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By D. J. Carney, R. L. Stephenson

A sampling technique has been developed for procuring a sample of sinter representative of the entire depth of the sintering bed. The sampling method involves the use of an open-bottom metal basket that rides on the grate of the sintering machine and when removed contains a sample of the sintered product. Additional data have been obtained to indicate that the tumbler test is a suitable means of measuring sinter strength. IN the last few years additional sintering facilities have been installed in both the Pittsburgh and the Chicago district of the United States Steel Co. Since the construction of these sintering plants made possible the use of higher percentages of flue-dust sinter in our blast-furnace burdens, it became important to study means of controlling the quality of sinter to obtain optimum results in the blast furnace. For controlling an operating process, it is necessary first to establish standards by which the quality of the product can be judged. For sinter, it appeared that an important property was its strength or its resistance to degradation during transportation and charging into the furnace. Consequently work was undertaken to establish a standard for sinter strength that could be used both for controlling sintering-plant operations and for correlating sinter quality with blast-furnace performance. The first problem in setting up a standard was that of procuring a sample that would be representative of the sinter made under any particular set of conditions at the sintering plant. Since the United States Steel Co. sintering plants discharge the finished sinter either into a large pit or onto a rotary cooler, the sinter becomes inseparably mixed with material sintered 2 hr before or 2 hr afterwards. For this reason the exact identity of the sinter is lost. A sample selected as the cooler is discharged, or as the sinter is removed from the pit, cannot be said to be truly representative of the sinter made at any specific time. Sampling The first attempt to procure a sample that would be representative of a specific sinter mix and of specific operating conditions was made by stopping the Dwight Lloyd sintering machine and removing an entire pallet full of sinter. This method, however, proved very difficult to perform and interfered considerably with the operation of the plant. To overcome this difficulty, a sampling method was devised by technologists at South Works enabling them to secure, without interrupting the sintering operation, a sample of about 1 cu ft of sinter, representative of sinter for the full depth of the sintering bed. The South Works method involves the use of a steel-frame-work basket. A typical basket is shown in Fig. 1. The basket has been used both with and without crossbars along the bottom. As long as the crossbars are in the same direction as the grate bars on the sintering machine they do not interfere with the sintering process. The basket is set on an empty grate of the Dwight Lloyd sintering machine before it passes under the swinging feed spout, see Fig. 2. When the basket is removed after it has travelled the length of the sintering machine, it contains the sample. Just before the basket is removed, the sinter is scored and chipped to facilitate removal of the sample from the sinter bed. A view of the basket after its removal is shown in Fig. 3. Although the sampling method was originally designed for use on a Dwight Lloyd sintering machine, it can also be used on the Greenawalt type of machine. When used on the Greenawalt-type machine, the basket is placed on the sintering grate before the charging car passes over it, and finally it is removed just before the pan is dumped. Testing After a method of obtaining a representative sample of sinter had been developed, the next step was to select a method of measuring its strength. The irregular shape and size of the sinter pieces precluded the use of a simple compression test for determining strength; consequently, the shatter test and tumbler test were investigated. To perform the shatter test, a sample of sinter, approximately 5 lb, is dropped from a hinged-bottom box at a height of 3 ft onto a steel plate. The broken sinter is sieve-analyzed after a specified number of drops. The tumbler test is performed with the use of a standard ASTM coke-tumbling drum. The drum is 3 ft in diam and is equipped with two lifter bars diametrically opposite one another on the inner periphery of the drum. The drum is rotated at a speed of 24 rpm for 200 revolutions, and after tumbling the sample is sieve-analyzed. To express as single numbers the results of sieve analyses after shattering or tumbling, the method suggested by R. E. Powers1 was employed. This method involved plotting the size of the sieve openings on a logarithmic scale and the cumulative per cent larger than each sieve on a probability scale as described by J. B. Austin.' By interpolating from the plotted data, which in most cases approximated

