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Generation of Numerical Models of Anisotropic Columnar Jointed Rock Mass Using Modified Centroidal Voronoi Diagrams

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Generation of Numerical Models of Anisotropic Columnar Jointed Rock Mass Using Modified Centroidal Voronoi Diagrams S S symmetry Article Generation of Numerical Models of Anisotropic Columnar Jointed Rock Mass Using Modified Centroidal Voronoi Diagrams Qingxiang Meng 1,2,* , Long Yan 2,3,*, Yulong Chen 4,* and Qiang Zhang 5,6 1 College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China 2 Research Institute of Geotechnical Engineering, Hohai University, Nanjing 210098, China 3 Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing 210098, China 4 Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China 5 China Institute of Water Resources and Hydropower Research, Beijing 100048, China; [email protected] 6 State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, China * Correspondence: [email protected] (Q.M.); [email protected] (L.Y.); [email protected] (Y.C.) Received: 21 October 2018; Accepted: 7 November 2018; Published: 9 November 2018 Abstract: A columnar joint network is a natural fracture pattern with high symmetry, which leads to the anisotropy mechanical property of columnar basalt. For a better understanding the mechanical behavior, a novel modeling method for columnar jointed rock mass through field investigation is proposed in this paper. Natural columnar jointed networks lies between random and centroidal Voronoi tessellations. This heterogeneity of columnar cells in shape and area can be represented using the coefficient of variation, which can be easily estimated. Using the bisection method, a modified Lloyd’s algorithm is proposed to generate a Voronoi diagram with a specified coefficient of variation. Modelling of the columnar jointed rock mass using six parameters is then presented. A case study of columnar basalt at Baihetan Dam is performed to demonstrate the feasibility of this method. The results show that this method is applicable in the modeling of columnar jointed rock mass as well as similar polycrystalline materials. Keywords: anisotropic columnar jointed rock; numerical model; centroidal Voronoi diagram; coefficient of variation 1. Introduction Columnar jointed rock is a typical geological structure formed by a lot of ordered colonnades like the world famous natural wonders of Fingal’s Cave and Giant’s Causeway [1,2]. Being a miraculous natural phenomenon, the study on the geological structure of columnar jointed rock can be dated back to the 17th century [3]. Nowadays, the formation of columnar jointing is reasonably understood as a result of cracks propagation in cooling lava flows [4–7]. In situ observations and laboratory tests have also been employed in the study of columnar jointing [8–10]. With the increase of human activities, some engineering projects have encountered columnar jointing, and a large hydropower station is even founded on it [11]. Because of its adverse geologic conditions, study on the mechanical properties of columnar jointed rock mass is very important in engineering projects. King et al. studied elastic-wave propagation in columnar joints with a series of cross-hole acoustic measurements made between four horizontal boreholes drilled from a near-surface underground opening situated in a basaltic rock mass [12]. In connection with the Baihetan Symmetry 2018, 10, 618; doi:10.3390/sym10110618 www.mdpi.com/journal/symmetry Symmetry 2018,, 10,, 618x FOR PEER REVIEW 2 of 15 Hydropower Station project, several studies have been implemented for the mechanical property of Hydropowercolumnar jointed Station rock. project, Meng severalsystematically studies analyzed have been the implemented anisotropic forproperties the mechanical of columnar property jointed of columnarbasalt using jointed analytical rock. and Meng numerical systematically methods analyzed [13]. Zheng the anisotropic et al. proposed properties a 3-D of modeling columnar method jointed basaltfor columnar using analytical basalt using and numericalrandom Voronoi methods tessellation [13]. Zheng [14]. et al.In proposedReferences a 3-D[15–17], modeling 2-D and method 3-D fordiscrete columnar element basalt simulation using random methods Voronoi of column tessellationar are [14 implemented,]. In References and [15 the–17 ],representative 2-D and 3-D discreteelementary element volume simulation scale and methods equivalent of columnar mechan areical implemented, parameters and are the obtained. representative Basedelementary on meso- volumestructural scale analysis, and equivalent macro-anisotropic mechanical constitutive parameters model are obtained. is developed Based and on 3-D meso-structural numerical simulation analysis, macro-anisotropicis performed for the constitutive dam foundation