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By R. M. Hardgrove

PULVERIZED coal was first used for firing cement kilns about 45 years ago, with such success that it has continued in general use. Based on cost, pulverized coal is usually the most economical fuel. In some localities natural gas or oil is cheaper, but 85.5 per cent of all cement kilns at present are fired with pulverized coal. The pulverized-coal flame is well suited to kiln conditions. Its soft, luminous nature gives it a high radiating property and the length of the flame can be readily varied to suit conditions in a kiln of any length. Ash and slag cause most of the difficulties incident to the burning of coal. In a cement kiln, however, the ash, being similar in composition to the raw mix and being deposited uniformly in the mix and clinker, occasions no difficulty as long as it remains constant in quality and quantity. Until 6 years ago almost all cement kilns using pulverized coal employed the bin system. The coal was dried and pulverized in a separate plant and delivered into a bin at the kiln. Thus the bin system requires driers, storage-type pulverizers, conveying systems, individual kiln bins, pulverized-coal feeders and primary air fans. The direct-fired system requires only a pulverizer and a blower. Fig. I shows typical plants. The first installations of direct-fired pulverizers were of the impact type but fineness of pulverization was both inadequate and variable. About 1927 a few direct-fired installations employing tube-type pulverizers were made, which were much more satisfactory. As more efficient pulverizers were developed from 1934 on, equipment for direct firing was installed at a large number of plants, until now 196 of about 900 kilns in use in the United States are direct-fired. Most of these replaced antiquated bin systems. PROCESS OF BURNING CEMENT CLINKERS The process of burning cement clinkers is fundamentally one of heat transfer. In boilers and other heat-exchange apparatus the area of the surface receiving heat from the gases is the best unit for comparing output. This seems to be equally true for kilns. The inside kiln surface is the heat-receiving surface and its area is used as the unit of kiln size in the following comparisons. Heat input is based on British thermal units in the fuel. E. G. Bailey,1 in his work on boiler-furnace heat absorption, has used the unit "B.t.u. available." This is the practical calorific power of the fuel, plus the heat in the air used for combustion. In other words, it is the heat available for useful work in the furnace or kiln. The unit of kiln output is barrels per hour. The composition of the raw mix, the percentage of water in the slurry of wet plants and hardness of burning all affect the output. These conditions, however, are hard to evaluate and probably have less effect on the output than irregularities of raw feed and clinker discharge and inexpert control of combustion. WET VS. DRY PLANTS The study presented here covers 88 plants, of which 41 are wet and 47 dry. Of

Jan 1, 1941

MORE than 5000 miners and suppliers descended upon Denver to make the American Mining Congress' four-day metal mining show one of the most extravagant equipment displays ever assembled in one place at one time. The most recent developments in equipment for mining, milling, and processing ore were shown by the 146 exhibitors at Denver's new City Auditorium which covers 57,000 sq ft of floor space. Other displays were placed in the open air opposite the auditorium site. Registration desks were opened Sunday, Sept. 21. A field trip to the Colorado School of Mines at Golden was the featured activity of the day. Official opening of the show took place on Monday, and presided over by Howard I. Young, president of American Zinc, Lead & Smelting Co.; Otto Herres, vice president, Combined Metals Reduction Co.; William J. Coulter, vice president, Climax Molybdenum Co.; Charles B. Stainback, Westinghouse Electric Corp.; and Merrill E. Shoup, president, Golden Cycle Corp. Convention welcoming addresses were delivered by a representative of Mayor James Q. Newton, of Denver, and Gov. Daniel V. Thorton of Colorado. Following the formal opening ceremonies a round table discussion dealt with the "Paley Commission Report" formally called Resources for Freedom. Participating in the panel were: Arthur H. Bunker and George R. Brown, members of the Paley commission; H. I. Young, president of the American Mining Congress; Horace M. Albright, president of U. S. Potash Co.; Evan Just, vice president, Cyprus Mines Corp.; Donald H. McLaughlin, president, Homestake Mining Co.; E. H. Snyder, president Combined Metals Reduction Co.; Simon D. Strauss, vice president, American Smelting & Refining Co.; Sen. H. C. Dworshak (R.-Idaho) ; Ira B. Joralemon, consulting engineer; Jess Larson, De fense Materials Procurement Agency; and Andrew Fletcher, president of St. Joseph Lead Co. The panel discussed the report for six hours, adding a great deal of information to the already voluminous amount which has been gathered concerning it. Three objections to the report frequently mentioned were: 1-the multilateral contract; 2-international buffer stocks; 3-international buffer stocks combined with limited quota arrangements. Points on which the panel commended the report were its recommendations for acceleration of topographic and geologic mapping of the United States and Alaska; recognition of the principle of percentage depletion for metals and minerals; and advocation of removal of the present limitations on the expensing of exploration costs applicable to minerals. Technical sessions were held throughout the four days of the meeting. Included in the program were: Progress in Mechanication; Small Mines; Public Lands; Open Pit Mining; Manpower-Labor Relations; Shaft Sinking; Roof Support; Taxation, Gold, Silver and Monetary Policy; Advances in Mill Practices; Special Conference on Strategic Materials; Uranium Mining-Atomic Energy; Prospecting and Exploration; New Metallurgical Processes; and Rock Breaking. About 45 papers and several panels were presented during technical sessions. Formal and informal social functions consumed the remaining time of the convention. Three locations were needed to handle the huge throng attending the Mining Jamboree Monday evening. The only other sponsored social function took place Thursday evening. A great many informal gatherings, chiefly convivial, were held in manufacturers' courtesy rooms-called wet stopes. Field trips included Cripple Creek, Pueblo, Climax -Leadville-Gilman, Colorado Plateau-Grand

