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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/

By L. W. Holm, A. K. Csaszar

A series of laboratory experiments was conducted in which oil was displaced from a porous medium by water-driven slugs of alcohols or similar solvents. The solvents used were soluble to some degree in both oil and water and covered the range of solubilities from complete solubility in oil to complete solubility in water. Displacement experiments were conducted on 2-and 3%-in. in diameter consolidated cores and 1-and 2-in. in diameter unconsolidated sand packs. The cores and sand packs ranged in length from 1 to 30 ft, and they were saturated with brine and crude or refined oils. The solvents used included ethyl alcohol, isopropyl alcohol (IPA), tertiary butyl alcohol (TBAI, secondary butyl alcohol (SBA), n-amyl alcohol (NAA), methyl ethyl ketone, acetone and others. It was found that all of the oil present in a porous medium could be miscibly displaced by injecting a slug of mutual solvent and driving it with water. The oil-recovery efficiency was dependent upon (1) the relative solubilities of the solvent in oil and water, and (2) the distance traversed by the flood. For complete oil recovery from cores, a smaller amount of a preferentially oil-soluble solvent was required, compared to the amount of preferentially water-soluble solvent needed. The size of the solvent slug required varied inversely with the linear flooding-path length raised to the 0.65 power. Water-driven dual solvent combinations (an oil-soluble solvent slug followed by a water-soluble solvent slug) were found to effect complete oil recovery with less total solvent than any single solvent used. In these dual-solvent displacement experiments, the slug size required varied inversely with length raised to the 0.55 power. Based upon the experimental results, a theory was developed to describe the displacement of oil and water by mutual solvents, and equations are presented to predict the production history in a linear system. These equations take into account the properties of the solvents and the porous medium. INTRODUCTION Oil-recovery processes which utilize displacing fluids that are miscible with the reservoir fluids have been studied extensively in recent years. Because of the poor contact efficiency and high pressure requirements of the LPG-gas displacement process there has been considerable interest in the alcohol-water process, and a number of studies have been made on the recovery of oil through the use of solvents which are mutually soluble in oil and water. An investigation by Sievert, Dew and Conleyl indicated that the use of mutual solvents would be limited because the presence of water in a porous medium would cause a phase break in the leading edge of the displacing solvent. Their study also showed that, in consolidated cores containing. only oil, the displacement of oil by a water-driven mutual-solvent slug of tertiary butyl alcohol (TBA) was affected by the viscosity ratios of the fluids involved. Gatlin and slobod3 concluded that an isopropyl alcohol (IPA) slug acts as a miscible piston, completely displacing both oil and water until the alcohol content of the mixing zone falls below the concentration necessary to maintain miscibility. Their study was conducted on uniform unconsolidated sand packs. They concluded further that IPA could be used effectively to recover oil from a watered-out sand. In a paper by Taber, Gamath and Reed4 relating an investigation on sandstone cores, it was stated that the displacement of oil by mutual solvents, particularly IPA, was not a miscible displacement and that no improvement in efficiency could be expected with increase in flooding-path length. However, their analytical analysis of the displacement mechanism using TBA is, in fact, one which indicates that the displacement is controlled by miscible mixing. They suggested that the lack of improvement in efficiency with flooding - path

