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By W. M. Rutherford

A knowledge of the limits of miscibility between reservoir oil and possible injection fluids is required for selection of the optimum miscible-injection fluid. Limits of miscibility can be estimated from the results of equilibrium phase-behavior experiments. They can also be determined by means of displacement experiments conducted in a high-pressure sandpack. This paper describes the equipment and procedure which have been developed for determining miscibility conditions by stable displacement. A systematic series of displacements of a West Texas reservoir oil was carried out. The results indicate that, at constant pressure, miscibility is a function only of the pseudocritical temperature of the injection gas. This fact, together with improved experimental methods, makes the displacement technique a rapid, reliable means for determining miscibility conditions. In conjunction with the displacement experiments, phase diagrams were constructed for the oil with dry gas and propane and with dry gas and ethane. Phase behavior of the methane-ethane-propane system was determined at 110°F. The experimental work demonstrates the feasibility of using ethane-rich gases to reduce cost and pressure requirements for miscible displacement. INTRODUCTION In recent years, interest in the miscible displacement of oil by light hydrocarbon mixtures has been high. Many pilot and a few field scale projects have been started.1 These projects have made use of various methods for achieving miscibility: (1) the LPG-slug process, (2) the enriched-gas-drive process and (3) the high-pressure gas-drive process. Some field projects have been successful; the results of others are debatable. In general, projects which have performed best have involved the injection of an appreciable fraction of a pore volume of miscible material. Economical application of miscible displacement depends strongly on the cost of the miscible-injection fluid. If an appreciable fraction of a pore volume of material is required for successful application of these methods, a precise knowledge of the minimum requirements for miscibility in terms of composition of injection fluid is essential. Therefore, reliable experimental methods for determining miscibility conditions are important, and a procedure for estimating these conditions from the composition of the reservoir fluid is highly desirable. The subject of this paper is the problem of determining conditions which result in miscible displacement of oil by light hydrocarbon mixtures. Miscibility conditions can be estimated by means of equilibrium experiments conducted in a PVT cell, or they can be determined by means of high-pressure displacement experiments. This paper describes the equipment and procedure which have been developed for the determination of miscibility by high-pressure displacement experiments. These methods have been applied to the displacement of a West Texas reservoir oil with mixtures of dry gas, ethane and propane. In conjunction with the displacement experiments, triangular phase diagrams have been constructed for mixtures of the oil with dry gas and propane and with dry gas and ethane. The effect of injection-gas composition on conditions for miscible displacement in high-pressure sandpacks and cores has been the subject of several published papers.2-5 The experimental methods used in these investigations were such that displacements were unstable, and the effects of fingering and/or gravity layover are clearly evident in the results. Miscibility conditions were probably correct in spite of the instability phenomena, but the experiments evidently were time-consuming, and limited data were reported. Systematic high-pressure flow studies which would support a correlation of miscibility conditions have not been reported; however, Wilson5 has proposed the use of pseudocritical temperature of the injection gas as a parameter and Benham, et al,6 have based a miscibility correlation on observed and calculated equilibrium data.

By J. Geertsma

The relationship between porosity and the speed of propagation of acoustic waves in fluid-saturated porous rocks as measured by the Sonic log and by ultrasonic tecbniques is analyzed. Biot's continuum theory 1 is used to explain the difference in acoustic wave propagation between a dry and a liquid-saturated porous material. The porosity is a variable in this theory. However, the acoustic wave propagation in the dry rock depends too on porosity, and this dependence is not predicted by the theory. Frequently in dry sandstones, a nearly linear relationship between reciprocal acoustic wave velocity and porosity is observed in the low-porosity range. The physics behind this behavior is outlined. An empirical relationship of the form, 1/v = A + F, applies accordingly for many porous dry rocks, provided the porosity is the only variable. The presence of a liquid in the pores changes the value of B, and this change is found to be in agreement with the Biot theory. The time-average relation introduced some years ago2 results in an equation of the same type — 1/V = F/Vf + (1 - F)/Vr — but is not based on a sound physical picture. Still, this relation sometimes predicts approximately correct A and B values. Carbonate rocks with their complicated pore structures sometimes show a different relationship between wave velocity and porosity, unfavorable for log interpretation. Examples are presented. The simultaneous presence of calcite, dolomite and anhydrite, with their different grain densities and matrix compressibilities, complicates acoustic-log interpretation in carbonate rocks still further. Other complicating effects of acoustic-log interpretation are discussed. They are related to the influence of shale streaks and natural fractures on the average wave velocity observed by the logging tool and to the effect of adsorption phenomena on wave propagation in unstressed rocks particularly in sandstones. INTRODUCTION The velocity of propagation of sound waves in porous sediments as a function of depth is a quantity frequently measured. Velocity loggers of various d-sign may be used, for which velocity is defined as the. distance between a wave generator and a wave detector (one-receiver system) or the distance between two wave detectors (two-receiver system), divided by the shortest time required for a vibrational pulse to cross this distance. Velocity loggers ale used to assist in seismic prospecting, to differentiate between the various types of sedimentary rock layers, and also to determine rock porosity and pore fluid content. This paper is especially concerned with the relation between velocity and porosity and the effect of the pore fluid. Most velocity-log interpretation in terms of porosity is based on a formula advocated by Wyllie, et al,2 suggested earlier by Hughes and Jones,3 and known as the "time-average relation". This formula app1ies for a model consisting of a layered system of parallel alternating slices of a solid and a liquid, crossed by the wave path perpendicular to the interfacrbs. This must be an unsuitable model for the derivation of wave propagation properties of liquid-saturated porous media. It suggests that only rock matrix and fluid properties influence the wave velocity. The surprising fact, nevertheless, is that applications of this formula to clean sandstones under specific conditions (sufficient effective stress) results in porosity predictions which are close to reality. Other authors, such as Gassmann,4 White and Sengbush,5 Brandt,6 and Hicks and Berry,7 make use of more realistic models to arrive at a wave velocity formula for porous sediments. Their models consist of various liquid-saturated packings of frictionless spherical grains and are, therefore, also highly idealized. A number of experimental results obtained in the laboratory can be better explained with these model theories than by the time-average relation. A study of a general character, not depending a priori on a special model of the porous structure, was still lacking.

By J. O. Brittain, E. Lautenschlager

Yield point measurements were made on polycrys-talline specimens of a iron deformed various amounts at 770, 1950, or 273°K, aged, and redeformed at 77: 195°, or 273°K. When the prestrain temperature is lower than that subsequent to aging, the yield point return is suppressed relative to specimens deformed at the same temperature both before and after aging. The degree of suppression is a function of both the magnitude of prestrain at the lower temperature and the difference between the deformation temperatures before and after aging. It is proposed that the suppression is caused by the formation of dislocation barriers during the initial low-temperature deformation. The aging kinetics of polycrystalline a-iron wires deformed at 77o or 303oK were studied at 44 " and 64 "C using internal friction measurements. No change was noted as a result of the difference in prestrain temperature. IN an earlier investigation1 it was observed that the return of the yield point was suppressed when a poly-crystalline specimen of a iron was plastically deformed at 77oK, aged at room temperature, and subsequently tested at room temperature. In order to investigate this observation, a research program was undertaken to measure the effect of prestrain temperature on the strain aging of a iron.' According to Thomas and Leak,3 return of the yield point and internal friction are similar measures of strain aging since both are dependent on solute atoms leaving the solid solution. The decrease in internal friction as a function of aging time is indicative of the removal of solute atoms from the solution, while the return of the yield point as a function of aging time gives the rate of arrival of solute atoms at the dislocations. According to Cottrell and Bilby,4 the solute atoms create an "atmosphere" which locks the dislocations and causes a return of the yield point upon restraining. In this experiment, both methods of measuring strain aging were employed in order to have two independent measurements of the strain aging phenomenon. In a previous study,5 the strain aging kinetics of a similar lot of a iron were measured for specimens which were prestrained and tested at the same temperature over the range of 77" to 273°K. Although some differences were found in the correlation of the two methods employed for measuring strain aging, the kinetics of strain aging were unaffected by the change of prestrain and test temperature. In another study,6 dislocation line tension and the density of the solute atmosphere were found to contribute to the return of the yield point when ingot iron specimens were prestrained at 303°K and tested at a lower temperature, namely 77°K. This paper is concerned with changes in strain aging kinetics produced by pre-straining at 77" to 303°K and testing at either the prestrain temperature or at higher temperatures up to 273°K. EXPERIMENTAL PROCEDURE The polycrystalline a iron specimens used in both the tensile and internal friction tests were 0.033-in. diam ingot iron wires, wet hydrogen treated and re-carburized. A more detailed procedure for processing a 3/16 in. ingot iron rod (nominal wt. pet composition: Fe - 0.013; C - 0.0125; Mn - 0.023; Si -0.005; P - 0.0023; S - 0.11; Cu - 0.004 N) to 0.033 in. diam specimens containing approximately 0.020 wt pet C and no detectable nitrogen with a grain diameter of 0.033 to 0.040 mm is given elsewhere.5 Immediately prior to testing, specimens were solution treated for 15 min at 700o ± 2°C in a static vacuum of less than 50 ?. of Hg and rapidly quenched into an ice-water bath while still enclosed in a 3 mm quartz vial. The internal friction specimens had 8-in. gage lengths and the tensile specimens had 2-in. gage lengths. The wires used as tensile specimens were first run as internal

