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HYDRAULIC

DESIGN

HANDBOOK

Larry W. Mays, Editor in Chief Department of Civil and Environmental Engineering Arizona State University Tempe, Arizona

McGraw-Hill New York • San Francisco • Washington, D.C.» Auckland • Bogota • Caracas • Lisbon • London • Madrid • Mexico City • Milan • Montreal • New Delhi • San Juan • Singapore • Sydney • Tokyo • Toronto

Library of Congress Cataloging-in-Publication Data Hydraulic design handbook/Larry W. Mays, editor-in-chief. p. cm. Includes bibliographical references and index. ISBN 0-07-041152-2 1. Hydraulic structures—Design and costruction Hanbooks, manuals, etc. I. Mays, Larry W. TC180.H94 1999 627—dc21 99-20240 CIP McGraw-Hill ^n /9liq 5. High boiling point—c.f., H2 (20 K), O2 (90 K) and H2O (373 K) 6 Good conductor of heat relative to other liquids and nonmetal solids. Chemical and Other Properties 1. Slightly ionized—water is a good solvent for electrolytes and nonelectrolytes 2. Transparent to visible light; opaque to near infrared 3. High dielectric constant—responds to microwaves and electromagnetic fields Note: The values are approximate. All the properties listed are functions of temperature, pressure, water purity, and other factors that should be known if more exact values are to be assigned. For example, surface tension is greatly influenced by the presence of soap films, and the boiling point depends on water purity and confining pressure. The values are generally indicative of conditions near 1O0C and one atmosphere of pressure.

table —especially density and viscosity values—are used regularly by pipeline engineers. Other properties, such as compressibility and thermal values, are used indirectly, primarily to justify modeling assumptions, such as the flow being isothermal and incompressible. Many properties of water depend on intermolecular forces that create powerful attractions (cohesion) between water molecules. That is, although a water molecule is electrically neutral, the two hydrogen atoms are positioned to create a tetrahedral charge distribution on the water molecule, allowing water molecules to be held strongly together with the aid of electrostatic attractions. These strong internal forces—technically called 'hydrogen bonds'—arise directly from the non-symmetrical distribution of charge. The chemical behavior of water also is unusual. Water molecules are slightly ionized, making water an excellent solvent for both electrolytes and nonelectrolytes. In fact, water is nearly a universal solvent, able to wear away mountains, transport solutes, and support the biochemistry of life. But the same properties that create so many benefits also create problems, many of which must be faced by the pipeline engineer. Toxic chemicals, disinfection byproducts, aggressive and corrosive compounds, and many other substances can be carried by water in a pipeline, possibly causing damage to the pipe and placing consumers at risk. Other challenges also arise. Water's almost unique property of expanding on freezing can easily burst pipes. As a result, the pipeline engineer either may have to bury a line or may need to supply expensive heat-tracing systems on lines exposed to freezing weather, particularly if there is a risk that standing water may sometimes occur. Water's high viscosity is a direct cause of large friction losses and high energy costs whereas its vapor properties can create cavitation problems in pumps, valves, and pipes. Furthermore, the combination of its high density and small compressibility creates potentially dramatic transient conditions. We return to these important issues after considering how pipeline flows respond to various physical constraints and influences in the next section. 2.4.2 Laws of Conservation (How?) Although the implications of the characteristics of water are enormous, no mere list of its properties will describe a physical problem completely. Whether we are concerned with water quality in a reservoir or with transient conditions in a pipe, natural phenomena also obey a set of physical laws that contributes to the character and nature of a system's response. If engineers are to make quantitative predictions, they must first understand the physical problem and the mathematical laws that model its behavior. Basic physical laws must be understood and be applied to a wide variety of applications and in a great many different environments: from flow through a pump to transient conditions in a channel or pipeline. The derivations of these equations are not provided, however, because they are widely available and take considerable time and effort to do properly. Instead, the laws are presented, summarized, and discussed in the pipeline context. More precisely, a quantitative description of fluid behavior requires the application of three essential relations: (1) a kinematic relation obtained from the law of mass conservation in a control volume, (2) equations of motion provided by both Newton's second law and the energy equation, and (3) an equation of state adapted from compressibility considerations, leading to a wavespeed relation in transient flow and justifying the assumption of an incompressible fluid in most steady flow applications. A few key facts about mass conservation and Newton's second law are reviewed briefly in the next section. Consideration of the energy equation is deferred until steady flow is discussed in more detail, whereas further details about the equation of state are introduced along with considerations of unsteady flow.

2.4.3 Conservation of Mass One of a pipeline engineer's most basic, but also most powerful, tools is introduced in this section. The central concept is that of conservation of mass and its key expression is the continuity or mass conservation equation. One remarkable fact about changes in a physical system is that not everything changes. In fact, most physical laws are conservation laws: They are generalized statements about regularities that occur in the midst of change. As Ford (1973) said: A conservation law is a statement of constancy in nature—in particular, constancy during change. If for an isolated system a quantity can be defined that remains precisely constant, regardless of what changes may take place within the system, the quantity is said to be absolutely conserved. A number of physical quantities have been found that are conserved in the sense of Ford's quotation. Examples include energy (if mass is accounted for), momentum, charge, and angular momentum. One especially important generalization of the law of mass conservation includes both nuclear and chemical reactions (Hatsopoulos and Keenan, 1965). 2.4.3.1 Law of Conservation of Chemical Species "Molecular species are conserved in the absence of chemical reactions and atomic species are conserved in the absence of nuclear reactions". In essence, the statement is nothing more a principle of accounting, stating that the number of atoms or molecules that existed before a given change is equal to the number that exists after the change. More powerfully, the principle can be transformed into a statement of revenue and expenditure of some commodity over a definite period of time. Because both hydraulics and hydrology are concerned with tracking the distribution and movement of the Earth's water, which is nothing more than a particular molecular species, it is not surprising that formalized statements of this law are used frequently. These formalized statements are often called water budgets, typically if they apply to an area of land, or continuity relations, if they apply in a well-defined region of flow (the region is well-defined; the flow need not be). The principle of a budget or continuity equation is applied every time we balance a checkbook. The account balance at the end of any period is equal to the initial balance plus all the deposits minus all the withdrawals. In equation form, this can be written as follows: (balance)^ = (balance), + ^ deposits - ^ withdrawals Before an analogous procedure can be applied to water, the system under consideration must be clearly defined. If we return to the checking-account analogy, this requirement simply says that the deposits and withdrawals included in the equation apply to one account or to a well-defined set of accounts. In hydraulics and hydrology, the equivalent requirement is to define a control volume—a region that is fixed in space, completely surrounded by a "control surface," through which matter can pass freely. Only when the region has been precisely defined can the inputs (deposits) and outputs (withdrawals) be identified unambiguously. If changes or adjustments in the water balance (AS) are the concern, the budget concept can be expressed as AS = Sf- S1 = (balance), - (balance). = V1 - V0

(2.1)

where V1 represents the sum of all the water entering an area, and V0 indicates the total volume of water leaving the same region. More commonly, however, a budget relation such as Eq. 2.1 is written as a rate equation. Dividing the "balance" equation by At and taking the limit as At goes to zero produces

& = JjL = I-O (2.2) at where the derivative term S' is the time rate of change in storage, S is the water stored in the control volume, I is the rate of which water enters the system (inflow), and O is the rate of outflow. This equation can be applied in any consistent volumetric units (e.g., m3/s, ftVs, L/s, ML/day, etc.) When the concept of conservation of mass is applied to a system with flow, such as a pipeline, it requires that the net amount of fluid flowing into the pipe must be accounted for as fluid storage within the pipe. Any mass imbalance (or, in other words, net mass exchange) will result in large pressure changes in the conduit because of compressibility effects. 2.4.3.2 Steady Flow Assuming, in addition, that the flow is steady, Eq. 2.2 can be reduced further to inflow = outflow or 7 = O. Since the inflow and outflow may occur at several points, this is sometimes re-written as E V A = S VA inflow 1 1 outflow

(2-3)

Equation (2.3) states that the rate of flow into a control volume is equal to the rate of outflow. This result is intuitively satisfying since no accumulation of mass or volume should occur in any control volume under steady conditions. If the control volume were taken to be the junction of a number of pipes, this law would take the form of Kirchhoff's current law—the sum of the mass flow in all pipes entering the junction equals the sum of the mass flow of the fluid leaving the junction. For example, in Fig. 2.2, continuity for the control volume of the junction states that G1 + Q2 = C3 + Q4

2.4.4

(2-4)

Newton's Second Law

When mass rates of flow are concerned, the focus is on a single component of chemical species. However, when we introduce a physical law, such as Newton's law of motion, we obtain something even more profound: a relationship between the apparently unrelated quantities of force and acceleration. More specifically, Newton's second law relates the changes in motion of a fluid or solid to the forces that cause the change. Thus, the statement that the resultant of all external forces, including body forces, acting on a system is equal to the rate of change of momentum of this system with respect to time. Mathematically, this is expressed as x^ d(mv) ix=v

(2 5)

-

where t is the time and Fext represents the external forces acting on a body of mass m moving with velocity i). If the mass of the body is constant, Eq. (2.5) becomes

FIGURE 2.2 Continuity at a pipe junction Q1 + Q2 = Q3 + Q4

2Fex, = «f = mfl

(2.6)

where a is the acceleration of the system (the time rate of change of velocity). In closed conduits, the primary forces of concern are the result of hydrostatic pressure, fluid weight, and friction. These forces act at each section of the pipe to produce the net acceleration. If these forces and the fluid motion are modeled mathematically, the result is a "dynamic relation" describing the transient response of the pipeline. For a control volume, if flow properties at a given position are unchanging with time, the steady form of the moment equation can be written as 2X*,= | PV(V n)dA ./CS

(2.7)

where the force term is the net external force acting on the control volume and the right hand term gives the net flux of momentum through the control surface. The integral is taken over the entire surface of the control volume, and the integrand is the incremental amount of momentum leaving the control volume. The control surface usually can be oriented to be perpendicular to the flow, and one can assume that the flow is incompressible and uniform. With this assumption, the momentum equation can be simplified further as follows: 2>« = (P 4000. These flow regime have a direct influence on the head loss experienced in a pipeline system. 2.6.1 Turbulent Flow Consider an experiment in which a sensitive probe is used to measure flow velocity in a pipeline carrying a flowing fluid. The probe will certainly record the mean or net component of velocity in the axial direction of flow. In addition, if the flow in the pipeline is turbulent, the probe will record many small and abrupt variations in velocity in all three spatial directions. As a result of the turbulent motion, the details of the flow pattern will change randomly and constantly with time. Even in the simplest possible system—an uniform pipe carrying water from a constant-elevation upstream reservoir to a downstream valve—the detailed structure of the velocity field will be unsteady and exceedingly complex. Moreover, the unsteady values of instantaneous velocity will exist even if all external conditions at both the reservoir and valve are not changing with time. Despite this, the mean values of velocity and pressure will be fixed as long as the external conditions do not change. It is in this sense that turbulent flows can be considered to be steady. The vast majority of flows in engineering are turbulent. Thus, unavoidably, engineers must cope with both the desirable and the undesirable characteristics of turbulence. On the positive side, turbulent flows produce an efficient transfer of mass, momentum, and energy within the fluid. In fact, the expression to "stir up the pot" is an image of turbulence; it implies a vigorous mixing that breaks up large-scale order and structure in a fluid. But the rapid mixing also may create problems for the pipeline engineer. This "down side" can include detrimental rates of energy loss, high rates of corrosion, rapid scouring and erosion, and excessive noise and vibration as well as other effects. How does the effective mixing arise within a turbulent fluid? Physically, mixing results from the random and chaotic fluctuations in velocity that exchange fluid between different regions in a flow. The sudden, small-scale changes in the instantaneous velocity tend to cause fast moving "packets" of fluid to change places with those of lower velocity and vice versa. In this way, the flow field is constantly bent, folded, and superimposed on itself. As a result, large-scale order and structure within the flow is quickly broken down and torn apart. But the fluid exchange transports not only momentum but other properties associated with the flow as well. In essence, the rapid and continual interchange of fluid within a turbulent flow creates both the blessing and the curse of efficient mixing. The inherent complexity of turbulent flows introduces many challenges. On one hand, if the velocity variations are ignored by using average or mean values of fluid properties, a degree of uncertainty inevitably arises. Details of the flow process and its variability will be avoided intentionally, thereby requiring empirical predictions of mean flow characteristics (e.g., head-loss coefficients and friction factors). Yet, if the details of the velocity field are analyzed, a hopelessly complex set of equations is produced that must be solved using a small time step. Such models can rarely be solved even on the fastest computers. From the engineering view point, the only practical prescription is to accept the empiricism necessitated by flow turbulence while being fully aware of its difficulties—he averaging process conceals much of what might be important. Ignoring the details of the fluid's motion can, at times, introduce significant error even to the mean flow calculations. When conditions within a flow change instantaneously both at a point and in the mean, the flow becomes unsteady in the full sense of the word. For example, the downstream valve in a simple pipeline connected to a reservoir might be closed rapidly, creating shock waves that travel up and down the conduit. The unsteadiness in the mean values of the

flow properties introduces additional difficulties into a problem that was already complex. Various procedures of averaging, collecting, and analyzing data that were well justified for a steady turbulent flow are often questionable in unsteady applications. The entire situation is dynamic: Rapid fluctuations in the average pressure, velocity, and other properties may break or damage the pipe or other equipment. Even in routine applications, special care is required to control, predict, and operate systems in which unsteady flows commonly occur. The question is one of perspective. The microscopic perspective of turbulence in flows is bewildering in its complexity; thus, only because the macroscopic behavior is relatively predictable can turbulent flows be analyzed. Turbulence both creates the need for approximate empirical laws and determines the uncertainty associated with using them. The great irregularity associated with turbulent flows tends to be smoothed over both by the empirical equations and by a great many texts. 2.6.2

Head Loss Caused by Friction

A basic relation used in hydraulic design of a pipeline system is the one describing the dependence of discharge Q (say in m3/s) on head loss hf (m) caused by friction between the flow of fluid and the pipe wall. This section discusses two of the most commonly used head-loss relations: the Darcy-Weisbach and Hazen-Williams equations. The Darcy-Weisbach equation is used to describe the head loss resulting from flow in pipes in a wide variety of applications. It has the advantage of incorporating a dimensionless friction factor that describes the effects of material roughness on the surface of the inside pipe wall and the flow regime on retarding the flow. The Darcy-Weisbach equation can be written as hf,DW =fj>T% = °-0826 § L/

(2-16)

where hf,DW = head loss caused by friction (m),/= dimensionless friction factor, L = pipe length (m), D = pipe diameter (m), V = QIA = mean flow velocity (m/s), Q = discharge (mVs), A = cross-sectional area of the pipe (m2), and g = acceleration caused by gravity (m/s2). For noncircular pressure conduits, D is replaced by 4R, where R is the hydraulic radius. The hydraulic radius is defined as the cross-sectional area divided by the wetted perimeter or, R = AIP. Note that the head loss is directly proportional to the length of the conduit and the friction factor. Obviously, the rougher a pipe is and the longer the fluid must travel, the greater the energy loss. The equation also relates the pipe diameter inversely to the head loss. As the pipe diameter increases, the effects of shear stress at the pipe walls are felt by less of the fluid, indicating that wider pipes may be advantageous if excavation and construction costs are not prohibitive. Note in particular that the dependence of the discharge Q on the pipe diameter D is highly nonlinear; this fact has great significance to pipeline designs because head losses can be reduced dramatically by using a large-diameter pipe, whereas an inappropriately small pipe can restrict flow significantly, rather like a partially closed valve. For laminar flow, the friction factor is linearly dependent on the Re with the simple relationship/= 64/Re. For turbulent flow, the friction factor is a function of both the Re and the pipes relative roughness. The relative roughness is the ratio of equivalent uniform sand grain size and the pipe diameter (e/D), as based on the work of Nikuradse (1933), who experimentally measured the resistance to flow posed by various pipes with uniform sand

grains glued onto the inside walls. Although the commercial pipes have some degree of spatial variance in the characteristics of their roughness, they may have the same resistance characteristics as do pipes with a uniform distribution of sand grains of size e. Thus, if the velocity of the fluid is known, and hence Re, and the relative roughness is known, the friction factor/can be determined by using the Moody diagram or the Colebrook-White equation. Jeppson (1976) presented a summary of friction loss equations that can be used instead of the Moody diagram to calculate the friction factor for the Darcy-Weisbach equation. These equations are applicable for Re greater than 4000 and are categorized according to the type of turbulent flow: (1) turbulent smooth, (2) transition between turbulent smooth and wholly rough, and (3) turbulent rough. For turbulent smooth flow, the friction factor is a function of Re: -^= = 21og (ReV/)

(2.17)

For the transition between turbulent smooth and wholly rough flow, the friction factor is a function of both Re and the relative roughness e/D. This friction factor relation is often summarized in the Colebrook-White equation: ^ = -21Og(^ + ^SLl V^ I3-7 ReVf)

(2.18)

When the flow is wholly turbulent (large Re and elD), the Darcy-Weisbach friction factor becomes independent of Re and is a function only of the relative roughness: -L= 1.14-21og(*/D)

(2.19)

In general, Eq. (2.16) is valid for all turbulent flow regimens in a pipe, where as Eq. (2.19) is merely an approximation that is valid for the hydraulic rough flow. In a smoothpipe flow, the viscous sublayer completely submerges the effect of e on the flow. In this case, the friction factor/is a function of Re and is independent of the relative roughness e/D. In rough-pipe flow, the viscous sublayer is so thin that flow is dominated by the roughness of the pipe wall and/is a function only of e/D and is independent of Re. In the transition,/is a function of both elD and Re. The implicit nature of/in Eq. (2.18) is inconvenient in design practice. However, this difficulty can be easily overcome with the help of the Moody diagram or with one of many available explicit approximations. The Moody diagram plots Re on the abscissa, the resistance coefficient on one ordinate and/on the other, with elD acting as a parameter for a family of curves. If e/D is known, then one can follow the relative roughness isocurve across the graph until it intercepts the correct Re. At the corresponding point on the opposite ordinate, the appropriate friction factor is found; e/D for various commercial pipe materials and diameters is provided by several manufacturers and is determined experimentally. A more popular current alternative to graphical procedures is to use an explicit mathematical form of the friction-factor relation to approximate the implicit ColebrookWhite equation. Bhave (1991) included a nice summary of this topic. The popular network-analysis program EPANET and several other codes use the equation of Swanee and Jain (1976), which has the form /=0.25 H pe ^ 5.74"{t (2.20) [log (TW + Re^l

To circumvent considerations of roughness estimates and Reynolds number dependencies, more direct relations are often used. Probably the most widely used of these empirical head-loss relations is the Hazen-Williams equation, which can be written as Q = C11 C£>2-63S0-54

(2.21)

where C11 = unit coefficient (C11 = 0.314 for English units, 0.278 for metric units), Q = discharge in pipes, gallons/s or m3/s, L = length of pipe, ft or m, d = internal diameter of pipe, inches or mm, C = Hazen-Williams roughness coefficient, and S = the slope of the energy line and equals hf/L. The Hazen-Williams coefficient C is assumed constant and independent of the discharge (i.e., Re). Its values range from 140 for smooth straight pipe to 90 or 80 for old, unlined, tuberculated pipe. Values near 100 are typical for average conditions. Values of the unit coefficient for various combinations of units are summarized in Table 2.2. In Standard International (SI) units, the Hazen-Williams relation can be rewritten for head loss as (n}o^4 i /I7^ =10.654 [^J -^L

(2.22)

where hfHW is the Hazen-Williams head loss. In fact, the Hazen-Williams equation is not the only empirical loss relation in common use. Another loss relation, the Manning equation, has found its major application in open channel flow computations. As with the other expressions, it incorporates a parameter to describe the roughness of the conduit known as Manning's n. Among the most important and surprisingly difficult hydraulic parameter is the diameter of the pipe. As has been mentioned, the exponent of diameter in head-loss equations is large, thus indicating high sensitivity to its numerical value. For this reason, engineers

FIGURE 2.4 Flow in series and parallel pipes.

