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By L. Russell Van Wert

DURING an investigation in which the solubility relations of the phases in Muntz metal (60 per cent. copper, 40 per cent. zinc) were under study, certain phenomena that had no immediate connection with the main problem were observed, which were thought sufficiently interesting to study further. Because of lack of time, only a preliminary survey was made of two of these; namely, the cleavage structure developed under certain conditions in a 60:40 brass and abnormal grain growth in this alloy. CLEAVAGE STRUCTURE IN HEAT-TREATED MUNTZ METAL Carpenter noted many years ago that cold-working a brass of the Muntz metal type, which had been quenched previously from within the beta field, results in a partial breakdown of the supersaturated beta solution and the precipitation of the solute (the alpha solution) in the cleavage planes of the beta grains. This manner of rejection of a solute from a metastable solid solution is not peculiar to this particular alloy, for among others, it occurs in the cold-working of certain austenitic steels; for example, the austenitic nickel steels. In these ferrous alloys, cold deformation produces a partial decomposition of the austenite, with the formation of martensitic "needles" and zigzags in the austenitic poly-hedra. It is interesting to note that while the two solid solutions-austenite and quenched Muntz metal- are metastable and in their mode of decomposition behave quite alike when deformed in the cold, the metastability of each results from a different cause; viz., that of the austenite from the fact that it is a supercooled solution (its solvent, gamma iron, at ordinary temperatures is below its normal stable range) whereas the metastability of the quenched Muntz metal is the result of supersaturation.

Jan 1, 1929

By James F. Kemp

Jan 1, 1914

By R. H. Widmyer, G. M. Wood

This investigation was prompted by the absence of a simple, accurate method to predict optimum second stage separation pressure in three stage separation operations. Three stage separation as used here includes two separators plus the stock tank. The first stage pressure is usually fixed by gas sales conditions and the third stage is at atmospheric pressure. Thus, the second stage pressure is the only one that may be varied in-discriminantly during production operations. No variations in temperature were considered in this study, even though it will vary among systems, for a change in temperature has little effect on the optimum pressure. Optimum pressures may be determined from flash calculations, but these are tedious unless a computer is available. An early attempt to develop a simpler method&apos; yielded an equation which gave the optimum pressure only as a function of initial and final pressure. Experience, however, confirms theoretical relationships which show that composition as well as pressure and temperature must be a factor. This indicates that the basic relationship should be an equation of the form, where P, is second stage pressure, P, is first stage pressure, A is a function of P. (stock-tank pressure) and the system composition, and n is a function of the number of stages and the system composition. Fig. 1 shows the basic variation of P2 with P for a fixed system composition. A series of such figures clearly showed the effect of composition. The peak of each curve reflects that second stage pressure at which the stock-tank recovery (gal/MMcf) residue gas is a maximum value. DEVELOPMENT OF NEW CORRELATION Wellstreams having 17 different analyses, ranging from low gravity crude to condensate, were used to determine the effect of composition. Each wellstream was flashed through three stages to determine the optimum second stage pressure for each first stage pressure. Equilibrium vaporization ratios proposed by Latham and Campbell were used throughout. The heptane plus fraction, for simplicity, was assumed to have the properties of octane. The results were subjected to an empirical analysis in an attempt to develop a correlation- that would be (1) simple, (2) accurate, and (3) eliminate the need for any flash vaporization calculations. This analysis yielded two equations, one for wellstreams having a specific gravity greater than one (air = 1.0) and one for those less than one. For crude streams (specific gravFor condensate (specific gravity < 1.0),

