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By R. W. Nelson

Methods for determining permeability presented by W. D. Kruger3 in a paper entitled "Determining Areal Permeability Distribution by Calculation" have considerable merit. They can be expected to contribute significantly to analysis of flow in heterogeneous systems. Some rather subtle yet important conditions must be met, however, to assure that the computed permeabilities are correct. The conditions which insure a unique permeability distribution may have been overlooked in the method as proposed and presented in the subject paper. The subtlety of the special requirements may permit good agreement between computed and measured pressures in the reservoir; but the computed permeability may still be in error. The general theory and discussion of the requirements for a unique determination of the permeability distribution are presented elsewhere in more detail.7'8 Accordingly, only major features will be presented here in the notation of Kruger. GENERAL DESCRIPTION OF REQUIREMENTS A brief mention of the route to be followed in showing the special requirements may be helpful. The equation to be solved is a quasilinear, first-order, partial differential equation in the unknown permeability. This equation can be solved through an extension of Lagrange's method by reduction to a system of subsidiary ordinary differential equations. Through consideration of the identity of one of the Lagrange subsidiary differential equations with the stream function, a special interrelationship can be shown. The interrelationship has special significance with respect to the boundary condition. If the boundary function satisfies part of the subsidiary differential equation, no unique solution for the permeability distribution exists. In the physical problem this requirement for uniqueness indicates that the permeability measurements used as a boundary condition can not be along a stream tube. The boundary condition must be along a line, or series of lines, which completely traverse the region and pass through every stream tube.

By E. W. Hough, G. L. Stegemeier

Empirical equations for surface tension of propane and normal butane as functions of reduced temperature are obtained from experimental data. Another correlation relating surface tension to enthalpy of vaporization is given for these two compounds. In addition, new paracbor numbers are calculated for the normal paraffin hydrocarbons. These numbers are utilized for the calculation of interfacial tension of two-component systems as functions of pressure and temperature, using a modified form of Weinaug-Katz equation. The experimental data for two binary systems are approximated by the correlation. From these results it is found that the interfacial tension in the high-pressure region remains extremely low at large pressure decrements below the critical pressure. Thus, it appears that condensate systems may have flow characteristics almost like single-phase conditions even though the pressure is within the two-phase region. Experimental data have shown that interfacial tension divided by density difference approaches zero as the critical pressure is approached. A calculation of wetting-phase saturations indicates that the saturation gradient at the two-phase contact becomes increasingly abrupt as the critical pressure is approached. DISCUSSION Prediction of the surface and interfacial tension of the light hydrocarbons and of two-component hydrocarbon mixtures at various temperatures and pressures may be made if other physical properties are known. Extensive experimental work on single-component and binary systemsl is the basis for the correlations outlined in this paper. Interfacial tension is defined as the specific surface-free energy between two phases of unlike fractional composition, while surface tension is defined as the specific surface-free energy between two phase:; of the same fractional composition. The usual definitions relating interfacial tension to a liquid-liquid interface and surface tension to a gas-liquid interface are not clearly defined when the critical region is included, and there is no sharp distinction between a gas and a liquid phase. Interfacial tension is probably the most important single force that makes one-half to one-third of the total oil actually in place in a reservoir rock unrecoverable by conventional gas-drive or waterflood methods. A rough estimate of this figure for the United States is 100 billion bbl. Interfacial tension presently is used by petroleum engineers in the estimation of saturation gradients at the gas-oil contact and at the oil-water contact. The data in this paper should prove useful for estimates of reserves involving gas-oil contacts. Relative permeability undoubtedly is influenced by interfacial tension, for sufficiently small values. These data should be useful in determining how small the values are In addition, these data should eventually add to our fundamental knowledge of surfaces. At the critical point, all surface excesses approach zero and the thickness becomes very large. SINGLE-COMPONENT SYSTEMS It has been observed1 that the following relationships are good approximations to the physical properties of propane and n-butane.

By W. D. Kruger

Methods for analyzing flooding or cycling projects by means of two-dimensional flow calculations are presented in the literature. The use of these methods allows the determination of optimum operating conditions for such projects and provides information on pattern efficiencies, relative flow rates, pressure distributions and floodout histories. The accuracy of such analyses depends on the basic reservoir data employed, especially the absolute permeability distribution in the reservoir; however, absolute permeability data normally are determined from well flow tests and/or core measurements and are so limited in quantity and quality that they do not adequately describe the reservoir flow properties. A calculation procedure is presented for determining the areal permeability distribution in the reservoir. Use of the procedure allows verification of the basic reservoir data through the matching of past reservoir conditions, thereby providing proper data for reliable predictions of future operations. The calculation procedure is a numerical technique based on a mathematical model of the reservoir and is designed for solution by means of an electronic digital computer. Results from field application of this procedure are presented. Permeability distributions from measured data are compared to those calculated to indicate the need for the analysis. Comparisons of field-measured and calculated pressure distributions are given to establish the validity of the calculation procedure. The conclusions of this paper are (I) use of the calculation provides data that allow a match of past reservoir conditions and thus reliable prediction of future operations; and (2) the procedure should, where applicable, be used to provide adequate reservoir data for use in two-dimensional flow calculations and related reservoir analyses. INTRODUCTION One means of providing more precise engineering control over reservoir operations during all stages of depletion is to develop some type of model to simulate the reservoir, using information developed from the behavior of the prototype to predict the performance of the actual reservoir. The use of mathematical rather than physical models for this purpose has been gaining in importance recently because of the application and availability of electronic computers. Mathematical models avoid instrumentation difficulties and allow easier handling of large-scale problems. At present, analyses employing two-dimensional models are common, and thought is being given to the use of models in three dimensions. The use of multi-dimensional models is necessary to show not only what is occurring in the reservoir, but also where in the reservoir the phenomena of interest are taking place. Methods for analyzing flooding or cycling projects by means of two-dimensional numerical models are presented in the literature. The use of these methods allows the determination of optimum operating conditions for such projects and provides information on pattern efficiencies, relative flow rates, pressure distributions and floodout histories. The applicability of numerical models and the results of using such analyses have been presented through field examples. The references indicated here discuss fully the data necessary to build the mathematical reservoir model and the way the model is used to study or predict the future performance of the reservoir. The accuracy of the analysis of such projects depends on the basic reservoir data employed in the construction of. the model. In order to be sure the model represents the reservoir, it must be used to match some known set of reservoir conditions, i.e., to match past reservoir performance. The accuracy of the model is directly dependent upon the absolute permeability distribution in the reservoir; however, absolute permeability data are normally determined from well flow tests and/or core measurements, and are so limited in quantity and quality that they do not adequately describe the reservoir flow properties. There is always the need for better data and for verification of existing data. The purpose of this paper is to show the use of the mathematical model to match past performance in flooding or cycling projects, the use of numerical techniques to adjust the basic absolute permeability data if necessary so that a match is obtained, and in this way to generate the effective areal permeability distribution of the reservoir. A review of the basic concepts of the numerical model will be given, followed by a discussion of the data-adjusting technique for matching past performance. TWO field examples will be cited to show

