2.1 PRECIPITATION
2.2 HYDROLOGIC ABSTRACTIONS
2.3 CATCHMENT PROPERTIES
Surface runoff in catchments occurs as a progression of the following forms, from small to large:
Overland flow is runoff that occurs during or immediately after a storm, in the form of sheet flow over the land surface [Fig. 230 (a)]. Rill flow is runoff that occurs in the form of small rivulets, primarily by concentration of overland flow. Gully flow is runoff that has concentrated into depths large enough so that it has the erosive power to carve its own deep and narrow channel (b). Streamflow is concentrated runoff originating in overland flow, rill flow, or gully flow and is characterized by well defined channels or streams of sizable depth (c). Streams carry their flow into larger streams, which flow into rivers to constitute river flow (d).
A catchment can range from as little as 1 ha (or acre) to millions of square kilometers (or square miles). Small catchments (small watersheds) are those where runoff is primarily controlled by overland flow processes. Large catchments (river basins) are those where runoff is controlled by storage processes in the river channels. Between small and large catchments, there is a wide range of catchment sizes with runoff characteristics falling somewhere between those of small and large catchments. Depending on their relative size, midsize catchments are referred to as either watersheds or basins. Regardless of their size, catchments can drain either inwards, into lakes (or seasonally dry lakes in arid regions), or outwards, toward the ocean. Catchments draining inwards have endorheic (or inland continental) drainages (Fig. 231). Catchments draining outwards have exorheic (or peripheral continental) drainages. Exorheic drainages have a catchment outlet or mouth at the point of delivery to the next largest stream, and ultimately, to the ocean.
The hydrologic characteristics of a catchment are described in terms of the following properties:
Catchment Area Area, or drainage area, is perhaps the most important catchment property. It determines the potential runoff volume, provided the storm covers the whole area. The catchment divide is the loci of points delimiting two adjacent catchments, i.e., the collection of high points separating catchments draining into different outlets. Due to the effect of subsurface flow (groundwater flow), the hydrologic catchment divide may not strictly coincide with the topographic catchment divide (Fig. 232). The hydrologic divide, however, is less tractable than the topographic divide; therefore, the latter is preferred for practical use.
The topographic divide is delineated on a quadrangle sheet or other suitable topographic map. The direction of surface runoff is perpendicular to the contour lines. All peaks and saddles are identified at the outset (Fig. 233). Runoff from a peak is in all directions; runoff from a saddle is in the two opposing directions perpendicular to the saddle axis. The catchment divide is delineated by joining peaks and saddles with a line which remains perpendicular to the topographic contour. The area enclosed within the topographic divide is measured to determine the catchment area.
In general, the larger the catchment area, the greater the amount of surface runoff and, consequently, the greater the surface flows. Several formulas have been proposed to relate peak flow to catchment area (Chapter 7). A basic formula is:
in which Q_{p} = peak flow, A = catchment area, and c and n are parameters to be determined by regression analysis. Other peak flow methods base their calculations on peak flow per unit area, for instance, the TR55 method (Chapter 5). Catchment Shape Catchment shape is the outline described by the horizontal projection of a catchment. Horton [28] described the outline of a normal catchment as a pearshaped ovoid. Large catchments, however, vary widely in shape. A quantitative description is provided by the following formula [26]:
in which K_{f} = form ratio, A = catchment area, and L = catchment length, measured along the longest watercourse. Area and length are given in consistent units such as square kilometers and kilometers, respectively. An alternate description is based on catchment perimeter rather than area. For this purpose, an equivalent circle is defined as a circle of area equal to that of the catchment. The compactness ratio is the ratio of the catchment perimeter to that of the equivalent circle. This leads to:
in which K_{c} = compactness ratio, P = catchment perimeter, and A = catchment area, with P and A given in any consistent set of units. Hydrologic response refers to the relative concentration and timing of runoff (Fig. 234). The role of catchment shape in hydrologic response has not been clearly established. Other things being equal, a high form ratio (Eq. 250) or a compactness ratio close to 1 (Eq. 251) describes a catchment having a fast and peaked catchment response. Conversely a low form ratio or a compactness ratio much larger than 1 describes a catchment with a delayed runoff response. However, many other factors, including catchment relief, vegetative cover, and drainage density are usually more important than catchment shape, with their combined effect not being readily discernible.
Catchment Relief Relief is the elevation difference between two reference points. Maximum catchment relief is the elevation difference between the highest point in the catchment divide (Fig. 235) and the catchment outlet. The principal watercourse (or main stream) is the central and largest watercourse of the catchment and the one conveying the runoff to the outlet. Relief ratio is the ratio of maximum catchment relief to the catchment's longest horizontal straight distance measured in a direction parallel to that of the principal watercourse. The relief ratio is a measure of the intensity of the erosional processes active in the catchment.
The overall relief of a catchment is described by hypsometric analysis [52]. This refers to a dimensionless curve
showing the variation with elevation of the catchment subarea above that elevation (Fig. 236). To develop this curve, the
elevation of the highest or maximum point in the catchment divide, corresponding to 0 percent area, is identified.
Also, the elevation of the lowest or minimum point of the catchment, corresponding to 100 percent area, is
identified. Subsequently, several elevations located between maximum and minimum are selected, and the subareas
above each one of these elevations determined by measuring along the respective topographic contour lines. The
elevations are converted to height above minimum elevation and expressed in percentage of the
maximum height. Likewise, the subareas above each one of the elevations are expressed as percentages of total
catchment area. The hypsometric curve shows percent area in the abscissas and percent height in the
ordinates (Fig. 236). The median elevation of the catchment is obtained from the percent height corresponding to 50 percent area.
