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By W. A. Krivsky, R. Schuhmann

Jan 1, 1962

By G. R. St. Pierre, A. J. Wilhelem

In connection with the reduction of hematite or magnetite to metallic iron, it appeared desirable to study the rate of reduction of each oxide to the next lower oxide under conditions which excluded all other condensed phases. For this purpose, pellets of Fe,O,, Fe,O,, and oFeOo were prepared. A gas train for the preparation of Ar-Hz-H20 mixtures was employed to establish a continuous flow of a controlled atmosphere over a pellet suspended on a balance. In this manner the following individual reduction steps could be studied: For example, pellets of hematite were reduced solely to magnetite by adjusting the gas atmosphere such that it was oxidizing to wustite. In other words, the reaction was complete when the hematite pellet had been totally converted to magnetite. The pellets were prepared from Baker Chemical ferric oxide. The ferric oxide powder was moistened and handrolled. Air sintering at 950°C produced Fe,0, pellets with a density of 3.65 g per cm3. The pellets ranged in size from 0.6 to 1.2 cm diam. The magnetite and

Jan 1, 1962

Adams, Benjamin C., Jr., Student, Univ. of Oklahoma Norman, Okla. '36 Adams, Ernest C., Student, Univ. of Illinois 908 W. Green St., Urbana, Ill. '35 Adams, George H., Student, Colorado School of Mines Golden, Colo. '36 Adams, Robert Keller Mitchell Hotel, No. 18, Golden, Colo. '35 Afshar, Hadji Khan A., Student Box 63, Univ. Sta., Moscow, Idaho. '36 Ahrenholz, Herman William, Jr., Student, Lehigh Univ Bethlehem, Pa. '35 Alger, Robert P., Student, Missouri School of Mines & Met.Rolla, Mo. '36 Allen, Paul W., Student, Mass. Inst. of Tech Cambridge, Mass. '35 Allen, R. M., Jr., Student, Virginia Polytechnic Inst., "G" Co.Blacksburg, Va. '35 Alley, John Arnold, Student, Montana School of Mines Butte, Mont. '36 Amberger, R. H., Student, Michigan College of Min. & Tech Houghton, Mich. '35 Ambrosiani, F. Peter, Student, Michigan College of Min. & Tech Houghton, Mich. '36 Andersen, Burton Lee, Student, School of Mines, Univ. of Washington Seattle, Wash. '36 Anderson, Carlos, Student, Univ. of Oklahoma Norman, Okla. '36 Anderson, J. Wendell, Student, Pennsylvania State College State College, Pa. '36 Anderson, Joseph S. Cornelia Hotel, Ajo, Ariz. '35 Anderson, R. C., Student, Univ. of Washington Seattle, Wash. '35 Anderson, Roy A., Student, Washington State College Pullman, Wash. '35 Anderson, Willard. Student. Univ. of Oklahoma Norman, Okla. '36

Jan 1, 1936

By Dozier Fircley

SOME nonmetallic minerals and their products, such as portland cement, common brick and hollow tile, sand, gravel, crushed rock, vitrified salt-glaze clay pipe, and the like, are a necessity in every community. They are mineral in nature but the establishment of plants for their production is usually a local and elementary problem. Yet it is probable that an economic study of the problem in every locality will show that such nonmetallic mineral commodities are not always obtained in the most economical way, or in the best quality. Other nonmetallic minerals are of such value that they may be shipped for long distances and sold in sufficient quantities to establish prosperous commercial ventures. Among these are phosphates, barium compounds, borates, nitrogen compounds, magnesium compounds, sulfur, asbestos, and potash. Success in marketing such products depends on quality, production cost, transportation, and similar conditions.

