effective stress mechanism, the theory of porous elastic medium, evolution of fractal dimensions, and the mechanism of fluid transportation, we establish an apparent permeability model of matrix and fracture system to couple the deformation of reservoir and gas flow in matrix. After that, the partial differential equation is solved by COMSOL and verified using the field shale gas production data of Barnett and Marcellus. Finally, the model is used to optimize the fracture parameters. The results show that by increasing the half‐length of hydraulic fracture, the gas production gets the maximum growth, followed by increasing the fracture diversion capacity and width of hydraulic fractures. For the shale gas reservoir with large‐fractured and relatively high permeability, increasing the length of hydraulic fractures can bring higher production, while for the micro‐fractured and low‐permeability, hydraulic fractures with complex shapes can contribute to production increase.

Evaluation of gully development over several years using GIS and fractal analysis: a case study of the Palmital watershed, Minas Gerais (Brazil)

Soil erosion affects agricultural production by reducing the area of rural properties and altering the dynamics of watersheds. Gullies are a complex and irregular type of erosion form that can reach large dimensions, and studying gullies to stabilize erosion processes is a challenge. This study aims to understand their environmental relationships as a basis for future research and to recover land degraded by soil erosion. We combined mapping techniques using geographic information system (GIS) with temporal evolution of land use and fractal dimensions of gullies using FracLac to determine the stability of gullies in a watershed in south Minas Gerais (Brazil). We used R software to apply linear regressions and tested the statistics to verify the compatibility of the model. Environmental characteristics of the watershed contribute to the formation of gullies, but human activities influence the deflagration of these features. The temporal evolution of the land use demonstrated that nearly 50% of the watershed was used as pastures throughout the years, which contributed to soil degradation. The evolution of the fractal dimension showed fluctuations over the years, indicating that the gullies are not stable. The regressions suggest that the vegetation, geology, land use, channel order, and curvature may influence the fractal dimensions of gullies. We concluded that human activity influences the evolution of gullies; the use of techniques such as vegetation measures can contribute to the stabilization of gullies. We also concluded that fractal analysis is an interesting tool for performing environmental evaluations of irregular and complex features, such as gullies.

Fractal evolution mechanism of rock fracture in undersea metal mining

Through rock mechanics test, similar simulation experiment, borehole photographic observation of rock fissure, numerical simulation calculation of plastic zone distribution and deformation monitoring of rock mass during undersea mining, the fractal evolution mechanisms of rock fracture in undersea metallic deposits of Sanshandao Gold Mine were studied by fractal theory. The experimental researches on granite mechanics test in undersea deposit indicate that with the increase of load, the granite deformation energy and the fractal dimension of acoustic emission (FDAE) increase gradually. However, after reaching the peak stress of specimen, the fractal dimensions of acoustic emission (FDAEs) decrease and the granite specimen fails. Therefore, the fractal dimension evolution of rock failure can be divided into four stages, which are fissure inoculation stage, fissure growth stage, fissure expansion stage and fracture instability stage, respectively. By calculating and analyzing the damage photographs of rock specimens in Sanshandao Gold Mine, the fractal dimension of rock fissure is 1.4514, which is close to the average value of FDAE during granite destruction, i.e., 1.4693. Similar simulation experiments of undersea mining show that with the excavation proceeding, the FDAE in rock stratum increases gradually, and when the thickness of the isolation roof is less than 40 m, the FDAE begins to decrease, and meanwhile the sign of water inrush emerges. The numerical simulation researches on the plastic zone distribution of undersea mining in Sanshandao Gold Mine indicate that the fractal dimension of plastic zone (FDPZ) where the failure characteristics occur is 1.4598, close to the result of similar simulation experiment of 1.4364, which shows the sign of water inrush. Meanwhile, the thickness of the isolation roof for undersea mining should be more than 40 m, which is consistent with the results of similar simulation experiment. In Sanshandao Gold Mine, the rock fissures in undersea mining were observed by borehole photography and the rock mass deformation was monitored by multi-point displacement meters, and at the same time the fractal dimensions of strata borehole fissure distribution and energy release ratio (ERR) of rock mass were calculated by fractal principle, which are 1.2328 and 1.2685, respectively. The results demonstrate that rock deformation and fissure propagation are both in the second stage of fissure growth, and have not reached the fourth stage of fracture instability. Therefore, the conclusion can be obtained that the undersea mining in Sanshandao Gold Mine is safe at present.

