NOTICE: this is the author’s version of a work that was accepted for publication in
Rock Mechanics and Rock Engineering. Changes resulting from the publishing
process, such as peer review, editing, corrections, structural formatting, and other
quality control mechanisms may not be reflected in this document. Changes may
have been made to this work since it was submitted for publication. A definitive
version was subsequently published in Rock Mechanics and Rock Engineering.
http://dx.doi.org/10.1007/s00603-011-0175-0. The original Publication is
available at www.springerlink.com
A method to estimate in-situ block size
distribution
M. K. Elmouttie
CSIRO Earth Science & Resource Engineering
Ph: +61 7 3327 4775
Fax: +61 7 3327 4455
Email: [email protected]
URL:
G. V. Poropat
CSIRO Earth Science & Resource Engineering
Ph: +61 7 3327 4425
Fax: +61 7 3327 4455
Email: [email protected]
URL:
Abstract
This paper presents a new technique for estimating the in-situ block size distribution in a jointed
rock mass. The technique is based on Monte-Carlo simulations using more realistic fracture
geometry as its input compared to other block size estimation methods described in the literature.
This geometry represents fractures as either polygons or triangulated surfaces and therefore
models persistence and truncation of fractures accurately. Persistence has been shown to be
critically important for the accurate prediction of block size and shape. We show that for rock
masses with relatively small discontinuities, the results of our predictions differ markedly from
previous methods which over-predict fragmentation.
1
Keywords
In-situ block size, discrete fracture network, DFN, polyhedral modelling
Introduction
The intersection of discontinuities or fractures within a jointed rock mass may
result in the presence of individual blocks of intact rock within the surrounding
rock matrix. The range of sizes of such blocks is represented as the in-situ block
size distribution (IBSD). IBSD is an important characteristic in many engineering
studies in underground and surface civil and mining operations including rock
mass classification, stability analysis, drilling, blasting design and hydrogeological analyses. The relationship of IBSD to the geometrical parameters of
the discontinuities, such as spacing, persistence and frequency, has long been
appreciated (ISRM 1978).
The development of methods to predict IBSD and related parameters has been
described extensively in the literature. Da Gama (1977) implemented a computer
algorithm to determine the volumes of in-situ blocks based on persistent
representations of individual joints mapped in the field. Miles (1972) examined
the partitioning of space via ‘sets’ of discontinuities, considering several
geometric configurations of infinite planes which were randomly oriented.
Hudson & Priest (1979) related this work to more typical geological geometries
involving three mutually orthogonal sets of persistent discontinuities. Wang et. al.
(1991) developed an algorithm to predict the volumes of blocks based on
daylighting joints using the block theory developed by Goodman & Shi (1985).
A matrix connectivity method was used by Young et. al. (1995) to study the effect
of orientation dispersion on IBSD for three orthogonal joint sets. They concluded
that as little as 10 degree dispersion in pole vectors was sufficient to fully ‘evolve’
the IBSD curve. Maerz & Germain (1996) used a software package to study IBSD
for some simple scenarios. Their algorithm was limited to using three sets of
persistent joints. Lu & Latham (1999) used equation-based methods to study the
effect of spacing distribution on IBSD. Their analysis was limited to three sets and
accounted for persistence indirectly, but they concluded that spacing distribution
was an important factor.
2
Wang et. al. (2003) provided a sophisticated software implementation of an
algorithm designed to handle any number of discontinuity sets and indirectly
account for persistence (assuming a persistence-spacing relationship). Their
algorithm randomly chose discontinuities from a previously generated
discontinuity ‘database’ and checked for the possibility of formation of polyhedral
blocks in the rock mass. Ahn & Lee (2004) attempted to account for nonpersistence via analogy with the two-dimensional geometries. Jern (2004)
developed an equation based on the product of spacing distributions for three
orthogonal sets. Latham et. al. (2006) expanded the use of equation-based
methods described in Wang et. al. (1991) to account for persistence and spacing
distributions for three orthogonal sets. Rogers et. al. (2007) described the use of
discrete fracture networks (DFNs) consisting of polygons accurately reflecting the
finite persistence of joints and other structures. Their algorithm utilized the
simulated two-dimensional trace map forming on the exposure (e.g. underground
cutting), identifying closed polygons on this map, and iteratively interrogating the
trace maps associated with the fractures responsible for each segment of the
polygons until the minimum-volume polyhedra were identified. Kim et. al. (2007)
utilized a commercially available polyhedral modeler to analyse IBSD for three
orthogonal sets of semi-persistent (i.e. persistent in one direction) discontinuities.