By J. W. Lacey, F. H. Brinkman, J. E. Faris

This paper presents an analysis of the high-pressure, gas-driven LPG-slug process, based on fluid flow tests in areal models. Two types of tests were made. One series was made in low-pressure models which permitted observation of fluid movement. Three completely misci-ble analog fluids were used. A second series of tests was made in high-pressure models using methane, propane and a light refined oil saturated with methane at room temperature and 1,550 psig. Under the test conditions of room temperature and a pressure level of 1,550 psig, the phase diagram for the fluids used is similar to those for many of the field systems where the process is considered for use. A method for using these laboratory data to calculate field performance of the process is outlined. As a result of this work, it is concluded that small banks of LPG (5 per cent HV or less) are not effective in increasing oil recovery in horizontal reservoirs. Znstead, where small banks are used, the driving gas quickly penetrates the LPG bank because of fingering and channeling; and from this point on, the process behaves essentially as an immiscible gas-injection project. The validity of this conclusion was substantiated by: (1) laboratory studies of the effect of rate, model size and mobility ratio on miscible displacement in areal models; and (2) calculation of field recovery, which compared closely with actual field recovery. INTRODUCTION Field applications and pilot tests of the gas-driven, LPG-bank, oil-recovery process are on the increase. Most of these tests are employing small banks (2% to 5 per cent hydrocarbon volume) of LPG, which is miscible with both the driving gas and the oil in place, in an effort to attain an effective yet economical miscible displacement of oil by gas. The expectation of miscible displacement with small banks is based on the concepts that: (1) lengths of solvent-oil mixed zones, measured during miscible displacements in long slim cores, are representative of those that will occur in the field; and (2) areal sweep efficiencies5 measured in electrolytic model studies are applicable to miscible displacement in reservoirs. Our experimental evidence indicates that the mixed-zone lengths and sweep efficiencies mentioned are not applicable to miscible displacement in reservoirs. In this paper we present an evaluation of the gas-driven, LPG-bank oil-recovery process based on fluid flow experiments in areal models. These results are used to predict the performance of a field pilot test of this process, and the results are compared with the actual test results. THE EFFECTIVENESS OF LPG BANKS The effectiveness of LPG banks of various sizes in accomplishing miscible displacement of oil by gas was determined by displacement tests in a model representing one-quarter of a confined five-spot pattern. The model, 12 X 12 X 1/4 in. in size, was packed uniformly with glass beads. It was operated at a pressure of 1,550 psig and at room temperature; the fluids used were methane, propane and refined oil saturated with methane at 1,550 psig. The saturated oil had a viscosity of 1.2 cp at room temperature. The results of these tests are shown in Fig. 1, where recovery is plotted as a function of the volume of fluid injected for: (1) an immiscible-gas drive with an oil-to-gas viscosity ratio of 85; (2) three sizes of LPG banks; and (3) two completely miscible displacements with mobility ratios of 85:1 and 10:1. (The miscible displacement data plotted are the averages of several tests.) The results show that a 2½ per cent bank of LPG does not increase oil recovery over that obtained by immiscible-gas drive. The 7 and 17 per cent banks are, respectively, about 30 and 50 per cent effective (with reference to the M = 85 miscible-displacement curve) in increasing oil recovery at the point where 2½ hydrocarbon volumes (HV) of fluid have been injected. The percentage effectiveness of the banks at a given volume of fluid injected is defined as Recovery by Bank — Recovery by Gas Drive Recovery by Miscible Recovery by Flooding Gas Drive We conclude from these results that banks of LPG smaller than about 5 per cent HV in an individual stratum will cause from little to no increase in oil recovery. This finding was substantiated by work in low-pressure models, which permitted visual observation of fluid movements. In these tests, three completely miscible fluids having the same viscosity ratios as those used in

By P. Hazebroek, M. Prats, E. E. Allen

Well locations in water floods frequently do not coincide with the ideal well locations associated with the flood pattern which is used. A question of importance is. "How much oil is lost because of this?". This is the problem treated in this paper. It is assumed that each pattern in a five-spot flood has the proper number of production and injection wells but that the wells may be off-pattern. The distance by which a well is off-patern is given in terms of the pattern spacing, defined as the length of one side of a five-spot. The influence of off-pattern wells on the performance of a five-spot flood has been investigated mathematically for unit mobility ratio. All production and injection rates are assumed to remain equal. The loss in recovery resulting from pattern irregularitie: has been calculated at a cumulative water injection corresponding to a 98 per cent water-cut in a regular five-spot. Displacing a single production or injection well by about one-third the pattern distance reduces the total field recovery by 4.4/N per cent of the recovery obtainable with regular spacing, where N is the number of five-spots in the flood (assumed large). When off-pattern wells are separated by at least one regular five-spot and are displaced by less than one-third the pattern distance, the total loss in recovery in a large-scale flood is, at most, 1 per cent of the recovery obtainable from that field with a regular five-spot pattern. However, much higher losses may occur if the off-pattern wells are not separated by a regular five-spot, if the rates are imbalanced, or if the mobility of the injected water is higher than that of the crude. Although the loss in oil recovery due to pattern irregularity often may be low, the off-pattern case will always require injection of more water to reach the same final water-cut. The volume of water injected per unit volume of oil displaced has been calculated for each five-spot adjacent to a five-spot containing a single 08-pattern well. The cumulative water injection at the time of 98 per cent water-cut may be increased by as much as 37 per cent when a production well is displaced one-third the pattern distance toward an injection well. The existence of differences between actual subsurface locations and the idealized well patterns used in secondary recovery operations is well recognized. However, the methods commonly used for predicting waterflood behavior ignore such differences, even when subsurface locations are known. The reason for this is simple. The idealized well patterns are highly symmetrical and, as such, the production history of the pattern (and even of the field) can be determined from that of a few production wells—usually from only one. Consideration of the many irregular well spacings found in a specific field would entail a large amount of work, and then the results would be applicable only to that specific field. In an attempt to provide an answer to the more general problem, this paper considers a water flood having an idealized well pattern, except for one well which is not at its regular location. When several off-pattern wells are present, their contributions are assumed to be additive unless the off-pattern wells are close to each other. Only four positions of the off-center well (at x, y. of Fig. 1) are considered in this paper. These are shown