By M. Prats, D. G. Russell

The performance of a well in a bounded, layered reservoir with interlayer crossflow has been investigated mathematically. The system studied comprises a centrally located well in a bounded cylindrical reservoir composed of two layers of contrasting physical properties. The reservoir contains a single fluid of small and constant compressibility. In the case of a well producing at constant pressure, the production rate of the crossflow system declines exponentially after the early transient behavior dies out. For wells which produce with constant rate, the pressure-time relationship is linear following the transient. For very early production time, the production rate of the two-layer crossflow system behaves as that of the two-layer stratified system (the case in which the layers are in communication at the well only). Except for early time, in most practical cases the performance of a two-layer reservoir with cross-flow can be duplicated by that of a single-layer reservoir. The equivalent single-layer reservoir must have the same pore volume and the same drainage and wellbore radii as in the two-layer case. In addition, the equivalent reservoir must have a "kh" (permeability x thickness) product equal to the sum of the "kh" products of the respective layers in the crossflow system. If the porosity and permeability distributions are known, the production and pressure behavior of a well with and without crossflow can be predicted. From a comparison of the observed pressure or production performance of a well and the theoretical behavior, the occurrence of interlayer crossflow can be detected. INTRODUCTION When flow is possible across the bedding plane from a low-permeability layer into an adjacent layer of moderate or high permeability, the earlier depletion of the more permeable layer will cause such flow to occur. If this situation exists, the oil recovery from tight layers can be appreciable; thus, the economic ramifications of interlayer crossflow are apparent. The possible importance and extent of flow between adjacent and connected layers — whether resulting from capillary, gravitational, or viscous forces — have been the subject of considerable speculation for many years. Jacquard1 presented a rigorous analysis of the crossflow due to single-fluid expansion occurring between two adjacent and connected layers when the well produces at constant rate. His presentation was entirely mathematical, however, and no numerical results from evaluation of the mathematical solutions were presented. Our objectives in the preparation and presentation of the present paper are essentially twofold— (1) to analyze the behavior of a well which produces a single compressible fluid from a bounded, layered reservoir with interlayer crossflow, and (2) to present practical numerical results and simplified formulas to facilitate engineering analysis of reservoirs with crossflow. The constant-terminal-pressure case was analyzed rigorously, whereas the constant-terminal-rate case was treated only for semisteady-state conditions. DESCRIPTION OF THE IDEALIZED TWO-LAYER RESERVOIR The reservoir to be studied is assumed to be bounded and cylindrical in shape, with a completely penetrating well at its center. The reservoir is composed of two adjacent horizontal layers between which fluid is free to flow. Each layer is assumed to be homogeneous, isotropic and completely filled with a fluid of small and constant compressibility. The total thickness of the reservoir is small enough

By J. B. Agan, R. J. Fernandes

This paper utilizes the layered-system approach, modified to include areal sweep efficiency, to determine the miscible-slug size required and to predict the perfort7zance of a miscible-slug process in a highly stratified reservoir having a permeability variation of 0.76. The slug size in each layer was determined from theoretical calculations and laboratory data which showed that, to maintain miscibility in the reservoir, the slug size had to be 8.2 per cent of the floodable hydrocarbon pore volume. Economic considerations, however, such as the value of the miscible product injected, the cost of injection and the cost of reprocessing the miscible product, limited the amount of miscible product which could be injected into the lower-permeability layers. This resulted in subjecting only 60 per cent of the reservoir to a slug size sufficient to maintain miscibility. Utilizing the relationship of producing gas-oil ratio and areal sweep efficiency as a function of pore volume injected (as obtained from data published by Dyes, Caudle and Erickson2), the theoretical performance was determined. This prediction is compared with actual performance to date. Actual performance reflects an increasing percentage of tniscible product in the produced fluid to a particular level, which remains static for a short period and then increases again to a higher level. This performance suggests that the reservoir has performed as a layered system. It is concluded that the layered-system approach can be used to predict performance in a highly stratified reservoir and that economic consideration is a limiting factor in determining the amount of the reservoir which can be miscibly swept. INTRODUCTION The Paloma zone in the Paloma oil field, Kern County, Calif., is comprised of several sand members separated by siltstone, shale and/or cherty sections. Rapid facies changes within the zone indicate that the sediments were deposited in a "near shore" environment. As a result, the Paloma sands have a high permeability variation, and the Paloma zone is comprised of several sand reservoirs. Since one of the sand reservoirs containing crude oil was not receiving the benefits of gas injection, consideration was given to recompleting certain wells to isolate the sand in order to perform a miscible-phase injection project and, thereby, to increase its ultimate oil recovery. Before proceeding with the project, a review of the literature was made to determine the miscible-slug size that would be required to maintain miscibility throughout the displacement process. Although the review showed that considerable information was available for homogeneous systems, insufficient information was available to determine slug-size requirements and to predict performances for a highly stratified reservoir. This paper has been prepared to describe the method used to determine the miscible-slug-size requirement and to predict the performance of the miscible-phase injection project initiated in the Paloma oil field. Although the Paloma project has only been in operation since Aug., 1958, sufficient production history has been obtained so that a comparison can be made with the predicted performance. DESCRIPTION OF RESERVOIR The reservoir in which the miscible-phase project was initiated in the Paloma field is shown on Figs. 1 and 2. Reservoir properties are as follows: average sand thickness, 43 ft; porosity, 19.1 per cent; water saturation, 47.2 per cent: oil viscosity, 0.328 cp; reservoir temperature, 250°F; reservoir pressure, 3,500 psi: gas saturation, 21.2 per cent at time of miscible product injection; and total net hydrocarbon pore volume, 19,993,000 reservoir bbl. LIMITATIONS The method described herein utilizes a layer-system approach, where injection into and production from each layer is assumed to be proportional to the layer's fractional capacity. The effects of increased fluid mobility on injection and production rates, in layers which have had breakthrough, are not reflected in this method. There has been considerable controversy over the slug size necessary to maintain miscibility throughout the displacement process. In this method, the slug-size

