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By Yi Luo, Jian Yang, Hua Jiang

Surface subsidence induced by underground coal mining activities, especially longwall mining operations, can cause various problems to interstate highways ranging from structural integrity, drivability to safety. Accurate subsidence prediction, correct influence assessments, and effective mitigation measures are the keys to ensure the safety of highway traffic. The surface subsidence prediction program, called comprehensive and integrated subsidence prediction model (CISPM), was developed based on the principle of influence function method and has been validated with a large number of subsidence measurements. In this paper, the techniques to use the predicted final surface movements and deformations for assessing their influence on the integrity and functionality of highway structures (e.g., road surface pavements, concrete slabs, transverse joints, etc.), vehicle dynamics, comfort level, and drivability are presented. A number of mitigation measures to protect the highway structures and to ensure traffic safety are proposed along with an application case.

By Manuel Pérez-Tello, María M. Salazar-Campoy, Óscar Rodríguez-Hoyos

A common problem during the operation of the Kaldo furnace is the formation of accretions and blockage of the injection lance due to splashing. This typically occurs during the processing of anodic slimes for the production of doré. In this work, a 1:7 scale model of the Kaldo furnace currently in operation at Metalúrgica de Cobre, Mexico, was built in transparent acrylic plastic. Water and colored oil were used to represent the doré and slag, respectively. The model was used to determine the maximum penetration distance of the injection lance under a variety of experimental conditions at room temperature. Input variables included two types of oil, rotation speed, inclination angle, and the fraction of the vessel volume occupied by the liquids (φ). A mathematical correlation for the dimensionless maximum penetration distance of the injection lance d ∗ as a function of φ, the Froude number of the liquids in the vessel (Fr), the Reynolds number of the gas jet (Re), and the oil-to-water kinematic viscosity ratio (η)was developed. Reasonable agreement between the calculated and experimental values was obtained. Digital images showed that splashing was caused by the turbulent gas jet impinging upon the free surface of the liquids, whereas the cascade effect was negligible. The results of this work have been useful to improve the operation of an industrial Kaldo reactor.

By Yang Ge, Yuzhu Pan, Mansheng Chu, Jue Tang, Cong Feng, Lihua Gao, Zhenggen Liu

The low-grade nickel laterite ore used in this study was successfully upgraded through selective reduction by industrialized rotary kilns followed by magnetic separation. The results indicated that both iron metallization extent and nickel metallization extent showed obvious linear correlation. In addition, rational roasting temperature and prolonged grinding time could enhance the overall fineness level of reduced laterite nickel ore. The moderate roasting temperature at a range of 1100–1150 °C should be suitable enough to a conversion of total nickel to metallic nickel and partly inhibited the conversion of total iron. Ultimately, the obtained ferronickel alloy with a nickel content of 7.32% and an iron content of 84.15% was prepared successfully from the lowgrade nickel laterite ore containing a nickel content of 1.92%. Nickel and iron recoveries were 96.72 and 89.08% respectively, which achieved the highly efficient recovery and utilization of iron and nickel from low-grade laterite nickel ore

By Chen-Luh Lin, Jan D. Miller, Sindhoora Puvvada

The depletion of good-quality phosphate resources, coupled by a need to supply acceptable products at a rapid rate, while meeting the specifications of fertilizer chemical plants, is of significant concern to the phosphate industry, particularly the production of concentrates with tolerable dolomite content. Tremendous efforts have been made in the past to develop an efficient and economical process that can effectively reject dolomite. However, dolomite removal continues to plague the phosphate industry due to several process-specific challenges being encountered with various beneficiation technologies. In this regard, detailed characterization of phosphate reserves is crucial to understand textural limitations and help design appropriate strategies to achieve quality products. In this paper, the development of procedures to establish dolomite rejection/francolite recovery curves for gravity preconcentration of pebble phosphate from Central Florida is described using high-speed X-ray computed tomography (HSXCT). The developed procedure is illustrated with the analysis of high MgO pebble phosphate samples (10.39 mm × 1 mm) received from South Pasture and Four Corners locations operated by the Mosaic company. The rejection/recovery curves were calculated for each sample for five different particle size classes to observe the trend in rejection/recovery with variation in particle size. In order to evaluate the reliability of the HSXCT data, analysis using the high-resolution X-ray computed tomography (HRXCT) technique was performed on one particular sample (FC-3-1). Finally, to assess the separation efficiency, the HSXCT and HRXCT rejection/recovery curves obtained were compared to experimental gravity preconcentration results reported in the literature. In this way, liberation limitations for gravity preconcentration in the processing of pebble phosphate have been established based on X-ray computed tomography analysis.

