2.0 Advective-Dispersive Equation


    The advective-dispersive equation for solute movement through a river forms the basis of the mathematical algorithm used by the riverine component. The surface-water flow is assumed to be steady and uniform; the algorithms are developed for the limiting case of unidirectional advective transport with three-dimensional (longitudinal, lateral, and vertical) dispersion. The advective-dispersive equation for solute movement in a river can be described by the following expression:

(2.1)



where
Equation 2.1 does not take into account the effects of contaminant adsorption to or desorption from sediment particles suspended in the water column or in the river bed. As explained in Chapter 1.0, this should result in conservative aqueous concentrations in most cases.

    Contaminant releases to the riverine pathway in the MEPAS methodology are generally of long duration relative to the travel time from the point of release to the receptor. Because transient solutions for contaminant migration and fate are most applicable for batch and infrequent releases over relatively short periods of time (Codell et al. 1982), a steady-state solution to the advective-dispersive equation is used in the riverine component of MEPAS. The steady-state, vertically-integrated, mass-balance equation for transport in a river (where longitudinal advection dominates longitudinal dispersion) can be written as follows:

(2.2)





(a) When two sets of units are provided, the first refers to chemicals, and the second refers to radionuclides.



 
in which

(2.3)



at y = 0 and y = B

where    B = width of stream channel (cm).






2.1 Contaminant Concentration Equation


When Equation 2.2 is solved with the appropriate boundary conditions (i.e., Equation 2.3), the riverine pathway is described by an analytical expression that characterizes the transport of contaminants through a river. For a point-source() contaminant release from the bank of a stream having a rectangular cross-section, the solution to Equation 2.2 employing the boundary conditions defined by Equation 2.3 is very similar to those outlined by Codell et al. (1982), Strenge et al. (1986), and Whelan et al. (1986):

       

(2.4)



where
    The concentration at y = 0 is used in computing contaminant levels for the exposure component of the MEPAS methodology. By assuming that y equals 0, Equation 2.4 reduces to

       

(2.5)





(a) The term "point source" refers to the source-term configuration, which reflects simplifying assumptions, and does not refer to the exact technical definition associated with the concentration equation.  Note, for example, that a vertically averaged point source represents a point source in the x and y directions, and  a line source in the z-direction.




    A line source along the edge of the stream can be represented as a series of point sources along the length of the line source. As the downstream receptor location is moved farther away, the line source resembles a point source located at the center of the line source (Receptor A in Figure 1.1). As the receptor location is moved closer to the center of the line source, only that portion of the source term upstream of the receptor has an opportunity to influence contaminant levels at the receptor; in effect, the strength of the source term is reduced. Under these circumstances, the line source can be approximated as a point source that is located at one-half the distance between the receptor location and the upstream end of the line source (Receptor B in Figure 1.1), and the reduced strength of the source is accounted for by multiplying Equation 2.5 by the fraction of the source term upstream of the receptor:

(2.6)



where
Therefore, only the point source solution is used in the surface-water component of MEPAS.

    Discharge in a river channel varies along the length of the river, generally increasing in the downstream direction due to inflowing tributary streams and groundwater. The riverine component accounts for an increase in discharge between the source location and the receptor location by using a dilution ratio:

(2.7)



where
Incorporating Equations 2.6 and 2.7 into Equation 2.5 gives the final form of the steady-state equation used to simulate concentrations in the riverine component of MEPAS:

(2.8)








2.2 Multiple River Receptors


    When more than one river receptor is specified in a MEPAS run, an estimation technique is employed to calculate concentrations at the second and subsequent receptors, rather than using Equation 2.8. The method estimates contaminant concentrations for the current receptor based on concentrations for the previous receptor as follows:

(2.9)



where





2.3 Lateral Mixing Length


    The mixing length is the distance over which contamination is considered fully mixed in the lateral direction and is used in Equation 2.9. The mixing length is estimated by employing the advective-dispersive equation and its associated Gaussian distribution solution. The one-dimensional advective-dispersive equation in the lateral direction is written as

(2.10)



The unit area solution to Equation 2.10 in a river of infinite lateral extent is described by

(2.11)



in which

(2.12)



where
With the assumption that contaminant dispersion does not occur through the river banks once the contaminant has entered the river, contaminant spreading is only in the lateral direction. Therefore, the lateral mixing length for the fully mixed condition is assumed as the dispersive distance associated with one-half the standard deviation:

(2.13)



where    lm = lateral distance over which the contaminant is assumed to be uniformly distributed (equivalent to one-half the standard deviation) (cm).

    To illustrate that Equation 2.13 has a physical basis, a similar mixing-length expression can be developed by defining a time scale associated with complete lateral mixing as similar to the one found in Codell et al. (1982):

(2.14)



where
By rearranging Equation 2.14, the effective length, which represents the fully mixed condition, can be solved for

(2.15)



in which

(2.16)



where    f = proportionality constant (dimensionless).

    Codell et al. (1982) note that when f# 3.3, the fully mixed condition can be assumed. When f is between 3.3 and 12, the release is considered to be neither fully mixed over the width of the river nor unaffected by the river boundary. Because Equation 2.13 represents a more conservative expression than Equation 2.15 when f equals 3.3, Equation 2.13 is used to describe the lateral mixing length in the riverine environment.
    The MEPAS methodology computes lm to identify the mixing width. If lm is larger than the width of the river, lm is set equal to the width of the river. This procedure ensures a continuous transition between the fully mixed condition and non-fully mixed condition.






2.4 Representative Contaminant Travel Time


    To use Equation 2.13, a representative travel time has to be identified. The MEPAS waterborne components recognize three travel times: 1) the advective travel time, 2) the travel time due to advection and dispersion, 3) the time to the peak flux or concentration (which includes advection, dispersion, and decay). Longitudinal dispersion is not considered by the riverine component, because advection is assumed to dominate dispersion in the flow direction. In addition, including the effects of decay produces an apparent travel time that is not indicative of the average time required for a particle to travel from the source to the receptor. Thus, the advective travel time is used in Equation 2.13:

(2.17)



The use of advective travel time in Equation 2.13 is consistent with the riverine mixing zone equation given in Mills et al. (1985). 






2.5 Lateral Dispersion Coefficient


    The transverse dispersion coefficient is required by Equations 2.8 and 2.13. Accurately defining this parameter for all riverine systems under all conditions is difficult. The coefficient is, therefore, defined such that representative properties of the water body are considered in the estimation.

    Fischer et al. (1979) note that dispersion in rivers is generally related to the characteristics of the river using the following relationship:

(2.18)



where
    Fischer(a) and Fischer et al. (1979) note that researchers (e.g., Orlob 1959; Sayre and Chamberlain 1964; Sayre and Chang 1968; Engelund 1969; Prych 1970; Elder 1959; Okoye 1970; Glover 1964; Fischer 1967; Yotsukura et al. 1970) have defined a range of values for b. In laboratory flumes, b ranges from 0.5 to 2.4. For practical purposes, Fischer (1967) suggests that b = 0.6.

    The shear velocity is estimated by Fischer (1974) by assuming that it is directly proportional to the average flow velocity of the stream (u):

(2.19)




(a) Fischer, H. B. Date Unknown. "Longitudinal Dispersion and Turbulent Mixing in Open Channel Flow." Working Paper. University of California at Berkeley. California.



 
Equation 2.19 was suggested for streams with Manning's roughness coefficients on the order of 0.04. By combining Equations 2.18 and 2.19 with b = 0.6, the dispersion coefficient in the lateral direction used by the riverine component is given by:

(2.20)