Subgrid Bathymetry

Subgrid Bathymetry. Modern advances in the field of airborne remote sensing can provide very high resolution topographic data. In many cases the data is too dense to be feasibly used directly as a grid for the numerical model. This situation presents a dilemma in which a relatively coarse computational grid must be used to produce a fluid simulation, but the fine topographic features should be incorporated in the computation.

The solution to this problem that HEC-RAS uses is the sub-grid bathymetry approach (Casulli, 2008). The computational grid cells contain some extra information such as hydraulic radius, volume and cross sectional area that can be pre-computed from the fine bathymetry. The high resolution details are lost, but enough information is available so that the numerical method can account for the fine bathymetry through mass conservation. For many applications this method is appropriate because the free water surface is smoother than the bathymetry; therefore, a coarser grid can effectively be used to compute the spatial variability in free surface elevation. In the figure above, the fine grid is represented by the Cartesian grid in gray and the computational grid is displayed in blue.

The derivation of variables follows the definition of variables excluding porosity and then new variables are defined which include porosity. This approach is convenient since the cell and face hydraulic property tables are computed without porosity and porosity is defined separately at cells and faces as a function of elevation.

Cell Hydraulic Property Curves

The cell volumes are computed from the horizontal wetted area as

$\begin{array}{l}\displaystyle \displaystyle \Omega_c (z_s) = \int_{z_b}^{z_s} A_c^W(z)dz\end{array}$

$\begin{array}{l}\displaystyle \displaystyle A_c^W(z_s) = \dfrac{d \Omega_c(z_s)}{d z_s}\end{array}$

where

$\begin{array}{l}\Omega_c(z_s)\end{array}$ : cell volume [L³]
$\begin{array}{l}A_c^W(z_s)\end{array}$ : cell wetted area [L²]
$\begin{array}{l}z_s\end{array}$ : water surface elevation [L]

The cell wetted areas and are defined as piece-wise constant curves

$\begin{array}{l}\displaystyle A_c^W (z_s) = \begin{cases} A_{c,i}^W & \forall \text{ $ z_{s,i} \le z_s < z_{s,i+1}$} \\ A_{c,N}^W & \forall \text{ $ z_{s,N} \le z_{s} $} \end{cases}\end{array}$

The cell volume curves can be described exactly from relatively simple piece-wise linear curves

$\begin{array}{l}\displaystyle \Omega (z_s) = \begin{cases} \Omega_{c,i} + A_{c,i}^W (z_s - z_{s,i}) & \forall \text{ $ z_{s,i} \le z_s < z_{s,i+1}$} \\ \Omega_{c,N} + A_{c,N}^W (z_s - z_{s,N})& \forall \text{ $ z_{s,N} \le z_{s} $} \end{cases}\end{array}$

Faces Hydraulic Property Curves

The face vertical area is computed as

$\begin{array}{l}\displaystyle \displaystyle A_f(z_s) = \int_{z_b}^{z_s} W_f (z)dz\end{array}$

$\begin{array}{l}\displaystyle \displaystyle W_f (z_s) = \dfrac{d A_f(z_s)}{d z_s}\end{array}$

where

$\begin{array}{l}A_f(z_s)\end{array}$ : face area [L²]
$\begin{array}{l}W_f(z_s)\end{array}$ : face wetted width [L]
$\begin{array}{l}z_s\end{array}$ : water surface elevation [L]

The wetted perimeter without porosity is defined as

$\begin{array}{l}\displaystyle \displaystyle P_f (z_s) = \int \sqrt {1 + \left ( \frac{dx}{dy} \right )^2 } dy\end{array}$

The face wetted width is described with the piecewise constant curves

$\begin{array}{l}\displaystyle W_f (z_s) = \begin{cases} W_{f,i} & \forall \text{ $ z_{s,i} \le z_s < z_{s,i+1}$} \\ W_{f,N} & \forall \text{ $ z_{s,N} \le z_{s} $} \end{cases}\end{array}$

Integrating the face wetted width leads to the following piecewise linear curve for the face area

$\begin{array}{l}\displaystyle A_f (z_s) = \begin{cases} A_{f,i} + W_{f,i} (z_s - z_{s,i}) & \forall \text{ $ z_{s,i} \le z_s < z_{s,i+1}$} \\ A_{f,N} + W_{f,N} (z_s - z_{s,N})& \forall \text{ $ z_{s,N} \le z_{s} $} \end{cases}\end{array}$

From these variables other hydraulic variables such as the hydraulic radius and conveyance can be easily computed.

In the figure below, the left figure represents the shape of a face as seen in the fine grid and the corresponding function for face area $\begin{array}{l}A_f\end{array}$ in terms of the water surface elevation $\begin{array}{l}z_s\end{array}$ .

Cell Face Terrain Data and Property Table.

Application of Porosity to Hydraulic Property Curves

The cell horizontal wetted area is modified to account for porosity as

$\begin{array}{l}\displaystyle \displaystyle A_c^W (z_s) = \phi_c (z_s) \tilde A_c^W(z_s)\end{array}$

where

$\begin{array}{l}A_c^W(z_s)\end{array}$ : cell wetted area [L²]
$\begin{array}{l}\tilde A_c^W(z_s)\end{array}$ : cell wetted area excluding porosity [L²]
$\begin{array}{l}\phi_c (z_s)\end{array}$ : cell porosity [-]

The face wetted width is modified to account for porosity as

$\begin{array}{l}\displaystyle \displaystyle W_f(z_s) = \phi_f (z_s) \tilde W_f(z_s)\end{array}$

where

$\begin{array}{l}W_f (z_s)\end{array}$ : face wetted width [L]
$\begin{array}{l}\tilde W_f (z_s)\end{array}$ : face wetted width excluding porosity [L]
$\begin{array}{l}\phi_f (z_s)\end{array}$ : face porosity [-]

Porosity is applied to the face wetted perimeter similarly to the face wetted area.

The cell porosity and are defined as piece-wise constant curves consistent with the definition of the cell wetted area and face wetted width.