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By Kalervo Rankama

UNTIL a few years ago, geological mapping, the study of ore boulders, and different geophysical methods were the principal means used in the systematic search for mineral deposits covered by a layer of disintegration products or soil. These methods have been developed to a high stage of perfection as far as the geometrical definition of the conductor is considered. Use of geophysical methods in prospecting is based on the fact that an ore body differs from the country rock not only geologically but also in its physical properties. The general drawbacks and complications included in the use of geophysical methods are well known to everyone who has been using them, so need not be entered into here in detail. One of the more important ones is the fact that geophysical methods are unable to furnish information concerning the chemical composition of the ore bodies responsible for the indications. Further, the ores of many metals, such as silver, gold, beryllium, zinc, mercury, tin, lead, vanadium, antimony, chromium. molybdenum, tungsten, manganese, cobalt, nickel, and the platinum metals cannot be directly located by known geophysical methods, owing to lack of physical characteristics suitable for distinguishing them from the country rock. Thus it is quite natural that efforts have been made to develop direct methods for prospecting and locating ore bodies of the elements named. Known as chemical, biogeo-chemical, and geobotanical methods of prospecting, these have been much improved during recent years. A brief review of these methods and of some results obtained with them will be presented in the following, on the basis of available information and my own experience.

Jan 1, 1947

By W. W. Anderson, D. G. Girton

The photoluminescence of Pr, Nd, Ho, Er, Tm, and Yb in CdS, and Ho, Er, Tm, and Yb in ZnSe has been observed from crystals Prepared by diffusion using rare earth metals and an excess chalcogen pressure. For a given temperature, time, and chalcogen pressure the spectral characteristics were very reproducible from run to run, and the emission intensity for Nd, Er, and Yb in CdS was as high or higher than the best vapor phase doped crystals we have grown. For a few rare earths it was found that certain conditions of diffusion tend to yield optimum rare earth emission intensity with respect to the background lattice emission. Photoluminescence measwements of Yb in CdS as a function of depth gave a profile which was neither a Gaussian nor complementary error function. Part of the profile appears to arise from a fast component of the diffusion and the other part from a slow diffusing component. At 960°C and 33 atm S pressure, a com -plimentary error function approximation of the slow diffusing component gave a diffusion coefficient of D = 1.3 x 10-9 sq cm per sec. MOST of the studies of emission from rare earth ions in II-VI compounds have been reported on crystals doped during growth,1,2 although Kingsley and Aven prepared ZnSe:Er by diffusion for paramagnetic resonance and fluorescence studies. Pappalardo and Dietz prepared CdS:Yb by diffusion, but they made optical absorption measurements., We know of no study on the properties of rare earth diffusion in the II-VI compounds. To date we have diffused Pr, Nd, Ho, Er, Tm, and Yb into CdS, and Ho, Er, Tm, and Yb into ZnSe and observed the rare earth emission spectra. For a given temperature and chalcogen pressure, the emission characteristics are very reproducible from run to run and for Yb, Nd, and Er in CdS, as good as the best crystals we had prepared by doping during vapor phase growth.2 The emission of Pr, Ho, and Tm has been observed in CdS prepared by diffusion for the first time. Previous attempts2 to prepare these later three materials by vapor phase growth were unsuccessful. The problem of obtaining reproducible characteristics in II-VI semiconductor compound work is well known.5 Not only is it difficult to reproduce results from one laboratory to another but it is sometimes difficult to reproduce results from one growth run to another under ostensibly identical conditions within one laboratory. This situation has been particularly bothersome in research on the luminescence of rare earth activated ZnS1 and Cds2. Crystals from one vapor phase growth run would show very strong rare earth line emission while crystals from a nearly identical run would show no rare earth emission. It was also observed on occasion that the intensity of the rare earth emission was not constant over the entire volume of a single crystal. MATERIAL PREPARATION AND INSTRUMENTATION Vapor phase grown boules of CdS were supplied by Dow Corning. This material was characterized by a free electron concentration of n - 3.5 x 1015 cm-3 and Hall mobility of 350 sq cm per v sec at room temperature. There were microscopic voids and decorated precipitates in some samples. The precipitates annealed out at diffusion temperatures but the voids remained. Single crystal rectangular samples of mm dimensions were sawed from the boules. The ZnSe was polycrystalline, UHP grade from Eagle-Picher. Poly-crystalline samples were sawed from the ingots. The samples were lapped, polished on one side, etched in a solution of 0.5 M K2Cr2O7 in 16 N H2SO4, and thoroughly washed in distilled water. A sample, excess sulfur (or selenium), and 5 mg of rare earth metal (turnings) were sealed in a 3.6 cm3 quartz ampoule at about 2 X 10-5 torr. The high chalcogen pressure used (1 to 30 atm) prevented thermal etching of the crystals and affected the diffusivity and solubility of the rare earth ions in the crystal lattice. For meaningful or reproducible results, it is thus necessary to specify the vapor pressure at which the diffusion was carried out. It is assumed that a negligible amount of the chalcogen was used in the formation of rare earth sulfides or selenides Our sulfur vapor pressure calculations are based on data assuming S2, S6, and S, molecules only in which case the equilibrium constants are given by6 where the pressures are expressed in torr. Selenium vapor consists of a mixture of Se2, Se4, Se6, and Se6 molecules. The selenium vapor pressure was calculated using equilibrium constants given by The status of the rare earth source during diffusion is unknown, i.e., the partial pressures of the rare earth metal and of the rare earth chalcogenides has not been determined. All emission spectra were recorded at 77°K on a Perkin-Elmer model 98-G spectrometer using a 640 line per mm grating. No correction was made for the spectrometer and detector spectral sensitivity. Excitation was by means of an XBO 1600 w xenon arc

