Breach flow discharge prediction - Analysis of USDA breach flow dataset

Integrated Flood Risk Analysis

and Management Methodologies

Breach flow discharge prediction

ANALYSIS OF USDA BREACH FLOW DATASET

Date April

2008

Report Number

T06-08-03

Revision Number 4_3_P01

Task Leader HR Wallingford

FLOODsite is co-funded by the European Community

Sixth Framework Programme for European Research and Technological Development (2002-2006) FLOODsite is an Integrated Project in the Global Change and Eco-systems Sub-Priority

Start date March 2004, duration 5 Years Document Dissemination Level

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OCUMENT

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NFORMATION

Title Breach flow discharge prediction: analysis of USDA breach flow _dataset

Lead Author Sylvain Néelz

Contributors [Click here and list Contributors]

Distribution [Click here and list Distribution]

Document Reference T06-08-03

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OCUMENT

H

ISTORY

Date Revision Prepared by Organisation Approved by Notes

27/06/08 4_0_P01 S. Néelz HR Wallingford

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CKNOWLEDGEMENT

The work described in this publication was supported by the European Community’s Sixth Framework Programme through the grant to the budget of the Integrated Project FLOODsite, Contract GOCE-CT-2004-505420.

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ISCLAIMER

This document reflects only the authors’ views and not those of the European Community. This work may rely on data from sources external to the FLOODsite project Consortium. Members of the Consortium do not accept liability for loss or damage suffered by any third party as a result of errors or inaccuracies in such data. The information in this document is provided “as is” and no guarantee or warranty is given that the information is fit for any particular purpose. The user thereof uses the information at its sole risk and neither the European Community nor any member of the FLOODsite Consortium is liable for any use that may be made of the information.

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UMMARY

This report concerns the analysis of flow/discharge measurements obtained from a set of 441 “rigid breach” models constructed between 22/07/1996 and 10/12/1998 at the Agricultural Research Service of the United States Department of Agriculture (USDA) in Stillwater, Oklahoma, US, by G.J. Hanson and colleagues. Each model consisted of a rigid plywood dam cut by a rigid breach. The program of experimental research was aimed at improving models of prediction of discharge through breached embankments, which is the first stage in the prediction of the hydraulic quantities of relevance to embankment erosion and sediment transport in breaches. In contrast with a previous analysis by Temple and Hanson of a subset of the data which consisted in calibrating existing weir discharge prediction equations or a combination of weir equations to obtain a best fit of the measured discharges, the work reported herein concentrates mainly on 1) verifying established weir equations and published guidance on the parameterisation of these equations using the USDA laboratory dataset, 2) testing an approach consisting in combining two weir equations and using the dataset to understand its limitations (the dataset also contains >5000 images which help understanding the physical processes involved in each combination of geometry and flow), 3) using the dataset to understand complex flow patterns observed that are of relevance to embankment breach flow, and 4) suggesting ways in which weir flow equations could be adapted to predict flow in various breach shapes.

Conclusions include the following: 1) The broad crested weir equation is adequate for any shape from the USDA experiments not involving a “headcut” overfall and with vertical breach side walls, independently of conditions on entry into the breach such as the upstream slope angle; 2) An established sharp crested rectangular weir flow equation adapted for contracted flow (Kindsvater and Carter) is adequate for the breach geometry with a headcut and a vertical upstream face. This also performed well when used additively with a V-notch equation for the cases with sloping breach sides. (although none of these shapes is directly relevant to real embankment breaches); 3) The effect of the upstream slope on the magnitude of the flow for the geometry involving a headcut overfall is explained; 4) An approach is suggested consisting in adding a rectangular and a V-notch flow component for all geometries involving non-vertical breach sides or/and a non-vertical upstream slope. In every case the appropriate angle between the breach lateral crest and the vertical needs to be considered in the calculation of the V-notch flow.

However, a number of limitations are also highlighted by the study, which are expressed as follows: 1) A simple approach (used in HR Breach) to predict the flow for breach shapes with a lateral slope (case with no headcut overfall) is found to consistently overestimate the measured flows; 2) The approach suggested in 3) and 4) above, relevant to breach flow over a headcut reaching the upstream slope of a dam is shown to deteriorate in accuracy with narrower breaches, higher flows, steeper upstream slopes and steeper (towards vertical) side angles. Reasons for this are suggested (convergence and/or downstream control), and it is demonstrated that a correction factor may be expressed as a function of the dimensionless head H/b, the upstream slope, and the angle of the breach lateral crest in relation to the longitudinal direction (which is 0 for vertical breach walls); and 3) the above approach overestimates flows in a case involving a very wide breach. An explanation for this last observation is suggested.

Practical implications of the study for future breach flow modelling research are mainly as follows: 1) It is highly important to distinguish between sharp weir type flow (headcut towards its final phase of development) where conditions on entry into the breach, such as the occurrence of a “lateral component”, are critically relevant to the discharge prediction, and broad-crested weir type flow, where the flow is primarily governed by the transverse geometry and dimensions of the breach (the study demonstrates that if the correct flow mode is not identified large errors - up to ~100% - are likely); 2) A more accurate flow prediction relies on a better understanding of phenomena such as contraction (when the nappe is narrower than the weir crest), convergence (when flow components coming from different directions converge into each other) and drowning (downstream control).

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ONTENTS Document Information ii Document History ii Acknowledgement ii Disclaimer ii Summary iii Contents v

