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By N. L. Svensson

The model developed by Tuylor for the calculation of Polycrystalline yield strength has been applied to the case of an aggregate hawing a preferred orientation. In general this procedure requires the specification of texture by means of weighting factors applied to specific orientations. The problem to which the model has been applied is that of the yield-strength aniso-tropy of cold-rolled aluminum whose rolling texture was described as a combination of (110)[112] and (311) [112] In this case yield-strength anisotropy is defined by the rutio of yield strength measured at an angle 8 to the rolling direction to that measured along the rolling direction. The method of calculation of yield-strength ratio as a function of ? is described and the results show good agreement with experimental values. The orthotropic yield criterion suggested by Hill has been applied to the results and the strain ratio R also calculated as a function of ?. This has been compared with calculations using the method suggested by Elias, Heyer, and Smith which does not exhibit suck good agreement with observation. one deficietlcy of the method presented is that the strain ratios used by are those applying to iso-Irobic materials. The method should therefore be reg-clrded only as a first abbroximation to the prediction of anisotropy. THE problem of calculating the stress-strain characteristics of polycrystalline aggregates from the properties of single crystals has attracted attention for a number of years. The most important contributions to this study have been those due to: Sachs,' Cox and sopwith,2 Taylor,3 Kochendorfer,4 Batdorf and Budiansky,5 Calnan and Clews,6 Bishop and Hill,7,8 Kocks,9 Budiansky, Hashin, and sanders, 10 Kroner,11 Cyzak, Bow, and payne, 12 Budiansky and Wu,13 and Lin.14 While the earlier work has been largely superseded, recent developments tend to support Taylor's solution" within the restriction imposed by his assumptions. The essential features of Taylor's approach were: 1) the material is rigid-plastic; 2) each grain experiences the same strain components as the aggregate as a whole (the problem was that of uniaxial deformation with principal strain components in the ratio 3) all regions of each grain deform uniformly; 4) work hardening occurs equally on all slip systems. While Bishop and Hill7 have generally validated this approach, there has been some criticism offered. Kocks? as pointed out that since multiple slip must occur the single-crystal data must be determined from orientations arranged such that polyslip takes place. Boas and Hargreaves,15 and others, have shown experimentally that the strain distribution within grains is not uniform, the strains in the vicinity of grain boundaries being less than those in the center of the grains. Both of these criticisms can be largely offset by the suitable choice of single-crystal critical shear stress. However, for the problem analyzed below, the critical shear stress is not directly used and, consequently, these criticisms lose their importance. The more recent contributions have attempted to obtain a more complete analysis by considering an elas-toplastic material and considering interactions between grains of differing orientations. Lin14 has considered the early stages of yielding for a polycrystalline aggregate having specific regions of defined slip plane orientations. On the other hand, Budiansky and Wu13 have allowed for these interactions for randomly disposed grain orientations and have calculated the polycrystalline stress-strain curves for crystals exhibiting either elastic-ideally plastic or kinematic hardening characteristics. This work has shown that yielding commences when the macroscopic stress is 2.2 times the critical shear stress for slip in a single crystal (7,). The yield stress-strain curve then rises becoming asymptotic to a value of 3.072 7,. This is close to the value obtained by Bishop and Hill (3.06) in their confirmation of Taylor's method. This, of course, is to be expected since, at large strain values, the elastic strains are negligible and the rigid-plastic model is satisfactory. The results of Budiansky and Wu indicate that the result obtained by Taylor is 7.7 pct high at a plastic strain which is two times the elastic strain at the initiation of yield. By defining the anisotropy in terms of relative values, the ratio of yield strength at orientation ?, to that measured in the rolling direction, the effect of the discrepancy in Taylor's solution is considered to be of lesser consequence. Therefore, it is anticipated that an analysis based on Taylor's solution, which can be quite straightforward, should provide a reasonable estimation of the anisotropy of materials having a preferred orientation texture. OUTLINE OF TAYLOR'S METHOD In fee metals there are four possible slip planes (the octahedral planes) and in each there are three possible slip directions (the edges of the octahedron), that is a total of twelve possible slip systems. von Mises16 has shown that at least five independent slip systems must become operative in each grain of the polycrystalline aggregate in order to preserve continuity of strain. With this geometrical requirement as basis and the assumptions previously listed, Taylor determined the operative slip systems for a number of orientations of the tensile stress axis specified in the unit stereographic triangle. For the ith slip system, the critical shear stress

Jan 1, 1967

By N. C. J. Ros

The paper deals with pressure gradients occurring in flowing and gas-lift wells, a knowledge of which can be applied to the determination of optimum flow-string dimensions and to the design of gas-lift installations. The study is based on a pressure-balance equation for the pressure gradient. It appears that a pressure-gradient correlation of general validity must essentially consist of two parts-—one part being a correlation for liquid hold-up and the other part being one for wall friction. Dimensional analysis indicates that both liquid hold-up and wall friction are related to nine dimensionless groups. It is shown that in the field of interest only four groups are really important. On the basis of these four groups a restricted experimental program could be selected that nevertheless covered practically all conditions encountered in oil wells. This experimental program has been carried out in a laboratory installation. Three essentially different flow regimes were found. The pressure gradients in these regious are presented in the form of a set of correlations. Comparison of these correlations with a few available oilfield data showed excellent agreement. INTRODUCTION Prediction of the pressure drop in the flow string of a well is a widely known problem in oilfield practice. Accurate data on the pressure gradient of a simultaneous flow of gas and liquid in a vertical pipe are especially useful for the determination of optimum flow-string dimensions. It is well known that with moderate gas and liquid flows such a vertical string acts as a "negative restriction". The pressure drop decreases (1) when the throughput through a given pipe increases, and (2) when at a given throughput the cross-sectional area is decreased. The reason is that, with increasing velocities, the flow becomes more agitated so that the gas slips relatively more slowly through the liquid. With the resulting increase in gas content in the string, the static head decreases. When the area becomes very small, however, the high velocities entail great wall friction, which causes an increase in pressure drop. For a given flow, therefore, minimal pressure drop is obtained by using a certain cross section. This means that, in principle, each well can be provided with an optimum flow string for minimum pressure drop and, hence, maximum possible production rate. The procedure for the selection of the optimum string has been discussed by Gilbert.' A necessary tool in the procedure, however, is accurate knowledge of the pressure gradient to be expected for various values of the governing variables. Another application of pressure-gradient data lies in the field of gas-lift practice: they provide a means of determining the optimum gas-injection rate, optimum injection pressure and optimum injection depth. Much work has already been done in the study of the pressure gradient of vertical gas-liquid flow. Poett-mann and Carpenter2 presented a pressure-gradient correlation based on measurements in wells. This correlation has been found to provide accurate predictions in high-pressure wells and in high-production wells for flow through both tubing and annuli.2-5 However, when their method is checked on low pressure-low production wells or on wells with viscous crudes, serious discrepancies are found. As we shall see in the next section, this is due to the fact that their correlation factor, representing all irreversible energy losses, is given as a function of only one correlation group. Some important variables, such as gas-liquid ratio and liquid viscosity, are not incorporated in this group so that their specific effects are not accounted for. To study also the mechanism of vertical gas-liquid flow outside the ranges covered by the Poettmann-Carpenter publication and extensions, a laboratory investigation has been carried out. This study is founded on a pressure-gradient equation that is based on a pressure balance. To reduce the number of test runs required, a dimensional analysis has been carried out, followed by a selection of relevant dimensionless groups. These groups guided a subsequent experimental study, and with their aid the experimental program could be minimized while still covering the majority of the situations encountered in oilfield practice. In this paper the choice of a formula for the pressure gradient is discussed first. This is followed by a brief description of the experimental setup. Subsequently, the dimensional analysis is discussed and the relevant dimensionless groups are selected, resulting in the experimental program required. The general relationships of pressure gradient and liquid hold-up are then described; various flow patterns and a certain flow instability (so-called "heading") are discussed and a set of correlations is presented which shows a good agreement with the measurements and a few available field

