1D Saint-Venant Equations

Most clear water hydraulic models compute the boundary friction force with a quasi-empirical formula that accounts for channel roughness, like the Manning's equation (SI units):

$\begin{array}{l}\displaystyle Q=\frac{AR^{2/3}}{n}S_f^{1/2}\end{array}$

The momentum equation incorporates this force by incorporating the dimensionless friction slope (Sf):

$\begin{array}{l}\displaystyle \frac{\partial Q}{ \partial t} +\frac{\partial (QV)}{ \partial x} + g A \left ( \frac{\partial z}{ \partial x} +S_f \right ) =0\end{array}$

Where Sf comes from the Manning equation:

$\begin{array}{l}\displaystyle Sf= \frac{ Q^2 n^2}{R^{4/3} A^2}\end{array}$

Representing empirical resisting forces as additive, dimensionless slopes allows developers to include additional forces that can collapse to one of these representative slopes. So, HEC-RAS includes unsteady contraction-expansion losses (S_CE) and wind forces (S_W) by including them as additive slopes in the momentum equation. Likewise, the Debris Library computes internal fluid forces in mud and debris flows as a new slope term (S_MD) that HEC-RAS, AdH, and GESSHA can incorporate into their momentum equation solutions:

$\begin{array}{l}\displaystyle \frac{\partial Q}{ \partial t} +\frac{\partial (QV)}{ \partial x} + g A \left ( \frac{\partial z}{ \partial x} + S_f + S_{CE} + S_W + S_{MD} \right ) =0\end{array}$

While the bed exerts a force on the fluid, the fluid also exerts a force on the bed. The bed shear stress is another way of describing the momentum exchange at the fluid boundary. Likewise, the internal forces can also be expressed as stresses. Thinking of these forces as stresses is useful because mud and debris flows depart from the relatively trivial stress-strain assumptions embedded in the clear water flow equations. Depending on the concentration and grain size, the Debris Library will assign a stress-strain model to the fluid and will compute internal shear stresses for the different internal resisting forces.
The library will then convert these internal shears into the mud and debris slope ( $\begin{array}{l}S_{MD}\end{array}$ ) to integrate these resisting process in the momentum equation by back calculating the slope from the shear:

$\begin{array}{l}\displaystyle S_{MD} = \tau_{MD}\end{array}$

Therefore, the mud and debris algorithms will identify the appropriate internal forces in the fluid, identify the appropriate stress-strain model for the fluid, compute an internal shear associated with these processes ( $\begin{array}{l}\tau_{MD}\end{array}$ ), and return as single mud and debris slope ( $\begin{array}{l}S_{MD}\end{array}$ ) that can integrate these forces into the momentum equation.