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By S. Ramachandran, R. G. Woolery

INTRODUCTION Union Carbide is currently operating an in-situ leach project on the Palangana Dome area in Duval county. This deposit meets all the requirements for in-situ leach in that the ore (1) is below the water table, (2) is in a permeable horizon, (3) is amenable to chemical leaching, and (4) is confined by impervious layers. This project has been under commercial production since 1976, and its capacity has been expanded on three occasions since going on-stream. Recently, additional uranium reserves were discovered on the Rogers-Cardenas (R-C) property about 32 km north of the Palangana operation. The ore is located within the Oakville sands and its characteristics are quite similar to those of the ore at Palangana. Both are an unconsolidated Arkosic sand high in clay and calcium carbonate. The R-C ore, however, is somewhat coarser with a mean particle size of 0.15 mm as compared to a mean particle size of 0.07 mm for the Palangana ore. In all respects it would appear that this ore would be a candidate for in-situ leach as a satellite operation to Palangana. Unfortunately, R-C ore is above the water table and, therefore, not amenable to the Palangana practice. Because of the limited known reserves in this deposit, it is readily apparent that conventional mining and milling are out of the question. However, because of its proximity to our Palangana operation, it seemed worthwhile to consider other options. The most viable route based on our past experience was to heap leach the ore. Our recent success at our Gas Hills facility and our Maybell operation, in employing a heap leach practice to our marginal reserves seemed to be a logical approach for processing this ore. Our experiences at both locations are described in "Heap Leaching - A Case History" by R. G. Woolery et al., Mining Engineering, March 1978. In both instances the process is an acid leach circuit and acid consumption averages 20 kg/t H2SO4. A preliminary feasibility study showed that because of the high strip ratio required for the R-C project to be successful, additional ore reserves must be located and that a method of heap leaching with an alkaline circuit would have to be developed. As a result of this paper study, the decision was made to proceed with a program of additional exploration drilling to determine the total ore reserves that could be mined economically. The Mining Department will evaluate each ore zone for cutoff grade, strip ratio, and expected mining cost. At the same time, a laboratory program to evaluate the available core samples for amenability to heap leaching with respect to an estimate of uranium recovery and processing costs was developed. This program is currently in progress, and at this time, we are just completing our process amenability study. BENCH-SCALE EVALUATION OF THE R-C ORE The initial bench-scale slurry leach tests on the R-C ore showed an acid consumption in excess of 200 kg/t H2SO4. These data, of course, discouraged us from considering this process route. Not only would the acid cost be prohibitive, but the gypsum generated by the reaction of the sulfuric acid with the calcium carbonate of the ore would severely effect the percolation of the lixivant. For this reason, the laboratory program was directed toward an alkaline circuit compatible with heap leaching. Because of the proximity of the R-C property to our Palangana operation, it seemed advisable to integrate the processing of this ore into the production at Palangana. Doing so would enable us to bring the R-C property into production by merely enlarging our present facilities at Palangana; otherwise, construction of a grass roots plant would be necessary. Ideally, the simplest method would be to construct the heaps at Palangana and employ an ammonium carbonate/bicarbonate leachant compatible with the in-situ production liquor. The product liquors could then be co-mingled or processed separately as desired. To determine if this goal was practical, samples of the R-C ore were obtained, and a laboratory program initiated. Heap leach amenability testing consisted of preliminary bench-scale evaluation to determine optimum solution strength and ultimate uranium recovery, followed by small column tests to confirm the bench-scale metallurgy and to determine percolation characteristics. These bench-scale tests are being followed by pilot-scale testing approximating field conditions. As expected, the bench-scale tests showed that the dissolution rate is considerably slower for alkaline leach than has been our experience in acid leaching. Because of the slower reaction rates, product liquor grades will be lower than for acid, as greater volumes of solution are required for satisfactory uranium extractions. The greatest influence on reaction times found in the laboratory was the carbonate/bicarbonate strength and oxidant addition. However, the higher salt concentration reduced the efficiency of the IX resin circuit and about 25g/L salt proved to an upper limit compatible with subsequent IX treatment. The oxidant contributed significantly to the early extraction rate but seemed to have only minimal effect on the total practical U308 extracted or the time required to achieve it. This variable will require larger scale testing to determine if the added cost of the oxidant is actually justifiable. Thus, the small-scale laboratory slurry tests, based on the 0.088% U308 sample available, indicate that leaching at 25g/L ammonium carbonate/bicarbonate, with or without oxidant, we might expect an 80-85% U308 extraction on this ore.

Jan 1, 1979

By E. T. Turkdogan, P. Grieveson

Molten silver is equilibrated with silicon nitride at 1400°C in nitrogen + hydrogen gas mixtures, and from the solubility data the activity coefficient of silicon is found to be 1.76 at silicon concentrations below 1 pct. Using this value, and that available from the silicon cmbide solubility measurements, together with those derived from the phase equilibrium diagvam, the activities of silver and silicon are estimated for the entire composition range. The main feature of the activity data is that there is a pronounced positive deviation from ideality. The activity of silver in molten silver containing up to 5 pct Si was determined by Schadel, Derge, and Birchenall1 by measuring the vapor pressure of pure silver and of the alloy over a temperature range of 850º to 1050°C. They deduced from their results that there is a positive deviation from Raoult's law and that the activity coefficient of silver increases with decreasing temperature. However, as pointed out by Kubaschewski and Catterall,2 their activity coefficients are much too high, giving silver activities greater than unity. At a later date McCabe and Birchenal13 showed that the previous vapor pressure data on silver4 were high by a factor of about 1.3; presumably a similar error must have occurred with the vapor pressure measurements on the Ag-Si alloys. The activities of silicon and silver calculated by tager' from the phase diagram and by Gibbs-Duhem integration, respectively, show a positive deviation from Raoult's law for silicon activity and positive and negative deviations for the silver activity. According to these computations, at infinite dilution of silicon ygi = 1.12 at 1420°C. d'Entremont and chipman6 measured the solubility of silicon carbide in molten silver at unit carbon activity; the equilibrium silicon content is found to be Nsi = 0.017 ± 0.002 atom fraction for 1420°C. Using the data of Grieveson and Alcock7 for the free energy of formation of silicon carbide from pure silicon and graphite, the activity of silicon for unit activity of carbon and of silicon carbide is asi = 0.0308 at 1420°c; therefore for the above solubility data on silver, YSi = 1.81, which is somewhat higher than that given by Hager. Use is often made of silver alloys in the measurements of activities of solutes in metals where silver has a very low solubility; in fact, Ag-Si and Ag-A1 alloys have been used in the past for such purposes. In an attempt to gain more thermodynamic data on molten Ag-Si alloys, measurements have been made in the present study on the solubility of silicon nitride in molten silver; the results on these solubilities are given in this paper together with a detailed discussion of activities of silver and silicon. SOLUBILITY OF SILICON NITRIDE The standard free energy of formation of silicon nitride is known with a reasonably high degree of accuracy. According to the measurements made by Hincke and Brantle8 and by Pehlke and Elliott,' the following are the free energy data for the standard states of solid Si3N4, liquid silicon, and Nz at 1 atm pressure:

Jan 1, 1963

By Fred H. Poettmann

An equation has been derived for use in calculating the sandface pressure of flowing gas wells in which the variation of the compressibility factor of the gas with pressure is taken into consideration. This variation due to compressibility has been put into both graphical and tabular form. Comparison of calculated results with field measured results were made on 20 dry gas wells from a given field and 11 distillate wells from different fields. The agreement between calculated and observed results is good. In addition to their use in the calculation of sandface pressures of flowing wells, the factors can be used in the direct calculation of the static bottom hole pressure of gas wells, the capacity of gas transmission lines, and in the calculation of the theoretical isothermal horsepower necessary to compress a natural gas. Examples demonstrating the use of various equations are given. INTRODUCTION Many of the equations used to calculate the sandface pressures of flowing gas wells from well head data do not take into consideration rigorously the deviation of the natural gas from ideal gas behavior. For low pressure wells this error is not serious. For high pressure wells flowing at high rates this error call be serious. Sandface pressures are used to determine the producing capacity of gas wells as shown in the U. S. Bureau of Mines' Monograph 7.' An equation has been derived for use in calculating the sandface pressure of flowing gas wells in which the variation of compressibility of the gas with pressure is taken into consideration. This variation due to compressibility has been put into both graphical and tabular form. In addition to their use in the calculation of sandface pressures, these "factors" can also be used to give a direct solution in the calculation of the static bottom hole pressure of gas wells, the calculation of the capacity of gas transmission lines, and the calculation of the isothermal horsepower necessary to compress a gas. CALCULATION OF SANDFACE PRESSURE OF FLOWING GAS WELLS The starting point in the derivation of any specific flow equation is an energy balance on the fluid flowing between any two points in the system. This energy balance based on one pound of flowing fluid is expressed by the well known general flow equation: jp~- vdP+ Ah + -— + wt + r. = o . . (i) P, 2 g. where V = specific volume of flowing fluid in cu ft/lb P = pressure in Ib per sq ft ah = difference in height above datum plane in, ft ------ = kinetic energy change of one lb of flowing fluid, v equals velocity in ft/sec and g. = 32.174 Ws = Work done by fluid while in flow (similar to shaft work in driving a turbine) W, = Znergy losses due to the irreversibilities of the flowing fluid This equation contains no limiting assumptions and can be made the basis of any fluid flow relationship by limitations or substitutions. The energy units are usually expressed in term* of foot-pounds of energy per pound of mass. Considering the vertical flow of gas through tubing or annu-lus, the external work W, done by the gas is zero. Let us also assume that the value of the kinetic energy function is small and can be neglected. Equation (1) is reduced to: /P: VdP + Ah+Wt = 0 .......(2) Pi The energy loszes PI can be expressed in terms of the well known Fanning equatinn: 4/Lf: rf = —— ............(3) 2g,D where L = the integrated average velocity in ft/sec of vertical flow in a uniform flow string -~h = L v = the integrated average velocity in ft/sec / = dimensionless correlating function

Jan 1, 1951

By Fred H. Poettmann

An equation has been derived for use in calculating the sandface pressure of flowing gas wells in which the variation of the compressibility factor of the gas with pressure is taken into consideration. This variation due to compressibility has been put into both graphical and tabular form. Comparison of calculated results with field measured results were made on 20 dry gas wells from a given field and 11 distillate wells from different fields. The agreement between calculated and observed results is good. In addition to their use in the calculation of sandface pressures of flowing wells, the factors can be used in the direct calculation of the static bottom hole pressure of gas wells, the capacity of gas transmission lines, and in the calculation of the theoretical isothermal horsepower necessary to compress a natural gas. Examples demonstrating the use of various equations are given. INTRODUCTION Many of the equations used to calculate the sandface pressures of flowing gas wells from well head data do not take into consideration rigorously the deviation of the natural gas from ideal gas behavior. For low pressure wells this error is not serious. For high pressure wells flowing at high rates this error call be serious. Sandface pressures are used to determine the producing capacity of gas wells as shown in the U. S. Bureau of Mines' Monograph 7.' An equation has been derived for use in calculating the sandface pressure of flowing gas wells in which the variation of compressibility of the gas with pressure is taken into consideration. This variation due to compressibility has been put into both graphical and tabular form. In addition to their use in the calculation of sandface pressures, these "factors" can also be used to give a direct solution in the calculation of the static bottom hole pressure of gas wells, the calculation of the capacity of gas transmission lines, and the calculation of the isothermal horsepower necessary to compress a gas. CALCULATION OF SANDFACE PRESSURE OF FLOWING GAS WELLS The starting point in the derivation of any specific flow equation is an energy balance on the fluid flowing between any two points in the system. This energy balance based on one pound of flowing fluid is expressed by the well known general flow equation: jp~- vdP+ Ah + -— + wt + r. = o . . (i) P, 2 g. where V = specific volume of flowing fluid in cu ft/lb P = pressure in Ib per sq ft ah = difference in height above datum plane in, ft ------ = kinetic energy change of one lb of flowing fluid, v equals velocity in ft/sec and g. = 32.174 Ws = Work done by fluid while in flow (similar to shaft work in driving a turbine) W, = Znergy losses due to the irreversibilities of the flowing fluid This equation contains no limiting assumptions and can be made the basis of any fluid flow relationship by limitations or substitutions. The energy units are usually expressed in term* of foot-pounds of energy per pound of mass. Considering the vertical flow of gas through tubing or annu-lus, the external work W, done by the gas is zero. Let us also assume that the value of the kinetic energy function is small and can be neglected. Equation (1) is reduced to: /P: VdP + Ah+Wt = 0 .......(2) Pi The energy loszes PI can be expressed in terms of the well known Fanning equatinn: 4/Lf: rf = —— ............(3) 2g,D where L = the integrated average velocity in ft/sec of vertical flow in a uniform flow string -~h = L v = the integrated average velocity in ft/sec / = dimensionless correlating function

Jan 1, 1951

By P. T. Carter, A. B. Murad, H. B. Bell

The distribution of manganese and oxygen between molten iron and FeO-MnO-SiO2 slags not saturated with SiO2 has been determined and used to calculate activities of MnO and SiO2 in MnO-SiO2 slags. These were found to be intermediate between the corresponding activities in FeO-SiO2 and CaO-SiO2 slags. THE equilibrium constant for the reaction between molten iron containing manganese and slags containing FeO and MnO, namely: FeO(slag) + Mn = MnO (slag) + Fe(1iquid) is most conveniently represented by: (MnO) Kmn== (FeOHMn] [Fe] is omitted, since for dilute solutions of manganese in iron, afe is approximately constant. When activities are used for (MnO), (FeO) and [Mn], Kmn is a true equilibrium constant and will apply to all types of slags, but when weight percentages are used, the values obtained for Kmn for different slags vary considerably. The equilibrium between iron containing manganese and relatively pure FeO-MnO slags has been investigated by several workers;1-4 the most recent investigation reported is that of Chipman, Gero, and Winkler.4 Their results, corresponding to slags containing up to 4 pct SiO2, were represented by the equation: 6440 log KMn = +2.95 In this case both slag and metal may be assumed to be ideal solutions, therefore the values of Kmn so derived represent the true constant for the manganese reaction. As the molecular weights of MnO and FeO are almost equal, weight concentration of these oxides may also be used in the above expression. Acidic oxides such as SiO2 and P2O5 affect the activities of FeO and MnO differently, so that when total concentrations by weight of FeO and MnO are used in the above expression for Kmn the values obtained are no longer constant. These will be denoted by K'mn. Körber and Oelsen5 investigated the manganese reaction using silica-saturated iron-manganese-silicate slags and found: 7940 logK'mn =3.172 This corresponds to Kmn values much higher than those obtained for pure oxide slags. The results of Körber and Oelsen, and Chipman, Gero, and Winkler are summarized in Table I. In the present investigation it was decided to study the manganese reaction using iron-manganese-silicate slags not saturated with silica, and which fell mainly between the orthosilicate join 2FeO . SiO2 - 2MnO.SiO2 and the FeO-MnO side of the ternary system FeO-MnO-SiO2. It has been shown" that the phases present in solidified slags of this part of the diagram are (FeO, MnO) and (2Fe0.Si02, 2Mn0-SiO,) solid solutions. Such an investigation would be of interest in deoxidation and also gives information on activities in liquid slags. This equilibrium has not as yet been investigated successfully. Hilty and Crafts7 attempted to examine this reaction in a rotating crucible furnace, but were unable to obtain suitable slag samples. In the present investigation one series of experiments was carried out in a rotating crucible furnace, because the success of Taylor and Chipman8 with such a furnace in their study of oxygen distribution between molten iron and (CaO, MgO) — FeO-Si0, slags suggested that it would minimize contamination of the slag by the crucible material. A second series was carried out using a wire-wound resistance furnace. Rotating Crucible Furnace Experiments In all cases the metallic charge consisted of 800 to 1000 g Armco iron containing 0.02 pct C, 0.03 pct Mn, and traces of Si. Manganese and silicon additions consisted of electrolytic manganese (99.9 pct)

