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By M. Lange

The postulation that primary platinum group metal (PGM) matte will chemically react with magnesia-chrome bricks when temperatures exceed 1500°C was tested. Magnesia-chrome brick samples were heated in contact with matte at 1300°C to 1750°C for 30 minutes, after which the refractory samples were analysed using reflected light microscopy and scanning electron microscopy. The samples were all completely penetrated by matte. As the temperature increased the matte also penetrated the fused aggregate grains and disintegrated them. The chromium concentration of the matte inside the refractory samples was found to be slightly higher than that of the bulk matte. At temperatures of 1500°C and higher, MgO, FeO, and magnesium-rich silicate crystals could be identified in the matte directly adjacent to the refractorymatte interface. Phase relations clearly indicated that chemical reactions take place between primary PGM matte and the magnesiachrome refractory material at temperatures above 1500°C, but that these reactions are more complex than expected from FactSAGE® calculations.

Jan 1, 2014

By E. M. W. Skinner

Various models of near-surface kimberlite pipe emplacement have been proposed over the years. These include a top-down, phreatomagmatic model, a bottom-up, embryonic pipe model and three top-down explosive dyke models. AU of these models consider kimberlites as essentially the same rock type. However, different kimberlites have different pipe shapes and contain different rock types with very specific mineral assemblages and textures and therefore are likely to have been emplaced by different processes. Some authors have considered local geological differences as the principal reason for the contrasting geology but others argue that, while geological differences might contribute locally the petrographic peculiarities of particular kimberlites may be due mainly to inherent compositional differences specifically in the ratios of juvenile CO2 and H2O.

Jan 1, 2010

By P. J. H. de Graaf, S. D. N. Wessels

"Rio Tinto’s Western Australian expansion, combined with mining within structurally complex geology and increasingly below the water table, presents challenges in effective slope management to ensure safe and economic mining.The Geotechnical Management System (GMS) was developed by Rio Tinto Iron Ore (RTIO) to manage geotechnical risks identified during the design process and implementation, and to ensure feedback based on the ‘as-found’ conditions. The GMS utilizes a risk-based approach to geotechnical risk management and is centred on the geotechnical risk and hazard assessment management system (GRAHAMS). GRAHAMS is used to assess pre- and post-control risk for future potential risks (planned slopes), current risks (as-built slopes), and actual geotechnical hazards (realized risks) identified in the pit. This serves as a core operational risk management tool in identifying and prioritizing key risk sectors and management of critical controls. The system’s database reporting functionality supports effective communication of operational risks to operational personnel, as well as reporting the risk profile across operations to management.A rigorous engagement process between design engineers and sitebased engineers is implemented to ensure that key design assumptions, limitations, risks, and opportunities are understood by the site teams. This information, together with mine plan schedule details, is used to assess the design risk and develop appropriate controls. These controls typically include slope performance monitoring and slope reconciliation. The design feedback loop is closed through sharing of key slope performance and reconciliation data with the design teams.The GMS has been successfully implemented in RTIO pits and is fundamental to successful geotechnical slope management. Improved characterization of design assumptions has allowed for re-assessment of the pit design and improved hazard management in high-risk pits. The GMS, GRAHAMS, and other processes reduce the incidence of unexpected slope instability. Improved understanding of rock mass conditions has allowed for economic optimization through redesign of slopes, allowing for an improved understanding of risk and fewer unexpected conditions (surprises), hence an increased realized value."

