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By C. J. Cutler

Metallurgical accounting for toll treatment smelters pose a number of challenges. Multiple feed stockpiles have to be accounted for as part of the monthly recovery estimation. Moreover, multiple metallurgical intermediates make up part of the in-process inventory, which also influence the monthly recovery calculation. Errors propagate from the measurements of volumes, assays, moisture fractions and bulk densities in the various material types through multiple steps up to the point where the final recovery is calculated. These errors contribute non-linearly to the variance in the final recovery estimate. This paper develops the mathematical formulation for variance propagation in toll smelting operations, including the effect of in-process inventory, assuming measurement biases have been eliminated beforehand. Operational data from a Southern African copper smelter is used for a case study. The method of propagation of variance showed that uncertainties in stockpile assays were the main contributors to variance in the recovery estimate. Variance in the volume and bulk density uncertainties contributed a secondary, yet significant, proportion to the overall recovery variance. It was determined, for the given case study, that the recovery variance depended on the calculation method used and that variance propagation via the two-product formula was smaller than recovery variances calculated via the standard recovery formula. However, the probability that the two product formula will give inaccurate (versus imprecise) results is significantly more due to the practical difficulty of equiprobable sampling of tailings streams from smelter plants (which includes materials such as slags and flue dust). Recommendations are made on how to achieve a reduction in overall uncertainty for toll treatment smelters. Keywords: metallurgical accounting, copper smelters, pyrometallurgy, variance propagation, errors, precision, sampling

Jan 1, 2006

By Wei Zheng

A pulverized coal fired power plant is being built in Osceola, Arkansas, along the Mississippi River. The subsurface soil conditions consist of a certain amount of very soft clayey silt and clay underlain with a medium dense sand layer and a dense sand layer. The site has been raised four to five feet with fill material consisting of sand and gravel. Consolidation settlement due to the fill is expected to cause negative shaft resistance (downdrag) on the pile foundations. The site is located in a high seismic hazard area - the New Madrid Seismic Zone (NMSZ). The liquefaction analyses indicated that there is a high liquefaction potential in the medium dense sand layer. Load tests are needed to determine the shaft resistance and the end bearing of each soil layer for design purpose. Two types of pile foundations ? augered pressure grouted displacement pile (APGD) and partial augered pressure grouted displacement pile (P-APGD) were considered for this project. Ten pile load tests, including five compression tests and five tension tests, were performed for three different areas. Strain gauges were installed for the compression tests. The APGD and P-APGD piles were installed side by side for comparison. In addition, three additional dynamic tests were performed to supplement the information from the static load tests. Load deflection curves for the compression tests were predicted, and the failure loads were interpreted using four different methods. The shaft resistance and the end bearing were determined from the strain gauge data and were used to back calculate the ultimate capacity. The back-calculated capacities are in good agreement with the predicted failure loads; they were further verified with the tension load test results.

Jan 1, 2007

By C McGuire, W Harris, S Arsenault, D Witow, B Rogers

Mining deep orebodies is increasingly becoming the norm as the ‘easy’ orebodies are depleted. Deep ore mining requires significant ventilation and cooling infrastructure to not only meet regulatory requirements for air quality but also to achieve acceptable workplace temperatures with the higher strata rock temperatures and increased auto-compression. At the same time, advances in battery equipment technology are turning the possibility of a battery powered fleet into reality. The switch from a diesel fleet to a battery powered fleet offers several positive opportunities for the design and operation of a deep mine.The potential to reduce ventilation and cooling requirements due to battery powered equipment results in significant cost savings for mine development and infrastructure. Battery powered fleets are not subject to the same dilution requirements associated with diesel fleets thus allowing lower ventilation rates. At the low ventilation volumes possible with a battery powered fleet, surface refrigeration plants are limited with respect to the amount of cooling that can be delivered to the workface. Considerations for calculating the optimum air flow and cooling requirements are presented. Considerations for locating the refrigeration plant are also discussed. This paper presents these opportunities and considerations; including a case study comparing the ventilation and cooling system design with a diesel fleet versus battery fleet for a planned mine in Northern Ontario, Canada.CITATION:Harris, W, Arsenault, S, McGuire, C, Rogers, B and Witow, D, 2017. Diesel to electric – creating a positive paradigm in underground ventilation and cooling, in Proceedings Australian Mine Vent Conference 2017, pp 169–174 (The Australasian Institute of Mining and Metallurgy: Melbourne).

Aug 28, 2017

By Upendra L. Karna

The pile capacity gain with time evaluated and implemented for the construction of the Route 21 Viaduct in Newark, NJ, U.S.A is discussed in this paper. The project consisted of the replacement of more than 3.2 km of viaduct and ramps. The project site consists of thick residual materials deposited by the Wisconsin Glacier over sedimentary rocks. The upper portion is glacial-lacustrine deposit underlain by marginal morainic till and stratified drift. Subsurface investigations identified significant variations in layer thickness and grain size consistency. The site was characterized into 4 Soil types, based on subsurface soil behavior. Closed end 600 mm and 450 mm diameter driven pipe piles were used for the foundation. Loading tests conducted during an advanced construction contract, established that a significant soil set-up could be expected. For the first construction contract, a pilot loading test program consisting of 112 dynamic loading tests and 77 re-strikes (at 2 to 4 weeks) using the PDA (Pile Driving Analyzer) with CAPWAP (Case Pile Wave Analyses Program) was conducted to establish the soil set-up behavior. Nine (9) quick static loading tests were conducted to substantiate the long-term pile capacity. Based on the loading test results, average set-up factors were established and utilized for developing production pile driving criteria. Bearing resistance with depth was analyzed and reviewed. The set-up factors varied significantly, ranging from 1.05 to 1.89, and averaged about 1.30. Conclusions are derived. Set-up factors behavior for each soil type plotted by linear regression are presented and discussed. An appropriate set-up with time relationship for a glacial soil is suggested.