Jan 1, 1952

• A new curriculum for combined liberal arts-engineering education has been announced by 6 middle western colleges. The plan involves three years of ;study at one of the 5 liberal arts colleges and two years and a summer of 'engineering work at Case; graduates receive both the A.B. and B.S. degrees. * A new kind of mining is being pioneered by a pest Virginia coal operator which enables four men to produce 800 tons of coal a day. The machine that makes this possible is a giant auger, 5 ft in diam and 100 ft long. Together with the carriage in which it is moved about, it weighs 60 tons. A mobile, but separate power plant of 1000 hp drives it. ' * The Sorel smelter, which will reduce Allard Lake titanium ores, will utilize five 6-electrode rectangular electric furnaces. Furnace hearth size will be about 18x50 ft, and the furnace will smelt about 300 tons of ore per day, producing 100 tons of iron and 130 tons of 70 pct Ti02 slag. The furnaces will be fed continuously by gravity from charge bins, and tap at intervals of 4 hr or more. Slag will be transferred to a high speed casting machine, using cast iron molds, then crushed for shipment to pigment producers. The iron will be treated to reduce its sulphur content to 0.03 to 0.05 pct, and then cast into small pigs or ingots for shipment to steel mills. • Flying of North Africa's first airborne mangetometer survey was completed this month, covering approximately 20,000 sq km in north central Tunisia for Gulf Exploration Co.'s program there. Mapping was completed in less than 30 days. •Yore than 9000 employees of 11 electric companies are enrolled in a 6-hr program that successfully informs workers on the principles of the American economic system. The course is given by the Middle West Service Co. which claims that before and after employee opinion polls show that the course instilled a better understanding of free enterprise and consequently a favorable response to it. •According to a report issued by the American Geological Institute there will be 2865 graduates in geology, including advanced degrees from U.S. colleges in 1950. This supply of geologists compares with requirements for 1278 at the professional and subprofessional levels. The report contains a detailed statistical study of the supply and demand situation for geologists in the 1949-50 period. •The dark cloud of over-production in the bituminous coal industry has a silver lining, according to Natural Resources Notes. Closing marginal mines, mostly strip and truck mines, will result in lower average costs and higher quality of the coal still produced. The producers that will then constitute the industry will be properly equipped and adequately financed to carry forward an intensive campaign to holdd and increase the markets for bituminous coal. •A light, compact, highly reliable blasting machine which does not rely for its effectiveness on the operator's skill is being tested by the Army Engineers. Operating under all climatic conditions from -65 to 125F, the 50-cap machine consists of a springwound do generator and is waterproof.

Jan 8, 1950

By D. F. Kaufman, E. D. Levine, L. R. Aronin

ThE primary slip plane in hcp metal crystals can usually be inferred from the c/a ratio. Basal slip is the primary system at room temperature for zinc, cadmium, magnesium, cobalt, and rhenium, all of which have a c/a ratio greater than 1.60. Zirconium and titanium, on the other hand, each with a c/a ratio of 1.59, deform primarily by (1010) [ll20] prismatic slip. The increasing tendency toward prismatic slip in these materials is generally thought to be associated with the increase in atomic density on (10i0) as c/a is reduced. Beryllium, with a c/a ratio of 1.57, would, according to the above scheme, be expected to exhibit prismatic slip. However, in beryllium of commercial purity, the primary deformation mode at room temperature is basal slip.' The ratio of critical shear stress for prismatic to that for basal slip at room temperature is approximately 5:l. The critical shear stress for prismatic slip decreases quite rapidly with increasing temperature, however, and at 500°C the ratio is approximately unity. The apparently anomalous behavior of beryllium has led to speculation that prismatic slip in this material may be restricted by an anisotropic effect of impurities,' and that, if beryllium could be purified to a sufficient extent, prismatic slip might predominate at room temperature. In such a case, the conditions under which basal cleavage can occur would be suppressed, and enhanced low-temperature ductility might result. With the advent of the floating-zone method for purification of beryllium,3 it has become possible to determine experimentally the effects of purification on basal and prismatic slip. Tensile tests were performed at room temperature on single crystals oriented favorably either for basal or prismatic slip. The crystals had been purified to various extents by zone-refining, employing procedures that have been described elsewhere.4 Relative purities of the crystals were estimated by measuring the ratio of electrical resistance at 4.2"K to that at 298°K. Typical values ranged from 0.40 for commercial-purity material to 7.7 X 10"4 for crystals prepared from vacuum-distilled beryllium and subjected to six zone-refining passes. The purity dependence of the critical shear stresses for basal and prismatic slip is shown in Fig. 1. In the purity range studied, the critical shear stress for basal slip is decreased from 1.4 to 0.14 kg per sq mm. The critical shear stress for prismatic slip is decreased from 6.7 to 5.3 kg per sq mm. The decreases in critical shear stress for each deformation mode, therefore, are approximately equivalent. No purification anisotropy is observed. Because of the absolute magnitudes of the critical stresses, the ratio of the critical shear stress for prismatic slip to that for basal slip actually increases from 5:l to 38:l at the highest purity level studied. It is evident, therefore, that basal slip is the primary deformation mode in high-purity beryllium. This fact must be accounted for in any rationalization of the deformation behavior of hcp metals. This work was performed at Nuclear Metals, Inc., under the sponsorship of the U.S. Atomic Energy Commission.

Jan 1, 1964

This issue presents the first of several articles making up the Bump Symposium, which was held at the 1958 Annual Meeting of AIME. Other Symposium papers will appear in the September issue of Mining Engineering. THE term mountain bump describes the sudden rupture of one or more coal pillars under excessive stress. These bursts occur with varying degrees of violence and sometimes include adjacent strata, especially the bottom rock. Many failures do not develop enough violence to be dangerous and are regarded as an aid to coal production, but severe occurrences are accompanied by tons of flying coal, dense clouds of dust, and excessive gas emission. Such bursts sometimes throw heavy machinery several feet. Major factors to be considered in the problem are the nature of the strata above and below the seam and the depth of cover. Bumps are classified as to their nature, extent of damage, and point of occurrence, e. g., face bump, coal bump, high side bump, low side pavement bump, and district bump. The problem is extremely complex, and it cannot be over-emphasized that preventive measures, if they are to be successful, must be based on a thorough study of conditions in the individual mine. In western Canada, for example, the McGillivray mine, operating in a 9-ft seam at 2000 ft, has experienced severe bumps, but there have been no occurrences in the International mine, which operates in the same coal seam at 2400 ft, where the coal averages 17 ft thick. Differences in conditions governing the operation of these two mines are described in one of the following articles (see page 883). Again, most bumps in coal mines take place along pillar extraction lines, but at the Sunnyside mine in Utah they also occur a long way from active workings and sometimes in virgin territory. It is usual, too, for such failures to occur under fairly deep cover. But at Sunnyside, to cite only one instance, there have been disturbances along the main fault zone under less than 500 ft of cover. All too often conditions favorable to bumps are revealed only by actual mining—at Springhill No. 2 mine, drilling in roof and pavement strata disclosed a 60-ft sandstone bed after operations had started— and where extraction has advanced too far for preventive measures to be incorporated in the mining plan, districts have had to be abandoned. But in an era when premium ores are being rapidly depleted in the face of rising demands, abandonment is not easily justified. There must be systematic planning to eliminate coal mine bumps, insofar as this is possible. As in the Gary district of West Virginia (see page 888) much has been accomplished by planned operations based on known factors in bump control. But even this is not enough. Precise instrumentation is needed to indicate where bumps are likely to occur, and in both the U. S. and Canada stress analysis studies are being carried out with this object in view. These pages reveal the progress being made.

