Surface Integral Equations for Low-Frequency Simulation in Well Logging Applications

Borehole and fracture in well-logging applications are usually simulated by the finite-element method (FEM) or volume integral equation (VIE). In this work, we propose the surface integral equation (SIE) to simulate them together. Specifically, the borehole is treated by the Poggio, Miller, Chang, Harrington, Wu, and Tsai (PMCHWT) formulation, whereas the fracture is handled by the thin dielectric sheet (TDS)-based SIE, i.e., TDS-SIE. The notorious low-frequency breakdown of PMCHWT is cured by using loop and tree basis functions for discretization. The hybridization of the loop-tree enhanced PMCHWT and TDS-SIE may be referred to as LT-TDS, which is reported for the first time. Moreover, we extend LT-TDS into the case where the background is a layered medium (LM), leading to the more universal and powerful solver entitled LM-LT-TDS. We validate the proposed (LM-)LT-TDS by simulating several low-frequency examples and comparing with the reference results obtained by FEM. Of particular concern is a practical well-logging case where both the borehole and the tilted fracture straddle a three-layer medium.