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By C. E. Brukl, E. Rudy, St. Windisch

The ternary-alloy system Nb-Mo-C was investigated by means of X-ray, melting point, DTA, and metallo-graphic techniques; a complete phase diagram for temperatures above 1500°C was established. Above 1960°C, niobium monocarbide and the cubic (Bl) high-tenzperature phase in the Mo-C system form an uninterrupted series of solid solutions. The ternary range of the pseudocubic q MoCl-, is very restricted. Dimolyb-denum carbide dissolves up to 44 mol pct Nb2C (2240°C), whereas the maximum solid solubility of MO2C in Nb2C does not exceed 5 mol pct. The order-disorder transformation temperatures in Mo2C and Nb2C are lowered by the mutual metal exchanges. Six invariant (p = const) reactzons occur in the ternary system; three correspond to class 11-type four-phase reactions involving a liquid phase, one to a class I (eutectoid)-type, and two further isotherms are associated with limiting tie lines. The results of the Phase diagram investigation are discussed, and the thermodynamic interpretation identifies the low relative stability of the binary sub-carbides in conjunction with the large stability diflerences between niobium and molybdenum carbides as the cause for the formation of a stable equilibrium between the monocarbide and the metal phase in the ternary reson. Due to their refractoriness, the carbides of the high-melting transition metals have received increased interest in recent years as base materials in composite structures for aerospace applications at high temperatures and for the development of self-bonded cutting tool materials; other novel fields of application include power reactors, where operation at high temperatures becomes essential for attaining high power efficiencies. In these applications, the increased reaction rates at high temperatures require a close consideration of the chemical interactions between the alloy constituents. As a consequence, a detailed knowledge of the phase relationships in the alloy systems is required in order to provide a sound basis for developmental -type work. Partly as a result of the considerable experimental difficulties associated with the investigation of this high-melting alloy class, no complete studies of ternary metal-carbon systems have been performed until recently. Even the high-temperature phase relationships in the binary transition metal-carbon systems have been delineated only during the past few years to a degree of accuracy required for a more detailed study of ternary or higher-order alloys. In recent investigations of binary and ternary systems of refractory transition metals with carbon, boron, and silicon,&apos; alloys from the ternary systems Nb-Mo-C became of interest because of the demonstrated possibility2,3 of obtaining compatible composites based on metal + monocarbide combinations. In the meantime, however, studies in other, but related, ternary metal-carbon systems, such as Ta-W-C, have shown that the solid-state equilibria may change significantly toward higher temperatures (>2000°C), and that extrapolations based on low-temperature equilibrium data are, in general, not very reliable. Although the lower-temperature (<2000°C) phase relationships in the Nb-Mo-C system are similar to those found in Ta-W-C, a cursory thermodynamic analysis of the equilibria indicated4 that complete solid-solution formation between Mo2C and Nb2C should not occur at higher temperatures. The present work was conducted in order to experimentally verify these expectations and, in addition, to provide phase equilibrium data in the melting range of the alloys. In the boundary systems, niobium and molybdenum are known to form a continuous series of solid solutions.576 The continuous solubility was also confirmed by Kornilov and Polyakova,7,8 who also observed a minimum melting point at 22 at. pct Mo and 2345°C. The phase diagram investigations of the Nb-C system by Storms and Krikorian9 and Kimura and Sasaki10 were recently supplemented by Rudy et al.11,12 The system contains a high-melting monocarbide with the B1 structure, Table I, and a subcarbide, Nb2C, which exists in at least two different states of sublattice order at low temperatures47"-&apos;3 and a disordered state above approximately 2500°C.11,12 The melting-point measurements by Rudy et al .11,14 are in close confirmation of the data by Kimura and Sasaki.10 The rather complex phase relationships in the Mo-C system were only recently Clarified.15,18 The system is characterized by three congruently melting, intermediate phases, MozC, ? MoC1-x and a Mol-,, Table I, of which only Mo2C is stable at temperatures below 1650°C. Substoichiometric MozC exists in several states of sublattice order which interconvert in homogeneous phase transitions. Hyperstoichiometric compositions cannot exist in the ordered state. Upon cooling through a critical temperature range, the

Jan 1, 1968

By Clarence O. Babcock

Model pillars of limestone, marble, sandstone, and granite, with length-to-diameter (LID) ratios of 3, 2, 1, 0.5, and 0.25 (0.286 for granite), were broken in axial compression to determine to what extent an increase in end constraint increased compressive strength. Radial end constraints of 13 to 23% of the average axial stress in the pillar, produced by solid steel rings bonded with epoxy to the ends of dogbone-shaped specimens, increased compressive strength somewhat above that of cylindrical pillars without ring constraint. However, when the results were compared with those obtained by other investigators for straight specimens of several rock types taken collectively, with LID ratios greater than 0.5, the resulting strengths were not significantly different. Thus, the amount of end constraint produced by the solid steel rings was about the same as that produced by the friction from the steel end plates. In other tests, a radial prestress of 3000 or 5000 psi was applied prior to axial loading by adjustable hardened steel rings to increase the constraint above that obtained for the solid rings. The average radial constraint stress, expressed as a percentage of the average axial pillar stress at failure for the 3000 psi prestress, was 54.3% for limestone, 40.3% for marble, 44.7% for sandstone, and 23.4% for granite. The average radial constraint stress, expressed as a percentage of the average axial pillar stress at failure for the 5000 psi prestress, was 74.2% for limestone, 51.2% for marble, 61.6% for sandstone, and 29.7% for granite. These constraints increased the compressive strength significantly above the strength of straight specimens and solid-ring constrained specimens. These results suggest that large horizontal stresses in orebodies mined by the room-and-pillar method should increase the strength of the pillars and allow an increase in ore recovery by a reduction of pillar size when major structural defects are absent. One important objective of the U.S. Bureau of Mines (USBM) mining research program is the rational design of mining systems. In the design of room-and-pillar mining operations, pillar strength is a fundamental variable. It is customary to estimate this strength from uniaxial compression tests of rock samples and to correct this value for the length-to-diameter (LID) ratio of the in-situ pillar. This method of estimating pillar strength corrects for pillar shape but does not consider the effect of a large horizontal in-situ stress field that sometimes exists in underground formations. The purpose of the work covered in this report was to determine to what extent the compressive strengths of model pillars of relatively brittle rock loaded in axial compression were affected by lateral end constraint. In previous work, Obert l used solid steel rings bonded to the ends of dogbone-shaped specimens to study the creep behavior of three quasi-plastic rocks -salt, trona, and potash ore - during a test period of 1000 hr. These rings provided radial constraint during the loading cycle of 20 to 50% of the axial stress for specimens with LID ratios of 2, 0.5, and 0.25. He concluded that (1) "for a quasi-plastic material the end constraint strongly affects the specimen strength, and (2) as D/L increases (length-to-diameter decreases), the specimens lose their brittle characteristics and tend to flow rather than fracture." He also concluded that model pillars constrained by rings were better for use in predicting the strength of mine pillars than either cylindrical or prismatic specimens. This conclusion appears to be valid where mine pillars, roof, and floor are a single structural element. In the present study, 460 specimens of four relatively brittle rocks — limestone, marble, sandstone, and granite - were tested to failure. The study consisted of two parts: (1) the effect on the compressive strength of end constraint produced during the axial loading cycle by solid steel rings bonded with epoxy to the ends of the specimens, and (2) the effect on the compressive strength of increased end constraint produced in part by a prestress applied prior to axial loading and in part by lateral expansion of the specimen during the loading cycle. The first part of this study was reported in some detail earlier.2 EXPERIMENTAL PROCEDURE AND EQUIPMENT Model rock pillars of the sizes and shapes shown in Fig. 1 were broken in axial compression when the ends were constrained as shown in Fig. 2. he straight specimens were broken without ring constraint. The specimens of dogbone shape were broken with (1) solid

