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Jan 1, 1900

By W. R. Hibbard, Ming-Kao Yen

Various rationalizations of preferred orientations in cold worked polycrystalline metals have been based on the operation of at least two three2 five, an indefinite number, and all,6 slip systems of the type occurring in single crystals. Pickus and Mathewson3 suggested that the cold rolling texture in face-centered cubic metals could be rationalized on the basis of two or more operating slip directions symmetrically disposed about the direction of flow in such a manner that the resolved shear stress factor is equal for all operating slip systems and lattice rotations cancel each other. The importance of the direction of flow, the necessity for the operation of slip systems capable of producing the required changes in external dimensions, the maintenance of a common surface between adjacent grains and a symmetrical arrangement of the operating slip systems in order to produce an end orientation which is stable under additional deformation lead to the following rationalization of textures based on the type of slip systems operative in single crystals. The rationalization is intended to explain why certain orientations are suitable for deformational flow without a change in orientation and thus are stable end orientations or ideal texture orientations. It does not attempt to explain how the grains of a randomly oriented polycrystalline aggregate will reach this orientation. The rationalization is based on the following four postulates: 1. The ideal textures for polycrystalline aggregates are the result of reorientations of individual crystals toward the same stable end orientation at which no further reorientation will occur even during additional deformation. Since, in simple slip, the relative orientations of the stress axis and the active slip system determine the change in orientation during slip, it is considered that the orientation of the stress axis with respect to the ideal texture determines what slip systems must be active to retain the stable end orientation. Therefore the stable end orientation should be a symmetrical one around the stress axis with respect to the active slip systems such that lattice rotations resulting from slip will cancel each other and prevent a resultant change in orientation. (Rotational symmetry only—not translation, inversion, nor others—is considered here.) Thus a metal under a pure stress should develop a texture consisting of one or more of the crystallographically symmetrical directions, <100>, <110> or <111> for cubic metals or <0001>, <1010> or <1120> for hexagonal close-packed metals, parallel to the stress axis. 2. Since the active slip direction is the direction of glide during simple slip and thus the direction of maximum shear movement, in the ideal texture the active slip

Jan 1, 1949

By Ming-Kao Yen, W. R. Hibbard

Various rationalizations of preferred orientations in cold worked polycrystalline metals have been based on the operation of at least two three2 five, an indefinite number, and all,6 slip systems of the type occurring in single crystals. Pickus and Mathewson3 suggested that the cold rolling texture in face-centered cubic metals could be rationalized on the basis of two or more operating slip directions symmetrically disposed about the direction of flow in such a manner that the resolved shear stress factor is equal for all operating slip systems and lattice rotations cancel each other. The importance of the direction of flow, the necessity for the operation of slip systems capable of producing the required changes in external dimensions, the maintenance of a common surface between adjacent grains and a symmetrical arrangement of the operating slip systems in order to produce an end orientation which is stable under additional deformation lead to the following rationalization of textures based on the type of slip systems operative in single crystals. The rationalization is intended to explain why certain orientations are suitable for deformational flow without a change in orientation and thus are stable end orientations or ideal texture orientations. It does not attempt to explain how the grains of a randomly oriented polycrystalline aggregate will reach this orientation. The rationalization is based on the following four postulates: 1. The ideal textures for polycrystalline aggregates are the result of reorientations of individual crystals toward the same stable end orientation at which no further reorientation will occur even during additional deformation. Since, in simple slip, the relative orientations of the stress axis and the active slip system determine the change in orientation during slip, it is considered that the orientation of the stress axis with respect to the ideal texture determines what slip systems must be active to retain the stable end orientation. Therefore the stable end orientation should be a symmetrical one around the stress axis with respect to the active slip systems such that lattice rotations resulting from slip will cancel each other and prevent a resultant change in orientation. (Rotational symmetry only—not translation, inversion, nor others—is considered here.) Thus a metal under a pure stress should develop a texture consisting of one or more of the crystallographically symmetrical directions, <100>, <110> or <111> for cubic metals or <0001>, <1010> or <1120> for hexagonal close-packed metals, parallel to the stress axis. 2. Since the active slip direction is the direction of glide during simple slip and thus the direction of maximum shear movement, in the ideal texture the active slip

