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Jan 1, 1957

By I. S. Parrish

Gold deposits in Hebei and Shanxi provinces in northeast China fall into three general classes: - Placer deposits - Mesothermal vein deposits - Disseminated and oxidized volcanogenic deposits Past production dating back thousands of years has all but exhausted the placers. The mesothermal vein deposits, incorrectly considered in the past as "greenstone" type, are small to moderate in both size and grade. Production from these granulite and amphibolite hosted veins, at present greater than from the newly discovered disseminated deposits, is expected to wane. Production from the disseminated deposits achieved only in the past few years is expected to equal or surpass that from the other classes of deposits. At least one "world-class" deposit is being developed.

Jan 1, 1989

By D. Harrington

"In at least three out of every four working faces in metal mines there is no attempt made to cause air circulation other than by use of compressed air, and in at least nine cases out of ten the compressed air is released merely by opening, or partly opening, the valve and allowing the compressed air to escape through open air hose used to convey the compressed air to the drilling machines. Usually no blowers are in use during periods of drilling, as the exhaust from the drill provides cool, comparatively fresh air for the driller; however, in some instances, the compressed-air line is provided with a tea connection, which permits taking air for the drill from one side of the tee, and air for ventilation through a separate hose used as a blower, simultaneously with drilling.Table 1 contains temperature and humidity data as to compressed-air blowers from open end of air hose at 36 working places in eleven metal mines of four States, embracing the mining of gold, silver, copper, lead and zinc ores, and at horizons ranging from 500 feet below the surface to as low as 4900 feet.It is noticed that the ordinary air surrounding the worker at the face gives an average wet bulb temperature of 81.7° F. for the 36 working faces, with 84.7° F. dry bulb and 87.2 per cent relative humidity. The average wet bulb tem¬perature of 36 readings at these working faces of compressed air at the end of the,hose as the air issues from the hose was 67.4° F., or 14.3 degrees lower than the average wet bulb readings of the air of the working places; however, the average dry bulb temperature of the 36 readings of compressed air as it issued from the end of the hose was 79.7° F., or only 5 degrees less than that of the general air (84.7°); the relative humidity was 51.1 per cent as against 87.2 per cent for the surrounding air. Readings were taken in five raises, seven crosscuts, ten drifts and 14 stopes.It will be noted that the velocity or movement of the general atmosphere at each of the entire 36 working faces, was found to be practically imperceptible; this might readily be inferred since compressed air is rarely used on blowers if there is adequate circulation of the general air of the place. Where rook temperatures could be obtained readily, they are shown; it is noticeable that the dry bulb temperature of the air of the working place varies little from the rock temperature, and that the dry bulb temperature of the compressed air as it issues from end of air hose varies only a few degrees from the temperature of the surrounding rock; in some instances the dry bulb temperature of the compressed air as it issued from the end of the air hose was higher than the rock temperature or the dry bulb temperature of the surrounding air."

May 1, 1921

By C. B. Manula

A method for the feasibility assessment of open-pit operations using conventional shovel/ truck systems to mine steeply pitched coal seams has been applied at The Pennsylvania State University. Three computer models were developed for the analysis: a geology and reserves model (GEOL), an open-pit materials handling simulator (OPMHS), and a cash flow model (CASH) (Manula, Albert, and Burgos, 1976). GEOL provides the basic mining data; OPMHS simulates shovel/truck performance related to pit profiles and average mining locations to provide productivity rates needed to size the system; and CASH provides either a selling price or rate of return (ROR) for the product. The general flow of information generated by these models is shown in Fig. 1. Since the ratio of overburden-extracted to coal-produced places a definite economic limit upon a mining venture, geological data must be analyzed with the least possible manual effort to minimize computational errors. This is especially true if the property contains several seams that are steeply pitched. In this situation, a computerized analysis is by far the most efficient and effective method to determine coal reserves and waste tonnage. This chapter describes the reserve model developed at Penn State and provides an example to demonstrate its application.