Jan 1, 1970

By Frederic L. Kadey

Diatomite is a siliceous, sedimentary rock consisting principally of the fossilized skeletal remains of the diatom, a unicellular aquatic plant related to the algae. Thus, it has been formed by the induration of diatomaceous ooze, and consists mainly of diatomaceous silica, a form or variety of opal which is first formed in the cell walls of the living diatom. Diatomaceous silica is not generally regarded as a synonym or the equivalent for diatomite, although it has been so used at various times. Accurately, diatomaceous silica is the preferred name for the principal mineral component of which the rock, diatomite, is composed. The terms diatomaceous earth and kieselguhr are used as synonymous with diatomite. The designations tripoli, tripolite, infusorial earth, etc., were used at one time but are now obsolete. With the changing nomenclature, these terms that were at one time correct when proposed and used for generations would be considered incorrect if used today in the light of current knowledge. The designation diatomite is reserved for those accumulations of diatomaceous silica that are of sufficient quality, size, and minability to be considered of potential commercial value. Processed diatomite possesses an unusual particulate structure and chemical stability that lends itself to applications not filled by any other form of silica. Foremost among these applications is its use as a filter aid, which accounts for over half of its current consumption. Its unique diatom structure, low bulk density, high absorptive capacity, high surface area, and relatively low abrasion are attributes responsible for its utility as a functional filler and as an extender in paint, paper, rubber, and in plastics; and as an anti-caking agent; thermal insulating material; catalyst carrier; and chromatographic support; polish, abrasive, and pesticide extender to name a few representative applications. The United States is the principal producing country, although diatomite is found in numerous other locations. Geology Composition and Morphology Diatomaceous silica qualifies as a. mineral of organic origin in much the same way that aragonite and collophane do. The silica of the fossilized diatom skeleton closely resembles opal or hydrous silica in composition (SiO2 • nH2O) .4, 12, 203 The silica is of acute biological significance, not only for the cell wall component, but also for the basic life process.45, 51 Without silica, cell development ceases .31 In addition to bound water, varying between 3.5 and 8%, the siliceous skeleton may also contain, in solid solution, or as part of the Si02 complex, small amounts of associated inorganic components-alumina, principally-and lesser amounts of iron, 46, 49, 55 alkaline earths, alkali metals, and other minor constituents.59, 63, 67 Boron is reported to be an essential element for diatom growth .52, 58 Since diatomaceous silica is not pure hydrous silica but contains other intimately associated elements, there is good reason to consider it a distinct type or variety.4 Associated with the diatomaceous silica, and integrated as part of the diatomite, may be variable amounts of organic matter, soluble salts, and particles of rock-forming minerals that were syngenetically deposited or precipitated with the diatom frustules. Sand, clay, carbonate, and volcanic ash are typical common contaminants. Other contaminating minerals may be present, such as feldspar, mica, amphiboles, pyroxenes, rutile, zircon-the result of weathering, then transporting, and subsequent redeposition of surrounding land masses. Commercial diatomite may also contain fragments and particles of

Jan 1, 1983

By C. G. Bieber, J. R. Mihalisin, R. T. Grant

A growing demand for longer service life of gas turbines has placed increasingly rigorous requiret~rents upon superalloys employed for that application. Long-titne testing at high temperature has revealed that phase transformations occur in all superalloys. A common one of particular interest is o formation. Presented here are studies made to identify a and to characterize its formation and effect on properties in three cast nickel-base superalloys—IN 100 alloy, alloy 713C, and alloy 713LC. Methods are discussed by which o can be eliminated or inhibited in IN 100 alloy and alloy 713C. Evidence was obtained to indicate that some types of o may be more detrimental than others. Limitations in the electron vacancy approach to o prevention are pointed out, and it is shown how alternative approaches, such as reducing a complex superalloy matrix to the form of a pseudo-ternary system permitting equilibrium diagram treatment, lead to additional insights into the formation of in these alloys. AROUND 1960. Beiber1 developed IN 100 alloy, which still remains one of the strongest commercially available nickel-base superalloys. The principle used in the design of this alloy was to produce large quantities of y&apos; phase in a y matrix through the use of copious amounts of aluminum and titanium. In 1963, ROSS&apos; showed that when certain heats of this alloy were held for a long time at 1650°F they formed an acicular phase, subsequently identified as a.3 a is a hard and brittle phase first discovered in the Fe-Cr system by Bain and Griffiths.4 They termed it the "B" constituent. Subsequently this same phase was found in other systems, primarily those of the transition elements, and acquired the name "a" by which it is now known. The crystal structure of the a phase was first determined in the Fe-Cr system in 1950.5 It was shown to be tetragonal with a c/a ratio of about 0.52. as is the case with a found in other systems. This characteristic crystal structure is now the means by which a is identified. In superalloys, such as IN 100 alloy. large amounts of o impair the high-temperature creep strength and drastically reduce room-temperature tensile ductility. Discovery of o phase in some heats of IN 100 alloy quickly led to investigations of other superalloys for similar transformations. It was found that many of the stronger, more highly alloyed. super-alloys were indeed susceptible to o formation. This investigation has been concentrated on three commercial alloys: IN 100 alloy, alloy 713C, and alloy 713LC. J.R.MIHALISIN,MemberAIME, and C.G.BIEBER are with The International Nickel Co., Inc., Paul D. Merica Research Laboratory, Sterling Forest, Suffern, N. Y. R. T. GRANT, Member AIME, is with The International Nickel Co., Inc., Pittsburgh, Pa. Manuscript submitted May 22. 1968. IMD A detailed study has been made of the phase transformations and their relation to a formation along with a consideration of electron vacancy approaches for predicting a-forming propensity in these alloys. EXPERIMENTAL PROCEDURE Phase transformations were studied by light and electron microscopy, electron diffraction, microprobe investigations, and X-ray diffraction. Specimens for light micrographic examination were prepared by conventional grinding and polishing followed by etching with glyceregia (2:l HC1/HNO3 + 3 glycerine by volume). Photomicrographs of stress-rupture specimens were taken adjacent to the fracture unless otherwise noted in the text. Negative replicas for electron microscopy were taken from surfaces electropolished with a solution of 15 pct H2SO4 in methanol. For carbon extraction replication, a solution of 10 pct HC1 in methanol was used. A Siemens Elmiskop I was used for all electron microscopy. Selected-area diffraction studies were made at 80 kv using evaporated aluminum for standardizing the patterns. A nondispersive electron microprobe attachment was used to analyze the extracted precipitates chemically. The fluorescent X-rays were recorded using a flow counter containing P10 gas (90 pct Ar-10 pct methane) with a beryllium window and a single-channel pulse-height analyzer. The pulses from the analyzer were passed to a scaler-ratemeter and differential curves of counting rate vs pulse amplitude were obtained. The base line of the analyzer was driven with a synchronous motor at 0.5 v per min and a channel width of 0.5 v. The time for 105 counts was printed out for each 0.5-v increment. The microscope was operated at 80 kv with beam currents of 1 to 20 pa. This equipment detects elements from atomic number 13 to 40. X-ray diffraction studies were usually made on residues electrolytically extracted in 10 pct HC1 in H2O, although in one case a pattern was obtained from an etched surface of a metallographic specimen. A Siemens Crystalloflex IV was used with iron-filtered CoKa radiation. X-ray patterns were recorded using a goniometer speed of : deg per min. The scintillation counter and pulse-height analyzer operated at a channel height of 10 v and a channel width of 12 v. The equipment was calibrated with a powdered gold standard. The residues usually contained a number of phases. several of which could not be found in the ASTM card file. In addition, as is shown for the case of a phase in IN 100 alloy, other phases had a somewhat different lattice parameter from that reported in the ASTM card file, making it difficult to separate and identify constituents by comparison with ASTM d spacings. For these reasons, phases were identified on the basis of the lattice parameter obtained by indexing the ob-