Jan 1, 2006

By Cooper R. J

The essential features of the zinc concentrate roasting installation at the Risdon Works by the Electrolytic Zinc Company of Australasia Ltd. are given.The problem of maintenance of ignition in down-draught zinc concentrate flash roasters is discussed, and a solution offered.A successful and economical method of using by-product oxygen for raising the roasting efficiency of hearth roasters is described. Three methods of utilizing oxygen for the same purpose in flash roasting are detailed, and their relative merit, considered.INTRODUCTIONThe zinc sulphide concentrate used at the Risdon Works is roasted autogenously for removal of sulphur in two hearth roasters, and four flash roasters. The hearth roasters are of modified Skinner type with eleven hearths. They were designed, and are used, for roasting mixtures of relatively low sulphide content. Their design and operation have been described elsewhere (Forster, 1949).The flash roasters are down-draughted furnaces with cross-section and rabbling equipment based on the hearth roaster design, and the process is similar to that developed by the Consolidated Mining and Smelting Company of Canada Ltd. (Cominco) at Trail, B.C. (Stimmel, Hannay, and McBean, 1936).These furnaces are 20 ft. in diameter. They have three drying hearths through which hot gas is drawn from the combustion chamber by the burner fan. The combustion chamber is 2'1 ft. high above the collecting hearth (27 ft. in No. 4 unit), and the discharge hearth below it is 6 ft. high, so that a large gas offtake is possible. Four rabble arms for each hearth are carried on a central column. Dried, ground concentrate, and the air for roasting it, are fed to a burner...

Jan 1, 1958

By S. Burks

This paper develops the themes explored by the authors at the Fourth International Platinum Conference,. Optimization techniques can be used to significantly increase the value of mining businesses by enabling better long-term planning decisions. Open pit and underground mine design, mine scheduling, cut-off grade and blending, stockpiling, and the linking of these to flexible elements of the metallurgical recovery processes are all evaluated together. Transport or sale of intermediate products and the requirements of the product metal markets can also be considered. Experience shows that net present value (NPV) can potentially be increased by 5-35 per cent or more, usually even before the expenditure of significant amounts of project capital. The case studies presented include: > An existing operation with multiple open pits and PGM concentrators linked to offsite downstream processing facilities, and with infrastructure constraints for which the scheduling of current operations and the potential for expansion were investigated > An existing operation with an integrated portfolio of mines, concentrators, smelter, and refinery, for which the various possible sources of ore for processing were prioritized to match the downstream process plant constraints and the product markets > An underground operation in Zimbabwe for which the optimum mining width was calculated and expansion plans were evaluated > An underground mine with many separate mining sections, for which a revised schedule was developed aiming to keep the shaft system and processing plant at capacity with the highest net value material available at any time throughout the life of mine. In the concluding section of the paper, a few brief notes are included suggesting how the strategic recommendations from enterprise optimization studies may be implemented by the owner's team.

Jan 1, 2013

This study was conducted to establish the effects of a diesel particulate filter (DPF) system with sintered metal substrate on the concentrations and size distributions of nano and ultrafine aerosols and concentrations of nitric oxides in underground mine air. The study focused on the formation and transformation of nucleation mode particles in mine air under prevailing test conditions. Experimental work was conducted in the NIOSH Diesel Laboratory at the Lake Lynn Laboratory experimental mine, a facility developed to allow evaluation of control technologies directly in underground conditions. The DPF system was examined for four repeatable steady-state engine modes using a naturally aspirated mechanically controlled Isuzu C240 diesel engine fueled with ultralow sulfur fuel. The engine coupled to an eddy-current dynamometer was operated in the underground drift with tightly controlled ventilation. Measurements upwind and downwind of the exhaust system discharge were used to quantify the effects of the DPF system on the mine air. A baseline was established using a standard muffler. The study showed that sintered metal DPF systems reduced the total mass of aerosols in the mine air by more than ten-fold when compared to a muffler. The nucleation mode aerosols (50dem<40 nm) in mine air were found to be strongly influenced by engine operating mode. The concentration of nucleation mode aerosols was found to be significantly higher for high-load modes than for low-load modes. The muffler and DPF system resulted in comparable total nitrogen oxides (NOx) concentrations. However, the nitrogen dioxide (NO2) fraction of the total NOx was found to be substantially higher for both systems for low-load modes than for high-load modes.

Jan 1, 2009

By C. C. Harris, A. Jowett, S. K. Ghosh

An equation derived from first-order kinetics and accounting for return from froth to pulp has correlated continuous flotation data. The concentration of floatable material in the pulp can be altered either by changing the feed concentration or the feed rate. The latter does not affect the flotation rate constant, the froth to pulp return constant, and the concentration of non-floatable material in both the pulp and concentrate. Both the return constant and the concentration of non-floatable material in the pulp are directly proportional to the feed concentration, while the concentration of non-floatable material in the concentrate is related to the feed concentration. The investigation of the froth flotation process by continuous test methods, previously used by Brown and smith,1 Jowett and Safvi2 and Jowett and West,3 is being continued. Recent tests, primarily concerned with flotation cell design, have yielded kinetic data suitable for testing an equation proposed by Arbiter and Harris4 which postulates that a flotation process is defined by two rate constants: 1) the rate constant for mass transfer from pulp to froth. Previous workers have regarded this as the flotation rate constant. 2) the rate constant for mass transfer from froth to pulp, which has previously been ignored. THEORY Under continuous flow conditions at the steady state in Fig. 1 the equation developed4 is Symbols are defined in Table I. The air-liquid ratio terms4 have been omitted, since they are substantially constant for a given set of conditions. Also, for a system operating at the steady state Vf and V, remain constant and Eq. 1 can be written: Cp/Cc=b&apos;/a,+Qc/a&apos; [2]

Jan 1, 1963

By Kenneth Eltschlager, Jim Ratcliff, Michael Mann, Edward Sheehan

The Appalachian Blaster Certification Delegation initiated a study to compare the consistency of measurements from blasting seismographs. Six blasts were monitored at surface coal mines in West Virginia, Maryland, and Ohio. A total of seven different makes were deployed during the tests, with as many as seven models from five different manufacturers compared during any one blast. Great care was taken to bury the geophones in identical fashion. The variations in seismograph readings were compared to the International Society of Explosives Engineers’ Performance Specifications for Blasting Seismographs. Between 4 and 125 Hertz, the ground vibration amplitudes should be within ±5% and airblast amplitudes should be ±1 decibel. For each of the three mutually perpendicular ground vibration channels and the acoustic channel, the median value of all the units was used as a baseline for comparison. The six blasts monitored had median peak particle velocities ranging from 0.42 to 3.47 inches per second (in/s) (10.7 – 88.1 mm/s) and median airblasts ranging from 124 to 135 decibels (dB). There were 46 seismograph recordings for the six blasts. Eighteen (39%) of the 46 maximum peak particle velocities varied more than ±5% from the median. A total of 138 (3 × 46) component particle velocities were recorded, with 50 (36%) of the readings varying more than ±5% from the median. For airblast comparison, 43 recordings were obtained, with seven (16%) varying more than ±1 dB of the median. Variances may be attributable to blasting seismograph differences, variations in field deployment, and relative locations of the geophones. Full waveform seismic and acoustic overlays for individual blasts showed encouraging conformity with some slight anomalies

