Block Size Distribution Research Articles

A Method to Estimate In Situ Block Size Distribution

This paper presents a new technique for estimating the in situ block size distribution in a jointed rock mass. The technique is based on Monte Carlo simulations using more realistic fracture geometry as its input compared to other block size estimation methods described in the literature. This geometry represents fractures as either polygons or triangulated surfaces and therefore models persistence and truncation of fractures accurately. Persistence has been shown to be critically important for the accurate prediction of block size and shape. We show that for rock masses with relatively small discontinuities, the results of our predictions differ markedly from previous methods which over-predict fragmentation.

Evaluating roadside rockmasses for rockfall hazards using LiDAR data: optimizing data collection and processing protocols

Highways and railroads situated within rugged terrain are often subjected to the hazard of rockfalls. The task of assessing roadside rockmasses for potential hazards typically involves an on-site visual investigation of the rockmass by an engineer or geologist. At that time, numerous parameters associated with discontinuity orientations and spacing, block size (volume) and shape distributions, slope geometry, and ditch profile are either measured or estimated. Measurements are typically tallied according to a formal hazard rating system, and a hazard level is determined for the site. This methodology often involves direct exposure of the evaluating engineer to the hazard and can also create a potentially non-unique record of the assessed slope based on the skill, knowledge and background of the evaluating engineer. Light Detection and Ranging (LiDAR)–based technologies have the capability to produce spatially accurate, high-resolution digital models of physical objects, known as point clouds. Mobile terrestrial LiDAR equipment can collect, at traffic speed, roadside data along highways and rail lines, scanning continual distances of hundreds of kilometres per day. Through the use of mobile terrestrial LiDAR, in conjunction with airborne and static systems for problem areas, rockfall hazard analysis workflows can be modified and optimized to produce minimally biased, repeatable results. Traditional rockfall hazard analysis inputs include two distinct, but related sets of variables related to geological or geometric control. Geologically controlled inputs to hazard rating systems include kinematic stability (joint identification/orientation) and rock block shape and size distributions. Geometrically controlled inputs include outcrop shape and size, road, ditch and outcrop profile, road curvature and vehicle line of sight. Inputs from both categories can be extracted or calculated from LiDAR data, although there are some limitations and special sampling and processing considerations related to structural character of the rockmass, as detailed in this paper.

Black dimensional stones: geology, technical properties and deposit characterization of the dolerites from Uruguay

Dimensional stones with a black color occupy a prominent place on the international market. Uruguayan dolerite dikes of andesitic and andesitic-basaltic compo- sition are mined for commercial blocks of black dimen- sional stones. A total of 16 dikes of both compositions were studied and samples collected for geochemical and petro- graphical analysis. Color measurements were performed on different black dimensional stones in order to compare them with the Uruguayan dolerites. Samples of the two commercial varieties (Absolute Black and Moderate Black) were obtained for petrophysical analysis (e.g. density, porosity, uniaxial compressive strength, tensile strength, etc.). Detailed structural analyses were performed in sev- eral quarries. Geochemistry and petrography determines the intensity of the black color. The Uruguayan dolerite Absolute Black is the darkest black dimensional stone analyzed in this study. The petrophysical properties of Uruguayan dolerites make them one of the highest quality black dimensional stones. Structural analyses show that five joint sets have been recognized: two sub-vertical joints, one horizontal and two diagonal. These joint sets are one of the most important factors that control the deposits, since they control the block size distribution and the amount of waste material.

A Statistical Representation of the Matrix–Fracture Transfer Function for Porous Media

In this article, the authors present a matrix–fracture transfer function where the statistical variation in geometric properties of the matrix blocks is considered. Several particular representations with hypothetical probability density functions (PDFs) for matrix block size distributions are presented, including: (a) the single-value distribution (the limiting case); (b) the uniform distribution; (c) the Gamma distribution; and (d) an approximate representation for arbitrary PDFs. An example using experimental data from the literature, along with the single-block based transfer function developed in this study, is presented demonstrating how the statistical procedure proposed in this text can be applied in practice. It is shown with this example that significant relative errors can be introduced when the statistical variance is ignored. Furthermore, two existing dual-porosity models, the Lim and Aziz model and the Zimmerman et al. model, are also considered using the experimental data. It is shown that considerable relative errors can be introduced with these two models when the effect of statistical variance is not taken into account.

