Stereological relationship between trace length and size distribution of elliptical discontinuities

This paper presents an approach for estimating the size distributions of elliptical discontinuities. This is an extension of approaches for estimating size distributions of circular discontinuities. By assuming an elliptical shape and changing the aspect ratio k of the ellipses (i.e. the ratio of the major to minor axes), many real discontinuities can be reasonably represented. For elliptical discontinuities, a general stereological relationship between distributions of trace length and distributions of discontinuity size (expressed by the major axis length, a, of the ellipse) is derived for area (or window) sampling. From this relationship, expressions are derived between the mean (μ_a) and standard deviation (σ_a) of the discontinuity size, a, and the mean (μ_l) and standard deviation (σ_l) of the trace length, l, respectively for log-normal, negative exponential and gamma distributions of discontinuity sizes. Using the derived expressions, the effect of sampling plane orientation on trace lengths is investigated. The results show that the sampling plane orientation has an important effect on both the mean and the standard deviation of trace lengths. Finally, the derived stereological relationship between the trace length distribution and the discontinuity size distribution was used in a simulation with the FracMan discrete fracture code. The results show that the approach for inferring the major axis orientation, aspect ratio and size distribution of discontinuities from observations on outcrops is satisfactory.