Buckley-Leverett Method Research Articles

Increasing the accuracy of oil recovery factor predictions by integrating lithology data

The object of research in the paper is the process of oil extraction during flooding. The Buckley-Leverett method, which is widely used to estimate oil production in flooding, has certain limitations that lead to uncertainty in the results. This paper proposes to extend the Buckley-Leverett algorithm by integrating lithological data. This approach allows to take into account the influence of geological characteristics of the formation on the process of displacement of oil by water, which leads to a significant increase in the accuracy of forecasting the oil production coefficient. The effectiveness of the proposed method is confirmed on the basis of data analysis of a real oil field. The methodology for calculating the oil recovery coefficient during flooding using lithological dissection is presented. In this work, the steps of determining the oil recovery coefficient were analytically determined, which achieves a certain degree of accuracy due to the inclusion of the lithological characteristics of the permeable zone of the formation. The basic calculation of the lithological distribution over the layer was performed using the Kriging method. To confirm the accuracy of the Buckley-Leverett method, taking into account lithological dissection, the use of data analysis, including an experimental histogram and a theoretical normal distribution plot, is proposed. For data analysis, one hundred cases of lithological distribution were generated using the Sequential Indicator Simulation method. The comparative analysis of the data of the experimental histogram and the theoretical graph of the normal distribution of the determination of oil recovery coefficients by the Buckley-Leverett method for cases with and without lithological dismemberment allows to quantitatively assess the accuracy of both studied options. On the basis of a real oil field, it is shown that the accuracy of oil recovery coefficients by the Buckley-Leverett method, taking into account lithological fragmentation, exceeds the similar method without taking into account lithological fragmentation by 11 %.

Investigating the vaporization mechanism's effect on interfacial tension during gas injection into an oil reservoir

In gas injection, which is one of the fascinating enhanced oil recovery techniques, the main mechanism involves decreasing interfacial tension (IFT). Although various mechanisms can affect the IFT of a system, in most experimental and numerical studies, condensation is considered the dominant mechanism among condensation-vaporization and vaporization. Investigating the impact of each mechanism is crucial as they can influence the IFT of the system and, consequently, the effectiveness of the gas injection method. This study introduces a novel model to assess the influence of different mechanisms on system IFT. The model defines system IFT, adjusts fluid relative permeability to represent miscible, immiscible, and near-miscible states, and utilizes the Buckley–Leverett method to analyze gas fractional flow and saturation profiles when injecting carbon dioxide (CO2), methane (CH4), and nitrogen (N2). Furthermore, the research explores the impact of injection pressure and IFT at minimum miscible pressure (IFT0) on gas injection efficiency. Based on our results, for both live and dead oil, the condensation mechanism reduces IFT and near-miscible pressure; switching to a condensing-vaporizing mechanism increases these parameters. This trend was consistent across all gases studied (N2, CO2, CH4), with a more significant effect observed on the CH4-live oil system compared to N2 and CO2. Controlling the condensing mechanism in IFT measurements enhances gas flow rate and relative permeability curve within the medium. Higher injection pressure in the condensing mechanism and IFT0 = 0.5 leads to faster fluid movement and improved relative permeability due to increased driving forces. Higher IFT0 accelerates the relative permeability of fluids and gas movement within the medium by promoting miscibility sooner. The impact of IFT0 was more pronounced on the dead oil–gas system compared to the live oil–gas system in this study.

Design and development of a hybrid (mpi+cuda) parallel program for solving the oil displacement problem

In this work, a numerical model of oil displacement was developed using parallel computing technology through the Buckley-Leverett method. Used the hybrid (MPI + CUDA (2 GPU)) high-performance parallel computing technologies. The main goal of this work is by using these two GPU to implement computing processes of distributed data through the MPI, as well as to make a comparative analysis of the computing time and acceleration of parallel algorithms. The MPI, CUDA, hybrid (MPI + CUDA) parallel computing algorithm and the parallel program were realized, and the results were analyzed.

