Search Documents

By Herbert Clark Hoover, Lou Henry Hoover

QUESTIONS as to the methods of smelting ores and of obtaining metals I discussed in Book IX. Following this, I should explain in what manner the precious metals are parted from the base metals, or on the other hand the base metals from the precious'. Frequently two metals, occasionally more than two, are melted out of one ore, because in nature generally there is some amount of gold in silver and in copper, and some silver in gold, copper, lead, and iron ; likewise some copper in gold, silver, lead, id iron, and some lead in silver ; and lastly, some iron in copper2. But I will begin with gold. Gold is parted from silver, or likewise the latter from the former, whether it be mixed by nature or by art, by means of aqua valens3, and by powders which consist of almost the same things as this aqua. In order to preserve the sequence, I will first speak of the ingredients of which this aqua is made, then of the method of making it, then of the manner in which gold is parted from silver or silver from gold. Almost all these ingredients contain vitriol or alum, which, by themselves, but much more when joined with saltpetre, are powerful to part silver from gold. As to the other things that are added to them, they cannot individually by their own strength and nature separate those metals, but joined they are very powerful. Since there are many combinations, I will set out a few. In the first, the use of which is common and general, there is one libra of vitriol and as much salt, added to a third of a libra of spring water. The second contains two librae of vitriol, one of salt-petre, and as much spring or river water by weight as will pass away whilst the vitriol is being reduced to powder by the fire. The third consists of four librae of vitriol, two and a half librae of saltpetre, half a libra of alum, and one and a half librae of spring water. The fourth consists of two librae of vitriol, as many librae of saltpetre, one quarter of a libra of alum, and three-quarters of a Mbra of spring water. The fifth is composed of one libra of saltpetre,

Jan 1, 1950

By Paul M. Tyler

Mineral raw materials, because they are essential to our industrial prosperity and military strength, must be made available in substantial quantities regardless of cost. Variations in the cost of procuring minerals, as compared with the cost of obtaining other commodities, and changes in the cost of production of different minerals relative to one another, will naturally influence the quantities of a given mineral item that can be marketed readily. However, in the last analysis, the cost of winning minerals from the ground usually constitutes only a small fraction of the value of the end-use products and services derived from them. The miner's share of the cost of a typewriter, telephone circuit, or even a modem office building, is almost insignificant. Far more important to the ultimate consumer, therefore, than any minor saving in cost is the assurance of ample supplies of the minerals needed to aliment our expanding economy. To create a successful metal mining operation involves (1 ) the discovery and systematic development of a mineral deposit, (2) extraction of the ore, (3) the mechanical and metallurgical processing to eliminate worthless or harmful impurities, and (4) the profitable disposal of the product. The fist step of the technological process belongs chiefly in the realm of the geologist, the second is a mining engineer's job, the third is mainly up to the millmen and extractive metallurgists; whereas the fourth and equally important step poses a commercial problem that has aspects in common with the marketing of other products. Ore finding is far from being an exact science. There is no mathematical criterion for evaluating the efficiency of prospecting and exploration. Lady Luck shares the honors or assumes part of the blame along with the prospector and geologist. Marketing like- wise involves factors that seemingly defy engineering analysis. Although the other steps can be measured with some degree of mathematical accuracy, even they tend to vary so widely in basic principle, as well as in

Jan 1, 1964

By Dever C. Ashmead

Anthracite was first mined in the Wyoming Valley and sold as an article of commerce in 1808. As some preparation has always been necessary to make it ready to burn, the preparation of anthracite must date back over a hundred years. Two vital factors have determined to a large extent the degree and the method of anthracite preparation. These are, first, the character of the beds worked and the methods by which they are mined and, second, the equipment used and practices followed in the burning of the coal. It is not the intention here to go deeply into these phases of preparation as a paper of no mean length could be prepared on the history of either. Rather, the intention here is to point out the main considerations and to show the influence they have exerted on the preparation of anthracitc. Mining Methods and Their Relation to Preparation In the beginning of the nineteenth century, the coal beds were virgin with the possible exception of some outcroppings that had been worked, to a slight extent, by the Indians. It is known that the American aborigines had a, knowledge of the use of this fuel, because when the Wyoming Valley was purchased from them, in 1754, they mentioned the fact that, by selling the land, they would lose their coal. Real mining of anthracite began about 1808 when Judge Jesse Fell, of Wilkes-Barre, discovered that the "common stone coal of the valley" could be burned in an open grate. For some years thereafter mining was conducted near the surface. The working p1aces were driven narrow; only the best of the coal was selected and the remainder was left in the ground. Only the lumps could be used, as no market existed for other sizes. As time passed, however, it became necessary either to go farther into the ground or to widen the working places. When these places were widened, falls of roof were apt to occur, making it necessary

Jan 1, 1922

By S. C. Miller, R. D. Carter, H. G. Riley

A merhod based on theoretical considerations is presented for estimating srabilized performance of gas wells from data obtained from short-term flow tests. Calculated results are presented for II gas wells evaluated with four-to five-hour flow tests. For seven wells on which comparison war available, satisfactory agreement between measured and calculated performance was obfained. Although the final results are favorable, diflerences in flow capacity k,h obtained from flow tests conducted at different rates were observed. This inconsistency did not seriously affect the calculated results in the test wells. INTRODUCTION In order to negotiate the sale of gas from newly drilled gas wells and to plan otherwise for the orderly exploitation of gas reserves, it is necessary to predict the long-term withdrawal rates physically possible for each well. Since low permeability gas wells do not stabilize except after a lengthy period of flow, the operator is unable, without wasting large amounts of gas, to measure directly the stabilized performance prior to contracting for sale of the gas and connecting the well to a pipeline. This problem is not new, but is one which has recently received considerable attention because of the rising importance of natural gas in the economy and new emphasis on the development of gas reserves. Thus, a method is needed for estimating stabilized performance from information obtained from short-flow tests in which stabilized flow is not attained. Such short tests would be suitable for wells not connected to a pipeline since the tests require only small amounts of time, labor and production of gas. Other requirements of such a method are short calculation time and use of a model of the gas well and its reservoir drainage area which is as consistent as possible with present theoretical, experimental and field knowledge. The method presented herein was developed in an effort to meet these requirements. It has been named the "two-Row method". In addition to the section on the evelopment and description of the method, a description of field application of the method and a discussion of well test results are in- cluded. Finally, there is a section on an additional consideration prompted by some of the test results. The Appendix contains certain details of the derivation of the two-flow method and a sample calculation for Test Well 1. MATHEMATICAL DEVELOPMENT AND DESCRIPTION OF THE METHOD The method is based upon the following assumptions: (1) The system can be described as a finite cylindrical gas reservoir produced by a single cylindrical well located at the center of the reservoir. Flow is strictly radial. (2) Flow obeys Darcy's law except very near the well where a quadratic flow law is assumed. (3) The system is also characterized by a laminar flow skin factor. The use of a quadratic or turbulent flow term is more fully justified in Refs. 1 and 2. When such a system is opened at the wellbore to constant rate flow from an initial condition of uniform pressure distribution, the radius or drainage quickly passes through the non-Darcy flow and "skin effect" region imrnediately surrounding the well. After the drainage radius or "transient" has passed beyond this region and before it reaches the external boundary, drawdown can be expressed by'." After the drainage radius reaches the external boundary, stabilized flow exists and drawdown is given by'," Eqs. 1 and 2 form the basis for the method. The validity of these equations will now be examined in the light of some well test data previously available. It has been found by actual measurements on gas wells that the relationship between drawdown and flow rate, that is the performance curve, is closely approximated by' Eq. 3 is a straight line on log-log graph paper. Eq. 2 and, for any lixed value of time, Eq. 1 plot as a curve which is slightly concave upward on log-log graph

