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By J. H. Frye, J. L. Scott, J. W. Woods

THERE are at least two effects of temperature on the stress required to produce plastic flow. 1) Metals plastically strained for the same amount at different temperatures possess different stress-strain curves and different internal structures (e.g., nature and density of imperfections). 2) There is both experimental and theoretical evidence that the stress necessary to produce plastic flow may vary with temperature even though the internal structure remains constant.1-5 Much of the data in the literature is difficult to interpret because these two effects are simultaneously present. In the work reported here, they are considered separately. Standard 0.505 in. tensile specimens were prepared from high-purity Mond nickel and OFHC copper. Nickel specimens were annealed in hydrogen at 1150°C for 30 min and furnace cooled. This produced an average grain diameter of 1.4 mm. Copper specimens were annealed in vacuum at 525°C for 6 hr and furnace cooled. This produced an average grain diameter of 0.11 mm. Nickel tensile specimens were prestrained in tension at 500°K, whereas copper tensile specimens were prestrained in tension at room temperature. All specimens were cooled to 74°K and heated to room temperature. After these treatments all of the tensile specimens of a given set possessed identical structures. All specimens were returned to the tensile machine and tested at various temperatures and constant cross-head speed. Many of the tensile curves showed well defined yields but, in order to minimize personal judgment, the yield was always chosen as the stress corresponding to 0.2 pct plastic deformation. Results are summarized in the graph of yield vs temperature of test, Fig. 1. All of the specimens corresponding to any one line have had identical treatments and, therefore, have identical internal structures. Thus, each line gives the relation between yield stress and temperature for one constant internal structure. The lowest line, which is for annealed nickel, shows no measurable variation of yield with temperature. This is similar to Blewitt's6 results on copper single crystals, but a variation in yield with temperature was found by Andrade and Henderson' in single crystals of nickel, and by Car-recker and Hibbard8 in annealed polycrystalline copper. The three other lines, which are for prestrained materials, show that there is a linear relation between yield stress and temperature when the internal structure remains constant. Such a relation might be expected from a rate equation such as that of Kauzmannl on the assump- tion that the activation energy is either independent of or linear with temperature. These results would also be expected if the shear moduli of copper and nickel are linear functions of temperature for, as Cottrell and Stokes have pointed out," in all dislocation theories of strength the flow stress appears proportional to the shear modulus of the material. In copper and nickel, since the modulus of elasticity decreases roughly linearly with increase in temperature," it is probable that the modulus of shear also decreases approximately linearly with increasing temperature. A detailed discussion of the effect of these and other factors on the temperature dependence of flow stress in face-centered-cubic metals has recently been given by Seeger.4

Jan 1, 1958

By A. T. Corey, C. H. Rathjens

INTRODUCTION Although the oil industry has been aware of the directional variability of permeability in porous rock, the directional variability of relative permeability has been largely ignored. Yet it is apparent that such an effect must be present in a system in which the distribution of oil and gas within the porous matrix is controlled by capillary forces. It is easy to visualize a rock composed of layers of fine and coarse material such that gas flow across the bedding planes would take place only after the average oil saturation had been reduced to a very low value. The fine layers, because of their greater capillarity, would remain saturated and act as barriers to the flow of gas after the coarse layers had been desaturated. Flow of gas parallel to the bedding planes would obviously take place at a much greater liquid saturation. Without more complete information concerning the geology of a reservoir than is generally available, it is not possible to predict exactly how such phenomena would affect the over-all performance of an oil field. It is possible, however, to predict qualitatively the effect of stratification on relative permeability measurements made on laboratory cores. In this investigation the effect of stratification was studied analytically by assuming that two porous materials with different capillary pressure-desaturation curves (but identical relative permeability curves) were in contact and in capillary equilibrium. As a qualitative check on the analytical results, cores having various degrees of visible stratification were used for relative permeability measurements made with fluids flowing both parallel and perpendicular to the bedding planes. A quantitative check was considered impractical because of the difficulty of devising models in which two materials of predetermined properties could be joined without the plane of contact becoming a discontinuity. THEORETICAL CONSIDERATIONS AND ASSUMPTIONS The assumption of capillary equilibrium in an oil-gas system implies that the difference in pressure between oil and gas is everywhere the same. This means that the curvature of the interfaces must be everywhere the same in order to satisfy the equation where PC is the pressure difference between phases, y the interfacial tension and r, and r2 are the major and minor radii of curvature. Depending on the pore size distribution of coarse and fine layers, the volumetric percentages of oil and gas in these layers will differ when equilibrium exists. The exact relationship can only be determined by obtaining the complete capillary pressure-desaturation curves for each of the porous materials in contact. It has been pointed out elsewhere' that the capillary pressure-desaturation curves of sedimentary porous materials can often be approximated by the relation where C is a constant and Soe is the effective saturation to oil based on a percentage of the pore volume effective to flow. In the same paper it was indicated that, as a first approximation, the values of oil relative permeability are given by and the values of gas relative permeability by For this analysis Eqs. 2, 3, and 4 were assumed to apply to each of two components of a hypothetical porous rock in capillary equilibrium. It was also assumed that each of the components had a residual wetting phase saturation of 20 per cent so that 80 per cent of the total pore volume was effective to flow. The permeability of the coarse stratum was taken as 100, and its displacement pressure was such that C in Eq. 2 had the numerical value of 1. The corresponding values for the fine stratum were 10 for the permeability and 10 for C. Units are not specified because they do not enter into the final results. The choice of the permeabilities and displacement pressure ratios was made to expedite the calculations. Any reasonable rock properties could have been chosen without changing the results qualitatively. Several arrangements of the two components were studied. Table 1 summarizes the resultant permeabilities obtained for four types of arrangement. RELATIVE PERMEABILITY CALCULATIONS The first step in the computation of relative permeability for the composite cores was the plotting of the capillary pressure-desaturation curves and the relative permeability curves for the individual components according to Eqs. 2, 3, and 4. At arbitrary

