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By R. W. Nelson

The theoretical basis is presented for energy dissipation methods of measuring permeability in saturated heterogeneous media. Analysis starts with the equation for single-phase flow in heterogeneous porous media which is a first-order partial differential equation in the unknown permeability. This equation is reduced to a system of characteristic equations useful in deriving a differential expression for the permeability as a function of the known potential distribution. For steady flow systems the expression is integrable, giving the permeability integral for direct calculation of the distribution along successive streamlines. The boundary condition in permeability is presented and includes a discussion of the special requirement to assure uniqueness. Theoretical results and necessary computations are incorporated into two general computer programs capable of determining the permeability distribution in two- and three-dimensional steady flow systems. Results are then presented for an energy dissipation analysis using the methods and soltware to determine a field permeability thickness distribution. INTRODUCTION Theoretical work leading to the description of fluid flow in heterogeneous porous media has developed slowly and somewhat sporadically, although the areas of engineering applications predominantly involve flow in nonhomogeneous media. The lack of accurate and economical methods for measuring field permeability distributions has been a major limitation to realistic description of natural flow systems. The economical measurement of permeability using energy dissipation methods may overcome this limitation. Among other things, any permeability measurement method involves consideration of energy dissipation; as used here, "energy dissipation method" denotes the following general approach. The areal energy distribution is measured on an existing flow system in porous media. Then through appropriate analysis of the dissipation of energy in the flow system, the permeability distribution is determined throughout the region using the minimum boundary conditions required to assure a mathematically unique result. This paper presents the theoretical basis for such a method, as well as the techniques and computer software needed to carry out the analysis. The method and techniques are then used in a preliminary field application. EQUATIONS DESCRIBING FLOW IN HETEROGENEOUS MEDLA Heterogeneous as used here implies differing in kind, having unlike quantities or possessing different characteristics. Accordingly, a heterogeneous porous medium has unlike quantities or differing characteristics at different locations. More precisely, if a porous medium is heterogeneous with respect to some property, then that property is functionally dependent on the spatial location. (Media properties are considered here as macroscopic.) Specifically considering permeability k,** then k = G(x, y, z) ,............(1) or the permeability is a scalar function G of the location (x, y, z). Spatial variation of several other properties, of which porosity, dispersivity and medium compressibility are but a few, is implied by the broad title of heterogeneous porous media. The term will be used hereafter to imply only heterogeneity with respect to permeability. The porous medium is considered isotropic as implied by G being only a scalar function. Anisotropy

Jan 1, 1969

By C. K. Eilerts

Finite difference equations were programmed and used to integrate the second-order, second-degree, partial differential equation with variable coefficients that represents the transient linear flow of gas-condensate fluids. Effect was given to the change with pressure of the compressibility factor, the viscosity, and the effective permeability and to change of the absolute permeability with distance. Integrations used as illustrations include recovery of fluid from a reservoir at a constant production rate followed by recovery at a declining rate calculated to maintain a constant pressure at the producing boundary. The time required to attain such a limiting pressure and the fraction of the reserve recovered in that time vary markedly with properties of the fluid represented by the coefficients. Fluid also is returned to the reservoir at a constant rate, until initial formation pressure is attained at the input boundary, and then at a calculated rate that will maintain but not exceed the limiting pressure. The computing programs were used to calculate the results that would be obtained in a series of back-pressure tests made at selected intervals of reservoir depletion. If effect is given to the variations in properties of the fluid that actually occur, then a series of back- pressure curves — one for each stage of reserve depletion — is required to indicate open-flow capacity and related flow characteristics dependably. The isochronal performnce method for determining flow characteristics of a well was simulated by computation. INTRODUCTION The back-pressure test procedurel is based on a derivation of the equation for steady-state radial flow of a gas, the properties of which are of necessity assumed to remain unchanged in applying the test results. The properties of most natural gases being recovered from reservoirs change as the reserve is depleted and pressures decline, and the results of an early back-pressure test may not be dependable for estimating the future delivery capacity of a well. The compressibility factor of a fluid under an initial pressure of 10,000 psia can change 45 per cent and the viscosity can change 70 per cent during the productive life of the reservoir. There are indications that the effective permeability to the flowing fluid can become 50 per cent of the original absolute permeability before enough liquid collects in the structure about a well as pressure declines to permit flow of liquid into the well. The advent of programmed electronic computing made practicable the integration of nonlinear, second-order, partial differentia1 equations2.3 pertaining to flow in reservoirs. Aronofsky and Porter4 represented the compressibility factor and the viscosity by a linear relationship, and integrated the equation for radial flow of gas for pressures up to 1,200 psi. Bruce, Peaceman, Rachford and Rice5 integrated the partial differential equations for both linear and radial unsteady-state flow of ideal gas in porous media. Their published results were a substantial guide in this study of integration of the partial differential equation of linear flow with coefficients of the equation variable. The computing program was developed to treat effective permeability as being both distance-dependent and pressure-dependent. In this study all examples of effective permeability are assumptions designed primarily to aid in developing programs for giving effect to this and other variable coefficients. The accumulation of data for expressing the pressure dependency of the effective permeability is the objective of a concurrent investigation. FUNDAMENTAL DIFFERENTIAL EQUATIONS DIFFERENTIAL EQUATION OF FLOW Linear transient fluid flow is represented by the partial differential equation,

Jan 1, 1965

By E. F. Sumner, C. K. Eilerts

A partial differential equation was developed and programmed to compute the cumulation and flow of the liquid phase of a gas-condensate fluid. It was found that such cumulation near the wellbore may attain saturations permitting liquid flow within periods of the order of 1 day. The radius at which this saturation obtains advances toward the external reservoir boundary with continued recovery, although it may move only 2 to 5 percent of that distance during the productive life of the well. This relatively high saturation can cause a serious dimunition in delivery capacity. The pressure dilferential required for flow of gas through a formation with such saturation may be as much as three times greater than the corresponding differential in a formation where saturation is less than 2 percent. Because of the radial geometry, total reservoir pressure differential required for flow with a zone of relatively high liquid saturation about the well was indicated to be two to three times greater than without the saturation. INTRODUCTION A computing program developed by the authors 1 to integrate the partial differential equation for transient radial flow of gas-condensate fluids required a prevalued function, W(Po) = kD(p0)/M(p0)z0(Po).......(1) related to cumulation of condensate in pores of the reservoir as pressure declines. Methods were available for relating mobility to saturation, saturation to liquid-vapor volume ratio, liquid-vapor volume ratio to pressure and, hence, mobility to pressure to obtain W(pD). However, certain assumptions had to be made in advance of program execution concerning movement of the liquid phase and change in radial distribution of liquid saturation as recovery progressed. Because reservoir fluid mobility and related pressure gradients are markedly dependent on liquid saturation, it was desirable to eliminate the necessity for these assumptions and in this respect improve the reliability of a result obtained by integration. Experimental research was begun to determine the relationship of variables that control the reservoir flow of low liquid-vapor ratio fluids. With the porous medium partly saturated, permeability to a two-phase flowing fluid could not be measured because the viscosity was unknown. Viscosity could not be measured without knowing permeability of the liquid-containing medium; but mobility, the ratio of permeability to viscosity, was measurable. This property, and liquid-vapor volume ratio of the flowing phases, have values determined by nature of the fluid and porous medium and, in addition, were found to be pressure, velocity and saturation dependent. A program developed by Kniazeff and Naville 2 for computing multicom-ponent, two-phase flow used polynomials to represent saturation dependence of permeabilities for gas and for liquid phases, and an effluent viscosity constant under reservoir conditions. The authors developed a program based on a partial differential equation of similar mass balance origin but designed to represent distribution of an unlimited number of components between gas and liquid phases utilizing the experimental information available as to variable dependence. Attention was given to liquid-vapor volume ratio of the flowing phases, a property which is essential as a liquid saturation boundary condition. A program for computing implicitly saturation increase at uniform intervals along the drainage radius was combined with an existing program for calculating pressures, and the two were used alternately in an iterative solution for both saturations and pressures. The function W(pD) needed for calculating pressures was replaced by the ratio of mobility to compressibility factor. BASIC EQUATIONS The partial differential equation for calculating

