- Email: [email protected]
Evaluation of the resources of a slate deposit using indicator kriging
Engineering Geology 81 (2005) 407 – 418 www.elsevier.com/locate/enggeo
Evaluation of the resources of a slate deposit using indicator kriging F.G. Bastante *, J. Taboada, L.R. Alejano, C. Ordo´n˜ez Mining Department, University of Vigo, Spain Received 14 May 2004; received in revised form 10 August 2005; accepted 26 August 2005 Available online 24 October 2005
Abstract The paper commences with a description of the mining resource studied (roofing slate), the particular features of slate deposits and slate mining techniques. We then describe the methodology used for drilling core studies, the most reliable method for ornamental rock mining exploration. From the drill-core data, applying statistical techniques (kriging) with the assistance of the powerful DATAMINE 5 software package ([MICL, 1999. bDatamine 5Q. Datamine Group. Mineral Industries Computing Limited.]) can make an assessment of the exploitable resources in a quarry. To do this, the research domain is first delimited – in terms of the top and bottom of the exploitable seam, as well as the sidewall boundaries – employing ordinary kriging. A wireframe model of the deposit (the domain) is constructed and is sub-divided into blocks (block model). The resources in the deposit are then evaluated using indicator kriging and taking into account the anisotropic directions of the slate deposit. This produces data in the form of locations, tonnages and percentages of exploitable slate. The final step compares results obtained from the model with results obtained from mining in order to arrive at conclusions respecting the methodology. D 2005 Elsevier B.V. All rights reserved. Keywords: Slate; Ornamental rock mining; Resources evaluation; Indicator kriging
1. Introduction In mining, all knowledge is inevitably linked to uncertainty, and this occurs in the form of risk when a mining deposit is assessed. In order to be able to obtain the necessary decision-making elements for each of the developmental phases of the mining project, the mining company has to quantitatively calculate the value for each parameter that intervenes in the mining evaluation along with the uncertainty associated with these values. ´ rea de Explotacio´n de * Corresponding author. E.T.S.I.Minas, A Minas, Campus Lagoas-Marcosende, 36200 Vigo (Pontevedra), Spain. Fax: +34 986812201. E-mail address: [email protected] (F.G. Bastante). 0013-7952/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2005.08.003
Innate to the mining deposit itself are the quantity, quality and spatial distributions of the different materials present, and from the outset it is on these primordial factors that the feasibility of a mining project will depend. The uncertainty associated with each will therefore have a significant bearing on the mining companyTs final decision as to whether to take on the risk of allocating human, economic and financial resources to the development of the project. The costliness of mining research combined with the fact that the geological variables (thickness, grades, accumulations of metal, etc) present a numerical structure and a certain spatial order – in many cases only statistically evident (Azca´rate, 1982) – have both made for a rapid development and widespread use of different kinds of geostatistical tools (Matheron, 1962, 1963),
408
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
such as, for example, techniques for the determination of the spatial correlation between variables and for the calculation of values and associated uncertainties in areas of potential interest. This paper describes the kriging procedure applied to an evaluation of the resources of a slate quarry located in Leon, Spain, with the assistance of the DATAMINE 5 software package (MICL, 1999). From the geological information available for the area and the logs of the boreholes made in the bed, the quantity of exploitable slate and its spatial distribution in supports of 1000 m3 have been obtained. We made use of this block model in the design and planning of the quarry using optimisation algorithms (Bastante et al., 2004). Fig. 1 which describes the overall process followed so as to attain our objective, refers to the following phases: ! Collection of topographical, geological and core log data
! Creation and checking of databases based on the above information ! Transformation and analysis of the data ! Extraction of the data necessary for the delimitation of the exploitable area ! Semivariogram modelling; Data kriging ! Creation of a wireframe model of the reference area based on the points obtained ! Data selection from within the model ! Semivariogram modelling ! Creation of a block model in the interior of the wireframe model; Block kriging ! Resource evaluation ! Validation of the results with real data from the quarry. We will now describe the general features of the deposit and the particular idiosyncrasies of slate core logging, before moving onto the evaluation proper.
Fig. 1. Research methodology.
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
2. Features of slate and slate mining On a general level, all fine-grained rock capable of being exfoliated in thin sheets of a large size relative to thickness is considered as slate. Our research is concerned with ornamental slate, normally dark in color, and exploited industrially for its applications in roofing, fac¸ade finishing, interior and exterior flooring, and stonework and decorative walls. Its main feature is its excellent fissility. Lithologically speaking, slate is clay shale that has undergone a low-grade metamorphism (below the isograde of biotite). Mineralogically speaking, we can describe its basic components as: Sericite (40–46%), Quartz (22–25%) and Chlorite (15–17%). The accessory minerals are very varied. They are not particularly important in terms of the physical and chemical properties of the rock but may, on the basis of their alterability, determine the commercial quality of the slate. The immense majority of the exploited slate quarries in Spain are Ordovician, although there are some isolated Devonian or Cambrian slate deposits that are commercially interesting (GarcVa-Guinea et al., 1997). During the different series, successive layers of clays were deposited on the sea floor and acquired a significant thickness. Heat and pressure provoked by Hercynian folding caused a metamorphism of the clay sediments that in turn produced mineralogical changes and a reordering of the crystals. The geological factors that have a bearing on the quality of a slate quarry are of three kinds: stratigraphic, structural and metamorphic. In this respect, the mineralogical composition and grain size are the first two factors to take into account. Iron sulfides such as pyrrhite, pyrrhotite and marcasite are not desirable in the rock, since they produce anti-aesthetic iron oxide spots on roofs on oxidation. Neither are carbonates desirable, given that these result in circular whitish spots on the slate surface when dissolved and weathered. In terms of the metamorphic process undergone by the rock, another conditioning factor is the slate’s microscopic homogeneity. Slate that has a homogeneous distribution of quartz grain size is generally more fissile than slate having greater variation of the grain size. Microtexture is the next factor to be taken into consideration. The desirable microtexture is lepidoblastic, in the form of penetrative slaty cleavage fabric. This consists of the planar orientation of all the phyllitose and inequigranular minerals, which constitute the preferred exfoliation plane for the rock (GarcVa-Guinea et al., 1998).
