Determination of the principal axes of elliptic cross sections in strain ellipsoids by a variational method

A method is presented for the determination of the shape and orientation of the ellipse on a central plane section across a triaxial ellipsoid (such as a strain ellipsoid) of known orientation and magnitude. Conditional extremal values of longitudinal strain (λ) are obtained in the plane section and these are the principal axes of the strain ellipse. In solving this variational problem, the arbitrary cosntants (two undetermined Lagrange multipliers) are found to have physical meaning in themselves as proved in a complementary geometric solution. The two roots of the first constant directly determine the magnitude of the principal axes of the strain ellipse in terms of the normal to the cross section. The two arbitrary constants together define shear strains and orientation of these axes.

Strain computation uses simple, single-line equations in either frames of strained or unstrained states.     

Light transmission method to explore the migration and distribution of Cr(VI) in a sandy aquifer

The migration and distribution of contaminants in heterogeneous aquifers are mostly investigated through numerical simulations. They are rarely examined through experimental simulations because of sampling difficulties. In this study, we used the light transmission method (LTM) to explore the migration and distribution of hexavalent chromium [Cr(VI)] in homogeneous sandy aquifers with or without a low-permeability bottom and lens (heterogeneous aquifer). Experimental phenomena were observed via images obtained with a high-power light source. The migration distances of pollution plume were also monitored. Under heterogeneous conditions, low-permeability aquifer materials considerably inhibited the migration of Cr(VI) and altered the Cr(VI) migration rate of horizontal and vertical plumes spread to a certain extent. Common horizontal and vertical migration curve equations in all three conditions were also obtained through distance–time curve fitting. The difference between high- and low-concentration zones was accurately revealed, and the experimental phenomenon was further explained. The proposed method could be used for the qualitative investigation and visualization of the migration and distribution of Cr(VI).     

多尺度非均质多孔介质中溶质运移的蒙特卡罗模拟

探讨了将蒙特卡罗(Monte Carlo)方法应用于多尺度非均质含水层中溶质运移模拟的方法。所研究的含水层由两种具有不同渗透系数统计特征的多孔介质所组成,每一种多孔介质是非均质的,且其渗透系数场符合平稳假设,而整个模拟区的渗透系数是非平稳的。Monte Carlo方法要求参数是平稳的,因此,分别对两种多孔介质产生若干随机渗透系数场后,用两种方法进行组合,并进行溶质运移的模拟计算。通过对计算结果的分析,综合考虑计算精度、计算时间等因素,得出了处理多尺度非均质多孔介质中溶质运移问题的较好方法。     

Pore‐scale modeling of fluid‐particles interaction and emerging poromechanical effects

A micro‐hydromechanical model for granular materials is presented. It combines the discrete element method for the modeling of the solid phase and a pore‐scale finite volume formulation for the flow of an incompressible pore fluid. The coupling equations are derived and contrasted against the equations of conventional poroelasticity. An analogy is found between the discrete element method pore‐scale finite volume coupling and Biot's theory in the limit case of incompressible phases. The simulation of an oedometer test validates the coupling scheme and demonstrates the ability of the model to capture strong poromechanical effects. A detailed analysis of microscale strain and stress confirms the analogy with poroelasticity. An immersed deposition problem is finally simulated and shows the potential of the method to handle phase transitions. Copyright © 2013 John Wiley & Sons, Ltd.     

用双椭圆方法确定反射点位置

在反射波地震勘探中使用动校正确定反射点位置用到很多假设,如勘探深度要远远大于炮检距;假定倾斜反射界面的反射点与水平反射界面反射点都位于炮检距的中点;近似认为倾斜反射界面的动校正量等于水平反射界面的动校正量;假定反射面倾角固定且较小等,上述假设一定会造成勘探误差。由于反射点位置和反射面倾角未知,所以理论上无法唯一地确定反射点位置。如果反射波传播的介质的波速一定,从炮点发出的地震波,经反射点后,在接收点被接收,其可能的反射点是椭圆的一部分,但还不能唯一确定反射点;再取炮点和另外一个接收点,其可能的反射点是另外一个椭圆的一部分。如果假定反射面是平面,可以是水平面,也可以是有固定倾角的倾斜平面,该平面在地震波射线平面内是一条直线,该直线一定是两椭圆的公切线。把两椭圆方程和切线方程联立,就可以求解出公切点位置,公切点位置就是反射点位置,这就是用双椭圆确定反射点位置的方法。通过建立模型对勘探方法进行了检验,证实了用双椭圆方法确定反射点位置的有效性。双椭圆方法有一个重要的副产品,就是在确定反射面位置的同时计算出反射面的视倾角。     

