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By N. J. Olson, G. W. Toop

A series of equations previously derived for calculating ternary thermodynamic properties using binary data has been applied to the problem of predicting ternary phase diagrams and quaternary excess free energy. The methods are considered to be rigorous for regular ternary and quaternary systerns and empirical for nonregular systems. The equations have been used to predict ternary phase boundaries in the Pb-Sn-Zn system at 926°K and the Ag-Pd-Cu system at 1000ºK. Calculated quaternary excess free-energy values are presented for the Pb-Sn-Cd-Bi system at 773°K. A method for predicting the location of ternary phase boundaries would be a useful supplement to experimental measurements in ternary systems. This has been recognized with the considerable work that has been done to find models to predict or extend thermodynamic properties and phase diagrams in binary and ternary systems1-18 for which direct experimental measurements are limited. With the access to highspeed digital computers and mechanical plotting devices, it is currently rather easy to compare mathematical models with experimental data. The regular-solution model is consistent with systems which exhibit negative heats of mixing, positive heats of mixing, and miscibility gaps, and therefore it is applicable to simple phase diagrams. The purpose of this paper is to illustrate the use of regular-solution equations to predict, empirically, phase equilibria in some types of nonregular ternary systems. Corresponding equations for regular quaternary systems are given and used to calculate empirical quaternary excess free-energy data. METHOD FOR PREDICTING THE LOCATION OF TERNARY PHASE BOUNDARIES USING BINARY DATA Meijerin1,6 has used the regular-solution model to calculate common tangent points to ternary free energy of mixing surfaces and hence to determine phase boundaries in ternary systems involving miscibility gaps. He used the following equation to calculate ternary excess free energy of mixing values: stants characteristic of the binary solutions, and Ni is the mole fraction of component i. An alternate expression which gives for regular solutions as a function of binary values of along composition paths with constant N1/N2, N2lN3, and N1/N3 may also be derived:15 ternary r xs 1 ?c-*n.Ti*.U*. This expression for is more useful for the empirical calculation of ternary excess free-energy values for nonregular systems because actual binary AFXS data may be used in the expression rather than attempting to find suitable constants for Eq. [I]. The results of this feature of Eq. [2] are illustrated in Table I where calculated excess free-energy values for the Ni-Mn-Fe system at 1232°K are compared with experimental data of Smith, Paxton, and McCabe.19 Although regular-solution equations have been shown to give calculated thermodynamic quantities which agree quite well with experiment for single-phase nonregular ternary systems,14,15 care should be exercised in the use of the equations to predict thermo-dynamic properties of multiphase ternary systems in which strong compound formation is suspected. This precaution is consistent with the simple regular-solution model which for negative values of ai_j will indicate a tendency toward compound formation but even for very large negative values of ai-jwill not give multiphase binary or ternary systems involving a distinct stable compound. Hence, calculated ternary free-energy data using Eq. [2] might be expected to vary between being rigorous and poor, in the following order, for ternary systems which are: a) regular solutions, b) nonregular single-phase liquids in which random mixing is nearly realized, c) nonregular single-phase solids, d) nonregular multiphase systems exhibiting miscibility gaps, e) nonregular multiphase systems with binary compounds but no ternary compounds, f) nonregular multiphase systems with highly stable binary and ternary compounds. The calculated data will be expected to be least accurate for the last two cases. The general method adopted in this paper involves two-dimensional plots of ternary activity curves. The principle used is that tie lines indicating two-phase equilibria join conjugate phases a and B for example, for which a1(a) = a1(B), a2(a) = a2(B), and a3(a) = a3(B). Tie lines may be determined by plotting the ternary activities of two components along an isoactivity line for the third component and the unique points where the above equalities hold may be found graphically.

Jan 1, 1970

By W. B. Dancy, A. Adams

Laboratory test work on heavy liquid separation of sylvite from halite is reported. Numerous tests were run on sylvite ore sized in the ranges of 4x20 mesh, 10x65 mesh, 8x100 mesh, -8 mesh and -10 mesh with heavy liquids in the range of 2.05 to 2.15 sp gr. From the test results, it was concluded that, with the type of ore under study and a size in the range of -8 mesh, a recovery as high as 90% could be achieved with a product grade of 70% KCl. However, a final product at an acceptable recovery cannot be made with one pass, and the float must either be further processed with heavy liquids or dried and sent to a conventional froth flotation circuit. Potash ores occurring in this country consist essentially of sylvite and halite plus minor amounts of magnesium sulfate salts and montmoril-lonite-type clays. Recovery of potash minerals from evaporite ores in the North American potash fields is accomplished almost exclusively by use of amine flotation. European practice involves froth flotation as well as solution-crystallization processes. Laboratory and pilot plant test work has been reported in Europe and the U. S. on the application of heavy media separation to potash ore beneficiation. Work was probably discontinued because of lack of ore with the required very coarse liberation characteristics (1/8 to 1/2 in. liberation size). Sylvite, with a gravity of 1.99, and halite, with a gravity of 2.17, appear to be ideal for separation by heavy liquids, which are now available in gravities from 1.59 to 2.95. This paper reviews preliminary results obtained from laboratory test work on heavy liquid separation of sylvite from halite. TEST WORK The heavy liquids used in the tests under discussion were chlorobromethane, with a specific gravity of 1.923, and dibromethane, with a gravity of 2.490. These liquids, completely miscible, were combined in the proportions needed to give a mixture having the desired specific gravity. Feed for the laboratory tests was mine-run ore screened to the desired mesh sizes. In conducting the tests, the sample was fed at a constant rate into a stream of heavy liquid and the mixture directed into a small separatory vessel. The float overflowed into a collecting pan while the sink collected in the bottom of the separatory vessel and was removed at the end of the test. Approximately 500 g of feed constituted a charge. Pulp density of the feed was kept low to prevent particle to particle interference in separation. With feed in the range of 8x100 mesh, a pulp density of under 10% solids by weight was found advisable. With coarser feed the pulp density could be carried as high as 15% solids. Time of separation was very rapid. In the case of 4x20-mesh material, separation was effected in 15 to 30 sec; with -10-mesh feed, separation required about 1 to 2 min. SPECIAL EQUIPMENT Since heavy liquids are toxic to varying degrees, all separatory work was carried out in a standard laboratory fume hood. It was noted that complete removal of fumes was not being effected; therefore the hood construction was modified, resulting in a completely satisfactory arrangement for heavy liquid test work. In the interest of safety, details of this fume hood are reported here. Unlike most fumes, heavy liquid fumes tend to settle and flow like water, rather than to rise like a gas. Working on this assumption, a standard water drain was installed in the hood. Across the front of the hood a 1-in. barrier was constructed. In the rear of the hood a false back was installed, with an adjustable sliding door on both the bottom and top of this panel. As shown in Fig. 1, the exhaust fan pulled a vacuum behind the barrier, sucking the heavy fumes from the bottom of the hood. Another addition was the drying box, shown to the right of the hood. This is simply a box covered on top with hardware cloth and connected by a 6-in. inlet to the hood. Sample trays made of fine mesh wire filter screens were found ideal for drying samples. With this arrangement, air flowed completely through the sample and all fumes were drawn into the hood. In use, it was found effective to cover with a

