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By S. A. Falconer

The benefication of non-sulfide or non-metallic minerals by froth flotation has been practiced commercially for at least 30 years. In the pioneer stages of development of this flotation art, most of the promoters investigated were fatty acids or soaps that had been found to float sulfide minerals. When commercial application of froth flotation started to assume importance in the late 1920' s and 1930's, a great deal of fundamental research work was done in the laboratory (11-14, 19, 22, 23, 26, 29, 32). Among other subjects, this work included studies of the relative effectiveness of the different fatty acids and their salts on the flotability of various non-sulfide and non-metallic minerals. The successful present-day use of fatty acid and petroleum sulfonate type promoters results from better understanding of the principles of non-sulfide flotation plus the developnent of effective selectivity modifying agents. Many investigators have contributed to a greater appreciation of the numerous interrelated factors involved in fatty acid and soap flotation of non-sulfide minerals. Of these, mention should be made of recent fundamental studies by Tagart and Arbiter (31); Wolstenholme and Schulman (35); Schulman and Smith (28); Wark and his co-workers(3,4) Du FIietz(l5); and J. Leja(24). According to J. Rogers, K. L. Sutherland and E. E. and I. W. Wark(27), "the basic requirements for a 'collector' in flotation are that it shall form a film over the surface of the mineral to be floated and that this film should be oriented so that hydrocarbon (non-polar) groups form the effective surface." When modified in that way, the mineral surface possesses some degree of air avidity. All investigators would agree with Taggart, Taylor and Ince (33)that such a film can be produced only if the collector compound contains an active (polar) group to be attracted to the mineral surface, as well as the non-polar group. The commonly used anionic-type promoters for non-sulfide, non-metallic and metallic oxide ores include: 1. Fatty acids, and their soaps and emulsions. 2. Fatty acids residues — "pitches" and "foots." 3. Sulfonated and sulfated fatty acids and so-called "petroleum sulfonates." 4. Paraffin chain salts — sulfonated and sulfated higher alcohols. 5. Cyclic organic acids or salts, such as naph-thenic acid and naphthenates. Certain fatty acids are more effective than others as promoters for particular minerals under certain specified conditions. Some of the more costly fatty acids or collectors of the paraffin chain salt type may give improved selectivity in some instances and are being used commercially in a limited number of operations. Local availability may dictate the choice of the fatty acid used — for example, the use of palmoil fatty acid for flotation of copper carbonate ores in the Belgian Congo. Coconut oil fatty acids have also found limited use.

Jan 1, 1961

By J. van Lookeren

In the case of a reservoir where the oil underlies a large gas cap and overlies bottom water, production can be inzproved considerably if wells are perforated below the water-oil contact rather than opposite the oil-bearing section. The theoretical arguments for this improvement, which in some cases may be tenfold, are supported by experimental evidence. INTRODUCTION Reservoirs frequently consist of one sand body containing an oil layer underlying a thick, gas-bearing section and overlying bottom water (Fig. 1). Wells in such reservoirs are customarily opened up opposite the oil-bearing interval or, rather, opposite the lower part of this interval (Fig. 2). More often than not, it is inefficient or prohibited to blow off large quantities of gas with the oil. This im-psses a severe limitation on the drawdown, which results in low oil-production rates. Because of the small difference in specific gravity between oil and water, the latter will easily cone upwards, so that the production of water together with the oil is not prevented by this completion method. In such a situation wells should be opened deeper, in the originally water-bearing zone. The gas can be shut off by not perforating over the upper part of the interval (Fig. 3) or by a tubing hung close to the bottom of the interval (Fig. 4). Initially much water will be produced, but when steady-state conditions have been established (as depicted in Figs. 3 and 4) larger drawdowns are permissible than in a conventionally completed well, and correspondingly higher oil-production rates will result. Water cuts are only slightly higher because of the small effect of gravity on the water-oil flow distribution. CALCULATIONS The ratio of improvement has been estimated by means of approximate calculations. The completion depicted in Fig. 4 is the most convenient basis for these calculations. By postulating that vertical flow gradients can be disregarded, the Appendix shows that the well-known Dupuit formula can be obtained except for some complication due to the presence of two fluids rather than one. The Appendix makes the reasonable assumption that, for suitable boundary conditions, the oil layer occupies a constant fraction of the total fluid height throughout the reservoir. The oil- and water-production rates can thus be related to the liquid levels in the borehole as follows:

