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By P. Gauvillé

The key driver for choosing a circulating fluidized bed (CFB) boiler is the ability to burn a wide range of fuels with highly efficient combustion while meeting low emission requirements. Reduced sorbent and water consumption compared with pulverized coal (PC) plants and the ability to deal with corrosive fuel constituents while still meeting local regulations make a CFB boiler-based power plant particularly attractive. CFB-based power generation is particularly relevant in coal mining areas. During mining, a portion of the roof and floor material may be extracted along with the coal seam in order to create adequate working height for the equipment and miners. Therefore, run-of-mine coal which comes directly from the mine contains impurities. The raw coal can undergo a washing or screening process in order to improve its quality by reducing the ash content in the fuel in order to sell it on the export market or to comply with domestic customer requirements. The coal discarded from the washing process usually has a high ash content and difficult mechanical properties resulting from the initial properties, the mining, and the coal cleaning processes. It can contain a large volume of stones or can be very fine as a result of the washing treatment. To create value from burning this discarded coal, the boiler supplier has to meet several technical challenges in developing the appropriate design. This paper highlights the performance achieved with Alstom's CFB boilers developed for power generation over the last two decades to extract value from discarded coals. Emile Huchet Power Plant is the first 125 MWe CFB unit developed for burning coal slurries (called schlamms in France). The Emile Huchet discarded coal fuel is a very fine high-ash residue with an average particle diameter by mass (d50) of around 80 µm as received from the washing plant. The paper describes the performances and the basic design of the CFB boiler to manage such a fine fuel. The performances of two other CFBs of approximately 300 MWe output in operation in the USA and PR China are also mentioned to demonstrate the capability of this technology to burn either low-volatile bituminous or anthracite waste coals. The significant issues experienced during commissioning are reported, and the conceptual choices for burning such fuels are mentioned.

Jan 1, 2012

By M. R. MacIver, M. Pawlik, H. Alizadeh

Treatment of tailings is an integral component of mineral processing. Tailings with a high clay content are a challenge for the mineral processing industry; e.g. the clay slimes produced from production of phosphate in the mid-western United States and the fine tailings produced from oil sands production in western Canada. Dispersants are a commonly used process aid, but their impact on the structure and settling properties of high clay content tailings is not well understood. The paper describes the changes in composition and bitumen release that occur when mature fine tailings (MFT) are treated with a polymeric dispersant (lignosulfonate.) Synchrotron-source computed tomography (CT) was used to obtain high-resolution 3D images of samples prepared at varying dosages of dispersant. The CT method permits a powerful visualisation of the internal structure and composition of the samples. These insights will enhance our understanding of the structural characteristics of mineral tailings relevant to solid-liquid separation by flocculation and thickening. Keywords: Dewatering, dispersion, dispersants, chemical treatment, computed tomography

Jan 1, 2020

By P. Pięta, A. Surowiak, T. Niedoba

In a flotation process of copper ore three factors were measured: copper grade in concentrate (β), copper grade in tailings (J) and recovery of copper in concentrate (e). The experiment was conducted by means of the Jameson cell. The factors influencing the quality of the process were particle size (d), time of separation (t), type of the collector (k) and dosage of the collector (s). The considered vector function was then [β(d, t, k, s), J(d, t, k, s), e(d, t, k, s)]. Optimisation achieved was based on determining the values of the adjustable factors (d, t, k, s). The goal was to obtain the highest possible values of functions β and e (maximum) and the lowest possible values of function J (minimum). To this purpose the taxonomic methods were applied. Keywords: Flotation, copper ore, lithology, flotation agents, particle size distribution, taxonomic methods

Jan 1, 2020

By W. J. Whiten

"Models of cone crushers and vibrating screens are described. These models have been constructed from actual plant data using a systematic model-building technique. Included in this technique is regression analysis using multi-dimensional spline functions.The models have been used to simulate complete crushing plants and hence to design and evaluate a control system for crushing plants."

