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By R. A. Greenkorn

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

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

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

By J. E. Warren

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

By J. E. Warren

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

By H. D. Outmans

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

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

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

By S. W. Churchill

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

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

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

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

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

By J. C. Deppe

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

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

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

By F. J. Fayers, S. A. Hovanessian

The one-dimensional displacement equation for a homogeneous porous medium, including the effects of gravity and capillaty forces, has been solved by a numerical method. A finite-difference scheme is developed for obtaining saturation, pressure and fractional flow profiles in waterflood recovety problems. From the numerical examples given, it is concluded that the gravitational forces have a pronounced effect on the saturation profiles and the pressure distribution curves of the system. INTRODUCTION Within the past 20 years, a number of papers have appeared in the literature dealing with the quantitative treatment of waterflood recovery problems. In their celebrated paper, Buckley and Leverett1 described a method for calculating saturation profiles when the effects of capillary pressure and gravity are excluded. Terwilliger, et al,2 included the effect of gravity in their theoretical and experimental investigation of oil recovery problems and obtained close correlation between experiment and theory. welge3 described a simplified method for obtaining the average saturation and the oil recovery from an oil reservoir. The effect of gravity can be included in this calculation. More recently (1958) Douglas, et al,4 presented a method for calculating saturation profiles which includes the effect of capillary pressure. The authors start with the one-dimensional displacement equation (which is nonlinear in the derivative) and, by a change of variable, transform this equation to a semi-linear partial differential equation. This equation is then solved by a finite-difference method on a high-speed digital computer. Fayers and Sheldon5 obtained the solution of the one-dimensional displacement equation by directly replacing the differential equation by a finite-difference equation. The effects of capillary pressure and gravity were in- cluded in their solution. Here, since the movement of the foot of the saturation front is governed by a separate equation, the elapsed time in attaining a particular saturation profile cannot be obtained. The results of these two papers indicate that the inclusion of capillary forces eliminates the triple-valued Buckley-Leverett saturation profiles. In the present paper, the one-dimensional displacement equation for a homogeneous permeable medium, including the effects of capillary and gravity forces, is solved. The method of solution resembles that described in the paper by Douglas, et al, since it involves a similar transformation of variable. However, it extends their work in that the effect of gravity is included and, in addition, pressure profiles may be calculated. Also, the functions of saturation [kro(S), krw(S) and Pc (S) ] required are entered in tabular form rather than as low-order polynomials. This simplifies data preparation and permits the use of a greater range of function types. Unlike the method of Fayers and Sheldon, the corresponding elapsed time required for the development of each saturation profile is calculated and, also, saturation profiles may be calculated after breakthrough. THEORETICAL CONSIDERATIONS The dimensionless form of the one-dimensional displacement equation for a homogeneous porous medium as obtained from Darcy's law and material balance relations can be written as follows.

