A study of the modelling error in two operator splitting algorithms for porous media flow

Operator splitting methods are often used to solve convection–diffusion problems of convection dominated nature. However, it is well known that such methods can produce significant (splitting) errors in regions containing self sharpening fronts. To amend this shortcoming, corrected operator splitting methods have been developed. These approaches use the wave structure from the convection step to identify the splitting error. This error is then compensated for in the diffusion step. The main purpose of the present work is to illustrate the importance of the correction step in the context of an inverse problem. The inverse problem will consist of estimating the fractional flow function in a onedimensional saturation equation.     

Heterogeneous strain of a phyllite as revealed by porphyroblast-matrix relationships

Garnet porphyroblasts in a homogeneous phyllite specimen from the central Norwegian Caledonides provide insight into the distribution and type of strain which followed the garnet growth. Matrix schistosity is traceable through the porphyroblasts in the form of inclusions which allow the measurement of two parameters: the flattening of the matrix schistosity around the porphyroblasts, and an angle of rotation of the porphyroblasts relative to the matrix schistosity. Both parameters vary considerably as studied in a large thin section cut parallel to mineral lineation and perpendicular to schistosity. Accepting some simplifications and assumptions, it seems necessary to consider the strain as composed of two components: a noncoaxial strain component of a simple-shear type, and an approximately coaxial strain component, both of which are heterogeneously developed on the scale of the thin section. The shear planes of the simple-shear-type strain are likely to lie parallel, or at a very small angle, to the matrix schistosity. The linear fabric of the phyllite seems to be a material expression of the coaxial strain component. The λ₁λ₂ plane of the finite strain ellipsoid most probably varies in orientation across the thin section and is only, by coincidence, parallel to the almost constantly oriented matrix schistosity.     

Modelling in-situ upgrading of heavy oil using operator splitting method

The in-situ upgrading (ISU) of bitumen and oil shale is a very challenging process to model numerically because of the large number of components that need to be modelled using a system of equations that are both highly non-linear and strongly coupled. Operator splitting methods are one way of potentially improving computational performance. Each numerical operator in a process is modelled separately, allowing the best solution method to be used for the given numerical operator. A significant drawback to the approach is that decoupling the governing equations introduces an additional source of numerical error, known as the splitting error. The best splitting method for modelling a given process minimises the splitting error whilst improving computational performance compared to a fully implicit approach. Although operator splitting has been widely used for the modelling of reactive-transport problems, it has not yet been applied to the modelling of ISU. One reason is that it is not clear which operator splitting technique to use. Numerous such techniques are described in the literature and each leads to a different splitting error. While this error has been extensively analysed for linear operators for a wide range of methods, the results cannot be extended to general non-linear systems. It is therefore not clear which of these techniques is most appropriate for the modelling of ISU. In this paper, we investigate the application of various operator splitting techniques to the modelling of the ISU of bitumen and oil shale. The techniques were tested on a simplified model of the physical system in which a solid or heavy liquid component is decomposed by pyrolysis into lighter liquid and gas components. The operator splitting techniques examined include the sequential split operator (SSO), the Strang-Marchuk split operator (SMSO) and the iterative split operator (ISO). They were evaluated on various test cases by considering the evolution of the discretization error as a function of the time-step size compared with the results obtained from a fully implicit simulation. We observed that the error was least for a splitting scheme where the thermal conduction was performed first, followed by the chemical reaction step and finally the heat and mass convection operator (SSO-CKA). This method was then applied to a more realistic model of the ISU of bitumen with multiple components, and we were able to obtain a speed-up of between 3 and 5.     

A correction model for conditional bias in selective mining operations

A nonlinear correction functionK(Z*) is proposed to transform any initial linear grade estimatorZ* into a conditional unbiased estimatorZ**=K(Z*) with reduced conditional estimation variance. Such a corrected estimator allows more accurate prediction of ore reserves at any level of selection performed during the mine lifetime. The correction is based upon an analytical or isofactorial representation of a bivariate distribution model of true gradeZ and its estimatorZ*. This correction model allows derivation of conditional estimation variances for both estimatorsZ* andZ** and provides a solution to the problem of change of support. A case study is presented and performance of the proposed correction model is evaluated in terms of actual conditional bias and mean squared errors. Results obtained stress the practical importance of the correction model in selective mining operations.     

