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By J. Mote, P. Ward Flynn, J. E. Dorn

The effect of strain and strain rate on the critical resolved shear stress for prismatic slip in specially oriented single magnesium crystals was determined over a range of temperatures in an effort to identify the thermally activated strain-rate controlling dislocation process. Below about 450°K prismatic slip was preceded by twinning and fracturing. Above 450°K extensive prismatic slip was obtained followed by fracturing. Neither the Peierls&apos; mechanism nor the dislocation intersection mechanism can account for the observations. The data, however, are in good agreement with the dictates of Friedel&apos;s theory for cross slip to the prismatic plane. PREVIOUS investigations on single and polycrys-talline magnesium have resulted in the identification of the mechanisms of plastic deformation summarized in Table I. The-only well-confirmed mechanisms are {0001}, (1010}, (1011) slip plus (1012) twinning. The other twinning modes were only tentatively established and additional investigations are required to further confirm their operation. As noted in Table I, slip in this hexagonal close-packed metal invariably occurs in the direction of greatest atomic density, revealing that the Burger&apos;s vector for-active slip dislocations is always given by b = a/3<1210>. Slip is easily induced on the basal plane, and recent investigations have shown that the strain-rate controlling mechanism at low temperatures is the thermally activated intersection of the glide dislocations with dislocations threading the basal plane. In contrast, very little is known about prismatic slip. It has been reported to be active at points of stress concentration in polycrystals below

Jan 1, 1962

By A. D. Rich

The term "bleaching clay" or "bleaching earth," as used in the oil industries, refers to clays that in their natural state, or after chemical or physical activation, have the capacity for adsorbing coloring matter from oil. There are three common types of bleaching clays: fuller&apos;s earth, activated clays, and activated bauxite. Fuller&apos;s earth, or naturally active clay, is prepared from bentonites (see Chap. 5) that possess natural activity. It is not activated commercially; in fact, earths of this type ordinarily do not respond satisfactorily to acid activation. Fuller&apos;s earth is prepared in pulverized as well as granular form for use in the contacting and percolation processes, respectively. Acid-activated clays also are of bentonitic origin. The basic raw clay, however, is of a type that has a very low natural activity but is highly activable by treatment with mineral acid. Commercial acid-activated clays generally possess several times the decolorizing power (bleaching efficiency) of the best quality of fuller&apos;s earth. The former are prepared commercially only in the pulverized grade because of the difficulties in obtaining a satisfactory granular product, and hence are used only in the contact process. Activated bauxite is made by heat-treating bauxite ore, which in its natural state has very little natural activity. The commercial product is prepared in granular form for use in the percolation process on petroleum oils. For the most part, bleaching clay is used commercially in the decolorization of animal, vegetable and petroleum oils, fats and waxes. It is applied in two ways: (1) by the percolation method and (2) by the contacting method. Percolation refers to the passage of oil through a bed of granular clay whereas contacting refers to direct agitation of oil and pulverized clay followed by removal of the clay from the oil by filtration. In either case, the decolorizing effect is obtained by intimate contact between the liquid oil and the solid adsorbent. Petroleum oils are treated commercially by either the percolation or the contact process. Animal and vegetable oils are treated by the contact process only. Composition The raw clays from which both fuller&apos;s earth and acid-activated clays are prepared are composed of hydrous aluminum silicates containing varying quantities of magnesium, iron, calcium and other elements. These raw materials are usually nonswelling bentonites derived from volcanic ash or lava by weathering and hydrothermal chemical reactions. The bentonites have as their chief mineral constituent montmorillonite, which has the formula [(OH)4Si6A14O20•nH2O], according to Grim. (See Chap. 5, ref. 6, p. 91.) The underlying reasons are not fully known for the natural activity of the fuller&apos;s earth type of clay and its usual failure to further activate with mineral acid in contrast to the natural inactivity of the activatable type of clay. However, it may be said that the bentonites that are responsive to the acid-activation treatment generally have not undergone any drastic metamorphism. The chemical composition is not a reliable means of discrimination between a nonactive, a naturally active, and a nonactive but activable clay, therefore it is not possible to forecast the ultimate quality of a clay on the basis of the raw-clay analysis alone. Table 1 gives typical analyses of representative clays of the foregoing classifications.

Jan 1, 1960

By C. M. Wayman, G. R. Srinivasan

A metallographic study of the isothermal products formed between 750° and 285°C was made in an Fe-7.9 wt pct Cr-1.11 wt pct C alloy, and a time-temperature-transformation diagram was determined. The study shows that the austenite decomposes into three different products: pearlite above 540°C, ferrite and carbide between 540° and 300°C, and lenticular bainite below 300°C. No upper bainite forms in this alloy. The results are discussed in terms of carbon diffusion models for upper bainite formation. AUSTENITE transforms into several products depending on the temperature, composition, and rate of cooling, and the various transformation products have been a source of controversy for a long time. Since the classical study of Davenport and ~ain&apos; extensive work has been carried out. The present study concerns the isothermal transformations in an alloy which was used for a detailed investigation of bainite crystallography and provides information supplementary to the work of Lyman and Troiano who studied austenite decomposition in steels containing 1 pct C and varying amounts of chromium. Their work showed that increasing the amount of chromium in high-carbon steels had the effect of depressing the B, temperature and of shifting the TTT curves to longer times and lower temperatures. They further found that if the chromium content is increased in a 1 pct steel to about 9 pct no bainite would form even after a period of 100 days. However, a preliminary investigation by the present authors showed that bainite plates do form in steels containing about 8 pct Cr and 1 pct C. In view of this and some results from a thin-foil electron microscope investigation of the bainite formed in the present alloy: it was considered desirable to investigate the other isothermal products as well. EXPERIMENTAL PROCEDURE The alloy used in the present investigation was obtained from a triply vacuum-melted ingot prepared from high-purity materials. Chemical analysis for the alloy verified the composition to be Fe-7.9 wt pct Cr-1.11 wt pct C. Metallographic observations were made on i by i by « in. specimens which were aus-tenitized at 12 50°C for 50 to 60 hr in evacuated quartz capsules and then water-quenched. Since the M, temperature for this alloy is below room temperature, the homogenizing treatment produced a completely austenitic structure consisting of very coarse austenite grains (ASTM #0-1). Several samples of the austenitized specimens were metallographically prepared and observed for any proeutectoid carbide separation. It was found that all the carbon was in solution. The water-quenched specimens were upquenched and given isothermal treatments at temperatures between 285" and 750°C for varying periods in molten salt or lead baths. The specimens were then water-quenched, and those showing less than 5 pct transformation were used for plotting the beginning of the austenite decomposition. To ensure the removal of the disturbed layer, the mechanically prepared specimens were electropolished in a 1:6 HzS04, CHSOH solution for 10 to 15 sec at 20 v. Final polishing was done with a vibratory polisher using 0.05-p alumina. Various etchants including nital, ferric chloride, and hot ferricyanide (Murakami&apos;s reagent) were used to bring out details in the different transformation products. The ferricyanide etch was used whenever the preferential etching of carbides was desired.

