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By W. F. Kieschnick, C. C. Miller, E. R. Brownscombe

The possibility has been mentioned that large pressure gradients in a solution gas driven field caused by high production rates might lead to a reduction in the ultimate recovery obtainable compared to that which would be obtained by a very slow rate of production. In the present study the reservoir conditions accompanying a high rate of production and the corresponding ultimate recovery were compared with those obtained if the reservoir were produced at some marginal rate throughout its entire life. The method of calculation involved the application of fluid flow - material balance analysis to a series of successive steady state conditions in the reservoir. In the case studied very little difference in recovery was obtained at the abandonment pressure. An analysis of the basic factors involved indicates that the sane results would hold for considerable variation in properties of reservoir fluids, permeability of the formation or well spacing. Even if the reservoir has a water drive it appears that no harm will be done by- open flow production if the rate is cut back before any appreciable free gas is produced. However, any condition which leads to disproportionate withdrawal rates and which causes large pressure differences over large areas might result in substantial loss in ultimate recovery. INTRODUCTION When the pressure declines in a reservoir, gas is evolved from solution causing a shrinkage in the reservoir oil so that a stock tank barrel of oil with its residual dissolved gas occupies a smaller volume than originally. For a given reservoir oil saturation, therefore. stock tank oil content will be higher at the lower pressure. At the same time the evolved gas which remains in the reservoir reduces the reservoir oil saturation, and this would be expected perhaps to cause a reduction in stock tank oil content. However, around the well the shrinkage occurs so rapidly that stock tank oil may actually accumulate in this region in spite of the simultaneous reduction in reservoir oil saturation. If this accumulation should continue, spreading into a larger zone, the loss might be appreciable and because larger pressure gradients accompany higher rates the accumulation might possibly be large if wells are produced at too high a rate. It is the purpose of this paper to determine the effect of rate of flow on ultimate recovery by solution gas drive. The problem under consideration involves the unsteady state flow of oil in a closed radial system from which oil is produced by the evolution of gas as a finite drawdown is estalllished and the reservoir pressure declines. A somewhat similar unsteady state production problem dealing, however, with a linear system has been considered by Muskat and Meres, but these authors do not discuss the effects which rate may have on recovery. Other methods have been reported for estimaling recovery as a function of pressure decline when producing at an infinitesimal rate, and Muskat and Taylor' have shown the effect of fluid and reservoir characteristics on recovery under these conditions. Assuming steady state to exist, Babson has given a procedure for computing the decline in well productivity as the field is depleted, and Moyer has combined these methods to estimate abandonment conditions and the resulting ultimate recovery. These latter approaches to the problem either do not consider the effect of rate or assume steady state can be established. In detail this paper presents the behavior of the reservoir producing under unsteady state conditions and considers the effect of production rate during this unsteady state flow on the recovery attained. The analysis also shows that there will exist at abandonment a type of steady state, and on this basis a method has been clevelopetl for rapid calculation of abandonment conditions.

Jan 1, 1949

By C. C. Miller, W. F. Kieschnick, E. R. Brownscombe

The possibility has been mentioned that large pressure gradients in a solution gas driven field caused by high production rates might lead to a reduction in the ultimate recovery obtainable compared to that which would be obtained by a very slow rate of production. In the present study the reservoir conditions accompanying a high rate of production and the corresponding ultimate recovery were compared with those obtained if the reservoir were produced at some marginal rate throughout its entire life. The method of calculation involved the application of fluid flow - material balance analysis to a series of successive steady state conditions in the reservoir. In the case studied very little difference in recovery was obtained at the abandonment pressure. An analysis of the basic factors involved indicates that the sane results would hold for considerable variation in properties of reservoir fluids, permeability of the formation or well spacing. Even if the reservoir has a water drive it appears that no harm will be done by- open flow production if the rate is cut back before any appreciable free gas is produced. However, any condition which leads to disproportionate withdrawal rates and which causes large pressure differences over large areas might result in substantial loss in ultimate recovery. INTRODUCTION When the pressure declines in a reservoir, gas is evolved from solution causing a shrinkage in the reservoir oil so that a stock tank barrel of oil with its residual dissolved gas occupies a smaller volume than originally. For a given reservoir oil saturation, therefore. stock tank oil content will be higher at the lower pressure. At the same time the evolved gas which remains in the reservoir reduces the reservoir oil saturation, and this would be expected perhaps to cause a reduction in stock tank oil content. However, around the well the shrinkage occurs so rapidly that stock tank oil may actually accumulate in this region in spite of the simultaneous reduction in reservoir oil saturation. If this accumulation should continue, spreading into a larger zone, the loss might be appreciable and because larger pressure gradients accompany higher rates the accumulation might possibly be large if wells are produced at too high a rate. It is the purpose of this paper to determine the effect of rate of flow on ultimate recovery by solution gas drive. The problem under consideration involves the unsteady state flow of oil in a closed radial system from which oil is produced by the evolution of gas as a finite drawdown is estalllished and the reservoir pressure declines. A somewhat similar unsteady state production problem dealing, however, with a linear system has been considered by Muskat and Meres, but these authors do not discuss the effects which rate may have on recovery. Other methods have been reported for estimaling recovery as a function of pressure decline when producing at an infinitesimal rate, and Muskat and Taylor' have shown the effect of fluid and reservoir characteristics on recovery under these conditions. Assuming steady state to exist, Babson has given a procedure for computing the decline in well productivity as the field is depleted, and Moyer has combined these methods to estimate abandonment conditions and the resulting ultimate recovery. These latter approaches to the problem either do not consider the effect of rate or assume steady state can be established. In detail this paper presents the behavior of the reservoir producing under unsteady state conditions and considers the effect of production rate during this unsteady state flow on the recovery attained. The analysis also shows that there will exist at abandonment a type of steady state, and on this basis a method has been clevelopetl for rapid calculation of abandonment conditions.