By Leon Dominian

(Wilkes-Barre Meeting, June, 1911.) I. INTRODUCTION. THE lack of Aryan roots for the names of metals commonly known among the Aryan settlers of Asia Minor, as well as the later colonizers of Europe, indicates that these races were generally ignorant of the use of metals until they came into contact with Semitic peoples. Practically all mining-terms in current use among the earliest Greeks resemble very strongly their distinctly Semitic equivalents, which can be traced all the way in a broad belt beginning in Lower Mesopotamia, and extending westwardly to the Syrian shores of the Mediterranean. The Greek word " "metallon," for instance, used indiscriminately to designate mine or ore, probably came from the earlier Semitic equivalent, " matal." Again, the Greek words " chrysos " (gold) and " chalkos " (copper) seem to be descended from the Semitic forms " throats " and a chalak." It is a natural inference that primitive mining-methods were evolved by the dwellers in the mineralized areas of Asia Minor, from whom later Greek, Roman, and even North European miners obtained their first notions of the reduction of metallic ores, by virtue of a general westward migration of mining and metallurgy. Some traces of its passage through Turkish territory will be noted in this paper. While European Turkey can boast of one ancient gold-field, the Asiatic dominions of the Sultan may lay claim to at least two well-defined and widely-separated gold-producing districts. These three regions may be distinguished as the Thracian, the Pontic, and the Anatolian gold-fields. II. TURKEY IN EUROPE. . The Thracian Gold-Field. The most conspicuous topographic feature of the lowland between Constantinople and Salonica is the uplifted Archaean

Nov 1, 1911

By T. R. A. Davey

The fundamental principles by which bismuth may be removed from lead by pyrometallurgical processes are enumerated. Qualitative discussion of the phase diagrams concerned is followed by presentation of quantitative diagrams. Brief mention is made of the practical aspects. Data presented show how chemical lead (<0.005 pct Bi) may be produced by the Jollivet, Dittmer, and Kroll-Betterton processes. BISMUTH is an element whose properties are very similar to those of lead, and its minerals are associated to a greater or lesser extent with nearly all the major lead ore deposits of the world. In consequence, the lead produced contains bismuth unless a special debismuthizing process is practiced. There does not appear to be any published comprehensive review of the effect of bismuth as an impurity on the properties of lead, and there is some uncertainty as to the beneficial or detrimental effects of small amounts of bismuth in lead used for particular applications. Standard specifications of most countries stipulate a maximum bismuth content of 0.005 pct for chemical lead. With one exception, this specification is met only by lead producers whose ores are relatively bismuth-free, or whose refineries are electrolytic. In lead for certain applications a higher bismuth content is preferred by some fabricators.&apos; In contrast to the refining for silver, copper, and antimony, the refining for bismuth does not usually result in a large overalil economic gain from the sale of byproduct. Thus, debismuthizing is customarily practiced only if market conditions demand a lead with a bismuth content lower than that produced by classical refining; i.e., softening, desilverizing, and zinc refining. Until this century, debismuthizing was carried out only a:; an incidental result of desilverizing, as later described. This century has seen the development of the Betts electrolytic process—the only successful one of many electrolytic processes proposed and patented —and a number of precipitation processes based on the pioneering work of Kroll,&apos; although only one of these, the Kroll-Better ton process,&apos; is in wide commercial use today. It is proposed in this paper to deal only with the pyrometallurgical processes, excluding those based on aqueous electrolysis. Historical Background Prior to 1833 silver (and incidentally bismuth) could not be removed from lead. To recover silver from rich bullion, the only course available to lead refineries was to cupel the bullion, forming a slag of litharge. The first slag tappings or skimmings, on reduction, yielded soft lead of very low silver and bismuth contents. Intermediate skimmings yielded a lead of somewhat higher silver and bismuth contents, which could if desired be recupelled. The last skimmings yielded a litharge very high in bismuth, which could by successive oxidation and reduction cycles be separated into high purity bismuth, and lead returns. This oxidation reduction process, with modern operating techniques, can still be used today to produce a portion of a lead refinery&apos;s output of very low bismuth content, as discussed later. In 1833 Pattinsone discovered that lead bullion could be purified of silver and bismuth by fractional crystallization, the primary lead crystals formed by partial freezing being purer than the melt. Repeated processing in pans produced a lead containing as little as 0.02 pct Bi. The simple explanation of this

