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By M. C. Flemings, G. E. Nereo

General expressions are given to describe macro-segregation in castings and ingots which results from mass flow of solute-rich liquid to feed solidification and thermal contractions. Analytical solutions are given for solidification with planar isotherms (unidirectional and bidirectional heat flow). These solutions describe inverse segregation and centerline segregation. Numerical examples are given for A1-4.5 pct Cu alloy. In 1540, Vannoccio Biringuccio published one of the first references to macrosegregation, describing the problem of exudation and inverse segregation in the manufacture of bronze gun barrels.' Since then, a great many studies of macrosegregation have been reported. Recent work of particular engineering interest includes that of Marburg 2 and others:" on centerline segregation and other types of segregation in large steel ingots, and work of Adams,5 Scheil, 6 and Kirkaldy and youdelis7 on inverse segregation in nonferrous ingots. In most studies on this subject, the various types of macrosegregation have been treated as separate phenomena. For example, quite different mechanisms have been proposed for formation of centerline segregation, negative cone of segregation, inverse segregation, banding, and so forth. In this, and in several papers to follow, it is proposed that these as well as other types of macrosegregation result from the same basic mechanism and can be quantitatively described by the same basic equation. It has often been suggested that some types of segregation (such as centerline segregation and banding) occur as a result of buildup of solute at the tips of growing dendrites.2-4 ,8-10 As example, banding can be induced in centrifugal castings by mechanical vibration, and Northcott 10 suggests this might be due to the influence of the vibration on the boundary layer at the dendrite tips of the growing dendrites. However, in view of the extremely small boundary layer known to exist at the tips of the dendrites,11,12 any interpretation of macrosegregation based on this boundary layer is clearly incorrect. Note that experiment has shown that significant buildup of solute does not occur at dendrite tips in ingot solidification, but even if such buildup occurred the thickness of the enriched zone would only be the order of D/U where D is liquid diffusion coefficient and U is dendrite tip velocity. For usual casting or ingot solidification this solute-enriched layer is 10"3 cm, far too small to account, for example, for centerline segregation.11'12 It has also been suggested that convection in the bulk liquid (ahead of the advancing dendrites) might cause segregation by sweeping away the solute-rich boundary layer in front of the dendrites.3 However, since there is no significant boundary layer, this explanation must be incorrect. The convection might directly cause segregation by penetrating between the dendrite arms, and presumably in a rimming steel ingot it is sufficiently vigorous to cause some segregation in this way. Centerline segregation, however, cannot be interpreted as resulting from this convection since a sharply segregated zone is usually found near the centerline, with the bulk of the ingot being nearly of nominal composition. If convection in the bulk liquid were the direct cause, a gradual change in ingot composition over the entire cross section would be expected. It is known that sudden changes in convection of the bulk liquid during solidification can result in banding,14 but it is not clear that the banding results directly from the effect of convection on solute distribution. Changes in convection also affect thermal conditions in the liquid-solid region of a solidifying ingot. It will be seen later in this paper, and in a paper to follow, that macrosegregation is strongly influenced by these thermal changes. Recognizing that in some cases of ingot solidification (as in solidification of rimming steels) convection in the bulk liquid may have a direct effect on solute distribution, we neglect it in this paper and in the several to follow. We then derive general expressions describing macrosegregation which results from the mechanism of interdendritic flow of solute-rich liquid

Jan 1, 1968

By Eugene A. Stephenson

THE year 1935 is noteworthy for the superior legislative measures passed by several states. Sincere efforts have been made to establish equity between the various producing tracts in oil and gas fields. Outstanding legislative measures include the New Mexico law, known as Senate Bill No. 79, approved on Feb. 23; the Oklahoma law on April 22; the Texas law on April 30; and the Kansas law, which became effective July 1. Primarily designed to prevent waste, but more or less the direct out- growth of the current excess of potential oil and gas production, these statutes fall in the class of "Proration Laws."