They concluded that the derived IBSD for several spacings and orientations are
log-normally distributed.
This review outlines the progressively more sophisticated approaches to
estimation of IBSD. However, all these studies have been hindered by one or
more of the following limitations in their approaches:
• Inability to accurately account for non-persistent discontinuities
• Inability to account for more than three discontinuity sets
• Inability to utilize fully detailed DFNs in a Monte Carlo simulation
• Detection of resulting polyhedra requires approximations and
simplifications such as assumed convexity or limited numbers of facets
These limitations result in estimations of IBSD over-estimating fragmentation of
the rock mass as well as being statistically inaccurate. Although some of the
previous methods are capable of generating computer simulations representing
3
‘realistic’ multi-faceted blocks (e.g. Wang et. al. 2003), the use of persistent
planes to partition space means concave polyhedra are impossible to detect (e.g.
Miles 1972). Further, the random sampling of a discontinuity data-base to provide
candidate block facets treats the blocks as independent and identically distributed
random variates. This is not the case since any generated block should be
conditional on the presence of surrounding blocks such that the sum of all
generated block volumes is that of the total rock mass. The technique outlined by
Rogers et. al. (2007) comes closest to addressing all the aforementioned
limitations however the polyhedral detection algorithm is an approximation and
yields minimal polyhedral volumes. Unfortunately, Rogers et. al. (2007) does not
outline the technique used in sufficient detail to comment on the impact of this
approximation on IBSD estimation.
This paper presents a new method based on Monte-Carlo simulations of DFN
geometry using more realistic and general representations of fractures combined
with a robust polyhedral modelling algorithm. The features of this technique
include the simulation of any number of discontinuity sets and random
discontinuities, the usage of a realistic DFN in its entirety, and the use of a robust
generalized polyhedral modeller so that all blocks associated with a model rock
mass are predicted simultaneously and consistently.
DFN Generation
The merits of the DFN approach to modelling heterogeneity in the rock mass over
equivalent medium (EM) approaches has been discussed extensively in the
literature (see Jing 2003 for a review). The DFN is usually constructed using both
deterministic data such as representation of large-scale bed and fault structures,
and stochastic data based on the sample of fractures observed in the field. These
data are usually acquired via core logging or exposure mapping. Providing that
sampling biases are properly accounted for (e.g. Mauldon 1998), the stochastic
strictures in the DFN should be statistically equivalent to those within the rock
mass. There are multiple possible fracture geometries although there are
suggestions that many of these variants are due to sampling biases and the fracture
size distributions are usually power law (Tonon & Chen 2007).
We have developed a DFN generator with facilities to simulate fractures as
polygons with size, spacing and orientations adhering to user-specified statistical
4
distributions. The generator is utilised by the Siromodel software package,
developed for the Large Open Pit Slope Stability Project (CSIRO 2011).
Polyhedral Modelling
Polyhedral modellers have traditionally required simplifications such as cubic or
hexahedral representations of the volume being simulated, very small numbers of
discontinuities (beds, faults and joint fractures), the assumption of planar
topologies for these discontinuities, the assumption of infinite persistence, and the
assumption that blocks can only form as tetra-, penta- or hexahedra. The third and
fourth simplifications are related and impose critical limits on the ability of a
modeller to represent jointed rock masses found in nature since such rock masses
consist of irregular rock blocks or concave polyhedra. The polyhedral modeller
utilised in this paper represents an evolution of modellers outlined by Lin et al.
(1987), Jing & Stephansson (1994), Jing (2000), Lu (2002) and is described in
detail in Elmouttie et al. (2010). In brief, the algorithm accepts a polygon ‘soup’
as input (such as that defined by a DFN) and detects all vertices, edges and faces
resulting from mutual intersections of the polygons or triangulated surfaces. A
face-tracing algorithm is then used to detect polyhedra within the volume of
interest.