The hypsometric curve is used when a hydrologic variable such as precipitation, vegetative cover, or snowfall shows a marked tendency to vary with altitude. In such cases, the hypsometric curve provides the quantitative means to evaluate the effect of altitude. Other measures of catchment relief are based on stream and channel characteristics. The longitudinal profile of a channel is a plot of elevation versus horizontal distance (Fig. 237). At a given point in the profile, the elevation is usually a mean value of the channel bed. Between any two points, the channel gradient (or channel slope) is the ratio of elevation difference to horizontal distance separating them.
In the absence of geologic controls, longitudinal profiles of streams and rivers are usually concave upward, i.e. , they show a persistent decrease in channel gradient in the downstream direction as the flow moves from mountain streams to river valleys and into the ocean (Fig. 237). The reason for this downstream decrease in channel gradient requires careful analysis. It is known that channel gradients are directly related to bottom friction and inversely related to flow depth. Typically, small mountain streams have high values of bottom friction (due to the presence of cobbles and boulders in the stream bed) and small depths [[Fig. 238 (a)]. Conversely, large rivers have comparatively lower values of bottom friction and larger depths [Fig. 238 (b)]. This interaction of channel gradient and bottom friction helps explain the typical decrease in channel gradient in the downstream direction.
Convex channel bed profiles (Fig. 239) are caused by tectonism, uplift, geologic controls, or rock outcrops predominating over an otherwise alluvial channel morphology in equilibrium. These convex stream profiles usually lead to sediment deposition upstream of the outcrop, and to erosion immediately downstream.
Channels gradients are usually expressed in dimensionless units. For convenience, they can be also expressed in m km^{1}, cm km^{1}, or ft mi^{1}. In nature, channel gradients vary widely, from higher than 0.1 in very steep mountain streams [see, for instance, Fig. 238 (a)], to less than 0.000006 in large tidal rivers [19].
In unusual geomorphological settings, inland rivers
may feature very small channel gradients; for instance, the Upper
Paraguay river near Porto Murtinho,
Brazil, which
has an average channel slope of
2 cm km^{1} (0.00002) (Fig. 240).
The channel gradient of a principal watercourse is a convenient indicator of catchment relief. A longitudinal profile is defined by its maximum (upstream) and minimum (downstream) elevations, and by the horizontal distance between them (Fig. 241). The channel gradient obtained directly from the upstream and downstream elevations is referred to as the S_{1} slope.
A somewhat more representative measure of channel gradient is the S_{2} slope, defined as the constant slope that
makes the shaded area above it equal to the shaded area below it (Fig. 241). An expedient way to calculate the
S_{2} slope is to equate the total area below it to the total area below the longitudinal profile.
A measure of channel gradient which takes into account the basin response time is the equivalent slope, or S_{3}. To calculate this slope the channel is divided into n subreaches, and a slope is calculated for each subreach. Based on Manning's equation (Section 2.4), the time of flow travel through each subreach is assumed to be inversely proportional to the square root of its slope. Likewise, the time of travel through the whole channel is assumed to be inversely proportional to the square root of the equivalent slope. This leads to the following equation:
in which S_{3} = equivalent slope, L_{i} = each i of n subreach lengths, and S_{i} = each i of n subreach slopes. Grid methods are often used to obtain measures of land surface slope for runoff evaluations in small and midsize catchments. For instance, the USDA Natural Resources Conservation Service determines average surface slope by overlaying a square grid pattern over the topographic map of the watershed [79]. The maximum surface slope at each grid intersection is evaluated, and the average of all values calculated. This average is taken as the representative value of land surface slope (Fig. 242).
Linear measures are used to describe the onedimensional features of a catchment. For instance, for small catchments, the overland flow length L_{o} is the distance of surface runoff that is not confined to any clearly defined channel. The catchment length (or hydraulic length) L is the length measured along the principal watercourse (Fig. 244). The principal watercourse (or main stream) is the central and largest watercourse of the catchment and the one conveying runoff to the outlet.
The length to catchment centroid L_{c} is the length measured along the principal watercourse, from the catchment outlet to a point located closest to the catchment centroid (point G in Fig. 244). In practice, the catchment centroid is estimated as the intersecting point of two or more straight lines that bisect the catchment area in approximately equal subareas. Basin Topology Basin topology refers to the regional anatomy of the stream network. Distributed rainfallrunoff modeling (Chapter 10) requires the hierarchical description of stream connectivity, i.e., of its topology. Stream Order. The concept of stream order classifies streams in a network following a hierarchical numbering system. Overland flow can be considered as a hypothetical stream of zero order. A firstorder stream is that receiving flow from zeroorder streams, i.e., overland flow. Two firstorder streams combine to form a secondorder stream. In general, two morder streams combine to form a stream of order m + 1. The catchment's stream order is the order of the mostdownstream main stem (Fig. 245).
Fig. 245 Concept of stream order. A catchment's stream order is directly related to its size. Large catchments have stream orders of 10 or more. The evaluation of stream order is highly sensitive to map scale. Therefore, considerable care is needed when using stream order analysis in comparative studies of catchment behavior. Pfasfstetter Coding System. The Pfafstetter coding system is a widely accepted methodology for the description of watershed/basin topology [**]. The system describes the regional anatomy of a stream network using a hierarchical arrangement of decimal digits. A Level 0 catchment corresponds to a continentalscale size or, alternatively, one that drains into the ocean. Higher levels represent progressively finer subdivisions of the Level 0 catchment. Theoretically, the system is not limited in the number n of levels. In practice, however, n = 6 to 8 levels are usually sufficient. At each level, each watershed is assigned a specific integer m, varying from m = 0 to 9, based on its location and function within the drainage network.
