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By M. A. Mahtab, J. D. Dixon

The results of a parametric study of the influence of natural features (stress field, rock strength, and strength and orientation of fractures) as well as the influence of induced features (undercut span, boundary slot, and other boundary-weakening measures) on the cavability of an undercut block of ore in mining by the block- caving method are presented. The two-dimensional finite-element analysis technique was used for this study. A coarse "mesh," along with linear homogeneous assumptions for rock properties, was used to model the problem of initiation of caving in a block of ore. Interpretation of the test results suggests that weakening the "abutments" of the block as well as certain orientations of natural fractures exercise significant influence on cavability. These findings hold promise for immediate application to known situations in block-caving mines.

Jan 1, 1977

By O. D. Sherby, J. E. Dorn

For elevated temperature-constant load creep tests of a solid solution alloys, the creep strain is a function of a temperature-compensated time parameter 0 = je H/RT dt. The activation energy H is equal to o constant of about 36,000 cal per mol. The substructures resulting from a given creep stress condition are functions of the creep strain independent of temperature. Each new creep stress gives a new unique set of grain substructures; this is one of the factors responsible for the failure of the mechanical equation of state for creep. IN this report an attempt is made to correlate the creep properties of dilute a: solid solutions in aluminum with the subgrain structures that are developed during creep. The possibilities of such correlations have already been suggested by Wood et al. and others1-" who demonstrated that the subgrain structures are functions of the creep stress, creep strain, rate of creep, and temperature. Investigations by Sherby and Dorn' have shown that the creep strain, E, at constant load for dilute a solid solutions of aluminum is given by the functional relationship: e = f(8,u,) [1] where: 0 equals te-AH/RT = temperature-compensated time; t, time; AH, activation energy, - 35,800 cal per mol; R, gas constant; T, absolute temperature, above 400°K; and u,, initial creep stress. Consequently, it was anticipated that the subgrain structure that develops during a constant load creep test should also correlate with any two of the three significant variables C, 8, and Another possible correlation between the subgrain structure and the creep variables is obtained by differentiating Eq. 1 with respect to time, whence the creep rate, 2, becomes: = (2L) (2) =f (, u,) e-AHRT [2] or: e = F (c e H/RT, 0) [3] For the minimum creep rate, ;,, 8 is solely a function of U, as revealed by Eq. 1. Consequently, at the minimum creep rate Eq. 3 reduces to: = F (eAH/RT) [41 Correlations between U, and i, eAH/RT were found valid for solid solution alloys of aluminum where AH was a constant of about 35,800 cal per mol. The parameter ;, eH/RT is identical to the Zener-Hollo-mon parameter8," used by them in evaluating the tensile properties of copper and other metals. Eq. 4 suggests that the subgrain structure developed during secondary creep might be correlatable with either of the two variables 2, e H/RT or u,. Some of the solid solution aluminum alloys which were discussed in an earlier report' were also used in the present investigation. Sheets of these alloys were homogenized, cold rolled from 0.100 to 0.070 in. in thickness and then recrystallized to about the same grain size. Their chemical composition and grain size are recorded in Table I. Creep specimens were selected with their tensile axes in the rolling direction. The details for creep testing have been described10 and will not be repeated here. All creep tests were performed under constant load conditions. Experimental Results and Discussion Metallographic Structure of Creep Specimens: In order to obtain a preliminary concept of creep on the metallographic structure of high purity aluminum, the various ruptured specimens previously used to obtain the creep data for pure aluminum were polished and etched electrolytically in a dilute solution of fluoroboric acid. Oblique illumination was used to reveal the substructures which, for convenience, are shown in correlation with the vs. ;, eH/RT curve in Fig. 1. It is necessary to emphasize that the data for the curve refer to the various secondary stages of creep whereas the micrographs refer to the fractured creep specimens. Nevertheless a regular and systematic correlation is observed between E, eAH/RT (or the creep stress) and the micro-structure at fracture. For In (e, eAH/RT) = 42.3, which refers to creep at a high creep rate and a low creep temperature, the fractured aluminum specimen, as shown by the micrograph in the upper right-hand corner of Fig. 1, exhibited extensive deforma-

