Rock Mech Rock Eng (2012) 45:159–181 DOI 10.1007/s00603-011-0136-7
ORIGINAL PAPER
Use of an Integrated Discrete Fracture Network Code for Stochastic Stability Analyses of Fractured Rock Masses V. Merrien-Soukatchoff • T. Korini A. Thoraval
•
Received: 11 February 2010 / Accepted: 22 February 2011 / Published online: 12 March 2011 Springer-Verlag 2011
Abstract The paper presents the Discrete Fracture Network code RESOBLOK, which couples geometrical block system construction and a quick iterative stability analysis in the same package. The deterministic or stochastic geometry of a fractured rock mass can be represented and interactively displayed in 3D using two different fracture generators: one mainly used for hydraulic purposes and another designed to allow block stability evaluation. RESOBLOK has downstream modules that can quickly compute stability (based on limit equilibrium or energy-based analysis), display geometric information and create links to other discrete software. The advantage of the code is that it couples stochastic geometrical representation and a quick iterative stability analysis to allow risk-analysis with or without reinforcement and, for the worst cases, more accurate analysis using stress–strain analysis computer codes. These different aspects are detailed for embankment and underground works.
V. Merrien-Soukatchoff (&) Laboratoire Environnement Ge´ome´canique Ouvrages (LAEGO), Ecole des Mines de Nancy, Nancy Universite´, Nancy, France e-mail:
[email protected] V. Merrien-Soukatchoff Institut de Recherche en Ge´nie Civil et Me´canique (GeM), CNRS—Ecole Centrale/Universite´ de Nantes, Nantes, France T. Korini Polytechnic University of Tirana, Faculty of Geology and Mining, Tirana, Albania A. Thoraval Institut National de l’Environnement Industriel et des Risques (INERIS), Ecole des Mines de Nancy, Nancy, France
Keywords Modelling Fractures Rock mass Blocks Discrete fractures network Stochastic Risk analysis
1 Introduction The study of fractured rock masses has led to the development of software that is able to represent their geometry; mechanical, hydraulic or thermal behaviour; and coupled actions. In particular, Discrete Fracture Network (DFN) codes have become popular during the last 20 years, because they take the close structure of the rock mass into account. Indeed, discontinuities play a major role in many analyses (stability analysis, water flow, and heat transfer). Progress in rock mass representation has led to better models of the geometry of the discontinuities in rock masses and more accurate consideration of discontinuity behaviour and rock masses. The geometric representation of a rock mass has an important influence on the subsequent mechanical, hydraulic or thermal computation. The ‘‘DEM (discrete element methods) model requires realistic numerical representation of a fracture-block system assuming that the input fracture data are reliable and detailed enough’’ (Jing 2000). According to Jing (2000), there are ‘‘three approaches for generating block structures for DEM models: the constructive solid geometry (CSG), successive space subdivision and boundary representation’’. However, choices are often made in geometric representation depending on subsequent concerns, i.e., hydraulic (including petroleum) or stability evaluation. For example, code such as Fracaflow (Sausse et al. 2008; Panien et al. 2010; Iding and Ringrose 2010) or Fracman (Dershowitz et al. 1998) appears to be designed more for hydraulic purposes. Even if Fracman allows stability analysis through its RockBlock downstream module (see
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for example Starzec and Andersson 2002), it was mostly used for hydraulic purposes and the stability analysis is restricted to the first ‘‘crown’’ around an excavation. This paper will focus on the DFN design for stability investigation purposes and on rigid block stochastic stability analysis in particular. In fact, even if rigid block stability studies are supplanted by more powerful methods that are able to take into account the deformability of rock masses and sophisticated behaviour of rock discontinuities, ‘‘In most cases we are concerned about the gross stability of a block rather than internal deformation and failure of the blocks’’ (Priest 1993) and, if the DFN concept is used widely as underlined by Starzec and Andersson (2002), relatively a few studies can predict the occurrence and size of key blocks based on a stochastic 3D fracture network. In the recent years, more rigid block softwares have been developed based on the previous work of Goodman and Shi (1985), Warburton (1981, 1985, 1987, 1993), Dershowitz et al. (1998), Priest (1993), Einstein (1993), Einstein et al. (1983), Hoek and Bray (1981), Lin and Fairhurst (1988), among others. Mene´ndez-Dı´az et al. (2009) cited 10 software programs (including their own program ASTUR) and we note that Fracman (Starzec and Andersson 2002) and BLOCSTAB (Song et al. 2001), also used for stability analysis are not quoted by Mene´ndezDı´az et al. (2009). More specialised landslide applications are also cited by Singh and Goel (2002). According to Mene´ndez-Dı´az et al. (2009), software can be classified into three main groups, but much software take a ‘‘ubiquitous approach’’, which mean that ‘‘given sets of discontinuities and an excavation surface are assumed to be able to occur everywhere and anywhere in space’’. This ubiquitous approach leads to identification of the maximum block size due to sets of discontinuities and an excavation. Since the work of Heliot (1988) a 3D Discrete Fracture Network software named RESOBLOK, which is able to provide a database that is representative of the deterministic or stochastic geometry of a rock mass, has been developed. Modules able to generate a quick stability analysis (Baroudi et al. 1990) and display this database and histograms of the volume of blocks (and of the unstable blocks) or the surface area have been added. First, this paper will present the general framework of RESOBLOK software, which is used for stochastic stability analyses. Because stability analyses are linked to block definition, fracture generation will be described in the third section, followed by the stability analysis principles with and without reinforcement. Possible stochastic analyses are then described using examples in Sect. 5, and emphasis is put on the ability of the code to provide statistical output (such as the distribution of unstable blocks, especially, their number, type, total volume, and mean volume). Finally, the implementation of the stochastic
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analysis and possible links with more sophisticated analyses are highlighted.
2 Presentation of RESOBLOK Software RESOBLOK is an integrated tool used to model a fractured rock mass, which is represented by a block assembly separated by joints. Its architecture is organised into modules as described below: – –
geometric modules able to represent the rock mass as blocks separated by joints; a set of downstream modules to study the block assembly (Fig. 1).