Jan 1, 2015

By Robert T. McKnight

"Traditional techniques for valuing resource projects companies have been primarily based on discounted cashflow methods (DCF) or by comparison to similar projects with an apparent market capitalization. Many resource projects are really forms of options to invest rather than a future cashflow stream. Employing the DCF method to value a resource project which is really an option to invest, can dramatically underestimate the real value of the project. The analytical tools required to evaluate options, while more complicated than the usual spreadsheets used for DCF, are becoming more easily available and usable by managers. This paper will discuss the background of options valuation and some of the tools which will make it easier for managers to estimate the value of projects and companies which have significant real option values.ResumeBob McKnight is a graduate of the University of British Columbia in Geological Engineering, supplemented by an MBA from Simon Fraser University. Following several years in the oil patch with Cities Service and Brascan, Bob returned to Vancouver in 1979 to work for Wright Engineers. He subsequently went on to work for Cassiar Mining, Getty Resources and Endeavour Financial before joining Pincock Allen & Holt Ltd. in 1998. His experience ranges from exploration to feasibility studies, financial evaluations of mining and oil & gas projects, due diligence, project finance, valuations and mergers and acquisitions. He has extensive international mining experience pertaining to such countries as Chile, Mexico, Uruguay, Thailand, Indonesia, Angola, Papua New Guinea, Kazakhstan.What are Options?Options have been around for centuries. They were familiar to Roman and Phoenician traders who were shipping cargoes from Mediterranean seaports. In the financial world, options are types of contracts that generally include the right, but not the obligation, to buy or sell a share, currency or commodity. The underlying commodity price or share price changes are assumed to be lognormally distributed. In this context, options are of two general types, ""call"" options and ""put"" options. Call options have the characteristic of having the right to buy a tradable commodity or currency at a predetermined price for a specified period of time. For the owner of call options, they become more valuable as the price of the underlying commodity increases. For the seller of call options, these options become an increasing liability as commodity prices rise, since there is the obligation to sell the commodity at a predetermined (perhaps lower-than-market) price. This characteristic of call options recently became high profile with the sharp rise in gold prices last year. The value of gold call options sold by certain producers increased their associated liabilities to the point where margin calls were made."

Jan 1, 2003

By A. V. Rasskazova

"The oxidized and mixed ores are processed by heap leaching. Primary ores with high gold content are floated. In recent years mining of primary ores with the low gold content (up to 4 ppm) has increased. Extraction of gold from these ores by floatation is technologically ineffective. Gravity concentration is ineffective for fine gold particles because of their floatability. The research objective was assessment of the viability of processing gold-bearing ore with combined gravitational and hydrometallurgical methods and testing of recovery of metals from a gold-copper oxidized ore chloride-hypochlorite leaching solution. The advantage of chloride-hypochlorite lixiviant is the low cost of reagents which is produced by the electrochemical synthesis of NaCl solution with the addition of HCl. The adsorption of chlorine compounds by a mineral matrix and its ability to oxidize the refractory ore components and allows for the application of chloride-hypochlorite systems for gold recovery for both oxidized and primary ores. The complexity and problem in using a gold–chloride system is the strong adsorption of gold–chloride complexes onto mineral surfaces, such as goethite, quartz, and alumina. In preliminary testing, gold recovery from gravity circuit gold bearing concentrates processed by agitation leaching in chloride-hypochlorite solution, gold recovery has reached up 82% after 4 hours of leaching. Chloride-hypochlorite leaching of copper-gold oxidized ore with preliminary processing by sulfuric acid provided for 94 % gold recovery to solution. Understanding of ecological hazard of cyanides is the motive power of researches in the field of search non-cyanic chemistries to provide for technologically and economically effective solvents of gold. Soluble compounds of chlorine with a suitable oxidizer are one of the alternatives to cyanides. Basic researches of gold extraction by chloride-hypochlorite solution have shown high rates of gold recovery.INTRODUCTION Because of the continued and growing environmental concern for the use of cyanide for gold processing, interest for the use of non-cyanide lixiviants, especially with halides has renewed recently (Tran et al., 2001; Ferron et al., 2003; Bhaghalha, 2007). Leaching solutions based on halides (chloride, bromide and iodide) are viable alternative to cyanide (Kwang et al., 2008). The major disadvantage of the chlorine–chloride system is the high reactivity of chlorine in reaction with sulfide and gangue carbonate minerals. It is believed that only high grade materials may be treated by chlorine–chloride systems economically (Baghalha, 2007). Another important problem of the application of gold–chloride complex is that it tends to be rather unstable and may re-precipitate upon contact with a reductant, such as sulfidic minerals of ore. It is well known that Au(III)–chloride hydrolysis species are strongly adsorbed by goethite, quartz, and alumina (Machfsky et al., 1991). To minimize adsorption of gold–chloro complexes, it works best to minimize the adsorption time."

