The static deflections of the membrane diaphragm of capacitance microphones due to electrical biasing are governed by a nonlinear differential equation. Since the governing equation is nonlinear, deflection and stability results are usually obtained using numerical methods. The static deflections and their stability values determine the operating range of certain microphone parameters and influence the sensitivity and frequency response. The results of two past investigations using different forms for the nondimensionalized nonlinear differential equations are reviewed and compared. [J. E. Warren, A. M. Brzezinski, and J. F. Hamilton, J. Acoust. Soc. Amer. 52, 711–719 (1972); F. W. Fraim et al., J. Acoust. Soc. Amer. 53 (to be published)]. Results are presented for circular, annular, and strip membranes using a third and perhaps more appropriate form for the nonlinear differential equation. Stability results for the three types of membrane systems for flat stationary electrodes with openings are also presented. Since the third nondimensionalized form is similar to that used by Teer, his results for an optimum stationary electrode curvature for circular membranes may now be applied to situations with two dielectric layers of different thicknesses and permittivities [K. Teer, Acustica 15, 256–263 (1955)].

Abstract An idealized model has been developed for the purpose of studying the characteristic behavior of a permeable medium which contains regions which contribute significantly to the pore volume of the system hut contribute negligibly to the flow capacity; e.g., a naturally fractured or vugular reservoir. Unsteady-state flow in this model reservoir has been investigated analytically. The pressure build-up performance has been examined in some detail; and, a technique for analyzing the build-up data to evaluate the desired parameters has been suggested. The use of this approach in the interpretation of field data has been discussed. As a result of this study, the following general conclusions can be drawn:Two parameters are sufficient to characterize the deviation of the behavior of a medium with "double porosity" from that of a homogeneously porous medium.These parameters can be evaluated by the proper analysis of pressure build-up data obtained from adequately designed tests.Since the build-up curve associated with this type of porous system is similar to that obtained from a stratified reservoir, an unambiguous interpretation is not possible without additional information.Differencing methods which utilize pressure data from the final stages of a build-up test should be used with extreme caution. Introduction In order to plan a sound exploitation program or a successful secondary-recovery project, sufficient reliable information concerning the nature of the reservoir-fluid system must be available. Since it is evident that an adequate description of the reservoir rock is necessary if this condition is to be fulfilled, the present investigation was undertaken for the purpose of improving the fluid-flow characterization, based on normally available data, of a particular porous medium. DISCUSSION OF THE PROBLEM For many years it was widely assumed that, for the purpose of making engineering studies, two parameters were sufficient to describe the single-phase flow properties of a producing formation, i.e., the absolute permeability and the effective porosity. It later became evident that the concept of directional permeability was of more than academic interest; consequently, the degree of permeability anisotropy and the orientation of the principal axes of permeability were accepted as basic parameters governing reservoir performance. More recently, it was recognized that at least one additional parameter was required to depict the behavior of a porous system containing regions which contributed significantly to the pore volume but contributed negligibly to the flow capacity. Microscopically, these regions could be "dead-end" or "storage" pores or, macroscopically, they could be discrete volumes of low-permeability matrix rock combined with natural fissures in a reservoir. It is obvious that some provision for the inclusion of all the indicated parameters, as well as their spatial variations, must be made if a truly useful, conceptual model of a reservoir is to be developed. A dichotomy of the internal voids of reservoir rocks has been suggested. These two classes of porosity can be described as follows:Primary porosity is intergranular and controlled by deposition and lithification. It is highly interconnected and usually can be correlated with permeability since it is largely dependent on the geometry, size distribution and spatial distribution of the grains. The void systems of sands, sandstones and oolitic limestones are typical of this type.Secondary porosity is foramenular and is controlled by fracturing, jointing and/or solution in circulating water although it may be modified by infilling as a result of precipitation. It is not highly interconnected and usually cannot be correlated with permeability. Solution channels or vugular voids developed during weathering or burial in sedimentary basins are indigenous to carbonate rocks such as limestones or dolomites. Joints or fissures which occur in massive, extensive formations composed of shale, siltstone, schist, limestone or dolomite are generally vertical, and they are ascribed to tensional failure during mechanical deformation (the permeability associated with this type of void system is often anisotropic). SPEJ P. 245^