用双椭圆方法确定反射点位置

在反射波地震勘探中使用动校正确定反射点位置用到很多假设,如勘探深度要远远大于炮检距;假定倾斜反射界面的反射点与水平反射界面反射点都位于炮检距的中点;近似认为倾斜反射界面的动校正量等于水平反射界面的动校正量;假定反射面倾角固定且较小等,上述假设一定会造成勘探误差。由于反射点位置和反射面倾角未知,所以理论上无法唯一地确定反射点位置。如果反射波传播的介质的波速一定,从炮点发出的地震波,经反射点后,在接收点被接收,其可能的反射点是椭圆的一部分,但还不能唯一确定反射点;再取炮点和另外一个接收点,其可能的反射点是另外一个椭圆的一部分。如果假定反射面是平面,可以是水平面,也可以是有固定倾角的倾斜平面,该平面在地震波射线平面内是一条直线,该直线一定是两椭圆的公切线。把两椭圆方程和切线方程联立,就可以求解出公切点位置,公切点位置就是反射点位置,这就是用双椭圆确定反射点位置的方法。通过建立模型对勘探方法进行了检验,证实了用双椭圆方法确定反射点位置的有效性。双椭圆方法有一个重要的副产品,就是在确定反射面位置的同时计算出反射面的视倾角。     

Intersections of ellipsoids and planes of arbitrary orientation and position

This paper considers an ellipse, produced by the intersection of a triaxial ellipsoid and a plane (both arbitrarily oriented), and derives explicit expressions for its axis ratio and orientation. The relationship between central and noncentral sections, and the problem of intersection probability, are discussed. The resulting equations have obvious applications in structural geology and optical mineralogy, and in stereological modeling of grain-size and grain-shape distributions.     

A micromechanical study of rolling and sliding contacts in assemblies of oval granules

The evolution of the microstructure of an assembly of cohesionless granular materials with associated pores, which carry the overall applied stresses through frictional contacts is a complex phenomenon. The macroscopic flow of such materials take place by the virtue of the relative rolling and sliding of the grains on the micro‐scale. A new discrete element method for biaxial compression simulations of random assemblies of oval particles with mixed sizes is introduced. During the course of deformation, the new positions of the grains are determined by employing the static equilibrium equations. A key aspect of the method is that, it is formulated for ellipse cross‐sectional particles, hence desirable inherent anisotropies are possible. A robust algorithm for the determination of the contact points between neighbouring grains is given. Employing the present methodology, many aspects of the behaviour of two‐dimensional assemblies of oval cross‐sectional rods have been successfully addressed. The effects of initial void ratio, interparticle friction angle, aspect ratio, and bedding angle on the rolling and sliding contacts are examined. The distribution of normals to the rolling and sliding contacts have different patterns and are concentrated along directions, which are approximately perpendicular to one another. On the other hand, the distribution of all contact normals (combined rolling and sliding) are close to that of rolling contacts, which confirm that rolling is the dominant mechanism. This phenomenon becomes more pronounced for higher intergranular friction angle. Characteristics of the rolling and sliding contacts are also discussed in the context of the force angle, which is the inclination of contact force with respect to the contact normal. Copyright © 2003 John Wiley & Sons, Ltd.     