Performance Analysis of Water Injection Well BEL-13 and Production Well BEL-01 Using Hall Plot and Buckley-Leverett Methods

At first, the FLO field had an ability to flow fluids to the surface in a natural flow, but the production capacity decreased with time. Therefore, production is increased using the waterflooding method. The purpose of this study was to analyze the performance of waterflooding carried out on injection well BEL-13 and production well BEL-01. There are two methods used in this study to achieve the desired aims and objectives, namely the Hall Plot method and the Buckley-Leverett method. The Hall Plot method was used to see the connectivity conditions between injection and production wells after being injected with water periodically. Whereas the Buckley-Leverett method was used to estimate the time of water breakthrough, production amount (Np), the injection rate (Wi), water cut value (WC), and recovery factor (RF). Based on the Hall Plot method, it was found that the waterflooding carried out was stable or normal because there was no negative skin or positive skin. In addition, based on relative permeability curve, rock wettability of the reservoir rock is water wet. Analysis using Buckley-Leverett shows that the cumulative production at the time of water breakthrough from the injection well BEL-13 to the production well BEL-01 is 364,733.64 STB. When the water breakthrough occurs, the predictions of WOR, water cut, and recovery factor are 2.13, 68.08%, and 29.76%.

A semi-analytical method for optimizing the gas and water bidirectional displacement in the tilted fault block reservoir

In tilted fault block reservoirs with edge water, the remaining oil is usually formed and located at the structurally highest. The gas and water bidirectional displacement process is an effective method to recover this type of remaining oil in these reservoirs. However, there is no optimizing method to design the injection-production schedule of this process for the maximum oil recovery. In this study, the imaging and superposition principle, Pollock method, equivalence principle, and Buckley-Leverett method were combined to develop a novel semi-analytical method for optimizing the injection-production schedule. An example study was carried out for verifying the proposed semi-analytical method and validated with the numerical simulation method. Furthermore, sensitive factors of this displacement process were quantitatively analyzed by the proposed method. The results showed that the optimized synergy of the gas and water can maximize the oil recovery for the bidirectional displacement process. Compared with no injection, inadequate injection, and excess injection of gas, the breakthrough recovery can be increased by 3.7%, 0.8%, and 4.88%, respectively, by controlling the gas-water synergy. Meanwhile, under the gas-water synergy displacement condition, the oil recovery is increasing with the tilt angle of the reservoir increasing and the distance of well from edge water decreasing. This enhanced oil recovery process is more sensitive to low-middle tilt angles and more effective for small distances. The results obtained from this study can be used to improve the exploitation strategies of tilted fault block reservoir with edge water.

Experimental Study on the Effect of Polymer Injection Timing on Oil Displacement in Porous Media

It has been proven that polymer injection at early times is beneficial to offshore heavy oil recovery. It is of significant importance to optimize the polymer injection timing and decide the residual oil distribution after polymer flooding. Aiming at a specific offshore heavy oil reservoir in Bohai, China, the optimum polymer injection timing is investigated through laboratory experiments. The influence of polymer injection timing on oil displacement and remaining oil distribution is analyzed by combining macroscopic and microscopic flooding experiments. The results reveal that the optimum polymer injection timing should be close to the water breakthrough, i.e., just before the waterflooding front reaches the outlet of the core. In addition, the waterflooding front position is analytically solved by using the Buckley–Leverett method and verified by experimental results, which supply an approach to predict the polymer injection timing. When polymer is injected before the waterflood front reaches the outlet of the core, the mobility control ability of polymer solution can reduce the fraction of bypassed volume of the reservoir by waterflooding. The early injected polymer mainly enters the high permeability zone, which works positively in two ways. Firstly, it improves the oil displacement efficiency of the high permeability zone. Secondly, the polymer establishes a flow resistance in the high permeable zones, thus improving the sweep efficiency in the low and medium permeability zones. However, our residual oil distribution experiments illustrate that there is still a large amount of oil remaining in the low and medium permeability zones. Therefore, it is necessary to explore additional EOR methods to recover the abundant residual oil.

