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By Gray L. L., Max Littlefield, Godbold A. C.

The West Edmond pool of Central Oklahoma, a limestone reservoir, has an area in excess of 29,000 acres and as of Sept. 15, 1946, had produced 53 million barrels of oil from 731 wells at an average depth of 6900 ft. Water has encroached into the reservoir along the west side of the pool and although the area of water invasion is large the net volume of water influx is small, because encroachment has been primarily into a fracture system. The most significant part of this study concerns the character of the Hunton reservoir. Geological study of cores of the producing section was made to supplement core-analysis data. It was determined that approximately 90 pct of the pore volume was contained in intergranular or sand-like porosity and 10 pct in fractures (intermediate porosity). The porous limestone is divided into blocks by the fractures, although in the strongly dolomitic parts of the producing section, fractures are less well developed. The low-permeability intergranular porosity, largely subsidiary to the high-permeability fractures, will produce oil into the fractures by evolution of solution gas. The fractures in effect serve as drainage channels for production of oil from the inter-granular porosity. Because of such a large difference in permeabilities of the two com-ponents, a high degree of by-passing will occur, and, accordingly, the economics involved in undertaking full-scale high pressure gas injection operations have been seriously questioned. Introduction The West Edmond Hunton pool, which covers portions of Canadian, Logan, Oklahoma, and Kingfisher Counties in Central Oklahoma, is the largest reservoir producing oil from a limestone in the state. The total area of the pool comprises about 29,240 acres and development is considered to be essentially complete. In all, 731 producing wells have been completed in the Hunton reservoir at an average depth of approximately 6900 ft, and have yielded a cumulative oil production in excess of 53 million barrels as of Sept. 15, 1946. In addition to wells completed in the Hunton limestone, 21 wells have been completed in the Bartlesville sand at a depth of approximately 6500 ft, which have produced a total of 730,000 bbl of oil. One small producer has been completed in the Cleveland sand at a depth of 5700 ft. However, a study of only the Hunton limestone reservoir will be presented in this paper. The most significant phase of this study concerns the special type of geological core examination that was made in an effort to ascertain the type and degree of porosity in the rock in accordance with lithologic types. Results of this geological study showed the west Edmond pool to be of such a complex nature as to appreciably affect the behavior normally associated with homogeneous reservoirs. Recognition of this condition in the

Jan 1, 1948

By Max Littlefield, Godbold A. C., Gray L. L.

The West Edmond pool of Central Oklahoma, a limestone reservoir, has an area in excess of 29,000 acres and as of Sept. 15, 1946, had produced 53 million barrels of oil from 731 wells at an average depth of 6900 ft. Water has encroached into the reservoir along the west side of the pool and although the area of water invasion is large the net volume of water influx is small, because encroachment has been primarily into a fracture system. The most significant part of this study concerns the character of the Hunton reservoir. Geological study of cores of the producing section was made to supplement core-analysis data. It was determined that approximately 90 pct of the pore volume was contained in intergranular or sand-like porosity and 10 pct in fractures (intermediate porosity). The porous limestone is divided into blocks by the fractures, although in the strongly dolomitic parts of the producing section, fractures are less well developed. The low-permeability intergranular porosity, largely subsidiary to the high-permeability fractures, will produce oil into the fractures by evolution of solution gas. The fractures in effect serve as drainage channels for production of oil from the inter-granular porosity. Because of such a large difference in permeabilities of the two com-ponents, a high degree of by-passing will occur, and, accordingly, the economics involved in undertaking full-scale high pressure gas injection operations have been seriously questioned. Introduction The West Edmond Hunton pool, which covers portions of Canadian, Logan, Oklahoma, and Kingfisher Counties in Central Oklahoma, is the largest reservoir producing oil from a limestone in the state. The total area of the pool comprises about 29,240 acres and development is considered to be essentially complete. In all, 731 producing wells have been completed in the Hunton reservoir at an average depth of approximately 6900 ft, and have yielded a cumulative oil production in excess of 53 million barrels as of Sept. 15, 1946. In addition to wells completed in the Hunton limestone, 21 wells have been completed in the Bartlesville sand at a depth of approximately 6500 ft, which have produced a total of 730,000 bbl of oil. One small producer has been completed in the Cleveland sand at a depth of 5700 ft. However, a study of only the Hunton limestone reservoir will be presented in this paper. The most significant phase of this study concerns the special type of geological core examination that was made in an effort to ascertain the type and degree of porosity in the rock in accordance with lithologic types. Results of this geological study showed the west Edmond pool to be of such a complex nature as to appreciably affect the behavior normally associated with homogeneous reservoirs. Recognition of this condition in the

Jan 1, 1948

By Mark D. Zoback, Bezalel C. Haimson

INTRODUCTION A Workshop on Hydraulic Fracturing Stress Measurements was convened by the authors in December 1981 in Monterey, California under the auspices of the U.S. Geological Survey and the U.S. National Committee on Rock Mechanics. There are now over a dozen groups of investigators around the world who are actively using the hydraulic fracturing method for making in-situ stress measurements. Because of the extensive recent experience of these different groups, the Workshop enabled forty investigators from eight countries to comprehensively assess the status of the method. Based on the presentations and discussions at the Workshop, in this paper we briefly discuss the current status of the hydraulic fracturing stress measurement method concentrating on problem areas where further research needs to be done. A complete account of the presentations at the Workshop will be available through the Proceedings of the Workshop which will be published by the U.S. National Committee on Rock Mechanics in early 1983. The most exciting aspect of the Workshop was the discovery by the various groups of investigators that others were having very similar experiences with the method. The consensus among the different groups of investigators at the Workshop was that they were recording very similar pressure-time data in the field, they were able to interpret them in similar ways, and they were achieving results that were quite encouraging because of several factors. First, the results at various locales were found to be internally quite consistent. This was reported by investigators from Germany (Rummel, Baumgartner, and Alheid, 1982), Japan (Tsukahara, 1982), Australia (Enever, 1982), and the U.S. (see, for example, Haimson, 1978, 1982a; Zoback