Jan 1, 1949

By H. E. Stamm, H. H. Spain, R. C. Rumble

This paper presents a reservoir analyzer study of the performance of the Woodbine formation in the East Texas basin. The study was made possible by the compilation of available information on the configuration. thickness and the pressure drawdown of the formation. The investigation was made of the pressure variations in the basin incident to production from the several Woodbine reservoirs. In addition, the apparent compressibility of Woodbine water was evaluated so that the potential water yield of the formation could be determined. The distribution of permeability of the Woodbine sand and the interference between producing areas were also investigated. In conjunction with this basic study of the Woodbine sand, an acceptable match of the production-pressure relationship in the East Texas Field was established on the analyzer. INTRODUCTION When production is taken from a reservoir contiguous to an aquifer, the resultant pressure gradient causes a water influx into the reservoir. For some years it has been the practice to predict the performance of water drive reservoirs with mathematical equations that relate the water influx into reservoirs to their pressure behavior. A rigorous solution of these equations is quite complex: hence, the conventional method requires the simplifying assumptions that the formation have constant thickness, permeability and poiosity, and that it have a known and regular shape. It is necessary to use performance records of the reservoir in question to determine the equation constants that make the results of the mathematical calculations duplicate the reservoir history. The values of the constants determined in this manner sometimes differ so greatly from theoretical or measured values that the constants appear unreasonable. The equation constants comprise such factors as the thickness, porosity and permeability of the formation, and the viscosity and compressibility of water. Usually the thickness, porosity and viscosity are known within a reasonable degree of certainty. whereas the formation permeability and the water compressibility are known with considerably less certainty. Hence. permeability and compressibility values are arbitrarily selected that will make the mathematical equations duplicate the reservoir behavior. In many reservoir studies the value of the compressibility of water thus selected has exceeded the reported value by a factor of from severalfold to manyfold. Whether this excessive value results from the inadequacy of the mathematical approach to take into account the shape irregularities and the heterogeneities of the aquifer, or is due to some type of formation compaction and decrease of porosity has not been established. Because of the significance of the potential water yield of formations with reference not only to oil reservoirs but to water resources as well, it is very important that a representative value for the effective water compressibility be determined. One of the best known examples of a field for which it has been possible in the past to predict the future reservoir pressure is the East Texas Field. Until about 1943 it was possible by means of mathematical equations to duplicate past pressure behavior and to predict with a high degree of accuracy the reservoir pressure of the field. Since 1943, however, the calculated pressures have deviated progressively from the observed pressures to such a degree that in recent years, it has

Jan 1, 1951

By H. H. Spain, R. C. Rumble, H. E. Stamm

This paper presents a reservoir analyzer study of the performance of the Woodbine formation in the East Texas basin. The study was made possible by the compilation of available information on the configuration. thickness and the pressure drawdown of the formation. The investigation was made of the pressure variations in the basin incident to production from the several Woodbine reservoirs. In addition, the apparent compressibility of Woodbine water was evaluated so that the potential water yield of the formation could be determined. The distribution of permeability of the Woodbine sand and the interference between producing areas were also investigated. In conjunction with this basic study of the Woodbine sand, an acceptable match of the production-pressure relationship in the East Texas Field was established on the analyzer. INTRODUCTION When production is taken from a reservoir contiguous to an aquifer, the resultant pressure gradient causes a water influx into the reservoir. For some years it has been the practice to predict the performance of water drive reservoirs with mathematical equations that relate the water influx into reservoirs to their pressure behavior. A rigorous solution of these equations is quite complex: hence, the conventional method requires the simplifying assumptions that the formation have constant thickness, permeability and poiosity, and that it have a known and regular shape. It is necessary to use performance records of the reservoir in question to determine the equation constants that make the results of the mathematical calculations duplicate the reservoir history. The values of the constants determined in this manner sometimes differ so greatly from theoretical or measured values that the constants appear unreasonable. The equation constants comprise such factors as the thickness, porosity and permeability of the formation, and the viscosity and compressibility of water. Usually the thickness, porosity and viscosity are known within a reasonable degree of certainty. whereas the formation permeability and the water compressibility are known with considerably less certainty. Hence. permeability and compressibility values are arbitrarily selected that will make the mathematical equations duplicate the reservoir behavior. In many reservoir studies the value of the compressibility of water thus selected has exceeded the reported value by a factor of from severalfold to manyfold. Whether this excessive value results from the inadequacy of the mathematical approach to take into account the shape irregularities and the heterogeneities of the aquifer, or is due to some type of formation compaction and decrease of porosity has not been established. Because of the significance of the potential water yield of formations with reference not only to oil reservoirs but to water resources as well, it is very important that a representative value for the effective water compressibility be determined. One of the best known examples of a field for which it has been possible in the past to predict the future reservoir pressure is the East Texas Field. Until about 1943 it was possible by means of mathematical equations to duplicate past pressure behavior and to predict with a high degree of accuracy the reservoir pressure of the field. Since 1943, however, the calculated pressures have deviated progressively from the observed pressures to such a degree that in recent years, it has

Jan 1, 1951

By William Hurst, A. F. van Everdingen

For several years the authors have felt the need for a source from which reservoir engineers could obtain fundamental theory and data on the flow of fluids through permeable media in the unsteady state. The data on the unsteady state flow are composed of solutions of the equation Two sets of solutions of this equation are developed, namely. for "the constant terminal pressure case" and "the constant terminal rate case." In the constant terminal pressure case the pressure at the terminal boundary is lowered by unity at zero time, kept constant thereafter, and the cumulative amount of fluid flowing across the boundary is computed, as a function of the time. In the constant terminal rate case a unit rate of production is made to flow across the terminal boundary (from time zero onward) and the ensuing pressure drop is computed as a function of the time. Considerable effort has been made to compile complete tables from which curves can be constructed for the constant terminal pressure and constant terminal rate cases, both for finite and infinite reservoirs. These curves can be employed to reproduce the effect of any pressure or rate history encountered in practice. Most of the information is obtained by the help of the Laplace transformations, which proved to be extremely helpful for analyzing the problems encountered in fluid flow. The application of this method simplifies the more tedious mathematical analyses employed in the past. With the help of Laplace transformations some original developments were obtained (and presented) which could not have been easily foreseen by the earlier methods. INTRODUCTION This paper represents a compilation of the work done over the past few years on the flow of fluid in porous media. It concerns itself primarily with the transient conditions prevailing in oil reservoirs during the time they are produced. The study is limited to conditions where the flow of fluid obeys the diffusivity equation. Multiple-phase fluid flow has not been considered. A previous publication by Hurst' shows that when the pressure history of a reservoir is known, this information can be used to calculate the water influx, an essential term in the material balance equation. An example is offered in the literature by Old² in the study of the Jones Sand, Schuler Field, Arkansas. The present paper contains extensive tabulated data (from which work curves can be constructed), which data are derived by a more rigorous treatment of the subject matter than available in an earlier publication.' The application of this information will enable those concerned with the analysis of the behavior of a reservoir to obtain quantitatively correct expressions for the amount of water that has flowed into the reservoirs, thereby satisfying all the terms that appear in the material balance equation. This work is likewise applicable to the flow of fluid to a well whenever the flow conditions are such that the diffusivity equation is obeyed. DIFFUSITY EQUATION The most commonly encountered flow system is radial flow toward the well bore or field. The volume of fluid which flows per unit of time through each unit area of sand is expressed by Darcy's equation as where K is the permeability, the viscosity and aP/ar the pressure gradient at the radial distance r. A material balance on a concentric element AB, expresses the net fluid traversing the surfaces A and B, which must equal the fluid lost from within the element. Thus, if the density of the fluid is expressed by p, then the weight of fluid per unit time and per unit sand thickness, flowing past Surface A, the surface nearest the well bore, is given as The weight of fluid flowing past Surface B, an infinitesimal distance 6r, removed from Surface A, is expressed as

