A generalized Archie’s law for n phases

Geophysics ◽  
2010 ◽  
Vol 75 (6) ◽  
pp. E247-E265 ◽  
Author(s):  
Paul W. J. Glover
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Reservoir Rock ◽  
Archie’S Law ◽  
Phase Volume Fraction ◽  
General Law ◽  
Special Cases ◽  
The Matrix ◽  
Definition Of

Archie’s law has been the standard method for relating the conductivity of a clean reservoir rock to its porosity and the conductivity of its pore fluid for more than [Formula: see text]. However, it is applicable only when the matrix is nonconducting. A modified version that allows a conductive matrix was published in 2000. A generalized form of Archie’s law is studied for any number of phases for which the classical Archie’s law and modified Archie’s law for two phases are special cases. The generalized Archie’s law contains a phase conductivity, a phase volume fraction, and phase exponent for each of its [Formula: see text] phases. The connectedness of each of the phases is considered, and the principle of conservation of connectedness in a three-dimensional multiphase mixture is introduced. It is confirmed that the general law is formally the same as the classical Archie’s law and modified Archie’s law for one and two conducting phases, respectively. The classical second Archie’s law is compared with the generalized law, which leads to the definition of a saturation exponent for each phase. This process has enabled the derivation of relationships between the phase exponents and saturation exponents for each phase. The relationship between percolation theory and the generalized model is also considered. The generalized law is examined in detail for two and three phases and semiquantitatively for four phases. Unfortunately, the law in its most general form is very difficult to prove experimentally. Instead, numerical modeling in three dimensions is carried out to demonstrate that it behaves well for a system consisting of four interacting conducting phases.

Instrumentation For Quantitative Microscopy

1977 ◽  
Vol 35 ◽  
pp. 206-209
Author(s):  
B. Ralph ◽  
A.R. Jones
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Automatic Data ◽  
Phase Volume Fraction ◽  
Statistical Confidence ◽  
Resultant Data ◽  
Automatic Data Collection ◽  
Third Dimension ◽  
Evolution Of Microstructure

In all fields of microscopy there is an increasing interest in the quantification of microstructure. This interest may stem from a desire to establish quality control parameters or may have a more fundamental requirement involving the derivation of parameters which partially or completely define the three dimensional nature of the microstructure. This latter categorey of study may arise from an interest in the evolution of microstructure or from a desire to generate detailed property/microstructure relationships. In the more fundamental studies some convolution of two-dimensional data into the third dimension (stereological analysis) will be necessary.In some cases the two-dimensional data may be acquired relatively easily without recourse to automatic data collection and further, it may prove possible to perform the data reduction and analysis relatively easily. In such cases the only recourse to machines may well be in establishing the statistical confidence of the resultant data. Such relatively straightforward studies tend to result from acquiring data on the whole assemblage of features making up the microstructure. In this field data mode, when parameters such as phase volume fraction, mean size etc. are sought, the main case for resorting to automation is in order to perform repetitive analyses since each analysis is relatively easily performed.

Export Performance: A Conceptualization and Empirical Assessment

1998 ◽  
Vol 6 (3) ◽  
pp. 59-81 ◽  
Author(s):  
Aviv Shoham
Keyword(s):  
Export Performance ◽  
Three Dimensional ◽  
Operational Definition ◽  
Three Dimensions ◽  
Relative Importance ◽  
Empirical Assessment ◽  
Conceptual Definition ◽  
Export Marketing ◽  
Definition Of

Export performance has been a central construct in the study of export marketing. Several studies have attempted to identify organizational and managerial antecedents of export performance and assess the relative importance of these antecedents. However, despite these fruitful efforts, there is little agreement in the literature about a conceptual definition of export performance, as well as about its operational definition. This article follows up on two previous articles that dealt with export performance conceptually (Madsen 1987; Shoham 1991) and uses a similar, three-dimensional conceptualization of export performance. This conceptualization is explored empirically with data from 93 Israeli exporters. The data indicate that there are indeed three dimensions to export performance, though the structure of these dimensions is somewhat different from expectations.