Abstract Because of the extensive utilization of hydraulic fracturing for the stimulation of low-productivity wells, the two related problems of fracture design and evaluation have become economically significant and, as a consequence, have motivated this investigation. The producing characteristics of horizontally fractured wells were studied to determine the fracture configuration that should be employed as the basis for the design of the treatment and to develop a method that can be used to establish the degree to which the design objectives have been achieved. The equations which describe the steady-state flow of a single-phase fluid into, and through, a finite-capacity fracture were solved numerically for an idealized reservoir-fracture model. The numerical results were used to obtain an apparent skin effect for each combination of the parameters considered. Based on the computed results, subject to the limitations implied by the assumptions that were made, the following general conclusions were drawn.For a radius of drainage at least four times as large as the radius of the fracture, an apparent skin effect that is independent of the radius of drainage can be calculated.The productivity of the hydraulically fractured system, relative to that of the unfractured well, can be determined from the apparent skin effect and can be used to establish design objectives.In the evaluation of a fracture job, it is not possible to determine both the radius of the fracture and its flow capacity uniquely from the apparent skin effect; an independent determination of one of the quantities is necessary. Introduction Although hydraulic fracturing has been employed as a method for stimulating the productivity of literally hundreds of thousands of wells during the past 10 years. It is only in the last few years that improvements in fracture design and fracturing technique have combined to increase the probability of obtaining a successful treatment to such an extent that the mechanics of the method may be considered to be standardized. From an economic point of view, however, two related questions must be satisfactorily answered before hydraulic fracturing can be used in the most profitable manner. The two questions are the following.For a particular well in a given formation, what are the optimum design specifications for the fracture treatment?Have the design objectives been achieved by the fracture treatment? The significance of these questions has been recognized, and some attempts to obtain answers have been made. Howard, et al, endeavored to determine the optimum treatment, based on maximizing profits, for any given formation; unfortunately, this work was based on a crude method for approximating the productivity of a well. Carter and Tracy utilized the same approximation to study the effect of fracturing on the behavior of a well producing by virtue of a solution-gas drive. Electrolytic models were used by van Poollen to investigate the variation in productivity due to fracturing; however, only a limited number of results were presented. Later, from the same model results, van Poollen, et al, attempted to justify an approximate expression for determining the productivity of a fractured well. It is quite apparent that there is a definite lack of the practical information necessary for specifying the optimum fracture configuration to be considered for design purposes. The only detailed attempt to develop a procedure for evaluating the result of a given fracture treatment appears to be that of van Dam and Horner. These authors described a technique for analyzing pressure build-down data, obtained immediately after fracturing, to determine the final fracture volume, the final fracture porosity, the fracture area, the fracture thickness and the in situ fluid loss of the fracturing fluid. While this approach should be useful whenever acceptable pressure measurements are available, it does not yield a value for the flow capacity of the fracture. Since the problems of fracture design and evaluation are inversely related, it should he sufficient to study the effect of the fracture configuration on the performance of a well. The primary objective of this investigation is to evolve a technique for computing the desired solutions. The secondary objective is to analyze these computed results in order to prescribe a method for evaluating fracture treatments. JPT P. 1050^

Abstract A technique for comparing core-analysis permeabilities to those determined from pressure build-up tests has been developed. This comparison has been used to obtain a qualitative indication of the spatial permeability distribution within the producing formation. Data from 14 wells, equally divided between the Bromide sands and the McKnight dolomite, have been analyzed; results have indicated that the permeabilities in those particular reservoirs are randomly arranged so that they appear to be homogeneously heterogeneous. It has been emphasized that more accurate core data, improved pressure build-up testing and independent determinations of anisotropy are needed before a more quantitative description of the reservoir is feasible. Introduction The purpose of this paper is to compare permeabilities determined by core analysis with those calculated from pressure build-up data. Since a single value for the effective permeability of the drainage area is obtained from a build-up test, it is necessary that the core analysis values be averaged in some manner to allow the comparison to be made. If an averaging method that reconciles the core and build-up permeabilities can be found, it is possible to infer, in a qualitative sense, the spatial distribution of permeabilities that exists in the reservoir. Since the suggested approach gives a gross indication of the type of reservoir heterogeneity that is based on a comparison, the inherent errors in the raw data must be carefully considered. Recent experimental and theoretical research on the nature of recovery mechanisms, developments in the field of numerical analysis and the evolution of truly high-speed digital computers have made it possible to predict the performance of virtually any reservoir that can be adequately described. Unfortunately, the necessary condition that demands an adequate description of the formation cannot be satisfied in general. As a result, the usefulness of the available numerical models is seriously impaired.

PUBLICATION RIGHTS RESERVED This paper is to be presented at the 36th Annual Fall Meeting of the Society of Petroleum Engineers of AIME in Dallas, October 8–11, 1961, and is considered the property of the Society of Petroleum Engineers. Permission to publish is hereby restricted to an abstract of not more than 300 words, with no illustrations, unless the paper is specifically released to the press by the Editor of JOURNAL OF PETROLEUM TECHNOLOGY or the Executive Secretary. Such abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in JOURNAL OF PETROLEUM TECHNOLOGY or SOCIETY OF PETROLEUM ENGINEERS JOURNAL is granted on request, providing proper credit is given that publication and the original presentation of the paper. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and considered for publication in one of the two SPE magazines with the paper. Abstract The process by which one miscible liquid displaces another liquid in a two-dimensional porous medium has been investigated by means of experimental digital computation. The effects of viscous and gravitational forces, molecular diffusion, microscopic dispersion and heterogeneous permeability have been explicitly included in the behavior equation's. A general numerical method for solving the difference analogues of the behavior equations has been developed and the occurrence of oscillatory numerical solutions has been explained. Based on the computed results, it has been concluded that, for the viscosity function chosen, gravity segregation does not materially affect the recovery at breakthrough in a homogeneous system. It has been observed that randomly heterogeneous permeabilities cause macroscopic dispersion which tends to damp out viscous fingers; it has been suggested that viscous fingering may be a laboratory phenomenon and not a reservoir problem. Introduction During the last ten years, almost one hundred papers on the subject of oil recovery by miscible displacement have appeared in the literature. These papers which have covered a wide range of interests—physical principles, mathematical theory, laboratory experiments, field tests and economics—have indicated the potential importance that has been assigned to this process by the petroleum industry. Despite the concerted efforts that have been reported, the future of miscible displacement as a practical method for improving the recovery of oil remains uncertain.

