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By R. C. Neavel

One of the most important requirements to be met by any coal producer is delivery of size consists tailored to specific applications. Generally these specifications include a minimum as well as a maximum size limit. The customers of a major Indiana coal producer noticed that some shipments of sized coal contained a large proportion of undersize pieces. Also, in these shipments, the pieces which remained within the specified limits were cracked and, when handled, crumbled readily. The producer, certain that these shipments had been properly sized at the tipple, suspected that degradation had occurred in transit. The Indiana Geological Survey was requested to examine the coal from one of these shipments to attempt to determine the cause of breakage. Two basic facts about coal and coal seams led to the employment of petrographic techniques in this investigation. First, all coal seams consist of accumulations of diverse organic and inorganic constituents; and second, most coal breakage, either desirable or undesirable, can be attributed to the inherent weakness of some of these constituents or to poor adhesion between certain constituents. Analyses of degraded coal Sampling: Samples were secured from 2 stock piles of 2-by 4-inch coal in a dealer's yard. The coal in both piles had received the same treatment from mining to storage, although it had been mined and shipped on two different days. The size consist of one pile, however, appeared smaller than that of the other. The sample from the degraded pile is referred to as the slack sample, and the other is called the unslack sample. Size analyses: In order to determine the actual size consist, the samples were screened, with a minimum of rough handling, on square 2-inch, 1-inch, 1/2-inch, and 12-mesh screens. It is assumed that 100 percent of the 2-by 4-inch coal is retained on a 2-inch round screen at the tipple. Analyses of several loads at the tipple indicate that approximately 26 percent of the 2- by 4-inch coal, 100 percent of which was assumed held on a 2-inch round screen, passed through a 2-inch square screen. Twenty-six percent may be taken as the normal difference between round and square screens. Results of the analyses of stockpile coal are shown in Table 1. Cumulative percent versus size are plotted on a Rosin-Rammler chart as Figure 1. Both samples approximately follow normal size distribution; this indicates that breakage is not size preferential and also that the samples are representative of the stockpiles. Although both samples exhibit the same general size distribution, the median size of the slack sample is considerably smaller than that of the unslack sample. Fifty-five percent of the unslack sample was undersize (29 percent greater than the normal 26 percent difference), and 83 percent of the slack sample was undersize (57 percent greater than the normal 26 percent difference). Both piles contained a greater percentage than normal of undersize coal; this indicates that they were both affected by degradation, but to a different degree. Bedding-plane analyses: Coal, because of its laminated nature, will break preferentially along bedding planes, especially where adjacent constituents are poorly cemented to each other. Subsequent breakage perpendicular to the

Jan 1, 1961

By D. E. Munson, P. R. Dawson

An empirical model is developed that predicts accurately the transient response of salt creep to incremental and decremental changes in stress and temperature. Even though the model is empirical, it is derived from a firm theoretical framework for the micromechanical deformation processes of both steady-state and transient creep. In the model, the transient creep functions modify the steady-state creep behavior, which in turn is based upon creep mechanisms and the def ormation-mechanism map. The model is applied successfully to experimental stress-drop results obtained for salt.

Jan 1, 1982

By A. R. Denison

The area discussed in this report comprises a group of 38 counties lying in the northeast corner of Texas. It covers all, and extends beyond, the borders of what is commonly described as the East Texas Basin. It includes two major oil fields, Van and East Texas; two major gas fields, Bethany and Waskom; and a number of fields, both oil and gas, of lesser importance. All commercial oil production comes from beds of Upper Cretaceous age and almost exclusively from the basal member, the Woodbine sand. Gas is produced from both Upper and Lower Cretaceous beds with the newer production from the older beds. East Texas Field This field continued as in 1932 and 1933 to dominate the drilling and production situation throughout the oil business. A total of 3681 producing wells was completed in this field during 1934. This is an increase of 1215 over the 2466 producers completed during 1933. Total wells

Jan 1, 1935

THE transformation, in a sample of levitation melted iodide titanium, has been studied with a Leitz hot-stage microscope. The specimen was etched in HF:HNO3: glycerine, (1:1:2) giving a structure shown in Fig. 1, and was then heated at approximately 5OC per min. At 870°C (±5OC) the etching product disappeared, the a-grain boundaries showed in relief, and some new grain boundaries (presumably of ß) appeared (Fig. 2, see arrow). Some evidence of ß-grain growth is indicated by the apparent longitudinal movement of a ß-grain boundary shown in Figs. 2 and 3 (see arrows). The specimen was later heated to 1000°C and then cooled at 2°C per min. At 845°C rumpling of the surface was first noticed, Fig. 4, and continued at constant temperature. The photomicrograph reproduced in Fig. 5 was taken 6 sec after Fig. 4, at 845°C. Fig. 6 shows the structure after cooling to room temperature. The room-temperature structure, showing a rumpled surface has been observed by previous investigators.' These structures presumably indicate a sheer transformation ß - a, athermally nucleated, but capable of isothermal progression.

