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By R. J. Sandrea, S. M. Farouq Ali

The results of an experimental and theoretical study of the effects of rectilinear impermeable barriers and highly permeable channels on the sweep efficiency and conductivity of a five-spot network are presented. The study was carried out using Hele-Shaw, and conductive sheet analogs for both. normal and inverted flood patterns. A functional relationship was developed which provides quantitative prediction of the above parameters; i.e., sweep efficiency and conductivity, for the nonhomogeneous system. To evaluate this, it is necessary only to measure the interference modulus of the obstacle using a simple expression. The results indicate that impermeable barriers, with an interference modulus greater than zero, always cause a decrease in the conductivity and the sweep efficiency of the pattern. On the other hand, permeability channels do not have a significant effect on these parameters, unless they are located along the streamlines. The study also discusses a method of formulating the difference equations pertaining to the two-dimensional Laplacian in a system containing rectilinear barriers. In addition, a technique is described for tracing the progress of a free surface, using marked particles, in a stream containing discontinuities SuCh as those occurring at the extremities of barriers. The above computational scheme was used in a digital computer program to simulate results of the Hele-Shaw analog. The computed values of sweep efficiency mere in good agreement with the experimental values. INTRODUCTION At the present time waterflooding remains the most widely used technique for the secondary recovery of oil. Among the numerous factors that affect the performance of a waterflood, the areal sweep efficiency constitutes one of the most critical parameters. It is a function only of the geometric distribution of the wells for a homogeneous medium and unit mobility ratio. Geological considerations, however, such as the presence of sealing faults or solution channels in the rock matrix, would generally exert an adverse effect on the areal sweep efficiency of a waterflood network. Surprisingly little information is presently available on the quantitative implications of such permeability interferences. It was the object of this investigation to consider the presence of rectilinear, impermeable and highly permeable interferences in an otherwise homogeneous five-spot pattern, and to analyze their effects on the resulting sweep efficiency and pattern conductivity. The experimental phase of the study was conducted using two different models: the Hele-Shaw flow cell and the field plotter. For experimental convenience, the geometric boundaries of the five-spot were assumed to constitute streamlines even in the case where the interference was asymmetrically placed with respect to a repeated five-spot pattern. A second object of the study was to develop a computational scheme for numerically calculating the pressure distribution in a five-spot flow pattern containing rectilinear, impermeable barriers, and to use this distribution to compute the sweep efficiency. EXPERIMENTAL PROCEDURE HELE-SHAW ANALOG The Hele-Shaw analog was used both as a visual model and for determining the sweep efficiency for systems involving impermeable barriers. The physical limitations of this model as a means of experimentally studying the motion of a fluid in porous medium have been adequately discussed elsewhere.2,7 Likewise, it can be easily verified that the analogy remains valid when a portion of the region between the plates contains an obstacle of the same thickness as the distance between the plates. The model used in this study was composed of

By R. L. Slobod, B. H. Caudle

In the past, the main emphasis in attacking the problem of the recovery of oil has been on the determination of fluid flow characteristics and residual oil saturation in the part of the reservoir which is contacted by a displacing phase. It is recognized that the determination of the fraction of the reservoir contacted in secondary recovery operations is also of great importance in predicting the ultimate recovery to be expected. A method is described in which radiographic techniques are used to determine these areal sweepout factors for any type of well spacing. Data showing the relationship of viscosity ratio and relative permeability to areal sweepout efficiency are presented for the five spot and the direct line drive well spacing patterns. Further applications of the method to studies of the displacement efficiency in the area contacted are mentioned. INTRODUCTION The problem of increasing oil recovery from typical reservoirs is generally attacked by trying to devise some means for reducing the residual oil in that part of the reservoir which is contacted. The fraction of the reservoir contacted in the displacement process. however, is also of major importance but has been given less consideration in the past. one reason for this neglect is the meager information available on the portion of the reservoir contacted and the magnitude of changes in this factor as operating conditions are varied. Until recently, the only means available for determining this quantity has involved the use of the electrolytic or poten-tiometric modelsl in which one is limited to working with a mobility ratio of unity. while means for calculating the volume of a reservoir contacted do exist. the calculations are very tedious for systems with mobility ratio different from unity. Recently several such determinations have been reported,213 and an experimental method involving a step-wise use of potentiometric models has been described.' A new method which has been developed for studying the areal distribution of phases in a porous medium is discussed below. This tool can be' used to determine the effect of operating variables such as viscosity, relative permeability, presence of gas, mobility ratio, rate and stratification on the fraction of the reservoir contacted by the displacing medium. This development based on the use of X-rays gives a pictorial representation of the location of the phases. Further, the amount of each phase at each point of interest can be determined quantitatively. X-RAY METHOD FOR AREAL STUDIES In this method the location of various phases distributed Over a test area are observed simultaneously by obtaining a photographic record of the transmission of X-rays through the test plate. This procedure is an adaptation of the radiographic techniques which have 10% been used in the medical and industrial fields. Just as the medical profession uses a chest plate to show spatial interrelationships, the location of an X-ray absorbing fluid and a nonabsorbing fluid in a porous medium can be determined by obtaining a photographic record of transmitted X-rays. Moreover, as changes occur in such a System, these changes can be observed by taking successive photographs. Previously, X-ray methods have been used in fluid flow studies to determine quantitatively the saturation at a given point in the system. The intensity of the transmitted X-rays was measured by an ionization chamber, so that a point by point survey of saturation could be made. In comparison with this earlier quantitative procedure, the photographic procedure which simultaneously observes the saturation at many

