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JOSEPH DANIELS, Seattle; Wash. (communication to the Secretary*). Mine models are of great value to the student and to the layman in visualizing underground conditions, and, in a measure, should be just as much a part of the equipment of a mining school as sets of ore specimens, catalogs, or drawings. The chief drawback, as Professor Stock has indicated, is the time and cost of making them. The greatest good in the development of the model comes to the person who makes it, and I have often wondered whether it might not be possible to have each student make one model during his undergraduate career. At the University of Washington we have been collecting sets of mine timbers, made of 6 by 6-in. lumber, illustrating shaft, level, and stope timbering. This work has been done by freshmen. Perhaps each educational institution could specialize in the creation of a certain type of model and exchange its product for that of another school, just as we now do with ore collections. In this way we could build up good museums at a relatively low cost. JOHN R. CHAMBERLIN, New York, N. Y. (communication to the Secretary?).-The type of model described by Professor Stoek is admirably suited to exhibits of coal mining in horizontal veins, and fulfills perfectly the three purposes mentioned. As an educational illustration in colleges, especially when the students themselves build the model under the direction of their instructors, it is of the highest value; and as each condition in mining requires special methods to depict properly in a model, the student is compelled to exercise considerable inventive resource to construct a true representation, stimulating the imagination and interest in his subject. Models built for exhibits in lawsuits are usually constructed to show prominently some particular feature of vein formation or geology, or the relation of underground workings to surface property lines, and as such their use is invaluable to the successful litigant. When the litigation is ended, it is natural that the model should be relegated to the basement and frequently destroyed, as data on underground workings are often considered of a private character by the operators; and it is

Jan 10, 1917

Jan 1, 1938

Jan 1, 1938

By Northcutt Ely

A year ago the petroleum code was in effect, and Congress had before it bills with powerful backing designed to extend and put on a permanent basis the Federal authority impliedly recognized in the petroleum code. Those men in the industry who were convinced of the unconstitutionality of Federal regulation, who believed in conservation, and who were seeking accomplishment of a conservation compact among the oil-producing states, were at that time encountering difficulties in bringing about such an agreement. Since then there has been a shift in the center of gravity. Today, by virtue of the Supreme Court's Schechter case decision, the petroleum code is not in existence, the industry is not operating under an oil administrator, and the Federal-control bills are not on Congress' calendar. In view of what the Supreme Court has said in the Schechter case and the Hoosac Mills case about the local character of mining and manufacturing, the problem today is not whether the Federal Government or the State Governments ought to regulate oil production, but how effective we can make the devices that are available to us under the constitution for the conservation of the nation's oil resources. The interstate oil conservation compact ratified during the past year by Texas, Oklahoma, Kansas, New Mexico, Colorado and Illinois, approved by Congress, but not yet in effect in California, is the mechanism now at work. Interstate Oil Conservation Compact What is this interstate compact? The basic theory is simple and logical. It has four elements to it. The first point is that in order to effect the maximum recovery of our oil arid gas reserves it is necessary for some governmental authority to regulate the way in which those resources are taken out of the ground, so that the reservoir energy is efficiently used, and used equitably by the various owners of the pool. Second, that the state, as distinguished from the national government, possesses the police power necessary to protect the public interest and to protect the correlative rights of the common owners of the oil pool. Third, that the state governments ought to cooperate with each other through an interstate advisory commission so that orderly production in one state is

