A recursive algorithm is adopted for the computation of dyadic Green's functions in three-dimensional stratified uniaxial anisotropic media with arbitrary number of layers. Three linear equation groups for computing the coefficients of the Sommerfeld integrals are obtained according to the continuity condition of electric and magnetic fields across the interface between different layers, which are in correspondence with the TM wave produced by a vertical unit electric dipole and the TE or TM wave produced by a horizontal unit electric dipole, respectively. All the linear equation groups can be solved via the recursive algorithm. The dyadic Green's functions with source point and field point being in any layer can be conveniently obtained by merely changing the position of the elements within the source term of the linear equation groups. The problem of singularities occurring in the Sommerfeld integrals is efficiently solved by deforming the integration path in the complex plane. The expression of the dyadic Green's functions provided by this paper is terse in form and is easy to be programmed, and it does not overflow. Theoretical analysis and numerical examples show the accuracy and effectivity of the algorithm. dyadic Green's functions, stratified uniaxial anisotropic media, recursive algorithm, Sommerfeld integralsThe dyadic Green's functions for stratified media are widely used in many areas of physics [1] , including electromagnetism. In electromagnetic theory, the dyadic Green's functions ( , ) EJ ′ G r r , HJ ( , ) ′ G r r and AJ ( , ) ′ G r r for stratified media represent the electric field, magnetic field and magnetic vector potential produced at field point r by three orthogonal unit electric dipoles located at source point ′ r in stratified media, respectively. Numerous documents have derived the dyadic Green's functions for stratified isotropic or anisotropic media because of their great significance in many areas such as geophysical prospecting, remote sensing and wave
The finite difference method based on staggered grids is applied to simulate the induction logging response in 3D media. The invariance property of the Krylov subspace is used to solve the resulting large sparse complex symmetric linear system. To improve iteration convergence the pseudo inverse of the coefficient matrix is employed when constructing the Krylov subspace instead of the coefficient matrix itself. In each iteration four 3D Poisson equations are computed to get a new Lanczos vector with the incomplete Cholesky decomposition conjugate method. Generally the desirable solution is achieved with the iteration less than 20 times. Also, a new material averaging formula is proposed to get a reasonable average of the conductivity, which guarantees the conservation of the electric current.
Three‐dimension (3‐D) Numerical Mode‐Matching (NMM) is selected to deal with resistivity logging responses in nonsymmetric conditions. First, an appropriate Descartes's reference frame is set up: the plane of deviated borehole and normal of formation is regarded as plane XOY , and formation strike is regarded as Z axis direction. In the plane XOY the finite‐element method (FEM) is used and a series of two‐dimension (2‐D) generalized eigenvalue problems are solved; whereas vertical analytic is adopted along Z axis. In analytical section, the medium is sliced into some layers artificially, and the relationship between adjacent layers is derived according to the electromagnetic field continuity, namely boundary transition matrices rather than reflection and transmission matrices theory in the 2‐D NMM method. Finally, a system of linear equations is constructed according to the natural boundary conditions that the topmost layer bed has no up‐going wave or the bottommost layer has no down‐going wave, and the potential of every point of formation is solved and the apparent resistivity is computed. In the given formation model, the results of 3‐D NMM agree well with previous ones of 2‐D NMM and FEM in symmetry conditions. Furthermore, the algorithm also applies to with formations titled angle and elliptical borehole; and some examples illustrate how all kinds of formation models with tilt angles affect logging response results.
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