Jan 1, 1953

By I. R. Kramer, H. Shen, S. E. Podlaseck

The tensile and creep properties of aluminum in vacuum have been investigated. It was found that the general effect of a vacuum environment was to reduce the rate of work hardening. Results obtained from creep tests showed that lowering the test pressure from 2 x 10-4 to 10-7 torr produced a decrease in the activation energy as much as 500 cal per mole. An attempt is made to explain these effects in terms of how the surface-stress field of a specimen may be affected by a vacuum environment during plastic deformation. In a previous paper,' the results were reported of some of our early work on the effect of vacuum environment on the stress-strain curve of aluminum single crystals. This effort has been extended to the studies of the tensile and creep properties in vacuum as a function of various metallurgical parameters. Some of the recent surface-effect studies2'3 have revealed that a region of high dislocation density, the so-called "debris" layer, is formed near the surface of the specimen as a result of plastic deformation. This layer acts as a surface barrier to oppose dislocation movement. The strength of this barrier depends on the surface condition such as oxidation or surface removal. Experimental results obtained from surface-removal work2, 3 indicate that such a surface barrier, which may be characterized by a stress field, affects the applied stress and the rate of work hardening. In the case of aluminum, this stress field was found to be directly proportional to the strain and the strain rate; however, the slope of this curve is independent of orientation of the crystal and may be altered by using different surface treatment. Since the escape of dislocations may be retarded by the presence of an oxide on the surface of a specimen, it is expected that a change in the rate of oxidation in vacuum will affect the process of dislocation accumulation in the debris layer. Consequently, it also affects the plastic behavior of a metal. This paper reports the changes in some of the creep and tensile properties of aluminum in vacuum and discussions of the observed effects in terms of the concept of the "debris" layer. EXPERIMENTAL PROCEDURE Tensile Tests. A sketch of the tensile apparatus is given in Fig. 1. This apparatus was contained in a bell jar (14 in. diam by 24 in. high) connected to a 10-in. vacuum system. The load was applied to the specimen by means of a gear train which was driven by a magnetic clutch located outside the vacuum chamber. The specimen, held by a pair of square shoulder grips, was attached to the load cell at the top and the drive gear at the bottom. A displacement transducer was placed between the two grips as shown in the figure to measure the elongation. Signals of elongation and load were amplified and recorded on an X-Y recorder. The system was capable of detecting loads as small as 0.025 lb and displacements of 0.0002 in. Shoulder-grip specimens of aluminum single crystals, having a nominal square cross section of 1/8 by 1/8 in. and a 3-1/2-in, gage length, were grown by the Bridgeman techniques. The original purity was 99.997 pet. The orientations of the crystals investigated are shown in Fig. 4(a). The crystals were annealed at 600°C for 2-1/2 hr, electropolished, and placed in the vacuum test chamber without any delay. All the crystals were deformed at a strain rate of 4 x 10-5 sec-1 at 24o ± 3°C. A study was made to find whether a chemically passive surface film would give the same effect as a vacuum environment. Specimens of aluminum 1100 (0.187 in. diam with a grain size of 0.1 mm)

Jan 1, 1965

By C. S. Matthews, M. J. F. Rosenbaum

The purpose of thir work wax to lcarn it~lzut infori~lation could he obtained from various typs of pilot water floods and to attempt to find the optunum pilot patter11, for a revervoir which had previously been depleted by a solution gas drive. The study was made in the laboratory with mathemetical methods a dynamic analog and a potentiotnetric analog. Results werp tested against the field llistorics of a nrrnlber of pilot water floods. At a reasonable valrre of currzulative injection, the total production rate for the one-injector five-spot should reach about 6.5 per cent of injection rate, and for a four-injector five-spot, about 9 per cent. Accurate estimates of ultimate recovery cannot be made on the basis of such snzall prorluction rates. However, with a pilot composed of nine ir1jector.s and 16 producers the production rate is approximately 50 pcr cent of injection rate at a reasonable value of camulative injection. Sonle inforn~ation for extended performance predictions might he obtained from such a large pilot. These conclusions were drawn on the basis of results obtained for unit mobility ratio, and a sturly using tlre potentiometric analog was made of the effect of other mobility ratios to determine the range of applicability of these predictions. For the four-injector, five-spot pilot with the ratio of production to injection rate (before water breakthrough) is about twice that for with it is about two-thirds; and with M0= 10, it is about one-third For high mobility ratios, it was found that the production rate increased considerably as water-cut increased. These result can be used to modify, qualitatively, the inter.pretntions based on curves for the unit rnobilit\. ratio CaSeS. It was found that the maximum ratio of production rate to injection rate obseriled in field pilot floods was of rhe scime order as that prerdicted by these methods. The time required to reach thisr maximum did not generally agree with the time predicted for a homogeti~orir reservoir. The differcrlce between predicted and observed time of response gives an indication of the permeability profile and of the condition of the producin,g wells. Pilot water floods of the pattern type are generally carried out in reservoirs which have been depleted by solution gas drive and are at low pressure. Under these conditions, oil and water can be considered incompressible. It is assumed that, as the water is injected, an oil bank forms ahead of it and that there is a distinct interface between the water zone (or bank) and the oil zone (or bank) and between the oil zone and the region ahead of the oil zone. It is further assumed that only gas is mobile in the unflooded (gas) region, only oil is mobile in the oil bank and only water is mobile in the water bank. The saturations and the mobilities associated with each zone are assumed uniform. We idealize our reservoir to be homogeneous, horizontal and of constant thickness. Effects of gravity within the producing layer are assumed negligible. If the actual time-dcpendent flow problem is approximated by a acries of steady-state problems. the potential and stream function in the oil bank and water hank satisfy Laplace&apos;s equation in two dimensions. We can therefore use a poteiitiometric analog of this system. Potentiometric models have yielded uscful results in this laboratory&apos; and clsewhere in the study of a variety of secondary recovery problems. For the case where M = I, we generally prefer to use theoretical mcthods as well as a simpler dynamic analog. Except where otherwise noted, the ratio, side of five-spot/wellbore radius. is taken to be 3,600. a figure which corrcsponds to a normal-size wellbore in a 10-acre well spacing. THEORETICAL EVALUATION OF VARIOUS PILOT PATTERNS, Mw0 = 1 <&apos;he theoretical models which we used to examine the performance of various pilots are shown in Fig. 1. Image theory was used to determine the ratio of production rate to injection rate as a function of the volumc of the flood. The ratio of production rate to injection rate was chosen because this is an easily measurable quantity which is characteristic of a pilot

