Relaxation time distribution from time domain induced polarization measurements

Abstract

SUMMARY We propose an algorithm for inverting time domain induced polarization data to a relaxation time distribution. The algorithm is based on the (Tikhonov) regularized solution of the 2 nd kind Fredholm integral equation. We test the algorithm on synthetic data, and show its robustness for a noise level, typical of laboratory time domain measurements. We also show that, for the inversion purpose, the time domain data must be obtained with the different current wavelengths. We then test the algorithm on the experimental data recently obtained on brinesaturated medium-grained quartz sand (average grain diameter of 4 × 10 −4 m), and on sand mixtures. For the medium-grained sand, relaxation time distribution contains a main peak at 25 s. Different amounts (3%, 8% and 12%) of fine-grained quartz sand (average grain diameter of 1.12 × 10 −4 m) were added to the medium-grained quartz sand. For the sand mixture, an additional peak is observed in the relaxation time distributions, in the time range from 1.0 to 2.5 s. The magnitude of the second peak increases with the increase of the fine-grained sand content. Therefore, the experimental data show that peaks in the relaxation time distributions are related to the grain size. On the basis of the known peak location, and of the known grain size value, we obtained the values of the diffusion coefficient, which were found to be of the same order of magnitude as those in the bulk solution.