By P. K. Foh, C. G. Dunn

IN an investigation of the rolling and recrystalliz-ing textures obtained from single crystals of Si-Fe, Koh and Dunn included specimens in the (111) [lie] orientation.1,2 This note reports some additional results obtained on samples of this orientation set aside for further study. After, primary recrystallization longer annealing treatments at 980°C produced only normal grain growth; after 16 hr at 980°C the largest grains were approximately 0.010 to 0.020 in. in diam. The orientations of 13 of the larger grains were found by the transmission Laue method. The unidentified (110) poles of these grains have been added to the (110) pole figure obtained after primary recrystallization2 to give the plot shown in Fig. 1. When grain identifications are also included, such a plot reveals the following. Two orientations agree almost exactly with the peak pole ,density positions of component B of the primary recrystallization texture, one with component A, and one with component C. All the remaining orientations except one have (110) poles near weaker and unidentified pole density peaks. In fact, the correlation is good enough for an identification of eight components. Previously only components A, B, and C were certain. The results are perhaps even more informative when combined with the (110) pole figure of the cold-rolled crystal to give the plot shown in Fig. 2. Each of the 13 orientations is identified and uniquely determined by selecting two (110) poles per crystal, which are 90" apart. The plot shows that every grain of the 13, except grain 6, has a (110) pole in common with a (110) pole concentration of the cold-rolled crystal. The orientation relationship with the cold-rolled orientation may be expressed as approximately 25" rotations about <110> axes, which is in agreement with the results found for coarse-grained primary recrystallization.2 If primary recrystallization had occurred according to the <110> rule without preference, there would have been 12 components. Actually our limited data reveals nine of the possible 12. The integrated pole figure also shows preferences among them, since three are considerably stronger than the rest. In addition to the above direct information, the following conclusions may be drawn. Primary recrystallization of the (111) [110] cold-rolled crystal produces a limited number of weak and strong components with the larger than average primary grains from these components related to the cold-rolled crystal by 25° <110> relationships. In arriving at this conclusion of limited orientations the (111) [110] orientation has to be ruled out as a source of large primaries for two reasons. First, growth in this orientation was not observed, despite an expected high grain boundary mobility (due to its 25" <110> relationship with every component of the texture). Secondly, primary recrystallization usually produces grains in new orientations, particularly in a single orientation matrix; but (111) [110] is not new. Other orientations besides those of the <110> type may be considered unlikely, of course, because the <110> relationship holds quite well for primary recrystallization2 and should produce the large primaries. There are exceptions to this rule, but probably not enough to provide large

Jan 1, 1958

By H. H. Podgurski

Above 200°C the observed increase in the apparent solubility of hydrogen in low alloy steels caused by cold work is attributed to the formation of methane in microvoids. This methane can be isolated quantitatively, and furthermore, the amount of methane is in accord with the equilibrium 2H2 + Fe3C = CH4 + 3Fe. The micro-void volume, calculated from the methane content via the equation of state for methane, corresponds approximately to that calculated from the measured density decrement produced by cold work. THIS investigation was undertaken with the intent of employing the hydrogen atom as a probe in detecting specific crystallographic defects in cold worked steel. It was expected that different degrees of chemical interaction of hydrogen with the various types of defects could be utilized. The original aim was to assign concentration values to defect sites. The success of this approach requires that experimental conditions can be attained under which lattice-dissolved hydrogen equilibrates primarily with only one type of defect. A set of experiments which appeared potentially to embody these requirements was described several years ago by Keeler and Davis.1 They noted in the case of a mild steel (low carbon) that the solubility of hydrogen between 200° and 400°C near atmospheric pressure was notably increased by cold working the steel. The observed solubility was an order of magnitude greater than the normal lattice solubility. * Although the Keeler and Davis studies suggested that the density decrement effected by their cold-working operation might be related to the anomalous solubility of hydrogen, they offered no explanation of the relationship between the two effects. Prior to this work, Darken and smith3 had noted a similar effect of cold work upon enhanced sorption of hydrogen by steel when charged by acid pickling. In the light of present-day concepts, interaction between lattice-dissolved hydrogen and dislocations was strongly suggested. More recently, Hill and Johnson,4 working with a

