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By W. C. Ellis, M. E. Fine

WHEN certain binary Fe-Ni alloys are worked cold and then stabilized by a stress-relief anneal, their Young's moduli are nearly invariant over a substantial temperature range determined by composition and work-anneal history.' In the present investigation addition of molybdenum to such alloys (besides increasing the yield strength) was found to diminish the sensitivity of the thermal coefficient of Young's modulus to composition changes. Such alloys are of interest for applications requiring 1 a metal with controlled, low temperature coefficient of Young's modulus and substantial magnetic permeability.' Young's modulus ordinarily decreases with rising temperature. In ferromagnetic alloys a change in modulus on heating due to loss of ferromagnetism modifies the .temperature dependence. Far below the Curie temperature ferromagnetism alters the modulus in two ways: a—Addition of the energy of magnetization, a function of interatomic distance, changes the relation between interatomic energy and distance.' This lowers the modulus in the Fe-Ni and Fe-Ni-Mo alloys of this investigation. b—An applied stress changes the ferromagnetic domain arrangement so that linear magnetostriction contributes to the strain and further lowers the modulUs.~ Since in the alloys of this investigation both effects lower the modulus, loss of ferromagnetism on heating changes the slope of the modulus-temperature curve in a positive direction. With increasing temperature the slope of the modulus-temperature curve, due to progressive loss of ferromagnetism, becomes less negative, goes through zero (the modulus is minimum), becomes positive, and resumes its normal negative value above the Curie temperature.' The increase in modulus and decrease in curvature in the modulus-temperature curve occurring on work hardening have been attributed to a reduction in effect (b).'. Vn the absence of ferromagnetism work hardening through introduction of residual strains would be expected to decrease the modulus.' Alloy Preparation—Test Methods Four series of alloys having nominal molybdenum contents of 5, 7, 9, and 10 pct and containing 47 to 58 pct Fe (the analyzed compositions are given in Table I) were melted in air and cast as bars weighing from 2 to 6 lb. The charge consisted of electrolytic nickel, Armco iron, molybdenum (99+ pct) or ferromolybdenum (62 pct Mo), manganese, and aluminum as a deoxidizer. The bars were swaged cold to the desired diameters with intermediate anneals. Alloys in this composition range are reported" to be face-centered cubic and ferromagnetic at room temperature. The Curie temperatures" decrease with molybdenum. For example, keeping the Ni/Fe ratio at unity and increasing the molybdenum from 0 to 17 pct decreases the Curie tempera- ture from approximately 500°C to room temperature. (The 17 pct Mo alloy requires quenching to retain a single phase.~) Young's modulus at a series of temperatures from —50" to 100°C was determined by a dynamic method previously described.', " In this method Young's modulus is calculated from the resonant frequency (in this case approximately 50,000 cycles per second) of a rod sample (approximately 2 in. long) vibrating longitudinally in the first mode. The densities of the samples at room temperature, Table I, were determined by the standard method of weighing in air and reweighing in redistilled bromobenzene. The linear coefficients of expansion, Table I, needed to calculate the sample length and density at the temperature of measurement, were obtained by observing the change in length (22" to 100°C) of an 8 in. gage distance with a two microscope cathetometer. The values of density and coefficient of expansion in the few samples checked are independent of treatment to the accuracy reported. This was assumed to be general for all the samples. The modulus values are accurate to &0.05x101' dynes per sq cm. However, values of thermal change in modulus are accurate to 20.005~ 10" dynes per sq cm. Young's Modulus Values at 25°C: The modulus values of O', 5, and 10 pct Mo alloys at 25 °C are plotted in Fig. 1 as functions of nickel. After working cold, the samples were either recrystallized at 950" to 1000°C or given a low temperature stabilizing anneal at 400° C. The 400°C anneal does not reduce the hardness nor cause recrystallization. Annealing at 400 °C does

