Fractal-based Model Research Articles

A New R35 and Fractal Joint Rock Typing Method Using MICP Analysis: A Case Study in Middle East Iraq

Reservoir characterization in carbonate formations plays a crucial role in understanding the complex pore structures and permeability properties. While absolute pore-throat radius (APTR) and R35 have been widely accepted in pore-scale rock typing, they fall short of providing detailed pore-throat distribution (PTD) information. To address this limitation and enhance PTD indication during rock classification, we introduce a novel approach—the R35 and fractal joint rock typing method—incorporating a new parameter, Dn (fractal dimension). This method is developed based on the analysis of mercury injection capillary pressure (MICP) data obtained from 20 carbonate samples in the Middle East, specifically Iraq. We delve into the discussion of APTR and R35 methods, employing both PTD and thin-section images to gain comprehensive insights into rock typing. Our approach incorporates a whole curve fractal-based model to determine the fractal dimension, Dn. The analysis reveals that Dn in each R35 rock type exhibits a decreasing trend with higher Hg intrusion peaks and wider pore-throat radius, providing valuable information on the distribution of pores within the rock samples. Furthermore, we extend our analysis to include the surface fractal dimension, Ds, obtained through two-dimensional (2D) fractal analysis on typical pores in binary thin-section images. The consistent decreasing trend of Dn aligns with the findings from Ds analysis, underscoring the effectiveness of parameter Dn in capturing the intricate pore structures within carbonate formations. Our results suggest that parameter Dn has the potential to serve as a standalone criterion in rock typing, eliminating the need for pretyping by R35 or APTR. The versatility of Dn as an indicator of pore distribution and complexity underscores its significance in advancing our understanding of carbonate reservoirs. As a recommendation, further exploration through additional fractal-based analyses is encouraged to solidify the role of parameter Dn in becoming a pivotal factor in the evolving field of rock typing.

Fractal-Based Analysis of Arteriovenous Malformations (AVMs).

Arteriovenous malformations (AVMs) are cerebrovascular lesions consisting of a pathologic tangle of the vessels characterized by a core termed the nidus, which is the "nest" where the fistulous connections occur. AVMs can cause headache, stroke, and/or seizures. Their treatment can be challenging requiring surgery, endovascular embolization, and/or radiosurgery as well. AVMs' morphology varies greatly among patients, and there is still a lack of standardization of angioarchitectural parameters, which can be used as morphometric parameters as well as potential clinical biomarkers (e.g., related to prognosis).In search of new diagnostic and prognostic neuroimaging biomarkers of AVMs, computational fractal-based models have been proposed for describing and quantifying the angioarchitecture of the nidus. In fact, the fractal dimension (FD) can be used to quantify AVMs' branching pattern. Higher FD values are related to AVMs characterized by an increased number and tortuosity of the intranidal vessels or to an increasing angioarchitectural complexity as a whole. Moreover, FD has been investigated in relation to the outcome after Gamma Knife radiosurgery, and an inverse relationship between FD and AVM obliteration was found.Taken altogether, FD is able to quantify in a single and objective value what neuroradiologists describe in qualitative and/or semiquantitative way, thus confirming FD as a reliable morphometric neuroimaging biomarker of AVMs and as a potential surrogate imaging biomarker. Moreover, computational fractal-based techniques are under investigation for the automatic segmentation and extraction of the edges of the nidus in neuroimaging, which can be relevant for surgery and/or radiosurgery planning.

Modification of cellulosic adsorbent via iron-based metal phenolic networks coating for efficient removal of chromium ion

Surface modification of durian rind cellulose (DCell) was done by utilizing the strong coordination effect of polyphenol-based metal phenolic networks (MPNs). MPNs from Fe(III)-tannic acid (FTN) and Fe(III)-gallic acid (FGN) were coated on DCell via a self-assembly reaction at pH 8, resulting in adsorbent composites of FTN@DCell and FGN@DCell for removal of Cr(VI). Batch adsorption experiments revealed that FTN coating resulted in an adsorbent composite with higher adsorption capacity than FGN coating, owing to the greater number of additional adsorption sites from phenolic hydroxyl groups of tannic acid. FTN@DCell exhibits an equilibrium adsorption capacity at 30°C of 110.9 mg/g for Cr(VI), significantly higher than FGN@DCell (73.63 mg/g); the adsorption capacity was increased at higher temperature (i.e., 155.8 and 116.8 mg/g at 50°C for FTN@DCell and FGN@DCell, respectively). Effects of pH, adsorbent dose, initial concentration, and coexisting ions on Cr(VI) removal were investigated. The kinetics fractal-based model Brouers-Sotolongo indicates the 1st and 2nd order reaction for Cr(VI) adsorption on FTN@DCell and FGN@DCell, respectively. The isotherm data can be described with a fractal-based model, which implies the heterogeneous nature of the adsorbent surface sites. The Cr(VI) adsorption via surface complexation with phenolic hydroxyl groups was confirmed by evaluating the functional groups shifting. FGN@DCell and FTN@DCell were found to have good reusability, maintaining over 50 % of their adsorption efficiency after four adsorption-desorption cycles. Environmental assessment with Arabidopsis thaliana demonstrated their potential in eliminating the Cr(VI) phytotoxic effect. Thus, this study has shown the efficient and economical conversion of durian waste into environmentally benign adsorbent for heavy metal treatment.