Particular issues have been considered in the development of the algorithm to
ensure robust and reliable performance. For example, spatial tolerance
management and checking for degenerate geometric scenarios (such zero-length
edges) is accomplished via efficient storage and use of a master vertex list. Partial
intersections of faces need to be processed to ensure that the ‘face tracing’
algorithm used for polyhedral detection is successful. Adjacency information for
edges, faces and polyhedra must be calculated for several tasks such as the
detection of partially intersecting and isolated polygons (representing fractures not
associated with rock blocks) and identification of embedded polyhedra and
internal voids (e.g. tunnels). Details on the techniques used to address these and
other issues are discussed in Elmouttie et al. (2010). The polyhedra can be
concave and contain arbitrary numbers of facets. The polyhedra are defined in
terms of oriented faces allowing volume calculations to be performed via
determinant calculations (Newson 1899).
5
Validation
The validation of the polyhedral modelling algorithm is described in detail in
Elmouttie et al. (2010). We have additionally performed some validation
simulations dealing specifically with fracture generation and IBSD prediction, and
these will now be discussed.
Simulations of the trivial cases of space partitioned by three orthogonal sets of
planes of constant spacings s1, s2 and s3 have been performed. The blocks interior
to the simulation volume (i.e. excluding blocks in contact with the boundaries of
the simulation volume) all show identical volumes equal to s1s2s3 to within the
numerical precision of the algorithm (approx 100 times the relative error in the
spacings which was around 10-15 for these simulations).
Miles (1972) derived an estimator for the volume moments associated with space
partitioned by three orthogonal sets of planes with negative-exponentially
distributed spacings. We have used this as further validation of the algorithm.
Figure 1 shows a screen-shot of the blocks formed from an example realisation of
these discontinuities.
Figure 1 Example of blocks formed from three orthogonal sets of exponentially-spaced
discontinuities
6
Table 1 outlines examples of the scenarios that were simulated using single
realisations, each comprising around 20 to 30 discontinuities per set and resulting
in around 10,000 to 30,000 blocks. In all cases, the agreement between the
predicted and measured mean block volumes is better that 0.05%. Agreement is
observed to improve as simulation volumes and numbers are increased, and as the
numbers of planes per set consequently increases.
Table 1 Predicted and simulated mean block volumes for space partitioned by three
orthogonal sets of planes with negative-exponentially distributed spacings
Scenario
Spacing of
Spacing
Spacing of
Mean block
Mean block
number
set 1
of set 2
set 3
volume
volume
(m)
(m)
(m)
(predicted)
(measured)
(m3)
(m3)
1
9.39
9.94
10.59
988.4
988.3
2
7.73
6.21
5.51
264.5
264.5
3
11.35
5.35
4.76
289.0
288.9
Comparison with other methods
The two most sophisticated IBSD prediction methods covered in the literature that
also include IBSD predictions are those described by Wang et. al (2003) and Kim
et. al. (2007). We will use these two references to detail the improvements offered
by our algorithm.
Wang et. al. (2003) used a method to predict IBSD that was based on Monte Carlo
sampling of a predetermined discontinuity database. Groups of discontinuities
were randomly sampled from this database and checked to see if polyhedral
formation was possible. Sensitivity of the IBSD to discontinuity density and
persistence were discussed. However, due to the limitations of the algorithms,
certain assumptions were made.
Discontinuity density was defined as the number of discontinuities detected in
each metre of a simulated sampling line. Since the orientation of the sampling line
was not specified, one assumes an ideal simulation scenario where the line is
7
oriented orthogonally to the mean orientation of the discontinuity set. For
persistent planes, the density, in effect, defines the spacing of the set.
Wang et. al. (2003) accounted for non-persistence indirectly by altering the
assumed discontinuity spacing values in proportion to persistence factor. We have
used the areal definition of persistence factor with pf representing the ratio of total
fracture area within a persistent plane to the plane area (Dershowitz & Einstein
1988). Thus infinite persistence is equivalent to a persistence factor of 1. The
overall persistence factor for the three sets combined is determined as pf =
(p1p2p3)1/3 where p1, p2 and p3, are the persistence factor for each joint set.
The structural data used by Wang et. al. (2003) are shown in Table 2. Two
dominant sets were considered. The assumed statistical distributions were normal
for orientations and trace lengths, and negative exponential for spacings. Wang et.
al. (2003) used these data to predict IBSD sensitivity to both discontinuity density
(frequency) as well as persistence.
