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By D. M. Kehn

A method has been devloped for chromatographic analysis Of the vapor and liquid phases Of a a system containing methane to components having 20 or more carbon atoms. The method uses a windowed equilibrium cell in which volumetric phase behavior of the system can be observed accurately and from which small samples of gas or liquid can be withdrawn for analysis. Analyses are made using two chromatographs, one for the lighter and one for the heavier components in a sample. Combination of the two analyses yields a detailed analysis of the gas or liquid sample. The complexity of the condensate heavy ends is evident from the chromatograms of these fractions, and the predominance of the paraffin hydrocarborn serves as a useful marker in interpreting the chromatograms. The K-values obtained in this analytical method are presented for a high-pressure condensate system and predict closely the observed volumetric behavior of the system. INTRODUCTION Quantitative analysis of hydrocarbons from natural gas reservoirs is necessary for several reasons—to calculate the amount of sales gas produced, to calculate the amount of natural gasoline produced, to plan a liquid recovery system, or to calculate the potential economic value of a reservoir produced under one or more of several different conditions. Analysis of natural gas fluids produced to the surface consists of identifying and computing the mol fraction of each component of the mixture. Although methane is the predominant component, varying amounts of ethane, propane, butane, pentanes and heavier components are also present. Materials containing up to 30 carbon atoms occur in amounts which decrease with increasing molecular weight. However, the quantities of components in the 20 to 30 carbon atom range are usually so small that their importance is negligible, and they are undetect-able in natural gas by ordinary analytical methods. All the components up to those having 20 carbon atoms may sigsficantly affect phase behavior, however. Commonly, only the methane-through-pentane fraction is analyzed quantitatively for each component, while components heavier than Pentane are lumped and repored as "hexane- plus". Expensive, tedious techniques are required for analysis of this fraction. Consequently the detailed analyses needed for prediction of reservoir behavior are usually undertaken only when major gas fields are being developed. The need for complete analyses of condensate systems is apparent when it is recalled that most gas fields are produced by pressure depletion. As the pressure declines, some of the heavier hydrocarbons are lost as liquids which are in the reservoir. In many instances the amount of liquid in equilibrium with the gas phase at high pressure constitutes only 1 or 2 mol per cent of the total system. Flash calculations generally must predict the actual amount of liquid with an accuracy of a few per cent in order to be useful. This retrograde condensation has been understood for years, but accurate correlation methods to permit quantitative prediction of phase behavior in the retrograde region are not presently available. The increasing importance of natural gas has made accurate prediction of phase behavior and composition of produced natural gas streams an economic necessity. The work reported here was undertaken to provide a rapid, economical method for obtaining the vapor-liquid equilibrium information needed to predict accurately the composition of the fluids produced from a gas reservoir throughout its life. TO develop this method, a pressure cell equipped with windows was designed and built for observing the volumes of liquid and gas present at reservoir pressures and temperatures. Use was made of established chromatographic methods for rapid and detailed analysis of both phases. This paper describes the equipment and techniques developed for obtaining vapor-liquid equilibrium data, presents the results of analyses of a condensate system, and indicates the usefulness of these data in predicting hydrocarbon phase behavior. DESCRIPTION OF EQUIPMENT USED The equipment used in obtaining the required information on phase behavior and the complete analysis of hydrocarbon mixtures through C, will be described first, followed by a discussion of the operation of the equipment. It will be helpful, however, to consider first a brief outline of the technique used. A sample of separator gas and liquid is charged to the windowed cell (see Figs. 1 and 2) where volumetric equilibrium phase behavior at reservoir pressures and temperatures can be determined. Then samples of the coexisting phases are withdrawn. The methane-through-pentane fraction is analyzed with a chromatograph equipped with a hot-wire detector, and the pentane-plus fraction is analyzed with a second chromatograph equipped

Jan 1, 1965

By H. K. van Poollen, H. C. Bixel

A treatment is given of the transient pressure behavior of a well located at the center of a circular region surrounded by a radial discontinuity. On either side of the discontinuity, the values of permeability, viscosity, compressibility and porosity are uniform but may be different from those on the other side of the discontinuity. The results are obtained by solving a pair of finite difference equations. The numerical solution to these equations is obtained using a digital computer. The results are compared with previously published, approximated analytical solutions to the same problem. Solutions, presented graphically, show pressure decline for constant rate fluid production. The range of variables studied include dimensionless time from '0.001 to 100 and storage capacity ratio from 0.001 to 1,000. The well behaves as if it were in an infinite reservoir for dimensionless times less than 0.25. Reservoir properties near the well can be estimated in the usual manner. An overlay technique is used to match an experimental curve with one of the theoretical curves. It is possible to estimate the distance to a discontinuity by substituting the actual time t and the corresponding dimensionless time tD at which a match occurs into the equation a = (0.000264 tk1/tD , µ1 cI)I/2 where kI/ I µI cI is the diffusivity near the well, and may be estimated from data taken at early time. Several buildup curves are computed. These curves show that for early shut-in times, correct values for transmissibility are obtained from conventional analysis. However, erroneous values of static reservoir Pressure are obtained unless data at large shut-in times are taken. INTRODUCTION A mathematical treatment of the transient pressure response of a well located at the center of a region bounded by a circular discontinuity is given. Within a region (Fig. 20) the properties of both the rock and the fluid are considered to be constant on either side of the discontinuity. While these properties are considered constant, they may be different on opposite sides of the discontinuity. A discontinuity of this type could be a fluid-fluid contact or a sudden change in rock characteristics such as thickness, porosity or permeability. Analytical solutions to this problem are available. 1-3 However, they are so involved that they are of little practical use. An approximate solutionl is available but the range of times over which it is valid has not been specified. A numerical solution to a pair of finite difference equations is used to obtain the solution given in this paper. PREVIOUS WORK One of the first solutions to the problem was published by William Hurst.1 Hurst considered unsteady-state flow of fluids through two sands in series with different mobilities in each sand. He used Laplace transforms to obtain a solution for a single well located at the center of the circle enclosing the first of the two sands. In this case, the solution for the pressure change in Region I (Fig. 20) is2