Jan 1, 1932

By Scott Turner

AS IN OTHER lines of engineering, progress in mining was influenced during 1931 by the world-wide economic depression. Low-metal prices ? resulted in active efforts to reduce production costs of base-metal mines. Probably the chief factors in reducing costs were the greater efficiency of labor, due to better relations between management and workmen, to the scarcity of jobs, reduced wages in most base-metal mines, and to lower overhead expenses secured by drastic economies. The use of mechanical equipment continued to increase, due to more efficient designs and a better practical understanding of the applicability of such equipment to requirements. With one exception, there were no startling innovations in stoping methods, but there was notable progress in the better adaptation of mining methods. In the United States, no large new base-metal or silver mines came into production during the year. At a number of mines, output was curtailed, and operations were discontinued at several important producers. On the other hand world-wide interest in gold was stimulated, as evidenced by increased output from producing mines, the reopening of some old properties, and intensive prospecting for new deposits. Mining methods at several gold mines in the United States and Canada were modified or changed to obtain a higher extraction of ore or to meet the need for additional support of the mine workings at depth. Permanent record of improvements in mining practices has been made during the year by the Federal Government; the industry thus has available for its use more detailed information than ever before. This work was done through cooperation of the mining companies and their representatives with the United States Bureau of Mines, whereby many papers descriptive of practices and costs at individual mines have been published. The Bureau's program of describing methods and costs at the more important mines (those producing in excess of $100,000 annually) is nearly completed, but papers will be issued concerning such additional mines as may possess particular interest because of novel methods, exceptional conditions or operating results, and others to supplement reports already issued. In this way, an interesting record of the results of changing conditions, or innovations in methods, will be made available.

Jan 1, 1932

By H. M. Howe

A Discussion of the paper of Professor Howe, presented at the London Meeting, July, 1906, and printed in Bi-Monthly Bulletin, No. 14, March, 1907, pp. 169 to 274. SECRETARY'S NOTE.-M. Beutter's introductory remarks, being simply a resume of Prof. Howe's paper, with a disclaimer of the intention to criticise it, have been omitted, and the remainder of this contribution has been somewhat condensed, without omitting any essential statements. M. BEUTTER, St. Etienne* (communication to the Secretary-+):-Professor Howe observes in paragraph 47 of his paper that the fluid-compression of steel. diminishes piping by forcing the liquid and viscous metal, while the pipe is forming, or after it has formed, into the space so produced, and proceeds to study the different systems of compression, alluding to the four principal ones: Whitworth, Illingworth, Williams, and Harmet. Having now for many years had in use the last-named process, that of M. Henri Harmet, I shall attempt to explain the difference between it and the three other processes named. The Harmet process does not diminish or close the pipe after it has formed, but prevents it from forming; and I may add that it is the only one of the four which produces this result. It is true that the Whitworth process diminishes the interior hollow by localizing it around the central axis of the ingot, which is afterwards removed by trepanning the hollow forgings for which this process has been, I believe, always exclusively used. The Illingworth and Williams processes, which, in my opinion, are applicable only to small ingots, of necessity allow the pipe to form, or (to use the expression of Professor Howe) they allow the surface of the lake of liquid steel to sink into the interior of the ingot, since they involve waiting a certain time before compression, so that (in the case of the Williams process) the ingot-mold can be opened without dan¬ger, or (in the case of the Illingworth process) so as to avoid the risk that the steel may enter the grooves produced by the removal of the distance-bars. During the time required for

Jul 1, 1907

By Stanley D. Michaelson, Ben. H. Slothower

The depletion of most high-grade ore reserves coupled with growing demands for metals and fuels has made it necessary to bring into production increasingly lower grade deposits. Improvements in technology have made lower grade ores economically recoverable in spite of major increases in labor and material costs and the initial investments required to open new mines and build their metallurgical and service plants. To remain competitive, managements of mining companies must keep abreast of and exploit these often complex technical developments. The proper application of these improvements in technology requires extensive and detailed study. In addition, managements of mining and other industries are becoming more deeply involved in public affairs and the myriad of other nontechnical problems common to all industries. These involvements, plus the demands of evaluating technological developments as well as the necessary work related to the routine administration of operations, have required mining managements to delegate additional study and review functions to staff services. To help overcome the problems involved in adopting the best technologies, managements have generally expanded their engineering and research staff services. Integrated, base-metal-producing companies, having open pits, concentrators, smelters, and refineries, are generally more research and engineering oriented than are most smaller mining companies and quarries. However, mining methods for even the simplest open pits and quarries are improving rapidly, and installations of these improvements are essential to combat increasing costs. Because of this, the managements of smaller operations may also require enhanced engineering staff services to realize the potential benefits.

