This paper presents an analytical solution to quantify the uncertainty of volumetric fraction (Vf) estimates in a heterogeneous material using 2-D probes. The analytical solution was derived based upon the concept of representative volume (RVE). The results show that the uncertainties of the estimates depend upon the size of blocks, measurement area, and level of Vf. The analytical solution has been verified via numerical simulation. The latter was carried out by first generating spherical blocks to various level of Vf, which was then sampled by 2D probes in obtaining the Vf estimates and the associated uncertainties. Application examples are given at the end.


The mechanical behavior of a heterogeneous geomaterial depends strongly on the volumetric fraction (Vf) of its various constituents [1-9]. To estimate Vf, one may take sample at random points, along a line, on a planar area, or in a three dimensional region. According to the basic principles of stereology, if features are distributed under isotropic, uniform and random conditions, the results obtained will be the same regardless of the dimension of the probes [10]. In geotechnical engineering, three geometric probes are often used in estimating the volumetric fractions of heterogeneous geomaterials (i.e. bimrocks), namely, the one dimensional scanlines or boreholes, the two dimensional cross sectional images or window mapping, and the three dimensional sieve analyses. 1-D probe is perhaps the most efficient and economical method for estimating volumetric fractions, and research on what constitute an adequate scanline length, or sample size, can be traced way back to 1961. That was when Hilliard & Cahn [11] made an assumption that intercept length of a scan line on the inclusion followed a Poisson distribution, and obtained an equation to calculate the standard deviation of Vf with respect to sample size.

This content is only available via PDF.
You can access this article if you purchase or spend a download.