Jan 1, 1957

By R. M. Genevray, M. B. Bever, E. J. Suoninen

THE martensitic transformation in the ß-phase of the Cu-Zn system has been the subject of several investigations. The transformation is known to be reversible and to be affected by stress. Its temperature range has been determined as a function of composition. In the investigation reported here, the effect of tensile stresses on the transformation was investigated quantitatively. Some information was also obtained on the thermoelastic behavior of the martensite formed in the first stages of the transformation. Most of the experiments were done with alloy E of an earlier investigation;' this alloy analyzed 60.49 pct Cu and 39.51 pet Zn by weight. The methods of shaping and heat treatment were also essentially the same as those previously used. The stress was applied to the specimen immersed in a cooling liquid. The transformation was followed by measuring the electrical resistance with a Kelvin bridge and the elongation with a cathetometer. Fig. 1 shows the M, temperature as a function of stress. Resistance and strain measurements gave essentially identical values. The results suggest a roughly linear relation between M. and in the range investigated, up to 12 kg s mm". At higher stresses, plastic deformation begins to interfere seriously with this relationship. The increase of M, with stress is consistent with published work on the effect of stress on the martensitic transformation. The slope of the curve, 4°C per kg mm ", is of the same order of magnitude as the corresponding value calculated for steel.' Fig. 1 also shows the difference, AM, between the temperature of 50 pct transformation on cooling, as measured by changes in length, and that of 50 pct reverse transformation on heating. This difference, which may be considered a measure of the hysteresis, increases with stress; the decrease at highest stresses is probably associated with plastic deformation. Preliminary work using only resistance measurements was done with an alloy containing 60.15 pct Cu and 39.79 pct Zn by weight. The results indicated higher values o-F M, in agreement with the known variation of M. with composition.' The effect of stress on M, (2°C per kg mm-') was of the same order of magnitude as that shown in Fig. 1 for composition E. An increase in hysteresis with stress was also found. The following experiment was made in order to investigate a partially transformed structure. A specimen of alloy E was cooled to — 85°C under a stress of 4.7 kg mm-'. Under these conditions, the martensitic transformation started but did not go to completion. The stress was then released and the specimen cooled to — 105°C. Fig. 2 shows the measured elongation c. The first change in the slope of the curve indicates the beginning of the transformation under stress. Removing the load at —85°C caused a decrease in length to the value corresponding to the elastic elongation of the parent phase resulting from the applied stress. Hence, the marten-site formed in the first part of the experiment apparently disappears completely and without hysteresis upon the release of the stress. The increase in length on further cooling indicates renewed formation of martensite. These conclusions are consistent with the concept of "thermoelastic" martensite," which has been confirmed by test." Acknowledgments The authors are greatly indebted to Professor M. Cohen for his advice and encouragement. They also thank F. Paxton for assistance. Thanks are due the American Brass Co. which supplied the alloys. References E. Kaminsky and G. V. Kurdjumov: Zhur. Tekhn. Fiziki SSSR (1936) 6, p. 984. A. B. Greninger and V. G. Mooradian: Trans. AIME (1938) 128, p. 337. "J. E. Reynolds, Jr. and M. B. Bever: Trans. AIME (1952) 194, p. 1065; Journal of Metals (October 1952). 'A. L. Titchener and M. B. Bever: Trai~s. AIME (1954) 200, p. 303; Journal of Metals (February 1954). " 3. A. Kulin, M. Cohen, and B. L. Averbach: Trans. AIME (1952) 194. p. 661; Journal of Metals (June 1952). "J. K. Pate1 and M. Cohen: Acta Metallurgica (1953) 1, p. 531. 'C. Crussard: Comptes Rendus (1953) 237, p. 1709. ' G. V. Kurdjumov: Zhur. Tekhn. Fiziki SSSR (1948) 18, p. 999. G. V. Kurdjumov and L. G. Khandros: Dokl Akad. Nouk. SSSR (1949) 66. p. 211.

