Transcript Slide 1
National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 1D MORPHODYNAMICS OF MOUNTAIN RIVERS: SEDIMENT MIXTURES Sediment in mountain rivers tends to be poorly sorted, including a wide range of grain size from sand to gravel and coarser. The bed and bedload should be characterized in terms of a grain size distribution rather than a single grain size. In characterizing grain size distributions, grain size is often specified in terms of a base-2 logarithmic scale (phi scale or psi scale). These are defined as follows: where D is given in mm, D2 2 Gravel and sand in cut bank, Las Vegas Wash, Arizona, USA n(D) og2 (D) n(2) 1 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 SAMPLE EVALUATIONS OF AND : SEDIMENT SIZE RANGES n(D) og2 (D) n(2) D2 2 Type D (mm) Notes D (mm) Clay < 0.002 < -9 >9 Usually cohesive 4 2 -2 Silt -9 ~ -4 4~9 2 1 -1 0.002 ~ 0.0625 Cohesive ~ noncohesive 1 0 0 Sand 0.0625 ~ 2 -4 ~ 1 -1 ~ 4 Non-cohesive 0.5 -1 1 Gravel 2 ~ 64 1~6 -6 ~ -1 “ 0.25 -2 2 Cobbles 64 ~ 256 6~8 -8 ~ -6 “ 0.125 -3 3 Boulder s >8 < -8 “ > 256 2 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 SEDIMENT GRAIN SIZE DISTRIBUTIONS The grain size distribution is characterized in terms of N+1 sizes Db,i such that ff,i denotes the mass fraction in the sample that is finer than size Db,i. In the example below N = 7. Sample Grain Size Distribution 100 90 Percent Finer 80 70 i Db,i mm ff,i 1 0.03125 0.020 2 0.0625 0.032 3 0.125 0.100 4 0.25 0.420 5 0.5 0.834 Grain Size mm 6 1 0.970 Note the use of a logarithmic scale for grain size. 7 2 0.990 8 4 1.000 60 50 100 x ff,4 = 42 40 30 Db,4 = 0.25 mm 20 10 0 0.01 0.1 1 10 3 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 SEDIMENT GRAIN SIZE DISTRIBUTIONS contd. In the grain size distribution of the last slide, the finest size (0.03125 mm) was such that 2 percent, not 0 percent was finer. If the finest size does not correspond to 0 percent content, or the coarsest size to 100 percent content, it is often useful to use linear extrapolation on the psi scale to determine the missing values. og2 (Db,2 ) og2 (Db,3 ) 0 ff,3 b,1 og2 (Db,3 ) ff ,2 ff ,3 i Db,i mm ff,i 1 0.03125 0.020 2 0.0625 0.032 3 0.125 0.100 4 0.25 0.420 5 0.5 0.834 6 1 0.970 7 2 0.990 8 4 1.000 b,i n(Db,i ) n(2) og2 (Db,i ) Note that the addition of the extra point has increased N from 7 to 8 (there are N+1 points). b ,1 Db,1 2 i Db,i mm ff,i 1 0.0098 0 2 0.03125 0.020 3 0.0625 0.032 4 0.125 0.100 5 0.25 0.420 6 0.5 0.834 7 1 0.970 8 2 0.990 9 4 1.000 4 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 SEDIMENT GRAIN SIZE DISTRIBUTIONS contd. The grain size distribution after extrapolation is shown below. Sample Grain Size Distribution (with Extrapolation) i Db,i mm ff,i 90 1 0.0098 0 80 2 0.03125 0.020 70 3 0.0625 0.032 60 4 0.125 0.100 5 0.25 0.420 6 0.5 0.834 7 1 0.970 8 2 0.990 9 4 1.000 Percent Finer 100 50 100 x ff,5 = 42 40 30 Db,5 = 0.25 mm 20 10 0 0.001 0.01 0.1 Grain Size mm 1 10 5 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 CHARACTERISTIC SIZES BASED ON PERCENT FINER Sample Grain Size Distribution (with Extrapolation) 100 D90 = 0.700 mm 90 Dx is size such that x percent of the sample is finer than Dx Examples: D50 = median size D90 ~ roughness height Percent Finer 80 To find Dx (e.g. D50) find i such that 70 60 50 ff ,i D50 = 0.286 mm 40 Then interpolate for x 30 20 10 0 0.001 x ff ,i1 100 0.01 0.1 Grain Size mm 1 10 b,i1 b,i x x b,i ff ,i ff ,i1 ff ,i 100 and back-calculate Dx in mm Dx 2 x 6 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 STATISTICAL CHARACTERISTICS OF SIZE DISTRIBUTION Sample Grain Size Distribution (with Extrapolation) 100 90 1 i b,i b,i1 2 1/ 2 Di Db,iDb,i1 Percent Finer 80 70 60 f5 = ff,6 - ff,5 = 0.414 50 40 D5 = (Db,5 Db,6)1/2 = 0.354 mm 30 20 fi ff ,i1 ff ,i i (Di) = characteristic size of ith grain size range 10 0 0.001 N+1 bounds defines N grain size ranges. The ith grain size range is defined by (Db,i, Db,i+1) and (ff,i, ff,i+1) 0.01 0.1 Grain Size mm 1 10 fi = fraction of sample in ith grain size range 7 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 STATISTICAL CHARACTERISTICS OF SIZE DISTRIBUTION contd. Sample Grain Size Distribution (with Extrapolation) 100 = standard deviation on psi scale N i fi 90 80 Percent Finer = mean grain size on psi scale i1 70 N i fi 60 2 50 40 i1 30 Dg 2 20 g 2 10 0 0.001 2 0.01 0.1 1 10 Grain Size mm Dg = geometric mean size Dg = 0.273 mm, g = 2.17 g = geometric standard deviation ( 1) Sediment is well sorted if g < 1.6 8 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 GRAIN SIZE DISTRIBUTION CALCULATOR Workbook RTe-bookGSDCalculator.xls computes the statistics of a grain size distribution input by the user, including Dg, g, and Dx where x is a specified number between 0 and 100 (e.g. the median size D50 for x = 50). It uses code in VBA (macros) to perform the calculations. You will not be able to use macros if the security level in Excel is set to “High”. To set the security level to a value that allows you to use macros, first open Excel. Then click “Tools”, “Macro”, “Security…” and then in “Security Level” check “Medium”. This will allow you to use macros. 