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1.
Fractures: Finite-size scaling and multifractals 总被引：1，自引：0，他引：1
The distributions of contact area and void space in single fractures in granite rock have been determined experimentally by making metal casts of the void spaces between the fracture surfaces under normal loads. The resulting metal casts on 52 cm diameter core samples show a complex geometry for the flow paths through the fracture. This geometry is analyzed using finite-size scaling. The spanning probabilities and percolation probabilities of the metal casts are calculted as functions of observation scale. Under the highest stresses of 33 MPa and 85 MPa there is a significant size-dependence of the geometric flow properties for observation scales smaller than 2 mm. Based on this data, the macroscopic percolation properties of the extended fracture can be well represented by relatively small core samples, even under normal stresses larger than 33 MPa. The metal casts also have rich multifractal structure that changes with changing stress. 相似文献
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Silicate melts impose systematic fracture patterns on their hosts and their own subsolidus portions as they crystallize and cool to ambient conditions. Weathering of these plutonic bodies accentuates the fractures and produces fragmented topographic surfaces whose geometric measures exceed those of a Euclidean surface. Because geometric measures of these surfaces are potentially indicative of the percolation properties of the fracture network, contours and vertical sections were digitized from 1250000 and 1240000 scale maps, and their statistical fractal dimensions,D, were computed. Regions underlain by granitic plutons with similar fragmentation patterns were found to haveD values ranging from 1.15 to 1.28 within a single pluton. This range of values is caused by noise and several geologic factors. The values contain noise introduced by the map-making and digitizing procedures that amounts toD1.05. However all values lie well above this threshold, and they correlate closely with local differences in weathering. SmallerD values derive from contours where accumulation of screen or glacial erosion has smoothed the surface, whereas largerD values derive from contours along aretes and in unglaciated terrains. In each of these regions, largerD values occur locally where fractures are most frequent and/or continuous. LargeD values were also derived from regions underlain by host rocks that lie in the near-field region of the stress field caused by the pluton. The fractral values derived are apparently related to the magma-hydrothermal event, they can potentially be used to derive properties of the percolation networks that were active during the hydrothermal events. 相似文献
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The fractal dimension of an individual floc is a measure of the complexity of its external shape. Fractal dimensions can also be used to characterize floc populations, in which case the fractal dimension indicates how the shape of the smaller flocs relates to that of the larger flocs. The objective of this study is to compare the fractal dimensions of floc populations with those of individual flocs, and to evaluate how well both indicate contributions of sediment sources and reflect the nature and extent of flocculation in streams. Suspended solids were collected prior to and during snowmelt at upstream and downstream sites in two southern Ontario streams with contrasting riparian zones. An image analysis system was used to determine area, longest axis and perimeter of flocs. The area–perimeter relationship was used to calculate the fractal dimension, D, that characterizes the floc population. For each sample, the fractal dimension, Di , of the 28 to 30 largest individual flocs was determined from the perimeter–step‐length relationship. Prior to snowmelt, the mean value of Di ranged from 1·19 (Cedar Creek, downstream) to 1·22 (Strawberry Creek, upstream and downstream). A comparison of the means using t‐tests indicates that most samples on this day had comparable mean values of Di . During snowmelt, there was no significant change in the mean value of Di at the Cedar Creek sites. In contrast, for Strawberry Creek the mean value of Di at both sites increased significantly, from 1·22 prior to snowmelt to 1·34 during snowmelt. This increase reflects the contribution of sediment‐laden overland flow to the sediment load. At three of the sampling sites, the increase in fractal dimensions was accompanied by a decreases in effective particle size, which can be explained by an increase in bed shear stress. A comparison of fractal dimensions of individual flocs in a sample with the fractal dimensions of the floc populations indicates that both fractal dimensions provide similar information about the temporal changes in sediment source contributions, about the contrasting effectiveness of the riparian buffer zones in the two basins, and about the hydraulic conditions in the streams. Nevertheless, determining the individual fractal dimensions of a set of large flocs in a sample is very time consuming. Using fractal dimensions of floc populations is therefore the preferred method to characterize suspended matter. Copyright © 2000 John Wiley & Sons, Ltd. 相似文献