By F. X. Tartaron

The comparison of actual energy of comminution with theoretical surface energy presents a wide gap. On the other hand, Solid State Theory presents a viewpoint that places actual energy of breakage in understandable relationship with the theoretical strength of materials. Therefore the physical background of comminution theory favors Kick&apos;s Law. A11 discussions of comminution theory appear farfetched when comparison is made of the actual energy of crushing with the energy input called for by thermodynamic theory. Quoting Taggart:&apos; "The work required to crush halite was of the general order of 0.10 kg cm per sq cm of new surface produced (based on air percolation) whereas the theoretical energy of this new surface, based on thermodynamic calculation by a number of investigators is 1 x 10&apos;4 kg cm per sq cm or the work expended was 1000 times the new surface energy — an indicated crushing efficiency of 0.10 pct." It is frustrating to probe into comminution theory with the prospect that the energy one is measuring is 99.9 pct waste energy. Do the laws of comminution deal with energy consumed by mineral breakage or with waste heat energy? There seems to be something wrong. Solid State Theory may provide the answer. The following statement is particularly apt to the situation: "The fundamental fact about metals and indeed all materials, is that they are much weaker than they should be. By this, we mean that if the stress required to cause a given plane of atoms to slip past a neighboring plane is calculated from basic atomic and Solid State Theory, the resulting number is some 1000 times too large to be accounted for by errors in approximations or assumptions. We know now that this is due to dislocations, and we even have a fair understanding of the details of the process by which dislocations make a crystal weak. An intriguing possibility which suggests itself, results from the discovery of very minute metallic crystals which are apparently free of dislocations and which have a strength approaching the ideal value for a perfect crystal." Here we have an extraordinary reversal of the situation. The actual energy for breakage is only one-thousandth the theoretical instead of 1000 times as given by Taggart. But the theoretical energy presented by Solid State Theory is the binding energy between atoms (or ions) in a crystal lattice. This is the kind of energy envisaged by Kick&apos;s law. The surface energy or the other hand corresponds to Rittinger&apos;s law. Comparison of theoretical with actual, on the basis of surface or on the basis of Rittinger&apos;s law, seems to present an absurdity; yet Rittinger&apos;s law is confirmed by an impressive array of experimental evidence. Comparison of theoretical and actual on the basis of Solid State Theory or, in other words, on the basis of Kick&apos;s law seems reasonable; yet there is not a scrap of evidence to verify Kick&apos;s law. The situation appears paradoxical, but the developments in the next section of this paper show that the paradox is a hint to a strange turn of events. ENERGY-WEIGHT FRACTION FUNCTION charles3 has presented a final equation showing the relation of energy consumed in comminution to size of product n{-1) a-n + 1 E is energy, C is proportionality constant, a and n are constants and k is the coarsest size in the ground product (size modulus). In the course of study of a large amount of experimental evidence, Charles discovered that in the general case, a-n + 1 approximated zero. For some reason, a and n differed only by unity. But if a—n + 1 = 0 within experimental error, then it is obvious that Eq. 1 states that it requires an infinite amount of energy to grind any sample to any size. This is a case where nature gives an absurd answer to a man-made mathematical equation, hence something is wrong with the equation. An adjustment must be made to fit the equation with experience. TO reveal the nature of the adjustment, the differential equation preceding the second integration in Charles&apos; group of equations is given When Charles integrated the quantity xa-n it was almost impossible for him to realize that he was dealing with a special case of integration and that

Jan 1, 1962

By W. A. Hockings, M. E. Volin, A. L. Mular

Batch grinding tests at short times were made in a laboratory rod mill with 10 x 14 mesh quartzite in corn syrup-water mixtures of varying viscosity. The weight fraction broken and size modulus were found to be independent of viscosity up to 20 cp, but at higher viscosities breakage of the feed decreased while the size modulus increased. The mechanisms of the effect of viscosity are discussed on the basis of particle dynamics. The effect of viscosity of the suspending medium on grinding has not been accorded much attention, although both wet and dry grinding are used in bene-ficiation processes and the viscosities of water and air differ greatly. schweyerl investigated the rates of grinding of quartzite in a pebble mill with air, water and glycerol as the suspending media and found the rate in terms of new surface developed per thousand revolutions of the mill to be constant and dependent on viscosity up to 20,000 rev, but to decrease and become independent of viscosity for longer grinding times. Based on considerations of the drag forces exerted on suspended particles and on the grinding media under turbulent and laminar flow conditions, viscosity can be expected to inhibit the rate of grinding in a given system as it changes the flow conditions from turbulent to laminar. The objective of this study was to determine the effect of viscosity of the suspending fluid in the batch grinding of a homogeneous feed of uniform size for short residence times, a region of more practical interest than the long times investigated by Schweyer. In actual grinding operations, the viscosity of the fluid does not change, but the consistency (apparent viscosity) of the pulp increases with additions of fines. In this study the consistency of the pulp was not measured because it was thought that the grinding times were too short to alter the apparent viscosity appreciably.* *Minus 200 mesh quartz at 65% solids has an apparent viscosity of about 25 cp as measured by a simple consistometer. Personal communication from D. F. Kelsall to A. L. Mular. For the purposes of this study, the effect of viscosity on grinding can be shown sufficiently well on the basis of the cumulative weight fraction finer than the feed size at any time, and the size modulus, ku, of the finer than feed sizes. The parameter, ku, is obtained from a form of the Gaudin-Schuhmann equation. METHOD AND MATERIAL The grinding tests were accomplished in a laboratory rod mill 10-1/2 in. long and 8 in. bore with a 20-lb rod charge consisting of two 1-in., eight 3/4-in., eight 1/2-in., and eight 1/4-in. diam rods. The mill speed was 43.2 rpm. Charges of 400 g 10 x 14 mesh Wisconsin quartzite** were ground for times of 60, **Courtesy of Minnesota Mining and Mfg. Co. 120, 180 and 240 sec in 500 cc of fluid made up of corn syrup and water in measured proportions. The temperature of the pulp was measured immediately upon completion of grinding, and the size distribution of the quartzite was determined by a wash-wet-dry screening technique. Corn syrup was chosen because of its high viscosity and ideal viscous behavior, and also because its density is not greatly different from that of water. The viscosity of each fluid mixture was measured with an Ostwald viscometer at 25 C. and a correction was made for the temperature of the pulp according to tabulated values for sucrose solutions. The density differences were so small that they probably did not significantly affect breakage. RESULTS AND DISCUSSION The conditions of the grinding tests, computed viscosities, weight fractions finer than the feed size and values of a and ku are listed in Table I. The values of a and ku were taken from the linear portions of the log-log plots of cumulative weight fraction finer vs size for the various grinding times. The plots for tests in fluids of average viscosity 8.6 cp are shown in Fig. 1. In general all of the sets of curves were linear throughout most of the size

Jan 1, 1965

By R. A. Wattenbarger, H. J. Ramey

A systematic study has been made of the application of the real gas pseudo-pressure m(p) to short-time gas well testing. The m(p) function can be used in real gas flow problems to account for the variation of viscosity and z-factor with pressure. A mathematical model was solved numerically to generate solutions of various real gas flow problems. The effects of turbulence, formation damage and wellbore storage were included in the model. The analysis of simulated well tests showed that the interpretation methods normally used for liquid flow are generally accurate when the m(p) is used. For practical rates without turbulence, the solutions are not rate-sensifive, and the flow capacity.. kh and skin effect can be determined accurately from either a buildup or a drawdown test. However, the kh calculated from a drawdown test can be significantly low when turbulence is present. A case was simulated in which this error wa.s 36 percent. Turbulence does not affect the determination of kh from buildup tests. The proper determination of kh from p and pa buildup and drawdown plots is developed by analyzing their relationship to the m(p) method. A simple equation is given that can be used for long-range gas well performance forecasting. This expression is compared with a method pre-sended by Russell et al." Introduction The formation flow capacity and wellbore damage condition can be determined for liquid-producing tests by means of buildup and drawdown tests. These tests make use of short-time pressure transient data rather than stabilized flow tests that are often used for gas well testing. Tracy&apos; presented a method of gas well testing that was based on ideal gas equations and utilized p2 buildup plots. This method was shown to be good for low pressure wells. MatthewsZ suggested plotting p and using an average slope and average gas properties. This method was more successful on high pressure wells. Al-Hussainy, Ramey and Crawfordh howed that variation in gas properties could be simplified by using the real gas pseu do-pressure m(p). Some cases showed that the use of the m(p) function Original manuscript received in Society of Petroleum Engineers office Aug. 8. 1967. Revised manuscript received June 11, 1968. Paper (SPE 1835) was presented at SPE 42nd Annual Fall Meeting held in Houston Tex., Oct. 1-4. 1967, and at SPE 4th Annual Eastern Regional ~eetin$ held in Pittsburgh. Pa., Nov. 2-3. 1967. 0 Copyright 1968 American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. &apos;References given at end of paper. provided an accurate method of interpreting buildup and drawdown tests. Because the work of Al-Hussainy et al. included only a few actual cases, there was a need to explore the use of the m(p) method for a greater variety of flow conditions. The occurrence of turbulent flow around the wellbore often is an important factor in gas well testing. Swift and Kiel,&apos; and Carter et a1.&apos; treated turbulence in gas well testing, but did not consider the variation of viscosity and z-factor with pressure. This paper presents the results of an investigation of the application of the m(p) method to buildup and drawdown testing. Vhe investigation included the effects of turbulence, formation damage and wellbore storage. Gas Flow Equations To formulate the mathematical model, many of the assumptions usually used in well testing theory are applied. The system has radial geometry with a closed outer boundary and is composed of a horizontal porous formation that has uniform and isotropic rock properties and uniform thickness. Allowance is made, however, for a radial region of reduced permeability near the wellbore. This region represents formation damage. The geometry of the system is shown in Fig. 1. Darcy&apos;s law does not always apply to gas flow. A more general expression is needed for non-Darcy, or "turbulent" REGION OF DAMAGED PERMEABILITY SEALED BOUNDARIES ClRCULAR WELL BORE Fig. 1—Radial flow model. AUGUST, BT7

