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By Morris Muskat

A theory is presented for calculating the performance history of complete water-drive systems producing from idealized stratified formations. The general equations are applied to systems where the permeability stratification is either of the exponential or linear type. Calculations were carried through for different degrees of permeability stratification, but with special emphasis on the effect of the mobility ratio between the produced oil and the invading water on the resultant performance. These results are also expressed graphically as curves for the initial water breakthrough recovery, for the different degrees of stratification, as a function of the mobility ratio, and of the composition of the produced fluid stream as a function of the cumulative oil recovery. For several typical cases the latter has also been plotted as a function of the cumulative oil and water throughflow. The general result is that when the mobility of the oil is lower than that of the invading water the channelling tendency resulting from the permeability stratification becomes aggravated as the higher permeability zones become flooded out. Situations of this type would obtain when producing low gravity or highly viscous oils. Conversely, if the mobility of the oil is high compared to that of the invading water, the flooding of the high permeability zones will lead to a retarding and choking effect, and the gross bypassing phenomena will be partially suppressed. These conditions would correspond to those of flooding high gravity or low viscosity oils. A discussion is given of the various basic assumptions made in the analysis, including that of ignoring the stripping phase of the production history as implied by relative permeability concepts. INTRODUCTION The physical ultimate recoveries from oil reservoirs are basically determined and limited by the physical oil displacement processes associated with the reservoir producing mechanism. In practice, however, the economic ultimate recoveries are further limited by the mobility of the reservoir fluids and the uniformity and continuity of the producing formation. In fact, it is the differential depletion between the component parts of the composite reservoir which ultimately determines the total recovery at the time of field abandonment. While this observation applies to both the solution gas drive and gravity drainage mechanisms, in which use is made only of the energy contained within the original oil-bearing reservoir, it is of even more paramount importance under operations wherein the energy associated with extraneous fluids provides the ultimate oil expulsion mechanism. Whether the invading fluid is the water from an edgewater drive, water injected for pressure maintenance, gas injected for pressure maintenance, or gas returned to the formation in a cycling program, it is often the continuity and uniformity of the producing section which will control the economic efficiency of the operations. The importance of the problem of reservoir non-uniformity does not, of course, lessen its complexity or the difficulties of its solution. In fact, these are inherently such that the concept of a "general" solution is virtually meaningless. About all that can be reasonably hoped for is the analysis of specific and well-defined types of non-uniformity which may give some degree of approximation to actual reservoir conditions. Since variations in the nature of the reservoir which depend only on the position along the streamlines will not lead directly to major differential depletion development within the reservoir, the types of non-uniformity considered thus far have involved stratification assumptions. That is, the producing section has been replaced by a multi-layer "sandwich," each uniform areally, and differing from the others only in its basic physical constants as to thickness, porosity and permeability. The fluid motion in the composite system is thus approximated by a parallel superposition of the independent fluid movements in the individual strata. For the specific application to cycling operations the theory of the effect of permeability stratification has been developed for both discontinuous' and continuous types of permeability stratification. Among the latter, treatments have been given of systems in which permeability distributions are governed by exponential, linear' or probability3 functions. In all these studies complete dynamical equivalence was assumed between the injected dry gas and the displaced wet gas. The overall effective permeability of each stratum was therefore considered as constant and independent of the degree of invasion of the injected fluid. In the case of the displacement of oil by water, the assumption of dynamical equivalence between the water and oil will be strictly valid only by accident. Even if the oil viscosity should be the same as that of the water, the effective permeability to the oil in the presence of the connate water will in general be quite different from that of the water behind the water-oil interface flowing past the trapped residual oil. As a result the effective permeability for the stratum as a whole and rate of water invasion will change with time as the intrusion continues. The differential fluid motion in the individual strata will thus also vary with time. Qualitatively, it is easy to predict the resultant effects. If the permeability to viscosity ratio of the invading fluid exceeds that of the fluid displaced, the stratification and bypassing effect of the perme-

Jan 1, 1950

By E. H. Jr. Dix

OF all the alloying elements used in commercial aluminum alloys, copper stands out as by far the most important, and it is perhaps for this reason that the constitution of the aluminum-copper system has received the attention of a number of experimenters covering a period of approximately 20 years. It would be expected that during this time the equilibrium relations had been thoroughly and exactly worked out. In a general way this is true, but there is a surprising lack of agreement in the details of the high aluminum end of the diagram, particularly in regard to the solid solubility of copper in aluminum, which has been given variously as 2 per cent. to 5 per cent. at the eutectic temperature. The importance of this is much greater than might at first appear to one unfamiliar with modern commercial aluminum alloys. This is true because, with one exception, the susceptibility of the strong alloys to heat treatment depends, in part at least, upon the solid solubility relations of copper in aluminum. In 1919, Merica, Waltenberg, and Freeman1 published the first detailed information on these solubility relations. They found that the solubility of copper decreases with falling temperature from about 4 per cent. at 5.25° C. to about 1 per cent. at 300° C. and is apparently still diminishing at lower temperatures. In 1921 there was published a brief account of the results obtained in the National Physical Laboratory, Teddington, England.2 This laboratory, however, reported that 5 per cent. of copper is soluble in aluminum at 540° C., and that this amount decreases to about 3 per cent. at 20° C.

Jan 2, 1926

By E. T. Turkdogan, P. Grieveson, J. F. Beisler

Experimental results are given for the rate of reduction of silica from silicate melts by paphite-saturated iron in the presence of carbon monoxide. It is shown that, when gas bubbles are present at the slag-metal interface, the interfacial chemical reaction is the rate-controlling process in the transfer of silicon from slag to metal. From the silicon-concentration profiles in the metal sample, the diffusivity of silicon in paphite-saturated iron is found to be (5.4 * 0.4) x 10 sq cm per sec at 1600°C. Some consideration is given to the transport of oxygen from the slug-metal interface to the walls of the gvaphite metal container as becoming the rate-controlling process in the absence of gas bubbles at the slag-metal interface. ATTEMPTS were made by Fulton and chipman1 to investigate the kinetics of reduction of silica from calcium alumino-silicate melts by graphite-saturated iron in an atmosphere of carbon monoxide. In these experiments with 400 to 500 g of metal and slag, both phases were stirred in such a manner that the interface was essentially undisturbed. Initially, the graphite-saturated iron contained about 15 pct Si and, after about 8 or 10 hr of reaction time at 1600°C, the silicon content of the metal in- creased by about 0.2 pct. In a written discussion on this paper of Fulton and Chipman, schuhmann2 pointed out the possibility that this observed slowness of the reduction of silica could be due to the transport of oxygen across the liquid metal boundary layer at the slag-metal interface. Although this explanation of Schuhmann is quite feasible for the experimental data of Fulton and Chipman on silica reduction, such a reasoning should not be extended to other rate data on slag-metal reactions unless sufficient evidence is available satisfying the boundary conditions for such a transport process. The hydrodynamics of stirred liquids are very complex and even for a simple method of stirring there is no exact mathematical solution which will predict the mass transfer coefficient accurately. The best available solution of this problem is that given by Lewis' and later modified by mayers~ for a two-liquid system where liquids are stirred by counter-rotating stirrer blades without disturbing the interface. In such a system the mass transfer coefficient is estimated from an empirical relationship using the data on viscosity and density of the two liquids, rate of revolution, the length of stirrer blades, and the diffusivity of the element which is transferred from one phase to the other. It was found in these investigations' that the estimated and observed mass transfer coefficients, in systems containing two immiscible aqueous solutions, agreed within *40 pct. Although this is not considered to be an unreasonable error for the study of industrial processes, uncertainties of this magnitude cannot be tolerated in any theoretical study of reaction kinetics. In view of the complexity of the

