Roussos Dimitrakopoulos

Geostatistical Simulation of Regionalized Pore-Size Distributions Using Min/Max Autocorrelation Factors

In many fields of the Earth Sciences, one is interested in the distribution of particle or void sizes within samples. Like many other geological attributes, size distributions exhibit spatial variability, and it is convenient to view them within a geostatistical framework, as regionalized functions or curves. Since they rarely conform to simple parametric models, size distributions are best characterized using their raw spectrum as determined experimentally in the form of a series of abundance measures corresponding to a series of discrete size classes. However, the number of classes may be large and the class abundances may be highly cross-correlated. In order to model the spatial variations of discretized size distributions using current geostatistical simulation methods, it is necessary to reduce the number of variables considered and to render them uncorrelated among one another. This is achieved using a principal components-based approach known as Min/Max Autocorrelation Factors (MAF). For a two-structure linear model of coregionalization, the approach has the attractive feature of producing orthogonal factors ranked in order of increasing spatial correlation. Factors consisting largely of noise and exhibiting pure nugget–effect correlation structures are isolated in the lower rankings, and these need not be simulated. The factors to be simulated are those capturing most of the spatial correlation in the data, and they are isolated in the highest rankings. Following a review of MAF theory, the approach is applied to the modeling of pore-size distributions in partially welded tuff. Results of the case study confirm the usefulness of the MAF approach for the simulation of large numbers of coregionalized variables.

Moving forward from traditional optimization: Grade uncertainty and risk effects in open-pit design

Abstract An economic argument is presented for the incorporation of quantitative modelling of the uncertainty of grade, tonnage and geology into open-pit design and planning. Two new implementations of conditional simulation—the generalized sequential Gaussian simulation and direct block simulation—are outlined. An optimization study of a typical disseminated, low-grade, epithermal, quartz breccia-type gold deposit is used to highlight the differences between the financial projections that may be obtained from a single orebody model and the range of outcomes produced when, for example, fifty deposit simulations are run. The effects on expectations of net present value, production cost per ounce, mill feed grade and ore tonnage are presented as examples and periods with a high risk of negative discounted cash flow are identified. Further integration of uncertainty into optimization algorithms will be needed to increase their efficacy.

A diversified Tabu search approach for the open-pit mine production scheduling problem with metal uncertainty

This paper presents a metaheuristic solution approach based on Tabu search for the open-pit mine production scheduling problem with metal uncertainty. To search the feasible domain more extensively, two different diversification strategies are used to generate several initial solutions to be optimized by the Tabu search procedure. The first diversification strategy exploits a long-term memory of the search history. The second one relies on the variable neighborhood search method. Numerical results on realistic large-scale instances are provided to indicate the efficiency of the solution approach to produce very good solutions in relatively short computational times.

Data-driven fuzzy analysis in quantitative mineral resource assessment

The integration of geo-information from multiple sources and of diverse nature in developing mineral favourability indexes (MFIs) is a well-known problem in mineral exploration and mineral resource assessment. Fuzzy set theory provides a convenient framework to combine and analyse qualitative and quantitative data independently of their source or characteristics.A novel, data-driven formulation for calculating MFIs based on fuzzy analysis is developed in this paper. Different geo-variables are considered fuzzy sets and their appropriate membership functions are defined and modelled. A new weighted average-type aggregation operator is then introduced to generate a new fuzzy set representing mineral favourability. The membership grades of the new fuzzy set are considered as the MFI. The weights for the aggregation operation combine the individual membership functions of the geo-variables, and are derived using information from training areas and L1 regression.The technique is demonstrated in a case study of skarn tin deposits and is used to integrate geological, geochemical and magnetic data. The study area covers a total of 22.5 km2 and is divided into 349 cells, which include nine control cells. Nine geo-variables are considered in this study. Depending on the nature of the various geo-variables, four different types of membership functions are used to model the fuzzy membership of the geo-variables involved.

Stochastic optimization for strategic mine planning: A decade of developments

Conventional approaches to estimating reserves, optimizing mine planning, and production forecasting result in single, and often biased, forecasts. This is largely due to the non-linear propagation of errors in understanding orebodies throughout the chain of mining. A new mine planning paradigm is considered herein, integrating two elements: stochastic simulation and stochastic optimization. These elements provide an extended mathematical framework that allows modeling and direct integration of orebody uncertainty to mine design, production planning, and valuation of mining projects and operations. This stochastic framework increases the value of production schedules by 25%. Case studies also show that stochastic optimal pit limits (i) can be about 15% larger in terms of total tonnage when compared to the conventional optimal pit limits, while (ii) adding about 10% of net present value to that reported above for stochastic production scheduling within the conventionally optimal pit limits. Results suggest a potential new contribution to the sustainable utilization of natural resources.