Jan 1, 1969

By V. J. Kniazeff, S. A. Naville

The problem of unsteady-state condensate-gas flow through porous media leads to a set of second-order non-linear partial differential equations. Such a set of equations is numerically solved in the case of radial two-phase flow around a well, taking into consideration both the thermodynamical properties of the fluid and the mechanical properties of the reservoir. The fluid properties, reflecting the PVT relationship of the gaseous and liquid phases, are expressed by using the partial specific masses of the two main separator products in these phases. The flow properties of the reservoir rock are expressed by the generalized Darcy&apos;s law for the liquid phase and by a quadratic relationship between the rate of flow and the pressure gradient for the gaseous phase. The numerical solution of the equations for pressure and saturation US radius and time is worked out through programs written for a computer. The evolution of bottom-hole pressures, well productivities or- deliverabilities and effluent compositions with the depletion of the reservoir is easily derived. The application to the Saharian gas-condensate field Hassi R&apos;Mel led to a better understanding of the drainage mechanism. A zone of fairly high liquid saturation develops around the wells, reducing the effective permeability, and represents a loss of condensible products in addition to the PVT-like retrograde condensation. lnside this zone, near the well, the deviation from Darcy&apos;s law for the flow of the gaseous phase governs the well deliver-ability. A back-pressure test has been computed and correlates with the field results. INTRODUCTION Two-phase flow of volatile hydrocarbons, like condensate gas or light crude oil, may be treated as the flow of a binary mixture by an arbitrary division of the chemical components into two groups. This is translated into two equations of mass continuity, which constitute a set of relation- ships for the pressure and the saturations vs the space coordinates and the time. The equations contain the laws governing the composition and the motion of the phases. The problem so defined is solved with the assumption that the compositions of the phases at any pressure are respectively the same as those observed in a PVT measuring cell under differential liberation. In a first series of computations, it was assumed that the flow obeys the generalized Darcy&apos;s law. A satisfactory representation of the retrograde condensation around the well was thus obtained. In addition, the trend toward decreasing effective permeabilities was obtained, and the computed composition of the effluent checked the laboratory values. However, it has not been possible within this basic assumption to represent the non-linear relationship between the production rate and the bottom-hole pressure drawdown as observed for gas wells in the field. Following the advice of A. Houpeurt it was decided to consider the relative permeability to gas as a function of the velocity of the gas Phase.1 The necessary physical determinations were made by E. COstaséque using a method devised by A. Houpeurt and R. Iffly. As numerical processing of the equations progressed, several difficulties were encountered which were overcome through collaboration with the computer manufacturer. This mathematical model of two-phase flow in porous media had been primarily intended for and extensively applied to the case of the Hassi R&apos;Mel gas-condensate field, operated by SN Repal for SEHR, a joint subsidiary of SN Repal and CFP (A). The programs have also found their applications to forecast the behavior of several fields in the Sahara area containing light and volatile hydrocarbons. BASIC EQUATIONS We will consider a zone in the porous reservoir where the flow properties and the in situ composition of the fluids can be assumed to be uniform. A part of the equations of transient flow can be written with the specific gravity of the fluids being taken equal to the sum of the contributions due to the

Jan 1, 1966

By J. J. Gilman

THEORIES of fracture in metals have been based primarily on the dependence of fracture strength on grain size. This dependence was first observed by Masing and Polanyi.l These workers, as well as 0rowan12 and Davidenkov and Wittman,&apos; thought the grain size could be equated with the crack size in the Griffith condition for crack propagation.4 In this way they could account for the observed proportionality between fracture stress and the reciprocal square root of the grain size. The chief results of recent investigations of the fracture strength of polycrystals are summarized in Fig. 1. Low for iron,5 Petch for iron,&apos; Greenwood and Quarrel for zinc, Petch and Zein for zinc,&apos; and Hauser, Landon, and Dorn for magnesium&apos;&apos; have all found that the fracture stress of a polycrystal is proportional to the reciprocal square root of the grain size. Also, they find that, for large grain sizes, the fracture stress lies near the extrapolation of the yield-stress curve determined at small grain sizes; where measurable plastic strain precedes fracture. Low9 has gone a step further and shown that, in Range 1, the fracture stress in tension is almost identical to the yield stress in compression for a given grain size. Homes and Gouzou, 10 and Low also have shown that, in Range 2 of Fig. 1, microcracks form when the yield stress is reached, but they do not cause failure until the stress reaches a much higher value. The fact that yielding always precedes fracture suggests that fracture is induced by plastic flow. A mechanism by which this might occur was first proposed by Zener.11 He pointed out that blocked dislocations would cause stress concentrations which might cause fracture. The blocking would be caused by grain boundaries. This idea has been developed by Eshelby, Frank, and Nabarro,12 by Koehler,13 and by Stroh.14 By means of etchpits blocked dislocations have been observed in brass by Jacquet,15 in zinc by Gilman,16 and in lithium fluoride by Gilman and johnston. The blocked-dislocation theory and the experiments that bear on it have been summarized by Mott.18 One of the strong points of the blocked-dislocation theory is that it can explain the grain-size dependence of the fracture stress in a natural way. However, Low9 has pointed out that the theory is not satisfying when it is considered quantitatively. Gilman10 also has criticized the theory and presented an alternative interpretation of the relation between fracture stress and grain size. Since the blocked-dislocation theory is not the only theory that can explain in the grain-size effect, it is necessary to find other evidence either to support or deny it. In this investigation the following points were studied: 1) If blocked dislocations normally cause fracture, then the fracture stress ought to be very high in the absence of plastic glide. Is it? 2) If dislocations must be blocked in order for metals to fracture, then certain types of grain

Jan 1, 1959

By Eugene S. Machlin

A classical theory of solid solutions involving neavest-nergkbor intevactions with both central and linked-central forces between atoms has been developed. It has been found that the theory, where it can be checked quantitatively, is in ageement with experiment. The theory encompasses a description of many diverse pkenomena, such as antiphase shift structures, size effect, relative stabilities of various solutions, lattice para,neters, and order-disorder transitions. In particulav. a quantitative prediction not involving adjustable pavameters is made concevning the deviation of the Au-Cu interatonlic distance in long-range ordered (Ll,) Cu-Au I fronl the average distance based on the distances in pure gold and copper. This prediction, which is in agreement with expel-intent, has not been encompassed by any preuious theory. The theory of order-disorder is fragmentary. That is, no one theory exists that can explain the variety of qualitative phenomena observed. Further, many theories are not in good quantitative agreement with experiment. This subject has been reviewed by Muto and Takagi, Tuttman, and Oriani.3 There exists no doubt that the quasi-chemical approximation is not a complete description and that the inclusion of strain energy using macroscopic elasticity theory concepts leads to results in disagreement with experimenL4 The observation of antiphase domains and ordering systems such as Cu-Pt has led to Brillouin zone treat-ment of the order-disorder transition as opposed to the classical Ising model. The objective of this paper is to demonstrate that it is possible to develop a pairwise approximation model that can explain many of the observed order-disorder phenomena that have puzzled investigators in the past. This theory is based upon an empirical model due to ergmman&apos; for the elastic constants of metals. This model is generalized for multicomponent systems. As will be shown, the theory yields a short-range ordering energy for the disordered solution which differs from the ordering energy calculated from the differences in energy of disordered and long-range ordered solutions. It will be demonstrated that there is no necessary correlation between heats of formation and the tendency to order or between size effect and the tendency to order. Also, the existence of antiphase domains and iso-short-range-order systems that form superlattices (Cu-Pt) is predicted on the basis of the theory. Further, the relative stability of competing superlattices is calculable from the theory. If single-crystal elastic-moduli data are available for the pure components and one superlattice then there exists but one adjustable parameter in the calculation of lattice parameters for both the disordered and ordered solid solutions. In one special case, no adjustable parameters are required and a quantitative prediction is made. For the calculation of energies and partial order, there exists but one additional adjustable parameter, the pair-exchange energy V used in the quasi-chemical approximation (or the Ising model.) However, in these calculations, much more precise values are required for the single-crystal elastic moduli than available if the quantitative uncertainties in the predicted values of the energies are to be sufficiently small. THEORY ~er~man&apos; has developed a model with which he was able to obtain fair agreement with experiment for the relations between the elastic constants for metals. This model which we shall call Bergman&apos;s model is a linear combination of his models I and 11. In effect, Bergman, in this model, considers that each interatomic distortion is composed of two components: a classical central force distortion with an associated central force constant and what we shall call a linked-central force distortion with an associated linked central-force constant. The linked-central force distortion component obeys the constraint that the sum of such distortions over all the bonds equals zero. No constraint is imposed on the classical central force distortion component. Bergman&apos; derives the constraint on the linked-central force distortion on the basis of application of Pauling&apos;s relation between bond distortion and bond number to metals.ga This assumption is not logically necessary, however, and the Bergman model may be taken as a mathematical model for elastic constants, e.g., a purely empirical model without a physical basis. In the present work, the method of Bergman has been applied to two-component systems (solid solutions). In place of an external strain—which would allow a calculation of the elastic constants for the two-component system—it is considered that internal interatomic distortions exist as a consequence of having three potentially unequal distortion-free interatomic distances and but one "average" interatomic distance. It is assumed that the distortion-free interatomic distances between atoms of the same element are those found in the pure element having the same undistorted crystal structure as the solid solution. The distortion-free interatomic distance between unlike atoms is in general not measurable except in the probably nonexistent

Jan 1, 1967

By J. E. 214-000-000-010 Warren, M. R. J. Wyllie, T. Meidav, L. Scharon, R. Uhley, A. J. deWitte