Jan 1, 1982

By Raymond B. Ladoo

Wollastonite is a calcium metasilicate, with the formula CaSiO3; containing theoretically 48.3 pct CaO and 51.7 pct Si02. It is one of many natural and synthetic silicates with varying CaO/SiO2 ratios, with and without water of crystallization. While it is a very common mineral, deposits of sufficient size and purity to be of commercial interest are unusual. There is only one producer (1958) of wollastonite for industrial use in the United States, Godfrey L. Cabot, Inc. at Willsboro, New York. There has been a small production of surface material for ornamental use from near Blythe, California. Carload tests have been made on California wollastonite for making mineral wool and a commercial plant for this purpose was projected in 1957. Properties Wollastonite is monoclinic and commonly occurs in masses or aggregates of bladed or needle-like crystals. It has a perfect cleavage parallel to the long prismatic axis. It is rather brittle and some types grind easily to a very finely fibrous product, but other types, due to interlacing of fibers, are tough and difficult to grind. Mobs hardness is 4.5 to 5; specific gravity 2.8-2.9; index of refraction 1.616 to 1.631; melting point 1,512°C. Color when pure is a brilliant white, but when less pure it may be gray or brown. Luster is vitreous, transparent to translucent. Common mineral associates are garnet (several varieties), diopside, calcite and quartz. The nature of the mineral impurities determines to a large degree the commercial value of a deposit. At Willsboro, New York, the impurities consist chiefly of iron garnet and iron diopside which are both magnetic and so are easily removable by magnetic separation, yielding a nearly theoretically pure product. Some deposits in California contain calcite and quartz which cannot be removed cheaply, thus limiting market possibilities. Occurrence Wollastonite is a contact metamorphic mineral occurring in most places within or associated with impure limestones. The deposits at Willsboro, New York1,2,3,5 consist of two or more beds dipping 15-50°E and striking NW-SE. The main bed, 30 to 70 ft thick, may be traced on the surface for at least 2 ½ miles. In this belt, the Pre-Cambrian Grenville limestone has been injected and partially replaced by the more liquid portions of an anorthositic magma. Buddington and Whitcomb3 have described the rock as "Mixed gneisses, pyroxenite and garnetiferous gneisses and skarns." Prospecting revealed the wollastonite at ten different places in the belt, overlain in most places by a garnet rock. Diamond drilling has proven several million tons with probable reserves around 15 million tons. Interbedded with the wollastonite are beds of garnet from a few inches to several feet in thickness. In the wollastonite ore itself the chief and almost only associated minerals are garnet (mostly almandite but some andradite and grossularite) with small amounts of green hedenbergite. Inasmuch as the almandite and diopside are magnetic they are easily separated from the wollastonite in high intensity magnetic separators. If grossularite should occur in quantity other means of separation would be needed. Wollastonite has been reported at 35 localities in California12 but only those in the Big Maria and Little Maria Mountains, 16 to 20 miles north of Blythe, Riverside County, are

Jan 1, 1960

By B. R. Banerjee

MATERIAL used throughout this investigation was high-purity aluminum (99.998 pct). The 1/2-in. cubes were cut out of a cold-rolled slab and annealed at 550°C for 1 hr before deformation. The single crystal specimen was 1 in. in diam x 0.33 in. thick, cut from a cylindrical crystal produced by cooling at about 55°C per hr in a furnace which had a convenient gradient of about 10°C in every 4.5 in. vertical crucible length. Specific data on the ten specimens studied are summarized in table I. Three types of deformation were employed: (I) compression in a hand-operated hydraulic press; (2) impact with a 7-lb sledge hammer; and (3) rolling, using a wedge-shaped specimen' to obtain a gradient of deformations varying from 0 to 50 pct along the length of the wedge. Specimens were either deformed at room temperature, or first cooled in a freezing mixture of dry ice and ether at a temperature of —72°C and then deformed between steel blocks cooled with dry ice. The specimens deformed at the low temperature were ground and polished through 000 emery paper and wheels impregnated with alundum and alumina No. 1. During this entire procedure the specimens were dipped frequently in the freezing mixture of dry ice and ether in order to maintain constantly a frosty surface as insurance against overheating. All specimens were given a final electrolytic polish, using two different electrolytes* to insure that the surface effects observed were not char- acteristic of the polishing technique. With both electrolytes, the annealed specimens gave even surfaces with good polish. Several standard etchants were tried without success. The best results were obtained with an immersion etchant, developed in this connection, containing: hydrofluoric acid, 4 parts; nitric acid, 1 part; glycerine, 3 parts. Discussion of Results Typical strain markings have been observed within deformation bands, as shown in figs. 1, 2, and 4. By stereographic plotting of metallographic measurements, the strain markings within a deformation band in a single crystal of aluminum deformed by compression were found to be traces of (111) planes within a maximum scatter of 6". The edges of the deformation band were found to be traces of the (100) planes also within the maximum scatter of 6" (in agreement with previous workers). At 30 pct deformation, this scatter was considered quite normal because of the fragmentation and rotation of the crystal. This solution, based on the traces on one plane of observation is not unique, but as the planes were observed to be those normally and logically to be expected, further work in this regard was not felt to be necessary. The study of strain markings in a wedge-shaped specimen, rolled to obtain a gradient of deformation from 0 to 50 pct, revealed prominent strain markings at 25 pct reduction in thickness (fig.5). Faint and weak strain markings were visually observable up to a minimum reduction in thickness of about 12 pct. Since strain markings have been observed in other face-centered cubic metals after much smaller deformations;1, 2 and since the rate of recovery in aluminum is high, it is believed that, at lower deformations, self-recovery removes the localized strains at the slip interfaces, thus removing the source of the etching effects that give rise to strain markings. At high magnifications, strain markings sometimes seem to appear as very thin, light-banded re-gions bounded by a pair of dark edges. This so-

Jan 1, 1951

By Harmon D. Nine

DISCONTINUITIES in the strain vs cycles to failure (S-N) fatigue curves have been reported for polycrystalline materials by Porter and Levy for copper,&apos; by Benham and Ford for mild steel,&apos; and by Williams and Stevens for 18/5 stainless steel.3 However, the considerable scatter of the points for polycrystalline materials have made a statistical analysis of the data necessary to confirm the discontinuities. By using oriented single crystals of high-purity material and low grown-in dislocation content, the reproducibility has been improved to a degree which makes statistical analysis unnecessary and clearly shows discontinuities in the S-N curves. A recent publication4 reports an abrupt change of slope in the S-N curve for ( 100) axially oriented copper single crystals fatigued in torsion. The reported discontinuities in the S-N curve, along with a different mode of failure in the two branches of the curve, have been interpreted as being due to two different amplitude-dependent mechanisms. Further work on copper single crystals of (111) axial orientation has now been completed which supports the existence of an amplitude-dependent change in the mechanism of fatigue. In the present study torsional-fatigue experiments were performed at room temperature in air on (111) axially oriented single crystals. The crystals were grown and prepared as reported in Ref. 4. Two different methods of growing the crystals also discussed in Ref. 4 produced crystals of two different perfections: Type I with dislocation density of >l06 per sq cm and Type I1 with dislocation density of <l05 per sq cm. In accordance with the accepted hypothesis that crystal flaws are primarily responsible for the scatter of fatigue lifetimes, particularly at low amplitudes, the values for the three lowest amplitudes for Type I crystal are believed to be uncertain. They are marked in braces in Fig. 1. On the other hand, at amplitudes above a surface shear strain (SSS) of *0.003, the data for both types of crystals are in agreement. Most remarkably the S-N curve for torsional fatigue of (111)-oriented single crystals is composed of three segments separated by rather large discontinuities, Fig. 1. Metallographic and X-ray measurements indicate that the lower branch corresponds to the low-amplitude (F) region of the (100) copper single-crystal results as discussed in Ref. 4, and the two upper branches correspond to the high-amplitude (H) region. The lower discontinuity in the (111) S-N curve and metallographic evidence give further support to the hypothesis that different mechanisms of failure operate for high and low strain amplitudes. Reference to the resolved shear stress diagram, Fig. 2, shows that the highest stressed systems in