By J. R. Kyte, V. O. Naumann, C. C. Mattax

Imbibition and waterflooding tests at reservoir conditions show that conventional tests at room conditions sometimes lead to recovery estimates that are too low. Core samples which are of intermediate wettability at normal test conditions may be strongly water-wet at reservoir conditions. This was found to be true for samples from two of three reservoirs investigated. For the other reservoir, less difference was observed between the wettability of samples at normal test conditions and at reservoir conditions. Samples which became more water-wet at reservoir temperature and pressure also water flooded more efficiently at reservoir conditions. These results show that core analysis tests on intermediate wettability samples should be performed at reservoir conditions to obtain representative data. INTRODUCTION Throughout the oil industry, core analysis tests normally are performed using refined oils and at room temperature and pressure. This procedure has always been subject to question because the possibility is generally recognized that the reservoir environment may influence displacement behavior. Burkhardt, et al,' have performed waterflood tests at reservoir conditions on East Texas cores. They found little difference between results obtained at reservoir conditions and results of conventional flooding tests at room conditions. As they pointed out, however, this situation might not be general since they obtained data on only one rock and crude-oil system. This paper presents data which demonstrate that it is sometimes necessary to perform core analysis tests in a simulated reservoir environment to properly represent the reservoir wettability and displacement characteristics. The work is specifically concerned with reservoirs which were thought to be of intermediate wettability on the basis of conventional test results at room conditions, i.e., core samples imbibed only small amounts of either oil or water and exhibited no decided preferential wettability characteristics. Tests were performed on cores from three such reservoirs to determine their behavior in the reservoir environment. lmbibition and waterflood tests were used to demonstrate the effect of reservoir environment on reservoir rock wettability and displacement characteristics. Results of tests performed at reservoir conditions were compared with results from conventional tests. Imbibition tests were performed on samples from all three reservoirs. Waterflood tests were performed on samples from only one reservoir. Fluid properties for the three reservoirs studied are shown in Table 1. Tests at reservoir conditions were performed using bottom-hole crude oil at reservoir temperature and pressure. The aqueous phase used in these tests was a synthetic brine approximating the reservoir brine composition. Conventional tests were performed at room conditions with refined oils and synthetic reservoir brine. The oil-to-water viscosity ratios were adjusted to the reservoir values. The cores used in all the tests were taken from the reservoirs using either brine alone or a bentonite-car-boxymethylcellulose mud without additives as coring fluids. Previous tests show that these fluids have no significant effect on the wettability of reservoir rock samples.1 The cores were preserved in brine at the wellsite to prevent wettability changes during shipment and storage. Wellsite imbibition tests have been performed on fresh samples from several reservoirs (including Reservoir A) and compared with later laboratory imbibition tests on samples preserved in brine. The comparison showed that the wettability of the preserved cores was the same as that of fresh, surfaced cores at the wellsite.2 IMBIBITION TESTS Conventional imbibition tests were performed on preserved cores using the apparatus and procedures described by Bobek, et al.2 A sketch of the apparatus for imbibition tests at reservoir conditions is shown in Fig. 1. The preserved samples were first centrifuged to expel excess water and then placed in a stainless-steel container which served

By W. R. Strickler, P. Hazebroek, M. Prats

The pressure and production behavior of a homogeneous cylindrical reservoir producing a single fluid through a centrally located vertical fracture of limited lateral extent was determined by using mathematical methods to solve the appropriate differential equation. It is assumed that there is no pressure drop within the fracture — that is, that the fracture capacity is infinite. It was found that the production-rate decline of such a reservoir is constant (except for very early times) when the flowing bottom-hole pressure remains constant. The production-rate decline increases as the fracture length increases. Thus, the lateral extent of fractures can be determined from the production-rate declines before and after fracturing or from the decline rate after fracturing when the properties of the formation and fluids are known. The production behavior over most of the productive life of such a fractured reservoir can be represented by an equivalent radial-flow reservoir of equal volume. The effective well radius of this equivalent reservoir is equal to one-fourth the total fracture length (within 7 per cent); the outer radius of this equivalent reservoir is very nearly equal (within 3.5 per cent) to that of the drainage radius of the fractured well. The effective well radius of a reservoir producing at semisteady state is also very nearly equal to one-fourth the total fracture length. It thus appears that the behavior of vertically fractured reservoirs can be interpreted in terms of simple radial-flow reservoirs of large wellbore. INTRODUCTION An earlier report1 has considered the effect of a vertical fracture on a reservoir producing an incompressible fluid. That investigation of the fractured reservoir producing an incompressible fluid was started because of its simplicity. Thus, pertinent behavior of fractured reservoirs was obtained at an early date, while experience was being gained of value in the solution of more complicated fracture problems. One of these more complicated problems, and the one discussed in this report, considers the effect of a compressible fluid (instead of incompressible fluids) on the production behavior of a fractured reservoir. In the incompressible-fluid work mentioned, it was shown that the production rate after fracturing could be described exactly by an effective well radius equal to one-fourth the fracture length whenever the pressure drop in the fracture was negligible. Because of the simplification in interpretation, it is a matter of much interest to determine whether the production behavior of reservoirs producing a compressible liquid could be described in terms of an effective well radius which remains essentially constant over the producing life of the field. The details of the mathematical investigation are given in the Appendixes. IDEALIZATION AND DESCRIPTION OF THE FRACTURED SYSTEM It is assumed that a horizontal oil-producing layer of constant thickness and of uniform porosity and permeability is bounded above and below by impermeable strata. The reservoir has an impermeable circular cylindrical outer boundary of radius re. The fracture system is represented by a single, plane, vertical fracture of limited radial extent, bounded by the impermeable matrix above and below the producing layer (reservoir). It is assumed that there is no pressure drop in the fracture due to fluid flow. Fig. 1 indicates the general three-dimensional geometry of the fractured reservoir just described. When gravity effects are neglected, the flow behavior in the reservoir is independent of the vertical position in the oil sand. Thus, the flow behavior in the fractured reservoir is described by the two-dimensional flow behavior in a horizontal cross-section of the reservoir, such as the one shown in Fig. 2. Production is due to the expan-

