scholarly journals On the stiffness of surfaces with non-Gaussian height distribution

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Francesc Pérez-Ràfols ◽  
Andreas Almqvist
Keyword(s):  
Weibull Distribution ◽  
Power Law ◽  
Linear Relation ◽  
Hurst Exponent ◽  
Height Distribution ◽  
Fractal Surfaces ◽  
Separation Curve ◽  
Load Separation ◽  
Dimensionality Analysis ◽  
Non Gaussian

AbstractIn this work, the stiffness, i.e., the derivative of the load-separation curve, is studied for self-affine fractal surfaces with non-Gaussian height distribution. In particular, the heights of the surfaces are assumed to follow a Weibull distribution. We find that a linear relation between stiffness and load, well established for Gaussian surfaces, is not obtained in this case. Instead, a power law, which can be motivated by dimensionality analysis, is a better descriptor. Also unlike Gaussian surfaces, we find that the stiffness curve is no longer independent of the Hurst exponent in this case. We carefully asses the possible convergence errors to ensure that our conclusions are not affected by them.

Roughness of fracture surfaces in Discrete Element Model triaxial deformation experiments

2020 ◽  
Author(s):  
Steffen Abe ◽  
Hagen Deckert
Keyword(s):  
Standard Deviation ◽  
Power Law ◽  
Hurst Exponent ◽  
Numerical Experiments ◽  
Discrete Element ◽  
Height Distribution ◽  
Confining Stress ◽  
Triaxial Deformation ◽  
Fracture Surfaces ◽  
Hurst Exponents

<p>The roughness of fracture surfaces is important for a range of geological processes such as the mechanical behaviour of faults or the fluid flow in jointed rocks or fault zones. However, the processes and parameters controlling the details of the fracture roughness are not fully understood yet. We therefore use numerical simulations based on the Discrete Element Method (DEM) to study the formation of fractures in triaxial deformation experiments under a wide range of stress conditions and to quantify the geometric properties of the resulting fracture surfaces. In the numerical experiments a DEM-model of a box-shaped rock sample is subjected to a displacement controlled load along its x-axis while a defined confining stress is applied to the other surfaces.</p><p>Based on the data from 131 numerical simulations the roughness of 388 fracture surfaces has been analysed. For this purpose the surface point clouds extracted from the Discrete Element models have been converted to height fields relative to a "best-fit" plane and the height distributions quantified. The results show that the heights are normally distributed. We observe no dependence on the confining stress except that models with equal confining stress in y- and z-direction show a higher standard deviation of the height distribution than those with differing y- and z-confinement. An analysis of the height-height correlation functions for those surfaces shows that they follow a power-law, demonstrating that the surfaces are self-affine. The Hurst exponent H describing the scaling of the roughness can be derived from the power-law relation. Values obtained are in the range H=0.2-0.6 for the full suite of experiments, while the mean of the Hurst exponents for each group of fracture surfaces generated under the same stress conditions is H=0.3-0.45. A weak decreasing trend of the Hurst exponent with increasing confining stress can be observed, but contrary to the standard deviation of the height distribution there is no dependence on the ratio of the confining stresses. There is also no difference between fractures generated in tensile (mode 1) or compressive conditions (mode 2).</p><p>Additionally, surfaces of rock samples fractured in triaxial tests in the laboratory have been analysed using the same methods. The surfaces show similar self-affine characteristics as those in the numerical experiments, although with significantly higher Hurst exponents H=0.6-0.8.</p><p>A comparison between our numerical models and laboratory experiments and data obtained from literature shows that natural and lab-created fracture surfaces and their numerically modelled counterparts are similar regarding the normally distributed heights and the self-affine scale, but the Hurst exponents do not match exactly. While the majority of field and experimental studies find significantly higher Hurst exponents of about 0.8, there are some studies, for example on Sandstone, which find H=0.4-0.5, falling into the range observed in our numerical experiments.</p>

scholarly journals Parameter Resolution of the Estimation Methods for Power Law Indices

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Zheng-Yun Zhou ◽  
Yi-Ming Ding
Keyword(s):  
Power Law ◽  
Linear Relation ◽  
Interval Estimation ◽  
Estimation Algorithm ◽  
Estimation Methods ◽  
Power Law Distribution ◽  
Significance Level ◽  
Asymptotically Normal ◽  
Non Gaussian ◽  
Rich Data