By considering the stoichiometry of the underground combustion process, an equrztion was derived relating the point velocity of the combustion front as a function of the air flux, fuel content, efficiency&apos; of oxygen uitilization, hydrogen:carbon ratio, and ratio of carbon dioxide In carbon monoxide produced. In addition, an equation for cnlcrllating the amount of injected air necessary 10 produce the oil from I Cu. ft of reservoir space was derived. This equation has a direct bearing on evaluating compression costs for the process. Values calculated from these equations are in good agreement with experimentnlly obscrvcd values obtained from laborator) tube runs reported in the literature. The equations are also u\eful for atlcrlyzinf: experinlental data for inconsistencie.r.. I NTRODUCTION The injection of energy in the form of heat into an oil-bearing formation as a means of increasing the recovery of oil has been the subject of much discussion and experimentation as far back as 1920. The ideas have ranged from the injection of steam to the direct use of electricity as a means of getting thermal energy into a formation. However, most laboratory and field studies have been confined to the process whereby a portion of the crude in the reservoir is burned to generate the necessary heat to increase the mobility of the oil by reducing its viscosity. A recent paper by W. L. Martin, et al.&apos;, described the results of a number of laboratory experiments in which the combustion process was studied in a sandpack. Studies such as this indicate that with the temperatures involved, the oil is partially coked in this process and it is this coke which provides the fuel for combustion. The combustion of this residual coke can be analyzed theoretically by applying simple stoich-iometric principles to the combustion process. It is the purpose of this paper to compare the results reported in these experiments with those obtained from theoretical considerations. The variables studied by Martin, et al, among others, include combustion front velocity, air flux, oxygen utilization, fuel concentration, ratio of carbon dioxide and carbon monoxide in combustion gases, and the hydrogen:carbon ratio of the residual fuel. All these variables come into consideration when the stoichiometry of the combustion process is studied. The experimental conditions and results are tabulated in Tables 3 and 4 of Ref. 1. For purposes of comparison with calculated results, it was necessary to change the units of a number of the variables. UNDERGROUND COMBUSTION STOICHIOMETRY The combustion or oxidation of the carbonaceous residue or "coke" on the sand can be described in terms of the equation: where n is atomic ratio of hydrogen to carbon, m is ratio of moles of CO2 to CO produced, and CH. is taken as the unit molecular weight, realizing that actually the coke has a large range of high molecular weights. Assuming that all the fuel laid down in the sand is burned, the ratio of moles of oxygen consumed to moles of coke can be written as follows. Moles 0MolesMolescoke In terms of moles O2 per pound of coke, where 12 + n is the molecular weight in pounds per mole. If the 0, consumed is measured in standard cubic feet (60°F and 1 atm) and the coke in pounds per cubic foot of formation, 2, then:

By Paul G. Shewmon

The migration of slightly solzrhle spherical particles through a solid under the infllrence of a temperature gradient is analylzed for the cases of various transport mechanisms. It is shown that the variation of the velocity of the particles with radius, r, depends on the dominant mechanism of matter transport around or through the inclusion. Thus the velocity varies as r-1 jor surface-dijf1~sion controlled migratzon. is independent of r for volume diffusion in either phase, and varies as if the rate is determined by an interfacial reaction (n is the order of the interfacia1 reaction). These same results hold tor the migration of a cvlinder of length much greater than its radius. and .for other types of potential gradients, e.g., an electrical field. These equations are combined with recent electron-microscopic ohservations to show that the rate of migratzolz of small helium-silled bubbles through copper is determined by surface diffusion of the metal atoms. With these equations and the temperature gradients attainahle in an electron-microscope joil, the dominant transpar/ mechanism (or any migrating pnrtzclos can he determined. AS a result of fission, rare-gas atoms are created in the hot fuel element of a nuclear reactor. These insoluble atoms precipitate to form bubbles of the gas in the fuel. The subsequent migration and coalescence of these bubbles is thought to play a dominant role in the swelling of fuel elements which in turn can limit the fuel-element life. As a result of this practical problem and because of the relative simplicity of the system, workers in several laboratories have imbedded rare-gas atoms in metals with the aid of an accelerator and studied the formation and behavior of the bubbles that form on annealing. Recently Barnes and Mazey have studied the migration of such helium-filled bubbles in copper foils using the electron microscope and the temperature gradient that can be induced in the foil by the electron beam. They found that the smaller bubbles moved faster than the larger ones. It is shown below that, if this is true, the rate-controlling step in their migration is surface diffusion of metal atoms instead of volume diffusion, vapor transport, or an interfacial reaction. The analysis given is in no way limited to gaseous inclusions so the variation of velocity with particle size for any relatively insoluble precipitate particles could be used to obtain information about the relative importance of surface diffusion, volume diffusion, and interfacial reaction in such two-phase systems. VOID MIGRATION IN A TEMPERATURE-GRADIENT ANALYSIS Barnes&apos; studies indicate that helium does not dissolve or diffuse in metals to a measurable extent, so that when the voids move they must do so through the movement of metal atoms from the leading to the trailing sides of the void.&apos; Thus voids might move by surface diffusion, diffusion of vacancies in the surrounding lattice, or vapor transport. We consider first the case of flow by surface diffusion alone. We assume that there is a force, in the sense of irreversible thermodynamics, tending to move the atoms around the bubble in a given direction. We designate this direction as the x axis and the force per atom as Fa. The net rate of flow of atoms from one side of the bubble to the other, under the influence of this force, is the surface flux, Js, times the cross-sectional area available for flow, A,. A, is taken as the circumference times the thickness of the high-diffusivity surface layer 6. Thus where 51 is the atomic volume. For bubbles to advance a distance dx, a volume equal to nr : dx must flow around the void so that The minus sign enters because the bubble moves in the direction opposite to that of the force on the atoms. Equations for volume diffusion and diffusion through the vapor can be obtained in the same manner. For vapor transport we have Ag = and the ratio of the densities of metal atoms in the gas and the solid (pg/pl) enters so that

Jan 1, 1964

By R. S. Wagner, R. G. Treuting, W. C. Ellis

Use of the Weisseizbevg goniometer for the study of the morphology of whisker crystals and transfer of the results to a stereograplzic projection are described. While in almost all instances a complete and rapid solution is obtained with a purely gaphical treatment, an absolute analytical treatment applicable to difficult cases is given. Examples are given for determination of growth direction and external features of a crystal, for detection of &apos;twintzittg, and for establislting the orientation relntionships in a kinked whisker crystal. We give here a method for the use of the Weissen-berg goniometerl" in the study of orientation relationships in "whisker" (micro) crystals3-8 and direct transference of the data to the stereographic projection. The operations are straightforward and require no involved calculations: in almost all cases solely graphical treatment is adequate. In a few complex instances, a more exact analytic method may be required. In the method, not only are equatorial reflections of an oscillation photograph used, but also those on higher layer lines, or those not occurring on a recognized layer line. Such reflections off the zero level are often essentialy to detect the presence of twinning. The method is particularly suitable for examination of whisker crystals since the axial or growth direction is almost always one of high symmetry, permitting a more restricted oscillation range to obtain the necessary reflections and an accordingly shorter exposure time. The method is also applicable when there is no lattice-related external symmetry, and is as well able to identify external crystal surfaces, when such exist, with the aid of an optical beam. Thus growth direction, identification of lateral faces, twinning, and orientation relationships in a kinked crystal can be established. A test for the Eshelby twist6-8 can readily be made. The method can be used for orientation, for structure analysis, and as an aid in the identification of an unknown crystal. Moving-film techniques have previously been employed variously for crystal-orientation studies.1,2,9-13 In particular, use of the Weissenberg method for orientation determination has been reported by Wooster and wooster.14 While their method derives the same data as here described, it differs in depending on the tracing of constant p streaks, a less than dependable phenomenon with small specimens, a few microns or less in cross section. The Laue method can and has been used for orientation studies of microcrystals,15 e.g., 2 p in cross section, but requires the extreme intensity and concentration from a microfocus tube to obtain reasonable exposure time. PRINCIPLES OF THE PRESENT METHOD Consider a crystal of random orientation with respect to the oscillation axis in a Weissenberg goniometer. Its diffraction pattern need not necessarily result in recognizable layer lines. In Fig. 1, a stereo-graphic projection of the reference sphere is given with the equatorial plane XZ normal to the oscillation axis Y, and the X-ray beam directed from X towards 2. The great circle ZZ is then normal to the beam, and the axes are right-handed. Thus the reflected X-ray beam lies at the intersection R of small circle ARB of radius 28 from 2, and of small circle RE of radius a from the rotation axis Y. a is the conventional semiapex angle of a cone of diffracted rays from the atoms aligned in the direction