Least-squares center-to-center and mean object ellipse fabric analysis

The subjectivity of ellipse fitting in many strain techniques has hindered the determination of fabric anisotropy and tectonic strain. However, many sets of x, y co-ordinates can be approximated as an ellipse using a least-squares algorithm to calculate a best-fit ellipse and associated average radial error. For instance, the two dimensional shape of many objects can be approximated as an ellipse by entering digitized co-ordinates of the object margin into the ellipse algorithm.The rim of maximum point density in a normalized Fry diagram is defined by normalized center-to-center distances between touching or nearly touching objects. The enhanced normalized Fry (ENFry) method automates ellipse fitting by entering center-to-center distances between these “touching” objects into the least-squares ellipse algorithm. For homogeneously deformed populations of 200 objects, the ENFry method gives an accurate and precise measure of whole-rock fabric anisotropy, particularly for low ellipticities. When matrix strain exceeds clast strain, manual ellipse fitting of normalized Fry plots gives more accurate matrix anisotropies.The mean object ellipse (MOE) method calculates the best-fit ellipse from the geometry of the objects. Three points from the margin of each object ellipse, centered at the origin and expanded or reduced to unit volume, are used to calculate the best-fit fabric ellipse. The MOE method is very precise for small data sets, making it a good method for mapping heterogenous object strain. However, least-squares calculations maximize the influence of distal and spurious ellipticities, causing the MOE method to overestimate the fabric ellipticity of most aggregates.     

Numerical simulation of NAPL flow in the subsurface

A three-dimensional, three-phase numerical model is presented for simulating the movement of immiscible fluids, including nonaqueous-phase liquids (NAPLs), through porous media. The model is designed to simulate soil flume experiments and for practical application to a wide variety of contamination scenarios involving light or dense NAPLs in heterogeneous subsurface systems. The model is derived for the three-phase flow of water, NAPL, and air in porous media. The basic governing equations are based upon the mass conservation of the constitutents within the phases. The descretization chosen to transform the governing equations into the approximating equations, although logically regular, is very general. The approximating equations are a set of simultaneous coupled nonlinear equations which are solved by the Newton-Raphson method. The linear system solutions needed for the Newton-Raphson method are obtained using a matrix of preconditioner/accelerator iterative methods. Because of the special way the governing equations are implemented, the model is capable of simulating many of the phenomena considered necessary for the sucessful simulation of field problems including entry pressure phenomena, entrapment, and preferential flow paths. The model is verified by comparing it with several exact analytic test solutions and three soil flume experiments involving the introduction and movement of light nonaqueous-phase liquid (LNAPL) or dense nonaqueous-phase liquid (DNAPL) in heterogeneous sand containing a watertable. This revised version was published online in August 2006 with corrections to the Cover Date.     

Developments in the characterisation of waste materials for environmental impact assessment purposes

The characterisation of waste is entering a new phase with the latest developments of characterisation leaching tests and associated modelling capabilities. The currently applied too simple testing approaches lead to poor choices in waste management. With the increased insight in release controlling processes chemical speciation aspects can be addressed even in the most complex and heterogeneous waste materials. From a composition perspective materials may be highly variable, but often materials are far more consistent in their leaching behaviour. This aspect should be exploited more extensively as it holds the key to treat waste materials in such a way that long term solutions are achieved rather than reaching temporary gains.     

Average Depth versus Average Strike-Length of Fissure Veins for Specified Geometries: A Quantitative Approach

Fissure veins are very important source of mineral deposits particularly that of hydrothermal deposits. The average depth vs. average strike-length of fissure vein, which is an interesting topic, was discussed in Bateman (1962). Some say that depth is equal to length while others opined that depth may be equal to half the length. Here the question arises whether the ratio depth/ length which is loosely used refers to depth to length at any given point of time or average depth vs. average length on a statistical basis. Infact these are empirical rules. The empirical rules hold good for an average basis only. Such problems are dealt under geometric probability. The ore body is more like a lens. Infact none of these hold good. The authors developed a method statistically for ore bodies of two geometries namely a circle and an ellipse. In the case of ellipse, three cases are considered namely, (i) the major axis of ellipse is horizontal to X–axis (ii) the major axis of ellipse is parallel to Y axis and (iii) the ellipse is inclined and whose major axis makes an angle q with the X–axis. In any case, only circular ore body is considered in Bateman (1962). In the case of circular body it is shown that the ratio is neither equal to one or ½ as pointed in Bateman (1962).     

