Drainage patterns in catchments vary widely. The more intricate patterns are an indication of high drainage density. Drainage patterns reflect geologic, soil, and vegetation effects (Fig. 2-47) and are often related to hydrologic properties such as runoff response or annual water yield. Types of drainage patterns that are recognizable on aerial photographs are shown in Fig. 2-48 [30].
Fig. 2-47 Drainage patterns as affected by local geology.
Fig. 2-48 Drainage patterns recognizable on aerial photographs.
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S
urface runoff, or simply runoff, refers to all the waters flowing on the surface of the earth, either by overland sheet flow or by channel flow in rills, gullies, streams, or rivers. Surface runoff is a continuous process by which water is constantly flowing from higher to lower elevations by the action of gravitational forces. Small streams combine to form larger streams which eventually grow into rivers. In time, rivers carry their flow into the ocean, completing the hydrologic cycle.Runoff is expressed in terms of volume or flow rate. The units of runoff volume are cubic meters, cubic feet, or acre-feet. Flow rate (or discharge) is the volume per unit of time passing through a given area. It is expressed in cubic meters per second or cubic feet per second. Flow rate usually varies in time; therefore, its value at any time is the instantaneous or local flow rate. The local flow rate can be averaged over a period of time to give the average value for that period. The local flow rate can be integrated over a period of time to give the accumulated runoff volume, as follows:
∀ =
∫
Q dt =Σ
Q Δt (2-55)in which ∀= runoff volume, Q = flow rate, t = time, and Δt = increment of time.
In engineering hydrology, runoff is commonly expressed in depth units. This is accomplished by dividing the runoff volume by the catchment area to obtain an equivalent runoff depth distributed over the entire catchment.
For certain applications, runoff is alternatively expressed in terms of either: (1) peak flow per unit drainage area, (2) peak flow per unit runoff depth, or (3) peak flow per unit drainage area per unit runoff depth. In the first case, the units are cubic meters per second per square kilometer; in the second case, cubic meters per second per centimeter; in the third case, cubic meters per second per square kilometer per centimeter.
Runoff Components
Runoff may consist of water from three sources:
1. Surface flow, 2. Interflow, and 3. Groundwater flow.
Surface flow is the product of effective rainfall, i.e., total rainfall minus hydrologic abstractions.
Surface flow is also called direct runoff. Direct runoff has the capability to produce large flow concentrations in a relatively short period of time. Therefore, direct runoff is largely responsible for flood flows.
Interflow is subsurface flow , i.e., flow that takes place in the vadose zone, i.e., in the unsaturated soil layers located beneath the ground surface (Fig. 2-49). Interflow consists of the lateral movement of water and moisture toward lower elevations, and it includes some of the precipitation abstracted by infiltration. It is characteristically a slow process, but eventually a fraction of the interflow volumes flow into streams and rivers. Typically, the quantities of interflow are relatively small compared to the quantities of surface and groundwater flow.
Fig. 2-49 The vadose zone.
Groundwater flow takes place below the groundwater table in the form of saturated flow through alluvial deposits and other water-bearing geologic formations located beneath the soil mantle (Fig. 2-50) (Chapter 11). Groundwater flow includes the portion of infiltrated volume that has reached the water table by percolation from the overlying soils. Like interflow, groundwater flow is characteristically a slow process. Like surface runoff, groundwater flow is a continuous process, with water constantly moving to lower elevations (or to zones of lower potential). Most groundwater flow is eventually intercepted by streams and rivers, discharging into them. A small portion of groundwater flow, particularly that flowing at great depths, slowly makes its way into the nearest ocean. The average global residence time of groundwater is 1400 years [**].
Fig. 2-50 General direction of groundwater flow (U.S. Geological Survey).
Stream Types and Baseflow
Streams may be grouped into three types:
1. Perennial,
2. Ephemeral, and 3. Intermittent.
Perennial streams are those that always have flow. During dry weather (i.e., absence of rain), the flow of perennial streams is baseflow, consisting mostly of groundwater flow intercepted by the stream.
Streams that feed from groundwater reservoirs are called effluent streams. Perennial and effluent streams are typical of subhumid and humid regions [(Fig. 2-51 (a)].
Ephemeral streams are those that have flow only in direct response to precipitation, i.e., during and immediately following a major storm. Ephemeral streams do not intercept groundwater flow and, therefore, have no baseflow. Instead, ephemeral streams usually contribute to groundwater by seepage through their porous channel beds. Streams that feed water into groundwater reservoirs are called influent streams. Channel abstractions from influent streams are referred to as channel transmission losses. Ephemeral and influent streams are typical of arid and semiarid environments [(Fig. 2-51 (b)].
Intermittent streams are those of mixed characteristics, behaving as perennial at certain times of the year and ephemeral at other times. Depending on seasonal conditions, these streams may feed to or from the groundwater [(Fig. 2-51 (c) and (d)].