9 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 GRAIN SIZE DISTRIBUTION CALCULATOR contd. When you open the workbook RTe-bookGSDCalculator.xls, click “Enable Macros”. The GUI is contained in the worksheet “Calculator”. Now to access the code, from any worksheet in the workbook click “Tools”, “Macro”, “Visual Basic Editor”. In the “Project” window to the left you will see the line “VBA Project (FDebookGSDCalculator.xls)”. Underneath this you will see “Module1”. Double-click on “Module1” to see the code in the “Code” window to the right. These actions allow you to see the code, but not necessarily to understand it. In order to understand this course, you need to learn how to program in VBA. Please work through the tutorial contained in the workbook RTe-bookIntroVBA.xls. It is not very difficult! All the input are specified in the worksheet “Calculator”. First input the number of pairs npp of grain sizes and percents finer (npp = N+1 in the notation of the previous slides) and click the appropriate button to set up a table for inputting each pair (grain size in mm, percent finer) in order of ascending size. Once this data is input, click the appropriate button to compute Dg and g. To calculate any size Dx where x denotes the percent finer, input x into the indicated box and click the appropriate button. To calculate Dx for a different value of x, just put in the new 10 value and click the button again. National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 GRAIN SIZE DISTRIBUTION CALCULATOR contd. This is what the GUI in worksheet “Calculator” looks like. 11 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 GRAIN SIZE DISTRIBUTION CALCULATOR contd. If the finest size in the grain size distribution you input does not correspond to 0 percent finer, or if the coarsest size does not correspond to 100 percent finer, the code will extrapolate for these missing sizes and modify the grain size distribution accordingly. The units of the code are “Sub”s (subroutines). An example is given below. Sub fraction(xpf, xp) 'computes fractions from % finer Dim jj As Integer For jj = 1 To np xp(jj) = (xpf(jj) - xpf(jj + 1)) / 100 Next jj End Sub In this Sub, xpf denotes a dummy array containing the percents finer, and xp denotes a dummy array containing the fractions in each grain size range. The Sub computes the fractions from the percents finer. Suppose in another Sub you know the percents finer Ff(i), I = 1..npp and wish to compute the fraction in each grain size range F(i), i = 1..np (where np = npp – 1). The calculation is performed by the statement 12 fraction Ff, f National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 WHY CHARACTERIZE GRAIN SIZE DISTRIBUTIONS IN TERMS OF A LOGARITHMIC GRAIN SIZE? Consider a sediment sample that is half sand, half gravel (here loosely interpreted as material coarser than 2 mm), ranging uniformly from 0.0625 mm to 64 mm. Plotted with a logarithmic grain size scale, the sample is correctly seen to be half sand, half gravel. Plotted using a linear grain size scale, all the information about the sand half of the sample is squeezed into a tiny zone on the left-hand side of the diagram. Grain Size Distribution: Half Sand, Half Gravel 0.0625 ~ 64 mm, linear scale 100 100 90 90 80 80 70 sand Percent Finer Percent Finer Grain Size Distribution: Half Sand, Half Gravel 0.0625 mm ~ 64 mm, Logarithmic Scale gravel 60 50 40 30 70 gravel 60 50 40 30 20 20 10 10 0 0.01 sand 0 0.1 1 10 D mm 100 0 10 20 30 40 50 60 D mm 13 Logarithmic scale for grain size Linear scale for grain size 70 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS The fractions fi(i) represent a discretized version of the continuous function f(), f denoting the mass fraction of a sample that is finer than size . The probability density pf of size is thus given as p = df/d. 1 The example to the left corresponds to a Gaussian (normal) distribution with = -1 (Dg = 0.5 mm) and = 0.8 (g = 1.74): p 1 1 exp 2 2 2 0.9 f() 0.8 0.7 0.6 0.5 0.4 0.3 p() 0.2 0.1 0 -4 -3 -2 -1 0 1 2 The grain size distribution is called unimodel because the function p() has a single mode, or peak. The following approximations are valid for a Gaussian distribution: D84 14 Dg D84D16 , g D16 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS contd. 