Example Application

In order to demonstrate how to estimate breach parameters, an example application for a fictitious dam is provided below. The event being evaluated in the example is a PMF scale event. This process for developing breach parameters needs to be performed for each failure mode/event (fully modeled hydrologic event or pool elevation for sunny day failures). The following is the necessary information required about a dam in order to develop breach parameter estimates as outlined in these guidelines.

Reservoir Data:

Important Pool Elevations	Elevation (m)	Volume (m³)
Stream Bed	1678.0	0.0
Multipurpose Pool	1692.1	15.81x10⁶
Top of Flood Control	1710.0	151.64x10⁶
Top of Dam	1720.9	327.01x10⁶
PMF Max Water Surface	1722.26	357.98x10⁶

Dam Embankment Data:
Crest Length:	4360 m
Crest Width:	9.15 m
Maximum Height above river bed:	42.9 m
Average Upstream Embankment slope:	3.3H:1V
Average Downstream Embankment slope:	3.3H:1V
Embankment Material:	Rolled earth, zoned
Embankment Core:	Impervious core, clay
Upstream slope Protection:	18" riprap
Downstream slope protection:	Topsoil and grass

Regression Equations:

For this example, the Froehlich (1995b), Froehlich (2008), MacDonald and Langridge-Monopolis (1984), Von Thun and Gillette (1990), and Xu Zhang (2009) regression equations for predicting breach size and development time were used. This dam is within the range of the data used to develop these regression equations, therefore the equations are considered to be an appropriate methodology for estimating the breach parameters. During the PMF event for this dam it is overtopped by 1.36 meters. The mode of failure for this example will be assumed as an overtopping failure. The failure location is assumed to be at the main channel centerline. The breach bottom elevation is assumed to be at an elevation of 1678 m (invert of the main channel). The water surface elevation at the initiation of the breach will be at an elevation of 1722.26 m (max pool for PMF event). The following are the calculations for each method.

Froehlich (1995a):

$\begin{array}{l}\displaystyle B_{ave} = 0.1803 K_o V_w^{0.32} h_b^{0.19}\end{array}$

$\begin{array}{l}\displaystyle B_{ave} = 0.1803 (1.4) (357.98 \times 106)^{0.32} (42.9)^{0.19}\end{array}$

$\begin{array}{l}\displaystyle \textbf{B} _{ave} \textbf{= 281.5 m}\end{array}$

$\begin{array}{l}\displaystyle t_f = 0.00254 V_w^{0.53} h_b^{-0.90}\end{array}$

$\begin{array}{l}\displaystyle t_f = 0.00254 (357.98 \times 106)^{0.53} (42.9)^{-0.90}\end{array}$

$\begin{array}{l}\displaystyle \textbf{t} _f \textbf{= 2.95 hrs}\end{array}$

The Froehlich (1995a) method assumes a side slope of 1.4H:1V for an overtopping breach. Given the breach height of 42.9 meters, this yields a bottom width for the breach of W_b = 221.4 m.

Froehlich (2008):

$\begin{array}{l}\displaystyle B_{ave} = 0.27 K_o V_w^{0.32} h_b^{0.04}\end{array}$

$\begin{array}{l}\displaystyle B_{ave} = 0.27 (1.3) (357.98 \times 106)^{0.32} (42.9)^{0.04}\end{array}$

$\begin{array}{l}\displaystyle \textbf{B} _{ave} \textbf{= 222.76 m}\end{array}$

$\begin{array}{l}\displaystyle t_f = 63.2 ( V_w /(gh_b^2))^{0.5}\end{array}$

$\begin{array}{l}\displaystyle t_f = 63.2 (357.98 \times 106 /(9.80665 \times (42.9)^2))^{0.5}\end{array}$

$\begin{array}{l}\displaystyle \textbf{t} _f \textbf{= 2.47 hrs}\end{array}$

The Froehlich (2008) method assumes a side slope of 1.0H:1V for an overtopping breach. Given the breach height of 42.9 meters, this yields a bottom width for the breach of W_b = 179.86 m.

