4.1 Rate Constants

Transformation rate constants are essential for the assessment of natural attenuation (an aquifer’s ability to reduce a plume of contamination without human intervention) as a strategy for site reclamation. For example, in some jurisdictions, the Risk Based Corrective Action (RBCA) approach is used for defining the level to which humans need to remediate a subsurface source of contamination depends on the natural attenuation factor (NAF), which is estimated as the ratio of the source concentration to the concentration reaching a receptor, at steady state (Begley, 1996). The NAF represents natural attenuation processes, usually biodegradation or abiotic degradation, and combined with the maximum concentration limit (MCL) permitted at the receptor, is used to back-calculate the maximum contaminant levels permissible at the source. Tabulated values of rate constants are available to assist with the parameterization of models, and where these values are unavailable or where concern exists that the available values may not be representative, laboratory tests may be undertaken to obtain them. However, neither of these data sources represents the site-specific dynamics of flow in the ground — they are concerned only with the chemical transformation rates, and some values may assume perfectly mixed solutions. As a result, no matter how accurate they are, they are incapable of anticipating the size of a contaminated zone resulting from the transport of pollutants by groundwater, i.e., the plume size, without accurate knowledge of the flow system, in particular the seepage velocity or Darcy flux. This concept is easily demonstrated by comparing two plumes of trichloroethane (TCA), which transforms to 1,2 dichloroethene and acetic acid through an abiotic reaction with water, with a well-documented half-life (t½) of about 2.3 years (assuming T ≅ 15 °C) as shown in Figure 15 (note: a half-life is the time required for the pollutant concentration to decrease to half its original value).

Graphs showing steady-state plume centerline profile of 1,1 trichloroethane in an aquifer
Figure 15a) Steady-state plume centerline profile of 1,1 trichloroethane in an aquifer with v = 10 cm/day and of 2.3 years. Plume size is sufficiently wide and deep that the boundaries do not influence the concentrations on the centerline. b) The same plume steady-state centerline profile with v = 30 cm/day. Prior knowledge of the rate constant (through ) is not deterministic of the plume size without good knowledge of the groundwater velocity. (In these calculations, longitudinal dispersivity was fixed at 0.1 m, horizontal transverse dispersivity was set to 0.01 m, and vertical dispersivity was set to 0.015 m.)

If such a plume is permitted to grow to its steady-state length, and a groundwater velocity of 10 cm/day is assumed to apply, the plume front (taken here to be C/Co = 0.01, for convenience, where C is the pollutant concentration at a specific place and time and Co is the pollutant concentration at the source) will reach a distance of about 550 m from the source. If the groundwater velocity is taken to be 30 cm/day — within the range of uncertainty typically afforded by seepage velocity estimated from Darcy calculations, which rely on hydraulic conductivity — the plume length will reach about 1500 m. This difference has profound implications for the risk experienced by receptors downgradient of the source area, and this is a case in which the transformation rate is highly reliable and predictable. In this case there appears to be a simple proportionality to the problem: tripling the seepage velocity tripled the length of the plume. This simplicity is, unfortunately, not generally assured.

A common scenario involving variable transformation rates involves the biodegradation of petroleum hydrocarbons in the presence of dissolved oxygen, or other so-called terminal electron acceptors (TEA). In these reactions, the hydrocarbons give up electrons to the TEAs as carbon leaves the hydrocarbon molecules and becomes carbon dioxide. The reaction rates in this case are limited by availability of the TEAs, and if TEAs are present the reactions can be regarded as instantaneous (Rifai and Bedient, 1990). TEA availability turns out to be dependent on the degree of mixing in the subsurface (this issue is discussed in more detail in the section “Subsurface Mixing” later in this book), which brings the dissolved hydrocarbons and TEAs into contact. Thus, the transformation rate depends on transport processes. Once the supply of TEA is exhausted, the transformation of hydrocarbons stops. The resulting non-proportional relationship between plume length and seepage velocity is illustrated with a simple calculation of such a biodegradation scenario (Figure 16). In this case, increasing the seepage velocity from 10 cm/day to 30 cm/day only advances the plume front from about 190 m to 255 m from the source. Other biodegradation rates, which depend on particular geochemical environments for transformation to occur, are affected — sometimes completely disrupted — in still more complex ways by the inflow of interfering chemicals (e.g., dissolved oxygen flowing into a zone of reductive dechlorination) and can be highly sensitive to groundwater velocity.

Graphs showing contaminant reacting with an electron acceptor in a flow system
Figure 16Two cases of a contaminant reacting with an electron acceptor in a flow system. In both cases the electron acceptor is present in the background water at a concentration of 0.5 M/L3 and has been injected at the source at 1 M/L3 for one year. The plumes are modeled in three dimensions. The contaminant is introduced at 1 M/L3 and has been released for 15 years. a) Plume center line profiles for an oxidizable contaminant and an electron acceptor in an aquifer with a v of 0.1 m/day. b) The same plume center line with a v of 0.3 m/day. Reactions between the contaminant and the electron acceptor are assumed to be instantaneous. The electron acceptor was assumed to move in the aquifer without retardation, and the contaminant was assumed to migrate with a retardation factor of 2.5, typical of simple hydrocarbons.


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