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By L. K. Nemeth, H. T. Kennedy

The investigation presented in this paper was performd to develop a relationship between the dewpoint pressure of a hydrocarbon reservoir fluid and its composition, temperature and characteristics of the heptanes-plus fraction. This is part of an over-all program to develop correlations for the essential properties of reservoir fluids with the composition, temperature and pressure. In all, 579 dewpoint pressures, determined experimentally on 480 different condensate systems along with their compositions and temperatures, were obtained from the literature and from previously unpublished sources. Compositions were expressed in terms of mol fractions of methane through heptanes-plus, nitrogen, carbon dioxide and hydrogen sulfide, and the molecular weight and specific gravity of the heptanes-plus fraction. Distribution of the samples assured a good representation of most of the major oil-producing areas in the world. A multiple regression analysis program written for the IBhl 7094 digital computer was used to compare numerous equations and to calculate their coefficients. The final equation containing 11 coefficients resulted in an average absolute deviation of 7.4 percent for all the samples. Segregation of data on the basis of one or more variables did not materially improve the results. The correlation developed is useful in predicting the dewpoint pressure of condensate reservoir fluids from variables normally measured in the laboratory. INTRODUCTION In any discussion of the dewpoint of hydrocarbon mixtures, it should be noted that there are two kinds of dewpoints which must be distinguished. The first type, or normal dewpoint, usually occurs at low pressures as dry gas is compressed to the point where liquid first forms. The second type, sometimes called the retrograde dewpoint, occurs when a gas mixture containing heavy hydrocarbons in solution is depressured until a liquid is formed. The first type, as applied to condensate mixtures, would normally be below atmospheric pressure and would be of no interest in reservoir performance. The second type is of great significance in interpreting reservoir behavior and is the property referred to throughout this paper. Laboratory measurement of dewpoints is laborious, expensive and subject to many errors, and it is desirable to find a reasonably accurate method to predict dewpoint pressures from the composition and other readily available properties normally measured on the fluids. A possible method of calculating dewpoint pressures would be to employ a set of equilibrium ratios (K-values). 1 However, the K-values themselves depend on composition as well as temperature and pressure; presently their use adds nothing to the accuracy of predicting dewpoints. If future correlations provide a significantly better method of estimating K-values for heavy hydrocarbons, they may be of greater use. REVIEW OF LITERATURE In 1944, Olds, Sage and Lacey 2 studied the behavior of fluids from the Paloma field. Properties of seven mixtures were studied at temperatures ranging from 100 to 250 F and at pressures up to 5,000 psia. From the information obtained, the influence of composition and temperature upon the dewpoint was determined. It was found that removal of the intermediate molecular weight components from the mixture resulted in considerable increases in the dewpoint pressures. Results indicated that the effect of changes in temperature was relatively minor when compared with the effect of modifying the mixture by removing the intermediate components. Thus, removal of propane and butanes resulted in an increase in dewpoint pressure more than 10 times greater than a change in temperature from 100 to 250 F. The authors stated that althouah

By S. R. B. Cooke, W. Philippoff, Donald E. Cadwell

THE importance of contact angles in flotation has long been recognized, but little has been done to get quantitative relationships between the surface coverage of the mineral by the reagent, the length of the hydrocarbon chain, and the contact angle. It is well known that for a complete monolayer on the surface of a mineral the contact angle 8 is determined only by the reagent1 and principally by the chain length of the hydrocarbon radical. Although for flotation the monolayer usually is regarded as being complete, early experiments of Wark and Cox&apos; showed that at very small concentrations of potassium ethyl xanthate the contact angle on galena changed in a continuous way from 0&apos; to the equilibrium value of 60". Further, Gaudin and Vincenta showed a similar continuity using heptylic acid and sphalerite. This shows that with probably incomplete monolayers, 0 varies with the degree of surface coverage. At the time these experiments were made it was impossible to measure the actual amount of surface coverage, especially since the decrease of the collector concentration in the liquor could be influenced by chemical side reactions. Through the use of the radioactive tracer method for the measurement of the amount of collector adsorbed on the surface of the mineral, Gaudin and Bloecher&apos; were able to determine accurately the actual surface coverage of dodecylamine acetate on quartz, but they did not measure any contact angles. It is believed that the relationship between the surface coverage and the contact angle is extremely important in flotation, especially as Gaudin and Bloecher showed that a 95 pct recovery is possible with a coverage of only 5 pct. Theoretical Considerations The work of adhesion W is the energy bound (in ergs per sq cm) when an interface is formed. It is determined by the difference of the free surface energies (surface tensions) of the respective phases 7j and yZ before contact and the interfacial free surface energy y, after contact. The definition of the contact angle permits the elimination of the unknown surface energy of the solid and hence the calculation of W from the surface tension of the liquid y and the contact angle 8: W = y (1 + COS0) [1] Consider a surface consisting of two components, identified as 2 and 3, 1 being the liquid. The amount of 2 in the composite surface is x, where 0 < x < 1. The respective contact angles with 1 and air are 8, and 8.. The total work of adhesion of such a surface is the sum of the values of the components multiplied by their respective areas. A formal relation can be written: Wm = x-W2+ (l-x)-W, [2] in which W, is the work of adhesion of the composite surface. Eq 1 can be introduced into Wt and Wa, but in the case of W, some discussion is needed. If the two components are segregated, each component forming an integral part of the composite surface there is no physical meaning for a contact angle 8m calculated from W,, by using eq 1. If, on the other hand, the area of each patch of the components is small in comparison with, say, the diameter of the bubble used in measuring, it could be expected that 0m would have a physical significance, because of the action of the surface tension of the liquid-gas interface tending to smooth out sharp corners. How small these patches would have to be is difficult to predict, but by using composite surfaces with microscopically