By Simón D. Strauss

El presente texto señala los factores que han influenciado el comportamiento de los precios de los metales no ferrosos desde el fin de la Segunda Guerra Mundial. Tras la finalización de este periodo bélico, se dio una inflación que afectó los niveles de precios de todos los productos, alterándose los costos de cuatro de los ocho metales no-ferrosos en la industria minera peruana. Para ello, se realiza un estudio comparativo de los precios estipulados durante la Segunda Guerra Mundial con los precios vigentes al 31 de octubre de 1969. Asimismo se analiza los problemas básicos de oferta y de demanda, encontrándose que la demanda es mayor a la oferta, pese al descubrimiento de nuevas reservas del mineral. En consecuencia, tanto el avance científico y tecnológico como el desarrollo de la competencia son factores de importancia en la tasa de mercado actual que necesitan evaluarse constantemente.

Nov 20, 1969

By Joseph Cervik

Gas occurs in coalbeds in an adsorbed and a free gas state. Adsorbed gas is stored in the micropore structure and its transport is governed by Fick's law. The free gas occurs in the fracture system and flows according to Darcy's law. These two modes of mass transport are interdependent. Production decline curves of coal-gas wells are of the constant percentage decline type and, thus, show no indications of flow characteristics peculiar to coal-gas reservoirs. The effectiveness of surface boreholes as a degasification scheme depends upon both good fracture permeability and a high fracture density. Conventional methods of reservoir engineering analysis are not applicable to coalbeds

May 1, 1969

By John V. Beall, William F. Haddon

A long the mighty Andean Cordillera, there is splendor beyond imagination-in the natural beauty of the mountains and in daring engineering and lavish investment in the mines. This is the story of the great copper mines of Chile and Peru. Each mine has a special quality which has tested the physical and spiritual resources of the founder. In the early days of the older mines, the mining engineer of song and story was more the rule than the exception. He was physically tough, ingenious and a leader of men. In the current political and social environment of Chile and Peru, the foreign contract mining engineer is still needed; but he must be a different type of man. To John J. Selters at Cerro's Rio Blanco, the principal ingredient is still "toughness," but it is "toughness of mind" that is needed. "He probably should be a teacher." The copper mines are not new. They were, almost without exception, worked by Indians (La Exotica might prove the rule), and later by the intrepid Spaniards. Modern mining began after the turn of the century. In Chile, the first of the big ones was the Braden Copper Co. founded on the El Teniente deposit. This development was followed by Chile Exploration Co. at Chuquicamata and Andes Copper at Potrerillos. These properties have been developed over the years by vast expenditures of money and a monumental amount of labor. None was free of technical problems-metallurgy at Chuqui, marginal grade of the old mine at Andes, avalanches and the great fire at El Teniente. The evolution which has brought the established mines of Chile, and the new ones coming on stream, to their present state of development is in reality the work of wonders of input-money, engineering and responsible management in an atmosphere of political stability. Without any one of these, the result would have been failure.