Counting error in petrographic point-count analysis: A theoretical and experimental study

The binomial model, commonly used to estimate counting error in point-count analysis, misestimates this error when the observation points on a grid are positively or negatively correlated. A model, called the cell model, is proposed as an alternative to the binomial model for use in studies, especially with coarse-grained rocks, in which such correlation is known or thought to exist. In the new model the thin section is conceptually partitioned into a number of cells (six is recommended), and the assumption is made that the proportions in the individual cells are statistically independent and that their variance does not differ from cell to cell. Empirical relations obtained from a suite of 200 thin sections of limestones are in reasonable support of the prediction that large particle size adversely affects counting error estimates based on the binomial model.     

The development of slaty cleavage, fleurieu peninsula, south australia

In low grade, biotite-rich metasiltstones and slates from the western side of Fleurieu Peninsula, the slaty cleavage is defined by elongate deformed old biotites and by the coincident elongate dimensions and (001) planes of thin, well aligned new biotites. Histograms of frequency versus (001)-S₁ angle (S₁ defined by aligned thin muscovite used as the reference plane) were determined in thin section for both populations. The old biotites show a symmetrical bimodal distribution of (001) about S₁, with maxima at around 20° either side ofS₁. This distribution, together with the preserved intracrystalline strain, indicates that these old grains deformed largely by slip on (001) with some modification of grain boundaries by diffusive transfer. The new biotite grains are very well aligned (standard deviation 3.8°) and show no sign of mechanical deformation. They have not been mechanically rotated into alignment but must have nucleated and grown in a specific orientation. The proportion of new to old grains increases with metamorphic grade, causing a rapid strengthening of the crystallographic alignment. In slates and phyllites with similar microstructure, the mica fabric determined by X-ray texture goniometry cannot be used as a quantitative measure of the geometry and magnitude of the bulk strain, as the intensity of the crystallographic fabric will be strongly influenced by the proportion of new mica, which is itself greatly affected by the metamorphic grade during slaty cleavage formation.     

Efficient Cable Shovel Excavation in Surface Mines

The cable shovel is widely used in surface mining. High operating and ownership costs necessitate efficient use of the cable shovel. Operator practices have long been suspected to contribute towards the inefficient use of the shovel. Crowd arm and hoist rope speeds are key measures of operator practices. The objective of this work is to find the crowd arm and hoist rope speeds for optimal shovel performance for given initial conditions and material properties. Shovel kinematics and dynamic modeling, using shovel geometry and the simultaneous constraint method, respectively, have been employed to build models of the excavation process. Dynamic models of the shovel payload and the material cutting resistance have also been developed using geometric simulation and passive soil pressures techniques, respectively. These models are solved numerically by combining Runge–Kutta and Gaussian elimination algorithms to compute the work done and the resistive forces during shovel excavation. The algorithms have been combined into a shovel simulator. The simulator has been used to simulate the P&H 2100BL shovel. The simulation results indicate that input energy and digging time increase with increasing crowd arm and decreasing hoist rope speeds. The input energy per unit loading rate is proposed as an appropriate measure of shovel performance. High energy per unit loading rate occurs for high crowd speeds and low hoist rope speeds. For the simulated conditions and crowd arm and hoist rope speeds ranging from 0.25 to 0.5 ms⁻¹ and 0.5 to 0.7 ms⁻¹, respectively, the optimal crowd arm and hoist rope speeds were found to be 0.25 ms⁻¹ and 0.7 ms⁻¹, respectively, and the objective function value was 0.21 KJs/kg. This work establishes, theoretically, the fact that operator practices have an effect on shovel performance and is useful in establishing optimum practices. The results are the initial steps towards full automation of the excavation process.     