By L. D. Mullins, W. R. Bartlett, R. H. Barham, W. L. Wahl

This paper describes an electrical model and its applicution to the analysis of four reservoirs in Saudi Arabia. The model has 2,501 mesh points and represents 35,000 sq miles of the Arab-D member. Details of modeling such as mesh size, control problems and standards of performance in matching reservoir history are discussed. The particular performance match achieved for the Arab-D rnember is presented. Details such as permeability barriers, aquifer depletion and iwerference between oil fields are given. The performance match realized in the Abqaiq pool is presented in detail. INTRODUCTION The resistor-capacitor network and associated control equipment described in this paper comprise an electrical analog of a reservoir system. Similar equipment has been used to study the transient response of reservoirs for many years. The unique feature of the model and application to be described is the extremely large size of the model and reservoir system, and the detail observed in simulating the reservoir with the model. The Arabian American Oil Co. first became interested in analog computers for simulation of oil reservoirs in 1949. Since that time, several models have been developed, each more elaborate and refined so that the reservoir system might be more closely simulated. The current model is the latest in a series designed, built and operated by the Field Research Laboratory of Socony Mobil Oil Co. in collaboration with Aramco. It has been and continues to be used to study the regional performance of the Arab-D member limestone reservoir. The Arab-D member is one of the Middle East's most prolific producing horizons.. THE MODEL The theory of simulating a reservoir system with an electrical system has been presented in the literature.1-5 Therefore, this paper will not discuss the theoretical aspect of the problem except to point out the correspondence between the fluid system and electrical system, as shown in Table 1. In general, the complete model is made up of input devices, output devices, central control and a resistance-capacitance (RC) network. At times, the RC network alone is referred to as the "model". However, it should be evident from the text which meaning is attached to the word "model". A discussion of the equipment follows. THE RESISTANCE-CAPACITANCE NETWORK The RC network consists of 2,501 capacitance decades interconnected through 4,900 resistance decades. The components are arranged to form a rectangular network of 2,501 mesh points in a 41- X 61-mesh array. Imposing the mesh grid system on the continuous reservoir system divides the reservoir into discrete areal segments. These discrete segments may be of various sizes. More precisely, the mesh size need not be uniform throughout the model. The RC network is fabricated in two sections which are connected at the top. An inside view of the "tunnel" formed by the two sections is shown in Fig. 1. The height and width of the tunnel are shown in the figure. Numerals appear along the bottom and along the back opening of the tunnel. These numbers denote the x and y co-ordinate positions of mesh points. Fig. 2 presents a rear view of one-half the model. The length dimensions of the model, as well as a rear view of the capacitor decades, are shown in this figure. The control dials used in adjusting the resistance and capacitance values on the model can be seen in the enlarged portion of the model shown in Fig. 3. The electrical capacity at any mesh point can range from 0 to 1.0 microfarads set to the nearest tenth of a microfarad. The electric resistance connecting any two mesh points can range from 0 to 9,990,000 ohms set to the nearest 1,000 ohms. External capacitors may be added to any or all mesh points if the need arises. The values of electrical resistance and capacitance are adjusted manually by manipulating the two types of decade units. INPUT EQUIPMENT A considerable quantity of equipment is used to control the input to the RC network. These input devices are

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 V. Carlson, J. A. Rowe, E. O. Bennett

This report concerns the microbial flora found throughout the surface facilities of six water-injection systems in Texas and Oklahoma. Each system is described in detail and water quality data are presented when available. Pseudomonads and sulfate-reducers (Desulfovibrio) were the predominate organisms found in the systems. The genera Sphaerotilus, Bacillus, Achromobacter, Minococcus, Clostridium, Flavobacteriurn, and Sarcina were also isolated in significant numbers. Molds, iron bacteria, sulfur bacteria and soil bacteria were found less frequently in the systems. It is concluded that a complex microbiological flora exists in oilfield water-injection systems. A tendency for waters to deteriorate in passage through solid bed filters is also noted. INTRODUCTION The injection of water into a subsurface formation is often accompanied by numerous problems, the origin and nature of which are only partially understood. During the past few years, investigators have indicated that bacteria are involved in some of these problems even though very little specific information is available concerning this matter. It is surprising that investigators working on microbiological problems in water injection have never made a complete study of the flora of these systems. The logical course to pursue is that of identifying the microbial flora and then examining each organism individually and in combination with others to determine if they produce a particular problem. The objective of the present investigation is to identify the microbial flora in water-injection systems. In addition. the quality of the water in most of the systems was determined and some observations made on the extent to which passage through surface-handling facilities affects water quality. The information derived from this study can serve as a foundation for additional basic investigations. EXPERIMENTAL PROCEDURE The microbial flora of six water-injection systems were studied at 50 sampling points. These systems are experiencing problems which may be attributable to micro-organisms, and two were being treated with antimicrobial agents at the time of this study. The samples were obtained by inoculating 1.0 ml of the water onto the different media at the sampling point. The media used for primary isolation included nutrient agar, Strokes agar, M-10-E medium and semi-solid thioglycollate medium. The cultures were returned to the laboratory, incubated at 30°C until growth occured; then, pure cultures were obtained from individual colonies. The pure cultures were identified according to standardbacteriologica1 procedures. Those organisms not fitting identification criteria exactly, even though exhibiting close similarities to known species, are listed as unidentified organisms. Membrane filters were used to make water-quality tests at intervals before and after collection of the microbiological samples. The filter discs were 47 mm in diameter with a 0.45-micron particle-size retention, and the tests were done under closed conditions at 20) psi. Membrane flow rates are plotted as increasing filtrate volumes, and the total solids contents arcs given when available. An effort was made to sample at each place in a system where a change in environment occurred. In this manner, the effect of passage through filters, tanks, hay sections, etc., on the microbial flora and water quality could be observed. RESULTS SYSTEM A This waterflood system (Fig. 1) is about four years old and daily pumps approx&ately 110,000 bbl of oil-free sour water from a 3,700-ft deep reef. The

