Depth-dependent Clustering Analysis of Fractures in Crystalline Basement Rocks

2021 ◽  
Author(s):  
Mohammad Javad Afshari Moein ◽  
Keith Evans ◽  
Benoît Valley ◽  
Kristian Bär ◽  
Albert Genter
Keyword(s):  
Fractal Dimension ◽  
Rock Mass ◽  
Scaling Laws ◽  
Tectonic Setting ◽  
Critical Role ◽  
Finite Size ◽  
Fracture Network ◽  
Crystalline Basement ◽  
Partial Fracture ◽  
Fracture Distribution

<p>Understanding the complex seismic, thermal, hydraulic and mechanical processes active during the hydraulic stimulation or continuous operation of Enhanced Geothermal Systems (EGS) requires an accurate description of the pre-existing fractures and faults. However, the three-dimensional characterization of the fracture network is challenging, as direct observation of the discontinuity network at great depth is limited. Fracture image logs and continuous core, which provide line samples through the fracture network, are most valuable in this regard as they provide the most precise option to place constraints on network attributes in stochastic realizations of the fractured rock mass. Among various geometrical attributes, the spatial clustering of fractures plays a critical role on the rock mass properties. </p><p>Here, we analyzed the spatial distribution of fractures derived from image log runs in six deep boreholes in crystalline basement rock. In one well, the fracture distribution from continuous core was also available. The wells were drilled to depths between 2-5 km, and were all located in the same tectonic setting of the Upper Rhine Graben, which is recognized for its high geothermal potential. The normalized correlation integral method was employed to define the scaling relationships of fracture patterns. This methodology is demonstrated to be less affected by the finite size effects, delivering reliable estimates of scaling laws.</p><p>Detailed analyses of image log datasets revealed fractal scaling with similar fractal dimensions (between 0.85 and 0.96), prevailed over almost two orders of magnitude of scale. The same was true for the fracture distribution derived from the continuous core, although this distribution was found to be more clustered than that derived from image logs in the same well (i.e. the fractal dimension was lower, which may be due to the partial fracture sampling of image logs which have a coarser resolution than continuous core analyses). Analysis of fractures in sub-sections of the core dataset from progressively increasing depths revealed no systematic depth-dependency for the fractal dimension, although a local variation at a scale of hundreds of meters was identified.</p>

scholarly journals Flow characteristics of fractal fracture with different fractal dimension and different fracture width

2021 ◽  
Vol 25 (6 Part B) ◽  
pp. 4477-4484
Author(s):  
Jun-Jun Liu ◽  
Jing Xie ◽  
Yi-Ting Liu ◽  
Gui-Kang Liu ◽  
Rui-Feng Tang ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Rock Mass ◽  
Flow Characteristics ◽  
Fracture Network ◽  
Basic Element ◽  
Single Fracture ◽  
Flow State ◽  
Complex Flow ◽  
Fracture Width ◽  
Fracture Models

Single fracture is the most basic element in complex fracture network of rock mass. Therefore, the study of flow characteristics of single fracture is an important way to reasonably predict the complex flow state in engineering rock mass. In order to study the flow characteristics of fractal single fracture, fracture models with dif?ferent fractal dimension and different fracture width are established in this paper. The results show that: the blocking effect of rough structure on fluid is obviously enhanced under high pressure. In addition, it is weakened and reaches a steady-state with the increase of fracture fractal dimension. The larger the fracture width is, the more obvious the phenomenon is. The hydraulic gradient index tends to 0.5 with the increase of fracture width when fractal dimension is greater than 1.3. It also could tend to 0.5 with the increase of fractal dimension when fracture width is greater than 1 mm.

On the multiscale quantification of fracture network geometry from lineament maps of crystalline basement units

2021 ◽  
Author(s):  
Alberto Ceccato ◽  
Giulia Tartaglia ◽  
Giulio Viola ◽  
Marco Antonellini
Keyword(s):  
Statistical Analysis ◽  
Scaling Laws ◽  
Numerical Models ◽  
Regional Scale ◽  
Fracture Network ◽  
Crystalline Basement ◽  
Fracture Networks ◽  
Spatial Organisation ◽  
Robust Analysis ◽  
Network Properties