Jan 1, 1970

By J. David Lowell

For the first time in several years, Chile is again opening its doors to foreign investment. In its new investment code, the Chilean government has announced its willingness to become a partner in joint Chilean-foreign projects for a number of government-controlled porphyry copper deposits at the developed, semi-developed, or prospect stage; and, that it will allow privately owned deposits or deposits discovered by new exploration to be developed with up to 100% foreign ownership. Three deposits of particular interest are: El Abra, with at least 300 million mt of +100% Cu; Los Pelambres, with at least 428 million mt of 0.78% Cu and 0.03% Mo; and Andacollo, with 350 million mt of 0.70% Cu and 0.015% Mo. (El Abra is almost entirely government- controlled; but both Los Pelambres and Andacollo are mostly under private ownership.)

Jan 11, 1974

By Wallace E. Johnson, Leslie R. Hill

As part of a continuing study of intense energy effects on the earth, such as explosive cratering for Plowshare, large scale hydro- dynamic computer codes which numerically integrate the two-dimensional mass, momenta, and energy conservation equations have been adapted for use on geological materials. Experimentally determined nonlinear dynamic equations of state and constitutive equations governing material strength are incorporated into a two-material Eulerian code, DORF. With the use of this code, impact studies have been made of projectiles with arbitrary shapes in the r,z cylindrical plane normally incident on semi-infinite half-space targets. The impact of projectiles on rocks has been studied empirically and experimentally; however, this paper employs a theoretical approach. Particular examples selected involve the impact of aluminum and iron projectiles on oil shale at velocities of 0.5 and 1.0 cm/µsec. These studies offer a quick and economic means to edit quantities unavailable in an experiment and to independently study the effect of varying parameters such as material shape, density, and strength, and impact velocity. Results reveal early time agreement with experimental high-speed photography as seen in the bowl-shaped crater and spray. Initial impact pressures at the projectile-target contact are in close accord with the one-dimensional Rankine-Hugoniot relations for the impact of dissimilar materials. These equations require local continuity of pressure and particle velocity across the interface and were found to hold locally, quite independent of projectile shape. Essentially all projectile energy is transferred quickly to the oil shale target (tens of microseconds). Over 75% of the original kinetic energy