By L. F. Elkins

In 1946 Littlefield, Gray and Godbold published a thorough geologic description of the West Edmond Hunton Lime reservoir, located in Central Oklahoma, and discussion of its early performance.l They stressed implications of the nature of the reservoir rock on performance to be expected with production both by natural means and by pressure maintenance with gas injection. The essence of their analysis was that presence of an extensive interconnected system of fractures and solution channels constituting about 10 percent of the total reservoir void space in otherwise very low permeability reservoir rock would result in severe channeling of naturally encroaching water or injected gas through the fractures with little or no benefit to ultimate recovery of oil. Considerations were in progress to unitize the reservoir for pressure maintenance by gas injection. Some other engineers involved in the project had considerably different opinions at the time regarding continuity of fractures and regarding effects of such fractures on performance. The field was unitized Oct. 1, 1947. Subsequently, pilot gas injection tests were conducted in four areas of the field; large scale waterflooding was conducted in two parts of the field in the Bois d'Arc or Upper Hunton; and pilot waterflooding was conducted in the Chimney Hill or Lower Hunton. Full scale water-flooding is now in progress in the Chimney Hill section where it is oil productive. The primary purpose of this paper is to divine some of the secrets of this reservoir through analysis of all aspects of its performance, with the hope that it will be of help to engineers and operators in developing and operating other low permeability fractured reservoirs. In part, this considers variance of actual performance from some widely used conventional reservoir engineering assumptions. For brevity, other aspects of unit operation and its benefits to the owners are not discussed. Oil and Gas in Place For unit participation, net pay in the Bois #Arc defined by the gross SP anomaly on electric logs totalled 2,100,000 acre-ft in the 30,000-acre reservoir. Development of West Edmond predated modem logging techniques that now would be used to determine "net" or "productive" pay. In Fig. 1 this log pick is compared with detailed core analysis and core descripfion by Littlefield et a1.l for one well identified as A on the index map in Fig. 2. Not all of the gross Bois #Arc section had oil staining, but all cores with measureable intergranular permeability were stained. (The committee selection of the Bois d'Arc SP interval also includes the Frisco section as identified from cores by Littlefield.) The Bois d'Arc is oil productive throughout the entire 600-ft monoclinal accumulation from the eastern upstructure erosional pinchout to the west-side oil-water contact. (See Littlefield et al.l and McGee and Jenkins for geologic detail.)

Jan 1, 1970

By H. S. Price, D. A. T. Donohue

The system of equations describing displacement of a hydrocarbon liquid by a hydrocarbon vapor in a porous medium where mass transfer takes place between the phases is solved numerically for a variety of gas injection processes. Even though the method of solution is quite general, only systems with three hydrocarbon components are considered. Computer simulations of displacement processes wherein mass transfer between phases is both considered and neglected are compared, and it is shown that neglecting mass transfer can give pessimistic displacement efficiencies. INTRODUCTION The role of the gas displacement process in the recovery of petroleum has been subjected to a series of detailed analyses; as a result, a number of predictive models have been published in the literature. However, because of major simplifying assumptions, most of these models do not completely represent the physical system. As a result, the effect of making the simplifying assumptions is unknown. Therefore, a complete representation of this process — one without major simplifying assumptions —should lead to a fuller understanding of the process, and perhaps to methods of improving it. The general method of developing a model for two-phase fluid flow in a porous medium is to solve simultaneously the continuity equation, the energy equations and the equation-of-state for each phase under the prescribed initial and boundary conditions. For an isothermal system, the energy equations reduce to the momentum equation. Darcy's law. However, since natural gas is the vapor state of the reservoir liquid, interphase mass transfer may take place with concomitant changes in both the intensive and extensive thermodynamic properties of each phase. It is this phenomenon that has often been omitted in previous mathematical models. An additional relation, then, which accounts for mass transfer between the phases, must be included with the other equations to specify a complete model. Completely formulating the equations to be solved is not a difficult task1 but obtaining their solution has been intractable up to now. Availability of large-memory, high-speed digital computers now makes an attack on this formidable problem possible. This paper presents a preliminary study of the problem. Since this investigation is intended to be exploratory, it is restricted to the linear, horizontal, isothermal, two-phase viscous flow of oil and gas in an oil reservoir. In the early development of predictive models of this process, the reservoir system was considered as a unit and various forms of the material balance equation were proposed.2-4 Pressure and saturation gradients were then added in the Buckley-Leverett model.5 The Buckley-Leverett formulation considered the fluids to be incompressible; thus, the mathematical model reduces to a steady-state system. In the 1950's, studies incorporating numerical techniques were being published.6-9 These mathematical models differed in the efficiency of finite difference techniques, the inclusion or exclusion of capillarity or the number of space dimensions considered. To solve these nonlinear, partial differential equations, each phase was considered to be homogeneous with time; therefore, mass transfer between phases was neglected. The effect of mass transfer on the gas displacement process was first reported by Attra. 10 He simulated the one-dimension flow system by a series of cells in each of which the fluids were equilibrated during a time step. In addition, the pressure throughout the system during each time step was predetermined and constant-phase velocities were calculated according to the Buckley-Leverett incompressible fluid flow model. Welge et al.11 developed a model for the displacement of oil by an enriched gas where composition is considered to be a dependent variable. Fluid

By R. H. Barham, G. W. Nabor

Linear aquifers, either limited or essentially infinite, may be encountered in reservoir engineering practice. In areas where faulting fixes reservoir boundaries, the fault block reservoir may have an aquifer of limited extent whose geometry is best approximated as linear. An infinite linear aquifer can occur as a regional feature whenever water movement through the aquifer member is constrained to one direction. Such constraints can arise from major faults, facies changes or pinchout of the member. Miller* pointed out that linear aquifers have received only meager attention in the past. He analyzed the performance of finite and infinite aquifers, developed working equations and curves, and presented examples. While Miller's curves may be used fairly easily, a separate one is required for each size of aquifer. In this paper, Miller's equations have been used as a starting point. By modifying them slightly, they can be reduced to a form which yields a single working curve, applicable to any size of aquifer. Thus. interpolation between curves is eliminated and accuracy is improved. Miller's results for finite aquifers covered only the boundary condition of no flow across the outer aquifer boundary. This paper also includes the case of constant pressure at the outer aquifer boundary.