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文中以甘肃北山花岗岩中发育的构造裂隙(主要指节理)为研究对象,通过野外裂隙调查,应用传统的概率统计方法与分形几何学理论,利用Mapinfo,ArcGIS平台进行裂隙几何学参数(方位、长度、密度等)的统计、计算和裂隙网络的空间结构分析,研究花岗岩岩体中裂隙的几何学特征。并以此为基础,对甘肃北山花岗岩岩体质量优劣进行初步评价。结果表明:在10～200cm范围内,裂隙网络是分形的;5个测点裂隙网络的分维值分别是1.636,1.548,1.596,1.724,1.604。分维数D不仅能刻画岩体中结构面发育的数量,而且能反映结构面在岩体中分布的均匀程度和交切方式。因此,可以表征岩体的质量优劣,对岩体质量进行分级。按照分维所划分的岩体质量分级,北山花岗岩属于裂隙较发育、岩体质量等级一般的岩体 相似文献
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We present a case of detailed analysis of fracture arrays spanning four orders of magnitude in length; all of them measured at a single natural site by acquiring images at progressively larger scales. There is a high dispersion of cumulative-length exponents, box dimensions and fracture densities. However, the fractal analysis supports the fractal nature of fracture arrays. Our data indicate the existence of an upper limit for the density parameters, as similarly reported by other authors. We prove that box dimension is in inverse relation with fracture concentration and in direct relation with fracture density. These relations are also observed in our data and additionally there is an upper limit for the box dimensions. We interpret the dispersion in our results as more fundamental than methodological problems. It represents a truncation in the complete evolution of the fracture systems because in natural cases strain initiates overprinting of previous fracture arrays. Considering that larger fractures accommodate strain more efficiently than small fractures, the generation of small fractures is inhibited in the presence of pre-existing larger fractures. Maximum values of fracture density prevent accommodating an excess of strain in a single or restricted range of scales; we claim this condition produces migration of fracturing to larger scales originating fracture scaling. 相似文献
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本文根据Okubo等人测量圣安德烈斯断层系所用的复盖维数法,对海原断层系进行了分形测量,求出海原断层系的整体维数D_0=1.137,其景泰段D_0=1.109,海原段D_0=1.182。计算中未得到邵家水段和李使堡段的分维数。此外,文中还着重探讨了断层几何与地震活动性及其力学环境的关系,进一步证明了自相似断层的几何复杂性与地震活动密切相关,剪切断裂带具较低分维数(1.1—1.3),而在张性环境中形成的断裂体系具较高的分维数(1.5—1.6)。最后本文讨论了断层迹线图等因素对分形测量精度的影响。 相似文献
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A systematic numerical method has been presented to investigate the constitutive relationships between two-phase flow properties of horizontal fractures and aperture distributions. Based on fractal geometry, single rough-walled fractures are generated numerically by modified successive random addition (SRA) method and then aperture distributions with truncated Gaussian distribution are formed by shear displacement between lower and upper surfaces. (The truncated Gaussian distribution is used to describe aperture evolution under different normal stresses.) According to the assumption of two-dimensional porous media and local parallel plate model, invasion percolation approach is employed to model the two-phase flow displacement (imbibition) in generated horizontal fractures, in which capillary forces are dominant over viscous and gravity forces. For truncated Gaussian distributions, constitutive relationships from numerical simulation are compared to closed-form relationships and a good agreement is obtained. The simulation results indicate strong phase interference with the sum of two phase relative permeability values being less than one in the intermediate saturations. It is found that fracture properties related to residual saturations depend on spatial correlation of aperture distributions. Based on the simulation results, we proposed an empirical relationship between the fracture residual-saturation-rated parameters and the corresponding aperture distributions. 相似文献
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Influences of Aquifer Properties on Flow Dimensions in Dolomites 总被引：1，自引：0，他引：1
by Timotej Verbov&#;ek 《Ground water》2009,47(5):660-668