TABLE 2.2 Unit Coefficient Cu for the Hazen-Williams Equation Units of Discharge Q

Units of Diameter D

Unit Coeficient Cu

MGD ftVs GPM GPD m3/s

ft ft in in m

0.279 0.432 0.285 405 0.278

and analysts must be careful to obtain actual pipe diameters often from manufacturers; the use of nominal diameters is not recommended. Yet another complication may arise, however. The diameter of a pipe often changes with time, typically as a result of chemical depositions on the pipe wall. For old pipes, this reduction in diameter is accounted for indirectly by using an increased value of pipe resistance. Although this approach may be reasonable under some circumstances, it may be a problem under others, especially for unsteady conditions. Whenever possible, accurate diameters are recommended for all hydraulic calculations. However, some combinations of pipes (e.g., pipes in series or parallel; Fig. 2.4) can actually be represented by a single equivalent diameter of pipe. 2.6.3 Comparison of Loss Relations It is generally claimed that the Darcy-Weisbach equation is superior because it is theoretically based, whereas both the Manning equation and the Hazen-Williams expression use empirically-determined resistance coefficients. Although it is true that the functional relationship of the Darcy-Weisbach formula reflects logical associations implied by the dimensions of the various terms, determination of the equivalent uniform sand-grain size is essentially experimental. Consequently, the relative roughness parameter used in the Moody diagram or the Colebrook-White equations is not theoretically determined. In this section, the Darcy-Weisbach and Hazen-Williams equations are compared briefly using a simple pipe as an example. In the hydraulic rough range, the increase in Ahf can be explained easily when the ratio of Eq. (2.16) to Eq. (2.22) is investigated. For hydraulically rough flow, Eq. (2.18) can be simplified by neglecting the second term 2.51 (Re^/f) of the logarithmic argument. This ratio then takes the form of hf,Hw ( £ V no.i3 i -j^ =128.94 1.14-2 log £ £B,~ n f'DW V ^) ^ *£

(2.23)

which shows that in most hydraulic rough cases, for the same discharge Q, a larger head loss hfis predicted using Eq. (2.16) than when using Eq. (2.22). Alternatively, for the same head loss, Eq. (2.22) returns a smaller discharge than does Eq. (2.16). When comparing head-loss relations for the more general case, a great fuss is often made over unimportant issues. For example, it is common to plot various equations on the Moody diagram and comment on their differences. However, such a comparison is of secondary importance. From a hydraulic perspective, the point is this: Different equations should still produce similar head-discharge behavior. That is, the physical relation between head loss and flow for a physical segment of pipe should be predicted well by

any practical loss relation. Said even more simply, the issue is how well the hf versus Q curves compare. To compare the values of hf determined from Eq. (2.16) and those from Eq. (2.22), consider a pipe for which the parameters D, L and C are specified. Using the HazenWilliams relation, it is then possible to calculate hf for a given Q. Then, the DarcyWeisbach/can be obtained, and with the Colebrook formula Eq. (2.18), the equivalent value of roughness e can be found. Finally, the variation of head with discharge can be plotted for a range of flows. This analysis is performed for two galvanized iron pipes with e = 0.15 mm. One pipe has a diameter of 0.1 m and a length of 100 m; and the dimensions of the other pipe are D= 1.0 m and L = 1000 m, respectively. The Hazen-Williams C for galvanized iron pipe is approximately 130. Different C values are used for these two pipes to demonstrate the shift and change of the range within which Ahf is small. The results of the calculated hfQ relation and the difference Ahf of the head loss of the two methods for the same discharge are shown in Figs. 2.5 and 2.6. If hf,DW denotes the head loss determined by using Eq. (2.16) and hf9HW that using Eq. (2.22), Ahf (m) can be Afy = hf,DW - hf,DW

(2.24)

whereby the Darcy-Weisbach head loss hf,DW is used as a reference for comparison. Figures 2.5 and 2.6 show the existence of three ranges: two ranges, within which hf,DW > hfDW , and the third one for which hf DW < hf DW. The first range of hf DW > hf DW is at a lower head loss and is small. It seems that the difference A/jy in this case is the result of the fact that the Hazen-Williams formula is not valid for the hydraulic smooth and the smooth-to-transitional region. Fortunately, this region is seldom important for design purposes. At high head losses, the Hazen-Williams formula tends to produce a discharge that is smaller than the one produced by the Darcy-Weisbach equation. For a considerable part of the curve—primarily the range within which hf DW > hf DW —A/iy is small compared with the absolute head loss. It can be shown that the range of small A/jy changes is shifted when different values of Hazen-Williams's C are used for the calculation. Therefore, selecting the proper value of C, which represents an appropriate

Darcy-Weisbach Hazen-Williams

L D e C

=100m = 0.1 m =0.15 mm = 122.806

FIGURE 2.5 Comparison of Hazen-Williams and Darcy-Weisbach loss relations (smaller diameter).

Darcy-Weisbach Hazen-Williams

L = 100Om D = 1.0 m e =0.15 mm C = 124.923

FIGURE 2.6 Comparison of Hazen-Williams and Darcy-Weisbach loss relations (larger diameter). point on the head-discharge curve, is essential. If such a C value is used, Ahf is small, and whether the Hazen-Williams formula or the Darcy-Weisbach equation is used for the design will be of little importance. This example shows both the strengths and the weaknesses of using Eq. (2.22) as an approximation to Eq. (2.16). Despite its difficulties, the Hazen-Williams formula is often justified because of its conservative results and its simplicity of use. However, choosing a proper value of either the Hazen-Williams C or the relative roughness elD is often difficult. In the literature, a range of C values is given for new pipes made of various materials. Selecting an appropriate C value for an old pipe is even more difficult. However, if an approximate value of C or e is used, the difference between the head-loss equations is likely to be inconsequential. Head loss also is a function of time. As pipes age, they are subject to corrosion, especially if they are made of ferrous materials and develop rust on the inside walls, which increases their relative roughness. Chemical agents, solid particles, or both in the fluid can gradually degrade the smoothness of the pipe wall. Scaling on the inside of pipes can occur if the water is hard. In some instances, biological factors have led to time-dependent head loss. Clams and zebra mussels may grow in some intake pipes and may in some cases drastically reduce discharge capacities. 2.6.4 Local Losses Head loss also occurs for reasons other than wall friction. In fact, local losses occur whenever changes occur in the velocity of the flow: for example, changes in the direction of the conduit, such as at a bend, or changes in the cross-sectional area, such as an aperture, valve or gauge. The basic arrangement of flow and pressure is illustrated for a venturi contraction in Fig. 2.7. The mechanism of head loss in the venturi is typical of many applications involving local losses. As the diagram indicates, there is a section of flow contraction into which the flow accelerates, followed by a section of expansion, into which the flow decelerates. This aspect of the venturi, or a reduced opening at a valve, is nicely described by the continuity equation. However, what happens to the pressure is more interesting and more important.

FIGURE 2.7 Pressure relations in a venturi contraction.

As the flow accelerates, the pressure decreases according to the Bernoulli relation. Everything goes smoothly in this case because the pressure drop and the flow are in the same direction. However, in the expansion section, the pressure increases in the downstream direction. To see why this is significant, consider the fluid distributed over the cross section. In the center of the pipe, the fluid velocity is high; the fluid simply slows down as it moves into the region of greater pressure. But what about the fluid along the wall? Because it has no velocity to draw on, it tends to respond to the increase in pressure in the downstream direction by flowing upstream, counter to the normal direction of flow. That is, the flow tends to separate, which can be prevented only if the faster moving fluid can "pull it along" using viscosity. If the expansion is too abrupt, this process is not sufficient, and the flow will separate, creating a region of recirculating flow within the main channel. Such a region causes high shear stresses, irregular motion, and large energy losses. Thus, from the view point of local losses, nothing about changes in pressure is symmetrical—adverse pressure gradients or regions of recirculating flow are crucially important with regard to local losses. Local head losses are often expressed in terms of the velocity head as A1 = * J

(2.25)

where k is a constant derived empirically from testing the head loss of the valve, gauge, and so on, and is generally provided by the manufacturer of the device. Typical forms for this relation are provided in Table 2.3 (Robertson and Crowe, 1993). 2.6.5

Tractive Force

Fluid resistance also implies a flux in momentum and generates a tractive force, which raises a number of issues of special significance to the two-phase (liquid-solid) flows found in applications of transport of slurry and formation of sludge. In these situations,

TABLE 2.3 Local Loss Coefficients at Transitions Description

Sketch

Additional Data

K

Source

Pipe entrance

Expansion

90° miter bend 90° miter bend

Threaded pipe fittings

Globe valve—wide open Angle valve—wide open Gate valve—wide open Gatevalve—half open Return bend Tee 90° elbow 45° elbow

the tractive force has an important influence on design velocities: The velocity cannot be too small or the tractive force will be insufficient to carry suspended sediment and deposition will occur. Similarly, if design velocities are too large, the greater tractive force will increase rates of erosion and corrosion in the channel or pipeline, thus raising maintenance and operational costs. Thus, the general significance of tractive force relates to designing

self-cleansing channel and pressure-flow systems and to stable channel design in erodible channels. Moreover, high tractive forces are capable of causing water quality problems in distribution system piping through the mechanism of biofilm sloughing or suspension of corrosion by-products. 2.6.6 Conveyance System Calculations: Steady Uniform Flow A key practical concern in the detailed calculation of pressure flow and the estimation of pressure losses. Because the practice of engineering requires competent execution in a huge number of contexts, the engineer will encounter many different applications in practice. Compare, for example, Fig. 2.4 to 2.8. In fact, the number of applied topics is so large that comprehensive treatment is impossible. Therefore, this chapter emphasizes a systematic presentation of the principles and procedures of problem-solving to encourage the engineer's ability to generalize. To illustrate the principles of hydraulic analysis, this section includes an example that demonstrates both the application of the energy equation and the use of the most common head-loss equations. A secondary objective is to justify two common assumptions about pipeline flow: namely, that flow is, to a good approximation, incompressible and isothermal. Problem. A straight pipe is 2500 m long, 27 inches in diameter and discharges water at 1O0C into the atmosphere at the rate of 1.80 m3/s. The lower end of the pipe is at an elevation of 100 m, where a pressure gauge reads 3.0 MPa. The pipe is on a 4% slope. 1. Determine the pressure head, elevation head, total head, and piezometric level at both ends of this pipeline. 2. Determine the associated Darcy-Weisbach friction factor/and Hazen-Williams C for this pipeline and flow. 3. Use the known pressure change to estimate the change in density between the upstream and downstream ends of the conduit. Also estimate the associated change

FIGURE 2.8 Flow in a simple pipe network.

in velocity between the two ends of the pipe, assuming a constant internal diameter of 27 in throughout. What do you conclude from this calculation? 4. Estimate the change in temperature associated with this head loss and flow, assuming that all the friction losses in the pipe are converted to an increase in the temperature of the water. What do you conclude from this calculation? Solution. The initial assumption in this problem is that both the density of the water and its temperature are constant. We confirm at the end of the problem that these are excellent assumptions (a procedure similar to the predictor-corrector approaches often used for numerical methods). We begin with a few preliminary calculations that are common to several parts of the problem. Geometry. If flow is visualized as moving from left to right, then the pipeline is at a 100 m elevation at its left end and terminates at an elevation of 100 + 0.04 (2500) = 200 m at its right edge, thus gaining 100 m of elevation head along its length. The hydraulic grade line—representing the distance above the pipe of the pressure head term P/y—is high above the pipe at the left edge and falls linearly to meet the pipe at its right edge because the pressure here is atmospheric. Properties. At 1O0C, the density of water p = 999.7 kg/m3, its bulk modulus K = pAp/ P/Ap = 2.26 GPa, and its specific heat C = 4187 J/(kg - 0C). The weight density is Y = pg = 9.81kN/m3. Based on an internal diameter of 27 in, or 0.686 m, the cross-sectional area of the pipe is A = 7^Di =-^ (0.686)2 = 0.370 m2 Based on a discharge Q= 1.80 mVs, the average velocity is Q 1.8OmVs „ 0 _ . T7 V = ¥ = O J 7 0 l r 7 = 4 - 87m/S Such a velocity value is higher than is typically allowed in most municipal work. 1. The velocity head is given by hv = ^- = 1.21 m Zg Thus, the following table can be completed:

Variable Pressure (MPa)

Expression P

Upstream

Downstream

3.0

0.0 0.0

Pressure head (m)

P/y

305.9

Elevation head (m)

z

100.0

200.0

405.9

200.0

407.1

201.2

Piezometric head (m) Total head (m)

P /y + z 2

P /y + z + v /2g

2. The head loss caused by friction is equal to the net decrease in total head over the length of the line. That is, hf = 407.1 - 201.2 = 205.9 m. Note that because this pipe is of uniform diameter, this value also could have been obtained from the piezometric head terms.

From the Darcy-Weisbach equation, we can obtain the following expression for the dimensionless/: hjD _ (205.9X0.686) U W/ v2 (2500)(1.21) ' 2* Alternatively, from the Hazen-Williams equation that Q = 0.278 C D263 (hf/L)°-54, we obtain the following for the dimensional C: J

c=

G 0.278 D2-63(h/L)0-54

=

1-8 _672 0.278(0.686)2-63(205.9/2500)°-54

These values would indicate a pipe in poor condition, probably in much need of repair or replacement. 3. In most problems involving steady flow, we assume that the compressibility of the water is negligible. This assumption is easily verified since the density change associated with the pressure change is easily computed. In the current problem, the pressure change is 3.0 MPa and the bulk modulus is 2200 MPa. Thus, by definition of the bulk modulus K, A p _ A P _ 3 ^00014 ~r7 T 2200 U'UU14 Clearly, even in this problem, with its unusually extreme pressure changes, the relative change in density is less than 0.2 percent. The density at the higher pressure (upstream) end of the pipe is P1 = p2 + Ap = 999.7 (1 + 0.0014) = 1001.1 kg/m3. Using the mass continuity equation, we have P(AV)1 - (pAV)2 In this case, we assume that the pipe is completely rigid and that the change in pressure results in a change in density only (in most applications, these terms are likely to be almost equally important). In addition, we assume that the velocity we've already calculated applies at the downstream end (i.e., at Location 2). Thus, the continuity equation requires D 999 7 ' = ^ = 487l(5oTT = 4-86m/s

y

Obviously, even in this case, the velocity and density changes are both negligible and the assumption of incompressible flow is an extremely good one. 4. Assuming that the flow is incompressible, the energy dissipated, Pd, can be computed using work done in moving the fluid through a change in piezometric flow (in fact, the head loss is nothing more than the energy dissipating per unit rate of weight of fluid transferred). Thus, Pd = lQhf Strictly speaking, this energy is not lost but is transferred to less available forms: typically, heat. Since energy is associated with the increase in temperature of the fluid, we

can easily estimate the increase in temperature of the fluid that would be associated with the dissipation of energy, assuming that all the heat is retained in the fluid. That is, Pf = pQ cAT = pgQhf. Solving for the temperature increase gives , (9.91m/s2)(205.9m) ^T= ^= 4187 J/(kg- 0 C) - 0-480C. We conclude that the assumption of isothermal flow also is an excellent one. 2.6.7 Pumps: Adding Energy to the Flow Although water is the most abundant substance found on the surface of the earth, its natural distribution seldom satisfies an engineer's partisan requirements. As a result, pumping both water and wastewater is often necessary to achieve the desired distribution of flow. In essence, a pump controls the flow by working on the flowing fluid, primarily by discharging water to a higher head at its discharge flange than is found at the pump inlet. The increased head is subsequently dissipated as frictional losses within the conduit or is delivered further downstream. This section provides a brief introduction to how pumps interact with pipe systems. Further details are found in Chap. 10. How exactly is the role of a pump quantified? The key definition is the total dynamic head (TDH) of the pump. This term describes the difference between the total energy on the discharge side compared with that on the suction side. In effect, the TDH HP is the difference between the absolute total head at the discharge and suction nozzle of the pump: that is, ( yi\ ( yi\ (2 26) M^H^i ' where hp = hydraulic grade line elevation (i.e., pressure-plus-elevation head with respect to a fixed datum), and subscripts d and s refer to delivery and suction flanges, respectively. Typically, the concern is how the TDH head varies with the discharge Q; for a pump, this H-Q relation is called the characteristic curve. What the TDH definition accomplishes can be appreciated better if we consider a typical pump system, such as the one shown in Fig. 2.9. In this relation, the Bernoulli equation relates what happens between Points 1 and 2 and between Points 3 and 4, but technically it cannot be applied between 2 and 3 because energy is added to the flow. However, the TDH definition spans this gap. To see this more clearly, the energy relation is written between Points 1 and 2 as

H5 = Hfs + hfs (2.27) where Hs is the head of the suction reservoir, HPS is the total head at the suction flange of the pump, and hfs is the friction loss in the suction line. Similarly, the energy relation is written between Points 3 and 4 as Hn = HD + hfd (2.28) where Hd is the head of the discharge reservoir, HPD is the head at the discharge flange of the pump, and hfd is the friction loss in the discharge line. If Eq. (2.27) is then added to Eq. (2.28), the result can be rearranged as Hpd - Hps = Hd-Hs + hfd + hfa

(2.29)

Pump FIGURE 2.9 Definition sketch for pump system relations. which can be rewritten using Eq. (2.26) as Hp = Hst + hf

(2.30)

where Hst is the total static lift and hf is the total friction loss. The total work done by the pump is equal to the energy required to lift the water from the lower reservoir to the higher reservoir plus the energy required to overcome friction losses in both the suction and discharge pipes. 2.6.8 Sample Application Including Pumps Problem. Two identical pumps are connected in parallel and are used to force water into the transmission/distribution pipeline system shown in Fig. 2.10. The elevations of the demand locations and the lengths of C = 120 pipe also are indicated. Local losses are negligible in this system and can be ignored. The demands are as follows: D1 = 1.2 m3/s, D2= 1.6 m3/s, and D3 = 2.2 m3/s. The head-discharge curve for a single pump is approximated by the equation H = 90 - 6Q1-70 1. What is the minimum diameter of commercially available pipe required for the 4.2 km length if a pressure head is to be maintained at a minimum of 40 m everywhere in the system? What is the total dynamic head of the pump and the total water horsepower supplied for this flow situation? 2. For the system designed in the previous questions the demand can shift as follows under certain emergency situations: D1 = 0.8 m3/s, D2 = 1.2 m3/s, and D3 = 4.2 m3/s. For this new demand distribution, can the system maintain a residual pressure head of 20 m in the system?

Solution. Total flow is Q1 = D1 + D2 + D3 = 1.2 + 1.6 + 2.2 = 5.0 mVs and, each pump will carry half of this flow: i.e., Qpump = Qt/2 = 2.5 mVs. The total dynamic head of the pump Hpump is Hpump = 90 - 6(2.5)" = 61.51 m which allows the total water power to be computed as Power = 2 (Qpump Hpumpj) = Q1 Hpumpj Thus, numerically, Power = fs.O —] (61.51m) f9810 m3 -^] = 3017 kW s ( ) ( J which is a huge value. The diameter J1 of the pipe that is 4.2 km long, the head loss Afy caused by friction for each pipe can be determined using the Hazen-Williams formula since the flow can be assumed to be in the hydraulic rough range. Because dl is unknown, A/*2, A/i3, and A^1 are calculated first. The site where the lowest pressure head occurs can be shown to be at Node 2 (i.e., the highest node in the system) as follows:

( 63 ]o34 _ ( 2.2 "p_ * ~ L3 (o.2780P.eJ ~ 8°° (o.278(120)(1.0672-«)J " 3'8° m

A/ 3

Because the head loss Ah3 is less than the gain in elevation of 10 m, downstream pressures increase; thus, Node 2 (at D2) will be critical in the sense of having the lowest pressure. Thus, if the pressure head at that node is greater than 40 m, a minimum pressure head of 40 m will certainly be maintained throughout the pipeline. Continuing with the calculations, XW54 1 N I OT54 1 I O fS SR M2 = L

* [o.278CcH

= 10

°° |o.278(120)(1.524M3)J

Pump

FIGURE 2.10 Example pipe and pump system.

= 23

°m

Now, the pressure head at Node 2 is hp2 = ZR + Hpump ~ AA1 - AA2 - Z2 = 40 m which implies that

AA1 = (z* - Z2) + Hpump - Ah2 - 40 - (240 - 255) + 61.51 - 2.30 - 40 = 4.21 m where z is the elevation and the subscripts R and 2 denote reservoir and Node 2, respectively. Thus, the minimum diameter J1 is A 5 '.°MM °( '°, 21 ..]*-"»• V 1^ n0.278(12O)^ T78M 9fh(4-505 JI y/

Finally, the minimum diameter (dl = 2.134 m) of the commercially available pipe is therefore 84 in. Under emergency conditions (e.g., with a fire flow), the total flow is Qt = D1 + D2 + D3 = 0.8 + 1.2 + 4.2 = 6.2 (m3/s). Note that with an increase in flow, the head lost resulting from friction increases while the head supplied by the pump decreases. Both these facts tend to make it difficult to meet pressure requirements while supplying large flows. More specifically, Hpump = 90- 6(3.iy.7 = 48.9Om and / 42 \ o34 3 = 800(o.278(120)(1.067H = ^ m Because this loss now exceeds the elevation change, Node 3 (at D3) now becomes critical in the system; minimum pressures now occur at the downstream end of the system. Other losses are / 54 \ 054 M =1000 * (o.278(120)(1.524p«) = 4'4 m and / 62 \o^4 M = 42 > °° (o.278(120)(2.134)^J = 4'6 m Thus, the pressure head at Node 3 is M

hp3 = fe - Z3) + Hpump - Ah1 - Ah2 - Ah3 = -5 + 48.9 - 12.6 - 4.4 - 4.6 = 22.3 m Clearly, a residual pressure head of 20 m is still available in the system under emergency situations, and the pressure requirement is still met, though with little to spare! 2.6.9

Networks—Linking Demand and Supply

In water supply and distribution applications, the pipes, pumps, and valves that make up the system are frequently connected into complex arrangements or networks. This topological complexity provides many advantages to the designer (e.g., flexibility, reliability, water quality), but it presents the analyst with a number of challenges. The essential problems associated with "linked" calculations in networks are discussed in Chap. 9.