Jan 1, 1962

By E. L. Dougherty

A phenomenological theory for a one-dimensional unstable miscible displacement similar in type to the Buckley-Leverett model but including the effects of mixing is proposed An equation giving the fractional flow of pure solvent through an oil phase containing dissolved solvent is derived including the effects of gravity and nonbomogeneities. Numer ical integration of the resulting pair of nonlinear hyperbolic partial differential equations gives volumes of dissolved and undissolved solvent as functions of space and time. Parameters in the model which characterize mixing due to dispersion were determined from experimental data; the parmeters correlated with viscosity ratio. The results indicate that the rate of dispersive mixing is proportional to volumetric flow rate Incorporated into our equations were Koval's heterogeneity factor H. which characterizes mixing due to channeling. This allowed satisfactory predictions of displacement behavior observed in horizontal floods in nonhomo-geneous cores; but in most cases the values of H which we used were considerably larger than those used by Koval. INTRODUCTION It has been shownl,2,3 that for favorable viscosity ratios, the diffusion equation with convection satisfactorily describes the behavior of a miscible displacement in a porous media. Numerous attempts to develop a satisfactory mathematical description for the case of unfavorable viscosity ratio have been reported, but they have met with only partial success. These attempts, which are reviewed by oval! have taken two approaches: (1) development of a flow model akin to the Buckley-Leverett approach for immiscible displacement neglecting the effects of mixing5,6 and (2) simultaneous application of Darcy's Law for flow and the diffusion equation with a convection term for mass transfer.7,8,9 We set out to construct a mathematical model of Type 1 including, though, the effects of mixing. Based upon a set of hypotheses, we derived a pair of nonlinear hyperbolic partial differential equations which describe in a one-dimensional system the combined effects of flow and mixing. To work with these equations it was necessary to develop a fractional flow formula. The combined system was integrated numerically using the method of characteristics. In our equations the mixing process is characterized by four parameters. Three of these, labeled ß, P1 and P2, account for dispersive type mixing. The fourth is the heterogeneity factor H, proposed by Koval to account for channeling due to nonhomogeneities in the porous media. Values of the parameters which would cause agreement between theory and experiment were determined by trial-and-error for miscible floods conducted in horizontal cores of both homogeneous and heterogeneous materials. Calculations were also performed for vertical floods in homogeneous cores. The purpose of this paper is (1) to present the details of the mathematical analysis, (2) to present the results of the calculations, (3) to consider what light the results shed on the mixing process in an unstable miscible displacement, and (4) to provide a firmer foundation for the correlation technique developed by Koval for predicting the behavior of unstable miscible floods. STATEMENT OF PROBLEM The problem is to construct a mathematical model which describes in one dimension the observed behavior in an unstable miscible displacement. The approach is phenomenological in that the equations are based on assumptions which violate certain physical precepts known to apply to the displacement phenomenon. However, the assumptions do allow us to account mathematically for the more essential phenomena. The work is of value if we can quantitatively predict experimental results which heretofore could not be predicted, even though the synthetic nature of the model belies complete explanation of the observations. We assume that the system is comprised of two contiguous flowing phases. One phase, which we call free solvent, has the properties of pure solvent. We call the fraction of the pore volume occupied by this phase the solvent saturation, designated s. The oil phase consists of oil containing dissolved

The authors of T.P. 4179 have conducted a large number of experimental water floods, and the data are reported in a very clear manner. These data show significant variations of residual oil saturation with changes in wettability, interfacial tension, flooding rate, and oil-water viscosity ratio. Furthermore, some of the data obtained cover a range of variables not previously reported in the literature. It is our opinion that at least some of the observed variation in oil recovery can be obtained only in floods made under conditions identical with those used by the authors. There are other systems which may be said to have the same wettability, but which exhibit an entirely different qualitative dependence of oil recovery on the experimental variables. In this regard, we would like to direct attention to several items which we feel are not given sufficient consideration in the paper. The first of these items has to do with wetting equilibrium. A survey of the current literature will reveal that little is known, and even less is understood, about wetting equilibrium. This is particularly true of the relation between contact angle measurements made under static conditions, and the wettability existing under the dynamic conditions of floods carried out at relatively high rates. We feel that this is a very important consideration, particularly in the floods the authors performed under what they consider to be "water-wet" conditions. It is our opinion that wetting equilibrium did not exist in these floods, except possibly at the very lowest rates. This opinion is in part based on some flooding tests we have made which are very similar to some of those described by the authors. These floods were made in unconsolidated cores, 6 in. long and 13/4 in. in diameter, made up of carefully graded glass spheres of the approximate size of the sand grains used in the subject paper. The cores were mounted vertically, and distilled water was used to displace a refined oil equivalent to the authors Soltrol C. The glass spheres had previously been carefully cleaned so that they were strongly wet by the water. The floods were made with and without an interstitial water saturation. When interstitial water was present, the solid surfaces were mostly covered with water prior to injecting the flooding phase. Consequently, a condition very near wetting equilibrium existed throughout the flood. The data in Fig. A show that under these eircumstances, rate had no influence on residual oil at flood-out, over the range of rates studied. On the other hand, in the absence of interstitial water, the surfaces are all covered with oil before the displacement. To achieve wetting equilibrium the injected water must displace the oil from the surfaces themselves. This is not a fast process. Therefore, the effective dynamic wetting condition in floods of this type must depend upon the flooding rate. When the flood rate is very low, compared to the rate at which wetting equilibrium is attained, the residual oil saturation at flood-out is essentially the same as the residual obtained when interstitial water is present. This is shown in Fig. A, and occurs because the dynamic wetting conditions are the same in both cases. However, at moderately high flooding rates, the rate has an important bearing on the amount of oil recovered because in this case the attainment of surface equilibrium is the slow step. We believe that, in the absence of an interstitial water saturation, oil recovery varies with rate because of the reasonably slow approach to wetting equilibrium which occurs during the displacement. In fact, as the rate is increased, the oil recovery performance indicates the core is becoming progressively less water-wet. At the highest rate there is even considerable simultaneous production of oil and water after breakthrough. This does not occur at the lower flooding rates, nor is there appreciable simultaneous oil and water produc-

