| An analytical expression of dyadic Green's functions in cylindrically stratified anisotropic media(transversely isotropic media)was obtained using a recursive method.The expression can be used to simulate the response of an arbitrary point source(including electric source and magnetic source)in cylindrically stratified anisotropic formations.The expression can be implemented for any number of formation layers,and with the source location and field location in any of the layers.The responses of multi-component induction logging tools with a metal mandrel and an insulating protection layer were simulated by the method in anisotropic formations with a borehole and an invasion zone.The shape of each component's coils was taken into account in order to increase the simulating precision,and the metal mandrel was taken as a layer whose conductivity can be both finite and infinite.The simulated results showed that the response of coplanar coil system has entirely different characteristics from that of coaxial coil system.The response of coplanar coil system is often more complex and more sensitive to the change in drilling liquid conductivity,invasion zone's conductivity,formation conductivity,and formation anisotropy,etc.The response often changes its sign with these varying parameters under many kinds of conditions.Furthermore,since the same response of coplanar coil system may be produced by entirely different formation conductivities in isotropic or anisotropic formations,it will become much complicated to interpret or process the measured data.The effect of eccentricity on the responses of electromagnetic well-logging tools in a borehole is of great value for precisely interpreting the measured data.For real electromagnetic well-logging tools such as array induction-logging tools,usually both a metal mandrel and an insulating protection layer are included in the tools.Only the existence of the metal mandrel was considered for simulating and analyzing the effect of tool eccentricity in conventional analytical methods,and the influence of the insulating protection layer has been neglected.Since the insulating protection layer has occupied part of the space of the drilling fluid,its influence on the responses should also be considered whether the tool is centric or eccentric in a borehole.The responses of centric and eccentric array induction logging tools in a borehole were simulated by the algorithm,and the difference between the responses of the tool with or without an insulating protection layer was compared.The observations include:(1)Given the borehole diameter and the conductivity of the drilling fluid and formation,the relationship between the receiver coil's response and the eccentric distance was obtained for different arrays.The results showed that the effect of the insulating layer on the tool's response is small and can be neglected when the fluid's conductivity is relatively low,whether the transmitter-receiver spacing is long or short.The effect is big for the condition of relatively high fluid's conductivity,and the shorter the transmitter-receiver spacing the bigger the effect,the bigger the eccentric distance the bigger the effect.(2)Given the conductivity of the drilling fluid and formation,the relationship between the coil's geometric factor and the borehole's diameter was obtained for different arrays with the tool being in the borehole center.Since the tool was positioned in the borehole center,the effect of the insulating layer was relatively small.(3)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.(4)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 angel was shown for different arrays of xx-component.The results showed that the influence of the azimuth angel 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 angel under the condition of big eccentricity ratio.(5)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.(6)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 zz-component 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. |
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