By T. L. Loucks, E. T. Guerrero

Pressure drop characteristics in a system composed of two adjacent concentric regions of different permeability were studied. The differential equations for continuity of mass flow in the two regions were solved using the Laplace transformation and the necessary boundary conditions to give the pressure distribution in the composite reservoir, the resulting equation for pressure drop at the inner boundary was evaluated for a variety of composite reservoirs and compared with the results for a uniform reservoir. From this study it was found that under certain conditions the permeability in both zones, as well as the size of the inner zone, can be determined from the pressure drop curve. INTRODUCTION The theory for the pressure distribution and pressure build-up behavior of a well producing a single, slightly compressible fluid from infinite and finite homogeneous reservoirs was presented by Hornerl and Miller, Dyes and Hutchinson.2 Extensions on this original work to provide improved and extended interpretations and better agreement between theory and observed results have been made by Matthews, et a1,3 van Everdingen,4 Gladfelter, et al5 Stegemeierand Matthews,6 Hurst and Guerrero,7 and Perrine.8 More recently Lefkovits, et al,9 studied pressure build-up behavior in bounded reservoirs composed of stratified layers. Houpeurtl0 has suggested various approaches to the general problem of variable permeability and porosity but presented no analytic solutions for particular permeability variations. Albert, Jaisson and Marionll studied the finite composite reservoir and presented numerical solutions to the unsteady-state case and an analytical solution valid only for large times. They also studied the so-called pseudo-steady state for several examples of radial permeability variations. Very similar examples have been treated in the unsteady state with application to pressure build-up by Loucks in an unpublished manuscript. More recently Hurst12 has presented the complete point-sink solution (valid for all times) for the infinite composite reservoir. He applied these solutions to interference between oil fields along with an even more elegant application of his explicit solution to the material-balance equation including water influx. Mortada 13 approaches the same application by avoiding the point-source limitation but gives the solution for the aquifer region which is valid only for large times. Hopkinson, Natanson and Temple14 have treated both the finite and infinite composite reservoir obtaining the pressure distribution for the inner zone valid for large times. This paper presents a theoretical study of the pressure distribution in an infinite composite reservoir composed of two adjacent concentric regions of different permeability. The object was to determine the manner in which pressure drop at the inner boundary of a composite reservoir depends upon time, the permeability of each zone and the size of the inner zone. Expressions for the pressure distribution in both zones are developed which take into account the radius of the sink and are valid for small times as well as large times. It was felt that an understanding of the pressure drop behavior in various composite reservoirs would be of assistance in the interpretation of some pressure build-up curves which do not behave according to the theory derived for uniform systems. Often the region surrounding the wellbore is either more permeable or less permeable than the reservoir because of the various drilling and completion practices. The effects of reduced permeability due to drilling-fluid invasion and of increased permeability due to fracturing or acidizing need to be more carefully defined. Therefore, an equation for the pressure drop in a composite reservoir was developed, and the effects of both the permeability in each zone and the size of the inner zone were studied. STATEMENT AND SOLUTION OF PROBLEM BOUNDARY CONDITIONS AND ASSUMPTIONS The physical system studied was a composite