Jan 1, 1963

By Elwood G. Haney, R. N. Parkins

Stress corrosion cracking of two heats of 18 pct Ni maraging steel in rod form immersed in an aqueous solution of 0.6N NaCl at pH 2.2 has been studied on un-notched specimens stressed in a hard tensilf machite. Austenitizing temperature in the range 1830 to 1400 F has been shown to have a marked influence on the propensity to crack, the loulest austenitizing- temperature producing the greatest resistance to failure. In the nzosl susceptible conditions, the cracks followed the original austenile grain boundaries; but when tlze steels zcere heal treated to inproze their resistance to stress corrosion, the cracks becatne appreciably less branched and slzouqed significant tendencies to become trans granular. Electron metallography of the steels indicated the presence of snzall particles, possibly of titanium carbide, along- the prior austenite grain boundaries and these particles u:ere more readily detectable in the structures that were most susceptible to cracking. Crack propagation rates, which appeared to be dependent upon applied stress and structure, were usually in tlze reg-ion of 0.5 mm per hr and may, therefore, be e.xplained on tlze basis of a purely electrochetnical ,nechanism. However, there is some ezliderzce from fractography that crack extension may be assisted by ttlechanical processes. Anodic stit)zulation reduced the tiwe to fracture, although cathodic currents of small magnitudes delayed cracking-; further increase in cathodic current resulted in a sharp drop i,n fracture litne, possibly due to the onset of hydrogen ewbrittlement. THE use of the high strength maraging steels, with their attractive fracture toughness characteristics, is restricted because of their susceptibility to stress corrosion cracking in chloride solutions. Although this limitation has resulted in investigations of the stress corrosion susceptibilities of these steels, there have been few systematic studies aimed at defining the various parameters that determine the level of susceptibility. It is the case that the usual tests have been performed with the object of defining some stress or time limit, on unnotched or precracked specimens, within which failure was not observed,' but while such results may be of some use in design considerations, they are necessarily concerned only with the steels as they currently exist and not with their improvement to render them more resistant to stress corrosion failure. This omission may be considered unfortunate because the indications are that stress corrosion in maraging steels shows dependence on structure in following an intergranular path, and since experience with other systems of intergranular stress corrosion crack- ing is that susceptibility may be varied by modifying heat treatments, a similar effect may be expected with maraging steels. It is sometimes from such observations that a fuller understanding of the mechanism of stress corrosion crack propagation begins to emerge, leading in time to the development of more resistant grades of material. The present work was undertaken to study only one aspect of the influence of heat treatment upon the cracking propensities of the 18 pct Ni maraging steel, namely the effect of austenitizing temperature, although certain ancillary measurements and experiments have been undertaken. EXPERIMENTAL TECHNIQUES Most of the measurements were made on a steel, A, having the analysis shown below, although a few results were obtained on a steel, B, having a slightly different composition. Both steels were supplied in the austenitized condition, A as 3/8-in-diam rod and B as 1/2-in.-diam rod. Cylindrical tensile test pieces were machined from the rods: the overal length was 2 1/2 in., the gage length 1 in. and the diameter 0.128 to 0.136 in. The stress corrosion tests were carried out with the specimens strained in tension in a hard beam testing machine, the necessary total strain being applied to the specimen over a period of about 30 sec, after which the moving crosshead was locked in position and the load allowed to relax as crack propagation proceeded; the load relaxation was recorded. The load was applied after the specimen had been brought into contact with the corrosive solution, the latter being contained in a polyethylene dish having a central hole through which the specimen passed, leakage being prevented by the application of a film of rubber cement. The specimen was in contact with the solution for over half of its gage length and the solution was exposed to the air during testing. The solution was prepared from distilled and deionized water to which NaCl was added, 0.6N, and the pH adjusted to 2.2 by HCl additions. The composition of the solution

Jan 1, 1969

By S. D. Baumer, P. M. Hulme

IN the late thirties, the National Carbide Co. cooperated with C. E. Wood, of the U. S. Bureau of Mines, in his investigation of the relative merits of various desulphurizers, including soda ash, caustic soda, and calcium carbide. Laboratory tests showed that carbide, when it could be made to react, is an excellent desulphurizing agent for molten iron. Sulphur content can be driven to lower levels and higher extractions obtained with carbide than with actionsany of the more common reagents. Wood's results1 are shown in Table I. Unfortunately, as the Handbook of Cupola Operation puts it, the chemical fact that carbide is a good desulphurizer was of only academic interest because it was found to be extremely difficult to devise a practical means to make it react with molten iron. Calcium carbide is formed in the electric furnace at 4000°F and above, and its softening point is probably at least 500 °F above the usual working temperatures encountered in iron and steel practice. Consequently, carbide does not form a true slag but floats as a dry powder on top of the metal and only a very small portion of it ever comes in actual contact with the iron. Stirring with a rabble, or pouring the metal over the carbide, increases the efficiency only slightly. Extractions of 20 to 30 pct can be obtained in this manner, but conventional soda slag treatment can do better than this and do it more cheaply. All attempts to lower the melting point of carbide in order to obtain a reactive, liquid slag have so far proved fruitless. Directly under the arc in a metallurgical electric furnace, carbide becomes highly reactive. Excellent sulphur removal can be obtained without any slag other than a thin layer of carbide." imilarly, good results are obtained by adding small amounts of carbide to the finishing slag in double-slag arc furnace practice. To react a liquid with a solid, it is axiomatic that the liquid has to wet the solid before anything can happen. If the solid is heavier than the liquid, the problem is easy, but it becomes more difficult when the solid is much lighter than the liquid, as in the case of carbide and liquid iron. Wood recognized this problem and solved it in a unique fashion. The results shown in Table I were obtained by spinning the carbide beneath the surface of the molten iron by means of a refractory centrifuge. This technique allowed each particle of the finely divided carbide to come into intimate contact with the metal and to be wetted thereby. Wood's centrifuge technique was successful in the laboratory where it achieved excellent and consistent results. Some attempts were made to expand this method to commercial practice, but serious difficulty was encountered in obtaining a refractory centrifuge head that would be economically feasible. About this time the war intervened and the project lay dormant for several years. In 1944, it was revived. It was suggested that the carbide could be blown into the metal with a carrier gas in an attempt to eliminate the necessity for the expensive and brittle centrifuge. The idea was first tried out in a fairly large ladle of iron using natural gas as the carrier. Considerable sulphur was removed, but it was quite obvious that the use of natural gas was not practical. Attempts then were made to blow carbide into molten iron using, in turn, nitrogen, argon, carbon dioxide, air, and oxygen. The latter two gases proved unsatisfactory. Calcium evidently prefers oxygen to sulphur because in the tests calcium oxide and carbon dioxide were produced, the sulphur still being untouched in the iron. Nitrogen, argon, and carbon dioxide gave much better results, although the efficiencies and extractions were erratic, and only a few isolated tests approached the results obtained by Wood. Table II shows typical results obtained with these gases. The sulphur removals were interesting, sometimes even encouraging, but it is evident that such erratic behavior could not be tolerated in commercial practice. A number of different types of equipment, such as sand blasting machines, refractory guns, and the like can used to blow the solid into the metal. All types required relatively large quantities of gas in order to maintain the flow of solid carbide through the system and into the metal. It was observed that the bubbles of gas breaking through the surface of the metal contained quantities of unreacted carbide. The liquid metal never came in contact with these particles and if it cannot wet them it cannot react with them. The initial work had shown that carbide had great possibilities as a desulphurizer. In practice

Jan 1, 1952

By A. B. Cook

Gas cycling is generally considered a much less efficient oil recovery mechanism than water flooding. HOWever, recoveries from some fields have been exceptionally high as a result of gas cycling. Recovery from the Pick-ton field, for example, was calculated to be 73.5 perceni of the stock-tank oil originally in place. In evaluating pressure maintenance projects, determining how much of the recovery is due to displacement by gas and determining how much is due to vaporization of the imrnohile oil in the flow path of the cycled gas is very difficrilt. Even though most of the oil is recovered by displacetr~ent, the success of a project may depend on the amount of oil vaporized. A limited number of experiments have heen performed with a rotating model oil reservoir that simulates gas cycling operations and allows a separation of the oil from, tile free gas flowing into the laboratory wellbore at reservoir conditions, thus revealing which is displaced oil and which is vaporized oil. It Iras been determined that the amount of varporizatio'n is .significant if proper conditions exist These experiments show that oil vaporization depends on pressure, temperature, volatility of the oil and amount of gas cycled. Increases in each of these conditions increase the volume of oil vaporized. Data from six experiments affecting vaporization are presented to illustrate reservoir condition that range from favorable to unfavorable. 111 these eaperitnenis recovery by vaporization ranged from 73.6 to 15.3 percent of /he immobile oil (oil not produced by gas displacerrlt). INTRODUCTION Between 1930 and 1950, gas cycling was a popular. oil recovery practice. especially for the deeper reservoirs. Later, with many case history-type studies published for both gas cycling and waterflooding, it was generally believed that waterflooding was far superior to gas cycling, even when gas cycling was conducted as a primary production procedure by complete pressure maintenance. A good example illustrating the advantage of water-flooding over gas cycling is given in a paper by Matthews' on the South Burbank unit where gas injection was followed by waterflooding. The author concluded in part that "Early application of water injection, without the intervening period of gas injection, would have recovered as much total oil as ultimately will be recovered by waterflooding following the gas injection, and total operating life would have been shortened". This appears to be a logical conclusion. However, it should not be applied to all fields. Pressure maintenance with gas in the Pickton field, as reported by McGraw and Lohec;' will result in a much larger percentage of oil recovery than was obtained in the South Burbank unit. The great success in the Pickton field resulted partly from vaporization of the immobile oil in the flow path of the cycled gas. The amount of vaporization is related to the following conditions: volatility of the oil as reflected by the APT gravity of the stock-tank oil; reservoir temperature; reservoir pressure during gas cycling; and the amount of gas cycled. Therefore, the U. S. Bureau of Mines is investigating these effects on vaporization in a research project using a model oil reservoir. Three different stock-tank oils having 22, 35 and 45" API gravities are being used as base stock to synthesize reservoir oils. Experiments are being performcd to determine vaporization at 100, 175 and 250F and at 1,100, 2,600 and 4,100 psia. This is a progress report showing the results from six experiments. Other Bureau of Mines reports"- concerning vaporization are listed. LABORATORY EQUIPMENT AND PROCEDURES The equipment ' consists of an internally chromium-plated steel tube packed with finely sifted Wilcox sand. The tube is approximately 44 in. long and has an ID of 13/4 in. The sand section contains approximately 570 ml of voids, has a porosity of 32 percent, and a permeability to air of 4.3 darcies. A unique feature of the laboratory reservoir (Fig. 1) permits the tube part to rotate at 1 rpm while the outlet and inlet heads are held stationary. The outlet end contains diametrically opposed windows to permit observatlon of the flowing fluids, and two valves, one on the top and the other at the bottom. Oil and free gas. when being produced simultaneously, can be separated by manipulating the two valves to keep a gas-oil interface in view through the windows. Thus, only gas is produced through the top valve and only oil flows through the bottom valve. The laboratory equipment was designed to study vaporization. Therefore, a uniform reservoir was made using dry sifted sand as opposcd to using a consolidated sand core with interstitial water. Furthermore. the reservoir was tilted to minimize fingering of gas. This tilting also in-

