AN OPTIMIZATION MODEL FOR THE DEVELOPMENT ANDRESPONSE OF ALLUVIAL RIVER CHANNELSByROBERT GARY MILLARB.Sc.(Hons), The University of Queensland, 1984M.A.Sc., The University of British Columbia, 1991A THESIS SUBMITfED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDepartment of Civil EngineeringWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1994© ROBERT GARY MILLAR, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department of 3 v vf€rThe University of British ColumbiaVancouver, CanadaDate S4q.r (9DE-6 (2/88)AbstractIn this thesis an optimization model has been developed to calculate the equilibrium geometryof alluvial gravel-bed rivers for a given set of independent variables. The independent variablesare the discharges, both the magnitude and duration which are represented by a flow-durationcurve; the mean annual load, both volume and grain size distribution, which is imposed on tothe channel reach from upstream; and the geotechnical properties of the bank sediment.The unknown dependent or decision variables to be solved for include the channel width, depth,bank angle, roughness, and grain size distribution of the bed surface. The dependent variablesadjust subject to the constraints of discharge, bedload, bank stability, and valley slope, todetermine a channel geometry which is optimal as defined by a maximization of , which is thecoefficient of sediment transport efficiency.The work in this thesis is an extension of earlier models that have predicted the geometry ofsand and gravel rivers with reasonable success, however the degree of scatter associated withthese models limited their application to quantitative engineering applications. The advances inthis thesis over the earlier optimization models are the inclusion of the bank stability analyses,modelling using the full flow-duration data, and calculating the grain size distribution of thebed-surface. The formulation presented in this thesis is specific to gravel-bed rivers, however itcan be reformulated for sand-bed rivers.11Only limited verification of the model has been attempted at this stage by testing the theory onpublished field data, and through comparisons with previously obtained qualitative relations.Good agreement between the modelled and observed channel geometries was obtained forchannels with weakly developed bank vegetation using a simplified version of the optimizationmodel. The effect of the bank vegetation on the channel width was shown to agree closely withempirical regime analyses.The potential uses of the optimization model are to interpret observed river adjustments, topredict future channel changes in disturbed catchments, and to gain further understanding andinsight into the behaviour of alluvial rivers.Recommendations are made for future verification of the model.111Table of ContentsAbstract iiTable of Contents ivList of Appendices ixList of Tables xList ofFigures xiList of Symbols xivAcknowledgement xx1 INTRODUCTION 11.1 INTRODUCTION 11.2 EXAMPLES OF ACTIVITIES THAT IMPACT RIVERS 41.2.1 Dams and Reservoirs 41.2.2 Urbanization and Agricultural Development 51.2.3 Logging 61.2.4 Other Activities 71.3 APPROACHES TO THE PROBLEM 71.3.1 Case Studies 71.3.2 Qualitative Relations 81.3.3 Empirical Regime Equations 91.3.3.1 Width 101.3.3.2Depth 111.3.3.3 Slope 111.4.4 Analytical Models 131.3.4.1 StaticModels 131.3.4,2 Dynamic Models 141.4 THESIS OUTLINE 16iv2 QUALITATIVE MODEL FORMULATION .192.1 iNTRODUCTION 192.1 TEMPORAL AND SPATIAL SCALES 192.2.1 Temporal Scales 192.2.2 Spatial Scales 202.3 DEFINITION OF EQUILIBRIUM 212.4 OPTIMAL HYDRAULIC GEOMETRY 222.5 OPTIMLZATION MODEL 242.5.1 Independent Variables 262.5.2 Dependent Variables 272.5.3 Objective Function 272.5.4 Constraints 312.6 SUMMARY 323 DISCHARGE CONSTRAiNT 363.1 INTRODUCTION 363.2 FLOW RESISTANCE 363.2.1 Subdivision offinto Grain and Form Components 393.2.2 Estimation off” for Gravel-Bed Rivers 423.2.2.1 Einstein and Barbarossa (1952) 423.2.2.2 Parker and Peterson (1980) 443.2.2.3 Prestegaard (1983) 453.2.3 Variation of f” with Discharge 463.3 BANKFULL DISCHARGE AND OVERBANK FLOW 493.3.1 Recurrence Interval ofBankfull Flow 523.4 SUBDIVISION OFf INTO BED AND BANK COMPONENTS 533.5 DISCHARGE CONSTRAINT 573.6 SUMMARY 584 BEDLOAD CONSTRAINT 684.1 INTRODUCTION 68v4.2 BOUNDARY SHEAR STRESS DISTRIBUTION .684.2.1. Cross-Channel Momentum Transfer .694.2.2 Analytical Solutions 704.2.3 Experimental Methods 714.2.3.1 Non-uniform bed and bank roughness 734.2.4 Gram and Bar Shear 754.3 MODELLING SEDIMENT TRANSPORTING CAPACITY ATBANKFULL 774.3.1 Generalized Equations for Mean Bed and Bank ShearStresses 784.4 BED SURFACE 804.4.1 Influence of the Armour Layer 824.5 PARKER SURFACE-BASED BEDLOAI) TRANSPORTRELATION 844.5.1 Calculation of Armour Grain Size Distribution 884.5.2 Modification of the Parker Surface-Based Relation forNatural Rivers with Variable Discharge 914.6 TOTAL BEDLOAD CONSTRAINT 944.7 SUMMARY 955 BANK STABILITY CONSTRAINT 1085.1 INTRODUCTION 1085.2 COHESIVE BANK SEDIMENT 1085.2.1 Mass Failure 1085.2.1.1 Bank Stability and Submergence 1125.2.1 2 Bank Height Constraint 1155.2.2 Fluvial Erosion 1155.2.2.1 Bank ShearConstraint 1175.3 NONCOHESIVE BANK SEDIMENT 1175.3.1 Mass Failure 1185.3.2 FluvialErosion 119vi5.4 SUMMARY .1216 EFFECT OF BANK STABILITY ON CHANNEL GEOMETRY 1276.1 INTRODUCTION 1276.2 BANKFULL : FiXED-CHANNEL-SLOPE OPTIMiZATIONMODEL 1276.2.1. Independent Variables 1286.2.2. Dependent Variables 1296.2.3. Objective function 1296.2.4 Constraints 1306.2.4.1 Continuity 1316.2.4.2 Bank Stability Constraint 1316.2.5. Optimization Scheme: Fixed-Channel-Slope 1326.3 NONCOHESIVE BANK SEDIMENT 1356.3.1 Low Density Bank Vegetation 1386.3.2 Effect of bank vegetation 1396.3.3 Influence of ç5is on Channel Geometry 1436.4 COHESIVE BANK SEDIMENT 1466.4.1 Bank Stability Routine 1476.4.2 Data From Charlton et al. (1978) 1486.4.3 Effect of Bank Vegetation 1536.4.4 Discussion ofMarch etal. (1993) 1536.5 BANKFULL MODEL:VARIABLE-CHANNEL-SLOPE 1576.5.1 Noncohesive Bank Sediment 1586.5.2. Cohesive Bank Sediment 1616.6 SUMMARY 1647 FULL MODEL FORMULATION 1817.1 INTRODUCTION 1817.2 MODEL FORMULATION 1817.2.1 Independent Variables 182VII7.2.2 Dependent Variables .1837.2.3 Objective Function 1847.2.4 Constraints 1847.2.4.1 Continuity 1847.2.4.2 Bedload 1857.2.4.3 Bank Stability 1867.2.5 Optimization Scheme 1867.3VARJABLEFLOWS 1877.4 ADJUSTMENT OF THE BED SURFACE COMPOSITION 1897.4.1 So - r*D50 Solution Curves 1907.4.2 Effect of Sediment Gradation on 550 1937.5 EFFECT OF SEDIMENT LOAD 1937.5.1 Valley Slope Constraint 1957.6 BANKFULL DISCHARGE AS A DEPENDENT VARIABLE 1967.6.1 Effect of Sediment Load on Bankfull Discharge 1997.7 APPLICATION OF THE MODEL 2007.8 HOW AND WHY DO ALLUVIAL CHANNELS OPTIMIZE? 2027.9 COMPARISON WITH OTHERNUMEBJCAL MODELS 2057.11 AN EXAMPLE OF RIVER BEHAVIOUR WITHIN ANOPTIMIZATION FRAMEWORK 2107.11 SUM14’IARY 2118 CONCLUSIONS AND RECOMMENDATIONS 2308.1 SUMMARY 2308.1.1 Inclusion Sank Stability Analysis 2318.1.2. Modelling Using The Full Flow-Duration Data 2338.1.3 Adjustment of the Bed Surface 2338.1.4 Bankftill Discharge as a Dependent Variable 2348.1.5 Verification of the Optimization Model 2358.2 RECOMMENDATIONS FOR FUTURE WORK 2368.2.1 Channel Form or Bar Roughness 236viii8.2.2 Surface-Based Transport Relation Based on t’bed 2378.2.3 Bank Stability 2378.2.4 Secondary Currents 2388.2.5 Formulation for Sand-Bed Rivers 2398.2.6 Model Verification 240List ofAppendicesA BANKFULL: FIXED SLOPE MODEL 261B BANKFULL: VARIABLE SLOPE MODEL 275C FULL MODEL FORMULATION 291D DATA FROM HEY AND THORNE (1986) AND ANDREWS (1984) 314E DATA FROM CHARLTONETAL. (1978) 319ixList of TablesTable 5.1. Parameters for 3 cases of bank stability 114Table 6.1. Ratios between the observed and modelled values of Wand Y 140Table 6.2. Summary of ratios of observed channel width divided by theunvegetated channel width 142Table 6.3. Summary of ç values obtained analytically for the data sets ofAndrews (1984) and Hey and Thome (1986) 145Table 6.4. Results from the analysis of data from Charlton et al. (1978) 151Table 6.5. Ratios between the observed and modelled values of W, Y, and S forvariable slope model 159Table 6.6. Effect of q on reach 13 from Hey and Thome (1986) 160Table 7.1. Summary of sediment size parameters from Figs 7.10 and 7.11 193Table 7.2 Comparison of flow parameters for Qbf= 65 m3/s and Qbf 135 m3/s 198Table D- 1. Data from Hey and Thorne (1986) 315Table D-2. Data from Andrews (1984) 318Table E-1. Data from Charlton et al. (1978) 320xList of FiguresFigure 1.1 Idealised fluvialsystem.18Figure 2.1 Definition sketch for a representative channel length 33Figure 2.2 A schematic representation of the optimal geometry in fluvialsystems 34Figure 2.3 Definition sketch for development of the coefficient of sedimenttransport efficiency i 35Figure 3.1 Variation offwith respect to Rh /D30 for selected gravel-bedrivers 60Figure 3.2 Variation off” with respect to ZI*D50 for selected gravel-bedrivers 61Figure 3.3 Variation off” with respect to the components of Z*D5O forselected gravel-bed rivers 62Figure 3.4 Variation offwith respect to Q 63Figure 3.5 Definition sketch of a prismatic channel 64Figure 3.6 Experimental velocity contours (from Seffin, 1964) 65Figure 3.7 Recurrence interval for Qbf (from Williams, 1978) 66Figure 3.8 Subdivision of cross-section into bed and bank sections 67Figure 4.1 Simplified velocity distribution in a straight open channel 97Figure 4.2 Area methods for calculating local boundary shear stress 98Figure 4.3 Boundary shear stress distribution from Lane (1955b) 99Figure 4.4 Boundary shear stress variation for variable Sand uniformboundary roughness 100Figure 4.5 Boundary shear stress variation for constant Sand non-uniformboundary roughness 101Figure 4.6 Grain shear stress versus total shear stress 102xFigure 4.7 Discretized flow-durationcue.103Figure 4.8 Bedload transport rates versus Q for Oak Creek, Oregon 104Figure 4.9 Variation of the function G 105Figure 4.10 Variation of the functions oj, and with gO afterParker (1990) 106Figure 4.11 Variation ofd50,with Q for Oak Creek, Oregon 107Bank stability analysis for planar failure surface 123Bank stability analysis for the method of slices 124Stability curves 125Three special cases for bank stability analysis 126Flow chart for bankfull:fixed-channel-slope model 166Variation of z and ed with 9 167Variation of i and 9 with bed 168Comparison of modelled and observed values of Wand Yfor low densities of bank vegetation 169Figure 6.5 Comparison of modelled and observed values of Wand Yfor all categories of bank vegetation density 170Figure 6.6 Comparison of modelled and observed values of Wand Yfor low densities of bank vegetation 171Figure 6.7 Comparison of modelled and observed values of Wand Yfor the data from Chariton et al (1978) 172Figure 6.8 The values of r calculated from the data of Chariton et al.(1978) for treed (T) and grassed (G) channel banks 173Figure 6.9 Bank stability curves from March et a! (1993) 174Figure 6.10(a) and (b) Photographs of stream bank erosion from asmall creek flowing across a beach at low tide 175Figure 6.10(c) and (d) Photographs of stream bank erosion from asmall creek flowing across a beach at low tide 176Figure 5.1Figure 5.2Figure 5.3Figure 5.4Figure 6.1Figure 6.2Figure 6.3Figure 6.4XIIFigure 6.11 Flow chart for bankfull: variable-channel-slope optimizationmodel 177Figure 6.12 Variation of ,7 and S with bed 178Figure 6.13 Variation of the optimal values of selected dependent variablesas a function of Qbf 179Figure 6.14 Variation of the optimal values of selected dependent variables as afunction of Gbf 180Figure 7.1 Flow-duration curve showing numerical approximation 213Figure 7.2 Sediment gradation curve showing numerical approximation 214Figure 7.3 Flow-chart for full optimization model formulation 216Figure 7.4 Flow chart for bedload subfunction 216Figure 7.5 Calculated sediment transport rates as a function of discharge 217Figure 7.6 Variation ofd501 with discharge 218Figure 7.7 The calculated ô., - r0 solution curve for Oak Creek, Oregon ... 219Figure 7.8 Armour and subarmour grain size distributions for Oak Creek 220Figure 7.9 c5-curve for data from Dietrich et a! (1989) 221Figure 7.10 Sediment gradation curves for a range of Ug values 222Figure 7.11 S- D50 curve for a range of 0g values 223Figure 7.12 Effect of sediment load on channel geometry 224Figure 7.13 Calculated variation of S with Qbf 226Figure 7.14 Calculated variation of W with Gb for constant and variable Qbf. .. 226Figure 7.15 Solution curve used to demonstrate channel optimization 227Figure 7.16 Solution curves used for a comparison of numerical models 228Figure 7.17 A solution curve used to interpret the behaviour of a meanderingriver that has been straightened 229xliiList of SymbolsA = Cross-sectional Areaii A = Change in A for one time stepa = Empirical Coefficientb = Empirical CoefficientC = Parameter from Eqn (4.11)C = Multiplier to Scale D to kC = Chezy Flow Resistance Coefficient (Eqn 3.16)c = Soil Cohesionc’ = Soil Cohesion in Terms ofEffective StressesD = Gram Diameter or SizeD = Bed Surface Grain Diameter where x% is FinerD,g = Geometric Mean Grain Diameter ofBed Surface from Eqn (4.35)D. = Repesentative Grain Diameter for Intervalj from the Sediment GradationCurve (Eqn 7.2)D50 = Median Grain Diameter ofNoncohesive Bank Sedimentbankd = Bedload or Sub Pavement Grain Diameter where x% is Finerd3g = Geometric Mean Grain Diameter of Subsurface Sediment= Total Potential Energy Expended by River (Eqn 2.3)= Potential Energy Expended by Water (Eqn 2.3)= Potential Energy Expended by Sediment (Eqn 2.3)e = Sediment Transport Efficiency (Eqn 2.2)= Volume Fraction ofDj in Bed-Surface Layer.xivF1 = Parameter from Eqn (6.2)FD = Driving Force for Mass Failure of Cohesive SoilFR = Driving Force for Mass Failure of Cohesive SoilF8 = Factor of Safety which Equals the Ratio FR / FDf = Darcy-Weisbach Friction Factor= Friction Factor for Q’= Fraction ofD in Subsurface SedimentG [1 = Bedload Transport Function (Eqn 4.32)Gb = Imposed Sediment Load (Generally in Units of kg/year)Gbf = Bankfull Sediment Transport Capacity (kg/s)G’b = Sediment Transport Capacity as a Rate for Q(Generally in Units of kg/s)gb = Dry Bedload Transport Rate per metre width in Mass Unitsg*= Dimensionless Bedload Transport Rate per metre widthg = Gravitational Acceleration (Assumed = 9.81 m/s2)go() = Reduced Hiding Function (Eqn 4.34)H = Vertical Bank Height (Fig 3.5)Hcrit = Maximum value ofH that is stable with respect to mass failureI = Volume of Sediment Input (Eqn 2.1)K = Conversion Factor (Eqn 2.7)k = Empirical Constant (Eqn 5.14)= Roughness Height which is the Equivalent Sand RoughnessL = Length ofRepresentative Channel Reach (Fig 2.1), or Length ofFailureSurface (Fig 5.1)1 = Basal Length of Soil Unitxvm = Number of Intervals in Flow-Duration Curve (Fig 7.1)n Number of Intervals in Sediment Gradation Curve (Fig 7.2)0 Volume of Sediment Output (Eqn 2.1)NR = Normal Upward Force at Base of Soil UnitN = Cohesive Bank Stability Factor (Eqn 5.2)P Channel Wetted Perimeterp1 Probability ofFlow Q’ within Discharge Interval iQ Discharge (m3/s)Qbf = Bankfull DischargeQd = Dominant or Characteristic Discharge= Represetative Discharge for ith Interval from Flow-Duration Curve(Eqn 7.2)Q = Arithmetic Mean Annual Discharge= Volumetric Sediment Transport Rate (L3 / T)‘lb = Volumetric Sediment Transport Capacity per Unit Channel Width (L2/T)Rh = Hydraulic Radius (without subscripts bed or bank refers to bankfull value)S = Energy, Water Surface, or Channel Slope (Uniform Flow Assumed)S = Valley SlopeSR = Soil Shearing Resistance4S = Change in Volume of Sediment Stored within a Channel Reach (Eqn 2.1)SF = Shear Forces = Specific Gravity ofNoncohesive Bed and Bank Sediment (Assumed2.65)U = Mean Channel Velocity (without subscript refers to bankfull value)(4 = Mean Shear Velocity (.J7)u = Vertically Averaged Local Velocity (Eqn 4.3)xW = Channel Surface Width (without subscript refers to bankfbll value)W = Dimensionless Bedload Transport Parameter (Eqn 4.28)V = Average Channel Depth (without subscripts bed or bank refers to bankfi.illvalue)Y = Normal Channel Depth (without subscripts bed or bank refers to bankfull value)z Cross-Channel Coordinate (Eqn 4.3)= Angle ofFailure surface (degrees)= Dimensionless Over-bank Flow Depthy Unit Weight of Water (Assumed = 9810 N/rn3)Unit Weight of Sediment= Drained Unit Weight of Cohesive Bank Soil= Total Unit Weight of Cohesive Bank SoilYb Bouyant Unit Weight of Cohesive Bank SoilRatio D50 / d50 (Eqn 4.25)Ratio D• / Dsg (Eqn 4.34)= Coefficient of lateral momentum transfer (Eqn 4.3)4. = Internal Friction Angle (degrees)= Friction Angle in terms of effective stresses (degrees)In Situ Friction Angle for Noncohesive Banks Sediment (degrees)4m = Modified Friction Angle (degrees) from Eqns (5.3) and (5.11)= Meander WavelengthCoefficient of Sediment Transport Efficiencye = Bank Angle (degrees)= Maximum Angle for which Banks are Stable (°)p = Density ofWater (Assumed = 1000 kg/rn3)a = Pore Water Pressure at Base of Soil Unit (Fig 5.2)= Standard Deviation Based on the Sedimentological Phi Scale (Eqn 4.37)xviiasg = Geometric Standard Deviation (Eqn 7.4).CYo = Value of o# from Oak Creekt = Mean Boundary Shear Stress (N/rn2)= Dimensionless Shear Stress for the Median Grain Diameter of theMean Annual Bedload, or Subsurface Sediment.Dimensionless Shear Stress for the Median Grain Diameter of theBed Surface.trD = Reference Dimensionless Bed Shear Stress (Eqn 4.31)t*bd = Critical Dimensionless Shear Stress for the Median BankGrain Diameter on Channel Bed.t*bank = Critical Dimensionless Shear Stress for the Median BankGrain Diameter on Sloping Channel Bank.tcrzt = Critical Shear Stress for Cohesive Soil.= Straining Function (Eqn 4.34)= Value of w from Oak Creekv = Kinematic Viscosity of Water (Assumed = 0.000001 m2/s)= Channel Sinuosity (S / S)= Non-Dimensional Bed Shear Stress (Eqn 4.31)= Non-Dimensional Bed Shear Stress Based on Dsg (Eqn 4.31)= Value of’I from Oak Creek assuming t*rD = 0.03 86‘1D35 = Intensity of the Grain Shear for Bed-Surface (Eqn 3.10) From Einstein (1950)r = Conversion Constant (Eqn 4.55)Subscriptsbed = Denotes Channel Bed Portionbank = Denotes Channel Bank PortionXVIII1 = Denotes i th flowj = Denotesj th grain size fractionobs = Denotes the Observed Valuemod = Denotes the Modelled Valuetotal = Denotes Total Valueunveg = Denotes the Unvegetated ValueSuperscriptsu = Denotes Upper bound of Interval (Eqns 7.1 and 7.2)1 = Denotes Lower Bound of Interval (Eqns 7.1 and 7.2)= Denotes Grain Component of r andj= Denotes Form or Bar Component of v andfxixAcknowledgementI am indebted to my research supervisor, Dr. Michael Quick, for continued support,encouragement, and trust shown to me throughout my graduate studies. We have spent manyhours discussing the ideas that are presented in this thesis, as well as many that are not. Inaddition the members of my supervisoly committee, Drs Olav Slaymaker, Bill Castleton, andDennis Russell, gave freely of their time and contributed much to this work.The readability of this thesis has been improved greatly by suggestions made by the UniversityExaminers, Drs Michael Church, and Dave McClung, and by the External Examiner Dr. DaleBray of the University ofNew Brunswick. They are to be thanked for the diligence with whichthey pursued their assigned task.This research has been funded by a University Graduate Fellowship, an Earl Peterson MemorialScholarship in Civil Engineering, and by the National Sciences and Engineering ResearchCouncil of Canada. I am grateful to all of these sources for providing to me the financialsupport to complete this work.Finally I would like to thank my wife Karen for her support throughout these years of graduatestudies.xxCHAPTER 1INTRODUCTION1.1 INTRODUCTIONRivers represent an integral component of our civilization and economy. Civilization hashistorically been concentrated along riparian zones and floodplains due to the availability offresh water and food resources, the presence of level arable land, and because rivers act astransportation corridors for travel or trade, and as receiving waters for domestic and industrialwaste.In a classic paper, Mackin (1948) emphasised that the river is part of drainage system and thatit cannot be understood apart from that system. That is changes which are observed in a riverchannel can only be understood if they are interpreted within the framework of larger basin-scale processes. Subsequent work such as Chorley and Kennedy (1971) and Schumm (1977)have reemphasised the system approach to fluvial and other geomorphic systems.The idealised fluvial system as presented in Schumm (1977) is shown in Fig 1.1. This simplemodel subdivides the fluvial system into 3 zones. Zone 1 is the production or sediment sourcearea where the flows are generated, and the sediment is derived primarily from hillslopeprocesses. Zone 2 is the transfer area where the principal activity is the passage of the flowsand the transport of sediment produced in Zone 1. Zone 3 is the deposition zone where1sediment is deposited on an alluvial fan, alluvial plain, delta, or in deeper waters. The work inthis thesis will deal primarily with channel processes in Zone 2.Recent increases in population and industrialization have resulted in increased developmentalong rivers and throughout the river basin. Water is controlled and regulated to serve a widevariety of purposes including hydroelectric power generation, irrigation, flood control andstormwater engineering, pollution control, navigation improvement, municipal and industrialwater supply, recreation, fish and wildlife, and the conservation of soil and water on watershedlands (Linsley et a?., 1992, p. 1-2). These activities can directly impact the river channels, ormay alter the basin hydrology and sediment yield.The work in his thesis will be restricted to single-thread, alluvial gravel-bed rivers. Alluvialrivers are defined as those with bed and banks composed of sediment which is similar to thesediment that is transported by the river. Gravel-bed rivers are by definition rivers with alluvialsediment in the bed of the channel that has a mean diameter greater than 2 mm. Typically mostgravel-bed rivers have a coarse surface layer (which is usually termed the “armour” or“pavement”) in which the sediment is much coarser than the subsurface material. Rivers that arebedrock controlled, or those that have incised into older sedimentary sequences will not beconsidered herein. The analysis in this thesis applies to alluvial rivers with mobile beds, that isthose rivers which actively transport bed-material sediment, and have the capacity to modiljtheir channel dimensions.It is an underlying assumption of equilibrium channel analysis that alluvial rivers develop amean hydraulic geometry in response to the water discharge, sediment load, and sediment.properties that are determined by the geology and hydrologic regime of the upstream areas.Blench (1969, p. 1) refers to the “basic principle of self-adjustment” and states that:2The fundamental fact of river science, pure and applied, is that (alluvial) channelstend to adjust themselves to average breadths, depths and slopes and meandersizes that depend on (i) the sequence of water discharges imposed on them, (ii)the sequence of sediment discharges acquired by them from the catchmenterosion, erosion of their own boundaries, or other sources and (iii) the liability oftheir cohesive banks to erosion or deposition.The term equilibrium is generally synonymous with the expression “in regime” which is widelyused in engineering circles (Blench, 1957).Changes in the imposed discharges and sediment load result in adjustments in the river channelsand the development of a new hydraulic geometry. The channel adjustments may result inchanges that are determined to be undesirable for economic, environmental, or aestheticreasons and include bank erosion and loss of riparian habitat and land adjacent to the river,degradation of the channel bed which can undermine bridge foundations and other hydraulicstructures, aggradation of the channel bed which can reduce the channel capacity and result inan increased frequency of over-bank flooding, and changes in the physical nature of the channelthat may impact the aquatic habitat.The basic goal of this thesis is to develop a mathematical model that can be used to calculatethe equilibrium hydraulic geometry of alluvial rivers with gravel beds. The potential uses ofsuch a model are to interpret observed river adjustments, to predict future channel changes indisturbed catchments, and to gain further understanding and insight into the behaviour of thisclass of rivers.31.2 EXAMPLES OF ACTIVITIES THAT IMPACT RIVERSIn this section a selection of land-use activities and their possible impacts on rivers channels willbe discussed. The discussion in this section is necessarily general, and is intended to introducesome of the more common effects of selected basin development activities.1.2.1 Dams and ReservoirsThe construction of dams and reservoirs can have large effects on the downstream channelgeometry as a result of changes to the characteristics of the discharge and sediment load. Thesestructures typically regulate the runoff such that the peak flows are reduced, and the flows aremore uniform. Furthermore they also act as sediment traps which capture a large proportion, ifnot all of the sediment yielded from upstream of the dam.The effect of the reduction in peak flows is to reduce the sediment transporting capacity of thechannel downstream, particularly in gravel-bed rivers where a threshold discharge may have tobe exceeded before bedload transport commences. Directly downstream of the damdegradation often results because the sediment supply is reduced close to zero, and the riverhas excess sediment transporting capacity despite the reduction in peak flows. The degradationzone can migrate downstream a considerable distance (Raynov et at., 1986). Bed degradation isparticularly common with sand-bed rivers. For gravel-bed rivers the degradation is oftenreduced considerably by the development of an immobile armour layer.In other cases aggradation can develop downstream where tributaries deposit sediment into themain channel that no longer has the capacity to transport the sediment. Dams and reservoirs canalso result in changes in the channel width. Wolman and Williams (1984) found that the effectof dams on channel width was highly variable from no change, to a 50% reduction, to a 100%increase in channel width.4Dams and reservoirs may not affect the total volume of water that is discharged to thedownstream channel, only the timing. This is particularly the case for dams constructedprimarily for power generation or flood control. However when a dam is used for water supplyor irrigation, or there is diversion into or out of the basin, the volume of water, as well as thetiming may be affected.1.2.2 Urbanization and Agricultural DevelopmentUrbanization can severely alter the runoff response of the affected area of the catchment.Impervious surfaces such as roof tops and paved areas reduce infiltration, and runoff isconcentrated and more efficiently routed along gutters and storm water drains. As a result thestorm hydrograph characteristically becomes “flashier”, that is the peak flows are much greaterand occur sooner than before urbanization. The low flows are also usually reduced so thatpreviously permanent streams may become ephemeral.The effect of urbanization on the sediment yield is less clear. Sediment yield may increaseduring the construction phase, but may ultimately be reduced below natural levels oncedevelopment is complete (Wolman, 1967). The bank vegetation may also be disturbed duringdevelopment.The most characteristic impact to channel morphology following urbanization is an increase inthe capacity of the channels as they erode to accommodate the increased peak flow (Hammer,1972; Park, 1977). Urbanization typically affects the lower-order channels which are often notfully alluvial.The conversion of forest to agricultural land may result in increased peak flows and a decreasein the hydrologic response time of the affected area. Although the changes to the runoffresponse are usually less pronounced than those due to urbanization when compared on a unitarea basis, the development of agricultural land typically covers much larger areas than urban5areas. Agricultural development may affect a significant proportion of the total catchment andcan therefore have a significant effect on the basin hydrology.The effects of the conversion from forest to agricultural land on catchment hydrology is long-term and when completed the alluvial rivers can be expected to develop a hydraulic geometrythat tends to equilibrium with the modified catchment hydrology.1.2.3 LoggingCertain logging practices, especially clear-cut logging and road construction can significantlyinfluence the watershed runoff and sediment yield characteristics.Hydrologic changes which may accompany logging are an increase in total runoff and peakflows. A loss of forest cover can reduce interception and evapotranspiration losses. Compactedareas such as roads and yarding areas can reduce infiltration and increase overland flow.Furthermore roads can intercept subsurface flow and route it more rapidly along drains.Sediment yield may increase following logging and road construction. The increase in thesuspended sediment yield due to surface erosion as a result of soil disturbance during loggingand burning activities is well documented (Beschta, 1978). The yield of coarse sediment mayalso increase due to accelerated mass-wasting processes on logged bill slopes, and due tofailures of road and landing fills, especially in steep mountainous areas.Removal of stream-side vegetation or disturbance of the stream banks can affect the bankstability and result in channel instability. The bank erosion which accompanies the channeldestabilization can increase the supply of sediment to the channel fl.irther downstream.The effect of forestry activities on runoff and sediment yield is dependent upon the size of thebasin. In medium to large watersheds timber harvesting may be carried out on a “sustainable”6basis over an 80 to 100 year rotation. Under these conditions only 1% or so of the catchmentarea may be affected during any given year, and the net effect on the catchment hydrology maybe slight. However for small catchments or individual sub-catchments within the watershed, inany given year logging may affect a significant proportion of the total area, which may have aprofound effect on runoff and sediment yield. At the small catchment or sub-catchment scalelogging may be viewed as an episodic disturbance with a period of between 20 to 100 yearsdepending upon regeneration rates.1.2.4 Other ActivitiesOther land-use activities that may affect the river channel stability include river training,dredging and mining. Also natural processes such as wild fire and climate change can have alarge impact on the channel geometry. The effect of climate change over shorter periods of timeare usually minor when compared to the short-term adjustments caused by human activities,however they may become more pronounced in the future as a result of global warming.1.3 APPROACHES TO THE PROBLEMThere are several approaches to the problem of interpreting and predicting channel adjustmentsdue to man-induced or natural causes.1.3.1 Case StudiesThis approach is based upon monitoring and recording observed channel adjustments in orderto gain a greater insight into river behaviour. The insight gained from such studies can then beused to assess the future response of the particular river under study, or can be applied to otherrivers which possess similar characteristics.Methods for monitoring river changes include calibrated river sections that are resurveyedperiodically over time, or remote sensing data such as sequential air photographs. Anotherapproach is to compare data from a modified catchment with data recorded from nearby7control catchments with similar physical characteristics. Longer term adjustments can bestudied from sedimentological or stratigraphic evidence, which may includedendrochronological or radiometric dating (Womack and Schumm, 1977).1.3.2 Qualitative RelationsQualitative proportionalities have been developed by river engineers and fluvialgeomorphologists to predict the response of alluvial channels. The best known examples arefrom Lane (1955a) and Schumm (1969).Lane (195 5a) suggested that the following relation is useflul when analysing changes in streammorphology:GbDcxQdS (1.1)where Gb is the bed material load, D is the sediment grain diameter, Qd is a characteristic (le.dominant) discharge, and S is the channel slope. For example (1.1) indicates that following anincrease in Gb, the river will tend to restore equilibrium by increasing S for given values of Dand Qd.Schumm (1969) developed this approach fhrther and proposed the following proportionalities:wY2sQd (1.2)W2S (1.3)where W = the channel width, Y = the mean channel depth, = sinuosity, and 2= the meanderwavelength.8Relations (1.1) to (1.3) are statements of indefinite proportions and therefore do not give anyquantitative information. These relations do however indicate general trends of riveradjustments.1.3.3 Empirical Regime EquationsEmpirical regime equations have been developed from the measurement and observation ofirrigation canals and natural rivers. Much of the early development of these equations wasbased on observations of irrigation canals in India and Pakistan. Kennedy (1895) and Lindley(1919) published the earliest works, however it was Lacey who in 1929 developed a set ofthree equations for calculating the velocity, depth and slope that could be used for canal design.This set of equations was modified in particular by Blench (1957) to yield three practical designequations used to calculate the width, depth and slope of a regime canal.This empirical approach was later extended to natural alluvial rivers by workers such asLeopold and Maddock (1953), Blench (1957), Nixon (1959), Kellerhals (1967), Bray (1982b),and Hey and Thorne (1986).The equations for width and depth are often expressed in the form:WcsQd (1.4)Qb (1.5)For both canal and river-based equations the value of the exponent a ranges between 0.45 -0.55 and takes a typical value of 0.5, and the exponent b ranges between 0.33- 0.41. Moresophisticated regime equations may include sediment size and load in the regime equations. Theequations of Hey and Thorne (1986) will be reviewed here as they were reportedly derived9from data from gravel-bed rivers with mobile beds and are therefore most similar in character tothe rivers that are modelled in this thesis. The Hey and Thorne study is the first to explicitlyinclude the effect of sediment load on the regime geometry.1.3.3.1 WidthThe general equation from Hey and Thorne (1986) for channel width is:W=3.67Q45 (1.6)where W is in metres, and Qbf = the bankfI.ill discharge in m3/s which is assumed to representthe dominant channel-forming discharge. The coefficient of determination for Eqn (1.6) is,r2=0.7884. Hey and Thorne determined that the channel width was independent of sedimentsize and load. Bray (1982b) has found that width varied slightly with grain diameter of the bedsediment.Hey and Thorne determined that the channel width was strongly influenced by the type anddensity of the bank vegetation, and that Eqn (1.6) could be improved by discriminating the dataon the basis of bank vegetation type. The rivers were subdivided into four bank vegetationcategories vegetation type I (grassy banks) to vegetation type IV (>50% tree/shrub cover). Theeffect of the increased density of trees and shrubs is to decrease the channel width.The revised equation for channel width from Hey and Thorne (1986) is:(1.7)where ct ranged from 2.34 for vegetation type IV, to 4.33 for vegetation type I. For Eqn (1.7)r2=O.9577 which represents a significant increase over Eqn (1.6). The bank vegetation is10interpreted as affecting the bank strength and thus its ability to withstand higher shear stressesexerted by the flowing water. This effect will be examined in Chapter 6.1.3.3.2 DepthThe simple hydraulic geometry relation gives:Y=0.33Q35 (1.8)The value ofr2= 0.8045.The coefficient of determination is increased tor2=0.8712 with the inclusion of bed materialgrain size:Y= 0.22 Qb37 D’ (1.9)where D50 = the median bed grain diameter in metres. The mean depth was not significantlyaffected by bank vegetation or sediment load.1.3.3.3 SlopeThe channel slope equation determined by the Hey and Thorne study is:S = 0.087Q3D0°9D G’° (1.10)where D84 is the grain diameter of the bed sediment for which 84% of the total sediment is finerthan. For Equation (1.10)r20.6285 which indicates considerable unexplained variance. Inaddition the exponent of 0.10 for Gb suggests that S is quite insensitive to the sediment load.However it has been demonstrated in Chang (1980) and Millar (1991) that S is stronglyinfluenced by the sediment load.11While the exponents for the width and depth regime equations are similar for canal and riverderived data, for the slope equations the exponents differ greatly. Lacey (1929) and Blench(1957) determined that:(1.11)while from analyses of natural rivers (Hey and Thorne, 1986) found that:SciQ°43 (1.12)Kellerhals (1967) found that the value of the exponent is equal to -0.4, and Bray (1982b) foundthat it was equal to -0.33. The exponent in Eqn (1.12) is very close to that obtained by theUSBR (-0.46) from their stable channel analysis (Lane, 1955b). Leopold and Wolman (1957)found that the function which separates meandering from braided rivers has essentially the sameexponent (-0.44). Therefore the value of this exponent of around -0.44 appears to be a verysignificant one for natural rivers.It is suggested here that the discrepancy between Eqns (1.11) and (1.12) is that for canals thevalue of S is largely imposed, while alluvial channels can adjust their value of S. Irrigationcanals are generally straight and their planform shape is usually affected more by economics andthe location of property boundaries and engineering structures, than as a consequence of anyprocess of self-adjustment. Natural rivers however can adjust their value of S, principallythrough adjustments of sinuosity, such that an equilibrium is established.The regime equations such as those presented in this section have been of considerable aid inthe design of canals or river training activities. However their usefulness for predicting theadjustments to altered hydrologic regime or sediment supply is limited. For example Eqns (1.7)12and (1.8) suggest that the values of W and 7 are independent of Gb. Yet Eqn (1.3) fromSchumm (1969) indicates that an increase in Gb results in wider and shallower channels.Furthermore the regime equations indicate that only W is affected by the bank stability.However from a consideration of continuity, a wide channel with easily erodible banks must besomewhat shallower than a narrow channel with resistant banks for the same value of Qbf1.3.4 Analytical ModelsAnalytic models are rational, process-based models which solve the governing equations thatdescribe various channel processes such as flow resistance, continuity, momentum, sedimenttransport and bank stability. These models can be broadly subdivided into static or dynamicmodels (Hey, 1982).1.3.4.1 Static ModelsStatic models attempt to model the steady-state, equilibrium hydraulic geometry. The transientperiod of adjustment is not considered. A solution is obtained by solving the governingequations for the prescribed values of the independent variables such as Qbf and Gb.The simplest static model is the threshold channel model developed by the United StatesBureau of Reclamation (USBR) which is presented in Lane (1955b). This model combinesequations for flow resistance, continuity, the threshold of sediment motion and bank stability toderive solutions for channels where the sediment is at the threshold of movement at every pointacross the channel perimeter. However an explicit solution is only possible for the narrowestType B channel, the wider Type A channel requires an additional relation in the form of anempirical regime equation for the channel width. Mobile-bed models require a sedimenttransport relation in place of the threshold condition.13Generally fluvial systems are recognised as being indeterminate (Hey, 1978; 1988) in that thereare more unknown variables than there are equations available for solution. The one and onlyexception is the Type B solution for the USBR threshold channel model discussed above.There are a number of other approaches that can be used to obtain solutions for theseindeterminate systems of equations. Parker (1978) has obtained solutions by using a resultobtained from a theoretical analysis whereby the dimensionless shear stress at the center of thechannel can only be 20% above critical value for stable, non-eroding banks to develop. Parker’sresults apply only to gently curved channels with banks composed of loose, unconsolidatedgravel sediment, and cannot be generalised to all gravel rivers.An additional approach is the formulation of the problem as an optimization model. Thegoverning equations, or constraints, are solved together with an additional condition that somecharacteristic of the channel is optimised. Examples are Chang (1980) who contends that thetotal streampower is minimized, and White et a!. (1982) who state that the channel adjusts suchthat the sediment transport capacity is a maximum.1.3.4.2 Dynamic ModelsDynamic models are used in an attempt to simulate the channel changes with time. The basicgoverning equations available for solution are momentum, continuity, sediment mass balancewhich includes a sediment discharge relation, and a bed elevation equation. Models have beenformulated for steady-uniform, and unsteady-non-uniform flow, and using kinematic waveapproximations, and full dynamic-wave formulations. The best known model of this type isHEC-6 (HEC, 1974; Thomas and Prasuhn, 1977). The HEC-6 model has undergone severalrevisions and later versions include provisions for bed armouring.The governing equations are formulated as partial differential equations which are solvednumerically by either finite-difference or finite-element schemes. The initial geometric and14hydraulic characteristics of the channel must be specified. Sediment routing is calculated on asize fraction basis to simulate bed coarsening and bed elevation changes.Models such as HEC-6 are one-dimensional and are principally used to calculate bed and watersurface elevation changes. The numerical models have been applied with reasonable success topredicting channel degradation following dam construction (eg Thomas and Prasuhn, 1977).However models such as HEC-6 do not consider width or planform changes which is a severelimitation for predicting channel adjustments.One of the first dynamic models to consider width adjustments is that of Chang (1982). Themodel is called FLUVIAL-14 and it can be termed a dynamic optimization model. It is similarin form to the basic HEC-6 model, except that width changes are calculated for each time stepusing the minimum stream power concept. The width changes at each cross-section are suchthat the stream power is minimised for the total reach length. A principal assumption of thisapproach is that during the transient channel adjustments the channel is in a state of dynamicequilibrium.Osman and Thorne (1988) and Thorne and Osman (1988) develop a bank stability analysis forcohesive bank sediment that could be incorporated into existing 1-D dynamic models. Theirbank stability analysis considers fluvial erosion and mass failure of cohesive channel banks. Therate of bank erosion and bed aggradation or degradation is calculated for each time step. Massfailure of the bank occurs when the bank exceeds a critical height following bed degradation orover steepening of the banks. The eroded bank material is included in the sediment massbalance.Further discussion of the various analytical modeling approaches is deferred to Chapter 7 wherecomparisons with the optimization model developed in this thesis will be undertaken.151.4 THESIS OUTLINEIn this thesis a static analytical model will be developed based upon an optimizationformulation. This work was commenced in Millar (1991) and is an extension of the modelsdeveloped by Chang (1980) and White el al. (1982). The formulation in this thesis will bedeveloped specifically for gravel-bed rivers, however the same concepts also apply to sand-bedrivers.This thesis can be subdivided into two sections Part A and Part B. Part A covers Chapters 2-5and deals with the theoretical development of the model, and formulation of the objectivefunction and the constraints. Part B includes Chapters 6-8 and covers the computationalscheme and the data analysis, together with the final conclusions and recommendations. Thecontent of each chapter is outlined below.In Chapter 2 the optimization approach to modeling the hydraulic geometry of alluvial rivers isdiscussed. Concepts of equilibrium and time scales are presented. The objective function isformulated, and the independent and dependent variables are defined.The development of the discharge constraint is presented in Chapter 3. The discharge constraintincludes two components, flow resistance and continuity. The flow resistance relation isdeveloped for gravel-bed rivers. The complete discharge record is utiised and is input into themodel as a series of quasi-steady flows obtained from the flow-duration curve.In Chapter 4 the bedload constraint is developed. This constraint requires estimation of themean bed shear stress to calculate the sediment transport. Empirical relations to estimate themean bed and bank shear stress values are presented. The Parker (1990) surface-based bedloadtransport relation is modified for inputs from the flow duration curve.16The bank stability constraint is formulated in Chapter 5 for both cohesive and noncohesive banksediment.Chapter 6 covers the major computer analysis in the thesis. A simplified optimization modelthat uses only the bank-full discharge as input is used to investigate the influence of the bankstability on the geometry of channels with both noncohesive and cohesive bank sediment. Thework relating to the noncohesive bank sediments has been published in Millar and Quick(1993a, b).In Chapter 7 the full model formulation is presented in which the sediment transporting capacityof the channel is calculated using the full range of flows. The model also calculates the grainsize distribution of the bed surface. The effect of sediment load on the channel geometry isexamined. Potential applications of the model are reviewed, and the optimization model iscompared with other numerical approaches.In Chapter 8 the final conclusions and recommendations are presented including limitations ofthis model, and proposals for future study.17Figure 1.1. Idealised fluvial system (after Schumm, 1977).Zone 1ProductionZone 2TransferZone 3Deposition18CHAPTER 2QUALITATIVE MODEL FORMULATION2.1 INTRODUCTIONIn this chapter the principal framework of the model will be discussed