Jan 1, 1959

By Louis Moyd

In the preparation of illustrations to accompany reports of investigations concerning particle shapes of various natural and manufactured materials proposed for use as he aggretates in concrete structures, it was found that stereoscopic views of such materials presented the information much more effectively than the usual two-dimensional photographs. With the whole-hearted cooperation of the Photographic Section, a technique was developed in the petro-graphic laboratory of the Concrete Research Division, U.S. Corps of Engineers, at Clinton, Miss., for simple and rapid preparation of paired stereoscopic photographs and micrographs. The method requires only equipment which is usually part of the general stock of any laboratory, and it is adaptable for all purposes in which three-dimensional views would be found superior to ordinary photographs. Stereoscopic Photographs Where the subject or subjects to be photographed are small but not of microscopic size, an ordinary camera having a ground-glass focusing screen is set up vertically over the field. The material to be photographed is placed on a card, the surface of which is suitable as a background. The card is mounted so that it can be tilted about 5' above and below the horizontal plane, on an axis that coincides with the north-south diameter of the viewing screen. A simple way of achieving this is to mount the card on an object having a convex lower surface. In our laboratory, a watch-glass has been used, with plasticene as the mounting medium. The convex surface should be mounted on a support having a round opening on top and a flat bottom. A roll of tape, a petri dish, a beaker, or any other piece of equipment having the proper shape to give stability to the setup will be satisfactory. The arrangement of the required equipment is shown in Fig 1. For the best results, the subject to be photographed should be mounted in the plane of the axis of rotation. To achieve this effect, the depth of the mass of plasticene between the background and the watch-glass is varied so that when the watch-glass is tilted back and forth on the axis, the subject to be photographed does not appear to move back and forth across the viewing screen. A permanent piece of equipment, embodying an adjustable vertical screw, with lock-nuts, could be made for this purpose. The subject may be illuminated in any manner desired. It was found that the presence of shadows intensifies the effect of depth. In taking the photographs, the concave surface is mounted so that one side of the background card is about 5' below the horizontal, while the axis remains in the horizontal plane. The subject is focused carefully on the ground glass and a picture is taken. The card is then tilted so that the opposite side is about 5' below the horizontal, while the axis still remains in the horizontal plane. The subject is again put in focus and another picture is taken. After both pictures are developed and printed, they are placed side by side in the same orientation and examined with a stereoscopic viewer. If depressions appear where there ought

Jan 1, 1950

By J. Trevor Eyton

This paper addresses some of the financial risk hedging mechanisms available to a corporation contemplating entering the mining business at a significant level of investment. The approach which I have adopted is that of a case study relating to the risk hedging aspects of Brascan Limited's acquisition of 46% of Noranda Inc., formerly Noranda Mines Limited, followed by some remarks on options open to mining companies for reducing financial risks. To appreciate the rationale for Brascan committing over (Can.) $800 million to the natural resource sector, some knowledge of Brascan's parent company, Edper Investments Ltd., is required at the outset. Edper Investments, prior to acquiring a 48% interest in Brascan, controlled Trizec Corporation, one of the world's largest property companies. Edper also was involved in the financial services sector through Continental Bank of Canada and Hees International Corporation. For some considerable time prior to Edper's investment in Brascan, Edper felt a need to provide a better balance to its portfolio of property and financial services. Brascan was rich in cash, relatively unleveraged, and with important existing holdings in natural resources and consumer products. The acquisition of a controlling interest in Brascan provided an opportunity to balance Edper's portfolio of property and financial services with world class acquisitions in natural resources, supplemented by consumer products and financial services. The end result of Brascan's activities over the last few years is illustrated by the following analysis of the book value of Brascan's assets at the end of December 1983. [Can.$% Millions _ Natural Resources81748 Consumer Products 654 38 Financial Services 157 9 Other Operations895$1,717100% ] Source: Brascan 1983 Annual Report. The large emphasis on natural resources came about from the fact that for Canadians operating mainly in Canada, natural resources were attractive. We were impressed by Canada's position in the world natural resource industry as illustrated by some of the following statistics: [CommodityCanadian Estimated Production In 1983 As % of World Total Newsprint 30 Potash 21 Uranium 18 Molybdenum 17 Zinc 17 Pulp 16 Nickel 16 Saw Logs 11 Silver 8 Copper 7 Lead 6 Gas 5 Oil3] Source: U.S. Bureau of Mines and Brascan estimates. In addition, the quality of Canada's reserves looked good to us. The polymetallic nature of many of Canada's orebodies offered a hedge against risk relating to any one metal. Put another way, the by-product metal credits offered a net operating cost advantage with respect to many of the metals mined. Our energy costs also appeared to provide Canada's natural resource industries with an advantage. We considered Canadian mining expertise as amongst the world's finest, as were the exploration equipment and technology developed in Canada. In addition, Canada offered a stable political climate and one we knew. Further, we felt the Canadian Government's future role in the industry would, on balance, be benign, partly because the Canadian content of the mining industry was very high in contrast to the petroleum industry. Granted, we could find cheaper sources of copper; however, to do so would involve venturing into regions where country risks were higher.

Jan 1, 1985

By H. O. Jahns, R. A. Greenkorn, R. E. Haring, L. K. Shallenbarger

This paper is a study of the effects of heterogeneity on flow in an analog of porous media, the Hele-Shaw model. l-3 A set of experiments in heterogeneous Hele-Shaw models showed streamlines through and around heterogeneities of various sizes, shapes and levels. (A level, we define as the ratio between the transmissibility of the heterogeneity and that of the rest of the model.) The heterogeneities were restrictions or expansions of the flow stream analogous to variations in the transmissibility of porous media. The experimental data agreed well with numerical results and with an analytical solution, which we derived for a circular heterogeneity in an infinite field. This study considers the flow-stream distortion due to the shape, size and level of heterogeneities. Size and level are much more important than shape provided the heterogeneity is not long and narrow. Our analytical solution shows that a circular heterogeneity in a large field can be replaced by an equivalent circle of either zero or infinite permeability. The radius of the. equivalent circle is a simple function of size and level of the actual circle. INTRODUCTION With the availability of high-speed computers and numerical procedures to predict reservoir behavior, we are faced with an important question. How much information about the reservoir do we need to justify the cost of the computer in any given case? To answer this question, we have to know how various reservoir parameters affect flow behavior. Reservoir heterogeneity is one of these parameters. In this study, we used a simple, two-dimensional model of porous media, the hele-shaw model, to investigate the effect of heterogeneity on flow behavior. We restricted ourselves to linear, single-phase, steady-state flow in a rectangular field with a single heterogeneity at its center. The analogy between flow in Hele-Shaw models and in porous media is easily verified. Let us first consider a homogeneous Hele-Shaw model with constant plate separation h. (The Hele-Shaw model is constructed by placing two plates, usually glass, very close together and allowing liquid to flow between them. Streamlines are made visible by introducing colored fluid into the space between the plates at a number of points across the model.) We assume a cartesian coordinate system with its origin in the middle between the plates and the z axis directed perpendicular to the glass plates (Fig. 1). The fluid flow is always in a direction parallel to the glass plates and varies from a maximum value to zero in the very small distance from the middle (z = 0) to either plate (z = h/2). For slow motion of an incompressible fluid, neglecting inertia and body forces, we have the viscous flow equation: where p is the fluid pressure, p the viscosity, and v the velocity vector with components u, v and w in the x, y and z directions, respectively. In our rase. and the derivatives of 3 with respect to x and y are small as compared with the derivative in the z direction. Therefore, approximately, with Vp and v being two-dimensional vectors in the x-y plane. Two integrations with respect to z give