Jan 1, 1949

By A. F. van Everdingen, William Hurst

For several years the authors have felt the need for a source from which reservoir engineers could obtain fundamental theory and data on the flow of fluids through permeable media in the unsteady state. The data on the unsteady state flow are composed of solutions of the equation Two sets of solutions of this equation are developed, namely. for "the constant terminal pressure case" and "the constant terminal rate case." In the constant terminal pressure case the pressure at the terminal boundary is lowered by unity at zero time, kept constant thereafter, and the cumulative amount of fluid flowing across the boundary is computed, as a function of the time. In the constant terminal rate case a unit rate of production is made to flow across the terminal boundary (from time zero onward) and the ensuing pressure drop is computed as a function of the time. Considerable effort has been made to compile complete tables from which curves can be constructed for the constant terminal pressure and constant terminal rate cases, both for finite and infinite reservoirs. These curves can be employed to reproduce the effect of any pressure or rate history encountered in practice. Most of the information is obtained by the help of the Laplace transformations, which proved to be extremely helpful for analyzing the problems encountered in fluid flow. The application of this method simplifies the more tedious mathematical analyses employed in the past. With the help of Laplace transformations some original developments were obtained (and presented) which could not have been easily foreseen by the earlier methods. INTRODUCTION This paper represents a compilation of the work done over the past few years on the flow of fluid in porous media. It concerns itself primarily with the transient conditions prevailing in oil reservoirs during the time they are produced. The study is limited to conditions where the flow of fluid obeys the diffusivity equation. Multiple-phase fluid flow has not been considered. A previous publication by Hurst' shows that when the pressure history of a reservoir is known, this information can be used to calculate the water influx, an essential term in the material balance equation. An example is offered in the literature by Old² in the study of the Jones Sand, Schuler Field, Arkansas. The present paper contains extensive tabulated data (from which work curves can be constructed), which data are derived by a more rigorous treatment of the subject matter than available in an earlier publication.' The application of this information will enable those concerned with the analysis of the behavior of a reservoir to obtain quantitatively correct expressions for the amount of water that has flowed into the reservoirs, thereby satisfying all the terms that appear in the material balance equation. This work is likewise applicable to the flow of fluid to a well whenever the flow conditions are such that the diffusivity equation is obeyed. DIFFUSITY EQUATION The most commonly encountered flow system is radial flow toward the well bore or field. The volume of fluid which flows per unit of time through each unit area of sand is expressed by Darcy's equation as where K is the permeability, the viscosity and aP/ar the pressure gradient at the radial distance r. A material balance on a concentric element AB, expresses the net fluid traversing the surfaces A and B, which must equal the fluid lost from within the element. Thus, if the density of the fluid is expressed by p, then the weight of fluid per unit time and per unit sand thickness, flowing past Surface A, the surface nearest the well bore, is given as The weight of fluid flowing past Surface B, an infinitesimal distance 6r, removed from Surface A, is expressed as

Jan 1, 1949

By R. I. Parsons, M. B. Standing, E. N. Lindblad

The recovery of the heavier components from a gas cap or retrograde pool is shown to be the greatest when the sand is cycled with a dry gas at a low pressure. This conclusion is in direct opposition to the belief that the most efficient production program is pressure maintenance and cycling at or near the dew point. The results are calculated from: (I) constant volume, variable composition pressure-volume-temperature tests on a mixture of trap gas and liquid from a producing well; (2) published equilibrium constant data and the measured composition of the dew-point material; and (3) the fact that in a sand section of homogeneous permeability, injected gas displaces reservoir gas nearly, quantitatively. The results are based on the simplifying assumption that variable permeability systems may be defined by the ratio of two statistical parameters, and that gas injected into an actual sand will behave as though the sand were composed of many layers, each of constant permeability. Introduction Pressure decline in gas-condensate type reservoirs is accompanied by the formation of a liquid phase throughout the reservoir. Over the past ten years the processing of the material from these types of reservoirs for the heavier hydrocarbon components and the return of the light fractions to the reservoir ("cycling") has become increasingly popular. It has often been stated that the purpose of such a program is to prevent the loss of the retrograde liquid phase formed in the reservoir. The purpose of this paper is to present the results of laboratory tests and computations concerned with several possible methods of producing a gas cap or condensate type of reservoir. The results show that the recovery of heavier hydrocarbons for this type of reservoir is not a maximum under conditions of cycling at or near the dew-point pressure. Instead, variations in permeability and the ability of the dry injected gas to revaporize condensate, point to cycling at a considerably reduced pressure as the optimum production-method. In addition the paper suggests a way of evaluating sands for their permeability variation. The calculations presented in the following sections of the paper are in terms of the production of the butanes and heavier fraction from the gas cap of a field which had an original pressure of approximately 3000 psi. Specifically, the calculations show: 1. To what extent the condensate formed in the reservoir after pressure decline can be recovered by the cycling of a dry absorber-plant gas through the sands. 2. Whether there would be greater recovery of hydrocarbons if cycling were instigated at the original reservoir pressure.

Jan 1, 1948

By M. B. Standing, R. I. Parsons, E. N. Lindblad

The recovery of the heavier components from a gas cap or retrograde pool is shown to be the greatest when the sand is cycled with a dry gas at a low pressure. This conclusion is in direct opposition to the belief that the most efficient production program is pressure maintenance and cycling at or near the dew point. The results are calculated from: (I) constant volume, variable composition pressure-volume-temperature tests on a mixture of trap gas and liquid from a producing well; (2) published equilibrium constant data and the measured composition of the dew-point material; and (3) the fact that in a sand section of homogeneous permeability, injected gas displaces reservoir gas nearly, quantitatively. The results are based on the simplifying assumption that variable permeability systems may be defined by the ratio of two statistical parameters, and that gas injected into an actual sand will behave as though the sand were composed of many layers, each of constant permeability. Introduction Pressure decline in gas-condensate type reservoirs is accompanied by the formation of a liquid phase throughout the reservoir. Over the past ten years the processing of the material from these types of reservoirs for the heavier hydrocarbon components and the return of the light fractions to the reservoir ("cycling") has become increasingly popular. It has often been stated that the purpose of such a program is to prevent the loss of the retrograde liquid phase formed in the reservoir. The purpose of this paper is to present the results of laboratory tests and computations concerned with several possible methods of producing a gas cap or condensate type of reservoir. The results show that the recovery of heavier hydrocarbons for this type of reservoir is not a maximum under conditions of cycling at or near the dew-point pressure. Instead, variations in permeability and the ability of the dry injected gas to revaporize condensate, point to cycling at a considerably reduced pressure as the optimum production-method. In addition the paper suggests a way of evaluating sands for their permeability variation. The calculations presented in the following sections of the paper are in terms of the production of the butanes and heavier fraction from the gas cap of a field which had an original pressure of approximately 3000 psi. Specifically, the calculations show: 1. To what extent the condensate formed in the reservoir after pressure decline can be recovered by the cycling of a dry absorber-plant gas through the sands. 2. Whether there would be greater recovery of hydrocarbons if cycling were instigated at the original reservoir pressure.

