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By Thomas B. Douglas

INSTRUMENTATION in the process industries can no longer be regarded as a convenience, but rather an absolute necessity. Although many chemical processes must already be conducted with instruments, every properly designed application of automatic process control will improve operation: 1) the product will be more uniform and of higher quality; 2) the manual labor saved will reduce cost; 3) uniformity of flow and uniform control of process will increase production; and 4) surges will be eliminated, making it possible to operate equipment near ultimate capacity at all times. Industrial instruments fall into two general classifications—those that measure and those that measure and control. The cement industry has been using instruments for a number of years, particularly in the burning process, primarily for indicatlng and recording. Automatic process control makes additional use of these instruments, and others, to control variables. Assisted by the instrument manufacturers, engineers at Ideal Cement Co. achieved considerable progress in determining how well the usual recording instruments could be made to control various phases of cement-making. The results of these experiments were used to make the new plant at Houston as wholly automatic as possible. The original Houston plant, built in 1937 by Gulf Portland Cement Co., consisted of only one 8 x 220-ft kiln and the necessary grinding mills and accessory equipment and had a capacity of 900 bbl per day (300,000 bbl annually). Ideal Cement Co. purchased this plant in 1940 and almost immediately started plans to increase its production. In 1941 a 9 x 220-ft kiln was installed and grinding and storage capacity were increased. The growing demand for cement in the Houston area necessitated a third kiln (9 ft 6 in. x 250 ft) in 1947. This brought the plant up to its present capacity of about 1.3 million bbl per year. Calcium silicates in cement are produced in a burning operation at high temperatures from raw mix of argillaceous and calcareous materials that have been crushed, mixed, and ground to a fine powder. The composition of this mixture must be kept consistent within narrow limits. Raw mixing and grinding may be done either wet or dry, but in both instances the materials must be correctly proportioned, finely ground, and thoroughly intermixed before entering the kiln. As the raw mix passes through a long horizontal rotary

Jan 1, 1959

By P. P. Gregory

ESTIMATION of re- serves in prorated sand fields has been discussed by S. A. Judson, H. D. Easton, Jr., and W. A. Schaeffer, Jr., in a paper that appears in Vol. 114 (1935), of the A.I.M.E. TRANSACTIONS. The purpose of this present paper is to extend the investigation to prorated flush limestone fields and to outline a method of procedure in determining the amount of oil originally in place and the amount of oil that can be recovered from such fields. Although it is possible to use the decline curves to estimate the amount of recoverable oil in certain limestone fields that are allowed to produce the oil at or near capacity production, such a method cannot be used in fields under severely restricted production rates. The problem is exceedingly complex due to fluctuation in the allowable rates of oil production over a wide range and at frequent intervals, and especially because of the heterogeneous nature of the reservoir itself. A well completed in a large solution cavity, capable of producing great quantities of oil per day, may be offset by a well of very limited capacity. Also, it is di5cult to arrive at an average figure for porosity in a given limestone reservoir due to abrupt variations in the character of limestone, in many instances even between two ad- joining wells. It is a safe statement that uniformity of porosity and permeability in limestone reservoirs .is nonexistent. It is evident, therefore, that the method of estimation of re- serves on the basis of per-acre content can give only approximate results.

Jan 1, 1935

By D. F. Lascelles

D. F. Lascelles (School of Earth Sciences, MacQuarie University, North Ryde, N. S. W., Australia)- The paper contains many misconceptions commonly held by both geologists and soil scientists with regard to soil processes. The analogy between supergene copper enrichment and "nickel laterite" deposits is quite apt and the use of nomenclature from the former in describing supergene nickel deposits is justified. However, care must be exercised with regard to the notable differences in