Jan 1, 1965

F. E. Thompson, Pottstown, Pa.: If Mr. Webster's endeavor to open up the subject of cast-iron should prove as prolific of results as did the discussion on " The Physics of Steel," he must certainly feel richly repaid. His remarks concerning the secretiveness and conservatism of the average foundry-man recall to my mind an instance of a blast-furnace which, for the last year, has been producing much foundry-iron. The manager, believing that much more of his product could be used in the foundry than the foundry-men were will-

Jan 1, 1896

By R. J. Blackwell, J. R. Rayne, W. M. Terry

This paper presents results of an experimental investigation of factors that control the efficiency with which oil is displaced from porous media by a miscible fluid. The study was made to elucidate the relevant processes both on microscopic level (within individual or between neighboring pore spaces) and on macroscopic level (within a large sand body). Mixing of miscible fluids on the microscopic level was studied in sand-packed tubes. It was found that molecular diffusion is the dominant dispersion mechanism for reservoir conditions of rate, length and pore sizes. Macroscopic channeling was studied for various mobility ratios in reservoir mode1s scaled to relate viscous, gravitational, and diffusional forces. The formation of channels was due to viscous firzgering, gravity segregation and variations in permeability. With adverse mobility ratios, it was found for reservoirs of realistic widths that diffusion will not be effective in preventing the formation and growth of fingers, even in homogeneous sands. At sufficiently low rates channeling was eliminated by gravity segregation in tilted reservoirs. The dependence of recovery on mobility ratio, length-to-width ratio, flow rate and angle of dip is presented. INTRODUCTION Oil recovery by solvent flooding is finding increasing application in the field. While the process promises high recoveries from the region swept by solvent, under adverse conditions only a small fraction of the reservoir volume may be swept at the time solvent breaks through to the producing well. Further, the high cost of the solvent encourages its use only as a bank whose size must be kept at a minimum. Thus, two important questions arise: (1) what fraction of the reservoir can be swept, practically, by solvent? and (2) what is the minimum size solvent bank that can be used to carry out the displacement? The answers to these questions require knowledge of both macroscopic channeling processes and microscopic mixing processes. The studies described here were carried out to gain this knowledge. Microscopic mechanisms which cause mixing will be discussed first, because an understanding of these mechanisms is necessary for proper interpretation of the experimental work on channeling described later. INVESTIGATION OF MICROSCOPIC DISPLACEMENT PHENOMENA FOR MISCIBLE FLUIDS The questions to be resolved concerning microscopic displacement behavior are: (1) how much mixing occurs between oil and solvent in the direction of flow? and (2) does the solvent completely flush the oil from the pore spaces in the region invaded by solvent? The classic work of Sir Geoffrey Taylor' and its extension by Aris2 have shown that it is possible from theory to describe the amount of mixing in single straight capillaries when solvent displaces a fluid of equal viscosity and equal density. Aris showed that the lengths of the mixing zones, 81, corresponding to the 10 per cent and 90 per cent concentration levels of the solvent can be calculated from the formula, 81 = 3.62 \/Kt, .........(1) where 8l is the length of the mixing zone, cm; t is the time, sec; and K is the effective dispersion coefficient, cm2/sec, given by K/D = 1 + (1/48) (au/D)2......(2) where a is the radius of the capillary, cm; D is the molecular diffusivity, cm2/sec; u is the average velocity

By R. M. Hays

At the present time, oxidized taconite and semitaconite are not processed on the Mesabi Range except in vely limited tonnages. However, it is estimated that by 1990 shipments from the Range will approach 75 million tons of iron ore per year. To achieve this and future production, either additional magnetic taconite reserves must be proven or oxidized taconite and semitaconite must be processed. One beneficiation process which uses high-intensity magnetic separation and flotation for concentrating oxidized taconite and semitaconite is described. Detailed metallurgical results are given and an economic evaluation of the process is presented with comparison to other processes. The results of the present study indicate that the high-intensity magnetic separation and flotation process appears nearer commercialization for the processing of Mesabi Range semitaconite. The disadvantages of the process are nearer solution and are being solved by well-defined programs. During the twentieth century, the Mesabi Range in northeastern Minnesota has been the main source of iron ore for American steel mills. The Mesabi Range, which is about 120 miles long and up to 3 miles wide, is the pre-glacial outcrop of a group of sedimentary iron-bearing rocks known as the Biwabik formation. The formation, which is 200 to 650 ft thick and dips 50 to 10° to the southeast, is divided into four stratigraphic units. From top to bottom, these are the Upper Slaty, Upper Cherty, Lower Slaty, and Lower Cherty. Since its deposition in Precambrian time, most of the original Biwabik formation has been subject to alteration. Thus, although the iron content of the original formation was quite uniform, major variations in mineral composition now occur both laterally along the length of the formation and vertically within its units. Details of the mineralogical variations in the Biwabik formation have been described by Wolff,' Gruner, 2 and White.3 The unaltered portion of the Biwabik formation includes the magnetite taconites which are being commercially processed by Reserve Mining, Erie Mining, Eveleth Taconite, National Steel, Butler Taconite, and U.S. Steel. These six operations have a capacity of 32 million tons of iron ore pellets per year and have produced approximately 170milion tons of pellets since 1956. Capacity for another 30 million tons of pellets per year is being considered by the industry for the next ten years. This increase will be obtained by constructing new plants and by expanding present operations. Thus, by ! 978, it is estimated that approximately 60 million tons of iron ore vellets will be produced each vear from the magnetite taconites of the Biwabik formation. The portions of the Biwabik formation subjected to intense alteration resulted in direct shipping, wash, and heavy-media ores. Since 1892, 1.8 billion tons of direct shipping ore have been shipped from the Mesabi Range. This ore has been the major source of iron ore for American steel mills. In 1967, about 11 million tons of direct shipping ore and 13 million tons of wash, jig, heavy-media, and other gravity concentrates were shipped from the Mesabi Range.4 These products generally have poorer physical and chemical properties than the iron pellets produced from magnetic taconite. It is predicted that the shipment of these materials will decrease from 24 million tons in 1967 to 6 to 8 million tons in 1975.' This decrease will