1 A sand-bed river has a characteristic size of bed surface sediment (D50 or Dg) 0.9 0.8 that is in the sand range. f() 0.7 Plateau 0.6 A gravel-bed river has a characteristic 0.5 bed size that is in the range of gravel or 0.4 coarser material. 0.3 The grain size distributions of most sand-bed streams are unimodal, and can often be approximated with a Gaussian function. Many gravel-bed river, however, show bimodal grain size distributions, as shown to the upper right. Such streams show a sand mode and a gravel mode, often with a paucity of sediment in the pea-gravel size (2 ~ 8 mm). Gravel mode Sand mode p() 0.2 0.1 0 -4 -2 0 2 4 6 8 A bimodal (multimodal) distribution can be recognized in a plot of f versus in terms of a plateau (multiple plateaus) where f does not increase strongly with . 15 10 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS contd. The grain size distributions to the left are all from 177 samples from various river reaches in Alberta, Canada (Shaw and Kellerhals, 1982). The samples from sandbed reaches are all unimodal. The great majority of the samples from gravel-bed reaches show varying degrees of bimodality. Note: geographers often reverse the direction of the grain size scale, as seen to the left. Figure adapted from Shaw and Kellerhals (1982) 16 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 VERTICAL SORTING OF SEDIMENT Gravel-bed rivers such as the River Wharfe often display a coarse surface armor or pavement. Sand-bed streams with dunes such as the one modeled experimentally below often place their coarsest sediment in a layer corresponding to the base of the dunes. River Wharfe, U.K. Image courtesy D. Powell. Sediment sorting in a laboratory flume. Image courtesy A. Blom. 17 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 EXNER EQUATION OF CONSERVATION OF BED SEDIMENT FOR SIZE MIXTURES MOVING AS BEDLOAD fi'(z', x, t) = fractions at elevation z' in ith grain size range above datum in bed [1]. Note that over all N grain size ranges: N f 1 i1 i qbi(x, t) = volume bedload transport rate of sediment in the ith grain size range [L2/T] s (1 p )fidzx 1 s qbi x qbi 0 t x x 1 x Or thus: qbi (1 p ) fidz t 0 x qbi qbi 1 z' 18 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 ACTIVE LAYER CONCEPT qbi qbi La x z' The active, exchange or surface layer approximation (Hirano, 1972): Sediment grains in active layer extending from - La < z’ < have a constant, finite probability per unit time of being entrained into bedload. Sediment grains below the active layer have zero probability of entrainment. 19 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 REDUCTION OF SEDIMENT CONSERVATION RELATION USING THE ACTIVE LAYER CONCEPT Fractions Fi in the active layer have no vertical structure. Fractions fi in the substrate do not vary in time. Fi ( x, t ) , La z fi( x, z, t ) fi ( x, z) , z La Thus La fidz fidz fidz fIi ( L a ) FiL a t 0 t 0 t La t t where the interfacial exchange fractions fIi defined as fIi fi L a describe how sediment is exchanged between the active, or surface layer and the substrate as the bed aggrades or degrades. 20 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 REDUCTION OF SEDIMENT CONSERVATION RELATION USING THE ACTIVE LAYER CONCEPT contd. Between q (1 p ) fidz bi t 0 x and La fidz fidz fidz fIi ( L a ) FiL a t 0 t 0 t La t t it is found that qbi (1 p )fIi ( La ) FiLa t x t (Parker, 1991). 21 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 REDUCTION contd. The total bedload transport rate summed over all grain sizes qbT and the fraction pbi of bedload in the ith grain size range can be defined as N qbT qbi , pbi i1 qbi qbT The conservation relation can thus also be written as q p (1 p )fIi ( La ) FiLa bT bi t x t Summing over all grain sizes, the following equation describing the evolution of bed elevation is obtained: qbT (1 p ) t x Between the above two relations, the following equation describing the evolution of the grain size distribution of the active layer is obtained: L q p q F (1 p )La i Fi fIi a bT bi fIi bT t x x t 22 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 EXCHANGE FRACTIONS fi zL , 0 a t fIi Fi (1 )pbi , 0 t where 0 1 (Hoey and Ferguson, 1994; Toro-Escobar et al., 1996). In the above relations Fi, pbi and fi denote fractions in the surface layer, bedload and substrate, respectively. That is: The substrate is mined as the bed degrades. A mixture of surface and bedload material is transferred to the substrate as the bed aggrades, making stratigraphy. Stratigraphy (vertical variation of the grain size distribution of the substrate) needs to be stored in memory as bed aggrades in order to compute 23 subsequent degradation. National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 ALTERNATIVE DIMENSIONLESS BEDLOAD TRANSPORT The generalized bedload transport relation of the type of Meyer-Peter and Müller (1948) was written in the form: q t ( c )nt where q qb RgD D , b RgD Recalling that b = u*2, the relation can be written in the alternative form nt W t 1 c where Rgq b qb W 3/ 2 3 ( ) u (Parker et al., 1982). The form W* versus * is often used as the basis for generalizing to sediment mixtures. 