Jan 1, 1952

By Wm. C. Madge, H. P. Allen

The chloridizing mill of the Standard Reduction Co. is located about 75 miles south of Salt Lake City on the Tintic branch of the Denver & Rio Grande Western R. R. and 12 miles from the Tintic Standard mine. The daily capacity is 200 toss of a siliceous, low-grade, silver-lead ore from this property. It has operated continuously since it was started in January, 1921. The process consists essentially of a chloridizing roast followed by a percolating leach with a nearly saturated solution of common salt, acidified with sulfuric acid, the precipitation of silver on sponge copper and of copper and lead on tin-plate cuttings. The precipitates are shipped to a smelter. Some of the general ideas involved are said to have been used by Augustin in England, in 1840. A number of textbooks treat of the subject, especially the chloridizing roast followed by a leach with sodium hyposulfite or amalgamation. The process was revived in this district by Theo. P. Holt, N. C. Christensen, the Bureau of Mines, and others. Nature of Ore Treated The average assay of the ore treated during 1924 is as follows: Gold, ounces per ton...................................... 0.025 Silver, ounces per ton..................................... 18.26 Copper, per cent......................................... 0.30 Lead, per cent........................................... 5.00 Silica, per cent........................................... 65.00 Iron, per cent............................................ 10.00 Lime, per cent........................................... 0.70 Sulfur, per cent.......................................... 3.00 Arsenic, per cent......................................... 0.70 The silver is finely disseminated and occurs as native, combined as a sulfide and, to a very small extent, as the chloride. The lead may be present as carbonate, sulfide, or sulfate.

Jan 1, 1926

By Oscar Weiss

An aerial magnetometer survey was carried out by the author's geophysical organization over the Vredefort dome, where Witwatersrand beds are wrapped around a granite plug 25 to 30 miles in diameter. The well- known magnetic shales of the Lower Witwatersrand system (to be referred to as LWW) give strong negative anomalies over the out-cropping portion of the dome. This is unusual as these same shales -cause positive anomalies over large areas near Johannesburg and along a strike of 130 miles between the East Rand and Klerksdorp. It is suggested that the negative polarization is caused by heat and stresses connected with the doming. This negative polarization of sedimentary beds, which usually cause positive anomalies, is a good example of how "magnetic lows" can be caused by "structural highs." Significant magnetic anomalies should be tested by gravity and seismic methods irrespective of the sign of the anomalies.

Jan 1, 1949

By Francis J. Dennis

Many factors govern the value of dredging-ground, and much capital can be wasted by the mistaken policy of contracting for the purchase of property arid the installation of machinery before a thorough examination has been made. To the uninitiated investor the presence of gold is generally the criterion, and very superficial evidence is necessary to satisfy him as to this point. he couuiclers the comprehensive report of a competent engineer as a wasteful extravagance, aild cannot understand why the engineer requires so much time and money to ascertain the information on which to base his conalusions, when the promoter can furnish him such pleaaing and satisfactory data with but little expenditure of time aud money. The uninitiated investor will often optimistically risk capital for purchase and equipment, and riot uiitil the veuture comes to grief does he learn that the condition8 are wholly unsuited for dredging. In many instances, a short preliminary examination by a competent engineer would have disclosed these facts. The mere presence of gold is by no means suficient; sonie of

Jan 1, 1912

By Charles Catlett

While the occurrence of phosphorus in coking-coal has assumed less importance with the development of the opeu-hearth method of steel-making, it may not be without interest to note the form in which phosphorus exists in one particular coal-seam. In the examination of what is known as the Big seam, which outcrops a few miles west of Columbiana, Ala., my attention was called to the distribution through the coal, in the form of minute veins and particles, of a resinous-looking substance. A small amount of this was selected, and was provisionally identified as evansite (Al6 Y2 O14, 18 Aq). Subsequently, through the courtesy of Dr. J. Sharshall Grasty, of the Geological Department of the University of Virginia, I was able to secure an additional amount of material, which was purified down to about 0.3 g. This was examined by Prof. John J. Porter, of the University of Cincinnati, and gave the following partial analysis: Per Cent. Loss on ignition,......... 37.43 Phosphoric anhydride,..10.33 Alumina,..........36.33 There was also a trace of silica, and quite a considerable quantity of lime and magnesia. Professor Porter was led to think that the material was not pure, but a mixture of several of the phosphates of aluminum carrying lime and magnesia. The material available did not permit of the convenient determination of the other ingredients. One form in which phosphorus occurs in coal is evidently as a hydrated phosphate of aluminum; and any coal which shows to the eye the occurrence of a light-colored resinous-looking material should be looked on with suspicion as being high in phosphorus.