Coherent and diffuse X-ray scattering in crystals modulated by a surface acoustic wave

Equations describing the coherent and diffuse scattering in a crystal modulated by a surface acoustic wave (SAW) are derived using the dynamical X-ray diffraction theory. The effect of depth attenuation of the Rayleigh surface wave amplitude on the crystal rocking curve profiles is investigated. Results of the numerical simulation of the dynamical diffraction in a mosaic crystal modulated by a SAW, taking into account a block size distribution, are presented. It is shown that the diffuse scattering is distributed in the reciprocal space not only in the vicinity of the main diffraction peak but also about the satellite diffraction peaks, and this distribution depends on the size fluctuations of the crystal defects. Theoretical reciprocal space maps and rocking curves are compared with the corresponding experimental results.

Influence of rock mass properties on blasting efficiency

The purpose of this paper is to determine the influence of rock mass properties on the blasting efficiency which is ratio of the block size distribution of the rock mass to the block size distribution of the muck-pile. The proposed methodology of blasting efficiency in this study is to compare physical and mechanical properties of the rock mass and block fragmentation under the same blasting conditions in Kirka borax mine. Intact rock properties, block size of rock mass before blasting and muck pile after blasting were found to measure blasting efficiency. Firstly, intact rock properties, which are unit volume weight, water absorption, uniaxial compressive strength, tensile (Brazilian) strength, cohesion and internal friction angle, were tested for each mining bench. Secondly, block sizes of rock masses in respect to discontinuity boundaries were measured and muck pile photos were taken in order to determine Block Fragmentation (BF) which is to separate the rock mass block size by blasting and that of the corresponding muck pile. Thirdly, statistical analysis between rock mass properties and block fragmentation were developed and these analysis test results have shown that a good relation between block fragmentation and Brazilian tensile strength and internal friction angle were found. As a result, block fragmentation in the same blasting conditions and other rock properties can be estimated from the best empirical correlations with the rock properties.

Prediction of fragmentation and yield curves with reference to armourstone production

Armourstone production involves aspects of blast design and yield prediction. How they differ from methods drawn from experience in mining and aggregates blasting operations is examined. A number of possible blast fragmentation models and associated prediction methods are described, several being outlined in full. Their applicability to armourstone production and yield curve prediction is discussed by comparing model results based on a hypothetical armourstone blast design in a rock mass with realistic properties for an armourstone quarry. It is suggested that appropriate models for armourstone yield prediction will require some form of an in-situ block size distribution assessment. Such approaches rule out the standard application of the Kuz–Ram model. The recently reported ‘Swebrec’ function and associated prediction model, developed by the Swedish Blasting Research Centre, provides a promising replacement for the Rosin–Rammler based models for representing armourstone blast yield curves.

Fragmentation energy in rock avalanches

Fragmentation is one of the mechanisms involved in rock avalanches. Quantifying the associated energy during a rock avalanche can help to assess the influence of fragmentation on post-failure mass movements. In this paper, in situ block size distributions of the intact rock mass and the debris deposits are presented and analyzed for nine rock avalanches, five in the Canadian Rocky Mountains and four in the European Alps. Degrees of fragmentation are estimated from these data. Two methods are examined to assess fragmentation energy, one based on the comminution theory, and one on the blasting energy used in the mining industry. The results show that, for the studied rock avalanches, there is a relationship between the reduction in diameter ratio, Rr = D50/d50 (where D50 and d50 are the mean diameter of the intact rock mass and the mean diameter of the debris, respectively), and the potential energy per unit volume normalized with respect to the point load strength of rock (γHG/σc), where γ is the unit weight of the material, HG is the vertical distance between the centres of gravity of the mass at the start and end positions, and σc is the point load strength). For the cases studied, fragmentation energy calculations average 20% of the potential energy. An empirical relationship between Rr and γHG/σc has been established and is used in the definition of a disintegration index (ID). This index seems to reflect the physics of the disintegration process, since it accounts for the fact that the reduction in diameter ratio is a function of the dissipated energy and the strength of the rock. These factors have long been known to affect fragmentation but have never been presented in a coherent manner for rock avalanches.Key words: rock avalanches, disintegration, fragmentation energy, Canadian Rocky Mountains, European Alps.