By K. A. Slagle

Hydraulic analysis of the wellbore has become increasingly inzportant for designing cementing operations and selecting equipment, materials and techniques to complenzent modern well-c-ompletion practices. Non-Newtonian fluid technology has advanced beyond the point where former empirical methods of analysis adequately define the hydraulic system and fluid properties. In view of these factors, this paper describes a series of rheological calculations which have been found practical, through field usage, for assistance in selecting a cementing program. A relatively simple laboratory method using standard viscometric equipment is suggested for determination of the rheological properties of slurries, and clrrta are presented on some of the more common cementitrg conzposition.A. A criterion for divergence from laminar-flow characteristics has been proposed. Usefulness of the calculations is indicated by examples of cementing operations where they have been used. INTRODUCTION With the changing aspects of well-completion practices during the past few years, it has been increasingly important to have a relatively simple method of analyzing the flow conditions existing in the well during cementing operations. This is particularly true in view of the improved economics toward which most of the changes have been directed. Rheological characteristics of slurries used for cementing should be a major consideration in the trend toward smaller casing sizes, either single or multiple strings. Receiving increased attention is the practice advocated in 1948 by Howard and Clark' of attaining turbulent flow with the fluids circulated during a primary cementing operation. While there may still be a difference of opinion concerning this technique, most available information indicates that superior primary-cementing results are generally obtained when high displacement rates are employed. Fluid properties of the slurry to be used must be available, as well as calculation methods, to determine what flow rates should be attained and the probable consequences in terms of frictional pressure and horsepower utilization. It would certainly be inappropriate to attempt high displacement velocities if sufficient pressure might be developed to create lost circulation. Since cementing slurries are non-Newtonian fluids, it is not possible to define their rheological or fluid properties by the single factor of viscosity and then make calculations for the quantities just described. Because the shear stress-shear rate ratio is not constant: it becomes necessary to establish at least two parameters for adequate fluid-flow calculations. It is not the purpose of this paper to delve into the mathematical development of non-Newtonian technology, nor to discuss the arbitrary classification system under which a single fluid may resemble two or three different classes depending upon experimental conditions. Rather, the intention is to present a useful series of calculations based on a concept applicable to both Newtonian fluids and to the preponderance of non-Newtonian fluids encountered in the oil-producing industry. Development of this approach was begun some 32 years ago,' and has most recently been brought to fruition by Metzner and his co-workers at the U. of Deleware. Some non-Newtonian fluids encountered in the petroleum industry, other than cementing slurries, have also had the benefit of this method of analysis."' The two parameters required to define the fluid are usually denoted by the symbols n' and K' and, for the purposes of this discussion, are called "flow behavior index" and "consistency index", respectively. These two slurry properties permit calculation of the Reynolds' number and the "critical" velocity, or the velocity at which departure from laminar flow begins. EXPERIMENTAL DETERMINATIONS The two principal instruments used for rheological studies are the pipeline (capillary-tube) viscometer and the rotational viscometer. When conveniently possible, a capillary-tube viscometer (where the pressure drop and flow rate of the material can be measured) is the better method for rigorous determination of the flow behavior index and consistency index for non-Newtonian fluids. With pressure-drop data at various flow rates, it is then possible to prepare a logarithmic plot of shear rate as the abscissa-shear stress as the ordinate. For fluids which do not exhibit time-dependency, these data will usually produce a straight line. The flow behavior index n' represents the slope of this line, while the consistency index K' becomes the intercept of this line at unity shear rate in accordance with the mathematical derivations associated with this concept of rheology. Due to the difficulties anticipated in maintaining a uniform, pumpable cement slurry for the time interval required to obtain measurements from the pipe viscometer, the n' and K' data reported herein were obtained using a direct-indicating rotational viscometer (Fig. 2). The

By A. H. Heim

A method and apparatus for measuring gas porosities of rocks are described. The apparatus can be assembled from commercially available components. In principle, measurements are made by volume substitution at constant pressure. The maximum error is not more than 0.3 porosity per cent. Typical results are given. INTRODUCTION Determining the porosity of rock samples is one of the most important and yet most varied types of measurement in core analysis. Among the many techniques devised are the so-called "gas porosity" methods. An old and well known example is the Washburn-Bunting method.' The U. S. Bureau of Mines2-' described and later improved the apparatus for a now widely used method generally known as the "Boyle's law" method. In the present form of the Washburn-Bunting method,' the volume of air in the pores of a rock sample at atmospheric pressure is extracted and then collected in a graduated burette at atmospheric pressure. The volume of air is read directly as the pore volume of the sample. The absolute error in reading the collected volume of gas is independent of the total volume; thus, the relative error is larger when the volume is small, as it is for rocks of low porosity. In addition, the sample after measurement contains mercury, which limits its use for other analyses. The Bureau of Mines (or Boyle's law) method measures directly the solids volume of a sample from which the pore volume and porosity are derived, using a separate measurement of the bulk volume. Gas at a few atmospheres pressure is introduced into a sample chamber of known volume containing the rock sample. The pressure is accurately measured. Following, the gas is expanded into a burette at 1 atm, and the gas volume is read directly. From the initial pressure p, and the final pressure p2 and volume v,, the initial gas volume v1 is calculated using Boyle's law; that is, p1v1 = p2v2. Volume v, minus the volume of the empty sample chamber is the solids volume of the sample. The accuracy of the method is limited, unless corrections are made, by deviations of the gas from the "ideal" gas-law behavior assumed in the simple form of Boyle's law. The purpose of the present paper is to describe a method for measuring the gas porosity of a rock which avoids many of these difficulties. Gas volumes are measured directly with the same accuracy as the bulk volumes. Pressures of at least an order of magnitude larger than those of previous methods are employed to insure rapid penetration of the gas into the sample. While special equipment may be built to apply the method, the porosimeter may be constructed as well from commercially available components. For simplicity, the apparatus described will be referred to as the "Constant-Pressure gas porosimeter". THE CONSTANT-PRESSURE METHOD Fig. 1 shows schematically the arrangement of components comprising the present Constant-Pressure porosimeter. Briefly, the method is one of volume substitution and may be considered a null measurement. Omitting (for the present) some of the operational details, the method of measurement consists of the following three steps. 1. After evacuation, the volume of the measuring system (a ballast chamber, a manifold, two gauges and their connections) up to the sample chamber is filled with gas to a high pressure (- 1,000 psi). A sample of the gas at this pressure is trapped in one side of a sensitive differential pressure gauge to serve as the reference pressure for subsequent steps. 2. The evacuated sample chamber containing the rock sample is opened to the measuring system. As the gas expands into the chamber, the resulting decrease in pressure unbalances the differential pressure gauge. 3. The pressure is restored by means of a mercury volumetric pump. The volume of mercury injected exactly equals the free or void volume of the sample chamber (volume of empty chamber minus the solids volume of the rock within). From the injected volume and the known empty chamber volume, the solids volume is obtained and the porosity calculated. The pressure and the volume occupied by the gas are the same before and after opening the sample chamber. Expansion and compression of the gas are incidental operations and do not enter into the calculation of porosity. By the pressure balancing or nulling, the free volume of the sample chamber is merely substituted by an equal and measured volume of mercury. Since the measurements are at constant pressure, there are no compressibility corrections necessary for the sample chamber.

