An Analytic Algorithm for Electromagnetic Field in Planar-Stratified Biaxial Anisotropic Formation

An analytic algorithm is developed to solve the electromagnetic (EM) field excited by orthogonal magnetic dipoles in the multilayered biaxial anisotropic medium. It allows forward modeling and inversion of EM well logging in the planar-stratified formation without borehole and invasion zones. First, the transverse components of the EM fields are divided into up/down-going waves in the frequency wavenumber domain. A novel concept of reflection/transmission is introduced to describe the propagation of the EM waves through the boundaries. Different from previous work, those reflection/transmission coefficients are concerning the propagation of up/down-going waves directly rather than the amplitude of the total EM field in each layer, which makes the derivation process clearer and easier to understand. Then, we adopt the cubic spline interpolation to calculate the 2-D Fourier integral that transforms the field from the spectral domain to the spatial domain, and an efficient sampling rule is also provided to achieve high accuracy and stability in numerical calculation. For avoiding the singularity when the field locations are close to the source along the vertical direction, the primary field is subtracted in the spectral domain to make the integrand decay fast. Finally, we validate our new formulas compared with another numerical method and analyze the responses of the triaxial induction tool in biaxial anisotropic formations.