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By G. Thodos, W. F. Stevens

The point-source function introduced by Horner' us a solution to the general unsteady-state equation for the flow of fluids through porous media has been utilized to calculate pressure profiles for adjacent wells as a function of time. By trial and error, the time at which the pressure waves meet to cause interference can then be calculated. For this calculation, the time of interference is arbitrarily defined as that time when both pressure waves exhibit a pressure decrease of 25 psi ar the same point within the formation. To illustrate the method, an example involving two adjacent wells producing at different rates has been worked out and is presented in detail. INTRODUCTION The initiation of production of a new well is usually accompanied by pressure drawdown measurements in order to establish the extent and characteristics of the formation. The interpretation of such pressure drawdown data is based on mathematical developments involving a single well in an infinite reservoir. If more than one well is present in a reservoir, the possibility of interaction between wells exists. Therefore, care must be taken in interpreting the pressure drawdown data. It becomes important to be able to determine when this interaction, or interference, becomes significant. This paper presents a method for the prediction of the approximate time of interference between two such adjacent wells. The basic differential equation describing the pressure distribution existing during unsteady-state flow of a slightly compressible fluid has been solved by Hor-ner.' for a constant production rate from a circular homogeneous reservoir of infinite extent. The solution utilizes the point-source function to define the pressure at various points within the formation as a function of time. The resulting expression is: q = production rate, STB/D ,;= formation volume factor, dimension-less µ = fluid viscosity, cp k = formation permeability. md H = formation thickness, ft C - fluid compressibility, psi' F = formation porosity, dimensionless R = distance from centerline of well, ft T = production time, hours Ei(—u) = Ei-function of U Eq. 1 may be used to calculate the pressure profiles generated during production from a single well in an infinite reservoir.

By E. B. Brauer, W. Hurst, J. D. Clark

Because of drilling, completion, and workover practices, the permeability around a wellbore generally is different from the permeability of the formation. The zone with the altered permeability is called "skin" and its effect on the pressure or flow behavior of the well is called "skin effect". Hawkinsl has shown that the radius and the permeability of this zone are related to the skin by: If the permeability in the skin zone is less than that of the formation, the skin is positive; if it is more than that of the formation, the skin is negative. If the two permeabilities are equal, s is zero; that is, there is no skin. Van Everdingen2 and Hurst3 have given mathematical solutions to the case of a zone of reduced permeability around the wellbore. This skin effect is illustrated in Fig. 1. They treat the positive skin as a zone of reduced permeability of infinitesimal thickness around the wellbore. When applied to a well with negative skin, however, their solutions lead to the calculated flowing well pressure which is smaller than the formation pressure, i.e., an injection situation. This is a physical contradiction. In this paper, it is shown that this mathematical difficulty can be overcome by assuming an effective wellbore radius larger than the actual wellbore. Existing solutions2"4 are modified to include the effect of an enlarged wellbore radius, and thus enable the engineer to deal with negative as well as positive skins. Theoretical Development Van Everdingen and Hurst,4 through the use of Laplace transformation, obtained the following expression for transient fluid influx into a wellbore: The values for QtD, reported extensively by van Everdingen and Hurst,4 are obtained from the following integral:

Jan 1, 1970

By F. A. Seward

A1though the expanding Eltran1 configuration has been used in induced polarization exploration for several years, there has been little application of this method to straight resistivity problems. A comparison of results of resistivity exploration by this technique versus those obtained from Wenner configuration surveys reveals a number of advantages in interpretative resolution as well as in field operation. The application of electrical resistivity to ground-water exploration is well known.= In the past a major portion of this work has been accomplished by use of one or more variations of a technique employing the Wenner3 electrode configuration. With the introduction of the frequency domain technique of induced polarization measurements, used primarily in the exploration for disseminated sulfide deposits, there arose the need for an electrode configuration by which potentials could be measured externally to the current electrode spread in order to minimize coupling between current sending and potential receiving circuits. It was also deemed desirable that the configuration would provide information laterally along profile as well as with depth. To reach these objectives the expanding dipole-dipole configuration (which probably first was used in the electrical transient method of exploration and from which the name 'Eltran' was derived) has been adapted to induced polarization exploration. The author has successfully applied the same techniques