Jan 11, 1969

By John V. Beall

Monotonous flood planes flanked either side of the road as we drove in the embassy car from the airport to downtown Seoul. We crossed the brown waters of the Hann River, meandering over mud flats. A great, gnarled tree bore beautiful fall colors of gold. "That is a Ginko Tree, the oldest living fossil, probably a hundred million years old;" said Jean W. Pressler. Compared to other cities of the Far East, the traffic in and around Seoul was sedate. Few motorcycles or bicycles, and little superfluous horn tooting. "This is the best season in Korea. We have long beautiful springs and falls," said Pressler, chief of the mining branch of USAID, and he continued with a briefing on the country.

Jan 8, 1969

By E. Pothas, P. R. Jochens, L. B. McRae, D. D. Howat

Discussion H. J. S. Kriek*: During 1956 Iscor became interested in the acid smelting in a blast furnace of siliceous ores containing 15 per cent silica. This interest was as a result of the successful acid smelting in Germany at a CaO/SiO2-ratio of between 0.4 and 0.8. Acid smelting can become an economic proposition due to a reduction in slag volume accompanied by a lowering of the coke rate. Paschke and Hannel1 found that a free flowing slag (i.e. viscosities of less than two poise) would not be obtained at C/S-ratios lower than 0.82 and that the addition of 3 to 4 per cent alkalies became necessary. In titanium bearing slags it was established that maximum fluidity is obtained with 10 per cent TiO2. Acid smelting resulted in increased sulphur pickup with decreasing O/S-ratio and desulphurization with soda-ash outside the blast furnace was necessary. At Iscor Mr B. B. Segal established that, for slags with (CaO+MgO)/SiO2-ratios of 0.5, 0.6 and 0.7, the addition of alkalies and TiO2 resulted in a substantial decrease in melting point and also of viscosity. At a (CaO+ MgO)/ ratio of 0.53 the addition of 10 per cent TiO2 is as effective as the addition of 3.75 per cent alkalies. This means that a blast furnace can be run without the addition of alkalies and consequently the detrimental effect of alkalies on the lining can be avoided; also that fluid slags with 5 to 15 per cent MgO and 8 to 10 per cent Al203 can be obtained with 10 per cent TiO present. (CaO+ MgO)/SiO2-ratios smaller than 0.5 the addition of titania resulted in a decrease in melting point and also in an increase in viscosity. In ceramic crucibles the slag did not become more viscous with time but thickened appreciably when held in graphite crucibles. No titanium carbonitride could be detected and only Ti20a was found. Prof Howat also did not obtain the TiC-TiN solid solution and therefore it appears that the thickening of the slag is not solely due to the formation of the titanium carbonitrides. This is in agreement with the work of Michailov and Belyakova2 who found that TiO2 decreased the viscosity of acid slags and increased the viscosity of basic slags; that titanium sesquioxide (Ti203) and titanium monoxide (TiO) increased the viscosity of both acid and basic slags and that titanium carbide, which is found in larger amounts in basic slags than in acid slags, considerably increases the viscosity of both acid and basic slags. Patzak et al3 describes the following oxygen deficient titanias, viz Ti2O, TiO, Ti203, Ti3O5, Ti4O7, Ti5O9, Ti6O11- Ti10O9,Ti7O13,Ti8O15 and Ti9O17. Until about 1959 a fair amount of data was available on the viscosity of normal slags to which titania was added2, 4, 5, 6, 7, 8, 9. At Iscor an electromagnetic viscometer based on the viscometer developed by Bockris10 was built. Progress was very slow as our main attention was centred around problems associated with the operation of the Rotor process. During 1962 and 1963 two papers containing fundamental work on the viscosities of slags in the systems CaO-SiO2-TiO211 and CaO-SiO2-TiO2 at 0 per cent A1203, 10 per cent Al203 and 20 per cent Al20312 appeared. The work of Ross and Ohno obviated the necessity of any further work by Iscor on the viscosity of blast furnace slags containing titania. Ross gave the following optimum compositions for blast furnace slags containing titania. Titania % Lime % Silica % Lime/Silica-ratio 20 30-42 38-50 0.6-1.10 30 27-36 34-43 0.63-1.06 40 24-32 28-36 0.66-1.14 The optimum temperature is 1,500°C because at this temperature the largest workable composition range exists. Higher temperatures result in narrower optimum composition ranges and in increased rates of thickening. The factors affecting the rate of thickening were also investigated by Ross and Ohno, as the main problem with slags containing titania is not finding compositions which are fluid at operating temperatures but the thickening due to a lack of oxygen. With these slags when there is insufficient oxygen to co-ordinate each silicon atom with its own individual shell of four oxygen ions, some sharing of oxygen ions must occur, resulting in the formation of large silicate ions and a rise in viscosity. The formation of oxygen deficient titania can be prevented as long as iron oxide or manganese oxide remain in the liquid slag. Iwase13 and also Viens, Camp bell and Rogers14 obtained in the experimental smelting of ilmenite ores fluid slags with more than 3 to 5 per cent manganese oxide and ferrous oxide in the slags. It appears that titaniferous ores can be smelted in blast furnaces under the following conditions: (1) Acid slag operation, i.e. at C/S-ratios from O.6-1.0. (2) Dolomite can be used as a flux as MgO is a network modifier. (3) The Al2O3-content must be less than 15 per cent as alumina promotes thickening of the slag and decreases the optimum composition range. (4) The hearth temperature must not exceed 1,500°C. According to Ross a definite amount of energy is required for each ton of iron produced and in order to increase production an increase in heat