Jan 1, 1949

By M. C. Leverett

THE dynamic behavior of a multiple fluid system is completely describable in terms of driving forces and resistances to flow. The latter are proportional to the vis-cosity of the fluid under consideration and inversely proportional to a property of the system termed the " effective permeability" to that fluid, a property therefore defined by this relation. Since three phases are usually present in oil reservoirs, it is essential to know the effective permeabilities to the several fluid phases, and the relationship between the composition of the fluid in the pore space and that of the flowing stream, in order to handle satisfactorily certain problems of oil production. In this regard, there are many unknown factors and parameters, and many that can be evaluated only indi-rectly. However, it is expected that work along lines similar to this present under-taking will help materially in solving these problems. Previous workers in this field have inves-tigated the two-phase systems, water-gas,1 water-oil2 in unconsolidated sands, and water-gas3 in consolidated sands. The ex-tension of the study of the flow of hetero-geneous fluid systems from two to three phases involved only slight changes in technique. The apparatus used in the present work, except for slight changes to be described, was that used by Leverett in his water-oil experiments.2 Correlation and interpretation of data were made by methods analogous to those used in two-phase work. The relative permeability to a phase-the effective permeability divided by the permeability of the system to a homogeneous fluid-was employed through-out in correlating the rate of flow of that phase with the composition of the pore space. Quantitative measurements of two inde-pendent physical properties of the system were required to determine the composition of the pore space. As in the two-phase work, the fraction of water was determined electrically. The gas fraction was deter-mined by measuring changes in volume accompanying measured pressure changes. Suitable account was taken of the solubility of the gas in the liquid phases, and of the vapor pressure of the water phase. Since this work was exploratory in nature, no attempt was made to obtain highly accu-rate numerical results. Reasonable precision and reproducibility were sought.

Jan 1, 1940

By G. W. P. Rengstorff, R. B. Fischer

A study was made of the effect of impurities on the bend ductility of cast molybdenum. High purity molybdenum was repared by re-melting under high vacuum. The ductility of "transverse-grain" specimens indicated that intergranular brittleness decreased with an increase in purity of the metal. THE production of as-cast molybdenum with high ductility at room temperature would make working of the metal easier and increase the usefulness of the as-cast metal. Furthermore, a .knowledge of the factors that influence the ductility of as-cast molybdenum might be applied to the problem of making fusion weldments having satisfactory room-temperature ductility. Individual grains of commercial as-cast molybdenum are slightly ductile at room temperature. However, the presence of carbides at the grain boundaries may cause the metal to fracture in a brittle manner. This behavior has been discussed.&apos; The amount and distribution of the impurities in as-cast molybdenum would be expected, from previous research on the metal, to have considerable influence on its ductility. Some effects of oxygen and carbon on the mechanical behavior of the metal were discussed by Parke and Ham,2 Woodside,3 and Zappfe. Landgraf, and Worden.4 Parke and Ham2 found that, if a sufficient number of molybdenum oxide particles were present at the grain boundaries, the intergranular cohesion was reduced to the extent that arc-melted molybdenum would rupture intergranularly during hot forging. They also found that if the oxygen content was less than 0.003 pct, the cast molybdenum was hot forgeable. It is suspected, however, that forgeable molybdenum may contain a small amount of oxide that could reduce the ductility of the as-cast metal at room temperature. Also, there is the possibility of the unfavorable distribution of other impurities having a detrimental effect on the ductility of cast molybdenum. The results of a study of the influence of purity on the room-temperature ductility of cast molybdenum are presented here. Vacuum Arc-Melting to Produce High Purity Mo One way of determining the effect of impurities on the ductility of molybdenum is to prepare and test metal as free of all impurities as possible. Accordingly, the present investigation was undertaken with this in view. Arc melting in high vacuum (less than 0.1 micron of mercury) was chosen as the method for reducing the impurities below the concentrations occurring in commercial molybdenum. Commercial arc melting of molybdenum is carried out at about 50 microns pressure. During the arc melting, carbon and oxygen in the metal react to form carbon monoxide. By virtue of the vacuum process, the partial pressure of carbon monoxide is maintained at such a low value that the reaction greatly lowers oxygen and carbon contents. An excess of at least 0.03 pct C is used in commercial arc melting to insure that the oxygen content will be below 0.003 pct. The excess carbon remaining after the deoxidation step forms carbides on freezing. Some of the carbides precipitate at the grain boundaries and, as will be shown, have an unfavorable effect on the room-temperature ductility. In the present work, only the small amount of carbon (0.003 pct) contained in the commercial powder-metallurgy molybdenum used for melting stock was present. Partial removal of oxygen from the melt probably occurred as the result of the volatilization of molybdenum oxides, or by distillation of oxygen at the high melting tem-