and the optimization modelwill be formulated in a qualitative manner. The modeling will treat the river channel as a systemthat tends to adjust to an equilibrium hydraulic geometry. The fluvial and alluvial channel systemwill be defined, and the condition for equilibrium discussed. The objective function for theoptimization model will be developed.2.2 TEMPORAL AND SPATIAL SCALESIt is necessary when discussing equilibrium in geomorphic systems to define the temporal andspatial scales over which the equilibrium can be considered to operate.2.2.1 Temporal ScalesCyclic or geologic time (Schumm and Lichty, 1965; Schumm 1977) refers to very long timespans. Over this time scale fundamental changes occur within the river basin, the topography isreduced by erosion, valley slopes are built up, sea levels and climatic conditions can change verydramatically. Graded time (Schumm and Lichty, 1965; Schumm 1977) represents much shortertime intervals. Over a graded time span the topography, valley slopes, sea level, and otherfundamental landforms are relatively constant.The actual time ranges of geologic and graded time scales are not absolute, but depend on therates of change of the controlling variables such as climate and topography. However in general19geologic time might be thought to have a time span in the order of 1,000,000 years, and gradedtime up to 1,000 years.Another time scale is referred to as engineering time, which is the typical life of an engineeringstructure and is generally considered to be about 100 - 200 years or less (Hickin, 1983; Newson,1986). Usually engineering time is a subset of graded time. Throughout this thesis, unlessotherwise stated, engineering time scales will be assumed.2.2.2 Spatial ScalesThe geometry of a river varies along its length. Typically in the downstream direction thedischarge increases, the sediment tends to become finer through selective transport and/orabrasion processes, and the bank sediment and vegetation may change. The term representativechannel reach will now be introduced (Fig 2.1). The representative channel reach is definedherein as the section of channel length L, along which the discharge, sediment load and calibre,and the bank material properties can be considered constant. This in turn implies that the meanchannel geometry will also be constant over this reach.The length of this representative channel reach in absolute terms depends upon the rate ofdownstream change of the channel properties mentioned in the previous paragraph. A gravel-bedriver in a mountainous area may experience a rapid downstream increase in discharge, and rapiddownstream decrease in the sediment size. In this case the length of the representative channelreach would be relative short, and may be of the order of ten times the channel width. At theother extreme, a large lowlands river, such as the Mississippi River along the mid to lowerreaches, may experience relatively constant values of discharge, and sediment load and calibre forextended distances. In the case the representative channel reach may extend for distances possiblyup to 500 times the channel width.202.3 DEFINITION OF EQUILIBRIUMA channel is defined herein as being in equilibrium when the following conditions are satisfied:1. The mean hydraulic geometry of the representative channel reach remains unchanged overan appropriate time scale for which a steady-state equilibrium can be assumed.2. There is no net erosion or deposition along the reach.3. Any perturbations from the equilibrium geometry will be offset and the equilibriumgeometry restored.Note that this definition applies to the reach-averaged and not to the local values of the hydraulicgeometry. The term steady-state refers to a time-invariant equilibrium. The absolute time periodover which this approximation is valid will depend upon the system being studied. It is validwhere there are no significant or systematic changes in the mean annual characteristics of thedischarge and sediment load, or in the competence of the bank sediment. In general the steady-state approximation may apply over engineering time scales up to 100 years or so.Sediment transport across the representative channel reach can be represented by the sedimentcontinuity equation for a specified time interval:I--O=ES (2.1)where I = input of sediment into the upstream end of the channel; 0 = the output of sedimentfrom the downstream end of the channel; and I4S = the change in the volume of sediment which isin storage along the reach (Fig 2.1). The value of iS’ can be positive or negative which representsnet deposition or net erosion along the channel reach. At equilibrium 0=1 and ES = 0.21The definition of equilibrium presented above is consistent with the so called regime theoryfavoured by engineers that was developed from the observation of stable, non-silting canalsconstructed in India and Pakistan. A canal was said to be “in regime” Wit was able to contain theflows and transport the sediment load without appreciable deposition or scour. This concept canbe traced back to Lindley in 1919 (See Blench, 1957, p. 20).The concept of regime was extended to natural rivers. Blench (1957, p. 3) defined the expression“in regime” as meaning that the average values of the quantities which define regime (namely thewidth, depth, and slope of the river) do not show a definite trend over some time interval, andsuggested that this may apply over a period of 20 to 40 years.2.4 OPTIMAL HYDRAULIC GEOMETRYAn alluvial channel is a relatively complex system with perhaps up to seven degrees of freedom(Hey, 1978). The system is indeterminate in that there are more unknown primary dependentvariables or degrees of freedom than there are equations for solution. Even for a simplifiedchannel geometry with only 3 degrees of freedom: width, depth, and slope, the solution remainsindeterminate as there are only two relations, flow resistance-continuity, and sediment transport,available for solution.The presence of an optimal geometry in fluvial hydraulics was first demonstrated by Gilbert in1914. In flume experiments under conditions of fixed discharge rate and channel slope, it wasdemonstrated that by varying the width of the flume an optimal value exists where the sedimenttransporting capacity of the flume was a maximum.The mechanisms responsible for the optimum in Gilbert’s experiments can readily be explained.For narrow flume widths much of the shear force is acting on the side walls, and this, togetherwith the narrow bed widths over which sediment transport can occur, results in a low totaltransport rate. Conversely for the large flume widths the depth of flow and the bed shear stress22both become small, and hence the total sediment transport rate also becomes small. Betweenthese two extremes lies an optimum where the sediment transport rate is maximized.Gilbert’s experimental result can be duplicated numerically. An example of a solution curve isshown schematically in Fig 2.2(a). The channel slope and discharge is constant at each point onthe solution curve. The numerical procedures for calculating these solution curves will bedeveloped in Chapters 6 and 7, or refer to White et al. (1982).In Fig 2.2(b) a solution curve is shown where the channel slope is a variable, and the dischargeand sediment transporting capacity are now fixed. In this case the optimum is the point where theslope is a minimum. This example is considered to be analogous to most natural alluvial riversover engineering time scales. The discharge and sediment load are imposed, and the width depthand slope of the channel are dependent variables that develop in response to these imposedvalues. This is discussed in Section 2.5.The experimental results of Gilbert (1914) together with the numerical analyses discussed abovecombine to lend significant support for the actual existence of an optimum in the fluvial system.The fundamental assumption employed in the optimization model is that a natural river channelwill tend to adjust to this optimum. and that this optimum corresponds to the equilibriumhydraulic geometry.The preceding assumption is referred to in physical sciences as a “variational principle”. The ideabehind a variational principle is that a physical system will select the most “economical” path ormode that requires the least “expenditure” (Konopinski, 1969; p. 169). A branch of mathematicscalled calculus of variations has been developed to solve variational problems.In many problems the existence of maxima or minima can be difficult to prove theoretically, andmost scientists who employ variational principles rely on their intuition in the initial formulation23(Kyrala, 1967; P. 119). The properties that the solutions must possess (when their existence hasbeen assumed), can be easier to investigate, both theoretically and experimentally, than thefunctional relation that describes the system. By employing a variational principle the complexityof the problem can be reduced, and it provides a point from which to commence an investigationinto the behaviour of a complex system.2.5 OPTIMIZATION MODELOptimization models have been developed previously by Chang (1980), Yang et a!. (1981), andWhite eta!. (1982) among others to predict the geometry, or aspects of the geometry, of alluvialchannels. When formulating these models it was recognised by the authors that the fluvial systemis indeterminate and that an additional relation is required to close the problem and to obtain asolution.An additional relation is often proposed as an “extremal hypothesis” whereby a selected channelparameter is either maximized or minimized. The term extremal hypothesis is synonymous withvariational principle. The principal extremal hypotheses which have appeared in the hydraulicsliterature are presented below. Only the original or significant publications are given.1. The Maximum Sediment Transport Capacity (MTC) hypothesis (Griffith, 1927; Singh,1962; White et al., 1982.) states that for a given discharge Q, and slope S, the channelwidth adjusts to maximize the sediment transport rate.2. The Minimization Hypotheses which include: (i) the Minimum Stream Power (MSP)hypothesis (Chang, 1980) whereby the total stream power, yQS is minimized; (ii) theMinimum Energy Dissipation Rate (MEDR) hypothesis (Yang and Song., 1979) whichstates that the rate of energy dissipation, yQS + y Q S (expressed here per unit channellength) is minimized; and (iii) the Minimum Unit Stream Power (MUSP) hypothesis (Yang,1976) which states that the stream power per unit weight ofwater, US, is minimized.24In the foregoing y and y are the unit weights of water and sediment, Q is the volumetricsediment transport rate, and U is the mean velocity.MEDR is the most general of the three minimization hypotheses. MSP is equivalent to MEDR ifQ << Q, which is generally the case for natural rivers, especially gravel rivers. MIJSP is a specialcase ofMEDR under a condition of constant channel width.3. The Maximum Friction Factor (MFF) hypothesis (Davies and Sutherland, 1983)states that a channel will adjust to maximize the friction factorfIt is recognised that general equivalence exists between the hypotheses, particularly in the case ofnatural channels where Q and Q are imposed, and the width, depth, and slope are free to adjust(White et al., 1982; Davies and Sutherland, 1983). In some instances certain extremal hypothesesmay not apply. The best example is MSP in the case of imposed Q and S such as in a laboratorystudy where equilibrium is attained primarily through width adjustment. Under these constraintsthe total stream power is fixed and therefore cannot be minimized, and hence MSP cannot apply.The extremal hypotheses have been used quite successfully to predict the geometry of alluvialchannels (Chang 1980; Yang et a!., 1981; White et al., 1982), meanders (Yang 1971), bedforms(Davies, 1980), and velocity profiles (Song and Yang, 1979). However the use of extremalhypotheses remains controversial, some suggest that an additional physically based relation suchas bank stability should be used (comments following Bettess and White, 1987; Hey, 1988), orthat the apparent progress in fiuvial hydraulics through the use of extremal hypotheses is anillusion (Griffiths, 1984). In Davies’ opinion (Davies, 1987) the empirical success of extremalhypotheses is “deeply significant” and may indicate the possibility of fundamental understandingof river behaviour. But until the reasons for the success of extremal hypotheses are known thisapproach to fiuvial hydraulics will remain unattractive. There have been attempts in the25references cited above to explain the extremal hypotheses from physical reasoning, but theseattempts have been unconvincing for one reason or another.The optimization model to be developed in this thesis will now be formulated in a qualitativemanner. It differs from the earlier models of Chang (1980), Yang et a!. (1981), and White et aL(1982) as the model includes constraints for the bank stability, and calculates the bedloadcapacity of the rivers using the fhll range of flows and not simply a single representative ordominant discharge. The objective function developed in Section 2.5.3 is equivalent to an“extremal hypothesis”.2.5.1 Independent VariablesThe independent variables represent the known, external controlling variables which are inputs tothe system. The variables which are considered independent with respect to the channel reachunder consideration and include the physiographic, geologic and hydrologic properties of thecatchment. These are the geology, relief; climate, runoff vegetation, and the volume and calibreof the sediment yield.Additional independent variables are related to the channel boundary, namely the bankvegetation, and bank sediment parameters such as cohesion and friction angle. The valley slope isalso considered to be independent over graded or engineering time scales. Although the valleyslope can be considered to be a dependent variable over geologic time scales, any significantchange in the valley slope requires the removal or deposition of large volumes of alluvium whichcan only occur over long periods of time.The channel will be modelled using the steady-state, mean values of the sediment yield and flowduration data, as well as the mean bank stability parameters for the representative channel reach.262.5.2 Dependent VariablesThe dependent variables (which are referred to as decision variables in optimization terminology)are the unknown channel geometry variables: width, depth, slope, bank angle, roughness,velocity, sinuosity, meander and pool-riffle wavelength, meander radius of curvature, andbedforms. In this thesis only a simplified channel geometry will be assumed. Secondary currentsand the planform variables will not be considered explicitly, however a comparison of the channeland valley slopes gives a measure of the channel sinuosity and therefore some limited informationon the planform geometry.Since the valley slope is assumed to remain constant over engineering time scales, any changes inthe channel slope are limited to changes in the sinuosity of the channel. Through adjustments ofits sinuosity, a channel can change its slope much more readily as only relatively small volumes ofsediment need to be eroded or deposited along the meander bends. This is in contrast to the largevolumes that need to be moved in order to change its slope through aggradation or degradation.2.5.3 Objective FunctionIn the following section the objective fhnction will be developed. Kirkby (1977) states that in acomplex system the criterion of maximum efficiency normally applies only to a single process,that is the limiting process. Therefore the objective flinction formulation should reflect thislimiting process. The principal processes operating in river channels is the passage of the flowsand the transport of sediment. The most efficient cross-section for passage of the flows is a semicircle where the hydraulic radius is a minimum. For a rectangular section the most efficient crosssection is that with a form ratio (W/ Y) of 2. However Gilbert (1914) found experimentally thatthe optimum form ratio for transporting sediment ranged from 2 to 20. Natural channels rarelyhave form ratios less than 7 or 8, and typically much greater. Furthermore analytical approachessuch as White et aL (1982) have been quite successful at predicting the channel width and depthby assuming a maximum sediment transport condition. This evidence supports the view ofKirkby27(1977) that it is the transport of sediment, rather than the accommodation of the flows, that is thelimiting process in alluvial channels.The concept of sediment transport efficiency will be developed from the arguments of Bagnold(1956, 1966) who described fluid flow in a river channel as a sediment transporting machine.From the principles of energy conservation, and analogies to mechanical systems Bagnold defineda sediment transport efficiency e:e= sediment transport work rate (2 2)available powerThe power available for transporting the sediment can be developed from Fig 2.3. Consider tworeservoirs A and B separated by a vertical distance AZ. In the upper reservoir A is a volume ofwater and sediment, V,., and V, respectively. The total potential energy E, expended in movingthe water and sediment from A to B is the sum of the potential energy expended by the water E,and the sediment E5:(2.3)E=yVAZ+y5VA (2.4)where y and y = the unit weights for water and sediment. For uniform flow along a channel oflength L, the kinetic energy of the fluid and sediment is constant. The power available isequivalent to the potential energy dissipated over duration Twhich is the travel time from A to B.The power available per unit channel length is given by:yVAZ y3VAZavailablepower= TL + TL(2.5)28which for steady flow conditions can be written as:available power =yQS+yQS (2.6)where Q and Q = the respective volumetric transport rates for the water and sediment.Bagnold (1966) defined the sediment transport work rate as equal to the transport rate byimmersed weight multiplied by a factor K:sediment transport work rate = K(r— r) Q (2.7)The factor K is necessary as the applied stress on the sediment is not in the same direction as thevelocity of the transported sediment. For bedload K = dynamic friction coefficient, for suspendedload K = ratio of the fall velocity to the mean sediment velocity.Therefore substituting Equations (2.6) and (2.7) into (2.2):K(y—r)Qe= (2.8)yQS+yQSFor natural rivers, particularly for gravel rivers Q<<Q, therefore Equation (2.8) can be simplifiedto:K(r8—y)Qe= 7QS (2.9)In alluvial rivers the numerical value of e is typically very small as most of the energy in a river isexpended overcoming the frictional resistance to the flow.29It will now be shown that the maximization of e is equivalent to the principal extremal hypothesesdescribed in the previous section. The denominator of Eqn (2.8) is the energy dissipation rate ofYang et a!. (1981) per unit channel length, Similarly the simplified denominator in Eqn (2.9) isthe total stream power (Chang, 1980). For systems where Q is independent, maximization of eimplies minimising the energy dissipation rate and total stream power, and is therefore equivalentto the MEDR and MSP hypotheses. Furthermore where both Q and Q are independent, themaximization of e implies a minimization of S.For a laboratory study where Q and S are imposed, and the channel width is free to adjust, amaximization of e implies a maximum value of which is the MTC hypothesis. Therefore thevarious extremal hypotheses are consistent with a maximization of e.The MFF hypotheses will not be considered further here, however Davies and Sutherland (1983)have shown that under conditions of independent Q and Q the MFF hypothesis agrees in generalwith the MEDR, MSP, and MTC hypotheses.For simplicity Equation (2.9) will be reduced to:Gb71=pQS (2.10)where Gb = the dry sediment transport rate in mass units, and p = the density of water which isalso in mass units. The values ofK and (y-‘y) I y are generally constant for a given sediment. Forbedload sediment K tan 30° = 0.58, and (y-y) I y 1.65. The product of K times (y-y) / y isapproximately equal to one, and therefore i is an index of the system efficiency which is not onlyproportional to e, but for bedload sediment is similar in numerical value.30The value of is always much less than one for rivers. Eqn (2.10) will be modified in Chapter 7to apply to variable flows.2.5.4 ConstraintsThe various constraints on the solution will now be discussed qualitatively. The mathematicalformulation specifically pertaining to gravel-bed rivers will be presented in Chapters 3 to 5. Theconstraints in the optimization formulation are discharge, bedload, bank stability, and the valleyslope.The discharge constraint represents the flows that are available to form the channel and transportthe sediment. Included in the discharge constraint is the flow resistance. The roughness of thechannel determines the velocity and depth of flow which has implications for sediment transportand bank stability.The bedload constraint represents the imposed bedload that must be transported by the flows. Itis assumed that the channel geometry develops in part as a response to the imposed bedload. Allsediment less than 2 mm in diameter is excluded from the analysis as the finer sediment is thoughtto travel essentially is suspension, and therefore does not directly influence the channel-formingprocesses. For true equilibrium all grainsizes must be transported through the representativereach at just the rate that they are supplied.Stable channel banks are an additional requirement for an equilibrium channel. This constraint hasnot been addressed in previous optimization formulations (Chang 1980, Yang et a!. 1981, andWhite et a!. 1982). The bank stability constraint will be formulated for noncohesive and cohesivechannels.31The valley slope constrains the optimal solution as the channel must develop a slope which is lessthan or equal to the valley slope. The maximum channel slope occurs when a straight channeldevelops along the valley axis.2.6 SUMMARYThe alluvial channel has been described as an indeterminate system. Experimental results andnumerical investigations indicate that an optimal hydraulic geometry can be observed in fluvialsystems. It is assumed that a natural river will tend to adjust to this optimal geometry. Thisassumption forms the basis for the optimization model. The equilibrium hydraulic geometry isconsidered to correspond to a solution where the discharge, sediment transport and bank stabilityconstraints have been satisfied, subject to the condition of maximum sediment transportefficiency.The solution obtained by the modeling represents the steady-state equilibrium geometry. Thetransient stages of adjustment will not be considered directly in this thesis. The discussion of theprocesses whereby a natural channel achieves the optimum configuration will be left until Chapter7.32Change in Storage = i.iSFigure 2.1. Definition sketch for a representative channel length which is the length ofchannel along which the mean channel geometry is approximately constant.I/0/33(a)GbWidth(b)s SolutionCurveWidthFigure 2.2. A schematic representation of the optimal geometry in fluvial systemsobtained from numerical analysis. (a) Constant discharge and slope (eg Gilbert, 1914). (b)Constant discharge and sediment load.SolutionCurve34AvwAZ BFigure 2.3. Definition sketch for development of the coefficient of sediment transportefficiency35CHAPTER 3DISCHARGE CONSTRAINT3.1 INTRODUCTIONThe discharge is a constraint on the channel system as it represents virtually the total energyinput. The channel development and sediment transport must be accomplished with theavailable flows. In this thesis the discharge constraint will refer to the condition that the channelmust have a discharge capacity equal to an imposed or trial value of the bankftill discharge, Qbf.In this chapter the flow resistance forms a large part of the discussion. Flow resistanceequations will be developed for gravel-bed rivers. The morphological and hydraulic significanceof the bankfull discharge and overbank flows will be addressed.Flow resistance is closely related to shear stress which is dealt with in Chapter 4. Some crossreferencing between Chapters 3 and 4 is necessary to avoid repetition.3.2 FLOW RESISTANCEThe following discussion will be limited to fully rough flow in channels with hydrodynamicallyrough boundaries, where the mass sediment transport rate is very small in comparison to themass discharge rate of water. It is generally assumed that flow in open channels obeys thePrandtl - von Karman logarithmic velocity distribution. Flow resistance is often expressed byany of the following equivalent equations:u (R= 6.25+5.75 Iog(,7-J (3.1)i.1* “S361 (12.2Rhy=2.O3logJç k, J (3.2a)/ f(12.2R 1)f=2.O3lokhJJ (3.2b)Swhere U = mean velocity; U = (g Rh S)O.5 which is known as the mean shear velocity;! = theDarcy-Weisbach friction factor; Rh = the hydraulic radius; S = the energy slope which is equalto the longitudinal channel slope under uniform flow conditions; g = gravitational acceleration;and k3 = the roughness height which is the equivalent sand roughness.Equation (3.1) was developed for open channel flow by Keulegan (1938) from the work ofNikuradse (1933) on flow resistance in closed conduits. Equation (3.2) is commonly known asthe Colebrook - White flow resistance equation. The Colebrook - White form as presented inEqn (3.2b) will be used in this thesis.The constant 2.03 in Equation (3.2a, b) is related to von Karman’s constant, and the coefficientvalue 12.2 has been obtained experimentally. The roughness height k3 was originally defined byNikuradse (1933) and Keulegan (1938) as the diameter ofuniform sand grains glued to the pipeor channel perimeter. Keulegan (1938) found that Equation (3.1) fitted the experimental data ofBazin (1865), and that k5 was approximately equal to the mean grain diameter of the sedimentsthat had been used to line the experimental trapezoidal flume.Natural gravel rivers are typically characterised by a coarse, graded, bed surface or armourlayer. Logically one would assume that ic, would take a value close to the mean grain diameterof the bed surface. However several investigators have concluded that D50 or even a coarser37characteristic grain diameter underestimated the value of k3. The following relation for k3 hasbeen proposed:k3=CD (3.3)where C is a constant corresponding to D. The values of C, were obtained by fittingexperimental or flume data to Equations (3.1) or (3.2). There is a wide range of values for Cfrom Ic = 1.25 1)35, (Ackers and White, 1973) to k3 = 3.5 D9, (Chariton et a!., 1978; Bray,1979, 1982a). Other examples from both field and flume studies are = 2 D90 (Kamphuis,1974), k = 6.8 D50 (Bray, 1979, 1982a), and k3 = 3.5 (Hey, 1979; Bray, 1982). The valueof/c3 is composed of the grain, form, and planform roughness which are discussed below.Fig (3.1) shows the friction factorf from the field data of Chariton et al. (1978), Bray (1979),Andrews (1984), and Hey and Thorne (1986) plotted against relative roughness, Rh ID50. Themean depth Y is generally published and used rather than Rh, however for these naturalchannels with width/depth ratios generally greater that 10, this introduces little error. Thepoints represent the values at the bankfill discharge with the exception of Bray (1979) whogives the 2-year flood which generally corresponds to near bankfiill conditions.The optimum value of C50 was obtained by minimizing the sum of the squares of the errorsbetween the observed and calculated values off The calculated value off were obtained fromEqn (3.2b) the value of k given by Eqn (3.3).D50 was selected as the characteristic grain size because it is most consistent with the originaldefinition of k3 by Keulegan (1938). For the data analysed, the values of C,0 ranged between0.40 and 56, and the optimum value was determined to be C,o = 5.8. This compares reasonablywith the value of C,o = 6.8 obtained by Bray (1979, 1982a).38Eqn 3.2 with the optimum value of k3 = 5.8 C50 is shown in Fig 3.1. Note the large degree of“random” scatter about this best-fit equation. This was also reported in the other investigationsmentioned previously. It will now be shown that this scatter should be interpreted as asystematic displacement off above a limiting value which is due to the grain roughness. Theadditional roughness is interpreted as the effect of bedforms, bars, other macro roughnesselements, and planform irregularities along the channel.3.2.1 Subdivision off into Grain and Form ComponentsOne of the principal contributions of Einstein (1950) and Einstein and Barbarossa (1952) tomobile-bed fluvial hydraulics was the assumption that the total channel shear can be expressedas a sum of the grain and form components:(3.4)where r is the shear force per unit perimeter area, or mean channel shear stress, that the flowexerts on the channel boundary, the value r’ is the component of the total shear stress due tothe grain roughness, while “is the component of the total shear stress due to bedforms andother channel irregularities.By definition the value of r is related tof by:r=pU2 (3.5a)where p is the density of the fluid. Equations analogous to Eqn (3.5a) can be defined for r’and r”:r’=_pU2 (3.5b)39rM=I_pU2 (3.5c)wheref’ is the friction factor due to the grains, and!” is the friction factor due to the bedformsand other irregularities.Therefore with Eqn (3.5 a-c), Eqn (3.4) can be rewritten:LpU2=LpU2+L_pU2 (3.6)Equation (3.6) simplifies to:(3,7)The friction factorf can therefore be subdivided linearly into grain and form components in amanner analogous to subdivision of zThe shear stress r is subdivided into the grain and form components here by a subdivision of thefriction factor. Einstein (1950) and Einstein and Barbarossa (1952) somewhat arbitrarilyassumed that the energy slope S is constant and subdivide Rh into the grain and formcomponents, while Meyer-Peter and Muller (1948) assumed a constant value of Rh andsubdivided S. The subdivision off as in Eqn (3.7) yields values for r’ and r” which areequivalent to the subdivision of 5, but differ from the approach of Einstein (1950) and Einsteinand Barbarossa (1952).It is proposed that the grain roughness height, ks’, is approximately equal to D50. This isconsistent with the original definition of k3 by Nikuradse (1933) and Keulegan (1938) who40developed their equations using data collected from straight pipes and channels with no surfaceirregularities other than uniform surface roughness.For the field data presented in Figure 3.1, the values of the ratio k I D50 (which is equal to C50)ranged from 0.4 to 56. If the premise that k8’ = 13 is correct, then the minimum theoreticalvalue of the ratio k3 I D50 is 1.0, which corresponds to the case where the total channelroughness is due to the grain component only. Of the 176 rivers analysed herein, only 4 hadvalues of k3 / D50 less than 0.98. This provides good support for the assumption that k3’ = Tho.Others have suggested larger values for ks’. Einstein and Barbarossa (1952) used k5’ =Parker and Peterson (1980) used k3’ = 2D90, and Prestegaard (1983) used k3’ = D84 (these 3studies are discussed below in Section 3.2.2). If any of these values were correct, then theminimum observed values of k3 / D50 would be much greater than 1. For example if Parker andPetersons’ value for k3’ were correct, then the minimum observed value of k / D50 would beexpected to be around 6 (as the ratio D90 /D50 is typically about 3).The friction factor due to the grain component in natural gravel channels should therefore begiven by:( (12.2Rh”Yf’=2.03loD(3.8)50Eqn (3.8) is plotted in Fig 3.1 and it is evident that, with the exception of very few points, thisequation forms a tight lower bound to the observed values. Therefore the scatter in Fig 3.1 isnot random about a mean value, but is more correctly interpreted as a displacement of the datapoints above the limiting grain roughness due to the presence of bars, bedforms, and otherchannel irregularities. The displacement of the data points above Eqn (3.8) in Fig 3.1 is equal to413.2.2 Estimation off” for Gravel-Bed RiversThe division of the flow resistance into grain and bedform components has been widely used instudies of sand-bed hydraulics where bedforms are well developed (Einstein and Barbarossa,1952; Shen, 1962; Engulund, 1966, 1967; Vanoni and Hwang, 1967; Alam and Kennedy,1969). However relatively little work has been undertaken on the contribution of bedformresistance in gravel-bed rivers. Three methods for estimating the value off” will now bereviewed, those of Einstein and Barbarossa (1952), Parker and Peterson (1980), andPrestegaard (1983).3.2.2.1 Einstein and Barbarossa (1952)Einstein (1950) and Einstein and Barbarossa (1952) argued that the bedform roughness must beexpected to be a function of the mobility of the bed sediment:(3.9)where Uk” = the form component of the mean shear velocity which equals.Jr “/ p, and ‘PD3Sthe intensity of the grain shear which is defined as:= pg(s—1)D35 (3.10)where s is the specific gravity of the sediment.The dimensionless parameter ‘1’ forms the basis for the Einstein bedload equation (Einstein,1950). In this thesis the dimensionless grain shear stress based upon the median pavement graindiameter, is the preferred index of bed sediment mobility:425O= pg(s-1)D50 (3.11)Unpublished data supplied to the author by R.D. Hey and C.R. Thorne on the sediment sizedistribution of the bed surface of several gravel-bed rivers in the UK indicates that D30 = 1.25D35 is a reasonable approximation. Therefore the relation between ‘P’D35 and t*D,o can beapproximated by:‘P(3.12)The form resistance coefficient of Einstein and Barbarossa (1952), U/ Us”, is related tof” bythe following relation:=9 (3.13)The relation of Einstein and Barbarossa (1952) for the form roughness can therefore beexpressed by the following equation which is equivalent to Equation (3.9):(3.14)The values off” from the field data in Fig 3.1 are plotted together with Eqn (3.14) againsttD in Fig 3.2. As Eqn (3.14) was derived by Einstein and Barbarossa (1952) from a limitedamount of data from sand-bed rivers, the poor agreement with the data from the gravel-bedrivers is not surprising. However despite the wide scatter there appears to be a fhnctionalrelation betweenf” and t*D,O of the type proposed by Einstein and Barbarossa.Assuming a wide channel approximation the value t*D,o can be expressed as (see Eqn (4.15)):43,* f’ RS —_____,* 315— f (s—l)D50 rD50 ( )From Eqn (3.15) it is evident that the relation (3.14) has the potential for spurious correlationas the form roughness, f “, appears on both sides of the equation. In Fig 3.3(a) & (b) the dataare subdivided such thatf” is plotted againstf’ / (f’ + f”), and against the total dimensionlessshear stress t*DJQ. It is evident that any functional relation such as Eqn (3.14) is due to therelationship between f” and f’ / (f’ + f”), as the relationship twf” and r*D$O appearstotally random. Furthermore the variation inf’ is small compared tof”, thereforef’ / (f’ +f”)can be approximated by a / (a +f”) , where a is a constant. When the flinctionf” = a I (a +f”)is plotted in Fig 3.3a with a = 0.043, which is the mean value off’ from the data set, thisfunction describes very accurately the variation betweenf” andf’ / (f’ +f”), and therefore itmust be concluded that Eqn (3.14), and therefore the bar resistance curve of Einstein andBarbarossa (1952), is largely a function of spurious correlation, and therefore does not give anymeaningful estimates of the form or bar roughness.3.2.2.2 Parker and Peterson (1980)Parker and Peterson (1980) performed an analysis similar to Einstein and Barbarossa (1952)except that for the division of r into grain and form components they used the method ofMeyer-Peter and Muller (1948) whereby S is subdivided, rather than Rh. The flow resistancedue to the grain component was calculated using an equation equivalent to Eqn (3.8) exceptk’=2 D90 was assumed, and the Chezy coefficient, C, rather thanf was used as a measure ofthe channel roughness. The equation developed by Parker and Peterson for the graincomponent of the Chezy roughness coefficient, Ci’, is:44f’ ( (iir”Y2C285751°D90JJ(3.16)where Y = the channel depth. The notation of the original equation as it appeared in Parker andPeterson (1980) has been changed to make it consistent with this thesis.An empirical bar resistance curve relation derived by Parker and Peterson relates C’ toThe Parker and Peterson bar resistance is analogous to Eqn (3.14), and therefore is as much aresult of spurious correlation as the Einstein and Barbarossa (1952) relation. The Parker andPeterson relation therefore gives no meaningfhl estimate of the bar resistance.In addition Parker and Peterson (1980) conclude that for high discharges close to bankfull thecontribution of bar and pool-riffle sequences to the total roughness becomes negligible.However from Fig 3.1 it is evident that f”is for many channels not insignificant at higherdischarges, but is often much greater thanf’. This is discussed fhrther in Section 3.2.3.3.2.2.3 Prestegaard (1983)Prestegaard (1983) subdivided v into grain and form components by subdividing S similar toMeyer-Peter and Muller (1948). To estimate the grain component of the channel roughness, thegrain roughness height, k5’, was assumed equal to D84 on the basis that the larger grainscontributed more to the channel roughness than the smaller grains. The grain slope, S’, wascalculated using the following equation:u2 ( (12.2Y*’YD84 JJ (3.17)Prestegaard originally expressed Eqn (3.17) in the Keulegan form (Eqn 3.1), however forconsistency it has been changed here to the Colebrook-White form (Eqn 3.2).45The values ofS” were obtained by subtracting the values of S’ estimated from Eqn (3.16) fromthe observed S. Prestegaard found that the bar roughness accounted for about 60% of the totalroughness at Qbf in the rivers studied.The bar roughness was shown to be independent of grainsize. A reasonable correlation wasdemonstrated between 5” and the dimensionless bar magnitude, which is the ratio of the baramplitude divided by the spacing of the bars, however no predictive equation was developed.The approach adopted by Prestegaard whereby the bar roughness, whether expressed as 5” orf “, is related to channel morphological features (apart from grainsize), represents a promisingmethod for estimating the bar roughness. However additional field-based research is required todevelop predictive equations.3.2.3 Variation of f” with DischargeThe channel roughness whether expressed as Manning’s n orf is known to vary with dischargein most rivers. For channels where bank vegetation is not a significant contributor to flowresistance, the friction factor generally decreases with increasing discharge up to bankfull(Kellerhals et aL, 1972; Parker and Peterson, 1980). The effect of overbank flow on the flowresistance will be addressed in the following section.The variation of f’ and f” for four rivers are shown in Fig 3 .4(a)-(d). The value off wasobserved, f’ was calculated from Equation (3.8), and f” =f - f’. The curve indicating thecalculated value off was determined with Eqn (3 .2b) using the value of k that gives the bestagreement between the observed and calculated variation offwith discharge. This is discussedbelow. The data for the Meduxnekeag, Northwest Miramichi, and Big Presque Isle Rivers fromNew Brunswick are given in Phinney (1975), and the values for the Oldman River from Alberta46are given in Kellerhals et a!. (1972, Reach 90). The data for the Oldman River were also usedby Parker and Peterson (1980) for their analysis.Two features are obvious from Fig 3.4 for all 4 rivers. Firstly the value of f’ decreases onlyslightly with increasing discharge. This decrease is expected as the relative roughness Rh / ]35increases with increasing Q. A much greater decrease f” with increasing Q is evident, andthis has a large influence on the totalf This reduction inf” is consistent with the interpretationby Kellerhals et a!. (1972) and Parker and Peterson (1980) that at low discharges the bars andpool-riffle sequences in gravel rivers contributed greatly to the channel roughness, however forhigher Q values these features become “drowned out” and contribute less to the totalresistance.Parker and Peterson (1980) went even fi.irther and concluded that the effect of the bars onchannel resistance at the higher flows when the bed becomes mobile are negligible, and theform resistance approaches zero. This appears to occur for the Northwest Mirarnichi River (Fig3.4(b)) for values of in excess of about 400 m3/sec. However Dr. Dale Bray (PersonalCommunication, 1994) has indicated that the Northwest Miramichi River is not highly mobile atthis location even for high discharges.The data from the Meduxnekeag, Big Presque Isle, and Oldman Rivers (Fig 3.4(a), (c), (d))indicate that for higher flows the value of f” approaches a constant, non-zero value. For theBig Presque Isle River the value off” is approximately equal to!’ at high discharges. For theMeduxnekeag and Oldman Rivers the value off” at high Q values is less than f’, butnonetheless makes a significant contribution to the totaif As mentioned previously, the data inFig 3.1 indicate that for many riversf” is greater thanf’ even at the bankfbll stage.The Oldman River data were used by Parker and Peterson (1980) to verilS’ their hypothesis thatf” approaches zero at higher Q, and yet as is indicated in Fig 3.4(d) this is not so. The reason47for this discrepancy is that the relation of Kamphuis (1974) where k3’ = 2 D90 was used byParker and Peterson, rather than k3’ = D50 used herein. It must be concluded that theKamphuis relation includes roughness elements in addition to grain roughness.The total channel friction is given by Eqn (3 .2b). For modeling purposes, or for computingsynthetic stage-discharge curves it is important to know how the values of k andf vary withdischarge. Eqn (3.2b) is plotted as the calculated f curve, together with the values of f’calculated from Eqn (3.8), and the observed variation off with Q for four rivers in Fig 3.4(a)-(d). In each example a value of k was selected to give the best agreement between theobserved and calculated values offwith emphasis on the higher discharge values because mostchannel adjustments occur at high discharges.The data from the Big Presque Isle river (Fig 3.4c) indicate that a constant value of k3 = 0.28 mproduces a good estimate of the observed variation off The data from the Meduxnekeag andOldman rivers (Fig 3 .4a, d) indicate that at high values of discharge a good agreement betweenthe observed and calculated values off can be realised with a constant value of k. Howeverdata from the Northwest Miramichi River (Fig 3.4(b)) indicate that the value of k is highlyvariable with discharge. Table 3.1 contains the values of D50 from each of the four riverstogether with the adopted value ofk8.For modeling purposes it is most important to accurately estimate the values off at highdischarges where most of the bedload transport and bank erosion, and therefore channeladjustments, occur. Three out of the four rivers shown here display stage-discharge relationswhich indicate relatively constant values of k3 for high discharges. Therefore for modelingpurposes it will be assumed that a river has a single value of k3 which remains constant for alldischarges and throughout any channel adjustments including changes in the value ofD50. InSection 3.4 the total roughness height will be divided into bed and bank components, andvalues of the separate components will be held constant. It is acknowledged that this is a48limitation of the modeling procedure, however this can only be resolved through furtherresearch into the sources of the form or bar channel roughness, and how this might be relatedto the channel geometry.Table 3.1. Values ofD50 from Phinney (1975) and Kellerhals et a!. (1972) together with theadopted value ofk for the Northwest Miramichi, Meduxnekeag, Big Presque Isle, and OldmanRivers.River D50 k3 k /D50(m) (m)Muduxnekeag 0.091 0.20 2.20Northwest Miramichi 0.08 1 0.10 1.23Big Presque Isle 0.064 0.28 4.38Oldman 0.040 0.20 5.003.3 BANKFULL DISCHARGE AND OVERDANK FLOWThe morphological significance of the bankfbll discharge and the implications of overbank flowwill be discussed in this section. The bankfitll discharge at a river cross section is defined as theflow that just fills the channel to the tops of the banks. A thorough review of the definitions ofbankfull discharge is given by Williams (1978). The floodplain is assumed to be the level of theactive floodplain which has developed from recent channel activity. In some instances the valleyflat may be equivalent to the active flood plain. A definition sketch of the channel andfloodplain is shown in Fig 3.5. The definition of Wolman (1955) will be used and this is thestage where the width to flow depth ratio (W / Y) is a minimum. In the simple case of aprismatic channel as is illustrated in Fig 3.5, the bankfhll discharge occurs when the flowreaches the height of the active floodplain (Y 11).The dimensionless floodplain depth of flow 3, is given by 13 = (Y -II) / H where Y is the totaldepth of flow, and H is the vertical height of the main channel banks, and the depth of flow onthe floodplain is equal to Y- H. At the bankfhll stage f3 = 0, and for discharges in excess of Qbf13>0.49For values of the discharge up to the bankfhll discharge Qbp Rh and the mean boundary shearstress t both increase with increasing Q. At Qbf the mean bank and bed shear stresses aregreater than for any lower Q values. This is important from bank stability considerationsbecause if the sediment which forms the channel banks is not being eroded at Qbf it will bestable at all lower discharges.A more complex case exits for flows in excess of Qbf The flow is spread over a wide area andthere are complex interactions between the overbank flow on the floodplain, and the channelflow. This floodplain-channel interaction was first demonstrated experimentally by Sellin (1964;although see Cruff (1965) for earlier unpublished reports) who identified vortices with verticalaxes at the interface between the channel and floodplain flows. These vortices result inmomentum exchange between the relatively fast channel flow, and the slower floodplain flowand produce additional flow resistance effects.The simplest case for analysis is the channel with an infinitely wide floodplain. The excess flowabove bankfi.ill will be spread across an infinitely wide area, and therefore the flow depth on thefloodplain is essentially zero, while the depth within the channel does not increase beyond thebankfull depth. For discharges in excess of the Qbp value of f3 remains equal to zero, and thefloodplain-channel interaction can be assumed to be negligible. The bank and bed shear stresseswhich developed at Qbf therefore represent maximum values which are not exceeded at higherdischarges. A channel which has stable banks at Qbf will be stable at all other discharges, andthe sediment transport rate also reaches the maximum value atFor channels with floodplains