Jan 1, 1967

By C. Gatlin, M. E. Chenevert

The effects of bedding plane orientation on the elastic constants and the yield strengths of three laminated rocks (one sandstone and two shales) and one isotropic rock (a limestone) were studied. The directional dependence of the elastic properties of these rocks was experimentally evaluated using a triaxial compression cell and auxiliary stress-strain measuring equipment. Symmetry of Poisson&apos;s ratios within the bedding plane suggested that horizontal isotropy exists, but the bedding planes do give rise to an appreciable difference between properties in the horizontal and vertical directions. For the three bedded rocks studied, Young&apos;s modulus was lower normal to bedding than along bedding. Yield strengths were determined at confining pressures from 0 to 12,000 psi in a triaxial compression cell. The rocks studied showed strength reductions as high as 40 per cent when the test specimen was oriented at 20°-30° to the bedding planes. The mechanical behavior of these rocks suggested that the rock properties of shear strength and/or coefficient of internal friction can vary with direction, depending on the particular rock tested. Tensile strengths were also measured and found to be lowest when failure occurred along bedding. This work shows that bedded formations exhibit sizable directional variations in both their elastic constants and yield strengths. It is suggested that these variations may be accounted for by using the "elastic laminate" model and the "variable coefficients" failure model. INTRODUCTION The nature of rock deformation at elevated pressures has been studied by many workers; papers by Handin,l,2 and Robinson3 illustrate the present state of knowledge. Most investigators have either chosen rocks which were as isotropic as possible (in order to avoid complications of data interpretation and analysis) or they have oriented their samples so that the effects of anisotropies (such as bedding planes) have been avoided. Of the studies performed, few were concerned specifically with mechanical anisotrop-ies. Griggs4 has presented limited data for specimens cut parallel, normal, and 45° to the bedding plane; he relates the strength anisotropy observed to the fabric (bedding) anisotropy. His tests were primarily concerned with large deformations (20 per cent strain); thus, no directional values of the elastic constants were reported. Handin has reported the results of similar experiments. Bott5has discussed rock strength anisotropies due to faults, cleavage, or bedding. He was concerned primari ly with determining the shear stress on such planes and did not mention the effect of friction. Jaeger6 later generalized Bott&apos;s work by taking friction into account and presented a limited theory for the failure of rocks having a "single plane of weakness", and also for rocks having a constant coefficient of friction, and a shear strength which varies with bedding plane orientation. Donath and Cohen,7 and Donathg have evaluated rock strengths from shale and slate specimens cut normal and parallel to bedding. A dependence of cohesive strength (7,) on the specimen orientation was also shown. Adler9 has also studied this problem and lists similar results. He assumes that all bedded rocks behave according to Jaeger&apos;s single-plane-of-weakness theory. Kalinin and Belorussov10 list results for strength tests parallel and perpendicular to bedding and use this information as a basis for hole deviation analysis. From the literature it is apparent that sedimentary rocks have been tested under widely varying conditions of stress; however, the assumption of isotropy is generally, but not always, made. Since geologic sedimentation often deposits sediments in very definite lavers.,. it seems that more svstema tic attention should be given to the possible effects of this natural bedding. Bedding, as used here, refers to visible regularities of grain size or orientation resulting from depositional processes. The mechanical strength anisotropies, which bedding may present, is not the only anisotronic property to be considered, since the elastic

Jan 1, 1966

By H. H. Rachford

Numerical solutions of immiscible flow problems in which dispersive effects of capillarity are dominated by convection require excessively fine grid spacing with attendant high computing costs. The use of coarser spacing reduces cost but often produces oscillation or undue dispersion associated with displacement fronts, A numerical formulation is proposed here which should be applicable to two-dimensional flow problems. it is in part analogous to an approach previously tested for miscible systems. The convective transport is approximated using a change of variables to yield a coordinate system moving approximately with the local characteristic velocity. The capillarity-induced dispersive terms in the differential system describing the process are approximated with respect to a fixed coordinate system by the usual implicit formulation. One-dimensional tests of the procedure yielded results in which the saturation profiles tended smoothly to the zero-capillary pressure solution as the ratio of viscous to capillary forces was successively increased in a sequence of calculations. This contrasted favorably with solutions by other numerical procedures which would require attendant grid refinements to approach the zero capillary pressure results. INTRODUCTION Numerical solution of displacement problems has until recently relied on applying methods developed primarily for transient heat-flow problems. Such problems are classified as parabolic in type, and where the heat transport is purely by diffusion their solutions are characterized by a high degree of smoothness. It is not surprising, therefore, that for approximating these solutions available finite difference methods are quite adequate. In flow problems the transport is partly by diffusion, partly by convection or flow. Although the problem remains of parabolic type because the dispersive effects of capillary forces or diffusion play some role in every displacement, at high flow rates the problem is dominated by convection, and solutions tend toward those of equations of the hyperbolic type. Solutions of hyperbolic problems are characterized by the translation of fronts, or discontinuities, that may progressively increase in sharpness. Numerical methods for treating parabolic problems become less and less satisfactory as displacement rates increase and the role of dispersion due to concentration or capillary pressure gradients becomes small relative to transport due to flow. In computation the difficulty manifests itself as an error associated with the grid size chosen. 1-6 In summary, if the heat-flow type approximations are to include the terms arising due to convection, one of several choices may be made: (1) an upstream (to the direction of flow) approximation for the convection terms may be used; (2) a centered-in-distance (CID) approximation may be used; or (3) a recently developed approximation based on the theory of oscillation matrices may be chosen.6 The last appears to have significant promise for one-dimensional flow problems; its extendibility to two or three dimensions is an open question. In either of the first two approaches, a suitably small ratio of v&/D must be maintained, where v is the velocity, & is the grid spacing and D the effective dispersivity in the direction of flow. In the first choice, the approximation of the convective part is only first-order correct and errors introduced appear as a numerically induced dispersivity of magnitude proportional to v?x. In the CID choice, the approximation can be second-order correct, but the difference formulation fails to satisfy the maximum principle unless a condition on v?x/D is met. Practically, this means that for high flow rates oscillatory solutions may result in the neighborhood of a front unless exceedingly small grid intervals are taken. While the procedure proposed by Stone and Brian4 permits a less severe limitation to be placed on this ratio, ultimately the flow rates increase relative to the dispersivity the oscillation obtains. Further, extensions of their approach to higher dimensional systems may be attended by considerable

Jan 1, 1967

By Charles R. Stewart, William W. Owens

Reservoir performance predictions based on laboratory core test data assume that fluid flow is laminar for the laboratory test. A study has been made to determine the validity of this assumption for laboratory tests on various types of porosity found in producing limestone formation. Data are presented which show that turbulence and slippage can occur during laboratory tests on hetero-geneous-type porosity limestones, thus causing serious errors in measured single-phase permeabilities and two-phase relative permeability characteristics. In single-phase flow tests it is possible to eliminate turbulence and correct for slippage or to eliminate both factors by controlling test conditions. It is not always possible to control test conditions and thereby eliminate turbulence and slippage in two-phase .flow tests. A correction method is presented which can be used to calculate the true two-phase laminar flow relative permeability characteristic even though furbulence and slippage exist. .INTRODUCTION It is customary to make use of Darcy&apos;s law and modifications of this law, together with laboratory data on formation core samples to predict the performance of producing reservoirs. Such predictions are based on an assumption that fluid flow is in the laminar or streamline region for the laboratory test. It was the purpose of this inves- tigation to determine the extent to which turbulent flow may occur in laboratory fluid flow tests on hetero-geneous porosity limestones. Considering that turbulent flow conditions might exist in some laboratory fluid flow tests, additional emphasis was placed on the development of a method to correct for turbulence when laminar flow conditions could not be attained. FLUID FLOW CONCEPTS FOR POROUS MEDIA The Influence of Pore Geometry on Fluid Flow One of the more important factors influencing fluid flow in porous media is the geometry of the pore space which includes such characteristics of the pores as size, shape, distribution, roughness, uniformity, etc. In general, oil- and gas-producing formations can be divided into two broad types on the basis of pore geometry. One has been called sandstone-type porosity media, which is characterized by a small range in pore size, uniformity in shape of the pores, smooth pore surfaces and a regular and uniform distribution of pores. The other type has been called heterogeneous porosity media and is usually limited to the dolomites and limestones. This type is characterized by a wide variation in the size, shape, and distribution of the pores and rough, irregular pore surfaces. It is therefore apparent that conditions are much more favorable for turbulent flow* in heterogeneous-type porosity media than in sandstone-type porosity media. Interrelationship Between Turbulence and Gas Slippage In studying the problem of turbulent flow in laboratory tests on porous media, it is necessary to be aware of the interrelationship between slippage and turbulence for gas flow. As a result of slippage or the Klinken-berg effecta, apparent perrneabilities to gas are greater than the true value because there is no stationary layer of gas in contact with the walls of the flow channels. Gas slippage decreases as the mean free path of the gas molecules decreases. Since the mean free path of any gas decreases with increasing density, increases in static pressure result in lower apparent gas permeabilities. However, a reduction in gas permeability can also be due to turbulence. Therefore, in studying only turbulent flow in porous media, it is necessary to hold gas density, and slippage, constant or to reduce slippage to a negligible value by operating at high static pressures. Presentation of Laminar and Turbulent Flow Data A graphical relationship between permeability and a pseudo-Reynolds number, N,, will be used to show the two types of fluid flow, i.e., laminar and turbulent. The usual graphical method for such a description has been the use of friction factor-Reynolds number charts4. On such a logarithmic diagram, the laminar region appears as a straight line having a slope of 45 degrees. As the friction factor decreases and the Reynolds number increases, the turbulent region is reached and appears as a deviation from the 45-degree slope line. However, in petroleum engineering literature resistance of por-