Jan 1, 1936

By Northcutt Ely

A year ago the petroleum code was in effect, and Congress had before it bills with powerful backing designed to extend and put on a permanent basis the Federal authority impliedly recognized in the petroleum code. Those men in the industry who were convinced of the unconstitutionality of Federal regulation, who believed in conservation, and who were seeking accomplishment of a conservation compact among the oil-producing states, were at that time encountering difficulties in bringing about such an agreement. Since then there has been a shift in the center of gravity. Today, by virtue of the Supreme Court's Schechter case decision, the petroleum code is not in existence, the industry is not operating under an oil administrator, and the Federal-control bills are not on Congress' calendar. In view of what the Supreme Court has said in the Schechter case and the Hoosac Mills case about the local character of mining and manufacturing, the problem today is not whether the Federal Government or the State Governments ought to regulate oil production, but how effective we can make the devices that are available to us under the constitution for the conservation of the nation's oil resources. The interstate oil conservation compact ratified during the past year by Texas, Oklahoma, Kansas, New Mexico, Colorado and Illinois, approved by Congress, but not yet in effect in California, is the mechanism now at work. Interstate Oil Conservation Compact What is this interstate compact? The basic theory is simple and logical. It has four elements to it. The first point is that in order to effect the maximum recovery of our oil arid gas reserves it is necessary for some governmental authority to regulate the way in which those resources are taken out of the ground, so that the reservoir energy is efficiently used, and used equitably by the various owners of the pool. Second, that the state, as distinguished from the national government, possesses the police power necessary to protect the public interest and to protect the correlative rights of the common owners of the oil pool. Third, that the state governments ought to cooperate with each other through an interstate advisory commission so that orderly production in one state is

Jan 1, 1936

By William Ide Pierce

It is often desirable to work an old mine that has not been in operation for some time arid that is filled with water. This is especially true in Nova Scotia, where no great depth has yet been reached in any of the mines. The following is a description of a self-dumping water-tank which was used by me at Tangier, for draining a mine, and which did the work of a large pump, while the original cost is merely nominal, in comparison with the price of a pump. The tank, A, Pig. 1, is 1 1/3 by 3 feet and 4 feet high, inside measurement, made of 2-inch stuff and securely bound with iron. There are two clacks in the bottom which open as the tank is lowered into the water, and which close and hold the water as the tank is raised. B is an iron rod running through the tank, below and back of the center, and projecting beyond the sides of the tank. C is one of two guides, and D, a movable iron rod in each guide. This rod, attached to an upright, E, is moved forward and close up

Jan 1, 1886

Jan 1, 1925

By W. E. Wickerham

THE Magnetic Storm Monitor is an instrument that continuously records variations in the earth's total magnetic field at a fixed location. It is intended for use in conjunction with airborne magnetometer surveys. There are stormy periods when changes in the earth's magnetic field are so violent that data obtained with airborne equipment would be of little value. The Monitor provides this information so that flights may be cancelled or terminated, thus eliminating useless flying. Moreover, when some unpredictable change in the earth's field occurs during an airborne survey, the Monitor record, showing a similar event, may serve to salvage the data. The utility of a base instrument is determined largely by its physical dimensions, power requirements, stability, and reliability. This instrument is small enough to be transported in any powered vehicle. It will operate from either a 110 v, 60 cycle power line or from an inverter powered by an automobile battery. The development of an electronic feedback circuit that measures the magnetic field strength and a fabricated buckout magnet having essentially zero temperature has contributed to instrument stability and reliability. A tripod supporting the magnetometer head, a chassis housing all electronic components, and an Esterline Angus Recorder comprise the equipment, shown in Fig. 1. The vibrator-type inverter shown is used when a 110 v, 60 cycle power line is inaccessible. Approxi- mately 60 w are required to operate the instrument and the circuit is insensitive to reasonable power line voltage changes or transients. The head unit contains a fluxgate-type magnetometer element and a special magnet mounted on a ceramic structure. Since this magnet is used to buck out most of the earth's field, adjustments are provided for properly positioning the magnet with respect to the element. This assembly is mounted in such a way that the element can be oriented into the total field. Fluxgate Element: The heart of the Storm Monitor is a fluxgate-type magnetometer element. The ability of this unit to detect changes in magnetic fields has been covered at length in the literature.1, 2 For this discussion it will suffice to say that the Gulf

