River Flow

Worldwide, rivers are the most important sources of water for cities and major industries. Hydroelectric power is a major source of electrical power, and transport of heavy, bulk goods by river barge is a vital link in most transportation systems. Understanding and predicting low, high, and average flows of rivers is therefore important to the people and industries that depend on them.

Types of Water Flow

There are two very different fundamental types of flow of water. Laminar flow is a smooth flow, in which particles suspended in the water will follow paths that are consistently parallel to each other throughout the flow. Turbulent flow is a complex, swirling flow in which the main flow separates into many different paths and the paths of suspended particles are not consistent with each other throughout the flow. Turbulent flow may have velocity components that are up or down from, or sideways or upstream of, the average flow direction, although the average flow direction is always in the downstream, or downslope, direction. Flow near the bottom and sides of a river or stream with a smooth bed is always laminar, but the zone of laminar flow is very narrow, as frictional forces due to differences of velocity in the flowing stream induce turbulence currents. Turbulent flow thus dominates throughout the cross section of stream flow.

Because flowing water exerts a shear stress, or viscous shear, along the bottom and sides of the river, average flow velocity is least at the bottom and increases upward into the body of the river. Because of energy losses to surface waves, the average turbulent downstream velocity (hereafter referred to simply as velocity) at the river’s surface is slightly below the maximum velocity. The maximum velocity occurs at about six-tenths of the river depth above the bottom of the river.

Because of viscous shear between flowing water and the bottom materials, the range of small and large sediment particles that make up most river bottoms are moved by the flowing water. Once in motion, small particles whose settling velocity is less than the upward component of turbulent flow move downstream with the water, settling out only in backwaters, such as the still water behind a dam, where flow velocities become very low. This continuously moving sediment is referred to as suspended load and is restricted, in most cases, to clay- and silt-sized sediments. Sand-sized and larger sediment grains are moved during periods of high flow velocity and dropped where flow velocity decreases. This coarse material is the “bed load.” Bed-load transport normally occurs only near the river bed. It is the sediment in transport that does the work of the river as erosion and sediment transport. Erosional features of river valleys, such as canyons and potholes, are produced by the “wet sandblast” of the suspended bed load, a process that is much more effective during periods of high flow velocity.

During periods of rainfall or snowmelt, direct precipitation into streams and runoff from adjacent land provide stream flow. Between periods of rainfall or snowmelt, stream flow comes from the slow seepage of groundwater into surface streams. Small streams, the smallest of which flow only during wet periods, join to form larger streams, which join to form rivers. Because rainfall or snowmelt occurs only occasionally in an area drained by a river (the river’s drainage basin), the quantity of water flowing past any point on a river, or any point on any of its tributary streams, varies with time.

Discharge

The quantity of water passing a point on the river is measured in cubic meters per second (or, in common North American and British practice, cubic feet per second) and is termed the discharge. For a period of time after a rainfall or snowmelt event, as, for example, during a flood, discharge increases at all points in the drainage basin. If the flood event occurs in only one part of the drainage basin—say the higher part of the basin with the smaller streams—the increase in discharge will occur first in the higher part of the basin and will occur in the larger streams lower in the basin at some later time.

If the discharge at a given point on a stream is measured continuously over a period of days and the discharge is plotted against time, with discharge on the vertical axis of the graph and time in days on the horizontal axis, the increase in discharge related to a storm or snowmelt (a flood) will appear as a hump in the discharge curve. A graph of discharge with time is called a hydrograph, and a hydrograph that shows a flood-related hump is a flood hydrograph. Flood hydrographs tend to be more pronounced, having a higher curve with steeper sides, on streams near the source of the flood water. Conversely, the flood hydrograph will be longer and less pronounced, having a lower curve with more gently sloping sides, farther downstream. Putting it another way, the flood hydrograph attenuates, or dies out, downstream. The low, flat portion of the hydrograph that measures flow between flood events is called baseflow. Experience with flood hydrographs for a river enables hydrologists to predict the effect of future rainstorms and snowmelts of varying intensity and to predict low flow during prolonged dry periods.

An increase in river discharge is accompanied by an increase in stream velocity. For rivers with sandy bottoms (sandy streambeds), the higher water velocity causes erosion, or scour, of the bottom. That is, the sand begins to move along the bottom with the water, and the river channel becomes deeper. The elevation of the water surface relative to some fixed point on the riverbank also increases. As a rule, the banks are not vertical but sloping, and the increase in elevation of the water surface causes a corresponding increase in width. In summary, an increase in discharge is accompanied by increases in flow velocity, stream width, water surface elevation, and depth of channel (the last two items add up to an overall increase in depth). A reduction in discharge has the opposite effects, including the deposition of new sand, arriving from upstream, in the channel as the velocity decreases.

