An analysis of the effects of panel markers on a cross-section’s conveyance curve

In this article, I present the effects of panelling a 1D cross-section. Panelling is the terminology used by the software Flood Modeller™ to divide the cross-section’s hydraulic properties (mostly the hydraulic radius) into a sum of them. The article contains a theory introduction about Manning’s n formula and 5 cross-sections are analysed with different panelling scenarios.

Introduction

What is panelling

Panelling is the action of locating one or more panel markers in a Flood Modeller™ cross-section. Thus, dividing its hydraulic properties, as if they were several cross-sections placed in parallel. The Flood Modeller’s help declares that A panel marker subdivides the channel into a number of vertical panels that should reflect the geometry of the channel. The conveyance of each of these panels will then be calculated individually as opposed to calculating conveyance across the full width of the channel as a whole. Then, summed across the channel. (Flood Modeller Team n.d.)

Although this article focuses on Flood Modeller’s way of panelling. It is important to note that other hydraulic software provides similar functionality. For instance, HEC-RAS auto-panels the cross-section when you set up a Left or Right Bank at those locations, subdividing the cross-section into 3 different sections. (Goodell 2017)

However, HEC-RAS limits the user to use two panel markers (divisors), hence 3 different zones, whilst Flood Modeller allows the user to set up or introduce as many panel markers as the user might want. Indeed, it is technically possible to panel every XZ pair of the cross-section.

Why is panelling used?

There are cases in which a cross-section holds information from parts that behave hydraulically different. For instance, a floodplain and the main channel will (likely) have different roughness; or in braided channels, each channel might have different roughness.

Figure 1: Flood Modeller’s showing a cross-section and its associated conveyance in function of the stage

During those cases, depending on the geometric shape of the cross-section, the conveyance curve can have a negative spike¹. An example of spike can be seen at an elevation of 4m in Figure 1. The ambiguity of having two possible water elevations for a given flow usually produce instabilities². Thus, panelling can (and should) be used to eliminate those spikes and have a well-behaved continuously growing conveyance curve³.

Problématique

Panelling is a powerful tool that is able to remove negative spikes that produce instabilities in our model. But it has a price. The price is that each panel marker alters the properties of the cross-section. Concretely, it reduces the roughness, which translates as a larger conveyance. A larger conveyance means more flow per stage.

I have encountered models with a large number of panel markers⁴ without realising the secondary effects of panelling. The mere act of (just) adding a panel marker at any location reduces the roughness of the cross-section, letting the cross-section hold and convey more water.

This article has the purpose of making the reader aware of the consequences of improper panelling. Supporting my point based on theory and 5 analysis cases. At the end of the article, the reader should understand how adding unnecessary panel markers tweaks the cross-section conveyance.

Theory

The Manning formula

Most of the free-flow computational hydraulics software uses the Manning’s n formula. The Manning’s n formula is an empirical formula to estimate the average velocity of a liquid flowing in an open channel flow (Chow 1985) and it is defined as:

\[ V = \frac{1}{n} R_h ^\frac{2}{3} \cdot S^\frac{1}{2} \]

Where \(V\) is the average velocity, \(n\) is a dimensional roughness coefficient, \(R_h\) is the hydraulic radius and \(S\) is the slope of the channel.

Before continuing, I need to highlight a few things about this widely used formula, that quite often gets overlooked. In order to get Manning’s equation, there are two important simplifications that have been done.

The first simplification is that the slope (\(S\)) must be substantially small to admit that \(S \simeq tan(S)\)(Charpentier 2018). Hence, be careful when using this equation for steep slopes.

The second simplification is that the term \(\frac{1}{n} R_h ^\frac{2}{3}\) is a roughness term obtained applying a Taylor series simplification over the event probability of energy contribution of all roughness terms (Charpentier 2018). There are two important things to keep in mind. The first one, this term is a Taylor approximation. The second one, the hydraulic radius \(R_h\) and the Manning coefficient \(n\) (both of them when) together represents the roughness.

The n term is not the only term representing the roughness, the hydraulic radius \(R_h\) term also represents roughness. This is important because each new panel marker alters the hydraulic radius of the cross-section. In other words, each new panel marker smooths the cross-section roughness. Each new panel marker increases the cross-section conveyance.