Jan 1, 1960

By O. E. KEOUGH

AT the Tooele custom lead-zinc ore concentrator,' two sections, each having a daily capacity of 500 to 600 tons, are operated on slightly different types of ores with but little difference in flow sheet and reagents. The ores comprising the two general classes are the coarsely crystalline type, with low pyrite and sphalerite low in iron content, from the Park City district, and the more finely disseminated higher pyrite ores from Bingham, which are more difficult to treat by flotation. Laboratory testing determines with which general class a new ore will work better. The ores are crushed to minus 1 1/4-in. size at either the Utah Ore Sampling Company's plant at Murray, Utah, the Tooele smelter sampling plant, or the crushing plant at the concentrator. The ores for each section are mixed to maintain as nearly as possible uniform lead and zinc contents.

Jan 1, 1930

By M. Deighton, R. N. Parkins

The metals Cd, ,Wg-, Sn, TI, Zn, and Zr reach a peak hardness after a criticfir1 deformation by rolling- and then soften with fwther rolling-, thereby exhibiting wovk softening. Optical metallography on Cd, Mg, and Zn shows that work softening is accompanied by a change in grain size occurring during deformation. The creep of zinc from -1l° to +60°C at stresses in the range of 7.9 to 17.3 kg per sq mm is given by € = e0 -t- Bt + Kt +at6. The third and second rate constcints are related by the equation a = K~ K6 and their stress and temperature dependence can be represented by the equations K = A, . exp - (u, - Bu)/kT. A model based upon the stress activated glide of sub-boundaries is proposed which qualitatively accounts for the metallog-raphic observations. Expressions, which are in reasonable quantitative agreement with the ex-pe~inzental observations, are derived for the creep of zinc. THE term "work softening" has been used previously by Polakowski1 and by Cottrell and stokes2 to describe phenomena where further strain of a deformed material leads to a decrease in flow stress. In both cases, however, the conditions were changed for the second straining. Here, the term "work softening" is intended to refer to a decrease in flow stress after continued straining in the same direction at the same temperature: work softening is the antithesis of "work hardening". Work softening of zinc was reported by chadwick3 and in discussion of that paper Jenkins4 indicated that cadmium also work softens. More recently work softening has been reported5 in two magnesium alloys, -99.5 pct Mg. Chadwick found that the hardness, 0.2 pct Proof Stress and the UTS of electrolytic zinc all increased with progressive cold reductions up to 30 pct and then progressively decreased with further rolling. Gay and Kelly6 used a back-reflection X-ray technique to study the effect of cold rolling zinc and found that although deformations greater than 2 to 5 pct reduction in thickness produced some recrystallized grains, deformations greater than 40 pct caused com-plete spontaneous recrystallization. At deformations greater than 60 pct the material was found to consist solely of recrystallized grains, -20 pm diam, the size of which decreased with increasing reduction and was much less than the initial grain size of the annealed material (-300 pm diam). Similar results were also reported for cadmium, tin, and lead.6 Gay, Hirsch, and Kelly' suggest that these experiments indicate recrys-tallization takes place when the dislocation density exceeds a certain value. However, no measurements of MATERIALS AND EXPERIMENTAL PROCEDURE The purity of the metals used in this work is indicated by the following figures: Zn (99.99); Cd (99.94); Mg (99.95); T1 (99.99). The zirconium was iodide crys tal bar with a probable purity of about 98 pct. The metals were obtained in the form of 0.25 in. thick strip or 0.425 in. diam rod by various fabrication methods and then annealed to ensure complete re-crystallization. Hardness-deformation curves were obtained at room temperature by rolling 4 in. thick strip under conditions which left a surface adequate for diamond pyramid hardness tests immediately after rolling. The hardness was taken as the niean of five impressions made using a 5 kg load and the time elapsing between rolling and making the last impression never exceeded 5 min. The zinc specimens for creep testing were from 3 by 3 by 4 in. cast slabs which were rolled to 0.10 in. thickness starting at 350°C and finishing cold. The resulting strip was cut into pieces 1 in. wide and annealed in batches. With suitable choices of annealing temperature between 100" arid 400°C five different grain sizes varying from 4.54 x 10' to 3.03 x l05 grains per sq cm (530 to 20 um diam) were obtained. Creep tests were done in compression using a sub-press, based on a design after Ford,8 in which the strip is compressed between dies 0.100 in. wide under conditions of plane strain. Since there is no lateral spread of the material, the area of contact between the dies and strip remains constant throughout the test and the application of a constant load, using the load maintaining device of a hydraulic testing machine, resulted in a constant stress. Covering the dies with strips of P.T.F.E. reduced frictional effects to a minimum. The creep strain was obtained by measuring the travel of the crosshead of the testing machine to a sensitivity of 0.1 pct reduction in thickness. The complete subpress assembly was contained in a steel box and for tests above the ambient this was filled with liquid paraffin and heated electrically. Temperatures below the ambient were obtained with a cooling mixture of acetone and solid carbon dioxide in the box. The liquids were stirred and the temperature of the specimen, which was controlled to ±0.5"C dur-ing a test, was measured by a thermocouple placed near the dies. Compression testing of cylindrical specimens was also carried out in the subpress using hardened flat discs separated from the test material by P.T.F.E. sheets which obviated barrelling of the specimens. Various initial strain rates were supplied by the hydraulic testing machine, and the deformation was measured by a clock dial gage resting on the cross-