Jan 1, 1957

By C. H. Mathewson

CONSIDERABLE progress has been made during the last few years in analyzing the action of plastic deformation in metallic crystals. This is due largely to the development of X-ray spectrographic methods for studying changes in the orientation of single crystals, or of conglomerates, as they occur during the process of plastic deformation, but partly to the widespread knowledge of suitable technique for preparing large (single or variously combined) crystals for direct crystallographic observation under stress or in various conditions of strain. These two methods of studying changes in the crystalline structure of metals may of course be used in conjunction with features of mutual support, but unfortunately the advantage of direct crystallographic observation is lost when we proceed to the study of severely cold-worked structures, owing to the occurrence of real or apparent crystalline discontinuities which become more and more complicated as the amount of plastic action increases. Furthermore, the X-ray methods do not appear to be adequate, at least in their present stage of development, to fix definitely the character of structural alteration in the parent crystal lattice by plastic deformation. The principal question at issue concerns the nature of the crystalline particles that remain after plastic deformation. These are generally known in our American metallurgical literature as crystalline fragments,&apos; and they seem, indeed, to diffract as individuals causing the Hull pattern of a severely cold-worked metal to resemble the pattern of a fine powder,2 with certain features of preferred orientation among the particles deter- mined by the manner and intensity of cold working. We must freely concede that there are conceptions of lattice alteration, apart from that of simple fragmentation, which would account for an increasing number of lattice positions according to X-ray spectrographic observation as the degree of deformation increased.

Jan 1, 1927

By D. W. Fuerstenau

In 1940 Schuhmann1 showed that the size distribution of a comminuted material can be expressed by the following empirical relationship: y = 100 (|f [i] where y is the cumulative percentage of material finer than size x, k is a size modulus, and a is a distribution modulus which is constant for a given material and machine. During the past few years, considerable emphasis has been given to the investigation of the extent of size reduction resulting from the expenditure of mechanical energy in crushing and grinding. Recently Schuhmann&apos; presented a derivation of the energy-size reduction relationship by postulating that comminution in a device such as a ball mill can be considered to be the summation of many single comminution events, each of which yields a size distribution of particles following the Schuhmann size distribution equation. By grinding

Jan 1, 1961

By C. G. Dunn, P. K. Koh

The primary recrystallization textures of 11 single crystals cold rolled 70 pct were determined. The effect of crystal orientation on recrystallization behavior was investigated also and found to be large. An exception was found to the relationship that strong components of the recrystolliza-tion texture are rotated from the cold rolled texture by 30° about <110> axes, but generally this relationship was found to hold quite well. Evidence supporting oriented nucleotion was found and discussed in terms of probable active slip systems and <110> edge dislocations. THE present paper is one of a series&apos;.&apos; on recrys-tallization phenomena in cold rolled 31/4 Si-Fe single crystals and is mainly concerned with the textures produced on recrystallizing from stable end orientations. Data for the cold rolled textures of the crystals used have been reported by Koh and Dunn." Their results showed that the sharpness of the cold rolled textures varied throughout the range of stable end orientations and that the tendency to re-crystallize was related to the sharpness of the cold rolled texture, wherein the rate of recrystallization increased and the size of recrystallization grains decreased with decreasing sharpness of the cold rolled texture. In the project under consideration the cold rolled crystals involved a range of initial orientations that, together with those studied earlier,&apos; provide information on several points. The first point is whether the recrystallization texture from a (111) [112] cold rolled texture, a stable end orientation that can be approached from different directions, correlates in any way with the initial orientation before cold rolling, or whether the recrystallization texture is entirely determined by the observable features of the cold rolled texture. The second point involves a preliminary study to see if the main components of the recrystallization textures correlate with the most probable active slip systems. The third point is to determine what effect departure from slip on one slip system has on the tendency to recrystallize. Experimental Procedure Nine specimens from a previous investigation and two additional ones from thicker stock (lot D) but of similar composition to the others were used in the present work. The latter two crystals, like the nine, were cold rolled to a reduction in thickness of 70 pct and the amount of widening was determined. Depending on their primary recrystallization behavior, the cold rolled crystals were given short or long anneals at temperatures of 980°C or higher. Table I summarizes the data on initial orientations, the textures obtained by cold rolling, and the anneals given to provide samples for the present work. Fine grained specimens were thinned to 0.002 in. thickness and placed in an integrating specimen holder. Chart recordings using a transmission X-ray method and MoKa radiation were obtained for diffraction from (110) planes. This method,&apos;,3 which gives data for all of the pole figure except a central region of 30" radius, proved sufficient to give generally five, and sometimes six, of the six possible pole concentration areas of any one component of the texture. These data, therefore, were generally adequate for a positive identification of each component present. The first texture obtained for specimen B-2 was poorly defined because of too coarse a grain structure and therefore too few crystals in each component for satisfactory X-ray reflections. The same

