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By L. A. Wilson, N. H. Harrison, R. R. Clay, R. L. Reed. D. W. Reed

Experiments have been performed in a linear near-adiabatic system for the purpose of extending data on reverse and forward combustion from atmospheric pressure to 1,000 psig. Results obtained from reverse combustion appear to conform qualitatively with the existing description of the process when reaction kinetics are suitably modified to account for the increase in pressure, i.e., increasing tbe pressure decreases the peak temperature and increases the combustion zone velocity. Forward combustion appears to be a fuel dominated process wherein peak temperature and combustion zone velocity are not very sensitive to changes in pressure. The moderate effects of pressure that do exist at low flux virtually disappear at high flux providing all oxygen is consumed. With this provision. increasing the pressure decreases the frontal velocity and increases the peak temperature. Results me shown graphicaIIy which demonstrate the effects of presswe on peak temperature, rate of advance, oil recovery, air/oil ratio, carbon oxides produced and temperature distributions. INTRODUCTION Numerous field tests of oil recovery using the technique of underground combustion are now in progress. Operating pressures used are always substantially in excess of atmospheric pressure. Nevertheless, the literature contains no laboratory data pertaining to the effects of pressure except for those of Martin, Alexander and Dew.1 Unfortunately, these data appear to reflect heat losses which may have obscured some of the effects of increased pressure. The underground combustion processes are exceedingly complex, and general concepts relating to them have necessarily been described in relatively simple terms. Particularly with regard to forward combustion, the interaction between multiphase fluid flow, heat transport and various rate mechanisms is so involved that to date a rigorous treatment is not available. In this paper a linear near-adiabatic system is described which was found capable of operating at pressures up to 1,000 psig and temperatures of at least 1,100 F. Using this equipment, experiments on both forward and reverse combustion were accomplished at elevated pressures and the data presented as a function of air flux or temperature. It will be assumed that the reader is familiar with the papers of Wilson, Wygal, Reed, Gergins and Henderson on forward combustion and that of Reed, Reed and Tracht3 on reverse combustion, so that it is not necessary to repeat the details of the two processes. EQUIPMENT AND PROCEDURE It was decided to study the effects of pressure over the interval 0-1,000 psig using a near-adiabatic system. Since satisfactory operation of such a system requires very thin tubing walls, it proved necessary to insert the entire combustion tube and heater assembly into a high pressure jacket in such a way that the internal tube pressure could be counterbalanced by pressurizing the annulus. This meant that the numerous heater and thermocouple wires had to pass through gas-tight seals that required considerable effort to maintain effective. An unexpected difficulty arose in connection with the nature of the annulus pressuring gas. It turned out that the composition of this gas critically affected the functioning of the adiabatic control system. Operation of the equipment at elevated pressures was not particularly difficult. The problems that did arise were a direct consequence of the large numbers of connections and seals involved. EQUIPMENT The equipment consisted of the combustion tube which fitted within the pressure jacket. This jacket was in turn mounted on a wheeled carriage which facilitated movement and provided for rotation of the

By F. E. Crane, H. A. Kendall, G. H. F. Gardner

Two effects of a difference in density may be observed in miscible displacement. As the difference is increased from zero there is first a change in mode from multiple fingers to a single finger. A further increase results in a more rapid growth of the single finger. Miscible displacement with multiple fingers has been discussed by Blackwell, et al, whereas miscible displacement with a single finger has been discussed by Craig, et al. In the present paper experiments are described which bridge the gap between these extreme cases. INTRODUCTION Craig, et al,1 investigated the effect of gravity on frontal displacements for circumstances in which the gravity forces were significant. Blackwell, et a1,2 dealt with the other extreme in which gravity forces were negligible. The considerable difference between their results is illustrated in Fig. 1 where the breakthrough recoveries are plotted against the mobility ratio. The experiments described in the present paper were conducted to examine how the transition from the one extreme case to the other occurs when the gravity forces are gradually reduced. The transition is well illustrated by the production history after breakthrough. Fig. 2 shows how the number of pore volumes which must be injected at a constant rate to displace 98 per cent of the original fluid depends on the gravity force. For each mobility ratio there is a transition range, and a very significant change in the number of pore volumes occurs when this range is traversed. This range of the gravity force is physically linked with the formation of viscous fingers. A single finger is predominant for larger gravity forces; a multiplicity of fingers is formed for smaller gravity forces. Improved recovery and the occurrence of multiple fingers is concomitant with a large increase in the amount by which the fluids are mixed within the porous medium. The transition range depends on all the dimension-less groups which characterize the problem; but this dependence has not been exhaustively studied as yet and only fragmentary results are discussed in this paper.

By P. J. Root, J. E. Warren

An idealized model has been developed for the purpose of studying the characteristic behavior of a permeable medium which contains regions which contribute significantly to the pore volume of the system but contribute negligibly to the flow capacity; e.g., a naturally fractured or vugular reservoir. Unsteady-state flow in this model reservoir has been investigated analytically. The pressure build-up performance has been examined in some detail; and, a technique for analyzing the build-up data to evaluate the desired parameters has been suggested. The use of this approach in the interpretation of field data has been discussed. As a result of this study, the following general conclusions can be drawn: 1. Two parameters are sufficient to characterize the deviation of the behavior of a medium with "double porositym from that of a homogeneously porous medium. 2. These parameters can be evaluated by the proper analysis of pressure build-up data obtained from adequately designed tests. 3. Since the build-up curve associated with this type of porous system is similar to that obtained from a stratified reservoir, an unambiguous interpretation is not possible without additional information. 4. Differencing methods which utilize pressure data from the final stages of a build-up test should be used with extreme caution. INTRODUCTION In order to plan a sound exploitation program or a successful secondary-recovery project, sufficient reliable information concerning the nature of the reservoir-fluid system must be available. Since it is evident that an adequate description of the reservoir rock is necessary if this condition is to be fulfilled, the present investigation was undertaken for the purpose of improving the fluid-flow characterization, based on normally available data, of a particular porous medium. DISCUSSION OF THE PROBLEM For many years it was widely assumed that, for the purpose of making engineering studies, two param- eters were sufficient to describe the single-phase flow properties of a producing formation, i.e., the absolute permeability and the effective porosity. It later became evident that the concept of directional permeability was of more than academic interest; consequently, the degree of permeability anisotropy and the orientation of the principal axes of permeability were accepted as basic parameters governing reservoir performance. 1,2 More recently, 3-6 it was recognized that at least one additional parameter was required to depict the behavior of a porous system containing regions which contributed significantly to the pore volume but contributed negligibly to the flow capacity. Microscopically, these regions could be "dead-end" or "storage" pores or, macroscopi-cally, they could be discrete volumes of low-permeability matrix rock combined with natural fissures in a reservoir. It is obvious that some provision for the inclusion of all the indicated parameters, as well as their spatial variations, must be made if a truly useful, conceptual model of a reservoir is to be developed. A dichotomy of the internal voids of reservoir rocks has been suggested.7,8 These two classes of porosity can be described as follows: a. Primary porosity is intergranular and controlled by deposition and lithification. It is highly interconnected and usually can be correlated with permeability since it is largely dependent on the geometry, size distribution and spatial distribution of the grains. The void systems of sands, sandstones and oolitic limestones are typical of this type. b. Secondary porosity is foramenular and is controlled by fracturing, jointing and/or solution in circulating water although it may be modified by infilling as a result of precipitation. It is not highly interconnected and usually cannot be correlated with permeability. Solution channels or vugular voids developed during weathering or burial in sedimentary basins are indigenous to carbonate rocks such as limestones or dolomites. Joints or fissures which occur in massive, extensive formations composed of shale, siltstone, schist, limestone or dolomite are generally vertical, and they are ascribed to tensional failure during mechanical deformation (the permeability associated with this type of void system is often anisotropic). Shrinkage cracks are the result