Jan 1, 1957

By J. H. Scaff, W. G. Pfann

The effects of impurities on the electrical properties of silicon are discussed in a companion paper by Messrs. Scaff, Theuerer, and Schumacher.' It was shown that an ingot of silicon which contained boron and phosphorus in certain concentrations consisted partly of p-silicon and partly of n-silicon2 and that the common boundary between these regions was the source of a photo-voltage. These features were ascribed to the segregation of boron and phosphorus during solidification of the ingot. Because of different rates of segregation the first portion of the ingot to freeze contained a molar excess of boron over phosphorus and hence was p-silicon. The last-solidified region contained a molar excess of phosphorus and was n-silicon. This paper is concerned with the microscopic examination of such ingots. The microstructures are rather unusual in certain respects and their study has helped to show how the segregation of minor elements modifies the electrical properties of silicon. The microstructures of ingots prepared from two lots of silicon, designated 1 and 2, will be described. These lots were obtained from the Electro Metallurgical Co. Both contain 99.8+ pct silicon, but they differ in impurity analysis. For each lot will be shown the microstruc-ture of a slowly cooled ingot, for which the time of solidification is 6 min., and that of a rapidly cooled ingot, for which the time of solidification is 2 min. The ingots were prepared from 45 g charges of silicon. When an alloy freezes a cored structure is generally produced because of insufficient diffusion to remove concentration differences established during freezing. As a result of coring the first portion of a crystal to freeze is usually poorer in solute than equilibrium requires, while the last material to solidify is richer. In a columnar grain, growth is principally unidirectional and coring can produce a concentration gradient along the entire grain. By carrying the picture of the coring process from a single grain to an entire ingot in which columnar grains are aligned more or less radially, one can easily see that a composition gradient could be established between the outside and center of the ingot. This, to a considerable extent, occurs during the solidification of a slowly cooled ingot of silicon. Slowly Cooled Ingot In slowly cooled ingots of both lots of silicon, columnar grains which are approximately radial in direction and longest at the top of the ingot enclose a region in which the shapes and align- ment of the grains are considerably less regular. The central region is last to solidify and contains cavities and inclusions. The sketch of Fig 1 illustrates some of these features, as well as others which will be described below. If a polished section of an ingot is etched in a mixture of 2 parts of 20 pct HF and 98 parts of HNO3 the p-n barrier becomes visible. The overall shape of the barrier in a slowly-cooled ingot is roughly oval, as may be seen in Fig 1. The proportion of n-silicon, that is, the region enclosed by the barrier, is considerably greater in ingots of lot 1 than in those of lot 2. A sample about 34 x 1/8 X 1/8 in. was cut from a slowly cooled ingot of lot 1 with its long direction extending from the top of the ingot to the porous region. Thus its upper half was psilicon, its lower half n-silicon. A longitudinal face was polished and etched in the acid mixture. The areas of interest are identified in Fig 2. A band of parallel markings extends across the sample in the columnar region. These striae are roughly parallel to the upper surface of the ingot and cross grain boundaries with only slight deviations. They are quite distinct in the n-silicon,

Jan 1, 1950

By Heribert Wiedemeier, Ali Khan

The tetnperature-conposition phase diagram of the syste,n MnS-CdS has been investigated over a range fro)n 600° to 1000° C. The small degree of solid solution on the MnS rich slde cornpared to the large CdS rich single-plzase region could be esplained by the difference in strzrctures of the cotnponents and the relative sizes of the metal ions. The change in lattice parameter as a function of cot)zposition is linear atzd jollous Vegavd's rule within experittzental limits of error. Mns and CdS have gained considerable attention in recent years because of their interesting solid state properties. In order to investigate such properties and thermodynamic quantities of mixed compositions of transition metal chalcogenides, it is necessary to establish their phase diagrams. Two earlier publications are concerned with the MnS-CdS system. Schnaase' found complete miscibility when MnS and CdS were co-precipitated in aqueous solutions at room temperature. However, there is evidence that the mixed crystals were not stable over the entire range of composition. roger's' investigation of the ternary system ZnS-CdS-MnS at 900" C indicates a solution of 48 mole pct MnS in CdS. During the course of this work which covers a temperature range from 600" to 100O° C, emphasis has been placed on the use of high-purity materials and well-defined modifications of starting products. EXPERIMENTAL Starting Materials. The stable green modification of MnS with NaCl structure was prepared by direct fusion of the elements and by chemical transport reaction. Stoichiometric amounts of manganese of 99.99+ pct purity and of sulfur of 99.999 pct purity from Gallard-Schlesinger Corp. were sealed in an evacuated quartz tube at a pressure of 10"5 mm Hg and annealed for 48 to 60 hr at 800°C. The quartz tubes used throughout this work were outgassed under high vacuum at a temperature of 90OU to 1000° C for 12 to 18 hr. For the synthesis of MnS by chemical transport3 stoichiometric ratios of elemental manganese and sulfur and small amounts of iodine were sealed under vacuum in a quartz tube, which was placed in the temperature gradient, 900/800-C, of a two zone furnace. Single crystals of MnS in form of platelets and octahedra were deposited at the cooler end of the tube. The manganese content was determined by EDTA titratin. A typical result of several analyses was 63.05 pct Mn which agreed well with the computed value, 63.14 pct. Commercially available CdS of 99.9+ pct purity, a mixture of the cubic and hexagonal forms, was converted into the hexagonal high-temperature modification by annealing CdS under vacuum in quartz tubes at 700° C for about 36 hr. As a second method, cadmium sulfide was transported in a temperature gradient, 70O° /650" C, with in vacuo sublimed iodine from the hot to the cooler zone of the tube. Large sized crystals in form of hexagonal columns and some platelets were obtained. An average result of the cadmium analysis by EDTA titration4 was 77.93 pct in good agreement with the computed value, 77.81 pct. Only the NaCl modification of MnS and the wurtzite modification of CdS were used for the solid solutions. Preparation of Solid Solutions. Finely powdered MnS and CdS samples were thoroughly mixed in molar ratios and compressed into pellets of about 6 mm diam and 2 mm in height at a pressure of approximately 13,500 atm.

Jan 1, 1969

By R. J. Grace

Primarily, this report presents the results of direct efficiency tests which were conducted on the 200-hp Coal-Pak, hot water generator, No. 3 unit, at the Otterbein College heating plant in Westerville, Ohio. Also included is a comparison of fuel consumption of the old plant with that of the new plant, a comparison of efficiencies obtained by the indirect method on a 300-hp Coal-Pak, and the cost of gas vs coal to show the savings effected by the latter. Tests were conducted with ten different bituminous coals selected by Mr. S. A. Frye, business manager at Otterbein College. The coals, of ten different suppliers, from four different states and eight different seams, represented competitive products economically available to the heating plant. The tests were conducted by the usual plant personnel which consisted of the plant superintendent and two assistants. No outside supervision of any kind was given during the tests. The tests were made from Dec. 13, 1960, through Mar. 4, 1961. During this period, 300.03 tons of coal were burned in the No. 3 unit. This tonnage represented 25.5 pct of the total coal requirements for the 1960-61 season. Operation during the period was typical for the comfort heat load which the plant supplies. Of the 78.12 operating days, the boiler was on bank or holdfire 14.31 days, or 18.32 pct of the time. The length of individual bank periods varied from 2 hr to three days. Run periods varied from about 2 hr to 7.26 days. DESCRIPTION OF PLANT AND HEATING EQUIPMENT The new Otterbein College heating plant, completed in the summer of 1957, is located in a 50x50 ft boiler room on the lower level of a combination maintenance building and heating plant. The building is situated away from the main college complex on a hillside, which allows for on-grade delivery of coal to the three bunkers. The bunkers are located directly above and connected with the individual stoker hoppers. Fig. 1 shows an in-plant view of the Coal-Pak installations. The No. 3, hot water, unit is at the extreme right. The heating plant consists of three 200-hp Coal-Paks. Unit No. 1 is operated at 4 psi and carries the steam heating load of 14,000 sq ft EDR (steam) installed in two buildings. Provision has been made to convert these buildings to hot water heating at some future date. Since the unit under maximum load conditions operates only at about 50 pct steaming capacity, a heat exchanger was installed to convert steam to hot water. Thus, during light loads in mild weather the unit can be employed more efficiently to supply both steam and hot water and eliminate the need for operating either of the two hot water units. Units No. 2 and No. 3 serve a hot water heating circuit for a large group of 21 buildings of the College. These boilers deliver hot water at temperatures up to 200°F to the various buildings through distribution lines in tunnels. The present calculated maximum heat loss for the College is 15.195 million Btu per hr. The Coal-Paks have individual design capacities of 6.700 million Btu per hr. Table I lists the design date for the Coal-Paks. The No. 3 unit was chosen as the test unit, primarily because of its bunker location at the far end of the delivery ramp. Because of this location any excess coal above bunker capacity could be conveniently stored and isolated from the adjoining bunkers. Each bunker has a capacity of 30 to 35 tons of coal. The No. 2 and No. 3 hot water generators are equipped with automatic controls which maintain the

