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By R. J. S. Brown, B. W. Ganison

A new logging method has been developed, based on measurement of the nuclear magnetism of formation fluids. The nuclear magnetism log (NML) is the only log that responds solely to formation fluids. It operates equally well in both oil-base and water-base muds and in empty holes, and can be used in all kinds of formations except strongly magnetic ones. Two separate NML measurements can be made, one of which provides a continuous formation fluid curve. This fluid curve is called the free fluid log (FFL) and is believed to indicate a minimum effective porosity in most formations. The FFL not only delineates fluid-containing zones, but provides an excellent correlation curve that can be obtained under conditions where conventional correlation logs are ineffective. Preliminary tests indicate that the second kind of NML measurement may help distinguish oil and water zones and provide information concerning permeability and wettability. (The FFL itself appears to provide some information on permeability.) The second kind of NML measurement requires stopping the logging tool for a short time opposite a zone of interest and taking more extensive NML data that can be displayed as nuclear magnetic relaxation curves. In some instances, oil and water saturations for the region immediately adjacent to the borehole can be read from these relaxation curves. INTRODUCTION In 1946, Bloch, Hansen and Packard and Purcell, Torrey and Pound3 independently announced the successful demonstration of the phenomenon of nuclear magnetic resonance. During the past 13 years, there have been many applications of nuclear magnetic resonance, including applications to the study of chemical structure and to the measurement of magnetic field strengths. Preliminary experiments on the feasibility of using nuclear magnetism measurements in well logging were made independently by California Research Corp. and Varian Assoc., the Varian work being sponsored by the Byron Jackson Tools, Inc. Since then a cooperative research program on nuclear magnetism logging has been carried out by the Byron Jackson Div. and Research Center of Borg-Warner Corp., and California Research Corp., subsidiary of Standard Oil Co. of California. The use of nuclear magnetism in well logging is of special interest because it offers a way of making direct measurements on the hydrogen in the formation fluids and not on the rock matrix. Within the past 1 1/2 years, successful measurements have been made with a research model logging tool in wells in California, Tex as, Utah, Louisiana and Wyoming. NUCLEAR MAGNETISM SIGNALS Polarization, Relaxation and Precession Many atomic nuclei possess magnetic moments and spins; that is, they are similar in some respects to bar-magnet and gyroscope combinations. Molecules and their nuclei are subject to thermal motion, which has a scrambling effect, tending to leave as many nuclear spins oriented in any one direction as in any other. However, if a magnetic field is applied, the magnetic nuclei tend to align in the direction of the field. The scrambling and aligning forces compete with each other, with the result that a few more spins are oriented parallel to the field than in other directions. This gives a net magnetization, or polarization, which is directly proportional to the strength of the applied magnetic field (aligning influence) and inversely proportional to the absolute temperature (scrambling influence). When the magnetic field in, or temperature of, a liquid sample containing protons is changed, the new equilibrium value of proton polarization is not established immediately but requires an amount of time which depends on the nature of the hydrogen-containing materials. The process of approaching the equilibrium value of polarization is called relaxation.'.' Polarization is a vector quantity, and the components parallel to and perpendicular to the magnetic field must be considered separately. Relaxation of the component parallel to the field is called "thermal relaxation", or "longitudinal relaxation", and the corresponding time for this component of non-equilibrium polarization to decay by a factor of e (natural log base) is denoted T. The relaxation of the perpendicular component is called "transverse relaxation", and the corresponding relaxation time is denoted T,. The potential energy of a magnet in a uniform field depends on the angle the magnet makes with the field; therefore, a change of the component of net polarization parallel to the magnetic field involves an exchange of energy between the spin system and the thermal motion of the molecules, leading to the term thermal relaxation for the relaxation of this component. Suppose we subject a sample to a strong magnetic field at right angles to the earth's field for a time greater than TI. A polarization, thus, is established at right an-