24 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 SURFACE-BASED BEDLOAD TRANSPORT FORMULATION FOR MIXTURES Consider the bedload transport of a mixture of sizes. The thickness La of the active (surface) layer of the bed with which bedload particles exchange is given by as La naDs90 where Ds90 is the size in the surface (active) layer such that 90 percent of the material is finer, and na is an order-one dimensionless constant (in the range 1 ~ 2). Divide the bed material into N grain size ranges, each with characteristic size Di, and let Fi denote the fraction of material in the surface (active) layer in the ith size range. The volume bedload transport rate per unit width of sediment in the ith grain size range is denoted as qbi. The total volume bedload transport rate per unit width is denoted as qbT, and the fraction of bedload in the ith grain size range is pbi, where N qbT qbi i1 qbi , pbi qbT Now in analogy to *, q* and W*, define the dimensionless grain size specific Shields number i*, grain size specific Einstein number qi* and dimensionless grain size specific bedload transport rate Wi* as 2 u b i RgD i RgD i qbi , qbi RgD i D i Fi Rgq bi q bi , Wi 3 / 2 ( i ) (u )3 Fi 25 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 SURFACE-BASED BEDLOAD TRANSPORT FORMULATION contd. It is now assumed that a functional relation exists between qi* (Wi*) and i*, so that qbi q fq ( i ) or RgD i Di Fi bi i W Rgq bi (u )3 Fi fW ( i ) The bedload transport rate of sediment in the ith grain size range is thus given as qbi Fi RgDi Di fq (i ) or u3 qbi Fi fW (i ) Rg qbi According to this formulation, if the grain size range is not represented in the surface (active) layer, it will not be represented in the bedload transport. qbi La x z' 26 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 BEDLOAD RELATION FOR MIXTURES DUE TO PARKER (1990) This relation is appropriate only for the computation of gravel bedload transport rates in gravel-bed streams. In computing Wi*, Fi must be renormalized so that the sand is removed, and the remaining gravel fractions sum to unity, Fi = 1. The method is based on surface geometric size Dsg and surface arithmetic standard deviation s on the scale, both computed from the renormalized fractions Fi. Wi 0.00218 Gi 0.0951 2 Di sg u i sgo , sgo , sg , ssrg 0.0386 D ssrg RgDsg sg 4 .5 0.853 for 1.59 54741 G() exp 14.2( 1) 9.28( 1)2 for 1 1.59 14 .2 for 1 N s s 1 O (sgo ) 1 Dsg 2 , s iFi O (sgo ) i1 N , i s Fi 2 s 2 i1 In the above O and O are set functions of sgospecified in the next slide. 27 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 BEDLOAD RELATION FOR MIXTURES DUE TO PARKER (1990) contd. 1.6 1.4 1.2 1 omegaO o o sigmaO O, O 0.8 0.6 0.4 0.2 0 0.1 1 10 100 1000 sgo It is not necessary to use the above chart. The calculations can be performed using the Visual Basic programs in RTe-bookAcronym1.xls 28 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 BEDLOAD RELATION FOR MIXTURES DUE TO WILCOCK AND CROWE (2003) The sand is not excluded in the fractions Fi used to compute Wi*. The method is based on the surface geometric mean size Dsg and fraction sand in the surface layer Fs. Wi* Gi 0.0027.5 4.5 G 0.894 14 1 0.5 for 1.35 for 1.35 b sg Di i ssrg Dsg ssrg 0.021 0.015 exp(20Fs ) b 0.67 1 exp(1.5 Di / Dsg ) 2 u sg RgDsg 29 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 AGGRADATION AND DEGRADATION OF RIVERS TRANSPORTING GRAVEL MIXTURES Results of a flood in the gravel-bed Salmon River, Idaho. Photo by author 30 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 MODELING AGGRADATION AND DEGRADATION IN GRAVEL-BED RIVERS CARRYING SEDIMENT MIXTURES Gravel-bed rivers tend to be steep enough to allow the use of the normal (steady, uniform) flow approximation. Here this analysis is applied using a Manning-Strickler formulation such that roughness height ks is given as ks nkDs90 where Ds90 is the size of the surface material such that 90% is finer and nk is an order-one dimensionless number (1.5 ~ 3; the work of Kamphuis, 1974 suggests a value of 2). No attempt is made here to decompose bed resistance into skin friction and form drag. The reach is divided into M intervals bounded by M + 1 nodes. In addition, sediment is introduced at a ghost node at the upstream end. Since the index “i” has been used for grain size ranges, the index “k” is used here for spatial nodes. x ghost k=1 2 3 M -1 L M k = M+1 M+1 31 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 