Jan 1, 1954

By H. M. St. John

MAINTAINING a satisfactory structure in brass and bronze castings has always been a foundry problem of great practical importance. While metallurgists and scientific investigators have not entirely ignored this matter, it has received less of their attention than has been given to iron, steel and the wrought copper and aluminum alloys. Investigation in this field has probably been discouraged to some extent by the complicated nature of the alloys involved. Hundreds of alloys are in more or less common use, most of them containing copper, zinc, lead and tin in-widely varying proportions. They also contain appreciable percentages of iron and antimony as impurities, and often phosphorus, which has been added as a deoxidizer and fluidifier. Small amounts of sulfur are usually present. It is not uncommon to find nickel, sometimes in rather substantial quantities. Traces of other metals are frequently present. So complicated and variable a mixture obviously offers difficulties in. any investigation of the various factors that influence the structure and other properties of the casting.

Jan 1, 1930

Jan 1, 1923

By Clarence Zener

Contents Page General Principles................................................................. 551 Equilibrium Diagrams......................................................... 551 Nucleation................................................................... 553 Kinetics of Phase-boundary Propagation......................................... 553 Kinetics of Growth of New Phase............................................... 556 Spheroidization............................................................... 560 Pure Iron-carbon System........................................................... 561 Formation of Pearlite.......................................................... 561 Interlamellar Spacing........................................................ 562 Pearlite C-curve............................................................. 563 Bainite Formation............................................................. 565 Formation of Martensite....................................................... 570 Loss of Tetragonality of Martensite.............................................. 570 Effect of Alloys upon Decomposition of Austenite..................................... 572 Pearlite...................................................................... 572 Bainite and Martensite......................................................... 574 Time-temperature-transformation Curves............................................. 576 Appendix A............................................................. ......... 579 Appendix B....................................................................... 580 Appendix C....................................................................... 580 The present investigation started in an attempt to understand certain details of the decomposition of austenite, and of the effect of alloying elements thereon. As the investigation proceeded it became apparent that not only these details, but all the general features of austenite decomposition, could be understood in terms of fundamental physical principles. Since the direct applicability of these physical principles has not been previously recognized, it was decided to incorporate the original results of the investigation into a review of the general subject of the kinetics of austenite decomposition. This review constitutes the present article. An extended summary of the results is given on pages 576 to 579. This article, which deals with the kinetics of phase transformations, may appropriately be considered as a sequel to two recent articles1,2 from this laboratory on the equilibrium relations in medium alloy steels.

Jan 1, 1947

By Clarence Zener

Contents Page General Principles................................................................. 551 Equilibrium Diagrams......................................................... 551 Nucleation................................................................... 553 Kinetics of Phase-boundary Propagation......................................... 553 Kinetics of Growth of New Phase............................................... 556 Spheroidization............................................................... 560 Pure Iron-carbon System........................................................... 561 Formation of Pearlite.......................................................... 561 Interlamellar Spacing........................................................ 562 Pearlite C-curve............................................................. 563 Bainite Formation............................................................. 565 Formation of Martensite....................................................... 570 Loss of Tetragonality of Martensite.............................................. 570 Effect of Alloys upon Decomposition of Austenite..................................... 572 Pearlite...................................................................... 572 Bainite and Martensite......................................................... 574 Time-temperature-transformation Curves............................................. 576 Appendix A............................................................. ......... 579 Appendix B....................................................................... 580 Appendix C....................................................................... 580 The present investigation started in an attempt to understand certain details of the decomposition of austenite, and of the effect of alloying elements thereon. As the investigation proceeded it became apparent that not only these details, but all the general features of austenite decomposition, could be understood in terms of fundamental physical principles. Since the direct applicability of these physical principles has not been previously recognized, it was decided to incorporate the original results of the investigation into a review of the general subject of the kinetics of austenite decomposition. This review constitutes the present article. An extended summary of the results is given on pages 576 to 579. This article, which deals with the kinetics of phase transformations, may appropriately be considered as a sequel to two recent articles1,2 from this laboratory on the equilibrium relations in medium alloy steels.

Jan 1, 1947

By C. E. Jr. Reistle

ALL crude oils become more viscous when chilled, but the only oils that congeal and precipitate paraffin to such an extent as actually to cause production troubles are those that contain an appreciable amount of wax. In handling these oils of high wax content, three distinct problems may be encountered before the oil reaches the refinery. The first problem is the precipitation and accumulation of paraffin in the well, either in the oil string or on the face and in the pores of the sand. The second problem is the accumulation of paraffin and the congealing of oil in the transportation lines. The third problem is the precipitation of paraffin and the formation of tank bottoms.