By S. Skowronski

OvErpolEd copper, as commonly defined,- is copper which has been excessively reduced during the poling period of the refining process. Owing to its porosity, such coppcr is unfit for commercial purposes. The cause of this phenomenon is undoubtedly the lack of oxides in the copper, yet this does not completely explain it. Refined copper of the same "pitch" and "set" varies considerably in oxygen contents, and analyses of overpoled bars also show similar wide differences; moreover, it is possible to melt copper in 'crucibles so as to obtain bars that are practically free from oxygen although they show a "tough pitch" and "level set" surface. Frederick Johnson advances the following theory: The regulation of the quantity of oxygen in copper is the keynote of all copper refining. It is the presence of a well defined proportion of oxygen in the form of cuprous oxide, forming a copper-cuprous oxide eutectic, which enablcs copper to be cast free from blowholes. It seems that when oxygen is present, other gases are less soluble in molten copper; when oxygen is absent, or is present in insufficient quantity, the copper dissolves other gases, such as hydrogen and carbon monoxide, which are insoluble in the copper when solid, and, being rejected during the process of solidification, causes internal porosity and external ridges and excrescences.1 The "level set" of wire bars, etc., is entirely the result of a compromise between the occluded gases and cuprous oxide. With excess of the former the metal would be "overpitch" and spongy, while an excess of the latter would result in the metal being "underpitch" and brittle.2 The result of the absorption and subsequent expulsion of reducing gases, carbon monoxide and hydrogen,' and the effect of this action on the overpoling of copper are in dispute; the literature on the subject is full of contradictory statements and will continue to be so until better analytical methods have been developed. Sulfur, however, is tangible. The refiner can see it in his button sample and consequently attributes to it any trouble that may happen during the melting of the copper.

Jan 1, 1919

By C. R. Sandberg, C. A. Connally, N. Stein

This paper describes the results of volumetric efficiency tests on oil well pumps handling gas oil mixtures. The work was performed in a large scale, above ground unit wherein test conditions could be accurately controlled and measured. The main variables studied were gas/oil ratio (including gas from solution and free gas mixed with oil), pump compression ratio, pump stroke length, pump speed, and clearance volume between the valves at their closest approach. Results are presented for two different pumps and for oils of two viscosities. Relatively small amounts of gas entering the pump resulted in large decreases in volumetric efficiency. Under conditions where the pump was operating at reduced efficiency because of the presence of gas, it was found that variation in the clearance volume between the standing and traveling valves had a considerable effect on pump efficiency level. This effect of the valve clearance volume was found to be significantly altered by the viscosity of the oil used in the tests. The effects on pump efficiency of the other variables studied were found to be relatively small over the range of conditions utilized. INTRODUCTION The production of oil by pumping is often hampered by low volumetric efficiency. A direct increase in lifting costs results from low volumetric efficiency. An indirect increase in lifting costs, probably greater than the direct increase, results from additional wear and tear on pumping equipment and from the down-time necessary for the repairs which can be traced to low-efficiency operation. Both increases in lifting costs tend to reduce economically recoverable oil. A number of different factors can contribute to low pump efficiency. A known basic cause of low efficiency is the presence of free gas in the pumped fluid. Pump volumetric efficiency is calculated only on the basis of liquid pumped and because any free gas pumped is discounted, this volume of free gas would represent a loss of pump efficiency. However, gas also causes a reduction in pump efficiency because it is a highly compressible fluid. It is known that pumps some- times "gas lock" because of excessive gas-to-liquid ratios in the pump barrel. Little is known of the role of gas compressibility in the intermediate case where the pump is operating at low efficiency. The opinion exists, however, that oil-well pumps tend to operate at higher efficiency with long stroke lengths at low speeds, but no quantitative studies of these pumping variables have been reported. It was believed that a much better understanding of the variables which control pump volumetric efficiency could be obtained and that possibly some suggestions as to the methods for increasing efficiency might be found from a study of the operation of pumps handling gas under closely controlled conditions. Previous investigators have studied the effects on pump efficiency of such factors as oil viscosity, oil temperature, slippage of oil. past pump plungers, pump submergence, valve size and spacing, pressure above pump plunger and fluid vapor pressure. However, none of these published investigations were conducted with pumps being subjected to large amounts of gas such as might be the case in a pumping well, nor did any of the investigations study the effect of variation in stroke length or pump speed. A large-scale teat unit was therefore constructed for studying the operation of pumps handling gas and for evaluating effects of such variables as pump stroke length and pump speed. PROCEDURE AND EQUIPMENT A schematic diagram of the pump testing equipment is given in Fig. 1. A 45-ft length of 6-in. casing is mounted vertically in a 65-ft tower. Sight ports are mounted in the casing at intervals near the location of the pump intake and the liquid level in the casing. These sight ports are fitted with Lucite windows sealed by neoprene "0" rings. The Lucite windows are machined to conform to the I.D. of the casing so that no obstruction to flow is present along the casing wall. The casing is fitted with a tubing head and 2-in. tubing is hung inside the casing. Pumps are seated in a shoe attached to the 2-in. tubing. A 1-in. polish rod is attacked directly to the pump without any intervening sucker rods. The top of the polish rod is attached to the weight carrier, which contains a number of weights to be used to force the polish rod in against tubing pressure on the down-stroke. This is necessary because a long string of sucker