Jan 1, 1966

By H. Kazemi, G. W. Thomas, M. S. Seth

The double-porosity model of Warren and Root7 for examining pressure drawdown and buildup phenomena in naturally fractured reservoirs has been extended to intetpret interference test results. Both analytical and numerical solutions are presented. The practicality of some of the assumptions in the Warren-Root model was also investigated. The standard procedure for interpreting interference tests in naturally fractured reservoirs beretofore employed an equivalent homogeneous reservoir model. A comparison of our results with the equivalent homogeneous model indicates the latter is inadequate for early time responses. For late time responses, however, the simpler equivalent homogeneous model yields favorable results. Finally, the effects of well spacing and the magnitude of the interporosity flow parameter are briefly cited. INTRODUCTION Interference tests can be used to provide reservoir information not obtainable from conventional pressure drawdown and buildup tests. The technique involves measuring the response in one or more wells arising from a pressure perturbation introduced into another well. It has been. successfully applied to single-layer, homogeneous, unfractured, isotropic and anisotropic formations to arrive at values of well storage capacity and a measure of directional permeability.l The techniques of interference testing usually can be extended to heterogeneous systems, provided the permeability contrast between any existing rock layers is not excessive.9 In such cases, the practical interference effects in an observation well of an interconnecting heterogeneous reservoir can be predicted using an equivalent homogeneous model. The calculated properties are then average values. Russell and Prats2 have demonstrated the utility of this approach for a single-phase, two-layer system. For naturally fractured reservoirs, some different approaches usually are necessary. The major departure from homogeneous or nearly homogeneous reservoir treatment is to characterize the fractured system as an entity having "double-porosity"; namely, we suppose the existence of both primary and secondary porosity, where the former is contained within a systematic array of identical rectangular matrix blocks and the latter is contained within an orthogonal system of continuous, uniform fractures (see Fig. 1). It is sufficient to assume that the permeability of the rock matrix is so small that movement of reservoir fluids to the producing wells is entirely through the fracture network. This assumption has been validated by Barenblatt3 and .employed by others?-7 Warren and Root7 used the double-porosity model to develop an interpretation of pressure drawdown and pressure buildup tests in naturally fractured

