An analytical solution on the uncertainty in the estimates of volumetric fraction of different compositions in a Bimrock is presented in this paper. The derivation was developed based upon the concept of representative volumes. To tackle the anisotropic orientation of blocks, this study developed the concept of a parallelogram representative volume element that contains an ellipse. Our results demonstrate that the uncertainties in the estimates depend upon the aspect ratio, orientation, and diameter of blocks and the level of volumetric fraction. We further simplified the analytical expression in terms of intercept length. Our results were verified through numerical simulation albeit preliminary.


We have previously developed an analytical solution to the uncertainty of volumetric fraction estimates of isotropic, Block-in-Matrix, ( Bimrocks) [1]. Bimrocks are defined as “a mixture of rocks, composed of geotechnically significant blocks within a bonded matrix of finer texture” [2]. Bimrocks thus encompass a wide range of geologic materials including, for example, melanges, fault rocks, landslide debris, and glacial till. Their overall mechanical behaviors are highly dependent on their volumetric block fraction [2][3][4].

Three categories of measurement methods have been used in estimating the volumetric fraction, Vf , of Bimrocks, namely, one-dimensional (linear measurement and borehole), two-dimensional (image analyses and window mapping) and three-dimensional (sieve analyses). Although sieve analysis is the most accurate method for laboratory-scale studies, separation of blocks from the weaker matrix is not always possible, affecting by factors such as the number and size of blocks, and the degree of contact strength between blocks and matrix [1].

According to the basic principles of stereology, if the sampling is under IUR - isotropic, uniform and random conditions- such that all portions of the structure are equally represented (uniform), there is no conscious or consistent placement of measurement regions with respect to the structure itself to select what is to be measured (random), and all directions of measurement are equally represented (isotropic) [5], namely, the results be the same regardless of the dimension of a measurement method. Or, simply put Vf = Af = Lf = Pf, where Vf is volumetric fraction, Af is area fraction, Lf is linear fraction, and Pf is point-count fraction. Thus, in this paper we interchange the use of Vf, Af and Lf. Scanline is one of the most efficient and economical method for estimating Vf , and may be the only way such as in the case of sampling through drilling. Scanlines estimate Vf by dividing the total cumulative intercept length, or block length, with the total scanlin length. Thus its operation and processing is rather straightforward. But scanline use has a caveat: How does one determine what constitute an adequate cumulative length of scanlines?

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