A STUDY OF FLUVIAL GEOMORPHOLOGY ASPECTS OF HYDRAULIC DESIGN

Transcription

1 A STUDY OF FLUVIAL GEOMORPHOLOGY ASPECTS OF HYDRAULIC DESIGN A. David Parr, Ph.D. and John Shelley CEAE Department University of Kansas (Funded by KDOT) 1

2 Acknowledgments Jim Richardson Brad Brad Rognlie Mike Mike Orth KDOT KDOT Bridge Section 2

3 Stable Channel Design KDOT is sometimes required to realign short reaches of alluvial channels to facilitate highway improvements or to provide protection for highway structures or roadway embankments. The new stream reaches should be dynamically stable and should have geomorphic properties that are characteristic of natural streams in similar settings. They should also be hydraulically and ecologically compatible with the contiguous upstream and downstream stream reaches. 3

4 Stream Modification - Road Project Old Road Old Road Old Stream Old Stream New Stream New Stream New Road New Road (b) (a) 4

5 Protection - Meanders on the Kansas River (a) (b) 5

6 Approaches to Stable Channel Design Regime Methods Empirical regression equations - not for natural channels Reference-Reach Reach Methods Rosgen-type methods The design channel geometry is scaled from a stable reference reach on the same stream network or from a stream of the same type with similar geologic and hydrologic characteristics. Analytical Methods Use hydraulic resistance equations and sediment transport equations to design a channel reach that has the same flow and sediment transport capacity as a representative stable upstream supply reach for Bank-full Discharge Conditions. 6

7 Bank-full Discharge Conditions Copeland* states Bank-full discharge is the maximum discharge that a steam can convey without overflowing into the floodplain. The water surface elevation for this condition is called the bank-full stage. Bank-full discharge is also referred to as the channel-forming discharge. * 7

8 Upstream Supply Reach Riffle on Stable Upstream Supply Reach Project Site Project Site Supply Reach Cross Section Flow Sinuosity = L stream /L valley 8

9 Bankfull Conditions for Supply Reach Cross Section (1.2 to 2 year recurrence interval) W bf Bankfull Stage d max A bf W bf = bankfull width A bf = bankfull area d bf = A bf /W bf = bankfull depth 9

10 Determination of Bank-full Stage ( Involves assessing the elevation where the channel, under bank-full discharge conditions, ends and the floodplain begins. The indicators used to assess this elevation are as follows: Top of the point bar A change in vegetation Slope change in channel cross section Top of the undercut slope Change in particle size (where soils end and sediments begin), Drift lines and water marks 10

11 University of Kansas Studies Guidelines for Stream Realignment Design KAM Method McEnroe, Young and Shelley Report No. K-TRAN K KU-08-2 Stream Realignment Design using a Reference Reach ARR Method McEnroe, Young and Shelley Report No. K-TRAN K KU-09-4 This Study A Study of Fluvial Geomorphology Aspects of Hydraulic Design (HEC-RAS applications) Parr and Shelley Report No. K-TRAN: K KU

12 KAM and ARR Methods Consider alluvial (noncohesive) and threshold (cohesive) channels. Strengths Include planform design for stream meanders and pool spacing Designs pool depth Uses a simple version of the Meyer-Peter Mueller sediment transport equation for analytical methods ARR incorporates features of both analytical and reference reach methods 12

13 KAM and ARR Methods (Cont.) Limitations Plane bed (no bedforms) Wide channels (Large width to depth ratios) No consideration of grain size distribution other than d 50 Does not allow for separation of bed and bank hydraulic roughness 13

14 Study Objectives Develop procedures to use HEC-RAS 4.0 in the design of stable channel reaches for alluvial streams using the Analytical Approach. Provide examples for streams with Sand beds Gravel/cobble beds. Compare HEC-RAS methods with McEnroe s KAM and ARR Methods. 14

15 Stable Channel Design in HEC-RAS Uses Steady Flow modeling to determine parameters needed for the sediment transport modeling components. Velocity Depth Area Only Manning s s n-values n can be used in the HEC-RAS steady flow model for resistance. Uses Hydraulic Design Functions to perform uniform flow and sediment transport capacity calculations. Brownlie, Strickler, Limerinos and Manning equations can be used to account for channel resistance. Brownlie and Limerinos resistance equations account for bed form resistance as well as resistance due to grains. 15