Jan 1, 1948

By Francis Collins, E. R. Brownscombe

A study has been made of the material balance-fluid flow method of estimating reserves and degree of water drive from pressure and production history data. By considering the effect of random pressure errors it is shown that in a particular example a standard deviation of three and one-half pounds in each of ten pressure survey? permits the determination of the reserves with a standard deviation of 8 per cent and the water drive with a standard deviation of 15 per cent, assuming that certain basic geologic data are correct. It is believed that this method of estimating reserves and water drive is useful and reliable in a number of cases. The method is particularly valuable when reservoir pressure data are accurate within a very few pounds, but may also be applied with less accurate pressure data if a relatively large reservoir pressure decline occurs early in the life of the field, as for example in an under-saturated oil field. INTRODUCTION A knowledge of the magnitude of reserves and degree of water drive present in any newly discovered petroleum reservoir is necessary to early application of proper production practices. A number of investigators have contributed to methods of relating reserves, degree of water drive, and production and pressure history. 1-8 Three types of problems of increasing complexity may be mentioned. If a reservoir is known to have no water drive. and if the ratio of the volume of the reservoir occupied by gas to the volume of the reservoir occupied by oil (which ratio permits fixing the overall compressibility of the reservoir) is known, then only one further extensive reservoir property remains to be determined, namely the magnitude of the reserves. A straightforward application of material balance considerations will permit this determination. The problem becomes very much more difficult if we wish to determine not only the magnitude of the reserves but also the magnitude of water drive, if any, which is present. In principle, a combination of material balance and fluid flow considerations will permit this evaluation. Finally, if neither the magnitude of reserves, the degree of water drive, nor the ratio of oil to gas present in the reservoir is known and it is desired to determine all three of these variables, the problem could in principle be solved by a fluid flow-material balance analysis which determines the overall compressibility of the reservoir at various points in its history. The change in compressibility with pressure would provide a means of determining the ratio of gas to liquid present, since the compressibilities of gas and liquid vary differently with pressure variation. However, in practice this problem is probably so difficult as to defy solution in terms of basic data precision apt to be available.' It is the purpose of this discussion to illustrate the second case, which involves the determination of two unknown variables, single phase reserves and degree of water drive, from pressure and production history and fluid property data, and to study the precision with which these unknowns can be determined in this manner in a particular case. Although an electric analyzer developed by Bruce as used in making the calculations to be described, numerical methods necessary in carrying out the process have been devised and have been applied for this purpose. Schilthuis,' for example, developed a comprehensive equation for the material balance in a reservoir. He combined this with a simplified water drive equation, assuming that the ratio of free gas to oil was fixed by geological data and that a period of constant pressure operation at constant rate of production was available to determine the constant for his water drive equation. On this basis he was able to compute the reserves and predict the future pressure history of the reservoir. Hurst developed a generalized equation permitting the calculation of the water drive by unsteady state expansion from a finite aquifer. He showed in a specific case how the water influx calculated by his equation, using basic geologic and reservoir data to fix the constants, matched the water influx required by material balance considerations. Old3 illustrated the simultaneous use of Schilthuis' material balance equation and Hurst's fluid flow equation for the determination of the magnitude of reserves and a water drive parameter from pressure and production history. He used this method to calculate the future pressure history of the reservoir under assumed operating conditions. As a basis for determining reserves, Old assumed a value for his water drive parameter and calculated a set of values for the reserves, using the initial reservoir pressure and each successive measured pressure. The sum of the absolute values of the deviations of the resulting reserve numbers from their mean value was taken as a criterion of the closeness of fit to the experimental data possible with the water drive parameter assumed. New values of the water drive parameter were then assumed and new sets of the reserves calculated until a set of reserves numbers having a minimum deviation from the average was established. The average value of- the re-

Jan 1, 1949

By E. R. Brownscombe, Francis Collins

A study has been made of the material balance-fluid flow method of estimating reserves and degree of water drive from pressure and production history data. By considering the effect of random pressure errors it is shown that in a particular example a standard deviation of three and one-half pounds in each of ten pressure survey? permits the determination of the reserves with a standard deviation of 8 per cent and the water drive with a standard deviation of 15 per cent, assuming that certain basic geologic data are correct. It is believed that this method of estimating reserves and water drive is useful and reliable in a number of cases. The method is particularly valuable when reservoir pressure data are accurate within a very few pounds, but may also be applied with less accurate pressure data if a relatively large reservoir pressure decline occurs early in the life of the field, as for example in an under-saturated oil field. INTRODUCTION A knowledge of the magnitude of reserves and degree of water drive present in any newly discovered petroleum reservoir is necessary to early application of proper production practices. A number of investigators have contributed to methods of relating reserves, degree of water drive, and production and pressure history. 1-8 Three types of problems of increasing complexity may be mentioned. If a reservoir is known to have no water drive. and if the ratio of the volume of the reservoir occupied by gas to the volume of the reservoir occupied by oil (which ratio permits fixing the overall compressibility of the reservoir) is known, then only one further extensive reservoir property remains to be determined, namely the magnitude of the reserves. A straightforward application of material balance considerations will permit this determination. The problem becomes very much more difficult if we wish to determine not only the magnitude of the reserves but also the magnitude of water drive, if any, which is present. In principle, a combination of material balance and fluid flow considerations will permit this evaluation. Finally, if neither the magnitude of reserves, the degree of water drive, nor the ratio of oil to gas present in the reservoir is known and it is desired to determine all three of these variables, the problem could in principle be solved by a fluid flow-material balance analysis which determines the overall compressibility of the reservoir at various points in its history. The change in compressibility with pressure would provide a means of determining the ratio of gas to liquid present, since the compressibilities of gas and liquid vary differently with pressure variation. However, in practice this problem is probably so difficult as to defy solution in terms of basic data precision apt to be available.' It is the purpose of this discussion to illustrate the second case, which involves the determination of two unknown variables, single phase reserves and degree of water drive, from pressure and production history and fluid property data, and to study the precision with which these unknowns can be determined in this manner in a particular case. Although an electric analyzer developed by Bruce as used in making the calculations to be described, numerical methods necessary in carrying out the process have been devised and have been applied for this purpose. Schilthuis,' for example, developed a comprehensive equation for the material balance in a reservoir. He combined this with a simplified water drive equation, assuming that the ratio of free gas to oil was fixed by geological data and that a period of constant pressure operation at constant rate of production was available to determine the constant for his water drive equation. On this basis he was able to compute the reserves and predict the future pressure history of the reservoir. Hurst developed a generalized equation permitting the calculation of the water drive by unsteady state expansion from a finite aquifer. He showed in a specific case how the water influx calculated by his equation, using basic geologic and reservoir data to fix the constants, matched the water influx required by material balance considerations. Old3 illustrated the simultaneous use of Schilthuis' material balance equation and Hurst's fluid flow equation for the determination of the magnitude of reserves and a water drive parameter from pressure and production history. He used this method to calculate the future pressure history of the reservoir under assumed operating conditions. As a basis for determining reserves, Old assumed a value for his water drive parameter and calculated a set of values for the reserves, using the initial reservoir pressure and each successive measured pressure. The sum of the absolute values of the deviations of the resulting reserve numbers from their mean value was taken as a criterion of the closeness of fit to the experimental data possible with the water drive parameter assumed. New values of the water drive parameter were then assumed and new sets of the reserves calculated until a set of reserves numbers having a minimum deviation from the average was established. The average value of- the re-

