Comparison of Fickian and temporally nonlocal transport theories over many scales in an exhaustively sampled sandstone slab

[1] It is not a compelling argument, solely on the basis of a better fit to solute breakthrough curve (BTC) data, that a temporally nonlocal model is necessary to simulate transport in an advection-dominated system. One may counter that the classical advection-dispersion equation (ADE) is a valid model at some small scale and that the detailed hydraulic conductivity (K) data must be well-represented: then the nonlocality is only a result of upscaling and loss of information. But is the nonlocal model demonstrably necessary at all scales? We examine the experiment conducted by Klise et al. (2008) in which a 30.5 × 30.5 cm slab of relatively homogeneous, cross-bedded sandstone was exhaustively sampled for K. The slab was sealed, saturated with potassium iodide, and X-rayed 10 times while being flushed with fresh water. The 8,649 air-permeameter measurements were down- and upscaled to make finer and coarser grids on which the velocity field was solved and the ADE applied. The optimized parameters in the ADE were found to scale predictably, most notably the longitudinal dispersivity (), which grew linearly with upscaling. But at all levels of up- and downscaling, including the original K measurement scale of 0.33 cm, the ADE did not adequately represent the late-time tails. The temporally nonlocal, time-fractional ADE (t-FADE) was applied and the optimized parameters ( and the immobile capacity ) did not depend on scale. The better fit provided by the t-FADE in the late BTC tails did not bring about a sacrificed fit elsewhere in the BTC. Furthermore, the optimized ADE and t-FADE solutions do not converge at the smallest scale, directly implying that the temporal nonlocality is a necessary model component. We conclude that the logical inference “if the ADE is valid in heterogeneous material, then there is tailing in the BTC” is not a proof that the reverse is true. We provide a clear counterexample. A corollary is that a mismatch between data and a discretized solution to the ADE does not imply that more data will improve fits or predictive ability.