Prediction of in-situ block size distributions with reference to armourstone for breakwaters

Understanding a quarry in terms of its potential for breakwater construction materials presents a special challenge for the engineering geologist. Unlike blasting in aggregates and mining operations, optimisation of the extraction process has a focus on the potential for production of large blocks for armourstone. These blocks weighing many tonnes are used for cover layers to resist wave action. The quarry-run is used for breakwater core. If the quarry has been developed as a source of materials dedicated to a breakwater construction project, the success of the project depends greatly on the blasting and production of rock sizes that are required and the avoidance of leaving a massive quantity of unused materials behind in the quarry after project completion. Prediction of in-situ block sizes such as from joint spacing data, provides the most critical input for the prediction of the blast pile block size distribution (BBSD), which in turn is a vital early design input if the constructed breakwater is to be economical as well as effective. This paper is part of a series of papers that introduces the coastal engineering motivation for this work on engineering geology, giving reasons why the prediction of the fragmentation curve of the blast products in a dedicated quarry is of such economic importance for breakwater projects. The first step towards blasted block size distribution (BBSD) prediction is the prediction of the in-situ block sized distribution (IBSD), the main subject of this paper. Drawing together research methods from the 1990s and the rock mechanics principles of discontinuity analysis, a practical step by step methodology for IBSD assessment that includes approaches that are not reliant on specialised computer software is presented. Continuing on the practical theme, a new extension of the volumetric joint count approach is suggested for IBSD prediction for the case when sparse borehole data is all that is available. A case study of IBSD assessment and the associated BBSD and blast assessment is presented from a Carboniferous limestone quarry. For clarity, details of blast design and yield curve prediction that are recommended for use in the context of armourstone production, have been presented in a companion paper. The Rosin–Rammler equation is used as an example form for the BBSD prediction of a dedicated quarry and the potential for breakwater project optimisation is illustrated. The final section sets out a method for directly comparing yield curves together with the demand for materials set by the breakwater design. On the same plot, sizes where there is a relative shortfall in production can be identified. The dependence of effective breakwater design on accurate quarry yield prediction and quarry blasting performance is discussed.

Mode of action and subsite studies of the guluronan block-forming mannuronan C-5 epimerases AlgE1 and AlgE6

AlgE1, AlgE5 and AlgE6 are members of a family of mannuronan C-5 epimerases encoded by the bacterium Azotobacter vinelandii, and are active in the biosynthesis of alginate, where they catalyse the post-polymerization conversion of beta-D-mannuronic acid (M) residues into alpha-L-guluronic acid residues (G). All enzymes show preference for introducing G-residues neighbouring a pre-existing G. They also have the capacity to convert single M residues flanked by G, thus 'condensing' G-blocks to form almost homopolymeric guluronan. Analysis of the length and distribution of G-blocks based on specific enzyme degradation combined with size-exclusion chromatography, electrospray ionization MS, HPAEC-PAD (high-performance anion-exchange chromatography and pulsed amperometric detection), MALDI (matrix-assisted laser-desorption ionization)-MS and NMR revealed large differences in block length and distribution generated by AlgE1 and AlgE6, probably reflecting their different degree of processivity. When acting on polyMG as substrates, AlgE1 initially forms only long homopolymeric G-blocks >50, while AlgE6 gives shorter blocks with a broader block size distribution. Analyses of the AlgE1 and AlgE6 subsite specificities by the same methodology showed that a mannuronan octamer and heptamer respectively were the minimum substrate chain lengths needed to accommodate enzyme activities. The fourth M residue from the non-reducing end is epimerized first by both enzymes. When acting on MG-oligomers, AlgE1 needed a decamer while AlgE6 an octamer to accommodate activity. By performing FIA (flow injection analysis)-MS on the lyase digests of epimerized and standard MG-oligomers, the M residue in position 5 from the non-reducing end was preferentially attacked by both enzymes, creating an MGMGGG-sequence (underlined and boldface indicate the epimerized residue).

Determination of the In situ Block Size Distribution in Fractured Rock, an Approach for Comparing In-situ Rock with Rock Sieve Analysis

Determination of the In situ Block Size Distribution in Fractured Rock, an Approach for Comparing In-situ Rock with Rock Sieve Analysis

Enumerative Problems Inspired by Mayer's Theory of Cluster Integrals

The basic functional equations for connected and 2-connnected graphs can be traced back to the statistical physicists Mayer and Husimi. They play an essential role in establishing rigorously the virial expansion for imperfect gases. We first review these functional equations, putting the emphasis on the structural relationships between the various classes of graphs. We then investigate the problem of enumerating some classes of connected graphs all of whose 2-connected components (blocks) are contained in a given class $B$. Included are the species of Husimi graphs ($B =$ "complete graphs"), cacti ($B =$ "unoriented cycles"), and oriented cacti ($B =$ "oriented cycles"). For each of these, we address the question of their labelled and unlabelled enumeration, according (or not) to their block-size distributions. Finally we discuss the molecular expansion of these species. It consists of a descriptive classification of the unlabelled structures in terms of elementary species, from which all their symmetries can be deduced.

