1064 S SEDIMENT TRANSPORT AND EROSION INTRODUCTION The literature on the subject of erosion and sediment transport is vast and is treated in the publication of such disciplines as civil engineering, soil science, agriculture, geography and geol- ogy. This article provides a brief introduction to the subjects of soil erosion, transport of detritus by streams and the response of a stream channel to changes in its sediment characteristics. The agriculturalist is concerned with the loss of fertile land through erosion. Sheet, gully and other erosion mecha- nisms result in the annual movement of about five billion tons of sediment in the United States. 1 By this process, plant nutrients and humus are washed away and conveyed to the streams, reservoirs, and lakes. The sediment characteristics of a stream also affect its aquatic life. Changes in the character of the sediment load will normally tend to change the balance of aquatic life. Fine sediment, derived from sheet erosion, causes turbidity in the waterways. This turbidity may interfere with photosynthesis and with the feeding habits of certain fish, thus favoring the less susceptible (often less desirable) varieties of fish. The resulting mud deposits may have similar selective results on spawning. The plant nutrients (phosphates and nitrates) that accompany erosion from farmlands may contribute to the eutrophocation of the receiving waters. Turbidity also makes waters less desirable for municipal and industrial use. Mud deposits may ruin sand beaches for recreational use. From the engineer’s point of view an understanding of sed- iment transport processes is essential for proper design of most hydraulic works. For example, the construction of dam on a stream is almost always accompanied by, a reservoir siltation or aggradation problem and a degradation problem. A reason- able prediction of the rate of reservoir siltation is necessary in order to establish the probable useful life and thus the econom- ics of a proposed reservoir. The degradation or erosion of the downstream channel and the consequent lowering of the river level may, unless properly accounted for, endanger the dam and other downstream structures (due to under-cutting). After construction of the Hoover Dam the bed of the Colorado River downstream from the dam started to degrade. In 12 years the bed level dropped about 14 feet (Brown 1 ). In addition the downstream channel may change its regime (i.e. its dominant stable geometry). For example, a wide braided channel or delta area may become a much nar- rower and deeper meandering channel thus affecting the prior uses of the stream. This appears to have happened as a result of the Bennett Dam on the Peace River in British Columbia. 2 CLASSIFICATION OF STREAMBORNE SEDIMENTS Terminology The materials transported by a stream may be grouped under the following type of load: 1) dissolved load, 2) bed load, 3) suspended load. The dissolved load, although a significant portion of total stream load, is generally not considered in sediment trans- port processes. According to Leopold, Wolman and Miller, 3 the dissolved load in US streams increases with increas- ing annual runoff, reaching a maximum of about 125 tonsր sq.mileրyear for runoffs of 10 inchesրyear or more. Bed load consists of granular particles, derived from the stream bed, which are transported by rolling, skipping or slid- ing near the stream bed. Einstein 4,5 defines the bed load as the sediment discharge within the bed layer which he assumes to have an extent of two sediment grain diameters from the bed. Suspended sediment load is that part of the sediment load which is transported within the main body flow, i.e. above the bed layer in Einstein’s terminology. Turbulent diffusion is the primary mechanisms of maintaining the sediment particles in suspension. The suspended load may be subdivided into: 1) Wash load which consists of fine sediments mainly derived from overland erosion and not found in C019_001_r03.indd 1064C019_001_r03.indd 1064 11/18/2005 11:05:59 AM11/18/2005 11:05:59 AM © 2006 by Taylor & Francis Group, LLC SEDIMENT TRANSPORT AND EROSION 1065 significant quantities in alluvial beds; often wash load is arbitrarily taken to be sediments finer than 0.062 mm, i.e. silts and clays. 