Jan 1, 1971

By Steven T. Mitchell

INTRODUCTION Mill production from the Homestake Mine in 1979 amounted to 1.29 Mt (1.43 million st) of ore containing 5.9 g/t of gold (0.172 oz per st) compared with 1978 production of 1.44 Mt (1.59 million st) containing 6.2 g/t of gold (0.180 oz per st). The average cost per ounce of gold produced in 1979 rose to approximately $250 contrasted with $176 in 1978. This increase in unit cost is the result of inflation, a diminishing ore grade, and productivity losses related to a shortage of experienced miners as well as compliance with federal regulatory requirements. Together, these factors prompted the investigation of several large-scale, highly mechanized underground mining methods. This analysis indicated Vertical Crater Retreat mining to be one of the safest underground methods known today, yet competitive with all other forms of stoping in terms of cost and productivity. Therefore, the feasibility of applying the VCR stoping method at the Homestake Mine was explored. Vertical Crater Retreat stoping is a recent innovation in large- scale underground mining based on C. W. Livingston's crater blasting theories. Figure 1 illustrates the method which consists of drilling 165.1-mm-diameter (6.5-in.) holes from a topsill down-dip to an under- cut on the level below. The bottom of each hole is then loaded from above with a predetermined charge weight and horizontal slices up to 4.27 m (14 ft) in thickness are blasted into the undercut. Broken ore is extracted from drawpoints located on or below the undercut level.

Jan 1, 1983

By A. F. Meston

Centrifugal force has been used for centuries for separating liquids but machines for doing this are a comparatively recent development. The use of these machines is being extended into many industries but this paper is confined to their use in the oil industry. The centrifugal machine is found at the well where "cut oil" is dehydrated for acceptance by pipe lines; and in the refinery and byproduct plant. In fact, wherever finely divided suspended particles should be removed from an oil, there is a possible use for a centrifugal machine, if the particles have, or can be made to have, a specific gravity different from that of the oil. Centrifugal machines are also used for cleaning fuel and lubricating oils and dehydrating transformer oils. Applications and Results When oil comes from the well mixed with water, it is spoken of as cut oil. Settling often does not remove sufficient water for the oil to be accepted by pipe lines, thus necessitating chemical or other treatment. Centrifuging alone, or following chemical treatment, effectively prepares such oil for pipe lines, as is shown by the following data: WateR, Etc. Wat~R. Etc. Temperature, Oil to Pipe Line, aT StUt at Finish Degrees F. Bel. Per HouR PIE Cent. PEECent Oklahoma lease.... 160 10 10 0.25 Kansas lease....... 125 4 20 0.25 to 2 On the Oklahoma lease, the machine was cleaned of solid material once in 12 hr. At first, it was thought that chemical treatment was necessary to start separation and 1 qt. of prepared compound was added to each 100 bbl. of the cut oil before centrifuging. Later it was found that

Jan 1, 1924

By C. L. Meyers, J. L. Lytton, T. E. Tietz

The recovery of tensile flow stress of 99.995 pct A1 under conditions of elastic strain and plastic creep straining was investigated using a fractional recovery parameter. Tensile straining was conducted at room temperature to a true strain of 0.10, and recovery treatments were made under stress in the temperature range from 80° to 198°C. Whereas elastic strains had no appreciable influence on flow-stress recovery, plastic creep straining was found to accelerate the recovery process. Although higher creep stresses gave greater recovery for a given time and temperature, it was found that lower stresses produced greater recovevy at equal ALTHOUGH many studies have been made to date on the recovery of metals from the cold-worked creep strains above about 0.005. This effect was attributed to the coarser, and therefore weaker, substructure developed during creep at the lower stresses. The activation energy for recovery during creep straining at 2075 psi between 160" and 198°C was found to be 25,000 cal per mole. This value was not significantly greater than the value previously obtained for no-load recovery. The activation energy for creep at 2075 psi between 160" and 198°C was 34,000 cal per mole, a value significantly greater than that for recovery under the same conditions. state, little is known concerning some important aspects of recovery behavior. One such area is the effect concurrent strain has on the recovery process. Since metals in structural applications at high temperatures are under stress, the possible effects of the resulting elastic and plastic strain on recovery may be of significant importance.

