Search Documents

By J. Douglas, W. T. Ford, G. E. Henderson, I. F. Roebuck

An implicit numerical method is presented for simulating the differential and algebraic relations governing one-dimensional three-phase flow in porous media. The method is based upon compositional representation of the hydrocarbon system. Variable physical properties and water-oil capillary forces are included in the formulation; however, the effect of gravity is ignored. The effects of changing composition and mass transfer are considered through the use of known phase behavior concepts and correlations. The differential and algebraic equations and the numeric approximations to these equations are presented, and the computing algorithm is discussed in detail. This method is used to simulate a single-phase, two hydrocarbon-component gas displacement laboratory experiment. The method is also applied in simulating a solution gas drive and a gas injection problem using a reservoir oil represented as a nine-component mixture. For comparative purposes, the latter two problems also are examined with a one-dimensional volumetric model. The results of the simulation of the laboratory experiment demonstrate the relevance of this method to problems wherein the effects of mass transfer significantly enter the displacement process. The comparison of the results of the volumetric and compositional methods applied to a reservoir system illustrates not only the advantages and benefits derived from examining the hydrocarbon system composition, but also the necessity of this approach in application to a general reservoir problem. INTRODUCTION The advent of large memory, high-speed digital computers enabled reservoir engineers and mathematicians to pool their knowledge in the development of sophisticated techniques for the prediction of oil and gas reservoir performance. These techniques have progressed from volumetric and trend analysis calculations to the development of "mathematical models", or reservoir simulators. These latter models, in general, utilize finite difference approximations to the rather complex partial differential equations that mathematically describe the physics and thermodynamics of fluid flow in porous media. Many forms of this "model" type are available today, and much of the recent literature has been devoted to either improving known methods or developing new techniques for use in these prediction methods. The more advanced techniques in use at this time consist of the solution of differential and algebraic relations that describe a reservoir geometry in one, two or three dimensions and that incorporate reservoir hydrocarbon and water fluid systems that obey specified physical laws, having properties that might easily be measured in the laboratory. This type of model utilizes formation volume factors and other volumetric data, expressed directly as functions of pressure, to represent the reservoir fluid system. Experience gained with this type of simulator, commonly known as a "beta-factor model", has indicated that the approach has many outstanding defects, most of them due to the assumption that those properties measured in the laboratory completely describe the behavior of the fluid system throughout the life of the project under study. The effects of varying hydrocarbon composition and mass transfer between the phases of this hydrocarbon system must, by definition, be ignored. These defects, which may often be ignored in

Jan 1, 1970

Jan 1, 1880

By R. P. Fischer

The Colorado Plateau region has been the principal domestic source of uranium, vanadium, and radium. The value of these commodities produced from the region through 1964 slightly exceeds $2 billion. Most of the deposits occur in streamlaid lenses of sandstone in the Chinle and Morrison Formations. The ore minerals below the zone of oxidation are low-valent oxides and silicates of uranium and vanadium; all of these, except the quite stable vanadium silicates, oxidize readily to a variety of high-valent uranium and vanadium minerals. The ore minerals partly replace fossil wood but mainly impregnate sandstone, typically forming tabular ore bodies that are nearly concordant to the bedding of the host rock. The deposits range in size from those containing only a few tons of ore to those containing more than 1 million tons. Stratigraphic, lithologic, and sedimentary structural factors obviously influenced the localization of these deposits, but igneous activity, hydrothermal mineralization, and tectonic structures show no similarly consistent influence on localization. The deposits are epigenetic, but the source of the ore metals, the nature of the ore-bearing solutions, and the time of ore emplacement are uncertain. A few uranium deposits in the region are dominantly discordant to bedding and occur in collapsed breccia pipes or along linear fractures. Their origins are also uncertain.