Jan 6, 1969

By C. A. Stiles, J. W. Nyhus

Erie Mining Co. is a 10.3-million-tpy pellet producer on the east end of the Mesabi Range 60 miles north of Duluth. Its complex includes facilities for mining, crushing, grinding and beneficiation in addition to a pellet plant and power plant. A 74-mile railroad links the mine and plant to Taconite Harbor on the shore of Lake Superior. The concentrate recovery ratio at Erie is about 1: 3, and this, along with stripping activity, makes it necessary for the company to mine nearly a million tons of material a week. At the outset, all crude ore mined at Erie was hauled entirely by rail, but with expansion into additional mining areas, the company made a gradual transition towards truck-to-rail haulage. Today, in Erie's 11th year of operation, over 50% of all crude taconite mined is transported by truck to loading pockets where it is transferred to rail cars. Future years will see more taconite moved in this manner. Thus, the company has an increasing interest in tire performance.

Jan 6, 1969

By T. D. Parris

Within a 30-mile radius of Sudbury, Ontario, the Ontario division of the International Nickel Co. of Canada, Ltd., operates nine underground mines and two open pits. Prior to 1966, ore removal from the underground workings was mainly by scraping, using 125-hp electric slushers in blasthole and caving mining, and 30-hp or less air-operated slushers for fill- method mining. The development and evaluation of low-profile, diesel-powered front-end loaders, designed specifically for underground operations, was followed with keen interest during the period 1964-1965.

Jan 6, 1969

By H. Morgolin, M. A. Greenfield, P. A. Farrar

In attempts to correlate mechanical properties with microstructure, it is desirable to determine the effect of ß grain size. Determination of 6 grain size is not a problem where a has precipitated at ß grain boundaries. However, in alloys which have been processed to produce an equiaxed a +ß structure the ß grain size is difficult to determine because ß grain boundaries do not etch. The ß grain size can sometimes be laboriously determined by searching for grains which have some a! outlining the 6 grains. This difficulty of ß grain size determination in equiaxed a + ß alloys can be overcome by heating below the ß transus temperature for a short time and water quenching. Fig. 1 shows the starting equiaxed a + ß structure of the Ti-5.25A1-5.5V-0.9Fe-0.5Cu alloy used in this investigation, and Fig. 2 shows the microstructure of the same sample water-quenched after heating 1 min at 1675"F, which is 25°F below the 0 transus. The heavily etched microstructure consists of some equiaxed a! in a structure showing considerable amounts of transformed ß. The ß grain

Jan 1, 1969

By M. O. Serbousek

To further investigate the potential of precast concrete sets as a support medium for underground mines, the Bureau of Mines designed an eight-piece precast reinforced concrete drift set for a small opening. Twenty-two sets of three different member sizes were tested to destruction under two loading conditions. A discussion of the test results and typical failure patterns is presented.