COMPUTATION OF BED SLOPE AND BOUNDARY SHEAR STRESS At any given time t in the calculation, the bed elevation k and surface fractions Fi,k must be known at every node k. The roughness height ks,k and thickness of the surface layer La,k are computed from the relations ks,k nkDs90,k La,k naDs90,k where nk and na are specified order-one dimensionless constants. (Beware: in the equation for roughness height the “k” in nk is not an index for spatial node.) Using the normal flow approximation, the boundary shear stress b,k at the kth node is 3 / 10 given from Chapter 5 as 1/ 3 2 b,k u 2 ,k k s,k qw 2 r g7 / 10 Sk7 / 10 where u,k denotes the shear velocity and bed slope Sk is computed as 1 2 , k 1 x Sk k 1 k 1 , k 2..M 2x Bed slope need not be computed at k = M + 1, where bed elevation is specified as a 32 boundary condition. National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 COMPUTATION OF BEDLOAD TRANSPORT Once Fi,k and b,k are known, the bedload transport rates qbi, and thus qbT and pi can be computed at any node. An example is given here in terms of the Wilcock-Crowe (2003) formulation. The surface geometric mean size Dsg,k is calculated at every node as N k iFi,k , Dsg,k 2k i1 where i = ln2(Di). The Shields number and shear velocity based on the surface geometric mean size are then given as sg,k k q g 1/ 3 2 s,k w 2 r 3 / 10 Sk7 / 10 RD sg,k u,k k q 1/ 3 2 s,k w 2 r 3 / 20 g7 / 20 Sk7 / 20 The same fractions Fi,k allow the computation of the fraction sand Fs,k in the surface layer at node k. This parameter is needed in the formulation of Wilcock and Crowe 33 (2003). National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 COMPUTATION OF BEDLOAD TRANSPORT contd. It follows that the volume bedload transport rate per unit width in the ith grain size range is given as qbi,k u3,k Fi,k Wi,k Rg N qbT ,k qbi,k , pbi,k i1 qbi,k qbT ,k where in the case of the relation of Wilcock and Crowe (2003), 0.002i7,k.5 4.5 Wi,k 0.894 141 0.5 i,k for i,k 1.35 for i,k 1.35 bi,k sg,k Di i,k ssrg,k Dsg,k ssrg,k 0.021 0.015 exp(20Fs,k ) bi,k 0.67 1 exp(1.5 Di / Dsg,k ) 34 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 MODELING AGGRADATION AND DEGRADATION IN GRAVEL-BED RIVERS CARRYING SEDIMENT MIXTURES contd. The discretized versions of the Exner relations are: k t t Fi,k t t If qbT k t 1 p x k La,k qbT pbi 1 If qbT Fi,k fIi,k t Fi,k fIi,k La,k t La,k (1 p ) x k x t k where fIi,k is evaluated from a relation of the type given in Slide 4: fs,i,int,k , 0 t k fIi,k Fi,k (1 )pbi,k , 0 t k In the above relation fs,i,int,k denotes the fractions of the substrate just below the surface layer at node k and is a user-specified parameter between 0 and 1. 35 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 MODELING AGGRADATION AND DEGRADATION IN GRAVEL-BED RIVERS CARRYING SEDIMENT MIXTURES contd. The spatial derivatives of the sediment transport rates are computed as qbT x k qbT ,k qbT ,k1 qbT ,k1 qbT ,k au (1 au ) x x qbT ,kpbi,k qbT ,k1pbi,k1 qbT ,k1pbi,k1 qbT ,kpbi,k qbTpi au (1 au ) x k x x where au is a upwinding coefficient equal to 0.5 for a central difference scheme. When k = 1, the node k – 1 refers to the ghost node, where qbi, and thus qbT and pi are specified as feed parameters. The term La,k/t t is not a particularly important one, and can be approximated as La,k t La,k La,k,old t where La,k,old is the value of La,k from the previous time step. In the case of the first 36 time step, La,k,old may be set equal to 0. National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 BOUNDARY CONDITIONS, INITIAL CONDITIONS AND FLOW OF THE COMPUTATION The boundary conditions are • Specified values of qb,i (and thus qbT and pbi) at the upstream ghost node; • Specified bed elevation at node k = M+1. The initial conditions are • Specified initial bed elevations at every node (here simplified to a specified initial bed slope Sfbl; • Specified surface and substrate grain size distributions Fi and fs,i at every node (here taken to be the same at every node). At any given time fractions Fi and elevation are known at every node. The values Fi are used to compute Ds90 Dsg, Ds50, ks, La and other parameters (e.g. Fs) at every node. The values of are used to compute slopes S and combined with the computed values of ks to determine the shear stress b at every node except M+1, where the information is not needed. The resulting parameters are used to compute qbi, qbT and pbi at all nodes except M+1. The Exner relations are then solved to determine bed elevations and surface fractions Fi at all nodes. At node M+1 only 37 the change in grain size distribution is evaluated. National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 INTRODUCTION TO RTe-bookAgDegNormGravMixPW.xls The workbook is a descendant of the