Jan 1, 1968

By G. Thodos, W. F. Stevens

The point-source function introduced by Horner' us a solution to the general unsteady-state equation for the flow of fluids through porous media has been utilized to calculate pressure profiles for adjacent wells as a function of time. By trial and error, the time at which the pressure waves meet to cause interference can then be calculated. For this calculation, the time of interference is arbitrarily defined as that time when both pressure waves exhibit a pressure decrease of 25 psi ar the same point within the formation. To illustrate the method, an example involving two adjacent wells producing at different rates has been worked out and is presented in detail. INTRODUCTION The initiation of production of a new well is usually accompanied by pressure drawdown measurements in order to establish the extent and characteristics of the formation. The interpretation of such pressure drawdown data is based on mathematical developments involving a single well in an infinite reservoir. If more than one well is present in a reservoir, the possibility of interaction between wells exists. Therefore, care must be taken in interpreting the pressure drawdown data. It becomes important to be able to determine when this interaction, or interference, becomes significant. This paper presents a method for the prediction of the approximate time of interference between two such adjacent wells. The basic differential equation describing the pressure distribution existing during unsteady-state flow of a slightly compressible fluid has been solved by Hor-ner.' for a constant production rate from a circular homogeneous reservoir of infinite extent. The solution utilizes the point-source function to define the pressure at various points within the formation as a function of time. The resulting expression is: q = production rate, STB/D ,;= formation volume factor, dimension-less µ = fluid viscosity, cp k = formation permeability. md H = formation thickness, ft C - fluid compressibility, psi' F = formation porosity, dimensionless R = distance from centerline of well, ft T = production time, hours Ei(—u) = Ei-function of U Eq. 1 may be used to calculate the pressure profiles generated during production from a single well in an infinite reservoir.

By R. V. Taborelli, P. B. Bucky

This investigation was carried on by means of models and the application of the principles of similitude to determine the effects of props, props and cribs and sand filling in longwall mining. The geologic structure and rate of face retreat were kept constant and the only change introduced was in the method of support. It is not felt necessary to repeat a description of apparatus, or to explain the fact that a scalar model made in all parts of the same material as the prototype will behave similarly to the prototype when rotated at speeds such that the centrifugal force is as many times the gravitational force as the prototype dimension is a similar linear model dimension. The reader is referred to earlier papers1,2 for further information regarding these statements. All models were made of sandstone and were as in Fig. 6. They are composed of a 3/16-in. ore body bed A, overlain by a 3/16-in. bed B, a ?-in. bed C and two ?-in. beds D and E. The model was rotated at + 2350 r.p.m. equivalent to a model ratio of 1268. It was worked back ¼ in. at a time before being replaced in the centrifuge and again rotated. Table 1 gives model and prototype dimensions occurring in this discussion, the prototype dimensions being 1268 times the model dimensions. Figs. 1, 2, 3 and 4 were taken while the model was rotating at 2350 r.p.m. One was taken for each face retreat of 26.4 ft., but for obvious reasons all are not presented. Fig. 1 (A-E) shows the behavior of this type of structure with face retreats of 26.4 ft. and no artificial support. Fig. 2(A-E) shows the behavior of this type of structure where breaker and face props were used. The props were exceptionally strong, being of steel ? in. square in cross section, and capped with 1/16 in. ± of soft wood and paper, to allow for roof deflection. The face props were

Jan 1, 1940

By P. B. Bucky, R. V. Taborelli

This investigation was carried on by means of models and the application of the principles of similitude to determine the effects of props, props and cribs and sand filling in longwall mining. The geologic structure and rate of face retreat were kept constant and the only change introduced was in the method of support. It is not felt necessary to repeat a description of apparatus, or to explain the fact that a scalar model made in all parts of the same material as the prototype will behave similarly to the prototype when rotated at speeds such that the centrifugal force is as many times the gravitational force as the prototype dimension is a similar linear model dimension. The reader is referred to earlier papers1,2 for further information regarding these statements. All models were made of sandstone and were as in Fig. 6. They are composed of a 3/16-in. ore body bed A, overlain by a 3/16-in. bed B, a ?-in. bed C and two ?-in. beds D and E. The model was rotated at + 2350 r.p.m. equivalent to a model ratio of 1268. It was worked back ¼ in. at a time before being replaced in the centrifuge and again rotated. Table 1 gives model and prototype dimensions occurring in this discussion, the prototype dimensions being 1268 times the model dimensions. Figs. 1, 2, 3 and 4 were taken while the model was rotating at 2350 r.p.m. One was taken for each face retreat of 26.4 ft., but for obvious reasons all are not presented. Fig. 1 (A-E) shows the behavior of this type of structure with face retreats of 26.4 ft. and no artificial support. Fig. 2(A-E) shows the behavior of this type of structure where breaker and face props were used. The props were exceptionally strong, being of steel ? in. square in cross section, and capped with 1/16 in. ± of soft wood and paper, to allow for roof deflection. The face props were