Study of Submicrocrystalline Materials by Conventional Powder Diffraction and Diffuse Scattering in Transmitted Wave

Two methods were used for the analysis of submicro-crystalline copper prepared by high-pressure (6 GPa) severe plastic deformation. Conventional powder XRD study was carried in terms of the modified Williamson-Hall (WH) plots and shown large line-broadening anisotropy. A new high-resolution small-angle transmission diffuse scattering method was applied for determination of crystallite size. Two methods were used for the evaluation of the diffuse scattering - calculation of the autocorrelation function of the crystalline shape by the transformation of the measured signal and by the direct fitting. Bimodal block size distribution has to be assumed for annealed samples. This was also confirmed by classical back-reflection film method as well as by the long tails of conventional diffraction profiles.

Nanosized Micelles Formed by the Self-assembly of Amphiphilic Block Copolymers with Luminescent Rhenium Complexes

We have synthesized a series of polystyrene-block-poly(4-vinylpyridine)s (PS-b-PVP) to which luminescent tricarbonyl(2,2‘-bipyridyl)rhenium(I) complexes were attached. The rhenium complexes could induce the self-assembly of the copolymers into nanosized micelles with different shapes and dimensions, depending on the block size distribution of the copolymers and the solvent system being used. In general, spherical micelles were observed when methanol, a nonsolvent for the polystyrene block, was added to a copolymer solution in dichloromethane. Micellization was observed when the added methanol concentration was approximately 30%. When toluene, a nonsolvent for the poly(4-vinylpyridine) block, was added to the copolymer solution in dichloromethane, micelles with different interesting shapes were observed. For copolymers with larger PVP block size, spherical micelles were observed. When the relative block size of the PVP block was reduced, the micelles gradually changed to disk or vesicle structures and then to rodlike structures. The rhenium complex can act as a luminescent probe in the resulting nanosized micelles and provide sufficient contrast for electron microscopic studies. Significant changes in luminescence spectra were observed after the micellization.

A Review of General Considerations for Assessing Rock Mass Blastability and Fragmentation

Blasting is the primary comminution process in most mining operations. This process involves the highly complex and dynamic interaction between two main components. The first is the detonating explosive and the second is the rock mass into which the explosive is loaded. The mechanical properties of the rock material (such as dynamic strength, tensile strength, dynamic modulus and fracture toughness) are important considerations in understanding the blasting process. However, it is the characteristics of the geological defects (joints, foliation planes, bedding planes) within the rock mass that ultimately determine how effectively a blast performs in terms of fragmentation, all else being equal. The defect characteristics include, but are not limited to, their orientation, spacing, and mechanical properties. During the blasting process, some of the geotechnical characteristics of the rock mass are substantially changed. From the blasting outcome point of view, the most notable and important is the change in fragment size distribution that the rock mass undergoes. The pre-blast in situ defect-bounded block size distribution is transformed into the post-blast muckpile fragment size distribution. Consequently, it is fundamental to our understanding of and ability to predict the blasting process that both the blastability of a rock mass and its transformation into the fragment size distribution can be appropriately quantified.

Distribution of the lateral correlation length in GaN epitaxial layers

GaN structures belong to the most popular wide-bandgap semiconductors. Large lattice mismatch existing between the layer and substrates (3.5% for SiC and even 16% in the case of sapphire substrate) results in structures with a large number of defects. The typical GaN epitaxial layer consists of dislocation-free regions with lateral dimensions equal to a few hundred nanometers. The dislocation density changes from 10 8 cm −2 inside the grains to 10 10 cm −2 in the grain boundaries. The average value of the lateral correlation lengths (coherence wavelength) seems to be not quite satisfactory. Particularly, it is connected with lateral direction, because the vertical length is approximately equal to the thickness of the epitaxial layer. This paper presents the new approach for the determination of GaN crystallites dislocation-free block size distributions. This method is based on the X-ray peak profile analysis and solution of the Fredholm integral equation. The necessary peaks are obtained from the high-resolution X-ray diffractometry measurements. The obtained results have been shown for the various samples: GaN layers grown on low temperature buffer layer (GaN or AlN). Very interesting results were obtained in the first case, where two different sizes of the blocks appear.