Accelerating Stochastic Lightning Attachment Simulations for the Estimation of Lightning Incidence to Overhead Lines

Lightning attachment can be modeled through a stochastic approach adopting a detailed representation of the lightning phenomenon. A fractal-based modeling technique can be used for this purpose, considering lightning discharge branched and tortuous behavior, as well as physical properties associated with downward and upward leaders’ inception and propagation. However, fractal-based simulations require substantial computational resources, especially for the accurate calculation of the electric field at all points of the discretized simulation domain at each simulation step. Thus, the considerable computational cost inhibits the extensive application of stochastic simulations for estimating lightning incidence to common structures and power systems. This work investigates optimization techniques for fractal-based simulations regarding total simulation time; these are applicable to both high-performance computing and personal computers. The proposed techniques consist of a C-MATLAB integration methodology, as well as a multi-color ordering algorithm enabling parallel execution using CPU and GPU programming. Applications associated with lightning incidence to overhead transmission lines are presented. The total simulation time is substantially reduced with respect to the original code. A reduction of up to 98% is achieved, enhancing the applicability of stochastic modeling to lightning incidence estimation problems.

Assessing models of viscous flow relative to fibres for obtention of the porosity of micro- and nano- fibrous mats

Theoretical and semi-empirical models of viscous flow relative to fibres were used to obtain the porosity of micro-fibrous and nano-fibrous mats applied in air filtration. Overall efficiencies of collection of sodium chloride particles (5.94–224.7 nm) were calculated for the mats considering the porosities obtained using different models. The lowest deviations for the collection efficiency were achieved using the model based on the fractal theory and the Davies’ model (-5.4% and 4.2% for the micro-fibrous mat; 2.0% and 2.9% for nano-fibrous mat, respectively). Analysis of the effect of the porosity on the structural properties of the mats was performed using the fractal-based model and Davies’ model, showing their suitability.

Hemodynamics of the right ventricle and the pulmonary circulation

Most cardiac diseases affect the left ventricle (LV), with its treatment being the focus of the majority of cardiovascular medical therapies and mechanical support. Many diseases also affect the right ventricle (RV) and pulmonary circulation, either directly or from secondary effects on the LV and systemic circulation. The development of therapies to treat RV failure is critical for the management of many patients with advanced heart and lung disease, but models to develop mechanical therapies to target the RV are currently lacking. While the RV and pulmonary vasculature have many similarities to the LV and systemic vasculature, there are notable differences between them at a morphological and functional level. This results in distinct patterns of coupling between the RV and pulmonary circulation. Models such as the Windkessel use lumped parameters that characterize the mechanical properties of the entire RV and pulmonary circulation. Contemporary computational approaches that have been used to model the pulmonary vasculature rely on patient-derived anatomy for the proximal pulmonary arteries and other approaches, such as fractal-based models for the distal vasculature. At this time, these models are still being optimized and significantly more work will be required to fully implement them and test their utility across multiple patient-derived anatomies. The development of these models for the RV and pulmonary circulation will be critical to address the unmet need of mechanical therapies for disease of the RV and pulmonary circulation.

A mathematical model for estimating effective stimulated reservoir volume

This study presents a model application for the evaluation of Effective Stimulated Reservoir Volume (ESRV) in shale gas reservoirs. This current model is faster, cheaper, and readily available for estimating ESRV compared to previously published models. Key controlling parameters for efficient ESRV modeling, including geomechanical parameters and time, are considered for the model development. The model was validated for both single and multi-stage fractured reservoirs. For the single fractured reservoir, an ESRV of 3.07 × 106 ft3 was estimated against 3.99 × 106 ft3 of ESRV-FEM field data. Whereas, 7.00 × 109 ft3 ESRV was estimated from the multi-stage fractured reservoir against 7.90 × 109 ft3 of fractal-based model results. Stress dependence, time dependence, and permeability dependence of shale gas reservoirs are found to be essential parameters for the successful calculation of ESRV in reservoirs. An ESRV determined using this method can obtain the estimated ultimate recovery, propped volume, optimal fracture length, and spacing in fractured shale gas reservoirs.