EVOLUTION OF FRACTAL DIMENSIONS AND GAS TRANSPORT MODELS DURING THE GAS RECOVERY PROCESS FROM A FRACTURED SHALE RESERVOIR

Previous studies ignore the evolutions of pore microstructure parameters (pore diameter fractal dimension [Formula: see text] and tortuosity fractal dimension [Formula: see text]) but these evolutions may significantly impact the gas transport during gas extraction. In order to investigate these evolutions of fractal dimension properties during gas extraction, following four aspects are studied. Firstly, surface diffusion in adsorbed multilayer is modeled for fractal shale matrix. Our new matrix permeability model considers the slip flow, Knudsen diffusion and surface diffusion. This model is verified by experimental data. Secondly, a new fracture permeability model is proposed based on fractal theory and the coupling of viscous flow and Knudsen diffusion. Thirdly, the multilayer adsorption and these permeability models are introduced into the equations of gas flow and reservoir deformation. Finally, sensitivity analysis is performed to determine the key factors on fractal dimension evolution. The results show that the multilayer adsorption can accurately describe the adsorption properties of real shale reservoir. Shale reservoir deformation and gas desorption govern the evolutions of fractal dimensions. The multilayer adsorption and adsorbed gas porosity [Formula: see text] play an important role in the evolutions of fractal dimensions during gas extraction. The monolayer saturated adsorption volume [Formula: see text] is the most sensitive parameter affecting the evolution of fractal dimensions. Therefore, the effects of gas adsorption on the evolution of fractal dimensions cannot be neglected in shale reservoirs.

Failure Prediction of Two Types of Rocks Based on Acoustic Emission Characteristics

The stability analysis of rock is an important basis to ensure the safe exploitation of underground resources and the reliable operation of space engineering. Uniaxial compression and acoustic emission (AE) tests were carried out on two common rock samples with strong rock burst tendency. The relationship between mechanical characteristics, AE characteristics, and rock burst tendency in the failure process of rock with different body types and the evolution of fractal characteristics of AE parameters were discussed. Based on the cusp mutation theory, the catastrophe model of AE characteristic parameters was established to quantify the instability mechanism of rock mass. The results show that the AE mutation rate (AEMR) of the cubic specimens increase from a low level to a high level gradually in the stable fracture stage, while that of the cylindrical specimens increase sharply to the maximum when the specimens are near failure. The AE cumulative energy curves of cubic specimens show a “step” rise, while that of cylindrical specimens show a “gradual” rise, and the rock burst process of cubic specimens is faster. The fractal dimension evolution mode of AE characteristic parameters of cubic specimens during uniaxial compression text is decline-rise-decline, while that of cylindrical specimen is decline-rise-decline-steeply rise. According to the periodic change of AE cumulative energy curve, combined with the rock failure cusp mutation model, the occurrence of rock burst can be well predicted, providing certain theoretical guidance for the stability analysis of underground engineering rock mass.

Neural network for fractal dimension evolution.

The coagulation/flocculation process is an essential step in drinking water treatment. The process of formation, growth, breakage and rearrangement of the formed aggregates is key to enhancing the understanding of the flocculation process. Artificial neural networks (ANNs) are a powerful technique, which can be used to model complex problems in several areas, such as water treatment. This work evaluated the evolution of the fractal dimension of aggregates obtained through ANN modeling in the coagulation/flocculation process conducted in high apparent color water (100 ± 5 PtCo), using alum as coagulant in dosages varying from 1 to 12 mg Al3+ L-1, and shear rates from 20 to 60 s-1 for flocculation times from 1 to 60 minutes. Based on raw data, the ANN model resulted in optimized condition of 9.5 mg Al3+ L-1 and pH 6.1, for color removal of 90.5%. For fractal dimension evolution, the ANN was able to represent from 95% to 99% of the results.

Self-Elongation with Sequential Folding of a Filament of Bacterial Cells

Under hard-agar and nutrient-rich conditions, a cell of $Bacillus$ $subtilis$ grows as a single filament owing to the failure of cell separation after each growth and division cycle. The self-elongating filament of cells shows sequential folding processes, and multifold structures extend over an agar plate. We report that the growth process from the exponential phase to the stationary phase is well described by the time evolution of fractal dimensions of the filament configuration. We propose a method of characterizing filament configurations using a set of lengths of multifold parts of a filament. Systems of differential equations are introduced to describe the folding processes that create multifold structures in the early stage of the growth process. We show that the fitting of experimental data to the solutions of equations is excellent, and the parameters involved in our model systems are determined.,

Monofractal and multifractal behavior of worn surface in brass–steel tribosystem under mixed lubricated condition

The dynamic evolution law of brass surface in brass–steel tribopair during wear process is studied by using monofractal and multifractal methods. Wear tests are performed on a ring-on-disc tribometer under mixed lubrication. The profile of brass surface is measured by the reposition-measurement apparatus. Both the maximum peak-to-valley height Rt and the width of multifractal spectrum Δα decrease in running-in wear stage, maintain at a small value in steady wear stage, and increase rapidly in severe wear stage. While the evolution of fractal dimension is opposite to that of Rt and Δα. The results are instructive to wear state identification.

