Fractal Power Law Research Articles

EVALUATION OF SHALE GAS EXPLORATION BY MICROSTRUCTURE BEHAVIOR AND SHALE PERMEABILITY BASED ON FRACTAL THEORY AND UNDER MULTI-FIELD EFFECTS

The key to shale gas exploration is the characterization of gas migration under the combination of multiple factors. To address the long-standing energy challenge of rapidly and accurately quantifying the behavior of natural fractures and matrix pores in shale at an engineering scale in interaction with gas migration. This study proposes an interdisciplinary model for shale gas extraction by adopting fractal theory. Five innovative microstructural parameters are developed to characterize the size and scale of natural matrix pores/fractures in shale, so as to investigate the contributions of fractal distributed pores and fractal power-law distributed fractures to shale gas extraction. The present results of the proposed model are consistent with the exploitation state of the UK Bowland Shale #114 well. The evolution of the shale microstructure will lead to changes in gas migration behavior throughout the reservoir and in turn affect shale stress, temperature and gas adsorption–desorption effect, and finally have a significant impact on permeability. It is found that in the present analysis of the entire Bowland Shale, the overall permeability changes by 10.8% with the evolution of fractal distributed pores and by 41.3% with the evolution of fractal power-law fractures. This work provides a new approach for rapidly exploring the behavior of shale fractures and matrix pores at engineering scales. This work also offers a new and practical baseline for shale gas extraction assessment and fossil energy management.

A fractal power-law approach for investigating the interactions between reservoir microstructure and gas migration

As one of the most significant clean energy sources, the migration characteristics and extraction processes of coalbed methane (CBM) have been extensively studied. The structural distribution and evolutionary characteristics of reservoir fractures, as the main conduit for gas migration, significantly affect the permeability and gas production. However, few models have been able to quantitatively and accurately explore reservoir micro–macro interactions under coupled thermal-fluid–solid effects. This work develops a new highly coupled model based on the widely adopted power-law function to quantify reservoir thermal conduction effect, gas pressure evolution, reservoir deformation, in situ stress, the adsorption–desorption effect, and reservoir microstructure evolution. Three parameters are adopted to quantitatively characterize the reservoir structure: (1) fracture power index αf (to characterize the fracture density), (2) fracture length ratio rf (to characterize the fracture size), and (3) the maximum fracture length l. The results demonstrate that the fractal network is a special kind of network in the power-law length distribution. The proposed power-law seepage model is able to accurately characterize the evolution of reservoir microstructure and the impact of microevolution on extraction under multi-field coupling effects, compared to the traditional power-law model. The proposed model can provide a good theoretical and practical support for the study of CBM migration and extraction.

AN INSIGHT ON THE FRACTAL POWER LAW FLOW: FROM A HAUSDORFF VECTOR CALCULUS PERSPECTIVE

In the paper, we suggest the Hausdorff vector calculus based on the Chen Hausdorff calculus for the first time. The Gauss–Ostrogradsky-like, Stokes-like, and Green-like theorems, and Green-like identities are obtained in the framework of the Hausdorff vector calculus. The formula is proposed as a mathematical tool to describe the real-world problems for the fractal power-law flow equations with the anomalous diffusion equation. A conjecture for the fractal power-law flow equations analogous to the Smale’s 15th problem and one of Millennium Prize Problems for the Navier–Stokes equations is also addressed.

Effects of Weakly Nonlinear Waves on Acoustic Scattering from the Ocean Surface

The elevation of ocean waves is always modeled in linear theory as a superposition of the sinusoidal components with crests and troughs of identical heights. However, under some circumstances, the wave amplitude is outside the linear range and presents as a weakly nonlinear asymmetrical waveform with sharper crests and shallower troughs. We studied the impact of the weakly nonlinear effect of ocean waves in deep and intermediate waters on acoustic scattering from the surface of the ocean using two rough surface models with fractal geometry and power law spectral behavior in the equilibrium range. The classic Weierstrass–Mandelbrot function was used to model the linear waves and a new fractal function, the fractional Weierstrass function developed in studies of electromagnetism, was used to model the weakly nonlinear waves. We evaluated these two models using the Pierson–Moskowitz spectrum and the incident wavelength. The bistatic scattering strength was obtained via a numerical method based on the “exact” solution of the integral equation. The weakly nonlinear phenomenon led to a very small reduction in the narrow area around the specular reflection angle and a small increase in the remaining wide area, including the backpropagation area with a scattering angle [Formula: see text]. The differences in backscattering strength between the two models were similar to the bistatic scattering strength in the backpropagation area and did not depend on the incident grazing angle.