By G. W. Doepel, W. P. Sibley

A three-dimensional, multilayer, mathematical model has been developed for predicting performance by mis-c.ible displacement. Areal and vertical coverage, total gas-oil ratio, propane-oil ratio and oil-recovery factor for the multilayer system are computed as functions of hydrocarbon pore volumes of solvent injected. The technique has been programed for use on the IBM-7090 digital computer and was used in calculating perfortnance for continuous miscible displacement of a five-spot, 50-layer .system. Results reported are based on the use of published laboratory data for a single, uniform five-spot layer. However, the technique is sufficiently generalized to calculate performance of any system of layers of any pattern configuration and mobility ratio, provided data are available to describe perfortilance of a single layer. The technique has also been adapted for predicting oil recovery resulting from various gas-driven slug sizes. The results presented for both continuous and slug-type injection show, that permeability variation has 0 significant effect on oil recovery but that solvent composition, within the range studied, has a lesser effect on oil recovery. INTRODUCTION Experience from many of the miscible-phase displacement projects currently in operation indicates that the injected solvent flows preferentially through the more permeable zones in the reservoir. This preferential solvent flow in the high-permeability streaks results in a reduction of vertical coverage (C,) and an attendant decrease in oil recovery. The multilayer technique presented in this paper uses areal coverage (C.,) data for a single layer from the literature and from laboratory measurements of displacement efficiency (herein referred to as microscopic coverage, C,,) to calculate areal coverage (C,) and vertical coverage (C,) for the multilayer system. Oil recovery for the multilayer system is obtained as the product of (Ca) (Cm) (Cv). It is assumed that C, in a single layer is 100 per cent; therefore, (C,) (Cm) is a measure of oil recovery from that layer. Data presented in the literature for a single layer generally deal only with areal coverage and/or microscopic coverage —but seldom specifically with oil recovery. Nevertheless, this technique is flexible in that either Ca and C,, of an individual layer or a direct measurement of oil recovery for the layer may be used. In developing this technique, the problem arose concerning a method of determining the relative amounts of solvent that will enter each layer during any increment of solvent injected. To solve the problem, a dimensionless equation was derived for expressing solvent injected into each layer as a fraction of total injection. A finite-difference method was applied in integrating the equation to determine oil recovery for the system of layers as a function of pore volumes injected. The technique subsequently was programed for solution on a digital computer. The method was applied to predict performance of a deep California reservoir under continuous miscible displacement and has been extended for calculating oil recovery as a function of slug size for gas-driven slug injection. The technique may be applied to any petroleum reservoir that can be adequately represented as a layered system, by considering permeability variation,1-3 permeability range and average permeability. Each layer is then assigned its appropriate permeability, porosity and hydrocarbon saturation. RESERVOIR VOLUME CONTACTED BY SOLVENT It has been recognized for some time that high microscopic coverage (C,) can be obtained by injecting a miscible agent, such as propane, into an oil reservoir. These high values of C,,, have been demonstrated in the laboratory and are normally between 80 and 100 per cent, depending on solvent and reservoir oil composition as well as on reservoir temperature and pressure. In miscible-phase displacement, because there is no interface between solvent and oil, effective permeability to solvent is the same as that to oil. (This assumes that oil saturation has not changed and that water is immobile.) Consequently, mobility ratio (M) simply reduces to that of viscosity ratio . Perrine4 and Habermann5 have shown that large

