Two of the main objectives in petrophysical log interpretation are to determine geologic bed boundaries and fluid contacts. For these, the induction log has several important properties: it is sensitive to fluid type and distribution in the pore space, the induction measurement is accurately modeled as a convolution of formation conductivity and the tool response function. The first property assures good discrimination of the reserves and at the same time delineates oil-water contacts. This information permits a fundamental zoning of the well log. The second property follows from the quasi-linear relationship between the induction log and formation conductivity. Thus it is possible to use linear system theory, and in particular digital filter design to adaptively deconvolve the original signal. The idea in this work is to produce an algorithm capable of identifying contacts between beds traversed by a borehole, given the apparent conductivity read by an induction tool. To simplify the problem, the formation model is assumed to be a distribution of plane-parallel homogeneous beds. This model corresponds to a rectangular formation conductivity profile. Using the digitized input log, inflexion points are obtained numerically as extrema of first derivatives. This generates a first approximation of the real formation profile. This estimated profile is then convolved with the tool response function giving an estimated apparent conductivity log. A conditioned least-mean-square cost function is defined in terms of the difference between measured and estimated apparent conductivity. Minimizing the cost function yields the bed conductivities. The optimization problem of finding the best rectangular profile for induction data is linear for amplitudes (bed conductivities), but non-linear estimation for bed boundaries. In this case amplitudes are estimated by linear least-squares maintaining fixed contacts. A second pass maintains fixed amplitudes and computes small changes in bed boundaries using a linearized approximation. This processes is iterated to obtain successive refinement until a convergence criteria is satisfied. The algorithm is applied on synthetic and real data showing the robustness of the method.
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