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

Abstract

[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 Â 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 ( L ), 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, timefractional ADE (t-FADE) was applied and the optimized parameters ( L 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.

Hydraulic conductivity fields: Gaussian or not?

“…The same value of d was effective in removing correlation in all four vertical HRK columns 111108A, 111108B, 111108C, and 121108A (see Figure 1) that formed the basis for our study. A fractional difference filter of order d = 0.89 was used in Lu et al [2002] to remove correlations in laboratory ln K data from a vertical sandstone core; a value of d = 0.9 was found suitable for a sandstone slab in Major et al [2011]; Meerschaert et al [2004] used d = 0.74 for lower resolution ln K values from three horizontal profiles in a sandstone facies at a site in Utah.…”

Final Report

“…This was used to analyze the data collected by Klise [15]. The details of the conditioning have only recently appeared [4], but allowed us to rigorously test the sufficiency of the classical, Fickian advection dispersion equation [16]. In that paper, we challenged the notion that, based solely on a better fit to solute breakthrough data, a temporally non-local model is necessary for transport in an advection-dominated system.…”

“…• Performed a quantitative analysis of the veracity and information requirements of several nonlocal versus local transport theories [27,28,29,30,16].…”

Fractional calculus in hydrologic modeling: A numerical perspective

“…This gives a simple method to simulate the fractal multi-rate mass transfer process in a particle tracking routine [54,58]: The mobile times are exponential with mean 1/( βλ ), where in this case λ is a tuning parameter to ensure enough transitions between mobile and immobile by the time of interest. If the particles’ locations are desired at some time t , choose λ so that the mean mobile step is approximately one tenth of this: λ > 10/( tβ ).…”