Jan 1, 1962

By Pablo Hadzeriga

Great Salt Lake brine was subjected, in laboratory scale, to conditions simulating solar evaporation. Solid phases and the variation in composition of the liquid phase throughout the potassium salts crystallization zone are presented. These experimental results are discussed and compare with stable equilibrium data of the quinary system Cl-, SO4, Na+, K+, Mg + + and H2O. The Great Salt Lake brine in Utah is a vast reserve of several potential chemicals which has attracted much attention in the recent past. From the standpoint of an ore reserve, it may be said that the lake contains a low grade ore which has to be concentrated by evaporation of water. Because of the fairly favorable weather conditions in the Great Salt Lake area, it seems reasonable to consider solar evaporation as an economical means of concentration. From the view of phase chemistry, the main constituents of Great Salt Lake brine make it belong to the five component system: C1-, SO4=, Na+, K+, Mg++ and H2O. Because of their low concentration, the presence of other elements may be regarded as having little effect on the equilibrium of the five main components. Equilibrium data for the quinary system of our interest has been available for many years. Using these data, it is possible to calculate how the brine of the lake will concentrate when subjected to solar evaporation. In practicing solar evaporation, the weather factor and supersaturation conditions cannot be ignored, but their introduction into calculations using equilibrium diagrams is almost an impossible task. With the experimental work presented in this paper, an attempt is made to compare the stable equilibrium data of the quinary system as applied to the case of the Great Salt Lake brine and the dynamic equilibrium when that brine is subjected to continuous solar evaporation. Sea water brine belongs to this quinary system when only the major constituents are considered. The study of this system has been carried on since the last decades of the nineteenth century. The reason for these intensive studies was to give a geological explanation to the underground potash deposits. AS a result, many theories were presented trying to relate the equilibrium data with the geological formation, paragenesis and metamorphism of these deposits. Two main approaches were followed for the equilibrium studies of this quinary system. One of them, and the best known to us, was the school started in Germany in the last century by J. H. Van't Hoff. The other one, which started about the same time, was the school of N. S. Kurnakov in Russia. Both approaches were different and as a consequence generated opposite theories about the geological formation of potash deposits. The Van't Hoff school studied the quinary system of sea water under careful laboratory control, resulting in stable phase diagrams. During these studies, Van't Hoff recognized the difficulties in crystallizing some double salts.' ". . . . we unexpectedly found that some compounds whose formation in the solutions under investigation was possible at 25°C, nevertheless totally failed to put in an appearance. These, not to mention the calcium salts, were leonite, MgSO4 . K2SO4 4 H2O; kainite, KC1-MgSO4 . 3 H2O; and kieserite, MgSO4 .H2O. Even exceedingly slow crystallization with the addition of the compounds themselves did not remove the condition of supersaturation in the case of these bodies." However, those salts (leonite, kainite, and kieserite) were included in the stable phase diagram at 25°C as shown in Fig. 1 using D'Ans data.3 The figure also shows the sequence of crystallization of the Great Salt Lake brine according to the stable diagram. In later studies by the Van't Hoff school, especially by D'Ans,3 Karsten,4 Autenrieth,5,6 attention

Jan 1, 1968

By G. R. Countryman, G. H. Thomas, I. Fatt

Procedures are described for scaling up turbulent friction pressure drops of inelastic fluids and laminar friction pressure drops of inelastic/elastic fluids in circular ducts. The laminar scale-up calculation method does not require the use of arbitrarily defined flow models, and can be performed directly from laminar capillary viscometry data, or by means of a numerical solution of the integral equation for Poiseuille flow, inputing true shear rate-shearing stress data from other simple shearing experiments. e.g. Couette and plane Poiseuille flows. The turbulent scale up procedure, based on the Dodge-Metzner correlation for inelastic fluids, requires evaluating the characteristic rheological parameters at the existing wall stress under laminar flow conditions. Since the wall stress is the quantity sought in a pipe flow scale-up problem, a trial-and-err or solution is indicated, and therefore a calculation method has been developed and programmed for a medium-size magnetic-drum-memory digital computer. Laminar flow data from other somces may also be adapted to this turbulent scale-upmetbod for inelastic fluids. Comparisons between predicted and experimentally observed friction losses for generalized Reynolds numbers ranging from 3 — 32.000 indicate an error band width of less than 10 per cent in circular tubes ranging in size from 0.5 to 1.89 in. in diameter. INTRODUCTION Considerable effort has been devoted to the problem of establishing an accurate procedure for predicting the laminar and turbulent flow behavior of non-Newtonian materials in pipes from data obtained with small scale laboratory or pilot plant equipment, as evidenced by the variety of reviews on the subject.1-3 Although there does not appear to be a universally accepted scale-up method for laminar flow, the various methods currently employed can be divided into two principal categories depending upon whether a functional relationship between shear rate and shearing stress is assumed, or whether a generalized correlation is adopted. The methods of Hedstrom4 and of Weltmann are typical of the first category, while the generalized correlation of Metzner and Reed6 illustrates the second category. Under appropriate conditions either approach yields a reasonably satisfactory description of laminar flow behavior. In contrast, the prediction of non-Newtonian transitional and turbulent friction losses is much more complex. For example, certain types of materials have been observed to sustain laminar flow at much higher flow rates than Newtonian liquids. When turbulence does develop, the level of agitation is damped to such an extent that the friction factor is lower than that of the original low viscosity Newtonian medium. Extremely broad transition regions and "diameter effects" have also been noted in the turbulent flow of certain chemically complex materials.' Aside from the latter complications, which are attributed in part to viscoelasticity, the existence of a variety of scale-up procedures for predicting transitional and turbulent friction losses is due principally to the chioce of viscosity parameter used in the definition of the Reynolds number. In contrast, Dodge and Metzner have demonstrated that the friction factorReynolds number relationship for inelastic fluids can be satisfactorily generalized provided the fluid property parameters are evaluated at the existing wall shearing stress.* However, the quantity usually sought in a pipe flow scale-up problem is the wall stress, and hence a trial-and-error procedure is required when dealing with non-power law materials. In this report we discuss two calculation methods: (1) for determining the wall stress, and hence the friction pressure drop, for inelastic fluids under conditions of turbulent flow by means of an iterative procedure, and (2) for predicting laminar friction pressure drops without recourse to arbitrarily defined flow models.