of finite width, when Q exceeds values of f3 > 0 can develop.Under this condition complex three dimensional flow patterns can develop due to interactionsbetween the floodplain and main channel flow. These complex flow patterns have been thefocus of a considerable research effort. Several investigators have demonstrated the influence of50the floodplain flow on the main channel hydraulics by constructing physical models, measuringthe hydraulic parameters of the main channel with floodplain flow, and then isolating thefloodplain from the main channel by smooth impervious barriers such as glass sheets. Anexample is shown in Fig 3.6 from Sellin (1964). The isovels for the channel with floodplainflow indicate the mean velocity and discharge within the main channel are reduced verysignificantly when compared to the same channel with the flood plain isolated from the mainchannel. From his experiments Sellin (1964) determined that the velocity and discharge in themain channel are reduced by approximately 30 %.It can be observed from the velocity gradients in Fig 3.6 that the bed and bank shear stressesare also reduced significantly due to the floodplain-channel interaction. Barishnikov (1967)performed experiments similar to those of Sellin (1964) to determine the effect of thefloodplain-channel interaction on the bedload transport capacity of the channel and determinedthat the bedload capacity of the main channel could be reduced by 75 - 80 % due to overbankflow interactions. Similarly Meyers and Elsawy (1975) investigated the effect of the floodplain-channel interaction on the boundary shear stress and found that the mean and maximum valuesof the bed and bank shear stresses could be reduced by 20 - 30% of the bankfiill values forsmall positive values of3.In general, it has been determined that the effects of the floodplain-channel interaction aregreatest when the difference between the flow velocities on the floodplain and channel isgreatest. The interaction effects increase with increasing floodplain roughness (Barishnikov,1967; Knight and Hamed, 1984), for larger ratios of floodplain width to channel width (Knightand Demetriou, 1983; Holden and James, 1989), and for small non-zero values of (3 (Knightand Demitriou, 1983). Several investigators have determined that the floodplain-channelinteraction effects on the main channel flow become negligible for values of (3 greater thanabout 0.3 - 0.5 (Posey, 1967; Meyers and Elsawy, 1975; Bhowmik and Demisse, 1982; Pascheand Rouve, 1985).51In conclusion it is evident that for values of 13 between 0 and about 0.5, the bed and bank shearstresses within the main channel are in general somewhat less than the values at Qbp and that achannel which is stable at the banlcfull stage will remain stable for larger discharges.Furthermore the sediment transporting capacity for discharges above bankfl.ill may be nogreater, or even less than at Qbf This conclusion does not necessarily apply to very large floodevents where 13 is greater than about 0.5 and the floodplain-channel interactions becomenegligible.In the present model formulation the infinite floodplain model will be assumed where (3 remainsequal to zero for discharges equal to and exceeding Qbf The values of the bed and bank shearstresses will be assumed to be equal to the bankfull values for discharges in excess of bankfiill.The stability of the channel will be assessed at Qbp and if the channel banks are stable at Qbfthey will be assumed to be stable with respect to fluvial erosion of the bank sediment for allother discharges. Furthermore the value of the bed shear stress calculated at Qbf will be used tomodel the sediment transporting capacity of the channel for flows which exceed Qbf3.3.1 Recurrence Interval of Bankfull FlowSeveral investigators in the past have concluded that Qbf corresponds to a characteristic returnperiod. Wolman and Leopold (1957) and Leopold et al. (1964) that an average recurrenceinterval of 1.5 years based on the annual maximum series is a good estimate of Qbf SimilarlyDury (1973) suggests a value of 1.58 years. Nixon (1959) based his analysis on the partialduration series and found a mean recurrence interval of about 0.5 years. According toHenderson (1966, p. 465) the result of Wolman and Leopold (1957) is equivalent to arecurrence interval of 0.9 years based on the partial duration series.Williams (1978) completed an analysis of 36 stations to compute the return period of Qbf forthe active floodplain and found a mean recurrence interval of 0.9 and 1.5 years based upon the52partial duration and annual maximum series, respectively. A plot of the frequency distributionfor the analysis ofWilliams (1978) is shown in Fig 3.7. Disregard the valley flat curve as Qbfhasbeen defined herein as the height of the active floodplain. While the mean recurrence values inFig 3.7 agree closely with the 1.5 year average of Wolman and Leopold (1957) and Leopold etaL (1964), there is a wide spread of values between 1.01 to 32 years based on the annualmaximum series (0.25 to 32 years for the partial duration series), and the standard errorassociated with this mean value is 0.277 log units. Williams (1978) concluded that because ofthe wide range in recurrence intervals an average recurrence interval has little meaning and is apoor estimate of QbfThe suggestion of a characteristic recurrence interval has resulted in the widespread assumptionthat Qbf is a primary feature of the watershed, and can be treated as an independent variablewhereby the value of Qbf is imposed on the channel. This has been a prime assumption inseveral analytical studies of river channel development (Parker, 1978; Chang, 1979, 1980;White et a!., 1982; Millar and Quick, 1 993b) and empirical regime studies of rivers (Charlton eta!., 1978; Andrews, 1984; and Hey and Thorne, 1986). However if one agrees with theconclusion of Williams (1978) that there is in fact no characteristic recurrence interval forthe assumption of Qbf as an independent variable becomes somewhat tenuous. Furthermore theactual value of Qbf is defined by the width, depth, slope, and roughness of the channel, all ofwhich are dependent variables. The optimisation model will be used in Chapter 7 to show thatthe value of Qbf can be viewed as a dependent variable.3.4 SUBDiVISION OFfINTO BED AND BANK COMPONENTSThe preceding discussion requires the assumption that the values off for the bed and banksections of the channel are equal. Alternatively the wide channel approximation may beassumed whereby the channel roughness is equal to the bed roughness only, and thecontribution of the bank sections to! is negligible. The distribution of the shear force (per unit53channel length) which is defined as the product of shear stress and wetted perimeter can bedivided into bed and bank components:bed.d bnk’3a (3.18)where P is the wetted perimeter, and the subscripts bed and bank indicate the bed and bankcomponents of the total (Fig 3.5b). Eqn (3.18) forms the basis for the empirical methoddeveloped by Knight (1981) and Knight eta!. (1984) for calculating the mean values of the bedand bank shear stresses, and is used in Chapters 4 and 5 to calculate the bedload transportcapacity and to assess the bank stability.By using Eqn (3.5) to express t in terms ofJ dividing throughout by P and cancelling liketerms, Eqn (3.18) can be expressed in terms offP Pff bedf bankJ jbed ibankwherefbed is the total bed friction factor, and fbaflk is the total bank friction factor.For the wide channels typical of natural rivers bed P, and bank <<F, therefore from Eqn(3.19) it is seen that f is approximately equal to fbed In many engineering studies the widechannel approximation can be used, and it can therefore be assumed that the contribution of thebank roughness to the total channel roughness can be ignored. However in the modeling workto be undertaken in Chapters 5 and 6 the sediment transporting capacities and bank stability ofnarrow channels will also be assessed, and the effect of the bank roughness on the total must beconsidered.54The values Offbed a.ndfbank can be calculated with Eqn (3 .2b) using the respective values of ksbedand ksbaflk together with the appropriate values for Rh. It is an important assumption that thebed and bank friction factors only influence a portion of the total cross-sectional area, and inorder to calculate fbed and fbank with Eqn. (3.2b), the bed and bank components of the totalhydraulic radius, Rhbd and Rhbk, must be determined. The side-wall correction procedure forflume studies developed by Johnson (1942) and Vanom and Brooks (1952; as reported inVanoni, 1975, p.l52l5z1) will be used to calculate the bed and bank components of Rh. Theprincipal assumption in this method is that the cross-sectional area, A, can be divided into bedand bank components:A=Ad+A (3.20)where Abed = the area of the total cross-section that is affected by the bed roughness, and A bank= the area of the total cross-section that is affected by the bank roughness. There is assumed tobe no interaction between the adjacent bed and bank areas, and therefore these areas must bedivided by a line of zero shear. Additional assumptions are that the mean velocity and energygradient are the same for each of the subsections.The subdivision of the cross-section is shown in Fig 3.8. The boundaries of the bed and banksections must be lines which pass through the junction of the bank and bed and are orthogonalto the isovels, an idea that is attributed to Leighly (1932). Rather than lines of zero shear, theorthogonals are more correctly viewed as lines of zero net momentum flux. Assuming zeroshear along the orthogonals, the only force that is resisting the stream-wise gravitational forceof each section is the shear force per unit length of channel acting along the respective channelperimeter, hence:bank’ank =yAbS (3.21)55which can be rewritten:bank rbank— P —. abank 7’Eqn (3.22a) can also be expressed in terms of the corresponding bed values:1? =-=- (3.22b)Jibed‘b.d 7’ SIn natural channels the presence of secondary currents complicates the isovel distribution.Furthermore in order to locate the lines of zero shear that divide the bed and bank sections ofthe cross-section, the isovel contours must be known. However an alternate approach forestimating Rhbed and Rhbank is suggested by Eqns (3 .22a, b). In Chapter 4 empirical relationsdeveloped by Flintham and Carling (1988) are used to calculate ;ed and Tbank in order tocalculate the bedload transport rates and to assess the stability of the banks. Therefore, thevalues ofRhbed and Rhbk can be calculated using Eqns (3.22a, b) with the values of rbCd andrbank being obtained from Eqns (4.8) and (4.9).For known values of bed’ bank’ ksbd and ksbk, the values of ‘1bed and ank can be calculatedwith Eqns (4.8) and (4.9), and RhbCd and Rhbank from Eqn (3.22a, b). Once these values areknown fbed and fbank can then be calculated from the following equations which aremodifications ofEqn (3.2b):/ ffb = 2.O3 lo kbank J (3.23a)Sb I/ f(12.2RJi ‘Ifb L2.03 lo kbedjJ (3.23b)abed56and the total! then calculated from Eqn (3.19).3.5 DISCHARGE CONSTRAINTThe discharge constraint is defined here as the requirement for a channel to have a dischargecapacity at bankfull equal to an imposed or trial value of Qbf The discharge constraint is usedto determine the size of the main channel.The discharge constraint is formulated through the definition of volumetric flow rate as follows:UA=Qbf (3.24)where U is the mean velocity, and A is the total cross sectional area, with both correspondingto the value at the bankfull discharge. The value of U is calculated using the Darcy-Weisbachequation:I8gRSU=1j (3.25)where the friction factor! is given by:(3.19)and the bank and bed components offgiven by:( (12.2 Rh -2fb =2.O3 lo kbflkJJ (3.23a)57( (12.2 Rh ‘fbed = 2.03 lo kbed JJ (3.23b)SbedIt will be assumed that the values of ksbaflk and ksbCd are independent variables whose values areknown and remain constant during stage changes and channel adjustment. This is known to beuntrue, however the current knowledge of channel roughness adjustment is insufficient toaccount for changes in ksbaflk and ksl,edThe values OfRhbk and Rhbed are determined from Eqns (3 .22a) and (3 .22b) respectively:A__41c . abank (LRh ==-- (3.22b)bed‘3bed )‘Swith the values of Thed and rbank being given by Eqns (4.8) and (4.9) respectively.The complete range of flows will be used in Chapter 5 to model the sediment transportingcapacity. The flow-duration curve will be used as input, and the discharge constraint modifiedto calculate the Tbed and rbank values for each flow.3.6 SUMMARYThe discharge constraint has been developed in this chapter. This constraint is composed oftwo components: flow resistance and the flow rate for uniform flow in a prismatic channel.Flow resistance is expressed by the logarithmic Colebrook-White equation. The total frictionfactor is subdivided into grain and form roughness. The friction factor due to the grainroughnessf’, is calculated from the Colebrook-White equation with the roughness height equalto the median bed grain size, D50 (for wide channels). No relationship describing the form58roughnessf” has been determined. For many gravel rivers the value off” is significant, andmay be greater than f’, even at high in-bank flows. It is demonstrated that the bar resistancecurve of Einstein and Barbarossa (1952) is largely a result of spurious correlation and thereforedoes not return meaningful values off”.The wide channel approximation is valid for most engineering purposes, however the modelingto be undertaken in this thesis requires that the friction factor be calculated for narrowchannels. In order to calculate the total friction factor the channel cross-section is subdividedinto bed and bank sections as suggested by Einstein (1942) and Johnson (1942) with the aid ofthe empirical boundary shear relations of Flintham and Carling (1988). Roughness values areassigned to the bed and bank portions of the channel, and the friction factor of each section isobtained by using the bed and bank values of the hydraulic radius.It is suggested herein that the bankfull discharge Qbf might be more correctly viewed as adependent variable, rather than as an independent variable as is commonly assumed. Theoptimisation model will be used to test this idea in Chapter 7.590.6f 0.40.20Figure 3.1. Variation off with respect to Rh / D50 for selected gravel-bed rivers forbankfull or near bankfull conditions.AHey and Thorne Andrews(1986) (1984)Equation (3.2b)k= 5.8 050Chariton et al. Bray(1978) (1979)0 *Equation (3.8)k= D50*0A0A *ELI0L)05 10 20 50 100Rh/DSQ600.70.60.50.40.30.20.1f..Hey and Thorne Andrews Charfton et at. Bray(1986) (1984) (1978) (1979)A 0 *Equation (3.14)Einstein and Barbarossa (1952)**A0A0 *A)*DDI I I I00 0.01 0.02 0.03 0.04 0.05 0.06 0.0750Figure 3.2. Variation off” with respect to zI*D,O for selected gravel-bed rivers forbankfull or near banlcfull conditions.610.70.2-BB0000.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16*D50Hey and Thorne Andrews Chariton et al.(1986) (1984) (1978)0 00.60.50.40.30.20.10(0.1)0.80.6f’ 0.4f’I(f’ +f)(a)(b)Figure 3.3. Variation off” with respect to the components of z*DSO for selected gravel-bed rivers for bankfiull or near bankfull conditions. (a)f’ If (b) rD30.00Bray(1979)*620.25 0.2f fobserved calculated calculated0 ----r--- --e-Figure 3.4. Variation off with respect to Q. (a) Meduxnekeag River. (b) NorthwestMiramichi River. (c) Big Presque Isle River. (d) Oldman River. The values ofk used tocalculateffor each river are (a) 0.20 m, (b) 0.10 m, (c) 0.28 m, (d) 0.20 m.(b)0C0I(a)0.20•t5 0.15C00.10.0500(c) 0.2013C013I.0.150.1f0.050100 200 300 400 500 600Discharge, Q (me/sec)100 200 300Discharge, Q (me/sec)4000.150.0.05013C01350 100Discharge, Q (m3/sec) Discharge, Q (m3/sec)63(a)\i w(b)FLOODPLAIN MAIN CHANNELwFLOODPLAINFigure 3.5. Definition sketch of prismatic channel. (a) Composite channel includingfloodplain. (b) Main channel only.°°bankbed64Figure 3.6. Experimental velocity contours (from Sellin, 1964). The upper figureindicates the velocity contours with overbank flow, and the lower with the floodplainisolated from the main channel.65RECURRENCE INTERVAL. IN YEARS (PARTIAL DURATION SERIES)Figure 3.7. Recurrence interval for Qbf (from Williams, 1978). In this thesis the activefloodplain values are of principal concern. The active floodplain data was collected from36 rivers, and the valley flat data from 26 rivers.RECURRENCE INTERVAL. IN YEARS (ANNUAL MAXIMUM SERIES)66O.5A O.5Abank bank vjnormalsisovelsFigure 3.8. Subdivision of cross-section into bed and bank sections. These subareascorrespond to the areas of flow that are affected by the bed and bank roughness.67CHAPTER 4BEDLOAD CONSTRAINT4.1 INTRODUCTIONThe bedload constraint is a key component of the model and ensures that the amount ofsediment that is imposed on the channel can be transported without appreciable net depositionor scour. In Chapter 2 it was argued that the transport of sediment is the limiting channelforming process. Since the model is being formulated for gravel-bed rivers it will be assumedthat it is only the transport of coarse gravel bedload sediment that is the limiting process; thefiner sand and silt fractions are assumed to move through the system essentially in suspension.This chapter addresses two principal topics, the boundary shear stress distribution and thebedload transport algorithm.4.2 BOUNDARY SHEAR STRESS DISTRIBUTIONThe boundary shear stress will be used to calculate the bedload sediment transport rates, and toassess the stability of the channel banks. Under conditions of uniform, steady flow the meanboundary shear stress z, can be represented byr=rRhs=pU2 (4.1)Eqn (4.1) can also be expressed in terms of grain and form components as in Chapter 3.68The boundary shear stress is not uniformly distributed across the wetted perimeter, and the localvalue of the boundary shear stress on the bed and banks can vary significantly fromSeveral mechanisms are responsible for the non-uniform boundary shear stress distribution andinclude the lateral transfer of longitudinal momentum, secondary currents, and the non-uniformdistribution of roughness elements across the channel perimeter. The influence of thesemechanisms and the various approaches to evaluating the boundary shear stress distribution willbe discussed below.4.2.1. Cross-Channel Momentum TransferAll shear flows are characterised by a net transfer of momentum from regions of highmomentum to regions of low momentum. In rectangular channels momentum is transferredtowards the bed and laterally from the center of the channel towards the banks (Cruff 1965). In2-D flow the transfer of momentum is towards the bed only. By analogy with the Prandtl theory,the lateral transfer of momentum is viewed as a diffusion process whereby the rate of lateralmomentum transfer is proportional to the cross-channel velocity gradient (Lundgren andJonsson, 1964). The velocity which result purely from momentum diffusion in turbulent flowsare indicated schematically in Fig 4.1(a).Superimposed upon the diffusion of momentum is the advective transport of momentum bysecondary currents. Einstein and Li (1958) proved theoretically that secondary currents arepresent even in straight channels. These secondary currents are sometimes referred to as ‘weak’secondary currents as opposed the “strong” secondary currents that are produced as a result ofchannel curvature. The strong secondary currents can be neglected in straight channels. Theeffect of the weak secondary currents on the velocity distribution are indicated in Fig 4. lb. Notethe depression of the maximum velocity contour below the free surface.694.2.2 Analytical SolutionsSolutions for the distribution of the boundary shear which neglect the influence of secondarycurrents have been determined for simple cases. Leighly (1932) assumed that the orthogonals tothe isovels represent lines of zero shear, and if the isovels are known, the boundary shear stresscan be calculated by:rdP=yS (4.2)where Tdp, dA and dP are respectively the local boundary shear stress, the area, and theperimeter bound by two orthogonals (Fig 4.2a).The area method of Leighly (1932) is of limited practical use as the velocity distribution must beknown in order to locate the orthogonals from which dA and dP, and hence dz can bedetermined.In a summary of the work of the USBR, Lane (1955b) gives the results of some analytical andnumerical studies to determine the values for the maximum bed and bank shear stresses forrectangular and selected trapezoidal channels. The solutions were obtained by assuming apower, rather than logarithmic, velocity distribution along lines normal to the boundary. Theresults are shown in Fig 4.3. For wide trapezoidal channels which are typical of natural rivers,the values of the maximum bed and bank shear stresses suggested for design purposes are yYSand 0.76 yYS respectively.Lundgren and Jonsson (1964) used Keulegan’s (1938) assumption that the velocity distributionis logarithmic along lines normal to the boundary. They developed the modified area methodwhich computes the value of ip bound by two normals to the boundary (Fig 4.2b). Since thenormals to the boundary do not represent lines of zero shear, the shear stress along these70normals must be considered. This shear stress is equivalent to the lateral diflhsion of momentumacross the normals.The modified area method of Lundgren and Jonsson (1964) can be expressed:dATdP_Y’p_P6t (4.3)where TdP, dA and dP are bound by the normals to the boundary, and e is the coefficient oflateral momentum transfer, and ôu / ãz is the local cross-channel velocity gradient.Lundgren and Jonsson (1964) showed that for practical purposes the value of Tdp fromEquation (4.3) agrees well with the method of Leighly (1932). The drawbacks of this approachare that secondary currents are neglected, and that it is applicable only to channels withgradually curving profiles. This method cannot be used for rectangular or trapezoidal crosssections.4.2.3 Experimental MethodsGiven the complexities of the effects of secondary currents and the non-uniform distribution ofroughness elements across the channel boundary, several investigators have approached theproblem of estimating the mean or maximum bed and bank shear stresses by formulatingempirical relations based on experimental data.In a number of investigations velocity distribution or Preston tube measurements in rectangularand trapezoidal flumes have been used to determine the distribution of the boundary shearstresses (Cruff 1965; Rajaratnam and Muralidham, 1969; Kartha and Leutheusser, 1970; Goshand Roy, 1970, 1972; Knight and Macdonald, 1979a, b; Knight, 1981; Knight et al., 1984;Flintham and Carling, 1988). The empirical approach developed by Knight (1981) and Knight et71aL (1984), with the subsequent modifications by Flintham and Caning (1988) will be used inthis thesis to determine the values of the mean bed and bank shear stresses, Tbed and ;ank Thismethod is based upon the distribution of the total shear force SFotai into bed and bankcomponents:= SFd + SF,ank (4.4)which is equal to:P=Tbed+TbF (4.5)where P is the total wetted perimeter, and the subscripts bed and bank indicate the bed and bankcomponents. The channel is defined in Fig 3.5.The proportion of the shear force acting on the banks SFbaflk is given by:— bankAL bankVelocity profile and Preston tube data collected from rectangular flumes were used by Knight(1981) and Knight et a!. (1984) to develop an empirical relation for SFbaflk. This relation wasinitially developed by Knight (1981) for smooth channels using his own data, and the publisheddata from Cruff (1965), Gosh and Roy (1970), and Meyers (1978). The general form of therelation was verified in Knight et a!. (1984), and the values of the coefficients refined withadditional experimental data collected by the authors, together with data from Noutsopoulosand Hadjipanos (1982) and Kartha and Leutheusser (1970).72Knights empirical relation for SFbaflk was expressed as a fl.inction of the aspect or form ratio W/ Y. This relation was modified by Flintham and Caning (1988) who reduced the scatter bycorrelating the ratio of the bed to bank perimeter lengths bed / ‘bank’ rather than the aspectratio W/ Y. Flintham and Caning included their own data from a trapezoidal channel, togetherwith the results of Gosh and Roy (1970) to develop the following equation for SFba, presentedhere in a power form, which is a modification of the empirical equation of Knight (1981) andKnight etaL (1984):(1d= 1.766 e + 1.5 (4.7)ban/cThe values of ;ed and tbaflk can be estimated from the following equations which are algebraicmanipulations of Equations (4.5) and (4.6):____I(w+p )sin8lbank_SFyJ’S banlc[ 4)7=(i-si)[2 +0.5] (4.9)where 9is the bank angle as indicated in Fig 3.5.The value of SFbank is obtained from Equation (4.7). Equations (4.8) to (4.9) are plotted againstthe ratio W / Y in Fig 4.4. A comparison with Fig 4.3 shows that the results are reasonablyconsistent with the results ofLane (1955b).4.2.3.1 Non uniform bed and bank roughnessThe effect on the shear stress distribution of the non-uniform distribution of roughness elementsacross the channel has long been recognised. Einstein (1942, 1950) allows for the influence of73bank resistance on the bed shear stress when calculating the sediment transport rates. Followingfrom Einstein’s work, Johnson (1942) developed a side wall correction procedure to be appliedto laboratory studies of bedload transport which were being undertaken in narrow glass (orother material) walled flumes. Knight (1981) analysed SFbaflk for channels with different bed andbank roughness values and developed an empirical correction relation for different bed and bankroughness. This correction relation is a function of the ratio of the bed to bank roughnessheights, ksbed / ksbank Flintham and Carling (1988) performed additional experiments withdifferential bed and bank roughness and developed the following equation which is amodification ofEqn (4.7):(Pi bedJFbJ I i’ D“1bankThe coefficient C is a function of the ratio of the roughness of the bed to the banks, k8bed /ksbank•For / ksbk = 1 the value of C = 1.5 and Equation (4.10) is equivalent to (4.7). Equation(4.10) was constrained to pass through the SFbank = 1.0 for “bed’ bank = 0.An additional relation was developed by Flintham and Carling (1988) for the parameter C fromexperimental values of ksbed / ksbank between 1 and 91.1. This function was forced to passthrough the point C = 1.5 for the value of the ratio / ksbank = 1:(C=1.5kdJ (4.11)The influence of the ratio k / k on the mean bed and bank shear stress values is indicatedSbed Sbankin Fig 4.5. As ksbed / ksbank increases, a greater proportion of the total shear force is carried bythe bed and the value of ed increases, together with a concomitant decrease in the value of r74bankS The value of rbank is more sensitive to the ratio ksbed/ ksbk than Tbed., especially for largevalues of WI I’.4.2.4 Grain and Bar ShearEinstein (1950) and Einstein and Barbarossa (1952) proposed that the total shear could bedivided into grain and bar (form) components:(3.4)where, as in Chapter 3, the superscripts’ and” refer to the grain and bar or form componentsrespectively.Eqn (4.5) can therefore be rewritten in terms of the grain and form components:= (ned + Tbed) ‘,ed +(r + (4.12)For a wide channel approximation bed F, and ‘3bank <<F, and Equation (4.12) can besimplified to:V = + Vbed (4.13)Einstein (1950) has argued that only r ‘bed contributes to the bedload transport. By definition(see Chapter 3):V’=pU2 (3.5b)Now:8gRS 8rU2= = (414)f fCombining Equations (3.5b) and (4.14) yields:75(4.15)fwheref is the total friction factor, and!’ is the grain friction factor given by Eqn (3.8).The value of r’ is plotted against r using the published field data of Chariton et a!. (1978), Bray(1979), Andrews (1984), and Hey and Thorne (1986) in Fig 4.6. The same data are used in Figs3.1 and 3.2.Fig 4.6 indicates that the value of r’ is bounded by z, and the scatter below the upper boundindicates that for most channels the value of ‘is significantly less than zThe exception is a fewchannels where r’ appears to exceed r which it cannot by definition. This anomaly is attributedto measurement errors as in practice it may be difficult to assign truly representative values ofD50, Rh etc to a river reach.For these data the mean value of the ratio t ‘i r is 0.49, and the minimum value is 0.11. Thisindicates that for some channels as little as 11% of the total bed shear stress is available forbedload transport even at or near Qbank. This contradicts Parker and Peterson (1980) whoconclude that for high in-bank flows, the total boundary shear stress is available for bedloadtransport. This result indicates that large variations in the calculated bedload transport rates canresult if Tbed rather than r bed is used.Despite this result, the Parker (1990) surface-based bedload transport relation, which uses iato calculate the bedload transport rates and not r ‘bed., will be used in a modified form inChapter 7 of this thesis to calculate the sediment transporting capacity of the modelled riverchannels. In order to reformulate the Parker (1990) relation on the basis of r ‘bed, considerableeffort is required.764.3 MODELLING SEDIMENT TRANSPORTING CAPACiTY AT BANKFULLOptimization procedures such as Chang (1980), White et a!. (1982), and Millar and Quick(1993b) have modelled the sediment transporting capacity of the channel as the transport ratecorresponding to the dominant or bankfull discharge. Similarly Hey and Thorne (1986) includedthe bankfI.ill sediment transport rate as an independent variable in their regression analysis. Thishas been justified as most of the transport, particularly for gravel-bed rivers, typically occurs ator near bankfull. However it will be argued that the duration of the transporting flows must bealso considered, and that the sediment transporting capacity should be defined by the totalbedload which can be transported over a significant duration T (eg. 1 year), and not by a singletransporting rate.Consider two channels which have identical values for both and the sediment transportcapacity at bankfull Gbf Furthermore assume that these channels are only just above the criticalthreshold for bedload transport at so that essentially no bedload transport occurs fordischarges less than Qbf Andrews (1984) determined from 24 gravel-bed rivers in Colorado thatduration for which the Qbfwas equalled or exceeded ranged between 0.12% and 6%. Thereforethe total bedload transported over duration T by two channels with identical Qbf and Gbf valuescould differ by a factor of 50 due to the differences in the duration of Qbfexceedence.It must be concluded that while Qbf is morphologically significant and that the stability of thechannel banks should be assessed at this value, it is not sufficient to model the sedimenttransport capacity of the channel at the single value of Qbp but rather the flow durations mustalso be considered. The complete flow-duration data will be used as input to the model. Thesedata are a representation of the complete range of flows over a period of record. Forcomputational purposes the flow-duration curve will be discretized into m segments (Fig 4.7) ofquasi-steady, uniform flow. The valuep1 is the probability of flow Q1, where:77p1=1.O (4.16)4.3.1 Generalized Equations for Mean Bed and Bank Shear StressesThe following equations are necessary to calculate the value of Tbedj which is in turn required tocalculate the bedload transport rate for each flow Q,. The equations used in the dischargeconstraint (see Sect 3.6) are modified to calculate the flow geometry and the mean bed shearstress values which correspond to each flow Q.The continuity requirement for uniform flow is given by:UA1=Q (4.17)where U1 is the mean velocity, and A1 is the cross sectional area for flow Q•. The value of U1 iscalculated using the Darcy-Weisbach equation:I8gR. Sf. (4.18)which is a modification ofEqn (3.25). The values ofRh, andf now take different values for eachvalue of Q,.The value off is given by:f (4.19)Note that the value of Pbed is constant for all flows. The bank and bed components off givenby:78( (12.2Rfb. =2.03 lohbJJ (4.20a)Sçj.J )(122 R= [2.03 lokhbed1 ITbed IThe values of and Rhbed. are determined from Eqns (4.21a) and (4.21b) respectively:Abed— 1bedj (4.21 a)Rhbed.=—V S_____— Tbank. (4.2 ib)Rhb,?,(.=—y SThe values of1•bed• and Tbankj are given by:Tbank=SFygS banki[ 4y j (4.22)JEwTbed. =(1—sF ,. +0.5 (4.23)yYS bank, bedThe value of SFbaflkj for each flow Q, is given by:‘iedy1.4o26S],I\C]Lk.+ (4.24)where the value C is given by Eqn (4.11).794.4 BED SURFACEGravel-bed rivers typically develop a layer on the bed surface that is significantly coarser thanthe underlying material. This layer is one grain thick and is often termed the pavement orarmour layer (Kellerhals and Bray, 1971; Parker et aL 1982), and the underlying finer sedimentis called the subsurface, subarmour, or subpavement material. In this thesis the term armour willgenerally be used to describe the surface layer, and subarmour to represent the subsurfacematerial.The armour layer plays an integral role in the channel dynamics of gravel-bed rivers. The coarsegrains which comprise the armour produce the grain roughness of the channel bed and thereforecontribute to the flow resistance. Furthermore the armour layer shields the finer subarmoursediment from the flow which has a strong influence on the transport of bedload sediment.Milhous and Klingeman (1973) conclude that the armour layer is the single most importantfactor in limiting the availability of sediment and controlling the relationship between streamfiowand bedload discharge.The grain size distribution of the armour layer can be most simply represented by a singlecharacteristic grain size such as the median grain diameter D50, or the geometric mean graindiameter D, together with a measure of the grain size dispersion about the mean or medianvalue such as the standard deviation based on the sedimentological phi scale a, or geometricstandard deviation a.The degree of armour development will be defined by the following relation:D(4.25)5080Where d50 is the median grain diameter of the subarmour sediment. The value of 350 has beenobserved to range from 1 with no armour development, to greater than 6 which indicates a largedegree of bed coarsening. The typical value of 3o for natural rivers is about 2.5 (Parker et aL,1982).The adjustment of the armour layer to changes in the flow regime or sediment supply has beenobserved both experimentally and from field observations. For instance Kellerhals (1967)demonstrated the coarsening of the armour layer under steady flow conditions when the supplyof bedload sediment was cut off This coarsening of the armour is commonly observeddownstream of a dam which has cut off the supply of sediment to a channel. Dietrich et a!.(1989), Lisle and Madej (1989), and Kuhnle (1989) have shown that channels with high bedloadtransport rates tend to have values of Sso which approach 1. Dietrich et a!. also noted that theeffect of reducing the sediment feed to a channel was for the channel to increase 6. It istherefore evident that both D50 and Dsg, as well as o or 0g’ must be considered to bedependent variables. The values ofD50 and Dsg can take minimum values ofd50 or d, when nosurface coarsening is developed, and theoretical maximum values ofd1, which is the maximumsubarmour grain diameter. This can be stated formally as:d50 D50 d100 (4.26)dsg Dsg d1 (4.27)Eqns. (4.26) and (4.27) are termed the armour grain size constraints. In reality the upper boundto D50 and D is probably closer to d90 rather than d100, as sufficient sediment of this graindiameter must be present in the subarmour in order to form an armour layer.In the proposed formulation D50 and D are treated as dependent variables which can adjustwithin the bounds given in Eqns. (4.26) and (4.27). The values of o and og will also be treated81as dependent variables. The grain size distribution of the subarmour sediment will be assumed tobe equivalent to the long-term grain size distribution of transported bedload sediment (Parker eta!. 1982). The values ofd50, d, d100, as well as other subarmour grain sizes are a result of thecatchment geology and sediment sorting and abrasion which occurs upstream, and thereforethese values are imposed on the channel reach, and are therefore considered to be independentvariables.To account for the effect of the armour layer in the model the surface-based bedload relation ofParker (1990) will be used. In this bedload relation Parker explicitly accounts for the influenceof the armour layer on the bedload transport rate. The armour layer grain size, which will berepresented by D50, is permitted to adjust together with the b’ ban1, S, 8, and otherdependent variables until an optimum solution is attained. The Parker surface-based bedloadtransport relation is presented in a Section 4.5.4.4.1 Influence of the Armour LayerThe simplest sediment transport model which accounts for the influence of the armour employsthe concept of an armour threshold. At discharges (or shear stresses) below the threshold, thearmour layer is essentially immobile, and the bedload transport rate is close to zero. Anymaterial which is transported above the immobile armour layer is considered as suspended orsandy throughput load which is considered to be more a function of upstream supply, ratherthan the local channel hydraulics. Only the load which is considered to result from an exchangebetween the bed and the flow will be considered in this paper. For many gravel-bed rivers thisapplies to sediment coarser than about 2 to 5 mm (Parker, 1990).For flows exceeding the threshold, the armour becomes mobile, exposing the finer subarmour tothe flow, and sediment transport rates increase dramatically as shown in Fig 4.8(a). The datafrom Fig 4.8(a) was collected from Oak Creek, Oregon, USA by Milhous (1972). The criticaldischarge is approximately 1 m3/sec which for Oak Creek corresponds to a value of that82is approximately equal to 0.03. Note that a threshold is evident only in Fig 4.8(a) which isplotted with a linear Y- axis. When the same data are plotted in Fig 4.8(b) with a logarithmic Yaxis no threshold is evident.Two models of armour behaviour at discharges above critical are recognised. The conventionalmodel assumes that when the discharge is significantly greater than the critical value the armourbreaks down and the subarmour sediment is fuily exposed to the flow. As the discharge waneson the falling limb of the flood hydrograph the armour layer reforms as the coarser particlesbecome increasingly immobile, while the more mobile fine sediment are winnowed out from thesurface layer.A more recent model has been developed by Gary Parker and others which contends that thearmour layer is a mobile bed phenomenon which is persistent even at relatively high discharges(Parker et a!., 1982b; Parker and Klingeman, 1982). In flume studies an equilibrium armourlayer was seen to develop which regulated the supply of fine material and resulted in the equalmobility of all size fractions (Parker et a!., 1 982a, b). There is also field evidence that thearmour layer is persistent in natural channels even during high discharges. For example Milhous(1972) found little variation in the composition of the armour layer following several stormevents where the bed became mobile, which suggests that the armour layer had remained intactduring these events. Andrews and Erman (1986) found that the armour layer of Sagehen Creekin California persisted during an extreme snowmelt event with a peak discharge which exceededthe bankfull discharge by 230%.The model of Parker for the mobilisation of the armour layer will be assumed in this thesis. Thearmour layer is regarded as a fundamental feature of the channel geometry which remainsrelatively constant once equilibrium has been achieved.834.5 PARKER SURFACE-BASED BEDLOAD TRANSPORT RELATIONThe Parker (1990) surface-based bedload transport relation was selected to model the sedimenttransporting capacity of the channel as it is the first relation to explicitly account for theinfluence of the armour layer on the bedload transport. An additional feature is that the relationcan be inverted and used to calculate the grain size distribution of the armour layer which mustdevelop in order to render all size fractions of an imposed bedload sediment “equally mobile”.In this section an outline of the development of the Parker relation will be given. The surfacebased relation is the result of a series of papers authored by Gary Parker and others which dealtwith the development of the armour layer in gravel-bed rivers and related bedload transportissues (Parker et a!., 1982a, b; Parker and Klingeman, 1982; Andrews and Erman, 1986;Andrews and Parker, 1987). The key concept in these papers is the “equal mobility” hypothesis.The sediment that is transported as bedload in gravel-bed rivers is typically well graded. Underthe low rates of bedload transport typical of natural gravel-bed rivers the coarse surficial armourlayer develops as a result of the difference in inherent mobility of the fine and coarse sediment.Any channel in which equilibrium has developed in such a way that there is no net deposition orscour, and hence must have developed a bedload transport capacity which is equal to the rate atwhich the bedload sediment is supplied to the channel. This applies to the total load, and also toeach size fraction. Therefore the coarsest 10% of the bedload sediment must be transportedthrough the equilibrium channel reach at exactly the same rate as the finest 10% of the sedimentover a significant duration which is generally taken to be one year.The following analysis applies only to gravel bedload sediment. The sand fractions finer than 2mm in diameter are considered to be transported as suspended or throughput load and the ratesof transport are governed more by supply from upstream, rather than local channel hydraulics(Parker, 1990).84Now a prime assumption in the Parker model is that the rate of transport of each size fraction isregulated by the volume fraction of the surface layer that it occupies. The higher inherentmobility of the fine sediment can be countered by reducing the volume fraction of the surfacelayer which it occupies. Conversely the coarse size fractions become over represented in thesurface layer to counter their inherent lower mobility. The net result is that a grain sizedistribution develops that, in an equilibrium channel, renders all size fractions equally mobile.For low bed shear stress values this results in an armour layer that is significantly coarser thanthe subarmour or bedload sediment grain size distribution. The theory predicts that the armourlayer disappears under conditions of high bedload transport rates which is in general agreementwith the experimental and field evidence of Dietrich et a!. (1989), Lisle and Madej (1989), andKuhnle (1989).The mathematical formulation of the Parker (1990) relation is presented below. The armour andsubarmour grain size distributions are subdivided into n intervals, and a representative grain sizeis assigned to each interval. The proportions of the total by volume which fall within eachinterval is denoted byf for the subarmour mixture, and 1, for the armour layer.The basic parameter of the relation is W which is the dimensionless transport rate per unit bedwidth of size fraction normalized per unit volumetric armour layer content(s—1)gq= 1.5 (4.28)‘ (i) ‘The value is the volumetric bedload transport rate per unit channel width of size fraction D.The total volumetric bedload transport rate per unit bed width, qb, is given by:85b (4.29)In order to understand the Parker relation, consider initially a channel composed of uniformsediment of diameter D; for which the value W’ is given by:= 0.00218G[ (4.30)where t is the dimensionless bed shear stress obtained by dividing the dimensionless (Shields)bed shear stress by a reference dimensionless bed shear stressrD which corresponds to a small,but non-zero transport rate:vbd/[Pg(s—1)D](4.31)The function G [c1J is shown in Fig 4.9 and is approximated mathematically by the followingsystem of equations:r 0.853Y554741— ) c1>1.59G[c1]=exp[14.2(c1_ 1) — 9.28(— 1)2] 1 1.59 (4.32)cF’42I-Equations (4.30) to (4.32) will now be generalised to apply to a bed comprised of gradedsediment. A key assumption is that the mobility of the armour layer is determined by themobility of the geometric mean grain diameter D. The function G is calculated as a function ofD. Further modifications are necessary to account for the hiding effects in a graded armour86layer, and the relation is generalised to a bed of arbitrary composition by an order-one strainingfunction. The value of W is given by:= 0.00218G[cI g0(o) c] (4.33)where the function G is unchanged from (4.32), 1 is calculated from Eqn (4.31) with DDsg divided by a dimensionless reference shear stress = 0.0386. The functiong0(4) is areduced hiding function, and a is a straining function. These two modifying functions will bediscussed below.The reduced hiding function accounts for the hiding of smaller grains and the over exposure oflarger grains relative to Dsg. The function 4. is a modification ofEqn. (4.25) and is defined as:D.Sf =_—‘ (4.34)The value ofDsg is given by:Dsg = expi1. lnDjJ (4.35)The function g0(4) was developed from Oak Creek data and was assumed to have the valueg0(1)1:go(S.) = (4.36)For grains smaller than Dsg the value ofg0(4) is less than one and the effective shear stressacting on the grain D. is reduced. The converse is true for grains larger than Dsg.87The order-one straining function a was introduced based upon an analysis of Oak Creek datato account for the grain size distribution of the armour layer sediment. From a numericalanalysis using Oak Creek data the function awas found to vary from an asymptotic value closeto 1 for low values ofc13g0 (and low values ofg) for which a coarse, well sorted equilibriumsurface prevails, to 0.453 at high values of where a finer and more poorly sorted surfacelayer is predicted. From this result it was postulated that a generalised straining function comight be a function of and some measure of the armour grain size dispersion. The measureof the grain size dispersion utilised is(4.37)In the case of uniform sediment = 0, and co should equal 1. For Oak Creek co = Co0. Thesimplest hypothesis assumes a linear variation in co which is consistent with uniform sedimentand Oak Creek:(4.38)where is the value of o obtained numerically from the Oak Creek data Fig 4.10.4.5.1 Calculation of Armour Grainsize DistributionThe usual application of the Parker surface-based relation is to input the values ofD3g, D1, aridand then together with TbedtO calculate the bedload transport rate per unit channel bed widthwhich is denoted g, when expressed in dry mass units, and qb in volumetric units. However ifDsg and o, together with the values of are unknown initially, it will be necessary to calculatethe value of these variables for imposed values of Gb. It is important to note here that while thetotal load Gb is an imposed value, the transport rates gb and q, are in effect dependent variables88as their values will be different for different channel geometries. By definition gb = Gb / Pbed, andsince the value of Pbed is considered in this thesis to be a dependent variable, the value of gb isalso a dependent variable. Furthermore changes in the channel geometry will also affect thevalues of md which in turn will affect the values ofgb and qb.By equating Eqns (4.28) and (4.33) and rearranging the following relation for is obtained:(s—i) gqb1,. = 15 (4.39)0.00218 G[sgo g0(s1)a] (Tb, I p)The fraction volume content of the transported bedload sediment in the jth range is denotedwhich is defined as:qb• (4.40)Combining Eqns (4.40) and (4.39) and rearranging yields:F—_____________(s—1)gq441— G[sgo go(s1)] 0.00218(rb / p)’5Since q andf are known for an imposed Gb and a known value ofPbed, the values ofF; and thecomposition of the equilibrium armour layer can be calculated for a value of r,ed from Eqn(4.41).However the optimization model to be developed in Chapter 7 evaluates a range of trial channelgeometries for which the value of q, is not known in advance. The calculation of an equilibriumarmour layer for this more general case will now be considered.89For a specific discharge the second term in Eqn (4.41) is constant, although the value of qb maynot be known initially. Eqn (4.41) can therefore be simplified to:(4.42)G[CI)so g0(31)a]Now by definition the following must hold:F;.