At each level,
watersheds are assigned into three types: (1) basin,
(2) interbasin, and
For each level, from 1 to n, the assignment of Pfafstetter codes is performed as follows:
Figure 246 shows a 3level example of the Pfasfstetter coding system. For each level, say Level 3, the assigned digits (XYm) are appended on to the Level 2 code (XY). For instance, watershed 849 is watershed 8 of Level 1 (coarser), watershed 4 of Level 2 (intermediate), and watershed 9 of Level 3 (finer).
Fig. 246 The Pfafstetter coding system for watershed identification (Click here to display). Drainage Density The catchment's drainage density is the ratio of total stream length (the sum of the lengths of all streams) to catchment area. A high drainage density reflects a fast and peaked runoff response, whereas a low drainage density is characteristic of a delayed runoff response. The mean overland flow length is approximately equal to half the mean distance between stream channels. Therefore, it can be approximated as onehalf of the reciprocal of drainage density:
in which L_{o} = mean overland flow length, and D = drainage density. This approximation neglects the effect of ground and channel slope, which makes the actual mean overland flow length longer than that estimated by Eq. 253. The following equation can be used to estimate overland flow length more precisely:
in which S_{c} = mean channel slope, and S_{s} = mean surface slope. Drainage Patterns 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. 247) 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. 248 [30].
Fig. 248 Drainage patterns recognizable on aerial photographs. 2.4 RUNOFF
Surface 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 or cubic 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:
in which ∀= runoff volume, Q = flow rate, and t = 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:
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 unsaturated soil layers (vadose zone) located beneath the ground surface (Fig. 249). 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.
Groundwater flow takes place below the groundwater
table in the form of saturated flow
through alluvial deposits and other waterbearing
geologic formations located beneath the soil mantle
Stream Types and Baseflow Streams may be grouped into three types:
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. 251 (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 regions [(Fig. 251 (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. 251 (c) and (d)].
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 versus 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, which ostensibly excludes indirect runoff, i.e., baseflow. Yet, at the basin scale, estimates of surface water yield are known to include both direct and indirect runoff. 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. 252).
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:
in which I_{i} = index for day i, I_{i1} index for day i 1,
and K = a recession factor taken normally in the range 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 soilmoisture index (SMI), with precipitation intensity as a third variable. Runoff percent is the ratio of runoff to rainfall, multiplied by 100. Such runoffmoisturerainfall relation is empirical and, therefore, is limited to the basin for which it was derived. 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 voluminous 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 rainfallrunoff is the following:
in which Q = runoff depth, P = rainfall depth, P_{a} = rainfall depth below which runoff is zero, and b = slope of the line (Fig. 253). Rainfall depths smaller than P_{a} are completely abstracted by the catchment, with runoff starting as soon as P exceeds P_{a}. To use Eq. 257 it is necessary to collect several sets of rainfallrunoff data and to perform a linear regression to determine the values of b and P_{a} (Chapter 7). The simplicity of Eq. 257 precludes it from taking into account other important runoffproducing 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.
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 rainfallrunoff 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:
in which Q_{e} = equilibrium flow rate; I_{e} = effective rainfall intensity; and A = catchment area. This equation is dimensionally consistent; however, a conversion factor is needed in the righthand side to account for the applicable units. For instance, in SI units, with Q_{e} in liters per second, I_{e} in millimeters per hour, and A in hectares, the conversion factor is 2.78. In U.S. customary units, with Q_{e} in cubic feet per second, I_{e} 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. 254. (In practice, due to nonlinearities, actual recession flows are usually asymptotic to zero). This type of response is referred to as 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. 255. This type of response is referred to as 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. 256. This type of response is referred to as 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. Time of Concentration. Hydrologic procedures for small catchments usually require an estimate of the time of concentration (Chapter 4). However, accurate estimates are generally difficult to make. 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 as a function of selected catchment parameters 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 openchannel 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 flowrate 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, 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 wellknown formula which relates time of concentration to length and slope parameters is the Kirpich formula, applicable to small agricultural watersheds with drainage areas of less than 8 acres (200 ha) [46]. In SI units, the Kirpich formula is:
in which t_{c} = 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 t_{c} in minutes, L in feet and S in feet per foot, the coefficient of Eq. 283 is 0.0078. The KerbeyHathaway formularelates time of concentration to length, slope, and roughness parameters as follows [22]:
in which n is a roughness parameter and all other terms are the same as in Eq. 259, expressed in SI units. Applicable values of n are given in Table 210.
The PapadakisKazan formula [62] relates time of concentration to length, slope, roughness, and rainfall parameters:
in which t_{c} 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 timetoequilibrium of kinematic overland flow (Eq. 450). Therefore:
in which t_{c} is given in seconds; L is in meters; n is a roughness parameter, i is in m/s, and m = exponent of the unitwidth dischargeflow depth rating (q = b h ^{m} ). For m = 5/3, applicable to turbulent Manning friction, the kinematic wave time of concentration is:
in which t_{c} 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. 263 and 261 is remarkable.