Fig. 2-51 (a) Perennial stream:
Indian Creek, California. Fig. 2-51 (b) Ephemeral stream:
Mojave river, California.
Fig. 2-51 (c) Intermittent stream:
Rosarito Creek, Baja California. Fig. 2-51 (d) Intermittent stream:
Gila river, Arizona.
Baseflow estimates are important in dry weather hydrology; for instance, in the calculation of the total runoff volume produced by a catchment in a year, referred to as the annual water yield. In flood hydrology, baseflow is used to separate surface runoff into: (a) direct, and (b) indirect runoff. Indirect runoff is surface runoff originating in interflow and groundwater flow. Baseflow is a measure of indirect runoff.
Surface runoff and Baseflow. In practice, surface runoff may or may not include baseflow. The term
"surface runoff" is often used at the watershed scale to refer to direct runoff, excluding baseflow. Yet,
at the basin scale, estimates of surface water yield typically include both direct runoff and baseflow.
The confusion is frequently a source of error in hydrologic analysis. For instance, the NRCS runoff curve number method (Chapter 5) was originally developed to calculate direct surface runoff from small watersheds. Yet, over the years since its original inception, the method has also been used to calculate surface runoff from larger watersheds, which may include baseflow.
Antecedent Moisture
Effective precipitation is the fraction of total precipitation that remains on the catchment surface after all the hydrologic abstractions have taken place. During rainy periods, infiltration plays a major role in abstracting total precipitation. Actual infiltration rates and amounts vary widely, being highly dependent on the initial level of soil moisture. Soil moisture varies with the history of antecedent rainfall, increasing with antecedent rainfall and decreasing with a lack of it. For a given storm, the history of antecedent rainfall, which may have caused the soil moisture to depart from an average state, is termed the "antecedent moisture" or "antecedent rainfall" condition. A catchment with low initial soil moisture (e.g., a catchment drier than normal) is not conducive to high surface flow and direct runoff. Conversely, a catchment with high initial soil moisture (e.g., a catchment wetter than normal) is conducive to large quantities of surface flow and direct runoff (Fig. 2-52).
Fig. 2-52 A catchment with high antecedent moisture: Campo Creek, California, on March 5, 2005, after a few days of heavy rain.
The recognition that direct runoff is a function of antecedent moisture has led to the concept of antecedent precipitation index (API). The average moisture level in a catchment varies daily, being replenished by precipitation and depleted by evaporation and evapotranspiration. The assumption of a logarithmic depletion rate leads to a catchment's API for a day with no rain:
Ii = K I i-1 (2-56)
in which Ii = index for day i, Ii-1 index for day i -1, and K = a recession factor taken normally in the range 0.85 ≤ K ≤ 0.98 [53]. If rain occurs in any day, the rainfall depth is added to the index. The index at day zero (initial value) would have to be estimated. Likewise, the applicable value of K is determined from either data or experience.
The API is directly related to runoff depth. The greater the value of the index, the greater the amount of runoff. In practice, regression and other statistical tools are used to relate runoff to API. These relations are invariably empirical and therefore strictly applicable only to the situation for which they were derived.
Other measures of catchment moisture have been developed over the years. For instance, the Natural Resources Conservation Service (NSCS) uses the concept of antecedent moisture condition (AMC) (Chapter 5), grouping catchment moisture into three levels: AMC I, a dry condition; AMC II, an average condition; and AMC III, a wet condition. Moisture conditions ranging from AMC II to AMC III are normally used in hydrologic design.
Another example of the use of the concept of antecedent moisture is that of the SSARR model (Chapter 13). The SSARR model computes runoff volume based on a relationship linking runoff percent to a soil-moisture index (SMI), with precipitation intensity as a third variable. Runoff percent is the ratio of runoff to rainfall, multiplied by 100. Such runoff-moisture-rainfall relation is empirical and, therefore, is limited to the basin for which it was derived.
Rainfall-Runoff Relations
Rainfall can be measured in a relatively simple way. However, runoff measurements usually require an elaborate streamgaging procedure (Chapter 3). This difference has led to rainfall data being more widely available than runoff data. The typical catchment has many more raingages than streamgaging stations, with the rainfall records likely to be longer than the streamflow records.
The fact that rainfall data is more readily available than runoff data has led to the calculation of runoff by relying on rainfall data. Although this is an indirect procedure, it has proven its practicality in a variety of applications.
