A self‐similar model for sedimentary rocks with application to the dielectric constant of fused glass beads

Geophysics ◽  
1981 ◽  
Vol 46 (5) ◽  
pp. 781-795 ◽  
Author(s):  
P. N. Sen ◽  
C. Scala ◽  
M. H. Cohen
Keyword(s):  
Dielectric Constant ◽  
External Field ◽  
Sedimentary Rocks ◽  
Glass Beads ◽  
The Self ◽  
Similar Model ◽  
Archie’S Law ◽  
Self Consistent ◽  
Self Similar ◽  
Coated Spheres

We develop a theory for dielectric response of water‐saturated rocks based on a realistic model of the pore space. The absence of a percolation threshold manifest in Archie’s law, porecasts, electron‐micrographs, and general theories of formation of detrital sedimentary rocks indicates that the pore spaces within such rocks remain interconnected to very low values of the porosity ϕ. In the simplest geometric model for which the conducting paths remain interconnected, each grain is envisioned to be coated with water. The dielectric constant of the assembly of water‐coated grains is obtained by a self‐consistent effective medium theory. In the dc limit, this gives Maxwell’s relation for conductivity σ of the rock [Formula: see text], where [Formula: see text] is the conductivity of water. In order to include the local environmental effects around a grain, a self‐similar model is generated by envisioning that each rock grain itself is coated with a skin made of other coated spheres; the coating at each level consists of other coated spheres. The self‐consistent complex dielectric constant [Formula: see text] is given in this model in terms of that of water [Formula: see text] and of rock [Formula: see text], by [Formula: see text] for spherical particles. This gives, in the dc limit, [Formula: see text]. For nonspherical particles, the exponent m in Archie’s law [Formula: see text] is greater than 3/2 for the plate‐like grains or cylinders with axis perpendicular to the external field and smaller than 3/2 for plates or cylindrical particles with axis parallel to the external field. Artificial rocks with a wide range of porosities were made from glass beads. We present data on the glass bead rocks for dc conductivity and the dielectric constant at 1.1 GHz. The data follow the conductivity and the dielectric responses given by the self‐similar model. The present theory fails to explain the salinity dependence of [Formula: see text] at lower frequencies.

Electrical properties of sedimentary rocks having interconnected water‐saturated pore spaces

Geophysics ◽  
2000 ◽  
Vol 65 (4) ◽  
pp. 1093-1097 ◽  
Author(s):  
Pham Duc Chinh
Keyword(s):  
Electrical Conductivity ◽  
Sedimentary Rocks ◽  
Open Water ◽  
Effective Medium ◽  
Multiphase Model ◽  
Archie’S Law ◽  
Effective Media ◽  
Self Similar ◽  
Model Components ◽  
Water Saturated

Permeable sedimentary rocks can often be modeled as an impermeable rock matrix cut by a system of an irregular system of interconnected, irregularly shaped, water‐saturated pore spaces. I represent this system by a multiphase effective medium that is compatible with Archie’s Law for electrical conductivity. My effective medium is an extention of the self‐similar Sen, Scalar, and Cohen model which characterizes sedimentary rocks as a water suspension of spherical solid grains. My generalized multiphase model includes two important components: open water spherelike pockets, which significantly increase the porosity but add little to the electrical conductivity, and thin films surrounding the grains and water‐filled cracks, which contribute little to the porosity but significantly to the electrical conductivity. By perturbing the relative balance between these two model components, I am able to represent a range of aggregates for which I can construct effective media that are consistent with the electrical conductivity predicted by Archie’s Law.