Jan 1, 1957

By Dean, R. S.

THE numerous theories of hardening which have been advanced in recent years are all satisfactory in accounting for some of the phenomena observed in hardening metals, but none so far presented account for all of the facts observed. Some years ago one of the authors, R. S. Dean,&apos; presented a theory to account for some of the properties accompanying work hardening. The present paper develops this theory further in the light of recent experimental work and extends it to other forms of hardening. Many of the ideas included have come from the writings and personally communicated thoughts of others. We have attempted to give proper credit in every case, but it is possible that we have not always done so. As an historical basis it is desirable here to review very briefly the hardening theories which have been current. The first comprehensive theory of work-hardening phenomena was due to Sir George Beilby.2 He contributed the idea that with cold work part of the structure of the metal was rendered amorphous and that-it was to this amorphous material that the cold-worked metal owed its difference in properties. According to Beilby&apos;s theory this amorphous material was a supercooled liquid and, therefore, had the characteristics of a vitreous material. That such a theory is not capable of explaining the property changes in metals has been repeatedly pointed out. This theory also allowed a slip to take place only once on a given plane, which, as has been pointed out by Jeffries and Archer13 is not consistent with the observed length of an, individual slip and the amount of elongation which can be obtained by wire drawing. The modifications, which have been proposed to this hypothesis to make it coincide with the experimental facts, have to a considerable extent robbed the theory of its clean-cut characteristics. The commonest of these modifications replaces the amorphous phase with an

Jan 1, 1927

By J. B. Cheatham, M. D. Biggs

This paper presents a systematic method for determining the forces on a simplified roller bit. The kinematics of an actual roller bit drilling in rock is complex; nevertheless, it is desirable to obtain an accurate analytical model.. As a preliminary study, the roller bit is represented here by a simple two-dimensional bit and the rock by a Coulomb plastic material. Results from previous work, plus additional assumptions, are incorporated into a computer program to determine the forces required to move the bit in a specified manner. Example problems are worked by use of this program. By studying similar drilling problems and by supporting laboratory work, one could eventually obtain an accurate analytical model for drilling by an actual roller bit. This could lead to the compilation of a catalog of results that, in turn, would allow selection of the optimum drill bit for any given situation. INTRODUCTION

Jan 1, 1970

By N. A. Gjostein

From their study of the anisotropy of grain boundary mobility in aluminum, the authors conclude that tilt boundaries have a higher mobility than twist boundaries because the atomic misfit at the pure-twist boundary is relatively small compared to the misfit at a tilt boundary for a given amount of misorientation. If this is true, then one would expect a twist boundary to have a lower grain boundary energy than the corresponding tilt boundary. Fig. 6, taken from the work of Gjostein and Rhines,10 shows that this is indeed the case for [001]-cu tilt and twist boundaries in the high-angle region, thus lending support to the authors&apos; conclusion.

