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By A. J. Lynch, J. A. Holmes, F. C. Bond, D. W. Fuerstenau

This is an outstanding paper on comminution theory and represents a considerable advance in mathematical formulation. It clears the way for a discussion that should ultimately decide whether the work index should be considered as a parameter according to the Third Theory or as a constant. In the empirical differential equation dE = C dx/xn either C or n must be treated as a variable to satisfy known crushing and grinding conditions. The author has chosen the exponent n as the variable, with C as a constant.

Jan 4, 1958

By John D. Gill

The purpose of this paper is to invite attention away from the obvious. direct monetary costs of oil-field overdevelopment (as distinguished from "social costs") to a consideration of the role played by the exercise of freedom to create even allegedly excessive1 plant facilities in contributing to the maintenance of equilibrium in the wholc economy and in promoting material progress. The obvious and most commonly discussed effect of overdevelopment of oil fields is illustrated by the allegation that already the excessive development of the East Texas field has cost upward of $162,000,000;2 in other words, the drilling of 12,500 wells more than the number estimated as the requirement for economical recovery of the reserve has unnecessarily raised the total cost of development that amount. Many other illustrations of so-called excessive development of oil pools could be presented. Some of them, like Cushing, Powell and Long Beach, might be drawn from the era antedating widespread discussion in the trade of such matters as optimum well spacing, reservoir energy, the conservation rate of depletion of reserves, etc. Numerous other cases can he found in the records of the current era, which began with the introduction of rationalized restriction of production. Besides East Texas, there may be mentioned Oklahoma City, Centralia and. more recently, the town-lot development of Bloomingdale, Michigan.

Jan 1, 1940

By C. D. Williams, C. E. Ells

The susceptibility of quenched and aged Zr-2.5 wt pct Nb alloy to embritt2ement during irradiation has been examined for a number of solution temperatures and aging times. Material quenched from temperatures approximately 40°C below the transus has high tensile ductility, and this ductility is insensitive to aging at 500°C or to irradiation. If, however, the material is quenched from temperatures above the transus it becomes highly susceptible to loss of ductility either from aging at 500 or from irradiation. Inter granular failure is characteristic of the materials having low ductility. The distribution of the equilibrium phase is found to control the susceptibility to embrittlement by restricting 6 grain growth during heat treatment and thus influencing crack propagation. IN zirconium, as in titanium, -stabilizing alloy additions are used to obtain high strengths via quench and age heat treatments, and the Zr-2.5 pct Nb alloy has been developed1 because of its strength advantage over the Zircaloys. Early in the development of the Zr-2.5 pct Nb alloy the problem of 13 embrittlement was appreciated, and for this reason the solution temperature was chosen below the p transus.' In the course of irradiation studies on quenched and aged Zr-2.5 wt pct Nb alloy it was found' that irradiation introduced an important aspect of p embrittlement, riz., material quenched from the phase and aged 24 hr at 500°C was severely embrittled by moderate doses of neutron irradiation. This effect had not been studied in titanium alloys. In titanium the metallurgical features leading to 0 ernbrittlement were found to be structures with: a) coarse a platelets at the grain bondaries, b) finely dispersed a uniformly distributed throughout the (0) matrix,6 c) Widmanstatten a-13 with more than 50 pct P, d) the presence of some metastable p transformation products,3 and e) large prior -phase grain size.5 Alternatively, the presence of a uniform distribution of coarse a was conducive to high ductility and a structure largely of equiaxed a was very dctile. The detailed mechanisms of the embrittlement have not been worked out for all of these conditions, although weakness at either a-matrix boundaries or prior p grain boundaries have been prominent in the eculation. It was proposed that acicular a might act as a mild notch, and low ductility has been associated with easy fracture along its boundary.' There have been two opposing suggestions for the source of the high ductility associated with equiaxed a phase. JaffeeB proposed that this a would accept a large por- tion of the oxygen, thus increasing the ductility of the matrix, whereas after study of a Zr-Nb-Cu alloy Weinstein and oltz proposed that the a phase, softer than the martensitic matrix, acted to blunt cracks formed in the matrix. In the present work we have studied the effect of neutron irradiation on the ductility, particularly the P embrittlement, of the Zr-2.5 wt pct Nb alloy. By a variation of solution temperature and aging time a variety of metallurgical conditions have been examined, and a range of resultant ductilities obtained. The ductility has been related to the material microstructure and mode of fracture. EXPERIMENTAL The alloy used in the present work came from two separate ingots fabricated into rod of 3/8 or i in. diam, Table I. For both batches the P transus temperature was approximately 890° C. Most of the heat treatments were done directly on lengths of the j} in. diam rod, after which the tensile test specimens were machined. Quenching was achieved by dropping rods from a dynamic vacuum into water, the cooling rate estimated to be 2 100°C per sec. For aging the rods were encapsulated in evacuated silica tubes. Round tensile test specimens, with gage diam and length 0.160 and 1.0 in., respectively, were used throughout and pulled at room temperature or 300°C on Instron tensile machines, at a crosshead speed of 0.05 ipm. Specimens were irradiated in the NRX and NRU reactors, in facilities described in previous publications.'0 The metallurgical conditions examined have been: All tensile test specimens were machined with axes in the axial direction of the swaged rod. Although the specimen had a degree of preferred crystallo-graphic orientation with basal plane normals both parallel with and perpendicular to the tensile axis, the material was comparatively isotropic." The techniques of thin foil examination in the electron micro-