By B. G. Matson, M. A. Whitfield, G. R. Dysart

This paper describes the development of u computer program to calculate changes that occur in the length of tubular goods due to temperature and pressure changes during stimulation operations. Due to the numerous variables involved and the uncertainty of all static and dynamic conditions that could exist, it becomes a staggering task for individuals charged with completions to perform the necessary mathematical calculations. The computer program permits advance calculations for several sets of conditions. INTRODUCTION In the Delaware basin of West Texas alone, 50 wells were contracted or drilled to 15,000 ft or deeper in 1965. Deep well activity is continuing in this and other areas on an expanding scale. Many of these deep wells require extensive stimulation for successful commercial production, and during these operations, pressures and temperatures are encountered that have a pronounced effect on the length of tubular goods. This length change during a large-volume, high-pressure stimulation treatment utilizing fluids considerably cooler than bottom-hole temperature can be of such a magnitude that permanent damage to casing and tubing will result unless mechanical design, pressures and fluid temperatures are evaluated and controlled. These pressure and temperature effects can be calculated. However, the process lends itself well to computer solutions because of the mathematical nature of the problem and the calculating hours involved in arriving at an answer. The engineering-hour demand becomes more severe as tapered strings are involved. On initial treatments on a given well, surface pressure and injection rate conditions are unknown, and offset well conditions have not proven to be a reliable method for making predictions. For these reasons, it has become rather standard procedure for operators to compensate for these uncertainties by placing unnecessary pressure and fluid temperature restrictions on stimulation design. On a number of occasions treating fluids have been preheated to as much as 160F as a means of compensating for thermal contmction resulting from pumping cool fluids. The maintenance of packer seals has been treated by Lubinski, Althouse and Logan',' and the problem of therma1 effects on pipe has been explored by Ramey." These works were expanded and the results made applicable to everyday oilfield terminology before submitting them to computer programming. The pressure and temperature effects on tubing movement previously mentioned occur simultaneously as fluid moves through the pipe. The pressure changes, for purposes of explanation, are categorized here as to the various effects these pressures have on a tubing string. These divisions are (1) the piston-like results of forces acting on horizontal surfaces exposed to pressure, (2) swelling or ballooning of the tubing along its entire length due to the forces of pressure acting against the tubing walls, (3) the elongation of tubing due to frictional drag and (4) corkscrewing of the pipe due to internal pressure. Thermal changes are also of great importance, as their results may be more significant than any of the pressure effects. Steel is an excellent conductor of heat and the earth is a relatively poor conductor. It has been calculated that pipe temperatures at depths of more than 20,000 ft approach within as little as 25" the temperfature of the surface fluid after pumping for 2 hours, or a drop in temperature in some treatments of more than 220F. The equations presented in this paper were developed for computer programming and simplicity of input information; therefore, numerical constants such as Young's modulus for steel (28 X 10\ si), the coefficient of thermal expansion of steel (6.9 X 10."IF) and Poisson's ratio for steel (0.3) are included with unit conversion factors. The moment of inertia of tubing cross-sectional area with respect to its diameter was changed to a constant times (D' — d') where D is outer diameter and d is inner diameter. Units in the equations are length in feet, diameter in inches, density in pounds per gallon, pressure in psi, rate in barrels per minute and time in hours. PISTON-LIKE REACTIONS A change in tubing internal dimensions and the exposure of other horizontal surfaces to different pressures on the inside and outside of the tubing result in a reaction much like a piston under pressure. Such is the case when the internal diameter changes in a combination string of pipe, when seals of a slick joint assembly are subject to pressure and in the end effects of a tubing string. The change in tubing length due to the piston effects of a slick joint packer is affected by the various diameters involved, the tubing pressure Ap,, the casing pressure ,Ap,, length of pipe L, densities of fluid in the tubing before and during pump-

By E. Kendall Cork

The traditional financial objective for a single mine company has been to operate as frugally as possible and to pay out most of the earnings as dividends. If the business is cyclical (as it is for most metals) the dividends might fluctuate quite widely. When the mine is exhausted the company disappears. This is still quite a viable strategy for a single mine company. It is not however a viable strategy for the world as a whole. The mining industry is built by mine development companies who can mobilize the people and capital to bring new mines into production. Their skills must include marketing, engineering, finance and other politics. It is very rare for a property to be brought in without the support of a major company that can provide all these services. The exceptions will usually have some other form of big brother support, for example the U.S. government uranium contracts at guaranteed generous prices. The mine development company will seek as a minimum to perpetuate itself by developing new mines in order to replace those which are running out. The more common and more ambitious objective is to grow -- that is to add to its ore reserves and current production by developing more new mines. The financial objectives for that company are very different. Obviously if all the earnings were paid out in dividends there would be nothing left to work with. The first financial policy then is to spend an appropriate amount on exploration for new properties. The next is to retain enough of the earnings to provide the capital for new projects at least sufficient for the equity. There is no magic formula as to what proportion of earnings should properly be distributed as dividends by a growth-oriented mine development company. As a rough rule of thumb distributing half or more will probably leave too little to work on and 30% or so is probably a good balance. However the circumstances differ widely from company to company. It may be useful to set an objective for the rate of growth of a company's earnings. Some have picked rates such as 15% per annum compounded. Others have set a target in real terms which might appear as 10 or 11% plus inflation. Obviously the arithmetic of compound interest is very attractive; however in practice there is much variation. Indeed current returns from existing operations swing widely with the business cycle and there is no assurance that economic new properties will be found according to someone's arbitrary time schedule. For example, Western Mining Corporation Limited in Australia explored for 30 years with little to show for it, but then found the great Australian nickel deposits and more recently the huge Roxby Downs copper. That long dry spell could not have fitted anyone's arbitrary calendar of growth and yet they would not have found such orebodies without that long period of effort. Should they have abandoned the search? Once a new property has been found or acquired there has to be a threshold rate of return on the new capital to be invested against which to evaluate the property's economics. Conventionally this seems to be 15% after tax, a number common in other heavy industries as well. In some cases it is expressed as a lower number plus allowance for inflation. Discounted cash flow analysis is a very useful tool but it does not make the decision. In the end a "go" decision depends on judgment of many factors some of which are numbers used in the DCF calculation whose credibility must be examined. It is curious how frequently investment proposals come in with the rates of return very close to 15%. The project advocates know that a number much less than 15% will not fly and that a number much more is not necessary. With much higher nominal and real interest rates of recent years, even though before tax, logic suggests that the hurdle rate should also rise. The power of compound interest is so great that 20% is very hard to achieve in any cash flow projection but 18% may be a sensible yard - stick. Once again it is remarkable how many project proposals come in with an 18% return. On the record the mining industry as a whole has not been overly restrictive in choosing its hurdle rates of return. This is shown by the abundance of metals in recent years and the failure of metal prices to keep up with inflation. All of the foregoing is standard text book stuff.