Jan 1, 1912

By Charles G. Yale

A few years ago somebody connected with one of those self-constituted bodies of unofficial character, like a Chamber of Commerce, Board of Trade, or State Development Board, started a catch-phrase referring to California as a The Land of Sunshine, Fruit, and Flowers," and the railroad magazines and folders keep it steadily in use, working day and night. Yet it altogether ignores the substance which brought the State into the Union, which peopled it, and which made it famous thronghout the world. You ladies and gentlemen who have come from what wc here call "the East," have in your own States, no matter which one, sunshine, fruit, and flowers. But your Eastern States, having these things as we do, have not the gold that we do. Therefore, the old designation of " The Golden State," applied to California, should be revived, as being the most distinctive term. It is worthy of remembrance, too, that during the dark days of the civil war this State handed over $172,000,000 in yellow gold, and saved the credit of the nation. Gold-mining has been carried on in California since '' The days of old, The days of gold, The days of '49," and it still continues. Since that historic year, and up to the end of 1910, the State has produced, in gold alone, $1,530,214,468. Since 1792 the entire United States production of gold has been $3,261,573,500, so that the single State of California has, in that period, produced within $201,144,564 of one-half of all the gold from Alaska, Arizona, Colorado, Idaho, Montana, Nevada, New Mexico, Oregon, South Dakota, Utah, Washington, and the Southern and scattering States. In other words, all the other 25 States of the United States combined

Jan 1, 1912

By Petri Bryk, John Ryselin, Rolf Malmstrom, Jorma Honkasalo

THE theoretical possibilities for the realization of flash smelting have been known for a long time. Calculations concerning the same can be found in previously published literature,1 and suggestions for its accomplishment in practice are described in patents on the subject. Likewise a smaller number of practical test results are known. Autogenous smelting is closely connected with flash smelting, since in flash smelting techniques it is possible to use the heat evolved by oxidation of the sulfides for the smelting process to such an extent that under certain conditions smelting can proceed without additional fuel. However, in the event that insufficient heat is available from the oxidation of the sulfide concentrates, then additional heat may be supplied by preheating the air, using oxygen enriched air, on burning additional fuel with the concentrate.

Jan 6, 1958

By H. J. Wasson

At the close of 1927 Venezuela was producing at the rate of 205,000 bbl, per day, making it the third country in the world in point of present daily production. For the year, the total was close to 64,000,000 bbl., representing an increase of about 70 per cent. over the 37,000,000 bbl. in round numbers produced in 1926. All of the 1927 production canie from fields opened up before the beginning of the year. While no new fields were added to the list of shippers, the past 12 months' wildcatting yielded a fair quota of new discoveries which, in the light of prcsent results, may be considered likely sources of production in due time. Altogether three such promising finds were made Of first importance, at least for the immediate future, is the completion by the Lago Petroleum Corpn. of the San Matins well, 4 miles north of the present edge of the Lagunillas field. Although rated as only a 125. bbl. well, at 3217 ft., it indicates a probable extension of the field over the intervening 4 miles between the two. The two other wildcat discoveries of importance were the Los Manueles well in the District of Colon, which has in all probability opened up a new field about 7 miles north of the present Tarra oprations; and the Honibre Pintado well of the Standard Oil Co. of Venezuela,. in the State of Falcon, which is considered to be a promising clc~vc~lop~ncnt. Neither of these last mentioned two discoveries will apprcciably affect production during 1928, as the Manueles area must await the carrying out of a general plan for the exploitation of the District of Colon, a problem requiring time. The Hombre Pintado area is being tested further, but doubtless several wells will be put down before. anything definite will be done towards making the pipe line. arrangements that rnust necessarily precede the commercial exploitation of the field. The San Matias Well The San Matias well is a highly significant dcvelopment,. It is located along the trend of the La Rosa-Ilagunillas producing formations, and when drilling started, production in the Lagunillas field was 4 miles

Jan 1, 1928

By F. E. Rizzo, L. R. Bidwell, D. F. Frank

THE use of galvanic cells for the determination of oxygen activities has spread rapidly since the suitability of calcia-stabilized zirconia as a solid electrolyte material was first demonstrated by Kiukkola and Wagner.' In many of these investigations:-7 including several at ARL, the free-energy changes for various oxidation-reduction reactions involving the oxides FexO, NiO, and Cu2O have been measured as a means of assuring that a cell apparatus is functioning properly. As a result, a large quantity of data pertinent to the calculation of the high-temperature standard free energy of formation (?GO) of these oxides has been accumulated. AGFexo has been carefully measured in many independent investigations, using a variety of techniques, and consequently is accurately known. chipmans has used this informa-