PASCAL code ACRONYM3 of Parker (1990a,b). It allows the user to choose from two surface-based bedload transport formulations; those of Parker (1990) and Wilcock and Crowe (2003). In the relation of Parker (1990) the surface grain size distributions need to be renormalized to exclude sand before specification as input to the program. This step is neither necessary nor desirable in the case of the relation of Wilcock and Crowe (2003), where the sand plays an important role in mediating the gravel bedload transport. The basic input parameters are the water discharge per unit width qw, flood intermittency If, gravel input rate during floods qbTf, reach length L, initial bed slope SfbI, number of spatial intervals M, time step t, fractions pbf,i of the gravel feed, fractions FI,i of the initial surface layer (assumed the same at every node) and fractions fsI,I of the substrate (assumed to be uniform in the vertical and the same at every node). The parameters Mprint and Mtoprint control output. Auxiliary parameters include nk for roughness height, na for active layer thickness, r of the Manning-Strickler relation, submerged specific gravity R of the sediment, bed porosity p, upwinding coefficient au and interfacial transfer coefficient . 38 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 INTRODUCTION TO RTe-bookAgDegNormGravMixPW.xls contd. One interesting problem of sediment mixtures is when the river first aggrades, creating its own substrate with a vertical structure in the process, and then degrades into it. The code in the workbook is not set up to handle this. The necessary extension is trivial in theory but tedious in practice; the vertical structure of the newlycreated substrate must be stored in memory as the calculation proceeds. A gravel-bed reach of the Las Vegas Wash, USA, where the river is degrading into its own deposits. Some calculations with the code follow. 39 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 CALCULATIONS WITH RTe-bookAgDegNormGravMixPW.xls The calculations are performed with the Parker (1990) bedload transport relation. The grain size distributions of the feed sediment, initial surface sediment and substrate sediment are all taken to be identical, as given below. Note that sand has been removed from the grain size distributions. Grain Size Distributions 100 90 Percent Finer 80 70 60 Feed Initial Surface Substrate 50 40 30 20 10 0 1 10 100 Size mm Dd,i mm Feed 256 128 64 32 16 8 4 2 1 0.5 0.25 0.125 100 95 80 50 25 10 5 0 0 0 0 0 Initial Surface Substrate 100 100 95 95 80 80 50 50 25 25 10 10 5 5 0 0 0 0 0 0 0 0 0 0 1000 40 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 CALCULATIONS WITH RTe-bookAgDegNormGravMixPW.xls contd. A case is chosen for which the bed must aggrade from a very low slope. Calculations are performed for 60 years, 600 years and 6000 years in order to study the evolution of the profile. Input parameters qw 6 qbTo 1.00E-04 Inter 0.05 etadI 3 SfbI 1.00E-04 L 20000 dt 73.05 M 25 Mtoprint 50 Mprint 6 60 The input cells are in gold water discharge/width, m^2/s These cells contain useful information Input "1" for Parker (1990) relation, "2" for gravel input rate, m^2/s Intermittency Wilcock-Crowe (2003) relation: initial base level, m 1 Input here to choose initial bed slope reach length,m Click to Run Program time step, days no. of intervals no. of steps until a printout of results is made no. of printouts after the initial one dt 6311520 sec years calculation time The software produces graphical output for the time development of the long profiles of a) bed elevation , b) surface geometric mean size Dsg and c) volume gravel bedload transport rate per unit width qbT. 41 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Bed Elevation 16 14 Parker relation After 60 years Elevation m 12 0 yr 10 yr 20 yr 30 yr 40 yr 50 yr 60 yr final w.s. 10 8 6 4 2 0 0 5000 10000 Distance m 15000 20000 42 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Surface Geometric Mean Size Surface Geometric Mean Size mm 100 Parker relation After 60 years 0 yr 10 yr 20 yr 30 yr 40 yr 50 yr 60 yr 10 1 0 5000 10000 Distance m 15000 20000 43 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 DownstreamVariation variationofofqbT/qbTo, qbT/qbTf,where where qbT Downstream qbT == Bedload Bedload BedloadTransport TransportRate Rateand andqbTo qbTf==Upstream Upstream Bedload Feed FeedRate Rate 100 Parker relation After 60 years qbT/qbTf qbT/qbTo 10 0 yr 10 yr 20 yr 30 yr 40 yr 50 yr 60 yr 1 0.1 0.01 0.001 0 5000 10000 Distance m 15000 20000 44 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Bed Elevation 35 30 Parker relation After 600 years Elevation m 25 0 yr 100 yr 200 yr 300 yr 400 yr 500 yr 600 yr final w.s. 20 15 10 5 0 0 5000 10000 Distance m 15000 