Jan 1, 2014

By L. M. Tavares, T. M. Campos, P. M. Faria, R. M. Carvalho

Recent advances in mechanistic ball mill modelling are expected to allow for more accurate predictions that overcome the well-known limitations of phenomenological approaches, including their inability to predict the effect of liner geometry and their wear, in grinding. Although able to describe continuous operation, the UFRJ model of ball mills has only been validated in the past in mills that operate in batch mode, either in laboratory or pilot scale. The present work demonstrates the application of the mechanistic UFRJ ball mill model to predict fine grinding of iron ore in a pelletising plant in Brazil, the first application of such a mechanistic modelling in continuous mode. It shows that the model provides accurate descriptions of flows after iterations in a closed-circuit with hydrocyclones. Keywords: Ball mill, iron ore, grinding, modelling, particle breakage

Jan 1, 2020

By T. G. Pedro, Q. Gonzalo, R. E. Roberto

Anionic surfactants, such as oleic acid and sodium oleate (OLE), and cationic surfactants, such as dodecyl amine-hydrochloride (DDA), are used as flotation collectors of spodumene. However, recovery is poor even if activating ions are used because gangue minerals are also activated. Some reports demonstrate that OLE/DDA mixtures act as a selective collector to enhance spodumene recovery. From mostly macroscopic tests, it has been argued that anionic OLE and cationic DDA form electroneutral complexes, which are adsorbed on the mineral surface through chemisorption; however, no molecular evidence has been presented. Therefore, the adsorption mechanisms of the collector mixtures remain mostly unknown. It seems necessary to first study the behaviour of these surfactants confined in salt solutions, to determine auto-associations when they are pure and cross-associations when mixed, the transition from monomers to micelles, and the structure of the micelles and the effects of pH, temperature and concentration. In this paper, ab initio calculations are first used to determine the surface charge distribution of OLE and DDA separately. Then molecular dynamics simulations are used to determine the conformation, aggregation, and diffusion of these pure and mixed surfactants in freshwater and saltwater with the ionic strength of seawater in a pH range. In particular, it is interesting to determine the various associations that result in mixtures of both surfactants and the impact of alkali and alkaline earth metal cations on the structures formedp. Cations influence the order of the structure and orientation of water dipoles, reducing the mobility of water, ions, and surfactant molecules. This knowledge is critical to improve the molecular understanding of the adsorption mechanisms on minerals. Keywords: Computer simulation, molecular dynamics, flotation, collectors, saltwater

Jan 1, 2020

By J. Petersen, K. Shaik

The traditional route utilised for the recovery of Platinum Group Minerals (PGMs) in South Africa involves the concentration of ore by flotation, followed by smelting to produce a PGM rich matte, which is then digested hydrometallurgically to solubilise base metals and Platinum Group Elements (PGE). HCl/Cl2 is conventionally used for PGE dissolution from concentrated material, which is an aggressive reagent associated with high maintenance costs as well as environmental concerns. Of interest is the leaching of PGMs directly from mineral ores, low grade concentrates and tailings in a cyanide-ferricyanide system as a more sustainable alternative. This system has shown favourable Pt recoveries among the Pt minerals, but with sperrylite (PtAs2) displaying highly refractory behaviour. The dissolution of sperrylite is investigated in a cyanide-ferricyanide system in the context of an electrochemical study involving OCP, chrono-amperometry and cyclic voltammetry. This approach allows the assessment of leaching characteristics as well as the investigation of surface passivation, which is believed to contribute to poor Pt recoveries. Some evidence is presented that such passivation is related to the formation of arsenide species on the mineral surface. Keywords: Sperrylite, PGMs, cyanide, ferricyanide, electrochemistry