Jan 1, 1970

Jan 1, 1969

By E. H. Boswell, E. M. Cushing

The Mississippi embayment east of the Mississippi River is part of a vast geologic and hydrologic province. The abundance of water available is largely responsible for the agricultural history of the area and is perhaps the key factor in the rapidly accelerating trend toward industrialization. The multiple artesian aquifer systems that underlie the region and the alluvial aquifer along the Mississippi River are virtually undeveloped sources of fresh water in that they are yielding only a fraction of the water available. The potential yield of the 12 aquifer systems is estimated to be about 25,000 mgd (million gpd) of which only about 600 mgd is presently being withdrawn. The amount of water in storage in these systems is about 1600 trillion gal. Ground water having a dissolved-solids content of less than 500 ppm is generally available at depths of less than 1000 ft, and water having a dissolved-solids content of less than 1000 ppm is available in some places to depths of more than 3000 ft. Iron is the most common troublesome chemical constituent in the ground water. Water in varying amounts is also available from streams within the region. The amount of water which originates within the region and which leaves it as streamflow during a year averages about 50 million acre-ft (about 45,000 mgd). An additional 375 million acre-ft (about 340,000 mgd), which originated outside the region, leaves the region as streamflow during an average year. Some streams have considerable flow during periods of no precipitation and the water during these periods generally has a low dissolved-solids content. Large quantities of water from streams and waterbearing sands (aquifers) are available in the Mississippi embayment east of the Mississippi River. Most of the region is underlain by aquifer systems that will yield large quantities of water to wells, making ground water the most readily available source of fresh water. About 90% of the water used in the region is obtained from wells. Streams have been used primarily to dispose of, and dilute, wastes that have been dumped into them. The Mississippi, Ohio and Tennessee rivers, and the lower parts of the Tombigee and Yazoo rivers also serve as a means of transportation. This nominally water-rich region receives from 48 to 56 in. of precipitation in average years and is crossed by several large rivers. The largest, the Mississippi River, is the western boundary of the region discussed in this paper. The economy of the region is basically agricultural, but industry has become increasingly important since World War 11. The diversification and mechanization of agriculture have resulted in a surplus of manpower, most of which is being absorbed by the expansion of industry. Further industrial expansion probably will occur because of the available manpower, natural resources and economical transportation. Increasing demands on available water supplies will be made and the future economic growth of the region will be largely dependent upon wise utilization and management of the water resources.

Jan 1, 1969

Jan 1, 1965

By H. F. Silliman, Daniel R. Hull, Earl W. Palmer

THE brass-rolling industry has not had a great deal of experience with antimony in its product. There have been some recent excursions with antimony as a corrosion inhibitor in tubes, but in sheet brass it has been regarded with apprehension and generally avoided. This is not to say that [ ] contaminations have never occurred, and probably a great many brass mill men have had their own ideas on the effect of such contamination. Our own views were based upon certain experiences some years ago with strip castings that split lengthwise in the early stages of rolling. The nature of this splitting is shown in Fig. I, from a photograph taken in 1922 as a record of 20 bars of 65-35 common brass made with an antimonial brand of copper. These bars were cast 6 ¾ in. wide by 1 in. thick by 45 in. long. In the first pass through the rolls, 19 of the 20 split lengthwise throughout their length, while each half curved outward in a wide arc as it emerged from the rolls. [ ] In those days, if splitting was not a frequent occurrence, it was not unfamiliar. One can recall periods of short duration when large amounts of common brass would behave in this way. There was nothing imaginary about these spells. The affected metal would split lengthwise in the breaking rolls, and no maneuvering of mill practice made any difference. The condition would descend like a plague, demoralize the mill for several days, and disappear. It is doubtful if anyone knew the cause-however many explanations may have been current. In 1921 a quantity of split brass was traced to the use of fired small-arms

Jan 1, 1943

By W. L. Saunders

WHEN work was begun in September, 1910, on the rock excavation for the foundation of the gigantic dam at Valhalla, N. Y., which is to convert Kensico lake into an important storage reservoir of the Catskill Aqueduct system, one of the first steps taken by the contractor, H. S. Kerbaugh, Inc., was the development of a quarry to supply the stone needed in the construction work.The importance of a bountiful supply of suitable rock in an accessible location is apparent when it is considered that nearly a million cubic yards of cyclopean masonry will be contained in the completed structure, not to mention dimension and facing stones, concrete blocks, and mass concrete. The dam structure will be over 1800 ft. long and about 170 ft. above the present river bed. However, for a distance of some 800 ft., it will be carried down to solid ledge rock, which will make the finished structure about 300 ft. in maximum height. It will be about 200 ft. thick at the base and directly beneath the coping it will be 28 ft. thick. This dam will raise the reservoir surface 110 ft. above old Kensico lake. The reservoir capacity will be 29,000,000,000 gal.-equal to New York's total needs for over two months. The contract price for this work is nearly $8,000,000 and is second only in amount to the cost, over $12,600,000, of the Ashokan Reservoir work. Fig. 1 is a view of the dam site.