409
The presence of more than a single cleavage in the rock is prejudicial. In the Spanish Hercynian Mass, the principal continuous cleavage – which causes the slaty cleavage denominated as S1 – is a consequence of the initial phase of Hercynian deformation. But many regions of the Mass are also affected by other, later phases, and especially by the third phase, which caused the slate to have microfolds or crenulation in the S1 planes, spaced crenulation cleavage denominated S3, and even compositional tectonic banding, due to the dissolution of the quartz in the short flanks of the asymmetric folds. The macroscopic geological aspects of the mass are also fundamental to the determination of whether a mass may or may not constitute a slate deposit. The thickness of the bed is the first aspect to take into consideration in this respect, and as a general rule, 10 m is the minimum thickness exploited, although this very much depends on the quality of the slate itself and the terrain morphology. The structural position of the bed is also influential. Hercynian Mass slate beds are tucked into asymmetric folds, and the flanks are preferable to the fold hinge zones, given that in the former bedding and cleavage form less of an angle than in hinges and the slate is consequently more homogenous. Nevertheless, deposits of slate to be found in hinges may in fact be larger, given the thickening that occurs in fold hinges. Another negative factor is the existence of contact metamorphism subsequent to very low-grade regional metamorphism, given that the metamorphic recrystallisation produced as a consequence welds the cleavage planes and prevents a satisfactory exfoliation of the rock. Undoubtedly, the definitive-limiting factor respecting the exploitability of a mass is its fracturation, with fractures understood not only as joints and faults but also as quartz dikes, kink-bands and other more or less planar discontinuities, other than bedding and cleavage. In that the system of fractures conditions the orientation of the working banks, and the state of fracturation conditions the size of block extracted from the mass and decisively determines the extraction methods to be employed as well as the yield of the workings. It may appear that any area with low-grade clayshale rock outcrops favours the existence of ornamental slate deposits, but this is only true to a certain extent. The French and Spanish Hercynian Mass deposits are concentrated above all in two geological formations, which is probably due to sedimentological causes that have not yet been sufficiently researched. As methods for slate quarrying, we can distinguish clearly between the more frequently utilised surface
410
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
mining method, and the underground method that is gradually becoming more popular. The first surface mining task is the clearing of the altered zone of the deposit or the overlying spoils in order to uncover the exploitable level. This operation is carried out using earth-moving equipment, or explosives for non-ripable areas. The difficulties will depend on the terrain orography and the spoils management provided for in the mining plan, an important aspect of planning given the great volumes of material shifted in modern slate mining processes. Once a working bank has been prepared, the task of extracting the blocks is undertaken using a variety of methods. These all have in common that (a) they should damage as little as possible the adjoining blocks and the residual rock mass; (b) that they should be applied according to the anisotropic directions and discontinuities of the mass, and (c) that the maximum block size takes into account, not only the natural limits imposed by the discontinuities, but also the dimensions of the treatment plant. Bearing these factors in mind, we can design our bank-cutting grid so as to obtain the required blocks, on the basis of one of three methods most utilised: low-grade explosives, diamond-wire cutter and diamond-disk or rock shearing machine. The method based on the use of low-grade explosives is the most typical. Of the mechanical cutting methods, the most popular method is the diamondwire cut since it maximises yield in almost all kinds of ornamental rock, although at a higher operational cost. Given the importance of the kind of cut from the point of view of yield, systems for the optimisation of this process on the basis of rock mass discontinuity data are being developed. The simplest system for calculating the quarry extraction yield is to consider an artificial cut as yet another discontinuity that is integrated into the structural study. On the basis of a trial-and-error approach that involves adaptations of the cutting grid applied, the optimal size for each bed can be determined (Nicieza et al., 1994). The blocks are extracted using the bucket teeth of a shovel loader or backhoe that extracts the blocks from the mass in accordance with the cleavage direction. These are then loaded onto trucks for their transportation to the plant. In the processing plant, commercial sheets of slate are obtained as follows: - Initial exfoliation of the blocks to a thickness of less than 30 cm - Cutting using a diamond-disk saw (1m in diameter) - Exfoliation in sheets of commercial thickness
- Cutting to final sizes and squaring in accordance with commercial templates - Selection and packaging. The deposit that is the subject of this study is in the Agu¨eira Formation (Middle-Upper Ordovician) and it is composed of a monotonous series of fine-grained slate with some sandy laminations. The principal cleavage is N110E in direction with a dip of 60 S. Overlaying the exploitable level there is a dense layer of kink-bands running N100E in direction with a dip of 30–50 N. The footwall of the deposit contains a layer of highly folded quartzite which joins the layer of kink-bands towards the north and thus restricts the possibilities for mining in that direction. The mining method used is the descending-terrace method, with 5 m high extraction banks; in-bank diamond wire cutting (horizontal cuts to break the cleavage plane continuity, and vertical cuts at 3 m perpendicular to the cleavage-bedding intersection); and backhoe cleavageoriented block extraction. 3. Core logging Research into slate quarries is based on outcrop data – which is, however, limited by the superficial alteration that generally affects slate masses – and above all, on continuous drill cores, which permits healthy rock to be studied and analyses its behaviour as ornamental slate (Taboada et al., 1998). In order to obtain all the geological–geotechnical data from the study of the slate drill cores, we need to employ very small-scale diagrams (1 : 50) that take into account the following parameter values, grouped into five main categories: Perforation Data ! Diameter, kind of perforation and depth This information is essential when interpreting the core logs, and at the same time serves to explain some of the results that may have been influenced by the mechanical perforation tasks. Geological Data ! Lithological description This section describes the several geological formations penetrated by the boreholes, with a special emphasis on the lithological characteristics of slate formations relevant to the exploitability potential of the deposit (fissility) or the quality of the rock (color, textural homogeneity, grain size, content in impurities or alterable minerals, laminations and cleavage plane roughness).
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
Geotechnical Data ! Core recovery This is the percentage of core samples obtained with respect to the total perforated length. ! R.Q.D. (rock quality designation) This geotechnical parameter is the percentage core recovery in pieces larger than 10 cm. In slaty formations it is very easy for the core samples to break along the cleavage plane when handled, for which reason it is important to distinguish the breaks due to natural discontinuities and those due to the slate structure itself or the handling of the samples. ! Number of fractures per meter This fracturation index per unit of borehole length has a value in conjunction with other parameters relative to the fracturation of the mass. In calculation, discontinuities such as bedding or cleavage will not be taken into account. ! Alteration degree This is defined according to the alteration scale for detrital sedimentary rock. Structural and Discontinuity Data ! Type of discontinuities We understand by discontinuity, any structure with a more or less planar geometry that breaks the continuity of the rock mass, such as stratification, cleavage, faults, joints, kink-bands, shear bands, dikes and veins. The type of discontinuity analysed has to be identified in this section. ! Joint dip This is preferably measured in the form dip direction/dip angle. In order to establish a joint dip direction, this should be measured in comparison with the joint that is always present in slaty formations, having a constant regional direction that can be measured at the borehole collar in outcrops. We refer to the flow cleavage of the axial plane, the basis for the exploitation and utilisation of ornamental slate. ! J.R.C. (joint roughness coefficient) This coefficient is important in order to describe and quantify the state of the surface, since it conditions the mechanical response of the joint to shearing efforts. ! Polarity In this parameter we refer to the location of the mass within more marked geological structures, which in the case of slate deposits, are folds. Polarity indicates the position of the mass in a direct or reverse flank, or in the hinge of a fold. This position can be determined on the basis of sedimentary criteria such as granulometric selection or on the basis of
411
structural criteria such as the regional shape of the folds and the relative positions of the bedding and cleavage planes (Taboada et al., 1997). Observations ! Opening, filling and weathering of the joints; water filtrations ! Discontinuity families With the joint orientation data and assisted by the stereographic templates we can define the different families of mass joints and classify the discontinuities. ! Spacing This is the distance between two adjacent joints of the same family. It can be measured on the basis of the joint data, once the families have been identified. It is a very important parameter since it affects block size, which is an important deposit feasibility selection factor (Ribeiro et al., 1997). Drill core logging is the most important method for investigating slate in depth since the indirect methods do not provide sufficient contrast in the slate beds. From the proposed core log a series of quality parameters is obtained for the mass in terms of its exploitability as ornamental rock: ! Jv (volumetric joint count): This is the number of discontinuities per unit of mass volume. ! Vb (useful volume): This is the relative volume of natural blocks greater than a given minimum size. It is indicative of the maximum theoretical yield of the exploitation. In the final analysis, and given the great number of parameters that influence the exploitability of a deposit, it is the slate expert who, following a study of the drill cores and an evaluation of the variables described above, identifies the part of the mass that is exploitable as ornamental slate, and the part that will go to the slag heap. 4. Delimitation of the domain The first step, prior to any assessment process, is to delimit the area of interest on which the study of the population is to be performed. In the interest of simplicity and meticulousness, the area must be broken up into independent zones in such a way that the universe we are examining will display simplicity or similarity. A geomorphologic interpretation of the genetic and structural information available was first carried out, followed by a detailed posterior analysis for spatial demarcation purposes, of the economically interesting sector in this case. Since our population is composed of the slate from the quarry, and our objective is the
412
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
estimation of useful or exploitable slate, our domain will be determined by the areas in which slate appears. Fig. 2 illustrates the division of the quarry into two independent zones on the basis of structural criteria, and according to the interpretation of the footwall quartzite on which the exploitable layer of slate rests. This paper focuses on a study of Zone A, for which more information is available and in which the greater part of the exploitation reserves are concentrated. 4.1. The bottom and the top of the exploitable layer The next step in the delimitation of the domain was the estimation of the bottom of the exploitable layer, understood as the surface below which, according to the cores, there is no slate that can be sent to the treatment plant. It has to be pointed out that although a technical criterion has been used for the definition of the separation area, this presupposes the confluence of a series of genetic-structural factors. On the other hand, the fact that a point corresponds to the said surface does not presuppose that in the upper adjoining area there is exploitable slate. The starting point was the identification of the coordinates (X, Y, Z) of the deepest extremity where exploitable slate was to be found for each drill core in Zone A. The vertical co-ordinate (Z) was considered as regionalized variable. With these data the omnidirectional variogram and the corresponding model were calculated, as illustrated in Fig. 3. The apparition of a monomic model of a degree close to two indicated a linear tendency, which means that the area approximated a plane. The corresponding equation was obtained using linear regression, with a resulting correlation coefficient equal to 0.929. Next, the points that remained below the previous mid-plane were selected and a linear regression was
Fig. 3. Semivariogram of the vertical co-ordinate (bottom points).