Jan 1, 1927

By Ronald D. Hill, Janet L. Auerbach

It is estimated that the mining industry generates over 2.1 petrograms per year of solid waste. These wastes are overburden, waste rock, low grade ore, concentrate (tailing), sludges from treatment facilities and wastes from support facilities. Concentrate wastes present the greatest environmental problem. Three recent federal laws impact the disposal of waste from mining: Resource Conservation and Recovery Act of 1976 (RCRA), Surface Mining Control and Reclamation Act of 1977, and Uranium Mill Tailings Radiation Control Act of 1978. RCRA specified a special study on mining waste which was to include an analysis of: (1) sources and volumes of waste produced, (2) present disposal practices, (3) potential dangers to health and the environment, (4) alternate disposal methods, (5) cost of disposal, and (6) potential for use of waste.

Jan 1, 1980

By J. Wright, R. J. Tyler

Changes in the pore structure of hematite pellets during reduction to iron were investigated under a variety of experimental conditions. Pore structures were determined by a mercury penetration technique. The reduction temperature was found to be the major influence on structure, the pore system being bimodal below 7.50°C and unimodal above 900°C. Pore sizes ranging front 0.04 to 20 m were observed in complete metabolized pellets produced over the temperature range 650 - 1050°C. Gas composition and reduction rates also influenced the pore structure of the metabolized pellets, although to a lesser extent than the reduction temperature. An apparent activation energy of 24 kcal/mole was derived for the pore formation process.

Jan 1, 1979

By M. Albertson

AT Cobalt, Ontario, Canada, a lake was drained to facilitate mining, by the Mining Corpn. of Canada, during the spring and early summer of 1915. Previous to pumping out the water, great quantities of sands and slimes from concentrating plants had been discharged into the lake and during and after the lake's drainage, its basin' Was a receptacle for tailing products. As the writer was at work along the shore line as the lake was being drained, he had a good opportunity to observe the changes that took place as the water was withdrawn. Some adjustments between the incoming sands and the mud in the lake had taken place before pumping was commenced. One of the most interesting results was the appearance of a small dome in a path the writer traversed twice a day for several months; he remembers distinctly the difficulty of crossing this. Cobalt Lake owed its existence to the gouging out of a rock basin by glaciation. The long axis of the basin closely follows the strike of a thrust fault of about 500 ft. (160 in.) vertical displacement. The lake was originally shaped somewhat as shown. The length was about 3000 ft. (914.4 in.), the width at the lower lobe was about. 1000 ft., and the, width at the narrows probably. 400 to 500 ft. The original depth of water varied from 20, to 30 ft. (6.1 to 9.1 m.), near where the island later was formed, to 60 to 70 ft. (23.6 to 27.6 m.) in the widest part of the lower lobe. At the narrows, the depth was 30 to 40 ft.. Above the bed rock was sand, with boulders near the bottom, and mud. During the building of the railroad in 1903, considerable filling was done along the right of way. With the commencement of mining operations, about 1905, waste rock and mill tailings were dumped into the lake. One of the mining companies operated a hydraulic giant to remove the glacial debris from several hundred acres of rock surface; much of the sand and clay from this operation was deposited in the. lake. About