Jan 1, 1953

Jan 1, 1890

By M. Prats, D. G. Russell

The performance of a well in a bounded, layered reservoir with interlayer crossflow has been investigated mathematically. The system studied comprises a centrally located well in a bounded cylindrical reservoir composed of two layers of contrasting physical properties. The reservoir contains a single fluid of small and constant compressibility. In the case of a well producing at constant pressure, the production rate of the crossflow system declines exponentially after the early transient behavior dies out. For wells which produce with constant rate, the pressure-time relationship is linear following the transient. For very early production time, the production rate of the two-layer crossflow system behaves as that of the two-layer stratified system (the case in which the layers are in communication at the well only). Except for early time, in most practical cases the performance of a two-layer reservoir with cross-flow can be duplicated by that of a single-layer reservoir. The equivalent single-layer reservoir must have the same pore volume and the same drainage and wellbore radii as in the two-layer case. In addition, the equivalent reservoir must have a "kh" (permeability x thickness) product equal to the sum of the "kh" products of the respective layers in the crossflow system. If the porosity and permeability distributions are known, the production and pressure behavior of a well with and without crossflow can be predicted. From a comparison of the observed pressure or production performance of a well and the theoretical behavior, the occurrence of interlayer crossflow can be detected. INTRODUCTION When flow is possible across the bedding plane from a low-permeability layer into an adjacent layer of moderate or high permeability, the earlier depletion of the more permeable layer will cause such flow to occur. If this situation exists, the oil recovery from tight layers can be appreciable; thus, the economic ramifications of interlayer crossflow are apparent. The possible importance and extent of flow between adjacent and connected layers — whether resulting from capillary, gravitational, or viscous forces — have been the subject of considerable speculation for many years. Jacquard1 presented a rigorous analysis of the crossflow due to single-fluid expansion occurring between two adjacent and connected layers when the well produces at constant rate. His presentation was entirely mathematical, however, and no numerical results from evaluation of the mathematical solutions were presented. Our objectives in the preparation and presentation of the present paper are essentially twofold— (1) to analyze the behavior of a well which produces a single compressible fluid from a bounded, layered reservoir with interlayer crossflow, and (2) to present practical numerical results and simplified formulas to facilitate engineering analysis of reservoirs with crossflow. The constant-terminal-pressure case was analyzed rigorously, whereas the constant-terminal-rate case was treated only for semisteady-state conditions. DESCRIPTION OF THE IDEALIZED TWO-LAYER RESERVOIR The reservoir to be studied is assumed to be bounded and cylindrical in shape, with a completely penetrating well at its center. The reservoir is composed of two adjacent horizontal layers between which fluid is free to flow. Each layer is assumed to be homogeneous, isotropic and completely filled with a fluid of small and constant compressibility. The total thickness of the reservoir is small enough