Jan 1, 2018

By G. F. Leaming

The paper by R.V. Ramani and R.J. Sweigard is a wonderful description of the labyrinthine web that has been spun about the mining industry by energetic bureaucrats and politicians over the past 50 years. The remedy for the problem, however, is not more of the same, but less. That may be difficult for the industry to achieve, for it is not a technical solution but a political one. And the current fervor for more detailed planning at all levels of government and private enterprise has become deeply ingrained. The authors recommend the provision of more information about mining and mineral resources to "macro" (i.e., government) land use planners. They apparently overlook, however, the already strong tendency on the part of most government land use planners to consider themselves omniscient. Thus, giving them more information about the technical problems of mining will only make them want to get more and more involved in the "micro" (private, site specific) mine development and production plans of the individual mining firm. In fact, this has already happened at all levels of jurisdiction from municipal to federal government. Examples are legion. The most effective way to ameliorate the adverse impacts of government land use planning on existing and potential mining operations is to: (1) introduce greater flexibility in the definition of land use zones by local and state governments; (2) adopt realistic and relevant ambient environmental performance standards in governing relationships between mineral land uses and concurrent or subsequent nonmining land uses; (3) allow greater leeway for economic considerations in land use decisions in contrast to the explicit legalistic approach now in vogue; (4) recognize that all minerals are not the same and that sand and gravel mining should not be treated the same as underground metal mining, coal stripping, oil field production, or in situ leaching; and (5) eliminate the notion that mining operators should be responsible for determining in detail the use of land by subsequent owners of mined land. This last bit of conventional ethic really makes no more sense than requiring the builders of every shopping center or government office complex to provide detailed plans for the use of that land when its use for shopping or government is ended. Did the builder of Ebbetts Field plan for Brooklyn after the Dodgers went to Los Angeles? Should the developer of the Bingham Pit plan for suburban Salt Lake City after the copper mining goes to Chile? The nation&apos;s mining industry must address these questions before further bankrupting itself to provide more data to planners and spending thousands of dollars per acre to create land that when reclaimed is worth only a few hundred dollars per acre. ? Reply by R.V. Ramani and R.J. Sweigard We thank Mr. Learning for his valuable contribution. His views on the problems of land use planning and mineral resources are most welcome additions to our paper. As the title indicates, our paper was more concerned with the impacts of land use planning on mineral resource conservation than with the details of the planning process. On the whole, his five recommendations would be helpful for mineral resource conservation. However, we would suggest that the argument he presents for his final recommendation does not address the differences between mining as a land use and commercial or institutional uses. We believe that this difference is the crux of the issue. We share Mr. Learning&apos;s desire to ameliorate the adverse impacts of land use planning. Possibly the most detrimental impact is the loss of mineral resources. Any development, whether mineral or community, that does not give proper consideration to other resources can result in permanent loss or sterilization of resources. With proper planning, some of these losses can be avoided. As our paper indicated, one factor that limits the consideration of mineral resources, and ultimately leads to their sterilization, is the generally inadequate levels of resource characterization and understanding of the unique nature of mineral resources and mining operations. The last point raised by Mr. Learning is also important. In terms of reclamation and land use planning in mining districts, we certainly do not advocate spending more than what the results are worth. The main thrust of the paper was to explore the avenues for conserving the mineral resources so that, at some appropriate time, the issue of mining and reclamation can still be addressed. ?

Jan 1, 1986

By J. H. Swisher

When an Fe-Mn alloy is internally oxidized, the inclusions formed are MnO which contains some dissolzled FeO. In the internal oxidation reaction, not all of the manganese is oxidized; some remains in solid solution as a result of the high Mn-0 solubility product in iron. Taking these factors into consideration, the rate of internal oxidation of an Fe-1.0 pct Mn alloy is computed as a function of temperature, using available thermodynanzic data and recently published data for the solubility and diffusivity of oxygen in iron. The predicted and experimentally determined rates for the temperature range from 950 to 1350°C are in good agreement. ThE rates of internal oxidation of austenitic Fe-A1 and Fe-Si alloys have been studied extensively.1"4 Schenck et al. report the results of a few experiments with Fe-Mn alloys at 854" and 956C, and Bradford5 has studied the rate of internal oxidation of commercial alloys containing manganese in the temperature range from 677" to 899°C. When Fe-Mn alloys are internally oxidized, the inclusions formed are solutions of FeO in MnO, the composition depending on the experimental conditions. Since the thermodynamics of the Fe-Mn and FeO-MnO systems have been investigated,6"9 and since the solubility and diffusion coefficient of oxygen in y iron have been determined recently,&apos; it is possible to predict the rate of internal oxidation from known data. The calculations used in predicting the rate of internal oxidation will first be outlined, then the results of the prediction will be compared with the experimental results of this investigation. PREDICTION OF PERMEABILITY FROM THERMODYNAMIC AND DIFFUSIVITY DATA Oxygen is provided for internal oxidation in these experiments by the dissociation of water vapor on the surface of the alloy. The dissociation reaction is: + H2(g) + [O] [1] where [0] denotes oxygen in solution. The equilibrium constant for this reaction is known as a function of temperature:&apos; log As oxygen diffuses into the alloy, oxide inclusions are formed which are MnO with some FeO in solid solution. The reactions occurring are: [Mn] + [0] = (MnO) [31 and [Fe] + [0] = (FeO) [41 where [ Mn] is manganese dissolved in iron and (FeO) is iron oxide dissolved in MnO. The overall reactions may be written as follows: [Mn] + HOte) = (MnO) + H2(£) [5] and [Fe] + H20(g) = (FeO) + Hz(R) [61 The standard free-energy changes and equilibrium constants for Reactions [5] and [6] are known.6 Therefore the equilibrium constants for Reactions [3] and [4] may be obtained by combining known thermodynamic data for Reactions [I], [5], and [6]. For Reactions [3] and [4]: K = and For the present purpose, both the Fe-Mn7,8 and FeO-~n0&apos; systems can be considered to be ideal, i.e., [amn] = [NM~] and (aFeO) = (NM~~) = 1 - (NFeO) where the Ns are mole fractions. These relations, together with Eqs. [I] and [8], permit us to compute both the oxide and metal compositions as a function of temperature and oxygen potential at any point in the specimen. For cases where the oxygen concentration gradient between the surface and the subscale-base metal interface is linear, the kinetics of internal oxidation is an application of Fick&apos;s first law: where dn/dt is the instantaneous flux of oxygen into the specimen, g-atom per sq cm sec; 6 is the instantaneous thickness of the subscale, cm; Do is the diffusion coefficient of oxygen in iron, sq cm per sec; p is density of iron, g per cu cm; h[%O] is the oxygen concentration difference between the surface and sub-scale-base metal interface, wt pct. B6hm and ~ahlweit" derived an exact solution to the diffusion equation for systems in which there is a stoichiometric oxide formed. They showed that the oxygen concentration gradient is given by a rather complex error function relation. For the Fe-Mn-0 system and for most other systems that have been studied, however, variations in oxide compositions are small and rates of internal oxidation are sufficiently slow that the deviation from linearity in the concentration gradient of oxygen is negligible. The mass of oxygen transported across a unit area of the specimen for the total time of the experiment is given by the mass balance equation:

