An Approximate Method for Determining Areal Sweep Efficiency and Flow Capacity in Formations with Anisotropic Permeability

10.2118/16-pa ◽  
1961 ◽  
Vol 1 (04) ◽  
pp. 277-286 ◽  
Author(s):  
M. Mortada ◽  
G.W. Nabor
Keyword(s):  
Principal Axis ◽  
Bedding Plane ◽  
Sweep Efficiency ◽  
Simple Method ◽  
Anisotropic Permeability ◽  
Preferred Direction ◽  
Flow Capacity ◽  
Principal Axes ◽  
Special Cases ◽  
Directional Permeability

Abstract The effects of anisotropic or directional permeability on the areal sweep efficiency and the flow capacity are examined. The paper points out the importance of taking directional permeability into consideration in planning a flood. It analyzes the two-dimensional flow pattern associated with the skewed line drive for a unit mobility ratio. The direct and staggered line drives are treated as special cases of the skewed line drive. Analytical expressions are developed for the areal sweep efficiency at breakthrough and the flow capacity. They are related to the spacing between like wells, the distance between a row of injectors and the nearest row of producers, and the degree of skewness of the line drive. The latter quantity is defined such that it is equal to zero for the direct line drive and equals one-half for the staggered line drive. The a real sweep efficiency and the flow capacity depend also on the orientation of the flood pattern with respect to the principal axis of anisotropy. The paper provides a simple method for determining the a real sweep efficiency and the flow capacity for a formation in which the permeability in the bedding plane is anisotropic. Introduction Directional or anisotropic permeability is manifested by the ability of the formation to conduct fluids more readily along certain preferred directions. This situation occurs in many producing formations and is usually attributed to depositional features in which the sand grains are oriented in a preferred direction. In some cases it results from the formation of a major and a minor fracture system. Directional permeability should be taken into account in many phases of the production and exploration activities. Recognizing its existence in the formation of interest and planning accordingly can lead to increased recovery and substantial savings. For instance, the areal sweep efficiency in a water flood depends to a great extent on the orientation of the flood pattern with respect to the principal axis of permeability. Anisotropic permeability is specified by the directions of its three principal axes and the permeability along each axis. The principal axes of permeability are mutually perpendicular. This paper deals with the areal sweep efficiency at breakthrough and the flow capacity for formations with anisotropic permeability. The flood pattern considered consists of alternate rows of injecting and producing wells. The rows of wells are parallel and form a developed, skewed line drive which is illustrated in Fig. 1. The staggered and direct line drives are treated as special cases of the skewed line drive.

Finding Principal Axes of Complex Plane Rigid Body with Rending-Image of MATLAB

2012 ◽  
Vol 490-495 ◽  
pp. 2156-2159
Author(s):  
Wu Gang Li
Keyword(s):  
Rigid Body ◽  
Complex Plane ◽  
Principal Axis ◽  
Flat Surface ◽  
Moment Of Inertia ◽  
Principal Axes ◽  
The Moment

In order to find the principal axes of inertia and calculate their moment of inertia to any plane homogeneous rigid body for calculating easily the moment of inertia to any axis of this rigid body, the principal axes could be found and their moment of inertia could be calculated automatically by using the reading-image of MATLAB to read the image messages about the flat surface of the rigid body and by the procedures which ware made according to the logic relation about the principal axis and the moment of inertia of the rigid body. Applying this method in a homogeneous cube, a result was acquired, error of which is small compared with the theoretical value. So this method is reliable, convenient and practical

scholarly journals Simple Method for Estimating the Contribution of Neighbouring n-beam Interactions to Two-beam Structure Factors

1988 ◽  
Vol 41 (3) ◽  
pp. 469
Author(s):  
HJ Juretschke ◽  
HK Wagenfeld
Keyword(s):  
Experimental Determination ◽  
Structure Factor ◽  
Beam Structure ◽  
Simple Method ◽  
Experimental Configuration ◽  
Structure Factors ◽  
Special Cases ◽  
Better Than

Unless special precautions are taken, the experimental determination of two-beam structure factors to better than 1 % may include contributions from neighbouring n-beam interactions. In any particular experimental configuration, corrections for such contributions are easily carried out using the modified two-beam structure factor formalism developed recently (Juretschke 1984), once the full indexing of the pertinent n-beam interactions is known. The method is illustrated for both weak and strong primary reflections and its applicability in special cases, as well as for less than perfect crystals, is discussed.

