Tomography of the Cross-Hole Electromagnetic Measurements

The regularized least-squares inversion scheme is used to reconstruct the formation resistivity within boreholes from the tomography of the cross-hole electromagnetic measurements. In the tomography, a two-dimensional conductivity variation profile with point dipole sources and receivers in it are assumed. The forward modeling was fulfilled by the iso-parametric finite element method in wave-number domain, and the 3-D problem was converted to a series of 2-D problems in K_y wave-number domain, thus make it possible to deal with the practical model. For the requisite number of K_y values, the coupled equations are solved by the finite element method on the iso-parametric element on the xoz plane. In order to improve the accuracy of the numeric solution, Gauss integral was applied to the wave-number numeric solution in the inverse Fourier transformation of the wavenumber domain solution to real space domain. The sensitivities of the electromagnetic measurements to the formation conductivity are effectively calculated by using of the reciprocity principle of the electromagnetic fields in source locations and receiver locations. Our method validated by using of the simulation data and field data. For the singularity of the source point in modeling of cross-hole electromagnetic measurement, a pseudo-delta function was adopted to distribute the dipole source current, and the numeric accuracy is enhanced. A quantitative test of accuracy is presented which compares the numeric results with the analytic solutions for a dipole source in layer medium for different wave number K_y to validate the code. Through the perturbation of the conductivity of a single block, the difference between two simulations was compared with that from the reciprocity principle, and the effectiveness has been verified. The sound tomography results of the simple block, slant fracture zones and the big character model, verify the effectiveness of the tomography procedures.