Extrapolation of O-cell drilled shaft tests for load and resistance factor design (LRFD) calibration

Load tests of drilled shafts are often performed using Osterberg cell (O-cell) testing, a popular load test method for drilled shafts, which measures both side and tip resistance. However, it is common that only one of the resistance components can be fully mobilized. Therefore, extrapolation of the partially mobilized resistance is often required to determine the total resistance or the equivalent top-down curve. The extrapolation tends to introduce errors to the constructed total resistance values, which subsequently affect the calibrated resistance factors required for the LRFD design of drilled shafts. In this study, eight O-cell tests of drilled shafts with total measured resistances close to the failure criteria defined by FHWA, 5% of the shaft diameter (B), were collected among 64 drilled shaft load tests from Louisiana and Mississippi. For each of the eight cases, extrapolation was performed on both tip and side movement curves for the construction of the equivalent top-down load-settlement (ELT) curves. Data points from the measured side or tip movement curve were removed systematically to create a total of 80 cases with partially mobilized movement curves, and extrapolation exercises were performed on each fabricated case to obtain its equivalent top-down curve. The error of bias for each fabricated case was determined for statistical analyses. Multiple linear regression analysis was performed on the bias errors to model the bias errors caused by extrapolation. Calibrated resistance factors were determined and compared between the original database and fabricated database needing extrapolation. A correction method is proposed, based on a linear regression relationship, to estimate and minimize the extrapolation error of bias for less mobilized databases.

     

Laser Ablation (U-Th-Sm)/He Dating of Zircons: Analytical Age Bias and its Consequence for Studying Detrital Zircon Populations

The in situ (U-Th-Sm)/He and U/Pb laser-ablation double-dating procedure is a valuable method that can provide a large dataset relatively efficiently in contrast with conventional bulk helium thermochronometry. In this study, we evaluate the potential age error associated with the double ablation procedure and report the in situ (U-Th-Sm)/He double-ablation dating of 249 zircons from the Fish Canyon Tuff locality. With LA-ICP-MS pseudo-depth profiling and 3D numerical modelling, we show that the concentric double-ablation procedure in minerals with U-Th-Sm zoning can generate a significant (U-Th-Sm)/He age error (positive or negative), resulting in over-scattering and/or an offset of the mean age. Pseudo-depth profiling is insufficient to predict the individual age error, partly because of the superimposed ablations. To evaluate the consequence of this inherent bias, we confront a synthetic age distribution to the error expected for U-Th-Sm zoned zircons analysed with double-ablation (U-Th-Sm)/He thermochronometry. As expected, a strong age bias causes the spreading of peak ages, downgrading the original signal. Yet, the throughput of the ablation-based method can allow intra- and inter-sample peak age identification and comparison, and the coupling of (U-Th-Sm)/He and U/Pb ages extends our ability to deconvolute a multimodal age spectrum.     

The Second-Order Stationary Universal Kriging Model Revisited

Universal kriging originally was developed for problems of spatial interpolation if a drift seemed to be justified to model the experimental data. But its use has been questioned in relation to the bias of the estimated underlying variogram (variogram of the residuals), and furthermore universal kriging came to be considered an old-fashioned method after the theory of intrinsic random functions was developed. In this paper the model is reexamined together with methods for handling problems in the inference of parameters. The efficiency of the inference of covariance parameters is shown in terms of bias, variance, and mean square error of the sampling distribution obtained by Monte Carlo simulation for three different estimators (maximum likelihood, bias corrected maximum likelihood, and restricted maximum likelihood). It is shown that unbiased estimates for the covariance parameters may be obtained but if the number of samples is small there can be no guarantee of good estimates (estimates close to the true value) because the sampling variance usually is large. This problem is not specific to the universal kriging model but rather arises in any model where parameters are inferred from experimental data. The validity of the estimates may be evaluated statistically as a risk function as is shown in this paper.     

Inner product matrices, kriging, and nonparametric estimation of variogram

Two important problems in the practical implementation of kriging are: (1) estimation of the variogram, and (2) estimation of the prediction error. In this paper, a nonparametric estimator of the variogram to circumvent the problem of the precise choice of a variogram model is proposed. Using orthogonal decomposition of the kriging predictor and the prediction error, a method for selecting, what may be considered, a statistical neighborhood is suggested. The prediction error estimates based on this scheme, in fact, reflects the true prediction error, thus leading to proper coverage for the corresponding prediction interval. By simulations and a reanalysis of published data, it is shown that the proposals made in this paper are useful in practice.     