A basic linear model of rainfall-runoff is the following:
Q = b ( P - Pa ) (2-57)
in which Q = runoff depth, P = rainfall depth, Pa = rainfall depth below which runoff is zero, and b = slope of the line (Fig. 2-53). Rainfall depths smaller than Pa are completely abstracted by the catchment, with runoff starting as soon as P exceeds Pa. To use Eq. 2-57 it is necessary to collect several sets of rainfall-runoff data and to perform a linear regression to determine the values of b and Pa (Chapter 7). The simplicity of Eq. 2-57 precludes it from taking into account other important runoff- producing mechanisms such as rainfall intensity, infiltration rates, and/or antecedent moisture. In practice, the correlation usually shows a wide range of variation, limiting its predictive ability.
Fig. 2-53 Basic linear model of rainfall-runoff.
The effect of infiltration rate and antecedent moisture on runoff is widely recognized. Several models have been developed in an attempt to simulate these and other related processes. Typical of such models is the NRCS runoff curve number model, which has had wide acceptance in engineering practice. The NRCS model is based on a nonlinear rainfall-runoff relation that includes a third variable (curve parameter) referred to as runoff curve number, or CN. In a particular application, the CN value is determined by a detailed evaluation of soil type, vegetative and land use patterns, antecedent moisture, and hydrologic condition of the catchment surface. The NRCS runoff curve number method is described in Chapter 5.
Runoff Concentration
An important characteristic of surface runoff is its concentration property. To describe it, assume that a storm falling on a given catchment produces a uniform effective rainfall intensity distributed over the entire catchment area. In such a case, surface runoff eventually concentrates at the catchment outlet, provided the effective rainfall duration is sufficiently long. Runoff concentration implies that the flow rate at the outlet will gradually increase until rainfall from the entire catchment has had time to travel to the outlet and is contributing to the flow at that point. At that time, the maximum, or equilibrium, flow rate is reached, implying that the surface runoff has concentrated at the outlet. The time that it takes a parcel of water to travel from the farthest point in the catchment divide to the catchment outlet is referred to as the time of concentration.
The equilibrium flow rate is equal to the effective rainfall intensity times the catchment area:
Qe = Ie A (2-58)
in which Qe = equilibrium flow rate; Ie = effective rainfall intensity; and A = catchment area. This equation is dimensionally consistent; however, a conversion factor is needed in the right-hand side to account for the applicable units. For instance, in SI units, with Qe in liters per second, Ie in millimeters per hour, and A in hectares, the conversion factor is 2.78. In U.S. customary units, with Qe in cubic
feet per second, Ie in inches per hour, and A in acres, the conversion factor is 1.008, which is often neglected.
The process of runoff concentration can lead to three distinct types of catchment response. The first type occurs when the effective rainfall duration is equal to the time of concentration. In this case, the runoff concentrates at the outlet, reaching its maximum (equilibrium) rate after an elapsed time equal to the time of concentration. Rainfall stops at this time, and subsequent flows at the outlet are no longer concentrated because not all the catchment is contributing. Therefore, the flow gradually starts to recede back to zero. Since it takes the time of concentration for the farthest runoff parcels to travel to the outlet, the recession time is approximately equal to the time of concentration, as sketched in Fig. 2-54. (In practice, due to nonlinearities, actual recession flows are usually asymptotic to zero).
This type of response is referred to as concentrated catchment flow.
Fig. 2-54 Concentrated catchment flow.
The second type of catchment response occurs when the effective rainfall duration exceeds the time of concentration. In this case, the runoff concentrates at the outlet, reaching its maximum (equilibrium) rate after an elapsed time equal to the time of concentration. Since rainfall continues to occur, the whole catchment continues to contribute to flow at the outlet, and subsequent flows remain concentrated and equal to the equilibrium value. After rainfall stops, the flow gradually recedes back to zero. Since it takes the time of concentration for the farthest runoff parcels to travel to the outlet, the recession time is approximately equal to the time of concentration, as shown in Fig. 2-55. This type of response is referred to as superconcentrated catchment flow.
Fig. 2-55 Superconcentrated catchment flow.
The third type of response occurs when the effective rainfall duration is shorter than the time of concentration. In this case the flow at the outlet does not reach the equilibrium value. After rainfall stops, the flow recedes back to zero. The requirements that volume be conserved and recession time be equal to the time of concentration lead to the idealized flat top response shown in Fig. 2-56. This type of response is referred to as subconcentrated catchment flow.
Fig. 2-56 Subconcentrated catchment flow.
In practice, concentrated and superconcentrated flows are typical of small catchments, i.e., those likely to have short times of concentration. On the other hand, subconcentrated flows are typical of midsize and large catchments, i.e., those with longer times of concentration. Figure 2-57 shows actual dimensionless hydrographs depicting the three types of catchment flows.
Fig. 2-57 Dimensionless hydrographs showing three types of catchment flow.
Time of Concentration. Hydrologic procedures for small catchments usually require an estimate of the time of concentration (Chapter 4). However, precise estimations are generally difficult. For one thing, time of concentration is a function of runoff rate; therefore, an estimate can only represent a certain flow level, whether it be low flow, average flow, or high flow.