Jan 1, 1963

By A. M. Gaudin

Jan 1, 1962

By N. A. D. Parlee, M. M. A. El-Naggar, G. 8. Horsley

The apparatus was of a modified Sieverts type1&apos;2 with a reaction tube designed to function also as the cell component of an oxygen pressure gage. The reaction tube assembly is shown in Fig. 1 and described in Table I. The main body is a 1-in.-OD 15 mol pct calcia-stabilized zirconia tube. The ground-glass joint, g, the male part of which carried a prism for optical pyrometer use, was connected to the remainder of the Sieverts apparatus. The calibration of the cell (at its 1000°K operating temperature) for internal oxygen pressures (PO2in) in the range 0.2 to 10 9 atm was made against internal air pressures measured by mercury manometer and McLeod gage. At very low po2in values down to l<Tn atm the performance was checked by comparing pQ calculated from electromotive force measurement when the apparatus was filled with a) pure CO gas and b) a CO-CO2 mixture of the ratio CO-CO2 = 1:0.398, with Po, values calculated from the equilibrium constants of the following reactions at 1000°K:

Jan 1, 1968

By J. Gurland

A method of measuring the degree of contact between grains of 1 phase in multiphase alloys is derived. It is shown that the number and the areas of contacts between grains can be determined by metallographic analysis. Relations between the area of contact, the dihedral angle, and the mean free path of spherical grains are developed. It is often desirable to interpret the mechanical behavior of alloys in terms of structural features observed under the microscope. A quantitative description of microstructures of multi-phase alloys requires consideration of geometrical and topological factors such as the amount, size, and shape of grains and the continuity of the constituent phases. The number and dimensions of grains may be metallographically determined by applying the methods described by Smith and Guttman1 and by Fullman2 in their papers on the measurement of particle size and internal boundaries in opaque bodies. The present work intends to show that the same methods can be used to evaluate some of the parameters of phase continuity; namely, contiguity, number of contacts, and areas of contact of grains of 1 phase. Application of these concepts may be found in such problems as the calculation of electrical conductivity of 2-phase alloys, or the evaluation of the length of fracture path through grains of a brittle phase more or less dispersed in a matrix. I. DEFINITION The following definitions and derivations are based on 2-phase alloys, of which the units of the microstructure are grains. Contiguity is defined as the average fraction of surface area shared by 1 grain of a phase with all neighboring grains of the same phase. In a 2-phase alloy of constituents A and B, the contiguity of phase B is given by: where Cb is the contiguity of phase B aBß is the average area of contact between a grain of phase B and adjacent grains of phase B, aAB is the average area of contact between a grain of phase B and adjacent grains of phase A. The contiguity is a measure of the average degree of contact between grains of phase B in the 2-phase mixture. The ratio varies from 0 to 1 as the distribution of phase B changes from a completely dispersed to a fully agglomerated structure. Smith and Guttman1 have shown that the relative areas of 2 kinds of interfaces in a volume are directly proportional to the number of interfaces of each type intersected by a random line. The lines can be drawn on metallographically prepared surfaces as long as these are random sections through the sample. The contiguity ratio of phase B then becomes: Cb = 2NBB/NAB + 2N BB [2] where: NBB is the number of interfaces between grains of phase B intersected per unit length of random line. Nab is the number of interfaces between grains of phase B and grains of phase A intersected per unit length of random line. The factor 2 arises in Eq. [2] because each interface B-B is associated with 2 grains of B-phase, whereas the interface A-B involves only 1 grain of B-phase. The calculations of contiguity do not require any assumption as to volume fraction, particle size, or particle shape. Fullman2 gives an expression for the determination of the mean free path between particles. The mean free path PB between grains of phase A, i.e., through grains or agglomerates of phase B, may be found from the relationship: PB =2fB/NAB [3] where fB is the volume fraction of phase B. The calculation of the mean free path is independent of grain shape and grain size distribution. The calculation of the number of contacts and of contact area requires assumptions of grain shape, grain size, and contact shape. In the case of uni-

Jan 1, 1959

By Jasper L. Stuckey

PYROPHYLLITE, first identified as soapstone,&apos; later as agalmatolite,2 and finally as pyrophyl-lite, has been known to occur in North Carolina for more than 130 years and has been produced intermittently in commercial amounts for the past 100 years. Pyrophyllite mining, however, has become a major industry in North Carolina only during the past 25 years. North Carolina and California are the chief producing states. Current annual production averages 125,000 tons, of which slightly more than 90 pct comes from North Carolina. Distribution and Geologic Setting: All known pyrophyllite deposits in North Carolina are associated with rocks of the Carolina Slate Belt,&apos; which is composed of a vast series of metamorphosed volcanic and sedimentary materials, occurring as two zones extending in a northeast-southwest direction across the state. The larger of these zones, lying in the central Piedmont region, varies from 8 to 50 miles wide. The smaller zone lies along the eastern edge of the Piedmont region and is partly covered by coastal plain sediments. The rocks in both zones consist of a great volcanic-sedimentary series, composed of air-laid and water-laid tuffs and breccias, lava flows, and slates. On the basis of composition, they may be classed into three groups: 1) tuffs, breccias, and lava flows of acid or rhyolitic composition; 2) tuffs, breccias, and lava flows of a basic or andesitic composition; and 3) water-laid slates composed of fine grained materials derived chiefly from the acid tuffs and breccias and