Jan 1, 1962

By F. W. Maclennan

PAPERS by D. B. Scott, E. G. Deane, and J. H. Hensley, Jr.1 describe the succession of mining methods used in the Miami mine—squarelset system, shrinkage stoping, top-slicing method, and the undercut caving system which was then being used, and which embraced the use of finger raises. The last-named system worked well but when the rich ore was exhausted it became necessary to reduce costs so as to make available a low-grade ore containing an average of less than 0.9 per cent. total copper with less than 0.8 per cent. in sulfide form and with no precious-metal values. There was then developed for this low-grade ore the system of separate large block-caving stopes, in lateral dimensions 150 by 300 ft., or 150 by 150 ft., with intervals between stopes of the same size so that subsequently these pillars may be similarly mined. The results have been most favorable. Under the previous system the production of ore per man-shift for all underground labor was 8 tons. The present production per man-shift underground is 28 tons. In starting this work the problems were: 1. How the ground would cave above the stopes. 2. Whether in the first stopes caving would encroach on the pillars as laid out. 3. Whether in subsequent mining of the pillars waste filling would be drawn from the previously caved adjacent stopes. Fig. 1 is a small key plan for showing the location of cross-sections as well as of ore mined and the limit of the escarpment and limit of cracking on the surface caused by the caving of the capping overlying the ore

Jan 1, 1929

By N. P. Pinto

Compacting below the recrystallization temperature was studied. Ideal density was attained at 550° to 600°C using 25 tsi. Compacts have strength and hardness higher than cold worked beryllium. The recrystallization temperature of this highly stressed metal is that of cast, cold worked beryllium and appears independent of pressing temperature. THE process metallurgy of beryllium is beset by many difficulties that hinder the production of sound beryllium shapes of controlled properties. Most of these obstacles are traceable directly to the low density, low ductility, and high affinity for oxygen of the metal. Castings are susceptible to cracking and porosity, and controlled grain size is not easily attained, yet is critical to further working. Any process must carefully avoid oxygen pickup. Conventional metallurgical processes, including melting, casting, forging, rolling, and extruding have been treated previously,'" as have the conventional powder metallurgical methods.' More recently, a method has been advanced for producing powder compacts of high density by hot pressing at 1000" to 1100°C, using pressures of 0.25 to 0.5 tons per sq in. (tsi).3 Here, Seybolt and coworkers attained theoretical density of 1.85 grams per cc using pres-sure-temperature combinations of 500 psi-1100°C, 750 psi-1075"C, and higher. The warm pressing of beryllium is a practical method for producing bars of controlled properties and appears to have certain advantages, particularly when high strength and hardness are required. Other investigators have studied the hot pressing of various pure metals, such as copper,"-" gold," iron,10-11 and aluminum." Their results indicate that ideal densities may be attained by compacting at temperatures less than 50 pct of the absolute melt-

Jan 1, 1955

By L. F. Reinartz

The furnace in which rimming steel is made has an important bearing on the quality of the steel produced. Furnace.—Particularly in the manufacture of low-carbon rimming steels, it is necessary to have an efficient, easily regulated furnace, because of the relatively high finishing temperature requirements. In recent years, the most important development in design of open-hearth furnaces has been the sealing and insulation of the greater part of the furnace to prevent air infiltration and loss of heat. Efficient, straight-line valves insure proper draft. Combustion controls guarantee correct gas-air ratios. Furnaces of all sizes ranging from 20 to 200 tons have been used successfully in the manufacture of rimming steels. Metal bath is varied from 20 to 40 in. in depth without appreciable variation in quality. Charging.—A normal open-hearth charge consists of pig iron (hot or cold), scrap of miscellaneous nature, and slag-making additions, such as limestone and burnt lime. Because of local conditions, some plants find it necessary to discard the use of pig iron and use some carbonaceolis substances with varying percentages of cast iron to replace the pig iron. It is questionable whether such practice will consistently produce the quality of rimming steel required for auto stampings and other severe drawing requirements. Pig Iron.—An open-hearth furnace under proper control should melt 150 tons of low-carbon rimming steel on about 35 to 38 per cent pig-iron charge basis. In some practices, best results are obtained when the total charge contains from 0.40 to' 0.50 per cent silicon. If reasonably low-

Jan 1, 1935

By William 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 bicarbonate.' The details of the procedure are as follows: Weigh out 0.25 gram of the roasted product into a trapped flask (Fig. 1), add 4 grams of powdered tartaric acid, 2 grams of potassium iodide and 40 c.c. of concentrated hydrochloric acid. Boil the solution gently for 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 con-