Jan 1, 1965

By H. C. Fiedler

The solubility of sulfur in Si-Fe was determined by the metallographic examination of heat-treated and quenched samples. In the absence of manganese, the solubility in 3.1 pct Si-Fe was found to be about 0.010 pct at 1000o C, 0.015 pct at 11000C, and 0.020 pct at 1150°C, which is substatntially less than hitherto reported. Increasing the silicon content was found to decrease the solubility of sulfur, which is consistent with the increase in the activity coefficient of sulfur with increasing silicon content that has been reported for Fe-Si alloys. Manganese additions were studied in alloys containing 3.1 pct Si and between 0.013 and 0.015 pct S. At low manganese levels the solubility product of manganese sulfide was found to be lower than previously reported. THE solubility of sulfur in pure polycrystalline iron has been reported by Rosenquist and Dunicz1 and by Turkdogan et al.2 Samples were equilibrated in atmospheres containing different ratios of H2S/H2. Both groups of investigators found a solubility limit in y iron of 0.013 pct at 1000oC and 0.031 pct at 1200°C. In addition, Rosenquist and Dunicz1 found a solubility limit in a iron of 0.020 pct at 900°C. Ainslie, Hoffman, and seybolt3 found significantly higher sulfur contents in polycrystalline iron samples than in single-crystal samples after heating at 890°C in a sealed tube containing a mixture of powdered iron and FeS made with s35. Whereas the single crystals contained 0.013 and 0.016 pct S (± 0.005 pct), the polycrystalline samples contained between 0.016 and 0.026 pct S. Autoradiographs suggested that the excess sulfur in the polycrystalline samples was due to the segregation of sulfur in solution at grain boundaries. Ainslie and seybolt4 have reported on the solubility of sulfur in 3 pct Si-Fe. Their samples were sealed in capsules containing hydrogen and sufficient sulfur as cobalt sulfide to saturate the sample. After heat treating, the sulfur-rich phase on the surface was removed by abrading, and the sulfur content of the sample was taken as the equilibrium value for that temperature. Before the equilibrating heat treatment, the samples were heated to 1300°C to coarsen the grains. The resulting grain size, which was about 3/18 in. in diam, was considered sufficiently large to preclude a sensible increase in sulfur solubility due to grain boundary segregation. They reported the solubility to be 0.029 pct at 900oC, 0.063 pct at 1000oC, 0.076 pct at 1100°C, 0.122 pct at 1200°C, and 0.098 pct at 1300°C.

Jan 1, 1968

By R. T. Webber, R. B. Owen

The vapor phase deposition of molybdenum and tungsten from the hexacar-bonyls has been thoroughly investigated by Lander and Germerl, and shown to yield well-bonded coatings on a variety of hot metallic surfaces. They also demonstrated that chromium coatings2&apos;3 could be similarly deposited, but did not investigate this metal extensively. Recently, conditions have been described by which this compound can be prepared in quantities sufficient for laboratory scale plating experiments, and it is the purpose of the present communication to outline the results of preliminary experiments upon the plating of chromium from its carbonyl. Apparatus and Technique The apparatus used in these experiments is shown in Fig I. At one end, connected by a short length of rubber tubing, is a Cenco Hypervac 20 pumping continuously. At the opposite end is a needle valve which allows a small stream of purified hydrogen to flow through the apparatus. Connected in series are a trap T1, maintained at the temperature of dry ice and acetone, the plating chamber, and two traps T² and T3 containing crystals of the carbonyl. T² is kept in a constant temperature bath (not shown), while T³ is at room temperature. Stopcock S6 allows the hydrogen flow to by-pass the carbonyl traps. A McLeod gauge is connected so that it can measure the pressure at the pump through Se, or the pressure at the plating chamber through S9 and S6. The standard (12 liter) volume connected by Sb is used to determine the volume included by S4, S6, SB, Slot the needle valve and the McLeod gauge. Knowledge of this volume allows calibration of the flow of hydrogen. The apparatus* is built entirely of pyrex, except for the needle valve, the rubber connection to the Hypervac, and the cover of the plating chamber. The plating chamber is water-jacketed, with a large brass top allowing insertion of the specimen to be plated. Set into the brass top is a ground glass stopper through which thermocouple leads are sealed. The thermocouple wires (pure Pt, and 87 pct Pt-13 pct Rh) serve the double purpose of measuring the specimen temperature and suspending it in the chamber. Surrounding the chamber is the coil of an induction heater, H. The apparatus is designed to allow variation of the following conditions: (I) the specimen temperature, from room temperature to about 1000°C, (2) the carbonyl vapor pressure, from 40 to 220 microns, and (3) the hydrogen flow. In several runs, 0.8 to 1.8 PCt of hydrogen sulfide was introuced through a needle valve that replaced S4. The vapor pressure of the carbonyl is