Jan 1, 1970

By H. E. Rowen

I hope my title is not confusing; even more so that I am not. Actually pelletizing and sintering, meaning both the formation of pellets and the hardening of them, are closely related to each other and to phenomena of nature. Figure. 1 This many-celled porous structure, frozen as it was at the fusion temperature and was being frothed by fast-moving gases, may be considered a typical piece of sinter. The same structures have been found in natural ores.

Jan 1, 1962

By C. S. Roberts, S. L. Couling

PLASTIC strain in polycrystalline metal as a result of bulk movement of one grain with respect to another along grain boundaries is not new. Rosenhain and Humphrey observed such effects shortly after the turn of the century.' Since then, observations of grain boundary deformation have been made on several metals. In recent years, intensive studies of the deformation at grain boundaries have been performed on polycrystalline metal and bicrystals.14-20 Aluminum or its alloys have been used in these latter studies. The subjects of controversy at present appear to be not whether grain boundary deformation exists, but 1) how quantitatively important it can be in polycrystalline metal, and 2) what its detailed mechanism is. Some investigators have concluded that such de- formation is minor in importance while others have concluded that it is a major consideration. Some have concluded that it involves shear of one grain over another at the interface while others have concluded that it operates by crystallographic deformation within the grains near ,the boundary. A categorical statement for electrolytic magnesium cannot be made on either of -these points. When polycrystalline magnesium is deformed in creep under conditions of high strain rate and low temperature, grain boundary deformation is relatively slight." Under these conditions, much deformation occurs within the grains and boundary contours change to a higher-energy jagged form. The result is shown in Fig. 1, where boundary jaggedness is pronounced upon polishing and etching after a plastic strain of approximately 6 .pct at 300°F and 3000 psi. No migration of the boundaries has been detected in this process. When the same metal is deformed the same amount at 600°F and 300 psi, resulting in a much lower strain rate, the boundaries migrate toward a lower energy configuration, where

Jan 1, 1958

As the Mount Morgan electrolytic chlorine plant is the first of its kind installed in Australia, and as the details of this method of producing chlorine have, up to the present time, been kept more or less secret all over the world, the author trusts that this paper may fill a gap in the technical literature of the subject.When an electric current is passed through a strong solution of sodium chloride the following reaction takes place -2 NaCl + 2 H 20 = 2 NaOH + C1 2 + H 2 Chlorine gas is liberated at the anode or entering pole of the current, while caustic soda and hydrogen are formed at the cathode or leaving pole of the current. If the two electrodes be simply immersed in a vessel containing salt and the current be turned on, there will be no result save slight smell of escaping chlorine gas. The passage of the current will have certainly split up the salt water according to the above equation, but as the products would be in juxtaposition they would immediately reunite, and the electrical energy would have been spent without any beneficial result having been obtained. Therefore, the all-important problem in designing an electrolytic chlorine cell is the finding of some effective means of isolating from each other the chlorine and caustic: soda as soon as liberated from the...

Jan 1, 1911

By Lloyd C. Venburg

Analytical and computer models are used to evaluate information obtained for the design and development of mine water control systems. Geologic, hydrologic, geophysical, biological, and chemical data must be gathered for use during all stages of mine development. Models allow the geohydrologist to use these data to predict the amount of ground-water inflow, the impact of dewatering and stream diversion on the hydrologic regime, equipment needs, and dewatering schedules. Stratigraphic and lithologic changes, faults, recharge and discharge areas, and aquifer parameters should be incorporated into the model to design and develop an effective mine water control system.