Jan 1, 1969

By C. H. Gorski

A method for preparing titanium tetrachloride is described which consists of reducing rutile with coke and chlorinating the reduced product at 200° to 500°C. The crude distillate is purified by treatment with copper powder and distillation. Recoveries, exceeding 90 pct of the titanium in the rutile, were obtained. IN recent years considerable interest in metallic titanium has been aroused because of its outstanding physical and chemical properties. The present commercial processes for preparing titanium are modifications of the Kroll1 process and consist essentially of the reduction of titanium tetrachloride with magnesium in an inert atmosphere to produce metallic titanium with magnesium chloride as a byproduct. At present,&apos; commercial titanium tetrachloride costs $0.32 a pound, which makes the price of the included titanium $1.28 a pound. As the titanium tetrachloride must be purified further before processing into metal, its ultimate cost obviously will be higher. Consequently, one of the prime requisites for the production of lower cost titanium metal is a source of low-cost titanium tetrachloride. Recently, Knickerbocker, Gorski, Kenworthy, and Starliper3 escribed a method for preparing titanium tetrachloride from low-grade Arkansas rutile by smelting the rutile with pyrite and coke to form a titanium matte and then chlorinating the titanium matte to prepare a crude titanium tetrachloride. This process had several inherent advantages. Low-grade rutile concentrates were utilized, the chlorination temperature was relatively low (approximately 200°C), and the reaction was strongly exothermic. However, this process had several disadvantages, chief of them being the rather low recoveries of titanium tetrachloride (65 to 81 pct) and the difficulty in purifying the crude titanium tetrachloride obtained. Because of the high-titanium and low-iron con- tents of rutile as compared to ilmenite, rutile should be of greater potential value as a source of titanium tetrachloride than ilmenite. Rutile, however, is not chlorinated directly in commercial practice, because reaction temperatures above 700 °C are required, at which point chlorine attacks most of the common materials of construction. The process now in use consists of converting the titanium in rutile to the cyanonitride by high-temperature reduction in an electric furnace, and then chlorinating this compound. The disadvantages of this process are the difficulty of producing the cyanonitride, the use of high-grade rutile concentrates in the charge, and the presence of several percent of silicon tetrachloride in the finished product. It would be of considerable interest to develop a new process for the preparation of titanium tetrachloride from rutile by chlorination at 200" to 500°C. Process Development Several investigators&apos;-&apos; have studied the reduction of titanium dioxide with carbon in an effort to prepare titanium monoxide. Invariably, the titanium monoxide formed in this manner was not pure, as it contained titanium carbide and one or more of the lower oxides of titanium. As these lower oxides of titanium are more reactive chemically than rutile, and as it is known that titanium carbide may be chlorinated readily, a study was made of the chlorination of carbon-reduced rutile. A sample of rutile concentrates from Hot Spring County, Ark., containing 76.19 pct TiO,, 5.65 pct Fe2O3, 4.18 pct SiO2, and 2.88 pct Al2O3, was used for this investigation. The experimental procedure consisted of making an intimate mixture of —100-mesh rutile and petroleum coke, adding water and molasses as a binder to foi-m a paste, charging into a graphite crucible that was covered, except for a vent to allow the

Jan 1, 1952

By Pedro L. Lizaur

INTRODUCTION The degree of success of an association between parties in establishing a new venture ultimately depends on three factors: a. the venture&apos;s underlying economic strength, which is based on the complementarity of the comparative advantages brought by each party to the negotiating table; b. the ability to reach an agreement which satisfies each party&apos;s objectives and constraints; and, c. the capacity to design and implement a funding structure which is consistent with the project economics. A thorough assessment of the venture&apos;s profitability and associated risks under alternative technical and economic scenarios is the basis for a sound "go" or "no go" decision. In-depth evaluation of the risks and rewards to each of the parties involved under alternative legal and financial structure must be the foundation for a long lasting "how to go" decision. The allocation of risks and rewards should be consistent with the objectives and constraints of all parties and the sharing mechanisms, established in the contract, should be flexible to assure the long-term stability of the agreement. Rapidly changing economic conditions during the last decade have had a dramatic impact on the envisaged allocation of risks and rewards between host countries and private investors. The effects of these changes, and the fact that the relative strengths of the parties involved may shift during the life of a project (e.g. before and after major capital investments have been made), have forced the renegotiation of a multitude of contracts. Therefore, the present and future stability of an economic agreement would depend not only on the originally designed provisions, but also on provisions which allow for the automatic adjustment of rewards to project sponsors under a variety of future economic situations. A pre-condition for the allocation and quantification of risks and rewards is the definition of standards of measurement used by each of the parties. This is critical in international negotiations when one of the parties represents the national interests of a country. The understanding and use of different standards for the measurement of rewards for risk taken between a government and private enterprise facilitate the negotiation of a satisfactory agreement. This is possible because certain trade-offs between the negotiating parties may have different relative values. STANDARDS OF MEASUREMENT OF REWARDS FOR RISK TAKEN The negotiating process takes a step forward when the parties are able to focus on their interests, explore different alternative solutions and then base results on a set of objective standards. Unfortunately, the parties frequently put more effort into "staking out" and defending positions, than into understanding their intrinsic interests. Moreover, when one party, such as a governemnt owned corporation, represents a country&apos;s national interests, the negotiation process increases in complexity. Private firms focus on commercial profitability, using discounted cash flow techniques such as NPV, IRR, or ROE. In contrast, the government is also concerned with the project&apos;s impact on the local socio-economic environment. Without entering into a discussion of alternative economic development theories and approaches, there exist several possible measures that could be pragmatically utilized by each party in evaluating the socio-economic impact of a project. The more complete measure is the National Profitability (NP). The NP is a discounted cash flow procedure similar in structure to IRR or NPV but in which the project&apos;s financial cash inflows and outflows and discount rates are adjusted to take into consideration the local conditions. The NP is calculated on an unlevered basis and all taxes and duties are removed from the project