The accuracy of parameter estimation plays an important role in economic and social models and experiments. Parameter resolution is the capability of an estimation algorithm to distinguish different parameters effectively under given noise level, which can be used to select appropriate algorithm for experimental or empirical data. We use a flexible distinguishing criterion and present a framework to compute the parameter resolution by bootstrap and simulation, which can be used in different models and algorithms, even for non-Gaussian noises. The parameter resolutions are computed for power law models and corresponding algorithms. For power law signal, with the increase of SNR, parameter resolution is finer; with the decrease of parameter, the resolution is finer. The standard deviation of noise and parameter resolution satisfies the linear relation; it relates to interval estimation naturally if the estimation algorithm is asymptotically normal. For power law distribution, parameter and resolution satisfy the linear relation, and experimental slope and theoretical slope tend to be consistent when significance level approaches zero. Last, we select an algorithm with finer resolution to estimate the Pareto index for the Forbes list of global rich data in recent 10 years and analyze the changes in the gap between the rich and the poor.

scholarly journals Extended power-law scaling of air permeabilities measured on a block of tuff

2012 ◽  
Vol 16 (1) ◽  
pp. 29-42 ◽  
Author(s):  
M. Siena ◽  
A. Guadagnini ◽  
M. Riva ◽  
S. P. Neuman
Keyword(s):  
Power Law ◽  
Structure Functions ◽  
Self Similarity ◽  
Scaling Exponents ◽  
Multi Scale ◽  
Sample Structure ◽  
Heavy Tailed ◽  
Non Gaussian ◽  
Nonlinear Fashion ◽  
Extended Power

Abstract. We use three methods to identify power-law scaling of multi-scale log air permeability data collected by Tidwell and Wilson on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on q-th-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents, ξ(q), of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. The finding by Tidwell and Wilson that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power-law scaling of our data, as well as of non-Gaussian heavy-tailed signals subordinated to tfBm, are extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders.

scholarly journals Observational evidence in favor of scale-free evolution of sunspot groups

2018 ◽  
Vol 618 ◽  
pp. A183
Author(s):  
A. Shapoval ◽  
J.-L. Le Mouël ◽  
M. Shnirman ◽  
V. Courtillot
Keyword(s):  
Weibull Distribution ◽  
Power Law ◽  
Large Range ◽  
Scale Free ◽  
Self Organized ◽  
Self Organized Criticality ◽  
Spatio Temporal ◽  
Computational Procedures ◽  
Definition Of ◽  
Sunspot Groups

Context. The hypothesis stating that the distribution of sunspot groups versus their size (φ) follows a power law in the domain of small groups was recently highlighted but rejected in favor of a Weibull distribution. Aims. In this paper we reconsider this question, and are led to the opposite conclusion. Methods. We have suggested a new definition of group size, namely the spatio-temporal “volume” (V) obtained as the sum of the observed daily areas instead of a single area associated with each group. Results. With this new definition of “size”, the width of the power-law part of the distribution φ ∼ 1/Vβ increases from 1.5 to 2.5 orders of magnitude. The exponent β is close to 1. The width of the power-law part and its exponent are stable with respect to the different catalogs and computational procedures used to reduce errors in the data. The observed distribution is not fit adequately by a Weibull distribution. Conclusions. The existence of a wide 1/V part of the distribution φ suggests that self-organized criticality underlies the generation and evolution of sunspot groups and that the mechanism responsible for it is scale-free over a large range of sizes.

scholarly journals Application of the Fractal Method to the Characterization of Organic Heterogeneities in Shales and Exploration Evaluation of Shale Oil

2019 ◽  
Vol 7 (4) ◽  
pp. 88 ◽  
Author(s):  
Bo Liu ◽  
Liangwen Yao ◽  
Xiaofei Fu ◽  
Bo He ◽  
Longhui Bai
Keyword(s):  
Organic Matter ◽  
Vertical Distribution ◽  
Power Law ◽  
Hurst Exponent ◽  
Hydrocarbon Exploration ◽  
Source Rocks ◽  
Organic Carbon Content ◽  
Shale Oil ◽  
Box Counting Dimension ◽  
The Vertical Distribution