Jan 2, 1914

By G. S. Baker, S. H. Carpenter

Langenecker has reported significant reduction in strength of high-strength metals during ultrasonic t'ibration. Decreases in static yield stress an order of magnitude greater than the calculated stress amplitude of vibrations were reported by him for a stainless steel and for tungsten. in the present work, attempts were nzadc to verify such large reductions in yield and flats stresses using resonant librations of measured amplitudes. The results show that both the soft metals (copper and aluminum) and the high-strength metals (sleels) hare reductions in yield and .flow stresses no larger than the stress amplitude of libration. BLAHA and Langenecker,' investigating the effect of vibratory stresses on the deformation of zinc and aluminum, found a decrease in the static yield and flow stresses approximately equal to the vibratory stress imposed. Langenecker later extending this work to high-strength metals such as stainless steels and tungsten2 found a much larger effect. In these metals the static yield and flow stresses decreased by an amount more than an order of magnitude greater than the calculated vibratory stresses. It was suggested that this phenomenon could be a serious hazard to high-strength structures in an acoustic environment, such as a large rocket at launch, or could be used advantageously in the working of hard-to-work materials such as the refractory metals. Langenecker and others3 have investigated the effect of high-amplitude vibrations on many metal-forming operations. The results have been quite favorable with usually a large decrease in the energy necessary for the forming operation. The processes, however, involve large frictional forces and it is not clear whether the decreases in forming forces are to be ascribed to decreased friction or to decreased flow stresses. The present paper reports an attempt to determine the effect of accurately known vibratory stresses on the static yield and flow stresses. I) EXPERIMENTAL TECHNIQUE The standard multicomponent resonator technique for measuring internal friction has been adapted to allow simultaneous high-amplitude vibration, internal-friction measurement, and tensile deformation. This has been described in detail elsewhere.4 Longitudinal oscillations of approximately 50 kc frequency are excited in a multicomponent resonator by a quartz transducer. A second transducer measures the amplitude of vibrations. Fig. 1 shows a schematic drawing of the resonator and grip arrangement and the displacement pattern produced. Since the centers of the dummy bars are at displacement nodes, they represent points where attachments will not materially affect the vibrations. The force is applied by a standard Instron tester. The sample may be deformed while simultaneously vibrating at up to 104 strain amplitude. 11) EXPERIMENTAL RESULTS The effect of high-amplitude vibrations on the yield and flow stresses of copper. aluminum, tantalum. a soft iron, and a stainless steel has been measured. In addition samples of tungsten vibrating at stress levels up to 10' dynes per sq cm have been subjected to static stresses near the static yield stress with no indication of macroscopic deformation. Fig. 2 gives typical results for 99.999 pct polycrys-talline copper. A vibratory stress of 7 x 107 dynes per sq cm is applied at points "a" and removed at points "h". The behavior of aluminum is identical. In agree-

Jan 1, 1967

By V. Koump, T. F. Perzak, R. G. Olsson

Jan 1, 1965

By Alfred H. Bell

Drilling activity increased in Illinois in 1935. There were 34 com-pletions as compared with 26 in 1934 and 18 wells were drilling at the end of 1935. Some large blocks of acreage were leased in Marion and Clay counties near the central part of the Illinois structural basin. Production of oil in the state totaled 4,305,000 bbl., a decrease of 4 per cent from that of 1934 There was some curtailment of production in the early part of 1935 but none after April 30 as shown in the following table. Percentage Pro-duced of Potential ProductioN (USING September 1934 Period as BaSis) Jan. .l-Feb. 20............................................ 79 Feb. 21-Feb. 25............................................ 85 Feb. 26-Feb. 38............................................ 79 Mar. 1-Mar. 11............................................ 85 Mar. 12-Mar. 31............................................ 100 Apr. 1-Apr. 9.......................................... 80 Apr. 10-Apr. 30............................................ 86 May 1-Dec. 31........................................... 100 This is equivalent to an average curtailment throughout the year of 5 per cent. The price of Illinois crude oil at the wells remained constant during 1935 at $1.13 per barrel. A small amount of oil was sold at a 1ower price, notably from the Dupo field, but to arrive at a value for the state's production there would be little error in assuming an average price of 81.13, giving a total value of $4,864,465. The following table shows production in Illinois by moths in 1935, according to the U.S. Bureau of Mines. Jan............. 332,000 May.......... 382,000 Sept ,............. 370,000 Feb............. 295,000 June. ........... 858,000 Oct.............. 391,000 Mar............ 370,000 July............ 377,000 Nov............. 369,000 April............ 338,000 Aug............. 379,000 Dec............ 344,000