=L0 (4.43)Therefore Eqn (4.42) can be presented as an equality by normalising over the summation of allfj/G[(I)sgo g0(o) a)](4.44)fj/G[sgo g0(o) wJEqn (4.44) permits the calculation of the values of F. without knowing the value of q,,beforehand.Since the values ofF, and therefore D and are unknown initially, an iterative procedure isrequired in order to solve Eqns (4.31), (4.36), (4.37), (4.38), and (4.44) in order to determinethe composition of the equilibrium armour layer, as well as q,. This will be discussed in Section7.2.5.By inspection between Eqns (4.44) and (4.41) it is evident that the value ofq is given by:90O.00218(r ip)15qb = (4.45)(s—i) gfj/G[go g0(o,) a]4.5.2 Modification of the Parker Surface-Based Relation for Natural Rivers withVariable DischargeThe discussion and theory presented in this section is an extension of the work by Parker(1990).The “equal mobility” hypothesis was derived originally from flume experiments which wereperformed using single, steady discharges (Parker et a!., 1982b). In its original form it impliesthat equal mobility occurs for all discharges above the critical for armour mobility, and thereforeassumes that the grain size distribution of the transported bedload is equal to the subarmour forall flows in excess of critical. This is known to be incorrect as is indicated by data from OakCreek (Milhous, 1972) which is shown in Fig 4.11. Note that below the critical Q of about 1m3/sec the value of d50 is approximately 2 mm and represents sandy throughput load movingabove an immobile armour. For flows in excess of 1 m3/s the value of d50 increases withincreasing Q.In its original form the equal mobility hypothesis supposes that d50 of the transported loadwould be a constant for discharges in excess of the critical, and for Oak Creek this constantvalue would be equal to 20 mm. As Fig 4.11 shows this is clearly incorrect. This wasacknowledged by Parker et aL (1982a) who state that equal mobility is at best a crude, butuseful, first-order approximation.However this hypothesis can be correctly applied to natural channels with varying discharge byrecognising that for a channel in equilibrium, the constraint of equal mobility must be fulfilledover a significant duration. Parker (1990) acknowledges that the size distribution of the annual91yield of gravel is similar to the subarmour. Similarly Church et al. (1991), although referring tofine sediment, state that “equal mobility is at best a statistical phenomenon which holds over a(significant duration)”.The experiments of Parker et a!. (1982b) and others have demonstrated that the development ofan equilibrium armour layer even under controlled conditions requires many hours. The flows inmany natural rivers, in particular ‘flashy’ gravel-bed rivers, are probably too variable for anarmour layer to develop an equilibrium state with a single discharge. However over a significantduration, which is usually at least one complete water year, a channel which is in equilibriummust have transported the bedload imposed on it without appreciable net deposition or scour,and therefore must have transported all size fractions at just the rate at which they were suppliedto the channel. That is all sediment sizes are rendered equally mobile. Therefore the armourlayer which develops in an equilibrium channel represents the grain size distribution necessary todevelop equal mobility for the complete range of flows experienced by the channel over asignificant duration. However any single discharge is likely to be in disequilibrium with thearmour because the sediment size distribution of the sediment being transported by a given flowat any given time is probably different than the subarmour.For each value Q, a value of 1•bed• is calculated from the equations presented in Section 4.3.2,from which can be determined. The value is the volumetric sediment transport rate perunit channel width for flow Q,. Eqns (4.28) and (4.33) must be modified as follows:(s—1)gq,.= (4.46)‘ (r,/p) ‘J =O.OO218G[cF0g0(81)a] (4.47)92where qb13 is the unit volumetric bedload transport rate for size fractionj for flow Q,, and thevalues Vbedj, Io1, and Co., and G are not constant, but now vary with Q.By equating Eqns (4.46) and (4.47) and rearranging the following relation for qb11 is obtained:O.00218F.qb= (s—l)gp’5 G[3goj g0(s) a]i, (4.48)By definition:= Pi (4.49)where p is the probability of Q,.Therefore from Eqn (4.48) and (4.49):O.00218F.(s— 1)g p15 1v G[goj g0(s) a] z, (4.50)Rearranging (4.50) together with (4.40) gives the following relationship for 13 which isanalogous to (4.41):F — fJ(s—1)gp5q (451)— p1G[0. g(o) ,] 0.002 18Since the second term of Eqn (4.51) is constant (although the value is not necessarily known inadvance), with Eqn (4.43) the following equation analogous to (4.44) is obtained:93f/P G[sgoj g0(o) c]r.F. = ‘‘ (4.52)G[so. g0(o) w}rBy inspection Eqns (4.51) and (4.52), it is evident that the total volumetric unit bedloadtransport rate per unit bed width q, is given by:0.00218qb = n m(4.53)(s- 1)g p’5 G[sgo. g0(o)€]If the SI system of units is used, qb will have units ofm2/sec.4.6 TOTAL BEDLOAD CONSTRAINTThe total bedload constraint for the channel subject to the complete range of flows defined inthe flow duration curve (Fig 4.7) is represented by:(4.54)where g,, is the total bedload transport rate per unit channel width in mass units, and Gb is anindependent variable and represents the total load imposed onto the channel from upstream. Thevalue ofgb is given by:g=Ispq (4.55)where r is a constant which is related to the time base of Gb. The value of qb in the totalbedload constraint is calculated using Eqn (4.53). The units ofg and Gb must reflect the loadimposed over a significant duration. For instance under steady flow conditions such as in alaboratory experiment Gb can have units kg / s as the rate is constant and will be the same for alltime scales. In natural rivers with highly variable and seasonal flows the logical time base would94be one water year, and therefore Gb would have units of kg / y. In general the modeling to beundertaken in Chapter 7 will assume a time base or significant duration equal to 1 year.Therefore, the value Gb represents the mean annual bedload supplied to the channel, and F takesa value equal to 365.25 24 3600 which is the number of seconds in one year.4.7 SUMMARYThe bedload constraint has been developed and consists of two components: boundary shearstress and sediment transport. In this thesis, the mean bed and bank shear stress (bank shearstress is necessary to assess the stability of the banks, see Chapter 5) are estimated using theempirical relations of Flintham and Caning (1988). These formulae were derived fromexperimental data in straight rectangular and trapezoidal flumes, and they permit the estimationof the mean values of rb€d and bank for trapezoidal channels with different bed and bankroughness. Flintham and Carling caution that these results may not be confidently applieddirectly to large scale natural channels or for bank angles less than 45°. This warning isacknowledged, however in the absence of anything better, the relations of Flintham and Caningwill be assumed to be valid for large natural channels with low angle banks.The effects on the shear stress distribution of weak, straight-channel secondary currents areimplicitly accounted for in the empirical shear stress relations. The strong secondary currentswhich are associated with curved channels are not accounted for in this thesis. The secondarycurrents are considered to exert only a second-order influence on the channel geometry and willresult principally in variations of the local geometry from the reach-averaged values. There issome support for this assumption in the regime equations ofHey and Thorne who found that thechannel width at pool and riffles sequences varied by less than 5% from the reach-averagedvalue.The Parker (1990) surface-based bedload transport relation will be used in this thesis to modelthe sediment transporting capacity of the channel in Chapter 7. This bedload transport relation95was selected because it explicitly accounts for the influence of the armour layer on the sedimenttransport, and the relation can be inverted to calculate the equilibrium armour layer grain sizedistribution that is required to transport an imposed bedload for given flow conditions. Thisallows the grain size distribution of the armour layer, together with the value of D50 to betreated as dependent variables in the optimization model. Therefore 135 can adjust together withW, Y, and S to define an optimum channel geometry.The bedload transporting capacity of the modelled channel is calculated using the completerange of flows represented by the flow-duration curve, rather than at a single discharge such asQbf The concept of “equal mobility” is restated whereby the armour layer adjusts such that allsize fractions are equally mobile over a given duration such as one year. The Parker (1990)relation is modified to apply to the full range of flows.96(a)(b) V/\\\‘s:==E— — —— Velocity ContoursWeak Secondary CurrentsFigure 4.1. Simplified velocity distribution in a straight open channel. (a) Momentum diffusiononly. (b) Momentum diffusion and secondary currents.I I I II I II II — I gf I g‘ /‘ — /I———“-‘97(a)(b)VdPFigure 4.2. Area methods. (a) Normals to isovels (Leighly, 1932). (b) Normals to bed(Lundgren and Jonsson, 1964).98OwyS’0.75OwySQ.97OwyS1.0 —0.90,V‘.‘ 0.78C0•00. 02__________DFigure 4.3. Boundaiy shear stress distribution from Lane (1955b). For Fig 4.3 only, b isequivalent to Pbed, y is equivalent to Y, and z is a measure of the bank angle which is equal tocotTropezois, z 2z I 50,0.8Q.0.7CCa.;V0.”‘iffl‘Troezods,z2ondI.5—/ecton9esI1 —“‘Tropezoids Z—,J.___‘——...l4— [IvLAH::iz:- - TflTTII —2 3 4 67b/y8 9 0 _0 2 .5b/y7 8 9 1099(a) 10.6a)C.0 0.4U,Ca)Ec 0.2C0z0(b) 10.80.6Cucr.iCC(0)C0)E0.2C0z0Figure 4.4. Boundary shear stress variation from Eqns (4.8) and (4.9) for variable bank angle6 and uniform boundary roughness in a straight trapezoidal channel. (a) Non-dimensional bedshear stress = Vbed / yJ’S. (b) Non-dimensional bank shear stress = Tbank / rYS.wIY100(a)10.80.6a)a)o 0.4Cl)Ca)E130.2e0z0(b) 1h 0.8Cl)Ca)a)C0Cl)Ca)UC0z0Figure 4.5. Boundary shear stress variation from Eqns (4.8) and (4.9) for constant 8(8 = 90°)and non-uniform boundary roughness. (a) Non-dimensional bed shear stress = id / yYS. (b)Non-dimensional batik shear stress = z / yYS.5 10 15 20 25w/Y101140zCl)C,)a)120100806040200Total Shear Stress, r (N/rn2)Figure 4.6. Grain shear stress versus total shear stress. The value of r is computed from theobserved channel geometry, and r’ is calculated from Eqn (4.15).0 20 40 60 80 100 120 140102C)a)EFigure 4.7. Discretized Flow-Duration Curve. The probability of exceedence refers to aspecified time base and period of record. The value Q is the characteristic discharge valueassigned to the interval, and Pi is the probability of a discharge being within a particularinterval.0Probability of Exceedence103(a) 0.120.1 -D‘ 0.08-0.06 -00.04- 0DDO0.02- DCB0 I0 1 2 3 4Q (me/s)(b) -0.01 -0.001 -0.0001 -1E-05-1E-06-f1E-07 -°1E-080 2 3 4Q (m3/s)Figure 4.8. Bedload transport rates versus Q for Oak Creek, Oregon, USA. Data from Milhous(1972). (a) Linear Y-axis. (b) Logarithmic Y-axis.104(a) 500400300G20010000.5(b)1001010.10.010.0010.00011E-050.5Figure 4.9. Variation of the function G from Eqn (4.32). (a) Linear Y-axis. (b) Logarithmic Yaxis.1 1.5 2•sgO1 1.5 2øsgO1052COo1.5—1 2 3 41’sgOFigure 4.10. Variation of the functions a and with q5 (from Parker, 1990). See text fordiscussion.10630002500d subarmour2000E 0000005 000E0,0:00023 4Q (m3/s)Figure 4.11. Variation of d501 d with Q for Oak Creek, Oregon, USA. Data from Milhous(1972). The value ofd50 from the subarmour sediment is approximately 20 mm.107CHAPTER 5BANK STABILITY CONSTRAINT5.1 INTRODUCTIONIn this chapter the bank stability constraints which apply to gravel-bed rivers will be developed.It is an obvious requirement that a stable channel must have stable banks. The bank stabilityconstraint will be formulated for both cohesive and noncohesive bank sediments.There are two fundamental processes of bank erosion which must be considered: mass failureand the fiuvial erosion of discrete grains or aggregates (Wolman, 1959; Thorne, 1982;Gressinger, 1982).5.2 COHESIVE BANK SEDIMENTCohesive channel banks are those composed of clay, silt and fine sand and have typicallyresulted from over-bank deposition of fine sediment from suspension. Cohesive sedimentscontain significant amounts of clay minerals which strongly influence the physical properties ofthe soil. According to Raudkivi (1990, p. 300) when the clay content of a soil is 10% orgreater, the physical properties of the soil are dominated by the clay fraction.5.2.1 Mass FailureMass failure of a river bank occurs when the driving force FD, which is due to the weight ofthe failed soil mass w; exceeds the resisting force FR, which is a result of the cohesion c andinternal friction angle çS of the soil acting along the failure surface. The unit shear strength of asoil is usually given by the Mohr-Coulomb failure criterion:108SR=c+NRtanØ (5.1)where SR is the shearing resistance for a unit area of soil, and is the normal upward forceacting on the base of the soil unit.Potential failure will occur along the surface where the factor of safety, F,, which is defined bythe ratio FR /FD, is a minimum. Failure will occur when F, is less than one. The critical heightof a bank corresponds to F, = 1, and can be expressed by the following equation fromTerzaghi and Peck (1948):Hmt=N (5.2)where y is the bulk unit weight of the soil and includes the weight of the pore fluid, and N isthe stability factor. Note that N as presented in Terzaghi and Peck (1948) and in Eqn (5.2) isthe inverse of Taylor’s (1948) stability number. The value of N, is a fi.mction of and thebank angle 6.The simplest model of slope failure is the Culmann analysis (Taylor, 1948; Spangler andHandy, 1982) which assumes that the failure surface is planar and passes through the toe of thebank (Fig 5.1). This method yields the following analytical solution for N:4 sinO cosçSN= 1—cos(6—.) (5.3)The slope of the failed surface, a, is obtained by differentiation:6+ça=2 (5.4)(See Fig 5.1).109Application of the Culmann solution is limited. It can only be applied with reasonableconfidence to completely drained slopes close to vertical. For lower angle slopes the failuresurfaces are strongly curved and the Culmann analysis under predicts N5, and thereforeThis will be demonstrated subsequently.No simple analytical solution exists for non-planar failure surfaces. The friction circle method(Taylor 1948), or the various methods of slices (Fellenius, 1936; Bishop, 1955; Morgensternand Price, 1965; Spencer, 1967) all require trial failure surfaces to be assessed. The frictioncircle method has in general been superseded by the method of slices and is seldom used today.For the method of slices the soil mass above each trial failure surface is divided into a numberof vertical slices of convenient width (Fig 5.2(a)), and the driving and resisting forces areresolved for each soil slice (Fig 5.2(b)). The factor of safety is established for that particularsurface by summation of the forces acting on all of the slices. A number of trial failure surfacesare examined until the surface with the lowest value of F5 is determined. If the value ofF3 isless than 1 the slope will fail.The two most commonly used methods of slices are the ordinary or Swedish method(Fellenius, 1936) and the Bishop simplified method (Bishop, 1955). For these and othermethods of slices, the number of unknowns exceeds the number of equations available forsolution and therefore some simplifying assumptions are necessary. In the ordinary method ofslices the vertical shearing forces between the slices are neglected, and the resultant force on aslice is assumed to act parallel to the base of the slice. This can lead to an under estimation ofthe factor of safety by as much as 60% (see Nash, 1987). In Bishop’s approach the verticalshearing forces between adjacent slices are considered in the analysis and are assumed to beequal and opposite. A key feature of the Bishop simplified method is an increase in the normalforce at the base of each slice which tends to compensate for the side forces on the slices.110The solution for the factor of safety using the ordinary method of slices and assuming acircular failure surface in terms of effective stresses is given by:FR (c’l+(wcosa—crl)tanç5’)SF wsinawhere o- is the mean pore pressure for the slice, and 1 is the basal length of each slice. The soilstrength parameters c’ and çS’ are given in terms of the effective stresses. The solution for theBishop simplified method is more complex and requires an iterative solution for F3.The ordinary and Bishop methods can also be applied to non-circular failure surfaces. Furtherdiscussion of slope stability methods can be found in soil mechanics texts such as Chowdhury(1978), Spangler and Handy (1982), or in a review by Nash (1987).The various methods of slices, as well as other more recent limit analysis methods whichrequire numerical techniques such as finite element analysis for solution (eg. Chen 1975; Chenand Lui, 1990), can be used to construct stability curves which show the variation ofN5 with &Examples of stability curves are shown in Fig 5.3(a). Those solutions indicated by the solidlines were derived from the graphs and tabulated data published in Taylor (1948, p.457) whichwere obtained using the friction circle method. The dashed lines are values of N from theCulmann solution, Eqn (5.3). For slopes close to vertical the Culmann solution agrees quiteclosely with the results of the friction circle method, however the two solutions diverge forlower values of 6The friction circle method agrees closely with results obtained from the ordinary method ofslices for simple failure geometries with homogeneous soils and no seepage (Taylor, 1948).However the friction circle method cannot be applied readily to more complex geometries,heterogeneous soils, or where seepage is significant, and therefore the method is not widely111used today. However the stability curves developed by Taylor (1948) can be applied to somespecial cases and these will be discussed in a following section.With the stability curves the value of can be calculated for any value of 9 and particularsoil values y c’ and ‘. Alternatively stability curves of versus 9 can be constructed forknown values of y,, and c’ as in Fig 5.3(b).To develop stability curves for an applied river bank analysis, field investigations are necessaryto determine the bank stratigraphy, the properties of the bank sediments, and the likely shapeof the failure surface. Also other variables such as the presence of surface tension cracks andthe influence of bank vegetation on the stability of the bank can be considered. Groundwatermonitoring might be necessary to determine the effect of pore pressures and seepage forces.Construction of the stability curves may require considerable effort, however once available thecurves can be approximated by empirical equations or piece-wise linearized segments, and thenthis information can be accessed quickly by the optimization model. This is necessary as theoptimization model must assess the stability of hundreds or thousands of H-9 combinationsbefore arriving at the optimum solution.5.2.1.1 Bank Stability and SubmergenceThe effect of submergence and emergence of the channel banks due to flow variability plays akey role in the stability of the channel banks. Bank submergence results in increased soil weightdue to saturation, and a decrease in the friction resistance due to increased pore waterpressures. These two effects may combine to reduce the stability of the banks. This decrease inbank stability may however be countered by increased hydrostatic loading along the banksurface.Three limiting cases will now be considered that can be analysed using the stability curvesdeveloped by Taylor (1948). These cases are shown in Fig 5.4 and will be used to demonstrate112the effect of submergence, saturation, and emergence on the stability of a river bank. Thevalues of c’ and ‘ to be used in the following discussion are assumed to equal the developedvalues, that is the factor of safety applied to the values c’ and qS’ equals 1, and the values c’ andare assumed to be known exactly.Case I corresponds to a condition of zero or low flow in the channel, together with drainedbanks. The value of N3 can be determined directly from the stability curves for a particularvalue of 0, and Hcrjt calculated for the known values of y and c’.Case II corresponds to that of a fhlly submerged bank. The soil mass is fully saturated and isbeing supported by buoyancy forces about its perimeter as indicated in Fig 5.3. For this casewhen the soil is fully saturated, and there is no seepage, the buoyancy forces are hydrostatic.The value ofN3 can be determined directly from the stability curves, and then calculatedusing the buoyant unit weight, Yb’ which is equal to saturated Yt minus y.Case III corresponds to a condition following complete and instantaneous drawdown of theflow in the channel. In this case the soil in the bank remains saturated, the pore pressuresunchanged from the fully submerged case. However the hydrostatic forces which werepreviously acting to stabilise the bank are now absent. Taylor (1948) determined that the valueofN for case III could be approximated by modifying the value of ‘to account for the porepressures acting along the failure surface which reduce the frictional resistance. It wasdetermined by Taylor (1948, p. 467) that the modified friction angle 9S,,, can be approximatedby:(5.6)Where y1 is the saturated value. This relation, which applies only to complete and instantaneousdrawdown, has been confirmed by Morgenstern (1963) who developed stability curves for113complete and partial instantaneous drawdown using the Bishop simplified method of slices. ForCase III the value of N., is determined from the stability curve corresponding to qSm, andcalculated using y1.The values ofN., and 11cr 1 will now be determined using reasonable soil values to demonstratethe relative stability of the three cases. A value of q’ = 25° together with the drained Yt = 20.80kN/m3, saturated y = 22.76 kNIm3,and Yb = 12.95 kN/m3 for 0= 700. From Eqn (5.6) thesevalues give a value of Øm = 14.2°. The values for ]V are read from the stability curve fromTaylor (1948) in Fig 5.3a. The results are presented below in Table 5.1.Table 5.1. Parameters for 3 cases of bank stability.Parameters Case I Case II Case IIIYr OT Yb (kN/m3) 20.80 12.95 22.76qY (°) 25.0 25.0 14.2c’ (kN/m2) 10.0 10.0 10.0M, 9.8 9.8 7.2(m) 4.71 7.56 3.16The values of in Table 5.1 indicate the role of submergence and emergence on the stabilityof the channel banks. Note from Case II that the effect of increasing the flow depth is tostabilise the channel banks and increase Therefore the limiting condition for bank stabilitymust be satisfied at low or zero flows. Cases I and ifi defined upper and lower limits for thebank stability. The critical period for bank stability is during flow recession. Case I correspondsto a river with very slow runoff response and/or very permeable bank sediment. As the flowrecedes the banks must be dewatering so that the bank above the water surface in the channelis fhlly drained, and the bank below the water surface fi.illy saturated. This continues until zero114flow condition and the banks are totally drained (Case I). The other limiting conditioncorresponds to a river with flashy runoff and/or highly impermeable channel banks. The flowrecedes at a rate at which no dewatering of the bank sediment occurs and the banks remainfl.illy saturated at the zero flow condition (Case III). The stability of natural rivers would liebetween the two limiting cases (I and III).5.2.1 2 Bank Height ConstraintThe bank height constraint requires that the bank height H, be less than or equal to the criticalbank heightHH (5.7)where H is equal to the bankfull flow depth 1’, the value for is calculated by Eqn (5.2) withN obtained from the stability curve. The value of should be evaluated for both Case I andCase III conditions. Examples of stability curves derived from field studies can be found inMarch et a!. (1993) and are shown in Fig 6.9. The stability curves and analysis of March et a!.are discussed in Section 6.4.5.5.2.2 Fluvial ErosionFluvial erosion is the process whereby individual grains or aggregates on the surface of thebank are removed and entrained by the flow. The driving force is the shearing action of thefluid on the bank sediment. The mechanics of fluvial erosion of cohesive sediment remainpoorly understood despite considerable research. Reviews of the research conducted on theerodibility of cohesive sediment are found in ASCE (1967), Graf (1971), Gressinger (1982),and Raudkivi (1990). The resultant interparticle force for cohesive sediment is the net result ofseveral forces of attraction and repulsion. These forces result from very complex electrochemical processes which include the clay mineralogy and content, and the temperature andchemistry of the pore and eroding fluids (Arulanandan et a!., 1980; Gressinger, 1982;115Raudkivi, 1990). Unlike noncohesive sediment, the weight component of the cohesive grains isquite insignificant when compared to the electro-chemical forces. The electro-chemical forcesmay be orders of magnitude larger than the weight forces of the individual grains (Raudkivi,1990, P. 310).Numerous experiments to determine the erodibiity of cohesive sediment have been performedusing a variety of techniques such as straight and circular flumes, rotating cylinders, submergedjets, impellers, and even portable flumes that can be used to test undisturbed soil in situ.Attempts to quantify the fluvial erosion of cohesive sediment often use the concept of criticalshear stress, critical velocity, or critical stream power which must be exceeded before erosioncommences.Attempts have been made to use the results from these erosion studies to develop correlationsbetween the erodibility of the cohesive sediment and other primary physical properties of thesoil which are more readily obtained. These physical properties include clay content, claychemistry such as the Ca/Na ratio and pH, grainsize, vane shear strength, plasticity index,organic matter content, as well as the temperature and chemistry of the eroding fluid. Ingeneral this approach has not been overly successful and has not yielded results which could bereadily used for predictive purposes. However the approach of Arulanandan et al. (1980) hasproduced charts which relate soil erodibility to the electrical conductivity and the magnitude ofthe dielectric dispersion of the soil, properties which reflect the type and amount of clay. Itaddition it was shown that the chemistry of the eroding fluid, expressed as the sodiumadsorption ratio, also significantly influences the soil erodibility. According to Arulanandan etal. (1980), while this study “did not yield a quantitative method to predict critical shear stress”,the charts “should give a reasonable estimate of r for a natural undisturbed soil”. Notethat the same soil can have different values of r for different eroding fluids, and therefore thevalue of corresponds to a particular soil-water system.116The critical shear stress concept will be adopted in this thesis. The critical shear stress of aparticular soil ;,,, can be evaluated directly using one of the erosion devices mentionedpreviously, preferably on undisturbed in situ samples, or it can be estimated indirectly bymethods such as Arulanandan eta!. (1980).5.2.2.1 Bank Shear ConstraintThe bank shear constraint requires that the bank shear stress be less than the critical valuerequired for fiuvial erosion of the bank sediment:;aflk (5.8)where the quantity Vbank is calculated from Eqn (4.8). This constraint assumes that the gravitycomponent of the eroded grains or aggregates is negligible compared to the cohesive forces,and therefore the value of z,rjt is independent of 6 and can be considered to be an independentproperty of the soil-water system.The assumption that a soil water system has a single definable value of belies thecomplexity of the erosion process. For example freeze-thaw action, ice wedging, anddesiccation and tension cracks can all result in aggregates of cohesive soil which may have asignificant gravity component and a much lower value of r than a smooth homogeneous soilmass. Nonetheless for modeling expedience it will be assumed that a soil-water system can becharacterised by a single value of z which is treated as an independent variable.5.3 NONCOHESIVE BANK SEDIMENTAs with cohesive sediments both mass failure and fiuvial erosion will be considered fornoncohesive sediment.1175.3.1 Mass FailureFor a bank composed of noncohesive sediment the resisting force that balances the drivingforce is due only to the friction term in Eqn (5.1). It is an elementary exercise to show that thefactor of safety for an infinite, drained slope of noncohesive sediment is given by:F = tanq5 (5.9)tan8from which the limiting stability occurs when the bank angle 8 is equal to the angle of internalfriction.Therefore for a bank to be stable the following constraint must be satisfied:8qS (5.10)If 8 exceeds 0, bank failure will result.When a bank is fully submerged as in Case II from the previous section, the stability isunchanged from the fully drained case as the buoyancy forces reduce the driving force and theresisting force by the same amount. For a condition of rapid drawdown seepage forces candevelop which reduces the stability of a noncohesive bank. Since only frictional resisting forcesare involved, the decrease in bank stability due to seepage forces can be simulated numericallyby modifying (ie. reducing) the value of ç. For the case where seepage is parallel to the slope,the modified value is Om is given by: (eg. see Spangler and Handy, 1982; p. 492):tan0m=tan0 (5.11)‘FtIn this case the value of is analogous that obtained by Taylor (1948) for cohesive sediment(Eqn 5.6).118However in many banks composed of noncohesive sediment, particularly those formed ofcoarse gravel, significant seepage forces will not develop due to their high permeability. Theriver with noncohesive banks will tend to behave as in Case I for cohesive banks, and Eqn(5 10) is sufficient to assess the bank stability with respect to mass failure.5.3.2 Fluvial ErosionIn flowing water the fluid forces exerted on the grains as well as the down slope gravitycomponent of the grains must be resisted by the frictional forces for a grain to remain stable.The well-known bank stability analysis described below was originally developed by the UnitedStates Bureau of Reclamation (USBR) to determine the stability of noncohesive banksediment. Its development is summarised in Lane (1955b) and it is presented in the followingsimplified form from Chow (1959, p.171) and Henderson (1966, p.419) for the limiting bankstability:Tbaflkc=—2 6 (5.12)Tbedc J sin2Øwhere rbaflkc is the critical shear stress for a grain located on a sloping bank, and Vbed is thecritical shear stress for the same grain located on the sub-horizontal channel bed.The original equation can be rewritten in a dimensionless form as follows:Vbank I S1fl6( _in 2 (5.13)r\s sinwhere s is the specific gravity of the sediment, D5Obaflk is the median bank grain diameter whichis assumed to be representative of the bank sediment, bed is the critical dimensionless(Shields) shear stress for a grain equivalent to D50bank on the channel bed.119Unlike the mobile bed sediment, the gravel comprising the channel banks may be consolidated,cemented by fine silt and clay, and stabilised by roots which have penetrated the gravel banks.Eqn (5.13) was developed for unconsolidated noncohesive sediment and must be modified toaccount for the properties of the bank sediment.USBR data (in Lane, 1955b) indicate that qS for coarse gravel approaches a maximum ofapproximately 400. However consolidation and imbrication of the bank sediments, togetherwith the effect of cohesive silt and clay cementing the grains and other stabilising influencescan increase the in situ friction angle above 0. Therefore çS in Eqn (5.13) can be replaced by thein situ friction angle, Ø, which can be allowed to take a maximum value up to 900.The value of t*bedc must also be adjusted because the critical shear stress for the stabilisedbank material would be higher. Henderson (1966, p. 414) has shown from the work of White(1940) that the dimensionless shear stress is related to 0:r=ktan (5.14)where k is an empirical constant. The value of k will be determined from field data in Chapter6.To include the influence of bank vegetation, consolidation, and cementation of the banksediments, 0 is replaced with Substituting Eqn (5.14) into (5.13), and using , theresulting constraint for noncohesive banks is:bank ktanqS1ll— Sifl 6 (5.15)sin120Note that Eqn (5.15) is presented as an inequality constraint with the mean bank shear stress rbank’ rather than ;ank in the numerator on the left hand side. The value 6,, can take a maximumvalue up to but not including 900, as tan 90° is infinity.By analogy Eqn (5.10) should be written in terms of the in situ angle of repose:(5.16)Eqn (5.15) automatically satisfies Eqn (5.16) for all values of çb< 90° because the square-rootterm becomes undefined for values of 9>. Eqn (5.15) therefore simultaneously constrainsthe bank stability with respect to both mass failure and fluvial erosion, unlike the noncohesivebanks where two separate constraints are required.The influence of the bank stability on the channel geometry will be demonstrated in Chapter 6.5.4 SUMMARYThe bank stability constraints were developed for both cohesive and noncohesive banksediment. For both sediment types there are two mechanisms of bank erosion that must beconsidered, namely mass failure and fluvial erosion of individual particles.For cohesive sediment the stability of the bank with respect to mass failure can be assessedusing slope stability techniques. To incorporate the slope stability data into the optimizationmodel stability curves are constructed. These curves can be accessed rapidly by theoptimization model. The banks are most susceptible to failure at low or zero flows when thehydrostatic supporting force from the flow is absent. Two limiting cases are recognised, fullydrained, and fully saturated bank conditions. For bank stability the cohesive bank sedimentmust also be stable with respect to fluvial erosion. The concept of critical shear stress forcohesive sediment is used. The bank shear constraint requires that the value of Tbank be less121than or equal to the critical shear stress for the bank sediment. The value ;ank represents themean bank shear stress. In practice it is the maximum bank shear stress that would likely occurnear the toe of the bank, rather than ank that would determine the bank stability with respectto fluvial erosion. Nonetheless ;ank will be used.To assess the bank stability of noncohesive bank sediment the USBR bank stability presentedby Lane (1955b) will be used in a modified form. The original equation is modified to reflectthe increase in the stability of the bank sediment due to packing, consolidation, imbrication,cementing by fines, and binding by root masses. This increase in stability is reflected in the insitu friction angle,that can take a value up to 900. The stability of noncohesive banksediment with respect to both mass failure and fluvial erosion is assessed by the single USBRrelation except in the case of large seepage forces following rapid drawdown. The influence oflarge seepage pressures in noncohesive sediment will not be considered as the high hydraulicconductivity of these sediments, particularly for coarse gravel sediment, would facilitate rapiddraining of the bank.Gravel-bed rivers commonly have channel banks that are composite in nature and typicallyhave a lower noncohesive unit which is overlain by an upper cohesive unit. In the case ofcomposite banks it must be determined which of the layers is controlling the bank stability.Often the lower noncohesive unit is eroded by fiuvial activity, and the overlying cohesive layerthen undergoes mass failure as it becomes undermined (Thorne, 1982). In this example itappears that the stability of lower noncohesive unit determines the stability of the bank.The influence of bank stability on the channel geometry will be examined in Chapter 6.122Figure 5.1. Bank stability analysis for planar failure surface.HFD = wsina= cL + NRtan123(a)(b)Figure 5.2. Bank stability analysis for the method of slices. (a) Definition sketch. (b) Forcesresolved on a single slice. The lateral forces not indicated.HL124(a) 252Oa,.0Ez 15.01o5(b)4Figure 5.3. Stability curves used to determine Hrit. (a) Stability number as a function of (b)Critical height as a function of Ofor prescribed values of c’ and rt.8(i)1086220 40 60 80e ()125Case IPore Water PressuresHydrostaticPressuresFigure 5.4. Three special cases for bank stability analysis. (a) Case I: Drained banks. (b) CaseII: Fully submerged and saturated. (c) Case ifi: Instantaneous and complete drawdown, bankfi.illy saturated. The hydrostatic and pore water pressures are indicated.Case IIACase III126CHAPTER 6EFFECT OF BANK STABILITYON CHANNEL GEOMETRY6.1 INTRODUCTIONIn Chapter 6 simplified versions of the optimization model will be presented whereby thesediment transporting capacity of the channel will be calculated at the bankfhll discharge only,and rather than the modified Parker (1990) relation developed in Chapter 4, a simple bedloadtransport relation will be used. This simplified model is inadequate to assess the response of anatural channel to variations in the discharge or sediment supply, but is sufficient to investigatethe influence of the bank stability on the channel geometry, and to introduce the optimizationmethodology to be fully developed in Chapter 7.6.2 BANKFULL : fiXED-CIIANNEL-SLOPE OPTIMIZATION MODELIn the bankfiill model the natural variation in the flows is reduced to a single dominant orchannel-forming discharge which is assumed to be equal to the bankfull discharge, Qbf Thesediment transporting capacity of the channel is indexed by the sediment transport ratecorresponding to the bankfull discharge, Gbf This simplification has been used by Chang127p(1980), White et a!. (1982), Millar (1991), and Millar and Quick (1993 a, b) in theiroptimization analyses of gravel rivers, and Hey and Thorne (1986) in their empirical regimeanalysis. The limitations of this approach with respect to calculating the sediment transportingcapacity of the channel were discussed in Chapter 4. The bankfiull model is however adequateto investigate the effect of the bank stability on the channel geometry.The banlcfull model will be developed for both noncohesive and cohesive channel banks. In thefirst version a further simplification will be made in that the channel slope S will be treated as anindependent variable. This is done to assess the effect of the bank stability on the values of Wand Y without the interference of varying channel slope. In Section 6.5 the model will begeneralised to include variable S.The bankfull:fixed-channel-slope model is analogous to an experimental setup where the slopeis fixed, and the channel width, depth, and sediment transport rate adjust to the imposeddischarge (eg Wolman and Brush, 1961). The model is formulated below.6.2.1. Independent VariablesThe variables that are assumed to be independent with known values are Qb S, d50,D50, k3bj,the bank stability variables which are D5Oba and ç for noncohesive banks, and c’, ‘, y,and for cohesive banks, and the following which are generally constant: g, p, s, v(kinematic viscosity), and y Since the value of S is specified, the value of Gb.,- is not required inthis formulation.1286.2.2. Dependent VariablesThe primary dependent variables to solve for are Pbed, Pbank, and 6 From these primaryvariables the secondary dependent variables such as j U Rh, W, Y, gb, and others can bedetermined.6.2.3. Objective functionThe objective function in this formulation is given by:maxf(,1,6)=77 (6.la)where:(61b)pQbfSAs the values of p, Qb, and S are prescribed, maximization of i is equivalent to themaximization of the product Pbed times gb. The value ofgb will be calculated using the Einstein-Brown sediment transport relation as presented in Vanoni (1975, p. 170). This relation requiresonly the value ofd50 to calculate gb:* 12.15 exp(_0.391 / <0.093gb (6.2a)L4o r 0.09350 50where gb*= dimensionless bedload transport rate per unit width given by:129* gbgb=1 ps,.J(s—1)gd50The dimensionless bed shear stress for the median bedload-grain diameter is given by:*— Tbed 62‘Cdr(sl)dCThe value of ‘Cbed is calculated from Eqn (4.9).The parameter F1 is related to the fall velocity of the sediment d,0 and is given by:12 36v2 I 36v21j3gd503(s—1) 1Jgd503(s—1) (6.2d)where v = kinematic viscosity of water. For gravel-sized sediment, F1 is approximately equal to0.82.6.2.4 ConstraintsThe only two constraints that are required in this formulation for uniform flow in a prismaticchannel are continuity and bank stability. As the value of S is prescribed the bedload constraintis not required.1306.2.4.1 ContinuityThe continuity constraint determines the dimensions of a channel which can just convey theimposed QbfThe continuity constraint for uniform floe in a prismatic channel is defined as:UA=Qbf (3.24)where U is the mean velocity, and A is the cross-sectional area of the flow. The value of U isobtained from the Darcy-Weisbach equation:I8gRSU=f (3.25)The value of the friction factorfis calculated using Eqn (3.19).6.2.4.2 Bank Stability ConstraintThe single bank stability constraint for noncohesive bank sediment is given by:Tbank I SiflOr(s—l)D501,ktanq1_Sfl2 (5.15)131where the value of Thank is given by Eqn (4.8). The value of the constant k is to be evaluatedfrom field data in Section 6.3.1.For cohesive bank sediments the bank stability constraints consists of the bank-height and thebank-shear components:HHmt (5.7)Tbank r, (5.8)Both constraints must be satisfied for cohesive banks to be stable.6.2.5. Optimization Scheme: Fixed-Channel-SlopeThe optimization flowchart is shown in Fig 6.1 and the source code for the computer programis given in Appendix A. The program is a step-wise iterative procedure that varies only onedependent variable at each stage. The program is initialised by selecting trial values of the threeprimary dependent variables, Pbed, Pbank, 0. The continuity constraint is satisfied by varying thevalue OfPbank for trial values OfPbed and When the continuity constraint has been satisfied forQbj the dimensions of the trial channel have been established.The bank stability is then assessed. The maximum value of 6 for which the banks are stable,6,,, is determined. Other values of 0 less than 0,,, may also satisf’ the bank stability132constraint, however these lower values do not represent the optimum bank angle. Fig 6.2shows the typical variation of Tj and r,ank for a range of values 6 for which the continuityconstraint has been satisfied. Note that the value of rd decreases monotonically withdecreasing 6 Therefore while values of 6 less than 9,, may satisfy the bank stability constraint,these values are associated with lower values of r, and is therefore the channel is less efficientwith respect to transporting sediment than a channel with a bank angle equal to O,,,. The finaloptimal solution must have a value of 8 that corresponds to a value of O,,,. Trial values of 8are assessed until 6m is determined. For each trial value of 6, a new value ofPbank must alsobe determined which satisfies the continuity constraint.Once the value of 8,,, has been determined, the sediment transport efficiency i is assessed. Thevariation of i over a range Of Pled values for a prescribed value of S is shown in Fig 6.3. Eachpoint on the solution curves in Fig 6.3 satisfies the continuity and bank stability constraints. Thevalue of i is reduced to zero either when Pbed equals zero, or when Pbed is very large and Ybecomes very small, and the value of r,,ed, and hence gb, both approach zero. The optimal valueof i is located between these two extremes, and is found by varying Pbed.The bisectrix method was found to be the most suitable for satisfying the constraints andlocating the optimum. This method requires that the initial upper and lower bounds of thesearch be initially specified for the variable in question. The midpoint between these twobounds is evaluated and, depending upon the outcome, becomes the upper or lower bound forthe next stage of the search, thus reducing the search area by half For example to satisfy the133continuity constraint upper and lower values of Pbank are established which are sure to containthe final value of Pbank. The midpoint between these two values is evaluated. If the calculateddischarge capacity of the channel at the midpoint is greater than Qj the midpoint Pbank value istoo large, and this value then becomes the upper bound for the next stage. Conversely if thedischarge capacity is less than Qb1 the midpoint value of Pbank then becomes the lower boundfor the next stage. Convergence is obtained when the calculated discharge capacity falls withina selected tolerance of Qb1 say ± 0.1%.A similar procedure is used to evaluate the optimal value ofPbed. The first derivative dr7 / dPbedis evaluated numerically by cental differencing at the midpoint between the upper and lowerbounds to Pbed. For values of di7 / dPd> 0 the midpoint becomes the lower bound, and forvalues ofdi7 / dPd <0 the