Comparison of Two Approaches to Model Cure-Induced Microcracking in Three-Dimensional Woven Composites

2012 ◽  
Author(s):  
Igor Tsukrov ◽  
Michael Giovinazzo ◽  
Kateryna Vyshenska ◽  
Harun Bayraktar ◽  
Jon Goering ◽  
...  
Keyword(s):  
Finite Element ◽  
Residual Stresses ◽  
Volume Fraction ◽  
Three Dimensional ◽  
Woven Composites ◽  
Finite Element Models ◽  
The Matrix ◽  
3D Woven Composites ◽  
The Difference ◽  
Resin Curing

Finite element models of 3D woven composites are developed to predict possible microcracking of the matrix during curing. A specific ply-to-ply weave architecture for carbon fiber reinforced epoxy is chosen as a benchmark case. Two approaches to defining the geometry of reinforcement are considered. One is based on the nominal description of composite, and the second involves fabric mechanics simulations. Finite element models utilizing these approaches are used to calculate the overall elastic properties of the composite, and predict residual stresses due to resin curing. It is shown that for the same volume fraction of reinforcement, the difference in the predicted overall in-plane stiffness is on the order of 10%. Numerical model utilizing the fabric mechanics simulations predicts lower level of residual stresses due to curing, as compared to nominal geometry models.

A Methodology for Synthesizing High-Performance Robots Fabricated With Optimally Tailored Composite Laminates

1987 ◽  
Vol 109 (1) ◽  
pp. 74-86 ◽  
Author(s):  
C. K. Sung ◽  
B. S. Thompson
Keyword(s):  
Composite Laminates ◽  
High Performance ◽  
Volume Fraction ◽  
Three Dimensional ◽  
High Strength ◽  
Fiber Volume ◽  
Cross Sectional ◽  
Fiber Properties ◽  
Robot Arms ◽  
The Matrix

An essential ingredient of the next generation of robotic manipulators will be high-strength lightweight arms which promise high-performance characteristics. Currently, a design methodology for optimally synthesizing these essential robotic components does not exist. Herein, an approach is developed for addressing this void in the technology-base by integrating state-of-the-art techniques in both the science of composite materials and also the science of flexible robotic systems. This approach is based on the proposition that optimal performance can be achieved by fabricating robot arms with optimal cross-sectional geometries fabricated with optimally tailored composite laminates. A methodology is developed herein which synthesizes the manufacturing specification for laminates which are specifically tailored for robotic applications in which both high-strength, high-stiffness robot arms are required which also possess high material damping. The parameters in the manufacturing specification include the fiber-volume fraction, the matrix properties, the fiber properties, the ply layups, the stacking sequence and the ply thicknesses. This capability is then integrated within a finite-element methodology for analyzing the dynamic response of flexible robots. An illustrative example demonstrates the approach by simulating the three-dimensional elastodynamic response of a robot subjected to a prescribed spatial maneuver.

scholarly journals Investigation of microvascular morphological measures for skeletal muscle tissue oxygenation by image-based modelling in three dimensions

2017 ◽  
Vol 14 (135) ◽  
pp. 20170635 ◽  
Author(s):  
B. Zeller-Plumhoff ◽  
K. R. Daly ◽  
G. F. Clough ◽  
P. Schneider ◽  
T. Roose
Keyword(s):  
Skeletal Muscle ◽  
Muscle Tissue ◽  
Cross Sections ◽  
Tissue Oxygenation ◽  
Volume Fraction ◽  
Three Dimensional ◽  
Capillary Density ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Contrast Imaging