火山岩相的岩屑识别——以松辽盆地南部东岭探区为例

以松辽盆地南部东岭探区为例,建立了用火山岩钻井岩屑来确定火山岩亚相的方法和工作流程。首先辨别真假岩屑,然后用真岩屑制片并进行薄片鉴定。根据火山岩显微镜下的岩性、结构和构造特征与火山岩岩相和亚相的对应关系确定其所对应的火山岩岩相和亚相。岩屑法对于厚度在一个采样间距（1 m）以上的火山岩层段可以准确定出火山岩亚相,与岩心鉴定结果的符合率大于90％。此外,该方法亦可校正录井岩性。用该方法对东岭地区15口探井火山岩段的录井岩性岩相进行了识别校正,提高了井间岩性岩相和储层对比的精度。     

Micropaleontological study (foraminifera, ostracods) and characterization of the paleoenvironment of middle Cretaceous deposits (Djebel Chemla area, north-eastern Algeria)

The Djebel Chemla section is situated in the north-east of Algeria, close to the town of Tébessa and just a few kilometers from the Tunisian border. This section is composed mainly of a sequence of marl and marly limestone levels, alternating with recurring oyster levels which range in age from Vraconnian to the Early Turonian. Firstly, the micropaleontological study of the content (counting of foraminifera and ostracods, thin sections) allows the subdivision of the section into four biozones, with Rotalipora appenninica, Rotalipora globotruncanoides, Rotalipora cushmani, and Whiteinella archaeocretacea as bio-indicators; and, secondly, it provides the opportunity to specify the paleoenvironment. The supposed depositional environment is rather calm and deep, of an external platform type, with a maximum depth of 200 m. Bottom waters seem to be commonly poorly oxygenated. The end-Cenomanian anoxic crisis was also characterized by the presence of Bahloul facies levels, rich in TOC, in filaments and of globulous planktic foraminifera, including a considerable proportion of Heterohelix. The genus Rotalipora has disappeared a few levels lower down.     

New biostratigraphic data for the Chikkim Formation (Cretaceous, Tethyan Himalaya, India)

The Chikkim Formation as exposed in the Tethyan Himalaya (India) has been studied at its type locality, using planktonic foraminifera for a detailed biostratigraphic elaboration. Divided into two members, the Lower and Upper Chikkim members, this formation ranges in age from Albian to early Maastrichtian(?), and reaches a maximum thickness of 150 m. Examination of thin sections has yielded 34 species of foraminifera in five genus-level assemblages. The Lower Chikkim Member is about 55 m thick; its basal portion is of Albian age based on the presence of Biticinella breggiensis and Planomalina buxtorfi. At 26 m above the base, Whiteinella archaeocretacaea documents OAE 2 (Oceanic Anoxic Event 2), and thus the Cenomanian/Turonian boundary in this section. The carbonate sequence is capped by a Santonian-age hardground with iron oxide crusts and bioturbation. Macrofossils, including belemnites (at the base) and irregular echinoids (upper part), are present. The basal carbonaceous marls of the Upper Chikkim Member yield both large (benthic) rotaliid as well as planktonic foraminifera (Globotruncanita elevata, Gl. stuartiformis, Gl. stuarti, Gansserina gansseri and others), indicating a Campanian age. The co-occurrence of Gl. elevata and G. gansseri in a single thin section results either from condensation or reworking in the basal part of the Upper Chikkim Member. Late Cretaceous index foraminifera such as Gl. elevata document deposition within the Tethyan Realm. The original thickness of the Upper Chikkim Member is uncertain, but would have been around 100 m; the unit appears markedly reduced through weathering at a height of about 5000 m above sea level. Equivalent sediments are exposed in the Zanskar area to the northwest, and in Nepal and Tibet. Cretaceous Oceanic Red Beds (CORBs) are probably missing due to the proximality of these pelagic settings.     