By V. M. Dobrynin

Experimental data demonstrate that physical properties of porous rocks change under pressure. In this paper an assumption is made and proved that under pressure the changes of physical properties such as porosity, density, permeability, resistivity and velocity of elastic waves are controlled to a large extent by the pore compressibility of rocks. It is also shown that the pore compressibility of rocks can be determined, within the range of pressures from 0 to 20,000 psi, by knowing the maximum pore compressibility and the magnitude of the pressure. Mathematical equations were developed which permit one to define changes in physical properties of porous rocks under pressure. These equations were verified by experimental data obtained from the study of sandstones. INTRODUCTION In studying the behavior of porous rocks under pressure in the field of petroleum technology, the most interesting aspect is the observation of those properties which characterize the rocks as possible reservoirs — for example, porosity, permeability, resistivity, density and the velocity of elastic waves. The literature dealing with this problem mainly contains data concerning the study of only one or at most two of these parameters, but not of the group as a whole. An attempt is made in this paper to find general equations involving each of these parameters, which will permit the study of the behavior of rocks under pressure. All experimental data used here were obtained from the investigation of consolidated sandstones. EXPERIMENTAL In addition to the use of published experimental data, an experiment was carried out which studied the main physical properties of sandstones under pressure. Two homogeneous quartz sandstones were chosen for this purpose: (1) the Torpedo sandstone from Kansas, and (2) the Medina sandstone from Ohio. The porosity of the Torpedo sandstone was 20.2 per cent, and that of the Medina 8.7 per cent. Permeabilities were 45 md and less than 1 md, respectively. Each sandstone contained about 5 per cent clay minerals, consisting mostly of kaolinite and chlorite, which were distributed quite evenly throughout the samples. One cylindrical sample 2 in. in diameter and 5 in. in length was cut from each sandstone and then saturated in a vacuum with a 3N solution of NaC1. This high concentration was used in order to obtain true formation factors and to decrease the swelling of the clay minerals. The methods of mounting the samples and measuring the changes in porosity and resistivity were practically the same as those described by Fatt and Mann.1,2 Changes of resistivity under pressure were studied for sandstones with 100 per cent water saturation, and for sandstones with the irreducible water saturation. The irreducible saturation was obtained by enclosing the saturated rock samples in relatively fine silicate powder so as to remove the water by capillary action. This procedure is described by Orkin and Kuchinski.3 Changes of permeability with pressure were determined at room temperature using nitrogen as the flowing medium. In studying the effects of pressure, one series of measurements was made using an internal pore pressure Pi equal to the atmospheric pressure, while the overburden pressure Pe ranged from 0 to 20,000 psi. A second series of measurements was used over the same range of overburden pressure, but with an internal pore pressure of 1,800 psi. When the results were compared on the basis of net overburden pressure (Pe — 0.85 Pi ), there was practically no difference for these two sandstones. The origin of the factor 0.85 in the expression for net overburden pressure is given by Brandt,4 Fatt5 and Geertsma.6 The relatively large bulk volumes of the sample (about 15 cu in.) made it possible to observe a

By Louis A. Rosales, Kanji Ono

The anomalous behavior of hydrogen in iron is examined and a new model is proposed It is postulated that hydrogen single and di-interstitials are in equilibrium with each other. When the binding enthalpy of the di-interstitial, B, is assumed to be about 10 kcal per mole, the predicted temperature dependence of the solubility agrees reasonably well with the reported data. Effective diffusiuity is calculated assuming the di-interstitials to be immobile. Since the number of mobile single interstitials is greatly reduced at lower temperatures, the effective diffusivity is decreased. A good agreement in the temperature dependence of diffusivity between theory and experiment is provided with B - 10 kcal per mole. The predicted diffusivity is also a function of total concentration of hydrogen and this appears to be Partly responsible for the large scatter in the reported data. HYDROGEN is known to form an interstitial solid solution with iron.''' At elevated temperatures, it has been well-established that hydrogen enters iron in the dissociated form and occupies the interstices, and that diffusion of hydrogen in iron is via the interstitial mechanism. The concentration, c, permeability, P, and diffusivity, D, of hydrogen in iron have been separately determined and are consistent with each other under the steady-state condition for temperatures above approximately 400°C; that is, the relationship P = c-D holds. At lower temperatures, however, anomalous behavior of hydrogen in iron has been observed by many investigators. The results of experimental studies4-'' of the diffusion of hydrogen in iron are summarized in Fig. 1, in which the values of D are plotted against the reciprocal temperature. Although there is an increasing divergence in observed values of D with decreasing temperature, the values of D at temperatures below approximately 200°C are definitely small as compared with the extrapolation from high-temperature data. The lines shown bracket the region containing the most probable data. The observed values of c6," are plotted against the reciprocal temperature in Fig. 2, where the dashed lines represent Geller and sun's'' most probable isobars at 1 and 100 atm. At high external hydrogen pressure and low temperatures, the values of c observed are higher than those expected from high temperature data. It is generally accepted that certain trapping mechanism is responsible for the anomalies, but the nature of the trapping mechanism has not been clarified. Various types of hydrogen traps were suggested in the literature, such as "rifts" in the lattice dislocations, voids, microcracks, and so forth. However, no quantitative account of the observed behavior has been given in terms of these trapping mechanisms. Formation of methane in hydrogen-charged steel was found by ~odgurski." This appears to explain the existence of the so-called "residual" hydrogen,4 but not the observed anomalies of hydrogen in iron. McNabb and Foster' proposed a mathematical analysis in terms of a trapping mechanism. However, the physical significance of the traps was not clarified. The necessity of evaluating five parameters in their theory made comparison with experiment difficult. In the present paper, a model is proposed in which the anomalous behavior of hydrogen is attributed to the formation of hydrogen di-interstitials at lower temperatures. In this model, it is postulated that single and di-interstitials are in equilibrium with each other. (The hydrogen di-interstitial is to be distinguished from a hydrogen molecule which may exist in voids.) It can then be shown that most of the anomalies can be explained by assuming that the di-interstitials are virtually immobile and that the binding enthalpy of the di-interstitial, B, is approximately 10 kcal per mole. Predictions of the theory agree with the available experimental data reasonably well. THEORY The basic postulate of the present analysis is that single and di-interstitials are in equilibrium with each

Jan 1, 1969

By P. G. Winchell, J. K. Jackson

The strtcture of body-centered Fe-Ni-C alloys (0 to 16.5 wt pct Ni) containing less than 0.015 wt pct C was investigated by measuring the carbon-diffusion peak at low frequencies with a torsion pendulurrr. By determining the Snoek damping in ferrite and in quenched alloys of identical composition, the effects of nickel content and heat treatment were separated, and the existence of a structure in which Snoek damping is greatly reduced was revealed in the quenched ternary alloys. The relative amount of this structure increases with increased nickel content and cooling rate. It is proposed that this structure may be highly imperfect or tetragonal. In Fe-Ni-C alloys containing only ferrite the addition of nickel (0 to 5.1 wt pet) decreases tke area of the carbon-diffusion peak and shifts its position to lower temperatures but broadens the peak only slightljl. The decrease in peak area with increasing nickel implies that short-range interactions with the nickel atoms prevent some of the carbon atoms from participating in the relaxation process. The shift in the peak position indicates a long-range effect of nickel on the movement of carbon atoms so that at 300°K the diffisivity of carbon in -ferrite is increased by about 35 pct by the addition of 5.1 tut pct Ni. The results are presented as distributions of relaxation strength 072 the activation energy under the assumption that the frequency factor is invariant. BODY-CENTERED iron-base solid solutions dissolve carbon interstitially in the octahedral sites. The occupation of an octahedral site produces a tetragonal distortion of the surrounding lattice. These sites fall into three sets according to the orientation of the tegragonal distortion associated with each and may be identified by their lattice coordinates as: type 1/2 0 0, type 0 1/2 0, type 0 0 1/2. At low carbon levels in cubic ferrite the carbon atoms are distributed at random1 among these sites. In high-carbon body-centered iron-base solid solutions, which are unstable with respect to several two-phase mixtures, a tetragonal structure occurs as the product of the diffusionless transformation of austenite. This transformation is commonly accompanied by appreciable microstrain and its product is called martensite. The tetragonality of martensite, a supersaturated solution of carbon in a iron, is maintained prior to decomposition even though local diffusion of carbon occurs as evidenced by carbide precipitation. The tetragonality is taken as evidence that one set of octahedral sites is preferentially occupied by carbon. (This preferential occupation is called ordering below.) Support for this view is presented by the absence of strain-induced carbon-atom diffusion in high-carbon solid solutions (martensite). Such diffusion occurs in ferrite and produces the 40°C internal-friction peak observed at 1 cps frequencies.' Thus, there appears to be qualitative evidence favoring the view that at sufficiently high carbon levels tetragonality arises spontaneously as a result of the ordering of the carbon atoms; however, no experimental evidence exists concerning an equilibrium between cubic and tetragonal body-centered solid solutions. Moreover, attempts to estimate the energy of ordering7-' predict a value too low to maintain the tetragonality observed at medium carbon levels, 0.3 to 0.6 wt pct C. In the present work an attempt using internal-friction measurements is made to establish the presence or absence of tetragonality in low-carbon body-centered solid solutions formed by quenching austenite. Fe-Ni alloys containing about 0.01 wt pct and 0 to 15 wt pct Ni were selected for the study. While the work was in progress Gilbert and Owen10 reported that the transformation which occurs in these alloys during quenching at rates up to 5000°C per sec is not martensitic but is more properly categorized as massive. Thus, the solid solution