Jan 1, 1982

By H. Ries

In the marketing of mineral products, it is always highly desirable for both the producer and the consumer to be able to discuss things in a common language, and this can only be done if there are standard methods for expressing the properties of the raw materials. Having developed such a form of expression, the next step would be the making of specifications, which may serve as a basis for selecting raw materials. Considering the large number of non-metallic minerals that are used, there are not a few for which there are no standard laboratory tests to be used in determining their properties, or even standard specifications on which grades or market price may be based. Indeed in many cases the consumer simply requests the producer to submit samples, and rejects or accepts them without any explanation. Could the producer know in advance just what the consumer needed, time and money might often be saved. It appears to the writer that it is of the highest importance to devise standard laboratory tests wherever possible, as this enables the consumer to state definitely what he needs, and is of great assistance to the producer. It should also work to reduce misunderstandings and disputes. In the case of sands, we find that they show a wide variation in their character, ranging from those that are so coarse as to be of gravelly nature to those that are so fine that they might be classed as silt; and from very clayey ones to those that are practically free from clay. For this wide range of sands, we have a corresponding variety of uses, which are based mainly on the physical properties of the sand, although these are not always very definitely stated. The chemical composition need rarely be considered, if the sand is of siliceous nature. To determine and express these properties there should be standard tests, which are to be followed in all cases where the material is tested in the laboratory. The use of sands for foundry work (making molds or cores) is old, and it has been recognized for some time that the important properties to be

Jan 1, 1926

By C. C. Clark, S. J. Jung

Determining the correct combination of mix components and water for high-density paste backfills that have the transport and deposition properties necessary for a mining operation is a difficult task. Rheologic properties cannot be determined without conducting full-scale pumping tests, and data from existing tests tire very limited. This paper presents the findings of investigations undertaken to determine if paste-backfill rheology can be determined using existing techniques and to determine which mix and system variables have the greatest influence on transport-pressure loss. Data from laboratory-scale pumping tests were used to produce a rheologic model of flow behavior for comparison, and a statistical analysis was conducted to determine the most significant factors affecting transport-pressure loss.

Jan 1, 1996

By Harrison Everett Ashley

Slimes are usually defined as all material passing a certain sized sieve, which is invariably the finest sieve employed by each metallurgist in his tests; 100-mesh and 200-mesh have been taken as the limits by different writers in recent volumes of the Transactions; 150-mesh is common practice. The reader of the first two volumes of Richards' is impressed by the inflexibility with which all grinding-arrangements are praised or condemned according as they make little or much slime. Until recently, one of the principal aims of the mill-man was to keep down the slimes. He was obliged to have separate apparatus for sand- and slime-treatment. The latter was sometimes puzzling and erratic. Often it was found most economical to omit treatment of the slimes entirely, or to store them in immense basins or heaps, waiting for possible future improvements in methods of treatment. Recently, improved processes of agitation and filtration have made possible the successful extraction of values from many slimes. The tendency, when the metal is exceedingly finely disseminated, is to cut out all sand-treatment and to slime everything. These processes have been almost entirely mechanical in their conception, and it is the object of the present paper to show that certain chemical considerations may greatly hinder or assist their successful operation. It has long been a source of dissatisfaction to chemists that their analyses fail to be of much help to the clay-worker; that it is a matter of common occurrence for clays of closely identical ultimate chemjcal constitution to show the widest possible divergence in physical properties.

