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By Robert L. Hoss

The application of Muskat's differential equations for predicting the performance of a solution gas-drive reservoir to the Fullerton field indicates that a recovery may be expected of 14,030 bbl per acre under primary depletion, and 15,910 and I7,190 bbl per acre, respectively, under the two pressure-maintenance programs analyzed. In this analysis it is assumed that fluid properties are the same throughout the reservoir, that the core data used are typical of the Clear Fork limestone in the locality, and that injected gas is completely diffused through the pay zones. The economic study of the three recovery programs considered indicates that the one in which 60 pct of the produced gas is returned to the formation is the most profitable. Introduction The purpose of this paper is to consider the effect of pressure maintenance by gas injection on the ultimate recovery of oil from a solution gas-drive reservoir. This prediction of reservoir behavior is the result of calculations based on the equations presented by M. Muskat in the Journal of Applied Physics, March 1945. Reservoir The reservoir analyzed is the Fullerton field in Andrews County, West Texas. It is situated on a broad anticlinal structure with approximately 330 ft of closure, and extends approximately eight miles north and south and five miles east and west. The areal extent is taken as 16,642 acres. The producing horizon is the Lower Clear Fork limestone of Permian age, occurring at depths from 6650 to 7350 ft. It is divided into four zones. For the purposes of this study, the fluid characteristics in all zones are assumed to be the same. Therefore an equivalent zone is assumed of thickness equal to the sum of the thicknesses of the four zones, or an average of 216 ft. The reservoir contains 3,586,991 acre feet of producing limestone, with an initial volume of stock tank oil in place of 918,-403,100 bbl. The average connate water saturation is 24.0 pct. The permeability ranges from 6.7 md. to 12.6 md., and the porosity ranges from 7.7 pct to 11.4 pct, except for occasional cavernous streaks. Based on these values, the limestone may be considered intergranu1ar.l Production histories and material-balance studies have indicated that the driving mechanism is solution gas. Water levels have been established. Theory It is possible to express mathematically the change in residual oil saturation with pressure decline in a solution gas-drive reservoir as being equal to the change in the physical properties of the fluids present in the reservoir as this pressure decline occurs. Essentially these equations, in differential form as presented by M. Muskat,2 express quantitatively the instantaneous net effect

Jan 1, 1948

By Robert L. Hoss

The application of Muskat's differential equations for predicting the performance of a solution gas-drive reservoir to the Fullerton field indicates that a recovery may be expected of 14,030 bbl per acre under primary depletion, and 15,910 and I7,190 bbl per acre, respectively, under the two pressure-maintenance programs analyzed. In this analysis it is assumed that fluid properties are the same throughout the reservoir, that the core data used are typical of the Clear Fork limestone in the locality, and that injected gas is completely diffused through the pay zones. The economic study of the three recovery programs considered indicates that the one in which 60 pct of the produced gas is returned to the formation is the most profitable. Introduction The purpose of this paper is to consider the effect of pressure maintenance by gas injection on the ultimate recovery of oil from a solution gas-drive reservoir. This prediction of reservoir behavior is the result of calculations based on the equations presented by M. Muskat in the Journal of Applied Physics, March 1945. Reservoir The reservoir analyzed is the Fullerton field in Andrews County, West Texas. It is situated on a broad anticlinal structure with approximately 330 ft of closure, and extends approximately eight miles north and south and five miles east and west. The areal extent is taken as 16,642 acres. The producing horizon is the Lower Clear Fork limestone of Permian age, occurring at depths from 6650 to 7350 ft. It is divided into four zones. For the purposes of this study, the fluid characteristics in all zones are assumed to be the same. Therefore an equivalent zone is assumed of thickness equal to the sum of the thicknesses of the four zones, or an average of 216 ft. The reservoir contains 3,586,991 acre feet of producing limestone, with an initial volume of stock tank oil in place of 918,-403,100 bbl. The average connate water saturation is 24.0 pct. The permeability ranges from 6.7 md. to 12.6 md., and the porosity ranges from 7.7 pct to 11.4 pct, except for occasional cavernous streaks. Based on these values, the limestone may be considered intergranu1ar.l Production histories and material-balance studies have indicated that the driving mechanism is solution gas. Water levels have been established. Theory It is possible to express mathematically the change in residual oil saturation with pressure decline in a solution gas-drive reservoir as being equal to the change in the physical properties of the fluids present in the reservoir as this pressure decline occurs. Essentially these equations, in differential form as presented by M. Muskat,2 express quantitatively the instantaneous net effect

