Rating Curves Part 1

Introduction

This article focusses on rating curves in 1D and 2D HEC-RAS models. We’ll start by extracting stage-discharge rating curves from HEC-RAS and then focus on refining the data that feeds into the rating curve. In the end, we’re looking for a graphical or tabular representation of the relationship between water surface elevation and discharge rate as shown here:

You may run across some rating curves that have the axes reversed, but I prefer to keep stage on the vertical axis since I find it simpler to picture the water surface rising and falling at a gauge like in the image above.

There are plenty of other publications about rating curves, so there’s no need to restate all of the theoretical background here, but I’ll cover a few items here with particular relevance to HEC-RAS:

  • Differences between rating curves in steady flow and unsteady flow models
  • Interpreting rating curves that have been extracted from HEC-RAS
  • Differences between rating curves in 1D and 2D models
  • Using HEC-RAS models to optimise gauging locations
  • Setting up sensitivity runs to estimate confidence levels for rating curve data
  • Calibrating HEC-RAS models to measured gauging station observations
  • Using HEC-RAS to interpolate and extrapolate rating curve data from measured calibration points

If you’re looking for more information on the background theory behind rating curves, the USGS has a number of detailed publications with a fairly concise online summary here. In Australia, the CRC for Catchment Hydrology published a technical report entitled “The Calculation of Streamflow from Measurements of Stage, authored by John Fenton and Robert Keller. The 2001 report provides a solid theoretical background for the issue of correlating stage and flow. The pdf version of the report is available for download here.


Definition of a rating curve

Most hydrologists understand the term “rating curve” to apply strictly to the relationship between stage and discharge; however, HEC-RAS takes a wider interpretation of the term. When you click on the “View Computed Rating Curves” icon in HEC-RAS, it will default to stage versus flow, but the chart can be customised to plot any single variable along the horizontal axis against any number of variables along the vertical axis.

If you click on the “Plot Stage and Flow Hydrographs” icon, however, you’ll see a tab called “Rating Curve” that is limited to stage vs. flow only; for the purposes of this article, we’ll do the same and limit our discussion of rating curves to stage-flow relationships only.

 

 


1D Rating Curves in HEC-RAS

Below is an example of the output options available from the “Plot Stage and Flow Hydrographs” button on the main HEC-RAS command window. This example is from the Bald Eagle Creek example data set that comes with the software and is also included in the HEC-RAS manuals. You can click on the Stage-Flow tab, the Table tab or the Rating Curve tab to see the same data in different ways.

The data in the Table tab can be selected, copied, and pasted into Excel or other spreadsheet/plotting program. The other two tabs show the stage and flow data graphically. Because this particular example is from an unsteady flow run, there will be some looping – although in this case it is almost indiscernible unless you zoom in.

The “View Computed Rating Curve” icon in HEC-RAS provides essentially the same data but as a single valued rating curve.

Now let’s cover how we read these curves. If there is no significant looping, any given stage will correspond to the same flow rate, regardless whether the stage occurs during the rising or falling limb of the hydrograph. In the figure below, each of the three stages shown corresponds to a single point on the rating curve (click to enlarge):

Both curves have the same primary vertical axis for stage. If I move the two charts together closer together, you can see how the rating curve essentially just turns the secondary vertical axis for flow on its side and removes the time axis (click to enlarge):

 


Tailwater curves

1D structures give you the option of displaying tailwater or backwater curves at each structure. Here is an example from the HEC-RAS manual:

As shown by the tailwater curves, any given stage can correspond to a wide range of discharge rates. Unless you add energy, pressure, or some driving force that affects your energy slope, your upper limit will be the free flow curve. The lower limit for any given stage, of course, is zero. You can have depth without flow, but you can’t have flow without depth if that makes sense. If your downstream boundary was a large, static body of water, for instance, you could back up the water surface and provide a static water level at the structure even long after the upstream inflow has receded to nothing. In our next article, we’ll show some examples of negative flows as well that can just as well correspond to any stage in your rating curve.


Steady Flow vs. Unsteady Flow Rating Curves

1D HEC-RAS models give you the option of using steady flow or unsteady flow, whereas 2D models require a hydrograph with a time axis. When you hit “compute” in a steady-state run, the model assumes that the given flow rate has been running at that rate forever and ever – to infinity and beyond. Even a tiny trickle of water will fill every available storage area (whether it’s a massive mining pit or a hole that is dug to the other side of the earth) and instantly appear at the downstream end the model, regardless whether there was ever enough water available to do so. If the trickle would take a billion years to actually fill the hole, your model just travelled a billion years into the future the instant you hit “compute.”

Despite these limitations, sometimes steady flow is good enough – when the peak flow actually occurs after all available storage has been filled, for instance (and if all you’re interested in is the instantaneous peak flow conditions rather than how the water actually got there and how it will recede). Steady flows will certainly give you a much cleaner-looking rating curve; because all storage is automatically filled in a steady-flow model, a rating curve developed from that steady-state run will give you a very nice-looking curve that follows the same path on the way up as on the way down. But does that reflect reality? In some cases it’s perfectly adequate, but might be worth modelling it both ways to check.

