**CHAPTER 9.-INDIRECT DETERMINATION OF**

**PEAK DISCHARGE**

**INTRODUCTION**

During floods, it is frequently impossible or impractical to measure the
peak discharges when they occur because of conditions beyond control.
Roads may be impassable; structures from which current­meter measurements
might have been made may be nonexistent, not suitably located, or
destroyed; knowledge of the flood rise may not be available sufficiently
in advance to permit reaching the site near the time of the peak; the
peak may be so sharp that a satisfactory cur­rent-meter measurement could
not be made even with an engineer present at the time; the flow of
debris or ice may be such as to prevent use of a current meter; or
limitations of personnel might make it impossible to obtain direct
measurements of high-stage discharge at numerous locations during a
short flood period. Consequently, many peak discharges must be
determined after the passage of the flood by indirect methods such as
slope-area, contracted opening, flow-over­dam, or flow-through-culvert.

Indirect determinations of discharge make use of the energy equa­tion for
computing streamflow. The specific equations differ for differ­ent types
of flow, such as unobstructed open-channel flow, flow over dams, and
flow through culverts. However, all the methods involve these general
factors:

1\. Physical characteristics of the channel; that is dimensions and
conformation of the channel within the reach used and boundary
conditions.

2\. Water-surface elevations at time of peak stage to define the upper
limit of the cross-sectional areas and the difference in elevation
between two or more significant cross sections.

3\. Hydraulic factors based on physical characteristics, water-surface
elevations, and discharge, such as roughness coefficients and dis­charge
coefficients.

This chapter provides only a brief general discussion of the proce-

dures used in collecting field data and in computing discharge by the
various indirect methods. That highly specialized subject is treated in
detail in the several manuals of the series \"Techniques of
Water­Resources Investigations of the United States Geological Survey\"
that are listed as references at the end of this chapter.

It should be remembered that the discharge that is determined by either
direct measurement or by indirect methods includes not only the water
but also any substances suspended or dissolved in the water (P. 79).

**COLLECTION OF FIELD DATA**

The data required for the computation of discharge by indirect methods
are obtained in a field survey of a reach of channel. The survey
includes the elevation and location of high-water marks cor­responding to
the peak stage; cross sections of the channel along the reach; selection
of a roughness coefficient; and description of the geometry of dams,
culverts, or bridges, depending on the type of peak-discharge
determination to be made. The selection of a suitable site is a most
important element in the application of the indirect method of discharge
determination.

It is recommended that a transit be used to make a \"transit-stadia\"
survey of the selected site. That method combines vertical and
hori­zontal control surveys in one operation and is accurate, simple, and
speedy.

Selection of a roughness coefficient remains essentially an \"art\" that
is developed through experience. The factors that exert the greatest
influence on the coefficient of roughness are the character of the
streambed material, cross section irregularity, the presence of
vegetation, and the alinement of the channel. In the Manning equa­tion
the roughness coefficient, n, ranges from as low as 0.012 for a
concrete-lined channel in excellent condition or for a smooth sand
channel of regular geometry to more than 0.1 for overbank areas having a
heavy cover of brush.

> **SLOPE-AREA METHOD**

The slope-area method is the most commonly used technique of indirect
discharge determination. In the slope-area method, dis­charge is computed
on the basis of a uniform-flow equation involving channel
characteristics, water-surface profiles, and a roughness or retardation
coefficient. The drop in water-surface profile for a uniform reach of
channel represents energy losses caused by bed and bank roughness.

In applying the slope-area method, any one of the well-known vari­ations
of the Chezy equation may be used. However the Manning equation is
preferred in most countries, including the U.S.A., because it is simple
to apply, and the many years of experience in its use have shown that it
produces reliable results.

where

> *Q* = discharge,
>
> A = cross-sectional area, *R* = hydraulic radius, *S* = friction
> slope, and n = roughness coefficient.

The Manning equation was developed for conditions of uniform flow in
which the water-surface profile and energy gradient are parallel to the
streambed and the area, hydraulic radius, and depth remain con­stant
throughout the reach. For lack of a better solution, it is assumed that
the equation is also valid for the nonuniform reaches that are
invariably encountered in natural channels, if the water-surface
gra­dient is modified by the difference in velocity-head between cross
sections. The energy equation for a reach of nonuniform channel be­tween
cross section 1 and cross section 2 shown in figure 134 is

where

> *h* =elevation of the water surface at the respective cross
>
> *h,,*
>
> *hf*
>
> Ah,. sections above a common datum,
>
> velocity head at the respective cross sections = aV2/2g, where a =
> velocity-head coefficient,
>
> energy loss due to boundary friction in the reach, upstream velocity
> head minus the downstream velocity head, used as a criterion for
> expansion or contraction of reach, and
>
> k(Ah,,) = energy loss due to acceleration or deceleration in a con-
>
> tracting or expanding reach, where
>
> *k* = energy loss coefficient.

The Manning equation, written in terms of discharge, is

The friction slope (S) to be used in the Manning equation is thus

defined as

where Ah is the difference in water-surface elevation at the two
sec­tions and L is the length of the reach.

In using the Manning equation the conveyance, K, is computed for each
cross section as (1.486/n)AR 2/3 in English units, or (11n)AR 2/3 in
metric units. The mean conveyance in the reach is then computed as the
geometric mean of the conveyance at the two sections. This proce­dure is
based on the assumption that the conveyance varies uniformly between
sections. The discharge is computed by use of the e ation

where S is the friction slope as previously defined,

**CONTRACTED-OPENING METHOD**

The contraction of a stream channel by a roadway crossing creates an
abrupt drop in water-surface elevation between an approach sec­tion and
the contracted section under the bridge. The contracted sec­tion framed
by the bridge abutments and the channel bed is in a sense a discharge
meter that can be utilized to compute floodflows. The head on the
contracted section is defined by high-water marks, and the geometry of
the channel and bridge is defined by field surveys.

In computations of peak discharge at a contraction, the drop in
water-surface level between an upstream section and a contracted section
is related to the corresponding change in velocity. The dis­charge
equation results from writing the energy and continuity equa­tions for
the reach between these two sections, designated as sections 1 and 3 in
figure 135.

> The discharge equation is

in which

> Q = discharge,
>
> *g* = acceleration of gravity,
>
> *C* = coefficient of discharge based on the geometry of the bridge
>
> and embankment,
>
> A,3 = gross area of section 3; this is the minimum section parallel
>
> to the constriction between the abutments, and it is not
>
> necessarily located at the downstream side of the bridge, Ah =
> difference in elevation of the water surface between sec­tions 1 and 3,

*a,* [V,\']{.underline} = weighted average velocity head at section 1,
where V, is

> \@9 the average velocity, Q/A,, and a, is coefficient that takes into
> account the variation in velocity in that sec­tion, and
>
> *hf* = the head loss caused by friction between sections 1 and 3.
>
> The friction loss, *hf,* as computed by the Manning equation, is only