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Advancements in rock block volume calculation by analytical method for geological engineering applications

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Abstract

The shape, the volume, and the distribution of the rock blocks represent important geomechanical factors of a rock mass behavior in engineering works. Several methods have been developed for estimating these parameters, including numerical models, as well as analytical and empirical methods. However, their determination in actual in-situ conditions can be quite challenging. The existing analytical methods show limitations in determining the in-situ rock blocks volume. Numerical models provide more reliable estimates of these parameters, but they are not accessible to all, and they require a good working knowledge. Increasing the accuracy of existing analytical methods, or developing more reliable and accessible methods, are more realistic approaches to obtain better estimates of rock block volumes. This paper presents a new method to obtain more accurate estimates of in-situ rock block volume. The method is developed for rock a mass consisting of three persistent joint sets, each set having constant spacing and orientation values. It is based on vector products to obtain exact block volumes, an improvement as compared to previous methods. The volumes of the rock blocks are calculated through the multiplication of the blocks’ edge vector. The results of the developed equation are validated with the output of numerical simulations using 3DEC version 7.0 software, and the results indicate that the developed method makes it possible to determine in-situ rock block volume more reliably than the existing methods.

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Data availability

All data, models, and code generated or used during the study appear in the paper (Tables and Appendix).

Abbreviations

(.):

The point is an inner product of a pair of vectors

(×):

The multiplication sign is a cross product

A :

The area of the observation zone (m2)

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {A} , \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B} , \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {C}\) :

Edge vectors of the intact block

a 1 :

Longest dimension of the block (m)

a 3 :

Shortest dimension of the block (m)

D 1, D 2, and D 3 :

Dips of the joint sets 1, 2, and 3

DD1, DD2, and DD3 :

Dip directions of the joint sets 1, 2, and 3

E doa :

Erosion, discontinuity orientation adjustment

J a :

Joint alteration number

J n :

Number of joint sets

J o :

Joint aperture (m)

J r :

Joint roughness number

J s :

Relative block structure

J v :

Number of joints intersecting a volume of 1 m3 of rock (m−3)

L :

Characteristic length of the rock mass (m)

l i :

Joint length (m)

N J 1, N J 2, and N J 3 :

Normal vectors to joint sets J1, J2, and J3 plane

N r :

Number of random joints in the real location

P i :

Joint persistence

RQD:

Rock quality designation

RMSE:

Root mean square error

S :

Average joint spacing measured along the drill core (m)

S a :

Average joint spacing of all sets (m)

S i :

Average spacing of joint set i (m)

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{u_{A} }}, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{u_{B} }}, \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{{u_{C} }}\) :

Unit vectors of \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {A} , \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {B} , \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {C}\)

V b :

Volume of blocks (m3)

V b A :

The analytically calculated block volume (m3)

wJd:

Weighted joint density

γ 1, γ 2, and γ 3 :

Angle between each pair of joint sets

β :

Form factor of the blocks

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Acknowledgements

The authors would like to thank the Natural Sciences and Engineering Research Council of Canada (NSERC) and Hydro-Québec for the funding provided through the RDC (RDC 537350) Program. Furthermore, the authors wish to convey their deep gratitude to the Itasca Consulting Group in Minneapolis for the IEP Research Program, especially Jim Hazzard, for his valuable technical support and advice.

Funding

Natural Sciences and Engineering Research Council of Canada (NSERC), Hydro-Québec for funding through the CRD program (CRD 537350).

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Authors and Affiliations

  1. Université du Québec à Chicoutimi, 555 Boulevard de l’Université, Chicoutimi, QC, G7H 2B1, Canada

    Aboubacar Sidiki Koulibaly, Alireza Shahbazi, Ali Saeidi, Alain Rouleau & Romain Chesnaux

  2. Hydro-Québec Production Unité Expertise en Barrages, 75 Boulevard René-Lévesque Ouest, Montréal, QC, H2Z 1A4, Canada

    Marco Quirion

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  1. Aboubacar Sidiki Koulibaly
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Contributions

ASK: formal analysis, conceptualization, resources, investigation, data curation, writing—original draft. AS: conceptualization, resources, data curation, software, validation, methodology, writing-original draft. AS: conceptualization, supervision, investigation, methodology, project administration, writing—review and editing. AR: conceptualization, supervision, writing—review and editing. MQ: funding acquisition, writing—review and editing. RC: supervision, writing—review and editing.

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Correspondence to Aboubacar Sidiki Koulibaly.

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Appendices

Appendix 1

See Table 6.

Table 6 Characteristic data of the database summarizing the case studies of Pells (2016)
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Appendix 2

This information is taken from: Itasca Consulting Group Inc, 2016. 3DEC: 3 Dimensional Distinct Element Code. Online Manual (http://docs.itascacg.com/3dec700/3dec/docproject/source/theory/3dectheory/theory_background.html?node2134)

«The space containing the system of blocks is divided into rectangular 3D cells. Each block is mapped into the cell or cells that its “envelope space” occupies. A block’s envelope space is defined as the smallest three-dimensional box with sides parallel to the coordinate axes that can contain the block. Each cell stores, in linked-list form, the addresses of all blocks that map into it. Following Fig. 9 illustrates the mapping logic for a two-dimensional space (as it is difficult to illustrate the concept in three dimensions). Once all blocks have been mapped into the cell space, it is an easy matter to identify the neighbors to a given block: the cells that correspond to its envelope space contain entries for all blocks that are near. Normally, this “search space” is increased in all directions by a tolerance, so that all blocks within the given tolerance are found. Note that the computer time necessary to perform the map and search functions for each block depends on the size and shape of the block, but not on the number of blocks in the system. The overall computer time for neighbor detection is consequently directly proportional to the number of blocks, provided that cell volume is proportional to average block volume. It is difficult to provide a formula for optimum cell size because of the variety of block shapes that may be encountered. In the limit, if only one cell is used, all blocks will map into it, and the search time will be quadratic. As the density of cells increases, the number of non-neighboring blocks retrieved for a given block will decrease. At a certain point, there is no advantage to increasing the density of cells, because all the blocks retrieved will be neighbors. However, by further increasing the cell density, the time associated with mapping and searching increases. The optimum cell density must therefore be of the order of one cell per block, in order to reduce both sources of wasted time.

Fig. 9
figure 9

Examples of block mapping to cell space, in to dimensions

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Koulibaly, A.S., Shahbazi, A., Saeidi, A. et al. Advancements in rock block volume calculation by analytical method for geological engineering applications. Environ Earth Sci 82, 344 (2023). https://doi.org/10.1007/s12665-023-11027-6

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