Jan 1, 1968

By P. B. Leavenworth

Development in the Upper Texas Gulf Coast during 1945 resulted in the discovery of 23 new fields; one Miocene, eight Frio, three Cockfield-Yegua and eleven along the Wilcox trend. The Wilcox trend appears to he the most active at this time. Producing depths of the new Wilcox discoveries range from 2050 to 10,050 ft. Drilling along the Frio trend has been, for the greater part, projected to the deeper downdip sands. Eighteen of the new discoveries are classified as oil or condensate and five are gas discoveries.

Jan 1, 1946

By Henry Souther

It is generally admitted that a god, practical and commercial set of specifications for the use of the many purchasers of cast-iron does not exist. A few good sets are in the hands of large buyers, but even these are very different, contradictory, and not intended to cover a wide range of product. Furthermore, most of them have been prepared to cover but one branch of the industry. It is strange that purchasers of cast-iron have not looked into this question long ago and demanded good, uniform cast-irou. Probably cast-iron has been discarded from many uses because of its failure in a few instances, even though the failure was due to ignorance of the quality of the iron, or perhaps to abnormally bad iron. Some foundrymen seem to consider all gray iron from a given foundry as uniform, especially if they know that it has been made from certain well-known brands of pig. It is likely that with standard specifications and the increase of knowledge necessary for their intelligent employ-

Jan 1, 1905

By R. Gilman Brown

Mines are interesting by reason of what they have done for man, or of what has been done for them by nature. Not all are interesting on both scores. Many profitable mines are commonplace to the geologist; and many presenting unique gcological conditions have been sad failures commercially. The Standard group at Bodie has produced in 25 years, $14,500,000, paying $5,000,000 in dividends.' On the other hand, this output has beeu mined (almost wholly above the 1,000-ft. level) from a system of more than 100 veins, ranging in thickness from 0.5 in. of ('specimen rock" to 30 ft, of clay and quartz; distributed through a zone about 2,000 ft, in width, and representing from three to five distinct periods of formation. This group, therefore, by reason of its industrial record as well as its geological features, may fairly be regarded as doubly interesting. The Bodie district occupies an island of recent (Tertiary) hornblende-andesite, surrounded by a complex of igneous rocks and breccias. It lies at the summit of the eastern foot-hills of the Sierra Nevada mountains, and within the Great Basin, a

Jan 1, 1908

By Gordon S. Duncan

(New York Meeting, February, 1912 THE U. S. Geological Survey, I believe, has almost completed a study of the Harney Peak quadrangle, preliminary to the publication of a report on that, district. As I was engaged for some months on an investigation of the mode of occurrence of the ore-bearing formation in this area, it has seemed to me timely to make public the results of my study, as a contribution to the consideration of this subject. It seems scarcely necessary to describe the locality and general geology of the Black hills of South Dakota, for since 1846 many have contributed information concerning this region, some of the reports bearing such well-known names as those of Walter P. Jenney, Henry Newton, S. F. Emmons, and C. R. Van Hise, while we have only to glance at Bulletin No. 4 of the South Dakota School of Mines, in which Prof. C. C. O'Harra, of that institution, has given us a bibliography of contributions to the geography and geology of the section, to find that other well-known men have studied the district, and have given us the benefit of their researches. The U. S. Geological Survey has made geological maps of nearly the whole of the Black hills, the only quadrangle yet to be completed being, as stated, the Harney Peak district. During the late summer and autnmn of 1910, it was my lot to examine and report. on a considerable area in this neighborhood, which once had been prospected, and in some cases exploited, for tin. This paper gives an account of certain geological phenomena, noted at that time, which greatly influenced my conclnsions. The predominating formation of the district is pre-Cambrian, represented by a highly-cleaved mica-schist, usually very garnetiferous, and sometimes hornblendic. Rising like an island from the surrounding schist is a large mass of granite, known as

