Verteilungsfunktion der Kluftkörpergröße von verwittertem oberkarbonischem Sandstein
Rock blocks are formed by natural processes, i.e. tectonics and weathering, and by man-made processes. These processes develop statistically as the rock blocks are “polydisperse”, i.e. of different size and of different shape. The distribution of the rock block size is of principal interest in engineering practice, e.g. for the stability of slopes, in mining and other underground excavations, for the fragmentation of rock for building purposes, as for instance embankment fills and also in ore dressing.
Two questions arise: firstly as to the methodical assessment of the distribution of the rock block size, and secondly as to the dependency of this distribution on lithology, tectonics, and weathering.
The analysis of rock block size is to be carried out on rock faces of both man-made and natural exposure, possibly also on rock cores. One principal problem arises from the fact that only sections of the actual rock blocks can be viewed, depending on the orientation of jointing, the Position of the blocks, and the orientation of the rock face. This has also been named the “tomato salad view”. Therefore, a method has been developed to derive a realistic distribution of the rock block size from simple geometric sections of the rock blocks which are easiliy reproducible. The “line analysis” after ROSIVAL is adapted as measuring method. The chord lengths, i.e. the distances between the boundaries of the rock blocks, i.e. between the intersections of the joints with the rock face, are measured along grid lines the directions of which depend on the orientation of the joint system.
The field investigation was carried out on a rock face exposing a weathered sandstone sequence of coal bearing Upper Carboniferous. The face was about 20 metres in height.
The distribution of the chord length measured was approximated by different Statistical functions, e.g. RRS, expontential, normal and log — normal distributions. The best fit was obtained by the RRS distribution developed by ROSIN, RAMMLER and SPERLING. More recently this distribution is also called the two-parametric Weibull distribution.
In the next phase the investigation is directed towards the statial change of the RRS distribution with changes of lithology, tectonics, and weathering as observed within the exposure. For this purpose theparameters and the characteristic values of the RRS distribution are studied. One of the two parameters, and also both mean and median, are of almost the same magnitude and vary synonymously with respect to space. They react, if tectonics remains unchanged, on changes of both weathering and lithology. They tend to decrease with increasing weathering and decreasing grain size, i.e. joint spacing. Most interesting is the reaction of the second parameter, the exponent of the RRS distribution. This parameter is barely affected by lithology provided the tectonics remain unchanged. However, it reacts significantly on changes in weathering observed towards the ground surface.
As an indicator of the Utility of the distribution of the rock block size an equi-distant joint spacing was developed based on integral geometry. A comparison of the mean chord length in RRS distribution with this joint spacing indicates that the mean of the chord length is always less than this mean joint spacing.
Therefore, a function had to be developed which would allow to derive the distribution of the actual rock block size from the distribution of the chord length. This was accomplished based on the well-known mathematical relationship between the distribution of the chord length of spheres with their actual diameters. A respective Solution was provided by CAHN and FULLMAN. If the RRS distribution and its derivative are inserted into the Cahn-Fullman function a new distribution is obtained which contains the two parameters of the RRS distribution. For the new distribution equations were developed for mean, variance, and mode. Depending on the spatial configuration of lithology, tectonics, and weathering within the exposure mean and mode of the new function describing the distribution of the rock block size change analogously with the mean of the RRS distribution of the chord lengths.
Striking agreement was observed in a comparison between the equi-distant joint spacing and the modified mean and mode of the new function describing the distribution of rock block size. This agreement proves that the proposed analytical approach yields the realistic distribution of the rock block size.
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