Jan 1, 1985

By R. A. Garling

INTRODUCTION Since the infancy of in situ uranium mining, a growing number of hydrometallurgical processes have been incorporated into pilot and commercial scale flowsheets. Although initial design efforts were geared toward maximizing uranium recovery and minimizing plant and wellfield flow circuit maintenance, recent emphasis has shifted to improved means of water conservation and aquifer restoration. As mining units approached depletion, evaporation ponds reached minimum freeboard, and state and federal agencies demanded proof of groundwater restoration, processes including mixed bed and conventional ion exchange, reverse osmosis and electrodialysis were adopted by the industry. These units served the additional function of reducing process bleed flows during mining in states where the deep disposal well permitting ice remains unbroken. This report concerns the use of electrodialysis as an alternative to the more conventional processes used in in situ mining. In addition to a brief history and description of the process, a comparison to reverse osmosis and operational data derived from testing an Ionics, Inc. 1.31 x 10-3 m /s (30,000 gallon/day) unit at the Teton-Nedco Leuenberger Research and Development pilot will be presented. HISTORY Commercially practicable electrodialysis was contingent upon the development of synthetic ion exchange membranes in 1940's. In 1952, Ionics Inc. demonstrated that the process was amenable to the treatment of salt and brackish water and, in 1954, made their first commercial sale. The following decade saw several major electrodialysis unit sales which were generally targeted for use on private or municipal potable water treatment. Major increases in membrane desalting unit capacities, facilitated by technological advances in the reserve osmosis industry, were noted during the 1970's. The development of polarity reversing electrodialysis equipment which reduced feed pretreatment requirements, increased water recovery rates, and simplified unit operation, kept Ionics Inc. competetive in the water treatment industry. Engineering advances which incorporated automated equipment, non-corrosive construction materials, and improved ion exchange membranes allowed the electrodialysis process to compete in industrial waste treatment among other commercial markets. PROCESS AND APPARATUS DESCRIPTION The electrodialysis process utilizes direct electrical current passed across a stack of alternating cation and anion selective membranes in order to achieve an electrochemical separation of ionized materials in an aqueous solution. The membrane stack has the appearance of a plate and frame filter press and auxilliary equipment includes solution pumps, electrically actuated valves, filters, piping and a direct current power source. The ion separation membranes are thin sheets of synthetic cation or anion selective resins. Attaching sulfonate or quaternary ammonium groups to the cross linked copolymer structure determines the ion selectivity of the membrane. The membranes are separated from each other in the stack by non-conductive spacers that house flow channels which route the flow tortuously and parallel to the membranes. Direct electrical current passing perpendicularly to the membranes and solution passages attracts cations toward the cathode and anions toward the anode (Figure 1). As the ions from the feed stream pass through the ion selective membranes, they become concentrated in the adjacent brine channel and are retained there by the combined attractive force of the electrode and the repelling force of the next membrane toward the electrode. Limiting factors on the degree of demineralization possible include chemical solubilities in the brine flow and the current density that will produce an unacceptable degree of polarization (Figure 1). Feed or brine solution treatment with complexing agents or acids has been successfully applied to prevent membrane scaling. Polarization can occur when sufficient current density is applied to dissociate water in the ion depleted region of the diluting compartments near the membrane surfaces. Significant polarization is evidenced by large electrical resistances across cell pairs and notable pH differences between diluting and concentrating streams. Limiting current densities have been increased in U.S. manufactured equipment by utilizing tortuous flow paths of relatively high linear velocities thereby promoting continous solution mixing. Energy consumption is due to separating electrolytes and solutions, oxidation and reduction reactions occurring in electrode compartments, overcoming electrical resistance, conversion from AC to DC power, solution pumping and auxiliary equipment actuation. A major improvement to the basic electrodialysis process was applied in 1970 which resulted in frequent, automatic cleaning and descaling of membrane surfaces. The process, polarity reversal, incorporates alternating the cathode and anode on a periodic basis while exchanging product and brine flow channels via electrically actuated values. The reversal reduces the potential of stack plugging with CaCO3 (calcite), CaSO4 (gypsum), and colloidal materials and, in most waters, eliminates feed pre-treatment requirements. For approximately two minutes during and following the reversal, off spec. water is flushed to waste or reintroduced to the feed supply. The usual feed treatment on polarity reversing electro-

Jan 1, 1982

By K. C. Hong, R. B. Jensen

The problem of determining the optimum set of steam volumes and cycle lengths for a single well undergoing multicycle steam stimulation in order to maximize the cumulative discounted net income has been formulated mathematically and programmed for a digital computer. The mathematical fonnulation of the problem and the method for its solution are discussed in this paper. The oil production performance during each stimulation cycle was simulated by either a constant percentage or a harmonic decline. Using simple analytical expressions for production performance, the cumulative discounted net income and cumulative time of operation were related to the pertinent process and cost parameters and two principal process control variables (cycle length and steam injection volume). The discrete maximum principles was used to transform the equations for cumulative time of operation and cumulative discounted net income into a set of simultaneous equations. The simultaneous equations then were solved by trial and error on a digital computer to determine the set of cycle lengths and stem injection volumes that gives maximum cumulative discounted net income over the project life. INTRODUCTION Steam stimulation is a process for improving the oil recovery rate from wells producing high-viscosity crudes. The process is applied on an individual well basis and is executed in a series of cycles, each consisting of three phases: steam injection, soaking (steam condensation), and production. Significant increases in production rate following the stimulation operation result from heating the reservoir around the wellbore. As heat is removed with produced fluids and dissipated into nonproductive formations, the production rate declines, usually to near the prestimulation value. Typical production responses are given in case histories reported by Owens and Suter,l and are depicted in Fig. 1. Duration of the production phase is equal to the time for the oil production rate to decline to some specified value and is called the cycle length. (Cycle length is defined as the producing portion of the cycle and does not include the downtime required for steam injection and soaking operations.) Termination of the production phase coincides with the start of steaming for the next cycle. The process is continued, cycle by cycle, until it becomes unprofitable. Models 2-4 have been developed to simulate behavior of a single well during one cycle of steam stimulation and have been used to investigate the effects of system and operating conditions on the production responses. However, these investigations have not shown how the models can be used to determine a most profitable set of operating conditions for a multicycle stimulation project. Perhaps these models are too complicated mathematically to be adapted readily to an optimization study. This paper presents a method for optimizing a multicycle steam stimulation project based on simple production performance models. First, the performance of a steam-stimulated well during each cycle was simulated by either a constant percentage or a harmonic decline model. Using these simple analytical expressions for production performance, the cumulative discounted net income (before tax) and cumulative time of operation for each cycle were related to the pertinent process and cost parameters and two process control variables—the cycle length and the steam injection volume. Finally, the profit function to be maximized was formed. The analogy between the above problem and problems of optimizing multistage decision processes that have been reported extensively in the chemical engineering literatures5-10 in recent years has led this investigation to the use of the discrete maximum principle.5-7 This principle was used to transform the equations for cumulative discounted net income and cumulative time of operation into a set of simultaneous equations characterizing the optimum operating conditions in terms of the two control variables. These equations

Jan 1, 1970

By I. H. Silberberg, R. L. Reed, O. K. Kimbler

interfacial films have frequently been observed at interfaces between certain crude oils and water. Several investigators have postulated that the presence of these films should influence the efficiency of oil recovery in water drive or waterflood operations. They may also influence the stability of emulsions which are sometimes a problem in petroleum production, and may be a factor in the fonation of paraffin deposits in oil well tubing and flow lines. This paper presents a technique with which a modified Langmuir film balance may be used to study the compressibility and collapse pressure of these natural interfacial films. Experimental data are presented for several nude oil-water systems. Data developed are used to infer the phase state of the film as a function of such variables as rate of reduction of interfacial area, ionic composition of the subtrate and pH of the subtrate. A film of known physical characteristics is shown to have a sinnificant effect on oil recovery from an unconsolidated sand pack. Possible applications of these results to petroleum production are discussed. INTRODUCTION The use of water to displace petroleum from reservoir rocks is of major importance both as a primary and a secondary recovery process. As water invades the rock, oil is completely displaced from some pores and left as a discontinuous phase in other pores. The manner in which water moves from pore to pore is strongly influenced by capillary forces. In view of the complexity of reservoir fluid systems, there can be little doubt that complicated interactions take place at both the liquid-solid and oil-water interfaces. One of the more interesting, and least understood, of the phenomena which take place at the oil-water interface is the formation of interfacial films. These films are believed to result from the adsorption of high molecular weight polar molecules at the interface.l.2 Presence of such molecules may cause a striking alteration in interfacial tension. When the oil-water interfacial area of certain crudes is rapidly reduced, a thin region (film) about the interface assumes the appearance of a solid membrane, and striations, wrinkles and gross distortions may occur. If such a film is solid, it should greatly alter the interfacial tension normally assumed to exist between the oil and water phases. If the membrane is continuous, a solid phase would separate the oil and water. Interfacial films between crude oil and water were observed in 1949 by Bartell and Niederhausers who commented upon the apparent rigidity of the films and their possible importance in the petroleum industry. Morrell and Egloff4 had earlier attributed the extreme stability of emulsions of sea water in fuel oil to very stable asphaltic films. Numerous investigators have observed rigid films in the course of crude oil-water interfacial tension determinations by the pendent drop method. Several investigators5,6,2 have separated inter-facially active materials from crudes and attempted to characterize them chemically. Reisberg and Doscher,2 using Ventura crude, showed the interfacial tension against water (as measured by the pendent drop method) to be affected by aging, contraction and expansion of the interface, and the pH of the water. These investigators attributed the adhesion of oil to a water-wetted surface and the distortions of flow paths in glass capillaries to the presence of rigid films. Dodd7 has studied the interfacial viscosity of adsorbed films and found them to be non-Newtonian in behavior. Craighead and Harvey8 reported a series of displacements in tubes packed with 60 mesh glass beads. They interpreted the results as indicating an effect of stearic acid films on waterflood recovery and imply that natural films may produce similar results.