Jan 1, 1953

By R. L. Smith, I. G. Greenfield

The binary system Ga-Sb has been investigated by thermal, X-ray, and metallo-graphic methods. &apos;The intermetallic compound GaSb melts at 705.9OC and forms a eutectic with antimony at 11.8 atomic pct Ga. This eutectic melts at 589.8OC. A eutectic appears to be formed at 29.8°C but is too close to the composition of pure gallium to be detected. IN the course of an investigation of the semiconducting properties of the intermetallic compound, GaSb, the phase equilibrium between gallium and antimony has been investigated. Recently, the phase diagram of similar systems, such as In-Sb,&apos; In-AS,&apos; and Bi-As3 have been determined. These diagrams have the same general appearance in that an intermetallic compound is formed at 50 atomic pct and a eutectic is formed on both sides of the intermetallic compound. Jenny,& who has been investigating the semiconducting properties of intermetallic compounds, has concluded from thermal data of several alloys that the phase diagram of the Ga-Sb system resembles those mentioned previously. This paper represents a more detailed study of the Ga-Sb equilibrium. Experimental Procedure The gallium that was used to prepare the alloys was obtained from A. D. Mackay Inc. and the Aluminum Co. of America; the latter reported the purity as 99.95 pct. The antimony was obtained from Bradley Mining Corp. who specified a purity of 99.997 pct. Since the intermetallic compound of GaSb was prepared primarily to be used for further studies of the semiconducting properties, additional purification by a standard zone-melting technique under an atmosphere of hydrogen was performed. Hall-effect measurements on this material indicated an impurity content of less than 3 ppm.5 The antimony also was purified by zone melting and used in conjunction with the compound GaSb to make the necessary alloys for the determinations of the antimony-rich side of the phase diagram. The antimony-rich alloys were chill cast in Vycor tubes which were evacuated to a pressure of less than one micron. Since these alloys expand on freezing, precautions were taken to prevent oxidation of the ingot in the event the tube fractured. To insure homogeneity, the brittle alloys were ground and then placed in the furnace for the thermal-arrest determinations. Since the cost of gallium is high, the experimental procedure was complicated somewhat because of the necessarily limited sizes of the melts. In order to economize, the gallium-rich side of the diagram was determined by using a 35 g ingot of pure gallium initially and adding increasing amounts of antimony in order to obtain additional alloys. The furnace used in the determination of the cooling curves is illustrated in Fig. 1. A 2¼ in. quartz tube, sealed at one end, was fitted with a brass flange which was attached with DeKhotinsky cement. The brass collar was water cooled to prevent the cement from softening. An "0" ring was used to provide a seal between the brass collar and the furnace cover. Nichrome ribbon was wound around a 2 ½ in. asbestos-covered copper core which was used to reduce the thermal gradient in the furnace. The insulation consisted of alternate layers of asbestos and aluminum foil surrounded by sil-o-cel refractory and

Jan 1, 1956

By Hermann Schmalzried, Carl Wagner

UPON heating a metal oxide or sulfide in H2, first only oxygen or sulfur is removed from the surface. Thus the metal/nonmetal ratio in the oxide or sulfide increases and the thermodynamic activity of the metal increases accordingly. For the formation of metallic nuclei, a certain super saturation is necessary, i.e., the activity of the metal must become greater than unity. Once nuclei of the metal have been formed, the supersaturation is supposed to decrease. Even under steady-state conditions of a reduction process, however, the activity of the metal inside an oxide or sulfide phase must be greater than unity in order to have a driving force for the individual steps of the reduction process. As an example, consider the reduction of Ag2S with a high mobility of cations and electrons. As shown schematically in Fig. 1, the reduction process involves the following steps:&apos; 1) sulfur removal along the entire Ag2S surface in form of H2S, 2) migration of Ag&apos; ions and electrons from the Ag2S surface to nuclei of metallic silver, 3) transfer of Ag+ ions and electrons from the Ag2S phase to the metal. For a finite rate of step 3, the activity of silver in Ag2S next to the Ag2S-Ag interface must be greater than unity. Moreover, for a finite rate of step 2, the activity of silver at the gas-Ag2S interface must be still somewhat greater. Theoretical considerations lead to the tentative conclusion that at least under steady-state conditions the supersaturation will be rather low.&apos; To test this conclusion, the average activity aAg of silver in a Ag2S tablet about 2 mm thick with a diameter of 5 mm was followed as a function of time by measuring the electromotive force E of the cell Pt I Ag | Agl I Ag2S I Pt where AgI is used as a solid electrolyte in which only Ag+ ions are mobile. The activity aAg is calculated as aAg=exp(-EF/RT) Since the diffusion coefficient for transport of silver in Ag2S is very high (- 3 . l0-2 sq cm per sec),3 the average activity aAg is representative for both the bulk and the surface of the Ag2S tablet. Experiments were conducted at about 400°C. The cell was surrounded by an aluminum block in order to secure a uniform temperature and to eliminate thermoelectric effects. With the help of a potentiometer, first a potential of 0.16 v was applied to the above cell in argon in order to obtain a well defined initial state with a Ag/S ratio close to 2.000.~ Then the cell was disconnected from the potentiometer and a gentle stream of H2 was passed along the Ag2S surface. In view of the removal of sulfur in form of H2S, the electromotive force E decreased within 20 to 30 min to zero corresponding to a Ag/S ratio of 2.003 and became negative, indicating a value of aAg> 1. A representative plot electromotive force E vs time is shown in Fig. 2. The minimum of the electromotive force characteristic of the onset of

Jan 1, 1963

By C. D. Stahl, E. E. Templeton, R. F. Nielsen

It has been customary, in predicting saturation changes, to use the Leverett fractional flow formula", obtained by eliminating the unknown pressure gradient from the generalized Darcy equations for the separate phases. The formula presents difficulties in the case of counterflow, since the "fractional" flow may be negative, greater than unity, or, in the case of a closed system, infinite. Recently, it has been shown by several authors that the corresponding equations (with capillary pressure and gravity terms) for actual flow of the phase may be used just as well. These equations are in agreement with Pirson&apos;s1 statement that, if the two mobilities differ considerably from each other in a closed system, the flow is largely governed by the lower value. The present study was undertaken because of an apparent lack of experimental data on gravity counter-/low with which to test the theory. A 4-ft sand-packed tube in a vertical position was employed. Electrodes for determining saturations by resistivity were spaced along the tube, one phase being always an aqueous salt solution. Air, heptane, naphtha, or Bradford crude oil was used for the other phase. A reasonably uniform initial saturation was set up by pumping the phases through the system, alter which the tube was shut in and saturation profiles obtained at definite intervals. Cumulative flows over certain horizontal levels were obtained by integration of the distributions; hence, differentiation of the cumulative flows with respect to time gave instantaneous flow rates. To compare experimental and theoretical flow values, capillary pressures were assumed given by the final saturation-distribution curve. The upper part corresponds to the "drainage" region and the lower part to the "imbibition" region, where trapping of the nonwetting phase occurred. While calculations indicated that the capillary pressure saturation function and, probably, the relative permeability saturation functions changed during the segregation, the relation of the measured rates to saturation distributions are in general accord with the frontal-advance equation. It appears that the Darcy equations, as modified for the separate phases, are generally valid for counterflow due to density differences. The usual method of predicting saturation changes, which involves a continuity equation and the elimination of the unknown pressure gradient from the flow equations, should therefore be applicable. However, the need for advance knowledge of drainage and and imbibition "capillary pressures" and relative permeabilities during various stages presents difficulties. INTRODUCTION The present study was undertaken because of a seeming lack of experimental data relating to vertical counterflow of fluids of different densities in porous media. In particular, it was desired to determine whether data obtained from these laboratory tests were in accordance with certain mathematical treatments of counterflow which have been proposed. The gravity "correction" has been incorporated into the flow equations (and, hence, into displacement theory) nearly as long as both have been used. field4 and laboratory5 data have generally borne out the validity of the theory as applied, for instance, to downward displacement by gas, with all fluids moving downward. However, the modifications for counterflow have only recently been pointed out. 2-3 It has been customary to use fractional flow rates instead of actual flow rates in displacement calcuIations. In the case of counterflow, this results in negative values, values greater than unity and, when rates are equal and opposite, in infinite values. As pointed out by Sheldon, et a1,3 and by Fayers and sheldon,2 actual flow rates may be used just as well. The fact that these may be of opposite signs for the two fluids does not present any difficulty.