By C. B. Manula, H. Gezik

In planning modem-day mining operations, management needs to pass from the area of subjective decision-making to an area of objective decision-makirlg. Planning procedures currently being practiced by many mine managers depend primarily on common sense and trial-and-error methods which oftentimes never reach optimum solutions. For this reason, an attempt is made to provide an understanding of linear programming to solve production planning problems. More specifically, a linear programming model is proposed to solve the production scheduling problem as it was found to exist in the crushed stone industry. Optimal production scheduling is of prime concern here, since the industry has to cope with seasonal demands for its products. The solution, as determined through the application of the model, provides management not only with an optimal production schedule but also with the following information regarding the improvement of existing facilities: (I) the location of bottlenecks which limit production and profit; (2) the directions in which the company i fixed equipment should be expanded or modified; and (3) the evaluation of unused resources. Future trends in stone production will quite likely parallel those in the construction industry. Continued record expenditures for structures such as dams, highways, and buildings are the reasons for this accelerated growth. Production forecasts for the next decade are predicted to be around one billion tons with an approximate 7% annual increase. The problem of increasing production from 570 million tons in 1966 to 1 billion tons by 1976 offers no small challenge. This situation is further compounded by other problems and economic factors. Population expansion, government regulation, labor agreement, and changes in technology wd cause managers and operators to be confronted by many more complexities than characterize present-day operations. Some of these. problems and economic factors are: (1) In some metropolitan and other urban areas, easily accessible sources of stone are almost depleted. Transportation costs will be a prime concern of the industry. (2) Population growth has forced residential expansion to a point where many quarry operators must conform to urban and suburban regulations. (3) Government regulation in regard to air and water pollution, noise, and reclamation as well as zoning developments necessitate that crushed stone producers keep abreast of changing conditions and requirements. (4) The crushed stone industry operates under pressure of a highly competitive situation. Every company's objective is to retain a favorable marketing position by exercising great attention to production and marketing costs. This is rather difficult, since the crushed stone industry is one segment of modern mining which has to cope with a seasonal demand for its products. Among the problems posed here, the one of exercising great attention to production and marketing costs is of current interest to management. Under seasonal demand conditions, this problem creates further difficulties for managers who need to plan and forecast. STATEMENT OF THE PROBLEM Various methods of Operations Research have been developed to handle a problem whose genesis is explained above. It is very seldom, however, that these quantitative means are used in practice by the mining industry. This paper describes the use of Operations Research methods to solve a production planning problem when demand is seasonal. The seasonal production planning problem is an important one because of the substantial costs associated with changing production, employment, and inventory levels to adjust to seasonal sales patterns. Most companies try to operate within reasonable levels of employment changes and hold inventories within storage capacities. In general, little or no attempts were made to develop minimum cosl schedules or to provide objective means to compare various production policies. When demand varies within a fixed time period, 11:-oduction requirements may be met by following one

Jan 1, 1970

By F. Keller, A. H. Geisler

ALTHOUGH the subject of precipitation from solid solution appears to be one of the more profitable fields in metallurgy for study with the electron microscope, few comprehensive studies have yet been made. Occasionally electron micrographs have been published that illustrated alleged precipitation in various alloys, but frequently the apparent size, shape and distribution of the particles do not fully agree with those expected from theory or from observations of the precipitate when the particles have grown to a size resolvable with the light microscope. The presence of a Widmanstätten pattern for submicroscopic precipitate particles has been demonstrated for only a few aged alloys.1 The initial problem has been the development of suitable techniques for applying the transmission-type instrument to a field that has been inherently associated with reflection-type microscopes. The various techniques have been the subjects of numerous publications, and instead of describing them individually here it will suffice to say that the oxide film method has proved to be the most satisfactory method yet found for studying the microstructure of aluminum alloys with the electron microscope. This method has the definite characteristic advantage that the actual surface layer of the specimen is examined and not a plastic or silica replica. Doubtless suitable methods for forming and removing the oxide film could be developed for alloys of other metals. The purpose of this report is to point out the characteristics of the oxide film method that have been observed during the studies of numerous aged aluminum alloys and to present the results of these studies. PREPARATION OF SPECIMENS The oxide film method for preparing specimens of aluminum alloys for examination in the electron microscope has been described in detail previously.1,2 Briefly, the procedure consists of preparing the metallographic specimen, forming the film by anodic oxidation and removing the film from the prepared surface of the oxidized specimen. The specimen is polished according to the usual procedure for microscopic examination.3 The specimen may then be etched to remove flowed metal but etching to attack the alloy constituents or to leave them in relief as in the replica processes is not necessary. Electrolytic polishing is not generally recommended. Deep-etched specimens are used frequently, however, since they provide information that is not revealed by polished specimens and frequently present the same information more clearly than do the polished specimens. The surface preparation of specimens to be deep-etched is not important, since specimens that have been subjected to the first wet polishing operation are generally used, but frequently the as-rolled surface is suitable. These specimens are then etched using a suitable macroetching reagent.