ANALYSIS OF FRACTIONAL FLOW AND RELATIVE PERMEABILITY OF HEAVY OIL AND KEROSENE DURING RECOVERY IN PETROLEUM RESERVOIR

This paper evaluates and compares the effect of fractional flow and relative permeability of heavy oil and Kerosene during recovery in a petroleum reservoir. Water fingering is one of the challenging problems during oil recovery and another comprehensive problem is, to exactly evaluate the amount of recover oil from a petroleum reservoir. To address these problems, the fractional flow and relative permeability of heavy oil and Kerosene are analyzed. The fractional flow approach is originated in the petroleum engineering literature and employs the saturation of one of the phases and a pressure as the independent variables. The fractional flow approach treats the multi-phases flow problem as a total fluid of a single mixed fluid and then describes the individual phases as fractional of the total flow. Laboratory steady state flow experiments are performed in two different types of oils (Heavy oil and Kerosene), to empirically obtain relative permeability and fractional flow curves, which have great influence in recovery efficiency calculation. Therefore, the famous Buckley-Leverett displacement mechanism has been used to calculate the performance of waterflooding. With Buckley-Leverett method, oil recovery from waterflooding is calculated and required water injection volume to achieve that oils recovery are estimated for heavy oil, which the total amount of oil produced up to the breakthrough is A φ B ×RF = 15901.92 m3 (= 99864.05 barrels) and for Kerosene is A φ B ×RF = 24992.06 m3 (156950.16 barrels). Additionally, the front flow of heavy oil is approximately spread to 80 m, and the front flow of light oil is approximately spread to 300 m. As a result indicates that fractional flow theory predicates a stable frontal displacement of all mobility ratios contrary to observed experimental facts. Therefore, fractional flow theory is suitable for describing the performance of stable displacement of oils by water.

Fractional flow in radial flow systems: a study for peripheral waterflood

The Buckley–Leverett displacement mechanism has been used to predict the performance of waterflood. With Buckley–Leverett method, oil recovery from waterflood is calculated and required water injection volume to achieve that oil recovery is estimated. This method does provide a very useful tool in waterflood design. Our experience in oil industry and a thorough literature review indicates that Buckley–Leverett method was used to analyze waterflood project directly without any adjustment based on the real reservoir and production situations. By doing so, errors are introduced into the analysis. Buckley–Leverett method assumed that displacement occurs in a linear system. This is true for some waterflood scenarios while for others it is not. For some waterflood scenarios, a radial system is more appropriate than a linear system. In this study, we investigated the fractional flow in a radial system and derived the solutions to predict the performance of water displacing oil in radial system. With this radial displacement model, design and prediction of waterflood can be achieved by Buckley–Leverett method and our model, whichever fits the waterflood pattern. Considering the fact that many waterflood scenarios follow radial displacement, our model is an important supplement to Buckley–Leverett method.

Use of Actual Dimension in the Percolation Model for the Calculation of Breakthrough Time in Oil Recovery: Case of Some Niger Delta Reservoirs

This paper calculates breakthrough time for some Niger Delta reservoirs using a percolation model. The model is less cumbersome and time saving when compared with traditional methods such as the Buckley-Leverett method. By incorporating some reservoir specifics, such as actual reservoir dimension, into the model used in this paper, better results which approximate empirical values closely are obtained. The fact that breakthrough time is reservoir specific is confirmed. Keywords: Oil recovery, Percolation model, actual reservoir length, Niger Delta, Water Drive Mechanism, Breakthrough time.

Displacement of a Newtonian fluid by a non-Newtonian fluid in a porous medium

This paper presents an analytical Buckley-Leverett-type solution for one-dimensibnal immiscible displacement of a Newtonian fluid by a non-Newtonian fluid in porous media. The non-Newtonian fluid viscosity is assumed to be a function of the flow potential gradient and the non-Newtonian phase saturation. To apply this method to field problems a practical procedure has been developed which is based on the analytical solution and is similar to the graphic technique of Welge. Our solution can be regarded as an extension of the Buckley-Leverett method to Non-Newtonian fluids. The analytical result reveals how the saturation profile and the displacement efficiency are controlled not only by the relative permeabilities, as in the Buckley-Leverett solution, but also by the inherent complexities of the non-Newtonian fluid. Two examples of the application of the solution are given. One application is the verification of a numerical model, which has been developed for simulation of flow of immiscible non-Newtonian and Newtonian fluids in porous media. Excellent agreement between the numerical and analytical results has been obtained using a power-law non-Newtonian fluid. Another application is to examine the effects of non-Newtonian behavior on immiscible displacement of a Newtonian fluid by a power-law non-Newtonian fluid.

