CIVE 445 - ENGINEERING HYDROLOGY
CHAPTER 4: HYDROLOGY OF SMALL CATCHMENTS

Catchments possessing some or all of the following properties are small in a hydrologic sense:

rainfall can be assumed to be uniformly distributed in time.
rainfall can be assumed to be uniformly distributed in space.
storm duration usually exceeds the time of concentration (small time of concentration)
runoff is primarily by overland flow.
channel storage processes are negligible (channels are steep and short).

Runoff response of small catchments can be described with relatively simple parametric methods.
In certain cases, the use of overland flow analysis may be required.
The upper limit of a small catchment is somewhat arbitrary.
Time of concentration and catchment area are used to describe the upper limit.
A time of concentration less than 1 hr describes a small catchment.
A catchment les than 2.5 km² (1 sq mi) is small.

1.1 RATIONAL METHOD

The rational method is the most widely used method fo tha analysis of runoff response from small catchments.
It has particular application in urban storm drainage.
The popularity of the rational method is attributed to its simplicity, although reasonable care is necessary in order to use it correctly.
The rational method takes into account the following hydrologic characteristics or processes:

rainfall intensity
rainfll duration
rainfall frequency
catchment area
hydrologic abstractions
runoff concentration
runoff diffusion

In general, the rational method provides only a peak discharge, although in the absence of runoff diffusion it also gives an isosceles-triangle-shaped hydrograph.
The peak discharge is the product of: (1) runoff coefficient, (2) rainfall intensity, and (3) catchment area, with all processes being lumped into these three parameters.
Rainfall intensity contains information on rainfall duration and frequency.
Rainfall duration is made equal to the time of concentration, i.e., to the runoff concentration property.
The runoff coefficient accounts for hydrologic abstractions and runoff diffusion, and may also indirectly account for frequency.
All major processes are embodied in the rational method.
The rational method does not take into account the following processes:

spatial or temporal variations in either total or effective rainfall
time of concentration being much greater than rainfall duration
a significant portion of runoff occurring in the form of streamflow
the catchment's antecedent moisture condition, although this may be implicitly incorporated into the runoff coefficient.

The above conditions dictate that the rational method be restricted to small catchments.
The assumption of constant rainfall in time and space is strictly valid only for small catchments.
For small catchments, storm duration usually exceeds the time of concentration (concentrated or superconcentrated catchment flow)
For small catchments, overland flow prevails over streamflow.
Upper limit for small catchments: 1.3 to 2.5 km².
The rational formula is:

Q = C I A

For rainfall in mm/hr, area in km², and peak discharge in m³/s:

Q = 0.278 C I A

For rainfall in mm/hr, area in ha, and peak discharge in L/s:

Q = 2.78 C I A

For rainfall in in/hr, area in acres, and peak discharge in cfs:

Q = 1.008C I A

The unit conversion coefficient 1.008 is usually neglected on practical grounds.

Methodology

The catchment should be small.
The catchment area is determined by suitable means (planimetering or CAD).
Determine the time of concentration:

using an empirical formula
assuming a flow velocity based on hydraulic principles and computing the travel time.

Procedures involve assumption of flow levels, channel shape, and friction coefficients.
The travel time, and therefore time of concentration, is a complex function of flow level (see Fig. 9-2).
Storm duration is made equal to time of concentration (assumption of concentrated catchment flow).
A rainfall frequency is chosen, to vary from 5 to 10 yr for storm sewers in residencial areas, 10 to 50 yr for commercial areas, and 50 to 100 yr for flood protection works.
Rainfall intensities are higher for shorter durations (see IDF curve).
Rainfall intensities are higher for longer return periods (see IDF curve).
Frequency of rainfall and runoff events may not be the same.
In practice, runoff coefficients are adjusted upward to reflect postulated decreases in runoff frequency.
Rainfall intensity is obtained from IDF curve.
Runoff coefficient is selected (see Table 4-1).
Variation of runoff coefficient with rainfall intensity (see Fig. 4-1).
Variation of runoff coefficient with rainfall frequency (see Fig. 4-2).
Variation of runoff coefficient with percent imperviouness and rainfall frequency (see Fig. 4-3).

Runoff concentration without diffusion

In the absence of diffusion, a catchment concentrates the flow at the outlet, attaining the maximum possible flow at a time equal to the time of concentration.
For duration equal to time of concentration, concentrated catchment flow is obtained at the outlet.
Since there is no diffusion, the method gives not only a peak flow but also a hydrograph in the shape of a isosceles triangle, with recession time equal to rising time.
The runoff coefficient is simply the ratio of effective rainfall to total rainfall.
A mass balance of effective rainfall and runoff leads to:

V_r = I_e t_r A = (C I A) t_r

from which:

C = C_a = I_e / I

where C_a = runoff coefficient due only to abstraction, C_a < 1 (a maximum value of 1).