An analysis of the variation between dairy-sire progeny-groups for milk yield and fat content

1958 ◽  
Vol 1958 ◽  
pp. 19-29 ◽  
Author(s):  
Alan Robertson ◽  
S. S. Khishin
Keyword(s):  
Milk Yield ◽  
Comparison Method ◽  
Fat Content ◽  
Fat Percentage ◽  
Simple Method ◽  
Factors Affecting ◽  
Special Cases ◽  
The Mean ◽  
Milk Marketing ◽  
The Relationship

The past few years have seen the development in Great Britain of the ‘contemporary comparison’ method for evaluating progeny tests of dairy sires (Macarthur, 1954; Robertson, Stewart and Ashton 1956). The final overall figure attached to a sire is the mean difference between the yield of his daughters and that of other heifers milking in the same herd in the same year, with due regard for the numbers of animals in the two groups. Although it has some imperfections in special cases, this is probably the most informative simple method of evaluating a sire for yield and, fortunately, one which could be easily integrated with the existing recording system. The method has been turned into a simple routine in the Bureau of Records of the Milk Marketing Board and several thousand bulls have now been evaluated. In this paper, we shall be mostly concerned to use this material to investigate the heritabilities of milk yield and fat content and the relationship between the two in the different breeds. The information that we shall use consists, for each bull, of the mean contemporary comparison, with its effective ‘weight’, and the average fat percentage of the daughters. Before we deal with the observed results, we should go into rather more detail into the nature of these two figures and into the factors affecting them.

The Behavior of Naturally Fractured Reservoirs

1963 ◽  
Vol 3 (03) ◽  
pp. 245-255 ◽  
Author(s):  
J.E. Warren ◽  
P.J. Root
Keyword(s):  
Pore Volume ◽  
Fractured Reservoirs ◽  
Sedimentary Basins ◽  
Reservoir Rock ◽  
Absolute Permeability ◽  
Additional Parameter ◽  
Porous System ◽  
Flow Capacity ◽  
Principal Axes ◽  
Naturally Fractured

Abstract An idealized model has been developed for the purpose of studying the characteristic behavior of a permeable medium which contains regions which contribute significantly to the pore volume of the system hut contribute negligibly to the flow capacity; e.g., a naturally fractured or vugular reservoir. Unsteady-state flow in this model reservoir has been investigated analytically. The pressure build-up performance has been examined in some detail; and, a technique for analyzing the build-up data to evaluate the desired parameters has been suggested. The use of this approach in the interpretation of field data has been discussed. As a result of this study, the following general conclusions can be drawn:Two parameters are sufficient to characterize the deviation of the behavior of a medium with "double porosity" from that of a homogeneously porous medium.These parameters can be evaluated by the proper analysis of pressure build-up data obtained from adequately designed tests.Since the build-up curve associated with this type of porous system is similar to that obtained from a stratified reservoir, an unambiguous interpretation is not possible without additional information.Differencing methods which utilize pressure data from the final stages of a build-up test should be used with extreme caution. Introduction In order to plan a sound exploitation program or a successful secondary-recovery project, sufficient reliable information concerning the nature of the reservoir-fluid system must be available. Since it is evident that an adequate description of the reservoir rock is necessary if this condition is to be fulfilled, the present investigation was undertaken for the purpose of improving the fluid-flow characterization, based on normally available data, of a particular porous medium. DISCUSSION OF THE PROBLEM For many years it was widely assumed that, for the purpose of making engineering studies, two parameters were sufficient to describe the single-phase flow properties of a producing formation, i.e., the absolute permeability and the effective porosity. It later became evident that the concept of directional permeability was of more than academic interest; consequently, the degree of permeability anisotropy and the orientation of the principal axes of permeability were accepted as basic parameters governing reservoir performance. More recently, it was recognized that at least one additional parameter was required to depict the behavior of a porous system containing regions which contributed significantly to the pore volume but contributed negligibly to the flow capacity. Microscopically, these regions could be "dead-end" or "storage" pores or, macroscopically, they could be discrete volumes of low-permeability matrix rock combined with natural fissures in a reservoir. It is obvious that some provision for the inclusion of all the indicated parameters, as well as their spatial variations, must be made if a truly useful, conceptual model of a reservoir is to be developed. A dichotomy of the internal voids of reservoir rocks has been suggested. These two classes of porosity can be described as follows:Primary porosity is intergranular and controlled by deposition and lithification. It is highly interconnected and usually can be correlated with permeability since it is largely dependent on the geometry, size distribution and spatial distribution of the grains. The void systems of sands, sandstones and oolitic limestones are typical of this type.Secondary porosity is foramenular and is controlled by fracturing, jointing and/or solution in circulating water although it may be modified by infilling as a result of precipitation. It is not highly interconnected and usually cannot be correlated with permeability. Solution channels or vugular voids developed during weathering or burial in sedimentary basins are indigenous to carbonate rocks such as limestones or dolomites. Joints or fissures which occur in massive, extensive formations composed of shale, siltstone, schist, limestone or dolomite are generally vertical, and they are ascribed to tensional failure during mechanical deformation (the permeability associated with this type of void system is often anisotropic). SPEJ P. 245^

