The Slope-Area Method for Peak Discharge Determination

I. Objective

The discharge of a stream is typically measured directly by stream gaging or a rating curve (see Lab 2). However, conditions sometimes prevent direct measurement of discharge, such as during flooding events, or in remote areas. The slope-area method is used to determine peak discharge along sections of a river or stream where gages are not present. It is particularly useful for (1) determining the discharge needed for flooding along a particular reach of stream after a flood has passed, (2) or to estimate the discharge necessary to cause flooding along a section of river.

II. Background

The slope-area method is based on the Manning's equation for determining discharge,

Q=(1.49AR2/3S1/2)/n

where A is the cross-sectional area (ft2), R is the hydraulic radius (cross sectional area/wetted perimeter (ft)), S is the slope (drop in elevation/length (dimensionless)) and n is the Manning roughness coefficient (dimensionless). Thus, the slope-area method is a function of (1) slope, (2) channel dimensions and (3) channel roughness, and therefore field data are required for estimation of peak discharge. These data include determining the elevation and location of high-water marks along the stream, measurement of channel cross section and wetted perimeter by surveying, tape and compass, or GPS, and selection of a roughness coefficient for the section of stream in question.

III. Procedure

A. Selection of Cross Sections

Select two cross sections along a reach that have at least 0.5 feet of elevation difference between their high water marks (McCuen, 1998). We recommend a minimum of 300 feet separation of these cross sections along a reach, and 2000' is optimal in low slope areas.

1. High water marks must be clearly evident on both sides of the river, at both cross sections.

2. The reach of river between the cross sections must have similar roughness characteristics

3. There must not be any bridges or other "disruptions" to the stream course between the cross sections.

4. If GPS is used to acquire cross section measurements, there must be no barriers that will block or degrade the satellite signals at both cross sections.

B. Water Surface Slope Determination

One of the most difficult concepts to understand when using the slope-area method is that 2 different slopes are calculated. The first is the water surface slope ((hu-hd)/L), which is simply the slope of the high water marks between the upstream (hu) and downstream (hd) sections, and would be the slope of the water surface during peak discharge. The water surface slope is used as a first approximation for the energy slope, which is ultimately the slope that the slope-area method relies on. The energy slope ((Hu-Hd)/L) is the slope between the upstream and downstream cross sections of the high water marks PLUS the velocity head (energy, or v2/(2g)), such that

Se = ((hu+ auvu2/(2g)) - (hd+ advd2/(2g)))/L

where Se is the energy slope, hu and hd are the high water marks, a is a correction factor that accounts for expanding or contracting reaches (and is typically ignored by giving it a value of 1 for both upstream and downstream sections), L is the distance between the two sections, and g is gravity.

For example, if the upstream section had a significant higher velocity than the downstream section, then the upstream velocity head (Hu = hu+ vu2/(2g)) would be much higher than the downstream head (Hd = hd+ vd2/(2g)) and the energy slope would be much greater (see diagram below). Alternatively, if the downstream section had a significantly higher velocity than the upstream section, than the energy slope would be lower (less difference between the upstream and downstream total heads). The diagram below shows the relationships between the water surface slope ((hu-hd)/L) and the energy slope ((Hu-Hd)/L).

 

The slope-area method ultimately uses the energy slope to determine peak discharge, but the energy slope is first approximated using the water surface slope (the "Calculation" section will explain this more in depth). Therefore, to determine the water surface slope, first identify the high water marks at each cross section (these marks should be at essentially the same elevation on both banks of the same cross section). Using GPS or surveying equipment, measure the elevation difference (hu-hd) and distance between the high water marks of the two cross sections (L) on the right edge (looking downstream), then repeat for the left edge. Calculate the slope between these water marks

S = (hu-hd)/L

Compare the calculated slope for the right and left edges. If these differ by greater than 5%, recheck your high water mark interpretation and your measurement of elevation differences and height.

If high water marks are not evident, or if there is a large discrepancy between your right edge and left edge slopes, it is possible to estimate the water surface slope from a topographic map, realizing that this method can introduce significant error in your estimate (the water surface slope and bed slope may be very different during flooding events). Select two topographic lines that cut across the streambed on the topographic map, and measure the distance the stream travels between these lines (not straight-line distance), then divide the elevation difference by the distance. Remember that using water surface slope is recommended when possible, but bed slope can be used when other methods of determining water surface slope fail.

