Advection-reaction-dispersion equation

MASS BALANCE IN AN INFINITELY SMALL CONTROL VOLUME THE ADVECTION-DISPERSION-REACTION EQUATION... 

So far, the concept of mass conservation has been applied to large, easily measurable control volumes such as lakes. Mass conservation also can be usefully expressed in an infinitesimal control volume, mathematically considered to be a point. Conservation of mass is expressed in such a volume with the advection—dispersion-reaction equation. This equation states that the rate of change of chemical storage at any point in space, dC/dt, equals the sum of both the rates of chemical input and output by physical means and the rate of net internal production (sources minus sinks). The inputs and outputs that occur by physical means (advection and Fickian transport) are expressed in terms of the fluid velocity (V), the diffusion/dispersion coefficient (D), and the chemical concentration gradient in the fluid (dC/dx). The input or output associated with internal sources or sinks of the chemical is represented by r. In one dimension, the equation for a fixed point is... 

Equation [1-5] is pertinent to a one-dimensional system, such as a long, narrow tube full of water, where significant variations in concentration may be assumed to occur only along the length of the tube. In a three-dimensional situation, the advection-dispersion-reaction equation can be represented most succinctly using vector notation, where V is the divergence operator ... 

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. 

If the Fickian transport coefficient is known, it is possible to predict the distribution of the tracer at any time and location after it is introduced into the column. At the time of injection of the tracer (f = 0), the concentration is high over a short length of column. At a later time fi, the center of the mass of tracer has moved a distance equivalent to the seepage velocity multiplied by fi, and the mass has a broader Gaussian, or normal, distribution, as defined in Eq. (2.6). For this one-dimensional situation, the solution to the advection-dispersion-reaction equation (Eq. 1.5) gives the concentration of the tracer as a function of time and distance. 

FIGURE 3.19 Solutions to the advection-dispersion-reaction equation (Eq. 1.5) for an ideal tracer. Cases for continuous input of mass beginning at time f = 0 are adapted from references cited, assuming x and/or r are much larger than D/v r equals x +y in two dimensions or... 

Equation 2.20 is the advection-dispersion (AD) equation. In the petroleum literature, the term convection-diffusion (CD) equation is used, or simply diffusion equation (Brigham, 1974). When a reaction term is included, the term advection-reaction-dispersion (ARD) equation is used elsewhere. When the adsorption term is expressed as a reaction term, the ARD equation is as discussed later in Section 2.4. Several solutions of Eq. 2.20 have been presented in the literature, depending on the boundary conditions imposed. In general, they are various combinations of the error function. When the porous medium is long compared with the length of the mixed zone, they all give virtually identical results. 

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). 

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. 

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. 

A natural response to the limitations of both geochemical equilibrium models and the solute transport models (see 10.3 for a discussion) is to couple the two. Over the last two decades, a number of models that couple advective-dispersive-diffusive transport with fully speciated chemical reactions have been developed (see reviews by Engesgaard and Christensen, 1988 Grove and Stollenwerk, 1987 Mangold and Tsang, 1991). In the coupled models, the solute transport and chemical equilibrium equations are simultaneously evaluated. 

At using these models chemical properties in water-dissolved components have great significance. Most nonpolar components do not participate in chemical reactions and mass exchange with rocks. For this reason modeling of their distribution processes of the chemical interaction, as a rule, are disregarded. Major factors in the change of their concentration in water turn out flow velocity and hydrodynamic dispersion. That is why the reviewed models for chemically passive nonpolar components often maybe solved analytically by equations of advective-dispersive mass transport. 

Analytical solutions are those whose precision depends only on the accuracy of the initial data. They do not contain errors associated with the approximation due to simplification of the computation process. This approach is applicable to the simplest models, which are often represented by a restricted number of relatively simple equations. These may be solved without specialized program software. Analytical solutions, as a rule, are used in modeling of processes with minimum participation of chemical reactions, in particular in the analysis of distribution of nonpolar components, radioactive decay, adsorption, etc. Under such conditions for modeling often are sufficient equations of advective-dispersive mass-transport, which are included in the section Mixing and mass-transport . 

From the large number of mathematical models for the transport of transformation products with kinetic reactions that can be considered in the Rockflow system we have chosen a first-order chemical nonequilibrium model to simulate the sorption reaction. It can be described by the governing solute transport equation with rate-limited sorption and first-order decay in aqueous and sorbed phases. This model includes the processes of advection, dispersion, sorption, biological degradation or radioactive decay of the contaminant in the aqueous and/or sorbed phases. Figure 6.1 illustrates the conceptual model for sequential decay of a reactive species. 

The model has a reactive module, which solves reaction kinetics and equilibrium reactions, and a transport module, which incorporates the advection-dispersion equation. The transport/reaction equation is formulated for each redox acceptor in the fluid phase as follows... 

The species continuity equation (CE) is an expression of the Lavoisier general law of conservation of mass. Equation 2.1 presents the CE in vector form and provides the proper context for the various types of chemical mass transport processes needed for chemical modeling and fate analysis. In Section 2.2.2, the mass accumulation portion of the CE is highlighted as the principal term for assessing chemical fate in the media compartments. This term includes reaction, advection, diffusion, and turbulent transport and dispersion processes. Because the magnitude and direction of this term reflect the sum total of all processes, this term uniquely defines chemical fate. In Equation 2.2, the steady-state CE minus the reaction term is commonly referred to as the advective-diffusive (AD) equation. It provides the appropriate starting point for addressing the various transport processes associated with the mobile phases in near-surface soils. 

Abstract Unsteady liquid flow and chemical reaction characterize hydrodynamic dispersion in soils and other porous materials and flow equations are complicated by the need to account for advection of the solute with the water, and competitive adsorption of solute components. Advection of the water and adsorbed species with the solid phase in swelling systems is an additional complication. Computers facilitate solution of these equations but it is often physically more revealing when we discriminate between flow of the solute with and relative to, the water and the flow of solution with and relative to, the solid phase. Spacelike coordinates that satisfy material balance of the water, or of the solid, achieve this separation. Advection terms are implicit in the space-like coordinate and the flow equations are focused on solute movement relative to the water and water relative to soil solid. This paper illustrates some of these issues. 

The first term on the right-hand-side of the equation accounts for contaminant dispersion in the x-,y-, and z-directions, while the second term accounts for contaminant advection. The third term on the right-hand-side of the equation is a sink term to account for sorption of dissolved contaminant to aquifer solids, and the fourth term is a sink term to account for loss of dissolved contaminant mass due to chemical and biological reactions. Note that one of the implicit assumptions in Eq. (18) is that the chemical/biological reaction sink applies only to dissolved phase contaminant. This is in accord with the frequently made observation that sorbed contaminant is not bioavailable [5-7]. 

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