Jan 1, 1965

By H. Kazemi

A pressure fall-off test on the injection well of a forward combustion project may permit us to calculate the distance to the burning front. In the mathematical description and analysis of test data, we have considered the effect of temperature on the thermodynamic properties of reservoir fluids. The temperature distribution predicted by Chu' for a combustion wave has been used in analyzing a field test. Numerical solutions were obtained by simultaneous solution of the flow equation, the continuity quation and the equation of state. Although this analysis conccntrated on isotropic systems, the results can be extended to systems having directional permeability. INTRODUCTION Conventional pressure fall-off tests have been used extensively to determine reservoir characteristics in gas- or water-injection projects.- such tests consist of injecting fluid into the formation at a constant rate until steady state is nearly achieved, then discontinuing injection and measuring the bottom-hole pressure as a function of time. A plot of pressure vs the logarithm of time (measured from the instant of shut-in) yields a smooth curve characterized by three distinct regions (Fig. 1): Region I reflects the permeability alterations in the vicinity of the wellbore; Region II reflects the reservoir characteristics further away from the wellbore; and Region 111, the characteristics in the displaced fluid bank. The curves in Regions II and III are straight lines! the slopes of which are proportional to the reciprocal of the flow capacity of the formation behind and ahead of the interface as shown by Eq. 1, respectively:

Jan 1, 1967

By W. N. Hiatt

The mathematical theory of the flow of two-phase, incompressible fluid through porous media is clarified, and the development of a general fluid displacement equation for two-phase, incompressible vertical flow through porous media is outlined. The relationship between the three components of the equation — the "fluid drive effect", the "gravity effect", and the "capillary effect'' — is discussed, as well as implications of the analysis in connection with the study of certain types of petroleum reservoirs. INTRODUCTION The purpose of this paper is to clarify and summarize the mathematical basis of the theory of two-phase incompressible fluid flow through porous media, particularly as it pertains to vertical flow, and to present a qualitative* exposition of the controlling factors involved in circumstances where this type of flow may exist. Although the generally accepted theory of heterogeneous fluid flow through porous media was developed a number of years ago by Muskat,1 and Buckley and Leverett2, 3 using the basic premise of Darcy, (and since then expounded upon by many others) there is a need for an over-all summary and clarification of this theory, especially regarding certain types of applications in connection with so-called "gravity-drainage-type reservoirs". The source and derivation is outlined for what is referred to here as the "General Fluid Displacement Equation for Incompressible Vertical Two-Phase Flow through Porous Media''. This equation would apply under the condition where the pressure drop along the path of flow is negligible in comparison with the pressure level on the system, or where compressibility and solubility effects are negligible. These conditions exist ordinarily in gravity-drainage-type reservoirs. Specifically, it will be shown that this general equation may be developed directly from the basic equations, which, according to present theory, govern the general case of viscous heterogeneous flow through porous media. This general equation has appeared in the literature previously, but in most cases its development from basic theory is not indicated clearly, or the equation is presented in terms of groups of constants and variables involving reduced or dimensionless quantities. This presentation is expedient in cases of experimental or computer applications; but it may make it difficult for the practicing petroleum engineer to identify all the factors involved or to understand the relationships between them. Therefore, in the following development, only ordinary mathematical parlance will be used, and no terms involving groups of other equations or reduced or dimensionless quantities will be used. DERIVATION The following equations for two-phase oil and gas flow result from analogy with Muskat's general equations,** if solubility and compressibility effects are neglected, and if all volumes are considered in terms of reservoir conditions, and the water phase, if present, is considered to be immobile (or effectively a part of the porous medium.) In this case, the sum of the oil and gas saturations may be taken as unity, and the saturation of each phase is in terms of the total hydrocarbon saturation. Note that in Eqs. 1 and 2, and in the derivation following, z is taken as positive in the vertically downward direction,

Jan 1, 1969

By G. T. Davis, C. C. Mattax, M. O. Denekas

The effects of crude oil cornponents on the wellabil-ities of sandstone and limestone were investigated. Fractions containing cornponents differing in molecular weight and molecular structure were obtained from crude oils by distillation, extration and chromatography. Individual fractions were then tested for their effects on rock wettability. Tests indicate that sundstone wetta-bility may he changed by a complex variety of surfactants varying both in molecular structure and molecular weight. Limestone appears to be particularly sensitive to basic, nitrogenous surfactants. INTRODUCTION Investigations in recent years have shown that petroleum reservoir rock wettability can exert a significant influence on the efficiency with which oil can be produced by water flooding. While most reservoirs are presumably water-wet, they niay range in their degree of water-wettability from near-neutral to strongly water-wet.'" Reservoir wettabilities other than strongly water-wet are likely to be induced by adsorption of surface-active components froni the crude oil on the pore walls of reservoir rock.:' Little is known, however, about the nature of the surface-active materials which are likely to be adsorbed by the reservoir rock. Due to the complexity of crude oils. attempts made in the past90 isolate these surface-active components have met with only limited success. It is probable that many different types of surface-active materials arc indigenous to crude oils and that many of these may be adsorbed to varying degrees by reservoir rock. This was cxolored in the studies discussed in this paper. The over-all objective in these studies is to ascertain whether the wettability of a given reservoir can be determined by examining the surfactant content of the reservoir crude. To this end, crude oils were examined to determine the variability of indigeneous surfactants with regard to chemical type and molecular weight. Crude oils were separated by distillation into fractions differing principally in molecular weight, by chroma-tography into fractions containing compounds differing in polarity, and by solvent extraction into nitrogenous and non-nitrogenous fractions. Individual fractions were then tested for their effects on the wettabilities of sandstone or limestone rock samples. EXPERIMENTAL PROCEDURES Fractionation of the Crude Oils Samples of Miocene, Eocene and Jurassic crudes were distilled at temperatures not exceeding 200°C. The final stages of distillation were completed in a molecular still at pressures down to three microns of mercury. Fifteen to 30 fractions were obtained from each crude oil. These cuts were sufficiently broad that separation can be considered to have been effected principally on the basis of the molecular weights of the constituents of the crude oil. A considerable portion (20 to 40 per cent) of the crudes would not distill under these conditions. The residues were recovered and tested with the other fractions. Fractions differing in polarity were separated from a crude of Pennsylvanian age and an extracted sample of Miocene oil by chromatography, using a solid adsorbent. Since surfactants are, for the most part, polar compounds, chromatography should separate many of the surfactants from the crude oil. Such a separation should provide fractions containing compounds differing in molecular structure. Nitrogeneous compounds were extracted from Miocene crude oil with a solution of sulfuric acid in meth-anol. The residual oil was further processed by chonia-tography. Each of the fractions obtained by thesc procedures was dissolved in a non-polar solvent (xylene) and diluted to its original concentration in the crude oil. No attempt was made to maintain an anaerobic atmosphere above the samples while they were being dissolved. These solutions of the fractions were then tested for their effects on the wettability of sandstone and limcstone as discussed in the next section. Measurement of the Effects of Crude Oil Fractions on Rock Wettability No entirely satisfactory method for measuring rock wettability has yet been developed. All methods used are empirical. The imbibition test was used in these studies. This test is based on the tendency of a rock to imbibe the wetting phase spontaneously. For example, if a strongly water-wet rock is first saturated with oil and then placed in water, the water will quickly invade the rock by capillarity and much of the oil will be displaced. If the rock is slightly water-wet, water irnbibition will proceed more slowly and, in many instances, considerably less oil will be displaced. A water-saturated, oil-wet rock will imbibe oil. The initial rate with which water (or oil) imbibition takes place indicates, qualitatively, the degree of water (or oil) wettability of the rock.