Jan 1, 1968

Pulverized Fuel Combustion The apparatus shown in the illustration is a complete equipment f r supplying coal to boiler and metallurgical furnaces, cement and lime kilns, dryers, and to all furnaces requiring a temperature and quality of fame under prompt and easy control. It is claimed that the Aero Pulverizer makes practicable the highest efficiency obtainable from burning coal. It makes coal burn like a gas, a flame, the physical and chemical character of which is regulable- a flame that may be elongated or shortened, thus placing the zone of highest temperature where needed- a flame that may be made oxidizing, reducing, or neutral, as occasion may require. The coal is burned as pulverized, and there is no storage of the powder with its attendant hazard. Artificial drying before pulverizing is not necessary if the coal supply be sheltered from rain and snow. Where the Aero is used, it is wholly a furnace question whether a dryer should be installed; it is not at all a pulverizing or storage question. Labor is reduced to a minimum. Slack coal at low cost yields its last B.T.U.

Jan 11, 1915

By S. G. Gibbs, A. B. Neely

Jan 1, 1967

By R. F. Shurtz

The problem of estimating the precision of systematic samples from a mineral deposit is attacked by interpolating the quality, or other attribute measured, by using Fourier approximation. Such approximation using trigonometric functions yields a particularly simple expression for the variance of the mean value of the attribute sampled. The increasing volume of publication in which the science of statistics has been applied to the problem of predicting the quality of an ore deposit from sets of samples taken from it might lead one to believe that engineering judgment can be replaced by statistics. There is, in fact, an implicit application of the principle of indifference or of "no advance knowledge" which has been discredited in modern statistical theory and practice. These pages will show that it is not always advantageous, efficient, or safe to apply this principle. This problem will be dealt with by using series of orthonormal functions to describe the distribution of the quality of ore in space as contrasted with the more commonly used distribution of quality as a frequency function. Mining engineers are rarely well acquainted with the use of such series or of the functions comprising them. These uses are commonplace in the fields of communication and information theory and it is not surprising that applications taken from these fields should be found appropriate for the analysis of the information contained in a set of samples from an ore deposit. It is hoped that presentation of the following basic, but incomplete, results will generate wider interest in this method and a fuller development of it as well as of the necessary computational techniques. FUNDAMENTAL PREMISES Random variables in the strictest sense are not very common in mine sampling. The sets of samples are almost always taken systematically and are, hence, not subject to all the laws governing random events. Therefore, unless there is indeed random experimental error in the determinations of quality, the treatment of systematic samples of n members as though they were random samples of n members often leads to calculation of confidence limits for the mean quality which are larger than they should be with the result that unnecessary additional sampling is done. The reason for this will emerge in the following discussion leading to Eq. 24 and its generalization. In order to inquire into such questions on a rigorous basis and at the same time avoid arithmetic complications that would obscure the concepts involved, a one dimensional distribution of quality will be assumed. The quality, Q, in suitable units is expressed as a function of the distance, x, along some straight traverse of the deposit. Such a function