Jan 1, 1916

By M. E. Shank, J. Wulff

In view of the current interest in magnetic materials having rectangular hysteresis loops, as for example those obtained with the grain oriented 50 nickel 50 iron alloys,t we wish to call attention again to the results obtainable with the perminvar (Co-Ni-Fe) alloys heat treated in a magnetic field. Data taken from direct current measurements made prior to and during World War II give the typical values shown below for two of the perrninvar alloys. Fig 1 and 2 show- the hysteresis loops for these samples. The first sample above consisted of small rings punched from 0.014 in. sheet stock. These rings were heat treated at 1000°C for one hour in a hydrogen atmosphere and then cooled with the furnace to 750°C at which point they were subjected to a magnetic field of 15 oersteds maintained during cooling to room telmperature. The molybdenum perminvar sample consisted of a spirally wound core of 0.001 in. tape. The heat treatment was the same except that box annealing methods were used and no hydrogen atmosphere was employed. Even better properties might be expected using hydrogen. The heat treating temperatures and cooling rates are not critical. Rectangular hysteresis loops produced in other perminvar alloys by heat treatment in a magnetic field were described earlier.',? Properties similar to those described above are realized in perminvars manufactured by conventional methods. Drastic cold working which in practice limits the final thickness and is gen- erally expensive may thus be avoided. This information may be of interest to those concerned with magnetic alloys and the development of contact rectifiers, magnetic amplifiers and so on.

Jan 1, 1950

By L. F. Reinartz

The furnace in which rimming steel is made has an important bearing on the quality of the steel produced. Furnace.—Particularly in the manufacture of low-carbon rimming steels, it is necessary to have an efficient, easily regulated furnace, because of the relatively high finishing temperature requirements. In recent years, the most important development in design of open-hearth furnaces has been the sealing and insulation of the greater part of the furnace to prevent air infiltration and loss of heat. Efficient, straight-line valves insure proper draft. Combustion controls guarantee correct gas-air ratios. Furnaces of all sizes ranging from 20 to 200 tons have been used successfully in the manufacture of rimming steels. Metal bath is varied from 20 to 40 in. in depth without appreciable variation in quality. Charging.—A normal open-hearth charge consists of pig iron (hot or cold), scrap of miscellaneous nature, and slag-making additions, such as limestone and burnt lime. Because of local conditions, some plants find it necessary to discard the use of pig iron and use some carbonaceolis substances with varying percentages of cast iron to replace the pig iron. It is questionable whether such practice will consistently produce the quality of rimming steel required for auto stampings and other severe drawing requirements. Pig Iron.—An open-hearth furnace under proper control should melt 150 tons of low-carbon rimming steel on about 35 to 38 per cent pig-iron charge basis. In some practices, best results are obtained when the total charge contains from 0.40 to' 0.50 per cent silicon. If reasonably low-

Jan 1, 1935

By Eugene McAuliffe

The management of The Union Pacific Coal Co., which operates a group of coal mines in Southern Wyoming, believes in the value of applied psychology as a productive factor in the work of getting out coal. Operating almost wholly through the medium of mechanical loading machinery, the company has for some years paid a bonus, equivalent to one-half the labor saving made above a determined average output per shift worked, on a given machine. No penalty is attached where the established output is not obtained, but in 1933 the men on one class of machines earned an average premium of $0.633 per shift on 56.4 per cent of the tonnage loaded. On another class of machine, the men earned an average premium of $0.686 per shift on 28.4 per cent of the tonnage loaded An increase of 36 per cent in tons loaded per man-shift has resulted since the program was instituted. This coal company has also been very successful in reducing mine accidents through prizes awarded in an unusual way. At the close of the year, every man whose particular mine section passes through the year without a lost-time accident is given a ticket entitling him to participate in a safety drawing where an automobile is awarded as first prim.. There are also a number of cash safety prizes which are similarly awarded. Some 1500 persons attend the drawing, which excites much interest and attention. The particular plan employed, the payment of production bonus awards and safety prizes, is foundationed upon collective. rather than individual effort, and, once gotten under way, has proved to be substantlally self-cnforcing.

