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By J. A. B. Forster

W. G. WOOLF*—The paper has a wealth of data that take careful, detailed study. As has been indicated the highlights can be only touched in the paper. The design and the arrangement of the rabble teeth as well as some of the other details of operation show a very careful study and a scientific approach to the problem which in most plants using hearth roasters are matters of trial and error with empirical conclusions and, perhaps too often, just arbitrary supervisory preferences which may be erroneous. Here the problem has been approached from a scientific angle which think well merits careful study. The ability to treat the tonnages they get through these furnaces think is made easier by reason of the high sulphide sulphur in the resulting calcines which the balance of the plant—the leaching and so on—can cope with. As indicated by Mr. Weldon in his dis- cussion earlier this morning, in most zinc plants an attempt is made to have as low a sulphide sulphur content in the roaster calcine as possible. But in the operation of the Risdon plants, they are able to cope with sulphide sulphur content up to 5 pct which is far beyond the amount of sulphide sulphur that can be handled elsewhere. I think the autogenous roasting is made easier because driving out the last percentages of sulphide sulphur brings on the necessity for extraneous heat, and that is further exemplified because even at Risdon the author shows the desirability of using some oil to better the operation, particularly for uniformity of roasting results and to obtain more regularity of furnace operation. A. A. CENTER*—Regarding the hard crusts in the beds of the furnace and autogenous roasting, the paper speaks of the former being due, at least to a considerable extent, to lead sulphide in the crusts. I wonder if the author has considered the hard crusts as being due in large part to zinc sulphate. While I was developing large zinc operations we ran into hard crusts which had to be dug up. I took several lumps which were so hard they showed the marks of the rabble teeth very plainly and kept them in my office for some weeks. A visitor came along one day, and as he was having similar troubles, I told him what we were doing about ours. I started to pick up one of these lumps and to my amazement the thing crumbled in my hands whereas originally it had been so hard it wore the rabbles down very rapidly in the furnace. I sent the sample to the laboratory and found that it analyzed very high in water-soluble zinc and sulphate-sulphur. The crumbling of the lumps was evidently due to the anhydrous zinc sulphate gradually taking on moisture from the air and forming crystalline zinc sulphate seven parts water. The increase in volume would, of course, result in disintegration of the lumps. As to getting a high tonnage in the autogenous roast, that is partly because of finishing at about 5 pct sulphide sulphur. It is a real job to get down to, say, ½ pct sulphide sulphur, which is what is usually desired, and still put through a good tonnage.

Jan 1, 1950

G. F. LOUGhlin,* Washington, D. C.—With reference to Mr. Bain's paper: The scarcity of adequate geologic data on building stone reflects, among other things, a lack of interest on the part of geologists. It has been one of my duties to attend to matters relating to building stone and other structural materials that have come to the U. S. Geological Survey, but for several years my principal administrative and other duties in the Section of Metalliferous Deposits have prevented me from giving structural materials the attention that they deserve. I have felt that investigations relating to structural materials should be assigned to someone else, but thus far nobody with a sufficient interest in the subject has come forward; further, the specific calls for these investigations have been so few and far between that it has not seemed practical to draft any one for the job. The general situation is such that steady and substantial progress in the application of geology to the utilization of structural materials is not likely to be made unless someone with keen interest in the subject takes the initiative. If the start is once made, I know from experience that cooperation from the more progressive members of the quarry and building industries will be readily forthcoming. Mr. Sales' suggestions are all constructive and most welcome. A thorough, detailed discussion of them might be as long as his manuscript, but could largely be summarized with the one word "amen" qualified by the alibi that a chronic shortage of funds and therefore of personnel has prevented us from carrying out some of his suggestions already. A few of them might call for an Act of Congress before we could undertake them, and one or two, such as the preservation of mine maps, with which I heartily agree, would call for much additional space as well as clerical and supervising help. Several items in Mr. Sales' paper—for example, the question of balance between the scientific and more directly practical studies and the surveys or resurveys of the larger and smaller mining districts—have been covered in my contribution to the symposium. The suggestion of frequent reviews of progress in the study of ore deposits is excellent and can be followed without serious diversion from projects already on the Survey's program. Brief reviews by Survey men have been published in periodicals, and the contributions by several of them to the Lindgren Volume1 may also be cited as serving this purpose to some degree. The practice of publishing short papers by Survey men in outside journals rather than in Government reports is largely the result of experience and expedience. Our printing funds are limited, and considerable time must elapse between the transmittal of a paper and its publication. Time is saved therefore by sending papers to outside journals, and the editors of those journals are usually glad to receive them. Members of the Survey would doubtless prefer to have the products and byproducts of their official work appear in official Survey publications if conditions were more favorable. If we were left entirely to ourselves the balance between the more scientific and the more directly practical studies would be somewhat better than it is, but the

Jan 1, 1935

G. F. LOUGhlin,* Washington, D. C.—With reference to Mr. Bain's paper: The scarcity of adequate geologic data on building stone reflects, among other things, a lack of interest on the part of geologists. It has been one of my duties to attend to matters relating to building stone and other structural materials that have come to the U. S. Geological Survey, but for several years my principal administrative and other duties in the Section of Metalliferous Deposits have prevented me from giving structural materials the attention that they deserve. I have felt that investigations relating to structural materials should be assigned to someone else, but thus far nobody with a sufficient interest in the subject has come forward; further, the specific calls for these investigations have been so few and far between that it has not seemed practical to draft any one for the job. The general situation is such that steady and substantial progress in the application of geology to the utilization of structural materials is not likely to be made unless someone with keen interest in the subject takes the initiative. If the start is once made, I know from experience that cooperation from the more progressive members of the quarry and building industries will be readily forthcoming. Mr. Sales' suggestions are all constructive and most welcome. A thorough, detailed discussion of them might be as long as his manuscript, but could largely be summarized with the one word "amen" qualified by the alibi that a chronic shortage of funds and therefore of personnel has prevented us from carrying out some of his suggestions already. A few of them might call for an Act of Congress before we could undertake them, and one or two, such as the preservation of mine maps, with which I heartily agree, would call for much additional space as well as clerical and supervising help. Several items in Mr. Sales' paper—for example, the question of balance between the scientific and more directly practical studies and the surveys or resurveys of the larger and smaller mining districts—have been covered in my contribution to the symposium. The suggestion of frequent reviews of progress in the study of ore deposits is excellent and can be followed without serious diversion from projects already on the Survey's program. Brief reviews by Survey men have been published in periodicals, and the contributions by several of them to the Lindgren Volume1 may also be cited as serving this purpose to some degree. The practice of publishing short papers by Survey men in outside journals rather than in Government reports is largely the result of experience and expedience. Our printing funds are limited, and considerable time must elapse between the transmittal of a paper and its publication. Time is saved therefore by sending papers to outside journals, and the editors of those journals are usually glad to receive them. Members of the Survey would doubtless prefer to have the products and byproducts of their official work appear in official Survey publications if conditions were more favorable. If we were left entirely to ourselves the balance between the more scientific and the more directly practical studies would be somewhat better than it is, but the

