A fast integral equation solver for 3D induction well logging in formations with large conductivity contrasts

摘要

Simulation of induction logging responses in formations with large conductivity contrasts is an important but challenging problem due to the singularity of a linear system caused by large contrasts. Also, three-dimensional (3D) analysis of complex geophysical structures usually encounters high computational demands. In this paper, a pre-corrected fast Fourier transform (pFFT)-accelerated integral equation method is applied to overcome these difficulties. In the approach, the entire formation is included in the solution domain. The volume integral equation is set up in the region based on the fact that the total field is the summation of the excitation field and the secondary field. The emitted field by the transmitter coil (treated as a magnetic dipole) is regarded as the excitation of the system. Then the method of moments (MoM) is used to solve the integral equation. To reduce the high computational requirements of the MoM, the pFFT method is used to speed up the solution of the matrix equation and reduce the memory requirement as well. The resultant method is capable of computing induction logging problems involving large and complex formations. For problems with high conductivity contrasts, the solution of the matrix equation usually converges very slow or even fails to converge due to the large condition number of the coefficient matrix. To overcome this difficulty, an incomplete LU pre-conditioner is used to significantly speed up the convergence of the matrix equation, thus further reducing the computation time. Numerical results show that the present method is efficient and flexible for 3D simulation of induction logging and is specifically superior for problems with high conductivity contrasts.