Jan 1, 1935

By C. R. McEwen

This paper presents a technique for calculating the original amount of hydrocarbon in place in a petroleum reservoir, and for determining the constants characterizing the aquifer performance, based on pressure-production data. A method for doing this based on a least-squares line-fitting computationwus proposed by vanEverdingen,Timmerman andMcMahon in 1953. We found that their method would not work when there is error in the reservoir pressure data — even moderate error. The technique presented here appears to give reasonable answers when pressure data are uncertain to the degree expected in reservoir pressure determinations. The major change introduced in the present analysis is to limit the least-squares line-fitting to yield only one constant— the amount of hydrocarbon in place. The water-influx constant is then taken as proportional to the oil (or gas) in place. The constant of proportionality can be computed from estimates of effective compressibility and reservoir water saturation. It is also pointed out that the commonly used least-squares analysis assumes all of the uncertainty to be in the dependent variable. The material balance should be arranged so that this condition is fulfilled if correct inferences are to be made from statistical calculations. Examples are shown of the application of the new technique to gas reservoirs—both hypothetical and real—and to the oil-reservoir example of van Ever-dingen, Timmerman and McMahon. INTRODUCTION The amount of hydrocarbon originally in place in a petroleum resenroir can be estimated by means of the material-balance calculation. Simultaneous observations of pressure and amounts of produced fluids are required, together with the PVT data applicable to the reservoir fluids. If water encroachment is occurring, it is desirable to try to infer the behavior of the aquifer, as well as the original hydrocarbon in place, from the pressure-production data. This imposes additional demands on the method of calculation, and uncertainty in the data can result in large uncertainty in the answer. In addition, if the size of a gas cap is to be established, the whole problem becomes indeterminate, as pointed out by Woods and Muskat.1 Brownscombe and Collins2 simulated a gas reservoir and its aquifer on a reservoir analyzer and derived quantitative information on the effect of uncertainty in pressure and aquifer permeability on computed gas in place. Among the various techniques of estimating the performance of an aquifer, the method of van Ever-dingen and Hurst? based on compressible flow theory, seems to have been the most generally successful (see Ref. 4, for example). In this paper we shall confine ourselves to their representation of the aquifer. In 1953,' van Everdingen, Timmerman and McMahon5 introduced a statistical technique for deriving the amount of oil originally in place and the parameters which describe the aquifer. (We shall refer to this technique as the "VTM method'', as suggested by Mueller.6) Their example reservoir had no gas cap. It has been our experience that the VTM method gives a reasonable answer when the data are very accurate, but that inaccuracy (particularly in pressure) can cause the method to break down. The effect was first observed in gas reservoirs, but has since been seen in oil reservoirs also. In this paper we present another statistical method which has been successful in achieving a reasonable answer where the VTM method has failed. In the new method, one less parameter is derived from material-balance computations. It is assumed that values can be established for effective compressibility in the aquifer and reservoir water saturation independently of the material-balance calculation. The water-influx constant can then be obtained from these data and the quantity of hydrocarbon in place.

By Donald L. Everhart

SOONER or later intelligent exploration for uranium leads to these questions: Where did the metallic ions that formed the orebodies come from? What processes and geologic factors were involved in ore replacement? These matters, of course, make a big difference in how best to look for orebodies. Before these questions are discussed it might be well to point out other factors bearing on the subject. The whole business of uranium exploration is only a little over ten years old, participation by the public is only about six years old, and real interest and activity by more than a few large mining companies has arisen only in the past two or three years. Throughout this short period, uranium exploration has been affected by the unique influences of necessary security measures and by division of the over- all job into individual projects by a large number of specialists in diverse fields. This has meant that a normal interchange of information on the nature of uranium deposits has really only just begun. Com- pared to the vast amount of published studies on iron, copper, lead, zinc, and gold deposits over the past 50 to 75 years, current knowledge of uranium deposits is in its infancy and much of the pertinent information is still scattered about in a number of unpublished sources. The volume of reports on all phases of uranium exploration is now beginning to mount at a remarkable rate, and geologists with widely varying specialized backgrounds in uranium geology are engaging in spirited debates over the origin of uranium and the relative importance of geologic controls of various deposits.

Jan 9, 1954

By E. W. Schilling

THE separation of hematite by hysteretic repulsion was first brought to the attention of the public in 1922, by W. M. Mordey1. Three years later another paper2 was published and after another four years a third paper3 appeared in the South African publications. Certain data were lacking in these papers. In order to continue the work intelligently a knowledge of the shapes of the hysteresis curves for certain minerals was very necessary. This information was not available until the present year4. It was known that reduction of certain ores rendered them amenable to this process of separation but definite data in the form of curves showing exactly the effect of such treatment were lacking. Prob-ably our best source of information on the separation of minerals by the use of alternating-current magnetic fields today is the reports from the U. S. Bureau of Mines 5-7. It is not the purpose of this paper to tell the reader in detail how to make a commercial separator. Experimental models have recently been made 7 and the results obtained have been so good that it is the opinion of the authors that the public will soon take an active interest in the subject and its development. The writers have constructed a separator that is considerably different in construction and operation from the two types described by Mr. Davis. This will be described, but the real purpose of this paper is to outline certain experiments and give curves showing data that have a fundamental bearing upon the practical construction of a separator of this type.

Jan 1, 1935

By E. W. Schilling

THE-separation of hematite by hysteretic repulsion was first brought to the attention of the public in 1922, by W. M. Mordey1. Three-years later another paper2 was published and after another four years a third paper3 appeared in the South African publications. Certain data were lacking in these papers. In order to continue the work intelligently a knowledge of the shapes of the hysteresis curves for certain minerals was very necessary. This information was not available until the present year4. It was known that reduction of certain ores rendered them amenable to this process of separation but definite data in. the form of curves showing exactly the effect of such treatment were lacking. Prob-ably our best source of information on the separation of minerals by the use of alternating-current magnetic fields today is the reports from the U. S. Bureau of Mines 5-7. It is not the purpose of this paper to tell the reader in detail how to make a commercial separator. Experimental models have recently been made7 and the results obtained have been so good that it is the opinion of the authors that the public will soon take an active interest in the subject and its development. The writers have constructed a separator that is considerably different in construction and operation from the two types described by Mr. Davis. This will be described; but the real purpose of this paper is to outline certain experiments and give curves showing data that have a fundamental bearing upon the practical construction of a separator of this type.