Jan 1, 1949

By A. D. K. Laird, John A. Putnam

This paper describes the application of x-ray theory to design procedures in connection with fluid saturation determinations during fluid flow experiments with porous media. A reliable and rapid method for calibrating the x-ray apparatuy is described. Extension of the method to fluid saturation determinations in three-fluid systems is described. INTRODUCTION In rerearch on oil production problems a method is required which will give quickly the quantity of each component of a fluid flow system present at any cross-section of a porous medium. The sample of porous medium under investigation is usually referred to as a core. The ratio of the volume of one component to the total fluid volume is defined as the saturation of the porous medium by that component. This ratio is generally given as per cent saturation. Some means of measuring saturation which have received consideration include: electrical conductivity of the fluids;1,2 emissions from radioactive tracers dissolved in the fluids; the radioactivity of silver caused by reflection of neutrons from hydrogen atoms in the fluids;' the attenuation of a microwave beam. the diminution and phase shift of ultrasonic wave trains.4,5 and the reduction in intensity of x-ray beams in passing through the fluids. X-rays have already been used with some success. Since every material has a different power to absorb x-rays, the reduction in intensity of an x-ray beam as it passes through a core depends on the fluids present. The strength of the emergent beam can be found by converting its energy into a measurable form such as heat or ionic current. or by its effect on a photographic plate or fluorescent screen. The beam strengths could be interpreted as quantities of known fluids in the core if, previously, these beam strengths had been identified with a known combination of the same fluids. With some fluid cornbinations it might be desirable to dissolve powerful x-ray absorbing materials in one or more of the fluids, to increase the differences in the beam strengths for various fluid saturations. Boyer, Morgan and Muskat6 have described a method of measuring two component fluid saturation. One component was air or water; the other. minerat seal oil in which was dissolved 25 per cent by weight of iodobenzene to increase its absorbing power. The x-ray source was a tungsten target tube operated at 43 kv potential. The beam emerging from the core was measured as ionic current flowing across an air-filled ionization chamber by means of an amplifying circuit and galvanometer. Another portion of the beam from the x-ray tube was passed through a metal plate and measured in another ionization chamber. This portion, called the monitor beam, was used as an indication of the performance of the x-ray tube. The galvanometer readings were calibrated against air-oil core saturations, gravimetrically determined. The method was apparently established by experimental means. In the present investigation the available theory of x-radia-tion was surveyed with a view to extending the usefulness of the method and to developing design procedures for its application to measurement of fluid saturation in porous media. Application of the theory permits prediction of relative meter readings to be expected for any combination of porous matrix, various saturating fluids and auxiliary filtering media. It is thus possible to calibrate the equipment in terms of fluid saturation by an indirect but rapid technique. The results of calculations based on x-ray theory indicate. and results of the saturation calibration technique confirm. that a valid measurement of the saturation of the core can be made for any two components and in some cases for three components. THEORY The strength of an x-ray beam, after it has passed through a distance. 1, of matter of density, p, and mass absorption coefficient, µ at a given wavelength, A, may be expressed by the absorption formula I = I0 e ...........(1) where I, represents the intensity of the incident x-ray beam and I is the intensity of the emergent beam. The expression e is called the transmission factor of the material. The variation of I,, with wavelength depends upon the materials through which the x-ray beam has previously passed and upon the spectral distribution of energy at the source of the x-radiation. A group of curves. called spectra. which show the variation of intensity with wavelength and x-ray tube voltage are given in Fig. 1. These curves represent the general radiation from a tungsten target tube. When the tube voltage is greater than 69.3 kv, the characteristic radiation of the tungsten is emitted and is superposed on the general radiation. At a given voltage the minimum wavelength A,,,,, at which energy can be emitted by an x-ray tube is given by the formula 12,340 xml. = ——..........(2) volts where A,.,,.. is in Angstrom units. The wavelength at which the spectra have maximum intensity a1so decreases with increasing x-ray tube voltaue. The area under each curve represents to an arbitrarv scale the total energy emerging from the x-ray tube for that voltage. The variation of µ with wavelength has been determined for many substances and may be found in such references as those by Compton and Allison7 and by Hodgman.8 The phenomenon of absorption is composed chiefly of the capture of photons by the atoms of the absorbing material with associated displacement of electrons, and of the scattering, or the deflection, of the photons by the atoms. Curves of these mass absorption coefficients show jump discontinuities. or absorption edges. at wavelengths which are short enough for the photons,