Jan 1, 1968

By J. M. Toguri, K. Grjotheim

It is known 1,2 that the equilibrium between titanium metal, TiCl2 , and TiCl3, in a solvent of molten metal chlorides, is influenced both by the total amount of dissolved titanium and by the type of solvent. Assuming that the trivalent titanium forms the complex ion Ti111 Cl4, the equilibrium reaction may be expressed by the following equation, (Me) TiIII cl4+ 1/2 Ti = 3/2 TiII cl2+ (Me) Cl The ideal ionic equilibrium constant for this reaction is, Where the N's are the molar ionic fractions in the equilibrium melt, and (Me) the different types of cations present in the equilibrium mixture. The following thermodynamic relation is derived between Kand the concentration of cations in the equilibrium melt, mix The N"S are the equivalent ionic fractions, and so forth are constants corresponding to the standard changes in free energy for reactions in melts where only the cation indicated is present, and f (y1) is a function of the activity coefficients in the equilibrium mixture. When this term vanishes, a simplified linear relationship between In Kideal and the concentration is obtained. The simplified relationship has been applied with satisfactory results to the experimental data of Mellgren and pie,' and Kreye and Kellogg.2 MANY existing methods for the production of light metals involve the reduction of their respective metal halides in the presence of salt melts composed of alkali or alkaline earth chlorides. In such a solution phase, the light-metal halide may undergo stepwise reduction from a higher, to a lower valence, and finally to a zero valent state. In this connection, the disproportionation equilibrium is of great interest. In the case where the higher valence is three and the lower valence is two, the disproportionation equilibrium may be written as follows, 3 Me11 = 2 Me111 + Me [I] If this equilibrium occurs in a salt melt containing a large excess of alkali chlorides, a typical ionic melt is obtained. It is then possible to formulate the cation equilibrium: 2 Me+h' + Me = 3 Me++ [II] Reaction [11] is analogous to a cation exchange reaction. It has been shown thermodynamically that the equilibrium constant for such a reaction in 2

Jan 1, 1960

AT THE dinner at the Cosmos Club in Washington, tendered to Herbert Hoover on Jan. 5, on the occasion of his retirement from the presidency of the Federated American Engineering Societies, he made an informal speech in which he pointed out not only the unique position which the engineer now occu-pies by virtue of the new Federated Societies but also the opportunity that is thus presented for national service. In this connection and in pointing out some of the reasons for the present economic maladjust-ment, Mr. Hoover said in part: This body has entered upon a path of public serv-ice that is unique in the whole United States. The engineer has devoted himself all these years to the upbuilding of the material values of the United States. He has done that effectively more effectively than-has been accomplished in any other of the great countries of the world. The birth of this society marks the evolution of the engineer into an interest in public affairs. With his intelligence, his experience and training and the unique knowledge that he possesses not only of the material but the intangible values among our people, he is now organized so that his united voice may become heard outside of his profession. We have probably 175,000 engineers in our country, representing an intellectual possibility for service possessed by no other group. It is therefore an augury of real social development when the engineers of the United States join together for purposes of public service.

Jan 3, 1922

By Carl E. Swartz

For many years kettles used in the melting and refining of lead and other nonferrous metals and alloys have been made of cast iron. The logic of this probably lies in the fact that cast iron has been known as a material of construction for vessels of this type much longer than steel, and satisfactory fabrication from it has been more easily accomplished and better understood. As the capacity of lead refineries has increased, it has been only natural to increase the size of kettles. Today kettles of 150 to 250 tons are not uncommon. As the size of the kettles increased and better shapes were designed, the uniformity of material of construction became more important. Records kept over comparatively long periods of time have shown that the length of service of kettles has not been uniform. Many causes contribute to this and the importance of each has not yet been evaluated. Kettle design, kettle setting, method of firing and material of construction have been changed from time to time to obtain better service, but the variation in length of service is still great. A kettle may fail during the first few heats or may last through more than one thousand heats. The purpose of this article is not to present the conclusions of a finished piece of research on kettle manufacture and use, but to present a series of observations made over considerable time in the hope that others will publish data and that all may be correlated into a logical scheme which will result in better service and longer kettle life.