Integrated Flood Risk Analysis ... 1

and Management Methodologies ... 1

Breach flow discharge prediction... 1

ANALYSIS OF USDA BREACH FLOW DATASET... 1

Date April 2008 ... 1

Report Number... 1

T06-08-03... 1

1. Introduction ... 1

2. Experimental setup... 1

2.1 Shape of the dam and breach ... 1

2.2 Details of parameter values tested ... 3

3. Relevant weir flow equations ... 4

3.1 Rectangular weir... 4

3.1.1 Broad -crested ... 4

3.1.2 Sharp-crested... 4

3.2 Sharp-crested V-notch weir ... 5

3.3 Comments on “fine tuning” ... 5

4. Previous dataset analyses ... 6

5. Sensitivity analysis... 6

5.1 Breach width b... 7

5.2 Upstream slope angle φ ... 7

5.2.1 For Hh=0.305m... 7

5.2.2 For Hh=0.152m... 8

5.2.3 For Hh=0 (no headcut) ... 8

5.2.4 Conclusion for Hh ... 9

5.3 Position of the upstream apron Hu ... 9

5.4 Headcut height Hh. ... 9

5.5 Breach side angle θ... 10

5.6 Downstream apron... 10

5.7 Misc. comments... 10

5.7.1 Nature of the dependency on φ ... 11

5.7.2 Case of Hh=0 (no headcut) ... 11

5.7.3 Incomplete flow contraction... 12

5.7.4 Total hydraulic head... 12

5.7.5 Low flows... 12

5.8 Conclusion: effective parameters considered ... 12

6. Breach with Hh=0 (no headcut) ... 13

6.2 Non-vertical sides ... 15

6.3 Conclusions for Hh = 0... 17

7. Breach with a headcut (Hh>0) ... 17

7.1 Vertical upstream slope (φ = π/2) ... 17

7.1.1 Vertical breach sides (θ = 0) ... 18

7.1.2 Non-vertical breach sides (θ > 0) ... 19

7.2 Non vertical upstream slope ... 20

7.2.1 Vertical breach sides (θ = 0) ... 20

7.2.2 Non-vertical breach sides (θ > 0) ... 22

7.2.3 Interpretation ... 23

7.2.4 Characterisation of Qc / Qm... 24

7.3 Conclusions for Hh > 0... 26

8. Conclusions and comments ... 27

1. Introduction

This report concerns the analysis of flow/discharge measurements obtained from a set of 441 “rigid breach” models constructed between 22/07/1996 and 10/12/1998 at the Agricultural Research Service of the United States Department of Agriculture (USDA) in Stillwater, Oklahoma, US, by G.J. Hanson and colleagues. Each model consisted of a rigid plywood dam cut by a rigid breach. This program of experimental research was aimed at improving models of prediction of discharge through breached embankments, which is the first stage in the prediction of the hydraulic quantities of relevance to breach erosion and sediment transport in breaches. The data had been partly analysed previously (see Section 4). A previous analysis by Temple and Hanson concerned only a subset of the data and consisted in calibrating existing weir discharge prediction equations or a combination of weir equations to obtain a best fit of the measured discharges. In contrast, the work reported herein concentrates mainly on 1) verifying established weir equations and published guidance on the parameterisation of these equations using the USDA laboratory dataset, 2) testing an approach consisting in combining two weir equations and using the dataset to understand its limitations (the dataset also contains >5000 images which help understanding the nature of the physical processes involved in each combination of geometry and flow), 3) using the dataset to understand complex flow patterns observed that are of relevance to embankment breach flow, and 4) suggesting ways in which weir flow equations could be adapted to predict flow in various breach shapes. It is emphasised that the observations presented and conclusions drawn are based on the analysis of the entire dataset. Illustrations concern specific examples which are representative of the main trends observed (unless otherwise stated).

The report starts with a complete description of the experimental setup (Section 2). This is followed by a reminder on standard broad-crested and sharp-crested weir flow equations (Section 3), a summary of previous work (Section 4), a sensitivity analysis aimed at simplifying the problem by a selection of effective parameters (Section 5), and the main analyses in Sections 6 and 7.

2. Experimental setup

2.1 Shape of the dam and breach

The experimental setup consisted of 441 dam models, all of different geometries, constructed between 22/07/1996 and 10/12/1998, for each of which 6 or 7 upstream head and discharge pairs at steady state were measured. The general shape of the dam and breach was as illustrated in Figure 2.1.

These models were constructed from 3/4 inch plywood and 2x4 lumber in hydraulic flumes. Pressure treated wood sealed with water seal was used to minimize swelling, twist and warp. Joints were sealed with silicone caulk to prevent leakage.

Fig 2.1: Definition of the dam geometry parameters. For clarity purposes this sketch is distorted and not to scale. The direction of the flow was from left to right.

Some of the geometrical characteristics of this dam were kept constant throughout the exercise, as follows:

- Slope of the downstream face of the dam: 1V for 3H or 0.322 rad - width of the flat top of the dam: 0.46m (1½ ft)

- width of the top face of the upstream (and downstream) aprons (“steps” fitted to the upstream and/or downstream dam slopes): 1.22m (4 ft)

- height of the dam above the flume floor: 0.76m (2½ ft)

- height of the weir crest above the flume floor (P): 0.305m (1 ft) The following parameters varied:

- the breach base width, referred to as : b

- the upstream slope angle relative to the horizontal: φ - the position of the upstream apron in relation to the crest of the base of the breach (hereafter

referred to as the “weir crest”: Hu

- the position of the model floor in relation to the weir crest (or “headcut height”): Hh

- the breach side angle relative to the vertical: θ

In addition, a downstream apron, similar to the upstream apron but fitted to the downstream slope of the dam, was applied in a very limited number of tests. Its position is also defined by the vertical distance from the weir crest, referred to as ds_ap.

The width of the flume B was as follows (most experiments were carried out in the same 6.71m wide flume, except a small number involving a larger breach width) :

- 3.05m or 10ft (small 0.203m (8in) and medium size 0.406 (16in) breaches) - 6.71m or 22ft (large 0.813m (32in) breach)

The longitudinal dimensions (in the direction parallel to the flow) of the dam and breach were completely defined by the above description.

The upstream water level is hereafter referred to as the upstream water head, H. It was measured by point gauge measurement in a stilling well and read to 0.3mm (0.001ft). H refers to the height above the weir crest.

The discharge Q was measured with calibrated orifice plates in a 12” pipe with the differential in pressure read by a water column manometer.

Vent holes were located below the weir crest and were used to aerate the nappe. Additional details:

- Distance from the weir crest to where H was measured: 3.66m (12 ft)

- Distance from the incoming flow pipe to where H was measured: 2.59m (8,5 ft) - Distance from the incoming flow pipe to the upstream toe of the model: 5.33m (17,5 ft)

Note on units: the metric system is used hereafter throughout the report. Constants in equations assume the use of the metric system, but non-dimensionalised constants are used where possible.