Jan 1, 1969

By R. I. Jaffee, R. M. Evans

IN recent years, the interest in liquid metals as heat-transfer media for power plants has been very great. The possibility of the development of nuclear power plants has increased this interest and served as the impetus behind much research on low melting metals and alloys for such purposes. The principal reasons for consideration of liquid metals as heat-transfer media lie in their high thermal conductivity and consequent high heat-transfer coefficients, stability at high temperatures, and the high ranges of temperature possible. The element gallium possesses some of the requisite properties for a heat-transfer liquid. It is a unique material, having a low melting point and a high boiling point. Pure gallium melts at 29.78oC, and suitable alloying will produce a metal which melts below room temperature. The boiling point is about 2000°C. As it is a liquid metal, the heat-transfer characteristics would be good. Gallium is not now readily available, due in part to a lack of uses for the metal. Nevertheless, it is not a rare element, and a sufficient supply of gallium exists to permit its consideration for this use. Since gallium has some promise as a heat-transfer liquid, owing to its unique properties, research on the subject was carried on at Battelle Memorial Institute at the request of the Bureau of Ships, U.S.N. The research had as its objectives the determination of the effect of alloying on the melting point of gallium, and the study of the corrosion of possible container materials. In this research, alloys were found which had significantly lower melting points than pure gallium, but none which simultaneously fulfilled other additional requirements, chiefly the corrosion problem. Neither was it found possible to reduce the melting point of certain otherwise suitable alloys appreciably by small additions of gallium or gallium alloys. The results gave little hope that gallium alloys can be developed which enhance the good properties and minimize the undesirable characteristics of elemental gallium. Thus, gallium now appears less promising than other metallic heat-transfer media. The experimental thermal-analysis techniques used in this work have been described.&apos; Experimental Results As a first approximation, the development of low melting gallium alloys was based on alloying elements suitable for use in a nuclear power plant, which also lowered the melting point of gallium. Information from the literature, summarized in Table I, indicates that. tin, aluminum, and zinc are the only suitable elements which cause a lowering of the melting point of gallium. Indium and silver also lower the melting point of gallium, but are of little interest for use in nuclear power plants. Of the elements reported not to lower the melting point of gallium, there is some ambiguity on the behavior of copper. Weibke3 obtained solidus arrest temperatures of 29°C for Cu-Ga alloys from 60 to 90 pct Ga, 0.8C lower than the generally accepted melting point. This may be the effect of a eutectic close to gallium, or, more likely, the result of impurities, or experimental error. The seven elements listed in Table I whose effects were not known were of potential interest if they lowered the melting point of gallium. Their effects were determined experimentally for this reason. Binary alloys containing nominally 2 pct of each of these elements were prepared in the form of 2-g melts by placing the components in a graphite crucible and holding them in an argon atmosphere at 370°C for 5 hr. These melts were then subjected to thermal analysis. In all cases. the solidus temperature was the melting point of gallium. Since these elements (As, Ca, Ce, Mg. Sb, Si, and T1) did not lower the melting point of gallium, they were not considered further as components of a eutectic-type alloy. Ga-Sn-Zn Alloys Preliminary considerations of this system for low-melting alloys were encouraging. All three binary systems were of the simple eutectic type. The composition and melting points of the eutectics were as follows: Sn-9 pct Zn (199°C), Ga-8 pct Sn (20°C), and Ga-5 pct Zn (25°C). Therefore, the probability of a ternary eutectic was high. For reasons to be discussed later, aluminum could not be used as an alloying constituent, leaving the Ga-Sn-Zn system as the only one of interest for low-melting gallium-

Jan 1, 1953

By M. Kalin, C. Steinberg

We have worked on several flooded pits from coal-mining activities in the former East Germany, as well as ones associated with hard- rock mining, including the B-zone pit discussed in the above technical paper. We found the paper to be a useful summary, but, unfortunately, it failed to give an adequate comparison of the physical limnology of the flooded pits, which is an essential component. While the title suggests that the primary focus of the review is physical limnology, it appears that it is essentially pit-lake chemistry being presented. Physical limnology requires that factors such as fetch, latitude, light penetration, relation to ground water table, methods of flooding and the physical shape of the pits be defined. These physical aspects of a pit interact with the chemical and biological processes taking place in it, all of which contribute to the character of a water body. Few of these physical aspects are presented, however. The conclusion that the authors reach suggests that meromixis may be a condition that would serve as an effective containment mechanism for contaminants in a pit. Although this may be desirable, such limnological conditions are not clearly supported by the data presented for any of the pits. These data should be summarized to facilitate comparison between the same structural units of the pit water - the epi- and metalimnion for example. The thermocline depth is a reflection of the physical forces mixing the water body, and pit dimensions affect these forces. Due to the use of different scales in Figs. 2 through 5, it is difficult to determine whether the thermocline is at the expected depth, because the fetch is not given. Moreover, the status of a water body cannot be determined unless measurements cover a period of at least one year, and depth profiles are completed to represent the entire depth of the pit. This shortcoming is most notable in the case of the Berkeley pit, where data are given for depths of only 20 and 35 m (66 and 115 ft), although the pit is reported to be 242 m (794 ft) deep. Limnological data to define the status of the pit water have to be collected at regular intervals, for the same parameters. The authors present temperature measurements for 1-m (3.3-ft) intervals, but fail to use that interval for other parameters, such as dissolved oxygen or, in some cases, for contaminant concentrations. Furthermore, the profiles for the deepest part of the pit display only part of the picture, because pits are rarely conical. Profiles can be considered to represent the status of a water body only after other stations in the pit have been monitored regularly and the consistency is determined. For example, fresh water, which can enter a pit at any depth, would interfere with the proposed meromictic conditions. Similarly, organic material at the bottom of a pit, such as the fish-waste deposited in the Gunnar pit, contribute to oxygen consumption. Oxygen depletion alone is not indicative of meromixis. It is interesting to note that the Dpit arsenic concentrations could possibly be slightly higher than the B-zone pit concentrations at depth, although this is difficult to determine accurately when a log scale is used for the D-pit and not for the B-zone pit. In our investigations, we noted arsenic removal in the B-zone pit bottom water, which was due to the formation of particles that are relegated to the newly forming sediment in the bottom of the pit. Particle-carrying contaminants form due to a combination of geochemical and biological factors and TSS contributed from erosion of the upper parts of the pit walls, whereas the settling out of particles from the water column is controlled by the physical conditions or turn over, for example. during ice cover in the B-zone pit. Although meromictic conditions for flooded pits may be desirable at decommissioning, this would depend largely on the physical conditions of the pit, because, under no circumstances, would this water be of desirable ground-water quality. Under meromictic conditions, acidity, if an environmental issue, may be reduced by microbial acid-neutralizing activity, and several heavy metals may form more or less stable sulphitic compounds. These may stay suspended in the water if conditions are such that they are not relegated to the sediments, i.e., in the absence of turnover. These processes do not take place in meromictic conditions only, but meromixis does require autochthonous and/or allochthonous organic substrate supplies, which are generated under aerobic conditions. Specific limnological (biological, chemical and physical) features of the pit lake under consideration have to be defined, such that water quality parameters can be predicted, and the objectives of the decommissioning activities, environ-