2.7 QUASI-STEADY FLOW: SYSTEM OPERATION The hydraulics of pressurized flow is modified and adjusted according to the presence, location, size, and operation of storage reservoirs and pumping stations in the system. This section discusses the criteria for and the approach to these components, introducing the equations and methods that will be developed in later chapters. A common application of quasi-steady flow arises in reservoir engineering. In this case, the key step is to relate the rate of outflow O to the amount of water in the reservoir (i.e., its total volume or its depth). Although the inflow is usually a known function of time, Eq. (2.2) must be treated as a general first-order differential equation. However, the solution usually can be approximated efficiently by standard numerical techniques, such as the Runge-Kutta or Adams-type methods. This application is especially important when setting operating policy for spillways, dams, turbines, and reservoirs. One simple case is illustrated by the example below. Usually, reservoir routing problems are solved numerically, a fact necessitated by the arbitrary form of the input function to the storage system and the sometimes complex nature of the storage-outflow relation. However, there are occasions when the application is sufficiently simple to allow analytical solutions. Problem. A large water-filled reservoir has a constant free surface elevation of 100 m relative to a common datum. This reservoir is connected by a pipe (L = 50 m, D = 6 cm, and / = constant = 0.02, hf = /LV2^gD) to the bottom of a nearby vertical cylindrical tank that is 3 m in diameter. Both the reservoir and the tank are open to the atmosphere, and gravity-driven flow between them is established by opening a valve in the connecting pipeline. Neglecting all minor losses, determine the time T (in hours) required to raise the elevation of the water in the cylindrical tank from 75 m to 80 m. Solution. If we neglect minor losses and the velocity head term, the energy equation can be written between the supply reservoir and the finite area tank. Letting the level of the upstream reservoir be /zr, the variable level of the downstream reservoir above datum being h and the friction losses being hf, the energy equation takes on the following simple form: hr = h + hf

This energy relation is called quasi-steady because it does not directly account for any transient terms (i.e., terms that explicitly depend on time). A more useful expression is obtained if we use the Darcy-Weisbach equation to relate energy losses to the discharge Q = VA:

h _ /^ v2 _ ^ Q2 _ 8/L f D 2g D 2gA2 gn2D5 * What is significant about this expression, however, is that all the terms involved in the last fraction are known and can be treated as a single constant. Thus, we can solve for Q and rewrite it as Q = CVhr - h, where C2 = gK2D5/8/L Thus far, we have a single equation involving two unknowns: the head h in the receiving tank and the discharge Q between them. A second relation is required and is given by the continuity equation. Because the flow can be treated as incompressible, the discharge in the tank (i.e., the tank's area At times its velocity of dh/dt) must equal the discharge in the pipe Q. in symbols,

*5-« Thus, using the energy equation, we have, *L = £Vh~^h dt A, ' Separating variables and integrating gives K * [C^ i, Vv^ -U and performing the integration and using appropriate limits gives

2 (Vh-^1 -V£^)-£r Finally, solving for t gives the final required expression for quasi-steadyflowconnecting a finite-area tank to a constant head reservoir: t = ^[Vh^7Ts-Vh^h^ The numerical aspects are now straightforward: 6) 2( 6)5 C = \K(O°,° m-/s = 3.068(1O)- m-/s '1 T / -C = > V 8°:° O*l If /ir = 100 m, H1 = 75 m, h2 = 80 m, than we have

/ 2 J (3m) 2 \ . . < = ^3.068(10)-3m*Vsj (V^™ ~ V20m) = 2432.6 s

Converting to minutes, this gives a time of about 40.5 minutes (0.676 hr). In problems involving a slow change of the controlling variables, it is often simple to check the calculations. In the current case, a good approximation can be obtained by using the average driving head of 22.5 m (associated with an average tank depth of 77.5 m). This average head, in turn, determines the associated average velocity in the pipeline. Using this "equivalent" steady velocity allows one to estimate how much time is required to fill the tank by the required 5 m. The interested reader is urged to try this and to verify that this approximate time is actually relatively accurate in the current problem, being within 6 s of the "exact" calculation.

2.8 UNSTEADY FLOW: INTRODUCTION OFFLUID TRANSIENTS Hydraulic conditions in water distribution systems are in an almost continual state of change. Industrial and domestic users often alter their flow requirements while supply conditions undergo adjustment as water levels in reservoirs and storage tanks change or as pumps are turned off and on. Given this dynamic condition, it is perhaps surprising that steady state considerations have so dominated water and wastewater engineering. The following sections provide an introduction to unsteady flow in pipe systems—a topic that is neglected too often in pipeline work. The purpose is not too create a fluid transients expert but to set the stage for Chap. 12, which considers these matters in greater detail.

2.8.1

Importance of Water Hammer

Pressure pipe systems are subjected to a wide range of physical loads and operational requirements. For example, underground piping systems must withstand mechanical forces caused by fluid pressure, differential settlement, and concentrated loads. The pipe must tolerate a certain amount of abuse during construction, such as welding stresses and shock loads. In addition, the pipe must resist corrosion and various kinds of chemical attack. The internal pressure requirement is of special importance, not only because it directly influences the required wall thickness (and hence cost) of large pipes, but also because pipe manufacturers often characterize the mechanical strength of a pipeline by its pressure rating. The total force acting within a conduit is obtained by summing the steady state and waterhammer (transient) pressures in the line. Transient pressures are most important when the rate of flow is changed rapidly, such as by closing a valve or stopping a pump. Such disturbances, whether caused intentionally or by accident, create traveling pressure and velocity waves that may be of large magnitude. These transient pressures are superimposed on steady-state values to produce the total pressure load on a pipe. Most people have some experience with waterhammer effects. A common example is the banging or hammering noise sometimes heard when a water faucet is closed quickly. In fact, the mechanism in this simple example typifies all pipeline transients. The kinetic energy carried by the fluid is rapidly converted into strain energy in the pipe walls and fluid. The result is a pulse wave of abnormal pressure that travels along the pipe. The hammering sound indicates that a portion of the original kinetic energy is converted into acoustic form. This and other energy-transformation losses (such as fluid friction) cause the pressure wave to decay gradually until normal (steady) pressures and velocities are once again restored. It turns out that waterhammer phenomena are the direct means of achieving all changes in fluid velocity, gradual or sudden. The difference is that slow adjustments in velocity or pressures produce such small disturbances that the flow appears to change smoothly from one value to another. Yet, even in these cases of near equilibrium, it is traveling pressure waves that satisfy the conservation equations. To illustrate why this must be so, consider the steady continuity equation for the entire pipe. This law requires that the rate at which fluid leaves one end of a conduit must be equal to the rate at which it enters the other end. The coordination between what happens at the two ends of the pipeline is not achieved by chance or conspiracy. It is brought about by the same physical laws and material properties that cause disturbances to propagate in the transient case. If waterhammer waves were always small, the study of transient conditions would be of little interest to the pipeline engineer. This is not the case. Waterhammer waves are capable of breaking pipes and damaging equipment and have caused some spectacular pipeline failures. Rational design, especially of large pipelines, requires reliable transient analysis. There are several reasons why transient conditions are of special concern for large conduits. Not only is the cost of large pipes greater, but the required wall thickness is more sensitive to the pipe's pressure rating as well. Thus, poor design—whether it results in pipeline failure or the hidden costs of overdesign—can be extremely expensive for large pipes. Despite their intrinsic importance, transient considerations are frequently relegated to a secondary role when pipeline systems are designed or constructed. That is, only after the pipeline's profile, diameter, and design discharge have been chosen is any thought given to transient conditions. This practice is troublesome. First, the pipeline may not perform as expected, possibly causing large remedial expenses. Second, the

line may be overdesigned and thus unnecessarily expensive. This tendency to design for steady state conditions alone has been particularly common in the water supply industry. In addition, there has been a widely held misconception that complex arrangements of pipelines reflect or dampen waterhammer waves. Although wave reflections in pipe networks do occur, attenuation depends on many factors and cannot be guaranteed. Networks are not intrinsically better behaved than simple pipelines are, and some complex systems may respond even more severely to transient conditions (Karney and Mclnnis, 1990). The remainder of this chapter introduces, in a gentle and nonmathematical way, several important concepts relating to transient conditions. Although rigorous derivations and details are avoided, the discussion is physical and accurate. The goal is to answer two key questions: How do transients arise and propagate in a pipeline? and under what circumstances are transient conditions most severe? Transient conditions in pressure pipelines are modeled using either a "lumped" or "distributed" approach. In distributed systems, the fluid is assumed to be compressible, and the transient phenomena occur in the form of traveling waves propagating with a finite speed a. Such transients often occur in water supply pipes, power plant conduits, and industrial process lines. In a lumped system, by contrast, the flow is considered to be incompressible and the pipe walls are considered to be inelastic. Thus, the fluid behaves as a rigid body in that changes in pressure or velocity at one point in the flow system are assumed to change the flow elsewhere instantaneously. The lumped system approximation can be obtained either directly or in the limit as the wavespeed a becomes unbounded in the distributed model. The slow oscillating water level in a surge tank attached to a short conduit typifies a system in which the effects of compressibility are negligible. Although the problem of predicting transient conditions in a pipeline system is of considerable practical importance, many challenges face the would-be analyst. The governing partial differential equations describing the flow are nonlinear, the behavior of even commonly found hydraulic devices is complex, and data on the performance of systems are invariably difficult or expensive to obtain. The often-surprising character of pulse wave propagation in a pipeline only makes matters worse. Even the basic question of deciding whether conditions warrant transient analysis is often difficult to answer. For all these reasons, it is essential to have a clear physical grasp of transient behavior. 2.8.2 Cause of Transients In general, any change in mean flow conditions in a pipeline system initiates a sequence of transient waves. In practice, we are generally concerned with changes or actions that affect hydraulic devices or boundary conditions attached to the conduit. The majority of these devices are used to provide power to the system or to control the flow in some way. The following list illustrates how some transient conditions can originate, although not all of the them are discussed further here: 1. 2. 3. 4.

Changes in valve settings (accidental or planned; manual or automatic) Starting or stopping of either supply or booster pumps Changes in the demand conditions, including starting or arresting a fire flow Changes in reservoir level (e.g., waves on a water surface or the slow accumulation of depth with time) 5. Unstable device characteristics, including unstable pump characteristics, valve instabilities, the hunting of a turbine, and so on

6. Changes in transmission conditions, such as when a pipe breaks or buckles 7. Changes in thermal conditions (e.g., if the fluid freezes or if changes in properties are caused by temperature fluctuations) 8. Air release, accumulation, entrainment, or expulsion causing dramatic disturbances (e.g., a sudden release of air from a relief valve at a high point in the profile triggered by a passing vehicle); pressure changes in air chambers; rapid expulsion of air during filling operations 9. Transitions from open channel to pressure flow, such as during filling operations in pressure conduits or during storm events in sewers. 10. Additional transient events initiated by changes in turbine power loads in hydroelectric projects, draft-tube instabilities caused by vortexing, the action of reciprocating pumps, and the vibration of impellers or guide vanes in pumps, fans, or turbines 2.8.3

Physical Nature of Transient Flow

In pipeline work, many approximations and simplifications are required to understand the response of a pipe system following an initialization of a transient event. In essence, this is because the flow is both unsteady in the mean as well as turbulent. Many of these assumptions have been confirmed experimentally to the extent that the resulting models have provided adequate approximations of real flow behavior. Yet, it is wise to be skeptical about any assumption and be cautious about mathematical models. As we have stressed, any model only approximates reality to a greater or lesser extent. Still, even in cases where models perform poorly, they may be the best way of pinpointing sources of uncertainty and quantifying what is not understood. An air of mystery often surrounds the development, role, and significance of transient phenomena in closed conduits. Indeed, the complexity of the governing differential equations and the dynamic nature of a system's response can be intimidating to the novice. However, a considerable understanding of transient behavior can be obtained with only the barest knowledge about the properties of fluid and a few simple laws of conservation. When water flows or is contained in a closed conduit so that no free surface is present—for example, in a typical water supply line—the properties of the flowing fluid have some direct implications to the role and significance of transient conditions. For a water pipeline, two properties are especially significant: water's high density and its large bulk modulus (i.e., water is heavy and difficult to compress). Surprisingly, these two facts largely explain why transient conditions in a pipeline can be so dramatic (see also, Karney and Mclnnis, 1990): 2.8.3.1 Implication 1. Water has a high density. Because water has a high density (~ 1000 kg/m3) and because pipelines tend to be long, typical lines carry huge amounts of mass, momentum, and kinetic energy. To illustrate, assume that a pipeline with area A = 1.0 m2 and length L = 1000 m is carrying fluid with a velocity v = 2.0 m/s. The kinetic energy contained in this pipe is then KE = \ mv2 = \ pLAv2 » 2,000,000 J Now this is a relatively ordinary situation: the discharge is moderate and the pipe is not long. Yet the pipe still contains energy equivalent to, say, 10,000 fast balls or to a pickup truck falling from a 30-story office tower. Clearly, large work interactions are required to change the flow velocity in a pipeline from one value to another.

In addition to kinetic energy, a pipeline for liquid typically transports large amounts of mass and momentum as well. For example, the above pipeline contains 2(106) kg m/s of momentum. Such large values of momentum imply that correspondingly large forces are required to change flow conditions. (Further details can be found in Karney, 1990.) 2.8.3.2 Implication 2. Water is only slightly compressible. Because water is only slightly compressible, large head changes occur if even small amounts of fluid are forced into a pipeline. To explain the influence of compressibility in a simple way, consider Fig. 2.11, which depicts a piston at one end of a uniform pipe. If this piston is moved slowly, the volume containing the water will be altered and the confining pressure will change gradually as a result. Just how much the pressure will change depends on how the pipe itself responds to the increasing pressure. For example, the bulk modulus of water is defined as K = -^- « 2,070 MPa

(2.31)

Thus, if the density of the fluid is increased by as little as one-tenth of 1 percent, which is equivalent to moving the imagined piston a meter in a rigid pipe, the pressure will increase by about 200 m of head (i.e., 2 MPa). If the pipe is not rigid, pressure increases are shared between the pipe walls and the fluid, producing a smaller head change for a given motion of the piston. Typical values are shown in the plot in Fig. 2.11. For example, curve 2 indicates typical values for a steel pipe in which the elasticity of the pipe wall and the compressibility of the fluid are nearly equal; in this case, the head change for a given mass imbalance (piston motion) is about half its previous value. Note that it is important for the conduit to be full of fluid. For this reason, many options for accommodating changes in flow conditions are not available in pipelines that can be used in channels. Specifically, no work can be done to raise the fluid mass against gravity. Also note that any movement of the piston, no matter how slowly it is accomplished, must be accommodated by changes in the density of the fluid, the dimension of the conduit, or both. For a confined fluid, Cauchy and Mach numbers (relating speed of change to speed of disturbance propagation) are poor indexes of the importance of compressibility effects. 2.8.3.3 Implication 3. Local action and control. Suppose a valve or other device is placed at the downstream end of a series-connected pipe system carrying fluid at some steady-state velocity V0. If the setting of the valve is changed—suddenly say, for simplicity, the valve is instantly closed—the implications discussed above are combined in the pipeline to produce the transient response. We can reason as follows: The downstream valve can only act locally, providing a relationship between flow through the valve and the head loss across the valve. In the case of sudden closure, the discharge and velocity at the valve becomes zero the instant the valve is shut. However, for the fluid mass as a whole to be stopped, a decelerating force sufficient to eliminate the substantial momentum of the fluid must be applied. But how is such a force generated? We have already mentioned that gravity cannot help us because the fluid has no place to go. In fact, there is only one way to provide the required decelerating force—the fluid must be compressed sufficiently to generate an increase in pressure large enough to arrest the flow. Because water is heavy, the required force is large; however, since water is only slightly compressible, the wave or disturbance will travel quickly. In a system like the one shown in Fig. 2.11, a pressure wave of nearly 100 m would propagate up the pipeline at approximately 1000 m/s. In many ways, the response of the system we have described is typical. For closed conduit systems, the only available mechanism for controlling fluid flows is the propagations of shock waves resulting from the elasticity of the fluid and the pipeline. In essence, tran-

Pressure head (m)

Piston movement (m) FIGURE 2.11 Relation between piston motion ('mass imbalance') and head change in a closed conduit sient considerations cause us to look at the flow of fluid in a pipeline in a new way: For any flow, we consider not only its present significance but also how this condition was achieved and when it will change because, when change occurs, pressure pulses of high magnitude may be created that can burst or damage pipelines. Although this qualitative development is useful, more complicated systems and devices require sophisticated quantitative analysis. The next section briefly summarizes how more general relations can be obtained. (Greater detail is provided in Chap. 12.) 2.8.4 Equation of State-Wavespeed Relations In pipeline work, an equation of state is obtained by relating fluid pressure to density through compressibility relations. Specifically, the stresses in the wall of the pipe need to be related to the pressure and density of the fluid. The result is a relationship between the fluid and the properties of the pipe material and the speed at which shock waves are propagated (wavespeed or celerity). The most basic relation describing the wavespeed in an infinitely elasticly fluid is usually written as follows:

a = 7^1

(2.32)

where a is the wavespeed, y is the ratio of the specific heats for the fluid, K is the bulk modulus of the fluid, and p is the fluid density. If a fluid is contained in a rigid conduit, all changes in density will occur in the fluid and this relation still applies. The following comments relate to Eq. (2.32): 1. As fluid becomes more rigid, K increases and, hence, a increases. If the medium is assumed to be incompressible, the wavespeed becomes infinite and disturbances are transmitted instantaneously from one location to another. This is not, strictly speaking, possible, but at times it is a useful approximation when the speed of propagation is much greater than the speed at which boundary conditions respond. 2. For liquids that undergo little expansion on heating, y is nearly 1. For example, water at 1O0C has a specific heat ratio (y) of 1.001. 3. Certain changes in fluid conditions can have a drastic effect on celerity (or wavespeed) values. For example, small quantities of air in water (e.g., 1 part in 10,000 by volume) greatly reduce K, because gases are so much more compressible than liquids are at normal temperatures. However, density values (p) are affected only slightly by the presence of a small quantity of gas. Thus, wavespeed values for gas-liquid mixtures are often much lower than the wavespeed of either component taken alone. Example: Elastic Pipe The sonic velocity (a) of a wave traveling through an elastic pipe represents a convenient method of describing a number of physical properties relating to the fluid, the pipe material, and the method of pipe anchoring. A more general expression for the wavespeed is

a=

K / '9W V I+C1KDIEe

„ ~~.

(233)

where K is the bulk modulus of the fluid, pw is the density of the fluid, E is the elastic modulus of the pipe material, and D and e are the pipe's diameter and wall thickness, respectively. The constant C1 accounts for the type of support provided for the pipeline. Typically, three cases are recognized, with C1 defined for each as follows (fl is the Poison's ratio for the pipe material): Case a. The pipeline is anchored only at the upstream end: q = 1 - £

(2.34)

Case b. The pipeline is anchored against longitudinal movement. C1 = 1 - j|2 Case c. The pipeline has expansion joints throughout.

(2.35)

C1 = 1

(2.36)

Note that for pipes that are extremely rigid, thick-walled, or both, C1KDIEe —> O and Eq. 2.33 can be simplified to a = VA7pw that which recovers the expression for the acoustic wavespeed in an infinite fluid (assuming y = 1). For the majority of transient applications, the wavespeed can be regarded as constant. Even in cases where some uncertainty exists regarding the wavespeed, the solutions of the governing equations, with respect to peak pressures, are relatively insensitive to changes in this parameter. It is not unusual to vary the wave celerity deliberately by as much as ± 15 percent to maintain a constant time step for solution by standard numerical techniques (Wylie and Streeter, 1993). (Again, further details are found in Chap. 12.) Wavespeeds are sensitive to a wide range of environmental and material conditions. For example, special linings or confinement conditions (e.g., tunnels); variations in material properties with time, temperature, or composition; and the magnitude and sign of the pressure wave can all influence the wavespeed in a pipeline. (For additional details, see Wylie and Streeter, 1993; Chaudhry, 1987; or Hodgson, 1983.) 2.8.5 Increment of Head-Change Relation Three physical relations—Newton's second law, conservation of mass and the wavespeed relation—can be combined to produce the governing equations for transient flow in a pipeline. The general result is a set of differential equations for which no analytical solution exists. It is these relations that are solved numerically in a numerical waterhammer program. In some applications, a simplified equation is sometimes used to obtain a first approximation of the transient response of a pipe system. This simple relation is derived with the assumption that head losses caused by friction are negligible and that no interaction takes place between pressure waves and boundary conditions found at the end of pipe lengths. The resulting head rise equation is called the Joukowsky relation: AH= ± --AV (2.37) 8 where AH is the head rise, AV is the change in velocity in the pipe, a is the wavespeed, and g is the acceleration caused by gravity. The negative sign in this equation is applicable for a disturbance propagating upstream and the positive sign is for one moving downstream. Because typical values of alg are large, often 100 s or more, this relation predicts large values of head rise. For example, a head rise of 100 m occurs in a pipeline if alg = 100 s and if an initial velocity of 1 m/s is suddenly arrested at the downstream end. Unfortunately, the Joukowsky relation is misleading in a number of respects. If the equation is studied, it seems to imply that the following relations are true: 1. The greater the initial velocity (hence, the larger the maximum possible AV), the greater the transient pressure response. 2. The greater the wavespeed a, the more dramatic the head change. 3. Anything that might lower the static heads in the system (such as low reservoir levels or large head losses caused by friction) will tend to lower the total head (static plus dynamic) a pipe system is subject to. Although these implications are true when suitable restrictions on their application are enforced, all of them can be false or misleading in more complicated hydraulic systems.