Use the Kirpich, Hathaway, PapadakisKazan, 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
ONLINE CALCULATION. Using ONLINE TIME OF CONCENTRATION, the answer
is:
Kirpich ⇒
In nature, catchment response shows a more complex behavior than that which may be attributed solely to runoff concentration. Theory and experimental evidence have shown that runoff rates are governed by natural processes of convection and diffusion. Convection refers to runoff concentration; diffusion is the mechanism acting to spread the flow rates in time and space. The net effect of runoff diffusion is to reduce the flow rates to levels below those that could be attained by convection only. In practice, diffusion acts to smooth out catchment response. The resulting response function is usually continuous, and it is referred to as the streamflow hydrograph, runoff hydrograph, or simply the hydrograph. Typical singlestorm hydrographs have a shape similar to that shown in Fig. 257. They are usually produced by storms with effective rainfall duration less than the time of concentration. Therefore, they resemble subconcentrated catchment flow, albeit with the addition of a small but perceptible amount of diffusion.
The various elements in a typical singlestorm hydrograph are shown in Fig. 2 58. The zero time (or starting time) depicts the beginning of the hydrograph. The hydrograph peak describes the maximum flow rate. The timetopeak is measured from zero time to the time at which the peak flow is attained. The rising limb is the part of the hydrograph between zero time and timetopeak. The recession (or receding limb) is the part of the hydrograph between timetopeak and time base. The time base is measured from zero time to a time defining the end of the recession. The recession is logarithmic in nature, approaching zero flow in an asymptotic way. For practical applications, the end of the recession is usually defined in an arbitrary manner. The point of inflection of the receding limb is the point corresponding to zero curvature. The hydrograph volume is obtained by integrating the flow rates from zero time to time base.
The shape of the hydrograph, showing a positive skew, with recession time greater than rising time, is caused by the essentially different responses of surface flow, interflow, and groundwater flow. Indeed, the runoff hydrograph can be thought of as consisting of the sum of three hydrographs, as shown in Fig. 259 (a). The fast and peaked hydrograph is produced by surface flow while the other two are the result of interflow and groundwater flow. The superposition of these hydrographs results in a runoff hydrograph exhibiting a long tail (positive skew), as depicted in Fig. 259 (b).
The feature of positive skew allows the definition of a few additional geometric hydrograph properties. The timetocentroid t_{g} is measured from zero time to the time separating the hydrograph into two equal volumes (Fig. 260). The volumetopeak V_{p} is obtained by integrating the flow rates from zero time to timetopeak. In synthetic unit hydrograph analysis, the ratio of volumetopeak to hydrograph volume is used as a measure of hydrograph shape (Chapter 5).
Hydrographs of perennial streams may include substantial amounts of baseflow. The separation of runoff into direct runoff and indirect runoff (baseflow) can be accomplished by resorting to one of several hydrograph separation techniques (Chapter 5). These techniques can also be used in the analysis of multiplestorm hydrographs, which typically exhibit two or more peaks and valleys. Analytical Hydrographs. Analytical expressions for streamflow hydrographs are sometimes used in hydrologic studies. The simplest formula is based on either a sine or cosine function. These, however, have zero skew (Chapter 6) and therefore do not properly describe the shape of natural hydrographs. An analytical hydrograph that is often used to simulate natural hydrographs is the gamma function, expressed as follows:
in which Q = flow rate; Q_{b} = baseflow; Q_{p} = peak flow; t = time; t_{p} = timetopeak; t_{g} = timetocentroid; and m = t_{p} /(t_{g}  t_{p}). For values of t_{g} greater than t_{p}, Eq. 264 exhibits positive skew.
Flow in Stream Channels Streamflow hydrographs flow in stream channels that are carved on the land surface. The following properties are used to describe stream channels:
The channel cross section has the following geometric and hydraulic elements: (a) flow area, (b) top width, (c) wetted perimeter, (d) hydraulic radius, (e) hydraulic depth, and (f) aspect ratio. The flow area A is the area of the cross section occupied by the flow. The top width T is the channel width at the elevation of the water surface. The wetted perimeter P is the perimeter of the flow area in direct contact with the land. The hydraulic radius R is the ratio of flow area to wetted perimeter: R = A/P. The hydraulic depth D is the ratio of flow area to top width: D = A/T. The aspect ratio, a measure of crosssectional shape, is the ratio of top width to hydraulic depth (T/D). Channel top widths vary widely, ranging from a few meters for small mountain streams to several kilometers for very large rivers. Mean flow depths range from as low as a fraction of a meter for small mountain streams to more than 50 m for very large rivers. [The maximum depth of the Amazon river, the largest in the world, is close to 90 m]. Aspect ratios vary widely in nature; however, most streams and rivers have aspect ratios in excess of 10. Very wide streams (e.g., braided streams) may have aspect ratios exceeding 100. The longitudinal channel slope is the change in elevation with distance. The mean bed elevation is generally used to calculate channel slope. For short reaches or mild slopes, slope calculations may be hampered by the difficulty of accurately establishing the mean bed elevation. A practical alternative is to use the water surface slope as a measure of channel slope. The water surface slope, however, varies in space and time as a function of the flow nonuniformity and unsteadiness. The steady equilibrium (i.e., uniform) water surface slope is usually taken as a measure of channel slope. Therefore, mean bed slope and steady equilibrium water surface slope are often treated as synonymous. Generally, the longer the channel reach, the more accurate the calculation of channel slope. Boundary friction refers to the type and dimensions of the particles lining the channel cross section below the waterline. In alluvial channels, geomorphic bed features such as ripples and dunes may represent a substantial contribution to the overall friction (Chapter15). Particles lying on the channel bed may range from large boulders for typical mountain streams (Fig. 261) to silt particles in the case of large tidal rivers.