Several formulas for the calculation of time of concentration are available. Most are empirical in nature and, therefore, of somewhat limited value. Nevertheless, a few are widely used in practice. An alternate approach is to calculate time of concentration by dividing the principal watercourse into several subreaches and assuming an appropriate flow level for each subreach. Subsequently, a steady open-channel flow formula such as the Manning equation is used to calculate the mean velocity and associated travel time through each subreach. The time of concentration for the entire reach is the sum of the times of concentration of the individual subreaches. This procedure, while practical, is based on several assumptions, including a flow-rate level, a prismatic channel, and Manning's n values.
A limitation of the steady flow approach to the calculation of time of concentration is the fact that the flow being considered is generally unsteady. This means that the speed of travel of the wavelike features of the flow (i.e., the kinematic wave speed, Chapters 4 and 9) is greater than the mean velocity calculated using steady flow principles (the Manning equation). For instance, for turbulent flow in hydraulically wide channels, kinematic wave theory justifies a wave speed as much as 5/3 times the mean flow velocity, with a consequent reduction in travel time and associated time of concentration. Yet, in many cases, the ratio between kinematic wave speed and mean flow velocity is likely to be less than 5/3:1. In practice, the uncertainties involved in the computation of time of concentration have contributed to a blurring of the distinction between the two speeds.
Formulas for Time of Concentration. Notwithstanding the inherent complexities, calculations of time of concentration continue to be part of the routine practice of engineering hydrology. The time of concentration is a key element in the rational method (Chapter 4) and other methods used to calculate the runoff response of small catchments. Most formulas relate time of concentration to suitable length, slope, roughness, and rainfall parameters [62]. A well-known formula which relates time of concentration to length and slope parameters is the Kirpich formula, applicable to small agricultural watersheds with drainage areas less than 8 acres (200 ha) [46]. In SI units, the Kirpich formula is:
L 0.77 tc = 0.06628 ___________
S 0.385
(2-59)
in which tc = time of concentration, in hours; L = length of the principal watercourse, from divide to outlet, in kilometers; and S = slope between maximum and minimum elevation (S1 slope), in meters per meter. In U.S. customary units, with tc in minutes, L in feet and S in feet per foot, the coefficient of Eq. 2-83 is 0.0078.
The Kerby-Hathaway formula relates time of concentration to length, slope, and roughness parameters as follows [22]:
( L n ) 0.467 tc = 0.606 _______________
S 0.234
(2-60)
in which n is a roughness parameter and all other terms are the same as in Eq. 2-59, expressed in SI units. Applicable values of n are given in Table 2-10.
Table 2-10 Value of roughness parameter n for use in Eqs. 2-60 to 2-63 [10].
Type of surface n
Smooth impervious 0.02
Smooth bare-picked soil 0.10 Poor grass, row crops, or moderately rough bare
soil 0.20
Pasture 0.40
Deciduous timber land 0.60
Coniferous timber land, or deciduous timber land
with deep litter or grass 0.80
The Papadakis-Kazan formula [62] relates time of concentration to length, slope, roughness, and rainfall parameters:
L 0.50 n 0.52 tc = 0.66 _________________
S 0.31 i 0.38
(2-61)
in which tc is in minutes; L is in feet; n is a roughness parameter; and i is the effective rainfall in inches per hour.
A physically based approach to the calculation of time of concentration is possible by means of overland flow techniques (Chapter 4). As a first approximation, time of concentration can be taken as the time-to-equilibrium of kinematic overland flow (Eq. 4-50). Therefore:
(L n) 1/m
tc = ______________________
S 1/(2 m) i (m - 1)/m
(2-62)
in which tc is given in seconds; L is in meters; n is a roughness parameter, i is in m/s, and m = exponent of the unit-width discharge-flow depth rating (q = b h m ).
For m = 5/3, applicable to turbulent Manning friction (in hydraulically wide channels), the kinematic wave time of concentration is:
(L n) 0.6 tc = ______________
S 0.3 i 0.4
(2-63)
in which tc is in seconds; L is in meters; n is a roughness parameter; and i is the effective rainfall in meters per second. The close resemblance of the exponents of Eqs. 2-61 and 2-63 is remarkable.
Example 2-10.
Use the Kirpich, Hathaway, Papadakis-Kazan, and kinematic wave formulas to estimate time of concentration for a catchment with the following characteristics: L = 750 m, S = 0.01, n = 0.1, and i = 20 mm hr -1.
After conversion to the proper units, the application of Eq. 2-59 leads to tc = 0.3127 hr = 18.76 minutes.
The application of Eq. 2-60 leads to tc = 0.531 hr = 31.86 minute. The application of Eq. 2-61 leads to tc = 45.13 minutes. The application of Eq. 2-62 leads to tc = 6716 seconds = 111.94 minutes.