Jan 1, 1935

By W. A. Rachinger, C. Graeme-Barber, R. L. Bell, R. C. Gifkins, T. G. Langdon

R. L. Bell. C. Graeme-Barber, and T. G. Langdon (Imperial College. London)— The internal-marker technique developed by Ishida, Mullendore, and Grant has enabled them to make some interesting observations on the role of grain boundary sliding in creep. Certain of their conclusions are supported by our own similar experiments on Magnox AL80 (Mg-0.8 pct Al) specimens containing several markers, full details of which will be published shortly. In particular, we would agree that: 1) Using the internal-marker method, there is little difference between Egb, the contribution made by grain boundary sliding to the total strain, measured at the surface and in the interior of a specimen. 2) The measurement of the change in grain shape during creep leads to an underestimation of the grain strain, and hence an overestimate of Egb.. However, the discrepancies between the values obtained by this method and those obtained using markers are not as great as Tables IV and V of Ishida, Mullendore, and Grant would suggest; see below. The authors' Eq. [I] which is used to compute Egb from the transverse displacement ITt of longitudinal marker lines is not given in the reference quoted,'' where in fact only relations for Egb in terms of the longitudinal displacements of markers were developed. As first pointed out by Gifkins and Gittins,12 a problem arises in the use of longitudinal markers. and we will show that Eq. [1] underestimates Egb in the interior by a factor of 2. In Fig. 12, two grains X and Y have become displaced by the vector AC which lies in the plane of the grain boundary ABCD separating them. A longitudinal marker has been split by this displacement into the two parts LA and CAW. If these were surface grains the transverse displacement, Ut. could be obtained by viewing normal to the "top" surface. If interior, a section through CD parallel to the top surface would show the same result, provided the longitudinal internal markers delineated planes normal to the top surface. (This is an essential feature of the markers not pointed out in the paper.) The important point is that the quantity Ut tan B (=BE=DF) only gives the elongation due to the one component AB of the sliding. The elongation V/tan 8 (=GD) due to the component AD must be added to produce the total elongation resulting from the boundary displacement AC. If the grain boundaries in a poly crystalline specimen are randomly oriented with respect to the stress axis it follows that In the interior of a specimen the two sums depend for any difference simply on the definitions "top'' and "side". On the surface it is possible that the two sums could differ somewhat. In either event it is clear that the authors' Eq. [I] underestimates Egb, and in the interior, and perhaps at the surface also, the appropriate formula is given by The suggested modification probably makes little difference to the comparison between surface and interior values of Egb where both were obtained from marker measurements. For the estimation of the correction becomes extremely important where the boundary contribution is large. In our own work on magnox, for example, we have obtained values of as large as 70 to 80 pct, under conditions of high temperature and low strain rate. The use of planar markers transverse to the stress axis would simplify the comparison between surface and interior behavior, because measurement of the longitudinal offsets Us of these markers could be summed to give the total elongation di-

Jan 1, 1965

By J. E. Gruzleski, O. Knacke, M. J. Meixner

The evolution of CO from small specimens of solidifying steel was investigated using a thermobalance technique. Gas evolution begins in the temperature range between 1400" to .1450°C. This behavior is independent of the initial carbon content and the cooling rate. At a constant temperature above 1220°C, the rate of gas evolution is temperature independent. Out-gassing of the specimen is thought to proceed through small channels formed by the coalescence of bubbles between dendrites. LIQUID metals have a higher solubility for gas than the solid. During freezing, these gases form bubbles which may either escape or remain trapped in the solid. Recent work1 on the evolution of oxygen from solidifying silver showed that the gas is not liberated until a large fraction of the melt has solidified, and that the gas is evolved in an almost explosive manner. Because of the commercial importance of CO evolution in solidifying steel, a similar investigation was undertaken to study the mechanism of gas liberation in steel. EXPERIMENTAL The thermogravimetric method of Knacke and eixner1 to follow gas evolution by weight changes was employed. Weight differences of 0.1 mg could be reproducibly measured on specimens weighing 2.5 g. These specimens were taken from alloys with varying carbon contents from <0.01 to 0.11 wt pct and <0.01 wt pct of each of silicon, manganese, and aluminum. Specimens contained in 8 mm ID magnesium alumi-nate crucibles were heated to 1550°C under 1 atm of very pure (99.995 pct) CO with a tungsten resistance furnace. The specimen temperature was measured with a Pt-Pt/Rh thermocouple located directly beside the crucible. It is felt that the temperature thus measured was within 5" to 10°C of the specimen temperature since on fusion the weight of the specimen increased due to uptake of CO, and this weight increase was always observed to begin within the temperature range of 1530" to 1540°C. Even after 3 hr there was a small weight increase with time indicating that the molten specimen was not completely saturated with CO. Therefore, complete saturation was not attempted, and all specimens were simply held for 3 hr at 1550°C before cooling. Two types of cooling experiments were performed. One consisted of continuous cooling at linear rates from 5°C to 20°C per min down to 1020°C. The second type of experiment involved continuous cooling at 10°C per min to a predetermined temperature, and then holding the specimen at this temperature for a certain time. During all of these experiments, the sample weight was recorded as a function of time and temperature. OBSERVATIONS Continuous Cooling Experiments. No gas evolution was detected above 1450°C. The initial weight decrease always occurred between 1400" and 1450°C with the average at 1430°C. With further cooling, gas evolution continued to about 1150°C. Between this temperature and 1020°C no further outgassing was observed. This behavior did not seem to depend on the initial carbon content nor on the cooling rate in the range investigated. Typical patterns of gas evolution are shown in Figs. 1 and 2 for different initial carbon levels and cooling rates. The total amount of gas liberated during solidification varied with the rate of cooling. The total weight change on continuous cooling at a rate of 10°C per min

