Search Documents

By R. Sun, R. G. Gillies, R. S. Sanders, J. Schaan

The pipeline transport of paste, backfill or thickened tailings slurries is an important component of the tailings disposal process in many mines around the world. These mixtures contain significant fractions of fine (clay) and coarse (sand) particles, causing them to exhibit non-Newtonian flow behaviour and high frictional pressure gradients. There is often economic incentive to operate these pipelines in the laminar flow regime. Irrefutable evidence has shown that coarse particle settling can occur during the laminar flow of these slurries even when static settling tests show no indication of such behaviour. Previously, a ?rule-of-thumb? minimum frictional pressure gradient of ~1.5 to 2 kPa/m was suggested to ensure effective sand transport, based on the results of tests conducted using viscous Newtonian oils. In this paper, we examine the results of laminar flow tests of non-Newtonian, coarse-particle laden slurries conducted with pipeline loops of 50, 100, 150 and 250 mm in diameter. The results indicate that the ratio of the mean wall shear stress to the mean surficial particle stress is abetter indicator of laminar flow settling tendency than frictional pressure gradient.

Jan 1, 2007

Good blasting practices based on the careful application of explosive energy lead to significantly safer mining operations. Case studies demonstrate how the selection of the correct charge mass, explosive type and round design are vital for extracting the exact amount of rock required and preventing spurious damage to the remaining rock mass. In other cases the explosives are applied to precondition the rock mass, design rockburst resistant support, prevent damage to important excavations and reduce the exposure of miners to unsafe conditions. The Hybrid Stress Blasting Model is used to understand the explosive rock interaction and to illustrate and contrast good and bad blasting practices. In all cases, good supervision is required to continue safe mining according to the design. Keywords Blasting, safety, production, preconditioning, explosives, numerical modelling, drop raising, vibrations, and presplitting.

Jan 1, 2011

By K. H. Berget, V. Andersen, H. Tveit, R. Jensen

This paper presents some of the work carried out in order to improve the tapping process in silicon production, with special focus on the environmental and the safety standards. Some of the modelling work that has been done to improve the understanding of the process is described, together with the resulting implementation to improve tapping standards.

Jan 1, 2014

By M. Verri

The paper describes the design options available for the simultaneous production of liquid sulphur dioxide, via cryogenic condensation, and sulphuric acid in a sulphur-burning sulphuric acid plant. The impact of the operating conditions of the cryogenic condensation on plant capital and operating costs are discussed, with the main focus on the optimization of the most important economic and operative drivers of the non-ferrous mining industry, such as energy efficiency, reliability, and availability. A case history relevant to an integrated sulphurburning plant producing liquid sulphur dioxide and sulphuric acid, recently started up in the Democratic Republic of Congo, is described. Keywords sulphur burning, sulphuric acid, liquid SO2, cryogenic SO2 condensation, liquid sulphur dioxide, sulphur dioxide.

Aug 1, 2013

[The following notations will be used: O an ore body. W a sample of size w located in O; the size of this sample. W a block or a sample, of size W, located in O; the size of this block or sample. z' a point in W. W' a block or a sample, of size W' greater than or equal to W, located in O; the size of this block or sample. z" a point in W'. y (h) point semivariogram in O. s (h) point covariogram in O. In the previous chapter, we saw that the point semivariogram y (h) usually cannot be estimated directly from the data available. Rather, the values of samples of size w are available in O, and from these a regularized semivariogram y w (h) can be estimated. The following relationship has been proved (5.8): Y w(h) = Y (w;w+h) - Y (w;w). y (w;w+h) is the average value of y (z' z") when z' takes all possible positions in a sample w and z" all possible positions in a sample w+h of the same size, h being the distance between w and w+h. y (w;w) is the value of y (w;w+h) for h = 0, i.e. the average value of y (z' z") when z' and z" take all possible positions in w. We also saw that the regularized covariogram aw(h) is related to the point covariogram a (h) as follows (5.9): w(h) Gr In the following chapters, it will be shown that calculation of the average value y of y, or -a of a, in the general case relating to samples or blocks W and W' which need not be of the same size, plays a very important part in all the practical applications of geostatistical theory. The purpose of this chapter is to introduce various methods of calculating y and a-. Furthermore, mathematical equations and graphs are provided which can be used when the point semivariogram can be represented by one of the more common equations, and when the blocks W and W' are of simple shape. By definition (Fig. 6.1): [ ] Y (W;W) = WW f dz' f y (z' z") dz" (6.1) in W z" in W']