Jan 1, 1946

By M. R. Tek, R. L. Nielsen

The scaling laws as formulated by Rapoport relate dynamically similar flow systems in porous media each involving two immiscible, incompressible fluids. A two-dimensional numerical technique for solving the differential equations describing systems of this type has been employed to assess the practical value of the scaling laws in light of the virtually unscalable nature of relative permeability and capillary pressure curves and boundary conditions. Two hypothetical systems — a gas reservoir subject to water drive and the laboratory scaled model of that reservoir — were investigated with emphasis placed on water coning near a production well. Comparison of the computed behavior of these par ticular systems shows that water coning in the reservoir would be more severe than one would expect from an experimental study of a laboratory model scaled within practical limits to the reservoir system. This paper also presents modifications of the scaling laws which are available for systems that can be described adequately in two-dimensional Cartesian coordinates. INTRODUCTION Present day digital computing equipment and methods of numerical analysis allow realistic and quantitative studies to be carried out for many two-phase flow systems in porous media. Before these tools became available the anticipated behavior of systems of this type could be inferred only from analytical solutions of simplified mathematical models or from experimental studies performed on laboratory models. To reproduce the behavior of a reservoir system on the laboratory scale, certain relationships must be satisfied between physical and geometric properties of the reservoir and laboratory systems. Where the reservoir fluids may be considered as two immiscible and incompressible phases, the necessary relationships have been formulated by Rapoportl and others.2-5 Rapoport's scaling laws follow from inspectional analysis of the differential equation describing phase saturation distribution in such systems. It will be recalled that these scaling laws presuppose three conditions: (1) the relative permeability curves must be identical for the model and prototype; (2) the capillary pressure curve (function of phase saturation) for the model must be linearly related to that of the prototype; and (3) boundary conditions imposed on the model must duplicate those existing at the boundaries of the prototype. These three requirements seldom if ever can be satisfied in scaling an actual reservoir to the laboratory system because: (1) The laboratory medium normally will be unconsolidated (glass beads or sand) while the reservoir usually is consolidated. Relative permeability and capillary pressure curves are usually quite different for consolidated and unconsolidated porous media. (2) The reservoir usually will be surrounded by a large aquifer which could be simulated in the laboratory only to a limited extent. (3) Wells present in the reservoir would scale to microscopic dimensions in the laboratory if geometric similarity is to be maintained. In view of these considerations, rigorous scaling of even a totally defined reservoir probably would never be possible. The purpose of this paper is to assess the practical value of the scaling laws in the light of the unscalable variables. This has been done by carrying out numerical solutions in two dimensions to the differential equations describing the flow of two immiscible, incompressible fluids in porous media for a field scale reservoir and a laboratory model of that reservoir. While both the reservoir and the laboratory model were purely fictional, each has been made as realistic and representative as possible. The field problem selected as the basis for the investigation was an inhomogeneous, layered gas reservoir initially at capillary gravitational equilibrium and subsequently produced in the presence of water drive. The laboratory model of this reservoir was designed to utilize oil and water in a glass bead pack. The numerical treatment employed was similar to that of Douglas, Peaceman and Rachford6 and it included both capillary and gravitational forces as

By E. L. Dougherty

A phenomenological theory for a one-dimensional unstable miscible displacement similar in type to the Buckley-Leverett model but including the effects of mixing is proposed An equation giving the fractional flow of pure solvent through an oil phase containing dissolved solvent is derived including the effects of gravity and nonbomogeneities. Numer ical integration of the resulting pair of nonlinear hyperbolic partial differential equations gives volumes of dissolved and undissolved solvent as functions of space and time. Parameters in the model which characterize mixing due to dispersion were determined from experimental data; the parmeters correlated with viscosity ratio. The results indicate that the rate of dispersive mixing is proportional to volumetric flow rate Incorporated into our equations were Koval's heterogeneity factor H. which characterizes mixing due to channeling. This allowed satisfactory predictions of displacement behavior observed in horizontal floods in nonhomo-geneous cores; but in most cases the values of H which we used were considerably larger than those used by Koval. INTRODUCTION It has been shownl,2,3 that for favorable viscosity ratios, the diffusion equation with convection satisfactorily describes the behavior of a miscible displacement in a porous media. Numerous attempts to develop a satisfactory mathematical description for the case of unfavorable viscosity ratio have been reported, but they have met with only partial success. These attempts, which are reviewed by oval! have taken two approaches: (1) development of a flow model akin to the Buckley-Leverett approach for immiscible displacement neglecting the effects of mixing5,6 and (2) simultaneous application of Darcy's Law for flow and the diffusion equation with a convection term for mass transfer.7,8,9 We set out to construct a mathematical model of Type 1 including, though, the effects of mixing. Based upon a set of hypotheses, we derived a pair of nonlinear hyperbolic partial differential equations which describe in a one-dimensional system the combined effects of flow and mixing. To work with these equations it was necessary to develop a fractional flow formula. The combined system was integrated numerically using the method of characteristics. In our equations the mixing process is characterized by four parameters. Three of these, labeled ß, P1 and P2, account for dispersive type mixing. The fourth is the heterogeneity factor H, proposed by Koval to account for channeling due to nonhomogeneities in the porous media. Values of the parameters which would cause agreement between theory and experiment were determined by trial-and-error for miscible floods conducted in horizontal cores of both homogeneous and heterogeneous materials. Calculations were also performed for vertical floods in homogeneous cores. The purpose of this paper is (1) to present the details of the mathematical analysis, (2) to present the results of the calculations, (3) to consider what light the results shed on the mixing process in an unstable miscible displacement, and (4) to provide a firmer foundation for the correlation technique developed by Koval for predicting the behavior of unstable miscible floods. STATEMENT OF PROBLEM The problem is to construct a mathematical model which describes in one dimension the observed behavior in an unstable miscible displacement. The approach is phenomenological in that the equations are based on assumptions which violate certain physical precepts known to apply to the displacement phenomenon. However, the assumptions do allow us to account mathematically for the more essential phenomena. The work is of value if we can quantitatively predict experimental results which heretofore could not be predicted, even though the synthetic nature of the model belies complete explanation of the observations. We assume that the system is comprised of two contiguous flowing phases. One phase, which we call free solvent, has the properties of pure solvent. We call the fraction of the pore volume occupied by this phase the solvent saturation, designated s. The oil phase consists of oil containing dissolved