The supply of oxygen in sufficient quantity is vital for the correct functioning of all organs in the human body, especially for skeletal muscle during exercise. Traditionally, microvascular oxygen supply capability is assessed by the analysis of morphological measures on transverse cross-sections of muscle, e.g. capillary density or capillary-to-fibre ratio. In this work, we investigate the relationship between microvascular structure and muscle tissue oxygenation in mice. Phase contrast imaging was performed using synchrotron radiation computed tomography (SR CT) to visualize red blood cells (RBCs) within the microvasculature in mouse soleus muscle. Image-based mathematical modelling of the oxygen diffusion from the RBCs into the muscle tissue was subsequently performed, as well as a morphometric analysis of the microvasculature. The mean tissue oxygenation was then compared with the morphological measures of the microvasculature. RBC volume fraction and spacing (mean distance of any point in tissue to the closest RBC) emerged as the best predictors for muscle tissue oxygenation, followed by length density (summed RBC length over muscle volume). The two-dimensional measures of capillary density and capillary-to-fibre ratio ranked last. We, therefore, conclude that, in order to assess the states of health of muscle tissue, it is advisable to rely on three-dimensional morphological measures rather than on the traditional two-dimensional measures.

scholarly journals Asteroidal Motion at Commensurabilities Treated in Three Dimensions

1979 ◽  
Vol 81 ◽  
pp. 207-215 ◽  
Author(s):  
Joachim Schubart
Keyword(s):  
Three Dimensional ◽  
Three Dimensions ◽  
Main Subject ◽  
Long Period ◽  
Period Effects ◽  
Special Cases ◽  
The Mean ◽  
Work Done ◽  
Asteroidal Motion

This paper consists of a review about work done on three-dimensional motion at commensurabilities of either the mean motions, or of secular periods, and of a report on the author's recent results on some special cases. Real and fictitious asteroidal orbits and the corresponding long-period effects are the main subject of interest. At first, methods are listed.

An Account of 4-Piece Mechanisms in Three Dimensions

1942 ◽  
Vol 26 (268) ◽  
pp. 5-20
Author(s):  
R. H. Macmillan
Keyword(s):  
Three Dimensional ◽  
Three Dimensions ◽  
Practical Applications ◽  
Special Cases ◽  
Modern Textbook ◽  
Theoretical Standpoint

The following paper is an attempt to explain the action of a class of three-dimensional mechanisms, first from a theoretical standpoint and afterwards to suggest some practical applications. One or two rather more general mechanisms are investigated than have previously been studied and well-known mechanisms are deduced as special cases of these. There is no modern textbook or recent publication dealing with these mechanisms and their properties are not generally known.

Thermo-mechanical Effects on Fracture Growth and Apertures in Three-Dimensional Subsurface Fractured Rocks 

2021 ◽  
Author(s):  
Adriana Paluszny ◽  
Robin N. Thomas ◽  
M. Cristina Saceanu ◽  
Robert W. Zimmerman
Keyword(s):  
Heat Transfer ◽  
Fluid Flow ◽  
Three Dimensional ◽  
Growth Patterns ◽  
Three Dimensions ◽  
Interaction Patterns ◽  
Flow Equations ◽  
The Matrix ◽  
Model Equations ◽  
Fracture Growth

<p>A finite-element based, quasi-static growth algorithm models mixed mode concurrent fracture growth in three dimensions, leading to the formation of non-planar arrays and networks. To model the fully coupled THM model, equations describing mechanical deformation as well as heat transfer in the matrix and in the fractures are introduced in the formulation, simultaneously accounting for the effect of fluid flow and stress-strain response. This results in five separate, but two-way coupled model equations: a thermoporoelastic mechanical model; two fluid flow equations, one for the rock matrix and one for the fractures; two heat transfer equations, similarly for both the matrix and fractures. Fractures are represented explicitly as discrete surfaces embedded within a volumetric domain [1]. Growth is computed as a set of vectors that modify the geometry of a fracture by accruing new fracture surfaces in response to brittle deformation. Fracture tip stress intensity factors drive fracture growth. This growth methodology is validated against analytical solutions for fractures under compression and tension [2]. Thermal effects on the apertures and growth patterns will be presented. Isolated fracture geometries are compared with selected experimental results on brittle media. Accurate growth is demonstrated for domains discretised by refined and coarse volumetric meshes. Fracture and volume-based growth rates are shown to modify fracture interaction patterns. Two-dimensional cut-plane views of fracture networks show how fractures would appear on the surface of the studied volume.</p><p><strong>REFERENCES</strong></p><p>[1] N. Thomas, A. Paluszny and R. W. Zimmerman. Growth of three-dimensional fractures, arrays, and networks in brittle rocks under tension and compression. Computers and Geotechnics, 2020. doi: 10.1016/j.compgeo.2020.103447</p><p>[2] Paluszny and R. W. Zimmerman. Numerical fracture growth modeling using smooth surface geometric deformation. Eng. Fract. Mech., 108, 19-36, 2013. doi: 10.1016/j.engfracmech.2013.04.012</p>