By A. M. Gaudin, C. S. Chang

Adsorption was measured for barium ion and laurate radical, using radioactively marked agents, over wide range of concentrations. Laurate adsorbed in absence of barium fails to float. With barium, flotation-effective laurate coating needs not exceed 5 pct of monolayer. Barium adsorption then is several fold larger. No barium laurate coating is obtained. IN general, fatty acids or their alkali salts do not cause clean quartz to float. However, the presence of multivalent ions, such as the alkaline-earth metal ions and heavy-metal ions, can activate quartz and thus make it respond to soap flotation. The mechanism of activation and soap flotation of quartz has been studied.&apos;-&apos; The problem was attacked by making flotation tests rather than by careful adsorption measurements on suspensions of ground quartz in solutions containing the activator and the collector. Attempts have been made to determine the adsorption of barium ion and calcium ion on quartz. The results have only qualitative value as the analytical methods were not adequately precise. Artificial radioactivity and progress in radioactivity detection have introduced a new analytical tool, which can have a sensitivity several orders of magnitude greater than that of other methods."-" In the experimental work described in this paper the radioactive tracers, barium 140 and carbon 14, were employed. Lauric acid was synthesized, having a radioactive carbon atom and was used with Ba" to study the adsorption of barium and laurate ions from aqueous solution on ground quartz. Flotation tests, using the vacuum flotation technique,&apos; were run in conjunction with the adsorption experiments. The object of this investigation was to understand better the reaction mechanism of barium activation and soap flotation of quartz. Methods and Materials Quartz Sample Preparation: Lumps of hand-picked quartz, washed free of stains, were reduced in size to —20 mesh in laboratory crushers. The product was boiled in concentrated hydrochloric acid to remove the abraded iron. The cleaned product then was ground dry in a laboratory porcelain mill, using siliceous pebbles, to pass through a 200 mesh screen. The ground product was deslimed by sedimentation at 37 microns. This —200 mesh +400 mesh material was subjected again to boiling with concentrated hydrochloric acid, followed by washing with distilled water and further desliming. The cycle was repeated until no trace of iron could be detected in the wash liquor. During the final washing, the pH of the wash water was checked to insure the complete removal of the acid from the mineral. The quartz then was rinsed with conductivity water. The resulting product was dried in a drying oven and stored in mason jars. The impurities (spectrographic) were as follows: Al, 0.01 to 0.001 pct; Li, Mg, Mn, Na, each less than 0.001 pct; Fe and Cu not detected. The specific surface of the quartz sample was determined by the krypton gas adsorption method.&apos; The average of two measurements was 881 sq cm per g. The calculated specific surface for 37/74-micron quartz particles, using a surface factor of 1.96; is 900 sq cm per g. This is in good agreement with the measured value of 881 sq cm per g. Beta-Counting Method: The radioactive isotope of barium used in the experimental work was barium 140. It has a sufficiently long half-life, 12.8 days, emits high-energy beta particles (up to 1.0

Jan 1, 1953

By E. R. Buchanan, R. E. Reed-Hill, F. W. Caldwell

Poly crystalline zirconium tensile specimens containing a sizable fraction of grains unfavorably oriented for slip were deformed at 77°K to strains as high as 9 pct. The contributions of the various twinning modes to the observed strain were deter~nlned quantitatively using measured twin volume fractions and Schrnid orientation factors. The results indicate that at very small strains twinning can account for almost all of the observed strain. At higher strains slip becomes the dominant means of deformation. The most significant form of twinning was {1121}. PLASTIC deformation in polycrystalline aggregates is poorly understood even today. This is primarily because slip cannot be easily observed or studied in the interior of a metal. On the other hand, twinning can be discerned inside a specimen after deformation. Twinning studies can therefore often yield quantitative information about polycrystalline deformation not otherwise obtainable. This investigation uses the above aspect of twinning in determining the contributions of twinning to the strain during tensile deformation at a cryogenic temperature (77°K). Polycrystalline zirconium was used as the object of study. There are several reasons why this information should be of value. In contrast to many metals, both the strength and ductility of zirconium increase with a corresponding decrease in temperature. This may be related to the fact that twinning becomes more profuse and complex with decreasing temperature. Secondly, it has been discovered that certain room-temperature properties of zirconium can be greatly improved by about 1 pct prestrain at 77°K. These include an increase in tensile elongation in critical directions by over 50 pctl and a large mechanical hysteresis effect.2 Both effects are believed to be related to twins formed by prestraining: plastic twin-boundary movements accounting for the increased ductility, and anelastic twin-boundary movements for the internal friction. Thirdly, hexagonal metals which twin primarily on only the (1012) mode (such as beryllium, zinc, and magnesium), unlike zirconium, are very limited in ductility at cryogenic temperatures. Therefore, the significance of the various types of twinning in zirconium at cryogenic temperatures is of more than academic interest. Three twinning mechanism-s have beenobserved in quantity at WOK, {1012), {1121), and {1122). Each must be considered as a unique deformation mechanism because of their markedly different twinning shears and basically different angles between twinning and basal planes, as shown in Table I. It is therefore necessary to separately consider the role of each of these twins during plastic deformation at 77°K. Fortunately, each twin possesses unique morphological features which makes its identification possible on a properly prepared metallographic section. The use of the polarized microscope3 can also greatly aid in identification of the various forms of twins. EXPERIMENTAL PROCEDURE Arc-melted sponge zirconium plate, previously describedY4 was machined into 1/8 by 1 1/4 in. threaded-end, round tensile test specimens. These specimens were deformed to the desired strains in a liquid-nitrogen cryostat mounted in an Instron testing machine. The tensile axis was the transverse plate direction. This specimen direction resulted in large numbers of grains in orientations unfavorable for deformation by prism or basal slip. Deformed specimens were sectioned, mounted in epoxy cold-mount resin, ground on metallographic papers, electropolished, and anodized to provide surfaces sensitive to polarized light. Standard quantitative metallographic procedures5