Jan 1, 1969

By David C. Jonson, W. Bruce MacKenzie, Arthur A. Bookstrom, Vaughn E. Surface, Neil K. Muncaster, Stewart R. Wallace

In mid-Tertiary time a wet silici-alkalic magma penetrated the Precambrian rocks of what is now the Tenmile Range of Central Colorado and formed the Climax Stock. The stock is a composite one and was intruded in four separate irruptions, each giving rise to a major hydrothermal event. The first three of these produced large stockwork ore bodies that cap and flank the tops of their respective source intrusives; the fourth, and last, yielded almost no commercial mineralization. The ore bodies and their related zones of alteration are circular to ring-shaped in plan and arcuate in section. The different phases of the stock are so positioned with respect to each other that many of the circular and arcuate zones of mineralization and alteration overlap or are juxtaposed. Evidence of a time separation between the different ore bodies is found in certain dikes that were intruded between periods of mineralization and are well mineralized in one ore body but are barren in another. If it is assumed that the parent magma at Climax contained 10 per cent "water," that the "water" contained 0.1 grams per liter of molybdenum, and that the processes of extraction were 100 per cent efficient, then approximately 10 cubic miles of magma would be required for the formation of each ore body. The volume requirements, taken in conjunction with the cross-sectional areas of the different phases of the stock to explored depths, suggest to the authors that each magma chamber was columnar, extended to a depth of thousands of feet, and was connected with a much larger master reservoir. The repetition of similar intrusive, structural, and hydrothermal events, indicates periodic tapping of this reservoir. A regular chemical evolution of the parent magma is suggested by the systematic change in metal ratios and differences in the sites of deposition, relative to the source intrusive, in each of the four separate igneous hydrothermal stages. The preparation of ground was due to changes in magmatic and/ or hydrothermal pressures within the igneous-hydrothermal columns; small, repeated, up and down movements of the rock enclosing the tops of the columns formed the stockwork fractures into which the mineralizers penetrated. The forceful emplacement of each successive phase of the Climax Stock resulted in a slight doming and rotation to the westward of the earlier-formed geologic features. Postore normal displacement along the Mosquito fault is estimated at about 9000 feet. Part of one ore body was cut by the fault and lies at depth below the Paleozoic sedimentary strata on the hanging wall. On the footwall side of the fault, subaerial erosion and glaciation removed a large part of one ore body and uncovered the top of a second.

Jan 1, 1968

By C. S. Sivaramakrishnan, Manjit Singh, Rajendra Kumar

Regarding liquid nzetals structurally as a suspm-sion of clusters , having derivated solid-state coordination, in truly liquid atoms, the recently developed Kuvlar-Samarin technique of centrifuging- in liquid state enabled the determination of the cluster sizes inAl-Cu mid Pb-Sn systems. It is shown that the colutne fraction of the clusters does not exceed 9 pct and the energy of their formation in Al-Cu is about 5.5 kcal per g-atom and in Pb-Sn eutectic alloys about 25 kcal per y-atoni. STRUCTURAL investigations of liquid state have principally followed the following three courses: i) studies with X-ray, electron, or neutron diffraction; these investigations have shown that there is a certain amount of regularity in the structure of liquid metals which can be defined by a coordination number and that the structure is a derived function of that in the solid state; ii) thermodynamical investigations which are based on the concept of ideal behavior; these describe the liquid state in terms of free-energy values and other thermodynamic functions; although these investigations are of help in the study of the general effects of alloying, they do not provide any structural insight into the precise atomic distribution in liquid state; iii) measurements of surface tension and viscosity; although it is natural to expect that the viscosity is related to the structure in liquid state, these investigations have so far only provided information which can be used by the foundry technologists and has been little utilized in formulating models of the structure of liquid state. As it happens, investigators in the three groups have worked almost independently of each other and there is practically no structural correlation between the results of one group with those of another. The purpose of the present paper is to indicate that the experimentally measured parameters of these three groups of research are closely related to the structure in the liquid state. STRUCTURE OF LIQUID METALS Although atomic distribution in solid and gaseous states is rigorously known, that of the liquid state is only appreciated on the fringes. There is no universal model of atomic distribution in liquid state, but two diverse models are at present hotly contested. The first, largely expounded by ~ildebrand,&apos; regards the liquids as condensed gas since many of their properties and much of their behavior can be adequately described by regarding them as fluids. The second mode12j3 considers that some form of near-solid as- sociation of large number of atoms exists in the liquid state. On the other hand, ~ernal&apos; was able to predict rather precisely the radial distribution functions in liquids on the basis of statistical geometric approach which considered that liquids are "homogeneous, coherent, and essentially irregular assemblage of molecules containing no crystalline regions or holes large enough to admit another molecule". He introduced the concept of pseudonuclei in the otherwise random structure as aggregates of closely packed tetrahedra which gradually merge into irregularity and continually replace each other. To what extent the pseudonuclei can be regarded as regions of near-solid association is indefinite but Bernal suggested that the concept of pseudonuclei can be compromised with the latter model if the near-solid associations are regarded as extremely dense and not necessarily crystalline. The difficulty in projecting the structure of liquid metals arises because they exhibit duplicity of character as some of their properties are closer to those of crystalline solids and others to fluids. There is an increasing tendency to discuss the structure of liquid metals in terms of the second concept according to which the structure of liquid metals may be conceived as consisting of i) clusters of atoms where the aggregation is a close derivative of that in the crystalline state, ii) individual atoms which behave like true liquids in respect to degrees of freedom and iii) excess number of vacancies. It is noteworthy that the introduction of only 5 pct vacancies is sufficient to transform crystalline matter into the liquid state. At any instant of time thermodynamic equilibrium exists between i, ii, and iii, but the relative proportion of the clusters and random atoms is not known. That this is so can be appreciated by the fact that, when liquid metal is rapidly cooled, liquid state vacancies may condense in the form of dislocation loops and vacancies in excess of their equilibrium number in solid state. These dislocation loops have been observed in thin foils of aluminum prepared from rapid cooled aluminum. As temperature increases above melting point the number and volume fraction of clusters decrease but those of vacancies and random atoms increase. Clusters are transient in nature. In pure metals the cluster is an aggregation of the metal itself. In alloys, however, the nature of the cluster largely depends on the interaction between solvent and solute atoms. If the interaction between unlike atoms is greater than between like atoms: the clusters are then aggregates of unlike atoms. Examples of this kind of system are A1-Cu, Mg-Pb, and so forth, i.e., systems which exhibit negative departures from the Raoult&apos;s law. In systems where the interaction between unlike atoms is smaller than be-