Jan 1, 1954

By Roscoe C. Clark, Henry F. Coffer, J. J. Reynolds

A cement slurry lightweight additive has been adapted in the Conoco laboratories for use in oil well cements. This additive makes possible the use of air to lighten oil well cement slurries. Specifically, the additive consists of small clay bubbles of air having sufficient strength to resist crushing by bottom hole pressures. Clay bubble cement slurry weights can be safely reduced to 10 Ibs/gal. In addition, the clay bubbles tend to act as lost circulation materials and to provide a cement texture which should produce good perforating characteristics. Clay bubble cement is relatively impermeable. INTRODUCTION In many areas in Oklahoma, Texas, Wyoming, and California, difficulties are experienced during casing cementing operations with lost returns caused by the high density of the cement slurries. Wells in these areas are ordinarily drilled with 9 to 11 lbs/gal. drilling mud while neat cement slurries may range in densities from 14.5 to 17 lbs/gal. In these areas, cement slurry weights are at present being successfully reduced to around 12 lbs/gal. (under bottom-hole conditions) by the use of additives in the cement. Since the use of additives will decrease the strength which the cement will develop, the amount of additives which can be used must be based on the minimum cement strength which is acceptable; and thus the maximum slurry weight can be reduced. At present, the minimum tensile strength generally acceptable is 50 lbs/sq in. cured for 24 hours under bottom-hole temperature. This figure is based on laboratory tests1 and on a large number of cement jobs which have been conducted using cement which will not develop more than 50 psi tensile strength in 24 hours under bottom-hole conditions. Two additives are now in use for decreasing cement slurry weights. These are expanded perlite and/or bentonite. The density of both of these materials is greater than that of the cement slurry; however, large quantities of water may be added with these materials without producing a slurry which will settle out of suspension. It is these large quantities of water which contribute to the reduction in the slurry weight. The minimum slurry weight at present obtainable with expanded perlite and/or bentonite is approximately 12 lbs/gal. using a standard Portland cement slurry. In many areas, this weight reduction is insufficient; and returns are still lost during casing cementing operations. In many cases, the pay formation itself is fractured; and a considerable quantity of this cement is lost to the pay zone. It is therefore believed that if cement slurry weights can be reduced to as low as 10 lbs/gal., two advantages will accrue: 1. The lighter densities (10 to 12 lbs/gal.) should mitigate or eliminate lost returns during casing cementing operations. 2. Cement slurries above 12 Ibs/gal. (the present minimum limit ) should develop considerably more strength than is now being obtained. A tensile strength of 100 psi is desirable.' It is the object of this paper to describe a cement slurry having a weight of 10 lbs/gal. and capable of

Jan 1, 1955

By Arthur F. Taggart

RECENT jig practice has shown such marked departures from the pronouncements of the textbooks, particularly as to particle size recovered and size range of feed, as to make it desirable to reexamine the mechanism of jig action. This paper presents the results of such a study. The general conclusions of the paper are: I. A jig bed is a bi-functioned as well as bi-parted body. The upper layer comprises a plastic stream moving slowly in a substantially horizontal direction from feed lip to tail board. It serves to rough out tailing and light middling and to pick up heavy middling and lift it out of the jig box. The lower layer acts as the ultimate separating means by absorbing concentrate and rejecting heavy middling. 2. The separating function of the lower layer is performed in three definitely different ways, depending upon the relative sizes of the grains presented to it and those composing it. EXPERIMENTAL The experimental work consisted of close and long-continued visual observations of operating beds in a plunger jig, eccentric driven, and in determinations of the settling rates of various metal spheres through a quartz bed under various conditions of operation. Falling velocities were measured by an ingenious method devised by A. C. Dorenfeld, a student in the Mining Department at Columbia. Spheres of different metallic alloys, Y8-in. diameter, were cast with a short piece of fine copper wire frozen in to serve for suspension. A short piece of black cotton thread was looped to the top of the wire to serve as an indicator. The set-up for a test is shown in Fig. I. The thread should be boiled, if necessary, to make it freely water absorbent. The piece of clean plate glass D is supported rigidly at the proper height in a vertical position. The white scale E is similarly supported behind the glass. At the beginning of a test, D is filmed with water by holding it (best with clean rubber finger cots) under running tap water. It is then clamped in approximate position and C, already wet, is laid across it and drawn straight and vertical with the upper end at or near the top of D, and the ball A slightly above the top of the expanded bed. The water film holds the string to the glass. For a test the string is released, its upper end position noted just after the top of the ball becomes submerged, when the corresponding time is read, and thereafter simultaneous readings of position and time are taken as desired. Results of the visual studies of the bed are given at the pertinent places in the following discussion. Results of the settling tests are summarized in Figs. 2 and 3. DEFINITIONS The name bed as used herein refers to the entire mixture of solids and water filling the jig box. A layer is a stratum of a bed composed of particles of substantially the same specific gravities. A granular product is long-range when it approximates a crushed or ground product limited as to upper size only; a shorlrange product is one closely sized. A bed is loaded when operating with continuous feed