Jan 1, 1970

By E. E. Wahlstrom

The Harold D. Roberts tunnel, in Summit and Park Counties, Colorado, is a concrete-lined pressure tunnel finished to a circular cross section of 10.25 ft diam. The tunnel is 23.3 miles long and is designed to transport water for domestic use by the City and County of Denver from reservoir storage at Dillon to a tributary of the South Platte River at Grant. The tunnel passes beneath the Continental Divide at the crest of the Front Range and intersects rocks of diverse types and origins and geologic structures of extreme complexity. Geologic investigations preceded and were continued during construction and contributed materially to the safe and economical advance of the tunnel headings. Geological data obtained and evaluated during construction include plans and sections on a scale of 1 in. to 50 ft, rock temperatures at 2000-ft intervals, records of progress related to geology, records of ground-water conditions, summaries of the use of steel and timber supports, and summaries of grouting operations. Adverse conditions required use of steel and timber supports for 71.74% of the total length of the tunnel. Most supported sections reached a condition of equilibrium within a few hours to a few months after being excavated, and the concrete lining serves chiefly to provide an additional safety factor and to provide necessary hydraulic characteristics to the tunnel. Efforts to attain a maximum rate of advance of the tunnel headings necessitated use of more steel and timber supports than would have been required in a less accelerated tunneling operation. This report summarizes geologic investigations prior to and during construction of the Harold D. Roberts tunnel in Summit and Park Counties, Colorado (Fig. 1) and analyzes the tunneling operation as it is related to the various kinds of geologic conditions encountered in the tunnel. The tunnel is concrete lined to a circular cross section of 10.25 ft and is designed to transport 788 sec ft from reservoir storage at Dillon 23.3 miles along a dog-leg course passing under the Continental Divide to outlet and control works at Grant, Colo. The primary purpose of the tunnel is to augment the domestic water supply of the City and County of Denver, but plans include provisions for power generation at the outlet portal at a future date. Location of the tunnel was made after extensive preliminary geologic and engineering investigations by the Board of Water Commissioners of the City and County of Denver, and the U.S. Bureau of Reclamation. In 1946 the Board of Water Commissioners started tunnel driving on a limited scale from the east (outlet) portal and by 1955 had completed 9986 ft of tunnel. An urgent need for additional sources of water supply required acceleration of the tunneling operation, and, in 1955, Tipton and Kalmbach Inc. of Denver were assigned the responsibility of reviewing the project and preparing plans and specifications for the tunnel and appurtenant features. Additional geologic studies were made, and the geology of several possible alternate tunnel lines were compared before determining that the tunnel should be driven along the course previously adopted by the Board of Water Commissioners. On July 12, 1956, a contract for construction was awarded to Blue River Constructors Inc., a combine

Jan 1, 1962

By L. G. Jones

An approximate method of calculation is developed in this paper which allows duplication of radial unsteady-state gas flow computer results where Darcy's law applies, such as those reported by Aronofsky and Jenkins and Bruce, Peaceman, Rachford, and Rice. Moreover, the new calculation method can be used to obtain results for radial unsteady-state gas flow obeying the quadratic flow law proposed by Duwez and Green. Means are discussed for predicting well behavior at single or superimposed flow rates in finite or infinite reservoirs, determining reservoir rock properties from well-test data, reproducing and interpreting back-pressure test data, and determining the radial extent and reserves of gas reservoirs from well-test data. Example calculations are presented for gas flow following both Darcy's law and the quadratic flow law. INTRODUCTION Since the publishing of U. S. Bureau of Mines Monograph 7,1 most gas-well testing methods have been based on the equation q = y (pl2 - pw2)a, where q = production rate, pl and pw, are formation pressure and sandface pressure, respectively, and y and a are constants to be obtained from test data. These methods, used for predicting both deliverability and "open-flow" capacity of gas wells, have been useful and accurate in many cases but unsatisfactory in others. Even at best, however, they do not supply information about the formation or lead to an understanding of nonsteady-state gas flow in porous media. Many theoretically based studies2-4 of gas flow obeying Darcy's law have been made.' Since the partial differential equations which result from combining Darcy's flow law with the continuity equation are nonlinear, most of the published research consists of either numerical solutions or analytical solutions for linear approximating equations. Such solutions have been of limited value in field work due to their unhandy form and their failure to correlate most field data. There is evidence5,6 which indicates that Darcy's law is inadequate to describe gas flow at some flow rates of practical interest. A quadratic flow law: which reduces to Darcy's law at low rates, is more successful in accounting for experimentally observed behavior. This flow law has been applied successfully to a few hypothetical reservoir cases in work which has recently been published.10 However, these numerical solutions of the equations involved have been successful only on a special analog computer. Routine applications to field cases would be awkward and have not been attempted. The present paper describes an approximate method for computing nonsteady-state gas flow solutions which has been completely successful in predicting the results for both Darcy flow and quadratic flow obtained by elaborate numerical methods. The new calculation method allows determination of the observable variables in gas-well testing at constant rates. It is similar to the scheme of using a succession of steady states suggested by Muskat7 in that it makes use of steady-state and material-balance equations. It also is similar to the work of Aronofsky and Jenkins2 in that the new method includes Aronofsky and Jenkins' stabilized flow equation as a special case. It improves upon both of these calculation schemes in that it accurately describes all portions of reservoir history and suggests means of determining reservoir rock properties from well-test data. This paper deals only with production from a reservoir, in which case the rate is defined as being negative. The reservoir studied here is a homogeneous, disk-shaped porous body of uniform thickness, with all boundaries sealed except the inner radial boundary of the well.