Note if using GPS. Make sure you measure the distance along the banks of the stream in order to determine L, and not just the straight line distance between the high water marks. Thus, it is better to select a "line" rather than 2 "points to determine the elevation difference and distance between the cross sections.

C. Measurement of the Hydraulic Radius (R)

The hydraulic radius (R) is the cross-sectional area divided by the wetted perimeter of the section, and thus has units of ft. A simple topographic profile of the cross-section measured by GPS, surveying, or tape and compass provide the distance along the stream bed from the left high water mark to the right high water mark (wetted perimeter), and cross-sectional area is easily determine by multiplying the average depth of the section (elevation of the high water mark minus the average stream bed elevation along the profile) by the width of the profile (straight-line distance between the high water marks. Therefore, one must measure the following to obtain the necessary data at both the upstream and downstream cross-sections.

1. Measure elevation (relative or absolute) of the high water marks (they should be essentially the same elevation).

2. Measure the straight line distance between the high water marks (width).

3. Measure depth of the river bed below the high water marks at set intervals (every 1-5 feet or so) and calculate the average depth. The area (A) is calculated by multiplying the width of the river by the average depth.

4. Measure the wetted perimeter (P) by measuring the distance along the river bed from one high water mark to the other. The hydraulic radius (R) = A/P.

Note if using GPS. Choosing a "line" setting rather than a "point" setting will enable you to easily dump the data into an Excel Spreadsheet and calculate both the wetted perimeter (3-dimensional line) and the straight line distance between the high water marks (2-dimensional line). You will also be able to reasonably estimate cross-sectional area IF you walk across the section with a constant velocity (straight line positions are fairly evenly spaced). See the GPS tutorial for calculating 2 and 3-d distances and cross-sectional areas for more help.

D. Estimation of Manning's Roughness Coefficient (n)

Inspection of the riverbed will reveal characteristics related to roughness. An excellent treatment of the use of Manning's coefficients is found on pages 128-136 of McCuen (1998). Below is a first-approximation of Manning's coefficients for some widely observed beds.

n = 0.04 - 0.05 Mountain streams

n = 0.035 Winding, weedy streams

n = 0.028 - 0.035 Major streams with widths > 100 ft at flood stage

n = 0.015 Clean, earthen channels

E. Calculations

1. Compute values of cross-sectional area (A), hydraulic radius (R), and roughness (n) for each cross section, and water surface slope (S) between the cross sections.

2. Calculate the upstream and downstream conveyance values (Ku and Kd), such that

Ku = 1.49AuRu2/3/nu and Kd = 1.49AdRd2/3/nd

3. Calculate the average conveyance of the reach of river between the cross-sections

K = (KuKd)0.5

4. Calculate the first estimation of peak discharge by multiplying the average conveyance (K) by the square root of the water surface slope.

Qp = KS0.5

This answer is in ft3/s, and gives a rough approximation of the peak discharge. However, a closer approximation is calculated if the slope includes the velocity heads, which can now be estimated from this initial discharge calculation.

5. Calculate the difference in the velocity heads (dhv) of the upstream and downstream sections according to

where g is gravity (32 ft/s2), and Au and Ad are the upstream and downstream cross-sectional areas. Also remember that a = 1. Note that the area values (and units) are squared, giving dhv in ft.

Note: If the downstream area (Ad) is less than the upstream area (Au) then dhv will be a negative number. Simply put, if the downstream area is less than the upstream, the downstream velocity and its velocity head will be greater than upstream. Thus, the greater downstream velocity head will reduce the energy slope (thus the negative number, see below).

6. Compute a new energy slope

Se = ((hu - hd) + dhv)/L

If the dhv is negative (downstream velocity is greater) than the energy slope is less than the water surface slope.

7. Compute a new peak discharge using the energy slope

Qp = KSe0.5

8. Repeat steps 6 and 7 above using this new peak discharge until Qp does not vary.

Reference: McCuen, R.H., 1998. Hydrologic Design and Analysis; Prentice Hall, New Jersey, 814 pages.

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