4.3 Subsurface mixing

For the purposes of this discussion, the term (groundwater) ‘mixing’ will refer to the blending of solutions in the subsurface at the molecular scale, i.e., a scale at which mass transfer limitations imposed on chemical reaction rates, even between neighboring pores, can be completely discounted. Historically, the mixing of pollutants and ambient groundwater has been ascribed to a process that hydrogeologists refer to as ‘hydrodynamic dispersion’, or sometimes simply ‘dispersion’. In the absence of dispersion, plumes are transported by the advective process alone, referred to as plug flow (Figure 18), and no mixing with the background water occurs. So, dispersion might simply be regarded as the combination of transport processes that causes plume to depart from plug flow behavior (Anderson, 1984).

Typically, dispersion is quantified in models based on a differential equation called the advection-dispersion equation. Unfortunately, the mixing represented by dispersion in this equation is geared at describing plume shape and extent empirically — as determined by the monitoring network in use — and is not necessarily indicative of mixing in the aquifer at the molecular scale. Confusion over this subtlety has implications for activities that depend on mixing. For example, mixing should increase the volume of an aquifer affected by a pollutant, making plumes easier to locate and delineate for remediation purposes. Complete molecular scale mixing in an aquifer is not an absolute requirement for this outcome, as long as monitoring wells can intercept detectable concentrations of the contaminant. The same may be true for the assessment of first arrival times of pollutants at receptors. Increased mixing — even mixing that is not complete at the molecular scale — leads to a plume front that extends beyond the plug flow front, resulting in pollutants that arrive sooner than predicted by average velocities, i.e., sooner than expected, at receptors (Figure 18). Molecular scale mixing will result in the dilution of dissolved substances, i.e., the uniform lowering of concentrations (mass per unit volume) due solely to replacement of solute mass by water mass in a given volume. Note that mixing at scales larger than the molecular scale may give the appearance of dilution in samples but may actually leave some zones in the subsurface unmixed and others solute-free — the blending of these zones occurs in the well or during sample collection. Therefore, molecular scale mixing is necessary for the dilution of pollutants, which in cases of chemicals that pose low risk to health or ecosystems might mean that humans need not actively respond to the release. Furthermore, as mentioned in the previous section, pollutant degradation by natural attenuation, or via engineered systems, may depend on the presence of dissolved electron acceptors (e.g., dissolved oxygen, nitrate or sulphate), or other reaction-enhancing solutions that humans inject into the subsurface to ameliorate a contamination problem. To be effective, these substances must mix with the polluted water volume at the molecular scale; high rates of mixing lead to the highest degradation rates the chemistry allows. Finally, contaminant plumes do not grow in length indefinitely. The maximum length a plume will grow is determined by contaminant mass loss rates due to radioactive decay or transformations — affected by mixing, as discussed above — and dilution achieved by dispersive mixing (particularly along the plume margins) at molecular scales. Mixing by dispersion depends on variability in both groundwater velocity (Figure 19) and contaminant (or other solute) concentration (Figure 20) (Cherry, 1990).

Plan view of the growth of a simple plume of groundwater contamination
Figure 18 – Plan view of the growth of a simple plume of groundwater contamination with and without the process of dispersion. The plume growing without dispersion illustrates ‘plug flow’ and is purely advective transport. The plume growing with dispersion illustrates the diluting effects dispersion imposes on a plume, as well as extending the volume of aquifer that is contaminated. In the longitudinal direction (the direction of flow), this extension results in slightly earlier arrival times of the pollutant at receptors. These simulations were performed with the same transport parameters as those given in Figure 14 except that the ‘no dispersion’ calculations used dispersivities < 0.001 m, and the ‘dispersion’ calculations used a longitudinal dispersivity of 1 m, a horizontal transverse dispersivity of 0.1 m and a vertical transverse dispersivity of 0.015 m.
Equipotential surface (with water level contours) for a hypothetical homogeneous flow system
Figure 19 – a) Equipotential surface (with water level contours) for a hypothetical homogeneous flow system with hydraulic conductivity of 2.5 m/d and an overall gradient across the domain of 0.001 causing flow from right to left. b) Traces of 12 particles transported for 40 years (porosity assumed to be 0.3). Note all particles travel the same distance in the same direction, indicating a plume in this setting would remain intact and undergo minimal mixing from causes related to groundwater velocity. c) Same flow system as (a) showing the water level surface in three dimensions for the case where the aquifer is heterogeneous with lenses of K ranging from 0.25 (blue) to 25 m/d (red) (shown on the x-y plane). d) The traces of the same 12 particles from (b) released into the heterogeneous flow system shown in (c) for 40 years. Note the variation in particle pathways suggesting considerable plume distortion and splitting, enhancing the conditions that promote mixing at the molecular scale. This effect is solely due to variations in groundwater velocity.
Figures illustrating conceptualization of dispersion
Figure 20 – Historically, dispersion has been visualized as the result of the processes illustrated above. These processes do not directly cause molecular scale mixing, but they can promote it by creating conditions favorable for diffusion. a) Velocity variations may occur within pores, or between pores resulting in enhanced pollutant concentration gradients (i.e., the difference of concentrations at two points divided by the distance between the points, ΔC/Δx) in directions transverse to flow. These gradients drive mixing by diffusion, which can occur over short time scales at the pore scale. b) In addition, the compression of streamlines within some pores increase the transverse concentration gradients by decreasing the magnitude of the Δx in the ΔC/Δx term. c) Variations in velocity at larger scales can also create zones with high concentration gradients. In such zones, diffusion occurs at the maximum rates for those scales, and mixing is promoted. Note: the diffusion profile shown is provided to illustrate the tendency for pollutant mass to mix with surrounding groundwater by diffusion. A fully developed diffusion profile such as the one shown would be repeatedly disrupted in a transient flow system, or as a plume evolved, and never be achieved. Velocity variations at the pore scale are important over centimeter lengths or less. Velocity variations at the centimeter scale or higher may be of practical concern for characterization and remedial design purposes.

The link between these two factors begins when a plume distorts, splits or fragments over time due to variations in velocity (e.g., Figure 19c and d) arising from such causes as aquifer heterogeneity, seepage to the ground surface (e.g., seeps, streams, lakes), or pumping. This phenomenon can occur at any scale where the flow variations occur. Once the plume has been deformed in this way, sharp concentration gradients can develop within and around the plume perimeter, leading to enhanced diffusive mixing between the plume and the ambient groundwater (Figure 20). Since diffusive mixing occurs at the molecular scale, any velocity variations that promote it are also drivers of the micro-scale mixing process. This view of mixing in the subsurface is sometimes referred to as the advection-diffusion mechanism.

In general, the geological variations that lead to flow variability and molecular scale mixing occur at scales smaller than the measurement scales used to characterize flow systems, making predictions of mixing rates a challenge. Recent research is re-examining the way dispersion is handled in the advection-dispersion equation and re-assessing the nature of the link between groundwater velocity and mixing rates. Nevertheless, the link itself is not in question. Therefore, the issues, discussed above, that make subsurface mixing important also make groundwater velocity measurement important, at the appropriate scale.


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