By M. R. Tek

A new method for correlating the data on multiphase flow through vertical pipe is presented. The correlation is based on a "two-phase f factor" concept which was developed and successfully applied to horizontal multiphase flow by Bertuzzi, Tek and Poettmann. Field data previously published on several flowing and gas-lift wells have been used as the basis for the developed correlation. The application of the method to actual two-phase flow problems indicate that this method is capable of predicting the pressure distribution in vertical multiphase flow strings well within the accuracy range usually desired for common engineering and design calculations. A new working chart developed for calculation of two-phase pressure gradients and a graphical step-by-step procedure for the computation of pressure distribution are presented along with an example problem. INTRODUCTION Multiphase flow through vertical pipe is encountered in many engineering installations. In petroleum, chemical process, nuclear engineering and many other industries, problems associated with simultaneous flow of two or more phases through vertical pipe have been of interest for a long time. This interest has increased considerably during recent years due to applications to new processes in petroleum production and refining and to problems of steam generation and heat removal from nuclear reactors. One prominent example of vertical two-phase flow is provided by the gas-lift process where oil, water and gas flow simultaneously. If the pressure profile in a gas-lift well can be predicted within reasonable accuracy, it would be possible to get good estimates of the power required to lift the oil, the optimum depth, the pressure and the rate at which to inject gas. Furthermore, the effect of production rate and tubing sizes on these quantities can be evaluated before any design decision is made on the installation and operation of the flow string. The majority of experimental work available in the literature deals with two-component systems where individual phase flow rates in and out of the pipe remain constant. The general problem of prediction of pressure drops in multiphase flow systems is very complicated. The co-existence of numerous flow patterns of widely different geometry and mechanism, conditions of surface instability and the nature of force fields acting upon the system are among the major difficulties commonly encountered. The classical approach of fluid dynamics which would be based upon the formulation and solution of Navier-Stokes equations has been found by many investigators completely devoid of any hope of success-—not only because of inherent nonlinearities but also because of insurmountable analytical difficulties standirng in the way of setting up the boundary conditions. The presence and effect of interfacial forces on mu1tiphase flow systems further complicate the theoretical approach. For these reasons, many investigators choose to adopt semi-empirical if not purely empirical approaches in order to obtain solutions of engineering utility. A prominent practical solution has been given by Poettmann and Carpenter' in the form of an empirical correlation. In their paper, total flowing densities of fluids and solubility effects of gas in oil have been taken into account for the correlation of field data covering a wide range of operating conditions for oil wells. They treated the gas. oil and water as a single phase of combined propcrties and correlated the multiphase friction factor as a function of the product (pvd), i.e., density X velocity X diameter of the flow string. However, because the product (pvd) is dimensional while the frictior factor is dimensionless, the generality of their result's is somewhat restricted. It seems that the omission 01 viscosity effect may be one of the reasons for the scatter of data as shown in their correlation chart. The concepts of the "two-phase f factor" and the "two-phase Reynolds' number function" were recently developed and successfully applied to correlate horizontal multphase flow by Bertuzzi, Tek and Poettmann.' Recently, two new methods of correlation by Tek and Chan' have been presented on simultaneous flow of liquid and gas through vertical pipe. These two methods of correlation and the working equations in field units necessary for their application are included in this paper. The extension of the conccpt of two-phasc Reynolds' number function successfully developed for horizontal flow into vertical multiphase flow systems, development and evaluation of working charts permitting calculation of two-phase pressure gradients and

By J. E. Miller, W. E. DeVaney, L. Stroud

During a recent phase study of a natural gas, two stable equilibrium liquid phases were observed at temperatures below —200°F and pressures above 200 psi. This paper reviews the published literature on the occurrence of multiple equilibrium liquid phases and presents analytical data for the vapor and two equilibrium liquid phases of the liquefied natural gas at five experimental conditions. In addition, data for 30 conditions of two-phase equilibria are included. INTRODUCTION The low-temperature phase behavior of gases, most of which contained helium, has been investigated in the laboratories of the Helium Activity for many years. Since 1952, experimental studies of these systems have been continuous as part of the research program at Amarillo, Tex. Because of their value to private industries interested in participating in the Helium Conservation Program, several "Open File" reports containing phase equilibria data for heliurn-bearing natural gases have already been released by the Helium Activity. A paper containing information on the general phase behavior, operating criteria and extensive vapor-liquid data for two helium-containing systems was recently published.2 Additional publications presenting experimental data on the phase relationships of various gas systems are now in process and will be available in the near future. PREVIOUS EXPERIMENTAL WORK Although the formation of multiple liquids has been reported for various systems, to our knowledge this paper is the only substantiated evidence of a vapor-liquid-liquid equilibria in a naturally occurring gas. In 1940, Vink, Ames, and others13 reported the presence of two liquid phases in a hydrocarbon system consisting of mixtures of crude oils, solvents and natural gas. Eilerts and co-workers; published data on the recombined fluids from a gas-condensate well. This condensed gas, containing approximately 76 per cent methane and 24 per cent ethane-plus, exhibited two distinct liquid phases. Weinaug and Bradley 14 observed "unusual" phase behavior in a reservoir mixture. These workers postulated that the anomalous phase behavior was due to the "imminent formation of a second liquid phase". Botkin, Reamer, Sage and Lacey 1 studied two California crude oils that exhibited multiple phases. Kohn and Kuratal0 recently reported two equilibrium liquid phases in the methane-hydrogen sulfide system. Roof and Crawfordl1 and Eakin, et a16 also have reported experiments with binary systems that formed two stable equilibrium liquid phases. APPARATUS AND PROCEDURE A U. S. Bureau of Mines Phase Equilibrium Apparatus was used in conducting this study. The apparatus and procedures employed in its operation have been previously described3 and will not be repeated in detail ill this report. Briefly, the apparatus consists of a windowed cell which can be maintained within ±0.5°F for temperatures between room temperature arid -320°F. Pressure within the cell can be maintained within 0.1 per cent of gauge reading up to 800 psig. Equilibrium vapor and liquid samples are obtained in special containers4 for analysis by a mass spectrometer. Although the accuracy of the analyzer is about ±0.1 mol per cent, the reproducibility of the phase-equilibrium apparatus is considered relatively poor. Values reported from methane and nitrogen are considered accurate to ±l.0 and ±0.6 mol per cent. Data for ethane-plus in the vapor are accurate within 0.2 mol per cent; liquid-phase data for this aggregate compolnent are accurate within 1.5 mol per cent. For helium in the vapor phase, the analytical data are accurate within 0.2 mol per cent; liquid-phase analyses for this component were obtained by the charcoal adsorption method described by Frost8 and are accurate within 0.006 mol per cent. All of these references to the accuracy of reported values are conservative estimates based upon a statistical treatment of reproducibility data obtained with the apparatus.