Jan 1, 1953

By J. M. W. Mackenzie, C. J. Wood

In attempting to describe the size classification performance of a hydrocyclone, most workers have elected to use either an equilibrium orbit theory or an non-equilibrium orbit theory. The equilibrium orbit theory has been used by the majority of workers including Lilge,' Bradley; and Yoshioka and Hotta. In applying this theory, it is argued that particles in the body of a hydrocyclone attain an equilibrium radial position where the drag force on the particle resulting from the inward radial fluid velocity is balanced by the outward centrifugal force caused by the tangential component of fluid flow. When considered over the full height of the hydrocy-clone, attainment of this radial equilibrium orbit results in the particle following a conical equilibrium envelope. It is then argued that if this envelope lies outside the envelope of "zero vertical velocity," the particle will report to the underflow, while if the equilibrium envelope lies inside the envelope of "zero vertical velocity," the particle will report to the overflow or vortex finder product. The d50-sized particle which reports in equal quantities to the underflow and overflow is assumed to correspond to particles whose equilibrium envelope is coincident with the envelope of "zero vertical velocity." In considering the equilibrium orbit theory, it is apparent that the horizontal position of the particles in the feed inlet pipe should have no effect on their ultimate destination on the hydrocyclone. Each particle should attain an equilibrium position which depends on the density, size, and shape of the particle; the density and viscosity of the fluid; and the flow patterns within the hydrocy-clone. The nonequilibrium orbit or unsteady state theory has been largely developed by Rietema4 and Mizrahi.6 Mizrahi has listed four main objections to the equilibrium orbit theory. These objections center on the short residence time in the hydrocyclone, the fact that the experimental classification curve is much less sharp than is theoretically predicted, and the absence of negative efficiency conditions in hydro cyclones operating on a feed material which is much finer in size than d50. Proponents of the nonequilibrium orbit theory argue that for a particle to discharge with the underflow it must have sufficient outward radial velocity to reach the downward-flowing region close to the hydrocyclone wall in which the flow lines are parallel to the wall and the ratio of vertical to radial velocity is constant. It is then postulated that a d50 particle entering the cyclone at the center of the feed inlet will just reach this downward-flowing region as it reaches the apex. Thus for uniform distribution of particles across the feed inlet, half the d50 particles—that is, those injected in the half of the inlet area nearest the cyclone wall —will report to the underflow while those injected in the other half will not reach the downward-flowing region and will be carried inward to the center of the cyclone and thus report in the overflow. The exact thickness of the down-ward-flowing region of fluid adjacent to the outer wall of the hydrocyclone is uncertain but Mizrahi considers it to be equal to the apex radius minus the air core radius. Particles larger than d50 have a greater outward centrifugal force acting on them than the d50 particles and may reach the wall even if injected at a distance from the wall greater than Di/2 (Di is inlet diameter). Conversely, particles smaller than d50 may not reach the wall even if injected at a distance less than Di/2 from the cyclone wall. Since the equations put forward by the proponents of both theories yield approximately the same values of d50, it is not possible to decide between these theories by measurement of d50. It should be possible however to examine the theories by injecting a small stream of solids into the feed inlet of a hydrocyclone running on clear water. If the efficiency or classification curve is measured for various horizontal injection positions, then the curves should be coincident if the equilibrium orbit theory holds. If, however, the unsteady state theory describes the cyclone operation, then the classification curves should show finer d50 sizes for particles injected close to the cyclone wall. Experimental A 6-in.-diam hydrocyclone with geometry as in Figs. 1 and 2 was used. Quartz particles were injected as a 50% by wt pulp via an 1/8-in. steel probe. For each in-