16 HEC-RAS Hydraulic Design Functions used in Analytical Design Stable Channel Design Uniform Flow Sediment Transport Capacity Sand Beds Sand Beds Gravel/Cobble Beds Gravel/Cobble Beds 16

17 Resistance Formulas Used in Hydraulic Design Functions 17

18 Equation Applicability Strengths Limitations Manning Stricker Limerinos Brownlie All natural and artificial streams. Cobble bed streams dominated by grain size friction. Stream beds with sediment sizes from coarse sand to cobble under an upper flow regime. Sand bed streams of either an upper or a lower regime. Easy to use and to understand. Required for HEC-RAS hydraulic modeling. Quantified the hydraulics losses due to grain size friction based on measurable parameters. Includes losses due to both grain roughness and bedforms. Based on measurable parameters. Requires a high level of engineering judgment to choose an appropriate value from a table or from a book of reference streams. Does not include losses due to bed forms. May be unrealistically low. Not applicable to other sediment sizes or to the lower regime flow. Includes losses due to both grain Not applicable to other roughness and bedforms. Based sediment sizes. on measurable parameters. Can be used for either the upper or the lower flow regime. Correlates with the Brownlie sediemnt transport function. HEC-RAS Resistance Formulas for Alluvial Channels 18

19 Manning s s Equation 1.49 n= R S V 2/3 1/2 V= Mean Velocity in ft/sec, 1.49 = coefficient for English Units (1.0 for Metric), n = Manning s s n value, R = Hydraulic radius, ft. = Area/Wetted Perimeter, S = Slope of the Energy Grade Line. (Bed slope for uniform flow) 19

20 Strickler Equation n R = φ ks k 1/6 s where k = Nikuradse equivalet sand roughness, ft or m, = d s for natural channels and d for riprap lined channels. R φ = Strickler function = for natrual channels k s = for velocity and stone size calculations in riprap channels = for discharge calculations in riprap channels R = hydraulic radius 20

21 Limerinos Equation n = R 1/ d log R d 84 = the particle size, ft, for which 84% of the sediment mixture is finer. Data ranged from to ft (1.5 to 250 mm). BIG STUFF n = Manning s s n value. Data ranged from 0.02 to R = Hydraulic radius, ft. Data ranged from 1 to 6 ft (0.35 to 1.83 m). 21

22 Brownlie Resistance Equations Sand Only Lower Regime R n= S σ d d 50 Upper Regime ( ) R n= S σ d d 50 ( ) d 50 = the particle size, ft, for which 50% of the sediment mixture is finer by weight, s = the geometric standard deviation of the sediment mixture

23 Brownlie Resistance Equations (Cont.) Lower Regime F Transition F < F < F ' g ' g ' ' 0.8 g g 1.25 g Upper Regime S > or F > F F F g 1.74 = 1/3 S V = = ( s 1) gd where s 50 g s = sediment specific gravity ( 2.65 for sand) s < F g ' g grain Froude number 23

24 HEC-RAS Sediment Transport Equations 24

25 Sand Beds - Brownlie Sediment Transport Equation Used in the Stable Channel Hydraulic Design Function for sand bed channels only.. The method is called the Copeland Method. Based on dimensional analysis and regression of a very large body of field and laboratory sediment transport data for sand beds. Only applied to the movable bed does not consider sediment transport from the main channel banks. 25

26 Brownlie Sediment Transport Equation (Sand Bed Natural Channels) C = F F S r d ( g go) / ( ) where C = bed material concentration in ppm by weight r = representative grain roughness height d * o ( ) F = V / gd ρ ρ / ρ = grain Froude number g go = geometric mean grain size of bed material S = slope of energy grade line F τ σ Y * o g = 4.596τ s S g (10) 7.7Y = Y + = ( R ( )/ ) g ρs ρ ρ R = gd / ν = grain Reynolds number g σ = critical grain Froude number critical shear stress = geometric standard deviation of bed material = 26

27 Gravel/Cobble Beds Sediment Transport Potential Functions Ackers-White Engelund-Hansen Laursen-Copeland Meyer-Peter Meyer-Peter Muller Toffaleti Yang 27

28 Temp, Temp, o F Ranges for Sediment Transport Functions 28 d range, range, mm mm Mean Mean d, mm mm Spec Spec Gravity Gravity Velocity, Velocity, fps fps Depth, Depth, ft ft Energy Energy Grad Grad Width Width ft ft