Jan 1, 1974

By L. C. Reese

The ultimate resistance against a rigid cylinder which is moved laterally in a cohesionless soil is a function of the geometry of the cylinder and the properties of the soil. An approximate method is developed for computing this resistance and is tested against results of laboratory experiments. Satisfactory agreement between the method and experiment was obtained. Not only was the ultimate resistance against the cylinder measured, but measurements were made of the shape of the rupture surfaces. These measurements should allow the development of a more rigorous computation procedure. INTRODUCTION A critical aspect in the design of offshore drilling platforms is assuring the stability of the platform during a hurricane. The large horizontal loads from waves and wind make a severe loading condition. Piles are generally employed as the foundation since they can be effective in resisting both horizontal and vertical loads. Spuds are sometimes used with a mat foundation, where the spuds are designed principally to resist horizontal loads and the mat designed principally to resist vertical loads. The research reported in this paper is related to one aspect of the problem of laterally loaded piles or spuds in sand. The complete solution to the problem of the laterally loaded pile in sand would require the prediction of soil resistance against the pile as a function of pile deflection; only the ultimate resistance against short piles is considered in this paper. A considerable amount of additional research will be necessary in order to obtain a complete solution to the problem of the laterally loaded pile in sand; however, the work reported here should be useful as a guide in the performance of some of the additional research. ULTIMATE LATERAL RESISTANCE AGAINST A LONG FALL IN SAND To aid in the understanding of the theory which is developed for a cylinder, the theory for earth pressure against a wall is reviewed. In Fig. l(a), a long wall of height H is shown embedded in soil. Soil resistance will develop as the wall is deflected, and the soil resistance will increase with deflection until some limiting value is reached. In this discussion, and those following, it is assumed that all points on the wall deflect equal amounts. A possible shape of the real curve which relates soil resistance to deflection is shown by the dashed line in Fig. l(b); however, as a means of simplifying the discussion, an idealized curve (shown by the unbroken lines) is drawn. If such a wall as this were furnishing the horizontal support for an offshore structure, the dashed curve in Fig. 1 would be the information needed in performing a foundation analysis, with the idealized curve possibly being an acceptable substitute. For the idealized curve, the soil resistance is linear with deflection for the early portion of the curve such that

By J. L. Gillson

The word "other" in paragraph 4, line 11, column 1, page 205, should read "over;" the correct version of reason 2, this paragraph, is: 'L. . . 2) to evaluate the usefulness of certain geo-chemical techniques, V. C. Kennedy and S. W. Hobbs of the USGS had already studied the distribution of copper, lead, and zinc in the soils (both in background areas and over veins) and in the water and plants, providing a criterion for measuring the magnitude and effect of soil contamination; ...9 On page 206, in the top left hand corner of Fig. 1 (in the area bounded by 47°35' and 116°10), the reference to Fig. 3 is incorrect. The area of Pine Creek, South Fork Coeur d'Alene River, and portion of Big Creek (the area which includes, from left to right, Kingston, Pinehurst, Smelterville, and Kellogg) should be identified as that of Fig. 2A. The headings for columns 1, 3, and 4, Tables I and 11, page 209, column 2, were reversed when the paper was printed. The corrected tables are:

Jan 1, 1961

Jan 1, 1968

By W. F. Kieschnick, C. C. Miller, E. R. Brownscombe

The possibility has been mentioned that large pressure gradients in a solution gas driven field caused by high production rates might lead to a reduction in the ultimate recovery obtainable compared to that which would be obtained by a very slow rate of production. In the present study the reservoir conditions accompanying a high rate of production and the corresponding ultimate recovery were compared with those obtained if the reservoir were produced at some marginal rate throughout its entire life. The method of calculation involved the application of fluid flow - material balance analysis to a series of successive steady state conditions in the reservoir. In the case studied very little difference in recovery was obtained at the abandonment pressure. An analysis of the basic factors involved indicates that the sane results would hold for considerable variation in properties of reservoir fluids, permeability of the formation or well spacing. Even if the reservoir has a water drive it appears that no harm will be done by- open flow production if the rate is cut back before any appreciable free gas is produced. However, any condition which leads to disproportionate withdrawal rates and which causes large pressure differences over large areas might result in substantial loss in ultimate recovery. INTRODUCTION When the pressure declines in a reservoir, gas is evolved from solution causing a shrinkage in the reservoir oil so that a stock tank barrel of oil with its residual dissolved gas occupies a smaller volume than originally. For a given reservoir oil saturation, therefore. stock tank oil content will be higher at the lower pressure. At the same time the evolved gas which remains in the reservoir reduces the reservoir oil saturation, and this would be expected perhaps to cause a reduction in stock tank oil content. However, around the well the shrinkage occurs so rapidly that stock tank oil may actually accumulate in this region in spite of the simultaneous reduction in reservoir oil saturation. If this accumulation should continue, spreading into a larger zone, the loss might be appreciable and because larger pressure gradients accompany higher rates the accumulation might possibly be large if wells are produced at too high a rate. It is the purpose of this paper to determine the effect of rate of flow on ultimate recovery by solution gas drive. The problem under consideration involves the unsteady state flow of oil in a closed radial system from which oil is produced by the evolution of gas as a finite drawdown is estalllished and the reservoir pressure declines. A somewhat similar unsteady state production problem dealing, however, with a linear system has been considered by Muskat and Meres, but these authors do not discuss the effects which rate may have on recovery. Other methods have been reported for estimaling recovery as a function of pressure decline when producing at an infinitesimal rate, and Muskat and Taylor' have shown the effect of fluid and reservoir characteristics on recovery under these conditions. Assuming steady state to exist, Babson has given a procedure for computing the decline in well productivity as the field is depleted, and Moyer has combined these methods to estimate abandonment conditions and the resulting ultimate recovery. These latter approaches to the problem either do not consider the effect of rate or assume steady state can be established. In detail this paper presents the behavior of the reservoir producing under unsteady state conditions and considers the effect of production rate during this unsteady state flow on the recovery attained. The analysis also shows that there will exist at abandonment a type of steady state, and on this basis a method has been clevelopetl for rapid calculation of abandonment conditions.