MacDonald and Langridge-Monopolis:*

The MacDonald and Langridge-Monopolis equation for an earthfill dam with a clay core is:

$\begin{array}{l}\displaystyle V_{eroded} = 0.00348 (V_{out} * h_w)^{0.852}\end{array}$

Since the outflow volume through the breach is unknown before performing the analysis, a good starting estimate is the volume of water in the dam at the peak stage of the event.

$\begin{array}{l}\displaystyle V_{eroded} = 0.00348 (357.98 \times 106 * 44.26 )^{0.852}\end{array}$

$\begin{array}{l}V_{eroded} = 1.70556 \times 10^6 m^3\end{array}$ of material

To compute the bottom width of the breach, the method says to use side slopes of 0.5H:1V. The user must also estimate an average side slope for both the upstream and downstream embankment of the dam. For this example average side slopes of 3.3H:1V were used for both upstream and downstream. Then using the bottom width equation (State of Washington, 1992):

$\begin{array}{l}\displaystyle \displaystyle W_b = \frac{V_{eroded} -h^2_b (CZ_b +h_b Z_b Z_3 /3)}{h_b (C+ h_b Z_3 /2)}\end{array}$

$\begin{array}{l}\displaystyle W_b = (1.70556 \times 10^6 – 42.9^2(9.15*0.5 + 42.9*0.5*6.6/3))/(42.9(9.15 + 42.9*6.6/2))\end{array}$

$\begin{array}{l}\displaystyle \textbf{W}_b \textbf{= 249.0 m}\end{array}$

$\begin{array}{l}\displaystyle t_f = 0.0179 (V_{eroded})^{0.364}\end{array}$

$\begin{array}{l}\displaystyle t_f = 0.0179 (1.70556 \times 10^6)^{0.364}\end{array}$

$\begin{array}{l}\displaystyle \textbf{t}_f \textbf{= 3.32 hrs}\end{array}$

Note

Once an actual breach hydrograph is computed with the MacDonald and Langridge-Monopolis parameters, the volume of water coming out of the breach should be calculated, and the parameters should be re-estimated using that volume of water for V_out.

Von Thun and Gillette:

The Von Thun and Gillette equation for the breach average width is:

$\begin{array}{l}\displaystyle B_{ave} = 2.5 * h_w + C_b\end{array}$

$\begin{array}{l}\displaystyle B_{ave} = 2.5 * 44.26 + 54.9\end{array}$

$\begin{array}{l}\displaystyle \textbf{B} _{ave} \textbf{= 165.6 m}\end{array}$

Von Thun and Gillette suggest using breach side slopes of 0.5H:1V for earthen dams with a clay core. Given the dam height of 42.9 meter, the Breach bottom width will be W_b = 144.2 m.

Von Thun and Gillette show two equations for predicting the breach failure time. One equation is a function of the depth of water only, while the other is a function of depth of water and the computed average breach width. Both equations are used below.

$\begin{array}{l}t_f = 0.02 * h_w + 0.25\end{array}$	$\begin{array}{l}t_f = B_{ave}/(4*h_w)\end{array}$
$\begin{array}{l}t_f = 0.02 * 44.26 + 0.25\end{array}$	$\begin{array}{l}t_f = 166/(4*44.26)\end{array}$
$\begin{array}{l}\textbf{t} _f \textbf{= 1.14 hrs}\end{array}$	$\begin{array}{l}\textbf{t} _f \textbf{= 0.94 hrs}\end{array}$

Both of the Von Thun and Gillette equations yield similar answers for the breach time. Reviewing the Von Thun and Gillette paper showed that the data they used in their experiments were mostly earthen embankments with slightly cohesive materials. Given that the example dam we are studying has an engineered clay core, the longer time estimate is probably more appropriate. Therefore the selected failure time is t_f = 1.14 hrs.