Jan 1, 1911

By C. O. Tarr, G. V. Smith, R. F. Miller

METALLURGISTS have long recognized that the Fe3C type of carbide is not a stable phase in steel and that, given sufficient time, it will decompose with formation of graphite, at least at temperatures below about 2050°F.1 This slow graphitization has never been of much concern to users of hypoeutectoid steels, for in such steels it occurs only during prolonged exposure at elevated temperature. In 1935, Kinzel and Moore2 reported complete graphitization of the carbide of a 0.1 5 per cent carbon steel that had been in senrice for about 26,000 hr. at a temperature of about 1100° to 1200°F. In discussing this paper, both Mathews and Sauveur reported graphitization of slightly hypereutectoid steels. In an extensive investigation of graphitization of high-purity iron-carbon alloys, Wells1 produced graphite at 7o0°C. (1290°F.) in an alloy containing as little as 0.13 per cent carbon, and showed that graphite may precipitate either directly from austenite or from decomposition of Fe3C, also that the rate of graphitization tends to increase with increase of temperature, at least up to a certain point, and with the presence of graphite nuclei. Jenkins, Mellor and Jenkinson,3 studying the structural changes in carbon steels ranging from 0.17 to 1.14 per cent carbon during stress-rupture tests at elevated temperature, obsenred graphite in an aluminum-killed 0.40 per cent carbon steel after fracture in 4340 hr. under a stress of 17,900 lb. per sq. in. at 840°F. and in many of the steels after stress-rupture test at 1200°F. Unstressed samples of 0.60, 0.86 and 1.14 per cent carbon steels graphitized almost completely in 360 hr. at 1200°F., while samples of 0.90 and 1.10 per cent carbon steels were somewhat graphitized, and steels containing 0.24, 0.40 and 0.57 per cent carbon showed no sign of graphite. Their results indicate that graphite precipitates the more readily the higher the carbon content, though this tendency is influenced to some extent by deoxidation practice, and that graphitization is accelerated by plastic deformation. In a recent study of the graphitization of 0.4 and 0.6 per cent carbon steel strip, made by M. A. Hughes,4 it was found that at 1200°F. graphite appeared after 72 hr. in aluminum-killed but not in silicon-killed steel, and that the rate of graphitization was increased by slow cooling from 1600°F. and by cold-working prior to annealing. In creep tests of normalized o. I 5 per cent carbon steel killed with 2.5 lb. of aluminum per ton, at the U. S. Steel Corporation laboratory at Kearny, spheroidization and some graphitization were found after 3000 hr. at 1000°F. under a stress as low as 15m lb. per sq. inch.

Jan 1, 1944

By Reamy Joyce

CONDITIONS in underground mining are so variable that in approach-ing the problem of the economies effected by the use of pressure-treated mine ties and mine timbers, it is necessary to secure specific data for each mine. Facts must first be dug out to show the service life of untreated timber and the reason for its failure. Generally speaking, decay is the usual cause of failure, but in some mines mechanical destruction is an important factor. Occasionally the two causes combine to produce fail-ures, and it is difficult to determine positively which factor is responsible. In making comparisons, it is always necessary to start with the cost in place and for this reason there must be accurate information covering the labor cost of installation and the labor cost of renewals. When treated timber is used, it is necessary to know the specifications for the treating process, whether the timber was painted, dipped, treated open tank, or impregnated under pressure according to standard specifica-tions. It is also necessary to know the preservative used and the amount of the preservative that was retained by the timber after treatment. It is also well to know whether the timber was thoroughly air-seasoned before treatment or was green at the time it was treated. The kind of timber available, both untreated and treated, must be known and all of the factors listed above must be taken into consideration in the study of any problem.

Jan 1, 1939

By Levi de Oliveira Bueno

Constant stress testing was used for characterization of creep behavior of a 1Cr0.5Mo steel plate in the virgin condition. The tests were performed on creep machines equipped with Andrade-Chalmers type profiles, high sensitivity SLVC transducer extensometry, high stability temperature controllers, and facilities for monitoring the creep curves with an automatic data logging system. Ten creep machines of this type were used in this research, in an experimental work that lasted for one year. Three temperature levels were selected: 538, 565 and 593°C, with stress in the range from 69 to 258 MPa producing rupture times varying from 8 to about 4908 h. The creep curves were all followed with great scrutiny using data logger scanning rates which enabled storing thousands of readings in each case. The experimental analysis involved measurements of minimum creep rate, rupture time and final elongation at rupture. The data were analyzed according to the classical phenomenological / physical relations (Norton, Arrhenius, Zenner-Hollomon, Monkman-Grant) and also considering five traditional parameterization methodologies (Larson-Miller, Orr-Sherby-Dorn, Goldhoff-Sherby, Manson- Haferd and Manson-Succop). Comparisons are presented among the possibilities of data extrapolation from five different methodologies. Comparison is also made with creep results mentioned in literature for the same material in the virgin condition.