Jan 1, 2016

By H. Von Michaelis

The world is rapidly becoming more ?green? i.e. energy conscious. Since mining and mineral processing are massive energy users, the mining industry is seeking ways to reduce its energy footprint. In mining and mineral processing operations, energy is often the most expensive cost item. Comminution, i.e. grinding, is frequently the most energy-intensive step between mine and metal. High pressure grinding rolls (HPGR) are being installed in a rapidly growing number of ?hard rock? mineral processing operations. In the energy conscious cement industry, HPGR grinding has been standard practice for decades. HPGR is known to be energy-efficient; the question is how energy efficient? How can the energy efficiency of HPGR be enhanced? This paper reviews recent HPGR applications for the processing particularly of copper, gold, platinum, PGM-Ni-Cu and iron ores. HPGR technology is discussed with respect to energy efficiency. The harder the ore the greater the energy savings are likely to be. Typical energy savings of 10?20% can be expected when installing HPGR vs. a SAG mill. Wipf(2005) showed why conventional Bond Work index tests on HPGR product is likely to under-estimate the energy savings that can be achieved by installing HPGR. Energy efficiency of HPGR is, however, ore-specific. The biggest energy savings of HPGR tested thus far is 9.5 kWh/tonne claimed for Vista Gold?s Mt. Todd gold ore from Western Australia, a savings of over one-third of conventional SABC comminution energy. Several commercial scale iron-ore pellet feed plants install HPGR for fine-grinding of concentrate to increase the surface area of pellet feed in a manner that reduces overall energy consumption. Energy-consumption in comminution is, however, only part of the energy savings benefit story. HPGR typically also reduces the amount of steel lost through wear of mill liners and media. Marsden(2008) showed that when the energy that would have been consumed to produce the steel that is saved by applying HPGR is factored in, then the overall energy savings of HPGR is considerably greater. Johnson et al.(1988, 2005) tested HPGR in a flowsheet in which energy savings of around 50% can be expected if HPGR product screen oversize is recirculated to the HPGR. Innovative flowsheets have been proposed by Rule et al.(2008) and by Morley (2008) which are expected to result in significantly increased energy savings. HPGR applied in flowsheets in conjunction with coarse ore separation devices, e.g. ore sorting or DMS to remove barren waste from HPGR feed promises significantly greater energy efficiencies. Fine-grinding and ultra-fine grinding of ore or concentrates in conventional ball mills results in energy consumptions that increase exponentially with product fineness. Wipf (2009) presented a flowsheet in which he proposed the installation of HPGR ahead of the Aerosion ?Disintegrator? for ultra-fine grinding of ores and/or concentrates. This dry grinding arrangement is expected to achieve ap80 = 7 µm using up to 100 kWh/t less energy than could be achieved by wet ball milling. Using conventional comminution, an ore might require grinding to p80 = 45µm (? 325 mesh), for example, to liberate the valuable components, e.g. magnetite, from gangue, e.g. silica. Inter-particle comminution in HPGR may break some ores along grain boundaries thereby liberating valuable minerals from gangue at a much coarser particle sizes. Early magnetic separation rejection of silica gangue liberated from magnetite by HPGR at coarser grain sizes could further reduce downstream comminution energy consumption.

Jan 1, 2009

By Z. Szczepanik, D. Milne, K. Kostakis

Abstract This paper presents the results of a series of creep tests performed on a total of nine samples of Lac du Bonnet granite from the Underground Research Laboratory near Pinawa, Canada. Each of the nine samples were subjected to stresses in excess of 80% of the average uniaxial compressive strength of the granite and were maintained at that level of constant loading for periods of time ranging between a few minutes and several months. Acoustic emission was recorded in order to monitor the development and accumulation of brittle microfracture damage in the samples throughout the long-term loading. According to previously published theoretical predictions and field results, as well as similar experimental results of creep tests in granite, samples should fail when loaded to 70% of their UCS for an extended period of time. Six of the samples did not fail when subjected to stresses in excess of 80% of their UCS over periods of several months and were subsequently subjected to uniaxial compression to failure. The average strength of the samples that had previously been subjected to long-term loading were comparable to the average published strength of this rock. Three of the nine samples had increased end confinement and these three samples failed under long term loading. Résumé Cet article présente les résultats d'une série d'essais de fluage de roche exécutés sur neuf échantillons de granite de Lac du Bonnet, du laboratoire souterrain de recherches près de Pinawa, Canada. Chacun des neuf échantillons a été alors soumis aux charges au-dessus de 80% de la résistance à la UCS moyenne du granit et a été maintenu à ce niveau du chargement constant pendant des périodes s'étendant entre quelques minutes et plusieurs mois. L'émission acoustique a été enregistrée pendant les essais afin d'obtenir des indications des dommages pendant le chargement à long terme. Selon des résultats théoriques précédemment édités et les résultats expérimentaux semblables des essais à long terme en granite, échantillons devrait échouer si chargé à 70% de leur UCS pendant une période prolongée. Six des échantillons n'ont pas cassé quant ils ont été soumis aux charges au-dessus de 80% de leur UCS et ont été plus tard soumis à la compression uniaxiale à la rupture. La force moyenne des échantillons qui avaient été précédemment soumis au chargement à long terme étaient comparable à la force éditée moyenne de cette roche. Trois des neuf échantillons avaient augmenté le confinement de fin et ces trois échantillons ont cassé sous le chargement à long terme. Zusammenfassung Dieser Artikel beschreibt die Resultate von Kriechversuchen, die an neun Proben aus dem Lac du Bonnet Granit des "Underground Research Laboratory" bei Pinawa in Kanada durchgeführt wurden. Jede der neun Proben wurde Drücken unterworfen, die mehr als 80% der einaxialen Druckfestigkeit des Granites entsprechen. Diese Drücke wurden für verschiedene Zeitspannen von wenigen Minuten bis zu einigen Monaten auf einem konstanten Niveau gehalten. Während der gesamten Langzeitbelastung wurden akustische Emissionen aufgezeichnet, um die Entwicklung und gesamthafte Beschädigung der Proben durch Sprödbruch aufzuzeichnen. Entsprechend bisher veröffentlichter theoretischer Vorhersagen, Feldversuchen und auch experimentellen Resultaten von Kriechversuchen an Granitproben wäre ein Versagen der Proben zu erwarten gewesen, wenn sie über längere Zeit einer Last von 70% ihrer einaxialen Druckfestigkeit unterworfen werden. Wider Erwarten trat bei sechs der Proben kein Versagen ein, als sie mit über 80% ihrer einaxialen Druckfestigkeit über mehrere Monate belastet wurden; daher wurden sie im Anschluss bei einaxialer Kompression zum Versagen gebracht. Die mittleren Festigkeiten der im Langzeittest verwendeten Proben sind vergleichbar mit den normalerweise für diesen Gesteinstyp angegebenen Festigkeiten. Drei der neun Proben versagten unter der Langzeitbelastung; diese drei Proben hatten eine erhöhte Endflächenreibung.