By H. M. Manasevit, R. L. Nolder, L. A. Moudy

The observation that silicon-on-sapphire crystallo-graphic relationships exist which are unrelated to one another implies that there are regions of sapphire orientations containing these relationships which are separated by boundaries. The presence of these boundaries is demonstrated, initially on rounded cylindrical surfaces md more definitively on flat sapphire substrates with orientations along primary crystallograPhic zones. By means of reflection electron and X-ray diffraction techniques the crystallographic relationships have been studied. Two new relationships have been found, viz., (111) Si (1014) sapphire and (001)Si (1012) sapphire. Also found were multicrystalline relationships. The approximated boundaries between the crystallographic regions for each relationship are shown by means of a stereo-graphic projection. These studies also clarify discrepancies in the literature concerning the silicon-sapphire relationships on the (1120) sapphire substrate. SINCE the first report of the deposition of single-crystal silicon on sapphire in 1963' this epitaxial system has been the object of study by many semiconductor laboratories. In particular, there has been strong interest in the crystallographic relationships between the two materials. In 19652,3 we demonstrated that at least four distinct modes of orientation relationships can occur, namely, (100)Si (0112) sapphire, (111)Si 11 (1124) sapphire, (111)Si (1120) sapphire, and (111)Si (0001) sapphire. Each relationship could be obtained when the substrate plane was at the particular orientation indicated. Other workers as well as our laboratory concurred in general with the initial results. However, two discrepancies were observed. First it was observed by Joyce et a1.4 that on a (1120) substrate a (110) silicon deposit could be obtained. This was actually consistent with the (111)Si (1124) sapphire mode. Second, two other laboratories,5,6 as well as our own,7 have found a deposition on the basal plane which is different than our initial findings. Even though a (111)Si deposit was obtained, the crystallographic relationships were not identical to those previously found. Subsequent work in our laboratories has confirmed that the form found in these later investigations of the basal plane is by far the more predominant, and the original finding was an anomaly. The relationships which have been previously estab1.ished are summarized in Table I. It is seen that at present a rather small number of possible substrate orientations has been investigated. Therefore, it seemed appropriate that a systematic investigation of silicon epitaxy on various sapphire substrate orientations would be of value. It was expected that the answers to three questions could be obtained from such an investigation. These are as follows: 1) Since we know of at least four distinct modes of orientation relationships, over what range of substrate orientation do these modes occur and what is the form of the boundary expected between the different regions in which each mode occurs? 2) Are there other modes which occur on substrate orientations not previously investigated? 3) What is the explanation for the discrepancy between our laboratory's findings and those results reported by Joyce et al.4 concerning the relationships occurring on (1120) substrates? APPROACHES AND EXPERIMENTAL METHODS Two different approaches were used. Initially, depositions were carried out on rounded cylindrical surfaces which lay perpendicular to the (0112) plane.* By this means the presence of boundaries and their approximate positions could be determined. Finally and more definitively, depositions on flat sapphire substrates with a variety of orientations, mostly along primary crystallographic zones and about the (1120) plane, were studied by means of reflection electron diffraction and X-ray diffraction (using back-reflection Laue patterns and full circle goniometer techniques) in order to determine crystallographic relationships. The sapphire orientations investigated by these means

Jan 1, 1969

By A. L. Barnes

A number of reservoirs have been partially invaded by bottom water. Water flooding such reservoirs can be especially inefficient if the oil has a high viscosity because injected water will under-run the oil and emerge at producing wells without having displaced much oil. A possible means of improving water sweep efficiency would be to precede the water flood with a slug of water purposely thickened with a chemical additive. This possibility has been studied with a scaled laboratory flow model. The investigation was carried out on a model representing a 20-acre five-spot with one set of reservoir condition~. Displacements were performed with water, with viscous-water slugs of 10 and 25 per cent of the pore volume followed by water, and with all viscous water. Displacement tests were made at several rates of injection. For all cases, compared at the same producing water-oil ratio, an increase in rate resulted in an increase in volumetric sweep. The larger the quantity of viscous slug injected, the greater the crossflow of oil ahead of the displacing front and, thus, the lower the WOR during that period of a displacement. This investigation indicates that the injection of aqueous viscous slugs in this type of system would give some additional benefit over conventional flooding by (I) reducing the life of the flood, (2) reducing lifting costs and (3) increasing ultimate recovery. The benefits of a viscous-slug displacement would warrant its use if the viscous water could be created for a few cents per barrel. INTRODUCTION The most appropriate method for water flooding oil reservoirs which have been partially invaded by bottom water has been a problem of considerable interest. Such reservoirs, having good vertical permeability and unfavorable mobility ratios, are difficult to produce efficiently under normal waterflood procedures. Even though the injection wells have been plugged back above the oil-water contact and the water zone has been squeezed off, injection water will enter the formation and not only displace oil, but also much of it will displace the bottom water, thus creating a high producing water-oil ratio at the start of a flood (see Fig. 1). Since the rate of injection and the viscosity of the displacing fluid are the more easily controlled physical parameters of the reservoir, it is quite natural to study their effects on the volumetric sweep of the oil zone and on the producing water-oil ratio. Increasing the viscosity of the displacing fluid has two disadvantages—it reduces the maximum rate of injection into the reservoir, and it costs more than water. It was thought that the injection of a viscous-fluid slug, followed by water, would minimize these two disadvantages, increase the volumetric sweep of the oil zone and reduce the producing water-oil ratio. Water with an additive to increase viscosity would probably be the most effective. The primary purpose of this investigation was to determine the feasibility of injecting viscous-water slugs followed by water to improve or hasten oil recovery while reducing the producing water-oil ratios. The effect of injection rate upon recovery was also investigated. A laboratory model, scaled to a 20-acre five-spot prototype with one set of reservoir conditions, was constructed. Laboratory studies of displacements with (1) water, (2) viscous-water slugs of 10 and 25 per cent of the pore volume followed by water and (3) all viscous water were performed at several rates of injection. THEORY OF SAND MODEL SCALING There are two conventional means of determining scaling criteria. These are dimensional analysis and inspec-tional analysis. Determining scaling criteria by dimensional

By S. F. Frederick, W. A. Bailey

The relationship between microstructure and mechanical properties of cast 356-type aluminum alloys was studied to determine the cause of the variations in properties resulting from differences in solidification rate. It was found that variations in strength are a consequence of variations in ductility and that ductility is inversely proportional to the dendrite cell size. A mechanism is proposed to account for this correlation based on the fracture strain of the inter-dendritic silicon particles and the differential strain across dendrite cell boundaries. IT has long been assumed that permanent mold castings of aluminum are superior to sand castings because of their finer grain structure and reduced porosity.' A systematic variation of static properties was noted during unpublished studies of fatigue-strength variations with solidification rate of A356-T6. Of particular interest was the observation that for a given composition and heat-treatment, both ductility and the ultimate strength increased with higher solidification rate. Because the A1-Si system forms the basis of many alloys used for castings and welding wire, a study was initiated to determine the details of how higher cooling rates improved mechanical properties of these alloys. The results of this study are presented here. EXPERIMENTAL PROCEDURE Specimens were prepared from 356-T6 to supplement the A365-T6 alloy data from earlier studies. This allowed a comparison of behavior of alloys with substantially different strengths but with only small differences in alloy composition. The composition and heat treatment of both alloys are given in Table I. The specimens were cut from a chill plate casting,' which is a 6 by 9-in. plate, 1 in. thick, which has one end heavily chilled, and a large riser at the other so that solidification is essentially unidirectional from the chill end to the riser. Although not measured here, solidification times measured on similar specimens by embedded thermocouples are a few seconds at the chill end and up to 5 min near the riser. Specimens were obtained by cutting slabs from the plate, parallel to the chill; these represented different cooling rates, with essentially the same composition. For the tests, either 0.125 or 0.250-in. thick slabs were cut for standard 2-in. gage length specimens. The thicker specimens were tested to failure at 2000 lb per min while the thinner specimens were tested in an Instron machine at a constant cross-head speed of 0.02 in. per min. These thin specimens were intended for trans- mission electron microscopy. Some thin specimens were tested to failure while the remainder were given various amounts of prestrain. Dendrite cell sizes were measured by the linear intercept method on photomicrographs at either X100 or X50, depending on the cell size. It was assumed that the dendrite cells were outlined by the silicon particles. Determinations were made on two mutually perpendicular planes and an rms average taken. Electron fractographs of representative tensile specimens were made in the conventional manner using the two-stage replica technique.3 Samples for transmission electron microscopy were prepared by chemical milling followed by electropolishing. RESULTS AND DISCUSSION The results of the tensile tests and the dendrite cell-size determinations are given in Figs. 1 and 2. All data show the same trends, namely, a decrease in ultimate strength and elongation with increasing distance from the chill (and solidification time), and a corresponding increase in dendrite cell size. The yield strength remains essentially constant. The constancy of the yield strength was particularly significant since it indicated that variation in the solidification rate does not affect the stress required to initiate plastic flow. By plotting true stress-true strain curves as in Fig. 3, it can be seen that both