The geometric representation is computed by a block generator, ‘‘bg’’ (LaEGO 2008b). This generator requires a ‘‘scenario file’’ as input to gather the information representing the studied rock mass and supplies an output file that describes the generated block assembly. The output file is the ‘‘database’’ and depicts the geometry of the blocks, the joints and the mechanical properties associated with both the joints and the rock mass. The studied zone (the ‘‘zone of interest’’) is defined as a rectangular parallelepiped with coordinate extension in three spatial directions. A special reference, called the global reference, is first specified in the geo-reference using two orientations that represent the x- and z-axes. The y-axis is set to be normal to both the x- and z-axes (Heliot 1988). The ‘‘zone of interest’’ is then subdivided into blocks by the progressive insertion of discontinuities (faults, strata, etc.) that are defined deterministically or stochastically. In the latter case, a set of fractures is defined using one statistical law that defines orientation and a second that defines spacing. The fracture persistence is taken into account by terminating extension into another set of fractures or by defining the fracture as polygonal (Bennani 1990; Thoraval and Renaud 2003). To aid in data input and to manage the data errors, a processing-specific language, BGL (Block Generation Language), has been developed (Heliot 1988). BGL allows a scenario file to be written, which will be interpreted by the ‘‘bg’’ command to generate the block assembly as a database file. Use of a scenario file may be less convenient in the first stage than use of an interactive mode, but it allows the input to be revised and facilitates parametric studies. Moreover, it does not require the development of a heavy and costly error management system. When the discontinuities are defined stochastically, the same scenario file allows an infinite number of possible geometries to be generated. The ‘‘bg’’ command uses a series of random numbers to generate different block assemblies from the same scenario file. The choice of random number series is
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Fig. 1 The modular organisation of RESOBLOK
Stress&strain analysis (3DEC)
Representation of blocks
2D Cross section, stress&strain analysis (UDEC) … … ...
bd
Scenario File in BGL language
bg Block generator = BGL interpreter
bh Database of blocks
Data analysis
bsa
? Stability analysis of isolated blocks
specified using a parametric number, the ‘‘simulation number’’. RESOBLOK has the following downstream modules (Fig. 1): –
– – –
BD (block display) allows an interactive, rotatable 3D display of the fracture zone, the unstable blocks, crosssections, etc. BH (block histogram) displays various histograms on the blocks database, for example block size distribution. BSA is used for stability analyses (see Sect. 4). Interfaces with other codes (see Sect. 6).
3 Fracture Generation and Realisation of the Block Database Although parts of the RESOBLOK software were described some years ago by Heliot (1988), this paragraph summarises the principal features and improvements in the fracture generation. Two types of fracture generators are implemented in RESOBLOK; both consider fractures as portions of planes. In fact, even if deviation from planarity can be observed, complete observation in 3Ds is difficult (Dershowitz and Einstein 1988) and generally does not provide enough data to restore non-planarity. The first fracture generator generates ‘‘unbounded joints’’, i.e., fractures with an ‘‘undefined’’ extension (Dershowitz and Einstein 1988), and allows hierarchic links between discontinuities to be restored. It can be classified as a ‘‘successive space subdivision’’ according to Jing (2000) and is used subsequently for stability analysis. The second fracture generator generates ‘‘bounded joints’’ (i.e., fractures of finite extension) and is ‘‘hydraulic-oriented’’, but prevents stability analysis. It can be considered to be a
‘‘boundary representation’’, even if it does not exactly correspond to the description proposed by Jing (2000). The respective advantages of each generator will be discussed in this section. 3.1 Hierarchic Fracture Generator The aim of this generator is to generate fractures ‘‘from geological evidence’’ (Heliot 1988) by taking into account the chronology of tectonic events (natural events or events induced by anthropogenic action) that have occurred at the site, assuming that the fracture pattern is the result of several tectonic episodes, each of which is characterised by its own stress field that produces one or several sets of fractures (Heliot 1988). RESOBLOK allows the following: (a) The zone of interest (i.e., the entire-modelled region) may be pre-cut into sub-zones, which are delimited by potential major discontinuities recorded in situ. Major discontinuities can be faults or stratigraphic planes. (b) The entire zone of interest, sub-zones or set of subzones may be cut up to their boundaries without consideration of other fractures, which allows simultaneously generated discontinuities (e.g., conjugate faults) to be simulated. (c) The entire zone, sub-zone or set of sub-zones may be cut by a set of joints; the extension is limited by other sets that are previously defined, constituting ‘‘restricted joints’’ (Zhang and Einstein 2010) that allow the fracture hierarchy to be taken into account. The fractures are generated according to either a deterministic or a stochastic process. For deterministic generation, the user must introduce the orientation of the plane
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(which can be defined in several ways) and a point belonging to the plane. For stochastic generation, the required inputs are the parameters of the statistical law of orientation (the Fisher law is used), the statistical law of spacing (uniform, normal, lognormal or exponential laws are available) and their parameters. We note that the last stage of hierarchic fracture is often the generation of planes that define an excavation. This excavation can be identified by a series of planes or, for a tunnel, its centre, radius, direction and the number of segments used to represent the circle. Sample results of the fracturing process are shown in Figs. 2, 3. The first example (Fig. 2) depicts a tunnel in the French Pyrenees (Merrien-Soukatchoff et al. 2007; Gasc-Barbier et al. 2006), and the second (Fig. 3) is a chalk cliff near the sea. This generator allows convex blocks to be identified whose stability can be checked through the BSA downstream module (see Sect. 4). 3.2 Limited Extension Fracture Generator The generation of joints using the first generator leads to limited joints of polygonal shape that can be questionable as discussed by Zhang and Einstein (2010) or previously by Priest (1993). Termination of one fracture on another is one of many scenarios. A second polygonal generator has been implemented in RESOBLOK to take the extension of fractures into account. Fractures are represented as polygons (see Fig. 4) whose vertices are points on a circle, according to the methodology proposed by Cacas (1989); Cacas et al. (1990). This circle belongs to a plane defined by its equation. The polygon characteristics are as follows:
Fig. 2 Hierarchic fracture generator: two simulations from a stochastic scenario file of a fracture zone mass around a tunnel. The orientation data are those presented in Table 1
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– –
– – –
the number of vertices is defined by the user; the circle radius is fixed by the user or defined stochastically. For stochastic generation, the required inputs are the statistical law for the radius (uniform, normal, lognormal, exponential or power laws are available) and its parameters; the orientation is fixed by the user or defined by a statistical law of orientation (the Fisher law is used); the spacing between the vertices can be uniform (equidistant vertices) or random; the coordinates of the centre are fixed (i.e., defined by the user) or stochastically generated according to a Poisson process.
We note that a joint, in this case, does not necessarily split the domain into blocks; thus, the joints are termed free joints (see Fig. 4). Moreover, this type of fracture has led to the development of a hydraulic structure in RESOBLOK. Currently, this generator is not hierarchic unlike the previous one. This generator has been used to generate 3D simulations as a contribution to the European project BENCHPAR (Thoraval and Renaud 2003). In this case, the following assumptions are made: –
–
–
fracture density: the number of fracture centres per unit volume of rock—P30 (see Appendix) is assumed to follow Poisson’s law. Fracture orientation is assumed to follow a Fisher law. Average dip, dip direction and the Fisher coefficient (k) must be defined for the different fracture sets. Fracture length is assumed to follow a power law distribution of the form N = CL-D, where N is the number of fractures that are equal to or greater than a given trace length L, D is the fractal dimension and C is a constant.