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 F. G. Miller, T. D. Mueller, L. B. Davidson

A mathematical model of reservoir behavior during one cycle of the intermittent steam injection process is presented in this paper. A cycle of the process is considered to be composed of three stages: injection, condensation and production. The model concentrates on behavior during the production stage. The influence of several parameters on production improvement is described, using the results obtained with a numerical equivalent of the model. Reservoir system parameters considered are thickness, reservoir thermal conductivity, oil viscosity, solution gas-oil ratio and the change in oil effective permeability with temperature. Operating parameters investigated are heat injection rate, steam temperature and cumulative injection time. In the face of considerable uncertainty, the model appears to give a fair prediction of the behavior observed during an actual test. This makes the model a useful means of selecting projects. It is shown that the model may also be utilized to select an optimum set of operating conditions for a particular project. Results of this study show the extent to which production improvement depends on each of the parameters investigated and give an indication of the range of system conditions within which the process would be effective. INTRODUCTION In recent years the intermittent steam injection or steam soak process of oil recovery has become a popular method of improving recovery in reservoirs bearing high-viscosity hydrocarbons. Production improvement due to the process results from heating the reservoir near the wellbore. The process is applied on an individual well basis and is executed in a series of cycles, each consisting of three stages: injection, condensation (the soaking period) and production. During the injection stage, high-pressure steam is injected into the reservoir surrounding the well-bore. This study assumes that injection is carried out over the entire vertical section of the reservoir rock that can be exposed to fluid injection. The condition of steam entering the reservoir is considered to fall within the following limits: (1) steam pressures less than 1,000 psig, (2) steam temperatures between 300 and 700F and (3) steam quality less than 90 percent. The last restriction is due to equipment limitations. Normally, injection re maintained for less than 3 weeks. After termination of injection, the condensation stage begins. The well is shut in for several days, supposedly to make certain that the injected steam condenses. This condensation would insure the retention of injected heat. The production stage is initiated when the well is opened again. Reservoir response during this stage determines the effectiveness of the process (response is defined as the oil production rates, the water production rates and the wellbore temperatures observed during the entire production stage). The immediate response of the reservoir is sometimes quite good. Sets of case histories by Longl and by Owens and Suter2 give some typical figures for response. Duration of the production stage is equal to the time needed for the oil production rate to decline to some predetermined value. This usually requires 4 to 6 months. Termination of the production stage coincides with the start of the next cycle. The process is continued, cycle by cycle, until it becomes unprofitable to continue. One simple model2 and two more sophisticated models3,4 of the process have been presented. These are based on fundamentally different assumptions than those used in this investigation. Although the three models might be modified to fit the assumptions used in this study, it is felt that greater rigor would result by developing a

By Francis Collins, E. R. Brownscombe

A study has been made of the material balance-fluid flow method of estimating reserves and degree of water drive from pressure and production history data. By considering the effect of random pressure errors it is shown that in a particular example a standard deviation of three and one-half pounds in each of ten pressure survey? permits the determination of the reserves with a standard deviation of 8 per cent and the water drive with a standard deviation of 15 per cent, assuming that certain basic geologic data are correct. It is believed that this method of estimating reserves and water drive is useful and reliable in a number of cases. The method is particularly valuable when reservoir pressure data are accurate within a very few pounds, but may also be applied with less accurate pressure data if a relatively large reservoir pressure decline occurs early in the life of the field, as for example in an under-saturated oil field. INTRODUCTION A knowledge of the magnitude of reserves and degree of water drive present in any newly discovered petroleum reservoir is necessary to early application of proper production practices. A number of investigators have contributed to methods of relating reserves, degree of water drive, and production and pressure history. 1-8 Three types of problems of increasing complexity may be mentioned. If a reservoir is known to have no water drive. and if the ratio of the volume of the reservoir occupied by gas to the volume of the reservoir occupied by oil (which ratio permits fixing the overall compressibility of the reservoir) is known, then only one further extensive reservoir property remains to be determined, namely the magnitude of the reserves. A straightforward application of material balance considerations will permit this determination. The problem becomes very much more difficult if we wish to determine not only the magnitude of the reserves but also the magnitude of water drive, if any, which is present. In principle, a combination of material balance and fluid flow considerations will permit this evaluation. Finally, if neither the magnitude of reserves, the degree of water drive, nor the ratio of oil to gas present in the reservoir is known and it is desired to determine all three of these variables, the problem could in principle be solved by a fluid flow-material balance analysis which determines the overall compressibility of the reservoir at various points in its history. The change in compressibility with pressure would provide a means of determining the ratio of gas to liquid present, since the compressibilities of gas and liquid vary differently with pressure variation. However, in practice this problem is probably so difficult as to defy solution in terms of basic data precision apt to be available.' It is the purpose of this discussion to illustrate the second case, which involves the determination of two unknown variables, single phase reserves and degree of water drive, from pressure and production history and fluid property data, and to study the precision with which these unknowns can be determined in this manner in a particular case. Although an electric analyzer developed by Bruce as used in making the calculations to be described, numerical methods necessary in carrying out the process have been devised and have been applied for this purpose. Schilthuis,' for example, developed a comprehensive equation for the material balance in a reservoir. He combined this with a simplified water drive equation, assuming that the ratio of free gas to oil was fixed by geological data and that a period of constant pressure operation at constant rate of production was available to determine the constant for his water drive equation. On this basis he was able to compute the reserves and predict the future pressure history of the reservoir. Hurst developed a generalized equation permitting the calculation of the water drive by unsteady state expansion from a finite aquifer. He showed in a specific case how the water influx calculated by his equation, using basic geologic and reservoir data to fix the constants, matched the water influx required by material balance considerations. Old3 illustrated the simultaneous use of Schilthuis' material balance equation and Hurst's fluid flow equation for the determination of the magnitude of reserves and a water drive parameter from pressure and production history. He used this method to calculate the future pressure history of the reservoir under assumed operating conditions. As a basis for determining reserves, Old assumed a value for his water drive parameter and calculated a set of values for the reserves, using the initial reservoir pressure and each successive measured pressure. The sum of the absolute values of the deviations of the resulting reserve numbers from their mean value was taken as a criterion of the closeness of fit to the experimental data possible with the water drive parameter assumed. New values of the water drive parameter were then assumed and new sets of the reserves calculated until a set of reserves numbers having a minimum deviation from the average was established. The average value of- the re-