Jan 1, 1969

By P. R. PaeIay, A. Slibar

Using three-dimensional, stress-deformation rate equations for a Bingham plastic, an approximate solution for the laminar flow of drilling mud between the drill pipe and casing is given for the case when the velocity gradients due to axial flow are large compared to velocity gradients due to rotation. INTRODUCTION The fact that drilling mud can be treated, to a first approximation, as a Bingham plastic has been established by several investigators12,3,4,5. A Bingham plastic is a material that is rigid until a certain combination of stresses exceeds a critical value. When the stresses are sufficiently high to initiate flow, the reduced stresses have a constant ratio to the corresponding deformation rates at each point. Prediction of pressure and torque requirements to produce the desired flow of the material have been made by analytical solutions due to Bingham6, Reiner7, and Buckingham8. One part of the present means of calculation that requires clarification is the flow between the drill pipe and casing. Present methods treat the flow resulting from the rotation of the drill pipe separate from the flow produced by the axial pressure gradient. Since the non-linear behavior of a Bingham plastic implies that solutions cannot be superposed, the present analytical investigation was undertaken to obtain a solution to the problem of combined axial and tangential flow. Equations have been proposed for the three-dimensional flow of a Bingham plastic9,10,11 and it has been shown that the proposed equations lead to the usual solutions from the one-dimensional theory". Applying these equations to the flow between two concentric cylinders the problem of axial flow only13 and tangential flow only12 have been solved. In addition to the separate problems of axial and tangential flow between concentric cylinders, the authors have determined the relationship between external torque on the inside cylinder and axial pressure gradient necessary to initiate flow". The solution given in Ref. 14 points out the complexity of the present problem. The problem considered here is the case of simultaneous axial and tangential flow. The axial flow, necessary to keep the drill bit free of removed material, results from the axial pressure gradient. Tangential flow is present since the drill pipe rotates in the hole. The solution given here is valid when the shear stresses in the drilling mud due to rotation are small compared to the shear stresses resulting from the axial flow. This case for laminar flow can be encountered in oilfield drilling. STRESS-DEFORMATION RATE EQUATIONS A detailed derivation of the equations governing the behavior of a Bingham plastic is given in Ref. 10. This section briefly outlines the physical reasoning leading to this three-dimensional formulation of the stress-deforma-tion rate law. Experiments on a certain class of materials have demonstrated the following properties: (1) when a certain combination of the stresses is below a critical value no deformation of the material occurs, (2) to a first approximation this combination of stresses is independent of the hydrostatic pressure (negative average normal stress), (3) if flow occurs the ratio of the reduced stress to the corresponding deformation rate is constant at each point of the flowing material, and (4) incom-pressibility. For the one dimensional case of shear deformation, Bingham6 formulated the governing equation as dxx, = 0 for sxx < to and µdxx = s xx = to for sxx to , . where dxx is the shear deformation rate, sxx the shear stress, to the critical shear stress, and p the slope of the sxx vs dxx curve for dxx > 0. Fig. 1 shows the behavior for pure shear between two parallel plates. Taking into account the physical properties given above and assuming the critical combination of stresses to be measured by the octahedral shear stress, the fol-lowing three-dimensional equations have been proposed.

Jan 1, 1958

By B. Trafton, H. L. Hartman

An earlier paper1 introduced a rapid solution for mine ventilation network problems, employing the digital computer. In this paper, refinements and additions to the computer program which expand its scope of application and a solution to a ventilation network having a coal-mine configuration are presented. The revised program-written in FORTRAN for the IBM 7074 computer in Penn State&apos;s Computation Center-is applicable to any mine ventilation network employing natural splitting and having its fan located externally. It accomplishes the following: 1) Distributes an assumed total quantity of air flow among all airways of the network, performing iterations until an acceptable level of error is attained and the flow in each airway is known. 2) Computes the pressure drop (head loss) in all airways and the overall network and mine head losses. 3) Determines the operating point for the mine and a given fan, adjusting all air flows and head losses to the actual values. The basis of analysis used is the Hardy Cross method of successive approximation for solution of any fluid network, explained in detail in the earlier article. In brief, corrections to assumed air flows are computed for all the loops of a network; following distribution to the airways of each loop, the process is repeated until acceptable accuracy is attained. When a hand or desk-calculator solution is used, there is advantage to close approximation of the actual flows in the original assumption of air-flow distribution in the network. Since publication of the previous paper, in which assumption of initial quantities was also necessary in the computer solution, the program has been modified to make the first distribution itself, assigning the total quantity to one specified path in a split and then proceeding to correct the flows. The number of iterations required for a high-accuracy solution is little affected, at least in networks of few loops, 10 to 15 being sufficient to reduce the error to 1 pct or less. The principal limitation of the original program was that it contained no provision for handling a fan. Further, it did not determine the overall head loss in the network and was not designed for multiple parallel openings, such as occur in coal mines. The revised program was written to overcome these disadvantages and also to print out the adjusted flows for all airways, corrected for the system operating point. At the present time, the program can handle only an external fan. The procedure employed in solving a ventilation network is now as follows: 1) Input data consisting of airway characteristics and loop designation along with assumed direction of flow, for each branch of the network and any trunk airways, are fed into the computer. 2) Employing the resistance relationship, where R is resistance, K is friction factor, P is perimeter, L is length, A is area, and n is number of airways (where multiple intakes and returns are employed); values are computed for all airways. 3) An arbitrary, continuous route of flow through the circuit—from beginning to end—is selected and the path identified by airway numbers and read in. 4) Based on a total assumed quantity of flow Q (usually 100,000 cfm.) the computer employs the procedure described in the previous article in distributing quantities of flow among all the airways until a satisfactory level of accuracy has been attained (in the current program, the machine is turned off when the cumulative corrective error in cfm, is reduced to five times the number of airways). 5) Airway head losses H are calculated by the relation H = R Q2 and cumulated along the original designated flow route to find the overall head loss for the network. Trunk airways are then included to obtain the mine head.