Fractal-based model for maximum penetration distance of grout slurry flowing through soils with different dry densities

Existing penetration models for grouting slurry seldomly consider the relationship between dry density and complexity of soil. A novel penetration model, in which the complexity of the flow path in soil was described by fractal theory, has been proposed for quantifying the maximum penetration distance of grout slurry flowing through soils with different dry densities. The relationship among dry density, fractal dimension and fractal tortuosity of flow path was identified by SEM analysis, showing that fractal tortuosity increases but fractal dimension is almost unchanged with increase of dry density. The results calculated by the penetration model were validated against experimental results for different soils under various pump pressures and volume flow rates. The analysis results show that the grout slurry tends to penetrate to a long distance under high pump pressure, low flow rate, low consistency factor and low fluidity index. High dry density and small average particle size of soil reduce the penetration distance. The proposed model is helpful to achieve a better design in soil ground stabilization via grouting.

A new multifractal-based grain size distribution model

Previous works related to the application of the multifractal theory for analyzing the grain size distribution (GSD), showed the potential of this approach to deal with this complex issue. However, absence of the practical application of this kind of statistical analysis raised some doubts among the soil scientists. Compared to the experimental dry sieving method, which is based on mass representations of different grain sizes, the approach presented in this work relies on the analysis of grain densities (density indicators) scanned by means of X-ray CT (Computed Tomography). By reducing the resolution of the scanned soil image(s), the cumulative representation of solid particles equal to or larger than the actual discretization element can be determined, and described analytically by means of the universal multifractals (UM).For validation of the new UM approach, the X-ray CT results of three different soils were used: the volcanic substrate covering Green Wave (a green roof of Champs-sur-Marne in France), and two horizons of the soil collected from the low land mountain area of Sierra de Guadarrama in Spain. Comparison between the proposed UM model and the experimental data of these three materials confirms that the GSD can be reasonably well predicted from the scanned images of soils covering wide range of grain sizes. The UM model, unlike the fractal-based models, accounts for fractal dimension that depends on grain size, and hence, based on the preliminary results presented in this work, it could be rather useful in case of multi-modal soils whose GSD curves are described with multiple fractal dimensions.

Machine Learning-Derived Fractal Features of Shape and Texture of the Left Atrium and Pulmonary Veins From Cardiac Computed Tomography Scans Are Associated With Risk of Recurrence of Atrial Fibrillation Postablation.

[Figure: see text].

Fractal-Based Modeling and Spatial Analysis of Urban Form and Growth: A Case Study of Shenzhen in China

Fractal dimension curves of urban growth can be modeled with sigmoid functions, including logistic function and quadratic logistic function. Different types of logistic functions indicate different spatial dynamics. The fractal dimension curves of urban growth in Western countries follow the common logistic function, while the fractal dimension growth curves of cities in northern China follow the quadratic logistic function. Now, we want to investigate whether other Chinese cities, especially cities in South China, follow the same rules of urban evolution and attempt to analyze the reasons. This paper is devoted to exploring the fractals and fractal dimension properties of the city of Shenzhen in southern China. The urban region is divided into four subareas using ArcGIS technology, the box-counting method is adopted to extract spatial datasets, and the least squares regression method is employed to estimate fractal parameters. The results show that (1) the urban form of Shenzhen city has a clear fractal structure, but fractal dimension values of different subareas are different; (2) the fractal dimension growth curves of all the four study areas can only be modeled by the common logistic function, and the goodness of fit increases over time; (3) the peak of urban growth in Shenzhen had passed before 1986 and the fractal dimension growth is approaching its maximum capacity. It can be concluded that the urban form of Shenzhen bears characteristics of multifractals and the fractal structure has been becoming better, gradually, through self-organization, but its land resources are reaching the limits of growth. The fractal dimension curves of Shenzhen’s urban growth are similar to those of European and American cities but differ from those of cities in northern China. This suggests that there are subtle different dynamic mechanisms of city development between northern and southern China.

