Fractal Characterization and Petrophysical Analysis of 3D Dynamic Digital Rocks of Sandstone

It is a crucial issue to comprehensively study the relations between microstructure and seepage capacity of porous media. Several physical-based parameters of fractal geometry can analyze the pore structure of rocks, while permeability and electrical conductivity are used to study seepage capacity. In this paper, we first created 3D dynamic digital models of nine different sandstones with varying clay content, cements, and intragranular pores in feldspar. These nine models were divided into three groups. Then, fractal dimension, lacunarity, and succolarity, permeability, and electrical properties of the models were calculated, and their relationships were investigated. We used fractal parameters to interpret the correlation between fluid flow and pore structure as one of the main petrophysical properties of a rock. Results showed that the coefficient of determination for cementation exponent m and fractal dimension is 0.869, while between m and porosity, and succolarity, it is 0.784 and 0.781, respectively. This indicates that the fractal dimension and cementation exponent describe the complexity of pores. The coefficient of determination between permeability and succolarity is 0.975, which is higher than that between permeability and the fractal dimension or porosity. The coefficient of determination between formation factor and succolarity is 0.957, which is higher than that between formation factor and the fractal dimension or porosity. Overall, a stronger relationship between petrophysical parameters, permeability in particular, and succolarity allows this lesser-used fractal parameter to be a good measure for characterizing the connectivity of pore space and pore network.

Painlevé analysis of Fokas–Lenells equation with fractal temporal evolution

This paper presents new optical solitons of a fractal Fokas–Lenells equation that models the dynamics of optical fibers. The Painlevé technique is employed to identify kink soliton solutions. The constraint conditions guarantee the existence of these solitons. The outcomes of this research give new solutions that are not achieved using some already defined algorithms. The derived method is efficient and its applications are promising for other nonlinear problems. The 3D graphical illustrations of obtained results are depicted for various values of the fractal parameter.

Battery discharging model on fractal time sets

This article is devoted to propose and investigate the fractal battery discharging model, which is one of the well-known models with a memory effect. It is presented as to how non-locality affects the behavior of solutions and how the current state of the system is affected by its past. Firstly, we present a local fractal solution. Then we solve the non-local fractal differential equation and examine the memory effect that includes the Mittag-Leffler function with one parameter. For that aim, the local fractal and non-local fractal Laplace transforms are used to achieve fractional solutions. In addition, the simulation analysis is performed by comparing the underlying fractal derivatives to the classical ones in order to understand the significance of the results. The effects of the fractal parameter and the fractional parameter are discussed in the conclusion section.

Fractal model equation for spontaneous imbibition

A new analytic model of fractal imbibition in porous media is derived. The topological Hausdorff dimension is used as a fractal parameter inthe proposed model. The fractal formulation is based on the model introduced by Li and Zhao to predict the production rate by spontaneousimbibition. Cantor Tartans and Menger sponge fractals are used to simulate fractal porous media with different ramifications. Results ofillustrative examples are presented in the form of a set of curves, which reveal the features of enhanced oil recovery of the model underconsideration. The results are compared with the experimental behaviour found on core samples of previous publications.

Estudio fractal de la superficie de la hoja de la especie vegetal Copaifera sp. haciendo uso del Microscopio de Fuerza Atómica-AFM