Jan 1, 1964

By F. Gordon Smith

ALTHOUGH several geological indicators of the critical type are known, including quartz inversions and decomposition of hydrous minerals such as serpentine, there are very few of the general type. Solid solutions are excepted, but the limitations of use are very restricted and interpretations are sometimes ambiguous. General methods for determining temperature and pressure conditions during the crystallization of minerals would have considerable scientific and economic value. It is not the purpose of this paper .to, discuss the various methods of geological thermometry and barometry, but to present one general method, applicable to all minerals, and to describe what progress has been made in the methods of measurement. The general method, in. brief, is a study of the stress conditions in and around various types of foreign inclusions which are trapped in minerals during growth. The method depends upon the fact that any homogeneous gas, liquid, or solid will in general have coefficients of thermal expansion and volume compressibility different from those of any given mineral. Therefore stress must develop in and around all types of inclusions in minerals if the temperature or the pressure, or both, are changed from the conditions which prevailed during the growth of the mineral. The methods of measurement consist of determinations of the temperature-pressure conditions of fit of the inclusions in the host mineral. The types of inclusions in minerals are: 1-gas, or liquid plus vapor, when observed at room temperature, due to crystallization under pneumatolytic conditions; 2-liquid, or liquid plus vapor when observed at room temperature, due to crystallization under hydrothermal conditions; 3-glassy solid, or devitrified glass, due to crystallization under magmatic or high temperature metamorphic conditions, in a siliceous liquid; and 4-crystals, due to overgrowth of other minerals crystallizing simultaneously or of other minerals which crystallized previously. A survey of the literature shows that much valuable earlier work on inclusions, especially that carried out in England in the last century, has dropped out of current knowledge. The following is a brief summary of the significant, contributions to the problem up to the present day. Davy in 1822 asserted that fluid inclusions in minerals consist of an aqueous solution of salts and a gas bubble, the whole being either at lower or higher pressure than atmospheric.1 At intervals from 1823 to 1862, Brewster contributed information concerning other types of inclusion consisting of 1-aqueous solution, a much more expansible liquid, and a gas; 2-aqueous solution, salt crystals, and a gas; and 3-the very expansible liquid and a gas. The very expansible liquid fills the gas space between 20° and 30°C. Compression strain effects were seen around inclusions in diamond, topaz, and other minerals.2-6 Sorby in 1858 and 1869 further advanced the study begun by Davy, stating that fluid inclusions represent a sample of the mother liquor of crystallization and that the degree of filling of aqueous inclusions at room temperature defines the temperature-pressure relations during formation of the host. The degree of filling may be measured by determining the minimum temperature of filling of the inclusion by the liquid phase. The very expansible liquid in some fluid inclusions is liquid. carbon dioxide. The temperature at which salt crystals in fluid inclusions completely dissolve in the fluid is the minimum temperature of formation. Inclusions of glass or devitrified glass indicate crystallization from a melt. Inclusions of crystals in minerals are often centers of strain, which may be seen by optical effects or by radial tension cracks. Sorby realized that an analysis of stress-strain relations about inclusions could be used to provide precise data on the temperature of crystallization, but the matter was never pursued.7,8 Hartley (1876, 1877),9,10 Hawes (1881),11 Wright (1881),12 Johnsen (1920),13 and Holden (1925)14

Jan 1, 1952

By C. L. Meyers, A. V. Levy, G. Y. Onoda, R. J. Kotfila

This paper presents the hypothesis that solid-solution hardening of regions in the order of tens of angstroms thick along grain boundaries is the most important mechanism controlling the ductile -brittle transition behavior of poly crystalline bcc refractory metals. Further, a physical model is developed and a general equation is derived to describe the dependence of the solute concentration at grain boundaries on temperature, nominal impurity concentration, thermal treatments, vain size, and prior plastic deformation. Analytical curves of the solute concentration at grain boundaries us temperature, calculated by means of the derived equation, are found to have the same general form as curves of ductility us temperature plotted from experimental data. Senziquantitative relationships between the transition temperature and metallurgical variables can he determined by using the physical model. The room-temperature brittleness of the poly-crystalline refractory metals (tungsten, molybdenum, and chromium) is a result of their high ductile-brittle transition temperatures. Recently studies on single crystals have revealed that, when sufficiently pure, these crystals have much lower transition temperatures than polycrystalline specimens of similar purity.'-3 For example, the transition temperature (measured by reduction in area in tensile tests) for very high-purity tungsten crystals has been reported to be as low as -50°F whereas polycrystalline specimens of similar purity had a transition temperature near 400" ~.' These results suggest that the ductile-brittle transition behavior of these polycrystalline metals can be attributed principally to the influence of grain boundaries on plastic flow and fracture. This paper contains a discussion of the manner in which grain boundaries influence the ductility of the poly-.crystalline bcc refractory metals and proposes that the segregation of impurities to grain boundaries is the most important mechanism for controlling the ductile-brittle transition behavior of these metals. Further, this paper contains an analysis of the effect of the compositional and structural variables on the grain boundary-ductility relationship, and introduces a semiquantitative model that can provide a foundation for more theoretical treatment to describe the influence of these variables on ductile-brittle transition behavior. PHYSICAL MODEL FOR PREDICTING DUCTILE-BRITTLE TRANSITION BEHAVIOR Grain Boundary Segregation of Impurities. Based on their own observations and on theoretical grounds, numerous investigators have suggested that impurities segregate to the grain boundaries. Such segregation would be expected because regions exist where both large and small foreign atoms fit with less strain than within the undistorted crystal structure. Therefore, the concentration of impurities should be higher at boundaries than within the grain body. TO estimate the degree (i.e., the ratio of grain boundary to nominal bulk concentration) to which an interstitial solute tends to segregate at boundaries within a solvent metal, the solubility rather than the size misfit could be used as a guide because solubility expresses the joint effect of both elastic and electronic factors.5 The solubilities of interstitial elements such as carbon, hydrogen, nitrogen, and oxygen are low in the group V-A metals and very low in the group VI-A metals.6 These low solubilities suggest that segregation to grain boundaries should occur for each of these solutes in both refractory metal groups. The difference between the ductility of single-crystal and polycrystalline bcc refractory metals can be described in terms of two distinct effects.4 The first of these is the general observation that grain boundaries hinder slip within grains and slip transfer between grains, and the second is that deformation is more complex in each grain of a polycrystalline specimen than it is in a single crystal. Mc Lean has pointed out, furthermore, that strong slip-hindrance effects at grain boundaries should occur in metals having strong segregation of impurities to grain boundaries. Therefore, the effect of grain boundaries on the hindrance of slip is expected to be the dominating influence on the ductile-brittle transition behavior of these polycrystalline refractory metals. cahn5 has suggested that temperature and solute mobility should influence solute equilibrium segregation at grain boundaries, and Fig. 1 illustrates