Jan 1, 1952

By A. L. Jr. Pozzi, A. O. Garder, D. W. Peaceman

A new numerical method is proposed for the solution of multidimensional miscible displacement problems. Besides the usual stationary grid associated with numerical procedures, the method uses moving points for which positions and concentrations are computed each time step. Based on the method of characteristics for treating combined transport and dispersion, the method applies equally well to any number of dimensions and, in contrast with other numerical procedures, properly takes into account the physical dispersion, no matter how small. Extensive tests show that accurate solutions of one-dimensional problems can be obtained over a wide range of values of the dispersion coefficient, including zero. They also show that the moving points do not need to be uniformly spaced, and that increasing the number of moving points beyond 2/grid interval does not significantly improve the accuracy of the solution. Two-dimmsional calculations using the new method were carried out to simulate laboratory model displacements in which gravity caused overriding of solvent. Good agreement between observed and computed profiles and measured and computed concentrations of produced fluid was obtained for a viscosity ratio of 1.89; at higher viscosity ratios, fair agreement was obtained Computed results were found to be significantly affected by vertical dispersion, and negligibly affected by dispersion in the direction of flow, in agreement with conclusions based on experimental observations. INTRODUCTION Although miscible displacement processes are potentially of great economic significance, limited progress has been made in developing mathematical methods for predicting the outcome of a solvent flood. Because of the complexity of the partial differential equations which describe multidimensional miscible displacement, numerical methods are a natural approach to their solution. The task of finding suitable numerical approximations to one of the partial differential equations, which involves both dispersion and convective transport, has been particularly difficult. Dispersion in the absence of transport is described by the heat flow equation, a second-order equation of parabolic type. This equation has been very successfully treated by numerical methods. Transport in the absence of dispersion is described by a first-order equation of hyperbolic type. This equation has been treated numerically with some success in one dimension, both by Lagrangian and Eulerian techniques, 1.2 but extension to two or more dimensions has not been satisfactorily accomplished. Usually one of two things happens: either the numerical solution develops oscillations or it becomes smeared by artificial dispersion resulting from the numerical process. When transport and dispersion are considered simultaneously, this numerical dispersion may dominate the low physical dispersivicy that generally characterizes miscible displacement. Stone and Brian2 have developed a new difference equation for treating combined transport and dispersion in one dimension. By suitable choice of certain parameters, they succeeded in markedly reducing both oscillation and numerical dispersion. However, a satisfactory extension to higher dimensions has not yet been found. Peaceman and Rachford3 presented a centered-in-time and centered-in-distance difference equation combined with a 1,transfer of overshoot" procedure which was demonstrated to work well in one dimension. The technique was applied to the two-dimensional miscible displacement problem, but subsequent calculations with zero dispersivity showed little change in results compared with the calculations they presented. This indicates that their method involved a numerical dispersion of the same order as the physical dispersion. The present paper describes an alternate method for solving the combined transport-dispersion

Jan 1, 1965

By L. Guttman, C. S. Barrett, J. S. Bowles

THE transformation from the face-centered cubic (Al) to the face-centered tetragonal (A6) structure in certain alloys of the indium-thallium system reported in the preceding paper1 exhibits many interesting crystallographic and metallo-graphic features, most of which presumably are similar to those of other cubic to tetragonal transformation such as, for example, the one which occurs in the chromium-manganese system,2 the one which, judging by microstructures and X-ray pat-terns,3 probably occurs in the copper-manganese system, the ones accompanying ordering in FePt,4 Copt,&apos; and AuCu,5 and the transformation near 115°C in barium titanate.6,7,8 If, as mentioned in ref. 1, the transformation is second order, the phases in equilibrium do not differ in composition, and it is both possible and likely that the mechanism is a diffusionless one. The present paper reports the results of a detailed study of the indium-thallium transformation and presents a theory for the atomic movements of the transformation which accounts for the observations. Experimental Crystallography of Transformation Markings The transformation from cubic to tetragonal in the indium-thallium alloys produces microstruc-tures of the type illustrated in fig. 1. These structures are observable either as relief effects produced on a smooth surface by the transformation, or after polishing and etching. These microstruc-tures are markedly similar to those found in copper-manganese alloys: iron-platinum alloys," and barium titanate,7,8 where it has been demonstrated that the lamellae are parallel to the (101) planes of the original cubic crystal. That the observed lamellae in indium-thallium alloys are also parallel to the (101) planes was proved by the following analyses performed on single grains in polycrystal-line samples containing 20.75 atomic pct T1. A study of the microstructures of a number of specimens revealed the fact that in many specimens the transformation markings continued unchanged in direction across the (111) interfaces of annealing twins. This at once demonstrates that the lamellae are parallel to a plane that is common to both parent and twin orientations, i.e., a plane in the zone <112>. This conclusion is compatible with the lamellae being parallel to (101) planes. To make a more complete determination of the habit plane of the lamellae, two grains were selected in which annealing twins ({Ill) twins) had been present in the high-temperature modification (cubic). The traces of the annealing twins, and of the transformation markings in both parent and twin crystals, were plotted together in stereographic projection, and a parent-twin pair of orientations was found that accounted for all lamellae as (101) planes. The projection of one of these twinned grains is reproduced in fig. 2. In view of these results the probability of there being another solution for the habit plane remote from (101) is very small, but to explore the possibility that the habit plane deviates slightly from (101), a more precise analysis was undertaken. In this, the orientation of a grain in the cubic modification was determined from a Laue back-reflection photograph taken with the sample at 90°C. The traces of the lamellae that formed in this grain on cooling were plotted in the stereographic projection

Jan 1, 1951

By Joseph L. Gillson

BRAZIL has had an industry based on ocean beach deposits of heavy minerals containing monazite, zircon, rutile, and ilmenite for well over 40 years, but except at the very earliest period, prior to 1906, and again during World War 11, has this industry been at all comparable in size with similar operations in other countries. Limiting factors have been neither a lack of reserves nor lack of markets (except for zircon). Although poor local transportation and shipping facilities have handicapped the development in Brazil, these conditions were perhaps equally bad (when mining operations started) in the Indian State of Travancore, and in New South Wales and Queensland, Australia, in which states the principal production of beach deposits of heavy minerals has occurred. The outstanding restraints on the Brazilian industry have been the lower quality of the Brazilian minerals, the relative small size of individual deposits that were known until recently, the lack of interest of Brazilian investors in mining ventures, and the unfavorable climate for investment of foreign capital under Brazilian mining law. The lower quality of the minerals is indicated by the fact that the Brazilian monazite carries only 5 to 6 pct Tho2*, and the ilmenite 561/2 pct TiO2, whereas the corresponding Travancore minerals carry 9 pct Tho, and 60 pct TiO, respectively. The Brazilian zircon is a dirty brown color (which can be removed readily) as compared with the gleaming whiteness of the Australian zircon. An inadequate market for the abundant zircon that will be available in any operation in Brazil is certainly a heavy restraint, since otherwise it could be an important byproduct. The oxide of zirconia, baddeleyite, that has come from Brazil is from other sources.&apos; Since the Brazilian beach sands carry only a little rutile, that mineral has not been available as a valuable byproduct to help finance the operations. Substantial tonnages of rutile have been exported from Brazil. The sources of this mineral are stream placers and residual deposits over gneisses in the States of Goyaz, Minas Geraes, Ceara, etc. (Leonardos,&apos; and Chambers". With these handicaps, Brazilian production from beach sands has been marginal, and there has been only one consistent producer. This is a company, originally French, which was called Companie de Franco-BrazilliCne, but, with the downfall of France in 1940, became Brazilian under the name Monazita e Ilmenita do Brasil ("Mibra"). This company has operated several deposits in the vicinity of Guara-pary, in the State of Espirito Santo. Its office and plant has been at Guarapary. Recently a company called, "Fomil" (Fomento Monazita-Ilmenita) started operations. This company was organized by the Foote Mineral Co. of Philadelphia, and such share as a foreign company can hold under Brazilian law has now been acquired by the Lindsay Light and Chemical Co. of Chicago. When the writer went to Brazil in the fall of 1940, he found that no general survey of the beaches had been made except for a thorough study of part of the area by Abreu.4 The paper by Miranda6 had not then been written. The federal Geological Survey and Department of Mines knew only of the existence of the deposits in Espirito Santo south of Vitoria, and had a hazy idea that others might be found north of Vitoria. One deposit in Bala was known in Rio, but the larger one at Guaratiba was not known, although a Frenchman had prospected