Jan 1, 1963

By G. R. Speich, D. J. Mack

The transformation of was studied by isothermal methods. At all temperatures, the ß transforms quickly to fine grained ß" which develops silver-rich striations. At higher temperatures the striations disappear, the final structure being Widmanstatten a in ß'. At lower temperatures, the striated ß" is consumed by pearlite nodules of a + ß' which in turn form the Widmanstatten a in ß'. The transformation is unlike any previously reported for a eutectoid. AS part of a general program of study on the eutectoid reaction, the Ag-Cd eutectoid seemed particularly attractive because the ß phase which undergoes the reaction is another of the typical "electron compounds," AgCd, having the disordered body-centered cubic structure. This work deals with the eutectoid which occurs at 50.5 wt pct Ag, and not with the eutectoid which has been reported at 42.9 pct Ag.' There has been no work reported specifically on the Ag-Cd eutectoid which was studied. The only information available on this eutectoid is that work done in the determination of the phase diagram. For the most part, the terminal portions of the diagram, Fig. 1, are well established; however, the region from 40 to 60 pct Cd is still somewhat in doubt. The X-ray evidence'..' in this region is in conflict with the results of thermal analysis. X-ray analysis3 does not show a polymorphic transformation at about 240°C as indicated by thermal analysis.1,5 Even the X-ray evidence, within itself, is confusing in this region of the diagram. One investigator2 reports the ß phase as hexagonal close-packed and has found three phases in the one-phase ß field. Other X-ray investigators"' ' report the ß phase as body-centered cubic. The authors believe that much of the confusion resulted from incomplete homogenization of the alloys. It was found that it takes almost a week to produce a completely homogeneous two-phase structure at 370°C. Another possible source of error is the volatilization of cadmium from the surface and cracks of the specimen during annealing with the possible formation of a cadmium-poor phase. The only metallographic data available on this eutectoid alloy are those of Durrant1 and Fraenkel and Wolff.5 his information is very limited. The phase diagram chosen as most nearly correct is that in the Metals Handbook," which is almost identical with the phase diagram suggested by Durrant.1 Materials and Procedure The alloy was prepared from high purity silver (999.5 plus fine) and high purity cadmium (0.02 Pb, 0.005 Cu, trace Fe). The metals were melted in carbon under a nitrogen atmosphere. Some cadmium was lost by volatilization but by remelting several times and adding cadmium a 200 g ingot of approximately eutectoid composition was obtained. Because the CdO fumes are quite toxic, the exit gas from the furnace was led through a water trap to condense the CdO. The ingot was homogenized at 650°C for 24 hr, no attempt was made to prevent decadmiumization. This resulted in a narrow decadmiumized rim on the ingot which was removed. Filings for analysis were obtained from cross sections of the ingot. Silver was determined by the Volhard thiocyanate method,? cadmium by difference. The ingot analyzed 50.8 pct Ag at the top of the ingot and 50.7 pct at the middle and at the bottom of the ingot. This alloy was only slightly hypo-eutectoid compared to the accepted value of 50.5 pct Ag and was used for the isothermal studies. It was felt that further attempts to adjust the composition might be futile because of the volatility of the cadmium. The ingot was sectioned to specimens approximately 3/4 x 3/4 x 1/8 in. These specimens were "austeni-tized" for 3 hr in air at 650°C to yield homogeneous ß and then quenched into a salt bath held at constant subeutectoid temperatures where the transformation

Jan 1, 1954

By R. A. Burton, J. T. Hansford, D. A. T. Donohue

Substantial evidence indicates that many petroleum producing horizons contain naturally occurring, ordered fracture systems and that within a particular geologic zone, vertical fractures induced in wellbores often will be directed along a particular compass direction. Both conditions will seriously alter the fluid displacement behavior within reservoirs. In this study the effect of induced fracture orientation and length on sweep efficiency is determined for a five-spot pattern. In general, it is assumed that all wells are fractured and directed along the same compass direction. Using the electrical analog to steady state, two-dimensional fluid flow in porous media, boundary conditions are obtained from which flood fronts are tracked numerically. The numerical computations require a particle tracking routine for approximating flood front histories. It is shown that recovery is sensitive to the length and orientation of fractures for the pattern studied. with the proper design of fracture-pattern systems, recovery can be enhanced considerably. INTRODUCTION Hydraulic fracturing introduced in 1949l gave the industry a rather inexpensive means of increasing the fluid injection or production capacity of wells. It has been used with particular success to increase the production rate of wells completed in tight formations, such as in western pennsylvania2 where producers have fractured in depleted or near-depleted fields and observed economic responses. Once the natural energy declines in such a reservoir where all wells have been fractured, waterflooding is generally suggested as means of further increasing recovery. Of the dual objective sought in waterflooding — high injectivity and high breakthrough sweep efficiency — the former condition can be obtained if all wells in the flood pattern are fractured; the latter condition should depend on the nature of the fracture system. Considerable theoretical work has been published on the nature of fractures induced in boreholes.3-5 Although discussion persists concerning the possibility of forming a horizontal fracture at a given point within the wellbore, it is generally conceded that only vertical fractures will develop below a given depth, i.e., where the fracturing Pressure is less than the overburden load. Given the fact that fractures will be vertical in most cases of interest, it is also important to know whether there is order to fracture orientations within a given geological region. Kehle has suggested that in tectonically relaxed areas of uncomplicated geology, the stresses are fairly uniform and all fractures in the region should be parallel.5 Dunlap arrived at a similar conclusion in a theoretical investigation of localized stress conditions surrounding the borehole.6 He concluded that most vertical fractures are propagated in a preferred azimuthal direction. Fraser and Pettitt, in extending these theoretical suggestions to a specific field case, used an impression packer to record both a vertical fracture and the orientation of this fracture in the wellbore of a well in the Howard Glasscock field, Tex.7 Use of this information enhanced the waterflood recovery of the field. Anderson and Stahl also used impression packers on three fractured wells in the Allegheny field. N. Y.. and found that the fractures were oriented more or less along the same compass direction.8 Orientation of the fractures in this manner depends on the stress condition within the formation during fracturing. Elkins and Skov have demonstrated that a natural, oriented, vertical fracture system exists within the Spraberry field? Heck suggests