2) Suspended bed material load which is the portion of the suspended load derived primarily from the channel bed; generally the bed material is assumed to be the sediment coarser than 0.062 mm, i.e. sands and gravels. Table 1 indicates the terminology used by the American Geophysical Union in describing various sizes of sediment. Properties of Sediments An excellent review of the properties of sediments is presented by Brown. 1 He discussed the determination and significance of the following: a) properties of the individual particle, b) particle size distribution and c) bulk properties of sediments. Properties of the Particle Neglecting interaction effects, the behavior of an individual particle in a stream depends on its size, specific weight, shape, and the hydraulics of the stream. Two commonly used methods of determining particle size are: (1) mechanical sieve analysis and (2) the fall velocity method. The sieve analysis method differentiates particle size on the basis of whether or not the particle will pass through a certain standard square opening in a sieve or mesh. This method is applicable for sands or coarser particles. Except in the case of spheres, “sieve size” will only be an approxima- tion to the true equivalent diameter of the particle since the results depend to some extent on the particle shape. The fall velocity method of determining the effective sedi- ment size is gaining popularity in sediment transport research. On the basis of the particle’s terminal velocity, in a specified fluid (water) at a specified temperature, the particle is assigned a fall or sedimentation diameter equal to the diameter of the quartz sphere which has the same terminal velocity in the same fluid at the same temperature. 1 This particle size inte- grates the effects of grain size, specific weight and shape into a single meaningful parameter for sediment transport studies. Researchers at Colorado State University have developed a Visual Accumulation Tube to aid in the determination of the fall diameter distributions for sediment samples. Particle Size Distribution On the basis of a sieve analysis of fall diameter analysis, of a sediment sample, a cumulative frequency curve for the particle size can be drawn. Figure 1 shows typical particle size frequency curves for a sample taken from a sandy stream bed and for a sample of suspended load over the same bed. 6 The frequency curves are usually plotted on logarithmic-probability paper. TABLE 1 Sediment grade scale Group Particle size range, mm Boulders 4096–256 Cobbles 256–64 Gravel 64–2 Sand 2–0.062 Silt 0.62–0.004 Clay 0.004–0.00024 FIGURE 1 Typical particle size frequencies curves for stream sediments (after Bishop et al.). GRAIN SIZE (mm.) IN TRANSPORT (by dunes) PERCENT FINER (by weight) ON THE BED 1.0 .01 .05 .1 .2 .5 1 2 5 10 20 30 40 50 60 70 80 90 95 96 99 99.5 99.9 99.99 .10 .08 .20 .30 .40 .50. .60 .80 C019_001_r03.indd 1065C019_001_r03.indd 1065 11/18/2005 11:05:59 AM11/18/2005 11:05:59 AM © 2006 by Taylor & Francis Group, LLC 1066 SEDIMENT TRANSPORT AND EROSION Some important descriptors of the frequency distribution are: (1) the median size or d 50 , that is, the size for which 50% by weight of the sample has smaller particles; (2) the scatter of particle size as indicated, for example, by the standard devia- tion or perhaps the geometric deviation; (3) the characteristic grain roughness which has been associated with the d 65 ; 5,7,8 (4) the d 35 has also been used as characteristic sediment size. 9 Bulk Properties The determination of bulk, in place specific weights of sediments is discussed under Reservoir Sedimentation. EROSION Most of the sediment in streams is produced by the following processes: 1 1) Sheet erosion, 2) Gully erosion, 3) Stream channel erosion, 4) Mass movements of soil (e.g. landslides and soil creep), 5) Erosion to construction works, 6) Solids wastes from municipal, industrial, agricul- tural and mining activities. Morisawa, 10 using a system diagram, similar to Figure 2, sum- marizes the inter-relation of climatic and geologic factors that influence soil erosion and runoff. Figure 2 also shows man’s influence on the system. Langbein and Schumm 3,17 proposed the correlation shown in Figure 3 between annual sediment yield and effec- tive annual precipitation for the United States. The effec- tive precipitation is the adjusted precipitation which would have produced the observed runoff for an annual mean temperature of 50°F. A recent paper by Saxton et al. 11 relates total runoff, surface runoff and land use practices to the sediment yield from loessial watersheds in Iowa. This paper compares erosion and surface runoff