A support vector regression approach to predict geotechnical properties of soils from electrical spectra based on Jonscher parameterization

Electrical-conductivity spectra of soils contain valuable information about their texture, structure, and composition that can be linked to their geotechnical properties. Concurrent measurements of electrical spectra in the frequency range of 0.01 Hz to 10 kHz and geotechnical properties, that is, the dry unit weight [Formula: see text], modulus of elasticity [Formula: see text], and the hydraulic conductivity [Formula: see text], are performed on natural soil samples in a laboratory environment. The electrical spectra are modeled with the Jonscher fractal power law model characterized by three parameters: DC conductivity [Formula: see text], transition frequency [Formula: see text], and an exponent [Formula: see text]. We explore a machine-learning technique, the support vector regression (SVR) methodology, to model and predict the geotechnical properties from the Jonscher parameters, and we compare our results with the predictions of multiple linear regression (MLR). For model training and testing, the Jonscher parameters are used as the input, and a geotechnical parameter is used as the output. Model comparisons indicate that the developed SVR models predict [Formula: see text] with an [Formula: see text], predict [Formula: see text] with [Formula: see text], and predict [Formula: see text] with [Formula: see text]. In comparison, MLR models predict [Formula: see text] with an [Formula: see text], [Formula: see text] with [Formula: see text], and [Formula: see text] with [Formula: see text]. The results illustrate that the SVR models are more accurate, reliable, and achieve better performance for predicting the geotechnical properties from the electrical parameters in comparison to the predictions of the MLR models. Our study offers an opportunity in our quest in using noninvasive electrical geophysical methods to obtain geotechnical properties of soils, and it has broad implications in engineering geophysics.

Assessing Relations Between Electrical and Geotechnical Properties of Sand-Clay Mixtures Using Jonscher Fractal Power Law Model

The geotechnical properties of unconsolidated geo-materials such as soils are influenced by modifications of their micro-structure, texture, mineralogy, water content and imposed effective stress levels. Fundamental relations between the characteristic electrical parameters describing the electrical responses soils based on a fractal power law model with scaling properties, and parameters influencing their geotechnical behavior are investigated. Low frequency electrical conductivity laboratory measurements were performed on sand and clay mixtures subjected to varying effective stress levels with concurrent measurements of their geotechnical properties. The conductivity spectra of the mixtures were described using a Jonscher fractal power law model characterized with three characteristic parameters, the dc conductivity ( σdc), the characteristic frequency ( fc) and an exponent ( n). Changes in effective stress, water content, clay content, and other engineering properties of the mixture such as dry density, porosity, pore size and intergranular void ratio are discussed with respect to changes in the electrical parameters. The dc conductivity and characteristic frequency decrease with an increase in effective stress levels. The exponent, however, has the opposite behavior and increases with an increase in effective stress. As the water content increases, σdc and fc increase while n decreases for all mixtures. With increasing stress levels, the average pore size of the mixtures decreases which results in a decrease in σdc and fc but an increase in the values of the exponent. An increase in dry density of the mixtures leads to a decrease in σdc and fc whilst n increases. Both σdc and fc increase with increase in the intergranular void ratio of the mixture whilst the exponent values decrease with an increase in the intergranular void ratio. This study serves as a contribution to our quest in utilizing electrical geophysical methods, to assess and monitor non-invasively, the geotechnical properties of the subsurface in a less expensive and faster manner.

A diffusivity model for predicting VOC diffusion in porous building materials based on fractal theory

Most building materials are porous media, and the internal diffusion coefficients of such materials have an important influences on the emission characteristics of volatile organic compounds (VOCs). The pore structure of porous building materials has a significant impact on the diffusion coefficient. However, the complex structural characteristics bring great difficulties to the model development. The existing prediction models of the diffusion coefficient are flawed and need to be improved. Using scanning electron microscope (SEM) observations and mercury intrusion porosimetry (MIP) tests of typical porous building materials, this study developed a new diffusivity model: the multistage series-connection fractal capillary-bundle (MSFC) model. The model considers the variable-diameter capillaries formed by macropores connected in series as the main mass transfer paths, and the diameter distribution of the capillary bundles obeys a fractal power law in the cross section. In addition, the tortuosity of the macrocapillary segments with different diameters is obtained by the fractal theory. Mesopores serve as the connections between the macrocapillary segments rather than as the main mass transfer paths. The theoretical results obtained using the MSFC model yielded a highly accurate prediction of the diffusion coefficients and were in a good agreement with the VOC concentration measurements in the environmental test chamber.