Jan 1, 1951

By Joseph L. Gillson

BRAZIL has had an industry based on ocean beach deposits of heavy minerals containing monazite, zircon, rutile, and ilmenite for well over 40 years, but except at the very earliest period, prior to 1906, and again during World War 11, has this industry been at all comparable in size with similar operations in other countries. Limiting factors have been neither a lack of reserves nor lack of markets (except for zircon). Although poor local transportation and shipping facilities have handicapped the development in Brazil, these conditions were perhaps equally bad (when mining operations started) in the Indian State of Travancore, and in New South Wales and Queensland, Australia, in which states the principal production of beach deposits of heavy minerals has occurred. The outstanding restraints on the Brazilian industry have been the lower quality of the Brazilian minerals, the relative small size of individual deposits that were known until recently, the lack of interest of Brazilian investors in mining ventures, and the unfavorable climate for investment of foreign capital under Brazilian mining law. The lower quality of the minerals is indicated by the fact that the Brazilian monazite carries only 5 to 6 pct Tho2*, and the ilmenite 561/2 pct TiO2, whereas the corresponding Travancore minerals carry 9 pct Tho, and 60 pct TiO, respectively. The Brazilian zircon is a dirty brown color (which can be removed readily) as compared with the gleaming whiteness of the Australian zircon. An inadequate market for the abundant zircon that will be available in any operation in Brazil is certainly a heavy restraint, since otherwise it could be an important byproduct. The oxide of zirconia, baddeleyite, that has come from Brazil is from other sources.&apos; Since the Brazilian beach sands carry only a little rutile, that mineral has not been available as a valuable byproduct to help finance the operations. Substantial tonnages of rutile have been exported from Brazil. The sources of this mineral are stream placers and residual deposits over gneisses in the States of Goyaz, Minas Geraes, Ceara, etc. (Leonardos,&apos; and Chambers". With these handicaps, Brazilian production from beach sands has been marginal, and there has been only one consistent producer. This is a company, originally French, which was called Companie de Franco-BrazilliCne, but, with the downfall of France in 1940, became Brazilian under the name Monazita e Ilmenita do Brasil ("Mibra"). This company has operated several deposits in the vicinity of Guara-pary, in the State of Espirito Santo. Its office and plant has been at Guarapary. Recently a company called, "Fomil" (Fomento Monazita-Ilmenita) started operations. This company was organized by the Foote Mineral Co. of Philadelphia, and such share as a foreign company can hold under Brazilian law has now been acquired by the Lindsay Light and Chemical Co. of Chicago. When the writer went to Brazil in the fall of 1940, he found that no general survey of the beaches had been made except for a thorough study of part of the area by Abreu.4 The paper by Miranda6 had not then been written. The federal Geological Survey and Department of Mines knew only of the existence of the deposits in Espirito Santo south of Vitoria, and had a hazy idea that others might be found north of Vitoria. One deposit in Bala was known in Rio, but the larger one at Guaratiba was not known, although a Frenchman had prospected

Jan 1, 1951

By Leonard R. Weisberg

HE procedure of directional freezing by the Bridgman technique1 is frequently used in crystal preparation. On those occasions where the crystal is regrown, it can be advantageous to Crop part of the ingot before regrowth to achieve purification. This note describes some simple, yet interesting, observations concerning the effects of cropping an ingot between successive freezes. First, it is not effective to Crop the front end of an ingot, whether or not the distribution coefficient, k , of the impurities is greater or less than unity. This can be shown as follows. The concentration C, of an impurity in a directionally frozen ingot of length d with a uniform cross section is given by where x is the distance along the ingot, and C0 is the initial concentration per unit length. It is assumed in deriving this equation that diffusion is negligible in the solid, is complete in the melt, and that k is constant. Consider that a section of the ingot Ad is removed from the front end. Then the average impurity concentration in the remaining (d- Ad) section of the ingot can be shown by integration to be given by When the ingot is remelted and refrozen, the new impurity distribution is given by where xl is the distance along the refrozen ingot. Substituting the value of c into Eq. [3], one finds This is identical to Eq. [11 when x = (x, Ad), which represents the original impurity distribution in the (d - Ad) section of the ingot after the first directional freeze. Therefore, the cropping and re-freezing accomplished nothing. This can also be seen physically by realizing that the average impurity content in the melt after the first Ad section is frozen, is identical to that after the resulting ingot has been cropped and remelted. Hence, it is assumed in the following sections that only the tail end of the ingot is cropped. However, it must be realized that for impurities with k greater than one, the pieces cropped from the tail end are purer, and are therefore saved. After cropping an amount Ad from the tail end of an ingot, the average impurity concentration of impurities in the remaining portion of the ingot is given by wheref= d/d. For the case of k < 1, the purification is defined as c,/c. It is obvious from Eq. [5] that the purification increases asf increases, but the increase in purification is not large. For example, iff is doubled from 0.1 to 0.2, the purification increases by only 10 pct for k = 0.5, by only 24 pct for k = 0.1, and by 32 pct for k =0.01. Much smaller values off than 0.1 are not desirable since insufficient purification is achieved. These figures indicate that much larger values off do not lead to substantial improvement, especially considering that much of the original ingot is lost. Therefore, it is generally useful to Crop about 1/10 of the ingot to achieve purification. As compared to doubling the value off, it is much more helpful to remelt, refreeze, and crop the ingot twice. Using Eq. [51 reiteratively, and assuming f= 0.1, it is found that the purification after this process is found to be increased over the purification after a single freeze and cropping by 33 pct for k= 0.5, 450 pct for k = 0.1, and 4100 pct for k = 0.01. This indicates that considerable purification can be obtained by this procedure. However, if purification is the main goal, one should resort to zone refining,3

