An Analytic Model for Electromagnetic Wireline Tools

Abstract

Summary. A mathematical model for electromagnetic tools is presented that accounts for the finite size of the transmitter and receiver coils, tool mandrel, a shielding conductor internal to the tool, and the concentrically stratified media surrounding the tool. The model is applied to various tools in the 20 × 10 3 to 1x10 9-cycle/sec [20-kHz to 1-GHz] frequency range and results indicate that this model will be useful in designing future tools. Introduction The response of an electromagnetic tool surrounded by an arbitrary number of coaxial zones of media has been the subject of several earlier publications. None of these analyses take into account the details of the tool itself. In the present model of the tool, we consider not only the finite radial and axial extensions of the coils, but also the effects of the copper tubing inside the mandrel and coaxial with it. Thus the problem to which an analytic solution is sought can be stated as follows. A coil carrying an alternating current is wound on an insulating mandrel centered coaxially in a borehole. A cylindrical conductor is placed inside the mandrel and concentric with it to find the electromotive force (EMF) induced in another coil. also wound on the mandrel, because of the current-carrying coil. Both coils have finite radial and axial extensions. This tool system is placed in a borehole surrounded by an arbitrary number of cylindrical zones. The results obtained are applied to both induction and dielectric logging systems, and some of the previously published results are discussed as special cases. Special attention is paid to the influence of the copper tubing on the EMF induced in the receiver and other quantities of interest (e.g., tool constant, amplitude ratio, etc.). Theory Initial calculations are performed for a single transmitter/receiver pair with a six-zone model (Fig. 1). The innermost zone is the region pair with a six-zone model (Fig. 1). The innermost zone is the region inside the copper tubing. The regions outside the copper tubing are the mandrel, the borehole, the invaded zone, and the formation. Let ro, rl = inner and outer radii of copper tubing. respectively, r, = radius of mandrel on which coils are wound, r3 = radius of borehole, r4 = radius of invasion, n = electrical conductivities, en = dielectric constants, and un, = magnetic permeabilities of different media. The subscripts n = 0, 1, 2, 3, 4, and 5 refer to the innermost zone, copper tubing, mandrel, borehole, invaded zone, and formation, respectively. The main purpose of this mathematical formulation is to evaluate the induced EMF, V, at the location of the receiver coil as a result of the current in the transmitter. The induced EMF is obtained from the line integral of the electric field: ..........................................(1) The electric field is related to the magnetic vector potential, by the relation ..........................................(2) where we have assumed the time dependence of the fields to be of the form e -iwt. Note that there is complete cylindrical symmetry about the axis of the borehole. Further, the source is a coil carrying an alternating current and coaxial with the borehole. This means that the only nonzero current component is J, leading to single components, A and E, for the vector potential and the electric field. The vector potential, is a solution of the Helmholtz equation, ..........................................(3) where Js is the current density of the source and k is the propagation constant. Thus at source-free points of the field, the nonzero propagation constant. Thus at source-free points of the field, the nonzero component A. satisfies the equation ..........................................(4) Application of the method of separation of variables yields a general solution of the form ..........................................(5) where ..........................................(6) and I1(Br) and K1(Br) are the modified Bessel functions of the first and second kind and Order 1, respectively. Upon applying the necessary boundary conditions at the different cylindrical surfaces, the following expression for the induced EMF at the receiver location is obtained: ..........................................(7) where ..........................................(8) ..........................................(9) SPEFE P. 17