Advection-dispersion model

Wilson and Liu showed that both location and travel time probabilities can be calculated directly, using a backward-in-time version of traditional continuum advection-dispersion modeling. In addition, they claimed that by choosing the boundary conditions properly, the method can be readily generalized to include linear adsorption with kinetic effects and 1st order decay. An extension of their study for a 2D heterogeneous aquifer was reported in Liu and Wilson [39]. The results for travel time probability are in very close agreement with the simulation results from traditional forward-in-time methods. 

Tritium measurements are frequently used to calculate recharge rates, rates or directions of subsurface flow, and residence times. For these purposes, the seasonal, yearly, and spatial variations in the tritium content of precipitation must be accurately assessed. This is difficult to do because of the limited data available, especially before the 1960s. For a careful discussion of how to calculate the input concentration at a specific location, see Michel (1989) and Plummer et al. (1993). Several different approaches (e.g., piston-flow, reservoir, compartment, and advective-dispersive models) to modeling tritium concentrations in groundwater are discussed by Plummer et al. (1993). The narrower topic of using environmental isotopes to determine residence time is discussed briefly below. 

Up to the present, Kr studies of shallow ground waters are rare and limited to small numbers of samples. An early assessment of the feasibility of the method with a few examples was presented by Rozanski and Florkowski (1979). In a study of ground-water flow based exclusively on Kr conducted in the Borden aquifer in Canada a monotonic increase of the Kr age along the ground-water flow path was found and compared with the results of a two-dimensional advection dispersion model of Kr transport (Smethie et al. 1992). It was found that the modeled Kr distribution was insensitive to dispersion and that therefore Kr could be used in a straightforward manner to estimate the ground-water residence time. 

The desorption of solvents from soil has not been extensively measured. In the application of advection-dispersion models to predict solute movement, it is generally assumed that adsorption is reversible. However, the adsorption of the solutes in T able 17.1.1 may not be reversible. For example, hysteresis is often observed in pesticide adsorption-desorption studies with soils. The measurement and interpretation of desorption data for solid-liquid systems is not well understood.Once adsorbed, some adsorbates may react further to become covalently and irreversibly bound, while others may become physically trapped in the soil matrix. The non-singularity of adsorption-desorption may sometimes result from experimental artifacts. ... 

SCRAM (28) is a TDE dynamic, numerical finite difference soil model, with a TDE flow module and a TDE solute module. It can handle moisture behavior, surface runoff, organic pollutant advection, dispersion, adsorption, and is designed to handle (i.e., no computer code has been developed) volatilization and degradation. This model may not have received great attention by users because of the large number of input data required. 

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) ... 

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... 

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... 

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...

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. 

As contaminant transport occurs over times much greater than the times over which groundwater flow fluctuates, steady flow is frequently assumed. For steady groundwater flow in three dimensions, the following vector equation, developed based on mass conservation principles, is typically used to model advective/ dispersive transport of a dissolved reactive contaminant (after [53]) ... 

BIOPLUME III is a public domain transport code that is based on the MOC (and, therefore, is 2-D). The code was developed to simulate the natural attenuation of a hydrocarbon contaminant under both aerobic and anaerobic conditions. Hydrocarbon degradation is assumed due to biologically mediated redox reactions, with the hydrocarbon as the electron donor, and oxygen, nitrate, ferric iron, sulfate, and carbon dioxide, sequentially, as the electron acceptors. Biodegradation kinetics can be modeled as either a first-order, instantaneous, or Monod process. Like the MOC upon which it is based, BIOPLUME III also models advection, dispersion, and linear equilibrium sorption [67]. 

In the RT3D simulation, advective/dispersive transport of each contaminant is assumed. Sorption is modeled as a linear equilibrium process and biodegradation is modeled as a first-order process. Due to the assumed degradation reaction pathways (Fig. 2) transport of the different compounds is coupled. In the study, four reaction zones were delineated, based on observed geochemistry data. Each zone (two anaerobic zones, one transition zone, and one aerobic zone) has a different value for the biodegradation first-order rate constant for each contaminant. For example, since PCE is assumed to degrade only under... 

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,... 

Zheng C (1990) MT3D, A modular three-dimensional transport model for simulation of advection, dispersion, and chemical reactions of contaminants in groundwater systems. SS Papadopulos Associates, Rockville, MD... 

Mathematical models for mass transfer at the NAPL-water interface often adopt the assumption that thermodynamic equilibrium is instantaneously approached when mass transfer rates at the NAPL-water interface are much faster than the advective-dispersive transport of the dissolved NAPLs away from the interface [28,36]. Therefore, the solubility concentration is often employed as an appropriate concentration boundary condition specified at the interface. Several experimental column and field studies at typical groundwater velocities in homogeneous porous media justified the above equilibrium assumption for residual NAPL dissolution [9,37-39]. 

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. 

Mass balance within an arbitrarily chosen biofilm section, or slice, taken parallel to the surface of attachment, is described by the one-dimensional, advection-dispersion-reaction equation, Eq. [1-5], with steady-state conditions and no advection. The sink term is microbial uptake, modeled using the parameters discussed in Section 2.6.3 see Eqs. [2-71 a] and [2-72],... 

The smallest spatial scale at which outdoor air pollution is of concern corresponds to the air volume affected by pollutant chemical emissions from a single point source, such as a smokestack (Fig. 4-24). Chemicals are carried downwind by advection, while turbulent transport (typically modeled as Fick-ian transport) causes the chemical concentrations to become more diluted. Typically, smokestacks produce continuous pollutant emissions, instead of single pulses of pollutants thus, steady-state analysis is often appropriate. At some distance downwind, the plume of chemical pollutants disperses sufficiently to reach the ground the point at which this occurs, and the concentrations of the chemicals at this point and elsewhere, can be estimated from solutions to the advection-dispersion-reaction equation (Section 1.5), given a knowledge of the air (wind) velocity and the magnitude of Fickian transport. 

M3TD Groundwater MT3D is a transport model that simulates advection, dispersion, source/sink mixing, and chemical reactions of contaminants in groundwater flow systems in either two or three dimensions. 

If transport occurs much faster than sorption, sorption processes may not reach equilibrium conditions. Nonequilibrium sorption may result from physical causes such as intraparticle rate-limited diffusion, chemical causes such as rate-limiting reaction kinetics, or a combination of the two. One approach used to model rate-limited sorption is bi-continuum models consisting of one region where transport is described by the advection-dispersion equation with equilibrium sorption, and another region where transport is diffusion limited with equilibrium sorption, or another region where sorption is chemically rate limited. 

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