The hydraulic radius

We have determined that this term \(\frac{1}{n} R_h ^\frac{2}{3}\) represent the cross-section roughness. But what exactly is the hydraulic radius?

The hydraulic radius is defined as:

\[ R_h = \frac{A}{W_p} \] Where \(A\), is the Cross-Sectional area of flow, and \(W_p\) is the wetted perimeter, or in other words, the total wetted length across the cross-section. When we increase the water depth (\(y\)) on a cross-section, we are increasing both terms, the area and the wetted perimeter. But they do not necessarily increase at the same rate. If the increase of \(W_p\) is much rapid than the increase in \(A\), the hydraulic radius \(R_h\) decreases. Producing the aforementioned spikes.

For instance, check Figure 2. When the water depth reaches the floodplain (\(y = 4m\)) a small increase in the water depth barely increases the area (\(A\)), but it increases substantially the wetted perimeter (\(W_p\)). This situation produces that hydraulic radius (green line) decreases. Spikes in the hydraulic radius will likely produce spikes in the conveyance of the cross-section.

Figure 2: Cross-Section and its calculated hydraulic radius per water depth

It is fairly complex to understand how the hydraulic properties of a cross-section change by looking at the hydraulic radius. This is because the hydraulic radius is not an intuitive variable. It has length dimensions, but it does not measure length. It measures roughness, the roughness is expressed as a coefficient between area (amount of water) and wetted perimeter (part of the water with friction).

The cross-section conveyance

Multiplying the Manning’s right and left term by the Area we can obtain the flow \(Q(y)\) as a function of the water depth \(y\):

\[ V \cdot A = Q = \frac{1}{n} \cdot R_h ^\frac{2}{3} \cdot A \cdot S^\frac{1}{2} \] In this equation we call conveyance \(K\) to the term( USACE Hydrologic Engineering Center n.d.): \[ K = \frac{1}{n} \cdot R_h ^\frac{2}{3} \cdot A \] So the flow can be defined as:

\[ Q(y) = K(y) \cdot S^\frac{1}{2} \]

The conveyance \(K(y)\) will vary based on two parameters the hydraulic radius \(R_h(y)\) and the area \(A(y)\), both are geometric functions that depend on the water depth \(y\).

When a panel marker is placed, the hydraulic properties of the cross-sections are divided into 2 zones. In figure 3 those zones have been called B (floodplain) and A (channel). Then the conveyance of each zone is calculated (independently) for a given depth, and the sum of them produces the conveyance of the panelled cross-section. Adding more panel markers add more terms (zones) to the summation.

\[ K =\sum_{i=1}^{n}K_i = K_A + K_B \] \[ K= \frac{1}{n_A} \cdot (R_h)_A ^\frac{2}{3} \cdot A_A + \frac{1}{n_B} \cdot (R_h)_B ^\frac{2}{3} \cdot A_B \]

Figure 3: Conveyance value of a cross-section with 1 panel marker

Methodology

Five cross-sections will be analysed. Case 1 will analyse the cross-section seen in figure 3. Case 2 will analyse a symmetrical trapezoid section that does not need any panel marker. Case 3 is based on a real-life cross-section with a well-defined channel. Case 4 is based on a real-life cross-section that is oddly shaped and contains partial information from its floodplains. Case 5 is a n extense real-life cross-section.

For simplification, the Manning’s n value used on all cases is \(n=0.035\).

Each case will have different scenarios. Each scenario will place the panel markers at different locations. The conveyance curve of each scenario will be obtained and compared against the no panel markers scenario.

Each case will have 4 tabs, the first tab contains the plot graph of the cross-section, each scenario conveyance and each scenario panel markers location. You can click on the legend to show and hide them. The second tab contains a small table comparing each scenario against the no panel markers scenario. The third and fourth tabs contain tabular data for QA purposes.

The best way to check where the panel markers are placed for each scenario is to click on the plot’s legend to see where the panel markers are.

For easiness, I have tried to come with some common scenarios names, not all the named scenarios are in all cases. In my opinion, the first three are the most important scenarios, but we can obtain some conclusions from the other scenarios.

