Advection-dispersal equation

The effect of advection and dispersion on the distribution of a chemical component within flowing groundwater is described concisely by the advection-dispersion equation. This partial differential equation can be solved subject to boundary and initial conditions to give the component s concentration as a function of position and time. 

The advection-dispersion equation follows directly from the transport laws already presented in this chapter, and the divergence principle. The latter states that the time rate of change in the concentration of a component depends on how rapidly the advective and dispersive fluxes change in distance. If, for example, more of component i moves into the control volume shown in Figure 20.1 across its left and front faces than move out across its right and back, the component is accumulating in the control volume and its concentration there increasing. The time rate of... 

We will concentrate on adsorption-desorption. The one-dimensional form for homogeneous saturated media of the advection-dispersion equation can then be written as... 

The traditional, advection-dispersion equation, a generalization of Eq. 10.4, is then written in one-dimensional form, as... 

Note that Eq. 10.5 is written to allow the velocity to vary as a function of location typical application of the advection-dispersion equation assumes the velocity and the hydrodynamic coefficients to be constant. Moreover, the time dependence of these parameters arises when flow (infiltration) is unsteady or transient in these cases, the contact time between contaminants and the solid matrix (and any immobile water within it) is too short to allow an equilibrium to be reached. 

While the advection-dispersion equation has been used widely over the last half century, there is now widespread recognition that this equation has serious limitations. As noted previously, laboratory and field-scale application of the advection-dispersion equation is based on the assumption that dispersion behaves macroscopically as a Fickian diffusive process, with the dispersivity being assumed constant in space and time. However, it has been observed consistently through field, laboratory, and Monte Carlo analyses that the dispersivity is not constant but, rather, dependent on the time or length scale of measurement (Gelhar et al. 1992),... 

To quantify such transport, the advection-dispersion equation, which requires a narrow pore-size distribution, often is used in a modified framework. Van Genuchten and Wierenga (1976) discuss a conceptualization of preferential solute transport throngh mobile and immobile regions. In this framework, contaminants advance mostly through macropores containing mobile water and diffuse into and out of relatively immobile water resident in micropores. The mobile-immobile model involves two coupled equations (in one-dimensional form) ... 

The previous sections discussed the advection-dispersion equation and variants such as the mobile-immobile conceptualization, which are based on the key assumption that mechanical dispersion is Fickian. In other words, the advection-dispersion equation (Eq. 10.5) is strictly valid only under perfectly homogeneous... 

A variety of specific mathematical formulations of the CTRW approach have been considered to date, and network models have also been applied (Bijeljic and Blunt 2006). A key result in development of the CTRW approach is a transport equation that represents a strong generalization of the advection-dispersion equation. As shown by Berkowitz et al. (2006), an extremely broad range of transport patterns can be described with the (ensemble-averaged) equation... 

Fig. 10.4 Measured breakthrough curve of bromide with CTRW and advection-dispersion equation (ADE) fits. Here, the quantity j represents the normafized, flux-averaged concentration (top) Complete breakthrough curve, (bottom) Region identified by the bold-framed rectangle in the top plot. Note the difference in scale units between the plots. Pressure head h=-10cm water velocity v=2.82 cm/h. The dashed tine is the best advection-dispersion equation solution fit. The soUd line is the best CTRW fit. (Cortis and Berkowitz 2004)...

Sorption of contaminants can be included in the advection-dispersion equation by introducing a retardation factor ... 

In this situation, transport equations similar to those discussed previously can be applied. For example, by assuming sorption to be essentially instantaneous, the advective-dispersion equation with a reaction term (Saiers and Hornberger 1996) can be considered. Alternatively, CTRW transport equations with a single ti/Ci, t) can be applied or two different time spectra (for the dispersive transport and for the distribution of transfer times between mobile and immobile—diffusion, sorption— states can be treated Berkowitz et al. 2008). 

The transport behavior of colloids commonly is modeled by colloid filtration theory (CFT) (Yao et al. 1971), which is based on extension of the common advection-dispersion equation. The one-dimensional advection-dispersion-filtra-tion equation is written... 

Loss or gain of dissolved chemical species from soil water by precipitation or dissolution, respectively, nsually is accounted for by adding a simple sink-source term in the advection-dispersion equation ... 

Quantification of NAPL transport usually is considered by using advection-dispersion equations for each of the water and NAPL phases and defining relative permeabilities (as noted previously). Alternatively, if the emphasis is on contamination... 