Jan 1, 1979

By Pura Alfonso, Sarbast Hamid, Hernan Anticoi, Eduard Guasch, Josep Oliva

"Av. Bases de Manresa 61-63, Manresa (08242) Barcelona, SpainTantalum is a strategic metal and an improvement of the mineral processing is an important issue. The main objective of this work is to optimize the grinding technologies for tantalum ores with high pressure grinding rolls. The selected ore for this study was an altered leucogranite from the Penouta Open Pit Mine (NW Spain). In the present work mono-size experiments have been carried out using grinding rolls with controlled pressure. The particle size distribution has been monitored in the input and output of the high pressure grinding rolls for each experiment. The cylinder pressure that is transmitted to the particles has been determined. New population balance models are presented for high pressure grinding rolls with parameter adjust for tantalum ore. The obtained model uses the physical description of particles under the comminution action of the grinding rolls. The obtained results are in agreement with other models and the errors in the adjustment are low. The selection function proposed for this model describes adequately the breakage probability of the particles inside the high pressure grinding rolls. The presented model can be a good alternative for simulating a high pressure grinding roll process. INTRODUCTIONTechnology for grinding have been improved for many years, introducing new variables and parameters involving energy consumption, mechanical stress analysis of the compression particles or pressure application for optimizing the comminution (Morrell et al, 1997). High pressure grinding rolls is a modern technology that is proven to reduce the operating costs in full scale plants when compared with other milling technologies. Its benefits on energy saving and simplicity in the process have been studied and applied, first in the cement clinker industry (Abouzeid & Fuerstenau, 2009) and after in mining activities, especially in metallic ore deposits (Austin et al, 1991; Guevara & Menacho, 1992; Morrel et al, 1997; Torres & Casali 2009; Numbi & Xia, 2015). Many author structures of the High Pressure Grinding Rolls modelling in three main branches; (a) throughput modelling, (b) Power and energy model and (c) particle size distribution modelling (Austin et al, 1991; Guevara & Menacho, 1992; Austin & Trubelja, 1994; Morrel et al, 1997; Torres & Casali, 2009)."

Jan 1, 2016

By James Terry Duee

No important producing developments took place in Colombia during the year 1928. This was at least partly due to the paralysis of operations by the passage, during the latter part of 1927, of a law which was regarded by the oil companies as confiscatory and unjust, and the promulgation of exceedingly stringent regulations of this law under decree No. 150, published Jan. 28, 1928. This decree, which contained 24 articles, practically prevented not only producing operations by the oil companies but even such activities as geological explorations and surveying, as severe fines were imposed for the possession of maps not registered with the Ministry of Industries. Both this regulatory decree and the law were subject to attack in the courts by several of the oil companies. The regulatory decree was suspended May 22 by Presidential proclamation, until the suits against it are decided by the Supreme Court and Council of State. Though another law was introduced into the Colombian Congress at a later date, no action was taken on it, and at the time of the writing of this paper, it would appear that no oil bill will pass the Colombian Congress during the present session. The Barco concession, which was revoked in February, 1926, still continues an unsettled matter, although there have been rumors that the government was attempting to come to some decision on this rather difficult question. A hopeful sign toward the close of the year was the settlement of negotiations with the Tropical Oil Co. over the acceptance of the payment for royalties due the government from that company. This settlement, which covered a royalty on crude oil produced from June 30, 1926 to June 30, 1928, represented a cash payment for approximately 3,000,000 barrels. It is to be hoped that during the coming year the Colombian government will see its way clear to clarify its position on petroleum legislation. A good augury for the future was the statement made by President Abadia Mendez on New Year's Day, in which he courageously admitted the errors of previous legislation and said, regarding oil, "Very soon