Jan 1, 1985

By E. M. MacKevett

The only source of uranium ore in Alaska that has been mined commercially is the Ross-Adams deposit, a gently inclined, fusiform orebody in alkali granite in which uranothorite and uranoan thoria-nit? ale the chief ore minerals. In mode of occurrence, apparent genetic association with the granite, and to a lesser extent in mineralogy, this type of deposit is uncommon. The mine is in the southern part of Prince of Wales Island (Fig. 1) at an altitude of about 950 ft on the southeast flank of Bokan Mountain. A steep unpaved road 1 3/4 miles long leads to the dock on the West Arm of Kendrick Bay, which can be reached by boat or seaplane from Ketchikan, some 35 miles to the northeast. The southern section of Prince of Wales Island is covered by dense forests interspersed with swampy tracts of muskeg. Most of the higher peaks support only sparse vegetation, and in areas underlain by alkali granite bare rocky slopes prevail to as low as 800 ft. Numerous fiord-like embayments penetrate this part of the island, and their shorelines provide near-continuous rock exposures. Rainfall is heavy, about 150 in. annually. Throughout the year the climate is relatively mild. History and Production: In May 1955 two members of a Ketchikan prospecting group, Don Ross and his wife, found the deposit while working with an airborne geiger counter. Later they confirmed their discovery by ground investigations. During the summer Climax Molybdenum acquired controlling interest in the property and started exploratory diamond drilling, resuming the drilling on a larger scale in the summer and fall of 1956. Climax began mining in July 1957, shipping the ore by barge and train to the Dawn Mining Co. mill at Ford, Wash. The open pit workings, about 370 ft long, between 25 and 75 ft wide. and 25 ft deep (shown in Fig. 2) were shut down in October 1957. As of September 1958 the pit was still inactive, although a small tonnage of known reserves remains in the southern part. The operators believe the 15,000 tons of uranium ore already produced will average 0.80 pct. GEOLOGY The Ross-Adams mine is within a small stock or boss of alkali granite, about 1000 ft from the southeast margin of the granite (Fig. 3). The area around the boss is underlain chiefly by plutonic rocks of intermediate composition and by metamorphosed sedimentary and volcanic rocks. The oldest rocks in the area probably are the metamorphosed volcanic rocks that are well exposed adjacent to the South Arm of Moira Sound. They are considered by Buddington and Chapin1 to be of Devonian age. These light gray to dark gray rocks mainly represent a layered volcanic sequence. They are commonly porphyritic with plagioclase pheno-crysts in an altered, fine-grained groundmass. The

Jan 1, 1960

By A. J. de Witte

INTRODUCTION The importance of determining dips of subsurface formations in wells, especially in wildcat wells, is self-evident and need not be enlarged upon. Various dipmeter instruments have been described in technical literature1,2,3,4 over the years. The introduction of the continuous dip-meters3,4 has made the earlier types obsolete. Continuous dipmeter logs are run by at least two different major service companies. Included as part of the service are a limited number of dip calculations—usually 10—made at different levels from the log. Very frequently this is the total result of a dip survey. It is believed that dipmeter surveys would be of considerably more advantage if geologists and engineers were able to make their own calculations from the original records. Determination of dips from the dipmeter record starts with the correlation of three log curves taken at different sides of the borehole. It cannot be over-emphasized that the quality of the results hinges primarily on this correlation. Nobody should be better qualified to correlate logs than a subsurface geologist or field engineer. The next requirement is to have an easy yet accurate way of computing the dip from the correlated points. Service companies use a mechanical device for this purpose. Directions for a graphical method were given in a recent article by B. 0. Prescotts. This paper describes a different graphical technique which, in our experience, is simpler, faster and at least as accurate. The entire manipulation is carried out on the stereonet in a number of easy moves. The paper is divided into two parts. This first part contains the introductory sections and full explanation of the geometry involved. The second part reviews the reading of data from an actual field log and summarizes the practical rules for computation of dips, the whole process being illustrated by an example. Following the outlined procedure, one could calculate a good many levels in a relatively short time, thus being able in particular to check suspected anomalies such as fault zones or unconformities in as close detail as the log will permit. GENERAL EXPOSITION OF THE METHOD Prnciple of the Dipmeter The position of a plane in space is defined by the location of three points, not on a straight line. Referring to Fig. I, if a borehole traverses a bedding plane, we can determine the strike and dip of the bed by finding three points which are not in line along this plane. Such points as A, B, and C are obtained with the dipmeter. The device consists of three electrodes (or caliper arms)1, 2, and 3 spaced at 120 angles, as schematically indicated, whose readings are separately recorded (Curves 1, 11, 111, to the right). When electrode 1 passes the boundary of the bed, Curve I will kick, etc. From these curves, we know the relative distances of points A, B, C, measured along the axis of the hole. If A is at zero, then B and C are at levels d1 and d2 up the hole,

Jan 1, 1957

By T. D. Yensen, R. K. McGeary

While we have not been primarily interested in the determination of hydrogen in the alloys that we have been dealing with, we arc very glad to cooperate in this symposium on sampling and analysis for hydrogen, as we can never tell when we may find it desirable to perform such analyses. Vacuum-fusion Method Our primary interest has been oxygen and carbon, in the analysis of which we think we have become very proficient. Our oxygen-analyzing apparatus is of the vacuum-fusion type (Fig. I). This employs Dr. Derge&apos;s type of graphite cruciblc and slit sleeve, with graphite powder between, for heat insulation (Fig. 2) supported in a clear quartz bulb connected to the Pyrex glass system by means of a carefully made ground joint. The latter is kept cool by means of an air blast from a circular copper tube surrounding the joint. The crucible is heated by high-frequency induction and the 24 one-gram samples successively dropped in from the horizontal tube at the top by means of a magnet. The gases (H², CO, N2) are passed through the CuO catalyst held at 300°C. (converting CO to CO²I, H² to H²O) and the CO² and H2O frozen out in the main liquid air trap, while N2 is evacuated. At the end of the operation, shown by the termocouple vacuum gauge indicating a pressure of 10-4 mm. Hg, the analyzing system is isolated by means of the mercury cutoffs, and the CO² and H2oare determined by transferring first the CO2 and then the HzO to the small trap at the right. This is done by placing liquid air on the small trap, then removing the liquid air from the large trap for one minute, which is sufficient to evaporate the Co2 and transfer it to the small trap. The CO2 is then measured by closing the mercury cutoff at the extreme right, removing the liquid air from the small trap, warming it up to room temperature, adjusting the mercury level to one of the three marks on the glass tube (corresponding to three volumes calibrated so that I mm. Hg of CO² corresponds to 0.001, 0.002 and 0.010 mg., respectively, of oxygen) and finally reading the pressure. Liquid air is now replaced on the large trap, the mercury cutoff at the right opened, transferring the COz back to the large trap. The cutoff to the pumps is then opened, the liquid air removed for one minute, thus evacuating the CO², the pumps again cut off and the H2 O transferred to the small trap as before, except that it is now necessary to warm the large trap with a Bunsen burner in order to evaporate all the H2O. This requires about three minutes, as the inner tube must be heated to at least 100°C. In order to prevent H²O from being adsorbed on the glass walls, the glass tubing beyond the catalyst (i.e., to the right in Figs. I and 3) is kept heated to about 150°C. by means of Nichrome ribbon as shown in Fig. 3. For a I-gram sample the above amounts of oxygen (0.001, 0.002 and 0.010 mg.) 35

