TEM responses in an inhomogeneous medium are modeled in 3D using the vector finite element method for induction logging in petroleum wells. The algorithm is applied to HFIL diagrams obtained in wells that tap thinly laminated formations with regard to the true position of logging tools.
In the edge vector finite element solution of the frequency domain Maxwell equations, the presence of a large kernel of the discrete rotor operator is known to ruin convergence of standard iterative solvers. We extend the approach of [R. Hiptmair, Multigrid method for Maxwell's equations, SIAM J. Numer. Anal. 36 (1) (1999) 204-225] and, using domain decomposition ideas, construct a multilevel iterative solver where the projection with respect to the kernel is combined with the use of a hierarchical representation of the vector finite elements.The new iterative scheme appears to be an efficient solver for the edge finite element solution of the frequency domain Maxwell equations. The solver can be seen as a variable preconditioner and, thus, accelerated by Krylov subspace techniques (e.g. GCR or FGMRES). We demonstrate the efficiency of our approach on a test problem with strong jumps in the conductivity.
Water salinity at shallow sea depths in shelf areas changes with depth, which causes respective conductivity changes. We discuss algorithms for computing monochromatic electric fields using the vector finite-element method with different realizations of depth dependence of conductivity. The algorithms have been applied to compute vertical conductivity patterns and to explore its influence on the electric field measured on the surface.
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