from contoured-corn watersheds and from pastured- grass and level-terraced areas. In a 6-year study the contoured- corn areas yielded, annually, about 19,000 tonsրsq. mile of sheet erosion plus 3000 tonsրsq. mile of gully erosion while the level terraced and grassed watersheds yielded about 600 tonsր sq. mile. Similarly the surface runoff from the contoured-corn areas was approximately 5 inches compared with 1.5 inches for the level-terraced and grassed areas. The experimental watersheds were of the order of 100 square miles. Other land use factors are discussed in a paper by Dawdy 12 who presents sediment yields for the state of Maryland. The annual sediment yield from heavily wooded areas is about 15 tonsրsq. mile compared with 200 to 900 for crop land. The annual sediment yields from urban development areas (usually only a few acres) varied from about 1000 to 140,000 tonsրsq. mile. Curtis 13 obtained annual sediment yields of 390 and 290 for two watersheds (264 and 651 square miles) in the Miami Conservancy District, Ohio. A number of empirical formulae have been devel- oped 1,14,15 to permit estimation of rates of overland erosion. The US Department of Agriculture developed the universal soil-loss equation (for upland areas), E ϭ RKLSCP, (1) where E ϭ soil lossրunit area; R ϭ rainfall runoff factor; K ϭ soil erodibility factor; L ϭ slope length factor; S ϭ slope steepness factor; C 1 ϭ crop management factor; and P 1 ϭ erosion control practice factor. Details for estimating the above factors are given by Meyer. 15 MEASUREMENT OF SEDIMENT DISCHARGE Samplers have been developed to measure both suspended and bed load in streams. However bed-load samplers are not temp, rain rock type topography excavations fills reservoirs CLIMATE GEOLOGY SOIL CHARACTER SOIL EROSION (RUNOFF) RAINFALL VEGETATION MAN farming lumbering amount intensity duration MAN eg. eg. FIGURE 2 The relationship of climate and geology to soil erosion (adapted from Morisawa). C019_001_r03.indd 1066C019_001_r03.indd 1066 11/18/2005 11:05:59 AM11/18/2005 11:05:59 AM © 2006 by Taylor & Francis Group, LLC SEDIMENT TRANSPORT AND EROSION 1067 widely used because of their doubtful accuracy. Generally, only suspended load samples are collected in samplers of the type shown schematically in Figure 4. This sampler is designed so that the intake velocity is nearly the same as the local stream velocity. The extent of the suspended sampled zone is limited by the size of the sampler. Methods or extrapolating these measurements and estimating bed load are discussed in the next section. For more details of sediment measurement techniques and equipment, the reader is referred to Nordin and Richardson, 15 010 20 30 40 50 60 200 400 600 800 1000 EFFECTIVE PRECIPITATION (inches/year) SEDIMENT YIELD (tons/sq.mi/year) T = 50°F FIGURE 3 Sediment yield in the United States (after Langbein and Schumm). FLOW AIR WATER SEAL INTAKE SAMPLE BOTTLE SPRING EXHAUST VENT FIGURE 4 Sketch of a suspended load sampler. C019_001_r03.indd 1067C019_001_r03.indd 1067 11/18/2005 11:05:59 AM11/18/2005 11:05:59 AM © 2006 by Taylor & Francis Group, LLC 1068 SEDIMENT TRANSPORT AND EROSION Shen, 15 Karaki, 15 Graf, 16 Brown, 1 Simons, and Senturk and the ASCE Sedimentation Engineering Manual. In some instances 15 turbulence flumes (a concrete lined reach with baffles to create severe turbulence) have been con- structed across a stream channel in order to suspend the bed load and thus to sample it by suspended load techniques. THE MECHANICS OF SEDIMENT TRANSPORT IN A STREAM General The nature of sediment transport in a stream depends on the shear intensity of the flow and the type of bed material. The diagram in Figure 5 shows the sequence of bed forms (waves) associated with increasing levels of shear on a fine granular bed material. 