Image‐based modelling and analysis of microstructures for two‐scale problems in geomechanics

This paper focuses on the geometry modelling and numerical analysis of microstructures of geomaterials employing the concept of image‐based engineering. The novel modelling and analysis techniques with digital images are incorporated with the mathematical homogenization method to study the interaction between individual phases, each of whose shape and spatial distribution are irregular. Owing to the distinctive features of these computational techniques, the evaluation of homogenized properties for geomaterials provides the reliable information about the micro‐ or macroscopic mechanical behaviours for engineering practice. It is, naturally, inevitable that engineers' demands on safety and efficient design place emphasis on quantitative estimates for these values. Thus, calibration accompanied with actual measurements comes within the scope of this study so that these properties would be realistic and practical from the engineering viewpoints. Copyright © 2002 John Wiley & Sons, Ltd.     

Quantitative X-Ray diffraction of carbonate sediments: mineralogical analysis through fitting of Lorentzian profiles to diffraction peaks

A new automatic method for quantitative mineralogical analysis of carbonate sediments by X-Ray diffraction is described. Percentage analysis has been carried out by means of the reference intensities method from integrated areas of the diffraction peaks. Diffraction peaks were fitted by means of Lorentzian profiles. The addition of an internal standard allows the determination of data on the stoichiometric disequilibrium of the carbonate phases and the existence of non-carbonate phases and/or non-crystalline phases simultaneously. Relative errors of the quantitative estimates of carbonate phases are always less than 15%, even though real percentage differences between actual and calculated values are less than 3·8% in all cases.     

Crystal Size Distributions (CSD) in Three Dimensions: Insights from the 3D Reconstruction of a Highly Porphyritic Rhyolite

Growth histories and residence times of crystals in magmaticsystems can be revealed by studying crystal sizes, size distributionsand shapes. In this contribution, serial sectioning has beenemployed on a sample of porphyritic rhyolite from a Permo-Carboniferouslaccolith from the Halle Volcanic Complex, Germany, to reconstructthe distribution of felsic phenocrysts in three dimensions inorder to determine their true shapes, sizes and three-dimensionalsize distributions. A model of all three phenocryst phases (quartz,plagioclase, K-feldspar) with 217 crystals, and a larger modelcontaining 1599 K-feldspar crystals was reconstructed in threedimensions. The first model revealed a non-touching frameworkof crystals in three dimensions, suggesting that individualcrystals grew freely in the melt prior to quenching of the texture.However, crystal shapes are complex and show large variationon a Zingg diagram (intermediate over long axis plotted againstshort over intermediate axis). They often do not resemble thecrystallographic shapes expected for phenocrysts growing unhinderedfrom a melt, indicating complex growth histories. In contrast,the three-dimensional size distribution is a simple straightline with a negative slope. Stereologically corrected size distributionsfrom individual sections compare well with stereologically correctedsize distributions obtained previously from the same sample.However, crystal size distribution (CSD) data from individualsections scatter considerably. It is shown that CSDs can berobustly reproduced with a sampling size of greater than 200crystals. The kind of shape assumed in stereological correctionof CSDs, however, has a large influence on the calculation andestimation of crystal residence times. KEY WORDS: 3D reconstruction; crystal shapes; CSD; porphyritic rhyolite; quantitative petrography     

The importance of surface area in metal sorption by oxides and organic matter in a heterogeneous natural sediment

This study provides empirical validation of current trace metal sorption theory in a small urban river. We demonstrate that trace metal complexation reactions occur predominantly at the suspended particulate surface involving surface layers of Fe oxides and organic matter. Associated surface areas of these geochemical fractions were calculated where possible, using the total surface area (TSA) of the suspended particulate matter pool (SPM) in conjunction with estimates of suspended iculate Fe and Mn oxides (SPOX) and organic matter (SPOM) concentrations. Iron and Mn oxides concentrations were estimated using an extraction scheme. For two samples where no SPOM or Mn oxides were present, estimates of Fe oxides associated surface area were determined which compared favourably to literature estimates, providing further evidence for acceptance selectivity of extraction schemes. The utility of literature estimates of surface areas for single component sediments in heterogeneous sediments was also assessed. In mixed sediment samples, exposed surface areas of discrete phases are probably reduced due to mixed layering effects of the coatings, and the use of constants to estimate the surface areas of individual fractions does not work, since the relationship between the concentration of a given sedimentary fraction and its exposed surface area is no longer predictable.     

