Multi-exponential inversion of induced polarization relaxation signal of shaly sand

Abstract

The induced polarization (IP) decay curve of natural shaly sand can be modeled as a weighted superposition of exponential relaxations with different relaxation time constant. The IP relaxation time spectrum, which is defined as plot of weight versus the relaxation time constant, has been previously demonstrated as a significant tool for capillary pressure curve, pore size distribution and permeability of reservoir. Earlier works successfully used singular value decomposition (SVD) method to extract the relaxation time spectra from the decay data. However, those works were obtained from the measured data with very high signal to noise ratio (SNR). The developed algorithm is suitable for these data. But for the practice use downhole, the obtained decay data have low SNR and then the obtained spectra from these decay data using the algorithm may be unstable and invalid. In this work, the method of regularization is applied to extracting continuous IP relaxation time spectra from decay data. The regularization operator is a unit matrix. To illustrate the influences of regularization parameters on the inversion of IP relaxation time spectra, the used decay data is generated from starting relaxation time spectra. Varying levels of random noise are added to these data sets and the IP relaxation time spectra are calculated for comparison to the starting distribution. Results show that for the data contains noise, the obtained spectra of the simulated decay data become smoother with increasing the regularization parameter. There exists an optimum regularization parameter which can be used to get the most reasonable spectrum. The logarithmical optimum parameter decreases linearly with increasing the logarithmical SNR. The developed algorithm and the prediction of the optimum parameter are very reasonable for the real rock sample. The results also show that the best number of relaxation distribution points ranges from 16 to 64.