Jan 1, 1969

By Walter Rose, W. A. Bruce

Improved apparatus, methods, and experimental techniques for determining the capillary pressure-saturation relation are described in detail. In this connection a new multi-core procedure has been developed which simplifies the experimental work in the study of relatively homogeneous reservoirs. The basic theory concerning the Leverett capillary pressure function has been extended and has been given some practical application. Some discussion is presented to indicate the relationship of relative permeability to capillary pressure, and to provide a new description of capillary pressure phenomena by introducing the concept of the psi function. INTRODUCTION For the purposes of this paper the capillary character of a porous medium will be defined to express the basic properties of the system, which produce observed results of fluid behavior. These basic properties may be classified in the following manner, according to their relationship to: (a) The geometrical configuration of the interstitial spaces. This involves consideration of the packing of the particles, producing points of grain contact, and variations in pore size distribution. The packing itself is often modified by the secondary processes of mineralization which introduces factors of cementation, and of solution action which causes alteration of pore structure. (b) The physical and chemical nature of the interstitial surfaces. This involves consideration of the presence of interstitial clay coatings, the existence of non-uniform wetting surfaces; or, more generally, a consideration of the tendency towards variable interaction between the interstitial surfaces and the fluid phases saturating the interstitial spaces. (c) The physical and chemical properties of the fluid phases in contact with the interstitial surfaces. This involves consideration of the factors of surface, interfacial and adhesion tensions; contact angles; viscosity; density difference between immiscible fluid phases; and other fluid properties. Fine grained, granular, porous materials such as found in petroleum reser~oir rock possess characteristics which are expressible by (1) permeability, (2) porosity, and (3) the capillary pressure-saturation behavior of immiscible fluids in this medium. These three measurable macroscopic properties depend upon the microscopic properties enumerated above in a manner which defines the capillary character. Systems of capillary tubes or regularly packed spheres may be thought of as ideal and numerous references can be cited in which exact mathematical formulations are developed to show the relationships governing the static distribution and dynamic motion of fluids in their interstitial spaces. The capillary character of non-ideal porous systems such as reservoir rock also is basic in determining the behavior of fluids contained therein; although, in general, the connection is not mathematically derivable but must be approached through indirect experimental measurement. This paper gives consideration to the evaluation of petroleum reservoir rock capillary character. The methods employed may be applied to the solution of problems in other fields, and the conclusions reached should contribute to the basic capillary theory of any porous system containing fluid phases. In this paper, a modification of the core analysis method of capillary pressure is employed and it is intended to show that the capillary character of reservoir rock can be expressed in terms of experimental quantities. A very general method of interpretation correlating the capillary pressure tests with fundamental characteristics such as rock texture, surface areas, permeability, occasionally clay content and cementation is introduced. Eventually an attempt is made for establishing a method of deriving relative permeability to the wetting phase from capillary pressure data. The experimental evaluation of capillary character must be approached in a statistical manner if reservoir properties are to be inferred from data on small cores. This is implied by the heterogeneous character of most petroleum reservoirs, and suggests that considerable intelligence should be applied in core sampling. Finally, this paper supports the view that once the capillary character of a given type of reservoir rock has been established by core analysis, fluid behavior can then be inferred in other similar rock. Although no great progress has been made in establishing what variation can be tolerated without altering the basic fluid behavior properties, evidence will be presented to indicate that certain reservoir formations are sufficiently homogenous with respect to capillary character that the data obtained on one core will be useful in predicting the properties of other cores of similar origin. Tests have shown that cores under consideration can vary widely with respect to porosity and permeability and still be considered similar in capillary character. EXPERIMENTAL METHODS AND TECHNIQUES Various types of displacement cell apparatus for capillary pressure experiments have been described in the literature. Bruce and Welge; Thornton and Marshall; McCullough, Albaugh and Jones3; Hassler and Brunner; Lever-

Jan 1, 1949

By C. A. Zapffe

BOTH the foundryman and the theoretical metallurgist are now generally agreed that the anomalous "rising" or "bleeding" of certain ferrous castings of killed metal is primarily attributable to hydrogen. How and why the phenomenon occurs, however, have not been made clear. The following discussion is intended to harmonize foundry experience with metallurgical theory so that the foundryman may possibly improve his ability to prognosticate whether a certain casting may bleed or not. BLEEDING, RISING, AND PIPING The "rising" to be discussed here is not to be confused with the action in rimming ingots as a result of the carbon-oxygen reaction, although a conclusion will be drawn that may apply to the art of controlling the rimming action. Only killed metal is considered, which includes both steels and irons. In Fig. I, three killed, chill-cast ingots are shown: a porous ingot whose evolution of gas during late stages of solidification has ejected metal through the top; an ingot in which moderate gas evolution during solidification has lifted the mushy central metal, i.e., it has "risen"; and, third, a sound ingot with good piping characteristics. The first type is referred to as "bleeding," and generally infers that a solid skin had first formed through which the liquid metal later erupted. There is no fundamental distinction between bleeding and rising in reference to killed metal.* POROSITY Hydrogen appears as a cause of bleeding because the gas evolution simultaneously leaves a porosity of a type traceable to hydrogen. Twenty years ago, Melmoth1 noted that killed steel that rose unexpectedly in the mold, if returned while liquid to the furnace and heated further under a reducing lime slag, would again rise in the mold; but that if the ingot were allowed to solidify and was then remelted no rising occurred. A critical quantity of the responsible gas is therefore removed by the rising action. Melmoth associated rising with high silicon (0.7 to 0.9 per cent) and concluded that silicon increased gas absorption. Because silicon is a strong deoxidizer, oxygen must, therefore, be eliminated from consideration of the three common gases in steel-oxygen, nitrogen, and hydrogen. Butterworth2 later decided that pinhole porosity was caused by gases dissolved in the metal and that hydrogen was obtained from steam in the mold. Allen's classical work on copper castings,3 soon followed by

Jan 1, 1942

By R. H. McLean

Velocity, kinetic energy and shear in crossflow beneath three-cone jet bits may influence cleaning of the bottom of the borehole and the teeth of the bit. Laboratory investigation shows that each of these parameters is a function of the diameter of the borehole and the product of the volume rate of flow and velocity through the nozzles (QVn). Increasing QV, or decreasing the diameter of the borehole increases each parameter. These functions provide means of predicting the magnitude of each parameter and of scaling the cleaning forces. INTRODUCTION In drilling operations using conventional jet-type rock bits, the impinging jels create an important flow mechanism. Called crossflow, this flow mechanism originates in the impact area of the jets, spreads across the bottom of the hole and supplies the principal source of energy to clean the teeth of the bit and most of the bottom.' Besides providing the means of cleaning cuttings from the bit and the hole bottom, crossflow may also have other, less direct, effects on the rate of penetration. The shear stress generated on the bottom by crossflow will inIluence the thickness and permeability of any filter cake of mud solids or crushed material which forms on the bottom.' These factors may affect the rate of penetration. A previous pulblication introduced some fundamental concepts of crossflow.' Crossllow was shown to occur in a thin layer adjacent to the bottom, and to cover the bottom completely. The maximum velocity in the crossflow above any position on the bottom was found to be directly proporlional to the square root of the product of the volume rate of flow and velocity through the nozzles—the jct QV,,—and inversely proportional to the diameter of the borehole. This information indicated that the effectiveness of crossllow in scavenging the bottom can be improved by maximizing the jet QV. The investigation reported herein amplifies the definition of crossflow. Complete velocity profile data above a representative position on the bottom are analyzed. These data better illustrate control of the capacity of crossflow to scavenge the bottom, and also relate shear stress on the bottom to known, controllable parameters. The conclusion reached in the previous publication that maximization of the jet QV. produces the maximunl cleaning beneath current jet bits is unchanged by these new data; rather, it is strcngthened. Data presented here show that the kinetic energy flux above a representative position on the bottom is maximized by maximizing the jet QV. The shear stress on the bottom will also be shown to be maximized in the same manner. Since the functions relating the jet QV. to velocity, shear stress and kinetic energy also involve the diameter of the borehole, means of equating, or scaling, these quantities in different sizes of boreholes will be illustrated. EXPERIMENTAL EQUIPMENT AND TECHNIQUE JET BIT MODEL Data were recorded from the same laboratory model as used in the aforementioned investigation of the flow around a jet bit.' The model consisted of a 43/4-in. three-cone, jet-type rock bit in a smooth, flat-bottomed borehole constructed of lucite. The bit had a shape and nozzle placement closely resembling larger three-cone jet bits commonly used in field operations. Fig. 1 illustrates the impact area on the bottom of a jet from this bit. Details of the orientation of the jets may be found in the previous publication. TECHNIOU'E OF MEASUREMENT Measurements of the crossflow were made by inserting a very small Pitot tube through the bottom of the simulatetl boreholc. Extreme thinness of the layer of crossflow necessitated accurate measurements of the height of the Pitot tube above the bottom to achieve close definition of the velocity profile. A cathetometer, which could be read to the nearest 0.005 cm, was used to make this measurement.