Jan 1, 1948

By P. E. Baker

The commercial neutron-gamma log measures the intensity of gamma-rays produced by the interaction of neutrons with nuclei in the formation and in the bore hole. This measurement gives information concerning the distribution of neutrons around the neutron source, and in particular gives information on porosity or fluid content. Another type of information is contained in the frequency or energy spectrum of the neutron capture gamma-rays. The capture gamma-rays of each element, in fact of each isotope, have only those energies characteristic of that isotope.'" The neutron capture gamma-ray spectrum, if obtainable, would permit the identification of the elements producing the spectrum. Such information would have considerable value in oilfield exploration work if it could be applied practically in well logging. Experiments have been carried out at the La Habra Laboratory of California Research Cop. to determine the feasibility of a method of well logging in which neutron capture gamma-ray spectra are measured down hole. In these experiments, spectra were obtained for artificial formations under simulated borehole conditions, using actual borehole geometry. This paper will describe the experimental apparatus and methods and will present the spectrum curves for several formations. THEORY Basic Formulas The neutron capture cross section of an element is a measure of its tendency to capture neutrons. The macroscopic cross section, or cross section per cubic centimeter, of a material composed of several elements is the sum of the cross sections of the nuclei making up 1 cc of the material. This is expressed in the equation SB = Sn1s2, ..........(1) where SB is the macroscopic cross section, s, is the cross section of a nucleus of type i, and ni is the number of nuclei of type i/cu cm. If N is the number of neutrons absorbed per second by the composite material, then the number captured by element j is Ni = N nisi/SB = N nisi/Si ni si ......(2) When the nucleus captures a neutron, a new nucleus is formed, which is usually in an excited state and falls to its ground or unexcited state by emitting one or more gamma-ray quanta.* The emission spectrum has been measured for many elements (see Table 1). The intensity of gamma-rays produced by an element is proportional to the number of neutrons captured per unit time by the element, which in turn depends on the capture cross section of that element and the capture cross sections of the other elements present which are competing with the first element for thermal neutrons. In a small sample of material the number of neutrons, N, captured per second is N = ?SB, ..........(3) where $ is the thermal neutron flux and SB is the macro-scopic cross section defined above. The number of neutrons captured by element j is N1 = ?ni si = N ni si/SB . ......(4) The number of gamma-ray quanta of energy, E, pro-duced per unit time by element, j, is I(E)=Nni si/SB P(E) .......(5) where P(E) is the emission probability of element, j, for this gamma-ray energy. Borehole Effects The capture gamma-ray spectra of many elements have been measured accurately in academic and government laboratories. These measurements are made on small samples, and from these measurements the emission spectra of the elements are obtained.

Jan 1, 1958

By B. H. Caudle, L. H. Silberberg

Reservoir depletion by natural water drive is typified by the movement of water from an aquifer into the adjacent oil-bearing formation. Prior studies of this type 01 water movement have generally neglected the resistance to flow in the aquifer and its effect on the movement of water into the oil bearing zone. A method for designing and operating scaled models of such reservoir systems is presented. Experimental data on a model of an edge-water-drive reservoir are shown and discussed. INTRODUCTION Most petroleum reservoirs derive at least a part of their productive capacity from water influx. This water may be injected from the surface or it may come from an aquifer adjacent to the oil zone. In either case, the reservoir engineer must be able to estimate the advance of the water as a function of either the elapsed time or the fluid produced. This paper describes a type of fluid flow model which includes the effect of viscous and gravitational forces in the reservoir and the surrounding aquifer. Natural water influx can be divided into three general (but widely overlapping) categories according to the direction of flow in the aquifer. These are shown schematically in Fig. 1. Fig. l(a) illustrates an edge-water-drive mechanism. In this case, water advances updip along the stratum but the movement of water is essentially horizontal and very little of the oil is actually underlain by water. The bottom-water drive (Fig. l(b)) is characterized by a thick aquifer underlying the oil zone. The water movement is generally vertical in the aquifer. The third category is illustrated in l(c). Referred to as the "thin oil column", this type of oil accumulation consists of oil over water in a relatively thin stratum. Fluid movement is horizontal in both the oil and water zones, except close to the producing well. One characteristic which all three of these types have in common is that part of the water flow takes place in the water invaded region of the oil reservoir while the remainder of the water flow occurs in the aquifer. The resistance to the flow of water usually will not be the same in these two regions. This occurs because the porous rock normally contains only water in the aquifer, while the water-invaded region always contains microscopic globules of bypassed oil (residual oil) which interfere with the flow of water through the rock. The production history of any type of a water-drive system is a function of two phenomena — where the water goes, and how it displaces the oil in the area invaded by the water. It is usually impractical to study these two phenomena at the same time. In most cases one type of analysis is used to predict the gross movement of water in the reservoir, while a different type of analysis is employed to determine the amount of oil to be recovered from the gross volume contacted by the water. The portion of the oil reservoir which is invaded by water is mainly a function of the resistances to fluid flow in the several parts of the flow system. This gross water influx in the three types of natural water drive typically results in water cusping to the production wells either from the side or from below (water coning). In 1747, Muskat described a greatly simplified mathematical model for predicting the area invaded by water in a bottom-water-drive reservoir.1 This model used the Laplace equation and suitable boundary conditions to describe the isopotentials and streamlines in the flow system. The use of this model assumes that (1) the reservoir rock is homogeneous in nature, (2) the oil and water mobilities are equal, (3) there is no oil flowing in the water invaded region, and (4) external forces — such as gravity — do not affect the flow. This method has been used for both natural water drives and water injection projects. In this, as in most models, the method of images is used to reduce the size of the model necessary to describe the reservoir. The assumption of equal viscosities and permea-