Jan 1, 1970

By Alan Stanley

The MacIntyre Development of the National Lead Co. is located at Tahawus, N. Y. The operations involve the mining and concentrating of a titaniferous iron ore to produce an ilmenite concentrate and a magnetite concentrate. Construction of the MacIntyre plant was commenced during the summer of 1941, when world conditions threatened to cut off the supply of Indian ilmenite. An open pit mining operation was developed and the crushing and milling equipment put in operation in July 1942. A general description of the operation was given in the Adirondack Issue of Mining and Metallurgy for November 1943. The metallurgy of the mill operation was described by Mr. Frank R. Milliken,* Plant Manager, National Lead Company, MacIntyre Development, and presented at the AIME New York Meeting, February 1948.

Jan 1, 1949

By M. L. Fuller

DURING the last several years many papers have appeared dealing with the structure of highly polished metal surfaces. The awakening of interest in this subject is due to the applicability of the electron diffrac-tion technique to the study of surfaces. The original theory of Beilby1 considered the surface of polished metals as of the nature of a super-cooled liquid or a vitreous, amorphous solid. The electron diffraction effects from polished metals have been interpreted by many to be a direct experimental proof of the correctness of the Beilby theory. This inter-pretation has been strongly opposed by others, so that at the present time the subject is in an uncertain and controversial state. In the present paper new, and hitherto unpublished, electron diffrac-tion effects from polished zinc are described. These observations indicate that the polished surface of zinc is crystalline.

Jan 1, 1938

By E. R. Darby

When bronze is melted in open-flame furnaces a considerable amount of slag is formed during the melting operation. This slag may be incidental to the melting practice or it may be formed intentionally by slag-forming constituents added to the furnace charge to provide a covering for the melted metal When this slag is removed from the furnace, it contains, in addition to the free metal carried with it by the skimming operation, a considerable amount of metal as oxide either combined or mechanically mixed with it. By the usual crushing and washing methods this metallic oxide generally goes to the tailings pile, so diluted by other foundry waste that its recovery is economically impossible. If this furnace slag is not inixed with other foundry waste, practically all of the metal oxide as well as the free metal may be recovered by some form of smelting operation. The first side of the problem to be considered is the amount of slag produced. It may not be true that the majority of foundries have no record of the amount of slag made, but it is certainly true that a very great many either have no record or pay little attention to it. Usually the collection of furnace slag for one normal week's run will be sufficient to determine the advisability of its being handled in a different manner from the crushing and concentrating generally employed. If a large representative sample of the accumulation is crushed and concentrated and the recovery compared with the results of a careful assay, it will be found that a large percentage of the total metal content was lost in the washing process. The percentage of loss will be greater than the metallic oxide content, as some free metal in extremely fine particles always is carried away in grains of slag. From such a comparison, it may be determined whether or not the slag should be melted, sold or treated as before. An example of an actual test of this kind is given below. In a foundry melting 50,000 to 80,000 Ib. of bronze per day, a week's accumulation of furnace slag and ladle skimmings amounted to 16,500 1b.—over three times as much as the foundry superintendent estimated before the accumulation was made. Two tons of this material was taken as representative of the lot. This was ground through a thoroughly cleaned ball-mill and concentrated over a Wilfley table. The

