Bridge Modeling in 2D

There are two general approaches for modeling bridges in 2D: (1) Simplified 1D/2D Bridge Modeling, and (2) Detailed Bridge Modeling. The simplified 1D/2D approach is designed for cases where the details of the bridge hydraulics are not important and only the overall energy/head losses are important for modeling areas around the bridge. The simplified 1D/2D approach is faster, requires less resolution of bathymetry around the bridge, but can be more difficult to calibrate since there are more parameters in the 1D modeling approaches. The simplified 1D/2D approach can be applied for all types of bridge flows including pressurized flow and bridge overtopping. The detailed modeling approach applies a fully 2D mesh to resolve the details of the bridge hydraulics. The detailed bridge approach is more expensive and requires detailed geometry of the bridge. Another limitation of detailed bridge modeling is that it is currently only applicable to low flow conditions. Fully 2D pressure bridge flow and bridge overtopping will be added in future version of HEC-RAS. Further details on the two approaches are described in detail in the sections below. The simplified 1D/2D bridge modeling approach is available for all of the 2D solvers including the Diffusive Wave Solver. This is considered acceptable since, the purposes of the simplified 1D/2D bridge modeling approach is to capture the overall bridge head losses and not the details of the flow hydraulics through the bridge. The detailed bridge modeling approach should only be applied with the non-linear shallow water equations solvers SWE-ELM and SWE-EM.

Simplified 1D/2D Bridge Modeling

In this approach, the bridge curves for modeling bridges in 1D are utilized to enforce flows through the bridge as a function of the computed head loss through the bridge. The bridge curves are generated using automatically generated 1D cross-sections upstream and downstream of the bridge. This approach is useful when only the head losses are of interest for the purposes of modeling areas other than the bridge itself. It is not appropriate for detailed modeling of bridges since the approach requires a relatively coarse mesh. The approach can be utilized for all hydraulic flow regimes including low flow with and without weir flow and pressure flow with and without overtopping. The bridge geometry including piers and deck are included in the 1D bridge geometry.

Drag Factor

Additional head losses are incurred inside the bridge region which are not represented explicitly in the standard momentum equation and need to be included in an additional term denoted by a slope $\begin{array}{l}\textbf{S}_B\end{array}$ . Without the additional losses the 2D model will generally under-predict the headwater water surface elevation compared to the 1D hydraulic bridge curves.

1)	$\begin{array}{l}\displaystyle \displaystyle \boldsymbol{\tau}_b = \rho \lambda_B C_D \| \boldsymbol{V} \| \boldsymbol{V}\end{array}$

where

$\begin{array}{l}\rho\end{array}$ : water density
$\begin{array}{l}\lambda_B\end{array}$ : bridge drag factor
$\begin{array}{l}C_D\end{array}$ : drag coefficient

The drag factor is computed using the Manning’s roughness coefficient as

2)	$\begin{array}{l}\displaystyle \displaystyle C_D = \frac{n^2 g}{R^{1/3}}\end{array}$

where

$\begin{array}{l}n\end{array}$ : Manning's roughness coefficient [T/L^1/3]
$\begin{array}{l}R\end{array}$ : hydraulic radius [L]
$\begin{array}{l}g\end{array}$ : gravitational acceleration [L/T²]

The factor $\begin{array}{l}\lambda_B\end{array}$ represents the additional drag needed for the 2D solution to match the 1D bridge hydraulic curves. From the above equation, it is clear that the square root of the factor $\begin{array}{l}\lambda_B\end{array}$ is proportional to the modified or equivalent Manning's roughness coefficient:

$\begin{array}{l}\displaystyle n' = \sqrt {\lambda_B} n\end{array}$

where

$\begin{array}{l}n'\end{array}$ : modified or equivalent Manning's roughness coefficient including the enhanced drag of the bridge [T/L³]
$\begin{array}{l}n'\end{array}$ : Manning's roughness coefficient [T/L³]
$\begin{array}{l}\lambda_B\end{array}$ : bridge drag enhancement factor (O(1)) [-]

The drag is increased in all of the faces under the bridge uniformly. In reality, the increased drag under the bridge is not uniform and certain areas under a bridge experience more drag than others. However, it is not clear how to distribute the increased flow under the bridge and this would significantly complicate the method. For simplicity, in HEC-RAS 6.5, the drag enhancement factor is constant for each bridge. The relaxation para is set to 0.1 for the Shallow Water Equation (SWE) models, and 0.01 for the Diffusive Wave Equation (DWE) model. The main reason for the difference is because the DWE reacts a lot faster to changes in $\begin{array}{l}\lambda_B\end{array}$ because of the lack of the unsteady term in the momentum equation. Another important distention between the DWE and SWE implementations is how frequently $\begin{array}{l}\lambda_B\end{array}$ is updated. In the case of the DWE solver, $\begin{array}{l}\lambda_B\end{array}$ is updated only once per time step while in the case of the SWE models, it is updated every outer loop iteration.

PID Controller

The drag enhancement factor is computed with a Proportional-Integral-Derivative (PID) Controller. The PID Controller is a common algorithm which employees a closed loop feedback mechanism.

