Non-Erodible Surfaces

Non-erodible surfaces (also known as hard bottoms) may be simulated in HEC-RAS and are specified using a minimum bed elevation,$//<![CDATA[ \begin{array}{l}z_{b,\text{min}}\end{array} //]]>$ , or maximum depth, $//<![CDATA[ \begin{array}{l}h_{\text{max}}\end{array} //]]>$ h_max. In the 2D model the non-erodible surfaces are specified at both cells and faces independently. Non-erodible surfaces are modeled at the subarea scale by limiting the erosion rate so that the minimum bed elevation is preserved (i.e. $//<![CDATA[ \begin{array}{l}z_{bi}\end{array} //]]>$≥ $//<![CDATA[ \begin{array}{l}z_{b,\text{min}}\end{array} //]]>$). The time stepping scheme is given by $//<![CDATA[ \begin{array}{l}z_{bi}^{n+1} = z_{bi}^n+\Delta z_{bi}\end{array} //]]>$ in which $//<![CDATA[ \begin{array}{l}z_{bi}^n\end{array} //]]>$ is the previous bed elevation, $//<![CDATA[ \begin{array}{l}z_{bi}^{n+1}\end{array} //]]>$ is the new bed elevation, and $//<![CDATA[ \begin{array}{l}\Delta z_b\end{array} //]]>$ is the bed change. Assuming that the bed elevation at the current time is higher than the hard bottom (i.e. $//<![CDATA[ \begin{array}{l}z_{bi}^n\end{array} //]]>$≥$//<![CDATA[ \begin{array}{l}z_{b,\text{min}}\end{array} //]]>$). Inserting the above equation into the bed change equation leads to the hard-bottom limited erosion rate

where

$//<![CDATA[ \begin{array}{l}E_{tki,hb}\end{array} //]]>$ : hard-bottom limited fractional erosion rate [M/L²/T]

$//<![CDATA[ \begin{array}{l}D_{tki}\end{array} //]]>$ : fractional deposition rate [M/L²/T]

$//<![CDATA[ \begin{array}{l}\Psi_{na,i}\end{array} //]]>$ : non-alluvial erosion limiter [-]

$//<![CDATA[ \begin{array}{l}ρ_{d1}\end{array} //]]>$ : dry density of active layer [M/L³]

$//<![CDATA[ \begin{array}{l}f_{1ki}\end{array} //]]>$ : grain fractions by weight [-]

$//<![CDATA[ \begin{array}{l}f_{M}\end{array} //]]>$ : morphologic acceleration factor [-]

$//<![CDATA[ \begin{array}{l}\Delta t\end{array} //]]>$ : computational time step [T]

The non-alluvial erosion limiter $//<![CDATA[ \begin{array}{l}\Psi_{na,i}\end{array} //]]>$  is function which limits the maximum erosion rate based on the proximity of the mobile bed surface to the non-erodible surface and is computed here as 

where

$//<![CDATA[ \begin{array}{l}\delta_{1i}^n\end{array} //]]>$ :  active layer thickness [L]

$//<![CDATA[ \begin{array}{l}\delta_{1i}^*\end{array} //]]>$  : potential or maximum active layer thickness if there was no non-erodible surface present [L]

When $//<![CDATA[ \begin{array}{l}\delta_{1i}^n\end{array} //]]>$ = 0, no erosion is possible and the maximum erosion rate becomes equal to the deposition rate (i.e. $//<![CDATA[ \begin{array}{l}E_{tki,hb}= D_{tki}\end{array} //]]>$) . Likewise when $//<![CDATA[ \begin{array}{l}\delta_{1i} = \delta_{1i}^*\end{array} //]]>$, the maximum erosion rate is equal to the amount of material in the active layer available to erode in a timestep plus the deposition rate. 

The hard-bottom limited erosion rate is therefore

The bed-slope term and avalanching algorithm are also modified so that only deposition may occur over non-erodible surfaces following an approach similar to that described above.

When simulating multiple grain classes, the bed gradation can significantly change at non-erodible surfaces during a time step. This means that the model may require at least one to two iterations to converge when simulating non-erodible surfaces with multiple grain classes.