Jan 1, 1966

By A. Lubinski, J. L. Logan, W. S. Althouse

A4ost gas twells and flowing oil wells are completed and treated through a string of tubing and a packer. Changes in temperature and in pressure inside or outside the tubing will: (I) if free motion of the tubing inside the packer is permitted, increase or decrease the length of the rubing; or (2) if free motion is prevented, induce forces in the tubing and on the packer. If pressure inside the tubing is greater than outside, the tubing may buckle helically even in the presence of a packer-to-tubing tension. The tubing will always buckle, and much more severely, if free motion is permitted. The buckling may be prevented by pulling the tubing in sufficient tension. The prediction of these forces and tubing movement has here to fore been based upon calculations that did not inciude helical buckling. This paper preseitts means with which these length or force changes can be calculated while taking into account the effect of helical buckling. To avoid damaging formations, failure of remedial operations, or damage to the tubing or packer, application is made to practical problems involving calculations of the required length of seals, amount of sluckoff or tension, and prevention of permanent corkscrewing. INTRODUCTION Leakage of a packer may result in costly failures of such operations as squeeze cementing, hydraulic fracturing, etc. To avoid such failures, the authors are often asked questions pertaining to the length of necessary seals, the amount of necessary slackoff, etc. Published work' does not take into account helical buckling of tubing. Investigation of helical buckling was prompted by the fact that allowance must be made for this phenomenon in order to provide relevant answers. In the past, theoretical work on helical buckling was confined to conditions for which such buckling does not The mathematical treatment of behavior in a buckled condition, given in the Appendix, is novel. Assumptions upon which this investigation is based are listed and discussed in a special section. The following are a few of the many kinds of problems which may be solved with this paper. 1. Consider a packer in which tubing may move. Such movement will occur after pressures and temperature are changed. The paper provides means for: (a) calculating the amount of such movement and, therefore. the required length of seals; and (b) calculating the necessary amount of initial slackoff, for which there is no danger of unsealing the packer, if the length of seals is given. Such calculations, as well as those which follow, tully take into account the fact that part of the movement ib due to elastic helical buckling of the lower part of the string. This buckling may occur even in tubing under tension. Insufficient initial slackoff for a given length of seals, or insufficient length of seals for a given slackoff, may result in costly failures. A field case of such a failure will be given further in this paper. 2. In the case in which tubing cannot move in the packer, changes of pressure and temperature result in tubing-to-packer forces and forces in the tubing above the packer, both of which may be calculated. This knowledge is important because, if these forces are too large, the) could damage the packer or the tubing. 3. For wells in which wireline tools are to be run through the tubing, the paper provides means to keep the tubing from buckling, thereby permitting free passage of tools. 4. In deep wells, mainly in the presence of large casing, tubing may become "corkscrewed", i.e., take a permanent helical set. A field case is described further in the text. Using this paper, one may calculate in advance conditions under which permanent corkscrewing would occur, and then take preventive steps. HELICAL BUCKLING Consider a string of tubing, freely suspended in the absence of any fluid inside casing, as shown in Fig. I(a). Now consider an upward force F applied at the lower end of this tubing. This force compresses the string; and if the compression is large enough (which is always the case in actual problems), the lower portion of the string will buckle into a helix, as shown in Fig. 1 (b). The lower end of the tubing is subjected to a compression F. This compression decreases with the distance from the bottom and becomes nil (neither compression nor ten-

By N. Mungan

A study was made of the effects of wettability and interfacial tension on the immiscible displacement of a Iiquid by another Iiquid from porous media. The influence of viscosity ratio was also investigated. Porous media used were polytetrafluoroethylene (TFE) cores prepared by compressing TFE powder under different pressures. lt is shown that displacement of a wetting by a nonwetting Iiquid is always less efficient than the displacement of a nonwetting by a wetting fluid, all other things being equal. In the former case, the recovery efficiency can be increased substantially by either reducing the interfacial tension or increasing the viscosity of the displacing fluid. A qualitative discussion is given on the implications of this work to the recovery of crude oil by water flooding. INTRODUCTION The high cost of oil exploration and new recovery schemes makes it imperative that waterflooding be conducted under conditions favoring most efficient oil recovery. To improve oil recovery by waterflooding, it is essential that the role played by interfacial forces in the entrapment of residual oil be studied and understood. Interfacial phenomena in natural rock, connate water and crude oil systems are very complicated because of the complexity of the natural liquids found in petroleum reservoirs, because of our inability to adequately describe the geometrical structure of the porous media and because of a lack of understanding of physical and chemical interactions between the liquids and surface of the pores. The problem becomes further complicated when one tries to elucidate the role of interfacial phenomena in fluid flow. Numerous studies of the displacement of oil by water under different interfacial tension or wet-tability conditions have been made.lm3 These studies have been performed in silica, alundum or sandstone systems using water and paraffin oil and also some surface active material to control the interfacial tension s and the contact angle 0. Unfortunately, the high energies of various interfaces involved favor adsorption and orientation of the surface active material at the interfaces. Also the surface active material concentration at the interfaces exceeds that in the bulk of the liquid phases. Such surface excess may cause the surfactant distribution, the contact angle and the interfacial tension to differ from their measured static equilibrium values and makes interpretation of the displacement experiments difficult. Furthermore, as changes in s also lead to changes in cos 0, the role played individually by one of these parameters in the displacement becomes obscured by the effect of the other. To circumvent these difficulties, a low surface energy solid and true solutions or pure liquids should be used. Use of a low surface energy solid minimizes adsorption and orientation effects at the solid-liquid interfaces. By controlling s and cos 0 through use of selected pairs of pure liquids or true solutions rather than by surfactants, the adsorption effects at liquid-liquid interfaces are eliminated. In the present study TFE cores were used as the porous media. Liquids used were water, sucrose solutions, paraffin oils and benzyl, n-butyl and isobutyl alcohols. The interfacial tension was varied from 40 to 1.1 dynes/cm by suitably choosing the liquid pair. A surface active material was added to the water-oil system only in the case where interfacial tension of 0.5 dyne/ cm was desired. No precise changes of cos 0 were attempted. However, either the displaced or the displacing liquid could be made the one which preferentially wets the TFE surface. Using sucrose solutions and blends of paraffin oils proved to be a convenient way of changing the viscosity ratio between the displaced and displacing liquids. The present investigation examines the effect of interfacial tension, wettability and viscosity ratio on the immiscible liquid-liquid displacement from porous media. Observed data permit a

