Effect of Dipping Beds on the Response of Induction Tools

Abstract

Summary. This paper describes a computer model for studying the effect of dipping beds on the response of induction tools. Simulated tool response in an arbitrary number of dipping beds is obtained from calculations of the magnetic fields of an equivalent magnetic dipole with an axis arbitrarily oriented with respect to the direction of stratification. The effects of the borehole and the invaded zone are neglected to solve the problem in a closed form. In addition to describing the model, we also study the effect of dip angle on the apparent resistivity and the apparent bed thickness as read by induction tools in several typical logging configurations. Introduction The interpretation of induction logs in dipping formations has customarily been based on qualitative estimates only. Because of the typical dipole distribution of the electromagnetic fields generated by induction tools (see Figs. 1A and 1B), the effect of dip in beds thick enough to contain producible amounts of hydrocarbons has been assumed to be small and predictable at formation dip angles usually encountered. Although this assumption may be adequate for thick beds, the interpretation of tool response to multiple dipping layers and laminated formations is much more complex. Furthermore, in the case of deviated wells drilled from offshore platforms, apparent dip can be much more severe than naturally occurring geologic dip. This paper provides a model that allows a precise analysis of the effects of dip on the response of induction tools. The resultant computer code can generate a simulated log for any induction tool in an arbitrary number of thin beds for dip angles ranging from 0 to 90 degrees [0 to 1.57 rad]. (A 90 degrees [1.57-rad] dip is also equivalent to tool response in horizontal boreholes.) The model neglects the presence of the borehole mud. This does not limit its application, however, because most commercial induction tools currently in use are designed to have minimal borehole effect in resistive muds where they are ordinarily run. Several examples given in this paper demonstrate the usefulness of the model. They area systematic comparison of deep investigation and medium investigation (ID and IM, respectively) induction sonde response in resistive and conductive thin beds with symmetrical shoulders for various dip angles,an example showing how dip deteriorates the vertical resolution of ID and IM in several adjacent conductive and resistive beds, andan example illustrating how dip affects ID response in laminated zones. Formulation Induction tools operate in the kilohertz frequency range. Because of this low frequency of operation, we will model the response of the induction tool as an equivalent magnetic dipole of moment M. As mentioned, we neglect borehole effect. Under these assumptions, the problem reduces to solving for the magnetic fields of an arbitrarily oriented magnetic dipole in a planar stratified medium as depicted in Fig. 2. We will formulate the response of the inclined dipole as the superposition of a vertical magnetic dipole (VMD) of moment Mv and a horizontal magnetic dipole (HMD) of moment Mh. These dipole moments are related to each other and to the dip angle, theta, by the following relationships: .......................................(1a) and .....................................(1b) The fields of each dipole are represented as the sum of the contributions from two orthogonal polarizations, i.e, the transverse eltric (TE) and transverse magnetic (TM) polarizations. In the former, the z component of the electric field, Ez, does not exist, whereas in the latter, the z component of the magnetic field, Hz, is zero. Following the approach of Kong, it can easily be shown that all field components can be represented in terms of the z components of the electric and magnetic fields. Ez and Hz are represented in the form of the following integral transform: ..................................(2a) and ,......................(2b) where E(k rho, rho, z, theta) and H(k rho, rho, z, theta) are the spectral amplitudes of Ez(rho, z, theta) and Hz(rho, z, theta), respectively. The former corresponds to the TM polarization, whereas the latter corresponds to the TE polarization. The transverse components of the electric, ET, and magnetic, HT, fields are derived from Eq. 3 as follows: .............(3a) and ,.............(3b) where .................(4) The explicit expressions for the magnetic field components are given in the Appendix. SPEFE P. 29^