Jan 1, 1971

By R. E. Collins, Paul B. Crawford

Mathematical equations have been derived to show the effect of the slippage phenomomenon (Klinkenberg effect) on unsteady-state gas flow through porous media. The assumption is made that the slippage may be represented by an equation of the type It is pointed out that most of the existing data on unsteady gas flow may be corrected for slippage by a simple change in definition of the dimen-sionless groups. INTRODUCTION Several solutions to the various problems associated with the unsteady-state flow of gas through porous media have been recently presented in the literature.','" The cited publications have resented solutions for flow under ideal conditjons. The principal assumptions involved include: (a) the density of the gas obeys the perfect gas law; (b) the viscosity of the gas is constant; and (c) the gas permeability is constant. Few, if any, actual gases under reservoir conditions are believed to conform with the above, but the ideal solutions do provide the basis for making improved estimates on the performance of gas reservoirs. The purpose of this paper is to show the performance characteristics when assumption (c) is corrected for the slippage phenomenon or the Klinkenberg effect. This is believed to be a step nearer to reality. MATHEMATICAL DEVELOPMENT Case 1. Gas Flow — Ideal Conditions The equations describing the flow of gas under ideal conditions have been clearly presented in the cited literature; however, for completeness of this work they are repeated here. The basic differential equation describing the flow of gas is Equation (6). It may be derived as follows: The continuity equation, the perfect gas law and Darcy's law are combined to give: Manuscript received in the Petroleum Branch office Aug. 3, 1963 or If — is constant, Equation (5) is rewritten to form the basic Equation (6). or for the linear case A dimensionless equation is obtained by dividing through by (pl)z and making the substitutions shown below: The resulting equation is A similar procedure for the radial case leads to the following equations: where and Equations (11), (12), and (13) are those used by Bruce, et al., and Jenkins and Aronofsky for their computational work.

Jan 1, 1953

By W. E. Bron, E. S. Machlin

GRAIN boundary migration in V-shaped thin specimens of high purity silver was studied. Each specimen contained two crystals having the relative orientation: (111)A/(111)B; (ll0)A/(101)B. The boundary between these two crystals varied between a continuous curved and an alternating two plane interface. The plane interfaces were oriented parallel to (101)A/(011B) and (121)A/(211)B. Traces of these planes are shown in Fig. 1. Evidently the two plane interface, although of higher area, comprises lower total interface energy than the average curved boundary it replaced. The plane boundaries were found to move at 650 °C to consume the broad segment of the V opposite to the direction of interface area and energy decrease. On investigation, using the thermal etching technique of Hendrickson and Machlin,1 it was found that the outer grain contained 8.3x108 more dislocations per cm" than the grain at the apex of the V. Consequently, the boundary moved to decrease the strain energy associated with the difference in dislocation density across the boundary. The rates of motion of specific {110} and {112} boundaries were measured during interrupted anneals at 650°C by referring the position of the boundaries to each other and a fiducial line on the specimen surface. The measured rates of motion are 0.075 and 0.025 mm per hr for the {110} and {112} boundaries, respectively. Plots of grain boundary motion vs time of annealing show straight lines with little deviation of data points from these lines.

Jan 1, 1957

By Edmund Davenport

Tier; production of thorium of high purity by the Ca-CaCl2 reduction has been described by Marden and Rentschler,1 who also reported some of the properties of the coherent, ductile metal obtained from this pure thorium powder. The present paper describes the properties of thorium produced by the Ca-CaCl2 process modified in some details to meet commercial conditions. The principal effect of these changes in process has been an unavoidable increase in the oxide content of the thorium.