The paper focuses on analyses and correlations of flow dimensions in different dolomite aquifers in Slovenia. Flow dimensions are obtained through the reinterpretation of 72 pumping tests with the generalized radial flow model, based on the fractional flow dimension. The average value of flow dimensions is 2.16 for all dolomites. A study of flow dimensions in individual aquifers categorized according to their lithological properties shows that higher dimensions occur in massive late-diagenetic Cordevolian and Anisian dolomites compared with bedded Main, Bača, and especially Lower Triassic dolomites, which contain a greater proportion of noncarbonate minerals. Partially penetrating wells have higher flow dimensions than fully penetrating wells. Flow dimensions are poorly correlated with hydraulic conductivities of fractures. When comparing the quantities of major dissolved minerals, obtained by hydrogeochemical inverse modeling, with the values of flow dimensions, the Cordevolian and Anisian dolomites are found to exhibit the highest values of both dissolved dolomite and flow dimensions, indicating that greater dissolution occurs at higher flow dimensions. For other aquifers, data points are more scattered and the correlation is mostly poor. When compared with three-dimensional fractal dimensions of fracture networks, there is no correlation with flow dimensions. However, almost all the values of flow dimensions are lower than the corresponding fractal dimensions in dolomites (average D = 2.77), possibly indicating the channeling of flow within the available space of the fracture networks, consequently reducing the flow dimensions. 相似文献
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We have fitted field measurements of fracture spacings (from the vicinity of Lake Strom Thurmond, Georgia, U.S.A.) to the Weibull, Schuhmann and fractal distributions. The fracture spacings follow a fractal and Weibull distribution which implies that they were formed as a result of a repetitive fragmentation process. The limited variation of the fracture density with orientation in the study area suggests that the stress distribution generating these fractures may be uniform. 相似文献
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Fluvial suspended sediment typically consists of a variety of complex, composite particles referred to as flocs. Floc characteristics are determined by factors such as the source, size and geochemical properties of the primary particles, chemical and biological coagulation processes in the water column and shear stress and turbulence levels in the stream. Studies of floc morphology have used two contrasting methods of sampling and analysis. In the first method, particles settle on a microscope slide and are observed from below using an inverted microscope. The second method uses filtration at no or low vacuum and particles deposited on the filter are observed with a microscope. Floc morphology can be quantified using fractal dimensions. The aims of the present study were to examine the effect of the two sampling methods on the fractal dimensions of particle populations, and to evaluate for each method how well the fractal dimensions at the various sampling sites reflect basin conditions. Suspended solids were collected in triplicate on inverted microscope slides and on 0·45 μm Millipore HA filters in two southern Ontario streams with contrasting riparian zones during a minor runoff event resulting from the melt of a freshly fallen snowpack. An image analysis system was used to determine area, longest axis and perimeter of particles. The morphology of the particle population of each sample was characterized using four fractal dimensions (D, D1, D2 and DK). Systematic differences in fractal dimensions obtained with the two methods were observed. For the settling method, outlines of larger particles were frequently blurred because of the distance between the focal plane (the top of the inverted microscope slides) and the plane of the particle outline. In this method, the blurring of large particles can cause an increase in the projected area and length of the particle. The effect on the particle perimeter is unpredictable because it depends on the amount of detail lost through blurring and its effect on the apparent increase in particle size. Because of blurring, D and D1 tend to be systematically lower for the settling method, whereas the net effect on D2 is unpredictable. Particle size distributions derived from settling are typically coarser because small, low density particles may remain in the water column and all particles may not deposit on the slides. This loss of fines results in systematically lower DK values for the settling method compared with the filtration method. Fractal dimensions and particle size distributions obtained with the filtration method were sensitive to and clearly indicated differences between drainage basins and between sites within each basin. These differences were explained by basin characteristics and conditions. Fractal dimensions and particle size distributions obtained with the settling method were less sensitive to drainage basin characteristics and conditions, which limits their usefulness as process indicators. Copyright © 1999 John Wiley & Sons, Ltd. 相似文献
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Robert W.D. Lodge David T. Lescinsky 《Journal of Volcanology and Geothermal Research》2009,185(4):298-310
