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By J. C. Bowling

The services provided by consulting engineers to the mineral process industry are briefly described. The organization of a multi-discipline consulting firm is outlined. Finally, the role of a consulting engineering firm is amplified by a description of its activities in the design and construction of a mineral process plant from inception to start-up.

Jan 1, 1978

W. O. OWEN,* Washington, D. C. (written discussion?). Few people appear to realize that the time to reach the crippled soldier is when he is first hurt. In my own judgment, the best time to reach him is before he is hurt, by showing him what other cripples have themselves clone of their own initiative, without Government-support, rendering themselves self-supporting, able to bear a little bit more than their own weight, so that they may lay aside a little bit for a rainy clay and some for a little fun in

Jan 5, 1918

By H. R. Banks

Pondering the action of a ball mill employed to carry on comminution from the product of fine crushing to the point where flotation, cyanidation or other concentrating methods are applicable, one wonders where this machine had its origin. When Henry Wadsworth Longfellow reiterated the observation of that older poet Von Logau, and wrote "Tho' the mills of God grind slowly, yet they grind exceeding small," he probably visualized the type of mill that was employed in medieval times for the production of flour, and can therefore not be considered as having gained admittance to the fraternity of millmen, on behalf of either poet. He does, however, make it quite clear that the efficiency of even these Mills of Destiny was low, and to that extent voices a characteristic of fine-grinding equipment that has been the bugbear of all metallurgical engineers. Principle of Ball Mill The principle of the ball mill probably is traceable to the action along the coast line of oceans or lakes, or in the mountain streams. Wave action on rocks and pebbles, close to the shore line, gives the potential energy to the grinding media, which, in their rolling and tossing, break down the sands, and the resultant "pulp" carried out from the shore is classified and settles out in sizes ranging downward to extremely he mesh. Surely, these are the real mills of God. If we allow our minds to dwell on the total tonnage that has been handled in this way, adding to the sands and clays at present in existence those enormous tonnages of metamorphosed sedimentaries we sense the vast quantity of rock that has been so reduced. Is it not likely, therefore, that the answer to the beginnings of pebble milling, and eventually ball milling, came through the perception of this natural mill by one of our engineering forebears? Methods of Grinding The relatively low return of mechanical input to a ball mill, in terms of useful work accomplished, has always intrigued me immensely. The potential energy absorbed by a ball raised to a high point in its path is apparently released with a very small return in useful work. Is the ball mill the most desirable means of he grinding, or is there some other method through which we can recover a reasonable amount of the energy input in terms of useful work? Some years ago, on his retirement, H. W. Hardinge was questioned as to his thoughts on the trend of fine-grinding equipment. As I recall, his prophecy at that time was that the most likely field for improvement in grinding media would be toward a charge of flat disks with some mechanical scheme to cause them to slide over each other. Undoubtedly, Mr. Hardinge went through the usual experience of the muller and bucking board, the customary introduction of the budding millman to his calling. I have often wondered, but have never got beyond the wondering stage, as to the

Jan 1, 1947

By H. R. Banks

Pondering the action of a ball mill employed to carry on comminution from the product of fine crushing to the point where flotation, cyanidation or other concentrating methods are applicable, one wonders where this machine had its origin. When Henry Wadsworth Longfellow reiterated the observation of that older poet Von Logau, and wrote "Tho' the mills of God grind slowly, yet they grind exceeding small," he probably visualized the type of mill that was employed in medieval times for the production of flour, and can therefore not be considered as having gained admittance to the fraternity of millmen, on behalf of either poet. He does, however, make it quite clear that the efficiency of even these Mills of Destiny was low, and to that extent voices a characteristic of fine-grinding equipment that has been the bugbear of all metallurgical engineers. Principle of Ball Mill The principle of the ball mill probably is traceable to the action along the coast line of oceans or lakes, or in the mountain streams. Wave action on rocks and pebbles, close to the shore line, gives the potential energy to the grinding media, which, in their rolling and tossing, break down the sands, and the resultant "pulp" carried out from the shore is classified and settles out in sizes ranging downward to extremely he mesh. Surely, these are the real mills of God. If we allow our minds to dwell on the total tonnage that has been handled in this way, adding to the sands and clays at present in existence those enormous tonnages of metamorphosed sedimentaries we sense the vast quantity of rock that has been so reduced. Is it not likely, therefore, that the answer to the beginnings of pebble milling, and eventually ball milling, came through the perception of this natural mill by one of our engineering forebears? Methods of Grinding The relatively low return of mechanical input to a ball mill, in terms of useful work accomplished, has always intrigued me immensely. The potential energy absorbed by a ball raised to a high point in its path is apparently released with a very small return in useful work. Is the ball mill the most desirable means of he grinding, or is there some other method through which we can recover a reasonable amount of the energy input in terms of useful work? Some years ago, on his retirement, H. W. Hardinge was questioned as to his thoughts on the trend of fine-grinding equipment. As I recall, his prophecy at that time was that the most likely field for improvement in grinding media would be toward a charge of flat disks with some mechanical scheme to cause them to slide over each other. Undoubtedly, Mr. Hardinge went through the usual experience of the muller and bucking board, the customary introduction of the budding millman to his calling. I have often wondered, but have never got beyond the wondering stage, as to the