Jan 1, 1957

By L. Mair

A STUDY by J. R. Low, Jr.1 on the effect of quench aging on the Charpy-impact specimens of semikilled 1020 steel disclosed that a decrease in cooling rate from 1275°F raised the transition temperature. This was demonstrated in his investigation by controlling the cooling rate of impact specimens to duplicate that of a heavier plate and com- paring the properties with air-cooled specimens and quenched specimens. The transition temperature of the samples that had been cooled at a rate estimated to be similar to that of an air-cooled 3/4 in. plate was approximately 45ºF higher than the quenched samples. The transition temperature of the air-cooled series was between that of the slow-cooled and the quenched series. On the basis of this work, it appeared that the cooling rate of plates after rolling in a hot mill would have a bearing on the brittle fracture of the

Jan 1, 1955

By P. Stark, B. S. Lement

ACCORDING to Hanemann1 the hardness of retained austenite in a 1.7 pct C steel is increased by subsequent tempering. He reported that the hardness of this constituent increases only slightly on tempering to 300°F, rises more rapidly above this temperature, and attains the same value as the tempered martensite present in steel at approximately 480°F. The variation of hardness of both retained austenite and martensite with tempering temperature as obtained by Hanemann1 is shown in Fig. 1. Hanemann attributes the increase in hardness of the retained austenite essentially to continued formation of fine needles of martensite on tempering. Although he did not explain the actual mechanism involved, work hardening of retained austenite by the plastic deformation associated with such martensite formation appeared to be a possibility. Recently Lement, Averbach, and Cohen2 reported the existence of subgrains, about l000Å in diameter, in retained austenite and suggested that they may be delineated by a subboundary carbide. This raised the possibility that carbide precipitation might be responsible for the hardening reported by Hanemann. In view of these conflicting possibilities, an investigation" was carried out on a high purity, Fe-C alloy containing 1.7 pct C in order to clarify the nature of the phenomenon involved in the reported hardening of retained austenite. This investigation involved correlation of changes in microhardness and microstructure of the austenite as a result of tempering. Microhardness tests were made with a Tukon tester using a 25 gram load and a Vickers diamond. It was found that two groups of austenite hardness values were obtained after tempering at a series of temperatures up to 450°F, which was the highest temperature at which retained austenite regions could be identified microscopically. One group represents a high hardness level which in-

Jan 1, 1955

By Paul B. Crawford, Bobby L. Landrum

Electrical model studies have been made of the effect of unsymmetrical vertical fractures on production cnpacity. The study war restricted to the case of two vertical fractures originating at the well and extending out into the reservoir for various distances and orientations. The ratio of the short fracture length to the long fracture length was varied from 0.1 to 1.0 in successive studies. The angle separating the two fractures was varied from 0 to 180 degrees. The most effective fractured patterns were obtained when the fractures were oriented 180 degrees. It was found that when the total fracture length was equal to 0.2 of the drainage radius, the increase in production capacity may vary from approximately 1.55 to 1.95. depending upon the ratio of the short fracture length to the long fracture length and the angle separating the two fractures. If the total fracture length were 0.5 of the drainape- radius. poroduction cavacitv., increases varving 1.95 to 2.65 were observed, depending on the angle between the fractures and the ratio of the .short fracture length to the long fracture length. The fractured well production capacity decreased aPproxin?ately I to 10 per cent as the angle separating the fractures was varied from 180 to 90 degrees. It was dependent on the particular fracture. The commercial application of reservoir fracturing techniques has caused an interest to be expressed in the effect of vertical and horizontal fractures on production capacities. Some data have indicated that substantially vertical fractures may be created in some reservoirs. It is believed that the fractures are possibly unsymmetrical, that is, one fracture may extend due north 100 ft and a second fracture on the opposite side of the well may extend south or southeast only 50 ft. The total fracture length, ratios of the lengths and angles of separation may vary considerably. For this reason variations in these factors were considered. The study was restricted to the case of two vertical fractures originating at the well and extending out into the reservoir. It is not known that such will occur in a reservoir, but the results may be of value in interpreting or studying fracturing applications. PROCEDURE An electrical model was used in this study of the effect of unsymmetrical vertical fractures on production capacity, A wooden tank 30 in. in diameter and approximately 2 in deep was coated with paraffin wax. A copper band was situated at the periphery and a single wire electrode was positioned at the center of the tank. The ratio of the reservoir radius divided by the