By J. C. Shyne, T. D. Gulden

The steady-state creep behavior, in compression, of indium containing a dispersion of atomized glass particles was studzed over a range of temperature, stress, and composition. The observed behavior was not consistent with a simple model of dislocation climb over dispersed particles. From room temperature to about 100°C the temperature dependence could be described by an activation energy about one and a half times that of self-dijfusion in indium while at higher temperatures a much higher ternperature dependence was observed. THE value of dispersion-hardened and precipitation-hardened alloys has long been recognized. The mechanism of hardening, however, is not completely understood, especially at elevated temperatures where dispersed particles increase creep resistance. Following the suggestion of Schoeck1 that the rate of creep in dispersion-hardened materials is controlled by dislocation climb over second-phase particles, Ansel and weertman2 derived steady-state creep equations for dispersion-hardened materials. Their derivation was based on dislocation climb over non-deformable spherical particles in a metallic matrix. For low stresses they predicted a linear relation between applied stress and steady-state creep rate. For stresses large enough to cause dislocations to pinch off around the precipitate particles, the theoretical creep rate was shown to be proportional to the fourth power of stress. For very high stresses an exponential stress dependence was predicted. In all cases the theoretical creep rate is linearly proportional to the diffusion coefficient. This provided the primary temperature-dependent factor. Subsequently, Ansel and Lenel3 reported some experimental data on SAP, a composite of aluminum and aluminum oxide, that was in good qualitative agreement with the above theories. Their experimental creep rates, however, were four orders of magnitude less than the theory predicts. This was attributed to a low density of active dislocation sources in the material; somehow the dispersed particles inactivated the dislocation sources instead of merely interferring with dislocation motion. Meyers and sherby4 have recently reported on the creep behavior of SAP. Their results were inconsistent with a simple dispersion-strengthening model and led them to conclude that a continuous network of oxide controls the high-temperature mechanical characteristics of SAP. Because of the rather complex mechanical behavior of SAP and the lack of agreement about the morphology of its oxide phase, it appeared desirable to study the creep of a dispersion-strengthened material of controlled morphology. For the present work a dispersion of spherical glass particles imbedded in a matrix of indium was chosen. The microstructure of this material was similar to the idealized uniform dispersion of hard spheres in a deforming matrix assumed by Ansel and Weertman in their theoretical treatment. SPECIMEN PREPARATION For experimental convenience, it was decided to use a low melting-point metal in this investigation. It was necessary that the metal not be oxidized readily up to its melting temperature and that it wet the second-phase particles. It was desired that the material chosen for the dispersed phase be available as a fine powder with closely controlled size and shape. The material chosen to satisfy these requirements consisted of a matrix of indium metal containing a dispersion of atomized particles of a soft soda-lime-silica glass ranging from 5 to 30 µ in diam. The glass particles had a highly regular spherical shape. The composite was made by stirring the glass powder into liquid indium maintained at a temperature just above the melting point of indium, 156°C. The creep specimens were prepared by a hot pressing operation. Five to ten g of the mixture were placed in a steel die and pressed at 125°C for 5 min under a pressure of 30,000 psi. The resulting specimens were cylinders about 3/8 in. in diam and ranged from 0.4 to 0.7 in. high. By following this procedure a uniform dispersion was obtained in the creep specimens. Typical microstructures of specimens containing twenty and forty volume percent glass powder are shown in Figs. 1 and 2. Unfortunately, grain size could not be determined by the metallo-graphic techniques used. CREEP TESTING PROCEDURE Creep tests were performed in compression under constant stress. The specimens were placed in

Jan 1, 1963

By O. D. Gonzalez, R. A. Oriani

The migration of hydrogen and of deuterium dissolved in a iron and in nickel induced by an applied electrical potential has been measured over a range of temperature. In all cases the intevstitial solute migrates to the cathode. The deduced charges of transport are positive, independent of or slightly dependent on temperature, and markedly larger for the beavier mass. These results are interpreted as showing predominating momenlum transfer between electron holes and the acticated complex of the elementary diffusion act. THE migration of components of a metallic system under the action of an applied electrical potential is called electromigration or electratransport. This phenomenon has been known since 1861 and is becoming of increasing interest to metal scientists because of the opportunity it offers to explore the connections between mass transport and electron transport. Until recently, reliable data have been very scarce because of the difficulties inherent in the measurements. In addition, theoretical understanding has been either nonexistent or rudimentary and at present this situation is only moderately improved. Whereas the prevalent notion had been (and is still widely held) that electromigration directly reflects the state of ionization of a component in its equilibrium state in the metal, it is now becoming clear that the major, if not the sole, factor is the momentum transfer between the charge carriers, themselves in motion because of the applied field, and the metallic components in their activated state. It becomes important therefore to obtain reliable data over a range of temperature on electrornigration in systems where only one component moves, that is, either in pure metals (vacancy motion) or in interstitial solid solutions. Our choice of systems was dictated by our measurements&apos; of the Soret effect in solid solutions of hydrogen and deuterium in iron and nickel. Indeed, the main reason for making measurements on electromigration at all is that we suspected a mechanistic connection between thermal diffusion and electromigration. In our former paper&apos; we proposed that what is common to both phenomena is the momentum transfer between the charge carriers in the metal and the atom as it jumps from one equilibrium position to another. In Soret diffusion, however, there is in addition the dissipation of the energy of the jumping atom via pho-nons. We have since found that similar ideas for this interconnection have been developed by Fiks.2 In the present paper we will discuss in more detail our understanding of the mechanism of electromigration itself, but we turn first to a presentation of the experiments and results. EXPERIMENTS AND RESULTS Let us consider a metallic system in which only one component, m , can diffuse at the temperature in question, and to which are applied simultaneously an electrical potential gradient, grad cp, and a chemical potential gradient, grad p,. One may write3 for the material flux Jm and the electronic charge flux Je by applying the condition, grad = 0, to Eq. [I] and evaluating grad(um/T) for the isothermal case and for activity coefficient independent of composition over the relevant composition range. ere, c is the concentration and D the diffusivity of the mobile component. The experimental technique that we have used to measure Z* for the isotopes of hydrogen individually dissolved in each of two transition metals has been developed from a method used by Wagner and Heller.4 (A somewhat similar method was also used by Herold.5) A wire or thin rod of the metal under study connects two volumes of gaseous hydrogen at identical pressures; hence grad um = 0. A direct current is made to pass through the specimen, serving to maintain the specimen at the desired temperature and also to produce the electromigration of the hydrogen dissolved in the metal in equilibrium with the gaseous hydrogen. Under these conditions, the flux of hydrogen from one volume to the other is given, from Eq. [I] for grad( um/T) =0by