Jan 1, 1965

By G. P. Marino

The "conductivity ", Kr, of a spherical void in a linear temperature gradient due to radiation is calculated by integration of the general expression for diffuse radiation between two gray surfaces of arbitrary orientation. The spherical geometry allows an exact evaluation of the phenomenon without resorting to up-proximate expressions for the radiant heat flux. The resulting expression for Kr is three quarters of the approximate value obtained by Loeb. IN the determination of the thermal conductivity of porous materials such as sintered UO2 and ThO2 fuel pellets, the magnitude of the thermal conductivity in the pores due to radiative heat transfer across the pore, Kr, may be important in assessing the total conductivity of the pellets. Loeb&apos; has calculated Kr for various pore geometries by approximating the net flux of heat between two surfaces of temperature difference AT as 4eT~AT and obtained: K,=4yoet3odp [1] where a is the Stefan-Boltzmann constant, E is the emissivity of the radiating surface, To is the average absolute temperature of the region across which radiation occurs, dp is the largest dimension of the gap in the direction of heat flow, and y is a geometric factor. That is: y = 1 for laminar pores and cylindrical pores with axes parallel to the heat flow direction, y = 2/3 for spherical pores, y = /4 for cylindrical pores with axes perpendicular to heat flow direction. It was the purpose of this work to obtain an exact expression for K, in spherical pores by integration of the general expression for the net heat transfer between two incremental areas on the surface of a spherical pore. This procedure yields a value of Kr 3/4 of that obtained by Loeb. I) CALCULATION OF Kr Fig. 1 illustrates the general geometry which defines the problem. The total emissive power for a gray body is given by: E = EoT4 [2] where E is the radiated energy per unit area of the radiation surface per unit time, T is the absolute temperature, o is the Stefan-Boltzmann constant (o = 5.67 X lo- &apos;&apos; watt per sq cm-"K1), and E is the emissivity of the surface. Although Eq. [2] gives the exact expression for the total emissive power of any gray body, all practical problems involve the net exchange of thermal radiation between a body and surrounding bodies. The two major considerations in the solution of such problems are a) shape and relative position of the surfaces and b) the emissivities of the surfaces. In this work it shall be assumed that all incremental areas on a pore have the same emissivity and, further, that little or no reflection occurs (i.e., the radiation hitting a surface is either absorbed or transmitted). Thus, the solution of the stated problem requires only that consideration a be satisfied. Consider two incremental surface areas dA 1 and dA2 of the same material in a general orientation to one another. Let dAl be at temperature TI and dAz be at T2. Fig. 2 illustrates the geometrical parameters involved. That is, 71, is the normal of dA, n2 the normal of dA2, r the line connecting the centers of each area, 81 the angle between nl and r, and Q2 the angle between 7,2 and r. Using Eq. [2] and the Lambert cosine law of diffuse radiation it is easily shown that:293 WdAf- dA, = ecr T*------^fpi------" dAidA2 [ 3a 1 and

Jan 1, 1970

By H. K. Najarian

This paper describes recent development work on the Wheaton-Najarian Vacuum Condenser for Zinc at Josephtown Zinc Smelter, wherein, to dissipate the heat of condensation, the "airlift" action of gases bubbling through the molten zinc in the condenser is utilized to circulate the zinc through an outside "cooling well" and cooling it continuously. AN earlier paper1 gave an account of initial development of the Wheaton-Najarian Vacuum Zinc Condenser at the Josephtown Zinc Smelter of the author&apos;s company. This paper is a progress report of further development work over the years directed toward improving the efficiency and increasing the capacity of the condenser. In this type of zinc condenser, zinc vapor with accompanying noncondensable gases from electro-thermic zinc smelting furnaces is passed upwardly through a mass of molten zinc confined in a refractory-lined U-tube. The zinc vapor condenses in the molten zinc and the noncondensable gases are drawn out of the condenser by suction applied over the surface of the molten zinc. The heat of condensation of zinc, amounting roughly to 1000 Btu per lb of zinc condensed, is absorbed by the molten zinc and, in turn, is dissipated continuously to keep the temperature of molten zinc in a range to permit effective condensation. At the time of publication of our first paper, we had at Josephtown Smelter five units of 69-in.-bore metal furnaces and one larger metal furnace having 96-in. bore. Each 69-in. metal furnace had a condenser with 22-ft overall length. The 96-in.-bore furnace had a condenser with overall length of 31 ft. Condensers were of circular cross section, with corrugated steel outer shell and carborundum lining. Diameter of these condensers was 32 in. inside the refractory lining. Molten metal in the con- denser was kept cool by spraying cold water on the outside of the condenser shell. Under good operating conditions, 22-ft condensers produced some 12 tons of metal per day; and the 31-ft condenser, about 22 tons. When the furnaces produced zinc vapor at higher rates, invariably the condenser temperature would rise to levels where excessive blue powder would form and accumulation of accretions in the condenser would be accelerated. We found that, with the system of cooling in use, we had to provide heat-exchange surface of some 150 sq ft to condense 1000 lb of metal per hr with reasonable efficiency. The wall of the condensers was made up of 5/8-in.-thick corrugated-steel shell having next to it several layers of mica and inner refractory lining of special carborundum shapes to fit the corrugation in the steel shell, and having a minimum thickness of about 2 in. Relatively low rate of heat transfer through the thick condenser wall and slow internal circulation of the molten zinc limited the rate of heat exchange between hot metal within the condenser and the cold water flowing over the outer steel shell. The thickness of the condenser wall was about minimum that we considered suitable for continuous operation. As we learned to operate the electrothermic furnaces with higher power input, producing zinc vapor at higher rates, it became apparent that we needed condensers having larger heat-exchange surfaces. Our 96-in.-bore furnace had 6 single-phase power circuits with transformers rated at 500 kva each and normally good for power input of 3000 kw.

Jan 1, 1959

By M. W. Legatski, D. L. Katz

The best currently available description of the longitudinal mixing properties of a porous medium is an equation of the form which relates the effective longitudinal dispersion coefficient Dp to the molecular diffusion coefficient Do, the electrical resistivity factor F, the porosity <f> and a Peclet number. If the parameters dpa and m are determined for a porous medium of known porosity and electrical resistivity factor, then a dispersion coefficient may be estimated for a given flow rate and a given gas pair. A new method, featuring on-line gas analysis by thermal conductivity and on-line data reduction by analog computation, was developed and used to determine these mixing parameters for eight naturally occurring sandstones and two dolomite samples. The exponent m of the above equation was found to vary between 1.0 and 1.5. The characteristic length dpa in the above equation was found to vary between 0.25 and 1.9 cm, with an average value of 0.4 cm for sandstones. Measurements were made on two cores in which paraffin wax had been deposited by evaporation from a pentane solution. They indicated that the presence of an immobile phase such as connate water could increase the dispersion coefficients significantly. INTRODUCTION While the petroleum and chemical industries have studied the mixing of miscible liquids flowing in consolidated porous media and of miscible gases flowing in unconsolidated porous media, relatively little data have been presented to describe the mixing of gases flowing through consolidated porous media. Such data are of particular interest to the gas storage industry. For instance, the U. S. Bureau of Mines is storing large quantities of a rich helium-nitrogen gas in contact with a natural gas in a dolomite reservoir. Since the rich gas occupies only 15 percent of the total reservoir volume, it is essential that the extent of rich gas-natural gas mixing be predicted and understood as a function of rock properties, pressure and rate of movement. This investigation was concerned only with the determination of longitudinal dispersion coefficients. It is understood that a transverse dispersion coefficient, which characterizes mixing perpendicular to the direction of flow, may be an order of magnitude less than the coefficient characterizing mixing in the direction of bulk flow.5719 It should also be recognized that the use of any dispersion coefficient is in itself a simplification. It is necessary to assume that mixing in a porous medium may be characterized by the equation for flow in a single direction. A number of authors1 l8,19 have pursued the mixing problem, not in terms of the so-called "dispersion model" described by Eq. 1, but in terms of a "mixing cell model". This model supposes that a porous medium is constructed of a large number of small mixing chambers and that the concentration of the diffusing component within each mixing chamber is uniform. Fick&apos;s law (Eq. 1) assumes that there is no gross by-passing of one fluid by another, and that there are not stagnant pockets of gas in the system under consideration as discussed by Coats and Smith.8 These assumptions are not always valid for flow through porous media and it is important to recognize the limitations upon Eq. 1. The most serious limitation upon the assumption of Fick&apos;s law mixing in a porous medium is