Cooling lava commonly develop polygonal joints that form equant hexagonal columns. Such fractures are formed by thermal contraction resulting in an isotropic tensional stress regime. However, certain linear cooling fracture patterns observed at some lava–ice contacts do not appear to fit the model for formation of cooling fractures and columns because of their preferred orientations. These fracture types include sheet-like (ladder-like rectangular fracture pattern), intermediate (pseudo-aligned individual column-bounding fractures), and pseudopillow (straight to arcuate fractures with perpendicular secondary fractures caused by water infiltration) fractures that form the edges of multiple columns along a single linear fracture. Despite the relatively common occurrence of these types of fractures at lava–ice contacts, their significance and mode of formation have not been fully explored. This study investigates the stress regimes responsible for producing these unique fractures and their significance for interpreting cooling histories at lava–ice contacts.Data was collected at Kokostick Butte dacite flow at South Sister, OR, and Mazama Ridge andesite flow at Mount Rainier, WA. Both of these lava flows have been interpreted as being emplaced into contact with ice and linear fracture types have been observed on their ice-contacted margins. Two different mechanisms are proposed for the formation of linear fracture networks. One possible mechanism for the formation of linear fracture patterns is marginal bulging. Melting of confining ice walls will create voids into which flowing lava can deform resulting in margin-parallel tension causing margin-perpendicular fractures. If viewed from the ice-wall, these fractures would be steeply dipping, linear fractures. Another possible mechanism for the formation of linear fracture types is gravitational settling. Pure shear during compression and settling can result in a tensional environment with similar consequences as marginal inflation. In addition to this, horizontally propagating cooling fractures will be directly influenced by viscous strain caused by the settling of the flow. This would cause preferential opening of fractures horizontally, resulting in vertically oriented fractures.It is important to note that the proposed model for the formation of linear fractures is dependent on contact with and confinement by glacial ice. The influence of flow or movement on cooling fracture patterns has not been extensively discussed in previous modeling of cooling fractures. Rapid cooling of lava by the interaction with water and ice will increase the ability to the capture and preserve perturbations in the stress regime. 相似文献
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The presence of fractures in fluid‐saturated porous rocks is usually associated with strong seismic P‐wave attenuation and velocity dispersion. This energy dissipation can be caused by oscillatory wave‐induced fluid pressure diffusion between the fractures and the host rock, an intrinsic attenuation mechanism generally referred to as wave‐induced fluid flow. Geological observations suggest that fracture surfaces are highly irregular at the millimetre and sub‐millimetre scale, which finds its expression in geometrical and mechanical complexities of the contact area between the fracture faces. It is well known that contact areas strongly affect the overall mechanical fracture properties. However, existing models for seismic attenuation and velocity dispersion in fractured rocks neglect this complexity. In this work, we explore the effects of fracture contact areas on seismic P‐wave attenuation and velocity dispersion using oscillatory relaxation simulations based on quasi‐static poroelastic equations. We verify that the geometrical and mechanical details of fracture contact areas have a strong impact on seismic signatures. In addition, our numerical approach allows us to quantify the vertical solid displacement jump across fractures, the key quantity in the linear slip theory. We find that the displacement jump is strongly affected by the geometrical details of the fracture contact area and, due to the oscillatory fluid pressure diffusion process, is complex‐valued and frequency‐dependent. By using laboratory measurements of stress‐induced changes in the fracture contact area, we relate seismic attenuation and dispersion to the effective stress. The corresponding results do indeed indicate that seismic attenuation and phase velocity may constitute useful attributes to constrain the effective stress. Alternatively, knowledge of the effective stress may help to identify the regions in which wave induced fluid flow is expected to be the dominant attenuation mechanism. 相似文献