Jan 1, 1945

By D. R. Spink

In the preparation of zirconium and hafnium metals, there are many indications that impurity levels control the physical and chemical properties of the metal. The purpose of this work was to develop methods of producing pure raw materials that could be reduced to highly pure metals. Early work by the Carbomcndum Metals Co. and others indicated that certain fused salts were highly efficient in the removal of impurities from gaseous zirconium tetrachloride. We have developed workable fused salts along with a continuous and higly efficient scrubber design which can be used to produce very pure zirconium tetrachloride. Metal made from the purified tetrachloride has been shown to approach crystal bar metal in purity. Scrubber designs are discussed with recommended materials of construction. VARIOUS methods have been used to purify zirconium (and hafnium) tetrachloride. Most of these methods involve thermal degassing initially (the removal of more volatile compounds at temperatures below that where zirconium tetrachloride exhibits a significant vapor pressure) followed by a sublimation step (which conveniently leaves the nonvolatiles behind.) While these procedures effect definite purification, such processing is batchwise and does not result in optimum purification. Early work by The Carborundum Metals Co. and others indicated that fused salts were highly efficient in the removal of impurities from gaseous zirconium tetrachloride. The objective of this work was to develop a continuous fused-salt scrubbing process that would yield high-purity zirconium and hafnium tetrachlorides. The term "fused salt" refers to the molten state of ionic compounds and as used here, specifically refers to alkali metal and alkaline earth halides. Fused salts, because of their many valuable properties, are becoming increasingly im- portant industrially. These valuable properties of the fused salts are: 1) stability at elevated temperatures; 2)low vapor pressure; 3) low viscosity; 4) ability to dissolve many different materials; and 5) good elctctrical conductivity. All but the last property enhances the use of fused salts as scrubbing agents for zirconium and hafnium tetrachlorides. Fused-salt scrubbing may be defined for our purpose as the intimate contact between a gaseous compound and a molten salt with the purpose of removing impurities from the gaseous compound. In this case, the gaseous compound is zirconium tetrachloride which submlimes at 331°C. When this gas is contacted with a molten salt, impurities such as aluminum, iron, and titanium are removed and the resulting product is considered fused-salt scrubbed. Zirconium and hafnium are sold to a purity specification. Generally speaking, normal production without scrubbing is capable of meeting specifications. There are, however, several considerations That make the knowledge of methods of additional purification highly desirable. These are: 1) the control of product analysis; 2) the production of a pure product for special applications; 3) the salvaging of off-specification material; and 4) the use of raw materials with higher impurity levels than 1 he present raw materials. You will note that items 1 and 2 are somewhat complementary to each other and that item 4 assumes that The present process has impurity removal limitations. To better understand the part that fused-salt scrubbing of zirconium tetrachloride plays in the manufacture of zirconium, Fig. 1 contains an illustrated flow sheet of the industrial production of zirconium metal. In referring to Fig. 1, we can point out the specific benefits derived from our ability to purify zirconium tetrachloride. Purification of the crude (hafnium-containing) zirconium

Jan 1, 1962

By Carl F. Floe, Michael B. Bever

he possibility that carbon may be soluble in copper to a limited extent has bten recognized for over a century. The quantitative investigation of this problem, however, requires more sensitive techniques than have formerly been available. Three recent developments have made possible more accurate determinations: the vacuum-fusion method of analysis, the production of copper of high purity and the availability of high-purity refractories, particularly graphite. In addition to its obvious theoretical interest, the problem of carbon solubility in copper is of practical importance. In the production of copper and of high-copper alloys, carbonaceous covers are freauerltlv used. If even minute amounts of carbon are dissolved, insoluble carbon oxide gases will be formed when such metals are poured under oxidizing conditions. A mechanism of this kind could well be one of the principal causes of gas porosity in casting copper-base products. Review of Literature In 1823 Vivian1 raised the question as to whether copper by overpoling becomes "combined with a minute portion of carbon." Karsten2 asserted, in 1832, that carbon reduces the high-temperature strength of pure copper, so that when the carbon content has reached 0.2 per cent the metal will be brittle at a dark red heat. According to Karsten, carbon contents of 0.05 per cent make it difficult to hot-work copper. He also reported that the specific gravity of copper is 8.8969 and that it increases to 8.9258 if the copper is cemented with carbon and then melted. He emphasized the difficulty of determining analytically the small amounts of carbon present in copper. The metal was removed by wet methods and the residue considered as carbon. At Percy&apos;s request, Dick,3 in 1856, carried out various working tests on copper that had been ill prolonged contact with carbon, and found that the working properties of the metal had not been affected. He also attempted to determine the presence of carbon analytically. In one experiment the copper was dissolved and the residue was fused with litharge, and in another essentially a combustion analysis was made on a copper sample. In a final experiment, a sample was dissolved and the residue subsequently subjected to a combustion analysis. Indications of as much as 0.005 per cent carbon were found, but Dick considered his work inconclusive largely because of the possible interference of sulphur. Percy4 concluded, from Dick&apos;s experiments and from some experimental work of his own, that the solubility of carbon in copper had not been established with