Discovery of the Heart in Mathematics: Modeling the Chaotic Behaviors of Quantized Periods in the Mandelbrot Set

The phenomenon of small changes leading to drastic effects is fundamental to Chaos Theory. Understanding the chaos in our world could provide more control over the systems that govern the universe. Therefore, this study aimed to predict and explain chaotic behavior in the Mandelbrot Set, one of the world’s most popular models of fractals and exhibitors of Chaos Theory. We hypothesized that repeatedly iterating the Mandelbrot Set’s characteristic function would give rise to a more intricate layout of the fractal and elliptical models that predict and highlight “hotspots” of chaos through their overlaps. While a novel method of discovering miniature versions of the Mandelbrot fractal was discovered and a statistically significant transformation function was developed, overlaps of the elliptical models were not supported to exhibit higher levels of chaos. Many biological and natural phenomena such as the heartbeat, lung vessels, neurons, weather, the stock market, and more, are both chaotic in nature and can be described using fractal-based models. The positive and negative results from this study may provide a new perspective on fractals and their chaotic nature, helping to solve problems involving chaotic phenomena.

A fractal-based model for soil water characteristic curve over entire range of water content

Soil water characteristic curve (SWCC) has been an important role in hydraulic engineering, civil engineer and petroleum engineering, etc. Most of SWCC models neglected the film flow in the dry state, so that they cannot accurately describe the SWCC over entire range of water content. In this work, an alternative fractal model is proposed to predict the SWCC over entire range of water content by combining Campbell and Shiozawa model and Tao model. The proposed model can well predict twelve sets of experimental data, and its parameters, including the fractal dimension, the saturated volumetric water content, the matric suction at oven-dry condition, and the air-entry value, accord with theoretical value. The results show that there is a strong linear relationship between volumetric water content and matrix suction in log-log scale for different fractal pore-size distribution of soils. In addition, good agreement is obtained between the experimental data and the model predictions in all of the cases. Cited as : Jin, T., Cai, X., Chen, Y., Jiang, S., Wei, W. A fractal-based model for soil water characteristic curve over entire range of water content. Capillarity, 2019, 2(4): 66-75, doi: 10.26804/capi.2019.04.02

BWM-ARAS: A new hybrid MCDM method for Cu prospectivity mapping in the Abhar area, NW Iran

Mineral prospectivity mapping (MPM) is reckoned as a lucrative tool regarding exploration programs that include lengthy and multi-branched decision pathways with variously conceivable outcomes. This paper outlines the prosperous application of a new hybrid multi-criteria decision-making (MCDM) method called BWM-ARAS for integrating multisource geological dataset to delineate highly Cu prospectivity areas in the Abhar area, NW Iran. In fact, the BWM-ARAS method is a compound of two distinct procedures, in which the BWM method is exploited for weighting to criteria and sub-criteria while ARAS method is responsible for ranking the varieties of alternatives. Herein, BWM-ARAS method is implemented on a set of multisource geoscience data as targeting criteria (derived from geological, geochemical, and remote sensing dataset) pertaining to the mineral deposit of type sought that are termed evidential maps. For fulfilling the integration process, the spatial values of evidential maps are initially transformed to a new space ranging [0, 1] by the logistic sigmoid function, and then discretized through C-A fractal-based model to ascertain thresholds corresponding favorable populations. In addition, a fuzzy MPM, a straightforward knowledge-based method, is accomplished to peruse the adequacy of the former method. The predication-area (P-A)plot and normalized density (Nd) as authoritative methods were conducted to evaluate the efficiency of abovementioned prospectivity models. Accordingly, the obtained results corroborated the higher authenticity of the BWM-ARAS model (Nd=7.33) compared with the fuzzy model (Nd=3.54) in delimiting the sought areas.

A fractal-based stepped downward leader model including branched channel charge distribution and branch fading

In this paper, a fractal-based model is proposed for lightning downward leader propagation through the air. The model simulates the zigzag movement and the branching nature of lightning close to reality in comparison to previous existing models. The probabilistic behavior of lightning is well modeled using the fractal approach. A new model is also introduced for charge distribution along different branches of the lightning downward leader. The model calculates the charge density of each branch based on the concept of electric field through the branches. Moreover, the probability of branch fading is considered in the simulations. The proposed model is evaluated for striking distance calculation of a transmission line structure and the results are compared to those of Leader Progression Model (LPM), and different models of Electrogeometric Model (EGM). Moreover, Shielding Failure Rate (SFR) of some structures is obtained by the proposed method and the results are compared to those of field observations. In addition, the electrical field on the ground near the striking point of lightning is calculated and compared to the measurements. The branch fading and the fractal dimension of simulated lightning are also discussed. The simulations show how the proposed model simulates the lightning in an acceptable manner.