By A. W. Adamson

A model for transport processes in liquid mixtures is discussed which supposes that the elementary act involves a position exchange between two species and that the exchange is so confined by the solvent cage as to occur nearly isosterically. The rate-determining step, thus, is likened to a bi-molecular reaction and is so treated, using absolute rate theory. The cage model has been applied to diffusion, thermal diffusion, sedimentation and viscosity, but only the first and last of these phenomena are emphasized in the present paper. The model leads to semi-empirical relationships between the absolute value for a digusion coefficient and the activation energy for diffusion, between mutual and self-diffusion coefficients and for the variation of the viscosity of a binary mixture with composition. These are discussed in relation to experimental data for various systems, including hydrocarbon mixtures. It is shown that the proposed viscosity equation and seven other commonly used ones all may be regarded as special cases of a single general relationship; a brief critical analysis is made of the basis of selection of one or the other for data fitting or interpolation. INTRODUCTION AND GENERAL THEORY The present paper covers a brief discussion of a cage model for transport processes in liquid mixtures and how this model may be useful in treating the diffusional behavior and the viscosity of such systems. Since diffusion requires the more detailed treatment, it will be taken up first, and the model then applied to viscosity. There are two types of diffusion coefficients that may be measured experimentally, apart from thermal diffusion quantities. The first is the mutual or binary diffusion coefficient, D which may be defined in terms of Fick's first law. This states that the permeation, or flux P, is proportional to the concentration gradient. In the usual experiment, P is measured relative to a frame of reference fixed with respect to the medium (e.g., the diaphragm in a diffusion cell); as a consequence, the same value of D is obtained regardless of whether P and C refer to Component 1 or to Component 2; i.e., there is only one independent mutual diffusion coefficient for a binary system. In addition to D there will be various self-diffusion coefficients. defined in terms of the gradient in labelled species i and its permeation in an otherwise uniform medium. The thermodynamic approach to mutual diffusion supposes that the actual driving force is the gradient of the chemical potential, i.e., that In the case of a dilute solution of solute, Eqs. 1 and 3 lead to the Einstein equation, If the solution is ideal and the friction coefficient is taken to be then the familiar Stokes- Einstein equation results. Mutual and self-diffusion coefficients can not be related on general thermodynamic grounds; it is necessary to invoke some additional assumptions, i.e., a model; several such have been proposed. Hartley and Crank' supposed the existence of separate, intrinsic diffusion coefficients (Dl and D2) for each component, essentially corresponding to the two self-diffusion coefficients. The two flows can not be independent, however, but must be coupled through the usual restriction that there be no net volume flow. For an ideal solution. one then obtains' Glasstone, et al' treated diffusion in terms of absolute rate theory, but their approach otherwise resembled the previously mentioned one in that each species was considered to move with respect to the general medium in a manner determined by its individual jump distance and specific rate constant. For other than dilute solutions, a coupling of flows leading to an equation such as Eq. 6 would again be present. However, as required by Eq. 6, one does expect that the self-diffusion coefficient for the solute and the mutual-diffusion coefficient for the system become identical at infinite dilution. Lamm4 recognized that there should be three distinctive interactions in a two-component system-1-1, 1-2 and 2-2 — and, therefore, proposed three rather than two fundamental friction coefficients. Mutual diffusion resulted from 1-2 interactions only, and self-diffusion resulted from 1-2 plus either 1-1 or 2-2 interactions. Again, a collective coupling between all motions was imposed to meet the condition of no net volume flow. Laity' has shown how to convert the Onsager equations to a form very similar to Lamm's. Cage Model For Diffusion Work in this laboratory on diffusion in aqueous sucrose solutions made it apparent that three, rather than two, interactions were indeed needed," but considera-

By D. H. Gray, G. Bergamini, I. Fatt

A method developed by soil scientists for measuring permeability anisotropy of soils has been adapted for use in studies of sandstone permeability aniso-tmpy. Studies on Berea and Bandera sandstone show that permeability of these materials decreases upon application of simulated overburden pressure. Permeability reduction of Berea and Grubb sandstone cores subjected to simulated overburden pressure is shown to be a function of the ratio of radial to axial stress. INTRODUCTION The effects of mechanical stress on reservoir rock parameters are now generally accepted as being important in reservoir engineering calculations. In several oil fields the unique behavior of the reservoir has been attributed to the influence of overburden pressure or internal fluid pressure on rock permeability or fluid content. l Although the gross effects of overburden pressure on permeability, porosity, waterflood efficiency, electrical resistance and sonic velocity have now been established,3-9 there are some very interesting, and perhaps important, details of rock behavior under stress that have not heretofore been studied. In this paper we will show that permeability anistropy of several sandstones is a function of overburden pressure. We will also show that permeability reduction of cylindrical samples of two sandstones, when subjected to mechanical stress, is a function of the ratio of radial to axial stress. Most studies of overburden pressure effects made in the past have attempted to duplicate reservoir conditions by applying hydrostatic external pressure on all surfaces of the sample. Assuming perfect elasticity, that is, no viscoelastic flow during geological time, reservoir rock in situ cannot be under hydrostatic stress as is apparent from the value of 0.25 for Poisson's ratio for most rock. In the absence of tectonic forces the horizontal stress component If there are directional permeability effects, as has been reported from field and laboratory studies, l0,11 then one would expect hydrostatic stress as used in the laboratory to give a different permeability reduction from that occurring under reservoir stress conditions. The influence of stress on directional permeability was measured by a modification of the method first described by Maasland and Kirkham for use on soil samples.12 Their method permits calculation of the permeabilities along each of the three mutually perpendicular axes of a cubic sample. The modification reported in this paper extends the method to observing these permeabilities when the sample is under hydrostatic stress. The Hassler sleeve type of core holder is satisfactory for studying overburden pressure effects on permeability if a known force is applied to the end plugs. Wyble6 used a sleeve core holder but did not load the end plugs. In this paper we combine sleeve pressure and axial force to give permeability changes which we believe are close to those occurring in reservoirs when reservoir pressure changes. The combination of sleeve pressure and axial force assumes, however, that the sleeve pressure simulates both vertical stress and horizontal stress in one direction. METHOD PERMEABILITY ANISOTROPY The Maasland and Kirkhaml2 method of measuring permeability anisotropy uses a cube of porous material which is Open On three faces and has discs with a Center hole for an air inlet cemented to the other three mutually perpendicular faces (Fig. 1). The rate at which air pressure falls in a bottle of known volume, and pressure slightly above atmospheric, when the bottle discharges through each disc, is