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Fig. 3 RESOBLOK representation of a chalk cliff and UDEC geometry obtained from a cross-section
(D) must be given for the different fracture sets. The simulations (Fig. 5) were performed based on the consideration that P30 can be estimated from P10 values. Average values of P21, computed from 10 RESOBLOK simulations, were compared with P21 measurements (5 m/m2 for a fracture length threshold of 0.5 m) to evaluate the realism of the simulations. The results for different model sizes are gathered in Table 3. We note that the computed P30 values are close to the input data (P10) and the total computed P21 value (4.39 m/m2 for a 37 m3 model and 5.01 m/m2 for a 64 m3 model) are close to the measured value (5 m/m2). 3.3 Combining both Fracture Generators
Fig. 4 Polygonal limited extension fracture (free-joint)
The data based on site characterisation of the area near Sellafield, Cumbria, England were collected by United Kingdom Nirex Limited (Andersson et al. 2003). The joint set properties are gathered in Table 2. The average dip, dip direction, Fisher coefficient (k), fracture density P30 (for a fracture length threshold of 0.5 m) and fractal dimension
The hierarchic generator allows the generation of joints that cut the space into blocks, and the stability of these blocks can then be analysed. The limited extension generator creates free joints that may or may not define blocks. As a result, stability analysis is not possible. The two generators can be partially combined as follows: –
– Table 1 Discontinuity orientation for the tunnel case presented in Fig. 2
Orientation Dip direction Dip
F1
F2
F3
F4
147
204
319
279
74
85
84
7
Spacing Mean (m)
0.5
1
1
1.2
k parameter of exponential law
2
1
1
0.83
–
–
the first stage cuts out the zone of interest using discontinuities of finite extent, which generates the first block assembly, it is then possible to generate joints with finite extension and, to compute new intersections with existing blocks and other finite extension joints; it also becomes possible to update the data structure; thus, a new list of blocks is obtained, together with a list of the joints of non-cutting blocks (free joints included).
This methodology has been used within the scope of the European program CAD-PUMA (Crassoulis et al. 2001; Thoraval 2005), whose objective was to develop a design for underground marble quarries (Fig. 6). RESOBLOK was
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164 Table 2 Joint set properties for one geological formation of Sellafield
V. Merrien-Soukatchoff et al.
Fracture set
Dip ()
D 1.2
8
145
5.9
0.16
88
148
9
0.31
0.7
3
76
21
10
1.0
1.1
4
69
87
10
0.5
1.1
The orientation of the fracture at site B is presented in Fig. 7 and Table 5. To represent this site, most of the measured fractures were introduced in the model by using the RESOBLOK-limited extension joint (polygonal joint) generator (Fig. 8) deterministically in the area where deterministic data were available. This facilitated an accurate comparison of the modelled and in situ traces of fractures through a cross-section that matches the planes of measurement. Fig. 9 shows good correspondence between the modelled and measured fracture traces. After this good coincidence was obtained, a second simulation was performed using the hierarchic generator such that a stability analysis could be performed, because the polygonal generator does not always define blocks. Other fractures were also introduced using the statistical hierarchic generator (the simulations assume statistical laws for orientation, spacing and extension adjusted from measurements) into the zones where no measurement was collected (Fig. 8). Determination of the hierarchical rules between the various sets of fractures required special care to avoid non-existing over-cutting of the marble. Marble mining consists of cutting out rock masses in the form of parallelepipeds whose sizes are determined by the owner. At the Dionyssomarble quarry, the dimensions are 4.5 m high by 6 m broad and 2 m deep (54 m3). The size of the recovered blocks depends both on the size of the gallery and the distribution of fractures that cut the rock mass. Thus, the size also depends on the angle between the
Model size
Joint set
Measured Pa10 (fracture nb/m)
Computed P31 (fracture nb/m3)
Computed P21 (fracture length/m2)
33 = 27 m3
1
0.16
0.19
0.11
2
0.31
0.41
1.37
3
1.0
1.19
1.87
4
0.5
0.60
0.97
Total 43 = 64 m3
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P30 (assumed equal to P10)
2
used to simulate the fracture network both deterministically and statistically, starting from in situ measurements. The purpose of the simulations was to possibly improve the profitability by determining the work geometry (gallery orientation, extension choices) that would maximise the size of the marble blocks and minimise the cost of the reinforcements needed to guarantee site safety.
a Measured values from borehole taking into account fractures whose length is greater than 0.5 m
k
1
Fig. 5 Polygonal fracture network generated using RESOBLOK (from Sellafield data)
Table 3 Verification and validation of RESOBLOK fracture network simulations
Dip direction d ()
2.39
4.39
1 2
0.16 0.31
0.18 0.33
0.26 1.27
3
1.0
1.05
2.57
4
0.5
0.55
0.97
2.12
5.01
Total
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Table 4 Impact of gallery orientation on the size of the recovered ‘‘big’’ marble blocks, example of site A (see Fig. 6) of Dionyssomarble quarry Gallery orientation
Gallery orientation (corresponding to the azimuth of principal fracture sets) Actual gallery orientations 240
Average (from several simulations) 6.8 (30.5%) number of blocks with a volume between 43 and 54 m3
300
188
306
36
13.4 (60.2%)
7 (31.4%)
12.6 (56.6%) 8.2 (36.8%)
(% total number)
Fig. 6 Location of the underground pilot site at Dionyssomarble quarry in Greece
Fig. 7 Fracture pole—site B of Dionyssomarble quarry with site B of Fig. 6
cutting plane orientation and the fracture set orientation. The block volume distribution must thus depend on the gallery orientation, because the galleries are parallel, perpendicular or oblique to the cutting plane. In general, galleries are positioned parallel to a major fracture to optimise block size a priori. RESOBLOK computes the distribution of the cut blocks by class of volume for various gallery orientations (defined by the angle a to the north see Fig. 10). The distribution shown (Fig. 10) is an average distribution computed from several stochastic simulations. As presented in Table 4, the optimal orientation corresponds to a gallery that is parallel to the family of azimuth 300. We note that this orientation almost corresponds to that selected by the owners (azimuth 306) for
Table 5 Impact of gallery orientation on the volume of unstable blocks, example of site B (see Fig. 6) of the Dionyssomarble quarry Gallery orientation
Average (from several simulations) volume of unstable blocks (m3)
Gallery orientation (corresponding to the azimuth of principal fracture sets)
Actual gallery orientations
110
250
212
160
306
36
826.2
473.7
253.7
168.4
883.7
225.1
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4 Stability Analysis Module Stochastic simulation Dé ter
m
ini
sti c
sim
ula
tio
n
Stochastic simulation Fig. 8 Cross-section of the simulation of the fracture network of the Dionyssomarble quarry with RESOBLOK combining both generators (site B of Fig. 6)
the direction of the current gallery. RESOBLOK also predicts that the size of the blocks recovered in the perpendicular direction (azimuth 36) is much less important.
