Mining Goes Digital – Mueller et al. (Eds)
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
177
Grade estimation in a tabular deposit using unstructured grids
M.A.A. Bassani, C.P. Araújo & J.F.C.L. Costa
Federal University of Rio Grande do Sul, Porto Alegre, Brazil
ABSTRACT: Tabular deposits are known to have two dimensions much larger than a third
one (e.g., bauxite deposits, gold reefs, coal strata). Grade estimation in these deposits is often
performed at regular Cartesian grids, whose blocks have the same size. The problem is that
regular Cartesian grids do not adapt well to the geological solid used to define the shape
and volume in this kind of mineral deposit. Better modeling of the geological solid may be
obtained with unstructured grids, whose blocks have different sizes. This work presents the
use of these grids for grade estimation in a bauxite deposit. The methodology to build the
unstructured grid is shown. Then the grades were estimated in the unstructured grid by ordinary kriging. The resulting grade model was checked by visual inspection, swath plot, and
cross validation. Results showed that the estimates reproduced the trend of the data and were
globally unbiased.
1 INTRODUCTION
Most geostatistical software tools work with regular Cartesian grids, whose blocks have the
same size and shape. Grids are often denoted as block models in the mining industry. The
problem is that regular Cartesian grids usually do not fit well the geological solid of tabular deposits. Tabular deposits are deposits whose dimensions along two directions are much
larger than the dimension in a third direction (i.e., bauxite and coal deposits). For grade estimation in these deposits, many users work with block models in two dimensions.
The estimates in 2D block models are performed with the variables accumulation (product
of grade and thickness) and thickness (Krige 1978, Bertoli et al. 2003, Marques et al. 2014).
The grade estimates are obtained by dividing the accumulation estimates by the thickness estimates. This approach eliminates the vertical component (Z), resulting in a two-dimensional (2D)
model. The problem with this method is that a 2D block model does not allow the mine planner
to take into account the vertical selectivity during mining planning. As a result, the mine planner
may not use a 2D block model to test mining scenarios with different vertical selectivity.
A common approach used in petroleum geostatistics involves transforming the coordinates to the depositional space and fitting a regular Cartesian grid in the depositional space
(Mallet 2002, Mallet 2004, Caumon et al. 2004, Pyrcz & Deutsch 2014). Geostatistical modeling is then performed in the depositional space. Finally, the grid is back-transformed to
the real space. This approach results in a grid whose number of cells (blocks) in the vertical
direction is the same. The problem with this approach is that the volumes of the blocks vary
a lot after the grid is back-transformed to the real space. This variation in the volume of the
blocks is not desired for mining planning when the vertical selectivity is roughly constant.
Another possibility is to use an unstructured grid, which may have a different number of cells
(blocks) in any direction. Moreover, the unstructured grid is very flexible, as its cells may have
different sizes and shapes. Unstructured grids have been used in the petroleum industry to obtain
more realistic models (Manchuk 2010). However, they have been little used in the mining industry.
This work presents a case study of grade estimation using an unstructured grid in a tabular
mineral deposit. The methodology for building the unstructured grid is presented. The case
study used data obtained from a bauxite deposit in Brazil. The estimates were performed
© 2019 Taylor & Francis Group, London, ISBN 978-0-367-33604-2
177
Grade estimation in a tabular deposit using unstructured grids
M.A.A. Bassani, C.P. Araújo & J.F.C.L. Costa
Federal University of Rio Grande do Sul, Porto Alegre, Brazil
ABSTRACT: Tabular deposits are known to have two dimensions much larger than a third
one (e.g., bauxite deposits, gold reefs, coal strata). Grade estimation in these deposits is often
performed at regular Cartesian grids, whose blocks have the same size. The problem is that
regular Cartesian grids do not adapt well to the geological solid used to define the shape
and volume in this kind of mineral deposit. Better modeling of the geological solid may be
obtained with unstructured grids, whose blocks have different sizes. This work presents the
use of these grids for grade estimation in a bauxite deposit. The methodology to build the
unstructured grid is shown. Then the grades were estimated in the unstructured grid by ordinary kriging. The resulting grade model was checked by visual inspection, swath plot, and
cross validation. Results showed that the estimates reproduced the trend of the data and were
globally unbiased.
1 INTRODUCTION
Most geostatistical software tools work with regular Cartesian grids, whose blocks have the
same size and shape. Grids are often denoted as block models in the mining industry. The
problem is that regular Cartesian grids usually do not fit well the geological solid of tabular deposits. Tabular deposits are deposits whose dimensions along two directions are much
larger than the dimension in a third direction (i.e., bauxite and coal deposits). For grade estimation in these deposits, many users work with block models in two dimensions.
The estimates in 2D block models are performed with the variables accumulation (product
of grade and thickness) and thickness (Krige 1978, Bertoli et al. 2003, Marques et al. 2014).
The grade estimates are obtained by dividing the accumulation estimates by the thickness estimates. This approach eliminates the vertical component (Z), resulting in a two-dimensional (2D)
model. The problem with this method is that a 2D block model does not allow the mine planner
to take into account the vertical selectivity during mining planning. As a result, the mine planner
may not use a 2D block model to test mining scenarios with different vertical selectivity.
A common approach used in petroleum geostatistics involves transforming the coordinates to the depositional space and fitting a regular Cartesian grid in the depositional space
(Mallet 2002, Mallet 2004, Caumon et al. 2004, Pyrcz & Deutsch 2014). Geostatistical modeling is then performed in the depositional space. Finally, the grid is back-transformed to
the real space. This approach results in a grid whose number of cells (blocks) in the vertical
direction is the same. The problem with this approach is that the volumes of the blocks vary
a lot after the grid is back-transformed to the real space. This variation in the volume of the
blocks is not desired for mining planning when the vertical selectivity is roughly constant.
Another possibility is to use an unstructured grid, which may have a different number of cells
(blocks) in any direction. Moreover, the unstructured grid is very flexible, as its cells may have
different sizes and shapes. Unstructured grids have been used in the petroleum industry to obtain
more realistic models (Manchuk 2010). However, they have been little used in the mining industry.
This work presents a case study of grade estimation using an unstructured grid in a tabular
mineral deposit. The methodology for building the unstructured grid is presented. The case
study used data obtained from a bauxite deposit in Brazil. The estimates were performed