Jan 1, 1959

By R. W. Drier

THE conductivity of copper wire is of prime importance to the electrical industry and consequently to the copper refiner and wire manufacturer. Annealed copper wire has a higher conductivity than hard-drawn wire, but, on the other hand, its tensile strength is lower. In an attempt to explain the reasons for these changes in property or at least to study the conditions existing in these two types of copper wire, the experiments described here were carried on in the Department of Metallurgy of the Michigan College of Mining and Technology. This is only a partial and preliminary report on some of the work done to date on a research program of which the object is an intensive study of copper. It is offered with the hope that, though it is not an exposition of the most rigorous of conclusions, the general trend of inferences which may be drawn will be of value and interest.

Jan 1, 1929

By J. L. Lytton

The flow-stress recovery of high-purity aluminurn following a 10 pct tensile prestrain was studied in terms of a fractional flow-stress recovery parameter fr. The flow-stress recovery behavior was related to the dislocation arrangements using transmissiort electron rnicvoscopy. During recovery at 120° and 160°C, nearly stable strength levels were attained, the strength level for 120°C recovery being greater than that for 160°C. This stabilization of flow stress was related to the formation of meta-stable dislocation networks bounding nearly perfect cells. Both network formation and the annealing of dislocation loops were found to be recovery mechanisms, and the stronger metastable strength level joy 120°C recovery was related to a smaller average cell size. Analysis of the networks was consistent with the reaction of sets of dislocations with a/2 (110) Burgers vector to give product Burgers DURING isothermal annealing of cold-worked metals, a degree of softening occurs without re-crystallization, and the processes responsible for this change are termed recovery processes. During recovery annealing, the dislocations generated during cold work become rearranged into low-energy configurations, and as much as 50 pct of the stored energy is released."&apos; The relationship between this rearrangement of dislocations and mechanical behavior is of considerable importance, since the dislocation substructures formed during recovery provide a significant degree of low-temperature strength3j4 and can favorably alter the course of primary creep at high temperatures.4 The purpose of the present study was to relate the flow-stress recovery behavior of high-purity aluminum to the substructural modifications that occur. Electron transmission microscopy was therefore employed for direct observation of dislocation substructures at various stages of recovery following 10 pct prestrain. Several investigators5-&apos; have proposed that dislocations can become rearranged or lost during the preparation of thin foils. It was found early in the present study that the dislocation structures in the thicker regions of the foils changed in a systematic way during the course of recovery. Therefore, it is unlikely that substantial losses or rearrangements occurred during thinning. This point will be discussed in more detail later. The results of various investigators indicate that recovery processes in fcc metals can result in release of 50 pct or more of the total energy stored in the as-prestrained metal, this percentage decreasing with increasing prestrain.&apos; Alloying elements or impurities tend to increase this percentage. The release of stored energy of fcc metals during recovery appears to occur by 1) annealing of point defects if the deformation occurred at sufficiently low temperaturesg and 2) rearrangement of dislocations into low-energy Configurations.1,2In bcc metals, the annealing of dislocation loops has also been observed during recovery after prestrain,10-12 and it has been proposed that this is a recovery mechanism for unidirectionally strained fcc metals.&apos; The density of dislocations in bcc metals has not been observed to decrease appreciably during recovery,10 although the formation of regular dislocation networks has been observed. has shown that the formation of such networks can result in elimination of long-range stresses and can lead to significant release of stored energy without dislocation annihilation. It might thus be expected that network formation is a significant recovery mechanism in fcc metals, although extensive network formation has apparently not been reported in controlled recovery experiments. The formation of sub-boundaries in aluminum has been observed by various investigators13&apos;17 during recovery studies of aluminum, but the structure of these boundaries was not resolved. In this regard, it is significant to note that Bailey and Hirsch,&apos; using transmission electron microscopy, detected no network or sub-boundary formation during recovery of 99.99 pct Ag. Furthermore, Clarebrough et al.2 found no such rearrangements during recovery of copper. EXPERIMENTAL PROCEDURE The high-purity aluminum used in this investigation was obtained in the form of 0.125-in.-thick sheet. Spectrographic analysis revealed that the sheet was 99.995 pct A1 with the following weight-percent impurities: 0.001 Cu, 0.002 Fe, 0.001 Si, and 0.0004 Mg. The as-received sheet was in the heavily cold-rolled condition. The as-received sheet was cold-rolled at room temperature to 0.040-in.-thick sheet in the longitudinal direction, which was the original rolling