Jan 1, 1956

By J. P. Hammond, C. J. Sparks, C. J. McHargue

IN a study of the effects of solid solution alloying on the deformation texture of titanium, an alloy containing 3.8 pct A1 was found to develop a (0001) [1010] texture instead of the texture characteristic of pure titanium.ls2 A short investigation has been made of the change in texture with aluminum content. Alloys containing 0.27, 0.47, 1.05, and 1.43 pct A1 were prepared by vacuum arc melting, using iodide titanium as the base material. Homogenized ingots were reduced 90 pct in thickness by cold rolling. Pole figures were obtained using the usual film techniques. Fig. 1 shows that the change from the tilted to the nontilted (0001) position is gradual. The dashed contour lines in Fig. 1 indicate the extent to which data was collected. The addition of 0.27 pct A1 resulted in a decrease in the transverse spread of basal poles. With increasing aluminum content, the amount of the tilt becomes less and the region of highest intensity takes the shape of a ring about the center of the pole figure. For aluminum contents greater than 1 pct, the (0001) [1010] texture is definitely present—the sharpness increasing as the amount of solute is increased.* It is interesting to note that Williams and Eppel-sheimer9 eported for commercial grade titanium a high intensity region 20" toward the rolling direction from the rolling plane normal in addition to. one 30" toward the transverse direction. Their basal pole figure was similar to Fig. 1, row inree. The observation that the texture changes gradually with solute concentration is similar to the results of similar studies on copper." At the present time these effects are not clearly understood. Acknowledgments This work was sponsored by the Aeronautical Research Laboratory (WCRRL), Wright Air Development Center, under Contract NO. AF' 33(038)-19574. The authors are grateful for the aid received and permission to publish these results.

Jan 1, 1958

By D. P. Koistinen, K. E. Beu

THE application of the integrated intensity X-ray diffraction method to the measurement of retained austenite concentrations in hardened steels has been fully described.'-' In developing this particular method, the assumption was made that the samples used would have flat surfaces which were free from voids and in which austenite was homogeneously distributed. The original calculation procedure' was modified to include the constant factor G which relates the intensity of a given diffraction line to the concentration of the component giving rise to that line. In order to include A (0) in the constant factor G, it was further assumed that the geometrical conditions relating the X-ray beam, camera cylinder, and sample surface were precisely met.V has been demonstrated that these geometrical conditions can be met experimentally without difficulty in many cases;' however, for those samples which have irregular surfaces or where it is impossible to locate the sample in the proper position with respect to the X-ray beam and film cylinder, it is desirable to evaluate experimentally what effect these nonideal conditions have on the reliability of the austenite determination. This note decribes the results of such experiments. SAE 1095 bar stock was used throughout and all samples were heat-treated as follows: 1600°F for 1 hr in a neutral atmosphere, brine quench, and 300°F temper for 30 min. To control heat-treatment conditions as closely as possible, all samples were plated with 0.002 in. copper to protect the immediate surface from the heat-treating atmosphere. After heat treating, the samples were deplated electroly-tically so that only the copper was removed. The heat-treated surfaces thus obtained were nearly as bright and polished as the original surfaces before copper plating although no mechanical work was done on the samples after heat treating.