Jul 1, 1912

By N. F. Fiore, M. F. McGuire

THIS communication describes a series of experiments in which dislocation etch pitting was used to ascertain the depth of damage in crystals which were mechanically polished or which were cut with either a commercial metallographic cut-off wheel,* a com- All specimens were cut from single crystals of American Smelting and Refining Co. 99.999+ pct pure copper which were grown from the melt by the Bridge-man technique. The test specimens were disks, with faces being (111) planes and the disk axis a (111) direction. The disks were annealed in a vacuum of 10-5 Torr for 24 hr at 950°C prior to cutting or polishing and were given a light electropolishing in an ortho-phosphoric acid bath before etch pitting. Both Youngs1 and Livingston's2 reagents were used for etch pitting and the reagents gave identical quantitative results; however Livingston's reagent produced less staining and so all photomicrographs shown are of {111) surfaces etched with this solution. The surface of the annealed crystals showed well-defined subboundaries and a density of dislocations of about 1 X 106 per sq cm excluding those present in subboundaries, Fig. 1. The depth of cold work produced during mechanical polishing was determined after an annealed specimen was mounted in a paraffin holding block to protect it from extraneous damage. The exposed (111) surface was ground on metallographic papers and then polished on billiard cloth through 0.05-p alumina abrasive. It was necessary to remove about 0.3 mm (-0.01 in.) to obtain a structure representative of the annealed sur- face, indicating that the crystal was damaged to this depth during mechanical polishing. The cold work caused by standard cutting methods was measured in another series of tests in which cuts

Jan 1, 1967

Discussion of the paper of C. A. HEBERLEIN, presented at the New York meeting. February, 1915, and printed in Bulletin No. 98, February, 1915, pp. 497 to 504. H. D. PALLISTER, State College, Pa. (communication to the Secretary).-Mr. Heberlein gives some valuable information concerning a branch of mining and metallurgy which has been neglected during the last few years. His results with fine grinding and Deister tables suggest the possibility of the mining and concentration of a number of low-grade cinnabar deposits. Can Mr. Heberlein give us some idea as to the loss of quicksilver during this concentration? The advances in the metallurgy of quicksilver have been few and far between in recent years. Most miners prefer to go on in the old way with the Scott furnace and the large brick condensers. The large brick condensers are expensive to build and to keep in repair, and on account of the thick walls are extremely slow in cooling the gases; and further, the joints between the bricks offer passages for the quicksilver vapors unless great care is used in building the condensers.. The escape of vapor at the top of the furnace is another source of loss and I see the people at the Oceanic mine have overcome this in a very ingenious way. I recently returned from the Terlingua quicksilver district of Texas, where the only operating company, the Chisos Mining Co., is operating a 20-ton furnace of the Scott type. They had considerable difficulty with getting a good extraction, clue to too high heating in the furnace with its .resulting diffIculties. Recently they have been running with ashpit and fire doors closed and have been able to keep the heat lower, thus getting much better results and practically verifying the results obtained at the Oceanic mine.

Jan 5, 1915

By Raymond Mancha

WHEN calculating the pressure-volume characteristics of projected mine-ventilat-ing circuits by orthodox methods, certain basic assumptions are required in order to employ the various available empirical data. It is assumed, for example, that the mine air is an incompressible fluid subjected to isothermal flow, an assumption suffi-ciently accurate for practical purposes since pressure and temperature differentials are small throughout the average circuit. A more erroneous assumption, however, is that all air is accounted for as it travels throughout the mine. No attempt is usually made to evaluate stopping leakage as it occurs; instead, the air volume required at the last crosscuts is assumed to enter the mine, travel the vari-ous intake air courses intact, sweep the workings and travel out by the return air courses to the point or points of exit from the mine. Actually, there is a leakage of air from intake to return at every stopping, the quantity of this leakage depending upon the tightness of the stopping and the pres-sure difference across the stopping. Unfortunately, a lack of empirical data and knowledge of the condition of individ-ual stoppings makes an exact analysis of underground stopping leakage impossible. Generally, leakage is most severe through the old stoppings outbye the circuit. These are also subjected to higher pressure differ-ences than the newer inbye stoppings. Therefore the circuit air volume diminishes at a decreasing rate progressing from out-bye to inbye the circuit.