Jan 1, 1965

By E. F. Koppe

The Upper Freeport coal in the Freeport and New Kensington quadrangles, Pennsylvania, varies from a bony streak to a thick coal deposit often exceeding ninety inches, the "Double" or "Thick Freeport". Distinctive basal partings, a central bone layer, and a canneloid top bench may be present. The appearance of a sandstone roof, sandstone rolls, and bony coal at the top of the main coal bench can be anticipated in thin Freeport areas when basal partings are absent. When the two basal partings are present, a shale roof is normal. These effects are correlated with the original topography upon which the coal was deposited. The original depositional high areas tend to coincide with present structural highs. Preliminary data indicate that coal quality, in part, is affected by the deposi-tional setting. In "high" areas, a tendency exists for the main coal bench to contain higher percentages of fusain and mineral matter. Within limits, some-what higher ash fusion temperatures appear to correlate with paleotopographic highs. As a result of these findings, critical examination of the paleotopography and related sediments is indicated in other areas and other coals. The Upper Freeport coal is quite variable both in thickness and composition in the Freeport and New Kensington quadrangles of Pennsylvania (Figure 1). The coal horizon in these quadrangles is marked either by a thin carbonaceous shale bed or by a single bed containing a variable number of partings (binders). The complex coal zone known locally as the "Double" or "Thick Freeport" contains the Upper Freeport Coal Bed at the base; but in addition, it has an overlying bone layer and an upper coal bed occasionally capped with a canneloid layer. The "Thick Freeport" often exceeds 90 inches in thickness, whereas the normal Upper Freeport coal usually varies between 34 and 50 inches.

Jan 1, 1961

By J. van Lookeren

In the case of a reservoir where the oil underlies a large gas cap and overlies bottom water, production can be inzproved considerably if wells are perforated below the water-oil contact rather than opposite the oil-bearing section. The theoretical arguments for this improvement, which in some cases may be tenfold, are supported by experimental evidence. INTRODUCTION Reservoirs frequently consist of one sand body containing an oil layer underlying a thick, gas-bearing section and overlying bottom water (Fig. 1). Wells in such reservoirs are customarily opened up opposite the oil-bearing interval or, rather, opposite the lower part of this interval (Fig. 2). More often than not, it is inefficient or prohibited to blow off large quantities of gas with the oil. This im-psses a severe limitation on the drawdown, which results in low oil-production rates. Because of the small difference in specific gravity between oil and water, the latter will easily cone upwards, so that the production of water together with the oil is not prevented by this completion method. In such a situation wells should be opened deeper, in the originally water-bearing zone. The gas can be shut off by not perforating over the upper part of the interval (Fig. 3) or by a tubing hung close to the bottom of the interval (Fig. 4). Initially much water will be produced, but when steady-state conditions have been established (as depicted in Figs. 3 and 4) larger drawdowns are permissible than in a conventionally completed well, and correspondingly higher oil-production rates will result. Water cuts are only slightly higher because of the small effect of gravity on the water-oil flow distribution. CALCULATIONS The ratio of improvement has been estimated by means of approximate calculations. The completion depicted in Fig. 4 is the most convenient basis for these calculations. By postulating that vertical flow gradients can be disregarded, the Appendix shows that the well-known Dupuit formula can be obtained except for some complication due to the presence of two fluids rather than one. The Appendix makes the reasonable assumption that, for suitable boundary conditions, the oil layer occupies a constant fraction of the total fluid height throughout the reservoir. The oil- and water-production rates can thus be related to the liquid levels in the borehole as follows:

Jan 1, 1966

By A. M. Khan, I. Fatt

A new type of steady-state thermal conductivity apparatus is presented and described. Results of measurements on Berea sandstone, Solehofen limestone, Bandera sandstone and Bakelite are presented. The measurements were made in a two-hour period at temperatures up to 80C and axial stress to 125 bars. Results show that the thermoelectric heat pump and tbermistor-controlled electric heater can be used in a steady-state apparatus for measuring thermal conductivity of mck. Steady-state is reached within two hours. With Urethane foam insulation, not more than 3 per cent of heat is lost by radial flaw. The apparatus can be used as a comparator or for absolute thermal conductivity determinations. INTRODUCTION Calculations of heat flow through the earth, both for scientific purposes and engineering design, require thermal-conductivity data on rock. However, the high variability of rock often makes handbook thermal-conductivity data of little value. For engineering purposes in particular, it is desirable to have measured data on samples of the rock mass under study. Laboratory methods for measuring thermal conductivity of rock use either time-dependent1-3 or steady - state temperatures .4,5 The time -dependent (transient) techniques give thermal diffusivity from which thermal conductivity can be calculated if the specific heat and density of the sample is known. Most transient techniques require lengthy and involved sample preparation. Also, the accuracy tends to be low. Steady-state techniques are more exact and are the most commonly used. The steady-state methods, however, also have their difficulties. A long time is often required to reach steady state, troubles have been encountered in establishing good thermal contact between the heating and cooling elements and the rock sample, and unaccounted-for heat losses are difficult to prevent. In this paper we present a description of a new type of steady-state thermal conductivity apparatus. With it, measurements have been made in two hours at temperatures up to 80C and axial stress to 125 bars. Sample preparation is simple and no thermal conductivity standard is needed.6 APPARATUS Our steady-state apparatus is an improvement . over the conduction pile used by Zierfuss and van der vliet4 and Somerton.5 A linear heat flux is established through a conduction pile consisting of the rock sample and a standard material of known thermal conductivity in series. The thermal conductivity of the sample is calculated from the ratio of the temperature drop across the sample

Jan 1, 1966

By B. A. Wilcox, A. H. Clauer

DURING the course of an investigation on the high-temperature creep behavior of TD Nickel* (Ni + 2) vol pct ThO2), it was observed that the creep fractures were similar in appearance to low-temperature ductile fibrous fractures and creep ruptures by cavitation. Microfractographs of fibrous ruptures are typically characterized by a dimpled appearance, with second-phase particles frequently found near the bottom of the dimples.&apos;-4 These somewhat resemble creep ruptures by cavitation, in that fracture occurs by void formation, with the subsequent growth and final merging of the voids to form a complete fracture.&apos; It is the purpose of this communication to describe the creep fracture of TD nickel and suggest a mechanism for the rupturing process. The discussion is relevant to material crept to fracture in vacuo at a constant stress of 18,000 psi, over the temperature range 900" to ll00°C. The material used in this investigation was 1/2-in.-diam bar stock supplied by Du Pont, contain- ing 2.3 vol pct Tho2. The final fabrication treatment consisted of -95 pct reduction by swaging followed by a 1-hr anneal at 1000°C. Transmission and replica electron microscopy revealed that the material had a fibered structure with a transverse grain size of -1 µ and a longitudinal grain size of 10 to 15 µ. Selected-area diffraction indicated that the fiber axis was parallel to (00l), in agreement with the results of Inman et a1.6 The thoria particle size varied from 80 to 1000Å, with an average size of 400Å, and the mean particle spacing was about 1200Å. The remarkable structural stability7 was confirmed by annealing for 6 hr at 1300°C (about &apos; 0.9 the absolute melting point). Transmission electron microscopy revealed that the only structural change was a slight decrease in dislocation density. Furthermore, recrystallization or grain growth during creep has not been observed. All specimens were annealed in vacuo at 1300°C for 3 hr prior to creep testing. The following times to rupture were obtained for vacuum creep tests at a constant stress of 18,000 psi: 900°C, 854 hr; 930°C,

Jan 1, 1965

By Edgar C. Bain, William E. Griffiths

The results of an inquiry into the structural nature of some 70 iron alloys containing both nickel and chromium over a considerable range of concentration are briefly described in this paper. This study is a part of a comprehensive program of investigation into alloys involving chromium in progress at the Union Carbide and Carbon Research 1aboratories. All the compositions studied contained carbon in amount comparable to commercial alloys. The B Constituent In addition to locating the limits of composition for the three principal types of structure (polyhedral ferrite, martensite and austenite) and their combinations, certain other features of the system were observed. A hitherto undescribed brittle constituent (designated here as the "B constituent") was discovered. It is very hard, non-magnetic and forms in two modes from chrome-ferrite by recrystallization. It is the stable low-temperature condition for the equal-proportion chromium-iron materials containing less than 10 per cent. nickel. Although it partakes of the nature of an inter-metallic compound, yet it may exist over a considerable range of composition. It is non-existent above a temperalure zone, varying with composition, from 800° C. to 950° C., and heating above this critical temperature develops chrome ferrite persistent after average cooling rate; a few hours just below this temperature will develop the complicated brittle crystalline type. The B constituent, when found in the grain boundaries of alloys of lower chromium content, contributes a serious brittleness removable, however, by heating and cooling rapid enough to prevent its re-format&apos;ion. By suitable means of precipitution, the hard B constituent can be made to strengthen the already stiff alloy matrix in the manner of submicro-scopic particles, as exemplified in duralumin or the Sykes iron-tungsten and iron-molybdenum alloys. A possible explanation is offered for the familiar brittleness developed in chrome iron at temperatures above about 475" C. and removable by