Jan 1, 1954

By P. W. Neurath, R. E. Waite

INFORMATION is available on the effect of stress on permeability, coercive force, and hysteresis for a variety of materials' but for the technically important grain-oriented 3 pct Si-Fe' the stress effects are not adequately covered in the literature; in addition, most of the stress data do not permit estimation of what would happen to the 60 cycle watt losses of the material under stress. Direct measurement of these effects therefore has been made and appears to be of considerable theoretical as well as practical interest. Losses were measured on the central 25 cm portion of 40~3~0.035 cm samples which could be stressed in a small tensile machine built for this purpose. Under tension the following effects, illustrated in the figures by a particular example, were observed: 1—"Unstrained" material of commercially acceptable loss level shows no change or a small increase in loss at low elastic stresses. For the sample shown in the figures, this is illustrated during the initial run, where a 7 pct increase is observed (Fig. 1). 2—There is no additional change in loss when the sample is further strained up to the elastic limit (Fig. 1). Within that limit the loss-stress curve is reversible. This is not illustrated in Fig. 1. 3—Even in the plastic range only small increases in loss are observed as long as the sample is kept under stress (Fig. 2, point N); often no increase can be observed (Fig. 1). 4—A large increase in watt losses is observed after removal of the stress subsequent to plastic deformation (Figs. 1 and 2). 5—All, or nearly all, of the foregoing increase in loss due to plastic deformation can be eliminated by a reapplication of tensile stress but returns upon removal of this stress. This is illustrated in the third run (Fig. 3). This run remains within the elastic limit and shows the reversibility of the stress-loss curve under this condition. These findings have been confirmed on large Samples (approximately 200~10~0.035 cm and larger) except that under 1 no initial increase in loss was ever observed on these. The accuracy of the watt loss measurements is of the order of 1 pct. Neither bending of the sample in the tensioning device nor a redistribution of flux along the width of the sample accounts for these results. They therefore must be explained on the basis of either a change in the domain configuration due to stress or a change in the domain wall mobility. Acknowledgment The assistance of Mr. Robert L. TenBroeck with the instrumentation is gratefully acknowledged.

Jan 1, 1956

By W. R. Hibbard

IN ref. 1, on the basis of tension and torsion experiments with single crystals of a brass, the values given in Table I were obtained for elastic coefficients. It was pointed out by Zener that the values for the 71.2 pct Cu were not satisfactory, since they resulted in a negative value for the bulk modulus, namely —4.17X1012 sq cm per dyne. In reevaluating the data of this investigation, it was determined that the reason for this negative bulk modulus lay in the extrapolation of the reciprocal Young's modulus to the zero value of the orientation function shown in Fig. 5 of ref. 1. This extrapolation was reevaluated by the method of least squares (50 pct probability) and a new value of the coefficient S,, was determined as 18.9220.42X 10M sq cm per dyne in lieu of the value originally reported of 18.82x10 sq cm per dyne. This new value of S, leads to a new value of S, of — 9.43 X101" sq cm per dyne. The values of the other coefficients are unchanged. results. This value compares with the value of 12.3X10 sq cm per dyne reported by Schmid and Boas? The new results are summarized in Table 11. These results follow similar qualitative trends, previously published,' relative to the effect of zinc content. However, the derived values of C11, C,, C,, and K are so sensitive to the uncertainty in the first decimal place of the S, values that they are probably not useful. This uncertainty must also apply to the previous results of Masima and Sachs," who obtained S, values from a similar extrapolation drawn wholly above their data points. Acknowledgments Advice and assistance on the statistical methods were obtained from R. W. Guard. The calculations were carried out by Miss E. Saacke.

Jan 1, 1958

By Wm. C. Phelps, R. E. Grace

b R.c c)nduciivity of slags and glasses is of particular interest when viewed as an activated ionic flow process. The ionic conductivity, for example, has been found to obey the Easch-

Jan 1, 1958

By E. A. Kaarsberg

The electron photomicrographs shown in this paper are of samples of the A.P.I. Reference Clay Minerals Montmorillonite No. 27 and Kaolinite No. 4 and of a shale from Spraberry Oilfield in Texas. These samples are part of a larger collection whose physical properties, composition and internal structure were previously studied by ultra-sonic and X-ray methods. ' Elastic anistropy was indicated in the montmorillonite and shale samples by sound velocity measurements parallel and perpendicular to the bedding. In the montmorillonite the velocities parallel and perpendicular to the bedding were 2.12 and 1.92 km/sec respectively, a difference of about 11%; in the shale the same velocity measurements were 4.18 and 3.75 km/sec, a difference again of about 11%. Within the bedding plane the sound velocity was also found to be independent of direction. The electron photomicrographs of the montmorillonite sample, which are shown in Fig. l(a) and (b), clearly indicate the preferred orientation of the mica-like clay particles parallel to the bedding. In Fig. 2(a) and (b) similar orientation in the shale sample can be seen although not quite as clearly. This is probably because, in addition to the clays illite and chlorite, the shale sample contains grains of quartz, calcite and dolomite which do not as readily show an alignment pattern as the clays do. This suggests that the elastic anisotropy of these two samples is derived from a preferred alignment of component mineral grains which are themselves elastically anisotropic. The bulk elastic properties of these two samples therefore appear to be much like those of mica. Velocity measurements were made on the kaolinite sample in three directions at right angles because no bedding could be discerned in it. The values ob-