The Prediction of Oil Recovery by Gravity Drainage

An approximate theory of free-fall gravity-drainage recovery is expanded to account for residual oil saturation. The recovery equation is derived and seven comparisons are made of observed gravity-drainage recovery with calculated recovery. Examples show how the theory can be applied. Introduction The gravity-drainage mechanism is one of the most efficient ways of producing an oil field. Unfortunately, most oil fields cannot be produced economically under free-fall gravity alone because the effective oil permeability is too low, the oil viscosity is too high, or the dip of the formation is too small. Nevertheless, many reservoirs take advantage of the gravity component, especially under pressure maintenance in areas where gas is available, such as overseas, or where inert gas can be used. The most ideal system would be one where oil is produced only under free-fall gravity drainage, giving the highest possible recovery at the most efficient rate. However, because a producing mechanism must be justified economically as opposed to another producing mechanism must be justified economically as opposed to another producing mechanism, an engineer must calculate gravity-drainage recovery producing mechanism, an engineer must calculate gravity-drainage recovery as a function of time.An approximate theory of strict gravity drainage has been presented by Cardwell and Parsons. They ignored capillary pressure terms and show how gravity-drainage equations can be solved. They give only one comparison of recovery calculated from theory with recovery obtained experimentally from a draining column.In addition to that study, many other papers have been published describing how to calculate reservoir performance where gravity effects influence the production mechanism. An early investigation describing the potentially high recovery with gravity drainage is Katz's discussion of potentially high recovery with gravity drainage is Katz's discussion of Oklahoma City Wilcox sand. Katz also presented data on the low (6 to 8 percent) residual oil saturations that could be obtained from a completely percent) residual oil saturations that could be obtained from a completely drained portion of the Wilcox sand. Lewis enlarged upon the gravity- drainage mechanism for oil recovery and gave four examples of gravity- drainage fields. Beginning in 1949, more studies discussed methods of calculating the effect of gravity component on oil recovery during depletion or pressure maintenance, and case histories of fields where gravity was an important factor in production. All the papers describing a calculating procedure used the Buckley-Leverett method. These studies also used historical rates of production or a pre-selected rate of production to get the advance of gas-oil front and to calculate recovery. More recently, high-speed computers were used to solve mathematical equations of various reservoir simulators. These reservoir simulators still required that rates or drawdown be pre-selected as a boundary condition to the solutions of mathematical equations. Thus, these simulators could not be used to determine the free-fall, gravity-drainage rate of production.One study attempted to determine production rate based on the location of the gas-oil front below a stated level. Essley et al. determined an approximate producing rate by arbitrarily assuming a maximum capacity for each well remaining on production below a given subsea level and determining the total production rate from that. They did not state how the arbitrary assumption was made, nor how this rate was related to a true gravity-drainage rate. JPT p. 818

A One-Dimensional Analytical Technique for Predicting Oil Recovery by Steamflooding

Abstract This paper describes a graphical method for calculating oil recovery by steamflooding that permits accurate prediction of fluid saturation permits accurate prediction of fluid saturation profiles and oil bank formation. profiles and oil bank formation. The oil displacement calculation is based on the Buckley-Leverett method for isothermal, two-phase flow in porous media. The fluid-flow theory is used with the Marx and Langenheim method to predict the size of a steam zone from energy balance and beat transfer considerations. The method is more complex than previous simplified methods, but permits band calculations of fluid saturation profiles and the delay of oil bank, arrival in producing wells for partially depleted oil reservoirs. These computations are in excellent agreement with those calculated numerically. Since uniform vertical displacement is assumed this method is suggested for use only for thin sands. Introduction The model for steamflooding described in this paper is based on the physical system illustrated paper is based on the physical system illustrated in Fig. 1. The reservoir is considered in vertical cross-section and is bounded above and below by impermeable strata. Heat losses to adjacent strata are accounted for by the method of Marx and Langenheim to describe the rate of steam front advance. This paper is concerned primarily with the prediction of linear fluid-flow behavior in the presence of temperature distributions that are time presence of temperature distributions that are time and distance dependent. The method may also be applied m hot waterflooding. In order to use the isothermal two-phase flow theory of Buckley and Leverett, the temperature distribution T(x, t) is represented by a series of n + 1 moving isothermal zones separated by discontinuities or "shocks" in temperature (Fig. 2a). The Marx-Langenheim method assumes only two isothermal zones (n = 1 in Fig. 2) -- one at injection temperature and the other at initial reservoir temperature - separated by a single shock. Since piston-like flow through the reservoir is assumed, the Marx-Langenheim method is not recommended for thick sands where gravity segregation would greatly reduce vertical sweep. A two-dimensional numerical method as described in Ref. 10 should be used in such a case. SPEJ P. 489