Jan 1, 1949

By E. R. Brownscombe, Francis Collins

A study has been made of the material balance-fluid flow method of estimating reserves and degree of water drive from pressure and production history data. By considering the effect of random pressure errors it is shown that in a particular example a standard deviation of three and one-half pounds in each of ten pressure survey? permits the determination of the reserves with a standard deviation of 8 per cent and the water drive with a standard deviation of 15 per cent, assuming that certain basic geologic data are correct. It is believed that this method of estimating reserves and water drive is useful and reliable in a number of cases. The method is particularly valuable when reservoir pressure data are accurate within a very few pounds, but may also be applied with less accurate pressure data if a relatively large reservoir pressure decline occurs early in the life of the field, as for example in an under-saturated oil field. INTRODUCTION A knowledge of the magnitude of reserves and degree of water drive present in any newly discovered petroleum reservoir is necessary to early application of proper production practices. A number of investigators have contributed to methods of relating reserves, degree of water drive, and production and pressure history. 1-8 Three types of problems of increasing complexity may be mentioned. If a reservoir is known to have no water drive. and if the ratio of the volume of the reservoir occupied by gas to the volume of the reservoir occupied by oil (which ratio permits fixing the overall compressibility of the reservoir) is known, then only one further extensive reservoir property remains to be determined, namely the magnitude of the reserves. A straightforward application of material balance considerations will permit this determination. The problem becomes very much more difficult if we wish to determine not only the magnitude of the reserves but also the magnitude of water drive, if any, which is present. In principle, a combination of material balance and fluid flow considerations will permit this evaluation. Finally, if neither the magnitude of reserves, the degree of water drive, nor the ratio of oil to gas present in the reservoir is known and it is desired to determine all three of these variables, the problem could in principle be solved by a fluid flow-material balance analysis which determines the overall compressibility of the reservoir at various points in its history. The change in compressibility with pressure would provide a means of determining the ratio of gas to liquid present, since the compressibilities of gas and liquid vary differently with pressure variation. However, in practice this problem is probably so difficult as to defy solution in terms of basic data precision apt to be available.' It is the purpose of this discussion to illustrate the second case, which involves the determination of two unknown variables, single phase reserves and degree of water drive, from pressure and production history and fluid property data, and to study the precision with which these unknowns can be determined in this manner in a particular case. Although an electric analyzer developed by Bruce as used in making the calculations to be described, numerical methods necessary in carrying out the process have been devised and have been applied for this purpose. Schilthuis,' for example, developed a comprehensive equation for the material balance in a reservoir. He combined this with a simplified water drive equation, assuming that the ratio of free gas to oil was fixed by geological data and that a period of constant pressure operation at constant rate of production was available to determine the constant for his water drive equation. On this basis he was able to compute the reserves and predict the future pressure history of the reservoir. Hurst developed a generalized equation permitting the calculation of the water drive by unsteady state expansion from a finite aquifer. He showed in a specific case how the water influx calculated by his equation, using basic geologic and reservoir data to fix the constants, matched the water influx required by material balance considerations. Old3 illustrated the simultaneous use of Schilthuis' material balance equation and Hurst's fluid flow equation for the determination of the magnitude of reserves and a water drive parameter from pressure and production history. He used this method to calculate the future pressure history of the reservoir under assumed operating conditions. As a basis for determining reserves, Old assumed a value for his water drive parameter and calculated a set of values for the reserves, using the initial reservoir pressure and each successive measured pressure. The sum of the absolute values of the deviations of the resulting reserve numbers from their mean value was taken as a criterion of the closeness of fit to the experimental data possible with the water drive parameter assumed. New values of the water drive parameter were then assumed and new sets of the reserves calculated until a set of reserves numbers having a minimum deviation from the average was established. The average value of- the re-