Jan 1, 1970

By Stephen Royce

[PAGE Ground Movement Adjacent to a Caving Block in the Climax Molybdenum Mine (TP 2000 by JOHN W. VANDERWILT, Min. Tech., May 1946. Discussion by STEPHEN ROYCE and the Studies of the Design of Shaped Explosive Charges and Their Effect in Breaking Concrete Blocks (TP 2157 by GEORGE B. CLARK, Min. Tech. May 1947. Discussion by PHILIP B. BUCKY, V. H. CLARK, E. D. GARDNER, M. H. HAWKINS, H. L. WALKER, and the author) 2 See TP 2377, Min. Tech., May 1948 for discussion of San Manuel ProsPect (TP 2255 by H. J. STEELE and G. R. RUBLY, Min. Tech., Sept. 1947. Discussion by L. KENNETH WILSON, ELDRED D. WILSON and the authors)] Ground Movement Adjacent to a Caving Block in the Climax Molybdenum Mine STEPHEN ROYCE•-It is interesting to note from B, Fig 2, how closely the subsidence limits approximate to the 60° angle of draw, which is so commonly found to rule in large scale rock subsidence. The tension cracks shown at B are commonly found in the initial stages of subsidence movement and spread like a halo about subsidence limits as movement extends. These tension cracks usually grow into escarpments later on. Apparently from the illustrations and description given, a system of vertical fractures is the predominating weakness in the wall rock at Climax, and therefore both tension cracks and escarpments are prevailingly vertical in attitude. The illustrations in Mr. Vanderwilt&apos;s article seem to conform to the general rule that yield by subsidence is heralded by tension cracks which foreshadow the appearance of the subsidence scarps which are both described in the paper and shown in Fig 2. The difference between the B side and the A side of Fig 2 is apparently caused mainly by the vertical fracturing which controls subsidence at Climax as compared to the relatively homogeneous rock formation at A upon which the original figure in Peele&apos;s Handbook was based. The draw at B is exactly 60°, while the draw at A is a little flatter. The influence of the vertical fractures explains the difference. The surprising feature is the relatively small deviation from the general 60° draw line, which is about the average of rock subsidence experience. The influence of local fracture systems, faults and bedding in controlling subsidence all the way from extremely low angles of draw to vertical is rather fully discussed in the Peele&apos;s Handbook article referred to. The more illustrations of subsidence there are published the better will be our knowledge. The general principles are pretty well established. The wide local differences reflect the effect of local conditions. This paper is a very valuable contribution to the subject, and for the benefit of the mining industry, it is to be hoped that there will be many more such clear and concise presentations. Particularly block caving is apt to be of greatly increasing importance in the future of a mining industry that must meet high labor costs. More examples of block caving from small scale to large scale operations like those at Climax will serve a vital need. JOHN W. VANDERWILT (author&apos;s reply)-I would like to thank Mr. Royce for his discussion; however, in his comments the meaning of subsidence, draw and tension cracks is not entirely clear. As used in my discussion of subsidence at Climax, subsidence refers to movements involving appreciable downward displacement. Tension cracks indicate only

Jan 1, 1947

By A. D. LeClaire

This paper claims to provide a description of diffusion processes in solids which is said to be a reformulation of Fick&apos;s Laws and therefore, presumably, is supposed to be as general as those laws. In fact, it appears to provide no more than an atomic model of a rather restricted type leading even to results in conflict with experiment. The restriction lies in the assumption, carried throughout the paper, that even in a chemical gradient the jump frequency i of an atom of species i is a function only of the composition on the plane from which it jumps, i.e., ?i = ?i(cl, c2 ...). This means that atom jumps in opposite directions from a site would occur with equal probability. There would then be no tendency for atoms to migrate preferentially in one direction along the gradient, so the model is incapable of accommodating the well established phenomenon of atom drift in a chemical gradient.10-" A more general model that does not suffer this difficulty was discussed in a paper13 I wrote in 1958, and the results therein already contain as a special case those given by Dr. DeHoff. The simplest sufficiently general assumption is to suppose that ?i is a function ?i(a1 a2,...; wl, w2, ..-) of the concentration aj and ? ?j at the site from which and to which, respectively, a jump is made. Any such function can always be expressed as a function of the sum and difference of its arguments. Thus d is the jump distance and g a geometrical constant. Obviously this is zero, i.e., no drift, if ??ri/??j = 0. X is written in this form, revealing two components of the total drift, because it can be shown that only the second component, depending on ??&apos;i/???(?j -aj), actually contributes to the net diffusion flux Ji. This component can be shown,13 by a straightforward ther-modynamic transcription, to be responsible for the so called "thermodynamic factor" in the Darken approximation for a chemical diffusion coefficient is the activity coefficient of i. Because the jump rate ?i in DeHoff&apos;s approximation has no dependence on the concentration gradient it does not seem possible to derive the Darken equation from his Eq. [6] without the further rather arbitrary assumption that ?i = const ri. It is to be noted too that the author&apos;s derivation of Eq. [12] is only valid when ?i is in fact a function only of the cj at the plane from which the atom jumps—for only then is ci?i a unique function of position and therefore of A. Author&apos;s Reply R. T. DeHoff I am indebted to Dr. LeClaire for his penetrating discussion of my paper, "A Reformulation of Fick&apos;s First Law". I concede that it does appear initially that the relationship presented in my paper could be derived from the detail atomic model of random walk and atom drift presented in Dr. LeClaire&apos;s approach. However, there are some important differences in concept and viewpoint which deserve further amplification. Dr. LeClaire&apos;s description is penetrating, detailed, and fundamental. His approach describes an extremely complex phenomenon, the displacement of atoms in a multicomponent gradient, at a sophisticated level. For precisely these reasons, his analysis has not proved to be useful for experimental studies that have practical objectives. For example, in order to obtain the information required to complete the analysis of the jump frequency distribution in a ternary system, the following series of experiments would, I believe, be required: 1) chemical (gradient) experiments, covering the composition range;

Jan 1, 1970

By M. E. Learmont, R. J. Lipp, I. Iwasaki

In the selective flocculation of iron ores, the concentration of alkaline earth ions in pulp solutions plays a critical role. Sodium silicate, alone or with sodium tripolyphosphate, has been used as a dispersant. When Ca+ + concentration exceeds 15 to 20 ppm, however, only a strong complexing agent, such as ethylenediaminetetraacetate (EDTA) was found to be effective (Heerema, Lipp, and Iwasaki, 1979). Some iron ores release Ca+ + in excess of 20 ppm. With ores that contain gypsum, the Ca+ + concentration may reach several hundred ppm. The purpose of this investigation was to compare the effectiveness of EDTA with nitrilotriacetate (NTA) in countering the adverse effects of Ca+ +, particularly at high Ca+ + concentrations. Emphasis was placed on the use of NTA since it is more cost-effective than EDTA. It is also used in detergents in some countries (Malaiyandi, Williams, and O&apos;Grady, 1979). NTA, like EDTA, forms a soluble complex with Ca+ + with a 1:1 molar ratio. Its structure and the value of log Kca are given below together with EDTA (Shibata, 1963). [ ] For experimental materials, an equal weight mixture of Lac Jeannine hematite and a high-purity quartzite from Thompson, MN, both ground to - 30 µm (- 560 mesh) was used. Selective flocculation test conditions and procedures were identical to those used in previous investigations (Heerema, Lipp, and Iwasaki, 1979). The optimum level of starch addition for the above mixture was determined to be at 300 g/t (0.6 lb per st). Initially, a series of selective flocculation tests were performed on the mixture pulped with 10-3M CaC12 (40 ppm Ca + +) at pH 11. With no NTA present, flocculation of the entire suspension resulted. The selectivities achieved in tests with 90, 100, and 150% of the stoichiometric amount of NTA were virtually identical to each other and also to the selectivity of separation obtained in the absence of Ca+ + Such an observation could have been expected since Ca+ + concentrations up to a few ppm may be tolerated in selective flocculation (Krishnan and Iwasaki, 1982). A similar series of tests were performed at 10-2M CaC12 (400 ppm Ca+ +). The results are shown in Fig. 1 as grade¬recovery curves. For comparison, the results obtained in the absence of Ca++ are included. At 90% of the stoichiometric amount of NTA, a large enough concentration of uncomplexed Ca+ + was left in solution to cause indiscriminate flocculation. Note in this figure that NTA is capable of countering the adverse effect of Ca+ + even at this high concentration when it is used in excess of stoichiometric amounts. The free Ca+ + concentrations at 100 and 150% are calculated to be 2.5 and 0 ppm, respectively. [ ] The relative effectiveness of NTA and EDTA was then compared at 100% of the stoichiometric amount using 10-2M and 10-3M CaCI2 (Fig. 2). EDTA appears to be slightly more effective than NTA. The free Ca++ of 0.02 ppm in 10-2M CaC12 with EDTA compared to 2.5 ppm free Ca+ + with NTA might account for the difference. As evident, prohibitively high additions of EDTA and NTA were required to complex high concentrations of calcium ion. The use of both of these agents would result in the accumulation of complexed Ca+ + in recycle water.