Local Field Effects Near Surfaces of Small Systems

1994 ◽  
Vol 366 ◽  
Author(s):  
David Beaglehole
Keyword(s):  
Dielectric Constant ◽  
External Field ◽  
Electric Field ◽  
Local Field ◽  
Computer Calculation ◽  
Local Fields ◽  
Small Systems ◽  
Shape Dependence ◽  
Self Consistent ◽  
Reflection Of Light

ABSTRACTThe interaction of light with a system of molecules depends upon the polarisation induced by an external electric field, which depends not only upon the external field but also upon the local fields due to neighboring polarised molecules. These local fields result in the traditional Clausius-Mossotti (CM) dielectric constant for a molecule deeply imbedded in a medium. Near the surface the local fields are altered, and the dielectric constant becomes anisotropic and dependent upon depth into the medium. The local fields are shape dependent in small systems and differ substantially from the CM value.A self-consistent computer calculation of the local fields has been implemented, and these effects will be shown using molecule positions and polarisabilities typical of liquids and crystals. The shape dependence of small systems, the reflection of light from liquids with fluctuating surfaces, and the effect of supporting substrates will be described.

scholarly journals A New Analytical Jet Model

1996 ◽  
Vol 175 ◽  
pp. 459-460
Author(s):  
C.R. Kaiser ◽  
P. Alexander
Keyword(s):  
Similar Model ◽  
Self Consistent ◽  
Self Similar

We present a self-consistent, self-similar model for classical radio double sources (FRIIs). This model depends only on quantities which can in principle be measured.

scholarly journals Evolution of X-Ray Clusters of Galaxies with Active Protogalaxies

1998 ◽  
Vol 188 ◽  
pp. 321-322
Author(s):  
H. Saga ◽  
S. Yachi ◽  
A. Habe
Keyword(s):  
Formation Process ◽  
The Self ◽  
Radiative Cooling ◽  
Clusters Of Galaxies ◽  
Similar Model ◽  
X Ray ◽  
Metal Abundance ◽  
Self Similar ◽  
The Difference ◽  
Intracluster Gas

We consider heating due to proto galaxies in the formation process of clusters of galaxies, since much metal is observed in the intracluster gas which must be ejected from protogalaxies vir strong galactic winds and the metal abundance in the intracluster gas correlates with the fraction of early type galaxies in clusters (Arnaud 1993). We also consider radiative cooling. From the difference between the observed LX - TX relation of X-ray cluster (Hatsukade 1989), LX ∝ T2.7-3.3X, and the prediction from the self-similar model (Kaiser 1986), LX ∝ T2X, it is pointed out that physical processes which are not taken into account in the self-similar model, e.g., effect of radiative cooling, and/or effect of proto galaxy heating, play a important role in the formation process (Evrard and Henry 1991, Kaiser 1991).

The effect of grain anisotropy on the electrical properties of sedimentary rocks

Geophysics ◽  
1982 ◽  
Vol 47 (2) ◽  
pp. 257-263 ◽  
Author(s):  
Kenneth S. Mendelson ◽  
Morrel H. Cohen
Keyword(s):  
Dielectric Constant ◽  
Electrical Properties ◽  
Sedimentary Rocks ◽  
Two Dimensions ◽  
Dc Conductivity ◽  
Three Dimensions ◽  
Complex Dielectric Constant ◽  
Archie’S Law ◽  
Principal Axes ◽  
Grain Distribution

We have examined the effect of particulate anisotropy on the electrical properties of sedimentary rocks by generalizing the treatment of Sen et al (1981) to the case of ellipsoidal grains with a distribution of orientations and depolarizing factors. Two distributions in orientation have been treated in detail—randomly oriented grains in three dimensions and grains with aligned principal axes in two dimensions. In the former case the conductivity is a scalar satisfying Archie’s law, [Formula: see text], with [Formula: see text] the conductivity of the pore fluid and ϕ the porosity. The exponent m has a minimum of 1.5 for spherical grains. The presence of highly oblate (disk shaped) grains raises m significantly. As long as grains with extremely large eccentricities (≳15) are not present, the exponent falls in the observed range [Formula: see text]. For aligned grains the conductivity is a tensor with principal values that satisfy a generalized Archie’s law of the form [Formula: see text], where [Formula: see text] is the jth principal value of the conductivity and [Formula: see text] can be expanded as a power series in ϕ with a constant leading term. For grain eccentricities in the range 0–0.95, the coefficients [Formula: see text] fall in the range 0.1–4. The exponent m has a minimum value of 2 for two dimensions, independent of grain shape, if all grains have the same shape, and it is larger for any distribution of grain shapes. If the distribution of grain shapes is chosen so that the rock is isotropic, m and a have the same values as for isotropic rock composed of grains with the same distribution of shapes but with random orientations. Since different distributions of grain orientation can lead to the same effective conductivity, it is clear that measurements of dc conductivity are not sufficient to determine the grain distribution. The model is also used to obtain the complex dielectric constant. If the dielectric constant of solid rock is small compared to the real part of the dielectric constant of water, the complex dielectric constant has the same dependence on porosity as the dc conductivity except at very small porosities.