Jan 1, 1949

By K. H. Coats

This paper presents the development and solution of a mathematical model for aquifer water movement about bottom-water-drive reservoirs. Pressure gradients in the vertical direction due to router flow are taken into account. A vertical permeability equal to a fraction of the horizontal permeability is also included in the model. The solution is given in the form of a dimensionless pressure-drop quantity tabulated as a function of dimensionless time. This quantity can be used in given equations to compute reservoir pressure from a known water-influx rate, to predict water-influx rate (or cumulative amount) from a reservoir-pressure schedule or to predict gas reservoir pressure and pore-volume performance from a given gas-in-place schedule. The model is applied in example problems to gas-storage reservoirs, and the difference between reservoir performances predicted by the thick sand model of this paper and the horizontal, radial-flow model is shown to be appreciable. INTRODUCTION The calculation of aquifer water movement into or out of oil and gas reservoirs situated on aquifers is important in pressure maintenance studies, material-balance and well-flooding calculations. In gas storage operations, a knowledge of the water movemenr is especially important in predicting pressure and pore-volume behavior. Throughout this paper the term "pore volume" denotes volume occupied by the reservoir fluid, while the term "flow model" refers to the idealized or mathematical representation of water flow in the reservoir-aquifer system. The prediction of water movement requires selection of a flow model for the reservoir-aquifer system. A physically reasonable flow model treated in detail to date is the radial-flow model considered by van Everdingen and Hurst.1 In many cases the reservoir is situated on top of the aquifer with a continuous horizontal interface between reservoir fluid and aquifer water and with a significant depth of aquifer underlying the reservoir. In these cases, bottom-water drive will occur, and a three-dimensional model accounting for the pressure gradient and water flow in the vertical direction should be employed. This paper treats such a model in detail — from the description of the model through formulation of the governing partial differential equation to solution of the equation and preparation of tables giving dimensionless pressure drop as a function of dimensionless time. The model rigorously accounts for the practical case of a vertical permeability equal to some fraction of the horizontal permeability. The pressure-drop values can be used in given equations to predict reservoir pressure from a known water-influx rate or to predict water-influx rate (or cumulative amount) when the reservoir pressure is known. The inclusion of gravity in this analysis is actually trivial since gravity has virtually no efFect on the flow of a homogeneous, slightly compressible fluid in a fixed-boundary system subject to the boundary conditions imposed in this study. Thus, if the acceleration of gravity is set equal to zero in the following equations, the final result is unchanged. The pressure distribution is altered by inclusion of gravity in the analysis, but only by the time-constant hydrostatic head. The equations developed are applied in an example case study to predict the pressure and pore-volume behavior of a gas storage reservoir. The prediction of reservoir performance based on the bottom-water-drive model is shown to differ significantly from that based on van Everdingen and Hurst's horizontal-flow model. DESCRIPTION OF FLOW MODEL The edge-water-drive flow model treated by van Everdingen and Hurst1 is shown in Fig. la. The aquifer thickness b is small in relation to reservoir radius rb. water invades or recedes from the field at the latter's edges, and only horizontal radial flow is considered as shown in Fig. 1b. The bottom-water-drive reservoir-aquifer system treated herein is sketched in Fig. 2a and 2b. Here the aquifer

By R. L. Perrine

A stability theory is developed for miscible liquid-liquid displacement within a porous medium. In the usual case considered, a high-density high-viscosity "oil" is displaced downdip by a low-density low-viscosity "solvent'! Perturbation methods are used to find the conditions under which the spreading mechanism changes from the stable dispersion process to unstable viscous fingering. We find that instability is conditional, that there is a dependence on the shape of a disturbance leading to a"diameter" effect and that very difficult experimental scaling problems may result. A useful consequence is the definition of a minimum "slug size" for stable miscible displacement. This should make possible optimum use of the solvent process for oil recovery. The results may apply to many situations in which one fluid displaces another of somewhat different fluid properties within a porous medium. INTRODUCTION The process of miscible liquid-liquid displacement looked very favorable when it first received serious consideration as a means to increase petroleum recovery. Some early laboratory results were interpreted to mean that only a small "slug" of solvent was needed, perhaps 2 to 3 per cent of a hydrocarbon pore volume. This "slug" could recover the oil in an entire reservoir provided the fluids remained miscible.1,2 Only a slow growth of the "mixing zone", or gradual transition from oil to solvent, should occur. This would follow naturally from mixing by a dispersion mechanism such as that described by Scheidegger.3 And, indeed, laboratory cores have shown this kind of behavior provided solvent viscosity was at least as great as that of the oi1.2,4-6 The same kind of behavior has also been observed with solvent viscosity lower than that of the oil after the distance traversed became large. Not all laboratory results have been this favorable, however.7-9 Mixing zone growth may become proportional to the distance traveled rather than the square root of this quantity. Such behavior accompanies high rates in short systems of large diameter, provided the viscosity ratio is adverse. If this more rapid spreading were to persist in a reservoir, the solvent requirement for miscible "slug" displacement could exceed 30 per cent of a pore volume. There is considerable economic importance in the difference between 3 and 30 per cent solvent. A 1ogical conclusion is that two different kinds of flow behavior are possible, with each kind leading to a different result. One kind gives efficient displacement, the other does not. In the efficient case, the solvent bank spreads out only by the mechanism we have termed dispersion. Under these conditions, displacement is as near piston-like as possible. In the second kind of flow, found only with adverse viscosity ratios, viscous fingering occurs. That is, permeability variations cause a small finger of low-viscosity solvent-rich fluid to move ahead of its average position within the mixing zone. This creates a path of low resistance to flow, and an even greater amount of solvent-rich fluid follows. Thus, the process is autocatalytic. Once started, the fingering mechanism rapidly becomes dominant. The question to be answered is this. Under what conditions will each of these two different kinds of flow occur? In particular, what conditions denote the transition from dispersion to viscous fingering as the solvent spreading mechanism? The answer may tell us whether or not the desirable, near piston-like miscible displacement process is practical. An answer to this problem can be obtained from theory by the use of perturbation methods. The procedure is as follows. We first formulate a mathematical representation of the system. Then a small disturbance (or perturbation) in solvent concentration profile is introduced and observed to see what happens. Unstable flow is indicated when a small disturbance will grow larger. This will lead to eventual viscous fingering. If on the other hand the disturbance dies out, the displacement is stable. In this latter case, with dispersion as the spreading mechanism, near piston-like displacement is possible. Thus, this paper presents the development of a stability theory for miscible liquid-liquid displace-