Jan 1, 1934

Insoluble slimes are present in nearly all sylvinite ore bodies and must be removed prior to potash flotation because they adsorb excessive amounts of potash flotation reagents and cause low potash recoveries and grades. The problems associated with insoluble slimes will become more pronounced as high grade-low insol potash ores are depleted and lower grade ores containing high insols are processed. The most common commercial practice for insoluble slimes removal is mechanical desliming (hydroclassification); however, most of these desliming circuits were designed to treat sylvinite ores containing less than 4 percent insoluble slimes. As Mr. Banks mentioned in his presentation, higher insoluble slimes concentrations (>4 pct) in the mill feed overtax the mechanical desliming circuit and result in excessive brine and potash losses. Increasing the capacity of the circuit would require redesigning of the entire potash circuit and additional costly equipment. Because of mineralization and liberation properties of the high insols sylvinite deposits, the expansion of the desliming circuit may not economically maintain high potash grades and recoveries. Selective flocculation-flotation is an excellent alternative for removing insoluble slimes from potash ores. Cominco's ore, containing 4.5 to 8 percent insoluble slimes, is particularly amenable to the slimes flotation process, probably because of the absence of expanding-type clays such as montmorillonite. The overall potash recovery of 86 percent and low slime flocculant and collector dosages of 0.03 to 0.04 pound reagent per ton of feed is good evidence of the success of the process. If appreciable montmorillonite were present, the metallurgy would not be as favorable and reagent consumptions would increase significantly.

Jan 1, 1979

By Herman Dykstra, R. L. Parsons

A numerical method for solving partial differential equations in steady state fluid flow is described. This method, known as the "relaxation method," has two advantages over analytical methods: (1) practically any problem can be solved, and (2) a solution can be obtained quickly. A disadvautage is that the solution is not general. The method is applied to core analysis and relative permeability measurement to calculate constriction effects and to calculate the true pressure drop measured by a center tap in a Hassler type relative permeability apparatus. Further applications are suggested. INTRODUCTION Many problems in fluid flow cannot be solved analytically because of the nature of the boundary conditions. For many problems, however. an exact answer is not necessary because boundary conditions are not exactly defined or the parameters describing the porous medium are not accurately known. The relaxation method can be used to obtain an approximate answer easily and quickly for the flow of incompressible fluids in porous media. The method can also be used for other types of problems, such as determining the stress in a shaft under load. or the temperature distribution during steady state heat flow. In this discussion only calculations concerned with the flow of fluids in porous media will be considered. The method was introduced by R. V. Southwell in 1935.' THEORY The treatment given here follows that given by Enimons.2 Consider a porous medium to be replaced entirely by a net of tubes of equal length and uniform cross-sectional area as shown in part in Fig. 1. Assume that the net of tubes behaves exactly like the porous medium which it replaces; that is, the net can be made fine enough to reproduce exactly the porous medium. Assume also that Darcy's Law can be used to calculate the flow from one point to another point through these tubes. The flow from point 1 to point 0 is KA . ------ P-P) .......(11 where a is the distance between points: K is the "permeability" of a tube; A is the cross-sectional area of a tube; is the viscosity of the liquid in the porous medium; and (P1 — P0) is the pressure difference between point 1 and point 0. In like manner the flow can be calculated from points 2, 3, and 4 to point 0. The net flow into point 0 is Qo = KA/µa (P1 + P2 + P33 + P4-4P0) . . (2) MB For an incompressible fluid the net flow into point 0 will be zero or, Q. = 0. This says that at point 0 fluid is neither being accumulated nor depleted. 'Therefore. P1 + P2 + P3 + P4 - 4P0 = 0 .... (3) . If. now. with specified boundary conditions. the pressure i.; known at a finite number of points in a given region, as at the points shown in Fig. 1, Equation (3) will be satisfied at every point. If, on the other hand, the pressure is not known, the pressure can be guessed at these points. Then. unless the guess is perfect. Equation (3) will not be satisfied at all of the points. When Equatiol~ (3,) is not satisfietl. let d = P1 + I?, + P, + P, - If' .,....(4) where 6 is an apparent error and is called the residual at point 0. Equation (4) shows how much the pressure guess is in error at point 0 with respect to the surrounding points. A positive residual means that the pressure is too low, and a negative residual means that the pressure is too high. To bring the residual, 6. to zero in order to satisfy Equation (3). it is necessary to make changes in the pressure guesses. Equation (4) shows that a +1 change in Po will change the residual at point 0 by -4. A +1 change in the pressure at any of the four surrounding points will change the residual at point 0 by +l. Thus it can be seen that a change at any point will affect the residual at that point and the four surrounding points. By changing the pressure from point to point, all of the residuals can eventually be brought nearly to zero and the problem will be solved. This procedure is the essence of relaxation methods and is used to relax the residuals so that Equation (3) is satisfied at every point. The procedure can be most easily explained in detail by solving a simple problem. as Southwell says, "To explain every detail of a practical technique is to risk an appearance of complexity and difficulty which may repel the reader. A