Jan 1, 1949

By C. C. Miller, W. F. Kieschnick, E. R. Brownscombe

The possibility has been mentioned that large pressure gradients in a solution gas driven field caused by high production rates might lead to a reduction in the ultimate recovery obtainable compared to that which would be obtained by a very slow rate of production. In the present study the reservoir conditions accompanying a high rate of production and the corresponding ultimate recovery were compared with those obtained if the reservoir were produced at some marginal rate throughout its entire life. The method of calculation involved the application of fluid flow - material balance analysis to a series of successive steady state conditions in the reservoir. In the case studied very little difference in recovery was obtained at the abandonment pressure. An analysis of the basic factors involved indicates that the sane results would hold for considerable variation in properties of reservoir fluids, permeability of the formation or well spacing. Even if the reservoir has a water drive it appears that no harm will be done by- open flow production if the rate is cut back before any appreciable free gas is produced. However, any condition which leads to disproportionate withdrawal rates and which causes large pressure differences over large areas might result in substantial loss in ultimate recovery. INTRODUCTION When the pressure declines in a reservoir, gas is evolved from solution causing a shrinkage in the reservoir oil so that a stock tank barrel of oil with its residual dissolved gas occupies a smaller volume than originally. For a given reservoir oil saturation, therefore. stock tank oil content will be higher at the lower pressure. At the same time the evolved gas which remains in the reservoir reduces the reservoir oil saturation, and this would be expected perhaps to cause a reduction in stock tank oil content. However, around the well the shrinkage occurs so rapidly that stock tank oil may actually accumulate in this region in spite of the simultaneous reduction in reservoir oil saturation. If this accumulation should continue, spreading into a larger zone, the loss might be appreciable and because larger pressure gradients accompany higher rates the accumulation might possibly be large if wells are produced at too high a rate. It is the purpose of this paper to determine the effect of rate of flow on ultimate recovery by solution gas drive. The problem under consideration involves the unsteady state flow of oil in a closed radial system from which oil is produced by the evolution of gas as a finite drawdown is estalllished and the reservoir pressure declines. A somewhat similar unsteady state production problem dealing, however, with a linear system has been considered by Muskat and Meres, but these authors do not discuss the effects which rate may have on recovery. Other methods have been reported for estimaling recovery as a function of pressure decline when producing at an infinitesimal rate, and Muskat and Taylor' have shown the effect of fluid and reservoir characteristics on recovery under these conditions. Assuming steady state to exist, Babson has given a procedure for computing the decline in well productivity as the field is depleted, and Moyer has combined these methods to estimate abandonment conditions and the resulting ultimate recovery. These latter approaches to the problem either do not consider the effect of rate or assume steady state can be established. In detail this paper presents the behavior of the reservoir producing under unsteady state conditions and considers the effect of production rate during this unsteady state flow on the recovery attained. The analysis also shows that there will exist at abandonment a type of steady state, and on this basis a method has been clevelopetl for rapid calculation of abandonment conditions.