Jan 1, 1964

By Herman Dykstra, Irving Fatt

Relative wetting phase permeabilities calculated from capillary pressure-saturation data are compared with measured relative permeability data. The equation relating relative permeability to capillary pressure-saturation data is derived by assuming that a porous medium is analogous to a bundle of capillary tubes. The equation includes a term to correct for the difference between fluid path length and length of core. Relative oil permeability data in the water-oil-gas system are presented, and it is shown that if the water is considered part of he rock matrix, the relative oil permeability curve is typical of a wetting phase relative permeability curve. Apparatus for measuring two and three-phase relative permeabilities are described. INTRODUCTION The difficulty in measuring relative permeability of cores has made it desirable to have a correlation between relative permeability and some more easily measured property of porous media. Such an easily measured property is the capillary preisure-saturation relation. In the past there has been a tendency to separate the capillary pressure concept from the very complex pore geometry and to consider that the capillary pressure-saturation curve give? only some characteristic distribution of interfacial curvature between two fluids.&apos; Recently, however, it has been realized that if capillary pressure data are to yield information concerning pore size distribution2 and fluid flow in porous media. the pore must be assumed to have a simple shape such as a cylinder or a sphere. Childs and George&apos; showed that relative water permeability could be calculated from the capillary pressure-water saturation curve by assuming that the pores are cylinders in which the fluid flowing obeys Poiseuille&apos;s Law, and that the capillary preisure curve indicated the size and number of pores. Purcell5 derived an equation relating the permeability of a porous medium to the capillary pressure curve by assuming the porous medium to he analogous to a bundle of capillary tubes. Gates and Lietz6 calculated relative permeability by an equa-tion derived from Purcell&apos;s equation. In this paper an equation is derived giving the relative water permeability as a function of water saturation by as-sulning that the core sample can be represented by a bundle of capillary tubes in which the fluid path length is not the same as the bulk length, and in which the fluid path length varies with saturation. THEORY For a bundle of N capillary tubes the flow through dN tubes will be dQ = qav dN ..........(1) where Q is the total flow rate through all the tubes and q., is the aberage flow rate through the tubes in the interval dN. If the interval dN is made small the average flow rate through a tube in the interval dN is given by Poiseuille&apos;s Law, 8 nl where ?P is the pressure drop across the tube of length 1 and radius r, and µ is the viscosity of the fluid. Substituting for qav in Equation (1), dQ =pr4 ? P/8µl dN.........(3) Darcy&apos;s Law for linear flow of an incompressible fluid in a porous medium is Q = KA?P/µL...........(4) where K is the permeability and ?P is the pressure drop across a bundle of tubes which has replaced the porous medium of length L and cross-sectional area A. Differentiating Darcy&apos;s Law with A, ?P, µ, and L constant, dQ = A?P/µL dK.........(5) Equating (3) and (5) gives dK------------dN.........(6) 8Al If the pores are assumed to be cylinders dV =pr2ldN..........(7) where V is the volume of flowing fluid in the pores. Substituting for dN in Equation (6) gives dK = —dV..........(8) 8AI&apos; By definition the saturation, S, of the core is v v S =V/Vp..........(9) Vv -PAL

Jan 1, 1951

By Irving Fatt, Herman Dykstra

Relative wetting phase permeabilities calculated from capillary pressure-saturation data are compared with measured relative permeability data. The equation relating relative permeability to capillary pressure-saturation data is derived by assuming that a porous medium is analogous to a bundle of capillary tubes. The equation includes a term to correct for the difference between fluid path length and length of core. Relative oil permeability data in the water-oil-gas system are presented, and it is shown that if the water is considered part of he rock matrix, the relative oil permeability curve is typical of a wetting phase relative permeability curve. Apparatus for measuring two and three-phase relative permeabilities are described. INTRODUCTION The difficulty in measuring relative permeability of cores has made it desirable to have a correlation between relative permeability and some more easily measured property of porous media. Such an easily measured property is the capillary preisure-saturation relation. In the past there has been a tendency to separate the capillary pressure concept from the very complex pore geometry and to consider that the capillary pressure-saturation curve give? only some characteristic distribution of interfacial curvature between two fluids.&apos; Recently, however, it has been realized that if capillary pressure data are to yield information concerning pore size distribution2 and fluid flow in porous media. the pore must be assumed to have a simple shape such as a cylinder or a sphere. Childs and George&apos; showed that relative water permeability could be calculated from the capillary pressure-water saturation curve by assuming that the pores are cylinders in which the fluid flowing obeys Poiseuille&apos;s Law, and that the capillary preisure curve indicated the size and number of pores. Purcell5 derived an equation relating the permeability of a porous medium to the capillary pressure curve by assuming the porous medium to he analogous to a bundle of capillary tubes. Gates and Lietz6 calculated relative permeability by an equa-tion derived from Purcell&apos;s equation. In this paper an equation is derived giving the relative water permeability as a function of water saturation by as-sulning that the core sample can be represented by a bundle of capillary tubes in which the fluid path length is not the same as the bulk length, and in which the fluid path length varies with saturation. THEORY For a bundle of N capillary tubes the flow through dN tubes will be dQ = qav dN ..........(1) where Q is the total flow rate through all the tubes and q., is the aberage flow rate through the tubes in the interval dN. If the interval dN is made small the average flow rate through a tube in the interval dN is given by Poiseuille&apos;s Law, 8 nl where ?P is the pressure drop across the tube of length 1 and radius r, and µ is the viscosity of the fluid. Substituting for qav in Equation (1), dQ =pr4 ? P/8µl dN.........(3) Darcy&apos;s Law for linear flow of an incompressible fluid in a porous medium is Q = KA?P/µL...........(4) where K is the permeability and ?P is the pressure drop across a bundle of tubes which has replaced the porous medium of length L and cross-sectional area A. Differentiating Darcy&apos;s Law with A, ?P, µ, and L constant, dQ = A?P/µL dK.........(5) Equating (3) and (5) gives dK------------dN.........(6) 8Al If the pores are assumed to be cylinders dV =pr2ldN..........(7) where V is the volume of flowing fluid in the pores. Substituting for dN in Equation (6) gives dK = —dV..........(8) 8AI&apos; By definition the saturation, S, of the core is v v S =V/Vp..........(9) Vv -PAL