Jan 1, 1949

By J. C. McKinnell

The appraisal of an oil lease may often be mad,? through production decline analysis, which requires description of the functional relationship between the oil production rate and either cumulative production or time. Heretofore, attempts to define the function appropriate to a given set of production data have depended to some extent on subjective procedures. The purpose here is to introduce an objective solution, obtained by combining the principle of least squares with the theory of equations. MATHEMATICAL ANALYSIS The fundamental differential equation appropriate to a general class of decline curves may be written. Solution of Eq. 1 for general boundary conditions may be written in terms of either the cumulative oil production or time: These equations constitute the mathematical expression of the so-called hyperbolic decline. Their complete description for a given set of production data requires the numerical evaluation of the three parameters1.&apos;, n, D, and (go),. For the special case of n = 0, Eq. 2 becomes q. = (q.), - DiN with the corresponding expression in terms of time being q. = (qa),e...........(5) Similarly, for the special case of n = 1, Eq. 3 becomes with the corresponding expression in terms of cumula- tive oil production being q. = (q.),e " &apos;.......(7) Eqs. 4 and 5 constitute the mathematical description of the so-called exponential decline; Eqs. 6 and 7 are the mathematical description of the harmonic decline.&apos;,&apos; Returning to the hyperbolic expressions, Eqs. 2 and 3 may also be written, q: = sNp + b........ (8) and where a = 1 - n; s = (n— 1)D,(q0);"; b = (go):-"; a&apos;=-n;s/= nD,(qo);"; and b&apos; = (go);-. Note that both Eqs. 8 and 9 are linear in either time or cumulative production with respect to an exponential function of the production rate. In other words, a straight line relationship results for both variables when associated with the appropriate power of q,. This fact is of prime importance in the application of statistical analysis. . STATISTICAL ANALYSIS For a given oil lease, the time, production rate and cumulative production are recorded records. To establish predictive formulas from Eqs. 2 and 3, it remains to determine numerical values for the three parameters [n, D, and (q,) .] through an investigation of the past history. Upon interpreting a, s and b as parameters, Eq. 8 may be expressed for every time interval of interest, i, over the past history as follows. (q,): = s(Np), + b........(10) In general, Eq. 10 will not be satisfied exactly, because of the assumptions in the expression of Eq. 1, and the errors and anomalies universally associated with production data. It will instead define the residual, s(Np), + b- CO=d......(11) Squaring both sides and summing over all the data points, m, under consideration yields

BENEFICIATION AND CONCEN&apos;TRATION; FROTH FLOTATION U.S. 4,069,144 - In the froth flotation beneficiation of phosphate rock using at least an acid flotation step and an amine flotation step, the crude ore is dried either during transport or storage or by air circulation immediately prior to classification, and the dried ore mixed with water and classified prior to flotation The preliminary drying substantially eliminates the usual problems with clay slimes G. C. Johnson and A. E. Schweizer, assigned to Mobil Oil Corp. Jan 17. 1978. 4 pp (209- 11) . Filed 1 1-19-75 (633,535).

Jan 1, 1979

By W. L., Yerkes

AN interesting study of the application of air-, conditioning to the problem of mine ventilation and cooling can be found on the Witwatersrand in South Africa. Here there are a large number of deep mines in a relatively small area. Due to the nature of the reef, many mines are working at a depth of 6000 ft. and some have gone down to 8000 and 9000 ft. below the surface. At such depths, even in a country geologically as old as Africa, the virgin rock temperature approaches 110°F. The flow of heat from the- rock to the ventilation air is tremendous though the shaft or drift may have been open to ventilation for some years. In addition, there is the heat produced by machinery, lights, and men at work and the heat of adiabatic compression of air introduced into the mine. The difference in atmosphere pressure between the collar of a shaft and a level 6000 ft. below is nearly 3lb. per sq. in., resulting in an increase in heat content of nearly four B.T.U. per pound of air delivered to the mine at surface. Further the exclusive use of wet mining to reduce the incidence of miner?s phthisis increases the latent- heat content of the air.

Jan 1, 1941

By J. D. Verhoeven

The effectiue distriblltion coefficient has been Measured in a series of vertical, normal freezing experirtzents with large current densities passing through the solid-liquid interface. The results indicate that the effect of the current on solute redistribution is due to electvotransport in the liquid. The high currents produced extensive mixing in the liquid which was considerably reduced by solidifying in small lubing. Analysis of the convection indicates that it probably vesulls from a horizontal temperature gradient produced by the Joule heating- of the electvic crirver~t. Due to the strong dependence of this convection upon the specimen radius, it is conclnded that only for specit~ens with radii of less than somewhere between 1 and 2 mm wilt electric field freezing be getlerally effec-tire at enhancing the purification of zone-melting experittierats in loe-melting alloys. Experiments on 0.8-mm lubes illustrate some interesting potential applications of this techniqne as a means of solute control on smcrll specimens. In a number of papers published within the last few years&apos;-4 it has been found that passage of large dc currents through the solid-liquid interface of a freezing alloy produces a change in the solute redistribution accompanying the solidification. It had been predicted theoretically by Hucke et al.s and by Pfann and Wagner that such an effect would occur as the result of electrotransport within the liquid boundary layer at the solid-liquid interface. These authors independently derived the following relation between the effective distribution coefficient and the variables of the process: Where k0 is the equilibrium distribution coefficient, V is the mobility of the solute relative to the solvent, E is the electric field intensity, D is the solute diffusion coefficient, R is the rate of interface motion, and 6 is the boundary layer thickness. Both groups of authors show that this equation predicts the possibility of extending the purification ability of zone-melting experiments by the addition of an electric current. However, there are other effects in addition to electro-transport which could be produced by the high dc current and might, perhaps, invalidate the applicability of Eq. [I]. For example, high electric current might affect the interface reaction or perhaps the equilibrium distribution which was assumed to exist at the interface. It was the purpose of this work to determine the applicability of Eq. [I] by systematically investigating a simple system for which all of the necessary parameters were known. It was of particular interest for speculating on the potential usefulness of this technique to determine the maximum practical field intensities obtainable, and the effect of these high fields upon convective mixing in the liquid. The initial work in this study was published in a previous paper3 which will be referred to as I. In that work, a current density of 2000 amps per sq cm was used with specimen diameters of 5 mm. The results gave a qualitative confirmation of Eq. [I.], and also showed that a large amount of convective mixing was produced in the liquid by the current. The present study is an extension of that work to smaller specimen diameters and higher current densities. Two points were overlooked in I and will be emphasized here: a) the value of ko has not been well-established in the Sn-Bi system and b) the observed convection could be due to thermal convection produced from the Joule heating. EVALUATION OF k0 To evaluate Eq. [ I.] it is necessary to know the value of the equilibrium distribution coefficient, k,. Tn I the value of k, was taken as 0.3 from a published phase diagram.7 According to ansen&apos; the best data on the phase boundaries of the Sn-Bi system are those of Davidson.5 The data of Davidson give a k, value ranging from 0.21 to 0.25 for the compositions studied in this work, or an average value of around 0.23. In a recent publication, Rigaud and Tougas&apos;10 determined ko by means of a solidification technique. For an alloy of approximately the same composition as that used in this work they obtained a ko value of 0.39. Hence, there is considerable uncertainty in the value of ko and, consequently, a series of experiments were made to determine k, more accurately.