2.2 Details of parameter values tested

All the values taken by each parameter are given in Table 2.1:

Breach width b (m) 0.203 0.406 0.813 Upstream slope φ (rad) 0.165 0.245 0.322 0.381 0.464 0.785 (π/4) 1.571 (π/2) Upstream apron Hu (m) 0 0.152 0.305 Headcut height Hh (m) 0 0.152 0.305 Breach side angle θ (rad) 0 0.245 0.322 0.464 0.785 (π/4) 0.983 1.107 Downstream apron ds_ap (m) 0 0.152 0.305

Table 2.1: Values taken by the parameters.

It is reminded that the flume width B depended on the breach width as mentioned in Section 2.1. 441 combinations of the above parameters were tested. This can be detailed as follows:

Middle-size breach (b = 0.406m):

This was the breach width used in most tests. It was tested with all 378 combinations of the upstream slope angle φ, the upstream apron Hu, the headcut height Hh, and breach side angle θ (except θ = 0.322 rad), all with ds_ap = 0.305m (ie “no” downstream apron).

In addition 3 of the above combinations were repeated with a downstream apron (ie. ds_ap = 0 or ds_ap = 0.152m).

This was tested with a constant value of the upstream slope angle φ (0.322 rad), no downstream apron, and all 54 combinations of the upstream apron Hu, the headcut height Hh, and breach side angle θ (except θ = 0.322 rad).

Middle-size breach (b = 0.813m):

This was only tested with a constant value of the upstream slope angle φ (0.464 rad), no downstream apron, a constant value of the breach side angle θ (0.322) and 6 combinations of the upstream apron Hu and the headcut height Hh.

Flow and head measurements

6 or 7 discharge and head measurements were collected for each of the 441 geometries, resulting in a total of 2691 data points (head / discharge pairs).

3. Relevant weir flow equations

As mentioned in Section 1, one of the aims of this analysis is to assess the capability of established weir discharge equations, including those used by HR Breach, to accurately reproduce the head / discharge curves measured at USDA as part of the experiments detailed in Section 2. These equations are used individually or combined with each other wherever relevant.

This section is an introduction to the relevant equations.

3.1 Rectangular weir

The general form of a rectangular weir flow equation is as follows:

Q = Cd.b.√g.H3/2 _Equ.(3.1)

where Cd is the discharge coefficient. 3.1.1 Broad -crested

As detailed in Ackers et al (1978), an established expression for Cd for a broad-crested rectangular weir is:

Cd = (2/3)3/2_{.Cb.F Equ.(3.2)}

where Cb is a fixed coefficient depending on the geometry of the weir, and F is an adjustment factor depending upon head and geometry, including the length of the weir in the direction parallel to the flow. The reader is referred to Ackers et al. (1978) p136-143.

3.1.2 Sharp-crested

An established equation for a full-width weir (width equal to the width of the channel) is:

Q = 2/3.√2.(0.602+0.075.H/P).(b-0.001). √g.(H+0.001)3/2 _Equ.(3.3)

This is given in Ackers et al. (1978) p52 (originally Kindsvater and Carter 1957). It is effectively Equation 3.1 with a head-dependent value of the discharge coefficient and slightly modified values of the weir width and upstream head (this is most significant for small heads).

For a fully-contracted weir (this is a weir of width sufficiently smaller than the channel width for the flow to be independent of the channel width):

Q = 0.554.(1-0.0035.H/P).(b+0.0025).√g.(H+0.001)3/2 _Equ.(3.4)

This is given in Ackers et al. (1978) p60 (originally Kindsvater and Carter 1957). This is also Equation 3.1 with a head-dependent value of Cd, although much less so than in Equation 3.3.

3.2 Sharp-crested V-notch weir

The geometry of a V-notch weir is defined mainly by the notch angle Ω, see Figure 3.1.

Figure 3.1: V-notch weir

The general form of the flow equation for a V-notch weir is as follows:

Q = Cd.tan(Ω/2).√g.H5/2 _Equ.(3.5)

This is detailed for example in Ackers et al. (1978), p64-70. Cd is recognised to be equal to ~0.58 for a very wide range of values of Ω, although this can vary slightly with parameters such as P/B and Ω (particularly for values of the former in excess of 0.2, and values of the latter below π/6, which does not apply here). The literature also advocates the use of the “effective head”, which is a value of H increased by up to several mm for small values of Ω (relevant mainly for small water heads).

3.3 Comments on “fine tuning”

Guidance is available in literature (for example in Ackers et al 1978) on the subject of fine tuning of weir equation parameters to take into account details of weir geometries (such as the thickness or sharpness or shape of a sharp weir edge), weirs of unusual dimensions, the possibility of very low flows (where otherwise neglected physical processes such as surface tension become important), approach conditions, etc. Some of this is briefly approached above. However this is beyond the scope of this work because 1) Not all details of the set-up geometry are known to the authors of this report (such as the sharpness of weir edges, approach conditions, etc.); 2) There may have been measurement errors (there is some evidence of this in the data); 3) Only one set-up out of the 441 is fully documented in literature (contracted sharp-crested rectangular weir) whereas others are at best “reasonably” close to shapes documented in literature; and 4) in most cases combinations of shapes (rectangular and V-notch), rather than one or the other, have been considered (for which finely tuned parameters become irrelevant as they apply only to very specific shapes).

4. Previous dataset analyses

A previous analysis of part of the dataset was published in Temple and Hanson (2005), which consisted in fitting weir flow equations to the data. It was assumed that the observed flow could be modelled as the sum of two components modelled respectively by a rectangular weir flow equation and a V-notch weir flow equation. The discharge coefficients were determined by a least-square fit of the data from the set-ups with a vertical upstream face. The head taken into account for the V-notch flow component was assigned a coefficient to account for the reduction due to convergence of the V-notch and rectangular components. This coefficient was assumed to vary linearly with H/d, where H is the height of the headcut and d is the critical flow depth computed for the total breach outflow. This was done only with the set-ups with vertical breach walls (θ=0), which excluded the widest breach (b=0.813m). This resulted in a “reasonable” agreement as indicated in Figure 4.1.

Fig 4.1: Comparison of measured (Qm) and computed (Qc) discharges for idealised headcut in the upstream embankment face. From Temple and Hanson (2005).