Jan 1, 1947

By Harold Cook

Discussion of the paper of HAROLD EARLE COOK, presented at the New York meeting, February, 1916, and printed in Bulletin No. 110, February, 1916, pp. 375 to 400. ALBERT SAUVEUR, Cambridge, Mass.-I think we must all agree with Lieutenant-Commander Cook when he says that there is a great deal of room for improvement in the annealing of steel castings and that better results would be attained if the annealing were conducted in closer agreement with scientific requirements. Whether even a carefully annealed steel casting can ever be made the equal of a steel forging of good quality is still a debatable question. Nevertheless, we all realize that very greatly improved results can be attained by careful annealing. Lieutenant-Commander Cook refers to some requirements calling for annealing steel castings above the Ac1 point. He is undoubtedly right where he says that this should be the Ac3 point. Take, for example, a steel casting containing, say, 0.35 per cent. of carbon, having a very coarse structure made up of coarsely crystalline ferrite. If this is heated just above the Ac1 point the ferrite remains, of course, just as it was before, while the pearlite is converted into a solid solution, austenite. Upon cooling, all this austenite is again converted into pearlite, or sorbite, while the coarse structure of the ferrite remains unchanged and the annealing has accomplished very little. It is therefore surprising that the requirement of annealing above the Ac1 point should appear, in these specifications.

Jan 5, 1916

By W. L. Kerlie, J. H. Richards

The variation in hydrogen content of basic open-hearth steels during refining and the effect of changing from steam to air atomization were studied. The water content of the furnace atmosphere, the slag basicity, and the degree of oxidation of the slag-metal system largely control the hydrogen in the steel. SMALL amounts of hydrogen have an adverse effect on the ductility of steel and a wide range of hydrogen contents has an adverse effect on the internal soundness of both ferrous and nonferrous castings. This has been of major concern to the metal industry, both manufacturers and fabricators, for many years. As one of the results of continuing investigations of the hydrogen problem, much has been learned about methods of reducing the hydrogen content of liquid metals below the relatively high levels that are normally associated with such phenomena as rising ingots and excessive porosity in sand casting. However, it was not until a reliable and simple method for sampling and analyzing liquid metal was developed that much progress was possible in quantitatively evaluating the effect of steelmaking practice on hydrogen contents at levels below 6 to 8 ppm. Throughout all the work described in the following presentation the Geffner pintube technique was used to obtain samples of liquid metal1 * and the tin-fusion method, developed at Massachusetts Institute of Technology,2 was used to analyze the samples. Before attempting a systematic experimental program for minimizing the hydrogen content of open-hearth steels, it was necessary to determine the variation in hydrogen in the open-hearth bath during the normal processing of a heat. The logs of ten heats are illustrated in Figs. 1 through 7. The number at each point on the graphs refers to the number of pintube samples analyzed for hydrogen. An increase in the hydrogen content of the liquid metal during the early period of refining is apparent in all heats. There is evidence of a decrease in hydrogen later in the refining period, see Fig. 2, for example. Therefore, the existence of a maximum hydrogen content at some intermediate time during the refining period is anticipated. To test for the existence of a maximum, the data illustrated in Figs. 1 to 5 were grouped together so that they could be examined on a collectively comparable basis. The grouping was effected, as illustrated in Fig. 8, by superimposing Figs. 1 to 5 so that the apparent hydrogen maxima coincided at a fixed position, indicated by the zero on the time axis. Also, by vertical adjustment until the graphs appeared to the eye to be superimposed, the hydrogen scale was made to show the change in hydrogen content. That is, a standard amount was subtracted from the hydrogen values for each heat. Therefore, Fig. 8 represents a generalized presentation of the data and, for this reason, cannot be used to predict the actual hydrogen content of the bath during refining, but rather to show the probable rate of change in hydrogen during refining. Statistical examination of the grouped data in Fi.g. 8 shows that the apparent. increase and suhsequent decrease in hydrogen are real and confirms the existence of a maximum in general. The average increase In hydrogen during the early period of refining is between 0.8 and 1.3 ppm per hr, and the