Jan 1, 1999

By O. D. Sherby, J. E. Dorn

SEVERAL years ago Zener and Hollomon1 suggested that the flow stress of metals might be related to the temperature and strain rate in accord with the functional equation: s=s(eeh/rt) [1] at the same state. The Zener-Hollomon relation also contains the significant tacit implication that the energy for activation, AH, is substantially independent of the state of the material. The utility of this method for correlating creep data with tensile data was illustrated in a preliminary reportL on the effect of alloying additions on the secondary creep rates of binary a solid solutions in aluminum as shown for Al-Mg alloys in Fig. 1. Not only do the secondary creep and ultimate tensile properties for each alloy correlate well on this plot but in addition the activation energy, AH = 17,900 R cal per mol. is identical for the various compositions. Similar types of curves correlating the secondary creep rates at 422°K (300°F) with ultimate tensile data were also obtained for a series of dilute a solid solutions of copper, germanium, zinc, and silver in aluminum. In all cases the activation energy for creep, AH, was found to be equal to about 17,900 R cal per mol independent of the type of alloying element or its concentration. The coincidence of the activation energies for these alloys is probably due to the fact that the activation energies for creep are rather insensitive to small composition changes. Alloying additions, as shown in Fig. 1, however, increased the stress necessary to obtain equivalent values of ee?H/RT. Thus the stress level in curves of the type represented by Fig. 1 gives the relative creep strengths of the various alloys. Inasmuch as these correlations between creep and tensile data were obtained for creep tests at 422°K, it appeared advisable to ascertain the range of secondary creep rates and temperatures over which correlations between creep and tensile data could be made on the basis of the Zener-Hollomon relationship. If, for example, the activation energy changes with different ranges of creep temperatures, the utility of the proposed analysis would be severely weakened. But if AH is constant not only for all dilute alloys, but also over wide ranges of temperature and secondary creep rates, confidence will be developed not only in the broad utility of the Zener- Table I. Chemical Analysis&apos;:&apos; and Grain Size of Alloys Mean Grain Residual Impurities. Wt Pct Diam- Alloying Atomic-eter, Element Pct Si Fe Cu Mg Mn mm Aluminum 99.987 0.003 0.003 0.006 0.001 0.25 Magnesium 1.617 0.003 0.004 0.006 0.26 Copper 0.101 0.003 0.003 0.0006 0.001 0.29 Zinc 1.616 0.003 0.005 0.007 0.001 &apos;0.26 * The authors express their appreciation to the Aluminum Company of America for the preparation and chemical analyses of these alloys. Hollomon relationship, but also in its theoretical justification. The following investigation was instituted in order to ascertain the range of validity of the Zener-Hollomon relationship when it is applied to creep and tension data. Materials, Equipment, Technique In view of rather extensive correlatable data now available on the plastic properties of a series of binary a solid solutions of magnesium, copper, germanium, zinc, and silver in aluminum, these alloys were again selected for the present investigation. Previous reports have already covered the tensile properties of these alloys at subatmospheric temperatures," and at elevated temperature," as well as their fatigue properties at atmospheric temperatures, nd their creep properties at 422°K.&apos; Resistivity measurementsa as well as precision lattice constant determinations? evealed that these alloys are a solid solutions. After homogenization. 0.100 in. thick sheets were rolled to 0.070 in. in thickness and recrystallized to about the same grain size. Since the details of these treatments were reported previously," they will not be repeated here. In order to appropriately reduce the scope of creep tests in this investigation only the typical alloys shown in Table I were investigated. All creep specimens were machined such that their tensile axes were in the rolling direction of the 0.070 in. sheet stock. The gage section was 2 in. long and 0.500 in. wide with a reduced section of 3.50 in. Creep strains were measured by means of rack and pinion type extensometers with a sensitivity of 0.005 in. per division. Readings were estimated to 1/10 of

Jan 1, 1953

By Anton Sawatzky, Erich Vogt

By means of mathematical solutions to the appropriate diffusion equations, we describe the kinetics of the thermal diffusion of hydrogen in Zircaloy-2 for the various temperatures and concentrations encountered in a heavy water moderated reactor. When the hydrogen concentration is below terminal solid solubility only the a Phase is present. Redistributions are then described in terms of the characteristic functions of the difhsion equation. For higher concentrations both the a and 6 phases are present. We assume the two phases to be always in equilibrium. For moderately small hydrogen concentrations exact solutions of the two-phase equation approach the the approximate solutions derived by Sawatzky and for all concentrations the exact solutions exhibit the qualitative features of his result: the two-phase concentration increases with time, everywhere; in the absence of a hydrogen current at the hot end of the sample an a-phase region always exists there; the interface of the a + 6 , a-phase boundary moves toward the cold md of the sample and the hydrogen concentration is discontinuous at the interface. Simultaneous solulions of the a and a + 6 hydrogen distributims and of the concomitant interface motion are obtained and compared to the observations of Sawatzky and Markowitz. The kinetics of the hydrogen diffusion process are shown to lead to m apparent heat of transport of the a phase which is lower than the actual value (even for samples with long anneals) thus resolving at least partially the disparity between experimental measurements of this quantity. A number of recent papers1"4 have reported measurements on the diffusion and redistribution of hydrogen in Zircaloy-2 under temperature and concentration gradients. These studies were instigated by problems arising from increasing use of Zircaloy-2 as a fuel element cladding material in pressurized-water power-reactors. The Zircaloy-2 picks up hydrogen during the operation of the reactor: the consequent precipitation of zirconium hydride in the Zircaloy-2 has pronounced effects on its mechanical properties. The Purpose of the present Paper is to describe the kinetics of the thermal diffusion of hydrogen in Zircaloy-2 for the various hydrogen concentrations and temperatures likely to be encountered in reactors. When the hydrogen in the Zircaloy-2 is entirely in the solid solution phase (a phase), the differential equation for thermal diffusion is well known and the redistribution can be described by standard mathematical methods some of which are given in Section 11 below. The treatment of the a + 6 (hydride) region differs from the earlier treatment of sawatzky3 by taking into account several modifications of the two-phase diffusion equations as suggested to us by Markowitz4 and Kidson.5 Even with the modifications the results obtained in our more complete treatment are substantially the same as those found by Sawatzky. As we show, the difference between the results of sawatzky3 and kIarkowitz4 is largely due to the difference in the geometry of their experiments. In the next section we derive the modified two-phase equation for arbitrary geometry. The exact solution of this equation is given in Section III for linear and cylindrical geometry. It is shown that Sawatzky&apos;s approximate solution is quite accurate for almost all the temperatures and hydrogen concentrations which are actually encountered. In Sawatzky&apos;s approximate theory the hydrogen concentration everywhere in the two-phase region increases continuously. As his paper pointed out, this result, together with hydrogen conservation, implies that in a sample with no hydrogen current flowing into the hot end a two-phase region is always accompanied by a single-phase region at the hot end of the sample. The net hydrogen gain in the two-phase region is supplied by the decrease of hydrogen in the single-phase region and by the movement of the (a, a + 6) boundary toward the cold end of the sample. Hydrogen conservation at the (a, a + 6) boundary leads to a discontinuity in the concentration and its derivative there. In Section IV it is shown how these qualitative features of the thermal diffusion kinetics arise from the simultaneous solution of the a-phase differential equation and a + 6 phase equation. Methods derived by us previously for the solution of the redistribution in both phases and the accompanying motion of the boundary are used to obtain approximate solutions valid for large times. The solutions account well for the observed redistributions of Sawatzky and Markowitz. It is shown how the continuing motion of the (a, a + 6) boundary leads to measured values of the heat of transport lower than the actual value, even for specimens with relatively long anneals.