Jan 1, 1966

By W. G. Pfann

The simultaneous segregation of two solutes during the directional solidification of an ingot is treated mathematically on the basis of simplifying assumptions. Expressions are derived for the difference in concentration of two solutes, and for the location and concentration gradient of a pn barrier formed in a semiconductor by the segregation of a donor and an acceptor. THE problem of normal segregation of a single solute during the freezing of an alloy has been treated mathematically by a number of investigators. Gulliver' showed that coring increases the quantity of eutectic above that to be expected at equilibrium, for a system having limited solid solubility and, on the basis of simplifying assumptions, calculated the fraction of eutectic to be expected. Scheuer2 expressed composition of solid solution in terms of a distribution coefficient and the fraction solidified and compared experiment with calculation for the systems A1-Cu, Al-Zn, Cu-Sn, Cu-Zn. Hayes and Chipman,3 in a detailed study of segregation in a low carbon, rimming steel ingot, calculated distribution coefficients for a number of solutes in iron, compared calculated and experimental segregation curves, and discussed the effects of process variables such as rate of solidification and stirring. In the present paper a mathematical analysis is made of the simultaneous segregation of two solutes during the orderly freezing of a solid solution system, with particular emphasis on the difference in solute concentrations. Although the analysis is quite general and can be applied to the segregation of minor elements in alloys, it is directed in particular at the solidification of a semiconductor containing a donor and an acceptor. Normal segregation has unique aspects in a semiconductor, because of the ways in which donors and acceptors affect the electrical properties. The difference between two solute concentrations becomes of importance, as does also the gradient of the difference where the difference goes through zero, this being the concentration gradient of excess carriers at a pn barrier. A primary object of this paper is to extend the mathematical treatment of segregation to include these newer aspects which are of particular significance for semiconductors. One property of interest is the electrical conduc- tivity, which arises from the presence of donors 'or acceptors in solid solution. The conductivity may be either n-type or p-type depending on whether donors or acceptors, respectively, are in atomic excess. For both germanium and silicon, elements of Group V of the Periodic System, such as P, As, and Sb, are donors and elements of Group 111, as B, Al, In, and Ga, are acceptors.4-6 If a donor or acceptor is present alone in solid solution, the conductivity is proportional to its concentration." If both a donor and an acceptor are present, then the conductivity is proportional to the difference in their atomic concentrations. If the donor and acceptor segregate at different rates then a pn barrier in certain circumstances may form at some point in the ingot. Accordingly, equations are derived and illustrated which express the effect of segregation on: 1—the concentration of a single solute with a discussion of the assumptions; 2—the difference between two solute concentrations and means for minimizing its variation in an ingot; and 3—the location and concentration gradient of a pn barrier. The analysis is applicable to processes in which the entire charge is melted and then progressively frozen from one end. The method in which an ingot is solidified in a crucible7 and that in which a solidifying rod is pulled from the melt8 both fall into this category. Segregation of One Solute If a cylinder of molten alloy is caused to freeze slowly from one end, a normal segregation of solutes will usually occur, producing a lengthwise concentration gradient in the ingot. Depending on whether a solute raises or lowers the melting point of the solvent, it will become concentrated in the first or last regions, respectively, to freeze. If it is assumed that freezing is such that there is no diffusion of solute in the solid, complete diffusion in the liquid, and that k, the distribution coefficient, defined as the ratio of solute concentration in the just-freezing solid to that in the liquid, is constant, then the

Jan 1, 1953

By Fred Bond

THE process of comminution by grinding is properly classified as an art, rather than as a science. Like most other operations concerned in ore dressing, or in the treat-ment of nonmetallic minerals, the mathe-matical relationships of rock breaking are still obscure, and much disputed. If some basic formula exists, comparable for in-stance to E = RI in electricity, it has not yet been placed in common use. The absence of such a comprehensive mathe-matical background undoubtedly is the cause of many inefficient designs and installations, as well as of much poorly oriented experimental work. On the other hand, the mathematical immaturity of the process constitutes an opportunity for a real contribution on the part of those who enjoy mental pioneering. Some of the statements made in any such exploratory work may be little more than opinions, and it is to be expected that some will later be proved untrue; but such an evolutionary development is necessary if we are ever to arrive at the fundamental laws, the discovery of which can change the empirical art into a mathematically exact science. The most significant principle of com-minution that we possess is probably Rittinger's law, which states that the work done in crushing or grinding is proportional to the area of the new surface produced. There is an alternative theory, known as Kick's law, which formerly was preferred to Rittinger's but which now is rather dis-credited. According to Kick, the work done in comminution is proportional to the volume, or weight, of the pieces broken, or to the cube of their diameters, while accord-ing to Rittinger it is proportional to the square of the diameters.

Jan 1, 1940

By C. W. Allen, B. D. Cullity, H. S. Choi

The torsional deformation of aluminum and magnesium crystals is investigated, with particular reference to the dependence of proportional limit on crystal orientation. The proportional limit is found to be governed by the average value of the resolved shear stress on the most highly stressed slip systems, but the proportional limit does not strictly obey a critical-resolved-shear-stress law. The behavior of a cylindrical single crystal stressed in tension is well known: plastic flow begins when the resolved shear stress on the most highly stressed slip system reaches a critical value. There is much less understanding of the effect of torsion on such a crystal and on the way in which the critical-resolved-shear-stress law should be applied to torsional deformation. In the present paper particular attention is paid to the dependence of plastic yielding on crystal orientation and lesser attention to the mechanism of flow after yielding has occurred. New work on aluminum and magnesium crystals is presented, and previous results obtained in this laboratory on magnesium1 are re-examined. STRESS DISTRIBUTION The problems involved in torsional deformation of single crystals are due entirely to the complexity of the stress distribution. In tension, if loading conditions are perfect, the shear stress on any particular slip plane is completely uniform over the entire area of that plane. In torsion, on the other hand, the stress on any selected plane in a particular direction varies not only from center to edge of the specimen but also around its perimeter. It is convenient to express the stress at any point in terms of t0, which is the shear stress acting at the surface of a cylindrical specimen on a plane normal to the axis and in a direction tangential to the cylinder. At point P of Fig. 1 this stress acts in the direction of axis 2 and is given by To = 2T/pr3 [1] where T is the applied torque and r the specimen radius. This stress can be resolved on any chosen slip system, and details are given in the Appendix. Let P be the point on the specimen surface where the stress is to be evaluated; P is defined by its angular distance A from an arbitrary reference line through 0 normal to the axis. his line is usually taken as the direction of incidence of the X-ray beam used to determine the crystal orientation.) Then the shear stress t, acting at P in direction Dona plane whose normal is N is given by Ts/TO = sin ? 0, cos ?d sin (?0, - ?) + cos ?, sin ?d sin (?d - ?) [2] where 0, and Od are the angles between N and D, respectively, and the axis, ?0 and ?d are the angles between the projections of N and D, respectively, on the transverse plane and the reference line through 0. By means of Eq. [2], the stress 7, acting in any given slip direction can be computed. In the previous work on magnesium1 only gross slip tangential to the specimen surface was considered, and this