again carried out, resulting in a correlation coefficient of 0.975. Thus, the plane may be said to satisfy our delimiting mining criterion. Even so, a change was made in the co-ordinates system to one that was parallel to the plane. The initial system of co-ordinates was rotated to align the X, Y axes with the regression plane. In this new system was observed that the tendency disappeared. The vertical co-ordinate in the rotated system of co-ordinates was considered as a new regionalized variable. The omnidirectional variogram was calculated and modelled, ordinary kriging was applied and the values obtained were transformed into the initial system of co-ordinates. Fig. 4 illustrates the previous process and the wireframe model created on the basis of the krigged points. The results of the crossvalidation are described in Table 1. Obtained in a similar way were the data for the top of the exploitable layer, defined as the surface below
Fig. 2. Location plan of the boreholes and interpretation of the footwall quartzite.
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
413
Fig. 4. Change in co-ordinates, semivariogram (new vertical co-ordinate) and final result.
which exploitable slate begins to appear in the drill cores. Although a great part of this area has already been exploited it continues to be important from our point of view of delimiting the exploitable layer. In this case, in addition to the genetic and structural factors, the surface alteration that the slate has undergone over time plays an important role. Fig. 5 shows the omnidirectional variogram for the top points. We see again the appearance of a monomic model, this time of a lower degree but also close to two. The regression plane (correlation coefficient r = 0.85) has a dip of only 7.58, which explains both why the variogram is smoother and the points show less variability. Changing the co-ordinate points system to one parallel to the regression plane produced the structure depicted in Fig. 6. The top surface was modelled in a similar way to the bottom, on the basis of the points obtained.
the south and in depth was drawn – following the direction and average slope of the kink-bands – for an average distance of 25 m from the drill cores on the site plan, and also taken into account was the need for access to the banks. This resulted in the wireframe model illustrated in Fig. 7. The next stage of the research consisted of utilising the information provided by the drill cores for the estimation of the potentially exploitable reserves as well as the uncertainty of these results. For this we had to decide the support on the basis of which to make the calculation, and then to complete the wire-
4.2. Delimitation of the sidewalls and final modelling The eastern and western boundaries were vertical surfaces (see Fig. 2), in the area close to the edge of Zone A. The northern boundary was delimited by a vertical surface to an average distance of some 25 m from the drill cores on the site plan. The boundary to Table 1 Results of the cross-validation (search radius: 125 m) % error in mean estimate Mean kriging variance Mean estimated variance Mean differences Mean absolute difference
0.62 10.45 10.53 0.090 2.48
Fig. 5. Semivariogram of the vertical co-ordinate (top points).
414
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
Fig. 6. Semivariogram of the vertical co-ordinate in the rotated system (top points).
frame model with blocks representing physical production units, necessary for the subsequent planning of the mining works. 5. Estimation of the reserves How can we utilise the information supplied by the drill cores in order to evaluate the slate reserves? The answer is in indicator kriging. If we transform the exploitable/non-exploitable slate information into a 1/0 binary system, we can carry out an indicator kriging (Journel, 1983) on the blocks in such a way that the result – the mathematical expectation or expected value of the indicator for each block – is a calculation of the proportion of material that corre-
sponds to Class 1 (exploitable slate) within the support chosen. The procedure is as follows: we select the drill cores that are within our wireframe model, and we divide these into stretches of 1 m. We assign values to each stretch – 1 for 100% exploitable slate and 0 for 0% exploitable slate – on the basis of the drill core information, and we calculate the omnidirectional variogram depicted in Fig. 8. The resolution of 1 m was established in part from the need not to lose precision in the transformation. In our case, more than 92% of the continuous stretches of exploitable slate have a length in excess of one metre, representing 99% of the total accumulated length of the exploitable slate. On the other hand, mining requires a minimum block thickness in the order of 50 cm, and therefore, as thickness is given by the normal direction to the cleavage plane that dips 608, the support can be considered as representative of the minimum block size in that direction. The next question that arises is: can we work with the isotropic model that we have adjusted to the experimental omnidirectional semivariogram? Unfortunately the answer is no. The reason for this is the directional bias of the base information. The behaviour of the semivariogram near the origin is defined principally by the vertical direction; in other words, it does not represent average behaviour in all directions. And the same happens with the rest of the semivariogram; some direction–distance combinations are better represented than others are. The data, therefore, is not distributed in the space in a homogeneous way. That said, the semivariogram obtained is extremely important. It demonstrates that the data do have a spatial order and permits us to make an initial preliminary estimation of the reserves under the assumption that the directional bias
Fig. 7. Wireframe model of the exploitable area.
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
415
Table 2 Results of the cross-validation (search ellipsoid: 75 75 25 m) % error in mean estimate Corr. coeff of actual and estimate Kriging variance ratio Mean differences Mean absolute difference
1.09 0.89 1.00 0.005 0.11
adjustment was made, backed up by the results of an adjustment via weighted least squares, and contrasting the results via cross-validation. The results are shown in Fig. 9. The system of coordinates was rotated until located parallel to the main anisotropic directions: Axis X 95/-9, Axis Y 10/30, Axis Z 171/59. Several important observations may be made in view of the results:
Fig. 8. Omnidirectional indicator semivariogram for exploitable/nonexploitable slate.
does not introduce a significant bias in the results obtained. Our next step is to determine the principal anisotropic directions and the corresponding semivariograms. This is quite a complex task. There are many variables at play and much information to process. In this case, geology is undoubtedly an ally in our search for orientation and advice. On this basis a manual
- The X-axis is oriented according to the lineation produced by the cleavage-bedding intersection - The Y-axis is parallel to the planes formed by the kink-bands - The semivariogram in both these directions is quite similar - The semivariogram in the direction parallel to the Zaxis and perpendicular to the kink-bands has the greatest variability. These results tie in with the fact that it is precisely the kink-bands that are responsible for breaking the continuity of the mass. Used for the cross-validation was an ellipsoid of 75 75 25 m parallel to the new system of co-ordinates and employing the 50 nearest points for each estimation. The results were as described in Table 2. Table 3 groups the estimations in classes and describes the real value for each range of the estimation. Although the results are very positive, we should not forget that the points kriging has been carried out along the drill cores, where there is a great deal of information Table 3 Estimated vs actual by range
Fig. 9. Directional indicator semivariograms for exploitable/non-exploitable slate.
