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Extrapolation of rating curve

The following procedures may be considered to extrapolate the stage-discharge relation (ISO, 1982):

• Double logarithmic straight line extrapolation.
  If the hydraulic characteristics of the control section do not change beyond the measured range, the power type stage-discharge relation is assumed to remain valid in the lower and upper end.
• Extension of the stage-area (h-A) and stage-velocity (h-v) curve beyond the measured range.
  This procedure is based on the observation that the h-v relation has little curvature under normal circumstances. The product of A and v can be used to obtain values of Q in the lower and upper end.
• Extrapolation based on the Manning or Chezy equation for steady flow .

HYMOS includes graphical and numerical options to assist you in the generation of synthetic stage-discharge data in the extrapolated range by:

• Computation of cross-sectional data
• Display of geometrical and hydraulic quantities in the measured range
• Computation of the same quantities in the extrapolated range from cross-sectional data, slope and roughness with the Manning equation.

Computation of cross-sectional data

The contour of a cross-section is obtained from:

• distance (x) from an initial point (left or right bank)
• depth (y), and
• depth-correction (y_c )

The depth-correction y_c may be introduced to evaluate quickly the effects of changes in the cross-section on geometric and hydraulic parameters.

The actual depth y_a is computed from:

y_a = y + y_c

A plot can be made of the cross-section and for levels at fixed intervals, the following quantities are computed, see also figure:

• surface width, (B)
• wetted perimeter, (P)
• cross-sectional area, (A)
• hydraulic radius, (R): R = A/P
• area.(hydraulic radius)^2/3 , (A.R^2/3 )

These parameters may be determined for the whole cross-section or for parts of the cross-section, e.g. main river or flood plain.

figure 8.6: cross-section

Note that if the cross-section is divided, the wetted perimeter for each part may be determined in two ways:

• the water boundary is not considered:
  • for the flood plain : P_fp = abc
  • for the river : P_r = cefg
• the water boundary is treated as a wall:
  • for the flood plain : P_fp = abcd
  • for the river : P_r = dcefg

To account for lateral transport of momentum between river and flood plain the latter option appears to be more realistic. It reduces generally the discharge capacity of the main channel. However, to obtain consistency with hydraulic computations, where generally the first approach is used (e.g. wendy), both options are included.

Display hydraulic quantities in the measured range

Geometric and hydraulic quantities are obtained in one of the following ways:

• from the stage-discharge database, provided that the cross-sectional parameters are also available
• from a combination of the cross-section profile and the rating curve.

The following parameters are displayed, c.q. computed for various depths:

• surface width, (B)
• wetted perimeter, (P)
• cross-sectional area, (A)
• hydraulic radius, (R): R = A/P
• area.(hydraulic radius)^2/3 , (A.R^2/3 )
• discharge, (Q)
• average velocity, (v): v = Q/A
• K-Manning. (energy slope)^1/2 , (K_M .S^1/2 )

The various quantities can be displayed as a function of stage or discharge. The quantity K_M .S^1/2 is derived from the Manning equation for steady flow:

Q = K_M A R^2/3 S^1/2

where:
Q = discharge
K_{M {}} = roughness parameter according to Manning (note: K_M = 1/n, where n is Manning's-n)
A = cross-sectional area
R = hydraulic radius
S = energy slope.

This function uses the available stage discharge data in the HYMOS database for the selected processing period. The hydraulic quantities are computed from these available data and saved in the Report file.

There is an extra option available which computes the discharges from the water levels with a selected rating curve. All other computed hydraulic quantities will be based on these discharge values instead of the measured ones.

Computation of hydraulic quantities in the extrapolated range

For extrapolation purposes it may be assumed that for the river section the quantity K_M .S^1/2 remains approximately constant at higher stages.
Assuming that the same value for S applies for the entire cross-section the roughness ratio between river and flood plain is the only unknown. hymosrequires the entry of the slope S and the K_M value for the distinguished n parts in the cross-section. The total discharge is then computed from:

The total cross-section may be divided in at maximum 5 parts.
This hymosoption produces the same output quantities as presented in the previous section.

After synthetic stage-discharge data have been obtained for the extrapolated range, these data can be entered manually in the stage-discharge database. Subsequently, new attempts can be made to fit a rating equation to the measured and estimated stage-discharge data.

Application

Select the function <Extrapolation of Rating Curves> from the Flow Measurements functions map. To execute the function, the following selections must be made. Select the procedure you want to use, e.g. 'Computation of cross sectional parameters' or 'Display hydraulic quantities' by pressing the appropriate tab.

To execute the 'Computation of cross sectional parameters' function you must perform the following steps:

1. Select a station from the station list box.
2. Select a cross-section from the Cross Sectional Profiles list box.
3. Display a graph of the Cross Section to analyse the cross section.
4. Enter the number of Sections in the with the appropriate Left, Right and Manning values of a section.
5. Enter the energy slope.
6. Change the Minimum and Maximum Elevation values, by default the minimum and maximum levels of the cross section are given. Take care that the minimum and maximum values are not lower and higher than the levels in the cross section.
7. Enter a value for the distance between two levels where the parameters are computed for. In the above example the parameters are computed for levels 512, 513, 514, .., 521.
8. Select to include the water boundaries or not.

When the levels at Cross-section variation is selected, the parameters will be computed for all levels where there is a change in the cross-section data.

Station Codes

Select a station by clicking the checkbox in the stations list. When the 'Display hydraulic quantities' tab is selected, HYMOSwill extract all Stage-Discharge data and Stage-Discharge relations for this station within the specified period from the database. When the "Computation of Cross sectional Parameters' tab is selected, HYMOSwill extract all cross-sections available for the selected station in the database. The parameter of the series where the cross-section belongs to and the date of the 'Cross-Sectional Profiles' are presented in the list box.

Cross Sectional Profiles

In the 'Entry and Edit' functions you can enter measured cross sections for a series. These cross sections can be measured at the station itself or some upstream or downstream of the station. While entering a cross-section in HYMOSthe cross-section data must be attached to a time series, this can be the upstream water level or the downstream water level of a station. In the 'Extrapolation of Rating Curves' function all cross-sections attached to the series of the selected station are available. The parameters of the series and the dates of the measurements are displayed in the list box, one of these dates must be selected. A graph of the selected cross-section of that date can be displayed by pressing the <Graph Cross Sections> button.

Entering K-Manning values

The total cross-section may be divided in at maximum 5 parts with the spin button '# Sections'. For each section the left and right value for the selected section as well as the K-Manning value can be entered. All other values can be changed in the textboxes.

Rating relations

When the 'Display Hydraulic Quantities' tab is active and a station is selected, the existing rating relations will be presented in the 'Rating Relations' list box. If a rating curve is used in the computations the user must first check the "Use Rating Curve for Discharge Calculation " checkbox and then select a Rating relation from the list box. HYMOSwill now present all coefficients from the selected relation in the spreadsheet. These values can be changed by the user when desired.

Execute function

After you made your selections press <Execute> and analyse the results from the report. To analyse the report press <Report> from the functions tab.
From the 'Display Hydraulic Quantities' tab a 'Graph Option' can be selected from the list box and analysed by pressing the <Show Graph> button.