Jan 1, 1936

By Harwick Johnson, E. W. Schilling

THE separation of hematite by hysteretic repulsion was first brought to the attention of the public in 1922, by W. M. Mordey1. Three years later another paper2 was published and after another four years a third paper3 appeared in the South African publications. Certain data were lacking in these papers. In order to continue the work intelligently a knowledge of the shapes of the hysteresis curves for certain minerals was very necessary. This information was not available until the present year4. It was known that reduction of certain ores rendered them amenable to this process of separation but definite data in the form of curves showing exactly the effect of such treatment were lacking. Probably our best source of information on the separation of minerals by the use of alternating-current magnetic fields today is the reports from the U. S. Bureau of Mines5-7. It is not the purpose of this paper to tell the reader in detail how to make a commercial separator. Experimental models have recently been made7 and the results obtained have been so good that it is the opinion of the authors that the public will soon take an active interest in the subject and its development. The writers have constructed a separator that is considerably different in construction and operation from the two types described by Mr. Davis. This will be described, but the real purpose of this paper is to outline certain experiments and give curves showing data that have a fundamental bearing upon the practical construction of a separator of this type.

Jan 1, 1935

By Frank Gilbreth

THE problem, of the crippled soldier in industry is not a problem of war work only; it is a problem of industrial development. As individuals, each one of you is seeking to provide our maimed heroes with such teaching as will render them, in the shortest possible amount of time, productive and satisfied members of the community. As individuals you are interested in the organizations that are preparing to receive these men, to train them and to place them at that work for which they are best fitted. As an organization, it is now your duty to take up the industrial end of the problem, to investigate existing conditions in the industries, to discover where the opportunities for employment lie, and so readjust industrial practice that the cripples may be utilized to their own and the community's good with the least shock to themselves or to the industries, and with the greatest amount of permanent benefit to all concerned. We have divided industrial opportunities suitable for cripples roughly, into three classes. First, such as already exist in the industries, which can be set aside for cripples only, present occupants being transferred to other work for which they can be fitted but which is not suitable for cripple work; second, opportunities in the industrial world that, through changes in the working equipment or tools, or through appliances attached to the cripple himself, can make the cripple a competitor of the sound worker; third, opportunities not existing at present in the industries, but which could be introduced for the good of the community, without placing the cripples into competition with any existing class of workers.

Jan 4, 1918

By K. D. Jacob, W. L. Hill

THE bulk of natural phosphates is comprised of calcium phosphates, which are usually apatites;1 calcium aluminum phosphates such as pseudowavellite;2 and aluminum phosphates, which occur in extensive deposits, notably on Grand Connetable Island.3 Nearly all deposits of current commercial value are apatite, often with admixed material of other classes. In the trade coarsely crystalline material, mainly fluorapatite of igneous origin, moves as apatite, whereas the fine-grained material of sedimentary origin is marketed as phosphate rock. The latter classification includes the phosphate shales in western North America. Although several genetic varieties of phosphate rock are recognized,4 most known reserves are of marine origin. Calcium phosphate is concentrated from Florida land pebble and Florida hard rock deposits and from Tennessee brown rock deposits. Phosphate Rock as a Fluorine Carrier: More than two decades ago Reynolds and Jacob surveyed the fluorine content of commercial phosphate rock from various parts of the world.3,5 Later these samples were reanalyzed by a new and more reliable method of determining flourine.6,7 The revised results with interim additions of new samples, published a few years ago in a brief summary,' are shown in Figs. 1-3. Through the points representing the respective varieties of rock medial straight lines are drawn. In some instances the lines are well defined by the points; in others, where scattering renders the position of the best line uncertain, the greater weight is given points near the end of the covered range. Wherever distinct varieties are not recognized among regional deposits the custom is to identify the sample with the geographical location of the deposit.