By F. W. Jessen, N. Lamont

The effect of an existing fracture or joint plane, which may exist in a rock, on the extension of a hydraulically induced fracture through the rock has been investigated in the laboratory. By use of a series of models made from various outcrop rocks, the width and orientation of existing fractures do not alter the extension or direction of the hydraulic fracture. The results, which conform quite well with the Grifith theory of failure, are illustrated by a series of photographs of the various models. INTRODUCTION The process of hydraulic fracturing has been widely used in the oil industry since its introduction in 1948 and has made possible the production of many reservoirs which would have been uneconomical prior to this process. Numerous studies of the mechanics of the process2-h nd of the effects of oriented fractures on recovery'.' have appeared in the literature. The increased recoveries led to attempts to predict the orientation of hydraulic fractures at the wellbore and to the development of methods to control this orientation. Field tests have been devised"." which indicate the validity of theoretical predictions of fracture orientation at the borehole. The direction of extension of hydraulic fractures from the borehole has not received much attention since most studies have predicted, at least implicitly, that a fracture will continue in the same plane unless a change of state of stress in the rock occurs. Since such predictions are based on the assumption of rock homogeneity, the effects of rock heterogeneities are left unknown. Most sedimentary rocks are composed of layers which reflect the changing depositional conditions of geological time. In addition, the more competent rocks frequently are fractured and jointed as the result of structural deformations or tectonic movements. Evidence of such joint systems exists in many surface outcrops, and it can be assumed that similar systems occur in many subsurface rocks, although the individual joints may be rather tightly closed due to overburden forces. This paper investigates the effect that an existing fracture in a rock would have on the direction or orientation of an extending or advancing hydraulic fracture when it intersects the existing fracture. The study was conducted on small rock models under triaxial stress conditions. The results indicate that an existing fracture will have little effect on the hydraulic fracture. THE MODEL STUDY The rock model represents an elemental portion of the earth containing an existing fracture which is located at some distance from a borehole from which hydraulic fracture is being extended. A generalized view of a model is presented in Fig. 1. The models were constructed from cement blocks and natural rocks. The various materials tested and some of the physical properties are listed in Table 1. The rocks were cut into rectangular blocks with dimensions of 11/2 X3Y2 X4 to 8 in. An initial slot was cut into the end face of each block along the longitudinal axis normal to the top face. The slot was filled with plastic aluminum along its outer edge to a depth of 1/16 in. A hole of 3/8-in. diameter was drilled into the sealed slot at the center of the end face to serve as an entry port for the fracturing fluid. Existing fractures of various types were simulated in the models. A "hairline" fracture with essentially zero width was created by cutting the model into two parts with a diamond saw and replacing the two parts together along the cut. A finite-width fracture was created by placing a layer of sand grains between the two faces of the cut just described (see Fig. 12). A large open fracture (similar, perhaps, to a vug) was created by cutting the model in two parts and removing a portion of the downstream block to a depth of % in. See Fig. 13. For a fourth

By D. Havlena, A. S. Odeh

The material balance equation used by reservoir engineers is arranged algebraically, resulting in an equation of a straight line. The straight line method of analysis imposes an additional necessary condition that a successful solution of the material balance equatiott should meet. In addition, this algebraic arrangement attaches a dynamic ineuning to the otherwise static material balance equation. The straight line method requires the plotting of one variable group vs mother variable group. The sequence of the plotted points as well as the general shape of the resulting plot is of utmost importance. Therefore, one cannot progrm the method entirely on a digital computer ar is usually done in the routine solution of the material balance equation. If this method is applied, then plotting and anaIysis are asential. Only the appropriate equations and the method of analysis and interpremtion with comments and discussion are presented in this paper. Illustrative field examples for the various cases treated are deferred to a subsequent writing. INTRODUCTION One of the fundamental principles utilized in engineering work is the law of conservation of matter. The application of this principle to hydrocarbon reservoirs for the purpose of quantitative deductions and prediction is termed "the material balance method of reservoir analysis". While the construction of the material balance equation (MBE) and the computations that go with its application are not difficult tasks, the criteria that a successful solution of the MBE should fulfill have always been a problem facing the reservoir engineer. True and complete criteria should embody necessary and dcient conditions. The criteria which the reservoir engineer uses possess a few necessary but no sufficient conditions. Because of this, the answers obtained from the MBB are always open to question. However, the degree of their acceptability should increase with the increase in the number of the necessary conditions that they should satisfy. Generally, the necessary conditions commonly used are (1) an unspecified consistency of the results and (2) the agreement between the MBE results and those determined volumetrically. This second criterion is usually overemphasized. Actually, the volumetrically determined results are based on geological and petrophysical data of unknown accuracy. In addition, the oil-in-place obtained by the MBE is that oil which contributes to the pressure-production history,' while the volumetrically calculated oil-in-place refers to the total oil, part of which may not contribute to said history. Because of this difference, the disagreement between the two answers might be of paramount importance, and the concordance between them should not be overemphasized as the measure of correctness of either one. In this paper, a third necessary condition of mathematical as well as physical significance is discussed. It is not subject to any geological or petrophysical interpretation, and as such, it is probably the most important necessary condition. It consists essentially of rearranging the MBE to result in an equation of a straight line. This straight line method of the MBE solution has invalidated a few long time accepted concepts. For instance, it has always been advocated that if a water drive exists, but one neglects to take it into account in the MBE, the calculated oil-in-place should increase with time. The straight line method shows that in some cases, depending on the size of the neglected aquifer, the calculated oil-in-place might decrease with time. The straight line method requires the plotting of a variable group vs another variable group, with the variable group selection depending on the mechanism of production under which the reservoir is producing. The most important aspect of this method of solution is that it attaches a significance to the sequence of the plotted points, the direction in which they plot, and to the shape of the resulting plot. Thus, a dynamic meaning has been introduced into the picture in arriving at the final answer. Since the emphasis of this method is placed on the interpretation of the sequence of the points and the shape of the plot, one cannot completely automate the whole sequence to obtain "the best value" as normally done in the routine application of the MBE. If one uses the straight line method, then plotting and analysis are musts. The straight line method was first recognized by van Everdingen, et al,2 but for some reason it was never fully exploited. The advantages and the elegance of this method can be more appreciated after a few cases are carefully treated and worked out by it. SOLUTION OF THE MATERIAL BALANCE EQUATION SATURATED RESERVOIRS The MBE for saturated reservoirs written in AIME symbols is