The effects of interfacial debonding on the elastoplastic response of unidirectional silicon carbide—titanium composites

2010 ◽  
Vol 224 (2) ◽  
pp. 259-269 ◽  
Author(s):  
M J Mahmoodi ◽  
M M Aghdam ◽  
M Shakeri
Keyword(s):  
Volume Fraction ◽  
Yield Criterion ◽  
Fibre Volume Fraction ◽  
Three Dimensional ◽  
Fibre Volume ◽  
Interfacial Debonding ◽  
Successive Approximation Method ◽  
Interface Damage ◽  
The Matrix ◽  
Von Mises

A three-dimensional micromechanics-based analytical model is presented to investigate the effects of initiation and propagation of interface damage on the elastoplastic behaviour of unidirectional SiC—Ti metal matrix composites (MMCs) subjected to off-axis loading. Temperature-dependent properties are considered for the matrix. Manufacturing process thermal residual stress (RS) is also included in the model. The selected representative volume element consists of r× c unit cells in which a quarter of the fibre is surrounded by matrix sub-cells. The constant compliance interface model is used to model interfacial debonding and the successive approximation method together with von Mises yield criterion is used to obtain elastoplastic behaviour. Dominance mode of damage including fibre fracture, interfacial debonding, and matrix yielding and ultimate tensile strength of the SiC—Ti MMC are predicted for various loading directions. The effects of thermal RS and fibre volume fraction on the stress—strain response of the SiC—Ti MMC are studied. Results revealed that for more realistic predictions, both interface damage and thermal RS effects should be considered in the analysis. The contribution of interfacial debonding and thermal RS in the overall behaviour of the material is also investigated. Comparison between results of the presented model shows very good agreement with the finite-element micromechanical analysis and experiment for various off-axis angles.

Analysis of Interfacial Debonding in Three-Dimensional Composite Microstructures

2005 ◽  
Vol 128 (1) ◽  
pp. 96-106 ◽  
Author(s):  
Shriram Swaminathan ◽  
N. J. Pagano ◽  
Somnath Ghosh
Keyword(s):  
Cohesive Zone ◽  
Volume Fraction ◽  
Three Dimensional ◽  
Interfacial Debonding ◽  
Fiber Model ◽  
Unidirectional Fibers ◽  
Fiber Spacing ◽  
Commercial Code ◽  
Initiation And Propagation ◽  
The Matrix

This paper is aimed at analyzing stresses and fiber-matrix interfacial debonding in three-dimensional composite microstructures. It incorporates a 3D cohesive zone interface model based element to simulate interfacial debonding in the commercial code ABAQUS. The validated element is used to examine the potential debonding response in the presence of fiber–fiber interactions. A two-fiber model with unidirectional fibers is constructed and the effect of relative fiber spacing and volume fraction on the stress distribution in the matrix is studied. In addition, the effect of fiber orientation and spacing on the nature of initiation and propagation of interfacial debonding is studied in a two-fiber model. These results are expected to be helpful in formulating future studies treating optimal fiber orientations and payoff in controlling fiber spacing and alignment.

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