Jan 1, 1968

By E. H. Lee, F. J. Fayers

Problems in reservoir analysis can usually he cupressed in terms of a system of nonlinear partial difjerential equations. A rnethod for .setting physically reasonable boundary conditions for these systerrls by ayplication of the theory of characteristics is discussed. A short description i.c. given of the technique for finding chnmcteristics of general first order and second orrler equations. Specific examples are quoted jrorn eqrrations occurring in theoretical reservoir studies. The princip1e. ossocicred wit11 the use of chnracteristic to set physically realistic boundary conditions me developed, and thr method is discussed in terms of boundary condition for the noncupillary, compressible, two-phase flow problen: jor the cupillary, ir~cowlpressible, two-phase flow problems for the tlrtergent flooding prohlenl; as for the compressible. noncapillary, two-phase flow pro11letn. INTRODUCTION In most problcms in reservoir analysis the physical situation is described by a system of nonlinear partial differential equations. The general solution to the set of equations usually comprises a wide range of functions ancl a particular solution is normally selectetl by applying suitable boundary conditions to the problem. In some cases it is clear from the physical situation what form the boundary conditions should take, but it is usually desirable to have an independent mathematical check of the suitability of the chosen conditions. From the mathematical standpoint, a system of partial differential equations with boundary conditions is said to be properly set if the combination determines a unique solution which is continuously dependent on the equation and the boundary conditions. Such a solution is sometimes termed "reasonable". In investigating the properties of the mathematical representation, it is useful to determine the structure of the system by the application of the theory of characteristics. This theory determines in the space of the independent variables, lines or surfaces which play a special role in thc system&apos;s behavior. The characteristics determine the possibility and nature of the propagation of discontinuities in the derivatives of the dependent variables. Along the characteristics certain differential relations hold which for some problems allow a so- lution to be obtained analytically. In the case of hyperbolic equations the characteristics can bc used as 3 basis for a numerical solution. If a complicated nonlinear system is to be solved, a study of behavior of the characteristics will indicate what boundary conditions should be chosen. By using the method of characteristics, several advances have rccently been made in solving reservoir engineering problems. Sheldon, Zondek and Cardwell&apos; have discussed how the the method may be used grap-phically to solve the incompressible two-phase displaccmerit problem when the gravity term is included in the equations. Fayers and Perrine&apos; have applied the method to study the equations representing detergent flooding. When compressibility and solution-gas effects are introduced into the two-phase flow problem, the equations and their characteristics become more complicated. It will be seen that in this problem the characteristics are useful for indicating a reasonable set of boundary conditions but cannot readily be used to compute a numerical solution. This problem has been solved on the IBM 701 using normal finite difference methods hy West, Garvin and Sheldon." The material to be prcscntcd in this report is divided into three sections. The first discusses the determination of the characteristics of first and second order partial differential equations; the second describes the selection of suitable boundary conditions by the application of the principle that the future cannot affect the past; the third discusses the use of characteristics for setting the boundary conditions of the two-phase solution-gas flow problem. A useful introduction to the method of characteristics has been given by Hildebrand.&apos; A summary of the method will be given in this section of the paper. The First Order Equation The quasi-linear equation of first order is It is linear in the partial derivations of z. The properly linear equation has the form The characteristics of are curves along which the

By Richard H. Jahns

Terminology and Basic Specifications Minerals and closely allied natural substances that are used for personal adornment, as raw stock for the fashioning of ornamental objects, or for other decorative purposes can be referred to collectively as gem materials. Industrial gem materials are those employed directly in various areas of industry. Substances that are manufactured for gem or industrial uses are termed synthetic gem materials; most of these correspond in composition and crystal structure to certain minerals found in nature. Imitation materials are those whose use is based upon their resemblance to substances of different composition and greater value. Gem stone is a designation commonly restricted to those materials suitable for personal adornment, and the complementary term ornamental stone is applied to the remaining materials not used industrially. This distinction is arbitrary and somewhat inconstant, owing to numerous geographic differences and historical shifts in end uses. Further, there is a complete gradation between the category of ornamental stones and that of dimension stone. Although many building and monumental stones are undeniably ornamental in their uses, treatment of such material is beyond the scope of this chapter. Substances that lose their identity in use are excluded from the gem category. Thus a transparent crystal of beryl would not be classed as a gem stone if it were treated as a source of beryllium metal. The adjectives precious and semiprecious have been employed as a means of distinguishing different gem materials on the basis of their recognized value. Diamond, emerald, ruby, and sapphire consistently have been regarded as precious stones, and in cut form they commonly are referred to as the noble gems. Opal, pearl, turquois, and chrysoberyl also have been classed as precious at various times. In recent years, however, it has been increasingly recognized that no satisfactory distinction can be based upon assessed value or any group of intrinsic properties, and at the present time all gem stones normally are referred to as precious. Indeed, the tendency is to extend this term to include many of the so-called ornamental stones, and the designation semiprecious has been essentially discontinued. Unlike most mineral commodities of economic value, gem materials are little altered in processing for end uses. Some stones are used in essentially their natural form, and most are reshaped and polished without significant changes in their fundamental properties. Thus the preparation of gem and ornamental stones is intended chiefly to emphasize or enhance desirable characteristics that are present initially. The worth of most synthetic gem stones is founded upon the identity between their major properties and the natural characteristics of the corresponding minerals, just as the pleasing qualities of imitation stones depend upon how well certain of their properties simulate those of more valuable substances. The most important specifications of gem materials represent four basic factors: beauty, durability, rarity, and portability. Beauty, whether expressed as splendor, "purity," attractiveness, or in some other way, is mainly a function of personal response to optical properties such as color, transparency, indices of refraction, and dispersion. Freedom from visible imperfections is important for transparent stones. For some substances beauty depends also upon asterism, opalescence, textural and

Jan 1, 1960

By G. Purcell, S. C. Sun

In an attempt to determine the relative collecting ability of 18-carbon fatty acids, studies were performed on rutile in aqueous solutions of the fatty acid soaps. The preceding article reported the electro-kinetic properties; this article outlines the flotation behavior. Results of these flotation experiments are described and correlated with the electrokinetic results. Recent studies of the electrokinetic1-4 and the electrochemical5,6 properties of a few solid-liquid interfaces have provided a better understanding of the flotation process. A number of investigators have attempted to determine the relative collecting ability of 18-carbon fatty acids, but their published results (even for the same mineral) vary widely. Such lack of agreement indicated a need for further investigation. In an effort, therefore, to resolve the problem, electro-kinetic studies were made on rutile in aqueous solutions of the fatty acid soaps. Following the determination of the electrokinetic properties, the flotation behavior of rutile was studied. Results of these flotation experiments are described in this paper and are correlated with the electrokinetic results. MATERIALS AND EXPERIMENTAL PROCEDURE Materials: Pure -28, + 35-mesh rutile crystals were used for all flotation tests, and nitrogen-saturated conductivity water was used throughout. Sodium hydroxide and hydrochloric acid were added for pH regulation. The methods of purification for these materials and the preparation of the soaps have been described elsewhere. 7 Equipment: A Hallimond cell, constructed in a manner similar to that described by Fuerstenau, Metzger and Seele,8 was used for the flotation of mineral particles; however, a coarse pyrex frit was substituted for the capillary. Procedure: Purified nitrogen was led through a two- way stopcock into a graduated cylinder partly filled with water. The gas forced water from the cylinder back into a header tank until the water level in the cylinder coincided with a predetermined mark. Reversal of the stopcock then allowed water in the tank to force nitrogen into the cell until the water in the graduated cylinder rose to a second predetermined mark. In this way a constant volume of nitrogen under practically constant pressure was used for each flotation test. However, the flotation time varied, depending upon the amount of mineral floated and the pH of the solution. Rutile crystals were transferred under water from a storage bottle by a glass scoop holding 1.7 g. Consistency of the scoop samples was good, varying from a low value of 1.67 to a high of 1.74 g. The water added to the conditioning bottle during mineral transfer was drained off, and then 2 cu cm of either hydrochloric acid or sodium hydroxide were added. The concentration of pH regulator was adjusted to give approximately the final soap solution pH that was required. The order of addition of reagents did not affect the results within the limits of experimental error. Soap solution of the required concentration was added under nitrogen pressure to fill the bottles completely. Conditioning for 1 hr was carried out on rolls rotating at 35 rpm. After conditioning, 2.5 cu cm of solution were run off into the container of a Beckman (model G) glass electrode pH meter. The electrodes were rinsed in this solution, which was then discarded, and 2.5 cu cm more were added for pH determination. The remaining solution and the mineral were transferred to the cell; care was taken to ensure that all mineral was washed out of the bottle by the conditioning solution. With the cell back in its flotation position the liquid just reached the desired level, and the test was started immediately. The amount of agitation in the cell was kept constant, and flotation was carried out without the use of a frother. In each series of ten tests one was run with no collector added to ensure that the mineral had not become contaminated, and two were conditioned at the same pH to check reproducibility. After each test the cell was dismounted, soaked in cleaning solution, washed in hot water and then rinsed with distilled water.