Jan 1, 1968

By D. A. Flanagan, P. B. Crawford

A study has been made of the feasibility of storing liquid meihane at low pressures in undergrohd caverns. Methane liquefies at — 258°F at atmospheric pressure. It is shown that the methane evaporation rates will rapidly decrease and cool the surrounding rock so that at the end of one month they would be between 50 Mcf/hr for a cavern of 25-ft radius and 700 Mcf/hr for a cavern of 100-ft radius. At the end of 10 years, the evaporation rates would be 18 and 100 Mcf/hr, respectively, for caverns of the same radii. The evaporation rates may be reduced by a factor of 2 to 10 by the application of insulation. The cost for mining the caverns is estimated to be $.75 to $1.25/Mcf of storage. This is substantially less than surface storage; it is believed to be safer and to result in lower maintenance, savings in space and savings in strategic materials. INTRODUCTION During the past few years, there has been an increasing interest in the economic feasibility of liquefying methane. Methane liquefaction is being considered for tanker transport; in addition, liquefaction is being reconsidered for shaving peak gas demands. Several articles have described natural gas liquefaction and the progress of the tanker in making trial runs from the United States to Great Britain to determine the feasibility of tanker transport. At the present time, pipelines are not designed to supply peak gas loads during extremely cold periods such as are oiten encountered in the North and Northeast. Gas is being stored in underground reservoirs en-route to its destination, but in many instances satisfactory storage in porous reservoirs has not been practical, especially along the Eastern seaboard where few petroleum reservoirs have been found. In England and other foreign countries, it is unlikely that satisfactory porous structures could be found, and it may be desirable to mine or excavate the rock to obtain storage. By storing the methane near the consumption point, product availability can be increased during periods of need. Methane liquefies at - 258OF at atmospheric pressure, and 1 cu ft of liquid methane will make about 600 cu ft of gaseous methane. Methane liquefaction for shaving peak gas demands was conducted in Cleveland, Ohio in the early 1940&apos;s. The Cleveland plant was designed to liquefy 4 MMcf/D of natural gas.&apos; Development of several low-temperature processes has resulted in some improvements in the design of low-temperature storage vessels. Liquefied-methane storage vessels can be designed for various evaporation rates. The desired rate will depend on the reason for liquefaction. For shaving peak gas demands, the desired evaporation rates would be low, possibly between 0.2 and 1.5 per cent/day. For methane tanker transport and supply, the evaporation rates must be high and must equal the total daily rate of supply. It is apparent that the cost of storage facilities will depend on the objective and will vary appreciably. Surface storage of methane may cost between $3,000 and $20,000/MMscf, depending on the size and design. The failure of the surface storage vessels at Cleveland, with the resulting injury and loss of many lives, has caused considerable emphasis to be placed upon developing new, safer and improved methods of storing liquid methane. Storage of liquefied propane and butane in underground salt domes has been very satisfactory. The per-barrel cost of underground storage for any sizeable capacity is very small compared to the cost of surface storage for LPG mixtures. The larger the capacity of the underground storage, the less is the per-barrel cost. Numerous advantages appear to exist if liquefied methane could be satisfactorily stored underground, the principal advantages being increased safety, lower initial cost, lower maintenance cost, savings in space and savings in strategic materials. The purpose of this paper is to determine the feasibility of storing liquid methane underground in mined caverns. If a large spherical cavern is dug underground and is filled instantaneously with liquid methane, the surface of the sphere may approach the liquid methane temperature almost instantaneously. The temperature distribution for a sphere of radius r. in an infinite medium, initially at one temperature with spherical surface kept at T, from time t = 0, is given by Eq. 1.&apos;

By Arthur Lubinski, K. A. Blenkarn

It is explained why the bottom portion of freely suspended tubing in a pumping well buckles and straightens in succession during the pumping cycle. Field evidence of resulting rod-on-tubing wear, excessive polished rod load, and excessive horsepower are given. The possibility that buckling hastens pump wear is strongly suggested. Means for either prevention of buckling or for mini-mizing its effects are explained, their relative merits compared, and field results of their use given. These means are: tension anchors, tail pipe, sucker rod guides, and corrosion inhibitors. Charts and formulas for best use of tension anchors, tail pipe, and guides are given. Use of tension anchors, or compression anchors with a heavy tail pipe, prevents both buckling and breathing, thus improving volumetric efficiency. Without tail pipe, buckling above compression anchors precludes such improvement. INTRODUCTION It is well known that a column must be subjected to some compression in order to buckle; and also that a structural member does not buckle if subjected to a tension. When consideration is given to a tubular column, the question may be asked, "How is the buckling phenomenon affected by inside or outside pressure?&apos; At first, it seemingly should not be affected. However, this is not so. In order to arrive at a correct deduction one should first understand the basic reasons for buckling of a column under loading*. Then he should add to the effect of that loading the effect of pressure. Following this procedure, one may discover quite unexpected phenomena. In some cases, when subjected to more pressure inside than outside, a pipe may buckle under tension. In other cases, a pipe may remain straight, although subjected to a very large compression. In 1950 and 1951 it was explained why the bottom portion of a string of pipe freely suspended in a well does not buckle even when subjected to a large compression due to buoy- ancy.2,3 It was shown that the com-monly used concept of a neutral point, as being a point in the string with neither compression nor tension, is inadequate, and a more relevant definition of the neutral point was introduced. The findings were also generalized to cases of different pressures inside and outside a pipe.3a Later, in a further generalization, allowance was made for variations of stress and pressure with depth.&apos; The findings were applied to the new technique of hydrostatic high pressure testing of lengths of pipe in the mill.5 In 1952 it was observed that previous theoretical findings apply to tubing in pumping wells. The lower part of freely suspended tubing buckles during the upstroke portion of the pumping cycle. Later, it was ascertained that this phenomenon is responsible for wear and malfunctioning of the equipment. Various remedial measures were devised and successful field trials conducted. Only brief mentions on the subject have been published.6,7 In this paper, the phenomenon of tubing buckling is explained, field