Runoff concentration with diffusion

When diffusion is present, it can be accounted for in the runoff coefficient.
This would be applicable in very flat catchments such as those in Florida.
Diffusion increases the recession time and reduces the peak flow.
A hydrograph shape can no longer be obtained directly from the mass balance.
The reduction in runoff coefficient amounts to:

C = C_d C_a

in which C_d = runoff coefficient due to diffusion, C_d < 1.
Abstraction and diffusion processes (see Fig. 4-4).
In practice, no quantitative distinction is made between the two components.
Most of the time diffusion is negligible, leading to C_d = 1 (small catchment is the normal condition).

Relation between runoff coefficient and φ-index

The runoff coefficient can be related to total rainfall intensity and φ-index, provided the following assumptions are satisfied:

catchment response occurs with negligible diffusion
total and effective rainfall intensities are constant in time.

The first assumption is valid for small catchments.
The second assumption is implicit in the rational method.
For catchment response without diffusion:

C = C_a = I_e / I

For constant rainfall intensities:

I_e = I - φ

Combining these two equations leads to:

C = (I - φ) / I

Areal weighing of runoff coefficients

Values of runoff coefficient may vary within a catchment.
When a clear pattern of variation is apparent, a weighted value should be used.
Individual subcatchments are delineated, and their respective runoff coefficients identified.
Areal weighing leads to:

Q_p = 0.278 I (Σ C_i A_i)

Composite catchments

A composite catchment drains two or more adjacent subareas of widely differing characteristics.
Assume two subareas A and B, with times of concentration t_A and t_B, respectively (t_A < t_B).
Several duration are chosen between t_A and t_B.
To calculate the partial (subconcentrated) contribution of B, an assumption is made regarding the rate at which the flow is concentrated at the catchment outlet.
An assumption of linear concentration at the outlet is usually appropriate.
The duration that gives the highest peak is the design duration.
Example 4-1.

Effect of catchment shape

The rational method is suited to catchments where drainage area increases more or less linearly with catchment length.
If this is not the case, the peak flow may not increase with catchment area.
For the peak flow not to decreae, the drainage area must grow in the downstream direction at least as fast as the decrease in corresponding rainfall intensity (with increase in duration).
Example 4-2.

1.2 OVERLAND FLOW

Overland flow is surface runoff that occurs in the form of sheet flow on the land surface without concentrating in clearly defined channels.
Overland flow is the first manifestation of surface runoff.
Overland flow uses conceptual and deterministic methods to describe surface runoff in overland flow planes.
In overland flow applications, effective rainfall is referred to as rainfall excess.
Overland flow can account for runoff concentration and diffusion, and produces a hydrograph reflecting these two properties combined.
Rainfall can be allowed to vary in space and time if necessary.
Overland flow theory is more complex that the rational method.
It solves the problem of hydrograph shape very well.
For small catchments, the peak flow calculated by overland flow is the same as that of the rational method.
In a modeling framework, with many overland flow modules, the calculated peak flow has more detail than that of the rational method.
Overland flow applies to small catchments, but can also be applied with care to midsize catchments.
Small modules will retain topographic details.
Larger modules may mask the distributed character of overland flow.
Overland flow requires a computer to perform the calculations.

Overland flow theory

The equation of mass conservation is:

∂Q/∂x + ∂A/∂t = 0

Inclusion of sources and sinks leads to:

∂Q/∂x + ∂A/∂t = q_L

This equation is expressed on the overland flow plane per unit of width:

∂q/∂x + ∂h/∂t = i

Flow over the plane can be described as follows:

Fig. 4-9

As excess rainfall begins, water accumulated on the plane surface and begins to flow out of the plane at its lower end.
Flow at the outlet increase gradually from zero, while the total volume of water stored over the plane also increases gradually.
If rainfall excess continues, both outflow and total volume of water stored over the plane reach a constant value.
Outflow and storage volume remain constant and equal to the equilibrium value.
Immediately after excess rainfall ceases, outflow begins to draw water from storage, gradually decreasing while depleting the storage volume.
Eventually, outflow returns to zero as the storage volume is completely drained.

Fig. 4-10

At equilibrium state, the outflow must equal the inflow. Therefore:

q_e = (i/3600) L

in which:

q_e= equilibrium outflow in L/s/m
i = rainfall excess in mm/hr
L = plane length in m.

This equation is a statement of runoff concentration.
The equilibrium storage volume is the area above the rising limb and below a line q = q_e.
As a first approximation, it can be assumed to be equal to:

S_e = (q_e t_e)/2

in which:

S_e= equilibrium storage volume in L/m
q_e= equilibrium outflow in L/s/m
t_e = time to equilibrium in seconds.

Irregularities cause the equilibrium state to be approched asymptotically, and the actual time to equilibrium is not clearly defined.
A value of t correspoding to q = 0.98 q_e may be taken as an approximation.
Equations of continuity and motion: Equations 4-21 and 4-22.
Solutions can be attempted with:

Storage concept of Horton and Izzard (conceptual) (19450's)
Kinematic wave technique of Wooding (deterministic) (1960's)
Diffusion wave technique (Ponce) (1986; HEC-1, 1990; HEC-HMS, 1998)
Dynamic wave technique (Ben-Zvi). (1970's)

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