<p>Fractured crystalline basement units are attracting increasing attention as potential unconventional reservoirs for natural (oil, heat and water) resources and as potential waste (nuclear, CO<sub>2</sub>) disposal sites. The focus of the current efforts is the characterisation of the structural permeability of fractured crystalline basement units, which is primarily related to the geology, geometry, and spatial characteristics of fracture networks. Fracture network properties may be scale–dependent or independent. Thus, a multi–scale characterisation of fracture networks is usually recommended to quantify the scale–variability of properties and, eventually, the related predictive scaling laws. Fracture lineament maps are schematic representations of fracture distributions obtained from either manual or automated interpretation of 2D digital models of the earth surface at different scales. From the quantitative analysis on fracture lineament maps, we can retrieve invaluable information on the scale–dependence of fracture network properties.</p><p>Here we present the results of the quantification of fracture network and fracture set properties (orientation, length, spacing, spatial organisation) from multi– (outcrop to regional) scale 2D lineament maps of two crystalline basement study areas of Western Norway (Bømlo island and Kråkenes). Lineament maps were obtained from the manual interpretation of orthophotos and 2D digital terrain models retrieved from UAV–drone and LiDAR surveys.</p><p>Analyses aimed at the quantification of: (i) scaling laws for fracture length cumulative distributions, defined through a statistically–robust fitting method (Maximum Likelihood Estimations coupled with Kolmogorov–Smirnov tests); (ii) variability of orientation sets as a function of scale; (iii) spatial organisation of fracture sets among scales; (iv) fractal characteristics of fracture networks (fractal exponent). Results suggest that: (i) a statistical analysis considering variable censoring and truncation effects allows to confidently define the best–fitting scaling laws; (ii) the analysis of orientation variability of fracture sets among different scales may provide important constraints about the geometrical complexity of fracture and fault zones; (iii) the statistical analysis of 2D spacing variability and fracture intensity can be adopted to quantify fracture spatial organisation at different scales.</p><p>A statistically robust analysis of the scaling laws, length distributions, spacing, and spatial organisation of lineaments on 2D maps provides reliable results also where only partial or incomplete dataset/lineament maps are available. Such properties are the fundamental input parameters for conceptual (geologic) and numerical (discrete fracture network, DFN) models of fractured crystalline basement reservoirs. Therefore, a statistically robust analysis of fracture lineament maps may help to improve the accuracy of conceptual and numerical models.</p>

scholarly journals A Fractal Model for Characterizing Hydraulic Properties of Fractured Rock Mass under Mining Influence

Geofluids ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Xiaoli Liu ◽  
Tao Liang ◽  
Sijing Wang ◽  
Kumar Nawnit
Keyword(s):  
Fractal Dimension ◽  
Rock Mass ◽  
Permeability Coefficient ◽  
Fractured Rock ◽  
Length Distribution ◽  
Fracture Network ◽  
Fractal Model ◽  
Physical Model Test ◽  
Fractured Rock Mass ◽  
Fractal Characteristics

In this paper, two basic assumptions are introduced: (1) The number and length distribution of fractures in fractured rock mass are in accordance with the fractal law. (2) Fluid seepage in the fractures satisfies the cubic law. Based on these two assumptions, the fractal model of parallel seepage and radial seepage in fractured rock mass is established, and the seepage tensor of fracture network which reflects the geometric characteristics and fractal characteristics of fracture network under two kinds of seepage is derived. The influence of fracture geometry and fractal characteristics on permeability is analyzed, and the validity and accuracy of the model are verified by comparing the calculated results of the theoretical model and physical model test. The results show that the permeability coefficient K of fracture network is a function of the geometric (maximum crack length Lmax, fractured horizontal projection length L0, diameter calculation section porosity Φ, fracture strike α, and fracture angle θ) and fractal characteristics (fracture network fractal dimension Df and seepage flow fractal dimension DT). With the increase of fractal dimension Df, the permeability coefficient increases. With the increase of DT, the permeability coefficient decreases rapidly. And the larger the Df (Df>1.5), the greater the change of permeability coefficient K with DT.

scholarly journals Fracture System and Rock-Mass Characterization by Borehole Camera Surveying: Application in Dimension Stone Investigations in Geologically Complex Structures

2021 ◽  
Vol 11 (2) ◽  
pp. 764
Author(s):  
Ivica Pavičić ◽  
Ivo Galić ◽  
Mišo Kucelj ◽  
Ivan Dragičević
Keyword(s):  
Rock Mass ◽  
Fractured Rock ◽  
Low Cost ◽  
Fracture Pattern ◽  
Dimension Stone ◽  
Rock Mass Characterization ◽  
Size And Shape ◽  
Distribution Parameters ◽  
Borehole Camera ◽  
Fracture Distribution

