Effects of formation anisotropy on resistivity‐logging measurements

Abstract

Some developments in the mathematical analysis of resistivity well‐logging measurements in anisotropic beds are presented. The no‐borehole case in which two thick, anisotropic beds meet at a plane interface for bedding planes parallel to the interface has been discussed previously by Kunz and Moran (1958). 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‐logging 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 of 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 Laterolog™, 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 Kurtz 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 are 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 out‐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 boundary shows undesirable artifacts, particularly on the vertical resistivity curve. In borehole cases, the values found also are strongly effected when hole‐fluid conductivity is much larger than the horizontal conductivity of the formation.