Interior Boundary Conditions (for Reach Connections)

A network is composed of a set of M individual reaches. Interior boundary equations are required to specify connections between reaches. Depending on the type of reach junction, one of two equations is used:

Continuity of flow:

1)	$\begin{array}{l}\displaystyle \displaystyle \sum_{i=1}^l S_{gi} Q_i =0\end{array}$

Symbol	Description	Units
$\begin{array}{l}l\end{array}$	the number of reaches connected at a junction
$\begin{array}{l}S_{gi}\end{array}$	-1 if $\begin{array}{l}i\end{array}$ is a connection to an upstream reach, +1 if $\begin{array}{l}i\end{array}$ is a connection to a downstream reach
$\begin{array}{l}Q_i\end{array}$	discharge in reach $\begin{array}{l}i\end{array}$

The finite differences form of (1) is:

2)	$\begin{array}{l}\displaystyle \displaystyle \sum_{i=1}^{l-1} MU_{mi} \Delta Q_i +MUQ_m \Delta Q_K =MUB_m\end{array}$

Symbol	Description	Units
$\begin{array}{l}MU_{mi}\end{array}$	$\begin{array}{l}\theta S_{gi}\end{array}$
$\begin{array}{l}MUQ_m\end{array}$	$\begin{array}{l}\theta S_{gK}\end{array}$

Continuity of stage:

3)	$\begin{array}{l}\displaystyle z_k = z_c\end{array}$

Symbol	Description	Units
$\begin{array}{l}z_k\end{array}$	the stage at the boundary of reach $\begin{array}{l}k\end{array}$ , is set equal to $\begin{array}{l}z_c\end{array}$ , a stage common to all stage boundary conditions at the junction of interest

The finite difference form of (3) is:

4)	$\begin{array}{l}\displaystyle MUZ_m \Delta Z_k - MU_m \Delta z_c = MUB_m\end{array}$

Symbol	Description	Units
$\begin{array}{l}MUZ_m\end{array}$	0
$\begin{array}{l}MU_m\end{array}$	0
$\begin{array}{l}MUB_m\end{array}$	$\begin{array}{l}z_c - z_k\end{array}$

Typical flow split and combination.

With reference to the figure above, HEC-RAS uses the following strategy to apply the reach connection boundary condition equations:

Apply flow continuity to reaches upstream of flow splits and downstream of flow combinations (reach 1 in the figure above). Only one flow boundary equation is used per junction.
Apply stage continuity for all other reaches (reaches 2 and 3 in the figure above). $\begin{array}{l}Z_c\end{array}$ is computed as the stage corresponding to the flow in reach 1. Therefore, stage in reaches 2 and 3 will be set equal to $\begin{array}{l}Z_c\end{array}$ .