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...

We introduce and successfully test an efficient method to simulate triaxial borehole electromagnetic ͑EM͒ induction measurements acquired in axially symmetrical and transversely isotropic ͑TI͒ media. The method uses a Fourier series expansion to express the azimuthal dependence of EM fields and the source term whereby the essentially 3D problem collapses to a series of independent 2D problems. Each 2D problem is solved with a semianalytic method that uses normalized Bessel functions and normalized Hankel functions to express the radial dependence of EM fields, thereby improving numerical stability. In addition, use is made of amplitude and slope basis functions to describe the longitudinal dependence of EM fields to avoid grid refinement in the vicinity of horizontal formation boundaries. For validation, we compare the new simulation method to two 1D analytic methods in horizontally and radially layered formations, and to one 3D finite-difference method ͑3DFD͒ in multilayered formations that include borehole and invasion zones. Numerical results indicate that the method is accurate in formations with high conductivity contrasts compared to 1D methods and is more than ten times more efficient than the 3DFD method in multilayer formations.

This theoretical investigation presents the responses in dipping beds of an induction dipmeter and standard induction tools, neglecting the influence of the borehole and invaded zones as well as the mandrel and coil size. The analysis is sufficiently general to determine response characteristics of an induction coil arbitrarily oriented in an arbitrary number of beds. In an induction dipmeter log (IDL) the most useful information is obtained from the out‐of‐phase voltage induced in the receiver coils when they are orthogonally oriented to the transmitter coils. A simple algorithm is developed to convert the crosscoupling signals into the apparent dip and strike information. The agreement between the actual and apparent values is nearly perfect for small transmitter‐to‐receiver spacings, i.e., less than 10 cm. The theoretical results for standard induction tools were consistent with prior work indicating the influence of dip is strongly dependent on the shoulder effect of the device in question. Consequently, dipping beds have less influence on the standard ILm because its shoulder effect is less than the standard ILd. These conclusions are supported by numerous simulations consistent with conditions encountered in the field. Finally, a simple no‐skin‐effect theory is developed for the case of a single dipping interface separating two infinitely thick beds. The results with this approach agree with the exact theory in resistive formations. This simplified theory is a generalization of Doll’s geometrical factor theory.

Theoretical relations exist in the literature for calculating the responses of electrode‐type resistivity logging tools when they are centered in the wellbore and the formations are thick and homogeneous (Fok, 1933; Stefanesco et al., 1929–32). These analyses are usually restricted to devices using dc or low‐frequency surveying currents, and they generally make use of the approximation of point electrodes. An analysis has also been made comparing the responses of such tools in anisotropic and isotropic formations (Kunz and Moran, 1958). In this paper, relations are derived for calculating the responses of electrode‐type logging tools when the sonde is not centered in the wellbore. Although similar relations could be applied to tilted sondes, the discussion in this paper is mainly restricted to the simple eccentered case in which the axis of the sonde is parallel to the axis of the hole. As will be shown, eccentering has a comparatively small effect on the responses of a normal device. On the other hand, certain types of focused‐log devices, in which a large proportion of the focusing current flows parallel to the hole, may exhibit an appreciable eccentering effect; this is particularly true when the contrast between formation resistivity and hole‐fluid resistivity is large. Spherically‐Focused‐Log (SFL*) resistivity devices (Schuster et al., 1971) have this type of current distribution, and, depending on electrode spacings, they may be appreciably influenced by eccentering. The applications shown in this paper will largely be done from SFL computations and measurements. The theoretical relations have been used in the development of a spherically focused tool less affected by eccentering. Tool responses predicted on the basis of the theoretical relations were found to be consistent with the results of test‐tank measurements made for one of the devices studied. Some field experiments have been made with centered and eccentered spherically focused tools, and more are to be done.

This paper develops the analysis necessary for the computation of the response of a resistivity tool within a well bore as it traverses a thin‐invaded bed. The solution for the potential induced by a steady current source (point current electrode) is formulated in terms of both Fourier cosine and Fourier sine transforms with arbitrary coefficients. A suitable matching of the necessary boundary conditions results in a system of singular integral equations. An iterative solution (Neumann series) is obtained for these transform coefficients which, in turn, are used to determine the potential at an arbitrary point of measurement. The theory is applied to some typical focused resistivity tools, and the results are found to be in close agreement with similar results obtained via a resistor network (analog) solution.