Jan 1, 1963

By B. C. Allen

The interface energies of chronzium, molybdenunz. hugsten, and their solid-solution alloys Cv-35Re, MO-33Re, and UJ-25Re were studied at 0.6 to 1.0 of the absolllte liquidus ter)zpe,vature using fiz&apos;e )izethods. Liquid surface tension, yv , was deter mined clsing the pendant-drop and drop-weight methods. Results are, respectizlely, 1700, 2370, and 2480 +100 dynes per ct for the rhernium -containing alloys and essentially the same as tlwse reported for liquid chro)riln, trolybdenum, and tungsten. Average solid slrjace energy, rsv< xias ))zeasured using tlre fiber-extetlsion method. The ratio of ysS, the acerage high-angle grain-boundary energy, to ySV cclas jolnd fronz grain-bolzdary grooue angles fort)zed at the surface in an inert atrfizosphere. Absolllte iute?:face energies were deterawined using ?nultip/rase equilibria involzing suitable liquids of known surface tension (tin, silver). Interpretation of the experimented results in view of pvobable tenzperatzcre, orientation, and purity effects giz,e the follouling approximations in ergs per sq ctn: ysv (i2lo. Mo-33Re) - 2100, ySS (Mo, Mo-33Re) - 800, rr (defornzation twins in MO-33Re at 1200"C) - 800. ysV (Cr. Cr-35Re) - 2400. YSS (CY, Cr-35Re) - 1000. Probably Ylv- YSV- 2500 for tungsten and W-25Re, giving yss (It&apos;, W-25Re) - 900. The interface energies of solid and liqid ch?&apos;omiu?.z, nolybdenu?rr, and tungsten are not geatly aff&apos;ected by rhenium and therefore are not a ttlajor factor in the ductili zing rhenium effect in Croup VI-A metals. THE interface energies of the refractory Group VI-A metals, chromium, molybdenum, and tungsten, are not well-established. The objective of this investigation was to study the liquid surface tension, solid surface energy, and grain-boundary energy of these metals and compare them to those found for the bcc solid-solution alloys, Cr-35e,&apos; 0-33e,&apos; and -25e. Five techniques were used to measure interface energies in high-purity polycrystal rod, wire, and sheet at 0.6 to 1.0 of the absolute liquidus temperature. The alloys were chosen to see if there was any connection between interface-energy behavior and the ductilizing rhenium effecL4j5 EXPERIMENTAL WORK Materials. A description of the materials used is presented in Table I. Chromium rod was prepared by arc melting iodide process crystals supplied by Chromallo Cor., hot extruding, and warm swaging to 0.63-cm-diam rod.6 The sheet was prepared by rolling as-extruded rod to 95 pct reduction in area from a hydrogen furnace at 800" to 900°C and surface grinding off 0.02 cm from each face. Cr-35Re rod was prepared by arc melting sintered rhenium powder and iodide chromium crystals, warm rod rolling to 50 pct reduction in area in cans, and swaging to 60 pct reduction in area at 1100" to 1200°C. Some of the rod was warm-rolled to sheet and then surface-ground. Portions of swaged chromium and Cr-35Re were further reduced by swaging and drawing to 0.013-cm-diam wire by the General Electric Co. Mo-33Re and W-25Re rod, sheet, and wire were provided by Chase Brass and Copper Co. The molybdenum sheet consisted of two lots, both essentially the same except for the carbon content. Liquid Surface Tension. The liquid surface tension of Cr-35Re, Mo-33Re, and W-25Re was measured by a combination of pendant-drop and drop-weight methods using techniques already decribed." Following out-gassing, molten drops were formed on the ends of centerless-ground Mo-33Re and W-25Re rods by electron bombardment at 5 x 106 mm. Similar drops were formed on outgassed Cr-35Re rods by induction heating under 1 atm of 99.995 pct Ar. Solid Surface Energy. Solid surface energy was measured by conducting microcreep experiments on molybdenum, Mo-33Re, chromium, and Cr-35Re wires at 2350°, 2306, 1550°, and 180O°C, respectively. In preparation, gage marks -2.5 cm apart and -0.001 cm deep were circumferentially scribed on the wire with a razor blade. Weights of the wire material were then attached. Five to seven reasonably straight wires were hung in a container made out of the wire material. The free end was placed through a small hole in the removable top and secured by bending a small portion 90 deg. The containers not only tended to provide vapor-solid equilibrium for the wires but also protected them from gaseous impurities. They were nominally 2.5 cm in diam by 5 cm high and were made from extruded chromium rod, Cr-35Re arc casting, molybdenum bar stock, or welded Mo-33Re sheet. After deg re as ing, the assembly was outgassed at a relatively low temperature to 2 x 10"5 mm and then recrystallized 2 to 8 hr at the creep temperature in a rhenium-element resistance furnace. The static argon atmosphere was gettered by tantalum radiation shielding. Specimen temperature was measured optically to 25"C using calibration with known melting points and blackbody conditions. The wires generally developed a stable bamboo-type structure according to Fig, l(b), (c), and (d) and retained their gage marks [upper portion of Fig. l(d)]. One or two of the weights were clipped off to provide a low load for the creep anneal. To minimize the possibility of bending or breakage, the wires remained attached to the top of the annealing container which was held to keep the wires vertical. The distance between gage marks was

Jan 1, 1967

By R. E. Ogilvie, N. L. Peterson

Diffi-lsion measurements were conducted at all compositims in the bcc solid solution of the U-Nb system employing incremental couples at composition intemals of 10 at. pct. Diffusion coefficients were determined by the Matano method from concentration gradients obtained with the electron-probe microanalyzer. The activation energy for inter-diffi-lsion as a function of compositim shows three distinct regions: 1) 80 to 100 pct U.6= 30 kcal per mole; 2) 20 to 80 pct U, $ - 70 kcal per mole; 3) Oto 20 pet U, Q = i40 kcal per mole. The frequency factor, fi0 and the activation energy $ were found to be roughly related by the following equation: log Do ^9.7 X IO-5Q -6,6. The Kirkendall marker movement indicates that DU is larger than DNb between 16 and 100 pct U and DNb is larger than DU from 0 to 4 pct U. FOR practical as well as fundamental reasons, the rates of diffusion in alloys are of considerable consequence. Most solid-state reactions are largely dependent upon the diffusion of atoms through the lattice structure and along grain boundaries. The high-temperature strength and reasonable nuclear properties of niobium have prompted its use as a reactor material in contact with uranium fuel. Hence, diffusion data for the U-Nb system are of considerable importance. In the previous diffusion study1 on the U-Nb system using pure element couples, reliable data were obtained only in the range of 0 to 10 at. pct Nb due to the large variance of the diffusion coefficient with composition. Also, a large Kirkendall effect and considerable porosity in the uranium-rich areas of the specimen were reported, which suggests that the true diffusion coefficients are somewhat larger. The purpose of the present study was to obtain reliable diffusion coefficients at all compositions using incremental diffusion couples with intervals of 10 at. pct. In view of the abnormal self-diffusion be- havior of y uranium2-4 and some other bcc transition elements,&apos;-&apos; it was felt that a comparison of the interdiffusion coefficients in the bcc U-Nb system with those of Reynolds et al.9 for the fcc gold-nickel system might shed some light on the diffusion mechanism involved. Both systems have similar phase diagrams, in that complete solid solubility exists above a miscibility gap. EXPERIMENTAL PROCEDURE The uranium used in this investigation was obtained through the courtesy of Argonne National Laboratory. An analysis of this material detected only Si-30, A1-7, C-6, N < 10 and 0-18 ppm. The niobium was electron-beam melted material obtained from Stauffer-Temescal. The gaseous impurities were less than 50 ppm, and the spec troc hemical analysis showed Ta-500 and W-200 ppm. U-Nb alloys were prepared at composition intervals of 10 at. pct by melting the appropriate amounts of the pure elements in an arc furnace. The buttons were inverted and remelted 6 times to assure complete mixing. The buttons were then wrapped in molybdenum foil, canned in Zircaloy-2 or stainless steel, and hot rolled 30 pct reduction in thickness at temperatures between 850" and 1100°C. Alloys with 10, 20, 30, 40, and 90 at. pct Nb rolled quite easily under these conditions, but the 50, 60, 70, and 80 pct alloys remained brittle. After melting and rolling (when possible), the alloys were annealed for 24 hr at a temperature within 100°C of their melting point in a dynamic vacuum of better than 4 x 10-8 mm Hg. These treatments produced alloys which were homogeneous on a 1 p scale within the detectability limits of the electron probe. During fabrication, the alloys picked up as much as 100 ppm Mo and 100 ppm Zr. Other elements checked for but not found were Co, Cr, Fe, Mn, Ni, and Ti. The grain size of the annealed samples ranged from 3 mm for the uranium-rich alloys to 0.3 mm for the niobium-rich alloys. This permitted measurements of the concentration gradients in the diffusion samples without crossing more than one or two grains, thereby eliminating any grain boundary effects. The specimens were bonded by theU&apos;picture frame" technique as reported by Kittel.10 Specimens of composition b)U + (100 - x)Nb were sandwiched between two specimens of composition (x + 10)U + (90 - x)Nb after they were ground flat and parallel

Jan 1, 1963