Jan 2, 1921

By E. N. Aqua, C. M. Owens

The microstructure in cold-rolled Zircaloy-4 sheet has been characterized by transmission electron microscopy and X-ray diffraction line-broadening techniques. The rate of work hardening is measured in terms of microstrains, particle sizes, micro hardness, the tensile yield strength. The tensile yield strength of Prestrained (less than 10 pet) Zircaloy-4 sheet is found to be a linear function of the square root of the dislocation density, us predicted by work-hardening theories. ZIRCALOY is one of the most important structural alloys for nuclear reactor applications.''2 In order to understand the effects of the nuclear environment on Zircaloy structural members, the preirradiation microstructure must be characterized. This report will describe the substructure existing in cold-rolled Zircaloy-4 sheet as observed by electron microscopy and by X-ray diffractometry. I) EXPERIMENTAL PROCEDURE Commercial Zircaloy-4 strip was cold-rolled and annealed in argon at 760°C for 1 hr. The annealed sheet was again cold-rolled to a maximum of 50 pct reduction in thickness with segments of the strip being retained after 2.5, 5, 10, and 20 pct reductions. One-inch-square portions were chemically polished (50 parts HNO3, 50 parts H2O, 10 parts HF) and mounted onto lucite blocks using double-sided tape to prevent bending during handling. Thin foils suitable for transmission electron microscopy were prepared from each of the sheet samples. The sheets were mechanically polished through 600-grit paper to a 3 to 4 mil thickness and were chemically polished by immersion for 1 to 2 min in a solution containing: 50 parts HNO3, 50 parts H2O, and 10 parts HF, and rinsed in distilled water. The final metallographic procedure was to electropolish for less than 10 sec with an electrolyte of 10 parts perchloric acid and 90 parts glacial acetic acid, at 20 v, at room temperature using stainless-steel point cathodes and a vigorous stirring action. These foils were examined in a Philips EM 200 electron microscope using a 100-kv accelerating voltage.

Jan 1, 1968

By H. H. Stout

THE high copper price during the war stimulated the capacity to produce far beyond a possible normal consumption. The curves in Fig. 1 show this. The line YZ indicates, what the world production and consumption of primary copper nor-mally would have been, had not the 3 war upset the rhythm of industry. Granting that the maximum world production of 3,150,000,000 lb. in 1917 was obtained by crowding both the equipment and the grade of ore mined, yet in many instances addi-tional capital was invested, upon which the producer will be driven to earn at least a reasonable interest. During the period of deflation, or C liquidation, which we have entered, the consumption of copper will nat-urally be below the normal. Fig. 1 shows the actual consumption ex-pressed-in per cent. of normal : 1919, 96 per cent.; 1920, 94 per cent.; 1921, 70 per cent. The copper industry, consequently, faces: (1) Potential over-production capacity; I (2) Diminished consumption. To deny the existence of these factors would be illogical. To minimize the menace of this potential producing capacity might lead to regret. All past attempts at artificial control have not proved very successful. Obviously the remedy is to increase, if possible, the consumption of copper to a point where these factors cease to be dangerous. In ultimate analysis, this means: Produce an article so cheap and so good that no one can afford to use substitutes. The problem can be solved by: (1) Re-ducing price; (2) Improving product offered.

Jan 11, 1922

By Gilbert P. Moore

The production of petroleum in Argentina during 1932 was greater than in any preceding year. The total amount produced was 13,012,-756 bbl. This figure represents a decided increase in the production of oil in Argentina and can be attributed largely to the increased activities of the Royal Dutch operations at Kilometer 27 in the Comodoro Rivadavia district and those of the Standard Oil Co. of Argentina in the Salta region. Production from the Royal Dutch wells amounted to considerably more than 10,000 bbl. per day over quite a long period during the year. Of the 13,012,756 bbl. of oil produced during 1932, the Government Agency (Y. P. F.) was responsible for some 5,578,954 bbl., while 7,433,-802 bbl. were produced by private companies. Although in 1930 the Y. P. F. produced about 55 per cent of the total production and in 1930 50 per cent of the total, in 1932 its production had fallen to about 42 1/2 per cent of the total. This decrease in the proportion produced by the Y. P. F. was not due to a decrease in its production but to the decided increase in the production of private companies, particularly those two whose names are given above. One new discovery was recorded during the year. This was a well drilled by the Y. P. F. at Cacheuta in the province of Mendoza which came in pumping 80 bbl. of 15" gravity oil from a depth of 673 meters. This well was completed at approximately the same time that the provincial government of Mendoza granted a concession to the Y. P. F. and declared the whole province as a reserve in which no concessions may be granted for a period of 10 years. Under these conditions, the federally operated Y. P. F. will have no competition in the Cacheuta area from private companies such as they have in the other three fields of the republic. The Y. P. F. wildcat west of Comodoro Rivadavia at Pampa Maria Santissima on the Rio Senguerr was completed as a dry hole during 1932. This we11 was reported as a producer in 1930 by persons returning to this country from Argentina but it appears that they had been misinformed. The site of this test has been abandoned and a new location made north of the Rio Senguerr.