Jan 1, 1951

By R. L. Parsons, Herman Dykstra

A numerical method for solving partial differential equations in steady state fluid flow is described. This method, known as the "relaxation method," has two advantages over analytical methods: (1) practically any problem can be solved, and (2) a solution can be obtained quickly. A disadvautage is that the solution is not general. The method is applied to core analysis and relative permeability measurement to calculate constriction effects and to calculate the true pressure drop measured by a center tap in a Hassler type relative permeability apparatus. Further applications are suggested. INTRODUCTION Many problems in fluid flow cannot be solved analytically because of the nature of the boundary conditions. For many problems, however. an exact answer is not necessary because boundary conditions are not exactly defined or the parameters describing the porous medium are not accurately known. The relaxation method can be used to obtain an approximate answer easily and quickly for the flow of incompressible fluids in porous media. The method can also be used for other types of problems, such as determining the stress in a shaft under load. or the temperature distribution during steady state heat flow. In this discussion only calculations concerned with the flow of fluids in porous media will be considered. The method was introduced by R. V. Southwell in 1935.' THEORY The treatment given here follows that given by Enimons.2 Consider a porous medium to be replaced entirely by a net of tubes of equal length and uniform cross-sectional area as shown in part in Fig. 1. Assume that the net of tubes behaves exactly like the porous medium which it replaces; that is, the net can be made fine enough to reproduce exactly the porous medium. Assume also that Darcy's Law can be used to calculate the flow from one point to another point through these tubes. The flow from point 1 to point 0 is KA . ------ P-P) .......(11 where a is the distance between points: K is the "permeability" of a tube; A is the cross-sectional area of a tube; is the viscosity of the liquid in the porous medium; and (P1 — P0) is the pressure difference between point 1 and point 0. In like manner the flow can be calculated from points 2, 3, and 4 to point 0. The net flow into point 0 is Qo = KA/µa (P1 + P2 + P33 + P4-4P0) . . (2) MB For an incompressible fluid the net flow into point 0 will be zero or, Q. = 0. This says that at point 0 fluid is neither being accumulated nor depleted. 'Therefore. P1 + P2 + P3 + P4 - 4P0 = 0 .... (3) . If. now. with specified boundary conditions. the pressure i.; known at a finite number of points in a given region, as at the points shown in Fig. 1, Equation (3) will be satisfied at every point. If, on the other hand, the pressure is not known, the pressure can be guessed at these points. Then. unless the guess is perfect. Equation (3) will not be satisfied at all of the points. When Equatiol~ (3,) is not satisfietl. let d = P1 + I?, + P, + P, - If' .,....(4) where 6 is an apparent error and is called the residual at point 0. Equation (4) shows how much the pressure guess is in error at point 0 with respect to the surrounding points. A positive residual means that the pressure is too low, and a negative residual means that the pressure is too high. To bring the residual, 6. to zero in order to satisfy Equation (3). it is necessary to make changes in the pressure guesses. Equation (4) shows that a +1 change in Po will change the residual at point 0 by -4. A +1 change in the pressure at any of the four surrounding points will change the residual at point 0 by +l. Thus it can be seen that a change at any point will affect the residual at that point and the four surrounding points. By changing the pressure from point to point, all of the residuals can eventually be brought nearly to zero and the problem will be solved. This procedure is the essence of relaxation methods and is used to relax the residuals so that Equation (3) is satisfied at every point. The procedure can be most easily explained in detail by solving a simple problem. as Southwell says, "To explain every detail of a practical technique is to risk an appearance of complexity and difficulty which may repel the reader. A