Jan 1, 1956

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Jan 1, 1938

By John Blatchford, William T. Hall

The product obtained by roasting stibnite is likely to contain some unoxidized antimony trisulphide and a mixture of antimony trioxide and antimony tetroxide. It was desired to determine, as accurately as possible, the condition of the antimony as well as the total quantity present. Attempts were made to separate the trisulphide and the two oxides by methods based upon their varying solubilities in different solvents, but no satisfactory results were obtained in this way. It was found, however, that by determining the total antimony content, the antimony present as trioxide, and the antimony present as trisulphide, a good idea of the chemical composition of the roasted product could be obtained. The total antimony was determined by dissolving the sample in concentrated hydrochloric acid, reducing the antimony entirely to the trivalent condition by means of hydriodic acid and eventually titrating the antimony back to the pentavalent condition by means of iodine in the presence of sodium bicarbonate1* The details of the procedure are as follows: Weigh out 0.25 gram of the roasted product into a trapped flask (Fig. I), add 4 grams of powdered tartaric acid, 2 grams of potassium iodide and 40 c.c. of concentrated hydrochloric acid. Boil the solution gently f;r 5 min. and then cool to room temperature by shaking the flask while holding it under running water. Then, very carefully discharge the iodine color by the cautious addition of 0.05-normal sodium thiosulphate solution, adding a little starch toward the last. Nearly neutralize the acid with ammonia solution, but leave the solution distinctly acid. Pour the slightly acid solution into 200 c. c. of water containing an excess of sodium bicarbonate and titrate to a permanent blue color with standard 0.1-normal iodine solution.

Jan 1, 1916

By R. C. Beckstrom

Russia's first "Petaletka'' has ended. Technically it did not succeed but practically it has been a great achievement in the petroleum industry. Out of the chaos of revolution and civil war Russia has developed her oil industry to second place in the world's production. For 1932 production of crude oil is estimated at 150,000,000 bbl. and refined products at 145,000,000 barrels. Russia has been able to give the petroleum industry every advantage that a government can extend to a fast growing industry. She has obtained all of her property by confiscation. Under government ownership, operation and control, her petroleum industry is not handicapped in any way by state rulings or by unnecessary competition in production, refining and marketing. There are no damage suits resulting from offsetting, flooding, repressuring, laying of pipe lines, building of docks or other necessary work. Russian oil fields2 have been easy to develop and operate. They lie advantageously to transportation for foreign markets. The chief disadvantages have been a lack of credit and technical cooperation, but the former is being overcome by the sacrifice of the people. They go without the comforts of life that all salable articles may be exported for the imports of modern machinery. Modern machinery is fast replacing the obsolete and worn-out equipment. Technical cooperation has been difficult because it has been necessary to train thousands of adults to do technical and mechanical work. Great quantities of pipe, tools and electrical equipment were necessary for the rapidly developing fields and refineries. The supplies have been inadequate and the quality poor. This lack of equipment, along with the untrained technical supervision, has been the cause of the proposed quota falling short. Fig. 1 shows the production for 1931 and 1932. During 1932 a large percentage of the oil was produced in the proven fields of the Caucasus.

Jan 1, 1933

By Leo Dana

As a part of an investigation conducted by the Cement Section of the Bureau of Standards, at the plant of the Security Cement & Lime Co., Security, Md., the High-temperature Measurements Section was called upon to measure temperatures in a dry-process, coal-burning kiln. Although the measurements were made only under the conditions existing in one plant, the methods employed and the conclusions drawn apply, in a general way, to those existing in most plants. As far as we have been able to learn, no sufficiently thorough methods of making accurate pyrometric measurements in the sintering zone and in the rear end of rotary cement kilns have been described. It is believed that in this investigation, data were obtained that can serve as a foundation for further study of the problems of pyrometry in cement kilns. IMPORTANCE OF TEMPERATURE MEASUREMENTS IN ROTARY CEMENT KILNS The manufacture of Portland cement is today fairly well standardized. The various steps in the manufacture have been developed to such an extent that the process is almost automatic. In spite of these strides the efficiency of utilization of heat in the kiln is very low and an undue proportion is wasted; for example, about 50 per cent; ordinarily goes up the stack. In order to prevent these losses, not only should means be provided to utilize the waste heat but instruments, such as pyrometers, should be installed to study and control the operation with the view of minimizing the losses. At the same time, such control will produce a more uniform product. The saving in-fuel that may be effected by the proper use of pyrometers would be much more than their cost. Where these pyrometers should be installed and what they can show are very briefly indicated in the following paragraphs and discussed in greater detail later.