Jan 1, 1967

By R. T. DeHoff, L. J. Buteau, J. Kronsbein

Network topology is introduced and used to describe sinter bodies at various stages of the sintering process. The matrices of incidence, loop-branch. tree-branch, and cut-branch are constructed and their relationship to the three-dimensional void space and the common two-dimensional bounding surface is established. The sintering body is assunzed to remain connected but this is not so with the void space. The topological invariants known as Betti numbers are related to this observed fact and provide parameters which measure the progress of the sintering process. They are extracted from the matrices by means of known algebraic operations. Experimental procedures are given which permit actual construction of the various matrices and determination of the Betti numbers by serial sectioning of the sinter bodies. A sinter body is produced by exposing a large number of metallic particles (0.00001 to 0.3 in. diam) to a temperature that is near but below their melting point. When these particles are maintained at such a temperature for a sufficient length of time, it is found that the interparticle contacts become welded. As time increases, these welds, often referred to as necks, grow. All of the space between the particles constitutes a single connected void; an ultimate consequence of sintering is the breaking up of the void into separated parts. A mathematical apparatus for describing the geometric characteristics of a sinter body at each stage of the process was first proposed by F. N. Rhinesl when he introduced the application of topology to this problem. He described the sinter body in terms of one of its topological invariants, the genus. The most important advantage of a topological description is invariance with respect to continuous deformations. This means that particle sizes and shapes need not be considered, resulting in considerable simplification and generalization of the geometric description. In the case of a sinter body, wherein the internal geometry is particularly complex, the simplification provided by topological concepts is absolutely essential. The important parameters in the Rhines model are the number of particles and the interparticle contacts. The invariant considered called genus is given by Euler's formula for the connectivity of two-dimensional surface of the sinter body. This surface includes the pore-solid interface. The Euler formula is G = C - P + 1 [1] where C = number of interparticle contacts, P = number of particles, and G = genus of the surface bounding the sinter body. In topology the term genus may be used to mean the number of self re-entrant cuts that can be made on a closed surface without dissecting it into more than one connected surface. Its use is restricted to the surface and does not apply to the three-dimensional volume enclosed or excluded by it. In the case of sintering, the volume enclosed by the surface is defined as the sinter body; the volume excluded will be referred to as the void space. There is an intrinsic reason for using genus as a characteristic number associated with a sinter body. The genus is related to the Gaussian curvature by the integral of the product of the principal curvatures taken over the entire surface of the sinter body, as given by

Jan 1, 1965

By M. Kaufman, A. E. Palty

The phase structure and aging characteristics of two nickel-base alloys, Inconel 718 and 702, were investigated. Wrought and cast Inconel 718 showed ArisCb as the major hardening phase, as well as ture was positively identified. Inconel 702 had Ni3Al as the major hardening phase, as well as Cr7C3-of the phase amounts with temperature was determined. 1 HE value of phase structure and aging rate information in the utilization of alloys has been well established. As part of a continuing program, the results obtained on two nickel-base alloys, hconel 718 and Inconel 702, are presented here. Inconel 718 is a relatively new alloy considered for application in both sheet and cast form. Its novelty lies in the use of high columbium tantalum levels in a Ni-Cr-Fe base. No previous work on this type of alloy has been reported in the literature. Inconel 702 is being used now as an oxidation-resistant sheet alloy. Occasional fabrication problems spurred work on its basic structural behavior. The composition of 702 is essentially nichrome plus aluminum and titanium for strengthening and added oxidation resistance. This puts it in the basic class of other nickel-base alloys which have already been studied. The two alloys will be treated separately. INCONEL 718 Phase Study Methods. The phases present after the various conditions investigated were determined by X-ray diffraction analysis of two different elec-trolytically extracted residues and microscopic examination. The extraction methods, using 10 pct HC1 in alcohol and 10 pct H3PO, in water and the X-ray technique have been described previously.' The effect of temperature on the phase behavior was studied on sheet material using the following heat treatments: 2250°F, 2 hr, ice brine quench 1300 100 hr, water quench 2250°F, 2 hr, ice brine quench 1400°, 100 hr, water quench 2250°F, 2 hr, ice brine quench 1500°F, 100 hr, water quench 2250o 2 hr, ice brine quench 1700°F, 48 hr, water quench 2250°F, 2 hr, ice brine quench 1850°, 24 hr, water quench 2250°, 2 hr, ice brine quench 2000°F, 6 hr, water quench The high solution temperature was an attempt to dissolve as many phases as possible without causing melting. Aging exposures were made longer than normal heat treating times in order to approach equilibrium conditions more closely. Cast material was examined only in the as-cast condition and with the normal heat treatment (1700°F, 1 hr, air cool 1325oF, 16 hr,air cool). Materials. The full phase study was performed on sheet specimens approximately 1/2 in. by 2 in. by 1/16 in. The cast pieces were cut from the gating system of a vacuum-cast test piece. An all weld-metal specimen was made by running many weld beads on the sheet using filler made from the sheared edges of the sheet. All external contamination was removed by belt-sanding followed by an electro-polishing treatment. The typical composition of wrought Inco 718 is as follows: The alloy is air-melted. Cast hco 718 may have higher molybdenum. Due to the expectation of finding a Ni-Cb phase (Ni, Cb) for which no X-ray diffraction pattern was available in the ASTM Index, a special 5 Ib heat to the stoichiometric composition of this intermetallic

Jan 1, 1962

By Leon T. Eleiel

Ten years ago a broad knowledge of aerial mapping, coupled with a smattering of geology, qualified one to speak on the subject of the application of aerial mapping to geology. Today, with aerial maps a common tool in the hands of expert geologists on every continent throughout the world, it would be nothing short of presumptuous for the author of this paper, who is not a geologist, to give any geological statements on his own authority. Relatively, today we are only purveyors of general geological gossip, with a definite case here and there by way of illustration. But we can present a story of the really marvellous strides that have been made in the evolution of aerial mapping, built around a skeleton of geological tidbits, from which some useful conclusions may be drawn. The average geologist goes about reading his aerial maps much as one would approach the problem of deciphering a cryptogram. To begin with, typical key expressions appear in recurring sequences and under certain conditions. If once the meaning of these isolated bits of information can be ascertained, the message begins to take meaning. In a similar fashion, the geologist works out his key from aerial maps of territory with which he has some familiarity. He learns that a certain type of thin, white line may mean a vein of quartz. He sees that a certain change in the photographic tone of vegetation on the ground reflects two kinds of soils on opposite sides of a contact. He sees that a certain type of alignment of topographical features often means a fault. He learns that by the erosion pattern he can frequently distinguish soft formations from harder ones. And so, bit by bit, he ties together the information of the picture with his experience on the ground until he develops a reasonably creditable capacity of interpretation. Through the complete gamut of surface conditions, ranging all the way from flat, arid, exposed desert terrain to dense jungle, or rugged mountains, the evidence from which the geologist must make his interpretation varies with bewildering profusion. The quality of photographs which are obtainable may range from perfect pictures in the desert to almost useless smudges in the steaming jungles. One thing he has learned with certainty—the airplane carrying its mapping camera is a superb reconnaissance tool, only awaiting his ingenuity in correlating the pictures with the ground to simplify most exploratory problems.