The successful exploration of dimension stone mainly depends on the quality, size, and shape of extractable blocks of dimension stone. The investigated area is in the Pelješac Peninsula (Croatia), in the External Dinarides orogeny, built from thick carbonate succession, characterized by relatively small deposits of high-quality dimension stone. These conditions demand challenging geological investigations in the “pre-quarry” phase to find optimal quarry location. The size and shape of dimension stone blocks are mainly controlled by fracture pattern systems. In the rugged, covered terrains, it is very hard to obtain a satisfactory amount of fracture data from the surface, so it is necessary to collect them from the underground. Borehole camera technology can visualize the inner part of the rock mass and measure the fracture characteristics. The main conclusions are as follows: (1) the digital borehole camera technology provides a quick, effective, and low-cost geological survey of fractured rock mass; (2) statistical fracture distribution parameters, P10, fracture spacing, Volumetric Joint Count (Jv) based on borehole wall survey can reflect the integrity of rock mass, providing a solid decision-making base for further investment plans and dimension stone excavation method.

KINETIC RESPONSE OF SURFACES DEFINED BY FINITE FRACTALS

Fractals ◽  
2010 ◽  
Vol 18 (04) ◽  
pp. 461-476 ◽  
Author(s):  
PRADEEP R. NAIR ◽  
MUHAMMAD A. ALAM
Keyword(s):  
Scaling Laws ◽  
Finite Size ◽  
Critical Dimension ◽  
The Novel ◽  
Scale Invariant ◽  
Fractal Surfaces ◽  
Dimension Formula ◽  
Diffusion Limited ◽  
Engineered Systems ◽  
Size Limits

Historically, fractal analysis has been remarkably successful in describing wide ranging kinetic processes on (idealized) scale invariant objects in terms of elegantly simple universal scaling laws. However, as nanostructured materials find increasing applications in energy storage, energy conversion, healthcare, etc., one must reexamine the premise of traditional fractal scaling laws as it only applies to physically unrealistic infinite systems, while all natural/engineered systems are necessarily finite. In this article, we address the consequences of the 'finite-size' problem in the context of time dependent diffusion towards fractal surfaces via the novel technique of Cantor-transforms to (i) illustrate how finiteness modifies its classical scaling exponents; (ii) establish that for finite systems, the diffusion-limited reaction is decelerated below a critical dimension [Formula: see text] and accelerated above it; and (iii) to identify the crossover size-limits beyond which a finite system can be considered (practically) infinite and redefine the very notion of 'finiteness' of fractals in terms of its kinetic response. Our results have broad implications regarding dynamics of systems defined by the same fractal dimension, but differentiated by degree of scaling iteration or morphogenesis, e.g. variation in lung capacity between a child and adult.

scholarly journals A methodology of re-generating a representative element volume of fractured rock mass

2020 ◽  
Vol 71 (4) ◽  
pp. 347-358
Author(s):  
DANG Hong-Lam ◽  
THINH Phi Hong
Keyword(s):  
Rock Mass ◽  
Fractured Rock ◽  
Fracture Network ◽  
Hydraulic Calculation ◽  
Fractured Rock Mass ◽  
Fracture Characteristics ◽  
Mechanical Calculation ◽  
Representative Element ◽  
Representative Element Volume ◽  
Actual Fracture

In simulation of fractured rock mass such as mechanical calculation, hydraulic calculation or coupled hydro-mechanical calculation, the representative element volume of fractured rock mass in the simulating code is very important and give the success of simulation works. The difficulties of how to make a representative element volume are come from the numerous fractures distributed in different orientation, length, location of the actual fracture network. Based on study of fracture characteristics of some fractured sites in the world, the paper presented some main items concerning to the fracture properties. A methodology of re-generating a representative element volume of fractured rock mass by DEAL.II code was presented in this paper. Finally, some applications were introduced to highlight the performance as well as efficiency of this methodology.

