Why Fickian Transport Is so Rare

Abstract:

It should come as no surprise that transport in porous media is rarely, if ever, Fickian (Brownian with linear clock). In a Lagrangian setting for the particle trajectory, Fickian has been assumed synonymous with linear in time mean square displacement (msd), while in an Eulerian setting for the concentration, it’s synonymous with a constant diffusion coefficient in Fick’s law. For a Fickian process the two settings are related; the diffusion coefficient is proportional to the coefficient that relates the msd to time. A Fickian process must be continuous with stationary, independent and Gaussian increments. The biggest weakness of the Fickian model is the assumption of independent increments, followed by stationarity and to a lesser extent the assumption of Gaussian increments. The conductivity field in most geological media is correlated over space, many times over a multitude of scales. The conductivity is related to the velocity through Darcy’s law and that in turn is the derivative of the position process, so why would one expect increments in the position process to be independent? Moreover, the conductivity is often nonstationary or has nonstationary increments so again why would one expect the position process to have stationary increments. We present a multitude of examples with correlated and/or nonstationary and/or non-Gaussian increments to illustrate problems with the Fickian assumption and the Fickian classification scheme.