Jan 1, 1985

By M. C. Flemings, T. S. Piwonka

Pore formation is examined analytically in cellu-larly solidified single crystals, in cylindrical castings of pure metals and alloys, and in unidirectionally solidified castings. Effects are considered of fluid flow. dissolved gas, and surface tension. Experimental results show that flow through a partially solid "mushy" aluminum alloy obeys relations similar to those applicable for flow through other types of porous beds. Flow rate is proportional to the square of the fraction liquid (fL) for fL less than 0.3. Results indicate models chosen for mathematical description of pore formation in alloy solidification are valid. ONE of the solidification variables which is influential in determining mechanical properties of castings and of wrought material produced from cast ingots is porosity. This porosity may result from shrinkage, dissolved gases, or a combination of both, and may take the form of either macroscopic pores in localized areas of the casting or microporosity evenly distributed throughout large areas of the casting. In alloys which freeze over a range of temperatures some microporosity is generally found after solidification. There has been much discussion as to whether the micropores are caused primarily by shrinkage, by gas, or by a combination of both.&apos;-3 There has, however, been little experimental or analytical work performed to delineate the relative contributions of shrinkage and gas in causing microporosity or to quantitatively describe the influence of processing variables on microporosity. Porosity other than microporosity (e.g., center line shrinkage) has also received only limited quantitative treatment.495 The aim of this work is to examine analytically the general problem of pore formation in solidification and to perform limited confirmatory experiments. Consideration is first given to shrinkage porosity in pure and nearly pure metals (shrinkage in cellular growth, centerline shrinkage in cylinders). Next, a discussion is given of formation of shrinkage-caused microporosity in unidirectional solidification and in "mushy" solidification. Finally, influences of dissolved gas and surface tension on microporosity are examined. Numerical results are given for aluminum and an aluminum alloy. Typical examples of microporosity are shown in Figs. 1 to 3, for 1) cellularly solidified aluminum, 2) sand-cast aluminum alloy, and 3) low alloy steel. SHRINKAGE POROSITY IN PURE AND NEARLY PURE METALS 1) Cellular Growth. In cellular freezing of an impure liquid, liquid grooves may exist over a considerable length of the growing crystal.6 The cross-sectional area of the grooves decreases with increasing distance from the cell tips; the root of the groove is blunt if the liquid at this point reaches eutectic composition. As a result of volume contraction during solidification, flow occurs down the groove, parallel to the growth direction; hence, there must be a pressure drop from the groove entrance to points within the groove. The geometry of the groove is simplified to that of a circular cylinder of radius r and length L, Fig. 4, L » r, and a simple and useful limiting case is obtained. This is that, in the circular cylinder freezing at constant velocity, the pressure at the root of the groove is lower than that pressure at radius r in grooves of other geometries where r decreases continuously to a lower limit at L. As discussed below, a pore resulting from a given pressure difference

Jan 1, 1967

By M. W. Clegg

The application of the Laplace transformation to problems in the flow of compressible fluids in porous media has provided a large number of exact solutions. For plane radial flow, however, these solutions are either complex integrals or infinite series and are of little value to the field engineer. In the case of production at constant well pressure, the available approximate solutions me valid for large times only. In this paper it is shown that an approximate inversion formula for the Laplace transform, developed for the solution of viscoelastic problems, is applicable to radial flow problems and provides simple analytical solutions to constant terminal pressure problems. The method may be used to obtain approximate solutions to many problems, including media with radial permeability discontinuities, multi-layer formations and pressure buildup in wells after shut-in. The results are compared with the few available computer solutions as well as the large time solutions, and it is shown that this approximate method greatly extends the time interval over which a simple analytical solution is acceptable. INTRODUCTION The study of transient problems in the flow of fluids through porous media has benefited greatly from the application of transform methods. The use of the Laplace transformation for solving parabolic equations has been widely discussed in the field of heat conduction and diffusion as well as in the petroleum literature. Removal of the time variable with the Laplace transformation generally reduces the problem to a boundary value problem which may be solved by standard techniques. A much more formidable problem then faces the engineer, however, for frequently the transform does not possess a simple inverse. The result is that the general inversion integral must be used and this leads to either an infinite integral or an infinite series, both of which are difficult to handle from a computational standpoint. Asymptotic approximations for the inverse have been known for some time and these yield approximate inverse functions that are valid for very large or very small times — but frequently the times of interest lie somewhere between these two extremes. Therefore, some acceptable approximation valid over a larger interval of time is desirable. During the past few years a number of methods for achieving this have been developed and some of these are discussed briefly in this paper. The relative merits of the various methods are not evaluated here, but some general conclusions reached by other authors are given. One of these methods has been applied to problems associated with the radial flow of compressible liquids to producing wells. In the case of production at constant well pressure, the method leads to simple analytical solutions for a number of standard problems; e.g., homogeneous formation, permeability discontinuities, pressure buildup. These solutions greatly extend the range of validity of the asymptotic ones (valid for large times only) and should be of value in studying the behavior of wells producing under constant pressure conditions. METHOD FOR THE APPROXIMATE INVERSION OF LAPLACE TRANSFORMS Much of the work that has been done on the approximate inversion of Laplace transforms has originated in viscoelastic problems. Apart from the asymptotic approximations,1 one of the first approximate methods was due to Alfrey.2 If (s) is the Laplace transform of f(t) defined by

