Forward Calculation of the Resistivity Logging Response Through Casing by Transmission Line Equation for Multi-Layer Formations

Because difference of formation model affects the potential distribution on the metal casing, the resistivity logging response can be obtained through casing by measuring the potentials on the metal casing. In fact, the logging responses are the comprehension effects of complex formation and metal casing, so that in the coefficients of transmission line equation (TLE), we should consider the effects of formation model in radialdirection. In the following, for the formation taking on multi-layer in axial-direction, and step in radial-direction, the resistivity logging responses through casing are researched and discussed, and given the recurrence formulae to compute potential. Using the potential distributions which are obtained by resolving TLE we realize the calculation of conductivity of practice formation model through casing, and give numerical computing examples for difference formation models. The computing results show that: this method has a strong capability to distinguish layer-formations in axial-direction; while, in the radial-direction, for the uniform formation, the conductivity of formation is computed is very approximating to that of formation model, which is able to reveal characteristic of practical formation; for the high conductive formation and the low conductive cement layer, the method has an obvious logging response; for the formation taking on steps in radial-direction, the computing conductivity is comprehension reflect for every layer in radial. The computing examples also show that this method has a fast computing speed (the run time of computing example is less than 1s). For analysis the logging response of multi-layers and block formations, and the design of logging instruments, the research is important significance and useful.