It is important to be skeptical about simple rules for identifying "worst case" scenarios in transient applications. Karney and Mclnnis (1990) provide further elaboration of this point. However, before considering even a part of this complexity, one must clarify the most basic ideas in simple systems. 2.8.6

Transient Conditions in Valves

Many special devices have been developed to control and manage flows in pipeline systems. This is not surprising because the inability to control the passage of water in a pipeline can have many consequences, ranging from the minor inconvenience of restrictive operating rules to the major economic loss associated with pipeline failure or rupture. The severity of the problem clearly depends on the timing and magnitude of the failure. In essence, control valves function by introducing a specified and predictable relationship between discharge and pressure drop in a line. When the setting of a valve (or, for that matter, the speed of a pump) is altered, either automatically or by manual action, it is the head-discharge relationship that is controlled to give the desired flow characteristics. The result of the change may be to increase or reduce the pressure or discharge, maintain a preset pressure or flow, or respond to an emergency or unusual condition in the system. It is a valve control function that creates most difficulties encountered by pipeline designers and system operators. Valves control the rate of flow by restricting the passage of the flow, thereby inducing the fluid to accelerate to a high velocity as it passes through the valve even under steady conditions. The large velocities combine with the no-slip condition at the solid boundaries to create steep velocity gradients and associated high shear stresses in the fluid; in turn, these shear stresses, promote the rapid conversion of mechanical energy into heat through the action of turbulence of the fluid in the valve. The net result is a large pressure drop across the valve for a given discharge through it; it is this A/z- 1, the flow is in a supercritical state and the inertial forces are dominant. From a practical perspective, sub - and supercritical flow can be differentiated simply by throwing a rock or other object into the flow. If ripples from the rock

TABLE 3.1 Channel Section Geometric Properties Channel Definition

Rectangle

Trapezoid with equal side slopes Trapezoid with unequal side slopes Triangle with equal side slopes Triangle with unequal side slopes Circular

Area

Wetted Perimeter

Hydraulic Radius

Top Width

Hydraulic Depth

progress upstream of the point of impact, the flow is subcritical; however, if ripples from the rock do not progress upstream but are swept downstream, the flow is supercritical. Hydraulic depth (D). The hydraulic depth is the ratio of the flow area (A) to the top width (T)OtD= A/T (Table 3.1). Hydraulic radius (R). The hydraulic radius is the ratio of the flow area (A) to the wetted perimeter (P) or R = A/P (Table 3.1). Kinetic energy correction factor (a). Since no real open-channel flow is one-dimensional, the true kinetic energy at a cross section is not necessarily equal to the spatially averaged energy. To account for this, the kinetic energy correction factor is introduced, or

K^) "-"^*

and solving for a,

-^ S0

for

y < yN

Fr > 1

for

y < yc

SfyN

Fr < 1

for

y > yc

and

These inequalities divide the channel into three zones in the vertical dimension. By convention, these zones are labeled 1 to 3 starting at the top. Gradually varied flow profiles are labeled according to the scheme defined in Table 3.5. For a channel of arbitrary shape, the standard step methodology of calculating the gradually varied flow profile is commonly used: for example, HEC-2 (USAGE, 1990) or HECRAS (USAGE, 1997). The use of this methodology is subject to the following assumptions: (1) steady flow, (2) gradually varied flow, (3) one-dimensional flow with correction for the horizontal velocity distribution, (4) small channel slope, (5) friction slope (averaged) constant between two adjacent cross sections, and (6) rigid boundary conditions. The application of the energy equation between the two stations shown in Fig. 3.7 yields

TABLE 3.5 Classifications of Gradually Varied Flow Profiles Profile Designation Channel Slope

Zone 1

Zone 2

Zone 3

(1)

(2)

(3)

(4)

Mild O < S0 < Sc

Ml

y>yN>yc Ml

yN>y>yc M3

Critical S0 = SC>0

Cl

yN>yc>y y>yc = yN

C2

y = yN = yc C3

Steep S0 > Sc > O

Relation ofy to yN and yc (5)

Sl

yc = yN>y y>yc>yN

52

yc>y>yN

Type of Curve (6) Backwater (dy/dx > O) Drawdown (dy/dx < O) Backwater (dy/dx > O) Backwater (dy/dx > O) Parallel to channel bottom (dy/dx = O) Backwater (dy/dx > O) Backwater (dy/dx > O) Drawdown (dy/dx < O)

Type of Flow (7) Subcritical Subcritical Supercritical Subcritical Uniform critical Supercritical Subcritical Supercritical

TABLE 3.5:

(Continued)

Channel Slope

Zone 1

Zone 2

Zone 3

(1)

(2)

(3)

Horizontal

(4)

Relation ofy to yN and yc (5)

Type of Curve (6)

Type of Flow (7)

S3

yc>yN>y

Backwater (dy/dx > O)

Supercritical

yN>y>yc

Drawdown (dy/dx < O) Backwater (dy/dx > O)

Subcritical

Drawdown (dy/dx < O) Backwater (dy/dx > O)

Subcritical

None

So = ®

m m Adverse S0 < O

yN>yc>y

Supercritical

None A2

yN>y>yc A3

)w > X: > y

Supercritical

Datum

Section 1

Section 2

FIGURE 3.7 Energy relationship between two channel sections.

yi V2 Z1 + Gi1^- = z2 + a2^ + hf+he

(3.61)

where Z1 and Z2 = elevation of the water surface above a datum at Stations 1 and 2, respectively, he = eddy and other losses incurred in the reach, and hf = reach friction loss. The friction loss can be obtained by multiplying a representative friction slope, Sf, by the length of the reach, L. Four equations can be used to approximate the friction loss between two cross sections: Sf

=

( Q. + Q2 > ~v v~ (average conveyance) [K1 + K2 )

(3.62)

— S +S Sf = ^ —— (average friction slope)

(3.63)

— 2S S Sf = ——^ ,f (harmonic mean friction slope) Sn + Sj2

(3.64)

~Sf = VSy1 Sfl (geometric mean friction slope)

(3.65)

and

The selection of a method to estimate the friction slope in a reach is an important decision and has been discussed in the literature. Laurenson (1986) suggested that the "true" friction slope for an irregular cross section can be approximated by a third-degree poly-

nomial. He concluded that the average friction slope method produces the smallest maximum error, but not always the smallest error, and recommended its general use along with the systematic location of cross sections. Another investigation based on the analysis of 98 sets of natural channel data showed that there could be significant differences in the results when different methods of estimating the friction slope were used (USAGE, 1986). This study also showed that spacing cross sections 15Om (500 ft) a part eliminated the differences. The eddy loss takes into account cross section contractions and expansions by multiplying the absolute difference in velocity heads between the two sections by a contraction or expansion coefficient, or h =c

*

*^Ts~^Tg

(3 66)

'

There is little generalized information regarding the value of the expansion (C6) or the contraction coefficient (Cc). When the change in the channel cross section is small, the coefficients C6 and Cc are typically on the order of 0.3 and 0.1, respectively (USAGE, 1990). However, when the change in the channel cross section is abrupt, such as at bridges, C6 and Cc may be as high as 1.0 and 0.6, respectively (USAGE, 1990). With these comments in mind, H1= Z1+ U1^

(3.67)

H2 = z2 + Oc2 ^J

(3.68)

and

With these definitions, Eq. (3.61) becomes H1 = H2 + hf + he

(3.69)

Eq. (3.69) is solved by trial and error: that is, assuming H2 is known and given a longitudinal distance, a water surface elevation at Station 1 is assumed, which allows the computation of H1 by Eq. (3.67). Then, hf and he are computed and H1 is estimated by Eq. (3.67). If the two values of H1 agree, then the assumed water surface elevation at Station 1 is correct. Gradually varied water surface profiles are often used in conjunction with the peak flood flows to delineate areas of inundation. The underlying assumption of using a steady flow approach in an unsteady situation is that flood waves rise and fall gradually. This assumption is of course not valid in areas subject to flash flooding such as the arid and semiarid Southwestern United States (French, 1987). In summary, the following principles regarding gradually varied flow profiles can be stated: 1. The sign of dy/dx can be determined from Table 3.6. 2. When the water surface profile approaches normal depth, it does so asymptotically. 3. When the water surface profile approaches critical depth, it crosses this depth at a large but finite angle. 4. If the flow is subcritical upstream but passes through critical depth, then the feature that produces critical depth determines and locates the complete water surface profile. If the upstream flow is supercritical, then the control cannot come from the downstream.

5. Every gradually varied flow profile exemplifies the principle that subcritical flows are controlled from the downstream while supercritical flows are controlled from upstream. Gradually varied flow profiles would not exist if it were not for the upstream and downstream controls. 6. In channels with horizontal and adverse slopes, the term "normal depth of flow" has no meaning because the normal depth of flow is either negative or imaginary. However, in these cases, the numerator of Eq. (3.60) is negative and the shape of the profile can be deduced. Any method of solving a gradually varied flow situation requires that cross sections be defined. Hoggan (1989) provided the following guidelines regarding the location of cross sections: 1. They are needed where there is a significant change in flow area, roughness, or longitudinal slope. 2. They should be located normal to the flow. 3. They should be located in detail—upstream, within the structure, and downstreamat structures such as bridges and culverts. They are needed at all control structures. 4. They are needed at the beginning and end of reaches with levees. 5. They should be located immediately below a confluence on a main stem and immediately above the confluence on a tributary. 6. More cross sections are needed to define energy losses in urban areas, channels with steep slopes, and small streams than needed in other situations. 7. In the case of HEC-2, reach lengths should be limited to a maximum distance of 0.5 mi for wide floodplains and for slopes less than 38,550 m (1800 ft) for slopes equal to or less than 0.00057, and 370 m (1200 ft) for slopes greater than 0.00057 (Beaseley, 1973).

3.6

GRADUALLY AND RAPIDLY VARIED UNSTEADY FLOW

3.6.1 Gradually Varied Unsteady Flow Many important open-channel flow phenomena involve flows that are unsteady. Although a limited number of gradually varied unsteady flow problems can be solved analytically, most problems in this category require a numerical solution of the governing equations. Examples of gradually varied unsteady flows include flood waves, tidal flows, and waves generated by the slow operation of control structures, such as sluice gates and navigational locks. The mathematical models available to treat gradually varied unsteady flow problems are generally divided into two categories: models that solve the complete Saint Venant equations and models that solve various approximations of the Saint Venant equations. Among the simplified models of unsteady flow are the kinematic wave, and the diffusion analogy. The complete solution of the Saint Venant equations requires that the equations be solved by either finite difference or finite element approximations. The one dimensional Saint Venant equations consist of the equation of continuity * + * + „* = dt dx dx

(3.7Oa)

and the conservation of momentum equation ^v *

3v ^v Tx + 8irx-8(S°-s')

+v

=0

(3 71a)

-

An alternate form of the continuity and momentum equations is r|y +

dr

a^) = 0 dJC

and

I*+vav g 3f g 3*

* dx

f

0

By rearranging terms, Eq. (3.7Ib) can be written to indicate the significance of each term for a particular type of flow, or \ -^ ^fS=S ^-"steady ~\ - f ^ l ^ ~J lJ steady, nonuniform - f -p1 — /)f Iunsteady, nonuniform

(372) \^''^)

Equations (3.70) and (3.71) compose a group of gradually varied unsteadyflowmodels that are termed complete dynamic models. Being complete, this group of models can provide accurate results; however, in many applications, simplifying assumptions regarding the relative importance of various terms in the conservation of momentum equation (Eq. 3.71) leads to other equations, such the kinematic and diffusive wave models (Ponce, 1989). The governing equation for the kinematic wave model is f

+

^f = 0

(3.73)

where e = a coefficient whose value depends on the frictional resistance equation used (e = 5/3 when Manning's equation is used). The kinematic wave model is based on the equation of continuity and results in a wave being translated downstream. The kinematic wave approximation is valid when

t SV -^- > 85

(3.74)

where tR = time of rise of the inflow hydrograph (Ponce, 1989). The governing equation for the diffusive wave model is 3(2 dQ ( Q Id 2 G + v 4 « >£=(*k}i£

(3 75)

-

where the left side of the equation is the kinematic wave model and the right side accounts for the physical diffusion in a natural channel. The diffusion wave approximation is valid when (Ponce, 1989), **So (y) 0 ' 5 * !5

(3.76)

If the foregoing dimensionless inequalities ( Eq. 3.74 and 3.76) are not satisfied, then the complete dynamic wave model must be used. A number of numerical methods can be used to solve these equations (Chaudhry, 1987; French; 1985, Henderson, 1966; Ponce, 1989).

3.6.2 Rapidly Varied Unsteady Flow The terminology "rapidly varied unsteady flow" refers to flows in which the curvature of the wave profile is large, the change of the depth of flow with time is rapid, the vertical acceleration of the water particles is significant relative to the total acceleration, and the effect of boundary friction can be ignored. Examples of rapidly varied unsteady flow include the catastrophic failure of dams, tidal bores, and surges that result from the quick operation of control structures such as sluice gates. A surge producing an increase in depth is termed a positive surge, and one that causes a decrease in depth is termed a negative surge. Furthermore, surges can go either upstream or downstream, thus giving rise to four basic types (Fig. 3.8). Positive surges generally have steep fronts, often with rollers, and are stable. In contrast, negative surges are unstable, and their form changes with the advance of the wave. Consider the case of a positive surge (or wave) traveling at a constant velocity (wave celerity) c up a horizontal channel of arbitrary shape (Fig. 3.8b). Such a situation can result from the rapid closure of a downstream sluice gate. This unsteady situation is converted to a steady situation by applying a velocity c to all sections; that is, the coordinate system is moving at the velocity of the wave. Applying the continuity equation between Sections 1 and 2 (V1 + C)A1 = (V2 + c)A2

(3.77)

Since there are unknown losses associated with the wave, the momentum equation rather than the energy equation is applied between Sections 1 and 2 or

FIGURE 3.8 Definition of variables for simple surges moving in an open channel.

FIGURE 3.8 Definition of variables for simple surges moving in an open channel.

YA1Z1 - 7A2Z2 = 1 ^1(V1 + C)(V2 + c - V . - c ) (3.78) O where boundary friction has been ignored. Eliminating V2 in Eq. (3.78) by manipulation ofEq. (3.77) yields " ( A2 } ~|°-5 8\-fA (A 1 Z 1 -A 2 Z 2 ) V i ) V1 + c = Y^A L i A2

(3.79)

In the case of a rectangular channel, Eq. (3.79) reduces to Vl + c _ V 5 - [ i ( 1 + a ) ] «

When the slope of a channel becomes very steep, the resulting supercritical flow at normal depth may develop into a series of shallow water waves known as roll waves. As these waves progress downstream, they eventually break and form hydraulic bores or shock waves. When this type of flow occurs, the increased depth of flow requires increased freeboard, and the concentrated mass of the wavefronts may require additional structural factors of safety. Escoffier (1950) and Escoffier and Boyd (1962) considered the theoretical conditions under which a uniform flow must be considered unstable. Whether roll waves form or not is a function of the Vedernikov number (Ve), the Montuori number (Mo), and the concentration of sediment in the flow. When the Manning equation is used, the Ve is

TABLE 3.6 Shape Factor for Common Channel Sections Channel Definition

Rectangle Trapezoid with unequal side slopes

Circle

Ve = |rFr

(3.8Ia)

Ve = IfFr

(3.8Ib)

and if the Chezy equation is used

Fr should be computed using Eq. (3.11) and F = a channel shape factor (Table 3.6) or F = 1 -R^ dA

(3.82)

When Mo » 1, flow instabilities should expected. The Montuori number is given by Mo = ^

(3.83)

It is appropriate to note that in some publications (e.g., Aisenbrey et al., 1978) Mo is the inverse of Eq. (3.83). Figure 3.9 provides a basis for deciding whether roll waves will form in a given situation. In the figure, data from Niepelt and Locher (1989) for a slurry flow are also plotted. The Niepelt and Locher data suggest that flow stability also is a function of the concentration of sediment.

3.7

CONCLUSION

The foregoing sections provide the basic principles on which the following chapters on design are based. Two observations are pertinent. First, open-channel hydraulics is incrementally progressing. That is, over the past several decades, there have been incremental advances that primarily have added details, often important details, but no major new advances. Second, open-channel hydraulics remains a one-dimensional analytic approach. However, the assumption of a one-dimensional approach may not be valid in many situations: for example, nonprismatic channels, flow downstream of a partially breached dam, or lateral flow over a spillway. In some of these cases, the one-dimensional approach may provide an approximation that is suitable for design. In other cases, however, a two- or three- dimensional approach should be used. Additional information regarding two- and three- dimensional aDDroaches can be found in Chaudhrv (1993).

Flows with roll-waves Flows without roll-waves Clear Water

—

Slurry Curve - 43% Concentration by Weight

Montuor! Number = *&FIGURE 3.9 Flow stability as a function of the Vedernikov and Montuori numbers for clear water and slurryflow.(Based on data from Montuori, 1963; Niepelt and Locher, 1989

REFERENCES Ackers, P., "Resistance to Fluids flowing in Channels and Pipes," Hydraulic Research Paper No. 1, Her Majety's Stationery Office, London, 1958. Aisenbrey, A. J., Jr., R. B., Hayes, H. J., Warren, D. L., Winsett, and R. B. Young, Design of Small Canal Structures, U.S. Department of Interior, Bureau of Reclamation, Washington, DC 1978. Arcement, G. J., and V. R. Schneider, "Guide for Selecting Manning's Roughness Coefficients for Natural Channels and Flood Plains," Water Supply Paper 2339, U.S. Geological Survey, Washington, DC, 1989. Barnes, H. H., "Roughness Characteristics of Natural Channels," U.S. Geological Survey Water Supply Paper No. 1849, U.S. Geological Survey, Washington, DC. 1967. Beasley, J. G., An Investigation of the Data Requirements of Ohio for the HEC-2 Water Surface Profiles Model, Master's thesis, Ohio State University, Columbus, 1973. Chaudhry, M. H., Open-Channel Flow, Prentice-Hall, New York 1993. Chaudhry, M. H., Applied Hydraulic Transients, Van Nostrand Reinhold, New York, 1987. Chow, V. T., Open-Channel Hydraulics, McGraw-Hill, New York, 1959. Cox, R. G., "Effective Hydraulic Roughness for Channels Having Bed Roughness Different from Bank Roughness," Miscellaneous Paper H-73-2, U.S. Army Engineers Waterways Experiment Station, Vicksburg, MS, 1973. Coyle, J. J. "Grassed Waterways and Outlets," Engineering Field Manual, U.S. Soil Conservation Service, Washington, DC, April, 1975, pp. 7-1-7-43. Einstein, H. A., "The Bed Load Function for Sediment Transport in Open Channel Flows. Technical Bulletin No. 1026, U.S. Department of Agriculture, Washington, DC, 1950. Einstein, H. A., and R. B. Banks, "Fluid Resistance of Composite Roughness," Transactions of the American Geophysical Union, 31(4): 603-610, 1950. Escoffier, F. F., "A Graphical Method for Investigating the Stability of Flow in Open Channels or in Closed Conduits Flowing Full," Transactions of the American Geophysical Union, 31(4), 1950. Escoffier, F. F, and M. B. Boyd, "Stability Aspects of Flow in Open Channels," Journal of the Hydraulics Division, American Society of Civil Engineers, 88(HY6): 145-166, 1962. Fischenich, J. C., "Hydraulic Impacts of Riparian Vegetation: Computation of Resistance," EIRP Technical Report EL-96-XX, U.S. Army Engineers Waterways Experiment Station, Vicksburg, MS, August 1996. Flippin-Dudley, S. J., "Vegetation Measurements for Estimating Flow Resistance," Doctoral dissertation, Colorado State University, Fort Collins, 1997. Flippin-Dudley, S. J., S. R. Abt, C. D. Bonham, C. C. Watson, and J. C. Fischenich, "A Point Quadrant Method of Vegetation Measurement for Estimating Flow Resistance," Technical Report No. EL-97-XX, U.S. Army Engineers Waterways Experiment Station, Vicksburg, MS, 1997. French, R. H., Hydraulic Processes on Alluvial Fans. Elsevier, Amsterdam, 1987. French, R. H., Open-Channel Hydraulics, McGraw-Hill, New York, 1985. Guy, H. P., D. B. Simons, and E. V. Richardson, "Summary of Alluvial Channel Data from Flume Experiments, 1956-61," Professional Paper No. 462-1, U.S. Geological Survey, Washington, DC, 1966. Henderson, F. M., Open Channel Flow, Macmillan, New York, 1966. Hoggan, D. H., Computer-Assisted Floodplain Hydrology & Hydraulics, McGraw-Hill, New York, 1989. Horton, R. E., "Separate Roughness Coefficients for Channel Bottom and Sides," Engineering News Record, 3(22): 652-653, 1933.

