Calculating ore resources on complex geology using ...

1st International Conference of Underground Mining October 19-21, 2016, Santiago, Chile

Calculating ore resources on complex geology using a geometric restitution methodology: from modeling to the estimation Felipe Navarro, Department of Mining Engineering, FCFM, University of Chile, Chile

Advanced Laboratory for Geostatistical Supercomputing (ALGES), Advanced Mining Technology Centre (AMTC), FCFM, University of Chile, Chile Daniel Baeza, Department of Mining Engineering, FCFM, University of Chile, Chile

Advanced Laboratory for Geostatistical Supercomputing (ALGES), Advanced Mining Technology Centre (AMTC), FCFM, University of Chile, Chile Dafne Herreros, Yamana Gold Inc. – Southern Operations, Production Geology Department, Chile Marcos Valencia, Yamana Gold Inc. – Southern Operations, Production Geology Department, Chile

Department of Mining Engineering, Engineering Faculty, Andres Bello National University, Chile

Abstract Tectonic associated to the deposits formation is responsible of its final spatial geometry, make interpretations or reproduce or even figure it out what will be the model and build geological scenarios is a daily and simple task for geologists and it is possible reflect with any 2D or 3D tools showing a deterministic and static result. The fact of build a final stage of a deposit is simple but from the resources estimation point of view not always is the optimal way to develop estimation. Thus, the condition of the tectonic effects over certain deposit was a need that Yamana Gold Inc. wanted to have a solution and ALGES Laboratory works in develop an efficient but customizable solution. The big challenge, in terms of computational development was define an advanced work methodology to carry the deposit to the point of its natural geological deposition stage called geometrical restitution with the main goal of a better utilization of the information that will improve geostatistical procedures. That was the aim of Yamana Gold Inc. to invest in I+D due to company’s main deposits are hosted in complex and sometimes poorly understanding geology, this fact cause impact from the geological modeling to the resources estimations and were develop appropriated methodologies and tools focused on create value over the investment in geological drilling. Yamana and ALGES have created a software to solve complex geological geometries called U-Fo, which have capabilities to find customizable solutions and have the functionality to connect with any commercial software acting like a perfect complement to the current software suites that the companies have.

1

Introduction

One of the main issues in the mining industry is the Return on Investments (ROI) and generally companies use this indicator associated with the mineral discovery and the uncertainty of these resources discovered. UMining2016 Abstract’s Guidelines – © March2016

Geological features like faults and folds can be a difficult if we can’t figure out how we can use this information, coming from drill holes, properly. For that reason, develop geometrical restitution in complex geology is one of the solutions to improve ROI because and having the premise that faults and folds are pre-mineral and assume geological simplification due to the effects of these structures we can improve our processes from modeling to the resources estimations getting accurate uncertainty over those resources rather than traditional procedure that do not include geometrical restitution. In order to tackle the problem of resource assessment on presence of faults and folds, different techniques have been developed. Kriging using local search has been used on traditional geostatistics (Deutsch & Lewis, 1992) (Xu, 1996). Through spectral methods (Borgman, Taheri, & Hagan, 1983) using independent Gaussian random variables or modelling as nonstationary spatial process, but stationary in local small regions (Fuentes, 2002). Multiple point geostatistics avoids the explicit definition of a random function but directly infers the required multivariate distributions from training images (Guardiano & Srivastava, 1993) (Journel, 2004). Additionally, the incorporation of image analysis techniques as a new form of kriging (Stroet & Snepvangers, 2005) was used in order to reproduce curvilinear structures. Important multiple point algorithms (Mariethoz & Caers, 2014) that have been used are SNESIM (Strebelle, 2000) and Direct Sampling (Mariethoz, Renard, & Straubhaar, 2010). Unfolding techniques were used to perform stratigraphic transformations (Mallet, 2002) from nonlinear features to linear through space transformation. As a result, Mallet conclude that the optimal transformation is not unique. Finally, the incorporation of Locally Varying Anisotropy (LVA) fields allows the definition of anisotropic distances used on models with complex geometries (Boisvert & Deutsch, 2001), however the algorithm still requires significant computational effort, consequently his application to real scale deposits becomes difficult.

2

Methodology The proposed methodology consists of the following steps: 1. Modeling of geological units that require geological restitution. 2. Restoration to correct and restore displacement of ore bodies. 3. Geological Restitution applied to these units, geological criteria must be used to define temporality of structural movements. 4. Point support kriging estimation of grades in the unfolded and restored space. 5. After the grade and variance of blocks estimation are calculated, the points are backward transformed to the original block model.