Jan 1, 1962

By J. W. Marx, R. H. Langenheim

The injection of heat-bearing fluids may offer a wider application to secondary and tertiary recovery from conventional oil reservoirs than inderground combustion since the process is more easily controlled and the reservoir requirements, in general, are less critical. In all cares, conduction heat losses to overburden and un-derburden will impose an economic limit upon the size of the area which can be swept out from any one injection point for any given set of conditions. The present report describes a method for c.rtin~ating thermal invasion rates, cumulative heated area, and theoretical economic limits for sustained hot-fluid injection at a constant rate into an idealized reservoir. Full allow-ance is made for non-productive reservoir heat losses. Required solutions can be obtained directly from the numerical tables given here, as soon as all operating conditions are specified, withou extensive mathematica1 ~nanipulation on the part of the user. The injection of a heat-bearing fluid into a reservoir is frequently proposed as a means of secondary or tertiary oil production. This form of thermal recovery may offer a wider application to conventional oil reservoirs than underground combustion since the process is more easily controlled and the reservoir requirements, in general, are less critical. Because of its large gross heat capacity, steam appears to be the most efficient heat injection medium, although mixtures of steam and other gases, hot water, hot oil and hot non-condensable gases also have been employed. Whatever the nature of the injection medium, economic evaluation of such thermal recovery processes will depend upon thermal invasion rates, or rates of productive reservoir heating, at any given time after the start of heat injection. In all cases, conduction heat losses to overburden and underburden will impose an economic limit upon the size of the area which can be swept out from any one injection point, for a given set of reservoir conditions, at any given heat injection rate. The present report describes a method for estimating thermal invasion rates, cumulative heated area and theoretical economic limits for sustained heat injection at a constant rate into an idealized reservoir, making full allowance for non-productive reservoir heat losses. Required solutions can be obtained directly from the nu- merical tables given here, as soon as all operating conditions are specified, without extensive mathematical manipulation. It should be noted that ordinary water flooding may be used to move an annular-heated zone outward, even though the economic areal limit for straight heat injection has been reached. Thus well spacing need not be confined to the areal limits as determined here. However, the present calculations do not cover such supplementary heat scavenging programs. The operating element consists of a radial flow system, concentric about the point of injection, with the idealized step function temperature profile illustrated by the dashed line in Fig. 1. The solid curve shown in Fig. 1 gives a qualitative picture of the true radial temperature distribution. It will be shown later that the expressions given below, which are derived from the simplifying step idealization (dashed line in Fig. I), also can be interpreted in terms of the more realistic temperature distribution shown in Fig. 2. For this case, the thermal invasion radius is simply redefined as the distance from injection well to the mid-point of the temperature distribution.

By M. A. Cook

Sutherland used an ion-pair adsorption model to derive the author&apos;s hy-drolytic pee-acid) adsorption equation for the contact bubble curves of Wark and Cox. To do so it was necessary to postulate that the sum of coverages of surface anion sites by the collector and depressant anions is practically unity. It is here shown that the free energy of Sutherland&apos;s postulated collector ion (x-)-depressant ion (OH-) exchange reaction (for no other than the OH- to compete with X-), namely EX + (OH-) Z SOH- + X-, is in some cases positive and X- should not even adsorb. Also, in many cases the (OH-) adsorption must be quite negligible even though the collector covers only about a sixth of the mineral surface at the threshold of air bubble contact. This, however, contradicts the postulate that permits one to derive the correct bubble contact eqwztion for the ion-pair model. Furthermore, the requirement for charge neutrality leads in the ion-pair model to the (impossible) constancy of the ratio @+)/(OH-) along a contact bubble curve in which pH may change by more than seven units. HYDROLYTIC ADSORPTION EQUATION Hydrolytic (free-acid or free-base) and ion-pair (H&apos;, X-, or R&apos;, OH-) adsorption models of the collector in metal sulfide and similar flotation systems are superficially alike and somewhat difficult to differentiate. This is because they yield, under seemingly valid assumptions, equivalent forms of the equations describing the distribution of the collector between the aqueous solution and the mineral surface. This has been discussed by the author and associates1-4 and more recently by Sutherland,&apos; both for simple collector-mineral and for more complicated collector-depressant-mineral systems. In the hydrolytic adsorption model the equations describing the contact bubble curves of Wark and coxeJ7 are as follows :2 Here X is the collector anion, D- the depressant anion, 9, and On are the fractions of the free-acid sites S on the surface of the mineral covered by collector and depressant free acids, respectively, and @ is the fraction of uncovered sites. Kx is the free-acid collector dissociation equilibrium con- stant, K1 that for the free-acid collector adsorption, KD that for the free-acid depressant dissociation, and K2 the equilibrium constant for the free-acid depressant adsorption. Eqs. 1 to 5 yield upon simultaneous solution the result is the critical free-acid concentration for bubble contact in the absence of depressant-free acid. Eq. 8 expresses the conditions along a simple collector-mineral contact bubble curve for the case HD = 0, or V = 0. Last and Cook showed that Eq. 6 applies generally and with good accuracy in the collector-depressant-metal sulfide mineral systems. Inci-dentally, it has been shown by infrared absorption studies that in some cases the free-acid adsorption process splits out water.&apos; In this case, Eq. 2 might be written SHOH + HX Z SHX + H20, with a similar equation replacing Eq. 4 for depressant-free acid-water exchange. This possibility does not change, however, the basic equations or considerations of the hydrolytic adsorption model. Indeed, this type of free acid-water exchange reaction was predicted for fluorite and iron ore flotation by oleic acid in the dry grinding and boiling processes.410

Jan 1, 1961

By L. G. Jones

An approximate method of calculation is developed in this paper which allows duplication of radial unsteady-state gas flow computer results where Darcy&apos;s law applies, such as those reported by Aronofsky and Jenkins and Bruce, Peaceman, Rachford, and Rice. Moreover, the new calculation method can be used to obtain results for radial unsteady-state gas flow obeying the quadratic flow law proposed by Duwez and Green. Means are discussed for predicting well behavior at single or superimposed flow rates in finite or infinite reservoirs, determining reservoir rock properties from well-test data, reproducing and interpreting back-pressure test data, and determining the radial extent and reserves of gas reservoirs from well-test data. Example calculations are presented for gas flow following both Darcy&apos;s law and the quadratic flow law. INTRODUCTION Since the publishing of U. S. Bureau of Mines Monograph 7,1 most gas-well testing methods have been based on the equation q = y (pl2 - pw2)a, where q = production rate, pl and pw, are formation pressure and sandface pressure, respectively, and y and a are constants to be obtained from test data. These methods, used for predicting both deliverability and "open-flow" capacity of gas wells, have been useful and accurate in many cases but unsatisfactory in others. Even at best, however, they do not supply information about the formation or lead to an understanding of nonsteady-state gas flow in porous media. Many theoretically based studies2-4 of gas flow obeying Darcy&apos;s law have been made.&apos; Since the partial differential equations which result from combining Darcy&apos;s flow law with the continuity equation are nonlinear, most of the published research consists of either numerical solutions or analytical solutions for linear approximating equations. Such solutions have been of limited value in field work due to their unhandy form and their failure to correlate most field data. There is evidence5,6 which indicates that Darcy&apos;s law is inadequate to describe gas flow at some flow rates of practical interest. A quadratic flow law: which reduces to Darcy&apos;s law at low rates, is more successful in accounting for experimentally observed behavior. This flow law has been applied successfully to a few hypothetical reservoir cases in work which has recently been published.10 However, these numerical solutions of the equations involved have been successful only on a special analog computer. Routine applications to field cases would be awkward and have not been attempted. The present paper describes an approximate method for computing nonsteady-state gas flow solutions which has been completely successful in predicting the results for both Darcy flow and quadratic flow obtained by elaborate numerical methods. The new calculation method allows determination of the observable variables in gas-well testing at constant rates. It is similar to the scheme of using a succession of steady states suggested by Muskat7 in that it makes use of steady-state and material-balance equations. It also is similar to the work of Aronofsky and Jenkins2 in that the new method includes Aronofsky and Jenkins&apos; stabilized flow equation as a special case. It improves upon both of these calculation schemes in that it accurately describes all portions of reservoir history and suggests means of determining reservoir rock properties from well-test data. This paper deals only with production from a reservoir, in which case the rate is defined as being negative. The reservoir studied here is a homogeneous, disk-shaped porous body of uniform thickness, with all boundaries sealed except the inner radial boundary of the well.