Jan 1, 1958

By D. A. Rice, M. C. Fuerstenau

Flotation data indicate that sulfonate and amine adsorb physically on manganese oxide; oleate also adsorbs physically if the zero point of charge is sufficiently high. Chemisorption of oleate occurs on pyrolusite between pH 8 and 9, and possible mechan- isms of collector adsorption involving metal ion hydrolysis are discussed. The flotation characteristics of most metal oxides1-15 have been examined with standard collectors such as fatty acids and sulfonates. From these studies it would appear as though electrical phenomena are important when shorte r-chained collectors are involved. For example, Iwasaki, Cooke, and Choi9 have shown that dodecyl amine and dodecyl sulfate float iron oxide above and below the zero point of charge, respectively. In another system, D.W. Fuerstenau and Modi10 have shown that dodecyl sulfate, dodecyl

Jan 1, 1969

By John Sherbourne

Much valuable scientific research has been performed in recent years on the subject of phase behavior of hydrocarbons. ? 1-11 Engineers employed in petroleum production are interesting themselves in this work as well as in methods of applying the fundamental data available to the solution of their various problems. Recently a number of papers have been published in which applica-tions of phase behavior have been made to specific cases pertaining to critical phenomena.)12-16 Little effort, however, has been made in the literature to show the relation between changes occurring in the critical region and the more common phase behavior, therefore it is believed that a presentation of the fundamentals of phase behavior with reference to hydrocarbons is timely. Study of phase behavior is not new. In the metallurgical field, knowledge of heterogeneous equilibria has advanced tremendously, particularly with reference to solid-solid and solid-liquid behavior. Much is known about vapor-liquid equilibria too, but few engineers are familiar with this subject. ?

Jan 1, 1940

By Joel J. Lloyd

The storage and retrieval of geological information is represented by a "Babel" of systems. If the energy and expense that is being expended is to be fully exploited by the earth science community, a common language and common practise is required. Interfacing, applicable in the crossover of scientific disciplines and of polyglot data collections, is neither practical nor economic in the handling of English language information. Whether a single input/output center or a locus of interlocking subcenters is established, is irrelevant. The capture of fugitive information and recapture of lost data is paramount. Existing national committees and international commissions are cognizant of the problem and are engaged with it. The time required to reach agreement may merely complicate the problem. It may prove necessary for a group of individuals to face up to the situation and take the indicated action -- facing the criticism that will probably fall on their heads. The cost of rectifying immediate mistakes may be far less than the delayed execution of a long studied plan. System designs are considered in terms of input elements and output products, various mechanical operating modes are presented, and the economics of organization and maintenance are discussed.

Jan 1, 1969

By R. P. King

Flotation plant simulation, a relatively recent methodology in mineral processing, is defined and corresponding uses are discussed. Typical simulation structures are compared. Current problem areas exist although these are not insurmountable. Following a brief state of the art discussion, the N. I. M. simulation program is reviewed. In general, digital plant simulation is expected to be of substantial benefit in the near future.

Jan 1, 1976

By R. Haynes

ON the basis of compositional changes of a phase observed in a eutectoid Cu-Al alloy1 it has been suggested that mechanisms for the decomposition of the ß phase proposed by earlier workers2-5 were incorrect. These data are reinterpreted in conjunction with lattice parameter data obtained on an alloy containing 12.2 pet Al, whose isothermal transformations have been reported elsewhere.5 The a Lattice Parameters of a 12.2 Pet A1-Cu Alloy—~iffraction patterns of completely transformed specimens were obtained in a back reflection camera using copper Ka radiation. Lattice parameters of