By R. L. Smith, I. G. Greenfield

The binary system Ga-Sb has been investigated by thermal, X-ray, and metallo-graphic methods. &apos;The intermetallic compound GaSb melts at 705.9OC and forms a eutectic with antimony at 11.8 atomic pct Ga. This eutectic melts at 589.8OC. A eutectic appears to be formed at 29.8°C but is too close to the composition of pure gallium to be detected. IN the course of an investigation of the semiconducting properties of the intermetallic compound, GaSb, the phase equilibrium between gallium and antimony has been investigated. Recently, the phase diagram of similar systems, such as In-Sb,&apos; In-AS,&apos; and Bi-As3 have been determined. These diagrams have the same general appearance in that an intermetallic compound is formed at 50 atomic pct and a eutectic is formed on both sides of the intermetallic compound. Jenny,& who has been investigating the semiconducting properties of intermetallic compounds, has concluded from thermal data of several alloys that the phase diagram of the Ga-Sb system resembles those mentioned previously. This paper represents a more detailed study of the Ga-Sb equilibrium. Experimental Procedure The gallium that was used to prepare the alloys was obtained from A. D. Mackay Inc. and the Aluminum Co. of America; the latter reported the purity as 99.95 pct. The antimony was obtained from Bradley Mining Corp. who specified a purity of 99.997 pct. Since the intermetallic compound of GaSb was prepared primarily to be used for further studies of the semiconducting properties, additional purification by a standard zone-melting technique under an atmosphere of hydrogen was performed. Hall-effect measurements on this material indicated an impurity content of less than 3 ppm.5 The antimony also was purified by zone melting and used in conjunction with the compound GaSb to make the necessary alloys for the determinations of the antimony-rich side of the phase diagram. The antimony-rich alloys were chill cast in Vycor tubes which were evacuated to a pressure of less than one micron. Since these alloys expand on freezing, precautions were taken to prevent oxidation of the ingot in the event the tube fractured. To insure homogeneity, the brittle alloys were ground and then placed in the furnace for the thermal-arrest determinations. Since the cost of gallium is high, the experimental procedure was complicated somewhat because of the necessarily limited sizes of the melts. In order to economize, the gallium-rich side of the diagram was determined by using a 35 g ingot of pure gallium initially and adding increasing amounts of antimony in order to obtain additional alloys. The furnace used in the determination of the cooling curves is illustrated in Fig. 1. A 2¼ in. quartz tube, sealed at one end, was fitted with a brass flange which was attached with DeKhotinsky cement. The brass collar was water cooled to prevent the cement from softening. An "0" ring was used to provide a seal between the brass collar and the furnace cover. Nichrome ribbon was wound around a 2 ½ in. asbestos-covered copper core which was used to reduce the thermal gradient in the furnace. The insulation consisted of alternate layers of asbestos and aluminum foil surrounded by sil-o-cel refractory and

Jan 1, 1956

This volume summarizes characteristics and the geological setting of porphyry copper deposits of the Western Hemisphere. Grouped by characteristics, porphyry copper deposits are described in six regions: Andean orogen, Appalachian orogen, Alaska, northern Cordilleran orogen, southern Cordilleran orogen, and the Caribbean. Each region has its own peculiarities of geologic setting, and those deposit characteristics that may be projected from one chapter to another will be used in reaching the conclusions from which to draw genetic models for porphyry copper deposits. The term porphyry in this volume denotes a type of ore deposit, rather than a rock classification. The classification of intrusive igneous rocks used in this volume is shown in Figs. 36a and 42 (pp. 103, 130). Some modern classifications suggest mineral-ratio definitions other than those used in this volume; however, most literature references cited are based on the classification shown in Figs. 36a and 42. The reader should be aware, therefore, that quartz monzonite in this volume contains from 35% to 65% potassuim feldspar and at least 10% quartz. A term used in this volume is defined where introduced within the context of a specific chapter. Distensional tectonic conditions are used in the sense of Cady (1972) in the Appalachian orogen, for example. Porphyry copper deposits are defined as large bodies of mineralized rock, usually including a porphyritic intrusive phase, that have copper sulfide minerals disseminated through them. The definition has no commercial connotation and is used strictly in a noneconomic sense to encompass the spectrum of copper mineralization associated with these deposits. Terms commonly accepted in the literature are not defined. Porphyry molybdenum and porphyry copper deposits commonly are distinguished by most economic geologists. The basis for the separation is summarized in one chapter of this volume. To permit the reader to more easily peruse the bibliography for any particular region, references will be found at the end of each chapter. The bibliography is not intended to be complete, but is designed to provide adequate minimum coverage for descriptions of deposits in the chapter concerned. Each entry is properly referenced in the text of that chapter to facilitate the reader&apos;s research into any particular topic. This volume is written from the point of view of the exploration geologist whose task it is to evaluate the structural, petrographic, mineralogic, and metallogenic characteristics of a porphyry copper deposit. The goal of the evaluation is to determine if, and where, concentrations of metals may exist in an igneous- hydrothermal system that includes the most complex set of geologic factors found in any of the many types of ore deposits. The successful porphyry copper geologist is simultaneously a petrographer, geochemist, structural geologist, and mineralogist. Some porphyry copper deposits have developed in volcanic arcs underlain by a Benioff seismic zone with a trench paralleling the arc. The most convincing example of a deposit known to have developed in an environment where subduction is an active process is the 3.3 m.y. (million years) old Dry Creek occurrence in the Aleutian arc (this volume). On the other hand, the younger deposits in the Cascades appear to have formed in a volcanic arc lacking both the Benioff zone and the offshore trench. Subduction is not widely accepted as an active mechanism for the period when the younger Cascade deposits formed. Subduction has also been questioned for deposits formed at a great distance from a plate boundary (e.g., Bingham). Regardless of the pros and cons regarding subduction, most deposits are associated with regional faults. Strike-slip