By G. E. Perry, J. F. Bruskotter, D. G. Russell, J. H. Goodrich

The depletion performance of gas wells has been investigated by mathematical simulation techniques. The gas well model which was studied consisted of a single well located in the center of a bounded, cylindrical, homogeneous reservoir. Dependency of gas compressibility and viscosity on pressure was considered in studies of well performance at both constant mass flow rate and constant flowing pressure conditions. To carry out the investigation, the nonlinear, second-order, partial differential equation which describes Darcy flow of a nonideal gas through porous media was solved numerically. Some of the previous investigations of gas well performance have been of limited general use, because assumptions were introduced to simplify either the gas properties or the basic differential equation. Other studies have been rigorous in these respects but have presented a very limited set of calculated results. The present study was attempted to present a rigorous theoretical model and sufficient numerical results to permit meaningful conclusions to be drawn. It was found that all terms must be retained in the partial differential equation to make accurate predictions. The neglect of higher-order terms, e.g., terms of the order of the "gradient squared", leads to serious material balance errors at large times and to conservative estimates of gas well performance. The higher the gas flow rate and/or the lower the permeability-thickness product of the formation, the more pronounced are these deviations. For example, in a well draining 640 acres in a 25-md-ft formation (8,120 MMcf gas in place) at a constant rate of 993 Mcf/D, the rigorous solution predicts a bottom-hole pressure decline from 4,000 to 1,000 psia in 8.7 years. If higher-order terms are neglected in the differential equation, this decline in pressure is predicted to occur in 5.3 years. With the results of the numerical solution of the differential equation as a basis, simple, easy-to-use approximations for predicting gas well performance for Darcy flow conditions have been developed. These simple approximations are based on the familiar equations for flow of a single, slightly compressible fluid. The approximate methods possess a high degree of accuracy and enable the prediction of long-term gas well performance to be made quickly and accurately without the use of a digital computer. Both transient and stable flow period approximations were developed. INTRODUCTION In recent years income from the sale of natural gas and associated products has represented an ever-incre as ing fraction of the industry's total revenue from operations. To meet the surge in demand for natural gas, the industry has depended heavily upon established reserves and has actively pursued development of new reserves. The search has progressively led to reservoirs which yesterday were too tight and/or deep to yield the desired return on invested capital. More than ever before, evaluation accuracy is now required to forecast the criteria upon which engineering recommendations and management decisions are based. Considerable effort has been expended by both research and operations personnel on the development and application of methods for analyzing and predicting the performance of gas wells. Fundamentally, the problem is the familiar one of extracting data during the drilling, testing and early production life of a well and applying these data within an accurate simulation model to predict long-term behavior. During the past 30 or more years a voluminous literature dealing specifically with gas field problems has been generated. A recent book' lists a comprehensive bibliography of published material through 1959. Over 1,200 references are cited. Since then 39 additional articles on natural gas technology have been published in Transactions volumes of the Society of Petroleum Engineers of AIME. Most existing theory for predicting gas well performance requires that one or more idealizations (e.g., steady-state flow, ideal gas of constant viscosity, small and constant compressibility and constant-viscosity fluid) be applied. Although existing theory may apply directly or be adapted by various artifices to describe specific gas well and reservoir behavior, no widely applicable method is available, and existing methods appear to be subject to appreciable error unless better limits of applicability are defined. The objectives of this paper are (1) to present numerical solutions to the partial differential equation describing gas flow under conditions of general interest in gas well performance prediction work, (2) to present solutions which possess a high order of accuracy for both transient and stabilized flow periods of a well producing at constant rate or constant pressure, and (3) to develop and present simplified relationships which can be used as high-order approximations to the exact numerical results for fast and accurate predictions of gas well performance at the operating level. Combined, these objectives are designed gen-