Table 2 Structural data used by Wang et. al. (2003)
Dip
(º)
Set 1
Dip
Trace
Direction
length (m)
(º)
52.9±14.8 138.0±23.6 4.15±2.01
Spacing
(m)
0.198±0.1
97
Set 2
61.8±14.1 327.0±33.3 3.79±1.26
0.282±0.2
96
To compare our approach with that of Wang et. al. (2003), we have performed a
number of simulations using the data in Table 2 for DFN generation. Figure 2
shows a block model based on one realisation of these DFN. Due to the persistent
nature of the discontinuities, all blocks in this model are convex.
8
Figure 2 Example block model based on a realisation of DFN using data in Table 2
Figure 3 shows the influence of discontinuity density (a.k.a. spacings) on the
IBSD curves. Although there is good agreement with Wang et. al. (2003) for the
low density case (right most curve), the variation seen as the density increases is
almost three times larger than observed in that study. The 90th percentile value
ranges from around 0.5m to 3m.
9
Figure 3 Comparison of IBSD curves for various spacings. From left to right, curves shown
for 0.066, 0.083, 0.1, 0.125, 0.160, 0.25 and 0.5m spacings respectively.
The shape index was defined by Wang et. al. (2003) as the ratio of the block
volume V to that of a sphere with diameter equal to the maximum block size, lmax
γ =
6 ⋅V
3
π ⋅ l max
Figure 4 shows the histograms of the block shape indices for the data shown in
Figure 3.
There is broad agreement with the observations made by Wang et. al. (2003). The
shape of the histogram peaks around γ = 0.01, in agreement with their prediction
of peaks in the range 0.001<γ <0.077. Further, no dependence on discontinuity
frequency is seen, also in agreement with Wang et. al (2003), although our
simulations predict significantly more blocks with shape indices γ > 0.1.
10
Figure 4 Histograms of shape indices for data shown in Figure 3. For each bin, bars from left
to right correspond to 0.066, 0.083, 0.1, 0.125, 0.160, 0.25 and 0.5m spacings respectively.
Figure 5 shows the sensitivity of IBSD to the persistence factor of the
discontinuity sets. Persistence factors of 0.6, 0.8 and 1 have been simulated. The
variation in the IBSD curves associated with these simulations necessitated a
Monte Carlo approach, with 20 simulations per persistence factor being
performed. Once again, although there is qualitative agreement with the results
presented by Wang et. al. ( 2003), the variance in the curves is much greater,
particularly when the confidence intervals are taken into account. Further, Wang
et. al. predict no blocks larger than 4.5m, whereas for the pf = 0.6 case, blocks as
large as 20m are predicted by our algorithm. This is a further example of how
sensitive rock mass fragmentation estimates are to the assumed persistence of
discontinuities.
11
12
Figure 5 Comparison of IBSD curves for various persistence factors. From top to bottom,
curves are shown for values of 1, 0.8 and 0.6 respectively. Solid lines represent mean IBSD,
dotted represent 95% confidence limits.
Kim et al. (2007) applied a polyhedral modelling methodology using more
simplistic discontinuity geometries to study the effect of including finitepersistence on IBSD prediction. Their model consisted of three orthogonal joint
sets with each set containing ten equally-spaced or quasi-equally-spaced joints.
After analyzing a number of simulations, they concluded that the effect of finite
persistence could, in general, be adequately represented by an empirical equation
derived by Cai et al. (2004). However, the polyhedral modeler used by Kim et. al.
(2007) is a commercially available one which can only accommodate persistence
in an approximate way. One of the three joint sets is required to be fully persistent
so as to offer a set of planes for which the other two joint sets can terminate
against (i.e. semi-persistent hereafter).. Further, the finite persistence of the other
two sets could only be modelled in a statistical sense as opposed to our approach
which uses a polygonal representation of discontinuities in 3-dimensional space.