Traditional and New MIP Models for Production Scheduling With In-Situ Grade Variability

Mixed integer programming (MIP) has become a common approach for optimizing production schedules of open pit mines since the 1960s. However, MIP has been found to be limited by: (a) feasibility in generating optimal solutions with practical mining schedules; and (b) inability to deal with in-situ variability of orebodies. In looking into these shortcomings, this paper presents a general production scheduling method for multi-element deposits in open pit mines. The method is subsequently used to optimize the production schedule in a nickel laterite deposit. The application confirms the weaknesses of MIP formulations mentioned above. An alternative MIP formulation is then presented and applied to the same deposit. The results of the new formulations show that the new MIP model can overcome the above shortcomings and generate practical mining schedules with a higher chance of achieving planned production targets than traditional MIP schedules.

STOCHASTIC LONG-TERM PRODUCTION SCHEDULING OF IRON ORE DEPOSITS: INTEGRATING JOINT MULTI-ELEMENT GEOLOGICAL UNCERTAINTY

Meeting production targets in terms of ore quantity and quality is critical for a successful mining operation. In-situ grade variability and uncertainty about the spatial distribution of ore and quality parameter cause both deviations from production targets and general financial deficits. A stochastic integer programming formulation (SIP) is developed herein to integrate geological uncertainty described by sets of equally possible scenarios of the unknown orebody. The SIP formulation accounts not only for discounted cashflows and deviations from production targets, discounts geological risk, while accounting for practical mining. Application at an iron ore deposit in Western Australia shows the ability of the approach to control a risk of deviating from production targets over time. Comparison shows that the stochastically generated mine plan exhibits less risk in deviating from quality targets than the traditional mine planning approach based on a single interpolated orebody model.

Conditional simulation algorithms for modelling orebody uncertainty in open pit optimisation

Conditional simulation is a class of Monte Carlo techniques that can be used to generate equally probable representations of in-situ orebody variability. Contrary to the traditional smooth orebody models, conditionally simulated orebodies provide the tools to quantify uncertainty in grade variability and the resulting effects on various aspects of open pit design and planning. This paper outlines a general framework for modelling uncertainty and assessing geological risk, presents currently used geostatistical simulation algorithms, and presents examples.

A new approach for geological pattern recognition using high-order spatial cumulants

Spatially distributed natural phenomena represent complex non-linear and non-Gaussian systems. Currently, their spatial distributions are typically studied using second-order spatial statistical models, which are limiting considering the spatial complexity of natural phenomena such as geological applications. High-order geostatistics is a new area of research based on higher-order spatial connectivity measures, especially spatial cumulants as suitable for non-Gaussian and non-linear phenomena. This paper presents HOSC or High-order spatial cumulants, an algorithm for calculating spatial cumulants, including anisotropic experimental cumulants based on spatial templates. High-order cumulants are calculated on two- and three-dimensional synthetic training images so as to elaborate on their characteristics. Spatial cumulants up to and including the fifth-order are found to be efficient in characterizing patterns on both binary and continuous images. The behaviour of spatial cumulants is shown to characterize well the behaviour of the spatial architecture of geological data, including the degree of homogeneity and connectivity. The high-order cumulants are found to be relatively insensitive to the number of data used, and relatively small data sets are sufficient to provide cumulant maps. HOSC has been coded in FORTAN 90 and is easily integrated to the S-GeMS open source platform.

Algorithmic approach to pushback design based on stochastic programming: method, application and comparisons

Abstract Pushback design affects the way a mineral deposit is extracted. It defines where the operation begins, the contour of the ultimate pit, and how to reach such ultimate contour. Therefore, different pushback designs lead to differences in the net present value (NPV) of a project. It is important to find the optimal pushback design which maximises the NPV. Conventional approaches to designing pushbacks lead to not meeting production targets and NPV forecasts. This is mainly due to the lack of integrating uncertainty into the process. Recent efforts have shown that the integration of uncertainty into production scheduling results in NPV increases in the order of ∼25%. The purpose of this research is to make use of a stochastic integer programming model to integrate uncertainty into the process of pushback design. The approach is tested on porphyry copper deposit. Results show the sensitivity of the NPV to the design of starting and intermediate pushbacks, as well as the pushback design at the bottom of the pit. The new approach yielded an increment of ∼30% in the NPV when compared to the conventional approach. The differences reported are due to different scheduling patterns, the waste mining rate and an extension of the pit limits which yielded an extra ∼5500 t of metal.