Jan 1, 1929

By G. W. Groove, E. E. Quenon

"INTRODUCTION Uncontrolled mine fires originating in exposed coal beds through the acts of careless or irresponsible persons present a serious threat to buildings, surface property, and the life and health of persons near the affected area. Apparently, the seriousness of such fires is not fully understood by property owners or those residing in the vicinity, as practically nothing has been done to alleviate conditions. In many instances, such fires have been active for several years before property owners or civic organizations have shown any interest toward having the fire extinguished. Recommendations by Bureau of Mines investigators in the interest of controlling such fires are seldom followed, as the Bureau cannot provide the funds for doing the work.Several uncontrolled and widespread fires are burning along the outcrop of coal beds and in underground abandoned mine workings within a 25-mile radius of Pittsburgh, Allegheny County, Pa. If these fires are permitted to continue to burn unabated, the ultimate results may be far-reaching. Most of the outcrop fires investigated are believed to have originated from burning rubbish, forest fires, or workmen, hunters, or others building fires near the outcrop of ,the coal beds. In other instances, smoke and fumes have suddenly appeared in the vicinity of abandoned coal-mine openings, indicating that a fire has either burned its way along the outcrop of the coal bed or has traveled through the abandoned mine from another location. An underground or outcrop coal-mine fire will be very likely to continue to burn as long as an oxygen supply is present.; and, so far as is known, such fires do not stop burning automatically until virtually all combustible matter is consumed."

Feb 1, 1948

By J. J. McKetta, O. L. Culberson

INTRODUCTION The equilibrium constants for methane and for water, and for ethane and water have been calculated from experimental data for the two binary systems.2,3,11,12 These constants are for the two-phase systems for the temperature range of 100" to 340°F and the pressure range of 200 to 10,000 psia. The constants for ethane are greater than those for methane. Equilibrium constants for a natural gas from a natural gas-water system6 are about the same as those for methane. Equilibrium constants for water in all three systems are very nearly the same over the greater part of the range of temperatures and pressures studied. In ideal solutions, the vapor-liquid equilibrium constants for a component should be the same in all such solutions at a given temperature and pressure.5 Few solutions are ideal, and consequently the use of generalized constants often leads to error in situations where they do not actually apply. In decidedly non-ideal systems, it is desirable to have equilibrium constant data for the specific system. Vaporization equilibrium constants may be calculated either by thermodynamic methods or from experimental vapor-liquid equilibrium data. In the thermodynamic calculation. the equilibrium constant K may be shown to be" K = fP,/fl'..........(1) where fPi is the fugacity of the pure component i at the given temperature and at the vapor pressure of the pure component, and fp is the fugacity of pure i in the gaseous state at the temperature and total pressure of the system. At best the thermodynamic calculation results in a K applicable to ideal solutions. Equilibrium constants calculated from Equation (1) have been found to have only limited agreement with experimental values, even though the solutions for which they were calculated approached ideality."'" If the more volatile component is at a temperature above its critical, the accuracy of the equilibrium constant for the component is made more doubtful by the use of the vapor pressure term under such conditions. In order to make the thermodynamic calculation on a component above its critical temperature, one must resort to one of the empirical methods of extrapolation of the vapor pressure to conditions above its critical point.13 Application of the thermodynamic equilibrium constant then to the methane-water or the ethane-water systems would be risky since the systems are non-ideal, and the methane and ethane are above their critical in most petroleum operations. ETHANE-WATER SYSTEM In the temperature range of from 100° to 340°F and for pressures up to 10,000 psia, the vapor-liquid equilibrium data 'References given at end of paper. Manuscript received in the office of the Petroleum Branch March 16, 1951.

Jan 1, 1951

By R. D. Groves

Techniques were developed for preparing metallurgical-grade copper powder from industrially produced cement copper. Flotation was used for rejecting discrete gangue particles, acid leaching to remove residual iron and aluminum, hydrogen reduction to eliminate oxygen, sintering to consolidate the fine-sized copper, and grinding to obtain copper powder. Powders prepared from six different cement copper samples contained, in percent, 98.5 to 99.0 Cu, 0.10 to 0.05 Fe, 0.10 to 0.08 insoluble material and the remaining individual impurities less than 0.15 percent. The apparent density, flow characteristics, and other physical properties were suitable for powder metallurgy applications. An economic evaluation showed that this process was economically feasible. Processing costs appeared modest when considering that copper powder commands a price normally about 17.5 cents per pound over that of electrolytic copper.