Particle paths, displacement and progressive strain applicable to rocks

If the particle paths are known for deforming continuous media such as rocks, the strain is determined at all stages of the deformation. The particle paths are studied for various types of simultaneous combinations of pure shear and simple shear. Any kind of progressive plane homogeneous strain can be expressed as a simultaneous superposition of pure shear and simple shear by selecting the proper ratio between the two strain rates and the proper angle between the slide direction of the simple-shear part and the principal axes of the pure-shear part. In the cases studied — except one — the angle between the slide direction of the simple-shear part and the principal strain rate of the pure-shear part is 45°. Several combinations of the simple-shear rate, γ, and the pure-shear rate, , are tested. These combinations give particle paths varying from sets of straight parallel lines to orthogonal hyperbolas. Distorted hyperbolas, ellipses and circles constitute the particle paths at intermediate ratios. From the particle-path equations — which are found by integration of the rate-of-deformation equations — the strain ellipse is readily determined at any stage of the deformation. One particularly intriguing result is the rotating and pulsating strain ellipse found in the cases when the particle paths are closed curves (ellipses). Application of the results to various fold-, thrust- and inclusion structures is suggested. In an appendix the treatment of rotational deformation as a superposition of irrttational strain and rigid rotation is considered for comparison.     

扩底桩承载力空间屈服包络面的基本特性

通过扩底桩的土槽载荷试验和离心机仿真模型试验,对不同类型荷载之间的相互影响、组合荷载作用下桩的抵抗机理和承载力屈服包络面的特性进行了分析。结果表明：铅直、水平荷载作用下桩的承载力屈服包络线,弯矩、水平荷载作用下的承载力屈服包络线都可近似地用椭圆来表现。屈服包络线可用桩的单方向极限承载力的经验公式来生成。根据承载力空间屈服面的连续性,提出了用桩的单方向极限承载力Vm,Hm,Um和Mm求解桩的承载力空间屈服面的经验公式。在桩的设计计算过程中,必须考虑铅直荷载、水平荷载及弯矩荷载间的相互影响。     

丹麦法稳健估计在隧道椭圆拟合中的应用

隧道断面在设计时为一标准圆形,受到施工及外部环境的影响变形为偏心率很小的椭圆。减小粗差影响、精确确定椭圆的几何参数是其研究的重点。通过对上海某一隧道断面的测量试验,得出用丹麦法稳健估计可以有效减小粗差的影响,而且拟合值可以很好地逼近无粗差情况下的最小二乘拟合值。采用Matlab 语言实现算法,并详细介绍了用二项式变换求解椭圆参数的方法。     

An oversampling technique for the multiscale finite volume method to simulate electromagnetic responses in the frequency domain

In order to reduce the computational cost of the simulation of electromagnetic responses in geophysical settings that involve highly heterogeneous media, we develop a multiscale finite volume method with oversampling for the quasi-static Maxwell’s equations in the frequency domain. We assume a coarse mesh nested within a fine mesh that accurately discretizes the problem. For each coarse cell, we independently solve a local version of the original Maxwell’s system subject to linear boundary conditions on an extended domain, which includes the coarse cell and a neighborhood of fine cells around it. The local Maxwell’s system is solved using the fine mesh contained in the extended domain and the mimetic finite volume method. Next, these local solutions (basis functions) together with a weak-continuity condition are used to construct a coarse-mesh version of the global problem. The basis functions can be used to obtain the fine-mesh details from the solution of the coarse-mesh problem. Our approach leads to a significant reduction in the size of the final system of equations and the computational time, while accurately approximating the behavior of the fine-mesh solutions. We demonstrate the performance of our method using two 3D synthetic models: one with a mineral deposit in a geologically complex medium and one with random isotropic heterogeneous media. Both models are discretized using an adaptive mesh refinement technique.     

Numerical analysis of the failure process around a circular opening in rock

Damage and fracture propagation around underground excavations are important issues in rock engineering. The analysis of quasi-brittle materials can be performed using constitutive laws based upon damage mechanics. The finite element code RFPA2D (Rock Failure Process Analysis) based on damage mechanics was used to simulate a loading-type failure process around an underground excavation (model tunnel) in brittle rock. One of the features of RFPA2D is the capability of modeling heterogeneous materials. In the current model, the effect of the homogeneous index (m) of rock on the failure modes of a model tunnel in rock was studied. In addition, by recording the number of damaged elements and the associated amount of energy released, RFPA2D is able to simulate acoustic activities around circular openings in rock. The results of a numerical simulation of a model tunnel were in very good agreement with the experimental test using the acoustic emission technique. Finally, the influence of the lateral confining pressure on the failure mechanism of the rock around the model tunnel was also investigated by numerical simulations.