Integral equation approach based on contraction operators and Krylov subspace optimisation

Abstract

We report development of a new code for modelling electromagnetic fields in complicated 3D environments and provide examples of the code application to a number of typical borehole as well as some non-borehole problems. The code is based on an integral equation for the scattered electromagnetic field presented in the form used by the Modified Iterative Dissipative Method (MIDM). Such integral equation possesses contraction properties that allow the equation to be solved iteratively. The iterative sequence of approximations converges to the equation solution at any frequency, for an arbitrary earth model, and any source of the electromagnetic field. The approach can also be applied to media with anisotropy of electrical properties and displacement currents. The system of linear equations that represents a finite-dimensional counterpart of the continuous integral equation is derived using a projection definition of the system matrix. In this definition, the matrix is calculated by integrating the Green?s function over the ``source' and ``receiver' cells of the numerical grid. The system of linear equations preserves contraction properties of the continuous equation and can be solved using the same iterative technique. The condition number of the system matrix and, therefore, the convergence rate depend only on the physical properties of the model under consideration. These parameters remain independent of the numerical grid used for numerical simulation. Applied to the system of linear equations, the iterative perturbation approach generates a sequence of approximations converging to the equation solution. The number of iterations necessary to reach the specified accuracy of the numerical solution is significantly reduced by finding the best possible approximant inside the Krylov subspace, which spans the iterates accumulated in the preceding steps. The effect of the optimisation grows with the complexity of the model under consideration. 3D codes based on the above principles have been applied to a number of problems of subsurface and borehole geophysics. The applications include such problems as global electromagnetic induction, regional and local magnetotelluric and magneto-variational soundings with natural and controlled sources, through-casing resistivity, etc. We compare our results for a number of models with independent results obtained using different approaches. The robustness of the algorithm is demonstrated by running the same model with different numerical grids. Models used for the code demonstration include a vertical borehole in a stratified formation excited by co-axial induction and co-planar tools. We also consider tilted boreholes and decentralized tools.