Jan 1, 1929

By R. W. Raymond

CLARENCE King born January 6,1842, at Newport, R. I. His ancestors on both sides mere New Englanders, of English blood, and among them not a few distinguished themselves in art, science, politics or commerce. His grandfather, Samuel Vernon King (partner in 1803 in the house of Talbot, O1yphant & King, later known as King $ Co.), was a pioneer in the China trade, and four of his sons succeeded him in the business. Three of the four died in the far East; and one of

Jan 1, 1903

By John B. Pearse

The works heretofore used in carrying out the Bessemer process have been constructed substantially as follows: The whole works or plant has been divided into three parts. 1. The division in which the metals for conversion are melted. 2. The division in which the metals are converted into steel and cast into ingots, or the converting department. 3. The engine and boiler department which supplies the hydraulic power required to move the machinery, and which furnishes the blast used for converting the crude metal into steel.

By N. Arbiter, C. C. Harris

Analysis of ball and pulp flow in ball mills indicates that three factors may become critical with increasing mill diameters: ball size, fraction critical speed, and average pulp flow velocities. Ball diameters may need to be decreased and fraction critical speeds increased with ores which show decreased breakage rate coefficients above 1.2 mm (14 mesh). With 5.5-m (18 ft) and larger mills at higher circulating loads, pulp flow velocities may become capacity limiting and also may limit increases in ball mill diameters.

Jan 1, 1985

By F. Galasso, A. Paton

ThE recent emphasis on producing boron fibers by reacting boron trichloride and hydrogen on a hot tungsten wire has brought about a renewed interest in the tungsten boride phase diagram. The reaction of the boron with the tungsten core results in the formation of 6 WB, W2B5, and another tungsten boride phase, which was suspected of being the WB4 phase reported by Chrbtien and Helgorsky.1 While the powder-diffraction data for the boride in the fiber and those reported

Jan 1, 1967

Organization Place Date 1917 American Society of Sanitary Engineers Grand Rapids, Mich. Aug. 1-3 National Association of Stationary Engineers... Evansville, Ind. Sept. 10-15 National Safety Council New York City. Sept. 10-15 National Exposition of Safety and Sanitation. New York City Sept. 10-15 American Iron, Steel & Electrical Engineers..:. Philadelphia, Pa. Sept. 10-14 American Institute of Metals Boston, Mass. Sept. 24-29 American Foundrymen's Association Boston, Mass. Sept. 24-29 Chemical Industries, Third National Exhibition. New York City Sept. 24-29

Jan 8, 1917

By John B. Pearse

THE works heretofore used in carrying out the Bessemer process have been constructed substantially as follows : The whole works or plant has been divided into three parts. 1. The division in which the metals for conversion are melted. 2. The division in which the metals are converted into steel and cast into ingots, or the converting department. 3. The engine and boiler department which supplies the hydraulic power required to move the machinery, and which furnishes the blast used for converting the crude metal into steel.

Jan 1, 1876

A recent study estimates that a 10% improvement in the performance of wire rope used on draglines could save the US coal industry $2 million annually in rope costs. In addition, such an improvement could boost annual coal production by allowing the industry's draglines to remove an extra 1.5 million m3 (2 million cu yd) of solid overburden each year. These findings, although admittedly preliminary, suggest the magnitude of benefits that can be derived from a well-planned wire rope maintenance program.

Jan 3, 1978

By J. W. Still

CIRCULAR steel sets have proved to be more economical for supporting slusher drifts in the block-caving mining used by the Miami Copper Co., at Miami, Ariz. The first steel sets were installed in July 1950, and up to July 1, 1951, about 474 sets (approximately 1896 ft of drift) were placed; the bulk of this work was done in the latter six months. All of the places selected for the steel were in heavy ground. Based on this experience the following definite conclusions can be drawn:

Jan 11, 1951

By I. P. Pardee

In our mixture at Musconetcong Iron Works we have an exceedingly hard, dense, magnetic ore, which contains some iron pyrites. The sulphur runs from 0.5 to 1.5 per cent. To prepare this ore better for the furnace, and with a view of making a softer iron, we tried roasting the ore in a Gjers kiln. When it worked well, the improvement in the iron was very perceptible ; but the ore was continually sintering and forming scaffolds across the bosh of the kiln.