Jarrett, R. D., "Hydraulics of High-Gradient Streams," Journal of Hydraulic Engineering, American Society of Civil Engineers, 110(11): 1519-1539, 1984. Kouwen, N., "Field Estimation of the Biomechanical Properties of Grass," Journal of Hydraulic Research, International Association and Hydraulic Research, 26(5): 559-568, 1988. Kouwen, N., and R. Li, "Biomechanics of Vegetative Channel Linings," Journal of the Hydraulics Division, American Society of Civil Engineers, 106(HY6): 1085-1103, 1980. Lane, E. W., and E. J. Carlson, "Some Factors Affecting the Stability of Canals Constructed in Coarse Granular Materials," Proceedings of the Minnesota International Hydraulics Convention, September 1953. Laurenson, E. M., "Friction Slope Averaging in Backwater Calculations," Journal of Hydraulic Engineering, American Society of Civil Engineers 112(12),1151-1163 1986. Levi, E., The Science of Water: The Foundation of Modern Hydraulics, Translated from the Spanish by D. E. Medina, ASCE Press, New York, 1995. Linsley, R. K. and J. B. Franzini, Water Resources Engineering, 3rd ed., Mc-Graw-Hill, New York, 1979. Meyer-Peter, P. E., and R. Muller, "Formulas for Bed Load Transport," Proceedings of the 3rd International Association for Hydraulic Research, Stockholm, 1948, pp. 39-64. Montuori, C., Discussion of "Stability Aspects of Flow in Open Channels," Journal of the Hydraulics Division, American Society of Civil Engineers 89(HY4): 264-273, 1963. Nagami, M., R. Scavarda, G. Pederson, G. Drogin, D. Chenoweth, C. Chow, and M. Villa, Design Manual: Hydraulic, Design Division, Los Angeles County Flood Control District, Los Angeles, CA, 1982. Niepelt, W. A., and F. A. Locher, "Instability in High Velocity Slurry Flows, Mining Engineering, Society for Mining, Metallurgy and Exploration, 1989, pp. 1204-1209. Petryk, S., and G. Bosmajian, "Analysis of Flow Through Vegetation," Journal of the Hydraulics Division, American Society of Civil Engineers, 101(HY7): 871-884, 1975. Ponce, VM., Engineering Hydrology: Principles and Practices, Prentice-Hall, Englewood Cliffs, NJ, 1989. Richardson, E. V, D. B.Simons, and P. Y. Mien, Highways in the River Environment, U.S. Department of Transportation, Federal Highway Administration, Washington, DC, 1987. Silvester, R., "Theory and Experiment on the Hydraulic Jump," Proceedings of the 2nd Australasian Conference on Hydraulics and Fluid Mechanics, 1965, pp. A25-A39. Silvester, R., "Hydraulic Jump in All Shapes of Horizontal Channels," Journal of the Hydraulics Division, American Society of Civil Engineers, 90(HYl): 23-55, 1964. Simons, D. B., and E. V. Richardson, "Resistance to Flow in Alluvial Ch°annels," Professional Paper 422-J, U.S. Geological Survey, Washington, DC, 1966. Simons, Li & Associates, SLA Engineering Analysis of Fluvial Systems, Fort Collins, CO, 1982. Straub, W. O. "A Quick and Easy Way to Calculate Critical and Conjugate Depths in Circular Open Channels," Civil Engineering, 70-71, December 1978. Straub, W. O., Personal Communication, Civil Engineering Associate, Department of Water and Power, City of Los Angeles, January 13, 1982. Temple, D. M., "Changes in Vegetal Flow Resistance During Long-Duration Flows," Transactions of the ASAE, 34: 1769-1774, 1991. Urquhart, W. J. "Hydraulics," in Engineering Field Manual, U.S. Department of Agriculture, Soil Conservation Service, Washington, DC, 1975. U.S Army Corps Engineers, HEC-RAS River Analysis System, User's Manual, U.S. Army Corps of Engineers Hydrologic Engineering Center, Davis, CA, 1997. U.S Army Corps Engineers, "HEC-2, Water Surface Profiles, Useris Manual," U.S. Army Corps of Engineers Hydrologic Engineering Center, Davis, CA, 1990.

U.S Army Corps Engineers, "Accuracy of Computed Water Surface Profiles," U.S. Army Corps of Engineers Hydrologic Engineering Center, Davis, CA, 1986. U.S Army Corps Engineers, "Hydraulic Design of Flood Control Channels," EM 1110-2-1601. U.S. Army Corps of Engineers, Washington, DC, 1970. Yang, C. T., Sediment Transport: Theory and Practice, McGraw-Hill, New York, 1996. Zegzhda, A.P., Theroiia Podobija Metodika Rascheta Gidrotekhnickeskikh Modele (Theory of Similarity and Methods of Design of Models for Hydraulic Engineering), Gosstroiizdat, Leningrad, 1938.

APPENDIX 3.A VALUES OF THE ROUGHNESS COEFFICIENT n

Values of the Roughness Coefficient n* Type of Channel

Minimum

A Closed Conduits flowing partly full A-I Metal a. Brass, smooth 0.009 b. Steel 1 . Lockbar and welded 0.010 2. Riveted and spiral 0.013 c. Cast iron 1. Coated 0.010 2. Uncoated 0.011 d. Wrought iron !.Black 0.012 2. Galvanized 0.013 e. Corrugated metal 1. Subdrain 0.017 2. Storm drain 0.021 A-2 Non-metal a. Lucite 0.008 b. Glass 0.009 c. Cement 1. Neat, surface 0.010 2. Mortar 0.011 d. Concrete !.Culvert, straight and free of debris 0.010 2. Culvert, with bends, connections, and some debris 0.011 3. Finished 0.011 4. Sewer and manholes, inlet, etc, straight 0.013 5. Unfinished, steel form 0.012 6. Unfinished, smooth wood form 0.01 2 7. Unfinished, rough wood form 0.015 e. Wood 1. Stave 0.010 2. Laminated, treated 0.015

Normal

Maximum

0.010

0.013

0.012 0.016

0.014 0.017

0.013 0.014

0.014 0.016

0.014 0.016

0.015 0.017

0.019 0.024

0.030 0.030

0.009 0.010

0.010 0.013

0.011 0.013

0.013 0.015

0.011

0.013

0.013 0.012

0.014 0.014

0.015 0.013 0.014

0.017 0.014 0.016

0.017

0.020

0.012 0.017

0.014 0.020

Type of Channel f. Clay 1 . Common drainage tile 2. Vitrified sewer 3. Vitrified sewer with manholes, inlet, etc. 4. Vitrified subdrain with open joint g. Brickwork 1. Glazed 2. Lined with cement mortar h. Sanitary sewers coated with sewage slimes with bends and connections i. Paved invert, sewer, smooth bottom j. Rubble masonry, cemented k.Polyethylene pipe 1. Polyvinyl chloride B. Lined or Built-up Channels B-I Metal a. Smooth steel surface 1. Unpainted 2. Painted b. Corrugated B-2 Nonmetal a. Cement 1. Neat, surface 2. Mortar b. Wood 1 . Planed, untreated 2. Planed, creosoted 3. Unplaned 4. Plank with battens 5. Lined with roofing paper c. Concrete 1. Trowel finish 2. Float finish 3. Finished, with gravel on bottom 4. Unfinished 5. Gunite, good section 6. Gunite, wavy section 7. On good excavated rock 8. On irregular excavated rock

Minimum

Normal

Maximum

0.011 0.011

0.013 0.014

0.017 0.017

0.013

0.015

0.017

0.014

0.016

0.018

0.011 0.012

0.013 0.015

0.015 0.017

0.012

0.013

0.016

0.016 0.018 0.009 0.010

0.019 0.025

0.020 0.030

0.011 0.012 0.021

0.012 0.013 0.025

0.014 0.017 0.030

0.010 0.011

0.011 0.013

0.013 0.015

0.010 0.011 0.011 0.012 0.010

0.012 0.012 0.013 0.015 0.014

0.014 0.014 0.015 0.018 0.017

0.011 0.013

0.013 0.015

0.015 0.016

0.015 0.014 0.016 0.018 0.017 0.022

0.017 0.017 0.019 0.022 0.020 0.027

0.020 0.020 0.023 0.025

Type of Channel

Minimum

Normal

Maximum

0.015 0.017

0.017 0.020

0.020 0.024

0.016 0.020 0.020

0.020 0.025 0.030

0.024 0.030 0.035

0.017 0.020 0.023

0.020 0.023 0.033

0.025 0.026 0.036

0.011 0.012

0.013 0.015

0.015 0.018

0.017 0.023 0.013

0.025 0.032 0.015

0.030 0.035 0.017

0.013 0.016 0.030

0.013 0.016

0.016 0.018 0.022 0.022

0.018 0.022 0.025 0.027

0.020 0.025 0.030 0.033

0.023 0.025

0.025 0.030

0.030 0.033

0.030

0.035

0.040

0.028

0.030

0.035

0.025

0.035

0.040

0.030

0.040

0.050

0.025 0.035

0.028 0.050

0.033 0.060

d. Concrete bottom float with sides of 1 . Dressed stone in mortar 2. Random stone in mortar 3. Cement, rubble masonry, plastered 4. Cement rubble masonry 5. Dry rubble or riprap e. Gravel bottom with sides of 1. Formed concrete 2. Random stone in mortar 3. Dry rubble or riprap f. Brick 1. Glazed 2. In cement mortar g. Masonry 1. Cemented rubble 2. Dry rubble h. Dressed ashlar i. Asphalt 1. Smooth 2. Rough J. Vegetal cover C-I Excavated or Dredged C.I General a. Earth, straight and uniform 1. Clean and recently completed 2. Clean, after weathering 3. Gravel, uniform section, clean 4. With short grass, few weeds b. Earth, winding and sluggish 1 . No vegetation 2. Grass, some weeds 3. Dense weeds or aquatic plants in deep channels 4. Earth bottom and rubble sides 5. Stony bottom and weedy banks 6. Cobble bottom and clean sides c. Dragline-excavated or dredged 1 . No vegetation 2. Light brush on banks

0.500

Type of Channel

Minimum

d. Rock cuts 1. Smooth and uniform 0.025 2. Jagged and irregular 0.035 e. Channels not maintained, weeds and brush uncut 1. Dense weeds, high as flow depth 0.050 2. Clean bottom, brush on sides 0.040 3. Same, highest stage of flow 0.045 4. Dense brush, high stage 0.080 C.2 Channels with maintained vegetation and velocities of 2 and 6 ft/s a. Depth of flow up to 0.7 ft 1. Bermuda grass, Kentucky bluegrass, buffalo grass Mowed to 2 in 0.07 Length 4 to 6 in 0.09 2. Good stand, any grass Length approx. 12 in 0.18 Length approx. 24 in 0.30 3. Fair stand, any grass Length approx. 12 in 0.014 Length approx. 24 in 0.25 b. Depth of flow up to 0.7-1.5 ft 1. Bermuda grass, Kentucky bluegrass, buffalo grass Mowed to 2 in 0.05 Length 4-6 in 0.06 2. Good stand, any grass Length approx. 12 in 0.12 3. Length approx. 24 in 0.20 Fair stand, any grass Length approx. 12 in 0.10 Length approx. 24 in 0.17 D Natural streams D-I Minor streams (top width at flood stage < 100 ft) a. Streams on plain 1. Clean, straight, full stage no rifts or deep pools 0.025 2. Same as above, but with more stones and weeds 0.030 3. Clean, winding, some pools and shoals 0.033 4. Same as above, but with some weeds and stones 0.035

Normal

Maximum

0.035 0.040

0.040 0.050

0.080 0.050 0.070 0.100

0.120 0.080 0.110 0.14

0.045 0.05 0.09 0.15 0.08 0.13

0.035 0.04 0.07 0.10 0.16 0.09

0.030

0.033

0.035

0.040

0.040

0.045

0.045

0.050

Type of Channel

Minimum

Normal

5. Same as above, lower stages more ineffective slopes and sections 0.040 6. Same as no. 4, more stones 0.045 7. Sluggish reaches, weedy, deep pools 0.050 8. Very weedy, reaches, deep pools or floodways with heavy stand of timber and underbrush 0.075 b. Mountain streams, no vegetation in channel, banks usually steep, trees and brush along banks submerged at high stages 1. Bottom: gravels, cobbles and few boulders 0.030 2. Bottom: cobbles with large boulders 0.040 D-2 Floodplains a. Pasture, no brush !.Short grass 0.025 2. High grass 0.030 b. Cultivated areas 1. No crop 0.020 2. Mature row crops 0.025 3. Mature field crops 0.030 c. Brush 1. Scattered brush, heavy weeds 0.035 2. Light brush and trees in winter 0.035 3. Light brush and trees in summer 0.040 4. Medium to dense brush in winter 0.045 5. Medium to dense brush in summer 0.070 d. Trees 1. Dense willows, summer, straight 0.110 2. Cleared land with tree stumps, no sprouts 0.030 3. Same as above but with a heavy growth of sprouts 0.050 4. Heavy stand of timber, a few down trees, little undergrowth, flood stage below branches 0.080 5. Same as above, but with flood stage reaching branches 0.100

Maximum

0.048 0.050

0.055 0.060

0.070

0.080

0.100

0.150

0.040

0.050

0.050

0.070

0.030 0.035

0.035 0.050

0.030 0.035 0.040

0.040 0.045 0.050

0.050

0.070

0.050

0.060

0.070

0.110

0.070

0.110

0.100

0.160

0.150

0.200

0.040

0.050

0.060

0.080

0.100

0.120

0.120

0.160

Type of Channel

Minimum

Normal

Maximum

D-3 Major streams (top width at flood stage > 100 ft); the n value is less that for minor streams of similar description because banks offer less effective resistance a. Regular section with no boulders or brush 0.025 b. Irregular and rough section 0.035

— —

0.060 0.100

— — — —

0.020 0.030 0.040 0.025

— —

0.015 0.020

0.17 0.17 0.20 0.20 0.10 0.30 0.05

— — — — — — —

0.80 0.48 0.40 0.30 0.20 0.40 0.13

0.09 0.05

— —

0.34 0.25

0.008 0.06 0.06 0.30 0.04 0.07 0.17

— — — — — — —

0.012 0.22 0.16 0.50 0.10 0.17 0.47

0.10

—

0.20

0.10 0.08 0.04 0.02

— — — —

0.15 0.12 0.10 0.05

D-4 Alluvial sandbed channels (no vegetation and data is limited to sand channels with D50< 1.0mm a. Tranquil flow, Fr < 1 !.Plane bed 0.014 2. Ripples 0.018 3. Dunes 0.020 4. Washed out dunes or transition 0.014 b. Rapid flow, Fr > 1 1. Standing waves 0.010 2.Antidunes 0.012 E. Overland Flow (Sheetflow) E-I Vegetated areas a. Dense turf b. Bermuda and dense grass c. Average grass cover d. Poor grass cover on rough surface e. Short prairie grass f. Shrubs and forest litter, pasture g. Sparse vegetation h. Sparse rangeland with debris 1.0% cover 2.20% cover E-2 Plowed or tilled fields a. Fallow—no residue b. Conventional tillage c. Chisel plow d. Fall disking e. No till—no residue f. No till (20-40% residue cover) g. No till (100% residue cover) E-3 Other surfaces a. Open ground with debris b. Shallow flow on asphalt or concrete c. Fallow fields d. Open ground, no debris f. Asphalt or concrete

Source: From Chow (1959), Richardson et al. (1987), Simons, Li, & Associates (SLA), 1982, and others. * The values in bold are recommended for design

CHAPTER 4 SUBSURFACE FLOW AND TRANSPORT

Mariush W. Kemblowki and Gilberto E. Urroz Utah Water Research Laboratory Utah State University Logan, Utah

4.7

INTRODUCTION

This chapter begins with the mathematical description of the constitutive relationships for flow and transport in porous media. Following this, simple analytical solutions are presented for a variety of subsurface flow and transport problems. The principles of flow and transport are outlined, and solutions are provided for practical problems of flow and transport in both the saturated and the unsaturated zones. The latter includes problems of transport in the vapor phase. The major focus is on the processes that are relevant to subsurface mitigation.

4.2

CONSTITUTIVERELATIONSHIPS

This section presents the basic concepts and laws used to describe flow and transport in the subsurface. In particular, the constitutive relationships defining the fluid flow in fully and partially saturated media are given as well as the relationships that describe diffusive and dispersive mass fluxes in porous media. Finally, we show the relations used to describe partitioning of chemicals in the subsurface environment. 4.2.1 Darcy's Law Consider the flow of a fluid through a pipe filled with a granular material, as shown in Fig. 4.1. In the figure, Z1 and Z2 represent the elevations of the pipe centerline above a reference level at Sections 1 and 2, respectively, whereas p/y and p2/j represent the water pressure head at Sections 1 and 2, respectively. We define the piezometric head at any location in the porous media as h = z+p/J

(4.1)

where y = specific weight (weight per unit volume) of water, typically, y = 9810 N/m3 or 62.4 lb/ft3. Let q be the average water velocity in the cross section of the pipe: i.e.,

q = Q/A

(4.2)

Arbitrary datum FIGURE 4.1 Porous media flow. where Q = volumetric discharge (volume per unit time) and A = total cross-sectional area of the pipe (including the soil matrix). French hydrologist Henry Darcy discovered that the average flow velocity could be estimated from q = KQi1 -H2)IL

(4.3)

where L is the distance, measured along the pipe, between cross sections 1 and 2, and K is a parameter that depends on the nature of the porous media as well as on the properties of the transported fluid. For water, K is known as the hydraulic conductivity or the coefficient of permeability. Typical values of K are given in many references (e.g., Bureau of Reclamation, 1985) see tables 4.1 ard 4.2. Eq. (4.3) is known as Darcy1s law and is commonly used to model the flow of fluids in porous media. Notice that the velocity V is not the fluid velocity in the soil pores, it is an average velocity calculated over the entire area of the flow cross section. The average pore velocity is calculated as v = -j(4.4) vw where Qw is the volumetric water moisture content. Note that for saturated flow, Qw = n, where n is porosity. Darcy's law (i.e., Eq. 4.3) also can be written more concisely as

q = KI

(4.5)

where / is the hydraulic gradient defined as / = ^)

4.6)

Hydraulic conductivity K is a function of aquifer and fluid properties -specifically of the intrinsic soil permeability k, fluid viscosity ^, and fluid density p-and is given by K = kpg/\L

(4.7)

For saturated flow of a constant density fluid in isotopic porous media, the Darcy law can be written as

*=-*!) For anisotropic media, the Darcy law is written as

(«)

(4 9) * = -*»(f|) where A^ is the conductivity tensor. For saturated flow of a fluid of variable density in anisotropic porous media, we have

*-fe(t+^) where ksoilij is the intrinsic permeability tenser. Finally, for unsaturated flow of variabledensity fluid in anisotropic porous media, we have + aX • -HrHr Pfluid \.OXj *-!) j) where kr(Q) = relative permeability of the porous media. Relative permeability is a function of soil saturation, which in turn is a function of the capillary pressure. These relationships for partially saturated flow are discussed in the next section.

