CASE STUDIES ON THE APPLICATION OF GEOSTATISTICS
TO IN SITU NATURAL STONE CHARACTERISATION
Pereira, H.G., Sousa, A.J., Ribeiro, J., Albuquerque, T., Luís, G., Lamberto, V., Saraiva, J.
CVRM / IST – Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Fax 351-21-8417442 e-mail hpereira@alfa.ist.utl.pt
ABSTRACT
The planning of natural stone exploitation suffers from a serious drawback stemming from the
lack of quantitative tools for the prediction, ex ante, of the in situ value of the material to be
extracted. This value, apart from extrinsic market factors that can not be controlled, depends
on a series of geological characteristics that can be measured or observed in the faces of the
quarry, like fracture density and attitude, presence of other discontinuities and intrusions.
When the fracture pattern is the most important factor controlling the value of the natural
stone, the main problem is a geometric one and a 3-D model can be simulated, giving rise to
the evaluation of the block size distribution. This model is established by the geostatistical
simulation of the spatial distribution of the fracture system. The results predicted by the
model are cross-validated against the zoning of the region where the quarries are located.
This zoning is based on real data referring to the dimensions of blocks and slabs produced
by the quarries that are active in the area. Two case studies referring to marble are presented,
in relation with this geometric approach.
For the cases where there is a multiplicity of factors affecting the value of the natural stone, a
synthetic index can be calculated in order to express those factors in a single quantitative
scale. This index, calculated by a modified Correspondence Analysis procedure, is the
analogue, for natural stone, of grade for metallic ores. Once calculated this index, the standard
geostatistical estimation methodology based on Kriging can be applied, provided that
validation procedures are available. Two case studies regarding marble and granite are
presented in order to illustrate the methodology, emphasising the specific features of each
particular geological setting, as well as different end-uses of each type of material depending
on other features, besides dimension.
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INTRODUCTION
Natural stone characterisation for exploitation planning purposes rises a variety of challenging
problems to the application of geostatistics. In fact, the nature of the available basic
information, on which the value of the material to be extracted is based, is far more complex
than in metallic deposits, consisting of a blend of quantitative and qualitative attributes,
namely the fracture system occurring in the deposit and in its regional geological envelope.
Hence, the specific data available in natural stone quarries call for important adjustments in
geostatistical methodologies. These adjustments apply in a different way when the main issue
is a geometric model based on fractures or when the quality of the stone depends also on other
characteristics, beyond fractures. In the first case, the 3-D model of the deposit is obtained by
geostatistical simulation of the fracture density, combined with Monte Carlo simulation of
other characteristics of the fracture system, as proposed by Luis et al. 1999. In the second
case, an analogue of the grade in metallic ore bodies must be calculated, summarising the set
of observable attributes in a quality index, according to the methodology proposed in Pereira,
1988 and applied by Pereira et al., 1993, Ribeiro et al., 1997, Taboada et al., 1999 and
Saraiva et al., 1999. This methodology relies on the construction of a quantitative index by
extracting the eigenvectors of the inertia matrix derived from the extreme poles of quality,
being these poles defined by an expert guess on the weights to be given to the categories of
the available attributes in order to account for the extreme configurations (‘GOOD’ and
‘BAD’ poles). Based on this index, the estimation of the quality in non-sampled blocks can
be performed by kriging, including in same cases an external drift representing the influence
of regional accidents on the local value of the blocks to be evaluated. In the following case
studies, the adjustments performed to cope with the specific nature of natural stone data are
illustrated, emphasising the repercussion on the geostatistical methodologies of particular
issues arising from different real problems faced by the natural stone industry.
CASE STUDY 1 – PREDICTING BLOCK DIMENSION DISTRIBUTION
This case study illustrates a methodology for calculating the block dimension distribution of a
marble deposit located in Southeast Portugal. The proposed methodology relies on a 3-D
model of the fracture system occurring in the area, obtained by geostatistical simulation. The
data on which the case study is based were obtained by drilling 16 holes and photographing
104 faces of 6 quarries operating in the deposit, as sketched in Fig. 1. By this procedure, a
total of 5161 fractures were recorded in a geo-referenced file that constitutes the basic input
where the methodology is grounded on.
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Deposit outline
Quarries
Boreholes
Face m
Collar of
borehole
Fracture 1
Fracture 2
Face
height
Core 1
Core characteristics
Core
Length
Beginning End
Azim.
Dip
Fracture 1
“scanline”
Fracture n Fracture 3
Face width
Fracture 2
Fractures distances from
collar and their attitude
Fracture n
No fract. Dist.