By H. D. Attra

The effects of phase behavior on reservoir economics is a function of both fluid composition and flow properties of the reservoir rock. In some operations such as gas injection into volatile crude-oil resertoirs, the stock-tank recovery derived from vaporization of the reservoir oil by the displacing gas can approach the recovery derived from the displacement mechanism. In other cases of gas-cycling reservoirs, the dry injected gas can effectively vaporize a retrograde liquid phase, allowing a substantial reduction in cycling pressure with no appreciable loss in recoverable liquids. Because of these effects, it is essential that phase behavior be considered in developing the economics of any secondary recovery program in which gas is the injected fluid. The inclusion of phase behavior in conventional calculation procedures has been retarded because of the complexity of the calculation and the large volume of laboratory data required. However, by applying high-speed computers and new laboratory techniques, the phase-behavior displacement calculation has been developed to economically include this basic fundamental of gas displacement. The calculation is performed by reducing the reservoir to a series of one-dimensional segments. Phase changes, saturation distributions and changing fluid properties are evaluated for each segment as a function of time. The relative volumes of equilibrium liquid and gas displaced from each segment are determined by a trial-and-error procedure to meet the requirements of phase equilibrium and two-phase flow. The produced fluids are flashed through conditions of surface separation approximating those in the field to obtain actual values for stock-tank liquid recovery and producing gas-oil ratios. Based on the results of successful field studies, the technique appears to have application to reservoirs producing under a nonmiscible gas-displacement mechanism and should substantially increase the reliability of this type of analysis when phase behavior is an important factor. Predicting reservoir performance under gas-injection operations involves three basic principles of reservoir engineering. First, a material balance exists between the displacing gas and the displaced reservoir fluid. Second, the oil and gas flow in relationship with the two-phase flow property of the reservoir rock; and third, phase equilibrium exists between the displacing gas and the native reservoir fluid. To illustrate these three principles, consider the one-dimensional reservoir shown in Fig. 1. The reservoir is divided into a series of segments representing a one-dimensional cross section between an injection area and a producing area. So represents the oil saturation, Cw represents the connate-water saturation and Sg represents the displacing gas saturation. The gas is moving into Segment 1 and the reservoir oil is being produced out of Segment N. Consider the case when gas has just moved into Segment 1 and the displacing gas-oil contact is at position a-a The volume of oil pushed out of Segment 1 is a function only of mobility and the two-phase flow properties of the reservoir rock, that is, Kg/Ko and µo/µg. Behind the front if the injected gas is foreign to the reservoir oil (a nonequilibrium gas), there will either be vaporization of the oil into the gas or condensation of the gas into the oil. In either case the net result is a mass transfer and a re-distribution of composition and liquid and gas saturation. If the displacement is at constant pressure and is continued, the gas moves into new segments, proceeding succesively into Segment 2, 3, etc. The total volume of oil produced from the reservoir

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/