Jan 1, 1931

By Carl Swartz

FOR many years kettles used in the melting and refining of lead and other nonferrous metals and alloys have been made of cast iron. The logic of this probably lies in the fact that cast iron has been known as a material of construction for vessels of this type much longer than steel, and satisfactory fabrication from it has been more easily accomplished and better understood. As the capacity of lead refineries has increased, it has been only natural to increase the size of kettles. Today kettles of 150 to 250 tons are not uncommon. As the size of the kettles increased and better shapes were designed, the uniformity of material of construction became more important. Records kept over comparatively long periods of time have shown that the length of service of kettles has not been uniform. Many causes contribute to this and the importance of each has not yet been evaluated. Kettle design, kettle setting, method of firing and material of construction have been changed from time to time to obtain better service, but the variation in length of service is still great. A kettle may fail during the first few heats or may last through more than one thousand heats. The purpose of this article is not to present the conclusions of a finished piece of research on kettle manufacture and use, but to present a series of observations made over considerable time in the hope that others will publish data and that all may be correlated into a logical scheme which will result in better service and longer kettle life.

Jan 1, 1931

By James E. York

AT a meeting of the American Society for Testing Materials at Atlantic City, June, 1908, Dr.. C. B. Dudley, in his presidential address,' showed the vital necessity of not only making a steel rail as good as in the past, but of making it considerably better in order to meet the present and prospective require¬ments of railroads. Doctor Dudley requested the co-operation of all who from personal experience could give information that would result in bringing about the production of a steel rail that could be relied upon to meet the present requirements. I have had considerable experience in the practical manufacture of steel of almost every grade, during the past 40 years, and I shall suggest changes that will bring about the results desired. Doctor Dudley has had a long experience as chemical and metallurgical expert for the Pennsylvania railroad, but he frankly admits that up to the present time both he and other metallurgical experts lack the practical knowledge of steel-making that is necessary to state correctly the reasons for the unreliability of the steel rails now produced. The necessity is self-evident for more positive practical knowledge on the part of these experts of what is possible and what is impossible regarding the practical manufacturing of steel rails from the ingot to the, finished product. Chemical knowledge in steel-making is far in advance of what it was years ago, and might be said to cover every requirement (in that department) for producing reliable steel. The study of the metallography of steel has been of great value, also the physical testing of steel, but this knowledge will only de-

May 1, 1909

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 Frederick Laist

I. INTRODUCTION The new slime-concentrating plant at the Washoe Reduction Works, Anaconda, was put into operation during March, 1914. This plant, which has a capacity of 26,000,000 gal. of slime pulp carrying 2,500 tons. of solid matter per day, consists of Dorr thickeners and Anaconda multiple-deck slime concentrators.' This plant is the culmination of a great deal of experimental work extending over a considerable period of time, which work will be described briefly in the following pages.

Jan 8, 1914

By John Fritz

MEMBERS of the American engineering societies who were in London and Paris during the last days of. June and early July were present at many interesting gatherings. The official delegates of the Founder societies were the specially honored guests of the foreign engineering societies, but other American engineers took part in the meetings. The members of the delegation were: Arthur S. Dwight, Charles F. Rand, William Kelly, representing A. I. M. E.; Ambrose Swasey (general chairman of deputation), Ira N. Hollis, Jesse M. Smith, representing A. S. M. E.; F. B. Jewett, A. E. Kennelly, G. O. Squier, representing A. I. E. E.; Charles T. Main, Robert A. Cummings, John R. Freeman, D. C. Henny, representing A. S. C. E.