By H. Rumfelt

The author demonstrates an approach for analyzing overcasting requirements for a stripping project. This approach to the problem employs indicated trends in the relationship of the weight of the machine (only electropowered draglines are considered) to its ability to do stripping work. Strip mining in the domestic coal industry is contributing a greater proportion of overall production than at any time in the past. U.S. Bureau of Mines statistics show that the percentage of total production mined in 1958 by stripping methods increased to more than 30 pct, up from 22 pct in 1951. The tendency to favor strip and open pit mining exists in other mining activities, but such trends are not as clearly outlined as in the coal industry. Significantly, the depth of overburden in coal properties considered suitable for stripping is also increasing. A depth of 50 to 60 ft of overburden was once considered to be about the maximum amount that could be handled economically. Today, a depth of 80 to 100 ft is frequently taken into consideration for simple overcasting operations (i.e., excavating overburden and dumping it in the spoil position where it remains indefinitely). Accompanying these changes, there have been continuing studies searching for procedures to appraise deeper stripping problems. Although the only known practicable way to accurately evaluate a proposed stripping venture is to make studies (including cost estimates) step by step, it is nevertheless desirable to have a quick method for making preliminary evaluations of potential strip mining operations. A recently devised digital computer method of providing a preliminary simple overcasting analysis for a stripping prospect relates the indicated trends in the relationship of the weight of the machine to its ability to do stripping work. In addition, this approach employs situations where the geometry of each cut-and-spoil section is assumed to take certain defined relationships for varying overburden depths. Slopes are necessarily considered to be stable, which means such practical factors as the mechanics of soils are neglected for the sake of convenience. Definition of Terms: Photographs A and B show a shovel-type and a dragline-type of simple overcast operation, plus terms used to denote certain parts of a pit. Pits that follow a straight line when projected on a horizontal plane are referred to as straightaway. It has been assumed that all section drawings accompanying this article pertain to straightaway cuts which, in turn, means that areas can be compared by relative volume. The volume of overburden is measured in virgin cubic yards. Material displaced from its virgin state to a spoil area normally occupies a larger volume than when in situ; the difference is termed swell, and it is expressed as a percentage of the original volume. For example, if the original cut volume is denoted V cu yd, and the spoil volume is 1.2 V cu yd, the swell is a positive 20 pct. MAXIMUM USEFULNESS FACTOR (MUF) CONCEPT Men experienced in overburden removal give emphasis to the weight of the machine in relation to its value as an excavator. From the writer's understanding, value has been only loosely defined and discussed generally and vaguely. The concept is considered important, however, since it is based upon the observations of many men. It provides a clue for the approach taken to solve deep stripping problems. In the procedure described below, value has been arbitrarily established as the product of the nominal dipper size of the shovel (or dragline) times a functional dumping reach. MUFs—Shovel: A shovel's capability to handle thick overburden is usually limited by its ability to dispose of the spoil. Thus the dumping reach, as well as its dumping height, is significant. The angle of repose of spoil material must also be considered. Although such slope angles vary among different mines, jobs, and materials, a slope of 1.25 to 1, frequently found in both planning and practice, is used by this author in the illustration of this analytical method. The relationship of the shovel's geometry to a stripping operation is shown in Fig. 1. To gain maximum advantage in constructing spoil piles, the shovel is placed so that its tracks adjacent to the spoil are as close as possible to the rib of the coal deposit. A vertical plane passing through the line

Jan 1, 1961

By I. E. Rosenzweig

GEOPHYSICAL prospecting by earth-resistivity methods is frequently applied to investigation of structural problems in geology. Fig. 1 indicates a scheme of the general arrangement used in these methods of measurements. The battery (or generator) E is connected with two "current electrodes" II and 12. The current of intensity J, supplied by this battery, passes through the earth between these elec- FIG. 1.-GENERAL ARRANGEMENT OF APPARATUS. trodes. A potential difference U will be measured between two other electrodes V1 and V2, called "potential electrodes." According to the configuration of electrodes (Fig. 1), the resistivity of a homogeneous soil results from the formula p = 2pcaU/J [1] where c=1/a/b11 - a/b12 ? (a/b21-a/b22) [2]

Jan 1, 1938

By I. E. Rosenzweig

The problem considered in this paper is as follows: An arbitrary horizontally stratified area is given. The electrical properties of this area are characterized by a function p(z) (Fig. I), which shows the relationship between resistivity p of the soil and the depth z. For such general case the apparent resistivity p* will be computed, by earth-resistivity measurements carried out by any given configuration of electrodes. General solutions of this problem appear to be very important for interpretation of field results of resistivity measurements, concerning structural investigations. Such solutions have been given by many authors for the special case of a finite number of homogeneous layers (Fig. 2)lV4 as well as for the more general case of continuously ranging resistivities.5-8