Jan 1, 1962

By J. G. Eenink, R. A. Cunningham

One phase has been completed of a laboratory invesrigation of formations with relatively high permeability under conditions of overburden, formation and mud coltrmn pressures. The following statements are based on these tests within the limitations described. Drilling rate decreased when mud column pressure was greater than formation pressure. The decrease was primarily due to a layer of cuttings and mud particles held to the hole bottom by the difference in pressure. Much of the force of the bit was wasted in this layer and was unavailable for penetrating virgin formation. Adequate jet velocities helped clean the filter cake and chips away and resulted in increased drilling rate — the higher the jet velocity the faster the drilling rate. Drilling rate increased slightly when formation pres sure was greater than mud column pressure. The i11(reuse resulted from cleaning the hole bottom as formarion fluid flowed into the borehole. Overburden pressure had practically no effect on drilling rate. INTRODUCTION Some formations are difficult to drill in a wellbore hole but are easily drilled under atmospheric conditions in the laboratory. The reasons for this apparent difference in drillability are not fully understood. An increase in formation compressive strength causes a reduction in drilling rate. It has been shown by Griggs1, Handin2 and others3-1 that formation strength increases under conditions of high hydrostatic stress. Based on these results, a decrease in drilling rate would be expected with increased drilling fluid pressure in formations of low permeability having low formation fluid pressure. Such a decrease has been shown by Murray. et a1 and Eckel" This effect holds even when using nitrogcn under high pressure as the drilling fluid (unrc-ported tests from the Hughes laboratory). Poor bottom-hole cleaning may account for decreased drilling rates. Even though it may not be easily explained, evidence of poor cleaning can easily bc seen. One evidence is the increased drilling rate that accompanies decreased balling obtained with addition of oil in some muds used to drill shales.4,7,3 Many variables can affect drilling rates. These may include mud composition and properties, formation properties, bit types and operating conditions, formation fluid, mud column and overburden pressures. These must be studied a few at a time under carefully controlled conditions which simulate actual conditions as closely as possible. The purpose of this work was to investigate effect of overburden, formation and mud column pressures on drilling rates in permeable formations. These tests were suggested by conditions shown in Fig. I. Represented is a borehole drilled into a formation with relatively high permeability. In the vicinity of the hole bottom there exists a mud column pressure, P a formation fluid pressure, P and an overburden pressure, The mud has low water-loss properties and sufficient weight to more than overbalance the formation pressure. Laboratory investigation included a range with formation pressure greater than mud column pressure. NOMENCLATURE Pm mud column pressure = pressure of drilling fluid or mud at hole bottom Pf, formation pressure = pressure of fluid in intersticies of formation

By R. E. Jenkins

The Bell Canyon member of the Delaware Mountain group has yielded quite a large number of fields in which completion and production problems have been numerous and complex. Reserves are difficult to estimate due to the problem of evaluating the formation water saturation, and the feasibility of water flooding is questionable. Extensive laboratory investigations were undertaken to determine if normally measured rock and contained fluids properties could account for many of the peculiar characteristics of the formation. A statistical study or routine core-analysis data was made. Capillary-pressure data were obtained by several techniques. Fresh-water and brine permeabilities were measured and relative permeabilities of oil, water and gas were determined. Some qualitative wetta-bility tests were performed, and the petrography of a few thin sections was studied. The results of this work are presented, with typical or average data shown for each test. The permeability -to- water characteristic of the Delaware was found to be abnormally poor. Correlations of laboratory and field performance data indicate high water saturations in much of the formation. Pore geometry is highly uniform. INTRODUCTION The Delaware formation has undergone extensive exploration and development in the past few years. This has resulted in the discovery of quite a number of fields—producing oil for the most part. A large percentage of the productive wells have produced water along with the oil. Often there is no apparent pattern for the percentage of water cut experienced. In several instances vertical displacments of over 200 ft have been established between communicating wells in which the formation appears to be about the same, yet the structurally high wells yield a much higher water cut than the lower wells. Drill-stem tests have contributed little to well completion because they nearly always show small amounts of water (usually mud filtrate), regardless of what is produced later in the life of the well. Other formation evaluation methods have had no more than limited success in indicating ultimate productivity. Consequently, predicting the type of fluid productivity of Delaware wells has been very difficult. Hydraulic fracture treatments are almost universal, and unstimulated Delaware wells have not exhibited the productivity that normally would be expected of wells with the specific permeabilities encountered. Some operators have indicated that their Delaware wells need frequent re-stimulation. The uncertainty of reservoir water saturations reported for the Delaware has made difficult the task of evaluating oil in place in the reservoirs and estimating primary recovery. The unreliability of this data also will cause feasibility studies for water flooding or other secondary-recovery methods to be problematical. Several opinions have been voiced to explain why the Delaware performs as it does. Chronically poor well completions, dynamic water and tilted water tables, capillary inequi-libria, peculiar relative permeability characteristics and oil-wetness are among the suggested reasons. The dual purpose of this paper is (1) to describe the principal rock characteristics, and (2) to present an evaluation of those characteristics so that approaches can be made to explain the irrational behavior of wells completed in the Delaware. STUDY OF ROUTINE CORE-ANALYSIS DATA A statistical study of routine core-analysis data has been made to relate fundamental rock properties and characteristics of the formation to well productivity. For the most part, well production data were obtained from operators and augmented by production data from commercial information services. Only data which were available in detail and which were considered to be accurate were used. The core data used were obtained on samples from the perforated or open interval only and, thus, may not always represent the total productive interval in a well. All the core-analysis data included in this study are from the Bell Canyon member of the Delaware and were obtained by "plug-type" techniques, as opposed to whole-core or full-diameter techniques. Porosity values were obtained by the summation-of-fluids technique, permeability values are air permeabilities corrected for slippage to an equivalent liquid permeability, and the oil and water contents were determined by means of a high-temperature downdraft retort. All the procedures used are discussed at length in Ref. I. The core-analysis data were sorted

By Louis P. Gignac

INTRODUCTION The selection of trackless equipment for underground mining can be a complex engineering problem due to the wide range of equipment sizes and operating modes. Computer simulation is particularly useful in estimating the performance of different systems for specific rock- handling problems. A hybrid simulator, incorporating some of the features of deterministic and stochastic simulation, was developed in order to handle not only queuing delays at loading and dumping points, but a1 so traffic interferences. Any number of vehicles can operate in any one of four basic modes (LHD, LAF, FEL, HFC) in parallel or in series. If the units use common roadways, loading and dumping points, certain operating delays will occur and be registered by the simulator and thus give a better evaluation of the marginal productivity of each additional unit. Based on a typical layout of drawpoints and ore passes for the sub-level stoping method, productivity and operating costs of different rock-handling systems will be examined. 1 . COMPUTER SIMULATION Numerous applications of computer simulations are reported in the 1iterature for various mining problems. Depending on the complexity of the system to be studied, simulation models were conceived with different degrees of sophistication. Three different types of simulators are generally recognized: stochastic, deterministic, and hybrid. Stochastic or Monte Carl o simulation randomly generates items, transactions, or events from some population defined by a frequency distribution and produces some expected future situations. Because this type of simulation is governed by the input of probability distributions, it requires a detailed knowledge of the system to be simulated; it implies expensive and time-consuming studies and reports to gene- rate this input information. A major short- coming of stochastic techniques is in new equipment evaluation, where the lack of data is unavoidable, and in new system design where the conditions are outside the range of the known historical behavior of the equipment. However, probabilistic simulation is almost essential for the study of cyclic queues and traffic problems. Deterministic simulation studies a system by generating performances on the basis of the mechanical capabilities of the vehicles and the physical limitations of the mining scheme. It is based on the engineering principle that the engine converts its energy into a rimpull at the wheels, which is in turn opposed by the rolling resistance and the grade of the ground; the machine is accelerated or decelerated until the tractive and resistive forces are in equilibrium, at which point it moves at constant speed. The information required by this technique, such as rimpull charts and equipment weight, is readily available from equipment suppliers. However, equipment performance at the mining site is also dictated by human and environmental conditions and changes with time and usage. For this reason, deterministic simulation generally overestimates the system capabilities; these must then be adjusted by efficiency factors based on observation and experience. Recently, hybrid simulation models, using both stochastic and deterministic techniques, have been built with some of the events generated stochasticly and others being deterministic. This compromising approach originated at the Pennsylvania State University. O&apos;Nei1 (1966) designed a simulator for truck-and- shovel operations that allows for transportation from multiple faces to mu1tiple destinations. Each truck performs according to its mechanical capabilities while its loading and dumping time and its load fluctuate according to specific probability distributions. A major advance in the simulation of mining systems is due to Sanford&apos;s model (1969) of underground coal mining operations. The originality of the model is in the use of an Executive System Control which sets up the initial system, de- fines the operation conditions , and instructs continuously four sub-assemblies representing shuttle cars, trains, continuous miners , and conveyors, which in turn generate a feed-back of their movement to the Executive Control System for further instructions. The model has enough flexibility to simulate simultaneous and sequential jobs that characterize any dynamic system. Sanford&apos;s early work evolved slowly to what is known today as the Under- ground Materials Handling Simulator for coal mining (Manula, 1974).