Jan 1, 1940

By AIME AIME

AMONG the recent mining and metallurgical developments in England great interest has been shown in the development of an iron-ore field covering 26,0.00 acres in Northamptonshire, containing 500,¬000,000 long tons of ore; and the erection at Corby, by Stewarts & Lloyds, of a modern basic bessemer steel plant and tube works. The entire project was planned to cost about £3,300,000. Many improvements have been made in the varied equipment that is being installed, some of which has set mew records for massiveness. There is, for instance, a new roll crusher, which has been manufactured by Hadfields, Limited, of Sheffield, the largest ever built in England, and weighing 130 tons. This machine, shown in Fig. 1, as erected at the Hadfields shops, is primarily designed to take blocks of iron ore as they come from the steam shovel. The maximum size is 54 by 48 by 36 in., hut occasional pieces slightly larger than this may be accommodated. The capacity is upwards of 500 tons per hour, the product passing to a secon-

Jan 1, 1934

By C. David Mann

Metal prices continued to improve in 1964, resulting in the opening of new mines and re- activation of old ones. Larger and deeper shafts are being bored. At the AEC's Nevada Test Site, a 72-in. diam shaft is being bored to 3200 ft. In New Mexico a 1650-ft shaft was bored with a 48-in. bit, then reamed to 124 in. to 935 ft, and 102 in. to 1650 ft. Deviation was only 26 in. In California a shaft sinking machine has been developed which combines drilling and mucking equipment into one mobile, integral unit.

Jan 2, 1965

THE opening session was held on Tuesday evening, February 17th, in the house of the American Society of Civil Engineers. The President of the Institute, Mr. E. B. Coxe, after a few introductory remarks, congratulating the members on the general prosperity in all the branches of the professions and manufactures represented by the Institute, called on Mr. E. F. Loiseau, of Philadelphia, for the first paper, On the Successful Manufacture of Compressed Fuel at Port Richmond, Philadelphia. Professor Prime called attention to the fact that the work of the United States Testing Board had been left unfinished at a point when valuable results had just been reached, and offered the following resolutions: WHEREAS, The American Institute of Mining Engineers has seen with regret that the work of the United States Testing Board has been suspended through the lack of an appropriation by Congress, which appropriation dues not seem likely to be renewed, and WHEREAS, It seems most important to the vital interests of our iron and steel manufactures that the experiments commenced by the United States Testing Board should be carried to successful completion ; Resolved, That a committee of three be appointed, to act with a like, committee to he appointed by the American Society of Civil Engineers, to carry out to completion the work commenced by the United States Testing Board. Resolved, That the American Society of Civil Engineers be requested to cooperate with this society by the appointment of a like committee. Resolved, That the committee have authority to request, in the name of the American Institute of Mining Engineers, contributions to a fund to be devoted to the completion of the experiments aforesaid. Resolved, That the Council of this Institute be authorized, whenever requested by the committee so appointed, to petition Congress for the free use of the testing machine at the United States Arsenal at Watertown. Resolved, That a committee be appointed by the President, to select the afore-said committee of experts. Resolved, That the President of the Institute have authority to fill any vacancies which may occur in the committee. The resolutions were unanimously adopted, and President Coxe appointed the following committee in accordance with the resolutions : Messrs. A. L. Holley, James Park, Jr., and J. F. Holloway. The President also appointed Messrs. Heinrich, Mickley, and Spils-