Azim.
Dip
Core 2
Plant of the quarry
Table of fracture
identification of face m
Code Azim.
Dip
1
2
N
Face
m
Table of input data to the
simulation model
n
Code
X
Y
Z
Azim.
Dip
Fig. 1 - Sketch of the procedure used to record the basic information on the fracture system
Based on this file, the RV Linear Fracture Density (LFD – number of fractures per length
unit) was calculated for each one of the 4 families of fractures recognised in the deposit and
the correspondent variogram was modelled by a spherical scheme, as depicted in the example
of Fig. 2.
Experimental variogram of the linear fracture density
(h)
The estimation of the above
mentioned RV was then performed
Azimuth N 55ºE and dip 65º
by kriging a set of 50x50x50 m
0.36
0.32
cubes enveloping the area of the
0.28
0.24
0.20
study. The geostatistical simulation
C0 - 0.03
0.16
0.12
C1 - 0.115
A1- 25 m
C2 - 0.09
A2- 150 m
0.08
of the LFD was produced by the
0.04
0.00
0
---
20
40
60
THEORICAL VALUE
80
100
120
140
160
180
200
Distances (m)
EXPERIMENTAL VALUE
Fig. 2 - Example of variogram of the Linear
fracture density for family 1
turning bands method, on the
grounds of the histogram and
variogram of the real fractures, and
of the estimated average value of
the LFD within each 50x50x50 m
cube that constitutes the geometric model of the deposit. Using the results of this
geostatistical simulation, combined with a Monte Carlo simulation of the other relevant
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characteristics of the fractures (attitude, dimension, probability of fracture intersection), a 3-D
model of the global fracture system occurring in the deposit is obtained. Based on this model,
a cumulative histogram of blocks that do not intersect any fracture is calculated, providing the
expected block size distribution, which is the final result of the study (cf. Fig. 3).
Nº of cubes with 0.25 m face length
10000000
1000000
100000
10000
1000
100
10
1
0.25
0.75
1.25
1.75
2.25
2.75
3.25
3.75
4.25
4.75
5.25
5.75
6.25
6.75
7.25
Dimensions of face lengths of composite cubes
Fig. 3 – Expected cumulative histogram of the blocks that can be extracted in the marble
deposit based on the fracture model
CASE STUDY 2 – ZONING BASED ON BLOCK SIZE AND SHAPE DATA
For the same marble massif which the precedent case is referred to, data were collected in 15
operating quarries, providing the distribution of the following dimensional parameters of
extracted material: maximum block size and shape of the slabs produced. A data base
containing 6289 blocks was obtained in the quarries that represent the area of the massif (cf.
Fig. 4).
Fig. 4 – Location of the quarries where data were collected in relation with the marble massif
Based on this data, two RV were defined : the maximum dimension of blocks and the
proportion of slabs exceeding the standard linear dimensions accepted by the transformation
factory (2.5x1.6 m). An example of a variogram for the second variable is given in Fig. 5, as
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well as the parameters of the fitted spherical model.
Fig. 5 - Variogram of the RV ‘proportion of slabs exceeding the standard dimensions’
On the grounds of these two variables, an estimation by kriging was performed for the entire
area of the massif, producing the maps given in Fig. 6, where the zoning of the massif can be
visualised, from the standpoint of maximum size and shape of expected blocks and slabs.
Fig. 6 - Zoning of the marble massif based on kriging of maximum block size and slab shape
CASE STUDY 3 – INCORPORATING REGIONAL GEOLOGICAL INFORMATION
This case study, referring to a marble formation located in Southeast Portugal where 15
quarries were sampled for the relevant local attributes (fracture length, curvature and density,
number of joint intersections, veining and ‘running’), illustrates how to incorporate, in the
estimation of the in situ quality of the material to be extracted, the influence of geological
regional accidents, like faults, metavulcanites and doleritic veins (cf. Fig. 7)
The two scales, local and regional, are linked in what concerns the in situ quality of the
marble, since there is a significant correlation between the index based on local attributes and
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the distance of each support where the attributes were observed to the nearest regional
accident (cf. Fig 8).
N
Graphic Scale
Fig. 7 – Locations of the quarries where local attributes were captured in relation to regional
accidents
Fig. 8 – Correlation between the local index and the distance to the nearest regional accident
In order to account for this regional information in the evaluation of marble quality, the
distances to the nearest accident were included in the estimation procedure as an external
drift, giving rise to the results depicted in Fig. 9, where the improvement brought by the
regional information can be assessed by comparison with the corresponding output produced
by kriging the local index. This comparison is based on cross validation by the leave-one-out
procedure, providing the scatterplots of real vs. estimated points depicted also in Fig. 9.