Jan 1, 2003

By J. Trcková

This paper describes the method of modelling rock mechanical problems on experimental models. Based on an actual experiment used as an example, assessment of the surface deformation due to undermining, the extensive possibilities of using a 3-D physical model and application results in practice are demonstrated. A special method of modelling the underground mine progress by using paraffin melting is presented. Keywords: undermining, subsidence of surface, 3-D experimental model

Jan 1, 2009

By S. Kok, D. Delport, J. Calitz

Altering the microstructure in order to improve the tensile properties of bow shackles resulted in inconsistency in the fatigue performance. This raises the question whether the inconsistency in fatigue life can be attributed to microstructural changes along the profile of the shackle or to decarburization at the surface. Bow shackles forged from 080M40 (EN8) material were subjected to different heat treatments in order to alter the microstructure. The shackles were subjected to five different fatigue load cases, which represented typical loads experienced at termination points for an overhead power line with a span length of 400 m, with changes in conductor type, configuration, wind, and ice loading. Although the change in microstructure does improve both the tensile and fatigue performance, we found that the depth of the decarburization layer has a greater effect on the high cycle fatigue life of bow shackles than the non-homogeneous microstructure.

Oct 1, 2021

By T. De Lange

Extensions to the beneficiation plant at the Grootegeluk Coal Mine are currently being planned in order to provide power station coal for Escom's Matimba Power Station, presently under construction. A simulation model was developed on the Olivetti M-24 microcomputer to facilitate the development of the envisaged flowsheet. The computer program simulates the heavy-medium separation and screening unit operations. The model is based on the construction of the Tromp partition curve described by an arctangent function, allowing ideal and non-ideal separations. The paper discusses the determination of the parameters required by the arctangent curve from standard process parameters such as Epm, Wolf cut-point, Tromp cutpoint and screening efficiency. The accuracy and shortcomings of the model are discussed, while an overview of the application of the model to evaluate borehole and bulk sample analyses is also given. It is concluded that the model is an invaluable aid to general flowsheet development.

