Through-Casing Hydraulic Fracture Evaluation by Induction Logging I: An Efficient EM Solver for Fracture Detection

Hydraulic fracturing is an essential way to improve the production of unconventional shale oil and gas. It is important to characterize the produced fractures using either acoustic or electromagnetic (EM) methods. Conventional EM solvers in the low-frequency range face significant challenges by such multiscale problems where the fracture width is orders of magnitude smaller than its diameters. Furthermore, the cased borehole environment is extremely difficult to simulate with conventional EM solvers due to meshing difficulties and the multiscale nature of the problem. In this paper, we develop a hybrid distorted Born approximation and 3-D mixed ordered stabilized biconjugate gradient fast Fourier transform (DBA-BCGS-FFT) method to simulate the very challenging 2-D, 2-D-axisymmetric, and 3-D hydraulic fracture models under both open and cased borehole environments. Numerical examples show that this method has orders of magnitude higher efficiency than the finite element method. The capabilities of the DBA-BCGS-FFT method for the induction tool fracture mapping are demonstrated by comparing with laboratory experimental results and other reference results.