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 Roughness of Fracture Surfaces in Numerical Models and Laboratory Experiments

2021 ◽  
Author(s):  
Steffen Abe ◽  
Hagen Deckert
Keyword(s):  
Numerical Simulations ◽  
Laboratory Experiments ◽  
Numerical Models ◽  
Stress Conditions ◽  
Roughness Coefficient ◽  
Confining Stress ◽  
Rock Samples ◽  
Fracture Surfaces ◽  
Wide Range ◽  
Hurst Exponents

Abstract. We investigate the influence of stress conditions during fracture formation on the geometry and roughness of fracture surfaces. Rough fracture surfaces have been generated in numerical simulations of triaxial deformation experiments using the Discrete Element Method and in laboratory experiments on limestone and sandstone samples. Digital surface models of the rock samples fractured in the laboratory experiments were produced using high resolution photogrammetry. The roughness of the surfaces was analyzed in terms of absolute roughness measures such as an estimated joint roughness coefficient (JRC) and in terms of its scaling properties. The results show that all analyzed surfaces are self-affine, but with different Hurst exponents between the numerical models and the real rock samples. Results from numerical simulations using a wide range of stress conditions to generate the fracture surfaces show a weak decrease of the Hurst exponents with increasing confining stress and a larger absolute roughness for transversely isotropic stress conditions compared to true triaxial conditions. Other than that, our results suggest that stress conditions have little influence on the surface roughness of newly formed fractures.

STATISTICS OF FRACTURE SURFACES

Fractals ◽  
1993 ◽  
Vol 01 (04) ◽  
pp. 1059-1067 ◽  
Author(s):  
J. PLANÈS ◽  
E. BOUCHAUD ◽  
G. LAPASSET
Keyword(s):  
Aluminium Alloy ◽  
Power Law ◽  
Fracture Mode ◽  
Experimental Measurements ◽  
Height Distribution ◽  
Gaussian Distributions ◽  
Fracture Surfaces ◽  
Roughness Exponent ◽  
Theoretical Predictions ◽  
Branched Structures

Experimental measurements of the roughness exponent ζ of some fracture surfaces are reported. Whatever the fracture mode, for unbranched (aluminium alloy) or branched (Ni3Al) surfaces, ζ is found quite close to 0.8, in accordance with other experimental determinations, but not with theoretical predictions of standard 3-d models. The complete height distribution for the complex branched structures is shown to slowly decrease with increasing altitude. It implies a power-law behavior for the low order moments which is experimentally recovered. Exponents reveal different from that of gaussian distributions.

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.

scholarly journals Roughness of fracture surfaces in numerical models and laboratory experiments

Solid Earth ◽  
2021 ◽  
Vol 12 (10) ◽  
pp. 2407-2424
Author(s):  
Steffen Abe ◽  
Hagen Deckert
Keyword(s):  
Numerical Simulations ◽  
Laboratory Experiments ◽  
Numerical Models ◽  
Stress Conditions ◽  
Roughness Coefficient ◽  
Confining Stress ◽  
Rock Samples ◽  
Fracture Surfaces ◽  
Wide Range ◽  
Hurst Exponents

Abstract. We investigate the influence of stress conditions during fracture formation on the geometry and roughness of fracture surfaces. Rough fracture surfaces have been generated in numerical simulations of triaxial deformation experiments using the discrete element method and in a small number of laboratory experiments on limestone and sandstone samples. Digital surface models of the rock samples fractured in the laboratory experiments were produced using high-resolution photogrammetry. The roughness of the surfaces was analyzed in terms of absolute roughness measures such as an estimated joint roughness coefficient (JRC) and in terms of its scaling properties. The results show that all analyzed surfaces are self-affine but with different Hurst exponents between the numerical models and the real rock samples. Results from numerical simulations using a wide range of stress conditions to generate the fracture surfaces show a weak decrease of the Hurst exponents with increasing confining stress and a larger absolute roughness for transversely isotropic stress conditions compared to true triaxial conditions. Other than that, our results suggest that stress conditions have little influence on the surface roughness of newly formed fractures.

scholarly journals On the emergent system mass function: the contest between accretion and fragmentation

2020 ◽  
Vol 500 (2) ◽  
pp. 1697-1707
Author(s):  
Paul C Clark ◽  
Anthony P Whitworth
Keyword(s):  
Star Formation ◽  
Standard Deviation ◽  
Mass Distribution ◽  
Power Law ◽  
Accretion Rate ◽  
Mass Function ◽  
High Mass ◽  
New Model ◽  
System Formation ◽  
Low Mass