By Ralph H. Sweetser

C. H. Herty, Jr.,* Pittsburgh, Pa.—The procedure of the Open-hearth Committee in studying pig iron and its effects in the open hearth has been influenced by the discussions at a number of meetings of the Committee. Those discussions have brought out very clearly that, as far as the pig iron affects the quality, interference with quality should be studied for three definite types of pig iron: (1) low-silicon pig iron; (2) very high-silicon pig iron, and (3) pig iron in which the silicon varies over a wide range in short periods, such as 1 to 3 casts. If a blast furnace is running steadily at 1 per cent. silicon iron, practically no trouble, apparently, can be traced to the pig iron, but if the furnace is running from 0.60 up to 1.50 and then down to 0.60 per cent., trouble in the finished steel is liable to occur, which apparently cannot be corrected by open-hearth practice. The open-hearth discussions have shown fairly definitely that with low-silicon iron, one type of trouble arises on the finished product and with the high-silicon iron, another. Our experience with the high-silicon is that any non-metallic matter that may form, does so when the pig iron is put into the furnace; that is, silicates immediately form which are more liable to be high in silica than would happen with low-silicon iron. I think Mr. Sweetser will agree with mc that if a blast furnace is very irregular, different points in the furnace may be working irregularly, in other words, the temperature gradient all over the furnace is not uniform. We have made many determinations on that point which indicate that a varying blast furnace causes hot spots and cold spots around the periphery. If some of those spots are hot and others are cold, although the analysis of the iron may be constant, then unreduced oxide, going through on one side, and good iron, going through on the other, may mix and the average temperature of the iron is not high enough to get completely rid of the oxides. So our method of attack is directly on that point; it is our first objective. Our procedure is to make a number of consecutive tests at a blast furnace, making a detailed study of the blast-furnace operation. We follow the iron through the open-hearth heat, being sure that we know just how much of each kind of iron gets into the open hearth, and take out specified samples at certain times during the process, working on steel where no finishing additions are made in the furnace. We are following one type of iron through the billet mills and into the finished product and are obtaining complete data on open-hearth operation. Besides the ordinary chemical analyses on the samples of slag and metal, we are running samples of the plg iron, the steel in the open hearth and the finished steel, for oxygen and silicates. We are going to find out whether or not oxygen, as silicates or oxides, has some connection with this trouble. If not, we may be able to locate some other factor. I am inclined to think that the silicates are responsible, judging from the indications in some other work that the Bureau of Mines is doing in Pittsburgh. We have laid out a very definite method of testing to which we will adhere at the different plants where we are now working and at others in the future. Our aim is to

Jan 1, 1927

By Roshan B. Bhappu, Raul A. Deju

A method is proposed by which the chemical reactions between water and a silicate bed can be related to equations of flow. First the reaction mechanism is studied both theoretically and experimentally. Once the reaction mechanism is established, the equations of flow are solved, and a solution (under physically reasonable side conditions) is obtained. Experiments to substantiate the solution are also presented. The knowledge of surface reactions at the water-solid interface is of critical importance in extractive metallurgy. This is especially true in regard to the froth-flotation process of mineral concentration and all phases of hydro metallurgical recovery of metals. Similarly, the problem of ground-water contamination is one of growing concern, not only because of the increasing demands on our sources of supply but also because of the increasing necessity to find vehicles of waste disposal. The hydrochemistry of waste waters must be understood to exploit and develop such resources properly. To understand these problems, it is necessary to obtain a better insight into the reactions that take place between water and a solid packed bed. To do this, we will develop here in a theoretical model that predicts the fate of a chemical species in solution as it reacts with a silicate bed through which the liquid percolates. The treatment will be one-dimensional in nature and will not include dispersion. Experiments will also be presented to substantiate the theoretical model, using the silicate minerals listed in Table 1. This paper is preliminary in nature and is part of a larger study of the surface chemistry of the silicate-water interface. Mechanism of Surface Reactions In a previous paper,&apos; the authors discussed the properties of various silicate minerals from a mineralogical standpoint. Now efforts will be made to study the reactions that occur when these minerals are put in contact with water. The reactions occur almost entirely in the surface layer of the mineralsa and are termed sorp-tion reactions. In this study sorption denotes the composite of all physical adsorption and chemical adsorption occurring between the silicate surface and the water in contact with it. A model of sorption as seen from a chemical-bond standpoint is presented below. On the basis of this model, the surface reactions between deionized water and a silicate mineral can be studied. Sorption Model: It is known that as the oxygen-silicon ratio increases from quartz to the olivines, a greater percentage of the oxygen bonding power is available for bonding to cations other than silicon. Hence, with an increasing oxygen-silicon ratio, the oxygen-to-metal bonding increases. Upon the fracturing of a silicate-mineral crystal, the oxygen-metal bond breaks more easily than the stronger oxygen-silicon bond, resulting in a negatively charged surface. Then, if the mineral is immersed in a liquid containing hydrogen ions, the negatively charged surface should tend to be neutralized by hydrogen ions from solution, resulting in a change of the solution pH. Thus, an increase in the degree of sorption of hydrogen ions must occur as the oxygen-silicon ratio in the crystal structure increases. Also in a previous paper,&apos; the authors found that for a given silicate mineral the change of pH with time is given by

