Determination of permeability from spectral induced polarization in granular media

TLDR

Surface conductivity of porous rocks arises from a frequency‑independent diffuse layer and a frequency‑dependent Stern layer, the latter dominating spectral induced polarization in granular media between 1 mHz and 100 Hz. The authors link the Stern‑layer contribution to macroscopic complex conductivity by convolving the inverse‑grain‑size distribution with a uniform‑size response, deriving an equation that relates effective pore size, electrical formation factor, and the expected inverse grain size, thereby connecting permeability to the relaxation‑time distribution. The derived relationships agree with existing formulas for delta and log‑normal grain‑size distributions and experimental data, but limitations arise because the phase relaxation of small grains is obscured by Maxwell–Wagner polarization above 100 Hz and cemented aggregates mimic coarser grains.

Abstract

The surface conductivity of porous rocks has two contributions: the first is associated with the diffuse layer coating the grains and is frequency-independent as long as the diffuse layer is above a percolation threshold. The second contribution is associated with the Stern layer of weakly sorbed counterions on the mineral surface and is frequency-dependent if the Stern layer is discontinuous at the scale of the representative elementary volume. In the frequency range 1 mHz–100 Hz, this second contribution is also associated with the main polarization mechanism observed by the spectral induced polarization method in granular media (neglecting the contribution of other polarization processes like those associated with redox processes and membrane polarization). At the macroscale, we connect the Stern layer contribution to the complex conductivity and to the expectation of the probability distribution of the inverse of the grain size. This is done by performing a convolution between the probability distribution of the inverse of the grain size and the surface conductivity response obtained when all the grains have the same size. Surface conductivity at the macroscopic scale is also connected to an effective pore size used to characterize permeability. From these relationships, a new equation is derived connecting this effective pore size, the electrical formation factor, and the expected value of the probability distribution for the inverse of the grain size, which is in turn related to the distribution of the relaxation times. These new relationships are consistent with various formula derived in the literature in the limit where the grain size distribution is given by the delta function or a log normal distribution and agree fairly well with various experimental data showing also some limitations of the induced polarization method to infer permeability. One of these limitations is the difficulty to detect the relaxation, in the phase, associated with the smaller grains, as this polarization may be hidden by the Maxwell—Wagner polarization at relatively high frequencies (>100 Hz). Also, cemented aggregates of grains can behave as coarser grains.