Jan 1, 2020

Dr Hanekom I would like to thank Dr Hanekom for a very valuable -contribution; it summarizes careful statistical test work which was an omission from the original paper. Minor manipulative details, the application of a finer measuring graticule and the reproducibility of the procedure are described and the author accepts with gratitude this ready-made answer to the first and second points of Dr Finkelstein's comments. Mr Williamson The author is indebted to Mr Williamson for historical background to early experiments with captive bubbles and it is conceded that the original paper described another avenue to an old flotation goal. The field of application is restricted and we would not work with all the material illustrated by Klassen and Mokrousov.1 The particles in Figure 74(a) are far too large and irregular; material in Figures (b) through (d) could be employed for semi-quantitative work in spite of obvious disadvantages; Figure 74(f) represents ideal conditions. We have studied this mineral-collector association in the size range given and results were satisfactory. For larger pick-up values than that illustrated an induction period of more than 0.01 second was required (actually 90 seconds for a maximum bubble load at pH 7.2 with 25 g/t oleic acid!). REFERENCE 1. KLASSEN,V. I. ANDMOKROUSOV,V. A., 'An Introduction to the Theory of Flotation,' Butterworths, London, 128, Figure 74, (1963). Mr Slander and Mr Kooij Researchworkers in a practical mineral development laboratory may well ask what contribution yet another micro-technique can make to the problems of large scale ore flotation. The obvious answer is that there can be little or no direct translation of results obtained with a static system comprising air bubble and pure mineral concentrate to a dynamic multicomponent operation. However, for the rapid study of one mineral or variable some small scale apparatus must be employed. In the Anglo American Research Laboratory, the described semi-quantitative pick-up technique has filled a niche and requirement. For ourselves, the system represents a definite advance over the contact angle approach in that samples of minerals conventionally prepared in laboratory or plant may be used, which is a desired step nearer flotation. We bracket this pick-up apparatus with the Hallimond tube1,2 and evidence is now to hand that pick-up curves may often be related to flotation recoveries obtained from this tube with pure mineral samples and from the Fuerstenau glass cell3 with the total ore at low pulp densities. We would not wish to push the comparison of pick-up results past this stage for there is much published development work relating the Hallimond and Fuerstenau equipment to flotation on a larger scale. At the Anglo American Research Laboratory for example, a test in a Fuerstenau cell is often the final integrated step in laboratory investigations into the TORCO process. As these contributors rightly point out, pick-up tests can offer nothing towards the determination of optimum reagent dosage for large scale flotation. This is the function of bench-type flotation cells with up to 2 kg batches of ore. At this level, collector performance and consumption and physical properties of the froth are more comparable with large scale operation. The pick-up cell, Hallimond tube and Fuerstenau cell are essentially equipment from which the effect of control parameters can be derived. For the assessment of collector behaviour, the pick-up approach is rapid and economic and we offer here a pick-up contribution towards the flotation of Witwatersrand uraninite with anionic sulphonate collectors (Fig. 1). A large crystal of the local mineral was not available for contact angle measurements and owing to its low concentration in Witwatersrand gold-bearing reef, a small sample was laboriously obtained by tabling, superpanning and heavy medium separation. The pick-up values of a sized portion were determined by the described procedure in the presence of nine petroleum sulphonate collectors. Responses fell into two categories; "normal" curves (dotted lines) and a group of collectors which were very active at low concentration but showed depression of pick-up at the higher levels. The similarity in form of curves for collectors 1 and 3 was apparent and reagent 3 was found to be simply a diluent of 1. Otherwise, collector performance could not be related to either sulphonate content or molecular weight of the active constituent. However, reagent 2 gave the best performance in a bench flotation cell with regard to overall uranium distribution but, as predicted, reagent 1 yielded the best grades. Later pick-up tests with a similar class of collector have suggested that depression at high reagent concentrations may be due to "bubble armouring" whereby an hydrophilic air-liquid interface is produced which rejects mineral attachment. Prolonged investigations with the same mineral sample have not been required except for the study of uraninite which lasted for more than a year. Variation in pick-up response could not be detected over this period during which the mineral was stored under distilled water. The only visible change was a small efflorescence of lead sulphate, readily removed and decanted off after stirring with a glass rod in the sample tube. In its main role as a "trouble-shooter", the pick-up method is applied to fresh mineral samples taken from a flotation plant and soon discarded or from a current run in the ore-dressing laboratory. After production of a concentrate, a solvent cycle, as described in the original paper, may be required to remove heavy organics or other contaminants as when a sample has been taken from circuits containing detergents, grease or return flotation solutions. This solvent cycle will not remove chemisorbed collectors and we prefer not to work with minerals from flotation tailings or with pick-up cell discards. It has been thought preferable to answer the contributors' other apprehensions with regard to sample preparation, activators, depressants and the physical properties of reagents in another paper on offer to this Journal. It covers pick-up investigations relating to the flotation of nickeliferous pyrrhotite at two Anglo American Corporation Mines, which approach seemed the correct orientation for a reply to another contributor to this paper.