Jan 1, 1959

By Henry M. Howe

1. The purpose of the National Research Council as organized for war purposes is twofold, to stimulate those outside its own personnel to conduct researches of importance for winning the war and to carry on such researches through its personnel to a limited extent. How this is done is explained in Section 10. "Research" is used here in a very broad sense, including, for instance, inventing and developing mechanical and physical devices. The need of this work arises from the inevitable concentration of most of the governmental war agencies on the production and transportation of war materials of the types now adopted, as distinguished from devising new kinds of instrumentalities, such as armor, guns, aircraft, and apparatus for detecting, locating, observing, signaling, transporting, and many other military purposes. But it is important that we should devise new agencies. The war came near being won by an invention, the submarine, and its course has been affected greatly by two other inventions, tanks and aircraft. In spite of our numerical superiority victory may well be snatched from us by an invention, if we allow Germany to outstrip us in inventing. Hence the importance of the Council's work in mobilizing and organizing the services of the patriotic civilian experts throughout the country, so as to bring their powers to bear on important war problems and on inventing war devices, as an adjunct to the development work of the regular governmental bodies. 2. Status of the Council.—Though President Wilson's request caused the Council to be created in 1916 by the National Academy of Sciences under its congressional charter; though his executive order of May 11, 1918, directs the heads of "Governmental departments immediately concerned" to "continue to co-operate with it in every way that may be required;" and though the Government has contributed largely to its financial support; it is not a department of the Government but an independent research body, aiming to evolve the necessary mechanism for the novel work of systematic stimulation and guidance of research. Thus the Government, which instigated the creation of the Council, recognizes and collaborates with it, and in part supports it. The Council acts moreover as the Department of Science and Research of the Council of National Defense.

Jan 1, 1919

By John E. Sherborne

Much valuable scientific research has been performed in recent years on the subject of phase behavior of hydrocarbons.l-11 Engineers employed in petroleum production are interesting themselves in this work as well as in methods of applying the fundamental data available to the solution of their various problems. Recently a number of papers have been published in which applications of phase behavior have been made to specific cases pertaining to critical phenomena.l2-16 Little effort, however, has been made in the literature to show the relation between changes occurring in the critical region and the more common phase behavior, therefore it is believed that a presentation of the fundamentals of phase behavior with reference to hydrocarbons is timely. Study of phase behavior is not new. In the metallurgical field, knowledge of heterogeneous equilibria has advanced tremendously, particularly with reference to solid-solid and solid-liquid behavior. Much is known about vapor-liquid equilibria too, but few engineers are familiar with this subject. Definition of Terms In a discussion of this sort, a definition of terms used is most important. Such terms as "pressure," "temperature" and "volume" need little definition other than mention of the units in which they are considered. Pressure is expressed in pounds per square inch absolute. Temperature is usually expressed as degrees Fahrenheit or degrees Rankine. In considering thermodynamic and phase behavior, the use of the absolute, Rankine, scale is desirable. Volume is expressed as specific volume, such as cubic feet per pound. This will be recognized as the reciprocal of the specific weight. In considering systems composed of more than one component, it is sometimes desirable to consider molal

Jan 1, 1940

By John E. Sherborne

Much valuable scientific research has been performed in recent years on the subject of phase behavior of hydrocarbons.l-11 Engineers employed in petroleum production are interesting themselves in this work as well as in methods of applying the fundamental data available to the solution of their various problems. Recently a number of papers have been published in which applications of phase behavior have been made to specific cases pertaining to critical phenomena.l2-16 Little effort, however, has been made in the literature to show the relation between changes occurring in the critical region and the more common phase behavior, therefore it is believed that a presentation of the fundamentals of phase behavior with reference to hydrocarbons is timely. Study of phase behavior is not new. In the metallurgical field, knowledge of heterogeneous equilibria has advanced tremendously, particularly with reference to solid-solid and solid-liquid behavior. Much is known about vapor-liquid equilibria too, but few engineers are familiar with this subject. Definition of Terms In a discussion of this sort, a definition of terms used is most important. Such terms as "pressure," "temperature" and "volume" need little definition other than mention of the units in which they are considered. Pressure is expressed in pounds per square inch absolute. Temperature is usually expressed as degrees Fahrenheit or degrees Rankine. In considering thermodynamic and phase behavior, the use of the absolute, Rankine, scale is desirable. Volume is expressed as specific volume, such as cubic feet per pound. This will be recognized as the reciprocal of the specific weight. In considering systems composed of more than one component, it is sometimes desirable to consider molal