Jan 1, 1971

By H. S. Price, D. A. T. Donohue

The system of equations describing displacement of a hydrocarbon liquid by a hydrocarbon vapor in a porous medium where mass transfer takes place between the phases is solved numerically for a variety of gas injection processes. Even though the method of solution is quite general, only systems with three hydrocarbon components are considered. Computer simulations of displacement processes wherein mass transfer between phases is both considered and neglected are compared, and it is shown that neglecting mass transfer can give pessimistic displacement efficiencies. INTRODUCTION The role of the gas displacement process in the recovery of petroleum has been subjected to a series of detailed analyses; as a result, a number of predictive models have been published in the literature. However, because of major simplifying assumptions, most of these models do not completely represent the physical system. As a result, the effect of making the simplifying assumptions is unknown. Therefore, a complete representation of this process — one without major simplifying assumptions —should lead to a fuller understanding of the process, and perhaps to methods of improving it. The general method of developing a model for two-phase fluid flow in a porous medium is to solve simultaneously the continuity equation, the energy equations and the equation-of-state for each phase under the prescribed initial and boundary conditions. For an isothermal system, the energy equations reduce to the momentum equation. Darcy&apos;s law. However, since natural gas is the vapor state of the reservoir liquid, interphase mass transfer may take place with concomitant changes in both the intensive and extensive thermodynamic properties of each phase. It is this phenomenon that has often been omitted in previous mathematical models. An additional relation, then, which accounts for mass transfer between the phases, must be included with the other equations to specify a complete model. Completely formulating the equations to be solved is not a difficult task1 but obtaining their solution has been intractable up to now. Availability of large-memory, high-speed digital computers now makes an attack on this formidable problem possible. This paper presents a preliminary study of the problem. Since this investigation is intended to be exploratory, it is restricted to the linear, horizontal, isothermal, two-phase viscous flow of oil and gas in an oil reservoir. In the early development of predictive models of this process, the reservoir system was considered as a unit and various forms of the material balance equation were proposed.2-4 Pressure and saturation gradients were then added in the Buckley-Leverett model.5 The Buckley-Leverett formulation considered the fluids to be incompressible; thus, the mathematical model reduces to a steady-state system. In the 1950&apos;s, studies incorporating numerical techniques were being published.6-9 These mathematical models differed in the efficiency of finite difference techniques, the inclusion or exclusion of capillarity or the number of space dimensions considered. To solve these nonlinear, partial differential equations, each phase was considered to be homogeneous with time; therefore, mass transfer between phases was neglected. The effect of mass transfer on the gas displacement process was first reported by Attra. 10 He simulated the one-dimension flow system by a series of cells in each of which the fluids were equilibrated during a time step. In addition, the pressure throughout the system during each time step was predetermined and constant-phase velocities were calculated according to the Buckley-Leverett incompressible fluid flow model. Welge et al.11 developed a model for the displacement of oil by an enriched gas where composition is considered to be a dependent variable. Fluid

By P. S. Rudman

Crystal-structure occurrences in binary phase diagrams of fifth- and sixth-period elements, zirconium through palladium and hafniun through platinum, are surveyed with the conclusion that average .group number GN is the dominant correlative. This is interpreted in terms of curves of enthalpy differences between competitiue structures that are continuous func -tions of GN. It is further reasoned that the absolute formation energies may also be continuous functions of GN and if so can be simply obtained from the regatives of the heats of sublimation of the elements. It is shown that from such a heat of formation-GN plot the heats of mixing of alloy systems can be quantitatively predicted. For example, for sixth-period alloys very large exothermic heats of mixing are predicted for systems that span GN = 6 (i.e., Hf-Pt, Ta-Os, ...). Thermodynamic data with which to test these predictions do not appear to be available. If in the latter transition (A subgroup) elements we eliminate from consideration the fourth period where magnetic terms contribute to crystal-structure stability, as shown in Table I, there is a very regular sequence of structures. Using the group number (GN) assignments of Table I, which differ from the usual assignments of 8 to the elements listed as 8 through 10, it is perhaps trivial to say that the phase occurrence is a unique function of GN. However, the interesting aspect of alloys from among these elements is that the alloy structures also appear to correlate with GN. The GN for an alloy is given as GN = SixiGNi, where xi and GNi = atomic fraction and GN of component i, respectively. It is difficult to credit anyone in particular with having discovered this correlation, so many people have implicitly or explicitly employed it in the last 10 years. However, Bloom and Grant&apos; in their correlation of s-phase occurrence with total number of d + s electrons were certainly one of the first to appreciate the correlation and Hume -Rothery2 appears to be responsible for the use of average GN as defined above. Only in the last few years have the phase diagrams of enough alloy systems been determined for this correlation to be surveyed with any degree of completeness. We shall here, in an abbreviated format, summarize the available phase-diagram literature and discuss some consequences of the GN correlation. BINARY DIAGRAM SURVEY In Figs. 1 to 3 the phase occurrences are summarized. In Fig. 4 the occurrence limits for the more common structures are summarized. That the GN of an alloy falls within the occurrence limits of a particular structure is a necessary but not a sufficient condition for the appearance of that structure. The preceding figures certainly illustrate that GN is a dominant correlative for structures of these elements. Many of the diagrams have complications, with ordering and other phase changes occurring in intermediate phases, that cannot be clearly accommodated in such a schematic survey and have hence been left out. Such omissions cannot, however, negate the GN correlations discussed here. Relative atomic size is also an important crystal-structure determinant and since atomic size varies quite regularly with GN for these elements it is possible that the apparent GN correlation is basically a size correlation. In Fig. 5 relative size occurrence limits are given for intermediate phases and extended terminal solutions. The relative volume, ?v/(v) - 2(vA - vB)/(vA + uB), has been taken as the measure

Jan 1, 1968

J. M. Wadsworth,&apos; Okmulgee, Okla. (written discussion).—It seems to me that thc young man must first be absolutely sure that the work he has taken up is to his liking and then I would advise that he get into the practical end of it and learn as much about it as he can as fast as he can. For instance, in the refinery business the college man, graduated in chemical engineering, should ground himself by going through the various departments in a refinery and working in them until he is thoroughly familiar with the operating problems. In a short time he would be able to apply many things which appear in the abstract form in his college work and this would opcn up an opportunity for him to capitalize on them, promoting his own interest. I do not criticize the present-day college men as a class, but I have found that many of them come into your plant with the idea that mere possession of a diploma should insure rapid promotion; in a short time they are impatient and usually leave. We have found in employing men that the single attribute most difficult to find in them is a lot of energy and whole-hearted interest in the work at hand, and it seems to me that if a man combines these with a careful technical training he should be fairly well equipped to gamble on his future. They must learn that the biggest satisfaction to be gained in life is in their work. Procedure for Attaining Objective of Young Petroleum Engineer H. B. Fell,+ Ardmore, Okla. (written discussion).—Of course, the first thing to be ronsidered in any proposition is the ultimate objective that is desired to be attained. 1 presume that the desired objective of most of the young men graduating in petroleum engineering will be that of chief executive of one of the companies in the petroleum industry, or one of the allied industries, or to become an independent oil producer, or oil refiner, or possibly an oil distributor. The next phase then to be considered is the procedure which should most logically be followed to attain the objeckive of the indivitlual, since in these cases these young men will have been trained in a course in petroleum engineering and it would logically be presumed that the attainment of their objective should be through the operating end of the branch of the business with which they are connected. The operating end is, of course, the department of each branch of the business that develops the resources, or refines the products, or distributes the products, from which the profits are derived, and is a part of any business which is filled with enough interesting work to appease the desires of the most ambitious young men. In order to get the best possible training it would appear to me that the place for a young man to start after graduation would be at the bottom of that branch of the industry which he expects to follow. For example, if he is going in the producing end he can gct no better training than is secured by starting as a roustabout in the ficltl, where he romes in constant rontact with the character of work that is necessary