By S. A. Hovanessian

Analytical solutions are obtained for calculating the pressure distribution in rectangular fields due to injection and/or producing wells located anywhere within the field. The field is assumed to be homogeneous with either constant pressure or no-flow boundaries. These solutions are extended to include the calculation of average reservoir pressure and permeability from pressure build-up curves. Numerical examples are given to illustrate the application of equations to practical problems. INTRODUCTION Pressure distribution analyses are of considerable importance in the field of reservoir mechanics. A number of papers dealing with this subject have appeared in the petroleum literature. perrinel in 1956 presented a summary of the methods available for pressure build-up calculations and discussed the applicability of each in some detail. More recently (1958), Hazebroek, et a1,2 discussed a theoretical method for obtaining pressure distribution in a radial field. Also in 1958, Nisle3 gave a theoretical solution of the pressure distribution in a field extending to infinity with a partially penetrating line source at the origin. Mathews, et a1,4 applied the method of images to existing solutions for pressure distribution in radial fields and obtained expressions for calculating pressures in bounded reservoirs. However, a literature survey revealed that a general analytical solution for the pressure distribution in a rectangular field, with injection and/or production wells located anywhere within the field, was not available. Such a solution could be used to approximate the pressure distribution in a field the shape of which is nearly rectangular. Furthermore, it could be applied to pressure calculations pertaining to a single well taken from a system of wells which were drilled in a rectangular pattern. In this case the boundaries of the well drainage area are considered to be the perpendicular bisectors of the lines joining the well and its nearest neighbors. The basic problem considered in this paper consists of a rectangular field with either constant-pressure 01 no-flow boundaries and with either an injection or production well located anywhere within the field. The physical properties of the rock and fluids present are considered to be constant. Analytical expressions are obtained for the pressure distribution within the field during the production or injection periods. These expressions, which are infinite series, can be used to evaluate pressure at any point within the field as a function of time and position. For fields containing more than one well (production and/or injection), the solutions for each well acting independently are superposed. That is, at any point in the field, the pressures due to each injection 01. production well acting alone can be algebraically added to give the pressure resulting from all wells acting simultaneously. Evaluation of the series solutions given in this paper is readily accomplished by means of an electronic digital computer; and, in most cases, sufficient accuracy is obtained by setting the upper limits of summation at 20. These solutions are extended to include the calculation of average reservoir pressure and permeability from build-up data. A number of numerical examples are given to illustrate application of the solutions to actual field problems. ANALYTICAL SOLUTIONS The partial differential equation representing the pressure distribution in a homogeneous rectangular field (see Fig. I.), having a single source or sink of strength (2, can be written*

By V. J. Berry, J. D. Pendergrass

Well pressure transient tests provide a means for directly obtaining information about formation pressure and reservoir flow capacity. Such tests have also been proposed for determining presence and location of faults or other reservoir closures and for measuring oil in place. For mathematical convenience, most theoretical studies have considered the reservoirs to be homogeneous. Definitive information is not yet available to show whether the actual presence of nonuniformities will make pressure transient behavior different from that of a uniform reservoir. The conclusions reached from actual transient tests are questionable, therefore, insofar as they rely on the original assumption of homogeneity. One type of nonuniformity commonly assumed to exist is that of stratification. In most reservoirs the strata are thought to be in vertical communication. Equations for the transient flow of a single-phase, compressible fluid in a one-well, bounded, circular reservoir have been solved for several situations involving crossflow between multiple strata of various thicknesses and permeabilities. The results show that except for the very early flow period, which usually is too short to be analyzed, the transient performance observed at the well is substantially identical with that of a homogeneous reservoir having the same dimensions and having the same steady-state flow capacity. Thus, stratification does not adversely affect interpretation of well transient tests. This conclusion holds for all commonly encountered combinations of reservoir thickness and external radius. Deviations are observed for unusually thick reservoirs whose outer radii are relatively small. The results of these studies also show that the presence and the amount of stratification cannot be simply diagnosed from reservoir pressure transient data when there is crossflow between strata. INTRODUCTION The last decade has brought wide acceptance of the transient well pressure test for determining reservoir parameters. Following the original work of Hurst and van Everdingen,1 the mathematical theory was thoroughly explored. Numerous authors have suggested how to determine static reservoir pressure, permeability-thickness product, original oil in place and reservoir limits for different reservoir geometry. The same mathematical techniques have been used to predict the transient performance of a reservoir over a long period of time. Most of the theoretical work has been for homogeneous, isotropic systems. Some results have also been presented for a homogeneous, anisotropic reservoir.2 Petroleum reservoirs are not homogeneous. The deposition process seems to favor creation of a stratified formation. This concept is sufficiently well accepted so that the most natural extension of the transient flow theory beyond the homogeneous case is to a stratified formation. Results for a stratified reservoir with no vertical communication between layers can be obtained from the results for a homogeneous reservoir. Lefkovitz, et a1,3 have given a thorough treatment of the two-layer case. Two recent papers4,5 have treated the case of a two-layered reservoir with vertical communication or crossflow, between layers. Russell and Prats4 find that, after a relatively short time, the two-layered reservoir with crossflow exhibits a simple exponential pressure decline. From this time forward, the behavior is not distinguishable from the behavior of a homogeneous reservoir having the same steady-state flow capacity. The results of Katz and Tek 5 are equivalent. Russell and Prats4 also speculate that a multi-layered reservoir with crossflow will behave as a homogeneous system after long enough production time "... providing the contrast in kh between layers is not too great". They also suggest that, at intermediate times, "... the relative positions of the layers with respect to each other will have a great influence on the production behavior and on the time at which the previously mentioned large-time approximation might be valid". Katz and Tek5 remark upon the mathematical difficulty of treating a reservoir having many layers or strata. If relative position of the layers is also