1. Simple: Or no panel markers scenario. Other scenarios will be compared against this scenario.
2. Frugal-panelling: When a cross-section contains only 2 panel markers. This is the same limitation that HEC-RAS has.
3. Super-panelling: When a cross-section contains at least 2 or more panel markers than the frugal option.
4. Single: Locating one panel marker only, dividing the section into 2 zones. This case is especially important on (artificial) geometric cross-section to see the influence that a panel marker can produce.
5. Over: A middle situation between Frugal and Super. Used when the Frugal scenario is unable to remove all the spikes.
6. Bed: I have seen some cross-sections whose panel markers are not at the top of the embankment, but rather at the extreme of the bed levels. This scenario tries to represent this situation.
7. Every: In this extreme scenario a panel marker is placed on every single possible location.
8. Other: Scenarios that cannot be classified on the previous categories will receive a different name.

Case 1: Geometric shape of Canal and left flood plain

In this study case, we are analysing the cross-section used on previous figures to see how it reacts to different scenarios. The cross-section is formed by a flat floodplain and the main channel. The right bank of the cross-section is a vertical wall. The “Frugal” scenario is formed by only one panel marker dividing the floodplain from the main channel. The “Super” scenario adds another one on the left bed channel. The “Every” scenario produces an absurdly large conveyance, the reason is explained below.

Simple

Figure 4: Case 1, Geometric shape of Canal and left flood plain

Differences

Conveyance difference compared against Simple scenario.
	Max K [m³/s]	Difference [m³/s]	Difference %
Simple	877.533	0.000	0.000
Frugal	878.428	0.895	0.102
Super	889.664	12.131	1.382
Every	1623.632	746.099	85.022

Cross-Section

Conveyance

Frugal Scenario

The “Frugal” scenario can mimic the “Simple” conveyance curve and remove the negative spike that happens at \(4 \leq y \lesssim 4.4\) . The difference of maximum conveyance is negligible, being approximately 0.1% larger. This scenario has solved the problems on the “Simple” scenario with an insignificant increase in the cross-section conveyance.

Super scenario

This scenario has an additional panel marker at the bottom left part of the channel. The conveyance curve is shifted to the right (more conveyance) for any water depth \(1 \lesssim y \lesssim 5\). Hence, the cross-section is potentially carrying more flow than the “Simple” or “Frugal” scenario for those depths. Although, for water depths of \(y > 5\) the conveyance curve of this scenario gets close to the previous scenarios. The difference of maximum conveyance is negligible but larger than in the “Frugal” scenario, being larger than 1%.

Every Scenario

The conveyance plot of the “Every” scenario is absurdly different to any other scenario. The cross-section conveyance has increased an 85%. This scenario almost duplicates the amount of flow the cross-section can convey at any given depth. This situation is produced by the combination of a vertical wall on the right bank with a panel marker on each extreme. The two conditions have eliminated the roughness influence of the wall.

The conveyance of a panelled section is calculated by the sum of the conveyances.

\[ K = \sum_{i=1}^n{k_i} = k_1+k_2+k_3+k_4+k_5 \] However, in our case, the conveyance of the vertical wall (\(k_5\)) is going to be always 0. The conveyance is defined as \(\frac{1}{n} \cdot R_h ^\frac{2}{3} \cdot A \cdot S^\frac{1}{2}\) but the area \(A\) of a vertical section is always 0, independently of its depth. Indeed, we have created a “magical” frictionless wall duplicating the amount of flow that the cross-section can carry.

Although this case has been created artificially to show the danger of inappropriate panelling. This situation can happen in a real-life cross-section. Be mindful with those cross-sections that have a vertical concrete wall, as setting a panel marker on each extreme of the wall will “remove it,” providing completely unrealistic effects.

Case 2: Trapezoidal cross-section

The cross-section is a symmetrical trapezoid. A trapezoidal cross-section does not require any panel marker as its conveyance curve will not present any spikes. However, we will introduce them to see how the conveyance curve gets influenced by the introduction of unnecessary panel markers. In addition to the “Bed” and “Every” scenarios, we are modelling scenarios BA, BB and BC, by placing one additional panel marker⁵ at different locations. Scenarios BD and BE will have more than one panel marker.