Jan 1, 1971

By B. Anameric

Considerable effort has been directed towards understanding the carbothermic reduction mechanisms of self-reducing, fluxing dried green balls to produce pig iron nuggets. Given the geometry of the situation, some investigators believed that the shrinking core model was applicable. Hence, this study involved investigation of the applicability of the shrinking core model to pig iron nugget production. The experiments involved heat treatment of dried green balls utilizing either a laboratory-scale resistance box furnace or a gas-fired muffle furnace at various furnace temperatures and residence times. The products were analyzed for the following: (i) preferred reaction pattern, (ii) distribution of the metallized areas, (iii) % iron content variation in the pig iron nuggets from surface to center, (iv) carburization pattern and (v) physical and chemical properties of the pig iron nuggets. It was determined that the reduction of iron oxide occurred simultaneously throughout the sample and not beginning at the surface and propagating inward; thus the shrinking-core model was not applicable. Heat transfer and gas diffusion from surface to center were not the rate-limiting steps for reduction and carburization reactions.

Jan 1, 2011

By Valentin V. Shatalov, Irina V. Pastukhova, Alexander V. Kurkov

"We have developed a technology to improve the quality of low-grade concentrates containing minerals of tantalum, niobium and tin on the basis of given conception by the way of improving selectivity and using certain reagents composition. The basic collector presents a traditional or complex-forming collector used together with organic diphilic reagents – the secondary reagents regarding the basic flotation regiment, capable of forming associates with the main collector on the surface of the mineral, thus altering the interaction energy. In the given process the main collector is the fraction of hydroxamic acids C7-C9, the reagent of the secondary action – sulphosuccinamate, which added in moderate concentrations range provides for a synergistic effect.We have determined the parameters of the process stability and necessary modifying reagents for a selective recovery of tantalum, niobium and tin minerals from refractory products. Variants have been studied where the main collector and the secondary action reagent were substituted with some more efficient flotation reagents. A possibility of a more flexible reagent compounding for the process has been shown. The flotation process is highly stable regards the saline composition of the liquid phase in the pulp. It can operate under conditions of the complete water recycling with the total salt content up to 12 kg/m3, iron up to 1.2 kg/m3.The given results consider the efficiency testing of the suggested technology for the gravitation products, which were obtained from ores of various genesis and contain large quantity of the most refractory minerals, such as topaz (80%), gamet (55%), tourmaline (30%). High selectivity of the process has been registered with the metal recovery about 92-98%, which appears to be similar to hydrometallurgical operations under certain conditions."

Jan 1, 2003

By Dominic Orf, Gregory Poole

On December 12, 2018 at 4:14 AM EST, an earthquake registering a 4.4 magnitude on the Richter Scale occurred in central Tennessee. According to a survey posted on the USGS website, this earthquake was felt throughout the southeast and even as far away as Arnold Nebraska. That’s over 900 miles (1,448 kilometers) from the epicenter. We recorded the vibrations from this earthquake on 44 different seismographs that were set up with remote access to continuously monitor vibrations from blasting activities. Having this data gave us an opportunity to compare the vibrations caused by earthquakes and blasting. A simple observation of the amplitudes, frequencies, and durations is enough to notice that vibrations from earthquakes are very different than vibrations caused by blasting in mining and construction. We were also able to observe the dissipation of the vibrations from the earthquake and see if the low frequencies, that are typical of earthquakes, dissipate at a consistent rate. Regression calculations were performed in order to see if there was a good data fit and if a specific vibration prediction could be made based on the magnitude of an earthquake. “How does quarry/mining/construction blasting compare to the Richter Scale?” This is a very common question that we get from homeowners when discussing vibrations due to blasting activities. This paper is intended to provide some simple answers for such a complicated question. Both the Richter and Mercalli scales are compared to the particle velocities as they were recorded from this earthquake.

By R. Hummel

When the slope angles of the southern side of the Venetia open pit were required to be flattened by some 20 degrees after an analysis completed by De Beers geotechnical engineering in conjunction with SRK, the additional waste stripping tonnage generated was in the order of some 70 million tons. The potential cost of having to move this tonnage had a significantly negative effect on the Net Present Value of Venetia Mine, challenging the viability of the third pushback and rendering a potential fourth pushback completely uneconomic. This paper details some of the innovation that went into the design and planning process undertaken to virtually negate the instantaneous effect of the flattened slope angles and the associated waste tonnage as well as further improve the NPV of Venetia Mine and bring the fourth pushback back into economic viability. This process has become known as NPV+ and is anon going exercise in waste reduction/deferment and revenue advancement.