By D. Govender

Theoretical flotation models suggest that there is a positive relationship between bubble-particle collision rates and turbulent kinetic energy dissipation. Fine particle flotation performance is generally enhanced by increased collision frequency and hence higher energy dissipation. Contrarily, increased turbulence in the rotor-stator region is related to higher detachment frequency of the coarser size range. Therefore, the optimal modes of recovery for the ?fine? and ?coarse? size classes appear to be diametrically opposed. Industrial applications have previously confirmed that applying greater power to flotation slurries yields significant improvements in fine particle recovery. However, recovery of the coarser size class favours a different flotation environment. An improvement in the flotation kinetics of the fine and coarse size classes, provided there is no adverse metallurgical influence on the intermediate size ranges, is obviously beneficial to the overall recovery response. Managing the local turbulent kinetic energy dissipation, and hence the power imparted to the slurry, offers the benefit of targeting the particle size ranges exhibiting slower kinetics. FLSmidth recently introduced the practical implementation of this concept. In principle, it decouples flotation regimes where fine and coarse particles exhibit preferentially recovery. In the case of naturally aspirated machines (Wemco®), it is referred to as Hybrid Energy Flotation? and incorporates at least three phases: ? Standard flotation machines (standard energy input, rotor speed (r/min), rotor size/type) at the beginning of the row, where flotation is typically froth-phase limited and operational and set-up parameters have a limited influence on the recovery ? Higher-powered flotation machines (high rotor speed, high-power rotor size/type) at the end of the row to improve fine particle recovery ? Lower-powered flotation machines (low rotor speed, low power rotor size/type) to enhance coarse particle recovery. A CFD-based flotation model is used to highlight the effect of turbulent dissipation energy on attachment and detachment rates. Preferential collection zones for ?fine? and ?coarse? particles are predicted for both forced-air and naturally aspirated machines. The greater predominance of UG2 ore types, coupled with the skewed feed distribution of platinum group metals (PGMs) to the finer size fractions, suggests that PGM flotation circuits are not designed for optimal recovery across the size distribution. The application of the hybrid energy concept to PGM flotation offers a possible shift towards a more efficient flotation circuit solution through a managed distribution of energy.

Jan 1, 2012

By M. Steyn

Until 2000, pricing involved profit taking and cost covering combined with limited global published information. Market information was mostly subjective in nature and large resource companies benefited exclusively from aggregated company information and data. The market constituted producers and consumers only. The nineties saw massive cost cutting drives and re-engineering initiatives and this underwrote the structural changes evident in the late 1990s? economy. This supported South African producers on the international FOB cash cost curve, and ensured profitable coal mining operations. As the debate on relative scarcity of fossil fuels ensued, producers and buyers looked to the developing terminal coal market for direction. The much needed direction was provided when the API coal indices were developed and derivative trading was embraced to include financial institutions, trading houses and hedge funds. The liquidity of coal trades increased rapidly and vastly and market information became obtainable on a daily basis. Interpretation of current and future coal prices became available to all market players and contributed to the sophistication of the market that is now able to participate in risk management of prices, capital expenditure and brownfield developments.

Jan 1, 2010

By L. C. Woollacott, A. K. Tripathy, J. H. Potgieter

This paper presents the second of a two part study on density stratification in a laboratory batch jig. The first study, reported elsewhere, investigated the effect of operating conditions on the stratification profile achieved at equilibrium. The study reported in this paper investigated their effect on the kinetics of density stratification under 18 different sets of operating conditions - five for binary systems with different particle densities; and 13 for a ternary system with different configurations of the jig cycle. Although the study was academic in nature (it involved simple systems of 8 mm glass beads) it has a number of practical implications. It found that the most influential operating variables affecting stratification kinetics are the amplitude of water displacement, the pulsion hold time, and, to a lesser extent, the pulsion time. It also found that reasonable rates of stratification can be expected in systems with density differences as low as 237 kg/m3 but below this the kinetics deteriorate significantly. The two studies together suggest that, within a broad range of operating conditions, the equilibrium stratification profile is independent of control variables and it is the control of stratification kinetics that has the primary influence on the quality of density separations in a jig. Keywords: Jigging kinetics, stratification, equilibrium stratification, batch jig

Jan 1, 2020

By G. M. Dunn

Pressure leach autoclave circuits are employed in the leaching of ores, concentrates, mattes, alloys and intermediates for the recovery of metals into solution. Once the metals are extracted into solution, the value metals can be recovered by hydrometallurgical means such as by purification followed by electrowinning, hydrogen reduction, pyrohydrolysis, crystallization and other unit operations. In many of these integrated flowsheets the pressure leach step is pivotal to the recovery of the value metals from the host material. Furthermore, the autoclave circuit invariably is a high capital cost component of the plant and an area that is carefully scrutinized when debottlenecking and capacity increases are being considered. This paper identifies a unique proven way of increasing the capacity of existing or new exothermic pressure oxidative leach autoclave circuits by as much as two or three times.

