Finite-difference simulation of borehole EM measurements in 3D anisotropic media using coupled scalar-vector potentials

This paper describes the implementation and successful validation of a new staggered-grid, finite-difference algorithm for the numerical simulation of frequency-domain electromagnetic borehole measurements. The algorithm is basedonacoupledscalar-vectorpotentialformulationforarbitrary 3D inhomogeneous electrically anisotropic media. We approximate the second-order partial differential equations for the coupled scalar-vector potentials with central finite differences on both Yee’s staggered and standard grids. Thediscretizationofthepartialdifferentialequationsandthe enforcement of the appropriate boundary conditions yields a complex linear system of equations that we solve iteratively using the biconjugate gradient method with preconditioning. Theaccuracyandefficiencyofthealgorithmisassessedwith examples of multicomponent-borehole electromagnetic-induction measurements acquired in homogeneous, 1D anisotropic,2Disotropic,and3Danisotropicrockformations.The simulation examples consider vertical and deviated wells with and without borehole and mud-filtrate invasion regions. Simulation results obtained with the scalar-vector coupled potentialformulationfavorablycompareinaccuracywithresults obtained with 1D, 2D, and 3D benchmarking codes in the dc to megahertz frequency range for large contrasts of electricalconductivity.Ournumericalexercisesindicatethat the coupled scalar-vector potential equations provide a generalandconsistentalgorithmicformulationtosimulateborehole electromagnetic measurements from dc to megahertz in the presence of large conductivity contrasts, dipping wells, electrically anisotropic media, and geometrically complex modelsofelectricalconductivity.

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