Jan 1, 2001

By Graham Singleton

Around 235 million tonnes of marine aggregates have been dredged in the last decade from the UK continental shelf alone. The current distribution of marine aggregate production licence areas in North West Europe reflects the relationship between a variety of factors such as the natural distribution of deposits of sand and gravel, the demand for aggregates, the technical capabilities of the dredging fleet and environmental considerations. Whilst the sand and gravel deposits targeted by the marine aggregate industry are generally relict and therefore geographically fixed, the remaining factors have evolved over time and will continue to do so in future. Applications for new licences must not only reflect the current situation, but anticipate future trends. The main markets for marine aggregates are the construction industry, centred around the larger centres of population and coastal protection projects along the coastline of Europe. The aggregate industry as a whole does not create demand but simply meets it. Many factors influence the overall demand for aggregates and guidance exists for those in authority and industry who need to plan for the supply of aggregates. Adequate supplies should be maintained at the best balance of social, environmental and economic cost, whilst ensuring that extraction and development are consistent with the principles of sustainable development.

Jan 1, 2004

By Michael A. Trevits

Since the promulgation of the MINER Act and the follow-up changes to the regulations governing mine seal construction and maintenance, mine operators must be acutely aware of the atmosphere in sealed mine areas and prepared to deploy control technologies when the conditions are warranted. On Site Gas Systems1 in Newington, Connecticut was awarded a contract by the National Institute for Occupational Safety and Health (NIOSH) to construct a novel in-mine nitrogen (N2) generation plant using pressure swing adsorption (PSA) technology. PSA technology utilizes carbon molecular sieve material and pressure to adsorb oxygen (O2) molecules while allowing N2 molecules to pass through the sieve material. The prototype PSA unit was tested at the NIOSH Safety Research Coal Mine (SRCM) where a 62,000 ft3 area of the mine was rendered inert during a series of two tests. The resultant data were then used to construct a CFD simulation of both injection tests. In addition, O2 depletion and gas leakage rates were quantified in the model and were compared to actual values. Once the model was calibrated, the sealed mine area was doubled and simulations were made with various injection rates and injection site locations to determine the impact on the time needed to render this area inert. This paper presents an overview of the PSA plant, the details of the gas injection tests and CFD modeling work.

Jan 1, 2010

By C. G. Dunn, F. W. Daniels, M. J. Bolton

IN recent investigations1. a data on relative grain boundary energies in silicon iron have been obtained. The present investigation is a continuation of this work along similar lines for the purpose of obtaining additional information on boundary energies. One aim has been to extend the data for a (110) series, which was formerly studied,' to small differences in orientation, since a recent theory3,4 based on a dislocation model of grain boundaries predicts a E. ? (A-ln ? )* form of energy curve for small differences in orientation. Another aim has been to provide data for a (100) series—all grains having (100) planes parallel with the surface of sheet speci-mens—and to compare the results with the predicted form of energy curve. Finally, information on the nature of approach to equilibrium conditions was to be obtained through observations on the movements of grain boundaries. Recently published papers2,3,5 have treated the mathematical problem of expressing equilibrium conditions for grain boundaries for the general case when boundary energy depends upon boundary orientation. The equilibrium equations, which relate equilibrium angles and grain boundary -free energies, contain additional terms that express the variation of energy with boundary rotation. It has been necessary in the present investigation, however, to neglect these additional terms since no data for their evaluation were available. Dropping the additional terms leads to the following approximate equations which were used: E12/sin ?3 = E18/sin?2 =E28/sin?1 or those of the form: E28 + E13 cos ?3 + E12 cos ?2 = 0 Experimental Procedure Except for sample F1, all specimens were made from two lots of silicon iron called C and L respectively. The compositions of these are listed in table I. In the preparation of 12 three-grain specimens for the present investigation, a controlled grain growth technique, which has been described previously,' was used. After preparation, the specimens were notched at the boundaries to shorten them as a means of providing more rapid approach to equilibrium in the anneals. Initial grain boundary angles were determined from micrographs taken at X500. Seven anneals, totaling about two days at 1300°C and two to four days at 1400°C, were run in an atmosphere of pure dry argon as described in a previous paper.' Boundary changes generally could be followed without metallographic surface preparation, because thermal etching occurred during the anneals. These observations indicated angle changes of as much as 35" in some instances, with the major changes occurring during the first anneal. Annealing was discontinued when the angles reached stationary values. Fig. 1 gives a schematic diagram for purposes of defining crystallographic and boundary directions for three grains with common plane parallel with the sheet. Differences in orientations (expressed by A) are the angles between [00l] directions. They can be calculated with the aid of the ?'s. The grain boundary angles can be calculated from the w'S. Results Results are given in table 11. Types of crystal-lographic planes refer to those parallel with the sheet. Orientations are given in terms of [00l] direc-tions as shown in fig. 1. Except for grain 3 of specimen H6, which was unintentionally tilted 7" out of a (100) plane, all grains generally were within 1" or 2" of the plane specified. The directions of boundaries were measured generally to within 1" on both upper and lower surfaces, and the average direc- tions were determined for use in calculating the grain boundary angles. Variations in grain boundary directions refer to deviations from average directions. Boundaries near the junction point for all specimens generally were very close to perpendicu-