Jan 1, 1880

Bailey, Lewis Newton, Master Engineer, Senior Grade, 4th Regiment, U. S. Engineers, Headquarters Company, died of pneumonia at Camp Merritt, N. J., on April 30, 1918. Baird, Louis, Lieut., Royal Field Artillery, British Army, died on the battlefield in 1915. Cobeldick, William Morley, Royal Engineers, died from gas poisoning on October 7, 1915. Dougall, Ralph, killed in action. Evans, Alfred Winter, Lieut.-Col., New Zealand Rifle Brigade, D. S. 0., D. C. M., killed in action on October 12, 1917. Gorman, Thomas C., Lieut., Canadian Engineers, killed in France, Mar. 18, 1918. Hague, William, 1st Lieut., Engineer Officers' Reserve Corps, died in active service. Hall, William T., Capt., Royal Flying Corps, killed in action. Heine, Bernhardt E., Lieut., Aviation Service, killed at Fort Sill, Okla., Aug. 10, 1918. Irving, John Duer, Capt., 11th Engineers, A. E. F., died while on active service in France on July 26, 1918.

Jan 9, 1918

By William P. Shinn

It is almost exactly seven years since the last previous meeting of the Institute in this city. In a paper on " Pittsburgh, its Resources and Sorroundings," read at that meeting, I showed that Allegheny County had produced in 1878, " one-twelfth of all the pig-iron,.... two-ninths of all the rolled iron (except rails), over one-eighth of all the Bessemer and open-hearth steel, two-thirds of all the crucible blister and puddled steel, one-fourth of all the glass, one-eighth of all the coal, and one-sixth of all the pig-lead produced in the United States;" and the object of this paper is to show that, great as the record then was, it has been greatly surpassed since.* Population. In my former paper I expressed the opinion that the census of 1880 would show the population of Allegheny County to be in excess of 300,000. The census report made it 355,869. The following shows the growth of the county in population from 1790 to 1880: 1790,...........10,309 1800,........... 15,087 1810,...........25,317 1820,...........34,921 1830,...........50,552 1840.............81,235 1850............138,290 1860............178,881 1870,...........262,204 1880,...........355,869 This makes Allegheny County, in 1880, rank sixth in population of the counties of the United States, whereas in 1870 its rank was ninth. The population of Pittsburgh in 1880 was 156,359, being twelfth

Jan 1, 1886

By George W. Maynard

(Pittsburg Meeting, March, 1910.) AT the Pittsburg meeting of the Institute, May, 1879, I made the first announcement in America of the results obtained by Sidney Gilchrist Thomas and Percy C. Gilchrist, ill their efforts to eliminate phosphorus in the manufacture of steel in the Bessemer converter and the open-hearth furnace.1 The first published statement in an American newspaper was a reprint of the first Thomas-Gilchrist paper, 2 which I sent to J. C. Bayles, at that time the editor. At the suggestion of many friends who were present at the Pittsburg meeting, and others who are cognizant of the beginnings of the process in the United States, and for the information of the younger generation of the iron and steel fraternity, I have thought it eminently proper that the history of the process in detail should be given where I made the first announcement and where the process has had its largest development. During my residence in England, from April, 1873, to February, 1879, and as a member of the Iron and Steel Institute since 1874, I was present at many of its meetings during those years. At nearly -every meeting the burden of discussion was the question of the use of pig-iron containing phosphorus in the. manufacture of steel by the Bessemer or the open-hearth process. In my biographical notice of Thomas;3 read at the New York meeting of, the Institute, February, 1885, I show that the working-out of the process was not a hap-hazard or accidental inspiration, but the culmination of many years of investigation of experiments and theories which had been carried on and advanced by chemists and metallurgists at home and abroad. 1 Trans., viii., .5 (1879-80). 2 The Iron Age, vol. 22, No. 17, p. 3 (Oct. 24, 1878). 3 Trans.. xiii., 785 to 791.