By W. B. Gogarty, H. R. Bailey

Methods are presented for determining molecular diffusion coefficients by using data from, capillary flow experiments. These methods are based on a numerical solution (presented in a previous paper) of the partial differential equation describing the combined mechanisms of flow and diffusion. Results from this numerical solution are given and compared with the approximate analytical solution of G. I. Taylor. The numerical solution is valid over a much larger time range. These methods are applied to experimental results for the fluid pairs water-potassium permanganate solution and amyl acetate-orthoxylene. Both of these fluid pairs have approximately equal densities and viscosities. Graphical and numerical techniques are presented for deter mining diffusion coefficients from the flow data. Values obtained by these techniques are compared with values obtained by other methods. INTRODUCTION The molecular diffusion coefficient is known to be a variable in determining the amount of mixing in a miscible displacement process. The effect of molecular diffusion on dispersion in longitudinal flow through porous media -has been examined by different investigators. These investigators concluded that at low velocities of flow, the amount of dispersion is approximate1y proportional to the molecular diffusion coefficient. The influence of diffusion on fingering, channeling, and overriding has been mentioned by other investigators.4 Recent studies5,6 have been made on the effects of molecular diffusion in connection with the problem of gravity segregation. Many different methods have been developed for the experimental determination of molecular diffusion coefficients? These methods differ mainly according to boundary conditions selected and analytical procedures used. Nevertheless, all of these methods have the condition in common that the bulk fluids in which diffusion is occurring are stationary with respect to each other. In connection with a series of papers on mixing in capillary flow,8-10 Taylor suggested the use of a flow method for determining molecular diffusion coefficients. Additional studies 11-13 have been conducted on miscible displacements in capillary tubes, but the data from these studies were not used for the specific purpose of determining diffusion coefficients. The flow method proposed by Taylor results in a single value of the diffusion coefficient for the fluid pair used in the displacement experiments. This single value represents the true value for the fluid pair when the diffusion coefficient is independent of concentration. If the diffusion coefficient is a function of concentration, the single value obtained by the flow method gives an average value for the coefficient of the fluid pair. These average values are based on diffusion taking place over the entire range of concentration, i.e., from 0 per cent of one fluid to 100 per cent of that same fluid. In field applications of the miscible displacement process, gradients occur over the same range of concentration as are found in the displacements in capillary tubes. Molecular diffusion coefficients obtained from the capillary flow method should, therefore, be especially relevant to field operations. This investigation was undertaken to evaluate the feasibility of obtaining molecular diffusion coefficients from capillary -flow experiments. In making this evaluation, diffusion coefficients were first determined for two systems from data obtained in capillary flow experiments. These values of the diffusion coefficient were then compared to values obtained by other methods. MIXING IN CAPILLARY FLOW—THEORETICAL The theoretical basis for determining molecular diffusion coefficients from capillary flow experiments is the partial differential equation relating the mechanisms of flow and diffusion. For the case of laminar flow and with a fluid pair having equal densities and viscosities, the equation is given by

By M. Bevis, A. G. Crocker

A. G. Crocker and M. Bevis (Battersea College of Technology)— Richman and Richman and conard17 have recently published the very interesting results of their studies on the plastic deformation of Fe-Ni-C martensites and of an Fe-5 wt pct Be alloy. Other work on the range of Fe-Be alloys containing between 16 and 27 at. pct Be has also been reported by Richman." One of the fascinating features of this work is that the materials investigated deform primarily by means of twinning and that several twinning modes are operative in addition to the well-established (112 ) mode of bcc materials The composition planes of these unusual twins were determined using a single-surface analysis and reported as (0131, "{045)" or "{089)", and "{2,3,10)" or "(127)". (It is now thought1&apos; that the "{0,1,13)" habit reported for the Fe-Ni-C alloys&apos;&apos; is more likely to be near "{127)". Although the full twinning elements of these unusual twinning modes could not be determined directly from the experimental results, the authors attempt to deduce the elements by considering twin intersectionslQ and by using the projection technique of Jaswon and Dove.&apos; The basic assumption of their approach is that the (013) habits are truly rational planes but that the other habits are irrational and merely approximated to by the indices quoted above. They thus deduce that the twinning mode describing the (013) habit is either Type I or compound and that the modes describing the other habits are Type 11. This deduction must however be false as all twinning modes in cubic structures obeying the classical orientation relationships of deformation twinning must be compound. Type-I and Type-I1 modes can occur in all of the other crystal systems (indeed in the triclinic system compound modes are not allowed), but in the cubic system only compound modes can arise. This fact, which does not appear to be well-known, fol- lows immediately from the relations between the four twinning elements defining a twinning mode. Thus for a Type-I mode which is uniquely defined by the rational twinning elements K1 = hi and tj2 = vi (i = 1, 2, 3), the Kz qd 771 elements are given by: where the summation convention of the tensor calculus has been adopted. For cubic materials the metrics cij Iand cij reduce to the unit tensors 6ij and 6ij so that we have It is clear from these equations that both ki and ui must take on integral values; i.e., Kz, and ql must be rational, if hi and vi are integers, i.e., if K1 and 77~ are rational. Thus for cubic materials a Type-I mode must degenerate into a compound mode and the same result also arises for Type-II modes. The apparently irrational habits observed by the authors can thus not be explained in terms of Type-I1 twinning if the classical laws of deformation twinning are to be obeyed. An alternative explanation of the irrational habits is that they may arise from twinning modes with four irrational twinning elements. These modes, which do not obey the classical orientation relationships associated with deformation twinning, have been shown to arise as the result of double twinning processes in magnesium,22 but may also describe a simple shear on an irrational plane in an irrational direction. Indeed double twinning processes in bcc

Jan 1, 1963

By W. B. Gogarty, H. R. Bailey

Methods are presented for determining molecular diffusion coefficients by using data from, capillary flow experiments. These methods are based on a numerical solution (presented in a previous paper) of the partial differential equation describing the combined mechanisms of flow and diffusion. Results from this numerical solution are given and compared with the approximate analytical solution of G. I. Taylor. The numerical solution is valid over a much larger time range. These methods are applied to experimental results for the fluid pairs water-potassium permanganate solution and amyl acetate-orthoxylene. Both of these fluid pairs have approximately equal densities and viscosities. Graphical and numerical techniques are presented for deter mining diffusion coefficients from the flow data. Values obtained by these techniques are compared with values obtained by other methods. INTRODUCTION The molecular diffusion coefficient is known to be a variable in determining the amount of mixing in a miscible displacement process. The effect of molecular diffusion on dispersion in longitudinal flow through porous media -has been examined by different investigators. These investigators concluded that at low velocities of flow, the amount of dispersion is approximate1y proportional to the molecular diffusion coefficient. The influence of diffusion on fingering, channeling, and overriding has been mentioned by other investigators.4 Recent studies5,6 have been made on the effects of molecular diffusion in connection with the problem of gravity segregation. Many different methods have been developed for the experimental determination of molecular diffusion coefficients? These methods differ mainly according to boundary conditions selected and analytical procedures used. Nevertheless, all of these methods have the condition in common that the bulk fluids in which diffusion is occurring are stationary with respect to each other. In connection with a series of papers on mixing in capillary flow,8-10 Taylor suggested the use of a flow method for determining molecular diffusion coefficients. Additional studies 11-13 have been conducted on miscible displacements in capillary tubes, but the data from these studies were not used for the specific purpose of determining diffusion coefficients. The flow method proposed by Taylor results in a single value of the diffusion coefficient for the fluid pair used in the displacement experiments. This single value represents the true value for the fluid pair when the diffusion coefficient is independent of concentration. If the diffusion coefficient is a function of concentration, the single value obtained by the flow method gives an average value for the coefficient of the fluid pair. These average values are based on diffusion taking place over the entire range of concentration, i.e., from 0 per cent of one fluid to 100 per cent of that same fluid. In field applications of the miscible displacement process, gradients occur over the same range of concentration as are found in the displacements in capillary tubes. Molecular diffusion coefficients obtained from the capillary flow method should, therefore, be especially relevant to field operations. This investigation was undertaken to evaluate the feasibility of obtaining molecular diffusion coefficients from capillary -flow experiments. In making this evaluation, diffusion coefficients were first determined for two systems from data obtained in capillary flow experiments. These values of the diffusion coefficient were then compared to values obtained by other methods. MIXING IN CAPILLARY FLOW—THEORETICAL The theoretical basis for determining molecular diffusion coefficients from capillary flow experiments is the partial differential equation relating the mechanisms of flow and diffusion. For the case of laminar flow and with a fluid pair having equal densities and viscosities, the equation is given by