Jan 1, 1928

By D. E. Thomas

Oxidation rate constants were determined for Cu-Pd and Cu-Pt alloys as a function of alloy composition and temperature. Reaction products were identified. Relationship between oxidation rate constants and diffusion constants in the reaction zones is discussed. THE mechanism of oxidation of alloys is consider--&apos;¦ ably more complex and not so well understood as that of pure metals. The oxidation of a one-phase binary alloy of which one component is a noble metal should logically be the simplest case of alloy oxidation, for in this case one is concerned presumably only with the oxidation of the base metal and the diffusion process by which it reaches the reaction site. This had led a number of investigators to study such systems. Probably the earliest work was that of Tammann and Rienacker&apos; on the tarnishing of Cu-Au alloys at temperatures up to 300°C. These alloys oxidize according to the logarithmic law, that is, the film thickness is proportional to the logarithm of the time. The rate of film formation was found to decrease with increasing gold content. Raub and Engel² investigated the oxidation of Au-Cu, Cu-Pd, Au-Ag-Cu, Au-Pd-Cu, and Ag-Pd-Cu alloys at 750°C in air. These alloy systems form a continuous series of solid solutions at the temperature of oxidation with the exception of those containing silver. Au-Cu alloys oxidize parabolically, the rate constant decreasing slowly with increasing gold content up to about 15 atomic pct* Au, and more rapidly thereafter. An alloy with 14 pct Cu forms only a very thin layer of CuO on the surface and probably also a subscale.† Alloys containing more than 14 pct Cu have scales consisting of CuO and subscales of Cu,O embedded in nearly pure gold. The Cu-Pd alloys obey the parabolic law except for slight deviations for alloys containing 53 and 84 pct Cu. The overall oxidation rate decreases rather smoothly with increasing palladium content. A scale consisting of a thin layer of CuO and a thick layer of Cu²O and a subscale of Cu²O embedded in nearly pure palladium was observed for alloys containing 84, 53, and 30 pct Cu. An alloy with 19 pct Cu formed only a subscale of CuO embedded in nearly pure palladium, and an 8 pct Cu alloy formed only a thin film which was presumed to be PdO. The ternary alloys considered by Raub and Engel oxidized in much the same way as did the binaries. Wagner and Grunewald³ found that Ni-Au alloys, oxidized at 900°C in oxygen, formed a scale of NiO and a subscale containing NiO and gold. The overall oxidation rate increases with increasing gold content, passes through a maximum and then decreases to zero for pure gold. The fact that gold increases the oxidation rate was attributed to "pores" in the scale. Kubaschewski&apos; investigated the air oxidation of alloys of platinum or gold with up to 10 pct Cu or Ni at temperatures in the range 300" to 1550°C. Parabolic behavior was noted for all alloys except Au-Ni, with rates decreasing in the order Au-Cu, Au-Ni, Pt-Ni, Pt-Cu. The diffusion coefficients for the latter three systems are known, and they decrease in the same order. However, activation energies for the oxidation of these alloys do not agree with the activation energies for diffusion in the respective alloy systems. Kubaschewski and Goldbeck7 nvestigated the oxidation of Ni-Pt alloys in air at temperatures of 600"

Jan 1, 1952

By M. R. J. Wyllie

Eight laboratory-prepared aqueous base drilling muds representing common mud types, and 15 aqueous base drilling muds sampled in the field, have been used in an experimental investigation of the relationship between the streaming potential developed across a mud filter cake and the differential pressure causing filtrate flow. It has been established that the relationship is of the form, ES = kPy, where Es is the streaming potential, P the differential pressure and k and y are constants for any particular mud system at constant temperature. Beyond a certain minimum thickness this relationship is independent of filter cake thickness. Experiment has shown that both k and y are temperature dependent parameters and that in certain mud systems there is a critical value of P above which the simple relationship Es = kPY ceases to hold. The constant k tends to be a function of the mud resistivity at 77ºF, but the exponent y has not been found to be related to any commonly measured mud property. Certain field experiments are described which serve to demonstrate that the total SP (self potential) developed opposite a formation may be quantitatively accounted for as the algebraic sum of an electrochemical potential and an electrokinetic potential. A chart to permit the simple use of mean solution activities in the interpretation of electrochemical SP data is given and its use described. Various field methods for computing the electrokinetic SP component of the total SP are considered and rejected as being unduly complicated and inaccurate. It is shown that the employment of low resistivity drilling muds is essential if optimum accuracy is desired in the use of the S.P. curve for the purpose of computing connate water salinities by the electrochemical SP method. The use of such low resistivity drilling muds is accordingly recommended. INTRODUCTION In a previous paper&apos; a quantitative analysis of the electrochemical compovent of the SP (self potential) curve was made and it was shown that the electrochemical* emf, E, could be expressed as E = ZtRT/F In a,/a,, where t is the transference number of the chloride ion, R is the gas constant, T the absolute temperature of the formation water and mud, a, the mean activity of the connate water (assumed to be sodium chloride) and a, the mean activity of the mud (as sodium chloride). The essential accuracy of this expression was confirmed by an investigation 2 of the SP kicks opposite sands of known water salinity. The logs examined had been routine runs in oil wells in a number of Mid-Continent states. Further confirmation of the original analysis of the electrochemical SP is implicit in the paper by Tixier3 wherein a relationship between the SP and the ratio of connate water and mud resistivities, based only on field data, is given. Allowing for the fact that mud and connate water resistivities were used instead of activities, and that electrokinetic effects were ignored, the constant reported by Tixier is essentially 2tRT/F. The error involved in the universal substitution of resistivities for activities will receive further reference below. Possible reasons for the fact that the electrochemical properties of shales do not appear to be significantly dependent upon either the nature or the quantity of cation exchange materials, such as clays, they contain has been considered at some length by Wyllie and Patnode.4 In the same paper, the probable effects of shale thickness and compaction on electrochemical efficiency, and hence the inherent difficulty of obtaining reliable laboratory-scale data, were also pointed out. However, it is

Jan 1, 1951

By I. A. Lesk

The various materials utilized in the construction of integrated circuits, and the resultant materials interfaces, are discussed with emphasis on a materials system that is compatible with all types of integrated circuits and minimizes the number of different interfaces. Problems with interfaces may originate in the silicon, on its surface, and between other materials within the integrated-circuit package independent of the silicon. ThE difference between integrated-circuit theory and practice may generally be spelled out in terms of the difficulties in fabricating a desired structure. There are, of course, problems in forming the requisite materials, of the necessary composition and geometry, where needed. A distinct class of problem, however, is associated with interfaces— generally between different materials, but occasionally between regions of the same material. We require that each interfacial system perform its proper electrical, mechanical, and thermal function, and do so in an unchanging manner with time under the most severe operating conditions. In a practical sense, we deal in nonideal systems, such that it is generally not possible to form an interface exactly the way it was intended, even if we allow for a knowledge of the practical limitations of the fabrication system. Another way of looking at this situation is to realize that, when we form a group of interfaces under apparently the same experimental conditions, there are often differences of important consequence between them. Very high-magnification pictures, such as are obtained by an electron microscope, reveal fine structure that often differs over the area of a given sample, as well as from sample to sample. Low-magnification pictures look better, but don&apos;t eliminate the problem. The trend towards integrated circuits, and away from circuits fabricated utilizing separate components, is very strong in many areas of electronics. A typical silicon monolithic integrated circuit, still in place in the wafer on which it was fabricated, is shown in Fig. 1. The increased reliability which is being experienced through the use of integrated circuits can, to a large extent, be credited to the reduced complexity of the interface system that is present in the electronic system. A glance at Fig. 1, however, with all of its epitaxial and diffused p-n junctions, and interconnect system, shows that interfaces have been far from eliminated. Fig. 2, depicting a schematic cross-sectional view of a (partial) typical integrated circuit, illustrates in detail the numerous interfaces that occur in this type of structure. Besides the individual circuit elements, there are interfaces for isolation, and, Fig. 3, even a crossover requires a series of interfaces. What has been accomplished, however, irrespective of whether or not the total number of interfaces in the circuit is increased or decreased, is that the variety of interfaces is very much reduced. This is because there are fewer materials involved. Since the expense incurred in an R & D program to improve reliability is pretty much directly proportional to the variety of interfaces involved, it is economically advisable to work with integrated circuits for highest reliability. That is not to say that circuits built using standard components could not be made as reliable as integrated circuits, but that the expense involved because of the more complex over-all interface system would be much greater. Although electronic performance utilizing integrated circuits with all silicon elements (i.e., transistors, diodes, resistors, capacitors) is adequate for many applications, there are numerous instances where thin-film passive elements must