Jan 1, 1981

By C. W. Volek, G. Mandl

Steam-injection tests in the field have shoum that heat transport into the oil/water region, ahead of the steam zone, may have a significant effect on the production process. Earlier theoretical work on reservoir heating by hot-fluid injection can be used to describe the growth of the steam zone, but only if heat transport from steam zone into oil/water region is neglected. It therefore became necessary to re-examine and extend the earlier theory. It was found that this theory ceases to be consistent with the physical model of the process at a certain critical time, tc, which depends on reservoir thickness, temperature, and quality of the steam. The critical time marks an important change in the heat flow across the condensation frovt: the heat flow, which is purely conductive during 0 < t < tc&apos; becomes predominantly convective at t = tc. Accordingly, at t = tc, the equation which governs expansion of the steam zone changes. On the basis of the new equation, an approximate description for the steam-zone growth during t > tc is developed by making use of upper and lower bounds for the exact solution of the problem. The boundary conditions for heat and mass flow inside the oil/water region are established, and a simple method of detennining the saturation at the downstream side of the condensation front is presented. Experimental results show that the theory gives an accurate description of steam-zone expansion before the critical time. At times near critical time, and also after critical time, experimental data show theoretical steam-zone volumes to be high. Even so, the accuracy of the calculated steam-zone volumes is sufficient for practical purposes. THEORY OF STEAM-ZONE GROWTH INTRODUCTION The engineering evaluation of steam-drive processes is based mainly on a mathematical description of reservoir heating by hot fluid injection presented several years ago by Marx and Langenheim.l This theory can be used to determine the growth of the steam zone,2 provided the flow of heat from the steam zone into the liquid zone ahead of the condensation front (Fig. 1) is neglected. Since we must expect, however, that heat transport into and inside the liquid region (the region between condensation front and production wells) will affect both the water/oil flow and the growth of the steam zone, it is necessary to re-examine and to generalize the Marx and Langenheim theory as applied to steam drive, which will hereafter be termed the "old" theory. Observations made in several field tests emphasize the need to study the heat flow across the condensation front (CF). On several occasions it was discovered that heat was transported in the liquid zone far ahead of the advancing CF, resulting in an early breakthrough of warm water. This seems to occur most often when high injection pressures are applied, when "wet" kONINKLIJKE/SHELL EXPLORATIE EN PRODUKTIE LABORATORIUM RIJSWIJK, THE NETHERLANDS SHELL DEVELOPMENT CO. HOUSTON, TEX.

Jan 1, 1970

By Carl F. Lutz, Robert F. Mehl

A layer of brass was formed on 0 brass using a vapor-solid reaction technique. The variation in composition with distance within the phase layer and across the a -ß interface was determined by an electron probe. The diffusivities were calculated as a function of concentration; the diffusion coefficient in brass was found to change by a factor of twenty-five over a compositional range of 8 at. pct. An explanation is proposed to account for the large change in the diffusivity, based on possible stresses in the diffusion zone; it is suggested that the concentration dependence of the thermodynamic factor might decrease the activation energy with increasing zinc content. ALTHOUGH the body of data on diffusion coefficients (D- values) in terminal solid solutions and in isomorphous systems is quite large, there is but little quantitative information for intermediate alloy phases. This paper recounts measurements of D-values for the ? phase in the Cu-Zn system. The experimental method employed consists in the forming of alloy layers and determining and analyzing the concentration-distance (c-x) curve. The experimental method employed could have been applied to several systems; the Cu-Zn system was chosen because the D-values in the a and ß phases are well known, and can thus be used for purposes of comparison. The rates of growth, of intermediate alloy layers are known for a number of systems. In nearly all cases the thickness varies parabolically with time, as to be expected if the rate of growth is controlled by the D-values. In a few cases, nonparabolic behavior has been noted.1 Nonparabolic growth may be the result of a) a variation of the composition at the phase interface with time, b) interface-controlled kinetics, c) short diffusion times coupled with long vacancy lifetimes (in vacancy diffusion),2 and d) crystal reorientation during diffusion in anisotropic systems.3 In the Al-Ni system4 and in the Al-u5 system parabolic growth kinetics are slowly approached after an initial transient period. In general those phases stable at a given temperature in a system A-B appear as separate phase layers when A and B are put in contact and given a diffusion he at-treatment. In most cases the compositions at the interface of adjacent phase layers are those read from the phase diagrams for the termini of the respective phases. This discontinuity in concentration is taken as independent of time, and the growth of one phase at the expense of another is assumed to be independent of interface kinetics, i.e., the rate of interface movement is controlled by volume diffusion in the phases concerned. wagner6 has given many solutions for multi-phase diffusion processes, assuming that the chemical diffusion coefficient, D, is independent of concentration, as have others. Jost7 has pointed out that the familiar Boltzmann-Matano solution is as valid for a (c-x) curve exhibiting concentration discontinuities at phase boundaries as for the usual single-phase case. Heumann and associates8,9 have applied this solution to the multi-phase case, but lacking concentration data within the several layers were forced to assume linear concentration gradients, thus calculating only average D-values. The purposes of the present study were: 1) to determine the dependence of D on concentration, 2) to calculate the intrinsic diffusion coefficients D?Cu and D?Zn, 3) to establish the time-law for the movement of the interface, 4) to determine the concentration limits at the ß-? interface, and, 5) to attempt to clarify the mechanism of diffusion. EXPERIMENTAL PROCEDURE The ? phase is exceedingly brittle; conventional solid couples and conventional sectioning methods were thus inapplicable. Vapor-solid couples were accordingly employed. Unfortunately the equilibrium vapor pressures of Zn for the ? phase are unknown; to escape this awkwardness, samples were enclosed in a capsule containing powder of an alloy of known composition, sufficiently large in quantity as to be effectively an infinite source, maintaining the concentration of Zn at the sample surface constant with time. The sources of Zn-vapor which can be employed in reaction with Cu, or a brass, or ß brass, to form a layer of ? brass satisfactorily wide in concentration range, are alloys of the phases ?, or ? + E or ? + d (depending on the temperature).