Jan 1, 1946

By Henry J. Welge

The approximate methods which are now in use for calculating oil displacement from reservoirs by gas-cycling or gravity-drainage at constant gas pressure, or by water flooding, make use of fundamental relationships derived by Leverett&apos; and Buckley and Leverett.2 The mathematical equations needed are derived by applying Darcy&apos;s law to the flowing phases, and by material balance considerations. ln general, any treatment of this type gives. for any particular exploitation time considered, a plot of oil saturation against distance in the reser-voir. The oil recovery must then be obtained by integrating in some manner the area under the plot. A useful analytical method has been derived for computing the average saturation, and hence the oil recovery. Use of this method simplifies the calculations because it makes unnecessary any numerical integrations. and even the saturation distribution plots are not needed. A further advantage of the method is that knowledge of the relative permeabilities is required only for a limited and intermediate saturation range. In both the Buckley and Leverett method and the method discussed here, a linear sand section is assumed, and in the case of gas drive the gas pressure is assumed sufficiently constant both with respect to reservoir position and time so that changes in gas density. solubility, or reservoir volume factor are negligiklle. Thus, the exploitation contemplates oil displacement as by an immiscible phase. Examples are given to illustrate how the new method can be used. INTRODUCTION AND THEORETICAL BACKGROUND This paper treats a simplified method for computing oil recovery when the oil is displaced, from the reservoir sand by a fluid which, within the limits of accuracy desired, can be assumed to be incompressible and immiscible with the oil. The method makes use of two basic relations originally developed for the case of water displacing oil. However, the case of gas displacing oil saturated with gas at a constant (or nearly constant) pressure may also be considered a displacement by an immiscible fluid. This is possible for the reason that the concentration of the gas in the oil never changes if tile pressure is fixed. consequently, any additional free gas must remain undissolved in the oil, and so must act essentially as an immiscible phase. For convenience, the cape of gas drive will be considered first. Both of the two basic relations needed refer to a linear reservoir, or one which may be somewhat idealized so that a constant cross-section is exposed to fluid flow. Further, the displacing phase is injected at one end, or face; of the reservoir, while the produced fluids are understood to emerge at one point only. which is the opposite face of the reservoir. As indicated in Fig. 1, the reservoir may he inclined by some angle 8. The first of the required equations may be derived by applying Darcy&apos;s law to both displacing and displaced phases, and subtracting the two equations. The result for water drive has been given previously, as Equation (13) of the reference by

Jan 1, 1952

By David V. Ragone, James A. Craig, Richard E. Balzhiser

The Gibbs free energies of formation of UC2 and UC were measured by equilibrating two-phase mixtures of UC2 + C and UC, + UC with liquid bismuth. The measured equilibrium concentrations of uranium in the bismuth combined with available activity coefficient information yielded uranium activities in these two-phase regions. Measurements on UC2 were performed over the range 1015 to 1160 K and on UC over the range 1115° to 1165°K. The values obtained for UC2 are in excellent agveement with values determined using combustton calorimetry and low-temperature heat capacity measurements. The UC values are in fair agreement with the calorimetric values, differing by about 10 pct from the calorimetric determination. MUCH interest has been shown in the uranium carbides in recent years because of their potential as nuclear fuels. As nuclear reactor operating temperatures have been raised to increase thermal efficiency, new fuels have been required. The uranium carbides are of interest because of their high melting points, >2000°C, high thermal conductivity, and resistance to irradiation damage. The manufacture and use of uranium carbides are aided by a knowledge of their stability and reactivity under various conditions. In this regard the thermo-chemical properties of the carbides, i.e., heat capacity, heat of formation, free energy of formation, and so forth, are particularly important. A great amount of work has recently been done on the thermodynamics of the U-C system. Unfortunately, however, there has not been good agreement among the various investigations. This study of the free energies of formation of uranium monocarbide and uranium di-carbide in the temperature region 1100°-1200° K was undertaken to help clarify the situation. It should be noted that in the temperature range of this investigation, 1100°- 1200° K, the thermodynam-ically stable phase in equilibrium with graphite is U2C3, not UC2, Fig. 1. However, U2C3 is reported to be stable only under extremely low oxygen and nitrogen pressure. Although UC2 is metastable below 150O°C, it exists under most experimental conditions. It should also be noted that "UC," never exists as UC2 but with compositions near UC1.9. However, "UC2" will be used to designate the dicarbide phase except when the stoi-chiometry is necessary for clarity. METHOD The approach taken in this study of the free energies of formation of UC and UC2 was to determine the thermo- dynamic activities of uranium in the two-phase regions UC, + C and UC + UC, by equilibrating the two-phase mixtures with the liquid bismuth. When a mixture of solid UC2 + C is brought to equilibrium with liquid bismuth, the reaction that occurs is: Neither carbon nor UC, could be detected as dissolved material in the bismuth. The standard free energy for this reaction may be written: In the two-phase region UC, + C, Fig. 1, the thermo-dynamic activities of UC2 and carbon are unity. The activity of uranium in this region was determined by measuring the concentration of uranium in a liquid bismuth melt in equilibrium with the two-phase mixture. This concentration was then used in conjunction with the activity coefficient data of uranium in bismuth reported by Tien et al.&apos; to determine the activity of uranium in the two-phase region, UC, + C. All of the quantities in Eq. [2] were then known, and the AG; (uc,) was evaluated. UC + UC2. For the studies involving UC + UC,, the reaction taking place when the two-phase mixture is brought to equilibrium with liquid bismuth is: The standard free-energy change for this reaction is given by:

Jan 1, 1969

By R. J. Borg, C. E. Birchenall

Vapor pressures of eight compositions of Mg-Cd have been determined by the Knudsen effusion technique. Measurements are made at several temperatures for each alloy and the results interpolated to a common temperature for the calculation of the activities. It is not possible to obtain values for the partial molar enthalpies because of the narrow temperature range spanned by the measurements. The results show less precision than those obtained for pure Cd which are also reborted. These alloys are extremely susceptible to corrosion which provides a possible explanation for the lack of precision. Nevertheless, the values of the activity are in good agreement with the results of electromotive force measurements previously reported for the Mg-Cd system. MAGNESIUM and cadmium are mutually soluble in all proportions in the solid state and form a single phase which has the hexagonal close-packed structure. X-ray investigations of this system1,2 have revealed a rapid variation in the relative dimensions of the unit cell occurring at approximately 50 at. pct. The rate of change of the c/a ratio with composition increases markedly for alloys containing more than 50 at. pct Cd. The purpose of this investigation was to determine the activity of Cd in the neighborhood of 50 at. pct and to discover whether or not the thermo-dynamic properties undergo a similar rapid change. The activities are calculated from vapor pressure data obtained by the Knudsen effusion method. Unfortunately the precision of the data from which the thermodynamic values are calculated is insufficient to establish unequivocally a departure from the normal trend. The results are reported primarily because they substantiate the only previously reported activity measurements3 and because they include values for pure Cd which in contrast to those of the alloys show excellent precision. The Cd-Mg system is known to contain several ordered phases. However, this investigation was made at temperatures considerably above the transition temperatures, and the alloys may be regarded as nearly random solid solutions. EXPERIMENTAL Apparatus—The effusion cells are machined from solid tantalum rod. The lids consist of threaded, flanged rings of tantalum onto which are welded shallow cups of 4 mil molybdenum foil. The orifice, which is located at its center, is made by grinding off a small dimple produced with a sharpened punch. Such an orifice has been mounted in plastic, sectioned, and the thickness of the orifice edge measured in a microscope. The maximum thickness is 0.00016 in., and the Clausing factor is consequently equated to unity. The area of the orifice is determined by tracing its projection on the screen of the metallograph at a known magnification and measuring the area of the tracing with a planimeter. Excellent temperature control is achieved with a 4 by 18-in. alundum core, gradient-wound, furnace