Jan 1, 1967

By E. L. Anders

The Mile Six pool is located on the La Brea-Parinas Cullcession of International Petroleum Co., Ltd., in northwestern Peru on the west coast of South America. The reservoir pressure in this pool has been maintained within 200 psi of its initial value throughout its history, and gravity drainage has played an important role in the production behavior. It has now produced 95 per cent of its estimated ultimate recovery. It is estimated that this interesting oil pool will ultimately produce 67 per cent of the initial oil in place and that the resulting residual oil saturation may be as low as 19 per cent of the pore volume (29 per cent of the hydrocarbon pore volume). An evaluation of reservoir rock and fluid characteristics and ultimate oil recovery is presented. INTRODUCTION This study of Mile Six pool was made to evaluate its performance according to latest available information. The production performance of this pool has been discussed in various articles in the past. and the reported behavior has been used as an example for application of computation procedures for gravity drainage depletion' and as an illustration of field behavior under gravity drainage or expanding gas cap drive. There have been wide variations in reported values of initial oil in place, reservoir oil volume factor. connate-water sauration, volume of effective sand, and ultimate recovery because of the paucity of reliable basic data. These various factors have been determined as accurately as practicable with the latest available information, and this evaluation is presented herein. The production history of Mile Six is an excellent example of gravity drainage depleion with effective pressure maintenance by gar injection. GENERAL Mile Six pool was discovered by cable-tool drilling in November. 1927. when well 1996 was completed in the Parinas sand. After slow development with cable tools and sporadic production. the pool was opened to continuous pro(iuction in November. 1933. and develpment was completed with rotary rigs. Pressure maintenance was started in December, 1933. by returning gas to upstructure wells. Most of the development was 'completed by 1937, but some additional wells were drilled in the period 1939-1947. and several old wells were deepened. A total of 46 oil and gas wells and 4 dry holes were drilled on approximately 7-acre spacing. Of the producers. 21 are now flowing. 2 are pumping. 44 are gas input wells. 3 are abandoned. I is a gas well shut in. and 15 are shut in because of non-commercial production or high gas-oil ratio. The locations of all wells are shown on the map of Fig. I. Total oil production on Dec. 31. 1952, was 30,867,373 bbl: cumulative gas production was 22,023,777 Mcf; and 26,410,946 Mcf of gas had been returned to the reservoir. These figures do not include oil and gas lost ill a blowout in January. 1940. GEOLOGICAL DESCRIPTION Mile Six pool is located on the northern end of a structural spur projecting from the La Brea-Negritos uplift.' The spur is probably a reflection of a basement structure. It plunges gently to the north, i broken into a complex series of fault blocks. and contain.; the Verdun Alto. Section Sixteen. and Mile Six pools. The Parinas handstone (lower Eocene!. which is the producing formation in Mile Six. occurs at an average depth of 2,200 ft in the pool and dips north and east at from 15' to 20". The pool covers an area of approximately 350 acres. Mile Six is downfaulted about 600 ft from Section Sixteen pool to the soutb. and a major fault forms its western boundarv. The north and east boundaries are formed by the intersection of the sand top with the water-oil contact which occurs at approximately 2.440 ft subsea. An original gas-oil contact probably existed at about 1.875 ft subsea. Fig. 1 presents the latest structural interpretation of the pool. and Fig. 2 is an isopach map showing thickness of the total Parinas formation above the original water-oil contact. The heavy lints of Fig. 1 are contours on the sand top, and the fine lines are contours on the fault planes. This type of straight-line structural map was developed 1)) International's geologists to reflect structural conditions where the bedding planes dip and have no curvature. 'The La Brea-Parinas Concession is highly faulted by normal fault.. The beds are flat wherever exposed. The Parinas formation is approximately 635 ft thick. and it is etimated that 62 percent II the formation is effective sand. The original oil zone was about 565 ft thick. Fig. 211 presents an electric log showing typical Parinas sand development ill Mile Six pool. The Parinas band in Mile Six i. a well-sorted. medium- to-coal-se-grained. cross-bedded sand with minor lenses of shale and small lenses and pockets of pebble conglomerate. The sand grains are subangular to rounded and consist chiefly of quartz with feldspars. biotite hornblende, and augite as accessory minerals. Because of faulting of the Parinas- formation to the east and north of the pool. there is probably l possibility of a significant, natural water drive in Mile six. The faults within the pool. as indicated in Figs. 1 and 2. are of smaller disI,laceInent and seem to act only , partial barriers to fluid movement within the reservoir. RESERVOIR CHARACTERISTICS Core analysis data are available from five wells. The data were obtained from three wells (Nos. 3401. 3586 and 3719) at the time of their completion and from well 1996 when the original liner was sidetracked and the well was deepened ill 1946. Data from well 2779 were obtained in 1943 from old cores taken when the well was deepened in 1934. From these core analyses. the average porosity was estimated to be 22.6 per cent. and the average permeability to dry ail. was estimated to be 780 and Measured productility indices varied from 3.1 to 71.4 B/D per psi differential. Specific productivity indices varied approximately from 0.1 to 0.3 B./D per psi per ft of sand.

Jan 1, 1953

By H. H. Rachford

Numerical solutions of immiscible flow problems in which dispersive effects of capillarity are dominated by convection require excessively fine grid spacing with attendant high computing costs. The use of coarser spacing reduces cost but often produces oscillation or undue dispersion associated with displacement fronts, A numerical formulation is proposed here which should be applicable to two-dimensional flow problems. it is in part analogous to an approach previously tested for miscible systems. The convective transport is approximated using a change of variables to yield a coordinate system moving approximately with the local characteristic velocity. The capillarity-induced dispersive terms in the differential system describing the process are approximated with respect to a fixed coordinate system by the usual implicit formulation. One-dimensional tests of the procedure yielded results in which the saturation profiles tended smoothly to the zero-capillary pressure solution as the ratio of viscous to capillary forces was successively increased in a sequence of calculations. This contrasted favorably with solutions by other numerical procedures which would require attendant grid refinements to approach the zero capillary pressure results. INTRODUCTION Numerical solution of displacement problems has until recently relied on applying methods developed primarily for transient heat-flow problems. Such problems are classified as parabolic in type, and where the heat transport is purely by diffusion their solutions are characterized by a high degree of smoothness. It is not surprising, therefore, that for approximating these solutions available finite difference methods are quite adequate. In flow problems the transport is partly by diffusion, partly by convection or flow. Although the problem remains of parabolic type because the dispersive effects of capillary forces or diffusion play some role in every displacement, at high flow rates the problem is dominated by convection, and solutions tend toward those of equations of the hyperbolic type. Solutions of hyperbolic problems are characterized by the translation of fronts, or discontinuities, that may progressively increase in sharpness. Numerical methods for treating parabolic problems become less and less satisfactory as displacement rates increase and the role of dispersion due to concentration or capillary pressure gradients becomes small relative to transport due to flow. In computation the difficulty manifests itself as an error associated with the grid size chosen. 1-6 In summary, if the heat-flow type approximations are to include the terms arising due to convection, one of several choices may be made: (1) an upstream (to the direction of flow) approximation for the convection terms may be used; (2) a centered-in-distance (CID) approximation may be used; or (3) a recently developed approximation based on the theory of oscillation matrices may be chosen.6 The last appears to have significant promise for one-dimensional flow problems; its extendibility to two or three dimensions is an open question. In either of the first two approaches, a suitably small ratio of v&/D must be maintained, where v is the velocity, & is the grid spacing and D the effective dispersivity in the direction of flow. In the first choice, the approximation of the convective part is only first-order correct and errors introduced appear as a numerically induced dispersivity of magnitude proportional to v?x. In the CID choice, the approximation can be second-order correct, but the difference formulation fails to satisfy the maximum principle unless a condition on v?x/D is met. Practically, this means that for high flow rates oscillatory solutions may result in the neighborhood of a front unless exceedingly small grid intervals are taken. While the procedure proposed by Stone and Brian4 permits a less severe limitation to be placed on this ratio, ultimately the flow rates increase relative to the dispersivity the oscillation obtains. Further, extensions of their approach to higher dimensional systems may be attended by considerable

