A Hybrid Loop-Tree FEBI Method for Low-Frequency Well Logging of 3-D Structures in Layered Media

Electromagnetic simulation plays an essential role in formation conductivity determination by electromagnetic well-logging tools, especially when the nonvertical borehole and invasion zones are present in a layered earth. A hybrid finite-element boundary integral (FEBI) method is suitable for such well-logging simulations. The usage of layered medium Green’s functions allows the simulation background to be a planar stratified medium to characterize the earth formation. However, conventional FEBI using the Rao–Wilton–Glisson (RWG) basis function produces inaccurate results due to the low-frequency breakdown of the boundary integral equation solution. The new contribution of this work is to apply the loop-tree (LT) basis functions in the FEBI method to reduce the numerical error at low frequencies. Such an LT-FEBI method can handle all specific requirements of well-logging simulations at low frequencies. A direct solver is applied to make it more efficient to obtain logging curves with multiple source locations. The LT-FEBI simulation results are validated in several numerical examples, including a challenging well-logging model with a deviated borehole and invasion zones in the Oklahoma formation having 28 layers.