RESOBLOK has a downstream module BSA (for Block Stability Analysis) that allows the stability of blocks in contact with an excavation to be analysed by simple computations based on the limit equilibrium (Asof 1991) or energy evaluation developed by Korini (1988) and based on the algorithms of Belytschko et al. (1984). BSA (LaEGO 2008a) tests whether isolated blocks next to an underground or open pit excavation are detachable. If possible, the boundaries of excavation are modified and the computation continues iteratively, until no more blocks are unstable. These iterative computations are important (as will be shown using further examples), but many codes only examine the stability of blocks next to the excavation. As the geometric representation of the rock mass can be either deterministic or probabilistic in RESOBLOK (or a combination of these), we note that the stability computation can use a ‘‘specific’’ approach, a ‘‘probabilistic’’ approach or a combination of both, using the classification of Mene´ndez-Dı´az et al. (2009).
Fig. 9 Comparison of the survey (left side) and the deterministic simulation (right side) using a polygonal generator (site B of Fig. 6)
North Gallery
Block number
Fig. 10 Cutting block numbers per class of volume in m3 (for a = 36)
Class of volume
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Use of an Integrated Discrete Fracture Network Code for Stochastic Stability Analyses
In addition, blocks can be of any shape, and the analysis is not limited to tetrahedral or pentahedral blocks as in much of the software quoted by Mene´ndez-Dı´az et al. (2009) or to planar and wedge failure as in other probabilistic studies (e.g., Park et al. 2005).
•
• 4.1 Stability Analysis Principle The stability analysis is based on Warburton’s (1981) algorithm and the following two stages (see Fig. 11): •
BlBloc oc k B
B
•
the geometric analysis enables the exclusion of any further analysis of blocks that cannot be removed due to their geometry and the direction of the resultant driving force; the mechanical analysis examines the potential movement of geometrically removable blocks (or kinematic feasibility according Priest 1993) and, if applicable, computes the block safety factor by taking into account both the geotechnical properties of discontinuities and the density of blocks introduced into the scenario file. The potential movements examined in this manner are the following:
RR
Excavation
Geometric analysis
STABILITY OF BLOCK B
Possible geometric movement
No movement
Unstable block
Test rotation
Mechanical analysis
Stable to rotation
Movement parallel to one or several faces
Direct fall
0
Fig. 11 BSA analysis
Stable block because of friction
Unstable block
1
Factor of safety
•
167
Direct fall: if, due to gravity, a direct fall is possible, then the block is declared unstable. However, no safety factor can be computed (except in the case of the support), because the joint is presumed to have no tensile strength. Movement parallel to one or several faces: in this case, a safety factor can be computed. If the movement is parallel to one or two faces, a limit equilibrium analysis is then possible, and the only parameters needed are the cohesion and friction angle of discontinuities bordering the block as well as the block density. If the movement is parallel to more than two faces, limit equilibrium analysis is not feasible, but an energy-based analysis can be performed (the energy-based analysis requires additional parameters such as the normal and shear stiffness of the discontinuities). Rotation is also checked as an option using vectorial conditions proposed by Lin and Fairhurst (1988).
A BSA run has three stages: data input, computation and output analysis. When fractures are stochastically defined, several realisations of the block assembly (identified by different simulation numbers) exist for each scenario file. A BSA module performs a stability analysis for all block assemblies, and the results can be processed statistically, including the number and volume of unstable blocks, the minimum volume, and the maximum volume (see Fig. 12). We note that the boundary of the ‘‘zone of interest’’, as defined in Sect. 3.1, can influence the stability analysis. If this zone of interest is too small, the evaluation of the volume of unstable blocks can be underestimated. We must ensure that the boundaries are far enough from the excavation that they do not influence the stability evaluation. The process is quite similar to that adopted when computing a finite element model to ensure that the boundaries do not influence the results (Merrien-Soukatchoff and Omraci 2000). One of the advantages of RESOBLOK software is that the stability analysis can be performed statistically (Baroudi et al. 1990). Baroudi et al. (1992) showed that in a slope stability problem, a minimum of 50 geometrical simulations must be performed to achieve a reasonable outcome in the stability analysis. In most cases, running 50 simulations allows stable values of the mean and the standard deviation as well as the histogram of variables such as the mean volume of unstable blocks, the total unstable volume and the number of unstable blocks to be obtained. This is confirmed by the simulation run for the tunnel presented in Fig. 2. Figures 13, 14 show that the average number and volume of unstable blocks is quite stable and remains stable up to 140 simulations. This type of figure can be automatically displayed using BSA analysis to determine whether the number of simulations is
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Fig. 12 Stochastic aspects of RESOBLOK and BSA
* Zone of interest * Discontinuities sets -deterministic -statistical --> geological scenario
blocks data base
* Excavation realization n°1 bg realisation1
Data file with BGL
blocks data base
bg realisation 2
bg realisation i
bsa realisation 1
bsa realisation 2
realization n°2
bsa realisation i
Mean stability analysis
bsa realisation n
bg realisation n
blocks data base realization n°n
Average volume (in m3) of unstable blocks
Average number of unstable blocks
50,00
40,00
30,00
20,00
10,00
0 0
10 20 30 40 50 60 70 80 90 100 110 120 130 140
Number of simulations
20,00
15,00
10,00
5,00
0 0
10 20 30 40 50 60 70 80 90 100 110 120 130 140
Number of simulations
Fig. 13 Variation in the average number of unstable blocks vs. number of simulations used for the tunnel stability analysis of Fig. 2 and Table 1
Fig. 14 Variation in the average volume of unstable blocks vs. the number of simulations used for the tunnel stability analysis of Fig. 2 and Table 1
sufficient. At present, the random variables are limited to the fracture orientation and spacing. However, the future development of random mechanical characteristics for fractures is foreseen. It is possible to take given stresses around excavations into account. This option has been little used because, due to stress redistribution, it makes little sense after the first iteration. We note the possibility of choosing between the use of a fixed number of iterations or to iterate until no more blocks are unstable (see example Sect. 5.4).
4.2 Stability Calculation of Blocks Reinforced by Bolting
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If the number of unstable blocks is important, system stability can be maintained by using bolts or cables to reinforce the rock mass. This role is taken into account in BSA limit equilibrium computation by considering that bolt resistance may prevent the direct fall or sliding of an isolated block and may avoid the propagation of instability by stabilising key blocks. The way this is formulated is developed in the following sections.
