Some developments in the mathematical analysis of resistivity well-logging measurements in anisotroplc beds are presented. The no-borehole case in which two thick, anixotropic beds meet at a plane interface for bedding planes parallel to the interface has been discussed previously by Kunz and Moran (I 9%). The treatment here is extended to include arbitrary orientation of the bedding planes. Relations are derived for finding the potential distributions in the two media produced by a point current source in one of them. Numerically evaluated apparent-resistivity profiles across the boundary between two anisotropic beds are shown for a normal resistivity-lopgin g device for various conditions of anisotropy and dip.The effect on the responses of a normal device of introducing a borehole perpendicular to the bedding planes of a thick anisotropic bed was also treated by Kunz and Moran (1958). The added effect oC eccentering has been studied following the methods of Gianzero and Rau (1977) but, since the results added nothing of interest. no details are given. For the borehole axis not perpendicular to the bedding planes, no progress can be reported in the analysis.When alternating currents are used, the solutions acquire characteristics dependent on the frequency. Appropriate relations are developed starting from Maxwell' s equations.For electrode devices using alternating current, such as a normal or a LaterologT" . solutions are derived in terms of the vector potential for the case of a borehole penetrating a thick anisotropic bed normal to the bedding planes and for the no-borehole case where the sonde axis is perpendicular to a sequence of beds with all bedding planes parallel to the bed boundaries. Details of the frequency effects are considered only for the case of a homogeneous medium, but indications of the method of solution for heterogeneous media are given. One principal result is that the "paradox of anisotropy" (see Kunz and Moran, 1958) still remains valid.For induction-logging devices, the transmitter-coil sources of the electromagnetic (EM) field are treated as (alternating) magnetic dipoles. When the source-and receiver-coil axes both are oriented perpendicularly to the bedding planes, only the component of resistivity parallel to the bedding planes affects the responses. With the addition of coils oriented parallel to the bedding planes, it is theoretically possible to determine formation dip from the out-of-phase (reactive) voltages in the receiver coils. Analyses arc outlined for a homogeneous medium. for a thin bed. and for borehole cases usually considered.Values of the horizontal and vertical conductivities (and coefficient of anisotropy) can. in principle, be derived from the measured values of the induction-logging conductivity signal and the ou-of-phase signal from the formation. A difficulty with the method is the effect of heterogeneities. When true horizontal conductivity changes across a bed boundary, a plot of computed apparent values of horizontal and vertical resistivities across the b...
The basic idea of induction logging is reviewed and the use of the geometrical factor is explained. The rigorous theory based on Maxwell’s equations, is then formulated for the simple two‐coil sonde, which is the basic building block for multicoil sondes. For a homogeneous conducting medium this theory leads to simple solutions for the fields and current density. These simple solutions are exploited in detail to build up a firm understanding of the skin effect phenomenon. The theory is then applied to nonhomogeneous media and numerical results obtained through the use of the IBM 704 are presented. On the basis of this complete study some simple approximate results are presented which allow a rapid evaluation of skin effect in induction logging. The nonhomogeneous media considered in detail are two semi‐infinite beds, thin beds bounded by adjacent formations, and an infinite bed with a borehole or invasion. It is shown that the borehole gives a negligible contribution to the skin effect.
The pressure build-up technique is a recognized method of determining permeability from conventional drillstem tests. In this paper an effort is made to extend such techniques to the interpretation of data obtained from the wireline formation tester. Such a study is necessary because of the differences, for this case, in the magnitude of the flow parameters (rate of flow, amount of recovered fluids) and in the flow geometry (flow through a perforation vs flow across the face of the wellbore, etc.) involved in the solution of the equations of flow for compressible fluids. The perforation is replaced by a spherical hole, and the effect of the borehole is neglected, so that the flow can be considered to be radial in a spherical co-ordinate system. Arguments are presented to justify this idealization. Assuming single-phase flow, general relations between pressure and flow rate are developed for a homogeneous medium. The study is then extended to permeable beds of finite thickness. It is shown that the early stages of pressure build-up tend towards spherical flow, while the later stages tend towards cylindrical flow. The thinner the bed, the more quickly flow approaches the cylindrical model. The prevalence of thin beds in practical work makes this analysis quite important. Cases involving permeability anisotropy are treated. Introduction From wireline formation tester operation, two types of data are obtained:the nature and amount of recovered fluids, andthe pressure history recorded during the test. A number of papers have been written dealing with the interpretation of formation production on the basis of the recovered fluids. In general, the methods described have been quite accurate for both high- and low-permeability formations. The present paperwill deal with an analysis of the pressures observed. An analysis of the pressure build-up curves obtained in hard-rock country has already been attempted on the basis of the formula proposed by Horner. Although this approach has met with success in many instances, some questions have been raised as to its validity. It is the aim of the present study to place the analysis of pressure build-up in the formation tester on a firmer basis, from which more detailed methods of interpretation can evolve. Because of the great differences between the operation of the wireline formation tester and the conventional drillstem test, modifications are necessary in the interpretation. The major difference relates to the flow geometry. Once the flow geometry has been established other features such as multiphase flow, skin effect, afterflow, etc., well described in the literature, can he introduced. It will be assumed that the mechanical operation of the formation tester is already known to the reader. It will suffice here merely to state that the tester provides the means for taking a relatively small sample of the fluid immediately adjacent to the borehole, and for recording the subsequent pressure response. In comparison with conventional drillstem tests, the time required