Laboratory Observations of Non-Fickian Transport: The Effects of a Cutoff Time

The transport process in the framework of a continuous time random walk (CTRW) is portrayed as a sequence of transitions with displacements s and times t. A central focus of the CTRW approach is an accurate physical model of the entire spectrum ψ ( s, t) (or in the uncoupled case p( s)ψ(t)). Observations of non-Fickian transport in sandbox experiments1 have been analyzed using a power-law tail ψ(t)~ t-1- β with 0<β<2. For each sandbox medium a choice of β results in an excellent fit to the breakthrough curve (BTC) data. However, the value of β slowly increases with decreasing flow velocity. This is consistent with the shape of the full spectrum of ψ(t) gleaned from analytic calculations2, numerical simulations3 and permeability fields4. We represent the main features of this complete form with a truncated power-law (TPL), ψ(t)~ (t1 + t)-1-β \exp(-t/t2), where t1 and t2 are the limits of the power-law spectrum. An excellent fit to the entire BTC data set (including the changes in flow velocity) for each sandbox medium is accomplished with a single set of values of t1,β,t2. The use of the full spectrum of ψ(t) is not only necessary for the transition to Fickian behavior2 but to account for the dynamics of these laboratory observations of non-Fickian transport, especially the important role of the cutoff time t2. 1Levy, M. & B. Berkowitz, J.Cont.Hydrol. 64, 2003. 2Cortis, A., Y. Chen, H. Scher & B. Berkowitz, Phys. Rev. E 70, 2004. 3Bijeljic, B. & M. J. Blunt, Water Resour. Res. 42, 2006. 4Di Donato, G., E.O. Obi & M.J. Blunt, Geophys. Res. Lett. 30, 2003.