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4.2.1 Calculation of the Safety Factor of Blocks Reinforced by Bolting
– –
Two types of behaviour can be taken into account in the analysis: active reinforcement and passive reinforcement. The first corresponds to the use of pre-stressed bolts and the second corresponds to the use of a fully grouted anchor (Seegmiller 1982). In both cases, the effect of bolts is taken into account (Korini et al. 1993) by adding a force into the force balance whose module is the strength of the bolt and acts in the direction of the bolt. The force induced by the bolt is resolved by the following:
–
For the three possible cases of instability, the formulation of the safety factor calculation may be summarised as follows: Direct fall (Fig. 15): Pnb cp ½ð^ rbi r^Þ sgnð^ rbi r^Þ fs ¼ i¼1 i ~ jRj
–
– –
–
a force parallel to the movement (in the opposite direction); a force perpendicular to the direction of movement.
In the case of the direct fall of a block, only the first component is taken into account (suspension role). In the case of sliding along one or more faces, the force parallel to the movement direction is added to the shear strength of the discontinuity, and the force perpendicular to the movement direction increases the normal stress. Therefore, sliding also increases the shear strength limit. For active reinforcement, the force exerted parallel to the movement is numbered among the driving forces, whereas for passive behaviour, it is numbered among the resisting forces. Depending on the case (active or passive reinforcement), the safety factor is calculated as: ða) Active reinforcement u þ Fnb tan u fsa ¼ Fc þ FnwFtan tw Ftb
ðb) Passive reinforcement fsp ¼ Fc þ Fnw tan uFþtwFnb tan u þ Ftb
where fsa and fsp are the safety factors for active and passive reinforcement, respectively. – – –
Fc is the cohesion force on the sliding face; Fnw is the normal component of active forces; Fnb is the normal component of the force induced by the bolts;
Ftw is the tangential component of active forces; Ftb is the tangent component of the force induced by the bolts; u is the friction angle over the sliding face.
where cpi is the bolt-porting capacity of bolt i, r^bi is the unit vector of the bolt orientation, r^ is the unit vector oriented along the resultant force acting over the block, ~ – R is the vector of active forces acting over the block (e.g., weight, water pressure), and – nb is the number of bolts. Sliding along a single face (Fig. 16): ða) Active reinforcement fsa ¼
C0 A þ ðFn þ Fnb Þtan u Ft Ftb
ðb) Passive reinforcement fsp ¼ C0 A þ ðFn þ FFtnb Þtan u þ Ftb
P ~ n^ Fnb ¼ nb ðcpi ðr^bi n^ÞÞ where Fn ¼ R i¼1 R jr^ ðr^ n^Þ n^j Ft ¼ ~ Xnb Ftb ¼ cpi jr^bi ðr^bi n^Þ n^j i¼1 – – – – – – –
C0 is the cohesion on the sliding face, A is the area of the sliding face, u is the friction angle over the sliding face, Fn is the normal component of the active forces (e.g., weight, water pressure), Fnb is the normal component of the force induced by the bolts on the sliding face, Ft is the tangent component of the active forces, and n^ is the unit normal vector perpendicular to the sliding oriented to the interior of the block.
rˆbi
cpi
ur R
Fig. 15 Computation scheme for a block in direct fall reinforced by bolting
Fig. 16 Block in planar sliding reinforced by bolting
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– 2
–
1
R
m
–
– Fig. 17 The movement of a wedge reinforced by bolting
–
n^1 and n^2 are the unit vectors that are normal to the first and the second sliding faces, respectively, Fn1 and Fn2 are the normal components of the active forces on the first and the second sliding faces, respectively, Fnb1 and Fnb2 are the normal components of forces induced by the bolts on the first and the second sliding faces, respectively, Ft12 is the component of the active force in the sliding direction, and Ftb is the component of force induced by the bolts in the sliding direction.
Wedge sliding (Fig. 17): (a) Active reinforcement
fsa ¼
C01 A1 þ C02 A2 þ ðFn1 þ Fnb1 Þ tan u1 þ ðFn2 þ Fnb2 Þ tan u2 Ft12 Ftb
(b) Passive reinforcement
fsp ¼
C01 A1 þ C02 A2 þ ðFn1 þ Fnb1 Þ tan u1 þ ðFn2 þ Fnb2 Þ tan u2 þ Ftb Ft12
where ~ R½r^ ð^ r n^2 Þ n^2 n^1 Fn1 ¼ 2 ~ I 12 ~ R½r^ ð^ r n^1 Þ n^1 n^2 Fn2 ¼ 2 ~ I 12 Pnb rbi n^2 Þ n^2 n^1 j jcpi ½r^bi ð^ Fnb1 ¼ i¼1 2 ~ I 12 Pnb rbi n^1 Þ n^1 n^2 j jcpi ½r^bi ð^ Fnb2 ¼ i¼1 2 ~ I 12 ~ I^12 Þ sgn ð^ Ft12 ¼ ðR r I^12 Þ Pnb rbi I^12 Þ sgn ð^ rbi I^12 Þ with ~ I 12 ¼ n^1 ^ n^2 Ftb ¼ i¼1 cpi ð^ and I^12 the associated unit vector. – – –
C01 and C02 are the cohesion on the first and second sliding faces, respectively, A1 and A2 are the areas of the first and the second sliding faces, respectively, u1 and u2 are the friction angles on the first and the second sliding faces, respectively,
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4.2.2 Setting up a Bolting Pattern Generally, bolting is applied according to a regular pattern, and the problem for each specific excavation is to find an adequate bolting pattern to prevent block instabilities. Given that the distribution of fractures is known, we generally obtain various block databases by statistical simulation, and each geometrical model can be computed using a range of bolting patterns. The bolting pattern is applied over a given surface that corresponds to one or multiple faces that define the excavation. It can be the roof or the lateral faces of an underground working or the slope of an open pit. The necessary parameters to define a bolting pattern are (Figs. 18, 19): – – – – –
The plane equation corresponding to the face to be bolted, A starting point and a second point (to define the orientation of the pattern), Angles a1 and a2, The dimensions of the mesh d1 and d2, Bolt length,
Use of an Integrated Discrete Fracture Network Code for Stochastic Stability Analyses Fig. 18 Bolting data and various patterns
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second point second point
(xs, ys, zs)
(a)
(b)
1
2
=0 = 90˚
d1= d2=d d2 2
d
d1
1
2
d
(xf, yf, zf) first point
first point second point
(d)
(c)
orientation
`
1
d2
2
d1
1
= 2 = 45˚ d1= d2=d
Lb L b- bolt length
first point
Second point
4.2.4 The following methodology is adopted –
90° d2
d1
orientation
– First point
Fig. 19 Setting a bolting pattern in a slope
–
– –
–
Bolt orientation, and The porting capacity of the bolts.
The parameters above ensure that various patterns can be considered (Fig. 18b, c). 4.2.3 Optimisation of the Bolting Pattern Stability analysis consists of determining the safety factor of potentially unstable blocks that are possibly reinforced by bolting.