For small streams, particles on the channel banks may be as large as the particles on the bottom. River banks, however, are likely to consist of particles of much varied size than those on the channel bottom. The high aspect ratio of rivers generally results in the banks contributing only a small fraction of the total boundary friction. Therefore, the boundary friction is often taken as synonymous with bed or bottom friction. Uniform flow formulas. Flow in streams and rivers is evaluated by using empirical formulas such as the Manning or Chezy equations. The Manning formula is:
in which V = mean flow velocity, in meters per second; R = hydraulic radius, in meters; S = channel slope, in meters per meter; and n = Manning friction coefficient. In U.S. customary units, with V in feet per second, R in feet, and S in feet per foot, the right side of Eq. 265 is multiplied by the constant 1.486. In natural channels, n can take values as low as 0.02 and as high as 0.2 in some unusually high roughness cases (e.g., flood plains adjacent to rivers). A good working value for a clean, straight, fullstage stream of fairly uniform cross section is 0.03. Typical n values for natural streams and rivers are in the range 0.030.05. A U.S. Geological Survey study [4] has documented n values for natural streams ranging from as low as n = 0.024 for the Columbia River at Vernita, Washington (a large river with streambanks largely devoid of vegetation) [(Fig. 262 (a)], to as high as n = 0.079 for Cache Creek near Lower Lake, California (a small stream with large, angular boulders in the bed, and exposed rocks, boulders, and trees in the banks) [(Fig. 262 (b)].
The Chezy equation is
in which C = Chezy coefficient, in m^{1/2} s^{1}; and other terms are the same as for Eq. 265. Chezy coefficients equivalent to the preceding conditions range from about 80 m^{1/2} s^{1} for large rivers to about 10 m^{1/2} s^{1} for small streams. Typical C values for natural streams and rivers are in the range 2550 m^{1/2} s^{1}. Equation 266 can be expressed in dimensionless form as follows:
in which g = gravitational acceleration, and C/g^{1/2} = dimensionless Chezy coefficient. Dimensionless Chezy coefficients equivalent to the preceding conditions range from 25.5 for large rivers to 3.2 for small streams. Typical values for natural streams and rivers are in the range 816. For certain applications, Eq. 267 can be readily transformed into a formula with an enhanced physical meaning. For hydraulically wide channels, i.e., those with aspect ratio greater than 10, the top width and wetted perimeter can be assumed to be approximately the same. This implies that the hydraulic depth (D) can be substituted for the hydraulic radius (R), leading to:
in which f = a dimensionless friction factor equal to f =
g/C^{ 2}, and F =
Froude number, equal to Equation 268 states that for hydraulically wide channels, the channel slope is proportional to the square of the Froude number, with the friction factor f as the proportionality coefficient. In practice, Eq. 268 be used as a convenient predictor of any of these three dimensionless parameters, once the other two are known. Furthermore, it implies that if one of the three parameters is kept constant, a change in one of the other two causes a corresponding change in the third. Notwithstanding the theoretical appeal of Eqs. 266 and 268, the Manning equation has had wider acceptance in practice. This is attributed to the fact that in natural channels, the Chezy coefficient is not constant, tending to increase with hydraulic radius. The comparison of Eqs. 265 and 266 leads to:
Equation 269 implies that, unlike Chezy C, Manning n is a constant. Experience has shown, however, that at a given crosssection, n may vary with discharge and stage (Fig. 263). Moreover, as stage varies from low to high, alluvial rivers can move their beds and generate/erase ripples and dunes, increasing/decreasing channel friction (Chapter 15).
River Stages. At any location along a river, the river stage is the elevation of the water surface above a given datum. This datum can be either an arbitrary one or the NAVD (North American Vertical Datum), the national geodetic vertical datum, a standard measure of mean sea level. River stages are a function of flow rate. Flow rates can be grouped into: (1) low flow, (2) average flow, and (3) high flow. Low flow is typical of the dry season, when streamflow is largely composed of baseflow originating mostly in contributions from groundwater flow. High flow occurs during the wet season, when streamflow is primarily due to contributions from surface runoff. Average flow usually occurs midseason and may have mixed contributions from surface runoff, interflow, and groundwater flow. Low flows studies are necessary when determining minimum flow rates, below which a certain use would be impaired. Examples of such uses are irrigation requirements, hydropower generation, and minimum instream flows needed for fisheries protection and compliance with water pollution regulations. Excessive use of groundwater may lead to baseflow losses; thus, Increasingly, surface runoff studies are focusing on baseflow and low flows. Average flows play an important role in the calculation of monthly and annual volumes available for storage and use. Applications are usually found in connection with the sizing of storage reservoirs. High flow studies are related to the floods and flood hydrology. Typically, during high flows, natural streams and rivers have the tendency to overflow their banks, with stages reaching above bankfull stage. In such cases, the flow area includes a portion of the land located adjacent to the river, on one or both sides. In alluvial valleys, the land that is subject to inundation during periods of high flow is referred to as the flood plain (Fig. 264). The evaluation of high flows is necessary for flood forecasting, control, and mitigation.
Rating Curves. It is known that river stage varies as a function of discharge, but the exact nature of the relationship is not readily apparent. Given a long and essentially prismatic channel reach, a singlevalued relationship between stage and discharge at a cross section defines the equilibrium rating curve. For steady uniform flow, the rating curve is unique, i.e., there is a single value of stage for each value of discharge and vice versa (Fig. 265). In this case, the equilibrium rating curve can be calculated with either the Manning or Chezy equations. In openchannel hydraulics, this property of uniqueness of the rating qualifies the channel reach as a channel control.