By M. Prats

The effect of a sand-filled vertical fracture of limited radial extent and finite capacity (fracture capacity is the product of the permeability and width of the fracture) on the flow behavior of a cylindrical reservoir producing an incompressible fluid through a centrally located well has been investigated mathematically. The shape of the lines of equal pressure near the fracture is essentially independent of the size of the reservoir, provided that the field radius is of the order of the fracture length or larger. For reasonable values of production rates and of fluid, reservoir and fracture properties, the total pressure drop between the end of the fracture and the well is gelrerally negligible compared with the pressure drop in the reservoir. With regard to production response, the effect of vertical fractures can be represented by the production response of an equivalent or effective well radius. For a high-capacity fracture, the effective well radius is a quarter of the total fracture length, decreasing with the fracture capacity. When invasion effects are simulated by decreasing the width of the damaged zone with distance from the well, the effect of formation damage around a fracture on the production response is not so seriolls as indicated by the literature. This suggests that frac fluids with a conventional filter-loss response are better than high-spurt-loss frac fluids, provided the effective permeability of the damaged zone is the same. INTRODUCTION This paper considers the effect of the fracture capacity, as well as the formation damage which can result from fracture treatments, on the productive capacity of vertically fractured wells. Other publications, notably those of van Poollen,l,2 consider these same effects. In addition to providing more general results for vertical fractures than are available from the liter- ature, the present paper gives the equivalent well radius of fractures having different lengths and capacities and, also, includes pressure distributions in and around the fractures. The effect of a damaged zone around a fracture on the production response was not found to be so great as that reported by van Poollen. 2 This difference probably stems from the fact that we consider a damaged zone which is widest (but is still small) near the well and thins out toward the extremities of the fracture, whereas van Poollen considers a damaged zone having a uniform width for the entire fracture length. Simple, but adequate, equations which describe the effect of these variables on production response are presented (in Appendixes A and B). Thus, results can easily be extended to values of the variables not specifically considered here. IDEALIZATION AND DESCRIPTION OF THE FRACTURED SYSTEM It is assumed that a horizontal oil-producing layer of constant thickness and of uniform porosity and permeability is bounded above and below by impermeable strata. The liquid is incompressible. It is assumed that, after the well is fractured, the cylindrical outer "boundary" is at a uniform potential, provided that it is not too near the fracture. The fracture system is represented by a single, plane, vertical fracture of limited radial extent, bounded by the impermeable matrix above and below the producing layer (reservoir).3 Fig. 1 indicates the general three-dimensional geometry of the fractured reservoir described above. If gravity effects are neglected, the flaw behavior in the reservoir will be independent of the vertical position ill the oil sand. This means that the flow behavior in the fractured reservoir can be described by the two-dimensional flow behavior shown in the horizontal cross section of the reservoir, Fig. 2. PARAMETERS The effect of a fracture on the pressure distribution in the fracture itself and in an otherwise undamaged (no skin) reservoir indicates that the pressure distribution is a function of three parameters.

By M. D. Arnold, H. J. Gonzalez, P. B. Crawford

A method is presented for estimating the effective directional permeability ratio and the direction of maximum and minimum permeabilities in anisotropic oil reservoirs. The method is based on the principle that production from a well in an anisotropic reservoir results in elliptical isopo-tentials about the well, rather than circular. Bottom-hole pressure data from three observation wells surrounding a producing well are required to apply the method. The method involves fitting field pressure data to a set of general charts of isopotentials and making a few simple calculations until a solution is found. The method is based on a steady-state equation for homogeneorrs fluid pow. In addition to the method, a brief discussion of the theory underlying it is presented. INTRODUCTION The existence of a different permeability in one direction than another in oil reservoirs has been mentioned in several papers. Hutchinson' reported laboratory tests on 10 limestone cores and pointed out that one-half of them showed significant, preferential, directional permeability ratios, the average being about 16:1. Johnson and Hughesz reported a permeability trend in the Bradford field in the northeast-southwest direction with flow being 25 to 30 per cent greater in that direction. Barfield, Jordan and Moore -eported an effective permeability ratio of 144:1 in the Spraberry. Crawford and Landrum4 showed that sweep efficiencies could often vary by a factor of two to four, and sometimes considerably more, due to variations in flooding direction and patterns in anisotropic media. These findings indicate that the poss'bility of anisotropy may be worthy of consideration in the development of an oil field. In considering this, it should first be determined if anisotropy exists. If it does, the direction of the maximum and minimum permeabilities and the ratio of their magnitudes are quantities which can be of value in planning the most efficient well-spacing patterns. Past methods of determining these quantities have included analysis of oriented cores and analysis of flooding performance of pilot injection patterns. In recent work, Elkins and Skov5 resented an analysis of the pressure behavior in the Spraberry which accounted for anisotropic permeability. This work was based on the transient pres- sure distribution in a porous and permeable medium, with the solution expressed as an exponential integral function involving rock and fluid properties. The purpose of this study is to provide a method, based on steady-state equations, of estimating the direction and relative magnitude of permeabilities in an oil reservoir from field pressure data and well locations only. The method presented is based on work by Muskat6 which shows that Laplace's equation represents the steady-state pressure distribution for homogeneous fluid flow in homogeneous, anisotropic media if the co-ordinates of the system are shrunk or expanded by replacing x with it is desirable that data be obtained early in the history of a field because knowledge of an anisotropic condition would allow new wells to be spaced in such a manner that reservoir development and subsequent secondary recovery programs could be planned more efficiently. THEORETICAL CONSIDERATIONS A brief discussion of the theoretical basis on which the graphical solution was developed is presented in this section. Muskat's two-dimensional6 olution for the pressure distribution in an homogeneous, anisotropic medium with an homogeneous fluid flowing can be algebraically manipulated to show that the isobaric lines are perfect ellipses. The ratio of the major axis to the minor axis, a/b, is related to the permeability ratio, k,/k,, as follows. alb = dk,/k,--...........(1) It can also be shown that the pressure varies linearly with the logarithm of the radial distance from the producing well. However, the gradient along any ray is a function of the orientation of that ray, and a ..xiable is present when anisotropy exists which cancels out for a radial (isotropic) system. For a system such as that described, a dimensionless pressure-drop ratio was developed which is completely independent of the actual magnitude of the pressures. This was done by arranging Muskat's solution in such a way that aIl variables cancelled out except k,/k, and well positions. However, this solution depends on having a co-ordinate system with axes coinciding with the major and minor axes of the elliptical isobars. Thus, it was necessary to introduce a co-ordinate system rotation factor. The two unknown variables are then k,/k. and 0, and the two measured dimensionless pressure-drop ratios are related to the unknown variables as follows.