By J. S. Aronofsky, A. S. Lee

Results and experimental procedures are presented covering a preliminary laboratory investigation of the compaction of reservoir rocks and its effect on porosity and permeability. "Egective" compressibility was measured by subjecting core material to an overburden load and reducing pore pressure. Preliminary results showed relatively high effective compressibilities which varied from 1.8 X 10-' pore volume/pore volume /lb / in.~ for limestones to 7.7 X 10-6 pore volume/pore volume/lb/in.' and even higher for sandstone reservoirs conmining large amounts of fine clay material. It appeared from the initiol results that the effective compressibility of the material could not be correlated directly with porosity and that other factors, such as the amount of clay material present might have to be taken into account. The work to date has indicated that effective compressibility factors will have to be measured for the particular reservoirs heing studied. Oil permeability of core materid was measured under various effective overburden loads from 0 to 8,000 psi. It has been found than reduction of permeability in clean sands was relatively small and the results agreed with published data. For sandstones containing large amounts of clay, preliminary tests indicated very large reductions in permeability with increasing effective overburden pressures. It is planned to continue laboratory work on this phase because of its importance in low-permeability reservoirs. INTRODUCTION During the past few years increasing interest has been shown in the differences obtained between conventional laboratory measurements on cores at atmospheric pressure and those when reservoir conditions are simulated. Since these data are used as the basis for calculations and the prediction of reservoir behavior, their evaluation under reservoir conditions is most important. The effect of the compressibility of the rock upon the reservoir behavior, as fluid pressure declines, is an important subject which has been insufficiently studied. The total pressure on any plane through a reservoir is the result of the grain-to-grain rock pressure due to the overburden and the pressure of the interstitial fluid on the plane. As reservoir fluid pressure declines, the net load on the rock phase must increase, and the result is the compaction of the rock structure which produces a decrease in pore volume. Simultaneously, the rock grains expand into the pore space as the fluid pressure declines. The result of these two effects is defined as the effective compressibility of the rock and is expressed in units of pore volume/original pore volume/psi. Work was started by Imperial in 1954 on the effects of pressure on the porosity of reservoir rock. These studies established effective compressibility factors for some limestones and dolomites. Since that time effective compressibilities have been obtained for other Canadian and some U. S. reservoir rocks. In conjunction with the study of porosity, an investigation has also been initiated to determine the effects of pressure on permeability. One outcrop and several reservoir sandstone cores have been tested to date. ROCK COMPRESSIBILITY Apparatus anD Procedure A diagram of the apparatus used in these studies is shown in Fig. 1. The equipment was constructed so that a uniform pressure, equivalent to the reservoir overburden, could be applied by the oil pump and transmitted on all sides of the core through the Lucite sheath. The sample was completely saturated by circulating brine through the mounted core. Pressure in

By L. E. Mannion

Sodium carbonate or soda ash is one of the two principal alkalis of commerce, the other being sodium hydroxide. The use of sodium carbonate is recorded in ancient Egypt, where naturally occurring brines or solid salts were the source of impure soda for glass manufacture. In early modern times until the 19th century, soda ash, as well as potash, was recovered mainly by leaching the ashes of plants. In Europe, during the 18th century, a brown lye for laundering was produced by pouring hot water over burned seaweed. One of the better grades of this kind of material was a product from Spain termed "barilla," which analyzed 24 to 30% Na2CO3 (Hou, 1942). The birth of the modern chemical industry may be said to start with the LeBlanc process for making sodium carbonate from salt, sulfuric acid, and lime. Although invented in France in 1691, England made the most extensive use of it. In 1863 Ernst Solvay devised the ammonia-soda or Solvay process, and by 1874 the first of many Solvay plants was built. These were to supplant the LeBlanc method and dominate soda-ash production until the present time. Although of lesser importance, natural sodium carbonate deposits continued to support operations in various parts of the world. Shortages during World War II stimulated production of "natural" soda ash, particularly at Searles Lake, Calif., and at Lake Magadi, Kenya. Within the last ten years in the United States, a notable surge in production has occurred based on the vast trona deposits in Wyoming. Table 1 lists some principal producers of natural soda ash and the character of the source deposits. Uses and Specifications Sodium carbonate (Na2CO3) is a white, crystalline hygroscopic powder, an aqueous solution of which is strongly alkaline. Most commercial soda ash is a product that must meet high standards of purity and uniformity. Two forms are commonly sold: a fine, powdery material called light ash, and a more coarsely crystalline dense ash; these weigh 32 to 39 and 60 to 66 lb per cu ft, respectively. Dense ash may be produced by adding a little calcium chloride to the sodium carbonate solution to control crystal size and prevent agglomeration. Specifications and typical analyses of light and dense ash in the United States are shown in [Table 2]. Sizing generally achieves about 75% of light ash and 90% of dense ash between +30 and -100 U.S. screen mesh. World use of sodium carbonate now exceeds 24 million tpy, of which the United States produces more than 7 million tons. The market for soda ash is dominated by the glass industry, which uses almost half the supply. Inorganic chemicals, including phosphates and silicates, consume about 30%. Other principal uses include the manufacture of soaps and detergents, petroleum, caustic soda, organic chemicals, and paper; and nonferrous and ferrous metallurgy. [Fig. 1] shows the percent consumption by use of soda ash in Europe, Japan, and the United States. The price of soda ash in the United States remained relatively stable during the 1960s at about $33 per ton; in 1972 it reached $35 per ton. The price in 1975 was $42 per ton. Geology Mineralogy [Table 3] is a list of the more common sodium carbonate-bearing minerals. Anhydrous sodium carbonate is almost never found in nature; only natron, thermonatrite, and trona have been used directly as sources of sodium carbonate. At Searles Lake, burkeite and hanksite are important in the reconstitution of

Jan 1, 1975

By D. H. Forbes, I. B. Cadoff, H. M. Manasevit

Single-crystal silicon films have been obtained on several natural crystal faces of BeO using the thermal decomposition of silane and the hydrogen reduction of silicon tetrachloride. From an analysis of X-ray diffraction data it was determined that the following film-substrate orientation relationships exist: (111) Si 11(0001) BeO, (111) Si 11(1011) BeO, (100) Si 11f1010) BeO. and (110) Si 11(1010) BeO. An analysis of the crystallographic relationships between the film and substrate indicates that a match occurs between the silicon atoms in the film and the berylliuni ion sites in the substrate. This correspondence is in agreement with previous work on the deposition of silicon on sapphire.1-3 silicon on spinel: and tungsten on sapphire. The high resistivity, thermal conductivity, and radiation resistance of beryllium oxide make it an important substrate material for the deposition of thin films of silicon or other semiconductors. The importance of semiconductor-insulator composites to the field of integrated microelectronics has been well-documented in the case of silicon on sapphire.'-' The successful growth of large-area single-crystal silicon on an insulator, sapphire, was first reported in 1963.1,2 Since that time studies have been made of the crystallographic relationships between the deposited film and the sapphire substrate. On the basis of the observed relationships, a model has been proposed relative to the crystallographic requirements for epitaxy between the semiconductor film and the metal oxide substrate.3 It is proposed in this model that epitaxy is the result of a match between atoms of the deposited film and the metal ion sites of the substrate plane. This match is not limited to a 1 to 1 correspondence between sites in the two lattices. Sub- sequent studies of the deposition of silicon on spinel4 and tungsten on sapphire5 have yielded data which support this model. On the basis of this previous work it was felt that silicon on BeO should also fit this model. In the present study, the choice of substrate orientation was limited to the natural faces of synthetic BeO crystals. Of the faces available, the (0001) plane seemed the most favorable for epitaxial growth. EXPERIMENTAL PROCEDURE Silicon films were deposited from silane and silicon tetrachloride in the apparatus shown in Fig. 1. The hydrogen ambient gas was purified by passage either through a DEOXO unit, molecular sieves, and a liquid nitrogen cold trap, or through a heated Pd-Ag thimble. The gas was then mixed with the silicon source materiai and passed over the substrate which lies on a spacer above an inductively heated silicon pedestal. The following deposition conditions were used: a hydrogen flow rate of 3 liters per min with silane and silicon tetrachloride concentrations of 0.2 mole pct and a pedestal temperature of 1175" * 5°C as observed with an optical pyrometer assuming an emissivity of 1.0. Deposition was preceded by a 15- to 30-min hydrogen-etch period conducted at a temperature of 1275°C (obs). Adherent silicon films from 2 to 10 u were grown on the various BeO faces. The substrates were synthetically grown beryllium oxide single crystals ranging from 5 to 10 mm in