Jan 1, 1951

By John A. Putnam, A. D. K. Laird

This paper describes the application of x-ray theory to design procedures in connection with fluid saturation determinations during fluid flow experiments with porous media. A reliable and rapid method for calibrating the x-ray apparatuy is described. Extension of the method to fluid saturation determinations in three-fluid systems is described. INTRODUCTION In rerearch on oil production problems a method is required which will give quickly the quantity of each component of a fluid flow system present at any cross-section of a porous medium. The sample of porous medium under investigation is usually referred to as a core. The ratio of the volume of one component to the total fluid volume is defined as the saturation of the porous medium by that component. This ratio is generally given as per cent saturation. Some means of measuring saturation which have received consideration include: electrical conductivity of the fluids;1,2 emissions from radioactive tracers dissolved in the fluids; the radioactivity of silver caused by reflection of neutrons from hydrogen atoms in the fluids;' the attenuation of a microwave beam. the diminution and phase shift of ultrasonic wave trains.4,5 and the reduction in intensity of x-ray beams in passing through the fluids. X-rays have already been used with some success. Since every material has a different power to absorb x-rays, the reduction in intensity of an x-ray beam as it passes through a core depends on the fluids present. The strength of the emergent beam can be found by converting its energy into a measurable form such as heat or ionic current. or by its effect on a photographic plate or fluorescent screen. The beam strengths could be interpreted as quantities of known fluids in the core if, previously, these beam strengths had been identified with a known combination of the same fluids. With some fluid cornbinations it might be desirable to dissolve powerful x-ray absorbing materials in one or more of the fluids, to increase the differences in the beam strengths for various fluid saturations. Boyer, Morgan and Muskat6 have described a method of measuring two component fluid saturation. One component was air or water; the other. minerat seal oil in which was dissolved 25 per cent by weight of iodobenzene to increase its absorbing power. The x-ray source was a tungsten target tube operated at 43 kv potential. The beam emerging from the core was measured as ionic current flowing across an air-filled ionization chamber by means of an amplifying circuit and galvanometer. Another portion of the beam from the x-ray tube was passed through a metal plate and measured in another ionization chamber. This portion, called the monitor beam, was used as an indication of the performance of the x-ray tube. The galvanometer readings were calibrated against air-oil core saturations, gravimetrically determined. The method was apparently established by experimental means. In the present investigation the available theory of x-radia-tion was surveyed with a view to extending the usefulness of the method and to developing design procedures for its application to measurement of fluid saturation in porous media. Application of the theory permits prediction of relative meter readings to be expected for any combination of porous matrix, various saturating fluids and auxiliary filtering media. It is thus possible to calibrate the equipment in terms of fluid saturation by an indirect but rapid technique. The results of calculations based on x-ray theory indicate. and results of the saturation calibration technique confirm. that a valid measurement of the saturation of the core can be made for any two components and in some cases for three components. THEORY The strength of an x-ray beam, after it has passed through a distance. 1, of matter of density, p, and mass absorption coefficient, µ at a given wavelength, A, may be expressed by the absorption formula I = I0 e ...........(1) where I, represents the intensity of the incident x-ray beam and I is the intensity of the emergent beam. The expression e is called the transmission factor of the material. The variation of I,, with wavelength depends upon the materials through which the x-ray beam has previously passed and upon the spectral distribution of energy at the source of the x-radiation. A group of curves. called spectra. which show the variation of intensity with wavelength and x-ray tube voltage are given in Fig. 1. These curves represent the general radiation from a tungsten target tube. When the tube voltage is greater than 69.3 kv, the characteristic radiation of the tungsten is emitted and is superposed on the general radiation. At a given voltage the minimum wavelength A,,,,, at which energy can be emitted by an x-ray tube is given by the formula 12,340 xml. = ——..........(2) volts where A,.,,.. is in Angstrom units. The wavelength at which the spectra have maximum intensity a1so decreases with increasing x-ray tube voltaue. The area under each curve represents to an arbitrarv scale the total energy emerging from the x-ray tube for that voltage. The variation of µ with wavelength has been determined for many substances and may be found in such references as those by Compton and Allison7 and by Hodgman.8 The phenomenon of absorption is composed chiefly of the capture of photons by the atoms of the absorbing material with associated displacement of electrons, and of the scattering, or the deflection, of the photons by the atoms. Curves of these mass absorption coefficients show jump discontinuities. or absorption edges. at wavelengths which are short enough for the photons,

Jan 1, 1951

By C. R. McEwen

The displacement equations of Buckley and Lever-ett' have been successfully applied to the prediction of oil recovery in frontal drives for a number of years. Commonly, the capillary pressure term is omitted from Leverett's fractional flow formula. The resulting satura-tion-vs-distance curves become multi-valued, however. Buckley and Leverett, recognizing that the neglect of capillarity was most likely the cause, proposed that this part of the solution he ignored. An ingenious method of propagating a discontinuity in place of the multi-valued portlon of the curve was devised by Cardwell using "non-capillary displacement theory". However, it would seem that a completc description of the displacement pracess would require the inclusion of capillary forces. Three publications on displacement theory have appeared in which capillary pressure was included. Ter-williger. et al', calculated saturation profiles which matched those which they observed when gas displaced water vertically downward. Based on their study, Jones-Parra and Calhoun5 roposed a method of computing saturation distributions sufficiently removed in time from the initial conditions so that all saturations in the flood front region were assumed to move with the same velocity. This latter assumption corresponds to the tangent construction on the flowing fraction-vs-satura-tion curve proposed by Welge." A numerical solution to the problem of water displacing oil from a water-wet porous medium was recently presented by Blair, ef at'. The initial conditions included a uniform saturation slightly above interstitial water saturation. that is, the permeability to water ahead of the front was not zero. This appeared to result in a region of slowly rising water saturation ahead of the front, at least for the particular set of relative permeability and capillary pressure data used to illustrate the computation. The authors also postulated an "end-effect" at the outflow face of the system, whereby no water could pass out of the system until the water saturation had built up to a high value. The saturation profiles computed under these assumptions show water accumulating at the downstream end of the system while the flood front proper is still traveling toward the outlet. The purpose of this paper is to present the results of a numerical solution to the one-dimensional water-flooding equation in which no outlet end-effect is considered and where there is no flow of water ahead of the flood front. It should be noted that the contemporary paper of Fayers and Sheldon8 deals with the problem under discussion. For a thorough treatment including both capillarity and gravity the reader is referred to their paper.

By H. T. Kennedy, J. E. Little

An empirical equation for the prediction of the viscosity of several pure paraffin hydrocarbons and nitrogen is presented. It involves temperature, pressure and six constants of the material, and it applies reliably to both liquids and gases. The equation is similar in form to van der Waal's equation of state. For the paraffin hydrocarbons methane through r-hexane and nitrogen, an average absolute deviation of 1.9 percent was obtained on 1,006 data points described in the literature by 14 authors. When this equation is extended to complex, liquid hydrocarbon mixtures, a correlation was obtained with an average absolute deviation of 9.9 percent. INTRODUCTION Equations describing the flow of gas and liquid through porous media contain the viscosity coefficient of the fluid. If other pertinent variables remain constant, the volume rate of flow is inversely proportional to this coefficient. In dealing with condensate fluids and volatile oils, however, the compositional effects resulting from changing pressure materially affect the viscosity. The effect of compositional changes also may be significant in certain secondary recovery or pressure maintenance processes, notably miscible displacement or gas injection. Early attempts to describe the performance of reservoirs utilized a volumetric material balance method wherein gas and liquid in the reservoir were identified as produced gas and liquid at the surface. This method of analysis proved adequate for reservoirs at moderate temperature and pressure that contained gas with moderately low amounts of condensable materials. The volumetric material balance procedures for "black oil" reservoirs leave much to be desired when applied to condensate and volatile oil reservoirs because phase behavior and compositional changes are relatively more important in these cases. The alternative is a compositional material balance, which in turn, requires a correlation of properties of the reservoir fluid with composition. This paper supplies this correlation in regard to viscosity, for reservoir crude oils. REVIEW OF LITERATURE The literature contains many empirical equations describing the effects of composition, temperature and pressure on the viscosities of pure liquids and binary liquid mixtures. However, the applicability of a majority of these equations is limited to very low pressures and to a small number of systems. Most of them, when applied to complex hydrocarbon systems, are of little value. The lack of utility of the majority of equations results from the fact that they were developed to show the separate effect of temperature, pressure or composition on viscosity, but not to predict the viscosity as a function of all three variables. And with the few exceptions noted below, they were developed to apply to much simpler systems than hydrocarbon mixtures. Eakin1 developed an equation for the viscosity of pure light hydrocarbons applicable in the gas, liquid and dense fluid regions, and in its most general form is written as µ - µo =A(ebP - e-cpk) The pure component viscosity at atmospheric pressure and temperature T, µo, was calculated using the Sutherland equation: µo = BT3/2/T + S The constants in Eakin's equation permitted the term e-CPk to become quite small at high densities (above 0.215 gm/cc), and the general equation would simplify to