midpoint becomes the upper bound for the next stage. Convergenceis attained when the separation between the upper and lower bounds of the search is reducedbelow a preset limit, say 0.01 m.The bisectrix method was found to be the most robust technique for obtaining convergence,although it is computationally intensive. Other convergence schemes such as the NewtonRapson method were generally more computationally efficient, however they were prone toinstability and occasionally caused the program to crash by returning negative values for thedependent variables.This iterative search scheme assumes that the functions are well behaved with no local optima.134This is a necessary requirement for this type of iterative search procedure, as well as othernonlinear optimization techniques such as the reduced gradient method, which can get caughtup in local optima. At a local optimum all of the convergence criteria can be satisfied and yet itnot possible to determine whether a local or global optimum has been found. Different optimalsolutions can be obtained from different starting points for the search procedure. The variationof the value of the objective function i with Pbed for the fixed slope bankfull model is shown inFig 6.3. The objective function is smooth with only a single global optimum. Evidence of localoptimum has not been observed in any of the modelling undertaken.In addition to , the variation of 6 with Pbed is also shown in Fig 6.3. The values of 6 are smallfor low values OfPbed because the high shear stresses acting on the banks. As Pbed increases, thevalue of Tbank decreases and banks with higher values of 6 are stable. In the example used in Fig6.3 the value q = 400 was used. As Pbed becomes very wide the value of 6 approaches .The optimization model will now be tested using observed gravel river data from channels withnoncohesive and cohesive bank sediment.6.3 NONCOHESWE BANK SEDIMENTThe analysis to be presented in this section is similar to that described in detail in Millar andQuick (1 993b), however there are significant differences which result in differences between thenumerical values obtained herein, and those from Millar and Quick. The general conclusions arehowever unchanged.135The model will be tested on the published gravel river data collected by Andrews (1984) andHey and Thorne (1986). The published data will be used as input to the model, and the outputgeometry will be compared to the observed geometry. The relevant data and modeffing resultsare tabulated in Appendix D.These rivers are described as stable single-thread channels with mobile beds that activelytransport sediment at higher stages. The banks are either comprised of noncohesive gravelsimilar to the material being transported, or have composite banks with a lower noncohesiveunit, overlain by a cohesive silty unit. For the composite banks it will be assumed in this analysisthat the stability of the banks is determined by the lower noncohesive unit. The banks arecharacterised in terms of their vegetation density. Andrews (1983) subdivided his data set intothose channels with thin and thick vegetation. Similarly, Hey and Thorne (1986) subdividedtheir data into four bank Vegetation Types ranging from Vegetation Type I, grass with no treesor bushes, to Vegetation Type IV, with> 50% trees and bushes.The empirical regime analyses of Andrews (1984) and Hey and Thome (1986) determined thatthe effect of the bank vegetation was to increase the stability of the channel banks whichresulted in narrower channels for the same value of Qbf In the first stage of the present analysisonly the channels which have low densities of bank vegetation will be assessed. It will beassumed that the bank stability of these channels is determined by the bank sediment propertiesalone, and is unaffected by the bank vegetation. This preliminary analysis will permit the136estimation of the bank stability parameters. In subsequent analyses (Section 6.3.2 and 6.3.3) thechannels with higher densities of bank vegetation will be examined and the effects of increasedbank stability on the channel geometry due to the bank vegetation will be demonstrated.The values of several key parameters were not explicitly given in the data sets. The value ofd50was not given by Hey and Thorne, and as an approximation D50 I 3 will be used. Thisapproximation will affect the absolute value of g calculated for the channel, but will notinfluence the location of the optimum significantly.Neither Andrews nor Hey and Thorne specified the values of D50k. In this analysis the valueOfD5Oba, will be assumed to equal D50, the median grain diameter of the armour layer sediment.Banks composed of sediment similar in size to the transported bedload sediment would beexpected to have a value of D5Oba,,k similar to d50, the median bedload or subarmour graindiameter. However previous work (Millar, 1991; Millar and Quick, 1993b) has shown that verypoor results were obtained using D5ob = d50, and much better results were obtained usingD5Qba = D50. This is consistent with the development of a coarse static armour layer on thechannel banks as the finer sediment is preferentially removed during bank erosion.Alternatively, the banks may be stabiised through the accumulation of coarse gravel at the toeof the bank.The assumption ofD5o0,g = d50 will be shown to be valid in Section 6.3.1.137The relative roughness values of the bed and banks are not known. Therefore since the bed andbanks are composed of similar sized gravel, the values of ksd and ks,flk will be assumed to bethe same, and both equal to the value of k which was calculated from the observed channelgeometry by inverting Eqn (3.2).6.3.1 Low Density Bank VegetationThe channels which will be analysed in this section are the channels described as having thinbank vegetation from Andrews (1984), and the Vegetation Type I from Hey and Thorne(1986). There are a total of 27 channels.The bank stability constraint, Eqn (5.15) contains two stability parameters: k which is related tothe critical dimensionless shear stress V*be4 by Eqn (5.14), and the insitu friction angle of thebank sediment.A value of çS, = 400 will be used as input which assumes that the banks arecomprised of loose, noncohesive gravel. A wide range of values for VCbe are cited in theliterature, with most faffing between 0.03 (Neil, 1968) and 0.06 (Egiazaroff 1965). For a valueof ç = 400 this translates into a range of values for k between 0.036 and 0.072. The value of kwithin this range that gives the best agreement between the observed and modelled channelwidths for the 27 channels will be determined, and then assumed to be a constant for allchannels.The model was run for a range of values of k to determine the best fit between the observedand modelled channel widths. The optimum value was determined to be k = 0.048 which is138equivalent to a value of Vbe4 = 0.04, for the value Ø = 400.The value of k obtained in the preceding analysis supports the assumption that D5Oba D50,rather than D5obQ,,k d50. Typically in gravel-bed rivers the value of 1),o of the armour layer isapproximately three times d50 from the subarmour sediment. If the assumption that Dsob d50is valid, then the value of k required to force an agreement between Wobs and would beapproximately three times as large than the value that is obtained when using D,0= D50. Thisvalue of k would correspond to a value of equal to about 0.12, which is well outside therange of values usually reported in the literature.Comparisons between the modelled and observed channel widths and mean depths are shown inFig 6.4(a, b). The agreement is good, the mean value of W0b I Wmod is 1.00 together with acoefficient of variation of 28.4%, and the mean value of Y0b/ Ymod is 1.05, and the coefficientof variation is equal to 14.5%.6.3.2 Effect of bank vegetationThe model was run with = 40° for the remaining 58 channels; those described as havingthick bank vegetation by Andrews, and the Vegetation Type II- IV channels from Hey andThorne. Reach 9074800 from Andrews’ data set was excluded from the analysis because astable channel width could not be obtained with çz5 = 40°.The results including the channels with the low density of bank vegetation are shown139graphically in Fig 6.5(a, b). Considerable scatter is evident away from the line of perfectagreement. Note the strong asymmetry in the scatter. The modelled channels are consistentlywider and shallower than the observed channels. The tendency for the modelled channels to bewider and shallower than the observed channels increases with the density of the bankvegetation. The channels with thin bank vegetation and those classified as Vegetation Type Iare scattered symmetrically about the line of perfect agreement as they were forced to do by theselection of the value for k in the previous section. The channels classified as having thick bankvegetation, and the Vegetation Type IV channels tend to lie furthest from the line of perfectagreement.The values of the ratios W0b/ Wmod and Yobs / Ymod for each Vegetation Type are summarisedin Table 6.1. Despite the wide scatter within each group it is evident that these two ratios showa systematic variation with the density of the bank vegetation with W0b/ W,,10d decreasing, andY0b3/ Ymod increasing with increasing bank vegetation density.Table 6.1. Ratios between the observed and modelled values of W and Y for fixed-slopeoptimization model with q = 400.Vegetation W,b / W I Y / Y, , No. ofType Mm Mean Max Mm Mean Max RiversI 0.64 0.97 1.77 0.70 1.05 1.29 13II 0.48 0.83 1.34 0.83 1.13 1.50 16III 0.28 0.72 1.16 0.90 1.28 2.05 13IV 0.25 0.59 1.00 0.98 1.48 2.16 20Thin 0.74 1.03 1.44 0.74 0.99 1.16 14Thick 0.17 0.58 0.82 1.11 1.31 1.44 9140The results obtained from the optimization modelling will now be compared to the those resultsobtained by Andrews (1984) and Hey and Thorne (1986) using empirical regime analyses.These researchers used regression analyses to derive empirical regime equations for thehydraulic geometry. Separate equations were obtained for each bank vegetation density. Thewidth equations were of the form:W=aQb/ (6.3)where Qbf is the bankfull discharge in Hey and Thorne, and is the dimensionless bankfulldischarge in Andrews. The exponent b shows little variation and typically takes a value ofapproximately 0.5. The coefficient a was found to vary with the density of the bank vegetationbecoming smaller as the bank vegetation density increased.The ratios formed by dividing the coefficient a from each of the regime equations, by thecoefficient a from the equation which represents the channels with the lowest density of bankvegetation is equal to W0b3 / Wunveg. where the subscript unveg refers to the unvegetated channelwidth. The ratio W0b / Wunveg is an index of the influence of the bank vegetation on the channelwidth. For example the value of a for the Vegetation Type I channels of Hey and Thorne is4.33, and the value of a for the heavily vegetated Type IV channels is 2.34. Therefore theVegetation Type IV channels are on average 2.34 / 4.33 = 0.54 times as wide as the weaklyvegetated Type I channels.141The ratio W0b / Wunveg is directly analogous to the preceding ratio W0b3 / Wmod where Wmod isthe calculated width assuming qS = 400. The values of W0b / Wunveg for the Andrews’ data setand the Hey and Thorne data set are summarised in Table 6.2 together with the values for W0bI Wmoj obtained from the modelling in this thesis with = 40°. Clearly there is reasonably goodagreement between the results of the optimization modelling with = 40°, and the empiricalregime analyses of Andrews and Hey and Thorne.Table 6.2. Summary of ratios of observed channel width divided by the unvegetated channelwidth. These ratios are an index of the effect of bank vegetation on the channel width. Thevalues from Thorne et a!. (1988) were calculated using their four regression equations and anassumed value of Wmod equal to 25m. See text for discussion.Vegetation W0b/ Wmod W0b/ Wunveg Wob / Wunveg Wobs / WmodType This Thesis Andrews (1984) Hey and Thorne Thome et a!.(1986) (1988)I 0.97 - 1.00 1.45II 0.83 - 0.77 1.20III 0.72- 0.63 0.96IV 0.59- 0.54 0.84Thin 1.03 1.00 - -Thick 0.58 0.79 - -Also in Table 6.2 are results from Thorne et a!. (1988) who tested the optimization model ofChang (1980) on the data set of Hey and Thorne (1986). The essential difference between theoptimization model developed by Chang, and the fixed-slope model from this chapter, is thatChang assumes a constant value for the bank angle Ofor a given channel, and does not considerbank stability.142A direct comparison between the results obtained from Chang’s optimization model and theresults obtained in this thesis is not possible as the values of Wmod were not given by Thorne etal. However an indirect comparison is possible. To account for the influence of the bankvegetation Thorne ci al. developed four regression equations, one corresponding to eachVegetation Type. This allowed Wmod obtained from Chang’s model to be corrected for theinfluence of the bank vegetation. Although the values of Wmo€i from Chang’s model are notknown, the four regression equations can be used to back calculate a value of Wmod for eachVegetation Type for a selected value of W0b3. The values of W0b / Wmod for a selected value ofWmod = 25 m are listed in Table 6.2.The results from the Thorne et al. analysis using Chang’s optimization model are significantlydifferent from the results of Hey and Thorne and in this thesis. This demonstrates that theinclusion of the bank stability constraint makes a very significant improvement in theperformance of the optimization model.6.3.3 Influence of ç6 on Channel GeometryThe preceding section indicates that the bank vegetation has a large influence on the bankstability, which in turn has a correspondingly large influence on the channel geometry. Theeffect of the bank vegetation on the value of will now be examined.One of the effects of the bank vegetation is to stabiise the bank sediment by binding of thegrains by the root masses. A simple method of accounting for this effect is by modifying the143value of which would take larger values for channels more strongly affected by the bankvegetation. This approach ignores the cohesive effects that would be introduced by the rootmasses, and is simply a device to account for the influence of the roots that can be incorporatedinto the noncohesive bank-stability constraint (Eqn 5.15).To demonstrate this effect the optimization model was programmed to run for a series of trialvalues of ç& for each of the channels from Andrews (1984) and Hey and Thorne (1986) untilthe value of Wmod was equal to W0b within a tolerance of ±1%. For example, Reach 13 fromHey and Thome has a value of Wobs = 18.4 m; when the optimization model was run with q5 =400 a value of Wmod = 74.4 m was obtained, which is over four times the observed value. Aftersuccessive trials it was determined that a value of = 73.10 forced an agreement betweenWmod and Wobs.The values of Ø obtained from this analysis depend on the value ofD50 assumed. Becausethe bank stability constraint (Eqn 5.15) is a function of two parameters, qS,. and D50 (thevalues of k, s, and yare assumed constant), the value of can only be determined if D5obO isknown, or in this case is assumed to equal D50. Using different values ofD50 will result indifferent estimates of Ø,.The maximum value of çS., obtained was 90.0°. From the bank stability constraint, Eqn. (5.15)the value of tan 90° is equal to infinity. However rounding-off errors in the computer program144result in a large real value being returned for tan 900.The values of ç5, obtained from the analysis are summarised in Table 6.3. Despite the largescatter within each vegetation grouping there is an observed increase in the mean values of q5with increasing bank vegetation density. This supports the general approach of accounting forthe effect of the bank vegetation by adjusting.The optimization model was rerun for all of the channels using the mean values of g fromTable 6.3 for each vegetation subdivision. The results are shown in Fig 6.6(a, b). There is areduction in the scatter when compared to the results shown in Fig 6.5(a, b) for = 40°. Usingthe mean values of for each Vegetation Type, for the complete set of channels the ratio W0b/ Wmod has a mean value of 1.00 and a coefficient of variation of 28.4%, and Yobs / Ymod has amean value of 1.05 and a coefficient of variation of 14.5%.Table 6.3. Summary of çt values obtained analytically for the data sets of Andrews (1984) andHey and Thorne (1986).Vegetation No. of çS valuesType Rivers Mm (°) Mean (°) Max (°I 13 21.1 42.0 52.8II 16 31.5 47.0 61.0III 13 34.2 53.0 72.5IV 20 39.7 60.1 90.0Thin 14 30.4 40.2 48.4Thick 9 45.7 55.7 67.6The residual scatter in Fig 6.6(a, b) can be attributed to several reasons. These include:1451. The bank vegetation categories are subjective, and a continuous gradation of the vegetationdensity exists within and between the vegetation types, and therefore a corresponding gradationin q5,. would be expected.2. The effect of the bank vegetation is not simply a function of the vegetation density, but isundoubtably dependent upon vegetation type, age, and rooting depth, as well as the thicknessof the overlying cohesive unit.3. Imbrication, packing, and cementing of the gravel by fine sediment is independent of theVegetation Type.4. The assumption that D5Obaflk = D50 is an additional source of uncertainty.6.4 COHESIVE BANK SEDIMENTThe model will now be formulated for cohesive bank sediments and tested on hypothetical andreal river data. The only change from the formulation for cohesive sediments is the bankstability constraint.As was discussed in Section 5.2 the bank stability constraint for cohesive sediment is composedof two individual constraints, namely the bank-height and the bank-shear constraints whichcorrespond to the erosion mechanisms of mass failure and fluvial erosion respectively.1466.4.1 Bank Stability RoutineThe bank stability routine satisfies the bank-height and bank-shear constraints consecutively.First the bank-height constraint is satisfied. The values ofH and Hrrt are initially assessed for 8= 900. For 6= 90°, and for subsequent values of 8, the stability number N3, is obtained fromstability curves such as those in Fig 5.3(a), and the value of Hcr,t is calculated from Eqn (5.2).The value H is by definition equal to the flow depth Y which is obtained by satisfying thecontinuity constraint for Qbf If the value ofH Hcrjt then the bank-height constraint is satisfied,and the routine then moves on to the bank-shear constraint. If the bank-height constraint is notsatisfied, the value of 8 is reduced until H = Hcrjt which gives the maximum bank angle 8 forwhich the bank-height constraint is just satisfied. For values of 9 < 8,,, the bank-heightconstraint is satisfied, but the channel will be less efficient with respect to transporting sedimentas was discussed in Section 6.2.5.Unlike the noncohesive case, the stability of cohesive banks with respect to fluvial entrainmentdoes not necessarily increase with a reduction of 8. Fig 6.2 shows the typical variation of Tbankwith 8 where continuity has been satisfied. As the value of 8 is reduced from the maximumvalue, Tbank increases to a maximum, and then decreases for low values of 9. Therefore exceptfor small values of 8, a reduction in 8 is accompanied by an increase in Tb07, and a reduction inthe stability of cohesive banks with respect to fluvial erosion. The bank stability routineassesses the bank-shear constraint for the value which satisfies the bank-height constraint,and if Tb0J r then the bank-stability constraint has been satisfied. If rit for 8,,,, thevalue of 8 is reduced until Thank = T. In practice it has been found that if the bank-147shear constraint is not satisfied for then the value of 8 which satisfies the constraint isabout 200 or less.The full data requirements for model testing were not available, however the model will betested on the published data from Chariton et a!. (1978) which lists many of the required inputvalues.6.4.2 Data From Chariton et aL (1978)Fourteen rivers from the Chariton et a!. (1978) data set which were described as having bankscomposed of fine sediment will be used. The rivers, together with the hydraulic geometry andother relevant information are listed in Appendix E. The value of d50 for the subarmoursediment which is used to calculate gb was not given in the data set, and as with thenoncohesive channels in Section 6.3, the value d50 = D50 / 3 will be used as an approximation.The value of ksbed was set equal to k5 calculated from the observed channel geometry. Unlikegravel-bed rivers with noncohesive gravel banks, those with cohesive banks may havesignificantly different values of and ksbank.The value of is unknown, so a value equalto 0.1 m was assumed for all channels. This assumption will not significantly affect the result asthe total channel roughness is determined largely by ksbed•The value of the unconfined compressive strength q was given for each channel. This value isan average of several samples from each river. The friction angle for the bank sediment was not148given and a value g’ = 25° is assumed for all channels which is a reasonable value for themoderately plastic silty soils described by Chariton et a!. (1978). With the values of qu and 0’,the value of c’ can be estimated from the following equation (eg. Spangler and Handy, 1982):c’=-tan(45__J (6.4)The values for the bulk unit soil weight were not given. A value of 7d 20.0 kNIm3 will beassumed for all channels for drained bank conditions, and rt 22.5 kN/m3 for saturated bankconditions. For the saturated bank conditions the value of ‘ will be modified by Eqn (5.6) toreflect the reduction in bank strength. This gives a modified value of the bank friction angle=(22.5-9.8)122.5 *250= 14.1°.The value ofN will be calculated from the stability curves in Fig 5.3(a). These stability curveswere approximated by piece-wise linearized segments over the range 0’ 6 90°. Althoughthese curves were developed for homogeneous, drained soils which are unlikely to exist inactual field situations, these curves will be used here for illustrative purposes. More realisticstability curves can be developed following field investigations. The stability curve for ‘ = 15°will be used for modelling the saturated conditions as it is sufficiently close to 14.10.The values for r were not given by Chariton et aL (1978), and it is not possible to directlyestimate the values for z. However in a manner similar to the estimation technique for for149the noncohesive channels in Section 6.3.3, the optimization model can be used to indirectlyestimate values forBecause the value of Trif is not known, initially the model will be run with a very large value forTcrjt (crit = 1000 N/rn2) to ensure that the bank-shear constraint is not influencing the solutionand the bank-height constraint alone is analysed. From this analysis the channels that arepotentially bank-height constrained can be determined. For the remaining bank-shearconstrained channels the values of rrjt can be estimated by forcing an agreement between Wmodand W0b5 as was done for Ø in Section 6.3.3 for the channels with noncohesive banks.The values obtained from the model assuming rrgt = 1000 N/rn2 are presented in Table 6.4 andFig 6.7(a, b). The model was run for both drained and saturated bank conditions. From thevalues of H / Hcrjt it is evident that channels 5, 7, 10, 12, and 13 (and 8 for saturated bankconditions) are constrained by the bank height. The remaining channels are bank-heightdegenerate. (The term degenerate is used in optimization to denote constraints that are notactively constraining the solution.) The bank-shear constraint is degenerate for all of the 14channels due to the large imposed value of Thrit. The bank-height constrained channels will nowbe considered. Note that different solutions are obtained for the drained and saturated bankconditions. The saturated banks are inherently less stable than the drained banks due to theincreased unit weight of the soil, and the increased pore pressures. This effect is well knownand is discussed in Section 5.2.1.1. The second feature evident from this analysis is that thebank-height constrained channels tend to be the larger channels with values of Qbf greater than150Table6.4.Resultsfromtheanalysisof datafromCharitonetaL(1978).Themodellingwith=1000N/m2forcesthebank-shearconstraint tobedegenerate, andonlythebank-height constraint isanalysed.TheestimatedvaluesofwereobtainedbyvaryingtrialvaluesofVcrjtuntilanagreementwasobtainedbetweenW,,,andW0b3.Reachnumbers10and13appeartobebank-height constrainedandthereforenoestimateof rrircouldbeobtained.ReachObservedModelledwith=1000N/rn2Estim-BankNo.DrainedSaturatedatedVegeWW‘H/Vbak/WjH/VbankrFtation(m)(m)(m)(m)HcrgtTent(m)(m)/rnjt(N/rn2)Type517.61.7911.92.361000.0412.22.241.000.0532.8T731.01.7712.53.081.000.0215.82.591.000.0210.0G828.71.6315.12.630.950.0215.02.601.000.0211.9T1019.02.4717.12.601.000.0220.72.261.000.02-T1239.32.6420.53.891.000.0225.83.291.000.0210.8G1359.44.1948.04.831.000.0263.34.001.000.02-G117.41.7811.72.380.650.0411.72.380.890.0429.7T214.00.736.11.180.500.036.11.180.680.0313.9G39.80.736.10.980.460.056.10.980.620.0532.4T413.71.348.91.820.380.038.91.820.520.0322.3T619.01.369.32.190.330.099.32.190.460.0950.0T916.70.696.81.260.530.026.81.260.720.029.4G115.20.653.30.930.280.013.30.930.380.017.0G1419.51.6712.72.230.610.0212.72.230.830.0211.1about 65 m3/s. This is to be expected as the smaller or lower discharge channels will developvalues ofH that are lower than Hcrjt.The remaining channels which are bank-height degenerate have the same optimal solution fordrained and saturated bank conditions. It is assumed that the value of Tcrjg is unaffected by banksaturation. For these channels the modelled values are all narrower and deeper than theirobserved counterparts. This suggests that the values of Thank from the modelled channels aregreater than can be sustained by the observed channel. By reducing the value of rrgt, andtherefore reducing the resistance of the banks to withstand applied shear, a wider and shallowerchannel will result. In this way the modelled channels can be brought into agreement with theobserved geometries, and estimates of zrjt obtained.Also note that channels 5, 7, 8, and 12, which appear to be bank-height constrained when thevalue r = 1000 N/rn2 is used, are also much narrower and deeper than their observedcounterparts. For smaller values of z,rit, these channels will become bank-shear constrained andthe value of Wmod can be brought into agreement with W0b. Therefore only channels 10 and 13are probably truly bank-height constrained as the values of W0b and Tobs lie within the rangedefined by the limiting cases of fully saturated and fully drained bank conditions.The value of TCrIt was varied until Wmod = W0 within a tolerance of ±1% for all channels withthe exception of numbers 10 and 13. These estimated values of are also shown in Table 6.4.The values range between 7.0 - 50.0 N/rn2,with a mean value of 20.1 N/rn2.1526.4.3 Effect of Bank Vegetation.Chariton et a!. (1978) have categorised the rivers in their data set on the basis of bankvegetation into channels with grassed (G) or treed (T) banks. The values of TCrjt obtained fromthe previous analysis show a strong influence by the bank vegetation. The mean value of Thrit forthe channels described as having grassed bank vegetation is 10.4 N/rn2, and 29.8 N/rn2 for thetreed banks.The values of Tcrjt are plotted in Fig 6.8 and this indicates a strong division between the twoVegetation Types. With the exception of one treed channel all of the treed channels plot above20 N/rn2, and all of the grassed channels below 14 N/rn2. This result suggests that the effect ofthe bank vegetation is to increase the value of Thrit either by binding the sediment by the rootmasses, or conversely by affording protection of the bank and effectively reducing the value ofThank acting on the bank sediment. Furthermore the bank vegetation may bind the bank sedimentas to increase the stability of the banks with respect to mass failure. In this way the roots act asinternal reinforcement, and have the effect of increasing the effective values of c’ and qS’ abovethe values obtained from the analysis of small samples of the bank material.Clearly there is a need for additional field work to identify the role of vegetation in stabiisingthe cohesive banks, a conclusion which also applies to noncohesive banks.6.4.4 Discussion of March etaL (1993)A recent paper published by March et a!. (1993) dealing with bank stability will now be153discussed as it highlights several important aspects of the above analysis. In this study stabilitycurves analogous to Fig 5.3(b) were constructed for the Long Creek drainage basin in northernMississippi using averaged values of c, Yt, and qS which were measured during previous fieldsurveys (Thorne er a!., 1981). Stability curves were developed for drained bank conditionsusing the measured averaged values of c, , and and for “worst case” soil conditions whichreflects saturated bank conditions. For values of 9 greater than 600 the Osman - Thorne slabfailure analysis (Osman and Thorne, 1988; Thorne and Osman, 1988) was used to determinethe stability curves that correspond to F = 1.0, from which Hcrjt can then be determined for anyvalue of The Osman - Thorne slab failure analysis is similar to the Culmann analysispresented in Fig 5.1 and Eqns (5.3) and (5.4), but is modified to include the effect of tensioncracks. For values of 0 less than 60° the stability curves were developed using the Bishop(1955) simplified method of slices. The stability curves from March et a!. (1993) arereproduced in Fig 6.9.With the exception of one data point, the points all plot below the “worst case” stability curvewhich represents saturated bank conditions at failure and corresponds to the CASE Ill examplein Chapter 5 (Fig 5.4). Three of the points lie on, or very close to the saturated curve. Thisdistribution of the observed H - 9 combinations is consistent with the bank-height constraint,Eqn (5.7), in that the bank heights must be less than or equal to the critical bank height.However note that 13 out of the 16 data points which plot in the stable field actually failedduring the previous winter period. Since these channels are all stable with respect to the “worst154case” saturated bank conditions, this may at first seem puzzling. However the bank-shearconstraint has not been directly addressed in the March et a!. study. A bank can only beconsidered to be stable when it satisfies both the bank-height and the bank-shear constraints.The role of the bank-shear constraint will now be addressed and it will be shown that this canexplain the failure of the supposedly stable banks given in March et a!.The two erosion mechanisms, mass failure and fluvial erosion, do not operate independently.The interrelation of the two mechanisms is illustrated in Figs 6. 10(a-d). These photographs areof the banks of a small creek flowing across a beach at low tide. The bank material is wellsorted fine to medium sand which is normally non-cohesive, but has developed apparentcohesion due to the moisture content and surface tension phenomena.Fig 6.10(a) was taken at an upstream location where the vertical, stable channel bank isapproximately 480 mm high. The flow at this location was not actively undercutting the bank,ie. Tbank There was no evidence of mass instability of the bank even when an additionalloading of 75 kg (the author’s body weight) was applied, and therefore H < Both bankstability constraints are satisfied.Figs 6. 10(b-d) are a sequence of photographs that were taken at a site approximately 20 metresdownstream from the location in Fig 6.10(a). This downstream location was experiencing rapidlateral channel shifting, and the bank is unstable. The time between each photograph is of theorder of 10 seconds. At this location the channel bank is approximately 100 mm high, much less155than at the stable upstream location.In Fig 6.10(b) the shallow, supercritical flow is beginning to undercut the vertical bank. In Fig6.10(c) the bank has been destabilised by undercutting and mass failure results. In Fig 6.10(d)the failed block has been washed away and a new cycle of undercutting is about to commence.If a bank stability analysis was conducted on the bank geometry in Fig 6.10(d), which may bepreserved at low flows, the bank-height constraint would appear satisfied as the bank is muchlower than the stable, vertical bank upstream. However the bank at this location is clearlyunstable as the bank-shear constraint is not satisfied, that is Zbank > z, and bank is beingundercut which leads to eventual mass failure. Therefore the primary erosion mechanism is infact fluvial erosion of sediment from the toe of the bank, and the observed mass failure is only asecondary effect.March er a!. do not explicitly recognise the requirement of the bank-shear constraint, althoughthey do state that the observed H - 8 combination may have been different at the time of failureand that channel migration may have over steepened the banks until failure occurred.Alternatively bed scouring may have increased H until Hcrit was exceeded.In conclusion it has been demonstrated that a bank composed of cohesive sediment can only beconsidered to be stable when both the bank-height and bank-shear constraints are satisfied.1566.5 BANKFULL MODEL: VARIABLE-CHANNEL-SLOPEThe model will now be modified to allow for variable channel slope. In this formulation thechannel slope S will be treated as a dependent variable. The sediment discharge capacity of thechannel at the bankfull discharge Gbf will be used as an independent variable. The continuity andbank stability constraints remain unchanged from the previous formulation in Section 6.2.There is an additional requirement for a bedload constraint:ed b = Gbf (6.5)where gb is calculated from the Eqn (6.2), and the value of Gbf is prescribed as an independentvariable.The objective function is modified to:maxf(Jd,],O,S)=1l (6.6a)where:Gb(6.6b)PQbfsince Gb1 p, and Qbf are all prescribed in this formulation, the maximization of i is equivalent to157a minimization of S.The flowchart for the modified optimization model is shown in Fig 6.11, and the source codefor the computer program in Appendix B. The only difference between Fig 6.11 and theflowchart for the previous formulation in Fig 6.1 is addition of the bedload constraint whichmust be satisfied before assessing the objective function. The bedload constraint is satisfied byvarying S for trial values OfPbed.An example of the variation of i with Pbed is shown Fig 6.12. Each point on the solution curvesin Fig 6.12 satisfies the continuity, bank stability, and bedload constraints. As with the fixedslope model the objective function curve is smooth with only a single global optimum. Alsoshown in Fig 6.12 is the variation of 5, which indicates the optimal solution corresponds to aminimum slope condition.6.5.1 Noncohesive Bank SedimentThe variable slope optimization model will now be run using the data from Andrews (1984) andHey and Thorne (1986). The value of Gbf was calculated from the observed geometry usingEqns (6.2) and (6.5). All other input values and assumptions are unchanged from Section 6.2.The program was run for all the data for Ø = 400 to determine the effect of the bank vegetationon the channel geometry, including the variation of S. As in Section 6.3.2 modelled valuesobtained assuming çS = 40° are assumed to represent the unvegetated channel dimension. Theresults are summarised in Table 6.5.158The results summarised in Table 6.5 indicate that as the density of the bank vegetation increasesthe observed channels become progressively narrower, deeper, and less steep. The observedand modelled channels summarised in Table 6.5 have the same sediment transporting capacities,therefore if more resistant banks permit narrower and deeper channels to be stable, the bedloadconstraint is satisfied by reducing the channel slope.Table 6.5. Ratios between the observed and modelled values of W, Y, and S for variable slopemodel with çb 400.Vegetation W0b5I Wmod Y0b5I Ymoci Sobs / Sm€jjType Mm Mean Max Mm Mean Max Mm Mean MaxI 0.59 0.98 1.84 0.64 1.05 1.43 0.84 0.99 1.26II 0.43 0.80 1.35 0.81 1.19 1.71 0.81 0.93 1.08III 0.23 0.70 1.17 0.88 1.39 2.61 0.66 0.90 1.03IV 0.20 0.55 0.97 1.01 1.67 2.60 0.68 0.85 0.96Thin 0.70 1.03 1.51 0.75 1.00 1.24 0.88 0.98 1.10Thick 0.16 0.54 0.78 1.16 1.37 1.59 0.86 0.95 1.38This effect of the bank stability will now be demonstrated using Reach 13 from Hey andThorne. The optimization model was run for Reach 13 using the values of ç& = 40° and c =73.1°, the latter value was found in Section 6.3.3 to force an agreement between the modelledand observed channel widths. The output together with the observed channel dimensions areshown in Table 6.6. Note the large influence that exerts on the channel geometry, includingthe channel slope.The results of this analysis indicate that W is most sensitive to variations in the bank stability,159followed by Y, with S the least sensitive. For the combined Vegetation Type IV and thickvegetation channels the channel widths, depths and slopes are in the order of 0.5, 1.6, and 0.9times their respective unvegetated channel dimension.Table 6.6. Effect of SL on Reach 13 from Hey and Thorne (1986). The effect of increasing thevalue of is for the channel to become narrower, deeper, and less steep.Channel Surface Width Mean Depth Channel Slope(m) (m)Observed 18.4 1.14 0.0133Modelled qS=40° 91.2 0.44 0.0183Modelledq5,=73.l° 18.4 1.18 0.0133By forcing an agreement between Wmôd and W0b, the values of Ymoci and Vobs must also showclose agreement as once Wmod is fixed, the value of Ym,ci is constrained by continuity. Similarlythe value of S,,, is determined largely by the bedload constraint, when the channel width isforced to agree with the observed value the modelled channel slope must also show closeagreement with the observed value. In other words when one of the values of any of thedependent variables is fixed, there is only one combination of the remaining dependent variablesthat can satisfy the constraints and fulfil the objective function.Recall from Section 6.3.3 that when the value ç5,. = 40° was used to model reach 13 with thefixed-slope-model, the computed surface width was equal to 73.5 m, in contrast to 90.0 m withthe variable-slope-model. The larger value of Wm obtained with the variable-slope-model is160due to the increase in the channel slope to S = 0,0183 from S = 0.0133 in order to satisf’j thebedload constraint. This illustrates the interrelationship between the dependent variables, andthat the adjustment of one variable such as W, cannot be viewed in isolation from theadjustments of the other dependent variables.6.5.2. Cohesive Bank SedimentThe model was run using hypothetical data; firstly Qbf was varied, and the value of Gbf heldconstant; then the value of Qbf was held constant and Gbf varied. For both analyses thefollowing values for the independent variables were used: D50 = 0.075 m, d50 = 0.025 m, ksbed =0,1 m, ksb nk = 0.1 m, c’ = 10 kN/m3, = 20 kNIm3, ç’ = 25°, and rjt = 25 N/rn2. When Qbfwas varied the value Gbf = 5 kg/s was held constant, and when Gbf was varied the value Qbf =100 rn3/s was held constant. The results are presented in Figs 6.13 and 6.14. The key resultfrom this analysis is the change in the active bank stability constraint.For small values of Qbf approximately less than 100 m3/s the optimum geometry is bank-shearconstrained and bank-height degenerate, and those in excess of 250 rn3/s or so the channelbecomes bank-height constrained and bank-shear degenerate (Fig 6.13(c)). Between about 100to 250 m3/s the channel is actively constrained by both the bank-shear and the bank-heightconstraints.When Qbf was held constant and Gbf varied, a similar result was obtained whereby the channelswith values of Gbf less than about 3.0 kg/s are bank-height constrained and bank-shear161degenerate, while those in excess of 5.0 kg/s are bank-shear constrained and bank-heightdegenerate (Fig 6.14(c)). Within the range between 3.0 to 5.0 kg/s the modelled channels bothbank-height and bank-shear constrained.To explain this change in the active constraint, first consider the case where Qbf is held constantso the results do not become confused by the different channel sizes which are associated withdifferent Qbf values. For low values of Gbf the values of S, ; and Thank are at their minimum, andY is a maximum (Fig 6.14). The channels corresponding to the low values of Gbf are bank-height constrained due to the relatively large values of Y, and the small values of Thank.As the slope increases with increasing Gbf, the channels become shallower due to the increasein both the mean velocity, U and W. The depth decreases at a slower rate than the value of Sincreases, and therefore the values of r and Thank both increase with increasing Gbf Thiscontinues until the value of Thank becomes equal to Tcrjt, at which point the channels becomebank-shear constrained. The value of S continues to increase more quickly than the depthdecreases, and therefore the channel remains bank-shear constrained.When Qbf is varied and Gbf is held constant, the values of S decrease with increasing Qbf (Fig6.13 (c)), which is a feature of natural rivers that is well known from field observations. Forsmall values of Qbf the channels are bank-shear constrained due to the combination of the high Tand Thank values that are associated with steep channel slopes, in addition to the small values ofH which are well below Hcr:t.162As Qbf is increased the value of Y increases and S decreases, however the values of v and rnkremain initially constant because the channel is bank-shear constrained. Eventually the value ofY (and by definition H as H = Y for Qbf) increases to the point where the channel becomes bank-height constrained. For increasing Qbf beyond this point, Y tends to increase at a lower rate thanthe decrease in S and therefore the shear stress values decrease, and this ensures that thechannel remains bank-height constrained.The variation of 6 has not been considered up to this point. In Fig 6.13 the value of Y is equalto 2.95 m at Qbf= 100 m3/s. At this point H Hrit. Yet for larger Qbf, the values of Y increaseand approaches 12.7 m for Qbf 2,500 m3/s where H is still equal to Hcrjt. This is possible dueto the change in 8 and the influence that this has on the value of Hcrjt. At Qbf = 100 m3/s thevalue of 6 = 90° and Hcrjt = 2.95 m. For Qbf = 2,500 m3/s the value of 6 has decreased to 44°,and has increased to 12.7 m. The influence of 9on the value ofHcrit is evident from Fig 5.3and Eqn (5.2).The general conclusion from the modelling in this section is that channels with large values of Stend to be bank-shear constrained. The large S may be associated with large sediment loads, orsmall values of Qbf, or a combination of both. Bank-height constrained channels tend to beassociated with low S values.1636.6 SUMMARYBank stability has been shown to exert a strong control on the optimal geometry of alluvialgravel-bed rivers. The bank stability constraint was formulated for noncohesive channel banksand the theory tested on the published data of Andrews (1984) and Hey and Thome (1986).The results indicate that the bank stability procedure significantly improves the modelperformance. The results are in good agreement with the results that Andrews and Hey andThorne obtained from empirical regime analyses. The influence of the bank vegetation appearsto stabilise the bank sediment, allowing the banks to withstand higher shear stresses. Invegetated channels this results in channels that are narrower, deeper and less steep than theirunvegetated counterparts. The change in W is the largest, followed by Y, and the smallest is inS.It was proposed that the effect of bank vegetation on channels with noncohesive banks can berepresented by,which was found to increase consistently with the bank vegetation density.Model formulations were also developed for channels with cohesive banks. The model wastested on data from Charlton et al. (1978). From this analysis it was shown that channels can beeither bank-shear or bank-height constrained. The values of TCr,t was shown to be higher inchannels with treed banks, than for there grassed counterparts.Modelling results using hypothetical data indicate that bank-shear constrained channels areassociated with large values of S which can result from large imposed sediment loads, and/or164small values of Qbf Conversely bank-height constrained channels tend to be associated withlower values of S which can result from large values of Qbf, low sediment loads, or acombination of both.In general only one bank stability constraint is active in channels with cohesive banks, andrecognition of this is essential when assessing channel stability.165Figure 6.1. Flow chart for the bankfull:fixed-slope optimization model.16614e(°)Tbank bedFigure 6.2. Variation of Thank and Tbed with 6 Each point on the curves satisfies the continuityconstraint for a constant value ofPhd.2018oJE161220 40 60 801670.2 401Figure 6.3. Variation of i and 6 with Fbed. Each point on the solution curves satisfies thecontinuity and bank-stability constraints.C.)C.)tCCl,0.180.160.140.120.10.080.060.0430102010010020 40 60 80bed (m)168(a) 1005020Va,:i0V52.(b)Ea,UVa,a)V0PerfectType I Thin Agreement Io •Figure 6.4. Comparison of Modelled and observed values of W and Y for rivers withnoncohesive banks and low densities of bank vegetation. The data points denoted “thin” arefrom Andrews (1984), and those denoted “Type r’ are from Hey and Thorne (1986). Themodelled values were calculated using the bankfull: fixed-slope optimization model.Observed Width (m)0.3 0.5 1Observed Depth (m)169(a) 100ro20.1o0522(b)2-Ea)DE0.5a)oO.30.20.101 0.2 0.3 0.5 1 2 3Observed Depth (m)Ripe I Type II Type III Type IV Thin Thick Perfect IAgreemento * CFigure 6.5. Comparison of Modelled and observed values of W and 7 for rivers withnoncohesive banks and variable densities of bank vegetation. The data points denoted “thick”are from Andrews (1984), and those denoted “Type r’ to “Type IV” are from Hey and Thome(1986). The modelled values were calculated using the bank1ill: fixed-slope optimizationmodel.Observed Width (m)0.,. 