Jan 9, 1919

By Henry Louis

The object of the present paper is to describe a primitive method of gold-extraction, practiced by a small colony of Chinese in the district of Tomoh, one of the Siamese-Malayan States. This district has been worked for gold about 150 years, but has been only twice or thrice visited by Europeans, who are looked upon with a good deal of jealousy by the miners. The gold-district, covering not more than 10 square miles, is situated about Lat. 5' 47l N. and Long. -101' 39 E., on the headwaters of the river Teluban. It is a very mountainous region, lying on the northern flanks of the great range that forms the watershed between the rivers Patani, Kelantan, Teluban, etc., on the north and Perak on the south. The gold occurs in narrow veins and leaders of quartz, intersect- ' ing and intercalated among irregularly upheaved and contorted, highly metamorphosed, micaceous and chloritic schists. These veins vary between 3 inches and 3 feet in thickness, and are sometimes very rich. The geology of this region is thus very similar to that characterizing gold-regions all over the world; if must, however, be.

Jan 1, 1892

By H. E. Howe, A. A. Smith

The creep properties of lead alloys have been studied by a number of investigators but most of the work has been done on extruded material and the amount of data available on rolled alloys is surprisingly limited. Greenwood1 and his collaborators investigated various rolled commercial leads containing relatively small amounts of impurities, Phillips2 likewise studied rolled commercial leads, although he did include a few alloys— such as tellurium lead and one per cent antimonial lead. But as far as the authors are aware no data on the creep properties of so-called "hard leads"-—i.e., containing from 4 to 8 per cent antimony—have been published. In this paper some data are presented on a few antimonial lead alloys and compared with a copper-bearing lead. Preparation OF Specimens All the alloys used in this investigation were cast at a temperature of 400°C. into an open cast-iron mold preheated to Ioo°C., the resultant casting being % by I by 10 in. The copper-bearing alloy was cold-rolled by 10 per cent passes to a thickness of 0.100 in. Creep specimens 16 in. long were prepared by clamping the strips between wooden blocks and machining the edges of the strips parallel by means of a plane. Careful manipulation gave parallel specimens with no reduced section and with a minimum of distortion. The antimonial lead castings were hot-rolled in 10 per cent passes at a temperature of I50°C. to a thickness of 0.400 in., the heating being carried out in an electric oven after each pass through the rolls. After the castings had been broken down to 0.400 in., they were allowed to cool to room temperature and then cold-rolled to 0.100 in. with the usual Io per cent passes. Sufficient time was allowed after each pass for the strips to cool to room temperature. The apparatus and method of measuring strain have been adequately described in a previous paper3 and essentially the same practice was used in obtaining the present data. Time-extension curves for only two alloys are given, since these are typical of all. Figs. I and 2 show the time-extension Curves for the copper-bearing alloy A at 30°C. and loo°C., and Figs. 3 and 4 show