4.2.2 Unsaturated Flow-Constitutive Relationship In unsaturated flow, the concern is water movement in the zone above the water table. In this case, the water saturation Sw is a function of the difference between air and water pressures because the water is resulting from held by capillary forces resulting from surface tension. This difference is known as the capillary pressure and is defined as haw = ha-hw

(4.12)

Typically, in unsaturated flow theory we assume negligible resistance to the gas-phase flow in porous media; as a result, we also can assume that the gas-phase pressure is uniform and equal to the atmospheric pressure. Hence haw = — hw. The aqueous pressure in the unsaturated zone is lower than the atmospheric pressure; thus, the capillary pressure is positive. The negative pressure head hw also is known as the soil matrix suction *F. Thus, the total head in the aqueous phase is h = — *¥ + z. To remove water from the pore space i.e., to reduce the water content we have to apply more negative pressure to the aqueous phase, i.e., increase the capillary pressure haw. This relationship is typically called the soil-water retention curve, and can be expressed by the following commonly used parametric models: the Brooks-Corey (BC) model and the van Genuchten (VG) model The BC model is 9- 0

(m>

"•-^-Rj for ¥ > hb and is otherwise (capillary fringe zone), e. = i

(413)

(4.14)

where n = porosity, G = volumetric moisture content (equal to n SJ, Qe = effective volumetric moisture content, 9r = residual (irreducible) moisture content, X = Brooks-Corey parameter, and hb = capillary fringe height. The van Genuchten model is

ee = e - -^ - er = [i + (-&l:s!r**-&'««"» where M(r

'° = Wt

(4 44)

'

and S = storativity (for confined aquifers) or porosity (for unconfined aquifers) and W(u) is known in subsurface hydrology as well function and in mathematics as exponential integral. This function is tabulated in almost every groundwater hydrology textbook. It also is available in many engineering mathematics software packages, such as Mathematica®, as a library function. 4.3.2 Superposition and Convolution For a time-variable pumping rate, the principle of convolution can be used to estimate the transient drawdown. This approach is strictly valid for linear systems: i.e., systems in which the response (drawdown) is a linear function of the excitement (pumping rate). The linearity assumption is strictly valid for confined aquifers only; however, as long as the drawdowns do not exceed 20% of the initial aquifer thickness, it also may be used for unconfined aquifers. Using the convolution approach, the transient drawdown for a pumping rate changing in a step-wise fashion is given by n S = ^ -^T ^ (Q* + Q* - 1)^«^ - 1» - VAt& + £)

'4^

W (f, £) = leaky well function (see, for example, Hunt, 1983, p. 100). 4.3.7.4 Point slug injection into a uniform flow field—3-D transport and retardation. In this case, a slug of contaminant of the mass M = C0V is injected at point (0,0,0). The transient distribution of concentration is described by C(x y Z t}=

' ' '

eJ_(*-V) L 4axvRt

VC %(nvRt)m°(axa^m 2

_^i___jM 4ayv/?r 4a2v/?rJ

(4.66)

4.3.7.5 Continuous injection from a finite-sized source with retardation and degradation. In this case, consider transport from a rectangular source that is perpendicular to the direction of flow. The source width is F, and its depth below the water table is Z. The transient concentration distribution in the presence of retardation and degradation is given by . C0 \{ x V 1 fi + 4V*H1 cfc^o^m-ji-HH}] 172 j-, /• fjc-v^l A ^ +4^) Kv' I l 2(CW)\

ErfC

I

\y + Y/2]

F_J

y - YI2\

M^^rM^HI {E*[^}-E*[^}

(4 67)

'

4.4 FLOW AND TRANSPORT IN UNSATURATED ZONE — AQUEOUS PHASE In this section, we briefly discuss the flow continuity equations and present some simple solutions to selected flow problems. That discussion is followed by a presentation of mass transport in the water phase of unsaturated zone. 4.4.1 Flow in an Unsaturated Zone The continuity equation for an unsaturated flow system can be written as U = -V.q

(4.68)

» = -gi + & + ^i] 3r (dx dy dz)

(4.69)

or

Combining Darcy's law with the mass continuity equation, we can write the final flow equations is ^ = V(KmVh) at

(4.70)

The flow equations can be simplified for horizontal and vertical flow conditions. a. One-dimensional horizontal flow: w = a {K?rg£\ Bt dx( dx)

(4.7i)

where 0 = volumetric water content. In this, the contribution of the elevation head, z, vanishes, since 3z/3jc = O. b. One-dimensional vertical flow: W = JL(Ww + 1I ot oz { (oz Jj

(472)

Note that the flow equations are characterized by the presence of two dependent, albeit related, variables: namely, 0 and 1F. To simplify this situation, we describe the relationship between 0 and 1F by a term called soil diffusivity D(0) as ™ =m where C(0) is called specific moisture capacity and is defined as C(0) = U

«•«> (4.74)

Using these definitions, the flow equation can be written as follows: c. One-dimensional horizontal flow: ^ = |W^] dt dx[ dx)

(4.75)

d. One-dimensional 1 -D vertical flow: r)ft _. ~\Q ^ = IOW-5- + ^))

(4-76)

As can be seen, we now have only one dependent variable: namely, 6. The only limitation of this formulation is that specific moisture capacity C(Q) becomes zero in the capillary fringe zone, thus making the solution impossible. Therefore, this formulation is valid only in the partially saturated zone (water content less than saturated value), not in the capillary fringe. Another way to solve this problem, which does not have the limitation discussed above, is to formulate the flow equations in terms of soil suction \(/. For the vertical flow, we obtain C(e)

? = i (^(XF)(lr+ l)j

(4 74)

*

Exercise. Consider steady-state vertical infiltration from the soil surface to the water table at depth L. The relative hydraulic conductivity of the soil is described by the following exponential law: kr(V) = exp[-aV]

(4.78)

Derive an expression for the vertical distribution of h. After the first integration of the flow equation, we obtain ~\\TJ \ +1 a£- J = « (4-79)

f

where q = infiltration rate and A" is a function of capillary pressure y. Kf¥) = Kjxp\-d¥\

(4.80)

where h = z — x|/. Substitution of these relationships into the flow equation leads to

or

ext\a(h - z)l^ = -?L -№ KSAT

(4.81)

exp[ah]dh = -/-exp[ax]dx ^-SAr After the second integration, we obtain

(4.82)

±exp(ah] = -J- exp[az] + C1 (4.83) oc «ASAr Substituting the boundary condition /z(0) = O and solving for C1 yields the following expression for the total head h: h(z) = ±-ln \J-(exp[az] - 1) + ll a L^5Ar J

(4.84)

4.4.2 Transport in an Unsaturated Zone The mass continuity equation for an unsaturated flow system with advection and diffusion/dispersion in the aquoeus phase, diffusion in the vapor phase, partitioning between four phases (soil, water, vapor, and (D)NAPL), and first-order degradation in the aqueous phase can be written as

wW 4= T?-^fk + ^N¥^l ^ - XC d/ =3x, ' +6£ Qw ) to,J- v &, W

(4.85)

where Cw = aqueous phase concentration; v = pore-water velocity; 6W = volumetric moisture content; a, = longitudinal, transverse horizontal, and transverse vertical macrodispersivities; D = Millington-Quirk dispersion coefficient; Dv = Millington-Quirk dispersion coefficient in the vapor phase; K = first-order degradation rate in the aquoeus phase; and R = retardation factor; R = 1 + ^SOIL^B _^_ ^NAPL^NAPL _j_ ^AIR^AIR vw vw Uw

/^ g^x

where kSOIL = water-soil partitioning coefficient, kSOIL = S/CW, S = weight/weight concentration of absorbed compound in soil, kNAPL = NAPL-water partitioning coefficient, ^VAPL = CNAPI! Cw> CNAPL = concentration of compound in the WAPL phase, kAIR = vaporwater partitioning coefficient, kAIR = V/CW, V = concentration of compound in the vapor phase, J8 = bulk density of dry soil, QAIR = volumetric vapor content, and QNAPL — volumetric NAPL content. Assuming again that the diffusive fluxes are negligible compared with the macrodispersive and advective fluxes, the transport equation can be simplified to yield T'l-hfh-f-v.

whereas for retarded compounds, we have f—f

where the retarded velocity is given by VR = ^ = R

j +1 ^w K(I - Sw)

(4.119)

Exercise. Consider diffusive vertical transport of a compound in vapor phase. The compound is subject to first-order degradation in the aqueous phase at rate X and to partitions between vapor and aqueous phases according to the following relationship: V=KC

(4.113)

where V = concentration in vapor phase, C = concentration in aqueous phase, and K = partitioning coefficient. At the depth of 100 cm below the ground surface, the vapor concentration of the compound was measured to be V0, whereas at the ground surface the concentration was Vs. Given the compound's diffusion coefficient in vapor phase D0, porosity n, and water saturation Sw, estimate the diffusive flux of the compound at the soil surface. The relevant mass transport equation is given by

DJfV-I SC = Q dx2 n

(4.120)

where Q3.33 D - D0-^n2 Substituting C = VIK into the mass transport equation leads to 32V ^JL-X2V=O dr where

(4.19)

(4.121)

X" = ^

We solve the modified mass transport equation to obtain V(Jt) = C1BXpI -Xx] + C2exp[Kx\

(4.123)

where constants C1 and C2 are obtained from the boundary conditions V(O) - V0, and V(IOO) = V,

(4.124)

The compound's mass flux at the soil surface is estimated from q = -D^VQ djt

@x = 10Q

(4.125)

REFERENCES Bear, J., and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer Academic, Dordrecht, The Netherlands, 1990. Bear, J., Dynamics of Fluids in Porous Media, Dover Publications, New York,k 1972. Bear, J., Hydraulics of Groundwater, McGraw Hill, New York, 1979. Bouwer H., Groundwater Hydrology, McGraw Hill, New York, 1978. Bruce, G. H., D. W. Peaceman, and H. H. Rachford, Jr., 1953. "Calculations of Unsteady-State Gas Flow Through Porous Media," Petroleum Transactions, AIME, 198: 1953. Bureau of Reclamation, Ground Water Manual (Reprint), U.S. Department of the Interior Washington, DC, 1995. Cedergren H. R., Seepage, Drainage, and Flow Nets, 3rd ed., John Wiley & Sons, Inc., New York, 1989. Charbeneau, R. J., "Kinematic Models for Soil Moisture and Solute Transport," Water Resources Research, 20: 699-706, June, 1984. Chirlin, G. R., "A Critique of the Hvorslev Method for Slug Test Analysis: The Fully Penetrating Well," Ground Water Monitoring Review, 130-138, 1989. Cho, J. S., 1991. Forced Air Ventilation for Remediation of Unsaturated Soils Contaminated by VOC., Publication No. EPA/600/S2-91/016, U.S. Environmental Protection Agency, Washington, D.C.

Dagam, G., "Solute transport in heterogeneous porous formations," Journal of Fluid Mechanics, 145: 151-177, 1984. De Josselin Jong, G. "Singularity Distribution for the Analysis of Multiple-Fluid Flow Through Porous Media," Journal of Geophysical Research, 65: 3739-3758, 1960. De Marsily G., Quantitative Hydrogeology—Groundwater Hydrology for Engineers, Academic Press, San Diego, CA, De Smedt R, and P. J. Wirenga, "Solute Transport Through Soil With Nonuniform Water Content," Soil Science Society of America Journal, 42. (1): 1978. Domenico, P. A., and F. W. Schwartz, Physical and Chemical Hydrogeology, John Wiley & Sons, New York, 1990. Dullien F. A. L., Porous Media: Fluid Transport and Pore Structure, 2nd ed., Academic Press, San Diego, CA, 1992. Edelman J. H., Groundwater Hydraulics of Extensive Aquifers, 2nd., International Institute for Land Reclamation and Improvement, Bulletin No. 13, The Netherland, 1983. Fetter, C. W., Contaminant Hydrogeology, Macmillan, New York, 1993. Fetter, C. W., Applied Hydrogeology, Simon & Schuster, Company Englewood, NJ, 1994. Freeze, R. A., and J. A. Cherry, Groundwater, Prentice-Hall, Englewood Cliffs, NJ, 1979. Gelhar, L. W. and C. L. Axness. "Three-Dimensional Stochastic Analysis of Macrodispersion in Aquifers", Water Resources Research, 19 (1): 161-180, 1983. Germann, P. R, M. S. Smith, and G. W. Thomas, "Kinematic Wave Approximation to the Transport of Escherichia coli in the Vadose Zone," Water Resources Research. 23 (7), 1281-1287, 1987. Girinsky, N. K. Determination of the Coefficient of Permeability, Gosgeolizdat, 1950. Grubb, S. "Analytical Model for Estimation of Steady-state Capture Zones of Pumping Wells in Confined and Unconfmed Aquifers," Ground Water, 31(1) 27-32, 1993. Hantush, M. S., Hydraulics of Wells, "in Advances in Hydroscience," V. T. Chow, ed., Academic Press, New York, 1964. Hantush, M. S. "Growth and Decay of Groundwater-mounds in Response to Uniform Percolation." Water Resources Research, 3 (1): 227-234, 1967. Harr, M. E., Groundwater and Seepage, Dover Publications, New York, 1990. Haverkmp, R., M. Vauclin, J. Touma, P. J. Wierenga, and G. Vachaud, A "Comparison of Numerical Simulation Models for One-Dimensional Infiltration," Soil Science Society America Journal, 41: 285-294, 1977. Hinchee, R. E. ed., Air Sparging for Site Remediation, Lewis Publishers, Boca Raton, FL, 1994. Huisman, L., Groundwater Recovery, Winchester Press, New York, 1972. Hunt, B. "Seepage to Collection Gallery Near Seacoast," Water Resources Research, 21: 311-316, 1985. Hunt, B., Mathematical Analysis of Groundwater Resources, Butterworths, London, UK, 1983. Jaffe, P. R., and R. A. Ferrara, "Desorption Kinetics in Modeling of Toxic Chemicals," Journal of Environmental Engineering, American Society of Civil Engineers, 109: 859-867, 1983. Javandel, I., and C. F. Tsang, "Capture-zone Type Curves: A Tool for Aquifer Cleanup." Ground Water, 24: 616-625, 1985.

L, C. Doughty, and C. F. Tsang, Groundwater Transport: Handbook of Mathematical Models, American Geophysical Union, Washington, DC, 1987. Johnson, P. C., M. W. Kemblowski, and J. D. Colthart, "Quantitative Analysis for the Cleanup of Hydrocarbon-Contaminated Soils by in-situ Soil Venting," Ground Water, 1990. Johnson, P. C., C. C. Stanley, M. W. Kemblowski, D. L. Byers, and J. D. Colthart. A "Practical Approach to the Design, Operation, and Monitoring of in situ Soil-Venting Systems." Ground Water Monitoring Review, Spring, 1990. Jury, W. A., "Chemical Transport Modeling: Current Approaches and Unresolved Problems," Chemical Mobility and Reactivity in Soil Systems, 1983, pp. 49-64. Jury, W. A., R. Grover, W. F. Spencer, and W. J. Farmer, "Modeling Vapor Losses of Soil Incorporated Triallate," Soil Science Society of America Journal, 44: 445^-50, 1980. Jury. W. A., W. F. Spencer, and W. J. Farmer, "Use of Models for Assessing Relative Volatility, Mobility, and Persistence of Pesticides and other Trace Organics in Soil Systems." Hazard Assessment of Chemicals: Current Developments, Vol. 2, 1983. Keely, J. F. and C. F. Tsang, "Velocity Plots and Capture Zones of Pumping Centers for GroundWater Investigations." Ground Water, 21: 701-714, 1983. Kishi, Y. and Y. Fukuo, "Studies on Salinization of Groundwater," L Journal of Hydrology, 35: 1-29, 1977. Kool, J. B., J. C. Parker, and M. T. van Genchten, "Parameter Estimation for Unsaturated Flow and Transport Modles—A Review." Journal of Hydrology, 91: 255-293, 1987. Kozeny, J., Thorie und Berchnung der Brunnen. Wasserkraft und Wasserwirtschaft, Nos. 8-10, 1933. Marino, M. A., "Artificial Groundwater Recharge: I. Circular Recharging Area," Journal of Hydrology, 25: 201-208, 1975. Marshall, T. J., J. W. Holmes, and C. W. Rose, Soil Physics, 3rd ed., Cambridge University Press, Cambridge, UK, 1996. McElwee, C., and M. Kemblowski, "Theory and Application of an Approximate Model of Saltwater Upconing in Aquifers," Journal of Hydrology, 115: pp 139-163, 1990. McWhorter, D. B. Steady and Unsteady Flow of Fresh Water in Saline Aquifers, Water Management Technical Report No. 20, Council of U.S. Universities for Soil and Water Development in Arid and Sub-Humid Areas, 1972. Musa, M. and M. W. Kemblowski. "Effective Capture Zone for a Single Well," Submitted to Ground Water July 1994. Newsom, J. M., and J. L. Wilson, "Flow of Ground Water to a Well Near a Stream - Effect of Ambient Ground-water Flow Direction." Ground Water, 25: 703-711, 1988. Oberlander, P. L. and R. W. Nelson, "An Idealized Ground-Water Flow and Chemical Transport Model (S-PATHS)," Ground Water, 22: 441^49, 1984. Ostendorf, D. W, R. R. Noss, and D. O. Lederer, "Landfill Leachate Migration through Shallow Unconfined Aquifers," Water Resources Research, 20: 291-296, 1984. Palmer, C. M., Principles of Contaminant Hydrogeology, Lewis Publishing, Chelsea, MI, 1992. Pankow, J. E, R. L. Johnson, and J. A. Cherry, "Air Sparging in Gate Wells in Cutoff Walls and Trenches for Control of Plumes of Volatile Organic Compounds (VOCs)," Ground Water, 31: 654-663, 1993.

Parker, J. C., and M. T. van Genuchten, Determining Transport Parameters from Laboratory ad Field Tracer Experiments, Virginia Agricultural Experiment Station Bulletin No. 84-3, Virginia Polytechnic Institute and State University, Blacksburg, 1984. Parker, J. C., and M. T. van Genuchten, "Flux-averaged and Volume-Averaged Concentrations in Coninuum Approaches to Solute Transport," Water Resources Research, 20: 886-872, 1984. Parker, J. C., K. UnIu, and M. W. Kemblowski, "A Monte Carlo Model to Assess Effects of Land Disposed E & P Waste on Groundwater," SPE Annual Technical Conference & Exhibition, 1993. Philip, J. R. "The theory of infiltration: 1. The infiltration equation and its solution." Soil Science. 83: 345-357, 1957. Raudkivi, A. J., and R. A. Callander, Analysis of Groundwater Flow, Edward Arnold, London, UK, 1976. Rosenshein, J., and G. D. Bennet, eds., Groundwater Hydraulics, American Geophysical Union, Washington, DC, 1984. Rubin, H., and G. F. Pinder., "Approximate Analysis of Upconing." Advances in Water Resources, 1 (2): 97-101, 1977. Sallam, A., W. A. Jury, and J. Letey, "Measurement of Gas Diffusion Coefficient Under Relatively Low Air-Filled Porosity," Soil Science Society of America Journal 48:3-6, 1983. Schiegg, H. O., "Considerations on Water,. Oil and Air in Porous Media," Water Science Technology, 17: 467^76. 1984. Shafer, J. M., "Reverse Pathline Calculation of Time-Related Capture Zones in Nonuniform Flow," Ground Water 25: 283-289, 1987. Sikkema, P. C. and J. C. Van Dam, "Analytical Formulae for the Shape of the Interface in a SemiConfined Aquifer," Journal of Hydrology, 56: 201-220, 1982. Sposito, G., and W. A. Jury, "Inspectional Analysis in the Theory of Water Flow Through Unsaturated Soil," Soil Science Society of America Journal, 42 (1): 1985. Sposito, G., "Chemical Models of Inorganic Pollutants in Soils," CRC Critical Reviews in Environmental Control, 15 (1): 1-24, undated. Strausberg, S. I. "Estimating Distances to Hydrologic Boundaries from Discharging Well Data," 19th Annual Meeting of the Rocky Mountain Section of the Geological Society of America, Las Vegas, NV, 1966. Thornton, J. S., and W. L. Wootan, Jr., "Venting for the Removal of Hydrocarbon Vapors from Gasoline Contaminated Soil," Journal of Environmental Science and Health, All (1), 31—44, 1982. Todd, D. K., Groundwater Hydrology, 2th John Wiley & Sons, New York, 1980. Todd, D. K., "Salt-Water Intrusion and Its Control." Journal of the American Water Works Associations, 180-187, 1973. U. S. Department of Agriculture Agricultural Research Service, Analytical Solutions of the OneDimensional Convective-Dispersive Solute Transport Equation, Technical Bulletin No. 1661, UnIu, K., M. W. Kemblowski, J. C. Parker, D. Stevens, P. K. Chong, and I. Kamil, "A Screening Model for Effects of Land-Disposed Wastes on Groundwater Quality" Journal of Contaminant Hydrology, 11: 27-^9, 1992. Washington, DC, 1982 Van Genuchten M. T. and W. J. Alves, "Analytic Solution of the One-Dimensional Convective Solute Transport Equation, Technical Bulletin. 1661, U.S. Departament of Agriculture, Washington D.C., 1982. Ward, C. H., M. B. Tomson, P. B. Bedient, and M. D. Lee, "Transport and Fate Processes in the Subsurface," Water Resources Symposium, VoI 13, WARSAG, 1987.

Warrick, A. W. , J. W. Biggar, and D. R. Nielsen, "Simultaneous Solute and Water Transfer for an Unsaturated Soil," Water Resources Research. 7: 1216-1225, 1971. Watson, K. K., and M. J. Jones, "Algebraic Equations for Solute Movement During Absorption," Water Resources Research, 20: 1131-1136, 1984. Wilson, J. L., and L. W Gelhar, "Analysis of Longitudinal Dispersion in Unsaturated Flow 1: The Analytical Method," Water Resources Research, 17 (1): 122-130, 1984. Wilson, J. L., and P. J. Miller, "Two-Dimensional Plume in Uniform Ground-Water Flow Discussion," Journal of the Hydraulics Division, American Society of Civil Engineers, 103 (HY12): 1567-1570, 1979. Wilson, J. L., and P. J. Miller., "Two-Dimensional Plume in Uniform Ground-Water Flow," Journal of the Hydraulics Division, American Society of Civil Engineers, 104 (HY4): 503-514, 1978. Wirojanagud, P., and R. Charbeneau., "Saltwater Upconing in Unconfined Aquifers," Journal of Hydraulic Engineering, American Society of Civil Engineers, 111: 417^-34, 1985. Yeh, G. T., Analytical Transient One-, Two-, and Three-Dimensional Simulation of Waste Transport in the Aquifer System, Environmental Sciences Division Publication No. 1439, Oak Ridge National Laboratory, Oak Ridge, TN, 1981.

CHAPTER 5 ENVIRONMENTAL HYDRAULICS

Richard H. French Water Resources Center Desert Research Institute University and Community College System of Nevada Reno, Nevada Steven C. McCutcheon Ecosystems Research Division National Exposure Research Laboratory U.S. Environmental Protection Agency Athens, Georgia James L Martin AScI Corporation Athens, Georgia

5.1

INTRODUCTION

The thermal, chemical, and biological quality of water in rivers, lakes, reservoirs, and near coastal areas is inseparable from a consideration of hydraulic engineering principles; therefore, the term environmental hydraulics. In this chapter we discuss the basic principles of water and thermal budgets as well as mixing and dispersion.