Jan 1, 1951

By J. H. Fowler

Canada's gypsum industry began over two hundred years ago in the Maritime Provinces when farmers attempted to supplement their incomes by selling "land plaster" to traders bound for the United States. Rising from such humble beginnings, gypsum now contributes significantly to the economies of nine of Canada's ten provinces. Thirteen open-pit and three underground mines produced a total of 8.38 million tonnes of crude gypsum and seventeen processing plants manufactured 2.49 million tonnes of finished products in 1985. Recent economic growth in North America has resulted in steady increases in gypsum production, however, competition from foreign producers and by-product gypsum poses a growing threat to Canada's gypsum mining industry.

Jan 1, 1987

The New South Wales Department of Mineral Resources has, since 1974, administered the Derelict Mines Rehabilitation Program (DMRP) to reduce the environmental impacts of abandoned and orphan mine sites. The application of the DMRP is presented in case studies of sites from the South Maitland Coalfield featuring residual drainage of acidic and saline water in a semi urban and rural community setting. The acidity here is typically derived from iron oxidation in shallow underground coal mine workings and in surface emplacements of coal washery reject materials. This paper presents summary information on AMD investigation, monitoring and rehabilitation. The experiences of community expectation, stakeholder consultation and peer review are discussed.

Jan 1, 2003

Water and wind erosion of toxic zinc and lead smelter waste is one of the most urgent environmental problems to be solved in the Silesia region of Poland. Over 87 x 106 Mg of various mining wastes were disposed of in uncontrolled piles leading to soil, water, and air pollution in highly populated areas. Welz and Doerschel process Pb/Zn smelter wastes were stabilized in the summer of 1994 by application of heavy loads of CaCO3 (30 Mg/ha), CaO (1.5 -15 Mg/ha), and municipal biosolids (150 -300 Mg/ha), followed by seeding with a mixture of acid-and salt-tolerant grass species selected in a pot experiment. Vegetation was successfully established on 85% of the Welz material demonstration plot area (0.5 ha), even though it was initially high in water-soluble Zn, Pb, and Cd ranging from 0.4 to 1311,0.1 to 5.8, and 0.05 to 55 mg/kg, respectively. Standing biomass averaged 3.3 Mg/ha on the stabilized Welz waste zone by the end of the third growing season (1997). The revegetation of much of the Doerschel material area failed initially because of extremely high salinity (16 dS m-1), high content of water soluble metals (Cd in particular), and heavy compaction. The area was capped (15 cm) with waste lime and retreated with bio-solids (300 Mg/ha) in 1995, and standing biomass averaged 2.9 Mg/ha in the fall of 1997. The observed plant response to bio-solids and lime indicates this approach to be an environmentally beneficial technology as compared to the traditional methods utilizing topsoil covers.

Jan 1, 1999

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