Jan 10, 1954

By W. E. Showalter

This paper discusses some of the results of combstion-drive tests which were made in a test cell using a sand bed 10 in. in diameter x 10-ft long. The test method is illustrated and described. The relationship between the API gravity of the in situ oil and the amount of air required for combustion drive is discussed in detail. Other things constant, the air requirement for combustion drive increases as the APl gravity of the in situ oil decreases. If the test results apply to actual reservoirs, the lowest-priced oils may cost the most to recover by this method. Information is shown which indicates that the effect of pressure on the amount of hydrocarbon burned is not large. A method of predicting air requirements from the API gravity of the in situ oil is presented. INTRODUCTION Combustion drive is the term used to identify the process of interstitial or in situ burning as an oil recovery method. Part of the in situ oil is burned to generate the energy needed to produce the remainder of the oil. Combustion drive as an oil recovery mechanism remains an economic uncertainty in spite of all the work that has been done by the industry in both laboratory and field?-6 This paper will show some of the results of tests which were made in a test cell for the purpose of studying the nature of the combustion-drive process. It will present data which indicate that the API gravity of the in situ oil is a significant indicator of the amount of air required to drive a burning front through oil sand. Air requirement varies inversely as the API gravity of the in situ oil. EXPERIMENTAL The tests were performed in a cell which utilized a cylindrical sand section 10 in. in diameter x 10-ft long. The thin-walled metal pipe which held the sand was wound with twenty external electrical resistance heaters which, by means of an automatic controller, maintained adjacent sections of the wall of the pipe at temperatures equal to the temperatures of the contained sand. Each heater covered 6 linear in. of the pipe. By this means lateral heat loss from the sand section was minimized, thereby causing the sand section to simulate more closely a horizontal increment of a combustion-drive reservoir. Fig. 1 shows a schematic diagram of the test assembly. Thermocouples to measure the temperature in the sand were located every 6 in. along the length of the sand section. The pipe containing the sand was enclosed in a cell designed for an operating pressure of 500 psig. The inlet air pressure was controlled at the inlet, and the gas flow rate was controlled and measured at the outlet of the cell. The oil sand used for the tests was prepared by mixing first water and then oil with the noncon-solidated sand using a closed mixer similar to a cement mixer. Table 1 shows a screen analysis of the sand. Ninety percent of the sand was 100 mesh or finer. This sand was a mixture of 80 per cent No. 120 Nevada White Sand and 20 per cent Tennessee Hi-Fusion Moulding Sand No. 3. The Nevada sand was a clean silica sand. The Tennessee sand con-

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 J. Lohrenz, R. J. Francis, G. C. Clark

Compositional material balance methods have been extended to solution gas-gas cap-water drive reservoirs. Partial or complete pressure maintenance as well as reservoir repressuring by gas and/or water injection may be considered. Increrments of reservoir withdrawals and injections equivalent to a given time increment are used as the basic iterative step to allow the extension. An example field study shows how the reservoir volumes. extent of the water drive, and relative permeability ratio function may be estimated. Knowing these, the methods can be used to predict reservoir behavior. INTRODUCTION The compositional material balance is similar to conventional material balances in many aspects. The assumption of a pressure-homogenous reservoir with no transients is common to all conventional material balances1-1 although numerical methods are different. The compositional material balance also contains this assumption. Inclusion of phase equilibria directly in reservoir calculations is the key difference and improvement in the compositional material balance. A continuous account of the compositions of the reservoir fluids and effluents is determined. Calculations of fluid phases and properties can be based on these compositions. Excepting fluid phases and properties determined in this manner, the compositional material balance uses procedures exactly analogous to conventional methods. Brinkman and Weinaug5 and Jacoby and Berry6 have given compositional material balances for simple solution gas drive reservoirs; subsequently Jacoby and Berry' extended their procedure to include pressure maintenance by gas injection at constant reservoir pressure. This paper extends compositional material balance techniques to a more general combination drive reservoir with various types of pressure maintenance operations. The procedure for this extended compositional material balance and an example of its field application are presented. PROCEDURE The numerical calculations of this compositional mate- rial balance are performed on a high-speed digital computer. The most comprehensive case the present compositional material balance program can handle is a solution gas-gas cap-water drive reservoir with (1) water production, (2) water influx, (3) separate or commingled production from the oil and gas zones, (4) any surface processing arrangement including LPG and gasoline production, and (5) gas injection into the gas and/or oil zones." This description of the procedure applies to this most comprehensive case. Appropriate deletions in the procedure can be made for more simple mechanisms. ASSUMPTIONS In common with conventional material balances, the compositional material balance makes the following assumptions: (1) the total reservoir is at a uniform pressure at a given time; (2) the volumetric ratio of gas to liquid withdrawal at reservoir conditions is directly proportional to their mobilities; and (3) instead of assuming a pressure-dependent total gas-oil equilibrium, the cornpbsitional material balance makes a sianificantly different and more rigorous assumption as follows. Thermodynamic equilibrium exists for each component in the liquid-gas mixtures in the reservoir and in surface processing. These three assumptions are fundamental in the compositional material balance procedure. Other assumptions involve arbitrary simplifications of the reservoir mechanisms. This compositional material balance uses the following simplifying assumptions: (1) no bulk mixing occurs between the gas and oil zone; (2) total mixing occurs between any injected gas and the contents of the injection zone; (3) no in situ hydrocarbon liquid is produced from the gas zone; (4) no reservoir gas is retained behind any net bottom-water influx; (5) the residual saturation of hydrocarbon liquid at reservoir conditions behind any net bottom-water influx is constant; and (6) fluids retained behind any net bottom-water influx exert no further influence on the reservoir. These six assumptions are not fundamental. Modifications of the present procedure can be made for different assumptions of the reservoir mechanisms. THE TIME INCREMENT Previous compositional material balances have used a pressure increment for simple solution gas drive reservoirs"" and an increment of injected gas for pressure maintenance.' These increments are suitable for these simple cases. They are, however, inadequate for reservoirs with time dependent drive mechanisms including water drive. A time increment removes this inadequacy and is