29 Sand Beds HEC-RAS Stable Channel Design 29

30 HR Stable Channel Design Copeland Method using Brownlie Resistance and Sediment Transport Eqs. Sand Bed Channels Only. Resistance Due to Sidewall Roughness, Grains of the Bed Material and Bed Forms. Sediment Transport from Bed Only. Sidewall Roughness Method Applied. Does not specify channel plan form geometry or profile features. (See McEnroe KAM and ARR methods.) 30

31 HR Stable Channel Design Requirements Upstream Supply Channel: (Trapezoidal channel geometry required.) Bottom width, depth, channel slope, side slopes, discharge, Manning s s n for sidewalls, sediment gradation or sediment conc. Design Channel: Manning s s n for sidewalls, side slopes, sediment gradation,and either the bottom width, depth, or channel slope. Both: Need d 16, d 50 and d 84 31

32 Procedure for Stable Channel Design of Sand Bed Channels Establish the bank-full properties of an upstream reference reach riffle cross section. Discharge, cross section geometry via station-elevation data, stage, bed material (d 16, d 50 and d 84 ),, longitudinal energy grade line slope. Open the Uniform Flow function with the Manning for the bank resistance and Brownlie for the movable bed resistance. Input the slope and discharge. By iteration, determine the bank n-values n needed to obtain the desired bank-full water surface elevation (stage) for the given bank-full discharge and slope. 32

33 Procedure for Stable Channel Design of Sand Bed Channels (Cont.) Using the bank n-values n from the previous step, change the resistance formula for the bed to Manning then by iteration determine the appropriate n-value n for the bed to obtain the desired bankfull stage. Create an upstream supply reach that has three of the natural channels using the bank and bed n-values determined above for the bankfull channel. Run the HEC-RAS model. 33

34 Procedure for Stable Channel Design of Sand Bed Channels (Cont.) Determine an equivalent trapezoidal channel that has the same conveyance as the natural supply reach. Open the Stable Channel Design function. Input the side slopes, base width, bank n-values n and the energy grade line slope of the equivalent upstream supply channel. Input the side slopes and bank n-values n for the design channel. Run the Stable Channel Design model. 34

35 Sand Bed Example Bank-full Conditions d 16, d 50, d 84 = 1.33, 2 and 3 mm, respectively Q = 325 cfs Stage = 7 ft Slope = Elevation (ft) Elevation (ft) Station (ft) Station (ft) Sta-Elev Sta-Elev Bankfull Elevation Bankfull Elevation Movable Bed Movable Bed 35

36 Uniform Flow with Final Manning s s n values (Initially Brownlie for movable bed, unknown for banks) 36

37 Natural Supply Reach 37

38 Equivalent Trapezoidal Channel h A bnk P bnk Equivalent Trapezoidal Channel Elevation (ft) Elevation (ft) Station (ft) Station (ft) Sta Elev Points Bankfull Water Surface Equivalent Sta Elev Points Channel Movable Bankfull Bed Water Surface Equivalent Channel Movable Bed 38

39 Trapezoidal Channel Supply Reach 39

40 Stable Channel Design Function 40

41 Compute 41

42 Select Design Channel for b = 20 ft 20 FT 20 FT 42

43 Stability Curve, Width vs. Slope ppm ppm 43

44 Gravel/Cobble Beds HEC-RAS Sediment Transport Capacity Function 44

45 HR Sediment Transport Capacity (STC) Grain size classes are input as grain size and percent finer. Computes STC for each size class, g si Total STC is computed by the equation g s,total = p i g si where p i = fraction of size class i in the bed. Can compute the total STC for all six Sediment Transport Potential functions. 45

46 Gravel/Cobble Example The stream has the following bank-full conditions Water surface elevation = 11.7 feet Discharge = 3,100 cfs Slope = Elevation (ft) Elevation (ft) Station Station (ft) (ft) Sta-Elev Sta-Elev Bankfull Bankfull Elevation Elevation Movable Movable Bed Bed

47 Pebble Count for Gravel/Cobble Stream INCHES PARTICLE MILLIMETER SIZE CLASS COUNT % CUM % D top (mm) Silt/Clay < S/C Very Fine S Fine A Medium N Coarse D Very Coarse S Very Fine Fine G Fine R Medium A Medium V Coarse E Coarse L Very Coarse S Very Coarse Small C Small O L Large B E Large B S Small B D Small O E Medium U R Lrg to Very Lrg L S BEDROCK BDRK NOTE 47