Jan 1, 1978

By M. Gericke

The microbial leaching of metal sulfides is now an established biotechnological technology. Over the past 25 years, refinements in the engineering design of bioleaching processes have paralleled advances in our understanding of the diversity and role of the micro-organisms driving the process and the mechanisms by which micro-organisms enhance metal sulfide oxidation. Commercial success started with the treatment of refractory gold concentrates using mesophilic micro-organisms, followed by the development of tank bioleaching processes for the treatment of base metal concentrates. This was, initially, a mesophilic process with limited potential for recovery of copper from chalcopyrite concentrates due to slow rates and low copper extractions. The exploitation of thermophiles represents a major breakthrough in the development of bioleaching technology for the treatment of chalcopyrite-containing ores and concentrates. This development also opened the route to heap bioleaching of chalcopyrite ores, which is now a major focus of research programmes and piloting campaigns. This paper reviews the historical development of minerals bioleaching processes and gives an update on the current status of commercial tank and heap bioleach operations around the world.

Jan 1, 2009

By S. M. Graham, G. Mellis

"The determination of the oxidation state(s) of an element present in a solid sample may be required for a number of reasons, including the assessment of the degree of reduction which has taken place during a multi-step production of a metal. In addition, possible flaws in a metal may result from small amounts of oxides included in the solid. The analyte of interest may be present as a major component of a sample of interest, or as a minor impurity - beneficial or requiring removal - in another matrix.Numerous methods have been established for the analysis of species in solution samples - for example, Fe2+ and Fe3+ by titration, Cr6+ by colourimetric means, etc. With appropriate knowledge of the solution being analysed, many of these methods are fairly straightforward, extremely reproducible and accurate. In those instances where the analyte in a solution is known to be in only one form, the analysis is even more straightforward, since it can be achieved instrumentally (by AAS, rCP-OES or rCP-MS, for example).However, the situation with respect to the analysis of solids is not as simple. Various techniques - such as X-ray diffraction - are routinely used by our mineralogists to obtain information about the various minerals present in solid samples, and thus about the oxidation states of the analyte elements present. However, these techniques usually give results that are qualitative, or semi-quantitative at best. They are also usually time-consuming and expensive. One of our colleagues at Mintek, Deshenthree Chetty, has done extensive work in developing a method for the quantification of manganese species in solids using mineralogical methods, and this is no small achievement."

Jan 1, 2014

By M. J. Kanda

"The design of rock slopes in the planning and operation of open pit mines and for road and rail cuttings requires the evaluation of their stability. This is commonly done using two well-established methods – limit equilibrium and numerical modelling. The research described in this paper compares the outputs of analyses using these two methods with regard to factor of safety and probability of failure, taking into account technical factors associated with each method and other factors such as material parameters and their variabilities. Results show that technical factors such as number of slices and slip circles, number and type of finite elements, and the stress reduction factor can all affect the results of stability analyses. Factors not taken into account in limit equilibrium analyses, such as dilation, in situ stress, and locked-in stresses can affect the outputs significantly. The research has focused particularly on the predicted location of the failure surface, the resulting volume of failure and, most significantly dependent on this volume, the predicted risk associated with slope failure. The results presented show that there is good agreement between the outputs of the two methods regarding factor of safety and probability of failure. However, the same is not the case with regard to volume of failure and hence risk, which can be very dependent on the methods of analysis and their input parameters. The research is not exhaustive, and there are many other factors that could affect the results of slope stability analyses that were not investigated. Some such factors are mentioned in the paper.IntroductionThe design of rock slopes in the planning and operation of open pit mines and for road and rail cuttings requires the evaluation of their stability. The traditional method of stability analysis used limit equilibrium (LE) methods, and this approach continues to be used. Numerical modelling (NM) packages are now also commonly used in slope stability analyses. The outputs from these analyses provide the factor of safety (FOS) and probability of failure (POF) of the slope. Since the capabilities of the two approaches are different, there can be some variations in FOS and POF outputs and in the predicted location of the failure surface. These variations can affect the predicted volume of failure, and this could in turn have a significant effect on the risk associated with the potential failure (Read and Stacey, 2009). Researchers such as Chiwaye and Stacey (2010) have found that LE analyses underestimate the predicted failure volume, unlike NM methods."