Jan 1, 1963

By Andrew Allen

WHEN a mine is changed from hand, to me-chanical loading, the character of the coal dumped at the tipple is changed in many ways. Selective mining is not so easy with mechanical load-ers, and where much picking is done at the face it is apt to be wasteful anyway. Much recoverable coal is apt to be lost by too free gobbing and, in my opinion, picking at the face should be confined to removal of heavy rock, such as pieces of roof or bottom or sulfur balls. and to such clearly defined and persistent bands as can be readily recognized and segregated. With mechanical loaders of the shovel type, even this amount of gobbing is almost impossible. When coal is shoveled into a conveyor there should be no difficulty in doing as much picking as now by hand loading. The size of lumps may be greatly increased, on ac-count of the ease of handling them with a shovel, as the miners will not break them up in order to lift them into cars. Fines may be increased. This is, of course, not universally true, yet it is my opinion that a shovel breaks the coal more, as a rule, than hand loading. This might not be true with all types of shovels and is cer-tainly not true with a conveyor, either with or without the "duck-bill." The average car load will be greatly reduced in fields such as Illinois and Indiana, in which it is the practice to build up the coal above the sides. Where "level full" regulations prevail there should be no material change. When the coal comes to the tipple these differences usually result in the discovery, that many changes are necessary in the preparation plant. The increase in the percentage of fine sizes will probably not bother the screen, but may overload the conveying apparatus. The larger size lump may clog the chute, but as a rule the efficiency of the sizing screens is not greatly affected. In one completely mechanized mine the proportions of 6-in. lump and 2-in. screenings, when loading by hand, were 16.9 and 51.3 per cent respectively. When loading mechanically, these proportions changed to 15 per cent lump and 53.5 per cent screenings. The picking force on the tipple has been increased from 14 on hand loading to 33 men on mechanically-loaded coal. This is for an output of about 5000 tons.

Jan 4, 1927

By Albert S. Trube

The purpose of this paper is to clarify the definition of compressibility and to present a uniform basis upon which instantaneous compressibilities of liquids and gases can be compared. The equations gaverning the instantaneous compressibilities of imperfect gases are derived and the concept of pseudo-reduced compressibility is introduced. Part of the data presented by Brown, Katz et a1 on compressibility factors for natural gases has been rearranged. A graph of pseudo-reduced compressibility vs pseudo-reduced pressure for various pseudo-reduced temperatures is presented. The need for additional work in relating the compressibilities of liquids and gases is discussed. This information should be of value to reservoir engineers in making non-steady state performance calculations in gas reservoirs. It should be of further use irz pointing the direction for additional research in the nature of liquid and gas compressibilities. INTRODUCTION With the increasing use of steady and non-steady state well and reservoir data, there is a corresponding increase in the importance of the various factors entering into such calculations. Increasing emphasis is being placed on the necessity for obtaining reasonably accurate estimates of the physical properties of the reservoir fluids well in advance of the more accurate laboratory data. One such factor is the isothermal coefficient of expansion of the media which are transmitting and attenuating the non-steady state pressure waves. The average isothermal coefficient of expansion, or "compressibility" is a complex function controlled by the physical properties of the formation and the fluids contained therein. The isothermal expansion coefficients for reservoir gases are usually quite variable, in many cases being highly-pressure sensitive. The coefficients for reservoir liquids tend to be pressure sensitive, but not nearly so much as reservoir gases. The coefficients for solids, usually expressed in terms of a "modulus of elasticity" are relatively insensitive to pressure variations within their elastic limits. For this reason, and also because many previous applications have been limited to rel- atively small pressure ranges, there has been a tendency to ignore the variable nature of isothermal expansion coefficients and treat them as constants. Also, the term "compressibility" by which these coefficients are generally designated is commonly confused with a similar term, z, used to define the deviation of an imperfect natural gas from the perfect gas laws. A clear distinction should be made at the outset between the term "compressibility", which is an isothermal coefficient of expansion of a substance, and the term "compressibility factor", z, which refers to the deviation of a gas from the perfect gas laws. Although the scope of this paper is limited to the compressibility of single phase natural gases, it is definitely related to the problem of accurately estimating the compressibilities of single phase hydrocarbon reservoir liquids, which will form the basis of a future presentation. BASIC PRINCIPLES The coefficient of isothermal compressibility of a substance, c, is usually determined from pressure-volume or pressure-length -measurements depending upon whether the substance is single phase gas, liquid, or solid. A convenient method for making such estimates for a finite change in pressure and volume at constant temperature is to use the well known equation V1-V2/V1 (p2 - p1) .....(1) Eq. 1 is negative because the volume of a confined substance decreases as the pressure is increased. In this case V1 > V2 and p2 > p1. This equation is useful in approximating the compressibilities of single phase gases and liquids undergoing small pressure changes. It is evident, however, that this equation is almost identical with the determination of Young's modulus of elasticity for solids. If the assumption is made that change in length is directly proportional to change in volume, as would very nearly be the case for a steel rod in tension within its elastic limit, then E5=-L1 (p2 - p1)/L1 - L2 .......(2) in which E. is the isothermal expansion coefficient, or Young's modulus of elasticity, for a solid. And further, for this special case L1 (p2 - p1)/L1 - L2 .......(3)