3 This figure also shows, schematically, the typi- cal changes in the Darcy friction factor and the sediment con- centration with increasing flow velocity. The primary mode of transport of particles, in the case of ripples, 17 is discrete steps along the bed; however with increasing shear more of the bed material becomes suspended until the particle motion is nearly continuous for anti-dunes. Dunes and ripples are triangular in shape with relative flat upstream slopes and sleep downstream slopes. The water surface waves are out of phase with the dune forma- tion while ripple formations appear to be independent of the free surface. Dune wave lengths, ␭ d , are related to the depth of flow and in general, l d Ͼ 3 feet (2) whereas ripple wave lengths ␭ r are shorter, 2" Շ l r Շ 18" (3) Dune heights, H d , are related to the depth of flow, with the limiting height approaching the average flow depth. The ratio of dune length to height is given by 17 815ՇՇ l H d . (4) The maximum ripple height is about 0.1 feet. Both ripples and dunes progress downstream. The tran- sition from dunes to anti-dunes occurs at a Froude number close to 1.0. Anti-dunes, as indicated by Figure 5, are in phase with the surface wave. The may be stationary or move upstream. The maximum height of an anti-dune is approximately equal to the flow depth at the trough of the surface wave. Simons, 17,18 on the basis of experimental data, developed the relationship shown in Figure 6 between stream power and bedform for varying fall diameters. Simons and Richardson 19 also studied the variation of Chézy’s C with bed form. Their results are summarized in Table 2. 0 12 34 56 V feet/sec. Friction Factor f 0.02 0.04 0.06 0.08 0.1 1 10 100 1,000 10,000 100,000 Concentration of Bed Material C ppm. Flat Bed Ripples Dunes Transition Antidunes C f FIGURE 5 The behavior of a mobile stream bed (adapted from Leopold, Wolman, and Miller). C019_001_r03.indd 1068C019_001_r03.indd 1068 11/18/2005 11:06:00 AM11/18/2005 11:06:00 AM © 2006 by Taylor & Francis Group, LLC SEDIMENT TRANSPORT AND EROSION 1069 Initial Motion White, in 1940, 20,21 using an analytical approach, showed that, for sufficiently turbulent flow over a granular bed, the critical shear or shear to initiate grain movement is t c ϭ k c g f ( S s –1) d, (5) in which k c ; 0.06; g f ϭ fluid specific weight; S s ϭ specific gravity of sediment grain; d ϭ grain diameter. Shields, 21 using an experimental approach, obtained the more general equation t c ϭ g f ( S s –1) d f ( R * ), (6) in which R * ϭ U * dր n; U * ϭ friction velocity; n ϭ kinematic viscosity; and f ( R * ) is defined in Figure 7. Permissible or allowable tractive stresses for use in chan- nel designs with granular or cohesive boundaries are given by Chow. 7 Bed Load Formulae When the bed shear, t o , due to the flowing stream exceeds the critical shear, t c , a part of the bed material starts to move in a layer of the stream near the bed, i.e. the bed layer. Experimental 0.2 0.4 0.6 0.8 1.0 1.2 0 Median Fall Diameter in mm. 0.001 0.002 0.004 0.006 0.008 0.01 0.02 0.04 0.06 0.08 0.1 0.1 0.2 0.4 0.6 0.8 1.0 1.0 10 4.0 2.0 Stream power, tV, lbs/ft – sec. Stream power, tV, gms/cm – sec. Upper Region Transition Dunes Ripples Plane FIGURE 6 Relation of stream power and median fall diameter to bed form (after Simons). TABLE 2 Chézy C in sand channels Regime Bed Form Cl √g (where C is Chézy C) Lower regime ripples d 50 Ͻ 0.6 mm 7.8 to 12.4 dunes 7.0 to 13.2 transition 7.0 to 20 Upper regime plane bed 16.3 to 20 anti-dune {standing wave 15.1 to 20 {breaking wavechutes and pools“slug” flow 10.8 to 16.3 9.4 to 10.7 — C019_001_r03.indd 1069C019_001_r03.indd 1069 11/18/2005 11:06:00 AM11/18/2005 11:06:00 AM © 2006 by Taylor & Francis Group, LLC 1070 SEDIMENT TRANSPORT AND EROSION studies 3 indicate that this sediment discharge, known as the bed load, q B , is a function of the excess of t o above t c or q B ϰ fcn ( t o — t c ). (7) Figure 8 illustrates a typical experimental relationship between q B and ( t o — t c ). DuBoys in 1879 22 treated the bed material, involved in the bed load, as it if consisted of sliding layers which respond to and distribute the applied stress t o . He proposed the relation q B ϭ C s t o ( t o – t c ) (8) Both C s and t c depend on particle size as indicated in Table 3. Chang, 1 Schoklitsch, 1,16 MacDougall, 1 and Shields 1 have presented bed load formulae similar to Eq. (8). The theoretical bed load model developed by Einstein 4,5,8,24 has formed the basis for a number of researches in sediment transport. 