Jan 1, 2009

By F. Smit

Mining and plant practices can benefit from preconcentration of UG2 platinum ores in a dense medium separation (DMS) plant. The main benefit of DMS is the selective discard of waste from the feed stream, thus improving project economics while reducing the power and water requirements for processing. On the mining side, with the correct combination of variables, implementing DMS as a pre-concentration step can reduce cut-off grades and, as a result, increase ore reserves. In addition, the use of DMS technology gives the mine team the opportunity to consider alternative mining methods for an operation. The challenges of balancing the mining tonnage, methods, and resulting grades with the process plant size, recoveries, and DMS discard volumes have resulted in the development of a model to optimize value on new projects or existing operations. This paper will focus on these benefits from a mining and plant perspective, quantifying the water and power savings, and look at the economic benefits of preconcentration and the effect of metal prices, plant feed grade, and DMS efficiency in different mining environments.

Jan 1, 2013

By Paul Fouchè

Contents ?Purpose ?Introduction ?Why manage a mineral resource ?Sampling or tracking from source ?How can RF tracking help ?Project Goals ?RF System ?Tracer Layout ?Antenna

Jan 1, 2005

By T. Tamenti

The purpose of the project was to conduct a feasibility study to determine if the current operating bottom dump trucks that haul coal should be replaced with rear dump trucks. The engineering availability of the bottom dump trucks was lower than targeted while the operating costs are the highest of all coal haulers within the Anglo Coal opencasts. To determine if it is feasible to replace the current bottom dump trucks with rear dump trucks, a production profile simulation was conducted to determine the number of rear dump trucks that would be required to meet the annual coal hauling requirements. The total capital and operating costs of these rear dump trucks were also determined using a total cost of ownership model. These were compared with the current costs of the bottom dump trucks to determine if it is financially feasible to replace the bottom dump trucks with rear dump trucks. Having considered the capital outlay that would be required to replace the bottom dump trucks with rear dump trucks as well as their respective operating costs, it was concluded that the current bottom dump trucks should be replaced with rear dump trucks. Other factors were considered in coming to this conclusion.

Jan 1, 2007

By T. Tamenti

The purpose of the project was to conduct a feasibility study to determine if the current operating bottom dump trucks that haul coal should be replaced with rear dump trucks. The engineering availability of the bottom dump trucks was lower than targeted while the operating costs are the highest of all coal haulers within the Anglo Coal opencasts. To determine if it is feasible to replace the current bottom dump trucks with rear dump trucks, a production profile simulation was conducted to determine the number of rear dump trucks that would be required to meet the annual coal hauling requirements. The total capital and operating costs of these rear dump trucks were also determined using a total cost of ownership model. These were compared with the current costs of the bottom dump trucks to determine if it is financially feasible to replace the bottom dump trucks with rear dump trucks. Having considered the capital outlay that would be required to replace the bottom dump trucks with rear dump trucks as well as their respective operating costs, it was concluded that the current bottom dump trucks should be replaced with rear dump trucks. Other factors were considered in coming to this conclusion.

Jan 1, 2007

By A. Chiappone, M. Barla, U. Rabagliati, L. Repetto

ABSTRACT: The paper describes the application of both continuum and discontinuum modelling to the study of the stability conditions of the Alpetto mine, a high rock cut located in Cesana Brianza (Italy). The main objective is to provide valuable information for future mine planning and for the assessment of the stability conditions of the abandoned high rock cuts. Continuum modelling involved the applications of the finite difference method, whereas the distinct element method was used for discontinuum modelling. RESUME: L'article décrit l'application des méthodes de modélisation continue/discontinue pour l'étude des conditions de stabilité de la mine d' Alpetto, située à Cesana Brianza (Italie). Le but du travail consiste à determiner des paramètres utiles vis-à-vis du projet et d'estimer les conditions de stabilité de l'amas rocheux au terme de l'exploitation de la mine susmentionnée. La modélisation continue a impliqué l'application de la méthode des différences finis alors que la méthode des éléments distincts a été utilisée pour la modélisation discontinue. KURZFASSUNG: Der Artikel beschreibt die Verwendung der Modellierung von continuum und discontinuum zur Untersuchung/Studium der Stabilitätsbedingungen von der Alpetto Mine, hohe Aushubsfront in einer Felsmasse die sich in der Gemeinde von Cesana Brianza befindet. Das Hauptziel dieser Arbeit ist die Festlegung von Parametern die zur Planung nützlich/behilflich sein können sowie auch die Möglichkeit zur Analyse/Auswertung der Stabilität der Felsmasse-Front nach Aufhebung der Nutzung derselben. Die continuum Methode/Analyse wurde verwendet durch Unterstützung der Finite Differenz Methode da man für die Analyse der discontinuum (Fels als discontinuum beschrieben) der Distinct Elemente Methode verwendet hat.