Jan 1, 1958

By R. R. Wilson, Robert Henderson

The Cassidy Colliery operated by the Granby Consolidated Mining Smelting & Power Co., Ltd., is situated about 9 miles in a southerly direc tion from the city of Nanaimo on Vancouver Island. The coal seam worked, which is called the Douglas bed, outcrops in the western part of the property, dipping easterly, and is opened by a slope 7 by 14 ft. in the clear, and is timbered by 12 to 14-in. framed sets of timber, 4 ft. center to center. A separate manway provides for a traveling road and at the same time forms an intake airway. The general average of the dip of the seam at Cassidy is 18º, and varies in thickness from 1 to 25 ft., averaging about 8 ft. The main slopes follow straight down the dip a distance of 4845 ft. At the bottom of the main slope two rock tunnels were driven a distance of 1500 ft. to the south, cutting through a fault, and opening up an area in the same seam which is known as No. 7 South Section. The coal is found in this section at a vertical depth of 1560 ft., and is very soft and friable. The immediate roof is a strong, sandy. shale overlain by massive sandstone while the floor is a light-colored shale, hard when freshly exposed. It does not soften materially on exposure. In the folding of the coal measures a great thrust caused a movement along the coal seam as indicated by the slickensided nature of the coal. The floor has buckled and this has rolled and squeezed the coal so that with rare exceptions it shows no normal bedding but curved slip planes, which frequently start at right angles to the floor, curving upward and then in the direction of the dip. As such planes are slickensided it is probable that there is a coating of clay which is more or less impervious to gas. Outbursts of Gas and Coal Outbursts of gas and coal have occurred in this mine previous to 1921, but not until June 14 of that year was a complete record kept of them. No outbursts occurred above the fourth levels, or, in other words, until a vertical depth of about 900 ft. was reached. Since that time quite a

Jan 1, 1927

FREQUENTLY the careful engineer puts the word porphyry in quotation marks or precedes it with "so-called," when he writes of the Porphyry Copper mines. This is done by way of serving notice that he is aware that the term is not a precise one, though no one who is informed on the subject would in any event misinterpret it. The name originated from the fact that the rock containing the copper at Utah Copper, Morenci, Nevada Consolidated, and Braden-the first four to be developed-was in each instance known as porphyry. The term was convenient; and when Ray, Miami, and inspiration came along they fell into the Porphyry category even though the ore consisted essentially of a rock called schist instead of porphyry. The word is not inclusive enough; nor as a matter of fact is it sufficiently exclusive. Numerous copper deposits occur in porphyry rocks that are not Porphyry Coppers, paradoxical as it may seem. Desiring to be more accurate and at the same time to use a word that is in some degree descriptive, various engineers have adopted the expression "disseminated copper deposits." This alludes to the characteristic scattering of grains or specks of copper minerals more or less uniformly throughout the mass of the rock. Yet here again the implication is not entirely accurate, for the following reason: in some of the ores, the proportion of copper minerals "peppered" in the rock is small, compared with that which coats the walls of cracks and minute fissures; and that which is found as veinlets along planes of fracture. Many geologists have extended their interpretation of the term disseminated to include both types of mineral occurrence. One of the most striking features of the Porphyry ores is the intense shattering and fracturing that the rock underwent at some period or periods prior to the deposition of the copper

Jan 1, 1933

By James C. M. Li

The Petch relation between the flow stress and the gain size is derived from a consideration of gain boundary sources of dislocations without the need of dislocation Pile-ups. Three mechanisms for inierpreting the yield stress: the gain boundary strength, the unpinning of Frank-Read source near a grain boundary, and the generation of dislocations from the grain boundary are compared and the condition of their equivalence is shown. The effect of the average angle of misfit of pain boundaries is found to be sma11 and so is that of the average angle of misfit of subboundaries having impurities. The effect of impurities on the ledge density in the grain boundary is treated thermodynamically and a relatwn is proposed for the variation of Petch slope with impurity activity. The effect of temperature on the Petch slope is interpreted as due to the change of ledge structure in the grain boundary. It is indicated that the effect of annealing temperature may be more important than that of the test temperature and therefore should be studied. The effect of plastic strain on the Petch analysis is deduced from a work-hardening equation in which the generation of dislocations has first-order kinetics and the annihilation of dislocations has second-order kinetics. It is concluded that the Petch slope will decrease with plustic strain if the rate of annihilation of dislocations is sufficiently large. Critical experiments which may shed light on the mechanism for the Petch relation are suggested. THE relation between the yield or flow stress, 0, and the grain size, l, was first proposed by all' and later studied more extensively by Petch and co-workers, who also proposed a similar relation for the fracture stress and deduced from these a grain-size effect of the ductile-brittle transition temperature. The microscopic mechanism used by all' and petch2 involves a pile-up of dislocations of like sign generated from a Frank-Read source. The yielding or flow takes place when the pile-up exerts sufficient stress at the grain boundary so that the plastic deformation can propagate from one grain to another. If the average strength of the grain boundary is ai and the average length of the pileup is lp, the Petch slope, k, is given by" where p is the shear modulus, b the Burgers vector, and v the Poisson ratio. This slope will be independent of the grain size if l/lp is a constant. This is possible, since, if the Frank-Reed source is situated near the grain boundary, lp = 1, and if it is situated in the middle of the grain, Ip = 1/2. cottrell,12 also using the pile-up mechanism, proposed that the stress concentration at the grain boundary will initiate Frank-Read sources near the grain boundary and in this manner a Lüders band can propagate from one grain to another. Assuming that the average distance between the Frank-Read sources and the grain boundary is 1, and the unpinning stress of the Frank-Read sources is op, Cottrell obtained the following Petch slope: This slope will be independent of the grain size if ls is independent of the same, which is not as obvious as the condition, Ip = I for Eq. [2]. In addition to this assumption, the direct relation between the Petch slope k and the unpinning stress, up, was recently questioned by Johnsonon grounds that it is inconsistent with the following observations: the independence of k with temperature and strain rate, and the small k in columbium, which, like iron, has a sharp yield point. As pointed out by ohnson," the most important objection both to the Hall-Petch mechanism, in which the strength of the grain boundary plays the role in yielding, and to the Cottrell mechanism, in which the unpinning of Frank-Read sources plays the role in yielding, is the lack of direct observation of the pile-ups. The dislocation structure in deformed iron has been examined recently in the electron microsope.'-' Dislocations appear to be generated from grain boundaries or other interfaces; they form clusters and tangles within the grain at very early stages of deformation, even in the Lüders band, if the deformation is slow or at normal and elevated temperatures. Although it is still too early to interpret bulk properties from thin-film observations, it does seem worthwhile to look for a mechanism for the Petch relation which does not require dislocation pileup. SUBBOUNDARY SOURCES In order to show that a consideration of grain boundary sources can lead to the same Petch relation as does the consideration of the strength of the grain boundary, we shall first discuss the case of a simple tilt boundary whose elastic properties have been studied in detail.17 The strength of a partially