Jan 1, 1969

By A. Tanaka, K. G. Carroll

The existence of local room-temperature plastic flow in germanium, which has been a subject of controversy for nearly two decades, has recently received renewed attention, albeit no more agreement than hitherto. Phenomenologically one major facet of the problem has been reduced to the question whether or not indentation can be effected in germanium without the introduction of cracks. That plastic flow can be induced at room temperature by fracture appears to have been rather convincingly demonstrated recently by Suzuki and Kamada and by Noble and Henisch.1 The literature contains both direct and incidental references to Vickers and Knoop indentations showing no cracks, within resolution of optical microscopy. Recently, however, following a controversy on the subject,2,3 johnson4 made a study of indentations produced by spherical indenters and concluded that these are accompanied by cracking. This study in-

Jan 1, 1969

A new test coke oven of seven cubic foot capacity is described. The oven was designed to simulate conditions in a central section of a battery oven and a special feature is the achievement of unidirectional heating from the flues on either side of the charge. The method of charging permits close control of charge conditions'such as bulk density and moisture content. These conditions, as well as coking conditions (e.g. flue temper- atures., coking time) must be varied to simulate conditions in different batteries, The effects of several variables on coke quality are discussed. Using test oven conditions corresponding to those of Newcastle steelworks batteries, good agreement was obtained between strength indices of test oven and battery cokes; for 17 tests, the average A.S.T.M. stability and hardness indices for the test oven coke were 38,5% and 69.7%, while those for the battery coke were 37.1.% and 67.6%.

Jan 1, 1969

By A. U. Seybolt

Experiments on both suljidation and hot corrosion have been carried out using ternary Ni-Cr-X alloys and commercial nickel-base superalloys. It has been shown that lhere are certain micro structural similarities between sulfidation using H,S/H2 mixtures and hot corvosion using Na,S04 in "crucible" experiments or hot corrosion tests using a flame tunnel apparatus. It is proposed that two key processes in the hot cowosion process are: 1) formation of chromium sulfides which deplete the alloy matrix, and 2) oxidation which can be accelerated because of chromium depletion. Other processes play an important role in the hot corrosion process, such as accelerated oxidation attack due to the presence of a N& S04 or N&SO,- NaC2 film which inhibits the formation of a protective external oxide scale. However, this latter phenomenon was not studied in this work. THE term hot corrosion has come to mean a combination of sulfur and oxygen attack on nickel-base or cobalt-base superalloys used as gas turbine blades or partitions. This attack can be initiated by any combustion process in air involving fuel containing sulfur if the sulfur level is sufficiently high. Levels of 5 pet SO2l for example, in the combusting gas have been observed to produce hot corrosion attack on nickel-base alloys. The important immediate corroding medium in aircraft gas turbines, however, has been established to be Na2S04, formed from reaction between NaCl from sea air and sulfur in the fuel.' The Na2S04 concentrates the sulfur sufficiently so that superalloy parts heated in the range 1400" to 1900°F can become sulfided by reducing the Na2S04 film which forms on the parts, and this reaction allows sulfur to gain access to the alloy. The sulfur thus released diffuses into the superalloy and forms primarily chromium sulfides of various compositions, but frequently Cr2S, has been identified. Figs. 1 and 2 show the morphology typical of hot corrosion of a cobalt-base superalloy X-45 (25 Cr, 10 Ni, 7.5 W, 0.25 C) and a nickel-base alloy, Udimet 500 (19 Cr, 19 Co, 4 Mo, 3 Ti, 2.9 Al, 0.07 C). Both alloy samples were subjected to the same hot corrosion conditions: 616 hr at 1600°F in a combusted 2.9 pct S distillate oil containing 125 ppm Na as NaC1. It will be noted at once that the nickel-base alloy is much more severely attacked. The depth of corrosion is much greater and the number and size of the sulfides present are much greater. Both alloys show a duplex oxide scale, which is more noticeable in the X-45 alloy. The external scale is probably a spinel in both cases, with CrQ3 lying next to the alloy. Many investigators1 -"ve been concerned with hot corrosion phenomena, but there has been a lack of unanimity of opinion as to the details of the hot corrosion mechanism. More than one definition of hot corrosion is possible. Some investigators con- sider as a measure of hot corrosion only the weight gain due to an accelerated oxidation. However another point of view is that the total depth of affected alloy may be used as a measure. While it is recognized that surface oxidation attack is greatly accelerated by the presence of salts like Na2S04 or Na2S04-NaC1 mixtures, sulfur penetration into the alloy also undoubtedly produces very important effects. The region of sul-fide penetration which is so prominent in Fig. 2 is one of the outstanding characteristics of hot corrosion. For highly stressed parts, such as first-stage buckets, this sulfur-penetrated region can be more significant in mechanical property loss than the reduction of cross section produced by the formation of the massive external scale. In addition, evidence will be presented that this sulfided region can be less resistant to subsequent oxidation attack. Since it appears that the sulfided region below the surface oxide scale plays an important role in the overall corrosion process,