Jan 1, 1987

By C. A. Young, J. L. Bowden

"Cyclic voltammetry experiments were conducted on copper and chalcopyrite (CuFeS2) in the absence and presence of ethyl xanthate, and compared to mass-balanced EH–pH diagrams. The results for copper duplicate those found in the literature and confirm xanthate chemisorption. However, the, results indicate that chalcopyrite oxidizes to chalcocite (Cu2S) and only afterwards is chemisorption observed due to small currents appearing with xanthate. This phenomenon suggests that the mineral's hydrophobicity is induced by more than dixanthogen and copper xanthate. Hydrophobicity was found to be pH-dependent over a range of alkaline conditions (pH 7–11) at narrow potentials (0 to –200 mV). An EH–pH diagram for chalcopyrite with xanthate is presented to illustrate the conditions under which chalcopyrite would be hydrophobic and thereby more thoroughly explain the results in the literature. IntroductionSulphide minerals are generally concentrated from ores by froth flotation, a physiochemical process that induces separation based on differences in hydrophobicity. In this process, a gas is bubbled into a slurry and the minerals that are hydrophobic (water-repelling) adhere to the bubbles and float to the surface while the hydrophilic (water-loving) minerals remain in the slurry. With sulphide minerals, the hydrophobicity is generally established in two ways. The first is by the addition of a collector, which is a heteropolar molecule with a reactive inorganic head group and an inert organic tail. Typically, the head group bonds with the sulphide mineral surface by adsorbtion either chemically (chemisorption) or physically (physisorption) or by complexing with atoms that solubilize somewhat at the surface (surface precipitation). These interactions leave the organic tail protruding from the mineral surface, thereby inducing hydrophobicity. The second method is by oxidation of the mineral surface to form a layer that is metal-deficient/sulphide-rich or comprised of elemental sulphur. Because the layer mimics naturally hydrophobic sulphur, this enables ‘collectorless’ flotation. Furthermore, this oxidation phenomenon adds an electrochemical dimension to the understanding of sulphide mineral flotation."

Jan 1, 2016

By M. B. C. Brandt

The near-surface wave attenuation factor k (kappa), which describes the attenuation of seismic waves with distance in the upper 1–3 km of the Earth, was determined for the Far West Rand gold mining area using seismic data recorded by station Parys (PRYS) of the South African National Seismograph Network. Twenty micro-events in the magnitude range 0.7 ? ML ? 1.8 for the period 1 July to 15 November 2015 were analysed. For the analysis a 10-second window for the S phase portion of the vertical component seismogram was selected. The result was an average k = 0.048 ± 0.014 s, which is much higher than for a stable continental region where k = 0.006 s. This is because the assumed Brune source model is inappropriate for describing mining-related events that often have moment tensors with a volumetric (implosive) component. The average k was calculated to be 0.098 ± 0.038 s for explosions of a similar magnitude. This higher value derived for mining-related k may be employed as a means of correcting the Brune source spectrum to calculate seismic moment as well as corner frequency for events in the mining areas of South Africa.

Jan 1, 2017

By T. P. Letsebe, L. Magwaza, T. A. Sethu, S. M. Rupprecht

"The authors were tasked with introducing safe and efficient drilling and blasting practices to a small artisanal mine located in the Eastern Province of Rwanda. The current mining methods employed at the mine are inefficient as rockbreaking is conducted utilizing hammer and chisel, or by drilling and blasting with no understanding of the rockbreaking process. The paper discusses the drill and blast cycle, as well as the associated activities necessary to support the mining cycle. The challenges and critical aspects in implementing drill and blast activities are discussed, as well as some of the technical aspects, i.e.drilling pattern, initiation and timing of the blast, and handling of misfires. Ancillary issues such as ventilation, training, communication, effective planning, logistics, and human relations are also examined. IntroductionRockbreaking accounts for 60% of the direct mining costs in small-scale mining. When drilling and blasting practices are poorly implemented the process becomes inefficient and dangerous. The introduction of a proven drill and blast system to artisanal mining offers a number of challenges, with safety being considered the backbone of any sustainable operation and a standard that cannot be compromised.The paper is based on a project at an artisanal cassiterite (tin) mining operation in a small village in the Kayonza District in the Eastern Province of Rwanda. A drill and blast programme utilizing industry best practice was introduced at an underground development end; a raise developed at an inclination of 23 degrees. The underground working is accessed via a 17 degree incline shaft with an operating length of 105 m and serviced by a single drum winder. The winder operates at a speed of about 3.5 m/min hoisting two 1 t wagons from the bottom of the incline to surface for tipping (Figure 1)."