Jan 1, 1967

By H. S. Price, J. C. Cavendish, R. S. Varga

A numerical formulation of bigh-order accuracy, based on variational methods, is proposed for the solution of multidimensional diffusion-convection-type equations. Accurate solutions are obtained without the difficulties that standard finite difference approximations present. In addition, tests show that accurate solutions of a one-dimensional problem can be obtained in the neighborhood of a sharp front without the need for a large number of calculations for the entire region of interest. Results using these variational methods are compared with several standard finite difference approximations and with a technique based on the method of characteristics. The variational methods are shown to yield higher accuracies in less computer time. Finally, it is indicated how one can use these attractive features of the variational methods for solving miscible displacement problems in two dimensions. INTRODUCTION The problem of finding suitable numerical approximations for equations describing the transport of heat or mass by diffusion and convection simultaneously has been of interest for some time. Equations of this type, which will be called diffusionconvection equations, arise in describing many diverse physical processes. Of particular interest here is the equation describing the process by which one miscible liquid displaces another liquid in a one-dimensional porous medium. The behavior of such a system is described by the following parabolic partial differential equation:

Jan 1, 1969

By R. W. Snyder, W. H. Guilinger, B. S. Gottfried

Two numerical methods are presented JOT solving the equations Jor one-dimensional, multiphase /low in porous media. The case oF variable physical properties is included in the Formulation, although gravity and capillarity are ignored. Both methods are analyzed mathematically, resulting in upper and lower bounds for the ratio of time step to mesh spacing. The methods are applied to two- and three-phase waterflooding problems in laboratory-size cores, and resulting saturation and pressure distributions and production histories are presented graphically. Results oJ the tuo-phase flow problem are in agreement with the predictions of the Buckley-Leuerett theory. Several three-phase flow problems are presented which consider variations in the water injection rate and changes in the initial oil-and ujater-saturation distributions. The results are different physically from the two- phase case; however, it is shown that the Buckley-Lez~erett theory can accurately predict fluid interlace velocities and displacing-fluid frontal snturations for three-phase flow, providing the correct assumptions are made. The above solutions are used as a basis JOT et~aluuting the numerical methods with respect to machine time requirements and allowable time step for a fixed mesh spacing. INTRODUCTION Considerable progress has been made in recent years in obtaining numerical solutions of the equations for two-phase flow in porous media. Douglas, Blair and Wagner2and McEwenll present different methods for solving the one-dimensional case for incompressible fluids with capillarity (the former using finite differences, the latter with an approach based upon characteristics). Fayers and Sheldon4 and Hovanesian and Fayers8 have extended these studies to include the effects of gravity. West, Garvin and Sheldon, l4 in a pioneer paper, treat linear and radial systems with both capillarity and gravity and they also include the effects of compressibility. Douglas, Peaceman and Rachford3 consider two-dimensional, two-phase, in compressible flow with gravity and capillarity and Blair and Peaceman1 have extended this method to allow for compressible fluids. No one, however, has examined the case of three-phase flow, even for the relatively simple case of one-dimensional flow of incompressible fluids in the absence of gravity and capillarity. In obtaining a numerical technique for simulating forward in situ combustion laboratory experiments, Gottfried5 has developed a method for solving the one-dimensional, compressible flow equations with any number of flowing phases. Gravity and capillarity are not included in the formulation. The method has been used successfully, however, for two- and three-phase problems in a variable-temperature field with sources and sinks. This paper examines the algorithm of Gottfried more critically. Two numerical methods are presented for solving the one-dimensional, multi-phase flow equations with variable physical properties. Both methods are analyzed mathematically, and are used to simulate two- and three-phase waterflooding problems. The numerical solutions are then taken as a basis for comparing the utility of the methods.

Jan 1, 1967

By E. H. Lee, F. J. Fayers

Problems in reservoir analysis can usually he cupressed in terms of a system of nonlinear partial difjerential equations. A rnethod for .setting physically reasonable boundary conditions for these systerrls by ayplication of the theory of characteristics is discussed. A short description i.c. given of the technique for finding chnmcteristics of general first order and second orrler equations. Specific examples are quoted jrorn eqrrations occurring in theoretical reservoir studies. The princip1e. ossocicred wit11 the use of chnracteristic to set physically realistic boundary conditions me developed, and thr method is discussed in terms of boundary condition for the noncupillary, compressible, two-phase flow problen: jor the cupillary, ir~cowlpressible, two-phase flow problems for the tlrtergent flooding prohlenl; as for the compressible. noncapillary, two-phase flow pro11letn. INTRODUCTION In most problcms in reservoir analysis the physical situation is described by a system of nonlinear partial differential equations. The general solution to the set of equations usually comprises a wide range of functions ancl a particular solution is normally selectetl by applying suitable boundary conditions to the problem. In some cases it is clear from the physical situation what form the boundary conditions should take, but it is usually desirable to have an independent mathematical check of the suitability of the chosen conditions. From the mathematical standpoint, a system of partial differential equations with boundary conditions is said to be properly set if the combination determines a unique solution which is continuously dependent on the equation and the boundary conditions. Such a solution is sometimes termed "reasonable". In investigating the properties of the mathematical representation, it is useful to determine the structure of the system by the application of the theory of characteristics. This theory determines in the space of the independent variables, lines or surfaces which play a special role in thc system's behavior. The characteristics determine the possibility and nature of the propagation of discontinuities in the derivatives of the dependent variables. Along the characteristics certain differential relations hold which for some problems allow a so- lution to be obtained analytically. In the case of hyperbolic equations the characteristics can bc used as 3 basis for a numerical solution. If a complicated nonlinear system is to be solved, a study of behavior of the characteristics will indicate what boundary conditions should be chosen. By using the method of characteristics, several advances have rccently been made in solving reservoir engineering problems. Sheldon, Zondek and Cardwell' have discussed how the the method may be used grap-phically to solve the incompressible two-phase displaccmerit problem when the gravity term is included in the equations. Fayers and Perrine' have applied the method to study the equations representing detergent flooding. When compressibility and solution-gas effects are introduced into the two-phase flow problem, the equations and their characteristics become more complicated. It will be seen that in this problem the characteristics are useful for indicating a reasonable set of boundary conditions but cannot readily be used to compute a numerical solution. This problem has been solved on the IBM 701 using normal finite difference methods hy West, Garvin and Sheldon." The material to be prcscntcd in this report is divided into three sections. The first discusses the determination of the characteristics of first and second order partial differential equations; the second describes the selection of suitable boundary conditions by the application of the principle that the future cannot affect the past; the third discusses the use of characteristics for setting the boundary conditions of the two-phase solution-gas flow problem. A useful introduction to the method of characteristics has been given by Hildebrand.' A summary of the method will be given in this section of the paper. The First Order Equation The quasi-linear equation of first order is It is linear in the partial derivations of z. The properly linear equation has the form The characteristics of are curves along which the