By C. Chu

An investigation has been made of the radial heat-wave process using a mathematical model in two-dimensional cylindrical coordinates. This model considers combustion, convection and conduction inside the reservoir, but only conduction in the bounding formations. From a study of the general features of the process, an important phenomenon has been revealed, namely, the feedback of heat into the reservoir on the trailing edge of the heat wave. The effects of various process variables on the performance characteristics of the process have also been investigated. It was found that up to the time when the combustion front reaches a given point, the per cent heat loss, provided it is not higher than 40 per cent, is approximately directly proportional to the square root of thermal conductivity arid fuel content, but inversely proportional to the square root of gas-injection rate and oxygen concentration. The effecr of reservoir thickness is more pronounced, since halving the thickness doubles the per cent heat loss. The most decisive factor in determining the center-plane peak temperature is the fuel content of the reservoir. Within the temperature range investigated, doubling the fuel content doubles the peak temperature in the early stage, but the rate of decline of the peak temperature is high. Reservoir thickness is also a very influential factor. The peak temperature is lowered when the thickness is reduced; however, the effect of thickness becomes less pronounced when the thickness is high. Reduction of oxygen concentration increases the peak temperature in the early stage but lowers it afterwards because of the higher rate of decline of the peak temperature. Increase in gas injection rate or decredse in thermal conductivity geives a higher peak temperature which stays high for a longer period. The propagation range of the heat wave is chiefly governed by the fuel content of the reservoir. An increase of 0.2 1b/cu ft in the fuel content increases the propagation range by 100 per cent. The propagation range is more than doubled by doubling the gas injection rate, or reservoir thickness, or by reducing the thermal conductivity by 50 per cent. Comparatively, oxygen concentration has less effect on the propagation range. INTRODUCTION Several investigators have conducted theoretical studies of a radial heat wave. Vogel and Krueger1 studied a system with a moving cylindrical heat source of constant temper- ature, considering conduction in the radial direction only. Ramey2 included conduction in the vertical direction in his studies. Bailey and Larkin2 attacked a more general problem where initial well heating, vertical heat losses and arbitrary frontal velocities were included. In all these studies, however, conduction was considered to be the only means of heat transfer. Bailey and Larkin in a later paper included the effects of convection in a study involving both linear and radial geometries. Vertical heat losses were neglected in the radial case. Katz5 studied a similar problem in a one-dimensional radial model, using a heat-loss coefficient to account for vertical heat losses. Selig and Couch6 mployed a cylindrical model and investigated two limiting cases. In one case they considered no heat loss from the reservoir whereas in the other they assumed a constant temperature at the interface between the reservoir and its bounding formations. Thomas' studied a more general case but assumed a permeable bounding formation so that the convection effect is not confined to the reservoir. In the present work a more realistic and more generalized model is used. It involves a two-dimensional cylindrical system with combustion, convection and conduction inside the reservoir, but only conduction in the bounding formations. The purpose is to establish the temperature distribution both inside and outside the reservoir, to study the general features of the radial heat wave process, and to investigate the effects of various process variables on the performance characteristics of the process. THEORY We fist consider a circular porous reservoir of thickness H extending vertically from y = — H/2 to y = + H/2. The reservoir extends from a well bore radius r, to an external radius re. A stream of oxygen-containing gas is introduced into the reservoir through the wellbore. The oxygen-containing gas reacts with the fuel contained in the reservoir and forms a combustion front wherever the prevailing temperature can support the combustion. It is here assumed that this combustion front constitutes a cylindrical surface source of heat having an infinitesimal thickness in the radial direction and extending vertically throughout the whole thickness H. This is designated as Region I. We next consider a Region II corresponding to the upper and lower formations bounding the reservoir, extending from y = — - to y = — H/2 and from y = + H/2 to y = + m. Since r, is very small, we may assume that the two bounding formations have the same dimensions, symmetric with respect to the center plane of the reservoir. In this way, we may take the upper half of the system alone into consideration. In contrast with

By R. G. Hawthorne

This report is concerned with fluid displacement in porous media, in those cases where viscous and gravitational forces control the displacement. Such a system would usually be found in a sand body of large physical dimensions such as an oil reservoir, although it is possible to create such a system in the laboratory. It is shown that the position of the fluid interface can be predicted by numerical calculations using a basic idea presented by Dietz. Fluid flow is considered in a vertical plane in a homogeneous, porous medium of sufficient thickness that the capillary transition zone is small in comparison with the total reservoir. A theory developed by Dietz' is used to make numerical calculations of the position of the fluid interface. The results for several conditions are compared with scaled model experiments. The results show that, for gas drive in a reservoir of steep dip, a relatively low flow rate can displace large volumes of oil before gas breakthrough. On the other hand, water injection at favorable mobility ratio and low dip may show best performance at high rates. Water tends to underride the oil and, given sufficient rime, will break through without much oil displacement. For certain conditions, which include relatively low flow rate, the interface is a straight line and its behavior is simple to calculate. At higher flow rates, the interface is unstable, and a numerical solution was programed for an automatic computer. In general, good agreement is shown between the fluid model and the computed results so long as gravitational forces have control. For a water drive at very unfavorable mobility ratio, many small water fingers appear. These viscous fingers are not controlled by the relatively small gravitational forces. When viscous fingering becomes the controlling factor, the mathematical model is oversimplified, and results do not check the fluid flow model. INTRODUCTION Present methods of reservoir analysis depend upon certain simplifying assumptions to obtain mathematical descriptions of practical use. Material-balance methods (Muskat2 or Tarner2) assume uniform fluid saturations in the entire reservoir, or in a few subdivisions of the reservoir. An unsteady-state flow calculation by West, er at considered pressure and saturation changes in flow to a well during solution gas drive, and neglected gravity effects. Results showed only a 4 per cent difference for ultimate oil recovery by the Muskat method, even though the case chosen for study was one in which unsteady-state effects should be high. The Buckley-Leverett5 method commonly assumes a one-dimensional flow system. It is applicable at high flow rates where viscous forces predominate over gravity forces. Simultaneous, parallel flow of the two fluids is assumed, and the concept of a fluid interface is not introduced. Permeabilities to each fluid for a given saturation must be known. The method is not applicable for a two-dimensional system where cross flow becomes possible. Less well known is the displacement equation derived by Dietz. This method is designed for two-dimensional flow systems and assumes a definable fluid interface within the porous medium. Dietz showed that, for a range of low flow rates, the interface would be stable. straight and at an angle of inclination which could be simply calculated. At a certain critical flow rate, the calculated interface tilt would equal the formation dip. For higher flow rates, a finger of displacing fluid would invade the displaced fluid. Dietz indicated that his method applied only to macroscopic reservoir behavior, while the Buckley-Leverett method applied to the small transition zone at the fluid interface. The examples worked out in this report are based on the fluid-displacement theory of Dietz. It is shown that the Dietz theory may be used to derive equations analogous to the Buckley-Leverett equations. In contrast to the Buckley-Leverett method, flow is considered in a plane rather than being limited to a line. Rather than a frontal advance, the movement of a fluid interface is followed. For flow rates substantially exceeding the critical rate and for high viscosity ratio, many fingers of invading fluid occur-—rather than the single finger assumed by Dietz. On the other hand, so long as some gravitational influence remains, the flow is not entirely parallel to the bedding planes as assumed by Buckley and Leverett; therefore, both methods fail to give an adequate descrip-

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

Flow of slightly compressible fluids through porous media can be described by linear equations. Many mathematical solutions- are available for the boundary conditions of constant terminal pressures or constant flow rates, or a combination of both. Various techniques of formation evaluation use drastic rate changes, such as sudden shut-in. Other methods require that the well be produced at a constant rate. In practice, one is not always able m fulfill these rigid requtirements. This report shows a methor1 by which varying-rate data can be converted to a constant-rate pressure response, called the unit response function. After obtaining the unit response function, one may analyze it, using techniques described in the literature. The method uses tile principle of superposition. Necessary data is a time history of pressure and production. If pressure changes and/or rate changes are very drastic, the unit response function becomes erratic and oscillates The erratic and oscillating results are obvious, and under those circumstances, no answer can he obtained. It appears, however, that a response function without oscilla-tion will not contain any appreciable errors —- one gets either good results or no results. This report includes a mumber of examples of converting constant-rate data to constant terminal-pressure data and vice versa. In addition, a number of field-test applications are given. This method should be useful in converting mathematical solutions from one boundary condition to another. It already has proven useful in converting varying-rate data to constant-rate conditions. INTRODUCTION Pressure drawdown tests are performed on production wells to determine various reservoir characteristics, including permeability, distances to discontinuities, reservoir continuity and wellbore conditions. An ideal drawdown test is performed at a constant production rate with pressure observation during the drawdown period. A typical drawdown curve for a constant rate of production is shown in Fig. 1. In practice, a constant production rate cannot always be maintained. Fig. 2 shows a pressure drawdown curve for a varying rate. Usual methods of analysis are not applicable to tests such as this. It is, therefore, necessary to convert the varying-rate data somehow, before usual analytical techniques can be applied. This report describes a method of obtaining a response function. A response function can be defined as the relationship between pressure and time for a well producing at a constant unit rate. Such a response function can be analyzed by the usual analytical techniques. Hutchinson and Sikora' first approximated a resistance function (response function), and by applying the superposition principle, pressures were calculated using the varying rates. If no match was obtained, adjustments were made to the response function and the closer approximation was used again in an attempt to match calculated and observed pressure behavior. If a good match was obtained: the response function was then used to extrapolate the future aquifer behavior.