Jan 1, 1933

By A. S. Adams

A GLANCE at the list of papers1 that have been published since 1920 on the general subject of flotation suggests the variety of ideas that exist regarding the underlying cause of the phenomenon. Among these ideas, we find several coherent theories, but none that adequately and completely explain the phenomenon. Ralston2 and Bains3 suggested an electrical theory which has to do with the attachment of mineral particles to a bubble by the attraction of unlike electrical charges. This theory has its root in the fact that a bubble of air in pure water is electrified in such a way that the surface is negatively charged while the body of the water is positively charged. This result, which was first discussed by Helmholtz, also obtains when the water contains anything except a high concentration of hydrogen ions, or ions which are trivalent or tetravalent. The theory also holds that the mineral particles have positive charges on their surfaces, and so there is attachment between bubble surface and mineral particles by the old law of electrostatics, which states that unlike charges attract.

Jan 1, 1928

By M. F. Goudge, R. V. Tomkins

Sodium sulfate is an important chemical raw material that is being used in ever increasing quantities in modem industry. It comes on the market mainly in three forms: (1) Salt cake (Na2SO4), a relatively crude form of the anhydrous salt containing up to 3 pct impurities, and comprising about three-quarters of the North American production; (2) anhydrous sodium sulfate (Na2SO4), a refined form containing less than 0.3 pct impurities; (3) Glauber's salt, the decahydrate (Na2SO4.10H2O) containing 55.9 pct water of crystallization. Competing with salt cake from natural sources is byproduct salt cake from a number of industrial operations. For instance there is Mannheim cake resulting from the manufacture of hydrochloric acid by the reaction of salt (NaCl) and sulfuric acid in Mannheim furnaces; rayon cake, a high-quality product, resulting from the process of making rayon by the viscose process; chrome cake from the processing of sodium chromate to dichromate; phenol cake from the manufacture of phenol; also byproduct sodium sulfate resulting from the conversion of lithium sulfate to lithium carbonate; from the processing of sodium tetraborate to boric acid; from converting sodium formate to formic acid; and in the manufacture of cellophane. Sodium sulfate is also obtained as a byproduct in the manufacture of isosebasic acid. In addition strong competition has resulted from imports of byproduct salt cake from the chemical industries of a number of highly industrialized European countries. However in recent years due to technological changes the output of byproduct salt cake is rapidly diminishing and the natural deposits are becoming the most important source of sodium sulfate used in North America. For instance, the production of hydrochloric acid in Mannheim plants has become obsolete and thus the greatest single source of byproduct salt cake will disappear. Also viscose rayon is meeting strong competition from nylon and other synthetic fibers and production of byproduct salt cake from this source may do little more than remain at the present level. These factors together with increasing demand for sodium sulfate point to increased production from natural sources. In 1958, 36 pct of the total United States production was derived from natural sources and in addition 97,000 short tons was imported so that over 40 pct of the total consumption was supplied by natural sources whereas in 1953 the proportion was only 24 pct. The rapid growth in production is shown in Table 1. Principal Sodium Sulfate Minerals There are many minerals in which sodium sulfate is a constituent but those of commercial importance at the present time, and consisting wholly of sodium sulfate in either the hydrous or anhydrous states, are mirabilite and thenardite. Minerals of potential importance but consisting in part of compounds other than sodium sulfate are glauberite in which sodium sulfate is combined with calcium sulfate, and blödite in which sodium sulfate is combined with magnesium sulfate. Mirabilite (Na2SO4.10H2O) the decahydrate, consists of 24.8 pct SO3, 19.3 pct Na2O, and 55.9 pct H20. It is a colorless to opaque, soft, water-soluble mineral with a slightly bitter taste. It has a hardness of less than 2 and a specific gravity of 1.48. It forms long needle-

Jan 1, 1960

By D. M. S. Bhatia

The intimate association of iron formation and massive sulfide deposits has been documented by many investigators (McAllister, 1960; Hutchinson, 1972). In a classic example of such an association, four distinct facies (oxide, carbonate, silicate and sulfide) were identified in an Algoma-type iron formation occurring in Paleozoic rocks, but exhibiting physical features (banding) and chemical composition (SiO2 + A1203 + Fe2O3) typical of Precambrian iron formations. The geochemical variation of the iron formation was studied by analyzing 62 samples for Si, Al, Fe, P, S, and Cop. Following a model proposed by James (1954), the geochemical and mineralogical variation of the iron formation can be used as an excellent tool for the exploration of massive sulfide deposits.