By R. A. Greenkorn

This project originated in a practical problem—we needed five tracers that could be used together to locate flow paths in a pilot flood. While tracers for subsurface liquids have been used since the turn of the century,1-15 none of those reported in the literature appeared to be either consistent or quantitative enough for our purposes. Most were used in field systems without controlled experiments to determine the accuracy and precision of analysis, and many were tracers that could not be used collectively. The ideal tracer, of course, would follow the fiuid of interest exactly, traveling at the same velocity as the fluid front. But the ideal is impractical to attain because adsorption-desorption effects cause the tracer to lag behind the front; these effects, plus diffusion-dispersion effects, cause the tracer front to spread more than the fluid front. Thus, our objective was not to locate a tracer that would be ideal for all circumstances but rather, to find one that would approximately follow the fluid, or one that under controlled conditions could be corrected to calculate the movement of the fluid front. We considered tracers satisfactory—(1) if they were easy to analyze; (2) if their breakthrough-elution curves were not too different from those for the chloride ion, a tracer believed to follow the fluid front closely; and (3) if we could calculate from the curves a material balance, at 1.25-PV (pore volume) injected, within 5 per cent of that calculated from chloride curves. Of a possible 35 materials, we selected 13 tracers that could be quickly and easily identified and whose analysis was claimed to be accurate within 5 per cent. Only one of these, tritiated water, was a radioactive tracer. Radioactive tracers are easy to detect even in small quantities, but they require special handling and special equipment. Also, those that can be used together are limited because special equipment is required to separate emissions from the various tracers. Two of the original 13 tracers were eliminated in static tests to determine how accurately and precisely they could be analyzed, and to check on gross adsorption. The remaining 11 were flowed through a 9-ft linear sandstone model, and breakthrough-elution curves were obtained. Finally, three tracers were field-tested as breakthrough tracers. These tests are described in the following sections. The 13 tracers considered in these experiments were EDTA (ethylene diamine tetra acetic acid), fluo-rescein, picric acid, salicylic acid and ammonium, boron (as borate), bromide, dichromate, iodide, nitrate and thiocyanate ions, plus chloride ion and tritiated water. All but the chloride ion and tritiated water were subjected to static sand tests to eliminate the tracers that could not be analyzed quickly and accurately and to eliminate those that showed excessive adsorption. All but two of the tracers, EDTA and salicylic acid, qualified for flow tests on this basis. The procedures used in the static tests (see Appendix A for analytical details) were as follows. Each tracer (in the amount shown in Table 1) was dissolved in 1,000 ml of water. Then, 800 ml of this solution was added to 500 gm of sand, and the remaining 200 ml was reserved as a control standard. The sand-tracer mixture was agitated once a day over a 10-week period, and during this period we analyzed five duplicate samples from this mixture, along with five duplicate samples from the control solution. From the control-sample analyses, we constructed a control chart on which the sand-tracer analysis results were plotted. The control chart, and its use to determine accuracy and precision of analytical results, as well as the amount of adsorption, is described in Appendix B. The results of the static tests are summarized in Table 1, which lists the concentration c (concentration at time t) divided by c, (initial concentration). If there was no adsorption, c/c, = 1. The acceptable deviation from this value varied from tracer to tracer, depending upon the limits of analytical precision established for the different tracers. The best results were obtained with boron, bromide and dichromate, with all values falling within limits of accuracy and precision. Ammonium, iodide, nitrate and picric acid also were satisfactory, although the ammonium results were erratic. Flourescein and thiocyanate were unsatisfactory ini-

By H. S. Price, J. E. Warren

Techniques for studying the performance characteristics of heterogeneous reservoirs have been developed. The effect of permeability variation on both the steady-state and the transient flow of a single fluid has been investigated. Limited comparisons with field and laboratory data have been made. The physical models studied consist of random three-dimensional arrays of homogeneous porous blocks. The permeabilities of the individual elements are assigned according to a specific distribution function; uniform anisotropy is introduced by varying the relative dimensions of the blocks. A particular model is perturbed simply by re-arranging its elements at random. The behavior of a physical model is determined by digitally solving its numerical analogue. Based on computational experiments, subject to the restrictions implied by the assumptions that were made, the following general conclusions have been drawn. 1. The most probable behavior of a heterogeneous system approaches that of a homogeneous system with a permeability equal to the geometric mean of the individual permeabilities. 2. The effects of flow geometry and anisotropy on the most probable value of the effective permeability of a heterogeneous medium are finite but not significant. 3. The permeability determined from a pressure build-up curve lor a heterogeneous reservoir gives a reasonable value for the effective permeability of the drainage area. 4. A qualitative measure of the degree of heterogeneity and its spatial configuration are obtained from a comparative study of core analysis and pressure build-up data. It has been indicated that these conclusions are predicated on the assumption that the core analysis and the pressure build-up data represent true reservoir characteristics. Common sources of error in these data have been discussed. One of the most significant problems of reservoir engineering is that of determining the nature and the disposition of the heterogeneities that inevitably occur in petroliferous formations. It is with this problem that this paper is concerned. Extensive research, both theoretical and experimental, has provided reasonable descriptions of the physical processes that are associated with primary and secondaryrecovery mechanisms. Similar progress in the fie1d of numerical analysis and the evolution of truly high-speed digital computers have made it feasible to calculate the behavior of these mechanisms under almost any environmental conditions. It appears to be logical, then, to conclude that the mechanistic performance of any reservoir can be predicted with some confidence. Unfortunate1y, a necessary condition for the validity of this conclusion is that the formation be adequately described in a macroscopic sense. This condition cannot be satisfied ill general; therefore, a frustrating situation currently exists — it is possible to compute the effect of reservoir heterogeneity on behavior, but it is not possible to specify the heterogeneity itself. In many cases, the predicted behavior of a project is so completely dominated by irregularities in the formation's physical properties that the gratuitous assumptior, of a particular form for the variation can reduce the solution of the problem to a mere tautological exercise. While all porous media are microscopically heterogeneous, only macroscopic fluctuations need be considered since the fundamental concepts of reservoir mechanics are based on macroscopic quantities. For the simplest case in which only one physical property varies from point to point, the degree of heterogeneity that is apparent depends on three distinct factors, i.e., the nature of the property's variation (frequency distribution), its disposition (spatial distribution) and the inherent stability of the mechanism being studied. Consequently, any attempt to devise an unambiguous scale of heterogeneity must take these factors into account. Since this investigation is intended to be exploratory, it will be restricted to single-phase flow because of the simplification implied; both steady-state and unsteady-state behavior will be considered. Furthermore, to keep the number of parameters