Jan 1, 2016

By J. L. Porter

This paper takes a personal look, through the author?s own experiences, at the implementation of innovations in the narrow-seam, hard-rock mining environment of South Africa?s mining industry over the past 40 years. A number of examples are used to illustrate some of the successes and failures over this period. More importantly, the examples are used to illustrate that the bulk of the investment for these projects should have been applied in a manner that could have achieved more sustainable results. The use of the word ?technology? in this context is also examined, as it is the author?s contention that only measuring the performance of machinery fails to encapsulate the potential of the entire system describing the scope of the innovation being implemented. These experiences and observations are projected onto the current backdrop of upheaval in the relationship between key industry stakeholders and the sustainability challenges facing areas of the mining sector in the South African economy. The paper concludes with some opinions on where attention can be focused in order to develop new thinking that can be applied for the safe and economic extraction of the nation?s mineral wealth.

Jan 1, 2014

By Z-P. Song, Z-B. Jiang, A-N. Jiang

"In deep excavations under conditions of high ground stress, even hard rock undergoes creep and other time-dependent effects that may result in instability. This study involves an analysis of the typical damage to the surrounding rock in the deep tunnels of the Jinchuan nickel mine in China. In order to understand the creep mechanism, triaxial creep laboratory tests were conducted under high stress. The creep characteristic curves and the Nishihara model parameters of typical rock were obtained from these tests. The numerical simulations, with which the creep model is developed, were performed using Flac3D. The creep model parameters were back-analysed on the basis of field monitoring data. Finally, appropriate support schemes for deep tunnels are suggested. The creep deformation of rock increases with time and axial load. At the same load level, the creep deformation of unloading is greater than that of loading. The simulated deformations of the surrounding rock are coincident with those from in situ monitoring, indicating the applicability of the presented rock creep model in situations of high ground stress. The results constitute a reference for support design in deep mining tunnels.IntroductionThe excavation of underground caverns can alter the natural stress field of the surrounding rock. In deep engineering projects under conditions of high ground stress, even hard rock shows obvious creep properties and time effects (Yao, 2005; Liu, 1994). The time effect of excavation not only causes decompression of the floor and increases sidewall stress, but also leads to stress concentration at the corners of the excavation, which influences the stability of the surrounding rock in a very complex manner.Many researchers have studied the creep properties of rock. ‘Griggs (1939) showed that creep occurs in sandstone, shale, and silty sandstone rock under the load of 12.5%–80% of the failure load.’ Subsequently, much research has been published on rock creep tests, including the uniaxial compressive creep test, the triaxial compressive creep test, the discontinuity rock shearing test, and field tests (Wawersik, 1974; Amadei et al., 1980; Okubo et al., 1991; Jin et al.,1995). A rheological model is used to describe the complex rheological deformation of material. This model is based on various fundamental parameters including rock elasticity, plasticity, and viscous properties, which can be combined into a constitutive model to reflect the rheological properties of various types of rocks. Wang et al. (2014) studied the creep– damage–rupture characteristics of rock salt by applying a creep-damage model. Munson (1997) developed a multi-mechanism constitutive model of creep of polycrystalline rock salt based on steady-state creep and modified it to incorporate transient creep through work hardening and recovery. An elasto-viscoplastic approach was developed to simulate the timedependent nature of the fracture zone (Malan, 1999). Rheological models were proposed to understand the collapse of iron ore mines in Lorraine by Dragan et al. (2003)."

Jan 1, 2016

By J. R. de Wet

The Skorpion Zinc Project demonstrated the viability of the production of zinc from non-sulphide (commonly called ?zinc-oxide?) ores, using the leach/solvent extraction/electrowinning process route. The success of Skorpion raised the profile of a number of other zinc oxide deposits, considered to be similar. However, a study of three such deposits demonstrates that the application of whole-ore-leaching may be limited and that the concentration of zinc, or at least the separation from acid-consuming gangue minerals, prior to leaching is essential to the successful recovery of zinc from these deposits. The recovery of zinc is dependent on an understanding of the geology of the deposit, the mineralogical association of zinc minerals with gangue minerals (Boni, 2005) and limitations of mining practice. Flowsheet development and the associated metallurgical testwork to derive an economically viable treatment route from two of these deposits are discussed. Key parameters affecting zinc recovery by gravity separation and flotation are presented, demonstrating that even minor differences in mineralogy have a profound impact on zinc recovery and reagent consumption. The study also deals with the determination of appropriate design parameters through metallurgical mapping of the dense media separation process.