Jan 1, 1962

By D. K. Crampton, H. I. Burghoff, J. T. Stacy

The effect of cold-working upon electrical conductivity of copper and of copper alloys appears not to be generally known in detail. Although several papers on the subject have been presented, showing variable effects according to alloy composition, it is often assumed that the 2 to 3 per cent lowering of the conductivity of ordinary tough-pitch copper by severe cold-working is typical for copper alloys. Because of this situation it was decided to carry out the present work upon several types of copper, copper-zinc alloys over the commercial range of brasses, other binary alloys, and numerous age-hardening alloys in the quenched, and quenched and aged conditions. The large number of alloys investigated made it impossible to study every one in the detail it may have merited. Inasmuch as several interesting points that could not be satisfactorily explained came to light, further work is contemplated by the authors. It is usually considered that the normal effect of cold-work on ductile metals and unordered alloys is to decrease conductivity slightly. In alloys that are in the ordered state, as annealed or otherwise heat-treated, cold--working tends to disorder the structure, resulting in very marked lowering of conductivity. The random state of atomic arrangement produces lower con- ductivity than the ordered state for the same composition.1 Previous Work Previous work in the authors&apos; laboratory has shown the electrical conductivity of several copper alloys to be decreased by greater amounts than might have been expected, in view of the general knowledge on the subject, when cold-worked. Guillet and Ballay2 investigated the influence of cold-work upon the resistivity of several metals and alloys, including copper and some copper alloys. The increase of resistivity due to severe working w is less than 4 per cent for all pure metals tested, while the increases for the alloys were generally higher. In the copper-zinc alloys, they found that the increase of resistance due to drawing increased with zinc content, and reported a 21.7 per cent increase of resistance for an alloy contaiiiing 67.9 per cent Cu as severely worked. Masima and Sachs3 studied the change in electrical resistance of three brass crystals when stretched in the tensile test, obtaining increases in resistance of the order of 3 per cent for 30 to 40 per cent reduction of area of the specimens. The resistance decreased again when the cold-worked crystals were subsequently annealed. Bardenheuer and Schmidt4 found that the conductivity of copper was decreased by 3 to 4 per cent when reduced 50 per cent by working. The conductivity of 72-28

Jan 1, 1941

By Howard Thompson, H. N. Hall, Frank Nuss

Removal of bit cuttings is an important function of drilling muds. In an effort to obtain better understanding of the factors influencing the removal of cuttings, an extensive series of laboratory tests were made in which slip velocities of various sizes and shapes of particles were measured in muds of different physical properties. Empirical equations were then derived from these experimental data. These equations show that slip velocity is dependent on cutting size and shape, mud flow constants, and flow state of the mud. Applicability of these equations for field use is demonstrated by comparing computed slip velocities with slip velocities obtained from field tests. Drilling muds have many functions, but perhaps the most important is the removal of bit cuttings from the well. In a 10,000-ft well, the weight of rock which is drilled and must be carried to the surface is over 150 tons. In addition to this, the mud must remove any sloughings from the well bore. Failure of the mud to promptly remove either bit cuttings or sloughs from the well bore causes re-drilling and excessive wear on bit teeth, slows down drilling rate thereby increasing well costs, and greatly increases possibilities for stuck drill pipe when circulation is stopped for any length of time. Drilling muds lift bit cuttings to the surface by means of the upward velocity of the drilling mud in the annulus. Because a density difference exists between the rock being drilled and the drilling mud, there is a difference between mud velocity and the rate of bit cutting travel, called the net rise velocity. To evaluate the ability of any type of drilling mud in respect to lifting cuttings, some method must be known for predicting the rate of fall of cuttings through mud traveling upwards in the annulus. Equations for the rate of fall of particles through a true fluid have been satisfactorily determined by several authors. 1-6 Since drilling muds are not true fluids, but obey the laws of Bing-ham,7 it should not be expected that equations for rate of fall of cuttings in a true fluid would be applicable to plastic fluids such as drilling muds. Several authors have attempted to compute cutting slip velocities in muds using a viscosity value obtained with a Stormer viscosimeter; however, unsatisfactory results were obtained. Pigott8 took cognizance of the phenomenon of variation of viscosity with velocity in laminar Row, and calculated laminar slip velocities using a variable viscosity with Stokes law, and turbulent slip velocities using Rittingers law. Drilling muds obey the laws of plastic flow which involve factors not affect-

Jan 1, 1950

By D. K. Crampton, H. I. Burghoff, J. T. Stacy

The effect of cold-working upon electrical conductivity of copper and of copper alloys appears not to be generally known in detail. Although several papers on the subject have been presented, showing variable effects according to alloy composition, it is often assumed that the 2 to 3 per cent lowering of the conductivity of ordinary tough-pitch copper by severe cold-working is typical for copper alloys. Because of this situation it was decided to carry out the present work upon several types of copper, copper-zinc alloys over the commercial range of brasses, other binary alloys, and numerous age-hardening alloys in the quenched, and quenched and aged conditions. The large number of alloys investigated made it impossible to study every one in the detail it may have merited. Inasmuch as several interesting points that could not be satisfactorily explained came to light, further work is contemplated by the authors. It is usually considered that the normal effect of cold-work on ductile metals and unordered alloys is to decrease conductivity slightly. In alloys that are in the ordered state, as annealed or otherwise heat-treated, cold--working tends to disorder the structure, resulting in very marked lowering of conductivity. The random state of atomic arrangement produces lower con- ductivity than the ordered state for the same composition.1 Previous Work Previous work in the authors&apos; laboratory has shown the electrical conductivity of several copper alloys to be decreased by greater amounts than might have been expected, in view of the general knowledge on the subject, when cold-worked. Guillet and Ballay2 investigated the influence of cold-work upon the resistivity of several metals and alloys, including copper and some copper alloys. The increase of resistivity due to severe working w is less than 4 per cent for all pure metals tested, while the increases for the alloys were generally higher. In the copper-zinc alloys, they found that the increase of resistance due to drawing increased with zinc content, and reported a 21.7 per cent increase of resistance for an alloy contaiiiing 67.9 per cent Cu as severely worked. Masima and Sachs3 studied the change in electrical resistance of three brass crystals when stretched in the tensile test, obtaining increases in resistance of the order of 3 per cent for 30 to 40 per cent reduction of area of the specimens. The resistance decreased again when the cold-worked crystals were subsequently annealed. Bardenheuer and Schmidt4 found that the conductivity of copper was decreased by 3 to 4 per cent when reduced 50 per cent by working. The conductivity of 72-28