Jan 1, 1940

By I. E. Rosenzweig

The problem considered in this paper is as follows: An arbitrary horizontally stratified area is given. The electrical properties of this area are characterized by a function p(z) (Fig. I), which shows the relationship between resistivity p of the soil and the depth z. For such general case the apparent resistivity p* will be computed, by earth-resistivity measurements carried out by any given configuration of electrodes. General solutions of this problem appear to be very important for interpretation of field results of resistivity measurements, concerning structural investigations. Such solutions have been given by many authors for the special case of a finite number of homogeneous layers (Fig. 2)lV4 as well as for the more general case of continuously ranging resistivities.5-8

Jan 1, 1940

By W. S. Gillam

A need for new supplies of fresh water exists today and in many specific areas that need is urgent. One solution lies in saline water conversion, a problem complicated by cost factors. The principles involved in saline water conversion, the status of development, and the estimated costs (present and future) of several processes are presented. Among the methods discussed are distillation, electrodialysis, and freezing. In general, the costs presented are based on a standardized procedure for estimating conversion costs, permitting a valid comparison among the various processes. The need for new supplies of fresh water and the potential benefits to be derived from an abundant supply of converted water are recognized by practically everyone concerned with water problems. The water supply problem exists today; it is urgent in many specific areas in this country and also in the world, and it will become more acute in the future. One answer to the growing problem of adequate water supplies is the development of new sources. Very significant quantities of brackish underground and surface waters exist in certain areas and an inexhaustible supply of ocean water is available. Thus in many areas water resources can be extended through saline water conversion. Congress recognized the need for new sources of fresh water in 1952 and passed the Saline Water Act, Public Law 448, amended it in 1955, and in September 1958, enacted Public Law 85-883, calling for the construction of at least five demonstration plants. The program is administered by the Dept. of Interior through the Office of Saline Water, and its primary objective is to reduce the cost of converted water produced, whether it be by development of new processes or improvement of known processes. This is a most difficult problem and one that will require several years of prodigious effort. It is difficult—not because of any intricate or new chemistry, engineering, or physics involved—but because of the difficulty in converting water at low cost. Whatever the sources of the saline water, the salts which are held tenaciously in solution must be removed before the water becomes suitable for industrial or domestic uses. Saline water is a relatively simple system of salts dissolved in water. It has certain chemical and physical properties that determine the various methods by which the salts may be separated from the water. The system, although not complex, in most instances, has had countless years in which to reach equilibrium and is, therefore, comparatively stable. Because of its stability, separation of saline solutions requires relatively large quantities of energy. The unique properties of water depend on the fact that its molecules are chemically active. The chemical and physical properties of water are associated with the type of bonding involved in the water molecule. Chemical changes such as hydrolysis, or rusting of iron, involve the breaking of chemical bonds between the hydrogen and oxygen atoms. Physical changes, such as evaporation in a boiler, the melting of ice, or the viscous resistance to flow in a pipe, involve breaking of the hydrogen bonds. (The hydrogen nucleus is so small that it can attract two negative atoms.) Thus water molecules not only combine with molecules of other compounds but even with one another; e.g., each molecule may be bounded to four other molecules. Water molecules cling to the ions of dissolved salt to form water-encumbered hydrated ions and they cling to one another to form entangling networks through which hydrated ions can be propelled only by tearing the networks apart. That is one reason why considerable energy still needs to be expended in our simplest procedures for purifying water. If water molecules did not have this habit of clinging so tenaciously to other molecules, and to one another, it would be easy to push salt ions past the water molecule and get a separation. But the water would not then dissolve salt, so the problem would not exist.' Water when heated evaporates very slowly, relative to other liquids having simple molecules. Vaporization involves the separation of molecules from the liquid, and this means overcoming the attraction between molecules which is due to the hydrogen bonding. The heat of vaporization for water is high; consequently, the boiling point of water is also high. Water boils at 100" C; hydrogen sulfide (H2S) at -60" C; oxygen (02) at -183" C; nitrogen (N2) at -196" C; and methane (CH4) at-161°C, even though the latter has about the same molecular weight as water. Because of these peculiar properties of water, it exists as a liquid on earth instead of a gas such as hydrogen sulfide or nitrogen and oxygen.