Jan 1, 1967

By F. P. Bullen

Empirical relationships between fracture stress, orientation angle, and diameter of crystal have been determined at 77°K. Orientation ranges of markedly different behavior were found—a law of constant normal stress&apos; of value a (diameter)-1/2 for the fracture of ductile crystals, and a condition of shear stress (or strain) for more brittle crystals. The observations are not consistent with current theories. An interpretation is advanced which is also applicable to observations on the effect of prestrain at room temperature on the subsequent fracture stress at 77°K and to the effect of cyclic stressing on the cleavage strength.&apos;&apos; The law of constunt normal stress&apos; and the brittle ductile transition are also explained. The interpretation is more consistent with the initiation of cracks by intersecting dislocations than with theories based on stress -concentration by dislocation arrays. ZINC single crystals are particularly suited to the study of cleavage because fracture occurs on the basal plane over a wide range of crystal orientations. Analysis of the conditions of stress and strain at fracture in crystals of different orientations should indicate which parameters control the cleavage process. Unfortunately, controversy has arisen over the correct empirical relationship between tensile fracture stress and orientation. Schmid&apos;s observations1,2 favored a &apos;law of constant normal stress&apos;, as observed in other materials.2 For zinc, however, the observed values are far below the theoretical strength and cannot represent the true limit of cohesion between neighboring atomic planes. Hence, the interpretation of such a &apos;law&apos; is not straightforward. Deruyttere and Greenough3,4 found a complex variation between tensile fracture stress and orientation; this variation did not agree with a &apos;law of constant normal stress&apos;. Two theories have been advanced to account for their observations: a) the propagation of cracks from low-angle boundaries,5 and b) the release of energy from piled-up dislocations during crack-propagation.4 The present work resolves the apparent discrepancy between the observations and shows that neither of the above theories are applicable to the tensile fracture of zinc single crystals. A phenomenological explanation, along the lines suggested by Gilman,&apos; is advanced and successfully applied to previously unexplained effects. EXPERIMENTAL DETAILS &apos;Crown Special&apos; redistilled zinc was used, except for one comparison series op tests using &apos;Tadanac&apos; electrolytic zinc. Crystals of 1 mm diam, subsequently called &apos;1 mm crystals&apos;, were grown from the melt in vacuo in precision-bore Pyrex tubes internally coated with graphite. Several specimens 1 in. long were cut from each crystal and chemically polished. Jigs were used to minimize handling strains, and crystals were mounted in the Polanyi machine the day prior to testing to allow recovery from any such strains. One-mm crystals were chemically polished for long periods to obtain 0.1 mm (approx) crystals. One-mm crystals were cemented into miniature gimbals by &apos;Araldite&apos; casting resin. The Appendix gives the reasons for using gimbals and the results obtained by other methods. More complete details of all techniques are given elsewhere.7 The symbols and terminology used are as follows: X = orientation angle (angle between tensile axis and line of greatest slope in the basal plane). T = tensile stress (on true cross-section) S,N = shear and normal stress (components of T with respect to the basal plane) ? = shear strain D = crystal diameter. The subscript &apos;f&apos; will be used to denote values at fracture. PART I-ANNEALED CRYSTALS EXPERIMENTAL OBSERVATIONS One-mm crystals were used to establish the variation of fracture stress at 77°K with orientation at fracture (Xf), Fig. 1. For 18 deg = Xf = 55 deg, a &apos;law of constant normal stress&apos; was observed. For Xf > 55 deg, the fracture condition approximated to a constant shear stress. At Xf< 18 deg, twinning occurred before fracture so that the results were not typical of homogeneous single crystals,4,8— such specimens will not be considered herein. The dependences of fracture stress upon Xf were of similar type for 6 mm,* 1 mm, and 0.1 mm crystals,

Jan 1, 1963

By Milton E. Wadsworth, W. Martin Fassell, William H. Dresher

A study of the rate of dissolution of molybdenite (MoS2) in alkaline solution was carried out under carefully controlled conditions. Effects of temperature, oxygen over-pressure, and KOH concentration were evaluated. Studies were made in the temperature range 100°C to 175°C and in the pressure range 0 to 700 psia of oxygen. Under these conditions molybdenite was found to leach according to a linear mechanism. Both oxygen over-pressure and KOH concentration were found to control the rate of leaching. The mechanism has been explained in terms of adsorption of oxygen at the molybdenite surface followed by configurational rearrangement of the adsorbed molecules. The hydroxyl ion dependency is believed to be diffusion-controlled. Laboratory batch studies have shown that molybdenite can be readily dissolved in alkaline solutions under moderate conditions of temperature and pressure. Commercial application of this process to the production of ferro-molybdenum and molybdenum chemicals is promising in view of the ease of dissolution of molybdenite and the relatively noncorrosive conditions involved in the process. HIGH temperature-high pressure techniques have long been used to great advantage in the organic chemical industry, the petroleum industry, and the paper industry. Only recently, however, have these methods been used to extract metals from their ores on a commercial scale. The Chemical Construction Corp., together with interested producers of nickel and cobalt, has done much to develop methods of producing nickel and cobalt powders by the use of high temperature-high pressure techniques.&apos; The Howe Sound Co. is currently operating a plant near Salt Lake City in which the new Chemical Construction Corp. technique is applied to cobalt-arsenic sulfide concentrates. In this method the concentrates are leached in a dilute solution of sulfuric acid under air pressure at a high temperature. Near Edmonton, in Alberta, Canada, the Sherritt Gordon Co. is also using the new method in conjunction with its ammoniacal leach process.&apos; National Lead Co., at Fredricktown, Mo., is operating a process similar to that of Howe Sound. In spite of the rapid growth of high temperature-high pressure hydrometallurgical processes, there has been very little work done on fundamental principles involved in this type of process. Conse- quently the literature lacks any significant data in this field. Oxidation of pyrite,3 phalerite,4 galena,5 and recently molybdenite" in alkaline solutions has been reported to some extent. This investigation was initiated, therefore, to broaden the application of high temperature-high pressure hydrometallurgy and also to contribute some insight into the basic mechanisms of oxidation in an aqueous medium under these conditions. In addition to the immediate consequences of these processes much valuable information remains to be learned about fundamental reactions when they are carried out at temperatures and pressures slightly higher than those normally encountered. Leaching of molybdenite by alkaline solutions under oxygen pressure has recently been reported in the Russian literature by E. S. Usataya,8 who found that leaching rate increased with an increase in the pH of the solution—the strongest effects being observed at a pH of 10—and also with an increase in temperature. Usataya also has reported the formation of a protective film on the surface of the molybdenite when the leaching solution was weakly alkaline, neutral, or acidic. He believes that this oxidation process explains the migration of Mo+ + in the oxidation zones of molybdenite ore deposits and the subsequent impoverishment of the ores. Equipment: The high temperature-high pressure reaction unit used in this work was especially designed for kinetic studies of this nature, emphasis being placed on accuracy of measurement and precision of control. As details of the unit&apos;s construction and operation have been discussed elsewhere,7 Only basic features and principles of operation will be discussed at this point.