Use of Simple Mathematical Models for Predicting Reservoir Behavior

Here is an approach for solving reservoir flow problems where behavior is dominated by a rate-limiting step. Simple models are developed for gravity drainage where vertical flow occurs, for water underrunning of viscous oils, for gravity segregation of water banks in gas caps, and for control of coning by injection of oil. Introduction Forecasting the behavior of a reservoir is one of the more important but complicated tasks of engineers in the oil industry. Knowledge of reserves remaining in a reservoir is vital to planning optimum depletion of a field. Unfortunately, the engineer assigned the task of predicting reserves often faces a difficult choice. For the most accurate answer, he can use a computer program that takes into account all of the pertinent factors, but this approach is usually pertinent factors, but this approach is usually expensive and time consuming, and requires a detailed knowledge of the reservoir. On the other hand, he can use conventional one-dimensional displacement calculations that are easily applied but that in some cases do not adequately describe the reservoir flow system. The purpose of this paper is to describe a middle-ground approach that in special situations has many of the advantages of the above methods with out their more serious drawbacks. This approach uses mathematical models that describe the principal flow mechanisms and can be quickly applied by hand calculations. The technology of predicting reservoir behavior has grown steadily since the pioneering work of Muskat and of Buckleys and Leverett. Muskat's tank-type or zero-dimensional method of predicting behavior in dissolved gas drive reservoirs has been invaluable to the industry. Another milestone was reached with the Buckley-Leverett method of predicting linear displacement of oil by water or gas when flow was principally along the bedding plane. The classic work of Hurst, Muskat, and van Everdingen and Hursts laid a firm foundation for problems involving unsteady-state flow of fluids. Later, progress was made by Welge in solving one-dimensional progress was made by Welge in solving one-dimensional displacement equations more easily. The advent of digital computers led to the development of methods of solving problems of greater and greater complexity. Indicating the progress being made with computers, Douglas et al. in one paper, included the effect of capillary pressure in one-dimensional flow, and in another paper dealt with the flow of two phases in two dimensions. The utility of computers in predicting reservoir behavior has continued to grow as programs have become more user-oriented and as computers have become faster and more economical to employ. But even today, the time, effort, and money required to use computers to solve reservoir problems cannot always be justified. Thus, other tools are needed. An excellent example of another approach is given in a paper by Joslin. In analyzing a gas injection project in a large Venezuelan reservoir, Joslin realized project in a large Venezuelan reservoir, Joslin realized that gas had overridden the entire oil sand because production was above the critical rate. However, the production was above the critical rate. However, the presence of pancake-like shale members penetrated presence of pancake-like shale members penetrated by the wells prevented coning of gas into perforations located below the shale near the base of the sand. Oil recovery was predicted by assuming that gas displaced oil vertically downward throughout the producing area. Another paper demonstrating the practical use of simple mathematical models is that of Matthews and Lefkovits for predicting producing rates for wells in depletion-type reservoirs. JPT P. 1145