Analytical solutions of a spherical nanoinhomogeneity under far-field unidirectional loading based on Steigmann–Ogden surface model

2020 ◽  
Vol 25 (10) ◽  
pp. 1904-1923
Author(s):  
Youxue Ban ◽  
Changwen Mi
Keyword(s):  
Flexural Rigidity ◽  
Elastic Stiffness ◽  
Surface Model ◽  
Far Field ◽  
Stress Concentrations ◽  
Equilibrium Equations ◽  
Legendre Series ◽  
Simple Method ◽  
Shear Moduli ◽  
Special Cases

For a solid surface or interface that is subjected to transverse loading, the influence of its flexural resistibility to bending deformation becomes significant. A spherical inhomogeneity or void embedded in an infinite elastic medium under the application of nonhydrostatic loads represents a typical example. In this work, we consider the most fundamental loading of a far-field unidirectional tension. Analytical displacements and stresses are developed by the coupling of a Steigmann–Ogden surface mechanical model, the simple method of Boussinesq displacement potentials, the semi-inverse method of elasticity, and Legendre series representations of spherical harmonics. The problem is then solved by converting the equilibrium equations of displacement into a linear system with respect to the Legendre series coefficients. The developed solutions are general in the sense that they may reduce to their classical or Gurtin–Murdoch counterparts as special cases. Analytical expressions reveal that the derived solution depends on four dimensionless ratios from among surface material parameters, shear moduli ratio, and inhomogeneity or void radius. In particular, instead of depending on both flexural parameters in the moment–curvature relation, one fixed combination is sufficient to represent the surface flexural rigidity. This is in contrast with the influence of the in-plane elastic stiffness, in which both surface Lamé parameters matter. Parametric studies further demonstrate that, for metallic inhomogeneities or voids with radii between 10 nm and 100 nm, the effects of surface flexural rigidity on stress distributions and stress concentrations are significant.

On the Relation Between Object Shape and Grasping Kinematics

2004 ◽  
Vol 91 (6) ◽  
pp. 2598-2606 ◽  
Author(s):  
Raymond H. Cuijpers ◽  
Jeroen B. J. Smeets ◽  
Eli Brenner
Keyword(s):  
Principal Axis ◽  
Grip Aperture ◽  
Object Shape ◽  
Hand Orientation ◽  
Principal Axes ◽  
Movement Distance ◽  
Maximum Grip Aperture ◽  
The Many ◽  
Elliptical Cylinders ◽  
Maximum Grip

Despite the many studies on the visual control of grasping, little is known about how and when small variations in shape affect grasping kinematics. In the present study we asked subjects to grasp elliptical cylinders that were placed 30 and 60 cm in front of them. The cylinders' aspect ratio was varied systematically between 0.4 and 1.6, and their orientation was varied in steps of 30°. Subjects picked up all noncircular cylinders with a hand orientation that approximately coincided with one of the principal axes. The probability of selecting a given principal axis was the highest when its orientation was equal to the preferred orientation for picking up a circular cylinder at the same location. The maximum grip aperture was scaled to the length of the selected principal axis, but the maximum grip aperture was also larger when the length of the axis orthogonal to the grip axis was longer than that of the grip axis. The correlation between the grip aperture— or the hand orientation—at a given instant, and its final value, increased monotonically with the traversed distance. The final hand orientation could already be inferred from its value after 30% of the movement distance with a reliability that explains 50% of the variance. For the final grip aperture, this was only so after 80% of the movement distance. The results indicate that the perceived shape of the cylinder is used for selecting appropriate grasping locations before or early in the movement and that the grip aperture and orientation are gradually attuned to these locations during the movement.