20000 45 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Surface Geometric Mean Size Surface Geometric Mean Size mm 100 Parker relation After 600 years 0 yr 100 yr 200 yr 300 yr 400 yr 500 yr 600 yr 10 1 0 5000 10000 Distance m 15000 20000 46 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream qbT == DownstreamVariation variationofofqbT/qbTo, qbT/qbTf,where where qbT Bedload Bedload BedloadTransport TransportRate Rateand andqbTo qbTf==Upstream Upstream Bedload Feed FeedRate Rate 100 Parker relation After 600 years qbT/qbTf qbT/qbTo 10 0 yr 100 yr 200 yr 300 yr 400 yr 500 yr 600 yr 1 0.1 0.01 0.001 0 5000 10000 Distance m 15000 20000 47 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Bed Elevation 70 60 Parker relation After 6000 years Elevation m 50 0 yr 1000 yr 2000 yr 3000 yr 4000 yr 5000 yr 6000 yr final w.s. 40 30 20 10 0 0 5000 10000 Distance m 15000 20000 48 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Surface Geometric Mean Size Surface Geometric Mean Size mm 100 0 yr 1000 yr 2000 yr 3000 yr 4000 yr 5000 yr 6000 yr Parker relation After 6000 years 10 1 0 5000 10000 Distance m 15000 20000 49 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream qbT == DownstreamVariation variationofofqbT/qbTo, qbT/qbTf,where where qbT Bedload Bedload BedloadTransport TransportRate Rateand andqbTo qbTf==Upstream Upstream Bedload Feed FeedRate Rate 100 Parker relation After 6000 years qbT/qbTf qbT/qbTo 10 0 yr 1000 yr 2000 yr 3000 yr 4000 yr 5000 yr 6000 yr 1 0.1 0.01 0.001 0 5000 10000 Distance m 15000 20000 50 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 CALCULATIONS WITH RTe-bookAgDegNormGravMixPW.xls contd. The next case is one for which the bed which the bed must degrade to a new equilibrium. The input grain size distributions are the same as the previous case. Again, the Parker (1990) relation is used. The input parameters are given below. The calculation shown is over a duration of 240 years. 51 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Bed Elevation 120 Elevation m 100 Parker relation After 240 years 0 yr 40 yr 80 yr 120 yr 160 yr 200 yr 240 yr final w.s. 80 60 40 20 0 0 5000 10000 Distance m 15000 20000 52 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Surface Geometric Mean Size Surface Geometric Mean Size mm 100 0 yr 40 yr 80 yr 120 yr 160 yr 200 yr 240 yr Parker relation After 240 years 10 1 0 5000 10000 Distance m 15000 20000 53 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 DownstreamVariation variationofofqbT/qbTo, qbT/qbTf,where where qbT Downstream qbT == Bedload Bedload BedloadTransport TransportRate Rateand andqbTo qbTf==Upstream Upstream Bedload Feed FeedRate Rate 100 qbT/qbTf qbT/qbTo 10 0 yr 40 yr 80 yr 120 yr 160 yr 200 yr 240 yr 1 Parker relation After 240 years 0.1 0.01 0.001 0 5000 10000 Distance m 15000 20000 54 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 CALCULATIONS WITH RTe-bookAgDegNormGravMixPW.xls contd. Sand is excluded from the input grain size distributions when using the Parker (1990) relation. The Wilcock-Crowe (2003) relation explicitly includes the sand. Two calculations follow. In the first of them, the input data are exactly the same as that for the calculations using Parker (1990) of Slides 51-54 (degradation to a new equilibrium). In particular, sand is excluded from the input grain size distributions. In the second of them, 25% sand is added to the grain size distribution. The WilcockCrowe (2003) relation predicts that the addition of sand makes the gravel more mobile. It will be seen that the bed elevation at the end of the 240-year calculation is predicted to be significantly lower when sand is included than when it is excluded. Input parameters qw 6 qbTo 1.00E-04 Inter 0.05 etadI 3 SfbI 5.00E-03 L 20000 dt 7.305 M 25 Mtoprint 2000 Mprint 6 240 The input cells are in gold water discharge/width, m^2/s These cells contain useful information compu Input "1" for Parker (1990) relation, "2" for gravel input rate, m^2/s Intermittency Wilcock-Crowe (2003) relation: initial base level, m 2 Input here to choose relation initial bed slope reach length,m Click to Run Program time step, days no. of intervals no. of steps until a printout of results is made no. of printouts after the initial one dt 631152 sec 55 years calculation time National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Bed Elevation 120 Wilcock-Crowe relation Sand excluded After 240 years Elevation m 100 80 60 40 0 yr 40 yr 80 yr 120 yr 160 yr 200 yr 240 yr final w.s. 20 0 0 5000 10000 Distance m 15000 20000 56 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Surface Geometric Mean Size Surface Geometric Mean Size mm 100 0 yr 40 yr 80 yr 120 yr 160 yr 200 yr 240 yr 10 Wilcock-Crowe relation Sand excluded After 240 years 1 0 