Jan 1, 1967

By J. W. Spretnak, H. R. Ogden, A. G. Imgram

The effect of working reduction, stress-relief annealing, and recrystallized grain size on the tensile and notch-tensile properties of molybdenum and Mo-0.5 Ti was studied. It was found that increasing the amount of working reduction increased the strength and reduced the tendencies toward brittle behavior. Stress -relief annealing did not alter the strength but lowered the transition temperature. Increasing the recrystallized grain size lowered the strength and raised the transition temperature. The notch -unnotch strength ratio began to decrease with decreasing temperature at temperatures below those at which the notched ductility fell to zero for all of the micro structural conditions investigated. BEING bcc metals, the refractory metals in Groups Va and VIa (V, Cb, Ta, and Cr, Mo, W) undergo a ductile-to-brittle transition and exhibit brittle fractures at low temperatures. Previous research has shown that molybdenum and tungsten not only have higher transition temperatures than columbium and tantalum but show a marked sensitivity of transition temperature to structure as well.' Recrystallized materials become brittle at considerably higher temperatures than is the case with material having a worked or fibered micro-structure. Evidence of structure sensitivity has also been detected in columbium- and tantalum-base alloys containing appreciable amounts of molybdenum or tungsten such as F-48 (Cb-15W-5Mo-1Zr) and Ta-10w.2 This paper describes the results of research conducted to further evaluate the effects of metal. lurgical structure on the low-temperature tensile and notch-tensile properties of molybdenum and Mo-0.5Ti. The effects of rolling reduction, stress-relief annealing temperature, and recrystallized grain size were studied. Considerable progress has been made in explaining the fundamental mechanisms of the flow and fracture characteristics of bcc metals. Several of these concepts are helpful in interpreting the data obtained in this study. Brittle fractures occur when the highly temperature-sensitive yield strength equals, or exceeds, the temperature-insensitive brittle cleavage fracture stress.3 The rapid increase in yield strength with decreasing temperature, characteristic of all bcc metals, is therefore of primary concern. petch4 has described the yield strength, ay, (stress required to initiate plastic flow) of bcc metals by the expression where ai is the frictional resistance to slip, Ky is a constant numerically equal to the slope of a a vs curve, and 2d is the average grain diamefer. The oi term itself is thought to be composed of a temperature-insensitive component dependent on impurity content and second-phase particles and a temperature-sensitive component dependent on the Pierls-Nabarro force.5 This equation has also been demonstrated to hold for the flow stress (stress required to maintain continued plastic flow).' Brittle-crack nucleation is thought to occur by the coalescence of dislocations piled up in slip planes at sessile dislocations or twin and grain boundaries.7,8 These proposed mechanisms require that a large number of dislocations be released into the active slip planes so that sufficiently large stress concentrations are produced at the dislocation pile-ups to enable coalescence into a micro-crack. Therefore, at least a microscopic amount of slip must occur in some grains to facilitate initial crack formation. Cottrell 7 has predicted that brittle fracture, nucleated by slip, should occur when where ty is the shear yield strength, ß = 1 in un- notched xension, u is the shear modulus, and y is the effective surface energy. This equation incorporates the Griffith concept of fracture that a crack will propagate if the decrease in strain energy equals or exceeds the increase in surface energy.

Jan 1, 1964

By William E. Galloway, Gary E. Smith, Christopher D. Henry

INTRODUCTION Studies relating to the Oakville aquifer as presented in this report and by Henry et al. (this volume), are part of a larger , comprehensive examination of regional and local stratigraphic, hydro- geologic, and geochemical parameters within the Oakville Formation recently completed by the Bureau of Economic Geology. The Oakville serves both as a major Gulf Coast aquifer and an important uranium host (Fig. 1). Regional and local controls on hydrodynamics and hydrochemistry have important implications for hydrochemical exploration, the establishment of premining baseline parameters, and the design of post mining restoration procedures. Investigations of the Oakville Formation were de- signed for the purpose of determining the response of the aquifer to uranium extraction and were funded by the U.S. Environmental Protection Agency under grant numbers R-805357-01 and R-805357-02. Physical Stratigraphy Physical stratigraphic analysis shows that the Oakville Sandstone, a lithostratigraphic unit deposited by a major coastal-plain f luvial system, consists of five mappable, component depositional elements (Figs. 2 and 3). Each major component element is the product of one or more rivers traversing the Miocene coastal plain. In turn, each river possessed a unique sediment load, discharge characteristics, and source terrane. Consequently, each element exhibits distinguishing geometric and compositional parameters that potentially influence the flux and hydrochemical evolution of contained ground waters (Galloway, Henry, and Smith, in preparation). The Oakvil le Sandstone, as mapped and interpreted as a lithostratigraphic unit in the study area, corresponds closely to the Jasper aquifer system, an equally well-defined Gulf Coastal Plain hydrogeologic unit (Baker, 1978). Principal surrounding hydrogeologic units include the underlying Catahoula confining system, the Jasper aquifer system itself, and the overlying Burkeville confining system (lower Fleming mudstones). The Evangeline aquifer system includes much of the Fleming Formation. The Jasper aquifer system is bounded above and below by relatively less transmissive, finer grained confining systems, although absolute transmissivity varies greatly within both the Jasper and confining units. Lithostratigraphic and hydrogeologic boundaries coincide almost exactly within the Hebbronville flu- vial axis (Fig. 3). Correspondence of geologic and hydrostratigraphic boundaries is good within the George West axis; farther to the northeast the correlation of the Jasper aquifer with units mapped as Oakville Formation diverges to varying degrees. A complete analysis of these trends and a study of the depositional facies comprising the Oakvillle are included in Galloway, Henry, and Smith (in preparation). REGIONAL HYDROGEOLOGY The stratigraphic, compositional, and structural framework of the Oakville Formation and the mosaic of interrelated depositional facies that are combined in this framework play a key role in defining the hydrogeology of the Oakville. Furthermore, a combination of climate and hydrogeology, as well as past and present socioeconomic conditions, has determined water and land use patterns within the region of Oakville outcrop. Utilization of Oakville Ground Water Stratigraphic control for determining which wells produce ground water from the Oakville aquifer was established by preparing structural and isopach maps of the Oakville Formation using regional electric log cross sections. Surface elevation and screened interval data (or total well depth, if open-hole completion techniques were used) from each water well, combined with the stratigraphic data, allowed accurate assessment of the producing aquifer for individual wells. Information on well and screen depths, water levels, and usage of water from