Jan 1, 1969

By B. H. Caudle, M. D. Witte

In predicting the performance of a pattern injection operation, the engineer needs to know both the amount of oil to be recovered and the rate at which the recovcry will take place. This paper- describes fluid flow model studies on the effect of mobility ratio on the rate of oil recovery in a five-spot. The results show the change in fluid conductivity (total flow rate/pressure drop) as the sweep-ont pattern increases for mobility ratios between 0.1 (and 10. There data, when cornbitted with a knowledge of reservoir permeabilities and sweep-orit pattern efficiencies, cata he used to predict the cLrmulntivr oil production as a function of time for homogeneous five-spot injection systems. INTRODUCTION The flow potential in a pattern flood element is a function of the mobility ratio and how far the injected flood front has gone.1,2 For a waterflood project having a mobility ratio of one, the conductance—which is the flow rate per unit pressure drop (q/Ap)—-will remain the same throughout the life of the flood. If the mobility ratio is favorable ( < I), the conductance of the project will drop continuously throughout the operation. If, on the other hand, the mobility ratio is unfavorahlc. the conductance ratio will increase continuously. Therefore, if the reservoir engineer wishes to predict either the rate at a specified pressure or the pressure at some rate for a secondary recovery project, he must know how the conductance will vary during the flood. Other investigators have studied this problem using potentiometric models&apos; or analytical solution&apos;. The

By A. J. Teplitz, R. J. Goodwin

A new type water shut-off for use in air drilling has been developed. The method has been 99 to 100 per rent effective in several different formations of inter-ranular-type porosity. Since costs for materials, time and equipment used are relatively small, the process should extend economies of air drilling to many new areas. The treatment involves a new technique for forming a water-inroluble precipitate within the pores of a rock. Advantage it taken of the difference in mobility of a liquid and a gas in porous media to achieve the greatest volume of precipitate in a given pore. Reactants selected for the laboratory (2nd field tests were aluminum sulfate solution and gaseous ammonia. This process has been tested in both intergranular and fractured-type porosities. It has consistently shut-of between 99 and 100 per cent of the water flowing from formations having intergranular-type porosity and perrlreuhilities ranging from a few millidarcise to over a darcy The treatment is not particularly suitable for shutting-off water from fractured formations although plugging of the order of 50 per cent or greater has been achieved in all such applications. Among advantages of the treatment are: (I) no reaction waiting rime after a relatively short placement time, (2) a low mate-rials cost of about $3/f1 of hole treated, (3) use of common service-type equipment for treatment, and (4) no unusual requirements for mixing and placement. INTRODUCTION In rotary drilling with air or gas it has been possible to increase bit life by tenfold and more than double rate of penetration over corresponding performance when drilling with mud. These factors have amounted to as much as 22 per cent savings on total cost of drilling a well. However, these advantages are rapidly lost when water is encountered in an air- or gas-drilled well because water interferes with the removal of cuttings from the hole and also gives rise to other difficulties. In attempting to combat water troubles, operators have commonly resorted to such measures as mist drilling, treatment with cement or chemicals, converting to conventional mud drilling, or setting casing. In many cases these measures have either been costly or ineffective. In an effort to overcome this problem, studies were undertaken to develop an effective and economical water shut-off technique for use in formations possessing inter-granular-type porosity. Extensive laboratory experimentation finally led to a unique method for producing a water-insoluble precipitate within the pores of a sandstone. The unique part of the treatment is use of a liquid and a gas to form the precipitate, and in this case the reactants used were aluminum sulfate solution and gaseous ammonia. The reaction between these two materials is shown in the following equation. Al2 (SO1)3 - 18 H20 6NH2? saturated Water- Solution Cat 2A1(OH)3 . n H20 3(NH4)2SO • n H20 Solid Solid That is, a solid precipitate consisting of aluminum hydroxide and ammonium sulfate is formed when ammonia reacts with aqueous aluminum sulfate. The aluminum hydroxide is insoluble in water and the ammonium sulfate, though soluble, can only be removed from the formation pores by diffusion—an extremely slow process. Although the precipitate formed by the reaction is not particularly firm, great strength is not needed for a precipitate to bridge and plug intergran-ular-type porosity. A flow path in such a system is extremely tortuous and involves a great variety of pore

By G. E. Perry, R. D. Seba

The steam soak process is the most widely applied and most successful thermal supplemental recovery process in use today. This process, which consists of injection of steam in various quantities into a well that is subsequently used as a producer, can be repeatedly applied to the same well. The treatment, if successful, will increase the production rate of the well for a given period. For this reason, the steam soak process should be considered primarily as an acceleration method rather than as a method for producing oil that could not be recovered by primary means. However, application of this process to reservoirs containing viscous crude may produce oil that otherwise would have zero present value or would be produced at such low rates by primary methods that it would not be profitable. A major necessity for the successful application of the steam soak process is a source of natural energy. This source of energy may exist in one or more of the following forms: expansion of fluids, compaction, gravity drainage. This study was undertaken to determine the effect of gravity drainage on steam soak production and recovery and to develop a method for predicting production rates, ultimate recovery, and heat requirements when applying this process to a reservoir in which the primary driving mechanism is gravity drainage. (For additional discussion of some of the factors that affect recovery of viscous oils by the steam soak process, see Ref. 5.) Model The model developed for analyzing the performance of the steam soak process in gravity drainage reservoirs is a modified form of that developed by Matthews and Lefkovits.1,2 The model used in this study consists of two concentric cylindrical volumes of productive formation with cap and base rock of impermeable material. The inner cylinder, extending from the radius of the wellbore to some predetermined radius, is referred to as the hot zone; and the outer cylinder extending from the exterior boundary of the hot zone to the outer drainage boundary will be referred to as the partially heated zone. Fig. 1 is a schematic cross-section of the model, indicating its thermal properties. Horizontally the reservoir is divided into two regions, the upper region having a high gas saturation and residual liquid saturation, and the lower region containing a high liquid saturation. The boundary between these two regions is referred to as the free surface, which drops and changes shape as fluid is produced from the reservoir. The model was used to analyze the steam soak process by applying the following assumptions: 1. The horizontal formation is composed of one or more homogeneous noninterconnected layers containing incompressible fluids. Each layer is treated separately. 2. The hot zone is heated by the initial soak and is maintained at the steam temperature corresponding to atmospheric pressure by resoaking. 3. The height of the free surface at the producing well is at all times zero (producing well is pumped off). 4. The position of the free surface at the start of