Jan 1, 1936

By Alfred H. Bell

Drilling activity increased in Illinois in 1935. There were 34 com-pletions as compared with 26 in 1934 and 18 wells were drilling at the end of 1935. Some large blocks of acreage were leased in Marion and Clay counties near the central part of the Illinois structural basin. Production of oil in the state totaled 4,305,000 bbl., a decrease of 4 per cent from that of 1934 There was some curtailment of production in the early part of 1935 but none after April 30 as shown in the following table. Percentage Pro-duced of Potential ProductioN (USING September 1934 Period as BaSis) Jan. .l-Feb. 20............................................ 79 Feb. 21-Feb. 25............................................ 85 Feb. 26-Feb. 38............................................ 79 Mar. 1-Mar. 11............................................ 85 Mar. 12-Mar. 31............................................ 100 Apr. 1-Apr. 9.......................................... 80 Apr. 10-Apr. 30............................................ 86 May 1-Dec. 31........................................... 100 This is equivalent to an average curtailment throughout the year of 5 per cent. The price of Illinois crude oil at the wells remained constant during 1935 at $1.13 per barrel. A small amount of oil was sold at a 1ower price, notably from the Dupo field, but to arrive at a value for the state's production there would be little error in assuming an average price of 81.13, giving a total value of $4,864,465. The following table shows production in Illinois by moths in 1935, according to the U.S. Bureau of Mines. Jan............. 332,000 May.......... 382,000 Sept ,............. 370,000 Feb............. 295,000 June. ........... 858,000 Oct.............. 391,000 Mar............ 370,000 July............ 377,000 Nov............. 369,000 April............ 338,000 Aug............. 379,000 Dec............ 344,000

Jan 1, 1936

By Y. Oka, H. Majima

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

Jan 1, 1970

By Cyril S., Smith

WE cannot but marvel at the fact that fire is necessary for almost every operation. It takes the sands of the earth and melts them-now into glass, now into silver, minium or other lead or some substance useful to the artist or physician. By fire, minerals are disintegrated and copper produced; in fire is iron born and by fire is it subdued; by fire, gold is purified; by fire, stones are burned for the binding together of the walls of houses.... Fire is the immeasurable, uncontrollable element, concerning which it is hard to say whether it consumes more or produces more." Thus wrote Pliny in his "Natural History," that great collection of fact and error which constitutes the principal contemporary description of Roman technology and which until the seventeenth century was the authority most often quoted and requoted by those who wrote on metals, as on many other subjects.