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Natural fracture patterns of producing geothermal formations in south-western Turkey are mapped at different scales. The fractal dimensions of different fracture network properties, such as spatial distribution, density, connectivity, orientation, and length are measured by different methods. Analysis of the natural fracture patterns from giga to microscales identifies the descending behavior of box-counting fractal dimension with respect to the scale. It is observed that the fracture networks represent scale-invariant properties, but fractal dimensions might notably differ when the mass dimension is measured applying different methods. Anisotropic nature of fracture networks is also included in the fractal analysis. 相似文献
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S. Thevanayagam 《地震工程与工程振动(英文版)》2007,6(2):123-134
Mechanical behavior such as stress-strain response, shear strength, resistance to liquefaction, modulus, and shear wave velocity of granular mixes containing coarse and fine grains is dependent on intergrain contact density of the soil. The global void ratio e is a poor index of contact density for such soils. The contact density depends on void ratio, fine grain content （Cv）, size disparity between particles, and gradation among other factors. A simple analysis of a two-sized particle system with large size disparity is used to develop an understanding of the effects of Cv, e, and gradation of coarse and fine grained soils in the soil mix on intergrain contact density. An equivalent intergranular void ratio （ec）oq is introduced as a useful intergrain contact density for soils at fines content of less than a threshold value Crth. Beyond this value, an equivalent interfine void ratio （ef）eq is introduced as a primary intergrain contact density index. At higher values of Cv beyond a limiting value of fine grains content CVL, an interfine void ratio ef is introduced as the primary contact density index. Relevant equivalent relative density indices （Drc）eq and （Drf）eq are also presented. Experimental data show that these new indices correlate well with steady state strength, liquefaction resistance, and shear wave velocities of sands, silty sands, sandy silts, and gravelly sand mixes. 相似文献
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This paper presents a dual-percolation model coupling the percolation theory and the fracture percolation theory to study the conductivity of the fractured porous media. The Monte-Carlo method is used in the numerical simulation. First an appropriate computing scale by considering the calculation precision and elapsed time together is validated. Then, two parameters, A 0 and D are presented in this model to determine the conductivity of the media. Generally the media can be blocked by itself in the condition of D > 2. However, the increase of pore connection and the randomness of fracture direction may release the selfblockage, increase the conductivity and make the dual porous media dissipated. A few long fractures can play a great role in the connection of media. 相似文献
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Haruo Sato 《Pure and Applied Geophysics》1988,128(1-2):43-47
Supposing that the distribution of scatterers in a three-dimensional medium is not uniform but fractally homogeneous with fractal dimensionD, we have made the dimensional analysis for the temporal decay of the multiple scattering energy density at the hypocenter.The number of scatters in a sphere of radiusR is assumed to be proportional toR
D
. Then, the energy density of thekth order scattering decays according to the [(D–2)k–3]th power of lapse time. A fractal dimension ofD=3 corresponds to the uniform distribution. If 2<D3, multiple scattering terms of orderk2 dominate over the single scattering term (k=1) at long lapse time. IfD=2, energy density of every order decays according to the — 3rd power of lapse time. The single scattering model survives on conditionD<2; the single scattering term dominates over the higher order multiple scattering terms even at long lapse time, since the negative power of lapse time fork=1 is the smallest of all. 相似文献
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S. T. Hatamian 《Pure and Applied Geophysics》1996,146(1):115-129
Three cellular automaton toy-models of fragmentation in two-dimensional lattices are explored. Of the three models, two can be considered in the class of simple bond percolation, and one as correlated bond percolation. Fractal fragment-size distribution in all models is found away from criticality, providing a certain fraction of the bonds is designated with considerably larger strengths than the rest in the system. As the fraction of these bonds is raised from zero, the fragment-size distribution transforms smoothly from exponential forms into a power law. Though each model takes a different path to the fractal distribution, they all show the same fractal exponent of 1.85(5). As might be expected in one dimension, the same models of their variants, failed to produce fractal distributions. 相似文献