Aug 16, 2017

By T. L. Joseph

Sulfur balances calculated from operating data covering thirty-five blast furnaces indicate that approximately 92 per cent. of the total sulfur charged enters with the coke, 7 per cent. with the ore, and 1 per cent. with the stone.. Most of the sulfur is carried from the furnace by means of the slag, the total amount eliminated being the product of the slag volume and the percentage of sulfur it contains. Additional sulfur in the charge must be eliminated either by increasing the percentage of sulfur in the slag or the quantity of slag. From relations found between slag volume and coke consumption and sulfur in the coke and slag volume, curves have been prepared showing the relative value of blast-furnace coke as its sulfur varies from 1 per cent. CHARCOAL was the predominant blast-furnace fuel until 1838, when it was found, by the operation of a 2-ton experimental furnace, that anthracite could also be used. This information was a stimulus to the iron industry, particularly in eastern Pennsylvania where anthracite was available at low cost. The production of anthracite pig iron increased steadily and, in 1855, its tonnage exceeded that of charcoal iron. Of the 680 furnaces in operation in 1874, the 226 using anthracite produced nearly one-half the total tonnage. In 1869, the production of coke furnaces exceeded that of charcoal furnaces and, in 1875, coke surpassed anthracite as a metallurgical fuel. In 1907, the output of anthracite pig dropped well below a million tons and its present significance is largely historical. In recent years, the production of coke has gone hand in hand with the production of pig iron and it has gradually attained the position it now occupies as a universal blast-furnace fuel. Charcoal, however, has great merit as a blast-furnace fuel chiefly because of its freedom from ash and sulfur. The harmful effect of the latter and the necessity of keeping the percentage of this element in pig iron at a minimum are well recognized. Other contaminating elements can be reduced to specified limits in the steelmaking process, but the control of sulfur belongs essentially to the blast furnace. The purpose of this paper is to point out the distribution of this impurity in blast-furnace

Jan 9, 1924

By J. W. Rinard

This paper is intended to introduce "non engineers" to one of the more misunderstood pump performance parameters, NPSH (Net Positive Suction Head). This basic presentation and resultant review should provide a practical understanding of NPSH and what its importance is to pump operation and reliability.

Jan 1, 1996

By J. J. Donos

IN the operation of a smelter, continuous and accurate determination of smoke losses is essential for purposes of metals inventories and as a check on the efficiency of smoke recovery apparatus. Previous to the development of a continuous and automatic smoke sampler by R. MacMichael, in 1924, two methods were in vogue, and still are in some smelters, for measuring solids passing out of a flue or stack. These are: (1) the so-called dust concentration method whereby the gas volume passing through the flue is first measured and then the total amount of solids carried in the smoke stream calculated from the weight of suspended matter filtered out of a measured volume of gas; (2) the balanced tube method is independent of gas volume flowing through the flue, and relies for its accuracy on the fact that a true smoke sample may be taken if the velocity of the gas in the sampling tube is constantly maintained at a known ratio of the average flue gas velocity. The total amount of solids passing through the flue is then proportional to the weight of filtered solids and to the ratio of the area of the flue to the area of the sampling tube, corrected for the velocity ratio. The limitations of the first method lie in the uncertainty of the varying volume of gas arising from metallurgical operations and the unreliability of ordinary gas meters to hold their accuracy under

Jan 1, 1951

By D. K. Fountain

"High metal prices and advanced technology in open-pit mining methods have resulted in considerable interest in the development of large, low-grade mineral deposits. Brenda Mines Limited is presently testing such a deposit of copper-molybdenum mineralization near Penticton, British Columbia.The mineralization of economic interest at Brenda has a total metallic sulphide content of between 1 and 1.5 per cent. This sulphide content approaches the limits of detectability of standard geophysical methods. How-ever, due to a favourable geological environment, it has been possible to utilize the Induced Polarization Method to outline the mineralization at the Brenda property."

