In My Experience: The Lessons from Dispersion—Don't Believe Everything You Read

This note is about my experience investigating and thinking about hydrodynamic dispersion, simply known as dispersion. In keeping with the tried and true practice of academic reductionism, I will constrain the discussion to the relatively simple and idealized case of point-source groundwater plumes in granular deposits with the aim of illustrating some persistent myths and insights about this process, which has attracted so much interest over the past several decades. Dispersion is the most enigmatic topic in hydrogeology and for good reason.

To most hydrogeologists, dispersion refers to mixing of solute concentration during transport of the solutes. It occurs at the plume periphery and, in some cases, internally within the plume. The partial differential equation (the advection-dispersion equation… the ADE), on which nearly all mathematical models (analytical and numerical) for representing solute transport and fate in groundwater are based, does not represent fundamentally the way in which advection and mixing happen in the field. This equation can be useful when simulating bulk plume spreading at a specified point in time but is deeply flawed for the intended purpose of representing plume mixing with background groundwater at the field scale. Although, when used carefully, these models can capture the nature of a contaminant plume for particular purposes at a moment in time, they cannot correctly represent the actual processes that govern the evolution of plumes, in the past or future.

The derivation of the ADE is founded on assumptions that cannot be expected to be met for heterogeneous geologic media and all such media in this context are substantially heterogeneous in their hydraulic conductivity distribution. Solute transport is dependent on the velocity field at a fine spatial scale that can be as small as millimeters or centimeters where molecular diffusion, caused by the local concentration gradients, is the process driving mixing with groundwater at lower solute concentration. This scale is too small for measuring the full velocity field and would require frequent temporal measurements for realistic assessment. Hence, the dispersivity values used in advection-dispersion models are bulk black-box parameters intended to capture the overall effects of the media heterogeneity. But, when viewed in the context of the spatial and temporal scale diffusive mixing, there are important deviations from the assumptions inherent in the derivation of the ADE. This is a serious matter because a common reason to use mathematical models is for realistic representation of the past as a basis for predicting the future. Moreover, the failures of the ADE to accurately represent the processes, limit model usefulness in exploring parameter interdependencies and sensitivities. In essence, what is published in textbooks about these for “transport and fate” models is, at best, misleading and at worst simply wrong. This has been perpetuated from textbook to textbook, starting with the Freeze & Cherry GROUNDWATER, 1979.

A key parameter in the ADE is the dispersion coefficient that contains a few parameters, including one known as dispersivity with the dimension of length. In the derivation of the ADE, dispersivity is assumed to be a fixed property of the geologic medium, like hydraulic conductivity and porosity. Values of dispersivity are obtained by fitting ADE models to tracer test data or actual plumes. However, these fitting values increase with the time and distance traveled by the solute cloud, implying ever increasing rates of mixing—recall that in the simple case under consideration here, accelerated mixing with time and distance traveled is beyond justification. The reason for this outcome derives from the way in which the diffusion coefficient is represented in the ADE. In other words, diffusion is incorrectly incorporated in the ADE for heterogeneous granular media, even for minimal spatial variability of grain size layers and lenses for which there are only weak variations in the hydraulic conductivity distribution.

I regret that it has taken me 43 years to get around to writing this warning about the established nonsense written about dispersion in the literature; unfortunately, this nonsense only became evident to me with the great advantage of hindsight. I am thankful that this journal has provided me with the opportunity to unburden myself by correcting this long-standing erroneous thinking that became so embedded in the literature. However, as I try to set the record straight concerning the problems of the ADE, I wish to be clear that I do not advocate its abandonment. The ADE models can be useful and even necessary given lack of practical alternatives, if used with judgment and care. Ironically, I believe that they likely should be used much more than they are. But that tangent is the subject for another essay.

So how did the ADE become so predominant in groundwater education? When groundwater contamination was first recognized as a topic worthy of attention in the 1960's, the research community conducted laboratory tracer tests in columns and boxes packed with homogeneous sand or glass beads. A mathematical model was needed to simulate the laboratory results, otherwise the experimental results could have no generality. The ADE was imported from the chemical engineering literature and it was found that analytical solutions fitted the laboratory data well enough to support adoption of the ADE in contaminant hydrogeology. There were no competing choices. Soon, the literature provided many analytical solutions to the ADE in 1, 2 and 3 dimensions.