Jul 1, 1910

By K. W. John, L. Mueller

The rock mechanics approach to stability problems of steep rock slopes dealt with in this article has particular reference to the concept and methods of analysis developed by the Salzburg, Austria, school of rock mechanics. These theories are based on the principal postulate of the anistropy of the technological properties of in-situ rock masses. The present approach, which is more in line with geological engineering concepts, differs distinctively from the soil mechanics and engineering geological concepts commonly practiced in the United States. PRINCIPAL ENGINEERING APPROACH TO GEOTECHNICAL PROBLEMS AVAILABLE ENGINEERING CONCEPTS At the present time two different concepts are available for engineering analysis of underground structures and foundations in different earth material: soil mechanics and rock mechanics. Of these two branches, soil mechanics is, after 40 years of initially revolutionary and more recently evolutionary development, well established. Its principal concept of replacing in-situ soils or rock masses by perfectly homogeneous substitues has resulted in many practicable methods of analysis which, if applied with judgement, have resulted in successful applications throughout the world. Conversely, rock mechanics is still in the early stages of revolutionary development. Few, if any, standard concepts or methods of analysis have been established. Opinions vary widely. At present the professional background of an investigator very often determines the engineering concept to be applied to a particular problem, rather than its actual requirements if considered objectively. This paper stresses the rock mechanics or geological engineering point of view advocated by the Salzburg group. A review of the soil mechanics concepts is considered beyond the scope of this paper. THE "SALZBURG APPROACH" TO ROCK MECHANICS The rock mechanics concept as outlined here was initiated by Stini, based upon work by sander.1 Since

Jan 1, 1963

By A. T. Shurick

THE NRA Administrator's casual reference to the coal code as the next "pineapple" to be fixed was a conservative estimate of his job. This thorny and adamantine morsel now looms as a critical test of NRA. A failure in coal supplies would have the gravest potentialities in the Administration's program at this time. And the loss of prestige from a breakdown of the enforcement machinery in this initial large-scale test of strength would accentuate the incipient indications of insurrection in other directions. General Johnson has had a difficult task. It involves problems festering for a quarter of a century. A succes¬sion of Presidents have been glad to tide over their respective regimens with inconclusive expedients; they have been baffled by the complexities of the technical angles and hopelessly irreconcilable concepts of powerful conflicting interests with large issues at stake.

Jan 1, 1933

By A. Raja, C. C. Harris

Apparatus for maintaining constant — but adjustable — pulp level and rate of froth removal are described. The results of preliminary experiments into the effect of scraping speed on the rate of solid and liquid recovery with a semi-batch system are presented. Glass spheres restricted to a narrow size range is the solid under investigation. Graphical methods for estimating the mass of recoverable material and the zero error are given. The semi-batch procedure1 is extensively used in laboratory flotation testing, though the method is subject to variability of moderate severity. Raffinot2 concluded from a statistical study of the difference between replicate tests, that the ptincipal source of errot was the test itself and not sampling or chemical analysis. These differences can be ascribed principally to changes in pulp volume and reagent concentration, and uncontrolled rate and depth of scraping. Semi-batch testing has been criticized1 on these grounds as a means of investigating flotation kinetics. This paper discusses and describes modifications to laboratory flotation equipment which reduce the number of manual operations and lead to improved control. Kinetic studies were the primary reason for developing the apparatus described. It should, however be a useful aid in normal flotation testing. Some preliminary results of experiments are presented.