By Paul A. Beck, M. N. Parthasarathi

By determining the (220) pole figure for OFHC copper rolled to 96 pct R. .A., the occurrence of four texture components of the type (135) [211] was confirmed. It was found that the total volume fraction of material close to the four symmetrical orientations of this type is larger than the volume fraction of the other components put together. THE rolling texture of copper was described by early investigators1&apos;" as a mixture of components of the (110)[l12] and (112) [11l] types. Other studies, still using the qualitative photographic method of texture determination arrived, instead, at orientations of the type (135) [533] or (135) [211] for copper3&apos;4 and at the closely related "Z-texture" for a fee nickel-iron alloy.5 The prevalence of this type of orientation, near (123) [412], in the rolling texture of aluminum and of copper was later confirmed by means of quantitative diffractometer methods.&apos; However, in a recent paper7 Jones and Fell stated that the (110) [112] and (112) [111] double texture was in better agreement with the measurements of Young&apos;s modulus for cold-rolled copper.&apos; These authors also expressed their belief that the "roll texture" can be determined by means of two photographic quasi-fiber texture patterns taken with sheet specimens rotated around the rolling direction and the transverse direction, respectively. Jones and Fell found7 that the results obtained from such patterns for cold-rolled nickel did not agree with any of the orientations mentioned above, but they suggested that the observed diffraction spots may have been brought about by the overlapping of reflections due to the (110) [112] and (112) [11l] double texture components. Jones and Fell concluded that the (123) 14121 type orientation is inconsistent with their patterns. They also stated, without offering any supporting evidence, that this orientation is inconsistent with the (220) pole figure. "Ideal orientations" are often used for a very rough description of the principal orientations encountered in a texture, It is, however, well to remember the fact that, no matter how "accurate", these ideal orientations are only schematic representatives of a continuous distribution9, which often (as in the case of rolled copper) includes extensive orientation spreads. The attempt by Jones and Fell7 to use elastic modulus data in deciding between various ideal orientations in the rolling texture of copper neglects this fact. The inadequacy of two-dimensional "axis density figures" in describing orientation distributions in sheet textures was previously discussed.9 It was also pointed out9 that there is no reliable evidence to prove that orientation distributions derived by the "quasi-fiber method" from rotated sheet specimens are fully equivalent to the corresponding information obtainable from quantitative pole figures. The view that a sheet texture could be adequately determined by means of two photographic quasi-fiber texture patterns is obviously erroneous9. The question as to the consistency with the (220) pole figure of ideal orientations derived from the (111) and (200) pole figures is, on the other hand, certainly very relevant. Since no quantitative (220) pole figure for cold-rolled copper appears to be available in the literature, the following work was undertaken. EXPERIMENTAL METHOD An extruded rod of OFHC copper, 1 in. in diam, was given alternate rolling (30 to 35 pct R.A.) and

Jan 1, 1962

By Herbert Banks Johnson

DURING the past 10 or 12 years very little information has been made generally available concerning the commercial possibilities of separating materials by means of static electricity; and yet during that same period important improvements have been made in various branches of the science of electricity, which, when taken together, and applied to electrostatic separation, have resulted in important improvements in that art as well. For example, much improvement has been made in high-voltage electrical equipment such as full-wave vacuum-tube rectification and improved mechanical rectification. Great strides have likewise been made in the development of insulators for high-voltage work, not only in the materials used but also in the design of the insulators themselves. Again, in another field, the development of air-conditioning equipment has improved the technique of controlling the surface condition, through drying or humidifying, of the various materials that are passed through the electrostatic separator. In many authoritative publications and textbooks, the electrostatic separation process is declared to require a balance far too sensitive for general commercial application. For example, one publication1 in the past has referred to an unfavorable effect on the process caused by varying atmospheric conditions such as humidity. Some2 have even gone so far as to state that sunlight and shadows may seriously affect the separating efficiency of the process. Such statements may have been true in the early development of the art, when the source of static electricity was derived from a friction disk machine, such as the Wimshurst. The Blake Mosher equipment and the work of Huff in applying motor- generator sets, transformers and mechanical rectification overcame much of this kind of objection to the commercial application of the electrostatic separating process. Another objection made periodically is that it is generally considered necessary to feed the particles into the electrostatic field in a stream that