Dispersion in point-source plumes is typically considered in three directions, first in the direction of flow, known as longitudinal dispersion, and in the two principal directions transverse to the longitudinal flow, known fittingly as transverse dispersion. Advection refers to the transport of the solute with the bulk motion of the water. For clarity and ease of discussion, we will limit this note to the circumstances wherein there is a contaminant plume migrating primarily in the horizontal direction so that longitudinal dispersion refers to dispersion in the horizontal direction. Vertical transverse dispersion occurs at the top and bottom of the plume and horizontal transverse dispersion occurs at the sides of the plume. Dispersion operates most importantly at the periphery of plumes but becomes increasingly important in the interior of plumes as the simple homogeneous case (but unrealistic in the field) gives way to plumes in “real” heterogeneous aquifers.

My research on contaminant transport and fate began in 1968, focusing on field investigations mostly in sandy deposits. First, I investigated landfill and radionuclide plumes where high-resolution plume characterization methods were first developed and applied. This was early in the transport game. The first paper in a peer review journal about groundwater plumes appeared in 1966 (Water Resources Research, WRR). In1977, colleagues and I began to conduct natural gradient field tracer tests in the Borden aquifer (Sudicky et al., 1978), which is a weakly heterogeneous sand deposit. These tests were found to be essential in the development of understanding solute transport and especially dispersion. They are labor and sample analysis intensive and require much time for the tracer to travel far enough for diffusion to result in meaningful results, which is why such tests had not been done earlier. We initiated more of these tracer tests (e.g., Mackay et al. 1985, WRR) because of claims in the literature that showed a surprisingly large range of transverse dispersivity values. Also, questions arose because the early numerical models used to simulate plumes, without exception, showed strong transverse spreading when actual plumes characterized by high resolution measurements showed the opposite. Hence, the published record for the effects of transverse dispersion had contradictions between field observations and numerical simulations.

While the first Borden experiment was in progress in 1977-1978 (Figure 1), I was writing the chapter on groundwater contamination in the Freeze & Cherry book. In retrospect, the dispersion discussion in the transport chapter is scientifically schizophrenic in that two conflicting lines of evidence are presented in the chapter's figures without recognition that any conflict is there. One, based on high resolution sampling, shows the weak strength of transverse vertical dispersion (small vertical spreading) of the landfill plume at the Borden site. On the other hand, there is a figure also in this chapter, produced by numerical simulations, for a longitudinal cross section through a hypothetical sand aquifer that shows strong vertical transverse dispersion (large vertical spreading entirely unlike the actual Borden plume; Figure 1). This simulation was done using one of the earliest numerical models for solute transport. Like all the early numerical models for transport, it suffered from strong numerical dispersion. Revisiting these figures, it is apparent to me now that both represented plumes misleadingly because both suggested what appeared to be strong transverse vertical dispersion. As I was writing the Freeze-Cherry chapter, the results of the first Borden tracer experiment arrived. They showed weak transverse dispersion, as was the case for the nearby landfill plume, but the significance of this did not adequately register in my mind. I continued to think of plumes as they were depicted in the first numerical models of solute transport, such as one that appeared in a paper in 1973 (Pinder, WRR) in which the simulations were made to fit a chromium plume mapped using only low-resolution methods (i.e., conventional monitoring wells). Ironically, in this case the erroneous simulations based on numerical dispersion fitted the erroneous field plume based on the blended samples typical of conventional monitoring wells where the water intake interval is too long. These two wrongs resulted in an even bigger wrong in conceptualizing plumes.