Simple

Figure 5: Case 2, Trapezoidal section

Differences

Conveyance difference compared against Simple scenario.
	Max K [m³/s]	Difference [m³/s]	Difference %
Simple	4402.178	0.000	0.000
BA	4677.494	275.316	6.254
BB	4505.396	103.218	2.345
BC	4413.395	11.217	0.255
BD	4718.288	316.110	7.181
BE	4866.107	463.929	10.539
Bed	5201.228	799.050	18.151
Every	5373.643	971.465	22.068

Cross-Section

Conveyance

BA, BB and BC scenarios

The key information on these scenarios is that adding any⁶ panel marker has increased the conveyance. In symmetric shapes, the closer that the panel marker is to the symmetry axis, the less influence it has. When adding panel markers into cross-sections that do not need them, the roughness is decreased and consequently, the conveyance is increased.

BD and BE scenarios

Adding more panel markers increases the conveyance of the cross-section considerably. BD with 2 panel markers obtains a 7% difference, and BE with 3 panel markers obtains a 10.5% difference.

Bed scenario

The “Bed” scenario increases significantly its conveyance (18%). The conveyance has been increased for a whole range of depths as the panel marker are placed at low elevation levels. Therefore, the modeller should be extra careful when panel markers are placed at bed levels. Not only decreases the roughness of the cross-section, but it does for a large range of depth values.

The Bed scenario is equivalent to running the “Simple” scenario with a Manning’s n value of 0.0296 instead of the 0.0350 used in all scenarios⁷.

Every scenario

The “Every” scenario has increased the conveyance by 22%. The Bed scenario is equivalent⁸ to running the “Simple” scenario with a manning’s n value of 0.0287 instead of the 0.0350 used in these scenarios.

Case 3: Cross-section with manmade wall

In this study case, a real-life cross-section has been used. The cross-section presents a set of structures on both sides, but the ones on the right bank are more pronounced. The cross-section’s conveyance curve presents two conveyance spikes, the first one near 9m, and the second one at 11.5m. Both spikes can produce stability issues. Usually, panel markers need to be located at the same height that spike occurs to remove it.

Simple

Figure 6: Case 3, Cross-section with manmade wall

Differences

Conveyance difference compared against Simple scenario.
	Max K [m³/s]	Difference [m³/s]	Difference %
Simple	40590.86	0.000	0.000
Single	45531.98	4941.114	12.173
Frugal	46356.79	5765.925	14.205
Over	47279.61	6688.743	16.478
Super	49386.63	8795.771	21.669
Bed	47696.99	7106.127	17.507
Every	50247.37	9656.503	23.790

Cross-Section

Conveyance

Single

The “Single” scenario has only one panel marker located at the bottom of the wall. This action has removed all the spikes of the conveyance plot, and also is the scenario with the smaller difference, 12%. However, is not always practical to fiddle with the cross-section to solve all spikes with only one panel marker⁹.

Frugal

This scenario uses 2 panel markers, one on the left side at 9m and one on the right at 11.5m. The larger spike that happens at 11.5m has been removed, but the use of only one panel marker on the left at 9m is not enough to remove the smaller spike. The conveyance difference is larger than in the “Single” scenario, being 14% larger.

Over

In the “Over” scenario, we fix the smaller spike of the frugal scenario by adding a third panel marker on the right side at 9m. This produces a continuously growing well-behaved conveyance curve, at a price of a 16.5% increase on the conveyance. This is 2.5% larger than the “Frugal” scenario.

Super

The “Super” scenario contains several panel markers at different locations. They have been placed more randomly than with logic. It has managed to remove any spike from the conveyance plot, with an increase of 21%, the second-largest, just behind the “Every” scenario. This proves that with enough panel markers we can remove any negative spike, at a price of decreasing the roughness of the cross-section.

Bed

The “Bed” scenario uses panel markers to divide the bed from the walls, it increases the max conveyance by 17.5% which does not differ much from the “Over” scenario. But unlike the “Over” scenario, it is also unable to remove the largest spike.

Every

The “Every” scenario increases the conveyance a 23.8% when compared against the “Simple” scenario.

Case 4: Cross-section with undeveloped floodplain

This cross-section has surveyed the main channel and a small part of the floodplain. The conveyance curve increases until reaching a water level of 39m, after that the conveyance gets more or less stuck in a set of spikes around 7500 m³/s. This situation occurs when the water level reaches an elevation of 39m or more.

The best option would be to trim the cross-section to the embankments (Ch: 28 - 96m) or include the missing part of the floodplain into the cross-section. By trimming we will be avoiding the extra area we will obtain once we apply the panel markers that inevitably would increase the conveyance over the 7500 m³/s value.