5.2

WATERANDTHERMALBUDGETS

5.2.1 Water Budget A water budget is a statement of the law of conservation of mass or (change in storage) = (input) — (output)

(5.1)

and the expressions of the water budget can range from simple to very complex. For example, consider the lake or reservoir shown in Figure 5.1. For this situation, a generic water budget could be written as follows ^ = (/, + /„ + /, + Pr + Rr) ~(Ev + Tr+Gs + Oc+W) (5.2)

Aquatic vegetation Water table

Water table

Storage, 5s

FIGURE 5.1 A hypothetical lake illustrating the variables in the water budget.

where Ic = channel inflow rate, I0 = overland inflow rate, Ig = groundwater inflow rate, Pr = precipitation rate, Rr = return flow rate, Ev = evaporation rate, Tr = transpiration rate, G5 = groundwater seepage rate, O0 = channel outflow rate, W = consumptive withdrawal, and Ss = lake/reservoir storage rate at time t (volume). The solution of Eq (5.2) quantifies the terms, and, in many cases, the goal of the modeling effort is to estimate the value of a single term or group of terms: for example, evapotranspiration (Ev + Tr). The reliability of using a water budget is directly related to the accuracy of the prediction techniques used, the availability and quality of gauged data, and the time period involved. Among the methods of evaluating the individual terms in Eq. (5.2) are the following: • • • • •

Channel inflow and outflow ( I c and Oc)—gauging, statistical simulation. Overland inflow (I0)—gauging, rainfall-runoff relationships. Groundwater inflow and seepage rate (Ig and Gs)—seepage equations, gauging. Precipitation (P1)—gauging, statistical simulation (Smith, 1993). Evaporation and transpiration (E and T)—gauging, evaporation/transpiration prediction relationships (Bowie et al. 1985; Shuttleworth, 1993). • Return flow and withdrawal (Rr and W)-gauging.

5.2.2 Thermal Budget The total thermal budget for a body of water includes atmospheric heat exchange at the air water interface (usually the dominant process), the effects of inflows (tributaries, wastewater, and cooling water discharges), heat resulting from chemical-biological reactions, and heat exchange with the stream bed. In the following sections, the primary components of the air-water interface heat budget will be briefly discussed; for further details the reader is referred to Bowie et al., (1985), McCutcheon (1989), or Shuttleworth (1993). Atmospheric heat exchange at the air-water interface is given by H=Qs-Qsr + Qa- Qar - Qbr ~Qe±Qc

(5.3)

where H = net surface heat flux, Q5 = shortwave radiation incident to the water surface [30-300 (kcal/m2)/h], Qsr = reflected shortwave radiation [5-25 (kcal-m2)/h], Qa = incoming longwave radiation from the atmosphere (225-360 kcal/m2/h), Qar = reflected longwave radiation [5-15 (kcal-m2)/hr], Qbr = longwave back radiation emitted by the water body [220-345 (kcal-m2)/h], Qe = energy utilized by evaporation [25-900 (kcal-m2)/h], and Q0 = energy converted to or from the body of water (—35-50 kcal-m2 /hr). Note that the ranges given are typical for the middle latitudes of the United States (Bowie et al., 1985). The equations for estimating the terms of the thermal budget use a mixed set of units, and appropriate conversions among the different units used are provided in Table 5.1. 5.2.2.1 Net atmospheric shortwave radiation (Qs — Qsr). The net shortwave radiation (Qsn) is that portion of the incident shortwave radiation captured at the water surface, taking into account losses caused by reflection. Although solar radiation can be measured with specialized meteorological stations equipped with radiometers, these instruments require painstaking calibration and maintenance. In most cases, measured values of solar radiation are not available at the location of interest and must be estimated from equations. Among the formulations for estimating net shortwave solar radiation is Qsn = QS- Qsr = 0.94fcc(l - 0.65C2C)

(5.4)

2

where Qsc = clear sky solar radiation [kcal Tn Xh) and C0 = fraction of sky covered by clouds (Anderson, 1954; Ryan and Harleman, 1973). It is pertinent to note that Eq. (5.4)

TABLE 5.1 Useful Energy Conversions for Energy Budget Calculations IBtu/ftVday lwatt/m 2 1 Ly/day 1 (kcal-m2)/hr

= = = =

0.131 W/m2 = 7.61 Btu-ft2)/day = 0.485 W/m2 = 1.16 W/m2 =

0.271 Ly/day = 2.07 Ly/day = 3.69 (Btu/-ft2)/day = 2.40 Ly/day =

0.113 (kcal-m2)/h 0.86 (kcal-m2)/h 0.42 (kcal-m2)/h 8.85 (Btu-ft2)/day

1 kpa lmb lmm Hg I m Hg

= = =

10 mb 0.1 kpa 1.3mb 33.0mb

7.69 mm Hg 0.769 mm Hg 0.13 kpa 25.4 mm Hg

0.303 in (Hg) 0.03 in (Hg) 0.039 in (Hg) 3.3 kpa

= = = =

= = = =

Abbreviations Ly = Langleys; mb = millibar; and Btu = British Thermal Unit

assumes average reflectance at the water surface and uses clear sky solar radiation. In some situations, the effects of atmospheric attenuation are much greater than normal and more complex equations are required (e.g., 1972). Clear sky radiation (Qsc) can be estimated as a function of calendar month and latitude from Fig. 5.2. Shortwave solar radiation is absorbed at the water surface and penetrates the water column, depending on the wavelength of the radiation, the properties of the water, and the matter suspended in the water. The degree of penetration of shortwave solar radiation (sunlight) into the water column has a significant effect not only on water temperature but also on the rate of photosynthesis by aquatic plants and the general clarity, color and aesthetic quality of the water. The penetration of shortwave solar radiation is described by / - I0exp (-k e y)

(5.5)

where / = light intensity at depth v, K6 = extinction coefficient, and I0 = light intensity at the surface (y = O). Values of the extinction coefficient can be estimated by several methods. For example, measurement of total light penetration into a water column can be made by using a pyreheliometer positioned at the surface that measures the total incoming solar radiation. Simultaneously, an underwater photometer is lowered and the radiation is recorded at each of a series of depths throughout the water column. Then, a value of Ke can be estimated by linear least-squares regression. An alternative but traditional, simpler, and less accurate method to estimate Ke is to lower a target into the water column until, by eye, the target just disappears. A standardized target (Secchi disk) is commonly used, and a number of investigators (Beeton, 1958; French et al., 1982; Sverdrup et al, 1942;) have developed empirical relationships between the Secchi disk depth (ys) and the extinction coefficient of the form ^ = M^ (5.6) y, Finally, the depth (ye) at which 1 percent of the surface radiation still remains (the euphotic depth) is given from Eq. (5.5) as

4.61 ye = -jp

(5.7)

5.2.2.2 Net atmospheric long-wave radiation (Qa - Qar). Atmospheric radiation is characterized by much longer wavelengths than solar radiation because the major emitting elements are water vapor, carbon dioxide, and ozone. The approach generally used to estimate this flux involves the empirical estimation of an overall atmospheric emissivity and the use of the Stephan-Boltzman law (Ryan and Harleman, 1973). Swinbank (1963) developed the following equation, which has been used in many water quality models Q0n = Qa~ Qar = U6 X 1013(1 + 0.11C2J(T0 + 46O)6

(5.8)

where Qan = net long-wave atmospheric radiation (Btu/ft2/day), Cc = fraction of sky covered by clouds, and T0 = dry bulb air temperature (0F). 5.2.2.3 Long-wave back radiation (Qbr). The long-wave back radiation from a water surface in most cases is the largest of all the fluxes in the heat budget (Ryan and Harleman, 1973). The emissivity of a water surface is well known; therefore, this flux can be estimated with a high degree of accuracy as a function of the water surface temperature:

FIGURE 5.2 Clear sky solar radiation. (From Hamon et al. 1954)

Qbr = 0.9707?

(5.9)

2

where Qbr = long-wave back radiation (cal/m /s), Ts = surface water temperature (K), and a = Stefan-Boltzman constant (1.357 X IQ-8 cal-m2/s/K4) 5.2.2.4 Evaporative heat flux (Qe). Evaporative heat loss (kcal/m2/s) occurs as a result of the change of state of water from a liquid form to vapor and is estimated by Q. = P^A

(5.10)

where Lw = latent heat of vaporization (= 597 — Q.51TS, kcal/kg), Ts = surface water temperature (0C), Ev = evaporation rate (m/s), and p = water density (kg/m3). A standard expression for evaporation from a natural water surface is Ev = (a + bW)(es - ea)

(5.11)

where Ev = evaporation rate (m/s), a and b = empirical coefficients, W = wind speed at some specified distance above the water surface (m/s), es = saturation vapor pressure at the temperature of the water surface (mb), and ea = vapor pressure of the overlying atmosphere (mb). In many cases, the empirical coefficient a has been taken as zero with 1 X ICh9 ^ b < 5 X ICh9 (Bowie et al., 1985). The saturated vapor pressure can be estimated (Thackston, 1974) by (5 12)

^^H^Tr^o)

'

0

where es is in inches of Hg, and Ts = water surface temperature ( F). There are a number of ways of estimating ea, depending on the available data. For example, if the relative humidity (Rn) is known, then RH = ^

(5-13)

and then if the wet bulb temperature and atmospheric pressure are known (Brown and Barnwell, 1987) ea = e,- 0.000367Pa(7; - T.'Jl + ^32J

(5.14)

where all pressures are in (in Hg), all temperatures are in (0F), P0 = atmospheric pressure, and Twb = wet bulb temperature. The relationship among the air and wet bulb temperatures (0F) and relative humidity (Thackston, 1974) is Twb= (0.655+ 036RH)Ta

(5.15)

There are many equations for estimating the rate of evaporation. For example, Jobson (1980) developed a modified formula that was used in the temperature modeling of the San Diego Aqueduct and subsequently was modified for use on the Chattahoochee River in Georgia (Jobson and Keefer, 1979). McCutcheon (1982) noted that, in many models, the wind speed function is a catchall term that compensates for many factors, such as (1) numerical dispersion in some models, (2) the effects of wind direction, fetch, channel width, sinuosity, bank, and tree height, (3) the effects of depth, turbulence, and lateral velocity distribution; and (4) the stability of air moving over the stream. (Fetch is the distance over which the wind blows or causes shear over the water's surface.) Finally, it is

important to note that evaporation estimators that work well for lakes or reservoirs will not necessarily provide the same level of performance when used in streams, rivers, or constructed open channels. 5.2.2.5 Convective heat flux (Qc). Convective heat is transferred between air and water by conduction and is transported to or from the air-water interface by convection. The convective heat flux is related to the evaporative heat flux (Qe) by the Bowen ratio (Bowie et al., 1985), or RB = ^JT = (6.19 X 1(M)P, eTs~eTa Qe s~ a

(5.16)

where all temperatures are in (0C), all pressures are in (mb), and R8 = Bowen ratio. 5.2.2.6 Conclusion. The foregoing is a brief summary of the approaches used most frequently to estimate surface heat exchange in numerical models. The reader is referred to other publications for a more detailed discussion of the approaches (Bowie et al., 1985) and meteorological data requirements (Shanahan, 1984). Note that each situation should be considered carefully from the viewpoint of specific factors that must be taken into account. For example, in most lakes, estuaries, and deep rivers, the thermal flux through the bottom is not significant. However, in water bodies with depths less than 3 m (10 ft), bed conduction of heat can be significant in determining the diurnal variation of temperatures within the body of water (Jobson, 1980, Jobson and Keefer, 1979).

5.3

EFFECTS AND CAUSES OF STRATIFICATION

5.3.1 Effects The density of water is strongly affected by temperature and the concentrations of dissolved and suspended solids. Regardless of the cause of differences in water density, water with the greatest density is found at the bottom, whereas water with the least density resides at the surface. When density gradients are strong, vertical mixing is inhibited. Stratification is the establishment of distinct layers of water of different densities (Mills et al., 1982). Stratification is enhanced by quiescent conditions and is destroyed by wind stress, turbulence caused by large inflows, and destabilizing changes in water temperature. In many bodies of water (rivers, lakes, and reservoirs), stratification is the single most important phenomena affecting water quality. When stratification is absent, the water column is mixed vertically and dissolved oxygen (DO) is present in the vertical water column from the top to the bottom: that is, fully mixed water columns do not have DO deficit problems. For example, when stratification occurs, in reservoirs and lakes mixing is limited to the epliminion or surface layer. Since stratification inhibits, vertical mixing is inhibited by stratification, and reaeration of the bottom layer (the hypoliminion) is inhibited if not eliminated. The thermocline (the layer of steep thermal gradient between the epiliminion and hypoliminion) limits not only mixing but also photosynthetic activity as well. The hypolimnion has a base oxygen demand and benthic matter and the settling of particulate matter, from the epiliminion only adds to this demand. Therefore, while the demands of DO in the hypoliminion increase during the period of stratification, inhibition of mixing between the epiliminion and the hypolimnion and the lack of photosynthetic activity deplete the DO concentrations in the

hypolimnion. Finally, a rule of thumb suggests that when water temperature is the predominant cause of differences in water density a temperature gradient of at least l°C/m is required to define the thermocline (Mills et al., 1982). The density of water can be estimated by p = pr + Aps

(5.17)

where p = water density (kg/m3), pr = water density as a function of temperature, and Ap5 = increments in density caused by solids. 5.3.2 Water Density as a Function of Temperature A number of formulations have been proposed to estimate pT and among these are pr = 999.8452594 + 6.793952 X 10 -2T6 - 9.095290 X 10-3 77 + 1.001685 X IQ-4T4,3 (5.18) 6 9 - 1.120083 X 10- T/ + 6.536332 X H)- T/ where Te = water temperature in 0C (Gill, 1982). 5.3.3 Water Density as a Function of Dissolved Solids or Salinity and Suspended Solids In most cases, data for dissolved solids are in the form of total dissolved solids (TDS); however, in some cases, salinity may be specified. The density increment for dissolved solids can be estimated by A

PTDS = CTDS(8.221 X 10"4 - 3.87 X 10'6Te + 4.99 X 10-8 Te2)

(5.19)

(Ford and Johnson, 1983), where CTDS = concentration of TDS (g/m3 or mg/L). If the concentration of TDS is specified in terms of salinity (Gill, 1982) ApSL = CSL(0.824493 - 4.0899 X 10"3 Te + 7.6438 X 10~5 T2 -8.2467 X 10~7 T/ + 5.3875 X 10~9 Te4) + C81L (-5.72466 X 10"3 + 1.0277 X 104 Te -1.6546 X 10"6 Te2) + 4.8314 X 10"4 CSL~2

(5.20)

3

where CSL = concentration of salinity (kg/m ). The density increment for suspended solids is Ap88 = CJl. --1 X l O 3 (5.21) I *GJ where SG = specific gravity of the suspended sediment (Ford and Johnson, 1983). The total density increment caused by solids is then Ap8 - (AoTDS or ApSL) + Apss

(5.22)

5.4

MIXINGANDDISPERSIONINOPENCHANNELS

Turbulent diffusion (mixing) refers to the random scattering of particles in a flow by turbulent motions, whereas dispersion is the scattering of particles by the combined effects of shear and transverse turbulent diffusion. Shear is the advection of a fluid at different velocities at different positions within the flow. When a tracer is injected into a homogeneous channel flow, the mixing process can be viewed as composed of three stages. In the first stage, the tracer is diluted by the flow in the channel because of its initial momentum. In the second stage, the tracer is mixed throughout the cross section by turbulent transport processes. In the third stage, longitudinal dispersion tends to erase longitudinal variations in the tracer concentration. In some cases, the second stage is eliminated because the tracer discharge has a significant amount of initial momentum associated with it; however, in many cases, the tracer flow is small and the momentum associated with it is insignificant. In the latter case, the first transport stage is eliminated. In this section, only the second and third transport stages will be treated, with the implied assumption that if there is a first stage, it can be treated separately. Section 5.6 details how excessive initial momentum must be analyzed. The reader is cautioned that, in this chapter, y is the vertical coordinate direction and z is the transverse coordinate direction. 5.4.1 Vertical Turbulent Diffusion To develop a quantitative expression for the vertical turbulent diffusion coefficient, consider a relatively shallow flow in an infinitely wide rectangular channel. It can be shown that the vertical transport of momentum in such a flow is given by ^ ^ Ty

^

where T = shear stress at a distance y above the bottom boundary, p = fluid density, ev = vertical turbulent diffusion coefficient, and v = longitudinal velocity (French, 1985). Because the one-dimensional vertical velocity profile and shear distribution are known, it can be shown that * = 4 v *(»)N)

(5 24)

-

where k = von Karman's turbulence constant (0.41), yd = depth of flow, v* = shear velocity (= Vgy^S), and S = longitudinal channel slope (French, 1985). The depth-averaged value of ev is e; = 0.067^v*

(5.25)

When the fluid is stably stratified, mixing in the vertical direction is inhibited, and one often quoted formula expressing the relationship between the unstratified and stratified vertical mixing coefficient was provided by Munk and Anderson (1948) * - = l + 3.MRi)»

(5 26)

'

where evs = the stratified vertical mixing coefficient and Ri = the gradient Richardson number.

5.4.2 Transverse Turbulent Diffusion In the infinitely wide channel hypothesized to derive Eq. (5.24), there is no transverse velocity profile; therefore, a quantitative expression for et the transverse turbulent diffusion coefficient, cannot be derived from theory. The following equations to estimate et derived from experiments by Fischer et al., (1979), and Lau and Krishnappen (1977). In straight rectangular channels, an approximate average of the results available is e, - 0.15y^v* ± 50%

(5.27)

where the ±50 percent indicates the error incurred in estimating et. In natural channels, e, is significantly greater than the value estimated by Eq. (5.27). For channels that can be classified as slowly meandering with only moderate boundary irregularities e= 0.6Oy^v* ± 50%

(5.28)

If the channel has curves of small radii, rapid changes in channel geometry, or severe bank irregularities, then the value of et will be larger than that estimated by Eq. (5.28). For example, in the case of meanders, Fischer (1969) estimated that V 2V3J ^.

e =2

~>

where a slowly meandering channel is one in which TV — .2

(5 29)

-

(5.30)

and V = average channel velocity, T = channel topwidth, and Rc = radius of the curve. A comparison of Eqs. (5.25) and (5.27) shows that the rate of transverse mixing is roughly 10 times greater than the rate of vertical mixing. Thus, the rate at which a plume of tracer spreads laterally is an order of magnitude larger than is the rate of spread in the vertical direction. However, most channels are much wider than they are deep. In a typical case, it will take approximately 90 times as long for a plume to spread completely across the channel as it will take to mix in the vertical dimension. Therefore, in most applications, it is appropriate to begin by assuming that the tracer is uniformly distributed over the vertical. In a diffusional process in which the tracer is added at a constant mass flow rate (M*) at the center line of a bounded channel (dC/dz = O at z — O and 3C/3z = O at z = 7), the downstream concentration of tracer is given approximately by

^-vfc.iH-^^^)

+ e5p (_x

(6.21)

Suggested values of OC5, which have appeared in the literature, are listed in Table 6.1 (Yen, 1992). Different sizes of sediment have been suggested for Dx in Eq. (6.21). Statistically, D50 (the grain size for which 50% of the bed material is finer) is most readily available and meaningful. Physically, a representative size larger than D50 is more meaningful to estimate TABLE 6.1 Ratio of Nikuradse Equivalent Roughness Size and Sediment Size for Rivers. Investigator Ackers and White (1973) Strickler (1923) Keulegan (1938) Meyer-Peter and Muller (1948) Thompson and Campbell (1979) Hammond et al. (1984) Einstein and Barbarossa (1952) Irmay (1949) Engelund and Hansen (1967) Lane and Carlson (1953)

Measure of Sediment Size, Dx D35 D50 D50 D50 D50 D50 D65 D65 D65 D75

(X^ = ks/Dx 1.23 3.3 1 1 2.0 6.6 1 1.5 2.0 3.2

TABLE 6.1. (Continued) Investigator

Measure of Sediment Size, Dx

Gladki (1979) Leopold et al. (1964) Limerinos (1970) Mahmood(1971) Hey (1979), Bray (1979) Ikeda(1983) Colosimo et al. (1986) Whiting and Dietrich (1990) Simons and Richardson (1966) Kamphuis (1974) van Rijn (1982)

D80 D84 D84 D84 D84 D84 D84 D84 D85 D90 D90

ens = ks/Dx 2.5 3.9 2.8 5.1 3.5 1.5 3~6 2.95 1 2.0 3.0

SOURCE: Adapted from Yen (1992) flow resistance because of the dominant effect by large sediment particles. In flow over a geometrically smooth, fixed boundary, the apparent roughness of the bed ks can be computed using Nikuradse's approach. However, once the transport of bed material has been instigated, the characteristic grain diameter and the thickness of the viscous sublayer no longer provide the relevant length scales. The characteristic length scale in this situation is the thickness of the layer where the sediment particles are being transported by the flow, usually referred to as the bedload layer. Once the bed shear stress ib exceeds the critical shear stress for particle motion Tc, the apparent bed roughness ka can be estimated as follows (Smith and McLean, 1997): (6 22) *• = * £^ + k° where OC0 = 26.3, ks is Nikuradse's fixed-bed roughness, and p5 is the bed sediment density. This approach is particularly suitable for sand bed rivers. Under intense sediment transport conditions, bedforms, such as dunes, can develop. In this situation, the apparent roughness also will be influenced by the form drag caused by the presence of bedforms. Nikuradse's approach is valid only for grain-induced roughness. Methods for flow resistance in the presence of both bedforms and grain roughness are presented later.