Jan 1, 1943

By AIME AIME

THE name of Dr. Henry Sturgis Drinker has been added to the short list of Honorary Members of the American Institute of Mining and Metallurgical Engineers, a signal honor rightfully bestowed. To quote the Philadelphia Inquirer, "It is essentially a recognition of the highest achievements, the greatest benefits, and most deserving personalities among engineers themselves." The regard in which Doctor Drinker is held by his friends and associates is reflected in the following comments of the press: Doctor Drinker is distinguished far beyond the confines of Bethlehem and of Lehigh University, to the development of

Jan 1, 1921

By S. A. Allison, N. P. Finkelstein

After a brief consideration of the properties of zinc sulfide minerals, their flotation characteristics, and their reactions with thiol collectors, the literature on the activation of the minerals by cupric ion, their deactivation by cyanide, and their depression is reviewed in detail. The distinction between deactivation and depression is stressed. The contradictions in the published results and theories are discussed and attempts are made to resolve them. It is concluded that, whereas all aspects of the subject require further investigation, the study of depression is most likely to yield benefits in practice.

Jan 1, 1976

By F. L. Bosqui

A. L. BlomFIeld, Denver, Col. (communication to the Secretary.*) — Mr. Caldecott says on p. 67: "The Dorr thickener shown, while a useful device when crushing with cyanide solution for removing surplus solution from slime pulp prior to air-lift agitation or vacuum filtration, is not adapted to effect as complete a separation of water from settled slime as the ordinary intermittent settlement and decantation." This statement might have passed muster in this country several years ago when. the Dorr thickener was a new device, but its use in hundreds of mills in the United States has demonstrated that as thick a product is being discharged from it as was obtained from the intermittent settling which it has supplanted. • It is interesting to note that recent investigation, especially the work of 11. 6. Coe and G. H. Cleveilger,' in the March Bulletin of the Institute, under the title: Laboratory Method for Determining the Capacities of Slime Settling Tanks, has developed the general theory of settlement and explains the results obtained in practice. Just as free settlement is a function of surface area, so thickening after the zone of free settlement is passed is a function of time. The time required is obtained by the capacity or volume provided in the settler for this thickening zone. Each pulp is a problem in itself, both in regard to free settlement and to time required to reach a commercially finished thickness. Some pulps require 48 hr. and more to reach a density of 40 per cent. solids; others no coarser to the screen test reach a density of 65 per cent.' solids in a fraction of that time. In thickening any pulp a point is reached where the increase in density is so slow as to be commercially unprofitable. When any part of the pulp has reached this point it should pass on for treatment—its place in the thickener should be occupied by pulp which has not yet reached the commercial limit. By doing this the whole volume of the settling tank is kept in active use. Thus the continuous thickener keeps the whole tank at work to its commercial limit, while the intermittent settler has much of its volume occupied at times by pulp which is either quite finished thickening or at least thickened past the commercial point. The mechanical efficiency of the continuous thickener can be the only argument for limiting this statement. In my own experience I have never

Jan 1, 1916

By Eckley B. Coxe

The plummet lamp, to which I called the attention of the Institute, at the Bethlehem meeting, in August, 1871, with the improvement of the compensating ring, suggested by Mr. Heller, having come into general use for mine surveying in this country, and my attention having been called to the fact that it could not be employed in mines in which there was danger of meeting with fire-damp, I have so modified it that it can be used in all cases with safety. Mining engineers, who are in active practice in those districts where fire-damp is met with in the mines (and I am sorry to say it is becoming much more common in the anthracite regions), know how dangerous their work is when they are called upon to survey such mines, particularly when they are obliged to run lines in those parts of a mine which have been unworked, since by opening of doors, brattice, etc., to get the sights, they are liable to disarrange the currents of air, and cause any fire-damp that may have accumulated in the old workings to find its way into the levels, etc., where the party is at work. It is, therefore, very dangerous to leave an open light (such as the original form of plummet lamp) in an old breast or gangway, even when the safety lamp has shown that no fire-damp is present, for the opening of the doors, etc., may drive the fire-damp to the light, and thus endanger the lives of the engineers. The improved form of plummet lamp, now shown to you, can be used either with or without the safety apparatus. Upon the top of the plumb-bob portion of the old lamp a screw is cut, upon which a ring A is screwed. (See Plate 11.) On the external cylindrical part of this ring two small conical holes are drilled, 180' apart. The compensating ring, by which the plumb-bob part of the lamp is suspended, passes over the ring A, and is fastened to it by two small screws having conical points, which fit into the conical holes in the ring A. The safety apparatus resembles, to a certain extent, that of the Mueseler lamp. It consists of a ring B and plate C, which are united by four rods. The plate C has a cylindrical hole in the middle, and four apertures distributed radially around it. In the central cylindrical hole is fitted a conical brass chimney, which projects below the plate C, and is fastened to the latter, being kept ver-