Jan 1, 1941

By C. R. Taylor, John Chipman

THE study of chemical reactions of liquid steel and basic open-hearth slag involves a complex slag system of at least eight important components, and often a number of others. In initiating an experimental program on the equilibrium relationship of slag and metal, it became evident that but little progress could be expected in the interpretation of the complex slags until an understanding had been gained of the simpler liquid systems composed of a limited number of slag constituents. The simplest system that in any way resembles open-hearth slag must contain lime, silica and iron oxide. Attempts to produce in an induction furnace slags containing only these constituents have met with serious limitations because of the solubility of refractories in the slag. Thus in acid-lined furnaces the slag is always saturated with silica and consequently contains between 48 and 63 per cent of this component. In basic-lined furnaces an undesired additional component, magnesia, is introduced to the extent of some 3 to 15 per cent, depending upon the acidity, sometimes with disastrous results to the furnace. Thus in a magnesia-lined furnace, Fetters and Chipman1 found it impossible to prepare slags in which the ratio (CaO + MgO)/SiO2 was less than that corresponding to the metasilicate (approximately 0.8: I). Between this composition and that of the true acid slags lies a field of composition about which very little is known and which cannot be explored by any ordinary means. This circumstance, as well as the desire to study the more basic slags in the absence of magnesia, calls for the development of a method by which slags may be brought into equilibrium with molten metal without coming in contact with refractory. Such a technic has been provided by the rotating induction furnace developed by Barrett, Holbrook and Wood.2 Three modifications of the rotating furnace were described by these authors. The first was essentially a rotating refractory cylinder containing the magnesia crucible, which was itself heated by a graphite crucible. In the second, only the magnesia crucible rotated, the cylindrical graphite heater being stationary. In the third, the coil and crucible revolved, and the metal was heated directly by induction. EXPERIMENTAL METHOD Furnace The furnace used in this investigation differed from previous designs and may be considered a fourth modification of the original rotating furnace. Its essential features are shown in Fig. I. The outer case A is a fire-clay flue liner for control of the atmosphere. The top B and base C are alberene stone. The coil supports D are of "Formica," a material formed from phe-

Jan 1, 1942

By Edward Saibel

THE various theories that have been advanced to explain or predict the conditions under which a metal fractures may be divided into two categories: First, there are the macroscopic theories generally used in engineering practice. These are empirical in nature and limited in application. For example, the maximum normal stress criterion, which is quite satisfactory for a mild steel in simple tension at atmospheric pressure, no longer suffices when this steel is pulled under hydrostatic pressure.2 In all of these empirical theories, the critical condition characteristic of the material and of the loading conditions must be obtained experimentally. Secondly, there are the theories based on interatomic forces,3,4 or on the energy required to form new surface.5,6 These have all dealt with the theoretical evaluation of the tensile stress necessary to bring about rupture in a purely brittle manner; that is, without plastic flow preceding fracture. All of these attempts have led to values of the tensile strength of crystals or of polycrystalline bodies that are 100 to 1000 times higher than observed values. On this account, it has been thought necessary by some investigators to conceive of a microcrack structure or distribution of flaws throughout the specimen such that the resulting stress concentrations provide the factor necessary to bring about agreement between calculation and experiment. It has been pointed out by several observers7 that this point of view is highly unrealistic, in that the weakening effect of these flaws or microstructures must be very uniform from specimen to specimen, since the fracture stresses found for various specimens of the same material tested under similar conditions are so - nearly constant. Furthermore, since calculations based on atomic forces or atomic energy are for purely brittle fracture, a condition probably never satisfied in tests, agreement is hardly to be expected. Thermodynamic Theory Of Fürth In order to resolve this apparent discrepancy between the theoretical cohesive strengths and the observed fracture strengths, Fürth8 advanced a thermo- dynamic theory of the tensile strength of isotropic bodies based on work of Born9 relating to the melting of crystals. Assuming an isotropic material free from imperfections and flaws, behaving elastically up to the breaking point, and assuming a connection between rupture and the melt- ing phenomenon, Fürth obtains an expression for the rupture stress u, in the form [ ] where Q is the melting energy per unit mass, ? is the density, and µ is Poissons’ ratio. Fürth compares fracture stresses computed from Eq I with the fracture stresses of metals as given in the Landolt-Bernstein Tables (1927) and in the Hand-

Jan 1, 1947

By B. D. Thomas

GRAVITY concentration is a general term designating processes for separating and sorting granular material by means of forces that depend on the density, size and shape of the particles. When these forces are applied to the particles the latter are caused to move, those of a given density along one path and those of a different density along another, until ultimately they are sufficiently separated to be removed from the operation of the process. The forces producing this separation are in general complex and, unfortunately, no completely rigorous analysis of them has as yet been devised. The science of the hydrodynamics of viscous fluids deals with the little that is known about the problem. Every observer of nature is familiar with the fact that an object moving in a fluid encounters a force of resistance. We have only to walk against a brisk wind or stir a cup of coffee, or observe the slow fall of a grain of sand in water, to experience directly the fact that such forces exist. The exchange of force between an object and a fluid in which it is moving operates on the fluid as well as on the object. Thus a sailboat may be set in motion by the force exerted by the wind on the sail; but the sail also deflects the wind-furthermore, the water in which the boat floats is set in motion by the force exerted by the boat as it cleaves through the water. The wind is thus enabled to do work on the water through the intermediary of the boat. In general, an object moving through a fluid encounters a force that opposes the motion and also acts to change the motion of the fluid. In gravity concentration we are dealing with this force of resistance acting - on the granular particles that are to be separated. Furthermore, the motion induced in the fluid by a particle will in turn affect the motion of a neighboring particle. This effect will be reciprocated and in consequence there will be an exchange of forces between all the particles in the material. These effects are complex and the most that can be expected is that we may be able to determine in a qualitative way how they operate. In gravity concentration, then, the following types of forces may be distinguished as acting on the particles: 1. The force of gravity, which acts on every particle in the system in proportion to its mass. 2. Buoyant forces, which arise from the force of gravity acting on the fluid.