Jan 1, 1940

By C. A. Wert, M. G. Coleman

The Zener relaxation in alloys in the a! solid-solution region of the Cu-Ni-Zn ternary system has bear investigated. Of the binary solid solutions that bound this region, Cu-Zn and Ni-Zn exhibit measurable Zener peaks whereas Cu-Ni does not. At constant atomic percent copper, the Peak height drops uni.forinly from the value in the Cu-ZTL binary to zero in CIA-Ali. The cariation of peak ternperatuve with composition is linear. At constant atomic percent zinc, the peak height in the ternary alloys goes through a large maximum as compared with the binary values. The temperature of these peaks varies smoothly with composition. The half-width of the peaks in the ter nary alloys, however, are much larger than those in the binaries. A large enhancement of the relaxation strength of the Zener relaxation in ternary Au-Ag-Zn alloys as compared to the relaxation strengths in Au-Zn and Ag-Zn binary alloys was observed by Firson and ert.' Since this effect was unexpected, the present study was devised to investigate whether enhancement of the Zener relaxation in ternary alloys is a general feature or is particular to the Au-Ag-Zn system. We have chosen the Cu-Ni-Zn alloy system for this investigation. Measurable Zener relaxations are observed in both the Cu-Zn and the Ni-Zn binary systems, but not in the Cu-Ni binary system. A large solid-solution region extends to large concentrations of zinc, making it possible to vary composition of the alloys over a wide range without crossing phase boundaries. EXPERIMENTAL PROCEDURE Two series of alloys were prepared. the first at constant atomic percent copper, the second at constant atomic percent zinc. The compositions of these alloys are shown by the circles in Fig. 1. The alloys were prepared by melting the individual components, each of which had previously been vacuum-melted, in an evacuated quartz capsule in an induction furnace at 1500 C. The capsules containing the molten alloys were quenched in water to reduce coring. Any individual melt which exceeded 0.001 pct weight loss was discarded. The ingots were swaged to a 50 pct reduction of area and annealed for 1 week in an evacuated quartz capsule to aid homogenization. The alloys were then

Jan 1, 1967

Executive Committee. CHESTER F. LEE, Chairman. C. R. CLAGHORN, Vice-Chairman. WC. BUTLER. M. K. LODGERS. Joseph DANIELS, Secretary-Treasurer. Meeting, Jai. 25, 1913.-Dean Milnor Roberts read a paper on The Equipment and Curricula of the College of Mines, University of Washington.

Jan 4, 1913

By T. L. Joseph

THE United States Bureau of Mines has been studying the blast-furnace process for, about ten years. An experimental furnace was developed by the bureau in cooperation with the Minnesota School of Mines Experiment Station, University of Minnesota. Many data have been obtained and published.1 Royster, Joseph and Kinney§(48) found that the particle size of ore had an important effect upon the blast-furnace, process. Kinney(28) stated: Economy in blast-furnace practice depends largely upon efficient gas-solid contact in the shaft of the furnace. If efficient work is not done in the shaft, the hearth and bosh will not function properly. A condition may be readily imagined where the particles of ore in the shaft are so large that the time entailed in the descent of these materials into the bosh is not sufficient to permit proper reduction and preparation prior to the time the ore enters the bosh of the furnace. Another condition may be pictured where the density of the stock column and the flow of stock is such that the ore will take one course and the gas another. Both of these examples will result in insufficient contact between the gas and the ore. Unreduced ore will reach the hearth, and the gas which has not been efficiently oxidized will reach the stock line. The result is the use of an excess of carbon in the smelting process. It is obvious that fine ore particles occurring in the ore as charged, or . formed in the furnace due to heat and mechanical action, place a definite limitation upon the efficiency of the operation. There is also an economic loss of fine material as dust.

Jan 1, 1930

By H. W. Camp

RESEARCH is defined, scientifically, as a "sys- tematic investigation of some phenomenon, and also a search for hidden treasures." Chemists tell us that the hidden treasures of petroleum are far richer than the .products which have been best known to us. However, these secrets will be discovered only through intensive and systematic, as well as scientific, industrial research. Indeed, the scope of the subject is so great that such an intensive research program should be assumed by a centralized organization supported by the petroleum industry as a whole. Such organization would assume the responsibilities for investigation of new products and development of markets for such products. Gasoline, the chief product of the petroleum industry, has become a cheap commodity. The continual un-controlled overproduction of crude petroleum, the in-tensive competitive condition in the marketing of petro-leum products and the increase of refining efficiency and installing of cracking facilities, have all combined to lower the market value of gasoline. The lower cost of gasoline has all been passed on to the purchaser in the form of lower prices so that today the most important product can scarcely be manufactured at a profit. In other fields, manufacturing industries have diversified their products when faced by a similar situation. Thus, they have been able to widen their marketing outlook. Why cannot the petroleum industry employ the same tactics ? Right now we are hearing much discussion concerning the development of a wider market for existing petro-leum products and new products for existing or prospec-tive markets. For instance, we can cite recent develop-ments for use of propane, butane, and ethylene. Butane has been found to be an efficient and inexpensive re-frigerant, especially adapted to home refrigerating sys-tems. Propane and butane fractions of natural gasoline may be shipped under pressure and may be used as a substitute for acetylene in cutting torches in connection with oxygen. There is a possibility of substituting butane for gas oil in the carbureting of coal gas, as we know that butane contains practically the gas enrich-ment as the same volume of gas oil and should prove to be a superior product for this function. Ethylene, a hydrocarbon present in cracked gases, is finding an outlet for the artificial ripening of green fruits. The following table . shows the more general use of petro-leum products as now established.