Analysis of the three-dimensional TGS crystal spatial inhomogeneity of the fractal structure

The new analysis of fractal spatial inhomogeneity of the fractal structure imitating evolution of fractal dimension during three-dimensional TGS crystal growth is provided. A final crystal structure was scanned layer-by-layer using X-ray tomography. Two-dimensional images were gradually imposed in order to calculate capacity fractal dimension. The spatio-temporal aspects of crystal growth were revealed by wavelet analysis and the main growth phases were determined.

Assessment of disintegrant efficacy with fractal dimensions from real-time MRI

An efficient disintegrant is capable of breaking up a tablet in the smallest possible particles in the shortest time. Until now, comparative data on the efficacy of different disintegrants is based on dissolution studies or the disintegration time. Extending these approaches, this study introduces a method, which defines the evolution of fractal dimensions of tablets as surrogate parameter for the available surface area. Fractal dimensions are a measure for the tortuosity of a line, in this case the upper surface of a disintegrating tablet. High-resolution real-time MRI was used to record videos of disintegrating tablets. The acquired video images were processed to depict the upper surface of the tablets and a box-counting algorithm was used to estimate the fractal dimensions. The influence of six different disintegrants, of different relative tablet density, and increasing disintegrant concentration was investigated to evaluate the performance of the novel method. Changing relative densities hardly affect the progression of fractal dimensions, whereas an increase in disintegrant concentration causes increasing fractal dimensions during disintegration, which are also reached quicker. Different disintegrants display only minor differences in the maximal fractal dimension, yet the kinetic in which the maximum is reached allows a differentiation and classification of disintegrants.

Analytical Model of Fractal Aggregate Stability and Restructuring in Shear Flows

We combine the analytical theory of nonaffine deformations of noncrystalline solids with numerical Stokesian dynamical simulations to obtain analytical closed-form expressions for the shear modulus of fractal aggregates in shear flows. The proposed framework also provides analytical predictions for the evolution of the fractal dimension df of the aggregate during the aggregation process. This leads to a lower bound on df below which aggregates are mechanically unstable (they possess floppy modes) and cannot survive without restructuring into more compact, higher-df configurations. In the limit of large aggregates, the predicted lower bound is df = 2.407. This result provides the long-sought explanation as to why all experimental and simulation studies in the past consistently reported df ≳ 2.4 for shear-induced colloidal aggregation. The analytical expressions derived here can be used within population balance calculations of colloidal aggregation in shear flows whereby until now the fractal dimension evolution was treated as a free parameter. These results may open up the possibility of developing new microscopic mechanical manipulation techniques to control nanoparticle and colloidal aggregates at the nanoscale.

A Fractal and Numerical Simulation Coupled Study of Fracture Network during Coal Mining Excavation

This paper features a numerical study that is carried out by using discontinuous deformation method (DDA) and fractal geometry. The configurations of rock strata calculated by DDA were imported into a code that is written by using VC++ called “Fractal” to calculate the fractal dimension of the rock strata. As illustrated, a long wall mining case in China was presented. The relationship of the fractal dimension, excavation length, stress, and movement of strata were discussed. The evolution of fractal dimension can be considered as an index of instability or failure. The method proposed in this paper can be employed to predict the period weighting in long wall mining engineering.

Fractal dimension evolution and spatial replacement dynamics of urban growth

This paper presents a new perspective of looking at the relation between fractals and chaos by means of cities. Especially, a principle of space filling and spatial replacement is proposed to interpret the fractal dimension of urban form. The fractal dimension evolution of urban growth can be empirically modeled with Boltzmann’s equation. For the normalized data, Boltzmann’s equation is just equivalent to the logistic function. The logistic equation can be transformed into the well-known 1-dimensional logistic map, which is based on a 2-dimensional map suggesting spatial replacement dynamics of city development. The 2-dimensional recurrence relations can be employed to generate the nonlinear dynamical behaviors such as bifurcation and chaos. A discovery is thus made in this article that, for the fractal dimension growth following the logistic curve, the normalized dimension value is the ratio of space filling. If the rate of spatial replacement (urban growth) is too high, the periodic oscillations and chaos will arise. The spatial replacement dynamics can be extended to general replacement dynamics, and bifurcation and chaos mirror a process of complex replacement.