2.5-D modeling of cross-hole electromagnetic measurement by finite element method

A finite element method is developed for simulating frequency domain electromagnetic responses due to a dipole source in the 2-D conductive structures. Computing costs are considerably minimized by reducing the full three-dimensional problem to a series of two-dimensional problems. This is accomplished by transforming the problem into y-wave number (K y ) domain using Fourier transform and the y-axis is parallel to the structural strike. In the K y domain, two coupled partial differential equations for magnetic field Hy and electric field Ey are derived. For a specific value of K y , the coupled equations are solved by the finite element method with isoparametric elements in the x-z plane. Application of the inverse Fourier transform to the K y domain provides the electric and magnetic fields in real space. The equations derived can be applied to general complex two-dimensional structures containing either electric or magnetic dipole source in any direction. In the modeling of the electromagnetic measurement, we adopted a pseudo-delta function to distribute the dipole source current and circumvent the problem of singularity at the source point. Moreover, the suggested method used isoparametric finite elements to accommodate the complex subsurface formation. For the large scale linear system derived from the discretization of the Maxwell’s equations, several iterative solvers were used and compared to select the optimal one. A quantitative test of accuracy was presented which compared the finite element results with analytic solutions for a dipole source in homogeneous space for different ranges and different wave numbers K y . to validate the code and check its effectiveness. In addition, we addressed the effects of the distribution range τ of the pseudo-delta function on the numerical results in homogeneous medium.