By J. E. Warren

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

By J. E. Warren

To clarify the specific questions raised by J. E. Warren's discussion, the following remarks are offered. 1. It is correct that stratification is assumed in the application of the statistical zonation technique, but this does not constitute a limitation. If, in fact, the reservoir is not stratified the zonation technique will show it. The reservoir is then considered as one zone with a particular distribution of permeabilities. 2. The statistical zonation technique was intended as wells may not be an adequate measure of the lateral permeability changes; e.g., if the data presented in Table A-4 of the subject paper are evaluated on the basis of two crossed classifications with interaction, the variation between zones is significant with a probability greater than .99; however, the variation betwen wells is also significant with a probability greater than .87. It is apparent that the technique suggested by the author can be generalized to provide a powerful method for describing petroleum reservoirs. This generalization can be achieved by treating the core data as a two-way classification to determine the between-zones and the between-wells variations; then, the build-up permeabilities can be analyzed, perhaps on at least-squares basis, to establish the probable distribution of the permeabilities within the zones associated with the individual wells. an additional tool for the reservoir engineer to use together with other techniques and all available information. If pressure build-up permeabilities are available, they should certainly be included. Pressure build-up tests were not available for the reservoir considered in the paper. 3. The proposed correlation of zones between adjacent wells was the best of the techniques that I examined. It is not so much a "measure" as an indication of lateral permeability changes, regardless of whether they are seemingly random or form an obvious trend.

By H. D. Outmans

The author applies the theory of hydrodynamic stability to fluid flow in porous media and concludes that the displacement of miscible liquids is stable if the stability coefficient, as defined by Eq. 19.2 or Eq. 27.0, is positive. The conclusion is invalid, however, since it is based on a faulty argument. To demonstrate this, the reader is referred to Eqs. 4.1 through 4.5 of the paper which describe, in differential form, the first-order disturbances c, u, v, w and p. In these same equations, cx, cz, gl, g3 and are functions of the stable concentration c. This concentration c is known and depends on x, z and t (the author assumes cy = 0). It follows that cx, cz, gl, g3 and µ are also functions of x, z and t. With this in mind, we turn to Eqs. 8.1 through 8.5 which were derived from Eqs. 4.1 through 4.5 by substituting Eq. 6.0. The significance of this substitution is that the author thereby limits the solutions of Eqs. 4.1 through 4.5 to equations of the form of Eq. 6.0. Eqs. 8.1 through 8.5 represent an infinite linear system with five unknowns: c, u, v, w and p. The coefficients cx, cz, gl, g3 and µ in this system are known at each point and at each instant; the coefficients containing derivatives of f are unknown. If this infinite linear system in c, u, v, w and p is to have a nontrivial solution, then it is indeed necessary (as the author states) that the determinant of the coefficient matrix of Eqs. 8.1 through 8.5, i.e., ( A 1, be equal to zero. This condition, however, is not sufficient. A non-trivial solution exists if, and only if, the rank of the coefficient matrix of the entire infinite system is smaller than five. In other words. all minors of order five must vanish and only if all the coefficients in Eqs. 8.1 through 8.5 are constant is this equivalent to the condition |A| =0. Constant coefficients means that gl, g3 and µ, for instance, are constant; and, since they depend on c it follows that c must be constant. We conclude that the condition I A I = 0, on which the author bases his calculations, is necessary and sufficient for a nontrivial solution only if the concentration c is constant. In other words, the author does not investigate the stability of a miscible displacement but, rather, only the stability of the flow of a homogeneous liquid. Using the initial condition and the fact that the coefficients are constant, we find

By J. M. Smith, D. Kunii, P. Adivarahan

Effective thermal conductivities were measured for seven samples of porous rocks through which gases or aqueous salt solution were flowing, parallel and countercurrent to the flow of heat. The results are analyzed in terms of the contributions of the solid and fluid phases to the total energy transfer. The solid contribution for the sandstone samples studied was found to be a Junction of the porosity and an effective thermal conductivity for the solid phase. The fluid contribution was a function of the Peclet number. Based upon the equations presented, the effective conductivity of the system can be estimated from the properties of the fluid, the flow conditions, and the porosity and pore-size distribution of the rock. INTRODUCTION Heat-transfer characteristics of porous media containing fluids are needed for the design and evaluation of thermal methods of petroleum production. The study reported here considers the problem of countercurrent flow of fluid and heat in sandstone rocks, and is a continuation of previous studies concerned with parallel and radial flow of energy and fluid in both rocks517 and beds of fine particles.6,10 In addition to these earlier works, Asaad,1 Kimura,4 Preston,8 Zierfuss and vlietll and waddams9 have reported effective conductivities for systems filled with stagnant fluids. There appears to be no published data on rocks through which fluids are flowing. The purpose of this study was to present such information and, particularly, to investigate the effects of properties of the rock and the flow rate on the results. The rock samples were naturally occurring sandstones from different locations and were similar to petroleum-bearing formations. The properties of the seven samples showing the range in porosities and permeabilities are given in Table 1. The mean pore sizes were calculated from mercury penetration data by averaging the radius with respect to pore volume. The details of this method and comparison with other procedures for estimating 7 are given elsewhere. Four fluids, nitrogen, carbon dioxide, helium and 10 weight per cent sodium-chloride solution were used with each sample. For Bartlesville sandstone, BA-5, runs were made at mean pressures from 15 to 250 psig using nitrogen. The other measurements were carried out at a mean pressure of 100 psig. The maximum temperature range eacountered was 70° to 300°F. The flow rate of fluid ranged from about 1.0 to 130 Ib/(hr)(sq ft), and this corresponded to a modified Reynolds-number range of 5 x 10-4 to 1.0. The variation in Prandtl number was 0.70 to 4.1. EXPERIMENTAL WORK The equipment consisted of apparatus to supply a constant flow of gas, or salt solution, at constant pressure to the bottom of a vertical test section, and metering and flow-regulation apparatus after the test section. The flow of salt solution was