Identifying Long-Memory Trends in Pre-Seismic MHz Disturbances through Support Vector Machines

In this paper, a novel algorithm is introduced for the analysis of long-memory patterns hidden in electromagnetic (EM) readings prior to earthquakes. The algorithm builds upon previous work on long-memory detection in EM measurements by fusing Support Vector Machine (SVM) classifiers with well-deployed power law fit tests and Rescaled-Range (R/S) time-series variability methods. To apply the algorithm, fractal power law in the wavelet domain is assessed so as to identify fractional Brownian motion (fBm) segments of continuously monitored pre-earthquake EM activity. The selected segments are then further processed through R/S Analysis in order to further refine the detection of prominent fBm behaviour. The combined output of the two methods is used to train a SVM classifier which is subsequently employed to verify similar fBm states in existing EM data and to allow for rapid fBm detection in large data sequences of unprocessed or newly incoming EM readings. The SVM classifier is added in a modular fashion, on top of pre-earthquake monitoring algorithms, and can be trained with a small fraction of a huge available dataset of EM readings. Three earthquake events in Greece, corresponding to different time occurrences and geographic locations, were investigated. For each of the three earthquakes, data collected by a nearby EM measurement station one month prior to the peak event were analysed by the proposed method. The results yielded an overall accuracy rate of at least 90% for the detection of specific, prominent fBm segments despite the fact that the fBm profile in the three investigated earthquake sequences was very different.

Abstract 17013: Post-traumatic Stress Disorder is Associated with Adverse Non-Linear Heart Rate Variability in Veteran Twins

BACKGROUND: Veterans with post-traumatic stress disorder (PTSD) have higher risk of coronary heart disease (CHD) incidence and mortality. Psychological conditions may have an adverse effect on the autonomic nervous system, and may in part have genetic origins. Novel non-linear measures of heart rate variability (HRV), such as fractal dimension (FD) and power law slope have been associated with mortality and may signify autonomic dysfunction. We hypothesized that PTSD is associated with reduced FD and increased power law slope, suggesting higher mortality risk. METHODS: FD and power law slope were measured from 24-hour Holter recordings in 416 predominantly healthy middle-aged male twins from the Vietnam Era Twin Registry. Current (symptoms within the past month) and remitted (past history of) PTSD were measured using the Structured Clinical Interview for Psychiatry Disorders. Mixed-effect regression models were used to adjust for twin clustering and account for genetic/familial influences. RESULTS: The mean age was 56 ± 3 years, 35 had current PTSD, and 46 had remitted PTSD. In models that analyzed twins as individuals, current PTSD was significantly associated with 22% reduced FD and 18% increased power law slope. This association persisted for power law slope after adjustment for traditional risk factors, lifestyle factors, depression, and anti-depressant medication use (table). After controlling for genetic/familial factors by comparing twin brothers discordant for current PTSD within pairs (20 pairs), both measures remained significantly associated with current PTSD. Remitted PTSD did not influence HRV measures. No interaction by zygosity was found. CONCLUSIONS: Vietnam era veterans with PTSD have reduced FD and increased power law slope persisting after multivariable adjustment. This suggests that autonomic mechanisms, independent of genetic and familial effects, may contribute to increased CHD and mortality in veterans with PTSD.

A New View of Financial Risk: What Officers Should Know to Minimize Risk?

This is an introductory highlight of the fractal theory and power law theory of investment analysis and with the findings of a research on using the above theory to predict crash and trough in Hang Seng Index.

Crash and Trough Prediction: Fractal Theory Modeling for Hang Seng Index

This research is about the application of Fractal Theory and Power Law Model to predict the possible crash of the stock market in a bubble. The stock market used in this research is the Hong Kong stock market, i.e. the Hang Seng Index. The research found strong power of the Fractal Theory and Power Law Model in predicting the possible crash of the market.This could be a good starting paper of employing Fractal Theory and Power Law Model to other investment vehicle such as commodity, gold, energy, and currency in Hong Kong.

Magnetic hydrophobic nanocomposites: Silica aerogel/maghemite

Magnetic hydrophobic aerogels (MHA) in the form of nanocomposites of silica and maghemite (γ-Fe2O3) were prepared by one step sol–gel procedure followed by supercritical solvent extraction. Silica alcogels were obtained from TEOS, MTMS, methanol and H2O, and Fe(III) nitrate as magnetic precursor. The hydrophobic property was achieved using the methytrimethoxysilane (MTMS) as co-precursor for surface modification. The so produced nanocomposite aerogels are monolithic, hydrophobic and magnetic. The interconnected porous structure hosts ∼6nm size γ-Fe2O3 particles, has a mean pore diameter of 5nm, and a specific surface area (SSA) of 698m²/g. Medium range structure of MHA is determined by SAXS, which displays the typical fractal power law behavior with primary particle radius of ∼1nm. Magnetic properties of the nanoparticle ensembles hosted in them are studied by means of dc-magnetometry.