Range
N8 samples
Av. estimate
Av. actual
0.00–0.10 0.10–0.20 0.20–0.30 0.30–0.40 0.40–0.50 0.50–0.60 0.60–0.70 0.70–0.80 0.80–0.90 0.90–1.00
383 7 3 0 71 89 0 2 9 429
0.02 0.13 0.22 0.00 0.48 0.52 0.00 0.79 0.88 0.98
0.00 0.14 0.00 0.00 0.51 0.57 0.00 0.50 0.78 0.98
416
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
parallel to the anisotropic directions and samples are required in at least three of the eight octants. The results are depicted in Fig. 11. For each block we have the following information: tonnage, percentage of exploitable slate, kriging variance, and finally, the number of samples used in the estimation. 6. Validation of the model Fig. 10. Search outside the internal ellipsoid.
near each point. If we apply a severe restriction to the search – excluding the information within a 25 25 8.5 m ellipsoid concentric to the previous ellipsoid (Fig. 10) – the results vary; for example, the percentage error in the mean is 4% and the ratio between the mean variance of the kriging and the mean squared error of the estimation is 0.82. For the estimation of the reserves we need to create a block model on which to carry out the kriging. In the selection of the support we need to take into account three conflicting interests: - The miner who will require a block size equal to the selective mining unit - The geostatistician who will recommend a correspondence between size and the spatial distribution of the samples in the domain, as well as with the variogram obtained - Computational factors which will limit the total number of blocks in accordance with the characteristics of the equipment and the time available. In this case we have adopted a standard block size of 14 15 5 m. In order to adequately delimit the surround of the wireframe model its subdivision has been permitted. The search ellipsoid (75 75 25 m) is
A comparison of the model of the deposit with the objective reality as observed by the mining expert is the crucial step that will indicate the usefulness of the model for mining planning. For the validation of the model the average overall bank yield was calculated for Zone A over a period of 10 months. This was done by dividing the production of saleable slate by the total volume of material extracted from the output level, with a result of 0.053 (5.3%). Also measured were average block/bank yield of exploitable slate and saleable-slate/block yield, with a view to obtaining the average saleable slate/bank yield for exploitable slate. The first of these was calculated on the basis of the in-bank volume of exploitable slate and the volume of blocks loaded onto trucks, and for various exploitation banks. Of a total of 511 m3 of exploitable slate, 269 m3 was loaded in the form of blocks, i.e., 52.5%. Even though the sample size was not large, these results coincided with the average values estimated by the mining experts (i50%), to which should be added a small dispersion in the measurements taken. Plant yield was calculated on the basis of block inputs to the plant and slate production for the period under consideration. A plant recovery value was thus obtained of 15.8%. The data would indicate that the variability in this recovery rate is very low for time
Fig. 11. Results of the block kriging.
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
417
Fig. 12. Wireframe and block models of the quarry.
intervals of a year. For measurements made in intervals of a month the standard deviation for average recovery is 1.5%, which would suggest a great uniformity in the monthly volumetric distribution of the input blocks, and therefore, a certain homogeneity in the spatial distribution of the discontinuities in the output layer. The required yield is therefore 8.3%; and if we bear in mind that the average overall layer yield is 5.3%, then for this period of 10 months, 63.8% of the material extracted from the layer was exploitable slate. In order to compare the model with the real results, the tonnage of material extracted from the output level and the average percentage of exploitable slate were calculated, this time with information supplied by the model. This was done on the basis of the differences between the block model of the output level and the topographic wireframe models of the quarry over the 10-month period (Fig. 12). The filled areas were naturally taken into account. The results can be seen in Table 4. With this data we calculated the average percentage of exploitable slate (62.1%), and the mean value of the block kriging variance (680%2). The variance of the overall estimation was 17%2. These results would indicate the solidity of the research method, and demonstrate its validity and utility in the yearly process of mining design and planning.
Table 4 Results of the evaluation
Tonnage (t) Average % exploitable slate
1st wireframe model
2nd wireframe model
Filling
514 265 67.8
679 438 65.7
23 603 82.8
7. Conclusions Throughout our research we have applied tools furnished by geostatistics in order to model the output layer of a slate deposit, and with positive results. Although widely used and proven in various mining sectors, they are not applied with great effectiveness yet to the mining of ornamental rock, and particularly to the extraction of roofing slate. The delay may be attributed to the following factors: the peculiarity of this kind of exploitation where the dgradesT are established visually and qualitatively; the difficulty of quantifying – in terms of concrete results – the combination of the wide range of factors that influence final slate quality; and finally, the difficulty in estimating the average yields in a reliable way until the mining operation is already underway. Developments in computerised applications geared specifically to mining and combining different techniques such as three-dimensional modelling of surfaces and irregular solids, statistical analysis and calculation, economic optimisation and maximisation, can only encourage the use of the different kriging techniques in these sectors, even at this late stage. SIMPLICITY may be an incentive for the gradual introduction of these estimation techniques in the many active workings in the northwest of Spain, given that all the data necessary for the estimations in this research is easily obtainable for mining operations and that the results are easily interpreted. Technically speaking we can conclude that: ! Evaluation of slate deposits should be made on the basis of continuous drill core logging ! Geostatistics will permit slate resources to be estimated as long as the drilling grid size is sufficiently closed
418
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
! Block models of the deposit facilitate extraction planning ! Software packages and calculation techniques employed in metals and energy mining can be applied to ornamental rock and particularly slate, as long as the particular features of the latter are taken into account ! The models obtained correlate to the real results obtained for the quarry. Acknowledgement Our thanks to the E.U.FEDER program for financing this research via its Project 1FD97-0091. References Azca´rate, J.E., 1982. Introduccio´n a la metodologVa de investigacio´n minera. I.G.M.E., Madrid. Bastante, F.G., Taboada, J., Ordo´n˜ez, C., 2004. Design and planning for slate mining using optimisation algorithms. Engineering Geology 73, 93 – 103 (Amsterdam). GarcVa-Guinea, J., Lombardero, M., Roberts, B., Taboada, J., 1997. Spanish roofing slate deposits. Transactions of the Institution of Mining and Metallurgy, B 106, B205 – B234.
GarcVa-Guinea, J., Lombardero, M., Roberts, B., Taboada, J., Peto, A., 1998. MineralogVa y microestructura de la pizarra de techar: comportamiento termoo´ptico y fisilidad. Materiales de Construccio´n 48, 37 – 48 (n8 251). Journel, A.G., 1983. Non-parametric estimation of spatial distributions. Mathematical Geology 69 (5), 673 – 687. Matheron, G., 1962. Traite´ de Ge´ostatistique Applique´e. Tome 1. Editions Technip, ParVs. Matheron, G., 1963. Traite´ de Ge´ostatistique Applique´e. Tome 2. Editions Technip, ParVs. MICL, 1999. Datamine 5. Datamine Group. Mineral Industries Computing Limited. Nicieza, C., Mene´ndez, A., Ordieres, J.B., Taboada, J., 1994. Optimisation of stripping in a slate bank. International Journal of Surface Mining, Reclamation and Environment, vol. 8. A.A. Balkema Publishers, Rotterdam, pp. 17 – 22. Ribeiro, J., Pereira, H.G., Sousa, A.J., Albuquerque, T., Tirabasso, F., Taboada, J., Rico, J.G., 1997. Geostatistical characterisation of natural stone quarries. In: Baafi, E.Y., Schofield, N.A. (Eds.), Geostatistics Wollongong’96. Kluwer, pp. 905 – 915. Taboada, J., Vaamonde, A., Saavedra, A., Alejano, L., 1997. Application of geostatistical techniques to exploitation planning in slate quarries. Engineering Geology 47, 269 – 277 (Amsterdam). Taboada, J., Vaamonde, A., Saavedra, A., Argu¨elles, A., 1998. Quality index for ornamental slate deposits. Engineering Geology 50, 203 – 210 (Amsterdam).
