A Model to Identify the Source for Contaminants Undergoing Non-Fickian Diffusion in Water.

[NNadir]

Last edited Sat Aug 6, 2022, 11:12 AM - Edit history (2)

Some time ago, in a rather long and perhaps boring post at DU, I discussed the diffusion of certain radionuclides, in particular the mobile element technetium, at the big, bad anti-nuke boogeyman at Hanford, Washington where I speculated on the release of the "missing" technetium into the Columbia River: 828 Underground Nuclear Tests, Plutonium Migration in Nevada, Dunning, Kruger, Strawmen, and Tunnels

I wrote:

I suppose one could go crazy doing calculations about this situation considering Fick's law of diffusion, or modifications of it to get diffusion advection equations in the case of flow. One could even go crazier, arguing all day whether a Stokes-Einstein formulation of the diffusion constant is appropriate with or without Langmuir or Freundlich corrections apply.

To be perfectly honest, I am not competent, to "go crazy" in this particular way. I have been peripherally involved in projects involving the determination of Langmuir constants in certain biological models, but the operative word is "peripherally."

Well, it appears that competent researchers, one at the University of Alabama, another at University of Arizona, a third at the University of Colorado, and a fourth at the Normal University in Nanjing, China have gone crazy doing this, as I came across this paper this morning: General Backward Model to Identify the Source for Contaminants Undergoing Non-Fickian Diffusion in Water, Yong Zhang, Mark L. Brusseau, Roseanna M. Neupauer, and Wei Wei Environmental Science & Technology 2022 56 (15), 10743-10753.

The paper contains a very beautiful, long, run on sentence which I will reproduce as a graphic object, with all kinds of references to mathematics.

Fick's law, at least in the simplest one dimensional case, is a straight forward differential equation based on concentration gradients.

First an excerpt from the paper giving a general statement of the problem:

Pollutant source identification (PSI) in surface and subsurface water focuses on using “results” (i.e., observed pollutant distributions) to find “causes” (such as the pollutant source location or release time). (1,2) PSI has remained a research topic and has been applied extensively in hydrology for four decades, including delineation of groundwater protection zones, (3,4) identification of responsible parties, (5,6) assessment of aquifer vulnerability, (7) recovery of the contaminant history, (8) calculation of groundwater ages, (9,10) and identification of pollutant sources in water (11,12) or soil. (13,14) Source-identification problems have also been popular in other disciplines related to water and environments, such as oceanic sciences where backward-in-time models were used to backtrack moving sea ice, ocean plankton, oil slicks, and marine debris, (15−17) atmospheric sciences where the models were used to track the source for airborne pollutants, (18,19) and other applications such as to track heat conduction or fish sources. (20,21)

PSI usually requires quantitative analyses, which involve chemical techniques (such as isotope signatures (22,23) and molecular markers (24,25) applicable for specific contaminants), statistical analysis, (26,27) or process-based physical/mathematical techniques (such as forward- or backward-in-time modeling of dissolved contaminant transport). The physical approach can incorporate the complex impact of geological media properties on PSI at various spatiotemporal scales, but it is computationally demanding and remains one fundamental challenge in environmental sciences (reviewed below). This study aims to quantify PSI using a generalized, computationally efficient mathematical model for contaminant transport in hydrologic systems with intrinsic physical heterogeneity by addressing the following three knowledge gaps in PSI.

First, PSI theories and applications mainly focused on pollutants undergoing Fickian diffusion, (1) while real-world pollutant transport in surface and subsurface water has been increasingly documented to be non-Fickian, characterized by the nonlinear growth of plume variance in time. (28) Non-Fickian diffusion generates pollutant plumes that are not uniformly distributed about the center of mass, with leading and/or trailing edges whose mass is orders of magnitude larger than that for Fickian diffusion. This discrepancy has important implications for water resources protection and remediation, challenging the Fickian diffusion-based PSI. For example, Zhang et al. (7) found that superdiffusion (where the plume variance increases faster than linear in time) must be accounted for in aquifers with preferential flow paths or rivers with turbulent flow when identifying the pollutant source position, since the standard Fickian diffusive models such as the classical advection–dispersion equation (ADE) cannot track the highly skewed probability density function (PDF) describing the pollutants’ initial locations. Neupauer and Wilson (29) developed the backward, single-rate mobile–immobile model to calculate the backward location PDF (which describes the possible location(s) of the pollutant source) for sorbing solutes in groundwater. To the best of our knowledge, these are the only works that have quantified the impact of non-Fickian transport on PSI using non-Fickian diffusive models with an upscaled, uniform velocity. We will expand upon them by considering a broader range of non-Fickian diffusion for dissolved contaminants in various hydrologic systems (i.e., rivers, soil, and aquifers).

Second, PSI has known mathematical challenges when using inverse or backward modeling. Inverse tools and backward models have been the two major physical methods in implementing PSI over the last four decades, each of which contains unsolved mathematical issues. The commonly used inverse tools can calibrate multiple unknown variables, including properties of pollutant sources (such as the location, number, mass, or release history of initial sources) and aquifer information (such as hydraulic conductivity), by repeatedly solving the forward-in-time model. (30) However, inverse problems are often ill-posed, (31,32) and the repeated simulation of forward-in-time models may sometimes be computationally prohibitive. Backward models, which backtrack pollutants by reversing the flow field, can directly identify a contaminant source; (33,34) hence, backward models do not suffer from the issues of solution existence, nonuniqueness, and instability...

Later the authors offer their approach:

...We propose the following tempered (meaning “truncation” of extremely long jump sizes) fractional-divergence (representing a fractional-order vector operator; see further explanation in Section S.1.4) advection–dispersion equation (TFD-ADE) with source/sink terms and chemical reactions (i.e., first-order decay) to model retention and superdiffusion for pollutants in aquatic systems at various scales...

The graphics object with the wonderful run on sentence, actually two beautiful run on sentences:




Another graphics object, table 1, with boundary conditions:



Yet another table, with some very justified bragging about the strength of the author's approach to this problem:


I note that this approach considers a case I made in my discussion of Hanford, linked above. Specifically it addresses the case I made with respect to the interaction of pertechnate with iron to form insoluble TcO2.

Of course, the rattle and gasps associated with Hanford by anti-nukes is based on the dangerous and ignorant assumption that the release of any amount of radioactivity justifies the rote acceptance of tens of millions of deaths every decade from air pollution. One sees this obscene and immoral claim in many settings, generally connected with things like Fukushima, Chernobyl and even - more obscenely - Three Mile Island.

The cost of the "clean up" of Three Mile Island in particular is patently absurd, since the "clean up" to a standard that almost no one applies to any other pollutant - the requirement that even badly educated people cannot imagine any risk to anyone at anytime - to will save very few lives, because very few lives are at risk. I personally drove through Harrisburg a few days back. It's still there, with lots and lots and lots of people living useful, productive, and healthy lives.

The fear and ignorance associated with this murderous risk fantasy is connected with the Linear No Threshold (LNT) Assumption, an assumption which has, a best, very, very, very, very weak scientific justification, the obsession with which is killing people on a vast scale, and in fact, killing the planet at an accelerating rate. In fact even if the the LNT assumption were justified - and I'm convinced it isn't based on some modern work involving molecular biology involving enzymatic analysis of DNA protein interactions - one doesn't need to have the mathematical sophistication of the fine scientists who authored this paper, to understand that the risk is almost vanishingly small in comparison to the risk of burning dangerous fossil fuels and dumping the waste directly into the planetary atmosphere.

The planet is burning. How about we wake up?

Have a pleasant weekend.

[2naSalit]

1. K&R