Calculation of Water Displacement by Gas in Development of Aquifer Storage

Abstract During the initial growth of a gas bubble in an aquifer storage reservoir the injected gas tends to override the water. The resulting low displacement efficiency and big rate of gas travel down-structure make it difficult to maintain water-free production. This paper illustrates the use of two-dimensional, two-phase calculations to simulate this gas-water displacement. Calculations were performed for a stratified aquifer and for several homogeneous sands differing in permeability, porosity and thickness. Results are discussed and compared with performance predictions obtained from the simpler Buckley-Leverett and Dietz formulas. This comparison indicates the necessity for the two-dimensional calculations in realistically simulating the gravity override of the water by injected gas. Introduction In recent years natural gas has been stored near markets in aquifers where insufficient storage capacity is available in depleted fields. Operators of these aquifer storage reservoirs have encountered technical problems relating to the gas-water displacement accompanying initial growth of these gas bubble. Injected gas tends to override the water, with a resultant low displacement efficiency and high rate of gas travel down-structure toward spill points. Low displacement efficiency makes if difficult to sustain water-free gas production. Sometimes the fingering of gas down-structure may be so pronounced that the injection rate must be severely curtailed, which excessively lengthens the time required for bubble growth. This paper is concerned with estimating - for given aquifer characteristics and fluid properties the displacement efficiency, rate of gas movement down-structure and rate of gravity drainage of water behind the gas front. Several published articles relate to simulative capability necessary to handle this problem. The Dietz formulae and Buckley-Leverett method have some applicability to the problem. Woods and Comer reported one - dimensional, radial calculations which accounted for the two-phase flow of gas and water. Most recent research directed toward understanding aquifer behavior in relation to gas storage has concentrated on single-phase flow in aquifer. Douglas, Peaceman and Rachfords presented a method for calculating multidimensional two-phase flow in reservoirs; their method has been applied in work reported by Nielsen and Tek, Blair and Peaceman and Goddin. Refs. 5 and 7 show comparisons between experimental data and calculations similar to those employed here. One purpose of this paper is to illustrate the use of two-dimensional, two-phase flow calculations in simulating the difficult problem of water displacement by gas in a vertical cross-section. The pronounced fluid density and viscosity, differences combine with the disparity in horizontal and vertical dimensions of the cross-section to pose one of the more difficult problems in reservoir simulation. The calculations described account for capillary and gravity forces, relative permeability and reservoir heterogeneity. An example reservoir is described and calculated results are presented for a variety of injection rates and values of permeability, reservoir thickness and dip angle. A second purpose is to compare the two-dimensional calculations with results obtained from the Buckley-Leverett and Dietz formulas. For this reason most calculations were performed for a simplified example reservoir to which the latter techniques night reasonably be applied However, the computer program employed applies equally well to cases involving arbitrary spatial variations of permeability porosity, reservoir thickness and dip angle. SPEJ P. 105ˆ

A Method for Predicting the Performance of Unstable Miscible Displacement in Heterogeneous Media

KOVAL, E.J., CALIFORNIA RESEARCH CORP., LA HABRA, CALIF. Abstract Practical miscible displacement processes will be characterized by fingering of the solvent into the oil. The fingering process is brought on by viscosity differences, and can be accentuated by channeling and longitudinal dispersion. The effects of these factors on the efficiency of unstable completely miscible displacements are combined in what is called the K-factor method. This method, analogous to the Buckley-Leverett method, predicts recovery and solvent cut as a function of pore volumes of solvent injected. Experimental data are included and show excellent agreement with theory for a wide variety of sandstone cores and viscosity ratios. Introduction Theoretical considerations, laboratory experiments, and pilot tests lead to the conclusion that miscible displacements in the field will be unstable. In an unstable miscible displacement, the solvent fingers through the oil. This fingering leads to early breakthrough of the solvent and an extended period during which both oil and solvent are produced.For such a system, there appear to be four principal factors which bring about or accentuate the effects of instability: longitudinal dispersion (including geometrical effects), channeling, viscosity differences and gravity differences. Other factors, such as diffusion and flooding rate, can also influence the effects of instability, but at the flooding rates considered in this report, circa 15 ft/D, they are unimportant. Longitudinal dispersion can be thought of as a spreading of the solvent front caused by the presence of microscopic inhomogeneities. Channeling of the solvent occurs when a porous medium has macroscopic inhomogeneities; i.e., gross permeability variations. Viscosity differences lead to fingering of the less viscous solvent. This difference in viscosity accelerates the growth of fingers along paths previously developed because of permeability variations. Gravity differences lead to overriding of the usually less dense solvent. Although gravity effects are generally small at a flooding rate of 15 ft/D, they would, nevertheless, unnecessarily complicate the interpretation of flooding experiments. For this reason, all the experiments reported herein were done with matched density fluids.Fingering and the resultant poor areal sweep were recognized early as the dominant influences on the efficiency and the economics of miscible displacement processes. Much research effort has been spent on ways to minimize fingering and increase areal sweep such as the use of graded viscosity slugs or water slugs. Some researchers attempted to work out ways to prevent fingering completely; i.e., to achieve a stable displacement through gravity or latitudinal dispersion stabilization. Others did not attempt to control fingering but obtained an economic process by merely recycling the solvent and sweeping pattern by pattern.During this period, all aspects of fingering came under close scrutiny. Some researchers reported on how fingering looks and how it is affected by viscosity ratio, geometry, and slug size. Peaceman and Rachford suggested a mathematical approach to the prediction of unstable miscible displacements in relatively homogeneous sand packs, but their work cannot be extended conveniently to heterogeneous systems. Hence for a heterogeneous system, no method is presently available for predicting solvent cut and recovery as functions of pore volumes of solvent injected.The purpose of this investigation was to attempt to fill in the gap in our knowledge concerning the prediction of performance of unstable miscible displacements. Necessarily, the system selected for study was a relatively simple one. The restrictions placed on the system were:The system was linear;The solvent was miscible in all proportions with the oil in place;The solvent was continuously injected into the porous medium;Gravitational effects were eliminated by matching densities andAll the flood rates were high and constant at 15 ft/D to avoid any small rate effect and to minimize any diffusion effects. To simplify and to indicate that both longitudinal dispersion and channeling arise from permeability variations, the effects which they cause or influence have been termed heterogeneity effects. SPEJ P. 145^