Accurate modeling of the semivariogram is an extremely important step in the geostatistical analysis of mineral deposits. Automatic fitting of a mathematical model is often possible but not necessarily desirable. Significant geologic and statistical information must be taken into account in the fitting process, which is not always available to the computer. Manual model fitting is, on the other hand, extremely time consuming. Interactive graphics are best suited to optimize the process. Many interactive methods are described, which are adapted to modeling semivariogram in one-, two-, or three- dimensional environments.

Jan 1, 1985

By Ottar Dragge, C. Danielsson, Bo Kalling

DISCUSSION, T. L. Joseph presiding L. F. Reinartz (Armco Steel Corp., Middletown, Ohio) —I would like to know, in the practical application of the Kalling process, what kind of a lining was used, how thick was the lining, and how much metal was treated at one time? S. Fornander (author's reply)—The rotary furnace is lined with a course of fireclay bricks 6 in. thick. This course is backed by 5 in. of insulation. The furnace has a capacity of about 15 tons. Mr. Reinartz—How was the ladle preheated? Mr. Fornander—As pointed out in the paper, the furnace was heated by a gas flame in the beginning of the experiments. During these first tests, however, the desulphurization was inconsistent. We think that this was due to the fact that iron droplets sticking to the furnace walls were oxidized by the gas flame. Now, the furnace is operated without preheating of any kind, and the results are much better. T. L. Joseph (University of Minnesota, Minneapolis, Minn.)—I might add one comment. This furnace was heated with a flame and for a time they had a little difficulty due to some residual metal in the rotating drum that would oxidize in between treatments and they found therefore, that it was very essential to drain the drum completely of metal so that they would not build up any ferrous oxide between treatments and they eliminated some of their erratic heats by maintaining those more reducing conditions. It was interesting to watch this operation. As soon as the drum started to rotate there was considerable flame, at least, at the time I saw it, that came out around the flanges, indicating there was quite a little pressure on the inside of the drum. W. 0. Philbrook (Carnegie Institute of Technology, Pittsburgh)—Is the reaction slag in the Kalling process liquid or solid, and how is it separated from the metal? Mr. Fornander—In the process there is no slag in the usual sense of the word. The lime powder does not melt during the treatment. After the treatment the lime is still in the form of a fine powder. It is separated from the metal by means of a piece of wood of suitable size placed within the furnace before it is emptied. D. C. Hilty (Union Carbide & Carbon Research Laboratories, Niagara Falls, N. Y.)—Dr. Chipman has given us some of his ideas in connection with a specific effect of silicon and silica on sulphur elimination and how silicon might interfere with desulphuriz- ing in the blast furnace. I wonder if he would like to elaborate on the possibility of a similar effect of silicon in the Kalling process? J. Chipman (Massachusetts Institute of Technology, Cambridge, Mass.)—Silicon does not interfere with the Kalling process. Anything that has strong reducing action is good for desulphurization. In these tests where the temperature was low compared to blast furnace temperatures, the silicon that is in the metal is a better reducing agent than the carbon. At high temperatures, carbon is the better. It is not the silicon in the metal that interferes with desulphurization, it is the silica in the slag. Mr. Joseph—I might add that the metal that was tapped from the drum after desulphurization was really at quite a low temperature. It was not measured, but I think it was well under 1300 °C, probably 1200" or a little above that. That was one of the difficulties, and I think there is no question about the fact that the Kalling process—in that it affects desulphurization between powdered lime, solid and liquid iron— is a reaction definitely between the solid lime and the liquid iron. E. Spire (Canadian Liquid Air, Montreal, Canada) — This Kalling process seems very interesting to us and after all it is only a mixing action that is taking place between the iron and the slag. We have attempted to do the same thing in another way. We have placed at the bottom of the ladle a porous plug through which we injected an inert gas. It can be nitrogen or argon. This plug is placed at the bottom of the conventional ladle and gas injected through the plug. That has appeared in our patent. To define this new type of treatment, I use the word gasometallurgy. I do not know if you like it, but it is a way of defining methods of treating metal using gases. What we do is exactly what is done in the exchange process in another way. We have a porous plug at the bottom with a high lime slag on top of the metal. Using this method, we have very good agitation of metal and slag, and with a small flow of gas, we can achieve a very strong agitation. For instance, in the 500 lb ladle, we use only 5 liters of gas a minute. We have an agitation compared to very rapidly boiling water in a pail. Moreover, the agitation can be controlled to create any amount of mixing desired. In a few minutes, with this method, the sulphur dropped from 0.58 to 0.11. These results have been improved since, and we have obtained results like 0.08

