Average Number Of Intersections Research Articles

A method to identify the connecting status of three-dimensional fractured rock masses based on two-dimensional geometric information

The hydraulic properties of three-dimensional (3-D) fractured rock masses are usually reduced to two-dimensional (2-D) problems, which may lead to unreliable conclusions. This study provides a theoretical method to identify the connecting status (connected or disconnected) of 3-D isotropic fractured rock masses based on 2-D geometric information on cutting planes. The percolation threshold of 3-D fractured rock masses can be judged by the average number of intersections per fracture I. The relationship between I, number of fractures in per sampled area P20 and trace length l are established. A connecting parameter P is defined to measure the average geometric information related to P20 and l. When P < 0.88, the 3-D fractured rock masses is not permeable; and when P ≥ 0.88, the fractured rock masses are connected and permeable. Numerical experiments are conducted, and the results show that the developed theoretical formulae are valid. In addition, the estimation methods of P are also discussed based on l and number of fractures in per sampled length P10 or Rock Quality Designation (RQD).

Distribution of the Proportionality Coefficient of Winter Route Accounting

In previous studies it was shown that the coefficient of proportionality of the winter route count (WRC) of animals included in the formula of WRC in the form of a constant multiplier π/2, is actually a random variable – the same as the average number of intersections account route traces per unit length, and the average length of the diurnal animals. The value π/2 is the mathematical expectation value of the proportionality factor, provided that the count route equiprobably crosses the daily footprint at any place and at any angle from 0 to 2π during a winter route counting of animals. At the same time, both the nature of the distribution of the coefficient as a random variable and the values of its variance as its other statistical characteristics remained unknown. In this study, it was found that when the above-mentioned count conditions are met, the distribution of the proportionality coefficient of WRC as a random variable will be exponential or power-like. This allows calculating the values of its variance and relative statistical error in advance without collecting additional count data.

A parallel construction method based on linear hierarchical bounding box without stack

The intersect operation of geometry is a common method of collision detection. To achieve a fast intersect operation, a hierarchical bounding box must be quickly constructed. This paper proposes an effective space division method for constructing high-quality linear hierarchical bounding boxes through ray tracing. This method can generate more regular axial bounding boxes into a complete binary tree. This structure can be completely parallel on the GPU, and effective parallelization can ensure the fastest tree-building time. The algorithm in this paper uses a non-stack ray tracing method to increase the frame rate of the 3D model rendering in the graphics processing unit and reduce the construction time of the bounding box. The experimental analysis of construction time, frame rate, average number of intersections and other performance indicators, the experimental results show that the use of stack-free midpoint splitting method and surface heuristic splitting method to ensure rapid construction time and efficient ray traversal performance, improve the object’s light traversal efficiency.

A predictive model correlating permeability to two-dimensional fracture network parameters

This study presents a predictive model of permeability with correlation to parameters of two-dimensional fracture networks, including mass density of fractures (dm), average number of intersections per meter at the inlet and outlet boundaries (din), and connectivity (Cr). The fracture networks were constructed by considering the influence of fracture number density, fracture length, orientation, and variance of fracture length. A total of 86 discrete fracture network (DFN) models were established using the Monte Carlo technique, and the relationships between permeability and dm, din, and Cr were analyzed. By fitting these calculated results, a multi-variable regression function was proposed for predicting permeability. The results show that the number density of fractures plays a more significant role in permeability than fracture length. Fracture orientation can change the connectivity of fracture networks robustly, and thereafter influence the permeability. Although variance of fracture length can affect the pattern of cumulative frequency – fracture length curves, the variance has negligible influence on the permeability of fracture networks. A necessary condition to form connected flow paths from inlet boundary to outlet boundary is: dm > 0.27 m/m2, din > 0.19 /m, and Cr > 1.20. The proposed regression function can predict permeability with a correlation coefficient larger than 0.87, and its validity is verified by comparisons with results reported in the literature. Finally, the potential future works that can facilitate the predictive models of permeability are pointed out as open questions.

Trace analysis for fracture networks with anisotropic orientations and heterogeneous distributions

Since only intersections with lines or planes are usually available to quantify the properties of real fracture networks, a stereological analysis of these intersections is a crucial issue. This article-the second of a series-is devoted to the derivation of the direct relations between the properties and the observable quantities. First, this derivation is achieved for anisotropic networks whose orientations obey a Fisher probability distribution function; second, it is extended to networks which are heterogeneous in space, i.e., whose density decays according to an exponential law. Five major quantities are determined: the excluded volume, the average number of intersections with a line and with a plane, the average trace length and the surface density of trace intersections. Some of these relations are valid for any convex fracture shape and some only for circular disks; however, numerical simulations show that excellent approximations are obtained by considering disks with the same area as the noncircular fractures.