Simple

Figure 7: Case 4, Cross-section with undeveloped floodplain

Differences

Conveyance difference compared against Simple scenario.
	Max K [m³/s]	Difference [m³/s]	Difference %
Simple	7799.11	0.000	0.000
Frugal	10187.36	2388.253	30.622
Super	10821.58	3022.474	38.754
Bed	10346.54	2547.433	32.663
Every	11022.44	3223.326	41.329

Cross-Section

Conveyance

Frugal

In cross-sections that are oddly shaped like this one, adding panel markers will remove the conveyance quirkiness produced by the strange geometric shape. But by doing so, we are increasing the area available and removing the macro-roughness of the cross-section. This is why trimming is a good option that removes conveyance problems without increasing the conveyance.

The “Frugal” option increases the conveyance a 30%.

Super

This scenario adds a few more panel markers, it increases the conveyance a 38.7%.

Bed

The “Bed” scenario manages to remove all the spikes with an increase in the max conveyance of 32%. This is a very similar figure to the “Frugal” scenario. I am usually opposed to panelling at bed levels, but I have to admit that in this case it does not really influence the results that much. Still we are shifting the conveyance curve to the right for a large range of depths, but this shift is rather small. The roughness decrease of panelling is equivalent¹⁰ to using a manning value of 0.024 on the simple section, instead of the 0.035 used on all scenarios.

Every

This scenario increases its conveyance a 41%. The roughness decrease of panelling is equivalent¹¹ to using a manning value of 0.020 on the simple section, instead of the 0.035 used on all scenarios.

Case 5: Large cross-section

In this case, a rather large cross-section from the Bald Eagle Creek is analysed. The data has been transformed from feet to meters. Given the size of the cross-section, the typical scenarios are not used. Instead, the modelled scenarios are EA (Panel markers around both channels), EB (Panel markers around 200m), EC (Panel markers at a high point located below 190m), Bed and Every.

When analysing a large cross-section, it is important not to lose the bigger picture. The cross-section is 30m tall. In reality, even during an extreme flood event, it is very rare to find a water depth larger than 10 meters. Therefore, the modeller has to be mindful when checking the maximum conveyance over a certain value of stage as it might not be representative of any real situation.

For this reason, the cross-section has been trimmed at 195m, which still is a rather large cross-section. The tabs for the trimmed plot and difference have been added.

Plot

Figure 8: Case 5, Large Cross-Section

Diff.

Conveyance difference compared against Simple scenario.
	Max K [m³/s]	Difference [m³/s]	Difference %
Simple	5556588	0.00	0.000
EA	5635338	78749.46	1.417
EB	6004320	447732.12	8.058
EC	5800060	243471.75	4.382
Bed	5842425	285837.01	5.144
Every	6197115	640526.54	11.527

XS.csv

K.csv

Plot (Tr)

Figure 9: Case 5, Trimmed Large Cross-Section

Diff. (Tr)

Conveyance difference compared against Simple scenario.
	Max K [m³/s]	Difference [m³/s]	Difference %
Simple	394299.9	0.00	0.000
EA	428926.2	34626.31	8.782
EC	418501.0	24201.09	6.138
Bed	427647.4	33347.46	8.457
Every	446181.8	51881.85	13.158

EA Scenario

This scenario is the closest one to the “Frugal” scenarios. This scenario with a panel marker at each side of the two main channels seems to behave well at the full cross-section (1.47%), however when the cross-section has been trimmed this scenario provides almost a 9% difference on the conveyance. It might not look that much but keep in mind that the simple scenario does not present any negative spikes and panel markers are not needed.

EB Scenario

This scenario is only appreciated on the full cross-section as the panel markers have been placed over the trimmed section. It increases the cross-section by a significant 8%.

EC Scenario

This scenario places panel markers at the high points located below 190m. It tries to represent a situation that I have seen frequently. For example, you have a floodplain consisting of grass with few bushes and a forest section. Then the modeller decides to divide each section with panel markers and assign a different n value for each section.

I have not been able to find a term for this situation so I called it “micro-assignation of n Manning’s” or in short “micro-n-assignation”¹².

In this example, the panel markers have been added to divide different sections of the flood plain, however, in this case, the manning n value remains 0.035 across the channel.