By B. K. Larkin, H. R. Bailey

One of the most complicated and potentially one of the most promising secondary recovery methods is that of underground combustion. A number of field tests1,2,3 have been performed recently, apparently indicating that the underground combustion process is technically feasible. Relatively little has been published concerning laboratory or theoretical investigations of the underground combustion process. A few results of laboratory tube runs have been published4,5 and these data are valuable in understanding the process. The tube run data also emphasize the complexity of the thermal recovery process. Under various conditions, problems in heat transfer, multiphase fluid flow and combustion rates arise. For example, at low temperatures the combustion process proceeds so slowly that the front cannot be propagated. It is a problem in heat transfer to predict what injection rate of air is necessary to sustain combustion in a particular reservoir. Certain fluid flow aspects of underground combustion have been studied by Wilson, et a13. In the development of some of the flow equations, assumptions of adiabatic combustion were made. Vogel and Krueger1 have devised an analog model of the heat conduction process for a constant temperature cylindrical source moving radially outward in a homogeneous conducting medium. Jenkins and Ramey7, in a discussion of the Vogel and Krueger paper, present an analytical solution of a heat conduction problem related to problems considered by Vogel and Krueger. In the present paper an idealized model of the heat conduction problem is considered, which is more general than the model of Vogel and Krueger or Jenkins and Ramey. In particular, initial well heating, vertical heat losses and arbitrary frontal velocities are included. Equations for the temperature are obtained in terms of integrals. These integrals have been evaluated numerically for the case of frontal movement corresponding to a constant air injection rate and the case of constant frontal velocity. These results are presented graphically. From these graphs one may estimate conditions under which nearly adiabatic temperatures are obtained. The application of these results requires some knowledge of reservoir thermal properties, combustion rates and residual fuel content. Assuming no vertical losses it is possible to obtain explicit evaluations of these integrals in the linear con- stant velocity case and in the radial case corresponding to a constant air injection rate. PRESENTATION OF RESULTS The geometry of the radial system is illustrated in Fig. 1 where the reservoir is represented as a slab of thickness h. By making an energy balance for an infinitesimal volume, one obtains the following equation. T= x T+F(z.r).........(1) where F (z,r) =HWv1/pc (r-r1), if z h/2 = 0, if z >h/2 . In Eq. 1, V- represents the Laplacian operator and 8 is the Dirac delta function. Eq. 1 is the well-known diff-sion equation with a source term and may be applied to a linear system. For example, if the source is moving in the direction of the x axis, then the argument of the delta function is x - 1,. In deriving Eq. 1 the following approximations were made concerning the actual combustion process: (1) all heat transfer is by conduction in a homogeneous medium, (2) the formations bounding the reservoir have the same thermal properties as the reservoir, (3) the combustion fuel is consumed instantaneously as the front arrives, (4) there is a constant fuel concentration

By F. J. Fayers, R. L. Perrine

For many years the problem of increasing ultimate recovery of oil from a reservoir has been a subject of interest to the oil industry. At present, a standard secondary recovery technique is to flood the reservoir with water once natural forces of the system are depleted. The permeability characteristics of the rock, together with interfacial forces which exist between the two fluids and the porous material. allow only part of the oil in the rock formation to be recovered by the water-flooding process. Laboratory experiments have shown that when detergents are added to the flooding water, the flow properties of oil in a porous rock arc improved. This improvement results in greater oil recovery. Even though higher recoveries are obtained using detergents, the extra cost associated with the method may make it economically unsatisfactory. To establish economic feasibility requires experimental laboratory measurement of fluid and rock properties together with a calculational procedure to predict field results. Pilot tests in the field may also be required. This paper presents a mathematical procedure for predicting detergent flood behavior. A number of reasonable simplifying assumptions are made to obtain a tractable system of equations. The resulting method may be regarded as a generalization of Buckley-Lev-crett' theory to solve simultaneously for water saturation and detergent concentration profiles. Consequently this type of calculation can be used in much the same manner as Buckley-Leverett theory is used to evaluate a water flood. The mathematical method employed is the so-called method of characteristics for solving hyperbolic partial differential equations. This application of the method is instructive because several other problems in reservoir analysis may be analyzed in a similar manner. FLUID FLOW EQUATIONS The flow behavior of oil and water without detergent through porous reservoir rock is usually expressed by Darcy's law and a continuity equation for each phase. If gravitational forces are neglected, vrg = kr (s)µr v p rg.........(1) vrg = kr (s)µr v p rg.........(1) ? pa va = F ?/?t (sd) where v is the volumetric flux, k represents permeability, S is saturation, and P the pressure. Fluid viscosity is denoted by 11, p is the density, 4 is porosity, and I the time. The subscripts w and o refer to water and oil. The capillary forces in the system lead to a difference in pressure between phases. This may be expressed by the relation: Pm -P0 =Pc(S).........(5) The capillary pressure? P,, is determined experimentally as a function of saturation. DETERGENT BALANCE EQUATION Addition of detergent to the water phase affects the flow properties of the system. Capillary pressure and relative permeability become functions of detergent concentration, c, as well as fluid saturation. That is: P--P (S,c) and k,,,.- - -- k,,,. At low oil saturation values, the presence of detergent reduces capillary pressure and reduces the relative permeability ratio. Detergents are surface-active and adsorb on rock surfaces. The quantity adsorbed, a, will depend on the concentration c of detergent solution and may also depend on the saturation; thus, a = a(S,c). In this analysis we have assumed that the detergents used are water soluble but not oil soluble, and their movement is therefore restricted to the water phase. The dispersion of detergent in water is neglected by comparison with transfer by flow within the system. Detergent movement is determined by the water flux and adsorption: ? cv m = -F ?/?t (cs + a) We will assume that the quantity of detergent in solution is too small to significantly change water density or viscosity. When the permeability and capillary pres-