Jan 1, 1966

By V. J. Driscoll

A method is given for checking pore volume reserve estimates and transient water inflzix factors utilizing early well performance. Principal applications are based on the observation that there is a convenient transient time-range in terms of well spacing in which observed pressure drop will give a good indication of fluid-in-place. Primary quantity determined is a porosity-feet value applicable in the vicinity of an observation well. Checks on porosity or net pay thickness and permeability values also are obtained. These are particularly valuable in improving quality of interpretive factors entering into pore-volume reserve determinations for larger fields. In smaller fields, direct determination of total field reserves and approximate location of boundaries may be made in some cases. Well interference and production data plus a build-up curve on the observation well are needed. The build-up curve may be taken at any time. Field applications are given. The method can be applied with the completion of the second well on a new discovery or to older reservoirs by major changes in withdrawal rates from a local area. Appropriate test length permits results to be obtained with equal accuracy on widely spaced wells. Criteria for optimizing test conditions are given. Provision is made for various boundary and well effects including incomplete observation well build-up. INTRODUCTION Due to low quality data, initial pore volume reserve and water influx estimates may often vary by a factor of two or more from results obtained at a later date. Unfortunately, by conventional methods, more reliable answers from performance usually are not available until after most significant economic decisions have been made. Furthermore, in older reservoirs, uncertainties may still exist as to original oil-in-place, recovery factors and residual oil saturations due to incomplete or inaccurate performance data. This paper outlines situations and appropriate arrangements of basic transient theory whereby well interference data can be utilized to improve the quality of estimates under certain conditions. In addition to production data on offset producers, key information required is a measured pressure drop at an observation well, corrected for any incomplete build-up due to previous production, plus a formation capacity value derived from a conventional pressure build-up plot on the observation well. The method, as outlined, is strictly applicable only to a horizontal single-layered system of uniform rock characteristics containing a homogeneous fluid of low and constant compressibility. However, under the set of test conditions devised, requirements can usually be relaxed so that it should be applicable to most low permeability gas, water and undersaturated oil reservoirs. This is an area where more reliable information is certainly needed. BASIC CONSIDERATIONS A, B, C and F as subscripts or in the text refer to producing Well A, observation Well B, a more remote producing Well C and field boundary F. Combinations such as AB refer to distances and dimensionless times in terms of distances between these points, while single letters refer to distances and dimensionless times in terms of the well-bore or field radius. Equations covering definitions of quantities and complex situations are given in the Appendix. Key equations in practical units are given in Table 2. Transient Pressure Behavior Fig. 1 illustrates pressure drop at Point B with time due to initiation of a constant rate of withdrawal from Point A. This is shown as since a quantitative under-

By M. A. Jones

Performance of a pilot flood in eastern Kansas indicates improved recovery and accelerated production resulting from mobility ratio control obtained by adding a high moleculur weight polymer to injected water. Ultimate pilot flood recovery of the 75-cp reservoir oil is anticipated to he 350 bbllacre-ft during the six-year flood life. Cumulative recovery to date totals 242 bbl/acre-ft after 30 months of injection. Reservoir parameters and flood performance are disc~rssed in detail to provide a case history relating to the feasibility of polymer flooding. INTRODUCTION The Vernon Polymer flood of the Brazos Oil & Gas Co., located in Sec. 2, 24 S., 16 E., Woodson County, Kans., was begun in late Oct., 1963. Injection of a 500 ppm solution of Pusher* polymer was initiated in a 15-acre pilot to determine feasibility of improving oil recovery by mobility ratio control. Performance to date and the anticipated recovery of 115,000 bbl from the 329 acre-ft pilot indicate efficient displacement of the viscous reservoir oil. Dependence of flood recovery on the displacing fluid-displaced oil mobility ratio has long been recognized. Displacement efficiency of the reservoir oil decreases with increasing mobility ratio. Therefore, economic reduction of this parameter is an attractive approach to improving oil recovery. particularly from viscous oil reservoirs where mobility ratios are often quite high. Waterflood mobility ratios can be controlled by dilute solutions of certain polymers'; which provide a displacing fluid with considerably reduced mobility. Reservoir parameters and flood performance are emphasized in this paper to provide data relating to polymer flood applicability. THEORY The polymer used in the Vernon pilot exhibits the property of unusually high resistance to flow through porous media.'.' A small concentration in injected water effects a mobility reduction considerably beyond that predicted from increases in bulk viscosity as measured in a capillary or rotating cup viscosimeter. Water mobility in the subject flood is reduced by a factor of 7.55 with the addition of 500 ppm Pusher chemical to injected water while the bulk viscosity is increased from 1.03 to only 1.40 cp at 75F. The exact flow mechanism causing resistance to flow has not been established; however, it is observed only in tortuous flow systems and appears to be influenced by brine properties, rock properties and oil saturation. Consequently, mobility reduction is estimated from flow studies for a given sample of reservoir rock and is expressed in terms of a "resistance factor".' This parameter is defined as the ratio of brine mobility to polymer solution mobility measured at residual oil saturation: RF= ?m/?p= (km/µm)/(kp/µp)M.......(1) where RF = resistance factor (dimensionless) k*. and A, = mobility of water and polymer solution, respectively k,,, and k,, = water and polymer permeability, respectively (darcies) µie and µp = viscosity of water and polymer, respectively (poise). Eq. 1 is similar to the expression commonly used to relate the relative mobility of the water bank at residual oil saturation to oil-bank mobility at irreducible water saturation: where M =mobility ratio of water to oil (dimension-less) A, = mobility of oil k, = permeability to oil (darcies) )i* = oil viscosity (poise). From Eqs. 1 and 2 it is apparent that the polymer-solution-to-oil-mobility ratio can be approximated by dividing the brine-to-oil ratio by the resistance factor RF: Mpn=M10/RF...........(3) where M,-, = mobility ratio of polymer solution to oil (dimensionless). This simplified presentation assumes no dilution of the polymer solution by connate water. Also, the presence of a connate water bank separating the polymer solution from the displaced oil is not considered. Displacement of oil by waterflooding is increasingly dominated by viscous forces as the viscosity of the oil displaced increases. When the oil mobility is less than that of the injected water, the oil-water interface is not a piston-like front.3,' Instead, instabilities occur at the flood front in the form of viscous fingers which tend to penetrate the less mobile oil bank. The magnitude of these protrusions, or the degree of channeling, increases

