Fault displacement modelling using 3D vector fields

In history matching and sensitivity analysis, flexibility in the structural modelling is of great importance. The ability to easily generate multiple realizations of the model will have impact both on the updating workflow in history matching and uncertainty studies based on Monte Carlo simulations. The main contribution to fault modelling by the work presented in this paper is a new algorithm for calculating a 3D displacement field applicable to a wide range of faults due to a flexible representation. This gives the possibility to apply this field to change the displacement and thereby moving horizons and fault lines. The fault is modelled by a parametric format where the fault has a reference plane defined by a centre point, dip and strike angles. The fault itself is represented as a surface defined by a function z = f(x,y), where x, y and z are coordinates local to the reference plane, with the z-axis being normal to the plane. The displacement associated with the fault outside the fault surface is described by a 3D vector field. The displacement on the fault surface can be found by identifying the intersection lines between horizons and the fault surface (fault lines), and using kriging techniques to fill in values at all points on the surface. Away from the fault surface the displacement field is defined by a monotonic decreasing function which ensures zero displacement at a specified distance from the fault. An algorithm is developed where the displacement can be increased or decreased according to user-defined parameters. This means that the whole displacement field is changed and points on horizons around the fault can be moved accordingly by applying the modified displacement field on them. The interaction between several faults influencing the same points is taken care of by truncation rules and the ordering of the faults. The method is demonstrated on a realistic synthetic case based on a real reservoir.     

The reliability of asymmetric c-axis fabrics of quartz to determine sense of vorticity

Asymmetric c-axis fabrics of quartz are commonly used to determine sense of vorticity in ductile shear zones. This method seems to work if the fabric pattern resembles a model fabric proposed by Lister and Hobbs (1980). Usually, however, c-axis fabrics are rather vague. The reliability of such vague fabrics was tested in a major shear zone with known sense of vorticity. Only 62% of the c-axis fabrics predict the correct sense. Great care should therefore be taken in applying this method to determine sense of vorticity.     

Analysis of triaxial ellipsoids: Their shapes, plane sections, and plane projections

A method is presented for the determination of a triaxial ellipsoid (such as a strain ellipsoid)from three nonparallel plane sections of the ellipsoid. The sections need be neither orthogonal nor central sections of the ellipsoid. Measurement errors are used to adjust the observed plane ellipses so that they are exact sections of the nearest true ellipsoid, whose dimensions and orientation are then found by solution of a system of six linear equations. A solution of the inverse problem is also presented: given a triaxial ellipsoid with known orientation, to determine the shape and orientation of the ellipse on a plane section. The problem is solved by expanding the equation of an ellipsoid with rotated coordinates, then setting one dimension to zero. Also, a method is presented for the projection of a triaxial ellipsoid onto a plane surface. This is solved by taking the derivative of the ellipsoid equation in the direction of the normal to the plane surface.     

Comparison of Fry strain ellipse and AMS ellipsoid trends to tectonic fabric trends in very low-strain sandstone of the Appalachian fold–thrust belt

In moderately to highly strained sandstones, both the long axis of the bedding-parallel finite-strain ellipse, as calculated by the normalized Fry method, and the projection of the long axis of the AMS ellipsoid on the plane of bedding, align well with local “structural grain” (trends of cleavage, folds, and faults). This relationship implies that results of both 2D Fry and AMS analyses represent the local layer-parallel tectonic strain component. Do both methods provide comparable results for very low-strain sandstone (e.g., <5%)? To address this question, Fry and AMS analyses were conducted in very low-strain sandstone from two localities in the Appalachian foreland fold–thrust belt: near Rosendale in New York and the Lackawanna synclinorium of Pennsylvania. We compared the map projections of both bedding-parallel Fry ellipses and AMS ellipsoids to the local structural grain. In both study areas, projections of the long axis of Fry strain ellipses do not cluster in a direction parallel to structural grain, whereas the projection of the long axes of AMS ellipsoids do cluster closely to structural grain. This observation implies that in very low-strain sandstone, AMS analysis provides a more sensitive “quick” indicator of tectonic fabric than does normalized Fry analysis.     

Average properties of ellipsoidal fabrics: Implications for two- and three-dimensional methods of strain analysis

Many rocks contain ellipsoidal objects (such as pebbles or reduction zones) which display a variety of shapes and orientations. In deformed rocks such objects may be used for strain analysis by using the concept of an average ellipsoid (here called the “fabric ellipsoid”). Two fabric ellipsoids are defined which are the results of two different algebraic averaging processes. During deformation of ellipsoidal distributions, the fabric ellipsoids change as if they were themselves material ellipsoids and are therefore of fundamental importance in strain analysis.In most studies to date, such 3-D fabric ellipsoids have been obtained from 2-D average ellipses determined on section planes cut through the rock sample. Previous work has assumed that the average ellipses will approximate to section through a single fabric ellipsoid. I show here that this is not the case as sectioning introduces a systematic bias into the section ellipse data. This bias is distinct from the statistical errors (due to finite sample size and measurement errors) discussed in other work and must be considered in any method of strain analysis using section planes.     