Jan 1, 1952

By T. H. Chin, R. W. Bartlett

A simple on-stream particle-size analysis method is presented that leads to a bulk-property single-channel device; although the concept can be extended to multi-channel sizing information. The method is based on continuously measuring and comparing the percent solids in the sample stream and in an internal-reference stream containing only undersize particles. A semipermeable hydraulic system is used such that the particle mass concentration in the internal stream can vary, and is always proportional to the undersize particle mass concentration in the sample stream.

Jan 1, 1975

By J. E. Nesset, J. A. Finch

A model of the fully-loaded condition of particle accumulation on the matrix of a high gradient magnetic separator is developed. The model accurately predicts the build-up profiles reported recently by Friedlaender et al. A powerful development from the model is that loading (volume of accumulated material per unit volume of matrix) is uniquely defined by a dimensionless group, called the loading number, NL. From direct measurements, NL proved superior to the dimensionless group proposed by Watson as a measure of loading. The fine size limit of hematite recovery is examined using NL.

Jan 1, 1980

By B. M. Larsen, T. E. Brower

IN two previous papersl,2 dealing with the carbon-oxygen reaction, and the simultaneous content of each, in liquid steel in the furnace, we have made use of the quantity 0[O], defined as the excess oxygen beyond what would be in equilibrium with carbon then present if the pressure of carbon monoxide were I atm,; that is, if [C] X [O] is taken as 0.00222, 0[O] = [O] - 0,00222/[C] where [O] is the percentage of oxygen by weight as measured by the method described earlier.' This quantity 0[O] proves to be substantially independent of carbon content over the range 0.08 to 1.2 per cent, as well as of the basicity of the slag, and is nearly always within the limits 0.015 to 0.025 per cent when the bath is boiling normally and is in what we have termed a steady state. Apparently it must be at least about 0.014 per cent before boiling occurs; but this threshold value seems to depend upon the ease of bubble formation at the metal-hearth interface, hence upon the condition of the surface of the hearth. In any case, 0[O], on the average, changes considerably following any disturbance of the bath, as by addition of ore. The present paper deals with the average changes in ?[O] level following addition of Spiegel or crop ends, or after procedures to promote stirring, or by tapping of the steel from the furnace. The observations were made on a large number of commercial heats, and consequently are subject to the influence of various disturbing factors; the results have the compensating merit of showing the consistency obtainable under the limitations imposed by practical operation. VARIABILITY OF EXCESS OXYGEN IN NORMAL OPEN-HEARTH OPERATION The preceding general statements refer to the average value of ?[O] particularly in heats that are boiling actively; on individual heats, which are less active, there may be a variable time lag between the disturbance and its effect, which complicates the interpretation of what is going on. For instance, if in the normal refining process boiling is sluggish, the slag usually is gaining oxygen from the gas phase above it, or from added ore, faster than it is losing oxygen to the metal; yet when the bath starts to boil vigorously, the opposite condition may prevail for some time. The consequence of this lag is a variation of ?[O], the interpretation of which in a single heat may not be immediately obvious, This condition is illustrated in Figs. I and 2. In Fig, I, ?[O] at the beginning of the test period was near the lower limit of its usual range (dashed lines) in the steady state as a consequence of the lime boil, which had ended about as indicated. The bath is now cold, carbon drop is slow, and the first small ore feed causes only a small rise in 0[O]; as the temperature rises and as lime dissolves in the slag, and the bath becomes more active, the