However it also appears on Figure 4.1 that prediction errors were not insignificant, reaching ~40% particularly for the higher flows. These errors may be due to a combination of factors, including:

- a misrepresentation of the physical processes involved. The fact that the V-notch and rectangular components flow perpendicularly to each other (as opposed to parallel) was taken into account through a reduction applied to the V-notch component only, whereas in reality this convergence effect affects both components.

- the above implies that the breach width b should be taken into account in the modelling of this convergence effect, which was not done.

- as will be seen in subsequent sections the flow (or part of) was partly downstream controlled (drown) in some cases.

In addition the analysis did not consider any set-up involving sloping breach sides, ie ~85% of the data.

5. Sensitivity analysis

The aim of the following preliminary analysis is to identify to which extent any dependency of the discharge on each of the parameters describing the geometric shape of the set-up (see Table 2.1) could be effectively identified from the USDA dataset, resulting in:

1) A simplification of the problem considering only those parameters to which the measured flows were significantly sensitive

2) A possible prioritisation of the parameters to include in the analysis

To this end, observed discharge vs. head curves were plotted on a graph for all configurations that had otherwise identical parameter values. The outcome of this sensitivity analysis is as follows (presented for each parameter in no particular order).

(Note: it is emphasised that if in the study below the discharge is observed to have a certain degree of sensitivity (or no sensitivity) to a parameter x (where x is the only parameter allowed to vary in the subset of data used to reach this conclusion), this will also apply if the discharge and x are considered as part of dimensionless coefficients.)

5.1 Breach width b

As expected the discharge was observed to be proportional in order of magnitude to b, the breach width. (no illustration of this is provided)

b is therefore included in the list of effective parameters.

5.2 Upstream slope angle φ

Patterns of sensitivity of the discharge to the upstream slope angle were found to be critically dependent on the headcut height, Hh. The outcome of the sensitivity study for φ is therefore reported separately for each of the three values taken by the headcut height.

5.2.1 For Hh=0.305m

For this case the sensitivity was typically up to ~100%, with the largest discharges found for the shallowest slopes as illustrated by an example in Figure 5.1 (out of 18 such plots).

Fig 5.1: Observed discharge vs. head curves for a range of upstream slope angle values, for Hh = 0.305m (1ft), and other parameter values as indicated in the figure title .

5.2.2 For Hh=0.152m

For this case the sensitivity was similar to that observed for Hh=0.305m. Observed discharge vs depth curves were similar to the ones for Hh=0.305m at least up to a certain water head. For larger heads the slope of the curve was reduced typically as shown in Figure 5.2, and the sensitivity to φ was therefore smaller than for Hh=0.305m.

Fig 5.2: Observed discharge vs. head curves for a range of upstream slope angle values, for Hh = 0.152m, and other parameter values as indicated in the figure title.

5.2.3 For Hh=0 (no headcut)

The sensitivity was found small, typically within a 0-5% range, with only the shallowest upstream slopes being more likely to cause a slightly higher flow, see Figure 5.3.

There was also often a lack of apparent consistency in the way the discharge depended on φ, and the curves for different values of φ appeared to cross each other. It is clear from Figure 5.3 that any noise (for example due to measurement errors) of magnitude 5-10mm would be sufficient for this to happen. In light of these comments it appears difficult to take into account φ in the study for Hh=0.

Fig 5.3: Observed discharge as a function of upstream head for a range of upstream slope angle values, for Hh = 0m. Other parameter values were as indicated in the figure title.

5.2.4 Conclusion for Hh

In light of the comments above φ is included in the list of effective parameters in the analysis of the breach flow data with a headcut, but not in the analysis of the breach flow data without a headcut. (At least in the first instance. Taking into account φ even in this latter case at a later stage cannot be completely excluded. However an appropriate approach, perhaps involving some form of regression, will be needed to take into account the noise in the data. See comment in 5.7.2)

5.3 Position of the upstream apron Hu

The sensitivity to Hu was found almost always negligible (up to a few % at most). Many plots (out of a total of ~150) revealed a slightly increased discharge for Hu=0 (apron at level of weir crest) compared to Hu=0.153m for the smaller head values (<0.05m or so). However this behaviour was not consistent (this may be explained by errors in the measurements of the upstream head of just a few mm) throughout the dataset.

This sensitivity reached 10-20% only with the largest breach width (0.80m), still only with the smaller discharges/heads.

In light of the above Hu is not included in the list of effective parameters.

5.4 Headcut height Hh.

As already implied above (Section 5.2) the sensitivity to Hh was found significant to large in some cases, with the following patterns prevailing (illustrated by a representative example in Figure 5.4, out of a total of ~150 such graphs):

- The flow was always significantly (up to say ~50%) smaller with Hh = 0 (than with Hh=0.15 or 0.30).

- There was no visible discrepancy in the discharge/head curves between Hh = 0.15m and Hh = 0.30m for a range of small flows (circled in Figure 5.4). For higher flows the discrepancy increased with increasing head. This behaviour is evidence of a certain amount of downstream control of the flow for Hh=0.15m and for flows higher than a certain threshold. (This threshold varied depending on other parameters, mainly b, θ and φ.)

Fig 5.4: Observed discharge as a function of upstream head for the 3 different values of Hh (and specific values of the other parameters, indicated in the figure title).

In light of the above (and Section 5.2), it appears that Hh should be considered in the analysis. The most appropriate approach is to treat Hh as a discrete parameter, taking either the value 0 (resulting in flow similar to that over a broad-crested weir), or the value Hh=0.305m (resulting in flow similar to that over a sharp-crested weir). The case Hh=0.153m is relevant only in the context of the study of downstream controlled sharp crested weir flow.

5.5 Breach side angle θ

As expected the effect of increasing θ was similar to that of widening the breach. The sensitivity of the discharge to θwas consistently significant. (this observation is not illustrated)

The largest flow increase caused by increasing θ from 0 to the largest value applied (1.107 rad) was of the order of 200% (case with a narrow breach width of 0.203m).

The angle θ is therefore included as a relevant parameter in the analysis of the data.

5.6 Downstream apron

Only one configuration was tested with 3 different positions of the downstream apron. The sensitivity was almost imperceptible. Any analysis of the sensitivity to ds_ap is therefore to be excluded.