Characterizing Fractal and Hierarchical Morphologies Beyond the Fractal Dimension

1994 ◽  
Vol 367 ◽  
Author(s):  
Raphael Blumenfeld ◽  
Robin C. Ball
Keyword(s):  
Asymptotic Behavior ◽  
Fractal Dimension ◽  
Finite Size ◽  
Size Analysis ◽  
Corrections To Scaling ◽  
Diffusion Limited Aggregation ◽  
Diffusion Limited ◽  
Markovian Processes ◽  
Branching Patterns ◽  
Admissible Solutions

AbstractWe present a novel correlation scheme to characterize the morphology of fractal and hierarchical patterns beyond traditional scaling. The method consists of analysing correlations between more than two-points in logarithmic coordinates. This technique has several advantages: i) It can be used to quantify the currently vague concept of morphology; ii) It allows to distinguish between different signatures of structures with similar fractal dimension but different morphologies already for relatively small systems; iii) The method is sensitive to oscillations in logarithmic coordinates, which are both admissible solutions for renormalization equations and which appear in many branching patterns (e.g., noise-reduced diffusion-limited-aggregation and bronchial structures); iv) The methods yields information on corrections to scaling from the asymptotic behavior, which is very useful in finite size analysis. Markovian processes are calculated exactly and several structures are analyzed by this method to demonstrate its advantages.

3-D fracture network dynamic simulation based on error analysis in rock mass of dam foundation

2018 ◽  
Vol 25 (4) ◽  
pp. 919-935 ◽  
Author(s):  
Deng-hua Zhong ◽  
Han Wu ◽  
Bin-ping Wu ◽  
Yi-chi Zhang ◽  
Pan Yue
Keyword(s):  
Rock Mass ◽  
Error Analysis ◽  
Dynamic Simulation ◽  
Fracture Network ◽  
Network Dynamic ◽  
Dam Foundation ◽  
Simulation Based

Micro Thermal Engines: Is There Any Room at the Bottom?

1999 ◽  
Author(s):  
Richard B. Peterson
Keyword(s):  
Heat Transfer ◽  
Scaling Laws ◽  
Critical Role ◽  
Heat Engine ◽  
Engine Performance ◽  
Characteristic Size ◽  
Operating Efficiency ◽  
Simple Scaling ◽  
Small Structure ◽  
Micro Heat Engine

Abstract Richard P. Feynman introduced the field of microscale and nanoscale engineering in 1959 by giving a talk on how to make things very small. Feynman’s premise was that no fundamental physical laws limit the size of a machine down to the microscopic level. Is this true for all types of machines? Are micro thermal devices fundamentally different than mechanically-based machines with respect to their scaling laws? This paper demonstrates that micro thermal engines do indeed suffer serious performance degradation as their characteristic size is reduced. A micro thermal engine, and more generally, any thermally-based micro device, depends on establishing a temperature difference between two regions within a small structure. In this paper, the performance of a micro thermal engine is explored as a function of the characteristic length parameter, L. In the development, the important features of thermal engines are discussed in the context of developing simple scaling laws predicting the dependency of the operating efficiency on L. After this is accomplished, a general model is derived for a heat engine operating between two temperature reservoirs and having both intrinsic and extrinsic sources of irreversibility, i.e. thermal conductances and heat leakage paths for the heat flow. With this model and typical numerical values for the conductances, micro heat engine performance is predicted as the characteristic size is reduced. This paper demonstrates that under at least one particular formulation of the problem, there may indeed be some room at the bottom. However, heat transfer does play a critical role in determining micro engine performance and depending on how the heat transfer through the engine is modeled, vanishingly small efficiencies can result as the characteristic engine size goes to zero.

scholarly journals Fractal dimension of lightning discharge

1995 ◽  
Vol 2 (2) ◽  
pp. 101-106 ◽  
Author(s):  
J. Sañudo ◽  
J. B. Gómez ◽  
F. Castaño ◽  
A. F. Pacheco
Keyword(s):  
Fractal Dimension ◽  
Stochastic Model ◽  
Size Effects ◽  
Dielectric Breakdown ◽  
Lightning Discharge ◽  
Finite Size ◽  
Finite Size Effects ◽  
3 Dimensional ◽  
Photographic Analysis

Abstract. Using a stochastic model, we simulate the process of dielectric breakdown in the atmosphere and calculate the fractal dimension of 3-dimensional lightning patterns. Finite-size effects have been studied. The projections of our patterns on vertical planes fit the experimental fractal dimension obtained from photographic analysis. This work is inspired by a previous work by A.A. Tsonis.

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