Jan 1, 1960

By F. Weinberg

In developing a mechanism for the solidification of metals from the melt, it has been proposed that solidification proceeds by the growth of platelets parallel to close packed planes. The evidence for this is based primarily on observations of decanted interfaces of lead1,2 and tin on which platelets 1 to 10 µ thick4 were observed. In recent observations of decanted eutectic alloys, 5,6 it was found that the eutectic structure on the decanted surface was markedly different than that of the bulk material, with the surface structure extending approximately 10 µ behind the interface.&apos; This suggested that, at least for these experiments, a residual liquid layer 10 µ thick had been left on the decanted interface, which solidified subsequent to decanting. If a similar thickness of residual liquid was left on the decanted surfaces in the lead and tin observations referred to above, then it is questionable to what extent the observed platelets can be related to the solid-liquid interface before decanting. The purpose of the present investigation was to measure the thickness of the residual liquid layer on a decanted surface of tin, using relatively standard decanting procedures. The technique used was to add approximately 200 ppm of Tl204 (PI) to the melted portion of a high purity (99.999 pet) bar of tin, which was being progressively melted from one end, and then decant during melting. Since only the melt contained the radioactive tracer, the activity emanating from the decanted surface could be used as a measure of the thickness of the residual liquid layer. Progressive melting was carried out in the same fashion as single crystal growth, namely, in a horizontal graphite boat, in an argon atmosphere, using a moving furnace. The decanting was done in two ways; 1) removing the furnace and suddenly dropping the melt end of the boat, and 2) suddenly jerking the solid away from the liquid by the action of a falling weight (estimated peak acceleration 3 x 105 cm per sq sec), using a split graphite container. In all cases the melt was vigorously agitated by bubbling argon through it. The activity of the decanted surface was measured through an aperture 1.5 cm by 0.5 cm in a lead shield with an Anton tube geiger counter. Assuming that the active layer on the surface had the same concentration of Tl204 as the decanted liquid, the thickness of the layer could be determined from the measured activity of the decanted material, and the relationship between activity and material thickness. The latter was determined by rolling some of the decanted material into foil and measuring the activity of multiple layers of the foil. The activity At of material of thickness t (in mm) was determined to be A, = Ao(1 - e-22.5t) where Ao is the activity of the bulk material. The results of the measurements of the thickness of the residual liquid layer for six decants are given in Table I. The variation in t is attributed to variations in the decanting procedure and the rate of melting, the latter being difficult to control because of the vigorous agitation of the liquid. The rate of melting was in the order of 2 cm per hr, estimated from the position of the solid-liquid interface during melting. The results indicate that a residual liquid layer of the order of 10 µ in thickness is left on the decanted interface in agreement with the observation on eutectic alloys referred to above. If the mixing of the liquid during melting is not complete, as was

Jan 1, 1962

By Richard Franchot

DISCUSSION of a paper is sometimes more in-teresting in what has not been said than in what is said. In the present case the paper is mainly directed to certain thermo-dynamic relation-ships in the blast furnace. Careful reading of the dis-cussion fails to show a single use of the term, heat unit, or its equivalent. If we wish to know very-much about the furnace action it is necessary to count calories or B.t.u. Nothing much is. said about the availability of heat at high temperatures, although it will be generally admitted that in seeking underlying limiting factors in the furnace economy the hearth heat relations offer-the most promising clues. Those discussing the paper, with exceptions of course, apparently have failed to recognize an outstand-ing fact. It is a fact that the heat actually generated by the air oxygen in CO formation (including the blast heat) is in excess, after charging off radiation, etc., by from 70 to .90 per cent over the heat absorbed in melting the iron and slag and calcining the stone reduction by CO absorbing no heat. The fact stands out as plainly as does the fact that the furnace functions largely as a gas producer. The surplus heat is there. It is in the picture. Why not take a look at it, face the fact? That the surplus heat is necessitated by the hearth conditions admits of very little if any doubt. The furnace is really two separate pieces of apparatus: the hearth, doing certain work, and the superimposed shaft, doing certain other work. The combustion of fuel in the hearth, primarily serving the hearth pur-poses, generates much more heat and gas than the shaft requires, or can use, in smelting work. The result is surplus heat makes still more gas in the heat-absorbing reaction between CO, and coke; what is left of the surplus going to top heat. Here we have a simple yet adequate sketch of the furnace action. Having obtained a fairly comprehensive view of the existing heat relations of both hearth and shaft, the way to improvement seems quite plain. Either do more of the work above the hearth or put more heat into the hearth. Since there is already a great surplus of heat above the hearth, probably little needs to be done by way of increasing the proportion of the work done in the shaft. The shaft can, however, probably do much more work than it is now doing provided that a way, can be found to get more work, done in the hearth. The hearth heat is nicely balanced. Losses by radiation and water cooling and other losses are already taken care of. Any increase of hearth heat development could be utilized in taking care of increased smelting work. Aside from the possibility of increased blast heat, there is one sure way of putting more heat into the hearth, and this is to blow more air. And since the shaft already has more gas than it needs, the increased gas resulting from the greater blast volume can be withdrawn from the bosh after it has done its work in the hearth. Now an increase of burden will cool the shaft, cut down solution loss and thereby supply addi-tional coke in the hearth to burn with the additional air blown. The hearth heat per pound of coke charged will be proportionately increased. The balance between hearth and shaft will be better. Gruener&apos;s ideal will be more nearly attained, if not exceeded. Most inter-esting of all perhaps, the more the burden is increased, the more coke will reach the hearth and the more burden can be added. The effect is to be cumulative.