Jan 1, 1950

By H. R. Ogden, E. S. Bartlett, A. G. Imgram

Unmodified disilicide coatirigs were applied to sheet-tensile specimens ofCb-Dg3 and Mo-TZM veJractovy- metal alloys. Coating thickness, degree of coating-substrate interdiffusion, and specimen geonzetry (notched and plain were included in the variables studied. Tensile tests were made to determine the ductile-lo-brittle transition temperature. The disilicide coating modestly increased the transition temperatlre of TZM, but had no effect on 043. Neither material condition (recrystallized or stress-velieved) nor specimen geometry (notched or unnotched) significantly altered the effects of coatings on the transilion temperatures of. the alloys. Cracks in the brittle coatings did not propagate into the substrate, and fracture modes appeared to be the same for both un-coated and coated specimens. MOST potential structural applications for refractory metals and alloys involve exposures to oxidizing environments at elevated temperatures. The general lack of oxidation resistance of these metals will require protective coatings to allow fulfillment of their potential. Currently preferred coatings for the oxidation protection of refractory metals are brittle intermetallic aluminides or silicides. These are typically formed on the surface of the refractory-metal substrate by a diffusion reaction between the substrate and a gaseous or liquid medium that is rich in aluminum or silicon. Because of the brittleness of these coatings, they will sustain no plastic deformation at low temperatures. They are frequently cracked by cooling from the coating temperature because of the thermal-expansion mismatch with the substrate alloy. Even if they survive cooling intact, they crack rather than sustain deformation under load at low temperatures. Thus, when a coated refractory metal is strained beyond the elastic limit of the coating at low temperatures, the mechanical environment of the substrate would include both static and dynamic cracks. These might be expected to influence the flow and fracture behavior of the substrate. This could be manifested in an altered fracture mode and/or an increase in the normal ductile-to-brittle transition temperature of the refractory-metal substrate. This paper presents the results of a research program that was conducted to determine the influence of the presence of a brittle surface coating on the low-strain-rate tensile behavior of typical refractory metals at low temperatures. EXPERIMENTAL PROCEDURES Material Preparation. Thirty-mil-thick sheets of molybdenum TZM alloy (Mo-0.5Ti-O.1Zr) and colum-bium D43 alloy (Cb-IOW-1Zr-O.1C) were obtained commercially. These alloys were selected as substrate materials representing two classes of materials important in current refractory-metal technology. The TZM was in the stress-relieved condition, and exhibited a heavily fibered grain structure. The D43 had been processed by the duPont "optimum" fabrication schedule,&apos; and exhibited slightly elongated grains typical of this process. Tensile specimens of two geometries were prepared from these materials: 1) plain specimens with 0.2-in.-wide 1.0-in.-long gage sections; 2) specimens similar to above, but with a 0.06-in.-diam hole drilled in the center of the gage section, providing a stress concentration factor, Kt, of 2.5. The "notch" geometry was selected to represent a typical condition of a rivet hole or other geometric discontinuities as might be encountered in various applications. Machined specimens were degreased, with a final rinse in acetone, prior to the application of coatings. Specimens of each substrate and configuration were pack-siliconizedin a particulate mixture of 80 pct A1203, 17 pct Si, and 3 pct NaF. Specimens were embedded in this mix (contained in graphite retorts) and coated in an electrically heated argon-atmosphere furnace under time-temperature conditions to effect nominal 1- and 3-mil-thick silicide coatings: Coating Thickness, mils Thermal Treatment 0.6 to 1.4 24 hr at 982°C 2.4 to 3.2 48 hr at 1093°C Coating kinetics were similar for both the TZM and D43 substrates. These treatments had little or no visible effect on the substrate microstructure as determined by optical metallography. The coatings on TZM were essentially single-phase unmodified disilicides, while those on D43 showed substantial evidence of modification by proportionate reaction with the respective substrate elements or phases, as shown in Fig. 1. It was recognized that these coatings might not be particularly desirable regarding protective capability. However, it was desired to circumvent possible inter -ferring chemical interaction with the substrate by pack additives such as chromium, titanium, boron, aluminum, and other elements that typify the better protective coatings for these materials.&apos; Thus, the results presented apply specifically to the simple silicide coatings investigated. They may not be rep-

Jan 1, 1967

By W. F. Kieschnick, C. C. Miller, E. R. Brownscombe

The possibility has been mentioned that large pressure gradients in a solution gas driven field caused by high production rates might lead to a reduction in the ultimate recovery obtainable compared to that which would be obtained by a very slow rate of production. In the present study the reservoir conditions accompanying a high rate of production and the corresponding ultimate recovery were compared with those obtained if the reservoir were produced at some marginal rate throughout its entire life. The method of calculation involved the application of fluid flow - material balance analysis to a series of successive steady state conditions in the reservoir. In the case studied very little difference in recovery was obtained at the abandonment pressure. An analysis of the basic factors involved indicates that the sane results would hold for considerable variation in properties of reservoir fluids, permeability of the formation or well spacing. Even if the reservoir has a water drive it appears that no harm will be done by- open flow production if the rate is cut back before any appreciable free gas is produced. However, any condition which leads to disproportionate withdrawal rates and which causes large pressure differences over large areas might result in substantial loss in ultimate recovery. INTRODUCTION When the pressure declines in a reservoir, gas is evolved from solution causing a shrinkage in the reservoir oil so that a stock tank barrel of oil with its residual dissolved gas occupies a smaller volume than originally. For a given reservoir oil saturation, therefore. stock tank oil content will be higher at the lower pressure. At the same time the evolved gas which remains in the reservoir reduces the reservoir oil saturation, and this would be expected perhaps to cause a reduction in stock tank oil content. However, around the well the shrinkage occurs so rapidly that stock tank oil may actually accumulate in this region in spite of the simultaneous reduction in reservoir oil saturation. If this accumulation should continue, spreading into a larger zone, the loss might be appreciable and because larger pressure gradients accompany higher rates the accumulation might possibly be large if wells are produced at too high a rate. It is the purpose of this paper to determine the effect of rate of flow on ultimate recovery by solution gas drive. The problem under consideration involves the unsteady state flow of oil in a closed radial system from which oil is produced by the evolution of gas as a finite drawdown is estalllished and the reservoir pressure declines. A somewhat similar unsteady state production problem dealing, however, with a linear system has been considered by Muskat and Meres, but these authors do not discuss the effects which rate may have on recovery. Other methods have been reported for estimaling recovery as a function of pressure decline when producing at an infinitesimal rate, and Muskat and Taylor&apos; have shown the effect of fluid and reservoir characteristics on recovery under these conditions. Assuming steady state to exist, Babson has given a procedure for computing the decline in well productivity as the field is depleted, and Moyer has combined these methods to estimate abandonment conditions and the resulting ultimate recovery. These latter approaches to the problem either do not consider the effect of rate or assume steady state can be established. In detail this paper presents the behavior of the reservoir producing under unsteady state conditions and considers the effect of production rate during this unsteady state flow on the recovery attained. The analysis also shows that there will exist at abandonment a type of steady state, and on this basis a method has been clevelopetl for rapid calculation of abandonment conditions.