Jan 1, 1967

By W. H. Boyer, J. W. Gabelman

The concept of metal zoning has long been recognized (Spurr, 1907, 1923; Emmons, 1924, 1926; Lindgren, 1933; Rateman, 1950), although it has been considered indistinct for most epithermal districts. The geometric relation of barren zones of hydrother-mally altered rock to ore deposits was emphasized in the work of Lovering (1941, 1949, 1950), Sales and Meyer (1948, 1949, 19501, Burbank (1941, 1950), Schwartz (1939, 1950), and Kerr (1947). Lovering (op. cit.) and Sales and Meyer (op. cit.) emphasized the consistent presence of barren alteration envelopes enclosing individual ore bodies, particularly of the vein type. Schwartz (1950) noted a difference between the distribution of alteration zones described by Lovering and by Sales and Meyer, as compared with those associated with the porphyry copper deposits of the Southwest. The disseminated deposits of the Southwest usually occur within large bodies of altered rock whereas most veins do not. In the following discussion the mineralization process is conceived of consisting of two distinct phases: pre-ore nonmetallic alteration and metallization. In the present investigation the position, size, and shape of all known hydro thermally altered areas and the position, size, and metal type of all known ore deposits were plotted on base maps of southwestern and west-central Colorado (Figure 1). The maps indicate a generally consistent geometric relation between altered and metallized areas. The addition of alteration to the pattern of metal deposits establishes a temperature zoning pattern of much greater clarity than that formed by the epithermal metal deposits alone, and allows a more precise definition of mineral belts. The center of strongest mineralization is indicated to be the area of greatest barren alteration, and metal deposits are grouped about it both under structural control and crudely in response to temperature zoning. The combination of an area of strong alteration and associated marginal clusters of metal deposits is defined as a mineral district. The strong similarity in individual deposits, in groups of different types of deposits, and in the alteration-metallization relation in districts throughout the region implies that a single metallogenic province is represented in southwestern and west-central Colorado. The occurrence of deposits in Tertiary voi-canics indicates a late Tertiary metallogenic epoci., although an early Tertiary epoch is known locally as at Ouray. Patterns formed by mineral districts can be

Jan 1, 1961

Jan 1, 1970

By N. O. Taylor

THE failure, by swelling, of several cast aluminum-zinc, spiral, pump rods, used to circulate water in a constant-temperature bath, brought up the question as to whether the presence of tin in varying quantities would have any appreciable influence in controlling this action. Previous workers had found the same difficulty with aluminum-zinc alloys, due to their swelling in water or solutions of electrolyte^.^ It was known that pure aluminum was practically insoluble in concentrated sulfuric and nitric acids, presumably due to the formation of an oxidized film on the surface. Concentrated sulfuric acid had little or no effect on pure zinc and tin and concentrated hydrochloric acid would attack all three metals. These facts aroused curiosity as to the probable reaction of the ternary alloys of these metals, and tests were made to determine their resistivity to the action of common laboratory reagents. Samples, covering the entire range of the ternary series of aluminum, zinc and tin alloys, were made up from good grades of the commercial metals.2 The materials were melted, at as low temperatures as possible, in graphite crucibles in a gas-fired crucible furnace. Graphite crucibles were used in preference to clay, to prevent the absorption of silicon by the aluminum, as found by previous workers. The melted materials were stirred thoroughly with pine sticks in order to ensure complete mixing and the greatest possible freedom from oxidation, and were cast in sectional, iron molds giving bars % in. in diameter and 5 in. long.

Jan 1, 1927

By J. C. Osmond

One of the least known geologic regions in the U.S. is the area now called the Basin and Range Province. It is paradoxical that so little geologic information has been compiled for a province that has yielded billions of dollars worth of metals and is currently producing a small amount of oil. Regional stratigraphic studies of parts of this large territory have been published only recently, and regional structural interpretations of even parts of the province are rare in the literature. This, of course, does great injustice to the earliest workers: Gilbert,1,2 Russell,9 Le Conte,4 Spurr,5 Louderback,8 and others, who recognized the area as distinctively different from the rest of North America. Their simplified picture of the structure of the ranges and the province as a whole is the groundwork for present understanding and has never been superseded. Since their early works, however, geologic data from the province has consisted largely of detailed reports of widely scattered mining districts. Each district has been described as though it were an island in an unknown sea. More recent syntheses of the geologic history have been prepared by Lindgren,7 Ransome,8 Nolan,9 Bill-ingsley and Locke,30 Longwell,11 and Eardley.12 The tentative nature of the present interpretation, which treats the structural evolution of the Utah and Nevada area, must be considered in light of the scarcity of geologic information from the province as a whole. The methods of interpreting tectonic history from stratigraphy are based largely on the concepts of Marshal Kay.23 Discussions with Hoover Mackin also have aided in organizing the Tertiary history. TECTONIC AND STRATIGRAPHIC SETTING Two approaches to the subject of the tectonic history are possible; they correspond to looking through opposite ends of a pair of binoculars. One involves an expansion and extension of detailed work and the other an interpretation based on regional geological relationships. It is believed that the second approach, used here, is not in disagreement with detailed observations. The area of the Basin and Range Province is shown in Fig. 1. Within this area middle Tertiary to Recent high-angle faulting has controlled development of mountain ranges and valleys. Only the part of the area north of Las Vegas, Nev., and south of the Snake River in Idaho will be considered in this article. Within this part of the province most of the ranges trend north-south. Approximately two thirds of the area is covered by late Tertiary to Recent valley fill and lake deposits.