Jan 1, 1940

By W. C. Maurer

A study of the mechanics of shear failure of rock under pressure has been made. The transition from brittle to ductile failure occurs when the friction along the fracture surfaces exceeds the shear strength of the rock. A machine has been developed for measuring both the shear strength of rock and the friction along rock fracture surfaces. In this shear machine, the fractures are forced along definite planes; as a result, the rock is stronger than in compression tests where the fractures choose the weakest paths through the specimens. Friction is composed of several forces: (1) the forces to slide irregulurities on surfaces over each other; (2) forces to shear some of these irregularities; (3) molecular attractions across the fractures; and (4) forces required to break bonds resulting from melting and welding at the tips of irregularities on the surfaces. Friction along rock fracture surfaces is not proportional to contact pressure as usually assumed; instead, it increases as the contact pressure raised to the 0.4 to 0.8 power. INTRODUCTION It is difficult to produce tensile stresses under the high compressive stresses present in the earth; therefore, in deep drilling and geological faulting, shear failure predominates. This shear failure mechanism is complicated because rock is a heterogeneous material containing pore spaces, microfractures, elastic discontinuities and other imperfections. A study of shear failure of rock under pressure has been made to obtain a better understanding of this complex mechanism. A machine has been developed and used to measure the shear strength of rock and friction along sheared surfaces. This machine has certain advantages over the compression test — shear properties can be measured under low normal stresses and shear stress and normal stress can be varied independently. The first part of this paper is a description of shear failure of rock under pressure. The role of shear strength and friction along fracture surfaces in the shear failure mechanism will be discussed. The second part consists of a description of the shear machine and a discussion of the shear strength and friction data obtained using this machine. It will be shown that shear strength depends upon the size of the stressed zone, and that friction does not increase linearly with contact pressure as usually assumed. Attempts are made to relate some of the new concepts to tectonic failure and to dril ling. SHEAR FAILURE IN ROCK TRIAXIAL TESTS The shear failure mechanism is illustrated by the triaxial test shown in Fig. 1. In this test, a cylindrical rock specimen is subjected to a confining pressure, PC, and an axial pressure, pa. These external pressures produce normal a and shear T stresses on inclined planes within the specimen equal to1 where 0 is the angle between the specimen axis and a normal to the inclined plane. Fig. 2 shows how triaxial specimens deform as the axial pressure is increased. The pressure is

Jan 1, 1966

By W. M. Peirce

COAL'S importance in the metallurgy of zinc may be gauged by the fact that approximately a million and a half tons is so employed annually in the United States. This brief paper will show in what way the various steps in the treatment of zinc ores utilize this fuel, either in its primary form or as coke or gas, and to what extent. The first step in zinc metallurgy that requires the use of fuel is roasting the sulphide ore, which is the type of ore in which zinc commonly occurs. Since the roasting is nearly or completely autogenous, however, this use of fuel is too minor to warrant further attention. Concentration of zinc ores is usually carried out by conventional milling methods, but a pyrometallurgical method, the Waelz process, is used to concentrate certain low-grade ores, and to recover the zinc content of certain residues in concentrate form. The Waelz process uses a continuous rotary kiln in which a mixture of coal and zinc-bearing material is heated to a

Jan 1, 1948

Jan 1, 1969

CONTENTS PAGE INTRODUCTION AND ACKNOWLEDGMENTS. By George M. Fowler 2 Question 1-Is Structural Deformation of Some Character Always Necessary for the Migration of Mineralizing Solutions, Especially in Limestone or Other Dense Rock? DISCUSSION by: A. M. Bateman 10 S. F. Kelly 15 C. H. Behre, Jr 8, 13 G. B. Langford 13 T. M. Broderick 11 H. E. McKinstry 6 R. H. Clarke 13 D. H. McLaughlin 8 J. D. Forrester for M. H. Gidel . . . 4 W. H. Newhouse 13 G. M. Fowler 2, 6, 11 R. H. Sales 6 L. C. Graton 11 Edward Sampson 12 R. D. Hoffman 10 J. S. Stevenson 7 I. B. Joralemon 11 L. B. Wright 2 Question 2-What Are Some of the Chemical and Physical Characteristics in Cer¬tain Beds within a Limestone Formation that Determine Their Favorable or Unfavorable Aspects as Reservoirs for Ore and Other Introduced Minerals? DISCUSSION by: G. W. Bain 17 G. M. Fowler 21 C. H. Behre, Jr 21 L. C. Graton 18 J. S. Brown 16 G. F. Loughlin 18 E. L. Bruce 19 R. H. Sales 19 R. H. Clarke 20 A. K. Snelgrove 20 Question 3-Are There Any Criteria by Which One Can Determine When Min¬ing Operations Have Reached the Roots of an Ore Deposit? DISCUSSION by: J. M. Boutwell 29 G. B. Langford 29 T. M. Broderick 28 G. F. Loughlin 25 E. Y. Dougherty 27 R. H. Sales 24 G. M. Fowler 22 F. A. Thomson 26 I. B. Joralemon 22 Edward Wisser 30 General S. F. Kelly 35 References 36