Jan 1, 1982

By R. W. Smith, T. J. Smolik

Recent work indicates that fluosilicate ions and or fluosilicate amine complex ions play important roles in fluoride activation of bery and feldspars in cationic flotation systems. In an attempt to further confirm conclusions reached in this earlier work, an infrared and X-ray diffraction study of the action of fluorides in cationic flotation of silicates was undertaken. It should be noted that in recent years considerable use has been made of infrared spectroscopy in studies of surface phenomena,2-4 and these studies have established the excellence of the technique. In the present study infrared spectra were obtained for various products of the reaction of HF and dode-cyclamine hydrochloride with quartz, microcline and beryl. These spectra were compared with one another and spectra of several inorganic and organic substances. In addition to infrared spectra, X-ray diffraction patterns were run on many of the reaction products obtained. A model 137 B Perkin Elmer Infracord double beam spectrophotometer containing sodium chloride optics was used to obtain the spectra. Infrared samples, except for the dodecylamine sample, were prepared by thoroughly mixing the substance to be analyzed with purified KBr in steel vessels on a Wig-L-Bug amalgamator. One-half in. pellets weighing 100 mg were than pressed under vacuum. Details concerning percentage of the substance to be analyzed in the pellets are indicated on the spectra. In all cases except for dodecylamine a KBr pellet was placed in the reference beam of the spectrophotometer to cancel bands due to possible impurities in the KBr. The dodecylamine sample on which an infrared spectrum was obtained was prepared by placing pure dodecylamine between two salt plates. X-ray diffraction patterns were obtained with a Norelco instrument, using a copper target tube, nickel filter and Geiger tube detector. The samples were ground into powders and either placed on glass slides with glue or placed into regular powder sample holders for diffraction studies. Goniometer scanning speed varied from 1°/min to 1/8°/min. EXPERIMENTAL RESULTS AND DISCUSSION Infrared spectra of microcline, beryl and quartz were obtained first and are shown in Fig. 1. These spectra agree reasonably well with previously reported spectra.5,6 The microcline (Fig. 1 a) and beryl (Fig. 1 b) were from good Black Hills crystals. The quartz (Fig. 1 c) was from a large, clear Brazilian crystal. It is not clear whether absorption bands of minerals are due to molecular vibrations, lattice vibrations or both. Fig. 2 shows infrared spectra of dodecylamine (Fig. 2 a) and dodecylamine hydrochloride (Fig. 2 b). The dodecylamine spectrum checks quite closely with that reported in Sadtler.8 Little is known of the spectra of hydrochloride salts of amines,9 and a spectrum of dodecylamine hydrochloride could not be

Jan 1, 1965

"Salt Lake City was founded July 24, 1847, by Mormons under the leadership of Brigham Young. It had a population of 118,110, according to the United States census of 1920, and of 151,968, according to a civic survey, January 1, 1925. The Salt Lake metropolitan district is estimated to have 180,079 inhabitants.The city's area is 51.8 square miles. Its altitude is 4,354 feet. It has 514 miles of streets, of which 91 miles are paved. The streets in the main portion of the city are 132 feet wide and the blocks are 660 feet long. It has 144 miles of street railway and 2,200 electric street lights. It has fifteen parks and playgrounds, forty public schools and six private schools. It has eighty-nine church edifices, of which forty-nine are devoted to denominations other than the Mormon Church.It is the capital of the state, the headquarters of the Mormon Church and the headquarters of most of the mining companies and many other business institutions of the state and adjacent territory.The State University, the State School of Mines and the Inter-Mountain Experiment Station of the U. S. Bureau of Mines are in the eastern part of the city. Fort Douglas is east of the University the country clubs are in the southeastern portion of the city. The State Fair Grounds are in the northwestern portion.The city is the center of a trade area about 700 miles in diameter and is in almost the exact center of a circle that embraces the principal western mining areas. It is the home of several widely-known mining supply companies and all the other leading concerns in that line are represented by agency or branch house."