Jan 1, 1978

By D. H. Thompson, A. W. Tracy

E. A. ANDERSON*—At the outset, I note that you are using a humid atmosphere containing ammonia but that you make no reference to the variable of carbon dioxide content. Edmunds in his work in this laboratory found very marked effects on the rate of failure due to variations in carbon dioxide in the environment. This point is, of course, strongly related to the overall corrosion rate problem since the role of the carbon dioxide seems to be to permit the formation of hygrogroscopic salts which give a controlled moisture film on the specimen surface, which in turn seems to support the intergranular penetration more consistently than would otherwise occur. (2) Your results with various copper binary systems are very striking. It is curious, however, that the time-to-failure minima all occur at percentages of the alloying element which are considerably below the solubility limits as given in the conventional equilibrium diagrams. If your theory regarding the presence of these impurities in some concentration in the grain boundaries is correct, then either the existing diagrams are not accurate in the room temperature region, or there is a difference in solubility in the grain boundaries as compared -with the grains. A study along these lines might be very revealing. (3) The undesirable effect of silicon in copper calls to mind our own observations on the influence of silicon on the stress corrosion cracking resistance of alpha brass. In this case, silicon was found to have a very marked beneficial effect, particularly when a heat treatment capable of producing a grain boundary constituent was used. I should appreciate it if in the discussion of your paper this point should arise, you would attempt to clarify the difference between the copper-silicon binaries and the copper-zinc-sili-con ternary alloys. D. H. THOMPSON and A. W. TRACY (authors&apos; reply)—(1) We have no data on the constancy of the carbon dioxide content of the air used for the test atmosphere but have believed that the normal carbon dioxide content of the air was as constant as could be obtained by using Edmunds&apos; method of removing carbon dioxide from the air and then adding a measured quantity. In any event, the test atmosphere was the same for all specimens in a test run. We agree with Mr. Anderson that the carbon dioxide content is important in the formation of a hygroscopic film on specimens. (2) We can imagine that there could be a concentration of solute atoms at the grain boundaries which would be so small that present methods would not detect it even if it exceeded the solubility limit of the alloy. There could be somewhat greater solubility at the grain boundaries for an interstitial solute. There is a possibility that phosphorus dissolves in this way but we believe that none of the other solutes studied does. In our present studies we hope to find out something about conditions at the grain boundaries of some of the systems mentioned in the paper. (3) Mr. Anderson points out that silicon brass, which has had a high temperature heat treatment followed by a quench. is highly resistant to stress-corrosion cracking. Although the test conditions used by Anderson* et al were different from our test conditions, it is noted that the silicon brasses which had not had the special heat treatment were less resistant to stress-corrosion cracking than copper containing approximately equal amounts of silicon. G. R. GOHN†—Nonferrous metallurgists and particularly, materials engineers, have been accustomed to thinking

Jan 1, 1950

By W. A. Backofen, M. L. Ebner

SINCE the early work of Gough with Hanson and Wright,l-3 the study of fatigue has been characterized by experiments on single crystals only in recent times.9-10 Now, increasing attention is given to this aspect of fatigue research for the insight that it may provide into details of mechanism. The investigations have concentrated to a large extent on the development of deformation markings on fatigued crystals, and have shown the cracks to originate in slip bands possibly preceded or accompanied by slip-band extrusions. Experiments of special interest to the present work were conducted by paterson5 on copper crystals and involved both metallographic examination and measurement of change in flow stress. Crystals were cycled in alternating tens ion-compress ion with a constant plastic shear-strain amplitude of approximately 0.8 pct, and were particularly revealing for their demonstration of hardening with accumulated strain similar to that in unidirectional straining, through an easy-glide stage I followed by a stage II of rapid hardening; deformation was not continued beyond 40 cycles, however, so that the eventual course of the hardening curve could not be decided. For the conditions used by Paterson, surface slip markings were similar to those observed in unidirectional straining, but there were no X-ray asterisms and no deformation bands on the surface. In current thinking about the nature of fracture in fatigue, two views relative to mechanism are generally acknowledged, with the reservation that both could apply simultaneously in some measure. As one possibility, fracturing across a slip plane is regarded as a result of loss of cohesion from the creation of many point defects by dislocation movement under the cyclic loading.&apos;&apos; On the other hand, fracture has also been taken to follow as a consequence only of the geometry of slip at a free surface, consisting of offsets and crevices which eventually become fatigue cracks.12, l3 The work of McCammon and Rosenbergl4 showing fatigue in polycrystals at 4.2oK makes clear that any long-range diffusion of point defects is unnecessary, yet studies such as those of Forsyth and stubington15 show that accelerated diffusion may be a characteristic of deformation under fatigue loading. Interest in obtaining data of possible use for resolving such questions led to the experiments described below. Copper crystals were conveniently loaded in alternating four-point bending at constant deflection, a test condition shown to approximate a constant plastic strain amplitude for a wide range of axial orientations. The method of testing was not readily adapted to extensive study of temperature effects, but an investigation of the geometry of crack formation could simply be made by orienting,the slip direction at different angles to the surface. EXPERIMENTAL PROCEDURES Single crystals were grown by a modified Bridgman method in a stationary gradient furnace under an atmosphere of purified dry nitrogen. Purity of as-grown crystals was 99.999 pct as determined by spectrographic analysis plus vacuum fusion and gravimetric analyses for oxygen and sulfur, respectively. Crystals were of reasonable perfection as evidenced by a critical resolved shear stress at 10 deg from (110) of 60 g per sq mm and half-width of a (400) X-ray diffraction line from one crystal of

Jan 1, 1960

By Jack A. Crichton, Henry J. Gruy

This paper discusses methods used in the estimation of natural gas reserves and the general conditions under which the various methods are applicable. The factors used in estimating natural gas reserves are reviewed. Errors which have been found to occur frequently are listed. Introduction The estimation of natural gas reserves has become of paramount interest because of the increasing importance of gas in the nation&apos;s economy. Gas is a most desirable fuel, a fact substantiated by the continued rise in the market demand. In addition, new uses for gas are being developed by the chemical, plastic, and associated industries, and plants are being built to make gasoline from natural gas. These are a few of the factors contributing to the importance of the nation&apos;s natural gas reserves. With the increased importance of gas there is an increase in the need for reliable estimates of the magnitude and availability of natural gas reserves. These estimates are being used currently: (I) to determine which fields contain sufficient available reserves to justify the construction of pipe-line outlets to serve particular markets; (2) to design pipe lines necessary to serve those fields adequately;] (3) to determine the location of industrial and chemical plants; (4) to finance the development of gas properties, and the construction of gas pipe lines; (5) to determine fair and adequate depletion allowances and depreciation rates; (6) to justify applications for gas pipe lines before various regulatory bodies; (7) to determine the number of wells required to exploit the reserves most economically; (8) to aid in establishing values for the purchase or sale of gas properties, and for purposes of inheritance taxes; (9) to determine equities under unitized operations; (10) to provide a basis for calculating the economics of gas-cycling operations. History or Estimation of Natural Gas Reserves The natural gas industry in the United States had its beginning during 1826 when natural gas was used for lighting the city of Fredonia, N. Y. The first natural gas pipe line was a 25-mi]le wooden line constructed from hollowed logs, connecting West Bloomfield and Rochester, N. Y. There is no record of an attempt to estimate the gas reserve at West Bloomfield but an attempt was made to determine the capacity of the well by measuring the time required to 611 a large balloon. One of the earlier publications concerning estimation of natural gas reserves is the "Manual for the Oil and Gas Industry," published by the Treasury Department, United States Internal Revenue, in i9i9. The following methods of computing "gas depletion" are listed: (I) production decline of the gas well or property; (2) decline in open-flow capacity; (3) comparison with the life history of similar wells or properties, particularly those exhausted Art

