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The use of fractal dimension to characterize irregular-shaped particles
Abstract
For several decades, sedimentologists have had difficulty in obtaining an efficient index of particle form that can be used to specify adequately irregular morphology of sedimentary particles. Mandelbrot has suggested the use of the fractal dimension as a single value estimate of form, in order to characterize morphologically closed loops of an irregular nature. The concept of fractal dimension derives from Richardson's unpublished suggestion that a stable linear relationship appears when the logarithm of the perimeter estimate of an irregular outline is plotted against the logarithm of the unit of measurement (step length). Decreases in step length result in an increase in perimeter by a constant weight (b) for particles whose morphological variations are the same at all measurement scales (self-similarity). The fractal dimension (D) equals 1.0-(b), where b is the slope coefficient of the best-fitting linear regression of the plot. The value of D lies between 1.0 and 2.0, with increasing values of D correlating with increasing irregularity of the outline. In practice, particle outline morphology is not always self-similar, such that two or possibly more fractal elements can occur for many outlines. Two fractal elements reflect the morphological difference between micro-scale edge textural effects (D1) and macro-scale particle structural effects (D2) generated by the presence of crenellate-edge morphology (re-entrants). Fractal calibration on a range of regular/irregular particle outline morphologies, plus examination of carbonate beach, pyroclastic and weathered quartz particles indicates that this type of analysis is best suited for morphological characterization of irregular and crenellate particles. In this respect, fractal analysis appears as the complementary analytical technique to harmonic form analysis in order to achieve an adequate specification of all types of particles on a continuum of irregular to regular morphology.
... More and more studies show that the optimal amount of sand is closely related to the particle size of silica sand, particle size gradation and curing temperature of cement slurry [22][23][24][25][26]. Therefore, with the help of fractal grading theory [17,27,28], the optimal theoretical ratio under different sand dosages is solved based on the particle size distribution characteristics of coarse sand and fine sand [8,14,[29][30][31][32]. At the same time, according to the change in the viscosity of the aqueous solution of polymer admixture before and after elevated-temperature curing [33][34][35], the high-temperature-resistant liquidphase admixture is selected. On the basis of this finding, the high-temperature cement slurry system is designed. ...
The efficient development of oil and gas resources is inseparable from the progress of drilling technology and the safety of the long life cycle of wellbore. At present, exploration and development is expanding to deep and ultra-deep areas. The long life cycle safety of deep and ultra-deep wells is mainly realized by the sealing performance of cement slurry. Additionally, the accumulation degree of cement slurry particles is closely related to sealing performance. Based on fractal theory, an accumulation model of continuous distribution of additive material particles was designed, which can determine the range of fractal dimension necessary to realize the tight stacking and guide the proportion of solid admixture. The formulation of high temperature-resistant cement slurry was prepared by designing the ratio of solid admixture and optimizing the high temperature-resistant liquid admixture. The evaluation of engineering and temperature resistance of the cement slurry proves the rationality of the accumulation model, which can be applied to the design of a high temperature cementing slurry system in deep and ultra-deep wells.
Sediment bed surfaces exist widely in natural rivers, and many aspects in river dynamics are closely relevant to bed surface roughness, such as flow structure, river resistance and sediment transport. As two important parameters for quantifying bed surface roughness, how average particle size and non-uniformity affect bed surface structure is unknown. Therefore, nine groups of sediment samples with different average particle sizes or different non-uniformities were firstly prepared by screening dry natural sediments. Then, the prepared sediment samples were used to manually pave nine groups of bed surfaces, and the high-precise bed surface digital elevations were obtained by a handheld 3D laser scanner. Finally, the effects exerted by the average particle size and non-uniformity on the bed surface fractal properties were discussed. The results showed that there is only a scale-free range in a profile or a two-dimensional specific direction of a bed surface with normal-distributed particle gradation. The averaged scale-free upper limit in the two-dimensional specific directions and that related to many profiles are less affected by the non-uniformity, but more affected by the average particle size. For the bed surfaces with the same non-uniformity, when the average particle size is smaller than 15 mm, the larger the average particle size is, the smaller the fractal dimension is, but the larger the scale coefficient is; when the average particle size is larger than 15 mm, the larger the average particle size is, the larger the fractal dimension and the scale coefficient are, while for the bed surfaces with the same average particle size, the non-uniformity has no significant effects on the fractal dimension and the scale coefficient. The averaged scale coefficient in the two-dimensional specific directions of an isotropic bed surface and that related to many profiles are approximately equal, but the averaged fractal dimension in the two-dimensional specific directions is obviously larger than that plus 1 related to many profiles.
