Algorithm of geometrical factor of the azimuthal electromagnetic wave logging while drilling with annular antenna recesses in layered anisotropic formation

In this paper, based on the geometrical factor theory of induction well logging and the first-order Born approximation, a novel algorithm of geometrical factors for the azimuthal electromagnetic (EM) wave logging while drilling (LWD) with annular antenna recesses is proposed. Firstly, we introduce the Green's functions of EM field for dual-component coaxial and transverse coils in inhomogeneous transversely isotropic (TI) formation. By applying the reciprocity theorem and Green's second integral formula, we advance the relationship of the changes in annular electric field at coaxial coils and radial magnetic field at transverse coils with the relative perturbations of both horizontal and vertical conductivity, and obtain the method to calculate the induced electromotive forces on different receiving coils and the expression of three-dimensional (3-D) geometrical factors (also called spatial sensitivity functions) of axial and cross components. Then, we further formulate both vertical and radial differential geometrical factors in the TI formation by using integral along the axial and radial directions, respectively. After that, we solve the input induced EM field and the Green' s function of EM field corresponding to the annular current source and radial magnetic dipole source based on 3-D finite volume method (3-D FVM) of Helmholtz equation of coupled potentials in cylindrical coordinate system. Finally, we give numerical results of the geometrical factors in various environments such as different frequency, different borehole dip, and different anisotropic coefficients to analyze and investigate the sensitivity and the detection ability of the tool.