Xu and Zhang (2009):

The Xu and Zhang equation for the breach average width is:

$\begin{array}{l}\displaystyle \displaystyle \frac{B_{ave}}{h_b} = 0.787 \left( \frac{h_d}{h_r} \right) ^{0.133} \left( \frac{V^{1/3}_w}{h_w} \right) ^{0.652} e^{B_3}\end{array}$

$\begin{array}{l}\displaystyle B_{ave} = (42.9)(0.787)(42.9/15)^{0.133}((357.98 \times 10^6)^{1/3}/44.26)^{0.652} e^{-0.283}\end{array}$

$\begin{array}{l}\displaystyle \textbf{B} _{ave} \textbf{= 178.67 m}\end{array}$

$\begin{array}{l}\displaystyle \displaystyle \frac{B_t}{h_b} = 1.062 \left( \frac{h_d}{h_r} \right) ^{0.092} \left( \frac{V^{1/3}_w}{h_w} \right) ^{0.508} e^{B_2}\end{array}$

$\begin{array}{l}\displaystyle B_t = (42.9)(1.062)(42.9/15)^{0.092}((357.98 \times 10^6)^{1/3}/44.26)^{0.508} e^{0.071}\end{array}$

$\begin{array}{l}\displaystyle \textbf{B} _t \textbf{= 220.64 m}\end{array}$

Based on the computation of B_ave and B_t above, the breach bottom width for this method is W_b = 136.7 and the side slopes are Z = 0.98H:1V.

The breach development time from the Xu and Zhang equation is as follows:

$\begin{array}{l}\displaystyle \displaystyle \frac{T_f}{T_r} = 0.304 \left( \frac{h_d}{h_r} \right) ^{0.707} \left( \frac{V^{1/3}_w}{h_w} \right) ^{1.228} e^{B_5}\end{array}$

$\begin{array}{l}\displaystyle T_f = (1.0)(0.304)(42.9/15)^{0.707}((357.98 \times 10^6)^{1/3}/44.26)^{1.228} e^{-0.327}\end{array}$

$\begin{array}{l}\displaystyle \textbf{T} _f \textbf{= 13.92 Hrs*}\end{array}$

Note

Please see note about the Xu Zhang method over estimating the breach time under the method description above

Physically-Based Breach Computer Models:

For this example, Dr. Fread's NWS-BREACH model was the only physically based breach model run to make an estimate of breach parameters. The physical dimensions of the dam, the soil properties, and the hydrologic event data were entered into the BREACH model. The results from the BREACH model for this example are:

Breach Bottom Width W_b	238 m
Breach Side Slopes	0.9H:1V
Breach Failure Time t_f	4.2 hrs

Summary Results for Breach Parameters:

Shown in the table below is a summary of the breach parameters computed from the regression equations and the NWS-BREACH model.

Summary of breach parameter estimates

Method	Breach Bottom Width (meters)	Breach Side Slopes (H:1V)	Breach Failure Time (hours)
Froehlich (1995a)	221.4	1.4	2.95
Froehlich (2008)	179.9	1.0	2.47
MacDonald and Langridge-Monopolis	249.0	0.5	3.32
Von Thun and Gillette	144.2	0.5	1.14
Xu and Zhang (2009)	136.7	0.98	13.92*
NWS-BREACH Computer Model	238	0.9	4.2

The data Xu and Zhang used in the development of their equation for breach development time includes more of the initial erosion period and post erosion period than what is generally used in HEC-RAS for the critical breach development time. In general, this equation will produce breach development times that are greater than the other four equations described above. Because of this fact, the Xu and Zhang equation for breach development time should not be used in HEC-RAS.

From here, all six sets of parameters should be entered into the HEC-RAS software and run as separate breach plans. This will result in six different breach outflow hydrographs. However, once the hydrographs are routed downstream, they will begin to converge towards each other. The selection of a final set of breach parameters for this event should be based on guidance provided above in the "Recommended Approach" section of this document.