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

Journal Article · Sat Oct 01 00:00:00 EDT 2011 · Water Resources Research

DOI:https://doi.org/10.1029/2011WR010857· OSTI ID:1051542

Major, Elizabeth ^[1]; Benson, David A. ^[2]; Revielle, Jordan ^[2]; Ibrahim, Hamed ^[2]; Dean, Arianne ^[2]; Maxwell, Reed M. ^[2]; Poeter, Eileen ^[2]; Dogan, Mine ^[3]

---

1. Hydrologic Science and Engineering, Colorado School of Mines, Golden, Colorado, USA; Colorado School of Mines
2. Hydrologic Science and Engineering, Colorado School of Mines, Golden, Colorado, USA
3. Department of Geological Sciences, Michigan State University, East Lansing, Michigan, USA

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 by 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 (α_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, time- fractional ADE (t-FADE) was applied and the optimized parameters (α_L and the immobile capacity beta) 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.

• Research Organization: Colorado School of Mines, Golden, CO (United States)
• Sponsoring Organization: USDOE SC Office of Basic Energy Sciences (SC-22)
• DOE Contract Number: FG02-07ER15841
• OSTI ID: 1051542
• Report Number(s): DOE/ER/15841-11
• Journal Information: Water Resources Research, Journal Name: Water Resources Research Journal Issue: 10 Vol. 47; ISSN 0043-1397
• Publisher: American Geophysical Union (AGU)
• Country of Publication: United States
• Language: English

Related Subjects

58 GEOSCIENCES