We used our approach to replicate the Kim et al. (2007) simulations using a cubic
simulation volume with a width of 10m and 1m joint spacings. This resulted in a
mean block volume for the persistent case of 1m3. The DFNs were then modified
13
repeatedly so as to reduce the radii of the joints associated with only two of the
sets. By requiring a persistent third set (i.e. p3 = 1) at least 10 blocks would form
from each DFN realisation. A Monte-Carlo approach was again adopted so that
each persistence factor was represented by multiple DFN realisations. An example
block model based on a DFN realisation is shown in Figure 6
Figure 6 Block model based on a realisation of DFN for the Kim et. al. (2007) scenario
The results obtained are shown in Figure 7. As the persistence factor approaches
1, the mean block volume ratio predicted by both Kim et. al. (2007) and our
algorithm approaches 1 as expected for the persistent case. Note that the error bars
in this figure represent the standard deviation of the sample means of the
parameter and the data-point represents the mean of the sample means.
Although the results in Figure 7 are consistent with the hyperbolic relationship
between block volume ratio and persistence factor seen by Kim et al. (2007), the
volume ratios are markedly different. The aforementioned limitations with their
14
method (including the semi-persistent representation) have resulted in predictions
of overly fragmented rock masses.
Figure 7 - Model sensitivity assessed similarly to Kim et al. (2007): solid blue curve
represents our simulations, solid green is a prediction assuming rectangular joints showing
maximal intersections, and solid red is our prediction of the maximally intersecting case
using semi-persistent joints.
By utilizing DFN based on polygonal approximations to joints, persistence has
been modeled realistically. This allows for the existence of partially throughgoing joints or fractures, which are present but not involved in block formation, as
well as the existence of concave blocks.
Figure 7 also shows the predicted curves for the most fragmented case for both
our joint representation and the semi-persistent joint representation used by Kim
et al. (2007). To derive the ‘most fragmented case’ we have assumed all
discontinuities from the two non-persistent sets have originated in the same part of
the simulation volume. This unlikely scenario would result in maximal
fragmentation of the rock mass. The curve corresponding to our simulations is
consistently above-scenario as expected. Further, our simulations show that the
ratio of mean block volumes to the non-persistent case approaches 100 when pf
approaches 0. This is consistent, since for low persistence factors, one would
expect only 10 ‘slabs’ to remain, compared to 1000 blocks for the high persistence
case (i.e. a ratio of 100:1). Therefore, even for the most fragmented cases, the use
15
of semi-persistent discontinuity geometries grossly over-estimates rock mass
fragmentation.
Conclusions
We have outlined a new method to predict IBSD in fractured rock masses. Using
realistic DFN, robust polyhedral modeling, and a Monte Carlo sampling approach,
the stochastic variability in the fracture geometry can be accounted for. The
method can deal with arbitrary numbers of discontinuity sets, finite persistence
representations of fractures, the consequent formation of concave polyhedral, and
fracture properties described via arbitrary statistical distributions.
We have compared our predictions of IBSD to those detailed in two previous
studies using methodologies limited by the absence of one or more of the
aforementioned features of our method. The results show that significant
differences in rock mass fragmentation estimates are possible, particularly when
persistence of fractures is not modeled in a geometrically or physically realistic
sense. These differences are maximally important when modelling rock masses
containing discontinuity sets with low persistence factors, but may not be severe
for high persistence factor (i.e. blocky rock mass) cases.
Acknowledgement
The authors would like to acknowledge the support of the Australian Coal Association Research
Program in the development of the prototype versions of the algorithms used in elements of the
structural modelling. The sponsors of the Large Open Pit Mine Slope Stability Project, which is
managed by the CSIRO, are acknowledged for their support in developing applications to utilise
the algorithm for the analysis of slope stability phenomena.
References
Ahn , S-H. & Lee, C-I. (2004) Removability Analysis of Rock Blocks by Block theory and a
Probabilistic Approach. In SINOROCK2004 Symposium, Paper 2A 11, Int. J. Rock. Mech.
Min. Sci. (41) 3, CD-ROM
16
Cai, M., Kaiser, P.K., Uno, H., Tasaka, Y., & Minami, M. Estimation of rock mass strength and
deformation modulus of jointed hard rock masses using the GSI system Int. Journal Rock
Mech. & Min. Sci., 2004, Vol 41(1): 3-19CSIRO (2011)
CSIRO (2011) Large Open Pit Slope Stability Project www.lop.csiro.au
Da Gama, C.D. (1977) Computer Model for Block Size Analysis of Jointed Rock Masses. In 15th
APCOM Symposium, Brisbane, Australia 305-315
Dershowitz, W.S. & Einstein, H.H. (1988) Rock Mech Rock Eng 21: 21-51
Elmouttie, M.K., Poropat, G.V & Krahenbuhl, G. (2010) Polyhedral modelling of rock mass
structure. Int. Journal Rock Mech. & Min. Sci., 47: 544-552
Goodman, R.E. & Shi, G. (1985) Block Theory and its Application to Rock Engineering. PrenticeHall, New Jersey.