A Connectivity Index for Discrete Fracture Networks

Connectivity is an important measure for assessing flow transport in rock, especially through fractures. In this paper, rock fracture systems are modelled by a discrete fracture model simulated by a marked point process. A connectivity index is then introduced to quantify the connectivity between any two points in space. Monte Carlo simulation is used to evaluate the connectivity index for stationary cases and relationships between the connectivity index and the parameters of the discrete fracture model are analysed. The average number of intersections per fracture, Xf, and the fracture intensity, P12 (P32), are calculated and the relationships between these parameters and the connectivity index are investigated, concluding that Xf is the more suitable parameter for the classification of rock mass flow properties. The relationships between the connectivity index and the percolation state of the fractured medium are also discussed. An edge correction is briefly discussed and a practical example is used to demonstrate the method of computing the connectivity index.

Analysis of the second-order statistical characteristics of a rapidly fading radio signal

The average number of intersections with a given level of the envelope of a rapidly fading signal and the mean duration of a fading are determined for different propagation paths.

Connectivity properties of two‐dimensional fracture networks with stochastic fractal correlation

We present a theoretical and numerical study of the connectivity of fracture networks with fractal correlations. In addition to length distribution, this spatial property observed on most fracture networks conveys long‐range correlation that may be crucial on network connectivity. We especially focus on the model that comes out relevant to natural fracture network: a fractal density distribution for the fracture centers (dimension D) and a power law distribution for the fracture lengths (exponent a, n(l) ∼ l−a). Three different regimes of connectivity are identified depending on D and on a. For a &lt; D + 1 the length distribution prevails against fractal correlation on network connectivity. In this case, global connectivity is mostly ruled by the largest fractures whose length is of the order of the system size, and thus the connectivity increases with scale. For a &gt; D + 1 the connectivity is ruled by fractures much smaller than the system size with a strong control of spatial correlation; the connectivity now decreases with system scale. Finally, for the self‐similar case (a = D + 1), which corresponds to the transition between the two previous regimes, connectivity properties are scale invariant: percolation threshold corresponds to a critical fractal density and the average number of intersections per fracture at threshold is a scale invariant as well.

Analysis of fracture network connectivity using percolation theory

Connectivity aspects of fracture networks are analyzed in terms of percolation theory. These aspects are of fundamental importance in characterization, exploitation, and management of fractured formations. General connectivity and power law relationships are determined that characterize the density of fractures and average number of intersections per fracture necessary to ensure network connectivity, the likelihood of a fractured formation being hydraulically connected, and the probability that any specific fracture is connected to the conducting portion of the network. Monte Carlo experiments with a two-dimensional fracture network model confirm the percolation theory predictions. These relationships may prove useful in formulating theoretically tractable approximations of fracture nerworks that capture the essential system properties.

Efficient measurement of microstructural surface area using trisector

It is shown that just three appropriately chosen metallographic sections can yield an efficient and reliable estimate of the total area of the internal microstructural boundaries per unit volume Sv in a microstructure having an arbitrary and unknown anisotropy and geometry. The following sampling scheme is required. (1) Choose a reference direction (called “vertical axis”) such that most of the surface elements of interest are not parallel to the selected reference direction (vertical axis). (2) Observe the microstructure on three metallographic sections that are mutually at an angle of 120 deg and contain the vertical axis. (3) Superimpose a set ofcycloid shape test lines on these metallographic sections (called vertical sections) such that the cycloid minor axis is parallel to the vertical axis, and estimate the average number of intersections between these test lines and the microstructural boundaries per unit test line length, [PL c]3. The surface area per unit volume Sv is given by Sv = [PL c]3 To emphasize the preceding sampling requirements, the composite test probe is called “trisector. ” It is also shown that independent random sections are not useful for estimation of Sv of anisotropic microstructural surfaces, in general. Further, two perpendicular vertical sections are also not generally useful for a reliable estimation of Sv in anisotropic microstructures.

Radionuclide transport in fractured rock a new model: Application and discussion

A model for predicting the concerntration of radionuclides transported through fractured rock was developed in several earlier papers as part of the CEC programme on management and storage of radioactive waste. In this report, we examine that theory in more depth and develop expressions for the system parameters in terms of the rock statistics. A review of related work on fracture statistics is given and its relevance to the present work highlighted. In particular, we note that the connectivity, i.e. the average number of intersections per fracture, plays a vital part in determining the ‘cross section’ and redistribution function at intersections. Expressions are developed for these cross sections in terms of fracture orientation distributions. For a single set of fractures, i.e. fractures of one particular type, we have obtained numerical solutions and have demonstrated how the concentration profile becomes more dispersive as the orientation distribution becomes more isotropic. The method shows considerable promise and has many advantages when compared with the classical advective-dispersive model. In particular, it enables microscopic rock parameters, e.g. fracture length and orientation, to be employed directly and avoids the effective continuum approximation common to many approximations. It also explains the scaledependent effect and can account in a consistent manner for dead-end pores and channelling.