The number of efficient bolts is determined for every removable block, meaning that the bolts, starting from the free face, cross the block and have sufficient length to be anchored to the neighbouring blocks. The movement type for each block is fixed based on Warburton’s algorithm (unremovable, direct fall, plan sliding, wedge sliding, and plurihedron sliding) without taking the bolts into account. Block safety factor is evaluated by taking into account the respective bolts. The stability of the tested blocks is determined.
For each simulation and bolting pattern, the results given are the number of unstable blocks, the volume and weight of each unstable block and the total weight of unstable blocks. It is also possible to obtain the results based on the number (or volume) of blocks corresponding to various types of instability (e.g., free-fall or plane sliding). Finally, for the set of simulations, it is possible to calculate the average volume (and/or the average weight) of unstable blocks for every bolting pattern. As a result, it is
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possible to choose the pattern that best minimises block instability. The choice can be confirmed using histograms that present the distribution of the number and volume of unstable blocks for all simulations (the histograms allow the determination of whether or not blocks that remain unstable after bolting have limited volume). Furthermore, small blocks falling through the bolting mesh are not significant, because, in practice, the bolting includes wire mesh that prevents the fall of small blocks. The optimalbolting pattern is not the best choice for each block. Rather, it is the choice that minimises the unstable block volume; it combines the best pattern geometry for deterministically or statistically known fracture geometries. This optimal-bolting pattern can differ from those chosen to support the maximum wedge, but the maximum wedge is rarely expressed. We note that the choice of an optimal pattern must also include economic considerations. 4.3 Advantages and Limitations of BSA As pointed out by Warburton (1993), although ‘‘many solutions have been published since the mid-1960s, it took almost 15 years for the block geometry to progress beyond tetrahedron with one free-face’’. In addition to this statement, we note that it took a number of years to take into account the possibility of rotation, the support and the scattering of the geometry in a single computer code. These improvements have been implemented progressively in BSA, which now allows a statistical analysis of block stability, including rotation, either with or without support. BSA analyses avoid some of the shortcomings of the previous rock block computer codes; the iterative analysis allows for more than a single ‘‘crown’’ of unstable blocks. However, BSA retains some limitations. It does not avoid the problem (as pointed out by Warburton 1993) of some declared unstable blocks becoming self-supporting by rotation, but the computation is thus conservative. BSA does not take into account groups of blocks that are possibly unstable, despite each block being stable in isolation, especially when bolted. This problem has been taken into account by Yarahmadi Bafghi and Verdel (2004), but only for 2D space. The algorithms are not yet written to represent 3D spaces. At present, the stability analysis assumes that the fracture cohesion and friction angle are constant along a given fracture and fracture persistence is not taken into account, which should be improved in the further development. 5 Stability Analysis Example RESOBLOK and BSA have been used by different authors for stability analyses (Korini et al. 1993, Deangeli et al.
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2001, Jaboyedoff et al. 2004, Gasc-Barbier et al. 2006, Segalini et al. 2006, Merrien-Soukatchoff et al. 2007, Ferrero et al. 2009, Korini and Merrien-Soukatchoff 2009). We present several typical cases below and make comparisons with well-known literature examples. 5.1 Marble Underground Quarry Referring to the marble quarry example presented in Sect. 3.3, BSA was used to compute the unstable block volume according to the gallery orientation in site B (see Fig. 6) of the Dionyssomarble quarry. The volume obtained in each case (Table 5) is an average volume computed from several simulations. For these calculations, no bolting was assumed at the gallery roofs. According to Table 5, the optimal orientation (minimal volume of unstable blocks) corresponds to a gallery that is parallel to the fracture set with an azimuth of 160. The results obtained for an orientation equal to that of the current galleries are in complete agreement with the onsite observations. We note that the owners reinforced many sectors of the N 306 galleries to prevent unstable blocks from falling whereas the reinforcement was much lighter in the N 36 galleries. The previous analysis was also applied to propose an optimized choice concerning an extension of the quarry. The alternatives (see Fig. 20) were a North-East (case 1) and a South-West extension. The South-West extension generates an average volume of unstable blocks of 427 m3, whereas the North-East extension induces a much higher volume of approximately 1,200 m3. High values are obtained in both cases, because of the extension, the gallery widens from 6 to 12 m. These values increase still more for widths of 15 m (3,033 m3) or 20 m (5,763 m3). Under these conditions, it is necessary to reinforce the stability of the roof by bolting in the unstable zones. According to the BSA computation, for a gallery width of 20 m (Fig. 21), a bolt density of 0.25 B/m2 (resp. 1 B/m2) makes it possible to reduce the unstable volume to 1,221 m3 (174 m3). 5.2 Road Cut and Model Size Figure 22 shows the case of a slope cutting for a road in southern France (detailed in Gasc-Barbier et al. 2008 and Godefroy et al. 2009). The studied slope is excavated in gneiss, and the cut has an average height of 48 m with a slope of 76. Scan lines and field observations determined four discontinuity families as well as their Langevin Fischer coefficient of dispersion (k), their spacing, and the fracture ends (see Table 6). The rock has a density of 25 kN/m3, and different mechanical properties were considered for the stability analysis. The computation presented here considered for the fractures:
Use of an Integrated Discrete Fracture Network Code for Stochastic Stability Analyses Fig. 20 Location of the quarry potential extensions (1–NorthEast extension, 2–South-West extension) of site A (see Fig. 6)
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1 – North-East extension
2 – South-West extension
1
Pillar let in place
Nord
2 Already excavated galeries
Fig. 21 Roof reinforcement using rock bolting
– –
a cohesion of 0 kPa (no bridges were considered), and a friction angle of 28.
RESOBLOK (Fig. 23) and BSA were run with and without reinforcement. Figure 24 shows the differences in both cases (resulting from 90 simulations) in the number of unstable blocks, total volume, and mean volume of unstable blocks. The computed-bolting scheme (Fig. 25), corresponding to those adopted in situ, led to a very few instabilities. The user of such software is faced with a choice of scale to be adopted for the model. In this case, we compare the following: – Fig. 22 Ax-les-Thermes City diversion road cut
the stochastic result by progressively increasing the width (the x-coordinate corresponding to a horizontal parallel to the slope) and the ‘‘depth’’ (the y-coordinate
Table 6 Discontinuity geometry for a cutting slope near Ax-Les-Thermes
Orientation
Spacing
F1
F2
F3
F4
Dip direction
249
173
111
21
Dip
83
44
89
54
K dispersion coefficient from the Fisher law
86
105
196.5
21.25
Mean (m)
4
2.5
3
2.5
k parameter of exponential law
0.25
0.4
0.33
0.4
Hierarchic rules
Stop on F4
Stop on F4
Stop on F4
Infinite
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unstable blocks is less than 1 m3, and 98% of the mean volume of unstable blocks is less than 10 m3. Without bolts, 100% of the mean volume of unstable blocks is greater than 8 m3, and 98% of the mean volume of unstable blocks is greater than 10 m3. Accurate comparisons to field data are difficult. As a result of bolting, no unstable block was observed on site, which confirms the effect of bolting predicted by the simulation qualitatively, but not quantitatively. 5.3 Harrison and Hudson Underground Excavation
Fig. 23 Ax-les-Thermes RESOBLOK
–
diversion
road
cut,
modelled
with
corresponding to a horizontal perpendicular to the slope). The height is fixed by the in situ case and also by the ‘‘largest deterministic geometrically unstable cluster of blocks,’’ which is defined by combining the four families of discontinuities and considering blocks from the bottom of the slope (see Fig. 26) with very low mechanical characteristics.