Jan 1, 1967

By A. J. Conelius, D. P. Sobocinski

This paper presents a correlation for predicting the behavior of a water cone as it builds from the static water-oil contact to breakthrough conditions. The correlation is partly empirical and involves dimensionless groups of reservoir and fluid properties and of production and well characteristics. The groups were deduced from the scaling criteria for the immiscible displacement of oil by water. The correlation is based on a limited amount of experimenfal data from a laboratory sand-packed model and on results from a computer program for a two-dimensional, incompressible system. Because the correlating groups are dimensionless, they can be used to estimate the performance of water coning cases not specifically considered in the correlation. However, despite its dimensionless nature, the correlation is not completely general and will not provide meaningful estimates of cone behavior in many situations. INTRODUCTION AND BACKGROUND The production of water from oil wells is a common occurrence which increases the cost of producing operations and may reduce the efficiency of the depletion mechanism and the recovery of reserves. We will deal with one cause of this water production, namely, coning. The coning of water into a producing well is caused by pressure gradients established around the wellbore by the production of fluids from the well. These pressure gradients can raise the water-oil contact near the well where gradients are most severe. Gravity forces that arise from fluid density differences counterbalance the flowing pressure gradients and tend to keep the water out of the oil zone. Therefore, at any given time, there is a balance between gravitational and viscous forces at points on and away from the completion interval. When the dynamic forces at the wellbore exceed gravitational forces, a cone of water will ultimately break into the well to produce water along with the oil. We can expand on this basic visualization of coning by introducing the concepts of stable cone, unstable cone and critical production rate. For instance, if a well is produced at a constant rate and the pressure gradients in the drainage system have become constant, a steady-state condition is reached. If, at this condition, the dynamic forces at the well are less than the gravity forces, then the water or gas cone that has formed will not extend to the well. Moreover, the cone will neither advance nor recede, thus establishing what is known as a stable cone. Conversely, if the pressure in the system is in an unsteady-state condition, then an unstable cone will continue to advance until steady-state conditions prevail. If the flowing pressure drop at the well is suffcient to overcome the gravity forces, the unstable cone will grow and ultimately break into the well. It is important to note that in a realistic sense, stable cones may only be "pseudostable" because the drainage system and pressure distribution generally change. For example, with reservoir depletion, the water-oil contact may advance toward the completion interval, thereby increasing chances for coning. As another example, reduced productivity due to well damage requires a corresponding increase in the flowing pressure drop LO maintain a given production rate. This increase in pressure drop may force an otherwise stable cone into a well. The critical production rate, well known in the literature, is the rate above which the flowing pressure gradient at the well causes water (or gas) to cone into the well. It is, therefore, the maximum rate of oil production without concurrent production of the displacing phase by coning. At the critical rate, a built-up cone is stable but is at a position of incipient breakthrough. Numerous papers have been published on critical rates. Some of the better known of these include the work of (1) Muskat and Wyckoff,' who first dealt with the coning problem; (2) Chaney, et al,' who developed expressions similar to those of Muskat but who presented results in a conven ient-to-use graphical form (the "Sun" method); and (3) Meyer and Garder,b hose analysis is based on radial-flow formulas. One assumption in critical production rate analyses is that the cone has built-up to just before its breakthrough into the well. But, these analyses reveal nothing directly about the time it takes for the cone to build up to this incipient breakthrough position. Thus, water-free oil can be produced from a well for prolonged periods at rates above the critical rate before the well reaches the condition to which the critical rate applies. The published literature contains little on the rate of growth of a cone. Experimentally, Meyer and Searcy studied the rate of rise of a cone in a Hele-Shaw model.' Additional related work on water breakthrough and produced water-oil ratios in water driven reservoirs was reported by Muskat," Hutchinson and Kemp," Henley, et al,' and Stevens, et a1.B Theoretically, the basic coning equations for a water-oil system can be developed by applying the conservation of mass to each of the phases, relating flow velocities with pressure by Darcy's law, and relating pressures across water-oil interfaces by capillary pressure. With the usual boundaries at the well and reservoir limits, the solution of the resulting equations for the time behavior of a water-oil interface constitutes a free-surface, boundary-value problem. The shape of the boundary depends on the internal potential distribution, while this

Jan 1, 1966

By R. J. Wagner, F. F. Craig, H. G. Riley, J. D. Griffith

Peripheral water injection has been underway in the Sholem Alechem Fault Block "A" Unit, Stephens County, Okla., since 1955. In the engineering planning of the flood, it was recognized that maintenance of pressure in the large, dry gas caps was vitally necessary to prevent the loss of an estimated 27 million bbl of oil through saturation of the gas caps by the oil banks moving updip on the peripheral flood pattern. The development of an abundant, low-cost water supply and the rising cost of makeup-gas purchases indicated that gas-cap control by water injection was desirable from an economic standpoint, provided an eflective water barrier could be placed without premature water breakthrough into downdip oil producers. A laboratory model scaled to the existing reservoir conditions was comtructed to evaluate the feasibility of gas-cap water injection. Model studies indicated that (I) water injected at the edge of the gar-oil contact would form a continuous barrier between the oil and gas zones, and (2) water would advance uniformly downdip without segregation due to gravity effects provided a specified, minimum water-injection rate was maintained. The model test results also indicated that it was not necessary to fill the gas cap completely with water. Water injection at the edge of one of the gas caps was initiated in March, 1958, to field test gas-cap water injection. Water breakthrough occurred at a downstruc-ture producing well in April, 1959, after approximately 2.5 million bbl of water had been injected into the gas cap. This was the volume calculated to be required to form the barrier between the oil zone and gas cap. This performance indicates that prevention of oil encroachment into dry gas caps in a peripheral water flood by water injection at the gas-oil contact appears practical and that the model studies provided a valid prediction of field performance. INTRODUCTION The Sholem Alechem Fault Block "A" Sims Sand Unit was one of the fist major waterflood projects in Oklahoma to utilize a peripheral injection pattern. Design of control measures to minimize oil encroachment into the large crestal gas caps was one of the greatest single problems during engineering planning of this flood. Engineering studies indicated that saturation of the gas caps with oil displaced by the peripheral water flood could result in a loss of about 27 million bbl of oil, or over one-half the estimated waterflood recovery. This estimate was based on a residual oil saturation after water flooding of 35 per cent pore volume, compared to the negligible oil saturation initially in the gas caps. The Sholem Alechem Springer field, located in Stephens County, Okla., is an elongated anticline with structural dips varying from 400 to 1,000 ft/mile. The structure contains numerous faults, some of which are sealing. The portion of the field considered in this paper is Fault Block "A", which is the largest of the five major fault segments in the field. A map of the unit is shown on Fig. 1. The unit is bounded on the east and west by major sealing faults, on the north by a water-oil contact and to the south by a deterioration of sand development. Two domes, each having gas caps, exist in the unit and are separated by a north-south fault as shown in Fig. 1. The sands being flooded are the Sims sand members of the Springer series of early Pennsylvanian age. The Sims sands average over 100 ft in thickness in Fault Block "A" and have a porosity of approximately 19 per cent. Core permeabilities average 160 md, with interstitial water saturation of about 23 per cent pore volume. Three different zones can be correlated over most of the area. However, they are commingled in most of the producing wells. The two upper zones, designated as First and Second Sims, are the most important. Both the First and Second Sims have gas caps in the east dome, while only the First Sims has a gas cap in the west dome. These gas caps, whose limits are also shown on Fig. 1, initially occupied nearly 11 per cent of the total reservoir volume. The major concern of this paper is the control of the two gas caps in the First Sims sand. Fig. 2 is a cross section, to scale, across the east dome of this formation. Notable features are the relatively shallow dip, the deterioration of the First Sims sand on the south side of the gas cap and the water-oil contact on the northern edge. The formation dip at the gas-oil contact in this cross section is approximately 5".