scholarly journals On self consistent electrostatic fields in inhomogeneous plasmas for the hydrodynamic approximation

1985 ◽  
Vol 3 (1) ◽  
pp. 79-91 ◽  
Author(s):  
R. Dragila
Keyword(s):  
Characteristic Time ◽  
The Self ◽  
Characteristic Time Scale ◽  
Free Expansion ◽  
Hydrodynamic Approximation ◽  
Self Similar Solution ◽  
Self Consistent ◽  
Self Similar ◽  
Fluid Hydrodynamics ◽  
Two Fluid

The self consistent electrostatic field due to plasma inhomogeneities and associated with free expansion of a plasma into a vacuum is evaluated in the framework of two-fluid hydrodynamics and the corresponding generalized Rieman's self-similar solution found. The case of an adiabatic expansion is also discussed and the results obtained are applied to a laser produced plasma with a characteristic electron temperature of 1 keV and a characteristic time scale of 1 nsec.

scholarly journals Hausdorff measure and Assouad dimension of generic self-conformal IFS on the line

2020 ◽  
pp. 1-31
Author(s):  
Balázs Bárány ◽  
Károly Simon ◽  
István Kolossváry ◽  
Michał Rams
Keyword(s):  
Hausdorff Measure ◽  
Similar Case ◽  
Iterated Function Systems ◽  
The Self ◽  
Dimensional Hausdorff Measure ◽  
Assouad Dimension ◽  
Iterated Function ◽  
Self Similar ◽  
Weak Separation ◽  
Topological Sense

This paper considers self-conformal iterated function systems (IFSs) on the real line whose first level cylinders overlap. In the space of self-conformal IFSs, we show that generically (in topological sense) if the attractor of such a system has Hausdorff dimension less than 1 then it has zero appropriate dimensional Hausdorff measure and its Assouad dimension is equal to 1. Our main contribution is in showing that if the cylinders intersect then the IFS generically does not satisfy the weak separation property and hence, we may apply a recent result of Angelevska, Käenmäki and Troscheit. This phenomenon holds for transversal families (in particular for the translation family) typically, in the self-similar case, in both topological and in measure theoretical sense, and in the more general self-conformal case in the topological sense.

scholarly journals Onset of the Mach Reflection of Zel’dovich–von Neumann–Döring Detonations

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 314
Author(s):  
Tianyu Jing ◽  
Huilan Ren ◽  
Jian Li
Keyword(s):  
Hot Spot ◽  
Mach Reflection ◽  
Near Field ◽  
The Self ◽  
Small Scale ◽  
Mach Stem ◽  
Self Similarity ◽  
Von Neumann ◽  
Similarity Problem ◽  
Self Similar

The present study investigates the similarity problem associated with the onset of the Mach reflection of Zel’dovich–von Neumann–Döring (ZND) detonations in the near field. The results reveal that the self-similarity in the frozen-limit regime is strictly valid only within a small scale, i.e., of the order of the induction length. The Mach reflection becomes non-self-similar during the transition of the Mach stem from “frozen” to “reactive” by coupling with the reaction zone. The triple-point trajectory first rises from the self-similar result due to compressive waves generated by the “hot spot”, and then decays after establishment of the reactive Mach stem. It is also found, by removing the restriction, that the frozen limit can be extended to a much larger distance than expected. The obtained results elucidate the physical origin of the onset of Mach reflection with chemical reactions, which has previously been observed in both experiments and numerical simulations.

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