By S. W. Churchill

A digital computer was used to obtain an exact numerical solution for the transient behavior of the insulation and earth adjacent to an isothermal, submerged flat surface for a single set of parametric values. Comparison of the computed results with analytical solutions for limiting conditions revealed that a complete and general solution for all parametric values could be constructed from these limiting solutions. Complete and general solutions for insulated spheres and cylinders can also be constructed from limiting solutions. The procedure is illustrated in part for an insulated spherical tank. The heat flow was found to fall rapidly to a pseudo-steady state; after some time, it then decreased slowly to zero for a flat plate and cylinder, and to a low steady-state rate for a sphere. The accumulative heat flow during the initial falling-rate period may be a significant fraction of the heat flow during the entire first year. INTRODUCTION Underground storage of liquefied natural gas and liquefied petroleum gas has received considerable recent attention. The rate of heat flow from the earth to the storage cavity and the resulting temperature field in the earth are important factors in the technical and economic evaluation of such storage facilities. The objective of this paper is to indicate how complete solutions can be developed for the transient flux and temperature field for various geometrical configurations. The representative properties and dimensions, and the resulting parameters utilized in the illustrative results, are indicated in Table 1. The results presented herein are for dry earth. The effect of the latent heat of solidification of water in the soiI will be described in a subsequent paper. The temperature field in the insulation and earth is determined by energy balances, boundary conditions and initial conditions. The physical problem can be described mathematically with virtually no idealizations insofar as physical properties are known. It appears possible to solve the equations by numerical integration on a high-speed digital computer for any geometrical configuration and conditions. For complex situations, however, the computations are expensive and the results are highly specific. Analytical solutions have been developed for a few simple but important geometrical configurations and conditions, including one-dimensional heat transfer from (or to) earth at an initially uniform temperature to (or from) flat plates, spheres and circular cylinders. The solution for an insulated flat plate has also been derived but is in the form of a slowly converging series involving taklated functions. It was planned to use a computing machine to evaluate this series for a number of representative conditions. However, upon examination of the results of preliminary computations it was discovered that a complete, general and accurate solution could be developed by interpolation between several much simpler solutions for limiting conditions. This technique then was used to develop a a complete solution for an insulated sphere. The equations presented herein are derived in Carslaw and Jaegerl and other books on applied mathematics, or they are simple extensions of these previous results. Hence, all derivations are omitted. Emphasis is instead placed on the results and their

By N. Marusov, D. K. Lowe, J. C. Karp

This paper considers, from an engineering viewpoint, several factors involved in creating, designing and locating horizontal barriers for controlling water coning. This is an effort to consolidate new concepts with previous information so that a reasonable selection of barrier materials, dimensions and vertical position can be made. Coning theories previously developed are briefly reviewed and an effort i,s made to reduce the results of coning-theory calculations to a point where routine calculations can be made with a desk calculator. It is expected that these simplified calculations will give usable predictions of the amounts of improvement attainable with barriers of various dimensions. Apparatus and procedures used for testing the suitability of commercial cements are described and test results are presented. INTRODUCTION Just about as soon as the phenomenon of water coning was recognized as an oilfield problem, there were suggestions that the production of water coning could be controlled or completely suppressed by means of horizontal barriers. It was also suggested that natural barriers such as shale streaks were helpful in restricting bottom-water production. The implication was that wells not having the benefit of continuous shale streaks to suppress bottom waters should, in some way, be supplied with an artificial barrier. With regard to the selection of barrier materials, there are many references which give the merits of liquid barriers.',' These liquids include surface-active agents," recipitates' and emu1sions. There is no real need to review the merits of these materials again. This paper deals with the use of solid barriers which could be made from various cements. The two major restrictions placed on these barrier cements are that (1) they must be commercially available in large quantities, and (2) they must be applicable by means of standard tools and equipment with techniques that have been fully developed and are in general use at the present time. A schematic drawing of a horizontal barrier in a reservoir is shown in Fig. 1. Two maximum stable water cones are illustrated-—-one for the wellbore diameter and the other for a barrier. A maximum stable cone is one that has attained its maximum volume at the critical production rate and is on the threshold of producing water. The volume displaced by each water cone and the rise in water table are indications of the amount of water-free oil produced. Corrections for porosity, connate water and residual oil, of course, must be applied. The water-oil interface which defines a maximum stable cone is called a free surface. It is a surface over which the pressure is constant. This free surface has the shape of a limiting streamline at and below which there is no movement or transport. The shape of the water cone describes the pressure drop in the reservoir. The effect of a completely impermeable barrier on the cone shape is essentially the same as extending the wellbore out to the barrier radius. The placement of a horizontal barrier requires that a horizontal fracture be created using a single-point entry technique. Two methods may be employed for filling this fracture. One method consists of propping the fracture before filling with cement. The other consists of fracturing with fracture fluid, then immediately following with cement

By J. R. Kyte, C. C. Mattax

Previous workers have developed differential equations to describe oil displacement by water imbibition, but have not explicitly defined the relationship between recovery behavior for a single reservoir matrix block and its size. In the present work, imbibition theory is extended to show that the time required to recover a given fraction of the oil from a matrix block is proportional to the square of the distance between fractures. Using this relationship, recovery behavior for a large reservoir matrix block is predicted from an imbibition test on a small reservoir core sample. The prediction is then extended to analyze recovery behavior for fractured- matrix, water- drive reservoirs in which imbibition is the dominant oil-producing mechanism. Experimental data are presented to support the basic imbibition theory relating matrix block size, fluid viscosity level and permeability to recovery behavior. INTRODUCTION Imbibition has long been recognized as an important factor in recovering oil from water-wet, fractured-matrix reservoirs subjected to water flood or water drive. l-3 Recently, two approaches have been published which might be used to predict imbibition oil-recovery behavior for reservoir-sized matrix blocks. Graham and Richardson4 used a synthetic model to scale a single element of a fractured-matrix reservoir. lair,' on the other hand, used numerical techniques to solve the differential equations describing imbibition in linear and radial systems. This latter method requires auxiliary experimental data in the form of capillary pressure and relative permeability functions. These two approaches, i.e., synthetic models and numerical techniques, have been used to study a variety of reservoir fluid-flow problems. One pur- pose of this work is to present, with experimental verification, a third method for predicting imbibition oil recovery for large reservoir matrix blocks. This method uses scaled imbibition tests on small reservoir core samples to predict field performance. The imbibition tests are easier to perform than the capillary pressure and relative permeability tests required to apply the numerical method. Furthermore, when suitably preserved reservoir-rock samples are used, the properties of the laboratory system are the same as those of the field. This offers an important advantage over the use of synthetic models because there is usually some question as to how accurately reservoir-rock properties can be duplicated in such models. Based on the recovery behavior for a unit matrix block, an analysis is presented to predict oil recovery for a fractured, water-drive reservoir made up of many such unit blocks. In the analysis, it is assumed that the flow resistance and the volume of the reservoir fracture system are negligible compared with that of the porous matrix. These assumptions are generally consistent with observed characteristics of many fractured-matrix reservoirs, and have been employed in previous studies.2,3,6-8 further assumed that the effect of gravity on flow in the matrix blocks is negligible. On first thought, the latter assumption might appear to seriously limit application of the method. However, in a fractured reservoir, the effect of gravity on flow in a matrix block will be restricted by the height of the block. Furthermore, matrix permeabilities are often very low (10 md or less) in such reservoirs. This means that capillary or imbibition forces will be large, thus tending to minimize the relative importance of gravity. For these reasons, imbibition should be the dominant oil-recovery mechanism in many fractured-matrix, water-drive reservoirs. The predictive method presented in this report is applicable to such reservoirs. SCALING IMBIBITION THEORY The "scaling laws" applicable to the operation.