Hydraulic Geometry

In 1953, Luna Bergere Leopold and Thomas Maddock, Jr., introduced a concept that describes the relationships among the variables that change when discharge changes. The concept is called hydraulic geometry, and it is embodied in the following three equations: w = aQb, d = bQf, and v = kQm, where w is stream width, d is average depth of the stream, v is the flow velocity, and Q is discharge. The coefficients a, b, and k and the exponents b, f, and m are determined empirically, from actual measurement in the field. Since the discharge is equal to the cross-sectional area of the stream (wd) times the flow velocity (that is, wdv = Q), wdv = aQb × bQf × kQm = Q. For this to be true, the product of a, b, and k must be 1 (abk = 1), and the sum of the exponents b, f, and m must also be 1 (b + f + m = 1). Many field studies have shown these equations to be good approximations of actual variation in width, depth, and velocity with variation in discharge.

The coefficients of the hydraulic geometry equations have little effect, relative to the powerful exponents, and are usually ignored. The exponents of the hydraulic geometry equations depend on the physical characteristics of the drainage basins and stream channels involved, but for a given point on a given stream, average values are b = 0.1, f = 0.45, and m = 0.45, which means that the increase in discharge during a flood event is expressed primarily in increases in depth and flow velocity.

The discharge of rivers increases downstream because of the larger length of stream receiving baseflow and the contribution from many tributaries, and hydraulic geometry equations may also be applied to downstream changes in discharge. Average values for the exponents in the downstream hydraulic geometry equations for width, depth, and velocity are b = 0.5, f = 0.4, and m = 0.1. This phenomenon is a paradox. Casual observation would suggest that small streams in the higher parts of the drainage basins flow faster than the large rivers in the lower parts of the drainage basins. Yet, appearances are deceptive: Water in the wider, deeper channels of the larger streams actually flows faster than does water in the smaller headwater streams.

When a flood flow exceeds the capacity of the river channel, the surface elevation of the river rises above the elevation of the riverbanks, and flooding of the surface adjacent to the stream occurs. This is what has generally been called a flood, or an overbank flood. Overbank floods do serious damage to homes, businesses, industrial facilities, and crops on the flooded areas, and much time and money are devoted to flood prevention. The principal method of flood prevention is the construction of levees, large earthen embankments or concrete walls along the stream bank, at locations where the potential economic loss because of flooding justifies the expense of their construction and maintenance.

Study of River Flow

Research on flow in rivers, or hydrology, almost invariably involves the determination of flow velocity and discharge. Because discharge (Q) equals stream width (w) times average depth (d) times average flow velocity (v), velocity and discharge are closely related. As implied by the world “average,” depth and velocity vary across the width of a river or stream. Velocity is lowest in the shallower parts of the river. Therefore, the cross section of the stream is divided into sections, and the discharge is taken as the sum of discharges of all the sections.

For a relatively small river, a rope or strong cord is stretched across the river and marked at regular intervals, typically 2 meters. The depth is measured at each marked point, and a current meter is used to measure the velocity at each point. The flow in rivers and streams is turbulent, and the current meter actually measures the average downstream component of velocity. Average downstream velocity varies with the depth of the channel and also through the vertical section at any point on the stream, the lowest velocities occurring at the bottom and at the surface. Average velocity at any given vertical section of flow occurs at about 0.6d above the bottom, which is where the current meter is positioned. Alternatively, two velocity measurements may be taken, at 0.2d and 0.8d above the bottom, and averaged. The equation Q = wdv, where w is the interval at which measurements were taken, is computed for each section, or interval, across the stream, and all the resultant discharges are summed to obtain the total discharge at the point on the stream, which is now called a station.

This process is laborious. If discharge at a station must be more or less continuously monitored, a quicker method is desirable, which is accomplished by developing a rating curve for the station. Discharge is measured in the manner already described at several different times, with as wide a range of discharges as practical. A staff gauge (a board mounted vertically in the stream and marked in units of length, usually feet in American and British practice) is erected at the station, and the level of the water surface, called the stage of the river, is determined each time the discharge is measured. The rating curve consists of a plot of stage against discharge. Once the rating curve has been determined, discharge is estimated from river stage by use of the rating curve. Because of the scour and fill of the river bottom that occur with each increase and decrease in discharge and velocity, the rating curve is not a straight line. Moreover, because the scour and fill may, over time, change the character of the river channel at the station, it is necessary to actually measure the discharge periodically to check the validity of the rating curve. If large changes in the channel occur, it is necessary to establish a new rating curve.

The U.S. Geological Survey maintains a large number of rating stations, or gauging stations, and periodically reports stream discharges. At most stations operated by the agency, the staff gauge is replaced by a vertical pipe driven into the streambed and perforated near the bed. Water is free to flow in and out of the pipe as river stage changes, but the water surface inside the pipe is not disturbed by surface waves. A cable with a weight at the end is attached to the float, and the cable is passed over a wheel near the top of the pipe. The float is free to move with the water surface inside the pipe and, as it moves, it turns the wheel. Sensitive instruments monitor the position of the wheel and therefore the river stage as well. In this way, the stage is periodically and automatically reported to a central office electronically.