6,15,24,36 Einstein utilized: (1) the statistical nature of turbulent flow; (2) the fact that in steady uniform flow there is an equilibrium between the processes of erosion and depo- sition, that is, (probability of erosion) ϭ (the probability of deposition); (3) the fact that grains near the bed are more in quick “steps” interrupted by “rest” periods; (4) a separate hydraulic radius, R ′, associated with grain roughness and another hydraulic radius, R ′′, associated with the bed form. Einstein obtained the erosion probability function by assuming that the lift force, on a grain, consists of an aver- age component [related to ( U * ) 2 ] and a normally distributed random component. Einstein thus obtained the “bed load” equation A A t B B o o ∗∗ ∗∗ −− − ∗∗ ∗∗ ∫ ⌽ ⌽1 1 1 1 ϩ ϭ p ch ch d (/ ) (/ ) (9) in which ⌽ϭ Ϫ ∗ iq igdS BB bs s g 3 1() (10) is Einstein’s bed transport function; A * ϭ 43.5; B * ϭ 0.143; h o ϭ 1ր2; cj ∗ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ ϭ ⌬ Ϫ ϪЈ Y X S d SR s log log . () 10.6 2 10 6 1 (11) in the Einstein flow intensity function; i B ϭ fraction of q B in the size range associated with d; d ϭ geometric mean of particle 1.0 0.01 0.1 1.0 10 100 1000 R s = U s f n f(R s ) Laminar flow of bed Turbient flow of bed FIGURE 7 Shields’ critical shear function (adapted from Henderson). 0.001 0.003 0.005 0.007 0.009 0.011 0.013 0.0 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.01 T 0 (gm/cm 2 ) SEDIMENT LOAD (gm./sec./cm.) T 0 FIGURE 8 Typical relation of shear and sediment load (adapted from Leopold, Wolman and Miller). TABLE 3 Typical values of C s and t c (after Straub 22,23 ) d mm 1 8 1 4 1 2 124 C s ft 6 2 16 Ϫsec 0.81 0.48 0.29 0.17 0.10 0.06 t c lb ft 2 0.016 0.017 0.022 0.032 0.051 0.09 C019_001_r03.indd 1070C019_001_r03.indd 1070 11/18/2005 11:06:00 AM11/18/2005 11:06:00 AM © 2006 by Taylor & Francis Group, LLC SEDIMENT TRANSPORT AND EROSION 1071 size range being considered; S e ϭ energy slope; i b ϭ fraction of bed sediment in specified range; j ϭ hiding factor; Y ϭ lift correction factor; ⌬ ϭ d 65 ր X; X ϭ correction factor for hydrau- lically smooth flow; dЈ ϭ 11.6 n ր U * ; UgRS e ЈϭЈ ∗ (12) X ϭ 0.770 if ⌬ր d Ͼ 1.80; X ϭ 1.39 d if ⌬ր d Ͻ 1.80. Schen 15 gives an up-to-date review of the modern stochas- tic approaches to the bed material transport problem. The Suspended Load Equations of Motion of the Fluid The flow in natural streams is almost always turbulent and may be assumed to be incom- pressible; consequently the applicable equations of motion for the fluid are the Reynolds 25 equations rs ∂ ∂ ∂ ∂ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ∂ ∂ U t U U xx F i j i jj ji i ϩϭ() ,ϩ (13) in which U _ i ϭ ensemble mean point velocity in the direction i; s ij ϭ stress tensor ϭ{ϪP _ d ji ϩmD _ ji Ϫru i u j } ϭ average pressure; d ji ϭ Kronecker delta; m ϭ dynamic viscosity; D _ ji ϭ deforma- tion tensor; Ϫ ru i u j ϭ turbulence or Reynolds Stresses; r ϭ fluid density ; u i ϭ random component of velocity in the i direction; F _ ϭ body force in the i direction. The first term in the stress tensor represents the normal stresses due to the aver- age pressure at a point; the second term represents the viscous shear forces; the last term or Reynolds stress has both normal and tangential components. A common method of simplify- ing equations involves the introduction of an eddy viscosity, ␧ m such that rr␧ϭϪ m ji i jji Duu () . ≠ (14) The requirement that ( i  j ) in Eq. (14) eliminates the normal stresses due to turbulence; in order to account for these normal stresses an average turbulence pressure P _ i is added to P _ thus yielding the simplified stress tensor sdmr ji t ji m i ji PP DϭϪϩϩϩ␧()( ). (15) The fluid continuity equation is ∂ ∂ U x i i ϭ 0. (16) Equations (13), (15), and (16) may be solved in a few cases by methods developed to solve the Navier-Stokes equations. Transport of a Scalar Quantity in Turbulent Flow In an incompressible turbulent fluid the conservation of a scalar quantity requires that the rate of change of a scalar (say c _ plus the rate of generation of c _ at the point or Dc Dt x h c x uc F ii i c ϭϪЈϩ ∂ ∂ ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (17) in which c ϭ c _ ϩcЈ; c _ ϭ ensemble average of c; c Ј ϭ random component of c; D ր Dt ϭ substantial derivative; F _ c_ is the gen- eration term; h ϭ molecular diffusion coefficient. It is usual to introduce, into Eq. (17), an “eddy” transport coefficient ␧ j , such that ϪЈϭ␧uc c x ic i ∂ ∂ . (18) Since in most practical problems ␧ c ϾϾ h, then Eq. (17) can be reduced to Dc tx c x F i c i c d ϭ␧ϩ ∂ ∂ ∂ ∂ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ . (19) Equations (3) and (19) are valid for low sediment concentrations. A review paper by Vasiliev 26 discusses the governing equation which account for various levels of sedi- ment concentrations. For example a first order correction to the Reynolds equations is rs DU Dt x rc F i i ji t ϭϩϩ ∂ ∂ ()( ).1 (20) The volume continuity equation is the same as Eq. (16) while the mass continuity