Application of the 1D Unsteady Flow Equations within HEC-RAS

The figure below illustrates the two-dimensional characteristics of the interaction between the channel and floodplain flows. When the river is rising water moves laterally away from the channel, inundating the floodplain and filling available storage areas. As the depth increases, the floodplain begins to convey water downstream generally along a shorter path than that of the main channel. When the river stage is falling, water moves toward the channel from the overbank supplementing the flow in the main channel.
Channel and floodplain flows

Because the primary direction of flow is oriented along the channel, this two-dimensional flow field can often be accurately approximated by a one-dimensional representation. Off-channel ponding areas can be modeled with storage areas that exchange water with the channel. Flow in the overbank can be approximated as flow through a separate channel.

This channel/floodplain problem has been addressed in many different ways. A common approach is to ignore overbank conveyance entirely, assuming that the overbank is used only for storage. This assumption may be suitable for large streams such as the Mississippi River where the channel is confined by levees and the remaining floodplain is either heavily vegetated or an off-channel storage area. Fread (1976) and Smith (1978) approached this problem by dividing the system into two separate channels and writing continuity and momentum equations for each channel. To simplify the problem they assumed a horizontal water surface at each cross section normal to the direction of flow; such that the exchange of momentum between the channel and the floodplain was negligible and that the discharge was distributed according to conveyance, i.e.:

1)	$\begin{array}{l}\displaystyle Q_c=\phi Q\end{array}$

Symbol	Description	Units
$\begin{array}{l}Q_c\end{array}$	flow in channel
$\begin{array}{l}Q\end{array}$	total flow
$\begin{array}{l}\phi\end{array}$	$\begin{array}{l}\frac{K_c}{K_c+K_f}\end{array}$
$\begin{array}{l}K_c\end{array}$	conveyance in the channel
$\begin{array}{l}K_f\end{array}$	conveyance in the floodplain

With these assumptions, the one-dimensional equations of motion can be combined into a single set:

2)	$\begin{array}{l}\displaystyle \displaystyle \frac{\partial A}{\partial t} + \frac{\partial (\Phi Q)}{\partial x_c} +\frac{\partial \left[ (1-\Phi ) Q \right]}{\partial x_f} =0\end{array}$

3)

$\begin{array}{l}\displaystyle \displaystyle \frac{\partial Q}{\partial t} + \frac{\partial \left( \frac{\Phi ^2 Q^2}{A_c} \right)}{\partial x_c} + \frac{\partial \left( \frac{ (1-\Phi ^2) Q^2}{A_f} \right)}{\partial x_f} +gA_c \left[ \frac{\partial Z}{\partial x_c} +S_{fc} \right] + gA_f \left[ \frac{\partial z}{\partial x_f} + S_{ff} \right] =0\end{array}$

in which the subscripts c and f refer to the channel and floodplain, respectively. These equations were approximated using implicit finite differences, and solved numerically using the Newton-Raphson iteration technique. The model was successful and produced the desired effects in test problems. Numerical oscillations, however, can occur when the flow at one node, bounding a finite difference cell, is within banks and the flow at the other node is not.

Expanding on the earlier work of Fread and Smith, Barkau (1982) manipulated the finite difference equations for the channel and floodplain and defined a new set of equations that were computationally more convenient. Using a velocity distribution factor, he combined the convective terms. Further, by defining an equivalent flow path, Barkau replaced the friction slope terms with an equivalent force.

The equations derived by Barkau are the basis for the unsteady flow solution within the HEC-RAS software. These equations were derived above. The numerical solution of these equations is described in the next sections.