Jan 1, 1965

By P. J. Root, J. E. Warren

An idealized model has been developed for the purpose of studying the characteristic behavior of a permeable medium which contains regions which contribute significantly to the pore volume of the system but contribute negligibly to the flow capacity; e.g., a naturally fractured or vugular reservoir. Unsteady-state flow in this model reservoir has been investigated analytically. The pressure build-up performance has been examined in some detail; and, a technique for analyzing the build-up data to evaluate the desired parameters has been suggested. The use of this approach in the interpretation of field data has been discussed. As a result of this study, the following general conclusions can be drawn: 1. Two parameters are sufficient to characterize the deviation of the behavior of a medium with "double porositym from that of a homogeneously porous medium. 2. These parameters can be evaluated by the proper analysis of pressure build-up data obtained from adequately designed tests. 3. Since the build-up curve associated with this type of porous system is similar to that obtained from a stratified reservoir, an unambiguous interpretation is not possible without additional information. 4. Differencing methods which utilize pressure data from the final stages of a build-up test should be used with extreme caution. INTRODUCTION In order to plan a sound exploitation program or a successful secondary-recovery project, sufficient reliable information concerning the nature of the reservoir-fluid system must be available. Since it is evident that an adequate description of the reservoir rock is necessary if this condition is to be fulfilled, the present investigation was undertaken for the purpose of improving the fluid-flow characterization, based on normally available data, of a particular porous medium. DISCUSSION OF THE PROBLEM For many years it was widely assumed that, for the purpose of making engineering studies, two param- eters were sufficient to describe the single-phase flow properties of a producing formation, i.e., the absolute permeability and the effective porosity. It later became evident that the concept of directional permeability was of more than academic interest; consequently, the degree of permeability anisotropy and the orientation of the principal axes of permeability were accepted as basic parameters governing reservoir performance. 1,2 More recently, 3-6 it was recognized that at least one additional parameter was required to depict the behavior of a porous system containing regions which contributed significantly to the pore volume but contributed negligibly to the flow capacity. Microscopically, these regions could be "dead-end" or "storage" pores or, macroscopi-cally, they could be discrete volumes of low-permeability matrix rock combined with natural fissures in a reservoir. It is obvious that some provision for the inclusion of all the indicated parameters, as well as their spatial variations, must be made if a truly useful, conceptual model of a reservoir is to be developed. A dichotomy of the internal voids of reservoir rocks has been suggested.7,8 These two classes of porosity can be described as follows: a. Primary porosity is intergranular and controlled by deposition and lithification. It is highly interconnected and usually can be correlated with permeability since it is largely dependent on the geometry, size distribution and spatial distribution of the grains. The void systems of sands, sandstones and oolitic limestones are typical of this type. b. Secondary porosity is foramenular and is controlled by fracturing, jointing and/or solution in circulating water although it may be modified by infilling as a result of precipitation. It is not highly interconnected and usually cannot be correlated with permeability. Solution channels or vugular voids developed during weathering or burial in sedimentary basins are indigenous to carbonate rocks such as limestones or dolomites. Joints or fissures which occur in massive, extensive formations composed of shale, siltstone, schist, limestone or dolomite are generally vertical, and they are ascribed to tensional failure during mechanical deformation (the permeability associated with this type of void system is often anisotropic). Shrinkage cracks are the result

By T. L. Loucks, E. T. Guerrero

Pressure drop characteristics in a system composed of two adjacent concentric regions of different permeability were studied. The differential equations for continuity of mass flow in the two regions were solved using the Laplace transformation and the necessary boundary conditions to give the pressure distribution in the composite reservoir, the resulting equation for pressure drop at the inner boundary was evaluated for a variety of composite reservoirs and compared with the results for a uniform reservoir. From this study it was found that under certain conditions the permeability in both zones, as well as the size of the inner zone, can be determined from the pressure drop curve. INTRODUCTION The theory for the pressure distribution and pressure build-up behavior of a well producing a single, slightly compressible fluid from infinite and finite homogeneous reservoirs was presented by Hornerl and Miller, Dyes and Hutchinson.2 Extensions on this original work to provide improved and extended interpretations and better agreement between theory and observed results have been made by Matthews, et a1,3 van Everdingen,4 Gladfelter, et al5 Stegemeierand Matthews,6 Hurst and Guerrero,7 and Perrine.8 More recently Lefkovits, et al,9 studied pressure build-up behavior in bounded reservoirs composed of stratified layers. Houpeurtl0 has suggested various approaches to the general problem of variable permeability and porosity but presented no analytic solutions for particular permeability variations. Albert, Jaisson and Marionll studied the finite composite reservoir and presented numerical solutions to the unsteady-state case and an analytical solution valid only for large times. They also studied the so-called pseudo-steady state for several examples of radial permeability variations. Very similar examples have been treated in the unsteady state with application to pressure build-up by Loucks in an unpublished manuscript. More recently Hurst12 has presented the complete point-sink solution (valid for all times) for the infinite composite reservoir. He applied these solutions to interference between oil fields along with an even more elegant application of his explicit solution to the material-balance equation including water influx. Mortada 13 approaches the same application by avoiding the point-source limitation but gives the solution for the aquifer region which is valid only for large times. Hopkinson, Natanson and Temple14 have treated both the finite and infinite composite reservoir obtaining the pressure distribution for the inner zone valid for large times. This paper presents a theoretical study of the pressure distribution in an infinite composite reservoir composed of two adjacent concentric regions of different permeability. The object was to determine the manner in which pressure drop at the inner boundary of a composite reservoir depends upon time, the permeability of each zone and the size of the inner zone. Expressions for the pressure distribution in both zones are developed which take into account the radius of the sink and are valid for small times as well as large times. It was felt that an understanding of the pressure drop behavior in various composite reservoirs would be of assistance in the interpretation of some pressure build-up curves which do not behave according to the theory derived for uniform systems. Often the region surrounding the wellbore is either more permeable or less permeable than the reservoir because of the various drilling and completion practices. The effects of reduced permeability due to drilling-fluid invasion and of increased permeability due to fracturing or acidizing need to be more carefully defined. Therefore, an equation for the pressure drop in a composite reservoir was developed, and the effects of both the permeability in each zone and the size of the inner zone were studied. STATEMENT AND SOLUTION OF PROBLEM BOUNDARY CONDITIONS AND ASSUMPTIONS The physical system studied was a composite