Jan 1, 2003

By S. C. Everett, I. S. Schache, P. A. Bannister, G. F. Balderson

The Pooncarie Project is located in the Murray Basin region of Australia. This region is a potential new mineral sands province. Due to the geological environment and the lithology of the deposit, these deposits have some technical challenges that need to be addressed prior to their successful development. Within the Pooncarie Project the Ginkgo deposit will be developed first. The technical challenges it presents include its geographical isolation, the requirement to remove substantial quantities of overburden as well as the processing issues raised by the highly weathered nature of the ilmenite family of minerals. Ore characterization bench scale testwork and bulk sample pilot plant programmes have been carried out and the data obtained used to design flowsheets for both the wet plant and mineral separation plant circuits with the flexibility to enable these plants to handle the variable heavy mineral assemblage. In addition to the solving of the processing challenges, technical issues such as removal of overburden for the minimum unit cost, the use of hyper saline water in the dredge pond and the provision of infrastructure, have had to be addressed. None the less, the orebodies making up the Pooncarie Project have many positive features including low slimes content, good rutile and zircon grades and the ability to recover most of the ilmenite family of minerals into acceptable TiO2pigment feedstocks. The development of the project is set to proceed with the establishment of mining operations on the Ginkgo orebody feeding a mineral separation plant at Broken Hill. The initial shipment of products is expected to occur in early 2005.

Jan 1, 2003

By P. Gonzalez

Minera La Parreña, S.A. de C.V., Milpillas operation is an underground mine designed to process 8 352 tons of ore per day for the recovery of copper through crushing, heap leaching, solvent extraction and electrowinning facilities. In order to optimize copper recovery during leaching, the ore is crushed to 13 mm and subjected to primary sampling every 15 minutes for metallurgical accounting. At the same particle size, primary sample goes to secondary sampling and then, the secondary sample is crushed to 2 mm before tertiary sampling. The tertiary sample is sent to laboratory where a final sample is obtained and pulverized to 106 microns for copper assay. This paper shows a review of sampling strategy focused on minimizing or eliminating sampling errors and an experiment designed to determine copper head grade profile as well as the best sampling frequency.

Jan 1, 2009

Statistics is essentially a study of variability, the use of some suitable mathematical model representative of such variability, and the application of this inferred pattern of behaviour to practical problems. Probability theory is the mathematical structure devised for providing models for chance happenings. Perhaps the simplest example of a chance happening is the result obtained in tossing a coin. Although the result of a single toss can never be predicted with certainty, there is no reason why in repeated tosses the expectancy of heads and tails should not be equal. In such a case the outcomes of the experiment are said to be equally likely, so that the probability of obtaining heads (or tails) is 1 in 2, that is, 0,5. This simple probability .model refers to a 'mathematical' or 'unbiased' coin. Whether such probabilities are meaningful for actual coins tossed in any particular way is a matter for experience to determine. Probability models have also provided the basis for representing observed variability in cases where the actual under- lying probabilities are only partly known or defy analysis. A typical example of the latter case is the variability observed in the values obtained from gold or other ore samples. The interacting natural forces which gave rise to this variability are obviously very complex and are unlikely ever to be unravelled completely. In spite of this, probability theory and the more recent geostatistical theories have provided us with models which can represent this variability on a reasonable basis. Applied statistics and geostatistics must therefore never be accepted as exact mathematical sciences. Furthermore, a basic and essential concept is that in the type of applications covered in these notes, no model must ever be accepted as necessarily the final and exact answer to the problem; thus, any model must prove and continue to prove its appropriateness in practice, preferably by follow-up comparisons.

Jan 1, 1981