By L. W. Emery

Undergrorlnd combustion operations were initiated in a 60-acre Bartlesville sand "shoe-string" reservoir in Allen Connty, Kans., in 1956. Tests in separate patterns were conducted using various co~nbinations of air and recycle gas to propagate combustion fronts from the injection toward the producing wells. These patterns were made up of 6 injection and 20 prodrrcing wells Gas and liquid prorluctiorz from each pattern was measured on an individual-well basis, and comparisons were made between the three patterns to ascertain the relative effects of injected gas composition on production behavior. Breakthrough of the combustion front at a well was characterized by an increase in water production from the well followed by an increase in bottomhole temperatrrre to approximately 250" F. After burning fronts had broken through at five producing wells, operations were terminated in 1960. From the total project approximately 79,000 bbl of oil were produced during thermal operations at a cumulative produced GOR of 23 Mcf/bbl. No appreciable change in the character of the produced crude was observed. Combustion in the reservoir was maintained with injected gas compositions ranging fronz 6 per cent oxygen in recycle gas to 100 per cent air. lnjectiotz of large quantities of recycle gas resulted in higher producing GOR's from offset wells than were measured from a pattern into ~vhich straight air ~vas injected. The air required to move the combustion front through I acre-ft of reservoir was computed to be 20 MMscf. This valrre was found to be relatively independent of the quantities of recycle gas injected. The recovery efficiency from the swept area was esti~izated to be about 59 per cent. Areas swept were similar in shape to tlzose obtained with a laboratory potentiometric model. Samples of sund taken from behind the burning front by coring indicated almost total oil removal from the sand. Petrographic analysis of the core samples indicated that the sand had been heated to peuk temperature of rlbout 1,200" F. No rignificant difference in peak temperature was forrnd in two areas where compositions of injected gas were quite different. Compression costs for thermal recovery were estimated to be $1,20/bhl of produced oil. INTRODUCTION The use of the "forward combustion" process as an oil recovery method has received a great deal of attention. This method involves ignition of the formation in an injection well, followed by propagation of a combustion front through the reservoir. Combustion is maintained by the injection of an oxygen-containing gas to react with reservoir hydrocarbons. As the flame front progresses through the reservoir, oil and formation water are vaporized, driven forward in the gaseous phase and recondensed in the cooler part of the formation. In turn, the condensed fluids push oil into the producing wellbores. Completed field tests of the process were first reported by Kuhn and Koch,' and by Grant and Szasz.' Results from other tests have since been reported by Walter,3 by Moss, White and McNeil,' and by Gates and Ramey." ach of these tests essentially utilized a single injection well surrounded by four or more producing wells. Sinclair Research, Inc., elected to do field experimental work using a number of test patterns in a single field in order that comparisons between various operating schemes could be made. The site selected and purchased in 1955 for this experimental work was a 60-acre Bartlesville sand reservoir located in Allen County, Kans. Combustion operations were initiated in mid-1956. Between that time and termination of the project in mid-1960, combustion fronts were propagated from injection wells to producers in three separate well patterns, using different mixtures of air and recycle gas. The test was terminated before sweep of the three patterns was complete so that information about the effect of combustion on the swept areas could be obtained by coring. Results from the test in the form of injection and producing well performance have been carefully recorded, and these form the general basis for this paper. DESCRIPTION OF RESERVOIR The reservoir in which the combustion tests were conducted is a Bartlesville sand "shoe-string", typical of a number of small reservoirs in Southeastern Kansas. Average reservoir characteristics are shown in Table 1. Fig. 1 is an isopachous map of the producing sand showing the reservoir to be approximately 400-ft wide and 2,500-ft long. Maximum net productive sand thickness is 21 ft. Fig. 2 shows a typical core analysis obtained by coring with water-base mud. The reservoir has no appreciable dip and is closed on the sides by degradation of sand into shale. The main body of sand is heavily laminated with shale stringers, which are not continuous between wells. The main reservoir is overlain by 30 to 40 ft of laminated low-permeability sand and shale streaks. No information is available on the original properties of

