Three-dimensional induction logging problems, Part 2: A finite-difference solution

A 3-D finite-difference solution is implemented for simulating induction log responses in the quasi-static limit that include the wellbore and bedding that exhibits transverse anisotropy. The finite-difference code uses a staggered grid to approximate a vector equation for the electric field. The resulting linear system of equations is solved to a predetermined error level using iterative Krylov subspace methods. To accelerate the solution at low induction numbers (LINs), a new preconditioner is developed. This new preconditioner splits the electric field into curl-free and divergence-free projections, which allows for the construction of an approximate inverse operator. Test examples show up to an order of magnitude increase in speed compared to a simple Jacobi preconditioner. Comparisons with analytical and mode matching solutions demonstrate the accuracy of the algorithm.