Evaluation of the resources of a slate deposit using indicator kriging F.G. Bastante *, J. Taboada, L.R. Alejano, C. Ordo´n˜ez Mining Department, University of Vigo, Spain Received 14 May 2004; received in revised form 10 August 2005; accepted 26 August 2005 Available online 24 October 2005
Abstract The paper commences with a description of the mining resource studied (roofing slate), the particular features of slate deposits and slate mining techniques. We then describe the methodology used for drilling core studies, the most reliable method for ornamental rock mining exploration. From the drill-core data, applying statistical techniques (kriging) with the assistance of the powerful DATAMINE 5 software package ([MICL, 1999. bDatamine 5Q. Datamine Group. Mineral Industries Computing Limited.]) can make an assessment of the exploitable resources in a quarry. To do this, the research domain is first delimited – in terms of the top and bottom of the exploitable seam, as well as the sidewall boundaries – employing ordinary kriging. A wireframe model of the deposit (the domain) is constructed and is sub-divided into blocks (block model). The resources in the deposit are then evaluated using indicator kriging and taking into account the anisotropic directions of the slate deposit. This produces data in the form of locations, tonnages and percentages of exploitable slate. The final step compares results obtained from the model with results obtained from mining in order to arrive at conclusions respecting the methodology. D 2005 Elsevier B.V. All rights reserved. Keywords: Slate; Ornamental rock mining; Resources evaluation; Indicator kriging
1. Introduction In mining, all knowledge is inevitably linked to uncertainty, and this occurs in the form of risk when a mining deposit is assessed. In order to be able to obtain the necessary decision-making elements for each of the developmental phases of the mining project, the mining company has to quantitatively calculate the value for each parameter that intervenes in the mining evaluation along with the uncertainty associated with these values. ´ rea de Explotacio´n de * Corresponding author. E.T.S.I.Minas, A Minas, Campus Lagoas-Marcosende, 36200 Vigo (Pontevedra), Spain. Fax: +34 986812201. E-mail address: [email protected] (F.G. Bastante). 0013-7952/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2005.08.003
Innate to the mining deposit itself are the quantity, quality and spatial distributions of the different materials present, and from the outset it is on these primordial factors that the feasibility of a mining project will depend. The uncertainty associated with each will therefore have a significant bearing on the mining companyTs final decision as to whether to take on the risk of allocating human, economic and financial resources to the development of the project. The costliness of mining research combined with the fact that the geological variables (thickness, grades, accumulations of metal, etc) present a numerical structure and a certain spatial order – in many cases only statistically evident (Azca´rate, 1982) – have both made for a rapid development and widespread use of different kinds of geostatistical tools (Matheron, 1962, 1963),
408
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
such as, for example, techniques for the determination of the spatial correlation between variables and for the calculation of values and associated uncertainties in areas of potential interest. This paper describes the kriging procedure applied to an evaluation of the resources of a slate quarry located in Leon, Spain, with the assistance of the DATAMINE 5 software package (MICL, 1999). From the geological information available for the area and the logs of the boreholes made in the bed, the quantity of exploitable slate and its spatial distribution in supports of 1000 m3 have been obtained. We made use of this block model in the design and planning of the quarry using optimisation algorithms (Bastante et al., 2004). Fig. 1 which describes the overall process followed so as to attain our objective, refers to the following phases: ! Collection of topographical, geological and core log data
! Creation and checking of databases based on the above information ! Transformation and analysis of the data ! Extraction of the data necessary for the delimitation of the exploitable area ! Semivariogram modelling; Data kriging ! Creation of a wireframe model of the reference area based on the points obtained ! Data selection from within the model ! Semivariogram modelling ! Creation of a block model in the interior of the wireframe model; Block kriging ! Resource evaluation ! Validation of the results with real data from the quarry. We will now describe the general features of the deposit and the particular idiosyncrasies of slate core logging, before moving onto the evaluation proper.
Fig. 1. Research methodology.
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
2. Features of slate and slate mining On a general level, all fine-grained rock capable of being exfoliated in thin sheets of a large size relative to thickness is considered as slate. Our research is concerned with ornamental slate, normally dark in color, and exploited industrially for its applications in roofing, fac¸ade finishing, interior and exterior flooring, and stonework and decorative walls. Its main feature is its excellent fissility. Lithologically speaking, slate is clay shale that has undergone a low-grade metamorphism (below the isograde of biotite). Mineralogically speaking, we can describe its basic components as: Sericite (40–46%), Quartz (22–25%) and Chlorite (15–17%). The accessory minerals are very varied. They are not particularly important in terms of the physical and chemical properties of the rock but may, on the basis of their alterability, determine the commercial quality of the slate. The immense majority of the exploited slate quarries in Spain are Ordovician, although there are some isolated Devonian or Cambrian slate deposits that are commercially interesting (GarcVa-Guinea et al., 1997). During the different series, successive layers of clays were deposited on the sea floor and acquired a significant thickness. Heat and pressure provoked by Hercynian folding caused a metamorphism of the clay sediments that in turn produced mineralogical changes and a reordering of the crystals. The geological factors that have a bearing on the quality of a slate quarry are of three kinds: stratigraphic, structural and metamorphic. In this respect, the mineralogical composition and grain size are the first two factors to take into account. Iron sulfides such as pyrrhite, pyrrhotite and marcasite are not desirable in the rock, since they produce anti-aesthetic iron oxide spots on roofs on oxidation. Neither are carbonates desirable, given that these result in circular whitish spots on the slate surface when dissolved and weathered. In terms of the metamorphic process undergone by the rock, another conditioning factor is the slate’s microscopic homogeneity. Slate that has a homogeneous distribution of quartz grain size is generally more fissile than slate having greater variation of the grain size. Microtexture is the next factor to be taken into consideration. The desirable microtexture is lepidoblastic, in the form of penetrative slaty cleavage fabric. This consists of the planar orientation of all the phyllitose and inequigranular minerals, which constitute the preferred exfoliation plane for the rock (GarcVa-Guinea et al., 1998).