By N. C. J. Ros

The paper deals with pressure gradients occurring in flowing and gas-lift wells, a knowledge of which can be applied to the determination of optimum flow-string dimensions and to the design of gas-lift installations. The study is based on a pressure-balance equation for the pressure gradient. It appears that a pressure-gradient correlation of general validity must essentially consist of two parts-—one part being a correlation for liquid hold-up and the other part being one for wall friction. Dimensional analysis indicates that both liquid hold-up and wall friction are related to nine dimensionless groups. It is shown that in the field of interest only four groups are really important. On the basis of these four groups a restricted experimental program could be selected that nevertheless covered practically all conditions encountered in oil wells. This experimental program has been carried out in a laboratory installation. Three essentially different flow regimes were found. The pressure gradients in these regious are presented in the form of a set of correlations. Comparison of these correlations with a few available oilfield data showed excellent agreement. INTRODUCTION Prediction of the pressure drop in the flow string of a well is a widely known problem in oilfield practice. Accurate data on the pressure gradient of a simultaneous flow of gas and liquid in a vertical pipe are especially useful for the determination of optimum flow-string dimensions. It is well known that with moderate gas and liquid flows such a vertical string acts as a "negative restriction". The pressure drop decreases (1) when the throughput through a given pipe increases, and (2) when at a given throughput the cross-sectional area is decreased. The reason is that, with increasing velocities, the flow becomes more agitated so that the gas slips relatively more slowly through the liquid. With the resulting increase in gas content in the string, the static head decreases. When the area becomes very small, however, the high velocities entail great wall friction, which causes an increase in pressure drop. For a given flow, therefore, minimal pressure drop is obtained by using a certain cross section. This means that, in principle, each well can be provided with an optimum flow string for minimum pressure drop and, hence, maximum possible production rate. The procedure for the selection of the optimum string has been discussed by Gilbert.' A necessary tool in the procedure, however, is accurate knowledge of the pressure gradient to be expected for various values of the governing variables. Another application of pressure-gradient data lies in the field of gas-lift practice: they provide a means of determining the optimum gas-injection rate, optimum injection pressure and optimum injection depth. Much work has already been done in the study of the pressure gradient of vertical gas-liquid flow. Poett-mann and Carpenter2 presented a pressure-gradient correlation based on measurements in wells. This correlation has been found to provide accurate predictions in high-pressure wells and in high-production wells for flow through both tubing and annuli.2-5 However, when their method is checked on low pressure-low production wells or on wells with viscous crudes, serious discrepancies are found. As we shall see in the next section, this is due to the fact that their correlation factor, representing all irreversible energy losses, is given as a function of only one correlation group. Some important variables, such as gas-liquid ratio and liquid viscosity, are not incorporated in this group so that their specific effects are not accounted for. To study also the mechanism of vertical gas-liquid flow outside the ranges covered by the Poettmann-Carpenter publication and extensions, a laboratory investigation has been carried out. This study is founded on a pressure-gradient equation that is based on a pressure balance. To reduce the number of test runs required, a dimensional analysis has been carried out, followed by a selection of relevant dimensionless groups. These groups guided a subsequent experimental study, and with their aid the experimental program could be minimized while still covering the majority of the situations encountered in oilfield practice. In this paper the choice of a formula for the pressure gradient is discussed first. This is followed by a brief description of the experimental setup. Subsequently, the dimensional analysis is discussed and the relevant dimensionless groups are selected, resulting in the experimental program required. The general relationships of pressure gradient and liquid hold-up are then described; various flow patterns and a certain flow instability (so-called "heading") are discussed and a set of correlations is presented which shows a good agreement with the measurements and a few available field