Jan 1, 2008

By T. Orupold

This paper covers the commercial development of gravity separation of fine particles using a Lamella High Shear Rate REFLUX? Classifier (REFLUX? Classifier), focusing primarily on coal applications. The REFLUX? Classifier is a fluidized bed device that incorporates a system of closely spaced parallel inclined channels located above the fluidized bed. These channels make it possible to achieve a significant suppression of the effects of particle size, resulting in a highly effective separation on the basis of density. The improved gravity separation performance is shown to be remarkably high, with a significant reduction in the variation of separation density with particle size, and a significant reduction in the change in Ecart probable error (Ep) with size. The first full commercial-sized units of the REFLUX? Classifier were field-tested in late 2009 in coal applications. More recently, the technology has been applied in fine particle separation in minerals applications and there are a number of full-sized units operating in chrome applications in South Africa. Initially, pilot-scaled units (typically the RC?300) were trialled in iron ore, mineral sands, and manganese plants amongst other minerals, typically after other technologies failed to achieve the desired results. Currently a number of laboratories globally are carrying out more testing in minerals applications. More than 50 RC? units are now operating in coal and minerals applications. This paper introduces the REFLUX? Classifier technology, identifies commercial applications, and gives some commercial results.

Jan 1, 2014

By H. A. D. Kirsten

The assessment of backfill as a support medium in gold mines has been based generally on its ability to reduce the amount of potential energy available to cause damaging rockbursts. While this regional support effect is very important, the methods of analysis commonly used to quantify it predict that, owing to the very compressible nature of the fill, the support provided in areas of small convergence will be negligible. This prediction, however, is contrary to the conditions observed in practice, namely that weak fill does provide a discernible beneficial effect. In addition to the regional support effect, several mechanisms of secondary support, particularly those applicable to areas of low convergence, are identified and discussed. These include the mechanism of deformation of the detached hangingwall beam, rotational and shear effects on bedding, joints and fractures, joint infill, the continuous support effect of the backfill under static and dynamic conditions, the preservation of the integrity of the hangingwall beam, and the potential modification of local stress fields. Confidence in the validity of some of these mechanisms is afforded by the results of theoretical investigations into the behaviour of backfill- supported stopes. The results dealt with are: ? the behaviour of the hangingwall as a beam on an elastic foundation; ? the behaviour of stope/backfill, modelling the surrounding rock as a jointed rock mass.

Jan 1, 1988

The successful commissioning of the Cerro Verde project in Peru at the end of 2006 marked the culmination of the efforts of many in the industry over many years to have high pressure grinding roll technology recognized and accepted as a legitimate alternative to the conventional approach to large-scale hard rock comminution circuit design. Other hard-rock projects that have recently adopted HPGR include Freeport (Indonesia), Boddington (Australia) and Amplats Potgietersrus (South Africa). With the increasing acceptance of this technology, the need for technical presentations and articles exhorting the industry to adapt has diminished somewhat, and it is considered appropriate now to shift the emphasis to the provision of information for those considering embracing change. This article endeavours to answer frequently asked questions on the subject of high pressure grinding roll technology, with particular reference to its application in the hard rock minerals processing sector.

Jan 1, 2010

By M. Bailey-McEwan

The application of refrigeration on deep South African gold mines is mostly achieved through the medium of chilled water which is produced in central installations either on surface or underground. The amount of refrigeration has been increasing rapidly over recent years, and the total installed capacity on all gold mines now exceeds 1000 MW(R), with annual running costs exceeding R100 million for electricity alone. Hence, there are strong incentives to design such installations for optimal cost-effectiveness. The duty required of these installations varies considerably, owing to daily and seasonal climatic changes and the varying nature of the water demand. It has long been recognized that a computer program to predict installation performance as a whole under any given conditions would be highly desirable. An interactive computer program, CHILLER, is described which enables the user to build an installation, consisting of any configuration of cooling towers, water storage reservoirs and packaged water chilling machines. Externally-dictated conditions such as the weather and chilled water demand must be specified, and all quantities under control of the installation, including internal water flow rates and compressor capacity regulators, must be set. The time-varying performance of the installation can then be simulated. Examples are given of the application of the program in the design of a new installation, and in evaluating the effect of any modifications to existing installations.

Jan 1, 1987

By J. O. Marsden

The copper industry is cyclical with the price determined by market forces (i.e. the balance between supply and demand for the metal.) The period of each cycle typically varies between 6 to 9 years. These cycles drive decision making for exploration, new project and mine development, mine closures, process improvement, research and development, and technological innovation. The gold industry is fundamentally different from copper in that supply and demand do not, alone, determine metal price and many other factors play a key role, including global and regional financial conditions, political stability, and the global future economic outlook. However, the gold price experiences trends and longer term cycles that similarly drive business decision making. This paper examines developments in the copper industry that have occurred over the past few price cycles (approximately 20 years) and explores how the lessons learned can be applied to gold extraction. A number of specific examples and opportunities are discussed in detail, including; technology development and innovation, cost control and business improvement, material characterization, comminution equipment, biological heap leaching, and concentrate pressure leaching.