00D*•4.00•0ci0 00ci• IIII———I170(a)(b)-504-20.V032E4-a)0.5a)Vo 0.30.20.101Type I Type II Type III Type IV Thin Thick PerfectAgreement0 *Figure 6.6. Comparison of modelled and observed values of W and 7 for rivers withnoncohesive banks and for all categories of bank vegetation density using the mean valuesfrom Table 6.3. The data points denoted “thin” are from Andrews (1984), and those denoted“Type r’ to “Type IV” are from Hey and Thorne (1986). The modelled values were calculatedusing the bankfull: fixed-slope optimization model.1100Observed Width (m)0.2 0.3 0.5 1 2 3Observed Depth (m)171(a)(b)7060-c0.20CCu0,0)V04030100 0 1054210070Figure 6.7. Comparison of Modelled and observed values of Wand Y for rivers with cohesivebanks from Chariton et a!. (1978). The modelled values were calculated using the bankfull:fixed-slope optimization model for values of rrjt = 1000 N/rn2.20 30 40 50 60Observed Width (m)31 2 3 4Observed Mean Depth (m)Bank-Shear Constrained Bank-Height ConstrainedDrained Banks4Bank-Height Constrained PerfectSaturated Banks Agreement1726050 TC.jE40Z T30----T-T20 -Mean value for grassed banks10 T----G eT G .0— I I I I I I I I I0 2 4 6 8 10 12 14Reach NumberFigure 6.8. The values of z calculated from the data of Chariton ci a!. (1978) for treed (T)and grassed (G) channel banks. The mean values for the treed and grassed banks are indicated.173LUwII — C STASLEDURINGI992Figure 6.9. Bank stability curves from March et a!. (1993). The lower “worst case” curvecorresponds to the Case III example from Fig 5.4, and the upper curve for average soilconditions corresponds to Case I from Fig 5.4.BANK ANcLE (DEcEES)174(a) : Z4 ir .. -‘•I:..A.__ ‘—.(b)Figure 6.10 (a) and (b). Photographs of stream bank erosion from a small creek flowing acrossa beach at low tide. The lens cap in the photographs is for scale and is approximately 45 mm indiameter. See text for discussion.175a‘a(c)(d)Figure 6.10 (c) and (d). Photographs of stream bank erosion from a small creek flowing acrossa beach at low tide. The lens cap in the photographs is for scale and is approximately 45 mm indiameter. See text for discussion.r --:a.-a. -•‘176Figure 6.11. Flow chart for the banlcfull: variable-slope optimization model.N1770.055 0.00477Figure 6.12. Variation of 77 and S with Pbed. Each point on the solution curves satisfies thecontinuity, bank stability, and bedload constraints.C)C)C)0cjHC)C)clD0.050.0450.040.0350.030.0250.00350.0030.00250.0020.00151000 20 40 60 80bed (m)178(a) 100 141280100CD60 8.3: 40 6420 20 0(b) 0.0035 450.003400.0025a)o. 0.002o0.0015 30 30.001250.00050 20Bank-Shear Constrained Duai Constrained : Bank-Height Constrained(c)_______________________________1 25-i0.g -.-----------0.8 H/H - 200.7bank’ crlt0.6-153tbank0.50.450 100 200 500 1,000 2,000Q (me/sec)Figure 6.13. Variation of the optimal values of selected dependent variables as a function ofQbf (a) Wand Y. (b) S and i (c) HIH0rjt, Thank / Zcrit, and z. The value of Qbf is constant andequal to 100 m3/s. The active bank-stability constraint is indicated in Fig 6.13(c).50 100 200 500 1,000 2,00050 100 200 500 1,000 2,000179(a) .101 2 51020501(b) 0.004 500.0035 - Slope0.0005 250.5 1 2 5 10 20 50DuConsqedBank-HeghtConstrairied Ban honsJnedG bf (kglsec)Figure 6.14. Variation of the optimal values of selected dependent variables as a function ofGbf (a) W and Y. (b) S and z (c) H I Thani I Tcrit, and Thank. The value of Qbf is constant andequal to 100 m3/s. The active bank-stability constraint is indicated in Fig 6.13(c).180CHAPTER 7FULL MODEL FORMULATION7.1 INTRODUCTIONIn this chapter the final formulation of the optimization model is presented. The equilibriumgeometry is modelled using the full range of flows, and not just the bankfull discharge.Furthermore the composition of the bed surface is permitted to adjust by using the modifiedParker (1990) surface-based bedload transport relation.7.2 MODEL FORMULATIONThe full model formulation is similar to the bankfull:variable-slope model presented in Chapter6 except that the sediment transporting capacity of the channel will be estimated using theParker (1990) surface-based bedload transport relation, rather than the Einstein-Brownrelation. The Parker surface-based relation was modified in Chapter 4 to apply to a naturalchannel with variable flows. This relation can be inverted to calculate the composition of thebed surface, or armour layer. The mathematical formulation of the full model is presentedbelow.1817.2.1 Independent VariablesIn this analysis the independent variables whose values are presumed to be known are the fillrange of flows and their durations (which can be represented by a flow-duration curve), thegrain size mixture of the imposed bedload sediment, Gb, kjb& ksbk, the bank stability variablesD5Obank, $, for noncohesive banks, and c 6’, Yt, and for cohesive banks, and the constantsg, p, s, v, yThe flow-duration curve and the grain size distribution are subdivided into a finite number ofintervals which are assigned representative values. The flow-duration curve is divided into mintervals and the grain size distribution curve into n intervals (See Figs. 7.1 and 7.2). Becauseboth the flow-duration curves and the sediment size distributions are typically approximatelylog-normal (Shaw, 1988 p. 276; Parker, 1990) the representative value for each interval isgiven by the geometric mean:Qj=JQ,1.Q (7.1)D1=JD’.D (7.2)where Q’ and D are the geometric mean values of intervals i and j respectively, and thesuperscripts 1 and u indicate respectively the lower and upper bounds of the interval. The valuesof i and j range from 1 to m and 1 to n respectively. For each interval the proportion of thetotal is determined. For the flow-duration curve the proportion of the total flows which fallwithin a specified discharge interval is denoted by i. Similarly for the grain size distribution182curve, the proportion of the sediment by volume, or the fraction content within a specified grainsize interval is denoted F. Summations of i and F over all i andj respectively are both equalto 1.0.The imposed bedload, Gb, represents the total mass of sediment in excess of 2 mm in diameter,which is supplied to the channel reach over some significant duration. In this chapter thesignificant duration is assumed to be one year, and therefore the Gb represents the mean annualbedload supply. The units of Gb will be kg/y, and the conversion factor, T will have a valueequal to 365.25 * 24 * 3600 = 31.56 X 106 s/y (see Section 4.6).Typically the value of the bankfull discharge Qbf is considered to be an independent variable.This was questioned in Chapter 3 where it was suggested that Qbf may be considered adependent variable. This will be investigated in Section 7.6.7.2.2 Dependent VariablesThe primary dependent variables which are to be solved are Pbed, Pbank, O S, and the grain sizedistribution of the bed surface which will be represented by D50. Other secondary dependentvariables which include Rh, (J W, and Y are readily obtained once the primary dependentvariables have been determined.1837.2.3 Objective FunctionThe objective function in the full-model formulation is give by:maxf(Jd,Pb,6,S,DSO) = (7.3a)where the coefficient of efficiency i is given by:Gb Gb77=m = — (7.3b)Tpp,S ‘°where the mean annual flow rate with units of m3/s. The total stream power which isrepresented by pQS for steady, uniform flow is modified in the denominator of Eqn (7.3b) forthe full range of flows as given by the flow-duration curve (Fig 7.1). The total stream powerexpended over one year per unit channel length is given by I’ p S.Since the values of Gb and are imposed, a maximization of i is equal to a minimization of S.7.2.4 ConstraintsThe constraints for the full model formulation are given below.7.2.4.1 ContinuityThe continuity constraint is unchanged from previous formulations and requires that thedischarge capacity of the channel is equal to the value of the bankfull discharge, Qbf:184UA=Qbf (3.24)where A is the channel cross-sectional area at bankflill, and U is the mean channel velocity atbankfull which is given by:(3.35)the value offis given by Eqn (3.19).7.2.4.2 BedloadThe bedload constraint requires that the bedload transporting capacity of the channel be equalto the imposed sediment load Gb:Fg=G (4.54)where:g=rspq (4.55)where qb is given by Eqn (4.53). The value qb is the average volumetric bedload transport rateper metre channel width, and has units of m2/s. The units of Gb and gb are kg/y and kg/y/m,respectively.The bedload constraint implicitly contains the “equal mobility constraint” which requires that allgrain sizes be transported equally over the one year duration (see Chapter 4).1857.2.4.3 Bank StabilityThe bank stability constraint can be formulated for both noncohesive and cohesive banks.The noncohesive bank stability constraint is given by:y(s—i)D5kfl1_s: (5.15)The constant k was evaluated from field data in Chapter 4 as 0.048.The cohesive bank stability constraint is comprised of two components namely the bank-heightconstraint, and the bank shear constraint, which are, respectively:HH1 (5.7)rbflk 1crit (5.8)7.2.5 Optimization SchemeThe optimization flowchart is shown in Figure 7.3 and is unchanged from Fig. 6.11. Howeverthe bedload constraint is substantially more complex than previous formulations and is shown inFigure 7.4.186In order to calculate qb the values of the geometric mean grain diameter of the bed surface Dsg(Eqn 4.35) and the standard deviation on the sedimentological phi scale o (Eqn 4.37) must beknown. However these are dependent variables whose values are not known in advance.Therefore an iterative scheme is necessary which is initiated with trial values. The values of Fare calculated which satisfy “equal mobility” using Eqn (4.52) from which updated values ofDsg and o are obtained. This is repeated until convergence is attained on the values ofD ando. From the values of13. at convergence, the value ofD50 can be readily determined.7.3 VARIABLE FLOWSThe Parker surface-based bedload transport relation was modified in Chapter 4 toaccommodate the variable flows which are inherent in natural river systems. The armour layerdevelops such that “equal mobility” applies to the total load transported over a significantduration (which will be taken as 1 year) as a result of the total range of flows. The sizedistribution of the annual transported sediment load is constrained to be equal to the subarmoursediment.To illustrate the process of modelling a range of flows a hypothetical channel with noncohesivebanks will be examined. The flow-duration curve presented in Fig. 7.1 will be used as input.The mean annual flow, , is equal to 17.5 m3/s. The grain size distribution of the bedload andsubarmour sediment is shown in Fig. 7.2, and both have values ofd50= 0.025 m and o = 0.43.The value ag is the geometric standard deviation which is given by:187ir (Dl2usg=a[logLJj F (7.4)The values of and ksbk will both be set equal to 0.40 m, D5Oba,, set equal to 0.07 m whichis about the anticipated value ofD50, and ç = 400. The total annual imposed sediment load is2.5 X 106 kg/year. The value of the bankfiill discharge, Qb., is set equal to 85 m3/s which isequalled or exceeded 1.7% of the time. This value of Qbf is not arbitrary but it will be shown inSection 7.6 that it corresponds to an optimum. The flows that exceed Qbf are set equal to Qbffollowing the assumption of an infinitely wide flood plain. This assumption is discussed inSection 3.3.The optimum values of W, 7, S, and D50 obtained from modelling are 37.6 m, 1.16 m, 0.00426,and 0.073 m respectively, and the bankfull value of is equal to 0.043. All of thesemodelled values are typical for natural gravel rivers with little bank vegetation (eg. see regimeequations and field data from Hey and Thorne, 1986).The sediment transport rate Gb1’ for each value of Q’ is shown in Fig. 7.5(a). Here the prime (‘)is used to denote the rate in kg/s, and Gb, without the prime denotes the total transported in oneyear, or the rate in kg/year. Note that the maximum value of Gb,’ at bankfiill is only 1.89 kg/s.In Fig. 7.5(b) the total load transported by each flow interval is shown. The lowest flow interval(which occurs 75% of the year) transports a negiigible volume of sediment (about 4 kg), while188the flows that equal or exceed Qbf, transport over 40% of the annual sediment load over 1.7%of the year, which is equal to about 6 days.The variation of the median grain diameter of the transported sediment diOload, with discharge isshown in Fig. 7.6 and illustrates the definition of “equal mobility” as applied herein. The valueof dSOJoad varies with discharge; the lower discharges are associated with values of d5Oload lessthan d50, while higher discharges result in values of d5Oload greater than d50. The net result is thatover the entire year the value of dSOIoad for the total annual bedload is equal to d50. Note thatbelow a discharge of about 35 m3/s the median grain diameter is constant. For these lowdischarges the bed surface is essentially immobile (Fig. 7.5(a)), and in most natural rivers anysediment transport would be restricted to “sandy throughput load”. In Oak Creek the sedimenttrapped for discharges less than the threshold value of about 1 m3/s had a mean grain diameterof about 2 mm (see Fig. 4.12). Once this threshold value was exceeded the value of d5Oloadshowed a consistent increase with discharge.7.4 ADJUSTMENT OF THE BED SURFACE COMPOSITIONIn previous model formulations (Chapter 6; Millar and Quick, 1993; Miflar, 1991; as well asWhite et a!., 1982; Chang, 1979, 1980) the armour layer grain size distribution as representedby D0 was considered to be a fixed independent variable. In the full-model formulation thegrain size distribution of the armour layer is able to adjust by using the Parker (1990) surface-based transport relation. The value ofD50 will therefore now be treated as a dependent variable.189The degree of armour layer development will be represented by the value of 650 which is theratio of the median armour, to the median subarmour grain diameters:(4.25)The value of 8, observed in natural gravel rivers ranges from 1.0 with no armour layerdevelopment, to greater than 6. Most rivers have values of 650 around 3 when measured at lowflows. It is assumed in this thesis that the composition of the armour layer which is observed atlow flows is maintained at high discharges (Sect 4.4.1). In addition to D50, a characterisation ofthe armour layer should also include a measure of the dispersion of grain sizes about the medianvalue, however this is of secondary importance and will not be considered herein.7.4.1 Sso - T*D50 Solution CurvesParker (1990) has shown that for an imposed bedload grain size distribution a solution curvecan be calculated to predict the variation ofD with q5sgo. In this section 5o - T.D50 solutioncurves, which are analogous to Parker’s, will be developed to demonstrate the adjustment ofthe armour layer. The 850-TD50 solution curve for the Oak Creek sediment is shown in Fig.7.7. For values of less than about 0.03, which is normally considered the “threshold”value for gravel rivers, a static, immobile armour layer is developed. For values of r’D50 greaterthan 0.03 a mobile armour is developed (Parker, 1990). The value of 8o is seen to decrease asT’D50 increases, and eventually disappears for very large values of V*D50 in excess of 0.3.190The value of ö observed at Oak Creek is about 2.2 which corresponds to a value of v”D50equal to about 0.06. The bed surface at Oak Creek, as with any natural river, is subject to awide range of shear stresses. Therefore the value r*D50 = 0.06 for Oak Creek can be consideredto be the “dominant” or “channel forming” dimensionless shear stress which is analogous to thedominant discharge concept. This dominant dimensionless shear stress is likely to correspond tobankfull, or near bankfhll values as most of the sediment transport occurs at these discharges.Once a channel has reached equilibrium it is presumed that the composition of the armour layerdoes not change appreciably despite the daily and seasonal variation in the flows and shearstresses.The grain size distribution curves for the Oak Creek subarmour and armour sediment areshown in Fig 7.8. These values are from Parker (1990) and exclude the sediment less than 2.0mm in diameter. The curve for the subarmour sediment has been slightly modified at the upperend because Parker reports that 0% of the subarmour sediment sampled was in the range 102-203 mm, whereas 8% of the armour layer falls within this range. The upper end of thesubarmour curve was smoothed to approximate a log-normal distribution with the maximumgrain size equal to 203 mm. This modified curve now has 3% of the subarmour sediment in therange 102-203 mm.The composition of the armour layer for an imposed bed shear stress of 45 N/rn2was calculatedusing Eqn (4.44). The selected value of the bed shear stress gives a value = 0.06 andtherefore observed and calculated values ofD50 match. The results are plotted along with theobserved grain size distribution in Fig. 7.8 and there is an excellent match. However the Parker191surface-based bedload relation that is used to calculate the composition of the armour layer wasdeveloped from bedload transport data collected on Oak Creek, and therefore the good matchis not surprising.As an independent test of the Parker theory the experimental data of Dietrich et aL (1989) willbe used. Dietrich et at. performed three flume experiments to investigate the effect of sedimentsupply on the development of the armour layer. They initially started with very high sedimentfeed rates which at equilibrium resulted in no armour layer development (8o = 1). The sedimentsupply rate was progressively reduced and the bed surface was observed to coarsen to values of850 = 1.16, and 1.32. The 850 V*D5 solution curve was calculated using the sediment feedgrain size distribution, and is shown together with the three observed armour-layer values inFig. 7.9. There is close agreement with the theory for two of the points, however the data pointcorresponding to the highest sediment feed rate plots well below the theoretical curve.The Parker surface-based bedload transport relation was developed over a fairly limited rangeof data. In particular the reduced hiding function (Eqn. 4.36) was developed for r’j50 0.05.At this value the armour layer is only moderately mobile. However for higher sustained shearstresses where the bed surface is much more mobile and the value of öo approaches 1, it isunlikely that Eqn (4.36) would still be valid. The results from the analysis of the data fromDietrich et a!., which is admittedly based only on three data points, suggest that the Parkertheory is applicable only up to moderate values of z*D50, say up to a maximum of 0.07. Beyondthis value the mobility of the bed sediment and the hiding relations appear to be considerablydifferent from those observed at Oak Creek. Fortunately natural gravel rivers generally have192bankfull values of T*D50 less than 0.07, with most falling in the 0.03 - 0.06 range where theParker theory can be applied.7.4.2 Effect of Sediment Gradation onTheoretical ö-r’D50 solution curves were calculated for a range of sediment mixtures fromuniform to poorly sorted which are shown in Fig. 7.10. The sediments all have the same mediangrain size (d50 = 25 mm). The geometric standard deviation, 0g, was used as a measure of thedispersion about d50. The resulting 650 - V*D50 curves are shown in Fig. 7.11. The sedimentgradation is shown to have a large influence on the shape of the curves and the values of 850.The sediment size parameters from Figs 7.10 and 7.11 are shown in Table 7.1. The value ofD50 is approximately equal to d85 in this example.Table 7.1. Summary of sediment size parameters from Figs 7.10 and 7.11. The values of Ugand d90 are from the subarmour sediment, and 850 and D50 are from the computed armour layer.The value ofD50 is approximately equal to d85.(Jcg d85 D50(mm) (mm)0 25 1 250.28 45 1.6 400.43 75 3.1 780.50 100 4.4 1107.5 EFFECT OF SEDIMENT LOADThe effect of increasing sediment load on the channel will be modelled in this section and thetheoretical results compared to experimental and field observations, and qualitative relationsobtained from these observations.193The effect of a 10-fold increase in the imposed sediment load from 2.5 X 106 to 2.5 X i07kg/year on the optimal hydraulic geometry for the set of independent variables described inSection 7.3 is shown in Fig. 7.12(a) and (b). The values of W and S increase in value withincreasing sediment load by 32% and 56% respectively, while the values of 7, and ]3 show adecrease of 24% and 15%.These modelled adjustments above are in general agreement with observed adjustments. Recallthe qualitative proportionality of Lane (1955a) which can be rearranged as follows:QdSG (1.1)where Qd is the characteristic or dominant discharge which is here assumed to be equal to Qb.,and D is a characteristic sediment grain diameter, which is presumed here to be D50.Since the value of Qd is assumed to be constant (cf Section 7.6.1), Formula (1.1) indicates thatan increase in Gb will produce an increase in the value of S, and a decrease in D. The decreasein the value ofTho with increasing sediment load was discussed in Section 7.4 and is supportedby numerous experiments and field observations (eg Dietrich et at., 1989; Lisle and Madej,1989; Kuhnle, 1989).A similar qualitative relation proposed by Schumm (1969):194W2ScxG,, (1.3)where 2= the meander wavelength, and = sinuosity, which is equal to the valley slope dividedby the channel slope, S / S.Formula (1.3) was derived from field observations, and indicates that an increase in Gb willresult in an increase in W, S, and 2, and a decrease in Y and . By definition an increase in S isequivalent to a decrease in as the value of S is assumed to remain constant. The adjustmentof 2 is not considered in the optimization modelling.For an increase in the imposed sediment load the results obtained from the optimizationmodelling show good qualitative agreement with the proportionalities developed by Lane(195 5a) and Schumm (1969). These relations are widely used as guidelines for interpreting andpredicting river adjustments, and the agreement between the modeffing results, and the abovequalitative formulas lend general support to the optimization approach.7.5.1 Valley Slope ConstraintThe adjustments to increasing sediment loading modelled above do not consider the valleyslope constraint which represents a physical bound that defines the maximum channel slope thatcan be attained over engineering time scales. The valley slope is considered to have developedover geologic time, and therefore over engineering time scales can be considered to be anindependent variable.195The initial hypothetical channel from Section 7.5 with a slope of 0.00426 and a sedimenttransporting capacity equal to 2.5 X 106 kg/year, would require a value of 1.57 if it were toadjust to an imposed load of 2.5 X kg/year which requires an increase in the channel slopeof 57% to 0.00668 (Fig. 7.12(b)). The 62 rivers listed in Hey and Thome (1986) have anaverage sinuosity of only 1.34, and therefore the majority of these could not increase theirchannel slope by as much as 57%. When the required channel slope exceeds the valley slope,the valley slope constraint is violated and the solution is not feasible.The valley slope defines the maximum channel slope that can be attained, and thereforedetermines the maximum sediment load which can be accommodated by a river system and stillmaintain a stable equilibrium channel. For example if the valley slope for the river system beingmodelled in Section 7.5 were 0.0056, then from Fig. 7.12(b) it can be seen that the maximumimposed load that can be maintained is about 1.0 X i0 kg/year. For a sediment load in excessof this value the channel cannot develop the required slope and therefore sediment continuitycannot be maintained. An aggrading, unstable, and possibly braided channel would be expected.7.6 BANKFULL DISCHARGE AS A DEPENDENT VARIABLEUp to this point in this thesis, and generally throughout the literature, the value of Qbf isconsidered to be an independent variable which is a function of the catchment hydrology. Thiswas questioned in Section 3.3. The optimization model will now be used to show that there isan optimum value of Qbf which suggests that, in the case of a river with an active floodplain, the196bankilill discharge should more correctly be considered a dependent variable. This analysis doesnot apply to incised channels.The equilibrium geometry for the hypothetical channel described in Section 7.3 was calculatedfor a range of Qbf values from 55 to 155 m3/s. The equilibrium values of S for the various Qbfvalues are shown in Fig 7.13(a). It is evident that an optimal value of Qbf which corresponds toa minimum value of S (and hence maximum ii), occurs at about Qbf= 85 m3/s.The reason for the existence of this optimum value of Qbf can be realised from examination ofthe transport rates and total annual loads transported over each flow interval. Contrast thetransport rates for values of Qbf equal to 65 and 135 m3/s which are given in Table 7.2. Forflow interval Q which equals 59.8 m3/s the sediment transport rates are 0.84 and 0.26 kg/s foreach channel respectively. Over one year Q transports 540,000 kg or 21.6% of the total for avalue of Qbf = 65 m3/s, while for Qbf = 135 m3/s the total transported over one year by Q4 isonly 170, 000 kg, or 6.8% of the total. The same flow in the smaller channel has a higher rateof transport because the depth of flow and therefore the value of Thed are greater. For Qbf = 65m3/s the depth of flow and value of Thed for flow Q are respectively 1.18 m and 46.1 N/rn2,while for a value of Qbf = 135 m3/s the respective values are 0.85 m and 35.6 N/rn2.As the value of Qbf increases the sediment transport rate at bankfull also increases. This is aresult of the greater discharge as well as the wider channel bed across which bedload transportcan occur. Compare the bankfl.ill sediment transport rates in both channels. For Qbf = 65 rn3/sthe bankflill transport rate is 1.13 kg/s, while for Qbf 135 m3/s it is 6.26 kg/s. However as Qbf197Table7.1Comparisonofflowparametersfor Qbf=65m3/sandQbf135m3/s.00I 2 3 4 58.130.448.159.865.0Qbf=65m3/s=135m3/s1PiYGb’GbIPiYGb’Gbms/smkg/skg/ym3/smkg/skg/y0.7530.1410.0500.0200.0370.040.81 1.041.181.2300.0260.3950.8391.1317.01.2X6.2X5.4Xi051.3X106I 2 3 4 5 6 7 8 9 10 11 128.130.448.159.869.879.889.999.9109.9119.9129.91350.7530.1410.0500.0200.0120.0080.0040.0030.0030.0020.0010.0040.270.590.760.850.93 1.001.071.131.191.251.311.3400.0010.0470.2640.5570.8801.331.982.833.965.396.260.60490074,0001.7X2.1X2.1X1.8X105 2.1X2.2X2.OX2.2X105 8.1Xincreases the duration which the bankflill discharge is equalled or exceeded also decreases.From Qbf= 65 to 135 m3/s the duration of the bankfull flow decreases from 0.0365 to 0.0041.Therefore the optimal value of Qbf results from the opposing tendencies of high sedimenttransport rates for low flows in the smaller channels, and higher effective discharges and greaterbed widths which combine to produce larger transport rates in the larger channels.7.6.1 Effect of Sediment Load on Bankfull DischargeThe effect of sediment load on the channel geometry will be used to demonstrate evidence thatQbf should be considered a dependent variable. It was shown in Fig 7.13(a) that for an imposedbedload of Gb = 2.5 X 106 kg/year the model returns an optimal value of Qbf = 85 m3/s. Theresults for an imposed load of 2.5 X i07 kg/year are shown in Fig. 7.13(b) and indicate anoptimal value of about Qi,= 135 m3/s. Therefore the value of Qbf is seen to adjust together withthe other dependent variables to define an optimal solution.There is some support for this result. The qualitative formula of Lane (1955a; see above Eqn(1.1)) indicates that an increase in Gb will cause an increase in Qd (or Qbf) which is in agreementwith the model predictions. Williams (1978) suggests that the recurrence interval (and hencethe magnitude) of the bankfull discharge is possibly affected by the sediment load. It is aninteresting result that requires follow up work to determine its significance.This result can have considerable influence on the solution. For instance Fig. 7.14 shows theeffect on the channel width of a 10 fold increase in sediment load from Gb = 2.5 X 106 kg/year199using the independent variables from Section 7.3. The variation in Wassuming a constant valueof Qbf = 85 m3/s, as well as variable Qbf are shown. A much wider channel is predictedassuming a variable QbrIt can be seen from Figs 7.13(a) and (b) that the optimum value of S, and hence i, is relativelyinsensitive to changes in Qbf For instance it can be seen from Fig. 7.13(a) that a 55% increasein Qbf from 55 to 85 m3/s produces a decrease in S (or an increase in i) of only 3.4%. A widerange of values of Qbf are therefore “near optimal”. It would seem reasonable that the “drivingforce” for a channel to attain the optimum value of Qbf would be proportional to ÔS / ôQbf.Since the value of this partial derivative appears to be small, the “driving force” for the channelto attain the optimum value would be correspondingly small. Therefore while a theoreticaloptimal value of Qbf can be determined, in reality there is a wide range of values of Qbf whichcome close to satisfying the objective function, and therefore in natural systems it could beexpected that the observed bankfull flows would be randomly scattered over a wide rangeabout the theoretical optimum.7.7 APPLICATION OF THE MODELThe formulation of the model allows it to be applied to any situation where there is alteration inthe volume and size distribution of the imposed sediment load, the volume and timing of theflows, or the properties of the bank sediment. For example the construction of a dam will affectthe sediment supply and flows. The sediment supply will be reduced dramatically, ofteneffectively to zero directly downstream from the dam. The area of interest may be somedistance downstream and the reduction of the sediment load from the dam may represent only a200fraction of the total transported at this point. The flows are usually affected to a large extent,typically by truncating the higher flows, and increasing the proportion of low flows. The totalrunoff volumes may or may not be affected. The bank stability parameters may not be affected.Other potential applications relate to land-use changes. For instance removal of forest cover bylogging or for agricultural development may increase the sediment production from thecatchment, and yet may or may not affect the runoff to any great extent. Often the riparianvegetation is affected by these developments, and as has been demonstrated in Chapter 6, thiscan have a profound influence on the bank stability.Regardless of the type of development or catchment disturbance, the input required for themodel is a flow-duration curve, an estimate of the volume of sediment load and grain sizedistribution, and estimates of the bank stability parameters. Hydrologic modelling and sedimentbudget studies may be necessary together with field observations to determine the appropriatevalues to use as input to the model. These values may be difficult to measure in an intact systemlet alone to predict the values for a disturbed catchment. Estimates of the current sediment loadof a stable river system can be obtained using the observed hydraulic geometry, which includesthe grain size distribution of the bed surface, together with the measured or estimated flowduration curve.The model can be used to perform sensitivity analyses to determine the effect of a potentialdevelopment on the river channel where the post-development inputs cannot be accuratelydetermined. For instance some studies have indicated that typical logging practises in coastal201B.C. and the Pacific north-west can result in an 8 to 10 fold increase in the yield of coarse bed-material sediment as a result of increased mass-wasting processes (eg Madej, 1978). The modelcan be calibrated using the observed hydraulic geometry, and then the sensitivity of the channelto increased sediment load can be determined.When using this optimization model it must be realised that the optimum value is only atheoretical value which the river may show a tendency to adjust towards. The optimizationmodel is only an adjunct to other techniques such as air photo and field monitoring of observedchannel adjustments of the river of interest and nearby channels that may have been subjectedto similar disturbances. Any modelling results must be tempered with sound engineeringjudgement, and must recognise the location of geologic controls that may limit the computedadjustment.7.8 HOW AND WHY DO ALLUVIAL CHANNELS OPTIMIZE?Up to this point the issue of how and why do alluvial channels reach the optimal solution hasnot been addressed. The value of the modelling approach has relied on its empirical success inpredicting channel geometry. It seems that the general reluctance of the engineering communityto more fully accept this approach to river adjustments is largely due to the “lack of a physicalbasis” for this approach.Yang and Song (1979, 1986) have attempted to explain their minimum energy dissipation rateand minimum unit stream power hypotheses by arguing from fundamental fluid mechanicsprecepts. Yang and Song suggest that many aspects of the behaviour of fluids, and not just202river channel adjustments, can be explained by their minimum energy dissipation ratehypothesis. However their explanations have not met with widespread acceptance.The argument to be presented in this section will attempt to demonstrate that river channelsdisplay the tendency to move towards the optimal configuration as a result of randomperturbations which exist in abundance in natural river systems. It is the asymmetry in riverchannel response to the random perturbations (which will be explained below) that drives thesystem towards the optimum.Figure 7.15 shows a single solution curve that satisfies the continuity, bedload, and bankstability constraints for a particular set of independent variables. The optimal geometry islocated at Point A. Any channels that are located in the area above the solution curve are overcapacity with respect to transporting sediment, and have the capacity to transport sediment inexcess of the imposed load. Similarly channels located in the area below the curve do not havethe capacity to transport the imposed sediment load. The further a point is above or below thesolution curve, the greater the sediment transport capacity of the channel diverges from theimposed load.Consider a channel that is initially located at Point B. This channel is narrower and steeper thanthe optimum. Ideally this channel could remain indefinitely at Point B in apparent stability as itsatisfies all of the constraints. However in nature there are abundant opportunities for thechannel geometry, at least locally, to be perturbed from B. Consider a local widening whichcould result from an event such as tree falling into the river which acts as a locus for deposition,203from which a gravel bar is developed. This could deflect the current locally against the bankresulting in local bank erosion and widening of the channel to Point C. The channel is now, atleast locally, over capacity, and will attempt to fulfil its sediment transporting capacity throughbed and bank erosion.It is a primary assumption throughout this thesis that bed erosion and degradation is negligiblein comparison to bank erosion, and that bank erosion is concentrated along the outer banks ofmeanders. When the sediment transporting capacity of the channel becomes greater than isrequired to transport the volume of sediment imposed from upstream, net erosion will occuralong the reach as the channel seeks to fulfil its sediment transporting capacity. Since theerosion is presumed to be concentrated along the outer banks of the meanders, this erosion willresult in further development of the meanders, therefore causing a reduction in the channelslope. There may also be some limited channel widening. The net result is that the channeleventually returns to the solution curve at Point D. The result of the initial perturbation istherefore to drive the channel geometry closer to the optimum.Perturbations which displace the channel geometry below the solution curve will produce alocal channel geometry that has insufficient capacity to transport all of the imposed sedimentload. The channel will not have the potential to modif,’ the channel boundary through erosion,and although the channel can modify its dimensions through deposition, it is difficult toenvisage how significant changes would result.204It is therefore possible that an asymmetry exists whereby perturbations which displace thechannel geometry above the solution curve result in a modification of the channel geometrywhich moves it towards the optimum. Perturbations which displace the channel geometry belowthe solution curve into the zone of under capacity will be damped out and have lesser effect onthe geometry.Once the channel reaches the optimum, any changes in the channel width will reduce thesediment transport capacity and these perturbations will be damped out. Any steepening of thechannel that produces an excess of sediment transporting capacity (such as a meander cut-off)will result in erosion principally along the outer bends of the meanders, which will result in areduction in the channel slope returning the channel to the optimal configuration.The above explanation of the optimization process is admittedly incomplete. The role of thesecondary currents is not considered. This explanation does indicate however, that the tendencyfor alluvial rivers to attain an optimum configuration is not a systematic process, but resultsfrom random heterogeneities in the channel.7.9 COMPARISON WiTH OTHER NUMERICAL MODELSThe modelling strategy developed in this thesis is fundamentally different from most othernumerical models which have been developed to predict river adjustments. The most widelyused modelling approaches are the dynamic models which were mentioned in Section 1.4.4,2.and summarised in Hey (1988). The HEC-6 model HEC, 1974; Thomas and Prashun, 1977) isthe most well known. These models assume that the channel width remains constant and205changes in the bed elevation are determined for each channel interval at each time step.Changes in the channel slope are assumed to occur as a result of bed aggradation ordegradation. Osman and Thorne (1988) and Thorne and Osman (1988) modified this approachto include a bank erosion algorithm for cohesive channels. Chang (1982) proposes that cross-sectional changes, which include a reduction or increase in the channel width, can bedetermined for each time step by using the minimum stream power hypothesis. These modelscan be run until steady state conditions have been attained.In contrast the optimization model proposed herein does not consider the transient, dynamicchannel adjustments and gives only the final steady-state channel geometry. Changes in channelwidth are modelled, and the slope adjustments over engineering time scales are assumed tooccur due to changes in the channel sinuosity, and not through aggradation or degradation.The results obtained by the different modelling approaches are illustrated in Fig. 7.16. Anydifferences that would arise between the models due to the different flow resistance andbedload transport equations used in each model are ignored. Two solution curves similar to Fig7.15 are shown. The two curves in this case can be taken to correspond to two values ofimposed bedload, the upper curve representing the higher value of Gb. Consider a stablechannel located at Point A on the upper curve which is subject to a reduction in sediment loadto the value represented by the lower curve. The optimization model predicts that the newstable equilibrium geometry will be given by Point B on the lower curve which represents areduction in the channel slope and a decrease in the width. There is no indication of thepathway or the time required for the channel to adjust from Point A to B.206A fixed width model such as HEC-6 will return a progressive decrease in the channel slope dueto bed degradation as the sediment transporting capacity of the channel will initially be greaterthan the sediment supply following the reduction. Assuming that the degradation is notterminated by the development of a static armour layer (which is almost certain to occur in agravel river), the HEC-6 type models will eventually arrive at a steady state condition close toPoint C on the lower curve. Since the degradation has not involved any changes in channelwidth the final channel must be deeper than the original and it is probable that the bank stabilityconstraint will not be satisfied at steady state. However with the exception of the OsmanThorne model to be discussed below, these dynamic models do not consider the stability of thebanks.The Osman-Thorne model recognises that the bank stability and erosion must be addressed. Intheir model they calculate the net bank erosion in each time step and add this volume of erodedbank sediment into the sediment continuity calculation for each time step. The eroded banksediment partially fulfils the sediment transporting capacity of the particular channel interval,and therefore reduces the net bed degradation. The Osman-Thorne model will reach a steadystate solution at some location between Points C and D on the lower curve. The exact locationwill depend upon the relative erodibility of the bed and bank sediment, for highly resistantbanks the final solution will be close to Point C (slope adjustment only), and for easily erodiblebanks relative to the bed the final solution will be closer to Point D (width adjustment only).207The numerical model of Chang (1982) uses sediment routing to compute the change in thecross-sectional area of the channel, AA, which can be positive or negative depending uponwhether the channel is aggrading or degrading. The minimum stream power (Section 2.2.3) isused to assign i4 to the bed, or the banks, or both. The channel width is first varied until S,and therefore i.QS is minimized. Once the banks are adjusted, the remaining i4 is applied to thebed. One feature of the model is that uniform flow is not assumed, and therefore the slope ofthe bed and the energy gradient are not necessarily the same. The dynamic optimization modelof Chang predicts that the channel in Fig 7.15 will move along a direct line from Point A toPoint B.Despite the capacity for width adjustment, the numerical model of Chang does not containalgorithms for calculating the distribution of the boundary shear stress, or for bank erosion ordeposition. For example the model formulation does not recognise that the rate of bank erosionis dependent upon bank shear stress and the erodibility of the bank sediment. Furthermore it isassumed that the channel adjustment is a continuous process of dynamic equilibrium, and thatthe channel maintains equilibrium at each time step. However in the static optimization modelproposed in this thesis, while the channel is assumed to ultimately reach an optimum, thetransient adjustments are not presumed to adhere to any principle of equilibrium.Clearly the models appear to give very different results, particularly the optimization model andthe Osman-Thorne model which both consider the bank stability and therefore one wouldassume that they would be in closer agreement than the fixed-width, HEC-6 type models.Importantly the Osman-Thorne model does not allow for bank deposition and therefore channel208narrowing in this case is not possible. The optimization model developed in this thesis does notexplicitly include bank deposition in its formulation, however bank deposition is implied whenthe value of falls below the critical value for bank stability since the banks of an optimalchannel develop at the limiting stability.One other important difference between the optimization model and the dynamic models is thelatter’s dependence upon the initial conditions. It is evident from Fig. 7.16 that the finalsolution for the dynamic models is dependent upon the initial conditions, different initialconditions will yield different final solutions. However the optimization model is independent ofthe initial conditions. It was argued in Section 7.8 that random perturbations in the course of ariver channel’s adjustment were necessary for it to attain the optimal geometry. These randomevents are not accounted for in the dynamic models and are important as they have the effect ofresetting to some degree the initial conditions. While final steady-state solution from thedynamic optimization model of Chang (1982) is not dependent on the initial conditions, thepath and the time taken during adjustment are.An interesting possibility would be to include a random component into the non optimizationdynamic models to see if there is a tendency for the channel to adjust to the optimalconfiguration.2097.10 AN EXAMPLE OF RiVER BEI{AVIOIJR WITHIN AN OPTIMIZATIONFRAMEWORKThe optimization framework can be used to interpret the response of an alluvial river followingriver training activities. A solution curve calculated using the optimization model is shownschematically in Fig 7.17. The valley slope constraint is also indicated. The feasible part of thesolution curve, that is the section of the curve that satisfies all of the constraints including thevalley slope constraint, is located below the valley slope constraint where S S.Assume that the channel is originally at the optimum (Point A). In order to reduce flooding,improve navigation, and to “stabilise” shifting channels, a meandering reach is commonlystraightened. If the straightened channel is oriented parallel to the valley axis, it will have aslope equal to S. The channellized reach can be designed such that it possesses approximatelythe same sediment and discharge capacity as the natural meandering channel, and also hasstable banks. A straight channel which is oriented parallel to the valley axis and also is designedso that is satisfies the discharge, bedload, bank stability and valley slope constraints is indicatedat Point B. Note that this represents the narrower of two possible options; a wider channel thatalso satisfies all of the constraints is located at Point C.However, although all of the constraints are satisfied at Points B and C, by straightening thereach the channel slope is increased, and therefore the channel is no longer at the minimumslope that corresponds to the optimal geometry. These straightened channelled reaches mustusually be maintained as any slight channel curvature or irregularity is often amplified by theflow, and this can result in the development of meanders which cause a reduction in the channel210slope. The tendency of a straightened channellized reach to develop meanders is interpreted asthe channel attempting to reestablish the optimum geometry.7.11 SUMMARYThe complete model formulation was presented in this chapter. The model was formulated touse the fill range of flows, which are represented by a flow-duration curve, as inputs to themodel. Furthermore the composition of the bed surface is able to adjust to the optimalconfiguration which is based on the maximization of the coefficient of sediment transportefficiency, i. This represents a significant advance over previous optimization formulationssuch as those of Chang (1980), White et al. (1982), and Millar and Quick (1993b).Rigorous verification of the model was not attempted, however some well known features ofgravel rivers were demonstrated. The effect of sediment load on the equilibrium channel wasshown to agree well with field observation. The model predicts that an increase in the sedimentload will result in the channel becoming wider, shallower, steeper, with a decrease in the valueofD50 of the bed surface.The common assumption of the bankfull discharge as a fixed independent variable wasexamined. It was demonstrated that there is a value of Qbf for rivers with active floodplains thatcorresponds to an optimum. However the value of i is relatively insensitive to different Qbfvalues, and it is probable that natural rivers show a range of Qbf values that are scatteredrandomly about this optimal value.211It is argued that natural channels tend towards an optimal geometry as a result of randomperturbations which exist in abundance in natural rivers. These random events are notconsidered in the dynamic, numerical models of river channel adjustment. Furthermore thedynamic models all attribute changes in the channel slope to either aggradation or degradation,and do not consider changes in the channel sinuosity.2122001501100500Fraction of Time Discharge is Equalled or ExceededFigure 7.1. Flow-duration curve showing numerical approximation. The flow-duration curve inthis example is subdivided into 18 discharge intervals. A maximum discharge value must beselected that is equalled or exceeded all of the time which is the maximum discharge on record,or an approximation.0.001 0.01 0.1 1213a)CLL100806040200Grain Size (mm)200Figure 7.2. Sediment gradation curve showing numerical approximation. The sedimentgradation curve is subdivided in this example into 7 sediment intervals.2 5 10 20 50 100214Figure 7.3. Flow-chart for fhll optimization model formulation.N215Initialize TrialD and a ValuessgCalculate TransportRates For AllGrainsizesCalculate all F That Use CalculatedSatisify Equal Mobility D and a ValuesEqn(4.52) sg •Calculate NewD and a ValuessgHavesg and a, ValuesConverged?CalculateGb D50End BedloadSub FunctionFigure 7.4. Flow-chart for bedload subfunction. An iterative scheme is required to calculate thevalues ofD and u.216(a)2a,t0Cl,I4-a,a,Co0100(b)1 E + 071E+061E+05- 1E+041E+031E+021E+01I-1 E + 00Figure 7.5. Calculated sediment transport rates as a function of discharge. (a) The values of Gb’which have units of kg/s. (b) The values of Gb which have units of kg/y.0 20 40 60 80Discharge (m3Is)Discharge (m3/s)2173025ENU)a). 