Calculation of the volume of an alluvial fan

Calculation of the volume of a lithostratigraphic unit within an alluvial fan allows a longterm denudation rate to be calculated for the fan's catchment, where sediment is not transported beyond the fan. The calculation of denudation rates in this case depends upon accurate estimates of the volume of sediment in a lithostratigraphic unit of known age. Two methods of volume calculation and their approximations are presented. The first method sums the volume of discrete blocks used to approximate a variety of fan shapes. The second method integrates a quadratic function that approximates both the surface and lower boundary of a lithostratigraphic unit where the edges of the fan are assumed to be vertical. For fans that satisfy these restrictive assumptions, the integration method should be more accurate than the summation method. However, the summation method is more generally applicable and calculations using the two methods, and their approximations, yield remarkably similar results.     

Strain and competence contrast estimation from fold shape

A new method to estimate strain and competence contrast from natural fold shapes is developed and verified by analogue and numerical experiments. Strain is estimated relative to the nucleation amplitude, A_N, which is the fold amplitude when the amplification velocities caused by kinematic layer thickening and dynamic folding are identical. A_N is defined as the initial amplitude corresponding to zero strain because folding at amplitudes smaller than A_N is dominantly by kinematic layer thickening. For amplitudes larger than A_N, estimates of strain and competence contrast are contoured in thickness-to-wavelength (H/λ) and amplitude-to-wavelength (A/λ) space. These quantities can be measured for any observed fold shape. Contour maps are constructed using existing linear theories of folding, a new nonlinear theory of folding and numerical simulations, all for single-layer folding. The method represents a significant improvement to the arc length method. The strain estimation method is applied to folds in viscous (Newtonian), power-law (non-Newtonian) and viscoelastic layers. Also, strain partitioning in fold trains is investigated. Strain partitioning refers to the difference in strain accommodated by individual folds in the fold train and by the whole fold train. Fold trains within layers exhibiting viscous and viscoelastic rheology show different characteristic strain partitioning patterns. Strain partitioning patterns of natural fold trains can be used to assess the rheological behaviour during fold initiation.     

Phase equilibria and physical properties

The average physical properties of multiphase aggregates, such as rocks, depend on the properties of the individual phases as well as the statistical spatial distribution of the phases, such as the relative concentrations, and the shape, orientation, and distribution of the grains of the individual phases. If the properties of the phases are similar, the aggregate properties depend mainly on volume averages of properties of the phases; this is the case for elastic properties of rocks. If the properties of the phases differ significantly, the geometric distribution of the phases becomes important; this is the case if a fluid phase is present, which may have a large effect on elastic and electrical properties. Laboratory measurements of properties of individual phases and aggregates, used with theoretical treatments of aggregate properties, permits the interpretation of seismic velocities, attenuation and electrical conductivity in the earth in terms of possible compositions, phases and distribution of phases, such as intergranular fluids.     

Dependence of thermal conductivity of snow on microstructure

A geometrical model, including different geometrical shapes influencing thermal conductivity of snow is proposed. The geometrical model has been assumed to comprise of unit cells having solid (ice) inclusion as an aggregation of spherical, cylindrical or cubical shapes with vertical connection, arranged in a cubic packing. From the geometrical model and one-dimensional heat transfer theory, the effective thermal conductivity has been computed. For this purpose, coupled one-dimensional heat transfer equations have been solved for steady-state condition to account for conduction in ice, conduction in air and latent heat transfer due to water vapour sublimation through air. The model demonstrates the dependency of thermal conductivity on density, grain-spacing, grain contact ratio and temperature. Spherical inclusions give highest conductivity while cubical inclusion estimates lowest value for the same density. Thermal conductivity has been found increasing sharply near to the packing density for all three shapes. Empirical model results and results obtained from existing microstructure based models have also been compared with the present model.     