48 Log-probability plot of Bed Material Sand Gravel Cobble 48

49 Make New HEC-RAS Model with one cross section and no discharge 49

50 Uniform Flow Bed uses Limerinos, banks use Manning (T&E gives n bank = 0.077) 50

51 Uniform Flow Bed uses Mannings, Banks use n = (T&E gives n bed = ) 51

52 Create and Run Natural Supply Reach 52

53 Sediment Transport Capacity Function LOB Input Grain Sizes (Fake size for banks) Main ROB Diam, mm % Finer Diam, mm % Finer Diam, mm % Finer

54 Compute, Sediment Rating Curve Plot, Generate Report 54

55 Select All Grains Sizes to see a more detailed Report Bed Material Fraction by Standard Grade Size A-W E-H Laur MPM Toff Yang Bed Material Fraction by Standard Grade Size A-W E-H Laur MPM Toff Yang Class dm (mm) Left Main Right (tons/day) (tons/day) (tons/day) (tons/day) (tons/day) (tons/day) Class dm (mm) Left Main Right (tons/day) (tons/day) (tons/day) (tons/day) (tons/day) (tons/day) All Grains All Grains

56 Meyer-Peter Mueller Function Results

57 Design 1 (b = 35 ft, m = 3:1 hor: vert) Assume slope Create 3 cross section model with trapezoidal xsecs with same n s n s and Q as supply reach Run steady flow model Run Sediment Transport Capacity function See if STC equals 1083 tons/day if not back to the top with a new slope 57

58 Design 1 (b = 35 ft, m = 3:1 hor: vert) T & E gives S = T & E gives S = S = = S = = b b = = Natural Natural Design Design b = 35 Natural Design Nat/Des b = 35 Natural Design Function tons/day Nat/Des Nat/Des Function tons/day Nat/Des Function tons/day Function tons/day A-W A-W NA NA A-W A-W NA NA E-H E-H E-H E-H Laur Laur NA NA Laur Laur MPM MPM MPM MPM Toff Toff Toff Toff Yang Yang Yang Yang S = = S = = b b = = Natural Natural Design Design b = 35 Natural Design Nat/Des b = 35 Natural Design Function tons/day Nat/Des Nat/Des Function tons/day Nat/Des Function tons/day Function tons/day A-W A-W A-W A-W NA NA E-H E-H E-H E-H Laur Laur Laur Laur MPM MPM MPM MPM Toff Toff Toff Toff Yang Yang Yang Yang

59 Final Design 59

60 Design 2 - Select slope and side slopes, find b S = , m = 2.5:1 hor: vert b b = = STC STC in in Tons/Day Tons/Day bnk bnk ht ht = = S S = = m = = Function Function All All Grains Grains RS RS 0 0 RS RS RS RS A-W A-W Station Station Elevation Elevation Station Station Elevation Elevation Station Station Elevation Elevation E-H E-H Laur Laur MPM MPM Toff Toff Yang Yang

61 S = , m = 2.5:1 hor: vert b = 15 ft, bank ht = ft Design Design Channel, Channel, b = 15', 15', hor:vert hor:vert = 2.5:1, 2.5:1, S = Elevation Elevation (ft) (ft) Station Station (ft) (ft) Design Design Channel Channel Bankfull Bankfull Elevation Elevation Movable Movable Bed Bed 61

62 Elevation Elevation (ft) (ft) Design 3 - S = , m = 2.5:1 hor: vert b = 76 ft, bank ht = 8.73 ft Station Station (ft) (ft) Design Design Channel Channel Bankfull Bankfull Elevation Elevation Movable Movable Bed Bed 4 4 b b = = MPM Gs (Tons/day) = Final Final Design Design 2 2 bnk bnk ht ht = = S S = = m = = Tons/day Tons/day RS RS 0 0 RS RS RS RS Function Function All All Grains Grains Station Station Elevation Elevation Station Station Elevation Elevation Station Station Elevation Elevation A-W A-W E-H E-H Laur Laur MPM MPM Toff Toff Yang Yang MPM Gs (Tons/day) = Final Design 3 62

