Efficient finite-volume simulation of the LWD orthogonal azimuth electromagnetic response in a three-dimensional anisotropic formation using potentials on cylindrical meshes

Abstract

In this study, the cylindrical finite-volume method (FVM) is advanced for the efficient and high-precision simulation of the logging while drilling (LWD) orthogonal azimuth electromagnetic tool (OAEMT) response in a three-dimensional (3D) anisotropic formation. To overcome the ill-condition and convergence problems arising from the low induction number, Maxwell’s equations are reformulated into a mixed Helmholtz equation for the coupled potentials in a cylindrical coordinate system. The electrical field continuation method is applied to approximate the perfectly electrical conducting (PEC) boundary condition, to improve the discretization accuracy of the Helmholtz equation on the surface of metal mandrels. On the base, the 3D FVM on Lebedev’s staggered grids in the cylindrical coordinates is employed to discretize the mixed equations to ensure good conformity with typical well-logging tool geometries. The equivalent conductivity in a non-uniform element is determined by a standardization technique. The direct solver, PARDISO, is applied to efficiently solve the sparse linear equation systems for the multi-transmitter problem. To reduce the number of calls to PARDISO, the whole computational domain is divided into small windows that contain multiple measuring points. The electromagnetic (EM) solutions produced by all the transmitters per window are simultaneously solved because the discrete matrix, relevant to all the transmitters in the same window, is changed. Finally, the 3D FVM is validated against the numerical mode matching method (NMM), and the characteristics of both the coaxial and coplanar responses of the EM field tool are investigated using the numerical results.