Jan 1, 1949

Jan 1, 1970

Jan 1, 1967

Jan 1, 1969

By D. Kuhlmann-Wilsdorf

A new theory of work hardening is developed which rests on only a few simple principles and is applicable to a wide variety of materials and dislocation structures. It explains, qualitatively, the general characteristics of the three-stage shear stress-shear strain curve of fcc metals, as well as the most important features connected with easy glide. The value of, the work-hardening coefficient in stage 11, is derived quantitatively, from first principles, and is shown to be insensitive to many changes in detailed dislocation behavior. The low rate at which energy is stored during mechanical working in stage 11, the dependence of slip line length on stress, and the so-called Cottrell-Stokes law are explained. Although the theory is primarily developed for fcc metals and alloys, it is largely applicable also to some other materiuls, and in particular apparently to poly crystalline simple steels in theh linear range of hardening. FOR pure fcc metals and alloys, after many different pretreatments and under a great variety of testing conditions, the well-known three-stage work-hardening curve is commonly observed. Moreover, the work-hardening coefficient in stage 11, 011, bears an almost constant relationship to G, the modulus of rigidity, such that K = G/Brr" = 300, within a factor of about 2 either way. The very fact that the three-stage curve and the value of K are so very persistent, can hardly be understood except on the assumption that some quite simple phenomena are responsible. In this light all available theories of work hardening must be judged unsatisfactory: Each of them is based on some quite specific dislocation model, while it is known that metals with similar values of K may have widely different dislocation structures, for example, dislocation tangles in the case of pure fcc metals, and piled-up groups of dislocations in a-brasstype alloys. Thus, while a particular theory might conceivably represent a special case with a reasonable degree of accuracy, it cannot possibly have illuminated the underlying principle. Another severe criticism applying to the presently most widely discussed theories is that they employ empirical or even quite unexplained parameters in order to arrive at a numerical value of K. This indicates that some vital aspect of the work-hardening process has remained unexplained. In the present paper, a new theory of work-hardening is developed which does not suffer from the above shortcomings. It is derived from first principles, and is founded on the realization that the underlying causes for the three-stage work-harden- ing curve and for the persistence of the numerical value of O,, must be few and simple. As a first step it is shown herein that the experimental oklservations on "easy glide" can be explained on the assumption that in this stage dislocations multiply, beginning in a number of restricted area:;, and from there penetrate into crystal regions still substantially free of glide dislocations until a quasi-uniform dislocation distribution has been established. During "easy glide", the resistance to the movement of the foremost dislocations, advancing into still largely a dislocation-free regions, is believed to control the flow stress. This resistance is determined by various factors, among them the reaction of the dislocations against bowing-out into loops, which is due to their line tension. Without such bowing-out no dislocation multiplication would occur, but the corresponding contribution to the flow stress stays practically constant all through stage I. According to the present theory, stage TI begins as soon as there are no more areas left into which newly formed dislocations have not yet penetrated, i.e., as soon :is a quasi-uniform dislocation density has been attaned. Regardless of how, in detail, the dislocations should happen to be arranged at the end of "easy glide", whether in pileups or tangles, on one glide system or on several, the stress necessary to overcome the line tension of the segments must now increase, because the dislocation density increiises while the average free dislocation length decreases. It is argued that all other contributions to the flow stress stay at about the same level as in stage I, and that the said increase in the stress necessary to bow out dislocation segments into loops is responsible for the major part of work hardening in stage 11. Stage TII of the stress-strain curve is explained by largely follouing the ideas of Seeger and his co-workers; however, it is pointed out that the start of stage m may on occasion indicate the onset of "conservative climb" instead of cross slip. Indirect evidence indicates that linear work hardening persists even under the profuse action of cross slip, but at a lower rate. Only when, in addition to cross slip, climb operates extensively, does the work-hardening rat,. drop sharply, ultimately down to zero. BRIEF SURVEY OF EXPERIMENTAL EVIDENCE The stress-strain curve of crystals of many different substances are qualitatively similar. No significant permanent plastic deformation takes place below the so-called "critical resolved shear stress". As the applied stress is raised beyond this level. yielding occurs with little or no work hardening and