By R. F. Nielsen, R. G. Bentsen

Jan 1, 1966

By G. L. Chierici

Starting from Muskat's theory of water and gas coning, maximum permissible oil production rates without water and/or free-gas production have been determined, in a broad range of reservoir and well parameters, using the potentiometric model technique. The main assumptions made are as follows: (1) the reservoir rock is homogeneeous (either isotropic or anisotropic); (2) the volume of the aquifer underlying the oil zone is very small, so that it does not contribute to reservoir energy; and (3) the gas cap expands at a very low rate, so that it can be assumed to be in quasi-static conditions. The results obtained are presented in the form of diagrams which can be used for solving two types of problems: (1) given the reservoir and fluids characteristics, as well as the position and length of the perforated interval, determine the maximum oil production rate without water and/or free-gas production; and (2) given the reservoir and fluids characteristics only, determine the position and length of the perforated interval which optimize the maximum permissible oil production rate, without water and/or free-gas production. INTRODUCTION In oil reservoirs where the oil-bearing formation is underlain by an aquifer which does not participate in the production mechanism, water-coning is a limiting factor to the flow rates of producing wells. Production rates are usually kept to a value that will prevent the water from entering the wells. The entry of water into a well lowers its productivity by increasing the weigbt of the fluid column; moreover, the separation of water from the effluent, at the surface, may constitute a very difficult problem in cases of heavy viscous oils. A similar situation is encountered in oil reservoirs with a gas cap overlying the oil-saturated zone; here a downward gas cone is induced by the flow of oil towards the producing wells. Production rates must be low enough to prevent the gas from being produced; producing gas from the gas cap would be a waste of energy. Of course, water-coning and gas-coning phenomena can occur at the same time in the same reservoir if the oil-producing formation is both overlain by a gas zone and underlain by a water zone. Due to its relevant practical importance, the mechanism of coning was studied by many people.2,3,5-8 Defining the conditions for getting the maximum water-free and/or gas-free oil production rate is a difficult problem, often encountered under one of the following aspects: 1. Predict the maximum flow rate that can be assigned to a completed well without the simultaneous production of water and/or free-gas. 2. Define the optimum length and position of the interval to be perforated in a well, in order to obtain the maximum water and gas-free production rate. A systematic study of these problems was made by means of the electrical analog technique. The results of this study are presented here, under the form of a set of curves providing solutions for the above stated problems. These curves are valid only for homogeneous forrnations, either isotropic or anisotropic. Should the formation be non-homogeneous (by horizontal or vertical variation of permeability, shale diaphragms, fractures, etc.), a specific potentiometric study would be required for each specific case. Especially when shale diaphragms of some radial extension are present, the critical rates observed are much larger than would be expected from the diagrams. STATEMENT OF THE PROBLEM In the present study the aquifer is supposed to be of such limited volume that it does not contribute to the energy of the reservoir. Moreover, the gas cap is supposed to expand at such a low rate that the potential gradient in the gas cap is negligible. Under static conditions water-oil and gas-oil interfaces (T1 and T2) are both horizontal. When the reservoir production starts, below each well these interfaces take a cone-like shape (Fig. 1) having as an axis the axis of the well. This shape results from the equilibrium between potential gradients in the oil zone and gravitational forces due to density differences between oil and water and between oil and gas. Assuming the oil-bearing formation to be homogeneous and the oil to be incompressible, the analysis of the problem (see Appendix) shows that the oil-water and gas-oil interfaces are stable only if the oil production rate of the well is not higher than the following values.

Jan 1, 1965

By R. W. Heins, N. Street

An investigation was made of the Rehbinder-Kuznetsov pendulum as an instrument to aid in the study of rock drillability and of the effect of surface active agents on rock drillability. The "critical weight", i.e., the pendulum weight at greatest sensitivity, was determined for several solids and it was shown that a change in surface environment changed the maximum value of the apparent hardness but not the critical weight, and the critical weight increased with increase of hardness of the solid used. Using silver halides as the solid material, and solutions of either silver nitrate or the appropriate alkali halide, it was shown that there is a hardness maximum at the zero point of charge of the particular silver halide. INTRODUCTION The development of relative hardness measurement by the method of damped oscillations, or what has been called the pendulum sclerometer, is due largely to the Russian workers P. A. Rehbinder and V. D. Kuznetsov and their co-workers.1,2 The apparent simplicity of the method has led to its use in a number of fields, particularly those where relative hardness has to be determined on a brittle solid. As will be shown, the simplicity of the method belies its complexities. THEORY OF THE PENDULUM SCLEROMETER The operation of the pendulum is quite simple. Make a sharp point such as a diamond, hardened steel or tungsten carbide, as the fulcrum of a pendulum, with the center of gravity of the system below the point of support. The attenuation of this pendulum is a function only of air friction, provided no fragmentation, breakage or other work is performed by the point. However, when the solid surface breaks or disintegrates under the point, and the latter penetrates the solid, part of the energy of oscillation is absorbed in the work of creating new surface, and the rate of attenuation increases. In the case of brittle solids, the point of the pendulum produces fragmentary particles, whereas the point produces plastic deformation in a plastic solid; but in both cases damping occurs. This pendulum damping is a function of the hardness or strength of the solid. Whereas a pendulum oscillating on a hard material attenuates primarily by air friction, in the case of a soft material, the points penetrate the surface of the solid, producing new surface, and the pendulum damps much faster. Thus the attenuation rate is a function of the hardness or strength of a material in both plastic and brittle substances. Russian workers initially used a single support for the pendulum, but this created many problems because of oscillation of the pendulum in more than one plane. A revised version used two points, or sometimes a prism as support, successfully confining oscillation to one plane. The version using two points was constructed and used throughout this investigation, making the recording of the attenuation as a function of time relatively easy. That both points should be in contact with the solid during the damping is implied. Rehbinder (c.f. Kuznetsov2) proposed that the hardness H as measured by this method could be expressed by a quantity which is inversely proportional to the relative initial rate of attenuation of amplitude. where H = hardness parameter (arbitrary units) a, = initial amplitude da/dt = rate of attenuation of amplitude with time. In the simplest case of damping with a constant logarithmic decrement, H can be regarded as a constant for the duration of the whole damped oscillation process. That is

Jan 1, 1966

By Caudle B. H., Cooper H. E. Jr., O. K. Kimbler

The results of studies of areal sweepout behavior in nine-spot injection patterns are presented in this paper. The ratio of the production rates of the two types of producing wells in a nine-spot pattern is considered in its effect on breakthrough. A model study was used to develop a method to predict the production histories of both side and corner wells and to develop the production history of the pattern or the field. INTRODUCTION The importance of areal sweepout behavior in injection operations has been recognized for a number of years. We now know that the fraction of a reservoir which will be reached by an injected fluid is a function of the geometric arrangement of the production and injection wells and of the mobility ratio of the fluids. Of these two, the mobility ratio (which is a measure of the relative ease with which the oil and the injected fluid can move through the reservoir) appears to have the greatest effect on recovery. The effect of mobility ratio on production history has been reported for most of the dispersed injection patterns (i. e., five-spot, staggered and straight-line drives).4-6 ere has been very little published on the nine-spot injection pattern. This may be because the two types of producing wells in the nine-spot pattern can—and sometimes should —be produced at different rates. This adds a second variable (the fist being mobility ratio) to be covered in a study of the nine-spot flooding network. Muskat1 has reported the sweep pattern efficiency at breakthrough for a mobility ratio of one at various ratios of producing rates for the two types of wells. For these conditions he found maximum oil production at breakthrough to occur when the corner well (the producing well farthest from the injection well) is produced 10.7 times as fast as the side well. Other investigatorsa2,3 have studied a few of the many possible operating conditions possible with this injection pattern. This paper presents the results of studies which are strictly applicable to a nine-spot injection pattern surrounded by other nine-spots in a homogeneous reservoir and which has no flowable gas present (no oil bank is formed). Although the full ranges of the variables mentioned above have not been covered, the results reported here should allow reasonable estimates of field operations to be made. EXPERIMENTAL The well spacing geometry of the nine-spot pattern is shown in Fig. 1. Here we see that each injection well is surrounded by eight production wells arranged in a square. Four of these producing wells are located at the corners of the square; the other four are in the midpoints of the four sides. These two types of producing wells are called "comer wells" and "side wells" respectively. It is the ratio of the producing rates of these two types of wells that affects the breakthrough recovery, as calculated by Muskat.' This ratio is defined as the total producing rate of a corner well divided by the total producing rate of a side well. The model used in this study represented one-eighth of a nine-spot pattern, as shown in Fig. 1. It contained the equivalent of one-fourth of a side well, one-eighth of a comer well, and one-eighth of an injection well. Because of these proportions, the actual producing rate ratios in the model were one-half of the ratios of producing rates as

