Markov-chain Monte Carlo estimation of distributed Debye relaxations in spectral induced polarization

Keery, John and Binley, Andrew and Elshenawy, Ahmed and Clifford, Jeremy (2012) Markov-chain Monte Carlo estimation of distributed Debye relaxations in spectral induced polarization. Geophysics, 77 (2). E159-E170. ISSN 0016-8033

Abstract

There is growing interest in the link between electrical polarization and physical properties of geologic porous media. In particular, spectral characteristics may be controlled by the same pore geometric properties that influence fluid permeability of such media. Various models have been proposed to describe the spectral-induced-polarization (SIP) response of permeable rocks, and the links between these models and hydraulic properties have been explored, albeit empirically. Computation of the uncertainties in the parameters of such electrical models is essential for effective use of these relationships. The formulation of an electrical dispersion model in terms of a distribution of relaxation times and associated chargeabilities has been demonstrated to be an effective generalized approach; however, thus far, such an approach has only been considered in a deterministic framework. Here, we formulate a spectral model based on a distribution of polarizations. By using a simple polynomial descriptor of such a distribution, we are able to cast the model in a stochastic manner and solve it using a Markov-chain Monte Carlo (McMC) sampler, thus allowing the computation of model-parameter uncertainties. We apply the model to synthetic data and demonstrate that the stochastic method can provide posterior distributions of model parameters with narrow bounds around the true values when little or no noise is added to the synthetic data, with posterior distributions that broaden with increasing noise. We also apply our model to experimental measurements of six sandstone samples and compare physical properties of a number of samples of porous media with stochastic estimates of characteristic relaxation times. We demonstrate the utility of our method on electrical spectra with different response characteristics and show that a single metric of relaxation time for the SIP response is not sufficient to provide clear insight into the physical characteristics of a sample.