However, other flow conditions, specifically nonuniformity (gradually varied steady flow) and unsteadiness (e.g., gradually varied unsteady flow), can cause deviations from the steady equilibrium rating. These deviations are less tractable. In particular, flood wave theory justifies the presence of a loop in the rating, as shown in Fig. 266. Intuitively, the rising limb of the floodwave hydrograph has a steeper water surface slope than that of equilibrium flow, leading to greater flows and lower stages. Conversely, the receding limb has a milder water surface slope, resulting in smaller flows and higher stages; thus the rationale for the loop's presence. The loop effect, however, is likely to be small and is usually neglected on practical grounds. Where increased accuracy is required, unsteady flow modeling can be used to account for the looped rating (Chapter 9).
Two other processes related to sedimentation have a bearing in the evaluation of stagedischarge relations: (1) the shortterm effects, and (2) longterm effects. The shortterm effects are due to the fact that the amount of boundary friction varies with flow rate. Rivers flowing on loose boundaries composed of gravel, sand, and silt constantly try to minimize their changes in stage. This is accomplished through the following mechanism: During low flow, the bed friction consists not only of grain friction but also of form friction, caused by bed features such as ripples and dunes (Chapter 15). During high flows, the swiftness of the current acts to obliterate the bed features, reducing form friction to a minimum, essentially with only grain friction remaining. The reduced friction during high flows gives rivers the capability to carry a greater discharge for a given stage. This explains the demonstrated shift from lowflow rating to highflow rating in natural river channels (Fig. 263). The longterm sedimentation effect is due to the fact that rivers continuously subject their boundaries to recurring cycles of erosion and deposition, depending on the sediment load they carry (Chapter 15). Some very active rivers may be eroding; others may be aggrading. Moreover, some geomorphologically active rivers may substantially change their cross sections during major floods. Invariably, shifts in rating are the net result of these natural geomorphic processes. Rating curve Formulas. In spite of the apparent complexities, rating curves are a useful and practical tool in hydrologic analysis, allowing the direct conversion of stage to discharge and vice versa. Discharge can be obtained from the rating by the simple procedure of measuring the stage. Conversely, if discharge is known, for instance, at a catchment outlet, stage at the outlet can be readily determined from a suitable rating. There are several ways to determine an equation for the rating. Invariably, they are based on curvefitting stagedischarge data. A widely used equation is the following [45]:
in which Q = discharge; h = gage height; h_{o} = reference height; and a and b are constants. Several values of reference height are tried. The proper value of reference height is that which makes the stagedischarge data plot as close as possible to a straight line on logarithmic paper. Subsequently, the values of the constants a and b are determined by regression analysis (Chapter 7). Streamflow Variability The study of streamflow variability is the cornerstone of engineering hydrology. Streamflow and river flow vary not only seasonally, but also annually, multiannually, and with climate and geographic location. Global climate change may also affect streamflow variability. Over the long term, the total amount of streamflow is directly related to the amount of environmental moisture, i.e., the moisture present in soil and air. The inland advection of water vapor supplies the moisture which eventually constitutes precipitation. Whether this moisture reaches the catchment outlet remains to be determined by further analysis. On an average global basis, mean annual runoff, measured at the mouths of peripheral continental basins, amounts to about 39 percent of total precipitation. Most of the remainder, about 59 percent, is accounted for by the longterm abstractive processes of evaporation and evapotranspiration, which include evaporation from water bodies, evaporation from soil and bare ground, and evapotranspiration from vegetation. A small percentage, about 2 percent, percolates deep enough into the ground to bypass the surface waters, eventually discharging into the ocean (Fig. 267).
Seasonal Variability. A typical catchment in a subhumid region may show runoff rates and volumes varying throughout the year, with a tendency to low flows during the dry season and high flows during the wet season. However, a catchment in a more extreme climate will show a quite different behavior. In the ephemeral streams typical of arid regions, runoff is nonexistent during periods of no precipitation; for these streams, runoff occurs only in direct response to precipitation. On the other hand, in humid and extremely humid climates, rivers show substantial amounts of runoff throughout the year, with relatively little variability between the seasons. The reason for the seasonal variability of streamflow lies in the relative contributions of direct (surface flow) and indirect runoff (mostly groundwater flow). In subhumid regions, indirect runoff is a small, but nevertheless measurable, fraction of total runoff. On the other hand, in arid regions, particularly for ephemeral streams, indirect runoff is either zero or negligible. Furthermore, in humid regions, indirect runoff is substantial throughout the year, often being a sizable fraction of total runoff. The phenomenon described above can be further explained in the following way: Groundwater reservoirs act to store large amounts of water, which are slowly transported to lower elevations. The bulk of this water (about 98% on a global basis) is eventually released back to the surface waters. With seepage being the dominant process, the flow of groundwater is slow and, therefore, subject to a substantial amount of diffusion. The net effect is that of a permanent contribution from groundwater to surface water in the form of baseflow, or the dryweather flow of rivers (Fig. 268). To evaluate the seasonal variability of streamflow, it is therefore necessary to examine the relationship between surface water and groundwater.
