Probability Density Function of Rock Mass Discontinuity Distances
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ORIGINAL PAPER
Probability Density Function of Rock Mass Discontinuity Distances Antonio Fernando Gorgulho Bruzzi . Pedro Alameda-Herna´ndez . Andre´ Monteiro Klen . Tiago Martins Pereira . Milene Sabino Lana
Received: 16 July 2020 / Accepted: 2 November 2020 Ó Springer Nature Switzerland AG 2020
Abstract Rock masses are presenting an enduring challenge since the first authors began trying to model these natural structures. This modelling needs a parametrisation and a latter value assignment for those parameter value series, which show high variability, and a lack of clear patterns, in nature. Understanding the statistical nature of these values is an essential goal of rock mechanics, searching for the most appropriate probability density function (pdf) for fitting each one of those parameter values. The identification of those pdf is an aid for a better understanding of the rock mass nature and a need for correct discrete fracture network generation, or any probabilistic calculation. The parameter studied in this work is the degree of fracturing, in particular: the distances between fractures in a borehole rock core, which determines the Rock Quality Designation
A. F. G. Bruzzi M. S. Lana School of Mines, Federal University of Ouro Preto, Ouro Preto, Brazil P. Alameda-Herna´ndez (&) School of Mines, Urban Engineering Department, Federal University of Ouro Preto, Ouro Preto, Brazil e-mail: [email protected] A. M. Klen Federal Institute of Ouro Preto-MG, Ouro Preto, Brazil T. M. Pereira Institute of Exact and Biological Sciences, Federal University of Ouro Preto, Ouro Preto, Brazil
(RQD) value. The most accepted hypothesis is that these values fit a negative exponential pdf, however, with some criticism. Through the analysis of 1985 m of borehole rock core from five lithologies from two rock masses, this work shows the lognormality of the data series. The negative exponential does not fit correctly; however, the old approach for RQD based on this hypothesis offers satisfying results. Furthermore, it has been observed that schist presents the unclearer random pattern. Keywords Rock borehole core Rock discontinuities Rock Quality Designation Goodness-of-fit tests Lognormal distribution List of symbols and abbreviations AD Anderson Darling goodness-of-fit test ADF Augmented Dickey–Fuller unit root test KS Kolmogorov–Smirnov goodness-of-fit test pdf Probability density function RQD Rock Quality Designation SW Shapiro Wilk goodness-of-fit test t Threshold value for RQD classical calculation X Random vector with the distances between joints xi Length of the i th fragment in a borehole core xit Length of i th fragment in a borehole core when it is bigger than t k Number of discontinuities per unit of length in a borehole or scanline
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Geotech Geol Eng
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Chi-square goodness-of-fit test
RQDð%Þ ¼ 100
N X
xit =L
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1 Introduction Rock masses are natural structures that host many infrastructure works within civil or mining engineering; hence
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