Jan 1, 1969

By Donald C. Strimbeck

The Bureau of Mines designed a down-fired refractory-lined combustor that efficiently burned pulverized coal to produce a gas suitable as working fluid for an open-cycle gas turbine. The gas contained relatively little carbon, so the load on the ash-separation equipment was significantly reduced. Moreover, reduction in carbon content of the gas increased the average density of the solid particles making them more amenable to removal by centrifugation. Centrifugal-type separators failed to remove sufficient ash; however, to limit blade wear enough to achieve operating periods generally deemed necessary for commercial practicability.

Jan 1, 1969

By R. S. Warfield

Several reconnaissance-type surface and drill core samples of Arctic Northwestern Alaska coals were taken during the summer field seasons of 1964 and 1966 for coking studies. Surface samples were taken from the Kokolik River and the Cape Beaufort areas; core samples were obtained from the Kukpowruk River and the Cape Beaufort areas. Only one Kokolik River surface sample possessed coking properties. This sample, representing an 11.6-ft coal seam, made coke approaching metallurgical quality when a 30-pct portion of selected blending coals was added. Four Kukpowruk River core samples, each representing the same 19-to 20-ft coal seam, were all of about the same coking quality. No differences of coke quality attributable to depth of permanently frozen overburden were found, but the data confirm previously reported information that the 19- to 20-ft coal bed is a potentially satisfactory base coking coal. In the Cape Beaufort area, a core sample with appreciable coking properties was taken at depth from a coalbed determined noncoking in a surface sample.

Jan 1, 1969

By L. R. Weisberg, E. A. Miller

Single-crystalline films of silicon of good quality were vacuum-deposited on sapphire. The improved crystallinity was achieved by the strict exclusion of oxygen from the evaporation system, including the use of a "oxygen-free" silicon source. To minimize possible contamination of the source, the silicon was sublimed from the solid at about 1390° C. using a simple electron bombardment system to heat the silicon. This resulted in deposition rates of about 5 to 7 µ per hr and film thicknesses of 4 to 20 µ. The vacuum during deposition was ordinarily about 5x 10-8 Torr, DESPITE the success achieved in the chemical vapor deposition of silicon on sapphire using Silane,1-4 vacuum deposition has not provided comparable properties. For example, Chu et a1.5 vacuum-deposited silicon on single-crystalline substrates of A12O3, Sic, ThO2, and other compounds, and, although some epi- but the use of ultrahigh vacuum (about 1 to 3 x10-9 Torr) during deposition did not significantly improve the results in the oxygen-free system. The crystalline quality of films deposited on (0001) sapphire was found to be considerably superior to (1102) sapphire surfaces. Using (0001) sapphire, single-crystalline films were deposited at temperatures as low as 780° C and as high as 1075°C, with the best crystalline quality obtained at about 1000°C. At this temperature, hole mobilities over 250 sq cm per v-sec were achieved. taxy was indicated, single-crystalline films were not produced. More recently, Salama, Tucker, and young6 evaporated silicon on single-crystalline sapphire, but only poor crystallinity was observed, and low carrier mobilities were measured. In the present experiments, good-quality single-crystalline films of silicon on sapphire have been achieved by specifically excluding oxygen from the evaporation. This included not only the use of a "clean" vacuum system but, of special importance, the use of an oxygen-free silicon source. This has also been shown to provide a dramatic improvement in silicon homoepi-taxy.7 As a result, silicon-on-sapphire films with hole mobilities over 250 sq cm per v-set were achieved, and single-crystal films were grown as low as 780°C and as