equation becomes Dc Dt x cu v c x i is ϭϪϩ ∂ ∂ ∂ ∂ ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ () 3 (21) in which r ϭ ( S s —1); c _ ϭ average ensemble concentration at a point (massրmass); n s ϭ settling viscosity; x 3 ϭ vertical coordinate (opposite to the direction of n s ). The Vertical Concentration Profile There is no general solu- tion for Eqs. (3), (16), and (19) or (18), (20), and (21); however a few special cases, of practical interest, have been solved. Using the simplifications which result for steady, uni- form flow in two dimensions (as shown in Figures 9 and 10), it is possible to obtain a solution for the vertical velocity, and concentration profiles. The following assumptions are typi- cal of those required to solve Eqs. (13), (14), and (21): a) c _ ϾϾ 1; b) ␧ c ϭ b␧ m where b ; 1; c) F _ x ; grs 0 ; C019_001_r03.indd 1071C019_001_r03.indd 1071 11/18/2005 11:06:00 AM11/18/2005 11:06:00 AM © 2006 by Taylor & Francis Group, LLC 1072 SEDIMENT TRANSPORT AND EROSION d) ∂ ∂ () ; PP x t ϩ ϭ 0 e) tr tro g and g );ϭϭSoD So (D yϪ Chang et al. , used the above assumptions to obtain: (a) the vertical velocity profile, U U U xx ϭϪ␧Ϫ ␧ ϪϪ␧ ϩϩ 2 1 11 1 3 12 * / k 1n ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ (22) and (b) the vertical concentration profile cy c a y a DDa DDy z () , / ϭ Ϫ Ϫ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − () − () ⎧ ⎨ ⎪ ⎩ ⎪ ⎫ ⎬ ⎪ ⎭ ⎪ 12 2 (23) in which U * ϭ friction velocity ϭ gDs 0 ; ␬ϭ fcn ( U * d ր v ) Ϭ 0.4; ␧ ϭ y ր D; Ú ϭ average velocity in vertical; a ϭ reference height; Z ϭ v s ր( bU * k ). Using the Keulegan velocity distributions U yU v x ϭ 575 905 10 .log . ∗ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (for smooth boundaries) (24a) U y d x ϭ 575 30 2 10 65 .log . ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (for smooth boundaries) (24b) Einstein and others 5,1 have obtained a slightly different equa- tion for c ( y ), i.e. cy c a aD y yD a z () . () () ϭ − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ (25) The Suspended Sediment Load The suspended sediment dis- charge q s (weightրunit timeրunit width) above a reference level y ϭ a is given by qUcdy s x a D ϭg ∫ , (26) where U – and c – are given by Eqs. (22) and (23) or (24) and (25). Longitudinal Dispersion Another problem which has received some attention is that of longitudinal diffusion and dispersion in natural streams and estuaries. Several research- ers 16,33,34,35 have sought analytical and numerical solutions for the longitudinal variation in the mean concentration in the vertical, c – . Considering two dimensional longitudinal dispersion, Eq. (17) can be approximated by 16 ∂ ∂ ∂ ∂ ∂ ∂ ˆˆ ˆ c t U c t E c x x L ϩϭ 2 2 (27) in which E L ; coefficient of longitudinal diffusion. A typical 16 value for E L is EUD L ϭ 59 ∗ The solution of Eq. (27) for an initial step change, M o , in concentration, is ˆ (,) . ()/ cxt M Et e o L xUt Et xL = −− 4 2 4 p (29) Other treatments of the dispersion problem may be found in the works of Holley 28 Householder et al. , 29 Chiu et al. , 30 Conover et al. , 31 Sooky, 32 Fischer, 33 Harleman et al. , 34 and Sayre. 35 The Total Sediment Load Einstein 5 developed a unified total bed material formulae by converting his computed bed load, q B to a reference concen- tration at y ϭ a ϭ 2 d. Inserting into Eqs. (25) and (26) he obtained an estimate of q sB the suspended bed material load. Hence the total bed material load per unit width, q TB is q TB ϭ q B ϩ q sB ϭ ⌺ i B q B (1 ϩ P e I 1 ϩ I 2 ) (30) all size ranges S 0 FLOW T 0 =gDS 0 g(D–y)S 0 1 y y D T FIGURE 9 Defining sketch for uniform flow. D U x U x C σ C y S 0 σ FIGURE 10 Defining sketch for velocity and concentration profiles. C019_001_r03.indd 1072C019_001_r03.indd 1072 11/18/2005 11:06:01 AM11/18/2005 11:06:01 AM © 2006 by Taylor & Francis Group, LLC SEDIMENT TRANSPORT AND EROSION 1073 in which I Ky yIKy z AA ee 1 1 2 1 11 11ϭϪϭϪ(); ();/d /lnd ∫∫ y y K ϭ 0.216 AA e Z e Z′′Ϫ Ϫ 1 1/( ) ; A e ϭ 2dր D ; P e ϭ 2.3 log 30.2 ␹d ր d 65 ; Z ′ ϭ v s ր( bU * ␬ ); ( X see Bed Load Formulae ). The total sediment load per unit width in a stream q T is q T ϭ q TB ϩ q w , (31) where q w ϭ wash load (fines) which must be obtained inde- pendently, e.g. by direct measurement. The Einstein method requires a knowledge of: grain size distribution in the bed; the grain density; the energy slope, S e ; and the water temper- ature, in order to compute both bed material load and water discharge for a given depth and width of flow. Colby and Hembree 9,36 modified Einstein’s method in order to compute total sediment load ( q T ). Their procedure utilizes: the sampled suspended load Q s ; measured discharge; measured depths and sampler depths, the extent of the sam- pled zone; and all the data used by the Einstein procedure except S e . Their main modifications are: 1) The finer portion of the total suspended load, Q s , is based on extrapolation of the actual sampled load Q′ s (using Eqs. (25) and (26)). 