Jan 1, 1927

By W. C. Lacy

This article is a progress report of a program designed to quantitatively evaluate geological parameters to predict stable slope angles. The author feels that, although there does not appear to be any one technique or characteristics that will suffice as an index to the determination of stable slope angles, a combination of seismic methods as outlined here, detailed geological mapping and logging of core, and a better understanding of the angle of internal friction in rock types will permit accurate estimation. A mining engineer friend once remarked, "If we don&apos;t have pit slope failures, the slopes are too flat." Since economic justification of an open pit project is based upon ultimate pit slopes, we cannot wait until the pit is developed and slopes established by trial and error. Some combination of geological and/or geophysical techniques must be developed to permit accurate prediction of stable slope angles and to suggest modification of pit outlines to result in slopes enabling maximum economic yield. This presentation is a progress report of a program designed to quantitatively evaluate geological parameters to predict stable slope angles. The principle of the weakest link applies to any slope stability evaluation and necessitates detailed geological examination and recording of observations. Physical failure will occur along any surface where shearing stress exceeds shearing resistance. Rock conditions are seldom uniform even within a single cut or pit face, so stability conditions will vary. Consequently, both plan outline and ultimate slope limits must be tailored to the specific situation. Economic advantages to be gained by slight increases in slope angles in deep pits are well recognized and stimulate efforts to obtain maximum permissible slope angles. MECHANICS OF SLOPE FAILURE Excellent reviews on the nature of slope failure have been presented+2,6,8,11 For the purpose of this study, the writer considers failure to take place by one or more of three processes: 1) ravelling, 2) translation or slope failure, and 3) rotation or base failure (Fig. 1). The effects of increased pore water pressure in soils were recognized in 1923 by Terzaghi in his development of the concept of effective stress, and he has applied the same principle to jointed rocks designating water pressure as cleft-water pressure.8 Cleft-water pressure in a rock slope reduces normal pressure on the plane of failure, thus reducing shearing resistance and resulting in slope failure. In the following discussion it is assumed that adequate drainage is supplied to prevent any build-up of water pressure. This is the first step in any program to establish maximum slopes. Ravelling: Ravelling occurs in rocks with low cohesion and entails the free movement of individual fragments ranging from single rock falls to running ground. Removal of lateral support wherever slope angle exceeds the angle of repose of the rock parti-

Jan 1, 1963

By J. C. Martin

Equations for three-phase, three-dimensional, compressible flow (including capillarity) are reduced to two-dimensional relations by a partial integration. This reduction allows three-dimensional flow problems to be treated with mathematics for only two spatial dimensions. The results can be used to formulate flow equations for two-dimensional reservoir simulators in which the effects of capillarity and fluid segregation in the third dimension are represented. Such reservoir simulators would retain many of the aduantages of two-dimensional simulators while simulating three-dimensional effects. The principal restriction of the method is that the thickness of the reservoir should be small, compared to the distance across the reservoir. INTRODUCTION In recent years, computers have been used to calculate the performances of many reservoirs. Most of the detailed calculations, however, are based on finite difference solutions of the flow equations, and present day computers are seldom able to handle a sufficient number of cells to produce entirely satisfactory solutions, even for reservoirs represented by two-dimensional arrays of cells. The situation becomes much worse when one wishes to approximate the reservoir by a three-dimensional array. A great saving in computation or a more detailed solution can be obtained for many reservoirs by using the partial integration of the equations of flow presented in this paper. The integration reduces the three-dimensional equations to two-dimensional relations; and, for studies of two-dimensional flows in vertical cross-sections, the equations are reduced to one-dimensional relations. Most reservoir performance calculations currently are based on one- or two-dimensional flow re1ations.l-3 In some cases flow in the third dimension is approximated by assuming a particular type of vertical saturation distribution, such as gravity segregation.l.2. The relations developed in this paper approach those for segregated flow as the capillary pressures approach zero, and they approach those for uniformly distributed saturations as the capillary pressures are increased. For this analysis, the ratio of the reservoir&apos;s thickness to the maximum distance across it must be small. The capillary pressures between the oil and water should also be small compared to the maximum pressure difference in the reservoir. This requirement is met by most reservoirs. It is assumed that the capillary-pressure curves are well defined, whether or not hysteresis effects are included. Also, the reservoir must have sufficient vertical permeability to allow the fluids to segregate. The results presented here provide a firm theoretical foundation to Coats&apos; et aL5 assumption of vertical equilibrium and extend the relations to three-phase flow. Coats&apos; assumption of vertical equilibrium, which he verified by calculations and experiment, is developed here mathematically from basic flow equations. DISCUSSION SATURATION AND PRESSURE DISTRIBUTIONS Appendix A presents a mathematical analysis of fluid flow in reservoirs where the ratio of thickness to maximum distance across the reservoir is small. The results indicate: (1) that the fluids along any line perpendicular to such a reservoir&apos;s upper surface are in static capillary equilibrium (vertical equilibrium); (2) that, to a first approximation, the fluid pressures and properties are functions of only areal position in the reservoir and time; and (3) that hydrostatic pressure gradients exist along any line perpendicular to the reservoir&apos;s upper surface. The results might be expected after studying several physical considerations. First, no flow is allowed normal to the upper and lower reservoir boundaries, which are relatively close together.