By E. F. Sumner, C. K. Eilerts

A partial differential equation was developed and programmed to compute the cumulation and flow of the liquid phase of a gas-condensate fluid. It was found that such cumulation near the wellbore may attain saturations permitting liquid flow within periods of the order of 1 day. The radius at which this saturation obtains advances toward the external reservoir boundary with continued recovery, although it may move only 2 to 5 percent of that distance during the productive life of the well. This relatively high saturation can cause a serious dimunition in delivery capacity. The pressure dilferential required for flow of gas through a formation with such saturation may be as much as three times greater than the corresponding differential in a formation where saturation is less than 2 percent. Because of the radial geometry, total reservoir pressure differential required for flow with a zone of relatively high liquid saturation about the well was indicated to be two to three times greater than without the saturation. INTRODUCTION A computing program developed by the authors 1 to integrate the partial differential equation for transient radial flow of gas-condensate fluids required a prevalued function, W(Po) = kD(p0)/M(p0)z0(Po).......(1) related to cumulation of condensate in pores of the reservoir as pressure declines. Methods were available for relating mobility to saturation, saturation to liquid-vapor volume ratio, liquid-vapor volume ratio to pressure and, hence, mobility to pressure to obtain W(pD). However, certain assumptions had to be made in advance of program execution concerning movement of the liquid phase and change in radial distribution of liquid saturation as recovery progressed. Because reservoir fluid mobility and related pressure gradients are markedly dependent on liquid saturation, it was desirable to eliminate the necessity for these assumptions and in this respect improve the reliability of a result obtained by integration. Experimental research was begun to determine the relationship of variables that control the reservoir flow of low liquid-vapor ratio fluids. With the porous medium partly saturated, permeability to a two-phase flowing fluid could not be measured because the viscosity was unknown. Viscosity could not be measured without knowing permeability of the liquid-containing medium; but mobility, the ratio of permeability to viscosity, was measurable. This property, and liquid-vapor volume ratio of the flowing phases, have values determined by nature of the fluid and porous medium and, in addition, were found to be pressure, velocity and saturation dependent. A program developed by Kniazeff and Naville 2 for computing multicom-ponent, two-phase flow used polynomials to represent saturation dependence of permeabilities for gas and for liquid phases, and an effluent viscosity constant under reservoir conditions. The authors developed a program based on a partial differential equation of similar mass balance origin but designed to represent distribution of an unlimited number of components between gas and liquid phases utilizing the experimental information available as to variable dependence. Attention was given to liquid-vapor volume ratio of the flowing phases, a property which is essential as a liquid saturation boundary condition. A program for computing implicitly saturation increase at uniform intervals along the drainage radius was combined with an existing program for calculating pressures, and the two were used alternately in an iterative solution for both saturations and pressures. The function W(pD) needed for calculating pressures was replaced by the ratio of mobility to compressibility factor. BASIC EQUATIONS The partial differential equation for calculating

By M. C. Fuerstenau, A. M. Gaudin

A homogeneous suspension is viewed by X-rays. The radiation density seen, affords a measure of the extra absorption due to the solids contained. This radiation density, at a predetermined depth, varies with time. The chart of this relationship gives the desired determination. s it is frequently necessary to know the distribu-rt tion of particle sizes in various products used in industrial applications, a number of means have been devised for these determinations. Some of them include direct microscopic measurement, screening of product, extraction of samples from a settling slurry, air and liquid elutriation, adsorption of gases and liquids, and light absorption. The absorption of light as a means for particle size analysis has been known and employed for some time. X-ray absorption has not received the attention that light has enjoyed primarily because of the complex and costly instrumentation involved with X-rays. In 1954, Brown and Skrebowskil suggested the use of X-rays for this type of determination but did not supplement their suggestion with experiment. In 1958, the authors conducted size analysis experiments on suspensions of quartz with the Trans-viewer: an X-ray unit specifically adapted for sedimentation research, and Rossa conducted experiments in a similar manner in 1959 with a stable gamma emitter, AmZ4l, on suspensions of uranium oxide. To illustrate the applicability of X-ray absorption for analyzing particle size distributions, an essentially mono-energetic and well collimated X-ray beam was used as a source. The beam was made mono-energetic by placing a zirconium filter before the counting window together with feeding the pulses from the proportional counter into a reverter, a device capable of discriminating against pulses of various heights. The sketch in Fig. 1 shows the collimation arrangement in more detail. As shown, the X-ray beam was collimated by inserting a 3/8-in. ID pipe into one of the ports of the X-ray tube and further by two small steel plates with slits 3/32-in. wide and 1/2-in. long placed in front of the window of the Geiger-Mueller (G-M) tube. One of the plates was located 1/2 in. and the other 2 1/2 in. from the window of the counter; both plates were contained in an aluminum holder mounted on the box containing the G-M tube. This type and degree of collimation should eliminate any intensity due to scattered radiation; the transmitted intensity would then be solely a functior. of the absorption by the suspension at that level. Experimentally, homogeneous suspensions of finely ground quartz were prepared, Daxad (a sulfo-nated compound made from wood) was added as dis-persant, and the transmitted X-ray intensity measured as a function of time at a fixed depth in the suspension. In more detail, the experimental procedure involved the following: 1) measuring the transmitted X-ray intensity through the Daxad solution at a predetermined position or settling depth in the absence of solids, 2) placing a known amount of dry solids into the vessel and agitating thoroughly, 3) measuring the transmitted X-ray intensity at time zero (i.e., immediately upon cessation of stirring), and 4) measuring the transmitted X-ray intensity as a function of time (a 10-sec counting interval being chosen and counting being started 5 sec before the desired time). Since a linear relationship exists between percentage solid in a slurry and the logarithm of transmitted intensity for a homogeneous material, the cumu-