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a)
b)
Fig. 9 – Results of the estimation by kriging (a) compared with the external drift (b)
CASE STUDY 4 – ACCOUNTING FOR OTHER LOCAL QUALITY ATTRIBUTES
APART FROM FRACTURES
This case study illustrates the prediction of in situ value of natural stones based on attributes
that do not depend on fractures, but on other characteristics of the rock, like enclaves, veins,
dykes and schlieren. To account for these characteristics, the above described methodology
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for the index calculation was generalised by the establishment of a third pole (besides the
GOOD pole and the BAD RECOVERY pole). This new pole, denoted BAD QUALITY pole,
encompasses the
observable attribute categories that are relevant for quality definition,
besides fractures. The case study refers to a granite quarry located in Southeast Portugal,
where important features contributing to bad quality occur (cf. Fig. 10)
Fig. 10 - Plant of the granite quarry showing the location of attributes that contribute to bad
quality, besides fractures
Since these attributes are included in the third pole (BAD QUALITY), the resulting index
contains two components: component 1 gives the transition from good to bad blocks and
component 2 gives the transition from bad quality blocks to bad recovery blocks.
Experimental variograms and spherical models for the two components are given in Fig. 11.
Component 1
EXPERIMENTAL VALUES
MODEL ADJUSTMENT
DISTANCE (m)
Component 2
c0 = 0.29
c1 = 0.085
a = 12 m
EXPERIMENTAL VALUES
MODEL ADJUSTMENT
c0 = 0.03
c1 = 0.033
a = 12 m
DISTANCE (m)
Fig 11 -Variograms of the component 1 and 2 of the index
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Results of the estimation by kriging are given in Fig. 12.
6750
N
6740
6730
6720
6710
6700
6690
6680
6670
GOOD BLOCK
6660
BAD RECOVERY BLOCK
BAD QUALITY BLOCK
6650
5980
5990
6000
6010
6020
6030
6040
6050
6060
Fig. 12 - Estimation of the in situ quality of granite based on kriging of the two components
of the index
CONCLUSIONS
The case studies presented in this paper demonstrate how to handle, in geostatistical terms,
the variety of variables that control, on geological grounds, natural stone in situ value. For
modelling the fracture system and predicting the expected block dimension distribution,
geostatistical simulation and estimation revealed to be valuable techniques to cope with
specific features of the problem. The zoning of a marble massif based on kriging real
production data referring to block dimension proved to be a sound basis for validation of
simulation results. Regarding the forecasting of in situ quality, the case studies reported here
show that it is possible to account for regional geological information and for local attributes,
other than fractures, that affect recovery. The validation of the methodology to summarise the
observable attributes depends on the availability of a training set where the estimations
provided by kriging are to be compared with real data on the commercial value of particular
blocks.
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ACKNOWLEDGEMENTS
The authors are indebted to FCT (Fundação para a Ciência e Tecnologia) for the M.Sc. and
Ph.D. grants supporting this research in the scope of VIROC project (Forecasting the in situ
value of Natural stones).
REFERENCES
Luís, G., Sousa, A.J., 1999 – Geostatistical simulation of fracture networks. Application to the
evaluation of the block dimension distribution in a marble deposit. Submitted to Engineering
Geology.
Pereira, H.G., 1988 – Case study on application of qualitative data analysis to an uranium
mineralization. C.F. Chung et al. (eds,), Quantitative Analysis of Mineral and Energy
Resources, Reidel, 1988, p. 617-624
Pereira, H.G., Brito, G., Albuquerque, T., Ribeiro, J., 1993 – Geostatistical estimation of a
recovery index for marble quarries. Geostatistics TROIA’92, Vol. 2, Kluwer 1993, p. 10291040
Ribeiro, J., Pereira, H.G., Sousa, A.J., Albuquerque, T., Tirabasso, F., Taboada, J., Rico, J.G.,
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Kluwer, 1997, p. 905-915
Saraiva, J., Ribeiro, J., Sousa, A.J., Pereira, H.G., 1999 – Definition of a quality index for
different types of natural stones. 28th APCOM, Colorado School of Mines, 1999, p. 85-91
Taboada, J., Vaamonde, A., Saavedra, A., 1999 – Evaluation of the quality of a granite
quarry. Engineering Geology 53, 1999, p. 1- 11