Twins of Marcasite in Regular Disposition Upon Cubes of Pyrites

1891 ◽  
Vol 9 (43) ◽  
pp. 209-210 ◽  
Author(s):  
C. O. Trechmann
Keyword(s):  
Principal Axis ◽  
Vertical Axis ◽  
Principal Axes ◽  
The Subject ◽  
Diagonal Axis ◽  
Lateral Axis

The subject of this note derives interest less from its novelty than from the beautiful manner in which the intergrowth is exemplified in Nature. As Sadebeek has shown, the intergrowth of marcasite and pyrites takes place according to two laws.(1) The vertical axis of marcasite is parallel with a principal axis of pyrites; one of the two other principal axes of the latter being parallel with the combination edge m (110) ∞P : c (001) 0P of the former ; or, in other words, the vertical and one lateral axis (ab) of marcasite coincide with two principal axes of pyrites.(2) The vertical axis of marcasite is parallel with a principal axis of pyrites; the braehydiagonal axis is parallel with a diagonal axis of pyrites.

Directional-Permeability Assessment in Formations With Complex Pore Geometry With a New Nuclear-Magnetic-Resonance-Based Permeability Model

SPE Journal ◽  
2016 ◽  
Vol 21 (04) ◽  
pp. 1436-1449 ◽  
Author(s):  
Lu Chi ◽  
Zoya Heidari
Keyword(s):  
Nuclear Magnetic Resonance ◽  
Magnetic Resonance ◽  
Pore Geometry ◽  
Pore Scale ◽  
Rock Permeability ◽  
Permeability Model ◽  
Pore Connectivity ◽  
Anisotropic Permeability ◽  
Rock Samples ◽  
Directional Permeability

Summary This paper proposes a new method for directional-permeability assessment with nuclear-magnetic-resonance (NMR) measurements. Conventional techniques for permeability assessment from NMR measurements include empirical correlations such as SDR (Schlumberger-Doll-Research) and Coates models. However, carbonate rocks are known for lack of good correlations between pore-body-size and pore-throat-size, which makes it challenging and often unreliable to estimate permeability from NMR T2 (spin-spin relaxation time) distribution in carbonate formations with complex pore structure. It also was proposed that conventional permeability models can be improved by incorporating an estimated pore-connectivity factor. However, none of the previously introduced techniques reflects the anisotropic characteristics of rock permeability. The new NMR-based directional-permeability model, introduced in this paper, incorporates a directional pore-connectivity factor into a conventional NMR-based permeability model. We introduce two approaches to quantify the directional pore-network connectivity of rock samples with pore-scale images. The first approach calculates directional pore connectivity in 3D pore-scale images with a topological technique. The second approach combines image analysis and electrical formation factor. The new NMR-based permeability model enables assessment of rock permeability in any desired direction. We successfully calibrated and tested the introduced NMR-based permeability model on carbonate, sandstone, and sandpack samples with complex pore geometry or anisotropic permeability. The anisotropic permeability used for calibration and test purposes was obtained by the lattice Boltzmann method (LBM) simulations on microcomputed tomography (CT) images of rock samples. The comparison between the permeability estimates with our new NMR model and conventional NMR models (e.g., SDR and Coates models) demonstrated that the NMR-based directional-permeability model significantly improves assessment of rock permeability, by reflecting rock's anisotropic characteristics and minimizing calibration efforts. The outcomes of this research can significantly improve permeability assessment in complex carbonate reservoirs and anisotropic sandstone reservoirs, and can be extended further to organic-rich mudrock formations.

ELECTRIC DIPOLE OVER AN ANISOTROPIC AND INHOMOGENEOUS EARTH

Geophysics ◽  
1967 ◽  
Vol 32 (4) ◽  
pp. 652-667 ◽  
Author(s):  
Ajit Kumar Sinha ◽  
Prabhat K. Bhattacharya
Keyword(s):  
Electric Dipole ◽  
Principal Axis ◽  
Low Frequency ◽  
Transversely Isotropic ◽  
Vector Potentials ◽  
The Earth ◽  
Longitudinal Conductivity ◽  
Asymptotic Expressions ◽  
Special Cases ◽  
Horizontal Electric Dipole

The electromagnetic fields of a low‐frequency horizontal electric dipole placed over a two‐layer earth have been derived. The overburden is considered to be transversely isotropic with respect to the conductivity. The unequal principal axis of the conductivity tensor is normal to the layers. The substratum is taken as isotropic. The vector potentials at the surface of the earth have been evaluated and expressed in such a way that the fields may be easily calculated. We have considered the special cases of a perfectly conducting, a perfectly insulating, and a substratum whose conductivity is very close to the longitudinal conductivity of the overburden. Asymptotic expressions for the fields have also been calculated. All the results are given in terms of known functions, and some numerical results have been included.

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