Jan 1, 1921

By Edward W. Parker

IF one is to place any credence at all in the reports published in the daily press, the subject of conservation has been a very lively topic of conversation during the past 60 days, and it does not appear that the temperature of the summer months has been in any way moderated by the discussion. It is a subject in which we are all vitally interested and to which, so far as our mineral resources are concerned, both the Institute and the Geological Survey have liberally contributed. The report of the National Conservation Commission, appointed by President Roosevelt, contains a series of papers on the conservation of mineral resources, all of which were prepared by members of the Geological Survey and were compiled largely from information previously collected by that Federal bureau in the performance of its regular duties. It is not the purpose of this paper to reiterate in extenso any of the material already published. The contributions of the Survey officials to the Commission's report have been published in a separate document as Bulletin No. 394. This docunient is for free distribution and may be obtained upon application to the Director of the Survey. In the preparation of this paper I desire merely to make a few suggestions regarding the possible necessity of some restraint upon or control of one branch of the mining industry with which I have been somewhat closely associated for the past 20 years-that of coal. Most of the members of the Institute are cognizant of the suits brought by the government against the anthracite-operators in Pennsylvania, or the combination of interests commonly known as the ? hard-coal trust." No defense of any illegal combination in restraint of trade is intended, but there are some facts which should not be lost sight of, and unfortu-\

Nov 1, 1909

By Pual M. Tyler

BLIZKRIEG tactics in the present war have consumed metals on such a profligate scale that some of the best-laid procurement plans for civilian and military needs of even a year ago seem in retrospect incredibly meager. Conservation of raw materials has reached an all-time climax in Germany, where metals are worth more than human life and throwing away an empty toothpaste tube may be legally punished by death. Even the United States and the British Empire, proud masters of nine tenths of the world?s known mineral reserves and processing plants, are being driven to metals and feverishly expanding production facilities that not long ago were deemed overbuilt. After the war is over and inevitable postwar readjustments are completed, it is reasonable to expect that purchasing power ? or at least the ability to procure the comforts and luxuries of life ? will be distributed among vastly larger population than ever before, accelerating the rising trend in demand for industrial raw materials in ever greater quantity and variety.

Jan 1, 1942

By A. S. Odeh

During the last few years, several authors1-6 have advanced mathematical solutions, both exact and approximate, to the pressure behavior of a well producing from a region bounded by a circular discontinuity. The most recent solutions were reported by Carter, and by Bixel and van Poollen. Varter solved the appropriate equations analytically. Bixel and van Poollen reported a numerical solution and provided correlation charts for the interpretation of flow tests; their method uses an overlay technique and is valid for wide ranges of mobility and storage capacity ratios. The mobility ratio is defined as the ratio between k/p of the two zones, and the capacity ratio is defined as the ratio between $c of the two zones. We have developed independently an analytic solution, programmed for the CDC 1604, that agreed with Carter's.'1 The same differential equation, assumptions, boundary, and initial conditions were used. We ran 30 drawdowns and 30 buildups of a unit capacity ratio and of varied mobility ratios. We also varied the radial distances to the discontinuity, and the radii of drainage. We examined the results and obtained correlations that were simple to apply and useful in flow test interpretation. The results and the correlations reported here in graphical form allow the determination of the mobility of the two zones and the radial distance to the discontinuity. They eliminate the need to fit field data to a curve as is required in the overlay technique.6 Thus, the advantages of the present correlations are that they require less effort and they provide a unique answer. They also provide guidelines for selecting the parts of the flow test plot to be used in the interpretations. Moreover, the results obtained from the programmed analytic solution define the time intervals where the approximate solution reported by Hurstl and discussed by Larkin2 are applicable. Normal buildup and drawdown plots resulted in an early straight line, a long transition zone and a late (second) straight line. However, the second straight line will not occur if the drainage radius is not at least 10 times the radial distance to the discontinuity. The mobility of the imer and outer zones can be calculated from the slopes of the first and second straight lines, respectively, by the normal methods of the flow test analysis. We found that the radial distance to the discontinuity may be determined from the abscissa value for the point of intersection of the extensions of the two straight lines. Results similar to some of these were reported by Adams et aL7 The approximate solution compared favorably with the exact one during the first and second straight-line periods. However, during the transition zone, deviation occurred between the two solutions. Discussion of Figures Figs. 1 through 7 summarize the results of this investigation. On Figs. 3 through 7 there appears the equation of each respective plot. In this paper the mobility ratio M is defined as klp2/kzpl, where subscripts 1 and 2 refer respectively to the inner and outer zones, and dimensionless time = t, = 6.32 X In the investigation, M ranged from 0.25 to 50;