5.7.1 Nature of the dependency on φ

The observations reported in 5.3 on the sensitivity to Hu clarify the behaviour reported in 5.2 (sensitivity to upstream slope angle φ): the significant sensitivity to φ cannot be explained by the effect of φ on the geometry of streamlines upstream from and below the crest level (if this was the case the sensitivity to Hu, especially for Hu=0 would be even larger than the sentivity to φ). Instead the effect of φ is to provide access to the breach laterally, in addition to the frontal component of the flow over the horizontal crest (Figure 5.5). This is further investigated in Section 6.2 below.

Figure 5.5:Flow over the breach crest for decreasing values of φ (π/2, π/4, 0.381, 0.165) illustrating the shift in flow behaviour from “frontal” to a combination of frontal and lateral flow (with the lateral flow dominating for shallow slope angles).

5.7.2 Case of Hh=0 (no headcut)

The observed patterns of sensitivity for Hh=0 suggest that the flow in this case is controlled mostly by the transverse geometry of the breach. Conditions on entry into the breach (for example due to φ) seem to have had a very small effect (Section 5.2.3). It should be pointed out that decreasing φ (making the upstream slope shallower) also had the effect of lengthening the breach (in the direction parallel to the flow). It cannot be excluded that this affected the discharge (broad crested weir flow is in theory affected by the weir length to a small extent, see 3.1.1), possibly counteracting the slight increase due to the change in entry conditions (and perhaps offering some explanation to the apparently inconsistent sensitivity to φ).

Figure 5.6:Example of flow over a broad-crested breach (Hh=0) for 2 values of φ (π/2 and π/4). The sensitivity analysis revealed that the resulting discharge was marginally sensitive to φ.

5.7.3 Incomplete flow contraction

The contraction ratio b/B (where B is the flume width) was respectively 0.067, 0.133, 0.121 for b=0.203m, b=0.406m and b=0.813m. This is well below the 0.3 threshold above which the contraction of the flow (Figure 5.5, top left) is considered “incomplete” (according to Ackers et al. 1978) ie the flow discharge is influenced by the distance from the flume sides to the breach. (Ackers et al. also mention the possible use of (B-b)/2H, which in our case is also found never to be in the range where the flow is incompletely contracted).

Therefore the flume width B is not a relevant parameter at least for the case of vertical upstream face with vertical breach sides. It is not considered in the study.

5.7.4 Total hydraulic head

Assuming a uniform cross-sectional velocity, it can be calculated that the largest flow velocity in the flume was ~0.1 m/s. This corresponds to a velocity head (v2_{/2g) of the order of 0.5 mm. This would}

have a negligible effect on the discharges (much smaller than the noise observed in the data). However it is not excluded that specific current patterns in the flume may have affected the flow. There is a lack of knowledge on the possible occurrence of this during the tests for this to be taking into account in the analysis.

5.7.5 Low flows

Low flows over weirs are recognised to be less accurately predicted because of a combination of factors including the effect of surface tension, the possibility that the nappe may not be suitably aerated on the downstream face of a sharp crested weir, and small-scale roughness and drag effects due to imperfections in the weir geometry. Published formulas (such as Equations 3.3 and 3.4) may take some of this into account by considering “effective” water heads and weir widths (modified by up to ~3mm).

In practice breach shapes will be different from the shapes considered in the USDA dataset. Discrepancies between real breach flows and flows predicted using any knowledge drawn from the present research may be reasonable for high flows, but will almost certainly be large for low flows. Therefore the emphasis is not put on low flows in the present analysis.

The consequence of the sensitivity analysis above is that the number of parameters included in the further analysis is effectively reduced to four, ie:

- the breach width, b

- the upstream slope φ (from the horizontal) - the model floor depth, Hh

- the breach angle, θ (from the vertical) This is illustrated in Figure 5.7.

Figure 5.7: Shape of the setup showing the effective parameters only

The “broad-crested” (Hh=0) case and the “sharp-crested” (Hh>0) case are analysed separately in Section 6 and 7.

6. Breach with Hh=0 (no headcut)

A further analysis of the subset of data corresponding to Hh=0 (no headcut) is presented in this Section. The conclusions of the sensitivity analysis (Section 5.8), are taken into account, implying that for this subset only the breach width b and the breach angle θ are assumed to have affected the flow discharge (at a later stage there may be scope for also considering the effect of φ). The special case of θ=0 is presented first.

Observed discharge/head curves were available with b=0.203m (φ=0.322 only) and b=0.406m (and 7 values of the upstream slope φ). With φ = π/2 (vertical upstream wall), the geometry was similar to that of an ideal broad crested weir (although with the weir width smaller than the channel width). As explained in 5.2.3 (see also 5.7.1), the sensitivity to φ for this case (no headcut) was very small and difficult to characterise. Therefore the observed discharge/head curves were compared to Equation 3.1 with Cd defined by Equation 3.2. The values of Cb and F were set as much as possible according to literature (FLOODsite technical note “Review of Flow Calculations in HR Breach using Weir Equations”, or Ackers et al. 1978 p136-143, original work by Singer and/or Crabbe), ie as follows:

- Cb = 0.85 - F = 1

resulting in Cd=0.463 in Equation 1.

(The literature suggests guidelines regarding setting values of F slightly higher than 1 depending on values taken by H/L and H/(H+P). However published charts are unclear for the range of values taken by these parameters in our case. It is possible that F = 1 is an underestimation by a few %)

Fig 6.1: Broad-crested weir: comparison between observed and calculated Q vs H curves for b=0.203m (available only with an upstream slope) and b=0.406m (here with a vertical upstream face).

6.2 Non-vertical sides

This was available with b=0.203m, φ=0.322 and 6 values of θ; with b=0.406m and 42 combinations of φ and θ; and with 0.813m and 1 combination of φ and θ.

A simple test was carried out consisting in taking into account non-zero values of θ by using the broad-crested weir equation (as in section 6.1) with an effective breach width b_eff depending on b, θ and H as follows:

b_eff = b + H.tan(θ) Equ. (6.1)

This is the approach used in HR Breach in its current version.

This resulted in consistently overestimated (by ~5 to ~15%) discharges compared to lab observations, except for small values of H, as illustrated in Figure 6.2 below (however the approach did appear to provide a useful estimate of the increase of discharge due to θ).