Jan 1, 1962

By W. H. Bassett

THE ease and value of the application of spectrum analysis to industrial chemistry appears to be appreciated in few of the large works laboratories of this country. For 8 years, this analysis has been of great value to the authors in both analytical and metallurgical work. To the analyst the complete qualitative analysis of an unknown alloy, revealed by a spectrogram, is a sure basis for the planning of the most direct and rapid method of attack. As the determination of each element proceeds, the purity of precipitates may be checked as often as desired. The spectrograph proves invaluable in the recognition of impurities, the separation of which would entail a lengthy and difficult procedure, or when the weight of an unknown is less than is necessary to complete the desired determinations. A few hundredths of a gram will usually suffice for spectrographic analysis. In the separation of the rarer elements, the spectrograph will show, with precision, the point where each step has been successful. In zinc of certain classes, indium, gallium, and germanium have been isolated in this way; the presence of antimony in lake coppers has been proved; and bismuth has been detected in zinc and in alloys where the presence of a few thousandths of a per cent. would be harmful. An antimony precipitate in a delicate determination may contain a considerable proportion of tin or arsenic, both of which will be revealed. The spectrograph is consequently important to the metallurgist as well as to the analyst.

Jan 2, 1922

Jan 1, 1970

Map

Jan 1, 1939

By E. Stamboliadis

Dithiophosphates are reaction products of phosphorous pentasulfide with various organic compounds such as phenols, alcohols, mercaptans, dioalcohols, amines, and nitrides. The products with alcohols are commercially known as "aerofloats" and these are investigated in this presentation. It appears that the formation of insoluble metal dithio phosphates is responsible for their collecting properties with minerals." In this aspect they are very similar to xanthates.3 The solubility of metal dithiophosphates depends on the number of carbon atoms in the organic chain of the corresponding alcohol. An empirical mathematical model is presented here. It allows one to predict the number of carbon atoms necessary to ensure that a certain metal dithiophosphate will form at a selected pH value of the water solution. The Nature of Dithiophosphates Dithiophosphoric acid is the product of the reaction of phosphorus pentasulfide with an alcohol, as shown in the following equation: [P2S3 + 4 ROH - 2 (RO)2 PS2H + H2S (1)] where R stands for a hydrocarbon chain. The acid so produced can be neutralized by a base to produce the corresponding salt as shown in the following: [(RO), PS2H + NH- (RO)2 PS, NH,. (2)] In this aspect dithiophosphates are similar to xanthates, the only difference being the substitution of the tetravalent carbon atom in the xanthate polar group by a pentavalent phosphorus atom in the dithiophosphate group. This difference implies at least two characteristics: (1) that there are two hydrocarbon chains associated with the polar group so that an increase in the number of carbons of the single chain is expected to have a pronounced effect on the flotation properties, and (2) that the bond between the sulfur and the central atom is stronger than in the xanthate group. This probably leads to a decrease in both the polarizability of the electronic cloud of the sulfur and the interaction with cations.3 However, what is lost in the dithiophosphates by having a weak polar group can be gained by an increase in the number of carbons as will be seen later. Taking into account the size of the atoms involved in a dithiophosphate molecule, as listed in Table 1, It is possible to draw an approximate diagram of its structure as shown in [Fig. 1]. It is assumed here that the length of the single hydrocarbon chain is equal to that of the corresponding alcohol. Preparation and Purification Acid dithiophosphates can be prepared, as previously mentioned, in accordance with Eq. 1. The rate of this reaction is too slow at room temperature and can be improved by increasing the temperature to about 60° or 80°C7 or by evacuating the system. In our investigation evacuation was performed by a water aspiration pump. The dithiophosphoric acid so produced is liquid and dissolves in the alcohol. At the end of the reaction the product is filtered from the remaining P2S3; and dissolved in petroleum ether. By blowing anhydrous ammonia gas into the ether phase the ammonium salt forms as a white precipitate and can be sepa-