63 Sidewall Correction Method W d n w A w /2 m d y d A A w /2 b A b 1 1 m d b d n b m d n w square } 4/3 2/3 1/2 2 A 2 4/3 1 1 Metric version of Manning ' s Equation V = R S V = S n n P Einstein assumed V and S are constant in bank area and sidewall area constant /3 2 3/4 3/4 2 3/4 V 1 A V 1 A 1 A V = 2 4/3 = 2 = 3/2 S n P S n P n P S 3/2 P Constant 3/2 Pw 3/2 Pb n = nw = nb = β A A A w b P P P A= A + A n = n + n n P= n P + n P β β β 3/2 3/2 w 3/2 b 3/2 3/2 3/2 w b w b w w b b 2/3 3/2 3/2 1 n P n P n = n P + n P also A = A and A = A 3/2 3/2 P n P n P 3/2 3/2 w w b b ( w w b b) w ( ) b ( ) 63

64 Sidewall Example 1 2 S = n w = n = w = n b = P = ft P = + = ft P= P + P = ft 2 2 b 50 ; w b w 72.4 (5)(10) A = 50*5 = 250 ft ; A = 2 = 50 ft A = A + A = 300 ft rect tria rect tria 2/3 2/3 1 3/2 3/2 1 3/2 3/2 n= ( nw Pw + nb Pb) = ((0.040) (0.025) 50) P 72.4 n = Q = 1.49 (300) (72.4) 5/3 2/ / (0.040) 22.4 (0.025) 50 Aw = = ft and A = = ft (0.030) 72.4 (0.030) /2 3/2 2 2 ( 300) 143 3/2 b 3/2 ( 300) 157 Geometric values A = 50 ft and A = 250 ft w = ft s 2 2 b 64

65 ARR Analytical Method W d n w y d n d = Manning s 1 composite n 1 m d m d b d n b m d n w 1.49 A Manning for Design Channel Qd = Sd ARR Eq.4 2 n P ( s ) ( ) 5/3 d 2/3 d 8b 3/2 MPM B = γys ( γs γ) d50 ARR Eq. 4 6 ρ y S G 1 d = =.4 8 m m 50 Bm Bd bd bm ARR Eq yd Sd Gs 1 d50 Given Q, m, n, S = S, G = 2.65, d, b and y. d d d m d s 50 m m Use iteration to solve Eqs.4 2and 4 8for b and y by iteration d Subscripts d and m denote design channel and match reach channels, respectively. d 3/2 d 65

66 ARR vs HEC-RAS Composite n from HEC-RAS Supply Reach ) ARR solution for Design 1 ( HR solution b d = 35 ft) d m (mm) ) ARR solution for Design 1 ( HR solution b d = 35 ft) d m (mm) m d = 3 S d = n d = m d = 3 S d = n d = y d b d A d P d W b Q b ΔQ y d b d A d P d W b Q b ΔQ b HR = 35 ft b ARR = 42.9 ft b HR /b ARR = 0.82 ) ARR solution for Design 2 ( HR solution b d = 15 ft) d m (mm) ) ARR solution for Design 2 ( HR solution b d = 15 ft) d m (mm) m d = 2.5 S d = n d = m d = 2.5 S d = n d = y d b d A d P d W b Q b ΔQ y d b d A d P d W b Q b ΔQ b HR = 15 ft b ARR = 22.8 ft b HR /b ARR = 0.66 ) ARR solution for Design 3 ( HR solution b d = 76 ft) d m (mm) ) ARR solution for Design 3 ( HR solution b d = 76 ft) d m (mm) m d = 2.5 S d = n d = m d = 2.5 S d = n d = y d b d A d P d W b Q b ΔQ y d b d A d P d W b Q b ΔQ ARR did not converge b HR = 76 ft b ARR = 58.8 ft b HR /b ARR =

67 ARR vs HEC-RAS Composite n values from HR Design Reaches (b) ARR solution for Design 1 ( HR solution b d = 35 ft) m d = 3 S d = n d = d m (mm) = 22.6 y d b d A d P d W b Q b ΔQ n = b HR = 35 ft b ARR = 33.2 ft b HR /b ARR = 1.05 (c) ARR solution for Design 2 ( HR solution b d = 15 ft) m d = 2.5 S d = n d = d m (mm) = 22.6 y d b d A d P d W b Q b ΔQ (d) ARR solution for Design 3 ( HR solution b d = 76 ft) ft) m d = 2.5 S d = n d = d m (mm) = 22.6 y d b d A d P d W b Q b ΔQ n = b HR = 15 ft b ARR = 17.8 ft b HR /b ARR = 0.84 ARR did not converge n = b HR = 76 ft b ARR = 58.8 ft b HR /b ARR =