2) The coarser part of the total load (including the bed load) is computed from a simplified Einstein procedure (using a modification of Eq. (30)). 3) Einstein’s grain shear velocity U′∗ is replaced by an equivalent shear velocity U m based on the Keulegan equations and the measured discharge. 4) Einstein’s flow intensity function ␺ * , is replaced by the larger of C m ϭ 1.65 gd 35 ր( U m ) 2 or C m ϭ 0.66 gd ր( U m ) 2 (32) 5) The modified term ⌽ m is used to enter Einstein’s Eq. (9) to obtain a bed transport function ␺ * ; the modified bed transport function is ⌽ * ϭ ⌽ * ր2. (33) The value of ⌽ m is used to compute the bed load associated with a size range d, i.e. i B q B ; 1200 d 3 ր2 i B ⌽ m lbրsecրft. (34) 6) Using the computed bed load for a certain size range, i B Q B , the measured suspended load in the same size range, I s Q′ s , and Einstein’s Eq. (30) one can obtain a value for Z ′ in Eq. (25) which should be better than a Z ′ based on an estimated v s . Bishop, Simons, and Richardson 6 simplified the Einstein procedure for determining total bed material load. They introduced a single sediment transport function ⌽ T which includes both suspended bed material and bed load. Their flow intensity term is y Ts e S d RS =− ′ ().1 35 (35) The experimental relationship shown in Figure 11 were estab- lished for actual river sediments of various median sizes. Using ⌽ T from Figure 11 the total bed material load per unit width, is q TB = ⌽ Trs (gd) 3/2 (S s –1) 1/2 . (36) The wash load must be added to q TB to obtain the total sediment load. Colby 37 analysed extension laboratory and field data to establish the empirical relationship, shown in Figure 12, for the determination of sand discharge. Figure 12 is valid for a water temperature of 60°F and a flow to moderate wash load (c ^ <10,000 ppm). Colby provides adjustment coefficients for water temperature and wash load. For example, at a flow depth of 10 feet a Ϯ20°F change in temperature would result in about ϩ25% change in the sand discharge and an increase in the concentration of fines from 0 to 100,000 ppm could cause up to 10 fold increase in the sand load. The reader is referred to Graf, 16 Shen, 15 and Chang et al. , 27 for other contributions to the determination of total bed material load. The Annual Sediment Transport In general it is not practical to continuously sample the sedi- ment in a stream; instead, representative samples are taken for various flow conditions and a sediment load versus water discharge or sediment rating curve is established. A typical sediment rating curve is shown in Figure 13. A number of factors contribute to the scatter of data points in Figure 13. The sediment load is out of phase with the discharge hydro- graph as illustrated by Figure 14. The sediment load depends on the season of the year. Using the available stream flows and the sediment rating curve an average annual sediment transport can be estimated. Often the bed load is not included in the sediment rating curve; if this is the case, the bed load may be computed as discussed under Bed Load Formulae and added to the annual sediment transport. THE RESPONSE OF A CHANNEL TO CHANGES IN ITS SEDIMENT CHARACTERISTICS Lane’s Model Lane 39 proposed the relationship (sediment load) ϫ (sediment size) ϱ (stream slope) ϫ (stream flow) or ( Q s ϫ d ) ϱ ( S ϫ Q ) (37) to describe qualitatively to behavior of a stream carrying sediment. Lane used the following terms in referring to streams: (1) “grade ϵ equilibrium or regime slope; (2) “aggrad- ing” ϵ rising of the stream bed due to deposition; (3) “degrading” ϵ losing of the stream bed due to scouring C019_001_r03.indd 1073C019_001_r03.indd 1073 11/18/2005 11:06:01 AM11/18/2005 11:06:01 AM © 2006 by Taylor & Francis Group, LLC [...]