Jan 1, 2017

By Z. Yang, X. Bian

"The synthesis of 3-hydroxy-2-naphthyl hydroxamic acid (H2O5) as a collector for bastnaesite is described. The flotation performance of H2O5 with respect to bastnaesite was investigated by flotation tests, measurements of the surface adsorption capacity, and polarizing microscopy. The results indicated that H2O5 exhibits superior collecting performance compared with direct flotation recovery of bastnaesite, with recoveries above 90% at pH 8–9. The adsorption mechanism of H2O5 on bastnaesite was studied by solution chemistry analysis and ?-potential tests. The results showed that the adsorption of H2O5 is associated with the types and concentrations of hydrolysis products of rare earth cations on the bastnaesite surface. At pH 8–9, the hydrolysis-dominant components RE(OH)2+ and RE(OH)2 + of the rare earth cations were adsorbed onto the bastnaesite surfaces, which were considered to be the major positive active points and were beneficial to chemical adsorption of the reagents. The pH range of 7–10 was optimum for bastnaesite floatability, and the adsorption of H205 on bastnaesite was by means of chemical adsorption through formation of a five-membered ring chelate. IntroductionThe rare earth elements (REEs) have long been known as ‘industrial gold’ and are widely used in the petroleum, chemical, ceramic, metallurgical, and permanent magnet material industries because of their special physicochemical properties (Chen, 2011; Krishnamurthy and Guptac, 2002; Esquivel, Bohe, and Pasquevich, 2002; Abrahami, Xiao, and Yang, 2015). Demand for REEs is growing in response to continuous advances in science and technology. Currently, most REEs are extracted from bastnaesite ((Ce,La)CO3F) (Chelgani et al., 2015; Yosry, 1990), which is recovered from rare earth ores by flotation. Typically, hydroxamic acids are used as bastnaesite collectors (Xu, 2015; Ren, 1997; Mousumi and Venugopal, 2016). Research into the mechanism of interaction between hydroxamic acids and bastnaesite is important for improving the recoveries of bastnaesite. However, there have been few studies that examine this mechanism."

Jan 1, 2017

By T. S. Kagogo

The factors affecting haul truck tyre performance and the effectiveness of the management system at Rossing Uranium Mine were investigated with the aim of increasing tyre performance in terms of running hours until failure. The main objectives were to identify the types of tyre failure, their causes and cost implications, and evaluate the effectiveness of the management system. A site severity survey, weight study, and TKPH studies were conducted to determine the pit conditions, and an analysis of failed tyres carried out. The results showed that tyre performance at the mine has declined from 2009 to date, and the increase in lost value amounted to R5.7 million in 2012 alone. The main cause of tyre failure is loose rocks in the pit. The present management system in the load and haul department is not effective enough due to operational constraints it is facing.

Jan 1, 2014

By G. C. S. Burns

"A system has been developed to simulate the mining of the Mufulira orebody. The main output from the system is a picture-gram from which the start date and effective life of all proposed mining faces can be measured. This information provides schedule dates for the completion of major capital development projects and stope preparation activities.The orebody must be divided horizontally into main levels and vertically into blocks of equal interval. Ore reserves are defined by specifying the tonnage contained in each block and main level. Mining faces are defined by their co-ordinates, and may themselves be split into sublevels. The simulation maintains an echelon between contiguous sublevels. Other face parameters control the speed of mining, the availability and relative priority of the face.The required production from each crusher/shaft system is specified. The technique of simulating the mining process in order to achieve the required production is described."