Jan 1, 1938

By K. S. Lane

The problem of stability of slopes in reservoirs is one of real interest in civil engineering. The problem is whether the valley walls will remain stable or whether they are likely to slide as conditions are changed by storage of water in the reservoir. If major slides are indicated, as on the walls of relatively steep canyons, one must estimate whether they will be so located and of sufficient size as to interfere with operation of the reservoir-as by blocking the outlets, or by displacing sufficient water to over-top the dam in a huge wave. Some cases involve problems in soil mechanics for slides in earth or weaker rock; other locations need the assistance of rock mechanics for slides in harder rock. While this is not a new problem, attention has recently been focused on its importance by the hugh landslide into the Vaiont reservoir in 1963, which claimed the lives of over 2000 people and forced abandonment of the dam and powerhouse. VAIONT RESERVOIR SLIDE Fig. 1 shows Vaiont Dam in Italy which is a double curvature arch, 875 ft high, located on a tributary of the Piave River, north and west of Venice. When completed in 1960 it was the highest arch dam in the world. In 1960 a sizable landslide (some 700,000 cu yd) occurred on the left abutment a short distance above the dam. Following this, the reservoir was held down for construction of a by-pass tunnel through the right valley wall to insure connection with the upper end of the reservoir in event of further slides. In October 1963 a giant landslide occurred on the left valley wall, over a width of some 14 miles (see Fig. 2). Over 300 million cu yd slid into the reservoir, displacing the water and forming a hige wave 700 ft above the top of the dam as it hit the opposite, or right, wall of the valley. The wave then passed some 500 ft over top of the dam, followed down the tortuous winding canyon and debouched into the

Jan 1, 1967

Jan 1, 1948

By J. D. Miller Discussion by N. P. Finkelstein, M. C. Fuerstenau

The point designating 10 -4 nole/per liter sulfonate addition in Fig. 1 has been presented incorrectly. This figure, with the corrected scale, is shown at right. N. P. Finkelstein (National Institute for Metallurgy, Johannesburg, South Africa) - The qualitative conclusion the authors of this paper have reached in evaluating the influence of hydrocarbon chain length on collector effectiveness agrees with conclusions in several published reports. However, in two other