CYRIL STANLEY SMITH, Chairman CARL E. SWARTZ, Past-Chairman ARTHUR PHILLIPS, Vice-Chairman E. E. SCHUMACHER, Vice-Chairman H. A. MALONEY, Treasurer FRANK. T. SISCO, Secretary 9 West 39th Street, New York, N. 1. Executive Committee 1944 1945 WILLIAM C. ELLIS C. S. BARRETT ALAN MORRIS T. A. WRIGHT KENT R. VAN HORN L. L. WYMAN

Jan 1, 1943

By James Bowron

CONSIDERING the importance of the steel trade and the strategic position occupied in it by the Birmingham District, it may be surprising to many to realize that even the first pig iron smelted with coke was made in Alabama on Feb. 28, 1876, although merchant pig iron had been produced in Rockwood, Tenn., in 1867, a plant which is still in active operation. The heavy deposit of Clinton ore extending from Pennsylvania into Alabama, varying greatly in accessibility on account of geological changes such as erosion or anticlinal upthrow, varies equally as much in thickness and in quality. The brown ores extensively scattered through the South also vary materially in quality, some being relatively low in phosphorus and high in manganese, and vice versa. The red ore varies materially in phosphorus, but the iron produced from these ores was always considered to be too high in phosphorus for treatment by acid Bessemer or open hearth and did not contain sufficient for the basic Bessemer process. Under these conditions there remained only the basic open hearth as a possibility, and for many years it appeared impossible to use it, because, with coke smelted iron, the silicon was too high to permit its use in basic lined furnaces.

Jan 1, 1924

By S. Frederick Ravitz

JUDGING from the inquiries that are constantly being received by the Utah Engineering Experiment Station as to the "Why," so to speak, of the flotation process of concentrating minerals, it occurred to me that perhaps a brief statement of the more important theories might interest those who have not followed the technical literature. Like many other industrial processes, flotation has hitherto been developed on an almost purely empirical basis, with relatively little practical contribution from the fundamental chemistry and physics on which the phenomena are based. In fact, its development by the trial - and - error method has been perfected to such a high state that it is becoming generally thought that further important advances can be brought about only by a thorough understanding of the actual mechanism of the process. Natur¬ally, such a complicated process has brought forth numerous theories, most of which have failed to withstand the test of time. It is significant that no theory has yet been advanced which is universally accepted.

Jan 1, 1933

By A. W. Asman

The problem of balanced design in a belt conveyor has been under constant study. Increased use of this method of transport in recent years has stimulated investigations. Conveyor belt, power transmission and other machinery should be adequate to perform the required service. All components of the belt conveyor should work as an efficient unit moving its load continuously between terminals. As part of a long term research project on belt conveyors, Hewitt-Robins established in 1954 a 2-year grant at Pennsylvania State University. Under terms of the grant, certain fundamental phenomena were to be studied. This work evolved a new concept in belt conveyor power determination based on the principle of belt tension. The Penn State contract furnished the foundation for later more detailed studies. From these most recent investigations, more specific factors were defined. In the final analysis, all of the data collected has been worthwhile in belt conveyor design. Units now operate over long distances at speeds of over 800 fpm. Long high lifts are commonplace. Where vertical or inclined shafts were the rule, belt conveyors receive first consideration. Although belt conveyors have moved forward rapidly, the future will see a tremendous gain in their use. The ideal test belt conveyor would be several thousand feet in length and have a maximum tension of at least 70,000 lb., Also, this conveyor should be horizontai and have a variable feed rate of more than one material. Data from such a field belt conveyor combined with laboratory information would permit accurate analysis. Since this type of conveyor was not available, many field locations were visited: An alternate would have been to erect a closed circuit system in the laboratory. Either method had limitations and the first was selected as having the best possibilities. Manufacturers of conveyor belting and machinery basically have been using the same horsepower formula. As more information became available from field installations, factors in the basic formula were modified. Practically all conveyors are designed today by adding together the horsepowers required to move the empty belt, move the load horizontally, and elevate the load. In selecting size of drive motor, the first two elements are relatively unimportant if the last is large. However, it is the combination of

Jan 1, 1961

The income of the Institute is derived from dues, subscriptions to MINING AND METALLURGY and sale of publications. These sources are fortunately supplemented by the interest from invested funds now amounting to over $500,000, a considerable portion constituting endowments for especial purposes. Aside from the medal funds the principal endowments are the James Douglas Fund for support of the Library, the Rocky Mountain Fund and the Seeley W. Mudd Memorial Fund. The income from the two last named is available for the support of research and a variety of other purposes. It is allocated by the Board of Directors upon recommendation of standing committees in each case. An active effort is being made to increase the endowments of the Institute, it being felt that unusual opportunities exist for research and public and professional service through the organization and at a minimum of expense. J. V. W. Reynders is chairman of the Endowment Committee.