An Experimental Verification of a Two-Dimensional Technique for Computing Performance Of Gas-Drive Reservoirs

Abstract The shape and position of the gas-oil transition zone during downdip displacement of oil by gas has been calculated using flow equations which include the effects of gravity, relative permeability, capillary pressure and compressibility of the fluids. The calculations treat the problem in two space dimensions, and results are compared with data from a laboratory model tilted at 30 degrees and 60 degrees from the horizontal on displacements near and above the maximum rate at which gravity segregation prevents channeling of the gas along the top of the stratum. The good agreement between calculated and experimental results demonstrates the validity of the technique as well as that of the flow equations. Introduction Knowledge of the fluid distribution and movement in and oil reservoirs important in producing operations and estimation of reserves. The history of the oil industry has included steady progress in improving the accuracy of calculations which provide the required knowledge. The earliest method of calculating reservoir performance consisted of material-balance equations based on the assumption that all properties were uniform throughout a reservoir. For many reservoirs such a simple formulation is still the most useful. However, when large pressure and saturation gradients exist in a reservoir, the assumption of uniform values throughout may lead to significant error. To reduce these errors, Buckley and Leverett introduced a displacement equation which considers pressure and saturation gradients. Methods available at that time permitted solutions to the Buckley-Leverett equation in one space dimension; these solutions have been very useful in solving many problems related to the production of oil. However, the one-dimensional methods are not adequate for systems in which saturations vary in directions other than the direction of flow. An example of such a system is the case of gas displacing oil down a dipping stratum in which the gas-oil contact becomes significantly tilted. Of course, the Buckley-Leverett displacement method cannot predict the tilt of the gas-oil contact. Recent improvements of the one-dimensional Buckley-Leverett method achieve some success in predicting the tilt of the gas-oil contact at sufficiently low flow rates. However, at rates high enough that the viscous pressure gradient nearly equals or exceeds the gravity gradient, even these improved one-dimensional methods incorrectly predict the shape and velocity of the contact. Further progress in estimating such fluid movements in a reservoir appears to require consideration of the problem in more than one space dimension. The recent two-dimensional method of Douglas, Peaceman and Rachford appears adaptable to calculate changes with time of the saturation distribution in a vertical cross-section of a reservoir. The movement of saturation contours should represent the moving fluid contacts and include the effects of crossflow due to gravity, as well as variations in the rock and fluid properties. The nonlinear nature of the equations used in the method has prevented proof of the validity of the solutions. Douglas, Peaceman and Rachford made some comparisons with experiment but did not include cases in which gravity was important nor cases involving displacement by the nonwetting phase. Forthesereasons, atestof the two-dimensional method for a case in which these factors are included would be very desirable. The test selected was a comparison of calculated results with those from a carefully controlled laboratory experiment on a model with measured physical properties. The model selected was one in which gas displaced oil down a tilted, rectangular sand pack. The model can be thought of as representing a vertical cross-section taken parallel to the dip of a reservoir. The displacement thus simulates gas displacing oil downdip that might result from gas-cap expansion or gas injection. SPEJ P. 19^