Jan 1, 1977

By E. R. Darby

When bronze is melted in open-flame furnaces a considerable amount of slag is formed during the melting operation. This slag may be incidental to the melting practice or it may be formed intentionally by slag-forming constituents added to the furnace charge to provide a covering for the melted metal When this slag is removed from the furnace, it contains, in addition to the free metal carried with it by the skimming operation, a considerable amount of metal as oxide either combined or mechanically mixed with it. By the usual crushing and washing methods this metallic oxide generally goes to the tailings pile, so diluted by other foundry waste that its recovery is economically impossible. If this furnace slag is not inixed with other foundry waste, practically all of the metal oxide as well as the free metal may be recovered by some form of smelting operation. The first side of the problem to be considered is the amount of slag produced. It may not be true that the majority of foundries have no record of the amount of slag made, but it is certainly true that a very great many either have no record or pay little attention to it. Usually the collection of furnace slag for one normal week's run will be sufficient to determine the advisability of its being handled in a different manner from the crushing and concentrating generally employed. If a large representative sample of the accumulation is crushed and concentrated and the recovery compared with the results of a careful assay, it will be found that a large percentage of the total metal content was lost in the washing process. The percentage of loss will be greater than the metallic oxide content, as some free metal in extremely fine particles always is carried away in grains of slag. From such a comparison, it may be determined whether or not the slag should be melted, sold or treated as before. An example of an actual test of this kind is given below. In a foundry melting 50,000 to 80,000 Ib. of bronze per day, a week's accumulation of furnace slag and ladle skimmings amounted to 16,500 1b.—over three times as much as the foundry superintendent estimated before the accumulation was made. Two tons of this material was taken as representative of the lot. This was ground through a thoroughly cleaned ball-mill and concentrated over a Wilfley table. The

By F. C. Wagner, E. P. Klier, D. E. Nulk

The presence or absence of plastic strain at a cleavarage surface is a question of much importance, and upon its answer depends future possible theoretical work. To investigate this question back-reflectinn X ray patterns were obained on the cleavage facets of ferrite grains. The test set-up consisting of FeK colli- mated radiation, single grain material and stationary cassette and specimen will not lead to a diffraction pattern, in general, but is an ideal arrangement for the exami-nation of surface effects. A pattern taken on hydrogen purified

Jan 1, 1949

By GERALD H. HUTTON

SINCE the inception of dredge mining, the attention of engineers and operators has been directed primarily toward mechanical improvements and refinements calculated to, reduce operating costs and to increase recoveries. The result has been to develop an efficient excavating and value-saving machine, which, in some instances, has an actual capacity of 15,000 cu yd. per day. Steel hulls and superstructures minimize the risk of destruction by fire and lengthen the life of the plant. Dredges now restore, to some extent, agricultural possibilities to mined ground. Large buckets, with correspondingly heavy parts, are able to handle heavy wash and massive boulders. Mechanical weaknesses have been strengthened. The use of manganese steel in bucket lines and tumblers permits the application too the bank of a digging force that can be withstood only by solid material, such as rock, cement or frozen ground. Various alloys of steel have been applied to certain duties and experiments in their heat treatment are still progressing. Haulage by tractor or truck is in common use. Discarded and worn steel parts are repaired or rebuilt by the use of the oxy-acetylene flame and the arc-welding machine. Gold and platinum saving is more efficient, although still subject to improvement. Dredging for tin ore is in successful operation. Crews are now highly specialized and experience has developed improved methods of work; particularly in the manipulation of dredges to build levees of tailings and to leave open waterways where desired. To sum up by the distribution of overhead expense over several units, ground under favorable conditions can now be dredge-mined for 5)2 c. per cu: yd., in spite of the increase in cost of supplies, power, and labor during, the past few years. Improvements in prospecting methods, however, have not kept pace with mechanical developments. Standard practices of ; testing ground today produce only wide approximations of values; although they obtain information concerning the character of the deposit and bedrock. It is interesting to note that few important changes in the methods of ten years ago are .to be recorded; the latest comparisons between pros¬pecting results and mined recoveries-have been at such variance as to cause confusion to the operator, and, in a degree, the sacrifice of the confidence of the capitalist. On this account, and because the growing scarcity of dredging ground. demands closer calculations on the part of the engineer, this appeal for collaboration and

Jan 1, 1921

By M. ROWE, H. E. Gilliland, L. BARNES

The authors of this paper are to be commended for a clear exposition of a timely subject. However, some important conclusions, in addition to those presented in the paper, can be drawn from the results of their elaborate and rather complete numerical computations. First of all, the major part of the calculations presented in the paper are for semi-infinite slabs of oil shale (the authors designate this as Model I) and a question must be posed — Are these elaborate computations needed to arrive at realistic estimates of the shale oil displaced. Consider the model of Lauwerier* that corresponds to the authors Model I if thermal conduction parallel to the fracture be neglected. If one also neglects the finite (but relatively small) heat capacity of the material in the fracture, it is easily shown from Lauwerier's solution that the cumulative oil production would be:

Jan 1, 1969