Jan 1, 1923

By Byron E. Eldred

(Pittsburg Meeting. March, 1010.) GENERALLY speaking, the practical study of combustion has been made mainly from the stand-point of the steam engineer. This. narrow view-point has left open a large field for scientific research on the application of combustion in other arts, where the conditions of heat-utilization are widely different from those presented in the steam-generator. The great economic importance of the boiler and the ease of measuring its performance, together with the comparative uniformity of conditions in the steam-generator, is, probably, quite largely responsible for the practical neglect of the study of the use of fuels in other arts save only on the basis of their thermal value. Yet these other arts show quite plainly that a thermal-value basis alone is Dot sufficient. For example, in burning lime, under ordinary conditions of firing, a cord of wood will do as much work as a ton of coal, though the latter affords more than twice the number of heat-units. Similar results are evidenced in burning brick, in the roasting of certain ores, in reheating billets, and in many other arts where heat is to be applied over extended areas to fragmental masses of more or less refractory materials. In these arts much is to be gained by a careful study of the application of the heating-agent in the form of the long flame, and in the production of the heat in contact with, or in direct proximity to, the object to be heated. Of necessity, in boiler-practice, combustion cannot be maintained in actual contact with, or in close proximity to, the object to be heated. Speaking in a general way, heat is imparted from burning bodies to things to be heated in two ways : directly, by radiation, and indirectly, by transfer of the sensible heat of hot gases by convection or contact; by conduction from the hot uses. Both ways are utilized in boiler-practice-directly, by

Jun 1, 1910

By Kenneth S. Ritchie

THE discovery well of the world's -deepest oil producing structure, University 1-B of Group No. One Oil Corporation, in the Big Lake oil field, Reagan County, Texas, has had a remarkable record. The well is outstanding for the following reasons: (1) It is the world's pioneer well to produce. oil in commercial quantities from a structure 8500 ft. below the earth's . surface; (2) this well was the only. producer from the 8500-ft. pay for a period of 17 months and 24 days; (3) the flow of the discovery well has been free from-the customary changes at the casinghead, with the exception of putting on a back-pressure of 185 lb.; and, (4) the well's daily production of oil and gas increased rather steadily for 17 months and 18 days from an initial of 40 bbl. per day to a maximum of 2926 bbl. per day, and then six months later, after a period of decline and self-readjustment to subsurface changes caused by six more deep producers, the well set its real record for one day's production of oil at 2950 bbl

Jan 1, 1931

By R. W. Waterfill

MANY existing ore deposits of valuable metals have been worked out in their upper surface levels and the continued productivity of these mines is dependent on their extension to greater depths in the earth. The maximum depths at present reached are about 6000 to 8000 ft. below the surface, with no evidence of an impoverished ore vein. These mines are widely distributed; some of the best known are in South Africa, Brazil, India and in the western sections of the United States. Among the factors which increase the difficulty of working these mines at increasing depths are hoisting, rock pressure and temperature. The manner and effectiveness with which these problems are met will, no doubt, determine in a large measure the ultimate depths to which any mine can be profitably worked. The most critical of these three factors at present is the increase in temperature with depth. The high temperature and humidity in many of these mines at present is severely detrimental to the health of the workers and greatly hampers their efficiency. Some have almost reached a condition in which the human system cannot survive. The problems involved in the cooling and ventilation of mines have all been ably analyzed and presented in previous articles. For this reason, only a resume of conditions affecting the cooling and conditioning of deep and hot mines will be given in this paper before attention is directed to possible methods of improving atmospheric conditions in the working levels.