By J. W. Marx, R. H. Langenheim

The authors are to he complimented for a timely presentation of useful information concerning application of heat to oil reservoirs to increase the rate and ultimate recovery of oil. The solution for heated area resulting from constant rate injection of a hot fluid, presented by the authors, and the fluid-flow analogy, solved by Carter,' are particularly interesting in that 110 restriction is placed on the direction of development of the heated area. It is not necessary that the heated area grow radially. Thus the solution could provide useful information for heat injection in any type of well pattern with any specified swept area data. However, it was assumed in the development that the heated region would remain at a constant clevatcd temperature. Vertical heat losses were computed on this basis. Physically. the assumption that the heated region will remain at a constant temperature (independent of distance from the injection point) appears to require that the heat injection medium be stcam, or other condensible gases near the boding point at injection pressure. If a hot liquid below the boiling point or a hot gas considerably above saturation temperature were used, the temperature of the heated region would decrease with distance from the injection well. Lauwerier2 has discussed this problem for a linear system. (See also Jenkins and Aronolsky; and McNiel and Nelson.') Thus, the authors' solution applies more nearly to steam injection than to hot water injection or hot, non-condensable gas injection. Assuming steam as the injection medium, it appears that steam would pass through the constant-temperature heated region and condense upon contact with the unheated sand. The tempcrature in the condensing region would probably be governed by the release of heat of vaporization required for phase equilibrium at the existing pressure distribution. Temperatures in the region would also be influenced by vaporization of hydrocarbon. Thus, it is doubtful that an expression for radial temperature distribution based on heat conduction only (such as the authors' Eq. 9) would have practical significance. As pointed out by Marx and Langenheim, the growth of the heated area under constant rate steam drive will eventually be limited economically by vertical heat loss. The possibility of continuing to move heat economically by cold water injection or by increasing the steam injection rate appears to be a very important consideration. (Movement of liquids away from the injection well should increase injectivity.) Fortunately, the solution presented by the authors may be generalized to thc case of a variable heat injection rate. The general solution is:

By J. R. Jargon, H. K. van Poollen

Non-Newtonian fluids may be injected into a reservoir during secondary recovery operations. The non-Newtonian fluid used in this work is a power-law type of fluid that is, the viscosity of the fluid decreases as the flow rate or rate of shear increases. This paper presents equations for steady-state linear and radial flow of such fluids, transient behavior results from a finite difference model of a radial system, and transient behavior results from a field test. The equations that describe the flow of a non-Newtonian fluid are non-linear and are solved numerically. Finite difference solutions are presented as curves of dimensionless pressure drop at the wellbore us dimensionless time for a constant injection rate. Solutions were obtained for 5-percent, 10-percent and 100-percent PV of a non-Newtonian fluid for injection rates of 1, 10, 100 and 1000 cc/sec and for a 5 percent PV of non-Newtonian fluid located at T = Tw' 3, 10, 20, 50 and 100 ft for a flow rate of 1 cc/sec. The buildup curves do not exhibit a stmight-line portion as is the case for Newtonian flow through porous media. Correlations also are shown for the productivity index vs rate for the computer model study and the field tests. INTRODUCTION During various secondary recovery operations non-Newtonian fluids are injected. Such fluids, in general, include thickened water and gelled fluids. The term "non-Newtonian" implies that the viscosity is not only dependent upon temperature and pressure, but also on the rate of shear that is applied to the fluid. For example, water, which is a Newtonian fluid, will have essentially the same viscosity no matter what rate of shear is applied. A pseudo-plastic fluid exhibits a decreasing viscosity when higher rates of shear are applied; a dilated fluid has an increasing viscosity with increasing rates of shear. The objective of studies performed and described in this report is to obtain relationships and mathematical and empirical descriptions of the flow of non-Newtonian fluids through porous media. Simple mathematical relationships, computer studies that include the unsteady-state behavior of such fluids, and field tests are used. PREVIOUS LABORATORY STUDIES Laboratory studies have been performed by several investigators. The fluids used in one investigationl were surfactant-stabilized dispersions of water in hydrocarbons. Its Fig. 6 is reproduced as our Fig. 1. This effective-viscosity vs frontal-velocity diagram shows that this fluid is of the power-law type over a fairly large range of frontal velocities. Gogartyl developed an equation for the effective viscosity as a function of shear rate that reduces to Eq. 2 for frontal velocities greater than about 2 ft/D. The fluid characteristics used in the present study are similar to those reported by Gogarty. Christopher and Middleman2 performed experiments with power-law fluids flowing through pipes packed with glass beads. They used a modified Blake-Koteny