Only adding the panel markers have increased the conveyance¹³ by a 6%. The modeller should keep this in mind when selecting different roughness for each section.

Bed

This scenario produces results very alike to scenario EA (8.5%) in the trimmed section, on the whole cross-section the influence is larger (5%).

Every

The “Every” scenario shows the largest difference, 12% larger than the “Simple” scenario on both trimmed and non-trimmed cross-section.

Conclusions

It is not an easy task to conclude a study using only a few cases, especially when the outcomes depend heavily on the unique cross-section shape.

We can confirm that adding a panel marker always reduces the roughness of the cross-section, allowing more water to be conveyed downstream. This situation can be appreciated in all cross-sections but it is especially visible in case 2. Adding panel markers is equivalent to reducing the Manning’s n value of the cross-section.

This has a second connotation if the modeller is performing “micro-n-assignations”¹⁴ within the cross-section. Flood Modeller’s warning W2044 might suggest entrapping those values (the micro-assignations) within panel markers. The modeller has to be especially careful when following this warning. The mere act of entrapping a zone of the cross-section will reduce the roughness of the cross-section as a whole, and might be necessary to revise the Mannings’ n values used are (still) achieving what it is intended. The modeller can assign the Mannings’ n values and ignore the warning¹⁵ or alternatively, I would suggest avoiding “micro-n-assignation” at all¹⁶.

We can also confirm that for any given configuration of panel markers, adding a panel marker will impact its conveyance. However, when the configurations of panel markers are different (they are placed at other locations like Case 4 Bed versus Super), it is not possible to know (beforehand) what configuration produces a larger increase in conveyance based on the number of panel markers.

The modeller should not forget that panelling is a powerful tool to remove conveyance problems, but it is not the only one. In certain cases, trimming the unused parts of a cross-section, connecting bank channels to something else such as floodplain units, reservoir units or 2d areas can solve the problem.

Finally, the last conclusion is that a panel marker only influences the conveyance curve for water levels higher than its location. This means that higher panel markers are desirable because they keep the conveyance curve unaltered for the lower water depths.

Disclaimer

This article has not been peer-reviewed. The author assures that the utmost care has been taken to ensure the accuracy of the content. However, errors may occur, if you found any, please report it to the author’s via Linkedin.

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1. I have called them spikes because they remind me of dragon spikes, but Flood Modeller refers to them as negative conveyance.↩︎

2. The instabilities are produced because the solver will eventually jump from one stage value to the other. This generates a wave that propagates downstream.↩︎

3. Sometimes called smooth conveyance curve.↩︎

4. I think a lot of people are misled by Flood Modeller’s Warning 2044, combined with micro-assignation of n values to a cross-section instead of assigning “global” values.↩︎

5. Click on the plot’s legend to activate/deactivate the location of the panel parkers.↩︎

6. The only exception is adding a panel marker on the symmetry axis.↩︎

7. Comparison only valid at the maximum stage.↩︎

8. Comparison only valid at the maximum stage.↩︎

9. It is also not always possible either. In this case, it has been achieved because placing the panel marker at the bottom of the wall removes the friction of the wall (8.93 and 7.23) that removes the first spike. The second spike is removed because the left zone weights more than the right, so the right is unable to produce a spike. But if you zoom at 11.5m you can see how the conveyance has a (positive) quirkiness.↩︎

10. Comparison only valid at the maximum stage.↩︎

11. Comparison only valid at the maximum stage.↩︎

12. Micro-n-assignation is the name I have given when a modeller uses precise manning values across the channel as opposed to using one global n manning value for the channel, another for the left floodplain and another value for the right floodplain. Limiting the cross-section to 3 different n values.↩︎

13. Or reduced the roughness↩︎

14. Micro-n-assignation is the name I have given when a modeller uses precise manning values across the channel as opposed to using one global n manning value for the channel, another for the left floodplain and another value for the right floodplain. Limiting the cross-section to 3 different n values.↩︎

15. This is, not entrapping the areas with different manning in between panel markers.↩︎

16. I would suggest using the full range of n values on the (Chow 1985) tables. Use a value located in between the minimum and normal bracket if the cross-section has features that decrease the roughness (v.g. concrete walls) and values located between maximum and normal bracket if the cross-section has features that increase the roughness (v.g. a tree).↩︎