Jan 1, 1887

By R. H. Richman, II Conard G. P.

AS part of a study of deformation twinning in bcc crystals, solid solutions of beryllium in iron have been found to twin profusely when strained slowly at room temperature. This note reports some crystal-lographic aspects of the operative mechanical twin modes in a quenched Fe-5 wt pct Be alloy. After melting twice under a partial pressure of helium, the alloy was homogenized, cut into pieces 7 to 10 mm on an edge, solution treated at 1130°C,1 and quenched. Debye-Scherrer patterns confirmed

Jan 1, 1963

By Stephen R. Holsinger

During 1972, Asarco's Silver Bell Unit embarked upon a project to test a digital computer-controlled grinding circuit against the performance of its other five grinding lines. The planning, implementation, and subsequent outcome of that project has been previously reported in a paper by Pitts 1, et al. This paper will describe the solutions to major problems incurred during the implementation of this control strategy on the remaining five primary grinding lines. Changes made in operating philosophy and methods during the expansion will also be discussed.

Jan 12, 1978

By L. B. Harris

Continuous cyclic unloading during tensile work hardening of polycrystalline zinc at room temperature enables specimens to sustain greatly increased extension. Such enhanced ductility is associated with creep-type extension, during which the cyclic component appears to promote relaxation of internal strains. It is possible to trace the transitions from tension, through enhanced cyclic extension, to fatigue and relate the observed behavior to simpler types of deformation. DISTINCTIVE deformation behavior has been shown to occur when loading includes both cyclic and unidirectional components. For aluminum and copper extended by a sequence of progressively biased strain reversals, coffin1 observed greater fracture ductility and lower flow stresses than during monotonic tension. This may be compared with the reduction in hardness obtained from conventional fatigue softening. But fatigue softening is accompanied by enhanced fatigue life,' whereas Benham3 found that fatigue life was reduced when copper continuously extended during axial load cycling. The corresponding specimen elongation, for large cyclic load amplitudes, was greater than during normal tension. Under different conditions, Bendler and wood4 found that torsional fatigue initiated extension in copper under low tensile loads previously in equilibrium with the specimen; fatigue life was again reduced. Thus there is a type of cyclic softening which is associated with reduced fatigue life and which occurs when the cyclic loading is accompanied by a unidirectional strain component. Further, the fracture ductility or extension obtained from the unidirectional component can exceed that obtained in the absence of cyclic load. Specimen extension was not initiated by axial fatigue at small load amplitudes;3 hence Benham's work distinguishes between fatigue at large and small amplitudes, the former often being linked with unidirectional deformation because of similarities in strain Structure. The initiation of tensile extension during torsional fatigue,4 however, occurred with small amplitudes. Comparable changes in behavior have been ob- served during creep. kennedy' found that combined cyclic and creep stresses on lead at 32°C increased the rate of elongation over that for simple creep, the consequence being a reduction in creep life. It was also demonstrated that small cyclic stresses applied after creep deformation initially increased recovery, but that prolonged application produced only normal fatigue hardening. It is clear that physical properties can be changed by interaction between the cyclic and unidirectional deformations, but owing to the number of different ways in which the deformations can be combined and the large number of possible variables for each combination, trends of behavior become easily obscured. One definite conclusion is that enhanced ductility or increased extension can be obtained by cyclic action, and it is this process that has been investigated for the tensile test by measurements of the extra tensile strain produced by the addition of a cyclic tensile component. EXPERIMENTAL PROCEDURE Material. Specimens were 99.99 pct Zn wire of 1.6 and 2.0 mm diam. Mean grain diameter was 0.02 mm, sufficiently small to give a ductile necked-down fracture under all loading conditions. All experiments were conducted at an average temperature of 20°C. Normal Tension. Unidirectional tensile curves at different strain rates were obtained from an In-stron testing machine, on which load-time curves were recorded autographically, using gage lengths of 5 and 10 cm. Straining was at constant cross-head velocity, and quoted strain rates are with reference to initial specimen length. Measurements of total elongation under simple tension were also made on the tensile machine that had been modified to incorporate an added fluctuating load. Combined Tension and Rapid Cyclic Load. An electromagnetic vibrator was coupled to the elastic beam of a Hounsfield ensometer- and driven sinus-oidally from a power amplifier at frequencies from 20 to 360 cps, the flexible connecting links on the tensometer being replaced by rigid fixtures to eliminate lateral vibration in the specimen. One end of the specimen was pulled by the normal extension drive of the tensometer while the other end, fixed to the elastic beam and vibrator, was subjected to rapid oscillations of load. It was thus possible to apply cyclic load to a tensile test without altering the rate of extension of the specimen