By J. C. Deppe

A method is presented for calculating approximate injection rates in secondary recovery operations. The method can be applied to cases of unequal fluid mobilities, irregular well patterns and boundary patterns. The steady-state pressure distributions for the four flood patterns reported by Muskat and for five additional patterns reveal that most of the difference in pressure between the injection and producing wells occurs in regions around the wells which can adequately be described as regions of radial flow. This leads to a method of calculating injectivity by approximating the flood pattern with radial flow elements (or a combination of radial- and linear-flow elements for some patterns such as the direct line drive). Irregular and boundary patterns can also be approximated by radial and linear elements. Each of these elements can be described by radial-and linear-flow equations and the results combined as series flow resistances to give an approximate equation for the initial injection rate. The mobility ratio does not affect the initial rate; therefore, if the well pattern is one of those regular patterns for which theoretical rate equations have been derived for unity mobility ratio, the approximate initial rate equation can be improved by adjusting it to match the theoretical equation. The available theoretical rate equations are listed, including five new cases. As the flood progresses, the injectivity changes because in general the flood front will divide the pattern into areas of different fluid mobilities. Simple shapes can be assumed for the flood front so that both areas can be divided into radial- and linear-flow elements. Radial- and linear-flow equations are applied to these elements to account for the change in flow resistance behind the front. To calculate injection rates after breakthrough, it is necessary to know the sweep efficiency at breakthrough. Areal sweep data are available in the literature for a number of patterns, and sufficiently accurate breakthrough sweep efficiency can be estimated from these data if it is not otherwise available. Again, simple shapes for the flood front can be assumed after breakthrough, and injection rates can be calculated for the remainder of the flood. When areal sweep data are available, these data can be used to cbeck. the calculated injection rates after breakthrough. Injection rates calculated by this approximate method compare favorably with available model data. INTRODUCTION When the fluid mobilities in the swept and un swept regions are equal, the injectivity will not change as the flood front advances; and for regular patterns it can be calculated by mathematical formu1as.l When the fluid mobilities in the swept and unswept regions are not equal, the injectivity will increase or decrease as the flood front advances. In this case, the injectivity has not been calculated by analytical means for any practical well pattern; and, furthermore, scale model and analog4,6 results have been published only for the five-spot pattern. The main object of this paper is to present an approximace method for calculating injectivity for the case of unequal mobilities. The method can be applied to regular, irregular and boundary patterns. Before the approximate method is discussed, the analytical formulas for equal mobilities will be summarized, and the analytical solution for radial flow with unequal mobilities will be used to show how the injectivity changes as the flood progresses. EQUAL MOBILITIES, REGULAR PATTERNS In a flooding operation the injectivity, the rate at which fluid can be injected per unit difference in pressure between injection and producing wells, depends on basic physical properties and, in addition, on three variables to be treated here—the area of the swept region, the fluid mobilities in the swept and unswept regions, and the well geometry (pattern,

By E. B. Brauer, B. F. Pennington, E. W. Hough, G. L. Stegemeier

Interfacial tension divided by the difference in density between the liquid and the vapor phases was determined experimentally by the pendant drop method on several isotherms in the two phase region below the critical point for the methane-normal decane system. The density difference data of Sage and Lacey was used in the calculation of interfacial tension. Both interfacial tension and and interfacial tension divided by density difference were found to vanish at the critical point. Interfacial tensions of less than one dyne/centimeter were observed as far as 1,000 pounds per square inch below the critical pressure. EXPERIMENTAL PROCEDURE The interfacial tension divided by the density difference for the methane-normal decane system was determined at the 100°, 130°, 160' and 190°F isotherms, from pressures of about 1,000 psi to the critical pressure, which is more than 5,000 psi for these isotherms. Particular emphasis was placed upon the investigation at pressures slightly below the critical pressure where the interfacial tension is less than 0.5 dyne/cm. Volumetric properties in the two-phase region, including the critical pressures and temperatures, were taken from the work of Sage and Lacey.1 The experimental pendant-drop technique used for the determination of interfacial tensions at high pressures incorporated the ideas of Michaels and Hauser,2 Hough, et a1,3 Walker4 and Heuer.5 In addition, the technique for determination of extremely small interfacial tension by Jennings6 was utilized in the region near the critical points. A detailed description of the apparatus is given in a dissertation by one of the authors.7 Cleaning operations on the stainless-steel sample system included successive washings with chromic acid, tap water and, finally, distilled water. Subsequent cleanings were performed with re-distilled normal pentane, which had an extremely low residue upon evaporation. Specific composition requirements necessitated a fairly precise sample introduction — although, fora two-phase, two-component system, the composition of each phase is completely determined if pressure and temperature are controlled. The normal decane was delivered into the evacuated sample system as a liquid from a burette. The methane was then introduced into the system from a calibrated isothermal container, so that pressure differentials could be used to determine the amount introduced. High pressures were obtained by compressing the sample with a mercury injection pump until the critical pressure was reached for the particular isotherm being studied. Experimental data were then obtained for specific pressures by first decreasing the pressure slightly so that two phases would appear, and then photographing a drop at that pressure. Subsequent photographs were made at increments throughout the pressure range. Calculation of interfacial tension divided by density difference was made from measurements