ABSTRACT
The concepts of conditional probability and geometrical probability are used to establish a relation between the probability density of the trace length and the probability density of the discontinuity diameter. The discontinuity is represented as a thin circular disc. A numerical solution scheme is used to obtain the probability density of the disc diameter from the probability density of the trace length. An example was used to illustrate the possibly large difference between the mean discontinuity diameter and the mean trace length.
Introduction
The engineering properties of most rock masses are controlled to a large extent by the characteristics of the discontinuities. To predict the behavior of structures in and on such rock masses, it is necessary to characterize the discontinuity geometry, the geomechanical properties of the discontinuities, and the geomechanical properties of the intact rock. The discontinuity geometry in three-dimensional space can be modeled by the number of discontinuities per unit volume and the location, orientation, shape and size of the discontinuities. The parameters that describe these characteristics need to be estimated from the data obtained from the discontinuity surveys. In discontinuity surveys, observations are usually made on intersections of discontinuities with lines, as in boreholes, or with planes as in exposures. Hence, the field data are in one or two dimensions, such as spacing between traces, trace lengths, or orientation of discontinuities. An important geometric property of discontinuities is the size. This paper describes a mathematical model which relates the trace length to the size of the discontinuities. In discontinuity modeling, joints have been represented as thin circular discs (Baecher et al., 1977; Bridges, 1976) and as Poisson planes {Glynn et al., 1978). Robertson (1970) reported that strike length and dip length of discontinuities were approximately equal. In this model, the assumption of circular discs is retained. Then the mean disc diameter provides a measure of discontinuity extent. The sampling domain is a vertical exposure of finite size. The concepts of conditional probability and geometrical probability are used to establish a relationship between the probability density of the trace length and the probability density of the disc diameter, D. A numerical solution scheme is developed. It allows one to obtain the probability distribution of D for any probability density function of the trace length. An example is given for trace lengths that have an exponential distribution. It is assumed here that sampling biases have been corrected (Bacher et al., 1977; Kulatilake and Wu, 1984).
Mathematical Model
Let f(x,I) be the probability density function of the corrected trace length (Kulatilake and Wu, 1984) on the infinite vertical plane, where I : event that the discontinuity intersects the plane and x = resulting trace length. The model discontinuity is a circular disc, whose diameter is represented by the probability density function, g{D), and g(D) is related to f(x,I) through(mathematical equation) (available in full paper)