By L. E. Snow, L. F. Pett

AT the Utah Copper Pit of Kennecott Copper Corp. a versatile mobile drill unit has been developed and tested. Through increased drilling speed and additional available drilling time, an improvement of 233 pct in drill performance has been realized. Operators using drill power can quickly and easily convert the unit from a vertical to a toe hole drill. Drilling is done with the machine in line with the unit or at right angles to it. Table I covers the 1950 drill record at the mine, showing the types of machines used and the footage obtained. Development of the new units was prompted by a desire for solution of the following problems: 1—The ever-expanding operations within the Utah Pit, requiring some 65 miles of transmission pipe lines radiating from a central compressed air plant, have now taxed this compressor plant beyond capacity. While 100 lb per sq in. pressure is maintained at compressors, transmission losses from innumerable small leaks result in low terminal pressure at the drills. Fig. 1 illustrates that while terminal pressure varies according to size of air hose and fittings used, the average drill is operating on air pressures within the 60 to 70-lb range. The compressor plant is operated primarily for air rock drills. All drilling is done on day shift, but it is necessary to run two of the three 3720 cu ft capacity compressors during night shift in order to prevent drastic loss of pressure through leaks. On day shift all three compressor units are operated at capacity, so that there is no reserve safeguard available for extensive repairs to any compressor. The calculated volume of compressed air required for operation of 25 drills is but 33 pct of three day-shift compressors. While compressed air is being used to a minor extent for purposes other than drilling, it is felt that satisfactory substitute facilities could be pro- vided in the event that air lines were no longer available. 2—Drill performance has increased beyond the capacity of present 3-in. drifter machines. 3—The installation, maintenance, and moving of air lines in routine mining is a growing problem. The 2-in. level lines are placed outside electrification towers and must be moved to new location upon completion of each shovel cut. In addition to these moves it is often necessary to dismantle level lines so that excessive shovel spillage, or material covering track following a blast, can be pushed in the clear with bulldozers. Where level room is limited, it is often necessary to remove air lines in giving bulldozers room to prepare grades for wagon drills. In spite of precautions taken, many level lines become buried, twisted, and constricted, with excessive air loss resulting. Excessive back break in blasting often leaves several sections of air line suspended in mid-air. Occasionally air lines come in contact with defective electric transmission towers, so that holes are burned through pipe. 4—Rail-handling of drilling equipment such as wagon drills from one location to another requires considerable locomotive crane crew time. Any new unit developed should travel under its own power. 5—Present practice requires accessory material including hose, water pressure tanks, water barrels, drill steel, and bits to be transported by means of push cars when drill locations are changed on any level. 6—In order to supply numerous drill locations with material, many stock piles of new drill rods, drill bits, and other essentials must be left at strategic points, usually in the vicinity of powder magazines. This practice makes for poor housekeeping and contributes to the loss of supplies. Planning the Unit It was felt that a mobile drill unit which would rectify the foregoing problems should meet certain demands: 1—It must furnish compressed air to operate a 4-in. rock drill at a pressure of 90 to 100 lb.

Jan 1, 1953

By L. E. Snow, L. F. Pett

AT the Utah Copper Pit of Kennecott Copper Corp. a versatile mobile drill unit has been developed and tested. Through increased drilling speed and additional available drilling time, an improvement of 233 pct in drill performance has been realized. Operators using drill power can quickly and easily convert the unit from a vertical to a toe hole drill. Drilling is done with the machine in line with the unit or at right angles to it. Table I covers the 1950 drill record at the mine, showing the types of machines used and the footage obtained. Development of the new units was prompted by a desire for solution of the following problems: 1—The ever-expanding operations within the Utah Pit, requiring some 65 miles of transmission pipe lines radiating from a central compressed air plant, have now taxed this compressor plant beyond capacity. While 100 lb per sq in. pressure is maintained at compressors, transmission losses from innumerable small leaks result in low terminal pressure at the drills. Fig. 1 illustrates that while terminal pressure varies according to size of air hose and fittings used, the average drill is operating on air pressures within the 60 to 70-lb range. The compressor plant is operated primarily for air rock drills. All drilling is done on day shift, but it is necessary to run two of the three 3720 cu ft capacity compressors during night shift in order to prevent drastic loss of pressure through leaks. On day shift all three compressor units are operated at capacity, so that there is no reserve safeguard available for extensive repairs to any compressor. The calculated volume of compressed air required for operation of 25 drills is but 33 pct of three day-shift compressors. While compressed air is being used to a minor extent for purposes other than drilling, it is felt that satisfactory substitute facilities could be pro- vided in the event that air lines were no longer available. 2—Drill performance has increased beyond the capacity of present 3-in. drifter machines. 3—The installation, maintenance, and moving of air lines in routine mining is a growing problem. The 2-in. level lines are placed outside electrification towers and must be moved to new location upon completion of each shovel cut. In addition to these moves it is often necessary to dismantle level lines so that excessive shovel spillage, or material covering track following a blast, can be pushed in the clear with bulldozers. Where level room is limited, it is often necessary to remove air lines in giving bulldozers room to prepare grades for wagon drills. In spite of precautions taken, many level lines become buried, twisted, and constricted, with excessive air loss resulting. Excessive back break in blasting often leaves several sections of air line suspended in mid-air. Occasionally air lines come in contact with defective electric transmission towers, so that holes are burned through pipe. 4—Rail-handling of drilling equipment such as wagon drills from one location to another requires considerable locomotive crane crew time. Any new unit developed should travel under its own power. 5—Present practice requires accessory material including hose, water pressure tanks, water barrels, drill steel, and bits to be transported by means of push cars when drill locations are changed on any level. 6—In order to supply numerous drill locations with material, many stock piles of new drill rods, drill bits, and other essentials must be left at strategic points, usually in the vicinity of powder magazines. This practice makes for poor housekeeping and contributes to the loss of supplies. Planning the Unit It was felt that a mobile drill unit which would rectify the foregoing problems should meet certain demands: 1—It must furnish compressed air to operate a 4-in. rock drill at a pressure of 90 to 100 lb.

Jan 1, 1953

By C. Wagner

When an alloy solidifies and the composition of the solid differs from the composition of the liquid, atoms of the alloying elements rejected at the solid-liquid interface have to diffuse toward the bulk liquid. Diffusion may be supported by convection. Theoretical calculations have been made for a liquid without convection, for a liquid involving natural convection, and for solidification of a liquid alloy at the surface of a rotating disk as an example of forced convection. WHEN an alloy solidifies, the equilibrium composition of the originating solid phase differs in general from the composition of the liquid phase. Thus, segregation may take place, i.e., the composition of the solid phase formed at different times differs from the bulk composition. The degree of segregation depends on several factors such as the partition ratio of the alloying elements between the liquid and the solid phase, the rate of solidification, and convection. To clarify, a theoretical analysis has been worked out. The following presuppositions are made: 1—Calculations are confined to the solidification of binary alloys in which the concentration of component A (solvent) is much greater than that of component B (solute). 2—The equilibrium concentration of the alloying element B is assumed to be lower in the solid phase than in the coexisting liquid phase. 3—At the solid-liquid interface, partition equilibrium is assumed. 4—The dependence of the partition ratio of the solute between the liquid and the solid phase on concentration and temperature is disregarded since only small concentrations of the solute are considered. 5—The concentration of the alloying element B in the solid phase is supposed to be lower than the saturation concentration for the formation of a second solid phase. 6—diffusion in the solid phase is disregarded, since diffusion rates in the solid state are comparatively low. When the partition ratio of the solute between the liquid and the solid phase is greater than unity, a part of the solute is rejected at the solid-liquid inter- face and has to diffuse toward the bulk liquid. Thus there is necessarily a concentration difference in the liquid between the solid-liquid interface and the bulk liquid as has been pointed out by Hayes and Chipman,' Rutter and Chalmers,' Tiller, Jackson, Rutter, and Chalmers, and others (see Fig. 1). In view of the concentration gradient in the liquid phase, the ratio of the concentration of the solute in the solid phase to that in the bulk liquid differs from the equilibrium ratio, although at the solid-liquid interface, partition equilibrium is assumed. Under certain limiting conditions, the solid may even have virtually the composition of the bulk liquid as is shown below. General Equations Let u be the linear crystallization velocity in cm per sec, D the diffusion coefficient of component B in the liquid phase in cm2 sec-', c',, and c", the concentrations of component B in the liquid and the solid phase, respectively, in mol per cm", and s the distance from the solid-liquid interface toward the bulk liquid at a given time t. The partition ratio is assumed to be equal to a partition constant K. Thus. The difference in the specific volume between the liquid and the solid phase may be disregarded. In view of the discontinuity of the concentration of the solute at the interface according to Eq. 1, u(c', — c",), , , = tic', (S = 0)(1 — K) gram-atoms B per unit area per unit time are rejected at the interface and have to diffuse toward the bulk liquid. From Fick's first law it follows that uc',,(s = 0) (1 ~K) == - D(dc',,/?>s). _-„. [2] The concentration profile in the liquid near the interface depends on the interplay of diffusion and convection as is discussed in the following sections. A schematic concentration profile is shown in Fig. 1. The "effective thickness of the diffusion boundary

Jan 1, 1955