Those steps will be applied to the corresponding block model and its samples, obtained from the geological model and the drill holes respectively. Therefore, the geological units (GUU) refers to that combination. A cross-section with an example of a faulted and folded mantle is showed in Figure 1. Here, the definition of GUU is obtained directly from the knowledge of the geological interpretation that consider fold and fault systems (Figure 1.a). The process of restoration will consider only the block model and samples associated to those units (Figure 1.b). In order to perform the unfolding step, algorithm requires the definition of a reference UMining2016 Abstract’s Guidelines – © March2016

surface, which is obtained from the restored body (Figure 1.c). Finally, the unfolding algorithm is applied to the defined units (Figure 1.d).



Figure 1

a) Initial model; b) After restoration performed; c) Reference surface; d) Unfolded model and surface

The definition of a reference surface gives a reduced dimensionality of the problem, which is known in literature as surface flattening problem. Surface flattening is an important process in many industries and is a critical aspect in the unfolding workflow of geological units (Horna, 2010) (Poudret, 2011) (Bennis, 2014). Restrictions like computational complexity and mathematical modeling has been studied for many years and is probed that is not possible to find the optimal solution (Chen, 2008). Many approaches to deal with this problem are in the academy (Wang, 2002) (Wang, 2005) (Wang, 2008) but the most attractive methods come to the heuristics algorithms and dynamics models. In this part we explain the new surface flattening implementation on unfolding workflow based on the energy minimization of the spring-mass system network. This method based on work of (Wang, 2002). To prevent overlapping a new function was implemented and to ensure the quality of this approach geometrics metrics and statistics measure were included in the experiments. Also were compared all flattening methods using on the unfolding workflow along the project. During this research we developed different algorithms to tackle this problem: projection, isomap (Tenenbaum, 2000), mds (Borg, 2005) and energy-based method.

2

Case Study

Geology of the case study is a calcareous sedimentary sequence mainly composed by limestone, shale and coquina rocks, last one bearing metal contents of gold associated to an amorphous gel composed by pyrite and marcasite called melnikovite. Main ore bodies use previous fractures to conduct mineralized fluids as feeders UMining2016 Abstract’s Guidelines – © March2016

and use primary porosity of coquinas to deposits mineralization. Structurally, the deposit is controlled first by a compressive system developing thrust faults and an associated fold made by the dragging action of compressive forces and later an extensive regime that is expressed as successive normal faults, like is showing in figures 1A, 2A, 2B and 2C.

Figure 2

a) Geological model including all units; b) Side view of fold to study; c) Top view of fold

After restoring the fault between green and purple GUU (as shown in Figure 2.b and 2.c), the corresponding reference surface must be obtained. Using a simple grid to calculate the maximum on each cell a triangulation is computed. The result of the reference surface triangulated is shown in Figure 3.

Figure 3

Different views of the surface used in experiments.

As mentioned in the previous section, after selecting and restoring the displaced GUU the next step will be unfold those units. This process is proven to be non-trivial and without an optimal solution (Mallet, 2002). Therefore, in our case the surface flattening will be the focus on the next section.

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3

Improving the surface flattening method on the unfolding workflow

3.1

Spring-mass model for a surface triangulation

In this method, surface triangulation is modeled as a spring-mass system, where nodes are representing for massive bodies and arcs for springs connections between them. The equilibrium position of each spring is the arc length in the original 3D surface. When flattening occurs, differences between the length on the flatten surface and the original 3D surface make displacement of springs respect to their equilibrium position suffering compression or stretching. The result of these deformations is a non-equilibrium dynamic system with nodes changing their positions to restore the balance. New positions, velocities and accelerations could be obtained solving kinematics and dynamics equations of each node. The elastic energy E and force f caused by deformations over a point is given by the follow equations,



















(1)















(2)

Where C is the spring constant, is the length of the arc ij on the planar surface and is the original length on the 3D surface. Figure 4.a shows a 3D surface and Figure 4.b shows a visualization of the energy distribution in the planar surface after the simulation.





(a) Figure 4

(b)

a) Original Surface; b) Visualization of the energy distribution

The mass of the node i is expressed like the sum of the k triangles which have the node i as a vertex, the equations for the mass of node i is as follow: where case is follow:



















(3)

is the area of the triangle k in the original surface and is a normalization parameter that in this . Finally, Lagrange equations for this model could be simplified and expressed like

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(4)

After doing the energy and force calculations over each node, the velocity of the i-node can in the can be expressed by:

















(5)













(6)

and position: To ensure a constant acceleration

3.2

must be very small, in this case 0.001.