68 ARR s s Simplified MPM Equation ARR simplified MPM equation 1 y d 8b } 3/2 B= ( RKR) γ R S γ γ d ρ ( ) }50 b s m 3/2 Meyer Peter Mueller ( MPM ) 8b B= RKR R S ( ) d ρ ' b b 3/2 ( ) γ b γs γ m ' n b RKR = = Nikuradse roughness ratio nb n n = Manning coefficient for grain size = total Manning coefficient 3/2 68

69 Conclusions McEnroe s s KAM and ARR methods provide very useful tools and should serve as references for all stable channel design projects. It is recommended that HEC-RAS be used in lieu of the analytical approaches of KAM and ARR if the grain size distribution is known for an alluvial channel. If the grain size distribution is unknown or if the channel has a cohesive bed, the KAM and ARR methods should be used in their entirety. The HEC-RAS methods reported herein only provide for design of the riffle cross section and do not include help for planform design aspects. The methods outlined in McEnroe s s reports should be used for the overall stream design. They provide guidance for the design of meanders, riffle pool spacing and pool dimensions. 69

70 Analysis of Bed Grain Size Distribution Sieve Analysis Visual-Accumulation Tube Pebble Count 70

71 Log-normal Distribution Probability Density Function PDF 1 1 log d log d f(log d) exp σlog d 2π 2 σ log d Cumulative Distribution Function CDF σ g 50 = 84.1 log σ log d d15.9 = = = log d d F(log d) = f(log d) d(log d) = P/100 StandardDeviation d d d d 1 1 d σ log d = (log d84.1 log d15.9 ) = log = log 2 2 d Geometric standard deveiation d d σ g 84.1 log σ log d d15.9 = = = d d d d d

72 Standardized Random Variable Mean = 0, standard deviation =1 2 log d log d 1 z = ( ) = exp 2π 2 50 z PDF f z σ log d z CDF F( z) = f ( z) dz where F( z) = 1 F( z) F(z) for Standard Normal Random Variable z F(z) for Standard Normal Random Variable z z z

73 Example - Sand Bed Material 73

74 Log-Probability Plot of Sand Bed Data Cumulative Distribution Function Expressed as a probability (%) d 84 =0.363 mm d (mm) d 50 =0.232 mm d 16 =0.158 mm σ σ d log d g 65 d = = = d log log = = ( ) ( ) 0.386σ + log d log = = = = log d mm 74

75 Log Probability Plot using NORMSINV Function in EXCEL -0.2 d 15.9 d 50 d 84.1 log(d 84 ) log(d 84 ) log(d 50 ) log(d 50 ) log(d 16 ) log(d 16 ) log 10 (d) NORMSINV(P/100) 75

76 NORMSINV and NORMSDIST EXCEL Functions d 50 = 0.48 mm log 10 (d 50) = d 50 = 0.48 mm log 10 (d 50) = σ g = 1.28 mm log 10 (σ g) = σ g = 1.28 mm log 10 (σ g) = log( di ) log( d50) zi = log( di ) log( d50) zi = log log ( ( σ g σ ) ) d d = = i i log ( d ) + z log ( σ ) ( ) ( ) 50 i g 10 log d50 + zilog σ g 10 g P (% finer) F(z) z=normsinv(f) d i (mm) F(z)=NORMSDIST(z) P (% finer) F(z) z=normsinv(f) d i (mm) F(z)=NORMSDIST(z) d 20 = d 20 = d 40 = d 40 = d 60 = d 60 = d 80 = d 80 = d = d = d 50 = d 50 = log( di ) log( d50) zi = log( di ) log( d50) zi = log log ( ( σ g σ ) ) g d i (mm) z F(z)=NORMSDIST(z) P (% finer) d i (mm) z F(z)=NORMSDIST(z) P (% finer)