The ultrafine component in Chinese loess and its variation over the past 7·6 Ma: implications for the history of pedogenesis

Grain‐size data from different climatic zones across the Chinese Loess Plateau show that the loess generally contains an ultrafine component, which has a consistent modal grain size of ca 0·37 μm and a variable proportion of 4 to 10%. The variation of the ultrafine component through a loess section is characterized by a high proportion and fine grain size in palaeosols, and by a low proportion and coarse grain size in loess layers. Its proportional content in a stratum roughly increases from north‐west to south‐east across the Loess Plateau. Quantitative X‐ray diffraction analysis indicates that the ultrafine component is composed mainly of clay minerals (ca 70%), which are mostly illite (ca 80%), and with significant amounts of kaolinite (ca 8%) and chlorite (ca 2%). The temporal and spatial variations of the ultrafine component, and the degree of illite crystallinity, suggest that this component in aeolian sediments is linked closely to the process of pedogenesis. Weakly altered loess on the north‐western margin of the Loess Plateau contains considerable amounts of detrital clay minerals derived from the aeolian source areas. Results from a loess section with a basal age of 7·6 Ma in the central Loess Plateau show that the ultrafine component increased from 7·6 to 5 Ma, and progressively decreased thereafter. This trend was punctuated by two abrupt changes at 2·6 and 0·6 Ma. These variations reflect to a considerable extent the history of pedogenesis during the Late Cenozoic.     

Secondary Ion Mass Spectrometry Methodology for Isotopic Ratio Measurement of Micro‐Grains in Thin Sections: True Grain Size Estimation and Deconvolution of Inter‐Grain Size Gradients and Intra‐Grain Radial Gradients

In a thin section, grains that were approximately spherical in situ appear circular in cross section, and the distribution of apparent diameters frequently assumed to be their size distribution. Scanning ion imaging by secondary ion mass spectrometry (SIMS) is capable of providing precise (< 1‰) stable isotope ratio measurements of such grains, but, importantly, also registers their rate of evolution in apparent size as they are ablated by the primary beam. By assessing rates of radius change with depth, the described methodology enables the ‘true’ size of grains to be estimated, as well as the distance of the sectioned surface from the original grain centre. Transects in three dimensions are made possible, and this capability enables better identification (and thus separation) of both inter‐grain chemical signatures as a function of grain size, and intra‐grain radial trends. In this example, we highlight the specific application to pyrite (FeS₂) minerals, which are frequently analysed by SIMS to determine their inter‐grain and intra‐grain geochemical variations, particularly in their sulfur stable isotopic ratios (δ³⁴S). Benefits of the new methodology over the Faraday cup ‘spot mode’ are described. Data correction algorithms and precision considerations are discussed.     

In situ U–Pb dating of zircon formed from retrograde garnet breakdown during decompression in Rogaland, SW Norway

Regionally metamorphosed metapelites from Rogaland, SW Norway, contain zircon formed during the decompression reaction garnet + sillimanite + quartz → cordierite. The zircon, which occurs as inclusions in cordierite coronas around garnet, is texturally, chemically and isotopically distinct from older zircon in other textural settings in the matrix. A SHRIMP U–Pb age of 955 ± 8 Ma based on analyses in thin section on the decompression zircon from the cordierite coronas, therefore dates a point on the retrograde path, estimated from garnet–cordierite equilibria to be 5.6 kbar, 710 °C. This population was under‐represented in conventional SHRIMP analyses of individual zircon in a mono‐mineralic grain mount and, in the absence of a textural context, its significance unknown. The dominant age identified from SHRIMP analyses of the grain mount, in combination with analyses from matrix zircon in thin section, was 1035 ± 9 Ma. Based on the lack of consistent textural relationships with any specific minerals in thin section, as well as rare earth element chemistry, the 1035‐Ma population is interpreted to represent zircon growth during incipient migmatization of the rocks at 6–8 kbar and c. 700 °C. This is consistent with previous estimates for the age of regional M₁ metamorphism during the Sveconorwegian Orogeny. The most important outcome of this study is the successful analysis of zircon grains in a specific, well‐constrained reaction texture. Not only does this allow a precise point on the regional P–T path to be dated, but it also emphasizes the possibility of zircon formation during the retrograde component of a typical metamorphic cycle.     