ABSTRACT We propose a new model for the evolution of a star cluster’s system mass function (SMF). The model involves both turbulent fragmentation and competitive accretion. Turbulent fragmentation creates low-mass seed proto-systems (i.e. single and multiple protostars). Some of these low-mass seed proto-systems then grow by competitive accretion to produce the high-mass power-law tail of the SMF. Turbulent fragmentation is relatively inefficient, in the sense that the creation of low-mass seed proto-systems only consumes a fraction, ${\sim }23{{\ \rm per\ cent}}$ (at most ${\sim }50{{\ \rm per\ cent}}$), of the mass available for star formation. The remaining mass is consumed by competitive accretion. Provided the accretion rate on to a proto-system is approximately proportional to its mass (dm/dt ∝ m), the SMF develops a power-law tail at high masses with the Salpeter slope (∼−2.3). If the rate of supply of mass accelerates, the rate of proto-system formation also accelerates, as appears to be observed in many clusters. However, even if the rate of supply of mass decreases, or ceases and then resumes, the SMF evolves homologously, retaining the same overall shape, and the high-mass power-law tail simply extends to ever higher masses until the supply of gas runs out completely. The Chabrier SMF can be reproduced very accurately if the seed proto-systems have an approximately lognormal mass distribution with median mass ${\sim } 0.11 \, {\rm M}_{\odot }$ and logarithmic standard deviation $\sigma _{\log _{10}({M/M}_\odot)}\sim 0.47$).

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.

Analysis of contact and bending stiffness for Curvic couplings considering contact angle and surface roughness

2019 ◽  
Vol 233 (6) ◽  
pp. 1257-1267 ◽  
Author(s):  
Younghun Yu ◽  
Bora Lee ◽  
Yongjoo Cho
Keyword(s):  
Surface Roughness ◽  
Contact Angle ◽  
Standard Deviation ◽  
Bending Stiffness ◽  
Height Distribution ◽  
Average Radius ◽  
Asperity Height ◽  
Surface Parameter ◽  
Tangential Stiffness ◽  
Nominal Contact Area

This paper develops a method for calculating the contact and bending stiffness of a Curvic coupling, and investigates stiffness changes according to the coupling shape and surface roughness characteristics. The surface of the on-site Curvic coupling is chosen as reference for a most accurate simulation. The three parameters representing the surface roughness characteristics—the standard deviation of the asperity height distribution, the average radius of asperities, and the density of asperity on the nominal contact area—are calculated with a profile of the coupling surface through a random process: the contact problem between rough surfaces is tackled using the Greenwood-Williamson model, the Curvic coupling is modeled assuming that it has as many teeth as possible within the machining limits depending on the contact angle, and the tangential stiffness resulting from the contact angle is calculated by dividing into the stick and slip regions, and is taken into account in terms of total stiffness. With this, results showed that using Curvic couplings reduces stiffness than using flat disc couplings because of the contact angle, and that the standard deviation of rough surface height is the most crucial surface parameter affecting stiffness.

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.

Dynamic roughness model for large-eddy simulation of turbulent flow over multiscale, fractal-like rough surfaces

2011 ◽  
Vol 679 ◽  
pp. 288-314 ◽  
Author(s):  
W. ANDERSON ◽  
C. MENEVEAU
Keyword(s):  
Large Eddy Simulation ◽  
Turbulent Flow ◽  
Power Law ◽  
Rough Surfaces ◽  
Scale Height ◽  
Small Scale ◽  
Height Distribution ◽  
Subgrid Scale ◽  
Eddy Simulation ◽  
Large Eddy

Many flows especially in geophysics involve turbulent boundary layers forming over rough surfaces with multiscale height distribution. Such surfaces pose special challenges for large-eddy simulation (LES) when the filter scale is such that only part of the roughness elements of the surface can be resolved. Here we consider LES of flows over rough surfaces with power-law height spectra Eh(k) ~ kβs (−3 ≤ βs < −1), as often encountered in natural terrains. The surface is decomposed into resolved and subgrid-scale height contributions. The effects of the unresolved small-scale height fluctuations are modelled using a local equilibrium wall model (log-law or Monin–Obukhov similarity), but the required hydrodynamic roughness length must be specified. It is expressed as the product of the subgrid-scale root-mean-square of the height distribution and an unknown dimensionless quantity, α, the roughness parameter. Instead of specifying this parameter in an ad hoc empirical fashion, a dynamic methodology is proposed based on test-filtering the surface forces and requiring that the total drag force be independent of filter scale or resolution. This dynamic surface roughness (DSR) model is inspired by the Germano identity traditionally used to determine model parameters for closing subgrid-scale stresses in the bulk of a turbulent flow. A series of LES of fully developed flow over rough surfaces are performed, with surfaces built using random-phase Fourier modes with prescribed power-law spectra. Results show that the DSR model yields well-defined, rapidly converging, values of α. Effects of spatial resolution and spectral slopes are investigated. The accuracy of the DSR model is tested by showing that predicted mean velocity profiles are approximately independent of resolution for the dynamically computed values of α, whereas resolution-dependent results are obtained when using other, incorrect, α values. Also, strong dependence of α on βs is found, where α ranges from α ~ 0.1 for βs = −1.2 to α ~ 10−5 for βs = −3.

Sign in / Sign up

Export Citation Format

Share Document