Jan 1, 1969

Jan 1, 1894

By M. F. Burr, F. V. Lenel, G. S. Ansell

The mechanism of materials transport was studied in a model experiment in which a silver sphere was sintered to a silver plate under an applied external stress near 800° C. For this purpose the rate of growth of the neck between the sphere and the plate which is proportional to the strain rate of the creep process involved in a neck growth was determined by measuring the change in electrical resistance of the away. This rate of neck growth will depend upon the magnitude of the applied external stress. By measuring the rate just before and just after a change in the external load and therefore the applied stress, the relationship between strain rate and stress can be derived. For material transport by slip in a pure metal it is expected that the strain rate should be proportional to the 4.5th power of the stress, while for transport by diffusional flow the strain rate should be proportional to the first power of the stress. It was found that for stresses above 5 X 10&apos; dynes per sq cm (70 psi) the exponent in the stvain rate-stress relationship was near 4.5 while for a stress of 0.7 x 108 dynes per sq cm (10 psi) an exponent of 1.8 was determined indicating that under the conditions of the experiment the transport mechanism for creep in silver shifts from diffusional flow to slip in the region of 0.7 x 106 dynes per sq cm (10 psi). The importance of these findings for an understanding of the mechanism of material transport in conventional sintering (without an external load) is pointed out. In a paper by Early et al.,&apos; the mechanism of material transport during sintering was treated as a creep process. It was pointed out that two possible types of creep processes could operate during sintering. In one of these material transport takes place by vacancy diffusion due to a gradient in chemical potential. The vacancy sinks in this diffusion process may be either grain boundaries, as in Herring-Nabarro microcreep, or regions beneath a surface of small radius of curvature. The diffusive transport may be by volume, grain boundary, or surface diffusion. In the other mechanism, creep takes place by slip, i.e., dislocation motion with its rate generally controlled by dislocation climb. One method of distinguishing between these two modes of materials transport is to investigate the creep rate, i.e., the rate of material transport as a function of stress. For material transport by vacancy diffusion, the creep rate is proportional to the stress.&apos; For creep by slip, controlled by dislocation climb, the creep rate is proportional to a power of the stress. For pure metals this power is 4.5;374 for dispersion-strengthened material a fourth power relationship can be derived.5 This difference in the stress dependence of the strain rate is illustrated schematically in Fig. 1, in which the logarithm of strain rate is plotted vs the logarithm of the stress for material transport in a pure metal by vacancy flux and by dislocation climb. The stress range in which the two types of material transport overlap would be expected to be relatively small. In the experiments described by Early and coworkers compacts pressed from spherical copper powder were sintered isothermally under an applied axial external load in a dilatometer. During sintering the magnitude of the applied external load was changed and the rate of axial shrinkage of the compact before and after the load change determined. In order to find the stress dependence of the strain rate the magnitude of the stress acting on the neck area between the spherical particles before and after the load change had to be calculated. This stress is the resultant of the stresses due both to the externally applied load and to surface tension. The stress due to surface tension is a function of the geometry of the powder compacts. It can be determined, at least for simple geometrical arrays,&apos;76 but it is difficult to obtain exact values of the stresses for actual powder compacts when only average values of particle diameters and of neck diameters between particles are available. The calculation of stress and the strain rate-stress relationship is therefore only very approximate.

Jan 1, 1968

Jan 1, 1944

Jan 1, 1949

Jan 1, 1943

By Henry D. Hibbard

FoR the purpose of this paper all steels will be divided into two divisions: effervescing and non-effervescing. This classification must be borne in mind as many statements true of one class are not true of the other. Effervescing steels are those that effervesce in the molds, evolving large volumes of gases, which, in escaping, throw up a spray of molten particles of steel and give the molten metal in the mold a rapid stirring or churning motion until it freezes, or, at least, becomes pasty. This effervescing is intended and proper. The non-effervescing steels, which are more or less completely killed, should not evolve any gas in the molds. Many years ago the writer proposed to call the two classes of steels evolution and solution steels, depending on whether the gases in the molds are evolved or kept in solution, but the present designations seem to be better. In America, at least, effervescing steels are in a class by themselves and include steels made for boiler, tank, and ship plates, steel pipe, soft steel wire, wire nails, soft machincry, and structural steels. These are all low-carbon steels with less than 0.40 per cent. carbon. Low-carbon steels that do not effervesce but are killed are also made; these are put into ingots and castings but they are outside the limits of this paper. Effervescing steel is cast into square or slab molds, either top or bottom poured. Most of the effervescing steel is made in the basic open-hearth furnace, though much is made by the acid Bessemer process and a little is made in the acid open-hearth. It is not made by the crucible or electric-furnace process, in which the volume of gas required cannot be produced in the steel and evolved from it in the mold to give the effervescence desired. Any steel ingot, when teeming is finished, is of a certain size or volume. Leaving out of consideration for the moment the lessening of the over-all dimensions of the ingot through contraction due to loss of heat, the ingot may be placed in one of three classes: It may settle, losing in volume; it may stand, that is, remain constant in volume; it may rise, that is, increase in volume. Into which class an ingot will fall depends on the gases of the steel,