Hudson, J.A. & Priest, S.D. (1979) Discontinuities and Rock Mass Geometry. Int J Rock Mech
Min Sci & Geomech Abstr 16: 339-362
International Society for Rock Mechanics (1978) ISRM Commission on Standardization of
Laboratory and Field Tests. Int J Rock Mech Min Sci & Geomech Abstr 15: 319-368
Jern, M. (2004) Technical Note: Determination of the In-Situ Block Size Distribution in fractured
Rock, an Approach for Comparing In-Situ Rock with Rock Sieve Analysis. Rock Mech. Rock
Engng. 37(5): 391-401
Jing L & Stephansson O. (1994) Topological identification of block assemblages for jointed rock
masses. Int J Rock Mech Min Sci Geomech Abstr 31:163–72
Jing L. (2000) Block system construction for three-dimensional discrete element models of
fractured rocks. Int J Rock Mech Min Sci 37:645–59
Jing, L. (2003). A review of techniques, advances and outstanding issues in numerical modelling
for rock mechanics and rock engineering. Int J Rock Mech Min Sci 40: 283-353
Kim, B.H., Cai, M., Kaiser, P.K., & Yang, H.S. (2007) Rock Mech Rock Eng 40(2): 169-192
Latham, J.-P., Van Meulen, J., & Dupray, S. (2006) Prediction of in-situ block size distributions
with reference to armourstone for breakwaters. Eng Geol 86: 18-36
Lin D, Fairhurst C & Starfield AM. (1987) Geometrical identification of three dimensional rock
block systems using topological techniques. Int J Rock Mech Min Sci Geomech Abstr
24(6):331–8
Lu, P. & Latham, J.-P. (1999) Developments in the Assessment of In-situ Block Size Distributions
of Rock Masses. Rock Mech Rock Eng 32(1): 29-49
Lu J. (2002) Systematic identification of polyhedral rock blocks with arbitrary joints and faults.
Comp. Geotech. 29:49–72
Maerz, N.H. & Germain, P. (1996) Block Size determination Around Underground Openings
Using Simulations. In Proceedings of the FRAGBLAST 5 Workshop on Measurement of Blast
Fragmentation, Montreal, Quebec, Canada, 23-24 Aug, 1996: 215-223
Mauldon, M. (1998). Estimating mean fracture trace length and density from observations in
convex windows. Rock Mech Rock Eng 31(4): 201-216
17
Miles, R.E. (1972) The Random Division of Space. Advances in Applied Probability, Vol. 4,
Supplement: Proceedings of the Symposium on Statistical and Probabilistic Problems in
Metallurgy (Dec., 1972): 243-266
Newson, H.B. (1899) On the volume of a polyhedron. The Annals of Mathematics, Second Series 1
(1):108-110
Rogers, S.F., Kennard, D.K., Dershowitz, W.S., Van As, A. (2007) Characterising the in situ
fragmentation of a fractured rock mass using a discrete fracture network approach. In Rock
Mechanics: Meeting Society’s Challenges and demands. Eberhardt, Stead & Morrison (eds)
Taylor & Francis Group, London.
Tonon F. & Chen S. (2007) Closed-form and numerical solutions for the probability distribution
function of fracture diameters. Int J Rock Mech Min Sci & Geomech Abstr 44: 332-350
Wang, H., Latham, J.-P., & Poole, A.B. (1991) Predictions of block size distribution for quarrying.
Quarterly Journal of Engineering Geology 24: 91-99
Wang, L.G., Yamashita, S., Sugimoto, F., Pan, C., & Tan, G. (2003) A Methodology for
Predicting the In Situ Size and Shape Distribution of Rock Blocks. Rock Mech Rock Eng 36(2):
121-142
Young, D.S., Boontun, A. & Stone, C.A. (1995) Sensitivity tests on Rock Block Size Distribution.
Rock Mechanics, Daemen & Schultz (eds). Balkema, Rotterdam
18