Jan 1, 1961

By H. D. Outmans

Present first-order theory for frontal stability and viscous fingering of immiscible liquids is improved by including the nonlinear terms in the equations describing conditions at the interface of the liquids. This leads to the revision of several conclusions which were based on first-order theory. They concern the growth rate and changing shape of a sinusoidal disturbance, particularly in relation to the wave number of this disturbance. Nonlinear aspects of effective interfacial tension are discussed and it is shown that this tension is not simply a positive proportionality constant in a linear relation between pressure difference and curvature at the interface. Scaling requirements are determined from the dimensionless groups which govern fingering. Gravity and interfacial tension invalidate a previously formulated conclusion that the shape of a finger after a given displacement is independent of the displacement velocity. Also, similarity of fingering (and, hence, of sweep efficiency in case of an unstable front) requires geometrical similarity of the initial disturbance in the model and the reservoir with a scale factor which is the same as the one for scaling down the dimensions of the reservoir. INTRODUCTION The critical velocity at which the transition zone in the vertical displacement of immiscible liquids becomes unstable was calculated by ill. More recently, the same was done for nonvertical displacements and it was also shown that the stability is affected by an effective interfacial tension between the liquids. 2-4 Although in the oil reservoir a stable front is always desirable from a recovery point of view, the necessary velocity may have to be so low that the corresponding production rate is not economical. In that case, a knowledge of the fingering unstable front is necessary for predicting recovery at break- through. Studies in this direction have not gone beyond the very moment at which fingering first occurs.5,6 Conclusions about stability and fingering in these references are all based on linear theory. In this theory the nonlinear terms in the equations describing conditions at the front are neglected. The usual justification for this lack of mathematical rigor is that the nonlinear terms can be made small relative to the linear terms and, supposedly, small causes produce only small effects. It has been recognized, however, that this is not necessarily true in the nonlinear problems of hydrodynamics.7 The hydro-dynamic stability and fingering of the front between two liquids, accelerated in a direction normal to the front, for instance, is strongly influenced by the nonlinearity of the boundary conditions.8-10 Since these boundary conditions are not unlike those arising in the problem of stability and fingering of the front during a slow immiscible displacement in porous media, it was thought that there, too, the nonlinear terms should be taken into consideration. A summary of the results obtained in linear theory and a comparison with experimental data precede the nonlinear theory developed in this paper. This summary serves to introduce some quantities which will be used in the further develop ment. It should also be pointed out that the method of higher-order approximation by which the nonlinear stability problem is solved has application in other reservoir studies.11 SUMMARY OF LINEAR THEORY 1. A plane interface remains stable if its velocity is smaller than a critical velocity. vc = (p2-p1)gn/(l/A2- l/A1 .........(1.1) This equation is derived for uniform flow in a thin layer inclined to the horizontal plane. The displacement takes place in a direction normal to the intersection of this layer and the horizontal plane. The undisturbed (line) interface is normal to the velocity. Fluid 1, the upper fluid (gas), is displacing Fluid 2 (oil). (VcQS would be the injection rate in cc/

By AIME AIME

AT bottom of this page is a photograph recently taken by a student-John E. Stewart-of the most northerly situated college in the world, the Alaska Agricultural College and School of Mines. It is situated at College, near Fairbanks, Alaska, approximately on the 65th parallel of latitude, and therefore just south of the Arctic Circle. It is farther north than Hudson Bay, and in about the same latitude as southern Greenland.

Jan 1, 1932

By F. H. McDowell

The increasing interest in the subject of electrical separation warrants the addition to what has already been written of a few lines covering the practical workings of the Tilly Foster mill under the direction of Mr. E. S. Moffat, General Manager. Since my last paper was presented to the Institute, Mr. S. P. Tomkins, the superintendent, has had thrust upon him for solution several problems complicated by the introduction of new conditions. 1. The richer portion of the lean ore dump, and that nearest the railroad track, having been worked up in 1890, the handling of the balance, including a largely increased amount of rock, could only be accomplished at an increased cost. 2. The tailings had accumulated close about the mill to such an extent that arrangements had to be perfected to move them further away.

Jan 1, 1893

By J. F. Abel

It can be concluded that rock mechanics instrumentation, geologic mapping, and operations research in combination will produce an accurate estimate of tunnel support requirements for establishing a steel-set load prediction model. This estimate can be used to increase safety and to reduce tunneling costs. Unnecessary supports can be eliminated; overloaded or insufficient supports can be replaced on a more scientific basis than previously possible. Only a limited number of measurements are required to develop a steel set load prediction model. The results of this study can be applied generally; but, every tunnel must be considered as a separate problem. The construction and geologic variables which were found to be significant in the prediction of rock loads in this tunnel were: steel section modulus, set blocking points, percentage of alteration, relative degree of faulting, relative water condition, distance to the nearest fault, and average joint spacing. INTRODUCTION A combined engineering geology and rock mechanics program was performed throughout the Straight Creek tunnel pilot bore in Colorado in order to develop a technique for estimation of steel tunnel support loads. The technique involves the construction of a statistical model from measured steel set loads, geologic and construction factors. The aim of the investigation was to produce a set of charts, based on these factors, which could be used by a trained man equipped with a hand lens and measuring tape, to specify a safe and economical steel size and set spacing. In order to be of practical value the measured parameters had to be rapidly measurable and capable of quantification. It was apparent from the beginning that quantitative values for many of the geologic factors would have to be arbitrary. It was impossible to ascertain in advance which of the readily determi-nable geologic and construction factors significantly affect steel set loading. As the investigation progressed the number of observations was systematically reduced in order to determine the minimum amount of data which would be necessary for a significant mathematical model. This paper presents an analysis of all the variables measured at each instrumented steel set. The statistical significance of the measured variables was determined. On the basis of this determination, a mathematical model was constructed and charts were prepared to select and space the steel tunnel supports at any given point in the tunnel. Once the effect of geologic and construction factors on set loading is sufficiently established, it is possible to exert close control over the selection of steel tunnel support size and spacing. The magnitude of this rock mechanics program can be appreciated when it is known that there were 44 instrumentation stations for which steel set loads were measured. For each of these sets there were three construction variables recorded, and nine geologic parameters determined (Table I). A rough approximation of the cost for both the collection and reduction of the data would be $50,000. This is in addition to the cost of geologic mapping.