Analytical expressions for the permeability of random two‐dimensional Poisson fracture networks based on regular lattice percolation and equivalent media theories

The permeability of random two‐dimensional Poisson fracture networks can be studied using a model based on percolation theory and equivalent media theory. Such theories are usually applied on regular lattices where the lattice elements are present with probability p. In order to apply these theories to random systems, we (1) define the equivalent to the case where p = 1, (2) define p in terms of the statistical parameters of the random network, and (3) define the equivalent of the coordination number z. An upper bound for permeability equivalent to the case of p = 1 is found by calculating the permeability of the fracture network with the same linear fracture frequency and infinitely long fractures. The permeability of networks with the same linear fracture frequency and finite fractures can be normalized by this maximum. An equivalent for p is found as a function of the connectivity ζ, which is defined as the average number of intersections per fracture. This number can be calculated from the fracture density and distributions of fracture length and orientation. Then the equivalent p is defined by equating the average run length for a random network as a function of ζ to the average run length for a lattice as a function of p. The average run length in a random system is the average number of segments that a fracture is divided into by intersections with other fractures. In a lattice, it is the average number of bonds contiguous to a given bond. Also, an average coordination number can be calculated for the random systems as a function of ζ. Given these definitions of p and z, expressions for permeability are found based on percolation theory and equivalent media theory on regular lattices. When the expression for p is used to calculate the correlation length from percolation theory, an empirical formula for the size of the REV can be developed. To apply the models to random length systems, the expression for ζ must be modified to remove short fractures which do not contribute to flow. This leads to a quantitative prediction of how permeability decreases as one removes shorter fractures from a network. Numerical studies provide strong support for these models. These results also apply to the analogous electrical conduction problem.

Directed percolation in the two-dimensional continuum.

We report the first study of directed percolation in the continuum. The percolation threshold is found to be at ${B}_{c}$=5.0\ifmmode\pm\else\textpm\fi{}0.1, where ${B}_{c}$ is the average number of intersections per diode at the threshold. This is to be compared with ${B}_{c}$=3.2\ifmmode\pm\else\textpm\fi{}0.1 in the corresponding nondirected problem. It is found that the critical exponents of this system are ${\ensuremath{\nu}}_{\mathrm{para}}$=0.74\ifmmode\pm\else\textpm\fi{}0.05, ${\ensuremath{\nu}}_{\ensuremath{\perp}}$=0.46\ifmmode\pm\else\textpm\fi{}0.08, \ensuremath{\beta}=0.33\ifmmode\pm\else\textpm\fi{}0.07, and \ensuremath{\beta}'=(2.00 \ifmmode\pm\else\textpm\fi{}0.05)\ensuremath{\beta}. The good agreement with values found for directed lattices appears to be a confirmation of universality for these systems as well as a demonstration that geometrical and physical properties of directed systems in the continuum can be computed.

Connectivity of fracture systems-a percolation theory approach

Critical percolation densities have been found numerically for various systems of lines uniformly distributed in the plane. The average number of intersections per line at percolation has also been found and varies only slightly over the cases considered. It may therefore provide a useful rule of thumb for deciding whether a system percolates. An estimate of the critical percolation density from the lattice percolation probability is presented. Possible extensions of the techniques described to three dimensions are discussed.

Measurement of the total area of secretory epithelium in the lactating mammary gland of the goat.

A method previously described for estimating the total surface area of secretory epithelium in the half-udder (single gland) of the goat, has been re-investigated and simplified. Fifty or more fields of lobule-alveolar tissue, at a constant magnification, are projected from histological sections taken from small blocks of fixed material selected at random, against a single line representing a diameter of the field. Twice the average number of intersections which the line makes with the epithelium lining the alveoli and intra-lobular ducts, multiplied by the volume of the half-udder, and divided by the length of the line corrected for magnification, gives an estimate of the total surface area of secretory epithelium. A highly significant correlation is obtained between the surface area and average weekly milk yields in a series of glands from ovariectomized goats brought into lactation by treatment with oestrogen alone, or in combination with progesterone, or from normal goats in lactation after kidding. An overall deficiency in surface area of epithelium is typical of the glands brought into lactation artificially by the administration of sex hormones. The ratio obtained by omitting the volume of the total gland from the formula provides a useful index of alveolar size, which is commonly much larger in glands developed in response to the injection of these hormones.