Some 2D models allow you to input a steady flow rate, but this rate is then run over a finite time axis; steady flows aren’t an option at all in HEC-RAS 2D models unless you generate a flat hydrograph to mimic steady flow conditions. Keep in mind that by generating a flat hydrograph, the volume of water flowing into your system (which can be calculated by integrating the hydrograph and taking the area under the curve) may far exceed any volume that the catchment could possibly deliver – possibly many times greater than the volume of water that could be generated from the PMP rainfall. Remember, a steady flow stays at that rate forever, so the area under that curve is effectively infinite!

Here is an illustration of that concept showing three steady flow rates and three unsteady flow hydrographs that were used for some recent HEC-RAS benchmarking efforts:

If a typical hydrograph is applied to an unsteady flow model (either 1D and 2D), the storage conditions may be different when a particular stage is reached during the rising limb than when that same stage is reached on the falling limb of the hydrograph. In that case, the rating curve will exhibit some form of looping, which is potentially more representative of what actually occurs at most gauging sites.


Reading a looped rating curve

Here is an example of a looped rating curve from the same unsteady flow model – this one comes straight out of the HEC-RAS manual:

When looping is involved, the stage-discharge relationship is different during the rising limb of the hydrograph than during the falling limb. The following figure shows one way of using the hydrographs and curves to show the flow rate corresponding to a given stage. The primary and secondary vertical axes have been reversed from the standard plot in HEC-RAS to show the elevation axis in each plot adjacent to each other (click to enlarge):

We’ll dive back into this example in the next article. Essentially we’ve taken the same steps as above, but this time, the same elevation corresponds to two different flow rates: one on the way up, and the other on the way down.

If you wanted use your unsteady flow hydrograph to generate a single value rating curve with no looping, you could run a set of plan files to represent individual points on the curve. Each flow file would have a single discharge rate that continues for as long as it needs to in order to fill all available storage and reach steady flow conditions. You could ramp it up the flows by, say, 10 m3/s increments, and run a new plan for each flow rate.

Essentially you have made a pseudo-steady flow file (not to be confused with quasi-unsteady since that’s just for sediment transport in the current version of HEC-RAS). When you record the water surface elevation corresponding to each flow rate, you’ll get a single point on the discharge vs. stage curve. By stitching together a series of points, you could then interpolate values to generate your rating curve.

If you are working with a large number of gauges, this process can be time consuming. Another alternative is to make a composite hydrograph with the time axis stretched out at each flow increment. Instead of a smooth or triangular hydrograph, you might have something like this:

The duration required to reach steady flow conditions at each interval would be different for every system; you may need to stretch each plateau out further than what is shown above. This composite pseudo-steady flow file is basically a series of steady flow rates with a time axis added. Using a hydrograph like the one shown above will generate a choppy rating curve where the flow ramp up, but the points on the curve that represent the steady-state flows will generally form a smooth line representing the steady-state, open channel flow curve.

This approach might be acceptable for systems where flood levels rise and fall slowly, allowing your rating curve to mimic what occurs naturally. If you’re real floods take five days to pass, for example, but you want to limit your simulation time to a maximum of 24 hours, this can save you some time. In “flashy” systems, however, the flood may arrive so quickly that you may never reach steady conditions. In that case, artificially spreading out the hydrograph can introduce errors, since the looping is a real phenomenon and not just a remnant of the modeling approach.

1D vs. 2D Rating Curves

In a 1D model, the water surface elevation will be computed and displayed perfectly flat along the entire extent of each cross section. Because the elevation is constant at every point along the section alignment at any given time step, the stage hydrograph will be identical at every point along the line, and a single stage hydrograph can be used to represent the whole section. In 1D, we can thus generate both stage and flow hydrographs for every cross section and easily plot the two parameters against each other; in fact, HEC-RAS will do that for you automatically.

Another inherent constraint of 1D models is that the computed discharge will cross the section line with a flow direction that is exactly perpendicular to its alignment. Even if we’re wrong about the flat water surface or the orientation of the flow as assigned by our selection of the alignment, HEC-RAS will force it to be so in the model.

In 2D models, on the other hand, water surface elevations can vary at each computational grid. See the image below for an example of a cross section that has been cut from a 2D model. The water surface elevation varies across the section:

Depending on your mesh resolution, any given cross section could have dozens or even hundreds of different stage hydrographs. To develop a stage-discharge relationship in a 2D model, we therefore need to measure the water surface elevation at the single computational grid element that lies closest to our gauge location.

While stage can be measured at this single point, there can be no flux through a dimensionless point, so we will need to draw a cross section along which to compute discharge. In a 1D model, this will already have been done; for 2D models, however, we do this by defining a profile line in RAS Mapper. The net flux across the profile line will then be computed based on the resultant velocity vectors for each face point along the profile alignment. We can orient the profile line any way we wish; the flow direction and the slope of the water surface will be computed independently of the alignment that we select for our profile line.

[Continue to Rating Curves Part 2]

 

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