Jan 1, 2014

By E. Rex, T. J. Goff, W. S. B. van Beek, P. E. Nel

"The PGM smelting industry is known for operating at significant high superheats. Both the slag and matte phases that separate via gravity need to be tapped out of the furnace in a safe and efficient manner, which can be achieved only through sound tap-hole management practices. This paper gives an overview of the tap-hole management approach followed in the operations at Lonmin’s Western Platinum Smelter.Five circular AC furnaces are operated, with each furnace comprising essentially a crucible and auxiliary equipment. Within the crucible, slag and matte tap-holes are situated at different elevations. The slag phase is tapped at a higher elevation than the matte phase. At the three older furnaces, tapping is done by the conventional method of lancing open the tap-hole and closing it with a dolly. However, at the two newer furnaces, tap-holes are opened and closed with the aid of a drill and mudgun.Tap-hole management can be divided in a number of aspects. In order to achieve a safe opening and closure every time, Lonmin distinguishes between three important aspects - the design of the tap-hole system, the operating practices followed, as well as care and maintenance. This paper briefly describes the tapping channel and cooling requirements for handling superheated matte, together with the difference in design of the matte and slag tap-holes. Mention is also made of the importance of auxiliary equipment, which includes the launders, fume extraction, granulation water, and mudgun and drill units. Improvements made to the matte tapblock design are discussed.The monitoring, alarm, and safety systems related to matte and slag tapping are described. Tap-hole life and different wear mechanisms are mentioned. Tapping can be done safely only if the tap-holes and equipment are maintained properly. Mention is made of the tap-hole refractory repair practices, and the improvements made in order to reduce repair time as well as to extend the time between deep repairs. The importance of the correct tap-hole clay selection in order to nurse the tapping channel after each tap is also discussed.The paper finally touches on some research and development work that was done in order to improve the monitoring of the condition of the tapblock by means of fibre optic temperature sensors"

Jan 1, 2014

Orange Free State Branch Minutes of a General Meeting held at the Welkom Mine Canteen on Wednesday, 21st November, 1973, at 4-.00 p.m. Present Mr E. T. Wilson (Chairman) in the Chair, 15 Fellows and Members, 12 Associates, Graduates, and Students, and 11 Visitors (Total 38). Book review Symposium on Environmental Engineering in Coal Mining The 20 papers presented at the symposium cover some 14 subjects related to environmental engineering in coal mines, such as dust, heat, gas emissions, ventilation, noise, and lighting. Competition for students Each year the Institute offers a prize (or prizes should the entries warrant it) of up to Rl00 for the best paper or dissertation on a topic appropriate to the interests of the Institute. A.8. & T.8. Trust The A.S. & T.S. Trust has been formed with the specific object of establishing a worthwhile source of funds which can be used to further the progress of science and technology in South Africa. Nim reports Report No. 1573 A pilot-plant investigation of a flotation model. Report No. 1597 The role of sulphur in the flotation of sulphide minerals. Report No. 1599 The analysis of silicate rocks and metallurgical slags by X -ray-fluorescence spectrometry. Report No. 1607 The analysis of ion-exchange resins by X-ray- fluorescence spectrometry. Report No. 1617 An electronic differential precision thermometer.

Jan 6, 1974

By M. David

This paper addresses the problem of technological transfer in geostatistics. Expert systems have become popular in certain fields because the needs of the industry are not provided for by traditional concepts and systems of education. The explicit knowledge approach in developing expert systems is presented through two experiments. The first describes a dedicated system written in GOLDEN LISP and the second a shell, using INSIGHT2, which has more general applications. Using expert systems forces on the user a new approach to parameter definition, and as an example an original method for the research of any anisotropy axes is developed which might well be useful in traditional programming. Two examples of application to a coal and a gold deposit are given.

Jan 1, 1987

By D. Harbottle, D. Rhodes, T. F. Jones, S. R. Biggs

Multiphase process design and optimization often depends upon the rheological properties of sludges and slurries. In practice the task of determining these important properties is usually carried out by removing small samples from a process line or process unit and performing tests with proprietary bench-top analysers near to site or at a remote laboratory. Sampling will remain an important procedure in industrial processes, but there is also a strong case for augmenting the routine information obtained using an in situ sensor, as suggested below. ?Sampling protocols can result in extensive turnaround periods from sample collection to the generation of results. The delay can have an effect on a variety of factors such as product quality, process operability and efficiency. ?Sampling of toxic or radioactive material can prove an even greater challenge to minimize human involvement. ?Removal often changes the rheology of a specimen, and might lead to incorrect conclusions being drawn about the condition of a sludge. ?Sludge and slurry in a containment suffers a gravitational gradient, so that a removed sample should be paired with its location, particularly in the vertical dimension.