77 Geometric Standard Deviation ( d84 d50 ) ( d50 d16 ) ( d d ) ( d d ) ( d d ) ( d d ) 2 logσ = log log + log log ( ) ( ) ( ) 2 logσ = log / + log / = log / logσ = log ( d / d ) logσ = log ( d / d ) = log d / d 2 σ = / d logσ = log d log d σ = d / d logσ top d Alternative Method in HEC bot RAS Manual top = log d log d σ = d / d bot d d σ = σ 0.5( ) ave = σtop + σbot = d d

78 Geometric Standard Deviation when P i for smallest sample d is greater than 0.16 Let P i = the lowest percent finer from the pebble count analysis and let z i = the standardized normal variable that gives F(z) = P i /100. ( 84 P ) (1 z)logσ = log d log d logσ σ ( log 84 log P ) = = = 1 (1 z) d 84 = d P 1 1 d log d d d d log (1 z) (1 z) d P (1 z) P 78

79 1 Example for Smallest d > d 16 Given d = 8mm and d = 2mm P = 32 F( z) = 0.32 F( z) = 1 F( z) = = 0.68 z = z = σ (1 z) d (1 ( )) = = = = 84 log d50 log d84 log 2.04 log 2.04 d 50 d P d = = d = = = 2.79 mm ( 79

80 Pebble Count for Gravel/Cobble Stream Sand Gravel Cobble 80

81 L A SOIL PARTICLES PORES Theoretical Justification for Pebble Count porosity = V / V ; A = na; V = A( x) dx pores total pores pores pores 0 As = (1 n) A = area of A occupied by soil Ai = area of A occupied by particles of a specified size range fi = Ai / As Wi γ svi Vi Vi pi = = = = W γ V V (1 n) V p p p i i i s, total s s, total s, total total L L L L i i ( i s) ( i[ (1 ) ]) A dx A dx f A dx p n A dx = = = = (1 nv ) (1 n) AL (1 n) AL (1 n) AL L total ( i[ ]) L f (1 n) A dx fi (1 n) Adx 0 0 fi (1 n) AL = = = (1 nal ) (1 nal ) (1 nal ) = f i Actual Pebble Count Shielding, settling, etc. Actual Pebble Count Shielding, settling, etc. L 81

82 Mixed Sand and Gravel Beds Watershed Institute, Inc. Pebble Count Data log10(d) log10(d) MT043442RR01 MT043442RR01 d D16 dd50 d D84 3 d D16 dd50 d D D d (mm) 0.5 D d (mm) 0.5 dd dd D -0.5 d dd dd dd normsinv(p/100) normsinv(p/100) Pebble Count Data D16 D50 D84 Extrapolated Pebble Count Data D16 D50 D84 Extrapolated log10(d) log10(d) SN321115RR SN321115RR dd16 dd50 dd84 3 dd16 dd50 dd d D (mm) 0 d D (mm) -1 dd dd 16 D d D dd d dd normsinv(p/100) normsinv(p/100) Pebble Count Data D16 D50 D84 Extrapolated Pebble Count Data D16 D50 D84 Extrapolated 82

83 Mixed Sand and Gravel Beds (cont.) W finer (gm) W finer (gm) Sand Gravel Sand Gravel Dd Dd σ g W σ g (gm) W (gm) Two Bed Materials Two Bed Materials log D log 10 (d) D log 10 (d) Gravel Sand Gravel Sand log log D log 10 D (d) log 10 (d) Two Bed Materials Two Bed Materials 3 3 Sand 2.5 Sand Gravel Gravel Sand 2.5 Dd Sand Gravel Gravel 10 2 Dd 50 σ g σ 1.5 W g (gm) W (gm) NORMINV(P/100) norminv(p/100) NORMINV(P/100) norminv(p/100) Sand Gravel Sand Gravel 83

84 Mixed Sand and Gravel Beds (cont.) W Finer (gm) W Finer (gm) Combined Combined Sand SandGravel Gravel 90 Sand SandGravel Gravel D dd σ g d σ W (gm) g g W (gm) σ g W (gm) W (gm) log D log D log log 10 (d) 10 (d) Combined Combined log log 10 (d) 10 (d) log D log D Sand SandGravel Gravel Sand SandGravel Gravel D dd σ g d σ W (gm) g g W (gm) σ g W (gm) W (gm) log D 50 D 50-1 log 10 (d 50 ) d 50 (mm) log log D 50 D (d ) d (mm) norminv(p/100) norminv(p/100) 84