Jan 1, 1949

By P. B. Leavenworth

Twenty-six fields were found in the Texas Gulf Coast in 1940 as compared to 17 fields in 1939. Of these, 14 were oil fields and 12 were gas and distillate. About 150 wells were drilling at the end of the year. Production for the year 1940 amounted to 104,127,247 bbl. as compared to 100,-253,937 bbl. for 1939. Considerable wildcatting for Wilcox production featured the exploration, but discoveries were few and the type of fields found mainly gas and distillate. On the whole, the program was discouraging. New Fields Alta Lomu. Galveston County.—The Alta Loma field was discovered on May 20, 1940, by the Stanolind Oil and Gas Company&apos;s No. I John A. Hulen, which blew out at a depth of 9166 it. and ran wild for several days, with an estimated production of 400 bbl. per hour of 36" gravity oil from a sand in the top of the Frio. Water soon appeared but the well produced more than I3o,ooo bbl. of oil between May 20 and June 10, together with a large amount of salt water. Later the well was killed and recompleted on Nov. 23, for 276 bbl. of 34" gravity oil per day. Probably this was the most sensational discovery of the year, although development to the end of the year was disappointing. The discovery was the result of reflection seismograph work of the Stano-End Oil and Gas Co., the principal leaseholders. Bailey&apos;s Prairie, Brazoria County.— Glenn H. McCarthy9s No. I W. B. Munson was completed in a lower Frio sand from perforations 11,345 to II,349 ft. for 8 bbl. of fluid (50 per cent black oil, 27" gravity, and 50 per cent B. S. and W.) on Sept. 24, 1940. It was finally recompleted as a gas well from perforations 11,631 to 11,651 ft., on Oct. 9, 1940. The well had been drilled to a total depth of 11,860 ft. This well produced for only a short time and the area is at present inactive. Blessing, Matagorda County.—The Texas Company&apos;s No. I A. B. Pierce was completed from a Frio sand from 8265 to 8300 ft. on July 8, 1940, for an initial production of 29 bbl. of distillate, I 1/2 million cu. ft. of gas. One other gas and distillate well was completed in 1940. Borden, Wharlon County.—J. F. Hutch-ins&apos; No. 2 A. P. Borden Estate was completed on Feb. 11, 1940, for 108 bbl. of 35,6° gravity oil, from a Frio sand at 4758 to 4771 ft. This is the second well drilled in the area by Hutchins. Two other wells in 1940 were failures and there was no activity at the end of the year. Dyersdale, Harris County.-The Dyers-dale field was opened by the completion, on Sept. 15, 1940, of H. C. Cockburn&apos;s No. I G. Burkitt from a sand in the top of the Frio, through perforations from 4067-4072 ft., for 240 bbl. of low-gravity oil. By the end of the year the field had been fairly well outlined and did not appear to be a major discovery. Duck Bay, Calhoun County.—Coronado Oil Company&apos;s No. I Welder, alter having drilled to 10,002 ft., perforated casing from 5654 1/2 to 5659 it. and was completed for 60 bbl. of 36.8° gravity oil On July 27, 1940, from a Miocene sand.

Jan 1, 1941

By P. B. Leavenworth

Twenty-six fields were found in the Texas Gulf Coast in 1940 as compared to 17 fields in 1939. Of these, 14 were oil fields and 12 were gas and distillate. About 150 wells were drilling at the end of the year. Production for the year 1940 amounted to 104,127,247 bbl. as compared to 100,-253,937 bbl. for 1939. Considerable wildcatting for Wilcox production featured the exploration, but discoveries were few and the type of fields found mainly gas and distillate. On the whole, the program was discouraging. New Fields Alta Lomu. Galveston County.—The Alta Loma field was discovered on May 20, 1940, by the Stanolind Oil and Gas Company&apos;s No. I John A. Hulen, which blew out at a depth of 9166 it. and ran wild for several days, with an estimated production of 400 bbl. per hour of 36" gravity oil from a sand in the top of the Frio. Water soon appeared but the well produced more than I3o,ooo bbl. of oil between May 20 and June 10, together with a large amount of salt water. Later the well was killed and recompleted on Nov. 23, for 276 bbl. of 34" gravity oil per day. Probably this was the most sensational discovery of the year, although development to the end of the year was disappointing. The discovery was the result of reflection seismograph work of the Stano-End Oil and Gas Co., the principal leaseholders. Bailey&apos;s Prairie, Brazoria County.— Glenn H. McCarthy9s No. I W. B. Munson was completed in a lower Frio sand from perforations 11,345 to II,349 ft. for 8 bbl. of fluid (50 per cent black oil, 27" gravity, and 50 per cent B. S. and W.) on Sept. 24, 1940. It was finally recompleted as a gas well from perforations 11,631 to 11,651 ft., on Oct. 9, 1940. The well had been drilled to a total depth of 11,860 ft. This well produced for only a short time and the area is at present inactive. Blessing, Matagorda County.—The Texas Company&apos;s No. I A. B. Pierce was completed from a Frio sand from 8265 to 8300 ft. on July 8, 1940, for an initial production of 29 bbl. of distillate, I 1/2 million cu. ft. of gas. One other gas and distillate well was completed in 1940. Borden, Wharlon County.—J. F. Hutch-ins&apos; No. 2 A. P. Borden Estate was completed on Feb. 11, 1940, for 108 bbl. of 35,6° gravity oil, from a Frio sand at 4758 to 4771 ft. This is the second well drilled in the area by Hutchins. Two other wells in 1940 were failures and there was no activity at the end of the year. Dyersdale, Harris County.-The Dyers-dale field was opened by the completion, on Sept. 15, 1940, of H. C. Cockburn&apos;s No. I G. Burkitt from a sand in the top of the Frio, through perforations from 4067-4072 ft., for 240 bbl. of low-gravity oil. By the end of the year the field had been fairly well outlined and did not appear to be a major discovery. Duck Bay, Calhoun County.—Coronado Oil Company&apos;s No. I Welder, alter having drilled to 10,002 ft., perforated casing from 5654 1/2 to 5659 it. and was completed for 60 bbl. of 36.8° gravity oil On July 27, 1940, from a Miocene sand.