Fractionation of particles in deep-water sediment gravity flows is an important factor in the resulting deposit and for discriminating sedimentary environments, but remains poorly understood. Quantitative characterization of particle shape was performed for more than ten-thousand particles of experimental gravity flow experiments (both of cohesive and non-cohesive nature) made using Accepted Article This article is protected by copyright. All rights reserved coal and kaolin particles. Eleven particle shape parameters were calculated and their distribution and trends within the experimental basin were evaluated. Results indicate the existence of non-normal distributions and observable correlations between particle shape parameters. Shape parameters such as circularity and roundness are dominant controls on shape variation. Strong correlations exist between mean shape parameters and along-flow distance from the source for particles in non-cohesive flow experiments. Important differences were observed between shape parameter distributions of particles sampled at different areas within the experimental basin, which can be grouped based on their depositional setting (proximal or distal) using multivariate statistical analysis, especially for the non-cohesive flow experiments. A tendency for more elongated and irregularly-shaped particles at the more distal and marginal areas of the studied experimental basin was observed and validated by previous field studies in real-world deep-marine deposits. Besides, fractionation of particles is less-pronounced in cohesive flows compared with non-cohesive ones suggesting the soundness of discrimination of depositional settings based solely on particle shape characteristics is strongly dependent on parent flow characteristics. Yet, results highlight the potential of particle shape analysis in revealing spatial particle shape trends due to hydrodynamic fractionation and discriminating different depositional settings within submarine fans. This methodology may be applied to seafloor and subsurface samples to help identify the flow process and depositional environment.
Karst aquifers are important sources of thermal and groundwater in many parts of the world, such as the Alpine–Dinaric–Carpathian region in Europe. The Upper Triassic dolomites are regionally recognized thermal and groundwater aquifers but also hydrocarbon reservoirs. They are characterized by predominantly fractured porosity, but the actual share of depositional and diagenetic porosity is rarely investigated. In this research, we presented the geometric characterization of the measured microporosity of the Upper Triassic dolomites of the Žumberak Mts (Croatia), through thin-section image processing and particle analysis techniques. Pore parameters were analyzed on microphotographs of impregnated thin sections in scale. A total of 2267 pores were isolated and analyzed. The following parameters were analyzed: pore area, pore perimeter, circularity, aspect ratio (AR), roundness, solidity, Feret AR, compactness, and fractal dimension. Furthermore, porosity was calculated based on the pore portion in each image. The effective porosity on rock samples was determined using saturation and buoyancy techniques as an accompanying research method. We analyzed distributions of each parameter, their correlation, and most of the parameters are characterized by an asymmetric or asymmetric normal distribution. Parameters that quantify pore irregularities have similar distributions, and their values indicate the high complexity of the pore geometry, which can significantly impact permeability.
Oral squamous cell carcinoma (OSCC) has complex molecular structure stimulated by certain chromatin architectural changes which are not easy to be detected by the naked eye. Prediction of OSCC can be done through various clinical and histological factors. As the OSCC treatment is dependent on histological grading the recent focus is to discover the morphological changes studied by computer-assisted image analysis. One of such more specific diagnostic and prognostic factor is the analysis of the fractal geometry. Fractal Dimension (FD) is the analysis and quantification of the degree of complexity of the fractal objects. FD provides a way to precisely analyse the architecture of natural objects. The purpose of this research is to estimate Minkowski fractal dimension of histopathological images for early detection of oral cancer. The proposed model is developed for analysis medical images to estimate Minkowski fractal dimension using a box-counting algorithm that allows windowing of images for more accurate calculation in the suspected areas of oral cancerous growth.
We present the first validation of the Swisens Poleno, currently the
only operational automatic pollen monitoring system based on digital
holography. The device provides in-flight images of all coarse
aerosols, and here we develop a two-step classification algorithm that
uses these images to identify a range of pollen taxa. Deterministic
criteria based on the shape of the particle are applied to initially
distinguish between intact pollen grains and other coarse particulate
matter. This first level of discrimination identifies pollen with an
accuracy of 96 %. Thereafter, individual pollen taxa are
recognized using supervised learning techniques. The algorithm is
trained using data obtained by inserting known pollen types into the
device, and out of eight pollen taxa six can be identified with an
accuracy of above 90 %. In addition to the ability to
correctly identify aerosols, an automatic pollen monitoring system
needs to be able to correctly determine particle concentrations. To
further verify the device, controlled chamber experiments using
polystyrene latex beads were performed. This provided reference
aerosols with traceable particle size and number concentrations in order to
ensure particle size and sampling volume were correctly characterized.
The discrete element method (DEM) has gained popularity for developing a qualitative
understanding of soil behaviour under a Critical State Soil Mechanics (CSSM)
framework. Most studies with a 3D assembly of particles have used spheres as
representative granular material to reduce computational demands. However, most
granular materials, e.g. sands are not rounded, but possess angularity features.