20a)CVa,Ca)Va,15100 100Figure 7.6. Variation Of d5Oload with discharge. The value of the subarmour d50 is 25 mm.20 40 60 80Discharge (m3Is)2182.62.42.220‘01.81.61.41.210.01Figure 7.7. The calculated 850-solution curve for Oak Creek, Oregon. Data fromMilhous (1972). The solution curve was calculated using the Parker (1990) surface-basedsediment transport relation.0.02 0.03 0.05 0.1 0.2 0.3T*D50219100ICU806040200Grain Size (mm)Figure 7.8. Armour and subarmour grain size distributions for Oak Creek, Oregon. Data fromMilhous (1972). The calculated values were determined using the Parker (1990) surface-basedsediment transport relation.2 5 10 20 50 100 2002201.51.41.30U,‘01.21.110.01Figure 7.9. Calculated öo-TD50 solution curve for data from Dietrich et a!. (1989). Thesolution curve was calculated using the Parker (1990) surface-based sediment transportrelation.0.02 0.03 0.05 0.1 0.2 0.3T*D502211008060a)CU402002Figure 7.10. Sediment gradation curves for a range of crg values.5 10 20 50 100Grain Size (mm)200222540210.01T*D50Figure 7.11. Calculated c5o - TD50 curve for a range of ag values. The curves were calculatedusing the Parker (1990) surface-based sediment transport relation.0.02 0.03 0.05 0.1 0.22230.0070.00650.0060)0.00550.0050.00450.004 3.OE+06 5.OE+06 1.OE+07Sediment Load (kg/y)1.20.90.80.080.0750.070.065 ,,0.060.0550.056055__50EV 45403530WidthDepth3.OE+06 5.OE+06 1.OE+07 2.OE+07Sediment Load (kgly)2.OE+07Figure 7.12. Effect of sediment load on channel geometry. (a) W, Y. (b) S, D50. The valueswere calculated using the optimization model.2240.00480.0046 -a)0.0•0.0042 —0.00460 80 100 120 140Bankfull Discharge (m3/s)0.00720.007 -::::0.006460 80 100 120 140 160Bankfull Discharge (m3/s)Figure 7.13. Calculated variation of S with Qbf (a) Gb = 2.5 X 106 kg/y. (b) Gb = 2.5 Xkg/y. The values were calculated using the optimization model.22570-. Variabre bfConstant bf602.•130 I I3.OE+06 5.OE+06 1.OE+07 2.OE+07Sediment Load (kgly)Figure 7.14. The calculated variation of W with Gb for constant and variable Qbf The valueswere calculated using the optimization model.226Over Capacity0C’)WidthFigure 7.15. Solution curve used to demonstrate channel optimization. The solution curve isshown schematically. Point A is at the optimum, Points B and D satisfy the constraints, but arenon-optimal, and Point C is overcapacity with respect to transporting sediment.CSolutionCurveUnder Capacity2270Cl)WidthFigure 7.16. Solution curves used for a comparison of numerical models. The solution curvesare schematic. The solution curves represent two different imposed sediment loads, with theupper curve corresponding to the higher imposed load. Points A and B are at the optima oftheir respective solution curves.CB228Non-Feasible\ \\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\a /B\ /C.2 s=S \ Feasible /CD VAWidthFigure 7.17. A solution curve used to interpret the behaviour of a meandering river that hasbeen straightened. The valley constraint is also indicated with shading on the non-feasible side.The initial equilibrium geometry is given by Point A. This solution is opimal. The straightenedchannel will be located at Points B or C, both ofwhich are non-optimal.229CHAPTER 8CONCLUSIONS AND RECOMMENDATIONS8.1 SUMMARYIn this thesis an optimization model has been developed to predict the stable, equilibriumgeometry of alluvial gravel-bed rivers for a given set of independent variables. The independentvariables are the discharges, both the magnitude and duration which are represented by a flow-duration curve; the mean annual bedload, both total mass and size distribution, which isimposed on to the channel reach from upstream; and the geotechnical properties of the banksediment.The unknown dependent or decision variables to be solved for include the channel width, depth,bank angle, roughness, and grain size distribution of the bed surface. The dependent variablesadjust subject to the constraints of discharge, bedload and bank stability to determine a channelgeometry which is optimal as defined by , which is the coefficient of sediment transportefficiency (Eqn 2.2).The work in this thesis is an extension of work by Chang (1979, 1980) and White et a!. (1982)whose models have predicted the geometry of sand and gravel rivers with reasonable success,230however the degree of scatter associated with these models limited their application toquantitative engineering applications. Others such as Yang (1971, 1976) and Yang et al. (1981)have proposed similar models which describe various features of alluvial rivers. The advances inthis thesis over the earlier optimization models are presented below:8.1.1 Inclusion Bank Stability Analysis.The bank stability analysis was introduced in Millar (1991) for channels with noncohesive banksediment that was unaffected by bank vegetation. It has subsequently been extended to includethe influence of bank vegetation and also for channels with cohesive banks.For noncohesive bank sediment the model (in a simplified form) was tested on the publisheddata of Andrews (1984) and Hey and Thome (1986). It was shown that the inclusion of thebank stability procedure significantly improves the model performance. The results of theinfluence of the bank vegetation on the channel stability are in good agreement with the resultsthat Andrews and Hey and Thorne obtained from empirical regime analyses. The influence ofthe bank vegetation is to stabilise the bank sediment, allowing the banks to withstand highershear stresses. This results in channels that are narrower, deeper, and less steep than theirunvegetated counterparts. The change in W is the largest, followed by Y, and the smallestchange is in S.The analysis of channels with cohesive banks indicates that channels tend to be either bankheight or bank shear constrained, and less commonly both bank stability constraints are active.In general steeper channels with high sediment loads and low bankfull discharges tend to be231bank shear constrained, while conversely channels with flatter gradients, low sediment loads,and large bankfhll discharges tend to be bank height constrained. The terms “low” and “large”in the previous sentence are relative, and depend largely on the properties of the bank sediment.In a typical channel with only one active constraint, the other constraint is degenerate in that isdoes not influence the channel geometry. It is therefore necessary to determine which of thetwo possible constraints is active in order to assess the stability of the channel. Failure to do socan lead to an assessment of the channel stability based on the degenerate constraint which willlead to erroneous results with regard to the stability of the banks.An analysis of published data from Charlton et al. (1978) for channels with cohesive banksindicates that the bank vegetation has a strong effect on the value of Tcrir (Fig 6.8). Channelswith treed banks have higher values of rcrit than channels with grassed banks. This resultsindicates that the effect of bank vegetation on channels with cohesive banks is similar to thenoncohesive case and that rivers with heavily vegetated banks would be narrower, deeper, andless steep than their unvegetated counterparts.These results regarding the bank vegetation have important consequences for streammanagement as the removal of the bank vegetation can reduce çS or zrjt and destabilise thechannel. Published observations have shown that streamside logging can result in increasedwidth and destabilisation of alluvial channels (Roberts and Church, 1986; Hartman andScrivener, 1990). The optimization modelling could be used to determine the sensitivity of achannel to streamside logging.2328.1.2. Modelling Using The Full Flow-Duration DataPrevious optimization models such as Chang (1979, 1980) and White et aL (1982), as well asthe simplified optimization models presented in Chapter 6 calculated the sediment transportingcapacity of the channel based only on the bankfull rate. It was argued in Chapter 4 that this isinadequate as flows less than bankfull can transport significant volumes of sediment.Furthermore not only is the bankfull sediment transport rate important, but the fraction of thetime where Qbf is equalled or exceeded determines the total volume of sediment that istransported by the channel over one year. The final version of the optimization model presentedin Chapter 7 uses the complete range of flows which are represented by a flow-duration curve.This curve is discretised and used as input to the model. Over-bank flows are also consideredand as a first approximation an infinitely wide flood plain is assumed. The depth of flow withinthe channel, and more importantly, the values of vE,d and Thank reach their maximum value atQbj; and are presumed to remain constant at this value for all larger discharges.8.1.3 Adjustment of the Bed Surface.The grain size distribution of the bed surface or armour layer, which is represented most simplyby D50, is considered to be a dependent variable. The size distribution of the bed surface and thebedload transporting capacity of the channel are calculated using the Parker (1990) surface-based bedload transport relation which is modified in Chapter 4 to apply to variable flows.In this thesis the concept of “equal mobility” is presumed to apply over some significantduration which is generally assumed to be one water year. Therefore the “equal mobility”233concept as applied herein is that the bed surface or armour layer forms so as to render all sizefractions equally mobile over a period of one year. Therefore the grain size distribution of themean annual load transported by a channel in equilibrium is exactly the same as the distributionof the imposed load. It is assumed that the grain size distribution of the mean annual imposedbedload is identical to the that of the subarmour sediment (Parker, 1990).The definition of equal mobility developed in this thesis is significantly different from theoriginal definition by Parker et a!. (1982a) who assumed that equal mobility was achieved foreach flow. This is known to be incorrect from field observations (eg Fig. 4.11), and the originaldefinition of equal mobility was acknowledged by Parker et a!. as a first-order approximationonly.8.1.4 Bankfull Discharge as a Dependent VariableIt was shown in Chapter 7 that there is an optimal value of Qbf for a given set of independentvariables. This result indicates that the value of Qbf can be considered a dependent variable,rather than an imposed value which is commonly presumed.However the limited modelling undertaken in Chapter 7 indicates that the value of i isrelatively insensitive to Qbf, and from this it would be expected that the observed values fromnatural rivers would be randomly scattered across a wide range about the optimal value. This isan interesting result which deserves fhrther attention to determine its significance.2348.1.5 Verification of the Optimization ModelModelling results of Chang (1979, 1980) and White et at. (1982) showed reasonably goodagreement between modelled and observed values of W and 7, although the degree of scatterwas too great for quantitative engineering projects. This scatter has been reduced in Millar(1991) and Millar and Quick (1993a, b) and in Chapter 6 through the inclusion of the bankstability constraint. In addition Yang et a!. (1981) have shown that their optimization modelyields the exponents ofwell-known empirical regime equations.Furthermore good quantitative agreement was obtained between the modelling in Chapter 6and Millar and Quick (1993a, b) for the effect of bank vegetation on channels with noncohesivechannel banks, when compared to the empirical regime equations of Andrews (1984) and Heyand Thorne (1986).In addition general qualitative agreement between the optimization modelling and the formulasof Lane (1955a) and Schumm (1969) has been demonstrated by Chang (1980), Millar (1991),and in Chapter 7.These results represent only partial verification of the optimization theory that has beenpresented in this thesis. This theory was advanced beyond that which could be fully verifiedwithin the scope of this thesis. A program for full verification of the optimization model isoutlined in Section 8.2.6.2358.2 RECOMMENDATIONS FOR FUTURE WORKThe optimization model presented herein is formulated using the most suitable equations thatwere available from the literature. Some modifications were made, however no experimental orfield work was carried out during this study to develop new, or improve existing relations.Several aspects of the optimization model that can benefit from additional research arepresented below.8.2.1 Channel Form or Bar RoughnessIn Section 3.2 it was argued that the grain roughness could be calculated using Eqn (3.8),however no existing methods were available to calculate the form or bar roughness component.The similar methods of Einstein and Barbarossa (1952) and Parker and Peterson (1980) wereshown to be a result of spurious correlation and therefore do not yield meaningful estimates ofthe form roughness. The approach adopted by Prestegaard (1983) is promising although nopredictive equations have yet been developed.The nature of the form roughness can be studied using an approach similar to that adopted byPrestegaard (1983). The total channel roughness,J can be determined from channel surveys ata location where good quality flow measurements have been obtained, preferably at ahydrometric station. The grain roughness,f’, can be calculated using Eqn (3.8) and the value off”is equal to!- f’ from Eqn (3.7). Data obtained from detailed geomorphic mapping of thechannel, such as bar amplitude and spacing, vegetation etc, can be then used in an effort todevelop empirical relations forf”.2368.2.2 Surface-Based Transport Relation Based on t’bdThe Parker (1990) surface-based transport relation uses the total bed shear stress Tbed,, and notthe grain shear stress r ‘L,ed,, to calculate the transport rate. The use of the total bed shear stressand not the grain component is a result of Parker and Peterson (1980) who concluded that theform component of roughness and shear stress in gravel rivers is negligible, particularly for highin-bank flows. This was shown to be incorrect in Section 3.2, and furthermore it was shown inSection 4.2.4 that as little as 11% of the total shear stress can be assigned to the graincomponent (See Fig 4.6).Because Thed appears in Parker’s dimensionless bedload transport parameter W (Eqn 4.28),conversion of the transport relation to one based on r is not simply a matter of changing acoefficient, but rather all of the relations which include the hiding and straining functions needto be recalculated.8.2.3 Bank StabilityThe results in Chapter 6 indicated that for both noncohesive and cohesive banks the bankvegetation exerts a strong influence on the bank stability, and hence the channel geometry. Theeffect of the bank vegetation on the bank stability is reduced to a single parameter which is çfor noncohesive bank sediment, or rent for cohesive bank sediment. In both cases the effect ofthe bank vegetation was interpreted as an increase in and znjt which resulted from thebinding and reinforcement of the sediment by the root masses. An alternative explanation is thephysical shielding of the bank sediment by the vegetation which reduces the magnitude ofacting on the bank sediment. Field investigation is necessary to determine the actual role of the237bank vegetation and how it relates to bank stability, and whether the effect is an increase in gor rrjt, or a decrease in Tbank. Furthermore it is necessary to develop field-based methods fordetermining the insitu values of g and rit8.2.4 Secondary CurrentsThe “strong” secondary currents that occur as a result of channel curvature have not beenconsidered in this thesis. The “weak” secondary currents which are present in straight channelsare implicitly accounted for in the empirical boundary shear stress equations of Flintham andCaning (1988) which were developed from experimental data.One important effect of secondary currents is the redistribution of the boundary shear stress. Inthis thesis the bank and bed shear stresses are assumed to be uniform across their respectivechannel perimeters. However redistribution of the shear stress can have a large effect forexample on the sediment transporting capacity of the channel.Consider a channel that at bankfull has a mean bed shear stress value that is just below thecritical value for bed mobilisation. This channel will, based upon the mean value, transport anegligible volume of sediment. However if secondary currents redistribute the shear stress suchthat one half of the channel bed has a shear stress value about 1.5 times the critical value, andthe other half is only about 0.5 times the critical value, significant sediment transport will occuracross one half of the channel.238From the optimization model it is assumed that a reduction in the channel slope implies adecrease in the sediment transporting capacity of the channel. However this reduction inchannel slope is presumed to be accompanied by increased sinuosity, resulting in strongersecondary currents which could conceivably increase the sediment transporting capacity of thechannel.The optimization model calculates the mean value of Vbank, and this value must be less than theshear stress required for fluvial erosion of the bank sediment in order to satisify the bank-stability constraint. Locally however, the value of Tbank may exceed the value required for bankerosion, particularly in areas such as along the outer bank of meanders. Along the convex innerbank the shear stress is lower and deposition of sediment can occur forming a point bar deposit.Therefore even though bank erosion and lateral channel migration is occurring, the channel isstill considered to be in equilibrium, and “statistically stable”, if there is no net change in themean channel geometry over a representative channel reach.8.2.5 Formulation for Sand-Bed RiversThe equations used in this thesis have been selected and developed for application to gravel-bedrivers, however the model can easily be reformulated for application to sand-bed rivers. Forexample the Einstein transport relation could be used in place of the modified Parker (1990)relation.2398.2.6 Model VerificationBefore the optimization model can be used for quantitative engineering studies to predict theimpact of engineering structures or land-use practices on the channel geometry a fhll programof model verification needs to be undertaken to determine the reliability in the model.Initially data should be collected from stable rivers which appear to be in approximateequilibrium. Existing data sets such as Kellerhals et a!. (1972) were found to lack certain keydata. However these data sets could be updated with some field work to determine the valuesof the bank stability parameters, and review of the original records such as flow data, sedimentanalyses etc.The values of the independent variables obtained from field surveys can be input into the modeland a comparison made between the modelled and observed value of the dependent variables.This should include an analysis of the bankfull discharge to determine if the optimal value of Qbffound from modelling has any physical significance.If the model was found to predict the geometry of existing rivers, this would then improve theconfidence of using these models for predicting the response of a river to catchmentdisturbance.Despite the limited quantitative verification that has been done, the model in its present form isvaluable as it gives the user some insight into, and understanding of the nature of river channel240adjustments which can be used to improve the judgement of a scientist or engineer interested inthe response of alluvial rivers.241REFERENCESAckers, P., and White, W.R., 1973: “Sediment transport: new approach and analysis.” I Hydr.Div., ASCE, 99 (11), 2041 -2060.Alam, A.M.Z., and Kennedy, J.F., 1969: “Friction factors for flow in sand bed channels.” IHydr. 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In this version of the model the channel slope is treated as an independent variable.The source code for the computer program (fixslope.bas) used in the thesis is presented below.The programming was encoded and run using Microsoft Quick Basic (Version 3.0 or later).This program can be used for channels with cohesive or noncohesive bank sediment.The program is designed to use an input data file named fixslope.dat, and will output theoptimal geometry in a data file named fixslope.out. The data file can contain the data for morethan one channel. The data for each channel must be input as follows:Qbf S ksbd ksbk d50 D50 D5Obkfor channels with noncohesive banks, and:Qbf S ksbd ksbk d50 D50for channels with cohesive banks.The data relating to each channel must start on a new line. The data values must be separatedby at least one space, although the number of spaces is not important. An example of an inputdat file is presented below:50 0.003100 0.0030.1 0.1 0.025 0.075 0.075 400.1 0.1 0.025 0.075 0.05 40261APPENDIX A‘isisrt C’ TcritThis file contains the input data for two channels with noncohesive banks. The user is requiredto input the bank sediment type (n for noncohesive or c for cohesive) as prompted. The outputdata will written to a data file named fixslope.out in the following order:W H 17* S 0The output file resulting from the example input file is:W H * S theta18.0 1.40 1.21 0.0030 30.037.7 1.28 1.17 0.0030 21.7The Bankfiull:Fixed Slope model was developed principally for illustrative purposes. It can beused in a field investigation to estimate or ‘calibrate’ the bank stability parameters that may bedifficult to measure. For example with noncohesive bank sediment the value of Ø., may bedifficult to estimate in the field. The measured independent variables can be input into themodel together with trial values of until the modelled width is equal to the observed channelwidth. This procedure was used in Chapter 6 to demonstrate the relationship between andbank vegetation. The value of which gives the best agreement between the modelled andobserved geometry can then be used as input into the fully developed model (Appendix C).The source code for the Bankfull: Fixed Slope Model is presented below.262‘BANKFULL:FIXED SLOPE MODEL FOR BOTH COHESIVE AND NONCOHESIVE BANK‘SEDIMENT. (FIXSLOPE.BAS)DECLARE SUB stabcurve 0DECLARE SUB bankstabilitycohesive 0DECLARE SUB optimum 0DECLARE SUB bankstabilitynoncohesive 0DECLARE SUB ebrown 0DECLARE FUNCTION bankshear! 0DECLARE FUNCTION bedshear! 0DECLARE FUNCTION shearforce! 0DECLARE SUB continuity 0DECLARE FUNCTION velocity! 0DECLARE FUNCTION hydrad! 0DECLARE FUNCTION meandepth! 0DECLARE FUNCTION area? 0DECLARE FUNCTION surfwidth! 0DECLARE FUNCTION depth? 0I*********************************************************************************program to calculate curves to demonstrate the optimumchannel geometiy for a given set of independent variables‘bankfull: fixed slope model for channels with‘both cohesive and noncohesive bank sediment‘Global Variables‘gravity!= gravitational constant gamma? unit weight of water‘ss! specific gravity of sediment viscosity?=kinematic viscosity of water‘density!=density of water slope?=channel slope‘pbed!=bed perimeter pbank?—bank perimeter‘theta!=bank angle (radians) dd5O?=median armour grain diameter‘d50!median subarmour grain diameter d5Obank?=median grain diameter of bank sediment‘discharge!=bankfull discharge ‘gbcalc!=valculated bedload transport rate ‘neta!=coefficient ofefficiency‘phipnme?=friction angle of bank sediment (radians)‘phidegrees!=friction angle of bank sediment (degrees)‘ksbed?=measure of bed roughness ksbank!=measure of bank roughness‘banktype$=type of bank sediment Wointerger counter‘gammat!=unit weight of cohesive bank sediment‘stabnum’=stability number for cohesive sediment‘cohesion!=soil cohesion taucrit?=critical shear stressCOMMON SHARED gravity?, gamma!, ss?, viscosity?, density!COMMON SHARED slope!, pbed?, pbank?, theta!, ddSO’, d5Obank?COMMON SHARED discharge!, gbcalc!, phiprime!COMMON SHARED d50!, bankcond$, pi?COMMON SHARED ksbed?, ksbank!, neta!, bankte$, f%COMMON SHARED ganimat?, stabnum?, cohesion!, taucrit?, phidegrees?OPEN “fixslope.dat” FOR INPUT AS #1OPEN “flxslope.out” FOR OUTPUT AS #2263‘SET VALUES OF INDEPENDENT CONSTANTSpi! = 3.14159gravity!= 9.81gamma! = 9810viscosity! = .000001density! = 1000ss! = 2.65CLSPRINTPRINTPRINT “BANKFULL:FIXED SLOPE OPTIMIZATION MODEL”PRINT : INPUT “Hit <ENTER> to Continue “,dummy$=0PRINT #2,” W H * S theta”CLS10PRINTPRINTPRINTPRiNT : INPUT “INPUT BANK SEDIMENT TYPE (n/c) “; banktype$iF banktype$ = “c” OR banktype$ = “C” THENbanktype$ = “cohesive”ELSEIF bankte$ = “N” OR banktype$ = “n” THENbanktype$ = “noncohesive”ELSEBEEPPRINT “Bank Type Unknown”GOTO 10END IFCLSDO WHILE NOT EOF(1)PRINTPRINT=1% + 1 ‘counterPRiNT “Channel Number“; f’%iNPUT #1, discharge!, slope!, ksbed!, ksbank!, d50!, dd5O!‘Input the independent variables from data fileIF banktype$ “noncohesive” THEN ‘input bank stability parametersINPUT #1, d50bank!, phidegrees!ELSE ‘banktype$=”cohesive”INPUT #1, gamxnat!, phidegrees!, cohesion!, taucrit!END IF264phiprime! = phidegrees! * 2 * pi! / 360 ‘convert friction angle from degrees to radiansCALL optimum‘calculates the optimal geometryPRINTPRINT” W H * S theta”PRINT USiNG “###.# ##.## ##.## #.#### ##.#“; surfwidth!; depth!; meandepth!; slope!; theta! * 360/2 /pi!PRINT #2, USING “###.# ##.## ##.## #.#### ##.#“; surfwidth!; depth!; meandepth!; slope!; theta! * 360 / 2/ pi!LOOP‘loops until end of data fileCLOSE #1CLOSE #2ENDFUNCTION area‘function to calculate the cross-sectional areaarea! = .5 * (pbed! + surfwidth!) * depth!END FUNCTIONFUNCTION bankshear‘calculates the bank shear stressbankshear! = gamma! * depth! * slope! * shearforce! * ((surfwidth! + pbed!) * SIN(theta!) / (4 * depth!))END FUNCTIONSUB bankstabilitycohesive‘satisfies bank stability constraint for cohesive banksheightcond$ = “unknown”shearcond$ = “unknown”bankcond$ = “unknown”thetamax! =90*2 *pi! /360thetanün! =0‘initialise search and convergence criteria‘First test ifvertical bank is stable wrt bank heighttheta! = thetamax!CALL continuity‘satisfy continuity265CALL stabcurve‘catculate stability numbercriticalheight! = stabnuml * cohesionl I gaxnmat’calculate the critical heightstability 1! = depth / criticaiheight!‘calculates stability criteria wit bank height‘if stability 1 I <= criticaiheight then bank is stable‘with respect to bank heightIF stabilityll <= 1.001 THENheightcond$ = “stable”ELSE ‘vertical bank is not stable therefore reduce the bank angle.‘Determine the maximum bank angle that is just stable.This is theta max from Chapter 6.DO UNTIL heightcond$ = ‘just stable”theta1 = (thetamax? + thetamin’) /2‘calculate midpoint of range for‘bisectrix convergence schemeCALL continuity‘satisfy continuity constraintCALL stabcurve‘calculate stability numbercriticaiheight! = stabnum! * cohesion! / gaimnat!stabilityll = depth / criticaiheight’‘calculates critical height and stabilty number‘for the bank height constraintIF stabilityl! >= 1.001 THENheightcond$ = “unstable”thetamax! theta?ELSEIF stabilityll <= .999 THENheightcond$ = “understable”thetamin! = theta!ELSEheightcond$ = ‘just stable”‘Bank Height = Critical HeightEND IFIF (thetamax! - thetamin’) / theta! <.001 THENheightcond$ = ‘Just stable”stabilityl! = 1‘Secondary Convergence Criterion in case of‘convergence problems due to numerical schemeEND IFLOOP266END IFbank height constraint now satisified‘now assess the bank shear constraintthetaniin?=0thetamax! = theta!‘thetamax is reset to maximum bankangle which satisfied the‘bank height constraint abovestability2! = bankshear! / taucritsatbility criterion for bank shear constraintIF stability2! < 1.001 THEN‘bank shear constraint is satisifledshearcond$ = “stable”bankcond$ = “stable”ELSE ‘must reduce theta!DO UNTIL shearcond$ = “just stable”theta! = (thetamax! + thetaniin!) /2‘calculate mid pointIFtheta! < 10*2*3.14159/36OTHEN‘bank angles less than 10 degrees‘nominally assumed to be unstableshearcond$ = “unstable”bankcond$ = “unstable”EXIT DOEND IFCALL continuity‘satisfy continuitystability2! = bankshear! / taucrit!‘calculate stability criterion for bank shear constraintIF stability2! >= 1.001 THEN‘bankshear > taucritshearcond$ = “unstable”thetamax! = theta!ELSEIF stability2! <= .999 THENshearcond$ = “understable”thetamin! theta!ELSEshearcond$ = “just stable”bankcond$ = “stable”267END IFIF (thetaxnax! - thetanun!) / theta! <.001 THENshearcond$ ‘just stable”bankcond$ = “stable”stability2? = 1END IFLOOPEND IFEND SUBSUB bankstabilitynoncohesive‘calculates theta where banks just stablebankcond$ = “unknown”thetamax! = phiprime!thetamin! =0‘initialise search bounds and convergence criteriaDO UNTIL bankcond$ = ‘just stable”theta! (thetaniax! + thetaniin!) / 2‘determine midpoint of search rangeIF theta! <5 * 2 * 3.14159/360 THENbankcond$ = “unstable”EXiT DO‘if theta is less than 5 degrees the channel is assumed‘to be unstableEND IFCALL continuity‘satisfies continuity constraintstability! = (bankshear! / (gamma! * (ss! - 1) * d5obank!)) I (.048 * TAN(phiprime!) * (1 - SIN(theta!) A 2 /SIN(phiprime!) A 2) A .5)‘calculates stability criterionIF stability! >= 1.001 THENbankcond$ = “unstable”thetamax! = theta!ELSEIF stability! <= .999 THENbankcond$ = “stable”thetamin! = theta!ELSEbankcond$ = ‘just stable”‘primaiy convergence criterionEND IF268IF (thetamax! - thetamin!) I theta! <.001 THENbankcond$ = ‘just stable”stability! = 1‘secondaiy convergence criterionEND IFLOOPEND SUBFUNCTION bedshear‘calculates the bed shear stressbedshear! gamma! * depth! * slope! * (1 - shearforcel) * ((surfwidth! / (2 * pbed!) + .5))END FUNCTIONSUB continuity‘varies pbank! for trial values of Pbed!, slope!, and theta! to‘satisfy the contiuity constraintpbankcond$ = “unknown”errorcalc! = 1000minpbank! =0maxpbank! =20 * discharge! A •35‘initialise search and convergence criteriaDO UNTIL pbankcond$ = “OK”pbank! = (minpbank! + maxpbank!) /2‘calculate midpointerrorcalcl = (area * velocity / discharge!)‘calculate the normalised errorIF errorcaic! > 1.001 THENpbankcond$ “too large”maxpbank! = pbank!ELSEIF errorcalc! <.999 THENpbankcond$ = “too small”minpbank! = pbank!ELSEpbankcond$ = “OK”END IFIF (maxpbank! - minpbank!) / pbank! <.0001 AND pbankcond$ = “too small” THEN‘resets maxpbank! if too smallmaxpbank! =2 * maxpbank!END IF269LOOPEND SUBFUNCTION depth‘function to calculate flow depth of a trapezoidal channeldepth! = .5 * SIN(theta?) * pbank!END FUNCTIONSUB ebrown‘calculate the sed trans capacity using em-brown formuladimlessbedshear! = bedshear! / (gamma! * (ss! - 1) * d50!)IF dinflessbedshear! <=0 THENdimlessbedload! =0ELSEIF dimlessbedshear! <.093 THENdimlessbedload! = 2.15 * EXP(-.391 / dimlessbedshear!)ELSEdimlessbedload! =40 * dimlessbedshear! “3END IFfi!= (2 / 3 + 36 * viscosity? “2/ (gravity! * d50! “3 * (ss? - 1))) “.5 -(36 * viscosity? “2/ (gravity! * d.50? “3*(ss!_1)))gbcalc! = pbed! * fi! * ss? * density! * ((ss! - 1) * gravity! * d50! A 3) A 5 * dimlessbedload!END SUBFUNCTION hydrad!‘function to calculate the hydraulic radiushydrad! = area! / (pbed? + pbank!)END FUNCTIONFUNCTION meandepth‘function to calculate the mean depthmeandepth! = area / surfwidthEND FUNCTIONSUB optimum‘determines the optimal geometzyoptcond$ = “unknown”‘initialises optimality test condition270lowerpbed! =0upperpbed? = 10 * discharge! “.5‘set bounds for Pbed based on Regime Eqns‘typical optimal value of Pbed is 2 to 5 * discharge” .5minpbed! = lowerpbed!maxpbed! = upperpbed!PRINTDO WHILE optcond$ <> “optimum”pbed! = (minpbed! + maxpbed!) /2‘calculate mid point of search rangeIF pbed! > .95 * upperpbed! THENupperpbed!2*upperpbed!maxpbed! = upperpbed!‘reset upper bound of searchEND IFPRINT “Assessing Trial Bed Perimeter “;PRiNT USiNG “####.##“; pbed’;PRiNT “ m “;pbed! = pbed! * .975‘calculate backwards difference value of PbedIF banktype$ = “noncohesive” THENCALL bankstabilitynoncohesive‘solve bank stability constraint for noncohesive banksELSE ‘banktype$ = “cohesive”CALL bankstabilitycohesive‘solve bank stability constraint for cohesive banksEND IFIF bankcond$ <> “unstable” THEN‘test conditions indicate that stable channel geometry‘has been determined and the bedload constraint‘has been satisifiedCALL ebrown‘calculate sdiment tarnsporting capacity of the channelnetal! = gbcalc! / (density! * discharge! * slope!)‘evaluate neta for backward difference pointpbed! pbed! 1.975 * 1.025‘calculate pbed for forward duffIF banktype$ = “noncohesive” THENCALL bankstabilitynoncohesive‘solve bank stability constraint for noncohesive banksELSE ‘banlctype$ = “cohesive”CALL bankstabilitycohesive‘solve bank stability constraint for cohesive banksEND IFCALL ebrown271‘calculate sdiment tarnsporting capacity of the channelneta2! = gbcalcl / (density’ * discharge! * slope!)‘calculates neta for forward difference valuepbed! =pbed! / 1.025‘reset pbed to midpoint valuednetabydpbed! = (neta2! - netal!) I (.05 * pbed!)‘calculate first derivative by finite difference‘to assess optimum conditionIF dnetabydpbed? <0 THEN ‘Trial Pbed is too demaxpbed! = pbed’ ‘Reduce upper bound of Pbedoptcond$ = “too wide”ELSEIF dnetabydpbed! >0 THEN ‘Trial Pbed is too narrowminpbed! = pbed! ‘Increase lower bound of Pbedoptcond$ = “too narrow”ELSEoptcond$ = “optimum” ‘Optimum AchievedEND IFIF (maxpbed! - minpbed!) I pbed! <.001 THEN optcond$ “optimum”‘Convergence attained. Second optimality criterionELSE ‘trial Pbed too small for stable geometiy: Increase minimum Pbedpbed! =pbed!/.975‘reset from backward difference to correct valueminpbed! = pbed!‘set lower limit at current trial value‘(as the optimum value must be greater)optcond$ = “too narrow”END IFPRiNT optcond$LOOP‘Evaluate geometiy at exact optimal Pbed‘(not at forward or backward difference value)IF banktype$ = “noncohesive” THENCALL bankstabilitynoncohesive‘solve bank stability constraint for noncohesive banksELSE banktype$ = “cohesive”CALL bankstabilitycohesive‘solve bank stability constraint for cohesive banksEND IFCALL ebrown‘calculate sdiment tarnsporting capacity of the channelneta! = (pbed! * gbcalc!) / (density! * discharge! * slope!)272‘evaluate objective functionEND SUB‘*********************************************************************************FUNCTION shearforce‘calculates the proportion of shear force on the banksshearforce! = 1.766 * (pbed! I pbank! + 1.5) A -1.4026END FUNCTION‘*********************************************************************************SUB stabcurve‘stability curves are from Figure 5.3thetadegrees! =theta! /2/3.14159 * 360IF phidegrees! <20 THEN‘use phiprime = 15 degree stability curve from Figure 5.3‘from Taylor (1948)SELECT CASE thetadegrees!CASE 48.65 TO 90stabnum! = -.141 * thetadegrees! + 17.46CASE 29.36 TO 48.6499stabnum! = -.533 * thetadegrees! + 36.53CASE 20.12 TO 29.3599stabnum! = -3.07 * thetadegrees! + 111CASE 15.001 TO 20.11999stabnum! = -189.6 * thetadegrees! + 3844CASEIS<= 15stabnum! = 1000000END SELECTELSE‘use phiprime =25 degrees curve from Figure 5.3‘After Taylor, (1948)SELECT CASE thetadegrees!CASE 58.62 TO 90stabnum! = -.212 * thetadegreesi + 24.86CASE 39.11 TO 58.61999stabnum! = -.8975 * thetadegrees! + 65.04CASE 25 TO 39. 10999stabnuml = -6.97 * thetadegrees! + 302.5273CASE IS <=25stabnuml = 1000000END SELECTEND IFEND SUB‘*********************************************************************************FUNCTION surfwidth‘function to calculate the surface widthsurfwidth! = pbed! + COS(theta?) * pbank?END FUNCTIONFUNCTION velocity!‘calculates the mean velocityrhbank! = bankshear! I (gamma! * slope!)rhbed! = bedshear! I (gamma! * slope!)Thank! = (2.03 * LOG(12.2 * rhbank! I ksbank!) I LOG(10)) A -2Ibed! = (2.03 * LOG(12.2 * rhbed! / ksbed!) I LOG(10)) A -2ifactor! = (ibed! * pbed! I (pbed! + pbank!) + Ibank! * pbankl / (pbank! + pbed!)) Avelocity! = (8 * gravity! * hydrad! * slope!) A 5 * ifactor!END FUNCTION274APPENDIX BBANKFULL: VARIABLE SLOPE MODELThe Bankfull:Variable Slope differs from the Fixed Slope model in that the channel slope isnow treated as a dependent variable.The source code for the computer program (varslope.bas) used in the thesis is presented below.The programming was encoded and run using Microsoft Quick Basic (Version 3.0 or later).This program can be used for channels with cohesive or noncohesive bank sediment.The program is designed to use an input data file named varslope.dat, and will write the outputto a data file named varslope.out. The data file can contain data for more tahn one channel. Thedata for each channel must be input as follows:Qbf Gbf ksd ksbk d50 D50 D5Obk sfor channels with noncohesive banks, and:Qbf Gbf ksbd k,bk d50 D50 rt c’ rcritfor channels with cohesive banks.The data relating to each channel must start on a new line. The data values must be separatedby at least one space, although the number of spaces is not important. An example of an inputdat file is presented below:50 5 0.1 0.1 0.025 0.075 0.075 40100 20 0.1 0.1 0.025 0.075 0.05 40275This file contains the input data for two channels with noncohesive banks. The user is requiredto input the bank sediment type (n for noncohesive or c for cohesive) as prompted. The outputdata will written to a data file named varslope.out in the following order:W H 17* S 6The output file resulting from the example input file is:W H * S theta16.5 1.66 1.39 0.0023 32.035.3 1.39 1.26 0.0027 22.2The Bankfiill:Variable Slope model was developed principally for illustrative purposes. Thesource code for the Bankfull:Variable Slope Model is presented below.276BANKFULL:VAR1ABLE SLOPE MODEL FOR BOTH COHESIVE AND NONCOHESIVE BANKSEDIMENT‘(VARSLOPE.BAS)DECLARE SUB stabcurve 0DECLARE SUB bankstabilitycohesive 0DECLARE SUB bedload2 0DECLARE SUB optimum 0DECLARE SUB bankstabilitynoncohesive 0DECLARE SUB bedload 0DECLARE SUB ebrown 0DECLARE FUNCTION bankshear! 0DECLARE FUNCTION bedshear? 0DECLARE FUNCTION shearforce? 0DECLARE SUB continuity 0DECLARE FUNCTION velocity! 0DECLARE FUNCTION hydrad! 0DECLARE FUNCTION meandepth! 0DECLARE FUNCTION area? 0DECLARE FUNCTION surfwidth! 0DECLARE FUNCTION depth! ()program to calculate curves to demonstrate the optimumchannel geometiy for a given set of independent variables‘bankfull: variable slope model for channels with‘both cohesive and noncohesive bank sediment‘Global Variables‘gravity!= gravitational constant gamma?= unit weight of water‘ss1= specific gravity of sediment viscosity!=kinematic viscosity of water‘density!=density of water slope!=channel slope‘pbedl=bed perimeter pbank!=bank perimeter‘theta!=bank angle (radians) dd5O!=median armour grain diameter‘d50!inedian subarinour grain diameter d5Obank!=median grain diameter of bank sediment‘discharge!=bankfull discharge gbload!=imposed bedload transport rate‘gbload!=calculated bedload transport rate ‘neta!coefficient of efficiency‘phiprime!=friction angle of bank sediment (radians)‘phidegrees!=friction angle of bank sediment (degrees)‘ksbcd!=measure of bed roughness ksbank!measure of bank roughness‘banktype$=type of bank sediment Wo=interger counter‘gammat!=unit weight of cohesive bank sediment‘stabnuxn!=stability number for cohesive sediment‘cohesion!=soil cohesion taucrit?=cntical shear stressCOMMON SHARED gravity!, gamma!, ss!, viscosity?, density?COMMON SHARED slope!, pbed?, pbank!, theta!, dd5O!, d5Obank!COMMON SHARED discharge!, gbcalc!, gbload!, phiprime?COMMON SHARED d50!, bankcond$, pi!COMMON SHARED ksbed!, ksbank!, neta!, slopecond$, banktype$, 1%COMN’ION SHARED ganimat!, stabnum!, cohesion!, taucrit?, phidegrees!277OPEN “varslope.dat” FOR iNPUT AS #1OPEN “varslope.out” FOR OUTPUT AS #2‘SET VALUES OF INDEPENDENT CONSTANTSpi! = 3.14159gravity! = 9.81gamma! = 9810viscosity! = .000001density! = 1000ss! =2.65CLSPRINTPRINTPRINT “BANKFULL:VARIABLE SLOPE OPTIMIZATION MODEL”PRINT : INPUT “Hit <ENTER> to Continue “, dummy$=0PRINT #2,” W H * S theta”CLS10PRiNTPRINTPRINTPRINT : INPUT “INPUT BANK SEDIMENT TYPE (a/c) “;banktype$IF banktype$ = “c” OR banktype$ = “C” THENbanktype$ = “cohesive”ELSEIF banktype$ = “N” OR banktype$ = “n” THENbanktype$ = “noncohesive”ELSEBEEPPRINT “Bank Type Unkno”GOTO 10END IFCLSDO WHILE NOT EOF(l)PRINTPRINTI% =1% + 1 ‘counterPRiNT “Channel Number “; WeINPUT #1, discharge!, gbload!, ksbed!, ksbank!, d50!, dd5O!‘Input the independent variables from data fileIF banktype$ = “noncohesive” THEN ‘input bank stability parametersINPUT #1, d50bank!, phidegrees!ELSE ‘banktypeS=”cohesive”iNPUT #1, ganunat!, phidegrees!, cohesion!, taucrit!END IF278phiprimel = phidegrees! * 2 * pit / 360 ‘convert friction angle from degrees to radiansCALL optimum‘calculates the optimal geometiyPRiNTPRINT” W H * S theta”PRiNT USiNG “##*.# #1.## #.## #.## #.#“; surfwidth!; depth!; meandepth!; slope!; theta! * 360 / 2 /pi!PRINT #2, USING “###.# ##.## #.#### ##.#“; surfwidth!; depth!; meandepth!; slope!; theta! * 360 /2/ pi!‘prints out the values of the dependent variablesLOOP‘loops until end of data fileCLOSE #1CLOSE #2ENDFUNCTION area‘function to calculate the cross-sectional areaarea! = .5 * (pbed! + surfwidth!) * depth!END FUNCTIONFUNCTION bankshear‘calculates the bank shear stressbankshear! = gamma! * depth! * slope! * shear.force! * ((surfwidth! + pbed!) * SIN(theta!) / (4 * depth!))END FUNCTIONSUB bankstabilitycohesive‘satisfies bank stability constraint for cohesive banksheightcond$ = “unknown”shearcondS = “unknown”bankcond$ = “unknown”thetaniax! =90*2*pi!/360thetamin! =0‘imtialise search and convergence criteria‘First test ifvertical bank is stable wrt bank heighttheta! = thetamax!CALL continuity‘satisfy continuityCALL stabcurve‘catculate stability number279criticalheight! = stabnum? * cohesion! I ganunat!calculate the critical heightstability!! = depth I criticalheight!‘calculates stability criteria wrt bank height‘if stabilityl! <= criticaiheight then bank is stable‘with respect to bank heightIF stabilityl! < 1.001 THENheightcond$ = “stable”ELSE ‘vertical bank is not stable therefore reduce the bank angle.‘Determine the maximum bank angle that is just stable.‘This is theta max from Chapter 6.DO UNTIL heightcond$ = ‘just stable”theta! = (thetamax! + thetanun!) /2‘calculate midpoint of range for‘bisectrix convergence schemeCALL continuity‘satisfy continuity constraintCALL stabcurve‘calculate stability numbercriticaiheight! = stabnum! * cohesion! / ganunat!stability 1! = depth / criticaiheight!‘calculates critical height and stabilty numberTor the bank height constraintIF stabilityl! >= 1.001 THENheightcond$ = “unstable”thetamax! =theta!ELSEIF stabilityl! < .999 THENheightcond$ = “understable”thetamin! = theta!ELSEheightcond$ = “just stable”‘Bank Height = Critical HeightEND IFIF (thetamax! - thetamin!) / theta! <.001 THENheightcond$ = “just stable”stability!! = 1‘Secondary Convergence Criterion in case of‘convergence problems due to numerical schemeEND IFLOOPEND IF280‘bank height constraint now satisified‘now assess the bank shear constraintthetamin! =0thetaniax! = theta!‘thetamax is reset to maximum bankangle which satisfied the‘bank height constraint abovestability2! = bankshear! / taucrit!‘satbility criterion for bank shear constraintIF stability2! <1.001 THEN‘bank shear constraint is satisifiedshearcond$ = “stable”bankcond$ = “stable”ELSE ‘must reduce theta!DO UNTIL shearcond$ = “just stable”theta? = (thetamax! + thetamin!) /2‘calculate mid pointIFtheta! <10*2*3.14159/360 THEN‘bank angles less than 10 degrees‘nominally assumed to be unstableshearcond$ = “unstable”bankcond$ = “unstable”EXIT DOEND IFCALL continuity‘satisfy continuitystability2! = bankshear! / taucrit!‘calculate stability criterion for bank shear constraintIF stability2! >= 1.001 THEN‘bankshear > taucritshearcondS = “unstable”thetamax! =theta!ELSEIF stability2! <= .999 THENshearcond$ = “understable”thetainin! =theta!ELSEshearcond$ = “just stable”bankcond$ = “stable”END IFIF (thetamax! - thetamin!) / theta! <.001 THEN281shearcond$ = “just stable”bankcond$ = “stable”stability2! = 1END IFLOOPEND IFEND SUBSUB bankstabilitynoncohesive‘calculates theta where banks just stablebankcond$ = “unknown”thetamax! = phiprime’thetamin! =0‘initialise search bounds and convergence criteriaDO UNTIL bankcond$ = ‘just stable”theta! = (thetamax’ ÷ thetamin!) /2‘determine midpoint of search rangelFtheta! <5 * 2 * 3.14159/360 THENbankcond$ = “unstable”EXIT DO‘if theta is less than 5 degrees the channel is assumed‘to be unstableEND IFCALL continuity‘satisfies continuity constraintstability! = (bankshear! / (gamma! * (ss! - 1) * d50bank!)) / (.048 * TAN(phiprime!) * (1- SIN(theta!) A 2/S1N(phiprime!) A 2) A .5)‘calculates stability criterionIF stability! >= 1.00 1 THENbankcond$ = “unstable”thetamax! =theta!ELSEIF stability! < .999 THENbankcond$ = “stable”thetamin! = theta!ELSEbankcond$ = “just stable”‘primary convergence criterionENDIFIF (thetamax! - thetainin!) I theta! <.001 THENbankcond$ = ‘just stable”stability! = 1282‘secondary convergence criterionEND IFLOOPEND SUBSUB bedload‘satisfies the bedload constraint using the Einstein Brown sedimenttransport relationgbcalc!