Advection-Dispersion

Concentrations of AD components are transferred explicitly between the river and surface depending on the direction of the flow. The surface model handles the coupling exchange of AD components in the following way: for river end links with flow from the river to the surface, the concentration of the AD components in the surface is imposed as with a single inflow source, i.e., as a flux of mass into the surface faces:

\[ \Delta\left( \frac{\partial V - C_{S}^{n + 1/2}}{\partial t} \right) = Q^{n + 1/2}C_{R}^{n} \]

Equation: Mass flux into the surface model where flow runs from the river to the surface.

Where \(C_{S}^{n + 1/2}\) represents the concentrations in the surface model at the link location for time step \(n + 1/2\) and \(C_{R}^{n}\) is the AD concentration for the river at the link location at time step \(n\).

When the flow direction is from the surface to the river, the modification to the AD equation is:

\[ \Delta\left( \frac{\partial V - C_{S}^{n + 1/2}}{\partial t} \right) = Q^{n + 1/2}C_{S}^{n} \]

Equation: Mass flux out of the surface model where flow runs from the surface to the river.

It is important to note that the first equation represents an inflow source point in the surface model, where the inflow \(Q\) has a concentration given by the river concentration at the link, \(C_{R}^{n}\). On the other hand, the second equation represents a sink source point in the surface model (water leaving the surface), where the sink \(Q\) has a concentration given by the surface concentration at the link location \(C_{S}^{n}\). Another relevant note is the fact that the inflow \(Q\) has opposite signs in the two equations.

The corresponding boundary condition in the river can be either a transport boundary or a concentration boundary as usual. Furthermore, a mixing coefficient can be defined as normal.

Tip

It is strongly recommended to use a transport boundary condition in the river model.

For lateral links the mass of the AD component being transferred is calculated from the lateral discharge and the concentration in the river or surface model (depending on the flow direction). This is then applied as a source or sink term in the branches and the elements. The source term into the surface is as for the standard links. For the river model, the one-dimensional advection dispersion equation is:

\[ \frac{\partial AC}{\partial t} + \frac{\partial QC}{\partial x} - \frac{\partial}{\partial x}\left( A \cdot D \cdot \frac{\partial C}{\partial x} \right) = -A \cdot K \cdot C + C_{2} \cdot q \]

Equation: One-dimensional advection dispersion equation used for the river model.

Where:

• \(C\) — concentration
• \(D\) — dispersion coefficient
• \(A\) — cross-sectional area
• \(K\) — linear decay
• \(C_{2}\) — source/sink concentration
• \(q\) — lateral inflow per river length