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.     

Quartz fabrics in shear-zone mylonites: Evidence for a major imprint due to late strain increments

Geometrical relations between quartz C-axis fabrics, textures, microstructures and macroscopic structural elements (foliation, lineation, folds…) in mylonitic shear zones suggest that the C-axis fabric mostly reflects the late-stage deformation history. Three examples of mylonitic thrust zones are presented: the Eastern Alps, where the direction of shearing inferred from the quartz fabric results from a late deformation oblique to the overall thrusting; the Caledonides nappes and the Himalayan Main Central Thrust zone, where, through a similar reasoning, the fabrics would also reflect late strain increments though the direction of shearing deduced from quartz fabric remains parallel to the overall thrusting direction. Hence, the sense of shear and the shear strain component deduced from the orientation of C-axis girdles relative to the finite strain ellipsoid axes are not simply related nor representative of the entire deformation history.     

Fabric development during shear deformation in the Main Central Thrust Zone, NW-Himalaya, India

Quartz microfabrics and associated microstructures have been studied on a crustal shear zone—the Main Central Thrust (MCT) of the Himalaya. Sampling has been done along six traverses across the MCT zone in the Kumaun and Garhwal sectors of the Indian Himalaya. The MCT is a moderately north-dipping shear zone formed as a result of the southward emplacement of a part of the deeply rooted crust (that now constitutes the Central Crystalline Zone of the Higher Himalaya) over the less metamorphosed sedimentary belt of the Lesser Himalaya. On the basis of quartz c- and a-axis fabric patterns, supported by the relevant microstructures within the MCT zone, two major kinematic domains have been distinguished. A noncoaxial deformation domain is indicated by the intensely deformed rocks in the vicinity of the MCT plane. This domain includes ductilely deformed and fine-grained mylonitic rocks which contain a strong stretching lineation and are composed of low-grade mineral assemblages (muscovite, chlorite and quartz). These rocks are characterized by highly asymmetric structures/microstructures and quartz c- and a-axis fabrics that indicate a top-to-the-south sense that is compatible with south-directed thrusting for the MCT zone. An apparently coaxial deformation domain, on the other hand, is indicated by the rocks occurring in a rather narrow belt fringing, and structurally above, the noncoaxial deformation domain. The rocks are highly feldspathic and coarse-grained gneisses and do not possess any common lineation trend and the effects of simple shear deformation are weak. The quartz c-axis fabrics are symmetrical with respect to foliation and lineation. Moreover, these rocks contain conjugate and mutually interfering shear bands, feldspar/quartz porphyroclasts with long axes parallel to the macrosopic foliation and the related structures/microstructures, suggesting deformation under an approximate coaxial strain path.On moving towards the MCT, the quartz c- and a-axis fabrics become progressively stronger. The c-axis fabric gradually changes from random to orthorhombic and then to monoclinic. In addition, the coaxial strain path gradually changes to the noncoaxial strain path. All this progressive evolution of quartz fabrics suggests more activation of the basal, rhomb and a slip systems at all structural levels across the MCT.     

The problem of the spatial variation of a spectral band ratio: a first order approach based on probability theory

Numerous efforts have been made to study how the spatial distribution of ground surface objects controls the image semivariogram. The present paper is centered on how the histograms and semivariograms of the individual bands x and y influence the spatial variation of a simple spectral ratio u = arctan(x/y). The image histogram of each separate band is described by a proper distribution. The exponential model is used to describe the semivariograms of x and y. Taking the first derivatives of the spectral ratio u for x and y and taking into account the mathematical behavior of the histograms of bands x and y, an approximate expression for the semivariogram γ _u of the spectral ratio is derived. This mathematical expression shows how the spatial variation of the spectral ratio depends on the standard deviations of the histograms, as well as the ranges of the semivariograms of x and y. Experimentation with multispectral images is then carried out and it shows that theoretical predictions agree, in qualitative terms, with real data. The results and conclusions of this paper may be useful in assessing the efficiency of various spectral band ratios and vegetation indices, which are often used in geological and environmental research (mapping of hydrothermal zones and land cover types).     

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.     