Jan 1, 1961

By A. M. Skov, L. F. Elkins

Inclusion of anisotropic permeability in mathematical analysis of pressure transients observed during development of the huge Spraberry field indicates a major fracture trend which is in good agreement with that observed by fluid-injection tests spread over a 12- by 17-mile area. Delineation of this trend is important in selecting a pattern of injection for the pending large-scale water flooding in this field. Determination of reservoir parameters yielding best agreement between calculated pressures and observed reservoir pressures in newly completed wells was made using an IBM 650 computer. INTRODUCTION The Spraberry field covering 400,000 acres is a tight sand of less than 1-md permeability cut by an extensive system of vertical fractures. Primary recovery dominated by capillary retention of oil in the fractured sand matrix blocks is less than 10 per cent of oil in place. Strong forces of capillary imbibition of water into the sand, coupled with water flow under dynamic pressure gradient, indicate considerable increase in oil recovery can be achieved through water flooding. Best results will occur if the pattern of water injection is selected to force the water flow across the grain of the major fracture system. Existence of an oriented vertical fracture system in the Spraberry, observed first in cores, was highlighted more recently by the 144-fold contrast in permeability along and at right angles to the major fracture trend required to match relative water breakthrough times in Humble Oil & Refining Co.&apos;s waterflood test there. Spraberry operators since have conducted two gas-injection tracer tests for further areal confirmation of the fracture trend. Re-analysis of early reservoir pressure transients for evidence of anisotropic permeability has permitted many more local determinations of major fracture trend without resort to further field tests. This paper is limited to updating analysis of reservoir pressure transients to include anisotropic permeability as a test for orientation of the major fracture trend in the Spraberry. The reader is referred to Refs. 1 and 2 for information about general Spraberry reservoir performance and to Refs. 3 and 4 for information about significance of fracture orientation in selection of the injection-well pattern for water flooding the Spraberry. RESERVOIR PRESSURE DATA—DRIVER AREA During early development of the Spraberry Driver area, Sohio Petroleum Co. made the extra effort to measure the initial pressure in each of the 71 wells in a 5-mile-long area immediately after completion. Progressively greater reductions in pressure ranging up to 400 psi were observed throughout the six-month development period. Detailed data are presented in Ref. 1. Since the reservoir oil was undersaturated some 300 psi initially, early reservoir performance involving 55 new well pressures is subject to analysis as flow of a single compressible fluid in a porous media. Assumption of uniform permeability in all directions yielded good agreement between calculated pressures and observed pressures of these wells in the earlier study,&apos; but subsequent, additional, mathematical development to include anisotropic permeability in the transient pressure considerations and present availability of electronic computers to perform the much more extensive arithmetical calculations now yield even better agreement. The previous analysis, assuming uniform permeability, consisted essentially of calculating pressure reduction expanding circularly around each producing well and summing these effects at the time and location of each newly completed well for comparison with the measured pressure reduction. Permeability, effective fluid and rock compressibility, and permeability X thickness were varied until the best match with measured pressures was obtained. The present analysis, assuming anisotropic

By N. E. Garner

Laboratory experiments were conducted to investigate the cutting of a single diamond on limestone and shale under simulated down-hole conditions. A high-pressure chamber was modified by adding a rock-rotating device so that planing tests could be run. Weight on the diamond, diamond geometry and differential pressure across the face of the rock were varied. It was found that (I) high differential pressure reduced the volume of rock cut by the diamond at a given bit weight by strengthening the rock and changing the mode of failure; (2) a finely powdered rock left in the bottom of the grooves reduced the volume cut by as much as 50 percent; and (3) orientation of a single diamond about the axis of a drill point had considerable influence on diamond cutting efficiency. INTRODUCTION Diamond bits consist of many diamonds held in a matrix provided with water courses for fluid passage to clean, cool and lubricate the diamonds as they drill formation. Each diamond contributes its share to the overall effectiveness of a bit, but a lack of understanding of the performance of a single diamond has hampered efforts to engineer improvements in diamond bits. The need for those improvements prompted this study. This paper describes the findings of a laboratory investigation of the outting action of a single diamond where the differential pressure between the wdlbore and formation, diamond geometry and vertical force on the diamond were independently varied. The tests were conducted in a pressure chamber that provided relative horizontal motion (planing) between a rock and diamond under simulated down-hole pressure conditions. EQUIPMENT HIGH-PRESSURE CHAMBER WITH ROTATING DEVICE The high-pressure chamber used in previous rock mechanics studies&apos; was modified for this study by addition of a rock-rotating device. The modified chamber (Fig. 1) differs from the old chamber in that the rock can be rotated to provide relative horizontal movement between the rock and the penetrator which may be a small element of a diamond or a drag bit. In this study the penetrator was a single diamond. The rock sample holder inside the chamber is mounted on a thrust bearing and is connected to the external drive mechanism by a stem that passes through two seal elements. The seal elements are needed to allow formation pressure to enter the bottom of the rock sample through a small hole in the stem. The Kapseal seal elements (Tefion boots backed by O-rings) provide low friction (hand free) even at differentials of 5,000 psi. The drive mechanism is a variable-speed transmission with a speed range from 0 to 30 rpm and an output torque of 2,800 in.-lb. A right-angle-drive gear box couples the drive to the stem. The new design has two independent pressure systems (formation and borehole) to simulate down-hole rock stress environment. Formation pressure enters from the bottom of the rock through a 6-in. diameter area, and borehole pressure acts over the remaining surface. A 14-in. layer of zero fluid loss oil-base mud poured over the top of the rock and an O-ring seal on bottom allow a differential between the borehole and formation pressures. Bayol 50 was used as the hydraulic fluid in the chamber. The differentials reported here were obtained by elevating the borehole pressure while keeping the formation pressure at zero.