Multiscale characteristic lengths of abraded surfaces: Three stages of the grit-size effect

The abrasion mechanisms in polishing titanium-alloy samples with different grades of silicon carbide-coated abrasives were characterized using a novel multiscale analysis of the extreme amplitudes of the peaks and valleys (EAPV) of surface roughness. Two stages of roughness were found: a fractal stage ( l∈10–160 μm), where EAPV values versus the observation length l were linked to the fractal dimension D (EAPV∝ l 2− D ), and a stochastic stage ( l>160 μm), where EAPV was modeled by the extreme-value theory, allowing the prediction of EAPV values versus observation length. Three regimes of abrasion were found: for grit particles of diameter d>100 μm, EAPV values did not depend on the observation scale and were consistent with Archard’s model. For particle sizes 10 μm< d<100 μm, the EAPV diminished with d regardless of scale, representing the “grit-size effect”. For particles of diameter d<10 μm, the EAPV dramatically decreased at all scales and was independent of d because of adhesive wear. We show that Regimes one and three were dominated by the valleys due to cutting and adhesive wear, respectively, whereas Regime two (the grit-size effect) was dominated by wear peaks due to clogging and deterioration of the abrasive surface, which led to a lower indentation of the abrasive. This contribution proves that the bifractal characteristic previously observed in abrasion can be explained by a single fractal power law and, above a threshold of l>160 μm, by a cumulative-damage model, with a probability proportional to the length of the sample but always uncorrelated with scale.

Complexity in the scaling of velocity fluctuations in the high-latitude F-region ionosphere

Abstract. The temporal scaling properties of F-region velocity fluctuations, δvlos, were characterised over 17 octaves of temporal scale from τ=1 s to &lt;1 day using a new data base of 1-s time resolution SuperDARN radar measurements. After quality control, 2.9 (1.9) million fluctuations were recorded during 31.5 (40.4) days of discretionary mode soundings using the Tasmanian (New Zealand) radars. If the fluctuations were statistically self-similar, the probability density functions (PDFs) of δvlos would collapse onto the same PDF using the scaling Ps (δvs, τ)=ταP (δvlos, τ) and δvs=δvlosτ−α where α is the scaling exponent. The variations in scaling exponents α and multi-fractal behaviour were estimated using peak scaling and generalised structure function (GSF) analyses, and a new method based upon minimising the differences between re-scaled probability density functions (PDFs). The efficiency of this method enabled calculation of "α spectra", the temporal spectra of scaling exponents from τ=1 s to ~2048 s. The large number of samples enabled calculation of α spectra for data separated into 2-h bins of MLT as well as two main physical regimes: Population A echoes with Doppler spectral width &lt;75 m s−1 concentrated on closed field lines, and Population B echoes with spectral width &gt;150 m s−1 concentrated on open field lines. For all data there was a scaling break at τ~10 s and the similarity of the fluctuations beneath this scale may be related to the large spatial averaging (~100 km×45 km) employed by SuperDARN radars. For Tasmania Population B, the velocity fluctuations exhibited approximately mono fractal power law scaling between τ~8 s and 2048 s (34 min), and probably up to several hours. The scaling exponents were generally less than that expected for basic MHD turbulence (α=0.25), except close to magnetic dusk where they peaked towards the basic MHD value. For Population A, the scaling exponents were larger than for Population B, having values generally in the range expected for basic MHD and Kolmogorov turbulence (α=0.25–0.33). The α spectra exhibited complicated variations with MLT and τ which must be related to different physical processes exerting more or less influence.

On the Gompertzian growth in the fractal space–time

An analytical approach to determination of time-dependent temporal fractal dimension b t ( t) and scaling factor a t ( t) for the Gompertzian growth in the fractal space–time is presented. The derived formulae take into account the proper boundary conditions and permit a calculation of the mean values 〈 b t ( t)〉 and 〈 a t ( t)〉 at any period of time. The formulae derived have been tested on experimental data obtained by Schrek for the Brown-Pearce rabbit's tumor growth. The results obtained confirm a possibility of successful mapping of the experimental Gompertz curve onto the fractal power-law scaling function y ( t ) = a t t b t and support a thesis that Gompertzian growth is a self-similar and allometric process of a holistic nature.

