Numerical modelling of the IP effect at the pore scale

Geophysicists have tried for a long time to correlate induced polarization data with parameters of the pore geometry, often with the overall aim to estimate the important parameter of hydraulic conductivity. However, no correlation has been found so far that is applicable to more than just a few special cases. Using empirical relationships and equivalent circuits often neglects the description of the processes in the pore space. One reason is that the mechanisms controlling the low-frequency polarization are still not completely understood. Only a few existing models try to explain the processes and derive relationships with geometry and most models need strong assumptions. Here, we aim at a deeper understanding by numerical modelling of the main processes responsible for the IP effect. Our approach is based on a 1D solution of the late 1950s that gives an expression for the maximum frequency effect for simplified geometries. The models depend on the lengths of active and passive zones and the corresponding ion mobilities in each zone for anions and cations respectively. The theory describes the ion diffusion along concentration gradients, the influence of an external electric field and the coupling between the two. We first verify our numerical results by comparison with analytical solutions and then extend the approach to flexible geometry, including higher dimension. This constitutes a considerable progress from I D models restricted to two alternating media with fixed lengths. Apart from arbitrary geometry, we can also simulate the full spectral behaviour. A relatively simple model is able to explain frequency-dependent magnitude and phase behaviour that is typically measured in the field. The Cole-Cole model can be considered as the result of a network of pores of varying lengths. From our modelling studies, we derive scaling laws for mobility and geometry and suggest an empirical equation for the relationship between the length of passive pores and the time of maximum phase shift.