Abstract Techniques for studying the performance characteristics of heterogeneous reservoirs have been developed. The effect of permeability variation on both the steady-state and the transient flow of a single fluid has been investigated. Limited comparisons with field and laboratory data have been made. The physical models studied consist of random three-dimensional arrays of homogeneous porous blocks. The permeabilities of the individual elements are assigned according to a specific distribution function; uniform anisotropy is introduced by varying the relative dimensions of the blocks. A particular model is perturbed simply by re-arranging its elements at random. The behavior of a physical model is determined by digitally solving its numerical analogue. Based on computational experiments, subject to the restrictions implied by the assumptions that were made, the following general conclusions have been drawn.The most probable behavior of a heterogeneous system approaches that of a homogeneous system with a permeability equal to the geometric mean of the individual permeabilities.The effects of flow geometry and anisotropy on the most probable value of the effective permeability of a heterogeneous medium are finite but not significant.The permeability determined from a pressure build-up curve for a heterogeneous reservoir gives a reasonable value for the effective permeability of the drainage area.A qualitative measure of the degree of heterogeneity and its spatial configuration are obtained from a comparative study of core analysis and pressure build-up data. It has been indicated that these conclusions are predicated on the assumption that the core analysis and the pressure build-up data represent true reservoir characteristics. Common sources of error in these data have been discussed. Introduction One of the most significant problems of reservoir engineering is that of determining the nature and the disposition of the heterogeneities that inevitably occur in petroliferous formations. It is with this problem that this paper is concerned.

Abstract An asymptotic solution to the equation that describes the flow of a slightly compressible fluid in an infinite porous medium has been used to estimate the interaction between two adjacent wells producing from a common reservoir. A direct method for approximating the interference time defined by Stevens and Thodos has been suggested. An alternative definition for the time of interference, based on the minimum pressure change in the interference region, has been proposed; also, a direct method of determination has been prescribed. Examples have been employed to illustrate the use of both methods. Introduction The pressure behavior associated with a completely penetrating well in a uniformly thick, homogeneous, horizontal reservoir containing a single, slightly compressible, mobile fluid is described by the radial form of the diffusion equation. A useful asymptotic solution to this equation is the continuous line-source solution obtained from the basic formulation of Lord Kelvin. ..................(1) In the derivation of Eq. 1, it is assumed that a well of infinitesimal radius is producing at a constant rate from an infinite reservoir. Because of the assumption of a vanishing well radius, the solution is valid only for large values of time. Horner discusses the limitations on the use of this equation; Elkins presents field data which support Horner's conclusions. If two or more wells are producing from the same reservoir, the resulting pressure distribution is obtained by the superposition of the solutions for the individual wells. The object of this paper is to develop approximate techniques for determining the time and the place at which pressure disturbances originating from two adjacent wells begin to interact, or interfere, significantly. Two distinct approaches will be followed. First, the effect due to each well will be considered separately; then, the cumulative effect of the two wells will be examined.

Introduction Over the years, technological development, expanding markets and competition have kept the petroleum industry dynamic and progressive, fulfilling a vital part in the nation's expanding economy. Looking to the future, indications point to a greater and greater demand for petroleum products to meet the growing needs of a larger population and an increasing per capita dependency upon energy. Expansion of facilities in all branches of the industry will be needed as demand grows. This expansion will increase financial requirements, for there is a continual need of capital to make it all possible. A major share of these capital requirements will be needed to explore for and develop increasing amounts of crude oil and natural gas. This paper will cover first, the banking concept as it applies to all oil property loans; second, current considerations concerning specific loan situations; and, finally, considerations of future political and economic forces as they are related to oil and gas loans. Banking Concept Concerning Oil Property Loans We have all heard in the oil country allegations such as "tight bankers" and "bankers only loan money on cinches". This is a natural human reaction, for all of us accept easily those things that conform to our own thinking and forward our own interests, and reject readily those contrary to our beliefs and interests. The misunderstanding arises because banking and oil producing, by their very natures, must operate in different business climates. Actually, banking and the oil business do not have conflicting objectives, for bankers are anxious to do all they can to build relationships and promote the welfare of their oil clients. They must, however, confine their judgment within limits imposed by their over-all responsibilities. So, perhaps it might be well at this point to compare the two businesses - banking and petroleum - and point out the differences in business climate in which each one is permitted to operate.