By R. L. Perrine

This paper shows how stability theory can be used to optimize solvent recovery of oil. Application of the theory leads to definition of the limiting conditions required for stable displacement to occur. One of these conditions is that the size of a solvent "slug" must at least equal a prescribed minimum. The criteria to be satisfied are miscibil-ity and stability. Stability implies that no viscous fingering will occur and that mixing will be caused only by a dispersion process. Miscibility implies that complete recovery will be obtained from the swept regions. Thus, for any specified set of reservoir conditions, an optimum use of solvent is defined. INTRODUCTION The early outlook for solvent flooding as a means to increase oil recovery was very favorable. Some laboratory results indicated that a "slug" containing perhaps 2 to 3 per cent of a hydrocarbon pore volume could be successful. However, other data suggested as much as 30 per cent would be required. The difference is of considerable economic importance. A theoretical explanation for these divergent results has been advanced by the author.1,2 Stability theory defines conditions for two distinct flow regimes. In stable displacement, solvent and oil become mixed by a dispersion process, and the solvent requirement is small. On the other hand, unstable displacement can degenerate into viscous fingering. The practical result is a considerable increase in the extent of mixing and, thus, in the solvent required. The present paper shows how to use stability theory to optimize solvent recovery of oil. Oil is usually considered to be displaced by a small amount, or a slug, of solvent. Gas in turn follows solvent. Any fingering will permit contact of nearly solvent-free gas and oil, leading to immiscibility and reduced recovery. Thus, our optimum is defined by two restrictions — stable flow and miscibility. Although an immiscible solvent process could prove more profitable than any miscible process, such processes lie outside the selected definition of an optimum. The usefulness of computations based on the present work will depend on the validity of the general theory. Because the important idealizations required tend to minimize the predicted solvent requirement, the results should state correctly the limiting conditions required. Therefore, theoretical calculations should have practical value. This paper first reviews several principles that result from stability theory. The following section describes use of these ideas in relatively simple systems which symmetry makes appear one-dimensional. Of particular importance is the step-by-step design of 3 stable minimum-solvent slug. The use of stability theory in cases of multi-dimensional displacement is discussed and, finally, suggestions are given for simplified practical application of the theoretical results. PRINCIPLES OF STABILITY THEORY The use of perturbation methods to derive stability conditions has been shown by the author.l The discussion is presented in terms of the fractional concentration of solvent in the single hydrocarbon phase. Quite generally, any disturbance can be represented as a Fourier series, of which only the least stable term need be considered. Denoting this term at time t = 0 by cn,