Jan 1, 1940

By John V. Beall

The forecast of continued growing demand for phosphate, chiefly for fertilizer, has caused a world-wide rush for deposits by a variety of companies many of which have never before mined phosphate rock. The newcomers are diversifying into this growth industry on the strength of certain capabilities. Among these are the availability of investment capital, mining or processing know-how and marketing experience. Product tie-ins have motivated some into the search. Companies with excess sulfuric acid or potential for making it are in this category. This acid is the most commonly used for dissolving phosphate rock to make phosphoric acid. Base metal smelting companies are prime candidates in this field with by-product acid from roasting processes. Producers of "sweet" natural gas, such as those in Louisiana and Texas, have a potential for recovering low cost hydrogen from excess gas production for the manufacture of ammonia, the main source of nitrogen for fertilizer. "Sour" gas producers, such as those in the Canadian fields, are making sulfuric acid that is combined with western U.S. phosphate to make fertilizer materials. The presence of one or two steel companies in the look-see for phosphate deposits is based on by-product ammonia from coke-oven gas.

Jan 10, 1966

By John R. Eckel

A Iaboratory drilling rig has been devised and placed in operation which permits the application of hydrostatic, terrastatic, and formation pore pressures to a rock sample for drilling under controlled conditions. Calibrated rests of this equipment in-dicate qualitative agreerment with field results on the effect of weight on bit, rotary speed, and rate of circ~~lation on drilling rate. Tests under pressure, using water and air as the drilling fluids, indicate that when drilling saturated limestone samples, a pressure differential between hydrostatic and formation pressure is the only pressure which affects drilling rate. A differential in either direction causes a reduction in rate, but the same diflerential from the wellbore into the formation causes the greater reduction. Also, in lime.stone, air favorably affected drilling rote only when compared to water at high hydrostatic pressurre, all other conditions remaining constant. Large increases in drilling rate with air could be obtained in the laboratory only while drilling a shale, which offered a limited permeability to nitrogen and which could be drilled with a differential from the formation into the wellbore. In general, the magnitude of the changes due to the pressure effect were small compared to changes in drilling rate obtainable by changing either bit weight or rotary speed. INTRODUCTION In the course of Humble's drilling research program, the desirability of investigating several factors which lend themselves to laboratory rather than field study became obvious. Principal reasons for this are the expense involved in field studies and the difficulty of exercising the required degree of control. Of primary importance are investigations of the effect of fluid pressure on rock rate, the effect of pressure on rock drillability, and a study of the fundamental mechanics of rock failure and chip generation. Several years ago some work in the laboratory on the effect of mud properties on drilling rate was re- ported to the API', but the completion of this work and the investigation of the other variables affecting drilling rate required equipment somewhat more elaborate than that used on these earlier tests. This equipment has been assembled to form Humble's Drilling Engineering laboratory. To date only the effect of pressure on rock drillability while using water and air has been investigated, and this work is not yet complete. Facilities are available to investigate the effect of fluid properties on rate of penetration, and information is being assembled on a current basis on the mechanics of drilling. The sequence of presentation in this progress report is somewhat different from that in which the data were obtained since the order of ex-

By R. J. Mechin

IN the early part of 1934, the St. Joseph Lead Co. purchased from the Kadco Corporation three dust-removal units, two suitable for raise work, and the other for drifting operations. The equipment was installed at the Balmat mine, in St. Lawrence County, New York. The Kadco dust-control system can be briefly described as one in which hoods, through which the drill steel is drawn, control the dust at the drill hole, the dust is conveyed through rubber hose or pipe to a collector, where it is removed from the air stream, and the air is then available for recircula-tion in the working place. As during the last few years a number of studies had been completed for the purpose of determining whether or not the dust was effectively removed by the Kadco system, and as the dust counts taken in these tests apparently indicate that it was, the St. Joseph Lead Co. was only inter-ested in obtaining data to show the efficiency of drilling, expressed in terms of minutes per foot drilled, when -using Kadco equipment, in comparison with the former method of drilling dry in a raise and drilling wet in a drift. Comparative dust counts were taken at Balmat, however, by the Metropolitan Life Insurance Co. to show the difference in the dust content of the air in which the operators worked when drilling dry, wet, or with the Kadco equipment.

Jan 1, 1935

By J. C. Martin

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

Jan 1, 1969