Induction response in dipping beds with invasion

A nonmatrix approach to the electromagnetic modeling of an induction sonde response in a dipping bed environment is presented. The method is approximate and is an extension of geometrical factor theory or the more rigorous Born approximations. The method uses a new electric field vector integral equation developed by C. C. Chew and R. L. Kleinberg (1988). The receiver coil voltage consists of two parts, a volumetric component v_g, and a surface component v_c. The term v_g reduces to the Born result when the dip angle is zero; v_c accounts for the surface charge effect and is only significant when the receiver coils are in close proximity to a bed boundary. Analysis shows that the double scattering events which produce the charge interaction at a bed boundary are proportional to the factor sin²(θ₀)(σ₂ - σ₁)²/ (σ₁ + σ₂), where θ₀ is the dip angle and σ₁ and σ₂ are the conductivities of the adjacent beds. Numerical comparisons with eigenfunction solutions in special-case geometries validate the theory.