Jan 1, 1963

By A. D. LeClaire

This paper claims to provide a description of diffusion processes in solids which is said to be a reformulation of Fick's Laws and therefore, presumably, is supposed to be as general as those laws. In fact, it appears to provide no more than an atomic model of a rather restricted type leading even to results in conflict with experiment. The restriction lies in the assumption, carried throughout the paper, that even in a chemical gradient the jump frequency i of an atom of species i is a function only of the composition on the plane from which it jumps, i.e., ?i = ?i(cl, c2 ...). This means that atom jumps in opposite directions from a site would occur with equal probability. There would then be no tendency for atoms to migrate preferentially in one direction along the gradient, so the model is incapable of accommodating the well established phenomenon of atom drift in a chemical gradient.10-" A more general model that does not suffer this difficulty was discussed in a paper13 I wrote in 1958, and the results therein already contain as a special case those given by Dr. DeHoff. The simplest sufficiently general assumption is to suppose that ?i is a function ?i(a1 a2,...; wl, w2, ..-) of the concentration aj and ? ?j at the site from which and to which, respectively, a jump is made. Any such function can always be expressed as a function of the sum and difference of its arguments. Thus d is the jump distance and g a geometrical constant. Obviously this is zero, i.e., no drift, if ??ri/??j = 0. X is written in this form, revealing two components of the total drift, because it can be shown that only the second component, depending on ??'i/???(?j -aj), actually contributes to the net diffusion flux Ji. This component can be shown,13 by a straightforward ther-modynamic transcription, to be responsible for the so called "thermodynamic factor" in the Darken approximation for a chemical diffusion coefficient is the activity coefficient of i. Because the jump rate ?i in DeHoff's approximation has no dependence on the concentration gradient it does not seem possible to derive the Darken equation from his Eq. [6] without the further rather arbitrary assumption that ?i = const ri. It is to be noted too that the author's derivation of Eq. [12] is only valid when ?i is in fact a function only of the cj at the plane from which the atom jumps—for only then is ci?i a unique function of position and therefore of A. Author's Reply R. T. DeHoff I am indebted to Dr. LeClaire for his penetrating discussion of my paper, "A Reformulation of Fick's First Law". I concede that it does appear initially that the relationship presented in my paper could be derived from the detail atomic model of random walk and atom drift presented in Dr. LeClaire's approach. However, there are some important differences in concept and viewpoint which deserve further amplification. Dr. LeClaire's description is penetrating, detailed, and fundamental. His approach describes an extremely complex phenomenon, the displacement of atoms in a multicomponent gradient, at a sophisticated level. For precisely these reasons, his analysis has not proved to be useful for experimental studies that have practical objectives. For example, in order to obtain the information required to complete the analysis of the jump frequency distribution in a ternary system, the following series of experiments would, I believe, be required: 1) chemical (gradient) experiments, covering the composition range;

Jan 1, 1970

By R. L. Perrine

This paper presents results obtained by consider ation of unstable miscible displacement within a porous medium as a quasi-turbulent displacement process, that is, a process characterized by fluctuations around a simpler, mean flow behavior. The instability usually results from the miscible displacing fluid having a viscosity lower than that of the displaced fluid. Such instability is evidenced in suitable experiments by the presence of viscous fingering. A unified theoretical relationship applicable to both stable and unstable displacement is derived. Parameters in the theory are obtained by empirical means. Although the theory describes an idealized conceptual model, agreement with experiment is quite reasonable up to moderately adverse viscosity ratios, such as 17:1. INTRODUCTION Many experimental studies of miscible displacement in porous media have been conducted with prediction of reservoir displacement efficiency as the ultimate objective. Most such studies have utilized lower displacing than displaced fluid viscosity, scaled to potential reservoir fluid pairs. Theoretical approaches have been largely limited to unit viscosity ratios, however, in spite of the necessity for an understanding of the mechanism of the unstable, adverse viscosity ratio displacement process. An obvious reason is the difficulty of describing in mathematical form the viscous fingering characteristic of these conditions. Observation of experiments conducted with dyed fluids in transparent systems suggests an analogy between unstable miscible displacement and turbulent flow in a pipe. In both cases there are fluctuations around a simpler, mean flow behavior. An important difference is that flow behavior of interest in the porous medium is entirely transient, contained within a transition zone between displacing and displaced fluids. Transient behavior complicates description in that coefficients in the equations become variables rather than constants. In the study reported here, the analogy with turbulent flow has been used in treating unstable miscible displacement as a quasi-turbulent displacement process. The purpose has been to derive, even if restricted to an idealized conceptual model, a unified theoretical relationship applicable to both stable and unstable displacement. A relationship meeting these specifications up to moderately adverse viscosity ratios, such as 17:1, is presented. One fluctuation parameter in the theory and dispersion coefficients are obtained by empirical means. The idealized theory is compared with experimental results. UNSTABLE MISCIBLE DISPLACEMENT AS QUASI-TURBULENT DARCY FLOW The miscible displacement of one fluid by another within a porous medium is considered in this study. Flow conditions are such that Darcy's law is applicable. It is further assumed that, by the stability criterion proposed by perrine,1 initial flow conditions lie well within the regime of instability. Thus substantial viscous fingering is assured. We wish to show how this flow regime can be represented as quasi-turbulent. That is, the Reynolds number for the established flow conditions is below that for inertial or turbulent flow within a porous medium, and lies within the regime for which Darcy's law is valid. Yet flow behavior can be described as the combination of some relatively simple average result, and characteristic fluctuations that are superimposed on the simpler behavior. Stated another way, flow behavior includes the movement of layers of fluids with different velocities past or over one another. Such descriptions are characteristic of turbulent pipe flow. The fluctuations in the present case are a direct consequence of the local viscous fingering which accompanies the unstable displacement process. Should displacement become stable, fluctuations would die out. It is of particular importance to note that fluctuations such as these can interact in a way that contributes to material transport by the basic flow. The basic transport equations required for engineering calculations must be modified to reflect this fact.

