Electromagnetic borehole fields in a layered, dipping‐bed environment with invasion

Abstract

Electromagnetic modeling of an induction sonde (1–100 kHz) in a dipping‐bed environment is a 3-D problem. The capability for such an analysis is necessary for interpretation of oil‐well logs in offshore environments where most holes are deviated. 3-D geometrical effects require vector field analysis. The method accounts for transverse magnetic mode (TM) coupling arising from surface charges deposited by eddy currents passing through bed boundaries. If borehole and invasion effects are included, the only available rigorous analytical methods are finite elements or finite‐difference techniques. These approaches require large‐scale computing. In contrast, our method is approximate and is an extension of the geometrical‐factor theory and Born approximation. The variational method does not require matrices and is numerically simpler than the more rigorous finite element method. The method uses a new electric field vector integral equation developed by Chew. The formulation accounts for low‐frequency behavior at bed boundaries where current channeling and surface charge phenomena dominate the interactions. The receiver voltage has two parts, a volumetric term [Formula: see text] and a surface term [Formula: see text]. The term [Formula: see text] reduces to the Born result when the dip angle goes to zero; [Formula: see text] accounts for the surface charge effect and is only significant when the receivers are in close proximity to a bed boundary. The local nature of the charge interaction results from double scattering events, which are necessary to produce this effect. The charge term is second order as explained intuitively in terms of polarization. The bed boundary interaction is proportional to the factor [Formula: see text], where [Formula: see text] is the dip angle, and [Formula: see text] and [Formula: see text] are the conductivities of the adjacent beds. Since the charge interaction is strongly nonlinear in conductivity, common induction log interpretation, which assumes linearity, is expected to fail near bed boundaries. Results for dip angles up to 60 degrees for variational results and eigenfunction solutions for the case of no borehole or invasion show good agreement. A few 3-D results are computed with simultaneous layering, dip, and invasion.