Therefore, ellipsoid and cluster particles with different degrees of eccentricity were
used in this study to evaluate the effect of the particle shape on the drained and
undrained triaxial loading behaviour after isotropic and K0- consolidation. The particle
numerical properties and grain size distributions were kept the same for all specimens
irrespective of particle shape. The critical state data points for spheres and ellipsoids
plotted on almost the same critical state line (CSL) in e-log(p') space, whereas the
CSLs of clusters plotted above them. Additionally, M lines shifted downward with
increasing sphericity. It was also found that the stress ratio at the triggering of static
liquefaction (η_IS=q/p') in η_IS-ψ space was affected by particle shape and
consolidation path. The dilatancy (d=(dε_v^p)⁄(dε_q^p )) was also affected by particle
shape. It was found that dilatancy parameters for the SANISAND constitutive model are
affected by particle shape, which may contribute to an improved understanding of particle
shape in constitutive modelling.
The influence of particle shape was evaluated under drained and undrained (constant volume) condition using 3D cubical assemblies of spheres, ellipsoids and cluster of spheres (combination of seven spheres with two different degrees of overlap) with same particles size distribution. It was found that the peak deviatoric stress, the minimum dilatancy, corresponding stress ratio (eta_dmin), the bounding surface dilatancy model, the location of critical state line both in the e-log(p') and the q-p' space were influenced by particle shape. Therefore, four corresponding sets of constitutive parameters for four different particle shapes were implemented in a bounding surface model to predict both drained and undrained (constant volume) DEM simulation. Good prediction, irrespective of particles shapes, indicates that the observed DEM behaviour can be adequately captured by the theories of continuum mechanics. Importantly, majority of the constitutive parameters were influenced by particle shape and can be correlated with simple shape descriptor of sphericity.
A series of particles, crushed quartz beach and pyroclastic grains, exhibiting various degrees of shape and roughness have been analysed by both methods. The instability/stability of both techniques is empirically assessed by altering the particle outline by tilting the SEM projection stage. Such variation in stability may prove hazardous to particle specification. It is suggested that Fractal and Fourier techniques are complementary in establishing particle form and roughness.-from Authors
Lack of a standard or generally accepted method of mathematically defining facies from a set of attributes in sedimentary rocks makes a priori selection of a particular method difficult. It is proposed that different combinations of alternative classificatory techniques, leading to competitive classifications, be used to arrive at the first approximations of the facies model. An assessment of these models then can be made, and the “best” one selected further improved on the basis of intuitively appealing objective criteria. These search procedures relieve the investigator of the rigidity of a particular method and also place his empirical facies model on a more rational basis in retrospect, allowing more meaningful interpretations. These principles are illustrated in a facies study of Pennsylvanian carbonate rocks (from southeastern Utah) in which principal component analysis, Euclidean distance function, standardization, and weighted and unweighted pair-group methods, are among the classificatory schemes employed in facies construction.
Grain shape may be described as precisely as needed by Fourier series expansion of the radius about the center of mass utilizing coordinates of peripheral points. Empirical results of grains from typical samples indicate that the series contains, in partitioned form, a large amount of geologically interesting information. Consequently, previous subjective evaluations of the worth of shape variations in the solution of geologic problems are justified. The shape variable is amenable to semiquantitative graphical evaluation or may be used as data for multivariate analytical schemes. Illustrative examples show that the shape variables easily discriminate grain differences arising from geographic, stratigraphic, and process factors.
The fractal dimension of a space-filling non-differentiable curve has recently been defined by Mandelbrot. It is shown that the fractal dimension of an indented or convoluted fine particle profile could provide an index for quantifying the ruggedness of the shape of the fine particle. An analogy between the work of Richardson on the problems of coastline specification in geography and the problems of the scientists seeking to specify fine particle structure is drawn. Three different methodologies for measuring the fractal dimension of a rugged and/or textured fine particle profile are suggested. These are (a) random walk strategies using coupled light pens, (b) line scan interception logic, and (c) digital set theory for binary image transforms. Preliminary data for the measurement procedures on the rugged profiles of fine particles and agglomerates are presented.It is shown that the curve specifying the fractal dimension of the rugged profile of an agglomerate made of clearly discernible subunits which have Euclidian profiles changes its nature when the exploratory step size in the random walk around the profile approaches the order of magnitude of the dimensions of the subunit. This could provide a basis of an algorithm for determining the structure of an agglomerate by an automated iconometric procedure. It is suggested that fractal dimensions along with descriptive parameters from the convex hull of a fine particle profile gives a comprehensive description of the profile. Other potential applications of fractals in fine particle science are briefly discussed.