Jan 1, 1969

By Arnulf Muan, Egil Aukrust

Activities of COO in the three solid solution series COO-MgO, COO-MnO, and COO-"FeO" have been determined at 1200°C by equilibrating oxide samples with a metal phase (cobalt or a Co-Fe alloy) in atmospheres of known oxygen activities. The systems COO-MgO and COO-MnO are ideal, within limits of experimental error, whereas the system COO-"FeO" shows a slight positive deviation from ideality at the temperature of this investigation. KNOWLEDGE of activities of oxide components in solid solutions is necessary for the calculation of the cation distribution among coexisting oxide phases in many important metallurgical, ceramic, and geochemical systems. Very few activity-composition curves are available for such systems at the present time. A start on the prodigious job of obtaining such data has been made by studies of solid solutions with simple sodium chloride type structure containing NiO or "FeO" as a component.1-3 The present paper describes similar studies for solid solutions with sodium chloride type structure containing COO as a component. For the systems COO-MgO and COO-MnO, the activity solid solution is determined by the equation Here po and are the oxygen activities of the gas phase in equilibrium with the phase assemblages metallic cobalt plus oxide solid solution, and metallic cobalt plus pure COO, respectively. In the system CoO-"Fe07&apos;, the metal phase is a Co-Fe alloy rather than pure cobalt. In this case the activity of COO in the oxide solid solution is calculated from the equation where a& is the activity of cobalt in the metal phase. Because the system Co-Fe is close to ideal4 and the metal phase appearing in the present work contains less iron than cobalt even at the lowest COO concentrations of the oxide phases used may be set equal to N . Hence, Eq. [2] becomes The value of Nc, can be estimated from the relative stabilities of the oxides COO (see below) and I) EXPERIMENTAL METHOD Two different but closely related procedures were used in the present investigation to determine the activity-composition curves. In one method, the oxide solid solutions were equilibrated against

Jan 1, 1963

By E. F. Johnson, V. O. Naumann, D. P. Bossler

A method is presented for calculating individual gas and oil or water and oil relative permeabilities from data obtained during a gas drive or a waterflood experiment performed on a linear porous body. The method has been tested and found both rapid and reliable for normal-sized core samples. INTRODUCTION Individual oil and gas or oil and water relative permeabilities are required for a number of reservoir engineering applications. Chief among these is the evaluation of oil displacement under conditions where gravitational effects are significant, such as a water drive or crestal gas injection in a steeply dipping oil reservoir. Numerous proposed methods of obtaining relative permeability data on reservoir core samples have been too tedious and time consuming for practical use, or have yielded questionable and sometimes inconsistent results. A method has been developed by which the individual relative permeability curves can be calculated from data collected during a displacement test. The method is based on sound theoretical considerations. Using this method, with a properly designed experimental procedure, relative permeability curves can be obtained using core samples of normal size (i.e., 2 to 3 in. in length and 1 to 2 in. in diameter) within a few days after receipt of the core. In a recent publication D. A. Efros1 describes an approach to the calculation of individual relative permeabilities that is based on the same theoretical considerations. We believe the approach described in the present paper is more adaptable to practical application than the method implied by Efros. In addition, comparisons with independently determined relative permeabilities are furnished to substantiate the reliability of the new method. DERIVATION Previously the theory of Buckley and Leverett as extended by Welge2 as been used to calculate the ratio of relative permeabilities. In the derivation which follows, this theory is further extended to permit calculation of the individual relative permeabilities. The theory assumes two conditions which must be achieved before the method is applicable. They are that the flow velocity be high enough to achieve what has been termed stabilized displacement,3 and that the flow velocity is constant at all cross sections of the linear porous body. In stabilized displacement the flowing pressure gradient is high compared with the capillary pressure difference between the flowing phases. The high pressure drop insures that the portion of the core in which capillary effects predominate will be compressed to a negligibly small fraction of the total pore space. The assumption of constant flow velocity at all