By B. Wilhelmi, W. H. Somerton

Simultaneous measurement of pore and elastic properties of socks under a wide range of triaxial stress conditions may be made with equipment and methods developed in this paper. Tests were run on three outcrop sandstones at several confining pressures and axial stresses up to approximately 80 percent of failure stress. The greater reduction in porosities and permeabilities occurred in the hydrostatic loading portion of the stress cycle, but substantial reductions in both properties did occur upon application of deviator (triaxial) stress. Triaxial elastic moduli were calculated from strain gauge measurements. Both Young&apos;s moduli and shear moduli were found to increase with increasing confining pressure. Poisson&apos;s ratios increased with axial stress but showed no consistent change with confining pressure. Only very general relations were observed between pore and elastic properties. The lesser decrease in porosity and permeability at higher stress values is related to increase in rigidity of rocks at higher stresses. INTRODUCTION Many investigators have studied the effects of hydrostatic stress on the properties and behavior of rocks. A few have studied the effects of triaxial stress fields on individual physical properties of rocks. To our knowledge no one has measured simultaneously pore and elastic properties of rocks under a wide range of triaxial stress conditions. This is the purpose of this work. Rocks exist in the subsurface under what Jaegerl refers to as "polyaxial" stress conditions (all principal stresses unequal). When two of the three principal stresses are equal, this is referred to as triaxial loading. The triaxial method of loading is most commonly used in modern testing work because experimental difficulties in measuring physical properties of rocks are greatly reduced over polyaxial testing. Although triaxial loading may not reproduce subsurface conditions precisely, subsurface stress conditions are never known precisely anyway. Triaxial testing would seem to be more realistic than testing under hydrostatic loading (all principal stresses equal) for application of results to subsurface problems. The objectives of this work were threefold: (1) to develop methods of measuring simultaneously pore and elastic properties of rocks under triaxial stress conditions, (2) to determine whether changes in these properties are of sufficient magnitude to be considered in subsurface calculations and (3) to attempt to relate changes in pore properties and elastic properties since they will have been measured at the same time under identical loading conditions. REVIEW OF EARLIER WORK PORE PROPERTIES Several investigators have studied the effects of stress on the physical properties of rocks. Fatt and Davis2 showed the reduction of permeability of rock with hydrostatic loading. McLatchie et al.3 and Knutson and Bohor4 related permeability reduction to compessibility of reservoir rock, again under hydrostatic loading. Wble,5 and later Redmond,6 evaluated the effects of applied radial pressure on conductivity, porosity and permeability of sandstones. In this latter work, no axial stress was applied. Dobrynin7 reviewed much of the previous work done . on the effects of overburden pressure on physical properties of rocks. He concluded that changes in physical properties are controlled to a large extent by pore compressibility of the rock. Several sets of curves developed from empirical relations are presented. These curves permit estimation of change of properties with net overburden pressure, knowing the pore compressibility and porosity of the rock. Net overburden pressure is defined as (Pe - 0.85 Pi ) where Pe is the

By R. D. Nelson, F. E. Bowman

Evidence has been found for a (010)a?(111)B[102]a /,11[341]/B crystallographic relationship between the monoclinic a and B phases of plutonium during the a?B transformation. The results of an X-ray diffraction study, where [010] fiber textured a samples were used in the absence of suitable single crystals indicate that the (OIO)a planes become (111)B planes upon transformation under near-equilibrium conditions. The transformation strain along the fiber axis, which was the [010] direction, was measured dila-tomekically; the measured 8.5 pct expansion compares favorably with the 8.44 pct expansion in the in-terplanar spacing represented by the (010)a?(111)B relationship. From plots of the atom positions in each of the related planes, a correspondence was deduced between the [102]a, direction and the [341]B direction. Further analysis of the two lattice structures in the light of this relationship revealed a rotation about the [102] direction of 1 deg 43 min and a slight shearing in this same direction of about 30 min. THIS paper discusses the (B crystal structure of plutonium and the orientation relationship existing between the monoclinic a and B phases during the a?ß transformation. The kinetics of the a?ß and (3 — a transformations have received extensive consideration, and because of the crystallographic complexity of these phases, the question of the mechanisms by which these transformations occur is especially intriguing. Nelson and Shyne1 evaluated the existing experimental evidence and concluded, from the indirect evidence available, that both the ß ? a and a ? ß transformations occur by a martensitic process. Their data reveal the anisotropic nature of these transformations which suggests the existence of specific crystallographic relationships between the parent and product phases. In this paper, we are concerned with the confirmation of the existence of such a relationship resulting from the a ? ß transformation and the derivation of the crystallographic details necessary to specify it. EXPERIMENTAL RESULTS X-ray diffraction and dilatometric data were obtained from samples of a plutonium having a [010] fiber texture. These cylindrical specimens, approximately 1 cm in diam, were prepared by transforming the ß phase to a under the influence of a uniaxially applied compressive load.&apos; The [0l0] direction is parallel to the direction of applied stress; it is also perpendicular to the (010) planes. This correspondence between the plane normal and the direction having the same indices is, in the monoclinic a plutonium structure, restricted solely to this (010)[010] combination. Such a specimen was mounted in an evacuated, elevated temperature specimen holder on an X-ray dif-fractometer so that the diffracting surface was perpendicular to the fiber axis. The a diffraction pattern, Fig. 1, taken at about 115ºC, revealed the predominance of the (020) reflection. This monoclinic, P 21/rn space group yields no (0k0) reflections when k is odd. The temperature was raised slowly to 160°C to effect the complete transformation to the phase; then it was lowered to 120°C so that a diffraction pattern of the ß could be obtained near the a = ß equilibrium temperature for direct comparison with the a at the same temperature. The (3 diffraction pattern, Fig. 2, obtained under these conditions showed a (222) reflection relatively much more intense than in the random pattern. The other reflections appearing with sig-

Jan 1, 1970

By D. A. Flanagan, B. L. Landrum, P. B. Crawford, B. C. Norwood

Herern is described a heat conduction model which has been developed to predict the pressure distribution within a reservoir being subjected to rrnsteady-state depletion. Although potentiometric models have been widely used to study reservoir pressure distribution, their application is limited to the study of steady-state cases. Electrical analyzers may be used to study unsteady-state phenomena, but their application and simulation have certain limitations. Much of the now accepted theory of unsteady-state fluid flow in porous media has evolved from previous heat conduction knowledge. An analogy between the flow of fluid in an unsymmetrical reservoir and con-duct ion of heat in a similarly shaped unsymmetrical metal plate is made to show adaptability of the hear conduction principle to transient reservoir cases. A description of the model and necessary instrumenrarion is included. Data show influence of well location on pressure distribution during unsteady-state deple[ion. Application of results is made to a hypothetical reservoir and to a reservoir located in an irregular aquifer. INTRODUCTION Development of methods for predicting behavior of petroleum reservoirs has resulted partially from adaptation of electrical and thermal flow concepts to the flow of fluids through porous media. Extensive investigation of electrical and thermal flow problems has resulted in mathematical solutions applicable to many reservoir problems. The relative ease of measuring electrical flow has brought wide acceptance of the use of the potentiometric model to study certain field problems. Potentiometric models are restricted to the study of steady-state conditions because of the instantaneous response of electrical flow to changes in the applied potential. The electrical reservoir analyzer has been one of our most useful instruments for studying certain phases of unsteady-state behavior1. Other than approximation techniques, no method exists for determining pressure distribution in an irregular reservoir during primary depletion. Heat flow, like fluid flow, is inherently transient. The model described here was developed to use this transient nature of heat flow, measurable in the laboratory, to simulate transient flow of fluids in the porous medium of the reservoir above the bubble point. A metal plate shaped like the reservoir to be studied was heated by a "well" source. Temperature increase in the plate is analogous to pressure decline in the field if the field is being depleted, or pressure increase if fluid is being injected. This study is intended to determine the feasibility of using the transient nature of the heat flow to study the transient behavior of fluid flow. It is hoped that this new technique will aid in the understanding of one of the more complex phases of reservoir engineering. THE FLUID FLOW-HEAT CONDUCTION ANALOGY The similarity of the equations which describe the flow of fluids through a porous medium and the conduction of heat through a solid is well known2. Rate of flow of fluids is given by Darcy&apos;s law3. while the rate of flow of heat is given by Fourier&apos;, (I,, - KA 4T........(2) Combining Eq. 1 with the equation of continuity for a compressible liquid", i.e.,results in Expressing y in terms of the compressibility, i.e., pr = peey ,..........(5) and by expanding the exponential and neglecting higher orders of cp one can obtain