Jan 1, 1937

By F. D. Rosi

The slip and twinning behavior in extended titanium crystals were studied in some detail. The formation and appearance of coarse kink bands are discussed. Their crystallographic geometry was determined by X-ray analysis. A phenomenological interpretation of the complexities in kink band development is also presented. The critical resolved shear stress, coefficient of shear hardening, and plane of fracture were determined for several crystals extended at room temperature. THE slip and twinning elements observed in the room-temperature deformation of titanium were enumerated in a previous paper1 in which considerations were advanced regarding the nature and selection of these elements and their effect on the known mechanical properties of this metal. The present study concerns the crystallographic, microscopic, and mechanical aspects of flow in relation to the slip and twinning elements, and includes a prediction of slip systems, nature of slip and twin markings, inhomogeneities of plastic flow, and stress-strain characteristics. Unless otherwise noted, arc-melted titanium sponge (99.77 pct) was used in these experiments. The method of production of crystals, their dimensions, and the surface preparation for micrographic examination have been reported.&apos; For obtaining stress-strain characteristics, only those crystals which traversed the entire width of the specimen and were at least 8 mm in length were used. Tensile deformation of the crystals was performed with conventional grips for sheet specimens and a constant-stress loading beam, designed after the method of Andrade and Chalmers.&apos; The specimens were loaded by allowing sand to flow from a reservoir into a bucket suspended from the longer end of a balanced 6:1 lever arm at a rate controlled to load the specimen approximately 2 kg per min. Strain measurements were made using the Baldwin SR-4, bonded, resistance-wire strain gage, Type A-8, which permitted a reading accuracy of 2 microinches per inch. The formulas used in the evaluation of shear stress and shear strain, as in deriving the coefficient of shear hardening, are given by Schmid and Boasv n terms of the original orientation and change in length of the crystal. For calculating the critical resolved shear stress, the standard equation was used. The crystallographic nature of the unpredictable slip observed in a number of specimens was determined by the single-surface X-ray method of analysis as described in ref. 1. Experimental Results Prediction of Slip System: It was reported&apos; that room temperature slip in titanium takes place predominantly on {10i0} and in a <1120> direction, giving three potential slip systems. Hence, it should be possible to predict the operative primary system in the manner used by Taylor and Elam&apos; for alu- minum (i.e., the one with the greatest component of shear stress in the direction of slip). The stereo-graphic construction in Fig. 1 shows that this is true. In all cases, slip was found to occur initially on the (0110) plane and, in a number of cases, in the [21f0] direction. (The direction of slip was not determined for all orientations.) It follows that duplex slip can be expected when two slip systems are geometrically equally favorable for slip, as demonstrated in Fig. 2. In both crystals slip took place simultaneously on the (1700) and (olio) prismatic planes, as indicated by Fig. 2a and b. In the case of crystal B the movement of the specimen axis with increasing extension was toward the (10i0) direction, which represents the stable end orientation resulting from alternate slip on the [2ii0] and [1120] directions. This is analogous to the duplex slip process in face-centered cubic metals." Unpredictable Slip: In a number of crystals whose orientations were well within the operation of a single slip system, secondary slip occurred on planes not predicted by the criterion of maximum shear stress. Examples are shown in Fig. 3. In addition to

Jan 1, 1955

By H. F. Silliman, Daniel R. Hull, John R. Freeman

IN the normal operation of a brass-rolling mill, sheet and strip has, for the most part, been finished in comparatively thin gauges, involving a substantial amount of coldwork and a considerable number of anneals. This condition has been disturbed by the prep0nderance of cartridge brass, which is not rolled to thin gauges, and particularly by brass for artillery cartridge cases, on which the mill may finish its work at a thickness between 0.400 and 0.900 in. Obviously, the amount of rolling and the number of anneals to which such a product can be subjected are small compared with strip that is carried to a thickness of a few hundredths of an inch. As a result, critical examination into the internal condition of metal has been made on an extensive scale at a stage of manufacture that would not, in normal times, have been chosen for complete inspection. Under these conditions, a type of defect has been recognized that is not seen at thinner gauges. It might be more accurate to call it the appearance of a defect, since it refers to an apparently laminated, discontinuous structure, which, in fact, does not exist as such, and disappears after suitable treatment. For this reason it has been called a phantom lamination, which the authors consider an appropriate term. It is the purpose of this paper to describe the phenomenon and the conditions of its occurrence and, at least tentatively, to discuss its nature and cause. Great numbers of phantoms may be found in 2: 1 brass, either hot-rolled or cold-rolled. They are also found in 70-30 brass, sometimes when hot-rolled and generally when cold-rolled. It is not believed that phantoms are confined to these alloys but this paper is limited to them for the purpose of description. HOT-ROLLED 2:I BRASS For illustration, therefore, a typical cake of hot-rolled 2: 1 brass has been chosen. This was rolled to a slab 7/8 in. thick, from a casting 5 in. thick by 24 in. wide by 18 in. high, which was believed to have been sound. After hot-rolling and cooling, the slab was sheared and a portion of such a shear cut is shown in Fig. 1. It would be difficult to convince anyone viewing this specimen that the metal was not unsound. The impression would be confirmed by the appearance of a tensile break, as shown in Fig. 2. This is an ordinary tensile-test piece from metal just described. Although there was nothing in its tensile strength or elongation that suggested unsoundness, it appeared run through with pipes and laminations. To question the reality of such a condition would imply questioning the obvious. But, considering the history of the metal, it w0uld be difficult to believe that it could be so bad as it appeared. It seemed reasonable, though, that if the defects were what they appeared to be, they could be traced back from the sheared or broken face.