Jan 1, 1920

By Richard P. Rothwell

Fires in mines are so serious in their consequences and of such frequent occurrence, that their causes and the means of extinguishing them are certainly questions of the greatest interest to a large part of the engineering profession. We have already, in this country, many mines which have been burning for years, and though our hard anthracite is so difficult to ignite that for a quarter of a century after it was first discovered it mas not known how to burn it except with the aid of an artificial blast, yet most of the fires which have occurred underground have been in anthracite mines, and in the coal itself. The injury which these fires have caused to properties in various parts of the anthracite regions, and the cost of extinguishing them, would amount to many millions of dollars, and they have also occasioned the loss of many valuable lives. It is not surprising, therefore, that the subject has already attracted much attention, and the exercise of great ingenuity; and the present communication is made less with the expectation of announcing anything new, than with the object of putting upon record the present state of our knowledge on this subject, or, in other words, of indicating how defective are the present means of combating underground conflagrations. Causes of Mine Fires.—Even upon the surface, fire is a terrible foe to contend with, and there is probably no other which inspires such thorough and well-founded alarm. With what feelings, then, must we look upon this destructive element, when its field is in the narrow galleries of a mine, where the poisonous products of combustion, spreading in every direction, protect, in the most effectual manner, the fire itself from the attacks of extinguishers, while, if the face of attack be ventilated so as to enable then] to approach the seat of the fire, the air current simply increases the extent and violence of the conflagration. Fires underground originate in as various ways as those on the surface, and it is scarcely necessary to say that, except in the rare cases where they are the result of design, they are invariably classed as " accidents," and most generally as " unavoidable accidents." In reality, by an intelligent understanding of their causes and with due precautions, they can almost always be prevented. Ignorance and

By D. L. Katz, M. J. Rzasa

The derivations of three methods of computing the static pressure gradients in natural gas wells have been presented to show the assumptions made. Charts were developed from which the pressure gradients may be read when the well-head pressure, the well fluid gravity, depth, and the average well temperature are given. A chart for estimating the well fluid gravity from the condensate content and separator gas gravity is included. The effect of the increased average well temperature after flow on the calculation of the static pressure gradient is discussed. Introduction Reservoir pressures have been calculated from well-head pressures for gas wells for many years. As the pressure measurements become more accurate, the need for a reliable calculation of the static pressure gradient often arises. This paper will develop the several methods for computing the pressure gradient in gas wells and make a comparison between them. The method of calculating static pressure gradients in common use is Eq. I (ref. I) or its counterpart which includes a factor for the deviation of the gas from the ideal gas law. P2 - P1 = P1(e0.0000347GX _ I) Eq. [I] in which P1 = pressure at well head, lb. per sq. in. abs. PP = pressure at bottom of well, Ib. per sq. in. abs. G = gas gravity X = depth of well, ft. An alternate method is to compute the average density of the gas in the well and multiply by the well depth in a manner similar to that used for liquid gradients. Derivation of Formulas A static pressure gradient is a special case of the general fluid-flow equation. Consider a pound of fluid flowing in a vertical column from point I to point 2, Fig. I. By an energy balance for fluid flowing from I to 2: U1 + P1V1 + u12/2g + X1 + q = U2 + P2V2+ u22/2g+X2 +W [2] in which U = internal energy, ft-lb. per lb. P = pressure, lb. per sq. ft. V = volume, cu. ft. per lb. u = average linear velocity, ft. per sec. X = height above datum, fl. q = heat absorbed by fluid, ft-Ib. pcr lb. W = work done by system, ft-lb. per lb. An energy balance on the fluid itself gives: U = ?TdS - ?PdV + etc. [3] in which T = temperature, deg- R- S = entropy, ft-lb. per deg. R. per lb.

Jan 1, 1945