- The fate and transport of the modeled solute can be described by a one-dimensional system.
- Model parameters may be spatially and temporally variable. Spatial variation is assumed to occur in the longitudinal direction only.
- The mechanisms shown in Table 1 influence solute behavior in the stream channel and the storage zone, as indicated.
- The volume of the storage zones does not change in time.

Mechanistic models such as the one presented in this report are based on a
fundamental engineering
principle, *conservation of mass*. This
principle states that for a finite volume of water, mass is neither created
or destroyed (see Chapra and Reckhow, 1983). In short, mass conservation
implies that:

Equation (A1) is often called a mass balance equation. This mass balance
must be applied to a finite volume of water, known as a control volume.
A control volume for our one-dimensional system is shown below. Note that
the terms 'segment' (introduced in Section 3.2) and 'control volume' can
be used interchangeably. In figure A1,
*x*
represents the segment length, while `1' and `2' are used to demarcate
the upstream and downstream boundaries of the control-volume.

The remainder of this appendix is devoted to the formulation of differential
equations based on Equation (A1).
To accomplish this task, we first define
`*Accumulation*' and then proceed to develop the `*in*'
and `*out*' terms for each process given in
Table 1. This procedure is carried out for both the stream channel and the
storage zone.

where

We can now expand Equation (A2) using the chain rule, yielding:

Cin-stream solute concentration [M/L ^{3}]Vvolume of the main channel segment [L ^{3}]ttime [T]

where

xsegment length [L] Astream cross-sectional area [L ^{2}]

where *Q* is the volumetric flowrate [L^{3}/T].

Next, the mass advected out of the control volume, i.e. across the downstream boundary, is given by:

Finally, we combine the `*In*' and `*Out*' terms, giving:

(A8)

(A9)

where *D* is the dispersion coefficient [L^{2}/T].

(A11)

(A12)

where

C_{L}solute concentration of the lateral inflow [M/L ^{3}]q_{LIN}lateral inflow rate [L ^{3}/T-L]q_{LOUT}lateral outflow rate [L ^{3}/T-L]

(A14)

(A15)

where

Note that the exchange coefficient, , is defined as the fraction of the main channel volume that is exchanged with the storage zone, per unit time.

storage zone exchange coefficient [/T] C_{S}storage zone solute concentration [M/L ^{3}]

Although Equation (A16) represents a mass balance for a one-dimensional stream, it may be simplified using the water balance shown below.

where _{w} is the
density of water [M/L^{3}].

The inflow term for the water balance is the sum of the water advected across the upstream interface and the water entering via lateral inflow:

Similarly, outflow is the sum of the water advected out and that leaving through lateral outflow:

Finally, the water balance is completed by combining Equation (A17)
through (A19), and dividing by
_{w}
*x*:

This equation is presented in Section 3.2.1 as Equation (1).

where

Note that the storage zone area,

V_{S}volume of the storage zone segment [L ^{3}]A_{S}storage zone cross-sectional area [L ^{2}]

(A24)

(A25)

where is the fraction of the main channel volume that is exchanged with the storage zone, per unit time.

This governing equation is presented in Section 3.2.1 as Equation (2).