川东北宣汉-达县地区长兴组-飞仙关组礁滩储层特征及其控制因素

川东北宣汉-达县地区是我国重要的气藏发育区块。为了深入探索储层特征、物性与岩石类型,成岩作用的相关性,运用薄片鉴定、扫描电镜等技术手段,结合物性测试数据,测井资料和压汞数据,对储层的岩石类型,孔隙类型,孔隙结构、物性特征以及储层展布进行分析。研究表明,川东北宣汉-达县地区优质储层为生物礁白云岩及鲕粒白云岩,经历过重结晶及多期溶蚀,粒间溶孔最为发育。长兴组储层孔隙度平均值为5．4％。渗透率平均值为93．41×10^-3μm^2;飞仙关组储层平均孔隙度为8％,平均渗透率为132,76×10^-3μm^2,孔喉结构良好。沉积相控制储层的岩石类型和展布,成岩作用尤其是溶蚀作用控制着储层的物性,而构造作用对储层的形成则有双重影响。     

Quantification of microcrack anisotropy in quartzite — a comparison between experimentally undeformed and deformed samples

In this paper, microcrack patterns in a quartzite are quantified using fractal geometry based methods. Since the quartzite does not show a mesoscopic foliation, the fabric was recognized using anisotropy of magnetic susceptibility (AMS) analysis. Microcracks were investigated in thin sections prepared along the three principal planes of the AMS ellipsoid. Point load tests were performed on cores drilled parallel as well as perpendicular to the magnetic foliation. After experimental deformation, thin sections were prepared in two orientations — (a) parallel to the plane of failure (i.e., parallel to the direction of loading), (b) perpendicular to the plane of failure (i.e., perpendicular to the direction of loading), and microcrack patterns in these sections were investigated. The box-counting method of fractal analysis was first applied to microcracks traced from SEM images from each thin section of the experimentally undeformed as well as deformed samples to establish the fractal nature of the microcrack pattern. It was found that in thin sections perpendicular to the direction of loading, the box (fractal) dimension tends to marginally increase. This is inferred as a manifestation of the increase in complexity of the pattern. The software AMOCADO, which is based on the modified Cantor Dust method of fractal analysis, was applied to microcrack pattern from each thin section in order to quantify the pattern anisotropy. It is noted that the anisotropy significantly reduces in sections perpendicular to the loading direction. SEM data are presented to demonstrate that this reduction in anisotropy is on account of generation and/or growth of new cracks in random orientations. It is envisaged that the approach adopted in this investigation maybe useful in rock mechanics and mineral-resource applications in future.     

Quartz <Emphasis Type="Italic">c</Emphasis>-axis texture mapping of mylonitic metapelite with rod structures (Calabria,southern Italy): Clues for hidden shear flow direction

This quantitative microstructural study deals with textures of quartz domains within a mylonitized metapelite collected near a thrust surface corresponding to the tectonic contact between two metamorphic units, which crop out in the Aspromonte Massif, southern Calabria (Italy). The sample investigated lacks a mesoscopic stretching lineation. Therefore, quartz c-axis fabrics were investigated in two mutually orthogonal thin sections (a) parallel to the quartz rod lineation and perpendicular to the foliation (YZ plane) and (b) perpendicular to the quartz rods and perpendicular to the foliation (XZ plane); the data were generated using classical (manual measurements of quartz c-axis using U-stage) and modern methods (Computer Integrated Polarization microscopy). Both these sections show oblique foliations at ca. 40° from the main shear plane, implying that the actual X direction (stretching lineation that is absent on the mesoscopic scale) must lie between these two sections. Quartz c-axis data from the YZ section when rotated by 90° are similar with those from the XZ section. Hence, the data from the two sections are merged. These data when rotated by an angle of 50° from the direction of quartz rod lineation, gives an asymmetrical pattern indicating top-to-the-North sense of shear. This was confirmed by investigating quartz c-axis patterns in a section striking NS and perpendicular to the foliation. Based on the study it is thus concluded that this method can be used to do kinematic analysis in rocks that are devoid of stretching lineations. Apart from the above, the advantages and disadvantages of the classical and modern methods of quartz c-axis analysis are discussed.