By K. C. Hong, R. B. Jensen

The problem of determining the optimum set of steam volumes and cycle lengths for a single well undergoing multicycle steam stimulation in order to maximize the cumulative discounted net income has been formulated mathematically and programmed for a digital computer. The mathematical fonnulation of the problem and the method for its solution are discussed in this paper. The oil production performance during each stimulation cycle was simulated by either a constant percentage or a harmonic decline. Using simple analytical expressions for production performance, the cumulative discounted net income and cumulative time of operation were related to the pertinent process and cost parameters and two principal process control variables (cycle length and steam injection volume). The discrete maximum principles was used to transform the equations for cumulative time of operation and cumulative discounted net income into a set of simultaneous equations. The simultaneous equations then were solved by trial and error on a digital computer to determine the set of cycle lengths and stem injection volumes that gives maximum cumulative discounted net income over the project life. INTRODUCTION Steam stimulation is a process for improving the oil recovery rate from wells producing high-viscosity crudes. The process is applied on an individual well basis and is executed in a series of cycles, each consisting of three phases: steam injection, soaking (steam condensation), and production. Significant increases in production rate following the stimulation operation result from heating the reservoir around the wellbore. As heat is removed with produced fluids and dissipated into nonproductive formations, the production rate declines, usually to near the prestimulation value. Typical production responses are given in case histories reported by Owens and Suter,l and are depicted in Fig. 1. Duration of the production phase is equal to the time for the oil production rate to decline to some specified value and is called the cycle length. (Cycle length is defined as the producing portion of the cycle and does not include the downtime required for steam injection and soaking operations.) Termination of the production phase coincides with the start of steaming for the next cycle. The process is continued, cycle by cycle, until it becomes unprofitable. Models 2-4 have been developed to simulate behavior of a single well during one cycle of steam stimulation and have been used to investigate the effects of system and operating conditions on the production responses. However, these investigations have not shown how the models can be used to determine a most profitable set of operating conditions for a multicycle stimulation project. Perhaps these models are too complicated mathematically to be adapted readily to an optimization study. This paper presents a method for optimizing a multicycle steam stimulation project based on simple production performance models. First, the performance of a steam-stimulated well during each cycle was simulated by either a constant percentage or a harmonic decline model. Using these simple analytical expressions for production performance, the cumulative discounted net income (before tax) and cumulative time of operation for each cycle were related to the pertinent process and cost parameters and two process control variables—the cycle length and the steam injection volume. Finally, the profit function to be maximized was formed. The analogy between the above problem and problems of optimizing multistage decision processes that have been reported extensively in the chemical engineering literatures5-10 in recent years has led this investigation to the use of the discrete maximum principle.5-7 This principle was used to transform the equations for cumulative discounted net income and cumulative time of operation into a set of simultaneous equations characterizing the optimum operating conditions in terms of the two control variables. These equations

Jan 1, 1970

By J. C. Martin

Equations for three-phase, three-dimensional, compressible flow (including capillarity) are reduced to two-dimensional relations by a partial integration. This reduction allows three-dimensional flow problems to be treated with mathematics for only two spatial dimensions. The results can be used to formulate flow equations for two-dimensional reservoir simulators in which the effects of capillarity and fluid segregation in the third dimension are represented. Such reservoir simulators would retain many of the aduantages of two-dimensional simulators while simulating three-dimensional effects. The principal restriction of the method is that the thickness of the reservoir should be small, compared to the distance across the reservoir. INTRODUCTION In recent years, computers have been used to calculate the performances of many reservoirs. Most of the detailed calculations, however, are based on finite difference solutions of the flow equations, and present day computers are seldom able to handle a sufficient number of cells to produce entirely satisfactory solutions, even for reservoirs represented by two-dimensional arrays of cells. The situation becomes much worse when one wishes to approximate the reservoir by a three-dimensional array. A great saving in computation or a more detailed solution can be obtained for many reservoirs by using the partial integration of the equations of flow presented in this paper. The integration reduces the three-dimensional equations to two-dimensional relations; and, for studies of two-dimensional flows in vertical cross-sections, the equations are reduced to one-dimensional relations. Most reservoir performance calculations currently are based on one- or two-dimensional flow re1ations.l-3 In some cases flow in the third dimension is approximated by assuming a particular type of vertical saturation distribution, such as gravity segregation.l.2. The relations developed in this paper approach those for segregated flow as the capillary pressures approach zero, and they approach those for uniformly distributed saturations as the capillary pressures are increased. For this analysis, the ratio of the reservoir's thickness to the maximum distance across it must be small. The capillary pressures between the oil and water should also be small compared to the maximum pressure difference in the reservoir. This requirement is met by most reservoirs. It is assumed that the capillary-pressure curves are well defined, whether or not hysteresis effects are included. Also, the reservoir must have sufficient vertical permeability to allow the fluids to segregate. The results presented here provide a firm theoretical foundation to Coats' et aL5 assumption of vertical equilibrium and extend the relations to three-phase flow. Coats' assumption of vertical equilibrium, which he verified by calculations and experiment, is developed here mathematically from basic flow equations. DISCUSSION SATURATION AND PRESSURE DISTRIBUTIONS Appendix A presents a mathematical analysis of fluid flow in reservoirs where the ratio of thickness to maximum distance across the reservoir is small. The results indicate: (1) that the fluids along any line perpendicular to such a reservoir's upper surface are in static capillary equilibrium (vertical equilibrium); (2) that, to a first approximation, the fluid pressures and properties are functions of only areal position in the reservoir and time; and (3) that hydrostatic pressure gradients exist along any line perpendicular to the reservoir's upper surface. The results might be expected after studying several physical considerations. First, no flow is allowed normal to the upper and lower reservoir boundaries, which are relatively close together.