model considering is developed the and effect 3-D of numerical columnar simulation joints [18]. isLaboratory performed tests for theare damcarried foundation out using considering experimental the effectanalogs of columnarof jointing joints under [18 ].uniaxial Laboratory compression tests are [19]. carried These out usingresearch experimental results indicate analogs that the of jointingcharacteristics under of uniaxial columnar compression jointed rock [ 19mass]. These are very research complex results with indicateanisotropic that and the nonlinear characteristics properties. of columnar jointed rock mass are very complex with anisotropic and nonlinearMost properties.of the modelling methods of columnar jointed rock mass use the program developed by ZhengMost et al. of the[14]; modelling however, methodsthis method of columnar has some jointed shortcomings, rock mass such use as the a programlow efficiency developed and byimmutable Zheng et columnar al. [14]; however,shapes, as this pointed method out has in someReference shortcomings, [20]. The suchobjective as a of low this efficiency paper is and to immutableovercome these columnar shortcomings shapes, asin the pointed generation out in of Reference columnar [20 joints]. The in objectiverock mass. of An this efficient paper isand to overcomecontrollable these method shortcomings for the generation in the generation of column of columnarar joints is joints proposed in rock based mass. on An a efficientconstrained and controllablecentroidal Voronoi method fortessellation the generation diagram. of columnar In Section joints 2, the is proposed characteristics based onof atypical constrained columnar centroidal rock Voronoimass are tessellation briefly introduced. diagram. The In Section properties2, the of characteristics Voronoi diagrams of typical and columnar the latest rock related mass research are briefly on introduced.this topic are Thediscussed properties in Section of Voronoi 3. Based diagrams on these and discussion the latest and related algorithms, research a detailed on this procedure topic are discussedfor the generation in Section of columnar3. Based jointed on these rock discussion mass is developed and algorithms, in Section a detailed4. A case procedurestudy on columnar for the generationjointed basalt of columnargeneration jointed for the rock Baihetan mass isHydropower developed in Station Section is4 .presented A case study in Section on columnar 5. Finally, jointed the basaltconclusion generation is given for in the Section Baihetan 6. Hydropower Station is presented in Section5. Finally, the conclusion is given in Section6. 2. Typical Shapes of Columnar Jointed Rock Mass 2. Typical Shapes of Columnar Jointed Rock Mass With the development of hydropower station, columnar jointing is often encountered in southwesternWith the China. development As a result of hydropowerof cooling lava station, flows and columnar ash-flow jointing tuffs, it is occurs often in encountered many types inof southwesternvolcanic rocks China.in the formation As a result of ofa regular cooling array lava of flows polygonal and ash-flow prisms tuffs,or columns it occurs [21]. in Several many types famous of volcanictypical columnar rocks inthe joints formation are shown of a in regular Figure array 1. of polygonal prisms or columns [21]. Several famous typical columnar joints are shown in Figure1. (a) (b) (c) Figure 1.1. TypicalTypical columnar columnar jointed jointed rock rock masses: masses: (a) Devil’s (a) Devil’s Postpile, Postpile, Yosemite Yosemite in California; in California; (b) Fingal’s (b) Cave,Fingal’s Staffa Cave, in Staffa Scotland; in Scotland; and (c) Giant’s and (c) Causeway, Giant’s Causeway, Antrim in Antrim Northern in Northern Ireland. Ireland. Although the characteristics of columnar jointed rock is typical, they they vary vary for for different different locations. locations. The column in in some some areas areas is is regular regular and and straight straight like like Giant’s Giant’s Causeway, Causeway, while while some some other other places places like likeBaihetan Baihetan Dam, Dam, it is it irregular. is irregular. Besides, Besides, the the diamet diametersers of ofcolumns columns vary vary from from meters meters to to centimeters. centimeters. Columns are usually parallel and straight, and and the length can can be as much as several meters. Furthermore, the number number of of sides sides of of an an individual individual column column also also varies varies from from three three to toeight. eight. In Inorder order to togive give a more a more specific specific illustration illustration of the of the columns, columns, a colorized a colorized O’Reilly O’Reilly map map is presented is presented in Figure in Figure 2. In2. this figure, hexagons are green, heptagons are blue, octagons are purple, pentagons
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