Jan 3, 1927

By Dr. O’Neil Thomas J., Donald W. Gentry

"There is nothing so disastrous as a rational investment policy in an irrational world. " John Maynard Keynes INTRODUCTION The principles of time value of money concepts were discussed and illustrated in Chapter 3. The importance of time value of money and the role it plays in the discipline of engineering economy have been well established. Chapter 9 further illustrates that project evaluation criteria that fail to consider this aspect are seriously deficient in providing appropriate capital budgeting decisions. Obviously future project receipts and expenditures must be discounted to permit valid comparisons with current cash flows. Although the concept of discounting is widely accepted, the selection of the appropriate discount rate has been the source of considerable debate and much disagreement. This chapter, then, provides an overview of the theoretical and practical considerations for determining or calculating the appropriate rate for discounting, the corporate cost of capital. Note specifically that the assumption of constant business risk among the various investment alternatives is maintained throughout this chapter. THE INTUITIVE APPROACH The fact that interest exists suggests that all money has a cost associated with its use. The cost of this money may be the result of either explicit or implicit charges or some combination of the two. Any time an individual or a firm consumes less than the total earnings generated in a given time period that individual or corporation has either consciously or unconsciously decided to "invest" these excess funds in some type of activity. For instance, the individual or corporation may decide to invest the excess funds in tangible property, place the funds in a money market account, or simply keep these funds in cash. In any case, these funds have been invested in some type of activity which promises the investor some return on the invested capital. Remembering that the firm&apos;s primary objective should be the maximization of shareholder wealth, then intuitively it seems reasonable to suggest that the firm should not invest in any project where the anticipated return does not exceed the cost of funds (capital) committed to the project. Indeed, if the firm always invests in projects having returns in excess of the cost of capital committed to them, then the wealth of the firm (as measured by the price of common stock) should be in-

Jan 1, 1984

By S. E. Buckley, M. C. Leverett

The production of oil is accomplished as a result of its displacement from the reservoir by either gas or water, and the amount of oil recovery is limited by the extent to which the displacing gas or water accumulates. This paper describes the mechanism by which the displacement is effected and the advantages of water over gas as a displacing agent. In the light of the results of experimental observations of the flow of mixtures of oil and/or gas and/or water through sands, certain conclusions are drawn relative to the changing character of the displacement as depletion proceeds, and on the effects of the properties of the fluids and of producing conditions on the ultimate oil recovery. Introduction Crude oil has no inherent ability to expel itself from the pores of the reservoir rocks in which it is found; rather, it must be forcibly ejected or displaced by the accumulation of other fluids. Accordingly, a knowledge of the mechanism by which one fluid is displaced by another is essential to an understanding of the fundamental process by which oil is recovered. The displacing fluids normally available are gas and water, either or both of which may exist originally associated with the oil in a potentially usable form or may be supplied to the reservoir from external sources. Gas is present in most oil reservoirs. If the quantity is relatively small, it may exist originally completely dissolved in the oil, but if the quantity exceeds that which may be held in solution by the oil at the prevailing pressure, the excess will be found in the free state. Most reservoir sands appear to contain some water,l and in the majority of fields the oil is trapped in the structure and held over water. In certain conditions the entrapping water may advance into the oil zone and displace the oil therefrom. This paper describes in a qualitative manner some of the characteristics of the displacement of oil by either gas or water, with an attempt to elucidate somewhat the mechanism by which such displacement is effected. Role of Dissolved Gas When the pressure on an oil containing dissolved gas is reduced to the saturation or bubble-point pressure, gas escapes from solution and expands, imparting to the oil-gas mixture a high degree of expansibility and causing it to flow in the direction of reduced pressure. This expansibility has frequently been attributed to the oil; closer analysis, however, reveals that as gas escapes from solution the oil phase actually. contracts. Hence, it is the accumulation and expansion of the liberated gas that supplies the propelling force for the oil. It has been shown experimentally2-5 that within certain limits a sand containing oil, gas and water will permit the simultaneous flow of all three; the readiness with which they may flow depends upon the characteristics of the sand, of each of the fluids, and upon the relative amounts of the fluids in the sand. Thus, in the initial stages of production by the mechanism of dissolved gas drive, the liberated gas is present in the sand in low concentration and is unable to

Jan 1, 1942

By G. D. Gemmell

Effects of solid-sdutidn alloying with titanium, molybdenum, and tungsten at concentrations up to 10 pct on the strength of pure columbium at elevated temperatures (mainly 2000°F) have been investigated by means of creep-rupture, hot-tensile, and hot-hardness testing and recrystallization studies. ThIS paper presents some results from the initial part of a study of the effects of alloying on the high-temperature strength properties of columbium. The elevated-temperature strength characteristics of pure columbium were first studied. Binary alloys of columbium with titanium, molybdenum, and tungsten containing 1, 5, and 10 pct of the solute elements were chosen on the basis of their high melting points, their known solid solubility in columbium, and results of separate oxidation-resistance studies of columbium alloys. EXPERIMENTAL PROCEDURE All the materials used were of highest purity. A typical analysis of the columbium is shown in Table I. Compositions of the alloys are also given in Table I. To evaluate the strength properties of pure co-lumbium and the alloys at elevated temperatures, creep-rupture, hot-tensile, and hot-hardness testing was done on recrystallized material. To prepare the pure metal and the alloys for these tests, 1- 1b buttons were made in a water-cooled copper crucible, inert atmosphere, tungsten arc furnace. Tapered ingots, 3 in. long, were made from these buttons by a skull-casting technique and these were machined to 1/2 in. diam, swaged to 50 pct reduction (pure columbium at room temperature, alloys at lowest temperature possible without cracking, generally in range 1100" to 1800°F) and heated in vacuum (lo-&apos; mm Hg) for 16 hr at 2550°F for re-crystallization and homogenization. The resulting grain diameter was 0.1 to 0.3 mm. The 10 pct Mo alloy casting was not successfully swaged at temperatures up to 1800°F without cracking and was not included in the test program. Creep-rupture testing was done at constant load in vacuum on cylindrical specimens of 0.16 in. diam, using Instron capsules. The pressure at test temperatures was less than 10-1 mm Hg. Hardneas readings taken on the test specimen before and after testing were used as a check on possible gaseous contamination. Hot tensile tests in vacuum on cylindrical specimens of 0.125 in. diam were also made with Instron equipment at a strain rate of 0.067 min.-l. Hot-hardness measurements were taken with a unit constructed at the Du Pont Experimental Station. The test piece, supported on an alumina anvil, and the molybdenum indenter shaft with sapphire tip is heated in a platinum-wound furnace. The load applied to the pyramid indenter is measured by strain gages attached to a proving ring located outside the hot zone. In the measurements reported here the load was I kg. The tester operates in a vacuum of about 10"5 mm Hg and is capable of temperatures up to 1400°C.