Jan 1, 1945

By E. J. Schmitz

This section was obtained last year during an examination of the iron-ore resources of the Rich Patch Mountain region and along Craig's creek valley. The geological members below the Oriskany and Lower Helderberg are exposed along the gap in a beautiful arch, while the higher formations appear in a more or less disturbed state on both sides of this arch, and are partly covered by dèbris. I have thought it of interest to the Institute to reproduce the drawing of this section at the gap which I made in my note-book at the time of my examination; those parts of the section, which lie at both sides of the arch, are only approximated. I give also a number of notes and measurements, referring to the Clinton formation and especially to the characteristics of the fossiliferous ore-seam opened in the various tunnels along the arch (see section). I. In Tunnel C: For the first 110 to 120 feet to second shute, we find the ore only here and there in small deposits, and interchanging with, or changing into, either a very siliceous limestone or a sandstone. About 10 feet beyond shute we have : Sandstone roof. Clay-slate, 211/2 inches Brown lean ore, 1 inch. Sandy clay, 3 to 31/4 inches. 1. Dark .brown ore, 3 inches. 2. Fossil-ore, changing into sandstone or very siliceous limestone, 2 inches. 3. Brown lean ore, 6 inches. Clay and slate below, 1 foot. 194 feet from entrance : Sandstone roof. Clay-slate, 4 inches. Clay 15 inches. Fossil-ore, 63 to 7 inches. Clay and ore, If inches. Lean brown ore, 2 inchea. Clay to bottom, 2 feet.

Jan 1, 1896

By M. C. Flemings, A. A. Tzavaras

Analytical and experimental procedures are descrihed for quantitative determination of volume-fraction microporosity in metals by mimorarlio-graphg. The analysis assumes spherical pores but allows for the observed fact that pores or portions of pores smaller than a given thickness do not project an image on the film. Volume-fraction microporosity was measuured in cast low-alloy steel and shown to vary from 0.6 x 10-4 to 10 x 10 depending on mode of solidification and on whether the material was air- or vacuum-cast. The aim of work described herein was to develop a method of quantitative measurement of the volume fraction of microporosity in cast metals. Microporosity, present in nearly all castings and ingots, is always interdendritic in nature and may be large enough to be seen on a polished surface with the naked eye. More generally, it is smaller in size, and its detection more difficult. Three techniques might be considered for measurement of microporosity: 1) lineal analysis of a polished metallographic surface, 2) density measurement, or 3) quantitative microradiography. As will be shown later, volume fractions of porosity in directionally solidified castings may be the order of 10-4 to 10-3; such small amounts of porosity are difficult to identify and measure on a polished surface. Density measurement is also not a suitable technique (for alloys) when the volume fraction of porosity is low. This is because the density of the base metal is often not known within the necessary accuracy, due to microsegregation or partial phase transformations. In general, it is not possible to completely eliminate microsegregation without also altering the porosity. Because of the limitations of the foregoing techniques, quantitative microradiography was employed to determine amount of porosity. In recent work, microradiography has been found to be useful for qualitative measurement of porosity, including determination of size and shape of individual pores;&apos;)&apos; however, quantitative determination of amount of porosity has not been attempted. I) CALCULATION OF VOLUME-FRACTION POROSITY The following analysis employs procedures similar to those of Cahn and Nutting3 and Hilliard.4 Major simplifications are that pores are assumed to be spherical and to be present in sufficiently small quantities that no overlapping occurs. A basic difference from the previous treatments is that the limited "sensitivity" of microradiography is considered. Assume first 1) cavities are spherical and of uniform diameter, 2) cavities are randomly distributed, 3) volume-fraction porosity and specimen thickness to pore radius ratio are sufficiently small that overlapping of the projected images of cavities is negligible, 4) sensitivity is limited (i.e., holes or portions of holes of less than a given thickness do not project an image on the film). The sensitivity, is defined as: where 5 = minimum thickness of void from which an image will be projected on the film and t = specimen thickness. Now, from Fig. 1, it is possible to distinguish between two different types of holes. These are 1) holes whose centers lie within the region t/2 > X > 6/2, where X is distance from one surface of the specimen (positive into the specimen), and 2) holes whose centers lie within the region (6 — r) < X < 6/2, Holes whose centers lie outside X = (6 -r) project no image on the film. These two types of pores are considered separately below. Type 1 Pores. When 6 is greater than 0, the image of a spherical cavity of radius r appears on the film as a circle of radius a which is smaller than r. From geometry, Fig. 1: a2=r2-52/4 [2] where a = radius of image of pore on film and r = radius of pore. Since the pores are randomly distributed and spherical V =4/3pr2Nv [3] where V = volume fraction of voids and Nv = number of pores per unit volume and the number of images of Type 1 pores per unit area of film, N&apos;A, is