Fig 6.2: Breach with no headcut and sloping sides: comparison between observed and computed Q vs H curves. This is an example representative of what was observed for all available combinations of b, θ and φ.

A more appropriate approach is needed for this case. It is likely that published literature exists on the subject (not investigated).

6.3 Conclusions for Hh = 0

The main conclusions of Section 6 are as follows:

- There is an adequate agreement between established formulas and discharge coefficients for broad-crested weir flow (ie the ones used in the current version of HR Breach) and the flows measured at USDA for the case of the breach geometry with no headcut. This applies to all shapes without a headcut and with vertical side walls.

- This type of flow is controlled by transverse dimensions of the breach. Conditions on entry into the breach (governed primarily by the upstream slope angle) do not need to be taken into account. The above conclusion therefore also applies independently of the upstream slope angle.

- A simple approach (used in HR Breach) to predict the flow for non-rectangular breach shapes is found to consistently overestimate the measured flows. There is scope for investigating the design of a better approach.

7. Breach with a headcut (Hh>0)

A further analysis of the subset of data corresponding to Hh>0 (with a headcut overfall) is presented in this Section. The conclusions of the sensitivity analysis (Section 5.8), are taken into account, ie for this subset only the breach width b, the breach angle θ and the upstream slope φ are assumed to have affected the flow discharge. (In contrast to the “no-headcut” case above, the sensitivity to the upstream slope φ was large.)

As made clear in Section 5.4, there is evidence that the flow was downstream controlled for at least the lower heads/discharges for Hh=0.152m. The following analysis was therefore done using the data for Hh=0.305m. (However it should not be assumed that there was not any downstream control of the flow particularly with the shallowest upstream angles and breach side angles, for which the highest discharges were observed. Unfortunately any analysis of this process would be difficult to carry out because downstream water levels are unavailable).

The following considers separately the cases where either the upstream slope or the breach sides (or both) were vertical, allowing the use of published equations wherever relevant, starting with the simplest geometries.

7.1 Vertical upstream slope (φ = π/2)

The breach crest geometry in this case was considered similar to the geometry of a sharp crested weir, and the relevant equations were used in (Sections 3.1.2 and 3.2).

It should be noted that this geometry effectively consisted in a thin vertical plate blocking the entrance to the breach. Its relevance to breaches in earth embankments is limited.

7.1.1 Vertical breach sides (θ = 0)

The only available case was with b=0.406m.

Predictions using both equations 3.3 (full width weir) and 3.4 (fully contracted weir) were compared to the only available Q vs. H curve for this case. This is shown in Figure 7.1 below.

Fig 7.1: Sharp-crested weir: comparison between observed Q vs H curves and the rectangular sharp-crested weir equations (both full-width and fully-contracted), for b=0.406m.

The figure shows that taking the contraction effect into account did result in a very good agreement with observations. It should be noted that the discharge coefficient (as defined in Equation 3.1) is much less dependent on H in Equation 3.4 (contracted flow) than in Equation 3.3. In this case it took the almost constant value of ~0.55.

Equation 3.1 was used in the remainder of this study with the same value of Cd (Cd=0.55).

(It is acknowledged that only the breach shape described here is significantly close to the vertical sharp weir shape for which this parameter value is derived. However 1) All subsequent breach shapes imply some form of flow contraction and 2) As mentioned in Section 3.3 there is no emphasis on “fine tuning” of parameter values).

7.1.2 Non-vertical breach sides (θ > 0)

This was available with b=0.406m and 6 values of θ.

This is a trapezoidal weir shape (with however a non-vertical drop downstream from the V-notch component) for which a classical approach is to apply the sum of the flow Qr predicted by a

rectangular weir flow equation and of the flow Qv predicted by a V-notch weir flow equation. This is

the approach taken here:

Q = Qr + Qv Equ.(7.1)

In this, Equation 3.1 was used for Qr, with Cd=0.55 as mentioned previously. Equation 3.5 was used

for Qv, with Cd=0.585 (see Section 3.2). It should be noted that Ω = 2θ.

An excellent agreement between computed and observed Q vs H curves was observed for all values of θ, as illustrated in Figure 7.2 below:

Fig 7.2 : Sharp-crested weir combined with a V-notch weir: comparison between observed and computed Q vs H curves for increasing values of θ (all for b=0.406m).Red: observed. Blue: computed

Comment: using the additive approach defined by Equation 7.1 is justified on physical grounds. However a practical difficulty would arise if one sought to optimise the values of the 2 discharge coefficients (Cd_v for the V-notch and Cd_r for the rectangular component). It may be that many combinations of Cd_v and Cd_r would result in an equally good fit of the data (the equifinality problem). Instead, a unique coefficient applied to the trapezoidal shape as a whole may be more relevant. However, such “fine tuning” is beyond the scope of this work(see Section 3.3).

7.2 Non vertical upstream slope

As mentioned in 5.7.1 the flow is allowed to penetrate sideways into the breach and can therefore be significantly increased. This lateral component of the flow is of triangular cross-section. The approach taken here is therefore similar to the one in 7.1.2 above which consists in adding a V-notch weir flow component to a rectangular weir flow component. The V-notch half angle to be considered needs to be the effective angle relative to the vertical. This is a function of both φ and θ, and can be shown to be expressed as follows:

Ω/2 = π/2 - arctan [tan φ / √(1+tan2_θ.tan2_{φ)] Equ.(7.2)}

For θ = 0 (vertical breach walls) this reverts to Ω/2 = π/2 – φ, as expected. The above is illustrated in Figure 7.3 below.

Fig 7.3: Angle used as part of the V-notch equation for the non-vertical upstream slope case (this sketch is not to scale)

This was available with b=0.406m and the 6 values of the upstream slope φ, as well as with b=0.203m and φ = 0.322 only. The results are illustrated in Figures 7.4 and 7.5 below.

It was observed that:

- The agreement between computed and observed Q vs H discharges improves with decreasing values of φ (although many of the curves showed a slight underestimation of the computed discharges)

- It deteriorates with increasing values of the upstream head / discharge. - It deteriorates much faster in the case of the narrower breach (b=0.203m).