Insights into photoacoustic speckle and applications in tumor characterization

In ultrasound imaging, fully-developed speckle arises from the spatiotemporal superposition of pressure waves backscattered by randomly distributed scatterers. Speckle appearance is affected by the imaging system characteristics (lateral and axial resolution) and the random-like nature of the underlying tissue structure. In this work, we examine speckle formation in acoustic-resolution photoacoustic (PA) imaging using simulations and experiments. Numerical and physical phantoms were constructed to demonstrate that PA speckle carries information related to unresolved absorber structure in a manner similar to ultrasound speckle and unresolved scattering structures. A fractal-based model of the tumor vasculature was used to study PA speckle from unresolved cylindrical vessels. We show that speckle characteristics and the frequency content of PA signals can be used to monitor changes in average vessel size, linked to tumor growth. Experimental validation on murine tumors demonstrates that PA speckle can be utilized to characterize the unresolved vasculature in acoustic-resolution photoacoustic imaging.

Fractal‐based lightning model for shielding failure rate calculation of transmission lines

In this study, performance of lightning shielding system of transmission lines is investigated by means of a new fractal-based model of simulated lightning. A new lightning downward stepped leader based on fractal approach is introduced in this study. The zigzag movement of lightning downward leader as well as branching nature are simulated using the proposed model. Moreover, a new charge distribution is proposed for branched lightning channel. In addition, the branch fading during lightning downward movement is also simulated in order to simulate the lightning closer to reality. The proposed method is applied for calculating the striking distance for a transmission line and the results are compared with those of leader progression model and different models of electrogeometric model. Furthermore, the shielding failure rate of some practical structures obtained by the proposed method is compared with the field observations. The results are discussed and superiority of the proposed simulated lightning model is concluded.

A Simple Fractal-Based Model for Soil-Water Characteristic Curves Incorporating Effects of Initial Void Ratios

In this paper, a simple and efficient fractal-based approach is presented for capturing the effects of initial void ratio on the soil-water characteristic curve (SWCC) in a deformable unsaturated soil. In terms of testing results, the SWCCs (expressed by gravimetric water content) of the unsaturated soils at different initial void ratios were found to be mainly controlled by the air-entry value (Ψa), while the fractal dimension (D) could be assumed to be constant. As a result, in contrast to the complexity of existing models, a simple and efficient model with only two parameters (i.e., D and Ψa) was established for predicting the SWCC considering the effects of initial void ratio. The procedure for determining the model parameters with clear physical meaning were then elaborated. The applicability and accuracy of the proposed model were well demonstrated by comparing its predictions with four sets of independent experimental data from the tests conducted in current work, as well as the literature on a wide range of soils, including Wuhan Clay, Hefei and Guangxi expansive soil, Saskatchewan silt, and loess. Good agreements were obtained between the experimental data and the model predictions in all of the cases considered.

Estimation of soil water retention curve using fractal dimension

The soil water retention curve (SWRC) is a fundamental hydraulic property majorly used to study flow transport in soils and calculate plant-available water. Since, direct measurement of SWRC is time-consuming and expensive, different models have been developed to estimate SWRC. In this study, a fractal-based model was developed to predict SWRC. A wide range of soil textures (130 soil samples) was used to determine the fractal dimension of SWRC (DSWRC). Moreover, the SWRC pedotransfer functions were established based on easily available soil properties such as particle size distribution and bulk density by applying multiple linear regression analysis. The measured DSWRC for 110 soil samples was considered for function parameterization and the remaining was used for model validation. The results illustrated that the DSWRC linearly correlates with clay and silt contents and soil bulk density (r2 = 0.909). The SWRC can, therefore, be easily and concisely estimated by the proposed fractal-based functions. Key words: Fractal model; Pedotransfer functions; Regression analysis; Soil water retention curve

Estimation of soil water retention curve using fractal dimension

The soil water retention curve (SWRC) is a fundamental hydraulic property majorly used to study flow transport in soils and calculate plant-available water. Since, direct measurement of SWRC is time-consuming and expensive, different models have been developed to estimate SWRC. In this study, a fractal-based model was developed to predict SWRC. A wide range of soil textures (130 soil samples) was used to determine the fractal dimension of SWRC (D SWRC ). Moreover, the SWRC pedotransfer functions were established based on easily available soil properties such as particle size distribution and bulk density by applying multiple linear regression analysis. The measured D SWRC for 110 soil samples was considered for function parameterization and the remaining was used for model validation. The results illustrated that the D SWRC linearly correlates with clay and silt contents and soil bulk density (r 2 = 0.909). The SWRC can, therefore, be easily and concisely estimated by the proposed fractal-based functions. Keywords: Fractal model; Pedotransfer functions; Regression analysis; Soil water retention curve