The model must have a width greater than 115 m and a ‘‘depth’’ greater than 45 m to include the ‘‘largest deterministic geometrically unstable cluster of blocks’’. Figure 27 shows the variation in the mean volume rate volume of unstable blocks defined by r ¼ Mean of total and Fig. 28 Volume of the model shows the variation in the volume of the unstable blocks. Each point in both figures corresponds to 80–100 simulations (100 simulations are run, but for some cases, the model is too large and the database is not created, because of a block-number limitation). The r ratio is stable when the model is wide enough, and the volume of unstable blocks is stable when the model is deep enough. For different depths, the ratio becomes stable for a model width of approximately 80 m (see Fig. 27), and the volume of unstable blocks becomes stable for a model depth of approximately 40 m (see Fig. 28). Thus, considering the size of the model, which includes the ‘‘largest deterministic geometrically unstable cluster of blocks,’’ (115 9 45 m) is a good order of magnitude for the size of the model to be adopted for a stochastic evaluation of stability. To show the dispersion resulting from the stochastic approach, Fig. 29 depicts the worst and best cases, in terms of unstable blocks, for simulations run without bolting. The results can then be exploited in a probabilistic way. For example, with bolting, 33% of the mean volume of
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We compute a case of underground excavation instability similar to that detailed in Question 19.1 in Harrison and Hudson (2000). Because no data were available on orientation dispersion and spacing, we chose a statistical distribution for each fracture family with a high k Langevin Fischer coefficient of dispersion (10,000, such that the variation around the mean direction was low) and a spacing fitting for an exponential law with a large distance between fractures (a mean of 20 m). Before running the stochastic case, we compared some cases of direct falls of tetrahedral blocks formed by the intersection of fracture sets and the roof. Figure 30 shows the tetrahedral blocks formed by fractures one, two and five; one, four and five; and two, three and five. We approximated the weight of the maximum wedge by progressively increasing the size of the block formed by the combination of three fractures. In fact, because RESOBLOK and BSA are not designed for this purpose, we rounded down the maximum wedge. The weights reported in Fig. 30 are close to those given by Harrison and Hudson (2000), but with slightly lower weight. The use of RESOBLOK and BSA allows the result to be analysed probabilistically. We present the result for a 100 m long-excavation; the model has a 200 m width (perpendicular to the tunnel) and a 300 m depth over the tunnel. Forty percent of the simulations (40 cases from 99 simulations) produce no instability. However, one case (Fig. 31) has a total volume of unstable blocks of 369,095 m3; this volume results from 46 stability iterations (removal of 46 successive blocks). This is more than 3,0009 greater than the volume of the maximum wedge. This volume is theoretical, because it is greater than that of the underground excavation, so in reality, it could not be achieved. The unstable volume is limited due to swelling, and in situ stresses (which have a stabilisation effect) have not been taken into account. We note that this result is quite general. Use of ‘‘the largest deterministic geometrically unstable cluster of blocks’’ approach generally leads to overestimation of the real observed largest block, but this approach could largely underestimate the total unstable volume of blocks. The
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Fig. 24 Histograms for 90 simulations (of 90 m width, 50 m depth models) of the number, total volume, and mean volume of unstable blocks, without (left) and with (right) bolting
histogram of total volume and block volume are obviously related to the input distance between fractures, which was not known in this case and was assumed to be 20 m. Of course, due to the equilibrium computation limits, the results of the BSA stability analysis do not take into account the depth of the tunnel (and, by correlation, the in
situ stress), and an accurate approach cannot be limited to these computations. However, such computations allow the worst configuration to be selected quickly and then run with stress–strain codes. We note that width or a depth smaller than those adopted has a significant influence on the results, which is contrary
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Fig. 25 Bolting scheme: RESOBLOK (left side) in situ (right side)
Fig. 26 The largest deterministic unstable cluster of blocks defined by the four families of Table 6
Fig. 28 Total volume of unstable blocks versus model depth for different widths (from Godefroy et al. 2009)
5.4 Wide Square Excavation (Unwedge Sample Problem #1)
Fig. 27 Rate of unstable blocks versus model width for different depths (from Godefroy et al. 2009)
to the results presented by Starzec and Andersson (2002) for CLAB-2 facility simulations performed using RockBlock. This difference is certainly due to the influence of successive iterations that appear not to exist in RockBlock.
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We performed a similar comparison with an example quoted by RockScience (2009), which examined the maximum wedge due to three discontinuity families in a wide square excavation. The maximum wedge (Fig. 32) computed with RESOBLOK has a volume of 29.182 m3 that is similar to the 29.177 m3 quoted by the RockScience Education site (2009), but the use of RESOBLOK allowed us to complete the determination of the maximum wedge using a stochastic analysis. Assuming a fracture spacing of 5 m for the three discontinuity families, we ran 100 geometrical simulations for a 10 m wide square excavation that was 100 m long. The total number of unstable blocks varied from 10 to 1,887 (0.1–19 blocks/m), and the total volume of unstable blocks varied from 15.95 to 61,004 m3 (0.15–610 m3/m), which
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Fig. 29 Unstable blocks for a model of 100 m width, 60 m depth without bolting: worst simulation (left) best simulation (right)
125 block instability W= 676.3 kN (691 kN for Harrison and Hudson computation)
245 block instability W= 45.9 kN (47.5 kN for Harrison and Hudson computation)
235 block instability W= 2548,9 kN (2615.6 kN for Harrison and Hudson computation)
Fig. 30 Some examples of tetrahedral blocks, represented with RESOBLOK, in direct fall, formed by the intersection of fracture sets and the roof for the underground excavation of Harrison and Hudson (2000)
represents an overestimation when compared with the volume of the maximum wedge. For a single iteration, the total number of unstable blocks varied from 6 to 39 (0.06–0.39 blocks/m), the total volume of unstable blocks varied from 4.35 to 793 m3 (0.04–8 m3/m) and the mean volume of unstable blocks varied from 0.62 to 62.44 m3 (0–0.6 m3/m).
6 Stochastic Implementation and Possible Downstream Analyses Stochastic analyses are of interest, but in the worst cases it could be of interest to have more accurate analyses that take rock mass deformation and fracture behaviour into account.