Jan 1, 1980

By F. V. Lenel, G. S. Ansell

The creep behavior of an aluminum -aluminum oxide alloy, A T 400, fabricated by compacting an atomized aluminum powder, extruding the compact, cold working, and recrystallizing the extrusion, was investigated. This alloy has a dispersion spacing of 1 . Previously Ansell and Weertman determined the steady-state creep rate of a recrystallized aluminum-aluminum oxide SAP-type alloy, MD 2100, with a 0.48-p dispersion. This alloy was found to have a steady-state creep rate of less than 10-6 min-1, too small to be measured accurately. By employing a more sensitive method of measuring creep rate, it was possible to determine the steady-state creep rate of AT 400 alloy, which is somewhat faster than that of MD 2100, at a series of temperature and applied stresses. Above a stress of Burger's vector, A = dispersion spacing, u = is a shear modulus) the rate follms an equation of the type: Steady-state creep rate K = A exp. where A is a constant, Q is the activation energy for sey-diffusion, and is the applied stress. At stresses lower than the steady-state creep rate drops off sharply and is no longer in agreement with this equation. This behavior is in good agreement with that predicted by the model for dispersion strengthened alloys in the stress and temperature range investigated. The absolute value of the creep rate is, however, four orders of magnitude lower than that predicted if the normal number of active dislocation sources are assumed to be present. It is concluded that in this SAP-type alloy the usual three-dimensional dislocation network is not present. Instead, short dislocation segments extending from one dispersed particle and terminating at a neighboring particle may act as dislocation sources. With this provision the creep model proposed by Ansell and Weertman reasonably accounts for the creep behavior which was observed. RECENTLY, Ansell and Weertman1 proposed a dislocation model from which they derived creep equations to describe the steady-state creep behavior of dispersion - strengthened alloys. Following Schoeck,2 they assumed that the rate controlling process for steady-state creep is the climb of dislocations over the dispersed particles. In order to verify their creep model, Ansell and Weertman determined the steady-state creep behavior of an aluminum-aluminum oxide SAP-type alloy, MD-2100, in both the fine-grained as-extruded, and coarse-grained recrystallized conditions. Since the grain size was the order of the dispersion-spacing, their model was not applicable for the as-ex- truded alloy. It should, however, be applicable to the coarse-grained recrystallized alloy. The results they obtained were rather unexpected. No meas-ureable steady-state creep was observed for the recrystallized MD-2100 alloy. If their model is correct, and if the second-phase particles act solely to hinder dislocation motion, then some measureable steady-state creep would have been expected. On this basis, they postulated that, in this SAP-type alloy, the main effect of the fine dispersion was to inactivate dislocation sources rather than to hinder the movement of dislocations. In order to understand more fully this unusual creep behavior, and to determine the validity of the model proposed,' the steady-state creep behavior of an aluminum-aluminum oxide recrystallized SAP-type alloy, with a somewhat coarser dispersion spacing, was investigated. The results of this study are presented in this paper.

Jan 1, 1962

By R. A. Greenkorn, R. C. Smith

Hele-Shaw cells are used to model creeping flow through porous media (where Darcy&apos;s law is valid). The effects of inertia on flow about obstructions in a Hele-Shaw cell can be calculated by a perturbation method if one can determine a solution to Laplace&apos;s equation. Results of a computer solution for flow about circular, square and elliptical obstructions are presented. These results show that or a modified Reynolds number of less than I, the inertia terms are small; and for values of less than 3, the average streamline predicts the ideal flow. Therefore, the analogy might be used for studying flow in porous media up to a modified Reynolds number of at least 3. INTRODUCTION The nature of fluid flow in porous media is of interest in the fields of soil mechanics, ground water flow, petroleum production, filtration and flow in packed beds. Because it is very difficult to study the phenomenological behavior of flow in porous media, homologs and analogs are used to study flow characteristics. A Hele-Shaw model, made of two closely spaced plates - usually glass — is often used as an analogy to two-dimensional flow in porous media. Hele-Shaw2 showed experimentally that the streamline configuration for creeping flow around an obstacle located between two closely spaced parallel plates is the same as for two-dimensional, ideal flow about the same obstacle. Stokes7 verified these observations mathematically. The usual equation of motion for flow in porous media is Darcy&apos;s law. The form of the mathematical statement of Darcy&apos;s law is identical, within a multiplicative constant, to the expression for the average velocity over the plate gap in the plane of a Hele-Shaw model. These models may be used to describe flow in both homogeneous and heterogeneous porous media. In the mathematical proof of the Hele-Shaw analogy it is assumed that the convective terms in the Navier-Stokes equations are negligible and that the equations of motion degenerate to Laplace&apos;s equation, with pressure the dependent variable. Whenever a Hele-Shaw model is used as an analogy to flow in porous media, .the validity of this assumption is in question. Riegels5 showed that if convection is not neglected, the velocity distribution around a cylindrical obstruction in the flow field depends on a Reynolds number, the plate spacing, and a dimension characteristic of the obstacle. Riegels&apos; solution, a perturbation solution, uses the boundary condition that the flow rate into the obstacle averaged over the plate gap at any point on the obstacle is zero. The method requires that a solution to Poisson&apos;s equation for the perturbation pressure be found. Riegels evaluated this solution for the case of the cylindrical obstruction. His method may be simplified by eliminating the need for solving Poisson&apos;s equation for the perturbation pressure. Instead, an analytic expression for the perturbation pressure gradient is obtained (valid for arbitrary shapes) and used to eliminate pressure from the equations for the perturbation velocities. The results show, for symmetrical shapes, that if NRe < 1, the convective acceleration terms are small, and that the average velocities represent ideal flow up to at least NRe = 3, where: L is a characteristic dimension of the obstacle perpendicular to flow, b is the plate spacing, u is viscosity, v, is velocity of approach and p is density. THEORY* Incompressible fluid flow through a Hele-Shaw model and a porous medium are analogous because the equations for the average velocity components in two-dimensional flow are identical if:

Jan 1, 1970