=0minslope’ =0maxslope! = .05‘set bound of slope searchslopecond$ “unknown”‘initialise slope conditionDO WHILE slopecond$ <> “satisfied”slope? = (minslope! + maxslope!) I 2‘calculate mid point of range for bisectrix convergence schemeIF banktype$ = “noncohesive” THENCALL bankstabilitynoncohesive‘solve bank stability constraint for noncohesive banksELSE ‘banktype$ = “cohesive”CALL bankstabilitycohesive‘solve bank stability constraint for cohesive banksEND IFIF bankcond$ <> “unstable” THEN‘bank stability is satisfiedCALL ebrown‘calculate sediment transporting capacity of trial channelIF gbcalc! <.999 * gbload! THEN‘compares calculated to imposed sediment loadslopecond$ = “too flat”minslope! = slope!ELSEIF gbcalc!> 1.001 * gbload? THENslopecondS “too steep”maxslope! = slope!ELSEslopecond$ = “satisfied”‘primary convergence criterion‘sediment transporting capacity of the channel‘equals the imposed load bedload constraint satisfiedEND IFIF (maxslope! - minslope!) / slope! <.0001 THENIF gbcalc! > .99 * gbload! THENslopecondS = “satisfied”‘secondary convergence criterion283‘when bounds of slope very small: convergence attained‘occasionally a problem with convergence due to numerical‘approximationsELSEslopecond$ = “not satisfied”EXIT DO‘indicates that trial Pbed value is too narrowEND IFEND IFELSEmaxslope! = slope!‘trial slope too steep: reduce upper boundslopecond$ = “unstable”END IFLOOPEND SUBSUB bedload2‘identical to SUB bedload except search range for slope!‘reduced to save on computations. Optimum slope! for forward‘difference will be very close to that determined forbackward differenceminslope! = 9 * slope!maxslope! = 1.1 * slope!‘set bound of slope searchslopecond$ = “unknown”‘initialise slope conditionDO WHILE slopecond$ <> “satisfied”slope! = (minslope! + maxslope!) /2‘calculate mid point of range for bisectrix convergence schemeIF banktype$ = “noncohesive” THENCALL bankstabilitynoncohesive‘solve bank stability constraint for noncohesive banksELSE banktype$ = “cohesive”CALL bankstabilitycohesive‘solve bank stability constraint for cohesive banksEND IFIF bankcond$ <> “unstable” THEN‘ifbank stability is satisfiedCALL ebrown‘calculate sediment transporting capacity of trial channelIF gbcalc! <.999 * gbload! THEN‘compares calculated to imposed sediment loadslopecond$ = “too flat”minslope! = slope!ELSEIF gbcalc!> 1.001 * gbload! THEN284slopecondS = “too steep”maxslope? = slope!ELSEslopecond$ = “satisfied”‘sediment transporting capacity of the channel‘equals the imposed load : bedload constraint satisfiedEND IFIF (maxslope! - minslope!) / slope! <.0001 THEN slopecond$ = “satisfied”‘secondary convergence criterion‘when bounds of slope veiy small: convergence attained‘occasionally a problem with convergence due to numerical‘approximationsELSE‘if bank stability constraint not satisfiedmaxslope! = slope!‘trial slope too steep: reduce upper boundslopecond$ = “unstable”END IFLOOPEND SUBFUNCTION bedshear‘calculates the bed shear stressbedshear! = gamma! * depth! * slope! * (1 - shearforce!) * ((surfwidth! / (2 * pbed!) + .5))END FUNCTIONSUB continuity‘varies pbank! for trial values of Pbed!, slope!, and theta! to‘satisfy the contiuity constraintpbankcond$ = “unknown”errorcalci = 1000minpbank! 0maxpbank! =20 * discharge! “.35‘initialise search and convergence criteriaDO UNTIL pbankcond$ = “OK”pbank’ = (minpbank! + maxpbank!) /2‘calculate midpointerrorcalc! = (area * velocity / discharge!)‘calculate the normalised errorIF errorcalc! > 1.001 THENpbankcond$ = “too large”285maxpbank! = pbank!ELSEIF errorcaic! <.999 THENpbankcond$ = “too small”minpbank! = pbank!ELSEpbankcond$ = “OK”END IFIF (maxpbankl - minpbank’) / pbank? <.0001 AND pbankcond$ “too small” THEN‘resets maxpbank’ if too smallmaxpbank? =2* maxpbank!END IFLOOPEND SUBFUNCTION depth‘function to calculate flow depth of a trapezoidal channeldepth! = .5 * STN(theta!) * pbank’END FUNCTIONSUB ebrown‘calculate the sed trans capacity using em-brown formuladimlessbedshear! = bedshear! / (gamma! * (ss! - 1) * d50t)IF dimlessbedshear! <=0 THENdimlessbedload? = 0ELSEIF dimlessbedshear! <.093 THENdimlessbedload! = 2.15 * EXP(-.391 / dimlessbedshear’)ELSEdimlessbedload! = 40 * dimlessbedshear! “ 3END IFfi! = (2/3 + 36 * viscosity! A 2 / (gravity! * d50’ A 3 * (ss! - 1))) A 5 -(36 * viscosity! A 2 / (gravity! * dSO? A 3*(ss!_1)))gbcalc! =pbed! *fl! *! *deusity! *((ssi - 1)*gravity! *d5o! A3)A.5 *dje55dicadfEND SUBFUNCTION hydrad!‘function to calculate the hydraulic radiushydrad! area! / (pbed! + pbank!)END FUNCTIONFUNCTION meandepth‘function to calculate the mean depth286meandepth! = area / surfwidthEND FUNCTIONSUB optimum‘determines the optimal geometryoptcond$ = “unknown”‘initialises optimality test conditionlowerpbedl =0upperpbedl = 10 * discharge? A‘set bounds for Pbed based on Regime Eqns‘typical optimal value of Pbed is 2 to 5 * discharge Aminpbed! = lowerpbed?maxpbed! = upperpbedlPRINTPRINTDO WHILE optcond$ <> “optimum”pbed? = (minpbedl + niaxpbed!) /2‘calculate mid point of search rangeIFpbed’ > .95 * upperpbed! TI-lENupperpbedl =2 * upperpbed?maxpbed! = upperpbed?‘reset upper bound if necessaryEND IFPRiNT “Assessing Trial Bed Perimeter “;PRINT USING “#####“; pbed?;PRINT “ mpbed’ =pbed? * .975‘calculate backwards difference value of PbedCALL bedload‘satisfies continuity, bank stability and bedload‘constraints for the trail Pbed valueIF slopecond$ = “satisfied” AND bankcond$ <> “unstable” THEN‘test conditions indicate that stable channel geometry‘has been determined and the bedload constraint‘has been satisifiednetal! = gbload! / (density! * discharge! * slope!)‘evaluate neta for backward difference pointpbed! =pbed! / .975 * 1.025‘calculate pbed for forward duff287CALL bedload2‘satisfies continuity, bank stability and bedload‘constraints for the trail Pbed value‘indentical to Sub Bedload except the bounds of‘the slope search has been reduced to reduce computationsneta2 = gbload! / (density! * discharge! * slope!)‘calculates neta for forward difference valuepbed’ pbed! / 1.025‘reset pbed to midpoint valuednetabydpbed! = (neta2! - netal!) / (.05 * pbed!)‘calculate first derivative by finite difference‘to assess optimum conditionIF dnetabydpbed! <0 THEN ‘Trial Pbed is too widemaxpbed! = pbed! ‘Reduce upper bound of Pbedoptcond$ = “too wide”ELSETF dnetabydpbed! >0 THEN ‘Thai Pbed is too narrowminpbedt= pbed! ‘Increase lower bound of Pbedoptcond$ = “too narrow”ELSEoptcond$ = “optimum” ‘Optimum AchievedEND IFIF (maxpbed! - minpbed!) / pbed! <.001 THEN optcond$ = “optimum”‘Convergence attained. Second optimality criterionELSE ‘tnal Pbed too small for stable geomeuy: Increase minimum Pbedpbed! = pbed! I .975‘reset from backward difference to correct valueminpbed! =pbed!‘set lower limit at current trial value‘(as the optimum value must be greater)optcond$ = “too narrow”END IFPRINT optcond$LOOP‘Evaluate geometiy at exact optimal Pbed‘(not at forward or backward difference value)IF banktype$ = “noncohesive” THENCALL bankstabilitynoncohesive‘solve bank stability constraint for noncohesive banksELSE ‘banktype$ = “cohesive”CALL bankstabilitycohesive‘solve bank stability constraint for cohesive banksEND IF288CALL bedload2neta = gbload! / (density! * discharge! * slope!)‘evaluate objective functionEND SUBFUNCTION shearforce‘calculates the proportion of shear force on the banksshearforce! = 1.766 * (pbed! / pbank! + 1.5) A -1.4026END FUNCTIONSUB stabcurve‘stability curves are from Figure 5.3thetadegrees! = theta! / 2 / 3.14159 * 360iF phidegrees! <20 THEN‘use phiprime = 15 degree stability curve from Figure 5.3‘from Taylor (1948)SELECT CASE thetadegrees!CASE 48.65 TO 90stabnum! = -.141 * thetadegrees! + 17.46CASE 29.36 TO 48.6499stabnum! = -.533 * thetadegrees! + 36.53CASE 20.12 TO 29.3599stabnum! = -3.07 * thetadegrees! + 111CASE 15.001 TO 20.11999stabnum! = -189.6 * thetadegrees! + 3844CASE IS <= 15stabnum! = 1000000END SELECTELSE‘use phiprime =25 degrees curve from Figure 5.3After Taylor, (1948)SELECT CASE thetadegrees!CASE 58.62 TO 90stabnum! = -.212 * thetadegrees! + 24.86CASE 39.11 TO 58.61999stabnum? = -.8975 * thetadegrees! + 65.04CASE 25 TO 39. 10999289stabnum = -6.97 * thetadegrees! + 302.5CASE IS <=25stabnum? = 1000000END SELECTEND IFEND SUBFUNCTION surfwidthTunction to calculate the surface widthsurfwidth? = pbed? + COS(theta’) * pbankEND FUNCTIONFUNCTION velocity!‘calculates the mean velocityrhbank! = bankshear! / (gamma! * slope!)rhbed! = bedshear! / (gamma! * slope!)Ibank! = (2.03 * LOG(12.2 * rhbank! / ksbank!) / LOG(10)) ‘ -2ibed! = (2.03 * LOG(12.2 * rhbed! / ksbed!) / LOG(10)) A -2ifactor! = (ibed! * pbed! / (pbed! + pbank!) + fipJç! * pbank! / (pbank! + pbed!)) Avelocity! = (8 * gravity! * hydrad! * slope!) A •5 * ifactor!END FUNCTION290APPENDIX CFULL MODEL FORMULATIONThe Full Model Formulation differs from the Bankfull:Variable Slope Model in that the fillflow duration data is used as input, the sediment transporting capacity of the channel iscalculated over one year using the modified Parker (1990) surface-based sediment transportrelation, and the size distribution of the armour layer is treated as a dependent variable.The source code for the computer program (rivmod6.bas) used in the thesis is presented below.The programming was encoded and run using Microsoft Quick Basic (Version 3.0 or later).This program can be used for channels with cohesive or noncohesive bank sediment.The program is designed to use an input data ifie named rivmod.dat, and will write the outputdata to a data file named rivmod.out. The input file contains data for one channel only and mustbe set-up as follows (for noncohesive bank sediment):291m n Qbf Gb k3d k$bk banktype1.0Q Probability of Q1U Being Equalled or ExceededQ Probability of Q Being Equalled or ExceededQ Probability of Q Being Equalled or ExceededQ. Probability of Q’ Being Equalled or Exceeded0D 0D1 Proportion Finer than DD Proportion Finer than DD Proportion Finer than D’D Proportion Finer than DD 1.0D5Ob,For noncohesive channel banks the banktype is equal to “n”, and for cohesive channels it isequal to “c”.For cohesive banks the last line of the data file becomes:Ti C’ critIn rivmod.dat m is the number of flow intervals in the numerical approximation of a flowduration curve (See Figure 7.1); n is the number of intervals in the numerical approximation ofa sediment gradation curve (See Figure 7.2); Q is the discharge at the lower bound of flowinterval 1 which is the lowest flow on record (or an approximate value), Q, is the discharge at292the upper bound of flow interval m and therefore the maximum flow on record (or anapproximation); D is equal to D0, and D is equal to D1. In both cases the superscript 1 or urefer to the lower or upper bound of the interval, and the subscript indicates the intervalnumber. The upper bound of one interval is equal to the lower bound of the next interval. Thedata for the input file is designed to be read directly from flow duration and sediment gradationcurves. The geometric mean values for each interval are calculated in the program.The other symbols used above have been defined numerous times in the text and are given inthe List of Symbols.The data values must be separated by at least one space although the number of spaces is notimportant. An example of an input file is given below:11 7 85 2500000 0.4 0.4 ii3 122 0.247342 0.106455 0.056965 0.036575 0.024785 0,017290 0.01505205 0.0095115 0.007125 0.0054205 00.002 00.0064 0.10.01 0.20.018 0.40.032 0.60.06 0.80.091 0.90.15 10.07 40The output data will be output to rivmod. out in the following order:293W H 17* S D50 9 T*D50The output data from the sample input data file given above is:W H * S D50 theta T*D5037.7 1.29 1.16 0.00427 0.073 19.1 0.043The source code for the Full Optimisation Model is presented below.‘OPTIMISATION MODEL BASED ON FULLY DEVELOPED MODELDESCRIBED IN PhD THESIS BY RG MILLAR‘UNIVERSITY OF BRiTISH COLUMBIA‘SEPTEMBER 1994‘CALCULATES THE OPTIMUM (OR EQUILIBRIUM) CHANNEL GEOMETRYDECLARE SUB stabcurve 0DECLARE SUB bedload2 0DECLARE SUB bankstabilitynoncohesive 0DECLARE SUB bankstabiitycohesive 0DECLARE SUB parkerl99O 0DECLARE SUB capG 0DECLARE SUB vaiytheta 0DECLARE SUB optimum 0DECLARE SUB bedload 0DECLARE FUNCTION bankshear! 0DECLARE FUNCTION bedshear! 0DECLARE FUNCTION shearforce! 0DECLARE SUB continuity (discharge!)DECLARE FUNCTION velocity! 0DECLARE FUNCTION hydrad! 0DECLARE FUNCTION meandepth! 0DECLARE FUNCTION area! 0DECLARE FUNCTION surfwidth! 0DECLARE FUNCTION depth! 0program to calculate curves to demonstrate the optimumCOMMON SHARED gravity!, gamma!, ss!, viscosity!, density!, meandischargelCOMMON SHARED slope!, pbed!, pbank!, theta!, ddSO!, d50bank!COMMON SHARED qbfl, gbcalc!, gbload!, phiprime!, stability!, phidegreesCOMMON SHARED thetacond$, xnincond$, alpha!, dSO!, bankcond$, pi!COMMON SHARED ksbed!, ksbank!, phisgo!, sigmaphi!, sigxnaphitiial!, dsgttial!COMMON SHARED capgamma!, capitalg!, dsg!, i%, j%, m%, n%, iefi%COMMON SHARED banktype$, slopecond$‘Global Variables‘gravlty!= gravitational constant gamma! unit weight of water‘ss!= specific gravity of sediment viscosity!=kinematic viscosity of water‘density!=density of water slope!channel slope‘pbed!=bed perimeter pbank!bank perimeter294‘theta!=bank angle (radians) ddSO!=inedian armour grain diameter‘dSO!—median subarmour grain diameter d5Obank!median grain diameter of bank sediment‘discharge!=bankfufl discharge gbload’=imposed bedload transport rate‘gbload!=calculated bedload transport rate ‘neta’=coefficient of efficiency‘phiprime!=friction angle of bank sediment (radians)‘phidegrees!=friction angle ofbank sediment (degrees)‘ksbed!measure ofbed roughness ksbank!=measure ofbank roughness‘banktype$=type of bank sediment Wo=interger counter‘gammat!=unit weight of cohesive bank sediment‘stabnum!=stability number for cohesive sediment‘cohesion!=soil cohesion taucrit!=critical shear stress‘meandischarge!=mean annual discharge‘phisgol=dimensionless bed shear stress (Eqn Eqn 4.31)‘sigmaphi!=grainsize dispersion of armour (Eqn 4.37)‘sigmaphitrial!=trial value of sigmaphil‘dsg!=geometric mean grain diameter of pavement layer (Eqn 4.35)‘capgamtna! = constant converts kg/sec to kg/year‘capitalgl=dimensionless bedload function Eqn (4.32)‘j%ith discharge j%jth sediment size‘m%=numder of flow intervals n%=number of sediment size intervals‘SET VALUES OF INDEPENDENT CONSTANTSpi! = 3.14159gravity! = 9.81gamma! = 9810viscosity! = .000001density! = 1000ss! =2.65capganuna! = 365.25 * 24 * 3600CLSPRINTPRINTPRINTPRINT” RIVERMOD AN OPTIMISATION MODEL FOR”PRINT” PREDICTING THE ADJUSTMENT OF ALLUVIAL RIVERS”PRINTPRINTPRINTPRJNT” WR1TEN BY”PRINT” ROBERT MILLAR”PRINTPRINT” UNIVERSITY OF BRITISH COLUMBIA”PRINTPRINTPRINTPRINTPRINT” HiT <ENTER> TO CONTINUE”SLEEPINPUT dummyOPEN “nvmod.dat” FOR INPUT AS #1295OPEN “rivmod.out” FOR OUTPUT AS #2iNPUT #1, m%, n%, qbfl, gbload!, ksbed!, ksbanld, banktype$‘input number of flow and sediment size intervals respectively‘bankfufl discharge, mean annual sediment loadbed and bank roughness heights‘and bank typeIF bankt,e$ = “c” OR banktype$ = “C” THENbanktype$ = “cohesive”ELSEIF banktype$ = “N” OR banktype$ = “n” THENbanktype$ = “noncohesive”ELSEBEEPCLSPRINTPRINTPRiNTPRiNT “ERROR iN DATA FILE”ENDEND IF‘*4c************************************4*******************************************DIM SHARED sediment(n%), f(n%), ff(n%), discharge(m%), qprob(m%), upperd(n%)DIM SHARED propfiner(n%)DIM SHARED qb(m%, n%), qbi(m%), upperq(m%), fload(n%), bedsheari(m%)DIM SHARED probexceed(m%), depthi(m%)‘declare shared array variablesi% =0DOWHILEi%<m%iNPUT #1, upperq(i%), probexceed(i%)‘input flow data into array‘upper bound of flow interval and probability of exceedence‘i%=O corresponds to lower bound, m% to the maximum flowi% = i% + 1LOOPj% =0DO WHILEj% < n%INPUT #1, upperd(j%), propflner(j%)‘input sediment data into array‘upper bound of sediment fraction an dvolumetric proportionj%() corresponds to lower bound ie dOj%% corresponds to dlOOj%=j%+ 1LOOPCLSPRINT “DATA SUMMARY”PRINTPRINT296PRiNT” Discharge Data”PRINT”i Qi pi”‘calculate geometric mean flow and probability for each intervali% = 1qprob(O) =0DO WHILE i% <= m%discharge(i%) = (upperq(i%) * upperq(i% - 1)) ‘ .5qprob(i%) = probexceed(i% - 1) - probexceed(i%)PRINT USiNG “## ##L# #.###“; i%; discharge(i%); qprob(i%)i% = i% + 1LOOPPRINTPRINTPRINT” HIT <ENTER> TO CONTINUE”SLEEPINPUT dununy$CLSPRINT “DATA SUMMARY”PRINTPRINTPRINT” Sediment Data”PRINT”j Di fl”‘calculate geometric mean grain diameter for each grain interval‘and volumetric proportion within each intervalj% = 1f(O) = 0DO WHTLEj% <= n%sediment(j%) = (upperd(j%) * upperd(j% - 1)) Af(j%) = propfiner(j%) - propfiner(j% - 1)PRINT USING “#41 #.###41 #.###“; j%; sediment(j%); f(j%)j%=j%+ 1LOOPPRINT‘calculate mean annual dischargei% = 1totalq!=ODOWHILEi%<=m%totalq! = totalq! + discharge’(i%) * ciprob(i%)i% = i% + 1LOOPmeandischarge! = totalq!PRINT USING ‘Mean Annual Discharge ##*#.# m”3/sec”; meandischargePRINT USING “Mean Annual Sediment Load II 111111111111 .41 kg/year”; gbload’‘input bank stability parametersIF banktype$ = “cohesive” OR banktype$ = “COHESIVE” THENbanktype$ = “cohesive”ELSEIF banktype$ = “NONCOHESIVE” OR banktype$ = “noncohesive” THENbanktype$ = “noncohesive”ELSEPRINT “Banktype unknoii: Error in Data File”BEEPCLOSE #1297CLOSE #2ENDEND IFPRINTPRINT “Bank Sediment Type:IF banktype$ = “noncohesive” THEN ‘input bank stability parametersINPUT #1, d5Obank!, phidegrees!PRINT USING “D50 bankPRINT USING “Phi#.### rn”; d50bank!##.# degrees”; phidegrees!ELSE ‘banktype$=”cohesive”INPUT #1, gzmmt!, phidegrees!, cohesion!, taucrit!PRiNT USING “ganunatPRINT USING “PhiPRINT USING “cohesionPRiNT USING “Tau critEND IF##.## kN/m”2”; gammat! / 1000#W.# degrees”; phidegrees!#### J/jA”; cohesion! / 1000##.# N/m’2”; taucrit!phiprime! = phidegrees! * 2 * pi! I 360PRINTPRINT : INPUT “IS INPUT DATA OK (y/n) “; booleanSIF boolean$ = “n” OR boolean$ = “N” THENCLOSE #1CLOSE #2ENDEND IFCLSPRINTPRINTdsgtrial! = (upperd(0) + upperd(n%)) /2‘set trial dsg value at midpoint of sediment gradationsigmaphitrial! = 1‘set trial sigamphi value‘COMMENCE OPTIMIZATION ROUTINECALL optimumPRINTPRINT” W H * S D50 theta T*D50”PRINT USING “###.# #4.## ##.#4 #.IIihf liii #.### ##.# #.###“; surfwidth!; depth!; meandepth!; slope!;dd5O!; theta! * 360 / 2 / pi!; bedshear! / (gamma! * dd5O! * (! - 1))PRINT #2,” W H Y* S D50 theta T*D50”298PRiNT #2, USING “##4t.# ##.## ##.## #.ihIIIiII( #.### ##.# #.###“; surfwidth?; depth!; meandepth!; slope!;ddSO!; theta! * 360 / 2 /pi!; bedshear / (gamma! * dd5O! * (ss! - 1))‘prints out the values of the dependent variablesCLOSE #1CLOSE #2END,**********************************************************************************FUNCTION area‘function to calculate the cross-sectional areaarea!=.5*(pbed!+surfwidth!)*depth!END FUNCTIONFUNCTION bankshear‘calculates the bank shear stressbankshear! = gamma! * depth! * slope! * shearforce! * ((surfwidth! + pbed!) * S1N(theta!) / (4 * depth!))END FUNCTIONSUB bankstabilitycohesive‘satisfies bank stability constraint for cohesive banksheightcond$ = “unknown”shearcond$ = “unknown”bankcond$ = “unknown”thetamax!=90*2*pi!/360thetamin! =0‘initialise search and convergence criteriaTirst test ifvertical bank is stable wrt bank heighttheta! = thetamax!CALL continuity(qbf!)‘satisfy continuityCALL stabcuive‘catculate stability numbercriticalheight! = stabnum! * cohesion! / gmmatI‘calculate the critical heightstabilityl! = depth / criticaiheight!‘calculates stability criteria wit bank height‘if stabilityl! <= criticaiheight then bank is stable‘with respect to bank heightIF stabilityl! <= 1.001 THENheightcondS = “stable”299ELSE ‘vertical bank is not stable therefore reduce the bank angle.Determine the maximum bank angle that is just stable.‘This is theta max from Chapter 6.DO UNTIL heightcond$ = ‘just stable”theta! = (thetamax! + thetamin!) /2‘calculate midpoint of range for‘bisectrix convergence schemeCALL continuity(qbfl)‘satisfy continuity constraintCALL stabcurve‘calculate stability numbercriticaiheight! = stabnum! * cohesion! / ganimat!stability 1! = depth / criticalheight!‘calculates critical height and stabilty number‘for the bank height constraintIF stabilityl! > 1.001 THENheightcond$ = “unstable”thetamax! = theta!ELSEIF stabiityl! <= .999 THENheightcond$ = “understable”thetamin! = theta!ELSEheightcond$ = ‘just stable”‘Bank Height = Critical HeightEN])IFIF (thetainax! - thetainin!) / theta! <.001 THENheightcond$ = ‘just stable”stabilityl! = 1‘Secondary Convergence Criterion in case of‘convergence problems due to numerical schemeEND IFLOOPEND IF‘bank height constraint now satisifled‘now assess the bank shear constraintthetamin!=0thetaniax! = theta!‘thetaniax is reset to maximum bankangle which satisfied the‘bank height constraint abovestabilily2! = bankshear! / taucrit!‘satbility criterion for bank shear constraint300IF stability2! < 1.001 THEN‘bank shear constraint is satisifiedshearcond$ = “stable”bankcond$ = “stable”ELSE ‘must reduce theta!DO UNTIL shearcond$ = “just stable”theta! =(thetaniax! +thetaniiu!)/2‘calculate mid pointIF theta! <10 * 2 * 3.14159/360 THENbank angles less than 10 degrees‘nominally assumed to be unstableshearcond$ = “unstable”bankcond$ = “unstable”EXIT DOEND IFCALL continuity(qbfi)‘satisfy continuitystability2! = bankshear! / taucrit!‘calculate stability criterion for bank shear constraintIF stability2! >= 1.001 THEN‘bankshear > taucntshearcond$ = “unstable”thetainax! =theta!ELSEIF stabiity2! <= .999 THENshearcond$ “understable”thetaniin! = theta!ELSEshearcondS = “just stable”bankcond$ = “stable”END IFIF (thetamax! - thetaniin!) / theta! <.001 THENshearcond$ = “just stable”bankcond$ = “stable”stability2! =1END iFLOOPEND IFEND SUB301‘**********************************************************************************SUB bankstabiitynoncohesive‘calculates theta where banks just stablebankcond$ = “unknown”thetamax! = phiprime’thetaniin! =0‘initialise search bounds and convergence criteriaDO UNTIL bankcond$ = ‘just stable”theta =(thetamax! +thetaniin!)/2‘determine midpoint of search rangeIF theta! <5*2*3.14159/360 THENbankcond$ = “unstable”EXIT DO‘if theta is less than 5 degrees the channel is assumed‘to be unstableEND IFCALL continuity(qbfl)‘satisfies continuity constraint for qbfstability! = (bankshear! / (gamma! * (ss! - 1) * d50bank!)) / (.048 * TAN(phiprime’) * (1 - S1N(theta!) “2/SIN(phiprime!) A 2) .5)‘calculates stability criterionIF stability! >= 1.001 THENbankcond$ = “unstable”thetamax! =theta!ELSEEF stability! <= .999 THENbankcond$ = “stable”thetamin! = theta!ELSEbankcond$ = “just stable”‘pnmaiy convergence criterionEND IFIF (thetamax! - thetamin!) / theta! <.001 THENbankcond$ = “just stable”stability! = 1‘secondaiy convergence criterionENDIFLOOPEND SUBSUB bedload‘satisfies the bedload constraint using the Parker 1990302‘surface-based sediment transport relationgbcalc! =0minslope! =0maxslope! = .05‘set bound of slope searchslopecond$ = “unknown”‘initialise slope conditionDO WHILE slopecond$ “satisfied”slope! = (minslope! + maxslope!) /2‘calculate mid point of range for bisectrix convergence schemeIF banktype$ = “noncohesive” THENCALL bankstabilitynoncohesive‘solve bank stability constraint for noncohesive banksELSE ‘banktype$ = “cohesive”CALL bankstabiitycohesive‘solve bank stability constraint for cohesive banksEND IFIF bankcond$ “unstable” THEN‘bank stability is satisfiedCALL parker 1990‘calculate sediment transporting capacity of trial channelIF gbcalc! <.999 * gbload! THEN‘compares calculated to imposed sediment loadslopecond$ = “too flat”minslope! = slope!ELSEIF gbcalc! > 1.001 * gbload! THENslopecond$ = “too steep”maxslope! = slope!ELSEslopecond$ = “satisfied”‘primary convergence criterion‘sediment transporting capacity of the channel‘equals the imposed load : bedload constraint satisfiedEN]) IFIF (maxslope! - minslope!) / slope! <.00001 THENlFgbcalc! >.99*gbload! THENslopecond$ = “satisfied”‘secondary convergence criterion‘when bounds of slope very small: convergence attained‘occasionally a problem with convergence due to numerical‘approximationsELSEslopecond$ = “not satisfied”EXIT DO‘indicates that trial Pbed value is too narrowEND IF303END IFELSEmaxslope! = slope!‘trial slope too steep: reduce upper boundslopecond$ = “unstable”END IFLOOPEND SUBSUB bedload2‘identical to SUB bedload except search range for slope!‘reduced to save on computations. Optimum slope! for forward‘difference will be very close to that determined for‘backward differenceminslope! = •9 * slope!maxslope! = 1.1 * slope!‘set bound of slope searchslopecond$ = “unknown”‘initialise slope conditionDO WHILE slopecond$ <> “satisfied”slope! = (minslope! + maxslope!) /2‘calculate mid point of range for bisectrix convergence schemeIF banktype$ = “noncohesive” THENCALL bankstabiitynoncohesive‘solve bank stability constraint for noncohesive banksELSE banktype$ = “cohesive”CALL bankstabiitycohesive‘solve bank stability constraint for cohesive banksEND IFIF bankcond$ “unstable” THEN‘ifbank stability is satisfiedCALL parker 1990‘calculate sediment transporting capacity of trial channelIFgbcalc! <.999*gbload! THEN‘compares calculated to imposed sediment loadslopecond$ = Htoo flat”minslope! = slope!ELSEIF gbcalc! > 1.001 * gbload! THENslopecond$ = “too steep”maxslopel = slope!ELSEslopecondS = “satisfied”‘sediment transporting capacity of the channel‘equals the imposed load : bedload constraint satisfiedEND IF304IF (maxslope! - minslope!) / slope! <.00001 THEN slopecond$ = “satisfied”‘secondary convergence criterion‘when bounds of slope very small: convergence attained‘occasionally a problem with convergence due to numerical‘approximationsELSE‘ifbank stability constraint not satisfiedrnaxslope! = slope!‘trial slope too steep: reduce upper boundslopecond$ = “unstable”END iFLOOPEND SUB‘**********************************************************************************FUNCTION bedshear‘calculates the bed shear stressbedshear! = ganuna! * depth? * slope! * (1 - shearforce!) * ((surfwidth! / (2 * pbed!) + .5))END FUNCTIONSUB capG‘calculate straining functionSELECT CASE phisgo!‘calculate omegao! based on piecewise linearisation of‘function given in Parker (1990) (See figure 4.10 in thesis)CASE IS <= 1.033omegao! = 1.011CASE IS <= 5.44omegao! = 1.027 * phisgo! A -.483CASE IS > 5.44omegao! = .453END SELECTSELECT CASE phisgo!‘calculate sigmaphio based on piecewise linearisation of‘function given in Parker (1990) (See figure 4.10 in thesis)CASE IS < .985sigmaphio = .816CASE IS <= 2.19sigmaphio! = .395 * phisgo! + .426CASE IS <=7sigiuaphio’ = .044 * phisgo! + 1.194CASE IS>7signiaphio! = 1.501END SELECT305straining! = 1 + sigmaphil I sigmaphio! * (omegao! - 1)‘Eqn4.38‘calculate hiding functionhiding! = (sediment(j%) / dsg!) A -.095 1‘Eqn 4,34 in thesismodifiedphi! = phisgo! * hiding! * straining!‘modifiedphi! is the product inside the square brackets‘in Eqn 4.33SELECT CASE modifiedphi!‘calculate Capital G: dimensionless bedad function‘Eqn 4.32 in thesisCASE IS < 1capitaig! =modifiedphi! A 14.2CASE IS <= 1.59capitalg! = EXP(14.2 * (modifiedphi! - 1) - 9.28 * (modifiedphi! - 1) A 2)CASE IS> 1.59capitaig! =5474*(1853/mjfipbj!)A45END SELECTEND SUBSUB continuity (discharge!)‘varies pbank! for trial values of Pbed!, slope!, and theta! to‘satisfy the contiuity constraintpbankcond$ = “unknown”errorcalcl = 1000minpbank! =0maxpbank! = 20 * discharge! A 35‘initialise search and convergence criteriaDO UNTIL pbankcond$ = “OK”pbank! = (minpbank! + maxpbank!) /2‘calculate midpointerrorcalc! = (area * velocity / discharge!)‘calculate the normalised errorIF errorcaic! > 1.001 THENpbankcond$ = “too large”niaxpbank! = pbank!ELSEIF errorcalc! <.999 THENpbankcond$ = “too small”minpbank! = pbank!306ELSEpbankcond$ = “OK”END IFIF (maxpbank! - minpbank!) / pbank! <.0001 AND pbankcond$ = “too small” THEN‘resets maxpbánk! if too smallmaxpbank! =2* maxpbank!END IFLOOPEND SUBFUNCTION depth‘ftmction to calculate flow depth of a trapezoidal channeldepth! = .5 * S1N(theta!) * phank!END FUNCTIONFUNCTION hydrad!‘function to calculate the hydraulic radiushydrad! =area! /(pbed! +pbank!)END FUNCTIONFUNCTION meandepth‘function to calculate the mean depthmeandepth! = area / surfwidthEND FUNCTIONSUB optimum‘determines the optimal geometryoptcond$ = “unknown”‘initialises optimality test conditionlowerpbed! =0upperpbed! = 10 * qbf! “.5‘set bounds for Pbed based on Regime Eqns‘typical optimal value of Pbed is 2 to 5 * discharge” .5minpbed! = lowerpbed!maxpbed! = upperpbed!PRINTPRINTDO WHILE optcond$ <> “optimum”307pbed! = (minpbed! + maxpbed!) /2‘calculate mid point of search rangePRINT “Assessing Trial Bed Perimeter “;PRINT USING “#### ##“; pbedPRINT “m “;pbed! = pbed! * .975‘calculate backwards difference value of PbedIFp > 95 * upperpbed! THENupperpbed! =2 * upperpbedlmaxpbed’ = upperpbed!‘reset upper bound if necessalyEND IFCALL bedload‘satisfies continuity, bank stability and bedload‘constraints for the trail Pbed valueIF slopecond$ = “satisfied” AND bankcond$ <> “unstable” THEN‘test conditions indicate that stable channel geometty‘has been determined and the bedload constraint‘has been satisifiednetal! = gbload! I (capgamma! * density! * meandischarge! * slope!)‘evaluate nets for backward difference pointpbed! =pbed! 1.975 * 1.025‘calculate pbed for forward duffCALL bedload2‘satisfies continuity, bank stability and bedload‘constraints for the trail Pbed value‘indentical to Sub Bedload except the bounds of‘the slope search has been reduced to reduce computationsneta2! = gbload! / (capgainma! * density! * meandischarge! * slope!)‘calculates nets for forward difference valuepbed! = pbed! / 1.025‘reset pbed to midpoint valuednetabydpbed! = (neta2!- netal!) / (.05 * phed!)‘calculate first derivative by finite difference‘to assess optimum conditionIF dnetabydpbed! <0 THEN ‘Trial Pbed is too widemaxpbed! =pbedl ‘ReduceupperboundofPbedoptcond$ “too wide”ELSEIF dnetabydpbed! >0 THEN ‘Trial Pbed is too narrowminpbed! = pbed! ‘Increase lower bound of Pbedoptcond$ = “too narrow”ELSE308optcond$ = “optimum” ‘Optimum AchievedEND IFIF (maxpbed - nunpbedl) / pbed’ <.001 THEN optcond$ = “optimum”‘Convergence attained. Second optimality criterionELSEtrial Pbed too small for stable geometry: Increase minimum Pbedpbed! =pbed’ 1.975‘reset from backward difference to correct valueminpbed! = pbed!‘set lower limit at current trial value‘(as the optimum value must be greater)optcond$ = “too narrow”END IFPRINT optcond$LOOP‘Evaluate geometry at exact optimal Pbed‘(not at forward or backward difference value)IF banktype$ = “noncohesive” THENCALL bankstabilitynoncohesive‘solve bank stability constraint for noncohesive banksELSE ‘banktype$ = “cohesive”CALL bankstabilitycohesive‘solve bank stability constraint for cohesive banksEND IFCALL bedload2‘satisfies continuity, bank stability and bedload‘constraints for the midpoint Pbed value‘indentical to Sub Bedload except the bounds of‘the slope search has been reduced to reduce computationsEND SUB‘*********************************************************************************,SUB parkerl99ODIM OMEGAIJ(m% n%), OMEGAJ(n%)‘OMEGAJ() and OmegalJ 0 are used in the snmmations in‘Eqn 4.53 onwards. See Below for explanation‘these are key functions in the modified Parker (1990)‘bedload equation= 1DOWH[LEi%<=m%discharge! = discharge(i%)IF discharge! > qbfl THEN discharge! = qbfl309‘sets all flows in excess of qbf equal to qbfl‘in order to calculate sediment transport rate‘for overbank flowsCALL continuity(dischargel)‘satisify continuity for flow i%bedsheari(i%) = bedshear!depthi(i%) = depth!‘calculate depth of flow and bedshear for each flow‘this saves recalculation below‘records depths and bedshear values in an array= i% + 1LOOPdsgcond$ = “unknown”sigmaphicondS = “unknown”‘initialise convergence criteria for dsg! and sigmaphi!DO WHILE dsgcond$ “converged” OR sigmaphicondS “converged”dsg!=dsgtrial!sigmaphi! = sigmaphitrial!OMEGA! =0j% = 1DOWIIILEj%<=n%= 1OMEGAJj%) =0DOWHILEi%<=m%‘calculate for sediment(j%) over range of flowsphisgo! = bedsheari(i%) / (gamma! * (ss! - 1) * dsg!) / .0386CALL capGOMEGAIJ(i%, j%) = qprob(i%) * capitaig! * bedsheari(i%) A 1.5‘OMEGAIJ(m%) is used in the summation over i% in the numerator‘of Eqn 4.52OMEGAJ(j%) = OMEGAJ(j%) + OMEGAIJ(i%, j%)‘summation for all flows for sediment(j%) Eqn 4.52‘OMEGAJ(n%) is the numerator in Eqn 4.52= i% + 1LOOPOMEGA! = OMEGA! + f(j%) I OMEGAJ(j%)‘OMEGA is the denomenator of Eqn 4.52j%=j%+ 1LOOP‘calculate Fj and Dsg and sigmaphi!j% = 1lndsg! =0310sigmaphi2! =0ff1 =0DOWHILEj%<=n%ff(j%) = f(j%) / OMEGAJ(j%) / OMEGA!Indsg! = lndsg! + ff(j%) * LOG(sediment(j%))sigmaphi2! = sigiuaphi2! + (LOG(sediment(j%) / dsgl) /LOG(2)) “2 * ff(j%)j%=j%+ 1LOOPdsg! = EXP(lndsg!)sigmaphi! = sigmaphi2! “.5‘test for convergence for dsg!IF (dsg! / dsgtrial!) < .99 THENdsgcond$ = “too small”ELSEIF (dsg! I dsgtrial!)> 1.01 THENdsgcond$ = “too large”ELSEdsgcond$ = “converged”END IF‘test for convergence for siginaphi!IF (sigmaphi! / sigmaphitrial!) < .99 THENsigmaphicond$ = “too small”ELSEIF (signiaphi! / sigmaphitrial!)> 1.01 THENsigmaphicondS = “too large”ELSEsigmaphicond$ = “converged”END IFsigmaphitrial! = sigmaphidsgtnal! dsg!LOOP‘loop until convergence for dsg! and signaphi!‘calculate the median grain diameterff(0) = 0sediment(0) =0j% =0totif! =0DO WHILE totifi <.5j%=j%+ 1totifi = totfP + ff(j%)LOOPdeldd50! = (totifi - .5) * (upperd(j%) - upperd(j% - 1)) / ff(j%)ddSO! = upperd(j%) - deldd50!‘calculate D50 by interpolationqbcalc’ = .00218 I ((ss! - 1) * gravity! * densityV 1.5 * OMEGA!)‘calculate the mean unit volumetric sediment transport rate‘in units of m”2/sec311gbcalc! = qbcalc? * pbed! * density! * ss! * capganuna!‘cacluate the total load in mass units‘transported over one year.END SUBFUNCTION shearforce‘calculates the proportion of shear force on the banksshearforce! = 1.766 * (pbed! /pbank! + 1.5) A -1.4026END FUNCTION,**********************************************************************************SUB stabcurve‘stability curves are from Figure 5.3‘NB ‘!!!? THE STABILITY CURVES USED IN ThIS SUB ARE‘FROM TAYLOR (1948) AND APPLY TO HOMOGENEOUS FULLY DRAINED‘SOIL:- THESE ARE FOR ILLUSTRATIVE PURPOSES ONLY!!!thetadegrees! = theta! / 2/3.14159 * 360IF phidegrees! <20 THEN‘use phiprime = 15 degree stability curve from Figure 5.3‘from Taylor (1948)SELECT CASE thetadegrees!CASE 48.65 TO 90stabnum! = -.141 * thetadegrees? + 17.46CASE 29.36 TO 48.6499stabnum! = -.533 * thetadegrees? + 36.53CASE 20.12 TO 29.3599stabnum! = -3.07 * thetadegrees! + 111CASE 15.001 TO 20.11999stabnum! = -189.6 * thetadegrees! + 3844CASE IS <=15stabnum! = 1000000‘bank of infinite height is stableEND SELECTELSE‘use phiprinie =25 degrees curve from Figure 5.3‘After Taylor, (1948)SELECT CASE thetadegrees!CASE 58.62 TO 90stabnum! = -.2 12 * thetadegrees! + 24.86312CASE 39.11 TO 58.61999stabnum! = -.8975 * thetadegrees! + 65.04CASE 25 TO 39.10999stabnum! = -6.97 * thetadegrees! + 302.5CASE IS <=25stabnuml = 1000000bank of infinite height is stableEND SELECTEND IFEND SUBFUNCTION surfwidth‘function to calculate the surface widthsurfwidth! = pbed’ + COS(theta!) * pbank!END FUNCTIONFUNCTION velocity!‘calculates the mean velocityrhbank! = bankshear! / (gamma! * slope?)rhbed? = bedshear! / (gamma! * slope!)Ibank! = (2.03 * LOG(12.2 * rhbank! / ksbank!) / LOG(10)) “-2Ibed! = (2.03 * LOG(12.2 * rhbed! / ksbed!) / LOG(10)) A -2ffactor2! = (Ibed! * pbed! / (pbed! + pbank!) + thank! * pbank! / (pbank! + pbed!)) Avelocity! = (8 * gravity! * hydrad! * slope!) A 5 * fl’actor2!END FUNCTION313APPENDIX DDATA FROM HEY AND THORNE (1986)AND ANDREWS (1984)Table D-1 contains the data from Hey and Thorne (1986). The “Observed” columns contain theactual published data: Qbfi W, Y, D50 and Vegetation Type. The values of k3 and Gbf arecalculated from the observed hydraulic geometry using Eqn (3.2) and the Einstein-BrownEquation (Eqn 6.2) respectively. The value Of is set equal to D50, and d50 is set equal toD50/3.Table D-2 contains the data from Andrews (1984). The “Observed” columns contain the actualpublished data: QbJ; W, Y, d50 and Vegetation Type. The values of k and Gbf arecalculated from the observed hydraulic geometry using Eqn (3.2) and the Einstein-BrownEquation (Eqn 6.2) respectively. The value of is set equal to 1)5o. Channel Number9074800 from Andrews (1984) was excluded from the analysis. (See Section 6.3.3).314TableD-1. DatafromHeyandThorne(1986).CalculatedChannelObservedFixedSVariableSNumber,=40°Est.Meanççb,—40°Qbfk3GbfWYSD50d50D5obOVegWIWIWIS(mi/s)(m)(kg/s)(m)(m)(m)(m)(m)Type(m)(m)(°)(m)(m)(m)(m)11040.16101.5525.21.520.00350.0570.0190.057III41.11.1555.527.11.4845.11.040.004121540.1194.4342.71.400.00310.0610.0200.061II45.41.3541.9361.5646.21.320.003231240.2640.9632.71.700.00250.0820.0270.082III33.91.6941.325.12.0434.21.660.00264170.604.9712.80.910.00450.1090.0360.109II11.60.9935.9101.0811.410.00455630.20293.8620.01.270.00420.0240.0080.024III71.70.6272.544.60.8189.80.480.00646680.0713.8321.61.360.00220.0610.0200.061IV21.41.3839.715.91.722.51.320.00237701.34450.7527.51.150.01270.1760.0590.176I37.60.9948.729.21.1339.40.950.013781480.0424.2929.41.940.00140.0410.0140.041IV381.6849.523.52.3139.21.60.0015976.10.24430.2415.81.470.00640.0460.0150.046IV59.70.6975.828.21.0772.60.560.0092101720.0722.7528.72.190.00160,0700.0230.070IV312.1246.323.52.5831.82,040.0017113040.1135.1248.52.490.00130.0660.0220,066III47.22.5837.536.13.0847.52.570.001312450.5427.5918.21.410.00300.0670.0220.067III25.61.185518.41.4426.41.140.003213500.851341.018.41.140.01330.0600.0200.060IV74.40,527333.20.7991.30.440.018414460.0719.9620.11.050.00270.0430.0140.043II240.964619.51.0924.60.920.002915530.1930.1725.21.020.00390.0750.0250.075II23.81.073819.81.223.91.060.003916530,63100.3224.41.160.00510.0710.0240,0711134.20.9750.126.81.12360.920.005617690.1061.0829.71.020.00360.0450.0150.045II35.90,9245.228.11.0637.40.870.003918480.12170.8713.01.070.00820.0650.0220.065IV32.20.646716.20.96370.550.010519240.2121.2417.70.800.00400.0480.0160.048II20.30.7544.116.20.8520.90.720.004220240.6641.8921.10.920.00340.0340.0110.034131.60.7551.229.10.7833.40.710.003821220.5330.6510.21.420.00220.0190.0060.019IV370.79018.21.0540.60.640.002622140.1450.4610.60.750.00390.0140.0050.014IV37.40.3772.517.60.5746.30.290.0058237.10.4613.0213.70.480.00610.0480.0160.048I13.70.4939.712.80.5113.70.480.006224100.2190.595.50.590.02140.1090.0360.109IV13.40.3673.97.20.5214.40.330.024925280.3433.0017.00.990.00360.0430.0140.043II25.40852,8200.922670.750.0040TableD- Icontinued.CalculatedChannelObservedFixedSVariableSNumber=40°Est.Meanç6,Qbfk,GbfWYSD50d.,0D50VegW1’WYWIS(mi/s)(m)(kg/s)(m)(m)(m)(m)(m)Type(m)(m)(°)(m)(m)(m)(m)26270.3240.8915.00.950.00480.0550.0180.055II23.10.7554.518.40.8624.30.710.005427340.328.3814.41.610.00130.0270.0090.027III26.31.1570.318.41.4426.71.120.0014281000.2161.9725.91.520.00350.0810.0270.081IV32.21.3547.319.71.8533.21.290.0038291000.1731.4023.01.700.00270.0840.0280.084IV27.11.5749.519.51.9628.91.490.0029307.50.456.9215.60.480.00520.0550.0180.055112.10.5630.911.60.5711.70.580.0048317.50.300.0712.60.560.00340.0860.0290.08617.10.821.16.90.8270.860.0027321700.17116.4231.81.820.00330.0790.0260.079IV43.11.5350.125.12.1645.41.440.0037337.10.211.919.60.750.00150.0170.0060.017II13.70.625511.30.714.10.60.0016349.70.3239.8312.30.480.01080.0660.0220.066115.40.4346.314.30.45160.410.011535350.44671.8212.50.930.01520.0680.0230.068IV46.70.4673.621.50.756.60.380.02123611.21.5122.7114.10.810.00550.0740.0250.0741180.7349.817.20.7518.10.720.005737530.38302.5815.21.180.00920.0770.0260.077IV39.30.767.719.11.0645.30.610.0117382371.1279.1357.82.600.00140.0560.0190.056I65.52.4544.161.62.5466.12.410.0015394241.47316.9176.53.100.00160.0500.0170.050I117.92.4651.71092.57126.82.290.001840610.1023.2922.91.200.00270.0530.0180.053IV261.1344.116.11.5426.41.10.002941610.3690.4827.71.290.00260.0280.0090.028II52.10.9157.740.21.0558.70.80.003242450.29107.%18.61.120.00460.0420.0140.042II38.80.746129.90.86440.650.005743750.5087.7320.41.580.00420.0810.0270.081IV33.51.2158.319.31.735.41.140.0047441200.59348.1041.71.250.00660.0910.0300.091I53.51.146.849.51.1456.41.040.0072453480.38438.0746.62.510.00300.0660.0220.066IV87.21.7658.343.82.6697.61.560.00374660.40.133.703221.310.00110.0360.0120.036I23.51.6130.422.31.6622.81.690.00104736.51.28126.7228.00.990.00570.0640.0210.064I380.8648.435.10.8939.70.820.00614874.70.87666.5819.31.530.00730.0580.0190.058IV69.40.7873.631.81.1882.60.660.009849950.4229.1332.21.400.00380.1300.0430.130II24.51.6631.522.31.7723.71.740.003550500.3498.1317.51.200.00550.0700.0230.070III30.40.8858.320.11.1334.20790.0064TableD-1continued.CalculatedChannelObservedFixedSVariableSNumberçó.=40°Est.Mean=40°Qbfk3Gk,WYSD50d50D50VegWY,..WYW‘(ms/s)(m)(kg/s)(m)(m)(m)(m)(m)Type(m)(m)(9(m)(m)(m)(m)5190.50.5732.6424.61.730.00330.1230.0410.123IV24.81.7639.719.82.0525.11.720.003452126.81.25227.4031.32.030.00360.0710.0240.071III59.11.4659.137.41.8863.91.340.004253358.30.1370.3177.12.010.00160.0600.0200.060I612.3233.457.42.4159.72.390.00155491.30.1026.7731.21.410.00190.0410.0140.041II34.91.3343.528.11.5235.61.290.002055153.30.099.5533.41.980.00140.0690.0230.069III28.62.2334.222.32.6428.42.260.001456192.30.6786.2145.22.080.00250.0920.0310.092111472.0640.833.62.5347.22.030.00265717.60.1218.189.10.730.00840.1140.0380.114IV10.70.6852.38.10.8211.20.650.0089581960.37334.7541.01.700.00470.0840.0280.084III60.51.3751.238.51.7965.11.270.005459190.2424.4111.61.050.00240.0180.0060.018111310.6170.319.40.7935.50.530.003060200.0523.2217.50.620.00350.0200.0070.020I27.40.4852.825.50.530.10.430.0041613.90.150.676.50.590.00190.0230.0080.023117.40.5646.36.40.627.60.540.002062280.052.2812.31.140.00180.0490.0160.049III131.1343010.31.3313.11.110.0019TableD-2.DatafromAndrews(1984).00CalculatedChannelObservedFixedSVariableSNumberçS.=40°Est.Mean—=400Qbfk3GbfWYSD.sod,0D50VegWY.WYWYS(mIs)(m)(kg!s)(m)(in)(m)(m)(m)Type(m)(m)(0)(m)(m)(in)(m)66119006.710.032.9010.40.480.00230.0230.0050.023Thin9.70.537.59.60.59.60.50.002466148000.70.177.852.30.200.02600.0490.0190.049Thick5.10.1367.630.185.40.130.0293662000085.20,0213.0147.20.970.00140.0240.0080.024Thin36.11.1633.1361.1634.71.220.001370830007.080.83129.829.10.520.01700.0790.0230.079Thick18.70.3760.510.50.4920.20.350.019070910009.770.45222.0111.50.460.01900.0850.0190.085Thick18.60.3652.810.50.49200.340.0213901350012.20.4012.1211.90.730.00460.0610.0190.061Thin140.6846.313.90.68140.670.004890180002.210.6210.617.30.340.01100.0520.0190.052Thin9.50.347.99.50.39.70.290.011590220002.690.2430.527.00.290.01400.0340.0110.034Thick130.2156.67.20.2914.40.190.016590348001.870.132.165.20.310.00610.0240.0110.024Thick7.30.2652.84.80.347.80.250.006790359008.360.18123.259.20.430.01500.0730.0130.073Thick13.40.3550.680.4714.40.330.0168903600022.60.1619.6117.50.730.00440.0430.0170.043Thick21.10.6645.7130.8922.10.630.004890781003.170.3410.106.30.390.01000.0610.0190.061Thick8.30.3450.65.50.448.40.330.010690782002.520.5010.885.60.410.00920.0370.0150.037Thick11.40.2963.86.70.38120.270.01029081600490.46169.6934.10.840.00580.0980.0150.098Thin25.21.0132.025.31.0124.91.030.0055911250037.50.8366.2226.00.910.00670.0910.0360.091Thin27.60.8941.927.50.928.10.880.006991155007.080.297.0611.60.520.00460.0460.0140.046Thin11.40.5339.711.40.5311.50.520.00479124500420.34151.6624.90.880.00580.0790.0140.079Thin25.40.8840.825.30.8825.70.860.006092425001010.64114.8336.61.450.00370.0640.0240.064Thin48.51.2547.947.91.26521.180.004092444101670.2146.7753.31.630.00180.0580.0170.058Thin43.41.8533.9431.8742.91.890.0017924950046.71.3916.8224.41.620.00200.0450.0220.045Thin32.61.4148.432.51.4135.41.30.002392510002550.1251.5383.81.850.00090.0340.0070.034Thin58.32.330.457.92.356.32.410.0008925300072.20.692184.7530.51.130.00710.1220.0070.122Thin32.41.141.932.31.1133.41.070.007692570001140.4342.6636.61.650.00240.0700.0250.070Thin37.51.6540.837.51.6538.21.610.0025APPENDIX EDATA FROM CHARLTON ETAL. (1978)Table E-1 contains the data from Chariton eta!. (1978) for channels with cohesive banks whichwas analysed in Chapter 6. The actual published data are: Q W, 7, S, D.50, q, and theVegetation Type. The values for k are calculated from the observed hydraulic geometry usingEqn (3.2), and c’ is calculated from qu using Eqn (6.4) with the assumption of ç4’ = 25° for allchannels. The value ofd50 is set equal toD50/3, and the drained and saturated values of the unitsoil weight are assumed to be 20.0 and 22.45 kN/m3respectively for all channels.The modelled output data is presented in Table 6.4.319t%)CTableE-1.DatafromCharitonela!. (1978).ReachRiverWYSD,d,oVegNumber(m’/s)(m)(m)(m)(in)(m)kN/m2kN/m2(°)kN/m3kN/m3Type1AforLwyd6417.41.780.00441.090.0750.02539.512.62522.4520T2Aiwen1-510140.730.00641.420.1060.03525.68.22522.4520G3Alwen5-910.79.80.730.01301.250.1130.03823.27.42522.4520T4Arrow29.513.71.340.00451.160.0410.01451.416.42522.4520T5Ceirog1-36617.61.790.00481.210.0710.02423.37.42522.4520T6Ceirog4-766191.360.01051.070.0820.02770.922.62522.4520T7Exe67311.770.00181.350.0430.01426.18.32522.4520G8IrfonC8128.71.630.00140.190.0550.01830.19.62522.4520T9Otter14.216.70.690.00320.290.0570.01925.88.22522.4520G10Teign66192.470.00141.410.0510.01715.75.02522.4520T11Trent2.75.20.650.00230.560.0330.01136.711.72522.4520G12Usk15739.32.640.00090.600.0720.02421.36.82522.4520G13Wye55059.44.190.00070.280.0280.00916.05.12522.4520G14Wrye3819.51.670.00201.540.0400.01339.912.72522.4520G
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An optimization model for the development and response of alluvial river channels Millar, Robert Gary 1994
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Title | An optimization model for the development and response of alluvial river channels |
Creator |
Millar, Robert Gary |
Date Issued | 1994 |
Description | In this thesis an optimization model has been developed to calculate the equilibrium geometry of alluvial gravel-bed rivers for a given set of independent variables. The independent variables are the discharges, both the magnitude and duration which are represented by a flow-duration curve; the mean annual load, both volume and grain size distribution, which is imposed on to the channel reach from upstream; and the geotechnical properties of the bank sediment. The unknown dependent or decision variables to be solved for include the channel width, depth, bank angle, roughness, and grain size distribution of the bed surface. The dependent variables adjust subject to the constraints of discharge, bedload, bank stability, and valley slope, to determine a channel geometry which is optimal as defined by a maximization of, which is the coefficient of sediment transport efficiency. The work in this thesis is an extension of earlier models that have predicted the geometry of sand and gravel rivers with reasonable success, however the degree of scatter associated with these models limited their application to quantitative engineering applications. The advances in this thesis over the earlier optimization models are the inclusion of the bank stability analyses, modelling using the full flow-duration data, and calculating the grain size distribution of the bed-surface. The formulation presented in this thesis is specific to gravel-bed rivers, however it can be reformulated for sand-bed rivers. |
Extent | 7646104 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-04-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0050425 |
URI | http://hdl.handle.net/2429/7052 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1994-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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