for a satisfactory pressure build-up response is much shorter, because of the relatively small quantity of fluid withdrawn by the wireline tester. This feature is highly desirable in the case of low-permeability formations. For an analysis of the pressure response within the formation, three simple flow geometries are considered-linear, cylindrical and spherical. The spherical and cylindrical flow geometries are most pertinent to the formation tester; therefore, they will receive the major emphasis. Since the configuration of the borehole and the perforation made by the tester complicate the flow geometry, it is necessary to allow for them in the drawdown response. However, because of the volume of formations contributing to the pressure- response, the details of the perforation shape are unimportant in the build-up period. Since relatively small amounts of fluid are withdrawn from the formation, in contrast to a conventional drillstem test, a study of the "depth of investigation" and the significance of drawdown as well as build- up data will be included. Because the "depth of investigation" will be shown to be rather large, the effect on the build-up curves of the finite thickness of the permeable bed is considered. It is this consideration that leads to the importance of cylindrical flow geometry. Also included is a discussion of permeability anisotropy and its effect on the interpretation of the tester results. The pressure curves recorded by the formation tester will follow two general patterns, depending upon whether the formation is of high or low permeability. Fig. 1 (a and b) schematically illustrates these two responses. In Fig. 1(a), the high pressure recorded during fill-up of the tool is essentially the pressure differential across the choke in the system. JPT P. 899^
It is shown that a wide class of potential problems involving anisotropic media can be transformed into equivalent problems involving only isotropic media. By means of such transformations it is possible, in a large number of cases, to determine the apparent resistivities which would be observed in anisotropic formations, using electrode‐type resistivity logging devices. Discussion is given of an infinite, anisotropic medium with and without borehole, of two semi‐infinite anisotropic beds (without borehole), and of a thin isotropic bed bounded by anisotropic adjacent formations (without borehole). An interpretation chart for the normal device is presented for thick, non‐invaded, anisotropic beds penetrated by a borehole.
A prototype equipment has been designed and built for the digital recording of well logs on magnetic tape at the same time that the regular film recording is made. The format of the digital tape produced is such that it can be used directly at the input of the IBM 704, 7090 or other models of IBM computers which accept digital magnetic tape. This apparatus has been used for the experimental field recording of dipmeter tape logs which were subsequently computed by means of an IBM 704 or 7090.In this paper the equipment and the digital tape are described briefly, and their application to the computer interpretation of dipmeter data is discussed. A principal element in the interpretation of the dipmeter log is the correlation of the three microresistivity dipmeter curves to determine the depth displacements between them. Several correlation methods for computer use are considered, with particular attention to their sensitivity to error and their consumption of computer time. The tape data were used to compute information content of the dipmeter microresistivity curves in terms of their frequency spectra. The results show that the sampling rate used in recording the digital information is quite adequate and illustrate a use of the digital tape in evaluating the characteristics of new tools. Some examples of field results are shown. It can be foreseen that, when digital tape recording becomes available for general field use, a whole new realm of possibilities will be opened up for the processing of other well logs through computations, which hitherto were not feasible because they were too laborious and time-consuming. Introduction The last few years have seen a revolution in the design and production of data-processing equipment. Stored-program digital computers have progressed from a research curiosity to the basis of a major industry. There are now hundreds of such machines in daily use in the United States. With the acceptance of a technique that was, in fact, already clearly described by John von Neumann in 1945, the last decade has seen great strides in the development of components, reliability, programming systems and, most spectacularly, in the sheer number of machines built and in use. In 1957 there were enough digital computers available to the oil industry to justify the suggestion that it would be worthwhile to investigate the possibility of using these machines in processing well log data. The first result of this investigation was the appearance of what may be referred to as the input-output bottleneck. Well logs are customarily recorded on film. To get these data into a machine required then (and still does): a time-consuming semi-automatic reading of the film; conversion of the log data to digital form; and recording these digital data in some medium acceptable for computer input, such as cards, magnetic tape, or punched paper tape. However, the recording, reading, and re-recording could only result in deterioration of the data. Therefore, it was concluded that the first step should be the development of a new, more direct recording technique supplemental to the film recording, which would provide easy access to the digital computer. There are many solutions to the problem of recording log data in an easily recoverable form. After careful consideration it was decided to adopt the boldest solution which, it was felt, was also the most elegant. It was decided to record well logs directly, in the field, on magnetic tape in such a way that this tape could be used without further modification as an input to the IBM 704 or 7090 computer. To realize practical field recording of magnetic tape logs, it became necessary to develop in a rather small package, an analog-to-digital converter, a tape recorder, and the necessary multiplexing and control circuits to allow the simultaneous recording of a multiplicity of logging signals. The magnetic tape recording was to be made simultaneously with the conventional logging operation in such a way as not to interfere with it. Along with the development of hardware, it was necessary to begin development of interpretation techniques and machine programs that would exploit the power of the digital computer. Here, again, there is a long list of possible applications. After much consideration it was decided to concentrate on the interpretation of the dipmeter log as a first application. It is the object of this paper to describe in some detail the developments sketched in the last three paragraphs. JPT P. 771^
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