5000 10000 Distance m 15000 20000 57 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 DownstreamVariation variationofofqbT/qbTo, qbT/qbTf,where where qbT Downstream qbT == Bedload Bedload BedloadTransport TransportRate Rateand andqbTo qbTf==Upstream Upstream Bedload Feed FeedRate Rate 100 qbT/qbTf qbT/qbTo 10 0 yr 40 yr 80 yr 120 yr 160 yr 200 yr 240 yr 1 0.1 Wilcock-Crowe relation Sand excluded After 240 years 0.01 0.001 0 5000 10000 Distance m 15000 20000 58 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Bed Elevation 120 Wilcock-Crowe relation Sand included After 240 years Elevation m 100 80 60 40 0 yr 40 yr 80 yr 120 yr 160 yr 200 yr 240 yr final w.s. 20 0 0 5000 10000 Distance m 15000 20000 59 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream Variation in Surface Geometric Mean Size Surface Geometric Mean Size mm 100 0 yr 40 yr 80 yr 120 yr 160 yr 200 yr 240 yr 10 Wilcock-Crowe relation Sand included After 240 years 1 0 5000 10000 Distance m 15000 20000 60 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 Downstream qbT == DownstreamVariation variationofofqbT/qbTo, qbT/qbTf,where where qbT Bedload Bedload BedloadTransport TransportRate Rateand andqbTo qbTf==Upstream Upstream Bedload Feed FeedRate Rate 100 qbT/qbTf qbT/qbTo 10 0 yr 40 yr 80 yr 120 yr 160 yr 200 yr 240 yr 1 0.1 Wilcock-Crowe relation Sand included After 240 years 0.01 0.001 0 5000 10000 Distance m 15000 20000 61 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 NOTES ON THE EFFECT OF SAND IN THE GRAVEL Comparing Slides 56 and 59, it is seen that the upstream end of the reach has degraded considerably more in the case of Slide 56, i.e. when sand is included in the Wilcock-Crowe (2003) calculation. Comparing Slides 52 and 59, it is seen that the bed profile at the end of the calculation using Wilcock-Crowe (2003) with sand included is almost the same as the corresponding profile using Parker (1990), in which sand is automatically excluded. The correspondence is not an accident. The field data used to develop the Parker (1990) relation did indeed include sand in the bed and load; sand was excluded in the development of the relation because of uncertainty as to how much might go into suspension. So the Parker (1990) relation implicitly includes a set fraction of sand in the bed. This notwithstanding, the Wilcock-Crowe (2003) relation has the considerable advantage that the quantity of sand in the feed sediment and substrate can be varied. As the calculations show, for all other factors equal the relation predicts that an increased sand content can significantly increase the mobility of the gravel. 62 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 REFERENCES Hirano, M., 1971, On riverbed variation with armoring, Proceedings, Japan Society of Civil Engineering, 195: 55-65 (in Japanese). Hoey, T. B., and R. I. Ferguson, 1994, Numerical simulation of downstream fining by selective transport in gravel bed rivers: Model development and illustration, Water Resources Research, 30, 2251-2260. Meyer-Peter, E. and Müller, R., 1948, Formulas for Bed-Load Transport, Proceedings, 2nd Congress, International Association of Hydraulic Research, Stockholm: 39-64. Parker, G., 1990, Surface-based bedload transport relation for gravel rivers,” Journal of Hydraulic Research, 28(4): 417-436. Parker, G., Klingeman, P. and McLean, D., 1982, Bedload and size distribution in natural paved gravel bed streams, Journal of Hydraulic Engineering, 108(4), 544-571. Shaw, J. and R. Kellerhals, 1982, The Composition of Recent Alluvial Gravels in Alberta River Beds, Bulletin 41, Alberta Research Council, Edmonton, Alberta, Canada. Wilcock, P. R., and Crowe, J. C., 2003, Surface-based transport model for mixed-size sediment, Journal of Hydraulic Engineering, 129(2), 120-128. For more information see Gary Parker’s e-book: 1D Morphodynamics of Rivers and Turbidity Currents http://cee.uiuc.edu/people/parkerg/morphodynamics_e-book.htm 63 National Center for Earth-surface Dynamics Short Course Morphology, Morphodynamics and Ecology of Mountain Rivers December 11-12, 2005 REFERENCES FOR CHAPTER 17 Parker, G., 1990a, Surface-based bedload transport relation for gravel rivers,” Journal of Hydraulic Research, 28(4): 417-436. Parker, G., in press, Transport of gravel and sediment mixtures, ASCE Manual 54, Sediment Engineering, ASCE, Chapter 3, downloadable at http://cee.uiuc.edu/people/parkerg/manual_54.htm . Toro-Escobar, C. M., G. Parker and C. Paola, 1996, Transfer function for the deposition of poorly sorted gravel in response to streambed aggradation, Journal of Hydraulic Research, 34(1): 35-53. Wilcock, P. R., and Crowe, J. C., 2003, Surface-based transport model for mixed-size sediment, Journal of Hydraulic Engineering, 129(2), 120-128. 64