Jan 1, 1949

By C. C. Miller, W. F. Kieschnick, E. R. Brownscombe

The possibility has been mentioned that large pressure gradients in a solution gas driven field caused by high production rates might lead to a reduction in the ultimate recovery obtainable compared to that which would be obtained by a very slow rate of production. In the present study the reservoir conditions accompanying a high rate of production and the corresponding ultimate recovery were compared with those obtained if the reservoir were produced at some marginal rate throughout its entire life. The method of calculation involved the application of fluid flow - material balance analysis to a series of successive steady state conditions in the reservoir. In the case studied very little difference in recovery was obtained at the abandonment pressure. An analysis of the basic factors involved indicates that the sane results would hold for considerable variation in properties of reservoir fluids, permeability of the formation or well spacing. Even if the reservoir has a water drive it appears that no harm will be done by- open flow production if the rate is cut back before any appreciable free gas is produced. However, any condition which leads to disproportionate withdrawal rates and which causes large pressure differences over large areas might result in substantial loss in ultimate recovery. INTRODUCTION When the pressure declines in a reservoir, gas is evolved from solution causing a shrinkage in the reservoir oil so that a stock tank barrel of oil with its residual dissolved gas occupies a smaller volume than originally. For a given reservoir oil saturation, therefore. stock tank oil content will be higher at the lower pressure. At the same time the evolved gas which remains in the reservoir reduces the reservoir oil saturation, and this would be expected perhaps to cause a reduction in stock tank oil content. However, around the well the shrinkage occurs so rapidly that stock tank oil may actually accumulate in this region in spite of the simultaneous reduction in reservoir oil saturation. If this accumulation should continue, spreading into a larger zone, the loss might be appreciable and because larger pressure gradients accompany higher rates the accumulation might possibly be large if wells are produced at too high a rate. It is the purpose of this paper to determine the effect of rate of flow on ultimate recovery by solution gas drive. The problem under consideration involves the unsteady state flow of oil in a closed radial system from which oil is produced by the evolution of gas as a finite drawdown is estalllished and the reservoir pressure declines. A somewhat similar unsteady state production problem dealing, however, with a linear system has been considered by Muskat and Meres, but these authors do not discuss the effects which rate may have on recovery. Other methods have been reported for estimaling recovery as a function of pressure decline when producing at an infinitesimal rate, and Muskat and Taylor&apos; have shown the effect of fluid and reservoir characteristics on recovery under these conditions. Assuming steady state to exist, Babson has given a procedure for computing the decline in well productivity as the field is depleted, and Moyer has combined these methods to estimate abandonment conditions and the resulting ultimate recovery. These latter approaches to the problem either do not consider the effect of rate or assume steady state can be established. In detail this paper presents the behavior of the reservoir producing under unsteady state conditions and considers the effect of production rate during this unsteady state flow on the recovery attained. The analysis also shows that there will exist at abandonment a type of steady state, and on this basis a method has been clevelopetl for rapid calculation of abandonment conditions.

Jan 1, 1949

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

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

By C. M. Loeb, A. M. Gaudin, J. H. Brown

THE object of most size reduction operations in the mineral industry is to liberate the grains of valuable minerals in the ore from those of the gangue. This is usually accomplished by crushing and grinding the entire mass of ore until there is only a small probability that any single particle contains more than one mineral. During this size reduction only limited control exists over size or composition of the particles exposed to the breaking action, and there is no control over the paths followed by cracks generated during the operation. This lack of control usually results in overgrinding and in production of large quantities of very fine material. The first detriment, overgrinding, is costly in itself, but when combined with the second factor it is doubly so. Not only is the fracture of a free particle unnecessary—the fracture of these particles may also make subsequent separation operations difficult, inefficient, and wasteful. It has been pointed out&apos; that if the object of size reduction is to liberate the valuable mineral component of the ore then, ideally, fracture should follow intergranular paths to the exclusion of trans-granular ones. This would result in liberation of the valuable minerals with as little size reduction as possible. This ideal comminution operation is referred to as intergranular comminution, and it was the object of the investigation to determine the extent to which it could be developed by heat treatments. There are many indications in the literature that heating rocks prior to crushing may be favorable. Reports by Holman,2 Yates3 and Myers&apos; are pertinent. These investigators showed that heating certain rocks prior to crushing them did, in fact, improve their crushing characteristics in that fewer fines were produced, although the fact that intergranular comminution was being effected apparently was overlooked. In addition, Sosman noted that if there is appreciable anisotropism in the thermal coefficients of expansion of even a pure mineral, then considerable permanent separation of the grains of the rock can be expected as a result of heating the rock to a high temperature.&apos; By the same token, if there are ap- preciable differences in the thermal expansion coefficients of the various minerals of a multi-component rock, similar results should be obtained by heating this rock. This has been tested, partially, by Brenner," who obtained patents covering the heat treatment of some pegmatitic rocks in order to facilitate comminution of these materials. It has also been demonstrated that this may occur in taconite." Also, the possibility of causing decomposition of one mineral in a rock as a means of promoting intergranular fracture has been considered. Seigle2 and Schiffman et al. have obtained patents on such processes as applied to calcareous iron ores. These reports all indicate that heat treatments prior to crushing may contribute materially to intergranular comminution, but they also indicate that no organized attempt has been made to determine the controlling factors of the method or to determine its applicability in general. The present article is a report on the initial phase of such an investigation. The authors have reviewed the claims of prior investigators and have attempted, also, to establish the factors that might determine the applicability of heat treatments in the mineral industry. In this work 2000-g samples of various rocks were heated in a small laboratory furnace and crushing and sizing operations were carried out in standard laboratory equipment. All samples of each rock were as nearly identical as possible in particle size, grain size, and composition and contained only lumps coarse enough to contain many grains each. Tests on Granite A number of tests were made on a coarse grained Finnish granite obtained in the form of coarse chips from a local monument yard. This rock exhibited little variation from piece to piece in either composition or grain size. The minerals contained were quartz, orthoclase, small amounts of hornblende, and minute quantities of mica. Grain size ranged from about 1 mm to about 3 mm. Temperature of the Heat Treatment: In some cases the granite was heated to a particular temperature and crushed, hot, immediately upon withdrawal from the furnace—in others the rock was allowed to cool before crushing, but without quenching to room temperature after heating. In most tests on granite the heating period was about 2 hr with the furnace at the highest temperature for about 1 hr. Cases in which these periods were varied greatly will be presented separately.

Jan 1, 1959