Jan 1, 1968

By Robert D. Pehlke, Weldon L. Daines

The reduction of manganese oxide from a basic slag by silicon dissolved in liquid iron at steelmaking temperatures was studied to determine the rate-controlling step for the process. The experiments were carried out under an argon atmosphere in a resistance-heated vertical-tube furnace. The two liquid phases, slag and metal, were held in zirconia crucibles. Sampling and chemical analysis of the slag phase led to time-concentration curves for the system. Results obtained by varying parameters such as temperature, stirring rate, concentration, and melt geometry defined the limiting reaction mechanism. The reduction of manganese oxide by silicon is a fast reaction and reached equilibrium within 90 min. The rate of stirring has a marked effect on the reaction rate, while, on the other hand, the effect of temperature is very small. This identifies the rate-controlling step as diffusion rather than interfaciul chemical reaction. The variation of other parameters revealed that a) diffusion of manganese in the iron phase is the slow step, b) the value of D/6 for manganese in iron at 1550°C is 7x 10-4 cm per sec, c) the energy of activation for the apparent rate constant is 7 kcal per mole, and d) increasing the amount of silicon in the iron decreases the diffusion coefficient of manganese. THE equilibrium relations pertinent to the reduction of manganese oxide from a steelmaking slag by silicon dissolved in liquid iron have been studied, but little is known about the mechanisms that control the rate of the process. The purpose of the present investigation was to study the rate of reduction by silicon and to determine the rate-limiting step. The effects of stirring rate, temperature, composition, and melt geometry were investigated as a means of identifying the controlling step. On the laboratory scale, relatively little work has been done in the area of slag-metal kinetics, primarily because of the problems confronted in attempting to contain liquid slags and metals in laboratory crucibles at steelmaking temperatures. Since graphite is an ideal material for a crucible, several investigations have been carried out on the reduction of various oxides from a slag by carbon in graphite-saturated iron and on the transfer of sulfur between a slag and graphite-saturated iron. Most of this work has been performed in induction furnaces. Studies where the container was not graphite are extremely limited. A single kinetic study of manganese transfer between molten iron and basic slags was carried out by Sano, Shiomi, and Matsushita.&apos; These investigators prepared calcium oxide and titanium oxide crucibles to contain the two-phase system. Only a few experiments were carried out and the results had to be treated qualitatively. Despite the paucity of kinetic data, a large quantity of information is available in the literature concerning the thermodynamics2-12 and physical properties&apos;3-21 of the slags containing manganese oxide and the dilute liquid iron alloys containing manganese and silicon which were involved in this research program. EXPERIMENTAL METHOD A vertical-tube resistance-heated furnace was used in this investigation to study the slag-metal reaction Although most previous slag-metal studies were carried out in induction furnaces, a resistance furnace has several advantages including better control of temperature and stirring rate. Hence, a resistance furnace was selected for this study, and its design is shown schematically in Fig. 1. Slip-cast, fine-grained, low-porosity, lime-stabilized cubic zirconia crucibles were used for the kinetic experiments. The crucibles were 4 1/2 in. high and 2 1/8 in. OD with a 1/16-in. wall. The crucibles were only slightly

Jan 1, 1969

By G. A. Vissac

The only fine coal washer with proved automatic controls, the feldspar jig is capable of good efficiencies even at low separating gravities, handles a variety of products, and treats 150 tph and over. IN continental Europe the feldspar jig is used almost exclusively for cleaning the fine sizes of coal; it operates with an artificial bed, made up generally of pieces of feldspar or of any other hard and heavy rock with good cleavages. A primitive form of this machine requiring close supervision was used in this country years ago, but output was small. The modern machines are generally air-pulsated Baum-type jigs, with capacities up to 150 tph, and treat sizes from % in. to 0, usually previously dedusted or deslimed. These jigs are equipped with very accurate and sensitive automatic controls insuring satisfactory operation even at low separating gravity and with feeds irregular in quality and quantity. This paper will deal more particularly with coal cleaning separation, but the technical studies apply equally to all minerals. Theory of the Feldspar Jig To understand fully the operation of the feldspar jig and to appreciate its potential value, it is necessary to measure and analyze the motion of small particles of various specific gravities, falling in various media, under various conditions. As no practical formulas were available for that purpose, the writer has developed new theories and established new sets of formulas, which cannot be demonstrated here. They are applicable to all minerals and to many problems of coal or mineral concentration. These formulas do not pretend to absolute accuracy, but results, tested by comparison with experimentally recorded data, are sufficient for practical purposes, as they give definite indications of trends, relative values, and order of magnitude of the many factors involved. For pulsating motions no earlier formulas are available. In the case of continuous currents or motions, such formulas as Rittinger&apos;s, Allen&apos;s, or Stokes&apos;, over a hundred years old, apply only to special shapes and sizes of particles, involve factors and coefficients difficult to measure or appreciate, and deal only with free motions. They are of little use in solving practical problems. Units used with the formulas presented here are in the CGS system. Lateral friction and suction have been taken into account according- to methods currently used in aerodynamics, and only hindered motion is dealt with. As the feldspar jig is primarily intended to create a fluid bed condition, the mixture of fluids and solids obtained is homogeneous, and the apparent specific gravity of the separating medium is calculated accordingly. For instance, if 20 pct of the volume is coal and shale, with an average of 1.60 sp gr, in a liquid of 1.10 sp gr, the actual specific gravity of the separating fluid is 0.80 x 1.1 + 0.20 X 1.6 = 1.20. The fundamental equations are as follows: The general motion of a particle falling in a medium of specific gravity equal to u, hindered motion. is dv/dt = g/Ls [ L (s-u) - u/g v2 [1] g = 981 cm per sec, acceleration of gravity, and L = thickness of the particle of specific gravity (s) and velocity (v). The initial acceleration dv/dt = g (1-u/s ) [2] is a function of the specific gravity of the particle, only for a given medium, and independent of its dimensions—an ideal separating condition. The terminal velocity Vt = gxL- (s-u)/u, [3] for a given value of u, is a function of s and L. Two particles, L, s, and L2 s2, are said to be equivalent if they have the same terminal velocity, and accordingly L1/L2=s2-u/s1-u. [4] Two such particles cannot be separated by continuous currents, in any concentrating device. Because the initial accelerated motion, which provides for efficient separation, is followed by a constant velocity, which has a limiting effect, it is essen-