Jan 1, 1967

By H. R. Skewes

IN the course of small seale experiments on the electrodeposition, of copper, extremely brittle deposits were obtained from electrolytes prepared by leaching calcined chalcopyrite with aqueous sulfuric acid. It was not possible to strip the metal without fracture after it had been deposited from such electrolytes onto smooth oiled copper sheets, the usual plant procedure for the production of so-called starting sheets. The embrittlement did not occur with electrolytes made either by leaching the calcine with water or by treating an acid leachate to remove iron as hydrous ferric oxide. The electrolytes which produced brittle deposits were found to contain molybdenum in concentrations of 10 to 25 mg per 1, whereas those which produced ductile deposits contained <0.5 mg per 1. Synthetic electrolytes having the composition: Cu", 40 gpl; Fe", 0.5 to 1.0 gpl; Fe3+, 0.5 to 1.0 gpl; and H,SO,, 10 to 50 gpl, were prepared from reagent-grade salts; by adding ammonium molybdate, the embrittling effect of the higher concentrations of molybdenum (VI) was strikingly confirmed. The brittle deposits contained up to 0.03 pct Mo. The importance of oxidation state of the molybdenum was demonstrated when the electrolyte containing it was circulated continuously through a bed of fine copper metal during deposi- tion. Under these reducing conditions, the cathode deposit was ductile and molybdenum could not be detected. Molybdenum has not been recognized as one of the metals causing embrittlement of copper nor is it listed among the usual impurities. In most elec-trowinning plants there is a purification step which removes iron from the leachate before electrolysis. In these cases molybdenum would be coprecipitated with hydrous ferric oxide. In the case of the large leaching plant at Chuquicamata, Chile, where the leachate is reported to contain considerable molybdenum, it would be reduced to a lower oxidation state during removal of chlorine by means of cement copper. This may explain why the embrittling effect of molybdenum has not been reported previously. Moreover, in copper refining plants where soluble anodes have been prepared by a high temperature smelting process, it is most unlikely that any molybdenum would remain with the copper, whereas the calcines in this work were prepared in a fluidized bed reactor at a temperature not exceeding 720°C. Acknowledgment This work is part of an investigation on the roasting and extraction of copper from sulfide ores sponsored jointly by Mt. Morgan Ltd., Queensland, and the Mt. Lyell Mining and Railway Co. Ltd., Tasmania.

Jan 1, 1958

By B. H. Bergstrom

The Gaudin-Meloy1 size distribution equation requires the evaluation of two constants, xo and r. The common log-log representation of a screen analysis as cumulative weight percentage passing versus size, readily yields the required constants as illustrated for curves A and B in Fig. 1. The 100% passing (y = 1.0) size is simply x,. If the straight portion of the curve be extrapolated to the 100% passing (y=1.0) abscissa, the intercept is the Schuhmann size modulus, k. The ratio of xo to the size modulus, k, is r as shown: The Gaudin-Meloy equation requires that the slope, a, of the size distribution approach unity in the finer sizes, as is the case for curves A and B. Often distributions are better characterized by slopes other than unity. The schuhmann2 equation is able to represent these distributions if the upper portions of the curves are ignored. Empirically, the Gaudin-Meloy equation can be modified to yield slopes other than unity by writing it as Here, a is the familiar Schuhmann slope, and xo is the 100% passing size. The value of r is still given by Eq. 2. Eq. 4 with its easily determined constants is more versatile than either the Schuhmann or the Gaudin-Meloy equations, since a wide variety of size distributions can be closely described. It is easier to fit than the Rosin-Rammler3 equation, and has the advantage that the top size, x,, is finite. REFERENCES A. M. Gaudin and T. P. Meloy: Model and a Comminution Distribution Equation for Single Fracture, AIME Transactions, 1962, Vol. 223, pp. 40 to 43. R. Schumann, Jr.: principles of comminution, I., size Distribution and Surface Calculations, AIME Technical Publication No. 1187, 1940. P. Rosin and E. Rammler: The Laws coverning the Fineness of Powdered Coal, J. Inst. Fuel, 1933. Vol. 7, p. 29.

