Simulation of multifrequency borehole resistivity measurements through metal casing using a goal-oriented hp finite-element method

The authors simulate multifrequency through-casing resistivity tool measurements operating at different frequencies in a borehole environment for the assessment of rock-formation properties. Rock formations are assumed to exhibit axial symmetry around the axis of a vertical borehole. The simulations are performed with a goal-oriented hp-adaptive finite-element method that delivers exponential convergence rates in terms of the quantity of interest (for example, the second vertical difference of the electric potential) against the CPU time. Numerical results confirm the efficiency and accuracy of the method, allowing for high-accuracy and reliable simulations of borehole logging measurements in the presence of highly conductive steel casing. The study of different tool configurations shows the advantages of using calibrated instruments with toroid antennas located on the borehole wall. The agreement between the numerical and analytical results, when the latter is available, is quantified. Errors on the simulations are consistently below 0.1%