The first member of the Qingshankou Formation, in the Gulong Sag in the northern part of the Songliao Basin, has become an important target for unconventional hydrocarbon exploration. The organic-rich shale within this formation not only provides favorable hydrocarbon source rocks for conventional reservoirs, but also has excellent potential for shale oil exploration due to its thickness, abundant organic matter, the overall mature oil generation state, high hydrocarbon retention, and commonly existing overpressure. Geochemical analyses of the total organic carbon content (TOC) and rock pyrolysis evaluation (Rock-Eval) have allowed for the quantitative evaluation of the organic matter in the shale. However, the organic matter exhibits a highly heterogeneous spatial distribution and its magnitude varies even at the millimeter scale. In addition, quantification of the TOC distribution is significant to the evaluation of shale reservoirs and the estimation of shale oil resources. In this study, well log data was calibrated using the measured TOC of core samples collected from 11 boreholes in the study area; the continuous TOC distribution within the target zone was obtained using the △logR method; the organic heterogeneity of the shale was characterized using multiple fractal models, including the box-counting dimension (Bd), the power law, and the Hurst exponent models. According to the fractal dimension (D) calculation, the vertical distribution of the TOC was extremely homogeneous. The power law calculation indicates that the vertical distribution of the TOC in the first member of the Qingshankou Formation is multi-fractal and highly heterogeneous. The Hurst exponent varies between 0.23 and 0.49. The lower values indicate higher continuity and enrichment of organic matter, while the higher values suggest a more heterogeneous organic matter distribution. Using the average TOC, coefficient of variation (CV), Bd, D, inflection point, and the Hurst exponent as independent variables, the interpolation prediction method was used to evaluate the exploration potential of the study area. The results indicate that the areas containing boreholes B, C, D, F, and I in the western part of the Gulong Sag are the most promising potential exploration areas. In conclusion, the findings of this study are of significant value in predicting favorable exploration zones for unconventional reservoirs.

A Microscopic Wear Measurement Method for General Surfaces

2002 ◽  
Vol 124 (4) ◽  
pp. 829-833 ◽  
Author(s):  
Yeau-Ren Jeng ◽  
Zhi-Way Lin ◽  
Shiuh-Hwa Shyu
Keyword(s):  
Surface Roughness ◽  
Measurement Method ◽  
Current Method ◽  
Current Approach ◽  
Height Distribution ◽  
Initial Surface ◽  
Roughness Parameters ◽  
Wear Measurement ◽  
Non Gaussian ◽  
Worn Surfaces

A method was developed to measure the wear of general engineering surfaces based on the roughness parameters of the worn surfaces. This method does not require any information of the initial surface. The surface height distribution is described using Johnson translatory system where the loss of surface height is attributed to wear. Experiments of engine running in were conducted to validate the method. The results show that the current method can determine wear comparable to surface roughness. The current approach simplifies the profilometrical wear measurement and extends such a measurement to non-Gaussian surfaces.

Growth Front Roughening of Room Temperature Deposited Oligomer Thin Films

2000 ◽  
Vol 648 ◽  
Author(s):  
D. Tsamouras ◽  
G. Palasantzas ◽  
J. Th. M. De Hosson ◽  
G. Hadziioannou
Keyword(s):  
Thin Films ◽  
Film Thickness ◽  
Room Temperature ◽  
Growth Front ◽  
Silicon Substrates ◽  
Height Distribution ◽  
Force Microscopy ◽  
Atomic Force ◽  
Roughness Amplitude ◽  
Non Gaussian

AbstractGrowth front scaling aspects are investigated for PPV-type oligomer thin films vapor- deposited onto silicon substrates at room temperature. For film thickness d~15-300 nm, commonly used in optoelectronic devices, correlation function measurement by atomic force microscopy yields roughness exponents in the range H=0.45±0.04, and an rms roughness amplitude which evolves with film thickness as a power law σ∝ dβ with β=0.28±0.05. The non-Gaussian height distribution and the measured scaling exponents (H and β) suggest a roughening mechanism close to that described by the Kardar-Parisi-Zhang scenario.

Power law behavior in diffraction from fractal surfaces

1998 ◽  
Vol 409 (1) ◽  
pp. L703-L708 ◽  
Author(s):  
Y.-P Zhao ◽  
C.-F Cheng ◽  
G.-C Wang ◽  
T.-M Lu
Keyword(s):  
Power Law ◽  
Fractal Surfaces

scholarly journals Log-Periodic Power Law and Generalized Hurst Exponent Analysis in Estimating an Asset Bubble Bursting Time

e-Finanse ◽  
2016 ◽  
Vol 12 (3) ◽  
pp. 49-58 ◽  
Author(s):  
Marcin Wątorek ◽  
Bartosz Stawiarski
Keyword(s):  
Power Law ◽  
Hurst Exponent ◽  
Financial Time Series ◽  
Power Law Model ◽  
Bubble Behavior ◽  
Bubble Bursting ◽  
Financial Time ◽  
Asset Bubble ◽  
Generalized Hurst Exponent ◽  
The Moment

Abstract We closely examine and compare two promising techniques helpful in estimating the moment an asset bubble bursts. Namely, the Log-Periodic Power Law model and Generalized Hurst Exponent approaches are considered. Sequential LPPL fitting to empirical financial time series exhibiting evident bubble behavior is presented. Estimating the critical crash-time works satisfactorily well also in the case of GHE, when substantial „decorrelation“ prior to the event is visible. An extensive simulation study carried out on empirical data: stock indices and commodities, confirms very good performance of the two approaches.

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