Simulating effect of tool eccentricity on responses of multi-component induction logging tools by dyadic Green's functions in cylindrically stratified anisotropic media

The multi-component induction logging tool has several arrays with each array being composed of three mutually orthogonal transmitter-receiver pairs, so it can simultaneously measure nine components of the magnetic field produced by the transmitters and can provide data on horizontal and vertical resistivities of the formation. The tool is of great importance for accurate evaluation of complex formation such as anisotropic (actually transversely isotropic, i.e. TI) formation. It is necessary to systematically investigate the influence of environments such as bed thickness, borehole and invasion zone on the responses of the tool. In this paper, an analytical algorithm was developed that can efficiently simulate the responses of a centric or eccentric multi-component induction logging tool with both a metal mandrel and an insulating protection layer in a borehole surrounded by TI formations, and the shape of all the component's coils of the tool was also taken into account. The algorithm was based on the spectral-domain dyadic Green's functions in homogeneous TI media. The spectral-domain dyadic Green's functions in cylindrically stratified TI media was then derived using a recursive method, which was realized by introducing local transmission and local reflection matrices under anisotropic condition and by using the continuity of the tangential components of the electromagnetic fields on interfaces. As for the situation of tool eccentricity in a borehole, the insulating protection layer where the coils are positioned was taken as the 0-th layer, while the borehole is taken as the 1-th layer. The fields in the borehole can be expressed by the fields in the insulating layer by using the continuity of their tangential components on the interface between the insulating layer and the borehole. Finally the new algorithm was developed for eccentric situation by using the relationship between the electromagnetic fields in the borehole under the cylindrical coordinates that are centered with respect to the borehole and are centered with respect to the tool. The responses of eccentric multi-component induction logging tools in a borehole were simulated by the algorithm. The observations include: (1) Given the conductivity of the drilling liquid and the horizontal and vertical conductivities of the TI formation, the relationship between the receiver coil' s response and the eccentricity ratio was shown for different arrays of zz-component and xx-component. The results showed that the influence of the eccentricity ratio on the tool's responses is much more obvious under the condition of relatively high conductivity of the drilling liquid, especially for short spacing arrays, so the eccentricity effect should be corrected. While for the condition of relatively low conductivity of the drilling liquid, the eccentricity effect can be neglected. (2) Given the conductivity of the drilling liquid, the horizontal and vertical conductivities of the TI formation and the tool' s eccentricity ratio, the relationship between the receiver coil's response and the azimuth angle was shown for different arrays of xx-component. The results showed that the influence of the azimuth angle is much more obvious under the condition of relatively high conductivity of the drilling liquid, and its influence can be neglected for the condition of relatively low conductivity of the drilling liquid. The results has also showed that when the tool is in the borehole with relatively high conductivity, the xx-component coil's response changes more obviously with the azimuth angle under the condition of big eccentricity ratio. (3) Given the horizontal and vertical conductivities of the TI formation and the tool' s eccentricity ratio, the relationship between the receiver coil' s response and,the conductivity of the drilling liquid was shown for different arrays of zz-component and xx-component. The results showed that the ratio between the coil's eccentric and centric responses is close to 1 when the conductivity of the drilling liquid is small. The ratio will deviate from 1 with increasing conductivity of the drilling liquid, especially for short spacing arrays. (4) Given the conductivity of the drilling liquid, the ratio between the horizontal and vertical conductivities of the TI formation and the tool' s eccentricity ratio, the relationship between the receiver coil' s response and the horizontal conductivity of the TI formation was shown for different arrays of zzcomponent and xx-component. The results showed approximately opposite patterns as in case 3. The ratio between the coil' s eccentric and centric responses will deviate from 1 when the horizontal conductivity of the TI formation is small for short spacing arrays, and it becomes more and more close to 1 with increasing horizontal conductivity of the TI formation. From all the results we can see that the effect of tool eccentricity on responses of multi component induction logging tools is big for short spacing arrays and under the condition of relatively high conductivity of the drilling liquid, and it should be corrected in order to accurately estimate formation vertical resistivity from tool measurements.