Stop criterion

Wang et al. (Wang, 2002) suggest stop the simulation when the minimum energy of the spring-mass system is reached. This meaning that the optimization problem can be expressed as follow: where





















(7)













(8)

is the elastic energy of whole system,









The Figure 5 shows the energy variations through the iterations and also that the minimum energy can be obtained in the first cycle. This property make that the stop criterion can be modeled as the lowest value obtained after the first cycle.

Figure 5

Elastic deformation energy v/s iterations

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3.3

Penalty function

To avoid the overlapping between triangles, we propose a function that penalizes nodes which their new position change the orientation of the normal vector of any triangle that this node belongs as a vertex. In these cases, the new position for the penalized node-i is going to be the center of mass calculated for all nodes belonging to the k-triangles without the node-i. If any triangle-k change the orientation of the vector normal then the position of will be

















(9)

where is the position of the node-j belonging to the triangle-k and is the mass of the node-j. Figures 6.a and 6.b shows the effects penalty function over the simulations results and the way it fix the overlapping problem. This penalty function has negligible effects over the system energy then so it does not affect in the final result.

(a) Figure 6

3.4

(b)

Effects of the penalty function over final result, a) Simulation without penalty function; b) Simulation with penalty function included

Metrics

There are three metrics that we use to the quality assurance (QA) and the comparison of this energy based method with others implemented methods used in the unfolding workflow. The first metric is related with the shape of the surface. The output gives the percentage difference between the total perimeter of the original 3D surface and the flatten surface. Figure 7 shows a graph of the Shape error along 100 iterations of the spring-mass system dynamic.

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Figure 7

Percentage Shape error v/s iterations

The second metric implemented is related with a perimeter measure. This metric gives a percentage difference between the area of the original surface and the area of the planar the surface. Figure 8 shows the area error along 100 iterations.

Figure 8

Percentage area error v/s iterations

The third metric it called Kullback-leibler divergence (Joyce, 2011) (Hershey, 2007). This is an informationbased measure and return a value related with the disparity of the points distribution of the original surface P and the points distribution of the planar surface Q. In other words, gives a quantification of the lost information in Q when Q is an approximation of P. Some properties of this metrics are:



only if

















(10)

















(11)

with respect to P











(12)

Figure 9 shows Kullback-leibler divergence measure along 100 iterations of the energy based method for surface flattening. UMining2016 Abstract’s Guidelines – © March2016

Figure 9

Kullback-leibler divergence v/s iterations

The stop criterion proposed in (Wang, 2002) has a perfect match with the minimum value of these three metrics, probing that is the best solution given by this method.

3.5

Differences between methods

(a)

(b)

(c) Figure 10







(d)

Results of the flattening processes of the different methods implemented. a) Isomap method; b) MDS method; c) Projection method; d) Energy-based method.

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Table 1

Metrics comparison between the methods implemented QAQC

Isomap

MDS

Projection

Energy

Shape error

15%

17%

3,9%

3.5%

Area error

28%

31%

7,3%

6,9%

Kullback-leibler

0.0017

0.0029

0.004

0.00089

Different flattening methodologies where compared to get any idea about the efficiency of each one and compared with the base case that is without geometrical restitution. Four types of flattening where developed and consider Isomap, MDS, Projection and Energy. Estimation parameters and samples are the same for each case. Ordinary Kriging was developed in three passes with different search radii. Simple categorization was done to define measured (40x20x2), indicated (60x30x4) and inferred (80x50x4) resources based on the passes when the block was estimated and also variograms were calculated in normal and in restituted spaces. Table 2

Block models estimated for the different flattening methodology compared with the base case

Isomap

MDS

Projection

Energy

Total blocks Estimated

-18%

-18%

-18%

+0.1%

Measured + Indicated

-21%

-13%

-7%

+13%

Final results show flattening by Energy improve the quality of estimations and overall categorization because a better use of the information (Figure 1 D) also indicate better performance rather than other methodologies of geometrical restitutions. +13% of measured and indicated resources will change the viability of a project and can define accurately new investments and also improve ROI significantly.

Conclusions Modelling of a deposits is an almost trivial task for geologist in every project and in many cases these deposits are affected by faults and folds that require proper algorithm to assess properly resource bearing and its uncertainty. In order to restore the deposit to its natural geological stage, we propose a simple methodology which uses geological interpretation and develop geometric restitution in the deposits. One of the most important steps is the methodology proposed to get the surface to be flattening. Was developed and compared four techniques of flattening, using different measurements and indicators. Energy based model algorithm show one of the most promising results, both as numerical and visual consistency. The presented approach shows that both Measured and Indicated resources increases in 13% from traditional methods, which impact directly in the valuation of a mining project and its corresponding ROI. UMining2016 Abstract’s Guidelines – © March2016



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