Jan 1, 1968

By E. A. Nesbitt

The important permanent-magnet alloys 15 years ago contained carbon and depended upon it for their permanent-magnet properties. In recent years great advances have been made in a number of new alloys substantially free of carbon. These include Remalloy,' an alloy of iron, cobalt, and molybdenum, the various iron-nickel-aluminum alloys of the Mishima2 type, and the more recent Alnico V3 in which the best properties are developed by heat-treating in a magnetic field. The Mishima and the Alnico types of alloys are very brittle and after casting must be ground to size. This article describes a permanent-magnet material4 that can be machined as cast, and when handled with some care can be rolled to thin sheet and drawn to fine wire. Optimum magnetic properties are developed by a simple heat-treatment, and at this time the ductility of the alloy disappears. If the cast alloy is heat-treated without cold-working the mag netic energy product is about 1.0 X 106. This material is called Vicalloy I (pronounced vicalloy, V for vanadium, I for iron, and C for cobalt). When the alloy is first hot-worked and then considerably reduced in area by cold-swaging, drawing or grooved rolling, it is found that after heat-treatment the magnetic energy product measured in the direction of previous extension can be increased to 2.0 to 3.5 X 106. These higher values have been obtained only after reductions in area of 75 to 95 per cent, and since the material hardens considerably during working, the reduction ordinarily can be accomplished only in fine wires, although larger sizes can be reduced by the use of powerful machinery. When the material has been cold-worked in such a manner as to improve its energy product substantially, it is called Vicalloy 11. Commercial use of Vicalloy has been made in the recording of speech. Tape 0.002 by 0.050 in. has been found superior in quality to all others tried, and is now used in the Western Electric Mirrorphone 5 In a variety of forms Vicalloy I or I1 is useful in many laboratory instruments, and in experiments in which the sizes and shapes required cannot be obtained with the other more brittle permanent-magnet materials. In 1936, during a study of other age-hardening magnetic systems, it occurred to the author that some of the iron-cobalt-vanadium alloys might be suitable for permanent magnets. The proportions of iron, cobalt, and vanadium were varied and a range of alloys of particular interest was found, as shown by the indicated area in Fig. I. The limits of this range of composition are 30 to 52 per cent iron, 36 to 62 per cent cobalt, and 4 to 16 per cent vanadium. The compositions indicated by points outside this area were also tested. Preparation and Properties OF Vicalloy I The alloys were usually cast in graphite molds into bars 3/4 in. in diameter and

Jan 1, 1946

By D. Tsweleng

In the Evraz Highveld production process, iron ore is prereduced in rotary kilns prior to charging into submerged arc furnaces (SAFs). Owing to the highly aggressive slags that are formed within the combustion zones of these kilns, chrome-oxide-containing lowcement castables have been used for some years in the kiln lining. Due to concerns around the potentially harmful effects should Cr(V1) be present in the spent linings, Highveld and its suppliers of kiln castables devoted time to developing alternative chrome-free lining materials. Such materials needed to be cost-effective and ensure good performance and improved kiln availability. This paper discusses all the parameters surrounding this development. Keywords Low-cement castable, chrome-oxide free, chrome (VI), titaniferous magnetite ore, rotary kiln, submerged arc furnace, open slag bath furnace, Evraz Highveld Steel & Vanadium Limited.

Aug 1, 2013

By J. Geertsma, F. de Klerk

During the hydraulic fracturing treatment of an oil or gas well the liquid pressure in the borehole is increased until tensile stress in the surrounding rock exceeds tensile strength. Once a tensile fracture is initiated, it is penetrated by liquid from the borehole and fracture propagation under continuous hydraulic action takes place. The fracturing liquid carries a propping agent to ensure a highly permeable flow channel after pressure release. Field results range from failure to obtain increased production to outstanding success. In all cases, however, it unfortunately remains uncertain whether the values chosen for the operational parameters, such as injection rate, pumping time and fluid viscosity, were in fact the ideal ones. Though experience provides a lead, a more satisfactory way to predict results would seem to be to subject the fracture propagation process to a theoretical analysis that (1) makes the maximum use of the relevant physical information and (2) so simplifies the resulting calculations that the field engineer gets practical data that he can handle comfortably. We are attempting here to do this in connection with the prediction of fracture width and areal extent before pressure release. What remains of the fracture afterwards depends on the distribution of the propping agent between the fracture walls, and that is a separate story. Idealization of the Problem To keep the problem tractable, a number of simplify- ing assumptions have had to be made: 1. The formation is homogeneous and isotropic as regards those of its properties that influence the fracture-propagation process. 2. The deformations of the formation during fracture propagation can be derived from linear elastic stress-strain relations. 3. The fracturing fluid behaves like a purely viscous liquid; i.e., any peculiar flow behavior due to the addition of gelling agents or other additives is neglected. Moreover, the effect of the propping agent distribution on the distribution of fluid viscosity in the fracture is not taken into account. 4. Fluid flow in the fracture is everywhere laminar. 5. Simple geometric fracture-extension patterns are assumed — either radially symmetrical propagation from a point source (Fig. 1A) or rectilinear propagation originating from a line source (Fig. 1B). In the first case the periphery of the fracture is circular, in the second case it is rectangular. 6. A rectilinear propagation mode can be accomplished only by injection over a large perforated interval, thus forming a line source. Such a rectilinear fracture must therefore be located in the vertical plane. A circular propagation mode might be expected from injection through a narrow band of perforations. This forms a point source. Because gravity effects are excluded from our considerations, the fracture propagation plane may in this case assume any angle with respect to the wellbore as far as the

Jan 1, 1970