Image analysis algorithms for estimating porous media multiphase flow variables from computed microtomography data: a validation study

Image analysis of three-dimensional microtomographic image data has become an integral component of pore scale investigations of multiphase flow through porous media. This study focuses on the validation of image analysis algorithms for identifying phases and estimating porosity, saturation, solid surface area, and interfacial area between fluid phases from gray-scale X-ray microtomographic image data. The data used in this study consisted of (1) a two-phase high precision bead pack from which porosity and solid surface area estimates were obtained and (2) three-phase cylindrical capillary tubes of three different radii, each containing an air–water interface, from which interfacial area was estimated. The image analysis algorithm employed here combines an anisotropic diffusion filter to remove noise from the original gray-scale image data, a k-means cluster analysis to obtain segmented data, and the construction of isosurfaces to estimate solid surface area and interfacial area. Our method was compared with laboratory measurements, as well as estimates obtained from a number of other image analysis algorithms presented in the literature. Porosity estimates for the two-phase bead pack were within 1.5% error of laboratory measurements and agreed well with estimates obtained using an indicator kriging segmentation algorithm. Additionally, our method estimated the solid surface area of the high precision beads within 10% of the laboratory measurements, whereas solid surface area estimates obtained from voxel counting and two-point correlation functions overestimated the surface area by 20–40%. Interfacial area estimates for the air–water menisci contained within the capillary tubes were obtained using our image analysis algorithm, and using other image analysis algorithms, including voxel counting, two-point correlation functions, and the porous media marching cubes. Our image analysis algorithm, and other algorithms based on marching cubes, resulted in errors ranging from 1% to 20% of the analytical interfacial area estimates, whereas voxel counting and two-point correlation functions overestimated the analytical interfacial area by 20–40%. In addition, the sensitivity of the image analysis algorithms on the resolution of the microtomographic image data was investigated, and the results indicated that there was little or no improvement in the comparison with laboratory estimates for the resolutions and conditions tested.     

Effect of Aberration on the Estimated Parameters of Relativistic Radio Jets

The influence of aberration on the observational parameters of radio jets and estimates of their physical properties is studied. Aberration distorts the apparent shapes of radio sources. Two identical relativistic jets (whose spectra have maxima) moving approximately along the line of sight could be observed as a compact GPS radio source (jet) and an extended source with a power-law spectrum (counterjet). The apparent flux densities, shapes, and spectra of relativistic radio jets are distorted even when the jets lie in the plane of the sky (across the line of sight). Exact formulas are derived for the estimated physical parameters of relativistic radio jets, taking into account aberration.     

Seismic hazard assessment under complex source size distribution of mining-induced seismicity

It is well-documented that a variety of factors controlling the rockmass fracturing process in mines often results in a complexity of mining event size distribution. In such cases, the estimation of the probability functions of source size parameterizations, with the use of presently known distribution models, brings about an unacceptable and systematic over- or underestimation of the seismic hazard parameters. It is, therefore, recommended that the non-parametric, kernel estimators of the event size distribution functions, be applied to stationary hazard studies in mining seismicity.These data-driven estimators, adapted to seismic source size characterization, accurately fit all kinds of data underlying distributions, regardless of their complexity. Recently, the non-parametric approach to size characterization was supported by a special method of uncertainty analysis based on resampling techniques. At present, it is a fully developed method, which provides point and interval estimates of size distribution functions and related hazard parameters. Two examples of its use in studying mining seismic data are presented and discussed in this paper. The analyzed data sets were recorded in two different copper mines in Poland. The smoothed bootstrap test for multimodality, which is a specialized tool for investigating the shapes of probability densities, provided highly significant proof that in both cases the probability densities of source size parameterization were complex thus implied the superiority of the non-parametric estimation to the classic, model-based approach in the studied cases. The data were then used to construct non-parametric, kernel estimates of the source size cumulative distribution function (CDF), the exceedance probability and the mean return period. Furthermore, confidence intervals for these quantities were also estimated. The intervals for CDF were narrow, showing that the procedures of non-parametric estimation and resampling based uncertainty analysis were precise. Due to the fact that the mean return period is very sensitive to values of the CDF, in particular for larger events sizes, the uncertainty of the return period estimates was not insignificant but remained manageable. The point and interval estimates of source size CDF and hazard parameters so obtained were compared with the respective point estimates achieved from the inappropriate in the case of complex magnitude distributions, model-based approach.     