Jan 1, 1951

By C. H. Li, J. A. Sarteli, W. Bonfield

The stress to cause a permanent micros train of 2 x 10-6 in. per in. (defined as the microscopic yield stress) in beryllium is found to be very sensitive to surface condition. The initiation of plastic flow in as-machined specimens, which contain a high density of twins and large residual compressive stresses to a depth of about 0.010 in. from the surface, occurs by the nucleation of slip from high stress fields around twin tips in the surface layers. Removal of the damaged surface layers by chemical polishing results in an appreciable increase in the microscopic yield stress, which is attributed to the remoual of stress raising twins rather than to the release of residual stresses. IT is well known that the macroscopic mechanical properties of beryllium are very sensitive to the nature of its surface finish. Matthews and coworkers1 have shown that the tensile strength of rolled beryllium sheet can be considerably enhanced by chemical polishing, presumably due to the removal of the high concentration of notches, twins, and large residual compressive stresses normally found in the surface layers of ground specimens. There was no similar effect of surface condition on the macroscopic yield stress (the stress to produce 0.2 pct strain), a point to which we shall return. To date there have been no extensive studies on the effect of surface condition on the microstrain region of beryllium. Recent measurements2 of the stress to cause a permanent microstrain of the order of 10-6 in. per in. on machined hot pressed QMV beryllium, gave values ranging from about 600 to 3500 psi. Reasons for the considerable scatter in the results were not discussed. In this paper, we wish to show that the microstrain region of polycrystalline beryllium is, in fact, very sensitive to surface condition. The stress to cause a permanent strain of 2 x 10-6 in. per in. which for convenience we will refer to as the microscopic yield stress, can be increased by almost a factor of 2 by minimizing the amount of surface damage. The variation of microscopic yield stress is attributed to the presence of twins in the surface layers, rather than to residual compressive stresses. EXPERIMENTAL PROCEDURE The starting material for this investigation was commercial purity hot pressed QMV beryllium, with an average grain size of about 20 . The effects of surface treatment were assessed by comparing the microstructure with residual stress measurements and microstrain properties. Specimens were prepared for metallographic examination by wet grinding with 240, 400, and 600 papers, followed by polishing with a suspension of fine alumina in warm 20 pct oxalic acid. After this treatment, the surface could be examined, either in the unetched condition with polarized light, or after etching in 2 pct hydrofluoric or boiling 20 pct oxalic acid with normal light. Thin films of beryllium suitable for electron transmission study were prepared from bulk specimens by chemical polishing with hydrofluoric and phosphoric acids, and were examined in an RCA-EMU-3 electron microscope. Residual stress measurements were made by a procedure similar to that of Treuting and Read.3 Specimens 4.0 in. by 0.625 in. by 0.080 in. were prepared by three different machining procedures, namely: turning, grinding, and sawing, in order to introduce different levels of residual stresses. An electrical resistance strain gage was attached to one side of the specimen, and that surface was then protected by an acid-resistant enamel, while the opposite surface was removed by etching approximately one thousandth of an inch at a time. The change in specimen curvature in the longitudinal direction was noted and the residual stress was calculated by the following formula:

Jan 1, 1963

By H. R. Hose

The total crude fluorspar reserves in Western Europe, including the USSR and the Soviet sphere, Thetotalare estimated to exceed 17 million metric tons. The total fluorspar production in 1951 in Western Europe, excluding the USSR and Soviet sphere, amounted to 371,109 metric tons. The chief fluorspar-producing countries in Europe, excluding Russia and the Soviet sphere, are France, Western Germany, Italy, Spain, and the United Kingdom. TOTAL crude fluorspar reserves in western Europe, containing more than 35 pct CaF2, are estimated at 12 million metric tons, while reserves in the USSR and Soviet sphere may exceed 5 million metric tons. The bulk of these reserves are concentrated in the USSR, Spain, United Kingdom, Western Germany, and Italy. Since published estimates of the tonnages of individual mines and explored deposits are not available for most countries in Europe, the estimates given in Table I for individual countries are necessarily approximate, as they have been compiled from various sources and modified and combined according to the author&apos;s judgment. The majority of fluorspar deposits in Europe belong to the late Paleozoic metallogenetic epochs and have been found within or along the borders of the Hercynian massifs. Deposits in the Central and Esterel massifs of France, those of Western and Eastern Germany, Sardinia, the Catalonian coastal range and the Sierra de los Santos in Spain, and those of Devonshire and Cornwall belong to the Hercynian metallogenetic epochs. The deposits in southern Norway and Sweden belong to the middle Paleozoic or later, those of northern England to the late Paleozoic or later, those of northern Spain and Greece to the late Mesozoic, and those of northern and probably central Italy and Switzerland to the early Tertiary. Fluorspar occurs typically in veins, lenses, and stringers, but also in bedded replacement deposits and pockets, and may be colorless to white, yellow, green, pink, brown, blue, or purple to almost black. Except for the colorless variety, the only kind which can be used for the optical industry, the color of the fluorspar does not affect its ultimate use. Crystalline cubic or octahedral fluorite of CaF2 composition, with 48.9 pct fluorine and 51.1 pct calcium, has a specific gravity of 3.18. The ore in place, however, usually averages 3.0 metric tons per cubic meter owing to the quantity of other minerals asso- ciated with the fluorspar in the veins and the physical state of the fluorspar, which may vary from sugary to massive crystalline or microscopic inter-growths with barite or quartz. Barite of 4.5 sp gr, quartz of 2.66 sp gr, and chert and calcite are commonly associated with fluorspar. Other minerals usually present in lesser amounts are celestite, wad, silver, lead, zinc, copper, iron, and uranium minerals. In some cavity fillings the rhythmic deposition of the fluorspar, barite, and quartz into successive crusts may make it possible to handpick the crude ore on belts or to separate the fluorspar from other minerals while the actual mining is carried on. Where metallurgical grade lump spar is required for foundries, separation of quartz and barite may be made also by jigs and heavy media, and in some mines the lead and zinc sulphides may be separated and recovered by vibrating tables. However, except in the case of unusually high grade deposits, acid grade fluorspar is obtained by flotation. By this process fluorspar deposits grading down to 35 pct CaF2 in the crude ore can be worked at a profit when fluorspar, lead, zinc, and silver prices are high, as the sulphides of these ores can be separated from the fluorspar by flotation and concentrated.

Jan 1, 1956

By H. B. Pulsifer

.ABOut 1.5 varieties, or tnodifications, of the best rnagnesiurn available were prepared and subjected to etching tests, then examined for micro-structure. Of the 30-udd etching reagents that were tried, nearly half, mostly ammonium salts, etched the metal satisfactorily. The surfacing and etching of magnesium is shown to be a very simple and quick operation. The density and hardness of magnesium were determined. The most interesting new observations relate to the finding of the hexagonal etch figures, the crystal laminations, and the fact that the metal is plastic, cold, when restrained or quickly deformed. Under slowly applied pressule cold metal deforms slightly, then shears to fracture without plastic flow. The chief structural features of magnesium arc presented in two reduced photographs and 30 photomicrographs. Materials Tested There are only two sources in the United States from which new metal can be procured: the American Magnesium Corpn. and the Dow Chemical Co. No pronounced structural or property differences in the metals from these two companies were disclosed by the work of this investigation. The following materials were obtained from the manufacturers: 1. Massive crystals of distilled metal (American Magnesium Corpn.). 2. Rods of hot-extruded metal, ½-in. squares and 5/8-in. rounds (American Magnesium Corpn.). 3. Sheet magnesium, 0.005-in. thick (American Magnesium Corpn.). 4. Cast stick metal, 13/8-in. dia. (Dow Chemical Co.). 5. Magnesium-aluminum alloy, hot-rolled plate, ½-in. thick (American Magnesium Corpn.). From these materials, were prepared: 6. Sections from a furnace-cooled ingot made from clistilled crystal. 7. Cold-strained pieces from (6), (4) and (2). 8. Pieces cold-squeezed to fracture from (6), (4) and (2). 9. Hammer-struck pieces from (6), (4) and (2). 10. Steam-hammer struck pieces from (6), (4) and (2). 11. Sections from furnace-cooled ingot of the magnesium-aluminum alloy.