Jan 1, 1965

By James W. Malcolmson

This paper treats of the history and development of the Sierra Mojada mines in the north of Mexico, the character and extent of the ore-deposits, and the methods of mining the ore. The growth of the smelting industry in Mexico in recent years is, to a large degree, due directly to the output of the lead-bearing ores from the Sierra Mojada mines, and at the present time (1901) new smelters are being built, looking to this district for their supply of lead-ores. These mines are located in the southern part of the State of Coahuila, in the Republic of Mexico, in latitude 27 24&apos; N. and longitude 103O 43&apos; W. of Greenwich. The last account of the Sierra Mojada published in our transactions was contributed by Mr. Richard E. Chism, in October, 1886.* Since that time some 3,000,000 tons of ore have been extracted from the mines, and much additional information has been developed regarding the mineral deposits. Since 1886, the Mexican Northern railroad, 78 miles in length, has been built from Escalon, connecting the camp with the Mexican Central railway, a trunk-line from El Paso, Texas, to Mexico City. The distances by rail from Sierra Mojada to the principal smelting-centers are as follows : To Torreon, Coahuila, Mex., 179 miles; to Monterey, Nuevo Leon, Mex., 422 miles; to El Paso, Tex., U. S., 494 miles; to Aguas Calientes, A. C., Mex., 522 miles; to San Luis Potosi, 8. L. P., Mex., 644 miles; to Pueblo, Colorado, U. S., 1158 miles; to Argentine, Kans., U. S.,

Jan 1, 1902

By Robley D. Evans, Gerald L. Schroeder

Advanced techniques for control of radon (Rn) daughter product concenrrations (working levels, WL) in the uranium mines are discussed. Understanding the physical laws which govern the flux of radon into a mine, the growth and decay of the radon daughters, and the particulate nature of the daughters when suspended in air allows ventilation personnel to improve the quality of mine air with no increase in ventilation rate. The task of controlling, in a mine atmosphere, the short-lived nuclides produced in the decay chain of radon is a unique problem and as such should be approached with ventilation techniques specific to the problem. For most mines this has not been the case. Rather, procedures for removing these airborne radioactive nuclides have been borrowed directly from ventilation experience in mines where dilution ventilation was the traditionally accepted approach for air purification. With radon daughter products, this has often led to frustrating difficulties and expense. Primary origin of the airborne radioactivity is the a-ray-emitting noble gas, radon (222Rn, Rn), a nuclide formed by the decay of radium (226Ra, Ra). Radon, being a gas, may escape from its parent crystal and diffuse through the parent mineral into the interstices of the rock body. In a sandstone structure, about 20% of all the radon formed may find its way into the interstices of the rock. If a cavity such as a mine is nearby, the radon may diffuse through the rock interstices into the mine. While radium is usually highly concentrated in uranium ores, it is also found in varying concentrations in all minerals. Radon is therefore present in all minerals. In normal glacial till, the equilibrium interstitial radon concentration often equals 5000 pCi (picocurie, also uuCi) Rn per liter of pore volume.&apos; In normal granitic rocks, the interstitial radon concentration often approaches 30,000 pCi per 1. Considering these typical, high-radon concentrations, radioactivity from airborne of daughter products is a potential problem unique not only to the uranium mining industry and should be considered as a potential hazard in all underground environments.