Crystallization conditions of the Wiborg rapakivi batholith, SE Finland: an evaluation of amphibole and biotite mineral chemistry

Summary The wiborgite and dark wiborgite rapakivi granite phases of the Wiborg batholith in southeastern Finland compose about 80% of the total batholith area. A new study of the dominant mafic silicate minerals, in comparison with mafic silicates from more evolved granite phases, hybridized granite and mafic magmatic enclaves provide insights into the overall petrogenesis of the Wiborg batholith. All of the mafic silicate minerals are iron-rich, reflective of the whole rock compositions. Biotite is annitic, calcic amphibole is ferro-edenite to hastingsite, and subsolidus Fe-Mg amphibole is found as accessory grunerite. Temperatures derived from amphibole-plagioclase thermometry suggest crystallization at about ∼ 740 °C. Pressure estimates derived from Al in amphibole barometry range between 2.5 and 5.4 kilobars. This is noticeably higher than the previous estimates of 1 kbar for the Wiborg batholith. Oxygen fugacity estimates from biotite suggest low fO₂ initial values and increase from FMQ to above NNO for late stage granite phases. Received February 29, 2000; revised version accepted December 27, 2000     

流体包裹体各相体积测定的现状和探索

流体包裹体研究是地质研究的重要手段之一，而流体包裹体各相的体积则是一个重要的参数。早期得不到体积数据，只能给出包裹体的二维面积。中国科学院地球化学的研究所包裹体实验室曾提出一条换算曲线，根据二维面积推测体积小，误差较大。     

Methods of Three-part Quantitative Assessments of Undiscovered Mineral Resources: Examples from Victoria,Australia

Quantitative mineral resource assessments following the 3-part form rely on grade and tonnage models and probabilistic estimates of the number of undiscovered deposits. Assessments completed in Victoria, Australia, indicate that undiscovered mineral resources can be effectively estimated using grade and tonnage sub-models constructed using only medium- and large-tonnage deposits. Numbers of undiscovered deposits can be estimated on the basis of expert judgement or entirely by statistical means. Appropriate mathematical aggregation of individual expert views, expressed at interactive expert workshops, provides robust estimates of the number of undiscovered deposits. Underestimation of uncertainty, which is common in expert judgement, can be compensated by the statistical modification of individual interval estimates. In this study, the linear opinion pool was used as a simple and robust method of mathematical aggregation of multiple expert estimates of the number of undiscovered deposits. A general regression model, which estimates numbers of undiscovered deposits based on the size of the geologically permissive area and the median deposit tonnage, provided results generally compatible with those based on expert judgement or local deposit density models.     

Three-dimensional imaging of a single fluid inclusion in sphalerite by nano X-ray tomography

This study first described three-dimension images of a fluid inclusion hosted within sphalerite based on transmission nano X-ray tomography. It shows multiple shapes of the fluid inclusion viewed from different angles. This fluid inclusion contains both vapor and liquid phases, with irregular, asymmetry shape and concaved boundary. A reconstructed three-dimension animation viewer makes the shapes clearly. This study offers a technique to penetrate inside the fluid inclusion and calculate the volume of the single fluid inclusion based on the 3D images and animation, which can be applied for the future single fluid inclusion compositional studies.     