Jan 1, 1958

By Josiah Royce

To evaluate chemically the sample of rock obtained by diamond drilling, it has long been recognized that the analyses of the two components of the sample, core and sludge, must be given appropriate influence in computing the average analysis of any unit of depth. The purpose of this investigation is to set forth what means are available for apportioning the effect of core and sludge on the final analysis, what variables affect the problem, and what combination of applied mathematics will closest approximate the truth under each condition as these variables proceed within their limitations. A drill hole is bored in iron ore exploration principally to test variations in rock composition with depth and is usually directed as nearly normal to the bedding of a horizon to be tested as possible. This practice has a tendency to minimize variation in composition laterally which in any event is not likely to be great. It is obvious that the opportunity for change in analysis of a particular rock is not statistically as great radially in a diamond drill hole where the distance in which such a change may occur is from 0.719 in. (EX bit) to 1.469 in. (NX bit) as there would be longitudinally even in a&apos; run as short as 5 ft. Variations in composition of bedded or layered rocks are usually greater normal to the bedding than parallel thereto. Even in massive rocks, like porphyries, variations are functions of distance. Hence. in either case, variations along the hole are of greater effect than across it. Let us examine Fig 1 briefly to observe a cross section of a unit of depth of a typical diamond drill hole. If we neglect radial change in chemical composition, which we have seen is small compared to lengthwise variation, we can see that if core recovery was 100 pct, the core and its surrounding area, which would be recovered as cuttings (assuming 100 pct sludge recovery), must analyze the same if the sampling is perfect. This is the foundation of this paper, namely, that if core recovery is complete we may assume that the core constitutes as nearly as possible a perfect sample of the ground drilled. Now let us pass to the methods of weighting core and sludge in the final analysis and then develop each in turn. One might employ core analyses only, sludge analyses only, or apportion the influence of each by one of several methods. The most common apportionment is by direct proportion to relative theoretical vol- &apos; ume occupied by each component in the cylinder hollowed out by the drill, known as the Longyear formula. Another logical treatment would be to consider core and sludge in the final analysis according to the weight of each recovered. This procedure was elaborately set forth and refined by R. S. Moeh1man.l Then to be sure that we have embraced all the possibilities, we must admit that some other method or methods may be devised empirically or by mathematical maneuvers. Should a drill machine be capable of recovering 100 pct of the core in all types of rock, certainly core alone need be analyzed and no complicated mathematics are required; but although some companies have had fair success in obtaining high percentages of core in certain homogeneous ores on the Mar-quette Range, their same methods have proved disappointing on the Menominee, Cuyuna, and Gogebic Ranges where the iron formation often varies widely from flinty chert beds interstratified with soft hematite to wholly leached ore material composed of an unpredictable mixture of hematite and limonite with occasional zones of sugary, recrystallized chert. In spite of numerous mechanical improvements in core barrels, few drilling programs can count on complete core recovery. The sludge alone might be analyzed but for practical reasons this is inadvisable when core is obtained due to the numerous opportunities for contaminating or losing portions of the cuttings. The Longyear formula involves a weighting of the core and sludge in proportion to the theoretical volume of each, that is, where the volume of the core recovered is one-third of the total cubic volume of the cylindrical hole made by the drill, the analysis of the core is given one-third of the weight in the final analysis. This approach to

Jan 1, 1950

By E. T. Hayes, A. H. Roberson, M. H. Davies

R. F. Dornagala and D. J. McPherson (Armour Research Foundation, Chicago)—I should like to compliment the authors for a workmanlike job in determining the partial phase diagram of a system comprised of two rnetals which are certainly not easy to work with. We are completing work at Armour Research Foundation on an Atomic Energy Commission-sponsored project for the determination of eight zirconium binary diagrams. Work on the Zr-Cr system has been completed and should be published within the next year. For our work, Westinghouse Grade 3 iodide crystal bar served as the zirconium melting stock. Johnson-Matthey, electrolytic chromium, specially treated for oxygen removal, was employed. The overall constitution of the system determined at Armour Research Foundation is in very good agreement with the present work. We found a eutectic at 18 pct Cr and 1280 °C, somewhat lower than the value reported. This temperature was confirmed by thermal analysis, incipient melting studies, and regular isothermal anneals. The eutectoid was located close to 1 pct Cr and 835°C by metallographic analysis of annealed specimens. Maximum solubility of chromium in /S zirconium was 4.5 pct at the eutectic temperature. Chromium solubility in a zirconium was less than 0.28 pct at all temperatures. We found the compound at 53 pct to melt around 1700°C, with an open maximum, but determined its crystal structure to be hexagonal close-packed (MgZn, type). The lattice parameters were in excellent agreement with those determined by Wallbaum in 1942. The diagrams are in substantial agreement, and .part of the differences are undoubtedly due to the use of different zirconium melting stock. M. K. McQuillan (Birmingham, England)—I read this paper with a great deal of interest, as it covered the same field as some work of my own.&apos; There are a number of points in the present paper on which I would like to comment. First, I should say that I, too, used zirconium prepared by magnesium reduction of the tetrachloride and electrolytically prepared chromium, and melted the alloys in a Kroll-type arc furnace. The purity of my alloys should, therefore, be comparable with the purity of those of the present authors, and any differences in our observations would not be expected to be attributable to this cause. The differences between my observations and theirs concern the presence of the eutectic, the temperature of the eutectoid, and the melting point of the compound. I would be very much interested in any further evidence the authors may have for the occurrence of the eutectic at 1380°C. During the course of my work I noted that a number of my alloys containing 60 to 90 pct Zr melted at about 1400 °C, and for a time assumed that a eutectic occurred at this temperature as described in the paper. On further investigation, however, I found that the structures of the as-melted alloys could not be made to fit in with this interpretation of the system. If a eutectic exists in this region of the system it would be expected that the as-melted alloys would show the usual type of cast structure, i.e., dendrites of the compound plus eutectic. This, however, does not occur, as may be seen from Fig. 9. The compound seen there is not dendritic in form, and the remaining material is by no means certainly eutectic. It may be argued that a compound such as ZrCrl would not form dendrites but would tend to crystallize in geometric shapes. In this case, however, I have evidence to the contrary, as on the chromium side of the compound, where a eutectic occurs at about 1545"C, the compound formed from the liquid takes on a conventional dendritic form, and the eutectic is observed in the interdendritic spaces in the usual way. There is no reason to suppose that the compound would behave differently in an alloy lying on the zirconium side of the compound composition if a eutectic existed there too.