Jan 1, 1970

By S. T. Yuster

An analysis has been made of the factors responsible for tilted fluid contacts in petroleum reservoirs. The factors are static and dynamic with the former being controlled by those variables responsible for the capillary rise of fluids, and the latter include rate of formation tilt and hydraulic circulation. The explanation that rate of formation tilt may be responsible for tilted contacts appears untenable. The role of some of these factors in the migration, fractionation. accumulation and possibly the prospecting for oil is indicated. INTRODUCTION While it is generally considered that the segregation of gas, oil, and water in a petroleum reservoir results in approximately horizontal planes separating these phases. there are factor; which modify this simplified picture to a very great extent. Some of these factors are static. and the departure of the fluid interfaces from horizontal may represent a static equlibriun. Some of these factors may be dynamic and the interface positions may be non-equilibrium and represent a steady .state condition. Of course combinations of these two situations may also exist. The reasons most commonly given for this behavior are hydraulic circulation of fluids, change in character of the formation, and a rate of tilt of the formations more rapid than the rate at which the fluids may re-establish an equilibrium. Russell's Principles of Petroleum Geology' contains an extensive discussion of tilted fluid interfaces and their importance in the migration and accumulation of petroleum. Russell cites examples of tilted water tables at Lance Creek, Wyo.; Rock River, Wyo.: Cut Bank. Mont.; Cushing, Okla.: and Graham; Okla. Also given was the Wasson Field of West Texas which has saucer shaped oil-water and gas-oil interfaces with the saucer being deeper in the case of the oil-water contact. The Bradford Field in Pennsylvania has a tilted water table. San Ardo. Coalinga None and some other California field.; have tilled contacts. A recent paper by King Hubbert discuses the effect of hydraulic circulation of the water underlying the oil in a given stratun and it- implications in the accumulation of and drilling for the oil. The foregoing would indicate that tilted fluid interfaces are not uncommon and that it would be of importance to underbtand the forces which are responsible. Further. it would be of interest to examine certain idealized models of situations which may cause tilting with the thought in mind of deriving relatinnships between the important variables. These relationships may only hold exactly for the model* assumed but the results may apply semi-quantitatively to actual formations and at least indicate trends. The discussion will be divided into the effect of static factors and the effect of dynamic factors. STATIC FACTORS It has already been mentioned that one reason given for tilted contacts is a variation in formation properties. Specifically this is a change in the pore size of the formation capilaries. If the internal surface of the reservoir is preferetially wet by water, capillarity will cause the water 10 rise to a greater extent above the free water table in a small capillary than in a large one. Consequently the water-oil contact will be higher in the lower permeability sands than in the higher permeability sand.;. The free water table may be defined as the position the oil-water interface would take if capillarity were absent. Since the phenomenon is the capillary rise of water in an oil environment, a derivation of this shold give a relationship between the variables. Fig. 1 is an idealized model of the situation with a uniform bore circular capillary ubstitttting for the tortuous non.circular formation pore. The upward force.; on the column of water will be equated to the forces downward to give the equilibrium situation. forces upward = force downward where r is the radius of the capillary in ceritimetera. a is the interfacial ten4on oil-water in dynes per centimeter, 0 is the

Jan 1, 1953

HUBERT INGERSOLL ELLIS Hubert Ingersoll Ellis, who met accidental death on Jan. 6, 1919, in eastern Washington, had already advanced far in the profession of mining engineering and gave promise of a brilliant future. He was born on the Washington coast in June, 1889, and had his preparatory schooling at Menlo, Wash. In 1905; he entered the College of Mines of the University of Washington, and at once made a high record of scholarship. After completing 3 years of his technical course, he went to the interior of Alaska, where he spent 4 years in mining. During this time he was field correspondent for technical-magazines and wrote several important articles. In 1915, Ellis completed his senior year in the College of Mines; his graduation thesis was a comprehensive paper on mining methods at Fairbanks, Alaska, which was published as a series of articles in the Engineering and Mining Journal. On July 14, 1915, he entered the employ of the Bunker Hill & Sullivan Mining & Concentrating Co., Kellogg, Ida., as an underground miner, and was soon promoted to the position of operative on experimental metallurgy and later chemist in the same work. Later, he did exhaustive work on flotation with particular attention to the use of various addition agents. For the past 212 years, his work was largely in the field, examining mining prospects and directing work of investigation, during which time, while located at Sumter Valley, Ore., in 1916, he was elected to membership in the American Institute of Mining and Metallurgical Engineers. During the same year, he was married to Miss Grace Wells, of Arkadelphia, Ark., and besides his widow leaves a young son. In Hubert I. Ellis, there was found the rare combination of executive, administrative, and business ability, together with a temperament exceptionally well adapted and trained for research work and for the careful and exhaustive examination of engineering problems, both practical, and academic. These qualities were in addition to a splendid physique and a personality that made him highly successful in the handling of sub-ordinates and in his contact with his associates and the public.