Jan 4, 1928

By L. Alan Weakly

The Bureau of Mines has pursued a program of research on mine illumination that has defined the coal miner's illumination needs. St. Joe Minerals Corporation, recognizing the need for improving the illumination of the working place in their mines, entered into a cooperative research agreement with the Bureau of Mines to utilize the Bureau’s expertise in this field. Some similarity in the environment, in which coal and hard rock miners work, has allowed the application of developed coal mine illumination technology and equipment to be utilized in hard rock, room and pillar mines.

Jan 1, 1979

By W. R. Strickler, P. Hazebroek, M. Prats

The pressure and production behavior of a homogeneous cylindrical reservoir producing a single fluid through a centrally located vertical fracture of limited lateral extent was determined by using mathematical methods to solve the appropriate differential equation. It is assumed that there is no pressure drop within the fracture — that is, that the fracture capacity is infinite. It was found that the production-rate decline of such a reservoir is constant (except for very early times) when the flowing bottom-hole pressure remains constant. The production-rate decline increases as the fracture length increases. Thus, the lateral extent of fractures can be determined from the production-rate declines before and after fracturing or from the decline rate after fracturing when the properties of the formation and fluids are known. The production behavior over most of the productive life of such a fractured reservoir can be represented by an equivalent radial-flow reservoir of equal volume. The effective well radius of this equivalent reservoir is equal to one-fourth the total fracture length (within 7 per cent); the outer radius of this equivalent reservoir is very nearly equal (within 3.5 per cent) to that of the drainage radius of the fractured well. The effective well radius of a reservoir producing at semisteady state is also very nearly equal to one-fourth the total fracture length. It thus appears that the behavior of vertically fractured reservoirs can be interpreted in terms of simple radial-flow reservoirs of large wellbore. INTRODUCTION An earlier report1 has considered the effect of a vertical fracture on a reservoir producing an incompressible fluid. That investigation of the fractured reservoir producing an incompressible fluid was started because of its simplicity. Thus, pertinent behavior of fractured reservoirs was obtained at an early date, while experience was being gained of value in the solution of more complicated fracture problems. One of these more complicated problems, and the one discussed in this report, considers the effect of a compressible fluid (instead of incompressible fluids) on the production behavior of a fractured reservoir. In the incompressible-fluid work mentioned, it was shown that the production rate after fracturing could be described exactly by an effective well radius equal to one-fourth the fracture length whenever the pressure drop in the fracture was negligible. Because of the simplification in interpretation, it is a matter of much interest to determine whether the production behavior of reservoirs producing a compressible liquid could be described in terms of an effective well radius which remains essentially constant over the producing life of the field. The details of the mathematical investigation are given in the Appendixes. IDEALIZATION AND DESCRIPTION OF THE FRACTURED SYSTEM It is assumed that a horizontal oil-producing layer of constant thickness and of uniform porosity and permeability is bounded above and below by impermeable strata. The reservoir has an impermeable circular cylindrical outer boundary of radius re. The fracture system is represented by a single, plane, vertical fracture of limited radial extent, bounded by the impermeable matrix above and below the producing layer (reservoir). It is assumed that there is no pressure drop in the fracture due to fluid flow. Fig. 1 indicates the general three-dimensional geometry of the fractured reservoir just described. When gravity effects are neglected, the flow behavior in the reservoir is independent of the vertical position in the oil sand. Thus, the flow behavior in the fractured reservoir is described by the two-dimensional flow behavior in a horizontal cross-section of the reservoir, such as the one shown in Fig. 2. Production is due to the expan-