Triaxial induction tools are used to evaluate fractured and low-resistivity reservoirs composed of thinly laminated sand-shale sequences. Thinly laminated and fractured reservoirs demonstrate transversely isotropic or fully anisotropic (biaxial anisotropic) electrical properties. Compared to the number of studies on transverse isotropy, relatively little work covers biaxial anisotropy because of the mathematical complexity. We have developed a theoretical analysis for the full response of a triaxial induction tool in a homogeneous biaxial anisotropic formation. The triaxial tool is composed of three mutually orthogonal transmitters and three mutually orthogonal receivers. The bucking coils are also oriented at three mutually orthogonal directions to remove direct coupling. Starting from the space-domain Maxwell's equations, which the electromagnetic (EM) fields satisfied, we obtain the spectral-domain Maxwell's equations by defining a Fourier transform pair. Solving the resultant spectral-domain vector equation, we can find the spectral-domain solution for the electric field. Then, the magnetic fields can be determined from a homogeneous form of Maxwell's equations. The solution for the EM fields in the space domain can be expressed in terms of inverse Fourier transforms of their spectral-domain counterparts. We use modified Gauss-Laguerre quadrature and contour integration methods to evaluate the inverse Fourier transform efficiently. Our formulations are based on arbitrary relative dipping and azimuthal and tool angles; thus, we obtain the full coupling matrix connecting source excitations to magnetic field response. We have validated our formulas and investigated the effects of logging responses on factors such as relative dipping, azimuthal and tool angles, and frequency using our code. We only consider conductivity anisotropy, not anisotropy in dielectric permittivity and magnetic permeability. However, our method and formulas are straightforward enough to consider anisotropy in dielectric permittivity.

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