Jan 1, 1956

By R. J. Reynik

A semiempirial small flunctation theory of diff- sion in liquids is presented, which employs a fluctuation energy assumed quadratic for a small atomic or molecular displacement and Einstein&apos;s random-iralh model. The resulting diffusion equation is given by In these equations. D is the diffusivity, is the average liquid shite coordination number (at interatomic distance d. cm. T is the absolute temperature, xu. em, is (the diffusive displacement. K, is the quadratic fluctuation energy force constant, and rg, cm, are the radii oj diffusing atoms A and B, respectively. The quantities Xn and K are calculated from the computer-filled values of the slope and intercept. respectively. The radius of self-diffusing atom or radii and of diffusing atoms A and B are eta United and compared with values reported in the literature.. The predicted linear variation of diffusivity with. It tempera lure htm been observed in approximately thirty-iire metallic liquid systems, and in over seventy-fiee other liquid systems, including the organic .alcohols, liquified inert gases, and the molten salts, ALTHOUGH the average density within a macroscopic volume element of liquid is constant for fixed total number of atoms. pressure. and temperature, there exist microscopic: density fluctuations within the respective volume element. As such the microscopic volume available to an atom and its Z first nearest neighbors at any instant of time fluctuates above and below the average volume available to these atoms. If one assumes that liquid state atoms vibrate as in a solid. and further postulates that the mean position of any atom in the liquid state is not stationary. but shifts during every .vibration a distance 0 5 j 5 xo. then every atom in the liquid state continuously undergoes diffusive displacements which vary in the range 0 5 j 5 ro. Mathematically. for a binary liquid system consisting of atcrms A and B. the maximum diffusive displacement. .YO, is defined by the equation: where d is the average liquid state interatomic distance at specified liquid state coordination number Z. and v~ \ and vg are the effective radii of diffusing atoms A and B: respectively. For self-diffusion. r^ equals rg , and Eq. [I.] reduces to: It is interesting to note that Eq. [l] or [2] can be used to compute the radii of the diffusing atoms, provided one had an experimental evaluation of xo. As such. the computed radii could be compared with metallic or crystallographic ionic radii to ascerlain the electronic character of the diffusing atoms. Thus it is proposed that in the liquid state the n~otion of an atom relative to its original equilibrium position of oscillation represents the thermal vibration of any atom and its Z first nearest neighbors. while the small and variable displacements. 0 5 1 5 xc,. of the centers of oscillation represent the complex diffusive motions of the atoms at constant temperature and pressure. This is consistent with data obtained from slow neutron scattering by liquids1 &apos; and resembles an itinerant oscillator model of the liquid state.&apos;" It is further postulated that the atomic displacements characterizing the liquid state diffusion process are essentially a random-walk process. As such. it nlay be described by Einstein&apos;s equation:&apos; where D is the diffusivity. sq cm sec-&apos;. j2 is the mean square value of the diffusive displacement. and i> is the frequency of density fluctuations giving rise to diffusion. FORMULATION OF DIFFUSION EQUATION The effective spherical volume occupied by an atom, as a consequence of a microscopic density fluctuation which enlarges the volume available to any atom, exceeds its average liquid state atomic volume by an amount: where AV is the enlarged spherical volume, v is the radius of the diffusing atom. and j is the elementary displacement distance from the original center of oscillation of the vibrating atom to a new center of oscillation position. For small atomic displacements. where c is a constant whose value depends upon the assumed geometry of the enlarged volume. For a spherical increase in volume, c equals 4nr2. Following the treatment of Furthl&apos; and ~walin." assuming the enlarged volu~nes AL7 for the diffusing atoms are distributed in a continuunl. the probability of finding a fluctuation in the size range 0 5 j 5 xo defined by Where c includes the geometric constant cl and Eij) is the fluctuation energy causing the volume change. But the proposed model assumes all the Z first nearest-neighbor atoms are centers of oscillation. and hence the probability that any of these atoms is adjacent to a fluctuation of magnitude 05j5xo is unity. Thus:

Jan 1, 1970

By C. K. Eilerts

Finite difference equations were programmed and used to integrate the second-order, second-degree, partial differential equation with variable coefficients that represents the transient linear flow of gas-condensate fluids. Effect was given to the change with pressure of the compressibility factor, the viscosity, and the effective permeability and to change of the absolute permeability with distance. Integrations used as illustrations include recovery of fluid from a reservoir at a constant production rate followed by recovery at a declining rate calculated to maintain a constant pressure at the producing boundary. The time required to attain such a limiting pressure and the fraction of the reserve recovered in that time vary markedly with properties of the fluid represented by the coefficients. Fluid also is returned to the reservoir at a constant rate, until initial formation pressure is attained at the input boundary, and then at a calculated rate that will maintain but not exceed the limiting pressure. The computing programs were used to calculate the results that would be obtained in a series of back-pressure tests made at selected intervals of reservoir depletion. If effect is given to the variations in properties of the fluid that actually occur, then a series of back- pressure curves — one for each stage of reserve depletion — is required to indicate open-flow capacity and related flow characteristics dependably. The isochronal performnce method for determining flow characteristics of a well was simulated by computation. INTRODUCTION The back-pressure test procedurel is based on a derivation of the equation for steady-state radial flow of a gas, the properties of which are of necessity assumed to remain unchanged in applying the test results. The properties of most natural gases being recovered from reservoirs change as the reserve is depleted and pressures decline, and the results of an early back-pressure test may not be dependable for estimating the future delivery capacity of a well. The compressibility factor of a fluid under an initial pressure of 10,000 psia can change 45 per cent and the viscosity can change 70 per cent during the productive life of the reservoir. There are indications that the effective permeability to the flowing fluid can become 50 per cent of the original absolute permeability before enough liquid collects in the structure about a well as pressure declines to permit flow of liquid into the well. The advent of programmed electronic computing made practicable the integration of nonlinear, second-order, partial differentia1 equations2.3 pertaining to flow in reservoirs. Aronofsky and Porter4 represented the compressibility factor and the viscosity by a linear relationship, and integrated the equation for radial flow of gas for pressures up to 1,200 psi. Bruce, Peaceman, Rachford and Rice5 integrated the partial differential equations for both linear and radial unsteady-state flow of ideal gas in porous media. Their published results were a substantial guide in this study of integration of the partial differential equation of linear flow with coefficients of the equation variable. The computing program was developed to treat effective permeability as being both distance-dependent and pressure-dependent. In this study all examples of effective permeability are assumptions designed primarily to aid in developing programs for giving effect to this and other variable coefficients. The accumulation of data for expressing the pressure dependency of the effective permeability is the objective of a concurrent investigation. FUNDAMENTAL DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATION OF FLOW Linear transient fluid flow is represented by the partial differential equation,

Jan 1, 1965

By M. H. Childs, K. E. Merklin

Blast furnace feed specifications are being revolutionized because of the large increase in iron pellet production from low-grade ores of the Lake Superior region. Recent laboratory findings aimed at a better understanding of factors influencing the physical quality of pellets produced from taconite concentrates are reviewed. Tests for determining better product quality, the effects of additives, and production factors are discussed. The current revolution in specifications for blast furnace feed is typified to a great extent by the recent tremendous growth in iron ore pellet production from low grade ores of the Lake Superior region. Indeed, pellets are responsible, at least in part, for many of the stringent new requirements that are being placed on the suppliers of furnace raw materials. As the tonnage of pellets going to the steel mills increases, more and more attention is being turned to the detailed characteristics that make this product an ideal blast furnace feed. Control of quality in pellets is similar to that of any other product. A basic understanding of the factors that are encountered in their production, from the crude ore through to the finished product, is required, along with careful process supervision and frequent quality checks. Normally, chemical grade is not a problem. This usually can be controlled within the desired specifications by minor operating variations in the con- centrator. While future changes in requirements may result in the need for self-fluxing pellets1 or other chemical changes, today&apos;s quality demands center primarily around the physical characteristics of the pellets. It is in this area that many of the recent laboratory studies have proven helpful to the operating plants. A better understanding of the physical factors involved in pelletizing has provided the basis for a continuing program of quality improvement. Tigerschiold and Ilmoni2 and Cooke with various associates are responsible for the development of much of the basic understanding of the pelletiz-ing process. These earlier findings have served as a most useful background for this work. LABORATORY PROCEDURES Magnetite concentrates produced from eastern Mesabi Range magnetic taconites were used for all of the work reported herein. The sizing and chemical analysis of a random sample are shown in Table t Additives usually are mixed with the concentrates in taconite plants before pelletizing to improve the strength characteristics of the unfired balls. Ben-tonite, soda ash, and coal are in common use. The bentonite used in these studies was of the Wyoming sodium type; the soda ash was a standard commercia1 brand. Anthrafines were used where coal was called for. The procedure for making batches of balls at the Hibbing Laboratory is as follows: In an especially designed mulling equipment 2800 g of dry ore are mixed for 10 min with the desired additives. Water is added and mixing continued for 1 1/2 min. The charge is removed from the muller and put through a mechanical shredder to break down any lumps that may have formed. sixty previously formed seed balls are placed in the balling drum, which is in fact a 6 x 16-in. diam airplane tire turning at 53 Wm. Two thousand grams of the shredded concentrate and additive mix are fed by hand into the tire at a con-

