Simulation of Full Responses of Triaxial Induction Logging in 1D Layered Arbitrarily Anisotropic Formations

Triaxial induction tools can be used to evaluate thinly laminated sand-shale sequences and fractured beds. This type of reservoirs exhibit transversely isotropy (TI) or arbitrarily anisotropy (also called as biaxial anisotropy (BA)). There have been several papers to study the responses in the TI model or the simplified BA model whose conductivity principal coordinate is always consistent with the formation coordinate. However, little work covers the most general biaxial anisotropic model whose conductivity tensor's orientation is arbitrary. We introduce the Euler angles, then the general biaxial anisotropic conductivity tensor can be determined by three principal components and three ordered Euler angles. To derive the electromagnetic (EM) fields in arbitrarily anisotropic medium, we first convert the Maxwell's equation of frequency-spatial domain into frequency-wavenumber domain by 2D Fourier transform, and obtain an ordinary differential system about horizontal components of EM fields. Using eigenvalue decomposition of the system matrix, this system can be decomposed into two group of equations associated with upward and downward eigen-waves respectively. We derive the solutions of EM fields in frequency-wavenumber domain by introducing transmission matrix, both local and generalized reflection matrix and propagator matrix After that, we use 2D Gauss-Legendre quadrature to calculate inverse Fourier transformation and obtain Green's function for simulation of the tri-axial induction responses. The numerical results are compared with 3D numerical method in both vertical and deviated wells and the agreement is satisfactory. Finally, we investigate the response characteristics in several formations with different Euler angles The results show that triaxial induction responses are remarkably influenced by Euler angles even if the values of three principal components of conductivity tensor remain unchanged. Compare to the responses of the simplified BA model, those of general BA model are more complex and contain more nonzero components. The results indicate that using TI model or the simplified BA model in complex environment may cause large errors. Our algorithm are more practical than algorithms based on the simplified model because the real depositional environments are usually complicated.