By Charles F. Park

CORRECTLY speaking, iron ore is limited to any naturally occurring rock from which iron may be extracted at a profit, but in practice the term is frequently used to indicate borderline material or inaccessible material that may in the future become commercially profitable. This distinction is much more than quibbling about terminology. When a layman hears that there are about 45 billion tons of iron ore in the world he naturally thinks that sounds like quite a bit of rock, and what is the use of worrying about the depletion of so plentiful a material. He does not realize, nor does he stop to analyze. why iron ore in Europe and Japan may carry no more than 30 per cent iron while tremendous quantities of 50-60 per cent iron ore in Brazil are not exploited. To people in the iron industry the answer is obvious: the Brazilian material is inaccessible, and there is no suitable fuel near-by. In places closer to home, submarginal ore in considerable quantity contains undesired impurities, such as sulphur, phosphorus, or titanium.

Jan 1, 1947

By Sven Fornander

An account is given of the way in which a new process for desulphurization of hot metal is carried out at a Swedish blastfurnace plant. In the process powdered burnt lime is used as a desulphurizing agent in a rotary furnace. Results are given. IN a paper by Kalling et al.* the fundamentals of a new process for desulphurization of hot metal using powdered burnt lime as a desulphurizing agent were outlined. The purpose of the present paper is to give a brief account of the way in which this process is being carried out in practice by Surahammars Bruks AB, Sweden, and of the results obtained. + The hot metal from the blast furnace is tapped into a ladle of 15 tons capacity and transferred to a rotary furnace, a simplified drawing of which is given in Fig. 1. The rotary furnace has a total length of about 160 in. and an outside diameter of the rings of about 130 in. The furnace is in a horizontal position, with its rings placed on four rollers, and can be rotated by a driving mechanism at various speeds of up to 34 rpm. The rotary furnace and its driving mechanism were designed and constructed at Surahammar. The hot metal is charged into the rotary furnace through the opening to the left in Fig. 1. See Fig. 2. An addition is then made to the furnace consisting of finely ground burnt lime amounting to 2 pct of the weight of the hot metal as well as 0.5 pct of coke breeze. After the additions are made, the furnace is sealed to the atmosphere by means of two lids and set in rotary motion at a rate of 34 rpm. The sulphur in the hot metal is rapidly absorbed by the lime added, the time of treatment needed is normally 30 min. The process is then stopped, the lid is removed, and the furnace is lifted up by a crane and used as a tilting ladle for pouring the metal into the pig-iron molds, Fig. 3. Process Metallurgy The hot metal charged into the rotary furnace has a composition which usually lies within the following limits: C, 3.70 to 4.10 pct; Si, 0.80 to 1.80; Mn, 0.40 to 0.80; P, 0.025 to 0.030; and S, 0.060 to 0.200. Figs. 4 and 5 give examples of the changes in hot metal composition taking place during the treatment in the rotary furnace. As will be seen from the diagrams, the sulphur content of the metal decreases very rapidly during the first 10 min of treatment from about 0.100 to about 0.020 pct S. During the following 20 min, the sulphur content goes down slowly to about 0.005 pct S. The carbon and silicon contents of the metal decrease by about 0.10 pct each during the treatment. The manganese and phosphorus contents remain unchanged. Thus, if the process is started with an iron of the composition given above, the finishing composition of the metal usually lies within the following limits: C, 3.60 to 4.00 pct; Si, 0.70 to 1.70; Mn, 0.40 to 0.80; P, 0.025 to 0.030; and S, 0.002 to 0.020. The chemical reaction taking place during the process can be written:

Jan 1, 1952

By H. I-Lieh Huang, O. D. Sherby, J. E. Dorn

RECENT investigations1-4 have suggested that the total plastic strain, C, for high temperature creep under a given stress can be correlated by means of a temperature-compensated time, te- where t is the duration of test, e is the base of natural logarithms, aH is the activation energy for creep, R is the gas constant, and T is the absolute temperature. A typical set of data on which this deduction was based is given in Fig. la. Inasmuch as the e-in t curves for the various temperatures are identical except for their displacement along the In t axis where f and are appropriate functions. If the increase in creep rate with an increase in temperature is exclusively ascribable to a single process involv- ing thermal activation, (T) should equal RT and therefore it = constant [2] where ?= te Assuming the validity of this relationship, ? = ?2 at the same strain, or where the subscripts refer to two alternate test conditions.