Jan 1, 1969

By J. C. Martin

Equations for three-phase, three-dimensional, compressible flow (including capillarity) are reduced to two-dimensional relations by a partial integration. This reduction allows three-dimensional flow problems to be treated with mathematics for only two spatial dimensions. The results can be used to formulate flow equations for two-dimensional reservoir simulators in which the effects of capillarity and fluid segregation in the third dimension are represented. Such reservoir simulators would retain many of the aduantages of two-dimensional simulators while simulating three-dimensional effects. The principal restriction of the method is that the thickness of the reservoir should be small, compared to the distance across the reservoir. INTRODUCTION In recent years, computers have been used to calculate the performances of many reservoirs. Most of the detailed calculations, however, are based on finite difference solutions of the flow equations, and present day computers are seldom able to handle a sufficient number of cells to produce entirely satisfactory solutions, even for reservoirs represented by two-dimensional arrays of cells. The situation becomes much worse when one wishes to approximate the reservoir by a three-dimensional array. A great saving in computation or a more detailed solution can be obtained for many reservoirs by using the partial integration of the equations of flow presented in this paper. The integration reduces the three-dimensional equations to two-dimensional relations; and, for studies of two-dimensional flows in vertical cross-sections, the equations are reduced to one-dimensional relations. Most reservoir performance calculations currently are based on one- or two-dimensional flow re1ations.l-3 In some cases flow in the third dimension is approximated by assuming a particular type of vertical saturation distribution, such as gravity segregation.l.2. The relations developed in this paper approach those for segregated flow as the capillary pressures approach zero, and they approach those for uniformly distributed saturations as the capillary pressures are increased. For this analysis, the ratio of the reservoir's thickness to the maximum distance across it must be small. The capillary pressures between the oil and water should also be small compared to the maximum pressure difference in the reservoir. This requirement is met by most reservoirs. It is assumed that the capillary-pressure curves are well defined, whether or not hysteresis effects are included. Also, the reservoir must have sufficient vertical permeability to allow the fluids to segregate. The results presented here provide a firm theoretical foundation to Coats' et aL5 assumption of vertical equilibrium and extend the relations to three-phase flow. Coats' assumption of vertical equilibrium, which he verified by calculations and experiment, is developed here mathematically from basic flow equations. DISCUSSION SATURATION AND PRESSURE DISTRIBUTIONS Appendix A presents a mathematical analysis of fluid flow in reservoirs where the ratio of thickness to maximum distance across the reservoir is small. The results indicate: (1) that the fluids along any line perpendicular to such a reservoir's upper surface are in static capillary equilibrium (vertical equilibrium); (2) that, to a first approximation, the fluid pressures and properties are functions of only areal position in the reservoir and time; and (3) that hydrostatic pressure gradients exist along any line perpendicular to the reservoir's upper surface. The results might be expected after studying several physical considerations. First, no flow is allowed normal to the upper and lower reservoir boundaries, which are relatively close together.

Jan 1, 1969