By H. D. Outmans

In steady vertical flow, the interface of an immiscible liquid-liquid displacement is horizontal for any flow rate below the critical. In nonvertical flow, however, the shape of the interface in the steady state does depend on the flow rate, and the purpose of this paper is to calculate the unsteady interfaces during the transition of one steady state of flow to another. A knowledge of these transient interfaces is of considerable importance in reservoir engineering where the calculation of breakthrough recovery depends on the instant the interface reaches the producing wells and on the shape of the interface at that time. Although the emphasis is put on transient interlaces, which eventually approach stable equilibrium, it is shown that if the displacement exceeds a critical rate no equilibrium is possible. The interface is then unstable and viscous fingers are formed during the displacement. The critical rate and the shape of the transient and equilibrium interfaces are affected by the effective interfacial tension; but since this effective interfacial tension appears in the calculations only in combination with the inverse square of the thickness of the medium, its effect in the reservoir would appear to be negligible compared to its significance in model experiments. INTRODUCTION Stability criteria and the early growth of interfacial disturbances in a plane parallel to the boundaries of a dipping formation in which oil is displaced by an incompressible fluid were described in a previous paper.l This type of instability is significant in thin reservoirs. However, if the reservoir has appreciable thickness, then interfacial stability in vertical planes, normal to the upper and lower boundaries, also becomes important (the displacement is supposed to be parallel to these vertical planes). The difference between the two stability problems is that, in the first case, the intersections of the interface with planes parallel to the boundaries are normal to the direction of the displacement; in the second case, the intersections, this time with vertical planes, are not normal to the displacement. Instead, they are tilted at an angle which depends on the displacement rate. The tilt of steady interfaces was calculated by Dietz2 who also determined the critical rate of displacement for stability in the vertical plane by assuming that this rate would coincide with an interfacial tilt equal to the dip of the formation. The critical rate thus calculated is the same as has been found for thin reservoirs (see Eq. 1.1 of Ref. 1 and of the present paper). Dietz&apos;s calculation of the stable tilt was verified by laboratory experiments and the agreement was found to be fairly good.3 It is doubtful, however, that stable tilts actually exist in the reservoir because a change in production rate is not followed by an instantaneous adjustment of the interface to the new rate but, rather, by a transition period during which the interface changes from one equilibrium tilt to the other. The principal objective of this paper has been to describe these transient interfaces without putting any restrictions on the flow conditions or the shape of the interface, as had been done previously. The second objective was to compute the critical velocity, taking into account capillary effects, and the third was to evaluate, at least qualitatively, the shape of the front at rates above the critical, again without making the simplifying assumptions introduced by previous investigators.2,3 In the following sections two examples are given of the calculation of interfacial motion. The first describes this motion for an initially horizontal interface in a dipping layer, and the second for a vertical interface in a horizontal layer. The mathematical formulation of the problem is nonlinear in the boundary conditions, and this prohibits its solution in closed form. Instead, the solution is obtained in the form of higher-order approximations. 1 Before proceeding to a description of the mathematical model, however, we define two quantities

By H. M. Myers, J. E. Funk

The two-phase flow of a drilling fluid in a diamond drill bit is investigated by deriving and solving the steady-state continuity and momentum equations. Parameters in this analysis are rotational speed of the bit, local rate of addition of drill solids, non-Newtonian nature of the drilling fluid and drilling rate. The describing equations relate the flow and pressure drop to the design characteristics of the bit. The equations are integrated numerically yielding the flow-rate pressure drop characteristics for the bit, flow distribution in the bit and velocity and static pressure as a function of position in each flow channel of the bit. Results are presented for a particular bit design. This paper shows that under normal drilling conditions certain parameters may be neglected in the hydraulic calculations for a diamond drill bit. Therefore, shortened forms of the equations presented can be used to fit the proper diamond bit to the drilling fluid hydraulic system. INTRODUCTION Like the conventional bit, the diamond drill bit is a key component in the circulating system of a well. The conventional rock bit is fitted to the circulating system by selecting the flow nozzles which provide the system with the proper flow-rate pressure drop relationship. Fluid behavior across the face of the diamond bit must be better understood so the bit can also be fitted to the circulating system. Data from field-operated diamond bits are presently available and do yield some information regarding the flow-rate pressure drop relationship. However, the number of factors affecting this relationship is quite large. Drilling parameters and conditions such as bit penetration rate, bit weight and rotational speed are important. Moreover, flow channels in the bit are part of the cutting head, and introducing drilled solids into the flow channels reduces the flow area, causes additional drag and influences the pressure drop. Many tests would have to be performed if field tests were required to study all parameters which might influence fluid behavior in the bit. A logical approach to this problem is to investigate analytically the fluid behavior in the bit, including parameters that are present under normal drilling operations. This approach, which seems to offer the best method for future studies, is presented in this paper. Following is the general procedure. 1. Equations describing the most general flow channel are written and include the addition of drilling solids (which is related to penetration rate), rotation of the bit, design parameters of the bit and non-Newtonian properties of the fluid. 2. A typical bit is designed and the bit parameters are substituted into the equations. 3. Parameters are assumed (penetration rate, rotational speed of the bit, mud weight, etc.). 4. The describing equations are numerically integrated to obtain the pressure drop across the face of the bit as a function of flow rate. 5. The pressure-drop contribution of each component (friction, acceleration, etc.) is obtained from the numerical solution, and those factors which largely influence the total pressure drop are identified. Classifying the drilling fluid is important in establishing the friction factor relationship since the frictional pressure losses are the main contributors to the total pressure drop. High-density drilling fluid is commonly assumed to be a Bingham plastic. 1-3 Govier4 presented a method for selecting a friction factor in the laminar flow region from the rheology properties of the mud and a modified Reynolds number. Metzner5 presented a criterion for selecting the friction factor in the turbulent flow region. The bit design used in conjunction with the solution of the equations has divergent flow channels. Stall, which is the backflow of fluid in the channels due to eddies that occur on the outer perimeter of the flow stream, can occur in these divergent channels (or diffusers). When stall does occur the static