Jan 1, 1961

By C. S. Chang

PEARL starch, a corn starch prepared with a minimum of alteration, is a good selective depressant for iron oxide minerals.1 From what is known of the complex chemistry of corn starch, there is evidence that it consists of a mixture of saturated giant polymeric molecules, both straight and branch- chained, the constituents consisting of glucose units with hydroxyl radicals as active or polar groups. To add to the existing knowledge of the chemistry of the flotation process, it is desirable to ascertain the effects on the corn starch as an iron oxide depressant when some of the hydroxyl radicals are replaced by other active groups that are common to flotation reagents. The ideal way to carry out this investigation would be to test reagents containing different active groups derived from corn starch. The preparation should be such that the reagents possess the same degree and location of substitution and are ac- companied by little or no degradation of the starch molecules. This, of course, is almost impossible to realize, as it is difficult to control the degree and location of substitution as well as to prevent degradation of the starch molecule during preparation.

Jan 9, 1954

By J. C. Wilhoit, J. B. Cheatham

Although an oil well is a long cylindrical hole with an irregular bottom, it appears likely that the nature of the stress concentration at the bottom of the hole can be ascertained from an analysis of the stresses around a short cylindrical cavity with rounded comers and smooth bottom. Such a cavity is studied primarily because it leads more readily to a solution to the problem by the use of stress functions. In this paper the stress distribution around a short cylindrical cavity subjected to bit loading, overburden and drilling-fluid pressures is determined by means of an analytical solution which approximately satisfies the boundary conditions of the problem. From this solution the stresses at the corner of the hole are calculated to be about 35 per cent lower than comparable results obtained by photoelastic and relaxation analyses. This difference is apparently due to the large radius of curvature at the corner of the cavity in the present analysis. Since good agreement is obtained between the results of this analysis and the stresses calculated for a similar loading on a semi-infinite elastic solid, it is concluded that the bit action in the region near the center of the hole is not appreciably affected by the presence of the sides of the hole. INTRODUCTION Much has been written concerning drilling "under down-hole conditions" and pertaining to the stress distribution in the rock at the bottom of an oil well.1-5 For example, it is known that identical rocks can be drilled more rapidly at the surface than under subsurface conditions of pressure and stress.6 Information on the behavior of rocks under loading can be obtained from triaxial test data.7-9 From such tests it is found that rocks exhibit brittle failure when the confining pressure and pore pressure are equal, but the mode of failure may change to ductile as the difference between the confining pressure and the pore pressure is increased. Brittle failure implies that there is very little permanent deformation before fracture, whereas ductile failure indicates that permanent deformation takes place before fracture. Some rocks are ductile at differential pressures of 5,000 psi , but other rocks are brittle even at differential pressures of more than 50,000 psi. Cuttings embedded in mud at the bottom of the hole may act as a plastic mass which the bit teeth must penetrate in order to attack the virgin rock below. 10 During drilling, the mud pressure acts as the confing pressure, and in many cases the difference between the mud pressure and the formation pore pressure IS sufficiently low so that the rock most likely falls in a brittle manner. Very little penetratiori by the bit teeth may be required for brittle failure of the rock. For these brittle materials, the elastic stress distribution is of practical interest in determicing the possible effects of a given loading on failure. The penetration of bit teeth into ductile or plastic rock has been analyzed previously ll,12 and will not be considered further in the present work. During actual drilling, many teeth act at various points ovel an irregular hole bottom. In the present analysis a solution is obtained for an idealized problem of determining the elastic stress distribution caused by only one tooth acting alone at the center of a smooth hole bottom. It obviously is not proposed that any driller should put a special one-tooth bit on bottom, but it is hoped that extensions of the simpler problems can eventually lead to a better understanding of more complex actual drilling phenomena. To obtain a solution by the use of stress functions, a short cylindrical cavity with rounded corners and smooth bottom is studied. The solution to this problem should give an insight into the nature of the stress concentration at the bottom of an oil well which is in reality a long cylindrical hole with an irregularr bottom. After selection of the proper curvilinear co-ordinate system, the stress functions are expressed as series of Legendre polynomials. The coefficient of each term in the series is then determined to satisfy the boundary conditions at a finite number of discrete points on the boundary of the cavity with least square error. Results are compared with bottom-hole stresses obtained by photoelastic and relaxation analyses.

By J. B. Cheatham, R. B. Paslay

Rock stresses and steady-state flow rates induced by the pressure gradient associated with the flow of formation fluid into a borehole have been analytically determined for a permeable, elastic material saturated with an incompressible fluid. In this analysis, the material properties and loading are considered to be symmetric about the axis of the borehole and independent of axial position. For Case I the material is assumed to have uniform permeability in the radial direction, whereas for Case 11 the permeability is assumed to have been reduced in a localized region adjacent to the hole by either normal well completion and production operations or deliberate plugging during air drilling. Results of a numerical example indicate that, in the absence of plugging, the rock shear strength must be approximately two-thirds the formation fluid pressure in order to prevent rock failure. The required rock strength is high for small radial zones of plugging and decreases as the region of reduced permeability becomes larger; however, a depth of plugging can be reached beyond which there is no real gain in strength, although the flow rate can be further reduced. INTRODUCTION During normal production of oil from a well, it is often desirable to increase the production rate of the formation fluid by increasing the pressure gradient through the formation adjacent to the borehole. Depending upon the magnitude of this pressure gradient and strength of the rock material, this production-rate increase can cause sloughing of the hole wall. In many cases, the production - rate increase can result in excessive sand production, increased wear of production equipment, lost production time and expensive workover jobs. In addition, the phenomenon of increased rock bit penetration rate with the use of a gaseous instead of a liquid drilling fluid has been observed in oilfield drilling operations and experimentally demonstrated by various investigators for several years. The improvement obtained by employing this technique can be quite significant and offers apromising method for reducing drilling costs. However, air drilling is currently limited to geographical locations where high-capacity water-bearing formations are not encountered. This limitation has prevented widespread adoption of air-drilling techniques, because the water influx into the borehole interferes with efficient removal of the drilling cuttings and usually results in a condition such that the bit becomes "balled-up" or stuck in the hole. In an attempt to remove the water - intrusion limitation from air drilling, various chemical and mechanical water shut-off methods have been proposed. Goodwin and Teplitzl suggested one such proposal whereby the permeability of the water-bearing rock structure was reduced in the vicinity of the borehole. Although the development of a shut-off method based upon this approach would certainly be welcomed by the oil industry, it is conceivable that, under certain conditions of the pressure gradient, strength of the rock material and depth of the modified permeability zone, a stress field can be created that will result in an unstable hole. As part of their study, an analytical solution is given for stresses in an idealized model of a hole and the surrounding rock material. The purpose of the present study is to extend the analysis of Goodwin and Teplitz to gain more insight into the details and consequences of excessive production rates and formation water shut-off. In particular, simplified models of these problems have been analytically examined, which makes possible the evaluation of the type of stress fields that can be anticipated as a result of these production and drilling practices. Both problems solved concern the determination of the steady-state volume flow rate of the formation fluid and the resulting steady-state stress and displacement distribution in a hollow, cylindrical configuration. The cylinder of Case I, corresponding to the production-rate problem, consists of a material with a constant permeability from the inside surface to the outside surface; the cylinder of Case 11, corresponding to the water shut-off problem, consists of a material with a constant permeability from the inside surface to an intermediate concentric cylindrical surface and a second constant permeability from the intermediate surface to the outside surface-