... Meandering The determination of annual sediment yields was discussed in Sections Erosion and The Mechanics of Sediment Transport in a Stream It is important to predict the possible effects of land development and or sediment control measures on future sediment yields.46 An estimate of the percentage sand, silt, and clay for the incoming sediment can be obtained on the basis of grain size analyses of. .. in the existing land use towards more: use of cover crops and crop rotation, maintenance of effective vegetative cover in critical areas, leaving of straw and stubble in the field, use of long-term hay stands, mulching, pasture planting, and re-forestation 2) Protecting existing vegetative cover involves protection of existing forest sands from excessive fire losses and the protection of all vegetated... practices are used in conjunction with (1) and (2) and include contour farming, contour furrowing of range land, contour strip-cropping, use of gradient and level terraces, use of diversions to divert runoff away from critical areas, use of grassed waterways and ditch and canal linings, and the use of grade stabilization structures in areas subject to possible gully erosion The ASCE Task Committee46 outlined... 11:06:01 AM 1076 SEDIMENT TRANSPORT AND EROSION 104 MEAN DAILY DISCHARGE (cfs) SNOW RUNOFF 103 SUMMER STORMS AND WINTER RUNOFF 102 101 102 101 103 105 104 106 SEDIMENT LOAD (tons/day) FIGURE 13 Typical sediment rating curve (adapted from Ref 38) Q Qs Q DISCHARGE Qs SUSPENDED SEDIMENT LOAD t FIGURE 14 Variation of sediment load with time (adapted from Graf) and fs depends on the cohesiveness of the material... correlation of stream geometry and M led Schumm to classify channels (see Table 6) using M as an index to the ratio of coarse load to total load Schumm45,17 associated meandering channels with high values of M and low bed load and braided channels (a relatively straight, steep main channel consisting of a maze of sub-channels sometimes separated by bars or islands) with low values of M and high bed... ASCE Sedimentation Tank Committee46 has classified sediment control measures under: (1) land treatment, (2) structural Land treatment measures are used to reduce wash load (fines) resulting mainly from sheet erosion Structural measures are most effective for reducing sediment load derived from stream-channel erosion, gully erosion, and sediment associated with mining and construction work The main land... H.A., The bed-load function for sediment transportation in open channel flows, US Dept of Agriculture Technical Bulletin No 1026, 1950 6 Bishop, A.A., D.B Simons, and E.V Richardson, Total bedmaterial transport, Proc ASCE, 91, HY2, March 1965, pp 175–191 7 Chow, and Ven Te, Open-Channel Hydraulics, McGraw-Hill Co., New York, 1959 8 Chow, and Ven Te, Handbook of Applied Hydrology, McGraw-Hill Co., New... 0.64; and y ϭ depth of flow Kennedy was followed by Lacey, Inglis, and Blench who developed equations for channel slope and width Lacey42,21 introduced the equations v = 1.17 fR (39) © 2006 by Taylor & Francis Group, LLC C019_001_r03.indd 1074 11/18/2005 11:06:01 AM SEDIMENT TRANSPORT AND EROSION 1075 10,000 DEPTH 0.1 ft DEPTH 100 ft DEPTH 10 ft Sand Size 1000 DISCHARGE OF SANDS, tons/day/foot of width... ␶c in Eq (62) SEDIMENT CONTROL The adverse effects of excessive sediment loads in reservoirs, navigation channels, harbors, and aquatic life may be alleviated by a sediment control program which may include measures such as prevention of over-hand erosion or containment of eroded soil near its source On the other hand the removal of an established sediment load from a stream may also lead to undesirable... 1080 11/18/2005 11:06:02 AM SEDIMENT TRANSPORT AND EROSION 9 US Bureau of Reclamation, Step method for computing total sediment load by the modified Einstein procedure, Sedimentation Section, Hydrology Branch, July 1955 10 Morisawa, M., Streams, Their Dynamics and Morphology, McGrawHill Book Co., New York, 1968 11 Saxton, K.E., R.G Spomer, and L.A Kramer, Hydrology and erosion of loessial watersheds, Proc . S SEDIMENT TRANSPORT AND EROSION INTRODUCTION The literature on the subject of erosion and sediment transport is vast and is treated in the publication of such disciplines as civil engineering, . Sections Erosion and The Mechanics of Sediment Transport in a Stream. It is important to predict the possible effects of land development and or sediment control mea- sures on future sediment. (1) and (2) and include contour farm- ing, contour furrowing of range land, contour strip-cropping, use of gradient and level terraces, use of diversions to divert runoff away from crit- ical