$\begin{array}{l}\displaystyle \lambda_B(t) = K_p e(t) +K_i \int_0^t{e(\tau)\text{d}\tau} + K_d \frac{\text{d} e(t)}{\text{d} t}\end{array}$

where

$\begin{array}{l}\lambda_B\end{array}$ : bridge drag enhancement factor (O(1)) [-]
$\begin{array}{l}K_i\end{array}$ : integral gain
$\begin{array}{l}K_d\end{array}$ : derivative gain
$\begin{array}{l}e(t) = z_{s,H} - \bar{z}_{s,H}\end{array}$ : error value [L]
$\begin{array}{l}z_{s,H} = f \left ( Q_B, \overline{z}_{s,T} \right )\end{array}$ : headwater computed from 1D bridge curves [L]
$\begin{array}{l}Q_B\end{array}$ : total bridge flow [L³/T]
$\begin{array}{l}\overline{z}_{s,T}\end{array}$ : average water surface at tailwater cross-section [L]
$\begin{array}{l}\overline{z}_{s,H}\end{array}$ : average water surface at headwater cross-section [L]

The headwater from 1D bridge curves ( $\begin{array}{l}z_{s,H}\end{array}$ ) is referred to as the setpoint (SP) and is the target variable which is being controlled. The computed average headwater ( $\begin{array}{l}\overline{z}_{s,H}\end{array}$ ) is referred to as the process variable (PV). The controller tries to minimize the error between the SP and PV based on proportional, integral, and derivative terms by adjusting the control variable (CV) which in this case is the bridge drag enhancement factor ( $\begin{array}{l}\lambda_B\end{array}$ ). The terms on the right-hand-side of the above equation are referred to as the proportional, integral, and derivative terms, respectively.

The above form of the PID Controller is referred to as the positional form. An alternative form is which is used here is the velocity form which is obtained by differentiating the positional form:

$\begin{array}{l}\displaystyle \frac{\text{d} \lambda_B(t)}{\text{d} t} = K_i e(t) + K_p \frac{\text{d} e(t)}{\text{d} t} + K_d \frac{\text{d}^2 e(t)}{\text{d} t^2}\end{array}$

where the terms in the above equation have been rearranged based on the order of the terms. The main advantage of the velocity form is that changes in the proportional and integral terms do no lead to sudden jumps in the controller output. The derivatives in the above equation are approximated by backward finite-differences and the equation is solved with an explicit Forward Euler time stepping scheme. The discrete form of the above equation is given by

$\begin{array}{l}\displaystyle \frac{\lambda_B^{n+1} - \lambda_B^n}{\Delta t} = K_i e^n + K_p E^n + K_d \frac{E^n - E^{n-1}}{\Delta t}\end{array}$

where

$\begin{array}{l}e^n = z_{s,H}^n - \bar{z}_{s,H}^n\end{array}$ : error value/term [L]
$\begin{array}{l}E^n = \frac{e^n - e^{n-1}}{\Delta t}\end{array}$ : error derivative [L/T]
$\begin{array}{l}n\end{array}$ : superscript indicating the time step level
$\begin{array}{l}\Delta t\end{array}$ : time step [T]

A low-pass filter with infinite impulse response (IIR) is applied to the error derivatives to improve stability:

$\begin{array}{l}\displaystyle E^n = (1 - r) E^{n-1} + r \frac{e^n - e^{n-1}}{\Delta t}\end{array}$

where

$\begin{array}{l}r\end{array}$ : smoothing factor [-]
$\begin{array}{l}n\end{array}$ : superscript indicating the time step level
$\begin{array}{l}\Delta t\end{array}$ : time step [T]

The improve the convergence of the solvers, the drag enhancement factor is only calculated once per time step and is not updating during the time step outer-loop iterations. The PID Controller has in total 4 parameters which need to be calibrated or tuned. In HEC-RAS these values are hard-coded based on the governing equations being solved.

Average Headwater and Tailwater

The average tail and headwater surfaces are computed using the the cell volumes and cross-section arc intersection lengths as weights for each cell as:

$\begin{array}{l}\displaystyle \overline{z}_{s,H|T} = \frac{\sum_i { \Omega_i l_i z_{s,i}}}{\sum_i { \Omega_i l_i }}\end{array}$

where

$\begin{array}{l}\overline{z}_{s,H}, \overline{z}_{s,T}\end{array}$ : average headwater and tailwater water surface elevations [L]
$\begin{array}{l}z_{s,i}\end{array}$ : cell water surface [L]
$\begin{array}{l}\Omega_i\end{array}$ : cell water volume [L³]
$\begin{array}{l}l_i\end{array}$ : cross-section arc length within cell [L]

The above computation method is only first-order and has some limitations. However, given the limitations and approximations of the simplified 1D/2D bridges, it is considered appropriate and sufficient. Future versions of HEC-RAS may have more sophisticated methods for computing the average tail and headwater surface elevations.

Detailed Bridge Modeling

In the detailed modeling approach, a high-resolution computational mesh is utilized to simulate the detailed bridge hydraulics. In version 6.5 and earlier, the method is limited to low flow conditions and cannot simulate pressured flow with or without overtopping. This approach is much more computationally expensive than the simplified 1D/2D approach. However, it provides the highest accuracy of bridge hydraulics. The bridge piers should be be included in the terrain.

Note

The detailed bridge modeling approach does NOT automatically modify the terrain to account for the bridge geometry. Therefore, the user must make sure the bridge geometry with the exception of the bridge deck is included in the terrain.