Jan 1, 1953

By O. R. Morris, G. L. Hawkes

A galvanic cell, using as electrolyte a fused salt solution of calcium carbide and as electrodes carbon and a Fe-C alloy of known composition, has been set up to study the thermodynamics of Fe-C alloys in the temperature rmzge 800" to 1000°C. Time independence and reproducibility of the cell electromotive force were taken as evidence of the reversible behavior of the cell. Carbon was believed to be present in the electrolyte as the so-called acetylide ion, C;-. The plots of the cell electromotive force us temperature for a specific alloy composition were straight lines within the limits of experimental error. The average Partial molar enthalpy of carbon in iron relative to pure carbon was found to be +10,610 i 93 cal per g-atom C. Thermodynamic analysis of the data has led to the following equation for the carbon activity, ac, based upon pure carbon as the standard state: In ac = In Zc + 10,560/RT + (10.02 + 77O/T)ZC - 2.350 where ZC is the lattice ratio [nC/(nF, - nc )] and T is the absolute temperature. This equalion gives carbon activity values generally slightly lower than those from gas equilibration studies reported in the literature. METAL LOGRAPHIC examination of a polished cross section of the steel anode used in the electrolysis studies of fused salt solutions of calcium carbide by Morris and Harry revealed extensive carburization of the steel by the electrodeposited carbon. This carburization was reflected in the variability, with time, of the applied potential to the electrolysis cell, necessary to maintain a constant current density at the electrodes. This observation suggested the setting up of a galvanic cell of the "alloy concentration" type to study the thermodynamics of some metal-carbon alloys. Cells of this general type have been widely used for the study of alloy systems.2 In view of the availability of published data in respect of the austenite phase of the Fe-C system, it was decided to carry out measurements upon these alloys before proceeding to studies of less well documented systems. The galvanic cell may be written: where [C] is carbon dissolved in iron. The electrolyte was a fused salt solution of calcium carbide, containing some 5 to 10 mol pct of carbide. The cell reaction is believed to be: C(s)-[CI [I1 Carbon forms an interstitial solid solution in iron, with the atoms located in the octahedral interstices. In the fcc crystal structure of austenite there is one octahedral interstice per iron atom. Thus, the lattice ratio, ZC, shown by Gurney3 to be the fundamental concentration parameter in the context of interstitial solutions, is given by: where nc and nFe are the number of carbon and iron atoms, respectively. chipman4 has recently shown empirically the advantages of using this concentration parameter instead of the more usual atom ratio or atom fraction. The cell electromotive force, E, assuming reversible behavior, is related to the carbon potential or the partial molar free energy of carbon in the solid solution relative to pure carbon at the same temperature and pressure, GP at the composition ZC, by the equation: where z is the carbide ion valency and F is the Faraday constant. An activity of carbon, ac, in the solution relative to the value of unity assigned to pure carbon, and an activity coefficient, qC , are defined such that: where R is the gas constant and T the absolute temperature. GF is further related to the relative partial molar enthalpy Hm, and the temperature coefficient of the cell electromotive force, (aE/aT)Zc, by the equations: Measurement of the cell electromotive force thus enables calculation of the relative partial molar thermodynamic properties of carbon in iron, if z is known. At E = 0, the solid solution is in equilibrium with pure carbon. More convenient for many purposes is the standard state based upon the infinitely dilute solution, Henry&apos;s law. The relationship between the activity coefficient of carbon based upon this standard state, and that based upon the pure carbon standard state, qC , may be obtained by considering the free energy of transfer of carbon from the latter standard state to the former. The relationship is: where +:H is the activity coefficient of carbon in the hypothetical standard state based on a reference of

Jan 1, 1969

By J. E. Drury, W. M. Byers, E. J. Lynch

Measurements made with starch ~~~~s indicate that the Presence of starch serves to depress the streaming potential of a clay mud system. Measurements with pure starch slurries show that the addition of NaCl or CaO reduces the streaming potential significantly. The addition of NaOH, however, causes only a minor reduction in potential even when there are large changes in mud resistivity. INTRODUCTION In their excellent study of the streaming potential, Hill and Anderson&apos; proposed that the presence of conductive solids within the filter medium will reduce the generated potential. This theory was demonstrated with measurements on shaly sandstone cores having various "b" factors; the cores with the higher negative b factors showed markedly lower potentials. (The factor b was originally defined by Hill and Milburn2 in an expression relating the resistivity of a shaly sandstone to the resistivity of the interstitial water: log F, = log F,, + b (log 100R,)&apos;. &apos;he factor b, therefore, is a measure of the contribution of the conductive solids to the total conductivity of the sample. It has been found experimentally to be directly related to the amount of chemically exchangeable ion per unit pore volume.) In the logical extension of this hypothesis to the streaming potential created by filtration through mud cakes, they suggested that the presence of nonconductive starch in a drilling mud would have the opposite effect; i.e., it would cause the streaming POtential to be increased. The high streaming potentials that they observed with lime muds were attributed to this cause. A cross-plot of some of their results which are relevant to this discussion is given in Fig. 1. The lime-mud curves and untreated-mud curve are taken from curves given in their paper. The gyp-mud line was estimated from their statement that the measured streaming potential for gyp muds in the range of 1.5 to 2.0 ohm-m was approximately 18 mv/1,000 psi. These gyp-mud potentials were the lowest that they reported for any of the bentonitic-clay muds, and were the only ones which were less than the untreated-mud potentials. This very low streaming potential of gyp muds has also been observed in field studies.&apos; By way of comparison, a U. of Texas publication4 gives the approxi-ate compositions for lime and gyp muds shown in Table 1. These materials are added to a bentonitic-clay mud base. It can be seen that both of these muds contain significant amounts of calcium salts and that the gyp mud has the higher concentration of starch. It would appear from this and from the data shown in Fig. 1 that starch may well be the cause of a streaming-potential reduction, rather than a potential increase. EXPERIMENTAL A series of streaming-potential experiments has been performed using a cell of a design similar to that reported by Wyllie.5 Calomel electrodes were used in both the mud slurry and the filtrate. Pressure in the cell was generated with high-pressure nitrogen. The primary objective of the investigation was to determine the effect of starch on the potential. Five different slurries were made which ranged in composition from untreated ben-tonitic mud (Baroid Aquagel) to a pure starch suspension (Baroid Im-permex). The intermediate slurries were prepared by mixing together a 9-lb/bbl clay mud with a 12-lb/bbl starch suspension in the ratios 1:3, 1:1 and 3:1. When starch was used alone, it was necessary to use an 18-Ib/bbl mixture, since leaner mixtures would not form a suitable filter cake. The clay mud used in the experiments was aged for three months before use. In addition to this series of measurements on the starch-clay mud mixtures, other experiments were performed to determine the effect on the potential of the absolute starch and Aquagel concentration, and the effect of adding various chemicals. RESULTS The effect of starch on the streaming potential is shown clcarly in Fig 2. Increasing percentages of starch resulted in n continually decreasing

