Fractal Size Research Articles

Fractal Size and Spatial Distribution of Hydrocarbon Accumulations--Implications for Resource Assessment and Exploration Strategies: ABSTRACT

Fractal Size and Spatial Distribution of Hydrocarbon Accumulations--Implications for Resource Assessment and Exploration Strategies: ABSTRACT

Fractal formation in a-Si:H/Ag/a-Si:H films after annealing

The crystallization behavior of a-Si:H/Ag/a-Si:H sandwich films has been studied in detail. Fractals of Si caused by metal enhanced crystallization appear after annealing at 350–600 °C. The fractal dimension decreases (the Si fractals become more open) with the increasing annealing temperature. The number density of fractals increases at 350–450 °C and turns to decrease at 500–600 °C. The average fractal size increases from 350 to 550 °C and shows a decreasing tendency at 600 °C. The formation of fractals can be explained by a random successive nucleation and growth model. The x-ray energy dispersive spectroscopy (EDS) microanalysis indicates that although there is lateral interdiffusion of Ag and Si atoms, the thicknesses of the fractal region and the matrix remain nearly the same. At the same time, EDS shows that there are also Ag aggregates extending out of the films. It is suggested that besides the preferred nucleation at the Ag/Si interface the break of Si—H bonds may also stimulate the crystallization of a-Si:H so that the crystallization temperature of an a-Si:H/Ag system is much lower than that of an a-Si/Ag system.

Amount of extension on "small" faults: An example from the Viking graben

Extension estimates based on palinspastic restoration of only the largest faults in a deformed terrane may be significantly deficient due to neglect of faults (i.e., unsampled faults, most of which have smaller displacements than do sampled faults). Small faults individually account for little extension, but there may be large numbers of them. The cumulative extension on small faults can be calculated from the fractal size distributions of fault displacements. Independent extension estimates support the inference from fault population statistics that small faults account for significant amounts of extension. For example, restorations of Mesozoic faults interpreted from seismic reflection profiles across the Viking graben in the North Sea yield extension estimates that are only 40%-75% of independent estimates. The fractal size distribution of fault displacements in the Viking graben is consistent with the hypothesis that the remaining 25%-60% of extension resulted from displacements on small faults.

Fractal crushing of ice and brittle solids

A significant ‘scale effect’ is observed when sea ice forces on structures are measured at field scale: the force per unit contact area is not independent of area, but decreases with increasing area. Fragments of broken materials are found to have a fractal size distribution, with a fractal dimension close to 2.5 over a remarkably wide range of fragment size. The research described in this paper brings these two observations together, and shows that they can be explained by a simple model of crushing, which incorporates the relation between fragment size and splitting force predicted by linear elastic fracture mechanics. The model indicates a special role for the fractal dimension of 2.5, and predicts a relation between force and area, consistent with field observations.

The Fractal Nature of Hydrocarbon Deposits, 1: Size Distributions

Fractal phenomena define scale invariant sets of objects that obey power-law size distributions. Thus, if oil deposits are fractal, the number of deposits of volume V, N(V), would, within a defined space, obey a relation N(V) = aV{sup {minus}D}, where the exponent D is known as the fractal dimension. Such a defined space might be the play, the basin, the province, or the world. The authors have examined a number of datasets for regions in mature states of exploration. In such regions one may expect that most of the larger deposits have been discovered so that the upper part of the distribution of discoveries approximately reflects the size distribution of original deposits. They find the size distributions of oil deposits to obey the fractal size distribution quite well, down to some field size where the distribution of discovered fields is truncated for economic reasons. This distribution appears to hold at all hierarchical levels they have studied to date: the distribution of pools in a small play - the Cardium Scour of Alberta; in a large play - the Frio Standplain of the Gulf Coast; fields in basins - the Permian basin and the western Gulf of Mexico; and fields in more » larger territorial units - fields in the lower 48 states and giant fields (globally). The misconception that oil are lognormally distributed appears to reflect an artifact of the economic truncation of the datasets. In contrast to a lognormal distribution, there is no mean field size for a fractal distribution. « less

Particle-size distribution and microstructures within simulated fault gouge

This paper presents an investigation of comminution mechanisms and microstructure development within simulated fault gouge. We sheared 4.0 mm thick layers of quartz sand between rough steel surfaces using a triaxial apparatus. The layers were sheared at constant effective normal stress of 100 MPa, under saturated drained conditions, and at 45° to the axis of cylindrical steel samples. Porosity changes were measured throughout shear and microstructural observations were carried out on the deformed layers. Two types of load paths were investigated for shear strains (γ) between 0 and 3.3; either the shear stress was repeatedly cycled from zero to failure or the sample was sheared in a single-load cycle. Multiple-cycle experiments exhibit significantly more compaction than single-cycle experiments deformed to similar strains. Gouge layers from both sets of experiments contain oblique zones of localized shear (Riedel shears bands) after γ = 1.3–1.5. Gouge particles obey a fractal size distribution for the range 12.5–800μm; i.e. particle density vs size follows a power law, N( n) IA = bn - D where N(n) is the number of particles in a size range, A is the area examined, n is the mean of the size range, b is a constant and D is the fractal dimension. D for particles within the bulk material increases with shear strain for γ < about 1.5 after which it remains 2.6 ± 0.15. This value of D agrees with that found for natural fault gouge and with that predicted by a comminution model in which fracture probability depends on the relative size of nearest-neighbor particles. Analyses of particles within shear bands indicate continued size reduction after γ = 1.5. These particles do not obey a fractal size distribution for the range 6.25–100 μm due to a lack of particles larger than 25–50 μm. The rate of comminution within the bulk layer decreases at about γ = 1.5, which coincides with the onset of shear localization. Our data indicate that comminution is driven by relative movement between particles and that gouge layers attain a steady state particle-size distribution at γ = 1.5. The porosity—strain data and microstructural observations show a correlation between the onset of shear localization and the rate of dilatancy with shear strain.

Quasielastic-scattering linewidths and relaxation times for surface and mass fractals.

A theory is given for the momentum-transfer dependence of the quasielastic-scattering linewidth and relaxation time for solutions of surface and mass fractals. The linewidth and relaxation time are found to be fractional powers of the momentum transfer if the distribution of fractal sizes is a power law, in agreement with recent experimental findings.