Molecular dynamics study of pressure-induced transformation of quartz-type GeO2

We simulated quartz-type GeO₂ and investigated its high-pressure transformation using the molecular dynamics (MD) simulation method with a model potential. The calculated results under hydrostatic compression indicated that a pressure-induced amorphization of quartz-type GeO₂ originated from the mechanical instability of the quartz lattice as, in previous theoretical studies of SiO₂. Furthermore, quartz-type GeO₂ directly transformed to a rutile-like structure with only subtle displacements of ions under σ _{x y} imposed shear stressed decompression. This is the first reproduction of the quartz-to-rutile transformation. A possible pathway of this transition is proposed in this study. Received: 14 April 1999 / Revised, accepted: 11 August 1999     

Solvothermal Synthesis of CdSxSe1-x Nanorods by Polymer Gel-Controlled Growth Strategy

Alloyed ternary CdSxSe1-x nanorods have been synthesized by the thermal treatment of Cd2 -dispersed polyethylene glycol 2 000 gel (PEG 2000) with ethylenediamine solution of sulfur and selenium in a sealed system at 180 ℃ for 24 h, in which the ratio of S to Se in the nanorods was controlled by adjusting the relative amounts of the starting materials. Based on the results of experiments, it is found that the CdSxSe1-x nanorods were also synthesized with several sulfur sources by the method. X-ray diffraction (XRD) shows that the alloyed ternary CdSxSe1-x nanorods are highly crystalline, and no other phase was observed in these nanorods.     

3D DEM simulation of principal stress rotation in different planes of cross-anisotropic granular materials

Three-dimensional Discrete Element Method simulations have been performed to study the deformation of cross-anisotropic granular materials under principal stress rotation (PSR), for rotation planes oriented at different angles θ with respect to the bedding plane. The simulations have been conducted with a novel technique for applying specified stresses at three-dimensional boundaries. The results are qualitatively in agreement with experimental results from literature. Cumulative volume contraction is always observed under continuous PSR and increases with increasing θ. The dilatancy rate decreases with increasing number of PSR cycles, tending to zero. The noncoaxiality angle between the strain increment and the stress in the PSR plane increases with increasing number of cycles, reaching the same asymptotic value for samples of various densities and for various θ. Periodic oscillations of the dilatancy rate and noncoaxiality angle within each PSR cycle are observed with an increasing oscillation magnitude with increasing θ, due to the larger fabric anisotropy within the PSR plane. When θ = 30 or 60°, significant noncoaxial strain accumulation occurs in the plane perpendicular to the PSR plane due to the oblique angle between the PSR plane and the bedding plane, echoing the major principal fabric direction's being neither parallel nor perpendicular to the PSR plane. The macroscopic behavior of the samples is related to the microscopic parameters including coordination number and fabric anisotropy. With increasing number of cycles, the difference between normalized stress/strain/fabric increment tensors tends to become constant, with only a small lag between each pair, irrespective of θ.     

Determination of fabric and strain ellipsoids from measured sectional ellipses—implementation and applications

Geologists examine fabrics to constrain models of formation or of deformation of rocks, and it is often convenient to summarise the results by a fabric ellipsoid. As fabric data are commonly collected on planar sections through the rock, estimating a fabric ellipsoid from sectional ellipses, often with arbitrary orientations, is an important task. An algebraic method to calculate such an ellipsoid, presented in an earlier paper, has been implemented with the program ellipsoid. It is used here on examples that illustrate questions and issues that arise when collecting, selecting and processing sectional fabric data, and when assessing the results. The quality of fit of the ellipsoid to the data is assessed in all cases. Examples include a case in which the average sizes of markers on individual sections can be used in the determination of the ellipsoid, and other cases in which such sizes are not useful; a case in which sectional ellipses are not obtained from closed markers; and a case in which data scatter and insufficient coverage of section orientations lead to a hyperboloid instead of an ellipsoid.     

Quartz c-axis fabric development and large-scale post-nappe folding (Wandfluhhorn Fold,Penninic nappes)

Quartz c-axis fabrics have been investigated within a suite of quartz veins and monomineralic layers around a major post-nappe fold hinge (the Wandfluhhorn Fold) in the Bosco area (Swiss-Italian border) within the lower Penninic nappes.Two kinematic domains which are separated by the axial plane trace of the Wandfluhhorn Fold are recognized; on the lower limb the measured quartz c-axis fabric asymmetry indicates a sense of shear in which the overlying layers move to the southwest (i.e. top-to-SW) whereas on the upper limb the shear sense is reversed with the top moving to northeast. The shear direction (N60°E–N80°E), however, is constant in both areas and oblique to an older stretching lineation as well as to the D₃ fold hinge. Such a distribution of asymmetric quartz c-axis fabrics and the constant orientation of their interpreted shear direction, which is apparent only from the fabric data and not from field evidence, indicates fabric development pre- or early syn-Wandfluhhorn folding, with subsequent folding and modification of the existing textures and possibly rotation of the initial fold axis.An overall westward-directed shear has been suggested for the whole of the Lepontine Alps. However, this study demonstrates that this simple general pattern has been modified locally by later folding. It also demonstrates that the dominant lineation may be a finite stretching lineation due to more than one phase of deformation and is not necessarily related to any particular transport direction.     