Jan 1, 1965

By C. D. Stahl, R. Sandrea

This investigation was made to evaluate the comparative effects of the viscosity and the phase behavior of the buffer fluid in the composite solvent displacement of Bradford crude from waterflooded sandstone cores. Buffer slugs of propane, naphtha and other refined hydrocarbons exbihiting increasingly favorable viscosity ratios and decreasing solubility relationships, respectively, with Bradford crude, were used in long Berea sandstone cores. The secondary slug was isopropyl alcohol. The results indicate that higher oil recoveries are obtained for increasingly favorable phase relationships even when these are accompanied by unfavorable viscosity ratios within the range studied. Moreover, when propane is used as a buffer slug with an adverse viscosity ratio of 36, it gives higher oil recoveries than at a similar size slug of an any1 alcohol having a viscosity ratio of 1.09. The investigation was extended to the study of the effect of flooding rate on oil recovery. Residual Bradford crude was displaced from a 6-ft Berea sandstone core at rates varying from 0.3 to 30 ft/ day. The results show that as flooding rates were increased above or decreased below a minimum range of 1 to 2 ft /day, displacement efficiency increased considerably. INTRODUCTION In recent years, the alcohol slug process as a means of tertiary recovery has been the subject of many investigations. Gatlin and Slobod showed the advantages of the combination solvent slugs over the previous single-slug process.1 The combination slugs require less total slug material to obtain similar oil recoveries. The basic combination process involves injecting a primary or buffer solvent which is preferentially oleophilic, followed in turn by a water-driven slug of isopropyl alcohol. This procedure ensures that the buffer slug, which is miscible only with the reservoir oil, displaces and replaces the oil and is in turn displaced, together with the water, by the isopropyl alcohol slug miscible with both fluids. Likewise, for those reservoir oils that show some degree of phase incompatibility with isopropyl alcohol, as does Bradford crude, a proper buffer solvent can artificially make the alcohol process technically feasible. From an economic standpoint, propane or LPG would be the most appropriate primary solvent to be used in the field; however, Taber, et a12 have indicated that the alcohol process becomes highly inefficient in the displacement of light hydrocarbons because of the entrapment of these hydrocarbons during the stabilized bank formation. Moreover, since materials such as propane have such low viscosities, adverse viscosity ratios with most reservoir oils would be penerated, thereby possibly further reducing the efficiency of the buffer solvent. This paper presents the results of an experimental study of the comparative effects of viscosity ratio and phase relationships on the displacement efficiency of Bradford crude by means of dual solvents. The work was extended to include the effect of flooding rate on oil recovery. Rates within the range of field and laboratory values were used. EXPERIMENTAL PROCEDURE APPARATUS The porous media used in this study consisted of two 10-ft and one 6-ft Berea sandstone cores, 3-in. in diameter. These were cast in galvanized steel pipe of 31/2-in. ID and the annulus was filled with Armstrong cement. The 10-ft cores were assembled by connecting two 5-ft units, which were joined together by 8-in. steel flanges fitted to the pipe. Intimate contact between the two sections was obtained by inserting a thin disk of the same sandstone between the core faces. The outlet and inlet ends of the cores were covered with threaded pipe caps. The injected fluid was gravity fed to two variable-rate positive-displacement pumps. The fluids were filtered before entering the core and the effluent was collected in an automatic fraction collector and

Jan 1, 1966

By B. D. Smith, K. H. Coats

Experiments in which calcium chloride displaced sodium chloride from four cores showed the extent of asymmetry in the resulting effluent concentration profiles. These results provided a check on how validly the mixing process is modeled by a differential (i.e., not finite-stage) capacitance mathematical model. The effluent concentration profile from two consolidated cores exhibited considerable asymmetry, while two unconsolidated cores yielded nearly symmetrical profiles. All runs resulted in breakthrough of the 50 per cent concentration significantly before one pore volume was injected. In addition, velocity appreciably affected the effluent concentration profile from a Torpedo sandstone core. The differential capacitance model matched the data significantly better than a simple diffusion model. The capacitance model allows determination of the amount of dead-end pore space in a porous matrix and the effect of velocity on the rate of diffusion into this space. An experimental program yielding insight into the physical validity of the capacitance effect is described. INTRODUCTION Axial dispersion — the mixing accompanying the flow of miscible fluids through porous media — has been the subject of many relativeiy recent studiesl-13 and a comprehensive review of the topic has been given by Perkins and Johnston.14 This dispersion is of practical interest in studies of the miscible displacement process, fixed-bed chemical reactors, and the adsorption of solutes from a flowing stream onto the surface of a porous medium. In the latter case, the effect of dispersion must be considered when adsorption parameters are determined from the nature of concentration profiles. In general, early studies of dispersion assumed applicability of a simple diffusion equation and were concerned with correlation of the experimentallv determined "effective" diffusion coefficient with system properties over a large range of the latter. Recent have been concerned with the deviations between the asymmetrical effluent concentration profiles observed and the symmetrical ones predicted by the diffusion model. In the present study, effluent concentration profiles were obtained from consolidated and unconsolidated cores. These profiles were compared with those predicted by a differential (i.e., not finite-stage) capacitance model. Solutions to the simple diffusion model, for three sets of boundary conditions, were compared with one another and with the experimental profiles. SUMMARY OF PREVIOUS WORK The reader is referred to Perkins and johnston14 for an extensive review of studies of dispersion in porous media. Many investigators have employed the simple diffusion model characterized by Eq. 1 below: The dispersion coefficient D for unconsolidated systems is correlated by u is interstitial velocity and dp is particle diameter. Since heterogeneity of the sand pack affects the mixing, this equation is also expressed as where a is proportional to the degree of heterogeneity and is about 3.5 for random packs of unconsolidated sand.l4 Eq. 3 holds for 2 For a Do homogeneous (regular) type of packing, should be 1 or less.6 Aris and Amundson6 and Carberry and Bretton consider o to be the number of particle lengths per mixing cell in the finite-stage model. Data from consolidated cores indicate dp to be about 0.36 cm for outcrop rocks, Torpedo sandstone having a reported value of 0.17 cm.15 For v /Do less than about 2, the effect of molecular diffusion must be considered and the total dispersion coeffi-

Jan 1, 1965