Jan 1, 1951

By N. Van Wingen

EFFective sand permeabilities can be ascertained from core analysis if the laboratory data are compensated to allow for the presence of connate or residual water. Such adjustments can be made by applying empirically derived correction factors. Effective sand permeabilities can be estimated also from analytical studies of field depth-pressure measurements. In one method the time rate of pressure build-up after a well is shut in from steady-state flow is required, and in the other the direct measurement of a steady-state rate of flow and the corresponding equilibrium pressure. The results of the investigation discussed in this paper demonstrate that these methods for determining mean effective sand permeabilities can be correlated. Introduction In recent years many investigators have demonstrated the practical value of an accurate knowledge of the physical and dynamical characteristics of reservoir sands containing oil and gas. Of these factors, which serve to completely define a porous medium as a container or carrier of fluids, the dynamical characteristics are of particular interest to the production engineer because they afford a direct measure of the rate at which the oil or gas content of a reservoir will become available. The constant that defines dynamically the porous medium as the carrier of a homogeneous fluid in viscous motion is the "permeability." Coke-analysis Method The generally accepted practice in com-piling permeability data has been to analyze in the laboratory core samples taken in the course of drilling through the productive horizon, and to obtain the mean measure of the productive ability or permeability for the interval open to production by graphical integration of the permeability profile. An average permeability ascertained in this manner, however, is useful to the production engineer only for comparisons of wells drilled in the same horizon, as permeabilities determined in the laboratory are measures of only the ability of a porous medium to transmit a single fluid. As a result of many investigations, it has been shown, however, that under actual reservoir conditions water is coexistent with the petroleum and that its presence greatly reduces the permeability of the sand to oil. Empirical relationships showing the effect of the presence of any immiscible fluid A (such as water) in a porous medium on the permeability of the medium to a fluid B (such as oil) have been established separately by Hassler,1 Dunlap,2 and the writer (Fig. I),3 so that the average specific or dry-core permeability as determined in the laboratory can be corrected to give the true measure of the ability of the horizon penetrated to transmit oil, provided the percentage of pore space occupied by connate water is known accurately.

Jan 1, 1942

By N. Van Wingen

EFFective sand permeabilities can be ascertained from core analysis if the laboratory data are compensated to allow for the presence of connate or residual water. Such adjustments can be made by applying empirically derived correction factors. Effective sand permeabilities can be estimated also from analytical studies of field depth-pressure measurements. In one method the time rate of pressure build-up after a well is shut in from steady-state flow is required, and in the other the direct measurement of a steady-state rate of flow and the corresponding equilibrium pressure. The results of the investigation discussed in this paper demonstrate that these methods for determining mean effective sand permeabilities can be correlated. Introduction In recent years many investigators have demonstrated the practical value of an accurate knowledge of the physical and dynamical characteristics of reservoir sands containing oil and gas. Of these factors, which serve to completely define a porous medium as a container or carrier of fluids, the dynamical characteristics are of particular interest to the production engineer because they afford a direct measure of the rate at which the oil or gas content of a reservoir will become available. The constant that defines dynamically the porous medium as the carrier of a homogeneous fluid in viscous motion is the "permeability." Coke-analysis Method The generally accepted practice in com-piling permeability data has been to analyze in the laboratory core samples taken in the course of drilling through the productive horizon, and to obtain the mean measure of the productive ability or permeability for the interval open to production by graphical integration of the permeability profile. An average permeability ascertained in this manner, however, is useful to the production engineer only for comparisons of wells drilled in the same horizon, as permeabilities determined in the laboratory are measures of only the ability of a porous medium to transmit a single fluid. As a result of many investigations, it has been shown, however, that under actual reservoir conditions water is coexistent with the petroleum and that its presence greatly reduces the permeability of the sand to oil. Empirical relationships showing the effect of the presence of any immiscible fluid A (such as water) in a porous medium on the permeability of the medium to a fluid B (such as oil) have been established separately by Hassler,1 Dunlap,2 and the writer (Fig. I),3 so that the average specific or dry-core permeability as determined in the laboratory can be corrected to give the true measure of the ability of the horizon penetrated to transmit oil, provided the percentage of pore space occupied by connate water is known accurately.

Jan 1, 1942

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Jan 1, 1961