Jan 1, 1967

By Y. Oka, H. Majima

When a rock particle is subjected to an external force, internal stresses and deformations are experienced by the particle. Assuming that the breakage of a rock is mainly due to tensile stresses generated by the action of a pair of concentrated loads, F, directed in compression towards the loading points at the shortest distance, as illustrated in Fig. 1, the stress and deformation experienced by a particle of irregular shape can be determined with the aid of the theory of elasticity. According to the photoelastic experiments made by one of the present authors,&apos; the stress states in a rock particle of irregular shape are quite similar to that of an inscribed sphere of diameter x, as indicated in Fig. 1, but accompanied by the faint stresses, in the vicinity of the particle outside of the inscribed sphere. Therefore, the stress components in an irregularly shaped particle can be estimated by referring to stress components in a sphere. Applying polar spherical coordinates to an elastic spherical body, as shown in Fig. 1, the stress components are all expressed as multiples of F /x2 and more or less influenced by the loading angle and Poisson&apos;s number m. In general, the energy w, acting on a rock particle is defined as the product of loading compression F

Jan 1, 1970

By L. M. Clarebrough, A. K. Head, G. W. West, M. E. Hargreaves

RECENTLY Welber and Webelerl reported that during the annealing of fatigued copper no energy was released, but that energy was absorbed in the range of temperature 250" to 400°C. In view of the fact that strain hardening occurs during the fatigue of copper, this result was surprising and it seemed desirable to check the measurements. This note describes measurements made by a different technique which indicates that energy is stored during the fatigue of copper. Two test bars with gage length ¾ in. in diameter and 6 in. long were machined from OFHC copper and annealed in vacuum for 1 hr at 600°C. These were fatigued to fracture in an Avery pulsator at a frequency of 2000 cycles per min. The first specimen fractured after 0.25x10 cycles under a push-pull

Jan 1, 1956

By Forrest F. Craig

Both early and more recent1 laboratory measurements of gas-oil relative permeabilities were made by subjecting oil-saturated cores to external gas drives. In these runs it was generally assumed that the flowing gas/oil ratio was constant at every point throughout the core. Using this assumption, the resulting equation for the gas-oil relative permeability ratio becomes A where Q is the flow rate at the producing end of the system. This relative permeability was then specified for the average saturation of the core. The assumption and resulting equation fail to recognize that in an external drive, the oil flow rate varies from zero at the injection end to a maximum at the producing end. The assumption also fails to consider the existence of a saturation gradient during an external drive. This saturation gradient is due solely to non-steady-state flow. Some authors2,3,4 have used this erroneous method of calculating relative permeability ratios from external drives. Other authors5,6,7 have used relative permeability ratios in calculating expected field performance by external drive, using Equation (1) and making the incorrect assumption that the saturation is uniform within the driven portion of the reservoir. To find the correct relation between the relative permeability. ratio at the average saturation and the produced gas/oil ratio, the insertion in Equation (1) of a factor, F, to adjust it for oil velocity gradient has been considered. The equation then takes the form Assuming that the oil velocity is directly proportional to the distance from the injection point leads to the conclusion that, for a linear system, the average oil velocity is one-half that at the producing end. Thus, the average gas/oil ratio is twice the produced gas/oil ratio, and Equation (2) is transformed into However, here again the existence of a saturation gradient is not recognized. The produced gas/oil ratio is related to the relative permeability ratio at the saturation of the outlet end of the system. It can be shown quite readily that, if the true log k,/k, saturation relationship is linear over small ranges of saturation, then d log kjko Where aaxv and u2 are the average and outlet saturations, respectively. Examination of Equation (3) shows that the factor F cannot have a single value over the whole saturation range. Another method of dealing with external drives is that of Buckley and Leverett.&apos; Their method, which assumes no gas expansion and outlet end effects, considers fully the saturation gradient present due solely to the displacement mechanism during an external drive, and applies Equation (1) to each point along this gradient. The calculation of production history using relative permeability ratio is straightforward. However, the calculation of relative permeability ratios from production history by this method is impractical. In a recent paper, Welge,1 using an approach similar to that of Buckley and Leverett, presents a simplified method for determining the saturation at the producing end of the system. At thi; saturation, the relative permeability ratio calculated by Equation (1) applies exactly, after breakthrough of the stabilizej front. By this method, relative permeability ratios or production history may be calculated with equal ease. Use of this method as a laboratory tool in determination of relative permeability ratios by external drive requires the knowledge of the total injected fluid volume; in addition to the average saturation and producing ratio. As a preliminary test of this method of calculation, laborntory external gas drives were run on a 3% in. diameter, 11 in. long sample of Nellie Bly consolidated sandstone. This core sample had a permeability of 824 md and a porosity of 28.1 per cent. The driving fluid was air. Three different oils were used, having viscosities of 1.4, 9.8, and 125 cp, respectively. The runs were made under identical pressure differentials,