By W. D. Ludemann, J. E. Dor, L. A. Shepard

Recovery of the creep resistance of 99.99 pct pure Al was studied at temperatures 540°, 573°, 600°, and 611°K. Poly-crystalline specimens crept under a stress of 950 psi to a strain of 5.5 pct were allowed to recover for periods of from 1 min to 16 days under a residual stress of 4.4 psi. Increased creep rates upon reapplication of the 950 psi stress evidenced softening of the material. The activation energy for the recovery process was found to be 64,000 cal-per mol. THREE major observations have clearly revealed that the creep of polycrystalline aluminum above about 510°K is controlled by the rate of climb of jogged edge components of dislocations past the barriers impeding their motion: 1) Extensive polygonization, characteristic of climb, takes place during creep.&apos;9&apos; 2) The activation energy for creep equals the estimated activation for self-diffusion. 3) The stress law for creep coincides with theoretical predictions based on the climb mechanism.4,5 The decreasing rate during the primary stage of creep must be ascribed to the introduction of additional barriers to the glide of dislocations. During secondary creep the density of barriers must remain constant, the rate at which new barriers are formed being equal to the rate at which they recover. For this reason the nature of the barriers, and their rates of formation and recovery, are significant to a complete understanding of creep. It is proposed here to study the kinetics of the recovery of barriers to creep of high-purity polycrystalline aluminum in the climb range. The technique will consist of creeping a series of tensile specimens under a prescribed stress to given state following which the specimens will be recovered for various times and temperatures under approximately zero stress. The amount of recovery will be determined by comparing creep curves following the recovery. EXPERIMENTAL PROCEDURE The material used for this investigation was a 99.99 pct pure Al supplied by the Aluminum Co. of America in sheets of 0.001-in. thickness having an H-18 temper. The spectroscopic analysis of impurities gave Cu-0.004 wt pct, Fe-0.002 pct, Si-0.001 pct, others-0.000 pct. Creep specimens were machined with their tensile axes in the rolling direction of the sheets. Annealing in a molten potassium nitrate-potassium nitrite bath for 1 hr at 686K, followed by air-cooling, resulted in a grain diameter of from 0.22 to 0.27 mm. Creep machines were equipped with constant stress lever arms of the type suggested by Fullman, Carreker, and Fishher,&apos; which maintained the applied stress to within 0.04 pct of the reported value of 950 psi. Creep strains were calculated from extensions over a 6-in. gage section to the nearest 3 X 10"5. The temperatures of testing, 540°, 573", 600°, and 611°K, were obtained by immersing the specimen and extensometer assembly in a molten KNO2-KNO3 bath. Temperatures during the periods of creeping could be maintained constant to within better than ± 1°K. Correlation of creep data at different temperatures of testing demanded the use of an appropriate temperature-compensated time 8. Previous investigations have shown that the creep of pure metals at temperatures above about 0.55 of their melting temperatures can be correlated by the functional relationship3" e=f(0), s= const. [1] where e = total plastic strain during creep f = a function that depends on stress a = stress 6 = te-Q/RT = a temperature-compensated time t = time under test Q = 35,500 cal per mol = activation energy for creep R = gas constant, and T = absolute temperature In the present series of tests all of the creep was conducted under the same stress of 950 psi. To provide identical initial conditions for the recovery all specimens were first crept to a temperature-compensated time of = 47 X 10-14 min which resulted in a creep strain of 0.055 + 0.005 regardless of the creep temperature. During recovery, a small stress (4.4 psi) was permitted to remain upon the specimens to maintain tautness in the pulling assembly and to prevent mechanical damage to the soft specimen. After each recovery period, the full stress was reapplied and

Jan 1, 1961

By M. R. Tek

General equations relating the pressure drop necessary to sustain the flow of a fluid through a porous matrix at a given rate have been developed. The results indicate that at high values of flow rate the pressure-flow behavior may not necessarily satisfy the usual Darcy equation. The mathematical analysis, carried through the micro-pore geometry and extended through the macro-reservoir scale, indicate that Darcy&apos;s law, of limited applicability to certain ranges of Reynolds numbers, can be generalized through the inclusion of some additional parameters. The "generalized Darcy equation" has also been formulated in dimen-sionless form permitting the evaluation of its predictive accuracy with regard to literature data. A comparison between predicted and experimental values indicates that the generalized Darcy equation predicts the pressure drops with good agreement over all possible ranges of Reynolds numbers. INTRODUCTION The limits and the nature of validity of Darcy&apos;s law&apos; has been a subject of every-day interest to the industry for many years. It is well known that as the Reynolds number, characteristic of the fluid flow through porous media, becomes large, Darcy&apos;s law gradually loses its predictive accuracy and ultimately becomes completely void. For the last 20 years much has been said and written on this subject. Unfortunately little has been accomplished to bring about a satisfactory agreement, at least on the nature of the threshold of validity of Darcy.&apos;s law. Fluid dynamists, geo-physicists, and engineers all had their individual views, explanations, interpretations and concepts on the subject. To some, a mechanistic analogy with pipe-flow proved a satisfactory explanation.&apos; To others,&apos; turbulence, in its random character, was incompatible with the geometric structure of consolidated porous systems. To some,4 turbulence merely represented a factor influencing the permeability measurements and again to others5,6,7 em-pirical or semi-empirical correlations proved satisfactory from an engineering viewpoint. Deviations from Darcy&apos;s law at high flow rates have been studied by systematic experiments by Fancher, Lewis, and Barnes.&apos; In an article on the flow of gases through porous metals, Green and Duwezs conclude that the onset of turbulence within the pores appears unsatisfactory to explain deviations from Darcy&apos;s law. This view is held by many others. While the subject remained controversial for many years, the development of vast natural gas reserves throughout recent years further justified considerable interest on this problem from the standpoint of gas reservoir behavior. As large amounts of field data became available from the operation of many gas fields, it became evident that the steady-state behavior of gas wells was not, in general, in agreement or compatible with Darcy&apos;s law. This suggested a careful reconsideration of all mechanisms which may account for pressure drops in addition to viscous shear. In a series of articles9,10 . Hou-peurt indicated that deviations from Darcy&apos;s law may be explained on the basis of kinetic energy variations and jetting effects without resorting to assumptions on turbulent flow conditions. Another article by Schneebeli11 indicates that special experiments by Lindquist clearly demonstrated that the onset of turbulence does not necessarily coincide with conditions of deviation from Darcy&apos;s law. This view is also held by M. King Hubbert.12 Starting with the basic pressure-flow relations suggested by Houpeurt, the derivation, development and extension of analytical expressions to -supplement and generalize Darcy&apos;s law has been the objective of this work. MATHEMATICAL ANALYSIS Derivation of Dimensionless Pressure-drop, Flow-rate Relations In considering the flow of a fluid through a porous matrix geometrically represented by a succession of capillary passages in the shape of truncated cones,810 an approximate expression may be derived relating viscous and inertial, i.e., total pressure drop to the physical properties of the fluid, geometric properties of the rock matrix and the rate of flow: ?P/?r = µ/k V [ 1 + c(m4 - 1) p V/16n" mµ w] ..........(1) Let us formally set: c (m4 - 1) / 16n" m = a d ......(2) Such a representation is equivalent to assert that the term [c(m4 — 1)/ 16n"m], variable with various porous media and probably highly variable within a given porous medium, may be macroscopically defined as equal to a lithology factor times the aver-age grain diameter d. In view of the usual grain and pore size distribu-

Jan 1, 1958