Jan 1, 1963

By P. Douglas, G. M. Leak, B. Mills

The sinusoidal surface relaxation technique has been used to measure the surface self-diffusion coefficient of spectroscopically pure nickel over a wide temperature range under a hydrogen atmosphere. A kink in the Arrhenius plot has been observed. In the temperature range T/T 0.98 to 0.80 (T in O K and T, is the melting temperature) the average self diffusion coefficient is given by Below the temperature T/T,- 0.80a decrease in the slope of the log Ds us 1/T plot is observed. This is associated with a diffusion process characterized by a lower activation energy (-20,000 cal mole&apos;&apos;) and smaller preexponential term (-10- sq cm sec"). A series of experiments were carried out at T/Tm = 0.61 under a hydrogen atmosphere of higher oxygen partial pressure than for the rest of the experiments. It was found that Ds was significantly depressed due to oxygen adsorption. This evidence supports the opinion that the low temperature process (activation energy -20,000 cal mole-&apos;) is unlikely to be due to oxygen adsorption. An interesting feature of the present data is that the transition temperature (T/Tm - 0.80) is a function of orientation. For a small number of crystals of measured orientation the transition temperature was observed to be higher towards the low index (100) pole. Theories of surface diffusion are briefly reviewed and it is concluded that the present reszuts are best explained by invoking a surface roughening process. GJOSTEIN has recently analyzed available surface diffusion data for a wide range of metals. He suggested that two mechanisms were operative for fcc metals, an adatom process at high temperatures and a vacancy process at low temperatures. Results for nickel can be summarized as follows. At low temperatures (T/T, - 0.3 to 0.44) under ultra high vacuum conditions, Melmed2 measured an activation energy Q of 21 kcal mole-&apos; using field electron emission microscopy. At higher temperatures (T/T - 0.7 to 0.9) under a vacuum of 10- &apos; torr, Maiya and lakel measured y as 39 kcal mole-&apos; using the multiple scratch smoothing technique. The present work was undertaken to try to find out if two distinct processes could be observed. High temperature results give Q about 47 kcal mole-&apos;: there is evidence also for a low temperature value of about 20 kcal mole-&apos;. These measurements were all made under a hydrogen atmosphere, in the temperature range 860" to 1412°C. Concurrent with the present study Bonze1 and jostein> have also observed a break in the Arrhenius plot for the (110) surface of nickel. These measure- ments under ultrahigh vacuum conditions using the laser diffraction technique are in excellent agreement with the work reported here under hydrogen annealing conditions. THEORY The available surface relaxation techniques include single and multiple scratch smoothing and grain boundary grooving. The processes have been compared in detail by Gjostein for conditions where surface diffusion dominates6 and Mills et al? where volume diffusion dominates. In summary the relevant points are as follows. Grain boundary grooving gives an average Ds for the two surfaces adjacent to the boundary and this can, to some extent, be simplified by using symmetrical bicrystals. This technique has been used to study the effect of environment on Ds for silver and copper.&apos;-&apos;&apos; Scratch techniques yield Ds values for the small orientation range exposed by the scratches (-2 deg). The multiple scratch process is preferable because the profile rapidly becomes sinusoidal and can then be interpreted theoretically in a relatively simple way. Also corrections for mass transport processes other than surface diffusion can be introduced easily. Mullins" considered a sinusoidal profile described the wavelength of the profile. After time t the profile can be described by the equation The terms A, A&apos;, C, and B which account, respectively, for contributions due to evaporation-condensation, diffusion through the gas phase, volume diffusion through the lattice, and surface diffusion are defined as: where Ds = the surface self diffusion coefficient ys = the surface energy per unit area p = the equilibrium vapor pressure over a flat surface pa = the equilibrium vapor density over a flat surface DG= the diffusion coefficient of vapor molecules in the inert gas DM = the mass transfer diffusion coefficient which for a pure cubic metal is Dv/f where Dv is the radiotracer diffusion coefficient and f is the correlation factor H = the molecular volume V = the surface density of atoms, il2&apos;3 M = mass of an evaporating molecule

