Fast 3-D Modeling of the LWD Ultradeep Resistivity Measurements Using the Field-Based Secondary-Field Finite Volume Method

In this article, to explore the efficiency and precision of the 3-D finite volume method (FVM) for the logging while drilling (LWD) ultradeep resistivity measurements, we compared four different schemes: field-based total-field FVM, coupled potentials total-field FVM, field-based secondary-field FVM, and coupled potentials secondary-field FVM. The fast and accurate discretization of scattered current density in the secondary-field method is another issue we focus on. On the one hand, we improve the discretization accuracy of the scattered current density near the source by extracting the direct waves in the background electric field. On the other hand, based on the dyadic Green’s functions (DGFs) of vector potentials, the number of Sommerfeld integrals in the background electric field is reduced as much as possible through the background field library and interpolation. The numerical results show that the accuracy and stability of the secondary-field method are better than those of the total-field method and the efficiency of the background electric field is greatly improved through the library and interpolation. Based on the premises of the LWD ultradeep resistivity measurements and the direct solver, the accuracy of the field-based and coupled potentials methods is almost the same, however, the field-based method is much more efficient. Overall, we believe that the field-based secondary-field FVM and the direct solver constitute a more efficient modeling scheme with high precision for LWD ultradeep resistivity measurements.