409
The presence of more than a single cleavage in the rock is prejudicial. In the Spanish Hercynian Mass, the principal continuous cleavage – which causes the slaty cleavage denominated as S1 – is a consequence of the initial phase of Hercynian deformation. But many regions of the Mass are also affected by other, later phases, and especially by the third phase, which caused the slate to have microfolds or crenulation in the S1 planes, spaced crenulation cleavage denominated S3, and even compositional tectonic banding, due to the dissolution of the quartz in the short flanks of the asymmetric folds. The macroscopic geological aspects of the mass are also fundamental to the determination of whether a mass may or may not constitute a slate deposit. The thickness of the bed is the first aspect to take into consideration in this respect, and as a general rule, 10 m is the minimum thickness exploited, although this very much depends on the quality of the slate itself and the terrain morphology. The structural position of the bed is also influential. Hercynian Mass slate beds are tucked into asymmetric folds, and the flanks are preferable to the fold hinge zones, given that in the former bedding and cleavage form less of an angle than in hinges and the slate is consequently more homogenous. Nevertheless, deposits of slate to be found in hinges may in fact be larger, given the thickening that occurs in fold hinges. Another negative factor is the existence of contact metamorphism subsequent to very low-grade regional metamorphism, given that the metamorphic recrystallisation produced as a consequence welds the cleavage planes and prevents a satisfactory exfoliation of the rock. Undoubtedly, the definitive-limiting factor respecting the exploitability of a mass is its fracturation, with fractures understood not only as joints and faults but also as quartz dikes, kink-bands and other more or less planar discontinuities, other than bedding and cleavage. In that the system of fractures conditions the orientation of the working banks, and the state of fracturation conditions the size of block extracted from the mass and decisively determines the extraction methods to be employed as well as the yield of the workings. It may appear that any area with low-grade clayshale rock outcrops favours the existence of ornamental slate deposits, but this is only true to a certain extent. The French and Spanish Hercynian Mass deposits are concentrated above all in two geological formations, which is probably due to sedimentological causes that have not yet been sufficiently researched. As methods for slate quarrying, we can distinguish clearly between the more frequently utilised surface
410
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
mining method, and the underground method that is gradually becoming more popular. The first surface mining task is the clearing of the altered zone of the deposit or the overlying spoils in order to uncover the exploitable level. This operation is carried out using earth-moving equipment, or explosives for non-ripable areas. The difficulties will depend on the terrain orography and the spoils management provided for in the mining plan, an important aspect of planning given the great volumes of material shifted in modern slate mining processes. Once a working bank has been prepared, the task of extracting the blocks is undertaken using a variety of methods. These all have in common that (a) they should damage as little as possible the adjoining blocks and the residual rock mass; (b) that they should be applied according to the anisotropic directions and discontinuities of the mass, and (c) that the maximum block size takes into account, not only the natural limits imposed by the discontinuities, but also the dimensions of the treatment plant. Bearing these factors in mind, we can design our bank-cutting grid so as to obtain the required blocks, on the basis of one of three methods most utilised: low-grade explosives, diamond-wire cutter and diamond-disk or rock shearing machine. The method based on the use of low-grade explosives is the most typical. Of the mechanical cutting methods, the most popular method is the diamondwire cut since it maximises yield in almost all kinds of ornamental rock, although at a higher operational cost. Given the importance of the kind of cut from the point of view of yield, systems for the optimisation of this process on the basis of rock mass discontinuity data are being developed. The simplest system for calculating the quarry extraction yield is to consider an artificial cut as yet another discontinuity that is integrated into the structural study. On the basis of a trial-and-error approach that involves adaptations of the cutting grid applied, the optimal size for each bed can be determined (Nicieza et al., 1994). The blocks are extracted using the bucket teeth of a shovel loader or backhoe that extracts the blocks from the mass in accordance with the cleavage direction. These are then loaded onto trucks for their transportation to the plant. In the processing plant, commercial sheets of slate are obtained as follows: - Initial exfoliation of the blocks to a thickness of less than 30 cm - Cutting using a diamond-disk saw (1m in diameter) - Exfoliation in sheets of commercial thickness
- Cutting to final sizes and squaring in accordance with commercial templates - Selection and packaging. The deposit that is the subject of this study is in the Agu¨eira Formation (Middle-Upper Ordovician) and it is composed of a monotonous series of fine-grained slate with some sandy laminations. The principal cleavage is N110E in direction with a dip of 60 S. Overlaying the exploitable level there is a dense layer of kink-bands running N100E in direction with a dip of 30–50 N. The footwall of the deposit contains a layer of highly folded quartzite which joins the layer of kink-bands towards the north and thus restricts the possibilities for mining in that direction. The mining method used is the descending-terrace method, with 5 m high extraction banks; in-bank diamond wire cutting (horizontal cuts to break the cleavage plane continuity, and vertical cuts at 3 m perpendicular to the cleavage-bedding intersection); and backhoe cleavageoriented block extraction. 3. Core logging Research into slate quarries is based on outcrop data – which is, however, limited by the superficial alteration that generally affects slate masses – and above all, on continuous drill cores, which permits healthy rock to be studied and analyses its behaviour as ornamental slate (Taboada et al., 1998). In order to obtain all the geological–geotechnical data from the study of the slate drill cores, we need to employ very small-scale diagrams (1 : 50) that take into account the following parameter values, grouped into five main categories: Perforation Data ! Diameter, kind of perforation and depth This information is essential when interpreting the core logs, and at the same time serves to explain some of the results that may have been influenced by the mechanical perforation tasks. Geological Data ! Lithological description This section describes the several geological formations penetrated by the boreholes, with a special emphasis on the lithological characteristics of slate formations relevant to the exploitability potential of the deposit (fissility) or the quality of the rock (color, textural homogeneity, grain size, content in impurities or alterable minerals, laminations and cleavage plane roughness).
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
Geotechnical Data ! Core recovery This is the percentage of core samples obtained with respect to the total perforated length. ! R.Q.D. (rock quality designation) This geotechnical parameter is the percentage core recovery in pieces larger than 10 cm. In slaty formations it is very easy for the core samples to break along the cleavage plane when handled, for which reason it is important to distinguish the breaks due to natural discontinuities and those due to the slate structure itself or the handling of the samples. ! Number of fractures per meter This fracturation index per unit of borehole length has a value in conjunction with other parameters relative to the fracturation of the mass. In calculation, discontinuities such as bedding or cleavage will not be taken into account. ! Alteration degree This is defined according to the alteration scale for detrital sedimentary rock. Structural and Discontinuity Data ! Type of discontinuities We understand by discontinuity, any structure with a more or less planar geometry that breaks the continuity of the rock mass, such as stratification, cleavage, faults, joints, kink-bands, shear bands, dikes and veins. The type of discontinuity analysed has to be identified in this section. ! Joint dip This is preferably measured in the form dip direction/dip angle. In order to establish a joint dip direction, this should be measured in comparison with the joint that is always present in slaty formations, having a constant regional direction that can be measured at the borehole collar in outcrops. We refer to the flow cleavage of the axial plane, the basis for the exploitation and utilisation of ornamental slate. ! J.R.C. (joint roughness coefficient) This coefficient is important in order to describe and quantify the state of the surface, since it conditions the mechanical response of the joint to shearing efforts. ! Polarity In this parameter we refer to the location of the mass within more marked geological structures, which in the case of slate deposits, are folds. Polarity indicates the position of the mass in a direct or reverse flank, or in the hinge of a fold. This position can be determined on the basis of sedimentary criteria such as granulometric selection or on the basis of
411
structural criteria such as the regional shape of the folds and the relative positions of the bedding and cleavage planes (Taboada et al., 1997). Observations ! Opening, filling and weathering of the joints; water filtrations ! Discontinuity families With the joint orientation data and assisted by the stereographic templates we can define the different families of mass joints and classify the discontinuities. ! Spacing This is the distance between two adjacent joints of the same family. It can be measured on the basis of the joint data, once the families have been identified. It is a very important parameter since it affects block size, which is an important deposit feasibility selection factor (Ribeiro et al., 1997). Drill core logging is the most important method for investigating slate in depth since the indirect methods do not provide sufficient contrast in the slate beds. From the proposed core log a series of quality parameters is obtained for the mass in terms of its exploitability as ornamental rock: ! Jv (volumetric joint count): This is the number of discontinuities per unit of mass volume. ! Vb (useful volume): This is the relative volume of natural blocks greater than a given minimum size. It is indicative of the maximum theoretical yield of the exploitation. In the final analysis, and given the great number of parameters that influence the exploitability of a deposit, it is the slate expert who, following a study of the drill cores and an evaluation of the variables described above, identifies the part of the mass that is exploitable as ornamental slate, and the part that will go to the slag heap. 4. Delimitation of the domain The first step, prior to any assessment process, is to delimit the area of interest on which the study of the population is to be performed. In the interest of simplicity and meticulousness, the area must be broken up into independent zones in such a way that the universe we are examining will display simplicity or similarity. A geomorphologic interpretation of the genetic and structural information available was first carried out, followed by a detailed posterior analysis for spatial demarcation purposes, of the economically interesting sector in this case. Since our population is composed of the slate from the quarry, and our objective is the
412
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
estimation of useful or exploitable slate, our domain will be determined by the areas in which slate appears. Fig. 2 illustrates the division of the quarry into two independent zones on the basis of structural criteria, and according to the interpretation of the footwall quartzite on which the exploitable layer of slate rests. This paper focuses on a study of Zone A, for which more information is available and in which the greater part of the exploitation reserves are concentrated. 4.1. The bottom and the top of the exploitable layer The next step in the delimitation of the domain was the estimation of the bottom of the exploitable layer, understood as the surface below which, according to the cores, there is no slate that can be sent to the treatment plant. It has to be pointed out that although a technical criterion has been used for the definition of the separation area, this presupposes the confluence of a series of genetic-structural factors. On the other hand, the fact that a point corresponds to the said surface does not presuppose that in the upper adjoining area there is exploitable slate. The starting point was the identification of the coordinates (X, Y, Z) of the deepest extremity where exploitable slate was to be found for each drill core in Zone A. The vertical co-ordinate (Z) was considered as regionalized variable. With these data the omnidirectional variogram and the corresponding model were calculated, as illustrated in Fig. 3. The apparition of a monomic model of a degree close to two indicated a linear tendency, which means that the area approximated a plane. The corresponding equation was obtained using linear regression, with a resulting correlation coefficient equal to 0.929. Next, the points that remained below the previous mid-plane were selected and a linear regression was
Fig. 3. Semivariogram of the vertical co-ordinate (bottom points).