Annual Variability. Yeartoyear streamflow variability shows some of the same features as those of seasonal streamflow variability. For instance, large catchments show runoff variability from one year to the next as a function of the state of moisture at the end of the first year and of the precipitation amounts added during the second year. As in the case of seasonal variability, annual streamflow variability is linked to the relative contributions of direct and indirect runoff. During dry years, precipitation goes on to replenish the catchment's soil moisture deficit, with little of it showing as direct runoff. This results in the low levels of runoff that characterize dry years. Conversely, during wet years, the catchment's moisture storage capacity fills up quickly, and any additional precipitation is almost entirely converted into surface runoff. This produces the high streamflow levels that characterize wet years. Annual streamflow variability is, therefore, intrinsically connected to the relative contributions of direct and indirect runoff. A line of inquiry that is becoming increasingly popular is to focus on the mechanics of surface flow, interflow, and groundwater flow, while accounting for the spatial and temporal variability of the various physical, chemical, and biological processes involved at the various scales. However, the dearth of reliable data for all phases of the hydrologic cycle makes the evaluation of streamflow using a purely mechanistic approach a rather complex undertaking. Recent progress has been made in the coupling of mathematical models with geographic information systems, digital elevation models, and other spatially related software. A practical alternative which has enjoyed wide acceptance in applications of flood hydrology is the reliance on statistical tools to compensate for the incomplete knowledge of the physical processes. Over the years, this has given rise to the concept of flow frequency, or commonly, flood frequency, expressed as the average period of time (i.e., the return period) that it will take a certain flood level to recur at a given location. An annual flood series is abstracted from daily discharge measurements at a given gaging station. This is accomplished by either selecting the maximum daily flow for each of n years of record (the annual maxima series), or by selecting the n greatest flow values in the entire nyear record, regardless of when they occurred (the annual exceedance series) (Chapter 6). The statistical analysis of the flood series permits the calculation of the flow rates associated with one or more chosen frequencies. The procedure is relatively straightforward, but it is limited by the record length. Its predictive capability decreases sharply when used to evaluate floods with return periods substantially in excess of the record length. An advantage of the method is its reproducibility, which means that two persons are likely to arrive at the same result when using the same methodology. This is a significant asset when comparing the relative merits of competing water resources projects. Methods for flood frequency analysis are discussed in Chapter 6. Recently, a complicating factor has arisen in flood frequency analysis. Global climate change promises to change the longterm depthdurationfrequency precipitation relations and, therefore, the magnitude and frequency of floods. Then, a historical flood record, however long, would have essentially lost its pristine character and could only serve as a rough indication for present and future analyses. Dailyflow Analysis. The variability of streamflow can also be expressed in terms of the daytoday fluctuation of flow rates at a given station. Some streams show great variability from day to day, with high peaks and low valleys succeeding one another endlessly. Other streams show very little daytoday variability, with high flows being not very different from low flows. The reason for this difference in behavior can be attributed to differences in the nature of catchment response. Small and midsize catchments are likely to have steep gradients and therefore to concentrate flows with negligible runoff diffusion, producing hydrographs that show a large number of high peaks and corresponding low valleys. Conversely, large catchments are likely to have milder gradients and therefore to concentrate flows with substantial runoff diffusion. The diffusion mechanism acts to spread the flows in time and space, resulting in a succession of smooth hydrographs showing low peaks and comparatively high valleys. Daily flow data may not be sufficient to allow calculation of the runoff volumes produced by small watersheds. In cases where accuracy is required, hourly flows (or perhaps flows measured at 3h intervals) may be necessary to describe adequately the temporal variability of the flow. In the past four decades, the development of stochastic models of streamflow variability has resulted in a substantial body of knowledge referred to as stochastic hydrology. For a detailed treatment of this subject, see [8, 70, 89]. Flowduration Curve. A practical way to evaluate daytoday streamflow variability is the flowduration curve. To determine this curve for a particular location, it is necessary to obtain daily flow data for a certain period of time. either 1 y or a number of years. The length of the record indicates the total number of days in the series. The daily flow series is sequenced in decreasing order, from the highest to the lowest flow value. with each flow value being assigned an order number. For instance, the highest flow value would have order number one; the lowest flow value would have the last order number, equal to the total number of days. For each flow value, the percent time is defined as the ratio of its order number to the total number of days, expressed in percentage. The flowduration curve is obtained by plotting flow versus percent time, with percent time in the abscissas and flow in the ordinates (Fig. 269).
A flowduration curve allows the evaluation of the permanence of characteristic lowflow levels. For instance, the flow expected to be exceeded 90 percent of the time can be readily determined from a flowduration curve. The permanence of low flows is increased with streamflow regulation. The usual aim is to be able to assure the permanence of a certain lowflow level 100 percent of the time. Regulation causes a shift in the flowduration curve by increasing the permanence of low flows while decreasing that of high flows (Fig. 267). Streamflow regulation is accomplished with storage reservoirs. The flowduration curve is helpful in the planning and design of water resources projects. In particular, for hydropower studies, the flowduration curve serves to determine the potential for firm power generation. In the case of a runoftheriver plant, with no storage facilities, the firm power is usually assumed on the basis of flow available 90 to 97 percent of the time. Flowmass Curve. Another way to evaluate daytoday (and seasonal) streamflow variability is the flowmass curve. A mass curve of daily values of a variable is a plot of time in the abscissas versus cumulative values of the variable in the ordinates. When using flow values, such a plot is referred to as the flowmass curve. For daily flow records in cubic meters per second, the ordinates of the flow mass curve are in cubic meters or cubic hectometers (1 cubic hectometer = 1 million cubic meters). For any given day, the ordinate of the flowmass curve is the accumulated runoff volume up to that day. According to Chow [10], the flowmass curve is believed to have been first suggested by Rippl [69]; hence the name Rippl curve. The shape of the flowmass curve resembles that of the letter S (Fig. 268); therefore, it is also referred to as the Scurve. Applications of flowmass curves are to reservoir design and operation, including the determination of reservoir capacity and the establishment of operating rules for storage reservoirs. Figure 270 shows a typical flowmass curve. At any given time, the slope of the mass curve is a measure of the instantaneous flow rate. The slope of the line PQ, drawn between the points P and Q, represents the average flow between the two points. The slope of the line AB, drawn between the starting point A and the ending point B, is the average flow for the entire period.