Principal Terms

discharge: the total amount of water passing a point on a river per unit of time

evapotranspiration: all water that is converted to water vapor by direct evaporation or passage through vegetation

hydraulic geometry: a set of equations that relate river width, depth, and velocity to discharge

hydrograph: a plot recording the variation of stream discharge over time

hydrologic cycle: the circulation of water as a liquid and vapor from the oceans to the atmosphere and back to the oceans

hydrology: broadly, the science of water; the term is often used in the more restricted sense of flow in channels

rating curve: a plot of river discharge in relation to elevation of the water surface; permits estimation of discharge from the water elevation

turbulent flow: the swirling flow that is typical of rivers, as opposed to smooth, laminar flow

Bibliography

Chorley, Richard J., Stanley A. Schumm, and David E. Sugden. Geomorphology. New York: Methuen, 1985. A well-referenced older textbook of geomorphology, it nevertheless provides a comprehensive treatment of the theoretical principles of water flowing in river channels. Contains numerous line drawings and working charts and graphs.

Christopherson, Robert W., and Mary-Louie Byrne. Geosystems: An Introduction to Physical Geography. Toronto, Ontario: Pearson Education Canada, 2006. An extremely readable and well-illustrated text that discusses water flow in rivers in general terms rather than mathematical specifics, in the overall context of systems functioning on the planet.

Collier, Michael. Over the Rivers: An Aerial View of Geology. New York: Mikaya Press, 2008. Discusses the dynamic landscape of the rivers. Explains the processes that shape the landscape and its influence on humans. Written in the popular style. Easily accessible to the general public. Filled with bits of information and extraordinary photographs. Presents multiple examples drawn from the Mississippi River.

Collins, B., and T. Dunne. Fluvial Geomorphology and River-Gravel Mining: A Guide for Planners. Sacramento, Calif.: California Department of Conservation, 1990. A thorough look at fluvial systems and geomorphology. Emphasizes design to avoid environmental problems associated with water in all its manifestations. Many worked problems, most involving only basic mathematics.

Darby, Stephen, and David Sear. River Restoration: Managing the Uncertainty in Restoring Physical Habitat. Hoboken, N.J.: John Wiley & Sons, 2008. Begins with theoretical and philosophical issues with habitat restoration to provide a strong foundation for decision making. Addresses logistics, planning, mathematical modeling, and construction stages of restoration in later chapters. Rounds out with post-construction monitoring and long-term evaluations to provide a full picture of the habitat restoration process. Highly useful for anyone involved in the planning and implementing of habitat restoration.

Dingman, S. L. Physical Hydrology. 2d ed. Long Grove, Ill.: Waveland Press, 2008. A thorough introduction to hydrology that offers a clear development of all the equations that are essential to fluvial hydrology. For full understanding, a year of college calculus is necessary, but about 80 percent of the material can be mastered by a student with knowledge only of algebra and trigonometry. An excellent treatment of the subject, with many answered problems and a helpful annotated bibliography. Accompanied by a CD.

Leopold, Luna B. A View of the River. Cambridge, Mass.: Harvard University Press, 2006. A lucid explanation of rivers and their processes. Clearly written and easy to follow. Highly recommended for both beginners and professionals.

‗‗‗‗‗‗‗‗‗‗. Water, Rivers, and Creeks. Sausalito, Calif.: University Science Books, 2009. A brief, very readable introduction to hydrology and the study of rivers. Highly recommended for all newcomers to the field and most professionals.

Manning, J. C. Applied Principles of Hydrology. 3d ed. Upper Saddle River, N.J.: Prentice Hall, 1997. An excellent short introduction to the principles of surface water and groundwater hydrology. Incorporates very little of the mathematics involved in the two fields, but the descriptions and illustrations of methods of making field measurements are clear and informative.

Newson, Malcolm. Land, Water and Development: Sustainable and Adaptive Management of Rivers. 3d ed. London: Routledge, 2008. Presents land-water interactions. Discusses recent research, study tools and methods, and technical issues, such as soil erosion and damming. Suited for undergraduate students and professionals, this text covers concepts in managing land and water resources in the developed world.

Richards, Keith. Rivers, Form, and Process in Alluvial Channels. Caldwell, N.J.: Blackburn Press, 2004. A college-level text requiring some familiarity with calculus for complete understanding. Its great strength is the very large number of research papers cited in the text. Conveys a sense of the quantity and type of research that has been done.

Wohl, Ellen. A World of Rivers. Chicago: University of Chicago Press, 2011. The Amazon, Ob, Nile, Danube, Ganges, Mississippi, Murray-Darling, Congo, Chang Jiang, and Mackenzie rivers each have a chapter in this book. Figure 1.1 contains more straightforward and organized information than some full textbooks. Discusses natural history, anthropogenic impact, and the future environment of these ten great rivers. The bibliography is organized by chapter.