Recharge characteristics of a phreatic aquifer as determined by storage accumulation

The cumulative storage accumulation curve (CSAC) is a tool for saturated-volume fluctuation (SVF) analysis of transient recharge to shallow phreatic aquifers discharging only to springs. The method assumes that little underflow or phreatic evapotranspiration occurs. The CSAC is a modified water-table hydrograph that distinguishes storage increase caused by recharge from loss due to springflow-induced recession. Required for the analysis are water-table fluctuations at a single representative location within the catchment of a single spring and either direct measurements or robust interpolations of springflows at different aquifer stages. The method employs empirical manipulation of head observations, varying spring catchment area to minimize CSAC water-level changes in late portions of long recessions. Results include volumetric estimates of recharge integrated over individual events and instantaneous rates of recharge to the water table, at the temporal resolution of the water-level sampling interval. The analysis may also yield physically realistic information on spring catchment and recharge focusing. In a test case in West Virginia, USA, recharge estimates by this technique were consistent with integrated springflow time series but greater than estimates based on potential evapotranspiration. Results give insight into dynamic recharge behavior over time as well as an indication of recharge catchment size. Electronic Publication     

Trace elements in the Allende meteorite—III. Coarse-grained inclusions revisited

New RNAA determinations of Ba, Sr, Zr, U, Re, Pd, Ag, Zn and Se and INAA measurements of Lu are added to published data for 21 other elements in the same suite of ten samples. On the average, 21 refractory elements are not significantly fractionated from one another. The mean of their enrichment factors relative to Cl chondrites is 17.5 ± 0.4, indicating that the high-temperature condensate inclusions represent 5.7 wt% of the total condensable matter. Os, Ir, Ru, Re and most of the W condensed in one or more refractory siderophile element alloys along with small fractions of the Pd, Co, Au and Ag. The bulk of the Eu and Sr condensed in solid solution in melilite. Sc, Zr, Hf, Ta, U and the remaining REE condensed in a phase whose abundance in the inclusions is negatively correlated with that of melilite, either diopside or one or more minor or trace phases, including perovskite. Ba condensed in a different phase, separately from all these elements. In individual inclusions, fractionations are common between elements which were carried in by different condensate phases. Smaller fractionations are also observed for elements which condensed together. These may be due to variable proportions of them in a common condensate phase in response to different nebular equilibration temperatures or to multiple condensate phases containing different proportions of these elements. Available evidence indicates that some trace elements no longer reside in the phases which carried them into the inclusions, indicating a post-accretion thermal event which redistributed some of them. From the minimal variation of the Zr/Hf ratio in the inclusions, the solar system ratio is estimated to be 29.6 ± 1.8. From the mean U content of the inclusions and estimates of the bulk terrestrial and lunar U abundances, the Earth and Moon are estimated to contain 21% and 22–30% high-temperature condensates, respectively.     

Late Weichselian environmental change in southern Sweden and Denmark

A synthesis is presented of the envronmental and climatic variations that are inferred to have occurred in southern Sweden (up to latitude 59°N) and Denmark during the Weichselian Late-glacial (14-9 ka BP). The chronology and characteristics of the main phases of deglaciation, sea-level change, periglacial activity, soil development, vegetation cover and climate change are summarised. A curve representing the main changes in temperature, including quantitative estimates based upon beetle data and using the ‘mutual climatic range’ method, is presented.     

Neoproterozoic Molar-tooth Structure and Constraint of Depositional Facies and Environment in the North China Platform in Jiangsu, Anhui and Liaoning, Eastern China

Molar tooth structure （MTS） represented by complex ptygmatical shapes is widely distributed in the Proterozoic of the world. MTS filled by fine, equant sparry calcite （or dolomite） displays an abrupt contact with hosting rocks, which are mainly composed of carbonaceous micrites and fine-grained carbonates with local silts and stormdominated deposits with graded, cross or wave beddings, numerous erosional surfaces and truncated and fills or guttered bases. Occurrence of MTS suggests a result of the constraint of sedimentary facies, and the storm-base in ramp settings is the maximum depth for the formation of MTS. Vertical succession of MTS-bearing carbonates shows a deposition stacked by high-frequency shallow subtidal and peritidal cycles. An individual cyclic MTS-bearing sequence is characterized by thinning, shallowing and dynamic decreasing-upward, and peritidal caps of purple red iron and organic carbonaceous sediments with more complicated shapes of MTS are common on the top of individual MTS-bearing sequences.