Jan 1, 2009

By E. L. Thyse, P. Bepswa

The Western Cape Mineral Processing Facility (WCMPF) is a collaborative educational initiative between the Department of Process Engineering at the University of Stellenbosch and the Chemical Engineering Departments at the University of Cape Town and the Cape Peninsula University of Technology. The primary objective of the WCMPF is to facilitate the education, development and graduation of suitably qualified engineering graduates and diplomats for the mining and mineral processing industries. In recent years the WCMPF has embarked on a number of initiatives aimed at improving the quality of mineral processing education which include, more importantly the introduction of novel ?end to end? or ?integrated? practicals. The practicals are developed around a specific mineral theme or route, which more than often includes the development of several new pilot plant practicals demonstrating key unit operations or processes in the selected mineral theme or route. The most recent mineral processing pilot plant practical programme is developed around the beneficiation of heavy mineral sands. The programme consist of the following developed pilotplant practicals: the online electrostatic separation of free flowing mineral sands using a Roche Mining Carrara high tension roll separator incorporating a Blue Cube Systems MQi in-line mineral quantifier; the production of titania-rich slag and pig iron from the smelting of ilmenite and anthracite in a 60 kW direct current (DC) plasma arc furnace as well as the mineralogical characterization of the feed and products; the continuous closed-circuit ball milling and screen classification of titania-rich slag. This paper illustrates the important role of a well-designed heavy mineral sands beneficiation pilot plant practical programme in facilitating the education and development of suitably qualified personnel for the mining and mineral process industries.

Jan 1, 2007

By Keith Rowley, Neal Harries, David Noon

The management of risks associated with slope instability is an essential process in the safe and economic operation of open cut mines. The ?slope stability radar? (SSR) has been developed to better manage those risks. The SSR remotely scans rock slopes to continuously measure any surface movement and can be used to detect and alert users of wall movements with sub-millimetre precision. The high level of movement precision and broad area coverage of the rock face can allow for a better understanding of the geomechanics of slope deformation, including magnitude of potential failures and additional warning time of impending instability. Additionally, radar waves adequately penetrate through rain, dust and smoke to give reliable measurements, 24 hours a day. SSR systems have been deployed in many mines in Australia, Indonesia, South Africa, Zambia, Chile and the United States. Greater than 70 rock falls and waste dump failures (from several to millions of tonnes) have been monitored, and on every occasion precursor ?warning? movements were recorded by the SSR. This technology enables a radical change in the management of risks in open cut mining. This paper will present recent case studies of the SSR operation in coal and metalliferous mines.

Jan 1, 2006

By B. Gamble

Solid Energy's project evaluation process utilises a standard staged gate approach, with five levels leading ultimately to project execution (level 6). For a project to progress through each level, specific financial criteria need to be satisfied. Solid Energy primarily use value based (discounted cash flow) metrics to support investment decisions. Within this approach Solid Energy utilise a number risk-based techniques to quantify project uncertainty. This includes the use of risk-based probabilistic analysis using Monte Carlo simulation and probability-based scenario analysis. Figure 1 shows the net present value (NPV) cumulative probability. Project risks are identified through comprehensive risk workshops. The output of these workshops (list of key risks, ranges of uncertainty and probability of occurrence), is quantified within the financial models. Incorporating this risk information on a probabilistic basis allows an investment decision to be made with a robust knowledge of the range of possible project outcomes along with the likelihood (probability) of those outcomes occurring, for example 80 per cent confidence that the project value will be greater than NZ$10 M, 50 per cent confident it will be greater than NZ$50 M and ten per cent confident it will be greater than NZ$70 M. Incorporating risk implicitly into the financial analysis allows the business to easily evaluate/measure projects with different risk profiles. The detailed risk analysis does away with the complexity of arbitrary risk-based hurdle (discount) rates, as projects can be evaluated based on their specific risk characteristics. Whether a project will progress to the next level of investment and ultimately into implementation depends firstly on the value assessment in comparison with the associated risk/uncertainty, and secondly matching this with the businesses appetite for risk. At Solid Energy standard criteria are used for specific project evaluations, but in addition projects are considered on a portfolio or strategic basis also. Solid Energy attempts to maintain a mix of low risk (low value) projects with some high risk (high value) opportunities. The mix of projects must remain in line with the shareholders' risk appetite. One of the keys to a successful investment decision is ensuring that the appropriate information is available to decision-makers so that they may proceed with a comprehensive understanding of the risks and impacts associated with the decision.

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