Jan 1, 1970

By H. H. Dreissen, F. J. Fontein

The operating principle of the classifying cyclone has been described often but some special aspects of the cyclone receive further clarification here. In addition, several uses of the classifiers are summarized and made clear by means of practical examples. In coal and ore washeries, the washing cyclone is widely used for the separation by specific gravity of sizes up to about 12 mm. This application has lately been extended to the separation by specific gravity of particles up to about 50 mm. This use makes special demands on the construction and wear resistance of the cyclone. The sieve bend, a development of the Dutch State Mines which is less well known than the hydrocy-clone, is dealt with at some length. The sieve bend is an apparatus designed for wet screening. Attention is given to the variables influencing the performance of the sieve bend. Hydrocyclones are now so widely applied that it would be difficult to find an industry not, directly or indirectly, using them or knowing how to use them since publications on this subject abound.1-3,5,9-11,13-15 However, because of the large number of variables influencing the operation of classifying cyclones, it is not entirely possible to calculate the results of this type of cyclone. Therefore, a number of lesser known variables will be discussed here along with some special applications of the classifying cyclone. FACTORS INFLUENCING CYCLONE CLASSIFICATION Shape of Particles: The shape of particles may cause a much coarser classification than the one expected, considering the sieve analysis and the cyclone dimensions chosen. This was clearly shown by feeding a mixture of spheres and disks in water into a cyclone. The spheres were 0.36 mm diam and the disks, 3 mm diam and 0.03 mm thick. These dimensions were chosen in order that the disks and spheres might have the same rates of fall under gravity. Both spheres and disks had a specific gravity of 1.05. The results obtained with these particles in a cyclone are summarized in Table I. Test No. 2 shows that the catch of spheres was

Jan 1, 1963

By M. Mortada

Nonsteady-state flow of slightly compressible liquids in porous media of non-uniform properties has been the subject of a number of recent studies. Most of these studies considered one-dimensional flow in systems with cylindrical geometry and with the properties of the porous medium varying with radial distance only. A. Houpeurt' studied the problem in its general form by examining the solutions of the differential equation, In Eq. 1, the permeability k and the porosity + are arbitrary functions of radial distance. Houpeurt concluded that, except for special forms of the functions k and the problem is not well suited for mathematical treatment, but lends itself to high-speed digital computation. Carslaw and Jaeger' discussed the solution of Eq. 1 for the case in which the permeability varies as a power of the radial distance. P. Albert3 studied analytically the pseudo steady-state solutions in systems of finite radial extent and non-uniform properties. He also obtained some numerical solutions for nonsteady-state flow using a high-speed computer. William Hurst4 examined the interference pressure drop due to a point sink located in an infinite system consisting of two permeability regions in series. In this case, the permeability changed at a radial distance r from the point sink. INTERFERENCE BETWEEN OIL FIELDS IN A NON-UNIFORM AQUIFER The interference pressure drop for oil fields located in an extensive aquifer of uniform properties is discussed in a number of places in the literature. This work considers the interference pressure drop for oil fields located in a non-uniform extensive aquifer comprising two regions of different properties as shown in Fig. 1. Region I of the aquifer extends from which is the radius of Oil Field A, to ar,,, where a>1. It has permeability k,, porosity +, and effective compressibility c,. Oil Field B is located in Region II which extends from ar, to infinity and has different properties from those of Region I, but the same formation thickness h. Region II has permeability k2, porosity and effective compressibility c2. The choice of this type of aquifer should not imply that it is of frequent occurrence. It is chosen primarily to illustrate the effects of non-uniform aquifer properties on oilfield interference. It is of interest to point out that a non-uniform radial aquifer in which both the permeability and the porosity are inversely proportional to the radial distance can be treated in a manner similar to a linear aquifer of uniform properties. The situation where only the permeability varies as the inverse of the radial distance is also amenable to analytical treatment. Both aquifers behave quite differently from a radial aquifer with uniform properties so far as the water influx and the interference pressure drop are concerned. METHOD OF SOLUTION The expression for the interference pressure drop in Oil Field B due to a constant rate of water influx in Oil Field A is developed in the Appendix. This is accomplished by the simultaneous solution of two partial differential equations describing the pressure in Regions I and 11. The two differential equations are solved subject to the following conditions: (1) at r,, the pressure gradient is constant; (2) at ar the pressure and