Jan 1, 2007

The theory of geostatistics covers a branch of .applied statistics aimed at a mathematical description and analysis of geological observations. Geostatistics can be used in pure geology (for example, for the analysis of trace elements in a metamorphic rock), in mineral exploration (for example, for the analysis of geochemical exploration data), as well as in mine valuation. This book is intended to provide a practical introduction to the theory of geostatistical methods of mineral evaluation. Over the years, various mathematical models have been developed to represent the distribution of values in mineral deposits. The simpler models are based on the assumption that the values are randomly distributed. Classical statistical methods, based on this assumption of the random distribution of values, are used to analyse mineral deposits to which these models apply or are assumed to apply. In all mineral deposits, however, one recognizes the presence of areas where the values are higher or lower than elsewhere. Also, the values of two samples in a mineral deposit are more likely to be similar if the samples are taken close together than if they are taken far apart. This indicates that there exists a degree of correlation between sample values, and that this correlation is a function of the distance between the samples. Models have been developed which take this correlation into consideration, with the degree of correlation between sample values being usually measured by the semivariogram function. In these models the fact that two samples taken next to each other will most probably not have the same value, must also be considered; even for very short distances the correlations are usually not perfect and a purely random component is present in the value distribution. The mathematical models will therefore assume the presence of two sources of variability in the values: a correlated component and a random component. Finally, one must consider the particular and very common case of mineral deposits in which the values present a systematic variation in space. This variation is usually referred to as a drift, or a trend. For example, the grade of an ore body may increase with depth of the ore, or it may decrease when one moves away from a central volcanic pipe. The earlier models did not give a satisfactory representation of drifts, and more complex models have been developed, in which three sources of variation are represented. These models are made up of: a deterministic component, a correlated component, and a random component. The deterministic component is used as a model of the drift. The correlated component explains regular changes in values which are not represented by the drift. The random component represents variations which cannot be explained by any of the above factors. The simpler models, based on the assumption of a single random component, will be described first (Chapter 2). The models based on the hypothesis of the superimposition of a correlated component on a random component, will then be analysed in detail. These models are most commonly used in the analysis of mineral deposits (Chapters 3-1 I). Finally, how to deal with the presence of a drift will be briefly described (Chapter 12). This book has been written essentially for students in mining engineering and for mining engineers who are interested in the background to the theory of geostatistics as well as its practical applications. The assumption is made that the reader has an elementary knowledge of statistics. Some knowledge of linear algebra is useful in part of Chapter 9, and is necessary to read Chapter 12. A proof is given of all the equations related to geostatistics, and which are not usually found in elementary textbooks on statistics. Understanding of these proofs is not necessary for practical application of the theory, and the reader may wish to skip them on a first reading, concentrating attention on the numerous simple practical examples given. Although the theoretical geostatistician will not find much new material in this publication, it is anticipated that he will develop some interest in the practical approach chosen to prove the various geostatistical equations. Many people and institutions contributed to the preparation and completion of this work. I am much indebted to Dr D. G. Krige and the Anglo Transvaal Consolidated In- vestment Company Limited, who gave me the opportunity to spend a considerable amount of time working on geostatistical problems, both theoretical and practical, during the time that I was in their employment. Dr Krige contributed greatly in developing my interest in studying both the theory and practice of geostatistics, always insisting that a correct balance be kept between theory and practice. I am grateful to Professor H. M. Wells and the Mining Department of the University of the Witwatersrand for inviting me to give lectures in a post- graduate course on geostatistics. The notes 1 prepared for that course became the foundation of the present work. I am also indebted to the Centre de Geostatistique of the École des Mines de Paris, where I received my first formal education in geostatistics during a summer course given by Charles Huijbregts, after many years of my lonely plodding through the published literature. Many graphs in the present volume are reproduced with the permission of the Centre de Geostatistique. The Department of Metallurgical and Mineral Engineering of the University of Wisconsin-Madison has also contributed in making this work possible, by allowing me to spend a considerable amount of time and resources in the writing, typing, and correcting of successive drafts. I am thankful to Lynn D. Kendall, who typed the entire manuscript under constant pressure.

Jan 1, 1978

Professor R.F. Sandenbergh matriculated at Germiston Afrikaans High School, and then he obtained the degrees of B.Eng. (Chem.) in 1972, M.Eng. (Chem.) in 1975, and D.Eng. (Met.) in 1983-all from the University of Pretoria. Mr J.A.D. Britz matriculated in 1958 at the Roodepoort Hoerskool, and joined the mining industry in 1959 as a student mine official at the Rand Mines Training Centre. Mr J.A. Cruise matriculated at Durban High School and joined Union Corporation as a Learner Official in 1962. He was awarded a scholarship to read for B.Sc. and M.Sc. degrees in mining engineering at the University of the Witwatersrand

Jan 1, 1990