Jan 1, 1967

By K. E. Merklin, F. D. DeVaney

Students of the modern blast furnace seem unanimously agreed that they are observing a major revolution in practice. Rather than changing construction and operation of the furnaces, most of the great advances now under way deal with the raw materials feed. Prepared burdens have gained increasing significance during recent years. Sizing, concentration, and agglomeration have all become bywords to greater production and reduced operating costs. The remarkable results with test furnaces in which self-fluxing sinter makes up part or all of the ore charge foretell even greater improvements. Other investigators have reported many of these advances, and it is enough to predict here that more and more operators will turn to the use of self-fluxing sinter. This change in the operator's requirements comes at a time when one major section of the iron ore regions, the Mesabi, is finding it difficult to maintain, let alone improve, the tonnage and grade of its shipping product. To meet the new demands with respect to quality and structure, the iron ore producers of this area turned first to screening, crushing, and concentrating intermediate ores. More recently, declining higher-grade reserves have led to multimillion-dollar investments in plants to treat the low-grade magnetic taconites.1,2 To yield a suitable concentrate, the Minnesota magnetic taconites must be ground to about 80 pct through 325 mesh, too fine to be charged directly to the blast furnace or to make acceptable feed to the sinter strand. The pelletizing process was developed to meet the structure requirements laid down by the blast furnace operators. Taconite producers are succeeding in their efforts just when the attention of the blast furnace man is being drawn to the great advantages of self-fluxing sinter.3 The taconite operators are ready with an alternative. Why not produce sclf-flus ing pellets? At the Hibbing laboratory of Pickands Mather & Co., where much of the early work was done that led to Erie Mining's 7.5-million ton taconite plant at Hoyt Lakes, Minn., preliminary investigations have been made. Present Taconite Operations: Details of this work will be clarified by a brief review of agglomeration practice in commercial taconite plants. After the taconite has been ground to about 80 pct through 325 mesh to liberate the magnetite grains, the ore is concentrated on magnetic separators, thickened, and filtered to about 10 pct moisture. This structure and moisture fit in closely with the conditions required to produce good balls. It has been found with taconite concentrates that at least 60 pct of the feed to the balling drum must be —325 mesh to obtain good ball compaction within reasonable time. The filter cake moisture usually permits a slight water addition at the balling drum, and further balling control is obtained by adding about 12 Ib of bentonite and 1 1/2 lb of soda ash per dry short ton of concentrate. These additives, and occasionally small quantities of fine coal, are thoroughly mixed with the filter cake ahead of balling. The swelling property of bentonite improves the green and dry strength of the balls so they can be handled without breakage until they are fired. Soda ash is added for pH control to insure maximum performance of the bentonite, and coal is added for more heat in the firing process when required. The balls, normally 3/8 to l-in. diam, are brought to a temperature of 2350" to 2500°F in the indurating furnace. Heat for this process comes from three sources: 1) fuel burned in the combustion chamber, 2) oxidation of the magnetite concentrates to hematite, and 3) coal added to the balls. Pellet strength is developed by grain growth and bridging and, if enough slag-forming constituents are present, by slag bonding. Conversion to hematite is usually close to 100 pct in the fired pellets. Batch Test Procedures: Initial studies of self-fluxing pellets were made by producing balls on a batch basis under carefully controlled laboratory conditions. The l-in. balls, uniform in size, were fired in an oxidizing atmosphere in a standard laboratory muffle furnace. The firing procedure consisted of

Jan 1, 1961

Meeting of Mar. 26, 1915.-The following Committee on Nominations was appointed: Fred W. Bradley, Chairman; James F. Kemp, Past President; Frank M. Smith, Chairman Montana Section; R. C. Gemmell, Chairman Utah Section; Frank A. Ross, Chairman Columbia Section; Louis S. Cates, Arthur Thacher. The Committee on Precious and Base Metals and the Committee on . Coal and Coke were appointed, the personnel of which is shown in another part of this Bulletin. The President was authorized to appoint a Committee on Welfare with J. Parke Channing as Chairman. W. P. Cutter, Librarian of the United Engineering Societies' Library, was appointed a representative of the Institute on-the Committee oh Classification of Engineering Knowledge to co-operate with similar committees of other engineering societies. A vote of thanks was extended-to Edward B. Sturgis for his excellent work in arranging for the Annual Dinner. in February, of this year. The suns of $57.30 was appropriated to the New' York Section. It was resolved that in response to the courteous invitation from 36 members of the Institute in Arizona and the neighborhood, the 1916 summer meeting of the Institute be held in Arizona.