Jan 1, 1925

By Jack R. Elenbaas, Donald L. Katz

A radial turbulent flow formula has been developed which permits the computation of the pressure drop for radial flow in gas wells whether the flow is laminar, turbulent, or partially laminar and partially turbulent. Using the formula a complete back-pressure curve has been calculated and analyzed by a comparison with existing back-pressure curves. A procedure is presented for computing the permeability of the porous media when the porosity, sphericity and average particle diameter are known. Introduction Recent studies on the flow of fluids through porous media have provided new methods for computing flow under turbulent conditions. The present study was initiated as an investigation of pressure -drop computations for flow through porous sands in gas wells and as an analysis of present-day back-pressure tests. Although laminar flow exists in the producing formation of gas wells under normal flow rates, turbulent flow does take place adjacent to the well bore. As the flow rate is increased, turbulent flow exists further and further into the producing formation and under open-flow conditions in relatively deep wells a considerable portion of the flow through the sand will be turbulent. In order to calculate a complete backpressure curve for relatively deep wells without resorting to extrapolation of curves drawn from data obtained in back-pressure tests or calculations made in the completely laminar region, it was necessary to have some means of calculating pressure drop for flow through porous sands under turbulent conditions.4.6 As a result, a radial turbulent flow formula was derived from the equations recently presented by Brownell and Katz.1 A rigorous solution of this equation involves a trial and error graphical integration. By making the assumptions of average viscosity, temperature and compressibility factor of the flowing gas, it was possible to graphically integrate the relation for the general case of radial flow of gases through the porous sand of gas wells. A chart has been prepared relating the value of the integral to the radius of the well bore and to the Reynolds number of the flowing gas. The chart actually performs the graphical integration and the solution of the formula is then reduced to a simple trial and error calculation. A typical back-pressure curve is calculated to show the transition from laminar to turbulent flow. This paper will briefly describe the pro-cedurcs for computing flow through porous media, including a method for predicting sand permeability from porosity and grain size and shape. Flow through Porous Media Brownell and Katzl recently reported a correlation for computing flow of fluids through porous media by the use of an enlarged Reynolds number and friction factor in which the porosity of the bed is included as an additional prime variable. The enlargcd Reynolds number com-

Jan 1, 1948

By Donald L. Katz, Jack R. Elenbaas

A radial turbulent flow formula has been developed which permits the computation of the pressure drop for radial flow in gas wells whether the flow is laminar, turbulent, or partially laminar and partially turbulent. Using the formula a complete back-pressure curve has been calculated and analyzed by a comparison with existing back-pressure curves. A procedure is presented for computing the permeability of the porous media when the porosity, sphericity and average particle diameter are known. Introduction Recent studies on the flow of fluids through porous media have provided new methods for computing flow under turbulent conditions. The present study was initiated as an investigation of pressure -drop computations for flow through porous sands in gas wells and as an analysis of present-day back-pressure tests. Although laminar flow exists in the producing formation of gas wells under normal flow rates, turbulent flow does take place adjacent to the well bore. As the flow rate is increased, turbulent flow exists further and further into the producing formation and under open-flow conditions in relatively deep wells a considerable portion of the flow through the sand will be turbulent. In order to calculate a complete backpressure curve for relatively deep wells without resorting to extrapolation of curves drawn from data obtained in back-pressure tests or calculations made in the completely laminar region, it was necessary to have some means of calculating pressure drop for flow through porous sands under turbulent conditions.4.6 As a result, a radial turbulent flow formula was derived from the equations recently presented by Brownell and Katz.1 A rigorous solution of this equation involves a trial and error graphical integration. By making the assumptions of average viscosity, temperature and compressibility factor of the flowing gas, it was possible to graphically integrate the relation for the general case of radial flow of gases through the porous sand of gas wells. A chart has been prepared relating the value of the integral to the radius of the well bore and to the Reynolds number of the flowing gas. The chart actually performs the graphical integration and the solution of the formula is then reduced to a simple trial and error calculation. A typical back-pressure curve is calculated to show the transition from laminar to turbulent flow. This paper will briefly describe the pro-cedurcs for computing flow through porous media, including a method for predicting sand permeability from porosity and grain size and shape. Flow through Porous Media Brownell and Katzl recently reported a correlation for computing flow of fluids through porous media by the use of an enlarged Reynolds number and friction factor in which the porosity of the bed is included as an additional prime variable. The enlargcd Reynolds number com-

Jan 1, 1948