Another modeling analysis is presented by Russo et al. (1998), who examined field transport of bromacil by application of the classical one-region, advection-dispersion equation (ADE) model and the two region, mobile-immobile model (MM) recall Sects. 10.1 and 10.2. The analysis involved detailed, three-dimensional numerical simulations of flow and transport, using in-situ measurements of hydraulic... 

Fig. 12.7 Profiles of means (a,b) and standard deviations (c,d) of the bromacil concentrations at four different time points. Solid curves denote simulated profiles obtained from the advection-dispersion equation (a,c) and the mobile-immobile model (b,d). The different symbols denote measured profiles at different times. Reprinted from Russo D, Toiber-Yasur I, Laufer A, Yaron B (1998) Numerical analysis of field scale transport of bromacil. Adv Water Resour 21 637-647. Copyright 1998 with permission of Elsevier...

The fate of the pollutant moving in the aquifer along the streamlines is determined by the advection-dispersion equation, Eq. 25-10 or 25-18. For Pe 1, that is, for locations x dis / if, the concentration cloud can be envisioned to originate from an infinitely short input atx = 0of total mass (a so-called5 input) that by dispersion is turned into a normal distribution function along the x-axis with growing standard deviation. Since the arrival of the main pollution cloud at some distance x is determined... 

Remember that from the nondimensional version of the advection-dispersion equation, Eq. 25-18, the Peclet Number Pe was identified as the only parameter that determines the shape of the concentration distribution in the aquifer. By introducing relative coordinates for space ( ,) and time (9) as defined in Eq. 25-16, Eq. 25-20 takes the form ... 

Wiedemeier and coworkers (1996) have suggested two methods to approximate biodegradation rates in groundwater field studies (a) use a biologically recalcitrant tracer (e.g., three isomers of trimethylbenzene) in the groundwater to correct for dilution, sorption, and/or volatilization and calculate the rate constant by using the downgradient travel time or (b) assume that the plume has evolved to a dynamic steady-state equilibrium and develop a one-dimensional analytical solution to the advection-dispersion equation. 

The transport of TCE through a porous medium, such as soil, can be described by the following ID advection-dispersion equation (Fetter, 1993). 

PFR models are limited, however, because of the slow velocities encountered in groundwater aquifers and the tendency for many contaminants (particularly hydrophobic organic compounds) to sorb. More appropriate but more complex models based on various forms of the advection-dispersion equation (ADE) have been used by several researchers to incorporate more processes, such as dispersion, sorption, mass transfer, sequential degradation, and coupled chemical reactions. 

Another approach has been to model sequential reactions by using multiple advection-dispersion equations [207]. The use of multiple ADEs provides a more realistic model where each reactant can degrade, sorb, and disperse. Simulations using this type of model reveal that breakthrough of degradation products could occur despite complete removal of the parent compound, TCE [207]. Additional simulations were used to explore the effect of slow sorption (i.e., nonequilibrium sorption), and the results suggest that it is reasonable to assume that an FePRB will reach steady-state conditions under typical field conditions. 

Carey et al. [70] used the BioRedox code to simulate the fate and transport of BTEX and chlorinated ethenes at a contaminated groundwater site at Plattsburgh Air Force Base. Transport of the compounds was modeled using the 3-D advection/dispersion equation, and sorption was assumed to be negligible. While BioRedox is capable of simulating oxidation of multiple electron donors,... 

Non-reactive contaminant transport may be represented by the Advection-Dispersion Equation (ADE) in three-dimensional (3D) form ... 

The numerical solution to the advection-dispersion equation and associated adsorption equations can be performed using finite difference schemes, either in their implicit and/or explicit form. In the one-dimensional MRTM model (Selim et al., 1990), the Crank-Nicholson algorithm was applied to solve the governing equations of the chemical transport and retention in soils. The web-based simulation system for the one-dimensional MRTM model is detailed in Zeng et al. (2002). The alternating direction-implicit (ADI) method is used here to solve the three-dimensional models. 

In order to obtain a comprehensive description of the transport of solutes in subsurface water, it is necessary to consider all three transport mechanisms. This can be accomplished by linking the three transport flux equations above through the continuity equation, yielding the advection-dispersion equation for steady-state flow ... 

FIGURE 3-19 Solutions to the advection-dispersion equation (Eq. [1-5]) for a conservative solute. Cases for continuous input of mass beginning at time t = 0 are adapted from references cited, assuming x and/or r are much larger than D/v r equals (x2 + y2Dx/Dy)m in two dimensions or (x2 + y2Dx/Dy + z2Dx/Dz)m in three dimensions. Note that the definitions of M and M vary with the number of dimensions. 

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