Jan 1, 1941

By B. C. Wonsiewicz, G. Y. Chin

THE presence of deformation bands in cold-worked metals is known to influence the development of deformation textures1 but has not been considered in any theoretical treatments. At the present time the most rigorous theory of deformation textures appears to be based on Taylor&apos;s concept of homogeneous deformation of a crystal within a polycrystalline aggregate.&apos; For each crystal the Taylor theory allows the operation of several equivalent combinations of five or more slip systems. Hence deformation bands can arise from the operation of a separate combination of slips in each of several portions within a crystal. On the other hand, a second type of band may occur as a consequence of slip on systems other than those selected on the basis on homogeneous deformation. This type appears to be associated with crystals of symmetrical orientations, such as the [loo] and [lll.] axial orientations in drawn wires of fcc single crystals studied by Ahlborn.3 In this note we propose a simple mechanism of deformation banding of the second type, and compare the theoretical results for wire drawing of [loo] and [Ill] with Ahlborn&apos;s observations. In outlini the model requires that the work to accomplish a given overall shape change by banding be less than or equal to the work to accomplish the same change by homogeneous deformation. While the work by homogeneous deformation, WH = ?T ;dri, can be calculated exactly by the equivalent methods of Taylor2 or of Bishop and Hill,4 the work by banding, WB, is less certain. The latter consists of 1) Wi, work by slip within a band, 2) Wb, work associated with formation of a boundary between bands, and 3) Wc, work to correct the difference in shape change arising from banding and homogeneous deformation. The Wi term can be evaluated explicitly for a band with an assumed set of slip systems, but a satisfactory analysis of Wb and Wc has not been obtained. Since WB = Wi + Wb + Wc WH, for banding to occur, however, a necessary condition is Wi = WH . Hence it seems reasonable to choose a band with the lowest value of Wi among competing bands consistent with Wi = WH, and then minimize Wb and Wc by arranging the physical configuration of the selected band. Wc is minimized by combining neighboring bands such that their net strain matches the macroscopic strain as closely as possible, and Wb is minimized by enforcing strain compatibility across the boundary between two neighboring bands, similar to the case of a grain boundary.5 It is realized that this procedure is not completely satisfactory, but it does seem to explain Ahlborn&apos;s observations reasonably well. The procedure is now outlined for the two corner orientations. a) [l00]. Fig. 1 shows a (100) stereographic projection. The results of the Taylor analysis for axi-symmetric flow by homogeneous deformation6 show that, while [loo] itself is unstable, the rotation of the tensile axis will nevertheless be confined to within about 15 deg from [loo]. Alternatively, less work may be expended by banding with slip on a pair of systems with primary-conjugate relationship, e.g., (ll1)[10l.I + (li1.)[110]. The latter represents minimum Wi for all possible bands.* Since there are four symmetrically

Jan 1, 1970

By Allan Martin, R. A. Swalin

Diffusion rates of manganese, aluminum, titanium, and tungsten in nickel were measured at temperatures between 1100° and 1300°C. Activation energies, Q, and values of the frequency factor, Do, were calculated for each system. Values of Q were found to be within 15 pet of that for self-diffusion of nickel in agreement with present theories of diffusion. ONE of the most interesting problems in diffusion concerns factors governing the order of diffusion coefficients of various solutes in a common solvent metal at low solute concentrations. An early suggestion by Rhines and Mehl&apos; was that diffusion coefficients might each approach that of the self-diffusion coefficient of the solvent itself. Later Thomas and Birchenall&apos; pointed out correlations between the order of the diffusion coefficients and the freezing point depressions of the solvent by the various solutes. Other correlations have been sought. To settle this question it seems likely that much more data will be needed and, in particular, data collected at low solute concentrations. This type can be obtained by special analytical techniques as well as by use of radioactive tracers. The following study was made of diffusion of manganese, aluminum, titanium, and tungsten in a common solvent, nickel, at relatively low solute concentrations: 1—Production of alloys of nickel with manganese, aluminum, titanium. and tungsten at concentrations of the order of 1 atomic pet. 2—Use of pressure-welded diffusion couples in cylindrical form. 3—Diffusion anneals in the temperature range of 1100" to 1300°C. 4—Lathe sectioning of the diffused couples. 5—Spectrophotometric analyses of lathe turnings for the solute being studied. 6—Calculation of the diffusion coefficient for each of the solutes as a function of temperature. 7—Interpretation of results with current theories of diffusion. Experimental Work Materials—Two types of nickel were used in this research. The grade designated as Vacuum melted Mond nickel was used for all diffusion systems except the Mn-Ni system for which the Vacuum melted electrolytic nickel was used. Analyses of the two grades are given in Table I. The manganese used in making alloys was 99.9 + pet pure, aluminum 99.99 pet pure, titanium 99.9 pet pure, and tungsten of C.P. grade powder. Alloys used in this investigation were prepared by melting in vacuo or under a low pressure of argon. After melting, all heats except those of nickel-manganese were poured into a cylindrical copper mold 3 in. long and 1 in. ID. In the case of nickel-manganese, the alloys were solidified in the melting crucible. For nickel-aluminum heats alumina crucibles were used and for all other heats, zirconia crucibles.

Jan 1, 1957

By Walter J. Karplus

By means of a finite difference expansion a fluid flow field in cylindrical coordinates with axial symmetry, is simulated by a network of electrical resistors. A series of DC analog computing units, connected to the network bounrlary corresponding to the free surface effectively adjust the network parameterr to satisfy simultaneously Laplace&apos;s equation and the borrndary conditions. The system automatica1ly "re1axes" to the correct boundary shape in a fraction of a second. A detailed application of the method to the water coning problem in oil wells is presented. INTRODUCTION The water coning problem in petroleum production is a prime example of a class of problems known as free surface problems. A free surface arises in fluid flow regions when the shape of the boundary of a single phase flow region is affected in some manner by the pressure or potential gradients within the region. When there exists an interface of two fluids of different densities, the shape of this interface will be determined by the relative velocities of the two fluids and will therefore comprise a free surface. Other free surface problems of interest to petroleum engineers include the gas coning, water injection, and gravity drainage probIcms. In each case it is desired to predict the shape of the boundary of the oil zone as a function of the flow rate, the well penetration and the physical parameters and dimensions of the system. Such problems do not lend themselves to the classical analytical treatments of boundary value problems because in this case the boundary of the potential field and the potential distribution within the field are interdependent and must be calculated simultaneously. A further difficulty arises from geometric considerations. Since the fluid flow is of the radial type in most cases, such effective methods as conformal transformation and the hodograph are not applicable. It is the purpose of this paper to present a method whereby free surface problems of this type may be solved rapidly to the desired order of accuracy by means of a combination of a network of electrical resistors and conventional analog computer units. The water coning problem is treated in some detail. Then the procedure for extending the technique to other free surface problems is indicated. FORMULATION Most commercial oil reservoirs consist of an oil-saturated sand, bounded above by an impermeable strata and below by a water-saturated sand. Gravimetric separation results in a horizontal oil-water interface. When an oil well is drilled so that it partially penetrates the oil zone, and production is initiated at a constant rate, the oil-water interface will assume a new shape as a result of the pressure gradients in the oil zone. It should be emphasized that this is a steady phenomenon. After an initial transient period the water phase will be completely static and only the oil will be in motion. The new steady-state shape of the interface, as shown in Fig. 1, depends upon the depth of penetration, d, the thickness of the oil zone, I, the production rate, q, the hydraulic permeability zone of the sand, k, the viscosity of the oil, µ, and the densities of the oil and water, p.,. and pw.. The problem is to determine this relationship. Darcy&apos;s law + = - (P f g p z).....(1) it.

Jan 1, 1957