Jan 1, 1952

By Paul B. Crawford, Bobby L. Landrum

An electrical model study has been mode to estimate the effect of horizontal fractures in thick reservoirs on pattern conductivity. For the patterns studied, it is shown that fracturing all the wells in the pattern in-creases the conductivity considerably more than frac-turing only single wells. For certain conditions one large fracture is more effective in increasing conductivity than several fractures of half the radius. Particularly in thick reservoirs and with large fractures, the center of the pay is found to be more responsive than other locations. For the seven-spot pattern fracturing the center well is more effective in increasing pattern con-ductivity than fracturing all outer wells with fractures of the same size. INTRODUCTION Although commercial fracturing has been used for some time very little quantitative information is known about the effects of fractures on reservoir behavior. The purpose of this work was to estimate the effect of horizontal fractures in thick reservoirs on the con-ductivity of various flooding and cycling patterns.2 Circular fractures originating at the wellbore and extending symmetrically into the pay were considered. ANALYSIS Electrical models were used in making this study. Horizontal fractures were simulated by copper sheets shaped like the fractures and of a proportional size; the wellbore was simulated by a copper wire. A dilute salt solution served as a conducting medium. A con-ductivity bridge was used to measure the conductivities of the various patterns studied. EFFECT OF HORIZONTAL FRACTURES ON CONDUCTIVITY Fig. 1 shows the effect on square and staggered line drive patterns of horizontal fractures located in the center of the pay. The model is shown schematically in the insert. The upper and lower groups of lines show the effect on conductivity when all wells are fractured, and when only one well is fractured, respectively. In general, pattern conductivities are substantially improved by fracturing all wells. Fig. 2 shows the effect of multiple horizontal frac-tures on the conductivity of a line-drive pattern. At

Jan 1, 1958

By J. J. Oberly

WHILE examining lineage boundaries in germanium single crystals as described by Vogel et al.,&apos; patterns of conical etch pits were noted which reveal interesting properties of dislocations. The crystals were grown by pulling in the <100> direction, and (100) surfaces perpendicular to the growth direction were etched 4 min in a mixture of 6 cu cm HF, 10 cu cm HNO3, 6 cu cm acetic acid, 3 drops Br,. Lineage boundaries extended in any direction, but favored the <110> direction. The patterns which were noted first were composed of two or three etch pits arranged symmetrically about the lineage boundary, so that the doublets were spaced equally on either side of the boundary line and the triplets had their center pit on the boundary line. The triplets occurred less often than did the doublets; occasionally combinations were found. Two such combination patterns are shown in Figs. 1 and 2. The orientation of the patterns was determined approximately by subsequently etching the same surfaces in a mixture of 5 cu cm HF, 5 cu cm super-0x01, and 20 cu cm distilled water which produced nearby random square etch pits having edges in the <110> direction. In all cases, the axis of symmetry was <110>, being parallel to the etch-pit edges. Several successive lapping and etching operations showed that the patterns tended to continue through the crystal in the <100> direction, but eventually changed or vanished. It can be seen in both photographs that the pits in the patterns are much further apart than the pits in the adjacent section of the lineage boundary. In Fig. 1, the 10 pits of the pat-

Jan 1, 1955

By F. D. Rosi

IT was recently shown1 that slip bands can be observed in bent germanium crystals after deformation, by means of etch pits along the slip traces. It is the purpose of this note to show how chemical etch pits can also be usedto examine the appear- ance of slip bands in silicon, as well as other deformation markings. A 30 ohm-cm, p-type single crystal bar, 2x1/4x1/8 in., was plastically bent at approximately 1200°C in a vacuum of 1x10-5 mm Hg. The axis of the crystal was coincident with a <III> direction, and the surface normal to the axis of bending was within 4" of a (110) plane. Prior to the deformation, the (110) surface was mechanically polished with

Jan 1, 1958