again carried out, resulting in a correlation coefficient of 0.975. Thus, the plane may be said to satisfy our delimiting mining criterion. Even so, a change was made in the co-ordinates system to one that was parallel to the plane. The initial system of co-ordinates was rotated to align the X, Y axes with the regression plane. In this new system was observed that the tendency disappeared. The vertical co-ordinate in the rotated system of co-ordinates was considered as a new regionalized variable. The omnidirectional variogram was calculated and modelled, ordinary kriging was applied and the values obtained were transformed into the initial system of co-ordinates. Fig. 4 illustrates the previous process and the wireframe model created on the basis of the krigged points. The results of the crossvalidation are described in Table 1. Obtained in a similar way were the data for the top of the exploitable layer, defined as the surface below
Fig. 2. Location plan of the boreholes and interpretation of the footwall quartzite.
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
413
Fig. 4. Change in co-ordinates, semivariogram (new vertical co-ordinate) and final result.
which exploitable slate begins to appear in the drill cores. Although a great part of this area has already been exploited it continues to be important from our point of view of delimiting the exploitable layer. In this case, in addition to the genetic and structural factors, the surface alteration that the slate has undergone over time plays an important role. Fig. 5 shows the omnidirectional variogram for the top points. We see again the appearance of a monomic model, this time of a lower degree but also close to two. The regression plane (correlation coefficient r = 0.85) has a dip of only 7.58, which explains both why the variogram is smoother and the points show less variability. Changing the co-ordinate points system to one parallel to the regression plane produced the structure depicted in Fig. 6. The top surface was modelled in a similar way to the bottom, on the basis of the points obtained.
the south and in depth was drawn – following the direction and average slope of the kink-bands – for an average distance of 25 m from the drill cores on the site plan, and also taken into account was the need for access to the banks. This resulted in the wireframe model illustrated in Fig. 7. The next stage of the research consisted of utilising the information provided by the drill cores for the estimation of the potentially exploitable reserves as well as the uncertainty of these results. For this we had to decide the support on the basis of which to make the calculation, and then to complete the wire-
4.2. Delimitation of the sidewalls and final modelling The eastern and western boundaries were vertical surfaces (see Fig. 2), in the area close to the edge of Zone A. The northern boundary was delimited by a vertical surface to an average distance of some 25 m from the drill cores on the site plan. The boundary to Table 1 Results of the cross-validation (search radius: 125 m) % error in mean estimate Mean kriging variance Mean estimated variance Mean differences Mean absolute difference
0.62 10.45 10.53 0.090 2.48
Fig. 5. Semivariogram of the vertical co-ordinate (top points).
414
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
Fig. 6. Semivariogram of the vertical co-ordinate in the rotated system (top points).
frame model with blocks representing physical production units, necessary for the subsequent planning of the mining works. 5. Estimation of the reserves How can we utilise the information supplied by the drill cores in order to evaluate the slate reserves? The answer is in indicator kriging. If we transform the exploitable/non-exploitable slate information into a 1/0 binary system, we can carry out an indicator kriging (Journel, 1983) on the blocks in such a way that the result – the mathematical expectation or expected value of the indicator for each block – is a calculation of the proportion of material that corre-
sponds to Class 1 (exploitable slate) within the support chosen. The procedure is as follows: we select the drill cores that are within our wireframe model, and we divide these into stretches of 1 m. We assign values to each stretch – 1 for 100% exploitable slate and 0 for 0% exploitable slate – on the basis of the drill core information, and we calculate the omnidirectional variogram depicted in Fig. 8. The resolution of 1 m was established in part from the need not to lose precision in the transformation. In our case, more than 92% of the continuous stretches of exploitable slate have a length in excess of one metre, representing 99% of the total accumulated length of the exploitable slate. On the other hand, mining requires a minimum block thickness in the order of 50 cm, and therefore, as thickness is given by the normal direction to the cleavage plane that dips 608, the support can be considered as representative of the minimum block size in that direction. The next question that arises is: can we work with the isotropic model that we have adjusted to the experimental omnidirectional semivariogram? Unfortunately the answer is no. The reason for this is the directional bias of the base information. The behaviour of the semivariogram near the origin is defined principally by the vertical direction; in other words, it does not represent average behaviour in all directions. And the same happens with the rest of the semivariogram; some direction–distance combinations are better represented than others are. The data, therefore, is not distributed in the space in a homogeneous way. That said, the semivariogram obtained is extremely important. It demonstrates that the data do have a spatial order and permits us to make an initial preliminary estimation of the reserves under the assumption that the directional bias
Fig. 7. Wireframe model of the exploitable area.
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
415
Table 2 Results of the cross-validation (search ellipsoid: 75 75 25 m) % error in mean estimate Corr. coeff of actual and estimate Kriging variance ratio Mean differences Mean absolute difference
1.09 0.89 1.00 0.005 0.11
adjustment was made, backed up by the results of an adjustment via weighted least squares, and contrasting the results via cross-validation. The results are shown in Fig. 9. The system of coordinates was rotated until located parallel to the main anisotropic directions: Axis X 95/-9, Axis Y 10/30, Axis Z 171/59. Several important observations may be made in view of the results:
Fig. 8. Omnidirectional indicator semivariogram for exploitable/nonexploitable slate.
does not introduce a significant bias in the results obtained. Our next step is to determine the principal anisotropic directions and the corresponding semivariograms. This is quite a complex task. There are many variables at play and much information to process. In this case, geology is undoubtedly an ally in our search for orientation and advice. On this basis a manual
- The X-axis is oriented according to the lineation produced by the cleavage-bedding intersection - The Y-axis is parallel to the planes formed by the kink-bands - The semivariogram in both these directions is quite similar - The semivariogram in the direction parallel to the Zaxis and perpendicular to the kink-bands has the greatest variability. These results tie in with the fact that it is precisely the kink-bands that are responsible for breaking the continuity of the mass. Used for the cross-validation was an ellipsoid of 75 75 25 m parallel to the new system of co-ordinates and employing the 50 nearest points for each estimation. The results were as described in Table 2. Table 3 groups the estimations in classes and describes the real value for each range of the estimation. Although the results are very positive, we should not forget that the points kriging has been carried out along the drill cores, where there is a great deal of information Table 3 Estimated vs actual by range
Fig. 9. Directional indicator semivariograms for exploitable/non-exploitable slate.