Jan 1, 1970

By F. G. Miller, T. D. Mueller, L. B. Davidson

A mathematical model of reservoir behavior during one cycle of the intermittent steam injection process is presented in this paper. A cycle of the process is considered to be composed of three stages: injection, condensation and production. The model concentrates on behavior during the production stage. The influence of several parameters on production improvement is described, using the results obtained with a numerical equivalent of the model. Reservoir system parameters considered are thickness, reservoir thermal conductivity, oil viscosity, solution gas-oil ratio and the change in oil effective permeability with temperature. Operating parameters investigated are heat injection rate, steam temperature and cumulative injection time. In the face of considerable uncertainty, the model appears to give a fair prediction of the behavior observed during an actual test. This makes the model a useful means of selecting projects. It is shown that the model may also be utilized to select an optimum set of operating conditions for a particular project. Results of this study show the extent to which production improvement depends on each of the parameters investigated and give an indication of the range of system conditions within which the process would be effective. INTRODUCTION In recent years the intermittent steam injection or steam soak process of oil recovery has become a popular method of improving recovery in reservoirs bearing high-viscosity hydrocarbons. Production improvement due to the process results from heating the reservoir near the wellbore. The process is applied on an individual well basis and is executed in a series of cycles, each consisting of three stages: injection, condensation (the soaking period) and production. During the injection stage, high-pressure steam is injected into the reservoir surrounding the well-bore. This study assumes that injection is carried out over the entire vertical section of the reservoir rock that can be exposed to fluid injection. The condition of steam entering the reservoir is considered to fall within the following limits: (1) steam pressures less than 1,000 psig, (2) steam temperatures between 300 and 700F and (3) steam quality less than 90 percent. The last restriction is due to equipment limitations. Normally, injection re maintained for less than 3 weeks. After termination of injection, the condensation stage begins. The well is shut in for several days, supposedly to make certain that the injected steam condenses. This condensation would insure the retention of injected heat. The production stage is initiated when the well is opened again. Reservoir response during this stage determines the effectiveness of the process (response is defined as the oil production rates, the water production rates and the wellbore temperatures observed during the entire production stage). The immediate response of the reservoir is sometimes quite good. Sets of case histories by Longl and by Owens and Suter2 give some typical figures for response. Duration of the production stage is equal to the time needed for the oil production rate to decline to some predetermined value. This usually requires 4 to 6 months. Termination of the production stage coincides with the start of the next cycle. The process is continued, cycle by cycle, until it becomes unprofitable to continue. One simple model2 and two more sophisticated models3,4 of the process have been presented. These are based on fundamentally different assumptions than those used in this investigation. Although the three models might be modified to fit the assumptions used in this study, it is felt that greater rigor would result by developing a

By B. Zondek, W. T. Cardwell, J. W. Sheldon

The fundamental equations that are used to describe two-phase fluid flow in porous media are Darcy&apos;s law for each phase and an equation of continuity for each component. The special case of one-dimensional, incompressible, two-phase flow has received much attention in the petroleum engineering literature.1-10 The basic paper on the subject is that of Buckley and Leverett.2 Buckley and Leverett showed that one could eliminate the pressure and obtain a single partial differential equation for the saturation. They solved this equation for the case when gravity and capillary forces are negligible. A frequently encountered property of the solution of the basic partial differential equation is that, as time progresses, the saturation becomes a multiple-valued function of the distance coordinate, x. Buckley and Leverett interpreted the formation of multiple values as an indication that the saturation-distance curve had be-come discontinuous. They developed a method of determining the position of the discontinuity from a material balance. We solve the Buckley-Leverett partial differential equation using the method of characteristics and the concept of shocks. Our treatment is analogous to treatments used in the theory of supersonic compressible flow." Although the numerical results obtained are the same as those obtained by previous methods, we believe that this approach is more logical and has broader applicability. It can, for example, be generalized to treat the case when the fluids are compressible. It is not clear how one would go about doing this using the method of Buckley and Leverett. To illustrate the power of the method of characteristics with shocks, we solve a problem in gravity segregation which has not been treated previously in the literature. A special case of the general method developed here was used by Welge, to solve a particular fluid displacement problem. We consider two-phase incompressible flow along the vertical direction x in a porous medium. We assume that there is no flow transverse to .x. We measure positive in the upward direction. The fundamental equations are obtained from Darcy&apos;s law and the mass conservation law. Since the two phases are incompressible, mass conservation is equivalent to volume conservation.

By C. S. Matthews, P. Hazebroek, F. Brons

A method has been developed for calculating the average pressure in a bounded reservoir. The reservoir is first divided into the individual drainage volumes of each well, by using the criterion that at steady state each individual drainage volume is proportional to a well&apos;s production rate. The average pressure in each drainage volume is then calculated by a method developed in the report. By volumetrically averaging these individual drainage volume pressures, the average pressure in the entire reservoir is obtained. To calculate the average pressure in each drainage volume, a correction is applied to the ordinary extrapolated pressure, i.e., the pressure obtained by extrapolating to infinite time the linear portion of the graph of closed-in pressure versus log[?t/(t + ?t)], where At is the closed-in time and t the production time. The correction, which is a function of the production time, is presented in graphical form for different shapes of the drainage area (horizontal cross section of the drainage volume). INTRODUCTION It is important to be able to find the volumetric average pressure in a reservoir so that the size of the reservoir may be determined from material balance calculations. It is also desirable to be able to find the approximate distribution of pressure within a reservoir for detection of fluid movement. &apos;The purpose of this paper is to present a method for calculating both the average reservoir pressure and the approximate distribution of pressure within a bounded reservoir — that is, a reservoir with no water drive. In reservoirs where the pressure builds up rapidly after wells are shut in, the determination of average pressure generally poses little problem, for one often need only average the final buildup pressures. It is when pressure buildup is slow that difficulties arise. For practical and economical reasons, the time allowable for closing in wells is limited. If at the maximum allowable closed-in time the pressure has not reached a con- stant value (and this is more often the case than is generally realized), calculation of average pressure presents difficulties. One method which has been used in the past for obtaining an "average" pressure involves the extrapolation to infinite closed-in time of curves of pressure as a function of closed-in time. Contour maps of these extrapolated pressures are used in arriving at an average pressure. In some cases wells are shut in for arbitrary Lengths of time, and contours of these arbitrary shut-in pressures are used for calculation. All these methods are somewhat arbitrary and suffer from lack of fundamental theoretical basis. For example, no precise meaning can be given either to the extrapolated or to the arbitrary shut-in pressure. Furthermore, there seems to be no reason why either set of pressures should be contourable. It was because of the vagueness of these old methods and their lack of a sound theoretical basis that, the present method for determining average reservoir pressure was developed. SYMBOLS A = drainage area of a well, cm&apos; A, = drainage area of well j, cm2 A , = total drainage area of all wells in a reservoir, cm2 c = coefficient of compressibility of fluid at reservoir temperature, atm-1 f = hydrocarbin-filled porosity, fraction h = net thickness of formation, cm k = permeability of formation, darcy p = pressure at a well, atm p0 = initial reservoir pressure, atm p* = pressure obtained by straight-line extrapolation of linear portion of plot of p versus log [At/(t+ At)] p = volumetric average pressure inside drainage volume of a well q = volume flow rate at a well, cc/sec a: prevailing reservoir conditions q, = total volumetric flow rate from a reservoir, cc/sec at prevailing reservoir conditions rw = radius of well bore, cm T = kt/fµcA t = corrected time of production of a well, seconds, defined by

Jan 1, 1955