Jan 1, 1957

By J. A. Herbst, D. W. Fuerstenau

This paper examines the zero order production phenomenon in the context of the size discretized batch grinding model. A restrictive interrelationship between the selection and breakage parameters of the model, which is mathematically sufficient to ensure zero order behavior, is delineated. Based on this interrelationship, a scheme is developed for predicting the values of the selection and breakage parameters for all size fractions from a minimum of experimental data. To test this scheme, dolomite was ground in a laboratory batch ball mill. Successful simulation of the comminution behavior of this system was achieved using the batch grinding model with the parameter values obtained from the proposed scheme. Historically, a preponderance of comminution research has centered on attempts to relate breakage energy to the performance of comminution machines. Concomitantly, grinding data were interpreted almost exclusively in terms of inherently empirical energy-size reduction relationships1-3 or "laws of comminution"4- 6 which were based on highly oversimplified descriptions of the fracture process. In some instances, these relationships provide a crude basis for the correlation of experimental data, but, invariably, this approach is inadequate for meaningful process simulation. The control and optimum design of comminution circuits require a mathematical model capable of depicting the size reduction behavior of every size fraction for grinding conditions of technological importance. Energy-size relationships do not provide this detailed information. For example, an energy-size analysis for batch ball-milled dolomite has recently been completed in the authors&apos; laboratories.7 This analysis provided a satisfactory correlation between the hypothetical size modulus, which characterizes only the finer portion of the size distribution, and the input energy. However, this relation did not constitute an adequate description of the complete system. In general, one of two distinct viewpoints toward simulation has been adopted in recent comminution research. The first approach focuses on the fracture of single mineral specimens, with the essential aim of representing the over-all process in terms of the breakage characteristics of individual particles and the characteristics of the stress field which the particles experience within the particular size reduction device. As illustrated by Harris&apos;s review,&apos; the actual incorporation of single-specimen fracture information into a description of the behavior of a multiparticle comminution system has seldom been attempted. The only model to accomplish this incorporation was the one developed by Schönert.8 The Schonert model treats single-passage grinding machines in terms of a distribution of effective loads acting on a particle, a distribution of required particle breakage energies, and the breakage product distribution. However, the use of single-particle fracture behavior to derive models of the size reduction process is at present limited to machines in which the particle residence time is approximately equal to the time required to apply stress. As a result, Schönert&apos;s analysis cannot be used to advantage to simulate the complex environment which prevails within a tumbling mill. For multiple-passage systems a second approach to the description of comminution processes has been more fruitful. This approach entails the formulation of a mathematical model which is phenomenological in nature in that it lumps together the entire spectrum of stress-application events which prevail in a system under a given set of operating conditions. The appropriately defined average of these individual events is then considered to characterize the over-all breakage properties of the device. Thus, to analyze the performance of a tumbling mill, the manner in which the particles of a particular size (or size fraction) are stressed need not be distinguished. Instead, a single parameter is assumed to represent the resistance of that size to fracture, given the average grinding environment which exists in the mill. The isolation of such a parameter and a related set of quantities, which constitute the breakage product size distribution for the average event in this size fraction, allows the formulation of physically meaningful descriptive equations capable of yielding precise and detailed information for simulation.

Jan 1, 1969

By I. Javandel, P. A. Witherspoon

The finite element method was originally developed in the aircraft industry to handle problems of stress distribution in complex airframe configurations. This paper describes how the method can be extended to problems of transient flow in porous media. In this approach, the continuum is replaced by a system of finite elements. By employing the variational principle, one can obtain time dependent solutions for the potential at every point in the system by minimizing a potential energy functional. The theory of the method is reviewed. To demonstrate its validity, nonsteady-state results obtained by the finite clement method are compared with those of typical boundary value problems for which rigorous analytical solutions are available. To demonstrate the power of this approach, solutions for the more complex problem of transient flow in layered systems with crossflow are also presented. The generality of this approach with respect to arbitrary boundary conditions and changes in rock properties provides a new method of handling problems of fluid flow in complex systems. INTRODUCTION Problems of transient flow in porous media often can be handled by the methods of analytical mathematics as long as the geometry or properties of the flow system do not become too complex. When the analytical approach becomes intractable, it is customary to resort to numerical methods, and a great variety of problems have been handled in this manner. One such method relies on the finite difference approach wherein the system is divided into a network of elements, and a finite difference equation for the flow into and out of each element is developed. The solution of the resulting set of equations usually requires a high speed computer. When heterogeneous systems of arbitrary geometry must be considered, however, this approach is sometimes difficult to apply and may require large amounts of computer time. The finite element method is a new approach that avoids these difficulties. It was developed originally in the aircraft industry to provide a refined solution for stress distributions in extremely complex airframe configurations. 27 Clough has recently reviewed the application of the finite element method in the field of structural mechanics.6 The technique has been applied successfully in the stress analysis of many complex structures.l, 27, 28 Recognition that this procedure can be interpreted in terms of variational procedures involving minimizing a potential energy functional7 leads naturally to its extension to other boundary value problems. In the field of heat flow, there recently have been introduced several approximate methods of solution that are based on variational principles.2-4, 17 By employing the variational principle in conjunction with the finite element idealization, a powerful solution technique is now available for determining the potential distribution within complex bodies of arbitrary geometry. In the finite element approximation of solids, the continuum is replaced by a system of elements. An approximate solution for the potential field within each element is assumed, and flux equilibrium equations are developed at a discrete number of points within the network of finite elements. For the case of steady-state heat flow, the technique is completely described by Zienkiewicz and Cheung.33 Since the flow of fluids in porous media is analogous to the flow of heat, Zienkiewicz et al. have employed the finite element method in obtaining steady-state solutions to heterogeneous and anisotropic seepage problems. 34 Taylor and Brown have used this method to investigate steady-state flow problems involving a free surface.25 The work of Gurtin has been instrumental in laying the groundwork for the application of finite element methods to linear initial-value problems.12 As a result, Wilson and Nickell have recently

Jan 1, 1969

By L. F. Mondolfo, B. E. Sundquist

The crystallographic orientation relationships resulting when lead is nucleated from the liquid by Ni, Cu, Ag, and Ge were determined. For each nucleating agent several definite orientatioz relationships were found. These relationships seemed to be controlled by good symmetry relations and low crystallographic disregistry between mating planes. For any given nucleating agent the under colling for nucleation was found fairly constant and independent of the orientation relationship and consequent disregistry. It was also found that, upon re melting and refreezing the Pb, the orientation relationship was changed. These findings prove that crystallographic disregistry is not the controlling factor in heterogeneous nucleation from the liquid. The results of this investigation tend to confirm the theory presented in a preceding paper that heterogeneous nucleation starts with the formation of an adsorbed layer of nucleated metal on the nucleat-ing impurity. Evidence is given that cavities in the nucleating agent act as centers of nucleation. IT has long been known&apos; that solid extraneous particles are active in catalyzing phase transformations that occur in a system, particularly condensation and crystallization. It is well established that these heterogeneities act as catalysts by providing surfaces upon which nuclei of the precipitating phase can form with activation energies smaller than those required for homogeneous nucleation. Numerous investigations have shown that in this process of heterogeneous nucleation: a) the nucleus forms with one, or several, definite crystallographic orientation relationships with the nucleating phase2-4 and b) that there is a small range of undercoolings or super saturations characteristic of the nucleation of a given solid on a given Substrate.5-10 Turnbull and vonnegut11 have developed a theory based on theories developed by Volmer12 and Turn-bull and Fisher1= for heterogeneous nucleation from gases and liquids, that relates the super saturation or undercooling required for nucleation to the dis-registry between the lattices of the nucleus and the nucleating agent. This theory predicts that nucleation should occur with the orientation relationship between the nucleus and nucleating agent that minimizes the disregistry. Further, it predicts that the undercooling or super saturation necessary for nucleation should be a function of the disregistry. Numerous investigations have dealt with the orientation relationships resulting from the condensation of vapors onto crystalline solid substrates2,3 and a few with the nucleation of one phase by a second phase in solidification4,14. Others have dealt with the supersaturation8-10 and undercooling5-7 associated with nucleation in condensation and solidification respectively. However, there is virtually no report that gives both of these factors for the same system. In this investigation a study was made of the undercoolings and orientation relationships resulting when Pb is nucleated from the liquid by Ni, Cu, Ag, and Ge. It was the purpose of this investigation to check the Turnbull-Vonnegut theory, i.e., the importance of crystallographic disregistry between nucleating catalyst and nucleated metal. The results indicate that disregistry is not an important factor in nucleation and that the nucleation process is probably somewhat more complex than current theories suggest. EXPERMENTAL PROCEDURE Small single crystals of nickel, copper, silver, and germanium were prepared from materials of four to five nines purity, and the Pb used was also 99.999+ pet pure. Cu and Ag single crystals were prepared by sealing small chips of Cu or Ag in an evacuated quartz capsule and heating the capsule at 2000°F for 1 hr before cooling. Nickel crystals of 200 diam were also prepared in evacuated quartz capsules, but melting was done by heating the capsules in an oxy-acetylene flame for a few minutes. These spheres were invariably polycrystalline so