Interactions of a Light Hypersonic Jet with a Nonuniform Interstellar Medium

We present three-dimensional simulations of the interaction of a light hypersonic jet with an inhomogeneous thermal and turbulently supported disk in an elliptical galaxy. These simulations are applicable to the GPS/CSS phase of extragalactic radio sources. The interstellar medium in these simulations consists of a conventional hot (T ~ 107 K) component together with a warm (T ~ 104 K) turbulently supported disk whose local density is described by a lognormal density distribution and whose spatial structure is realized from a fractal power law. We model the jet as a light, supersonic nonrelativistic flow with parameters selected to be consistent with a relativistic jet with kinetic power just above the FR1/FR2 break. We identify four generic phases in the evolution of such a jet with the inhomogeneous interstellar medium: (1) an initial flood and channel phase, where progress is characterized by high-pressure gas finding changing weak points in the ISM, flowing through channels that form and reform over time; (2) a spherical, energy-driven bubble phase, where the bubble is larger than the disk scale, but the jet remains fully disrupted close to the nucleus; (3) a subsequent, rapid, jet break-out phase where the jet breaks free of the last obstructing dense clouds, becomes collimated once more, and pierces the spherical bubble; and (4) a classical phase, where the jet propagates in a momentum-dominated fashion similar to jets in single-component hot halos, leading to the classical jet-cocoon-bow shock structure. Mass transport in the simulations is investigated, and we propose a model for the morphology and component proper motions in the well-studied compact symmetric object 4C 31.04.

Fractal power law in literary English

We present in this paper a numerical investigation of literary texts by various well-known English writers, covering the first half of the twentieth century, based upon the results obtained through corpus analysis of the texts. A fractal power law is obtained for the lexical wealth defined as the ratio between the number of different words and the total number of words of a given text. By considering as a signature of each author the exponent and the amplitude of the power law, and the standard deviation of the lexical wealth, it is possible to discriminate works of different genres and writers and show that each writer has a very distinct signature, either considered among other literary writers or compared with writers of non-literary texts. It is also shown that, for a given author, the signature is able to discriminate between short stories and novels.

On the nature, scaling and spectral properties of pre-seismic ULF signals

Abstract. Earlier work by the authors (Vallianatos and Tzanis, 1999b), has proposed a model for the propagation and scaling of electric earthquake precursors, according to which the pre-seismic electric field emission is due to some time dependent polarisation appearing in an ensemble of electrified crustal volumes within the seismogenic source, which are distributed according to a fractal power law. Herein, we extend this formulation to the analysis of ULF magnetic precursors. We calculate the resulting transient magnetic field, which turns out to be mainly vertical and observable only if the seismogenic process generates a source with polarization rate perpendicular to the vertical plane through the source and the receiver. Furthermore, a scaling law between the vertical magnetic field and the magnitude of the associated earthquake is provided. We also investigate the spectral distribution law expected from such a set of emitters. To this effect, we assume that the evolution of the precursory polarisation process is quasi-incoherent over the exited ensemble, i.e. there is no unique relaxation time, but rather a spectrum of these with energy dependence expressed by an Arrhenius law with uniformly distributed energies. We show that the macroscopic ULF field resulting from the superposition of such an ensemble of sources has a power density spectrum distributed proportionally to 1/f . The above theoretical prediction appears to be consistent with independent observations by other investigators.

A New Theoretical Determination of Yield Stress Based on Efficient Area of Stress and Universal Fractal Power Law

A New Theoretical Determination of Yield Stress Based on Efficient Area of Stress and Universal Fractal Power Law

Kinetics of carbon oxide evolution in temperature-programmed oxidation of carbonaceous laydown deposited on wet oxidation catalysts

The combustion kinetics of coke laydown on wet oxidation catalysts was studied by means of temperature-programmed oxidation and mass spectrometry within the temperature range (30–600°C). The coke deposits were formed over three different catalysts 1 wt.% Pt/Al 2O 3, MnO 2/CeO 2 and 1 wt.% Pt–MnO 2/CeO 2 during phenol deep oxidation in a three-phase slurry reactor at various reaction conditions (exposure time, temperature, oxygen pressure, catalyst loading). The carbon oxides, oxygen and water fluxes arising from the combustion of the carbonaceous deposits in a 5% O 2/He mixture, were continuously monitored. In all cases, unimodal quasi-Gaussian distributions were obtained for CO 2 while no CO was detected. These evolutions were successfully described by a modified “fractal power-law” grain model. The coke-dependence of the carbon dioxide profiles was related to the fractal dimension of the catalyst surface and to the oxygen partial order during coke burn-off. The corresponding change in O 2 partial order was ascribed to competition between three steps in the combustion mechanism: non-dissociative O 2 chemisorption, interaction of oxygen with undissociated dioxygen bearing surface species, physical desorption of the complex oxide as carbon dioxide.