By R. W. Vieth, S. A. Bradford

THE orientations of tetragonal tin crystals are commonly examined by either the transmission Laue method or the back-reflection Laue method. The predominant planes and zone axes of the pattern are plotted on a Wulff net and indexed by measuring the angles between planes and directions. The problem of identifying the Laue spots has been approached in several different ways. Nicholas 1 tabulated interplanar angles for ten planes but did not relate the angles to permutation of indices or sign. A table of angles between adjacent planes and of angles between adjacent zones was published by Chandrasekhar and veal.&apos; These tables are very helpful, provided the film shows all the low-order zones that occur within about 20 deg of each other. Frounfelker and Hirthe&apos;s compilation3 of angles between the (001) plane and twenty-nine other planes is useful for the indexing, only if the (001) spot can be found on the film. For all but the simplest orientations, however, more information is needed to index the diffraction spots. Standard stereographic projections, such as those published by Bicelli, 4 provide part of the information required. By their nature, these projections show the positions of only a few zone axes. In the tetragonal system many low-order zone axes are not normal to the planes having the same indices. Consequently, planes and directions can have entirely different positions on a stereographic projection. For example, the [011] zone axis is 33 deg from the normal to the (011) plane. The crystallographic orientation usually has to be deduced from a combination of the predominant zone axes and a few low-order planes that appear on the film. In addition to a standard stereographic projection of the planes, a stereographic projection of the directions is needed. Such a projection can be constructed from the angles given in Table I. To provide more information about angles between planes and directions, we have calculated interplanar angles up to (332) and angles between planes and directions up to [332] on an IBM 1620 computer. The angles are differentiated according to permutations of sign and indices. Those doing research on tin single crystals are welcome to request a copy of these tables from the authors.

Jan 1, 1967

By L. Moyd, P. Moyd

The first commercially available portable gamma ray-neutron beryllium detector, the Rerylometer, was developed by the same group that developed the first practical portable scintillation counter, the Scintillo-meter. This group includes Professor George Brownell, of the Geology Department of the University of Manitoba in winnipeg, and his associates, K. I. Roulston, R. w. Pringle, and R. L. Funt, of the Physics and Engineering Departments of the University of Manitoba. A prototype model was tested in the field by Dr. Brownell during the summer of 1958. Commercial production was undertaken shortly thereafter by Nuclear Enterprises Ltd., of Winnipeg, and a license for manufacture and distribution in the U.S. was granted to Isotopes Specialties Co., of Burbank, California, a division of Nuclear Corporation of America. Early in 1959, we were retained by the &apos;Mining Division of Minerva Oil Co. to Legin a search throughout North America for minable deposits of beryllium, and to train members of their permanent staff to continue the program. The project included literature sur-veys, interviews with members of various federal, state, and provincial agencies, scanning of our own collection of suites of specimens from various mineral occurrences, and the examination of deposits in the field. The value of a beryllium detector to this program was obvious and arrangements were made to procure one as soon as possible, which resulted in our obtaining and using the first instrument that was produced commercially in the U. S. In addition to the commercially available Berylometer, Mr. William Vaughan and his associates, of the U. S. Geological Survey Geophysics Division, in Denver, have developed and tested several experimental instruments based on the same principles. DETECTOR Principles The detection of beryllium is based on the fact that "hard" gamma rays (1.6 MEV and geater), such as those produced by Antimony convert ordinary beryllium (Be9) to Be8 , with the release of a neutron. The neutrons are counted by a scintillator designed to favor neutron responses while suppressing those of gamma rays. &apos;The instrument is completely selective for beryllium. No other element can interfere. Instrunlentation The Rerylometer itself, which is cylindrical, is about 8 inches in diameter, 13 inches high, and weighs about 11 pounds. (Figs. 1, 2). The principal components include: 1. A disc of wax to moderate (or "thermalize") neutrons. 2. A boron-enriched polyester-zinc-sulphide fluorescent screen. This material produces strong flashes when struck by thermalized neutrons but only weak flashes from gamma rays. 3. A photomultiplier tube. 4. An electronic circuit, complete with one srnall 71/2-volt dry battery. The photomultiplier tube and electronic circuit convert neutron flashes to electrical impulses, but gamma ray effects are almost

Jan 1, 1961