Jan 1, 1940

By N. A. D. Parlee, M. M. A. El-Naggar, G. 8. Horsley

The apparatus was of a modified Sieverts type1&apos;2 with a reaction tube designed to function also as the cell component of an oxygen pressure gage. The reaction tube assembly is shown in Fig. 1 and described in Table I. The main body is a 1-in.-OD 15 mol pct calcia-stabilized zirconia tube. The ground-glass joint, g, the male part of which carried a prism for optical pyrometer use, was connected to the remainder of the Sieverts apparatus. The calibration of the cell (at its 1000°K operating temperature) for internal oxygen pressures (PO2in) in the range 0.2 to 10 9 atm was made against internal air pressures measured by mercury manometer and McLeod gage. At very low po2in values down to l<Tn atm the performance was checked by comparing pQ calculated from electromotive force measurement when the apparatus was filled with a) pure CO gas and b) a CO-CO2 mixture of the ratio CO-CO2 = 1:0.398, with Po, values calculated from the equilibrium constants of the following reactions at 1000°K:

Jan 1, 1968

By H. K. VAN POOLLEEN, W. Hurst, H. C. Bixel

In an earlier publication* I showed the development of the instantaneous point source solution for a well producing at a constant rate at the center of a system of two radial, adjoining sands of different permeabilities. I also showed how a wellbore radius could be incorporated if this were the development required. For the purpose of that paper and the point I wished to make, this solution served. It yielded a simplifying solution of sands in series identified by Eqs. VI-26 and VI-27. (These equations are reproduced in part by Bixel and van Poollen as their Eq. 3, Page 302.) I am well aware that the simplified solutions are approximations. The facts that these Ei-functions satisfy the diffusivity equation and meet the boundary conditions both at the well and at the interface between sands, and that the earliest time considerations yield zero pressure drop, are evidence that the solutions are good approximations. This was the purpose for these formulas, which I interpret that where the permeabilities are of finite order of magnitudes in the respective media, the relations give satisfactory answers. However, from the authors&apos; point of view there are reservations, which leads me to consider their entire generalization as illustrated in their Figs. 1 through 14. What I understand from their storage capacity, relationship Fs, which is the ratio of the product of porosity and compressibility of Sand II to Sand I, that is employed here by the authors as a correlative means of identifying one figure from another, there is nothing in the exact mathematics that supports such a term as a means of correlation. This is taken out of context, unsustained that this applies. Here the mathematics are known, where the true correlation is the diffusivity term of one sand to another. This entails permeability that refers back again to the term M in their paper for the transmis-sibility of fluid from one sand to the other that should have also been a part of this storage capacity. In Figs. 8 through 14, which purported to show the simplified equation following this correlation, there are apparent discontinuities for M equal to or less than 0.1000 that differ from the trends for greater values of M. In this connection there is no reference radius given for this cumulative pressure drop in the paper that would permit me to make these calculations. However, what is clear for M equal to 0.1000 or less we are treated to almost a closed barrier at the interface; a condition entered by the writers without qualification. This across-the-board treatment of these simplified formulas, whatever the motive of the authors, enters into a range of permeabilities that even I as the contributor would have reservations that such could apply. Under comparable conditions I would be looking at a closed reservoir or some facsimile for such. The extent to which the approximation or the exact equations pertain is not evident in the generality offered by Bixel and van Poollen in their paper. Only when they or other writers show the comparison of the pressure distributions within the formations can one be able to judge the rational limits to subscribe to the simplified formulas.

Jan 1, 1969