By H. D. Yoo, M. R. Tek, D. L. Katz

The behavior of gas-storage reservoirs subject to water drive is investigated through analog simulation on an electronic differential analyzer. The simulation technique developed on an LM-10 computer permits the prediction of reservoir volume or pressure resulting from the movement of water in the surrounding aquifer. The method developed on the analog computer consists of setting up an appropriate transfer-function circuit and feeding the arbitrary time-varying boundary conditions as an input signal. The input may be specified as gas reservoir pressure, pore volume or the water flux. Several cases studied include an isolated gas reservoir on a limited aquifer, interference among three reservoirs adjacent to a common aquifer and the growth of gas-storage volume on an aquifer. It is concluded that the method developed on an electronic differential analyzer provides an excellent technique to simulate and investigate the behavior of gas reservoirs subject to water drive. The agreement between the reservoir performance as predicted from the simulation technique and as measured from actual field data is found to be better than the range usually encountered in predicting water-drive behavior. INTRODUCTION It is generally known that some gas-storage reservoirs are located on top of blanket sands of large extent, saturated with brine called aquifers. Because the volume of the body of water associated with aquifers is usually very large and water is compressible, the cyclic pressure variations encountered in normal storage service inevitably cause unsteady, compressible flow conditions in the adjacent aquifers. The solution to radial diffusivity equation for a limited aquifer for constant terminal conditions has been know111 since the early 1930's. The solution for the constant terminal conditions for an infinite aquifer was puhlished in 1949.2 The proldem of handling the unsteady flow of water through a porous medium with time-varying boundary conditions has been studied by many investigators through the use of graphical,3 digital4 computing techniques or on an R-C type of electrical analog called a reservoir analyzer.5 The digital computer performs a series of arithmetic calculations to superimpose the solutions of constant terminal conditions, while the R-C type of analyzer permits the simulation of the continuous medium by means of "lumped-cell" electrical resistance and capacitance elements based on finite-difference approximations. The graphical method is based on the convolution integral solution of the partial differential equalion. The purpose of the present study was to simulate the behavior of an aquifer on a general-purpose, electronic differential analyzer to obtain the influx of water or the change in pressure at the reservoir continuous1r and quickly. The use of the electronic differential analyzer was based on well known solutions for the constant terminal conditions already available in the literature.2,6-8 THEORETICAL DEVELOPMENT MODEL. The gas reservoir (R) is described in relation to an aquifer in Fig. 1. The permeability and porosity of an aquifer sand, the viscosity of the brine, and the compressibility of the brine and the formation are assumed to be constant. It further is assumed that the blanket sand is not thick enough to give appreciable vertical pressure distribution. The model in consideration is circular with radial geometry. The aquifer radius can be of limited or infinite extent. The gas-water interface is assumed to be fully segregated, and the displacement of the gas by

By P. E. Baker

Approximate solutions are presented for the heat-flow equations in a loss-free linear system with a moving source and with heat transfer by convection and conduction, representing in situ combustion in an oil reservoir. The rate of heat generation and the velocity vf of the source (flame front) are assumed constant. A cold front or convection front caused by the cooling effect of in-jected air develops behind the flame front and moves with velocity vc, which is determined by the thermal properties of the air and rock. Because of thermal conduction the fronts are not sharp; i.e., they do not represent temperature discontinuities. The following solutions apply after a sufficiently long time. where Q is heat generated per unit volume of formation, k is thermal conductivity, CF is volumetric heat capacity, a is thermal diffusivity, and subscripts 1 and 2 refer to values behind and ahead of the flame front, respectively. The parameter CF2 is contained in v . Te is initial reservoir temper- . C2 e ature and air-injection temperature. Other mathematically possible solutions are presented. The different solutions result from the use of different values for various physical parameters. The solution just presented is thought to be most commonly applicable. INTRODUCTION In this paper, approximate analytical expressions are derived for the temperature profile developed in the underground combustion method of oil recovery. A qualitative description of the process may be found in a paper by Tadema.1 The flame front is regarded as a moving source with a constant rate of heat generation. Heat transfer is by conduction and convection. Previous publications, of which a few are cited here, 2-5 have also attacked this problem and have presented solutions pertaining to different methods of approximating or idealizing the process studied. The present paper also treats an idealized case: linear flow is assumed for heat and fluids, and vaporization and condensation effects are neglected. The results obtained are similar to those derived by Bailey and Larkin,3 but are obtained by an entirely different procedure. The method used here should give considerably greater insight into the physics of the in situ combustion process. Four types of cases are considered, distinguished by thermal and fluid-flow properties of the system. A constant rate of heat generation and constant rate of flame-front advance are assumed. Flame-front temperature is determined for each case using a heat balance which involves the entire temperature profile. The applicability of the different solutions to real systems is discussed after they are all presented. Again, this is done for the purpose of understanding better the physics of the process. Consideration of the nonapplicable solutions and of the reasons for their not applying is significant. In a paper by Cooperman,4 the only case solved is one considered here as not applicable. Fluid flow, as such (pressures, saturations, oil recovery, etc.), is not treated. This subject has been covered in a paper by Wilson, et al.6 Also, we are not concerned with requirements for continuance of the process; we only attempt to determine the temperature history for cases where the process has continued for some time. PHYSICAL SYSTEM AND APPROACH TO PROBLEM A linear flow (heat and fluid) system is assumed and the x-axis is taken parallel to the flow direction. The combustion zone (flame front) is vertical,