Jan 1, 1965

G. D. Kneip, Jr., and J. 0. Betterton, Jr. (Union Carbide & Carbon Corp., Oak Ridge, Tenn.)—The authors have contributed to the theory of zone melting by considering the effects of the solidification of the final zone on the distribution curves for the finite length bar. However, they do not consider in sufficient detail the restrictions imposed by the particular metals involved. Eq. 4 has the analytical solution which is equivalent to Pfann&apos;s&apos; equation for normal solidification of the final zone. For distribution coefficients less than one, it is apparent from this form of the equation that, as the solidification approaches completion, the concentration attains extremely large values which are inconsistent with the density of physical materials. The necessary restrictions then are that the solute concentration cannot exceed the density of the alloy, or more frequently, the solubility limit in the solid phase. Secondly, the distribution coefficient cannot be constant from 0 to 100 pet solute unless the liquidus and solidus coincide over this whole region. It would thus be more correct to impose limits on the solubility such as would be indicated by a typical eutectic diagram. Furthermore, in this case the assumption that the distribution coefficient remains constant is more likely to be realized. Eq. 4 should then be replaced by the following C,,(x) =k/L-x [ ?11 3 C11 (x)Ddx - ?11 4 Cn(x)dx= k[Cn (L-l)/k] [ L-x/l]x-1 L-l = x = L-l [kCenterlie/Cn (L-l)]1/k-1 The typical shape for the distribution curve in the final zone is indicated schematically in Fig. 12 for the simple eutectic case. The solidification proceeds according to Pfann&apos;s&apos; expression until the concentration reaches the maximum solubility in the solid phasc. At this point, the concentration changes abruptly to the eutectic composition and remains at this value for the duration of the solidification. The width of the flat region, and the back reflection of this effect into the concentration curve on subsequent passes, depends upon the ratio of the eutectic concentration to the original concentration, upon the distribution coefficient, and upon the number of passes. Similar effects on the concentration curves in the first zone length would be expected in many systems for which the distribution coefficient is greater than one, and where the maximum solubility is not too much larger than the original concentration. The discussers agree with the authors that the assumption that the liquid is uniform in concentration should be considered with caution. Kneip and Better-ton&apos; have shown, however, that in floating zone refining of zirconium, using induction heating, the distribution of iron agrees with theory which assumes a uniform liquid composition. Hence, in this case, the experimentally realized distribution coefficient agrees with the present phase diagram within the experimental uncertainty. L. Burris, Jr., C. H. Stockman, and I. G. Dillon (authors&apos; reply)—The authors are happy to have received the comments of G. D. Kneip, Jr., and J. 0. Betterton, Jr., which provide additional insight into the zone refining process. It is true that the equations developed are inapplicable if the solute solubility in the solid phase is exceeded. However, this restriction on the use of the equations was clearly stated in the paper. A closer approach to

Jan 1, 1957

By Langtry E. Lynd

The predominant tonnage use of titanium is as a white pigment, in the form of rutile or anatase, which are allotropic modifications of titanium dioxide. Because of its whiteness, high refractive index and resulting light scattering ability, titanium dioxide is unequaled as a white pigment for paints, paper, rubber, and a wide variety of other materials. Consumption of titanium dioxide pigments for these uses has been rising steadily for many years, United States consumption in 1956 amounting to about 486,000 tons, worth $250,000,000. Production was off a little in 1957 and 1958. An outstanding development in the titanium field in recent years is the expansion of titanium metal production. Virtually nonexistent before 1948, United States production of pure titanium metal sponge reached a high in 1957 of about 17,200 tons, worth about $77,000,000. United States titanium sponge metal production in 1958 was cut back to about 4,600 tons because of reductions in military requirements. Another major use of titanium, in the form of the mineral, rutile, is for welding rod coatings. United States consumption of rutile for this purpose in 1958 amounted to about 11,000 tons, worth $1,100,000. Titanium-Bearing Minerals Although several minerals contain titanium as a major constituent, only a few of these occur in sufficient quantities and concentrations to be of commercial importance. These are ilmenite (FeTiO3), rutile (TiO2), and alteration products of ilmenite which have been referred to as weathered ilmenite, leucoxene, or arizonite. The other polymorphs of titanium dioxide, brookite and anatase, occur in a few deposits, of which only one has been of even minor importance. Considerable concentrations of titanite or sphene (CaTiSiO5) and perovskite (CaTiO3) are known, but do not compete with ilmenite and rutile deposits in grade or size. Rutile is very widely distributed as an accessory mineral in most rocks, but large deposits of commercial importance are rare. A few rock deposits have been worked for rutile but only one, in Virginia, is in operation (1959). Development of one in Mexico was discontinued recently. At present, the main sources of rutile are the beach and dune sands of Australia and Florida. A major use of rutile is for the manufacture of titanium tetrachloride, which is the intermediate compound universally used in titanium metal manufacture. Another main use of rutile is for welding rod coatings. Although the whiteness and brightness of titanium dioxide pigment is due to the high refractive index of rutile or anatase, natural rutile is too impure to be used directly as a white pigment, is too expensive to use for manufacturing pure oxide pigment, and is too scarce to provide the required tonnages. Instead, the cheaper, more abundant, mixed oxide of iron and titanium, ilmenite, which is readily decomposed by sulfuric acid, is used for manufacture of rutile and anatase pigments. Ilmenite is mined from both rock and sand deposits. Rock deposits contain ilmenite in massive or heavily disseminated form along with magnetite or hematite and are almost invariably associated with anorthositic and gabbroic rocks. Ilmenite also occurs as an accessory mineral in a great many igneous and metamorphic rock types. Concentrations of

Jan 1, 1960