Jan 1, 1964

By T. K. Perkins, L. R. Kern

A study of fluid mechanics, rupture of brittle materials and the theory of elastic deformation of rocks shows that, for a given formation, crack width is essentially controlled by fluid pressure drop in the fracture. Operating conditions which cause high pressure drop along the crack (such as high injection rate and viscous fluids) will result in relatively wide cracks. Conversely, operating conditions which cause low pressure drop (low injection rates and thin fluids) will result in relatively narrow cracks. Charts and equations have been derived which permit the estimation of fracture widths for a variety of flow conditions and for both horizontal and vertical fractures. INTRODUCTION There has been considerable speculation concerning the geometry of hydraulically created fractures in the earth's crust. One of the questions of practical importance is the width of fractures under dynamic conditions, i.e., while the fracture is being created and extended. Such width information could be used, for instance, to help estimate the area of a fracture generated under various conditions. Also, there has been a recent trend toward the use of large propping partiles.13, 15 Therefore is is desirable to know what factors can be varied in order to assure entry of the large particles into the fracture. There has been some work on fracture widths reported in the literature. In particular, there have been several Russian publications dealing with this sub-jeCt.1.31,3 These papers have dealt principally with the elastic theory and the application of this theory to hydraulic fractures. These studies have not led to an engineering method for estimating fracture widths under dynamic conditions. A recent paper3 has reviewed and summarized the Russian concepts. An earlier paper- from our laboratories also discussed the application of the elastic theory to hydraulic fractures. This first approach, based largely on photoelastic studies, has proved to be too simplified to accurately describe the fracturing process. However, these early thoughts have served as a guide during the development of more exact concepts. We would like to present in this paper our current concepts regarding fracture widths and some estimates of hydraulic fracture widths for several conditions. We believe that it is now possible to predict with fair accuracy the factors influencing fracture widths. Furthermore, the method of prediction has been reduced to a simple and convenient graphical or numerical calculation. CRACKS IN A BRITTLE, ELASTIC MATERIAL Many investigators2, 4, 30 have shown that competent rocks behave elastically over some range of stresses. Of course, if the tensile stress imposed upon a rock exceeds some limiting value, then the rock will fail in tension. In similar manner, there are some limiting shear stresses that can be imposed upon rocks. Hubbert and Willis11 have discussed the shear conditions which will lead to failure. Under moderate stress conditions (such as those likely to be encountered when hydraulically fracturing) and when stresses are rapidly applied, relatively, most rocks will fail in a brittle manner. Hence, for this discussion of hydraulic fractures in the earth's crust, we assume the rocks behave as brittle, elastic materials. Let us develop the discussion in the following way. (The following thoughts are applicable only to brittle materials.) 1. First we consider a brittle, elastic system. An energy balance will show the minimum pressure necessary to fracture rock, and from this pressure we calculate the minimum crack width resulting from extension of a hydraulic fracture. 2. Then we will show that, under ordinary fracturing conditions, fracture widths are appreciably greater than the minimum widths of extending fractures. In fact, we will find that crack width is controlled by fluid pressure drop in the fracture. 3. We will discuss pressure drops in fractures and resulting crack widths for various operating conditions and both vertical and horizontal fractures. 4. Finally, we will discuss the significance of these concepts, their relationship to fracturing pressures, etc. First, consider minimum fracture extension pressures. We can shed some light on this question by considering the theory proposed by Griffith7, 8 Yo explain the rupture of brittle, elastic materials. Griffith recognized that solid materials exhibit a surface energy8 (similar to surface tension in a liquid). The fundamental concept of the Griffith theory is that, when cracks spread without the application of external work (in the interior of an elastic medium which is stressed