Evaluation of the Part of Blasting Energy Used for Rock Mass Fragmentation

In the literature, the results of numerous studies were used to propose predictive tools for the evaluation of fragmentation efficiency in rock mass blasting. The methodology developed during several years at the Centre de Geologie de l’Ingenieur is based on the comparison of rock mass and corresponding muck piles block size distributions [1–4]. This paper introduces a new approach of the evaluation of the part of blasting energy used in fragmentation, based on the assessment of the energy devoted to the formation of crack surface. The fundamental assumptions of this method will be exposed and discussed in detail. The method is then applied to blasts performed in two Spanish mines in different geological contexts. The corresponding results are presented and the influence of some blast parameters (such as geological rock mass structure, specific energy and time delays) is discussed.

Assessment of harvest blocks generated from operational polygons and forest-cover polygons in tactical and strategic planning

Manually designing harvest units for strategic planning is expensive. This paper compares blocking methods based on forest-cover polygons and manually designed harvest units. Routines are used to split and aggregate polygons into three block size distributions: (i) uniform 40-ha blocks; (ii) uniform 120-ha blocks; and (iii) by area, one-third 20 ha, one-third 60 ha, and one-third 150 ha. Three harvest rules that influence adjacency and the cutting of polygons within a block are applied to each block size distribution to compare forecasts generated by forest-cover and operational blocks. Generally, volume flows from the two methods deviate by less than 5%, and the highest deviations usually occur during the first 20 years. Projected landscape structure, as measured by interior forest area, is also similar under the two blocking methods. The results indicate that forest-cover data provide a reasonable alternative to manual blocking in tactical and strategic plans. This is significant because it removes an important barrier to timely and cost-effective planning, especially for large geographic problems where manual blocking is not an option.

Concentration and temperature transients in heterogeneous porous media: Part I: Linear transport

The transport of solute/heat in porous media is modeled by the convection–dispersion equation. In the study of such transport processes, while the resident concentration has always been used to develop the governing differential equations, the flux concentration has been the most commonly measured quantity in experiments. Hence, earlier studies have emphasized that care must be exercised in the quantitative interpretation of experiments to avoid inconsistent use of solutions with the actual conditions of experiments. They have further demonstrated that performing an actual variable transformation in the homogeneous medium differential equation of resident concentrations shows that the flux concentration also satisfies the convection dispersion equation including those with linear reaction. Hence, the physical meanings of solutions with respect to these two concentrations are inferred from boundary conditions. The earlier studies have also provided a classification of solutions of homogeneous medium models based on these two concentration variables. This work first generalizes the theory of dependent variable transformation for dispersive transport and further extends it to include the heterogeneous medium models that assume a diffusive transport between the fracture and matrix phases. Then, two new solutions of the heterogeneous medium solutions are derived. Hence, the experimentalist is furnished with the heterogeneous medium model solutions in which at least one of them is consistent with the actual conditions of an experiment. Secondly, in this work, the concept of block geometry functions (BGFs) is extended to include frequency distributions of multiple block sizes more likely to exist in heterogeneous media than a single block. It was found that BGFs of various distributions with λ min less than 0.1 differ only slightly, and hence, may be represented by the BGF of the mean block size. Otherwise care must be exercised to include the block size distribution effect. In addition, numerical Laplace inversion of complete solutions having those slightly differing BGFs is found to lead to significant differences for cases where mobile and immobile phase fractions are close. Finally, interpretation of tracer return profiles in a heterogeneous system by employing a nonlinear regression technique is illustrated. Simulated field data are matched by a four-parameter theoretical model and sensitivity of results to parameter values is investigated.

Assessing the Martian surface distribution of aeolian sand using a Mars general circulation model

A sand transport model using White's [1979] sand flux equation and the Mars general circulation model [Pollack et al., 1990] was developed to understand the erosional sources, transport pathways, and depositional sinks of windblown sand on Mars. An initially uniform distribution of sand (4 mm over the entire surface) is regionally transported based on wind stress, saltation threshold, and percentage of topographic trapping, Results are consistent with the observed polar and Hellespontus dunes and Christensen's [1986] modeled block size distribution, but only for an extremely low saltation threshold (0.024 N/m2). Low thresholds generally result in transport of sand‐sized particles originating in the northern mid latitudes to the north pole, and transport from the northern lower latitudes to the southern hemisphere. Our results indicate that the polar dune fields could form in 50,000 years, consistent with the active polar dunes and lack of longitudinal dunes observed on the surface of Mars.