Jan 1, 1969

By H. J. de Haan, L. Schenk

Scope for Thermal Recovery in Shell's Heavy Oil Fields in Venezuela The main heavy oil reservoirs on the East coast of Lake Maracaibo (Fig. I), known as "Bolivar Coast", initially contained some 20 billion bbl of oil in place with gravities in the range of 10 to 15° API. The current total production rate is about 400,000 B/D. Since the recovery to date is on an average only 12.5 percent of the initial oil in place, these fields offer a vast potential for secondary recovery methods. The reservoirs are characterized by moderate depth (generally 1,000 to 3,000 ft; maximum 5,000 ft), good formation properties such as net oil sand thickness (50 to 300 ft), high porosity (30 to 40 percent) permeability (1 to 3 darcies) and oil saturation (initially about 80 percent) and high oil viscosity (100 to 10,000 cp in situ), all of which are favorable for the application of thermal recovery processes. Since 1957, steam drive, steam "soak" and underground combustion have been tested or are being tested in this area.1-3 This paper deals with a major steam drive test in the Tia Juana field. The test was commenced in Sept., 1961, on the basis of encouraging results of laboratory experiments and pilot field tests, which will be discussed briefly. Early Laboratory Investigation of the Steam Drive Process In 1956 a series of model experiments was carried out in the Koninklijke/Shell Laboratorium, Amster- dam, to investigate the displacement of heavy oil by steam.' The prototype studied was a horizontal, un-consolidated heavy oil reservoir, subjected to a linear steam drive. The model, representing a slab from this reservoir along the main flow direction, consisted of a steel tube filled with oil-saturated sand, provided with layers of rock at the top and bottom to simulate conductive heat losses to cap and base rock. Lateral heat losses, perpendicular to the main flow direction, were minimized by insulating the sides of the model. Heat flow and viscosity and gravity forces were scaled by taking the steam injection rate and sand permeability inversely proportional to the geometric reduction factor of the model, and using materials (oil, steam, cap rock, etc.) identical with those of the prototype. The main feature of the process was a frontal displacement mechanism that was observed within wide ranges of injection pressure and rate, oil viscosity, initial oil saturation and sand permeability. The high stability of this front can be attributed to the high volumetric steam flow rates, combined with the reduction in oil viscosity. High recoveries were obtained due to the low residual oil saturation of about 15 percent in the zone swept by steam. The latter value pertains to the specific test conditions (heavy oil, uncon-solidated sand) and may be lower for light oil, due to the effects of steam distillation. The applicability of this experimental approach is limited, as it does not take into account the fact that

Jan 1, 1970

By P. B. Crawford, W. L. Huskey

A study has been made to determine the effects of random vertical fractures existing in a reservoir matrix on the effective matrix properties and producing characteristics of a well. The study indicated that short vertical fractures may distort the isopotentials by 1 percent in the vicinity of the fracture. A correlation was developed which showed the effect of fracture density on the producing capacity of a well. The correlation indicated that fracture shape had little effect on producing capacity, but that total fracture length was closely correlated with the producing capacity. Effective values of reservoir permeability were found to correlate closely with fracture density when measured parallel to the direction of flow. INTRODUCTION In reviewing the petroleum literature, it is found that in most analytical solutions of hydrocarbon reservoir behavior the assumption is made that the rock is uniform throughout although core analysis and well logging surveys indicate that substantially all reservoirs are heterogeneous and some are fractured. Only in recent years have investigators studied the effects of reservoir hetrogeneity. Russell1 discussed the effect of fractures in reservoirs. The literature indicates that fractures greatly affect the exploitation of man reservoirs. This is described by Elkins and Skov2 and Little-field et al. 3 Several types of reservoir heterogeneity have been studied by unsteady-state methods. In 1959, Landrum et al. 4 showed that transient phenomena in hydrocarbon reservoirs could be studied by thermal models. pickering5 studied the flow of heat in stratified linear reservoirs consisting of plates of different metals separated by sheets of rubber which were suddenly exposed to a low-temperature source at one end. Cotman6 studied both laterally and vertically heterogeneous thermal model reservoirs with special reference to cross flow. Another type of reservoir heterogeneity involves the presence of highly permeable, circular lenses around or near the wellbore which was studied by Miesch7 using a thermal and a steady-state model. Many studies have been made of the effects of induced hydraulic fractures on well performance. 8-10 Work regarding vertical fractures that exist initially in the reservoir was reported by Givens, et al.11 Since there appears to be little published work on the importance and influence on reservoir performance of fractures that exist out in the rock matrix, this study was conducted to determine the effects of vertical fractures on pressure distribution, producing capacity and the effective permeability of the reservoir rock. The analogy between the flow of electrical current through a conducting body and the flow of fluid in a porous medium provides a convenient method of studying various aspects of petroleum reservoirs. For electrical radial steady-state flow,12 For steady-state fluid flow, With the above analogy, it is only necessary to establish the ratio of the values of the electrical units in terms of the corresponding flow units to make the analogy quantitative. The ratios are shown below.

By B. H. Caudle, B. M. Hickman, I. H. Silberberg

Secondary recovery projects often are not started in oil reservoirs until dictated by rising GOR's or declining oil production. Such circumstances require a well dispersed injection pattern to prevent the oil bank from bypassing the production wells. Many times this purpose is served by using the five-spot or line-drive patterns, in which half of the wells are made injection wells. A regular injection pattern is proposed that is based on the same well spacing as the five-spot but in which only a third of the wells need be converted to injection wells. Fluid flow model studies on this proposed pattern show that production histories will be as good as those for a five-spot pattern at mobility ratios of three and above. The model data are presented in a form that may be used to predict recoveries from both homogeneous and stratified formations. Introduction The injection of fluid into a petroleum reservoir usually is prompted by an actual, or predicted, loss in the productivity of the reservoir. Often, an oil reservoir is produced by pressure depletion until high GOR's dictate a change in policy. Fluid injection also causes increased ultimate recovery, but this usually is of less importance to the operator than is the maintenance of the production rate. It is a fact that the ultimate recovery in any particular fluid injection project is relatively independent of the number and spacing of the injection wells as long as the reservoir voidage through the production wells is balanced by the total injection rate. However, the response of a production well is directly affected by the distance it lies from an injection well. In places where the well productivities are declining, the injection wells should be spaced as evenly as possible among the producing wells ("dispersed" injection pattern). More particularly, if the production response depends upon the formation (and subsequent production) of an oil bank, each production well should be surrounded by injection wells. Otherwise, the oil productivity will increase very little as the oil bank moves past the well.'.' An evenly spaced injection pattern has the best chance of generating an oil kick in the production wells. If reservoir voidage and injection rates are to be equal, the ratio of injection wells to production wells must be equal to the ratio of the average injectivity to the average productivity (in reservoir volumes). This ratio is in strumental in choosing the injection pattern. Table 1 shows the production well to injection well ratio for the dispersed injection patterns that have been described in the literature.'-' All of these patterns will have comparable recoveries and response times (for the same basic distance between wells). The choice of pattern there fore depends upon the relative injectivity and productivity of the wells and upon the type of drilling pattern employed. Another important factor may be the ability of the proposed production wells to make up the lost productivity from those wells converted to injectors. These factors become more pronounced at the higher oil viscosities (low gravity oils). All other things being equal, the probable ratio of single well injectivity to single well productivity will be equal to the water-oil mobility ratio of the proposed flood. Thus, high mobility ratios (due to low gravity oil) should call for fewer injection wells — as long as these injection wells are "dispersed" to obtain more even pro duction response for the remaining wells. Secondary re covery projects in oil reservoirs seldom, if ever, will have mobility ratios requiring more injection wells than production wells. Thus, only the last five of the patterns listed in Table 1 will be of significant interest in oil recovery. Almost every development drilling program calls for the wells to be drilled along equally spaced rows and

Jan 1, 1969