RESOBLOK, strictly speaking, is a geo-modeller that is interfaced with – –
downstream modules (such as BD or BSA, as above), or other codes.
At present, RESOBLOK is interfaced with the distinct element method and, because Itasca products are widely used, it was interfaced with UDEC and 3DEC (Itasca 1998). However, it can also interface with other discrete elements methods such as NSCD (Non-Smooth Contact Dynamics), which was introduced by Moreau and Jean (Dubois and Jean 2006, Rafiee and Vinches 2008, Rafiee et al. 2008) and developed in LMGC90 software (Dubois 2010). Quick-stability analyses and discrete methods are complementary rather than competitive. In fact, both methods
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The interface RESOBLOK-3DEC allows a 3DEC file to be generated in which the cutting commands are transmitted. Because 3DEC does not allow fractures to be generated that terminate inside a block, polygonal fractures are extended to the next fracture (or block face). However, 3DEC automatically distinguishes two zones (from the information output of RESOBLOK): an ‘‘active’’ zone inside the polygon area (a real joint) and an ‘‘inactive’’ zone outside the polygon (a fictitious joint). Thus, the user will be able to subsequently allot different properties to real and fictitious joints in 3DEC. This interface has been used at the Sellafield site to determine the equivalent hydro-mechanical properties of a fractured rock-mass. RESOBLOK was used to generate the 3D fracture network (as previously described in Sect. 3.2), and 3DEC was used to perform the hydro-mechanical computations needed. It was shown (Thoraval and Renaud 2003) that the extension of the joint up to the next block face (which artificially increases the fracture network connectivity) could lead to overestimation of the equivalent permeability by a factor of two or three. The effect of this prolongation on the equivalent stiffness tensor is small (\20%). Fig. 31 Worst case (from 99 simulations) of the Harrison and Hudson underground excavation for a 100 9 200 9 350 m model
7 Conclusion
Fig. 32 Maximum wedge computed with RESOBLOK for Unwedge sample problem #1
can be used together in a complementary manner: quick, statistic rigid block analyses allow the detection of problematic cases that can then be analysed using more refined discrete methods. To interface RESOBLOK with UDEC, a cross-section is defined by its orientation and position, and a file containing all of the command lines to generate 2D geometry in UDEC is created. To run mechanical analyses, we must add appropriate command lines to the geometry in the UDEC command file. Figure 3 shows a 3D geometry generated using RESOBLOK and a cross-section transferred and displayed with UDEC.
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Much software has been developed to represent the fractured rock mass or to perform different types of stability analyses. However, many stability studies of surface works remain based in 2D and are purely deterministic, especially those that include bolting support. RESOBLOK software together with its downstream module, BSA, which runs iterative stability analyses of isolated blocks, allows both geometrical and mechanical 3D studies and can be coupled to more sophisticated programs if more accurate evaluation is needed. It is the only rapid stability software, to our knowledge, that analyses not only the unstable blocks on the excavation boundary (i.e., the first crown of blocks) but also the unstable blocks that remain after removing this first crown. For risk-analysis, block instability outputs can be coupled with trajectory computation. The rock mass geometries can then be controlled visually in 3D via geometrical indicators such as Pij. Interactive visualisation allows the visualisation in 3D of the fractured volume and the instable zone or of an individual block, and it allows us to rotate the visualisation. The safety factors are computed by limit equilibrium analysis for the possible cases of block movement: direct fall, rotation, plane and wedge sliding. For the particular case of multihedron sliding, an energetic method is used for stress calculation over the sliding faces, and the safety
Use of an Integrated Discrete Fracture Network Code for Stochastic Stability Analyses
factor is evaluated. Some of the included algorithms are used in other codes, but the association in the same code of all these different features including rotation, and the iterative analysis allow RESOBLOK to run a large panel of investigations, deterministically or statistically. RESOBLOK allows investigation of the impact of bolt support. The stability of bolted blocks is determined using a method derived from the Warburton vector methods. The bolt behaviour that is taken into account is only the tensile behaviour and excludes the shearing failure. The choice of the optimal-bolting pattern may be made by analysing and comparing several proposed patterns to minimise the unstable block volume either deterministically or statistically, if the fracture geometry is statistically known. Above, we have demonstrated the use of RESOBLOK for rapid 3D statistical stability analyses of fractured rock masses and risk-analyses. Naturally, the quality of these estimates will depend on the quality and accuracy of inputs (particularly parameters relating to the statistical laws of fracture orientation and spacing). The main obstacle to such analysis remains the lack of in situ data for modelling and for validating models for common works (i.e., not underground research laboratory works). Yet, even with small amounts of data, probabilistic stability analysis can demonstrate a wide range of possible scenarios of instability and narrow this range with increasing amounts of data. Ubiquitous approaches (Mene´ndez-Dı´az et al. 2009) focus on defining the maximum block size when probabilistic approaches describe the total volume and volume distribution of blocks in terms of risk. As shown in the examples, the total volume of unstable blocks is often greater than the volume of the maximum wedge due to iterative computation, despite the maximum wedge is often taken as a reference. In summary, the advantages of RESOBLOK include its high quality 3D geometric representation, the possibility of analysing results in statistical form, the speed of the limit equilibrium computation (which, when combined with statistical input allow a fast statistical analysis) and its interface with other more advanced codes. RESOBLOK is not in competition with discrete-element computer codes. The results are more global and rapidly give a rough and statistical result to a large-scale problem. This method is complementary to the stress–strain method, which can be used more precisely to analyse detected problematic cases after the first statistical estimation. For risk purposes, ‘‘extensive’’ (i.e., in a large zone with stochastic properties) block probabilistic stability analyses can be performed for the identification of the worst cases to be analysed further with a more complete ‘‘intensive’’ computation, such as the discrete elements method. We stress that the main problems commonly encountered when conducting these types of analyses remain
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sampling limitations, the information available on mechanical properties, the persistence of joint families’ accuracy, the precision of field measurements and the statistical uncertainty due to the deviations of the theoretical probability distributions from the observed data.
Appendix The Pij notation has been introduced by Dershowitz et al. (1998) and completed by Dershowitz and Herda (1992) and Mauldon and Dershowitz (2000) to distinguish between the various fracture densities. Mauldon and Dershowitz (2000) define the following: –
– – – – – –
P10 [L-1], P20 [L-2] and P30 [L-3] are the number of fractures per unit length (scan line or borehole), area (trace plane) and volume, P11 [-] is the sum of the fracture apertures per unit length (scan line or borehole), P21 [L-1] and P31 [L-2] are the sum of the fracture lengths per unit area (trace plane) and volume, P22 [-] is the sum of the fracture trace areas (trace length 9 aperture) per unit area (trace plane), P32 [L-1] is the sum of the fracture areas per unit volume, P33 [-] is the sum of the fracture volumes (area 9 aperture) per unit volume. This notation has been introduced into the RESOBLOK code.
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