Jan 1, 1962

By D. Turnbull, R. E. Cech

FISHER, Hollomon, and Turnbull have developed a theory for the nucleation of martensite. They first tested the theory on Fe-C alloys and low alloy steels. The major factor influencing nucleation of martensite was considered to be statistical composition fluctuations occurring in small regions at high temperature and frozen-in on quenching. These local regions of varying size and composition serve as nucleation centers. They become supercritical, one by one, as temperature is progressively lowered, resulting in temperature-dependent or athermal transformation. Fisher next applied nucleation theory to substi-tutional solid solution alloys. Detailed predictions were made for Fe-Ni alloys because of the availability of free energy data on this system. It was shown that composition fluctuations that were significant energy-wise did not occur, and nucleation frequencies could be calculated from average properties of the system. Nucleation was predicted to occur as time-dependent and having the functional relationship to give a C curve of nucleation frequency vs temperature. The analysis further predicted that the nucleation frequency was extremely sensitive to composition. Experimentally, it would be found that the transformation in some compositions is so slow that measurable amounts will not form in a reasonable length of time. With other compositions, only slightly different, the nucleation frequency becomes so great that the material becomes transformed while still distant in temperature from the maximum nucleation frequency. On quenching an alloy of such composition, the observed transformation kinetics would be similar to those found in Fe-C alloys. Cech and Hollomon repeated experiments of Kurdjumov n which the kinetics of transformation were similar to those predicted by Fisher for Fe-Ni alloys. The alloy studied in this investigation contained 73.3 pct Fe, 23.0 pct Ni, and 3.7 pct Mn. Fisher,&apos; using an idealized model for the extent of transformation as a function of the number of martensite crystals per grain of parent phase, derived nucleation frequencies from the transformation curves of Cech and Hollomon. Complicating influences such as coupling effects between grains in the polycrystalline specimens were neglected. Nevertheless, excellent agreement was found between the theoretically and experimentally derived nucleation frequencies. These experiments, however, could not provide a critical test of the theory. Experimental nucleation frequencies could vary widely from those calculated, depending upon the extent of deviation from ideal partitioning and the extent of coupling effects. Further, since the compositions of material theoretically analyzed and experimentally determined were different, the free energy changes involved in the experimental work could only be estimated. Also, the effect of heterogeneities on the transformation kinetics was not considered. For these reasons, it was decided that experiments designed to test the validity of the Fisher analysis must be conducted on binary Fe-Ni alloys, which were the ones considered theoretically by Fisher. The martensite transformation in Fe-Ni alloys has been the subject of considerable study. Machlin and Cohen have shown that transformation proceeds in a manner quite unlike that in any other ferrous alloy system. They found that single crystals would transform to a large extent in a single burst. In large grain polycrystalline specimens, frequently more than one grain and sometimes the whole specimen would transform at the same instant in this manner. Results on filings indicated that different particles would undergo the burst transformation at widely different temperatures. These results support the conclusion that the transformation behavior could not be described by a single nucleation frequency as would be the case if the nucleation were homogeneous. It appeared that further work was necessary to define the factors responsible for burst-type transformation, so that the conditions could be altered to favor homogeneous nucleation of martensite if such could be accomplished. This report summarizes the results of some experiments conducted with powdered Fe-Ni alloys for this purpose and the re-

Jan 1, 1957

By G. A. Stamboltzis, N. Arbiter, C. C. Harris

Fracture under low-velocity free-fall and double impact and under slow compression have been investigated. The pattern of breakage and the size distribution of resulting fragments of sand-cement and glass spheres have been determined. Photoelasticity methods were used to simulate the stress distributions in free-fall impact in order to explain the observed patterns of breakage. Oblique fracture planes, occurring only in free-fall impact, develop along lines which coincide with the trajectories of maximum compression as determined by mathematical analysis. Breakage efficiencies for different modes of fracture were compared for both types of spheres. For the same specimen and loading system, static loading and low-velocity dynamic loading induce geometrically similar stress fields resulting in reasonably similar fracture patterns and shapes of fragments. This work had its origin in a desire to understand three fundamental processes involved in autogenous comminution: namely; free-fall impact, double impact, and slow compression. To simplify the investigation, it is preferable to reduce the entire multi-stage fracture process to its most elementary form; that is, to study products resulting from a single-stage operation, such as the breakage of a specimen under single fracture conditions. The goal of this research is to study the energy utilization in the single fracture of brittle specimens and to relate the pattern of breakage and the resultant fragment size distribution with the nature of the material, the specimen size, the manner of load application, and the rate of loading. Published works on single fracture have mainly been concerned with tests on large (> 1-in.) irregular mineral pieces, especially coal and coke. These were mainly friability tests employing a single blow on closely sized pieces. More recently, small mineral specimens in the sieve range and glass spheres have been investigated.5-7 Kick8 has reported free-fall and double impact tests with large large cast-iron, cement, and clay spheres. The establishment of a minimum breaking height independent of size in the free-fall impact tests, and the direct proportionality between the minimum work required for fracture and the volume of the specimen in double impact tests were used by Kick as an experimental proof of his classical law of "Proportional Resistances." In the present work, spherical shapes were chosen because of their simple body geometry and consequent impact and stress field symmetry. Because the study involves several physical principles in connection with brittle-fracture, it may be of interest in fields where the strength of materials is of importance. APPARATUS The equipment for the free-fall impact testing is illustrated in Figs. 1 and 2. This consisted of a massive (17 x 17 x 3-in.) hard steel plate onto which specimens were dropped and a specimen release mechanism which could be set at any desired height up to 10 ft above the plate so that the impact velocity could be varied up to about 25 ft per sec. Double impact breakage is characterized by two points of loading situated at opposite poles. For this mode of breakage the apparatus was modified (Figs. 3, 4 and 5) so that a falling mass provided an impact of predetermined magnitude on a specimen resting on the plate. A spring-loaded device operated immediately after fracture to arrest the falling mass and to record its residual kinetic energy. Precautions were taken to avoid secondary breakage. Slow compression tests were performed on a conventional hydraulic testing machine, the specimen being held between two hardened parallel bearing blocks. In the case of glass spheres, carbide bits were used. Photoelasticity studies were performed on two-dimensional models held in a loading frame equipped with a force gauge (Fig. 6). The models were viewed in a polariscope and isochromatic fringe patterns were recorded photographically. SPECIMENS Sand-Cement Spheres: A series of sand-cement spheres were prepared by molding in spherical glass flasks.

Jan 1, 1970

By D. S. Rowley, F. C. Appl

Assuming that rock behavior, during cutting with a single diamond, may be approximated by that of a rigid, Coulomb, plastic material, a theory of single diamond cutting action has been developed. Using this theory, the stresses on the diamond cutting surface and the components of the cutting force have been determined. Theoretical results agree reasonably well with available experimental data. The theory of cutting of single diamonds may be used as the basis for subsequent study of the perfonnance of surface set diamond drill bits and other surface set diamond cutting tools. INTRODUCTION For more than twenty years, surface set diamond bits have been used successfully in the mining and petroleum industries to drill and core rock formations that range from medium hard to the hardest known formations. Most extremely hard rock formations cannot be economically drilled or cored by any other means. In view of the growing use and importance of surface set diamond drill bits, it is reasonable to study the way in which these tools cut, and the factors that influence wear of the diamonds. Such knowledge will contribute to better bit design and will improve operating procedures. To analyze the cutting performance of these tools it is appropriate to consider the way in which an individual diamond cuts. Once this is established, the over-all performance of the bits can be determined by considering the total effect of the individual diamonds. In recent years there has been a growing interest in rock mechanics and there now exist several publications dealing with the deformation and failure of rocks. Notable experimental work has been reported by Bredthauer,1 Handin and Hager,2, 3 and Gnirk and Cheatham.4 An extensive collection of data pertaining to rock properties at low confining pressures has been given by Wuerker.5 Cheatham and Gnirk have recently presented an excellent review of the state of knowledge regarding analytical and experimental rock deformation and failure.6 Generally rocks are extremely weak in uniaxial tension, and weak or only moderately strong in uniaxial compression. Both tensile failure and compressive failure are generally brittle in nature. This means that there is little or no deformation or "flow" of the material before shear or tensile fracture occurs. Thus, until fracture occurs, rock behaves very nearly like an elastic material in uniaxial stress. Significant changes in behavior of rock occur under conditions of triaxial compressive stress. There is generally a marked increase in compressive strength and in ductility. It has been demonstrated that the failure becomes plastic (in a macroscopic sense) at sufficiently high values of confining pressure. Sharp wedge chisel indentation studies conducted by Gnirk and Cheatham indicate that many rocks behave in a plastic manner at relatively low triaxial compressive stress states.4 If "failure" under triaxial compressive conditions is considered to be either complete rupture or macroscopic plastic flow, then the general failure characteristics of a rock can be reasonably well represented by the commonly known Mohr envelope. This certainly is not a perfect failure criterion (since the effect of the intermediate principal stress is neglected) but it has been found to be adequate for most engineering purposes. Mohr envelopes usually are constructed graphically using experimental triaxial test data. For purposes of analysis it is convenient to represent the envelope in equation form. Among the several approximate forms appearing in the literature, a parabolic form is suggested by Cheatham,7 and an exponential form is suggested by Paone and Tandanand.8 The exponential form has been adopted for this study, and based on published data, appropriate parameters have been established for 19 rock types. A typical plot of the data compared with the approximate envelope, and a summary of the rock properties and

Jan 1, 1969