Range
N8 samples
Av. estimate
Av. actual
0.00–0.10 0.10–0.20 0.20–0.30 0.30–0.40 0.40–0.50 0.50–0.60 0.60–0.70 0.70–0.80 0.80–0.90 0.90–1.00
383 7 3 0 71 89 0 2 9 429
0.02 0.13 0.22 0.00 0.48 0.52 0.00 0.79 0.88 0.98
0.00 0.14 0.00 0.00 0.51 0.57 0.00 0.50 0.78 0.98
416
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
parallel to the anisotropic directions and samples are required in at least three of the eight octants. The results are depicted in Fig. 11. For each block we have the following information: tonnage, percentage of exploitable slate, kriging variance, and finally, the number of samples used in the estimation. 6. Validation of the model Fig. 10. Search outside the internal ellipsoid.
near each point. If we apply a severe restriction to the search – excluding the information within a 25 25 8.5 m ellipsoid concentric to the previous ellipsoid (Fig. 10) – the results vary; for example, the percentage error in the mean is 4% and the ratio between the mean variance of the kriging and the mean squared error of the estimation is 0.82. For the estimation of the reserves we need to create a block model on which to carry out the kriging. In the selection of the support we need to take into account three conflicting interests: - The miner who will require a block size equal to the selective mining unit - The geostatistician who will recommend a correspondence between size and the spatial distribution of the samples in the domain, as well as with the variogram obtained - Computational factors which will limit the total number of blocks in accordance with the characteristics of the equipment and the time available. In this case we have adopted a standard block size of 14 15 5 m. In order to adequately delimit the surround of the wireframe model its subdivision has been permitted. The search ellipsoid (75 75 25 m) is
A comparison of the model of the deposit with the objective reality as observed by the mining expert is the crucial step that will indicate the usefulness of the model for mining planning. For the validation of the model the average overall bank yield was calculated for Zone A over a period of 10 months. This was done by dividing the production of saleable slate by the total volume of material extracted from the output level, with a result of 0.053 (5.3%). Also measured were average block/bank yield of exploitable slate and saleable-slate/block yield, with a view to obtaining the average saleable slate/bank yield for exploitable slate. The first of these was calculated on the basis of the in-bank volume of exploitable slate and the volume of blocks loaded onto trucks, and for various exploitation banks. Of a total of 511 m3 of exploitable slate, 269 m3 was loaded in the form of blocks, i.e., 52.5%. Even though the sample size was not large, these results coincided with the average values estimated by the mining experts (i50%), to which should be added a small dispersion in the measurements taken. Plant yield was calculated on the basis of block inputs to the plant and slate production for the period under consideration. A plant recovery value was thus obtained of 15.8%. The data would indicate that the variability in this recovery rate is very low for time
Fig. 11. Results of the block kriging.
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
417
Fig. 12. Wireframe and block models of the quarry.
intervals of a year. For measurements made in intervals of a month the standard deviation for average recovery is 1.5%, which would suggest a great uniformity in the monthly volumetric distribution of the input blocks, and therefore, a certain homogeneity in the spatial distribution of the discontinuities in the output layer. The required yield is therefore 8.3%; and if we bear in mind that the average overall layer yield is 5.3%, then for this period of 10 months, 63.8% of the material extracted from the layer was exploitable slate. In order to compare the model with the real results, the tonnage of material extracted from the output level and the average percentage of exploitable slate were calculated, this time with information supplied by the model. This was done on the basis of the differences between the block model of the output level and the topographic wireframe models of the quarry over the 10-month period (Fig. 12). The filled areas were naturally taken into account. The results can be seen in Table 4. With this data we calculated the average percentage of exploitable slate (62.1%), and the mean value of the block kriging variance (680%2). The variance of the overall estimation was 17%2. These results would indicate the solidity of the research method, and demonstrate its validity and utility in the yearly process of mining design and planning.
Table 4 Results of the evaluation
Tonnage (t) Average % exploitable slate
1st wireframe model
2nd wireframe model
Filling
514 265 67.8
679 438 65.7
23 603 82.8
7. Conclusions Throughout our research we have applied tools furnished by geostatistics in order to model the output layer of a slate deposit, and with positive results. Although widely used and proven in various mining sectors, they are not applied with great effectiveness yet to the mining of ornamental rock, and particularly to the extraction of roofing slate. The delay may be attributed to the following factors: the peculiarity of this kind of exploitation where the dgradesT are established visually and qualitatively; the difficulty of quantifying – in terms of concrete results – the combination of the wide range of factors that influence final slate quality; and finally, the difficulty in estimating the average yields in a reliable way until the mining operation is already underway. Developments in computerised applications geared specifically to mining and combining different techniques such as three-dimensional modelling of surfaces and irregular solids, statistical analysis and calculation, economic optimisation and maximisation, can only encourage the use of the different kriging techniques in these sectors, even at this late stage. SIMPLICITY may be an incentive for the gradual introduction of these estimation techniques in the many active workings in the northwest of Spain, given that all the data necessary for the estimations in this research is easily obtainable for mining operations and that the results are easily interpreted. Technically speaking we can conclude that: ! Evaluation of slate deposits should be made on the basis of continuous drill core logging ! Geostatistics will permit slate resources to be estimated as long as the drilling grid size is sufficiently closed
418
F.G. Bastante et al. / Engineering Geology 81 (2005) 407–418
! Block models of the deposit facilitate extraction planning ! Software packages and calculation techniques employed in metals and energy mining can be applied to ornamental rock and particularly slate, as long as the particular features of the latter are taken into account ! The models obtained correlate to the real results obtained for the quarry. Acknowledgement Our thanks to the E.U.FEDER program for financing this research via its Project 1FD97-0091. References Azca´rate, J.E., 1982. Introduccio´n a la metodologVa de investigacio´n minera. I.G.M.E., Madrid. Bastante, F.G., Taboada, J., Ordo´n˜ez, C., 2004. Design and planning for slate mining using optimisation algorithms. Engineering Geology 73, 93 – 103 (Amsterdam). GarcVa-Guinea, J., Lombardero, M., Roberts, B., Taboada, J., 1997. Spanish roofing slate deposits. Transactions of the Institution of Mining and Metallurgy, B 106, B205 – B234.
GarcVa-Guinea, J., Lombardero, M., Roberts, B., Taboada, J., Peto, A., 1998. MineralogVa y microestructura de la pizarra de techar: comportamiento termoo´ptico y fisilidad. Materiales de Construccio´n 48, 37 – 48 (n8 251). Journel, A.G., 1983. Non-parametric estimation of spatial distributions. Mathematical Geology 69 (5), 673 – 687. Matheron, G., 1962. Traite´ de Ge´ostatistique Applique´e. Tome 1. Editions Technip, ParVs. Matheron, G., 1963. Traite´ de Ge´ostatistique Applique´e. Tome 2. Editions Technip, ParVs. MICL, 1999. Datamine 5. Datamine Group. Mineral Industries Computing Limited. Nicieza, C., Mene´ndez, A., Ordieres, J.B., Taboada, J., 1994. Optimisation of stripping in a slate bank. International Journal of Surface Mining, Reclamation and Environment, vol. 8. A.A. Balkema Publishers, Rotterdam, pp. 17 – 22. Ribeiro, J., Pereira, H.G., Sousa, A.J., Albuquerque, T., Tirabasso, F., Taboada, J., Rico, J.G., 1997. Geostatistical characterisation of natural stone quarries. In: Baafi, E.Y., Schofield, N.A. (Eds.), Geostatistics Wollongong’96. Kluwer, pp. 905 – 915. Taboada, J., Vaamonde, A., Saavedra, A., Alejano, L., 1997. Application of geostatistical techniques to exploitation planning in slate quarries. Engineering Geology 47, 269 – 277 (Amsterdam). Taboada, J., Vaamonde, A., Saavedra, A., Argu¨elles, A., 1998. Quality index for ornamental slate deposits. Engineering Geology 50, 203 – 210 (Amsterdam).