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 based on a coupled scalar-vector potential formulation for arbitrary 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. The discretization of the partial differential equations and the enforcement of the appropriate boundary conditions yields a complex linear system of equations that we solve iteratively using the biconjugate gradient method with preconditioning. The accuracy and efficiency of the algorithm is assessed with examples of multicomponent-borehole electromagnetic-induction measurements acquired in homogeneous, 1D anisotropic, 2D isotropic, and 3D anisotropic rock formations. 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 potential formulation favorably compare in accuracy with results obtained with 1D, 2D, and 3D benchmarking codes in the dc to megahertz frequency range for large contrasts of electrical conductivity. Our numerical exercises indicate that the coupled scalar-vector potential equations provide a general and consistent algorithmic formulation to simulate borehole electromagnetic measurements from dc to megahertz in the presence of large conductivity contrasts, dipping wells, electrically anisotropic media, and geometrically complex models of electrical conductivity.