I now recognize that, at the time, there were extreme differences between high-resolution field plumes and model plumes and that the field evidence overwhelmingly supported weak transverse dispersion in sandy aquifers of weak to moderate heterogeneity. I recall that euphoria abounded in contaminant hydrogeology in that era due to the great possibilities of the relatively powerful computers of the time. This apparently clouded my perception of what the conflicting data should have been telling me. It was not until 1994 that Hoekanan and Fried (WRR) properly simulated the Borden landfill plume in 3D without appreciable numerical dispersion. Misunderstandings about dispersion were promulgated further in the 1980-1990 era when graphs appeared in the literature indicating that dispersivity values showed longitudinal dispersivity directly related to the distance traveled by the plume and most importantly these papers also claimed a clear relationship between the magnitude of longitudinal dispersivity and the transverse dispersivity values. These graphs are most reasonably attributed to artifacts arising from the blending of water drawn into monitoring wells that connect zones with different solute concentrations (low-resolution sampling) and the inclusion of results from different hydrogeologic circumstances (apples and oranges). Unfortunately, this relationship between the longitudinal and transverse dispersivity values has been reproduced in many textbooks. It represents the ultimate misleading conceptualization about dispersion. Unfortunately, this perception is widely held by practitioners. Instead, reasonable dispersivity values can be useful in some cases for contaminated site simulations when selected based on what has been learned cumulatively based on plumes characterized since the late 1950's using high-resolution methods and, after 1978, based on natural gradient tracer tests.

Not long after the publication of the Freeze & Cherry book (1979), we pointed out (Gillham and Cherry, GSA Special Vol., 1982) that the ADE representation of transport was flawed. We presented an alternative conceptual model, referred to as the advection-diffusion model. In this conceptualization, if we were to know the velocity distribution in enough detail, what passes for dispersion (i.e., mixing) in the ADE model is explained for mobile solutes based on advection combined only with solute diffusion. This thinking was to be the start of what could have been a paradigm shift for dispersion but the shift did not evolve to maturity. The literature supporting the advection-diffusion view grew by a few papers but was largely ignored, perhaps because there are disadvantages associated with such a paradigm shift. For example, acceptance of it requires site characterization at spatial scales much finer than what has become “standard practice.” Also, there is continuing convenience to the belief in strong transverse dispersion in the use of ADE models because this makes numerical simulations easier. Large dispersivity values stabilize the model numerics and the simulations produce wide, fan-shaped rather than narrow pencil-shaped plumes. Furthermore, the perceived fan-shaped plumes are much less expensive to monitor because they do not require closely spaced monitoring points for plume detection. For example, for a landfill with a leaky patch of liner that creates a 5-m wide contaminant input to a sand aquifer with steady flow, the simulated plume width at a distance of 100 m downgradient is only about 6-8 m wide when reasonably small lateral transverse dispersivity values are used. However, the lateral plume spread (width) is 20 m or even much more when the much larger values in common in use in plume modeling are assigned. Conveniently, plumes that go under-detected are less expensive to deal with in the regulatory realm, at least in the short term. In this case and many others, longitudinal dispersion is irrelevant. It is the transverse dispersion that matters.

In summary, in the conceptual and mathematical representations of plumes, dispersion should not be ignored when a particular understanding of contaminant plumes is needed. An example is when monitored natural attention (MNA) is proposed because mixing can govern rates of biodegradation. But as a parameter in models for predicting the behavior of plumes into the future, dispersivity is problematic because the values cannot be measured in any practical way at the appropriate spatial and temporal scales. The values must be assigned based on judgment derived from comprehensive knowledge of the site in the context of relevant dispersion literature.

When new or altered numerical models for transport are to be used, verification with analytical solutions that incorporate the appropriate processes and/or field data are essential to avoid excessive misrepresentations of reality. A practical challenge in the assessment of contamination plumes is the decision to apply ADE models or simply rely on assessment of the groundwater flow system, which may be all that is needed for the circumstance.

The overall lesson I have taken away from this, and a few other experiences during my career, is that there are concepts published in books that, although they appear to be valid at face value, are nonetheless wrong. We must maintain inquiring minds and be vigilant to recognize when simulations deviate from the field evidence and be willing to adjust the written record and academic training accordingly. Concerning dispersion, more thinking is still needed.