Jan 1, 1970

By R. L. Fleischer, W. F. Hosford

Tensile data for coarse grained aluminum Polycrystals suggest that the "grain size" effect is not due to dislocations piled up at grain boundaries but rather is primarily a relative size effect due to surface crystals being weaker and less confined. STUDIES directed at interpreting hardening of poly-crystalline metals normally identify their strain hardening properties with those in some particular type of single crystal.1"4 The recent recognition in face centered-cubic metals of a nearly linear stage with rapid hardening occuring at comparable rates for both polycrystals and single crystals, suggested that the same process or processes determine both cases and hence that there exists some justification for the use of single crystals to understand polycrystals. Further evidence for the above view may be found by an approach initiated by Chalmers:5 By using bicrystals of controlled orientation it is possible to begin to assemble a polycrystal and also to study grain boundary effects in detail. In this way it has been found that a single grain boundary affects easy glide but not the subsequent stage II hardening.6 This result suggests that a sensitive way to observe grain boundary effects in polycrystals would be to vary grain size and measure easy glide. As will be seen, easy glide is only possible for coarse-grained samples, and hence the results will serve to fill in the gap in measurements between single crystals and bicrystals on one hand and fine-grained polycrystals on the other. One problem inherent in comparing single crystals with polycrystals is the uncertainty as to what slip systems are acting in a polycrystal. To compare the two types of samples, rates of shear hardeninn---L. on the acting -planes are needed. and these may be computed only if it is known what particular systems are active. The acting systems were examined for a coarse-grained polycrystal and it will be shown that the systems supplying the preponderance of slip can be determined with little ambiguity. EXPERIMENTAL PROCEDURE Twelve samples of aluminum were prepared by chill casting into a heated graphite mold, followed by annealing at 635° ± 5°C for 24 hr with an 8-hr fur- nace cool, and finally either etching7 or electropol-ishing.&apos; The samples, with a 7 to 10 cm length between grips and 4.4 by 6.6 mm in cross section, were deformed at a strain rate of about 3 10 -3 . per min in a tensile device which has been described elsewhere.5 The composition was reported by Alcoa as 99.992 pct Al, 0.004 pct Zn, 0.002 pct Cu, 0.001 pct Fe, and 0.001 pct Si; nine samples were deformed while immersed in liquid helium and three in air at room temperature. The stress-strain curve for one of the samples (P-1) deformed at 4.2 "K has been reported previ~usl~.~ This sample was selected for determination of active slip systems. Eighteen of the crystals were examined by optical microscopy to determine the angles of slip line traces and by X-ray back reflection to determine orientation. By this means the slip planes were determined and the resolved shear stress factors for possible slip systems could be computed. Finally each sample was sectioned so that after etching, the number of crystals could be counted for each of ten newly exposed surfaces. The average of these ten values will be termed n, the number of crystals per cross section. Values of 11, varied from 1.9 (nearly bamboo structure) to 12.7. Sketches of typical cross sections appear in Fig. 1. RESULTS AND DISCUSSION: SLIP SYSTEMS 1) Determination of Acting Slip Planes—The stress axis orientation and operative slip planes in eighteen crystals of sample P-1, as determined by slip line traces and crystal orientation, are summarized in Fig. 2. For one of the crystals two planes had a common trace. so that the traces alone did not distinguish which plane or planes were slipping. However it was found that the stress resolving factor for the primary system was 0.386, .while that for the most stressed system in the other plane (indicated bv the dotted arrow) is 0.138. It will be assumed tgerefore that only the primary plane acted. Since the orientations were determined after extending the samples 4 pct, the stress axes may be rotated from their original value by as much as 2 deg in some cases. It is interesting to note that in five crystals only one slip plane acted, in eight two acted, and in five three planes were observed—an average of two slip

Jan 1, 1962

By G. W. Nabor, A. S. Odeh

The effect of production history of a well on the results of two-rate flow tests, and conventional build-up analyses was investigated. The effect was examined by means of digital computers and an R-C network model, respectively, for wells with infinite and finite radii of drainage. For systems which behave as infinite, it was found that production history and the duration of production at constant rate prior to the initiation of the test have important effects on the results. During build-up time equal to about one-fourth of the stabilized time, correct permeability-thickness product calculations can be made. For wells with finite radii of drainage, the time was determined during which the straight line can be satisfactorily used for permeability-thickness product calculations in case of drawdowns and build-ups. On build-ups, the dimensionless time (based on the external radius) during which the straight line gives reliable results was detertriined to be about 1/12. This is one-fourth as long as that of the drawdown. The investigation was done theoretically, and subsequently was verified by R-C network model runs. General interpretive rules were formulated which, if not followed, could lead to serious errors. Moreover, a recommended testing procedure is reported. INTRODUCTION The method used by most reservoir engineers for estimating formation characteristics in a producing well is the analysis of pressure build-up data. The method originally devised by Horner&apos; makes use of the point source solution to the diffusion equation. This solution is approximated by a logarithmic function and the superposition principle is employed to arrive at the well known pressure build-up equation:where q, the flow rate, is in reservoir B/D; ft is in cp; kh is in md-ft; At is the shut-in time; and t is the producing time. At and t are in any consistent time units. Ey. 1 is applicable to a well of unlimited drainage radius which produces at a constant rate q from zero to time t and is then shut in. Such a constant production rate seldom obtains in practice. Therefore. a correction term must be applied to Eq. I to account for the varying rate. Two theoretically accurate methods are available for treating the variable rate case. The first, originally derived by Horner,&apos; is based on the application of the superposition theorem. It requires knowlege of production history of the well as a function of time and results in lengthy and laborious calculations. The second t*q* niethod is suited for short production tests and requires that the shut-in time be at least one and one-half times the production time. A third method which is not based on any theoretical justification and which was suggested by Horner as a "good working approximation" is the one used by the majority of reservoir analysts. especially when the well has been producing for a long time and the t*q* method is not practicable. The key to this method is in choosing or determining the t that appears in Eq. 1. Horner suggested using a corrected time t, in place of t. t, is calculated by dividing the total cumulative production by the last established rate. Therefore, a normal procedure of pressure build-up testing is to stabilize the well at a constant rate for at least 24 hours before shut in and to use the stabilized rate to calculate t,. The analysis is then made by plotting either or P. and examining the resulting plot for the expected straight line to calculate kh and the original reservoir pressure. Recently, Russell" proposed a method for determining formation characteristics from two-rate flow tests. His method reduces to pressure build-up if the second flow rate is zero. Russell uses the Horner simplified procedure for calculating a corrected t,. His method also requires the stabilization of the well at a constant rate q which is used to calculate t,. Theoretically, the above procedure is valid for a well with an unlimited radius of drainage or with a limited radius as long as the boundary effect has not been felt by the well. Several authors&apos;&apos; derived formulas which allow the estimation of time during which limited reservoirs behave as infinite ones and, thus, can be treated by unsteady-state mechanics. One equation derived by Swift and Kiel&apos; terminates the application of unsteady-state theory when the drainage radius reaches one-half the reservoir radius. Thereafter, steady-state behavior obtains. Another equation derived by Jones&apos;" initiates steady-state

Jan 1, 1967