Estudio fractal de la superficie de la hoja de la especie vegetal Copaifera sp. haciendo uso del Microscopio de Fuerza Atómica-AFM Study fractal leaf surface of the plant species Copaifera sp. using the Microscope Atomic-Force-AFM Mario Omar Calla Salcedo, Robert Ronald Maguiña Zamora, y José Carlos Tavares Carvalho Universidade Federal de Amapá, Rodovia Juscelino Kubitschek de Oliveira, Km 02 - s/n, Bairro Jardim Marco Zero - Macapá -AP, CEP 68.902-280 DOI: https://doi.org/10.33017/RevECIPeru2016.0002/ Resumen Las especies de copaifera sp, que también son denominadas de copaíba y que son ampliamente utilizadas en la medicina popular debido a sus propiedades etnofarmacológicas. Este trabajo fue realizado con el objetivo de padronizar las hojas, mediante el estudio de la textura superficial da la hoja, para eso se necesita la obtención de los parámetros fractales como la dimensión Fractal, Lagunaridad y succolaridad, haciendo uso de los datos que proporciona el Microscopio de Fuerza Atómica, más conocido como AFM (por las siglas en inglés) se trabajó con la área óptima (25x25 mm2), con el procesamiento de datos y aplicando la geometría fractal, se desarrollaron los algoritmos haciendo uso del programa computacional Fortran 77, el estudio fue realizado a partir de la dificultad que se tiene al diferenciar una especie de otra de la Copaifera sp, ya que para hacer tal identificación se necesita la flor y hoja, esto es porque la planta solo florece una vez al año, y por eso se está proponiendo una manera más fácil, y efectiva da tal identificación solo haciendo uso de la hoja de la Copaifera sp, para el cálculo de la dimensión fractal se hizo uso del método de conteo de cajas (Box-Counting), se usó este método por su simplicidad y exactitud, la dimensión fractal va a servir para calcular la rugosidad y porosidad de la superficie de la hoja de la Copaifera sp., donde el valor de la rugosidad obtenido por medio de la dimensión fractal es más exacto que el cálculo de la rugosidad por medio de la geometría Euclidiana. La lagunaridad, es otro parámetro fractal, que sirve para medir el grado de uniformidad de los huecos en la superficie de la hoja de la Copaifera sp, para el cálculo de la lagunaridad se hizo uso de método conteo de Caja Diferencial (Differential Box Counting) que es un método basado en el conteo de cajá (Box-Counting), si la lagunaridad es mucho mayor que 1, existe mayor desorden de los huecos, si la lagunaridad es más próximo a 1, existe menor desorden, ahora si la lagunaridad es igual 1, la superficie es completamente uniforme, seria invariante a la rotación. La succolaridad es el último parámetro fractal que se aplicó al estudio de la superficie de la hoja, que mide la capacidad de un flujo de agua de atravesar toda la superfície en una determinada dirección, a este proceso se le llama percolación, se midió la succolaridad en las cuatro direcciones es decir de arriba hacia abajo, de abajo hacia arriba, de izquierda a la derecha, y por ultimo de derecha a la izquierda. Teniendo calculado los tres parámetros fractales: dimensión fractal, lagunaridad, y succolaridad, se tiene caracterizado completamente la superficie foliar. Descriptores: Copaifera, Dimensión Fractal, Lagunaridad, Succolaridad, Textura. Abstract The species of Copaifera sp. which are also called copal are widely used in folk medicine due to its ethnopharmacological properties. This work was accomplished with the purpose of the possibility of standardization of the leaves, on the study of the surface texture of the leaf, for this you need to obtain the fractal parameters as fractal dimension (roughness, porosity), lacunarity (rotational invariance of the holes ) and succolarity (percolation), making use of the data of the Atomic Force Microscopy (AFM) worked with the optimal area (25x25 mm2), with the data process and applying fractal mathematics, algorithms were developed with the computer program Fortran 77. The study was conducted from difficulty that one has to distinguish one species from another of Copaifera sp., and to make such identification is needed flower and leaf Copaifera sp., this is because the plant blooms only once a year. That's why it is proposing an easier and effective way to such identification, only making use of leaf Copaifera sp. for the calculation of the fractal dimension. It will make use of Box Counting method for its simplicity and exactitude, which will serve to calculate the roughness and porosity of the surface of the sheet Copaifera sp. It is expected that the value of roughness obtained by the Fractal geometry is more accurate, the calculation of roughness with Euclidean mathematics. The Lacunarity is another fractal parameter used to determine readily the uniformity of the holes for the calculation of lacunarity be made using the method of the counting boxes (Differential Box Counting) which is a method based on the counting boxes (Box-Counting), but the lacunarity is much greater than one, there is greater disorder of the holes.The lacunarity is closer to 1, there is less clutter, now the lacunarity is equal to 1, the surface is completely uniform, is down is invariant rotation, it is expected that lacunarity of Copaifera sp leaf is close to an a succolarity is the last fractal parameter that is doing applied to the study of surfaces, which measures the ability of a flow through the entire surface that serves to measure the percolation surface level. It is measured succolarity in the four directions is down from above down, bottom-up, from left to right, and finally from right to left. When it has calculated the three fractal parameters: fractal dimension, lacunarity and succolarity, it is possible to have fully characterized the leaf surface. Keywords: Copaifera. Fractal Dimension. Lacunarity. Succolarity. Texture