Jan 1, 1967

By H. H. Evinger, M. Muskat

A methoD of calculating productivity factors for oil, gas, and water systems in the steady state is presented as an illustration of the quantitative application of the fundamental data on the flow properties of heterogeneous fluid systems. While the numerical results of the calculations cannot be applied directly to field conditions, because they are based on permea-bility data for unconsolidated sands and involve specific assumptions regarding the nature of the reservoir fluids, the methods of analysis and interpretation should havc significance in considering certain specific field situations when the necessary field and laboratory data become available. Moreover, it is felt that even under the idealized conditions to which the numerical calculations refer they should serve to give at least the orders of magnitude of the effects of gas-oil ratio and connate water on observed productivity factors. Within the indicated limitations, the computations imply that the foregoing factors can explain only a part of the apparent discrepancy between the homogeneous fluid productivity factors and those obtained recently in field measurements on the West Coast. IntroductioN Sufficient laboratory research has been carried out in the last several years on the fundamental features of the flow of heterogeneous fluids through porous media to call for consideration of what can be done with this type of information. It has, of course, already led to a, much clearer general understanding of the mechanics of oil production. However, examples of the quantitative application of the results of such laboratory studies are few in the literature. part of the explanation for this fact lies undoubtedly in the lack of sufficient laboratory data relating to practical fluids and sand systems such as would occur ill actual oil reservoirs. However, unless it can be shown that if such data were obtained quantitative applications could be made, serious doubt may be cast upon the desirability of carrying on further extensive experimental research work on fluid flow. The purpose of this paper is to present one approach to the problem of quantitative application of laboratory data to practical field problems. Because the published results on the basic experiments with three-phase fluid systems in porous media are limited entirely to uncon-solidated sands, the numerical applications made here will not pertain directly to any actual field situation. However, they should serve to illustrate what could be done with fundamental laboratory data if they were extended to cover the field of consolidated sands. From this point of view, it is hoped it will serve to encourage further laboratory research on this problem. The particular application to be treated here relates to the calculation of the productivity factors for oil-gas-water systems in the steady state. This specific problem seems especially appropriate for discussion in view of the recent publication by Johnston1 of the results of produc-

Jan 1, 1942

By M. Muskat, H. H. Evinger

A methoD of calculating productivity factors for oil, gas, and water systems in the steady state is presented as an illustration of the quantitative application of the fundamental data on the flow properties of heterogeneous fluid systems. While the numerical results of the calculations cannot be applied directly to field conditions, because they are based on permea-bility data for unconsolidated sands and involve specific assumptions regarding the nature of the reservoir fluids, the methods of analysis and interpretation should havc significance in considering certain specific field situations when the necessary field and laboratory data become available. Moreover, it is felt that even under the idealized conditions to which the numerical calculations refer they should serve to give at least the orders of magnitude of the effects of gas-oil ratio and connate water on observed productivity factors. Within the indicated limitations, the computations imply that the foregoing factors can explain only a part of the apparent discrepancy between the homogeneous fluid productivity factors and those obtained recently in field measurements on the West Coast. IntroductioN Sufficient laboratory research has been carried out in the last several years on the fundamental features of the flow of heterogeneous fluids through porous media to call for consideration of what can be done with this type of information. It has, of course, already led to a, much clearer general understanding of the mechanics of oil production. However, examples of the quantitative application of the results of such laboratory studies are few in the literature. part of the explanation for this fact lies undoubtedly in the lack of sufficient laboratory data relating to practical fluids and sand systems such as would occur ill actual oil reservoirs. However, unless it can be shown that if such data were obtained quantitative applications could be made, serious doubt may be cast upon the desirability of carrying on further extensive experimental research work on fluid flow. The purpose of this paper is to present one approach to the problem of quantitative application of laboratory data to practical field problems. Because the published results on the basic experiments with three-phase fluid systems in porous media are limited entirely to uncon-solidated sands, the numerical applications made here will not pertain directly to any actual field situation. However, they should serve to illustrate what could be done with fundamental laboratory data if they were extended to cover the field of consolidated sands. From this point of view, it is hoped it will serve to encourage further laboratory research on this problem. The particular application to be treated here relates to the calculation of the productivity factors for oil-gas-water systems in the steady state. This specific problem seems especially appropriate for discussion in view of the recent publication by Johnston1 of the results of produc-

Jan 1, 1942

By R. W. Parsons

wben a displacement front encounters an isolated permeability heterogeneity, it will be perturbed. The details of this perturbation will depend on the beterogeneity size and shape, the Permeability contrast, and the mobility ratio. This paper deals with the displacement front behavior for one, simple, mathematically tractable model: a circular permeability discontinuity and unit mobility ratio. The progress of an initially planar front is shown for four permeability contrasts: k2/k1 = 0, 5/2, 2, m. The front distortion is more pronounced than streamline and isopotential distortion and it approaches a constant shape downstream from the discontinuity. INTRODUCTION The impact of reservoir heterogeneities on oil displacing processes depends upon the variation of the heterogeneous property (frequency distribution), its disposition (spatial distribution), and the inherent stability of the displacement mechanism.l The interplay of these factors plus the impracticability, if not impossibility, of describing a natural rock system in any detail makes exact displacement front movements difficult to determine. It is useful, however, to be able to qualitatively visualize what heterogeneities will do to a displacement front. In this respect certain simple models that can be treated analytically are of value. The following analysis will show the displacement front behavior for one such model. MODEL DESCRIPTION The flow field is two-dimensional and infinite in extent. The fluid flow is single phase, steady state and unidirectional at x, y = 2 m, or before insertion of the discontinuity. A circular permea- bility discontinuity is placed at the origin. Porosity and thickness are the same within and exterior to the heterogeneity. This is exactly the same model that was used by Greenkorn et al. to check analytically Streamlines as determined by Heie-Shaw flow models and finite difference computer techniques.2 FRONT POSITION CALCULATIONS The results of displacement front calculations are presented in Figs. 1 through 4 for four different permeability contrasts: k2/kl = O, %, 2, m. The times corresponding to the front locations are the Same for the 0 and cases! and for the 1/2 and 2 cases. The determination of front positions at various times is based on known analytical expressions for velocity potentials ().2 and stream functions ().2 These expressions are in terms of the x and y

Jan 1, 1969