To use the flowmass curve for reservoir design, two lines parallel to line AB and tangent to the flowmass curve are drawn (Fig. 268). The first one, A'B', is tangent to the mass curve at the highest tangent point C. The second one, A"B", is tangent to the mass curve at the lowest tangent point D. The vertical difference between these two tangent lines, in cubic meters, is the storage volume required to release a constant flow rate. This constant release rate is equal to the slope of the line AB. A reservoir with a volume equal to AA" at the start would be full at C and empty at D, with no spill (excess volume) or shortage (deficit). A reservoir that is empty at the start has water while the S curve remains above the AB line and is empty (show a deficit) when the S curve moves below that line. A reservoir that is full at the start will spill water (excess volume) as long as the inflow remains greater than the outflow (from A to C). The draft rate (or demand rate) is the release rate required to fulfill downstream needs, such as irrigation or power generation. A line having a slope equal to the draft rate is the draft line. The draft rate need not be necessarily constant. In practice, reservoir withdrawals are variable, leading to a variable draft rate and variable draft line, which amounts to an outflow mass curve. The superposition of inflow and outflow mass curves enables the detailed analysis of reservoir storage. The residual mass curve is a plot of the differences between the S curve ordinates and the corresponding ordinates from line AB. The ordinates of the residual mass curve can be either positive or negative. The residual mass curve accentuates the peaks and valleys of the cumulative flow record. Range is the difference between the maximum and minimum ordinates of the residual mass curve for a given period. Range analysis was pioneered by Hurst [31, 32] who proposed the following formula for the calculation of maximum range:
in which R = reservoir storage volume required to guarantee a constant release rate equal to the mean of the data (annual runoff volume) over a period of N years, and s = the standard deviation of the data (annual runoff volume) (Chapter 6) (Fig. 271).
Equation 271 was derived by curvefitting data for a wide variety of natural phenomena. The exponent 0.73 was the mean of values varying between 0.46 and 0.96. A theoretical analysis based on the normal probability distribution (Chapter 6) showed that the exponent of Eq 271 should be 0.5 instead of 0.73. This apparent discrepancy between theory and data, known as the Hurst phenomenon, has been the subject of numerous studies [47]. Geographical Variability of Streamflow. Streamflow varies from one catchment to another and from one geographical region of a certain climate to another of a different climate. Moreover, exorheic and endorheic drainages have quite different streamflow patterns. While the outflow from an exorheic drainage is finite (nonzero), that of an endorheic drainage is zero, i.e., in the latter, no surface flow has a chance to leave the basin. In exorheic drainages, the geographical variability of streamflow can be explained in terms of:
Intuitively, the volume available for runoff is directly proportional to the catchment area. This, however, is tempered by the available precipitation, which is conditioned by the prevailing climate. The temporal frame refers to whether the streamflow evaluation is for shortterm runoff (event, or storm, runoff) or longterm runoff (annual water yield). The catchment area is important in shortterm evaluations, not only because of the potential runoff volume, but also because larger catchments tend to have milder overall gradients. (While the upper limit to catchment relief is in the thousands of meters, the upper limit to catchment length is in the thousands of kilometers; a three orderofmagnitude difference). This causes increased runoff diffusion, while increasing the chances for infiltration and loss of surface water to groundwater. In flood hydrology applications, the net effect is a decrease in peak discharge per unit area. The above reasoning is supported by data showing peak flows to be directly related to catchment area, as shown in Eq. 249. Consequently, the peak discharge per unit area is:
in which q_{p} = peak discharge per unit area, in m^{3} s^{1} km^{2} (or alternatively, in ft^{3} s^{1} mi^{2}); A = catchment area, in km^{2} (or mi^{2}), and c and m are empirical constants, with m = 1  n. Since n is generally less than 1, it follows that m is also generally less than 1. Equation 272 confirms that peak discharge per unit area is inversely related to drainage area. An example of such a trend is given by the classical Creager curves, shown in Fig. 272 [14]:
Values of C in the range 30100 encompass most of the flood data compiled by Creager et al. [14]. This range can be taken as a measure of the regional variability of flood discharges. Equation 273, however, limits itself to providing a peak discharge per unit area, wIth no connotation of frequency attached to the calculated values.
For shortterm (event or storm) runoff evaluations, the precipitation rate and
catchment abstractive capability determine the streamflow variability from catchment to catchment. In
small catchments, runoff is characterized by the event runoff coefficient C, i.e., the ratio of storm runoff
depth to storm rainfall depth (Chapter 4). This ratio increases with the impermeability of the catchment
surface, from values close to zero For longterm (runoff or water yield) runoff evaluations, geographical
location and associated climate determine to a large extent the seasonal and annual variability of streamflow.
In the typical exorheic drainage, mean annual runoff increases with environmental moisture, i.e., the
moisture present in soil and air. The mean annual runoff coefficient K, i.e., the ratio of mean annual runoff
to mean annual rainfall, varies from QUESTIONS
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