Fig 7.4: Sharp-crested weir combined with a V-notch weir: comparison between observed and computed Q vs H curves for a selection of values of the upstream slope angle φ and θ = 0 (all for b=0.406m). Red: observed. Blue: computed

Fig 7.5: Sharp-crested weir combined with a V-notch weir: comparison between observed and computed Q vs H curves for φ = 0.322 and θ = 0 (b=0.203m). Red: observed. Blue: computed

7.2.2 Non-vertical breach sides (θ > 0)

This was available with

- b=0.406m and the 30 combinations of θ and φ - b=0.203m, φ = 0.322 only and 5 values of θ. - b=0.813m, φ = 0.464 only and θ = 0.322 only. The results can be summarised as follows:

- For b = 0.406m, most curves showed a very good agreement (within a few %), with the exception of the ones for very small angles θ (say θ < 0.2 rad), where some discrepancies appeared for increasing heads/discharges. This effect was also more pronounced (ie larger discrepancies) with the larger values of the upstream slope angle φ (ie for the steeper slopes). - For b = 0.203m, the behaviour was consistent with the behaviour illustrated in Figure 7.5, with

- For b = 0.813m, the behaviour was not consistent with the observations for other breach widths, with a significant discrepancy (underestimation) increasing with H (even for small values of H) and reaching ~15%.

The results for b=0.203m and b=0.406m are not illustrated here. The inconsistent result observed with b=0.813m is shown in Figure 7.6.

Fig 7.6: Sharp-crested weir combined with a V-notch weir: comparison between observed and computed Q vs H curves for φ = 0.464 and θ = 0.322. (only geometry available with b=0.813m).

7.2.3 Interpretation

The following interpretation is based on the data with b = 0.203m and b = 0.406m only (ie most of the data). The special case of h=0.813m (Figure 7.6) is commented on at the end of this section.

The improvement of the prediction for non-vertical breach sides is clarified in Figure 7.7: with a vertical breach side (left-hand side), the lateral (V-notch) components of the flow and the frontal (rectangular component) converge into a small area immediately downstream from the weir crest, resulting in a convergence (or/and downstream control) effect that may prevent the flow from reaching its full size. With the breach wall being at an angle (right-hand side), the extent to which this happens is much reduced and both components are able to flow mostly unhindered.

Fig 7.7: Photos of the breach flow for θ = 0 and θ = 1.11, and otherwise equal parameter values (including φ = 0.464)

It is suggested that the above effect may be characterised by the angle between the lateral ridge and the vertical plane in the direction parallel to the flow (Figure 7.8). This is referred to as β and is expressed by

β = arctan[tanθ.tanφ)] Equ.(7.3)

Fig 7.8: Angle β used as part of the V-notch equation for the non-vertical upstream slope case (this view from above is not to scale)

This convergence effect is expected to be more pronounced for narrow breaches and/or higher flows / heads, explaining the larger discrepancies observed for b = 0.203m and for the higher flows.

An interpretation of the inconsistent behaviour observed in the only test wide a wide breach (b = 0.813m) can be suggested. According to 7.1.1, taking into account contraction (resulting in Cd=0.55) improved the prediction significantly as shown by Figure 7.1. However, with the wide breach, the flow may have been wide enough to have a significant uncontracted component in its central part (no image is available to support this), resulting in an underestimation of the computed prediction. Also, the overestimation observed for b = 0.203m and b = 0.406m for larger flows (where the various components of the flow interacted with each other) may have been much less significant for b = 0.813m.

The aim of this section is to attempt to characterise patterns in the variations of the ratio of computed (Qc) vs measured (Qm) discharges as observed throughout Section 7.2, using a limited number of

dimensionless parameters.

As suggested above, the upstream slope angle φ and the angle β are used. In addition, it follows immediately from the conclusion of the previous section that H/b is a relevant parameter (rem: H is the upstream head).

Approximate contours of the ratio Qc / Qm in the H/b vs. β plane are plotted in Figure 7.9 for each

value of the upstream slope angle (except φ = π/2 for which a specific behaviour was observed as detailed in Section 7.1).

It is emphasised that the plot for φ = 0.322 rad (third from top in Figure 7.9) also contains all the data for the narrow breach (b = 0.203m) for which higher values of H/b were obtained (this plot therefore has different axis scales).

As mentioned before the data for the wide breach are not represented (they result in inconsistently low values of Qc/Qm, for which an explanation is suggested at the end of the previous section).

Fig 7.4: Approximate contours of the ratio Qc / Qm in the H/b vs. β plane for the 6 values of φ.The

stars represent the data points from which the contour lines were interpolated. The axis scales for φ = 0.322 are not the same as for the other graphs. For clarity the value taken by Qc / Qm at each point is not shown.

Figure 7.4 suggests that the discrepancy between the measured flows and the flows calculated using the proposed approach (introduction of 7.2) may be appropriately characterised using the dimensionless head H/b and the angles φ and β, as the Qc / Qm ratio shows consistent patterns when plotted as a function of the H/b, φ, β parameters.

At this stage an empirical relationship between Qc / Qm, H/b, φ and β will not be proposed.

It can be noted that for b=0.203m and the larger flows, the behaviour of the flow may be significantly different from the one implied by the approach used, as illustrated by Figure 7.4.

Fig 7.4 Constriction flow observed with b=0.203m

7.3 Conclusions for Hh > 0

The main conclusions of Section 7 are as follows:

- There is an adequate agreement between predictions based on established formulas and discharge coefficients for sharp crested rectangular weir flow and the flows measured at USDA for the case of the breach geometry with the highest headcut. (This is relevant in theory only to headcuts reaching the upstream slope of the dam profile.). In particular, applying a specific formula for contracted flows has been shown to be useful.

- The effect of the upstream slope on the magnitude of the flow has been explained. It has been shown that, critically, this applies to the headcut scenario only, where entry conditions into the breach play a critical role, whereas they are not in the “no headcut” case (This explains that although the discharge coefficient for a rectangular sharp-crested weir is only ~20% higher than for a broad-crested weir, the effect of the headcut was observed to increase the flow by up to ~100%)

- An adequate approach has been suggested consisting in adding a rectangular and a V-notch flow component for all geometries involving non-vertical breach sides or/and upstream slope. In every case the appropriate angle between the breach lateral crest and the vertical needs to be considered in the calculation of the V-notch flow.