Jan 5, 1915

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

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

Jan 1, 1969

By J. R. Kyte, L. A. Rapoport

The material balance equation for partiai or complete 1:trter-drivc reser1,oir.s been re-arranged to include a pressure irrferferencr ternr. This pressure interfernce term was ohtained from the theory established hy Morinda. A .successful progranl deissed to solve the resrclritlg c,qrration in a few hours on a nleelitr .sizeti, digital compuier. No prior knowdelge of the properties of the eqrrifer is required in the solution. 7' ri~rl calculations areatle using various assumed con!binuiions of value of the constant expretsing the qquifer properties correspoinds to the best fit ohtained between observed and calculated field history. Details of the method of calculaion are illus trated by an example. INTKCDUCTICN A method 01' calculating the material balance equation for a partial water-drive reservoir completed in an aquifer has been described in the literature." Pressure interference i'rc,m other pools iocated in the same nquiucr is not taken into account in this solution. When present, the pres- sure interference effect must be included in order to make a complete performance analysis. A correction for this effect has been developed using the theory cstahlished by Mor-tada.' Manual calculation of the material balance for water-drive reservoirs is a formidable task. Such manual ca1culation when interference eflects are includcd would be impossible to completc in any reasonable time period. However, a medium-sized stored program, digltal conlputer can he used satisfactorily. The method is described in the following sections. THECKETICAL CONSIDEKATIOhS The material balance equation for a reservoir producing by partial or total water drive, as proposed by Van Everdingen, This relationship is linear for Z/a from which thc slopc, Z, and the intercept, N, can be obtained by the least-squares method from the data points. It is not necessary lo know the aquifer properties to uktain oplimum values tor 2nd IV. Various assumptions are made tor At,,: where 11 is production time, seconds (oilfield between the equal successive time intervals into which the history is divided. t,, h It, times the number of equal time periods from i.e.. the total dimen-sionless time interval from It is noted that the dinlensionless time interval, At,,, involves all the relevant aquifer properties. Accordingly, any assunlption of is an assumption of the combined aquifer properties which pertain to the solution of the equation. Deviations of the data points from a straight line arc calculated for each assumption of and the value of A, correspond ing to the minimunl deviation is chosen as the optimum solution. This method is described in Ref. 1 and 2. Thc material balance solution is usually made with an initial assumption that the reservoir being studied is located in an infinite aquifer from which there are no other withdrawals. Deviation of the points by amounts greater than can be attributed to inaccuracies in field data may be the result of: (1) boundary effects of a limited aquifer, (2) pressure interference from othcr pools completed in the .sarne aquifer, and (3) a combination of these effects. Differentiation of these effects can be made only on the basis of geologic data. Pressure interference from other fields will result in a pressure drop at the original oil-water boundary having an interference component, The measured instantaneous pressure drops at the oil-water

By K. E. Brown, J. P. Brill, A. R. Hagedorn, T. C. Doerr

A recent correlation for multiphase vertical flow by Hagedorn and Brown1,7 has been used to examine the effect of numerous variables such as pipe size, gas-liquid ratio, liquid flow rate, liquid viscosity, liquid surface tension, oil API gravity, water cut, etc., on flowing pressure gradients. Most previous correlations were derived by using a form of the general energy equation which assumed negligible changes in kinetic energy. Under certain flow conditions this can lead to serious errors in predicting pressure traverses. Examples of traverses calculated both with and without the kinetic energy term are included here. The Chew and Connully correlation for calculating changes in viscosity with changing pressure and temperature has been extended so that the effect of a changing viscosity with depth on pressure gradients can also be predicted. Examples illustrating the effect of tubing size show the importance of this correlation in determining pipe sizes for initial well completions. Limitations on production rates in small pipe sizes from excessive friction losses can be predicted with greater accuracy. The combined use of a vertical-flow correlation with a horizontal-flow correlation provides a method for determining the maximum flow rate possible from both flowing and gas-lift wells. Examples are included to illustrate application of the combined use of vertical and horizontal-flow correlations in optimizing production rates and minimizing excessive costs from over injection of gas in gas-lift wells. INTRODUCTION The problems of multiphase vertical and horizontal flow are immediately encountered in any producing well. Fig. 1' shows three stages of production: flow in porous medium, vertical flow and horizontal flow. All three stages affect the production rate from a well. If we are confronted with a particular well and consider only the variables involved in the vertical and horizontal stages of Fig. 1, we should be capable of calculating the flowing bottom-hole pressure necessary to produce a particular flow rate. A change in any of the variables results in a new solution to the problem. Recent approaches in developing horizontal and vertical-flow correlations have improved to the extent that the solution can now be obtained with more confidence than previously. VERTICAL AND HORIZONTAL FLOW CORRELATIONS Since Poettmann and Carpenter's' original work in multiphase vertical flow, there have been many additional vertical-flow correlations developed which have greatly improved the accuracy of pressure-loss calculations. These

Jan 1, 1967