Jan 1, 1970

By James Hall

A geological map of a State involves very much, and its pre sentation to the public incurs grave responsibilities. It has been for a long time my earnest desire to bring out and place before the pub lic a geological map of New York which might be creditable to the State and worthy of a place among her scientific publications. We have done much for the advancement of geological science in this State—very much, and in many directions—but unfortunately, this feature, which should have been made the basis of all our work, has been too long neglected and delayed. For twenty-five years I have been laboring and looking forward to this consummation. I have recorded the field-work, systematic and desultory, upon such maps as were available for this use; and notwithstanding their imperfec tion, I have been able to transfer this recorded information to a more perfect base, which it is hoped will be completed during the early months of the next year. In 1883 I had combined, upon a small map of the State, the in formation which we possessed at that time, and communicated the same with my annual report to the Legislature. The report mas published in the regular course, but the special appropriation made for the map was vetoed by the Governor. Not discouraged by this untoward and unexpected circumstance, I continued my work, and made every effort to render the map more complete, preparatory to its communication with the next annual report. I brought together all the information within my reach, whether published or unpublished, reviewing and revising, as far as practicable, the work done by myself and assistants in southern and southeastern New York.

Jan 1, 1893

By H. H. Rachford

Numerical solutions of immiscible flow problems in which dispersive effects of capillarity are dominated by convection require excessively fine grid spacing with attendant high computing costs. The use of coarser spacing reduces cost but often produces oscillation or undue dispersion associated with displacement fronts, A numerical formulation is proposed here which should be applicable to two-dimensional flow problems. it is in part analogous to an approach previously tested for miscible systems. The convective transport is approximated using a change of variables to yield a coordinate system moving approximately with the local characteristic velocity. The capillarity-induced dispersive terms in the differential system describing the process are approximated with respect to a fixed coordinate system by the usual implicit formulation. One-dimensional tests of the procedure yielded results in which the saturation profiles tended smoothly to the zero-capillary pressure solution as the ratio of viscous to capillary forces was successively increased in a sequence of calculations. This contrasted favorably with solutions by other numerical procedures which would require attendant grid refinements to approach the zero capillary pressure results. INTRODUCTION Numerical solution of displacement problems has until recently relied on applying methods developed primarily for transient heat-flow problems. Such problems are classified as parabolic in type, and where the heat transport is purely by diffusion their solutions are characterized by a high degree of smoothness. It is not surprising, therefore, that for approximating these solutions available finite difference methods are quite adequate. In flow problems the transport is partly by diffusion, partly by convection or flow. Although the problem remains of parabolic type because the dispersive effects of capillary forces or diffusion play some role in every displacement, at high flow rates the problem is dominated by convection, and solutions tend toward those of equations of the hyperbolic type. Solutions of hyperbolic problems are characterized by the translation of fronts, or discontinuities, that may progressively increase in sharpness. Numerical methods for treating parabolic problems become less and less satisfactory as displacement rates increase and the role of dispersion due to concentration or capillary pressure gradients becomes small relative to transport due to flow. In computation the difficulty manifests itself as an error associated with the grid size chosen. 1-6 In summary, if the heat-flow type approximations are to include the terms arising due to convection, one of several choices may be made: (1) an upstream (to the direction of flow) approximation for the convection terms may be used; (2) a centered-in-distance (CID) approximation may be used; or (3) a recently developed approximation based on the theory of oscillation matrices may be chosen.6 The last appears to have significant promise for one-dimensional flow problems; its extendibility to two or three dimensions is an open question. In either of the first two approaches, a suitably small ratio of v&/D must be maintained, where v is the velocity, & is the grid spacing and D the effective dispersivity in the direction of flow. In the first choice, the approximation of the convective part is only first-order correct and errors introduced appear as a numerically induced dispersivity of magnitude proportional to v?x. In the CID choice, the approximation can be second-order correct, but the difference formulation fails to satisfy the maximum principle unless a condition on v?x/D is met. Practically, this means that for high flow rates oscillatory solutions may result in the neighborhood of a front unless exceedingly small grid intervals are taken. While the procedure proposed by Stone and Brian4 permits a less severe limitation to be placed on this ratio, ultimately the flow rates increase relative to the dispersivity the oscillation obtains. Further, extensions of their approach to higher dimensional systems may be attended by considerable

Jan 1, 1967