Jan 1, 1945

By R. C. Hassinger, D. V. von Rosenberg

Transverse dispersion has received considerably less treatment in the literature than has longitudinal dispersion. Different methods for determining transverse dispersion coefficients have been used in different investigations, and the results obtained have not been consistent enough to permit accurate generalizations as to the effect of various physical parameters on the magnitude of these coefficients. A numerical solution to the differential equation describing transverse dispersion in the absence of longitudinal dispersion was obtained to enable one to calculate the dispersion coefficient from experimental results. The more general dispersion equation including longitudinal dispersion also was solved numerically to give quantitative limits of a dimensionless group within which the assumption of negligible longitudinal dispersion is justified. Possible experimental procedures were examined, and one utilizing a cylindrical packed column was chosen for the determination of transverse dispersion coefficients. Values of these coefficients were determined for a system of two miscible organic fluids of equal density and viscosity, for two sizes of packing material over a wide range of flow rates in the laminar regime. The dispersion coefficient was found to decrease, for a constant value of the product of packing size and interstitial velocity, as the size of the packing material particles increased. INTRODUCTION Longitudinal dispersion has received extensive treatment in the literature, and consequently is better understood than its orthogonal counterpart, transverse dispersion. Many mathematical models of dispersion processes assume that transverse dispersion is rapid enough to damp out any radial** concentration gradients and therefore may be neglected. Laboratory and production results, however, indicate that this is a poor assumption.l,2 Various experimental procedures for determining transverse dispersion coefficients have been used in previous investigations, but the results have generally been expressed by similar correlations. The transverse dispersion coefficients obtained, however, have often varied considerably for given values of the correlation parameters. We feel that further experimental determinations of transverse dispersion coefficients will help alleviate some of the inconsistencies in these empirical correlations. One assumption implicit in all previous investigations is that of negligible longitudinal dispersion in the experimental system. An attempt to justify this assumption often is made using intuitive reasoning, but it is apparent that this reasoning must break down as the condition of zero flow rate is approached. A mathematical examination of the equations describing the system yields physical limits outside of which the assumption of negligible longitudinal dispersion is invalid. BACKGROUND In a porous medium, the "effective molecular diffusivity" De is less than the molecular diffusivity D measured in the absence of a porous medium, due to the tortuous path which a diffusing molecule must travel. Various authors3-5 have reported values of the ratio De/D in the range of 0.6 to 0.7. When there is fluid flow within the porous medium, mass transfer occurs by convective dispersion as well as by molecular diffusion. These are separate phenomena and can be treated as such on a microscopic scale. However, the mathematical complexity is such that only extremely simple geometries could be considered, and the results hardly would be applicable to the complex geometries existent in actual porous media. A more

Jan 1, 1969

By R. H. Oliver

The purposes, types, preparation, and testing of flocculants are discussed. A flocculation compendium is included, indicating choice of flocculant for a given set of conditions. An economic evaluation of the process is presented. Solid-liquid separation is a major expense in most mineral processing flowsheets today. Proper use of flocculation can often lower the overall cost of a solid-liquid separation by reducing the size of sedimentation or filtration equipment. PURPOSE OF FLOCCULATION When is use of a flocculant justified in terms of overall process economy? 1) when it results in process improvement, such as producing a clear liquor for electrolysis, precipitation, ion exchange, or solvent extraction; 2) when it results in satisfying requirements of pollution abatement ordinances; 3) when it results in increased recovery of values that would otherwise be lost; or 4) when it results in considerable savings in capital expenditure due to use of smaller equipment. When can the use of a flocculant adversely effect overall process economy? 1) when it results in increased filter cake moisture, 2) when it results in decreased filtration rates, 3) when it results in considerable bulking of sedimentation underflow, 4) when it contaminates either the supernatant or sludge, or 5) when the cost of using the flocculant is greater than the savings due to its use. First step in consideration of flocculants for a given application is establishment of desired goals in terms of increased clarity of effluent, increased recovery of solids, or decreased size of equipment. The most economical solution to a solid-liquid separation problem is that combination of equipment and flocculant costs which meets the established goals at the lowest total annual expenditure. This economic decision can be made only after technical data has been obtained as outlined in next four sections. TYPES OF FLOCCULATION Flocculation is a process wherein individual particles are united into more or less tightly bound agglomerates or flocs, thereby increasing effective particle size of solids suspended in a liquid. Degree of flocculation of a suspension of finely divided solids in a liquid is controlled by a combination of probability of collision between particles and probability of adhesion after the collision has occurred.&apos; Probability of collision can be increased commercially through use of a paddle-type flocculator, combination flocculator, and clarifier or flocculating-type feedwell. Probability of adhesion usually can be increased by addition of a reagent known as a flocculant. Reagents act as flocculants through one or a combination of three possible mechanisms. The first is electrolytic neutralization of inter molecular repulsive force due to Zeta potential. This neutralization enables Van der Waal&apos;s cohesive force to hold the particles together after they collide.&apos; The second is the precipitation, within a definite pH range, of voluminous metallic hydroxide flocs which entrap fine particles. This process is known as coagulation.14 Third is the bridging of two or more particles by either natural or synthetic long-chain, high-molecular-weight organic polymers. This bridging is accomplished by adsorption of two particles at different sites on the same molecule or by bonding of two molecules, each adsorbed to a different particle. Considerable effort has been expended in explaining the action of these polyelec-trolytes and detailed accounts are available in the literature. CHOICE OF FLOCCULANTS It is impossible, at the present time, to specify the optimum type or amount of flocculant for a given application from a knowledge of the material to be treated. The standard procedure is to choose reagents or combination of reagents most likely to be effective on the slurry; then test them.&apos; The accompanying Flocculation Compendium (Table I) contains details of all currently available reagents sold as flocculants and it should be used when selecting the group of reagents for tests. This selection can be based on information listed in columns 4 and 5. Column 4 presents industries in which each floc-

Jan 1, 1961

By J. L. Walter, W. R. Hibbard

A GREAT deal of work has been done on the rolling of single crystals of iron. Barrett and evensoon studied end orientations of heavily cold-rolled iron crystals and found that some crystals maintained a sharp single orientation during rolling. Others rotated into two or more orientations, and some fragmented and approached the entire polycrystalline texture depending on the original orientation of both the normal and rolling directions. Barrett, Ansel, and Mehl2 studied iron-silicon alloys and found that the addition of silicon to the iron had little effect on heavily rolled polycrystalline material. Dunn3 has extensively studied the cold-rolling and recrystallization of silicon-iron single crystals. Dunn found that for single crystals having initial orientation between (110) [001] and (111) [112], the principalcomponent of the cold-rolled texture is (111) [112]. Koh and Dunn4 studied the cold-rolled textures of single crystals having initial orientations in the range (001) [110] to (111) [IT01 and found that these are stable end orientations. They also found that all orientations having a [110] direction parallel to the transverse direction rotate to either the (001) [110] or the (111) [112] stable end orientation except for orientations near (112) [11], which divide and produce a final texture consisting of (001) [110]and (111) [112] components. Crystals having a (111) <112>-type texture re-crystallized to a (120) [001] or (110) [001]-type texture. Crystals having orientations with a range of (001) to (335) in the rolling plane and [110] in the rolling direction recrystallized5 to a wide spread of the initial orientations resulting from a multiplicity of 30-deg rotations about several (110) poles. The present study was undertaken to relate other orientations of crystals of silicon-iron to their cold-rolling and recrystallization textures, so that one might rationalize polycrystalline recrystallization data on the basis of the known deformation texture and the behavior of single crystals in these orientations. PROCEDURE Samples were obtained from a cylindrical ingot of 3 pct silicon iron prepared as follows: Armco iron was melted with high-purity silicon flakes under an atmosphere of argon to minimize oxidation. The alloy was poured into a special mold to form an ingot which consisted partially of large columnar crystals. Plates were cut from the columnar portions of the ingot with the longitudinal direction parallel to the columnar axes and ground to 0.040 in. thick. The plates were heavily etched to a thickness of 0.032 in. to remove the cold-worked surface. Orientations of seventeen crystals were determined by Laue back-reflection techniques keeping the mean columnar axis as the reference direction. The plate was then cut in two, parallel to the columnar axis. One portion of the plate was rolled at room temperature parallel to the columnar axis to a reduction in thickness of 84 pct. The remaining portion of the plate was rolled also at room temperature across the columnar axis to a reduction of 66 pct, at which point the crystals began to separate at the boundaries. After rolling, nine individual crystals were removed from the strip by cutting along the crystal boundaries with a jeweler&apos;s saw. Fig. 1 shows the original orientations of the six crystals that were straight-rolled and of the three crystals (4, 7, and 8) that were cross-rolled. Each rolled crystal was carefully etched to approximately 0.002 in. thick. Textures of the cold-rolled crystals were determined by use of the diffractometer with a non-integrating sample holder. The pole figures were made for the (200) poles using molybdenum Ka radiation and were plotted by an automatic stereographic projection recorder.&apos; Spurious poles were determined by rerunning portions of the pole figures using strontium oxide to filter out the Ka radiation. The central portions of the pole figures have not been determined. The pole figures were completed to about 55 deg from the periphery at which point the sample holder interfered with passage of the beam. Following the determination of the cold-rolled