Some applications of the Mohr diagram for three-dimensional strain

The Mohr diagram for strain is rarely used in its full form, as a representation of three-dimensional strain. Recent attention has focused on various uses of the Mohr circle to express two-dimensional strain tensors. This contribution redescribes the Mohr diagram for three-dimensional strain and illustrates some new applications. The Mohr diagram for any strain ellipsoid provides an immediate method for ellipsoid shape classification. However, its greatest new potential is considered to be in the representation of strain ellipses as sections of ellipsoids.Any plane section of a strain ellipsoid can be plotted on the ellipsoid's Mohr diagram: it is here called a ‘Mohr locus’ because it is constructed as a locus of points representing the sheaf of lines which can be considered to define the plane. Mohr loci for sectional ellipses have a variety of forms, according to their orientation in the strain ellipsoid. Generally oblique sections are represented by loops bounded by the three principal circles. Their most leftward and rightward points are the plane's principal axes. Any Mohr locus can be transformed into a Mohr circle for the sectional ellipse.Mohr diagrams with Mohr loci have considerable potential as a graphical method of deriving best-fit strain ellipsoids from natural strain data. This is illustrated in three examples.     

Strain analysis using quartz deformation bands

Quartz deformation bands are kink bands in quartz crystals. A deformation band develops as a region of localized crystal-plastic deformation with boundaries perpendicular to the slip plane and slip direction, which usually is along an -axis in the basal plane. Under cross-polarized light, the difference in crystallographic orientation between a deformation band and its host is indicated by a difference in extinction positions. The displacement between the c axis in a deformation band and the c axis in the host represents the angular shear of the deformation band in the direction of the c axis in the host grain. Assuming the deformation is homogeneous at the grain scale, the angular shear of the grain (the gauge) is calculated by multiplying the angular shear of the deformation band by the ratio of the sheared part to the whole grain. Using the strain-gauge method for three-dimensional infinitesimal strain analysis, a minimum number of five grains measured on universal stage is needed to solve for the deviatoric strain components of the aggregate if the strain is homogeneous in the aggregate. Data from more than five grains are used to find the best-fit strain components by a least-squares method. The principal strains and their orientations are found from these strain components by calculating the eigenvalues and eigenvectors. A 3-D strain ellipsoid also is obtained from strain ellipses in three perpendicular planes determined from the two-dimensional flat-stage measurements by the Wellman method. Both the strain-gauge method and the Wellman method are tested by using synthetic data sets and applied to a naturally deformed sample. Both methods give similar results; the established Wellman method thus confirms the strain-gauge calculation.     

Smoothing and interpolation by kriging and with splines

Let scalar measurements at distinct points x₁, , x_n be y₁, , y_n.We may look for a smooth function f(x)that goes through or near the points (x_i, y_i).Kriging assumes f(x)is a random function with known (possibly estimable) covariance function (in the simplest case). Splines assume a definition of the smoothness of a nonrandom function f(x).An elementary explanation is given of the fact that spline approximations are special cases of the solution of a kriging problem.     

Rapid determination of lineament and joint densities

Lineament, joint and drainage density values are obtainable from the number of intersections between the mapped lines and a superimposed test grid or array, using the stereological equation where x is the density (length per area) and n′ is the number of intersections per unit length of the test array. Circular test grid must be used in order to eliminate the effect of preferred orientation on the n′-values.     

非等轴颗粒付林图解

非等轴颗粒付林图解由直线a=1、b=1、a=b和曲线a2b=1、ab2=1分割成10个微区,并对应于不同的e_x、e_y、e_z组合和应变类型。这些直线和曲线分别对应于特殊的e_x、e_y、e_z组合和岩石应变类型。岩石的应变类型有18种。函数a.b的表征方程有4种类型,并分别表征直线a=1/k₁和b=1/k₂所分割的4个区间。a,b表征方程的确定取决于岩石总的变形行为。此图解可用于非等轴颗粒标志体应变测量和应变类型的确定和图示,文章给出了应用实例.