Continuity Equation0

The continuity equation describes conservation of mass for the one-dimensional system. From previous text, with the addition of a storage term, $\begin{array}{l}S\end{array}$ , the continuity equation can be written as

1)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial A}{\partial t} + \frac{\partial S}{\partial t} + \frac{\partial Q}{\partial x} = q_1\end{array}$

Symbol	Description	Units
$\begin{array}{l}x\end{array}$	distance along the channel
$\begin{array}{l}t\end{array}$	time
$\begin{array}{l}Q\end{array}$	flow
$\begin{array}{l}A\end{array}$	cross-sectional area
$\begin{array}{l}S\end{array}$	storage from non conveying portions of cross section
$\begin{array}{l}q_1\end{array}$	lateral inflow per unit distance

The above equation can be written for the channel and the floodplain:

2)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial A_c}{\partial t} + \frac{\partial Q_c}{\partial x_c} = q_f\end{array}$

and

3)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial A_f}{\partial t} + \frac{\partial S}{\partial t} + \frac{\partial Q_f}{\partial x_f} = q_c + q_1\end{array}$

where the subscripts c and f refer to the channel and floodplain, respectively, $\begin{array}{l}q_1\end{array}$ is the lateral inflow per unit length of floodplain, and $\begin{array}{l}q_c\end{array}$ and $\begin{array}{l}q_f\end{array}$ are the exchanges of water between the channel and the floodplain.

NOTE

The HEC-RAS Unsteady flow engine combines the properties of the left and right overbank into a single flow compartment called the floodplain (Finite Difference solution only, not the finite volume solution). Hydraulic properties for the floodplain are computed by combining the left and right overbank elevation vs Area, conveyance, and storage into a single set of relationships for the floodplain portion of the cross section. The reach length used for the floodplain area is computed by taking the arithmetic average of the left and right overbank reach lengths (L_L + L_R)/2 = L_F. The average floodplain reach length is used in both the continuity and momentum equations to compute their respective terms for a combined floodplain compartment (Left and right overbank combined together).

This is different than what is done in the Steady Flow computational engine (described above in the previous section), in which the left and right overbank are treated completely separately.

(2) and (3) are now approximated using implicit finite differences by applying (.Implicit Finite Difference Scheme v6.0:4) through (.Implicit Finite Difference Scheme v6.0:6):

4)	$\begin{array}{l}\displaystyle \displaystyle \frac{\Delta A_c}{\Delta t} + \frac{\Delta Q_c}{\Delta x_c} = \overline{q}_f\end{array}$

5)	$\begin{array}{l}\displaystyle \displaystyle \frac{\Delta A_f}{\Delta t} + \frac{\Delta S}{\Delta t} + \frac{\Delta Q_f}{\Delta x_f} = \overline{q}_c + \overline{q}_1\end{array}$

The exchange of mass is equal but not opposite in sign such that $\begin{array}{l}\Delta x_c q_c = -q_f \Delta x_f\end{array}$ . Adding the above equations together and rearranging yield:

6)	$\begin{array}{l}\displaystyle \displaystyle \Delta Q + \frac{\Delta A_c}{\Delta t} \Delta x_c + \frac{\Delta A_f}{\Delta t} \Delta x_f + \frac{\Delta S}{\Delta t} \Delta x_f = \overline{Q}_t\end{array}$

Symbol	Description	Units
$\begin{array}{l}\overline{Q}_t\end{array}$	the average lateral inflow
$\begin{array}{l}\Delta x_c\end{array}$	the length of the main channel between two cross sections
$\begin{array}{l}\Delta x_f\end{array}$	the length of the floodplain between two cross sections (computed as the arithmetic average of the left and right overbank reach lengths)