Jan 1, 1957

By M. M. Rao, P. G. Winchell

The growth rates of bainitic plates were measured at 400°C in Fe-Ni-C alloys containing 0.10 atom-fract~on nickel and 0.0012 to 0.0075 atonz-fraction carbon. The growth rates are adequately represented by where xc is nearly the atom fraction carbon of the bulk austenite and PXCy is nearly the carbon atom fractlon in the ferrlte of radiids p In equilibrium with austetzite. The form of the equation is that predicted by a model in which carbon diffusion in austernite controls the gvowth, but the numerical constatnt is two orders of magnitude below that suggested by the model. THE growth of bainitic plates in steel is often assumed to be controlled by the diffusion of carbon away from the advancing plate tip. This hypothesis predicts that the growth rate will increase as the carbon content of the austenite, xCz, is reduced toward the carbon content of the saturated ferrite comprising the plate tip, PxCY The growth rate should vary approximately as (xCg- pxCy)-1. Experimental observation of the growth behavior at low carbon levels should provide a significant test of this model. An alloying element in addition to carbon is required so that low-carbon austenite can be experimentally observed while undergoing bainitic transformation. Nickel was selected. The presence of nickel complicates the interpretation of the data in two ways: First, diffusion of nickel during the transformation would make analysis very difficult. Nickel is assumed immobile during the transformation. Second, nickel affects the solubility of carbon in ferrite and austenite in equilibrium. This effect has been evaluated.' At the completion of our experimental work Goode-now et al.2 published data in essential agreement with the observations to be reported here. Since their discussion is abbreviated and their data are scanty in the region of interest, we believe the present work is of significance. I) THE MODEL OF BAINITIC PLATE GROWTH The rate of lengthening of a plate is assumed to be controlled by the diffusion of carbon from the advancing ferrite-austenite interface into the surrounding austenite. The precipitation of carbides is assumed to be a secondary process. For ease of analysis the carbon-atom ratio,* pxCy, of austenite in equilibrium with ferrite which is convex with minimum curvature radius p, and the carbon-atom ratio, PxCY, of that ferrite in equilibrium with austenite are assumed independent of location on the ferrite-austenite interface. Since these carbon contents vary with the radius of curvature of the ferrite, p, their assumed positional independence must be held as an approximation. The consequences of these assumptions have been developed approximately by zener3 and Hillert,4 and the resulting equation for a platelet has been applied to bainite by Speich and cohen5 and Kaufman, Radcliffe, and Cohen.8 The Zener-Hillert equation* for plates is: The analysis of Hillert is supported by that of Hor-vay and cahn7 which involves no mathematical approximations but does include the assumption that the a/y interface coincides with an isoconcentration line. The solutions of Horvay and Cahn for elliptic paraboloids are replotted in Fig. 1. The shape of the paraboloid is expressed in terms of the ratio of the principal radii of curvature at its tip, A =p1/p2, which is also the ratio of the minor to the major axis of the elliptic cross section. The Zener-Hillert equation for plates is also plotted. The agreement is within a factor of two for (pxyaCr - xyC )/(xyC - PxCaY) between 0.5 and 100. This is the range of interest here and in most other work on bainite. The original form of the Zener-Hillert equation was the form given above with the right-hand side replaced by (pxCya -xCy)/(PxCya). This replacement is not appropriate here. 11) THE EXPERIMENTAL PROCEDURE Alloys were prepared and three kinds of experiments carried out. Continuous-cooling-transformation experiments were carried out on wires by measuring temperature and resistance during continuous cooling. Isothermal-transformation experiments were carried out on wires by measuring electrical resistance as a

Jan 1, 1968

By R. K. Agarwal

In this paper, a new technique developed to analyze two-dimensional nonhomogeneous models is described. A simple three-layered medium with a circular hole centrally located in the intermediate layer is considered. The layered sheet is loaded uniaxially perpendicular to the stratification. The Young's modulus of the intermediate layer is lower than that of the outer layers. The stress distribution around the circular hole in a stratified medium is significantly different when compared to the case of a continuous homogeneous medium. The distribution depends upon the ratio of the height of the intermediate layer and the hole diameter. A comparison of stress distribution and procedures for preparation and testing of the model are given. The potential of the technique is outlined. In the process of applying the techniques of stress analysis to problems of rock mechanics, the recent discernible trend has been to postulate highly idealized models of physical situations. As new techniques have been developed, more complicated models have been conceived to simulate mote accurately the physical environment to which the analysis results are eventually applied. In this investigation, a new technique is developed to analyze two-dimensional nonhomogeneous models. The technique is used to determine the stress concentration around a circular hole in a soft material overlain and underlain by a hard material. In mining practice, such situations arise commonly in coal deposits and frequently in bedded or vein-type metallic mineral deposits. DEFINITION OF THE PROBLEM A plate, consisting of three layers with a circular hole in the intermediate layer, is loaded uniaxially, Fig. 1. The variable in the experimental investigation is the height of the intermediate layer, H. The diameter of the hole, D, is kept constant. The stress concentration factors around the periphery of the hole, for various values of H/D, are evaluated using the technique of photoelasticity, Table I. TECHNIQUE OF EXPLORATION Photoelastic analyses of composite models have hitherto been limited to the study of lap and butt joints between glued metals and dissimilar materials.1-3 In mining, Haber and Oudenhoven used composite models to study the stress distribution around a circular hole in two-layered media and the behavior of elastic liners in shallow underground openings.4-5 They used the stress freezing technique devised by Dally et a1.6 The procedure used in this investigation is outlined below: Casting of Layer A: Flat sheets of plastic were cast in a silicon-coated mold using Araldite 502 with hardener 951 in the ratio of 100:8 by weight. After

Jan 1, 1969

By Ian William Wark

DIFFERENT parts of the world in which mineral separation by flotation is practiced experience vastly different natural temperatures, and in some districts there is a big difference between summer and winter tempera-tures. It will be shown in this paper that the temperature is of con-siderable importance where it is desired to separate pyrite or arsenopyrite from other minerals. It is probable that some of the differences obtained in the flotation of these minerals at plants working under otherwise similar conditions are due to temperature variations. The influence of temperature on the flotation of galena, chalcopyrite and sphalerite by xanthates is less marked than its influence on pyrite flotation.

Jan 1, 1938