6.3

SEDIMENTPROPERTIES

6.3.1 Rock Types The solid phase of the problem embodied in sediment transport can be any granular substance. In engineering applications, however, the granular substance in question typically consists of fragments ultimately derived from rocks—hence the name sediment transport. The properties of these rock-derived fragments, taken singly or in groups of many particles, all play a role in determining the transportability of the grains under fluid action. The

important properties of groups of particles include porosity and size distribution. The most common rock type one is likely to encounter in the river or coastal environment is quartz. Quartz is a highly resistant rock and can travel long distances or remain in place for long periods without losing its integrity. Another highly resistant rock type that is often found together with quartz is feldspar. Other common rock types include limestone, basalt, granite, and more esoteric types, such as magnetite. Limestone is not a resistant rock; it tends to abrade to silt rather easily. Silt-sized limestone particles are susceptible to solution unless the water is buffered sufficiently. As a result, limestone typically is not a major component of sediments at locations distant from its source. On the other hand, it often can be the dominant rock type in mountain environments. Basaltic rocks tend to be heavier than most rocks composing the earth's crust and typically are brought to the surface by volcanic activity. Basaltic gravels are relatively common in rivers that derive their sediment supply from areas subjected to vulcanism in recent geologic history. Basaltic sands are much less common. Regions of weathered granite often provide copious supplies of sediment. Although the particles produced by weathering are often in the granule size, they often break down quickly to sand size. Sediments in the fluvial or coastal environment in the size range of silt, or coarser, are generally produced by mechanical means, including fracture or abrasion. The clay minerals, on the other hand, are produced by chemical action. As a result, they are fundamentally different from other sediments in many ways. Their ability to absorb water means that the porosity of clay deposits can vary greatly over time. Clays also display cohesivity, which renders them more resistant to erosion. 6.3.2 Specific Gravity The specific gravity of sediment is defined as the ratio between the sediment density ps and the density of water p. Some typical specific gravities for various natural and artificial sediments are listed in Table 6.2. 6.3.3

Size

Herein, the notation D is used to denote sediment size, the typical units of which are millimeters (mm) for sand and coarser material or microns (JLI) for clay and silt. Another standard way of classifying grain sizes is the sedimentological scale, according to which TABLE 6.2 Specific Gravity of Rock Types and Artificial Material Rock type or materia/ quartz limestone basalt magnetite plastic coal walnut shells

Specific gravity p/p 2.60 ~ 2.70 2.60 ~ 2.80 2.70 ~ 2.90 3.20 ~ 3.50 1.00 ~ 1.50 1.30-1.50 1.30- 1.40

D = 2*

(6.23)

Taking the logarithm of both sides, it is seen that O = - log2(D) = - Igg-

(6.24)

Note that the size O = O corresponds to D = 1 mm. The usefulness of the O scale will become apparent upon a consideration of grain size distributions. The minus sign has been inserted in Eq. (6.24) simply as a matter of convenience to sedimentologists, who are more accustomed to working with material finer than 1 mm than they are with coarser material. The reader should always recall that larger O implies finer material. The O scale provides a simple way of classifying grain sizes into the following size ranges in descending order: boulders, cobbles, gravel, sand, silt, and clay. (Table 6.3). Note that the definition of clay according to size (D < 2JJL) does not always correspond to the definition of clay according to mineral. That is, some clay-mineral particles can be coarser than this limit, and some silt-sized particles produced by grinding can be finer than that. In general, however, the effect of viscosity makes it difficult to grind up particles in water to sizes finer than 2 JJL. In practical terms, there are several ways to determine grain size. The most popular way for grains ranging from O= -4 to O = -4 (0.0625 to 16 mm) is with the use of sieves. Each sieve has a square mesh, the gap size of which corresponds to the diameter of the largest sphere that would fit through it. Thus, the grain size D so measured corresponds exactly to the diameter only in the case of a sphere. In general, the sieve size D corresponds to the smallest sieve gap size through which a given grain can be fitted. For coarser grain sizes, it is customary to approximate the grain as an ellipsoid. Three lengths can be defined. The length along the major (longest) axis is denoted as a, the length along the intermediate axis is denoted as b, and the length along the minor (smallest) axis is denoted as c. These lengths are typically measured with a caliper. The value b is then equated to grain size D. For grains in the silt and clay sizes, many methods (hydrometer, sedigraph, and so forth) are based on the concept of equivalent fall diameter. That is, the terminal fall velocity vs of a grain in water at a standard temperature is measured. The equivalent fall diameter D is the diameter of the sphere having exactly the same fall velocity under the same conditions. Sediment fall velocity is discussed in more detail below. A variety of other more recent methods for sizing fine particles rely on blockage of light beams. The blocked area can be used to determine the diameter of the equivalent circle: i.e., the projection of the equivalent sphere. It can be seen that all the above methods can be expected to operate consistently as long as grains shape does not deviate too greatly from a sphere. In general, this turns out to be the case. There are some important exceptions, however. At the fine end of the spectrum, mica particles tend to be platelike; the same is true of shale grains at the coarser end. Comparison with a sphere is not necessarily an especially useful way to characterize grain size for such materials. 6.3.4 Size Distribution Any sample of sediment normally contains a range of sizes. An appropriate way to characterize these samples is by grain size distribution. Consider a large bulk sample of sediment of given weight. Let Pj(D)—or p/O)—denote the fraction by weight of material in the sample of material finer than size £>(O). The customary engineering representation of

TABLE 6.3 Sediment Grade Scale Class Name

Size Range Millimeters

Very large boulders 4,096 - 2,048 Large boulders 2,048 - 1,024 Medium boulders 1,024 - 512 Small boulders 512 - 256 Large cobbles 256 - 128 128-64 Small cobbles Very coarse gravel 64-32 32- 16 Coarse gravel Medium gravel 16-8 8-4 Fine gravel Very fine gravel 4-2 Very coarse sand 2.000 - 1.000 Coarse sand 1.000 - 0.500 Medium sand 0.500 - 0.250 Fine sand 0.250 - 0.125 Very fine sand 0.125 - 0.062 Coarse silt 0.062 - 0.031 Medium silt 0.031 - 0.016 Fine silt 0.016 - 0.008 Very fine silt 0.008 - 0.004 Coarse clay 0.004 - 0.0020 Medium clay 0.0020 - 0.0010 Fine clay 0.0010 - 0.0005 Very fine clay 0.0005 - 0.00024 SOURCE: AdaptedfromVanoni, 1975.

O

-9- -8 -8 7 -7 6 -6 --5 -5- -4 -4- -3 -3- -2 -2- -1 -1-0 0-1 1 -2 2-3 3-4 4-5 5-6 6-7 7-8 8-9

Microns

Inches 160 - 80 80-40 40-20 20- 10 10-5 5-2.5 2.5 - 1.3 1.3 - 0.6 0.6 - 0.3 0.3 - 0.16 0.16 - 0.08

2,000 - 1,000 1,000 - 500 500 - 250 250 - 125 125 - 62 62-31 31 - 16 16-8 8-4 4-2 2- 1 1 -0.5 0.5 - 0.24

Approximate Sieve Mesh Openings per Inch Tyler U.S. standard

2- 1/2 5 9 16 32 60 115 250

5 10 18 ac 35 60 120 230

C0 n>Qg § |.

I 2. § 3 OI 2 S

the grain size distribution consists of a plot of pf *100 (percentage finer) versus log10(D): that is, a semilogarithmic plot is used. The same size distribution plotted in sedimentological form would involve plotting /y 100 versus O on a linear plot. The size distribution p/O) and size density /?(O) by weight can be used to extract useful statistics concerning the sediment in question. Let x denote some percentage, say 50%; the grain size Ox denotes the size such that x percent of the weight of the sample is composed of finer grains. That is, O^ is defined such that (6 25) PW = jfe It follows that the corresponding grain size of equivalent diameter is given by Dx, where

Dx = 2 -**

(6.26)

The most commonly used grain sizes of this type are the median size D50 and the size Z)90: i.e., 90% of the sample by weight consists of finer grains. The latter size is especially useful for characterizing bed roughness. The density p(O) can be used to extract statistical moments. Of these, the most useful are the mean size Om and the standard deviation a. These are given by the relations. O1n = /Op(O)JO;

a2 = (O - OJ2p(O)DO

(6.27a, b)

The corresponding geometric mean diameter Dg and geometric standard deviation og are given as Dg = 2-^;

(5g = 2°

(6,28a,b)

Note that for a perfectly uniform material, a = O and ag = 1. As a practical matter, a sediment mixture with a value of Gg less than 1.3 is often termed well sorted and can be treated as a uniform material. When the geometric standard deviation exceeds 1.6, the material can be said to be poorly sorted (Diplas and Sutherland, 1988). In fact, one never has the continuous function p(O) with which to compute the moments of Eqs. (6.27a, and b). Instead, one must rely on a discretization. To this end, the size range covered by a given sample of sediment is discretized using n intervals bounded by n + 1 grain sizes O1, O2,..., On + l in ascending order of O. The following definitions are made from / = 1 to n: ®> = ^t + °/+i)

(6-29a)

P1=Pj(Q)-Pf(^+1)

(6.29b)

Eqs. (6.27a and b) now discretize to n n *m = E *tPt °2 = E (*i - ^m)2P, (6.30) I= 1 I= 1 In some cases, especially when the material in question is sand, the size distribution can be approximated as gaussian on the O scale (i.e., log-normal in D). For a perfectly Gaussian distribution, the mean and median sizes coincide: *„ = *50 = ^84 + *,«)

(6-31)

Furthermore, it can be demonstrated from a standard table of the Gauss distribution that the size O displaced one standard deviation larger that Om is accurately given by O84; by symmetry, the corresponding size that is one standard deviation smaller than Om is O16. The following relations thus hold: a = ^(O84 - O16)

(6.32a)

®m = |(84 + *16>

(6-32b)

Rearranging the above relations with the aid of Eqs. (6.28a and b) and Eqs. (6.31 and 6.32a), °.=®f Ds = (D84T)16)"*

(6.33b)

It must be emphasized that the above relations are exact only for a gaussian distribution in O. This is not often the case in nature. As a result, it is strongly recommended that Dg and Gg be computed from the full size distribution via Eqs. (6.3Oa and b) and (6.28a and b) rather than the approximate form embodied in the above relations. 6.3.5 Porosity The porosity Xp quantifies the fraction of a given volume of sediment that is composed of void space. That is, , _ volume of voids P volume of total space If a given mass of sediment of known density is deposited, the volume of the deposit must be computed, assuming that at least part of it will consist of voids. In the case of well-sorted sand, the porosity often can take values between 0.3 and 0.4. Gravels tend to be more poorly sorted. In this case, finer particles can occupy the spaces between coarser particles, thus reducing the void ratio to as low as 0.2. Because so-called open-work gravels are essentially devoid of sand and finer material in their interstices, they may have porosities similar to sand. Freshly deposited clays are notorious for having high porosities. As time passes, the clay deposit tends to consolidate under its own weight so that porosity slowly decreases. The issue of porosity becomes of practical importance with regard to salmon spawning grounds in gravel-bed rivers, for example (Diplas and Parker, 1985). The percentage of sand and silt contained in the sediment is often referred to as the percentage of fines in the gravel deposit. When this fraction rises above 20 or 26 percent by weight, the deposit is often rendered unsuitable for spawning. Salmon bury their eggs within the gravel, and a high fines content implies a low porosity and thus reduced permeability. The flow of groundwater necessary to carry oxygen to the eggs and remove metabolic waste products is impeded. In addition, newly hatched fry may encounter difficulty in finding enough pore space through which to emerge to the surface. All the above factors dictate lowered survival rates. Chief causes of elevated fines in gravel rivers include road building and clear-cutting of timber in the basin.

6.3.6

Shape

Grain shape can be classified in a number of ways. One of these, the Zingg classification scheme, is illustrated here (Vanoni, 1975). According to the definitions introduced earlier, a simple way to characterize the shape of an irregular clast (stone) is by lengths a, b, and c of the major, intermediate, and minor axes, respectively. If the three lengths are equal, the grain can be said to be close to a sphere in shape. If a and b are equal but c is much larger, the grain should be rodlike. Finally, if c is much smaller than b, which in turn, is much larger than a, the resulting shape should be bladelike. 6.3.7

Fall Velocity

A fundamental property of sediment particles is their fall velocity. The relation for terminal fall velocity in quiescent fluid vs can be presented as

*-[hy

where

Rff = ^VR^D

(6.35a)

RP=V-^-

(6.35b)

and the functional relation C0 = CD(Rp) denotes the drag curve for spheres. This relation is not particularly useful because it is not explicit in V5; one must compute fall velocity by trial and error. One can use the equation for C0 given below C0 = ^ (1 + 0.152/?/'2 + 0.015 lRp) P

(6.36)

and the definition Rep = ^j^

(6.37)

to obtain an explicit relation for fall velocity in the form of Rf versus Rep. In Fig. 6.2, the ranges for silt, sand, and gravel are plotted for v = 0.01 cm2/s (clear water at 2O0C) and R = 1.65 (quartz). A good summary of relations for terminal fall velocity for the case of nonspherical (natural) particles can be found in Dietrich (1982), who also proposed the following useful fit: Rf = expi-b, + b2\n(Rep) - b,[\n(RepW ~ b4[\n(RepW + b5\\n(Re^Y}

(6.38)

where b, = 2.891394, b2 = 0.95296, b3 = 0.056835, b4 = 0.002892, and b5 = 0.000245 6.3.8 Relation Between Size Distribution and Stream Morphology The study of sediment properties and, in particular, size distribution is most relevant to the context of stream morphology. The following discussion points out some of the more interesting issues.

FIGURE 6.2 Sediment fall velocitydiagram

PERCENT FINER OY HEIGHT

In Fig. 6.3, several size distributions from the sand-bed Kankakee River in Illinois, are shown (Bhowmik et al., 1980). The characteristic S shape suggests that these distributions might be approximated by a gaussian curve. The median size D50 falls near 0.3 to 0.4 mm. The distributions are tight, with a near absence of either gravel or silt. For practical purposes, the material can be approximated as uniform. In Fig. 6.4, several size distributions pertaining to the gravel-bed Oak Creek in Oregon, are shown (Milhous, 1973). In gravel-bed streams, the surface layer ("armor" or "pavement") tends to be coarser than the substrate (identified as "subpavement" in the figure). Whether the surface or substrate is considered, it is apparent that the distribution ranges over a much wider range of grain sizes than is the case in Fig. 6.3. More specifically, in the distributions of the sand-bed Kankakee River, O varies from about O to about

I linois - Indiana State Line 3 31/79 - 4/21/79

PERCENT FIHER OY WEIGHT

GRAIN SIZE, mm

Illinois - Indiana State Line 4/29/79 - 9/14/79

Pavement Sub pavement Composite of Pavement

and Subpavement

Sediment Size in Millimeters FIGURE 6.4 Size distribution of bed material samples in Oak Creek. Oregon. Source: (Milhous, 1973)

PERCENTAGE FINER (Per cent)

3, whereas in Oak Creek, O varies from about -8 to about 3. In addition, the distribution of Fig. 6.4 is upward-concave almost everywhere and thus deviates strongly from the gaussian distribution. These two examples provide a window toward generalization. A river can be loosely classified as sand-bed or gravel-bed according to whether the median size D50 of the surface material or substrate is less than or greater than 2 mm. The size distributions of sandbed streams tend to be relatively narrow and also tend to be S shaped. The size distributions of gravel-bed streams tend to be much broader and to display an upward-concave shape. Of course, there are many exceptions to this behavior, but it is sufficiently general to warrant emphasis. More evidence for this behavior is provided in Fig. 6.5. Here, the grain size distributions for a variety of stream reaches have been normalized using the median size D50. Four sand-bed reaches are included with three gravel-bed reaches. All the sand-bed distributions are S shaped, and all have a lower spread than the gravel-bed distributions. The standard deviation is seen to increase systematically with increasing D50(White et al., 1973). The three gravel-bed size distributions differ systematically from the sand-bed distributions in a fashion that accurately reflects Oak Creek (Fig. 6.4). The standard deviation in all cases is markedly larger than any of the sand-bed distributions, and the distributions are upward-concave except perhaps near the coarsest sizes.

River Miss, Tarbeii La. Niobran Miss, at Sl Louis Mountain free* Elbow Aart

DfMENSIONLESS GRAIN SIZE COMPOSITION

(0,/O3^

FIGURE 6.5 Dimensionless grain-size distribution for different rivers (White et al., 1973)

6A

THRESHOLD CONDITION FOR SEDIMENT MOVEMENT

When a granular bed is subjected to a turbulent flow, virtually no motion of the grains is observed at some flows, but the bed is mobilized noticeably at other flows. Factors that affect the mobility of grains subjected to a flow are summarized below:

I grain placement randomness ^ [turbulence f fluid 1/lift mean & turbulent forces on grain < I drag I gravity In the presence of turbulent flow, random fluctuations typically prevent the clear definition of a critical, or threshold condition for motion: The probability for the movement of a grain is never precisely zero (Lavelle and Mofjeld, 1987). Nevertheless, it is possible to define a condition below which movement can be neglected for many practical purposes. 6.4.1 Granular Sediment on a Stream Bed Figure 6.6 is a diagram showing the forces acting on a grain in a bed of other grains. When critical conditions exist and the grain is on the verge of moving, the moment caused by the critical shear stress Tc about the point of support is just equal to that of the weight of the grain. Equating these moments gives (Vanoni, 1975): Tc = — (Js ~ J) Dcos §(tan 9 - tanty) C2a2

(6.39)

in which ys = specific weight of sediment grains, y = specific weight of water, D = diameter of grains, is the slope angle of the stream, 6 = the angle of repose of the sediment, C1

FIGURE 6.6 Forces acting on a sediment particle on an inclined bed

and C2 are dimensionless constants, and av and a2 are lengths shown in Fig. 6.6. Any consistent set of units can be used in Eq. (6.39). For a horizontal bed, Eq. (6.39) reduces to Tc = — (Y, - J)D tan 6 C2#2

(6.40)

For an adverse slope (i.e., 4> < O), C CL Tc = -^- (Y, - J)D cos §(tan 0 + tan ) (6.41) C2#2 Equations (6.39), (6.40), and (6.41) cannot be used to give ic because the factors C1, C2, G1, and a2 are not known. Therefore, the relation between the pertinent quantities is expressed by dimensional analysis, and the actual relation is determined from experimental data. Figure 6.7 is such a relation, first presented by Shields (1936) and carries his name. The curve is expressed by dimensionless combinations of critical shear stress Tc, sediment and water specific weights Y5 and Y, sediment size D, critical shear velocity w*c = Vi/p and kinematic viscosity of water v. These quantities can be expressed in any consistent set of units. Dimensional analysis yields,

«-«^5-4^) The Shields values of Tc* are commonly used to denote conditions under which bed sediments are stable but on the verge of being entrained. Not all workers agree with the results given by the Shields curve. For example, some workers give Tc* = 0.047 for the dimensionless critical shear stress for values of R* = u*D/v in excess of 500 instead of 0.06, as shown in Fig. 6.7. Taylor and Vanoni (1972) reported that small but finite amounts of sediment were transported in flows with values of i* given by the Shields curve. The value of Tc to be used in design depends on the particular case at hand. If the situation is such that grains that are moved can be replaced by others moving from upstream, some motion can be tolerated, and the Shields values can be used. On the other hand, if grains removed cannot be replaced, as on a stream bank, the Shields value of ic are too large and should be reduced. The Shields diagram is not especially useful in the form of Fig. 6.7 because to find Tc, one must know w* = VT/P. The relation can be cast in explicit form by plotting TC* versus Rep, noting the internal relation _u*D 1_ = v

u _ VfltfD U s D L v VRgD

= (T*)i/2/fc

(6.43)

p —p where R = — is the submerged specific gravity of the sediment. A useful fit is given

byBrownlie(1981a): T* = 0.22Re-™ + o.06 exp(- 17.77/te;006)

(6.44)

With this relation, the value of Tc* can be computed readily when the properties of the water and the sediment are given. The value of bed-shear stress ib for a wide rectangular channel is given by ib = jHS, as shown earlier. The average bed-shear stress for any channel is given by ib = JR^S, in which Rh = the hydraulic radius of the channel cross section.

Amber

Fully developed turbulent velocity profile

Turbulent boundary layer Value of

Boundary Reynolds Number, R* FIGURE 6.7 Shields diagram for initiation of motion. Source Vanoni (1975)

**• (Shields) Granite Barite Sand (Casey) Sand (Kramer) Sand (U.S. WES.) Sand (Gilbert) Sand (White) Sand in air (White) Steel shot (White)

6.4.2 Granular Sediment on a Bank A sediment grain on a bank is less stable than one on the bed because the gravity force tends to move it downward (Ikeda, 1982). The ratio of the critical shear stress iwc for a particle on a bank to that for the same particle on the bed ic is (Lane, 1955) V L (tan Cb1 > 17 = 00 ^ 1 V 1 -MJ

where (J)1 is the slope of the bank and 0 is the angle of repose for the sediment. Values of 0 are given in Fig. 6.8 after Lane (1955) and also can be found in Simons and Senturk (1976). 6.4.3 Granular Sediment on a Sloping Bed

Angle of Repose, 9, in Degrees

Equation (6.39) shows that ic diminishes as the slope angle (|> increases. For extremely small