Jan 1, 1959

By L. J. Kashar

By means of a continuum-mechanics approach, the following equation for calculuting the pressure for axisymmetric cold forging has been derived: where P is the extrusion pressure; K and n are the strength coefficient and the strain-hardening exponent of the steel, respectively; a and b are strain factors, primarily dependent on the die design; and R is the extrusion ratio. Data from compression and backward-extrusion tests performed at a crosshead speed of 0.5 in. per min on a nonwork-hardening material pure lead) and on a work-hardening material (1018 steel) indicate that the predicted extrusion pressures obtained from the derived equation agree very closely with the actual extrusion pressures. Therefore, the equation can be used to predict backward-extrusion pressures from standard mechanical-test data. The decision to cold-forge a part is based on several considerations, most of them related to economics. However, the production feasibility is usually determined by the pressure needed to extrude the part and the availability of tooling that can repeatedly sustain this pressure for over several hundred thousand cycles. A major contribution to the cold forging of steel would result from the development of steels that could be forged at reduced pressures and from the development of techniques that permit the accurate prediction of the pressure necessary to cold-forge the part. The ability to predict extrusion pressures is necessary to properly design the tooling required for the cold-forging process. Attempts to theoretically predict the cold-forging pressure, especially for work-hardening materials, have been unsuccessful in previous investigations of the cold-forging process.1-3 Many empirical relations have, therefore, been proposed relating the extrusion pressure to the chemical composition,&apos; the extrusion ratio,2 the tensile strength,273 and hard- ness, 2 and the yield strength3 of the material being extruded. However, these empirical relations are valid only for the specific conditions used in their development, and, in some cases, the empirical relations had a scatter band greater than * 10 pct. A study of the extrusion process was undertaken with the objective of developing predictor equations for extrusion pressure by using a continuum-mechanics approach to the problem of metal flow. The derivation of such an equation is summarized in this paper, and data are presented that confirm the validity of the derived equation. DERIVATION OF THE EXTRUSION EQUATION If the extrusion of a material through a die is considered in terms of a work balance, it has been shown that it is possible to derive an equation relating the work necessary to move the material through the die to the deformation work undergone by the material. The work done, W, in moving the material is given by where F is the extrusion force, I, is the distance the material was moved, A, is the original cross-sectional area of the material, P is the extrusion pressure acting on A,, Ap is the cross-sectional area of the extrusion punch, and Pp is the punch pressure acting on Ap. The deformation work, WD, is given by where V is the volume of metal deformed, ay is the flow stress of the metal, and E is the total strain undergone in the extrusion. If there are no frictional losses, the work done, W, is equal to the deformation work, WD; therefore, Since the volume of a metal does not change during deformation,

Jan 1, 1968

By B. Chalmers

THE practical importance of the phenomena of melting and freezing must have been recognized for a very long time. The difference between ice and water, for example, has had a profound influence on the history of mankind and the evolution of society. The possibility of melting a metal and allowing it to freeze in a mold of chosen shape has been an essential ingredient in our mastery of the art of shaping metals, and therefore in the evolution of the: machine age in which we find ourselves. The importance of melting and freezing, as applied to metals and alloys, has been so great, in fact, that empirical solutions have been found for the multitude of practical problems that have arisen. This approach has been so successful that relatively little attention has been directed to arriving at an understanding of the fundamentals of the processes. But metallurgy has come to a stage at which we may expect that some, at least, of the more complex problems that have not yet been solved (or perhaps even recognized) may be handled more effectively by scientific study, theoretical understanding, and logical experimentation than by trial and error. In this lecture, therefore, I propose to describe in outline what I think really happens when a metal freezes. In doing so I hope to explain many of the phenomena which have been observed, and in particular to account for the structures that are obtained in actual ingots and castings. The basic problem, to which this lecture represents a tentative partial answer, is this: a mass of metal, containing known proportions of various elements, is melted, heated to a given temperature, and then allowed to freeze under specified conditions. What will be the "structure" of the resulting metal? The term structure includes: 1—crystal size, shape, and orientations, 2—distribution of chemical elements, and 3-—shape, including cracks, cavities, pores, etc. The Solid-Liquid Interface We will first consider what takes place if a single crystal of a metal in the form of a rod is heated, not uniformly, but so that one end is hotter than the other. If this heating process is continued long enough, the hotter end will eventually melt; we will suppose that the rod is in a containing vessel so that the molten metal does not run away, Fig. 1. When some of the metal has melted, we have some solid, some liquid, and an interface or surface of contact between them. If the source of heat is now removed, the interface will move so that some of the liquid freezes, and if the supply of heat is suitably adjusted the interface will remain at rest. This very simple arrangement allows us to study the basic processes of melting and freezing, and if we fully understand this simple case, we may be able to account for what takes place under practical conditions where the heat does not all flow in the same direction, and where the heat flow is determined not by a controllable source of heat but by the heat capacity and temperature of metal and mold, and by the heat loss from the mold surface. The solid-liquid interface is evidently the region of the greatest interest to us; on one side of it there is crystalline solid, and on the other, liquid. In the solid, each atom has a well defined position, around which it vibrates as a result of thermal agitation. It only leaves this position in the relatively rare event of a "diffusion jump." The liquid is much less systematically organized. The atoms are about as far from their neighbors as in the solid, but the arrangement is much less systematic and is continuously changing. The solid and the liquid are represented diagrammatically in Fig. 2. The average energy of the atoms in the liquid is greater than in the solid by an amount that corresponds to the latent heat of fusion, i.e., the amount of heat that has to be supplied to convert unit mass of solid into liquid at the same temperature. The Two Processes As has recently been shown by Jackson and Chalmers,3 many of the features of the processes of freezing and melting can be understood if it is assumed that a continuous and rapid interchange of atoms between solid and liquid always takes place at a solid-liquid interface." It is necessary to con- sider two distinct processes, that of melting, in which atoms leave the surface of the solid and become part of the liquid, and the converse process,

Jan 1, 1955