- The above approach was generally accurate, but the accuracy deteriorated with narrower breaches, higher flows, steeper upstream slopes and steeper (towards vertical) side angles. - Reasons for this have been suggested (convergence and/or downstream control) and a

preliminary study has shown that it may be possible to express a correction factor as a function of the dimensionless head H/b, the upstream slope, and the angle of the breach lateral

crest in relation to the longitudinal direction (which is 0 for vertical breach walls). However this may not apply to the more extreme cases (high flow in very narrow breaches) where the discharge may be governed by a constriction effect.

- In all cases above Cd==0.55 (contracted weir flow coefficient) was used for the rectangular component of the flow. However a significant underestimation of the measured flow using the approach above (adding a rectangular and a V-notch component) has been observed in the sole case involving a wider breach. It was suggested that this is due to a specific behaviour involving a significant central uncontracted and unhindered component.

8.

Conclusions and comments

Conclusions specific to the study of the “no headcut” and the “headcut” scenarios are detailed in 6.3 and 7.3. These refer to the adequacy of using published equations for broad crested, sharp rectangular and sharp V-notch (each on its own or in combination with each other where appropriate) to model the flows measured at USDA.

Some of these conclusions do not require any further investigation, ie:

1) The broad crested weir equation is adequate for any shape from the USDA experiments with “no headcut” and vertical side walls, independently of the conditions on entry into the breach (governed primarily by the upstream slope angle, whose effect was small and difficult to characterise).

2) The sharp crested rectangular weir flow equation adapted for contracted flow (Kindsvater and Carter) is adequate for the case of the breach geometry with the highest headcut and a vertical upstream face. This also performed well when used additively with a V-notch equation for the cases with sloping breach sides. (however none of these shapes is directly relevant to real embankment breaches).

3) The effect of the upstream slope on the magnitude of the flow has been explained. This applies to the headcut scenario only.

4) An adequate approach has been suggested consisting in adding a rectangular and a V-notch flow component for all geometries involving non-vertical breach sides or/and upstream slope. In every case the appropriate angle between the breach lateral crest and the vertical needs to be considered in the calculation of the V-notch flow.

However, the study also highlighted a number of limitations, expressed as follows:

1) A simple approach (used in HR Breach) to predict the flow for breach shapes with a lateral slope (no headcut case) was found to consistently overestimate the measured flows.

2) The approach suggested in 3) and 4) above, relevant to breach flow over a headcut reaching the upstream slope of a dam was shown to deteriorate in accuracy with narrower breaches, higher flows, steeper upstream slopes and steeper (towards vertical) side angles. Reasons for this have been suggested (convergence and/or downstream control)

3) In addition, the above approach overestimated flows in a case with a very wide breach. It is suggested that this was (at least partly) due to a) the use of a rectangular weir discharge coefficient for contracted flows, becoming inappropriate for wider breaches where there is a significant central uncontracted component b) reduced convergence / downstream control effects at higher flows (than for less wide breaches)

Further work using the dataset may include:

1) Characterisation of the (small) effect of the upstream slope for the broad-crested weir case. However it has been already suggested that this will be difficult (mainly due to a: small amplitude noise in the data and b: physical processes cancelling each other out).

2) It may be possible to express a correction factor for the generalised rectangle + V-notch approach (headcut case) as a function of the dimensionless head H/b, the upstream slope, and the angle of the breach lateral crest in relation to the longitudinal direction (which is 0 for vertical breach walls).

However item 2) above is made difficult by the wide variety of modes in which the flow appears to be flowing depending on the geometry. In the most general case combining a rectangular weir and a V-notch weir (with the appropriate angle) seems appropriate for the lower flows. However, for higher flows the rectangular and the V-notch components hinder each other, but there is also evidence of downstream control or drowning of the flow. For narrow breaches with vertical or nearly vertical side walls the flow may even be a “constriction” type flow. For wider breaches the rectangular weir coefficient relevant to contracted flows results in an underestimated prediction.

Practical implications of the study for future breach flow modelling research are mainly as follows: 1) It is highly important to be able to distinguish between sharp weir type flow (headcut towards

its final phase of development) where conditions on entry into the breach, such as the occurrence of a “lateral component” (as defined above), are critically relevant to the discharge prediction, and broad-crested weir type flow, where the flow is primarily governed by the transverse geometry and dimensions of the breach. Considering the weir crest geometry appropriately may be much more important than the choice of discharge coefficient (there is only a ~20% difference between sharp crested and broad crested weir discharge coefficients). 2) A more accurate flow prediction relies on a better understanding of phenomena such as

contraction (when the nappe is narrower than the weir crest), convergence (when flow components coming from different directions converge into each other) and drowning (although it may be argued that in practical applications this last item is unlikely to be relevant).

3) In the case of broad-crested type flow, an approach is needed to deal with non-rectangular cross-sections (breaches with side slopes).

To complement item 1), it should also be pointed out that an appropriate approach is needed to handle the transition between the 2 different modes (when the headcut has not reached the back slope, and/or if the embankment is narrow). If the correct flow mode is not identified large errors (up to ~100%) are likely, as implied by the study.

More generally, the following points can be made:

1) Breaches will not have ideal shapes such as those in the USDA dataset. Modelling approaches are needed to handle irregular shapes.

2) This study relates to flow discharge prediction. Appropriate methodologies are needed to derive velocities and shear stresses from discharge predictions.

9.

References

1. Ackers P, White W.R., Perkins J.A., and Harrison A.J.M (1978): Weirs and Flumes for Flow

Measurement. John Wiley.

2. FLOODsite technical note “Review of Flow Calculation in HR Breach using Weir Equations” 3. Kindsvater C.E. and Carter R.W.C. (1957). Discharge Characteristics of Rectangular Thin

Plate Weirs. Proceedings of the American Society of Civil Engineers, Journal of the

Hydraulics Division, Vol. 83, No. HY6, December 1957, pp. 1453/1-1453/36.

4. Temple, D.M., Hanson, G.J. (2005). Earth dam overtopping and breach outflow. In: Walton, R., editor. Proceedings of the World Water and Environmental Resources Congress, May 15-19, 2005, Anchorage, Alaska. American Society of Civil Engineers.