Jan 1, 1961

By P. van Meurs, C. van der Poel

From observations of the linear displacement of oil by water from a porous medium as visualized in transparent models, new insight into the mechanism of vircous fingering, as occurring in the case of unfavorable oil-water viscosity ratios, has been obtained. A mathematical description of the observed phenomena leads to formulas. expressing both oil production and pressure drop across the formation as a func-tion of cumulative water injection with the oil-water viscosity ratio as a parameter. Theoretical results are compared with those obtained from scaled laboratory experiments in sand-filled rubes in which water was injected at the one end and production was obtained from the other end. In the second part of the paper the same principles are used to calculate a more realistic production scheme in which production of oil and water is obtained from wells regularly distributed over the field. Results show satisfactory agreement with production data of an actual oil field. Application of the theory is extremely simple as only a limited number of readily available parameters have to be known and results are presented in an analytical form. INTRODUCTION When oil is displaced from a porous medium by water having a viscosity lower than the oil. the oil-water interface is essentially unstable and has the tendency to break up into what are called "fingers" or "streamers". This Viscous fingering has been noted before by Engelberts and Klinkenberg&apos; and has recently been demonstrated in transparent models by van Meurs-. It is highly questionable whether it is at all possible to perform relative permeability measurements In the laboratory reproducing the same distribution of water and oil in the cores as Present in practice. Therefore, we based our descriptive theory on an idealized model which was "tailored" to the fingering mechanism as observed in transparent models under scaled conditions. This approach was the more attractive, as a highly simplified model had already yielded an adequate description of the process. Though we are aware our theoretical approach can be extended in several ways, it is given here in its simplest form to explain the fundamental principles and to avoid complicated formulas. SIMPLEST CASE OF A LINEAR DRIVE Observations and Theoretical Model In Fig. 1 a series of photographs shows how water penetrates into a formation saturated with an oil which is considerably more viscous than the water. The technique for obtaining these photographs, using transparent models, has already been described?. The fingers can be clearly observed and are seen to be mainly parallel to the flow direction. Note that the contours of these fingers are not smooth but show protrusions of water enclosing oil pockets.

By H. S. Price, J. C. Cavendish, R. S. Varga

A numerical formulation of bigh-order accuracy, based on variational methods, is proposed for the solution of multidimensional diffusion-convection-type equations. Accurate solutions are obtained without the difficulties that standard finite difference approximations present. In addition, tests show that accurate solutions of a one-dimensional problem can be obtained in the neighborhood of a sharp front without the need for a large number of calculations for the entire region of interest. Results using these variational methods are compared with several standard finite difference approximations and with a technique based on the method of characteristics. The variational methods are shown to yield higher accuracies in less computer time. Finally, it is indicated how one can use these attractive features of the variational methods for solving miscible displacement problems in two dimensions. INTRODUCTION The problem of finding suitable numerical approximations for equations describing the transport of heat or mass by diffusion and convection simultaneously has been of interest for some time. Equations of this type, which will be called diffusionconvection equations, arise in describing many diverse physical processes. Of particular interest here is the equation describing the process by which one miscible liquid displaces another liquid in a one-dimensional porous medium. The behavior of such a system is described by the following parabolic partial differential equation:

Jan 1, 1969