Jan 4, 1919

By Edward Keller

(Canal Zone Meeting, November, 1910.) UNDER the title, Labor-Saving Appliances in the Works-Laboratory, I published a paper1 in which was described how multi-manipulations, in a works-laboratory, and in the furnace-room of an assay-laboratory, can be condensed into single manipulations by applying the proper mechanical devices. For example, I now deposit in, or withdraw from, a muffle a set of 48 cupels as one unit. The front row. (8) of these. are blanks or heaters; the other 40, when they have attained the proper temperature in the muffle, are charged simultaneously with the 40 lead-buttons. Three manipulations here accomplish an operation which by the generally customary method requires 136, and this number is only limited by the size of muffle and cupels. This operation is an enlargement of what I have already described in my former paper, and Fig. I shows the improved implements. In the paper referred to I described a gold-silver bead parting-bath, which is the last of the devices used in regular sequence in practice. This bath was designated as being convenient, but was not a labor-saver properly. Fig. 2 of the present paper shows a new device. Instead of the original tray, there are now sectional holders for the test-tubes, each having a wooden handle on either end, so that the holders may be removed from the boiling bath, and the acid or water poured off from each set of tubes (in this case, 7) without waiting to cool. The tubes, held in place by clutches, as shown in Fig. 3, rest in holes in the base-strip, having a smaller diameter than the tubes. Each holder is stamped at either end with a number, so that the bath becomes further useful by permitting several men to use it at the same time without interference. Apart from the handles, the holders are made of sheet-copper. ' Trans., xxxvi., 3 to 18 (1906).

Aug 1, 1910

By W. C. Hauber

Techniques previously published to predict the production performance of a water flood when the mobility ratio is not unity have been primarily restricted to a five-spot well pattern. When other types of well patterns are considered, the prediction of performance has required either a model study or a complex numerical solution of a multiple-fluid system with a high-speed digital computer. Simple methods are developed in this report for predicting the areal sweep efficiency and injectivity history of a water flood for arbitrary well patterns and mobility ratios. The effect of an initial gas saturation can be incorporated into this method of prediction. With the layer superposition concepts published by Prats, et al.,1 these methods can be used to predict the production performance of a water flood in a reservoir having a wide range in permeability. A reasonably good agreement is shown by comparisons of the results calculated by this method with those obtained either by model studies or by complex numerical solution of multiple-fluid systems. Therefore, it has been concluded that a reasonably accurate method has been developed for predicting the production performance of a water flood employing an arbitrary type of well pattern for any given mobility ratio. A medium-speed digital computer can be used in applying this method. INTRODUCTION Production performance of multi-fluid five-spot water floods can be predicted by the techniques published by Prats, et al.1 This technique takes into account the effects of initial gas saturation, mobility ratio and vertical variations in horizontal permeability. A basic assumption in applying this technique is that unit displacement efficiency does not vary with time after the passage of the flood front. Prats, et al. demonstrated that this assumption is valid for normal waterflood applications with small mobility ratios. To extend Prats' technique to other types of well patterns has required either an experimental model study or a numerical solution of a multiple-fluid system to describe the performance of an individual homogeneous layer. Sheldon and Dougherty2 presented an exact numerical solution employing any arbitrary well pattern. This technique is complex to apply, requiring (1) numerical solution of the single-fluid system, (2) conformal mapping of the streamlines and iso-potential lines to form a new grid network, and (3) a numerical solution of the multi-fluid system on the new grid network. A method was developed in this paper to approximate areal sweep efficiency and injectivity as functions of time for an individual homogeneous layer for arbitrary well patterns and mobility ratios. Analytic expressions were derived to directly apply this method to five-spot and direct line drive well patterns. For other types of well patterns, this method is similar to the approach of Higgids and Leighton,6 in that it utilizes the stream tubes and iso-potential lines obtained by the solution of differential equations when the mobility ratio is unity. However, this method is easier to apply than the Higgins-Leighton method since it does not require knowledge of the variation between oil and water relative permeabilities and saturation. Using the relationship between areal sweep efficiency, injectivity and the volume of water injected into an individual homogeneous layer as obtained by the method presented in this paper, and Prats' technique for the superposition of individual layers, one can predict the production performance of any water flood. ASSUMPTIONS GENERAL Basic assumptions made in this report are that fluids are incompressible and the layers are horizontal, homogeneous and uniformly thick. It is assumed that as water is injected into the layer, three distinct regions are formed —a water bank, an oil bank, and an unflooded region. Each region is assumed to displace other regions in a pis-tonlike manner, with only one fluid flowing in each region, so that there is no change in unit displacement efficiency with time after the flood front has passed in the reservoir. Therefore, the residual oil saturation in the water bank and the oil saturations in the unflooded region are constant throughout the layer. The ratio (F) of the bulk volume of a layer occupied by the oil and water banks at any time before oil breakthrough, to the corresponding bulk volume of a layer occupied by the water bank, is given' by the equation BREAKTHROUGH AREAL SWEEP EFFICIENCY To predict the behavior of any type of waterflood pattern, one must know the breakthrough areal sweep efficiency Eb(Mc,o, F) for the pattern for any water:oil mo-

Jan 1, 1965