Jan 1, 1929

By E. W. Parker

THE anthracite region, from which there is produced annually about 80,000,000 tons, or approximately 15 per cent. of the total coal supply of the United States, has a combined area of a little less than 500 sq. mi. The total area occupied by the bituminous fields (including the sub-bituminous and lignite fields) is in the neighborhood of 400,000 sq. mi., so that 15 per cent. of the total output comes from this small area of only a little more than 0.1 per cent. of the coal-producing territory. of the United States. The productive anthracite area is divided into a series of basins or troughs, roughly parallel, extending. in a northeast-southwest direction through the. Counties

Jan 1, 1921

By S. A. Hovanessian

The method of characteristics which reduces the Buckley-Leverett mnlineur, partial differential equation to two first-order, linear differential equations is applied to the solution of a waterffood recovery problem for multiple lines of producing wells and one line of injection wells. The application of the method is described by a numerical example, and a brief theoretical background is given. From the results of this study, it is concluded that the cumulative water injected and oil produced are the same for single and multiple sets of producing wells. INTRODUCTION In this paper, the method of characteristics is applied to the solution of a waterflood recovery problem with multiple lines of producing wells and one line of injection wells. The application of the method to this type of recovery problem considerably simplifies the procedure of obtaining a numerical solution. It differs from the method described by Kern1 because it eliminates the need for integration in obtaining saturation fronts. In addition, the saturation history at any location within the field is obtained from a single diagram (time vs distance with saturation as a parameter)'. The paper includes a brief theory of the method of characteristics and its application to the solution of oil recovery problems. A numerical example is worked out, and the cumulative oil recovery-vs-time curve for a system with multiple sets of producing wells is obtained. This curve is then compared with the recovery curve of a system with only one set of producing wells. From this study, it is concluded that the cumulative water injected and oil produced are the same for single and multiple lines of producing wells. However, the recovery time is considerably longer for multiple sets of producing wells.

By L. F. Stutzman, George Thodos

A new graphical method, which is a modification of that proposed by Buckley and Leverett', is presented for the determination of the displacing fluid saturation at breakthrough for frontal drive mechanisms. This method is applicable to all types of fluids, which may be either soluble or insoluble in the oil at reservoir and injection conditions. This adaptation has unique advantages in handling problems in which either liquids or gases are the displacing fluids. INTRODUCTION In the displacement of oil from a reservoir with water or gas injected into the formation or by the expansion of a gas cap, the determination of breakthrough or flood advance is important to define the initial and subordinate phases of the operation. Thus, the determination of the displacing fluid saturation at breakthrough is fundamental to the establishment of over-all reservoir production. Methods for analyzing this saturation and for calculating production rates have been previously reported1,'*".". All of these methods utilize the rate equation for linear flow developed by Buckley and Leverett1. xs = QT - Qn/A? (?fu/?Su) .....(1) Sp where f, represents the volumetric fraction of displacing fluid in the total flowing fluid within the reservoir as is defined by: fo = 1/1+ ku/kd µn/µn .......(2) Since the time, ?, necessary for the flood front to travel distance ?X is unknown, Eq. 1 cannot be solved directly for the determination of the breakthrough point. Therefore, it becomes necessary to solve Eq. 1 simultaneously with an equation representing a fluid bal-ance on the reservoir behind the flood front. To develop such a fluid balance equation, it has been assumed that the fluid composition is uniform in the reservoir behind the flood front. Based upon this assumption, the follow-ing equation can be derived from a material balance on the displacing fluid: x = (?X) s Dx ? x=0 [ Snx - Sn1 ] dx = QT - Qn/A? It is readily observed that the displacing fluid satura-tion behind the flood front, Snx, may be determined through the simultaneous application of Eqs. 1 and 3. Although the approach may vary, this objective re-mains the same for all of the previously mentioned graphical or analytical methods. After Sox is determined, it is then possible to calculate the gas-oil and/or water-oil ratios and the rates during the initial and subordinate phases of production, and thereby define the production history of the reservoir. In this presentation only the de-termination of Snx is considered, and the reader is re-ferred to the several texts in reservoir engineering2,4,5 for a complete discussion on the methods for establish-ing the production history of a reservoir. PROPOSED METHOD FOR DETERMINING Snx Since at any time the quantity, QT - Qn/A? is a constant, Eq. 1 may be expressed in differential form as follows: dxsn = QT - Qn/A? d (? fn/? Sn) sn .....(4) Eq. 4 may be substituted directly into Eq. 3 to obtain: ? fn/? Sn ? [ Snz - Sn] QT - Qn/A? d ( ? fn/? Sn) = QT - Qn/A? ...............(5) Eq. 5 simplifies to the expression: ? fn/? Sn ? [ Snz - Sn] QT - Qn/A? d ( ? fn/? Sn) = 1 ....(6)

Jan 1, 1958