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.

We use numerical simulations to investigate the possibility of enabling steel-cased wells as galvanic sources to detect and quantify spatial variations of electrical conductivity in the subsurface. The study assumes a vertical steel-cased well that penetrates electrically anisotropic horizontal layers. Simulations include a steel-cased vertical well with a finitelength thin wire of piecewise-constant electric conductivity and magnetic permeability. The steel-cased well is energized at the surface or within the borehole at an arbitrary depth with an electrode connected to a current source of variable frequency. Electromagnetic ͑EM͒ fields excited by the energized steel-cased well are simulated with an integral-equation approach. Results confirm the accuracy of the simulations when benchmarked against the whole-space solution of EM fields excited by a vertical electric dipole. Additional simulations consider a wide range of frequencies and subsurface conductivity values for several transmitter-receiver configurations, including borehole-to-surface and crosswell. The distribution of electric current along the steel-cased well is sensitive to vertical variations of electric conductivity in the host rock. In addition, numerical simulations indicate that crosswell and borehole-to-surface receiver configurations could reliably estimate vertical variations of electric conductivity within radial distances of up to 500 m for frequencies below 100 Hz and for average host rock electric conductivities below 1 S / m.