Jan 1, 1946

ZAY JEFFRIES, ? Cleveland, Ohio (written discussion?).-Mr. Miller's work on the path f rupture in fusion welds is fundamental. He is to be congratulated on the thoroughness with which he outlined and executed the experimental work. Following are some observations based on the data included in the papers and elsewhere. The iron nitride needles greatly increase the elastic limit of the ferrite-nitride complex. This is proved by the fact that the elastic limit of the weld material is very much higher than the elastic limit of pure

Jan 5, 1919

Stripping of the Exotica open-pit mine, which is proceeding at about 120,000 tpd, is dwarfed by its proximity to the 500-million-ton Chuquicamata tailing pile. The excavating work is being carried on smoothly and efficiently and, as of the end of October, accounts for the removal of 77 million tons of total material of which some 200,000 tons is stockpiled ore. Diamond drilling has outlined an orebody 5900 ft long and 3400 ft wide containing an estimated 153 million tons of ore at 1.35% recoverable copper based on 80 % extraction.

Jan 11, 1969

By Alan D. Hoagland

The Flat Gap mine, are important zinc deposit of the Appalachian type, is located in the Copper Ridge Belt in the Valley of Tennessee approximately 30 miles northeast of the Mascot-Jefferson City zinc district (Fig. 1). The ore is found in large stratigraphically linked deposits in solution and collapse breccias of the Kingsport, Longview and, to a minor extent, Mascot formations of Lower Ordovician age. The principal ore mineral is sphalerite and the principal gangue minerals are dolonmite and accessory pyrite. There is local minor galena and very rare barite and fluorite.

Jan 10, 1962

W. O. OWEN,* Washington, D. C. (written discussion?). Few people appear to realize that the time to reach the crippled soldier is when he is first hurt. In my own judgment, the best time to reach him is before he is hurt, by showing him what other cripples have themselves clone of their own initiative, without Government-support, rendering themselves self-supporting, able to bear a little bit more than their own weight, so that they may lay aside a little bit for a rainy clay and some for a little fun in

Jan 5, 1918

By H. L. Armstrong

SOLID-STATE diffusion is an important technique in certain aspects of metallurgy, and especially the metallurgy of semi-conductors and semi-conductor devices. If the diffusion is carried out under constant conditions for a time, t, the concentration, C, of the diffusant at a depth, x, below the surface from which the diffusion is carried out is given by an expression of the form. C = Cof(x,Dt) [I] Co being a constant, and D the diffusion constant of the diffusant in the material and at the temperature

Jan 1, 1959

By William Lodding

In leaching ores at atmospheric pressure it is often desirable to apply high reagent concentration, long exposure time, and elevated temperature. All of these conditions tend to increase the treatment cost. In conventional leaching, direct or countercurrent, the leaching fluid is needed to transport the solids through the process. This requires relatively large volumes of fluid and limits the reagent concentration economically feasible. The relatively high dilution also increases fuel cost if leaching is to be done above ambient temperature. Heating of water is expensive.

Jan 3, 1967

By F. Milton Lewis, Roshan B. Bhappu, James L. Coburn

FOREWORD-To comminute is to pulverize, but, in mining technology, "comminution" includes all the multiple crushing and grinding operations on ore, rocks, coal, and certain other bulk materials. The primary objective of this two-part article series is to provide the reader with basic information and guidelines for designing comminution systems. Part One is confined to ore processing operations, where size reduction is required to liberate the valuable minerals so that they may be concentrated and recovered. Part Two will illustrate various design techniques through practical examples.

Jan 9, 1976

By J. E. Warren

The mathematical problem considered by the author1 can be given another physical interpretation which is of some practical significance. The alternative physical problem involves the approximate behavior of a single well which is producing at a constant rate through an axially symmetric, horizontal fracture of infinite flow capacity which is located at the center, top or bottom of a uniformly thick, horizontal, homogeneous, anisotropic reservoir of infinite lateral extent which contains a single, slightly compressible fluid. Using Coats' asymptotic results, the wellbore pressure in the fractured system is given by the following equation.