Fractal-Based Computation of Heat Source Depths and Temperatures for the Soutpansberg Basin, South Africa

The fractal-based approach was used for computing magnetic depths and temperatures for the Soutpansberg Basin in South Africa. The average depth to the top Zt and basement depth Zo for the Soutpansberg Basin were 4.36 ± 0.28 km, 10.43 ± 0.65 km, respectively. The average temperature at depth Zt was 184.69 ± 7.66 ºC. Magnetic source depths and basal temperatures that were in the Curie point range were determined, to be within 20.35 km to 21.68 km and 549.34 ºC to 585.24 ºC, respectively. Increasing the value of the fractal parameter β from 0 to 4, had an effect of retaining deeper depths and higher temperatures. The fractal parameter values of β > 3 retained Curie point depths and temperatures that indicated basal rock types with an igneous predisposition. The fractal-based approach proved to be an improved technique as compared to the conventional centroid method.

Influence of Moisture Content on Physicomechanical Properties, Starch-Protein Microstructure and Fractal Parameter of Oat Groats

This study was performed to investigate the influence of moisture content on physicomechanical properties, starch-protein microstructure and fractal parameter of oat groats. Selected physical properties were determined as a function of moisture content. The results showed that moisture content had a significant effect on these characteristics. Majority of physical properties increased linearly with moisture content ranged from 11.8 % to 27.0 %, while mechanical properties decreased nonlinearly as third power function in the above range. Moreover, the increasing granule size, less gaps and more contact points can be observed in the microstructure of starch-protein network with high moisture. Meanwhile, high moisture content also resulted in that fractal parameter of oat section increased from 2.6891 to 2.8001 significantly. These moisture-dependent characteristics are useful in further study of oat groats and the heuristic methods used in this study may be extrapolated to other varieties of cereal.

Fractal Features of Proto-Plant’s Origin and their Possible Consequences

The extension of the sectional model of the spruce crown’s dynamics into diapason (0, 1) of the fractal parameter μ has demonstrated the existence of green biomass on branches of three orders in form of photosynthesizing (green) points. We investigated the growth of point sets on an interval as a model of the origin of proto-plants, which are formed due to endosymbiosis of cyanobacteria and protists. The fractal properties of the sets of evenly placed points and group sets were studied using the box-counting method. For the group sets, the character of dependence μ on the growing total number of points changes radically differently depending on whether the number of the points per group or the number of groups was fixed. As the host does not have the initial infrastructure needed for an increase in cyanobacteria per group, the first path is implemented and μ decreases from 1 to 0.25 when groups consist of two points per group. If and when the host develops necessary anatomical features (infrastructure), the second pathway is realized and μ grows to 1. The combined trajectory of μ initially demonstrates a slow growth of the size of the photosynthetic system and then an exponential growth after the development of the host’s infrastructure. Similar fractal peculiarity also characterizes trees and is an innate property of plants. Assumptions on the morphological recapitulation of proto-plant in higher plants’ ontogenesis (embryogenesis and seed germination) and also a possibility to fix the number of cyanobacteria per group are discussed.