Jan 1, 1960

By Denton R. Wieland, Harvey T. Kennedy

The frequency of bubble formation is measured for oils from the East Texas field and the Slaughter field in cores from these fields. East Texas oil was also tested in a Slaughter core. The data checked previous results in that a definite supersatrtration may exist without the formation of bubb1es in periods of observation totaling 158 hours. Measured frequencies varied from zero at supersaturations below 14 psi to 3.1 bubbles/sec/ cu ft of rock at 40 psi supersaturation INTRODUCTION The formation of bubbles is essential in a solution gas-drive reservoir in the displacement of oil. In formations where the basic pore structure consists of fine pores or fractures connected to larger pores or crevices, bubble formation in the fine pores may be the most effective means by which the oil may be displaced, even if secondary methods are applied. The most efficient oil displacement would occur where at least one bubble would form in every pore. As the pressure decreased, the bubbles would expand, due to the gas diffusing into the bubbles, and displace the oil toward the lower pressure area around the wellbore. Kennedy and Olson&apos; have shown that the total number of bubbles formed in a reservoir is primarily dependent upon the rate of pressure drop, the rate of diffusion of gas through oil, and surface area of the rock. The surface tension of the liquid and other characteristics of the rock may also exert an important effect, which would give a different frequency to different rocks and oils and various combinations of different rocks and oils. The purpose of this paper is to present frequency measurements on two rocks and two oils. A hydrocarbon liquid is at equilibrium with gas when it contains the maximum volume of gas in solution that it can have at the prevailing temperature and pressure. If the temperature is maintained constant and the pressure is reduced below the equilibrium pressure, the liquid is supersaturated to the extent of the difference in the two pressures until the gas comes out of solution as a free phase. As shown by previous work, and confirmed in the present investigation, the number of bubbles formed per second increases with an increase in supersaturation. Kennedy and Olson showed that a tenfold increase in the rate of pressure decline resulted in a tenfold increase in the number of bubbles formed per cubic foot of reservoir rock, e.g., for decline rates of 0.1, 1, and 10 psi/day, the number of bubbles formed per cubic foot would be 40, 400, and 4,000, respectively. For another hypothetical system, Stewart, Hunt, and Berry&apos; estimated that for pressure declines of 0.1, 1, and 10 psi/day, the number of bubbles formed would be approximately 10, 100, and 1,000 cu ft of reservoir rock. Their value of bubbles formed is lower than given by Kennedy and Olson; however, the order of magnitude agreed satisfactorily. The rate of diffusion will also affect the number of bubbles formed. If the diffusion coefficient is high, the gas in solution in the oil surrounding a single bubble will tend to diffuse into the bubble rather than forming other bubbles. On the other hand, the formation of more bubbles will tend to occur if the diffusion coefficient is very low. In his thesis dated Aug., 1953, Wood" reports on the frequency of bubble formation in a system consisting of oil and rock from the Rangely field. In the present work, the apparatus of Wood was redesigned to allow greater precision, especially at low supersaturations, where bubble growth is slower, and detection is more difficult.

Jan 1, 1958

By J. S. Aronofsky, J. P. Heller

This paper presents a mathematical analysis of the fluid mixing which occurs during flow through porous media. The analysis is based on the well-known diffusion equation with mass transfer term. It is pointed out that the use of this equation is justified by a single general assumption which does not specify any particular mechanism for the mixing process. Formulas are given for tracer fluid concentrations for two different boundary and initial conditions. Calculated numerical values compare closely with some published results of experiments in which no viscosity or density difference existed between displaced and displacing fluids. INTRODUCTION A phenomenological theory of the mixing and diffusion process is presented for the flow in a porous medium of two miscible liquids of equal viscosity and density. This is based on the classical diffusion equation which has been used to explain such processes as Brownian motion and heat conduction. More recently, this same differential equation has been derived (in a manner analogous to the work in this paper) for the problem of heat transfer during fluid flow in porous media.&apos; The purpose of this paper is to show that the available experimental evidence justifies the use of this simple diffusion equation. The recent data of Koch and Slobod&apos; and of von Rosenberg&apos; are used to test this diffusion model by comparing concentration curve shapes and calculating "effective diffusion coefficients". The same differential equation for miscible displacement, as well as some experimental confirmation, has been published in Japan by K. Yuhara,4 and recently in this country by Day.5 Yuhara arrives at his con-clusions through analogy with turbulent diffusion, de- scribing the microscopic velocities in terms of "eddy motion" — although unlike the ordinary eddies of hydraulics. Day proceeds directly from Scheidegger&apos;s6 concept of the dispersion of velocities. After a very clear description of Scheidegger&apos;s theory, he applies it to calculate the motion first of a drop and then of a thin layer of brine into a bed of sand. For the latter case, Day derives and solves the same differential equation as Eq. 1. In our presentation no detailed assumptions are made as to the mechanism of displacement. In fact, several microscopic processes may well be involved—the only general assumption made is that their cumulative microscopic effect may be described by the addition of a second-order term to the differentirl equation of continuity. DERIVATION OF EQUATIONS Let us consider the displacement in one spatial dimension of Liquid A from a sample of porous media by Liquid B. A and B are assumed completely miscible and of the same viscosity and density. Both fluids are also assumed to be incompressible and it is specified that the total flow is the same for any cross section. It is evident that the rate at which Liquid B would be carried across such a section would be given by the product of the fluid velocity, u, and the concentration, C, of B in the fluid mixture assuming the constancy of u and C over the cross section. Since the volume of Fluid B is conserved, this leads to the one-way wave equation - u ?C/?x = ? ?C/?l where x increases in the direction of u—the "Darcy velocity", Q/A (or velocity flux per unit area), and 9 is the fractional porosity. Consider the case where fluid B is displacing Fluid A and the concentration distribution is represented by a step funcLon (with a vertical discontinuity). The solution of Eq. 1 predicts no mixing

Jan 1, 1958